arXiv:0704.1252v2  [astro-ph]  23 May 2007
Astronomy & Astrophysics manuscript no. grs1915-corrected c© ESO 2018
October 30, 2018
LNRF-velocity hump-induced oscillations of a Keplerian disc
orbiting near-extreme Kerr black hole: A possible explanation of
high-frequency QPOs in GRS 1915+105
Zdeněk Stuchlı́k, Petr Slaný, and Gabriel Török
Institute of Physics, Faculty of Philosophy and Science, Silesian University in Opava, Bezručovo nám. 13,
CZ-74601 Opava, Czech Republic
Received / Accepted
ABSTRACT
Context. At least four high-frequency quasiperiodic oscillations (QPOs) at frequencies 41 Hz, 67 Hz, 113 Hz, and 167 Hz were re-
ported in a binary system GRS 1915+105 hosting near-extreme Kerr black hole with a dimensionless spin a > 0.98.
Aims. We attempt to explain all four observed frequencies by an extension of the standard resonant model of epicyclic oscillations.
Methods. We use the idea of oscillations induced by the hump of the orbital velocity profile (related to locally non-rotating frames–
LNRF) in discs orbiting near-extreme Kerr black holes, which are characterized by a “humpy frequency” νh, that could excite the
radial and vertical epicyclic oscillations with frequencies νr, νv. Due to non-linear resonant phenomena, the combinational frequen-
cies are allowed as well.
Results. Assuming mass M = 14.8 M⊙ and spin a = 0.9998 for the GRS 1915+105 Kerr black hole, the model predicts frequencies
νh = 41 Hz, νr = 67 Hz, νh + νr = 108 Hz, and νv − νr = 170 Hz corresponding quite well to the observed ones.
Conclusions. For black-hole parameters being in good agreement with those given observationally, the forced resonant phenom-
ena in non-linear oscillations, excited by the ”hump-induced” oscillations in a Keplerian disc, can explain high-frequency QPOs in
near-extreme Kerr black-hole binary system GRS 1915+105 within the range of observational errors.
Key words. Accretion, accretion discs – Black hole physics – Methods: data analysis
1. Introduction
Detailed analysis of the variable X-ray black-hole binary system
(microquasar) GRS 1915+105 reveals high-frequency QPOs ap-
pearing at four frequencies, namely ν1 = (41 ± 1) Hz, ν2 =
(67 ± 1) Hz (Morgan et al. 1997; Strohmayer 2001), and ν3 =
(113±5) Hz, ν4 = (167±5) Hz (Remillard 2004). In this range of
its errors, both pairs are close to the frequency ratio 3:2 suggest-
ing the possible existence of resonant phenomena in the system.
Observations of oscillations with these frequencies have differ-
ent qualities, but in all four cases the data are quite convincing;
see (McClintock & Remillard 2004; Remillard & McClintock
2006).
Several models have been developed to explain the kHz QPO
frequencies, and it is usually preferred that these oscillations are
related to the orbital motion near the inner edge of an accretion
disc. In particular, two ideas based on the strong-gravity prop-
erties have been proposed. While Stella & Vietri (1998, 1999)
introduced the “Relativistic Precession Model” considering that
the kHz QPOs directly manifest the modes of a slightly per-
turbed (and therefore epicyclic) relativistic motion of blobs in
the inner parts of the accretion disc, Kluźniak & Abramowicz
(2001) propose models based on non-linear oscillations of an
accretion disc that assume resonant interaction between orbital
and/or epicyclic modes. In a different context, the possibility of
resonant coupling between the epicyclic modes of motion in the
Send offprint requests to: Z. Stuchlı́k,
e-mail: zdenek.stuchlik@fpf.slu.cz
Kerr spacetime was also mentioned in the early work of Aliev &
Galtsov (1981).
In the case of near-extreme Kerr black holes, it was sug-
gested that the epicyclic oscillations in the disc could be excited
by resonances with the so-called “hump-induced” oscillations,
see papers of Aschenbach (2004, 2006) and Stuchlı́k et al. (2004,
2007). This idea was proposed so as to extend standard orbital
(resonant) models meant to explain high-frequency QPOs ob-
served in black-hole sources.
