A small contribution to the giant problem

We present a simple analytic model of a composite polytropic star, which exhibits a limiting Schonberg-Chandrasekhar core mass fraction strongly analogous to the classic numerical result for an isothermal core, a radiative envelope and a mu-jump (i.e. a molecular weight jump) at the interface. Our model consists of an n(c) = 5 core, an n(e) = 1 envelope and a mu-jump by a factor greater than or equal to 3; the core mass fraction cannot exceed 2/pi. We use the classic U, V plane to show that composite models will exhibit a Schonberg-Chandrasekhar limit only if the core is 'soft', i.e, has n(c) greater than or equal to 5, and the envelope is 'hard', i.e, has n(e) < 5; in the critical case (n(c) = 5), the limit only exists if the CL-jump is sufficiently large, greater than or equal to 6/(n(e) + 1).