# Laser-Accelerated Plasma-Propulsion System
## 6 LAPPS Analytical Model
### TNSA Acceleration
Figure 4: The plots consider the intensity range \(I_{0}=10^{18}-10^{21}\) W/cm\({}^{2}\) and a laser wavelength \(\lambda_{L}=0.8\) μm, typical of a TiSa laser system. The discontinuity in the subplot (**a**) corresponds to the saturation of conversion efficiency \(\Omega(I_{0})=0.5\). In subplot (**b**), low \(Z\) metals and typical constituent of organic polymers are considered. Except for the carbon, only first ionization is considered in order to present the lower velocity limit of each species. For the calculation of the specific impulse \(I_{sp}\), it is necessary to estimate the averaged relative velocity of the accelerated ions. This quantity, which is independent of \(\tau_{acc}\), is obtained from the integration of Equation (33):  
\[\left\langle v_{r}\right\rangle_{I}=C_{s}=\sqrt{\frac{Zk_{b}T_{hot}}{m_{I}}}. \tag{34}\]  
The ratio \(Z/m_{I}\) is the only parameter that characterizes the exhaust speed at fixed laser intensity. Typical ion velocities are presented in Figure 4b, where metals with low atomic number and organic components have been considered. For low laser intensities of the order of \(I_{0}\simeq 10^{18}\) W/cm\({}^{2}\), the condition \(\left\langle v_{r}\right\rangle_{I}\gtrsim 10^{6}\) m/s, assuring a specific impulse \(I_{sp}\gtrsim 10^{5}\) s, that is needed for realistic interplanetary applications, is reached only by low atomic number targets. The estimate of \(I_{sp}\), however, does not account for the efficiency of the system \(\eta_{LP}\); a more detailed calculation of the impulse will be given in the dedicated section. Considering a more realistic situation, the ensemble of accelerated ions \(n_{0I}^{ij}\) can include different species \(i\) and different ionization states \(j\), where  
\[n_{0e}=\sum_{j=1}^{max(i)}\sum_{i=1}^{species}\ Z_{j}n_{0I}^{ij}. \tag{35}\]  
Their thermodynamic state, and therefore the relative ratios \(n_{0I}^{ij}/n_{0I}^{lim}\), is here mainly determined by the intensity of the sheath field, by the Ohmic heating of thermal (cold) electrons, by the scattering of fast electrons and by the absorption of X-rays produced during laser-plasma interaction. Numerical simulations, describing laser-plasma interaction and collisional-radiative equilibrium, are therefore required to retrieve a realistic ionization distribution [35]. In general, the relative velocity of accelerated ions, averaged over all species, is given by:  
\[\left\langle v_{r}\right\rangle_{I}^{tnsa}\equiv\sqrt{k_{b}T_{hot}}\left( \frac{\sum_{i,j}n_{0I}^{ij}\sqrt{Z_{j}m_{i}}}{\sum_{i,j}n_{0I}^{ij}m_{i}}\right), \tag{36}\]  
which can be expressed in terms of the averaged atomic charge \(\left\langle\sqrt{Z_{i}}\right\rangle\equiv\sum_{j}n_{0I}^{ij}\sqrt{Z_{j}}/ \sum_{j}n_{0I}^{ij}\) as:  
${}^{2}$0  
It is important to observe that the averaged speed given by Equation (37) does not represent the propellant speed of massive particles to be considered in Equation (19). In fact, the angular distribution of the accelerated particles is not yet taken into account and, most importantly, TNSA is only one of the acceleration mechanisms involved in laser-target interaction. For these reasons, the propellant speed of massive particles can be expressed as:  
${}^{2}$1  
where \(\left\langle v_{r}\right\rangle^{All}\) is the average relative speed of massive particles accelerated through other mechanisms, while \(\varphi\) and \(\varphi^{\prime}\) (where usually \(\varphi^{\prime}>\varphi\)) are the values of the angular divergence of particle beams in the two cases. The dimensionless coefficients \(A\) and \(B\) indicate the weight of each component, where \(A+B=1\).