Spinning compact binary dynamics and chameleon orbits

Abstract

We analyse the conservative evolution of spinning compact binaries to second post-Newtonian (2PN) order accuracy, with leading order spin-orbit, spin-spin and mass quadrupole-monopole contributions included. As a main result we derive a closed system of first order differential equations in a compact form, for a set of dimensionless variables encompassing both orbital elements and spin angles. These evolutions are constrained by conservation laws holding at 2PN order. As required by the generic theory of constrained dynamical systems we perform a consistency check and prove that the constraints are preserved by the evolution. We apply the formalism to show the existence of chameleon orbits, whose local, orbital parameters evolve from elliptic (in the Newtonian sense) near pericenter, towards hyperbolic at large distances. This behavior is consistent with the picture that General Relativity predicts stronger gravity at short distances than Newtonian theory does.

Precessing Black Hole Binaries and Their Gravitational Radiation

“…The dynamics up to 2PN orders, including the SO, SS and QM couplings of compact binaries in terms of dimensionless osculating orbital elements and spin angles is given by Equations (36)-(42) of [40]. The two dimensionless shape variables characterizing the orbits of the object with reduced mass µ moving around another body having total mass m are introduced by…”

“…The full set of contributions can be found in [46]. In order to ascertain the validity of the secular dynamics we compare it with the instantaneous dynamics of [40]. We depict the evolution of the nine variables under 5600 orbital periods on Figure 1.…”

“…It has been also proven that (ii) the relative angle of the spins stays constant for equal mass black holes, irrespective of their orientation, found previously in [35] (with inclusion of only SO and SS contributions) and [36] (with all relevant contributions included, but only for circular orbits), and (iii) the special configuration of equal mass black holes with equal, but antialigned spins, both laying in the plane of motion (leading to large recoils found in numerical simulations [37,38], although the largest was found for binaries with partially aligned spins [39]) is preserved at the second post-Newtonian (2PN) level of accuracy, for generic eccentric, parabolic or hyperbolic orbits. Further, the closed system of first-order differential equations has been rewritten in a compact form, for a set of dimensionless variables encompassing both orbital elements and spin angles, and a consistency check run to prove that the constraints are preserved during evolution [40]. As an application, the existence of a new type of interesting orbit, the chameleon orbit, has been shown, whose local, orbital parameters evolve from elliptic near pericenter, towards hyperbolic at large distances [40].…”

“…Further, the closed system of first-order differential equations has been rewritten in a compact form, for a set of dimensionless variables encompassing both orbital elements and spin angles, and a consistency check run to prove that the constraints are preserved during evolution [40]. As an application, the existence of a new type of interesting orbit, the chameleon orbit, has been shown, whose local, orbital parameters evolve from elliptic near pericenter, towards hyperbolic at large distances [40].…”

“…The most comprehensive instantaneous, 2PN accurate non-circular dynamics with generic quadrupole parameters discussed in [40] with consistency condition checked and the number of independent variables identified has been orbit averaged in the eccentric case [46]. In this conference presentation we succinctly summarize these secular dynamics and indicate possible applications.…”