Orbital flips in hierarchical triple systems: Relativistic effects and third-body effects to hexadecapole order

We analyzed the secular evolution of hierarchical triple systems in the post-Newtonian approximation to general relativity (Will 2017). We expanded the Newtonian three-body equations of motion in powers of the ratio a/A, where a and A are the semimajor axes of the inner and outer orbits, respectively. The leading order ``quadrupole'' terms, of order (a/A)^3 are responsible for the well-known Kozai-Lidov oscillations of orbital inclination and eccentricity. The octupole terms, of order (a/A)^4 have been shown to allow the inner orbit to ``flip'' from prograde relative to the outer orbit to retrograde and back, and to permit excursions to very large eccentricities. We carried the expansion of the equations of motion to hexadecapole order, (a/A)^5, and included the leading orbital effects of post-Newtonian theory. Using the Lagrange planetary equations for the orbit elements of both binaries, we obtained the equations for the secular evolution of the elements, and employed them to analyze cases of astrophysical interest. We found that, for the most part, the orbital flips found at octupole order are robust against both relativistic and hexadecapole perturbations. We showed that, for equal-mass inner binaries, where the octupole terms vanish, the hexadecapole contributions can alone generate orbital flips and excursions to very large eccentricities.

Higher-order effects in the dynamics of hierarchical triple systems. Quadrupole-squared term

We analyzed the secular evolution of hierarchical triple systems to second-order in the quadrupolar perturbation induced on the inner binary by the distant third body (Will 2020). The Newtonian three-body equations of motion, expanded in powers of the ratio of semimajor axes a/A, become a pair of effective one-body Keplerian equations of motion, perturbed by a sequence of multipolar perturbations, denoted quadrupole, (a/A)^3, octupole, (a/A)^4, and so on. In the Lagrange planetary equations for the evolution of the instantaneous orbital elements, second-order effects arise from obtaining the first-order solution for each element, consisting of a constant (or slowly varying) piece and an oscillatory perturbative piece, and reinserting it back into the equations to obtain a second-order solution. After an average over the two orbital timescales to obtain long-term evolutions, these second-order quadrupole (Q^2) terms would be expected to produce effects of order (a/A)^6. However we find that the orbital average actually enhances the second-order terms by a factor of the ratio of the outer to the inner orbital periods, ~ (A/a)^{3/2}. For systems with a low-mass third body, the Q^2 effects are small, but for systems with a comparable-mass or very massive third body, such as a Sun-Jupiter system orbiting a solar-mass star, or a 100 solar mass binary system orbiting a 10^6 solar mass black hole, the Q^2 effects can completely suppress flips of the inner orbit from prograde to retrograde and back that occur in the first-order solutions.

A hidden friend for the galactic center black hole, Sgr A*

The hierarchical nature of galaxy formation suggests that a supermassive black hole binary could exist in our galactic center. We proposed a new approach to constraining
the possible orbital configuration of such a binary companion to the galactic center black hole Sgr A* through the measurement of stellar orbits (Naoz et al 2020). Focusing on the star S0-2, we showed that requiring its orbital stability in the presence of a companion to Sgr A* yields stringent constraints on the possible configurations of such a companion. Furthermore, we showed that precise measurements of time variations in the orbital parameters of S0-2 could yield stronger constraints. Using existing data on S0-2 we derived upper limits on the binary black hole separation as a function of the companion mass. For the case of a circular orbit, we ruled out a 10^5 solar mass companion with a semimajor axis greater than 170 astronomical units or 0.8 mpc. This is already more stringent than bounds obtained from studies of the proper motion of Sgr A*.Including other stars orbiting the galactic center should yield stronger constraints that could help uncover the presence of a companion to Sgr A*. Finally, if such a companion exists, it will emit gravitational wave radiation, potentially detectable with LISA.

Relativistic orbits around spinning supermassive black holes.

We studied the secular evolution of the orbital elements of a stellar-mass object orbiting a spinning massive black hole (Will and Maitra 2017). We used the post-Newtonian approximation of geodesic motion in the Kerr spacetime in harmonic coordinates, valid through 3PN order, augmented with radiation-reaction contributions through 4.5PN order, including the 4PN damping effects of spin-orbit coupling. The secular evolution equations for the osculating orbit elements were iterated to high PN orders using a two-timescale approach. We derived a criterion for terminating the orbit when its Carter constant drops below a critical value, whereupon the body plunges across the event horizon at the next closest approach. We then analyzed numerically the orbits of objects injected into high-eccentricity orbits of arbitrary inclinations via interactions within a surrounding star cluster, obtaining the number of orbits and the elapsed time between injection and plunge, and the residual orbital eccentricity at plunge as a function of inclination. We showed that, if the black hole is spinning rapidly, the flux of gravitational radiation during the final orbit before plunge may be suppressed by as much as three orders of magnitude if the orbit is retrograde on the equatorial plane compared to its prograde counterpart.

A new general relativistic contribution to Mercury's perihelion advance

We pointed out the existence of a new general relativistic contribution to the perihelion advance of Mercury that, while smaller than the contributions arising from the solar quadrupole moment and angular momentum, is 100 times larger than the second-post-Newtonian contribution (Will 2018). It arises in part from relativistic ``cross-terms'' in the post-Newtonian equations of motion between Mercury's interaction with the Sun and with the other planets, and in part from an interaction between Mercury's motion and the gravitomagnetic field of the moving planets. At a few parts in 10^6 of the leading general relativistic precession of 42.98 arcseconds per century, these effects are likely to be detectable by the BepiColombo mission to place and track two orbiters around Mercury, scheduled for launch around 2018.