Analysis of the size of Solar system close to the state with zero total angular momentum via Sundman's inequality.

In this paper, we present a new mathematical approach or solving procedure for analysis of the Sundman's inequality (for estimating the moment of inertia of the Solar system's configuration) with the help of Lagrange-Jacobi relation, under additional assumption of decreasing of the total angular momentum close to the zero absolute magnitude in the final state of Solar system in a future. By assuming such the final state for Solar system, we have estimated the mean-size of Solar system R via analysis of the Sundman's inequality. So, to answer the question "Does the ninth planet exist in Solar system?", one should meet the two mandatory criteria for such the ninth planet, first is that it should have the negligible magnitude of inclination of its orbit with respect to the invariable plane.

On the stability of Laplace resonance for Galilean moons (Io, Europa, Ganymede).

This paper presents the application of recent ansatz for estimation of stability of the Laplace resonance for Galilean moons (Io, Europa, Ganymede). We estimate over time the eccentricity + semi-major axis in a binary system experiencing the net tidal friction, including the additional tidal heating which comes from the transformation of net transfer of angular momentum between the Galilean moons of Jupiter (due to dynamical features of the Laplace resonance). Presumably, there should be a net transfer of angular momentum between Io and Europa (for the reason that tidal heating on Ganymede seems to be negligible with respect to Io and Europa).