**Publication date:** Mar 2007

**Abstract:**

Context: Change of sign of the LNRF-velocity gradient has been found for accretion discs orbiting rapidly rotating Kerr black holes with spin a>0.9953 for Keplerian discs and a>0.99979 for marginally stable thick discs. Such a „humpy“ LNRF-velocity profiles occur just above the marginally stable circular geodesic of the black hole spacetimes.

Aims: Aschenbach (2004) has identified the maximal rate of change of the orbital velocity within the „humpy“ profile with a locally defined critical frequency of disc oscillations, but it has been done in a coordinate-dependent form that should be corrected.

Methods: We define the critical „humpy“ frequency ν_{h} in general relativistic, coordinate independent form, and relate the frequency defined in the LNRF to the distant observers. At radius of its definition, the resulting „humpy“ frequency ν_{h} is compared to the radial ν_{r} and vertical ν_{v} epicyclic frequencies and the orbital frequency of the discs. We focus our attention to Keplerian thin discs and perfect-fluid slender tori where the approximation of oscillations with epicyclic frequencies is acceptable.

Results: In the case of Keplerian discs, we show that the epicyclic resonance radii r_{3{:}1} and r_{4{:}1} (with ν_v{:}ν_r=3{:}1, 4{:}1) are located in vicinity of the „humpy“ radius r_{h} where efficient triggering of oscillations with frequencies ν_{h} could be expected. Asymptotically (for 1-a<10^{-4}) the ratio of the epicyclic and Keplerian frequencies and the humpy frequency is nearly constant, i.e., almost independent of a, being for the radial epicyclic frequency ν_r{:}ν_{h} ∼ 3{:}2. In the case of thick discs, the situation is more complex due to dependence on distribution of the specific angular momentum ℓ determining the disc properties. For ℓ=const. tori and 1-a<10^{-6} the frequency ratios of the humpy frequency and the orbital and epicyclic frequencies are again nearly constant and independent of both a and ℓ being for the radial epicyclic frequency ν_r{:}ν_{h} close to 4. In the limiting case of very slender tori (ℓ∼ℓ_{ms}) the epicyclic resonance radius r_{4{:}1}∼ r_{h} for all the relevant interval of 1-a<2× 10^{-4}.

Conclusions: .The hypothetical „humpy“ oscillations could be related to the QPO resonant phenomena between the epicyclic oscillations in both the thin discs and marginally stable tori giving interesting predictions that have to be compared with QPO observations in nearly extreme Kerr black hole candidate systems. Generally, more than two observable oscillations are predicted.

**Authors:**

Stuchlík, Z.; Slaný, P.; Török, G.;