Photon rocket

Speed

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:

: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 m_\text{i} is the initial mass and m_\text{f} is the final mass.

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{\text{Mass of }\ce{2^3He}}\approx\frac{(2\cdot2809.41-3728.40-2\cdot938.27-2\cdot0.51)\text{ MeV}/c^2}{2\cdot2809.41\text{ MeV}/c^2}\approx0.002289 .}} – 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 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 cannot 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:

:\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 \frac{m_\text{f}}{m_\text{i}} is very nearly 0.9.

Derivation

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

: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:

: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 E^2-(pc)^{2}=(mc^{2})^{2}, the square of the four-momentum equals the square of the mass, and 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:

:{P}_\text{i} = \begin{pmatrix} \frac{c} \ 0 \ 0 \ 0 \end{pmatrix},

while the final four-momentum is:

:{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:

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

We can now solve for the gamma factor, obtaining:

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

Maximum speed limit

Standard theory says that the theoretical speed limit of a photon rocket is below the speed of light. Haug has recently suggested 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., 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 p, is given by

:E_\text{p} = \alpha\gamma\delta M c^2

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

:T_\text{p} = \frac{E_\text{p}}{c} = \alpha\gamma\delta M c

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

: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.

The beamed laser propulsion, such as photonic laser thruster, however, in principle can provide the maximum spacecraft velocity approaching the speed of light, c.

Notes

References

References

1. McCormack, John W.. "5. PROPULSION SYSTEMS". Select Committee on Astronautics and Space Exploration.
2. Tsander, F.A / K. (1967). "Tsander, K. (1967) From a Scientific Heritage, NASA Technical Translation TTF-541. - References - Scientific Research Publishing".
3. (March 1984). "Roundtrip interstellar travel using laser-pushed lightsails". Journal of Spacecraft and Rockets.
4. (February 1961). "A Photon Rocket".
5. "There will be no photon rocket".
6. (August 2019). "Comment on 'the ultimate limits of the relativistic rocket equation. The Planck photon rocket'". Acta Astronautica.
7. (1964). "On the mechanics of photon rockets". Bulletin of the Astronomical Institutes of Czechoslovakia.
8. (2012). "Prospective of Photon Propulsion for Interstellar Flight". Physics Procedia.
9. (2017). "The ultimate limits of the relativistic rocket equation. The Planck photon rocket". Acta Astronautica.