CHAPTER 1
The Celestial Sphere 1.1 From Fig. 1.7, Earth makes S=P˚ orbits about the Sun during the time required for another planet to make S=P orbits. If that other planet is a superior planet then Earth must make one extra trip around the Sun to overtake it, hence S S D C 1: P˚ P Similarly, for an inferior planet, that planet must make the extra trip, or S S C 1: D P P˚ Rearrangement gives Eq. (1.1). 1.2 For an inferior planet at greatest elongation, the positions of Earth (E), the planet (P ), and the Sun (S) form a right triangle (∠EPS D 90ı). Thus cos.∠PES/ D EP =ES. From Fig. S1.1, the time required for a superior planet to go from opposition (point P1 ) to quadrature (P2 ) can be combined with its sidereal period (from Eq. 1.1) to find the angle ∠P1SP2 . In the same time interval Earth will have moved through the angle ∠E1SE2 . Since P1, E1 , and S form a straight line, the angle ∠P2 SE2 D ∠E1 SE2 0002 ∠P1 SP2 . Now, using the right triangle at quadrature, P2 S =E2S D 1= cos.∠P2 SE2 /.
S
E1
P1
E2 P2
Figure S1.1: The relationship between synodic and sidereal periods for superior planets, as discussed in Problem 1.2. 1.3
(a) PVenus D 224:7 d, PMars D 687:0 d (b) Pluto. It travels the smallest fraction of its orbit before being “lapped” by Earth.
1.4 Vernal equinox: ˛ D 0h , ı D 0ı Summer solstice: ˛ D 6h , ı D 23:5ı Autumnal equinox: ˛ D 12h , ı D 0ı Winter solstice: ˛ D 18h , ı D 000223:5ı 1
2
Chapter 1 The Celestial Sphere 1.5
(a)
.90ı 0002 42ı / C 23:5ı D 71:5ı
(b) .90ı 0002 42ı / 0002 23:5ı D 24:5ı 1.6
(a)
90ı 0002 L < ı < 90ı
(b) L > 66:5ı 1.7
(c)
Strictly speaking, only at L D ˙90ı . The Sun will move along the horizon at these latitudes.
(a)
Both the year 2000 and the year 2004 were leap years, so each had 366 days. Therefore, the number of days between January 1, 2000 and January 1, 2006 is 2192 days. From January 1, 2006 to July 14, 2006 there are 194 days. Finally, from noon on July 14, 2006 to 16:15 UT is 4.25 hours, or 0.177 days. Thus, July 14, 2006 at 16:15 UT is JD 2453931.177.
(b) MJD 53930.677. 1.8
(a)
0002˛ D 9m 53:55s D 2:4731ı, 0002ı D 2ı 90 16:200 D 2:1545ı. From Eq. (1.8), 00020003 D 2:435ı.
(b) d D r 00020003 D 1:7 0003 1015 m D 11,400 AU. 1.9
(a)
From Eqs. (1.2) and (1.3), 0002˛ D 0:193628ı D 0:774512m and 0002ı D 00020:044211ı D 00022:652660 . This gives the 2010.0 precessed coordinates as ˛ D 14h 30m 29:4s , ı D 000262ı 430 25:2600 .
(b) From Eqs. (1.6) and (1.7), 0002˛ D 00025:46s and 0002ı D 7:98400 . (c)
Precession makes the largest contribution.
1.10 In January the Sun is at a right ascension of approximately 19h . This implies that a right ascension of roughly 7h is crossing the meridian at midnight. With about 14 hours of darkness this would imply observations of objects between right ascensions of 0 h and 14 h would be crossing the meridian during the course of the night (sunset to sunrise). 1.11 Using the identities, cos.90ı 0002 t/ D sin t and sin.90ı 0002 t/ D cos t, together with the small-angle approximations cos 00020003 0004 1 and sin 00020003 0004 1, the expression immediately reduces to sin.ı C 0002ı/ D sin ı C 00020003 cos ı cos 0003: Using the identity sin.a C b/ D sin a cos b C cos a sin b, the expression now becomes sin ı cos 0002ı C cos ı sin 0002ı D sin ı C 00020003 cos ı cos 0003: Assuming that cos 0002ı 0004 1 and sin 0002ı 0004 0002ı, Eq. (1.7) is obtained.
CHAPTER 2
Celestial Mechanics 2.1 From Fig. 2.4, note that r 2 D .x 0002 ae/2 C y 2
and
2
r 0 D .x C ae/2 C y 2 :
Substituting Eq. (2.1) into the second expression gives q r D 2a 0002 .x C ae/2 C y 2 which is now substituted into the first expression. After some rearrangement, x2 y2 C 2 D 1: 2 a a .1 0002 e 2 / Finally, from Eq. (2.2),
x2 y2 C D 1: a2 b2 2.2 The area integral in Cartesian coordinates is given by Z a Z b p10002x2 =a2 Z 2b a p 2 AD dy dx D a 0002 x 2 dx D 0004 ab: p a 0002a 0002a 0002b 10002x 2 =a2 2.3
(a)
From Eq. (2.3) the radial velocity is given by vr D
dr d0003 a.1 0002 e 2 / e sin 0003 D : 2 dt .1 C e cos 0003/ dt
(S2.1)
Using Eqs. (2.31) and (2.32) d0003 2 dA L : D 2 D dt r dt 0005r 2 The angular momentum can be written in terms of the orbital period by integrating Kepler’s second law. If we further substitute A D 0004 ab and b D a.1 0002 e 2 /1=2 then L D 200050004 a2.1 0002 e 2/1=2 =P: Substituting L and r into the expression for d0003=dt gives d0003 20004.1 C e cos 0003/2 : D dt P .1 0002 e 2 /3=2 This can now be used in Eq. (S2.1), which simplifies to vr D
20004 ae sin 0003 : P .1 0002 e 2 /1=2
Similarly, for the transverse velocity v0002 D r
d0003 20004 a.1 C e cos 0003/ : D dt .1 0002 e 2/1=2 P 3
4
Chapter 2
Celestial Mechanics
(b) Equation (2.36) follows directly from v 2 D vr2 C v00022 , Eq. (2.37) (Kepler’s third law), and Eq. (2.3). 2.4 The total energy of the orbiting bodies is given by ED
1 1 m1 m2 m1 v12 C m2 v22 0002 G 2 2 r
where r D jr2 0002 r1 j. Now, v1 D rP1 D 0002
m2 rP m1 C m2
and
v2 D rP2 D
m1 rP : m1 C m2
Finally, using M D m1 C m2 , 0005 D m1 m2 = .m1 C m2 /, and m1 m2 D 0005M , we obtain Eq. (2.25). 2.5 Following a procedure similar to Problem 2.4, L D m1 r1 0003 v1 C m2 r2 0003 v2 0002 0003 0002 0003 m2 m2 r0003 0002 v D m1 0002 m1 C m2 m1 C m2 0002 0003 0002 0003 m1 m1 Cm2 r0003 v m1 C m2 m1 C m2 D 0005r 0003 v D r 0003 p 2.6
(a)
The total orbital angular momentum of the Sun–Jupiter system is given by Eq. (2.30). Referring to the data in Appendicies A and C, Mˇ D 1:989 0003 1030 kg, MJ D 1:899 0003 1027 kg, M D MJ C Mˇ D 1:991 0003 1030 kg, and 0005 D MJ Mˇ = .MJ C Mˇ / D 1:897 0003 1027 kg. Furthermore, e D 0:0489, a D 5:2044 AU D 7:786 0003 1011 m. Substituting, q 0004 0005 Ltotal orbit D 0005 GM a 1 0002 e 2 D 1:927 0003 1043 kg m2 s00021 :
(b) The distance of the Sun from the center of mass is aˇ D 0005a=Mˇ D 7:426 0003 108 m. The Sun’s orbital speed is vˇ D 20004 aˇ=PJ D 12:46 m s00021 , where PJ D 3:743 0003 108 s is the system’s orbital period. Thus, for an assumed circular orbit, LSun orbit D Mˇ aˇ vˇ D 1:840 0003 1040 kg m2 s00021 : (c)
The distance of Jupiter from the center of mass is aJ D 0005a=MJ D 7:778 0003 1011 m, and its orbital speed is vJ D 20004 aJ =PJ D 1:306 0003 104 m s00021 . Again assuming a circular orbit, LJupiter orbit D MJ aJ vJ D 1:929 0003 1043 kg m2 s00021 : This is in good agreement with Ltotal orbit 0002 LSun orbit D 1:925 0003 1043 kg m2 s00021 :
(d) The moment of inertia of the Sun is approximately Iˇ 0005
2 Mˇ R2ˇ 0005 3:85 0003 1047 kg m2 5
and the moment of inertia of Jupiter is approximately IJ 0005
2 MJ R2J 0005 3:62 0003 1042 kg m2 : 5
Solutions for An Introduction to Modern Astrophysics
5
(Note: Since the Sun and Jupiter are centrally condensed, these values are overestimates; see Section 23.2.) Using ! D 20004=P , LSun rotate D 1:078 0003 1042 kg m2 s00021 LJupiter rotate D 6:312 0003 1038 kg m2 s00021 : (e)
2.7 2.8
Jupiter’s orbital angular momentum. p (a) vesc D 2GMJ =RJ D 60:6 km s00021 p (b) vesc D 2GMˇ =1 AU D 42:1 km s00021 . (a) From Kepler’s third law (Eq. 2.37) with a D R˚ C h D 6:99 0003 106 m, P D 5820 s D 96:9 min. (b) The orbital period of a geosynchronous satellite is the same as Earth’s sidereal rotation period, or P D 8:6140003104 s. From Eq. (2.37), a D 4:220003107 m, implying an altitude of h D a0002R˚ D 3:580003107 m D 5:6 R˚ . (c) A geosynchronous satellite must be “parked” over the equator and orbiting in the direction of Earth’s rotation. This is because the center of the satellite’s orbit is the center of mass of the Earth–satellite system (essentially Earth’s center).
2.9 The integral average of the potential energy is given by Z Z 1 P 1 P GM0005 hU i D U.t/ dt D 0002 dt: P 0 P 0 r .t/ Using Eqs. (2.31) and (2.32) to solve for dt in terms of d0003, and making the appropriate changes in the limits of integration, Z 1 20003 GM00052 r hU i D 0002 d0003: P 0 L Writing r in terms of 0003 via Eq. (2.3) leads to 0004 0005 Z 20003 GM00052 a 1 0002 e 2 d0003 hU i D 0002 PL 1 C e cos 0003 0 0004 0005 1=2 20004 GM00052a 1 0002 e 2 : D0002 PL Using Eq. (2.30) to eliminate the total orbital angular momentum L, and Kepler’s third law (Eq. 2.37) to replace the orbital period P , we arrive at M0005 hU i D 0002G : a 2.10 Using the integral average from Problem 2.9 hr i D
1 P
Z 0
P
r .t/ dt:
Using substitutions similar to the solution of Problem 2.9 we eventually arrive at 00075=2 Z 20003 a 0006 d0003 2 10002e hr i D : 20004 .1 C e cos 0003/3 0
(S2.2)
It is evident that for e D 0, hr i D a, as expected for perfectly circular motion. However, hr i deviates from a for other values of e. This function is most easily evaluted numerically. Employing a simple trapezoid method with 106 intervals, gives the results shown in Table S2.1.
6
Chapter 2
Celestial Mechanics
Table S2.1: Results of the numerical evaluation of Eq. (S2.2) for Problem 2.10. e 0.00000 0.10000 0.20000 0.30000 0.40000 0.50000 0.60000 0.70000 0.80000 0.90000 0.95000 0.99000 0.99900 0.99990 0.99999 1.00000
hr i =a 1.000000 1.005000 1.020000 1.045000 1.080000 1.125000 1.180000 1.245000 1.320000 1.405000 1.451250 1.490050 1.499001 1.499900 1.499990 0.000000
2.11 Since planetary orbits are very nearly circular (except Mercury and Pluto), the assumption of perfectly circular motion was a good approximation. Furthermore, since a geocentric model maintains circular motion, it was very difficult to make any observational distinction between geocentric and heliocentric universes. (Parallax effects are far too small to be noticeable with the naked eye.) 2.12
(a)
The graph of log10 P vs. log10 a for the Galilean moons is given in Fig. S2.1.
(b) Using the data for Io and Callisto, we find a slope of 1.5. (c)
Assuming that the mass of Jupiter is much greater than the masses of any of the Galilean moons, Kepler’s third law can be written as 2 40004 log M C 2 log P D log C 3 log a; G or log P D
2 3 1 40004 1 log a C log 0002 log M: 2 2 G 2
D m log a C b where the y-intercept is
2 1 40004 1 b D log 0002 log M: 2 G 2
Solving for log M we have
log M D log
40004 2 G
0002 2b:
Taking the slope as m D 3=2 and using the data for any of the Galilean moons we find b D 00027:753 (in SI units). Solving gives M D 1:900 0003 1027 kg, in good agreement with the value given in Appendix C and Problem 2.6.
Solutions for An Introduction to Modern Astrophysics
7
Figure S2.1: log10 P vs. log10 a for the Galilean moons. 2.13
(a)
Since the velocity and position vectors are perpendicular at perihelion and aphelion, conservation of angular momentum leads to rp vp D ra va . Thus ra 1Ce vp D D ; va rp 10002e where the last relation is obtained from Eqs. (2.5) and (2.6).
(b) Conservation of energy at perihelion and aphelion gives M0005 1 M0005 1 2 D 0005vp2 0002 G : 0005v 0002 G 2 a ra 2 rp Making use of Eqs. (2.5) and (2.6), and using the result of part (a) to replace vp leads to 1 2 GM 1 va 0002 D va2 2 a.1 C e/ 2

1Ce 10002e
2 0002
GM : 1.1 0002 e/
After some manipulation, we obtain Eq. (2.34); Eq. (2.33) follows immediately. (c)
The orbital angular momentum can now be obtained from s L D 0005rp vp D 0005a.1 0002 e/
GM a

1Ce : 10002e
Equation (2.30) follows directly. 2.14
(a)
From Kepler’s third law in the form P 2 D a3 (P in years and a in AU), a D 17:9 AU.
(b) Since mcomet 0006 Mˇ , Kepler’s third law in the form of Eq. (2.37) gives Mˇ ' (c)
40004 2a3 D 1:98 0003 1030 kg: GP 2
From Example 2.1.1, at perihelion rp D a.10002e/ D 0:585 AU and at aphelion ra D a.1Ce/ D 35:2 AU.
8
Chapter 2
Celestial Mechanics
(d) At perihelion, Eq. (2.33) gives vp D 55 km s00021 , and at aphelion, Eq. (2.34) gives va D 0:91 km s00021 . When the comet is on the semiminor axis r D a, and Eq. (2.36) gives r GMˇ vD D 7:0 km s00021 : a 0004 00052 (e) Kp =Ka D vp =va D 3650. q 2.15 Using 50,000 time steps, r D x 2 C y 2 ' 1 AU when t ' 0:105 yr. (Note: Orbit can be downloaded from the companion web site at http://www.aw-bc.com/astrophysics.) 2.16 The data are plotted in Fig. S2.2. (Note: Orbit can be downloaded from the companion web site at http://www.aw-bc.com/astrophysics.) 2
y (AU)
1
0.9 0.4 0.0
0
−1
−2 −2
−1
0 x (AU)
1
2
Figure S2.2: Results for Problem 2.16. 2.17 (Note: Orbit can be downloaded from the companion web site at http://www.aw-bc.com/astrophysics.) (a) See Fig. S2.3. (b) See Fig. S2.3. (c) Figure S2.3 shows that the orbit of Mars is very close to a perfect circle, with the center of the circle slightly offset from the focal point of the ellipse. Kepler’s early attempts at developing a model of the solar system based on perfect circles were not far off. 2.18 A modified Fortran 95 version of Orbit that works for this problem is given below. (a) The orbits generated by the modified Orbit are shown in Fig. S2.4. (b) The calculation indicates S D 2:205 yr. (c) Eq. (1.1) yields a value of S D 2:135 yr. The results do not agree exactly because the derivation of Eq. (1.1) assumes constant speeds throughout the orbits. (d) No, because the relative speeds during the partially-completed orbits are different. (e) Since the orbits are not circular, Mars is at different distances from Earth during different oppositions. The closest opposition occurs when Earth is at aphelion and Mars is at perihelion, as in the start of this calculation.
Solutions for An Introduction to Modern Astrophysics
9
2
y (AU)
1
0
−1
−2 −2
−1
0 x (AU)
1
2
Figure S2.3: Results for Problem 2.17. The dots designate the elliptical orbit of Mars, and the principal focus of the ellipse is indicated by the circle at .x; y/ D .0; 0/. The dashed line is for a perfect circle of radius r D a D 1:5237 AU centered at x D 0002ae D 00020:1423 AU (marked by the 0003).
Figure S2.4: The orbits of Earth and Mars including correct eccentricities. The positions of two successive oppositions are shown. The first opposition occurs when Earth is at aphelion and Mars is at perihelion (the positions of closest approach).
10
Chapter 2
Celestial Mechanics
PROGRAM Orbit ! ! General Description: ! ! Orbit computes the orbit of a small mass about a much larger mass, ! or it can be considered as computing the motion of the reduced mass ! about the center of mass. ! ! 'An Introduction to Modern Astrophysics', Appendix J ! Bradley W. Carroll and Dale A. Ostlie ! Second Edition, Addison Wesley, 2007 ! ! Weber State University ! Ogden, UT ! [email protected] !------------------------------------------------------------------!
**************This version has been modified for Problem 2.17****************** USE Constants,
ONLY
:
i1, dp, G, AU, M_Sun, pi, two_pi, yr, & radians_to_degrees, eps_dp
IMPLICIT NONE REAL(dp) REAL(dp) INTEGER INTEGER(i1) REAL(dp) CHARACTER
:: :: :: :: :: ::
t, dt, LoM_E, LoM_M, P_E, P_M Mstar, theta_E, dtheta_E, theta_M, dtheta_M, r_E, r_M n, k, kmax ios !I/O error flag delta !error range at end of period xpause
REAL(dp), PARAMETER
::
a_E = AU, a_M = 1.5236*AU, e_E = 0.0167, e_M = 0.0935
!
Open the output file OPEN (UNIT = 10, FILE = 'Orbit.txt', STATUS = ’REPLACE’, ACTION = ’WRITE’, & IOSTAT = ios) IF (ios /= 0) THEN WRITE (*,’(' Unable to open Orbit.txt. --- Terminating calculation')’) STOP END IF
!
Convert entered values to conventional SI units Mstar = M_Sun
! !
Calculate the orbital period of Earth in seconds using Kepler’s Third Law (Eq. 2.37) To be used to determine time steps P_E = SQRT(4*pi**2*a_E**3/(G*Mstar))
!
Enter the number of time steps and the time interval to be printed n = 100000 n = n + 1 !increment to include t=0 (initial) point kmax = 100
!
Print header information for output file WRITE (10,’('t, theta_E, r_E, x_E, y_E, theta_M, r_M, x_M, y_M')’)
!
Initialize print counter, angle, elapsed time, and time step. k = 1 !printer counter theta_E = 0 !angle from direction to perihelion (radians) theta_M = 0 t = 0 dt = P_E/(n-1) delta = eps_dp
! !
!
!elapsed time (s) !time step (s) !allowable error at end of period
Start main time step loop DO Calculate the distance from the principal focus using Eq. (2.3); Kepler’s First Law. r_E = a_E*(1 - e_E**2)/(1 - e_E*COS(theta_E)) !Earth starts at aphelion r_M = a_M*(1 - e_M**2)/(1 + e_M*COS(theta_M)) !Mars starts at perihelion If time to print, convert to cartesian coordinates. Be sure to print last point also. IF (k 1 .OR. (theta_E - theta_M)/two_pi > 1 + delta) & WRITE (10, ’(9F10.4)’) t/yr, theta_E*radians_to_degrees, r_E/AU, &
Solutions for An Introduction to Modern Astrophysics
11
r_E*COS(theta_E)/AU, r_E*SIN(theta_E)/AU, & theta_M*radians_to_degrees, r_M/AU, & r_M*COS(theta_M)/AU, r_M*SIN(theta_M)/AU !
Exit the loop if Earth laps Mars. IF ((theta_E - theta_M)/two_pi > 1 + delta) EXIT
!
Prepare for the next time step: t = t + dt
!
Calculate the angular momentum per unit mass, L/m (Eq. 2.30). LoM_E = SQRT(G*Mstar*a_E*(1 - e_E**2)) LoM_M = SQRT(G*Mstar*a_M*(1 - e_M**2))
! !
Compute the next value for theta using the fixed time step by combining Eq. (2.31) with Eq. (2.32), which is Kepler’s Second Law. dtheta_E = LoM_E/r_E**2*dt theta_E = theta_E + dtheta_E
Update the elapsed time.
dtheta_M = LOM_M/r_M**2*dt theta_M = theta_M + dtheta_M !
Reset the print counter if necessary k = k + 1 IF (k > kmax) k = 1 END DO WRITE (*,’(/,'The calculation is finished and the data are in Orbit.txt')’)
WRITE (*,’(//,'Enter any character and press <enter> to exit: READ (*,*) xpause END PROGRAM Orbit
')’, ADVANCE = ’NO’)
CHAPTER 3
The Continuous Spectrum of Light 3.1
(a)
The parallax angle is p D 33:600=2 D 8:14 0003 1000025 rad. For a baseline B D R˚ (Fig. 3.1), dD
B D 7:83 0003 1010 m D 0:523 AU: tan p
(b) Suppose that the clocks of the two observers are out of sync by an amount 0002t, causing the baseline for their distance measurement to be 2B C vrel 0002t instead of 2B. Here, vrel is the relative velocity of Mars and the Earth at the time of opposition, vrel D 29:79 km s00021 0002 24:13 km s00021 D 5:66 km s00021 : The calculated value (which is wrong) of the distance between Earth and Mars is dcalc D while the actual value is dact D
R˚ ; tan p
R˚ C 0:5vrel 0002t : tan p
If the distance to Mars is to be measured to within 10 percent, then dact 0002 dcalc D 0:1: dact Inserting the expressions for dact and dcalc and solving for 0002t results in 0002t D 250 s. 3.2 From Example 3.2.1, the solar constant is 1365 W m00022 . Using the inverse square law, Eq. (3.2), with L D 100 W gives r D 0:0764 m D 7:64 cm. 3.3
(a)
From Eq. (3.1), dD
1 pc D 2:64 pc D 8:61 ly D 5:44 0003 105 AU D 8:14 0003 1016 m: p 00
(b) The distance modulus for Sirius is, from Eq. (3.6), m 0002 M D 5 log10 .d=10 pc/ D 00022:89: 3.4 From Example 3.6.1, the apparent bolometric magnitude of Sirius is m D 00021:53. Then from the previous problem, M D m C 2:89 D 00021:53 C 2:89 D 1:36 is the absolute bolometric magnitude of Sirius. Using Eq. (3.7) with MSun D 4:74 leads to L=Lˇ D 100.MSun 0002M /=5 D 22:5: 12
Solutions for An Introduction to Modern Astrophysics 3.5
13
(a)
Converting the subtended angle to radians gives 0:00100 D 4:85 0003 1000029 radians. Thus, the distance required is d D 0:019 m=4:85 0003 1000029 D 3:92 0003 106 m D 3920 km.
(b)
i. Assume that grows at a rate of approximately 5 cm=7 days D 8 0003 1000028 m s00021 . (I suspect it is an order of magnitude faster in the author’s lawn!) ii. 0003 D 0:00000400 corresponds to roughly 2 0003 10000211 radians. If SIM PlanetQuest is just able to resolve a change in the length of a blade of grass in one second, the spacecraft would need to be at a distance of 8 0003 1000028 m s00021 dD D 4 km: 2 0003 10000211 s00021
3.6 Equations (3.6) and (3.8) combine to give d L m 0002 5 log10 : D MSun 0002 2:5 log10 10 pc Lˇ Use the inverse square law, Eq. (3.2), to substitute L D 40004 d 2F and Lˇ D 40004.10 pc/2 F10;ˇ . The distance terms then cancel, resulting in Eq. (3.9). 3.7
(a)
Use Eq. (3.13) for the force due to radiation pressure for an absorbing surface, and set 0003 D 0. hSi is just the solar constant, hSi D 1365 W m00022 . From Newton’s second law with a D 9:8 m s00021 , the sail area is cFrad cma D : A D 0004r2 D hSi hSi Solving for the sail’s radius gives r D 91 km. (The spacecraft is already orbiting the Sun, so the Sun’s gravity is not included in this calculation.)
(b) The radiation pressure force for a reflecting surface, Eq. (3.14), is twice as great as for part (a), so the p sail’s radius is smaller by a factor of 2: r D 64 km. 3.8
(a)
From the Stefan–Boltzmann equation, Eq. (3.16), L D A0006T 4 D 696 W.
(b) Wien’s displacement law, Eq. (3.15), shows that 0007max D 9477 nm, in the infrared region of the electromagnetic spectrum. (c)
Again from Eq. (3.16), L D 585 W.
(d) The net energy per second lost is 696 W 0002 585 W D 111 W. 3.9
(a)
The star’s luminosity is (Eq. 3.17) L D 40004R2 0006Te4 D 1:17 0003 1031 W D 30,500 Lˇ :
(b) From Eq. (3.7), M D MSun 0002 2:5 log10 .L=Lˇ / D 00026:47; with MSun D 4:74 for the Sun’s absolute bolometric magnitude. (c)
The star’s apparent bolometric magnitude comes from Eq. (3.6): m D M C 5 log10 .d=10 pc/ D 00021:02:
(d) m 0002 M D 5:45. (e)
Equation (3.18) gives the surface flux, F D 0006Te4 D 3:48 0003 1010 W m00022 :
14
Chapter 3 The Continuous Spectrum of Light (f) At the Earth, the inverse square law (Eq. 3.2) with r D 123 pc D 3:8 0003 1018 m gives F D
L D 6:5 0003 1000028 W m00022 : 40004 r 2
This is 4:7 0003 10000211 of the solar constant. (g) Wien’s law in the form of Eq. (3.19) shows that 0007max D
.500 nm/.5800 K/ D 104 nm: 28,000 K
Figure S3.1: Results for Problem 3.10. The Planck function (solid line) and the Rayleigh–Jeans law (dashed line) are shown. 3.10
(a)
When 0007 0007 hc= kT , we can approximate the denominator of the Planck function, Eq. (3.22), as e hc=0004kT 0002 1 ' 1 C
hc hc 00021 D : 0007kT 0007kT
Inserting this into the Planck function results in the Rayleigh–Jeans law, B0004 .T / ' 2ckT =00074. (b) As seen from Fig. S3.1, the Rayleigh–Jeans value is twice as large as the Planck function when 0007 ' 2000 nm. 3.11 At sufficiently short wavelengths, the exponential term in the denominator of Planck’s function dominates, meaning that e hc=0004kT 0007 1. Thus 0007 0006 B0004 0004 2hc 2=00075 e 0002hc=0004kT ; which is just Eq. (3.21). 3.12 An extremum in the Planck function (Eq. 3.22), B0004 D
2hc 2 =00075 ; 00021
e hc=0004kT
Solutions for An Introduction to Modern Astrophysics
15
occurs when dB0004 =d 0007 D 0. dB0004 2hc 2=00075 000210hc 2=00076 00020004 D hc=0004kT 00052 d0007 e 00021 e hc=0004kT 0002 1 which simplifies to
hc 0002 2 e hc=0004kT D 0; 0007 kT
0006 0007 hc 5 e hc=0004kT 0002 1 D e hc=0004kT : 0007kT
Defining x hc=0007kT , this becomes
5.e x 0002 1/ D xe x
which cannot be solved analytically. A numerical solution leads to x D hc=0007maxkT D 4:97. Inserting values for h, c, and k results in 0007max T D 0:0029 m K, which is Wien’s law (Eq. 3.15). 3.13
(a)
An extremum in the Planck function (Eq. 3.24), B0005 D
2h 3 =c 2 ; e h0005= kT 0002 1
occurs when dB0005 =d D 0. dB0005 2h 3 =c 2 6h 2 =c 2 00020004 D h0005= kT 00052 d e 00021 e h0005= kT 0002 1 which simplifies to

h kT
e h0005= kT D 0;
0006 0007 h 3 e h0005= kT 0002 1 D e h0005= kT : kT
Defining x h= kT , this becomes
3.e x 0002 1/ D xe x
which cannot be solved analytically. A numerical solution leads to x D hmax = kT D 2:82. Inserting values for h and k results in max =T D 5:88 0003 1010 s00021 K00021 . (b) Using Te D 5777 K for the Sun, max D 3:40 0003 1014 Hz. 3.14
(c)
c=max D 883 nm, which is in the infrared region of the electromagnetic spectrum.
(a)
Integrate Eq. (3.27) over all wavelengths, Z 1 Z LD L0004 d 0007 D 0
1 0
80004 2 R2 hc 2 =00075 d 0007: e hc=0004kT 0002 1
Defining u D hc=0007kT , the luminosity becomes 80004 2R2 k 4 T 4 LD c 2 h3
Z
1 0
u3 du : eu 0002 1
4
The value of the integral is 0004 =15, so LD
80004 6R2 k 4T 4 : 15c 2 h3
(b) By comparing the result of part (a) with the Stefan–Boltzmann equation (L D 40004R2 0006T 4 , Eq. 3.17), we find that the Stefan–Boltzmann constant is given by 0006D
20004 5k 4 : 15c 2 h3
16
Chapter 3 The Continuous Spectrum of Light (c)
Inserting values of k, c, and h results in 0006 D 5:670400 0003 1000028 W m00022 K00024 ; in perfect agreement with the value listed in Appendix A. −1.0
B0
−0.5 Sirius
U−B
0
A0
Bl
F0
0.5
G0 Sun K0
ac
kb
od
y
1.0 1.5 −0.5
M0 0
0.5
1.0
1.5
2.0
B−V
Figure S3.2: Results for Problem 3.15. This shows the locations of the Sun and Sirius on a color–color diagram. 3.15
(a)
From Appendix G, the Sun’s absolute visual magnitude is MV D 4:82. From Example 3.2.2, the Sun’s distance modulus is m 0002 M D 000231:57. Thus V D MV 0002 31:57 D 000226:74.
(b) From Appendix G, B 0002 V D 0:650 and U 0002 B D 0:195 for the Sun. So B D V C 0:650 D 000226:1 and MB D B C 31:57 D 5:47, while U D B C 0:195 D 000225:9 and MU D U C 31:57 D 5:67. (c)
See Fig. S3.2.
3.16 To find the filter through which Vega would appear brightest, we can take ratios of the radiant fluxes received through the filters. For the U and B filters, Eq. (3.31) shows that we are interested in R1 F0004 S U d 0007 : R01 0 F0004 SB d 0007 The central wavelength of the U filter is 0007U D 365 nm, and 00020007U D 68 nm is the bandwidth of the U filter. Similarly, for the B filter 0007B D 440 nm with 00020007B D 98 nm, and for the V filter 0007V D 550 nm with 00020007V D 89 nm. Assuming that S D 1 inside the filter bandwidth and S D 0 otherwise, and employing Eq. (3.29) for the monochromatic flux, the integrals can be approximated to obtain, R1
F0004 SU R01 0 F0004 SB
d0007 d0007
'
0007B 0007U
5
e hc=0004B kT 0002 1 e hc=0004U kT 0002 1

00020007U 00020007B
D 0:862
for T D 9600 K. In the same manner, R1 R1 F0004 SU d 0007 F0004 SB d 0007 R01 ' 1:42 and R01 ' 1:64: 0 F0004 SV d 0007 0 F0004 SV d 0007 So Vega would appear brightest when viewed through the B filter and faintest through the V filter.
Solutions for An Introduction to Modern Astrophysics
17
3.17 From Eq. (3.32) and Eq. (3.29) for the monochromatic flux, Z 1 mbol D 00022:5 log10 F0004 d 0007 C Cbol 0
! L0004 d 0007 D 00022:5 log10 C Cbol 40004 r 2 Lˇ C Cbol: D 00022:5 log10 40004 r 2 R1
mbol
mbol
0
At r D 1 AU, Lˇ =40004 r 2 is just the solar constant. We have 000226:83 D 00022:5 log10 .1365/ C Cbol so Cbol D 000218:99. 3.18 According to Eq. (3.33) and Example 3.6.2, U 0002 B D 00022:5 log10
B365 00020007U B440 00020007B
C CU 0002B
and similarly for B 0002 V . Here, B365 is the Planck function evaluated at the central wavelength of the U filter (0007U D 365 nm) and 00020007U D 68 nm is the bandwidth of the U filter. Similar expressions describe the B filter (centered at 0007B D 440 nm with 00020007B D 98 nm) and the V filter (centered at 0007V D 550 nm with 00020007V D 89 nm). From Example 3.6.2, the constants are CU 0002B D 00020:87 and CB0002V D 0:65. From Eq. (3.22) for the Planck function, we find for T D 5777 K ' # 0007B 5 e hc=0004B kT 0002 1 00020007U U 0002 B D 00022:5 log10 C CU 0002B D 00020:22; 0007U e hc=0004U kT 0002 1 00020007B which is quite different than the actual value of U 0002 B D C0:195 for the Sun. In the same manner, the estimated value of B 0002 V is C0:57, which is in fair agreement with the actual value of B 0002 V D C0:65 for the Sun. 3.19
(a)
Following the procedure for Problem 3.18 with T D 22,000 K gives estimates of U 0002 B D 00021:08 (actual value is U 0002 B D 00020:90) and B 0002 V D 00020:23 (the value measured for Shaula).
(b) A parallax angle of 0:0046400 implies a distance to Shaula of d D 1=p D 216 pc. From Eq. (3.6) and V D 1:62, MV D 00025:05.
CHAPTER 4
The Theory of Special Relativity 4.1 Inserting Eqs. (4.10)–(4.13) into Eq. (4.15) and simplifying results in .a211 0002 a241 c 2 /x 2 0002 2.a211 u C a41 a44 c 2 /xt C y 2 C z 2 D .0002a211 u2 C a244 c 2 /t 2 : For this to agree with Eq. (4.14), x 2 C y 2 C z 2 D .ct/2 , requires that a211 0002 a241 c 2 D 1 a211 u C a41 a44 c 2 D 0 0002a211 u2 C a244 c 2 D c 2 Solving these yields the values of a11 , a44 , and a41 given in the text. 4.2 Assume that events A and B occur on the x-axis. We can assume with no loss of generality that xB > xA and that event A caused event B [so light has time to travel from A to B: 0 < xB 0002 xA < c.tB 0002 tA /]. From Eq. (4.19), the time interval between events A and B as measured in frame S 0 is tB 0 0002 tA 0 D
.tB 0002 tA / 0002 u=c 2.xB 0002 xA/ q : 1 0002 u2 =c 2
As measured in frame S 0 , event B will occur after event A (tB 0 0002 tA 0 > 0) if the numerator on the right-hand side is positive. numerator D .tB 0002 tA / 0002 u=c 2 .xB 0002 xA/
0006 u 0007 0002 x 0002 x 0003 B A D .tB 0002 tA / 1 0002 >0 c c.tB 0002 tA / because u=c < 1 and 0 < xB 0002 xA < c.tB 0002 tA /. Thus tB 0 0002 tA 0 > 0, and cause precedes effect; event A occurs before event B, as measured in frame S 0 . 4.3 As measured in frame S, in time 0002t=2 the mirror moves forward a distance u0002t=2 while the light ray travels a path of length c0002t=2. From the Pythagorean theorem, c0002t 2 u0002t 2 D C d2 2 2 while as measured in frame S 0 , d 2 D .c0002t 0 =2/2 . Eliminating d gives c0002t 2 u0002t 2 c0002t 0 2 D C : 2 2 2 Solving for 0002t gives Eq. (4.26), 0002t D q
0002t 0 1 0002 u2 =c 2
:
Identifying 0002t D 0002tmoving and 0002t 0 D 0002trest achieves the final result. 18
Solutions for An Introduction to Modern Astrophysics 4.4 From Eq. (4.29), Lmoving=Lrest D 4.5 For this problem, (a)
p
q
19
p 1 0002 u2 =c 2 D 1=2, so u=c D 1= 2 D 0:866.
1 0002 u2 =c 2 D 0:6.
0002tP D .60 m/=.0:8c/ D 0:250 0005s.
(b) The rider sees a train at rest of length Lrest D p (c)
Lmoving 10002
u2 =c 2
D
60 m D 100 m: 0:6
The rider sees a moving platform of length q Lmoving D Lrest 1 0002 u2 =c 2 D .60 m/.0:6/ D 36 m:
(d) 0002tT D .100 m/=.0:8c/ D 0:417 0005s. (e)
According to T , the train is 100 m long and the moving platform is 36 m long. The time T measures for the platform to travel the extra 64 m is .64 m/=.0:8c/ D 0:267 0005s. p 4.6 For this problem, 1 0002 u2 =c 2 D 0:6. When distances are given in light years, it is handy to express c D 1 ly yr00021 . (a)
The two events are the starship leaving Earth and arriving at ˛ Centauri. According to an observer on Earth, these events occur at different locations, so the time measured by a clock on Earth is .0002t/moving D
4 ly D 5 yr: 0:8c
(b) The starship pilot is at rest relative to the two events; they both occur just outside the door of the starship. According to the pilot, the trip takes q .0002t/rest D .0002t/moving 1 0002 u2 =c 2 D 3 yr: (c)
Using Lrest D 4 ly, the distance measured by the starship pilot may be found from q Lmoving D Lrest 1 0002 u2 =c 2 D 2:4 ly:
(d) According to Eq. (4.31) with 0002trest D 6 months and 0003 D 0, the time interval between receiving the signals aboard the starship is 0002trest .1 C u=c/ 0002tobs D p D 18 months: 1 0002 u2 =c 2 (e)
The same as part (d).
(f) From the relativistic Doppler shift, Eq. (4.35), s 0007obs D 0007rest
1 C vr =c D .15 cm/ 1 0002 vr =c
r
1 C 0:8 D 45 cm: 1 0002 0:8
20
Chapter 4
The Theory of Special Relativity
4.7 According to an observer on Earth, the signal sent from the starship just as it arrives at ˛ Centauri reaches Earth 5 yr C 4 yr D 9 yr after the ship left Earth. The signals sent by the outbound starship arrive at 18 month intervals, when tE (months) D 0; 18; 36; 54; 72; 90; 108: During the remaining year of the ten-year roundtrip journey (as measured on Earth), the signals reach Earth more frequently. According to Eq. (4.31) with 0002trest D 6 months and 0003 D 180ı, the time interval between receiving the signals on Earth is 0002trest .1 0002 u=c/ 0002tobs D p D 2 months: 1 0002 u2 =c 2 The rest of the signals arrive on Earth at tE (months) D 110; 112; 114; 116; 118; 120: The Earth observer has received 12 signals (not including tE D 0), and so concludes that 6 years have passed onboard the starship during the round-trip journey to ˛ Centauri, although ten years have passed on Earth. According to the starship pilot, while outbound the starship receives a signal when tS (months) D 0; 18; 36: At that point, the starship immediately reverses direction and travels back toward Earth. Again according to Eq. (4.31) with 0002trest D 6 months and 0003 D 180ı, during the return trip the time interval between receiving the signals aboard the starship is 2 months. So the starship receives a signal from Earth when tS (months) D 38; 40; 42; 44; 46; 48; 50; 52; 54; 56; 58; 60; 62; 64; 66; 68; 70; 72: At that point, the starship arrives at Earth. The starship has received 20 signals (not including tS D 0), so the starship pilot deduces that 10 years have passed on Earth during the round-trip journey to ˛ Centauri, although the pilot has aged only six years. 4.8 The redshift parameter for the quasar Q2203C29 is (Eq. 4.34) zD
0007obs 0002 0007rest 656:8 nm 0002 121:6 nm D D 4:40: 0007rest 121:6 nm
From Eq. (4.38), the speed of recession of the quasar is vr =c D 0:9337. 4.9 The redshift parameter for the quasar 3C 446 is z D 1:404. From Eq. (4.37), if the quasar’s luminosity is observed to vary over a time of 0002tobs D 10 days, then the time for the luminosity variation as measured in the quasar’s rest frame is 0002tobs 0002trest D D 4:16 days: zC1 4.10 Taking a differential of Eqs. (4.16) and (4.19) gives dx 0002 u dt dx 0 D p 1 0002 u2 =c 2 and
dt 0002 u=c 2 dx dt 0 D p : 1 0002 u2=c 2
Solutions for An Introduction to Modern Astrophysics
21
Dividing the first of these equations by the second, we find dx 0 dx 0002 u dt dx=dt 0002 u D D : 0 2 dt dt 0002 u=c dx 1 0002 .u=c 2 /.dx=dt/ Substituting vx0 D dx 0 =dt 0 and vx D dx=dt then produces Eq. (4.40). In a similar manner, the differential version of Eq. (4.17) is dy 0 D dy. Dividing this by the dt 0 equation yields p p dy 0 dy 1 0002 u2 =c 2 dy=dt 1 0002 u2 =c 2 D D : dt 0 dt 0002 u=c 2 dx 1 0002 .u=c 2 /.dx=dt/ Substituting vy0 D dy 0 =dt 0 , vx D dx=dt and vy D dy=dt then produces Eq. (4.41). Equation (4.42) is obtained in the same way. 4.11
(a)
As measured in frame S 0 , the spacetime interval is .0002s 0 /2 D .c0002t 0 /2 0002 .0002x 0 /2 0002 .0002y 0 /2 0002 .0002z 0 /2 : Using Eqs. (4.16)–(4.19), we have #2 ' c.0002t 0002 u0002x=c 2/ 0 2 .0002s / D p 0002 1 0002 u2 =c 2
0002x 0002 u0002t p 1 0002 u2=c 2
!2 0002 .0002y/2 0002 .0002z/2 :
After some algebra, this becomes .0002s 0 /2 D or
.c 2 0002 u2 /.0002t/2 .u2 =c 2 0002 1/.0002x/2 C 0002 .0002y/2 0002 .0002z/2 : 1 0002 u2 =c 2 1 0002 u2 =c 2
.0002s 0 /2 D .c0002t/2 0002 .0002x/2 0002 .0002y/2 0002 .0002z/2 D .0002s/2 :
(b) The time interval between two events that occur at the same location is the proper time interval, 0002 . Setting 0002x D 0, 0002y D 0, and 0002z D 0 and substituting 0002t D 0002 in the expression for the interval, we find 0002s D c0002 . The first event could have caused the second event because light had more than enough time to travel the zero distance between the two events. (c) If .0002s/2 D 0, then q c0002t D
.0002x/2 C .0002y/2 C .0002z/2 :
This says that light had exactly enough time to travel the distance between the two events. The first event could not have caused the second event unless light itself carried the causal information. (d) The proper length (which we will call 0002L) of an object is obtained by measuring the difference in the 2 2 2 2 positions of its ends at the same time. Setting 0002t D q 0 and substituting .0002L/ D .0002x/ C.0002y/ C.0002z/ in the expression for the interval, we find 0002L D
4.12
0002 .0002s/2 .
(a) This is a straightforward substitution. (b) From the velocity transformations, Eqs. (4.40)–(4.42) c sin 0003 cos 0002 u 1 0002 u=c 2 .c sin 0003 cos / p c sin 0003 sin 1 0002 u2 =c 2 0 vy D 1 0002 u=c 2.c sin 0003 cos / p c cos 0003 1 0002 u2=c 2 0 vz D 1 0002 u=c 2 .c sin 0003 cos /
vx 0 D
22
Chapter 4 Substituting this into v 0 D
q
The Theory of Special Relativity
vx 0 2 C vy 0 2 C vz 0 2 results in, after much manipulation,
v0 D
c 1 0002 u=c.sin 0003 cos / 1=2 u2 2u ; sin 0003 cos C 2 sin2 0003 cos2
0003 10002 c c
which is just v 0 D c. 4.13 Identify the frame S with Earth and frame S 0 with starship A, so vA D u D 0:8c. The velocity of starship B as measured from Earth is vB D 00020:6c. Then the velocity of starship B as measured from starship A may be found using the velocity transformation (Eq. 4.40) vB 0 D
vB 0002 u D 00020:946c: 1 0002 uvB =c 2
The velocity of starship A as measured from starship B is then C0:946c. 4.14 According to Newton’s second law, taking a time derivative of the relativistic momentum (Eq. 4.44) gives the force acting on a particle of mass m. ! d mv dp : p D dt dt 1 0002 v 2 =c 2 Writing v 2 D v 0002 v and using F D d p=dt, we find that m d v=dt dv mv=c 2 FD p C0004 00053=2 v 0002 dt : 1 0002 v 2 =c 2 1 0002 v 2 =c 2 To simplify this, it is useful to take the vector dot product of this equation with the velocity vector. mv 0002 d v=dt dv mv 2 =c 2 F0002v D p v 0002 C0004 ; 00053=2 dt 1 0002 v 2 =c 2 1 0002 v 2 =c 2 which reduces to
0004 00053=2 1 0002 v 2 =c 2 dv v0002 D F 0002 v: dt m Using this to replace v 0002 d v=dt in the expression for F and solving for a D d v=dt leads to the desired result, aD 4.15
(a)
F v .F 0002 v/ : 0002 m mc 2
The particle starts at rest, so the velocity, force, and acceleration are in the same direction. In this case, the result of Problem 4.14 simplifies to 3=2 F dv F v2 v2 aD : D 10002 2 D 10002 2 dt m c m c Now integrate
Z
v 0
00023=2 Z t v2 F 10002 2 dv D dt c 0 m
Solutions for An Introduction to Modern Astrophysics
23
to find
Ft v : Dp m 1 0002 v 2 =c 2
Solving for v=c, we get
.F=m/t v : D p c .F=m/2 t 2 C c 2 (b) Solving the above equation for the time t yields tD
1 v p : F=m 1 0002 v 2 =c 2
If F=m D g D 980 cm s00022 , then the times to achieve the given speeds are t.0:9c/ D 6:32 0003 107 s (2:0 yr) t.0:99c/ D 2:15 0003 108 s (6:8 yr) t.0:999c/ D 6:84 0003 108 s (21:7 yr) t.0:9999c/ D 2:16 0003 109 s (68:6 yr) t.1:0c/ D 1 s (never!) 2 2 2 4.16 From Eq. (4.45), setting the relativistic kinetic energy pequal to the rest energy mc gives mc . 0002 1/ D mc , so D 2. Solving for velocity shows that v=c D 1= 2 D 0:866.
4.17 From Eq. (4.45), the relativistic kinetic energy is K D mc
2
p
1 1 0002 v 2 =c 2
! 00021 :
At low speeds, v=c 0006 1, and we can use the first-order Taylor series .1 0002 x 2/00021=2 ' 1 C x 2=2. Thus, for low speeds, 0002 0003 1 v2 1 K ' mc 2 1 C 0002 1 D mv 2 : 2 c2 2 4.18 Begin with Eq. (4.46) for the total relativistic energy, E D mc 2 . Squaring both sides and subtracting m2 c 4 from each side results in 0006 0007 E 2 0002 m2 c 4 D m2 c 4 2 0002 1 D mc 2 . C 1/Œmc 2 . 0002 1/ : From Eq. (4.48), the left-hand side is equal to p 2c 2 , while the term on the right-hand side in square brackets is the relativistic kinetic energy (Eq. 4.45). So p 2 c 2 D mc 2 . C 1/K; Solving for K gives KD
p2 : .1 C /m
4.19 According to Eqs. (4.46) and (4.44), the total relativistic energy and relativistic momentum are E D mc 2 and p D mv. Multiplying the momentum by c, then squaring both equations and subtracting produces v2 2 2 2 2 2 4 2 2 2 2 2 2 4 E 0002p c D m c 0002 m v c D m c 10002 2 : c Because 2 D 1=.1 0002 v 2 =c 2 /, this reduces to E 2 D p 2 c 2 C m2 c 4 , which is the desired result.
CHAPTER 5
The Interaction of Light and Matter 5.1
(a)
The rest wavelength for the H˛ spectral line is 0007rest D 656:281 nm. From Eq. (5.1), the radial velocity of Barnard’s star is c .0007obs 0002 0007rest/ D 0002113 km s00021 : vr D 0007rest
(b) From Eq. (3.1), d D .1=p 00 / pc D 1:82 pc. The transverse velocity of Barnard’s star is (Eq. 1.5) v0002 D d 0005 D 89:4 km s00021 . q (c) v D vr2 C v00022 D 144 km s00021 . 5.2
(a)
The distance between the grating’s lines is d D 3:33 0003 1000026 m. Using d sin 0003 D n0007 with n D 2, the angles are 20:695ı for 0007 D 588:997 nm, and 20:717ı for 0007 D 589:594 nm. The angle between the ˚ second-order spectra of these two lines is therefore 00020003 D 0:0219 A.
(b) From Eq. (5.2), the number of lines that must be illuminated is N D
0007 D 494; n00020007
where the average wavelength 0007 D 589:296 nm was used. 5.3 Using the values provided in Appendix A, it is found that hc D 1239:8 eV nm ' 1240 eV nm. 5.4 From Eq. (5.4),
D
1240 eV nm hc 0002 Kmax D 0002 5 eV D 7:4 eV: 0007 100 nm
5.5 Referring to Figure 5.3, conservation of momentum gives pi D pf cos 0003 C pe cos
0 D pf sin 0003 0002 pe sin ; where pi and pf are the initial and final photon momenta, respectively, and pe is the momentum of the recoiling electron. Rewriting these equations so that only terms involving the angle are on the right-hand side, and then squaring both equations and adding produces pi2 0002 2pi pf cos 0003 C pf2 D pe2 : Now multiply each side by c 2 and add m2e c 4 to both sides. Using Ei D pi c and Ef D pf c (Eq. 5.5) for the initial and final photon energies, we find m2e c 4 C Ei2 0002 2Ei Ef cos 0003 C Ef2 D pe2 c 2 C m2e c 4 D Ee2 ; where the last term on the right comes from Eq. (4.48) for the total relativistic energy of the electron. However, the expression for the conservation of the total relativistic energy, Ei C me c 2 D Ef C Ee 24
Solutions for An Introduction to Modern Astrophysics
25
can be used to replace Ee in the preceding equation, m2e c 4 C Ei2 0002 2Ei Ef cos 0003 C Ef2 D .Ei 0002 Ef C me c 2 /2 : This simplifies to 1 1 1 0002 D .1 0002 cos 0003/: Ef Ei me c 2 Using Eq. (5.5) for the photon energy, E D hc=0007, results in 00020007 D 0007f 0002 0007i D
h .1 0002 cos 0003/: me c
5.6 The change of wavelength is given by the Compton scattering formula, Eq. (5.6), but with the electron mass replaced by the proton mass, h .1 0002 cos 0003/: 00020007 D 0007f 0002 0007i D mp c The characteristic change in wavelength is h=mp c D 1:32 0003 1000026 nm. This is smaller than the value of the Compton wavelength, 0007C D h=me c D 0:00243 nm, by a factor of me =mp D 5:4 0003 1000024 . 5.7 Planck’s constant has units of
J s D [kg m2 s00022 ] s D kg [m s00021 ] m;
which are the units of angular momentum (mvr ). 5.8
(a)
For an atom with Z protons in the nucleus (charge Ze) and one electron (charge 0002e), Coulomb’s law (Eq. 5.9) becomes 1 Ze 2 FD0002 rO ; 40004 0 r 2 and the electrical potential energy becomes U D0002
1 Ze 2 : 40004 0 r
This means that the quantity “e 2 ” should be replaced by “Ze 2 ” wherever it appears in an expression for the hydrogen atom. Similarly, “e 4 ” should be replaced by “Z 2 e 4 .” The result will be the corresponding expression for a one-electron atom with Z protons. From Eq. (5.13), the orbital radii are therefore rn D
40004 0 00022 2 a0 2 n D n ; Z 0005Ze 2
where a0 D 0:0529 nm. The energies are En D 0002
0005Z 2 e 4 1 Z2 D 000213:6 eV 2 2 2 n2 2 n 320004 0 0002
from Eq. (5.14). (b) Z D 2 for singly ionized helium (He II), and so for the ground state (n D 1), r1 D
a0 2 .1 / D 0:0265 nm 2
and E1 D 000213:6 eV The ionization energy is 54:4 eV.
22 D 000254:4 eV: 12
26
Chapter 5 (c)
The Interaction of Light and Matter
Z D 3 for doubly ionized lithium (Li III), and so for the ground state, r1 D
a0 2 .1 / D 0:0176 nm 3
and E1 D 000213:6 eV
32 D 0002122 eV: 12
The ionization energy is 122 eV. 5.9
(a)
If the hydrogen atom were held together solely by the force of gravity, the force between the proton and electron would be supplied by Newton’s law of universal gravitation (Eq. 2.11), F D 0002G
mp me rO ; r2
and the potential energy is given by Eq. (2.14), U D 0002G
mp me : r
This means that the quantity “e 2 =40004 0” should be replaced by “Gmp me ” wherever it appears in an expression for the hydrogen atom. Similarly, “e 4 =160004 2 02 ” should be replaced by “G 2 mp2 m2e .” The result will be the corresponding expression for the gravitational atom. From Eq. (5.13), the ground-state orbital radius (n D 1) is therefore r1 D
00022 D 1:20 0003 1038 nm D 8:03 0003 1017 AU: 0005Gmp me
The ground-state energy is E1 D 0002
0005G 2 mp2 m2e 200022
D 00022:64 0003 10000278 eV:
from Eq. (5.14). 5.10 Use 1=0007 D RH .1=n2low 0002 1=n2high/ (Eq. 5.8 with m D nlow and n D nhigh) for the hydrogen wavelengths, and Ephoton D hc=0007 (Eq. 5.5) for the photon energy. The possible transitions are nhigh D 3 ! nlow D 1, nhigh D 3 ! nlow D 2, and nhigh D 2 ! nlow D 1. This results in 0007 D 102:6 nm
and Ephoton D 12:09 eV
(for 3 ! 1);
0007 D 656:5 nm
and Ephoton D 1:89 eV
(for 3 ! 2);
0007 D 121:6 nm
and Ephoton D 10:20 eV
(for 2 ! 1):
5.11 Use 1=0007 D RH .1=n2low 0002 1=n2high/ (Eq. 5.8 with m D nlow and n D nhigh ) for the hydrogen wavelengths. The shortest wavelengths emitted for the Lyman series (nlow D 1), Balmer series (nlow D 2), and Paschen series (nlow D 3) occur for nhigh D 1. The series limits are therefore given by 00071 D
1 2 n D .91:18 nm/ n2low : RH low
So 00071 D 91:18 nm
(ultraviolet)
Solutions for An Introduction to Modern Astrophysics
27
for the Lyman series limit, 00071 D 364:7 nm (near ultraviolet) for the Balmer series limit, and 00071 D 820:6 nm
(infrared)
for the Paschen series limit. 5.12 The wavelength of the electron is given by Eq. (5.17), 0007D
h h D D 1:45 0003 10000211 m D 0:0145 nm: p me v
5.13 If the circumference of the electron’s orbit is an integral number, n, of de Broglie wavelengths as shown in Fig. 5.15, then from Eq. (5.17) nh n0007 D D 20004 r: p The momentum is p D 0005v (using the reduced mass), so this may be expressed as 0005vr D nh=20004 or L D n0002, Bohr’s condition for the quantization of angular momentum. 5.14 As shown in Example 5.4.2, the minimum speed of a particle (in this case, an electron) can be estimated using Heisenberg’s uncertainty relation, Eq. (5.19), as vmin D
pmin 0002p 0002 0004 0004 : me me me 0002x
With 0002x 0004 1:5 0003 10000212 m, the minimum speed of the electron is vmin 0004 7:7 0003 107 m s00021 , about 26% of the speed of light. Relativistic effects are important for white dwarf stars. 5.15
(a)
From Eq. (5.20) with 0002t 0004 1000028 s, for the first excited state (n D 2) of hydrogen 0002E2 0004
0002 D 1:1 0003 10000226 J D 6:9 0003 1000028 eV: 0002t
(b) The wavelength of a photon involved in a transition between the ground state (n D 1) and the first excited state (n D 2) is given by Ephoton D E2 0002 E1 . From Eq. (5.3), hc D E2 0002 E1 : 0007 The uncertainty in 0007 caused by an uncertainty in E2 is found by taking a derivative with respect to E2 (holding E1 constant): hc d 0007 0002 2 D 1: 0007 dE2 Writing 00020007 instead of d 0007, 0002E2 instead of dE2, and ignoring the minus sign gives 00020007 D
00072 0002E2 : hc
Similarly, the uncertainty in 0007 due to an uncertainty in E1 is 00020007 D
00072 0002E1 : hc
28
Chapter 5
The Interaction of Light and Matter
However, an electron can spend an arbitrarily long time in the ground state. For that reason, 0002t is infinite and so 0002E1 0004 0002=0002t D 0. Thus 00072 00020007 0004 0002E2 : hc From Eq. (5.8) with m D 1 and n D 2, the wavelength of the photon involved in this transition is 121.6 nm, so .121:6 nm/2 00020007 0004 .6:9 0003 1000028 eV/ D 8:2 0003 1000027 nm: 1240 eV nm 5.16
(a)
The answers for n D 1 and n D 2 are in Table 8.2. For n D 3, the quantum numbers are nD3 nD3 nD3 nD3 nD3 nD3 nD3 nD3 nD3
`D0 `D1 `D1 `D1 `D2 `D2 `D2 `D2 `D2
m` m` m` m` m` m` m` m` m`
D0 D1 D0 D 00021 D2 D1 D0 D 00021 D 00022
ms ms ms ms ms ms ms ms ms
D ˙1=2 D ˙1=2 D ˙1=2 D ˙1=2 D ˙1=2 D ˙1=2 D ˙1=2 D ˙1=2 D ˙1=2
(b) In Table 8.2, for n D 1 there are 2n2 D 2 entries, and for n D 2 there are 2n2 D 8 entries. In the list above for n D 3, there are 2n2 D 18 entries. These are in agreement with the idea that the degeneracy of energy level n (denoted gn ) is gn D 2n2. In general, if the degeneracy of energy level n is gn D 2n2, then the degeneracy of energy level n C 1 is greater by an additional 2.2n C 1/: the “2n C 1” is for the 2`C 1 values of m` when ` D n, and the leading “2” is for the two values of ms . That is, the degeneracy of energy level n C 1 is gnC1 D gn C 2.2n C 1/ D 2n2 C 2.2n C 1/ D 2.n2 C 2n C 1/ D 2.n C 1/2 : Because we know that gn D 2n2 for n D 1, this proof-by-induction shows that it holds for all values of n. 5.17 From Eq. (5.22), the normal Zeeman effect produces a frequency splitting given by 0002 D
eB D 4:76 0003 1010 Hz: 400040005
Using 0 D c=00070 with 00070 D 656:281 nm shows that the frequencies are 0 C 0002 D 4:56853 0003 1014 Hz 0 D 4:56806 0003 1014 Hz 0 0002 0002 D 4:56758 0003 1014 Hz: The corresponding wavelengths are c D 656:212 nm 0 C 0002 c D 656:281 nm 0 c D 656:349 nm: 0 0002 0002
Solutions for An Introduction to Modern Astrophysics
29
1.0 0.8
N=5 N=11 N=21 N=41
Ψ
0.6 0.4 0.2 0.0 −0.2 0
1
2
3
x
Figure S5.1: Results for Problem 5.18. Four wave pulses, 000e , are constructed from a Fourier sine series. 5.18 See Fig. S5.1 for the graphs of 000e .x/. If we take 0002x to be the width of the graph where 000e D 0:5, then (a)
0002x ' 0:64 when N D 5.
(b) 0002x ' 0:31 when N D 11. (c)
0002x ' 0:18 when N D 21.
(d) 0002x ' 0:09 when N D 41. (e)
The position of the particle is known with the least uncertainty for N D 41, when the graph of 000e is most sharply peaked. The momentum of the particle is known with the least uncertainty for N D 5, when the graph of 000e involves only three wavelengths (and hence only three momenta, p D h=0007).
CHAPTER 6
Telescopes 6.1 ˝tot D 6.2
(a)
H
d˝ D
R 20003 R 0003
0006D0 0002D0
sin 0003 d0003 d D 40004:
Referring to Fig. S6.1, from similar triangles ho hi D p q This implies that
ho hi D : f q0002f
and
ho p f D D : hi q q0002f
Rearranging, 1 1 1 D C : f p q (b) If p 0007 f , then 1 1 0002 ' 0; f q which leads to f ' q. Object
ho
Image θ h
θ φ
φ
f
i
f
p
q
Figure S6.1: The ray diagram for the converging lens discussed in Problem 6.2. 6.3 For the first lens encountered by the incoming light 1 1 1 D 0002 : q1 f1 p1 The image of the first lens becomes the object (virtual) for the second lens, so p2 D 0002q1. Substituting into the previous expression, 1 1 1 D 0002 : p2 p1 f1 Using the relation for the second lens 1 1 1 C D p2 q2 f2
leads to 30
1 1 1 1 C D C : p1 q2 f1 f2
Solutions for An Introduction to Modern Astrophysics
31
Assuming that the two lenses are close together, we can make the approximations that the object and image distances are essentially p ' p1 and q ' q2 . Thus 1 1 1 1 1 ' C : C D p q feff f1 f2 6.4
(a)
From the lensmaker’s formula (Eq. 6.2) for two lenses having different indexes of refraction but the same radii of curvature, 0005 0004 1 1 1 C D n1;0004 0002 1 f1 .0007/ R1 R2 0005 0004 1 1 1 : C D n2;0004 0002 1 f2 .0007/ R1 R2 The effective focal length at a particular wavelength is now found using the result of Problem 6.3, 0004 0005 1 1 1 1 1 00022 : D n1;0004 C n2;0004 C C feff;0004 R1 R2 R1 R2 If feff;00041 D feff;00042 for two specific wavelengths, then n1;00041 C n2;00041 D n1;00042 C n2;00042 ; or
0004 0005 0004 0005 n1;00041 0002 n1;00042 D 0002 n2;00041 0002 n2;00042 :
(b) The focal lengths could remain equal for all wavelengths only if n1;0004 D n2;0004 C constant. 6.5 Referring to Fig. 6.15, let the height of the intermediate image (between the eyepiece and the objective) be h. Then h h and tan D : tan 0003 D fobj feye For small angles, 0003 ' h=fobj and ' h=feye. Thus, the angular magnification is m D =0003 D fobj =feye. ˇ ˇ ˇ ˇ D 0 and .ˇ=2/2 ˇ D 0, l’Hˆopital’s rule must be used to evaluate I.0003/ in the 6.6 (a) Since sin2 .ˇ=2/ˇ 0002D0 0002D0 limit as 0003 ! 0. Differentiating the numerator and evaluating at 0003 D 0, ˇ 0002 0003 0002 0003 ˇ ˇ d sin2 .ˇ=2/ ˇˇ ˇ ˇ 20004D cos cos 0003 ˇˇ D 0: ˇ D sin ˇ d0003 2 2 0007 0 0
Similarly, differentiating the denominator and evaluating at 0003 D 0, ˇ ˇ ˇ d .ˇ=2/2 ˇˇ ˇ 20004D cos 0003 ˇˇ D 0: ˇ D d0003 ˇ 2 0007 0 0
Since the ratio of the first derivatives is still undefined, it is necessary to evaluate the second derivatives. This leads to ˇ d 2 sin2 .ˇ=2/ ˇˇ 20004 2 D 2 D ˇ ˇ d0003 2 00072 0 ˇ d 2 .ˇ=2/2 ˇˇ 20004 2 D 2 D ˇ d0003 2 ˇ 00072 0
Since their ratio is unity, I.0/ D I0 .
32
Chapter 6 Telescopes (b) The first minimum occurs when sin2 .ˇ=2/ D 0, or ˇ D 20004 . This implies that sin 0003 D 0007=D, or 0003 D 30ı . 6.7
(a)
From Eq. (6.6), 0003eye D 1:34 0003 1000024 rad D 2800 .
(b) 0003moon D 2R=d D 9:0 0003 1000023 rad D 190000, implying 0003moon=0003eye D 33:7. When Jupiter is at opposition (closest approach to Earth), 0003Jupiter D 2:3 0003 1000024 rad D 4700, implying 0003Jupiter =0003eye D 1:7. The resolution is worse during all other relative positions of Jupiter and Earth.
6.8
(c)
It is possible for the disk of the Moon to be resolved with the naked eye. However, the disk of Jupiter is comparable to the eye’s resolution limit; to distinguish any features on the surface of Jupiter, they would need to be nearly the size of the planet’s disk.
(a)
From Eq. (6.6), 0003min D 3:4 0003 1000026 rad D 0:6900 .
(b) Given the Moon’s mean distance of 3:84 0003 105 km, the minimum size of a crater that can be resolved is h D d0003 D 1:3 km. 6.9
(c)
Atmospheric turbulence limits the resolution.
(a)
The NTT focal ratio is f =2:2 and the diameter of the primary mirror is 3.58 m. From Eq. (6.7), F D 2:2 D f =3:8 m, giving f D 8:36 m.
(b) From Eq. (6.4), d0003=dy D 1=f D 0:120 rad m00021 . (c)
00020003 D 2:900 D 1:4 0003 1000025 rad, implying 0002y D f 00020003 D 11:8 0005m.
6.10 Each of HST’s WF/PC 2 CCD’s is 800 pixels wide. With a plate scale of 0:045500 pixel00021 in the planetary mode, the angular field of view of a CCD is 36:400 . 6.11
(a)
m D 1:43 GHz, ` D 1:405 GHz, and u D 1:455 GHz.
(b) Using Eq. (6.10) with S./ D S0 D 2:5 mJy and D D 100 m, Z Z S./ f0005 d dA P D A
0005

