Main-sequence mass–luminosity scaling

Mass continuity gives a characteristic density proportional to mass divided by radius cubed. Hydrostatic balance gives pressure proportional to gravitational constant times mass squared divided by radius to the fourth power. Combining these with ideal-gas pressure gives a characteristic temperature proportional to mean molecular weight times mass divided by radius. Dimensionless structural coefficients remain fixed only within the assumed homologous family. They cannot be presumed identical for a star that has developed a composition gradient or a qualitatively different convective core.

Radiative diffusion estimates the luminosity as radiation constant times light speed times temperature to the fourth power times radius, divided by opacity and density. Substituting the mechanical scales cancels the radius and leaves a cubic mass dependence. This is a conditional theoretical scaling, not a fit to a modern observational sample. Opacity varying with temperature and density, radiation-pressure support, convection, partial ionization, or changed dimensionless profiles alter the result. The argument is useful because it reveals precisely which assumptions produce the exponent.

Luminosity transport alone does not specify the rate at which nuclei react. For a thermal-equilibrium sequence, the integrated nuclear heating must supply the luminosity after neutrino losses and any changing thermal reservoir are treated consistently. Adopt a local toy rate per unit mass proportional to density times temperature to a fixed exponent nu. This approximates a restricted reaction regime over a limited range; it is not an exact reaction-network law across all temperatures or compositions.

The integrated heating scale is then mass times that local rate. Equating it to the radiative homology luminosity produces a radius dependence involving nu, opacity, mean molecular weight, and the rate coefficient. Reactions thus help select the structural temperature and radius needed to deliver the transport-constrained luminosity. A higher reaction coefficient need not simply multiply the luminosity of an unchanged star: the star can readjust. This thermal regulation relies on the appropriate equation of state and fails in important degenerate-burning instabilities.

An approximate fuel lifetime divides the accessible nuclear reservoir by a representative luminosity. Let f be the fraction of stellar mass whose hydrogen can be consumed in the phase, X its initial hydrogen fraction, and q the effective rest-mass fraction released as usable heat. This counts accessible hydrogen once. Mixing, changing luminosity, and incomplete burning make the estimate different from a detailed age. A local nuclear reaction time instead describes consumption at specified local density, temperature, and abundance; it is not automatically the lifetime of the whole star.

The thermal time estimates how long gravitational and internal-energy reservoirs can sustain luminosity, while the dynamical time measures mechanical readjustment. Long-lived central burning generally separates these scales, but rapid thermal phases and collapse do not share the same hierarchy. Comparing two idealized models at fixed accessible fuel fraction and composition makes lifetime proportional to mass divided by luminosity. Rotation, overshooting, and changing convective-core size can change that fraction, so a cubic luminosity law cannot yield a universal lifetime without additional assumptions.