Astronomy 404 Fall 2012
Solutions to Homework Set #1
1. (a) The distance OA = radius of unit sphere = 1. The projection of OA onto
the x-y plane is therefore OAxy = cos A , and onto the z-axis OAz = zA =
sin A . Breaking OAxy into x- and y- components, we

Astronomy 404 Fall 2011
Solutions to Homework Set #5
1. From Eq. (8.7), use gn = 2n2 and E1 = 13.6 eV, E2 = 3.40 eV, and E3 = 1.51 eV
with T = 10, 000 K to get
ZI = g1 + g2 e(E2 E1 )/kT + g3 e(E3 E1 )/kT
= 2 + 5.79 105 + 1.45 105 .
Thus ZI 2.
2. (a) Using

Astronomy 404 Fall 2012
Solutions to Homework Set #3
1. As measured in frame S , the spacetime interval is
(s )2 = (ct )2 (x )2 (y )2 (z )2 .
Using Eqs. (4.16)(4.19), we have
2
(s ) =
c(t ux/c2 )
1 u2 /c2
2
x ut
2
1 u2 /c2
(y)2 (z)2 .
After some algebra

Astronomy 404 Fall 2012
Solutions to Homework Set #4
1. (a) Referring to Figure 5.3, conservation of momentum gives
pi = pf cos + pe cos
0 = pf sin pe sin ,
where pi and pf are the initial and nal photon momenta, respectively, and pe is
the momentum of t

Astronomy 404 Fall 2012
Solutions to Homework Set #2
2
1. (a) Equating v and v 2 ,
a v,0
r
2
2
2
= vr,0 + v,0 + 2GM
1
1
.
r a
Writing vr,0 and v,0 in terms of v and (see the solutions to Problems 5(d)
and 5(e) in Homework #1),
a
r
2
2
(v + v cos )2 = v +

Astronomy 404 Fall 2012
Solutions to Homework Set #6
1. (a) Dividing the specic blackbody energy density, Eq. (9.5), by the energy per
photon hc/ gives the specic number density,
n d =
8/4
d .
ehc/kT 1
(b) Integrating the expression for n over all wavelen

Astronomy 404 Fall 2012
Solutions to Homework Set #7
1.
4
R3
3
Ug = 3
P dV
0
is true, regadless of the source of pressure support. Substituting P = 1 u, we nd
3
4
R3
3
Ug = 3
0
1
u dV
3
= K.
Thus
Etot = Ug + K = 0.
2. (a) Substituting for Ug ,
P
1 GM
1

Astronomy 404 Fall 2011
Solutions to Homework Set #8
1. (a) The Sun produces 2 neutrinos for every E = 26.124 MeV in energy, so the
solar neutrino production rate is
L
(2)(3.839 1026 W)
N = 2
=
E
(26.124 MeV)(1.602176462 1013 J MeV1 )
= 1.834 1038 neutrin

Astronomy 404 Fall 2012
Solutions to Homework Set #13 (Extra Credit)
1. (a) Gravitational time dilation aects both the rate at which photons are emitted
by a blackbody and the frequency of each photon. Both eects decrease the
luminosity of the blackbody a

Astronomy 404 Fall 2012
Hour Exam Solutions
1. (a) Denoting the properties of the probes orbit by subscript p,
rp,min = ap (1 ep ) = a
and
rp,max = ap (1 + ep ) = a
Adding these equations, we nd:
ap =
1
a + a = 1.2618 AU.
2
(b) Dierencing the equations fo

Astronomy 404 Fall 2012
Solutions to Homework Set #10
1. For the specied composition (X = 0.70, Y = 0.28, Z = 0.02), the particle mean
molecular weight is
1/[2X + (3/4)Y + (1/2)X] = 1/1.62 = 0.62
and the electron mean molecular weight is
e 1/[X + (1/2)Y

Astronomy 404 Fall 2011
Solutions to Homework Set #11
1. (a) The total stellar deathrate is
N=
dN (Mi ) =
Mmin
Mmin
CMi2.80 dMi =
C
M 1.80 .
1.80 min
Setting N = 1.5 yr1 and Mmin = 1 M , and solving for C,
1.80
C = 2.7 M yr1 .
(b)
i.
Nwd =
8 M
CMi2.80 dMi

Astronomy 404 Fall 2012
Solutions to Homework Set #9
1. (a) According to Eq. (12.1), mV = MV +5 log10 d5+aV = 1.1+5 log10 7005 =
8.1 mag. (aV = 0.)
(b) Using aV = 1.1 mag, mV = 9.2 mag.
(c) If extinction is not considered, using mV = 9.2 mag leads to a di

Astronomy 404 Fall 2012
Solutions to Homework Set #12
1. Using a central core density of 0 1013 kg m3 for a 15 M star, the time for the
homologous collapse of the inner core is roughly given by
t =
3 1
32 G0
1/2
= 0.021 s.
2. (a) The neutrino energy ux at

