General relativistic effects on the orbit of the S2 star with GRAVITY

Context. The first observations of the GRAVITY instrument obtained in 2016, have shown that it should become possible to probe the spacetime close to the supermassive black hole Sagittarius A* (Sgr A*) at the Galactic center by using accurate astrometric positions of the S2 star. Aims. The goal of this paper is to investigate the detection by GRAVITY of different relativistic effects a ff ecting the astrometric and /or spectroscopic observations of S2 such as the transverse Doppler shift, the gravitational redshift, the pericenter advance and higher-order general relativistic (GR) e ff ects, in particular the Lense-Thirring effect due to the angular momentum of the black hole. Methods. We implement seven stellar-orbit models to simulate both astrometric and spectroscopic observations of S2 beginning near its next pericenter passage in 2018. Each model takes into account a certain number of relativistic effects. The most accurate one is a fully GR model and is used to generate the mock observations of the star. For each of the six other models, we determine the minimal observation times above which it fails to fit the observations, showing the effects that should be detected. These threshold times are obtained for di ff erent astrometric accuracies as well as for di ff erent spectroscopic errors. Results. Transverse Doppler shift and gravitational redshift can be detected within a few months by using S2 observations obtained with pairs of accuracies (sigma(A); sigma(V)) = (10 100 mu as, 1 10 km s(-1)) where sigma(A) and sigma(V) are the astrometric and spectroscopic accuracies, respectively. Gravitational lensing can be detected within a few years with (sigma(A); sigma(V)) = (10 mu as, 10 km s 1). Pericenter advance should be detected within a few years with (sigma(A); sigma(V)) = (10 mu as, 1 10 km s(-1)). Cumulative high-order photon curvature contributions, including the Shapiro time delay, a ff ecting spectroscopic measurements can be observed within a few months with (sigma(A); sigma(V)) = (10 mu as, 1 km s 1). By using a stellar-orbit model neglecting relativistic effects on the photon path except the major contribution of gravitational lensing, S2 observations obtained with accuracies (sigma(A); sigma(V)) = (10 mu as, 10 km s(-1)), and a black hole angular momentum (a; i'; Omega') = (0 : 99; 45 degrees; 160 degrees), the 1 sigma error on the spin parameter a is of about 0.4, 0.2, and 0.1 for a total observing run of 16, 30, and 47 yr, respectively. The 1 sigma errors on the direction of the angular momentum reach sigma(i') 25 degrees and sigma(Omega') approximate to 40 degrees when considering the three orbital periods run. We found that the uncertainties obtained with a less spinning black hole (a = 0 : 7) are similar to those evaluated with a = 0 : 99. Conclusions. The combination of S2 observations obtained with the GRAVITY instrument and the spectrograph SINFONI (Spectrograph for INtegral Field Observations in the Near Infrared) also installed at the VLT (Very Large Telescope) will lead to the detection of various relativistic e ff ects. Such detections will be possible with S2 monitorings obtained within a few months or years, depending on the e ff ect. Strong constraints on the angular momentum of Sgr A* (e. g., at 1 sigma = 0 : 1) with the S2 star will be possible with a simple stellar-orbit model without using a ray-tracing code but with approximating the gravitational lensing e ff ect. However, long monitorings are necessary, and we thus must rely on the discovery of closer-in stars near Sgr A* if we want to e ffi ciently constrain the black hole parameters with stellar orbits in a short time, or monitor the flares if they orbit around the black hole.