Chapter 2 Stellar Structure Equations 1 Mass conservation equation

The gas pressure P gas = P i + P e . For our Sun, it is approximately equal
to 1.61ρkT/m H .
We have not finished the discussions on EOS yet since we leave out radiation
pressure. Recall from the course Statistical Mechanics and Thermodynamics
that photons are bosons. Therefore, in thermal equilibrium, photons
obey Bose-Einstein statistics. That is, the distribution of photons is isotropic
and the number density of photons with frequencies between ν and ν + ∆ν
equals
n(ν) dν = 8πν2
c 3
dν
exp(hν/kT ) − 1 . (79)
To see why photon number density follows Eq. (79), one recall that the
energy of having s photons each with momentum ⃗p equals spc. As photon
is bosonic, s can take on any natural number. Moreover, the probability
P (s) that there are exactly s photons each with momentum ⃗p follows the
constraint
P (s + 1)
= e −pc/kT = e −hν/kT , (80)
P (s)
where ν is the frequency of the photon. Consequently,
P (s) =
e −shν/kT
∑ +∞
s ′ =0 e−s′ hν/kT = e−shν/kT (1 − e −hν/kT ) (81)
provided that ⃗p is an allowed momentum of a photon. Thus, the expected
number of photons with an allowed momentum ⃗p is given by
⟨N γ (⃗p)⟩ = 2
+∞∑
s=0
sP (s) =
2
e hν/kT − 1 . (82)
(Note that the 2 above reflects the fact that a photon has two possible polarizations.)
As a result, the total expected number of photons equals
⟨N γ ⟩ = 1 ∫ ∫
⟨N
h 3 γ (⃗x)⟩ dV d 3 ⃗p
= 1 ∫ ∫
2
dV
h 3 e hν/kT − 1 d3 ⃗p
Hence, Eq. (79) is valid.
= V
∫
8πν 2
dν . (83)
c 3 (e hν/kT − 1)
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