# Laser-Accelerated Plasma-Propulsion System
## 6 LAPPS Analytical Model
### LAPPS Thrust  
As discussed in the previous section, LAPPS is effectively able to generate an adequate specific impulse for interplanetary space missions. This condition is reached even for a relatively low fraction of TNSA accelerated particles suggesting the possibility of further significant improvements. In contrast, the realization of a sufficiently high thrust is a much more challenging task, representing the main limit of this scheme [39]. An order of magnitude for the thrust needed in deep-space missions can be obtained from the firing time Equation (17):  
\[T_{i}\simeq u[\text{m}/\text{s}]10^{b-6}\ \ \ \text{Days}, \tag{55}\]  
where \(b=\log_{10}(g_{0}M_{i}/F)\) is the weight-thrust ratio. Considering a relatively high acceleration of 100 m/s per day, a thrust of 1 N for each tonne (\(b\simeq 4\)) is roughly required for thruster based on a particle accelerator. The LAPPS thrust can be obtained directly from the definition (14) in the non-relativistic limit:  
\[F^{LP}\simeq(1-\epsilon)\bigg{[}\frac{2\eta_{0}\tau_{acc}}{\rho d\langle Z \rangle}(E^{tnsa}-E)+V\bigg{]}\dot{M} \tag{56}\]  
where Equation (48) has been used for the specific impulse. The infinitesimal mass \(\dot{M}\) expelled in every laser cycle can be obtained from Equation (39) and gives, in the limit of a thin target \(\dot{M}\approx\pi\
u_{s}d\rho u_{0}^{2}/(1-\epsilon)\), where \(\
u_{s}\) is the laser repetition rate. According to Equation (53), the thrust in the high intensity limit (HI) admits a pure TNSA interpretation:  
\[F^{LP}\simeq\frac{2\pi n_{0}\tau_{acc}\
u_{s}w_{0}^{2}}{\langle Z\rangle}E^{ tnsa}=\frac{2\pi n_{0}\tau_{acc}w_{0}^{2}\eta_{L}\langle P_{R}\rangle}{ \langle Z\rangle\epsilon_{L}}E^{tnsa}\ \ \ \ \ \ (\text{HI}) \tag{57}\]  
where \(\epsilon_{L}\
u_{L}=\eta_{L}\langle P_{R}\rangle\) has been used. As expected, density and thickness of the target are not relevant in this limit. Moreover, the term \(n_{0}w_{0}^{2}E^{tnsa}\sim\Omega(I_{0})\) does not depend on \(I_{0}\) above \(I_{0}=5.7\times 10^{19}\) W/cm\({}^{2}\). In other words, the TNSA contribution to the thrust becomes constant above this threshold. As it occurs in LAP engines [6], the thrust can be here varied independently of \(I_{sp}\) by modifying the laser repetition rate \(\
u_{s}\). This is limited by the laser pumping frequency \(\
u_{s}=\
u_{p}/n\), where \(n\geq 1\) is a positive integer. Within this range, it is possible, in principle, to regulate the thrust preserving the specific impulse. Relying on the expression in Equation (56), the dependence of \(F^{LP}\) on the laser intensity and the parameter \(\chi\) are reported in Figure 6b. For each laser intensity \(I_{0}\), the laser waist (\(w_{0}^{2}=4\epsilon_{L}\sqrt{\ln 2/2}/(\pi\tau_{L}I_{0})\)) has been calculated considering a fixed laser pulse energy of \(\epsilon_{L}=5\times 10^{3}\) J. Similarly to the specific impulse, the thrust is strongly affected by the non-TNSA accelerated particles in the low-mid laser intensity range. When the laser intensity grows, however, \(F^{LP}\) becomes almost independent from both the fraction \(\chi\) and the target ionization. The blue line reported in Figure 6b represents the plot of Equation (57) and, as expected, correctly describes the high-intensity behaviour. At fixed laser pulse energy \(\epsilon_{L}\) and repetition rate \(\
u_{s}\), the optimization of the thrust strongly depends on both specific orbital maneuvers and spacecraft mass. In particular, the mass of the power supply, electronics, power source, and laser system need to be considered within the calculation. Without going into details, considering a mass in the few tonne range for a \(\epsilon_{L}=10^{3}\), \(\
u_{L}=1\) kHz laser system [39], a thrust of \(\gtrsim 1\) N would represent an attractive result in the view of a realistic LAPPS implementation. As it generally occurs with electric propulsion, at fixed input power, the specific impulse and the thrust cannot be independently varied.