# Epicyclic oscillations of non-slender fluid tori around Kerr black holes

Odele Straub1 and Eva Šrámková2 1Copernicus Astronomical Centre PAN, Bartycka
18, 00-716 Warsaw, Poland  2Department of Physics, Silesian University in
Opava, Bezručovo nám. 13, 746-01 Opava, Czech Republic<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Considering epicyclic oscillations of pressure-supported perfect fluid tori
orbiting Kerr black holes we examine non-geodesic (pressure) effects on the
epicyclic modes properties. Using a perturbation method we derive fully
general relativistic formulas for eigenfunctions and eigenfrequencies of the
radial and vertical epicyclic modes of a slightly non-slender, constant
specific angular momentum torus up to second-order accuracy with respect to
the torus thickness. The behaviour of the axisymmetric and lowest-order
($m=\pm 1$) non-axisymmetric epicyclic modes is investigated. For an arbitrary
black hole spin we find that, in comparison with the (axisymmetric) epicyclic
frequencies of free test particles, non-slender tori receive negative pressure
corrections and exhibit thus lower frequencies. Our findings are in
qualitative agreement with the results of a recent pseudo-Newtonian study of
analogous problem defined within the Paczyński-Wiita potential. Implications
of our results on the high-frequency QPO models dealing with epicyclic
oscillations are addressed.

###### pacs:

95.30.Lz, 95.30.Sf, 95.85.Nv, 97.60.Lf

††: Class. Quantum Grav.

,

##  1 Introduction

Oscillations of black hole accretion discs have been studied extensively in
various astrophysical contexts. Investigations of hydrodynamic oscillation
modes of geometrically thin accretion discs revealed three fundamental,
discoseismic classes of modes: acoustic pressure p-modes, gravity g-modes and
corrugation c-modes. Kato et al (1998); Wagoner (1999); Kato (2001a) and
Wagoner et al (2001) give comprehensive reviews on the subject of relativistic
’discoseismology’. Geometrically thick discs (tori) have been examined rather
less and mainly in matters of their stability (e.g. Papaloizou & Pringle,
1984; Kojima, 1986). Only recently oscillatory modes of fluid tori were
explored in more detail in several numerical studies (e.g. Zanotti et al,
2003; Rezzolla et al, 2003a; Rubio-Herrera and Lee, 2005a, b; Šrámková et al,
2007; Montero et al, 2007) and in a purely analytic work by Blaes, Arras and
Fragile (2006). The latter presents a thorough analysis of oscillatory modes
of relativistic slender tori.

The analysis of disc oscillation modes is motivated by observations of quasi-
periodic oscillations (QPOs) in the light curves of Galactic low-mass X-ray
binaries (see McClintock and Remillard, 2003; van der Klis, 2004, for
reviews). In particular the understanding of high frequency (HF) QPOs, which
are presumably a strong gravity phenomenon, would provide a deeper insight
into the innermost regions of accretion discs and the very nature of compact
objects. Black hole HF QPOs occur at frequencies that are constant in time and
characteristic for a particular source. If more than one HF QPO is detected in
a given system, the frequencies typically appear in ratios of small natural
numbers, whereas in most cases the ratio is close to 3:2 (Abramowicz and
Kluźniak, 2001; McClintock and Remillard, 2003).

Several models explain HF QPOs in terms of disc (or blob) oscillations. Stella
and Vietri (1998) for instance consider an orbiting hot spot and propose that
HF QPOs arise due to a modulation of the spot radiation by its precessional
motion. Pointing out the observed rational ratios of HF QPO pairs Kluźniak and
Abramowicz (2000) and Abramowicz and Kluźniak (2001) suggest an underlying
non-linear resonance between some modes of accretion disc oscillation. In the
resonance model the mode pair is commonly represented by the radial and
vertical epicyclic oscillation. The importance of epicyclic oscillations is
also stressed by Kato (2001b) who attributes the origin of HF QPOs to non-
axisymmetric g-mode oscillations. The corotation resonance in a disc which is
deformed by a warp would be responsible for the excitation of g-modes and
general relativistic effects trap them near the inner edge of the accretion
disc (Kato, 2003, 2004). HF QPOs could also result from acoustic p-mode
oscillations of a small accretion torus orbiting close to the black hole
(Rezzolla et al, 2003b). Recently Blaes, Arras and Fragile (2006) discussed
the possibility that the vertical epicyclic and the lowest-order (acoustic)
breathing mode of a relativistic slender torus might represent the two black
hole HF QPOs in 3:2 ratio.

