Solution
= Solution
A <photon> reaching spacecraft position $x(t)$ at <Earth> time $t$ was emitted at $t_e=t-x(t)/c$. Differentiating gives $dt_e/dt=1-\beta$. Since $R\,dt_e$ <photons> arrive in $dt$,
$$
\boxed{R_{\rm hit}=R(1-\beta).}
$$
The emission rate and impact rate refer to different events, even though both are expressed in <Earth> coordinate time.