# Compact binary inspiral: Nature is perfectly happy with a circle

## TL;DR

This paper clarifies that the apparent growth of a certain eccentricity measure in inspiraling binaries is a coordinate effect, and the physical orbit remains circular, especially when considering the direction of the Runge-Lenz vector.

## Contribution

It demonstrates that the growth of the Runge-Lenz vector magnitude does not indicate true eccentricity increase, emphasizing the importance of its direction in orbital analysis.

## Key findings

- The Runge-Lenz vector magnitude can grow without indicating true eccentricity.
- The physical orbit remains circular despite the apparent growth in certain eccentricity measures.
- Including post-Newtonian effects confirms the orbit's circularization.

## Abstract

It is standard lore that gravitational radiation reaction circularizes the orbits of inspiralling binary systems. But in recent papers, Loutrel et al. have argued that at late times in such inspirals, one measure of eccentricity actually increases, and that this could have observable consequences. We show that this variable, the magnitude of the Runge-Lenz vector ($e_{\rm RL}$), is not an appropriate measure of orbital eccentricity, when the eccentricity is smaller than the leading non-Keplerian perturbation of the orbit. Following Loutrel et al., we use Newtonian equations of motion plus the leading gravitational radiation-reaction terms, the osculating-orbits approach for characterizing binary orbits, and a two-timescale analysis for separating secular from periodic variations of the orbit elements. We find that $e_{\rm RL}$ does grow at late times, but that the actual orbital variables $r$ and $dr/dt$ show no such growth in oscillations. This is in complete agreement with Loutrel et al. We reconcile this apparent contradiction by pointing out that it is essential to take into account the direction of the Runge-Lenz vector, not just its magnitude. At late times in an inspiral, that direction, which defines the pericenter angle, advances at the same rate as the orbital phase. The correct picture is then of a physically circular orbit whose osculating counterpart is indeed eccentric but that resides permanently at the orbit's latus rectum at $-90^{\rm o}$, therefore exhibiting no oscillations. Including first post-Newtonian effects in the equations of motion, we show that $e_{\rm RL}$ grows even more dramatically. But the phase of the Runge-Lenz vector again rotates with the orbit at late times, but now the osculating orbit resides at "perpetual apocenter", so again the physical orbit circularizes.