# On post-Newtonian orbits and the Galactic-center stars

## Miguel Preto(1), Prasenjit Saha(2)

(1) Astronomisches Rechen-Institut, Zentrum für Astronomie, University of Heidelberg, D-69120 Heidelberg, Germany

(2) Institute for Theoretical Physics, University of Zürich, Winterthurerstrasse 190, CH-8057 Zürich, Switzerland

### Paper: ApJ, Oct 2009, in press

**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 km 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 > 0.01 pc (or
250 mas). Nevertheless, the known and distinctive time
dependence of the relativistic perturbations may make it possible
to disentangle and extract both effects from observations.

Preprints available from the authors at miguelp@ari.uni-heidelberg.de
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