Polytropic

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Polytropic fluids are idealized fluid models that are used often in astrophysics. A polytropic fluid is a type of barotropic fluid for which the equation of state is written as

[P = K \rho^]

where [ P ] is the pressure, [ K ] is a constant, [ \rho ] is the density, and [ n ] is a quantity called the polytropic index.

This is also commonly written in the form

[P = K \rho^\gamma]

Note that [ \gamma ] need not be the adiabatic index (or ratio of specific heats), and in fact often it is not. This is sometimes a cause for confusion.

In the case of an isentropic ideal gas, [ \gamma ] is the ratio of specific heats, or adiabatic index:

[\gamma = \frac]

An isothermal ideal gas is also a polytropic gas. Here, the polytropic [ \gamma ] differs from the adiabatic index [ \gamma ], and in particular it is equal to one: [ \gamma = 1 ].

In order to discriminate between the two gammas, the polytropic gamma is sometimes capitalized, [ \Gamma ].

To confuse matters further, some authors refer to [ \Gamma ] as the polytropic index, rather than [ n ]. Note that

[n = \frac.]

A solution to the Lane-Emden equation using a polytropic fluid is known as a polytrope.

Polytropic process

There is a thermodynamic process called Polytropic process. It is a generalization of every process.

[p V^ = K ]

where [ p ] is the pressure, [ V ] is the volume, [ n ] is the polytropic index, [ K ] is a constant.

The main article is in Polytropic process

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