A Precise Determination of Angular Momentum in the Black Hole Candidate Gro J1655-40

We note that the recently discovered 450 Hz frequency in the X-ray flux of the black hole candidate GRO J1655-40 is in a 3:2 ratio to the previously known 300 Hz frequency of quasi-periodic oscillations (QPO) in the same source. If the origin of high frequency QPOs in black hole systems is a resonance between orbital and epicyclic motion of accreting matter, as suggested previously, the angular momentum of the black hole can be accurately determined, given its mass. We find that the dimensionless angular momentum is in the range 0.2 < j < 0.65 if the mass is in the (corresponding) range of 5.5 to 7.9 solar masses. We have previously suggested [1] that “twin” kHz QPOs in accreting neutron stars arise as a result of non-linear 1:2 or 1:3 resonance between the radial epicyclic motion and the orbital motion of matter in a nearly keplerian accretion disk, and we have noted that the same phenomenon should also arise in accreting black holes, where only a single high frequency had been observed. Strohmayer now reports [2] the discovery of a second QPO in GRO J1655-40, a well known black hole candidate in a low-mass X-ray binary, with the mass of the compact X-ray source determined from optical studies to be in the range 5.9 < M/M⊙ < 7.9 [3]. There are time intervals, when both QPOs are present at the same time [2]. The two QPOs now known in the source occur at frequencies 300 Hz and 450 Hz, i.e., in a 2:3 ratio, strongly supporting the notion of a resonance in the system. Of all rational ratios only 1:2 and 1:3 resonances are capable of giving a 2:3 ratio of frequencies. Specifically, if the lower frequency in the resonance is ω, and the higher frequency Ω, the only two possibilities are that Ω = 300Hz and Ω + ω = 450 Hz for the 1:2 resonance, or that Ω = 450Hz and Ω− ω = 300 Hz for the 1:3 resonance. In the spirit of ref. [1], we consider resonances between orbital and epicyclic motion in the Kerr metric. For the given mass range of the black hole, and for the frequencies of 300 Hz and 450 Hz, we can exclude that orbital frequency is in resonance with the epicyclic motion in “vertical” (meridional) direction. Only resonance with radial epicyclic motion can give the observed frequencies. So in the case of GRO J1655-40, ω is the radial epicyclic frequency and Ω is the orbital frequency for circular equatorial orbits in the Kerr metric. The formulae for the frequencies can be found e.g., in the review by Kato [4]. We find the following angular momentum range: 0.2 < j < 0.6 for the 1:2 resonance, and 0.3 < j < 0.65 for the 1:3 resonance. Here j = JG/(cM) = a is the Kerr parameter of the black hole, i.e., its dimensionless angular momentum.