= Solution
A <blackbody> grain absorbs starlight of flux $L_\star/(4\pi r^2)$ and reradiates isotropically in its own rest frame. <Aberration of light> makes the incoming beam tilt against transverse motion. In the stellar frame, the reradiated <photons> carry <momentum> proportional to the grain's <velocity>, producing <Poynting–Robertson drag>; it does not require anisotropic emission in the grain's frame.
Define the <radiation-pressure coefficient>
$$
\boxed{\beta=\frac{L_\star\sigma}{4\pi c\,GM_\star m}}.
$$
For a spherical grain of radius $s$ and <mass density> $\rho$, $\beta=3L_\star/(16\pi c\,GM_\star\rho s)$, assuming unit <radiation-pressure efficiency>. To first order in $v/c$, the <first-order stellar radiation force on a blackbody grain> is
$$
\boldsymbol F_{\rm rad}=\frac{m\beta\mu}{r^2}
\left[\left(1-\frac{\dot r}{c}\right)\widehat{\boldsymbol r}-\frac{\boldsymbol v}{c}\right].
$$
The incoming absorbed <photon> <momentum> contains the relative-flux factor $1-\dot r/c$; reradiation supplies the additional $-\boldsymbol v/c$. Separating the velocity-independent <radiation pressure> leaves
$$
\boxed{\boldsymbol F_{\rm PR}=\frac{m\beta\mu}{r^2}
\left[-\frac{2\dot r}{c}\widehat{\boldsymbol r}
-\frac{r\dot\theta}{c}\widehat{\boldsymbol\theta}\right]}.
$$
Thus there are two radial first-order contributions and one tangential contribution. The approximation assumes $v\ll c$, steady absorption/reradiation and constant grain properties; a non-blackbody requires its actual optical efficiencies.
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