= Solution
The <Schönberg-Chandrasekhar limit> is the largest mass fraction that an approximately isothermal inert core can have while remaining matched in hydrostatic and thermal equilibrium to a hydrogen-rich envelope. For a simple isothermal core and polytropic envelope,
$$
\boxed{\frac{M_c}{M}\lesssim0.37
\left(\frac{\mu_{\rm env}}{\mu_c}\right)^2}.
$$
Because a helium core has larger <mean molecular weight> than its hydrogen-rich envelope, the limiting fraction is typically near ten per cent. Once shell burning grows the core beyond this limit, no neighboring equilibrium with an isothermal core exists: the core contracts and heats by <Kelvin-Helmholtz contraction>, the hydrogen-burning shell brightens, and the envelope expands rapidly toward a red giant.
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