Recently, careful and detailed analysis of the spectral contin-
uum from GRS 1915+105 has put a strong limit on the black-
hole spin,1 namely 0.98 < a < 1 (McClintock et al. 2006),
indicating the presence of near-extreme Kerr black hole whose
mass has been restricted observationally to M = (14.0±4.4) M⊙,
see (McClintock & Remillard 2004; Remillard & McClintock
2006). Therefore, the microquasar GRS 1915+105 seems to be
an appropriate candidate to test the extended resonant model
with hump-induced oscillations.2
The idea of hump-induced oscillations and their possible res-
onant coupling with the epicyclic ones is briefly discussed in
Sect. 2. The related resonant model, assuming the excitation of
epicyclic oscillations by the hump-induced oscillations through
1 Units c = G = M = 1 (M is the total mass of the Kerr black
hole) and the Boyer-Lindquist (B-L) coordinates (t, r, θ, ϕ) are used
hereafter.
2 However, Middleton et al. (2006) refer to a substantially lower, in-
termediate value of black-hole spin, a ∼ 0.7, to which the model of
hump-induced oscillations cannot be applied.
http://arxiv.org/abs/0704.1252v2
2 Zdeněk Stuchlı́k et al.: LNRF-velocity hump induced oscillations in GRS 1915+105
non-linear resonant phenomena, is applied to GRS 1915+105 in
Sect. 3, concluding remarks are presented in Sect. 4.
2. Hump-induced and epicyclic oscillations in
Keplerian discs and possible resonant coupling
In order to describe the local processes in an accretion disc,
it is necessary to choose a local observer (characterized by its
reference frame). In general relativity there is no preferred ob-
server. On the other hand, if we want to study processes related
to the orbital motion of matter in the disc, it is reasonable to
choose the observers with zero angular momentum, so-called
ZAMOs, as their reference frames do not rotate with respect to
the spacetime, and thus ZAMOs should reveal local kinematic
properties of the disc in the clearest way. (In rotating–stationary,
axisymetric–spacetimes, they are dragged along with the space-
time.) In the Kerr spacetime, ZAMOs are represented by locally
non-rotating frames (LNRF); see Bardeen et al. (1972). Notice
that in the Schwarzschild spacetime, LNRF correspond to the
static observer frames.
Aschenbach (2004) finds that for near-extreme Kerr black
holes with the spin a > 0.9953, the test-particle orbital veloc-
ity V(ϕ) related to LNRF reveals a hump in the equatorial plane
(θ = π/2). This non-monotonicity is located in a small region
inside the ergosphere of the black-hole spacetime close to, but
above, the marginally stable orbit.3 Therefore, it can be relevant
for thin accretion discs around near-extreme Kerr black holes,
as the inner edge of the disc can extend down to the innermost
stable circular orbit (ISCO).
Moreover, Stuchlı́k et al. (2005) shows that for a > 0.99979
the similarly humpy behavior of the orbital velocity in LNRF
also takes place for the non-geodesic motion of test perfect fluid
in marginally stable barotropic tori characterized by the uni-
form distribution of the specific angular momentum, ℓ(r, θ) ≡
−Uϕ/Ut = const, where the motion of fluid elements is given
by the 4-velocity field Uµ = (U t(r, θ), 0, 0, Uϕ(r, θ)). Outside
the equatorial plane, the non-monotonic behavior of V(ϕ) in
marginally stable tori is represented by the topology change of
the cylindrical equivelocity surfaces in the region of the hump,
because the toroidal equivelocity surfaces centered around the
circle corresponding to the local minimum of V(ϕ) in the equa-
torial plane exist for a > 0.99979 (Stuchlı́k et al. 2005). This
suggests a generation of possible instabilities in radial and ver-
tical directions; see Stuchlı́k et al. (2004). In the following, we
restrict our attention to the case of Keplerian discs.
Heuristic connection between the positive part of the veloc-
ity gradient, ∂V(ϕ)/∂r, and the excitation of epicyclic oscilla-
tions in Keplerian discs was suggested by Aschenbach (2004,
2006), who defined the characteristic frequency of oscillations,
induced by the humpy profile of V(ϕ), by the maximum posi-
tive slope of the orbital velocity in terms of the coordinate ra-
dius, νcrit ≡ (∂V(ϕ)/∂r)max. This coordinate-dependent defini-
tion was corrected in Stuchlı́k et al. (2007), where the proper
radial distance dr̃ =
grr dr rather than the coordinate dis-
tance dr was used to define the characteristic (critical) frequency
νr̃crit ≡ (∂V
(ϕ)/∂r̃)max. Such a locally defined critical frequency
was further related to a stationary observer at infinity, obtaining
the so-called “humpy frequency”
−(gtt + 2ωgtϕ + ω2gϕϕ)r=rh ν
3 We stress that the Aschenbach effect is frame-dependent, as it is
related to LNRF, but recall the arguments for relevance of the LNRF
point of view at the beginning of the section.
