A small contribution to the giant problem

We present a simple analytic model of a composite polytropic star, which exhibits a limiting Schönberg—Chandrasekhar core mass fraction strongly analogous to the classic numerical result for an isothermal core, a radiative envelope and a μ-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 μ-jump by a factor ≥ 3; the core mass fraction cannot exceed 2/π. We use the classic U,V plane to show that composite models will exhibit a Schönberg—Chandrasekhar limit only if the core is ‘soft’, i.e. has n_c ≥ 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 μ-jump is sufficiently large, ≥ 6/(n_e + 1).