## 4 Discussion  
Considering the two-body problem with variable masses, we need to apply perturbation theory because [PERSON]'s Equation (1) describing the relative motion of the bodies is not integrable, in general. This means that, first of all, we need to modify the problem in such a way that it would have an exact solution. It is obvious that we need to find such an exact solution, which describes the system motion very close to its real motion. Here, we consider two models that are used in applications most often. In the framework of the Model 1, we rewrite Equation (1) in the form (5), which is integrable in the case of \(\vec{F}=0\). Its right-hand side is small enough because it is proportional to the derivative of the function \(\gamma(t)\) and may be considered as perturbation. Recall that in the case of the Eddington-Jeans law, for example, an expression for the time derivative of mass contains the small parameter \(\alpha\). The corresponding exact solution is given by (8) and describes aperiodic motion on a conic section that is characterized by six constant orbital parameters, similar to the well-known case of constant masses. The only difference is that in the case of elliptic trajectory (\(e<1\)), the time of one revolution changes over time, and the motion becomes aperiodic. Comparison of this exact solution with the corresponding solution of the original Equation (1) demonstrates (see Figure 5) that the semi-major axis in real motion increases with time, while the exact solution (8) provides a constant value of the semi-major axis. In addition, the increase in the period of revolution in real motion is not taken into account correctly. Of course, applying perturbation theory, we can find the secular perturbation of the semi-major axis in this model and improve it, but this complicates the calculations if we need to take into account interaction with other bodies in the framework of the many-body problem, where the two-body problem itself is used as an approximation. Another approach is based on Equation (23), where the right-hand side considered as perturbation contains the second derivative of the function \(\gamma(t)\). Note that the rates of mass change are usually small; in the case of the Eddington-Jeans law, for example, the second derivative of the mass is proportional to \(\alpha^{2}\), where \(\alpha\) is a small parameter. Recall that in Model 1, the perturbing force is of order \(\alpha\). The smallness of the perturbing force in Model 2 results in only small oscillating corrections to the orbital elements determined by solution (24). It means that the solution presented in Equation (24) may be successfully used as the unperturbed solution in many applications. Some examples of this approach may be found in our works [33; 34; 35; 36; 37; 38; 39; 40], where the problems of three and four bodies with variable masses are considered. Figure 6: Dimensionless coordinate \(x^{*}(t^{*})\) as a function of the dimensionless time (\(1~{}yr=2\pi\), \(\alpha^{*}=10^{-4}\)). Solid red curve corresponds to solution of Equation (1), magenta and blue dashed curves correspond to solutions of equations of the perturbed motion in Models 1 and 2, respectively. It should be noted also that Equation (24), which describes an aperiodic motion on a quasi-conic section, may be considered as the unperturbed motion in the study of other many-body problems with variable masses. Indeed, such an approach enables one to easily take into account the mutual interaction of the bodies and reactive forces; it is sufficient to add in the right-hand side of Equation (24), the corresponding perturbing forces, and repeat the calculations. Of course, in the case of \(N\) bodies, we have to consider a system of \(N-1\) differential equations, similar to solution (24), with more complicated right-hand sides. The calculations become much more complicated, but the application of a modern computer algebra system like Wolfram Mathematica enables one to solve such problems successfully. Conceptualization, A.P. and M.M. ; methodology, A.P.