**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 (ℓ̃ℓ_{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.;