Lane–Emden equation

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.

Solutions thus describe the run of pressure and density with radius and are known as polytropes of index

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.

Consider a self-gravitating, spherically symmetric fluid in hydrostatic equilibrium.

yields, in some sense, a dimensional form of the desired equation.

between 0 and 5, the solutions are continuous and finite in extent, with the radius of the star given by

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

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

This relation can be solved leading to the general solution:

This exact solution was found by accident when searching for zero values of the related TOV Equation.

Plugging this into the Lane-Emden equation, we can show that all odd coefficients of the series vanish

but numerical results showed good agreement for much larger values.

Chandrasekhar believed for a long time that finding other solution for

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.

In applications, the main role play analytic solutions that are expressible by the convergent power series expanded around some initial point.

One can prove [4][5] that the equation has the convergent power series/analytic solution around the origin of the form

The radius of convergence of this series is limited due to existence [5][7] of two singularities on the imaginary axis in the complex plane.

, and therefore, they are called movable singularities due to classification of the singularities of non-linear ordinary differential equations in the complex plane by Paul Painlevé.

A similar structure of singularities appears in other non-linear equations that result from the reduction of the Laplace operator in spherical symmetry, e.g., Isothermal Sphere equation.

[7] Analytic solutions can be extended along the real line by analytic continuation procedure resulting in the full profile of the star or molecular cloud cores.

Two analytic solutions with the overlapping circles of convergence can also be matched on the overlap to the larger domain solution, which is a commonly used method of construction of profiles of required properties.

It is used to shift the initial data for analytic solution slightly away from the origin since at the origin the numerical methods fail due to the singularity of the equation.

Many standard methods require that the problem is formulated as a system of first-order ordinary differential equations.

If one chooses variables that are invariant to homology, then we can reduce the order of the Lane–Emden equation by one.

The behaviour of solutions to these equations can be determined by linear stability analysis.

) and the eigenvalues and eigenvectors of the Jacobian matrix are tabulated below.