0 0.001 0.002 0.003 0.004
0 0.001 0.002 0.003 0.004
1 − a
a = (0.9998 ± 0.0001)
✉ GRS 1915+105
✛ 607
✛ 1.29
Fig. 1. Spin-dependence of the humpy frequency νh and the humpy ra-
dius rh. For completeness, the B-L radius of the innermost stable circu-
lar geodesic (ISCO) is plotted.
(rh − 2) −
rh(r2h + a
2) + 2a2
r5h + a
4(3rh + 2) − 2a3r
h (3rh + 1)
2∆3/2h
h + a)
2a2r2h(2rh − 5) − 2ar
h (5rh − 9)
2∆3/2h
h + a)
, (1)
where gµν are the metric coefficients of the Kerr geometry and
ω = −gtϕ/gϕϕ is the angular velocity of the LNRF; see, e.g.,
Bardeen et al. (1972); ∆h = r
h − 2rh + a
2. The analytic formula
is given for the equatorial plane (θ = π/2). The B-L radius rh
where the positive gradient of the velocity profile in terms of the
proper radial distance reaches its maximum, so-called “humpy
radius”, is given by the condition
∂V(ϕ)
= 0 (2)
leading to the equation
3a7(r + 2) + a6
r(21r2 + 18r − 4) − a5r(33r2 + 10r + 20)
r(45r3 − 62r2 − 68r + 16) − a3r3(83r2 − 122r − 60)
+a2r4
r(27r2 − 130r + 136) − 9ar5(7r2 − 26r + 24)
r(3r − 2) = 0, (3)
which must be solved numerically. The spin dependence of the
humpy radius and the related humpy frequency is illustrated in
Fig. 1. The humpy radius rh falls monotonically with increas-
ing spin a, while the humpy frequency νh has a maximum for
Zdeněk Stuchlı́k et al.: LNRF-velocity hump induced oscillations in GRS 1915+105 3
a = 0.9998, where νh (max) = 607 (M⊙/M) Hz, and it tends to
νh (a→1) = 588 (M⊙/M) Hz.
When particles following a Keplerian circular orbit are per-
turbed, they begin to follow, in the first approximation, an
epicyclic motion around the equilibrium Keplerian orbit, gen-
erally characterized by the frequencies of the radial and vertical
epicyclic oscillations νr, νv (Aliev & Galtsov 1981; Nowak &
Lehr 1998):
r = ν
K(1 − 6r
+ 8ar−3/2 − 3a2r−2), (4)
v ≡ ν
θ = ν
K(1 − 4ar
+ 3a2r−2), (5)
where νK is the Keplerian orbital frequency
2π(r3/2 + a)
. (6)
The ratios of the humpy frequency and the epicyclic frequencies
at the humpy radius were determined in Stuchlı́k et al. (2007)
revealing almost spin-independent asymptotic behavior for a →
1 represented closely by the ratios of integer numbers, νv : νr :
νh ∼ 11 : 3 : 2, which imply a possibility of resonant phenomena
between the hump-induced and epicyclic oscillations predicted
by Aschenbach (2004). The ratios of the epicyclic frequencies
and the humpy frequency are given in the dependence on the
black-hole spin in Fig. 2.
3. Application of the hump-induced resonance
model to high-frequency QPOs in GRS 1915+105
Primarily concentrating on the lower pair of frequencies, we as-
sume that the lowest frequency is directly the humpy frequency,
νh ≡ ν1 = (41 ± 1) Hz, (7)
while the second lowest frequency corresponds directly to the
radial epicyclic frequency at the same radius rh,
νr ≡ ν2 = (67 ± 1) Hz. (8)
These frequencies are close to a 3:2 ratio, therefore the forced
non-linear resonance can be relevant in such a situation. The
ratio of νr/νh = (1.63 ± 0.06) gives the black hole spin a =
(0.9998 ± 0.0001) (the uncertainty of the spin is implied by un-
certainties of the lower pair of frequencies being ∼ 1 Hz); see
Fig. 2. Notice that this spin corresponds to the maximal pos-
sible value of the humpy frequency νh (max) (Fig. 1). Since the
humpy frequency is 1/M-scaled, the absolute value of νh im-
plies the black hole mass M = (14.8±0.4) M⊙. The correspond-
ing humpy radius is rh = 1.29
+0.01
−0.02 (Fig. 1). At such a radius,
the vertical epicyclic frequency of a particle orbiting the Kerr
black hole with the mass and spin inferred above reaches the
value νv = (0.23 ± 0.01) kHz. Then the upper pair of observed
frequencies can be explained, within the range of observational
errors ±5 Hz, by combinational frequencies at the humpy radius
rh in the following way:
ν3 ∼ (νr + νh) = (108 ± 2) Hz (9)
ν4 ∼ (νv − νr) = (0.17 ± 0.01) kHz. (10)
4. Conclusions
The idea of epicyclic oscillations induced by the LNRF-velocity
hump in the region where the positive part of the velocity gra-
dient reaches its maximum is able to address all four high-
frequency QPOs observed in the X-ray source GRS 1915+105.
