Post-Newtonian effects in N-body dynamics

The increasing role of general relativity in the dynamics of stellar systems with central massive black holes, in the generation of extreme mass-ratio inspirals and tidal disruption events, and in the evolution of hierarchical triple systems inspires a close examination of how post-Newtonian effects are incorporated into N-body dynamics. The majority of approaches incorporate relativity by adding to the Newtonian N-body equations the standard two-body post-Newtonian terms for a given star around the black hole or for the close binary in a triple system. We argued that, for calculating the evolution of such systems over timescales comparable to the relativistic pericenter advance timescale, it may be important to include ``cross terms'' in the equations of motion (Will 2014a). These are post-Newtonian terms in the equation of motion of a given body that represent a coupling between the potential of the central black hole and the potential due to other stars in the system. For hierarchical triple systems, these are couplings between the potential of the inner binary and that of the distant third body. We wrote down the post-Newtonian N-body equations of motion including a central black hole in a truncated form that included all the relevant cross terms, in a format ready to use for numerical implementation. We did the same for hierarchical triple systems, and illustratd explicitly the effects of cross terms on the orbit-averaged equations of evolution for the orbit elements of the inner binary for the special case where the third body is on a circular orbit. We showed that including such cross-terms in the dynamics of a hierarchical triple system is essential in obtaining an orbit averaged energy that is manifestly conserved over a pericenter precession timescale (Will 2014b).

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.

Stellar dynamics of extreme-mass-ratio inspirals

We presented results from direct integration of the post-Newtonian N-body equations of motion describing dense clusters of compact stars around Schwarzschild massive black holes (MBH) (Merritt, Alexander, Mikkola and Will, 2011). The simulations embodied an essentially exact (at the post-Newtonian level) treatment of the interplay between stellar dynamical relaxation, relativistic precession, and gravitational-wave energy loss. The rate of capture of stars by the MBH was found to be greatly reduced by relativistic precession, which limits the ability of torques from the stellar potential to change orbital angular momenta. Penetration of this ‘‘Schwarzschild barrier’’ does occasionally occur, resulting in capture of stars onto orbits that gradually inspiral due to gravitational wave emission. We derived an approximate formula for the capture rate, which predicts that captures would be strongly disfavored from orbits with semi-major axes below a certain value; this prediction, as well as the predicted rate, were verified in the N-body integrations. We discussed the implications of the results for the detection of extreme-mass-ratio inspirals from galactic
nuclei with a range of physical properties.

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.


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