Lane–Emden equation on arvau

Wikipedia / 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. The equation reads . Solutions thus describe the run of pressure and density with radius and are known as polytropes. If an isothermal fluid (polytropic index tends to infinity) is used instead of a polytropic fluid, one obtains the Emden–Chandrasekhar equation.

Applications

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.

Derivation

From hydrostatic equilibrium

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

From Poisson's equation

Equivalently, one can start with Poisson's equation

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

Exact solutions

Finally, if the gas is ideal is the Boltzmann constant

For n = 0

For n = 1

The boundary condition

For n = 2

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

We consider a series expansion.

For n = 5

Differentiating with respect to

"is complicated and involves elliptic integrals".

Srivastava's solution

. 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.

Analytic solutions

In applications, the main role play analytic solutions that are expressible by the convergent power series at the centre of the star. One can prove

The radius of convergence of this series is limited due to existence of two singularities on the imaginary axis in the complex plane. These singularities are located symmetrically with respect to the origin. Their position change when we change equation parameters and the initial condition, 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.

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.

The series solution is also used in the numerical integration of the equation. 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.

Numerical solutions

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,

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

Homologous variables

Homology-invariant equation

. 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.

This is now a single first-order equation.

Topology of the homology-invariant equation

The behaviour of solutions to these equations can be determined by linear stability) and the eigenvalues and eigenvectors of the Jacobian matrix are tabulated below.