Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 317 / 4 / ii

The radiative-pressure equation is

\frac{d P _{rad}}{d r} = - \frac{κ ρ L _{r}}{4 π c r ^{2}} .

Dividing by hydrostatic equilibrium and substituting

κ = κ_{0} (M_{r} / L_{r}) (L / M)

gives

\frac{d P _{rad}}{d P} = \frac{κ _{0} L}{4 π c GM},

a constant. Since both pressures vanish at the surface,

P_{rad} = (1 - β) P

and

β

is constant throughout the star. Part i therefore has

P = K ρ^{4/3}, n = 3,

with spatially constant

K

ρ = ρ_{c} θ^{3}

and

α^{2} = \frac{K}{π G} ρ_{c}^{- 2/3}, r = α ξ,

gives the

n = 3

\frac{1}{ξ ^{2}} \frac{d}{d ξ} (ξ^{2} \frac{d θ}{d ξ}) = - θ^{3}, θ (0) = 1, θ^{'} (0) = 0.

The surface is the first zero

θ (ξ_{1}) = 0

. The

n = 3

equation has no elementary closed-form solution; among nonnegative indices, the standard analytic cases are

n = 0, 1, 5

. This Eddington standard model approximates chemically homogeneous massive main-sequence stars with substantial radiation pressure.

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