# Laser-Accelerated Plasma-Propulsion System
## 7 Radiation Contribution and High Intensity Limit  
As well known, high power laser experiments in the TNSA regime produce radiation. From the rocket dynamics point of view, the relevant contributions may come from both bremsstrahlung and recombination X- and \(\gamma\)-ray photons emitted from the plasma and produced by the interaction of high-energy electrons with the target. In addition, laser photons which are scattered or reflected during the interaction, can also be taken into account. The specific impulse for a LAPPS engine produced by the photons can be calculated by means of general Equations (13) and (20), obtaining a photon contribution to Equation (46) given by:  
\[\Delta I_{sp}^{R}\simeq\frac{(1-\delta)\epsilon\eta_{LP}}{g_{0}}c. \tag{58}\]  
The infinitesimal mass lost by the rocket in every laser cycle is then given by  
\[dM_{Tot}\simeq\frac{dM}{1-\epsilon}+\frac{E_{R}}{c^{2}}, \tag{59}\]  
Figure 7: Engine thrust as a function of laser intensity at fixed pulse energy \(\varepsilon_{L}=10^{3}\) J. A uniformly ionized \(C^{6+}\) target with a density of \(\rho=1\) g/cm\({}^{3}\) has been considered. Different target thickness and pulse duration have been analyzed. The laser repetition rate is fixed by \(\
u_{L}=1\) kHz.  
where \(dM\) is given by Equation (39), while \(E_{R}\) takes into account the total energy generated by the system (nuclear energy). The approximation of \(1-\epsilon\simeq(w_{0}/w_{eff})^{2}\) is still valid in this model. In the case of a thin target, we immediately find:  
\[\Delta I_{sp}^{R}\simeq\frac{(1-\delta)E_{R}\eta_{I}I_{sp}^{L}}{\left(w_{eff}^ {2}c^{2}\pi d\rho+E_{R}\right)}, \tag{60}\]  
where the definitions (22) and (28) have been used. As shown in Equation (60), as long as the term \(E_{R}/c^{2}\) is negligible with respect to the rest mass of the propellant, also \(\Delta I_{sp}^{R}\simeq 0\) and therefore can be considered negligible. The upper limit of \(\Delta I_{sp}^{R}\) corresponds to the photon rocket specific impulse and can be obtained by removing the solid target, i.e., for \(\delta=\rho=0\) and \(\eta_{I}=1\). In the same way, the total correction to Equation (48), which is given by  
\[\Delta I_{sp}^{LP}\simeq\frac{E_{R}\eta_{I}\Big{(}(1-\delta)I_{sp}^{L}+\delta \eta_{L}v_{r}^{m}/g_{0}\Big{)}}{\left(w_{eff}^{2}c^{2}\pi d\rho+E_{R}\right)}, \tag{61}\]  
can be considered negligible using current laser technology and solid targets. As already discussed, we are not able to extend the LAPPS model over arbitrary high laser intensities. However, the ultimate limit of high intensity interaction at fixed pulse energy must coincide with the photon rocket, which is independent from the laser intensity and well defined. For this reason, it is interesting to define the condition above which a photon rocket becomes convenient over a LAPPS rocket. Considering that, for a single laser shot, the spacecraft speed gain is proportional to \(I_{sp}dM_{tot}\), we find:  
\[I_{0}\geq\frac{\pi\sqrt{32\ln 2}\tau_{acc}n_{0}c}{\langle Z\rangle\tau_{\rm L}} E^{tnsa}, \tag{62}\]  
where \(I_{sp}^{LP}dM/(1-\epsilon)\leq I_{sp}^{L}E_{R}/c^{2}\) has been used with \(I_{sp}^{LP}\) given by Equation (53). The threshold (62), if calculated for typical laser-plasma accelerator parameters, exceeds the intensity range in which the TNSA model is well defined. In other words, an optimized LAPPS scheme is always convenient compared to a pure photon rocket at least for \(I_{0}\lesssim 10^{21}\) W/cm\({}^{2}\). This conclusion becomes evident in Figure 8, where the limit of the photon rocket is reported for a relatively high efficiency of \(\eta_{L}=5\%\).  
Figure 8: Thrust as a function of the specific impulse at fixed pulse energy \(\varepsilon_{L}=10^{3}\) J. A uniformly ionized C\({}^{6+}\) target with a density of \(\rho=1\) g/cm\({}^{3}\) has been considered. The laser repetition rate is fixed by \(\
u_{L}=1\) kHz. In case of photon rocket, the thrust is given by \(F^{L}=\
u_{L}\varepsilon_{L}/c\).