In astrophysics, the Lane-Emden equation is Poisson's equation for the gravitational potential of a self-gravitating, spherically symmetric polytropic fluid. It is named after the astrophysicists Jonathan Homer Lane and Robert Emden. Its solution provides the run of pressure and density with radius r:
- <math> \frac{1}{\zeta^2} \frac{d}{d\zeta} \left({\zeta^2 \frac{d\theta}{d\zeta}}\right) + \theta^n = 0 </math>
where
- <math> \zeta = r \left(\frac{4 \pi G \rho_c^2}{(n+1)P_c}\right)^{\frac{1}{2}} </math>
and
- <math> \rho = \rho_c \theta^n \,</math>
where the subscripts "c" refer to the values of pressure and density at the center of the sphere. Here <math> n </math> is the polytropic index in which the pressure and density of the gas are related by the polytropic equation
- <math> P = K \rho^{1 + 1/n}\, </math>
Note that solutions to the Lane-Emden equation for a given polytropic index <math> n </math> are known as polytropes of index <math> n </math>. 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. It should be clear then if we know nothing about the gas other than the way pressure and density vary with respect to one another, we can reach a solution, in principle. The particular choice of a polytropic gas as given above makes the mathematical statement of the problem particularly succinct, resulting in the Lane-Emden equation. This is a useful "zeroth order" solution for self-gravitating gaseous spheres such as stars. It is still a useful approximation in certain situations, but typically it is a rather limiting assumption.
Solutions of equation
It is known that the equation can be solved analytically when n = 0, 1 or 5:
| n = | 0 | 1 | 5 |
|---|---|---|---|
| ρ = | <math> \rho_c </math> | <math> \rho_c \frac{\sin\zeta}{\zeta} </math> | <math> \rho_c \left(1+ \frac{\zeta^2}{3}\right)^{-\frac{1}{2}} </math> |
| ζ = | <math> \sqrt 6 </math> | <math> \pi </math> | ∞ |
It is indeed constant density for n = 0, and the equation reduces to a Spherical Bessel differential equation which gives a sinc function when n = 1.
References
- 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 Known to Terrestrial Experiment", The American Journal of Science and Arts, 2nd series 50: 57–74.
External links
- Eric W. Weisstein, Lane-Emden Differential Equation at MathWorld.
- Horedt, George Paul ( 1986 ) 'Seven-digit tables of Lane-Emden functions' PDF ( 5.9MB ), Astrophysics and Space Science vol. 126, no. 2, Oct. 1986, p. 357-408. ( ISSN 0004-640X ). Collected at the Smithsonian/NASA Astrophysical Data System.
