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 Newtonian 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 . If an isothermal fluid (polytropic index tends to infinity) is used instead of a polytropic fluid, one obtains the Emden–Chandrasekhar equation.

ApplicationsEdit

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 spheres of plasma such as stars, but typically it is a rather limiting assumption.

DerivationEdit

From hydrostatic equilibriumEdit

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 the continuity equation has been used to replace the mass gradient. Multiplying both sides by   and collecting the derivatives of   on the left, one 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 equationEdit

Equivalently, one can start with Poisson's equation,

 

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

 

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

Exact solutionsEdit

For a given value of the polytropic index  , denote the solution to the Lane–Emden equation as  . In general, the Lane–Emden equation must be solved numerically to find  . There are exact, analytic solutions for certain values of  , in particular:  . For   between 0 and 5, the solutions are continuous and finite in extent, with the radius of the star given by  , where  .

For a given solution  , the density profile is given by

 .

The total mass   of the model star can be found by integrating the density over radius, from 0 to  .

The pressure can be found using the polytropic equation of state,  , i.e.

 

Finally, if the gas is ideal, the equation of state is  , where   is the Boltzmann constant and   the mean molecular weight. The temperature profile is then given by

 

In spherically symmetric cases, the Lane–Emden equation is integrable for only three values of the polytropic index  .

For n = 0Edit

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  . Therefore,

 

For n = 1Edit

When  , the equation can be expanded in the form

 

One assumes a power series solution:

 

This leads to a recursive relationship for the expansion coefficients:

 

This relation can be solved leading to the general solution:

 

The boundary condition for a physical polytrope demands that   as  . This requires that  , thus leading to the solution:

 

For n = 5Edit

We start from with the Lane–Emden equation:

 


Rewriting for   produces:

 

Differentiating with respect to ξ leads to:

 

Reduced, we come by:

 

Therefore, the Lane–Emden equation has the solution

 

when  . This solution is finite in mass but infinite in radial extent, and therefore the complete polytrope does not represent a physical solution. Chandrasekhar believed for a long time that finding other solution for   "is complicated and involves elliptic integrals".

Srivastava's solutionEdit

In 1962, Sambhunath Srivastava found an explicit solution when  .[2] His solution is given by

 

and from this solution, a family of solutions   can be obtained using homology transformation. Since this solution does not satisfy the conditions at the origin (in fact, it is oscillatory with amplitudes growing indefinitely as the origin is approached), this solution can be used in composite stellar models.

Numerical solutionsEdit

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 initial conditions are   and  . The first equation represents hydrostatic equilibrium and the second represents mass conservation.

Homologous variablesEdit

Homology-invariant equationEdit

It is known that if   is a solution of the Lane–Emden equation, then so is  .[3] Solutions that are related in this way are called homologous; the process that transforms them is homology. If one chooses 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 equationEdit

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.[4]

 

See alsoEdit

ReferencesEdit

  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 as known to terrestrial experiment". American Journal of Science. 2. 50 (148): 57–74. Bibcode:1870AmJS...50...57L. doi:10.2475/ajs.s2-50.148.57. ISSN 0002-9599.
  2. ^ Srivastava, Shambhunath (1962). "A New Solution of the Lane-Emden Equation of Index n=5". The Astrophysical Journal. 136: 680. Bibcode:1962ApJ...136..680S. doi:10.1086/147421. ISSN 0004-637X.
  3. ^ Chandrasekhar, Subrahmanyan (1957) [1939]. An Introduction to the Study of Stellar Structure. Dover. Bibcode:1939isss.book.....C. ISBN 978-0-486-60413-8.
  4. ^ Horedt, Georg P. (1987). "Topology of the Lane-Emden equation". Astronomy and Astrophysics. 117 (1–2): 117–130. Bibcode:1987A&A...177..117H. ISSN 0004-6361.

Further readingEdit

External linksEdit