# Laser-Accelerated Plasma-Propulsion System
## 2 Relativistic Propellant
However, according to the covariant derivation of Equation (7), an engine based on photons emission can be simply obtained in the limit of \(v_{r}\to c\). As expected, these results are in accordance with a non-covariant derivation [29]. Figure 1: A schematic image of a relativistic rocket system. The relativistic propellant, of infinitesimal mass \(dm\), is expelled to produce the rocket motion. In case of massless propellant, \(dmv^{\mu}\) is the photons four–momentum. We are interested in the case of non-relativistic rocket motion (i.e., \(\theta\simeq u/c\)) with a thruster based on a lapps. If we define \(\Delta M=M_{i}-M\) as the amount of the consumed fuel during the flight, Equation (7) can be expressed as  
\[u\simeq v_{r}\sum_{n=1}^{\infty}{(-1)^{n+1}\bigg{(}\frac{\Delta M}{M_{i}} \bigg{)}^{n}}, \tag{8}\]  
where the [PERSON]'s expansion of the logarithm has been used. Equation (8), in contrast to Equation (7), is valid only in the case of non-relativistic rocket motion, for which \(\sigma\simeq u\). For particle accelerators, differently from chemical fuel engines, the consumed propellant is expected to consist of a small fraction of the entire system mass, so that we may assume \(\Delta M/M_{i}\ll 1\). In this case, considering only the first term in Equation (8), we obtain  
\[u\simeq v_{r}\frac{\Delta M}{M_{i}}. \tag{9}\]  
Equation (9) admits a simple physical interpretation as the classical total momentum of both rocket and propellant are conserved. Here, the propellant mass is defined as the rocket \"lost\" mass \(\Delta M\) measured in the rest frame. Again, despite the classical aspect, Equation (9) remains valid in the case of relativistic (and eventually massless) propellant. Therefore, the laser pulse or the emitted radiation can be considered as a part of propellant. Realistically, in order to produce propulsive capabilities adequate for orbital maneuvering or interplanetary space missions, we need spacecraft velocities \(u\sim 10^{3}\)-\(10^{4}\) m/s. Considering a conservative fuel-mass ratio of \(\Delta M/M_{i}=0.01\), the above velocities can be obtained by expelling the propellant at relative velocities \(v_{r}\sim 10^{5}\)-\(10^{6}\) m/s. This suggests that a relativistic speed of the propellant is not mandatory as the condition \(v_{r}/c\sim 10^{-2}\) seems to fulfill the requirements. This example also fixes the expected order of magnitude for one of the most relevant engine parameters, the specific impulse (see formal definition in Equation (13)) \(I_{sp}=v_{r}/g_{0}\simeq 10^{4}\)-\(10^{5}\) s, where \(g_{0}=9.81\) m/s\({}^{2}\) is the gravitational acceleration measured on Earth at sea level. It is worth mentioning that Equations (3) and (4) do not include explicitly the mass wasted for the acceleration of the propellant; a more complete treatment, accounting for the dissipation channels and therefore including the efficiency of the process, will be tackled in a following section. It is however important to anticipate that, in a comprehensive approach, the specific impulse is obtained by substituting the propellant velocity \(v_{r}\) with an \"effective\" \(v_{r}^{*}<v_{r}\), accounting for the efficiency of the system (see Equation (19)).