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Lane–Emden equation
Dimensionless astrophysics equation / From Wikipedia, the free encyclopedia
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In astrophysics, the Lane–Emden equation is a dimensionless form of Poisson's equation for the gravitational potential of a Newtonian self-gravitating, spherically symmetric, polytropic fluid. It is named after astrophysicists Jonathan Homer Lane and Robert Emden.[1] The equation reads
![Thumb image](http://upload.wikimedia.org/wikipedia/commons/thumb/7/77/FinitePolytropes.svg/320px-FinitePolytropes.svg.png)
where is a dimensionless radius and
is related to the density, and thus the pressure, by
for central density
. The index
is the polytropic index that appears in the polytropic equation of state,
where and
are the pressure and density, respectively, and
is a constant of proportionality. The standard boundary conditions are
and
. Solutions thus describe the run of pressure and density with radius and are known as polytropes of index
. If an isothermal fluid (polytropic index tends to infinity) is used instead of a polytropic fluid, one obtains the Emden–Chandrasekhar equation.