Dynamical Evolution of Quasi-Hierarchical Triples

Abstract

We study the gravitational dynamics of quasi-hierarchical triple systems, where the outer orbital period is significantly longer than the inner one, but the outer orbit is extremely eccentric, rendering the time at pericentre comparable to the inner period. Such systems are not amenable to the standard techniques of perturbation theory and orbit-averaging. Modelling the evolution of these triples as a sequence of impulses at the outer pericentre, we show, by comparing with direct three-body integrations, that such triples lend themselves to a description as an analytical map between subsequent outer pericentre passages. This map exhibits secular oscillations, going beyond the von Zeipel--Lidov--Kozai mechanism. We show that the time to coalescence due to gravitational waves in such systems is modified. We then study the long-term evolution under this map, which lead to a random-walk-like behaviour of the inner eccentricity. While this behaviour is probably absent from isolated triples, it could exist in triples where the outer orbit is weakly coupled to a system with which it can exchange angular momentum, and we describe some properties of this random walk.

Figures (11)

• Figure 1: A depiction of the orientations of the three bodies (see text and equations \ref{['eqn:vectors']}). The $\mathbf{\hat{x}}$-axis points from the binary's centre-of-mass to the outer pericentre, and the $\mathbf{\hat{z}}$-axis points out of the page.
• Figure 2: A comparison between the analytical map (blue) $\Delta \mathbf{e}$, $\Delta \mathbf{j}$ given in equations (\ref{['eqn: changes per encounter']}--\ref{['eqn: evolution']}), and the results of a direct three-body simulation (in black, dash-dotted); see text for details. We see that the simulations matches the analytical formula well. We ascribe the disagreement in inclination to a different way of defining it in rebound.
• Figure 3: The evolution of a quasi-hierarchical orbit over a few secular times, from the map \ref{['eqn: evolution']} (blue) and rebound (black), for $m_1 = m_2 = m_3 = M_{\odot}$, $\omega_0 = \Omega_0 = \pi/4$, $r = 10a_{\rm in}$, $a_{\rm out} = 1000a_{\rm in}$, $i_0 = 7\pi/9$. The purple curve shows is the maximum eccentricity according to pure quadrupole ZLK evolution.
• Figure 4: Left: a diagram of the maximum eccentricity achieved by a quasi-hierarchical triple over an $\mathcal{O}(\tau_{\rm sec})$ time-scale, found as the first local eccentricity maximum, for equal masses. Right: a comparison with the maximum eccentricity according to pure (quadrupole) ZLK evolution; the colour-scheme is logarithmic. See text for details.
• Figure 5: The change in the time to coalescence from standard ZLK evolution, from equation \ref{['eqn:delta gw']}. The change is stronger around $i_0 = \pi/2$, where both $e_{\max}$ and $e_{\rm ZLK}$ are very high.