First-order stellar radiation force on a blackbody grain (${\mathbf{F}}_{\mathrm{rad}}$)

For a blackbody grain with geometric cross-section $\sigma$, mass $m$ and stellar-frame velocity $\mathbf{v}$, define the radiation-pressure coefficient $\beta =L_{\star }\sigma /(4\pi cG{M}_{\star }m)$. To first order in $v/c$, steady absorption and isotropic rest-frame reradiation giveThe relative incoming flux provides $1-\dot{r}/c$; photon reradiation carries stellar-frame momentum proportional to $\mathbf{v}/c$. Subtracting static radiation pressure gives radial drag $-2m\beta G{M}_{\star }\dot{r}/(c{r}^2)$ and transverse drag $-m\beta G{M}_{\star }{v}_{\theta }/(c{r}^2)$. Non-blackbody grains require their actual radiation-pressure efficiency.