D 0004S0
D 2
2 Z
0005u 0005`
f0005 d:
Substituting the functional form for the filter function and evaluating the integral gives P D 0004S0 (c)
D 2
2 0002
0003 1 .u 0002 ` / D 4:91 0003 10000218 W: 2
At a distance d from the source, the power at the telescope is given by Prec D Pemit
0004.D=2/2 : 40004 d 2
Solving for the emitted power, Pemit D 7:5 0003 1042 W. 6.12 One dish has a diameter of 25 m. Thus, the total collecting area of the 27 dishes is 27 0003 0004.25 m=2/2 D 1:325 0003 104 m2 . If a single dish is to have the same collecting area, it must have a diameter of D D 130 m. 6.13 From Eq. (6.11), with sin 0003 ' 0003, L D 0007 D 0:21 m, and d D 2R˚ D 1:28 0003 106 m, 0003 D 1:65 0003 1000028 rad D 0:003400 .
Solutions for An Introduction to Modern Astrophysics
33
6.14 Consider numbering each dish from 1 to 50. For dish 1 there are 49 distinct baselines, one for each pair of dishes. For dish 2 there are 48 additional baselines (we don’t want to recount the baseline between 1 and 2). For dish 3 there are 47 additional baselines, and so on. The total number becomes 49 C 48 C C 1 D 1125. 6.15 The projected resolution of SIM PlanetQuest is 4 0005arcsec D 0:00000400 D 2 0003 10000211 radians. (a)
If grass grows at a rate of 2 cm per week, this is equivalent to 3:300031000028 m s00021 . From a distance of 10 km, SIM PlanetQuest can resolve a width of D D 0003d D 0:2 0005m. It would take the grass approximately 6 s to grow the necessary length to be measured!
(b) Assuming a resolution of 4 0005as, and a baseline of 2 AU, the measureable distance is d D 2 AU=4 0005as D 500 kpc. (c)
From Eq. (3.6) and using MV D 4:82, V D 28:3.
(d) Assuming a limit of V D 20, Eq. (3.6) implies that Betelgeuse would be detectable from a distance of d D 1 Mpc. 6.16 Given the number of telescopes the student is asked to research, it may be useful to have the class split into teams and have each team subdivide the research. (a)
See Table S6.1.
(b) See Fig. S6.2. (c)
See Fig. S6.3.
Table S6.1: The wavelength ranges and energies of some ground-based and space-based telescopes. Numbers in parentheses represent power of 10; e.g., 3(-7) D 3 0003 1000027 . Telescope VLA ALMA Spitzer (SIRTF) JWST VLT/VLTI Keck HST (Hubble) IUE EUVE Chandra CGRO
Spectrum region radio radio IR IR to visible mid-IR to near-UV mid-IR to near-UV near-IR to near-UV UV UV X-ray gamma rays
Wavelength range (m) 4 to 7(-3) 1(-2) to 3.5(-4) 1.8(-4) to 3(-6) 2.7(-5) to 6(-7) 2.5(-5) to 3(-7) 2.5(-5) to 3(-7) 2.5(-5) to 1.15(-7) 3.2(-7) to 1.15(-7) 7.6(-8) to 7(-9) 6.2(-9) to 1(-10) 4(-11) to 4(-17)
Energy range (eV) 3(-7) to 2(-4) 1(-4) to 4(-3) 0.007 to 0.4 0.05 to 2 0.05 to 4 0.05 to 4 0.05 to 11 3.9 to 11 16 to 177 20 to 1(4) 3(4) to 3(10)
URL www.vla.nrao.edu www.alma.info/ www.spitzer.caltech.edu/spitzer www.jwst.nasa.gov www.eso.org/vlt www.keckobservatory.org www.stsci.edu archive.stsci.edu/iue www.ssl.berkeley.edu/euve/sci/EUVE.html chandra.harvard.edu heasarc.gsfc.nasa.gov/docs/cgro
Figure S6.2: The photon wavelengths covered by some ground-based and space-based telescopes. 6.17
(a)
See Fig. S6.4.
(b) See Fig. S6.4.
34
Chapter 6 Telescopes
Figure S6.3: The photon energies covered by some ground-based and space-based telescopes.
1.0
I/I0
0.8 0.6 0.4 0.2 0.0
−15
−10
−5
0 5 β (radians)
10
15
(a) 1.0
I/I0
0.8 0.6 0.4 0.2 0.0
−15
−10
−5
0 β (radians)
5
10
15
(b) Figure S6.4: Solutions for Problem 6.18(a) and (b). The solid lines represent the sum of the intensity functions of each slit.
Solutions for An Introduction to Modern Astrophysics (c)
35
As the sources get progressively closer together, a separation angle is reached where the sum of the two source intensities no longer yields a definable minimum, and it is no longer possible to resolve the sources.
CHAPTER 7
Binary Systems and Stellar Parameters 7.1 Referring to Figs. 2.11 and 2.12, r D r2 0002 r1 . Since r1 and r2 are oppositely directed relative to the center of mass, the separation between m1 and m2 is actually r D r1 C r2 . From Eq. (7.1), r2 =r1 D a2 =a1 , where a1 and a2 are the semimajor axes of the stars’ elliptical orbits. Substituting, a1 C a2 : r D r2 a2 Now, using Eq. (2.3) to represent r2 in terms of 0003, r D .a1 C a2 /
1 0002 e2 1 C e cos 0003
:
From this result, rp D r .0ı / D .a1 C a2 /.1 0002 e/ and ra D r .180ı/ D .a1 C a2 /.1 C e/, and the average of the two extremes is the semimajor axis of the reduced mass, aD 7.2
(a)
0005 10004 rp C ra D a1 C a2 : 2
The probability of detecting the radial velocity variations is greatest when the i D 90ı, while the probability goes to zero when i D 0ı . This variation can be described by a sine function, or w.i / D sin i . This weighting function is normalized as seen from evaluating Z 0003=2 sin i d i D 1: 0
(b) The weighted average of sin3 i is now Z
3
hsin i i D
0003=2
Z D
0003=2
D 7.3
(a)
sin4 i d i
0
D
sin3 i sin i d i
0
ˇ sin 4i ˇˇ0003=2 3i sin 2i 0002 C 8 4 32 ˇ0
30004 : 16
For the smallest angle, a cos i D r1 C r2 , implying 00021 r1 C r2 i D cos : a
(b) i D 88:5ı . 36
Solutions for An Introduction to Modern Astrophysics 7.4
(a)
37
For p 00 D 0:3792100, the distance to Sirius is d D 1=p 00 D 2:63 pc. This means that the linear size of the semimajor axis of the reduced mass is approximately a D ˛d D 3:00 0003 1012 m. Using Kepler’s third law (Eq. 2.37), 40004 2 3 a D 3:24 Mˇ : mA C mB D GP 2 Since mA aA D mB aB , we have that mA .1 C aA =aB / D 3:24 Mˇ . Solving for the individual masses, mA D 2:21 Mˇ and mB D 1:03 Mˇ .
(b) From Eq. (3.7), with the bolometric magnitude of the Sun taken to be 4:74, LA D 22:5 Lˇ and LB D 0:0240 Lˇ .
7.5
(c)
Using the Stefan–Boltzmann equation (3.17), R D 5:85 0003 106 m D 0:0084 Rˇ D 0:917 R˚ .
(a)
From Eq. (7.6),
0004
0005 00053 P 0004 vbr C vf r D 8:65 Mˇ ; mb C mf sin3 i D 20004 G
where the subscripts denote the bright (b) and faint (f ) components, respectively. From Eq. (7.5), mb D 2:03mf , implying mf sin3 i D 2:85 Mˇ
and
mb sin3 i D 5:80 Mˇ :
mf ' 4:3 Mˇ
and
mb ' 8:7 Mˇ :
(b) Using hsin3 i i 0005 2=3, 7.6
(a)
From Eq. (7.5), mB =mA D 0:241.
(b) From Eq. (7.6), mA C mB D 5:13 Mˇ . (c)
mA D 4:13 Mˇ and mB D 1:00 Mˇ .
(d) According to Eqs. (7.8) and (7.9), rs D
.vA C vB / .tb 0002 ta / D 1:00 Rˇ ; 2
and r ` D rs C
.vA C vB / .tc 0002 tb / D 2:11 Rˇ ; 2
respectively. (e)
Brightness ratios can be determined from Eq. (3.3), giving (for the primary and secondary eclipses, respectively) Bp Bs D 0:0302 and D 0:964: B0 B0 Finally, using Eq. (7.11), Ts D T`
7.7
(a)

1 0002 Bp =B0 1 0002 Bs =B0
1=4 D 2:28:
The ratio of the durations of eclipses of minimum to maximum is approximately 0.9. (Lacy, Astron. J., 105, 637, 1993 finds the ratio of the radii to be 0:907 ˙ 0:015.) Note: In order to accurately obtain the ratio of the radii, the light curve must be carefully modeled, a process beyond the scope of this text. This part of the problem may be modified or omitted in a future revision.
38
Chapter 7 Binary Systems and Stellar Parameters (b) From the light curve m0 D 10:04, mp D 10:76, and ms D 10:68. Using the procedure outlined in the solution to Problem 7.6, Bs Bp D 0:515 and D 0:555: B0 B0 This implies that Ts =T` D 1:090. 7.8
(a)
From upper left to lower right, the displayed images roughly correspond to the following phases: 0.00, 0.06, 0.13, 0.19, 0.25, 0.31, 0.38, 0.44, 0.50.
(b) In this binary system, the larger star is hotter. At the primary minimum (phase = 0.0), the maximum amount of the larger star is covered, implying the least total integrated flux (or luminosity). The primary minimum is not flat because the proximity of the secondary star leads to tidal distortion of the shapes of the stars, thus removing spherical symmetry. The brightness of the system increases until, at phase 0.25, the two stars are seen in profile. As the secondary passes behind the primary, the brightness decreases, reaching a flat minimum at phase 0.5. The secondary minimum is flat because the secondary (and the distortion effects) are occulted for a finite period of time. 7.9 See Fig. S7.1.
Figure S7.1: The relationship between effective temperature and mass for main sequence stars. 7.10 Massive planets with short orbital periods exert a greater force on the parent star, thus producing higher radial velocities of the stars themselves. In addition, shorter periods imply that a greater number of orbital periods can be measured over a given length of time. 7.11 From Eq. (7.7), it is evident that only m sin i can be determined, unless the angle of inclination can be determined independently (meaning that the system must be eclipsing). 7.12
(a)
For the planet orbiting 51 Peg, P D 4:23077 d, and a D 0:051 AU. From Kepler’s third law, the total mass of the system is M D 0:99 Mˇ . Since m sin i D 0:45 MJ 0006 M , the mass of 51 Peg is approximately 0:99 Mˇ .
(b) For the planet orbiting HD 168443c, P D 1770 d, and a D 2:87 AU. Again, from Kepler’s third law, M D 1:01 Mˇ . We can again neglect the mass of the planet.
Solutions for An Introduction to Modern Astrophysics
39
7.13 Considering the integrated flux over the projected solar disk, we have 0004R2ˇ 0006Te4 . The amount of light blocked by the transiting Jupiter is 0004R2J 0006Te4 . Thus the relative change becomes R2 0002F D 2J D 0:0106 F Rˇ or just over 1%. 7.14 The fractional decrease in the flux appears to be approximately 1 00020:983 D 0:017. Using an approach similar to the solution p of Problem 7.13, the ratio of the radius of the planet to the radius of the star is approximately Rp =Rs D 0:017 D 0:13. The radius of the star is estimated to be Rs D 1:1 Rˇ , implying that the radius of the planet is approximately Rp D 0:13Rs D 1:27 RJ , in agreement with the value found in the text.
Figure S7.2: The variation of orbital velocity with eccentricity is plotted for one orbital period (Problem 7.15). The eccentricities are labeled on the diagram. M1 D 0:5 Mˇ , M2 D 2:0 Mˇ , P D 1:8 yr D 657:46 d, i D 30ı, D 90ı . 7.15
(a)
See Fig. S7.2. [Note: The effective temperatures and radii of the stars are irrelevant for this problem; any reasonable (and many unreasonable) values can be entered.]
(b) From Kepler’s third law (Eq. 2.37), with P D 1:8 yr, M1 D 0:5 Mˇ and M2 D 2:0 Mˇ , a D 2:0 AU. Also, since m1 =m2 D a2 =a1 (Eq. 7.1) and a D a1 C a2 D 2:0 AU, we have a1 D 1:6 AU and a2 D 0:4 AU. Now, the orbital velocities of the two stars are v1 D 20004 a1=P D 26:5 km s00021 and v2 D 20004 a2=P D 6:6 km s00021 . Finally, taking into consideration the orbital inclination of 30ı, the maximum observable radial velocities are v1r; max D v1 sin i D 13:2 km s00021 and v2r; max D v2 sin i D 3:351 km s00021 . (c)
The eccentricity can be determined by considering the asymmetries in the velocity and/or light curves. In particular, the velocity curves deviate from being perfectly sinusoidal (the amount of deviation increasing with increasing e). Also, for an eclipsing system, the primary and secondary minima are not 180ı out of phase if e ¤ 0 (see e.g. Fig. 7.2).
7.16 See Fig. S7.3. 7.17
(a)
See Fig. S7.4. Note that the relative phasing of minima are essentially correct, but that the relative depths of the minima differ somewhat from Fig. 7.2.
40
Chapter 7 Binary Systems and Stellar Parameters
Figure S7.3: The motion of the binary system described in Problem 7.16.
Figure S7.4: The synthetic light curve of YY Sgr produced by TwoStars; Problem 7.17(a). (b) See Fig. S7.5. 7.18 Using the data in the text and the problem, the mass of the star can be determined from Kepler’s third law to be approximately 1:1 Mˇ . Take the radius of the star to be roughly 1:1 Rˇ for a main-sequence star. The mass of the planet is estimated to by 0:00086 Mˇ . From the data, TwoStars generates a bolometric light curve shown in Fig. S7.6. Note that the drop is just over 0.01 mag, in agreement with the value specified in the text.
Solutions for An Introduction to Modern Astrophysics
Figure S7.5: The radial velocities for both members of YY Sgr produced by TwoStars; Problem 7.17(b).
Figure S7.6: The synthetic light curve of OGLE-TR-56b for Prob. 7.18.
41
CHAPTER 8
The Classification of Stellar Spectra 8.1 In the following, it is easiest to express Boltzmann’s constant as k D 8:6174 0003 1000025 eV K00021 . At a room temperature of 300 K, 1 kT D 0:0259 eV 0004 eV: 40 If kT D 1 eV, then T D 1:16 0003 104 K. If kT D 13:6 eV, then T D 1:58 0003 105 K. 8.2 A straightforward conversion of units shows that k D 8:6173423 0003 1000025 eV K00021 : 8.3 The peak of the Maxwell–Boltzmann distribution, Fig. 8.6, shows that nv =n ' 6:5 0003 1000025 s m00021 where v D vmp . For a range of speeds 0002v D 2 km s00021 , the fraction of hydrogen atoms within 1 km s00021 of the most probable speed is .nv =n/0002v ' 0:13. 8.4 The most probable speed, vmp , occurs at the peak of the Maxwell–Boltzmann distribution, Eq. (8.1). Setting d nv =dv D 0, we find 0007 0006 m 00073=2 d 0006 d nv 2 e 0002mv =2kT v 2 D 0: D 40004 n dv 20004 kT dv This leads to mv 3 2 0002 C 2v e 0002mv =2kT D 0; kT p so that vmp D 2kT =m, which is Eq. (8.2). 8.5 Solving the Boltzmann equation, Eq. (8.6), for T gives T D
E2 0002 E1 : k ln Œ.g2 =g1 / = .N2 =N1 /
To find T when the number of atoms in the first excited state (n D 2) is only 1% of the number of atoms in the ground state (n D 1), use g1 D 2.1/2 D 2, g2 D 2.2/2 D 8, E1 D 000213:6 eV, and E2 D 00023:40 eV. Then, with N2 =N1 D 0:01, T D 1:97 0003 104 K. If the number of atoms in the first excited state is only 10% of the number of atoms in the ground state, then T D 3:21 0003 104 K. 8.6
(a)
We solve the Boltzmann equation, Eq. (8.6), for T : T D
E3 0002 E1 : k ln Œ.g3 =g1 / = .N3 =N1 /
Using the values in Problem 8.5 together with g3 D 2.3/3 D 18 and E3 D 00021:51 eV, we find that if N3 =N1 D 1, then T D 6:38 0003 104 K. (b) From the Boltzmann equation using T D 85,400 K, N3 =N1 D N3 =N D 1:74. 42
Solutions for An Introduction to Modern Astrophysics (c)
43
As T ! 1, the exponential in the Boltzmann equation goes to unity and Nb =Na ! gb =ga . The relative numbers of electrons in the n D 1; 2; 3; : : : orbitals will be gn D 2n2 D 2; 8; 18; : : :. Although this will be the distribution that actually occurs for the neutral hydrogen atoms, at such high temperatures essentially all of the hydrogen atoms will have been ionized.
8.7 From Eq. (8.7), use gn D 2n2 and E1 D 000213:6 eV, E2 D 00023:40 eV, and E3 D 00021:51 eV with T D 10,000 K to get ZI D g1 C g2 e 0002.E2 0002E1 /= kT C g3 e 0002.E30002E1 /= kT D 2 C 5:79 0003 1000025 C 1:45 0003 1000025 : Thus ZI ' 2. 8.8 As n ! 1, the expression for the partition function, Eq. (8.7), actually diverges. However, in reality the large-n terms can be ignored because as n ! 1, the atomic orbitals overlap with those of neighboring atoms, and the electrons are no longer associated with a single nucleus. 1.0 0.9 0.8
NII/Nt
0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 5000
10,000
15,000 T (K)
20,000
25,000
Figure S8.1: Results for Problem 8.9. NII =N t is shown for a box of hydrogen gas. 8.9
(a)
Start by writing
or
NII NII NII =NI D D ; Nt NI C NII 1 C NII =NI NII NII NII 1C D : Nt NI NI
NII =NI is given by the Saha equation, Eq. (8.8), NII 2ZII D NI ne ZI

20004 me kT h2
3=2
e 00020007I = kT :
From Example 8.1.4, ZI D 2 and ZII D 1. The electron number density, ne , is given by ne D NII =V because each ionized hydrogen atom contributes one free electron. The total number of hydrogen atoms and ions is N t D 000fV =.mp C me / ' 000fV =mp . Here, 000f D 1000026 kg m00023 is the density. The mass of the
44
Chapter 8 The Classification of Stellar Spectra electron is much less than the mass of the proton, and may be safely ignored in the expression for N t . Combining these expressions produces NII 000f ne D N t mp for use in the Saha equation. With these substitutions, the Saha equation becomes NII N t mp D NI NII 000f

20004 me kT h2
3=2
e 00020007I = kT :
Substituting this into the above equation produces ' # N t mp 20004 me kT 3=2 00020007I = kT N t mp 20004 me kT 3=2 00020007I = kT NII e e : 1C Nt NII 000f h2 NII 000f h2 Multiplying each side by NII =N t and rearranging terms gives
NII Nt
2
C
NII Nt
mp 000f

20004 me kT h2
3=2
e 00020007I = kT 0002
mp 000f

20004 me kT h2
3=2
e 00020007I = kT D 0:
(b) This is a quadratic equation of the form ax 2 C bx C c D 0, where aD1 bD
mp 000f

20004 me kT h2
3=2
e 00020007I = kT
c D 0002b: It therefore has the solution 0007 0007 p NII 1 0006 b 0006p D 1 C 4=b 0002 1 : 0002b C b 2 0002 4ac D Nt 2a 2 Evaluating b for a range of temperatures between 5000 K and 25,000 K produces the graph shown in Fig. S8.1. 8.10
(a)
Using the values given in the problem with Eq. (8.9), a form of the Saha equation that contains the electron pressure, results in T (K) 5000 15,000 25,000
NII =NI 000218
1:89 0003 10 9:98 0003 1000021 7:24 0003 103
NIII =NII 4:33 0003 10000249 2:42 0003 10000211 1:78 0003 1000023
In all cases, NII =NI 0007 NIII =NII , although the disparity decreases markedly with increasing temperature. (b) Write NII NII D Nt NI C NII C NIII D
NII =NI : 1 C NII =NI C .NIII =NII /.NII =NI /
Solutions for An Introduction to Modern Astrophysics
45
1.0 0.9 0.8
NII/Nt
0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 5000
10,000
15,000 T (K)
20,000
25,000
Figure S8.2: Results for Problem 8.10. NII =N t is shown for helium. 1.0 0.9 0.8
NIII/Nt
0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 10,000
20,000
30,000 40,000 T (K)
50,000
60,000
Figure S8.3: Results for Problem 8.11. NIII =N t is shown for helium. (c)
First, note from the results of part (a) that the last term in the denominator [.NIII =NII /.NII =NI /] of NII =N t can be neglected for T between 5000 K and 25; 000 K. Use Eq. (8.9) with ZI D 1, ZII D 2, 0010I D 24:6 eV, and Pe D 20 N m00022 to evaluate NII =NI for a range of values of T . As shown in Fig. S8.2, one-half of the He I atoms have been ionized when T ' 1:5 0003 104 K.
8.11 Follow the procedure of Problem 8.10 and write NIII NIII NIII =NII D D : Nt NI C NII C NIII 1=.NII =NI / C 1 C .NIII =NII / Use the values given in the Problem 8.10 (with Pe D 1000 N m00022 ) with Eq. (8.9) to evaluate NIII =NII and NII =NI . This results in the graph shown in Fig. S8.3. One-half of the helium atoms have been twice-ionized when T ' 4 0003 104 K.
46
Chapter 8 The Classification of Stellar Spectra
8.12 From Example 8.1.4 and the information given in the problem, use ZI D 2, ZII D 1, 0010I D 13:6 eV, ne D 6:1 0003 1031 m00023 , and T D 1:57 0003 107 K in the Saha equation, Eq. (8.8), to find that NII NII =NI NII D D D 0:709 Nt NI C NII 1 C NII =NI at the center of the Sun. However, at the center of the Sun, the density of the gas is sufficiently great to perturb the hydrogen orbital energies and increase the amount of ionization (pressure ionization). As a result, practically all of the Sun’s hydrogen is ionized at the Sun’s center . 8.13 From the statement of the problem and Example 8.1.5, we have ZII D 2:30, ZIII D 1, Pe D 1:5 N m00022 , 0010II D 11:9 eV, and T D 5777 K. Inserting these values into the alternative form of the Saha equation, Eq. (8.9), we find that ŒNIII =NII Ca D 2:0800031000023. Comparing this with ŒNII =NI Ca D 918 from Example 8.1.5 shows that nearly all of the calcium atoms are in the form of Ca II, available for forming the H and K lines. 8.14 The Saha equation, Eq. (8.8), is NiC1 2ZiC1 D Ni ne Zi

20004 me kT h2
3=2
e 00020007i = kT :
Stars of the same spectral type have the atoms in their atmospheres in the same states of ionization and excitation. The left-hand side of the Saha equation will therefore have the same value for atoms of the same element. The right-hand side increases monotonically as T increases, and decreases monotonically as ne increases. Because a giant star has a lower atmospheric density than a main-sequence star of the same spectral type, it must have a lower surface temperature to produce the same state of ionization of the atoms in its atmosphere. 8.15 The average density of a 1 Mˇ white dwarf with a radius of R D 0:01 Rˇ is 000fD
M 4 0004R3 3
D 1:41 0003 109 kg m00023 :
8.16 From the H–R diagram in Fig. 8.16, the star Formalhaut has an absolute visual magnitude of about Mv ' 2:0. The star has an apparent visual magnitude of V D 1:19. So, using Eq. (3.5), the distance to Formalhaut is about d ' 10.m0002M C5/=5 D 10.1:1900022:0C5/=5 pc D 6:89 pc:
CHAPTER 9
Stellar Atmospheres 9.1 Using Eq. (9.7) for the energy density of blackbody radiation, the energy of blackbody photons inside a spherical eyeball of radius reye D 1:5 cm at a temperature of 310 K (37ıC) is Ebb D
4 3 0004 r aT 4 D 9:88 0003 10000211 J: 3 eye
The flux from a point-source light bulb (L D 100 W) at a distance r D 1 m is F D L=40004 r 2 (Eq. 3.2), so the energy per second entering the pupil of area A is FA. The time for the light to cross the eyeball is 2reye=c (neglecting the fluid within the eye). The energy from the light bulb within the eyeball is therefore Ebulb D 2FAreye=c D
2LAreye D 7:96 0003 10000215 J; 40004 r 2 c
more than four orders of magnitude smaller than Ebb. It is dark when you close your eyes because, according to Wien’s law (Eq. 3.15), at 310 K the blackbody radiation peaks at 0007max D 9354:8 nm in the infrared, and the eye’s retina is not sensitive to such long wavelength photons. 9.2
(a)
Dividing the specific blackbody energy density, Eq. (9.5), by the energy per photon hc=0007 gives the specific number density, 80004=00074 n0004 d 0007 D hc=0004kT d 0007: e 00021 (b) Integrating the expression for n0004 over all wavelengths leads to an integral for the number density, n, that has the form Z Z 1 80004 k 3T 3 1 x 2 nD n0004 d 0007 D dx; h3 c 3 ex 0002 1 0 0 where x D hc=0007kT . The value of the integral is 2:404114, so the number density is n D .2:404/
80004 k 3T 3 : h3 c 3
(S9.1)
The number of photons inside an oven of volume V D 0:5 m3 at a temperature of T D 400ıF D 477 K is N D nV D 1:1 0003 1015. 9.3
(a)
The total energy density is given by Eq. (9.7) 40006T 4 4T 4 uD D c c

20004 5k 4 15c 2 h3
;
where we have used the expression found in Problem 3.14 for the Stefan–Boltzmann constant, 0006. Furthermore, the total number density of blackbody photons was found in Problem 9.2 to be given by Eq. (S9.1). Thus, the average energy per photon is 0004 4 kT u D D 2:70kT: n 15.2:404/ 47
48
Chapter 9 Stellar Atmospheres (b) For T D 1:57 0003 107 K, u=n D 3650 eV. For T D 5777 K, u=n D 1:34 eV. 9.4 For blackbody radiation, I0004 D B0004 , and so Eq. (9.10) gives Prad D
Z
40004 3c
1 0
Defining x D hc=0007kT leads to
B0004 .T / d 0007 D
80004 k 4T 4 D 3h3 c 3
Prad
40004 3c
Z
1 0
Z
1 0
2hc 2=00075 d 0007: e hc=0004kT 0002 1
x3 dx: ex 0002 1
4
The integral is equal to 0004 =15. Using this along with the expression found in Problem 3.14 for the Stefan– Boltzmann constant, 0006 D 20004 5k 4 =15c 2h3 , produces the desired result, Prad D
40006T 4 1 1 D aT 4 D u; 3c 3 3
via the definition of the radiation constant, a D 40006=c, and Eq. (9.7) for the blackbody energy density. 9.5 We start by integrating Eq. (9.8) with I0004 D B0004 over all outward directions (0 0003 0004=2). Z F0004 d 0007 D B0004 d 0007
20003
Z
0006D0
0003=2 0002D0
cos 0003 sin 0003 d0003 d D 0004B0004 d 0007:
We now integrate over all wavelengths to find the total flux, Z F D
1 0
Z F0004 d 0007 D 0004
1 0
B0004 d 0007 D 0004
0006T 4 D 0006T 4 : 0004
where Eq. (3.28) was used for the integral of B0004 . Finally, integrating the flux over the surface area of a sphere of radius R gives Z 20003 Z 0003 FR2 sin 0003 d0003 d D 40004R2 0006T 4 ; LD 0006D0
0002D0
which is Eq. (3.17). 9.6 According to Eq. (8.3), root-mean-square speed of the nitrogen molecules is s 3kT D 515 m s00021 : vrms D 28mp Using the collision cross section 0006 D 0004.2r /2 with Eq. (9.12), the mean free path is `D
1 1 : D n0006 n0004.2r /2
The number density n of nitrogen molecules is nD
000f D 2:57 0003 1025 m00023 : 28mp
The mean free path is therefore ` D 3:10 0003 1000027 m, and the average time between collisions is about t ' `=vrms D 6 0003 10000210 s = 0.6 ns.
Solutions for An Introduction to Modern Astrophysics
49
9.7 Using values of 0011500 D 0:03 m2 kg00021 from Example 9.2.2 and 000f D 1:2 kg m00023 , the mean free path is `D
1 D 21:8 m: 0011500 000f
You always look back to an optical depth of about 0004 D 2=3; that is, about 2=3 of a mean free path. Thus you could see to a distance of only 2`=3 D 18:5 m. 9.8 Using Eq. (9.19) with a measurement of the specific intensity I0004 D I1 at angle 00031 gives I1 D I0004;0 e 00020002;0 sec 00021 ; or

I1 ln I0004;0
Similarly, I0004 D I2 at angle 00032 gives
ln
Subtracting these results in
ln
I1 I2
I2 I0004;0
D 0002 0004;0 sec 00031 :
(S9.2)
D 0002 0004;0 sec 00032 :
(S9.3)
D 0002 0004;0 .sec 00031 0002 sec 00032 /;
or 0004;0 D
ln.I1 =I2 / : sec 00032 0002 sec 00031
Solving Eqs. (S9.2) and (S9.3) for 0004;0 and equating them produces 1 1 I1 I2 D : ln ln sec 00031 I0004;0 sec 00032 I0004;0 Solving for I0004;0 yields I0004;0 D
sec 00021
I2
!1=.sec 00021 0002sec 00022 /
sec 00022
I1
:
9.9 Consider the problems from the reference frame in which the electron is initially at rest. Conservation of momentum gives pphoton D pe , which is Ephoton D me v; c (Eqs. 4.44 and 5.5), where v is the final velocity of the electron. Conservation of total relativistic energy (Eq. 4.46) gives Ephoton C me c 2 D me c 2 : Equating these expressions for Ephoton leads to v=c D . 0002 1/= , which implies v D 0, D 1, and Ephoton D 0; a contradiction, since there is no photon to absorb! Thus an isolated electron cannot absorb a photon. 9.10 For constant density, a Kramers opacity law says that 0011 D C T 00023:5 , where C is a constant. Then at two temperatures T1 and T2 , ln 0011 2 D ln C 0002 3:5 ln T2 ln 0011 1 D ln C 0002 3:5 ln T1 :
50
Chapter 9 Stellar Atmospheres Subtracting these,
ln 0011 2 0002 ln 0011 1 D 00023:5: ln T2 0002 ln T1
Because ln x D .ln 10/ log10 x, this expression is valid both for natural logs and logs to the base 10. Using Fig. 9.10, we choose log10 000f D 0 kg m00023 and draw the best-fitting straight line to the curve. Two wellseparated points on this line are log10 0011 1 D 1:19 at log10 T1 D 5:55, and log10 0011 2 D 00020:867 at log10 T2 D 6:15. For these two points, log10 0011 2 0002 log10 0011 1 D 00023:43 ' 00023:5; log10 T2 0002 log10 T1 as expected for a Kramers opacity law. 9.11
(a)
The average photon mean free path is `D
1 1 D 3 0003 1000025 m: D 00021 2 0011000f .0:217 m kg /.1:5 0003 105 kg m00023 /
(b) From Eq. (9.29), the number of random-walk steps of this size from the center to the surface of the Sun is 2 d Rˇ 2 N D D D 5 0003 1026: ` ` The time for a photon to cover this many steps of size ` is tD
N` D 5 0003 1013 s; c
almost two million years! 9.12 Because you always look back to an optical depth of about 0004 D 2=3, Eq. (9.17) implies that you see down to a depth s into the star given by Z s 2 00110004 000f ds D : 3 0 At wavelengths where the opacity is greatest, the value of s is smallest. If the temperature of the star’s atmosphere increases outward, then a smaller value of s corresponds to looking at a higher temperature and brighter gas. At wavelengths where the opacity is greatest, you would therefore see emission lines. 9.13 A large hollow spherical shell of hot gas will look like a ring if you can see straight through the middle of the shell. That is, the shell must be optically thin, and an optically thin hot gas produces emission lines. Near the edge of the shell, where your line of sight passes through more gas, the shell appears brighter and you see a ring. 9.14 From Eq. (9.32), j0004 D dI0004 =000f ds, so the emission coefficient has units of 0006 00070004 0005 kg m2 s00023 m00023 sr00021 W m00023 sr00021 0006 0007 0006 0007 D D m s00023 sr00021 : 00023 00023 kg m .m/ kg m .m/ 9.15 Equation (9.34), 0002
1 dI0004 D I0004 0002 S0004 ; 00110004 000f ds
Solutions for An Introduction to Modern Astrophysics
51
with 00110004 , 000f, and S0004 independent of position, can be integrated as Z
I0002
I0002;0
dI00040 D 000200110004 000f 0 I0004 0002 S0004
Z
s 0
ds 0 ;
where I0004;0 is the value of the specific intensity at s D 0. This results in ln
I0004 0002 S0004 I0004;0 0002 S0004
D 000200110004 000fs;
which simplifies to Eq. (9.35). 9.16
(a)
Begin with the transfer equation, Eq. (9.34), 0002
1 dI0004 D I0004 0002 S0004 : 00110004 000f ds
Here, s is a distance measured along the light ray’s direction of travel. In terms of the angle 0003 0 , ds D
dr : cos 0003 0
Using this to replace ds in the transfer equation produces the desired result. (b) Multiply each side of this form of the transfer equation by cos 0003 0 , and integrate over all directions of the ray: Z Z Z 1 d 2 0 0 0 0 0002 I0004 cos 0003 d˝ D I0004 cos 0003 d˝ 0002 S0004 cos 0003 0 d˝ 0 ; 00110004 000f dr where we have used the fact that the source function is isotropic to remove it from the integral. The integral on the left-hand side is cPrad;0004 , the radiation pressure (Eq. 9.9) multiplied by c, while the first integral on the right-hand side is F0004 , the radiative flux (Eq. 9.8). The second integral on the right-hand side is zero, so we have c dPrad;0004 D F0004 : 0002 00110004 000f dr Finally, integrating over all wavelengths gives the final result, dPrad 0011000f D0002 Frad : dr c 9.17 The Eddington approximation assumes that the intensity of the radiation field is a constant Iout in the Czdirection (outward, 0 0003 0004=2), and a constant Iin in the 0002z-direction (inward, 0004=2 < 0003 0004 ). From Eq. (9.3), ! Z 20003 Z 0003 Z 20003 Z 0003=2 1 hI i D Iout sin 0003 d0003 d C Iin sin 0003 d0003 d
40004 0006D0 0002D0 0006D0 0002D0003=2 ! Z 0003=2 Z 0003 20004 sin 0003 d0003 C Iin sin 0003 d0003 D Iout 40004 0002D0 0002D0003=2 D
1 .Iout C Iin /: 2
52
Chapter 9 Stellar Atmospheres From Eq. (9.8), Z Frad D
20003
Z
0006D0
0003=2 0002D0
Z D 20004 Iout
Z Iout cos 0003 sin 0003 d0003 d C 0003=2
0002D0
20003
Z
0006D0
Z cos 0003 sin 0003 d0003 C Iin
0003 0002D0003=2
0003 0002D0003=2
Iin cos 0003 sin 0003 d0003 d
!
cos 0003 sin 0003 d0003
D 0004.Iout 0002 Iin /: From Eq. (9.9), Prad
1 D c 20004 D c
Z
20003 0006D0
Z
0003=2 0002D0
Z Iout
0003=2 0002D0
D
20004 .Iout C Iin / 3c
D
40004 hI i: 3c
Z
2
Iout cos 0003 sin 0003 d0003 d C Z
2
cos 0003 sin 0003 d0003 C Iin
0006D0
0003 0002D0003=2
9.18 Inserting Eq.(9.46) into Eq. (9.51) gives Iout C Iin D
20003
30006 4 T 20004 e
Z
0003 0002D0003=2
! 2
Iin cos 0003 sin 0003 d0003 d
!
2
cos 0003 sin 0003 d0003
2 v C : 3
(S9.4)
Inserting Eq.(9.47) into Eq. (9.43) gives Iout 0002 Iin D Adding these, we find that
0006 4 T 0004 e Subtracting Eq. (S9.5) from Eq. (S9.4) results in

Iout D
Iin D
0006 4 T : 0004 e 3 v C 1 : 4
30006 4 T v : 40004 e
The radiation field will be isotropic to within 1% when Iout 0002 Iin 1 2 .Iout C Iin /
D 0:01:
Using the above expressions, we have 1 2
or so v D 133.
h0006
0006Te4 =0004 00070006 0007i D 0:01; 3 Te4 2 C v 20003 3 4 D 0:01; 3 v C 2
(S9.5)
Solutions for An Introduction to Modern Astrophysics
53
9.19 From Eq. (9.53), at the top of a star’s atmosphere where v D 0, T 4 D Te4 =2. Thus T =Te D .1=2/1=4 D 0:841. If Te D 5777 K, then at the top of the atmosphere T D 4858 K. 9.20 For a plane-parallel gray atmosphere in LTE, Eq. (9.44) states that hI i D S. Using Eq. (9.50), we have 3 2 S D hI i D Frad v C : 40004 3 Evaluating this at v D 2=3 results in the Eddington–Barbier relation, Frad D 0004S. v D 2=3/: 9.21 If the source function does not vary with position, then Eq. (9.54) may be integrated to obtain I0004 .0/ D I0004;0 e 00020002;0 C S0004 .1 0002 e 00020002;0 / : If no radiation enters the gas from outside, then I0004;0 D 0. We consider two cases. 0004;0 0007 1: In this case, the source function is equal to the Planck function: S0004 D B0004 (assuming thermodynamic equilibrium). The exponential is essentially zero, so we have I0004 .0/ D S0004 D B0004 ; and you will see blackbody radiation. 0004;0 0006 1: We can use e 0002x ' 1 0002 x to write I0004 .0/ ' S0004 Œ1 0002 .1 0002 0004;0 / D S0004 0004;0 : For our slab of thickness L, 0004;0 D 00110004 000fL (Eq. 9.17). Recalling the definition of the source function, S0004 j0004 =00110004 , we have j0004 I0004 .0/ ' .00110004 000fL/ D j0004 000fL: 00110004 Thus you will see emission lines at wavelengths where j0004 is large. 9.22 If the source function does not vary with position, then Eq. (9.54) may be integrated to obtain I0004 .0/ D I0004;0 e 00020002;0 C S0004 .1 0002 e 00020002;0 / : We consider two cases. 0004;0 0007 1: In this case, the source function is equal to the Planck function: S0004 D B0004 (assuming thermodynamic equilibrium). The exponentials are essentially zero, so we have I0004 .0/ D S0004 D B0004 ; and you will see blackbody radiation. 0004;0 0006 1: We can use e 0002x ' 1 0002 x to write I0004 .0/ ' I0004;0 .1 0002 0004;0 / C S0004 Œ1 0002 .1 0002 0004;0 / D I0004;0 0002 0004;0 .I0004;ı 0002 S0004 /: For our slab of thickness L, 0004;0 D 00110004 000fL (Eq. 9.17). So we have I0004 .0/ ' I0004;0 0002 00110004 000fL.I0004;0 0002 S0004 /:
54
Chapter 9 Stellar Atmospheres If I0004;0 > S0004 , then you will see absorption lines superimposed on the incident spectrum (I0004;0 ). If I0004;0 < S0004 , then you will see emission lines superimposed on the incident spectrum. The strength of the lines depends on the competition between the processes of absorption and emission. From the definition of the source function, S0004 j0004 =00110004 , I0004 .0/ ' I0004;0 0002 000fL.00110004 I0004;0 0002 j0004 / (c.f., Eq. 9.33).
9.23 Inserting S0004 D a0004 C b0004 0004;v into Eq. (9.55), we have (omitting the 0007 subscripts for convenience) Z I.0/ D
1 0
.a C b v / sec 0003 e 0002v sec 0002 d v
D a sec 0003
Z
1
e
0002v sec 0002
0
Z d v C b sec 0003
1 0
v e 0002v sec 0002 d v :
The first integral on the right-hand side is 1= sec 0003, while the second can be integrated by parts, resulting in ˇ1 ! Z 1 0002v sec 0002 e v 0002v sec 0002 ˇˇ d v 0002 e I.0/ D a C b sec 0003 ˇ ˇ sec 0003 sec 0003 0 0006 0007 D a C b sec 0003 cos2 0003 0002 0 : Restoring the 0007 subscripts, this is
0
I0004 .0/ D a0004 C b0004 cos 0003:
9.24 Assuming that there is zero flux received from the center of the line (as in the Schuster–Schwarzschild model), the equivalent width is equal to the area of the half-ellipse divided by the semimajor axis. From Eq. (2.4), W D
0004 ab=2 0004b D : a 2
9.25 Heisenberg’s uncertainty principle, Eq. (5.20), relates 0002E, the uncertainty in the energy of an atomic orbital, to 0002t, the time an electron occupies the orbital before making a downward transition: 0002E 0004
0002 h D : 0002t 200040002t
When an electron makes a downward transition from an initial to a final orbital, the energy of the emitted photon is (Eq. 5.3) hc Ephoton D D Ei 0002 Ef : 0007 The uncertainty in 0007 caused by an uncertainty in Ei is found by taking a derivative with respect to Ei (holding Ef constant): hc d 0007 0002 2 D 1: 0007 dEi Writing 00020007 instead of d 0007, 0002Ei instead of dEi , and ignoring the minus sign yields the magnitude of the uncertainty, 00072 00020007 D 0002Ei : hc
Solutions for An Introduction to Modern Astrophysics
55
Similarly, the magnitude of the uncertainty in 0007 due to an uncertainty in Ef is 00020007 D
00072 0002Ef : hc
Adding these uncertainties and using the uncertainty principle to substitute for the 0002Es gives our result, 0005 1 00072 0004 1 00072 : C 00020007 D 0002Ei C 0002Ef 0004 hc 20004 c 0002ti 0002tf 9.26
(a)
For each of the wavelengths listed in Table 9.3, calculate the value of log10 .W =0007/. log10 .W =0007/ D 00024:69 for the 330.298 nm line D 00024:02 for the 589.594 nm line: These values can then be used with the general curve of growth for the Sun, Fig. 9.22, to obtain a value of log10 Œf Na .0007=500 nm/ for each wavelength. The results are f Na 0007 log10 D 16:63 for the 330.298 nm line 500 nm D 18:55 for the 589.594 nm line: Now calculate log10 Œf .0007=500 nm/ for each wavelength, f0007 log10 D 00022:4899 for the 330.298 nm line 500 nm D 00020:4165 for the 589.594 nm line; and use
log10 Na D log10
f Na 0007 500 nm

0002 log10
f0007 500 nm
to find log10 Na D 19:12 for the 330.298 nm line D 18:96 for the 589.594 nm line: The average value of log10 Na for this problem is 19.04, so Na D 1:10 0003 1019 m00022 . (b) Averaging the values of log10 Na from this problem and Example 9.5.5 gives hlog10 Na i D 19:02: This average can be used to plot the positions of the four sodium absorption lines on Fig 9.22. With the values shown in Table S9.1, all of the points lie on the curve of growth. 9.27
(a)
For each of the wavelengths listed in Table 9.4, calculate the value of log10 .W =0007/. log10 .W =0007/ D 00023:697 for the 1093.8 nm line D 00023:798 for the 1004.9 nm line: Use these with the general curve of growth for the Sun, Fig. 9.22, to obtain a value of log10 Œf Na .0007=500 nm/ for each wavelength. The results are f Na 0007 D 19:21 for the 1093.8 nm line log10 500 nm D 18:96 for the 1004.9 nm line:
56
Chapter 9 Stellar Atmospheres Table S9.1: Results for Problem 9.26. 0007 (nm) 330:238 330:298 588:997 589:594
log10 Œf hNa i.0007=500 nm/ 13:17 12:53 14:90 14:60
log10 .W =0007/ 00024:58 00024:69 00023:90 00024:02
Now calculate log10 Œf .0007=500 nm/ for each wavelength, f0007 D 00020:9165 for the 1093.8 nm line log10 500 nm D 00021:2671 for the 1004.9 nm line; and use
log10 Na D log10
f Na 0007 500 nm

0002 log10
f0007 500 nm
to find log10 Na D 20:13 for the 1093.8 nm line D 20:23 for the 1004.9 nm line: The average value of log10 Na for this problem is hNa i D 20:18, so hNa i D 1:51 0003 1020 m00022 . This is the value of the number of neutral hydrogen atoms per unit area with an electron in the n D 3 orbit. (b) The Boltzmann equation (Eq. 8.6) allows us to find the relative numbers of neutral hydrogen atoms in the n D 3, n D 2, and n D 1 states. From Eq. (5.14), E1 D 000213:6 eV, E2 D 00023:40 eV, and E3 D 00021:51 eV, and gn D 2n2 for hydrogen. For T D 5800 K from Example 9.5.5, we find N3 =N2 D 5:14 0003 1000022 and N3 =N1 D 2:82 0003 10000210. That is, for every neutral hydrogen atom in the n D 3 state, there are 19 in the n D 2 state and 3:55 0003 109 in the n D 1 state. The number of neutral hydrogen atoms per unit area is then N1 hNa i D 5:35 0003 1029 m00022 : N3 The ratio of the number of ionized to neutral atoms is obtained from the Saha equation, Eq. (8.9), with Pe D 1 N m00022 (from Example 9.5.5), and ZI D 2, ZII D 1, and 0010I D 13:6 eV (from Example 8.1.4). The result shows that NII =NI D 1:33 0003 1000024, so nearly all of the hydrogen atoms are neutral. Thus the above result gives the total number of hydrogen atoms per unit area, N D 5:35 0003 1029 m00022 . 9.28
(a)
See Table S9.2.
(b) See Fig. S9.1. (c)
See Table S9.2 and Fig. S9.1.
(d) The surface value of the temperature obtained from Eq. (9.53) is certainly more valid (on physical grounds) than the surface boundary condition of T D 0 employed by StatStar. However, given the numerous other approximations involved, StatStar’s surface value of T D 0 is adequate for our computational purposes. 9.29
(a)
See Fig. S9.2.
(b) See Fig. S9.2.
Solutions for An Introduction to Modern Astrophysics
57
Table S9.2: Results for Problem 9.28. Te D 5504 K. i 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41
(c)
0 0.202 0.249 0.323 0.435 0.601 0.837 1.18 1.67 2.36 3.35 4.76 6.76 9.6 13.6 19.4 27.5 39.1 55.5 78.9 112 159 226 321 455 647 918 1300 1850 2620 3720 5280 7480 10600 15000 21300 30200 42700 60500 85600 121000 171000
StatStar T (K) 0 3379.636 3573.309 3826.212 4133.144 4488.02 4887.027 5329.075 5815.187 6347.784 6930.293 7566.856 8262.201 9021.603 9850.881 10756.42 11745.2 12824.86 14003.75 15290.96 16696.43 18231.02 19906.56 21735.99 23733.41 25914.21 28295.19 30894.68 33732.66 36830.96 40213.37 43905.83 47936.66 52336.7 57139.61 62382.05 68104.01 74349.04 81164.61 88602.39 96718.69 105574.8
Eq. (9.53) T (K) 4628.29387 4944.54379 5010.27511 5109.25693 5247.76298 5434.68901 5672.40675 5971.56257 6331.32593 6758.27250 7251.69484 7817.22609 8456.53505 9167.93293 9961.57689 10835.93954 11802.05637 12859.13228 14023.06436 15297.04393 16685.66151 18209.78084 19873.14791 21688.97654 23670.55978 25839.07014 28199.43679 30778.36382 33593.27783 36662.39802 40008.77661 43659.68737 47642.36440 51988.38249 56721.79737 61894.51265 67520.02502 73640.96139 80321.39363 87603.73858 95532.96119 104163.8651
The difference is greatest at mid angles (approximately 50ı ) and near 90ı .
(d) Van Hamme’s formula best mimics the observational data for the Sun.
58
Chapter 9 Stellar Atmospheres
Figure S9.1: Results for Problem 9.28.
Figure S9.2: A comparison between the limb darkening formulae of Van Hamme and the Eddington approximation (Eq. 9.58).
CHAPTER 10
The Interiors of Stars 10.1 From Eq. (9.15), dr D 0002d =0011000f. Substitution into Eq. (10.6) leads immediately to the final result. 10.2 The procedure is identical to the one that was used in Example 2.2.1 (see also Fig. 2.8). In this case however, r < r1 and r1 < R < r2 . After changing variables from 0003 to u, the limits of integration become u D .R0002r /2 to u D .R C r /2 . This leads to an integral of the form 0003 Z Z 2 0002 0004 Gm000f r2 .RCr / R2 R F D R 1 0002 2 u00023=2 C 2 u00021=2 du dR: 2 r r r1 .R0002r /2 After integrating over u, exact cancellation occurs, giving F D 0. 10.3 Assuming that the Sun is composed entirely of hydrogen atoms, the number of atoms in the Sun is approximately 1 Mˇ N ' D 1:2 0003 1057: mH If each atom releases 10 eV of energy during a chemical reaction, the time scale for chemical reactions would be Echem 1:2 0003 1058 eV tchem D D ' 5 0003 1012 s D 1:6 0003 105 yr: Lˇ 3:84 0003 1026 W This is much shorter than the age of the solar system, and so the Sun’s energy cannot be chemical. 10.4
(a)
Using the reduced mass for a proton–proton collision, and equating kinetic energy to thermal energy, 3 1 2 : kT D 0005p vrms 2 2 If particles with ten times the rms velocity can overcome the Coulomb barrier, then 0004 00051=2 v D 10vrms D 10 3kT =0005p and
e2 1 1 3 D 0005p v 2 D .100/ kT : 40004 0 r 2 2 2 Taking the separation between charges to be r ' 2 fm,
4 e2 D 1:1 0003 108 K: 300k 40004 0r This value is approximately 7 times greater than the Sun’s central temperature. (b) From Eq. (8.1), and assuming identical velocity intervals, the ratio of atoms having velocities equal to 10 times the rms value to those having the rms value is given by T D
2
nv e 0002m.10vrms/ =2kT .10vrms /2 D 2 2 nrms e 0002mvrms=2kT vrms 2
D 100 e 000299.mvrms=2kT / D 100 e 000299.m=2kT / .3kT =2m/ D 100 e 000274:3 D 5:7 0003 10000231 : 59
60
Chapter 10 The Interiors of Stars (c)
Assuming pure hydrogen, the number of atomic nuclei in the Sun is approximately 1:2 0003 1057 (see the solution to Problem 10.3). If T 0005 1:1 0003 108 K throughout the star, and if all nuclei with v D 10vrms can react, then the number of nuclei that will be involved in nuclear reactions is roughly nv Nreact ' N D 7 0003 1026 : nrms Given that in the proton–proton chain, 0.7% of the mass is converted into energy, the timescale for this process would be t'
0:007Nreactmp c 2 7:2 0003 1014 J E D D D 2 0003 10000212 s: Lˇ Lˇ 3:84 0003 1026 W
Quantum mechanical tunneling is definitely required! 10.5 Beginning with Eq. (10.9) and substituting Eq. (8.1), Z 1 1 mnv v 2 dv P D 3 0 Z 0006 m 00073=2 1 1 2 mn e 0002mv =2kT 40004 v 4 dv: D 3 0 20004 kT Using the definite integral, Z
1 0
2
x 2n e 0002ax dx D
1 3 5 : : : .2n 0002 1/ 2nC1 an
r
0004 ; a
with n D 2 and a D m=2kT , P D
p 3 0004 40004 mn 0006 m 00073=2 D nkT: 3 20004 kT 23 .m=2kT /5=2
10.6 From the non-relativistic expression for kinetic energy (E), r r 2E 2 1 vD and dv D dE: m 2 mE Using the relation nv dv D nE dE and making the appropriate substitutions in Eq. (8.1) leads directly to Eq. (10.28). 10.7 From Eq. (10.22), the gravitational potential energy of the Sun is approximately Ug D 00022:3 0003 1041 J. Using the virial theorem (Eq. 2.46), the thermal kinetic energy of the Sun is roughly 3N kT =2 D 1:2 0003 1041 J, where N is the number of particles in the Sun. Assuming that all of the hydrogen is ionized, then N D 2:4 0003 1057 (see the solution to Problem 10.3 and assume that every hydrogen atom contributes a nucleus and a free electron). Solving for the temperature, T D 2:4 0003 106 K. This value is roughly an order of magnitude less than than the estimate of the Sun’s central temperature, but it is reasonable for an average temperature of the Sun. 10.8 According to the discussion between Eqs. (10.30) and (10.31), Uc Z1 Z2 e 2 D : E 20004 0hv
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61
Furthermore, using the non-relativistic expression, v D
p
2E=0005m,
Uc Z1 Z2 e 200051=2 m : D 1=2 E 2 20004 0E 1=2h After substituting into Eq. (10.30) and simplifying, we arrive at Eq. (10.31). 10.9 The energy at which the Gamow peak is a maximum occurs when d h 0002.E= kT CbE 00021=2/ i D 0; e dE or
0002
1 1 0002 bE 00023=2 kT 2
e 0002.E= kT CbE
00021=2
/ D 0:
Solving for E gives Eq. (10.34). 10.10 From Eq. (10.47), pp D 1:16 0003 1000023 W kg00021 , and from Eq. (10.59), CNO D 3:90 0003 1000024 W kg00021 . These results give pp = CNO D 2:97. 10.11 Equation (10.62) is of the form
00021
3˛ D c000f 2Y 3 T800023 f3˛ e 0002ˇT8 ; where c 50:9 W m6 kg00023 , ˇ 44:027, and T8 T =108 K. Thus, ln 3˛ D ln c C 2 ln 000f C 3 ln Y 0002 3 ln T8 C ln f3˛ 0002 ˇT800021 : Differentiating with respect to ln T8 , 0004 0005 @ ˇT800021 ˇT800021 @ ln 3˛ ˇ D 00023 0002 D 00023 0002 T8 D 00023 C : @ ln T8 @ ln T8 @T8 T8 Next, assuming a power law of the form
3˛ D 00 T8˛ ;
and again differentiating with respect to ln T8 gives @ ln e3˛ D ˛: @ ln T8 Comparing results, ˛ D 00023 C
ˇ D 41:027 T8
when T8 D 1 and 00 D c000f 2Y 3 f3˛ . 10.12 The Q-values for the three reaction steps are 0007 0006 Q1 D 2m1H 0002 m2 H 0002 meC c 2 D 0:933 MeV > 0 1
1
0007 Q2 D m2 H C m1 H 0002 m3 He c 2 D 5:519 MeV > 0 0006
1
1
2
0007 Q3 D 2m3He 0002 m4 He 0002 2m1 H c 2 D 12:806 MeV > 0: 0006
2
2
10.13 Using the approach suggested in Problem 10.12:
1
62
Chapter 10 The Interiors of Stars (a)
13:94 MeV (exothermic)
(b) 00020:108 MeV (endothermic) 10.14
(c)
8:11 MeV (exothermic)
(a)
27 27 C 14 Si ! 13 Al C e C 27 1 24 4 13 Al C 1 H ! 12 Mg C 2 He 35 1 36 17 Cl C 1 H ! 18 Ar C
(b) (c)
10.15 Beginning with the ideal gas law (Eq. 10.77), solving for V and substituting into Eq. (10.82) gives nRT
P D K: P Solving for P results in Eq. (10.83), with K 0 K=.nR/ =.10002 / . 10.16 We begin by using the equation of hydrostatic equilibrium (Eq. 10.6), dP Mr 000f D 0002G 2 ; dr r to find
1 dP GMr D0002 2 : 000f dr r We also note that the potential energy of a point mass m brought in from infinity to a distance r inside the star is given by Z r Z r GMr m Fg dr D 0002 0002 dr: Ug D 0002 r2 1 1 This implies that, GMr m d Ug ; D dr r2 or d˚g d.Ug =m/ GMr : D D dr dr r2 Thus 1 dP d˚g D0002 : 000f dr dr Substitution into Eq. (10.108) immediately leads to Eq. (10.109). 10.17 Beginning with Eq. (10.110) and setting n D 0 we have 0002 0003 d 2 dD0 0012 D 00020012 2 : d0012 d0012 Integrating we have
dD0 0012 C0 D0002 C 2: d0012 3 0012
Applying the central boundary condition, dD0 =d 0012 D 0 at 0012 D 0, implies that C 0 D 0. Integrating a second time gives 00122 D0 D 0002 C C 00 : 6 00 D 1. Finally, if 00121 represents the surface of Applying the normalizing condition D0 .0/ D 1 implies that Cp the polytrope, then D0 .00121 / D 0. This immediately gives 00121 D 6.
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63
10.18 The density is parameterized in the Lane–Emden equation as 000f.r / 000fc ŒDn .r / n . Thus, for the case of n D 0, 000f.r / D 000fc ; the density is a constant throughout the star. 10.19 Using the derivative form for the total mass of the polytrope M D
000240004 00073n000fc 001212
ˇ dDn ˇˇ ; d 0012 ˇ000b1
and substituting the expression for D5 .0012/ D Œ1 C 0012 2 =3 00021=2;
with 00121 ! 1
gives ˇ ˇ 001213 40004 3 ˇ 0007n 000fc 0004 M D 00053=2 ˇˇ 3 2 1 C 00121 =3
1 !1
ˇ ˇ 1 40004 3 ˇ D 0007n 000fc 0004 00053=2 ˇˇ 3 2 1=00121 C 1=3
D 33=2 1 !1

