For an isothermal gaseous star, the pressure  is due to the kinetic pressure and radiation pressure
whe
* is the density
* is the Boltzmann constant
* is the mean molecular weight
* is the mass of the proton
* is the temperature of the star
* is the Stefan–Boltzmann constant
* is the speed of light
The equation for equilibrium of the star requires a balance between the pressure force and gravitational force
where  is the radius measured from the center and  is the gravitational constant. 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