### 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 with a certain relation between density and pressure,
called **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 is:

1 d
—— —— (E^{2} dθ/dE) + θ^{n} = 0
E^{2} dE

where:

- n - a constant, determined by a function of the constant γ above.
- θ - a function of density and the constant n.
- E - a function of the radius and some cleverly-chosen constants.

As such, it spells out the density by radius, and from that, the pressure
at any radius can be determined by the earlier equation.

Solutions to the equation are known to astrophysicists as **polytropes**.
There are analytic solutions if n = 0, n = 1, or n = 5.
Otherwise, numerical methods are used.

(*equation,astrophysics*)
**Further reading:**

http://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 pages:**

specific heat

Index