Polytrope

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In astrophysics, a polytrope refers to a solution of the Lane-Emden equation in which the pressure depends upon the density in the form $P = K ρ (1 + 1 / n)$ , where $P$ is pressure, $ρ$ is density and $K$ is a constant. The constant $n$ is known as the polytropic index. This relation need not be interpreted as an equation of state, although a gas following such an equation of state does indeed produce a polytropic solution to the Lane-Emden equation. Rather, this is simply a relationship that expresses an assumption regarding the evolution of $P$ with radius, in terms of the evolution of $ρ$ with radius, yielding a solution to the Lane-Emden equation.

For example, given an ideal gas and given a polytropic index, the constant $K$ is

$K = \left({ \frac{N_A k_B T}{\mu}}\right) \rho^{-1/n}$

and the expression on the right hand side is, for whatever reason, assumed to be constant throughout the solution.

Sometimes the word polytrope may be used to refer to an equation of state that looks similar to the thermodynamic relation above, although this is potentially confusing and is to be avoided. It is preferable to refer to the fluid itself (as opposed to the solution of the Lane-Emden equation) as a polytropic fluid. The equation of state of a polytropic fluid is general enough that such idealized fluids find wide use outside of the limited problem of polytropes.