Polytrope

{{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, a '''polytrope''' is a thermodynamic system with pressure dependent upon density, leaving only one independent state variable. A polytropic process is intermediate between an isothermal process and adiabatic one.<ref name=Horedt-2010/>{{rp|3}} The dependence of pressure on density is a solution to the Lane–Emden 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>ρ</var>}} is density and {{math|<var>K</var>}} is a constant of proportionality.<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.<ref name=Horedt-2010/>{{rp|24}}

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. 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 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 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 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== 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 planets. 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 stars are well 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 star cores<ref name="StellarStructure"/><ref>C. J. Hansen, S. D. Kawaler, V. Trimble (2004). ''Stellar Interiors – Physical Principles, Structure, and Evolution'', New York: Springer. {{ISBN|0-387-20089-4}}</ref> (like those of red giants), brown dwarfs, giant gaseous planets (like Jupiter). With this index, the polytropic exponent is 5/3, which is the heat capacity ratio (γ) for monatomic gas. For the interior of gaseous stars (consisting of either ionized hydrogen or helium), this follows from an ideal gas approximation for natural convection conditions. *A polytrope with index {{math|<var>n</var> {{=}} 1.5}} is also a good model for white dwarfs of low mass, according to the equation of state of non-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 degenerate matter.<ref name = Sagert2006/> *A polytrope with index {{math|<var>n</var> {{=}} 3}} is usually also used to model main-sequence stars like the Sun, at least in the radiation zone, corresponding to the 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 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 exact solution. *A polytrope with index {{math|<var>n</var> {{=}} ∞}} corresponds to what is called an ''isothermal sphere'', that is an isothermal 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 ρ1/n, 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.

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

Category:Astrophysics

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