  The S-star cluster in the Galactic center allows us to study the physics
close to a supermassive black hole, including distinctive dynamical tests of
general relativity. Our best estimates for the mass of and the distance to Sgr
A* using the three stars with the shortest period (S2, S38, and S55/S0-102) and
Newtonian models are M_BH = (4.15+- 0.13 +- 0.57) x 10^6 M_sun and R_0 = 8.19
+- 0.11 +- 0.34 kpc. Additionally, we aim at a new and practical method to
investigate the relativistic orbits of stars in the gravitational field near
Sgr A*. We use a first-order post- Newtonian approximation to calculate the
stellar orbits with a broad range of periapse distance r_p. We present a method
that employs the changes in orbital elements derived from elliptical fits to
different sections of the orbit. These changes are correlated with the
relativistic parameter defined as {\Upsilon} = r_s/r_p (with r_s being the
Schwarzschild radius) and can be used to derive {\Upsilon} from observational
data. For S2 we find a value of {\Upsilon} = 0.00088 +- 0.00080, which is
consistent, within the uncertainty, with the expected value of {\Upsilon} =
0.00065 derived from MBH and the orbit of S2. We argue that the derived
quantity is unlikely to be dominated by perturbing influences such as noise on
the derived stellar positions, field rotation, and drifts in black hole mass.
