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Polytrope

## Summary

In astrophysics, a polytrope refers to a solution of the Lane–Emden equation in which the pressure depends upon the density in the form

The normalized density as a function of scale length for a wide range of polytropic indices
${\displaystyle P=K\rho ^{(n+1)/n},}$

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. 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[2] 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.

## Example models by polytropic index

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

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

## References

1. ^ Horedt, G. P. (2004). Polytropes. Applications in Astrophysics and Related Fields, Dordrecht: Kluwer. ISBN 1-4020-2350-2
2. ^ Weppner, S. P., McKelvey, J. P., Thielen, K. D. and Zielinski, A. K., "A variable polytrope index applied to planet and material models", Monthly Notices of the Royal Astronomical Society, Vol. 452, No. 2 (Sept. 2015), pages 1375–1393, Oxford University Press also found at the arXiv
3. ^ S. Chandrasekhar [1939] (1958). An Introduction to the Study of Stellar Structure, New York: Dover. ISBN 0-486-60413-6
4. ^ C. J. Hansen, S. D. Kawaler, V. Trimble (2004). Stellar Interiors – Physical Principles, Structure, and Evolution, New York: Springer. ISBN 0-387-20089-4
5. ^ a b Sagert, I., Hempel, M., Greiner, C., Schaffner-Bielich, J. (2006). Compact stars for undergraduates. European journal of physics, 27(3), 577.
6. ^ O. R. Pols (2011), Stellar Structure and Evolution, Astronomical Institute Utrecht, September 2011, pp. 64-68