A second-order ordinary differential equation arising in the study of stellar interiors, also called the polytropic differential equations. It is given by
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(1)
|
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(2)
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(Zwillinger 1997, pp. 124 and 126). It has the boundary conditions
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(3)
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![[(dtheta)/(dxi)]_(xi=0)](/images/equations/Lane-EmdenDifferentialEquation/Inline4.svg) |  |  |
(4)
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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
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(5)
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(Chandrasekhar 1967, pp. 91-92). Directly solving gives
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(6)
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(7)
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(8)
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(9)
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(10)
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(11)
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The boundary condition 
then gives 
and 
, so
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(12)
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and 
is parabolic.
For 
(
),
the differential equation becomes
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(13)
|
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(14)
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which is the spherical Bessel differential equation
![d/(dr)(r^2(dR)/(dr))+[k^2r^2-n(n+1)]R=0](/images/equations/Lane-EmdenDifferentialEquation/NumberedEquation13.svg) |
(15)
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with 
and 
,
so the solution is
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(16)
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Applying the boundary condition 
gives
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(17)
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where 
is a spherical Bessel function
of the first kind (Chandrasekhar 1967, p. 92).
For 
,
make Emden's transformation
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(18)
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(19)
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which reduces the Lane-Emden equation to
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(20)
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(Chandrasekhar 1967, p. 90). After further manipulation (not reproduced here), the equation becomes
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(21)
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and then, finally,
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(22)
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