Photon rocket

The speed an ideal photon rocket will reach (in the reference frame in which the rocket was at rest initially), in the absence of external forces, depends on the ratio of its initial and final mass:

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

where ${\displaystyle m_{\text{i}}}$ is the initial mass and ${\displaystyle m_{\text{f}}}$ is the final mass.[6]

For example, assuming a spaceship is equipped with a pure helium-3 fusion reactor and has an initial mass of 2300 kg, including 1000 kg of helium-3 – meaning, 2.3 kg will be converted to energy[lower-alpha 1] – and assuming all this energy is emitted as photons in the direction opposing the direction of travel, and assuming the fusion products (helium-4 and hydrogen) are kept on board, the final mass will be (2300 − 2.3) kg = 2297.7 kg and the spaceship will reach a speed of 1/1000 of the speed of light. If the fusion products are released into space, the speed will be higher, but the above equation can't be used to compute it, because it assumes that all decrease in mass is converted into energy.

The gamma factor corresponding to a photon rocket speed has the simple expression:

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

At 10% the speed of light, the gamma factor is about 1.005, implying ${\displaystyle {\frac {m_{\text{f}}}{m_{\text{i}}}}}$is very nearly 0.9.

We denote the four-momentum of the rocket at rest as ${\displaystyle P_{\text{i}}}$, the rocket after it has burned its fuel as ${\displaystyle P_{\text{f}}}$, and the four-momentum of the emitted photons as ${\displaystyle P_{\text{ph}}}$. Conservation of four-momentum implies:[7][8]

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

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

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

According to the energy-momentum relation ${\displaystyle E^{2}-(pc)^{2}=(mc^{2})^{2}}$, the square of the four-momentum equals the square of the mass, and ${\displaystyle P_{\text{ph}}^{2}=0}$ because photons have zero mass.

As we start in the rest frame (i.e. the zero-momentum frame) of the rocket, the initial four-momentum of the rocket is:

${\displaystyle {P}_{\text{i}}={\begin{pmatrix}{\frac {{m}_{\text{i}}c^{2}}{c}}\\0\\0\\0\end{pmatrix}},}$

while the final four-momentum is:

${\displaystyle {P}_{\text{f}}={\begin{pmatrix}\ {\gamma }{m}_{\text{f}}c\\{\gamma }{m}_{\text{f}}{v}_{\text{f}}\\0\\0\end{pmatrix}}.}$

Therefore, taking the Minkowski inner product (see four-vector), we get:

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

We can now solve for the gamma factor, obtaining:

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

Standard theory says that the theoretical speed limit of a photon rocket is below the speed of light. Haug has recently suggested[9] a maximum speed limit for an ideal photon rocket that is just below the speed of light. However, his claims have been contested by Tommasini et al.,[6] because such velocity is formulated for the relativistic mass and is therefore frame-dependent.

Regardless of the photon generator characteristics, onboard photon rockets powered with nuclear fission and fusion have speed limits from the efficiency of these processes. Here it is assumed that the propulsion system has a single stage. Suppose the total mass of the photon rocket/spacecraft is M that includes fuels with a mass of αM with α < 1.  Assuming the fuel mass to propulsion-system energy conversion efficiency γ and the propulsion-system energy to photon energy conversion efficiency δ ≪ 1, the maximum total photon energy generated for propulsion, E_p, is given by

${\displaystyle E_{\text{p}}=\alpha \gamma \delta Mc^{2}}$

If the total photon flux can be directed at 100% efficiency to generate thrust, the total photon thrust, T_p, is given by

${\displaystyle T_{\text{p}}={\frac {E_{\text{p}}}{c}}=\alpha \gamma \delta Mc}$

The maximum attainable spacecraft velocity, V_max, of the photon propulsion system for V_max ≪ c, is given by

${\displaystyle V_{\text{max}}={\frac {T_{\text{p}}}{M}}=\alpha \gamma \delta c}$

For example, the approximate maximum velocities achievable by onboard nuclear powered photon rockets with assumed parameters are given in Table 1. The maximum velocity limits by such nuclear powered rockets are less than 0.02% of the light velocity (60 km/s). Therefore, onboard nuclear photon rockets are unsuitable for interstellar missions.

Table 1  The maximum velocity obtainable by photon rockets with onboard nuclear photon generators with exemplary parameters.

The Beamed Laser Propulsion, such as Photonic Laser Thruster, however, can provide the maximum spacecraft velocity approaching the speed of light, c, in principle.

• Beam-powered propulsion
• Laser propulsion
• Nuclear photonic rocket

• Whatever happened to Photon Rockets?