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Treat each grain as an isolated compact sphere and take its wavelength-independent radiation-pressure efficiency to be $Q_{\mathrm{pr}}=1$. At distance $R$, its mass and radiation-force cross-section areThe inward gravitational force and outward radiation pressure force areThus the opacity per unit dust mass and the dust-grain Eddington limit areTo equal the Eddington luminosity based on electron-scattering opacity, ${\kappa }_d$ must equal ${\kappa }_{\mathrm{es}}$. ThereforeFor fully ionized hydrogen, ${\kappa }_{\mathrm{es}}={\sigma }_T/m_p\simeq 0.40\mathrm{c}{\mathrm{m}}^2{\mathrm{g}}^{-1}$, and the stated ${\rho }_d=1\mathrm{g}\mathrm{c}{\mathrm{m}}^{-3}$ givesUsing $0.34\mathrm{c}{\mathrm{m}}^2{\mathrm{g}}^{-1}$ for a roughly solar-composition gas instead gives $2.2\mathrm{cm}$. This composition-dependent normalization should be stated because the problem does not fix it. For a different constant radiation-pressure efficiency, the critical radius is multiplied by $Q_{\mathrm{pr}}$.

At $L=L_{\mathrm{Edd},\mathrm{es}}$, the grain's radiation-pressure coefficient isConsequently, smaller grains experience a net outward force, larger grains remain gravitationally attracted, and critical-size grains have zero net radial force. Grains released at rest are therefore size-sorted into expelled and infalling populations. With an initial inward velocity, an outward force does not guarantee reversal before the grain reaches the hole; trajectories also depend on initial conditions. Gas drag and dust destruction have been omitted from this isolated-grain calculation. Ordinary interstellar grains are much smaller than a centimetre, so their direct radiative acceleration is very strong in this idealized model.