---
tag: ANRZZMH2Q0
kind: concept
name: Lane–Emden equation
msc: [35J61, 85A15]
sources:
  - url: https://en.wikipedia.org/wiki/Lane%E2%80%93Emden_equation
same_as:
  wikidata: Q1804812
  wikipedia: Lane–Emden equation
---

## Summary

The Lane–Emden equation is the [[ordinary differential equation|0NT64JV96A]] $\xi^{-2} (\xi^2 \theta')' + \theta^n = 0$, which describes the density of a self-gravitating ball of gas in equilibrium, called a polytrope.
It is the equation $\Delta u + u^n = 0$ for radial functions in $\R^3$.
In PDE the name also refers to the semilinear elliptic equation $-\Delta u = u^p$ in any dimension, whose solutions depend on how $p$ compares with critical exponents.
It is a standard test case for existence, nonexistence and Liouville-type theorems.

## Explanations

### L1 human

A star made of gas is held together by its own gravity and pushed apart by its pressure.
If the pressure is a fixed power of the density, the balance of the two forces becomes the Lane–Emden equation.
The unknown $\theta$ measures the density as a function of the scaled distance $\xi$ from the centre, starting at $\theta(0) = 1$ with zero slope.
For $n = 1$ the solution is $\theta(\xi) = \sin \xi / \xi$, and its first zero $\xi = \pi$ marks the surface of the star.