Polytrope

In astrophysics, a polytrope refers to a solution of the Lane–Emden equation in which the pressure depends upon the density in the form $${\displaystyle P=K\rho ^{(n+1)/n}=K\rho ^{1+1/n},}$$ where P is pressure, ρ is density and K is a constant of proportionality.^[1] The constant n is known as the polytropic index; note however that the polytropic index has an alternative definition as with n as the exponent.

This relation need not be interpreted as an equation of state, which states P as a function of both ρ and T (the temperature); however in the particular case described by the polytrope equation there are other additional relations between these three quantities, which together determine the equation. Thus, this is simply a relation that expresses an assumption about the change of pressure with radius in terms of the change of density with radius, yielding a solution to the Lane–Emden equation.

Sometimes the word polytrope may 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 or polytropic gas. Specifically, the polytropic gas is a gas for which the specific heat is constant.^[2][3] 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.

The polytropic exponent (of a polytrope) has been shown to be equivalent to the pressure derivative of the bulk modulus^[4] where its relation to the Murnaghan equation of state has also been demonstrated. The polytrope relation is therefore best suited for relatively low-pressure (below 10⁷ Pa) and high-pressure (over 10¹⁴ Pa) conditions when the pressure derivative of the bulk modulus, which is equivalent to the polytrope index, is near constant.