A polytrope is a relation between gas pressure and density -- an equation of state -- where the temperature does not appear explicitly.

Polytropic equations of state take the form

P = K ργ = K ρ(1 + 1/n)

where P is the pressure, ρ is the density, γ is the polytropic exponent (note: not necessarily the adiabatic gamma), n is the polytropic index, and K is an independent "constant" which varies from model to model depending upon the initial conditions. For example, if the ideal gas law is used, K is simply

K = (RT)/μ

where R is the gas constant, T is the temperature, and μ is the mean molecular weight.

The polytropic equation of state is commonly used in building simple stellar models, particularly in models of very young stars, or in completely degenerate objects like white dwarfs. They also occur when using the ideal gas equation of state under adiabatic conditions. For the adiabatic case, γ takes its familiar value of 5/3 (n=3/2), but in the degenerate, relativistic limit, γ=4/3 (n=3).

Analytic stellar models using polytropes are relatively easy to build, as polytropes can be inserted into the equation of hydrostatic equilibrium

dP/dr = -ρ d(Φ)/dr

and the Poisson equation

1/r2 × d/dr (r2 d(Φ)/dr) = 4 π G ρ

with ease. (Here, r is the radius, and Φ is the gravitational potential.) In this case you wind up with an equation of the form

d2(Φ)/dr2 + ((2/r) d(Φ)/dr) = 4 π G (-Φ/((n+1) K))n

which reduces to the Lane-Emden Equation with some clever changes of variables.

The Nobel Prize-winning physicist Subrahmanyan Chandrasekhar derived the Chandrasekhar limit using the Lane-Emden Equation with a fully degenerate relativistic polytrope (n=3), realizing that stars with masses higher than this limit would lose pressure support in their cores and collapse (into what he didn't know at the time). Despite polytropes being a purely theoretical construct, no white dwarfs have been observed with masses higher than the Chandrasekhar limit of around 1.45 solar masses.

Source: Kippenhahn and Weigert, Stellar Structure and Evolution, Springer-Verlag (student edition), 1994.

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