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The Lane-Emden Polytrope

A star is a ball of gas held up against its own gravity by pressure, and the simplest model of one assumes the two are tied by a power law, $P = K\rho^{1+1/n}$, a polytrope of index $n$. Feed that into the equation of hydrostatic equilibrium and everything collapses to a single dimensionless equation for the scaled density $\theta$, the Lane-Emden equation, integrated outward from the centre until $\theta$ first hits zero, which marks the surface at $\xi_1$. The index $n$ is the whole story: it sets how steeply the density falls from core to edge. At $n=0$ the star is a uniform sphere of constant density with surface at $\xi_1 = \sqrt 6$; at $n=1$ the equation is solved exactly by $\theta = \sin\xi/\xi$ with surface at $\pi$; by $n=3$, the model for a star supported by relativistic electrons or radiation pressure, the density is already crushed into the centre, the core some fifty times denser than the average; and as $n\to 5$ the star formally swells to infinite radius. The scene paints the polytrope as a glowing disk, hot and bright where the gas is dense, fading to the cool surface, beside the density and enclosed-mass profiles, and a draggable cursor reads off how much of the star lies inside any radius. The bottom panel tracks the central concentration, the ratio of central to mean density, as it climbs from one at $n=0$ toward infinity, the same trend that makes a real evolved star a tiny dense core wrapped in a tenuous envelope.

Figure 1. The Lane-Emden polytrope. Top: a density cross-section of the star beside its interior profiles against fractional radius, the Lane-Emden function theta (which spans the whole star), the density theta^n, the pressure theta^(n+1), and the enclosed mass m/M, with a draggable cursor. Higher n crushes the density and pressure toward the centre while theta still runs the full radius. Bottom: the central concentration rho_c / mean density against polytropic index n, climbing from 1 at n=0 toward infinity at n=5. Method: RK4 integration of the Lane-Emden equation (shared engine). Source: Chandrasekhar, An Introduction to the Study of Stellar Structure, Ch. 4.
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WHAT TO TRY

  • Raise the index $n$ (or pick a preset): the density and pressure crush toward the centre, the core brightening and the envelope dimming, while the Lane-Emden function theta keeps spanning the whole radius.
  • Pick the $n=1$ preset: the solution is the exact analytic $\theta=\sin\xi/\xi$, reaching the surface at $\xi_1=\pi$. Compare the convective $n=1.5$ and the Eddington $n=3$ presets.
  • Drag the radius cursor: read theta, the density, the pressure, and the fraction of the star's mass inside that radius, ever more centrally concentrated as $n$ grows.
  • Watch the central concentration climb toward infinity as $n$ approaches 5, where the star has no finite radius.