Red-giant cores and shell burning

A hydrogen-burning shell adds helium ash to the hydrogen-exhausted core. Outside it, the envelope can be cool, extended, and largely convective. A high density contrast makes the compact core important for the shell’s gravitational environment, even though the photosphere lies far away. The surface luminosity is not always identical to instantaneous shell power: an expanding envelope can absorb energy, contraction can release energy, and neutrinos can remove it. The local energy equation must decide how these contributions combine.

A cold nonrelativistic degenerate-core approximation gives an inverse cube-root mass–radius relation at fixed electron abundance. A larger core mass then implies a smaller core radius. An ideal-gas shell just outside that core has a characteristic gravitational temperature proportional to core mass divided by core radius. Combining those approximations gives a rapidly rising shell temperature scale. This is a structural estimate with fixed profile factors, not an exact law for the core’s central temperature or a complete red-giant luminosity relation.

Let Q effective be the net heat supplied per kilogram of newly deposited helium-rich ash, already including the hydrogen fraction of the processed material and any stipulated reaction-neutrino loss. If shell burning alone supplies its own luminosity and its ashes all join the core, shell power equals Q effective times the core-growth rate. Counting a separate hydrogen fraction again would double count composition. The relation neglects changes in shell inventory and any ash removed or remixed rather than retained in the core.

Increasing core mass often drives rising shell luminosity in red-giant models, which in turn accelerates ash deposition. The exact core-mass–luminosity relation depends on reaction sensitivities, opacity, shell geometry, and surrounding structure. It is not obtained simply by raising the shell temperature estimate to one arbitrary nuclear exponent while holding density and burning mass fixed. Composition discontinuities left by envelope mixing can modify the shell’s burning conditions and produce recognizable features in an evolutionary track. Such details require an evolving structure and abundance model.

In a nondegenerate gas, heating tends to increase pressure and drive expansion, which can reduce density and temperature and regulate burning. Strongly degenerate electron pressure responds much less directly to temperature. When temperature-sensitive helium burning begins in a degenerate core, that immediate expansion response is weakened, enabling a thermal runaway called the helium flash. Thermal conduction and neutrino cooling influence the temperature profile and can lead to off-center ignition. The runaway eventually changes the thermodynamic conditions and lifts the degeneracy.

The helium flash is not automatically an explosion of the entire stellar envelope; much of its energy changes the core’s internal state and structure. A nondegenerate helium core can instead ignite more quietly. No universal initial-mass boundary between these behaviors is asserted here: composition, mixing, rotation and mass loss affect the transition. Likewise, a local reaction temperature exponent describes sensitivity in a stated regime, not the global luminosity response of an unconstrained evolving star. Pressure feedback, fuel supply, geometry, and transport determine whether that local sensitivity produces an instability.