Abstract
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. The second condition is that the orbit of the ninth planet should be located within the estimation for the mean-size of Solar system R.
Highlights
IntroductionWe 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
We present a new mathematical approach or solving procedure for analysis of the Sundman’s inequality 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
The main reason for such assumption is that the invariable plane is within circa 1° of the orbital planes for all the 4 jovian planets, whereas about 98% of the aforementioned total angular momentum is contributed by the orbital angular momentum of the four Gas giants (Jupiter, Saturn, Uranus and Neptune)
Summary
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. In the denotations above, K means the kinetic energy of the entire dynamical configuration of Solar system; J means the scalar product of the radius-vector of each object in the Solar system (including planets and Sun) onto appropriate velocity vector of the same object respectively; C denotes the pseudo-vector of total angular momentum of the system
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