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Lane-Emden Differential Equation

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LaneEmden

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

1/(xi^2)d/(dxi)(xi^2(dtheta)/(dxi))+theta^n=0
(1)
1/(xi^2)(2xi(dtheta)/(dxi)+xi^2(d^2theta)/(dxi^2))+theta^n=(d^2theta)/(dxi^2)+2/xi(dtheta)/(dxi)+theta^n=0
(2)

(Zwillinger 1997, pp. 124 and 126). It has the boundary conditions

theta(0)=1
(3)
[(dtheta)/(dxi)]_(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/(xi^2)d/(dxi)(xi^2(dtheta)/(dxi))+1=0
(5)

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

d/(dxi)(xi^2(dtheta)/(dxi))=-xi^2
(6)
intd(xi^2(dtheta)/(dxi))=-intxi^2dxi
(7)
xi^2(dtheta)/(dxi)=c_1-1/3xi^3
(8)
(dtheta)/(dxi)=(c_1-1/3xi^3)/(xi^2)
(9)
theta(xi)=intdtheta=int(c_1-1/3xi^3)/(xi^2)dxi
(10)
theta(xi)=theta_0-c_1xi^(-1)-1/6xi^2.
(11)

The boundary condition theta(0)=1 then gives theta_0=1 and c_1=0, so

theta_1(xi)=1-1/6xi^2,
(12)

and theta_1(xi) is parabolic.

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

1/(xi^2)d/(dxi)(xi^2(dtheta)/(dxi))+theta=0
(13)
d/(dxi)(xi^2(dtheta)/(dxi))+thetaxi^2=0,
(14)

which is the spherical Bessel differential equation

d/(dr)(r^2(dR)/(dr))+[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)=(sinxi)/xi,
(17)

where j_0(x) is a spherical Bessel function of the first kind (Chandrasekhar 1967, p. 92).

For n=5, make Emden's transformation

theta=Ax^omegaz
(18)
omega=2/(n-1),
(19)

which reduces the Lane-Emden equation to

(d^2z)/(dt^2)+(2omega-1)(dz)/(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)/(dt^2)=1/4z(1-z^4)
(21)

and then, finally,

theta_5(xi)=(1+1/3xi^2)^(-1/2).
(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.

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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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