A new radial, natural, higher order intermediary of the main problem four decades after the elimination of the parallax

TL;DR

This paper introduces a novel Lie transform method that simplifies the main problem in celestial mechanics by removing long-period effects without eliminating parallax, leading to a new higher-order intermediary.

Contribution

It presents a new Lie transform technique that achieves simplifications similar to classical methods but without removing parallax, and introduces a new higher-order intermediary of the main problem.

Findings

The new transformation removes long-period effects without eliminating parallax.

It generalizes the classical radial intermediary to higher orders.

The method maintains higher order J2 effects in the simplified model.

Abstract

Simplifications in dealing with the equation of the center when the short-period effects are removed from a zonal Hamiltonian are commonly attributed to the elimination of parallactic terms. But this interpretation is incorrect, and the simplifications rather stem from the removal of concomitant long-period terms, an outcome that can also be achieved without need of eliminating the parallax. To show that, a Lie transforms simplification is invented that augments the exponents of the inverse of the radius and still achieves analogous simplifications in handling the equation of the center to those provided by the classical elimination of the parallax simplification. The particular case in which the new transformation does not modify the exponents of the parallactic terms of the original problem, leads to a new intermediary of the main problem that, while keeping higher order effects of J2, is formally analogous to Cid's first order radial intermediary.