Emden–Chandrasekhar equation

In astrophysics, the Emden–Chandrasekhar equation is a dimensionless form of the Poisson equation for the density distribution of a spherically symmetric isothermal gas sphere subjected to its own gravitational force, named after Robert Emden and Subrahmanyan Chandrasekhar.

[1][2] The equation was first introduced by Robert Emden in 1907.

is the related to the density of the gas sphere as

The isothermal assumption is usually modeled to describe the core of a star.

The equation is solved with the initial conditions, The equation appears in other branches of physics as well, for example the same equation appears in the Frank-Kamenetskii explosion theory for a spherical vessel.

The relativistic version of this spherically symmetric isothermal model was studied by Subrahmanyan Chandrasekhar in 1972.

[5] For an isothermal gaseous star, the pressure

is due to the kinetic pressure and radiation pressure where The equation for equilibrium of the star requires a balance between the pressure force and gravitational force where

The equation is re-written as Introducing the transformation where

is the central density of the star, leads to The boundary conditions are For

, the solution goes like Assuming isothermal sphere has some disadvantages.

Though the density obtained as solution of this isothermal gas sphere decreases from the centre, it decreases too slowly to give a well-defined surface and finite mass for the sphere.

are constants which will be obtained with numerical solution.

This behavior of density gives rise to increase in mass with increase in radius.

Thus, the model is usually valid to describe the core of the star, where the temperature is approximately constant.

can be derived, This equation can be reduced to first order by introducing then we have There is another reduction due to Edward Arthur Milne.