Measuring mass and angular momentum of black holes with high-frequency quasi-periodic oscillations

For the three microquasars GRO J1655-40, XTE J1550-564 and GRS 1915+105 twin high frequency quasi-periodic oscillations (HFQPOs) with a ratio of 3:2 and/or 3:1 have been measured. For a test particle orbiting a rotating black hole on a stable circular orbit there exist two different orbits at which the vertical and radial epicyclic oscillations are in either a 3:1 or 3:2 parametric resonance for any choice of the black hole angular momentum a. If the two orbits are required to be frequency commensurable Keplerian orbits there is only one solution for the two orbit radii and a. This model predicts that the microquasars have the same a, and it predicts their black hole masses on the basis of the measured HFQPOs in agreement with the dynamically determined masses. Application of this model to the Galactic Center black hole Sgr A* using the recently measured QPOs (Genzel et al. [CITE]; Aschenbach et al. [CITE]) leads to a black hole mass of $(3.28 \pm 0.13)\times10\sp 6 ~M\sb{\odot}$, and the same a as for the microquasars. The possibility that all four sources have $a=0.99616$ suggests that this value is the upper limit of a imposed by general relativity. The same value for the lower orbit radius and the same value for a are also suggested by an analysis of the general relativistic expression for the radial gradient of the orbital velocity, which changes sign in a narrow annular region around the lower orbit when $a>0.9953$