-8 -7 -6 -5 -4 -3
log(1 − a)
a = (0.9998 ± 0.0001) ✲
GRS 1915+105 ✲
✻νr :νh
(νv − νr) :(νh + νr)
νv :νr
νv :νh
∼ 3:2
∼ 7:2
∼ 6:1
Fig. 2. Spin dependence of frequency ratios including the radial (νr)
and vertical (νv) epicyclic frequencies, and the humpy frequency (νh)
evaluated at the same radius rh where the humpy frequency is defined.
The range of the spin relevant for GRS 1915+105 Kerr black hole is
shaded. For the mean value a = 0.9998, the frequency ratios are close
to the ratios of integer numbers, suggesting a possibility of resonances
between hump-induced and epicyclic oscillations in GRS 1915+105.
The model implies a near-extreme spin of the central black hole
(a ∼ 0.9998), which agrees well with results from the spec-
tral continuum fits, and the black-hole mass M ∼ 14.8 M⊙ be-
ing well inside the interval given by other observational meth-
ods. Note that the orbital resonance model of Kluźniak &
Abramowicz, assuming the parametric resonance between the
vertical and radial epicyclic oscillations in frequency ratio 3:2
represented by the upper pair of observed frequencies, also gives
the spin a > 0.99 but for M ≃ 18M⊙ (Török et al. 2005). On the
other hand, the “Relativistic Precession Model” gives a substan-
tially lower value for the spin: a ∼ 0.3 (Stella et al. 1999).
In the presented model, we assume that all four observed
frequencies arise due to forced non-linear oscillations of the
Keplerian disc at the same radius rh, excited by the hump-
induced oscillations characterized by the humpy frequency νh.
The black-hole parameters a, M are fixed by the requirement
that the lower pair of observed frequencies is identified with the
humpy frequency and the radial epicyclic frequency, ν1 ≡ νh,
ν2 ≡ νr. Assuming non-linear resonant phenomena enabling
the existence of combinational frequencies and the possibility
of observing them, the upper pair of observed frequencies can
be explained as the combinational ones of the humpy frequency
and both epicyclic frequencies, ν3 ∼ (νr + νh), ν4 ∼ (νv − νr).
Moreover, both frequency ratios νr : νh, and (νv − νr) : (νr + νh)
are close to 3:2 ratio (Fig. 2), in which the resonant phenomena
can be strong enough. On the other hand, as 4νh = (164± 4) Hz,
which is also close to the uppermost frequency, there is an-
other possibility of explaining ν4 through a sub-harmonic reso-
nance forced by the humpy oscillations as well. Finally, note that
Strohmayer (2001) also reports another relatively weak QPO at
frequency of (56 ± 2) Hz. If this is the case (which, according
to our knowledge, has not been confirmed by other observations
yet), it could be related to the second harmonic of the combina-
tional frequency4 (νr − νh) = (26 ± 2) Hz.
4 Combinational frequency (νr − νh) corresponds to the same order of
nonlinearity as (νr + νh).
Note added in the manuscript: After the paper was accepted we ob-
tained an information that a weak QPO at frequency 27 Hz is referenced
in Belloni et al. (2001).
4 Zdeněk Stuchlı́k et al.: LNRF-velocity hump induced oscillations in GRS 1915+105
Generally, other harmonics and combinational frequencies
may occur in a non-linear oscillating system corresponding to
higher approximations, when the equation of motion describing
the non-linear oscillations is solved by the method of successive
approximations. The statement by Landau & Lifshitz (1976) that
“As the degree of approximation increases, however, the strength
of the resonances, and the widths of the frequency ranges in
which they occur, decrease so rapidly that in practice only the
resonances at frequencies5 ν ≈ pν0/q with small p and q can be
observed” can explain why a QPO near the frequency 237 Hz,
corresponding to the vertical epicyclic frequency νv at the same
radius rh as the previously mentioned humpy and radial epicyclic
frequencies νh, νr, is not directly observed, despite the commen-
surability of these frequencies represented by the frequency ra-
tios νv : νh ∼ 6 : 1 and νv : νr ∼ 7 : 2 (Fig. 2).
