S-stars

These equations provide differential equations for the four space-time components t(s), r(s), θ(s), φ(s), where s is an affine parameter (the proper time of the star, in our case), that can be numerically integrated once the initial conditions are specified. Initial conditions t(0), r(0), θ(0), φ(0) and numerical values of the parameters are set to integrate numerically the aforementioned equations. A convenient way to place the initial conditions is to assume that θ(0)= π/2 (the star initially lies on the equatorial plane of the reference system) and that its first and second derivatives are equally set to zero (i.e. its initial velocity is initially parallel to the equatorial plane). Initial conditions for r and φ, on the other hand, can be inferred from the orbital elements describing the Keplerian orbit of the star: the semi-major axis a, the eccentricity e, the inclination i, the angle of the line of node Ω, the angle from ascending node to pericentre ω, the orbital period T and the time of the pericentre passage tp. These orbital parameters are time-dependent. Thus, they should be interpreted as the ones of the osculating ellipse at the initial time. 

In particular, we set for simplicity the initial conditions at the time of passage at apocentre, where velocity is zero. Therefore, both the true anomaly φ and the eccentric anomaly Φ are equal to π. Thus, the Cartesian coordinates of the star, expressed in the orbital plane, at the initial time are:

Using these quantities, we retrieve the initial radial and angular coordinates on the orbital plane and the respective velocities. Finally, the initial condition for the first derivative of the time descends from the normalization of the metric to the speed of light. Once orbits are integrated, a projection in the reference frame of a distant observer is needed. Such a projection can be performed by means of the Thiele-Innes relations to obtain  positions and velocities.

A few effects (both relativistic and classical) need to be taken into account to compare the predicted positions of S2 in the reference frame of a distant observer, (x, y, z), and their velocities in it,  with the  observational data.

• The Rømer delay

• Frequency shift

where