Phys 132 Homework 16 Solutions
Professor Crystal Martin
TA: Ellie Hadjiyska
Problem 16 (5.2): (a) We now consider a family of chemically homogeneous stars similar in every
way but for their masses and density profiles which go as
ρ(r) = M/R3Fρ(x)
where x=r/R. We further assume energy is generated by p-p chain, stellar material obeys ideal
gas law, and opacity κ∝ρT −3.5. From
dP
dr =−ρ(r)GM(r)
r2
where
M(r) = 4πZr
0
dr r2ρ(r) = 4πMZx
0
dx x2Fρ(x)
we get
dP =M2
R4·−4πG dx Fρ(x)
x2Zx
0
dx x2Fρ(x)¸
P(r) = ZP(x=r/R)
P(x=1)
dP =M2
R4"4πG Zx=1
x=r/R
dxFρ(x)
x2Zx
0
dx x2Fρ(x)#,
where the expression in the last square bracket is FP(x). From the ideal gas equation of state,
T(r) = µmp
k
P(r)
ρ(r)=M
Rhµmp
kFρ(x)−1FP(x)i
where the expression in the square bracket is FT(r). We have the opacity as κ=κ0ρT −3.5and
Hrad =−4
3
acT (r)3
ρ(r)
1
κ
dT
dr =−4
3
ac
κ0
ρ(r)−2T(r)6.5M
R2
dFT
dx
=−4
3
ac
κ0
M5.5R−2.5Fρ(x)−2FT(x)6.5d
dxFT(x)
Then from Lrad(r) = 4πr2Hrad(r) we get
Lrad(r) = M5.5
R0.5·−16π
3
ac
κ0
x2µFρ(x)−2FT(x)6.5d
dxFT(x)¶¸
where the expression in the square bracket is Frad(x). Now with the nuclear fusion via the PP
chain, we derived in earlier problems the expression ²fusion =ξρT 4where ξis constant. We then
have
Lfusion =M6
R7"4πξ Zx=r/R
x=0
dx x2FT(x)4Fρ(x)2#
