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

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.

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.

In general as the polytropic index increases, the density distribution is more heavily weighted toward the center (r = 0) of the body.