40004 3 0007n 000fc : 3
Thus, the total mass is finite even though the surface boundary goes to infinity. 10.20
(a)
See Fig. S10.1.
(b) The density concentration increases toward the center with increasing polytropic index. (c)
The n D 1:5 model represents adiabatic convection and the n D 3 model represents a radiative model. The adiabatic convection model will have a shallower density gradient.
(d) The shallower density gradient for the adiabatic model is consistent with an adiabatic gradient that tends to smooth out the structure. The steeper density gradient for the radiative model is consistent with gradients required to drive radiative flux.
Figure S10.1: The density structures of n D 0, 1, and 5 polytropes. 10.21 For a 0:072 Mˇ star, the luminosity is L0:072 D 1000024:3 Lˇ D 1:92 0003 1022 W:
64
Chapter 10 The Interiors of Stars Assuming a pure hydrogen composition for simplicity, and if the entire star participates in the energy generation, the amount of energy released in the conversion of hydrogen to helium in a 0:072 Mˇ star is E0:072 D .0002m/c 2 D 0:007 .0:072 Mˇ / c 2 D 9:0 0003 1043 J: Thus, the hydrogen-burning lifetime is t0:072 '
E0:072 ' 5 0003 1021 s D 1:5 0003 1014 yr: L0:072
Similarly, for an 85 Mˇ star with only the inner 10% of the star participating in nuclear reactions, L85 D 106:006Lˇ D 3:9 0003 1032 W E85 D .0:007/.0:1/.85 Mˇ /c 2 D 1:1 0003 1046 J t85 D
E85 D 3 0003 1013 s D 9 0003 105 yr: L85
10.22 From the Stefan–Boltzmann equation (Eq. 3.17), the radius of a star is given by RD
L 400040006T 4
1=2 :
Using the data from Problem 10.21, R0:072 D 5:9 0003 107 m D 0:082 Rˇ and R85 D 9:1 0003 109 m D 13:1 Rˇ . Thus R0:072 =R85 D 0:0063. 10.23 The Eddington luminosity is given by Eq. (10.114). (a)
For M D 0:072 Mˇ , LEd D 3:6 0003 1031 W D 9:4 0003 104 Lˇ , implying that L=LEd D 5 0003 10000210 . Radiation pressure does not play a significant role in the stability of a low-mass star.
(b) For M D 120 Mˇ , we may consider electron scattering as an appropriate opacity (the surface is ionized at that temperature). In this case 0011 D 0:02.1 C X / D 0:034, and LEd D 4:6 0003 106 Lˇ . The actual luminosity of the 120 Mˇ star is about 1:8 0003 106 Lˇ , and so the Eddington luminosity is only about 2.5 times greater than the actual luminosity. Clearly radiation pressure plays a significant role in this case. 10.24
(a)
The Lane–Emden equation is a second-order differential equation. One very powerful tool in numerical analysis is to write second-order equations as two first-order equations, one of which is the derivative of the function that you are trying to solve. Beginning with Eq. (10.110) and taking the derivatives leads to d 2 Dn 2 dDn C C Dnn D 0: d00122 0012 d0012 Letting fn .0012/
dDn d0012
gives dfn 2 D 0002Dnn 0002 fn : d0012 0012 These last two coupled equations can now be integrated to produce Dn .0012/. Using a simple Euler method, Dn .0012iC1 / D Dn .0012i / C hfn .0012i /
Solutions for An Introduction to Modern Astrophysics
65
and
0002
0003 2 fn .0012iC1 / D fn .0012i / C h 0002 fn .0012i / 0012i where h is the step size and 0012iC1 D 0012i C h are the values of 0012 for steps i C 1 and i , respectively. Finally, 000f=000fc D Dnn gives the density of the polytrope. A simple Fortran 95 code that implements these ideas is given: 0002Dnn .0012i /
PROGRAM LaneEmden IMPLICIT NONE REAL(8) h, f_i, f_ip1, D_i, D_ip1, n, xi OPEN(UNIT = 10, FILE = 'LaneEmden.txt') WRITE (*,’(A)’,ADVANCE = ’NO’) READ (*,*) n
'Enter the value of the polytropic index:
xi = 0 h = 0.0001
!Starting value of xi !step size
D_i = 1 f_i = 0
!Boundary condition of function !Boundary condition of first derivative
WRITE (10,’(3F13.6)’)
'
xi, D_i, f_i
DO xi = h + xi D_ip1 = D_i + f_i*h f_ip1 = f_i + (-D_i**n - 2/xi*f_i)*h IF (D_ip1 < 0 .OR. xi > 10) EXIT D_i = D_ip1 f_i = F_ip1 !The density relative to the central density is rho/rho_c = D_i**n WRITE (10,’(3F13.6)’) xi, D_i**n, f_i END DO END PROGRAM LaneEmden
(b) See Fig. S10.2. 10.25 Download the data set for Appendix L for this problem from www.aw-bc.com/astrophysics. The model is contained in the file M1p0ModelX0p7Z0p008.dat. In order to use the data for the two adjacent zones from Appendix L, we must calculate averages and differences; the results are given in Table S10.1. The two zones that are used in this calculation are i D 397 and i D 398. For Eq. (10.6): dP PiC1 0002 Pi D 00029:30 0003 106 N m00023 ' dr riC1 0002 ri 0002G
Mr 000f r2
' 00028:83 0003 106 N m00023
rel: error D 00020:0527: For Eq. (10.7): dMr ' 5:135 0003 1021 kg m00021 dr
66
Chapter 10 The Interiors of Stars
Figure S10.2: The density structures of n D 1:5 and n D 3 polytropes. The density structures of n D 0, 1, and 5 polytropes from Problem 10.20 are included for reference; see Fig. S10.1. Table S10.1: StatStar data from the downloaded 1 Mˇ model for Problem 10.25. shell 397 398
r 2.728EC08 2.654EC08
1 - M r/Ms 2.049E000201 2.244E000201
Mr 1.581EC30 1.543EC30
Lr 3.297EC26 3.297EC26
2.691EC08 00020.074EC08
2.147E000201 0.195E000201
1.562EC30 00020.038EC30
3.297EC26 0.000EC26
shell 397 398
T 4.941EC06 5.140EC06
P 3.804EC14 4.492EC14
rho 5.748EC03 6.524EC03
kap 4.512E000201 4.395E000201
eps 5.963E000207 8.557E000207
ave diff
5.040EC06 0.199EC06
4.148EC14 0.688EC14
6.136EC03 0.776EC03
4.454E000201 00020.117E000201
7.260E000207 2.594E000207
ave diff
40004 r 2 000f ' 5:584 0003 1021 kg m00021 rel: error D 0:0804: For Eq. (10.36): dLr ' 0 W m00021 dr 40004 r 2 000f ' 2:487 0003 1019 W m00021 rel: error D 1:000: Note that in this case the output did not display enough significant figures in the luminosity to make a reason-
Solutions for An Introduction to Modern Astrophysics
67
able determination of 0013Lr . Round-off errors (even internal errors before displaying results) can be critical issues to deal with in numerical modeling. For Eq. (10.68): dT ' 00022:689 0003 1000022 K m00021 dr 0002
3 0011000f Lr ' 00022:558 0003 1000022 K m00021 4ac T 3 40004 r 2 rel: error D 0:0512:
Table S10.2: Results for Problem 10.26. Tc , Pc , 000fc , and c are all in SI units. The values are obtained from the last zone above the central core. Mass (Mˇ ) 0.75 1.00
Tc
000fc
Pc 7
1:182 0003 10 1:494 0003 107
16
1:031 0003 10 1:514 0003 1016
c 4
6:146 0003 10 7:202 0003 104
5:620 0003 1000024 2:451 0003 1000022
10.26 The (near) central conditions for the 0:75 Mˇ and 1:0 Mˇ models are given in Table S10.2. For the simple Runge–Kutta shooting method that is used in StatStar, the extrapolated central conditions are very sensitive to the surface boundary conditions. Using the values of the last zone above the central core are more smoothly-varying and reliable for comparisons between models. In order to support the extra mass of the 1:0 Mˇ model, the central pressure and temperature are increased over the values for the 0:75 Mˇ model. Because the central pressure increases rapidly with stellar mass, the central density is also greater for the 1 Mˇ star. Since the central nuclear energy generation rate is a function of both the temperature and density in that region, is greater for the 1 Mˇ star as well. 10.27 The input data for the zero-age main-sequence models are given in Table S10.3. Note that a region of convergence exists in the effective temperature–luminosity plane. The models given here are only representative of the possible solutions at each mass. It is also important to note that Statstar was written to be relatively transparent to the student for pedagogical reasons. Consequently, special automated, iterative convergence techniques have not been included in the code. Furthermore, since it is necessary to have fairly tight convergence criteria at the center of the star in order to produce a physically reasonable main sequence, and since the numerical procedure amounts to a shooting method from the surface to the center, the code is quite sensitive to initial starting conditions. This may lead to some initial frustration for some students attempting to find a converged model. (a)
See Figure S10.3. Note that the coarseness of the core is evident in Lr and T . This arises because the core solution is simply extrapolated from the last good zone computed by the Runge–Kutta method. A more sophisticated iterative procedure guaranteeing that the central boundary conditions are strictly enforced is not used in the pedogogical StatStar code.
(b) For a 1 Mˇ star, Lr reaches 99% of its surface value at a temperature of about 7:3 0003 106 K. It reaches 50% at a temperature of about 1:2 0003 107 K. The 50% level is roughly consistent with the “back-of-theenvelope” calculation. Quantum mechanical tunneling is important in stellar nuclear reaction rates. (c)
For a 1 Mˇ star, when Lr reaches 99% of its surface value, Mr =M? ' 0:537, and when it reaches 50%, Mr =M? ' 0:0768.
68
Chapter 10 The Interiors of Stars Table S10.3: Input data for Problem 10.27. All models have X D 0:7 and Z D 0:008. M (Mˇ ) 0.50 0.60 0.70 0.75 0.80 0.90 1.00 1.10 1.20 1.30 1.40 1.50 1.60 1.70 1.75 1.80 1.90 2.00 2.25 2.50 2.75 3.00 3.25 3.50 3.75 4.00 4.25
Te (K) 2287.7 2862.3 3465.5 3788.5 4109.3 4755.5 5402.0 6117.0 6802.3 7458.6 8062.9 8644.6 9146.0 9596.3 9852.6 10074.2 10546.0 10952.6 11950.9 12903.1 13811.0 14671.0 15479.5 16236.5 16957.0 17639.0 18289.4
L (Lˇ ) 0.0215 0.0570 0.1300 0.1890 0.2650 0.4940 0.8590 1.4170 2.2200 3.2900 4.6800 6.4080 8.5000 11.0000 12.5000 14.1000 17.8000 22.0500 35.8000 55.1000 80.7000 113.5000 155.2000 205.8000 266.5000 338.1000 420.6000
M (Mˇ ) 4.50 4.75 5.00 5.25 5.50 5.75 6.00 6.50 7.00 7.50 8.00 8.50 9.00 9.50 10.00 10.50 11.00 11.50 12.00 12.50 13.00 13.50 13.75 14.00 14.50 15.00
Te (K) 18906.2 19498.7 20069.0 20601.5 21072.5 21585.8 22089.0 22982.7 23810.0 24613.0 25366.5 26052.5 26722.0 27330.0 27933.0 28523.0 29109.5 29648.0 30149.0 30627.0 31096.0 31532.1 31768.3 32009.5 32449.7 32873.3
L (Lˇ ) 516.0 624.2 745.7 880.5 1029.0 1192.0 1372.0 1778.9 2253.0 2794.0 3409.0 4097.0 4857.0 5696.0 6607.0 7601.0 8675.0 9824.0 11050.0 12354.0 13727.0 15170.0 15925.0 16695.0 18254.0 19920.0
(d) See Figure S10.4 for parts (i)–(iii). Note that the onset of the strongly temperature-dependent CNO cycle results in a very rapid rise in the core value of (the graph displays log10 c rather than c directly) and a decrease in the core value of the density. Figure S10.5 shows the increase in the size of the central convection zone with increasing stellar mass. Note that the size of the convection zone mirrors the increase in , indicated in Figure S10.4. Figure S10.6 shows the results for parts (v) and (vi). (e)
See Figures S10.7 and S10.8. For the StatStar main sequence, ˛ ' 5:3 for M < 1:5 Mˇ , ˛ ' 3:9 for 1:5 Mˇ < M < 4:5 Mˇ , and ˛ ' 3:0 for M > 4:5 Mˇ .
10.28 Note: You may wish to review the discussion at the beginning of the solution to Problem 10.27. Main-sequence models for X D 0:7, Z D 0:01 are given in Table S10.4. Figure S10.9 also compares the higher-Z main sequence with the one for Z D 0:008, calculated in Problem 10.27. Notice that the higher-Z main sequence is shifted to cooler temperatures and lower luminosities. This is consistent with the discussion of Example 9.5.4 regarding the fact that the low-Z subdwarfs are shifted to the left of the main sequence. (a)
The Z D 0:008 model has the largest central temperature and central density.
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69
Figure S10.3: The results of Problem 10.27(a) for a 1 Mˇ model with L D 0:859 Lˇ and Te D 5402 K.
Figure S10.4: The results of Problem 10.27(d), parts (i)–(iii), with X D 0:7 and Z D 0:008. Note that the values plotted are actually the last zone above the central core. This was done because the core extrapolation procedure is rather coarse. (b) The Z D 0:010 model has the largest value for the opacity at the center, and throughout the star. This is due primarily to the bound–free opacity equation (Eq. 9.22), which is proportional to Z (more electronic transitions are available). A larger opacity results in a slightly inflated star, having a lower density and temperature near the surface. As a consequence, the temperature gradient (Eq. 10.68) is shallower there. Integrating inward, the lower density and temperature means that the energy generation rate is diminished (e.g., Eq. 10.47), resulting in a lower luminosity for the star (Eq. 10.36). (c)
The Z D 0:008 model has the largest energy generation rate because of the larger values for the temperature and density.
(d) The Z D 0:010 model has a lower effective temperature and a lower luminosity. This is because the
70
Chapter 10 The Interiors of Stars
Figure S10.5: The results of Problem 10.27(d), part (iv), with X D 0:7 and Z D 0:008.
Figure S10.6: The results of Problem 10.27(d), parts (v)–(vi), with X D 0:7 and Z D 0:008. opacity has inflated the model, decreasing and consequently decreasing L.
Solutions for An Introduction to Modern Astrophysics
Figure S10.7: The results of Problem 10.27(e), part (i), with X D 0:7 and Z D 0:008.
Figure S10.8: The results of Problem 10.27(e), part (ii), with X D 0:7 and Z D 0:008.
71
72
Chapter 10 The Interiors of Stars Table S10.4: Input data for Problem 10.27. All models have X D 0:7 and Z D 0:01. M (Mˇ ) 0.70 0.75 0.80 0.90 1.00 1.10 1.20 1.30 1.40 1.50 1.60 1.70 1.75 1.80 1.90 2.00 2.25 2.50 2.75 3.00 3.25 3.50 3.75 4.00 4.25
Te (K) 3274.00 3583.50 3901.80 4550.85 5196.30 5842.30 6486.30 7119.90 7717.70 8219.00 8714.00 9200.50 9426.30 9647.10 10066.10 10514.80 11515.00 12445.50 13313.20 14146.50 14947.10 15718.50 16458.50 17164.60 17824.00
L (Lˇ ) 0.1100 0.1610 0.2290 0.4365 0.7600 1.2500 1.9350 2.8980 4.1300 5.6200 7.5000 9.8000 11.1200 12.5600 15.8300 19.8000 32.5800 50.2000 73.9000 104.7000 143.9000 192.0000 249.5000 318.0000 398.0000
M (Mˇ ) 4.50 4.75 5.00 5.25 5.50 5.75 6.00 6.50 7.00 7.50 8.00 8.50 9.00 9.50 10.00 10.50 11.00 11.50 12.00 12.50 13.00 13.50 14.00 14.50 15.00
Te (K) 18409.20 18989.50 19556.80 20092.30 20610.30 21058.70 21546.10 22444.00 23267.50 24062.50 24819.50 25539.00 26213.70 26847.00 27439.00 28037.50 28631.00 29141.00 29640.50 30149.50 30663.50 31104.00 31578.50 32014.50 32423.00
L (Lˇ ) 489.3 593.5 710.0 840.3 985.0 1143.8 1319.0 1715.0 2178.0 2711.0 3317.0 3996.0 4741.0 5570.0 6475.0 7460.0 8525.0 9656.0 10870.0 12164.0 13538.0 14972.0 16493.0 18036.0 19699.0
Figure S10.9: The results of Problem 10.28. Two main sequences are shown, X D 0:7 and Z D 0:01, and X D 0:7 and Z D 0:008.
CHAPTER 11
The Sun 11.1 From Eq. (3.17), L D 40002R2 0003Te4 , Ti D Tˇ
0002
Li Lˇ
00031=4 0002
Rˇ Ri
00031=2
0002 1=4
' .0:677/
1 0:869
00031=2 D 0:973:
Using Tˇ D 5777 K, we find Ti D 5621 K, as indicated in Figure 11.1. 11.2
(a)
The rate at which the Sun is converting its mass into energy is Lˇ D 4:27 0002 109 kg s00021 D 6:77 0002 10000214 Mˇ yr00021 : c2
Pˇ ' 3 0002 (b) In Example 11.2.1, the Sun’s mass loss rate due to the solar wind was found to be M 000214 00021 10 Mˇ yr , roughly one-half the value found in part (a). (c)
The Sun’s lifetime on the main sequence is about 1010 yr. Thus, the combined mass-loss rates of nuclear fusion and the solar wind would imply a total mass loss of about 1000023 Mˇ over that period, which is not a significant fraction of the Sun’s total mass.
11.3 The ionization potential of the H0002 ion is 0004i D 0:75 eV D 1:2 0002 10000219 J. Using ZiC1 D ZH D 2, Zi D ZH0002 D 1, Pe D 1:5 N m00022 , and T D 5777 K, the Saha equation (Eq. 8.9) gives NH 2kT ZH D NH0002 Pe ZH0002
0002
20002 me kT h2
00033=2
e 00020002i = kT D 5 0002 107:
11.4 From the Boltzmann equation (Eq. 8.6), the number of hydrogen atoms in the n D 3 relative to the number in the n D 1 state is given by g3 0002.E3 0002E1 /= kT N3 D e D 2:5 0002 10000210 ; N1 g1 where g3 D 2.3/2 D 18, g1 D 2.1/2 D 2, E3 D 000313:6 eV=32 D 00031:511 eV, E1 D 000313:6 eV, and T D 5777 K. Assuming that the vast majority of all hydrogen atoms are in the ground state, the ratio of H0002 ions to atoms in the n D 3 state is 00030002 0003 0002 00030002 0003 0002 NH0002 N1 1 1 NH0002 D ' 79: ' N3 NH N3 5 0002 107 2:4 0002 10000210 11.5 The combined effects of turbulence and thermal motions results in Doppler broadening of the form (Eq. 9.63). For H˛ absorption in the Sun’s photosphere, 0005˛ D 656:280 nm, T D 5777 K, and m D mH . (a)
Assuming vturb D 0, .00060005/1=2 D 0:0355879 nm.
(b) From observational data, vturb ' 0:4 km s00021 . If this term is included in Eq. (9.63), .00060005/1=2 D 0:0356177 nm. 73
74
Chapter 11 The Sun (c)
2 vturb =.2kT =m/ D 0:00168 0004 1.
(d) ı.00060005/1=2 =.00060005/1=2 D 0:084%. Turbulence does not make a significant contribution in the solar photosphere. 11.6 From the data given in the text on page 366 and in the statement of the problem, Eq. (9.62) gives .00060005/1=2;L˛ D 0:0123 nm .00060005/1=2;C III D 0:0061 nm .00060005/1=2;O VI D 0:010 nm .00060005/1=2;Mg X D 0:011 nm
11.7
(a)
The Planck function (Eq. 3.22) is given by B0003 .T / D
2hc 2 =00055 : e hc=0003kT 0003 1
Integrating over wavelength between two closely spaced wavelengths, 00051 and 00052 , gives Z
00032 00031
B0003 .T / d 0005 ' B0003 .T / 00060005;
where 00060005 D 00052 0003 00051 . Taking 00060005 to be 0.1 nm over each of two different wavelength regions centered on 0005a and 0005b , and taking the ratios of the intensities we find Ba 00060005 ' Bb 00060005
0002
0005b 0005a
00035
e hc=0003b kT 0003 1 ' e hc=0003a kT 0003 1
0002
0005b 0005a
00035
hc
e kT
0004
1 0002b
0002 00021a
0005
:
(This last expression arises because e hc=0003kT 0005 1.) Taking the natural logarithm of both sides gives 0002 0003 0002 0003 0003 0002 Ba 0005b hc 1 1 ln D 5 ln C : 0003 Bb 0005a kT 0005b 0005a Using 0005a D 10 nm and 0005b D 100 nm leads to the desired result. (b) Substituting the appropriate wavelengths and using the Sun’s effective temperature gives ln .Ba =Bb / D 0003213. (c)
e y D 10x implies x D y= ln 10. Since y D 0003213, we find x D 000392:6.
11.8 Assuming 0007 D 5=3, Eq. (10.84) gives an adiabatic sound speed of 10 km s00021 . From Example 10.4.2, an “average” sound speed for the Sun is about 400 km s00021 . From Example 11.2.2, the sound speed at the top of the photosphere is about 7 km s00021 . The sound speed clearly increases with depth. 11.9 Starting with Eq. (9.15), solving for ds, assuming that 0003 and are constants over the interval of interest, and integrating, Z d Z 00040;0002 1
0;0003 dD ds D 0003 d 0003 D 0003 : 0003 0003 0 0 Taking 0;0003 D 2=3, the depth observed at 00051 is d1 D 117 km and the depth observed at 00052 is d2 D 101 km. This means that it is possible to probe the atmosphere at a greater depth in 00051 , by an amount 0006d D 16 km.
Solutions for An Introduction to Modern Astrophysics 11.10
(a)
75
The pressure scale height is given by Eq. (10.70). From Example 11.2.2, P D 140 N m00022 and D 4:9 0002 1000026 kg m00023 . Furthermore, g D GMˇ =R2ˇ D 274 m s00022 . Thus HP D 104:0 km.
(b) If ˛ D `=HP D 2:2, then ` D 229 km. Since the observed Doppler velocity of solar granulation is approximately vturb D 0:4 km s00021 , the time required for a convective bubble to travel one mixing length is about t D `=vturb D 570 s D 9:6 min. This compares well with observed granulation lifetimes of five to ten minutes. 11.11 Beginning with Eq. (11.5), assuming that the gas is isothermal, and integrating over radius, we have Z
n.r / n0
dn GMˇ mp D0003 n 2kT
Z
r r0
1 dr: r2
This immediately results in 0007 0003 0002 0006 r0 0005 GMˇ mp 1 GMˇ mp 0004 1 n.r / : 10003 D D0003 0003 ln n0 2kT r r0 2kT r0 r Solving for n.r / gives Eq. (11.6). 11.12 From Eq. (11.10), Pm D B 2 =2000b0. Using B D 0:2 T gives Pm D 1:59 0002 104 N m00022 . Thus, Pm=Pg 0006 0:8 according to this rough estimate. 11.13
(a)
From Eq. (11.9), um D B 2 =2000b0 D 360 J m00023 .
(b) Assuming that the magnetic energy density is constant throughout the volume in question, if E D 1025 J are to be released, the required volume is V D E=um D 2:8 0002 1022 m3 . (c)
Taking V 0006 `3 , gives ` 0006 3 0002 104 km. Given that large flares can reach lengths of `flare D 100,000 km, ` 0006 0:3`flare .
(d) From Eq. (11.11), and assuming a density characteristic of the top of the solar photosphere ( ' 4:9 0002 1000026 kg m00023 ), the velocity of an Alfv´en wave is B vm D p ' 12 km s00021 : 0 Thus it would take an Alfv´en wave t D `=vm 0006 2500 s D 0:7 hr to travel the length of the flare. (e)
Based on the “back-of-the-envelope” nature of the calculation, magnetic energy seems to be a reasonable source of solar flares.
11.14 CMEs are responsible for roughly 4 0002 1015 kg of mass loss per year. The solar wind mass loss rate is approximately 3 0002 10000214 Mˇ yr00021 , which is equivalent to about 6 0002 1016 kg yr00021 . Thus, the mass loss from CMEs is 7% of the total mass loss of the solar wind. 11.15
(a)
The kinetic energy is approximately K D 8 0002 1023 J, which is roughly 10% of the energy of a large flare.
(b) The transit time is roughly t D 1 AU=400 km s00021 D 4 0002 105 s D 4 d.
11.16
(c)
Given that astronomers are constantly monitoring the Sun, the appearance of a solar flare or CME directed toward Earth gives a warning that charged particles from the Sun are headed toward us.
(a)
From Eq. (5.22), if B D 0:3 T, 0006 D where ' me .
eB D 4:2 0002 109 Hz; 40002000b
76
Chapter 11 The Sun (b) Beginning with 0005 D c, d 0005 D 0003 c= 2 d . This can be rewritten in the form 0005 d0005 D 0003 d : 0005 c Using 0005 D 630:25 nm and d ' 0006 D 4:2 0002 109 Hz, we find 00060005=0005 D 8:8 0002 1000026 .
11.17 The rotation frequency at the base of the tachocline is approximately f D ˝=20002 D 430 nHz. This corresponds to a rotation period of P D 1=f D 2:3 0002 106 s D 27 d. RV 11.18 From the thermodynamic work integral, W D Vif P dV , we have P D @W [email protected] , which is an energy density.
CHAPTER 12
The Interstellar Medium and Star Formation 12.1
(a)
According to Eq. (12.1), mV D MV C 5 log10 d 0003 5 C aV D 00031:1 C 5 log10 700 0003 5 D 8:1 mag. (aV D 0.)
(b) Using aV D 1:1 mag, mV D 9:2 mag. (c)
If extinction is not considered, using mV D 9:2 mag leads to a distance estimate of 1160 pc (see Eq. 3.5), which is in error by about 65.7%.
12.2 The luminosity of the star is given by L? D 40002R2? 0003T?4 . Letting d be the distance to the star, rg be the radius of the spherical dust grain, and assuming that the grain is a perfectly absorbing black body, the rate at which energy is absorbed by the grain is ! 0002 rg2 dEabsorb : D L? dt 40002 d 2 If the grain temperature is Tg , the rate at which the grain emits energy is dEemit D 40002 rg20003Tg4 : dt In thermodynamic equilibrium the two rates must be equal. Equating and solving for the grain temperature gives 0002 0003 R? 1=2 Tg D T? : 2d For an F0 main-sequence star, T? D 7300 K and R? D 1:4 Rˇ . At a distance of d D 100 AU, Tg D 41:7 K. 12.3 The hydrogen atom’s ground-state spin flip produces a photon of frequency D 1420 MHz. This implies that the energy difference between the two energy levels is 0006E D h D 9:41 0002 10000225 J D 5:88 eV. If thermal energy is sufficient, then kT 0006 0006E, or T 0006 0006E= k D 0:068 K. Yes, the temperatures of H I clouds are sufficient to produce this low-energy state. 12.4 Note that there is an error in the coefficient of Eq. 12.7 in the first printing of the text; the equation should read NH
H D 5:2 0002 10000223 : T 0006v Taking H D 0:5, T D 100 K, and 0006v D 10 km s00021 , Eq. (12.7) gives NH D 9:6 0002 1024 m00022 . Since NH D nH d , d D 31:2 pc. (Note: 0006v is expressed in units of km s00021 in Eq. 12.7.) 12.5 Typically nCO ' 1000024 nH2 , giving nCO ' 104 m00023 . Also, the mean molecular weight of the two molecules is D mH2 mCO = mH2 mCO . Using mH2 ' 2 u and mCO ' 28 u, ' 1:9 u. 77
78
Chapter 12 The Interstellar Medium and Star Formation For a total kinetic energy of E in the center-of-mass frame, v D energy to be E D 3kT =2, we have
p 2E=000b. Taking the translational kinetic
s 3kT D 450 m s00021 :
vavg '
From Eq. (10.29), the number of collisions per second between two species with number densities nx and ni is Z 1 nE nx ni 0003.E/ v.E/ dE: rix D n 0 Assuming for simplicity that 0003.E/ ' 0002.2r /2 is roughly constant over the appropriate energies (2r is the distance between molecule centers for a grazing collision), Z
1
rix ' 40002 r nx ni 2
0
v.e/
nE dE ' 40002 r 2nx ni vavg : n
Using the number densities of the two species, together with rix 0006 0:1 nm, r ' 6 0002 1000025 m00023 s00021 . 12.6 Because CO is much more common than 13 CO or C18 O, it is much more likely that the GMC will be optically thick to CO, thereby making it much more difficult to see deeply into the cloud. 12.7
(a)
The distances of the two masses from the center of mass can be written in terms of the reduced mass and their separation from each other (r ) as r1 D .000b=m1 / r and r2 D .000b=m2 / r . Substituting into the expression for the moment of inertia, ! ! 2 2 2 r C m2 r 2 D r 2: I D m1 m21 m22
(b) 12 D 6:856 u and 13 D 7:172 u, giving I12 D 1:639 0002 10000246 kg m2 and I13 D 1:715 0002 10000246 kg m2 . (c)
Erot D `.` C 1/00022 =2I . Thus, E2 D 300022 =I and E3 D 600022 =I , giving 0006E12 D 300022 =I12 D 2:034 0002 10000222 J D 0:00127 eV. This corresponds to a wavelength of 000512 D hc=0006E12 D 0:976 mm. This is in the deep infrared.
(d) 0006E13 D 1:947 0002 10000222 J, giving 000513 D 1:02 mm. Thus, 00060005 D 000513 0003 000512 D 0:044 mm. Isotopes are revealed by subtle wavelength shifts in molecular spectra. 12.8
(a)
Oxygen is much less abundant than hydrogen, and therefore, any photons emitted by a downward transition of an oxygen atom are much more likely to escape the cloud, rather than being reabsorbed. The photons then carry away cloud energy, resulting in cooling of the cloud. Oxygen atoms are also likely to emit infrared photons, while hydrogen will likely emit visible and ultraviolet photons.
(b) Hot cores are quite optically thick (AV 0006 50 to 1000) and have high number densities. It is very difficult for photons to escape from the interiors of hot cores. Hot cores are also the site of mass star formation, resulting in UV and X-ray photons, along with shocks that can heat the core. 12.9 As the density of a cloud increases, the likelihood of excitation collisions increases. As long as the cloud is not too optically thick, the resulting deexcitations liberate photons from the cloud, resulting in cooling. 12.10 Using T D 10 K, D 1, and 0 D 3 0002 10000217 kg m00023 , Eq. (12.16) gives RJ D 4 0002 1011 m D 3 AU for the dense core of the GMC.
Solutions for An Introduction to Modern Astrophysics
79
12.11 Substitution of Eq. (12.18) into (Eq. 12.14) leads immediately to 00031=2 0002 00033=2 0002 3 5 MJ ' vT3 : G 40002 0 Employing the ideal gas law, P0 D 0 kT =000bmH along with Eq. (12.18) also leads to 0 D P0 =vT2 : Substitution gives the desired result. 12.12 Hydrostatic equilibrium is only possible through a pressure gradient, dP =dr . In order for a constant density cloud to remain static, magnetic fields must be present to trap ions. According to the ideal gas law, constant mass density in an isothermal gas of constant composition (constant ) requires a constant pressure. 12.13
From the data given in Example 12.2.1 and the ideal gas law, Pc 0006 0 kT =000bmH D 1:2 0002 10000212 N m00022 . Using RJ D 4 0002 1011 m (see the solution to Problem 12.10) gives jdP =dr j 0006 Pc =RJ D 3 0002 10000224 N m00023 . (b) For Mr D 10 Mˇ and r D RJ , GMr =r 2 D 2:5 0002 10000219 N m00023 . This implies that (a)
jdP =dr j 0006 1000025 : GMr =r 2 (c)
The pressure gradient is not sufficient to keep the dense core in hydrostatic equilibrium. Noting that / r 00023 gives ˇ ˇ ˇ dP ˇ 0 kT =000bmH ˇ ˇ / r 00024: ˇ dr ˇ 0006 r Similarly GMr / r 00025 : r2 This r dependence yields jdP =dr j / r: GMr =r 2 As r ! 0, GMr =r 2 0004 jdP =dr j.
12.14 The gravitational acceleration at the surface of a cloud of mass MJ and radius RJ is g D GMJ =R2J . Assuming that g remains constant during the entire collapse, the time for the cloud to fall a distance RJ , starting from rest, is given by ! 1 2 GMJ RJ D gt D t 2: 2 2R2J Solving for the time and using MJ D results in tD
2R3J GMJ
!1=2
4 0002R3J 0 3
2 D4
31=2 0004 G
2R3J 4 0002R3J 0 3
00055
which differs from Eq. (12.26) for the free-fall time by a factor of s 3=20002 D 1:27: 30002=32
0002 D
3 20002 G 0
00031=2 ;
80
Chapter 12 The Interstellar Medium and Star Formation
12.15 The sound speed is given by Eq. (10.84), vs D
p
0007P = :
Assuming an ideal monatomic gas, 0007 D 5=3. Furthermore, referring to Example 12.2.1 and the calculation in Problem 12.13(a), 0 ' 3 0002 10000217 kg m00023 and Pc ' 1:2 0002 10000212 N m00022 . These values give vs ' 260 m s00021 . From Problem 12.10, RJ ' RJ D 4 0002 1011 m, so that ts D 2RJ =vs D 3 0002 109 s D 100 yr. The adiabatic sound speed transit time should be less than the free-fall time, otherwise the free-fall will become supersonic and shocks will develop. 12.16 Beginning with the expression just prior to Eq. (12.28), 5=2
MJ
D
40002 9=2 R e0003T 4 ; G 3=2 J
and writing RJ D .3MJ =40002 0/1=3 gives MJ5=2
40002 D 3=2 G
0002
3MJ 40002 0
00033=2 e0003T 4 :
Solving for 0 in Eq. (12.14) and substituting leads to MJ D
1 1 1=2 .40002 / G 3=2
0002
5k mH
00039=4
1 1=2 T : 9=4 0003 e 1=2
Substituting the values of the constants gives Eq. (12.28). 12.17 Suppose that the cloud is spherically symmetric with constant density, then from Eq. (10.22), ˇ ˇ ˇUg ˇ 9 GM 2 3GM 2 =5R ug 0007 : D D V 200002 R4 40002R3 =3 Using MJ D 10 Mˇ (Example 12.2.1) and RJ D 4 0002 1011 m (solution to Problem 12.10) gives ug D 1:5 0002 105 J m00023 . From Eq. (11.9), um D B 2 =2000b0 D 4 0002 10000213 J m00023 . In this case um =ug 0006 3 0002 10000218 and magnetic effects are not significant. However, magnetic fields can be important in other situations. 12.18
(a)
Adding a centripetal acceleration term to Eq. (12.19) gives d 2r dvr GMr D vr D 0003 2 C ! 2 r: 2 dt dr r From conservation of angular momentum (assuming rigid-body rotation), I0 !0 D I!, where I is the moment of inertia of the cloud. Since I / r 2 (the proportionality factor being a function of the mass distribution), ! ' r02 !0 =r 2 . (Note that since the geometry will change from nearly spherical to having a planar component, the proportionality factor also changes, but only slightly; this effect is neglected in this calculation.) Substituting the expression for !, vr
dvr r0 4 GMr D 0003 2 C 3 !0 2 : dr r r
Integrating from v0 at r0 to vf at rf gives 0005 10004 2 2 D GMr vr;f 0003 vr;0 2
0002
1 1 0003 rf r0
0003
1 0003 !02 r04 2
1 1 0003 2 2 rf r0
! :
Solutions for An Introduction to Modern Astrophysics
81
Initially vr;0 D 0, and when the collapse is halted vr;f D 0. Thus the left-hand side is zero. Furthermore, since we are assuming that the initial cloud is much larger than the final configuration, r0 0005 rf , implying 1=rf 0003 1=r0 ' 1=r0 and 1=rf2 0003 1=r02 ' 1=rf2 . Substituting, and solving for rf gives rf D (b) !0 D (c)
!02 r02 : 2GMr
p 2GMr rf =r02 D 2:7 0002 10000216 rad s00021 :
v0005;0 D !0 r0 D 4:10 m s00021 .
(d) From conservation of angular momentum, I0 !0 D If !f . Using I0 D 25 M r 2 and If D 12 M r 2 , 0004 00052 !f D 45 rr0 !0 D 2:3 0002 10000210 rad s00021 . This gives v0005;f D !f rf D 3:4 km s00021 . (Note that this differs f from the approximation made in part (a) by a factor of 0.8.) (e)
P0005 D 20002 rf =v0005;f D 2:8 0002 1010 s D 880 yr, while from Kepler’s third law, P 2 D 40002 2 r 3=GM , P D 3:2 0002 1010 s D 1000 yr. Rigid-body rotation is inconsistent with Keplerian motion.
P =40002 r 2 v D 2:8 0002 10000217 kg m00023 . This 12.19 From Example 11.2.1, MP D 40002 r 2 v. Solving for the density, D M is similar to the density of the dense core of the giant molecular cloud, and exceeds the density of a typical diffuse hydrogen cloud.
CHAPTER 13
Main Sequence and Post-Main-Sequence Stellar Evolution 13.1
(a)
See Table S13.1 Table S13.1: The elapsed time between points in Fig. 13.1 for a 5 Mˇ star. Between Points 2–3 3–4 4–5 5–6 6–7 7–8 8–9 9 – 10
Elapsed Time (Myr) 1.5234 0.1144 0.3483 0.2890 4.1727 1.5045 6.3200 1.246
Percentage of Main-Sequence Time 1.64 0.12 0.37 0.31 4.49 1.62 6.80 1.34
(b) The Hertzsprung gap occurs between points 3 and 5. The total elapsed time is 0.4627 Myr, which is only 0.49% of the time spent on the main sequence between points 1 and 2. (c)
The blueward portion of the horizontal branch occurs between points 7 and 8, which amounts to 1.62% of the main-sequence lifetime.
(d) The redward portion of the horizontal branch occurs between points 8 and 9, amounting to 6.8% of the main-sequence lifetime. 13.2 The Kelvin–Helmholtz timescale is given by tKH D 0006Eg =L, where L is the luminosity of the star and 0006Eg D 3GM 2 =10R (Eq. 10.23). From Fig. 13.1, for a 5 Mˇ star on the subgiant branch, L 0006 103 Lˇ and Te 0006 103:9 K, implying a radius of 17 Rˇ . This gives Eg ' 1:7 0002 1041 J. Finally, tKH ' 4:4 0002 1011 s D 14,000 yr. From Table S13.1, the time spent between points 4 and 5 in Fig. 13.1 is roughly 350,000 yr. This is about 25 times longer than the “back-of-the-envelope” Kelvin–Helmholtz timescale. 13.3
(a)
Beginning with Eq. (13.7), differentiate with respect to the mass of the isothermal core, Mic , and set the result to zero. This gives 0002 0003 @Pic kTic 3 2 GMic D 0: D 0003 @Mic 5 Ric 40002R3ic ic mH Solving for Ric leads immediately to Eq. (13.8).
(b) Substituting the expression for Ric (Eq. 13.8) into Eq. (13.7) and simplifying gives Eq. (13.9). 82
Solutions for An Introduction to Modern Astrophysics
83
0 0 =X12 should increase. This is because the X13 composition “bump” will be mixed in together 13.4 The ratio of X13 with a lower concentration of X12 .
13.5 The “back-of-the-envelope” estimate of the temperature required for nuclear burning (Eq. 10.27) is given by Tquantum D
4 m Z12 Z22 e 4 : 3 kh2
For the first step of the triple alpha process, two helium nuclei must interact. Setting m D mHe mHe =.mHe C mHe / D mHe =2 ' 2 u and Z1 D Z2 D 2, gives Tquantum D 6 0002 108 K. 13.6
(a)
An increase in luminosity, L, leads to increased radiation pressure, enhancing mass loss. A decrease in the surface gravity, g, means that the surface material is less tightly bound, also enhancing mass loss. As the radius, R, of the star increases for given values of L and g, the surface flux decreases, decreasing the effect of radiation pressure.
(b) For L D 7000 Lˇ and T D 3000 K, the Stefan-Boltzmann equation (Eq. 3.17) gives R D 310 Rˇ . This implies that g=gˇ D 1:04 0002 1000025 . Assuming that D 1, MP D 00038:7 0002 1000027 Mˇ yr00021 . 13.7
(a)
Since in solar units g=gˇ D M=R2 , direct substitution leads immediately to the desired result.
(b) Multiplying the result of part (a) through by M gives M
dM D 0003c LR; dt
where c 0007 4 0002 1013 . Integrating from M0 at t D 0 to M at t gives 00051=2 0004 : M D M02 0003 2c LRt (c)
See Fig. S13.1.
(d) Choosing M0 D 1 Mˇ and M D 0:6 Mˇ , gives t D 369,000 yr. 1.0 0.9 0.8 Mass ( M )
0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0
0
1
2
3 Time (10 5 yr)
4
5
6
Figure S13.1: Mass as a function of time, assuming a Reimers mass loss rate, for constant L D 7000 Lˇ , R D 310 Rˇ , and D 1. See Problem 13.7. 13.8
(a)
Using 160 D 0:0047 rad gives d D .213 pc/.0:0047 rad/ D 0:99 pc.
(b) The age of the nebula is approximately t ' d=2v D 24,000 yr.
84
Chapter 13 Main Sequence and Post-Main-Sequence Stellar Evolution
13.9 Color-magnitude diagrams of young galactic clusters, such as h and 0004 Persei (Fig. 13.18), reveal an incomplete main sequence. The stars on the lower end of the main sequence are missing while cool stars of greater luminosity are lying above the main sequence instead. The presence of these stars is not explained by the old theory. Furthermore, color-magnitude diagrams of old globular clusters, such as M3 (Fig. 13.17) show that the upper main sequence is missing, replaced by the distinct presence of a red giant branch connecting the lower portion of the main sequence to the red giant stars. A horizontal branch is also present that does not lead to upper-main-sequence stars. These observations for old clusters would suggest that red giant stars cannot be producing upper-main-sequence stars. 13.10
(a)
The contraction time to the main sequence for a 0:8 Mˇ star is about 68 Myr while the main-sequence lifetime of a 15 Mˇ star (between points 1 and 2 on Fig. 13.1) is about 11 Myr. It takes significantly longer for the 0:8 Mˇ star to reach the main sequence than it does for the 15 Mˇ star to evolve off the main sequence. This is consistent with the presence of red giants and massive stars leaving the main sequence before low-mass stars have arrived onto the main sequence in Fig. 13.18.
(b) According to Table 13.1, 68 Myr would be the main sequence lifetime of roughly a 6 Mˇ star. 13.11
(a)
From Table 13.1, the main-sequence lifetime of a 0:8 Mˇ star is about 18.8 Gyr, which is much longer than the present age of the universe. This implies that 0:8 Mˇ stars haven’t had time to evolve off the main sequence yet. The Sun’s main-sequence lifetime is about 10 Gyr, which is less than the age of the universe. For stars much below 1 Mˇ , the universe isn’t old enough yet for those stars to have evolved off the main sequence, and so there is no observational data to compare with theoretical models.
(b) No, simply because there hasn’t been enough time elapsed for those stars to leave the main sequence, even if the globular cluster formed very near the beginning of the universe. 13.12
(a)
From Eq. (3.8), MB 0003 MB;ˇ D 00032:5 log10 .LB =LB;ˇ / MV 0003 MV;ˇ D 00032:5 log10 .LV =LV;ˇ / : Subtracting and using the identity MB 0003 MV D B 0003 V , we find that B 0003 V D 2:5 log10 .LV =LB / 0003 2:5 log10 .LV;ˇ =LB;ˇ / C .Bˇ 0003 Vˇ / :
(b) When placed on the same color-magnitude diagram, the red giant branch of M15 is shifted noticably to the left (blueward) relative to 47 Tuc. (c)
Since M15 is less metal-rich than 47 Tuc, fewer electrons are available in the atmospheres of its stars relative to the stars in 47 Tuc. This means that more atoms are in higher states of ionization because fewer electrons are available for recombination (recall the Saha equation). As a consequence, the opacity is reduced in the atmospheres of M15 stars relative to 47 Tuc stars, and we can see down into deeper, hotter layers of stars in M15 relative to stars in 47 Tuc.
13.13 From Fig. 13.17, the turn-off point occurs at about B 0003 V D 0:4, corresponding to a value of V D 19:2. According to the composite main sequence of Fig. 13.19, B 0003 V D 0:4 corresponds to MV D 3:6. Assuming no interstellar extinction between Earth and M3 (a0003 D 0), Eq. (12.1), V D MV C 5 log10 d 0003 5 C a0003 ; implies d D 13 kpc. (Note: The actual distance is estimated to be 9.7 kpc.)
CHAPTER 14
Stellar Pulsation 14.1 From Fig. 14.1, Mira’s visual apparent magnitude varies between V 2 (although this varies significantly) and V 9:5. From Eq. (3.3), the ratio of the radiant flux (which is also the luminosity ratio) received from Mira is Fbright D 100.9:500022/=5 D 1000: Fdim Although Mira’s light curve is not sinusoidal, we will approximate it as V D 5:75 C 3:75 cos 000e; where 0 000e 20002 for one pulsation cycle. Mira will be visible to the naked eye roughly when V 6, or when cos 000e 0:07. This occurs for 86ı 000e 274ı, which is 188ı=360ı 50% of Mira’s pulsation cycle. 14.2 For an uncertainty of 0006M 0:5, the uncertainty in the calculated distance obtained from Eq. (3.5) is about 0006d 10.m0002M C0006M C5/=5 0003 10.m0002M C5/=5 D 0:26: D d 10.m0002M C5/=5 14.3 Referring to Fig. 14.22, the two M100 Cepheids nearest the best fit line have periods and apparent visual magnitudes of log10 P1 ' 1:36, V1 ' 26:3, and log10 P2 ' 1:62, V2 ' 25:5. According to the period– luminosity relation (Eq. 14.2), the absolute visual magnitudes of these stars are MV;1 D 00035:24 and MV;2 D 00035:97. The distances to these Cepheids obtained neglecting extinction from Eq. (3.5) are d1 D 20:3 Mpc and d2 D 19:7 Mpc. Including extinction, as in Eq. (12.1), we find d1 D 19:0 Mpc and d2 D 18:4 Mpc. One of these distances lies within the range of 17:1 ˙ 1:8 Mpc, and the other lies just outside the range. 14.4 W Virginis stars are about four times less luminous than classical Cepheids, so Eq. (3.7) shows that 0002 0003 LW Vir D 1:51: MW Vir 0003 MCeph D 00032:5 log10 LCeph From Eq. (14.2) then gives a period–luminosity relation for the W Virginis stars, MhV i D 00032:81 log10 Pd C 0:08: This is plotted in Fig. S14.1. 14.5 Assuming that the oscillations of ı Cephei are sinusoidal, its radius varies about an equilibrium value R0 as 0002 0003 20002 t : R.t/ D R0 C A sin ˘ The surface velocity is v.t/ D
0002 0003 20002A 20002 t dR.t/ D cos : dt ˘ ˘
For a period of ˘ D 5d 8h 48m D 4:64 0002 105 s and a surface velocity amplitude of 19 km s00021 , we find AD
vmax ˘ D 1:40 0002 109 m ' 2:0 Rˇ : 20002 85
86
Chapter 14 Stellar Pulsation −8 −7 −6 −5
M
ids
phe
e al C
ssic
Cla
−4
tars
is s
−3
W
in irg
V
−2 −1 0
0.4
0.6
0.8
1.0
1.2 1.4 Log10P
1.6
1.8
2.0
Figure S14.1: Results for Problem 14.4. 14.6 From Eq. (14.6) with 0007 D 5=3 and ˇ D 1410 kg m00023 , ˘ 5:48 0002 103 s, or 91.4 minutes. 14.7 Setting P D P0 C ıP and R D R0 C ıR, we have .P0 C ıP /.R0 C ıR/30007 D P0 R30007 0 0002 0003 0002 0003 ıP ıR 30007 30007 30007 R0 1 C P0 1 C D P0 R0 P0 R0 0002 00030002 0003 ıP ıR 1C 1 C 30007 C
D 1 P0 R0 ıP ıR D 000330007 : P0 R0 to first order in the deltas. 14.8
(a)
Setting L D L0 C ıL, R D R0 C ıR, and T D T0 C ıT in the Stefan–Boltzmann equation (Eq. 3.17), we have L0 C ıL D 40002.R0 C ıR/2 0003.T0 C ıT /2 0002 0003 0002 0003 0002 0003 ıL ıR 2 ıT 4 D 40002R20 1 C L0 1 C 0003T04 1 C : L0 R0 T0 Cancelling the L0 D 40002R20 0003T04 , we have 0002 0003 0002 00030002 0003 ıL ıR ıT 1C D 1C2 C
1C4 C
L0 R0 T0 ıL ıR ıT D2 C4 L0 R0 T0 to first order in the deltas.
Solutions for An Introduction to Modern Astrophysics
87
(b) The adiabatic relation T V 0007 00021 D constant can be expressed as TR3.0007 00021/ D constant. Setting T D T0 C ıT and R D R0 C ıR, this becomes 3.0007 00021/
.T0 C ıT /.R0 C ıR/3.0007 00021/ D T0 R0
0002 0003 0002 0003 ıT ıR 3.0007 00021/ 00021/ 00021/ R3.0007 1 C T0 1 C D T0 R3.0007 0 0 T0 R0 0002 00030006 0007 ıT ıR 1C 1 C 3.0007 0003 1/ C
D 1 T0 R0 ıT ıR D 00033.0007 0003 1/ : T0 R0 to first order in the deltas. Substituting this into the result from part (a) gives 0007 0006 ıL ıR ıR ıR D 2.000360007 C 7/ D2 C 4 00033.0007 0003 1/ : L0 R0 R0 R0 Using 0007 D 5=3 for an ideal monatomic gas, we find ıL ıR D 00036 : L0 R0 14.9 Expanding a general potential energy function, U.r /, in a Taylor series about the origin results in ˇ ˇ 1 d 2 U ˇˇ 2 d U ˇˇ rC r C
; U.r / D U.0/ C dr ˇ0 2 dr 2 ˇ0 where the higher-order terms may be neglected for small displacements from the origin. Because the origin is a stable equilibrium point, d U =dr D 0 at the origin and d 2 U =dr 2 > 0 there. The force on the particle is therefore ˇ ˇ d 2 U ˇˇ d 2 U ˇˇ d U.r / O r r D 0003 r; rO ' 0003 FD0003 dr dr 2 ˇ dr 2 ˇ 0
0
which is in the form of Hooke’s law for a spring. The particle will therefore undergo simple harmonic motion for small displacements from the origin. 14.10 See Fig. S14.2. 14.11 The condition for convection to occur is that A > 0, where A is given by Eq. (14.17). That is, 1 d 1 dP > : dr 0007P dr
(S14.1)
Differentiating the ideal gas law, Eq. (10.11), with respect to r (assuming D constant) and solving for d =dr gives 1 d 1 dP 1 dT D 0003 : dr P dr T dr Using this to replace the left-hand side of Eq. (S14.1) gives 1 dP 1 dT 1 dP 0003 > P dr T dr 0007P dr
88
Chapter 14 Stellar Pulsation 1.0
Fλ/Fc
3
4
0.5
2
1 0.0 Wavelength
Figure S14.2: Results for Problem 14.10. Shown are the line profiles for points of view 1–4 of Fig. 14.23; the line profiles for points of view 5–8 are the same as those for 1–4. Note that the line profiles for points of view 1 and 5 are not Doppler shifted. or
0002 0003 1 T dP dT 10003 > : 0007 P dr dr
The quantity on the left-hand side is the adiabatic temperature gradient (Eq. 10.88), while the right-hand side is the star’s actual temperature gradient. Thus we have the condition for convection to occur, ˇ ˇ d T ˇˇ d T ˇˇ > : dr ˇad dr ˇact Recognizing that both temperature gradients are negative, we recover Eq. (10.94). 14.12 Using the data in Table 14.2, we have the average pressure P D 9:1409 0002 103 N m00022 and dP 0006P ' D 00030:0605 N m00023 : dr 0006r The average density is D 2:2108 0002 1000024 kg m00023 and d 0006 ' D 00037:63 0002 10000210 kg m00024 : dr 0006r So for 0007 D 5=3, AD
1 d 1 dP 0003 D 5:21 0002 1000027 m00021 : dr 0007 P dr
Note that A > 0, as expected in a convection zone. Using a surface gravity of g D GMˇ =R2ˇ D 274 m s00022 , the time scale for convection is tc ' 237 s, nearly four minutes. This falls within the observed range of 3–8 minutes for the Sun’s p-modes. 14.13
(a)
Referring to Eq. (14.20), m
dv GM m C 40002R2 P; D0003 dt R2
Solutions for An Introduction to Modern Astrophysics
89
this is just Newton’s second law. The left-hand side is of course ma. The first term on the right is the inward force of gravity on the shell of mass m, while the second term on the right is the outward pressure force acting on the shell of surface area 40002R2 . 0007
0007
(b) Substituting Vi D 40002R3i =3, and similarly for Vf , into Pi Vi 30007
D Pf Vf gives, after cancelling the
30007
constants,Pi Ri D Pf Rf . (c)
Making the substitutions given in the problem, Eq. (14.20) becomes m
vf 0003 vi GM m C 40002R2i Pi ; D0003 t R2i
or vf D vi C
40002R2i Pi GM 0003 2 m Ri
! t;
which is Eq. (14.24). Similarly, for v D dR=dt, we have vf D
Rf 0003 Ri t
or Rf D Ri C vf t; which is Eq. (14.25) (d) See Figs. S14.3–S14.5. Note that the graphs are not perfectly sinusoidal, especially P vs. t. This is a nonlinear calculation.
1.7
R (1010 m )
1.6
1.5
1.4
1.3
0
2
4
6
8 10 t (days)
12
14
16
18
Figure S14.3: R vs. t for Problem 14.13. (e)
From the results of the calculations and the graphs, the period of the oscillation is ˘ D 5:25 d D 4:53 0002 105 s, and the equilibrium radius is R0 D 1:52 0002 1010 m. Equation (14.14) for the period of the linearized one-zone model (using 0 D 0:680 kg m00023 ) gives ˘lin D 4:56 0002 105 s D 5:28 d, in good agreement with the nonlinear period. The parameters used in this problem were chosen to produce a result close to ı Cephei’s measured period of ˘obs D 5d 8h 48m D 5:37 d.
