Chapter 2 Stellar Structure Equations 1 Mass conservation equation

Note that the first fraction in Eq. (79) is called phase space factor while
the second fraction is the Bose-Einstein distribution factor. Moreover, this
distribution n(ν) is sometimes called the blackbody spectrum.
For a photon of momentum ⃗p hitting a wall and then reflected back elastically,
the magnitude of the change in momentum equals 2p cos θ = 2hν cos θ/c
where θ is the angle between ⃗p and the normal of the wall. The radiation
pressure due simply the momentum transfer to the wall per unit time per
unit surface area, i.e. P rad = ∫ dN γ (p,θ)
2p cos θ= ∫ c cos θ dN γ(p,θ)
2p cos θ, in
dAdt
dV
other words
P rad = 1 h 3 ∫ +∞
=
0
∫ +∞
0
∫ π/2 ∫ 2π
0
0
hν
3 n(ν)dν
c cos θ
2hν cos θ
c
2
e hν/kT − 1
( ) 2 ( )
hν
hν
sin θ dϕdθd
c
c
= 4σ
3c T 4 , (84)
where σ = 2π 5 k 4 /15c 2 h 3 is the Stefan’s constant.
Thus, our ideal gas EOS for stellar matter is
P = P i + P e + P rad . (85)
Finally, note that the energy density of a photon gas at temperature T is
given by
u rad =
Hence, u rad = 3P rad .
∫ +∞
0
hνn(ν)dν = 4σ c T 4 . (86)
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