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  "body": "\n## Summary\n\nThe 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.\nIt is the equation $\\Delta u + u^n = 0$ for radial functions in $\\R^3$.\nIn 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.\nIt is a standard test case for existence, nonexistence and Liouville-type theorems.\n\n## Explanations\n\n### L1 human\n\nA star made of gas is held together by its own gravity and pushed apart by its pressure.\nIf the pressure is a fixed power of the density, the balance of the two forces becomes the Lane–Emden equation.\nThe unknown $\\theta$ measures the density as a function of the scaled distance $\\xi$ from the centre, starting at $\\theta(0) = 1$ with zero slope.\nFor $n = 1$ the solution is $\\theta(\\xi) = \\sin \\xi / \\xi$, and its first zero $\\xi = \\pi$ marks the surface of the star.\n",
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