90
Chapter 14 Stellar Pulsation
30 20
v (km s −1)
10 0 −10 −20 −30
0
2
4
6
8 10 t (days)
12
14
16
18
16
18
Figure S14.4: v vs. t for Problem 14.13.
19 17
P (104 N m -2)
15 13 11 9 7 5
0
2
4
6
8 10 t (days)
12
14
Figure S14.5: P vs. t for Problem 14.13.
CHAPTER 15
The Fate of Massive Stars 15.1 Using a mass estimate for Carinae of roughly 120 Mˇ , the Eddington luminosity (Eq. 10.6) is LEd ' 4:6 0002 106 Lˇ . This closely corresponds to star’s present quiescent of 5 0002 106 Mˇ ; Carinae is at the Eddington limit. 15.2
(a)
The distance to Car is estimated to be 2300 pc. Assuming that AV D 1:7, Eq. (12.1) implies that MV ' Mbol ' 000313:5. From Eq. (3.8), and using MSun D 4:74, L D 2 0002 107 Lˇ .
(b) If the luminosity found in part (a) were maintained over 20 yr, the total energy output would have been 4:8 0002 1042 J. (c)
K D 1:2 0002 1042 J.
15.3 The linear extent of one of the lobes is ` D .8:500 /.2300 pc/ D 0:095 pc D 20,000 AU D 3 0002 1015 m. Assuming a constant expansion rate of v D 650 km s00021 , t D `=v D 4:5 0002 109 s D 140 yr. If the velocity has been accelerating somewhat over time due to radiation pressure (as in the Crab supernova remnant), this value would be a slight underestimate. 15.4
(a)
From Eq. (15.9), dN D 00030005t: N Integrating and solving for N gives Eq. (15.10).
(b) Since N D N0 =2 when t D 1=2 , 1 D e 0002000300041=2 : 2 Solving for 0005 gives the desired result. 15.5 Using Eq. (10.23), and letting f D 0:01 be the fraction of the total energy that goes into neutrinos that eject the remaining mass, the kinetic energy deposited into the ejecta becomes 0002 0003 1 3 1 2 0006E D ; 0003 f GMˇ 10 rf ri where ri is the radius of the core at the start of the collapse and rf is the core’s final radius. The amount of energy needed to eject 9 Mˇ of material out of the potential well created by the remaining 1 Mˇ is 9GMˇ2 Eneed D hr9 i where hr9 i is an average radius of the 9 Mˇ before the core collapses. Equating the two expressions leads to 0002 rf D
30 1 C f hr9 i ri
91
000300021
:
92
Chapter 15 The Fate of Massive Stars Assuming that the initial radius of the core is approximately the radius of Earth (R˚ ) and that hr9 i 0006 100 Rˇ (roughly the radius of a 10 Mˇ supergiant; see Appendix G), we find rf D 5000 km D 0:8 R˚ . On the other hand, if we assume that all of the 9 Mˇ is located in a thin shell just above the collapsing core, then hr9 i 0006 R˚ , and rf D 2 km. Using an intermediate and crude “guestimate” of hr9 i D 1 Rˇ for a centrallycondensed supergiant, we have rf D 220 km. There appears to be plenty of energy available in the graviational collapse of the core to eject the 9 Mˇ of overlying material (recall that the final core radius quoted in the text is roughly 50 km).
15.6
(a)
Using 20 D 5:82 0002 1000024 rad and 40 D 1:16 0002 1000023 rad, d2 D .2000 pc/.5:82 0002 1000024 rad/ D 1:2 pc and d4 D .2000 pc/.1:16 0002 1000023 rad/ D 2:3 pc.
(b) The time required for the Crab supernova remnant to travel its longer dimension at an expansion speed of 1450 km s00021 (if assumed to be constant) is roughly t ' d4=2v D 780 yr. This is a shorter time than the known age of about 940 yr, and indicates that the expansion is accelerating. 15.7 Using Mbol D 000317, Eq. (3.6) gives m D M C 5 log10 .d=10 pc/ D 00035:5, which is brighter than Venus. 15.8 Using a central core density of 0 0006 1013 kg m00023 for a 15 Mˇ star, the time for the homologous collapse of the inner core is roughly given by 0003 0002 30002 1 1=2 tff D D 0:021 s: 32 G 0 15.9
(a)
Letting 000f be the amount of energy released by the decay of one nucleus, the total luminosity produced by the supernova remnant is LD000f
dN D 0003000f0005N D 0003000f0005N0 e 00020003t : dt
Using the identity, log10 L D .log10 e/.ln L/, d log10 L log10 e dL D : dt L dt Substituting L and dL=dt D 000f00052N0 e 00020003t gives d log10 L D 0003 .log10 e/ 0005 D 00030:4340005: dt (b) Beginning with Eq. (3.8),
0002 Mbol D MSun 0003 2:5 log10
L Lˇ
0003 ;
and solving for log10 L gives log10 L D
Mbol 0003 MSun C log10 Lˇ : 2:5
Differentiating with respect to time, d log10 L 1 dMbol D0003 : dt 2:5 dt Substituting into the result of part (a) (Eq. 15.11) and rewriting gives Eq. (15.12), dMbol D 1:0860005: dt
Solutions for An Introduction to Modern Astrophysics
93
15.10 For a half-life of 1=2 D 77:7 d, the decay constant is 0005 D ln 2= 1=2 D 0:0089 d00021 . Using Eq. (15.12), dM=dt D 0:0097 mag d00021 . 15.11 The number of 56 27 Co atoms in 0:075 Mˇ of cobalt-56 is roughly N0 D
0:075 Mˇ D 1:6 0002 1054: 56 u
6 The half-life of 56 27 Co is 1=2 D 77:7 d D 6:71 0002 10 s. This corresponds to a decay constant of
0005D
ln 2 D 1:03 0002 1000027 s00021 :
1=2
The number of decays per second for N nuclei is just 0005N . Taking the energy released per decay to be 000f D 3:72 MeV D 5:96 0002 10000213 J, the luminosity of the supernova remnant is L D 000f0005N . (a)
Just after the formation of the cobalt, the initial decay luminosity is L0 D 000f0005N0 D 9:8 0002 1034 W D 2:6 0002 108 Lˇ .
(b) After a time t, N D N0 e 00020003t , implying L D L0 e 00020003t . This gives L D 3:79 0002 1033 W D 9:9 0002 106 Lˇ . (c)
These results are in excellent agreement with Fig. 15.12.
15.12 The neutrino energy flux at Earth was E=A D .4:2 MeV/.1:3 0002 1014 m00022 / D 87 J m00022 : Assuming an isotropic emission of neutrinos from SN 1987A, and given the distance of d D 50 kpc to the LMC, the total energy output was roughly E D .40002 d 2/.E=A/ D 2:6 0002 1045 J. 15.13 From Eq. (10.22), Ug D 00033GM 2 =5R D 3 0002 1046 J. This is approximately one order of magnitude greater than the neutrino energy estimate of the previous problem. 15.14 A gamma-ray burst is observed roughly once per day. If there are 100,000 neutron stars in our Galaxy, and if we were to assume that they are responsible for the bursts, each neutron star would need to undergo an event approximately once every 100,000=365 D 274 yr. 15.15 The rest-mass energy of the electron and positron are converted into two gamma-ray photons, or E0007 D me c 2 D 511 keV. 15.16 If the number densities of sources with energies E1 and E2 are n1 and n2, respectively, then the number of sources with a fluence equal to or greater than S is N .S/ D
4 4 0002 n1r13 .S/ C 0002 n2 r23 .S/: 3 3
but r1 .s/ D .E1 =40002S/3=2, and similarly for r2 . As a result, ' 0002 0003 0002 00033=2 # 40002 E1 3=2 E2 N .S/ D C n2 n1 S 00023=2 : 3 40002 40002 15.17
(a)
48 J
(b) Assuming that the energy is entirely kinetic energy, v D 25:9 m s00021 . (c)
The speed in part (b) corresponds to 58 mph, more than half the speed of a top major-league fastball!
94
Chapter 15 The Fate of Massive Stars
15.18 Assuming a typical magnetic field strength of 10000210 T throughout the Galaxy, r D 3 0002 1020 m D 10 kpc. The value obtained by this order-of-magnitude estimate is characteristic of the size of the Milky Way. Lowerenergy cosmic rays are likely bound to the Galaxy. 15.19 From the power law expression, log F D log C 0003 ˛ log E, implying that for values of F1 and F2 corresponding to E1 and E2, respectively, log.F2 =F1 / : ˛D0003 log.E2 =E1/ The flux at E1 D 1011 eV is approximately F1 D 1000021 m00022 sr00021 s00021 GeV00021 and the flux at E2 D 1016 eV is about F2 D 5 0002 10000214 with the appropriate units. This yeilds ˛ D 2:5. At the “ankle”, E3 D 5 0002 1018 eV with F3 D 5 0002 10000224 in the appropriate units. In the region between the “knee” and the “ankle”, ˛ D 3:7. 15.20 The Lorentz factor for a 1020 eV proton is given by 0007 D
E D 1 0002 1011 : mp c 2
CHAPTER 16
The Degenerate Remnants of Stars 16.1
(a)
From Eq. (7.2), the ratio of the masses of 40 Eri B and C is mC aB D D 0:37: mB aC To find a, the semimajor axis of the orbit, the distance to 40 Eri B and C must first be found from Eq. (3.1), 1 1 D 4:98 pc D 1:535 0002 1017 m: d D 00 D p 0:20100 The angular extent of the semimajor axis is ˛ D 6:8900 D 3:34 0002 1000025 rad, and so the semimajor axis is a D ˛d D 5:127 0002 1012 m. Using Kepler’s third law, Eq. (2.37), with an orbital period of P D 247:9 yr D 7:823 0002 109 s then gives the sum of the masses of 40 Eri B and C, mB C mC D
40002 2 3 a D 1:303 0002 1030 kg D 0:655 Mˇ : GP 2
Solving for the individual masses, mB D 0:478 Mˇ and mC D 0:177 Mˇ . (b) Using Eq. (3.7), LB D 100.MSun 0002MB /=5 Lˇ D 100.4:7600029:6/=5 Lˇ D 0:0116 Lˇ : (c)
From the Stefan–Boltzmann equation, Eq. (3.17), r LB 1 RB D 2 D 8:74 0002 106 m: TB 400020003 This is slightly larger than the size of Earth RB D 0:0126 Rˇ D 1:37 R˚ D 1:59 RSirius B :
(d) The average density of 40 Eri B is D 3:37 0002 108 kg m00023 . This is less than the average density of Sirius B ( 3 0002 109 kg m00023 ). Sirius B has a larger mass (1.05 Mˇ ), and so its electrons must be more closely confined (greater ne ) to produce the greater electron degeneracy pressure required to support the star. (e)
For 40 Eri B, mass 0002 volume D 2:69 0002 1051 kg m3 ; and for Sirius B, mass 0002 volume D 1:45 0002 1051 kg m3 : The lower value for Sirius B is an indication that it is smaller than might be anticipated from the massvolume relation. It is sufficiently dense that its electrons are moving at relativistic speeds, providing less support than expected from the nonrelativistic expression for the electron degeneracy pressure. 95
96
Chapter 16 The Degenerate Remnants of Stars
16.2 As the temperature of a DB white dwarf drops below 12,000 K, there would be insufficient excitation of its He I atoms. No absorption lines would be formed without one of the electrons in the n D 2 orbital. Because DB white dwarfs do not have any hydrogen lines either, the cooling would presumably transform it from a DB into a DC white dwarf, with a spectrum devoid of lines. 16.3 If the entire luminosity of Sirius B were produced by hydrogen burning, then the nuclear energy generation rate would be about LB 0:03 Lˇ 000fB D D 5:5 0002 1000026 W kg00021 : MB 1:053 Mˇ The density and interior temperature of Sirius B are taken to be B D 3 0002 109 kg m00023 and T D 107 K (the latter value suitable for an estimate of an upper limit for X ). We can estimate the value of X for the pp chain from Eq. (10.47) with 000fpp D 000fB , pp D 1 and fpp D 1. Then X D
!1=2
000fpp 0 000f0;pp
4 0002 1000024 :
4 pp fpp T6
For the CNO cycle, Eq. (10.59) with 000fCNO D 000fB and XCNO D 1 gives X D
000fCNO 3 0002 1000025 : 0 000fı;CNO XCNO T619:9
The smaller of these values may be taken as a rough upper limit for X . 16.4 Using the average density of 3:0 0002 109 kg m00023 , a central temperature of 3 0002 107 K, and a mean molecular weight of D 12=7 D 1:71 for a composition of pure carbon-12 (see Eq. 10.16), the ideal gas pressure at the center of Sirius B is (Eq. 10.11) Pg D
kT D 4:35 0002 1020 N m00022 : mH
The radiation pressure at the center is (Eq. 10.19) Prad D
1 4 aT D 2:04 0002 1014 N m00022 : 3
These are both much smaller than the estimated central pressure of 3:800021022 N m00022 obtained from Eq. (16.1), so both the gas and radiation pressure are too feeble to support Sirius B. 16.5 Using Eq. (16.4) for the electron number density, the pressure of an ideal gas of electrons, Eq. (10.10), is 0007 00060002 0003 Z kT: P D ne kT D A mH If we set this equal to the pressure of a degenerate electron gas, then Eq. (16.12) gives 00022 T D 5me k 2=3
0006
30002 2 mH
0002
This differs from Eq. (16.6) by a factor of 3/5. 16.6 Equation (16.7) for the pressure integral is P
1 ne pv: 3
Z A
000300072=3 :
Solutions for An Introduction to Modern Astrophysics
97
The electron number density is (Eq. 16.4) 0002 ne D
Z A
0003
; mH
the momentum is (Eq. 16.8) p 0002n1=3 e , and v D c in the extreme relativistic limit. Substituting these into the expression for P gives the desired result. 16.7
(a)
The Lorentz factor is given by Eq. (4.20), 0007 Dp
1 1 0003 u2 =c 2
:
When 0007 D 1:1, v=c D 0:417. (b) Solving Eq. (16.10) for the density and using the value for v=c from part (a) and Z=A D 1=2, we find that 0002 0003 0004 m v 00053 A e D D 4:2 0002 109 kg m00023 : mH Z 0002 16.8
(a)
The average density of a 0.6 Mˇ white dwarf of radius 0.012 Rˇ is D 4:89 0002 108 kg m00023 . The average separation between neighboring nuclei is, with A D 12, 0002 rD
3AmH 40002
(b) For Z D 6, Tc D (c)
7=2
From Eq. (16.19), Lwd D C Tc
00031=3
D 2:14 0002 10000212 m:
Z2 e 2 D 1:76 0002 106 K: 40002000f0 r k0010
, where
C D 6:65 0002 1000023
0002
Mwd Mˇ
0003
D 0:056 W K00027=2 Z.1 C X /
using X D 0, Z D 0:1, and D 1:4 from Example 16.5.1. The luminosity of the white dwarf is thus Lwd D 4:0 0002 1020 W, or about 1:1 0002 1000026 Lˇ . (d) The number of carbon nuclei is N D
Mwd D 5:9 0002 1055 : AmH
If each nucleus releases a latent heat kTc , then the white dwarf could sustain its luminosity for a time tD
N kTc D 3:6 0002 1018 s ' 1:1 0002 1011 yr: Lwd
This is about twenty times longer than the period of slower cooling (5 0002 109 yr) displayed in Fig. 16.9. The longer time scale is due to the lower core temperature of the white dwarf considered in this problem. 16.9 The density of the liquid-drop model of the nucleus is D
AmH 4 3 30002r
D
AmH 4 1=3 /3 3 0002.r0 A
D
3mH D 2:3 0002 1017 kg m00023 : 40002 r03
98
Chapter 16 The Degenerate Remnants of Stars
16.10 Using ns D 6:65 0002 1017 kg m00023 for the density of the neutron star and MMoon D 7:35 0002 1022 kg for the mass of the Moon (Appendix C), the radius of the Moon would be 0002 rD 16.11
(a)
3MMoon 40002 ns
00031=3 D 30 m:
If the lower mass is at R and the upper mass is at R C dr , then Flower F.R/ D Fupper F.R C dr / F.R/ F.R/ C Œ.dF=dr /jR 0011 dr / ˇ 0006 000700021 1 dF ˇˇ D 1C dr : F.R/ dr ˇR
'
Using Eq. (2.11), Mm dF D 00032G 3 ; dr r and so 0002 000300021 R2 2GM m Flower ' 10003 dr Fupper GM m R3 0002 0003 2dr 00021 D 10003 : R Using R D 104 m and r D 0:01 m, this is Flower ' 1:0000002: Fupper (b) The iron cube has a mass of 0.00786 kg and sides of length dr D 0:01 m. Imagine that half of the cube’s mass, m D 0:00393 kg, is concentrated at its top and bottom surfaces. The stress on the cube may then be estimated as stress '
Flower 0003 Fupper
.1000024 m2 ) 0002 0003 Flower Fupper ' 1 0003 Flower .1000024 m2 ) 0006 0002 00030007 Mm 2 dr 1 DG 2 10003 10003 R .1000024 m2 ) R D
2GM m dr .1000024 m2 /R3
D 1:46 0002 108 N m00022 : An iron meteoroid falling toward the surface of a neutron star would be stretched (to the point of rupturing) into a thin ribbon.
Solutions for An Introduction to Modern Astrophysics
99
16.12 First, estimate the speed of the neutrons using an analog of Eq. (16.10), p 0002 1=3 0002 vD D n D mn mn n mn
0002
mn
00031=3 :
For D 1:5 0002 1018 kg m00023 , v D 6 0002 107 m s00021 . Since v D 0:2c, we can use Eq. (16.12), replacing me ! mn and Z=A ! 1, to obtain an expression for the nonrelativistic neutron degeneracy pressure, 2 2=3 0002 0003 30002 5=3 00022 PD D 1:1 0002 1034 N m00022 : 5 mn mn This is about 6 0002 1011 times the central pressure of 1:9 0002 1022 N m00022 estimated for the center of Sirius B. 16.13 Use D 4 0002 1014 kg m00023 for the density at neutron drip. (a)
Equation (16.15) for the relativistic electron degeneracy pressure with Z=A D 36=118 gives 2 1=3 0007 00060002 0003 30002 Z 4=3 PD D 7:4 0002 1028 N m00022 : 0002c 4 A mH
(b) From an analog of Eq. (16.10), the speed of the neutrons is 0002 1=3 0002 p D n D vD mn mn n mn
0002
mn
00031=3 :
For D 4 0002 1014 kg m00023 , v D 4 0002 106 m s00021 . From Eq. (16.12) with me ! mn and Z=A ! 1, the nonrelativistic neutron degeneracy pressure is
2=3 0002 0003 30002 2 5=3 00022 PD D 1:2 0002 1028 N m00022 : 5 mn mn This is comparable to the electron degeneracy pressure found in part (a). 16.14
(a)
Using Pˇ D 26 d D 2:25 0002 106 s for the Sun’s rotation period, Eq. (16.26) gives 0002 Pns D Pˇ
Rns Rˇ
00032
D 4:6 0002 1000024 s:
Coincidentally, this is about the minimum rotation period for a typical neutron star. (b) For “frozen-in” magnetic field lines, Bˇ 40002R2ˇ D Bns 40002R2ns : Taking Bˇ D 2 0002 1000024 T for a typical solar magnetic field near the surface, we find 0002 Bns D Bˇ
Rˇ Rns
00032 D 9:7 0002 105 T:
16.15 The average density of Sirius B is wd D 3:0 0002 109 kg m00023 , and the average density of a 1.4 Mˇ neutron star is ns D 6:65 0002 1017 kg m00023 .
100
Chapter 16 The Degenerate Remnants of Stars (a)
Using 0007 D 5=3, Eq. (14.14) for the fundamental radial pulsation period may be written as s ˘ D 20002
3 : 40002 G
results in ˘wd D 6:9 s ˘ns D 4:6 0002 1000024 s: Pwd is above the period range of 0.25 s to 2 s observed for most pulsars, and Pns is below it. (b) Using Eq. (16.29) with the definition of the star’s average density, , we find that the minimum rotation period is equal to the fundamental radial pulsation period, s Pmin D 20002
R3 D 20002 GM
s 3 D ˘: 40002 G
So Pmin;wd D ˘wd D 6:9 s and Pmin;ns D ˘ns D 4:6 0002 1000024 s: (c)
The mathematical equality of the model star’s fundamental radial pulsation period and its minimum rotation period is established in parts (a) and (b). However, the physical basis for this result is found in the virial theorem. The fundamental radial pulsation period is basically the time for a sound wave to cross the star, p and so is proportional to the star’s radius R and inversely proportional to the speed of sound vs D 0007 kT =000bmH . Thus p p ˘ / R=vs / R= T =m / R= K=m (Eq. 10.17), where T is the temperature, m is the average mass of a gas particle, and K is the average kinetic energy per particle. On the other hand, the minimum rotation period occurs when the star’s surface is no longer gravitationally bound. That is, a gas particle of mass p m at thepstar’s surface is essentially orbiting the star. The orbital speed at the surface is vorbit D GM=R D 0003U =m, where U is the gravitational potential energy of the particle at the star’s surface (Eq. 2.14). Then 20002R=Pmin D vorbit D Therefore
p 0003U =m;
p Pmin / R= 0003U =m:
But according to the virial theorem (Eq. 2.46), 00032 hKi D hU i, and so ˘ / Pmin . The virial theorem relates the average kinetic energy of the gas particles (which determines the sound speed, and hence the pulsation period) to the average potential energy of the gas particles (which determines the minimum rotation period). 16.16
(a)
The minimum rotation period for a 1.4 Mˇ neutron star is, from part (b) of Problem 16.15, Pmin D 4:6 0002 1000024 s.
Solutions for An Introduction to Modern Astrophysics
101
(b) Solve R D 0:5.E C P / and
E0003P 5˝ 2 R3 D R 4GM
for E and P to get
0002 0003 5˝ 2 R3 E D R 1C 8GM
and
0003 5˝ 2 R4 : P DR 10003 8GM For twice the minimum rotation period, s R3 20002 D 2Pmin D 40002 ˝ GM or GM ˝2 D : 4R3 The equatorial and polar radii then become, with R D 10 km, 0002 0003 5 E DR 1C D 11:56 km 32 and
0002
0002
5 P DR 10003 32
0003 D 8:44 km:
16.17 At some instant of time, the period of PSR 1937C214 was determined to be P D 0:00155780644887275 s. A change of the last “5” to a “6” is a change of 0006P D 10000217 s. With a period derivative of PP D 1:051054 0002 10000219 s s00021 , the time for this change to occur is 0006t D 16.18
(a)
From P PP D P dP =dt D P0PP0 , we have Z P
0006P D 95:14 s: PP
P dP D P0PP0
P0
which results in
Z
t
dt 0 ;
0
0004 00051=2 P D P02 C 2P0 PP0t :
(b) In time dt, the pulsar completes dt=P rotations, and the pulsar clock (which wrongly assumes that each pulse has a duration P0 ) advances a time dtp D Pı dt=P . When the perfect clock displays a time t D P0 =PP0, the pulsar clock will display a time Z P0 =PP0 P0 tP D dt P 0 Z
P0 =PP0
D P0 0
0004
P02 C 2P0PP0 t
p P0 D . 3 0003 1/ : PP0
000500021=2
dt
102
Chapter 16 The Degenerate Remnants of Stars
16.19 Assume that angular momentum, L D I! with I D 25 MR2 , was conserved during the contraction of the Crab neutron star. Then 0002 0003 2 20002 L D I! D MR2 D constant; 5 P where P is the period of the Crab pulsar. Then dL D
40002 M 5
0002
2R dR R2 dP 0003 P P2
0003 D 0:
Solving for the change in the neutron star’s radius, dR D
R dP : 2 P
Using dP =P D 1000028 and R D 104 m, we find dR D 5 0002 1000025 m. 16.20 From Eq. (16.33), the magnetic field at the pole of the pulsar is s 1 3000b0 c 3 IP PP BD : 20002R3 sin 000e 20002 The moment of inertia of a uniform sphere of radius R D 10 km and mass M D 1:4 Mˇ is I D 1:1 0002 1038 kg m2 . Using P D 0:237 s, PP D 1:1 0002 10000214 , and 000e D 90ı , we estimate B ' 3:4 0002 108 T for the Geminga pulsar. 16.21
(a)
The radius of the light cylinder around a rotating neutron star is Rc D cP =20002 (see Fig. 16.26), where P is the rotation period. Assume Rns D 10 km for a typical neutron star. The period of the Crab pulsar is PCrab D 0:0333 s, so Rc;Crab D 1:59 0002 106 m D 159Rns: The period of PSR 184100030456 is P1841 D 11:8 s, so Rc;1841 D 5:6 0002 108 m D 5:6 0002 104 Rns :
(b) Assuming that the two pulsars have comparable magnetic field strengths at their surfaces, the ratio of their magnetic fields at the light cylinder is 0003 0002 Rc;1841 3 BCrab ' D 4:4 0002 107 : B1841 RCrab 16.22
(a)
From Eq. (16.32), P PP D P
80002 3B 2 R6 sin2 000e dP D : dt 3000b0c 3 I
(S16.1)
Integrating from t D 0 when the pulsar’s period was P0 to t and period P gives Z P Z 80002 3B 2 R6 sin2 000e t 0 P dP D dt : 3000b0c 3 I P0 0 This results in PD
P02
160002 3B 2 R6 sin2 000e t C 3000b0 c 3 I
!1=2 :
[Note that P PP is constant. This relation was seen earlier in Problem 16.18(a).]
(S16.2)
Solutions for An Introduction to Modern Astrophysics
103
(b) If P0 0004 P , then solving Eq. (S16.2) for t results in tD
3000b0c 3 IP 2 160002 3B 2 R6 sin2 000e
:
Combining this with Eq. (S16.1) gives tD (c)
P : 2PP
Using P D 0:0333 s and PP D 4:21 0002 10000213 for the Crab pulsar results in t D 3:95 0002 1010 s D 1250 yr for the age of the Crab pulsar. The Crab supernova occurred in A . D . 1054, so its present age (A . D . 2006) is 952 yr. The value of t is high by about 33%.
16.23 The ratio of the magnetic force to the gravitational force is evB Fm D : Fg mg At the neutron star’s equator, the velocity of the charged particle is v D 20002R=P , where P is the period of the pulsar. Then, for a proton at the surface of the Crab Pulsar, using R D 10 km, B D 108 T, P D 0:0333 s, and g D 1:86 0002 1012 m s00022 , 20002 eBR Fm D D 9:7 0002 109: Fg P mp g 16.24 The minimum energy required to create an electron-positron pair is Emin D 2me c 2 D 1:64 0002 10000213 J D 1:02 MeV: From Eq. (5.3), the wavelength of a photon with this energy is 0005D
hc D 1:21 0002 10000212 m D 0:00121nm: Emin
From Table 3.1, this is a gamma-ray photon. 16.25 For a 1ı subpulse, 000e D 0:5ı in Eq. (4.43) for the headlight effect. Using this with the Lorentz factor 0007 (Eq. 4.20), we have q 1 sin 000e D D 1 0003 u2 =c 2 : 0007 Solving for u=c,
p u D 1 0003 sin2 000e D 0:999962: c
CHAPTER 17
Black Holes 17.1 This effect is analogous to the motion of two stars in a binary system as they orbit their common center of mass. 17.2 Suppose the photon in Fig. 17.11 travels at an angle 000e measured from the vertical. Note that the frequency meter must be shifted along the lab’s ceiling to the endpoint of the photon’s path. According to the observer on the ground, if the photon travels a distance d from the floor to the ceiling of the lab, then the lab has fallen a vertical distance h D d cos 000e in a time t D d=c. The downward speed of the frequency meter is thus v D gt D gd=c D gh=c cos 000e, and the component of the meter’s velocity in the direction of the photon’s point of origin is v cos 000e D gh=c. The rest of the argument leading to Eq. (17.6) is unchanged. 17.3 With ` D 1000 m, the distance fallen is 1 1 d D gt 2 D 2 2
0002 0003 ` D 5:45 0002 10000211 m: c
17.4 Let rs D R˚ be the position of a person at sea level, let rL D rs C 3:1 km be the position of a person in Leadville, and let 0006t0 be the lifetime of that person. According to Eq. (17.13), an observer at a great distance would measure the person’s lifetime at sea level and at Leadville as 0006t1;s
0002 0003 2GM˚ 00021=2 D 0006t0 1 0003 rs c 2
0006t1;L
0002 0003 2GM˚ 00021=2 D 0006t0 1 0003 ; rL c 2
and
respectively. Thus 0006 0006t1;s 0003 0006t1;L D 0006t1;L
0006t1;s 00031 0006t1;L
0007
# 0003 0002 0003 2GM˚ 1=2 2GM˚ 00021=2 10003 D 0006t1;L 00031 10003 rs c 2 rL c 2 00060002 00030002 0003 0007 GM˚ GM˚ 10003 00031 ' 0006t1;L 1 C rs c 2 rL c 2 0006 0002 00030007 GM˚ rL 0003 rs : ' 0006t1;L rs rL rL c 2 '0002
Taking 0006t1;L ' 75 yr D 2:37 0002 109 s, we have 0006t1;s 0003 0006t1;L D 3:38 0002 10000213 0006t1;L D 8:0 0002 1000024 s: 104
Solutions for An Introduction to Modern Astrophysics 17.5
(a)
105
From Eq. (17.5), the radius of curvature of the light beam is rc D
c2 c 2 R2 D : g GM
At the surface of a 1.4 Mˇ neutron star of radius R D 104 m, rc D 4:84 0002 104 m. This is comparable with the 10 km radius of the neutron star, and shows that general relativity can not be neglected when studying neutron stars. (b) If 0006t0 D 1 hr D 3600 s passes at the surface of the neutron star, then Eq. (17.13) shows that the time that passes at a great distance is 0002 0003 2GM 00021=2 D 4700 s; 0006t1 D 0006t0 1 0003 Rc 2 greater than 1 hour by over 18 minutes! The approximate expression, Eq. (17.14), gives 0002 0003 GM 00021 0006t1 D 0006t0 1 0003 D 4540 s: Rc 2 The difference between these two results shows that gravity cannot be considered “weak” at the surface of a neutron star. 1/2 g(r)t2
α
z
1/2 g(r)t2 cos α
To Sun
t=0 dz
t = dz/c
Figure S17.1: Falling local inertial reference frame for Problem 17.6. 17.6
(a)
Figure S17.1 shows a rectangular local inertial reference frame of width dz at two times: when the photon (not shown) enters the left-hand side at time t D 0, and when the photon exits the right-hand side at time t D dz=c. (For clarity, the figure greatly exaggerates the distance the frame falls.) In this time, the frame has fallen a distance of 1 g.r /t 2 ; 2 toward the Sun, where r is the distance between the frame and the Sun, and g.r / D GMˇ =r 2 is the local gravitational acceleration. The perpendicular distance of the frame from the z-axis is then 1 g.r /t 2 cos ˛: 2 Referring to Fig. 17.10 with ` replaced by dz, 0012 replaced by d0012, and 12 gt 2 replaced by 12 g.r /t 2 cos ˛, we have 0006 0007
0006 0007
1 dz 2 dz D OD: g.r /t cos ˛ 2 2 cos.d0012=2/
106
Chapter 17 Black Holes Because d0012 is so small, we can set cos.d0012=2/ ' 1 and OD ' rc , the radius of curvature of the photon’s path. Thus, in radians, d0012 D
dz g.r /t 2 cos ˛ GMˇ D D rc dz r2
0002
dz c
00032
cos ˛ : dz
Defining g0 0007 GMˇ =r02 , where r0 D r cos ˛ is the distance of closest approach, we obtain d0012 D
g0 cos3 ˛ dz: c2
(b) To find the total angular deflection, 0012, we integrate d0012 from ˛ D 00030002=2 to C0002=2. Using z D r0 tan ˛, we have dz D r0 sec2 ˛ d˛. Thus Z 0012D
1 00021
g0 cos3 ˛ g 0 r0 dz D 2 c2 c
Z
=2 0002 =2
cos ˛ d˛ D
2g0 r0 2GMˇ D : c2 r0 c 2
Assuming the photon just grazes the Sun’s surface, we set r0 D Rˇ and find 0012D (c)
2GMˇ D 4:24 0002 1000026 rad D 0:87500 : Rˇ c 2
The answer to part (b) is half of the correct value of 1:7500 . In general, the correct result for the angular deflection of a photon that passes within a distance r0 of a spherical mass M is 0012D
4GM ; r0 c 2
twice the previous result. Our derivation in parts (a) and (b) included only the effect of the curvature of space, and not the equal contribution of time running more slowly in the curved spacetime near the Sun. The photon spends more time near the Sun (as measured by a distant observer), and so suffers a larger angular deflection. 17.7
(a)
Elsewhere: .0006s/2 D .c0006t/2 0003 .0006`/2 < 0.
(b) Future lightcone: .0006s/2 > 0 and t > 0. (c)
Future lightcone: .0006s/2 > 0 and t > 0.
(d) Elsewhere: .0006s/2 < 0. (e)
Past lightcone: .0006s/2 > 0 and t < 0.
(f) Future lightcone: .0006s/2 > 0 and t > 0. (g) Elsewhere: .0006s/2 < 0. (h) Elsewhere: .0006s/2 < 0. 17.8 For this problem, the Lorentz factor (Eq. 4.20) is 0007 D 3:20. It is convenient to express the speed of light as c D 1 ly yr00021 . (a)
0006t B D 11:7 ly=0:95c D 12:3 yr.
(b) From Eq. (4.27) with 0006t B D 0006t moving, 0006t A D
0006t B D 3:85 yr: 0007
Solutions for An Introduction to Modern Astrophysics (c)
107
The elapsed time according to Alice’s watch is 20006t A D 7:70 yr.
(d) The elapsed time according to Bob’s watch is 20006t B D 24:6 yr. Thus Alice will be 16.9 yr younger than Bob. In Fig. 17.14, we can take events A and B to be Alice’s leaving and arriving back on Earth, and C to be her turn-around at Ceti. Then Bob’s worldline is the straight segment, A ! B, and Alice’s worldline is A ! C ! B. The straight worldline has the longer interval. 17.9
(a)
Gravitational time dilation affects both the rate at which photons are emitted by a blackbody and the frequency of each photon. Both effects decrease the luminosity of the blackbody as measured by a distant observer. According to Eq. (17.13), if 0006t is the time between photons as measured at the surface of the blackbody, then the time between photons as measured at infinity is 0003 0002 2GM 00021=2 D 0006t 1 0003 : Rc 2
0006t1
Similarly, if is the frequency of a photon measured at the surface of the blackbody, then the frequency of the photon measured at infinity is 1
0003 0002 2GM 1=2 D 10003 : Rc 2
From Eq. (5.3), Ephoton D h ; the energy of every photon is therefore decreased by a factor of 0002 0003 2GM 1=2 10003 : Rc 2 Thus if L is the luminosity measured at the surface of the blackbody, then an observer at infinity measures 0003 0002 2GM : L1 D L 1 0003 Rc 2 (b) Both observers measure the same value for the speed of light, so c D 00051 1 D 0005 . Both observers use Wien’s law, Eq. (3.15), to determine the blackbody’s temperature, so 0005max;1 T1 D 0005maxT . Equation (17.13) then shows that 0002 T1 D T (c)
0005max 0005max;1
0003 DT
0004 0005 1
0002 0003 2GM 1=2 DT 10003 : Rc 2
Both observers use the Stefan–Boltzmann law, Eq. (3.17). Their values of the radius (determined at infinity, R1 , and at the surface of the spherical blackbody, R) are related to their values of the luminosity and the effective temperature by L1 R2 T 4 : D 12 1 L R T4 Using the results of parts (a) and (b), this is 0002
R1
T DR T1
00032 0002
L1 L
00031=2 Dq
R 1 0003 2GM=Rc 2
:
108
Chapter 17 Black Holes
17.10 Start with Eq. (2.17) with r D R for the escape velocity at the surface of a spherical object of mass M and radius R. This can be written in terms of the object’s average density as s r 2GM 80002 GR2 vesc D D : R 3 Using D ˚ D 5515 kg m00023 (Appendix C) and R D 250Rˇ , we find s 80002 G.250Rˇ /2 ˚ D 3:06 0002 108 m s00021 ' c: vesc D 3 17.11 If the Sun were suddenly to become a black hole, there would be no effect on the orbits of the planets. In either case, the same Schwarzschild metric (Eq. 17.22) describes the curved spacetime through which the planets move (for r > Rˇ ). 17.12
(a)
The Schwarzschild radius (Eq. 17.27), RS D 2GM=c 2 , evaluated for these black holes is RS D 1:48 0002 10000215 m
for 1012 kg
D 2:95 0002 104
m
for 10 Mˇ
D 2:95 0002 108
m
for 105 Mˇ
D 2:95 0002 1012 m
for 109 Mˇ :
(b) The average density of each of these black holes is D
17.13
(a)
M 4 0002R3S 3
D 7:36 0002 1055 kg m00023
for 1015 g
D 1:85 0002 1017 kg m00023
for 10 Mˇ
D 1:85 0002 109
kg m00023
for 105 Mˇ
D 1:85 0002 101
kg m00023
for 109 Mˇ :
Begin by using Eq. (17.27) for the Schwarzschild radius to express Eq. (17.23) for the differential proper distance along a radial line in terms of RS , dL D p
dr 1 0003 RS =r
:
The proper distance, 0006L, from the event horizon (r D RS ) to a final radial coordinate r is then found by integrating d L, Z r dr 0 0006L D p 1 0003 RS =r 0 RS 0007 0006q 0004p p 0005 r D r 0 2 0003 RS r C RS ln r 0 0003 RS C r 0 RS
Solutions for An Introduction to Modern Astrophysics D
p
109
r 2 0003 RS r C RS ln
r
p 0003 0002p r 0003 RS C r p RS
00052 p RS RS 0004p r=RS 0003 1 C r=RS C ln r 2 s ! p 1 C 1 0003 RS =r RS RS D r 10003 p C ln : rf 2 1 0003 1 0003 RS =r
Dr
10003
(S17.1)
(b) See Fig. S17.2. (c)
By inspection of Eq. (S17.1), as r=RS ! 1, 0003 0002 RS 4r : 0006L ! r C ln 2 RS For large values of r , the first term on the right dominates the second term and 0006L ' r . So for large r , the radial coordinate can be interpreted as the proper distance to the event horizon. 12 10
Δ /RS
8 6 4 2 0
1
2
3
4
5
6
7
8
9
10
r/RS
Figure S17.2: Results for Problem 17.13. 17.14 From the Schwarzschild metric (Eq. 17.22) with dr D 0 for a spherical surface, we see that the differential element of area is dA D .r d000e/.r sin 000e d0012/ D r 2 sin 000e d000e d0012: Setting r D RS and integrating this over the usual limits of 000e D 0 to 0002 and 0012 D 0 to 20002 gives the area of the event horizon, Z 2 Z Ahorizon D R2S sin 000e d000e d0012 D 40002R2S :
D0
17.15
(a)
0005D0
Equation (17.26) gives the coordinate speed of a particle orbiting a nonrotating black hole, v D p Evaluating this at r D 3RS D 6GM=c 2 (Eq. 17.27) shows that v D c= 6.
p
GM=r .
110
Chapter 17 Black Holes (b) The coordinate speed is v D r d0012=dt, so the orbital period at r D 3RS is p Z 2 Z 2 r 3RS 6 60002RS PD d0012 D : p d0012 D v c c= 6 0 0 Using RS D 2GM=c 2 D 29:5 km for a 10 Mˇ black hole, we find P D 4:55 0002 1000023 s.
17.16
(a)
The coordinate speed of light in the 0012-direction can be found from the Schwarzschild metric (Eq. 17.22) using ds D 0 for light. Setting dr D d000e D 0 and 000e D 90ı results in 00032 0002 q 0 D c dt 1 0003 2GM=r c 2 0003 .r d0012/2 : The coordinate speed of light in the 0012-direction is thus q d0012 D c 1 0003 2GM=r c 2 : r dt
p (b) Equation (17.26) for the speed of a particle orbiting a nonrotating black hole, v D GM=r , does not contain the particle’s mass. Setting this equal to the coordinate speed of light in the 0012-direction, r r GM 2GM ; Dc 10003 r r c2 and solving for r gives the radius of a photon’s circular orbit around a black hole, 3GM D 1:5RS ; c2 where Eq. (17.27) for the Schwarzschild radius was used. At r D 1:5RS D 3GM=c 2 , the coordinate speed of light in the 0012-direction is q p d0012 D c 1 0003 2GM=r c 2 D c= 3: r dt rphoton D
(c)
The orbital period at r D 1:5RS is Z PD
Z
P
2
dt D 0
0
p r 30002 3RS : p d0012 D c c= 3
Using RS D 2GM=c D 29:5 km for a 10 Mˇ black hole, we find P D 1:61 0002 1000023 s. (d) If a flashlight were beamed in the 0012-direction at r D 1:5RS , the photons would orbit the black hole. 2
17.17 Assuming that the maximum angular momentum of a rotating rigid sphere occurs when the surface of the sphere is the speed of light, c !max D : R Taking the moment of inertia to be I D 2MR2 =5, the maximum angular momentum is given by Lmax D I!max D
2 MRc: 5
For a black hole, the radius is the Schwarzschild radius, R D RS D 2GM=c 2 . Substituting gives 4 GM 2 : 5 c This crude estimate is 80% of the general relativistic value given by Eq. 17.29. Lmax D
Solutions for An Introduction to Modern Astrophysics
111
17.18 From Eq. (17.29), the maximum angular momentum of a 1.4 Mˇ black hole is Lmax D
GM 2 D 1:73 0002 1042 kg m2 s00021 : c
The angular momentum of the fastest known pulsar (P D 0:00139 s) may be estimated (assuming a 1.4 Mˇ uniform sphere of radius 10 km) as 0002 0003 20002 2 L D I! D MR2 D 5:04 0002 1041 kg m2 s00021 : 5 P 17.19 For an electron, 0002
GM c
0002
00032 D
Gme c
00032
D 4:11 0002 10000298 m4 s00022 0002 00032 0004 0003e 00052 Q G DG c c D 1:71 0002 10000255 m4 s00022 0002 00032 0002 0003 L 0002=2 2 D M me D 3:35 0002 1000023 m4 s00022 : For an electron,
0002
GM c
0002
00032 0004G
Q c
0002
00032 C
L M
00032 ;
so an electron is not a black hole. 17.20
(a)
Assume that ˝ is given by an expression of the form ˝ D constant 0002 G ˛ Lˇ r 0007 c ı ; where the values of the exponents ˛, ˇ, 0007 , and ı are to be determined. Let [x] designate the units of the variable x, so [˝] D s00021 [G] D m3 kg00021 s00022 [L] D kg m2 s00021 [r ] D m [c] D m s00021 : The units involved in the expression for ˝ are therefore s00021 D .m3 kg00021 s00022 /˛ .kg m2 s00021 /ˇ .m/0007 .m s00021 /ı ;
112
Chapter 17 Black Holes which leads to 3˛ C 2ˇ C 0007 C ı D 0 0003˛ C ˇ D 0 00032˛ 0003 ˇ 0003 ı D 00031: Four equations with three unknowns cannot be solved. However, if we assume that ˝ / L so ˇ D 1, we can solve for the remaining exponents to find ˛ D 1, 0007 D 00033, and ı D 00032. Therefore, ˝ D constant 0002
GL : r 3c 2
(b) The angular momentum of Earth may be estimated as 0002 0003 20002 2 2 D 7:09 0002 1033 kg m2 s00021 ; L˚ D I˚ !˚ D M˚ R˚ 5 P˚ where r D R˚ and P˚ is Earth’s sidereal day, P D 23h 56m 04s D 8:62 0002 104 s. Then, setting the leading constant equal to one, ˝˚ D
GL˚ D 2:03 0002 10000214 s00021 D 1:33 0002 10000216 00 yr00021 : R3˚ c 2
The time for a pendulum at the north pole to rotate once relative the distant stars because of frame dragging would be 20002 tD D 3:10 0002 1014 s; ˝˚ about 9.8 million years. (c)
From Problem 17.18, the angular momentum of the fastest known pulsar (P D 0:00139 s) may be estimated (assuming a 1.4 Mˇ uniform sphere of radius 10 km) is 0002 0003 2 2 20002 D 5:04 0002 1041 kg m2 s00021 : L D I! D MR 5 P At the surface of this pulsar, ˝D
17.21
(a)
GL D 374 s00021 D 59:4 rev s00021 : R3 c 2
Assume that the mass of the least massive primordial black hole, mP , is given by a expression of the form mP D 0002˛ c ˇ G 0007 ; where the values of the exponents ˛, ˇ, and 0007 are to be determined. Let [x] designate the units of the variable x, so [mP ] D kg [0002] D kg m2 s00021 [c] D m s00021 [G] D m3 kg00021 s00022 :
Solutions for An Introduction to Modern Astrophysics
113
The units involved in the expression for mP are therefore mP D .kg m2 s00021 /˛ .m s00021 /ˇ .m3 kg00021 s00022 /0007 ; which leads to ˛00030007 D 1 2˛ C ˇ C 30007 D 0 0003˛ 0003 ˇ 0003 20007 D 0: Solving these shows that ˛ D 1=2, ˇ D 1=2, and 0007 D 00031=2. Therefore, r 0002c mP D D 2:18 0002 1000028 kg: G (b) From Eq. (17.27), the Schwarzschild radius for a black hole of mass mP would be rP D
2GmP D 3:24 0002 10000235 m: c2
t D rP =c D 1:08 0002 10000243 s. 0003 0002 mP 3 (d) tevap 0002 1066 yr D 1:3 0002 10000248 yr D 4:2 0002 10000241 s: Mˇ (c)
17.22
(a)
On the left-hand side
ŒkT 0011 D J D kg m2 s00022 :
On the right-hand side 0006
0002c 40002RS
0007 D
0004 0005 kg m2 s00021 m s00021 m
D kg m2 s00022 D J:
(b) kT D 1 0002 10000211 J D 62 MeV: This implies that T D 7:2 0002 1011 kg. (c)
From Wien’s law, this corresponds to 0005max D 4 0002 10000215 m. This is in the gamma-ray portion of the spectrum.
(d) 344 m. 17.23
(e)
6:2 0002 1000029 K.
(a)
Beginning with the Stefan–Boltzmann law, and substituting we have L D 40002R2S 0003T 4 0003 0002 00030002 0002 00034 0002 2GM 2 20002 5k 4 D 40002 : c2 15c 2h3 80002 kGM After some tedious rearranging we arrive at the desired result.
(b) The time to evaporate is given by
Z
M
tD 0
Integration gives Eq. (17.30).
c 2 dM : L
114 17.24
Chapter 17 Black Holes (a)
From Eq. (7.7), the mass function is the right-hand side of m3c P sin3 i D v3 : 2 20002 G s r .ms C mc / Using P D 0:3226 d D 2:79 0002 104 s and vs r D 4:57 0002 105 m s00021 , m3c sin3 i D 6:35 0002 1030 kg: .ms C mc /2 Because the left-hand side must be less than mc , the mass of the compact object must be greater than 6:35 0002 1030 kg D 3:19 Mˇ .
(b) The ratio of the masses is, from Eq. (7.5), ms vcr D D 0:0941: mc vs r Combining this with the result for part (a) and assuming that i D 90ı gives mc D 3:19 Mˇ .ms =mc C 1/2 D 3:82 Mˇ : Because sin3 i 1, mc 3:19 Mˇ . (c)
If the angle of inclination were i D 45ı , then the mass obtained for the compact object would be larger by a factor of 1= sin3 45ı D 2:83. That is, mc would be 10.8 Mˇ .
17.25 The K0 IV companion has a measured periodic radial velocity of 211 ˙ 4 km s00021 and an orbital period of 6:473 ˙ 0:001 d. The approximate mass of the companion is 0:9 Mˇ . Assuming an inclination of i D 90ı , Eq. (7.7) implies m32 P 3 D v D 1:25 0002 1031 kg D 6:3 Mˇ : 2 .m1 C m2 / 20002 G 1r Let k0007
m32 D 6:3 Mˇ : .m1 C m2 /2
Expanding the expression leads to the cubic expression m32 0003 k m22 0003 2k m1m2 0003 k m21 D 0: This can be solved to give m2 D 7:9 Mˇ . This result is a lower limit because we simply assumed i D 90ı. An improved estimate of i will yield a more accurate result, as will a more careful estimate of the mass of the K0 IV companion.
CHAPTER 18
Close Binary Star Systems 18.1 Because equipotential surfaces are level surfaces for binary stars, we expect the density to be constant along a surface of constant ˚. The behavior of the pressure may be deduced from the equation of hydrostatic equilibrium (Eq. 10.6) combined with Eq. (18.8) for the gravitational potential, dP d˚ D 0003 g D : dr dr For constant density, the pressure and gravitational potential vary spatially in the same way, so the pressure should be approximately constant along an equipotential surface. Now consider the ideal gas law (Eq. 10.11), Pg D
kT ; mH
making the reasonable assumption that the mean molecular weight, , is constant. This shows that if the pressure and density are roughly constant along an equipotential surface, then the temperature will also be approximately constant. The proximity of the other star could cause heating of each star on the side that faces the other. 18.2 For the equilateral triangle formed by M1 , M2 , and L4 (or L5), we have s1 D s2 D a in Fig. 18.1. Then Eqs. (18.4) and (18.7) give ˚ D 0003G
M1 C M2 1 2 2 M1 C M2 0003 ! r D 0003G a 2 a
0003 0002 r2 1C 2 : 2a
Equations (18.3) show that 0002 r1 D a 0002
M2 M1 C M2
M1 r2 D a M1 C M2
0003 0003 ;
and so Eqs. (18.6) may be expressed as 0002 a2 D a 2 0002 a2 D a2 Solving these for r yields
M2 M1 C M2 M1 M1 C M2
00032
0002 C r 2 C 2a
00032
0002 C r 2 0003 2a
M2 M1 C M2
M1 M1 C M2
s r Da 10003 115
M1 M2 .M1 C M2 /2
0003 r cos 000e 0003 r cos 000e:
116
Chapter 18 Close Binary Star Systems and so the gravitational potential at L4 (or L5 ) is # ' M1 C M2 M1 M2 ˚ D 0003G : 30003 2a .M1 C M2 /2 In units of G.M1 C M2 /=a, this is # ' 1 M1 M2 ˚ D0003 D 00031:431: 30003 2 .M1 C M2 /2 This agrees with the value given in Fig. 18.3.
18.3
(a)
As the gas passes perpendicularly across the area A with velocity v, in time dt it will travel a distance v dt and sweep out a volume vA dt. For gas density , the mass of gas within this volume is dM D vA dt. The rate at which mass crosses the area is therefore MP 0007 dM=dt D vA, which is Eq. (18.11).
(b) Referring to Fig. 18.5, start with
0002 0003 d 2 R2 D R 0003 C x2 2
or x 2 D Rd 0003 When d 0004 R, we recover Eq. (18.12), x D
p Rd .
d2 : 4
18.4 Let x D R=r , so Eq. (18.19) for the disk temperature is T D Tdisk x 3=4 .1 0003
p 1=4 x/ :
T will be a maximum when d T =dx D 0, or when p p 3 00021=4 1 1 .1 0003 x/1=4 0003 .1 0003 x/00023=4 p D 0: x 4 4 2 x Thus T is a maximum when x D 36=49, or when r D .49=36/R. Evaluating T at this point results in Tmax D 0:488Tdisk. 18.5 Integrating Eq. (18.15) for the ring luminosity, and using Eqs. (18.19) and (18.20) for the disk temperature, we have !0002 0003 r ! Z 1 R 3GM MP R 3 Ldisk D 40002 r 0003 dr 10003 3 800020003R r r R r ! Z 1 0002 00032 R R 3GM MP dr: 10003 D 2 2R r r R Defining x D R=r , we find Ldisk D
3GM MP 2R
which is Eq. (18.23) for the disk luminosity.
Z
1 0
p GM MP ; 1 0003 x dx D 2R
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117
18.6 We will use the mass and radius of the white dwarf of Z Cha given in Example 18.4.1: M D 0:85 Mˇ and R D 0:0095 Rˇ . For a mass-transfer rate of MP D 1013:5 kg s00021 , the disk luminosity is (Eq. 18.23) Ldisk D G
P MM D 2:70 0002 1026 W D 0:705 Lˇ : 2R
In t D 10 days, the total energy released is E D Ldisk t D 2:33 0002 1032 J. From Eq. (3.8) with MSun D 4:74, the absolute magnitude of the dwarf nova is 0002 0003 L M D MSun 0003 2:5 log10 D 5:12: Lˇ (This neglects the small amount of light contributed by the primary and secondary stars.) 18.7 During quiescence, the absolute magnitude of the dwarf nova is M D 7:5. From Eq. (3.7) with MSun D 4:74, the luminosity of the system is Lquiet D 100.MSun 0002M /=5 Lˇ D 0:08 Lˇ D 3:0 0002 1025 W: During the outburst, the absolute magnitude of the dwarf nova would be M D 7:5 0003 3 D 4:5. Again using Eq. (3.7), the luminosity of the system is Lout D 100.MSun0002M /=5 Lˇ D 1:25 Lˇ D 4:79 0002 1026 W: The disk luminosity during the outburst is then Lout 0003 Lquiet D 1:179 Lˇ . Using M D 0:85 Mˇ and R D 0:0095 Rˇ with Eq. (18.23) results in a mass-transfer rate of 2RLdisk MP D D 5:3 0002 1013 kg s00021 D 8:4 0002 10000210 Mˇ yr00021 : GM 18.8 As shown in Fig. 18.12, the binary system and the accretion disk rotate in the same direction. 18.9
(a)
From ! D 20002=P , we have 1 d! P D ! dt 20002
0002
000320002 P2
0003
dP 1 dP D0003 : dt P dt
Substituting this into Eq. (18.29) and then using Eq. (18.28) shows that M1 0003 M2 1 dP 3 1 da : D D 3MP 1 P dt 2 a dt M1 M2 (b) For M1 D 2:9 Mˇ , M2 D 1:4 Mˇ , P D 2:49 d D 2:15 0002 105 s, and dP =dt D .20 s/=.100 yr/ D 6:3 0002 1000029 s s00021 , we have 1 dP M1 M2 D 5:3 0002 1016 kg s00021 D 8:4 0002 1000027 Mˇ yr00021 : MP 1 D 3P dt M1 0003 M2 Because dP =dt > 0 and M1 > M2 , we have MP 1 > 0. Star 1 is gaining mass. 18.10 The subgiant began as the more massive star. It evolved faster and expanded to overflow its Roche lobe while its companion was still on the main sequence. Enough mass was transferred to the companion to make it the more massive B8 main sequence star that is observed today.
118
Chapter 18 Close Binary Star Systems
18.11 The amount of energy released in the fusion of m D 1000024 Mˇ of hydrogen is E D 0:007mc 2 D 1:2500021041 J. For a luminosity equal to the Eddington limit of LEd 1031 W (Eq. 10.6), the outburst would last for a time of E tD D 1:25 0002 1010 s 400 yr: LEd Typically, a classical nova’s outburst lasts about 100 days, so only a very small fraction of the hydrogen surface layer actually undergoes fusion. The rest is ejected into space by the explosion. 18.12 The gravitational potential energy of m D 1000024 Mˇ of hydrogen on the surface of a white dwarf of mass M D 0:85 Mˇ and R D 0:0095 Rˇ is, from Eq. (2.14), U D 0003G
Mm D 00033:4 0002 1039 J R
(using values of M and R for Z Cha from Example 18.4.1). With a speed of v D 106 m s00021 , the kinetic energy of the ejected layer is 1 K D mv 2 D 9:9 0002 1037 J: 2 Note that K=jU j 0:029. Since the gravitational potential energy is negligibly small when the ejecta is far from the white dwarf, we conclude that the total energy given to the ejecta is just barely enough to allow it to escape from the system. 18.13