Acknowledgements. The authors are supported by the Czech grant
MSM 4781305903.
References
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5 p, q are integers.

  At least four high-frequency quasiperiodic oscillations (QPOs) at frequencies
41Hz, 67Hz, 113Hz, and 167Hz were reported in a binary system GRS 1915+105
hosting near-extreme Kerr black hole with a dimensionless spin a>0.98. We use
the idea of oscillations induced by the hump of the orbital velocity profile
(related to locally non-rotating frames - LNRF) in discs orbiting near-extreme
Kerr black holes, which are characterized by a "humpy frequency" f_h, that
could excite the radial and vertical epicyclic oscillations with frequencies
f_r, f_v. Due to non-linear resonant phenomena the combinational frequencies
are allowed as well. Assuming mass M=14.8M_sun and spin a=0.9998 for the GRS
1915+105 Kerr black hole, the model predicts frequencies f_h=41Hz, f_r=67Hz,
(f_h+f_r)=108Hz, (f_v-f_r)=170Hz corresponding quite well to the observed ones.
For black-hole parameters being in good agreement with those given
observationally, the forced resonant phenomena in non-linear oscillations,
excited by the "hump-induced" oscillations in a Keplerian disc, can explain
high-frequency QPOs in GRS 1915+105 within the range of observational errors.

Introduction
Detailed analysis of the variable X-ray black-hole binary system
(microquasar) GRS 1915+105 reveals high-frequency QPOs ap-
pearing at four frequencies, namely ν1 = (41 ± 1) Hz, ν2 =
(67 ± 1) Hz (Morgan et al. 1997; Strohmayer 2001), and ν3 =
(113±5) Hz, ν4 = (167±5) Hz (Remillard 2004). In this range of
its errors, both pairs are close to the frequency ratio 3:2 suggest-
ing the possible existence of resonant phenomena in the system.
Observations of oscillations with these frequencies have differ-
ent qualities, but in all four cases the data are quite convincing;
see (McClintock & Remillard 2004; Remillard & McClintock
2006).
Several models have been developed to explain the kHz QPO
frequencies, and it is usually preferred that these oscillations are
related to the orbital motion near the inner edge of an accretion
disc. In particular, two ideas based on the strong-gravity prop-
erties have been proposed. While Stella & Vietri (1998, 1999)
introduced the “Relativistic Precession Model” considering that
the kHz QPOs directly manifest the modes of a slightly per-
turbed (and therefore epicyclic) relativistic motion of blobs in
the inner parts of the accretion disc, Kluźniak & Abramowicz
(2001) propose models based on non-linear oscillations of an
accretion disc that assume resonant interaction between orbital
and/or epicyclic modes. In a different context, the possibility of
resonant coupling between the epicyclic modes of motion in the
Send offprint requests to: Z. Stuchlı́k,
e-mail: zdenek.stuchlik@fpf.slu.cz
Kerr spacetime was also mentioned in the early work of Aliev &
Galtsov (1981).
In the case of near-extreme Kerr black holes, it was sug-
gested that the epicyclic oscillations in the disc could be excited
by resonances with the so-called “hump-induced” oscillations,
see papers of Aschenbach (2004, 2006) and Stuchlı́k et al. (2004,
2007). This idea was proposed so as to extend standard orbital
(resonant) models meant to explain high-frequency QPOs ob-
served in black-hole sources.
Recently, careful and detailed analysis of the spectral contin-
uum from GRS 1915+105 has put a strong limit on the black-
hole spin,1 namely 0.98 < a < 1 (McClintock et al. 2006),
indicating the presence of near-extreme Kerr black hole whose
mass has been restricted observationally to M = (14.0±4.4) M⊙,
see (McClintock & Remillard 2004; Remillard & McClintock
2006). Therefore, the microquasar GRS 1915+105 seems to be
an appropriate candidate to test the extended resonant model
with hump-induced oscillations.2
The idea of hump-induced oscillations and their possible res-
onant coupling with the epicyclic ones is briefly discussed in
Sect. 2. The related resonant model, assuming the excitation of
epicyclic oscillations by the hump-induced oscillations through
1 Units c = G = M = 1 (M is the total mass of the Kerr black
hole) and the Boyer-Lindquist (B-L) coordinates (t, r, θ, ϕ) are used
hereafter.
