Proper-time acceleration by reflected laser light ($\beta (\tau )=\tau /(\tau +{\tau }_0)$)

An ideal receding mirror reflecting a source beam of constant power $P$ receives force $(2P/c)(1-\beta )/(1+\beta )$ after including both Doppler-shifted reflected momentum and reduced impact rate. For fixed rest mass $M$, use $dt/d\tau =\gamma$ in the momentum equation. It reduces to $d\beta /d\tau =(1-\beta {)}^2/{\tau }_0$, where ${\tau }_0=M{c}^2/(2P)$. Starting from rest gives $\beta =\tau /(\tau +{\tau }_0)$. This continuum mirror model neglects individual-photon recoil and finite beam interception.