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The Lane-Emden equation describes a gas ball bound together by gravity in hydrostatic equilibrium that is polytropic, i.e., that adheres to the following relation between density and pressure:
P = Kργ
In some circumstances, an ideal gas can act in this manner. The Lane-Emden equation, which relates these to distance from the center of such a body is:
1 d —— —— (E2 dθ/dE) = -θn E2 dE
where:
The pressure at each radius is easily available through the earlier equation. The "change of variables" allows the equation to be concise.
This relation, termed a polytrope, is used in modeling stars and gas planets. The Lane-Emden equation has the advantage that one can solve it to model the body: in some cases, analytically (for n = 0, n = 1, or n = 5), and in other cases, with numerical methods. Polytropes produce an idealized stellar structure: n = 1 (or slightly higher) approximates fully-convective stars, and n = 3 approximates fully-radiative stars. n = 0 models a non-compressible material (constant density) and n = 5 is not a useful solution.