where is the dimensionless radius and is the related to the density of the gas sphere as , where is the density of the gas at the centre. The equation has no known explicit solution. If a
polytropic fluid is used instead of an isothermal fluid, one obtains the
Lane–Emden equation. 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]
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
Limitations of the model
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. It can be shown that, as ,
where and 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.[6]
Singular solution
Introducing the transformation transforms the equation to
^Emden, R. (1907). Gaskugeln: Anwendungen der mechanischen Wärmetheorie auf kosmologische und meteorologische Probleme. B. Teubner.
^Kippenhahn, Rudolf, Alfred Weigert, and Achim Weiss. Stellar structure and evolution. Vol. 282. Berlin: Springer-Verlag, 1990.
^Chandrasekhar, S. (1972). A limiting case of relativistic equilibrium. In General Relativity (in honor of J. L. Synge), ed. L. O'Raifeartaigh. Oxford. Clarendon Press (pp. 185-199).
^Henrich, L. R., & Chandrasekhar, S. (1941). Stellar Models with Isothermal Cores. The Astrophysical Journal, 94, 525.