Lane-Emden equation

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

• P - pressure.
• ρ - density.
• γ, K - constants.

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:

• n - a constant such that γ = 1/(1+n).
• θ - a function of density ρ and the constant n such that ρ = c θⁿ.
• E - dimensionless number that is proportional to the radius by a cleverly-chosen constant.

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.