## 3 Dynamics of Spacecraft Acceleration
Generally, the kinetic energy of an exhaust particle is determined by the expression  
\[K=\mu_{\Sigma}c^{2}\left(\frac{1}{\sqrt{1-\big{(}\frac{u_{ex}}{c}\big{)}^{2}} }-1\right). \tag{10}\]If we consider only elastic head-on collisions with the spacecraft's surface and assume that the energy of the coming particles is spent on the deceleration of the spacecraft, one can write the power balance relation for absorbed radiation as  
\[\zeta WS_{sp}-Sv\mu_{\Sigma}n_{\Sigma}\frac{v^{2}}{2}=N\;(K+\varepsilon_{i}), \tag{11}\]  
where \(S_{sp}\) is the area of the solar panel and \(\zeta\) is the conversion coefficient of solar radiation into power going to the production and acceleration of plasma flow. We introduced this factor to take into account some energy losses in solar batteries in the processes of ionization and acceleration, etc. The use of such an integral value makes it possible to account for all energy losses in a parametrical way without considering the physical essence of these parasitic processes. On the other hand, the choice of \(\zeta\) requires a detailed analysis of all technical elements determining the operation of the plasma thruster. However, while this question emerges from the framework of the present paper, we hope to discuss this point later. Here, we shall limit ourselves to choosing some acceptable value for \(\zeta\). As an example of using expression (11), we consider the analytically simplest case when \(u_{ex}\gg v\), which allows us to neglect the second term on left-hand side in Equation (11). Also, we assume that \(K\gg\varepsilon_{i}\), i.e., \(v\gg\sqrt{2\varepsilon_{i}/\mu_{\Sigma}}\), and the ionization losses are negligibly small (in our case, this means \(v\gg 10^{4}\) m/s). In this range of spacecraft speeds, setting \(S_{sp}=S\) for simplicity, from Equation (11), we derive  
\[m\frac{d\mathbf{v}}{dt}=(\mathbf{v}_{+}-\mathbf{v})\frac{dm_{+}}{dt}-(\mathbf{v}_{-}-\mathbf{v})\frac {dm_{-}}{dt}, \tag{3}\]0  
Then, from (10) and (12), we arrive at  
\[m\frac{d\mathbf{v}}{dt}=(\mathbf{v}_{+}-\mathbf{v})\frac{dm_{+}}{dt}-(\mathbf{v}_{-}-\mathbf{v})\frac {dm_{-}}{dt}, \tag{3}\]1  
where  
\[m\frac{d\mathbf{v}}{dt}=(\mathbf{v}_{+}-\mathbf{v})\frac{dm_{+}}{dt}-(\mathbf{v}_{-}-\mathbf{v})\frac {dm_{-}}{dt}, \tag{3}\]2  
and the value  
\[m\frac{d\mathbf{v}}{dt}=(\mathbf{v}_{+}-\mathbf{v})\frac{dm_{+}}{dt}-(\mathbf{v}_{-}-\mathbf{v})\frac {dm_{-}}{dt}, \tag{3}\]3  
is the characteristic velocity scale for the problem under consideration and reflects the dynamics of the transformation from radiation energy into the kinetic energy of the accelerated plasma flux. At this stage, we would again like to emphasize that \(u_{ex}\) is in fact the specific impulse for the problem under consideration. To understand the order of this value, we shall make a rough estimate for typical conditions. Assuming that the energy conversion factor \(\zeta\) is \(10^{-3}\) (this value is used in all further calculations), and if as a characteristic estimate of radiation flux near the Earth we take \(\mathrm{W}=(1\div 2)\times 10^{3}\) W/m\({}^{2}\) (see Figure 2), then, using these values in Equation (15), we obtain \(U_{*}=(20\div 40)\) m/s. In our further estimations we shall use the value \(U_{*}=20\) m/s as a characteristic magnitude. That is, the value \(U_{*}\) is very small for the dynamic scales of the problem under consideration. The relation (13) shows how \(u_{ex}\) depends on the spacecraft's velocity \(v\) and the characteristics of the interplanetary medium via the parameter \(y\). As an illustration of this, Figure 3 presents a graph of \(\beta_{ex}=u_{ex}/c\) for relatively small values of \(y\). As can be seen in Figure 3, for greater values \(y\gg 10^{3}\), a nonrelativistic case is realized in which \(\beta_{ex}<10^{-1}\). Such dynamics result from the growing number of particles captured \(N\) with an increase in velocity \(v\) when the absorbed energy per particle falls. We now return to the analysis of the generalized Meshchersky equation.