# Laser-Accelerated Plasma-Propulsion System
## 5 LAPPS Performance Limits  
We are interested in a rocket in which the engine is based on a LAPPS. As presented in Figure 3, this implies that a laser system must be installed and powered directly on the rocket. From Equation (21), we can now define the specific impulse \(I_{sp}\equiv v_{r}^{*}/g_{0}\), which is a generalized form of the impulse already defined in Equation (13), accounting for all possible channels of energy dissipation; as already said, the effective velocity \(v_{r}^{*}\) is here lower than the propellant speed \(v_{r}\). In case of a photon rocket (\(\delta=0\) and \(\epsilon=1\)), in which the laser pulse is directly used as propellant, the specific impulse assumes the form  
\[I_{sp}^{L}=\frac{c\eta_{L}}{g_{0}}, \tag{22}\]  
where \(\eta_{L}\) is the energy conversion efficiency of the laser system calculated considering all the energy released by the nuclear reactor. The parameter \(0\leq(1-\eta)\leq 1\) indicates the fraction of the available energy wasted by the system. The wasted energy takes into account, among other factors, of heating dissipation and of the orthogonal component of particle velocity, which does not contribute to rocket acceleration. For this reason, in the case of a well collimated laser beam, we obtain \(\eta\simeq\eta_{L}\). On the other side, if the laser pulse is focalized (and almost stopped) on a target, as it occurs in a LAPPS, we expect that the photons contribution to the rocket motion can be considered negligible at first order (the EM radiation will be considered later in detail). In this case \(\delta\simeq 1\), and differentiating Equation (21), we find that the maximum specific impulse is reached for  
\[\epsilon_{max}\simeq\frac{1}{2-\eta_{LP}}\simeq\frac{1}{2}\Big{[}1+\frac{\eta _{LP}}{2}\Big{]}, \tag{23}\]  
where usually \(\eta_{LP}<\eta_{L}\ll 1\). Equation (23) indicates that the best performance is obtained when the infinitesimal mass \(dM\) is almost equally distributed between the rest mass of the accelerated particles \((1-\epsilon_{max})dM\) and the available energy \(\epsilon_{max}dM\). Consequently, the kinetic energy acquired by the massive propellant becomes \(\epsilon_{k}\simeq(\eta_{LP}/2)c^{2}dM\). This quantity is related to the averaged power generated by the nuclear reactor \(P_{R}\) calculated in the rest frame through  
\[\langle\dot{\epsilon}_{kin}\rangle\simeq\frac{c^{2}\eta_{LP}}{2g_{0}}\left( \frac{F^{LP}}{I_{sp}^{LP}}\right)=\eta_{1}\eta_{L}\langle P_{R}\rangle, \tag{24}\]  
Figure 2: Energy scheme calculated in the rocket space frame. The total infinitesimal mass \(dM\) lost by the rocket is related to the rest mass of the accelerated particles, their energy and the energy of the massless expelled particles. Energy that is not used to accelerate the rocket is considered wasted energy. where Equation (14) has been used, while \(0\leq\eta_{I}\leq 1\) characterizes the laser-plasma interaction efficiency. The parameter \(\eta_{I}\) indicates, for \(\delta=1\), the fraction of the laser pulse energy converted into \"parallel\" kinetic energy. Considering Equation (23), the maximum specific impulse which can be obtained in a LAPPS is given by  
\[I_{sp}^{max}\simeq\frac{c}{g_{0}}\sqrt{\frac{\eta_{LP}}{2-\eta_{LP}}}\simeq \frac{c}{g_{0}}\sqrt{\frac{\eta_{LP}}{2}}\Big{[}1+\frac{\eta_{LP}}{4}\Big{]}, \tag{25}\]  
and gives, once combined with Equation (24), the expression for the engine thrust  
\[F^{LP}\left(I_{sp}^{max}\right)\simeq\frac{\eta_{I}\eta_{L}}{c}\sqrt{\frac{2} {\eta_{LP}}}\langle P_{R}\rangle. \tag{26}\]  
For an ideal system (maximum specific impulse), the relative speed of the massive propellant becomes  
\[v_{r}^{m}\left(I_{sp}^{max}\right)\simeq c\sqrt{2\eta_{LP}}\Big{[}1+\frac{ \eta_{LP}}{2}\Big{]}. \tag{27}\]  
Equation (27) is intrinsically limited by \(\eta_{LP}\), which is the fractional amount of available energy that is actually utilized for propulsion.