On the analytic properties of the perturbing function in the PCR3Body Problem

TL;DR

This paper develops a new Fourier coefficient expansion for the Perturbing function in the PCR3Body problem, enabling precise asymptotic analysis and zero detection, which could improve KAM theory applications in celestial mechanics.

Contribution

It introduces a Hansen coefficient-based expansion of the Fourier coefficient, facilitating detailed asymptotic and zero analysis relevant for KAM theory in the PCR3Body problem.

Findings

Derived a new Fourier coefficient expansion in terms of Hansen Coefficients.

Performed numerical zero analysis up to order 60 in eccentricity and semi-major axis.

Potentially reduces the measure of the non-torus set in phase space for PCR3Body.

Abstract

We provide a new expansion of the Fourier coefficient of the Perturbing function of the PCR3Body problem in terms of Hansen Coefficients. This gives us a precise asymptotic formula for the coefficient in the region of application of KAM theory (i.e small value of eccentricity and semi-major axis see e.g. \cite{Celletti-Chierchia}). Moreover, in the above region, we study the presence of zeros of the Fourier coefficient for coprime modes $(m,k) \in \Z^2$ and the presence of common zeros between coefficients relative to modes $(m,k)$,$(2m,2k)$ and $(m,k)$,$(2m,2k)$,$(3m,3k)$. Thanks to the previous expansion, this numerical analysis is done up to order $60$ in the power of eccentricity and semimajor axis. This is a first step for a possible application of \cite{Singular KAM, BBCZ} to PCR3Body Problem that would imply a reduction in terms of measure in the phase space of the so called "non--torus" set from $O(1-\sqrt{\e})$ (implied by standard KAM theory) to $O(1-\e |\log\e|^c )$ for some $c>0$.