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 $n = 5$ .

Figure 1. Lane-Emden solutions for

n = 0, 1, 1.5, 3, 5

. Method: RK4 with

d\xi = 10^{-3}

n 1.5

WHAT TO TRY

Step n from 0 to 5: the star goes from a uniform-density ball with a sharp edge to a centrally concentrated, diffuse cloud. Higher n piles the mass into the core, as the cutaway and the isodensity rings show.
Compare the xi_1 markers: n=0 ends at sqrt(6), n=1 at pi, n=3 at 6.897. The rail checks each against its closed form.
Select n=5: theta never reaches zero, so the polytrope has infinite radius (the curve fades to zero only asymptotically and the star reads "diffuse").
Watch the enclosed-mass plot: for high n almost all the mass sits at small xi, the seismic signature of a centrally condensed star.