Polytropic mass-radius relation

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}

Entropy dependence of a polytropic mass-radius relation ( $R \propto K^{n / (3 - n)} M^{(1 - n) / (3 - n)}$ )

For a finite regular polytrope with fixed index, the scale

α^{2} \propto K ρ_{c}^{(1 - n) / n}

implies

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

. Thus, for

n \neq = 3

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

. For

n = 3/2

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

. Fixed composition makes the cold electron degeneracy pressure coefficient approximately fixed, but a convective ideal gas has

K

controlled by entropy. A stellar sequence with changing entropy therefore need not follow the fixed-

K

negative slope.

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