Lane-Emden Equation in Stellar Structure
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The Lane–Emden equation is , where dimensionless variables
and
relate to the density
and radius
(see Details). The equation is used in the study of polytropic gaseous spheres and the modeling of stars. Solutions of this second-order differential equation use polytropes to relate pressure
and density
as a function of
, the radial coordinate measured from the center of the polytropic sphere (the star);
and
have the polytropic relation
, where
is the polytropic constant and
is the polytropic index that you can choose with the control.
Contributed by: Brian Kent (August 2011)
Open content licensed under CC BY-NC-SA
Snapshots
Details
Typically we are interested in finite solutions at . If we take the values
and
to be the densities, then
.
The relationship between the radial coordinate and its dimensionless counterpart
is
,
where is the gravitational constant. The relationship between the density
and its dimensionless counterpart
is
.
Studies and derivations of these relations and the Lane–Emden equation can be found in [1] and [2].
References
[1] S. Chandrasekhar, An Introduction to the Study of Stellar Structure, Chicago: University of Chicago Press, 1939.
[2] R. Kippenhahn and A. Weigert, Stellar Structure and Evolution, Berlin: Springer-Verlag, 1994.
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