# Laser-Accelerated Plasma-Propulsion System
## 6 LAPPS Analytical Model
### Fraction of TNSA Particles  
When the laser pulse hits a thin solid target an amount \(dM\) of propellant is vaporized and expelled. Since the laser power supply causes a negligible decreasing of the rocket weight compared to the target vaporization, the infinitesimal mass \(dM\) can be considered, in practice, as the total mass lost by the rocket in a single shot. In a rocket powered by a nuclear reactor, this approximation can be considered always valid except in the case of a pure photon rocket. For a target characterized by a thickness \(d\) and a density \(\rho\), the infinitesimal mass expelled in every laser cycle can be approximated by  
\[dM\simeq\rho d\pi w_{0}^{2}\bigg{(}1+\frac{d\tan(\phi/2)}{2w_{0}}\bigg{)}^{2}, \tag{39}\]  
where a conical frustum volume with an average radius of \(w_{0}+(d/2)\tan(\phi/2)\) has been considered (see Equation (32)). Unfortunately, we expect that only a small fraction of \(dM\) will be accelerated through the TNSA mechanism. This quantity can be roughly estimated as:  
\[\frac{dM^{tnsa}}{\pi(w_{0}\sigma(\phi))^{2}}\simeq\tau_{acc}\sum_{i,j}m_{i}n_{0 \text{I}}^{ij}\sqrt{\frac{Z_{j}k_{\text{D}}T_{hot}}{m_{i}}}, \tag{40}\]  
which is the generalization of \(M/S_{r}\simeq m_{\text{I}}N\) for more than one ion species, where \(S_{r}=\pi(w_{0}\sigma)^{2}\) is the rear target surface with \(\sigma(\phi)=1+d\tan(\phi/2)/w_{0}\). Using Equation (36), it is possible to simplify Equation (40) as:  
\[\frac{dM^{tnsa}}{\pi(w_{0}\sigma(\phi))^{2}}\simeq\rho^{tnsa}\langle v_{r} \rangle_{\text{I}}^{tnsa}\tau_{acc}, \tag{41}\]  
where the TNSA particle density \(\rho^{tnsa}\) is defined by:  
\[\rho^{tnsa}\equiv\sum_{i=1}^{species}m_{i}n_{0\text{I}}^{i}, \tag{42}\]  
where \(n_{0\text{I}}^{i}\equiv\sum_{j}n_{0\text{I}}^{ij}\) is the whole particle density of the i-th ion species. The total density \(\rho^{tnsa}\), which is a function of both target and laser intensity, can be approximated in terms of averaged quantities as:  
\[\rho^{tnsa}\simeq n_{0e}\frac{\langle m_{\text{I}}\rangle}{\langle Z\rangle}, \tag{43}\]  
where \(\langle m_{\text{I}}\rangle\equiv\sum_{i}m_{i}n_{0\text{I}}^{i}/\sum_{i}n_{0 \text{I}}^{i}\) is the averaged ions mass while \(\langle Z\rangle\equiv\sum_{i}n_{0\text{I}}^{i}\langle Z_{i}\rangle/\sum_{i}n_ {0\text{I}}^{i}\) is the atomic charge averaged over both ionization level and ion species. In order to retrieve the explicit form of \(v_{r}^{m}\), it is necessary to estimate the fractions \((A,\,B)\in[0,1]\) introduced in Equation (38). By definition, we have \(A\equiv dM^{tnsa}/dM\) and \(B=1-A\). Using Equations (39) and (41), we immediately obtain:  
\[A\simeq\frac{\rho^{tnsa}\langle v_{r}\rangle_{\text{I}}^{tnsa}\tau_{acc}}{ \rho d}\bigg{(}1+\frac{\sigma(\phi)-1}{\sigma(\phi)+1}\bigg{)}^{2}. \tag{44}\]  
The fraction \(A\) of the TNSA accelerated particles represents a fundamental parameter affecting the performance of a TNSA engine. In fact, considering that usually \(\langle v_{r}\rangle^{All}\ll\langle v_{r}\rangle^{tnsa}\), the optimization of \(l_{sp}\) requires increasing \(A\) as much as possible. One of the easiest ways to maximise \(A\) is to consider the limit of ultra-thin targets. This is a consequence of the TNSA mechanism which extracts ions only from the rear side of the target. Indeed, in real experimental conditions, the lowest target thickness \(d_{min}\) is determined by the contrast of the laser pulse, which is a parameter quantifying the precursor radiation of the laser peak, as well as the mechanical and chemical properties of the target. In optimized setups, where an ultrahigh contrast laser is used, and a \(d_{min}\) of the order of a few hundreds nanometers can be utilized [25]. In addition to the minimum target thickness, the optimization of \(A\) requires to maximise the relative velocity of accelerated ions.