Lane–Emden equation
The astrophysical Lane -Emden equation describes the structure of a selbstgravitierenden sphere whose equation of state is that of a polytropic fluid. Their solutions describe the dependency of the pressure and the density of radius r, thus allowing conclusions to be drawn on the stability and expansion of the ball. It is named after the astrophysicists Jonathan Homer Lane (1819-1880) and Robert Emden; Lane struck before 1870 as a mathematical model to study the internal structure of stars. Lord Kelvin and August Ritter were also instrumental in the development of this equation.
Physical context
A polytropic fluid satisfies the equation ( P: pressure: density). However, instead of using the usually Polytropenindex n, which is defined as follows:. Star- matter can be considered a good approximation as the polytropic fluid, such as degenerate gas. Depending on whether it is or not relativistic - relativistic, has a Polytropenindex of 3 and 1.5
Derivation
The equilibrium condition is generally for isentropic balls. It is the gravitational potential, the enthalpy. After application of the Laplace operator on both sides results.
With the definition and accordingly is the enthalpy.
The Poisson's equation is calculated from the equilibrium condition. With the scale transformation and a conveniently chosen so that one obtains
If one identifies with the density in the center, and results.
Solutions
The initial conditions are and. The zero of, noted as, ie defines the boundary of the ball, in the application of the boundary of the star, firmly.
The Lane -Emden equation can be solved analytically for n = 0, 1 and 5. While the first two cases lead to simple equations to be solved, all other more complicated. The solution for n = 5 was found in 1885 by Arthur Schuster, and later regardless of Emden itself. The three analytical solutions are shown in the table:
For n = 1, the equation of a spherical Bessel differential equation with the sinc function as a solution.
Radius
With the definition applies to the radius of the star (in equilibrium )
For n = 1 is independent of the total mass or density in the center because of the radius. The star contains the same volume of any amount of mass that satisfies the equilibrium condition.