How Does a Relativistic Photon Rocket Calculate Acceleration and Speed?

[unscientific]

Homework Statement

(a) Show $a = \frac{a_0}{\gamma^3}$.
(b) Find proper acceleration of rocket
(c) Find speed as a function of time.
(d) Find acceleration of second rocket.

Homework Equations

The Attempt at a Solution

Part(a)
4-vector acceleration is given by $\gamma^2 \left[ \frac{\gamma^2}{c}(\vec u \cdot \vec a), \frac{\gamma^2}{c^2}(\vec u \cdot \vec a)\vec u + \vec a \right]$.
For acceleration parallel to velocity, using invariance we have $a_0^2 = a^2\gamma^6$. Thus we show that
$$a = \frac{a_0}{\gamma^3}$$

Part(b)
We know that $\frac{M(\tau)}{d\tau} = -\alpha M(\tau)$, so solving we have $M(\tau) = M_0 e^{-\alpha \tau}$. Considering the change in energy of the photon in time $d\tau$, we have $\frac{dE}{d\tau} = c \frac{dp}{d\tau} = cm_0 a_0$. Also in time $d\tau$, the change in mass converted to energy is $dE=c^2dM = -\alpha M_0 e^{-\alpha \tau} c^2 d\tau$. Thus we have the acceleration as
$$a_0 (\tau) = \alpha c e^{-\alpha \tau}$$

Part(c)
I'm not sure how to do this part. I have $v = c \tanh \left( \frac{a_0 \tau}{c} \right)$.

Part(d)
Same concept as part(b). In time $dt$, energy of photons produced is $\alpha M_0 c^2 dt = p c$. Thus upon reflection, the change in momentum in time $dt$ is $\Delta p = 2\alpha m_0 c dt$. Acceleration is $a = 2\alpha c$. Finally, we have $a_0 = \gamma^3 a$:
$$a_0 = \gamma^3 (2 \alpha c) = \frac{2\alpha a}{\left( 1 - \frac{v^2}{c^2} \right)^{-\frac{3}{2}}}$$

How do I do part (c)?

 

[TSny]

Part(b)
We know that $\frac{M(\tau)}{d\tau} = -\alpha M(\tau)$, so solving we have $M(\tau) = M_0 e^{-\alpha \tau}$. Considering the change in energy of the photon in time $d\tau$, we have $\frac{dE}{d\tau} = c \frac{dp}{d\tau} = cm_0 a_0$.

In the right hand side of the last equation, shouldn't $m_0$ be replaced by $M(\tau)$?

Part(d)
Same concept as part(b). In time $dt$, energy of photons produced is $\alpha M_0 c^2 dt = p c$. Thus upon reflection, the change in momentum in time $dt$ is $\Delta p = 2\alpha m_0 c dt$.

Due to the motion of the rocket relative to the launch pad, the rate at which photons arrive at the rocket according to launch-pad time is not the same as the rate at which photons are produced according to lauch-pad time. Also in the launch-pad frame, there is a doppler shift in the frequency (or energy) of the photons as they reflect off of the moving rocket.