= Solution
For a fixed <polytropic index> $n>0$ and fixed equation-of-state constant $K$, let $\xi_1$ be the first zero of the regular <Lane-Emden equation> solution. A finite-radius model requires such a zero; for the usual nonnegative indices this holds for $n<5$. The surface radius and mass follow from $r=\alpha\xi$ and $\rho=\rho_c\theta^n$:
$$
R=\alpha\xi_1,\qquad M=4\pi\alpha^3\rho_c\int_0^{\xi_1}\xi^2\theta^n\,d\xi=4\pi\alpha^3\rho_c[-\xi_1^2\theta'(\xi_1)].
$$
The last equality integrates the <Lane-Emden equation>; define the positive <Lane-Emden surface mass constant> $\omega_n=-\xi_1^2\theta'(\xi_1)$. Substituting $\alpha=C_1\rho_c^{(1-n)/(2n)}$ gives
$$
R=C_1\xi_1\rho_c^{(1-n)/(2n)},\qquad M=4\pi C_1^3\omega_n\rho_c^{(3-n)/(2n)}.
$$
The central-density exponents cancel when the required powers are taken. Thus the <polytropic mass-radius relation> is
$$
\boxed{M^{n-1}R^{3-n}=C_2=(4\pi C_1^3\omega_n)^{n-1}(C_1\xi_1)^{3-n}.}
$$
It is independent of $\rho_c$, with $n,K$ and composition held fixed. For $n=1$ the radius is fixed; for $n=3$ the mass is fixed. These limiting powers need no division by a vanishing exponent.
For $n=0$, the literal pressure-density power and the supplied expression for $\alpha$ are singular. Interpret this case separately as an <incompressible planetary interior> with constant density. Then $M=4\pi\rho R^3/3$, or \b[$R\propto M^{1/3}$] at fixed density. A weakly compressed rocky body, or a rough uniform-density approximation to <Earth>, is an example; realistic terrestrial planets are stratified and compressible.
For $n=3/2$, $P\propto\rho^{5/3}$. This describes a cold nonrelativistic degenerate electron gas of fixed composition, or a fully convective monatomic <ideal gas> at fixed <entropy>. Examples are a nonrelativistic <white dwarf>, a sufficiently cooled partly degenerate <brown dwarf> as an approximation, and an approximately fully convective low-mass star for the ideal-gas version. The fixed-$K$ scaling is \b[$R\propto M^{-1/3}$]. It must not be applied to an entire main-sequence stellar sequence with different entropies; the <entropy dependence of a polytropic mass-radius relation> explains the distinction.
Back to article page
