## 2 Models and Algorithms
### Model 2: Aperiodic Motion on Quasi-Conic Section  
There exists another possibility to reduce Equation (1) to a form appropriate for the application of perturbation theory. Let us rewrite Equation (1) in the form  
\[\frac{d^{2}\vec{\tau}}{dt^{2}}+Gm(t)\frac{\vec{\tau}}{r^{3}}-\frac{\dot{\gamma }(t)}{\gamma(t)}\vec{\tau}=\vec{F}. \tag{23}\]  
where the perturbing force \(\vec{F}\) is given by expression (3).  
One can readily see that Equation (23) coincides with Equation (1) because we have added the same term \(\vec{F}\) to both sides of (1). However, Equation (23) becomes integrable if the force \(\vec{F}\) in the right-hand side is replaced by zero. Indeed, one can readily check by substitution that in the case of \(\vec{F}=0\), Equation (23) has an exact solution given by  
\[x=a\gamma(t)\Big{(}(\cos E-e)(\cos\omega\cos\Omega-\sin\omega \sin\Omega\cos i)-\] \[\sqrt{1-e^{2}}\sin E(\sin\omega\cos\Omega+\cos\omega\sin\Omega \cos i)\Big{)},\] \[y=a\gamma(t)\Big{(}(\cos E-e)(\cos\omega\sin\Omega+\sin\omega \cos\Omega\cos i)-\] \[\sqrt{1-e^{2}}\sin E(\sin\omega\sin\Omega-\cos\omega\cos\Omega \cos i)\Big{)},\] \[z=a\gamma(t)\Big{(}(\cos E-e)\sin\omega+\sqrt{1-e^{2}}\sin E\cos \omega\Big{)}\sin i. \tag{24}\]  
Here, \(E\) is an analogue of the eccentric anomaly that is determined again by the Kepler Equation (9), in which the mean anomaly \(M\) is replaced by  
\[M=\frac{\sqrt{\kappa}}{a^{3/2}}(\Phi(t)-\Theta),\ \ \Phi(t)=\int_{0}^{t}\frac{dt}{ \gamma^{2}(t)},\ \ \Theta=\Phi(\tau), \tag{25}\]  
where constant parameter \(\tau\) is an analogue of the time of passage through the pericenter. As in solution (8), all six constants \(a,e,i,\Omega,\omega\), and \(\Theta\) in (24) are determined from the initial conditions of motion.  
In the case of constant masses (\(\gamma(t)=1\)), solution (24) reduces to the corresponding solution of the classical two-body problem and describes motion on a conic section. However, due to a time-dependent scaling coefficient \(\gamma(t)\) in (24), the conic section undergoes deformation and the trajectory becomes a spiral. Therefore, one can say that solution (24) describes aperiodic motion on a quasi-conic section; it is also quite natural to call the constants \(a,e,i,\Omega,\omega\) in (24) the analogues of the Keplerian orbital elements (see [33]). Equation (25) shows that the analogue of the mean anomaly \(M\) is an increasing function of time, which depends on the laws of mass variation \(\gamma(t)\) and becomes a linear function of time in the case of constant masses (\(\gamma=1\)).  
If the masses \(m_{0}(t),m_{1}(t)\) are given, the function \(\gamma(t)\) defines the mean anomaly \(M(t)\), and the Kepler Equation (9) enables one to find the eccentric anomaly \(E(t)\) as a function of time. Therefore, solution (24) defines the unperturbed motion of the body \(P_{1}\) in terms of the time and six constants of integration \(a,e,i,\Omega,\omega\), and \(\Theta\). The corresponding total time derivatives of the coordinates (24) may be written in the form  
\[\dot{x}=\frac{\dot{\gamma}(t)}{\gamma(t)}x-a\gamma(t)\dot{E}\Big{(} \sin E(\cos\omega\cos\Omega-\sin\omega\sin\Omega\cos i)+\] \[\sqrt{1-e^{2}}\cos E(\sin\omega\cos\Omega+\cos\omega\sin\Omega \cos i)\Big{)},\] \[\dot{y}=\frac{\dot{\gamma}(t)}{\gamma(t)}y-a\dot{E}\Big{(}\sin E (\cos\omega\sin\Omega+\sin\omega\cos\Omega\cos i)+\] \[\sqrt{1-e^{2}}\cos E(\sin\omega\sin\Omega-\cos\omega\cos\Omega \cos i)\Big{)},\] \[\dot{z}=\frac{\dot{\gamma}(t)}{\gamma(t)}z+a\dot{E}\Big{(}-\sin E \sin\omega+\sqrt{1-e^{2}}\cos E\cos\omega\Big{)}\sin i. \tag{26}\]  
where  
\[\dot{E}=\frac{\sqrt{\kappa}}{a^{3/2}(1-e\cos E)\gamma^{2}(t)}. \tag{27}\]