Lane-Emden polytrope

What you are seeing: the Lane-Emden polytrope shown as the star it describes. The dimensionless solution $\theta(\xi)$ of $\theta'' + (2/\xi)\theta' + \theta^n = 0$ ($\theta(0)=1$, $\theta'(0)=0$) is mapped to a density-shaded sphere with $\rho/\rho_c = \theta(\xi)^n$: a bright dense core fading to a faint envelope, a cutaway wedge exposing the interior run, and isodensity contour rings. Selecting the index $n$ restructures the star, from a compact uniform $n = 0$ sphere to the huge, formally infinite, diffuse $n = 5$ envelope.

A linked $\theta(\xi)$ strip with the $\xi_1$ surface marker and an animated radial probe ties the 1D solution to the 2D structure. Cross-checked: $\xi_1 = \sqrt{6}$ for $n = 0$, $\pi$ for $n = 1$, $\approx 3.6537$ for $n = 1.5$, $\approx 6.8969$ for $n = 3$, and $\xi_1 \to \infty$ for $n = 5$.