Chapter 2 Stellar Structure Equations 1 Mass conservation equation

Therefore near the centre
T (r) = T c − 1
8ac
k c ρ 2 cq c
r 2 . (24)
Tc
3
Note that these relations hold regardless of the functional dependence
P (ρ, T ), q (ρ, T ), and k (ρ, T ).
3 Simple stellar models
So far, we have 3 unknowns, namely, m(r), ρ(r) and P (r) but only two
equations, namely, the mass conservation equation and equation of hydrostatic
equilibrium. So, we need another equation to relate m, ρ and P .
Astronomers have proposed a number of extremely simplified equations to
fulfill this task. Since all these proposed equations are not derived from first
principle, they are nothing more than crude approximations of the realistic
situation. Yet, these approximations are sometimes quite useful to investigate
the approximate structure of a star.
The first such model is called constant density model, namely, we assume
ρ is a constant inside the entire star. The second one is called the linear
density model, namely, ρ(r) = ρ c (1 − r/R) where ρ c is a constant. You may
solve m(r) and P (r) in the above two models yourself.
4 The polytrope model
The last such model I am going to introduce, which is the most important
model of this kind, is called the polytrope model. We assume that
P = Kρ γ (25)
for some constants K and γ. Note that this model is based on the observation
that P and ρ follow the above equation for ideal gas in various situations such
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