Instability of Embedded Polytropes

Abstract

The equivalende of two stability criteria (decreasing pressure with increasing radius versus increasing central density with increasing mass) is shown analytically for spherical polytropes without background medium, and numerically for embedded ones. Previous results concerning the instability under external pressure of incomplete (truncated) polytropes within a uniform background medium are shown to be not true. The spherical incomplete embedded polytrope is stable/unstable for the same polytropic indices and for the same background densities as the corresponding compete embedded polytrope.