Quasi-stars, giants and the Schönberg-Chandrasekhar limit

The Schönberg-Chandrasekhar (SC) limit is a well-established result in the understanding of stellar evolution. It provides an estimate of the point at which an evolved isothermal core embedded in an extended envelope begins to contract. We investigate contours of constant fractional mass in terms of homology invariant variables U and V and find that the SC limit exists because the isothermal core solution does not intersect all the contours for an envelope with polytropic index 3. We find that this analysis also applies to similar limits in the literature including the inner mass limit for polytropic models of quasi-stars. Consequently, any core solution that does not intersect all the fractional mass contours exhibits an associated limit and we identify several relevant cases where this is so. We show that a composite polytrope is at a fractional core mass limit when its core solution touches but does not cross the contour of the corresponding fractional core mass. We apply this test to realistic models of helium stars and find that stars typically expand when their cores are near a mass limit. Furthermore, it appears that stars that evolve into giants have always first exceeded an SC-like limit.