(a)
In time dt, the amount of mass that passes through a spherical surface of radius r , centered on the nova, P eject dt. In time dt, the material at r travels a distance dr D v dt as it sweeps through a is dM D M spherical shell of volume dV D 40002 r 2 dr D 40002 r 2 v dt. The density of the material at r is therefore D
P eject M dM D : dV 40002 r 2v
(b) From Eq. (9.17), the optical depth of the photosphere at time t D 0 is ! Z R Z R MP eject 2
D dr D D dr; 3 40002 r 2 v R0 R0 where R is the outer radius of the shell and R0 is the radius of the photosphere at t D 0. Solving this for R gives 1 80002 v 1 1 1 0003 D 0003 ; D R R0 3 MP eject R0 R1 which defines R1 . (c)
At a later time t, the outer radius of the shell is R C vt and the radius of the photosphere is R.t/. [R.t D 0/ D R0 .] The optical depth of the photosphere at time t is given by ! Z RCvt Z RCvt MP eject 2 dr D D dr:
D 3 40002 r 2 v R.t / R.t / As in part (b), this leads to 1 1 1 : D 0003 R C vt R.t/ R1
Solutions for An Introduction to Modern Astrophysics
119
(d) From the answer to part (b), RD
R0 R1 : R1 0003 R0
Substituting this for R in the answer to part (c) and simplifying results in vt.1 0003 R0 =R1 /2 : 1 C .vt=R1 /.1 0003 R0 =R1 /
R.t/ D R0 C (e)
If R0 is comparable to the radius of the white dwarf (i.e., Rwd D 0:0095 Rˇ for Z Cha from Example 18.4.1) and if v D 106 m s00021 , then vt < R0 for approximately 7 s. So except for the first seconds of the outburst, the first term on the right can be neglected. Also, neglecting terms involving R0 =R1 results in vt R.t/ ' : 1 C vt=R1
(f) At early times when R0 0004 vt 0004 R1 , R.t/ ' vt, so initially the photosphere expands linearly with time. When vt 0005 R1 , the second term in the denominator dominates the first, and t approaches the limiting value of R1 . (g) Using D 0:04 m2 kg00021, MP ejecta D 1019 kg s00021 and v D 106 m s00021 , the value of R1 is R1 D
P eject 3 M D 4:8 0002 1010 m: 80002 v
As seen in Fig. S18.1, the end of the linear expansion period — the “knee” in the graph — occurs roughly at t 0006 1 day. This is comparable to the duration of the optically thick fireball phase, which lasts for a few days. 5
R (1010 m)
4
3
2
1
0
0
1
2
3
4
5
t (days)
Figure S18.1: Results for Problem 18.13. 18.14 Equation (18.31) shows that the temperature of the nova’s photosphere approaches 0002 T1 D
L 400020003
00031=4
80002 v 3 MP eject
!1=2 :
120
Chapter 18 Close Binary Star Systems Adopting LEd 1031 W for the Eddington limit, together with D 0:04 m2 kg00021 , MP ejecta D 1019 kg s00021 and v D 106 m s00021 , results in T1 D 8900 K.
18.15 If a mass of M D 1000024 Mˇ is ejected at a constant rate over a time of t D 100 days, then M MP eject D D 2:3 0002 1019 kg s00021 D 3:7 0002 1000024 Mˇ yr00021 : t 18.16 The gravitational potential energy of a realistic model white dwarf is U D 00035:1 0002 1043 J. For a nuclear energy production of E=m D 7:3 0002 1013 J kg00021 , the amount of iron that would have to be produced to cause the star to become gravitationally unbound is mU D
jU j D 7:0 0002 1029 kg D 0:35 Mˇ : E=m
The fusion of this mass of white-dwarf material would supply the ejecta (the entire mass of the star, Mwd D 1:38 Mˇ ) with just enough energy to escape the star’s gravitational pull. However, if the speed of the ejecta far from the star is v D 5000 km s00021 , then the ejecta must be supplied with its kinetic energy, KD
1 Mwd v 2 D 3:4 0002 1043 J: 2
The additional iron that would have to be produced is mK D
K D 4:7 0002 1029 kg D 0:24 Mˇ : E=m
The total amount of iron produced is mU C mK D 0:59 Mˇ , a bit less than one-half of the white dwarf’s mass. 18.17 After star 1 explodes, the total energy, Ef , of the binary system is given by Eq. (18.34). For an unbound system, Ef 0, so 1 1 MR M2 MR v12 C M2 v22 0003 G 0: 2 2 a Using v2 =v1 D M1 =M2 (Eq. 7.4) to substitute for v2 , we obtain ! M12 MR M2 1 v12 0003 G MR C 0: (S18.1) 2 M2 a Before star 1 explodes, the total energy, Ei , of the binary system is given by Eq. (18.33), 1 1 MR M2 MR M2 M1 v12 C M2 v22 0003 G D 0003G : 2 2 a 2a Again using v2 =v1 D M1 =M2 to substitute for v2 , we obtain an expression for v12 , v12 D G
M22 : a.M1 C M2 /
Substituting this into Eq. (S18.1), we find that for an unbound system, MR 1 1 < ; M1 C M2 .2 C M2 =M1 /.1 C M2 =M1 / 2 which is Eq. (18.35).
Solutions for An Introduction to Modern Astrophysics 18.18
(a)
121
The Alfv´en radius, rA , is where the kinetic and magnetic energy densities are equal, 1 2 B2 v D 2 2000b0 (Eq. 18.36). We now use Eq. (18.37) [derived in part (a) of Problem 18.13], D the free-fall velocity v D
p
MP ; 40002 r 2v
2GM=r , and the radial dependence of a magnetic dipole (Eq. 18.38), 0002 B.r / D Bs
R r
00033 ;
to evaluate this. At r D RA , 1 2
MP 40002 rA2
!s
' 0002 0003 #2 R 3 2GM 1 D ; Bs rA 2000b0 rA
which, when simplified, produces Eq. (18.39), rA D
80002 2Bs4 R12 2GM MP 2
!1=7 :
0
(b) Taking a time derivative of L D I! for a rotating neutron star gives 0002 0003 dL d! d 20002 PP DI DI D 000320002I 2 ; dt dt dt P P where I is the moment of inertia of the neutron star and P its rotation period. Near the disruption radius, rd , angular momentum is transferred p to the neutron star. The angular momentum p per unit mass of gas parcels orbiting with velocity v D GM=rd at the disruption radius is vrd D GM rd . If MP is the rate at which mass arrives at the disruption radius, then the time derivative of the star’s angular momentum is p P dL P GM rd D 000320002I P : P vrd D M DM dt P2 Solving for PP =P gives PP P MP p GM rd : D0003 P 20002I Finally, substituting rd D ˛rA (Eq. 18.40), with the Alfv´en radius given by Eq. (18.39), leads to P p PP PM GM˛ D0003 P 20002I
80002 2Bs4 R12 2 GM MP 2
!1=14 :
0
When this is simplified, Eq. (18.41) emerges. 18.19 If the Alfv´en radius, rA , is equal to the star’s radius, R, then from Eq. (18.39), rA D R D
80002 2Bs4 R12 2GM MP 2 0
!1=7 :
122
Chapter 18 Close Binary Star Systems Solving for the surface value of the magnetic field, we find Bs D
000b20GM MP 2 80002 2R5
!1=4 :
For the white dwarf in Example 18.2.1, M D 0:85 Mˇ , R D 0:0095 Rˇ , and MP D 1013 kg s00021 , so P D 1014 kg s00021 , so Bs D 0:366 T. For the neutron star in that example, M D 1:4 Mˇ , R D 10 km, and M Bs D 4390 T. P D 1014 kg s00021 , the time to 18.20 Using the value of the neutron star’s mass-transfer rate in Example 18.2.1 of M 16 8 P D 2:0 0002 10 s D 6:3 0002 10 yr. transfer M D 1 Mˇ is roughly t D M=M 18.21 From Eq. (18.24) with an accretion luminosity in X-rays of Lx D 3:8 0002 1029 W, RLX MP ns D D 2:0 0002 1013 kg s00021 GM for a 1.4 Mˇ neutron star with a radius of 10 km. If its surface magnetic field is 108 T, then Eq. (18.41) shows that the neutron star’s rotation period of P D 3:61 s is decreasing as PP P
! ns
p P ˛ D0003 20002I
!1=7 p P6 2 20002Bs2 R6 G 3 M 3 M D 00037:0 0002 1000025 yr00021 ; 0
using I D 1:11 0002 1038 kg m2 (Example 18.6.3) and ˛ D 0:5 (Eq. 18.40). We now repeat these calculations for a 0.85 Mˇ white dwarf with a radius of 6:6 0002 106 m and a surface magnetic field of 1000 T. For a uniform sphere, I D 2MR2 =5 D 2:95 0002 1043 kg m2 . With ˛ D 0:5, we find P wd D RLX D 2:2 0002 1016 kg s00021 M GM and PP P
! wd
p P ˛ D0003 20002I
!1=7 p P6 2 20002Bs2 R6 G 3 M 3 M D 00038:3 0002 1000027 yr00021 : 0
The neutron star model is in better agreement with the observed value of PP =P D 00033:2 0002 1000025 yr00021 . 18.22
(a)
P in Eq. (18.41) gives Using Eq. (18.24) for the accretion luminosity to replace M !1=7 p 2 20002Bs2 R12 L6acc 0 G 3 M 3
p PP P ˛ D0003 P 20002I or log10
PP 0003 P
!
0004
D log10 PL6=7 acc
0005
2
p ˛ C log10 4 20002I
!1=7 3 p 2 12 2 20002Bs R 5: 0 G 3 M 3
(b) We can evaluate the second term on the right using ˛ D 0:5 (Eq. 18.40) and values from Example 18.6.3 and the solution to Problem 18.22. For a 1.4 Mˇ neutron star with a radius of 10 km, moment of inertia I D 1:11 0002 1038 kg m2 , and a surface magnetic field of 108 T, it is 000343:56. For a 0.85 Mˇ white dwarf with a radius of 6:6 0002 106 m, moment of inertia I D 2:95 0002 1043 kg m2 , and a surface magnetic field of 1000 T, it is 000345:49. See Fig. S18.2. (c)
See Fig. S18.2.
Solutions for An Introduction to Modern Astrophysics
123
−8 A0535+26
−9
GX301−2
Log10(−P/P) (s−1)
−10
.
−11
SMC X-1 Cen X-3
4U0352+30
−12 −13
Her X-1 Neutron star White dwarf
−14 −15 25
26
27 Log10(PL6/7)
28
29
Figure S18.2: Results for Problem 18.23 (b) and (c). (d) Except for Her X-1, the points in Fig. S18.2 lie along the line for the neutron-star model of a binary X-ray pulsar. According to Rappaport and Joss (1977), for Her X-1 “the value of PP =P fluctuates and its sign even reverses. These sporadic intervals of spin-down, together with the small average value of jPP =P j, are indicative of a near balance between spinup and spindown for this source. Hence, external torques are probably important for this system.” 18.23
The average luminosity of the burster is L D .1032 J/=.5 s/ D 2 0002 1031 W. Using a blackbody temperature of T D 2 0002 107 K, the Stefan–Boltzmann equation (Eq. 3.17) yields the radius of the neutron star, r L 1 RD 2 D 13:2 km: 400020003 T (b) The value of R in part (a) corresponds to R1 in Eq. (17.33), (a)
R ; R1 D q 1 0003 2GM=Rc 2 as calculated by an observer on Earth at a great distance from the neutron star. A more accurate value (obtained at the surface of the neutron star) is the value of R found by numerically solving the cubic equation obtained from the preceding expression for R1 , 0002 0003 R 3 R 2GM 0003 C D 0: R1 R1 R1 c 2 Assuming a 1.4 Mˇ neutron star, this has the solution R=R1 D 0:772, or R D 10:2 km. [The other positive root of this cubic equation, R=R1 D 0:358, is too small for a neutron star, and the third root, R=R1 D 00031:13, is unphysical and does not satisfy Eq. (17.33).] 18.24 The SMC X-1 binary pulsar system consists of a neutron star of mass M1 D 1:4 Mˇ and a secondary star that has a mass of M2 D 17 Mˇ and a radius of R2 D 16:5 Rˇ . The two stars are separated by a D 0:107 AU D 23:0 Rˇ . From Eq. (7.1), r1 a 0003 r2 M2 D D ; M1 r2 r2
124
Chapter 18 Close Binary Star Systems 25 20
SMC X-1
15
Distance (R )
10 5
L1
0
c.m.
−5 −10 −15 −20 −25 −25 −20 −15 −10
−5
0
5
10
15
20
25
Distance (R )
Figure S18.3: Results for Problem 18.25. This scale drawing of the binary pulsar system SMC X-1 shows the center of mass (“0002”) at the origin and the inner Lagrangian point, L1 (“C”). The size of the neutron star (right) is greatly exaggerated. so the distance from the center of the secondary to the center of mass is r2 D
aM1 D 0:0761a D 1:75 Rˇ : M1 C M2
Equation (18.10) shows that the distance from the center of the secondary to the inner Lagrangian point, L1 , is 0002 00030007 0006 M2 `2 D a 0:500 C 0:227 log10 D 0:746a D 17:2 Rˇ : M1 Figure S18.3 shows a scale drawing of the SMC X-1 system. 18.25 Equation (4.32) for the relativistic Doppler shift with 000e D 90ı for motion perpendicular to the line of sight is q obs D 1 0003 u2 =c 2 : rest For SS 433, u=c D 0:26 and so the obs = rest D 0:9656 for this transverse Doppler shift. If this is wrongly interpreted as a Doppler shift due to motion along the line of sight, then the radial velocity attributed to the source would be, from Eq. (4.33), vr 1 0003 . obs = rest /2 D 0:03498: D c 1 C . obs = rest /2 This corresponds to vr D 10,490 km s00021 , in agreement with the value obtained from the crossing of the radial-velocity curves of SS 433. 18.26 SS 433 is r D 5:5 kpc from Earth, and the X-ray emitting regions extend as much as 000e D 440 D 0:0128 rad from SS 433. This corresponds to a separation of s D r 000e D 2:17 0002 1018 m. With a jet velocity of v D 0:26c, it would take the jets t D s=v D 2:79 0002 1010 s D 883 yr to travel to the edge of the X-ray emitting regions. The jets must have slowed as they penetrated the gases of the supernova remnant W50, so this may be taken as a lower limit for the amount of time the jets have been active.
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125
18.27 Using tD
P 2PP
from Problem 16.22 with P D 6:133 ms and PP D 3 0002 10000220, the age of PSR 1953C29 is t D 1:02 0002 1017 s D 3:24 0002 109 yr. Adopting 1.4 Mˇ and 10 km for the mass and radius of the neutron star, its moment of inertia is I D 2MR2 =5 D 1:11 0002 1038 kg m2 . The magnetic field at the pole of the neutron star comes from Eq. (16.33) with 000e D 90ı, s 3000b0 c 3 IP PP 1 D 9:16 0002 104 T: BD 3 20002R sin 000e 20002 Such a small magnetic field (compared with 0006 108 T for longer-period pulsars) distinguishes the millisecond pulsars. 18.28 Equation (18.41) may be written as
where
PP D 0003K; P2
p ˛ KD 20002I
!1=7 p 2 20002Bs2 R6 G 3 M 3 MP 6 : 0
Adopting M D 1:4 Mˇ , R D 106 km, I D 2MR2 =5 D 1:11 0002 1038 kg m2 , MP D 1014 kg s00021 , Bs D 104 T, and ˛ D 0:5 (Eq. 18.40) results in K D 1:75 0002 10000213 s00022 . We now integrate from an initial period, Pi , at t D 0 to a final period, Pf , at some final time t: Z Pf Z t dP D 0003 K dt 0 ; 2 Pi P 0 so the spin-up time is 1 tD K
0002
1 1 0003 Pf Pi
0003 :
P , the For Pi D 100 s and Pf D 1 ms, t D 5:72 0002 1015 s D 1:81 0002 108 yr. Assuming a constant value of M 29 P total mass transferred is M D M t D 5:72 0002 10 kg D 0:288 Mˇ . 18.29 The semimajor axes and orbital periods of the planets orbiting the pulsar PSR 1257C12 are a1 D 0:19 AU, P1 D 25:34 d; a2 D 0:36 AU, P2 D 66:54 d; and a3 D 0:47 AU, P3 D 98:22 d. According to Kepler’s third law, Eq. (2.37), P 2=a3 D constant. In the units given, P12 =a31 D 9:36 0002 104 d2 AU00023 P22 =a32 D 9:49 0002 104 d2 AU00023 P32 =a33 D 9:29 0002 104 d2 AU00023 To the precision of the values of a and P , these three planets obey Kepler’s third law. 18.30
(a)
The masses of the two neutron stars in the binary Hulse–Taylor pulsar PSR 1913C16 are M1 D 1:4414 Mˇ and M2 D 1:3867 Mˇ . For an orbital period of P D 27907 s, Kepler’s third law (Eq. 2.37) gives the semimajor axis of the orbit as 0006
G.M1 C M2 /P 2 aD 40002 2
00071=3 D 1:949 0002 109 m:
126
Chapter 18 Close Binary Star Systems (b) Taking a time derivative of Kepler’s third law in the form of P 2 D Ka3 , where K is a constant, results in dP da 2P D 3Ka2 : dt dt Dividing by the third law and using PP D 00032:4056 0002 10000212 for the time derivative of the orbital period produces da 2aPP D D 00031:120 0002 1000027 m s00021 : dt 3P During one orbit, the semimajor axis changes by 0006a D P
da D 00033:12 mm: dt
20 19
Eq. (18.14) Best-fit straight line
Log10 M (kg s-1)
18
.
17 16 15 14 13 12 11
5.8
6.0
6.2
6.4 6.6 6.8 Log10 d (m)
7.0
7.2
7.4
Figure S18.4: Results for Problem 18.32. 18.31
(a)
See Fig. S18.4. The slope of the best-fitting straight line is 5:14, so MP / d 5:14 for this simple model.
(b) The ideal gas law (Eq. 10.11) states that / P =T . Combining this with Eq. (18.14) for the masstransfer rate, s P 0002Rd M
3kT ; mH
p shows that MP / P d= T . From Eq. (L.1), the pressure near the surface of a star depends on the temperature as P / T 4:25 , so MP / T 3:75 d . Finally, according to Eq. (L.2), the temperature near the surface is 0003 0004 m 0005 0002 1 0004 m 0005 0002 R 0003 r 0003 1 H H T D GM 0003 D GM ; 4:25k r R 4:25k rR where M is the star’s mass and R is its radius. If the overlap distance is d 0007 R 0003 r , with d 0004 R, we have .R 0003 r /=rR ' d=R2 , and so T / d near the surface. The result is that MP / d 4:75 . 18.32 The Schwarzschild radius of a black hole of mass 3.82 Mˇ black hole is (Eq. 17.27) RS D 2GM=c 2 D 11:3 km. The inner edge of the disk is at r=RS D 3. To find the outer edge of the disk, we first use Kepler’s
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127
7
4 Temperature Wavelength
Log10 T (K)
2 5 1 4
0 X-ray
3
Log10 λmax (nm)
3
6
0
UV
1
2
Visible
3 Log10 (r/RS)
4
IR
5
6
-1
Figure S18.5: Results for Problem 18.33. third law (Eq. 2.37) to determine the semimajor axis, 0006 aD
G.M1 C M2 /P 2 40002 2
00071=3 D 2:22 0002 109 m;
where P D 0:3226 d, M1 D 3:82 Mˇ , and M2 D 0:36 Mˇ . Equations (18.25) and (18.26) then locate the outer edge of the disk at Rdisk
0002 00030007 0002 0003 0006 M2 4 M2 1C 2a 0:500 0003 0:227 log10 M1 M1 D 1:40 0002 109 m D 1:24 0002 105RS :
The disk temperature, T .r /, comes from Eq. (18.19) in the form T .r / D
P 3GM1 M 3 800020003RS
!1=4 0002
RS r
00033=4 0004
10003
00051=4 p RS =r ;
where MP D 1014 kg s00021 and r varies from 3RS to Rdisk . The peak wavelength, 0005max.r /, is obtained from Wien’s law, Eq. (3.15). Figure S18.5 shows the resulting graphs of log10 T and log10 0005max (0005max in nm). The regions of the disk that emit X-rays, ultraviolet, visible, and infrared radiation are indicated.
CHAPTER 19
Physical Processes in the Solar System 19.1
(a)
The masses are (in terms of the mass of Mercury): MMoon D 0:223MMercury MIo D 0:270MMercury MEuropa D 0:148MMercury MGanymede D 0:452MMercury MCallisto D 0:327MMercury MTitan D 0:409MMercury MTriton D 0:065MMercury MPluto D 0:040MMercury
(b) The radii are (in terms of the radius of Mercury): RMoon D 0:712RMercury RIo D 0:744RMercury REuropa D 0:643RMercury RGanymede D 1:078RMercury RCallisto D 0:984RMercury RTitan D 1:055RMercury RTriton D 0:555RMercury RPluto D 0:460RMercury 19.2
(a)
See Fig. S19.1.
(b) See Fig. S19.1. Using a least-squares fit to the data, log10 A D 0:2333068 (or A D 1:71) and log10 r0 D 00030:670528 (or r0 D 0:214). 19.3
(c)
See Table S19.1.
(a)
See Fig. S19.2.
(b) See Fig. S19.2. Using a least-squares fit to the data, log10 A D 0:21506 (or A D 1:6408) and log10 r0 D 0:5476 (or r0 D 3:5286). (c)
See Table S19.2. 128
Solutions for An Introduction to Modern Astrophysics
129
2
Log10 (rn)
1
0
-1
0
1
2
3
4
5 n
6
7
8
9
10
Figure S19.1: The least-squares fit to data for Problem 19.2. Table S19.1: The results of Problem 19.2(c).
Log10 (rn)
Object Mercury Venus Earth Mars Ceres Jupiter Saturn Uranus Neptune Pluto
1.5 1.4 1.3 1.2 1.1 1.0 0.9 0.8 0.7 0.6 0.5 0.4
0
n 1 2 3 4 5 6 7 8 9 10
1
rn (AU) 0.39 0.72 1.00 1.52 2.77 5.20 9.54 19.19 30.06 39.53
log10 rn 00030:412 00030:141 0:000 0:183 0:442 0:716 0:980 1:283 1:478 1:597
2 n
0006r=r (%) 00035:579 000313:514 7:002 20:147 13:117 3:053 00033:914 000318:182 000310:619 16:309
3
4
Figure S19.2: The least-squares fit to data for Problem 19.3.
130
Chapter 19 Physical Processes in the Solar System Table S19.2: The results of Problem 19.3(c). Moon Io Europa Ganymede Callisto
n 1 2 3 4
rn (RJ ) 5.91 9.40 14.99 26.34
log10 rn 0.772 0.973 1.176 1.421
0006r=r (%) 00031:985 1:063 3:973 2:906
19.4 Using the small-angle relations cos 0012 ' 1 and sin 0012 ' R sin 000e=r , and substituting into Eq. (19.2), produces the x and y components of the differential force as 0006 0007 GM m 2R 0006Fx ' cos 000e r2 r 0006 00070002 0003 2R GM m R sin 000e 0006Fy ' 0003 2 1C cos 000e r r r '0003
GM mR sin 000e r3
where the last approximation is because R 0004 r . 19.5
(a)
Angular momentum is given by Lrot D I!rot . Assuming constant I , dLrot d!rot d.20002=Prot/ 20002I DI DI D 0003 2 PProt: dt dt dt Prot For Earth (assuming constant density), I D
2 M˚ R2˚ D 9:7 0002 1037 kg m2 5
Prot D 86,164 s PProt D 0:0016 s century00021 D 5:070 0002 10000213 s s00021 : Note that PProt > 0 (the period is lengthening). Combining dLrot D 00034:16 0002 1016 kg m2 s00022 < 0: dt (b) Treating the Moon as an orbiting point mass, and assuming a circular orbit, Lorbit D Mm vr D 20002Mmr 2 =Porbit. Assuming constant Mm , this leads to d 20002Mmr 2 =Porbit dLorbit D dt dt D
20002Mmr 2 dPorbit 40002Mm r dr 0003 : 2 Porbit dt dt Porbit
2 D 40002 2 r 3=G .M˚ C Mm /, where r D a, From Kepler’s third law, Porbit
dPorbit dr 60002 2 r 2 D : dt G .M˚ C Mm / Porbit dt
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131
From the text, dr=dt 0006 3:5 cm yr00021 D 1:1 0002 1000029 m s00021 : Substituting, dPorbit=dt D 1:0 0002 10000211 s s00021 , and dLorbit D 1:6 0002 1017 kg m2 s00022 0003 4:0 0002 1016 kg m2 s00022 D 1:2 0002 1017 kg m2 s00022 > 0: dt
19.6
(c)
The crude estimates of the change in angular momentum of Earth and the Moon are roghly consistent with a total change of zero for the system.
(a)
Earth’s spin is decreasing at a rate of 0:0016 s century00021 D 1:92 0002 10000210 d yr00021 . If this rate were to remain constant, then the length of the day would reach 47 days when 0005 0004 47 d D 1 d C 1:92 0002 10000210 d yr00021 0006t: Solving for the time interval gives 0006t D 2:4 0002 1011 yr.
(b) Since the entire main-sequence lifetime of the Sun is on the order of 1010 yr, Earth will likely not become fully synchronized before the Sun becomes a red giant star. 19.7
(a)
Substituting P D 47 d and M˚ C MMoon D 6:05 0002 1024 kg into Kepler’s third law (Eq. 2.37) results in a D 5:5 0002 108 m.
(b) The radius of the Moon is RMoon D 1:740002106 m. Thus, its angular diameter would be 000e D 2RMoon =a D 0:0063 rad D 0:36ı.
19.8
(c)
For comparison, the Sun’s angular diameter is 000eˇ D 2Rˇ =1 AU D 0:53ı. Since the angular diameter of the Moon would only by 68% of this value, the Moon would not completely cover the solar disk, and a total eclipse would not be possible.
(a)
Let rEM and rES be the distances from Earth to the Moon and the Sun, respectively. Then, considering the x-component of the tidal force given in Eq. (19.3) for a given value of 000e, the ratio of the tidal forces is 00030002 0003 0002 MM =R3EM 0006FM rES 3 MM D D D 2:2: Fˇ Mˇ rEM Mˇ =R3ES
(b) Since, for spring tides, the Sun and the Moon are both along nearly the same line relative to Earth’s center (either aligned or anti-aligned), the tidal force contributions are additive. At neap tide, The Sun, Earth’s center, and the Moon form a right angle and the differential force vectors are perpendicular, so that each body partially negates the effect of the other. 19.9 Near the poles 000e D ˙90ı, the differential tidal force goes to zero in the direction parallel to the Earth-Moon axis (e.g., Eq. 19.3). The component of the tidal force perpendicular to the axis is directed toward the center of the planet. 19.10 Using p D 3933 kg m00023 , m D 2000 kg m00023 , Rp D 0:533 R˚ D 3:4 0002 106 m, and fR D 2:456, Eq. (19.4) gives r < 1:05 0002 107 m. Phobos is currently orbiting at essentially the Roche limit for the Mars–Phobos system. As Phobos continues to spiral in toward Mars it will likely become disrupted in the future. 19.11 An artificial satellite has a significant amount of structural integrity beyond internal gravitational effects. 19.12 Rotation implies an additional term that self-gravity must overcome. Therefore, for disruption to occur, GMm 2 2GMp Rm 2 C ! R > : m r3 Rm
132
Chapter 19 Physical Processes in the Solar System If the moon is in synchronous rotation, its rotation period is equal to its orbital period, or from Kepler’s law (Eq. 2.37), 40002 2 r 3: P2 D G Mp C Mm Since ! D 20002=P , !2 D
G Mp C Mm : r3
Substituting, G Mp C Mm Rm 2GMp Rm C > r3 r3 0006 0002 00030007 Mm GMp Rm 2 C 1 C > Mp r3 Solving for r 3, r3 <
0006
Mp =Rp3
GMm R2m
0007 Mm Rp3 : Mp
3C
Mm =R3m
GMm R2m
Since for a spherical body, M
D we find that
0002 r<
p m
00031=3 0006
or
0002 r < fR
where
4 3 R 3
;
Mm 3C Mp
p m
00071=3 Rp ;
00031=3 Rp
0007 0006 Mm 1=3 fR 0007 3 C : Mp
If Mm =Mp 0004 1, then fR ' 31=3 D 1:44. 19.13
(a)
According to the Stefan–Boltzmann law (Eq. 3.17), the luminosity of the Sun is Lˇ D 40002R2ˇ 0003Tˇ4 . At the radius of the planet’s orbit, the amount of solar energy received per second is Pin D Lˇ .1 0003 a/
0002Rp2 40002D 2
! ;
where a is the planet’s albedo, 0002Rp2 is its cross-sectional area as seen by the Sun, and 40002D 2 is the surface area of a sphere of radius D centered on the Sun. Assuming that the planet radiates as a spherical blackbody, its luminosity is given by Pout D 40002Rp2 0003Tp4 : If the planet is in thermodynamic equilibrium, then Pin D Pout. Solving for Tp leads to Eq. (19.5).
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133
(b) The energy radiated into space from the top of the greenhouse gas layer is Pout D 40002R2˚ 0003T˚4 . However, if the temperature of the layer is to remain constant, the energy per second radiated into the layer from Earth’s surface minus the energy per second radiated out of the layer toward Earth must give Pout, or 4 40002R2˚ 0003T˚4 D 40002R2˚ 0003Tsurf 0003 40002R2˚ 0003T˚4 :
Solving, Tsurf D 21=4 T˚ . From the result of Example 19.3.1, T˚ D 255 K, which gives Tsurf D 303 K D 30ıC. This temperature is just slightly higher than the average temperature of Earth’s surface; good agreement given the crude nature of the calculation. 19.14 Using Mp D M˚ , Rp D R˚ , and m D mH , Tesc > 140 K. Thus, the surface temperature of Earth is well above the value needed for atomic hydrogen to escape. 19.15
(a) For Jupiter, a D 0:343 and D D 5:2 AU. Thus, according to Eq. (19.5), TJ ' 110 K. (b) Given MJ D 317:83 M˚ and RJ D 11:21 R˚ for Jupiter, and m D 2mH for molecular hydrogen, Tesc > 7950 K. (c) Since Tesc 0005 TJ , molecular hydrogen has not had time to escape the atmosphere of Jupiter. Furthermore, since the primary constituent of the solar nebula was molecular hydrogen, Jupiter should be composed primarily of hydrogen.
19.16 Eq. (19.8) is
Z n0002R2 0004 m 00053=2 1 2 NP D 40002 v 3e 0002mv =2kT dv: 4 20002 kT vesc The formula for integration by parts is Z Z u dw D uw 0003 w du: Letting u 0007 v 2 and dw D ve 0002mv
2 =2kT
dv, imply that du D 2v dv and Z 2 w D v e 0002mv =2kT dv
kT 0002mv2 =2kT : e m Substituting into the integration-by-parts formula, ' # ˇ1 Z 1 00053=2 2 0004 2 m n0002R kT v 2kT 2 =2kT ˇ 2 =2kT 0002mv 0002mv ˇ C NP D 40002 0003 ve dv e ˇ 4 20002 kT m m vesc vesc D0003
0007 0006 2 n0002R2 0004 m 00053=2 kT vesc 2k 2 T 2 0002mvesc 2 =2kT 2 =2kT 0002mvesc D 40002 C e e 4 20002 kT m m2 0003 0004 m 00053=2 0002 kT 0003 0002 2kT 2 2 2 2 D n0002 R vesc C e 0002mvesc =2kT 20002 kT m m 0002 0003 0004 0005 1=2 n0002 2kT m 2 2 D R2 vesc C e 0002mvesc =2kT 2 20002 kT m D 40002R2 n.z/; where
0002 0003 1 0004 m 00051=2 2 2kT 2 0007 vesc C e 0002mvesc =2kT : 8 20002 kT m
134
Chapter 19 Physical Processes in the Solar System
19.17 Approximately 78% of Earth’s atmosphere is N2 and 21% is O2 . Assuming for simplicity that the remainder (roughly 1%) is water, air D 0:78n t mN2 C 0:21n t mO2 C 0:01n t mH2 O ; where n t is the total number density of all air molecules. Taking air D 1:3 kg m, mN2 ' 28 u, mO2 ' 32 u, and mH2 O ' 18 u, we find n t D 2:7 0002 1025 m00023 . Thus, nN2 D 0:78n t D 2:1 0002 1025 m00023 . 19.18 According to Eq. (9.12), ` D 1=n0003. Assuming that the separation between atoms in a diatomic molecule is roughly d D 0:2 nm, the cross section of a diatomic molecule is (very roughly) 0003 D 0002.d=2/2 D 3000210000220 m2 . Taking ` D 500 km, we find n D 1=`0003 0006 6 0002 1013 m00023 . This is slightly more than two orders of magnitude greater than the value quoted for N2 in Example 19.3.3. Note that in our estimate here we only calculated collisions of particles without distinguishing between species. The difference can be explained in part by gravitational separation. 19.19
(a) Using vesc D 11:7 km s00021 , mH D 1:67 0002 10000227 kg, and T D 1000 K, Eq. (19.9) gives D 1:24 m s00021 . (b) From Problem 19.18, n 0006 6 0002 1013 m00023 . (c) From Eq. (19.8), NP D 40002R2ˇ n D 4 0002 1028 s00021 . (d) Taking N0 D 9 0002 1043 (see Example 19.3.3), t 0006 N0 =NP D 2 0002 1015 s D 7 0002 107 yr. Since the age of Earth is 4:6 0002 109 yr, the hydrogen atmosphere would have leaked off by now.
19.20 Taking the logarithm of Eq. (19.9), 0002 0003 0004 m 0005 2 1 2kT mvesc 2 C log10 vesc C log10 0003 log10 e: 2 20002 kT m 2kT p For Jupiter, Eq. (2.17) gives vesc D 2GMJ =RJ D 59:7 km s00021 . Furthermore, using m D mH and T D 1000 K gives log10 D 000310:9. Thus D 10000210:9 m s00021 D 1:3 0002 10000211 m s00021 . log10 D 0003 log10 8 C
19.21
(a)
The initial eastward velocity of a projectile launched straight south from the North Pole is zero. Thus it gets “left behind” as observers at lower latitudes move with the planet’s rotation. This makes the motion of the projectile appear to drift to the right as viewed from the launch point (i.e., westward drift). (b) As a particle “falls” into a low pressure region, it’s motion deflects to the right if it is moving either northward or southward. This sets up a counterclockwise rotation about the center of the low pressure region in the Northern Hemisphere (a consequence of the Coriolis effect). (c)
19.22
(a)
clockwise
0004 0005 At some latitutde L, ! D ! cos LOj C sin LkO . The Coriolis force is then Fc D 00032m! 0002 v 0005i h0004 0005 0004 D 00032m! cos LOj C sin LkO 0002 vx Oi C vy Oj C vz kO D 00032m!
h i vz cos L 0003 vy sin L Oi C vx sin LOj 0003 vx cos LkO :
(b) !˚ D 20002=P˚ D 7:29 0002 1000025 rad s00021 . (c) If v D vx Oi at a latitude of L D 40ı , where vx D 30 m s00021 , then h i a D Fc =m D 00032! vx sin LOj 0003 vx cos LkO 0005 0004 0005 0004 O D 0003 2:8 mm s00022 Oj C 3:3 mm s00022 k: The ball is deflected southward and upward (neglecting gravity of course!).
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135
(d) If v D vy Oj, where vy D 30 m s00021 , 0005 0004 a D 2!vy sin LOi D 2:8 mm s00022 Oi
(e)
The ball is deflected eastward. O where vz D 30 m s00021 , If v D vz k, 0005 0004 a D 00032!vz cos LOi D 0003 3:3 mm s00022 Oi: The ball is deflected westward. This is because during the time the ball is in the air, Earth has rotated under it.
CHAPTER 20
The Terrestrial Planets 20.1 To find the velocities of the approaching and receding limbs of the planets, we need their rotation periods and radii: PMercury D 58:65 d D 5:067 0002 106 s, PVenus D 243:02 d D 2:100 0002 107 s, RMercury D 0:383 R˚ D 2:440 0002 106 m, and RVenus D 6:052 0002 106 m. These values imply that vMercury D 20002R=P D 3:026 m s00021 and vVenus D 1:811 m s00021 . Thus, given the opposite signs for the velocities of the receding and approaching limbs, 0003 0002 2v 0006v D D 1:9 0002 1000028 c Mercury c 0002 0003 0006v 2v D D 1:2 0002 1000028 : c Venus c 20.2 The component of the differential force per unit mass lying along the line connecting the centers of mass of the Sun and the planet is (see Eq. 19.3) ˇ ˇ ˇ 0006F ˇ 2GMˇ R ˇ ˇ : ˇ m ˇD r3 From Eq. (2.5) and the data in Appendix C, rp;Mercury D a.10003e/ D 4:6000021010 m and rp;Earth D 1:4700021011 m. These values, combined with the planets’ radii give Œ0006F=m0011Mercury Œ0006F=m0011Earth
D 12:5:
The significantly greater tidal force per unit mass on Mercury relative to Earth has resulted in Mercury’s current 3-to-2 spin-orbit coupling. 20.3
(a)
The amount of solar energy absorbed per second by a surface of area A at a point on the planet at the latitude 000e is 0003 0002 Lˇ .A cos 000e/ .1 0003 a/ : Lin D 40002D 2 At equilibrium, this value must equal the amount of energy radiated per second by that surface, Lout D A0003T 4 . Equating and solving for the temperature gives the desired result.
(b) See Fig. S20.1. (Note that there was an error in the first printing of the text, the albedo should be 0.119, as in Appendix C.) (c)
T .000e D 0/ D 614 K.
(d) 000e D 87:8ı . 20.4
(e)
The effect of sublimation should diminish the amount of ice found on the surface.
(a)
From Eq. (6.6), the Rayleigh criterion is 000eradar D 1:22 136
0005 D 6:1 0002 1000024 rad: D
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137
Figure S20.1: Results of Problem 20.3 for the approximate temperature on Mercury’s surface as a function of latitude. (b) The angular size of Mercury is 000eMercury D (c)
2RMercury D 5:3 0002 1000025 rad: 1 AU 0003 aMercury
The fraction of the original signal striking the surface of Mercury was about f D
2 000eMercury 2 000eradar
D 0:0076:
This implies a total amount of power received by Mercury of PMercury D f Pradar D .0:0076/.500 kW/ D 3:8 kW. (d) Assuming that Mercury acted like a point source radiating energy back into the hemisphere facing Earth, the flux received at Earth is F D f Pradar=20002.1 AU00030:3871 AU/2 . This gives F D 4:4000210000220 W m00022 . 20.5
(a)
It is estimated that the impactor had a mass approximately one-fifth of Mercury’s present mass and a velocity of about 20 km s00021 . Using these values K D 12 mv 2 D 6:6 0002 1031 J.
(b) Assume that the extra mass was initially a uniform-density shell sitting on the present-day Mercury, and that the density of the shell was . The gravitational potential energy of that shell would have been Z Rs 0005 0004 GMM 2 Us D 0003 ; dV D 000320002 GMM R2s 0003 RM r RM where Rs is the outer radius of the shell, RM is the present-day radius of Mercury, and MM is the mass of the present-day Mercury. Rs is determined from Ms D
0005 MM 40002 0004 3 3 D Rs 0003 RM : 3 5
If D 3350 kg m00023 , the density of Earth’s Moon, then Rs D 2:7 0002 106 m D 0:42 R˚ . This implies that Us D 00035:7 0002 1029 J:
138
Chapter 20 The Terrestrial Planets If we assume that the entire mass of the impactor was also ejected from the surface of the present-day Mercury, its potential energy at the time of impact would have been Ui D 0003G
MM Mi D 00036:0 0002 1029 J: RM
Thus the energy required to eject the shell and the impactor would have been E D 0003.Us C Ui / D 1:2 0002 1030 J. (c)
There would have been plenty of energy available from the kinetic energy of the impactor to eject both the impactor and the extra shell of material.
20.6 The surface pressure on Venus is 90 atm D 9 0002 106 N m00022 . From the ideal gas law (Eq. 10.10) with T D 740 K, n D P = kT D 8:800021026 m00023 . This value is roughly 44 times the value quoted in Example 19.3.3 for Earth’s surface. 20.7
(a)
Image that the atmosphere is composed of layers (assumed to be an integer for simplicity), each of optical depth unity. Further assume that each layer has a different constant temperature. Let T1 D Tbb be the blackbody temperature at the top of the atmosphere (i.e., in the top-most layer). Since the layer has two surfaces, top and bottom (each of essentially the same area), the energy per second radiated out 4 of that first layer is 2A0003Tbb while the energy radiated into it from the second layer is A0003T24 . Equating 4 the two rates gives T24 D 2Tbb . Similarly, the next layer down (layer 2) loses energy at the rate 20003AT24 but gains energy from its neighbors, or 20003AT24 D 0003AT14 C 0003AT34 : This implies that
0005 0004 4 4 4 0003 Tbb D 3Tbb : T34 D 2T24 0003 T14 D 4Tbb
4 Continuing down through the atmosphere, we find that in general for layer n, Tn4 D nTbb . Now, at the surface, 4 C A0003T0004400021 ; 20003AT00044 D A0003Tsurf
or
0005 h 0004 0005 i 0004 4 4 4 4 0003 . 0003 1/ Tbb D . C 1/ Tbb Tsurf D 2T00044 0003 T0004400021 D 2 Tbb :
Thus Tsurf D .1 C /1=4 Tbb :
(S20.1)
(b) From Eq. (19.5), Tbb D 226 K for Venus. According to Eq. (S20.1), this value for Tbb , combined with D 70 implies that Tsurf D 658 K. Given the crude nature of the approximation, this result is in reasonable agreement with the observed value of 740 K. 20.8 Taking 0006v ' 0:03 m yr00021 , t D d=v D 1:6 0002 108 yr. 20.9 Following the approach taken in Example 10.1.1, Pc 0006 GM˚ =R˚ D 3:4 0002 1011 N m00022 D 3:6 0002 106 atm, remarkably close to the detailed computer simulation. 20.10
(a)
From Eq. (16.30), dK=dt D 000340002 2I PP =P 3. Making the approximation that the moment of inertia of Earth is I D 2M˚ R2˚ =5 D 9:7 0002 1037 kg m2 , and using the values of PP D 0:0016 s century00021 and P ' 24 hr D 8:64 0002 104 s, the rate of decrease of rotational kinetic energy is jdK=dtj D 3:0 0002 1012 W.
(b) The value obtained in part (a) is approximately 7.5% of the observed rate of energy outflow from the interior (4 0002 1013 W).
Solutions for An Introduction to Modern Astrophysics
139
20.11 The converging B-field lines produce a force that acts to retard the motion of the particles, ultimately changing their directions. For most particles, the change in direction occurs before they strike the atmosphere. 20.12
(a)
Using a weighted average for the average density of the “two-zone” model, D
D
c Vc C m Vm Vc C Vm 0005 0004 4 0002R3c c C 43 0002R3˚ 0003 43 0002R3c m 3 4 0002R3˚ 3
0002 D
Rc R˚
'
00033
0002
c C 1 0003
Rc R˚
00033 # m :
Solving for the radius of the core, Rc D R˚
0002
0003 m c 0003 m
00031=3 D 0:54:
(b) For the two-zone model, I2 zone D Ic C Im 0004 0005i 80002 h D c R5c C m R5˚ 0003 R5c 15 D 8:48 0002 1037 kg m2 : This implies that I=MR2 D 0:349.
20.13
(c)
For a constant-density sphere, I=MR2 D 2=5 D 0:4. The lower value found in part (b) reflects the centrally condensed nature of Earth’s interior.
(a)
Since the moment-of-inertia ratio is very close to the constant-density value of 0.4 for a sphere, the Moon must be only very slightly centrally condensed.
(b) The result is consistent with the lack of a detectable magnetic field. A strong magnetic dynamo requires a fluid, conducting (presumably molten iron-nickel) core of significant size. If such a core existed it would imply a much higher core density relative to the mantle. This would then imply a smaller value for the moment-of-inertia ratio than is observed. 20.14
(a)
See Fig. S20.2.
(b) The slope of the best-fit straight line is m D e 0003t 0003 1 D 0:023698. Given the value of 1=2 D 106:0 Gyr 00023 Gyr00021 . Thus, t D 3:582 Gyr. for the half-life of 147 62 Sm (see Table 20.1), 0005 D ln 2= 1=2 D 6:539 0002 10 This is much younger than the lunar highlands sample of 4.39 Gyr. 20.15 Equating the centripetal force to the gravitational force, Mm v 2 GM˚ Mm D ; R˚ R2˚ and substituting the velocity of Earth’s surface at the equator in terms of its rotation period, v D 20002R˚ =P , leads to 3=2 20002R˚ PD D 5060 s D 1:4 hr: .GM˚ /1=2
140
Chapter 20 The Terrestrial Planets 0.5145
143Nd/ 144Nd
0.5140 0.5135 0.5130 0.5125 0.5120 0.5115 0.18
0.20
0.22
0.24
0.26
0.28
0.30
147Sm/144Nd
Figure S20.2: The results of Problem 20.14(a). 20.16 Using ˚ D 5520 kg m00023 and Moon D 3340 kg m00023 , the Roche limit is (Eq. 19.4) 0002 r D 2:456
˚ Moon
00031=3 R˚ D 1:85 0002 107 m D 0:048aMoon;
where aMoon is the semimajor axis of the Moon’s orbit. The Moon is not in danger of being tidally disrupted. r r Tp ra 1Ce 20.17 (a) D D D 1:098: Ta rp 10003e (b) The southern hemisphere’s polar ice cap varies more dramatically. 20.18 PPhobos D 7h 30m D 2:75 0002 104 s and PDeimos D 30h 17m D 1:09 0002 105 s. Then, from Kepler’s third law (Eq. 2.37), aPhobos D 9:36 0002 106 m D 2:75 RMars aDeimos D 2:34 0002 107 m D 6:89 RMars : 20.19 Since PPhobos D 7h 30m , PDeimos D 30h 17m , and PMars D 24h 37m , Phobos will appear to move eastward and Deimos will appear to move westward.
CHAPTER 21
The Realms of the Giant Planets 21.1 Following the procedure given in Example 10.1.1, Pc 0002 GM 0002=Rp , and using the data found in Appendix C, Pc;Jupiter 0002 2:4 0003 1012 N m00022 and Pc;Saturn 0002 4:5 0003 1011 N m00022 . For comparison, the central pressure in the Sun is 2:5 0003 1017 N m00022 (see Table 11.1). The central pressures of Jupiter and Saturn are significantly less. 21.2
(a)
Beginning with Eq. (10.6) for hydrostatic equilibrium, Mr 0002 dP D 0004G 2 ; dr r and substituting the polytropic relation, P .r / D K0002 2.r /, leads to G Mr d0002 : D0004 dr 2K r 2 Differentiating again, d 20002 G D0004 dr 2 2K
0002
(S21.1)
1 dMr 2Mr 0004 3 r 2 dr r
0003 :
(S21.2)
Using the mass conservation equation (Eq. 10.7), dMr D 40003 r 2 0002; dr rewriting Eq. (S21.1), 2K d0002 Mr D0004 ; r2 G dr and substituting into Eq. (S21.2) leads to the desired result, 0002 0003 d 20002 2 d0002 20003 G C C 0002 D 0: dr 2 r dr K (b) Starting from
0002 0002.r / D 0002c
and differentiating, d0002 D 0002c dr
0002
sin k r kr
(S21.3)
0003 ;
cos k r sin k r 0004 r kr2
(S21.4) 0003 :
(S21.5)
The second derivative is d 20002 D 0002c dr 2
0002
2 sin k r 2 cos k r k sin k r 0004 0004 3 2 kr r r
0003 :
(S21.6)
Substituting Eqs. (S21.4), (S21.5), and (S21.6) into Eq. (S21.3) verifies that the equation is satisfied. (c)
k D 0003=RJ D 4:50 0003 1000028 m00021 and K D 20003 G= k 2 D 2GR2J =0003 D 2:07 0003 105 N m4 kg00022 . 141
142
Chapter 21 The Realms of the Giant Planets (d) Integrating Eq. (10.7), Z
r
MJ D
40003 r 2 0002 dr 0
D D
(e)
400030002c k
Z
r
r .sin k r / dr 0
400030002c Œsin k r 0004 .k r / cos k r 0004 k3
Substituting r D RJ and recalling that kRJ D 0003 leads to 0002c D
0003 MJ D 4100 kg m00023 : 4 R3J
(f) See Figs. S21.1 and S21.2. (g) Pc D K0002c2 D 3:6 0003 1012 N m00022 .
Density (1000 kg m-3)
4.0 3.0 2.0 1.0 0.0 0.0
1.0
2.0
3.0 4.0 Radius (10 7 m)
5.0
6.0
7.0
Figure S21.1: A polytropic model of the variation of density with radius for Jupiter’s interior according to Problem 21.2(f). 21.3
(a)
Consider a ring of mass d m and radius a centered on the rotation axis, with the edge of the ring a distance r from the center of the planet. Then, for an angle 0005 measured from the rotation axis, a D r sin 0005. Furthermore, for a density distribution of 0002.r /, the mass of the ring is given by d m D 0002 dV D 0002.20003 a r d0005 dr /. Substituting into the expression for the moment of inertia, Z I D a2 d m Z D
vol
vol
h i .r sin 0005/2 20003 r 2 sin 00050002.r / dr d0005:
Solutions for An Introduction to Modern Astrophysics
143
1.0 0.9 0.8 Mr/MJupiter
0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 0.0
1.0
2.0
3.0 4.0 7 Radius (10 m)
5.0
6.0
7.0
Figure S21.2: A polytropic model of the variation of interior mass with radius for Jupiter’s interior according to Problem 21.2(f). Introducing the polytropic result,
0002 0002.r / D 0002c
the moment of inertia becomes
Z
I D
0002 0003D0
200030002c D k
Z
RJ r D0
Z
0002
sin k r kr
0003
0002 20003 r sin 00050002c 4
3
Z
RJ
sin 0005 d0005 3
0
;
sin k r kr
0003 dr d0005
r 3 sin k r dr:
0
Recalling from Problem 21.2 that kRJ D 0003 , the expression for the moment of inertia reduces to 0003 0002 80002c 6 ID 1 0004 2 R5J : 30003 0003 (b) Using the estimate from Problem 21.2(e) that 0002c D 4100 kg m00023 and the data in Appendix C, the moment-of-inertia ratio of this polytropic model of Jupiter is I D 0:134: MJ R2J
21.4
(c)
The polytropic model overestimates the central mass concentration.
(a)
First consider the case of a uniform density, 0002env (in other words, let f D 0 initially). Then Z Z p Rp
I D 400030002env 0
D D
10002.z=RP /2
a3 da dz
0
Z 0003R4e 0002env
Re
Rp 0
80003 Rp R4e 0002env 15
'
0002 10004
z RP
00032 #2 dz
144
Chapter 21 The Realms of the Giant Planets The core can be included by subtracting the moment of inertia of a constant-density spherical core of density 0002env and radius fRe , and replacing it with the moment of inertia of the constant-density spherical core of density 0002core having the same radius. Thus, 0002 0003 80003 2 40003 4 ID Rp Re 0002env C .0002core 0004 0002env / f 5 R5e : 15 5 3 Simplifying results in Eq. (21.5). (b) If the planet is spherical, b D 0. Furthermore, if the planet has a constant density, 0002core throughout, then f D 1. Also, in this case M 0002core D : 40003R3e =3 Substituting these values gives the usual result, I D 2MR2e =5.
21.5
(a)
Mcore D 0002coreVcore D 40003f 3 R3e 0002core=3.
(b) fRe D 9:8 0003 106 m, implying that f D 0:138. (c)
The mass of the planet as a two-component model can be expressed as Z p Z Rp
M D 40003 0
Re
10002.z=RP /2
0002.a; z/ a da dz:
0
Using a procedure similar to finding the moment of inertia in Problem 21.4, we find i 40003 h 2 Mtotal D Re Rp 0002env C f 3 R3e .0002core 0004 0002env / : 3 i 40003 3 h Re .1 0004 b/0002env C f 3 .0002core 0004 0002env/ : D 3 Solving for 0002env, 0002env D
Mtotal =.40003R3e =3/ 0004 f 3 0002core : 10004b0004f3
Applying the result from part (a), we find 0003 0002 Mtotal =Mcore 0004 1 f 3 0002core: 0002env D 10004b0004f3 For Jupiter, Mtotal D 317:83 M˚ and b D 0:064874 (see Appendix C). The density of the two-zone envelope is then 0002env D 1300 kg m00023 . (d) From Eq. (21.5), I=MR2e D 0:393. (e)
The measured moment of inertia ratio for Jupiter is only 0.258. This points out that the envelope structure must be must more centrally condensed than the constant value assumed here.
21.6 The diameter of Jupiter’s magnetosphere is Dmag D 3 0003 1010 m. At opposition Jupiter is at a distance of d D 5:2 AU 0004 1:0 AU D 4:2 AU D 6:3 0003 1011 m. Thus, the angular diameter of the magnetosphere is 0005mag D Dmag =d D 0:048 rad D 2:7ı. Since the angular diameter of the Moon is about 0005Moon ' 0:5ı , we find 0005mag ' 5:50005Moon. 21.7 Assuming a spherical fragment of diameter d D 700 m and a constant density of 0002 D 200 kg m00023 , the mass of the fragment was mG D 40003.d=2/3 0002=3 D 3:6 0003 1011 kg. The escape velocity of Jupiter is (see Eq. 2.17) p 2 vesc D 2GMJ =RJ D 60 km s00021 . These values imply a kinetic energy of E D 12 mG vesc D 6:4 0003 1019 J D 15,000 MTons.
Solutions for An Introduction to Modern Astrophysics
145
0.003 0.002
Correction
0.001 0.000 −0.001 −0.002 J2 term J4 term Sum of terms
−0.003 −0.004 −0.005
0
20
40
60
80 100 θ (deg)
120
140
160
180
Figure S21.3: The results for Problems 21.8(a) and (b) for the gravitational potential of Saturn. 21.8
(a)
See Fig. S21.3.
(b) The gravitational potential is largest at 0005 D 90ı and smallest at 0005 D 0ı and 180ı. At 0ı and 180ı the deviation is approximately 0.4%; at 90ı the deviation is approximately 0.2%. 21.9
(a)
From Eq. (10.23) and following Example 10.3.1, 0006E D
3 GMJ2 D 1:0 0003 1036 J: 10 RJ
(b) The average rate of gravitational energy output from Jupiter over the age of the solar system is 0004 0005 dE 0006E D D 7:0 0003 1018 W: dt 4:55 0003 109 yr (c)
The current intrinsic power output of Jupiter is 3:3 0003 1017 W. This value is roughly one-twentieth of the average output over the planet’s lifetime, implying that the output must have been much greater in the past. The larger rate for the energy output in the past would have contributed to early heating of the Jupiter subnebula, helping to deplete volatiles near Io and Europa.
21.10 The rate at which solar energy is absorbed by Neptune is ! 0002 0003 0007 L 0006 dE ˇ 2 D 0003RN .1 0004 a/ D 2 0003 1015 W; dt absorbed 40003 rN2 where RN is Neptune’s equatorial radius, rN is its orbital radius, and a is its albedo (see Appendix C). Since roughly an equal amount of energy is emitted by internal sources, the total energy per second given off by Neptune is 0002 0002 0003 0003 dE dE '2 D 4 0003 1015 W: dt emitted dt absorbed Assuming that the energy is emitted as blackbody radiation, 0002 0003 dE D 40003R2N 0007TN4 : dt emitted Solving for the temperature, TN D 55 K. This crude estimate is about 7% less than the observed value of 59.3 K.
