# Photon rocket

A photon rocket is a hypothetical rocket that uses thrust from emitted photons (radiation pressure by emission) for its propulsion.[1]

Photons could be generated by onboard generators, as in the nuclear photonic rocket. The standard textbook case of such a rocket is the ideal case where all of the fuel is converted to photons which are radiated in the same direction. In more realistic treatments, one takes into account that the beam of photons is not perfectly collimated, that not all of the fuel is converted to photons, and so on. A large amount of fuel would be required and the rocket would be a huge vessel.[2][3]

In the Beamed Laser Propulsion, the photon generators and the spacecraft are physically separated and the photons are beamed from the photon source to the spacecraft using lasers.

In the Photonic Laser Thruster, collimated photons are reused by mirrors, multiplying the force by the number of bounces.

## Speed

The speed an ideal photon rocket will reach, in the absence of external forces, depends on the ratio of its initial and final mass:

$v = c \frac{\left(\frac{m_{i}}{m_{f}}\right)^{2}-1}{\left(\frac{m_{i}}{m_{f}}\right)^{2}+1}$

where $m_{i}$ is the initial mass and $m_{f}$ is the final mass.

The gamma factor corresponding to this speed has the simple expression:

$\gamma = \frac{1}{2}\left(\frac{m_{i}}{m_{f}} + \frac{m_{f}}{m_{i}}\right)$

## Derivation

We denote the four-momentum of the rocket at rest as $P_{i}$, the rocket after it has burned its fuel as $P_{f}$, and the four-momentum of the emitted photons as $P_{\text{ph}}$. Conservation of four-momentum implies:

$P_{\text{ph}} = P_{i} - P_{f}$

squaring both sides (i.e. taking the Lorentz inner product of both sides with themselves) gives:

$P_{\text{ph}}^{2} = P_{i}^{2} + P_{f}^{2} - 2P_{i}\cdot P_{f}$

According to the energy-momentum relation, the square of the four-momentum equals the square of the mass, and $P_{\text{ph}}^{2}=0$ because all the photons are moving in the same direction. Therefore the above equation can be written as:

$0 = m_{i}^{2} + m_{f}^{2} - 2 m_{i}m_{f}\gamma$

Solving for the gamma factor gives:

$\gamma = \frac{1}{2}\left(\frac{m_{i}}{m_{f}} + \frac{m_{f}}{m_{i}}\right)$