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

TL;DR

This paper introduces a new mathematical method using Sundman inequality and Lagrange-Jacobi relation to analyze the Solar system's size near a state of zero total angular momentum, with implications for the existence of a ninth planet.

Contribution

It develops a novel analytical approach combining Sundman inequality and Lagrange-Jacobi relation to estimate Solar system size under specific angular momentum assumptions.

Findings

Estimated the mean size of the Solar system R.

Provided criteria for the potential existence of a ninth planet.

Linked orbital inclination and position within Solar system size estimate.

Abstract

In this paper, we present a new mathematical approach or solving procedure for analysis of the Sundman inequality (for estimating the moment of inertia of the Solar system 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 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.