Assignment 4 — Solutions [Revision : 1.2]
1. (a) Rearranging the first-moment equation,
rad,λ
κc ρ dPdr
Fλ
λ
Substituting in the expression for Prad,λ , this becomes
Fλ
λ
3κ4πρ dB
.
dr
λ
The chain rule is used to write
dBλ
dr
λ dT
dB
,
dT dr
from which we obtain the result
Fλ
λ dT
.
3κ4πρ dB
dT dr
λ
(b) Integrating over all wavelengths,
F
»8
Fλ dλ 0
»8
4π dBλ dT
dλ.
3κλ ρ dT dr
0
Bringing the wavelength-independent terms out from under the integral sign,
4π dT
F 3ρ dr
»8
1 dBλ
dλ.
κλ dT
0
(c) Equating the two expressions for F ,
»8
4π dT
3ρ dr
0
1 dBλ
4acT 3 dT
dλ .
κλ dT
3κ̄ρ dr
Rearranging,
1
κ̄
Recognizing that
»8
0
³8
1 dBλ
dλ
0 κλ dT
.
3
acT π
{
Bλ dλ σT 4
π
acT
4π
4
,
the expression for κ̄ may also be written
1
κ̄
³8 1 dB
0³8κ dBdT
λ
λ
λ
0
dT
dλ
dλ
.
highlighting the fact that 1{κ̄ is a weighted average of 1{κλ .
2. (a) Because
R λn ξ1 ,
it follows that
λn
1 R@
6.897
1.01 1010 cm
(b) Using the equation
M
with M
dDn ,
dξ ξ1
4πλ3n ρc ξ12
1 M@ , it follows that
ρc
76.3 g cm3 .
(c) To find the central pressure, we use the polytropic relation:
Kργ Kρ1{n
P
The constant K follows from the relation
λn
pn
1q
{ 1 n 1
Kρc
4πG
1
1{2
;
rearranging,
K
and so
K
4πGλ2n
{ pn 1q
1 n 1
ρc
3.84 1014
(with funny units). Now using the polytropic relation,
Pc
1.24 1017 dyne cm2 .
(d) If the radiation pressure is negligible, the gas obeys the ideal equation of state,
P
ρkT
µm
H
Solving for the central temperature,
Tc
1.21 107 K
3. From the various equations relating the mass and radius of polytropes, we find
R 9 λn
but
λn 9 K 1{2 ρ1c {2n1{2 .
Due to the degenerate equation of state, K A is a constant — we don’t have the liberty to
choose it. Also, since γ 5{3, it follows that n 1{pγ 1q 3{2. Hence,
1{6
R 9 ρ
c
But
and so
M 9 λ3n ρc 9 R3 R6
R 9 M 1{3 .
4. See Fig. 1 for the plot. The gradient is estimated by eye as
obtain n 1{pγ 1q 4.
d log P
d log ρ
γ 10{8, from which we
5. See Fig. 3 for the plot.
6. See Fig. 3 for the plot. The two luminosities differ in the core of the star (r{R À 0.25), with
Lrad Lr , because there a fraction of the total luminosity Lr is transported by convection
rather than radiation.
Figure 1: Pressure-density plot for the 10 M@ model (Q4).
Figure 2: Temperature gradient plot for the 10 M@ model (Q5).
Figure 3: Luminosity plot for the 10 M@ model (Q6). The solid line shows the radiative luminosity
Lrad , and the dotted line the total interior luminosity Lr .