2 However, Middleton et al. (2006) refer to a substantially lower, in-
termediate value of black-hole spin, a ∼ 0.7, to which the model of
hump-induced oscillations cannot be applied.
http://arxiv.org/abs/0704.1252v2
2 Zdeněk Stuchlı́k et al.: LNRF-velocity hump induced oscillations in GRS 1915+105
non-linear resonant phenomena, is applied to GRS 1915+105 in
Sect. 3, concluding remarks are presented in Sect. 4.
2. Hump-induced and epicyclic oscillations in
Keplerian discs and possible resonant coupling
In order to describe the local processes in an accretion disc,
it is necessary to choose a local observer (characterized by its
reference frame). In general relativity there is no preferred ob-
server. On the other hand, if we want to study processes related
to the orbital motion of matter in the disc, it is reasonable to
choose the observers with zero angular momentum, so-called
ZAMOs, as their reference frames do not rotate with respect to
the spacetime, and thus ZAMOs should reveal local kinematic
properties of the disc in the clearest way. (In rotating–stationary,
axisymetric–spacetimes, they are dragged along with the space-
time.) In the Kerr spacetime, ZAMOs are represented by locally
non-rotating frames (LNRF); see Bardeen et al. (1972). Notice
that in the Schwarzschild spacetime, LNRF correspond to the
static observer frames.
Aschenbach (2004) finds that for near-extreme Kerr black
holes with the spin a > 0.9953, the test-particle orbital veloc-
ity V(ϕ) related to LNRF reveals a hump in the equatorial plane
(θ = π/2). This non-monotonicity is located in a small region
inside the ergosphere of the black-hole spacetime close to, but
above, the marginally stable orbit.3 Therefore, it can be relevant
for thin accretion discs around near-extreme Kerr black holes,
as the inner edge of the disc can extend down to the innermost
stable circular orbit (ISCO).
Moreover, Stuchlı́k et al. (2005) shows that for a > 0.99979
the similarly humpy behavior of the orbital velocity in LNRF
also takes place for the non-geodesic motion of test perfect fluid
in marginally stable barotropic tori characterized by the uni-
form distribution of the specific angular momentum, ℓ(r, θ) ≡
−Uϕ/Ut = const, where the motion of fluid elements is given
by the 4-velocity field Uµ = (U t(r, θ), 0, 0, Uϕ(r, θ)). Outside
the equatorial plane, the non-monotonic behavior of V(ϕ) in
marginally stable tori is represented by the topology change of
the cylindrical equivelocity surfaces in the region of the hump,
because the toroidal equivelocity surfaces centered around the
circle corresponding to the local minimum of V(ϕ) in the equa-
torial plane exist for a > 0.99979 (Stuchlı́k et al. 2005). This
suggests a generation of possible instabilities in radial and ver-
tical directions; see Stuchlı́k et al. (2004). In the following, we
restrict our attention to the case of Keplerian discs.
Heuristic connection between the positive part of the veloc-
ity gradient, ∂V(ϕ)/∂r, and the excitation of epicyclic oscilla-
tions in Keplerian discs was suggested by Aschenbach (2004,
2006), who defined the characteristic frequency of oscillations,
induced by the humpy profile of V(ϕ), by the maximum posi-
tive slope of the orbital velocity in terms of the coordinate ra-
dius, νcrit ≡ (∂V(ϕ)/∂r)max. This coordinate-dependent defini-
tion was corrected in Stuchlı́k et al. (2007), where the proper
radial distance dr̃ =
grr dr rather than the coordinate dis-
tance dr was used to define the characteristic (critical) frequency
νr̃crit ≡ (∂V
(ϕ)/∂r̃)max. Such a locally defined critical frequency
was further related to a stationary observer at infinity, obtaining
the so-called “humpy frequency”
−(gtt + 2ωgtϕ + ω2gϕϕ)r=rh ν
3 We stress that the Aschenbach effect is frame-dependent, as it is
related to LNRF, but recall the arguments for relevance of the LNRF
point of view at the beginning of the section.
0 0.001 0.002 0.003 0.004
0 0.001 0.002 0.003 0.004
1 − a
a = (0.9998 ± 0.0001)
✉ GRS 1915+105
✛ 607
✛ 1.29
Fig. 1. Spin-dependence of the humpy frequency νh and the humpy ra-
dius rh. For completeness, the B-L radius of the innermost stable circu-
lar geodesic (ISCO) is plotted.
