# Laser-Accelerated Plasma-Propulsion System
## 6 LAPPS Analytical Model
### LAPPS Specific Impulse
Thus, above a certain laser intensity (which depends on the target properties), the specific impulse becomes dominated by the TNSA mechanism and we obtain:  
\[I_{sp}^{LP}\simeq\frac{(1-\epsilon)}{g_{0}}\left(\frac{2n_{0}\tau_{acc}}{\rho d \langle Z\rangle}\right)E^{tnsa}\quad(\text{HI}). \tag{53}\]  
From Equation (53), we conclude that, for a pure TNSA engine, the specific impulse linearly increases with the mean kinetic energy released to the accelerated particles. This fundamental property characterizes the ideal LAPPS engine. The calculation of the specific impulse in the general form of Equation (48) requires to estimate the relative speed \(V\sim\langle v_{r}\rangle^{All}\). In addition to TNSA, several mechanisms may contribute to produce a local acceleration of ions, electrons or uncharged debris when an ultra-intense laser-pulse hits a thin target. For these reasons, an analytical estimation of \(V\) is generally difficult to obtain. However, we can quantify the effects of the non-TNSA accelerated particles by investigating the dependence of the specific impulse \(I_{sp}^{LP}\) on the parameter:  
\[\chi\equiv\frac{\langle v_{r}\rangle^{All}}{\langle v_{r}\rangle^{tnsa}}\ll 1. \tag{54}\]Realistically, the model of TNSA presented here cannot be extended to arbitrarily high laser intensities. As previously discussed, a conservative limit may be represented by a maximum laser intensity of \(l_{0}\simeq 10^{21}\) W/cm\({}^{2}\). Within this limit, it is interesting to define the conditions to reach the ideal range for the specific impulse discussed in Section 2. The dependence of \(I_{sp}^{LP}\) on the parameter \(\chi\) is shown in Figure 5(a). In the low-middle intensity range, any small variation of \(\chi\) induces a drastic difference in the specific impulse. This property indicates that the TNSA contribution is not dominant. Roughly speaking, the highest non-TNSA contribution coincides with an LAP engine [6]. Differently, for higher intensities close to \(l_{0}\gtrsim 10^{21}\) W/cm\({}^{2}\), a specific impulse \(I_{sp}^{LP}\simeq 10^{5}\) s is achieved, with a negligible dependence on \(\chi\) (always in the limit of \(\chi\ll 1\)). This condition corresponds to the high-intensity limit described by Equation (53), indicating that the contribution of non-TNSA ions is negligible here. For high intensities, the differences also induced by the ionization properties of the target vanish as the terms \(E^{tnsa}/\langle Z\rangle\) are independent from the ionization distribution in case of a single accelerated species. In view of the above results, despite \(\chi\ll 1\), an evaluation of \(V\) is mandatory in order to quantify \(I_{sp}^{LP}\) along the entire laser intensity range. The upper limit \(\chi_{max}=0.05\) reported in Figure 5(a) was estimated by considering that the non-TNSA particles leaving the target have a velocity of the order of the plasma expansion speed, i.e., \(\langle v_{T}\rangle^{All}\sim C_{I}\). This implies that \(\chi\sim\sqrt{T_{I}/T_{hot}}\), where \(T_{I}\) is the ion temperature in the plasma. The maximum value \(\chi(I_{0})\simeq 0.05\) was therefore obtained by taking a value of \(T_{I}\approx 2\)-\(2.5\) keV. A more accurate evaluation of the \(\chi\) parameter needs dedicated experimental measurements in the conditions of interest or extensive molecular dynamics/hydrodynamic numerical simulations. Compared to other advanced electric-propulsion schemes [11; 37; 38], an LAPPS is actually capable of far better performance in terms of the specific impulse, even in the mid-intensity range. These conclusions were already obtained in previous papers but with a general overestimation of several orders of magnitude. For instance, in Ref. [39], a laser pulse of intensity \(l_{0}=3\times 10^{20}\) W/cm\({}^{2}\) and a gold-foil target of thickness \(d=125\) m were considered. Despite the thick and the high-atomic number target, they found impressive values of \(l_{sp}=3.2\times 10^{6}\) s, which is well beyond the results presented in Figure 5(a).