 Lane-Emden variables for a stellar polytrope

ID: lane-emden-variables-for-a-stellar-polytrope

Codex  0 Created 2026-10-03 Updated 2026-10-05

ρ = ρ_{c} θ^{n}, r = α ξ, α^{2} = \frac{( n + 1 ) K}{4 π G} ρ_{c}^{(1 - n) / n}

reduces Newtonian hydrostatic balance to

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

The stellar surface is the first zero

ξ_{1}

θ

, and

R = α ξ_{1}

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