### Lane-Emden equation

(form of equation of state for gas ball in hydrostatic equilibrium)

The Lane-Emden equation is a form of an equation of state for a gas ball bound together by gravity, in hydrostatic equilibrium, and of a gas having this of relation between density and pressure, which is termed polytropic:

```P = Kργ
```
• P - pressure.
• ρ - density.
• γ, K - constants.

If the constants exist that make this equation hold, the gas is polytropic. The relation is used in modeling stars and gas planets. In some circumstances, an ideal gas can act in this manner. The Lane-Emden equation, which related 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.

Solutions to the Lane-Emden equation are known to astrophysicists as polytropes. There are analytic solutions for n = 0, n = 1, and n = 5, otherwise, numerical methods are used. 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.

(equation,astrophysics)