Comparison of various mathematical models on the example of solving the equations of the movement of large planets and the Moon

In this paper, we study the accuracy of solving various differential equations describing the motion of large planets, the Moon and Sun. On the time interval from 31 years BC to 3969 AD, the numerical integration of Newtonian relativistic differential equations and equations obtained on the basis of the interaction of the surrounding space with moving material bodies was carried out. The range of applicability of the considered differential equations for the investigated objects is revealed. By comparing of the coordinates of the Moon, found by solving various differential equations and the DE405 data bank, it is shown that the greatest accuracy in the elements of the orbits of large planets is achieved by solving differential equations obtained on the basis of the interaction of the surrounding space with moving material bodies. The solution of relativistic equations provides high accuracy of the orbit elements for Mercury and the outer planets throughout the integration interval. However, for the remaining inner planets and the Moon, the accuracy of the orbital elements obtained by solving relativistic equations is comparable to the accuracy obtained by solving Newton equations. It is believed that the use of the harmonic coordinate system is justified only for Mercury from the point of view of the velocity of the secular longitude displacement of its perihelion, but for other internal planets (the Venus, Earth & Moon, and Mars) the velocities of secular displacements of the longitude of the perihelion's are overstated. It is shown that the solution of differential equations obtained on the basis of the interaction of the surrounding space with moving material bodies ensures a high accuracy of obtaining orbital elements for all objects under consideration on the time interval under study.