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 Incoming links: Index-one plane-parallel polytrope

For a plane-parallel slab, all equilibrium quantities depend only on height

z

\frac{d p}{d z} = - ρ \frac{d ψ}{d z}, \frac{d ^{2} ψ}{d z ^{2}} = 4 π Gρ .

p = K ρ^{2}

and constant

K

2 K ρ \frac{d ρ}{d z} = - ρ \frac{d ψ}{d z} .

In the gas interior, where

ρ > 0

, divide by

ρ

and differentiate once:

2 K \frac{d ^{2} ρ}{d z ^{2}} = - \frac{d ^{2} ψ}{d z ^{2}} = - 4 π Gρ .

Consequently

\frac{d ^{2} ρ}{d z ^{2}} + \frac{2 π G}{K} ρ = 0.

2 π G / K

has units inverse square metres. The index-one plane-parallel polytrope illustrates the resulting structure: symmetry about

z = 0

gives

ρ (z) = ρ_{c} cos (k z)

, with

k = 2 π G / K

. A slab bounded by vacuum ends at its first zero,

∣ z ∣ = π / (2 k)

; the negative continuation of the cosine is not physical mass density.