Chapter 2 Stellar Structure Equations 1 Mass conservation equation

By substituting θ and ξ in the Lane-Emden equation back our equation of
mass conservation (also known as equation of (mass) continuity) and equation
of hydrostatic equilibrium, we obtain
[ ] 1/2 (n + 1)K
R =
ρ (1−n)/2n
c ξ 1 , (36)
4πG
where ξ 1 is the smallest positive value of ξ having θ(ξ) = 0,
[ ] 3/2 (n + 1)K
M = 4π
ρ (3−n)/2n
c
4πG
[
−ξ 2 dθ ]∣
∣∣∣ξ=ξ1
, (37)
dξ
and
[
P c = GM ( ) ] ∣ 2
2 −1∣∣∣∣∣ξ=ξ1 dθ
4π(n + 1)
, (38)
R 4 dξ
¯ρ =
M [
4πR 3 /3 = ρ c − 3 ]∣
dθ ∣∣∣ξ=ξ1
, (39)
ξ dξ
Ω = − 3 GM 2
5 − n R . (40)
Note that R is independent of ρ c for n = 1, Ω > 0 for n > 5, Ω is
undefined for n = 5. Therefore, the polytropic models for n = 1 and n ≥ 5
are not physical. (Can you verify that M is finite, P c is infinite and ¯ρ is 0
for case of n = 5?)
Interestingly, for n = 3, the mass of the star is independent of its central
density. This particular polytropic index is sometimes also called the Eddington
standard model. (We shall say more about the Eddington standard
model later in the next chapter.)
Example. For a given mass M and central pressure P c , which polytrope
yields a bigger star: that of index 1.5 or that of index 3?
The central pressure of the star can be written as
P c = (4π) 1/3 B n GM 2/3 ρ 4/3
c . (41)
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