  The Sch\"onberg-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.

Solar and Stellar Astrophysics