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Concept
Lane–Emden equation
Summary
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. The equation readswhere 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.
Physically, hydrostatic equilibrium connects the gradient of the potential, the density, and the gradient of the pressure, whereas Poisson's equation connects the potential with the density. Thus, if we have a further equation that dictates how the pressure and density vary with respect to one another, we can reach a solution. The particular choice of a polytropic gas as given above makes the mathematical statement of the problem particularly succinct and leads to the Lane–Emden equation. The equation is a useful approximation for self-gravitating spheres of plasma such as stars, but typically it is a rather limiting assumption.
Consider a self-gravitating, spherically symmetric fluid in hydrostatic equilibrium. Mass is conserved and thus described by the continuity equation
where is a function of . The equation of hydrostatic equilibrium is
where is also a function of . Differentiating again gives
where the continuity equation has been used to replace the mass gradient. Multiplying both sides by and collecting the derivatives of on the left, one can write
Dividing both sides by yields, in some sense, a dimensional form of the desired equation.
Official source
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