A second-order ordinary differential equation arising in the study of stellar interiors, also called
the polytropic differential equations. It is given by

(1)

(2)

(Zwillinger 1997, pp. 124 and 126). 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

(5)

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

(6)

(7)

(8)

(9)

(10)

(11)

The boundary condition then gives and , so

(12)

and
is parabolic .

For
( ),
the differential equation becomes

(13)

(14)

which is the spherical Bessel
differential equation

(15)

with
and ,
so the solution is

(16)

Applying the boundary condition gives

(17)

where
is a spherical Bessel function
of the first kind (Chandrasekhar 1967, p. 92).

For ,
make Emden's transformation

which reduces the Lane-Emden equation to

(20)

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

(21)

and then, finally,

(22)

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References Chandrasekhar, S. An Introduction to the Study of Stellar Structure. New York: Dover, pp. 84-182,
1967. Iyanaga, S. and Kawada, Y. (Eds.). Encyclopedic
Dictionary of Mathematics. Cambridge, MA: MIT Press, p. 908, 1980. Seshadri,
R. and Na, T. Y. Group
Invariance in Engineering Boundary Value Problems. New York: Springer-Verlag,
p. 193, 1985. Zwillinger, D. Handbook
of Differential Equations, 3rd ed. Boston, MA: Academic Press, pp. 124
and 126, 1997. Referenced on Wolfram|Alpha Lane-Emden Differential
Equation
Cite this as:
Weisstein, Eric W. "Lane-Emden Differential Equation." From MathWorld --A Wolfram Web Resource. https://mathworld.wolfram.com/Lane-EmdenDifferentialEquation.html

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