Schönberg–Chandrasekhar limit

Remarks

Let $T_c$ be the core temperature and $G$ the gravitational constant. The virial theorem applied to the core gives the pressure at its surface as a function of its mass and radius. Over all radii this pressure has a maximum proportional to $T_c^4 / (\mu_c^4 G^3 M_c^2)$. The envelope needs, at the edge of the core, a pressure proportional to $T_c^4 / (\mu_e^4 G^3 M^2)$. Comparing the two gives a bound on $M_c / M$ proportional to $(\mu_e / \mu_c)^2$. The factor $0.37$ comes from numerical models and is approximate. Example: for a helium core, $\mu_c \approx 1.3$, under an envelope of hydrogen-rich gas with $\mu_e \approx 0.6$, the limit is about $0.37 \cdot (0.6 / 1.3)^2 \approx 0.08$, so about a tenth of the mass of the star. The hypothesis of a non-degenerate core is necessary. In stars of low mass the helium core becomes degenerate, its pressure no longer obeys the ideal gas law, and it can exceed this fraction. The result was published by M. Schönberg and S. Chandrasekhar in 1942.

Going deeper

For a student

The limit explains the gap, in the Hertzsprung–Russell diagram, between the main sequence and the red giant branch for stars of a few solar masses. When the helium core passes the limit, it contracts on the Kelvin–Helmholtz timescale, heats up and drives the envelope to expand. The contraction ends either when helium ignites or when the core becomes degenerate.