Lane-Emden Differential Equation

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

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

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

It has the Boundary Conditions

$$\theta(0) = 1$$ (3)

$$\left[{d\theta\over d\xi}\right]_{\xi=0} = 0.$$ (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

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

(Chandrasekhar 1967, pp. 91-92). Directly solving gives

$${d\over d\xi}\left({\xi^2{d\theta\over d\xi}}\right)=-\xi^2$$ (6)

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

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

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

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

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

The Boundary Condition $\theta(0)=1$ then gives $\theta_0=1$ and $c_1=0$, so

$$\theta_1(\xi)=1-{\textstyle{1\over 6}}\xi^2,$$ (12)

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

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

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

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

which is the Spherical Bessel Differential Equation

$${d\over dr}\left({r^2{dR\over dr}}\right)+[k^2r^2-n(n+1)]R=0$$ (15)

with $k=1$ and $n=0$, so the solution is

$$\theta(\xi) = Aj_0(\xi)+Bn_0(\xi).$$ (16)

Applying the Boundary Condition $\theta(0)=1$ gives

$$\theta_2(\xi)=j_0(\xi)={\sin\xi\over\xi},$$ (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

$$\theta = Ax^\omega z$$ (18)

$$\omega = {2\over n-1},$$ (19)

which reduces the Lane-Emden equation to

$${d^2z\over dt^2}+(2\omega-1){dz\over dt}+\omega(\omega-1)z+A^{n-1}z^n=0$$ (20)

(Chandrasekhar 1967, p. 90). After further manipulation (not reproduced here), the equation becomes

$${d^2z\over dt^2}={\textstyle{1\over 4}}z(1-z^4)$$ (21)

and then, finally,

$$\theta_3(\xi)(1+{\textstyle{1\over 3}}\xi^2)^{-1/2}.$$ (22)

References

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