Editing Polytrope

{{short description|Solution of the Lane-Emden pressure-density equation for astrophysical bodies}}
{{Distinguish|Polytope}}
{{see also|Polytropic process}}
[[File:Polytropes.gif|thumb|350x350px|The normalized density as a function of scale length for a wide range of polytropic indices]]

In [[astrophysics]], a '''polytrope''' refers to a solution of the [[Lane–Emden equation]] in which the [[pressure]] depends upon the [[density]] in the form
<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 (mathematics)|constant]] of [[Proportionality (mathematics)|proportionality]].<ref>Horedt, G. P. (2004). ''Polytropes. Applications in Astrophysics and Related Fields'', Dordrecht: Kluwer. {{ISBN|1-4020-2350-2}}</ref> The constant {{math|<var>n</var>}} 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 &rho; 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 [[thermodynamics|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 ''' or '''polytropic gas'''. Specifically, the polytropic gas is a gas for which the [[specific heat]] is constant.<ref>Chandrasekhar, Subrahmanyan, and Subrahmanyan Chandrasekhar. An introduction to the study of stellar structure. Vol. 2. Courier Corporation, 1957.</ref><ref>Landau, Lev Davidovich, and Evgenii Mikhailovich Lifshitz. Fluid mechanics: Landau And Lifshitz: course of theoretical physics, Volume 6. Vol. 6. Elsevier, 2013.</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]] of the [[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]], 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]] has also been demonstrated. The polytrope relation is therefore best suited for relatively low-pressure (below 10<sup>7</sup>&nbsp;[[Pascal (unit)|Pa]]) 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==
[[Image:Polytrope3n.svg|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 [[Terrestrial planet|rocky 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]]s are well [[model (abstract)|modeled]] 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 [[Stellar core|star cores]]<ref>[[Subrahmanyan Chandrasekhar|S. Chandrasekhar]] [1939] (1958). ''An Introduction to the Study of Stellar Structure'', New York: Dover. {{ISBN|0-486-60413-6}}</ref><ref>C. J. Hansen, S. D. Kawaler, [[Virginia Louise Trimble|V. Trimble]] (2004). ''Stellar Interiors – Physical Principles, Structure, and Evolution'', New York: Springer. {{ISBN|0-387-20089-4}}</ref> (like those of [[red giant]]s), [[brown dwarf]]s, [[gas giant|giant gaseous planets]] (like [[Jupiter]]). With this index, the polytropic exponent is 5/3, which is the [[heat capacity ratio]] (&gamma;) for [[monatomic gas]]. For the interior of gaseous stars (consisting of either [[Ionization|ionized]] [[hydrogen]] or [[helium]]), this follows from an [[Adiabatic process#Ideal gas (reversible process)|ideal gas]] approximation for [[natural convection]] conditions.
*A polytrope with index {{math|<var>n</var> {{=}} 1.5}} is also a good model for [[white dwarf]]s of low mass, according to the [[equation of state]] of non-[[relativistic particle|relativistic]] [[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 particle|relativistic]] [[degenerate matter]].<ref name = Sagert2006/>
*A polytrope with index {{math|<var>n</var> {{=}} 3}} is usually also used to model [[Main sequence|main-sequence]] [[star]]s like the [[Sun]], at least in the [[radiation zone]], corresponding to the [[Radiation zone#Eddington stellar model|Eddington standard model]] of [[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 [[Infinity|infinite]] radius. It corresponds to the simplest plausible model of a self-consistent stellar system, first studied by [[Arthur Schuster]] in 1883, and it has an [[Lane–Emden equation#For n = 5|exact solution]].
*A polytrope with index {{math|<var>n</var> {{=}} &infin;}} corresponds to what is called an ''isothermal sphere'', that is an [[Isothermal process|isothermal]] [[Self-gravitation|self-gravitating]] sphere of gas, whose structure is identical to the structure of a collisionless system of stars like a [[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]]
* [[Equation of state]]
* [[Murnaghan equation of state]]

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

[[Category:Astrophysics]]

[[de:Polytrop]]