## 3 Dynamics of Spacecraft Acceleration  
We study the acceleration of a nonrelativistic object (i.e., for the spacecraft's speed, we have \(v\ll c\)) with an allowance for the particles collected from the interplanetary medium ejected after acceleration to relativistic velocities. We focus on the case in which the effects of external factors (e.g., frictional force in the atmosphere, gravitational fields near massive  
Figure 2: Dependence of solar radiation power as a function of distance between Sun and Ceres. Here, 1 corresponds to the Earth orbit (\(R_{E}=1\)_A.U._) and 2—the Ceres orbit (\(R_{C}=2.76\)_A.U._) [8; 9]. objects) on the movement of the spacecraft are negligible, i.e., we shall proceed from the generalized Meshchersky equation [2; 30]:  
\[m\frac{d\mathbf{v}}{dt}=(\mathbf{v}_{+}-\mathbf{v})\frac{dm_{+}}{dt}-(\mathbf{v}_{-}-\mathbf{v})\frac {dm_{-}}{dt}, \tag{3}\]  
where \(m\) is the mass of spacecraft, \(\mathbf{v}\) is the spacecraft's speed in the laboratory frame of reference, \(m_{+}\) and \(\mathbf{v}_{+}\) are the respective mass and speed of particles captured by the spacecraft, and \(m_{-}\) and \(\mathbf{v}_{-}\) are the respective mass and speed of particles emitted into space. Note that Equation (3) is valid only if the following conditions hold:  
\[\bigg{|}\frac{dm_{+}/dt}{m_{+}}\bigg{|}\gg\bigg{|}\frac{dv_{+}/dt}{v_{+}} \bigg{|},\quad\bigg{|}\frac{dm_{-}/dt}{m_{-}}\bigg{|}\gg\bigg{|}\frac{dv_{-}/dt }{v_{-}}\bigg{|}. \tag{4}\]  
We shall consider the simplest one-dimensional case in which the interplanetary medium is immobile, i.e., we set \(v_{+}=0\), and the velocity \(v_{-}\) of the ejected particles is considered in a nonrelativistic way in a laboratory frame. This may be accomplished due to the nonrelativistic spacecraft speed  
\[v_{-}=\frac{v-u_{ex}}{1-(u_{ex}v)/c^{2}}\approx v-u_{ex}. \tag{5}\]  
Herein, the exhaust plasma flow may be relativistic so far as the value of \(u_{ex}\) is defined by the absorption and dissipation of radiation flux, as well as the number of particles trapped per unit time:  
\[N=Svn_{\Sigma}, \tag{6}\]  
where the cross-section of the trap, \(S=S(D)\), through which interplanetary matter is captured by the spacecraft is defined by the inlet diameter of the trap, \(D\) (see Figure 1). In this case, the attached mass is created only by an incoming flow of captured particles:  
\[\frac{dm_{+}}{dt}=\mu_{\Sigma}N. \tag{7}\]  
Assuming all ejected mass \(dm_{-}/dt\) is the accelerated flow of incoming particles, we can write  
\[\frac{dm_{-}}{dt}=\frac{1}{\sqrt{1-\big{(}\frac{u_{ex}}{c}\big{)}^{2}}}\frac{ dm_{+}}{dt}. \tag{8}\]  
Note that the exhaust velocity \(u_{ex}\) depends on the solar radiation power \(W\) which, in turn, is a function of the distance from the Sun to the spacecraft (see Equations (13)-(15)). Such a consideration is necessary for flight over long distances (say, of the order of 1 _A.U._) when the radiation power varies greatly (see Figure 2). With allowance for immobile interplanetary medium, substituting (5), (7) and (8) into Equation (3), in one-dimensional geometry, we derive  
\[\frac{dv}{dt}=\frac{\mu_{\Sigma}}{m}Sn_{\Sigma}\Bigg{(}\frac{u_{ex}}{\sqrt{1- \big{(}\frac{u_{ex}}{c}\big{)}^{2}}}-v\Bigg{)}v. \tag{9}\]  
To use this equation, we need to define the relationship between the kinetic energy of the accelerating particles which form the outgoing plasma flux and the processes of the absorption and dissipation of radiation.