Modified method of round Gaussian rings. Application to the two-planetary problem

A scheme of the modified method of round Gaussian rings, designed to study the secular evolution of orbits in systems consisting of a central star and two planets, is presented. The reason for the secular evolution of the nodes and inclinations of the orbits of the planets is their mutual gravitational attraction. The orbits of the planets are modeled by homogeneous round Gaussian rings, to which the masses, sizes and angles of inclination of the orbits, as well as orbital angular momenta of the planets, are transferred. The method takes into account the fact that, in general, the ascending nodes of the orbits may not coincide. The mutual gravitational energy of the rings 𝑊𝑚𝑢𝑡 is represented as a series in the quadratic approximation in powers of small inclination angles. Using this function 𝑊𝑚𝑢𝑡, a closed system of four differential equations describing the secular evolution of the planets’ orbits is composed. The solution to the equations is obtained in finite analytic form, which simplifies the interpretation of the investigated planetary motions. The method was tested on the example of the Sun-Jupiter-Saturn system; for it, in particular, the difference in the longitudes of the nodes of the orbits of Jupiter and Saturn was calculated as a function of time. New approach is also used to study the precession of nodes in the exoplanetary system K2-36; graphs of all unknown quantities are obtained. It has been established that in the course of evolution the mutual inclination angle of the orbits remains constant, and the librations of the orbits in the inclination angle and in the motion of the nodes occur synchronously.