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Lane-Emden Differential Equation


A second-order Ordinary Differential Equation arising in the study of stellar interiors. It is given by

{1\over\xi^2} {d\over d\xi}\left({\xi^2 {d\theta\over d\xi}}\right)+\theta^n = 0
\end{displaymath} (1)

{1\over\xi^2}\left({2\xi {d\theta \over d\xi} +\xi^2 {d^2\th...
...ver d\xi^2} + {2\over \xi} {d\theta \over d\xi} +\theta^n = 0.
\end{displaymath} (2)

It has the Boundary Conditions
$\displaystyle \theta(0)$ $\textstyle =$ $\displaystyle 1$ (3)
$\displaystyle \left[{d\theta\over d\xi}\right]_{\xi=0}$ $\textstyle =$ $\displaystyle 0.$ (4)

Solutions $\theta(\xi)$ for $n=0$, 1, 2, 3, and 4 are shown above. The cases $n=0$, 1, and 5 can be solved analytically (Chandrasekhar 1967, p. 91); the others must be obtained numerically.

For $n=0$ ( $\gamma =\infty$), the Lane-Emden Differential Equation is

{1\over\xi^2}{d\over d\xi}\left({\xi^2{d\theta\over d\xi}}\right)+1=0
\end{displaymath} (5)

(Chandrasekhar 1967, pp. 91-92). Directly solving gives
{d\over d\xi}\left({\xi^2{d\theta\over d\xi}}\right)=-\xi^2
\end{displaymath} (6)

\int d\left({\xi^2{d\theta\over d\xi^2}}\right)= -\int \xi^2\,d\xi
\end{displaymath} (7)

\xi^2{d\theta\over d\xi}=c_1-{\textstyle{1\over 3}}\xi^3
\end{displaymath} (8)

{d\theta\over d\xi}={c_1-{\textstyle{1\over 3}}\xi^3\over \xi^2}
\end{displaymath} (9)

\theta(\xi)=\int d\theta = \int {c_1-{\textstyle{1\over 3}}\xi^3\over \xi^2} \,d\xi
\end{displaymath} (10)

\theta(\xi)=\theta_0-c_1\xi^{-1}-{\textstyle{1\over 6}}\xi^2.
\end{displaymath} (11)

The Boundary Condition $\theta(0)=1$ then gives $\theta_0=1$ and $c_1=0$, so
\theta_1(\xi)=1-{\textstyle{1\over 6}}\xi^2,
\end{displaymath} (12)

and $\theta_1(\xi)$ is Parabolic.

For $n=1$ ($\gamma = 2$), the differential equation becomes

{1\over\xi^2}{d\over d\xi}\left({\xi^2{d\theta\over d\xi}}\right)+\theta=0
\end{displaymath} (13)

{d\over d\xi}\left({\xi^2{d\theta\over d\xi}}\right)+\theta\xi^2=0,
\end{displaymath} (14)

which is the Spherical Bessel Differential Equation
{d\over dr}\left({r^2{dR\over dr}}\right)+[k^2r^2-n(n+1)]R=0
\end{displaymath} (15)

with $k=1$ and $n=0$, so the solution is
\theta(\xi) = Aj_0(\xi)+Bn_0(\xi).
\end{displaymath} (16)

Applying the Boundary Condition $\theta(0)=1$ gives
\end{displaymath} (17)

where $j_0(x)$ is a Spherical Bessel Function of the First Kind (Chandrasekhar 1967, pp. 92).

For $n=5$, make Emden's transformation

$\displaystyle \theta$ $\textstyle =$ $\displaystyle Ax^\omega z$ (18)
$\displaystyle \omega$ $\textstyle =$ $\displaystyle {2\over n-1},$ (19)

which reduces the Lane-Emden equation to
{d^2z\over dt^2}+(2\omega-1){dz\over dt}+\omega(\omega-1)z+A^{n-1}z^n=0
\end{displaymath} (20)

(Chandrasekhar 1967, p. 90). After further manipulation (not reproduced here), the equation becomes
{d^2z\over dt^2}={\textstyle{1\over 4}}z(1-z^4)
\end{displaymath} (21)

and then, finally,
\theta_3(\xi)(1+{\textstyle{1\over 3}}\xi^2)^{-1/2}.
\end{displaymath} (22)


Chandrasekhar, S. An Introduction to the Study of Stellar Structure. New York: Dover, pp. 84-182, 1967.

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© 1996-9 Eric W. Weisstein