Dust-grain Eddington limit ($L_{\mathrm{Edd},\mathrm{d}}$)

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.