## 3 Dynamics of Spacecraft Acceleration
Inserting (13) into (9), we derive  
\[m\frac{d\mathbf{v}}{dt}=(\mathbf{v}_{+}-\mathbf{v})\frac{dm_{+}}{dt}-(\mathbf{v}_{-}-\mathbf{v})\frac {dm_{-}}{dt}, \tag{3}\]4  
written in the dimensionless form using (13), and  
\[m\frac{d\mathbf{v}}{dt}=(\mathbf{v}_{+}-\mathbf{v})\frac{dm_{+}}{dt}-(\mathbf{v}_{-}-\mathbf{v})\frac {dm_{-}}{dt}, \tag{3}\]5  
where the value \(\tau_{0}=m/\left(\mu_{\Sigma}n_{\Sigma}cS\right)\) plays the role of a natural temporal scale for the process under study and may be considered the characteristic time of the spacecraft's acceleration. Thus, Equation (16) completely describes the acceleration of the spacecraft with an allowance for the capture and ejection of interplanetary matter and forms the basis of the following analysis. However, in the present study, we restrict ourselves to the analysis of the initial stage of an acceleration when the last term in Equation (16) is a negligibly small quantity. Such approximation shall be realized if  
\[m\frac{d\mathbf{v}}{dt}=(\mathbf{v}_{+}-\mathbf{v})\frac{dm_{+}}{dt}-(\mathbf{v}_{-}-\mathbf{v})\frac {dm_{-}}{dt}, \tag{3}\]6  
where \(y_{0}=y(\tau=0)\gg 1\). So, from this inequality, it follows that  
\[m\frac{d\mathbf{v}}{dt}=(\mathbf{v}_{+}-\mathbf{v})\frac{dm_{+}}{dt}-(\mathbf{v}_{-}-\mathbf{v})\frac {dm_{-}}{dt}, \tag{3}\]7  
In this case, Equation (16) transforms into  
\[m\frac{d\mathbf{v}}{dt}=(\mathbf{v}_{+}-\mathbf{v})\frac{dm_{+}}{dt}-(\mathbf{v}_{-}-\mathbf{v})\frac {dm_{-}}{dt}, \tag{3}\]8The solution of this equation is  
\[m\frac{d\mathbf{v}}{dt}=(\mathbf{v}_{+}-\mathbf{v})\frac{dm_{+}}{dt}-(\mathbf{v}_{-}-\mathbf{v})\frac {dm_{-}}{dt}, \tag{3}\]9  
This relationship describes the initial transient stage for spacecraft acceleration as long as condition (17) is satisfied. The use of Equation (19) makes it possible to estimate the characteristic time \(\tau_{m}\) for which our spacecraft shall reach its maximum velocity \(y_{m}\sim\sqrt[3]{2/\beta_{*}}\). Substituting this value \(y_{m}\) into (19), we obtain the following rude estimation:  
\[\bigg{|}\frac{dm_{+}/dt}{m_{+}}\bigg{|}\gg\bigg{|}\frac{dv_{+}/dt}{v_{+}} \bigg{|},\quad\bigg{|}\frac{dm_{-}/dt}{m_{-}}\bigg{|}\gg\bigg{|}\frac{dv_{-}/dt }{v_{-}}\bigg{|}. \tag{4}\]0  
which can be rewritten for clarity in a dimensional form as  
\[\bigg{|}\frac{dm_{+}/dt}{m_{+}}\bigg{|}\gg\bigg{|}\frac{dv_{+}/dt}{v_{+}} \bigg{|},\quad\bigg{|}\frac{dm_{-}/dt}{m_{-}}\bigg{|}\gg\bigg{|}\frac{dv_{-}/dt }{v_{-}}\bigg{|}. \tag{4}\]1  
As can be seen from this relation, there is a strong, linear dependence with respect to the characteristic time \(\tau_{0}\) and a very weak power dependence on the parameter \(\beta_{*}\). In the discussed case, the influence of varying parameters \(\zeta\) and \(W\) on the acceleration time is negligible. That is, the inaccuracy in determining these parameters shall have little effect on the value of \(t_{m}\). We now estimate the spacecraft thrust:  
\[\bigg{|}\frac{dm_{+}/dt}{m_{+}}\bigg{|}\gg\bigg{|}\frac{dv_{+}/dt}{v_{+}} \bigg{|},\quad\bigg{|}\frac{dm_{-}/dt}{m_{-}}\bigg{|}\gg\bigg{|}\frac{dv_{-}/dt }{v_{-}}\bigg{|}. \tag{4}\]2  
With allowance for (13)-(15), Equation (21) transforms into  
\[\bigg{|}\frac{dm_{+}/dt}{m_{+}}\bigg{|}\gg\bigg{|}\frac{dv_{+}/dt}{v_{+}} \bigg{|},\quad\bigg{|}\frac{dm_{-}/dt}{m_{-}}\bigg{|}\gg\bigg{|}\frac{dv_{-}/dt }{v_{-}}\bigg{|}. \tag{4}\]3  
For \(y\gg 1\), by again setting \((1+2y)^{1/2}\approx\left(2y\right)^{1/2}\), we rewrite this relation in dimensional form as  
\[\bigg{|}\frac{dm_{+}/dt}{m_{+}}\bigg{|}\gg\bigg{|}\frac{dv_{+}/dt}{v_{+}} \bigg{|},\quad\bigg{|}\frac{dm_{-}/dt}{m_{-}}\bigg{|}\gg\bigg{|}\frac{dv_{-}/dt }{v_{-}}\bigg{|}. \tag{4}\]4  
where we denote \(\kappa=(2\zeta W\mu_{E}n_{\Sigma})^{1/2}\), and the thrust \(T\) is defined in Newtons. As can be seen from this relation, the thrust is increased with respect to an increase in \(v\) values. This point is a fundamental othemes from chemical and nuclear thrusters for which there is no dependence of spacecraft speed.