Improved equations for eccentricity generation in hierarchical triple systems

TL;DR

This paper introduces an improved analytical method for estimating the inner eccentricity in hierarchical triple systems with initially circular, coplanar orbits, addressing previous limitations and validated through numerical simulations.

Contribution

It presents a new secular solution for inner eccentricity that corrects earlier weaknesses, applicable to a wide range of mass ratios and initial conditions.

Findings

New formula accurately predicts eccentricity evolution.

Numerical simulations confirm the formula's validity.

Enhanced understanding of short-term eccentricity behavior.

Abstract

In a series of papers, we developed a technique for estimating the inner eccentricity in hierarchical triple systems, with the inner orbit being initially circular. However, for certain combinations of the masses and the orbital elements, the secular part of the solution failed. In the present paper, we derive a new solution for the secular part of the inner eccentricity, which corrects the previous weakness. The derivation applies to hierarchical triple systems with coplanar and initially circular orbits. The new formula is tested numerically by integrating the full equations of motion for systems with mass ratios from 10^(-3) to 10^(3). We also present more numerical results for short term eccentricity evolution, in order to get a better picture of the behaviour of the inner eccentricity.