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Date: 21-5-2018
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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
(3) |
|||
(4) |
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
(18) |
|||
(19) |
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) |
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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