Astronomy 404 Fall 2012
Hour Exam
Due in class: Friday, Oct. 19
1. In Homework Sets #1 and #2, we approximated the orbits of Earth and Mars as
circular and coplanar. You found that the most ecient orbit for a probe going from
Earth to Mars was one with pe

Astronomy 404 Fall 2012
Hour Exam Solutions
1. (a) Denoting the properties of the probes orbit by subscript p,
rp,min = ap (1 ep ) = a
and
rp,max = ap (1 + ep ) = a
Adding these equations, we nd:
ap =
1
a + a = 1.2618 AU.
2
(b) Dierencing the equations fo

Astronomy 404 Fall 2012
Solutions to Homework Set #1
1. (a) The distance OA = radius of unit sphere = 1. The projection of OA onto
the x-y plane is therefore OAxy = cos A , and onto the z -axis OAz = zA =
sin A . Breaking OAxy into x- and y - components,

Astronomy 404 Fall 2012
Solutions to Homework Set #2
2
1. (a) Equating v and v 2 ,
a v,0
r
2
2
2
= vr,0 + v,0 + 2GM
1
1
.
r a
Writing vr,0 and v,0 in terms of v and (see the solutions to Problems 5(d)
and 5(e) in Homework #1),
a
r
2
2
(v + v cos )2 = v +

Astronomy 404 Fall 2012
Solutions to Homework Set #3
1. As measured in frame S , the spacetime interval is
(s )2 = (ct )2 (x )2 (y )2 (z )2 .
Using Eqs. (4.16)(4.19), we have
2
(s ) =
c(t ux/c2 )
1 u2 /c2
2
x u t
2
1 u2 /c2
(y )2 (z )2 .
After some algeb

Astronomy 404 Fall 2012
Solutions to Homework Set #4
1. (a) Referring to Figure 5.3, conservation of momentum gives
pi = pf cos + pe cos
0 = pf sin pe sin ,
where pi and pf are the initial and nal photon momenta, respectively, and pe is
the momentum of t

Astronomy 404 Fall 2011
Solutions to Homework Set #5
1. From Eq. (8.7), use gn = 2n2 and E1 = 13.6 eV, E2 = 3.40 eV, and E3 = 1.51 eV
with T = 10, 000 K to get
ZI = g1 + g2 e(E2 E1 )/kT + g3 e(E3 E1 )/kT
= 2 + 5.79 105 + 1.45 105 .
Thus ZI 2.
2. (a) Using

Astronomy 404 Fall 2012
Solutions to Homework Set #6
1. (a) Dividing the specic blackbody energy density, Eq. (9.5), by the energy per
photon hc/ gives the specic number density,
n d =
8/4
d .
ehc/kT 1
(b) Integrating the expression for n over all wavelen

Astronomy 404 Fall 2012
Solutions to Homework Set #7
1.
4
R3
3
Ug = 3
P dV
0
is true, regadless of the source of pressure support. Substituting P = 1 u, we nd
3
4
R3
3
Ug = 3
0
1
u dV
3
= K.
Thus
Etot = Ug + K = 0.
2. (a) Substituting for Ug ,
P
1 GM
1

Astronomy 404 Fall 2011
Solutions to Homework Set #8
1. (a) The Sun produces 2 neutrinos for every E = 26.124 MeV in energy, so the
solar neutrino production rate is
L
(2)(3.839 1026 W)
N = 2
=
E
(26.124 MeV)(1.602176462 1013 J MeV1 )
= 1.834 1038 neutrin

Astronomy 404 Fall 2012
Solutions to Homework Set #9
1. (a) According to Eq. (12.1), mV = MV +5 log10 d 5+ aV = 1.1+5 log10 700 5 =
8.1 mag. (aV = 0.)
(b) Using aV = 1.1 mag, mV = 9.2 mag.
(c) If extinction is not considered, using mV = 9.2 mag leads to a

Astronomy 404 Fall 2012
Solutions to Homework Set #10
1. For the specied composition (X = 0.70, Y = 0.28, Z = 0.02), the particle mean
molecular weight is
1/[2X + (3/4)Y + (1/2)X ] = 1/1.62 = 0.62
and the electron mean molecular weight is
e 1/[X + (1/2)Y

Astronomy 404 Fall 2011
Solutions to Homework Set #11
1. (a) The total stellar deathrate is
N=
d N ( Mi ) =
Mmin
Mmin
CMi2.80 dMi =
C
M 1.80 .
1.80 min
Setting N = 1.5 yr1 and Mmin = 1 M , and solving for C ,
1
C = 2.7 M.80 yr1 .
(b)
i.
Nwd =
8 M
CMi2.80