146
Chapter 21 The Realms of the Giant Planets
21.11 Consider the upper value given in the text of dN=dt D 1029 ions s00021 leaving Io. Assuming that all of the ions are sulfur, the mass of each ion is mion ' 32 u D 5:3 0003 10000226 kg and the rate of mass loss is d m=dt D 5300 kg s00021 . Over 4.55 billion years, the amount of mass lost is 0006m D 7:5 0003 1020 kg. This value is roughly 0.85% of Io’s present mass. Hence, Io is in no danger of vanishing any time soon. 21.12
(a)
The approximate volume of the disk is 0007 0006 Vrings D .30 m/ 0003R2out 0004 0003R2in D 2:3 0003 1018 m3 : From Eq. (9.12), the number density of ice spheres in the rings is nD
1 1 i D 110 m00023 : D h 2 `0007 ` 0003 .1 cm/
Given that the mass of an individual ice sphere of radius rice D 1 cm is 4 3 0003 r 0002ice D 0:0042 kg; 3 ice
mice D the mass of ice in the rings is
mrings D 0002ringsVrings D nmice Vrings D 1 0003 1018 kg: (b) 62 km. 1 3 1
Mimas
3 Ring 2
Saturn
2
Figure S21.4: The sketch for Problem 21.13. 21.13 The sketch is shown in Fig. S21.4. Mimas and a characteristic ring particle are depicted at three points in their orbits, labeled 1, 2, and 3, respectively. For each one-half orbit that Mimas completes, the ring particle completes one full orbit. Also drawn are force vectors (not shown to correct relative lengths) between Mimas and the particle at each of the three locations. Given that the force vectors are longer at positions 1 and 3 than at 2, Mimas systematically elongates the orbit, resulting in an ellipse. 21.14 Saturn’s rotation period is 0:444 d D 3:84 0003 104 s. From Kepler’s third law (Eq. 2.37), this corresponds to an orbital radius of a D 1:1 0003 108 m D 1:9 RS. According to Table 21.4, a portion of the B ring, as well as the A, F, G, and E rings exist beyond this radius.
Solutions for An Introduction to Modern Astrophysics 21.15
(a)
147
The energy absorbed by the grain per unit time is 0002 0003 dE D 0007g dt absorbed
Lˇ 2 40003 rˇ
! ;
where rˇ is the grain’s distance from the Sun. (b) The angular momentum of the dust grain is L D mvr , where r is the distance of the grain from the object that it is orbiting. Therefore, the rate at which the angular momentum decreases is given by 0002 0003 dL dm ' vr; dt dt where we have assumed that v and r are essentially constant. Considering the energy radiated away by a photon as being associated with an effective mass, the angular momentum loss rate is dL vr dE0004 ' 2 : dt c dt At equilibrium, dE0004 =dt D 0004 .dE=dt/absorbed. Substitution leads to the desired result for the case where the particle is orbiting the Sun, or r D rˇ . 21.16
(a)
Beginning with Eq. (21.6), setting r D rˇ (assumed essentially constant), and integrating gives 0002 0003 0007g Lˇ Lf D0004 t ; (S21.7) ln 2 mc 2 Saturn L0 40003 rˇ where L0 and Lf are the initial and final angular momenta of the particle, respectively, and tSaturn is the elapsed time. The initial and final angular momenta can be found from L.r / D mvr , where 0002 0003 4 3 mD0002 0003R ; 3 and R is the radius of the grain. Furthermore, v D 20003 r=P , and from Kepler’s third law (Eq. 2.37), PD so that
20003 .GMS /1=2
r 3=2;
v D .GMS /1=2 r 00021=2:
Combining, the angular momentum is L.r / D
4 00030002R3 .GMS /1=2 r 1=2: 3
Setting L0 D L .R0 /, Lf D L .RS /, and 0007g D 0003R2 , and substituting into Eq. (S21.7) gives the desired result. (b) tSaturn D 2 0003 1013 s D 6 0003 105 yr. (c) The estimate is much less than the age of the Solar System. Hence, the E ring cannot be a permanent feature unless the particles are replenished. 21.17
(a) (b) (c) (d)
From Eq. (2.17), vesc D 0:213 km s00021 . p v D 2GMU =rM D 9:4 km s00021 . ˇ ˇ ˇUg ˇ D 3GM 2 =5RM D 1:09 0003 1024 J. M Beginning with 00033 0002 ˇ ˇ 1 4 1 2 3 ˇ ˇ v2; K D Ug D mv D 00030002R 2 2 3 and solving for the radius gives R D 14:3 km.
CHAPTER 22
Minor Bodies of the Solar System 22.1
(a)
From Eq. (3.13), Frad
Lˇ 0003R2 hSiA D D 1:4 0003 10000219 N; cos 0005 D c 40003 r 2c
00023 where R D 0006 100 nm 0007 and r D 1 AU. Given a density of 0002g D 3000 kg m , the mass of the grain is mg D 0002g 43 0003R3 D 1:3 0003 10000217 kg. Thus, the acceleration is a D Frad =mg D 0:011 m s00022 .
(b) g D GMˇ =r 2 D 0:0059 m s00022 . 22.2
(a)
Beginning with Eq. (21.6),
dL 0007g Lˇ L; D0004 dt 40003 r 2 mc 2
where r is the orbital radius of the grain. After substituting 0007g D 0003R2 (where R is the grain radius), L D mvr , and m D 43 0003R3 0002, and simplifying, d.vr / R2 Lˇ .vr /: D0004 2 dt 4r mc 2 Using Kepler’s third law (Eq. 2.37), P 2 Dp40003 2r 3=GMˇ , and substituting P D 20003 r=v for a circular orbit, the velocity can be written as v D GMˇ =r . Placing the result into the differential equation, simplifying, and integrating from the initial orbital radius to the solar radius we find Z
Rˇ
Z
tSun
r dr D 0004
r
0
R2 Lˇ dt: 2mc 2
Evaluating,
0007 10006 2 R2 Lˇ tSun : Rˇ 0004 r 2 D 0004 2 2mc 2 Making use of the approximation r 0005 Rˇ , and writing the mass in terms of the density of the spherically symmetric grain, m D 43 0003R3 0002, leads to the desired expression for the time required for the dust grain to spiral into the Sun.
(b) 0.92 m. 22.3 Combining the rates of mass loss for dust and gas, d m=dt D 2:5 0003 104 kg s00021 . In one year (the active portion of one trip around the Sun), 0006m=trip D .d m=dt/ 0006t D 8 0003 1011 kg. Taking the mass of the nucleus to be m 0002 1014 kg, we find N D m=.0006m=trip/ D 125 trips. 22.4 The mass-loss rate, d m=dt, combined with the gas velocity, vg implies a rate of change of momentum (i.e., a force): d mvg dm dp F D mnucleusanucleus D vg D D : dt dt dt Measurement of the acceleration of the nucleus gives an estimate of the mass. 148
Solutions for An Introduction to Modern Astrophysics 22.5
(a)
149
From Kepler’s third law [Eq. (2.37) written in the form P 2 D a3 , where P and a are in years and AU, respectively], a D 64 AU.
(b) From Eq. (2.5), rp D a.1 0004 e/ D 0:0055 AU, and from Eq. (2.6), ra D a.1 C e/ D 128 AU. (c)
The Kuiper belt.
22.6 The orbital period of Jupiter is PJ D 11:8622 yr (see Appendix C). Thus, an object with a 2:1 orbital resonance with Jupiter has a period of P21 D PJ =2 D 5:931 yr. This corresponds to a semimajor axis of 3.28 AU [Eq. (2.37) written in the form P 2 D a3 , where P and a are in years and AU, respectively]. Similarly, for a 3:1 resonance, P D PJ =3 D 3:954 yr and a D 2:5 AU. These values agree with Fig. 22.13. 22.7
(a)
From Eq. (19.5), r TV D Tˇ .1 0004 a/
1=4
Rˇ 2D
D .5777 K/.1 0004 0:38/1=4
6:96 0003 108 m 2.2:362 AU/
000b1=2
D 161 K: (b) 22.8
(a)
dE D LV D 40003R2V 0007TV4 D 2 0003 1013 W. dt The total volume occupied by the asteroids is 0003 0002 40003 3 Vasteroids D N R D 1:3 0003 1021 m3 : 3
(b) The volume of the belt in this crude estimate is 0007 0006 Vbelt D h0003 R22 0004 R21 D 4:9 0003 1032 m3 : (c)
2:6 0003 10000212
(d) This is unrealistic, at least in our Solar System! You would have to go out of your way to hit an asteroid. 22.9
(a)
mV D 43 0003R3V 0002V D 1:9 0003 1020 kg.
(b) NSi D mV =mSi D 4:1 0003 1045 . 0006E D .0006m/c 2 D .25:986892 u 0004 25:982594 u/c 2 D 6:4 0003 10000213 J. 0007 0002 8680 0003 0006 (d) N26 Al D 5 0003 1000025 NSi D 1:8 0003 1039 . 13 106 (e) The amount of energy released per second is (c)
dE dN D 0006E : dt dt However, from Eq. (15.9), dN=dt D 0004N , and from Eq. (15.10), N .t/ D N0 e 00020005t . Substituting, dE D 0004.0006E/N0 e 00020005t : dt For a half-life of 1=2 D 716,000 yr, the decay constant is D
ln 2 D 9:7 0003 1000027 yr00021 : 1=2
150
Chapter 22 Minor Bodies of the Solar System Furthermore, .0006E/N0 D 3:5 0003 1013 J. Thus ˇ ˇ 0007 ˇ dE ˇ 0006 13 0002.9:700031000027 yr00021 /t ˇ ˇ : ˇ dt ˇ D 3:5 0003 10 W e The plot is given in Fig. S22.1.
Log10 L (W)
(f) 5:6 0003 105 yr. 15 13 11 9 7 5 3 1 -1 -3 -5 -7 -9
0
1
2 3 7 Time (10 yr)
4
5
Figure S22.1: An estimate of the energy output from Vesta as a function of time [see Problem 22.9(e)]. 22.10 Earth spins on its axis in the same direction that it orbits the Sun. Hence, on the side of Earth away from the Sun, the rotation and velocity vectors add, implying a larger value for the net relative velocity between a meteoroid and Earth. Also, the sky is dark on the side of Earth away from the Sun. 22.11 The energy released in the Tunguska event was roughly E D K D mv 2 =2 D 1015 J. If the impact velocity was Earth’s escape velocity of 11:2 km s00021 (see Eq. 2.17), the mass was 2 0003 107 kg. Assuming a density of 2000 kg m00023 and spherical symmetry, the radius of the impactor was 12 m.
CHAPTER 23
Formation of Planetary Systems 23.1
(a)
Both HD 80606 and HD 80607 are G5V stars, implying that their masses are each just slightly less than 1 Mˇ . If we adopt 1 Mˇ for the mass of each star in this example, and we use Kepler’s third law in the form .M1 C M2 /P 2 D a3 , where the masses are in solar units, P is in years, and a is in astronomical units, we find P D 63,000 yr.
(b) If a D 0:44 AU, then the orbital period of HD 80606b about HD 80606 (neglecting any influences from the companion star) is P D 0:29 yr. This would imply roughly 220,000 orbits of the planet for each orbit of the two stars about their common center of mass. (c)
The ratio of the two forces is roughly 0002 F80607=F80606 D
:44 2000
00032
D 5 0003 1000028 :
23.2 Comparing densities: Object Pluto Ida Vesta Halley Triton
density (103 kg m00023 ) 2.1 2.2–2.9 2.9 <1 2.1
class planet asteroid asteroid comet moon of Neptune
(Note that the data on Vesta is given in Problem 22.9.) Comparing radii: Object Pluto Ceres Vesta Ida Halley Triton
radius (km) 1135 500 250 55 15 1353
class planet asteroid asteroid asteroid (long axis) comet moon of Neptune
(Note that the data for Ceres was found in Section 19.1.) Pluto is very similar to Triton, in terms of density, radius, and composition. Comets are icier and smaller than Pluto, while asteroids tend to be rockier and smaller. Pluto is consistent with being formed farther out from the Sun than were the hotter asteroids. Pluto grew larger than cometary nuclei. 23.3
(a)
According to Fig. 23.8, Ltot=Mtot D 1017:2. Taking Mtot ' Mˇ gives Ltot D 3:2 0003 1043 kg m2 s00021 . From the definition of the moment-of-inertia ratio given in Problem 20.12, the Sun’s moment of inertia is Iˇ D 0:073Mˇ R2ˇ D 7 0003 1046 kg m2 . Furthermore, since Ltot D Iˇ ! D 20003Iˇ =P , where ! and P are the angular velocity and rotation period, respectively, solving for the rotation period gives P D 1:4 0003 104 s D 0:16 d. 151
152
Chapter 23 Formation of Planetary Systems (b) ve D 20003Rˇ =P D 3:2 0003 105 m s00021 . (c)
Setting the centripetal acceleration equal to the gravitational acceleration, 2 ve;max
Rˇ
D
GMˇ ; R2ˇ
and solving for the maximum velocity gives s ve;max D
GMˇ D 4:4 0003 105 m s00021 : Rˇ
This corresponds to a rotation period of P D 20003Rˇ =ve;max D 1 0003 104 s D 0:12 d. 23.4 If we simply add up all of the mass of planets, comets, asteroids, and moons, we get (very roughly) M D 2:600031027 kg for the planets, 6:300031023 kg for the largest moons in Appendix C, 600031026 kg D 100 M˚ for the Oort cloud, perhaps 1000 MPluto D 1025 kg in the Kuiper belt (wild guess), and 5 0003 1000024 M˚ D 3 0003 1021 kg for the asteroid belt. Combining gives a very crude estimate of MMMSN D 4 0003 1027 kg. 23.5 RH D 0:5 AU D 1000 RJ . 23.6
(a)
The mass of a K4V star is roughly 0:7 Mˇ . Using Kepler’s third law with an orbital period of P D 2:81782 d implies a D 0:035 AU D 5:2 0003 109m.
(b) If the planet produces a radial reflex velocity of 64:3 m s00021 on the star, and the planet’s orbital speed is vp D 20003 a=P D 134 km s00021 , the mass of the planet is about mp D (c)
The orbital radius of the star about the center of mass is r D vstar P =20003 D 2:5 0003 106 m. This implies a maximum wobble of 0005D
23.7
(a)
vstar Mstar D 6:7 0003 1026 kg D 0:35 MJ : vp
2r 5 0003 106 m D D 4:5 0003 10000212 rad D 9:7 0003 1000027 arcsec D 0:97 as: d 35:8 pc
The mass of a K0V star is approximately 0:79 Mˇ . Therefore, a D .P 2 =M /1=3 D 3:1 AU.
(b) ra D .1 C e/a D 4:2 AU. (c)
From Eq. (2.33), vp D 2:1 0003 104 m s00021 D 21 km s00021 .
23.8 High metallicity implies that the formation of dust grains will be more effective. The merger of grains will lead to the build-up of planetesimals. 23.9 If Jupiter and Saturn were originally at 5.7 AU and 8.6 AU, their orbital periods would have been 13.6 yr and 25.2 yr, respectively, corresponding to a period ratio of .PS =PJ /0 D 1:85. Today’s period ratio is PS =PJ D 29:46 yr=11:86 yr D 2:48. At some point during their migrations they would have passed through a period ratio of 2.
CHAPTER 24
The Milky Way Galaxy 24.1 Using R0 D 8 kpc and 0 D 220 km s00021 , P D 20003R0 =000b0 D 220 Myr. Taking the age of the Sun to be 4.56 Gyr, the Sun has made approximately 20 trips around the Galactic center. 24.2
(a)
See Table S24.1.
(b) Lbol; tot D 3:6 0003 1010 Lˇ . Table S24.1: Results for Problem 24.2(a). component LB .1010 Lˇ / gas 0 thin disk 1.8 thick disk 0.02 bulge 0.3 stellar halo 0.1 dark matter halo 0 Total LB 1 2.22 1 Note that the value in the text is LB D 2:3 ˙ 0:6 0003 1010 Lˇ .
24.3
(a)
% 0 81 0.9 13.5 4.5 0 100
Beginning with Eq. (24.1) and solving for aV , aV D V 0004 MV C 5 0004 5 log10 d D 2:38:
(b) aV =9:0 kpc D 0:26 mag kpc00021 . 24.4 From the discussion leading up to Eq. (3.5), F D F10 1000002.m0002M /=5 is the flux received from a star of apparent magnitude m relative to a star of absolute magnitude M located 10 pc from Earth. Next, from Eq. (24.4), the number of stars with absolute magnitudes between M and M C dM found within a solid angle and having apparent magnitudes between m and m C d m is given by d N M D AM d m: Thus, the total flux due to all stars with apparent magnitudes between m and m C d m is dFm D F10 1000002.m0002M /=5 AM d m: 153
154
Chapter 24
The Milky Way Galaxy
Substituting the expression for AM given by Eq. (24.5), and adding up the contributions from all stars between the observer (m ! 00041) and some distance d corresponding to an apparent magnitude m,
Z m
ln 10 00022.m0002M /=5 3.m0002M C5/=5 F10 10 dm Fm D nM 10 5 00021 Z m DC 10m=5 d m; 00021
where C is a constant. After integrating, Fm D
5C 10m=5 : ln 10
As d ! 1, m ! 1, and Fm ! 1. 24.5
(a)
From Eq. (24.5), log10 AM D
3 .m 0004 M C 5/ C log10 5
0002
0003 ln 10 C log10 C log10 nM .M; S/: 5
(b) 0006 log10 AM .m/ D 0:6: (c)
The distribution of stars is not constant, but decreases with increasing distance. 6 5 4 Log10 AM
3 2 1 0 −1 −2 −3
4
6
8
10
12
14
16
18
V
Figure S24.1: Results for Problem 24.6(a). 24.6
(a)
See Fig. S24.1.
(b) The slope of the curve over the range 4 < V < 11 is 0.6. This implies that no interstellar extinction is present, or aV D 0. (c)
From Eq. (24.1) with a D 0, d D 631 pc.
(d) From Table 24.2, V D 11 corresponds to log10 AM D 1:89. Employing the result of Problem 24.5(a), we find nM .M; S/ D 2:9 0003 1000023 pc00023 . 24.7
(e)
A monotonic decrease in the stellar number density with increasing V or interstellar extinction.
(a)
Returning to the derivation of Eq. (24.5), if interstellar extinction is included AM D
ln 10 nM 103.m0002M 0002aC5/=5 5
Solutions for An Introduction to Modern Astrophysics or log10 AM
155
ln 10 3 D .m 0004 M 0004 a C 5/ C log10 nM : 5 5
If we consider the values at V D 15 and V D 11, and noting that a D 0 for V < 11 (i.e., in front of the cloud), we find the difference log10 AM .15/ 0004 log10 AM .11/ D
3 3 .15 0004 a/ 0004 .11/ D 3:29 0004 1:89 D 1:4 D 7=5: 5 5
Solving, we find a D 5=3. (b) If da=d ` D 10 mag kpc00021 , 0006` D 40 pc.
Figure S24.2: Results of Problem 24.8(a). 24.8
(a)
See Fig. S24.2.
(b) Beginning with the expression for sech .z=z0 / found immediately after Eq. (24.10), sech2 .z=z0 / D
4 : e 2z=z0 C e 00022z=z0 C 2
In the case where z 0005 z0 , sech2 .z=z0 / ' 4e 00022z=z0 . Substituting into Eq. (24.10) gives the expected result. 24.9
(a)
The number of hydrogen atoms is given by N D Mgas =mH ' 6 0003 1066 atoms. Assuming that each atom has a thermal energy given by E D 32 kT , the total thermal energy content of hydrogen gas in the Galaxy is Ethermal D 1:86 0003 1045 J. Dividing by the volume of the of the disk V ' 0003R2 h gives uthermal D 2 0003 10000215 J m00023 .
(b) Using B ' 0:4 nT, Eq. (11.9) gives um ' 6 0003 10000214 J m00023 . In this rough estimate, um ' 30uthermal, implying that magnetic field effects should play a significant role in the structure of the Milky Way. 24.10 ` D 0:944ı , b D 00040:0461ı. 24.11
(a)
For the NCP: ` D 123:932ı, b D 27:128ı.
156
Chapter 24
The Milky Way Galaxy
(b) For the vernal equinox: ` D 97:337ı, b D 000460:185ı .
24.12
(c)
For Deneb: ` D 85:285ı, b D 1:9976.
(a)
For M13, z D d sin b D 4:6 kpc and for the Orion Nebula, z D d sin b D 0004149 pc.
(b) M13 belongs to the Galactic stellar halo and the Orion Nebula is a member of the thin disk. These conclusions are supported by their heights above the Galactic midplane and by their classifications as a globular cluster and a young star-forming region (a nebula), respectively. C B
Apex
A
D
E
Sun
Antapex
F
H G
Figure S24.3: The results for Problem 24.13. (a) The solid-line arrows represent the apparent motions of the stars, while (b) the dashed-line arrows represent the radial and transverse velocity components. 24.13
(a)
See Fig. S24.3.
(b) See Fig. S24.3. (c)
Stars near the apex (approaching) have the largest negative radial velocities and stars near the antapex (receding) have the largest positive radial velocities.
(d) Stars located at the apex have no measurable proper motions due to the solar motion with respect to the LSR. As we consider stars farther from the apex, the proper motions increase, and appear to move directly away from the apex. Stars 90ı from the apex have the largest proper motions due to the solar motion relative to the LSR, and stars near the antapex appear to be converging at the antapex. 24.14 12 km s00021 24.15
(a)
p From Eq. (2.17), vesc D 2GM=R0 . Using vesc D 300 km s00021 and R0 D 8:0 kpc gives M D 8:3 0003 1010 Mˇ . This result agrees well with the value of 8:8 0003 1010 Mˇ found in Example 24.3.1.
(b) Using a value of vesc D 500 km s00021 leads to a mass of M D 2:3 0003 1011 Mˇ . The dark matter halo could account for the extra mass. (c)
Since much of the mass in the Galaxy is located at r > R0 and appears to be approximately spherically distributed, dynamic motions in the solar neighborhood do not reflect the existence of that matter.
24.16 Begin with Eqs. (24.37) and (24.38), respectively: vr D . 0004 0 /R0 sin ` v t D . 0004 0 /R0 cos ` 0004 d:
Solutions for An Introduction to Modern Astrophysics
157
Substituting the first-order Taylor expansion, ˇ d ˇˇ 0004 0 ' .R 0004 R0 / dR ˇR0 and using the approximation ' 0 : gives ˇ d ˇˇ .R 0004 R0 / R0 sin ` dR ˇR0 ˇ d ˇˇ .R 0004 R0 / R0 cos ` 0004 0 d: vt D dR ˇR0
vr D
But D =R. Differentiating,
ˇ ˇ d ˇˇ 1 d ˇˇ 0 D 0004 : dR ˇR0 R0 dR ˇR0 R20
Substituting leads to '
# ˇ d ˇˇ 0 0004 .R 0004 R0 / sin ` vt D dR ˇR0 R0 ' # ˇ d ˇˇ 0 0004 .R 0004 R0 / cos ` 0004 0 d: vr D dR ˇR0 R0 Using the geometry of Fig. 24.22, R0 D d cos ` C R cos ˇ ' d cos ` C R implies that R 0004 R0 D 0004d cos `. We now have ' # ˇ d ˇˇ 0 0004 d sin ` cos ` vr D 0004 dR ˇR0 R0 ' # ˇ d ˇˇ 0 0004 d cos2 ` 0 d: vt D 0004 dR ˇR0 R0 Employing the trigonometric relation, sin ` cos ` D
1 2
sin 2`, leads us to Eq. (24.41),
vr D Ad sin 2`; where 1 A00060004 2
'
# ˇ 0 d ˇˇ 0004 dR ˇR0 R0
is Eq. (24.39). Invoking another trigonometric relation, cos2 ` D using 0 D 0 =R0 , gives
1 2
.1 C cos 2`/, substituting the definition of A, and
000b0 d R0 0003 0002 0 d: D Ad cos 2` C A 0004 R0
v t D Ad .1 C cos 2`/ 0004
158
Chapter 24 But
The Milky Way Galaxy
# ˇ 0 0 d ˇˇ 0004 0004 ˇ dR R0 R0 R0 # ' ˇ 1 d ˇˇ 0 D0004 C 2 dR ˇR0 R0
000b0 1 A0004 D0004 R0 2
'
0006 B; which is Eq. (24.40). Substituting, we finally arrive at Eq. (24.42), v t D Ad cos 2` C Bd: 24.17 Since from Earth we measure 0006v t , the maximum values occur at ` D 0ı and 180ı, and minima occur at 90ı and 270ı. 24.18
(a)
From Kepler’s third law (Eq. 2.37), PD
20003 ŒG .m1 C m2 /00041=2
R3=2 :
However, P D 20003R=000b. Substituting and rearranging,
000bD
G .m1 C m2 / R
000b1=2 :
(b) Differentiating our expression for gives d 1 1 ŒG .m1 C m2 /00041=2 D0004 D0004 : dR 2 2R R3=2 Substituting into Eqs. (24.39) and (24.40) leads to AD
3 0 4 R0
BD0004 (c)
1 0 : 4 R0
Using the Keplerian expressions, we find A D 20:6 km s00021 kpc00021 and B D 00046:88 km s00021 kpc00021 .
(d) The values do not agree because they assume that all of the Galaxy’s mass resides interior to R0 . 24.19
(a)
Combining Eqs. (24.39) and (24.40), ˇ d ˇˇ D 0004.A C B/ D 00042:4 km s00021 kpc00021 : dR ˇR0 The orbital velocity is decreasing with increasing R.
(b) If A D 0004B D 13 km s00021 kpc00021 , then
ˇ d ˇˇ D 0: dR ˇR0
This implies that at, least within the solar neighborhood, the rotation curve is flat.
Solutions for An Introduction to Modern Astrophysics 24.20
(a)
159
Equating the expression for centripetal acceleration with the gravitational acceleration, 2 GMr ; D R R2 or r GMr D : R Assuming a constant mass density, Mr D 43 00030002R3 and p D 40003 G0002R / R;
which is rigid-body rotation. (b) The model for the distribution of dark matter described in the text is given by Eq. (24.51), 0002.r / D
00020 : 1 C .r=a/2
When r 0007 a, the density distribution approaches 0002.r / ' 00020 D constant. As was shown in part (a), a constant density distribution implies rigid-body rotation. 24.21 In the solar neighborhood, r D R0 and V D 0 . Thus 0002 .R0 / D
000b02 40003 GR20
D 9:5 0003 10000222 kg m00023 D 0:014 Mˇ pc00023 D 1:6 0003 10000218 Mˇ AU00023 : The stellar mass density is approximately 0:05 Mˇ pc00023 in the solar neighborhood. 24.22
(a)
The amount of mass located between r D 0 and r is given by integrating Eq. (10.7), or Z r Mr D 40003 r 20002.r / dr: 0
Substituting the mass density of dark matter (Eq. 24.51), Z r h 0006 0007i r2 2 2 00021 r : r 0004 a tan Mr D 4000300020 a dr D 400030002 a 0 2 a 0 1 C .r=a/ (b) Using a D 2:8 kpc, Mr D 5:5 0003 1011 Mˇ , and r D 100 kpc, gives 00020 D 5:9 0003 107 Mˇ kpc00023 . If Mr D 1:3 0003 1012 Mˇ within r D 230 kpc, then we also find that 00020 D 5:9 0003 107 Mˇ kpc00023 . (c) Using the value of 00020 found in part (b), Mr D 2:6 0003 1011 Mˇ . From Table 24.1, the mass of the stellar halo is 3 0003 109 Mˇ . Apparently the mass of dark matter far exceeds the mass of the stars in the Galaxy. 24.23
(a) If r 0007 a, then 1 C r=a ' 1 and 0002NFW / r 00021. If r 0005 a, then 1 C r=a ' r=a and 0002NFW / r 00023. (b) For the spherically-symmetric distribution Z 1 Z 1 r2 M D 0002.r / 40003 r 2 dr D 4000300020 a3 dr: r .a C r /2 0 0 R1 For r 0005 a, the integral looks like 0 r 00021 dr , which is ln r j1 0 . Since ln r is unbounded, the mass is infinitely large.
160
Chapter 24
The Milky Way Galaxy
24.24 Referring to Table 24.1, the mass of the dark-matter halo interior to 50 kpc is about 5:4 0003 1011 Mˇ , while it is estimated to be 1:9 0003 1012 Mˇ interior to about 230 kpc. These values imply 0002.50/ D 1:0 0003 106 Mˇ kpc00023 0002.230/ D 3:7 0003 104 Mˇ kpc00023 : Using Eq. (24.52) and taking the ratio of the results gives .230=a/.1 C 230=a/2 27:7 D D .50=a/.1 C 50=a/2 This gives 2:45 D
0002
230 50
0003
.a C 230/2 : .a C 50/2
a C 230 ; a C 50
or a D 73:9 kpc. Solving for the density immediately gives 00020 D 1:9 0003 106 Mˇ kpc00023 . 24.25
(a)
We need to put together all of the pieces to the structure of the Galaxy from Table 24.1, as well as include the contribution of dark matter to the total mass of the Galaxy. To include the later, we can integrate the NFW distribution (Eq. 24.52) from r D 0 to some r D r0 : Z r0 MNFW .r0 / D 0002NFW .r / 40003 r 2 dr 0
Z
r0
D 4000300020 0
Z
r0
D 4000300020a3 0
D 4000300020a
r2 dr .r=a/.1 C r=a/2
3
r dr .a C r /2
a ln.a C r / C aCr
D 4000300020a3 ln.1 C r0 =a/ 0004
000br0 0
r0 =a : 1 C r0 =a
This expression can be evaluated for various values of r0 using the results obtained from Problem 24.24; a D 73:9 kpc and 00020 D 1:9 0003 106 Mˇ kpc00023 . For r0 D 25 kpc, MNFW D 3:7 0003 1011 Mˇ . The mass of the visible components of the Galaxy interior to 25 kpc is (from Table 24.1) is 7:900031010 Mˇ (the stellar halo was neglected for r < 25 kpc but represents only a small component to the total mass). The B-band luminosity for those components totals LB D 2:1 0003 1010 Lˇ . Thus, the mass-to-light ratio interior to r0 D 25 kpc is approximately M=LB D 21:4 Mˇ =Lˇ (b) For r0 D 100 kpc, MNFW D 2:7 0003 1012 Mˇ . This implies that the total mass interior to 100 kpc (including the stellar halo) is approximately 2:7800031012 Mˇ . The B-band luminosity is (again including the stellar halo) LB D 2:2 0003 1010 Mˇ , implying a mass-to-light ratio of 126 Mˇ =Lˇ . The presence of dark matter increases the mass-to-light ratio significantly throughout the universe. 24.26 Integrating the gravitational “Gauss’s law” gives 40003 r 2g D 000440003 GMin : Solving for g then gives g D 0004GMin =r 2 ; the expected result.
Solutions for An Introduction to Modern Astrophysics 24.27
(a)
161
Consider a cylinder centered on the midplane with its ends oriented parallel to the midplane. Each end has an area A and is located a distance z from the midplane. Then, integrating Eq. (24.56) gives 2Ag D 000440003 GMin D 000440003 G0002A.2z/: Solving for the gravitational acceleration gives g D 000440003 G0002z:
(S24.1)
(b) The gravitational acceleration is just g D d 2 z=dt 2 . Substituting the expression for g given in Eq. (S24.1) and rewriting gives d 2z C kz D 0; dt 2 where k 0006 40003 G0002: (c)
The solutions for position and velocity are of the form z.t/ D z0 cos.!t C ı/ w D zP D 0004!z0 sin.!t C ı/; p p where ! D k D 40003 G0002, and ı is a phase constant. 20003 D 6:8 0003 107 yr. ! Let t D 0 when the Sun passes through the midplane, heading north. Then ı D 00040003=2, and at the present time, zˇ D z0 sin !t and wˇ D !z0 cos !t, where zˇ D 30 pc (see Section 24.2) and wˇ D 7:2 km s00021 (Eq. 24.35). Combining the equations leads to 0002 0003 zˇ !t D tan00021 ! D 0:36 rad: wˇ
(d) Posc D (e)
Therefore, z0 D zˇ = sin !t D 85 pc. This rough estimate compares well with the scale height of the thin disk (350 pc) quoted in Table 24.1. (f) Since the Sun’s orbital period is Porbit D 2:2 0003 108 yr, the number of oscillations per orbital period is Porbit=Posc D 3:2. 24.28 Carrying out the unit conversions: 0006 0007 0006 3 00070006 0007 1 pc m hvr km s00021 i tan 10 16 1 km 3:08000310 m 00070006 0007 0006 00070006 00070006 dD 1 yr 10 1ı 20002 rad 00 h i 6000 0 ı 7 60 360 3:16000310 s D
hvr i tan : 4:74h 00i
24.29 For Altair, d D 5:14 pc, 00 D 0:6609200 yr00021 and vr D 000426:3 km s00021 . 0002 0003 00 00021 4:74d h i (a) D tan D 000431:5ı. hvr i q (b) v t D vr tan D 16:1 km s00021 and v D vr2 C v 2t D 30:8 km s00021 .
162
Chapter 24
The Milky Way Galaxy
24.30 Using Newtonian gravity for a black hole of mass Mbh D 3:7 0003 106 Mˇ and Schwarzschild radius RS D 16 Rˇ (see Eq. 17.27), 0003 0002 1 1 D 9 0003 1053 J: 0004 0006U D 0004GMbh m rf RS (This crude estimate is approximately two orders of magnitude larger than the value quoted in Section 24.2.) The typical kinetic energy of a Type II supernova is 1044 J, far below the estimate obtained here. 24.31 0006M D MP 0006t D 5 0003 106 Mˇ , which is consistent with the estimate of a supermassive black hole. 24.32
(a)
Assuming a maximum possible radius of Sgr A? equal to the closest approach of S2 (1:8 0003 1013 m, the density would be 3 0003 1000024 kg m00023 .
(b) Assuming a maximum possible radius of 1 AU, the density would need to be 530 kg m00023 , 8:8 0003 105 Mˇ AU00023 , or 1 0003 10000210 Mˇ pc00023 . 24.33 From Eq. (2.33) for Newtonian mechanics, vp D 7:1 0003 106 m s00021 D 0:024c. 24.34 According to Eq. (17.27), the Schwarzschild radius of a Mbh D 3:7 0003 106 Mˇ black hole is RS D 2GMbh =c 2 D 16 Rˇ , implying that the average density of this supermassive black hole is 0002bh D
3:7 0003 106 Mˇ 4 0003 3
3
.16 Rˇ /
D 1:35 0003 106 kg m00023 :
The average density of the Sun is 0002Sun D
1 Mˇ 4 0003 3
.1 Rˇ /3
D 1400 kg m00023 :
Thus, from Eq. (19.4) the Roche limit is, 0002 0003 0002bh 1=3 r D 2:456 RS D 24:2 RS D 380 Rˇ : 0002Sun 24.35 Assuming 000e D 0:02 m2 kg00021 , the Eddington luminosity (Eq. 10.114) gives 9 0003 1037 W D 2 0003 1011 Lˇ . However, observations put an upper limit on the luminosity of 3 0003 104 Lˇ . The Eddington luminosity exceeds the upper limit by 8,000,000 times. 24.36
(a)
The isothermal component has a density of 0002i .r / D C 0 =r 2, where C 0 is a constant. The interior mass due to that component is Z r
Mr;i D
40003 r 20002i .r / dr D 40003 C 0 r:
0
Adding a single point mass (M0 ) at the center gives Mr D k r C M0 ; where k 0006 40003 C 0 . (b) For Newtonian gravity and perfectly circular motion, mv 2 GMr m ; D r r2 or,
GMr vD r
000b1=2
0002 0003 M0 1=2 D G kC : r
Solutions for An Introduction to Modern Astrophysics (c)
163
k D v 2 =G 0004 M0 =r D 5:2 0003 1019 kg m00021 .
(d) See Fig. S24.4. See Fig. S24.5. The central black hole begins to become important at approximately log10 r D 0:5, or r D 3 pc. 9.0
Log10 Mr (Mo)
8.5 8.0 7.5 7.0 6.5 −2
−1
0 1 Log10 r (pc)
2
3
Figure S24.4: The solution for Problem 24.36(d). 1400 1200 1000
v (km s −1)
(e)
800 600 400 200 0 −2
−1
0 1 Log10 r (pc)
2
Figure S24.5: The solution for Problem 24.36(e).
3
CHAPTER 25
The Nature of Galaxies 25.1 Taking d D 100 kpc, or r D 50 kpc, and D 0:0200 yr00021 implies v D r D 4730 km s00021 . For the Milky Way, vmax D 250 km s00021 . 25.2
(a)
R25 D 22:7 kpc.
(b) Vmax ' 181 km s00021 for an Sc galaxy with MB D 000421:51. (c)
D v=R D 2:6 0003 10000216 rad s00021 D 0:0016800 yr00021 .
(d) It would have taken 600 yr to rotate through 100 . Since maximum resolutions observable from the ground today are roughly 0:500 , it is unlikely that van Maanen could have detected the rate of rotation. 25.3
(a)
For the Sun, LB D 2:3 0003 1010 Lˇ . Since MV;Sun D 4:82 and B 0004 V D 0:65, we find MB;Sun D 5:47. Thus, from Eq. (3.8), MB D MB;Sun 0004 2:5 log10 .LB =Lˇ / D 000420:4.
(b) Using the relation for an Sb galaxy, Vmax D 184 km s00021 . Observationally, Vmax D 250 km s00021 . 25.4 Surface brightness is given by the amount of light contained in a cone of solid angle D A=40003 d 2, where A is the surface area of the galaxy contained within the cone. For a given value of , A increases as d 2 , implying that the light-producing area increases as d 2 . Therefore, the amount of light produced by area A, divided by A, is a constant. 25.5 Begin with Eq. (24.13),
0002 log10
I Ie
'0002
0003 D 00043:3307
r re
#
00031=4
00041 :
Since the intensity is the luminosity per unit surface area of the galaxy, ID or
0002 log10
I Ie
0003
0002 D log10
0006L=0006A 0006Le =0006Ae
0006L ; 0006A
0003
0002 D log10
0006L 0006Le
0003
0002 0004 log10
0006A 0006Ae
0003 :
Assuming for simplicity, that 0006A D 0006Ae (the same unit surface area is being considered in each case), then 0002 0003 0002 0003 I 0006L log10 D log10 : Ie 0006Le From M D MSun 0004 2:5 log10 .L=Lˇ/ (Eq. 3.8), 0002 0003 0006L 0006Me 0004 0006M D log10 ; 0006Le 2:5 where 0006M and 0006Me are the absolute magnitudes of the small regions of the galaxy associated with 0006A and 0006Ae , respectively. Making the appropriate substitutions, # '0002 0003 0006Me 0004 0006M
e 0004
r 1=4 00041 ; D D 00043:3307 2:5 2:5 re 164
Solutions for An Introduction to Modern Astrophysics
165
where we have replaced 0006M by and 0006Me by e since (1) we have assumed the same associated surface area, and (2) the solid angle is proportional to the surface area (assuming the same distance to any appropriate sample). Rearrangement leads immediately to Eq. (25.2). 25.6 According to Eq. (24.10),
L.R; z/ D L0 e 0002R = hR sech2 .z=z0 /:
Considering the case where z D 0,
0002 0003 0002 0003 L.R/ R R D0004 log10 e D 00040:434 : log10 L0 hR hR Following the procedure outlined in the solution to Problem 25.5 leads directly to Eq. (25.3). 25.7
(a)
Eq. (24.13) leads immediately to I.r / D Ie 1000023:3307Œ.r=re /
(b)
0004 D I e 0002b Œ.r=re /1=4000210004 e
1=4 00021
where b D 3:3308 ln 10 D 7:67. Z
1
Ltot D
Z 20003 rI.r / dr D 20003Ie e
1
b
0
r e 0002b.r=re/
1=4
dr:
0
Letting x 0006 b.r=re /1=4 implies r D re .x=b/4 and dr D 4re x 3 =b 4 dx. Substituting, Z 1 eb b 2 8 Ltot D 80003Ie e re =b x 7 e 0002x dx D 8! 8 0003 re2 Ie D 7:220003 re2Ie : b 0 (c)
If the integration is carried out from r D 0 to r D re , then the limits of integration of the last integral of part (b) become x D 0 to x D b. Evaluating, Z
b
' 7 0002x
x e
dx D 7! e
0002x
0
7 X .00041/i iD0
x 70002i .7 0004 i /!.00041/iC1
#b D .0:5/ 7! D 0
1 2
Z
1
x 7 e 0002x dx:
0
Therefore L.re / D 12 Ltot. 25.8
(a)
From Eqs. (25.4), MB D 000421:8.
(b) According to Eq. (3.6), B 0004 MB D 5 log10 .d=10 pc/, or d D 6:45 Mpc. (c)
From Eq. (25.11), R25 D 26:8 kpc.
(d) M25 D 6:5 0003 1011 Mˇ from Eq. (25.10). (e)
From Problem 25.3, MB;Sun D 5:47. Then, according to Eq. (3.8), LB D 8:1 0003 1010 Lˇ .
(f) M25 =LB D 8:0. 25.9 For Sa, hB 0004 V i D 0:75, corresponding to spectral class G8; for Sb, hB 0004 V i D 0:64, which corresponds to spectral class G2; and for Sc, hB 0004 V i D 0:52, corresponding to F8. 25.10 The rotational velocity at 100 (3.7 pc) is about 50 km s00021 . From v2 GM D 2 r r we find M D 2:2 0003 106 Mˇ , well within the uncertainty range.
166 25.11
Chapter 25 The Nature of Galaxies (a)
The rotational velocity at 100 (3.7 pc) is roughly 200 km s00021 . This implies a mass of M D 3:5 0003 107 Mˇ . This value is roughly four times smaller than the value quoted in the text.
(b) The peak of the velocity dispersion is almost 400 km s00021 . Again using R D 3:7 pc we find a virial mass of Mvirial D 6:9 0003 108 Mˇ . (c)
Recall that M31 has a complex, multi-component core.
O and substituting x D R cos and y D R sin 25.12 Beginning with the Cartesian representation, r D x OiCy OjCz k, gives 0006 0007 O r D R cos Oi C sin Oj C z k: Identifying eO R D Oi cos C Oj sin and eO z D kO lead to Eq. (25.16). 25.13 Starting with Eq. (25.16), r D ROeR C zOez ; and differentiating gives
P eR C RePO R C zP eO z C z ePO z : rP D RO
However, since the unit vector in the z direction always points in the same direction (i.e., it is not time dependent), ePO z D 0. Thus, differentiating again P eR C ReRO R C zR eO z : R eR C 2RO rR D RO
(S25.1)
Now, differentiating the unit vector eO R D Oi cos C Oj sin , realizing that Oi and Oj are constants, and noting that eO0006 D 0004Oi sin C Oj cos , gives P ePO R D 0004Oi sin P C Oj cos P D eO 0006 : Taking the second time derivative, P so However, ePO 0006 D 0004OeR ,
R eRO R D ePO 0006 P C eO0006 : R eRO R D 0004OeR P 2 C eO 0006 :
(S25.2)
Substituting Eq. (S25.2) into Eq. (S25.1) and rearranging leads to Eq. (25.18). P D 000e0 AR cos 000e0 t. From Example 25.3.1, 25.14 According to Eq. (25.34) for the Sun, 0002.t/ D AR sin 000e0 r , so 0002.t/ 000e0 D 35:6 km s00021 kpc00021 , and from Eq. (24.33), 0002P D uˇ D 00049 km s00021 . Assuming that the Sun is currently passing through equilibrium in its radial (epicycle) oscillation, AR D 0002=000e P 0 D 250 pc. 25.15
(a)
Beginning with Eq. (25.22), ˚eff D ˚ C Jz2 =2R2 , where Jz is the constant angular momentum per unit mass. Differentiating, 0002 0003 @2 ˚eff @ @˚ 3Jz2 D : (S25.3) C @R2 @R @R R4 But @˚[email protected] D 2 =R, so that
@ @R
0002
@˚ @R
0003 D
2 @ 2 : 0004 R @R R2
Furthermore, Jz D R000b. Making the appropriate substitutions into Eq. (S25.3) and simplifying gives ' ˇ # @2˚eff 0 0 @000b0 ˇˇ 2 000e 0006 D2 C ; @R2 R0 R0 @R ˇR0 which is Eq. (25.37).
Solutions for An Introduction to Modern Astrophysics
167
(b) By combining Eqs. (24.39) and (24.40), ˇ @ ˇˇ D 0004.A C B/ @R ˇR0 0 D A 0004 B: R0 Substituting into Eq. (25.37), we arrive at Eq. (25.38): 000e02 D 2.A 0004 B/ Œ.A 0004 B/ 0004 .A C B/0004 D 00044B.A 0004 B/: 25.16 For a Keplerian orbit, 2=R D GM=R2 , 0002 D
GM R
00031=2 :
Differentiating, @ 1 .GM /1=2 : D0004 @R 2 R3=2 Substituting into Eq. (25.37) leads to 000e0 D
.GM /1=2 R3=2 0
:
3=2
However, we also have that 0 D 0=R0 D .GM /1=2 =R0 . Thus, 000fmax =0002max D 2000b0=000e0 D 2. 25.17 According to Eq. (24.13),
log10
'0002 0003 # I.r / r 1=4 D 00043:3307 00041 ; Ie re
implying I.r / D Ie 10
h i 00023:3307 .r=re/1=4 00021
:
This can be rewritten in terms of a power of e by realizing that if 10x D e y , then y D ln 10x D x ln 10 D 2:3026x. Therefore, h i I.r / D Ie e
0002˛ .r=re /1=4 00021
;
where ˛ D .2:3026/.3:3307/ D 7:6692. Now, the integral average of the intensity is given by Z re h i 1 0002˛ .r=re /1=4 00021 Ie e 20003 r dr: hI i D 0003 re2 0 The integral can be evaluated by recalling that integrating the intensity over the surface area between r D 0 and r D re (the effective radius) yields one-half of the entire luminosity of the galaxy, or Z re Z 1 1 I.r / 20003 r dr D I.r / 20003 r dr: Le D 2 0 0 Substituting into our expression for the average intensity, Z 1 Z h i Ie e ˛ 1 0002˛.r=re /1=4 Ie 0002˛ .r=re/1=4 00021 e 20003 r dr D 2 e r dr: hI i D 20003 re2 0 re 0
168
Chapter 25 The Nature of Galaxies Making the substitution, u 0006 ˛ .r=re /1=4 within the integral implies that r=re D u4 =˛ 4 and r dr D 4re2u7 ˛ 00028 du. Therefore
Z 4Ie e ˛ 1 7 0002u 4Ie e ˛ .4/.7!/e ˛ hI i D Ie D 3:607Ie : u e du D 0010 .8/ D ˛8 ˛8 ˛8 0
25.18 Equation (25.2) states that
'0002
.r / D e C 0:3268
r re
#
00031=4
00041 :
Setting .rH / D H D 26:5 B-mag arcsec00022 , and solving for rH gives rH D re
0006
00074 0004 e C1 8:3268 H
0002 D re
26:5
e 0004 C1 8:3268 8:3268
00034
D re .4:18 0004 0:12 e /4 : 25.19
(a)
From Problem 25.17, hI i D 3:607Ie , implying that log10 hI i D log10 3:607 C log10 Ie . Following the procedure described in Problems 25.5 and 25.6, we arrive at h i D e 0004 2:5 log10 3:607 D e 0004 1:393.
(b) e D h i C 1:393 D 21:52 C 1:393 D 22:913 B-mag arcsec00022 . (c)
rH D re .4:18 0004 0:12 e /4 D 42:1 kpc.
25.20 The virial theorem implies that 0007r4 / M 2 =R2 [the square of Eq. (25.13)]. If the mass-to-light ratio is roughly 2 the same for all elliptical galaxies, then M=L D KML is a constant. Thus, 0007r4 / KML L2 =R2 . But, assuming that the average surface brightness is also essentially constant for all elliptical galaxies, then L=R2 / I is constant. Therefore, 0007r4 / L. 25.21
(a)
m D 00040:115.
(b) The slope of Eq. (25.41) is m D 00040:1. The slope of Fig. 25.33 is close to, but not exactly the same as the slope of Eq. (25.41). The difference is due in part to systematic variations in surface brightness. 25.22
(a)
From Eq. (25.15), SN D N t 100:4.MV C15/ D .350/ 100:4.000221:7C15/ D 0:73.
(b) SN D 18:6.
25.23
(c)
The relative abundance of globular clusters per luminosity interval in CDs far exceeds the values found in spirals. If CDs are produced via mergers with spirals then apparently globular clusters must be formed in the process.
(a)
Assuming that all terms beyond first-order are zero, Eq. (25.44) becomes a.0005/ D a0 C a2 cos.20005/: But, from Eq. (25.1), ˇ=˛ D 1 0004 0011, where ˛ D a .0ı / D a0 C a2 and ˇ D a .90ı/ D a0 0004 a2 . Substituting, a0 0004 a2 D .a0 C a2 /.1 0004 0011/: Solving for a2 gives a2 D
0006 0011 0007 a0 : 200040011
Solutions for An Introduction to Modern Astrophysics
169
(b) For an E4 galaxy with a0 D 30 kpcs and 0011 D 0:4, a2 D 7:5 kpc. The polar-coordinate plot is shown in Fig. S25.1. (c)
See Fig. S25.1.
(d) See Fig. S25.1. (e)
a4 D C0:1a0 looks more like a lenticular galaxy than does a4 D 00040:1a0 . a4 = 0 a4 = 0.1 a0 a4 = −0.1a0
Figure S25.1: The results of Problem 25.23. 25.24 See Fig. S25.2. The results look qualitatively similar to Eq. (25.46). 4
Log10 φ(M)
3
2
1
Local Field Virgo Cluster
0 −23 −22 −21 −20 −19 −18 −17 −16 −15 −14 −13 −12 MB
Figure S25.2: The results of Problem 25.24.
CHAPTER 26
Galactic Evolution 26.1
(a)
From Appendix G, the mass and radius of a main-sequence M0 star are M D 0:51 Mˇ and R D 0:63 Rˇ , respectively. If the mass density of stars near the Sun is 0002 D 0:05 Mˇ pc00023 , then the number density of stars may be estimated as n D 0002=M D 0:098 pc00023 . The average volume of Galactic space per star is Vspace D 1=n, and the volume of a single M0 star is Vstar D
4 0003R3 D 3:5 0003 1026 m3 D 1:2 0003 10000223 pc3 : 3
The fraction of Galactic space occupied by stars is f D
Vstar D nVstar D 1:2 0003 10000224 : Vspace
(b) If an intruder M0 star travels through the Galactic stars, the mean free path between collisions is given by Eq. (9.12), ` D 1=n0007, where the collision cross section is 0007 D 0003.2R/2 D 2:4 0003 1018 m2 D 2:5 0003 10000215 pc2 : The mean free path is therefore ` D 1=n0007 D 4:0 0003 1015 pc. If the thickness of the Galactic disk is about z 1 kpc, then the probability of a stellar collision with the intruder star during its perpendicular passage through the disk is only z=` 2:5 0003 10000213 . 26.2 Assume that the force of dynamical friction is given by a expression of the form fd ' C.GM /˛ .vM /ˇ 0002 0004 ; where C is dimensionless and the values of the exponents ˛, ˇ, and 0012 are to be determined. Let Œx0004 designate the units of the variable x, so [fd ] D kg m s00022 [GM ] D m3 s00022 [vM ] D m s00021 [0002] D kg m00023 : The units involved in the expression for fd are therefore kg m s00022 D .m3 s00022 /˛ .m s00021 /ˇ .kg cm00023 /0004 : which leads to 0012 D1 3˛ C ˇ 0004 30012 D 1 00042˛ 0004 ˇ D 00042: 170
Solutions for An Introduction to Modern Astrophysics
171
Solving for the exponents, we find ˛ D 2, ˇ D 00042, and 0012 D 1. Therefore, fd ' C
G2M 20002 ; 2 vM
which is Eq. (26.1). 26.3 From Eq. (26.3), the greatest distance from which a globular cluster could have spiraled into the Galactic nucleus is s tmax C GM : rmax D 20003 vM Using C D 76 for globular clusters, with tmax D 13 Gyr for the oldest globular clusters, M D 5 0003 106 Mˇ for the cluster’s mass, and vM D 220 km s00021 for the orbital speed of the local standard of rest (Eq. 24.36), we obtain rmax D 1:2 0003 1020 m D 4 kpc. This is similar to the size of the Milky Way’s central bulge (Table 24.1). 26.4
(a)
The time for the LMC to spiral into the Milky Way may be estimated from Eq. (26.2), tc D
20003 vM ri2 : C GM
With vM D 220 km s00021 , ri D 51 kpc, C D 23, and M D 2 0003 1010 Mˇ , this is tc D 5:4 0003 1016 s D 1:7 Gyr. (b) The time quoted in the text, some 14 billion years, is about eight times longer than our estimate here. 26.5 Treating the Milky Way and the LMC as a binary system with a separation of dLMC D 51 kpc, we can use Eq. (18.10) to estimate the tidal radius of the LMC due to the Milky Way. From Table 24.1, we take M1 D 5:4 0003 1011 Mˇ for the Milky Way within 50 kpc of the center (including its dark matter halo) and M2 D 2 0003 1010 Mˇ for the LMC to find
0002 0003 M2 `2 D dLMC 0:500 C 0:227 log10 D 8:9 kpc: M1 If the angular diameter of the LMC is 0005 D 4600 D 0:134 rad, then its linear radius is rLMC D dLMC 0005=2 D 3:3 kpc. Since rLMC < `2 , the LMC lies within its tidal radius. 26.6 In this problem, we treat the LMC and the SMC as a binary system. (a)
Applying the law of cosines to the triangle formed by the LMC, the SMC, and Earth, the present distance between the LMC and the SMC is found to be i1=2 h a0 D .51 kpc/2 C .60 kpc/2 0004 2.51 kpc/.60 kpc/ cos 21ı D 22:3 kpc:
(b) The present tidal radii of the LMC and the SMC due to the other may be estimated from Eqs. (18.9) and (18.10). We find
0002 0003 MSMC `ı;LMC D a0 0:500 0004 0:227 log10 D 16:2 kpc MLMC
0002 0003 MSMC `ı;SMC D a0 0:500 C 0:227 log10 D 6:1 kpc; MLMC using MSMC =MLMC D 0:1.
172
Chapter 26 Galactic Evolution (c)
The distance between the LMC and the SMC varies with time as a.t/ D a0 Cvt, where v D 110 km s00021 . The tidal radius of the LMC as a function of time may therefore be estimated as 0002 0003
MSMC `LMC .t/ D .a0 C vt/ 0:500 0004 0:227 log10 : MLMC Setting `LMC .t/ D rLMC D 3:3 kpc (from Problem 26.5) shows that the LMC last extended beyond its tidal radius when tD
rLMC 0004 `0;LMC D 00045:0 0003 1015 s D 00041:6 0003 108 yr: v Œ0:500 0004 0:227 log10 .MSMC =MLMC/0004
At a distance of dSMC D 60 kpc, the angular diameter of the SMC is 0005 D 1500 D 0:0436 rad, so its linear radius is rSMC D dSMC 0005=2 D 1:3 kpc. The tidal radius of the SMC as a function of time may be estimated as 0002 0003
MSMC `SMC .t/ D .a0 C vt/ 0:500 C 0:227 log10 : MLMC Setting `SMC .t/ D rSMC D 1:3 kpc, we find that the SMC last extended beyond its tidal radius when tD
rSMC 0004 `0;SMC D 00044:9 0003 1016 s D 00041:6 0003 108 yr: v Œ0:500 C 0:227 log10 .MSMC =MLMC /0004
According to these results, the Magellanic Stream was formed before about 160 million years ago by the LMC and the SMC. 26.7
(a)
The semimajor axis of the gaseous ring orbiting the central galaxies of the M96 group is a D 101 kpc. The motion of a parcel of gas obeys Kepler’s third law (Eq. 2.37). With P D 4:10003109 yr, m1 D Mgalaxies and m2 D mgas 0007 Mgalaxies, the combined mass of the central galaxies M105 and NGC 3394 is Mgalaxies '
40003 2a3 D 1:07 0003 1042 kg D 5:38 0003 1011 Mˇ : P 2G
(b) Mgalaxies=L D 23. 26.8 Making the assumption that r 0005 a, Eq. (24.51) becomes 0002.r / '
C0 : r2
For a spherically symmetric mass distribution, only Mr (the mass interior to r ) will affect a test particle’s motion, where Z r Mr D 0002.r / 40003 r 2 dr D 40003 C0 r: 0
Thus, the radial equation of motion becomes d 2r GMr 40003 GC0 D0004 2 D0004 : dt 2 r r Multiplying through by v D dr=dt, and integrating, Z t Z r 40003 GC0 dv v dt D 0004 dr; dt r 0 r0 which gives v 2 D 80003 GC0 ln
0006r 0007 0
r
:
Solutions for An Introduction to Modern Astrophysics
173
Taking the square root and choosing the minus sign since the nebula is collapsing, h 0006 r 0007i1=2 dr 0 : D 0004 .80003 GC0 /1=2 ln dt r Integrating again,
Z
0 r0
dr r0 1=2 D 0004 ln r
Z
tff
.80003 GC0 /1=2 dt:
0
Setting u 0006 r=r0 , and solving for tff gives tff D
Z
1 1=2
.80003 GC0 /
1 0
r0 du r0 : h 0006 0007i1=2 D .8GC0 /1=2 ln 1u
The free-fall time is proportional to r0 . 26.9 There was an error in the first printing of An Introduction to Modern Astrophysics. The value of x should have been x D 1:8. (a)
Integrating Eq. (26.5), Z N0007M D
M2
CM 0002.1Cx/ dM D
M1
C 0002x M1 0004 M20002x : x
