The Lane-Emden Equation is a form of an Equation of State (EoS) 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 = Kργ
Assuming such a constant exists, the gas is polytropic. The model 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 —— —— (E2 dθ/dE) + θn = 0 E2 dE
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 where n = 0, n = 1, or n = 5. Otherwise, numerical methods are used.