Chapter 2 Stellar Structure Equations 1 Mass conservation equation

The ion pressure is given by
where n i is the ion number density.
P i = n i kT (75)
What are the values of n e and n i ? To answer this question, we denote the
mass fraction of H, He and “heavy elements” (that is, those elements heavier
than H and He) in a star by X, Y and Z = 1 − X − Y respectively. (More
precisely, X and Y are the mass fraction of 1 H and 4 He respectively. Do
you know why we can forget about the contribution of 2 H, 3 H, 3 He?) Then,
number density of hydrogen and helium nuclei equal Xρ/m H and Y ρ/4m H ,
respectively. Therefore,
n i =
ρ (
X + Y m H 4 + Z )
, (76)
⟨A⟩
where ⟨A⟩ denotes the average mass number of heavy elements in a star.
Once again by virial theorem and the above analysis, we know that the
thermal energy of a star is of the order of GM 2 /R. That is,
GM 2 /R ∼ k ¯T M/m H , (77)
where ¯T ≈ 10 6 K is the average temperature of a star. Hence, for most
part of a star, the temperature is so high that most atoms are completely
ionized. (As we have already seen, this is not a valid assumption for stellar
atmospheres. Nonetheless, this assumption has little effect in determining
the overall structure of a star. Similarly, this assumption is not valid for
high atomic number atoms, but their numbers are small compared to that of
hydrogen and helium for a typical main sequence star.) With this assumption
plus the charge neutrality in mind,
n e = ρ (X + 2 Y ⟨ z
⟩ )
m H 4 + Z A
≈
ρ (X + Y m H 2 + Z )
2
ρ(1 + X)
= , (78)
2m H
where ⟨z/A⟩ ≈ 1/2 is the ratio of atomic number to mass number averaged
over all heavy elements.
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