Secular Evolution of a Two-Planet System of Three Bodies with Variable Masses

Comments 1: First of all, it would be good if the authors could explain to the reader why their orbital elements are osculating. This is a nontrivial issue, certainly not apparently evident. The authors may want to remind the reader that a physical trajectory consists of points donated by a continuous set of “simple” instantaneous orbits, and that these orbits may be contributing these points (instantaneous positions) in either an osculating or nonosculating manner. In the former case, the instantaneous orbits are tangent to the physical trajectory, in the latter not. From the material provided in the authors’ previous paper published in MNRAS, I observe that in their studies the authors employ instantaneous orbits that are not Keplerian. Nonetheless, the way how the resulting trajectory is assembled of those orbits is osculating. The authors should elucidate this to the reader, perhaps with a reference to the MNRAS paper. Specifically, it would be good to point out an analogy between the osculation condition imposed in the authors’ theory and the more standard Lagrange constraint usually (but not always) imposed on the Keplerian or Delaunay elements.

Response 1: We agree with this comment, many thanks for pointing this out. This is really nontrivial issue. The problem is that in the case of variable masses a general solution to the two-body problem does not exist, in contrast to the case of constant masses. From the other side, to apply the perturbation theory we need to find some exact solution that could be considered as unperturbed one. Such solution should contain 6 parameters which are constants of integration depending on the initial conditions. In the case of constant masses these parameters are known as the Keplerian orbital elements and completely define an elliptic orbit. These elements become functions of time if the perturbations are taken into account, and their instantaneous values at time t define an ellipse tangent to the physical trajectory at time t. And just such orbital elements are called the osculating elements. Solution (5) that is used as unperturbed one in our paper does not correspond to physical trajectory in the absence of perturbations, it only approximates the physical trajectory. But it depends on 6 constants which are analogs of the Keplerian orbital elements. It is important that geometrically they are almost identical to Keplerian orbital elements, at least over the finite time intervals over which observations are made, since the masses of celestial bodies change very slowly in reality. These parameters also become functions of time it perturbations are taken into account. It should be noted that differential equations determining time evolution of the orbital parameters are obtained with application of the method of constants variation and the assumptions that at any instant of time the coordinates and velocity components determined by solution (5) coincide with the actual values on the physical trajectory. Thus, the instantaneous values of the orbital parameters at time t define a curve determined by solution (5) tangent to the physical trajectory at time t. According to definition, just such orbital elements are called the osculating elements, in spite of the fact that they are not the Keplerian orbital elements. Due to this definition we say that we define equations of the perturbed motion in terms of the osculating elements of aperiodic motion on quasi-conic sections.

We have added necessary explanations in the paper shown in red color at pages 5-6.

Comments 2: Second, I would recommend the authors to cite the recent activities in the field, and perhaps to briefly juxtapose the approaches. I specifically imply the work by Dosopoulou and Kalogera (2016) ApJ 825:70, who addressed a similar problem but chose to model it by Keplerian elements — which within their treatment turned out to be nonosculating.

4. Response to Comments on the Quality of English Language

Point 1: he English is fine and does not require any improvement

Response 1:    No comment

5. Additional clarifications

We have taken into account all comments of the reviewers and made necessary corrections and improvements. All the changes in text are shown in red color.

Thanks a lot for your work!!!

Comments 1: On page 2, lines 45-46, one reads “Although this discrepancy was finally accounted for by Einstein’s theory of general relativity it does not exclude an existence of other reasons.” The “discrepancy” refers to the difference between observational and theoretical values for the advance of Mercury’s perihelion. I think that most, if not all, readers will disagree with the authors. The superiority of General Relativity with respect to Classical Mechanics is unquestionable. Therefore, the authors must rephrase their statement. They might say that their approach may provide a refinement of results obtained in the realm of Classical Mechanics, although recognizing the higherranking in the framework of General Relativity.

Response 1: We completely agree with this comment, many thanks. We do not question the superiority of General Relativity with respect to Classical Mechanics, we only want to note that the general theory of relativity does not exclude a possibility to improve such classical model as the many-body problem. Each theory has some domain of application and taking into account some new factors enables to refine the model and to get more accurate results. We have changed the lines 45-46 in the manuscript, new parts of text are shown in red color.   

Comments 2: What is the physical meaning, beyond mathematical convenience, of the mass function γ, introduced between equations (3) and (4) on page 3?

Response 2:

As in the case of constant masses, dynamics of the two-body system is determined by a sum of two masses m_0 and m_j and due to this the corresponding dimensionless function γ is introduced. This function γ(t) is determined by the laws of masses variation. An advantage of such definition is that equation (3) becomes integrable in the case of zero right-hand side and its general solution may be written for arbitrary function γ, the only condition should be fulfilled – this function must be twice differentiable. As a result, one can investigate dynamics of the system for different laws of masses variation.

Comments 3: Why do the authors choose specifically the Eddington-Jeans law, equation (33), on page 9, in order to describe the mass decrease of the central object?

Response 3: Thank you for pointing this out. The Edington-Jeans law was chosen as a model dependance because we made the calculations for the Sun-Mercury-Venus system and the Sun exists in a main-sequence phase of its evolution. For main-sequence stars the Edington-Jeans law is usually considered to describe their mass loss. But the calculations may be easily repeated for other law of the mass loss.

Comments 4: Actually, instead of have chosen the Eddington-Jeans law, the authors should treat the mass decrease of the central object as a free parameter. Then, in possession of observational data, such a parameter must be adjusted in order to describe them satisfactorily. As it stands with the Eddigton-Jeans law, the manuscript is one of a purely mathematical character, not reflecting a truly physical situation.

Response 4: We agree with this comment. We’d like to note only that the main aim of this work was to demonstrate an essential influence of masses variation on the secular evolution of 2-planet system. The calculations were performed for the systems Sun-Mercury-Venus and Sun-Mercury-Jupiter because it is known that there is the discrepancy between the observed value of the advance of Mercury's perihelion and the theoretical value predicted from Newtonian theory with constant masses of the bodies. The obtained results have shown that variability of mass leads to additional noticeable changes in the long-term behavior of the orbital elements. At the same time it is difficult to say whether the Sun loses its mass according to the Eddington-Jeans law and which values of parameters should be chosen for simulation. We have used this law as a model and so one could not expect to get the results corresponding to real physical situation. We can state only that taking into account the variability of masses enables to refine the results obtained in the framework of the many-body problem.

4. Response to Comments on the Quality of English Language

Point 1: The English is fine and does not require any improvement.

Response 1:    No comment

5. Additional clarifications