en>Steelpillow: /* See also */ wholly unrelated

2015-01-05T16:33:40Z

en>K6ka: Reverted edits by 103.12.119.86 (talk) to last revision by 164.85.68.11 (HG)

2014-01-13T13:48:18Z

Reverted edits by 103.12.119.86 (talk) to last revision by 164.85.68.11 (HG)

New page

{{Distinguish|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 <math>P = K \rho^{((n+1)/n)}</math>, where <math>P</math> is pressure, <math>\rho</math> is density and <math>K</math> is a [[Constant (mathematics)|constant]]. The constant <math>n</math> 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 <math>P</math> with [[radius]] in terms of the change of <math>\rho</math> 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 [[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 process|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==
[[Image:Polytrope3n.svg|thumb|left|Density (normalized to average density) versus radius (normalized to external radius) for a polytrope with index n=3.]]

*[[Neutron star]]s are well [[model (abstract)|modeled]] by polytropes with index about in the range between <math>n=0.5</math> and <math>n=1</math>.

*A polytrope with index <math>n=1.5</math> is a good model for fully convective star cores (like those of [[red giant]]s), [[brown dwarf]]s, [[gas giant|giant gaseous planets]] (like [[Jupiter]]), or even for [[terrestrial planet|rocky planet]]s.

*Main sequence [[star]]s like our [[Sun]] and relativistic [[degenerate matter|degenerate]] cores like those of [[white dwarf]]s are usually modeled by a polytrope with index <math>n=3</math>, corresponding to the [[Eddington standard model]] of [[stellar structure]].

*A polytrope with index <math>n=5</math> has an infinite radius. It corresponds to the simplest plausible model of a self-consistent stellar system, first studied by [[Arthur Schuster|A. Schuster]] in 1883.

*A polytrope with index <math>n=\infty</math> 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

[[Category:Astrophysics]]

[[de:Polytrop]]

Phase rule - Revision history

en>Steelpillow: /* See also */ wholly unrelated

en>K6ka: Reverted edits by 103.12.119.86 (talk) to last revision by 164.85.68.11 (HG)