Precession of the orbital nodes of Jupiter and Saturn triggered by the mutual perturbation: A model of two rings

Abstract

The problem of the precession of the orbital planes of Jupiter and Saturn under the influence of mutual gravitational perturbations was formulated and solved using a simple dynamical model. Using the Gauss method, the planetary orbits are modeled by material circular rings, intersecting along the diameter at a small angle α. The planet masses, semimajor axes and inclination angles of orbits correspond to the rings. What is new is that each ring has an angular momentum equal to the orbital angular momentum of the planet. Contrary to popular belief, it was proved that the orbital resonance 5 : 2 does not preclude the use of the ring model. Moreover, the period of averaging of the disturbing force (T ≈ 1332 yr) proves to be appreciably greater than a conventionally used period (≈900 yr). The mutual potential energy of rings and the torque of gravita tional forces between the rings were calculated. We compiled and solved the system of differential equations for the spatial motion of rings. It was established that a perturbing torque causes the precession and simulta neous rotation of the orbital planes of Jupiter and Saturn. Moreover, the opposite orbit nodes on the Laplace plane coincide and perform a secular movement in retrograde direction with the same velocity of 25.6′′/yr and the period yr. These results are close to those obtained in the general theory (25.93′′/yr), which confirms the adequacy of the developed model. It was found that the vectors of the angular velocity of orbital rings move counterclockwise over circular cones and describe circles on the celestial sphere with radii β 1 ≈ 0.8403504° (Saturn) and β 2 ≈ 0.3409296° (Jupiter) around the point which is located at an angular dis tance of 1.647607° from the ecliptic pole.

Secular Evolution of Rings around Rotating Triaxial Gravitating Bodies

“…However, this opinion cannot be agreed upon unconditionally. Real orbits have deviations from strict resonance, which makes possible the addi- tional averaging of local perturbations [19]. Thus, if the resonance is sharper, then the averaging period of the additional perturbations are longer.…”

“…OF THE CENTRAL BODY After substituting expression (18) into the equations for osculating elements [15], we obtain the following system of differential equations: (19) where is the mean motion of the point mass m in orbit and the coefficients and are given in (12).…”

The Mechanism and Timescale of Nodal Precession: Two Nuclear Stellar Disks in the Galactic Center

“…The angular velocity of the line of nodes for the "counter-clockwise" disk in the invariable plane is then equal to (Kondratyev 2014b) …”