= Dust-grain Eddington limit
{title2=$L_{\rm Edd,d}$}
An isolated compact dust grain of radius $r_d$ and material density $\rho_d$ has opacity per unit dust mass $\kappa_d=3Q_{\rm pr}/(4\rho_dr_d)$, where $Q_{\rm pr}$ is its <radiation-pressure efficiency>. Its <Eddington luminosity> is $4\pi GMc/\kappa_d$. Matching the <electron-scattering opacity> gives $r_{d,\rm crit}=3Q_{\rm pr}/(4\rho_d\kappa_{\rm es})$, approximately $1.9\,\mathrm{cm}$ for $Q_{\rm pr}=1$, $\rho_d=1\,\mathrm{g\,cm^{-3}}$, and $\kappa_{\rm es}=0.4\,\mathrm{cm^2\,g^{-1}}$. 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.
Back to article page
