Chapter 2 Stellar Structure Equations 1 Mass conservation equation

whose solution that matches the correct boundary conditions above is θ(ξ) =
sin ξ/ξ. Hence,
ρ = ρ c
sin ξ
ξ
. (31)
For the case of n = 5, Lane-Emden equation can be rewritten as
d 2 z
dt = z(1 − z4 )
2 4
, (32)
where ξ = e −t and θ = z/(2ξ) 1/2 . Multiplying Eq. (32) by dz/dt, we obtain
1
2
( ) 2 dz
= 1 dt 8 z2 − 1 24 z6 + C . (33)
Boundary conditions demand that C = 0 and hence
dz
dt = −z 2
) 1/2 (1 − z4
. (34)
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After making the substitution z 4 /3 = sin 2 ζ and upon integration, we have
e −t = C ′ tan(ζ/2) where C ′ is a constant of integration. Applying the boundary
conditions and after simplification, we obtain
( ) −5/2
ρ = ρ c 1 + ξ2
. (35)
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(Could you express ρ c as a function of M and R? Besides, could you
express ρ and P as a function of r?)
It is straight-forward to check that the smallest positive root for the
solution θ(ξ) of the Lane-Emden equation equals ξ = ξ 1 ≡ √ 6 and ξ = ξ 1 ≡ π
when n = 0 and 1 respectively. This value of ξ 1 corresponds to the boundary
of a star. On the other hand, θ ≠ 0 for any real-valued ξ for n = 5. In
other words, the solution of the Lane-Emden equation for n = 5 does not
correspond to a physical star. In fact only n < 5 and n ≠ 1 can correspond
to a physical star (cf. table 1 in supplementary notes).
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