The main interest of our work lies in the epicyclic modes of oscillations.
Most models that are dealing with them consider geodesic flows and are based
on free test particles. However, non-geodesic effects related to e.g. magnetic
fields, viscosity or pressure forces may play a certain role in this concern.
The aim of this paper is to investigate such non-geodesic effects on the two
epicyclic modes by means of a pressure-supported perfect fluid torus and to
find the consequential pressure corrections to the mode eigenfunctions and
eigenfrequencies.

For an infinitely slender torus, the frequencies of epicyclic oscillations are
consistent with the epicyclic frequencies of free test particles on a circular
orbit in the equatorial plane, where the torus pressure exhibits a maximum
value (Abramowicz et al, 2006). It has been shown that the epicyclic modes may
be retained also for thicker tori (Blaes et al, 2007). Numerical simulations
(Rezzolla et al, 2003a; Rubio-Herrera and Lee, 2005a, b; Šrámková et al, 2007)
as well as analytic calculations in pseudo-Newtonian approximation (Blaes et
al, 2007) showed that with growing torus thickness the (axisymmetric)
epicyclic frequencies decrease. What we have in mind here is a slightly non-
slender torus (with a radial extent that is very small in comparison to its
distance from the central object) in hydrostatic equilibrium in Kerr
spacetime. We perform a second-order perturbation analysis of the
eigenfunctions and eigenfrequencies of both modes with respect to the torus
thickness and derive exact analytic formulas for the pressure corrections.

Blaes et al (2007) have studied an analogous problem in the framework of
Newtonian physics using the pseudo-Newtonian potential of Paczyński and Wiita
(1980) to model the relevant general relativistic effects. Our work represents
a generalisation of their results into Kerr geometry. Like Blaes et al (2007)
we assume a non-self-gravitating, non-magnetic, stationary torus with a
constant specific angular momentum distribution. We neglect self-gravity
because the mass of a black hole exceeds the mass of a torus many times over.
Although effects of magnetic fields are neither negligible nor irrelevant to
our problem, we study here a purely hydrodynamical case. Our calculations are
only valid as long as magnetic fields are unimportant, i.e., as long as the
torus pressure dominates the magnetic pressure. The stability of these
hydrodynamic modes in presence of magnetohydrodynamical turbulence is an issue
that needs yet to be investigated. The above assumptions are supported by
numerical simulations. Proga & Begelman (2003) show for instance in an
inviscid hydrodynamical simulation that the inner axisymmetric accretion flow
settles into a pressure-rotation supported torus with constant specific
angular momentum. Once magnetic fields are introduced the angular momentum
distribution gets a different profile, torus-like configurations are, however,
still seen as ”inner tori” in global MHD simulations (e.g. Machida et al,
2006; Fragile et al, 2008). Although the described torus setup is not the most
likely to be found in nature, we think it is reasonable to assume such a
configuration as a first approximation.

In this work we focus on a mathematical description of the problem, whereas
the astrophysical applications shall be presented separately. The paper is
outlined in the following manner: In sections 2 and 3 we give a brief
introduction to the problem writing down the equations that describe the
relativistic equilibrium tori and the relativistic Papaloizou-Pringle
equation. Then, in section 4, we describe the perturbation method and derive
formulas for the radial and vertical epicyclic mode eigenfunctions and
eigenfrequencies. The results are presented for different values of the black
hole spin parameter in section 5 and discussed in section 6.

