On post-Newtonian orbits and the Galactic-center stars

Abstract

Stars near the Galactic center reach a few percent of light speed during pericenter passage, which makes post-Newtonian effects potentially detectable. We formulate the orbit equations in Hamiltonian form such that the $O(v^2/c^2)$ and $O(v^3/c^3)$ post-Newtonian effects of the Kerr metric appear as a simple generalization of the Kepler problem. A related perturbative Hamiltonian applies to photon paths. We then derive a symplectic integrator with adaptive time-steps, for fast and accurate numerical calculation of post-Newtonian effects. Using this integrator, we explore relativistic effects. Taking the star S2 as an example, we find that general relativity would contribute tenths of mas in astrometry and tens of $\mathrm{km}{\mathrm{s}}^{-1}$ in kinematics. (For eventual comparison with observations, redshift and time-delay contributions from the gravitational field on light paths will need to be calculated, but we do attempt these in the present paper.) The contribution from stars, gas, and dark matter in the Galactic center region is still poorly constrained observationally, but current models suggest that the resulting Newtonian perturbation on the orbits could plausibly be of the same order as the relativistic effects for stars with semi-major axes $\gtrsim 0.01$ pc (or 250 mas). Nevertheless, the known and distinctive {\it time dependence} of the relativistic perturbations may make it possible to disentangle and extract both effects from observations.

25 pages, 6 figures, accepted for publication in Astrophysical Journal. Section 3 (on symplectic integrators) rewritten with more details, new section 5 on comparison with other post-Newtonian formulations, new figure 6, corrected typos, added references

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• Investigating the Relativistic Motion of the Stars Near the Supermassive Black Hole in the Galactic Center
• Perturbed stellar motions around the rotating black hole in Sgr A* for a generic orientation of its spin axis
• Scalar field effects on the orbit of S2 star
• Construction of Explicit Symplectic Integrators in General Relativity. I. Schwarzschild Black Holes
• Construction of explicit symplectic integrators in general relativity. II. Reissner-Nordstrom black holes
• Construction of explicit symplectic integrators in general relativity. IV. Kerr black holes
• On Testing the Kerr Metric of the Massive Black Hole in the Galactic Center via Stellar Orbital Motion: Full General Relativistic Treatment
• Construction of explicit symplectic integrators in general relativity. III. Reissner-Nordstrom-(anti)-de Sitter black holes
• Long-term classical and general relativistic effects on the radial velocities of the stars orbiting Sgr A*
• Towards relativistic orbit fitting of Galactic center stars and pulsars
• Relativistic redshift effects and the Galactic-center stars
• The Klein-Gordon-Fock equation in the curved spacetime of the Kerr-Newman (anti) de Sitter black hole
• Gravitational redshift/blueshift of light emitted by geodesic test particles, frame-dragging and pericentre-shift effects, in the Kerr-Newman-de Sitter and Kerr-Newman black hole geometries
• Applying explicit symplectic integrator to study chaos of charged particles around magnetized Kerr black hole
• A Note on the Construction of Explicit Symplectic Integrators for Schwarzschild Spacetimes
• The Galactic Center as a laboratory for theories of gravity and dark matter
• Galactic-center S-Stars as a prospective test of the Einstein Equivalence Principle
• Clocks around Sgr A*
• Post-Keplerian effects on radial velocity in binary systems and the possibility of measuring General Relativity with the S2 star in 2018
• Explicit symplectic methods in black hole spacetimes
• On the Newtonian and Spin-induced Perturbations felt by the Stars Orbiting around the Massive Black Hole in the Galactic Center
• Symplectic integrators for spin systems
• Electromagnetic field and chaotic charged-particle motion around hairy black holes in Horndeski gravity
• Dynamics of charged particles moving around Kerr black hole with inductive charge and external magnetic field
• The short-period S-stars S4711, S62, S4714 and the Lense-Thirring effect due to the spin of Sgr A${}^{\ast }$
• Charged particle motions near non-Schwarzschild black holes with external magnetic fields in modified theories of gravity
• Influence of bulk mass distribution on orbital precession of S2 star in Yukawa gravity
• Estimating the parameters of Hybrid Palatini gravity model with the Schwarzschild precession of S2, S38 and S55 stars: case of bulk mass distribution
• PPN motion of the S-stars around Sgr A*
• Energy-conserving integrator for conservative Hamiltonian systems with ten-dimensional phase space
• Explicit symplectic integrators with adaptive time steps in curved spacetimes
• Impact of electric charges on chaos in magnetized $Reissner-Nordstr\ddot{o}m$ spacetimes
• Relativistic effects in orbital motion of the S-stars at the Galactic Center
• Is it possible to measure the Lense-Thirring orbital shifts of the short-period S-star S4716 orbiting Sgr A${}^{\ast }$?
• Adjustment of force-gradient operator in symplectic methods
• An analytic solution for weak-field Schwarzschild geodesics
• The detection of relativistic corrections in cosmological N-body simulations
• Construction of second-order six-dimensional Hamiltonian-conserving scheme
• Testing GR with Galactic-centre Stars