# Laser-Accelerated Plasma-Propulsion System
## 4 Specific Impulse  
Besides the general considerations on the rocket dynamics, it becomes important to introduce a more complete approach, including a practical parametrization to calculate the specific impulse of a generic particle accelerator. In this way, it will be possible to analyze the fundamental properties of the LAPPS engine with the appropriate formalism. The calculation of \(I_{sp}\) requires determining the relative propellant speed \(v_{r}\) that can be interpreted as the first moment of the velocity distribution of an heterogeneous variety of particles with different masses. For this reason, the optimization of \(I_{sp}\) may not coincide with the classical performance optimization of a particle accelerator, independently on the specific acceleration mechanism.  
We start considering more in detail the infinitesimal mass \(dM\) lost by the rocket. We will use the convention introduced by [PERSON] [30]. According to the scheme presented in Figure 2, if we define the fraction \((1-\epsilon)dM\) as the mass of the massive particles actually expelled, \(ec^{2}dM\) defines the energy available for utilization. Of the total energy available, in general, only a fraction \(\eta ec^{2}dM\) is utilized to increase the energy of the propellant of which only a fraction \(\delta\eta ec^{2}dM\) is used to accelerate massive particles. Thus, in the rocket rest frame, we find:  
\[\delta\eta ec^{2}dM\equiv(\gamma(v_{r}^{m})-1)(1-\epsilon)c^{2}dM, \tag{18}\]  
which simply defines, in terms of the [PERSON]'s parameters \((\epsilon,\eta,\delta)\), the fraction of available energy converted into kinetic energy of massive particles. From Equation (18), we can easily retrieve the propellant speed of massive particles [30]:  
\[\frac{v_{r}^{m}}{c}=\sqrt{1-\bigg{(}\frac{1-\epsilon}{1-\epsilon(1-\delta \eta)}\bigg{)}^{2}}. \tag{19}\]  
The general invariant solution \(Md\theta=-dMv_{r}/c\) of the rocket motion is linear in \(v_{r}\), thus it can be used to find the effective exhaust speed in terms of a classical weighted average over velocities. The mass lost by the rocket due to massive particles is \(((1-\epsilon)+\delta\eta\epsilon)dM\); this fraction includes the rest mass of the expelled particles \((1-\epsilon)dM\) and the mass-equivalent energy used to accelerate them \(\delta\eta edM\). Note that in the original paper [30] this term was omitted leading to an error. In the same way, the mass lost by the rocket which is used to \"create\" massless particles is \((1-\delta)\eta edM\). We can therefore calculate a weighted velocity, including both massive and massless particles, as:  
\[v_{r}^{*}\equiv[1-\epsilon(1-\delta\eta)]v_{r}^{m}+(1-\delta)\eta ec. \tag{20}\]  
Equation (20) represents the effective propellant speed measured from the rocket rest frame. Basically, in the general Equation (7), \(v_{r}\) must be replaced with \(v_{r}^{*}\) to take into account the rocket global efficiency. In terms of [PERSON]'s parameters, i.e., using Equations (19) and (20) can be written as:  
\[v_{r}^{*}=c\bigg{[}\sqrt{\delta\eta\epsilon(2-2\epsilon+\epsilon\delta\eta) }+(1-\delta)\eta\epsilon\bigg{]}, \tag{21}\]  
which corresponds to the corrected solution given by [PERSON] [31]. The expression (21) is valid for all the rocket configurations in which both propellant and energy are stored inside (closed cycle). This list does not include, for example, spacecraft propelled by directed energy, jet engines and atmospheric LAP engine as the [PERSON]'s lightcraft [32].