# 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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