(rh − 2) −
rh(r2h + a
2) + 2a2
r5h + a
4(3rh + 2) − 2a3r
h (3rh + 1)
2∆3/2h
h + a)
2a2r2h(2rh − 5) − 2ar
h (5rh − 9)
2∆3/2h
h + a)
, (1)
where gµν are the metric coefficients of the Kerr geometry and
ω = −gtϕ/gϕϕ is the angular velocity of the LNRF; see, e.g.,
Bardeen et al. (1972); ∆h = r
h − 2rh + a
2. The analytic formula
is given for the equatorial plane (θ = π/2). The B-L radius rh
where the positive gradient of the velocity profile in terms of the
proper radial distance reaches its maximum, so-called “humpy
radius”, is given by the condition
∂V(ϕ)
= 0 (2)
leading to the equation
3a7(r + 2) + a6
r(21r2 + 18r − 4) − a5r(33r2 + 10r + 20)
r(45r3 − 62r2 − 68r + 16) − a3r3(83r2 − 122r − 60)
+a2r4
r(27r2 − 130r + 136) − 9ar5(7r2 − 26r + 24)
r(3r − 2) = 0, (3)
which must be solved numerically. The spin dependence of the
humpy radius and the related humpy frequency is illustrated in
Fig. 1. The humpy radius rh falls monotonically with increas-
ing spin a, while the humpy frequency νh has a maximum for
Zdeněk Stuchlı́k et al.: LNRF-velocity hump induced oscillations in GRS 1915+105 3
a = 0.9998, where νh (max) = 607 (M⊙/M) Hz, and it tends to
νh (a→1) = 588 (M⊙/M) Hz.
When particles following a Keplerian circular orbit are per-
turbed, they begin to follow, in the first approximation, an
epicyclic motion around the equilibrium Keplerian orbit, gen-
erally characterized by the frequencies of the radial and vertical
epicyclic oscillations νr, νv (Aliev & Galtsov 1981; Nowak &
Lehr 1998):
r = ν
K(1 − 6r
+ 8ar−3/2 − 3a2r−2), (4)
v ≡ ν
θ = ν
K(1 − 4ar
+ 3a2r−2), (5)
where νK is the Keplerian orbital frequency
2π(r3/2 + a)
. (6)
The ratios of the humpy frequency and the epicyclic frequencies
at the humpy radius were determined in Stuchlı́k et al. (2007)
revealing almost spin-independent asymptotic behavior for a →
1 represented closely by the ratios of integer numbers, νv : νr :
νh ∼ 11 : 3 : 2, which imply a possibility of resonant phenomena
between the hump-induced and epicyclic oscillations predicted
by Aschenbach (2004). The ratios of the epicyclic frequencies
and the humpy frequency are given in the dependence on the
black-hole spin in Fig. 2.
3. Application of the hump-induced resonance
model to high-frequency QPOs in GRS 1915+105
Primarily concentrating on the lower pair of frequencies, we as-
sume that the lowest frequency is directly the humpy frequency,
νh ≡ ν1 = (41 ± 1) Hz, (7)
while the second lowest frequency corresponds directly to the
radial epicyclic frequency at the same radius rh,
νr ≡ ν2 = (67 ± 1) Hz. (8)
These frequencies are close to a 3:2 ratio, therefore the forced
non-linear resonance can be relevant in such a situation. The
ratio of νr/νh = (1.63 ± 0.06) gives the black hole spin a =
(0.9998 ± 0.0001) (the uncertainty of the spin is implied by un-
certainties of the lower pair of frequencies being ∼ 1 Hz); see
Fig. 2. Notice that this spin corresponds to the maximal pos-
sible value of the humpy frequency νh (max) (Fig. 1). Since the
humpy frequency is 1/M-scaled, the absolute value of νh im-
plies the black hole mass M = (14.8±0.4) M⊙. The correspond-
ing humpy radius is rh = 1.29
+0.01
−0.02 (Fig. 1). At such a radius,
the vertical epicyclic frequency of a particle orbiting the Kerr
black hole with the mass and spin inferred above reaches the
value νv = (0.23 ± 0.01) kHz. Then the upper pair of observed
frequencies can be explained, within the range of observational
errors ±5 Hz, by combinational frequencies at the humpy radius
rh in the following way:
ν3 ∼ (νr + νh) = (108 ± 2) Hz (9)
ν4 ∼ (νv − νr) = (0.17 ± 0.01) kHz. (10)
4. Conclusions
The idea of epicyclic oscillations induced by the LNRF-velocity
hump in the region where the positive part of the velocity gra-
dient reaches its maximum is able to address all four high-
frequency QPOs observed in the X-ray source GRS 1915+105.
