### 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 = Kρ^{γ}

- 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
—— —— (E^{2} dθ/dE) = -θ^{n}
E^{2} 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 θ
^{n}.
- 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*)
**Further reading:**

https://en.wikipedia.org/wiki/Lane-Emden_equation

http://www.astro.caltech.edu/~jlc/ay101_fall2015/ay101_polytropes_fall2015.pdf

http://www.nicadd.niu.edu/~bterzic/PHYS652/Lecture_23.pdf

**Referenced by page:**

specific heat

Index