# Polytrope

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{{short description|Thermodynamic concept imporant in astrophysics}}
{{about|constrained thermodynamic model|the geometric object|Polytope}}
thumb|350x350px|The normalized density as a function of scale length for a wide range of polytropic indices

In [astrophysics](/source/astrophysics), a '''polytrope''' is a thermodynamic system with [pressure](/source/pressure) dependent upon [density](/source/density), leaving only one independent state variable. A [polytropic process](/source/polytropic_process) is intermediate between an [isothermal](/source/isothermal_process) process and [adiabatic](/source/Adiabatic_process) one.<ref name=Horedt-2010/>{{rp|3}} The dependence of pressure on density is a solution to the [Lane–Emden equation](/source/Lane%E2%80%93Emden_equation):
<math display="block">P = K \rho^{(n+1)/n} = K \rho^{1 + 1/n},</math>
where {{math|<var>P</var>}} is pressure, {{math|<var>&rho;</var>}} is density and {{math|<var>K</var>}} is a [constant](/source/Constant_(mathematics)) of [proportionality](/source/Proportionality_(mathematics)).<ref name=Horedt-2010>{{cite book | last=Horedt | first=Georg P. | title=Polytropes: Applications in Astrophysics and Related Fields | publisher=Springer | publication-place=Dordrecht | date=2010 | isbn=978-90-481-6645-9 }}</ref>{{rp|28}} The constant {{math|<var>n</var>}} is known as the [polytropic index](/source/polytropic_index).<ref name=Horedt-2010/>{{rp|24}}

This relation need not be interpreted as an [equation of state](/source/equation_of_state), which states ''P'' as a function of both &rho; and ''T'' (the [temperature](/source/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](/source/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](/source/thermodynamics) relation above. It is preferable to refer to the [fluid](/source/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](/source/specific_heat) is constant.<ref name="StellarStructure">{{cite book | last1=Chandrasekhar | first1=Subrahmanyan | title=An Introduction to the Study of Stellar Structure | publisher=Dover | publication-place=New York | date=1957 | orig-date=1939}} {{isbn|978-0-486-60413-8 }}</ref><ref name="LandauLifshitz">{{cite book | last1=Landau | first1=L D | last2=Lifshitz | first2=E. M. | title=Fluid Mechanics | publisher=Elsevier | publication-place=Amsterdam Heidelberg | date=2013 | isbn=978-1-4831-6104-4 }}</ref> 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](/source/derivative) of the [bulk modulus](/source/bulk_modulus)<ref name="mnras">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](/source/Monthly_Notices_of_the_Royal_Astronomical_Society), Vol. 452, No. 2 (Sept. 2015), pages 1375–1393, Oxford University Press also found at [https://arxiv.org/abs/1409.5525 the arXiv]</ref> where its relation to the [Murnaghan equation of state](/source/Murnaghan_equation_of_state) has also been demonstrated. The polytrope relation is therefore best suited for relatively low-pressure (below 10<sup>7</sup>&nbsp;[Pa](/source/Pascal_(unit))) and high-pressure (over 10<sup>14</sup>&nbsp;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==
thumb|bottom|Density (normalized to average density) versus radius (normalized to external radius) for a polytrope with index n=3.

*An index {{math|<var>n</var> {{=}} 0}} polytrope is often used to model [rocky planet](/source/Terrestrial_planet)s. The reason is that {{math|<var>n</var> {{=}} 0}} polytrope has constant density, i.e., incompressible interior. This is a zero order approximation for rocky (solid/liquid) planets.
*[Neutron star](/source/Neutron_star)s are well [modeled](/source/model_(abstract)) by polytropes with index between {{math|''n'' {{=}} 0.5}} and {{math|''n'' {{=}} 1}}.
*A polytrope with index {{math|<var>n</var> {{=}} 1.5}} is a good model for fully convective [star cores](/source/Stellar_core)<ref name="StellarStructure"/><ref>C. J. Hansen, S. D. Kawaler, [V. Trimble](/source/Virginia_Louise_Trimble) (2004). ''Stellar Interiors – Physical Principles, Structure, and Evolution'', New York: Springer. {{ISBN|0-387-20089-4}}</ref> (like those of [red giant](/source/red_giant)s), [brown dwarf](/source/brown_dwarf)s, [giant gaseous planets](/source/gas_giant) (like [Jupiter](/source/Jupiter)). With this index, the polytropic exponent is 5/3, which is the [heat capacity ratio](/source/heat_capacity_ratio) (&gamma;) for [monatomic gas](/source/monatomic_gas). For the interior of gaseous stars (consisting of either [ionized](/source/Ionization) [hydrogen](/source/hydrogen) or [helium](/source/helium)), this follows from an [ideal gas](/source/Adiabatic_process) approximation for [natural convection](/source/natural_convection) conditions.
*A polytrope with index {{math|<var>n</var> {{=}} 1.5}} is also a good model for [white dwarf](/source/white_dwarf)s of low mass, according to the [equation of state](/source/equation_of_state) of non-[relativistic](/source/relativistic_particle) [degenerate matter](/source/degenerate_matter).<ref name = Sagert2006>[https://arxiv.org/abs/astro-ph/0506417 Sagert, I., Hempel, M., Greiner, C., Schaffner-Bielich, J. (2006). Compact stars for undergraduates. European journal of physics, 27(3), 577.]</ref>
*A polytrope with index {{math|<var>n</var> {{=}} 3}} is a good model for the cores of white dwarfs of higher masses, according to the equation of state of [relativistic](/source/relativistic_particle) [degenerate matter](/source/degenerate_matter).<ref name = Sagert2006/>
*A polytrope with index {{math|<var>n</var> {{=}} 3}} is usually also used to model [main-sequence](/source/Main_sequence) [star](/source/star)s like the [Sun](/source/Sun), at least in the [radiation zone](/source/radiation_zone), corresponding to the [Eddington standard model](/source/Radiation_zone) of [stellar structure](/source/stellar_structure).<ref>O. R. Pols (2011), Stellar Structure and Evolution, Astronomical Institute Utrecht, September 2011, pp. 64-68</ref> 
*A polytrope with index {{math|<var>n</var> {{=}} 5}} has an [infinite](/source/Infinity) radius. It corresponds to the simplest plausible model of a self-consistent stellar system, first studied by [Arthur Schuster](/source/Arthur_Schuster) in 1883, and it has an [exact solution](/source/Lane%E2%80%93Emden_equation).
*A polytrope with index {{math|<var>n</var> {{=}} &infin;}} corresponds to what is called an ''isothermal sphere'', that is an [isothermal](/source/Isothermal_process) [self-gravitating](/source/Self-gravitation) sphere of gas, whose structure is identical to the structure of a collisionless system of stars like a [globular cluster](/source/globular_cluster). This is because for an ideal gas, the temperature is proportional to &rho;<sup>1/n</sup>, so infinite ''n'' corresponds to a constant temperature.

In general as the polytropic index increases, the density distribution is more heavily weighted toward the center ({{math|<var>r</var> {{=}} 0}}) of the body.

== See also ==
* [Polytropic process](/source/Polytropic_process)
* [Equation of state](/source/Equation_of_state)
* [Murnaghan equation of state](/source/Murnaghan_equation_of_state)

==References==
{{reflist|colwidth=35em}}

Category:Astrophysics

[de:Polytrop](/source/de%3APolytrop)

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Adapted from the Wikipedia article [Polytrope](https://en.wikipedia.org/wiki/Polytrope) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Polytrope?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.