##  2 Equilibrium configuration

Consider an axisymmetric, non-self-gravitating perfect fluid torus in
hydrostatic equilibrium on the background of the Kerr geometry. The flow of
fluid is stationary and in a state of pure rotation. Generally, the line
element of a stationary, axially symmetric spacetime is given in Boyer-
Lindquist coordinates ($t,r,\theta,\phi$) by

$ds^{2}=g_{tt}dt^{2}+2g_{t\phi}dtd\phi+g_{rr}dr^{2}+g_{\theta\theta}d\theta^{2}+g_{\phi\phi}d\phi^{2}.$
(1)

We take the ($-+++$) signature and units where $c=G=M=1$. The explicit
expressions for the covariant and contravariant coefficients of the Kerr
metric then write

$\displaystyle g_{tt}=-\left(1-\frac{2r}{\Sigma}\right),\qquad$ $\displaystyle
g^{tt}=-\frac{\Xi}{\Sigma\Delta},$ (2) $\displaystyle
g_{t\phi}=-\frac{2ar}{\Sigma}\sin^{2}\theta,\qquad$ $\displaystyle
g^{t\phi}=-\frac{2ar}{\Sigma\Delta},$ $\displaystyle
g_{rr}=\frac{\Sigma}{\Delta},\qquad$ $\displaystyle
g^{rr}=\frac{\Delta}{\Sigma},$ $\displaystyle g_{\theta\theta}=\Sigma,\qquad$
$\displaystyle g^{\theta\theta}=\frac{1}{\Sigma},$ $\displaystyle
g_{\phi\phi}=\left(r^{2}+a^{2}+\frac{2a^{2}r}{\Sigma}\sin^{2}\theta\right)\sin^{2}\theta,\qquad$
$\displaystyle
g^{\phi\phi}=\frac{\Delta-a^{2}\sin^{2}\theta}{\Sigma\Delta\sin^{2}\theta},$

where $\Sigma\equiv r^{2}+a^{2}\cos^{2}\theta$, $\Delta\equiv r^{2}-2r+a^{2}$,
$\Xi\equiv(r^{2}+a^{2})^{2}-a^{2}\Delta\sin^{2}\theta$ and $M$ is the mass and
$a$ the specific angular momentum (spin) of the black hole.

Because the flow is assumed to be purely azimuthal, the four-velocity has only
two non-zero components, $u^{\mu}=(u^{t},0,0,u^{\phi})$. One may derive the
fluid specific angular momentum $l$, angular velocity $\Omega$, specific
energy $\mathcal{E}$ and the contravariant $t$-component of the four-velocity,
often denoted as $A$, in the form

$\displaystyle l\equiv-$ $\displaystyle\frac{u_{\phi}}{u_{t}}$
$\displaystyle=-\frac{g_{t\phi}+\Omega g_{\phi\phi}}{g_{tt}+\Omega
g_{t\phi}},$ (3) $\displaystyle\Omega\equiv$
$\displaystyle\frac{u^{\phi}}{u^{t}}$
$\displaystyle=\frac{g^{t\phi}-lg^{\phi\phi}}{g^{tt}-lg^{t\phi}},$ (4)
$\displaystyle\mathcal{E}\equiv-$ $\displaystyle u_{t}$
$\displaystyle=(-g^{tt}+2lg^{t\phi}-l^{2}g^{\phi\phi})^{-1/2},$ (5)
$\displaystyle A\equiv$ $\displaystyle u^{t}$ $\displaystyle=(-g_{tt}-2\Omega
g_{t\phi}-\Omega^{2}g_{\phi\phi})^{-1/2}.$ (6)

The perfect fluid is characterised by the stress-energy tensor
$T^{\mu\nu}=(p+e)u^{\mu}u^{\nu}+pg^{\mu\nu}$. We restrict our consideration to
polytropic flows such that, measured in the fluid’s rest frame, pressure $p$,
internal energy density $e$ and rest mass density $\rho$ are related by
$p=K\rho^{(n+1)/n}$ and $e=np+\rho$, where $n$ is the polytropic index and $K$
the polytropic constant.

We assume the specific angular momentum to be constant throughout the torus,
i.e., $l(r,\theta)\equiv l_{0}=const$. Such a configuration is governed by the
relativistic Euler equation which may be written as

$-\frac{\partial_{\mu}p}{p+e}=\partial_{\mu}\left(\ln\mathcal{E}\right),\qquad\mu\in\\{r,\theta\\}.$
(7)

Introducing the enthalpy $H\equiv\int{dp/(p+e)}$, the integration of (7) leads
for a barotropic fluid, for which $p=p(e)$, to the following form of the
Bernoulli equation

$H+\ln\mathcal{E}=const.$ (8)

that determines the structure of the torus in the $r-\theta$ plane. The
subscript zero refers to the special location $r=r_{0}$ in the equatorial
plane where the pressure gradients vanish ($p$ has a maximal value) and the
fluid moves along a geodesic line.

