# Laser-Accelerated Plasma-Propulsion System
## 6 LAPPS Analytical Model
### LAPPS Specific Impulse  
The specific impulse for a LAPPS engine can be calculated starting from the general Equations (13) and (20). Neglecting the photon contribution, we obtain:  
\[I_{sp}^{LP}\simeq\frac{(1-\epsilon(1-\delta\eta_{LP}))}{g_{0}}v_{r}^{m}. \tag{46}\]  
The term (\(\delta\eta_{LP}\ll 1\)) can be omitted at first order while \(v_{r}^{m}\) is given by Equation (38). In the previous section, we assumed that the rocket mass lost coincides with that of the massive particles expelled, so that \((1-\epsilon)\simeq 1\).This approximation relies on the fact that the expelled particles are non-relativistic, at least considering their ensemble average. Therefore, their kinetic energy is negligible if compared to their rest mass. However, without loss of generality, we may now account for a certain amount of mass that is wasted in the process, contributing to \(\epsilon\). By using a simple model (see Figure 5b), we consider that a fraction of the solid target, with a radius larger than the laser waist \(w_{0}\), detaches without any significant initial acceleration, therefore without producing any thrust. This region is  
Figure 5: On the left, the fraction \(A\) of the TNSA accelerated particle as a function of laser intensity. Uniformly ionized \(C^{+}\) and \(C^{6+}\) targets, with thickness \(d=5\) μm and density \(\rho=1\) g/cm\({}^{3}\), irradiated by a laser pulse of duration \(\tau_{L}=100\) fs, have been considered. On the right, the scheme of a laser-solid target interaction. The propellant includes a variety of contributions including radiation. heated by electron conduction into the target and by the shock propagation. The simplest model is then:  
\[1-\epsilon\simeq\left(\frac{w_{0}}{w_{eff}}\right)^{2}, \tag{47}\]  
where \(w_{eff}\geq w_{0}\) is the effective radius of the destructive laser-target interaction. Considering that \(w_{eff}\) can be limited by the target-holder conic guide, \(\epsilon\) can be approximated as a small constant in many realistic situations. As previously discussed, we are interested to maximize the specific impulse at fixed laser intensity. In the limit of a thin target, combining Equations (38), (45) and (46) and considering \(\eta_{LP}\simeq 0\), we can obtain a general expression for the LAPPS specific impulse \(I_{sp}^{LP}\):  
\[I_{sp}^{LP}\simeq\frac{(1-\epsilon)}{g_{0}}\left[\frac{2n_{0}\tau_{acc}}{\rho d \langle Z\rangle}\left(E^{tnsa}-E\right)+V\right] \tag{48}\]  
where we have introduced the parameter \(E^{tnsa}\), given by  
\[E^{tnsa}\equiv\frac{1}{2}\langle m_{I}\rangle\left[\langle v_{r}\rangle_{I}^{ tnsa}\right]^{2}\cos(\frac{\varphi}{2}), \tag{49}\]  
and representing the mean kinetic energy of the forward-accelerated TNSA particles. The cross term \(E\), which is generally characterized by \(E\ll E^{tnsa}\), is defined by  
\[E\equiv\frac{1}{2}\langle m_{I}\rangle\langle v_{r}\rangle^{All}\langle v_{r }\rangle_{I}^{tnsa}\cos(\frac{\varphi^{\prime}}{2}), \tag{50}\]  
and can be interpreted as a corrective term. Finally, the parameter \(V\) accounts for the effect of the longitudinal averaged velocity of the residual (non-TNSA) accelerated particles:  
\[V\equiv\langle v_{r}\rangle^{All}\cos(\frac{\varphi^{\prime}}{2}). \tag{51}\]  
Differently from \(E\), the velocity \(V\) may represent the dominant term in Equation (48) in the Low Intensity (LI) limit, which is defined by:  
\[I_{sp}^{LP}\simeq\frac{(1-\epsilon)}{g_{0}}V\quad(\text{LI}). \tag{52}\]  
The specific impulse in the High Intensity (HI) regime can be obtained considering that \(n_{0}\) and \(E^{tnsa}\) are monotonically increasing functions of \(I_{0}\).