Lane-Emden equation

Solutions of the Lane-Emden equation for n = 0, 1, 2, 3, 4, 5, 6.

The astrophysical Lane-Emden equation describes the structure of a self-gravitating sphere whose equation of state is that of a polytropic liquid. Their solutions describe the dependence of pressure and density on the radius and thus allow conclusions to be drawn about the stability and expansion of the sphere. It is named after the astrophysicists Jonathan Homer Lane (1819–1880) and Robert Emden (1862–1940); Lane proposed it in 1870 as a mathematical model for studying the internal structure of stars. Lord Kelvin and August Ritter were also instrumental in developing this equation. ${\ displaystyle r}$

Physical context

A polytropic liquid satisfies the equation ( : pressure ,: density). However, using instead mostly the polytropic index , which is defined as follows: . To a good approximation, stellar matter can be viewed as a polytropic fluid, such as degenerate gas which, depending on whether it is relativistic or non-relativistic, has a polytropic index of (i.e. ). ${\ displaystyle \ textstyle P = K \ rho ^ {\ gamma}}$ ${\ displaystyle P}$ ${\ displaystyle \ textstyle \ rho}$ ${\ displaystyle \ textstyle \ gamma}$ ${\ displaystyle \ textstyle n}$ ${\ displaystyle \ textstyle \ gamma = 1 + {\ frac {1} {n}}}$ ${\ displaystyle \ textstyle n = 3 \, \ mathrm {or} \, 1,5}$ ${\ displaystyle \ textstyle \ gamma = {\ frac {4} {3}} \, \ mathrm {or} \, {\ frac {5} {3}}}$

Derivation

The equilibrium condition is common for isentropic balls: . Here is the gravitational potential, the enthalpy . Applying the Laplace operator on both sides results in . ${\ displaystyle \ textstyle \ varphi + H = const}$ ${\ displaystyle \ textstyle \ varphi}$ ${\ displaystyle \ textstyle H = (n + 1) {\ frac {P} {\ rho}}}$ ${\ displaystyle \ textstyle \ Delta (\ varphi + H) = 0}$

With the definition and accordingly is the enthalpy . ${\ displaystyle \ textstyle \ rho = \ lambda \ theta ^ {n}}$ ${\ displaystyle \ textstyle P = K \ lambda ^ {1 + {\ frac {1} {n}}} \ theta ^ {n + 1}}$ ${\ displaystyle \ textstyle H = (n + 1) K \ lambda ^ {1 / n} \ theta}$

With the Poisson equation is obtained from the equilibrium condition: . With the scale transformation and one favorably chosen so that one obtains ${\ displaystyle \ textstyle \ Delta \ varphi = 4 \ pi G \ lambda \ theta ^ {n}}$ ${\ displaystyle \ textstyle 4 \ pi G \ lambda \ theta ^ {n} + (n + 1) K \ lambda ^ {1 / n} \ Delta \ theta = 0}$ ${\ displaystyle \ textstyle r = \ alpha \ xi}$ ${\ displaystyle \ textstyle \ alpha}$ ${\ displaystyle \ textstyle 4 \ pi G \ lambda = {\ frac {(n + 1) K \ lambda ^ {1 / n}} {\ alpha ^ {2}}}}$

{\ displaystyle {\ frac {1} {\ xi ^ {2}}} {\ frac {d} {d \ xi}} \ left ({\ xi ^ {2} {\ frac {d \ theta} {d \ xi}}} \ right) + \ theta ^ {n} = 0}

If one with the density in the center identified results and . ${\ displaystyle \ textstyle \ lambda}$ ${\ displaystyle \ textstyle \ rho _ {c}}$ ${\ displaystyle \ textstyle \ xi = r \ left ({\ frac {4 \ pi G \ rho _ {c} ^ {2}} {(n + 1) P_ {c}}} \ right) ^ {\ frac {1} {2}}}$ ${\ displaystyle \ textstyle \ rho = \ rho _ {c} \ theta ^ {n}}$

solutions

The initial conditions are and . The zero of , noted as , defines the limit of the sphere, in the application the limit of the star. ${\ displaystyle \ textstyle \ theta (0) = 1}$ ${\ displaystyle \ textstyle \ left. {\ frac {{\ rm {d}} \ theta} {{\ rm {d}} \ xi}} \ right | _ {\ xi = 0} = 0}$ ${\ displaystyle \ textstyle \ xi}$ ${\ displaystyle \ textstyle \ xi _ {1}}$

The Lane-Emden equation can be solved analytically for n = 0, 1 and 5. While the first two cases lead to equations that are easy to solve, all others are much more complicated. The solution for n = 5 was found in 1885 by Arthur Schuster , later also independently by Emden himself. The three analytical solutions are shown in the table:

n =	0	1	5
${\ displaystyle \ theta}$ =	${\ displaystyle 1 - {\ frac {\ xi ^ {2}} {6}}}$	${\ displaystyle {\ frac {\ sin \ xi} {\ xi}}}$	${\ displaystyle \ left (1 + {\ frac {\ xi ^ {2}} {3}} \ right) ^ {- {\ frac {1} {2}}}}$
${\ displaystyle \ xi _ {1}}$ =	${\ displaystyle {\ sqrt {6}}}$	${\ displaystyle \ pi}$	∞

For n = 1, the equation becomes a spherical Bessel differential equation with the sinc function as the solution.

radius

With the definition for , the radius of the star (in equilibrium) applies ${\ displaystyle \ textstyle \ xi}$

{\ displaystyle R = {\ sqrt {\ frac {(n + 1) P _ {\ rm {c}}} {4 \ pi G \ rho _ {\ rm {c}} ^ {2}}}} \; \ xi _ {1}}

.

For n = 1, the radius is independent of the total mass or the density in the center. The star contains any amount of mass in the same volume, which fulfills the equilibrium condition. ${\ displaystyle \ textstyle P _ {\ rm {c}} = K \ rho _ {\ rm {c}} ^ {2}}$

literature

Ya.B. Zel'dovich, SI Blinnikov, NI Shakura: Physical Grounds of Structure and Evolution of Stars . Moscow University Press, 1981
Bradley W. Carroll, Dale A. Ostlie: An Introduction to Modern Astrophysics . 2nd edition. Pearson, 2007
George Paul Horedt: Seven-digit tables of Lane-Emden functions . In: Astrophysics and Space Science. Volume 126, No. 2, October 1986, pages 357-408, bibcode : 1986Ap & SS.126..357H

Web links

Eric W. Weisstein : Lane-Emden Differential Equation . In: MathWorld (English).