Chapter 2 Stellar Structure Equations 1 Mass conservation equation

7.1 Derivation of Saha Equation
Recall from the course Statistical Mechanics and Thermodynamics that
for a particle (or system of particles), the probability of a particle is in state
s is proportional to g s e −E s/kT where g s is the degeneracy of state s and E s is
the energy of state s relative to some fixed reference energy level (sometimes
taken to be the ground state). More importantly, the proportionality constant
is the same for all the states. Thus, the ratio of the probability that
the particle is in state (s + 1) to the probability that it is in state s is given
by the Boltzmann formula
g s+1
g s
e −(E s+1−E s)/kT . (53)
The Statistical Mechanics and Thermodynamics course also tells us that the
thermodynamic properties of a system with fixed number of particles N and
fixed temperature T and fixed volume V is encoded in the so-called partition
function
Z ≡ ∑ ( ) −Ei
g i exp , (54)
kT
i
where the sum is over all possible energy states of the system. Furthermore,
the ratio between the mean number of particles N s in state s and the total
number of particles N ≡ ∑ N j in a sample is given by
N s
N =
g se −E s/kT
∑
j g je −E j/kT
≡ g se −E s/kT
Z
. (55)
For an ideal classical non-relativistic gas particle with ground state energy
E 0 and degeneracy g, its partition function is given by
∫ ∫
g
Z(1, V, T ) =
e −(E 0+p 2 /2m)/kT d 3 ⃗x d 3 ⃗p
(2π) 3
=
gV
(2π) 3 e−E 0/kT
= gV
λ 3 e−E 0/kT
∫ +∞
0
4πp 2 e −p2 /2mkT dp
(56)
where m is the mass of the particle and
√
2π
λ ≡
mkT . (57)
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