= Moment of inertia of a polytropic star
{title2=$I/(MR^2)$}
For a spherical <stellar polytrope>, the axial <moment of inertia> is $I=(8\pi/3)\int_0^R\rho(r)r^4\,dr$. With <Lane-Emden variables for a stellar polytrope>, first zero $\xi_1$ and <Lane-Emden surface mass constant> $\omega_n$, its dimensionless form is
$$
\frac{I}{MR^2}=\frac{2}{3\omega_n\xi_1^2}\int_0^{\xi_1}\xi^4\theta(\xi)^n\,d\xi.
$$
A <polytrope of index zero> gives $2/5$, while a <polytrope of index one> gives $(2/3)(1-6/\pi^2)$. The latter is smaller because more of the mass lies near the center. The scalar second mass moment $\int r^2dm$ is $3I/2$, not the axial <moment of inertia>.
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