Polytrope

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Not to be confused with polytope.

In astrophysics, a polytrope refers to a solution of the Lane–Emden equation in which the pressure depends upon the density in the form $P=K\rho ^{{((n+1)/n)}}$ , where $P$ is pressure, $\rho$ 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 relation that expresses an assumption about the change of $P$ with radius in terms of the change of $\rho$ with radius, yielding a solution to the Lane–Emden equation.

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.

Example models by polytropic index

Density (normalized to average density) versus radius (normalized to external radius) for a polytrope with index n=3.

Neutron stars are well modeled by polytropes with index about in the range between $n=0.5$ and $n=1$ .

A polytrope with index $n=1.5$ is a good model for fully convective star cores (like those of red giants), brown dwarfs, giant gaseous planets (like Jupiter), or even for rocky planets.

Main sequence stars like our Sun and relativistic degenerate cores like those of white dwarfs are usually modeled by a polytrope with index $n=3$ , corresponding to the Eddington standard model of stellar structure.

A polytrope with index $n=5$ has an infinite radius. It corresponds to the simplest plausible model of a self-consistent stellar system, first studied by A. Schuster in 1883.

A polytrope with index $n=\infty$ corresponds to what is called isothermal sphere, that is an isothermal self-gravitating sphere of gas, whose structure is identical with the structure of a collisionless system of stars like a globular cluster.

Note that the higher the polytropic index, the more condensed at the centre is the density distribution.

References

Chandrasekhar, S. [ 1939 ] ( 1958 ). An Introduction to the Study of Stellar Structure, New York : Dover. ISBN 0-486-60413-6
Hansen, C.J., Kawaler S.D. & Trimble V. ( 2004 ). Stellar Interiors - Physical Principles, Structure, and Evolution, New York : Springer. ISBN 0-387-20089-4
Horedt, G.P. ( 2004 ). Polytropes. Applications in Astrophysics and Related Fields, Dordrecht : Kluwer. ISBN 1-4020-2350-2

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