 Polytropic mass-radius relation

ID: polytropic-mass-radius-relation

Codex  0 Created 2026-09-28 Updated 2026-10-03

Eliminating central density from the Lane-Emden scalings gives

R \propto K^{n / (3 - n)} M^{(1 - n) / (3 - n)}

for

n \neq = 3

. In particular,

R \propto K M^{- 1/3}

for

n = 3/2

With

Y_{n} = - ξ_{1}^{2} θ^{'} (ξ_{1})

, the exact rearranged relation is

M = A_{n} K^{n / (n - 1)} R^{(3 - n) / (1 - n)},

where

A_{n} = 4 π Y_{n} (\frac{n + 1}{4 π G})^{n / (n - 1)} ξ_{1}^{(3 - n) / (n - 1)} .

For

n = 3

, the radius dependence disappears and the mass is proportional to

K^{3/2}

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