Mass-luminosity relation

The mass-luminosity relation connects a star's mass to its luminosity. Its slope depends on opacity, nuclear burning, pressure support, composition, and evolutionary state.

Balancing outward radiative acceleration against gravity givesIt is an approximate upper luminosity for a hydrostatic star of opacity $\kappa$.

This factor is the ratio of acceleration due to electron-scattering opacity to gravitational acceleration in a spherical luminous star. The remaining effective gravity is approximately $GM(1-{\Gamma }_e)/R^2$ when other radiative forces are neglected. Being below the electron-scattering limit does not preclude a line-driven stellar wind, because additional opacity can provide more acceleration.

An isolated compact dust grain of radius $r_d$ and material density ${\rho }_d$ has opacity per unit dust mass ${\kappa }_d=3Q_{\mathrm{pr}}/(4{\rho }_d{r}_d)$, where $Q_{\mathrm{pr}}$ is its radiation-pressure efficiency. Its Eddington luminosity is $4\pi GMc/{\kappa }_d$. Matching the electron-scattering opacity gives $r_{d,\mathrm{crit}}=3Q_{\mathrm{pr}}/(4{\rho }_d{\kappa }_{\mathrm{es}})$, approximately $1.9\mathrm{cm}$ for $Q_{\mathrm{pr}}=1$, ${\rho }_d=1\mathrm{g}\mathrm{c}{\mathrm{m}}^{-3}$, and ${\kappa }_{\mathrm{es}}=0.4\mathrm{c}{\mathrm{m}}^2{\mathrm{g}}^{-1}$. At that electron-scattering luminosity, smaller grains feel a net outward force and larger grains remain attracted. This is an isolated-grain force balance, not the opacity of an arbitrary gas-dust mixture.

A radiative force multiplier is the ratio of total radiative acceleration to a reference electron-scattering acceleration. Spectral lines, bound-free absorption, and dust can make $\mathcal{M}\gg 1$, whereas highly ionized gas approaches $\mathcal{M}=1$.