User:Warrickball/Lane–Emden equation

Solutions of Lane–Emden equation for n = 0, 1, 2, 3, 4, 5.

In astrophysics, the Lane–Emden equation is a dimensionless form of Poisson's equation for the gravitational potential of a self-gravitating, spherically symmetric, polytropic fluid. It is named after astrophysicists Jonathan Homer Lane and Robert Emden.[1] The equation reads

where is a dimensionless radius and is related to the density (and thus the pressure) by for central density . The index is the polytropic index that appears in the polytropic equation of state,

where and are the pressure and density, respectively, and is a constant of proportionality. The standard boundary conditions are and . Solutions thus describe the run of pressure and density with radius and are known as polytropes of index .

Applications

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Physically, hydrostatic equilibrium connects the gradient of the potential, the density, and the gradient of the pressure, whereas Poisson's equation connects the potential with the density. Thus, if we have a further equation that dictates how the pressure and density vary with respect to one another, we can reach a solution. The particular choice of a polytropic gas as given above makes the mathematical statement of the problem particularly succinct and leads to the Lane–Emden equation. The equation is a useful approximation for self-gravitating gaseous spheres such as stars, but typically it is a rather limiting assumption.

Derivation

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From hydrostatic equilibrium

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Consider a self-gravitating, spherically symmetric fluid in hydrostatic equilibrium. Mass is conserved and thus described by the continuity equation

 

where   is a function of  . The equation of hydrostatic equilibrium is

 

where   is also a function of  . Differentiating again gives

 

where we have used the continuity equation to replace the mass gradient. Multiplying both sides by   and collecting the derivatives of   on the left, we can write

 

Dividing both sides by   yields, in some sense, a dimensional form of the desired equation. If, in addition, we substitute for the polytropic equation of state with   and  , we have

 

Gathering the constants and substituting  , where

 ,

we have the Lane–Emden equation,

 

From Poisson's equation

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Equivalently, one can start with Poisson's equation,

 

We can replace the gradient of potential using hydrostatic equilibrium, via

 

which again yields the dimensional form of the Lane–Emden equation.

Solutions

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Exact Solutions

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There are only three values of the polytropic index   that lead to exact solutions.

 

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If  , the equation becomes

 

Re-arranging and integrating once gives

 

Dividing both sides by   and integrating again gives

 

The boundary conditions   and   imply that the constants of integration are   and  .

 

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When  , the equation can be expanded in the form

 

Multiplying both sides by   gives a spherical Bessel differential equation with   and  . The solution, after application of the boundary conditions, is

 

 

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After a sequence of substitutions, it can be shown that the Lane–Emden equation has a further solution

 

when  . This solution is infinite in radial extent.

Numerical Solutions

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In general, solutions are found by numerical integration. Many standard methods require that the problem is formulated as a system of first-order ordinary differential equations. For example,

 
 

Here,   is interpreted as the dimensionless mass, defined by  . The relevant boundary conditions are   and  . The first equation represents hydrostatic equilibrium and the second mass conservation.

Homologous Variables

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Homology-invariant equation

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It is known that if   is a solution of the Lane–Emden equation, then so is  .[2] Solutions that are related in this way are called homologous; the process that transforms them is homology. If we choose variables that are invariant to homology, then we can reduce the order of the Lane–Emden equation by one.

A variety of such variables exist. A suitable choice is

 

and

 

We can differentiate the logarithms of these variables with respect to  , which gives

 

and

 .

Finally, we can divide these two equations to eliminate the dependence on  , which leaves

 

This is now a single first-order equation.

Topology of the homology-invariant equation

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The homology-invariant equation can be regarded as the autonomous pair of equations

 

and

 .

The behaviour of solutions to these equations can be determined by linear stability analysis. The critical points of the equation (where  ) and the eigenvalues and eigenvectors of the Jacobian matrix are tabulated below.[3]

Critical point Eigenvalues Eigenvectors
         
         
         
     

Further reading

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Horedt, Georg P. (2004). Polytropes - Applications in Astrophysics and Related Fields. Dordrecht: Kluwer Academic Publishers. ISBN 978-1-4020-2350-7.

References

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  1. ^ Lane, Jonathan Homer (1870). "On the Theoretical Temperature of the Sun under the Hypothesis of a Gaseous Mass Maintaining its Volume by its Internal Heat and Depending on the Laws of Gases Known to Terrestrial Experiment". The American Journal of Science and Arts. 2. 50: 57–74.
  2. ^ Chandrasekhar, Subrahmanyan (1939). An introduction to the study of stellar structure. Chicago, Ill.: University of Chicago Press.
  3. ^ Horedt, Georg P. (1987). "Topology of the Lane-Emden equation". A&A. 117 (1–2): 117–130. Retrieved 27 June 2012.{{cite journal}}: CS1 maint: date and year (link)
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Category:Astrophysics Category:Ordinary differential equations