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.