A second-order Ordinary Differential Equation arising in the study of stellar interiors. It is given by
![\begin{displaymath}
{1\over\xi^2} {d\over d\xi}\left({\xi^2 {d\theta\over d\xi}}\right)+\theta^n = 0
\end{displaymath}](l1_544.gif) |
(1) |
![\begin{displaymath}
{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}](l1_545.gif) |
(2) |
It has the Boundary Conditions
Solutions
for
, 1, 2, 3, and 4 are shown above. The cases
, 1, and 5 can be solved analytically
(Chandrasekhar 1967, p. 91); the others must be obtained numerically.
For
(
), the Lane-Emden Differential Equation is
![\begin{displaymath}
{1\over\xi^2}{d\over d\xi}\left({\xi^2{d\theta\over d\xi}}\right)+1=0
\end{displaymath}](l1_552.gif) |
(5) |
(Chandrasekhar 1967, pp. 91-92). Directly solving gives
![\begin{displaymath}
{d\over d\xi}\left({\xi^2{d\theta\over d\xi}}\right)=-\xi^2
\end{displaymath}](l1_553.gif) |
(6) |
![\begin{displaymath}
\int d\left({\xi^2{d\theta\over d\xi^2}}\right)= -\int \xi^2\,d\xi
\end{displaymath}](l1_554.gif) |
(7) |
![\begin{displaymath}
\xi^2{d\theta\over d\xi}=c_1-{\textstyle{1\over 3}}\xi^3
\end{displaymath}](l1_555.gif) |
(8) |
![\begin{displaymath}
{d\theta\over d\xi}={c_1-{\textstyle{1\over 3}}\xi^3\over \xi^2}
\end{displaymath}](l1_556.gif) |
(9) |
![\begin{displaymath}
\theta(\xi)=\int d\theta = \int {c_1-{\textstyle{1\over 3}}\xi^3\over \xi^2} \,d\xi
\end{displaymath}](l1_557.gif) |
(10) |
![\begin{displaymath}
\theta(\xi)=\theta_0-c_1\xi^{-1}-{\textstyle{1\over 6}}\xi^2.
\end{displaymath}](l1_558.gif) |
(11) |
The Boundary Condition
then gives
and
, so
![\begin{displaymath}
\theta_1(\xi)=1-{\textstyle{1\over 6}}\xi^2,
\end{displaymath}](l1_562.gif) |
(12) |
and
is Parabolic.
For
(
), the differential equation becomes
![\begin{displaymath}
{1\over\xi^2}{d\over d\xi}\left({\xi^2{d\theta\over d\xi}}\right)+\theta=0
\end{displaymath}](l1_565.gif) |
(13) |
![\begin{displaymath}
{d\over d\xi}\left({\xi^2{d\theta\over d\xi}}\right)+\theta\xi^2=0,
\end{displaymath}](l1_566.gif) |
(14) |
which is the Spherical Bessel Differential Equation
![\begin{displaymath}
{d\over dr}\left({r^2{dR\over dr}}\right)+[k^2r^2-n(n+1)]R=0
\end{displaymath}](l1_567.gif) |
(15) |
with
and
, so the solution is
![\begin{displaymath}
\theta(\xi) = Aj_0(\xi)+Bn_0(\xi).
\end{displaymath}](l1_568.gif) |
(16) |
Applying the Boundary Condition
gives
![\begin{displaymath}
\theta_2(\xi)=j_0(\xi)={\sin\xi\over\xi},
\end{displaymath}](l1_569.gif) |
(17) |
where
is a Spherical Bessel Function of the First Kind (Chandrasekhar 1967, pp. 92).
For
, make Emden's transformation
which reduces the Lane-Emden equation to
![\begin{displaymath}
{d^2z\over dt^2}+(2\omega-1){dz\over dt}+\omega(\omega-1)z+A^{n-1}z^n=0
\end{displaymath}](l1_575.gif) |
(20) |
(Chandrasekhar 1967, p. 90). After further manipulation (not reproduced here), the equation becomes
![\begin{displaymath}
{d^2z\over dt^2}={\textstyle{1\over 4}}z(1-z^4)
\end{displaymath}](l1_576.gif) |
(21) |
and then, finally,
![\begin{displaymath}
\theta_3(\xi)(1+{\textstyle{1\over 3}}\xi^2)^{-1/2}.
\end{displaymath}](l1_577.gif) |
(22) |
References
Chandrasekhar, S. An Introduction to the Study of Stellar Structure. New York: Dover, pp. 84-182, 1967.
© 1996-9 Eric W. Weisstein
1999-05-26