-8 -7 -6 -5 -4 -3
log(1 − a)
a = (0.9998 ± 0.0001) ✲
GRS 1915+105 ✲
✻νr :νh
(νv − νr) :(νh + νr)
νv :νr
νv :νh
∼ 3:2
∼ 7:2
∼ 6:1
Fig. 2. Spin dependence of frequency ratios including the radial (νr)
and vertical (νv) epicyclic frequencies, and the humpy frequency (νh)
evaluated at the same radius rh where the humpy frequency is defined.
The range of the spin relevant for GRS 1915+105 Kerr black hole is
shaded. For the mean value a = 0.9998, the frequency ratios are close
to the ratios of integer numbers, suggesting a possibility of resonances
between hump-induced and epicyclic oscillations in GRS 1915+105.
The model implies a near-extreme spin of the central black hole
(a ∼ 0.9998), which agrees well with results from the spec-
tral continuum fits, and the black-hole mass M ∼ 14.8 M⊙ be-
ing well inside the interval given by other observational meth-
ods. Note that the orbital resonance model of Kluźniak &
Abramowicz, assuming the parametric resonance between the
vertical and radial epicyclic oscillations in frequency ratio 3:2
represented by the upper pair of observed frequencies, also gives
the spin a > 0.99 but for M ≃ 18M⊙ (Török et al. 2005). On the
other hand, the “Relativistic Precession Model” gives a substan-
tially lower value for the spin: a ∼ 0.3 (Stella et al. 1999).
In the presented model, we assume that all four observed
frequencies arise due to forced non-linear oscillations of the
Keplerian disc at the same radius rh, excited by the hump-
induced oscillations characterized by the humpy frequency νh.
The black-hole parameters a, M are fixed by the requirement
that the lower pair of observed frequencies is identified with the
humpy frequency and the radial epicyclic frequency, ν1 ≡ νh,
ν2 ≡ νr. Assuming non-linear resonant phenomena enabling
the existence of combinational frequencies and the possibility
of observing them, the upper pair of observed frequencies can
be explained as the combinational ones of the humpy frequency
and both epicyclic frequencies, ν3 ∼ (νr + νh), ν4 ∼ (νv − νr).
Moreover, both frequency ratios νr : νh, and (νv − νr) : (νr + νh)
are close to 3:2 ratio (Fig. 2), in which the resonant phenomena
can be strong enough. On the other hand, as 4νh = (164± 4) Hz,
which is also close to the uppermost frequency, there is an-
other possibility of explaining ν4 through a sub-harmonic reso-
nance forced by the humpy oscillations as well. Finally, note that
Strohmayer (2001) also reports another relatively weak QPO at
frequency of (56 ± 2) Hz. If this is the case (which, according
to our knowledge, has not been confirmed by other observations
yet), it could be related to the second harmonic of the combina-
tional frequency4 (νr − νh) = (26 ± 2) Hz.
4 Combinational frequency (νr − νh) corresponds to the same order of
nonlinearity as (νr + νh).
Note added in the manuscript: After the paper was accepted we ob-
tained an information that a weak QPO at frequency 27 Hz is referenced
in Belloni et al. (2001).
4 Zdeněk Stuchlı́k et al.: LNRF-velocity hump induced oscillations in GRS 1915+105
Generally, other harmonics and combinational frequencies
may occur in a non-linear oscillating system corresponding to
higher approximations, when the equation of motion describing
the non-linear oscillations is solved by the method of successive
approximations. The statement by Landau & Lifshitz (1976) that
“As the degree of approximation increases, however, the strength
of the resonances, and the widths of the frequency ranges in
which they occur, decrease so rapidly that in practice only the
resonances at frequencies5 ν ≈ pν0/q with small p and q can be
observed” can explain why a QPO near the frequency 237 Hz,
corresponding to the vertical epicyclic frequency νv at the same
radius rh as the previously mentioned humpy and radial epicyclic
frequencies νh, νr, is not directly observed, despite the commen-
surability of these frequencies represented by the frequency ra-
tios νv : νh ∼ 6 : 1 and νv : νr ∼ 7 : 2 (Fig. 2).
Acknowledgements. The authors are supported by the Czech grant
MSM 4781305903.
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5 p, q are integers.
