Polytropic Gaseous Spheres

Abstract

As we have seen in §9.1 the temperature does not appear explicitly in the two mechanical equations (9.1,2). Under certain circumstances this provides the possibility of separating them from the “thermo-energetic part” of the equations. For the following it is convenient to introduce once again the gravitational potential Φ, as it was defined in §1.3. We here treat stars in hydrostatic equilibrium, which requires [see(l.ll),(2.3)]

$$ \frac{{dP}}{{dr}} = - \frac{{d\Phi }}{{dr}}\varrho $$

(19.1)

, together with Poisson's equation (1.10)

$$ \frac{1}{{{r^2}}}\frac{d}{{dr}}\left( {{r^2}\frac{{d\Phi }}{{dr}}} \right) = 4\pi G\varrho $$

(19.2)

. We have replaced the partical derivatives by ordinary ones since only time-indenpendent solutions shall be considered.