# Laser-Accelerated Plasma-Propulsion System
## 3 Engine Thrust  
In general, in order to meet the needs of interplanetary robotic (or manned) missions, the travel time \(T_{t}\) should be limited to years or tens of years according to the classical optimized Hohmann or bi-elliptic transfer orbit maneuverings. A desirable rocket configuration, however, should meet the relation \(T_{i}\ll T_{t}\), where \(T_{i}\) is the total engine firing time. In this case, all the interplanetary space maneuverings can be considered local. Equation (7), however, does not contain any information about the time needed to accelerate the rocket up to the desired speed \(u\sim 10^{3}\)-\(10^{4}\) m/s. For this reason, it is important to derive the relativistic expression of the engine thrust. In the case of relativistic systems characterized by a variable rest mass, some of the standard covariant relations must be reformulated. In order to avoid any confusion, the simplest way is to consider only kinematic quantities as, for example, the rocket four-acceleration \(a^{\mu}=du^{\mu}/d\tau\), where \(d\tau=dt/\gamma(u)\) is the proper time in the rocket frame. In a 1D system, the quantity \(a^{\mu}a_{\mu}=-\gamma^{6}(u)a^{2}\), where \(a=du/dt\) is the ordinary acceleration, is invariant under Lorentz transformations. In general, the rapidity and the four-acceleration are related through the expression:  
\[\frac{d\theta}{d\tau}=\frac{1}{c}\gamma^{3}(u)a=\frac{1}{c}\sqrt{-a^{\mu}a_{ \mu}}, \tag{10}\]where the definition \(\theta\equiv\tanh^{-1}(u/c)\) has been used. Equation (10) can also be expressed in terms of the proper velocity \(d\sigma/d\tau=\sqrt{-a^{\mu}a_{\mu}}\), which clearly shows the physical meaning of this cinematic relation. Finally, from Equations (7) and (10), we can find the invariant relation for the relativistic rocket acceleration:  
\[\sqrt{-a^{\mu}a_{\mu}}=-\frac{v_{r}}{M}\frac{dM}{d\tau}. \tag{11}\]  
The engine is the only mechanism able to accelerate and decelerate the rocket; thus, the thrust can be defined as the total force \(F\) acting on the system. From the definition \(F=Md(\gamma(u)u)/dt\), and considering that force and acceleration are here parallel to the rocket velocity, we need only the expression of the longitudinal acceleration: \(a_{\parallel}=\gamma(u)^{-3}(F_{\parallel}/M)\). Combining \(a_{\parallel}\) with Equation (11), we obtain the engine trust measured in the laboratory space frame:  
\[F=-v_{r}\frac{dM}{d\tau}. \tag{12}\]  
The specific impulse can be defined, starting from Equation (12), according to the classical definition:  
\[I_{sp}\equiv\frac{F}{g_{0}}\bigg{(}-\frac{dM}{d\tau}\bigg{)}^{-1}=\frac{v_{r }}{g_{0}}, \tag{13}\]  
where \(-dM/d\tau\) represents the rocket lost mass as it appears from the rocket space frame. As expected, \(I_{sp}\) is an invariant that does not contain any relativistic factor. This property can be directly obtained considering that \(\theta\) is linear in \(v_{r}\) (see Equation (7)). In terms of the specific impulse, the engine thrust calculated in the laboratory rest frame is given by:  
\[F=-I_{sp}g_{0}\frac{dM}{d\tau}, \tag{14}\]  
which is analogous to the classical non-relativistic equation in the limit of \(u/c\ll 1\). In Ref. [14], the alternative relativistic definition of the specific impulse \(I_{sp}^{\prime}=\gamma(v_{r})v_{r}/g_{0}\) diverges for \(v_{r}\to c\), making this choice ambiguous in the case of comparison with other advanced propulsion schemes, for which the def. (13) is generally used. Basically, considering the linear increasing of \(I_{sp}\) with \(v_{r}\), we do not obtain a dramatic gain when the propellant approaches the light speed. However, \(I_{sp}\) ranges through several orders of magnitude and the limit \(I_{sp}\to I_{sp}^{max}\equiv c/g_{0}\) allows for obtaining spacecraft velocities which are compatible with interplanetary (or even interstellar) space missions for \(\Delta M/M_{i}\ll 1\). In this case, the fuel is fully used to accelerate the entire payload overcoming the so-called \"rocket tyranny\".