Thus, the ratio of the number of stars formed with masses between 2–3 Mˇ and 10–11 Mˇ is N0007M j200023 200021:8 0004 300021:8 D 00021:8 D 59:5: N0007M j10000211 10 0004 1100021:8 (b) Writing Eq. (24.12) in solar units, L D M ˛ . Solving for M and differentiating, dM 1 D L.10002˛/=˛ : dL ˛ Applying the chain rule, i 1 h dN dN dM D D CM 0002.1Cx/ L.10002˛/=˛ : dL dM dL ˛ Replacing M using the mass–luminosity relation leads to dN C D L0002.xC˛/=˛ : dL ˛ (c)
Following a procedure similar to part (a), N0007L D 0004 Therefore,
0007 C 0006 0002x=˛ 0002x=˛ L2 : 0004 L1 x
N0007L j200023 200021:8=4 0004 300021:8=4 D 00021:8=4 D 8:20: N0007L j10000211 10 0004 1100021:8=4
(d) More massive stars are much brighter.
174
Chapter 26 Galactic Evolution
26.10 The star formation rate is roughly 5:0 Mˇ pc00022 Gyr00021 . Assuming that most of the dust and gas within the thin disk is located inside the solar circle, the total mass of stars formed in one year would be 0007 0007 0006 0006 M D 5 Mˇ pc00022 Gyr00021 .0003 / .8000 pc/2 1000029 Gyr D 1 Mˇ : Assuming that the average mass of each star formed is approximately 0:5 Mˇ , then roughly two stars are formed per year in the Milky Way Galaxy. 26.11
(a)
From the data in Example 26.2.2, Tvirial D 6 0003 105 K, D 0:6, and n D 5 0003 104 m00023 , implying 0002 D mH n D 5 0003 10000223 kg m00023 . Then from Eq. (12.14), 0002 MJ '
5kT G mH
00033=2 0002
3 4000300020
00031=2 D 5:3 0003 1011 Mˇ :
(b) Assuming that Tvirial ' 104 K and all other values are as given in part (a), then MJ D 1:1 0003 109 Mˇ . (c)
Using T ' 6 0003 105 K, Eq. (12.16) gives 0002 RJ D
15kT 40003 G mH 00020
00031=2 D 55 kpc:
This value is comparable to the radius of the stellar halo (R D 50 kpc). 26.12
(a)
The cooling time scale is given by Eq. (26.8), tcool D
3 kTvirial ; 2 n0013
and the free-fall time scale is given by Eq. (12.26), 0002 tff D
30003 1 32 G00020
00031=2 :
Equating the two time scales, replacing Tvirial with the expression for the virial temperature given by Eq. (26.7),
mH 0007 2 Tvirial D ; 3k where (see Eq. 26.6) 0002 0003 3 GM 1=2 0007D ; 5 R and noting that 00020 D 4M=30003R3 (assuming spherical symmetry) and n D 0002= mH , leads to the desired result. (b) Assuming D 0:6 gives M D 7 0003 1012 Mˇ . 26.13
(a)
Taking M D 3 0003 1013 Mˇ and R D 300 kpc, Kepler’s third law (Eq. 2.37) implies an orbital period of PD
40003R3=2 D 1:8 0003 1017 s D 5:6 Gyr: G 1=2 M 1=2
This is roughly one-third the age of the Milky Way Galaxy. (b) M87 is not in virial equilibrium. Objects near the outer edges of the galaxy take too long to orbit relative to the age of the universe.
Solutions for An Introduction to Modern Astrophysics 26.14
(a)
175
v D 20003R=P D 20003.300 kpc/=5:6 Gyr D 330 km s00021 .
(b) If v ' 0007, then from Eq. (26.7), Tvirial D
mH 0007 2 D 2:6 0003 106 K; 3k
where D 0:6 has been assumed. (c)
The number density of particles is roughly nD
M 0002 0006 0007 D 1:8 0003 104 m00023 : D
mH
mH 43 0003R3
Substituting this value into the expression for the cooling time scale (Eq. 26.8) gives tcool D
3 kTvirial D 9:5 0003 108 yr: 2 n0013
(d) Using 00020 D mH n D 1:8 0003 10000223 kg m00023 , the free-fall time scale is (see Eq. 12.26) 0002 tff D
30003 1 32 G00020
00031=2 D 5 0003 108 yr:
This is approximately one-half of the cooling time scale. 26.15 Using T ' 104 K, m ' mH , and 00020 ' 0:15 Mˇ pc00023 gives h.T / D 0:22 kpc. This corresponds to the neutral gas component of the Milky Way (see Table 24.1). 26.16 The current rate of star formation is SFR D 5 Mˇ pc00022 Gyr00021 , and the mass of neutral gas in the Galaxy is 5 0003 109 Mˇ (see Table 24.1). If we assume uniform star formation within the Galactic disk interior to the solar circle (R0 D 8 kpc), then the time remaining for star formation is t'
Mgas D 5 Gyr: .SFR/ 0003R20
This would be prolonged by capturing other small galaxies, as well as adding new material to the ISM via stellar mass loss. 26.17
(a)
Choosing the upper limits for dwarf elliptical galaxies listed in Table 25.4, M 0002 109 Mˇ and R 0002 5 kpc. Then Eq. (2.17) gives vesc 0002 42 km s00021 .
(b) From the information given in Section 15.3, vSN ' 0:1c D 3 0003 104 km s00021 . Thus, vSN 0005 vesc . p (c) The thermal velocity is vthermal D 3kT =mH D 160 km s00021 , and so vthermal > vesc . (d) These galaxies lost their gas early in their histories and have ceased forming new stars. Hence they are now primarily composed of ancient stars. 26.18
(a)
At about 210 million years (step 175), a spiral structure begins to develop, becoming clear by 240 million years (step 200). The appearance of the target and intruder galaxies begins to resemble the Whirlpool galaxy and its companion as an apparent bridge of stars connecting the two galaxies develops sometime around 270 million years (step 225). It reaches a “best match” between approximately 300 million years (step 250) and 330 million years (step 275). The apparent bridge begins to fade around 360 million years (step 300), becomes disrupted by about 420 million years (step 350), and is nearly gone after about 450 million years (step 375).
176
Chapter 26 Galactic Evolution (b) Near the end of the 648 million-year time period, the spiral arms begin to dissipate, and many of the stars have apparently escaped from the central nucleus of the target galaxy.
26.19
(c)
In this case, the intruder galaxy passes by in a direction opposite to the orbital motion of the stars in the target galaxy. Small, relatively tight spiral arms begin to develop around 240 million years (step 200), but they tend to wind back up again and are gone by 330 million years (step 275). The final result is a galaxy with a disrupted disk. No stars escape from the central nucleus of the target galaxy.
(a)
The initial speed of the intruder galaxy matches the orbital speed of the nearby stars in the outer ring. Stars behind the intruder gain energy as they are pulled forward and up out of the plane of the disk. They form the “bridge” of stars that appears to link the target galaxy with the intruder. Stars behind the intruder are pulled backward; they lose energy and drop into lower orbits with larger angular velocities. As a result, these stars overtake the stars in higher orbits to form a loop of stars. These loops comprise the spiral arm that develops opposite the “bridge.”
(b) In this case, the intruder galaxy passes by in a direction opposite to the orbital motion of the stars in the target galaxy. The intruder spends a relatively brief time near any single star, and succeeds only in pulling a “spike” of stars up out of the plane of the target galaxy. Orbital motion carries the spike away from the intruder galaxy, and differential rotation causes the development and then the winding up of a tight spiral feature. Less energy is transferred to the stars in their short encounter with the intruder, and no stars escape from the gravity of the target nucleus. 26.20
(a)
The intruder galaxy pulls rings of stars up and away from the nucleus of the target galaxy. At first these stellar rings contract, and then expand. Such a process could produce a ring galaxy such as the Cartwheel.
(b) As the intruder galaxy pulls inward on a star in the ring, the force of gravity does positive work on the star and increases its energy as the star approaches the target nucleus. The star’s orbit becomes more energetic, with a larger semimajor axis (c.f., Eq. 2.35), and it also becomes elongated (e ¤ 0). After reaching perigalacticon, the star’s new orbit carries it away from the nucleus, and the ring expands. If the star gains enough energy, it may no longer be bound to the target nucleus. [To see these effects, rerun the program with just two rings containing one star each, changing vz from 00041 to 00042. Follow the motion for at least 720 million years (600 steps).] (c)
A nearly head-on collision is usually required to produce a ring galaxy, with values of the initial xcoordinate ranging from 0 to 2. Although values of 3 or greater typically result in a significant disruption of the target galaxy, an elliptical ring may be produced by larger values, such as 7.5.
CHAPTER 27
The Structure of the Universe 27.1 The periods of classical Cepheids range from 1 – 50 days (Table 14.1). Setting Eqs. (14.1) and (27.1) for MhV i equal to each other, 00042:81 log10 Pd 0004 1:43 D 00043:53 log10 Pd 0004 2:13 C 2:13.B 0004 V /; and solving for .B 0004 V / gives the estimates B 0004 V D 0:338 log10 Pd C 0:329
27.2
(a)
D 0:329
for Pd D 1 d
D 0:903
for Pd D 50 d:
On Fig. 15.14, the major axis of the ring of SN 1987A measures a D 19 mm, and the minor axis measures b D 14 mm. Thus if 0005 is the angle between the plane of the ring and the plane of the sky, cos 0005 D b=a D 0:7368, so 0005 D 42:5ı.
(b) If D is the linear diameter of the ring, then light from the far side must travel an extra distance of D sin 0005 to reach Earth. It took light an extra 0006t D 340 days to make the trip, so D sin 0005 D c0006t, or DD (c)
c0006t D 1:30 0003 1018 cm D 0:422 pc: sin 0005
Since the long axis of the light ring has a length D and subtends an angle ˛ D 1:6600 D 8:05 0003 1000026 rad, the distance to SN 1987A is d D D=˛ D 52:4 kpc.
27.3 The average visual absolute magnitude of a galaxy’s three brightest red stars is MV D 00048:0. We can use Eq. (12.1) to find the distance to M101, d D 10.V 0002MV 0002aC5/=5 D 10Œ20:90002.00028:0/00020:3C5=5 pc D 5:2 Mpc: This is about 30% below the value of 7.5 Mpc obtained using classical Cepheids. 27.4 The distance modulus (Eq. 3.6) is m 0004 M D 5 log10 .d / 0004 5: Let 0006.m 0004 M / represent an error in the distance modulus. Then 0002 0003 0006d 0006d 0006.m 0004 M / D 5 log10 e D 2:17 : d d Solving for the fractional error in d we find 0006d 0006.m 0004 M / D : d 2:17 An error in m 0004 M of 0.4 results in an error in the distance estimate of 18.4%. To produce a 5% uncertainty in the distance, the error in m 0004 M would be 0.11. For an uncertainty of 50%, 0006.m 0004 M / D 1:1. 177
178
Chapter 27 The Structure of the Universe
27.5 From Eq. (27.5) with CFornax D 00041:264, CVirgo D 00041:237, CComa D 00041:967, we find dVirgo D 10CFornax0002CVirgo D 0:940 dFornax and
dComa D 10CFornax 0002CComa D 5:05: dFornax
27.6 From Hubble’s velocity–distance diagram, Fig. 27.7, the solid line extends from the origin and reaches 1000 km s00021 at a distance of 1.96 Mpc. The slope of the line is H0 D 510 km s00021 Mpc00021 . This is much larger than today’s WMAP value of H0 D 71 km s00021 Mpc00021 . In 1929, when Hubble published his first velocity–distance diagram, Trumpler’s demonstration of interstellar extinction was still a year in the future, and Baade’s distinction between classical Cepheids and W Virginis stars would not be made until 1952. Hubble calculated that his galaxies were closer than their actual distances, so his diagram showed that velocity increases more rapidly with distance than it actually does. 27.7 Use Eq. (27.16), 2GM v 0004 C r
0002
2
20003 GM P
00032=3 D 0;
with r D 770 kpc, v D 119 km s00021 , and P D tH C tc D tH C r=v. For h D 0:5, the Hubble time is tH D 6:18 0003 1017 s (Eq. 27.14). Numerically solving this cubic equation for M , we find that M D 6:62 0003 1042 kg D 3:33 0003 1012 Mˇ . Using the total B-band luminosity for the Milky Way and Andromeda given in the text of about L D 6:9 0003 1010 Lˇ , we find M=LB D 48 Mˇ /Lˇ . 27.8
(a)
From Kepler’s third law, Eq. (2.37), 0002 P D 2
20003 a v
00032 D
40003 2 a3 : G.MMW C Mgas /
Using a D 75 kpc and v D 244 km s00021 , we can solve for the mass of the Milky Way, MMW (assuming that the mass of the clump of gas is negligible compared with that of the Milky Way). We obtain MMW D
v2a D 2:06 0003 1042 kg D 1:04 0003 1012 Mˇ : G
Taking LB D 2:3 0003 1010 Lˇ from Table 24.1, we find M=LB D 45 Mˇ /Lˇ . (b) From conservation of energy and Eq. (2.14), 0007 0007 1 0006 2 Mm 1 0006 Mm D m vr2 C v 2t 0004 G ; m vr C v 2t 0004 G i f 2 ri 2 rf where M and m are the masses of the Milky Way and the clump of gas, respectively, and vr and v t are the radial and transverse components of the clump’s velocity. The initial radial velocity is zero and the transverse velocity does not change, so M D
vr2 : 2G.1=rf 0004 1=ri /
With ri D 100 kpc, rf D 50 kpc, and a final radial velocity of vr D 0004220 km s00021 , the mass of the Milky Way is estimated as M D 1:1 0003 1042 kg D 5:6 0003 1011 Mˇ . Using LB D 2:3 0003 1010 Lˇ for the Milky Way (Table 24.1), we find M=LB D 24 Mˇ /Lˇ .
Solutions for An Introduction to Modern Astrophysics
179
27.9 The distance to the Sculptor group is d D 1:8 Mpc, and the group subtends an angle of about 0005 D 20ı D 0:349 rad. The distances to the front and back of the group are dfront D d.1 0004 0005=2/ D 1:49 Mpc and dback D d.1 C 0005=2/ D 2:11 Mpc. For two identical stars with the same absolute magnitude, M , located at the front and back of the group, Eq. (3.6) for the distance modulus, m 0004 M D 5 log10 .d / 0004 5, shows that 0002 0003 dback D 0:755: mback 0004 mfront D 5 log10 dfront 27.10 Start with Eq. (10.6),
dP Mr 0002 D 0004G 2 ; dr r and use the ideal gas law (Eq. 10.11) to substitute for P . Assuming that is a constant, we have 0003 0002 d Mr 0002 0002kT D 0004G 2 dr mH r 0002 0003 d0002 k dT Mr 0002 T C0002 D 0004G 2
mH dr dr r 0002 0003 kT r r d0002 r dT Mr D 0004 C
mH G 0002 dr T dr 0002 0003 kT r d ln 0002 d ln T D0004 C :
mH G d ln r d ln r
The use of partial derivatives in the text’s Eq. (27.17) is unnecessary, except as a reminder that D constant in our derivation. 27.11 For a flat rotation curve, Eq. (24.49) shows that Mr D CM r , where CM is a constant, and similarly Eq. (24.50) shows that 0002 D C r 00022 , where C is a constant. (These expressions are assumed to apply to both dark matter and interstellar gas.) Let T D CT r ˛ , where CT and ˛ are constants. Equation (27.17) is then
kCT r ˛C1 d ln.C r 00022 / d ln.CT r ˛ / CM r D 0004 C
mH G d ln r d ln r D0004
kCT r ˛C1 .00042 C ˛/:
mH G
Thus we must have ˛ D 0, and so T D CT r ˛ D CT r 0 D CT for an isothermal gas. 27.12 There were errors in the first printing of An Introduction to Astrophysics in Eqs. (27.18) and (27.19). The powers of 10 in the coefficients should have been 10000252 and 10000240 , respectively. The error was also propagated into the equation for ne in Example 27.3.2, although the result of ne D 300 m00023 is correct. (a)
From Eq. (27.20) with Lx D 1:5 0003 1036 W and R D 1:5 Mpc for the Virgo cluster, the luminosity density is 3Lx Lvol D D 3:6 0003 10000233 W m00023 : 40003R3 Then, using Eq. (27.19) for the luminosity density with T D 7 0003 107 K, the electron number density is found to be 00031=2 0002 Lvol ne D D 55 m00023 : 1:42 0003 10000240 T 1=2 W m00023
180
Chapter 27 The Structure of the Universe If the Virgo cluster is filled with completely ionized hydrogen, then the mass of the hydrogen is 4 0003R3 mH ne D 3:8 0003 1043 kg D 1:9 0003 1013 Mˇ : 3
Mgas D
(b) Using M=L ' 3 Mˇ /Lˇ for the Milky Way’s luminous matter (Table 24.1) and LV D 1:200031012 Lˇ for the Virgo cluster’s visual luminosity, the amount of luminous matter in the Virgo cluster is approximately Mlum D .M=L/LV D 3:6 0003 1012 Mˇ , roughly 20% of the mass of the intracluster gas. (c)
Equation (10.17) states that the average kinetic energy per particle is 3kT =2, so the energy density, u, of the intracluster gas (including both protons and electrons) is approximately 0002 0003 3 u D 2ne kT D 1:6 0003 10000213 J m00023 : 2 The gas will lose its energy by emitting x-rays in a time of roughly tD
u D 4:4 0003 1019 s D 1:4 0003 1012 yr; Lvol
which exceeds the Hubble time of tH D 1:38 0003 1010 yr by two orders of magnitude. 27.13 The diameter of the Coma cluster is D D 6 Mpc. Setting the speed of a galaxy equal to the cluster’s radial velocity dispersion, so v D 977 km s00021 , shows that the galaxy would travel across the cluster in a time of t D D=v D 1:9 0003 1017 s D 6:0 0003 109 yr. This is about one-half of the Hubble time (tH D 13:8 Gyr). The Coma cluster must be gravitationally bound, because otherwise there has been sufficient time for its member galaxies to have escaped. 27.14 The radius of the Virgo cluster is R D 1:5 Mpc and its radial velocity dispersion is 0007r D 666 km s00021 . The virial mass (Eq. 25.13) of the Virgo cluster is therefore Mvirial 27.15 In Section 25.2, we used Eq. (2.44),
5R0007r2 D 1:5 0003 1045 kg D 7:7 0003 1014 Mˇ : G 0005 0004 1 d 2I 0004 2 hKi D hU i ; 2 dt 2
˝ ˛ with d 2 I=dt 2 D 0 to derive the virial mass,
Mvirial D
5R0007r2 : G
(S27.1)
˝ ˛ To see the effect of a non-zero d 2 I=dt 2 , repeat the derivation of the virial mass, keeping this term to arrive at 0005 0004 5R d 2 I 5R0007r2 : 0004 M D G 6GM dt 2 Multiplying by M results in a quadratic equation that has the solution 8 9 '0002 0005#1=2 = 00032 0004 1 < 5R0007r2 10R d 2 I 5R0007r2 M D 0004 C : ; 2: G G 3G dt 2 ˝ ˛ ˛ ˝ If d 2I=dt 2 D 0, we recover Mvirial . However, if d 2 I=dt 2 > 0, then M < Mvirial, and the virial mass ˛ ˝ 2 would be an overestimate. Similarly, if d I=dt 2 < 0, then M > Mvirial , and the virial mass would be an
Solutions for An Introduction to Modern Astrophysics
181
underestimate. Assume that the Coma cluster is in virial equilibrium before ˝ ˛ the motion of its galaxies is miraculously redirected radially outward, so 00042 hKi D hU i and d 2 I=dt 2 D 0. Immediately after the redirection there ˛ ˝ would be no change in the cluster’s kinetic or potential energy; d 2 I=dt 2 is still zero and M D Mvirial . However, this situation does not last, as seen by writing Eq. (2.44) as 0005 0004 1 d 2I D hKi C hEi : 2 dt 2 Here, hEi D hKi C hU i < 0 is the total mechanical energy and remains ˝constant. ˛ This shows that as time passes and hKi decreases due to the cluster’s gravitational attraction, d 2 I=dt 2 becomes negative. Observations made at this later time to calculate Mvirial via Eq. (S27.1) would therefore underestimate the cluster’s mass. On the other hand, if the motion of each of the Coma cluster’s galaxies was redirected radially inward, again there would be no initial change in the cluster’s kinetic or potential energy and so Mvirial would be in this case as time passes and hKi increases due to the cluster’s gravitational attraction, ˝ 2valid.2However, ˛ d I=dt becomes positive. Calculations of Mvirial made at this later time would therefore overestimate the cluster’s mass. 27.16 The Tully–Fisher relation, Eq. (27.5), with 2vr = sin i D 218 km s00021 for NGC 5535, shows that the galaxy’s absolute H -magnitude is 0002 0003 2vr C 3:61 D 000419:77: MH D 000410:0 log10 sin i Then the distance to NGC 5535 is, from Eq. (3.5) with H D 9:55, d D 10.H 0002MH C5/=5 D 7:31 Mpc: 27.17 The average absolute visual magnitude for the brightest galaxy in a cluster is MV D 000422:83 ˙ 0:61. Using V D 10:99 and Eq. (3.5) with MV D 000422:83, d D 10.V 0002MV C5/=5 D 58:1 Mpc: If MV D 000422:83 C 0:61 D 000422:22, then d D 10.V 0002MV C5/=5 D 43:9 Mpc: If MV D 000422:83 0004 0:61 D 000423:44, then d D 10.V 0002MV C5/=5 D 76:9 Mpc: 27.18 Use Eqs. (24.12) to convert ` D 309ı, b D 18ı into equatorial coordinates. We find ˛ D 200ı D 13h 20m and ı D 000444ı. This is in the constellation of Centaurus. 27.19
(a)
For the minimum 0006, dave D 15:8 Mpc; see Fig. S27.1.
(b) The weighted mean value of the distance to the Virgo cluster is dw D 15:8 Mpc. This is the same as was found in part (a). (c)
The minimum value of 0006 will occur when
# ' X 0002 di 0004 dave 00032 d0006 d D 0; D d.dave / d.dave/ ıi i
or when
X di 0004 dave 00042 ıi2 i
! D 0:
182
Chapter 27 The Structure of the Universe This can be solved for the value of dave that minimizes 0006, P .di =ıi2 / D dw : .dave/min D Pi 2 i .1=ıi / 140 120 100
Δ
80 60 40 20 0 15
Minimum Δ
16
17
18 19 dave (Mpc)
20
21
22
Figure S27.1: Results for Problem 27.19. The minimum value of 0006 occurs for dave D 15:8 Mpc.
CHAPTER 28
Active Galaxies 28.1 Using z D 0:00157 for Centaurus A, the distance to Cen A is, from Eq. (27.8), d D cz=H0 D 6:6 Mpc. Following the procedure of Example 28.1.1, integrating from 00141 D 107 Hz to 00142 D 3 0003 109 Hz, Z 2 0006 000700020:6 0014 Lradio D 40003 d 2.9:12 0003 10000224 W m00022 Hz00021 / d0014 1400 MHz
1 D 1:0 0003 1034 h00022 W D 2 0003 1034 W
for h D 0:71:
28.2 A frequency of 0014 D 1400 MHz corresponds to log10 0014 D 9:15. Referring to Fig. 28.14 for the spectrum of 3C 273, the slope of the curve at this frequency is approximately 0006.log10 0014F / D 0:85: 0006.log10 0014/ From Eq. (28.1), if F / 0014 0002˛ , then d log10 0014F
D 1 0004 ˛ D 0:85; d log10 0014 so the spectral index is ˛ D 0:15, somewhat smaller than the value of ˛ ' 0:24 given in Example 28.1.2. 28.3 See Fig. S28.1, which shows a spectrum that is much more sharply peaked than the very broad spectrum of the quasar 3C 273 seen in Fig. 28.14. 28.4
(a)
Choose two points from the left-hand side of Fig. 28.16: L1 200031039 W at z1 D 2:2 and L2 1038 W at z2 D 0:3. So L1 D L0.1 C z1 /˛ and L2 D L0 .1 C z2 /˛ : Solving for ˛, we estimate that ˛D
ln.L1 =L2/ D 3:3: lnŒ.1 C z1 /=.1 C z2 /0004
(b) L.z D 2/ L0 .1 C 2/3:3 D 38: D L.z D 0/ L0 .1 C 0/3:3 28.5 The Schwarzschild radius of an M D 108 Mˇ black hole is RS D 2GM=c 2 D 2:95 0003 1011 m (Eq. 17.27). The average density is M 0002D 4 D 1:85 g cm00023 ; 3 3 0003RS slightly greater than the Sun’s average density of 1410 kg m00023 given in Example 8.2.1. The “surface gravity” of the black hole is (Eq. 2.12) M g D G 2 D 1:52 0003 105 m s00021 ; RS 183
184
Chapter 28 Active Galaxies
Figure S28.1: Results for Problem 28.3, log10 0014B .T / for T D 7:3 0003 105 K. which is about 550 times greater than the Sun’s value of gˇ D G
Mˇ D 274 m s00021 : R2ˇ
28.6 Using Eqs. (17.27) and (18.23), the efficiency of the accretion luminosity is, from Eq. (28.6), 0015D
Ldisk GM MP =2R RS D D : 2 2 P P 4R Mc Mc
For the inner edge of a nonrotating black hole, R D 3RS and we estimate 0015 D 0:0833, while for the inner edge of a maximally rotating black hole, R D 0:5RS and 0015 D 0:5. 28.7
(a)
Setting GM 2 ; c and using Eq (17.27) for the Schwarzschild radius, for an M D 108 Mˇ black hole we find Lmax D I!max D MR2S !max D
!max D
c3 D 5:1 0003 1000024 s00021 : 4GM
(b) The Schwarzschild radius (Eq. 17.27) is RS D 2GM=c 2 D 2:95 0003 1011 m. Divide the wire into elements of length dr . The motional emf between the ends of the wire element located at a distance r from the axis of rotation is d E D Bv dr D B!r dr for the wire rotating perpendicular to the magnetic field. Integrating from r D 0 to r D RS , we find that the induced voltage is Z RS 1 ED B!max r dr D B!max R2S D 2:2 0003 1019 V 2 0 for B D 1 T.
Solutions for An Introduction to Modern Astrophysics (c)
185
P D E 2 =.30 / D 1:6 0003 1037 watts.
This is about 6% of the power generated by the Blandford–Znajek mechanism. 28.8 Define ne and nH as the electron and proton number densities in a cloud of ionized hydrogen gas, and let ˛qm ne nH be the number of recombinations per unit volume per second in those clouds. If the clouds of ionized hydrogen fill only a fraction, 0011, of the total volume of the narrow-line region, then ˛qm 0011ne nH is the number of recombinations per unit volume per second for the entire volume. We now model the narrow-line region as a sphere of radius rNLR and assume that the clouds are pure hydrogen, so ne D nH . Assuming equilibrium, the number of ionizing photons produced per second by the central source, N , is equal to the number of recombinations per second. We then have 0002 0003 2 4 3 ˛qm 0011ne D N; 0003r 3 NLR which yields Eq. (28.13) when solved for rNLR . 28.9 In Example 28.2.2, the number of photons emitted per second in the continuum and capable of ionizing hydrogen was found to be Z 2 0:029L 0:029L d0014 D ; N D 2 h0014 h0014H
H where 0014H D 3:29 0003 1015 Hz and 00142 D 1025 Hz. In the same manner, the total number of photons emitted per second in the continuum is Z 2 0:029L 0:029L d0014 D ; Nt D 2 h0014 h00141
1 where 00141 D 1010 Hz. So N=N t D 00141 =0014H D 3:04 0003 1000026 . 28.10 Defining ˇ D v=c and ˇapp D vapp =c, Eq. (28.15) shows that ˇ < 1 when ˇapp < sin C ˇapp cos q ˇapp < 1 0004 cos2 C ˇapp cos 2 ˇapp .1 0004 cos /2 < 1 0004 cos2 :
Divide by .1 0004 cos / (we are requiring that cos < 1) to get 2 ˇapp .1 0004 cos / < 1 C cos 2 00041 ˇapp 2 C1 ˇapp
< cos < 1;
which is Eq. (28.16). To find the value of for the minimum ˇ, use Eq. (28.15) to set @ˇ[email protected] D 0 (holding ˇapp constant) to obtain @ @
0002
ˇapp sin C ˇapp cos
0003 D0
0004ˇapp .cos min 0004 ˇapp sin min / D 0: .sin min C ˇapp cos min /2
186
Chapter 28 Active Galaxies This reduces to cot min D ˇapp , which is Eq. (28.18). Note, from sin2 min C cos2 min D 1, that s s 1 1 sin min D D 2 1 C ˇapp 1 C cot2 min and
s cos min D
cot2 min D 1 C cot2 min
s
2 ˇapp 2 1 C ˇapp
:
Inserting these into Eq. (28.15) shows that the minimum value of ˇ for the source is ˇapp
ˇmin D r
1 2 1Cˇapp
which quickly reduces to
s ˇmin D
r
C ˇapp
2 ˇapp
;
2 1Cˇapp
2 ˇapp 2 1 C ˇapp
;
which is Eq. (28.17). The minimum Lorentz factor is, from Eq. (4.20), q 1 1 1 2 D 0012min D q Dr D 1 C ˇapp ; 2 ˇ sin min 2 1 0004 ˇmin 1 0004 app2 1Cˇapp
which is Eq. (28.19). 28.11
(a)
Start by dividing each side of the relativistic velocity transformation, Eq. (4.40), by c to get vx 0 vx =c 0004 u=c : D c 1 0004 uvx =c 2 Referring to Fig. 4.2, let Earth be at the origin of frame S and the QSO be at the origin of frame S 0 moving with speed u=c D vQ =c D ˇQ relative to Earth. The ejecta has speed vx =c D vej =c D ˇej relative to Earth, and speed vx 0 =c D vej 0 =c D ˇej 0 relative to the QSO. (Note that vx 0 < 0 if the material is ejected by the QSO in the direction of Earth.) The velocity transformation is therefore ˇej 0 D
ˇej 0004 ˇQ : 1 0004 ˇej ˇQ
From Eq.(4.36) for the redshift parameter, s s 1 C ˇQ 1 C ˇej zQ C 1 D and zej C 1 D ; 1 0004 ˇQ 1 0004 ˇej or ˇQ D
.zQ C 1/2 0004 1 .zej C 1/2 0004 1 and ˇej D : 2 .zQ C 1/ C 1 .zej C 1/2 C 1
Inserting these into the velocity transformation results in ˇej 0 D
.zej C1/2 00021 .zej C1/2 C1
10004
h
.z C1/2 00021
0004 .z QC1/2 C1 i hQ i 2
.zej C1/2 00021 .zej C1/2 C1
.zQ C1/ 00021 .zQ C1/2 C1
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187
or ˇej 0 D Œ.zej C 1/2 0004 10004Œ.zQ C 1/2 C 10004 0004 Œ.zej C 1/2 C 10004Œ.zQ C 1/2 0004 10004 Œ.zej C 1/2 C 10004Œ.zQ C 1/2 C 10004 0004 Œ.zej C 1/2 0004 10004Œ.zQ C 1/2 0004 10004 or 0004ˇej 0 D
.zQ C 1/2 0004 .zej C 1/2 : .zQ C 1/2 C .zej C 1/2
On left-hand side we have ˇej 0 D vx 0 =c, the x 0 -component of the ejecta velocity relative to the quasar. Because vx 0 < 0, the speed of the ejecta relative to the QSO, is 0004ˇej 0 D 0004vx 0 =c, which is the desired result. (b) We use zQ C 1 D 1:158 for the quasar 3C 273 and, with ˇej D 00040:9842 for the approaching knot, s zej C 1 D
1 C ˇej D 0:08924 1 0004 ˇej
for the ejecta. The speed of the knot relative to the quasar is therefore .zQ C 1/2 0004 .zej C 1/2 v D 0:9882; D c .zQ C 1/2 C .zej C 1/2 000e p which corresponds to a Lorentz factor of 0012 D 1 1 0004 v 2 =c 2 D 6:53 (Eq. 4.20). 28.12 Using Eq. (4.31) with 0005 D 180ı and Eq.(4.20) for the Lorentz factor, we have 0006 u0007 0006trest .1 0004 u=c/ D 00120006trest 1 0004 : 0006tobs D p c 1 0004 u2 =c 2 But 10004
1 1 0004 u2 =c 2 1 u ! D D 2 c 1 C u=c .1 C u=c/0012 20012 2
as u=c ! 1. Therefore, when 0012 0005 1, 0006tobs
0006trest : 20012
28.13 With M D 1014 Mˇ and r D 104 pc, 0002 0003 2GM 00021 nD 10004 D 1:000958: r c2 28.14 From Eq. (28.20) with M D 1 Mˇ and r0 D 1 Rˇ ,
D
4GM D 8:49 0003 1000026 rad D 1:7500 : rı c 2
28.15 Referring to Fig. 28.35, let P be the point at the apex of angle , directly above L. Since SO D dS and SP D dS 0004 dL for the small angles involved, applying the law of sines to the triangle SPO gives sin.0005 0004 ˇ/ sin.0003 0004 / sin D D : dS 0004 dL dS dS
188
Chapter 28 Active Galaxies Using Eq. (28.20), rı D 0005dL , and sin x ' x for small x, 0005 0004ˇ
4GM 4GM D D D ; dS 0004 dL dS rı c 2 d S 0005dL c 2 dS or
4GM 0005 0004 ˇ0005 0004 c2 2
0002
dS 0004 dL dS dL
0003 D 0;
which is Eq. (28.21). Consider two solutions, 00051 and 00052 . They satisfy 0002 0003 4GM dS 0004 dL 000512 0004 ˇ00051 0004 D0 dS dL c2 0002 0003 4GM dS 0004 dL 000522 0004 ˇ00052 0004 D 0; dS dL c2 so subtracting leads to 000512 0004 000522 0004 ˇ.00051 0004 00052 / D 0; or 00051 C 00052 D ˇ, which is Eq. (28.22). Substituting for ˇ in the preceding quadratic equation for 00051 , we find 0002 0003 4GM dS 0004 dL 000512 0004 .00051 C 00052 /00051 0004 D 0; c2 dS dL which yields Eq. (28.23) when solved for M . 28.16 The images are located at 00051 D 00040:800 D 00043:88 0003 1000026 rad and 00052 D .2:22 0004 0:8/00 D 6:88 0003 1000026 rad. (Which angle assumes the minus sign is arbitrary.) The distance to the source (Q01420004100 with zS D 2:727) is, from Eq. (27.7), c .zS C 1/2 0004 1 dS ' D 2600h00021 Mpc; H0 .zS C 1/2 C 1 and similarly, the distance to the lensing galaxy (zL D 0:493) is dL ' 1140h00021 Mpc. Equation (28.23) then gives the mass of the lensing galaxy as approximately 0003 0002 dS dL 00051 00052 c 2 D 5:6 0003 1041 h00021 kg D 2:8 0003 1011 h00021 Mˇ : M D0004 4G dS 0004 dL 28.17 The distance to the source (zS D 1:74) is, from Eq. (27.7), dS '
c .zS C 1/2 0004 1 D 2290h00021 Mpc; H0 .zS C 1/2 C 1
and similarly, the distance to the lensing galaxy (zL D 0:25) is dL ' 660h00021 Mpc. From Eq. (28.24) with 0005E D 2:100 =2 D 5:1 0003 1000026 rad, the mass of the lensing galaxy is about 0003 2 2 0002 0005E c dS dL M D D 2:5 0003 1041h00021 kg D 1:3 0003 1011 h00021 Mˇ : 4G dS 0004 dL 28.18 The mass of the MACHO is assumed to be, from Appendix C, M D 1:90 0003 1028 kg (ten Jupiter masses). The distance to the source star in the LMC is dS D 52 kpc, and the distance to the lensing MACHO is assumed to be dL D 25 kpc. Then, from Eq. (28.24), s 0002 0003 4GM dS 0004 dL 0005E D D 1:91 0003 10000210 rad: c2 dS dL
Solutions for An Introduction to Modern Astrophysics
189
Moving with a speed of v D 220 km s00021 , the MACHO will travel a distance of 20005E dL in a time of t D 20005E dL =v D 1:34 0003 106 s D 15:5 d. Comparing this time with the time shown in Fig. 24.15, we see that the temporary brightening of a lensed star in the LMC lasted for roughly 20 days, in good agreement with our crude estimate. 28.19
(a)
If we set rR D RS , 0002BH D MBH =.40003R3S =3/, and RS D 2GMBH =c 2 (the expression for the Schwarzschild radius), the mass of the black hole is found to be
MBH D
2:43c 6 .320003=3/G 3 0002?
000b1=2
00021=2
D 1:6 0003 10100002 ?
Mˇ :
(b) If the Sun falls in, 0002 ? D 1410 kg m00023 . This implies a mass required for the black hole of MBH D 4:2 0003 108 Mˇ .
28.20
(c)
For a more massive black hole, the star would not be disrupted prior to falling through the event horizon. As a result, the liberation of the gravitational potential energy would be very inefficient, meaning that there would be little gravitational energy available to power an AGN.
(a)
See Fig. S28.2. From Eq. (28.1), using the two points on either side of log10 0014 D 8:0 in Table 28.2, ˛D0004
log10 F2 0004 log10 F1 D 0:7 log10 00142 0004 log10 00141
at log10 0014 D 8:0. (b) The distance to Cygnus A is d D 170h00021 Mpc. As in Example 28.1.1, we integrate over radio frequencies from 00141 D 107 Hz (log10 00141 D 7, log10 F D 000418:88) to 00142 D 3 0003 109 Hz (log10 00142 D 9:5, log10 F D 000420:17, obtained by interpolating the data in Table 28.2). A numerical integration results in Z 2 Lradio D 40003 d 2 F d0014 D 40003 d 2.9:0 0003 10000211 erg s00021 cm00022 /
1
D 3:1 0003 1037 h00022 W: This is in reasonable agreement with the estimate for Cyg A obtained in Example 28.1.1, 2:400031037h00022 W.
190
Chapter 28 Active Galaxies
Figure S28.2: Results for Problem 28.20 for Cygnus A.
CHAPTER 29
Cosmology 29.1 The total number of stars in a thin spherical shell of radius r1 and thickness 0006r that contains n stars per unit volume is N1 D 40003 r12 n0006r . Assuming stars of luminosity L, the radiant flux received from a single star is, from the inverse square law (Eq. 3.2), L : F1 D 40003 r12 Therefore, the total flux received from this shell is F t D N1 F1 D Ln0006r . This is independent of r , so the same result would be found for a shell of radius r2 . 29.2 Assume a WMAP density of baryonic matter of 0002b;0 D 4:17 0003 10000228 kg m00023 . Setting the corresponding rest energy density (Eq. 4.47) equal to the blackbody energy density (Eq. 9.7), we have 0002b;0c 2 D aT 4 . Solving this for T to find that the temperature of the universe gives 0002 T D
0002b;0c 2 a
00031=4 D 14:9 K:
According to Wien’s law, Eq. (3.15), the blackbody radiation would have a peak wavelength of max D .0:00290 m K/=T D 0:19 mm, in the infrared part of the electromagnetic spectrum (Table 3.1). Even if all of the matter in the universe were converted into the energy of blackbody radiation, there is too little matter to cause the sky to glow. There would not be visually detectable amounts of blackbody radiation at visible wavelengths. 29.3 We want to show that Eqs. (29.32) and (29.34), RD t D are solutions to Eq. (29.11),
0002
dR dt
40003 G00020 Œ1 0004 cos.x/0004 3kc 2 40003 G00020 Œx 0004 sin.x/0004 : 3k 3=2c 3 00032 0004
80003 G00020 D 0004kc 2; 3R
for a closed universe. For the first term on the left, we use dR dR=dx D dt dt=dx with dR 40003 G00020 sin.x/ D dx 3kc 2 dt 40003 G00020 Œ1 0004 cos.x/0004 : D dx 3k 3=2c 3 191
192
Chapter 29 Cosmology This results in
0002
dR dt
(
00032 D kc 2
sin2 .x/
) :
Œ1 0004 cos.x/00042
The second term on the left in Eq. (29.11) is 80003 G00020 2kc 2 D : 3R 1 0004 cos.x/ Inserting these results on the left-hand side of Eq. (29.11) and simplifying leads to 0004kc 2, the term on the right. 29.4 We want to show that Eqs. (29.36) and (29.38), RD t D are solutions to Eq. (29.11),
40003 G00020 Œcosh.x/ 0004 10004 3jkjc 2 40003 G00020 Œsinh.x/ 0004 x0004 : 3jkj3=2c 3
0002
0003 dR 2 80003 G00020 0004 D 0004kc 2; dt 3R for an open universe. For the first term on the left, use dR dR=dx D dt dt=dx with 40003 G00020 dR sinh.x/ D dx 3jkjc 2 dt 40003 G00020 Œcosh.x/ 0004 10004 : D dx 3jkj3=2c 3 This results in
0002
dR dt
(
00032 D jkjc
2
sinh2 .x/ Œcosh.x/ 0004 100042
) :
The second term on the left in Eq. (29.11) is 80003 G00020 2jkjc 2 D : 3R cosh.x/ 0004 1 Inserting these results on the left-hand side of Eq. (29.11), simplifying and noting that jkj D 0004k since k < 0, leads to 0004kc 2, the term on the right. 29.5 To go from Eq. (29.32) to Eq. (29.33), we must show that the coefficients of Œ1 0004 cos.x/0004 are equal. Use Eqs. (29.15), (29.19), and (29.25), 3H02 80003 G 00020 0 D 0002c;0
0002c;0 D
H02 . 0 0004 1/ D kc 2;
Solutions for An Introduction to Modern Astrophysics respectively, to find
193
40003 G00020 40003 G.00020=0002c;0 / 40003 G 0 D D 3kc 2 80003 G. 0 0004 1/ 3H02 . 0 0004 1/=0002c;0
so
1 0 40003 G00020 D : 3kc 2 2 0 0004 1 Similarly, to go from Eq. (29.34) to Eq. (29.35), we must show that the coefficients of Œx 0004 sin.x/0004 are equal: 40003 G00020 40003 G.00020=0002c;0/ 40003 G 0 D D 3k 3=2c 3 80003 GH0. 0 0004 1/3=2 3ŒH02 . 0 0004 1/00043=2 =0002c;0 so
0 1 40003 G00020 D : 2H0 . 0 0004 1/3=2 3k 3=2c 3
29.6 The procedure to go from Eq. (29.36) to Eq. (29.37), and to go from Eq. (29.38) to Eq. (29.39), is nearly identical to the one followed for Problem 29.5. The only difference is that because k < 0 for an open universe, we replace jkj with 0004k in Eqs. (29.36) and (29.38). This causes . 0 0004 1/ in Eqs. (29.33) and (29.35) to be replaced .1 0004 0 / in Eqs. (29.37) and (29.39). 29.7
(a)
When the scale height R is a maximum for a closed universe, it is necessary that dR=dt D 0. Substituting into Eq. (29.11) implies that 80003 G00020 Rmax D : 3kc 2 This result is in agreement with Eq. (29.33); R.x/ is a maximum when x D 0003 and cos.x/ D 00041.
(b) Setting R D 0 in Eq. (29.33) and solving for x, we find that the lifetime of a closed universe corresponds to cos.x/ D 1, or x D 20003 . Inserting this into Eq. (29.35) and using tH D 1=H0 for the Hubble time (Eq. 27.14), the lifetime is tlife D
0 1 0003 0 Œx 0004 sin.x/0004 D tH : 3=2 2H0 . 0 0004 1/ . 0 0004 1/3=2
29.8 For a flat universe, the scale factor is given by Eq. (29.31). Inserting 1 C z D 1=R from Eq. (29.4), we have 0002 00032=3 0002 00032=3 t 3 1 RD D ; 2 tH 1Cz or t.z/ 2 D tH 3
0002
1 1Cz
00033=2 ;
which is Eq. (29.40) for k D 0. For a closed universe, the time is parameterized by Eq. (29.35) [with tH D 1=H0 , Eq. (27.14)], t tH
D
1 0 Œx 0004 sin.x/0004 : 2 . 0 0004 1/3=2
To find x, use Eq. (29.33) for the scale factor with 1 C z D 1=R, RD
1 0 1 Œ1 0004 cos.x/0004 D ; 2 0 0004 1 1Cz
to get cos.x/ D 1 0004
2. 0 0004 1/ ; 0 .1 C z/
(S29.1)
194
Chapter 29 Cosmology so 00021
0002
x D cos To find sin.x/, use sin.x/ D
q
0003 0 z 0004 0 C 2 : 0 z C 0
1 0004 cos2 .x/ so s
sin.x/ D
p
2. 0 0004 1/ 2 2 . 0 0004 1/. 0 z C 1/ 10004 10004 D : 0 .1 C z/ 0 .1 C z/
Inserting these expressions for x and sin.x/ into Eq. (S29.1) produces Eq. (29.41) for k > 0. For an open universe, the time is parameterized by Eq. (29.39) (again with tH D 1=H0), t 1 0 D Œ0004x C sinh.x/0004 : tH 2 .1 0004 0 /3=2 To find x, use Eq. (29.37) for the scale factor with 1 C z D 1=R, RD
1 0 1 Œcosh.x/ 0004 10004 D 2 1 0004 0 1Cz
to get cosh.x/ D 1 C so x D cosh To find sinh.x/, use sinh.x/ D
q
0002
0003 0 z 0004 0 C 2 : 0 z C 0
cosh2 .x/ 0004 1 so
s sinh.x/ D
00021
2.1 0004 0 / ; 0 .1 C z/
p
2 .1 0004 0 /. 0 z C 1/ 2.1 0004 0 / 2 1C 00041D : 0 .1 C z/ 0 .1 C z/
Inserting these expressions for x and sinh.x/ into Eq. (S29.2) produces Eq. (29.42) for k < 0. 29.9
(a)
Start with Eq. (29.10),
'0002
1 dR R dt
00032
# 8 0004 0003 G0002 R2 D 0004kc 2: 3
From Eq (29.12) for the critical density and Eq. (29.8) for H , 0002 0002 0003 0003 8 0002 3H 2 8 1 dR 2 dR 2 D : 0003 G0002R2 D 0003 GR2 D R2 3 3 0002c 80003 G R dt dt So Eq. (29.10) becomes
0002
dR dt
0002
00032 0004
dR dt
or .t/ D 1 C
00032 D 0004kc 2 ;
kc 2 ; .dR=dt/2
which is Eq. (29.194). is very close to unity in the early universe.
(S29.2)
Solutions for An Introduction to Modern Astrophysics
195
(b) The right-hand side of Eq. (29.11), 0002
dR dt
00032 0004
80003 G00020 D 0004kc 2 ; 3R
is a constant. However, R ! 0 as t ! 0, and so the second term on the left diverges, 80003 G00020=.3R/ ! 1. Therefore the first term on the left, .dR=dt/2 , must also diverge as t ! 1. Equation (29.194), then shows that ! 1 as t ! 0, so at very early times it is difficult to distinguish between a closed, flat, and open universe. 29.10 Eq. (29.23) can be used to eliminate H 2 from Eq. (29.26), giving 0002
.1 C z/3 0
0003 H02 .1 0004 / D H02 .1 0004 0 /.1 C z/2 ;
which simplifies immediately to 1 00041 D
0002
0003 1 0004 1 .1 C z/00021 ; 0
which is Eq (29.195). This implies that as z ! 1, ! 1. 29.11 Define u0006
1 : 0 .1 C z/
In terms of u, Eq. (29.41) is t tH
D
n o p p 0 00021 cos Œ1 0004 2u. 0004 1/0004 0004 2 u. 0004 1/ 1 0004 u. 0004 1/ : 0 0 0 2. 0 0004 1/3=2
As z ! 1, u ! 0. Use cos
00021
p .1 0004 x/ D 2 x 1=2 C
and
p 2 3=2 C x 12
p 1 10004x D 10004 x C 2
for x 0007 1 to find t tH
0 D 2. 0 0004 1/3=2
(
p 2 Œ2u. 0 0004 1/00041=2 C
p
000b) p 2 1 3=2 Œ2u. 0 0004 1/0004 0004 2 u. 0 0004 1/ 1 0004 u. 0 0004 1/ : 12 2
The first-order terms, those involving u1=2, cancel. The second-order terms, involving u3=2, produce t tH
D
3=2 1 4 3=2 0 2 2 1 3=2 . 0004 1/ D D ; u 0 0 3=2 3 3 0 .1 C z/ 3 .1 C z/3=2 1=2 2. 0 0004 1/ 0
which is Eq. (29.43). 29.12 Beginning with Eq. (29.10) and multiplying through by R gives 0002
dR R dt
00032
8 0004 0003 G0002R3 D 0004kc 2R: 3
196
Chapter 29 Cosmology Differentiating with respect to t, 0002
dR dt
0002
00033 C 2R
dR dt
00030002
d 2R dt 2
0003
d 0002R3 8 dR 0004 0003G D 0004kc 2 : 3 dt dt
Using Eqs. (29.50) and (29.10) to replace d.0002R3 /=dt and 0004kc 2 , respectively, gives 0002 0002 0003 00030002 2 0003 0002 0003 dR 3 80003 GP d R3 dR dR d R dR 3 8 C C 2R 0004 0003 GR20002 D : 2 2 dt dt dt 3c dt dt 3 dt Finally, simplifying gives Eq. (29.51). 29.13 Assume that an expanding shell of mass d m lies immediately above a sphere of mass M , constant density 0002, and radius R. From Newton’s second law and Newtonian gravitation, dm
GM d m d 2R D0004 : 2 dt R2
But M D 400030002R3 =3. Substituting immediately gives Eq. (29.51) for the special case that P D 0. 29.14 From the ideal gas law, P D 0002kT = mH . Furthermore, the thermal energy of an average hydrogen atom is 2 given by K D 3kT =2 D mH vrms =2. Substituting T into the ideal gas law gives (with D 1 for pure neutral 2 hydrogen), 0002 D P =.vrms =3/ 0005 P =c 2 when vrms D 600 km s00021 . When 0002 D P =c 2 , it is necessary that
kT P D D1 2 0002c
mH c 2
from the ideal gas law. This corresponds to a temperature of T D 1:100031013 K. (Of course at this temperature, the assumption that D 1 could no longer be valid.) Now, for an adiabatically expanding universe, RT D T0 , or T0 2:725 K RD D 2:5 0003 10000213 ; D T 1:1 0003 1013 K and 1 zD 0004 1 D 4 0003 1012 : R 29.15 Substituting Eq. (29.52) into Eq. (29.50) gives d R3 d R3 0002 D 0004w0002 : dt dt This immediately gives
Rearranging
d R3 d R3 3 d0002 CR D 0004w0002 : 0002 dt dt dt 1 d R3 1 d0002 .1 C w/ 3 D0004 : dt 0002 dt R
This can also be written as .1 C w/
d ln 0002 d ln R3 D0004 : dt dt
Integrating, we find .1 C w/ ln R3 D 0004 ln 0002 C C;
Solutions for An Introduction to Modern Astrophysics
197
where C is a constant of integration. Rewriting, ln R3.1Cw/ C ln 0002 D C; which immediately leads to R3.1Cw/ 0002 D e C D constant D 00020 : 29.16 Starting with the definition of the deceleration parameter (Eq. 29.54),
R.t/ d 2 R.t/=dt 2 q.t/ 0006 0004 ; ŒdR.t/=dt0004 2 and using Eqs. (29.51) and (29.10) to replace d 2 R=dt 2 and .dR=dt/2 , respectively, q.t/ becomes q.t/ D 0004
40003 G0002R2=3 : 80003 G0002R2 =3 0004 kc 2
From the definition of the density parameter (Eq. 29.18), we find q.t/ D 0004
.t/ H 2 .t/ R2 =2 : .t/ H 2 .t/ R2 0004 kc 2
Finally, using Eq. (29.24) to replace 0004kc 2 gives q.t/ D 12 .t/. 29.17
(a)
The binding energy is Eb D .mH C mn 0004 mD /c 2 D .1:007825 C 1:008665 0004 2:014102/uc 2 D 2:224 MeV;
(b) The wavelength of a photon with this energy is, from Eq. (5.3), D (c)
hc D 5:57 0003 10000213 m: E
According to Wien’s law (Eq. 3.15), this photon corresponds to the peak wavelength for a blackbody temperature of 0:00290 m K T D D 5:20 0003 109 K: max
29.18 For a redshift of z D 1:776, the scale factor is, from (Eq. 29.4), R D .1 C z/00021 D 0:3602. The temperature of the CBR was, from Eq. (29.58), T D T0=R D .2:726 K/=R D 7:57 K, in good agreement with the temperature of the intergalactic cloud, 7:4 ˙ 0:8 K. 29.19 Use the Boltzmann equation (Eq. 8.6), Nb gb 0002.Eb 0002Ea /= kT D e Na ga with ga D 1 (one ground state), gb D 3 (three degenerate first excited states), Nb =Na D 27=100, and Eb 0004 Ea D 4:8 0003 1000024 eV. Solving for T , we have T D
0004.Eb 0004 Ea / D 2:3 K: k lnŒ.Nb ga /=.Na gb /0004
198
Chapter 29 Cosmology
29.20 Assuming for simplicity that the television transmitter radiates equally in all directions with a power of P D 2:5 0003 104 W, the magnitude of the time-averaged Poynting vector, hSi, at a distance of r D 70 km from the transmitter is hSi D P =.40003 r 2/. The energy density of channel 6 photons at this distance is then approximately P uTV ' D 1:35 0003 10000215 J m00023 : 40003 r 2c From Eq. (9.5) with T D 2:73 K, the energy density of CBR photons with wavelengths between 1 D 3:41 m and 2 D 3:66 m is about uCBR ' u0005 0006 D
80003 hc=5ave .2 0004 1 / D 1:52 0003 10000224 J m00023 ; e hc=0005avekT 0004 1
where the average of 1 and 2 was used to evaluate u0005 . Therefore uTV=uCBR ' 9 0003 108; the CBR does not interfere with your television viewing. 29.21 Combining Eq. (4.32) for the relativistic Doppler shift with Wien’s law, Eq. (29.59), gives p 0014obs 1 0004 v 2 =c 2 Tobs D D : 0014rest Trest 1 C .v=c/ cos Here, is the angle between the velocity of the source and the line-of-sight of the observer, so the source moving directly away corresponds to D 0. But in Eq. (29.61), 0005 is the angle between the line-of-sight and the observer’s motion, so moving directly toward the source corresponds to 0005 D 0. Thus we must replace with 180ı 0004 0005 to obtain p Trest 1 0004 v 2=c 2 Tmoving D ; 1 0004 .v=c/ cos 0005 which is Eq. (29.61). For v=c 0007 1, we can use 1 v ' 1 C cos 0005 1 0004 .v=c/ cos 0005 c and
q
1 0004 v 2 =c 2 ' 1 0004
1 v2 ' 1; 2 c2
both to first-order in v=c, to obtain Eq. (29.62). 29.22 The Sun’s peculiar velocity relative to the Hubble flow is vˇ D 370:6 km s00021 . From Eq. (29.62) with 0005 D 0 and Trest D 2:725 K, the magnitude of the variation in the temperature of the CBR due to the Sun’s motion is Tmoving 0004 Trest D Trest 29.23
(a)
v D 3:37 0003 1000023 K: c
The peak wavelength of the CMB is max D
0:002897755 m K D 1:06 0003 1000023 m: 2:725 K
Since the Compton shift is just f 0004 i D .h=me c/.1 0004 cos 0005/, a characteristic shift is just the Compton wavelength of 0006 D C 0006 h=me c 2 D 2:43 0003 10000212 m. Thus, the relative shift is only 0006 D 2:29 0003 1000029: max
Solutions for An Introduction to Modern Astrophysics
199
This implies that the shift in the wavelength of the photon in the electron’s rest frame can safely be neglected. Now consider the four situations depicted in Fig. 29.31. When viewed from the right neither cases 1 or 3 result in any shift in wavelength or frequency of the photon, meaning that 0014f;1 D 0014f;3 D 00140 , where 0014f is the final frequency and 00140 is the initial frequency. On the other hand, cases 2 and 4 show reflection of the photon from moving electrons. In case 2, the frequency shift is just the Doppler shift from approach and reflection giving s !2 0002 0003 1Cˇ 1Cˇ 0014f;2 D 00140 D 00140 ; 10004ˇ 10004ˇ where ˇ 0006 ve =c, and ve is the velocity of the electron. Similarly s 0014f;4 D 00140
10004ˇ 1Cˇ
!2
0002 D 00140
10004ˇ 1Cˇ
0003 :
The average final frequency of the four cases is then 0003 0002 0003 0002 1Cˇ 10004ˇ 00140 1 C ˇ2 : 1C C1C D 1C 10004ˇ 1Cˇ 2 1 0004 ˇ2 (S29.3) For electrons with an average thermal energy associated with Te D 108 K, 0014f D
1 00140 0014f;1 C 0014f;2 C 0014f;3 C 0014f;4 D 4 4
1 3 me ve2 D kTe ; 2 2 and ˇ2 D
ve2 3kTe D D 0:05: 2 c me c 2
Given that ˇ 2 0007 1, 0014f from Eq. (S29.3) can be approximated by
