A small contribution to the giant problem

We present a simple analytic model of a composite polytropic star, which exhibits a limiting Scho¨nberg–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 Scho¨nberg–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).