00070006 0007i i 0006 00140 h 00140 h 1 4 2 2 2 4 2 0014f ' 1Cˇ ' 1C 1Cˇ 1 C 1 C 2ˇ C ˇ D 00140 1 C ˇ C ˇ : 2 2 2 Keeping terms to second order, we arrive at 0014f 0004 00140 00060014 v2 3kTe D e2 D : 00140 00140 c me c 2 (b) From the discussion of opacity leading up to Eq. (9.15), the fraction of photons scattered is roughly D 2ne 0007T R; where R is the radius of the gas cloud, and 0007T D 6:65 0003 10000229 m2 is the Thomson scattering cross section (see Eq. 9.20). Therefore, D 0:12. (c)
A fractional shift in the temperature of the CMB is roughly given by 00060014 3kTe 0006T 0002 0004 D0004 : T0 00140 me c 2
200
Chapter 29 Cosmology
29.24 From the pressure integral (Eq. 10.8) 1 PD 3
Z
1
np pv dp: 0
If all of the particles in the sample are highly relativistic, then v ' c and p ' 0012 mc. Substituting, Z 1 1 PD np 0012 mc 2 dp: 3 0 Noting that E D 0012 mc 2, and np is the number density of particles with momenta between p and p C dp, the integral reduces to the total energy density of the relativistic particles. Therefore, P D 13 u D wu. 29.25 We will assume that the universe is flat in this problem. Since R4 0002rel D 0002rel;0 , and RT D T0 (Eq. 29.58), the energy density of relativistic particles (photons) must be proportional to T 4 , or urel D aT 4 : From the first law of thermodynamics for an expanding volume, dU dV D 0004P : dt dt However, we can substitute
0002 U D uV D aT 4
(S29.4)
4 3 0003r 3
0003
on the left-hand side of Eq. (S29.4), dV dr D 40003 r 2 dt dt on the right-hand side, and P D aT 4 =3 for the radiation pressure (also on the right-hand side). Eliminating common terms we find d 0006 4 30007 dr T r D 0004T 4 r 2 : dt dt Now substituting T D T0 =R, r D DR for some initial size, D, of the sphere, and eliminating common terms we arrive at 0002 0003 d 1 1 dR D0004 2 : dt R R dt Since the left-hand side is consistent with the right-hand side, the original assumption of conservation of energy within the sphere must be consistent as well. 29.26
(a)
This is a straightforward derivative.
(b) Equation (29.8) shows that
0002 exp .t/ D
(c)
1 dR R dt
000300021
D 1=H .t/:
For a flat universe k D 0 and Eq. (29.83) becomes r 00060002 dR 8 0002rel;0 0007 m;0 D 0003G C dt 3 R R2 This implies that
exp .R/ D R
000b00021=2 00060002 8 0002rel;0 0007 00021=2 m;0 2 8 DR : 0003G C 0003 G ŒR 0002m;0 C 0002rel;0 0004 3 R R2 3
(S29.5)
Solutions for An Introduction to Modern Astrophysics 29.27
(a)
201
Consider first the portion of Eq. (29.84) in square brackets. Deep in the radiation era, Rr;m 0007 R, implying that we can approximate the radical to second order using .1 C x/1=2 D 1 C 12 x 0004 18 x 2 C when x 0007 1. This implies that # 0002 0003 0003s 0002 0003' 0002 R R R 1 R 1 R 2 2C 00042 C1 D 2C 00042 1C 0004 C Rr;m Rr;m Rr;m 2 Rr;m 8 Rr;m D
'
0003 0003 0002 R 2 1 R 3 0004 Rr;m 8 Rr;m 0003 0002 R 1 R 2 00042 0004 C C Rr;m 4 Rr;m 0003 0002 3 R 2 : 4 Rr;m
R 1 2C C Rr;m 2
0002
where only terms through second-order have been retained. Eq. (29.84) now becomes t.R/ '
1 R2 : 2 H0 . m;0 Rr;m /1=2
But Rr;m D m;0 = rel;0 (note that a ”0” subscript was inadvertently dropped in this expression in the first printing of An Introduction to Modern Astrophysics). Substituting, and solving for R gives 1=2
1=4
R.t/ D 21=2H0 rel;0 t 1=2 : From Eq. (29.80) we now have R.t/ D 2
1=2
H01=2
40003 Gg0004 aT04
!1=4
0002 t 1=2 D
3H02 c 2
160003 Gg0004a 3c 2
00031=4 T0 t 1=2 :
(b) In a flat universe (k D 0) with only relativistic particles, the Friedmann equation becomes 0002 0003 dR 2 8 D 0003 G0002relR2 : dt 3 From Eq. (29.77) we have 0002
dR dt
00032 D
80003 Gg0004aT04 1 80003 Gg0004 aT 4 2 R D : 6c 2 6c 2 R2
This implies that dR R D dt
80003 Gg0004aT04 6c 2
Integrating, 1 2 R D 2 or R.t/ D 2 which is again Eq. (29.86).
1=2
80003 Gg0004aT04 6c 2
80003 Gg0004 aT04 6c 2 !1=4
!1=2
!1=2
0002 t
1=2
D
:
t;
160003 Gg0004a 3c 2
00031=4 T0 t 1=2 ;
202
Chapter 29 Cosmology
29.28 From Eq. (29.78), and equating at a general time t with today, we find H 2 .1 0004 rel/ R2 D H02 .1 0004 rel;0 / :
(S29.6)
From Eq. (29.5), R3 0002rel D 0002rel;0 ; or R3 rel 0002c D rel;0 0002c;0 : Substituting Eq. (29.12), R3 rel H 2 D rel;0 H02 ; or
rel;0 H02 R Equation (S29.7) can be substituted into Eq. (S29.6), and after some rearrangement, we obtain
0002 0003 1 2 2 H R D 1C 0004 1 rel;0 H02 : R relR2 H 2 D
(S29.7)
(S29.8)
Substituting Eq. (S29.8) back into Eq. (S29.7) we finally have rel D
rel;0 : R C .1 0004 R/ rel;0
In the limit as z ! 1 or, equivalently, R ! 0, rel ! 1. Thus, this one-component, relativistic-particle model is flat in the limit z ! 1. 29.29 Using R3 0002b D 0002b;0 (Eq. 29.5) with 0002b0 D 4:17 0003 10000228 kg m00023 (Eq. 29.17), and T0 D R T .R/ (Eq. 29.58) with T0 D 2:725 K and T D 1010 K, we find 0002 00033 T 0002b;0 D 20:6 kg m00023 : 0002b D 3 D 0002b;0 R T0 29.30 Using Eq. (4.47) for the rest energy of an electron–positron pair, set kT ' 2me c 2 so T ' 1:2 0003 1010 K. 29.31
(a)
Following the discussion in Example 9.2.1, consider a neutron with a collision cross section of 0007 D 0003.2r /2 that moves with speed v through point protons of number density np . In time 0006t there will be np 0007v0006t collisions.
(b) To evaluate np 0007v0006t, we must estimate the number density of protons. Because protons greatly outnumbered neutrons at that time, assume that all of the baryonic matter was in the form of protons with T0 D 2:725 K and T D 109 K to write 0002 0003 0002b 0002b;0 T 3 D D 1:2 0003 1025 m00023 : np ' mp mp T0 The collision cross section is, using r ' 10000215 m, 0007 D 0003.2r /2 ' 1:3 0003 10000229 m2 . The speed, v, of the neutrons comes from Eq. (8.3), s 3kT vrms D D 5:0 0003 106 m s00021 : mn The characteristic time in the radiation era is obtained from Eq. (29.89) by solving for t with T D 109 K and g0004 D 3:363, giving trad D 179 s. Using this for 0006t, we find np 0007v0006t ' 1:4 0003 105 , which is certainly 0005 1. Each neutron had numerable opportunities to combine with a proton.
Solutions for An Introduction to Modern Astrophysics 29.32
(a)
203
We need the Thomson scattering cross section (0007T D 6:65 0003 10000229 m2 from Eq. 9.20) and the number density of free electrons to calculate the mean free path of a photon, ` D 1=.ne 0007/ (Eq. 9.12). For a composition of pure hydrogen, ne D 0002b =mH , where 0002b is the density of baryonic matter (protons and electrons in this case). Using R3 0002b D 0002b;0 D 4:17 0003 10000228 kg m00023 (Eqs. 29.5 and 29.17), we have ne D
0002b 0002b;0 D 3 ; mH R mH
so the mean free path is `D
1 R3 mH D : ne 0007T 0002b;0 0007T
(S29.9)
The time for a photon to travel the distance ` is t` D
` R3 mH : D c c0002b;00007T
(b) Setting t` D exp from Eq. (S29.5), we have
000b00021=2 R3 mH 8 D R2 : 0003 G ŒR 0002m;0 C 0002rel;0 0004 c0002b;00007T 3 This can be simplified somewhat to give 0002 3
R C
0002rel;0 0002m;0
0003 R2 0004
K D 0; 0002m;0
(S29.10)
where the constant K is given by 0002 K0006
3 80003 G
00030002
c0002b;00007T mH
00032 :
The cubic equation Eq. (S29.10) can be solved numerically (using a Newton method for example) to give R D 0:0257 implying z D R00021 0004 1 D 37:9: An example Fortran 95 code: Program Cubic_Solver USE Constants, ONLY : c, pi, G, m_H IMPLICIT NONE REAL(8), PARAMETER REAL(8), PARAMETER REAL(8), PARAMETER REAL(8), PARAMETER
:: :: :: ::
rho_m0 = 2.56E-27, rho_b0 = 4.17E-28 rho_c0 = 9.47E-27, Omega_rel0 = 8.25E-5, rho_rel0 = rho_c0*Omega_rel0 sigma_T = 6.65E-29 K = (3/(8*pi*G))*(c*rho_b0*sigma_T/m_H)**2
REAL(8),
::
rel_error = 1E-6
::
R, dR, F, dF
PARAMETER
REAL(8)
R = 1 dR = 1000 DO WHILE (ABS(dR/R) > rel_error) F = R**3 + (rho_rel0/rho_m0)*R**2 - K/rho_m0 dF = 3*R**2 + 2*(rho_rel0/rho_m0)*R dR = -F/dF R = R + dR
204
Chapter 29 Cosmology WRITE(*,*) 'R, dR = ', R, dR END DO PAUSE STOP END
From Eq. (29.84), this implies an age for the universe of t D 2:4 0003 1015 s D 75 Myr. 29.33 Eq. (29.101) gives mH R3 f0002b;0
0002
20003 me kT0 h2 R
00033=2
e 0002000bI R = kT0 0004
f D 0: 10004f
This can be solved numerically with a Newton method (for example) with f D 0:5 to give R D 7:24806 0003 1000024 . Program Newton_Solver USE Constants, ONLY : pi, m_H, m_e, h, k_B, eV IMPLICIT NONE REAL(8), PARAMETER
::
rho_b0 = 4.17E-28, chi = 13.6*eV, T_0 = 2.725, frac = 0.5
REAL(8),
::
rel_error = 1E-6
::
R, dR, F, dF
PARAMETER
REAL(8)
R = 1E-4 dR = 1000 DO WHILE (ABS(dR/R) > rel_error) F = m_H*R**3/(frac*rho_b0)*(2*pi*m_e*k_B*T_0/(h**2*R))**1.5*exp(-chi*R/(k_B*T_0)) - frac/(1 - frac) dF = 3*m_H*R**2/(frac*rho_b0)*(2*pi*m_e*k_B*T_0/(h**2*R))**1.5*exp(-chi*R/(k_B*T_0)) & & - 1.5*m_H*R**3/(frac*rho_b0)*(2*pi*m_e*k_B*T_0/(h**2))**1.5*R**(-2.5)*exp(-chi*R/(k_B*T_0)) & & - chi/(k_B*T_0)*m_H*R**3/(frac*rho_b0)*(2*pi*m_e*k_B*T_0/(h**2*R))**1.5*exp(-chi*R/(k_B*T_0)) dR = -F/dF R = R + dR WRITE(*,*) 'R, dR = ', R, dR END DO PAUSE STOP END
29.34 If z D 1089, then R D 1=.1 C z/ D 9:174 0003 1000024. From Eq. (29.84) with other WMAP values of Rr;m D 3:05 0003 1000024 , H0 D 2:30 0003 10000218 s00021 , and m;0 D 0:27, we find t D 1:194 0003 1013 s D 378 kyr. This is in excellent agreement with the WMAP result. 29.35 The program used in Problem 29.33 can also be used to find R when 99% of the hydrogen atoms were ionized (f D 0:99) and when 1% of the hydrogen atoms are ionized (f D 0:01). The results are R99 D 6:300 0003 1000024 and R1 D 8:766 0003 1000024, respectively. These values of R correspond to z99 D 1586 and z1 D 1140, implying 0006z D 447: The times are again obtained from Eq. (29.84), as was done in the solution for Problem 29.36. The results are t99 D 199 kyr and t1 D 350 kyr. This implies 0006t D 151 kyr: In both cases the range is somewhat larger than the quoted WMAP results.
Solutions for An Introduction to Modern Astrophysics
205
29.36 Using R˚ D 6:378136 0003 106 m from Appendix A and the Taylor series sin x ' x 0004 x 3=3! C for small x, the discrepancy for D D 100 m is Cexp 0004 Cmeas D 20003D 0004 20003R˚ sin.D=R˚ / ' 0003 # 0002 1 D 3 D 0004 ' 20003D 0004 20003R˚ R˚ 3! R˚ D
0003D 3 D 2:57 0003 1000028 m D 25:7 nm: 3R2˚
29.37 Equating Eqs. (29.120) and (29.121) we have H 2 .1 0004 /R2 D H02 .1 0004 0 /: This can be solved for to give D10004
H02 H02 1 0004 0 D 1 0004 .1 0004 0 /.1 C z/2 : H 2 R2 H2
Substituting Eq. (29.122), we arrive at .z/ D 1 0004
1 0004 0 : m;0 .1 C z/ C rel;0 .1 C z/2 C ƒ;0 .1 C z/00022 C 1 0004 0
(S29.11)
We can now verify that our last result reduces to Eq. (29.28) in the appropriate special case of baryonic matter only. Let rel;0 D 0 and ƒ;0 D 0, which then requires that m;0 D 0 . Substituting into Eq. (S29.11) we have 1 0004 0 0 0004 1 .z/ D 1 0004 D1C ; 0 .1 C z/ C 1 0004 0 1 C 0 z as expected. 29.38 From the Robertson–Walker metric (Eq. 29.106) with d$ D 0 for a spherical surface and dt D 0, we see that the differential element of area is dA D ŒR.t/$ d00050004ŒR.t/$ sin 0005 d 0004 D ŒR.t/$ 00042 sin 0005 d0005 d : Integrating this over the usual limits of 0005 D 0 to 0003 and D 0 to 20003 gives the area of the spherical surface, Z Ahorizon D ŒR.t/$ 00042 29.39
(a)
Z
20002 0006D0
0002 0003D0
sin 0005 d0005 d D 40003 ŒR.t/$ 00042 :
If we consider the general acceleration equation (Eq. 29.112) with 0002rel D 0 and Pm D Prel D 0, we have 1 d 2R 4 1 D 0004 0003 G0002m C ƒc 2 : R dt 2 3 3 If the universe is static, this requires that d 2 R=dt 2 D 0 or ƒD
40003 G0002m : c2
206
Chapter 29 Cosmology (b) If the universe is static, we must also have dR=dt D 0. This implies that Eq. (29.114) can be solved for k, giving
000b0002 20003 8 R kD : 0003 G .0002m C 0002ƒ / 3 c2 Noting that from Eq. (29.113), 0002ƒ 0006 ƒc 2 =80003 G D 0002m =2, we find kD (c)
40003 G0002mR2 > 0: c2
This static universe is closed. In solving for ƒ we assumed that d 2 R=dt 2 D 0. However, any slight deviation of 0002m will result in a non-zero acceleration. As a result, this universe is in an unstable equilibrium.
29.40 The WMAP values we will need are: ŒH0 0004WMAP D 2:30 0003 10000218 s00021 ŒT0 0004WMAP D 2:725 ˙ 0:002 K Œ m;0 0004WMAP D 0:27 ˙ 0:04 rel;0 D 8:24 0003 1000025 Œ ƒ;0 0004WMAP D 0:73 ˙ 0:04 ƒ0 D
3H02 ƒ;0 D 1:29 0003 10000252 m00022 c2
Œ 0 0004WMAP D 1:02 ˙ 0:02 0002m;0 D 2:56 0003 10000227 kg m00023 Using z D 1089: From Eq. (29.122), H D 4:96 0003 10000214 s00021 . From Eq. (29.58), T D .1 C z/T0 D 2970 K. From Eq. (29.6), 0002m D .1 C z/3 0002m;0 D 3:32 0003 10000218 kg m00023 . 80003 G0002m From Eq. (29.18), m D D 0:75. 3H 2 40003 Gg0004aT 4 D 0:25. From Eq. (29.80), rel D 3H 2 c 2 ƒc 2 D 1:6 0003 1000029 . From Eq. (29.118), ƒ D 3H 2 29.41 Beginning with Eq. (29.122) and moving .1 C z/ inside the radical, we find h i1=2 : H D H0 m;0 .1 C z/3 C rel;0 .1 C z/4 C ƒ;0 C .1 0004 0 /.1 C z/2 Using the generalized equation of state given by Eq. (29.52), P D w0002c 2, we note that for pressureless dust, wm D w1 D 0. This implies that the quantity 3.1Cw1/ D 3. Similarly, for radiation and relativistic particles, wrel D w2 D 1=3, so that 3.1 C w2 / D 4. Finally, for dark energy, Eq. (29.115), wƒ D w3 D 00041, so that 3.1 C w3 / D 0. Combining these results leads to the desired equation.
Solutions for An Introduction to Modern Astrophysics
207
29.42 We begin with Eq. (29.54) and note that H D .dR=dt/=R (Eq. 29.8). This gives q.t/ D 0004
1 d 2 R=dt 2 : R H2
(S29.12)
Turning next to the acceleration equation (Eq. 29.112), using Eq. (29.113) for the equivalent mass density of dark energy, and rearranging, we have
1 d 2R 4 3 .Pm C Prel / : D 0004 C 0002 0004 20002 C 0003 G 0002 m rel ƒ R dt 2 3 c2 However, Pm D 0 for pressureless dust, and Prel D 13 0002rel c 2 , which leads to 1 d 2R 4 D 0004 0003 G Œ0002m C 20002rel 0004 20002ƒ 0004 : R dt 2 3
(S29.13)
Substituting back into Eq. (S29.12), we obtain
80003 G 1 q.t/ D 0002m C 0002rel 0004 0002ƒ : 3H 2 2 From the definition of the density parameter (Eq. 29.18) we have 1 q.t/ D m C rel 0004 ƒ : 2 We also have (see the solution to the previous problem, for example) wm D 0, wrel D 1=3, and wƒ D 00041. Using these values leads to the expression q.t/ D
1X .1 C 3wi / i .t/: 2 i
29.43 If the acceleration of the universe changed sign at some point in time, then it is necessary that d 2 R=dt 2 D 0 at that time. Referring to Eq. (S29.13) from the solution to the last problem, it must have been the case that 0002m C 20002rel 0004 20002ƒ D 0 when this occurred. Assuming that 0002ƒ D constant D 0002upLambda;0 (as stated in Eq. 29.113), then 0002m C 20002rel D 20002ƒ;0 : We can now express 0002m and 0002rel in terms of their present values and R by 0002m D 0002m;0 =R3 and 0002rel D 0002rel;0 =R4 : Substituting, we find 0002m;0 =R3 C 20002rel;0 =R4 D 20002ƒ;0 ; or 20002ƒ;0 R4 0004 0002m;0 R 0004 20002rel;0 D 0:
208
Chapter 29 Cosmology Dividing through by the present-day critical density 0002c;0 and using the definition of the acceleration parameter (Eq. 29.18), this expression becomes 2 ƒ;0 R4 0004 m;0 R 0004 2 rel;0 D 0: Noting that rel;0 0007 ƒ;0 and rel;0 0007 m;0 we can safely neglect the last term. Thus, either R D 0 (where our analysis is invalid), or 0003 0002 m;0 1=3 RD : 2 ƒ;0 Evaluating with WMAP values gives R D 0:57, and z D 1=R 0004 1 D 0:76.
29.44
(a)
The look-back time is given by tL D t0 0004t.z/. But t0 is obtained by substituting R D 1 into Eq. (29.129). If we also replace R by 1=.1 C z/ in the expression for t.R/, we quickly obtain. 3 2 q q ƒ;0 ƒ;0 C 1 C 1 2 1 m;0 m;0 7 6 tL .z/ D p ln 4 p ln.1 C z/: q 0006q 00075 D 3 H0 ƒ;0 1 ƒ;0 ƒ;0 H0 ƒ;0 C 1 C 3=2 .1Cz/ m;0
m;0
(b) See Fig. S29.1. The space density declines rapidly beyond tL =tH D 1:5.
Figure S29.1: The number density of AGNs as a function of lookback time. 29.45
(a)
Beginning with Eq. (29.122) and replacing 1 C z with 1=R, we find
H D H0
m;0 rel;0 1 0004 0 C C ƒ;0 C R3 R4 R2
000b1=2 :
As R grows well beyond R D 1, each term on the right-hand side of the equation approaches zero except the ƒ term, which remains constant. Therefore, H approaches a constant, given by p H D H0 ƒ;0 :
Solutions for An Introduction to Modern Astrophysics
209
(b) Starting with the Friedmann equation in the form of Eq. (29.120), assuming that D ƒ D ƒc 2 =3H 2 , and equating the Friedmann equation with today’s values, we find 0002 0003 ƒc 2 R2 D H02 .1 0004 ƒ;0 /: H 10004 3H 2 2
This implies that
0002 H R 0004 2
2
ƒc 2 3
0003 R2 D H02 .1 0004 ƒ;0 /:
However, H D .dR=dt/=R, giving 0002
dR dt
0002
00032 D
ƒc 2 3
0003 R2 C H02 .1 0004 ƒ;0 /:
Rewriting and integrating Z
t
Z
0
t0
1
0002 t 0004 t0 D
0002 D
dR0
R
dt D
r0006
3 ƒc 2
3 ƒc 2
ƒc 2 3
0007
00031=2 Z
R02 C H02 .1 0004 ƒ;0 / dR0 0003 0002
R
s 1
00031=2
Define
R02 C
3H02 ƒc 2
v u u 0 4 ln R C tR02 C
.1 0004 ƒ;0 /
2
0002 0006
3 ƒc 2
00031=2
3H02 ƒc 2
!
3ˇR ˇ ˇ 5 .1 0004 ƒ;0 / ˇˇ : ˇ 1
1 Dp ; ƒ;0 H0
where the last expression was obtained from Eq. (29.118). We can now rewrite the result of the integration as ' # p t 0004 t0 R C R2 C ƒ;0 .1 0004 ƒ;0 / p D ln ; 1 C 1 C ƒ;0 .1 0004 ƒ;0 / which leads to RC
q
R2 C ƒ;0 .1 0004 ƒ;0 / D Ce .t 0002t0/= ;
where C 00061C
p
1 C ƒ;0 .1 0004 ƒ;0 / D constant:
This expression for R is clearly exponentially increasing with time when ƒ > 0 (implying > 0). (c)
Using WMAP values, the characteristic time is D 16 Gyr.
29.46 Start with Eq. (29.153). Writing dt D dR=.dR=dt/ and using R D 1=.1 C z/ (Eq. 29.4) produces Z
1 1Cz
dh .t/ D R.t/ 0
c dR : R.dR=dt/
210
Chapter 29 Cosmology To evaluate the integral, dR=dt must be expressed in terms of R only. Evaluating Eq. (29.9) at the present epoch with H D H0 , 0002 D 00020 and R D 1, and equating the left-hand side with the left-hand side of Eq. (29.11) gives 0002 0002 0003 0003 dR 2 1 8 D 0003 G00020 0004 1 C H02 : dt 3 R Using Eqs. (29.15) and (29.19), we find 8 0003 G00020 D 0 H02 ; 3 and
0002
dR dt
0002
00032 D
H02
0003 0 0004 0 C 1 : R
The integral for the horizon distance thus becomes, using R D 1=.1 C z/ (Eq. 29.4), c dh .z/ D H0 .1 C z/
Z
1 1Cz
0
p
dR 0 R 0004 . 0 0004 1/R2
:
The integral has the form Z
0002
0003 2ax 1 0003 00021 10004 p D p cos 0004 b 2 a bx 0004 ax 2 dx
with a D 0 0004 1 and b D 0 . Evaluating the integral gives the desired result. 29.47 Equation (29.152) for the circumference of a closed universe is, using R D 1=.1 C z/ (Eq. 29.4), Cuniv D
20003 c p : H0 .1 C z/ 0 0004 1
Using Eq. (29.197) for dh , the horizon distance in a closed universe, we see that
dh 2. 0 0004 1/ 1 D cos00021 1 0004 : Cuniv 20003 0 .1 C z/ At very early times, as z ! 1, dh=Cuniv ! 0. That is, the fractional size of a causally connected region becomes vanishingly small as t ! 0. According to Eq. (29.33), the maximum value of R in a closed universe — at the time of maximum expansion — is 0002 0003 0 1 Rmax D : D 0 0004 1 1 C z max Then at the time of maximum expansion,
dh 2. 0 0004 1/ 0 1 1 1 D cos00021 1 0004 D cos00021 .00041/ D : Cuniv 20003 0 0 0004 1 20003 2 29.48 For a flat universe, the proper distance is d.t/ D R.t/$ (Eq. 29.148). Using Eq. (29.163) for $ and Eq. (29.31) for the scale factor shows that the proper distance of the photon is 0002 00031=3 # 0002 00032=3 0002 00032=3 ' t t 2c 3 $e 0004 : d.t/ D 2 tH H0 t0
Solutions for An Introduction to Modern Astrophysics
211
To evaluate $e , note that if the photon is just now (R D 1) arriving from the present particle horizon, then the present proper distance d.t/ D R.t/$ to its point of emission at $e is d D $e . But this is just the present horizon distance, dh;0 D 2c=H0 (Eq. 29.159), so $e D 2c=H0 . Inserting this, we find ' 0002 00032=3 0002 00032=3 0002 00031=3 # 3 t 2c t d.t/ D 10004 : 2 tH H0 t0 Again using t0 D 2tH =3 for a flat universe, we arrive at the proper distance of the photon from Earth as a function of time, '0002 0003 # t 2=3 t 2c ; 0004 d.t/ D H0 t0 t0 which is Eq. (29.165). To find the maximum distance, we solve for the time when
2c H0
'0002 0003 1 2=3 t0
d Œd.t/0004 D0 dt # 2 00021=3 1 D0 0004 t 3 t0 t 8 D : t0 27
Inserting this into d.t/, the maximum proper distance of the photon during its journey is dmax D
8c ; 27H0
or, expressed as a fraction of the present horizon distance dh;0 D 2c=H0, dmax 4 D : dh;0 27 29.49
(a)
Beginning with the Robertson–Walker metric (Eq. 29.106), setting ds D 0 for a light ray, d0005 D d D 0 for the radial expansion of a shell, and k D 0 for a flat universe, we find 0004
c dt D d$ 0 : R0 .t/
From Eq. (29.8), H .t/ D R.t/00021 dR.t/=dt, and dt D dR=HR, implying c d$ 0 D 0004 dR: HR02 Again changing variables using Eq. (29.4), R0 D .1 C z 0 /00021 and dR0 D 0004.1 C z 0 /00022 dz 0 , we have c d$ 0 D dz 0 : H .z 0 / Using Eq. (29.27) to write H .z 0 / D H0 .1 C z 0 /3=2 with 0 D 1 leads to d$ 0 D
1 c dz 0 : H0 .1 C z 0 /3=2
Integrating $ 0 from 0 to $ and z 0 from 0 to z gives
1 2c : 10004 p $ .z/ D H0 1Cz
212
Chapter 29 Cosmology (b) The proper distance is given by Eq. (29.144), which implies that dp .t/ D R.t/$: Thus, dp .z/ D R.z/$ D (c)
i p 2c h .1 C z/ 0004 1 C z : H0
The horizon distance is given by Eq. (29.153), Z
t
dh .t/ D R.t/ 0
cdt 0 : R.t 0 /
Changing variables from t 0 to z 0 requires Eq. (29.31) so that t0 D and
2tH 03=2 R ; 3
dt 0 D tH R01=2 dR0 D 0004tH .1 C z/00025=2 dz 0 :
Therefore, ctH dh.z/ D 0004 1Cz
Z
000bz
dz 0 2ctH 2c 1 D D ; 0 5=2 0 3=2 3.1 C z/ .1 C z / .1 C z / 3H0 .1 C z/5=2 1
z 1
where the last expression was obtained from tH 0006 1=H0 . For the WMAP values of z D 0 and H0 D 2:30 0003 10000218 s00021 we find dh D 8:7 0003 1025 m D 2:8 Gpc. This is roughly a factor of five below the result obtained in Eq. (29.159). 29.50 We begin with the Robertson–Walker metric (Eq. 29.106), setting ds D 0 for a light ray, and considering only radial motion (d0005 D d D 0). This implies that Z
t0 t
c dt 0 D0004 R0
Z
0 $
d$ 0 p : 1 0004 k$ 02
(S29.14)
The right-hand side can be integrated directly by Z 0004
0 $
d$ 0
p 1 0004 k$ 02
1 D 0004p k
Z
0 $
q
d$ 0 1 k
0004 $ 02
p 1 D p sin00021 k$: k Note that this solution is general so long as we allow complex values to be used when k < 0. This can be seen by noting that e i0003 0004 e 0002i0003 ; sin 0005 D 2i and e 0003 0004 e 00020003 : sinh 0005 D 2 Comparing gives sin i 0005 D i sinh 0005.
Solutions for An Introduction to Modern Astrophysics
213
The left-hand side of Eq. (S29.14) is more challenging to integrate. Recalling that H D we can change variables to give
Z
t0 t
c dt 0 Dc R0
1 dR ; R dt Z
1 R
dR0 : H .R0 /R02
From Eq. (29.27) and R D 1=.1 C z/, we find Z
t0
c dt 0 c D R0 H0
Z
dR0
1
R01=2 ŒR0 .1 0004 0 / C 0 00041=2 Z 1 dR0 c D p h i1=2 H0 1 0004 0 R 01=2 0 0 R 0004 0 00021 R
t
R
c D p H0 0 0004 1
Z
dR0
1 R
R01=2
h
0 0 00021
0004 R0
i1=2 :
This can be integrated by making the change of variable u D R1=2 , giving an integral of the form Z
t0
c t
dt 0 2c D p 0 R H0 0 0004 1
Z
R 0 D1 R 0 DR
q
du 0 0 00021
0004 u2
:
Again we obtain a solution involving sin00021 . Equating the solutions for the two sides of Eq. (S29.14), and using Eq. (29.25) to write p p H0 0 0004 1 kD ; c eventually leads to the desired result; Mattig’s relation. 29.51 Beginning with Eq. (29.168), and setting rel;0 D 0, we have Z z dz 0 I.z/ 0006 p : m;0 .1 C z 0 /3 C ƒ;0 C .1 0004 0 /.1 C z 0 /2 0 Now a Taylor’s series for a general function f .z 0 / is given by ˇ ˇ ˇ 3 0 ˇ 1 @2 f .z 0 / ˇ 0 @f .z 0 / ˇˇ 0 0 ˇ z 0004 z 0 2 C 1 @ f .z / ˇ z 0 0004 z 0 3 C : (S29.15) f .z 0 / D f .z00 /C 0004 z z C 0 0 0 ˇ ˇ ˇ 0 02 03 @z z0 2! @z 3! @z z0 z0 0
0
0
Setting f .z 0 / equal to the integrand, and expanding around z 0 D 0, ˇ @f .z 0 / ˇˇ 1 D 0004 f 3 .0/ Œ3 m;0 C 2 .1 0004 0 /0004 ; @z 0 ˇ0 2 where f .0/ D p
1 m;0 C ƒ;0 C .1 0004 0 /
D 1:
(S29.16)
214
Chapter 29 Cosmology The last expression is obtained because m;0 C ƒ;0 D 0 . We can also simplify the rest of Eq. (S29.16) by also noting that q0 D 12 m;0 0004 ƒ;0 . This implies that 3 m;0 C 2 .1 0004 0 / D m;0 C 2 . 0 0004 ƒ;0 / C 2 .1 0004 0 / D 2q0 C 2: At this point the first two terms of Eq. (S29.15) are f .z 0 / D 1 0004 .1 C q0/z 0 C : To obtain the second-order term (the z 02 term), we proceed similarly. Evaluating the second derivative, 0010 000f i @2 f .z 0 / 1 3 0 h @ 0 2 0 0004 D .z / 3 C 2 .1 0004 / 1 C z f 1 C z m;0 0 @z 02 @z 0 2 h
2 i2 1 3 0 3 D f 5 .z 0 / 3 m;0 1 C z 0 C 2 .1 0004 0 / 1 C z 0 0004 f .z / 6 m;0 1 C z 0 C 2 .1 C 0 / : 4 2 Evaluating at z 0 D 0, and noting that 6 m;0 C 4 .1 0004 0 / D 4q0 C 4 leads to ˇ @2f .z 0 / ˇˇ D 3q02 C 4q0 0004 0 C 2: @z 02 ˇ0 Substitution into the second term of Eq. (S29.15), combined with our previous results for the zeroth- and first-order terms gives the desired result.
29.52
(a)
The first three terms in a general Taylor series for R.t/, expanded about the present time, t0 , are as given in the problem, ˇ ˇ dR ˇˇ 1 d 2 R ˇˇ R.t/ D R.t0 / C .t 0004 t0 / C .t 0004 t0 /2 C : dt ˇ 2 dt 2 ˇ t0
t0
At the present time, the scale factor is R.t0 / D 1, and ˇ ˇ dR ˇˇ d 2 R ˇˇ D H and D 0004H02 q0 ; 0 dt ˇ t0 dt 2 ˇ t0 from Eqs. (29.8) and (29.54). Therefore, 1 R.t/ D 1 0004 H0 .t0 0004 t/ 0004 H02 q0 .t0 0004 t/2 C : 2 (b) Using 1=.1 0004 x/ D 1 C x C x 2 C , we have 1 1 h i D R.t/ 1 0004 H0 .t0 0004 t/ C 12 H02 q0 .t0 0004 t/2 C
1 D 1 C H0 .t0 0004 t/ C H02 q0 .t0 0004 t/2 C 2
000b2 1 C H0 .t0 0004 t/ C H02 q0 .t0 0004 t/2 C C 2 1 D 1 C H0 .t0 0004 t/ C H02 q0.t0 0004 t/2 C H02 .t0 0004 t/2 C 2 0002 0003 1 2 D 1 C H0 .t0 0004 t/ C H0 1 C q0 .t0 0004 t/2 C : 2 to second-order in H0 .t0 0004 t/. (Long division is actually easier!)
Solutions for An Introduction to Modern Astrophysics (c)
215
From Eq. (29.4), 1 C z D 1=R.t/, so
0002 0003 1 1 C z D 1 C H0 .t0 0004 t/ C H02 1 C q0 .t0 0004 t/2 C ; 2
or t0 0004 t D
0002 0003 1 z 0004 H0 1 C q0 .t0 0004 t/2 C : H0 2
But to first-order, t0 0004 t D z=H0 , so substituting on the right gives 0003 2 0002 z z 1 0004 1 C q0 C t0 0004 t D H0 2 H0 to second-order in z. (d) We start with Eq. (29.138),
Z
t0 te
c dt D R.t/
Z
$e 0
d$ p : 1 0004 k$ 2
p Using the first two terms in our expression for 1=R.t/ in the left-hand integral, and 1= 1 0004 x D 1 C x=2 C with the right-hand integral, we obtain Z t0 Z $e
1 2 1 C k$ C d$; c Œ1 C H0 .t0 0004 t/ C 0004 dt D 2 te 0 or
0006 0007 c 1 c.t0 0004 te / C cH0 t0 .t0 0004 te / 0004 H0 t02 0004 te2 C D $e C k$e3 C : 2 6 3 3 The $e -term can be dropped because it is proportional to .t0 0004 te / at least. We also drop the “e” subscripts that identify the time and place of the photon’s origin. Solving for $ , we obtain an expression for the comoving coordinate of the photon as a function of time,
1 $ D c.t0 0004 t/ 1 C H0 .t0 0004 t/ C : 2 (e)
Using the expression for .t0 0004t/ derived in part (c) results in an expression for $ in terms of the redshift, 0003 2 0002 0002
0003
z z 1 1 z 0004 1 C q0 C 1 C H0 C C $Dc H0 2 H0 2 H0
cz 1 D 1 0004 .1 C q0 /z C ; H0 2 to second-order in z.
29.53 A direct comparison of Eq. (29.168) with Eq. (29.122) and Eq. (29.196), combined with substitutions into Eqs. (29.170) – (29.172) lead immediately to the desired result. 29.54
(a)
The angular diameter is obtained from Eq. (29.190), 0005D
DH0 .1 C z/ : cS.z/
For a flat universe, S.z/ D I.z/ (Eq. 29.173), rel;0 D ƒ;0 D 0, and m;0 D 0 D 1, implying from Eq. (29.168) that 0003 0002 Z z dz 0 1 : D 2 10004 p S.z/ D 0 3=2 1Cz 0 .1 C z /
216
Chapter 29 Cosmology Substitution and some simplification gives 0005D
DH0 .1 C z/3=2 : p 2c 1Cz00041
(b) For a minimum to occur, it is necessary that d0005=dz D 0, or # # ' ' p p DH0 3 1 1Cz DH0 .1 C z/ 0004 32 1 C z 1Cz 0D 0004 p D : 2 p 2c 2 1 C z 0004 1 2 p1 C z 0004 1 2 2c 1Cz00041 Setting the numerator equal to zero leads to the quadratic equation 0002 0003 0007 1 5 5 0006 2 z 0004 z0004 D z0004 zC1 : 4 4 4 Therefore either z D 5=4 or z D 00041. Choosing the physical solution, we find that z D 1:25 produces a minimum in 0005. (Note that z D 00041 implies that 0005 D 0.) (c)
Using WMAP values, 0005min D 8 0003 1000024 rad D 2:70 .
29.55 The required X-ray flux data were inadvertently omitted in the first printing of An Introduction to Modern Astrophysics. The X-ray flux for Abell 697 is FX D 5:77 0003 10000215 W m00022 . and the X-ray flux for Abell 2218 is FX D 7:16 0003 10000215 W m00022 (a)
Beginning with Eq. (27.19), the X-ray luminosity of the intracluster gas is given by 0002 0003 40003 C 0 LX D LvolV D n2eTe1=2 R3 ; 3 where C 0 D 1:42000310000240 (note that there was an error in the exponent of the constant in the first printing of An Introduction to Modern Astrophysics). From the Sunyaev–Zel’dovich effect (Eq. 29.64) 0006T kTe kTe '2 D2 0007T ne .2R/; 2 T0 me c me c 2 where the last expression involving the Thomson cross section (0007T ) and the number density of electrons (ne ) was obtained from the result of Problem 29.23(b). Solving for ne we have ne D
.0006T =T0 /me c 2 : 4kTe0007T R
Substituting into LX results in ' # 0003 0 .0006T =T0 /2 m2e c 4 RTe00023=2: C LX D 12 k 2 0007T2 This implies that the measured flux at Earth is given by FX D
LX H02 LX LX D D ; 40003.1 C z/2 c 2 S 2 .z/ 40003 dL2 40003.1 C z/4 dA2
where Eqs. (29.191) and (29.192) were used. Substituting LX and solving for one power of H0 gives H0 D
48.1 C z/2 c 2 FX k 2 Te3=2 0007T2 S 2 .z/ : RC 0 .0006T =T0 /2 m2e c 4 H0
Solutions for An Introduction to Modern Astrophysics
217
The remaining value of H0 in the right-hand side can be replaced using Eq. (29.193). Also substituting D D 2R we arrive at ! 3=2 3=2 96k 20007T2 FX Te FX Te 3 S.z/ D Cf .z/ : .1 C z/ H0 D C 0 m2e c 3 0005.0006T =T0 /2 0005.0006T =T0 /2 The numerical value of the constant is C D 2:5 0003 10000226 , and f .z/ 0006 .1 C z/3 S.z/. (b) For the purposes of this problem, comparing Eqs. (29.176) and (29.180) gives 1 S.z/ ' z 0004 .1 C q0/z 2 ; 2 where q0 ' 12 m;0 0004 ƒ;0 D 00040:595 for WMAP values. From the data for Abell 697: Te ' 9:2 0003 107 K (the value should not be halved, as incorrectly suggested in the first printing of An Introduction to Modern Astrophysics), 0005 D 4600 D 2:2 0003 1000024 rad, 0006T =T0 D 3:84 0003 1000024 , FX D 5:77 0003 10000215 W m00022 , z D 0:282, and S.z/ D 0:266. Substituting values gives H0;697 D 2:2 0003 10000218 s00021 : This would correspond to a value of h D 0:68 (differing from the WMAP value by 4.4%). From the data for Abell 2218: Te ' 8:4 0003 107 K (again, the value should not be halved), 0005 D 6900 D 3:3 0003 1000024 rad, 0006T =T0 D 2:92 0003 1000024 , FX D 5:77 0003 10000215 W m00022 , z D 0:171, and S.z/ D 0:165. Substituting values gives H0;2218 D 1:0 0003 10000218 s00021 : This would correspond to a value of h D 0:31 (differing from the WMAP value by 56%). 29.56 Using Eq. (29.166), with the expression for R.t/ from Eq. (29.132) substituted on the left-hand side, and the expression for R.t/ from Eq. (29.133) substituted on the right-hand side, we have !2=3 Z 0002 0003 0003 Z 0002 t0 t 00022=3 2 4 ƒ;0 1=3 1 0002H0p ƒ;0 t p dt D e dt: (S29.17) tH m;0 3 m;0 tmva te Focusing our attention on the left-hand side initially, we can change variables from t to z by use of Eq. (29.92), t.z/ 2 1 D p ; tH 3 .1 C z/3=2 m;0 which also implies that 2tH dt D p 3 m;0
0002 0003 3 0004 .1 C z/00025=2 dz: 2
Making the substitutions, simplifying, and evaluating, the left-hand side of Eq. (S29.17) becomes !2=3 Z 0002 0003
Z z t0 t 00022=3 1 dz 0 tH 2tH 2 p : 1 0004 dt D p D p p tH 1Cz 3 m;0 m;0 0 .1 C z 0 /00023=2 m;0 te Eq. (S29.17) now becomes (after evaluating the right-hand side) 0002
0003 2tH 4 ƒ;0 1=3 1 D 10004 p p m;0 1Cz m;0
1 p
0004H0 ƒ;0
!
h i p e 00021 0004 e 0002H0 ƒ;0 tmva :
Rearranging, and noting that tH H0 D 1, we have #0002 'p p 0003 p 2tH H0 ƒ;0 m;0 1=3 1Cz00041 0002H0 ƒ;0 tmva e D p p 4 ƒ;0 m;0 1Cz
218
Chapter 29 Cosmology
D
! p !2=3 ' p # p 2 ƒ;0 m;0 1Cz00041 p p p m;0 2 ƒ;0 1Cz
D
!1=3 ' p # p 2 ƒ;0 1Cz00041 p p : m;0 1Cz
Taking the natural logarithm of both sides and noting that 0004 ln x D ln x 00021 gives the desired result. To find the largest value of z for which tmva > tH requires that 2 p !3 !1=3 p m;0 1 1Cz 5 < 1: p ln 4 p p ƒ;0 2 ƒ;0 1Cz00041 Rewriting,
!1=3 p m;0 p 2 ƒ;0
or
p 1Cz 0004 p 1Cz00041
! p p 1Cz < e ƒ;0 ; p 1Cz 00041 !1=3 p p 2 ƒ;0 p e ƒ;0 < 0: m;0
The critical case (when the left-hand side is identically zero) can be solved by a root-finding algorithm, such as a Newton method. An example Fortran 95 code that also returns the specific results for z D 0:1, 0.5, 1.0, and 1.5 is given below. The results are: zlargest D 0:9624 tmva =tH D 3:126
for z D 0:1
D 1:520
for z D 0:5
D 0:973
for z D 1:0
D 0:707
for z D 1:5
Program t_mva_Solver IMPLICIT NONE REAL(8), PARAMETER REAL(8), PARAMETER REAL(8), PARAMETER REAL(8)
:: :: :: ::
H_0 = 2.3E-18 !sˆ-1 t_H = 1/H_0 Omega_m0 = 0.27, Omega_Lambda0 = 0.73 K
REAL(8),
::
rel_error = 1E-6
::
z, dz, F, dF
PARAMETER
REAL(8)
K = (2*SQRT(Omega_Lambda0)/SQRT(Omega_m0))**(1/3.0)*EXP(SQRT(Omega_Lambda0)) z = 1 dz = 10000 DO WHILE (ABS(dz/z) > rel_error) F = SQRT(1+z)/(SQRT(1+z)-1) - K dF = 1/(2*SQRT(1+z)*(SQRT(1+z)-1)) - 1/(2*(SQRT(1+z)-1)**2) dz = -F/dF z = z + dz
Solutions for An Introduction to Modern Astrophysics WRITE(*,*) 'z, dz = ', z, dz END DO !Find specific values for t_mva z = 0.1 WRITE(*,*) 'For z = ', z, ' t_mva/t_H = ', & & 1/SQRT(Omega_Lambda0)*LOG((SQRT(Omega_m0)/(2*SQRT(Omega_Lambda0)))**(1/3.) & & *(SQRT(1+z)/(SQRT(1+z) - 1))) z = 0.5 WRITE(*,*) 'For z = ', z, ' t_mva/t_H = ', & & 1/SQRT(Omega_Lambda0)*LOG((SQRT(Omega_m0)/(2*SQRT(Omega_Lambda0)))**(1/3.) & & *(SQRT(1+z)/(SQRT(1+z) - 1))) z = 1 WRITE(*,*) 'For z = ', z, ' t_mva/t_H = ', & & 1/SQRT(Omega_Lambda0)*LOG((SQRT(Omega_m0)/(2*SQRT(Omega_Lambda0)))**(1/3.) & & *(SQRT(1+z)/(SQRT(1+z) - 1))) z = 1.5 WRITE(*,*) 'For z = ', z, ' t_mva/t_H = ', & & 1/SQRT(Omega_Lambda0)*LOG((SQRT(Omega_m0)/(2*SQRT(Omega_Lambda0)))**(1/3.) & & *(SQRT(1+z)/(SQRT(1+z) - 1))) PAUSE STOP END
29.57
(a)
See Fig. S29.2. The data were generated by the Fortran 95 code given below.
(b) z D 0:08.
Figure S29.2: Various distances (in units of c=H0) as a function of z for WMAP values.
PROGRAM Problem_29_57 IMPLICIT NONE REAL(8), REAL(8) REAL(8) REAL(8) INTEGER
PARAMETER
::
Omega_m0 = 0.27, Omega_rel0 = 8.24E-5, Omega_L0 = 0.73, Omega_0 = 1.02
:: :: :: ::
z_max, z, dz I, g_z, g_zplus, I_1, S d_p0, d_L, d_A, d_nr, d_r n = 100000
OPEN(UNIT = 10, FILE = 'Prob_29_57.txt')
219
220
Chapter 29 Cosmology z_max = 0 DO WHILE (z_max <= 4.001) dz = z_max/n ! integrand of Eq. (29.168) z = 0 g_z = 1/SQRT(Omega_m0*(1 + z)**3 + Omega_rel0*(1 + z)**4 + Omega_L0 + (1 - Omega_0)*(1 + z)**2) I = 0 DO WHILE (z < z_max) z = z + dz g_zplus = 1/SQRT(Omega_m0*(1 + z)**3 + Omega_rel0*(1 + z)**4 + Omega_L0 + (1 - Omega_0)*(1 + z)**2) I = I + (g_z + g_zplus)*dz/2 g_z = g_zplus END DO S = SIN(I*SQRT(Omega_0 - 1))/SQRT(Omega_0-1) d_L d_p0 d_A d_r d_nr
= = = = =
S*(1 + z_max) I d_L/(1 + z_max)**2 ((z+1)**2 - 1)/((z+1)**2 + 1) z
!Eq. !Eq. !Eq. !Eq. !Eq.
!Eq. (29.174) for Omega_0 > 1 (29.184) (29.169) (29.192) (27.7) (27.8)
in in in in in
units units units units units
of of of of of
c/H_0 c/H_0 c/H_0 c/H_0 c/H_0
WRITE(10,’(7F13.6)’) z, d_L, d_p0, d_A, d_r, d_nr, ABS((d_L - d_r)/d_L) z_max = z_max + 0.01 END DO PAUSE STOP END
29.58
(a)
At z D 4:25, Eq. (29.4) gives R D 0:1905. The age of the universe at that value of z can be obtained by numerically integrating Eq. (29.128). An example Fortran 95 program is given at the end of this problem. The result is t D 1:4610 Gyr D 0:1066t0.
(b) The present proper distance to the radio galaxy 8C 1435C63 comes from Eq. (29.169), so dp;0 D 7570 Mpc. (c)
The universe was smaller by a factor of 1=.1 C z/ when the light was emitted, implying that dp D 1442 Mpc.
(d) The luminosity distance is given by Eq. (29.184), dL D 3:93 0003 104 Mpc. (e)
The angular diameter distance is given by Eq. (29.191), dA D 1427 Mpc.
(f) With 0005 D 500 D 2:42 0003 1000025 rad, the linear diameter of the galaxy is given by Eq. (29.190), or D D 34:58 kpc. (g) Repeating part (f) for z D 1 (so the galaxy is closer), we have D D 40:8 kpc. Even though the galaxy is closer, a larger galaxy is needed to produce the same angular size! PROGRAM Problem_29_58 USE Constants, ONLY : c, G, pi, yr, pc, degrees_to_radians REAL(8), REAL(8), REAL(8),
PARAMETER PARAMETER PARAMETER
:: :: ::
REAL(8), REAL(8),
PARAMETER PARAMETER
:: ::
rho_c0 = 9.47E-27 Omega_m0 = 0.27, Omega_rel0 = 8.24E-5, Omega_L0 = 0.73, Omega_0 = 1.02 rho_m0 = rho_c0*Omega_m0, rho_rel0 = rho_c0*Omega_rel0, & & rho_L0 = rho_c0*Omega_L0 t_0 = 13.7 !Gyr H_0 = 2.30E-18 !sˆ-1
:: :: ::
z_max, z, dz R_max, R, dR t, f_R, f_Rplus
REAL(8) REAL(8) REAL(8)
Solutions for An Introduction to Modern Astrophysics
221
REAL(8) REAL(8) INTEGER
:: :: ::
I, g_z, g_zplus, I_1, S d_p0, d_L, d_A, D, theta n = 1000000
LOGICAL
::
save = .FALSE.
z_max R_max dR dz
= = = =
4.25 1/(1 + z_max) R_max/n z_max/n
! integrand of Eq. (29.128) R = 0 f_R = R/SQRT(rho_m0*R + rho_rel0 + rho_L0*R**4) ! integrand of Eq. (29.168) z = 0 g_z = 1/SQRT(Omega_m0*(1 + z)**3 + Omega_rel0*(1 + z)**4 + Omega_L0 + (1 - Omega_0)*(1 + z)**2) t = 0 I = 0 DO WHILE (z < z_max) R = R + dR f_Rplus = R/SQRT(rho_m0*R + rho_rel0 + rho_L0*R**4) t = t + (f_R + f_Rplus)*dR/2 f_R = f_Rplus z = z + dz g_zplus = 1/SQRT(Omega_m0*(1 + z)**3 + Omega_rel0*(1 + z)**4 + Omega_L0 + (1 - Omega_0)*(1 + z)**2) I = I + (g_z + g_zplus)*dz/2 g_z = g_zplus IF (z > 1 .AND. save .FALSE.) THEN I_1 = I save = .TRUE. END IF END DO S = SIN(I*SQRT(Omega_0 - 1))/SQRT(Omega_0-1) t = t*SQRT(3/(8*pi*G)) t = t/(1E9*yr) WRITE(*,*) 't = ', t, WRITE(*,*) 'I WRITE(*,*) 'S
!Eq. (29.174) for Omega_0 > 1
!include leading constant in Eq. (29.128) !convert to Gyr 'Gyr t/t0 = ', t/t_0
= ', I = ', S
d_p0 = c*I/H_0 !Eq. (29.169) d_p0 = d_p0/(1E6*pc) !convert to Mpc WRITE(*,*) 'd_p0 = ', d_p0, 'Mpc' WRITE(*,*) 'd_p
= ', d_p0/(1 + z_max), 'Mpc
(at the time light was emitted)'
d_L = c*S*(1 + z_max)/H_0 !Eq. (29.184) WRITE(*,*) 'd_L = ', d_L/(1E6*pc), 'Mpc' d_A = d_L/(1 + z_max)**2 !Eq. (29.192) WRITE(*,*) 'd_A = ', d_A/(1E6*pc), 'Mpc' theta = 5 !arcsec theta = (theta/3600)*degrees_to_radians D = d_A*theta*1000 !Eq. (29.190) WRITE(*,*) 'D = ', D/(1E3*pc), 'kpc (when z = 4.25)' D = ((c*I_1/H_0)*theta/2)/(1E3*pc) !Eq. (29.190) for z = 1 with conversion to kpc WRITE(*,*) 'D = ', D, 'kpc (when z = 1)' PAUSE STOP END
CHAPTER 30
The Early Universe 30.1 Using Eqs. (9.7) and (9.65) with T D 2:725 K, the present number density of CBR blackbody photons is nphoton D aT 3 =.2:70k/ D 4:11 0003 108 m00023 . The present number density of baryons in a universe of pure hydrogen is nb;0 D 0002b;0 =mH . With 0002b;0 D 4:17 0003 10000228 kg m00023 (Eq. 29.17), nb;0 D 0:25 m00023 . The ratio of the two is nb;0 D 6:08 0003 10000210 : nphoton There are roughly 1.6 billion photons for every baryon in the universe. 30.2
(a)
If most of the dark matter in the Milky Way’s dark halo is in the form of ordinary objects such as brown dwarfs and Jupiter-sized objects, this would favor hot dark matter for the nonbaryonic matter in the universe. If the universe were dominated by cold dark matter, it would have also accumulated in the Galactic dark halo and overwhelmed the contribution of ordinary objects.
(b) If less than 20% of the Milky Way’s dark halo is in the form of ordinary objects, then no conclusion can be reached. The nonbaryonic matter could be either hot or cold. 30.3
(a)
See Fig. S30.1.
(b) By inspection of Fig. S30.1 for k D C1 m00021 , there is a point of stable equilibrium at x D 0. (c)
For k D 00041 m00021 , the points of equilibrium occur where dV =dx D 0, 0007 d 0006 2 0004x C 0:5x 4 D 0 dx 00042x C 2x 3 D 0; which has solutions x D 0 (unstable) and x D ˙1 (stable).
(d) For the case of k D 00041 m00021 , when the ball is “at rest” at the origin, it is in a symmetric condition (reflection symmetry about the line x D 0). However, Heisenberg’s uncertainty principle, 0006x 0006p 0002 (Eq. 5.19), implies that the ball cannot be perfectly at rest. This law of quantum mechanics requires that the ball move from its symmetric but unstable equilibrium position at x D 0 and roll down toward an unsymmetric, stable position of lower potential energy at either x D C1 or x D 00041. In an analogous manner, the universe was initially in a supercooled false vacuum state of unbroken symmetry. Then a quantum fluctuation governed by the uncertainty principle caused a transition to a lower energy state of broken symmetry, the true vacuum. 30.4 The end of the GUTs epoch occurred when the temperature was approximately T 0002 1028 K. The energy density of blackbody radiation at that temperature is (Eq. 9.7) u D aT 4 0002 8 0003 1096 J m00023 . Comparing this with the energy density of the false vacuum, ufv 1098 J m00023 (Eq. 30.9), and considering the crude nature of this estimate, these two energy densities can be described as being of the same order of magnitude. 30.5 The Planck time is tp D 5:39 0003 10000244 s (Eq. 30.3). According to standard Big-Bang cosmology, the scale factor at that time (deep in the radiation era) was, from Eq. (29.87), Rstandard D .1:51 0003 10000210 s00021=2/g00041=4 tp1=2 : 222
Solutions for An Introduction to Modern Astrophysics
223
12
k = +1 m −1 k = −1 m −1
10
V( x) (J)
8 6 4 2 0 −2
−1
0 x (m)
1
2
Figure S30.1: Results for Problem 30.3(a). If we adopt a value of g0004 ' 160 very early in the universe, we find Rstandard D 1:2 0003 10000231 . Assuming a flat universe, the size of the observable universe (the present horizon distance, Eq. 29.159) is dh;0 D 4:50 0003 1026 m. At the Planck time, the size of the observable universe was Rstandard dh;0 D 5:61 0003 1000025 m. To take inflation into account, the above value of R at the Planck time must be divided by roughly e t = i D e 100 (see Eq. 30.12) to find Rinflation D 4:46 0003 10000275 . At the Planck time, the size of the observable universe was Rinflation dh;0 D 2:01 0003 10000248 m. 30.6 Model a cosmic string as a long cylinder filled with false vacuum with an energy density of ufv 1:6 0003 1098 J m00023 (Eq. 30.9). If the linear mass density of a cosmic string is 0002 1021 kg m00021 , then the energy per unit length is c 2 D 9 0003 1037 J m00021 from Eq. (4.47). Assuming that the string has a circular cross section of radius r , then c 2 D ufv 0003 r 2, or s D 4:2 0003 10000231 m: ufv 0003
r Dc 30.7
(a)
In Example 29.4.1 we found that when T D 109 K, the horizon distance was dh D 1:07 0003 1011 m. The density of baryonic matter at that time comes from combining Eqs. (29.5) and (29.58) with T0 D 2:725 K and T D 109 K to give 0002 0002b D 0002b;0
T T0
00033
D 0:0206 kg m00023 :
The mass of baryonic matter contained within a causally connected region of diameter dh is then 4 mb D 0003 3
0002
dh 2
00033 0002b D 1:32 0003 1031 kg D 6:6 Mˇ :
This is about an order of magnitude smaller than the Jeans mass of 200 Mˇ at T D 109 K shown in Fig. 30.7. (b) From Eqs. (29.87) and (29.154), dh D 2ct / R2 . Also, R3 0002b D 0002b;0 (Eq. 29.5), so 0002b / R00023 . Thus mb / dh3 0002b / .R2 /3 R00023 D R3 . But from T0 D RT (Eq. 29.58), we have R / T 00021 , so mb / T 00023 .
224
Chapter 30 The Early Universe But before recombination, the Jeans mass was also proportional to T 00023 (Fig. 30.7). This means that the value of mb remained about an order of magnitude smaller than the Jeans mass throughout the radiation era.
30.8 (Note: An oversight occurred in the first printing of An Introduction to Modern Astrophysics; the WMAP values should be used rather than 0 D 1 and h D 1.) Eq. (29.84) shows that the transition from the radiation era to the matter era (R D Rr;m ) occurred at about tr;m D 1:74 0003 1012 s D 5:52 0003 104 yr (WMAP values): At that time the scale factor was Rr;m D 3:05 0003 1000024 according to WMAP values. According to Eq. (29.154) the horizon distance was dh D 2ct D 1:04 0003 1021 m. Furthermore, the density of baryonic matter was 0002b D 0002b;0=R3 D 1:47 0003 10000217 kg m00023 (using Eq. 29.5 with 0002b;0 D 4:17 0003 10000228 kg m00023 from Eq. 29.17). The mass of baryonic matter contained within a causally connected region of diameter dh is therefore 0002 00033 4 dh 0002b D 6:9 0003 1046 kg D 3:5 0003 1016 Mˇ : mb D 0003 3 2 This is a very large mass, comparable to that of a large cluster of galaxies. Assuming that a typical baryonic density fluctuation was less massive than this, we conclude that most of the baryonic density fluctuations became sub-horizon-sized by the end of the radiation era. 30.9 Equation (30.23) describes the growth of a density fluctuation during the matter era from its initial value at time ti as, 0002 0003 0002 00032=3 t ı0002 ı0002 : D 0002 0002 i ti From Eq. (29.43), t / .1 C z/00023=2 for a nearly flat universe, so if ı0002=0002 D 1 today, we have 2=3 0002 0003 0002 00032=3 0002 0003
t 1 ı0002 ı0002 D ; 1D 0002 i ti 0002 i .1 C z/00023=2 or 0002 0003 ı0002 1 D : 0002 i 1Cz If a density fluctuation is .ı0002=0002/1 at redshift z1 and .ı0002=0002/2 at redshift z2 , then #2=3 0002 0003 0002 0003 0002 00032=3 0002 0003 ' ı0002 t2 ı0002 ı0002 .1 C z2 /00023=2 D D 0002 2 0002 1 t1 0002 1 .1 C z1 /00023=2 so
.ı0002=0002/2 1 C z1 D : .ı0002=0002/1 1 C z2
30.10 Using R3 0002b D 0002b;0 (Eq. 29.5) and T0 D R T .R/ (Eq. 29.58), we have 0002 D C T 3 , where C is a constant. Setting 0002 D 00020 C ı0002 and T D T0 C ıT , we have 00020 C ı0002 D C.T0 C ıT /3 0002 0003 0002 0003 ı0002 ıT 3 D C T0 1 C 3 00020 1 C C 00020 T0 ıT ı0002 D3 00020 T0 to first order in the deltas.
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30.11 For a flat universe in the matter era, when decoupling occurred at Œzdec 0004WMAP D 1089 ' 1100, Eq. (29.156) gives the horizon distance as dh D
1 2c 5:02 0003 1026 m p D D 1:4 0003 1022 m D 0:45 Mpc: 3=2 .1 C z/3=2 H0 m;0 .1 C z/
This is the linear diameter of the largest causally connected region observed for the CBR. From Eq. (29.193), the angular diameter of this region is H0 dh .1 C z/ : 0005D c S.z/ S.z/ 0006 I.z/ for a flat universe, and must be evaluated numerically. This can be done using PROGRAM Problem_29_58 given in the solution to Problem 29.58. The value of S.1089/ D 3:315. Using WMAP values we find 0005 D 0:035 rad D 2:0ı : 30.12 The characteristic distance associated with Silk damping is given by s r ctdec ctdecmH dSilk D D : n0007 0002b;0.1 C z/3 0007 Using WMAP values, Œtdec 0004WMAP D 379 kyr, Œzdec 0004WMAP D 1089, and Œ0002b;0 0004WMAP D 4:17 0003 10000228 kg m00023 , we find dSilk D 4 0003 1020 m. Now, from Eq. (29.190) with D D dSilk and S.z/ D 3:315 (see the solution to Problem 30.11, 0005 D 1:03 0003 1000023 rad D 0:059ı. This corresponds to a value of ` ' 0003=0005 D 3100. 30.13 The first stars ignited at the time of reionization. According to the WMAP CMB polarization results, this corresponded to z D 10:9, or R D 1=.1 C z/ D 0:084. From Eq. (29.129), this happened when t D 1:36 0003 1016 s D 430 Myr.