For a small torus cross-section, when the adiabatic sound speed defined at the
pressure maximum $c^{2}_{s0}=(n+1)p_{0}/(n\rho_{0})$ satisfies
$c^{2}_{s0}<<c^{2}$, one may write $H\approx(n+1)p/\rho$ (Abramowicz et al,
2006). Following Abramowicz et al (2006) and Blaes et al (2007) we introduce
the function $f(r,\theta)$, which takes constant values at the isobaric and
isodensity surfaces, by

$\frac{p}{\rho}=\frac{p_{0}}{\rho_{0}}f(r,\theta).$ (9)

Form the Bernoulli equation (8) with the constant evaluated at the pressure
maximum $r_{0}$ and the above form of $H$ we get

$f=1-\frac{1}{nc_{s0}^{2}}\left(\ln\mathcal{E}-\ln\mathcal{E}_{0}\right).$
(10)

###  2.1 Epicyclic oscillations

It is advantageous to introduce the effective potential

$U=g^{tt}-2l_{0}g^{t\phi}+l_{0}^{2}g^{\phi\phi}$ (11)

that has its minimum at the torus pressure maximum $r_{0}$. A small
perturbation of a test particle orbiting on a geodesic line $r=r_{0}$ with
$l=l_{0}$ results in radial and vertical epicyclic oscillations around the
equilibrium point $r_{0}$ at a radial $\omega_{r0}$ and vertical
$\omega_{\theta 0}$ epicyclic frequency given by (e.g., Abramowicz et al
(2006))

$\omega_{r0}^{2}=\frac{1}{2}\left(\frac{\mathcal{E}^{2}}{A^{2}g_{rr}}\frac{\partial^{2}U}{\partial
r^{2}}\right)_{0}\qquad\textnormal{and}\qquad\omega_{\theta
0}^{2}=\frac{1}{2}\left(\frac{\mathcal{E}^{2}}{A^{2}g_{\theta\theta}}\frac{\partial^{2}U}{\partial\theta^{2}}\right)_{0}.$
(12)

In Kerr geometry (2) the above definitions lead to (Aliev and Galtsov, 1981;
Nowak and Lehr, 1998; Török and Stuchlík, 2005)

$\omega_{r0}^{2}=\Omega_{0}^{2}\left(1-\frac{6}{r_{0}}+\frac{8a}{r_{0}^{3/2}}-\frac{3a^{2}}{r_{0}^{2}}\right)\qquad\textnormal{and}\qquad\omega_{\theta
0}^{2}=\Omega_{0}^{2}\left(1-\frac{4a}{r_{0}^{3/2}}+\frac{3a^{2}}{r_{0}^{2}}\right),$
(13)

where $\Omega_{0}$ is the angular velocity at the pressure maximum $r_{0}$
that in Kerr geometry reads $\Omega_{0}=1/(r_{0}^{3/2}+a)$.

In order to investigate the behaviour of the equipotential function $f$ in
close vicinity of the equilibrium point $r_{0}$, Abramowicz et al (2006)
introduced local coordinates,

$x=\sqrt{g_{rr0}}\left(\frac{r-r_{0}}{r_{0}}\right)\qquad\textnormal{and}\qquad
y=\sqrt{g_{\theta\theta 0}}\left(\frac{\pi/2-\theta}{r_{0}}\right),$ (14)

satisfying $x=y=0$ at $r_{0}$. For small $x$ and $y$, and a constant specific
angular momentum torus, the equipotential function $f$ can be expressed as

$f=1-\frac{1}{\beta^{2}}\left(\bar{\omega}_{r0}^{2}x^{2}+\bar{\omega}_{\theta
0}^{2}y^{2}\right),$ (15)

where $\bar{\omega}_{r0}\equiv\omega_{r0}/\Omega_{0}$, $\bar{\omega}_{\theta
0}\equiv\omega_{\theta 0}/\Omega_{0}$, and $\beta$ is a dimensionless
parameter given by

$\beta^{2}\equiv\frac{2nc_{s0}^{2}}{r_{0}^{2}A_{0}^{2}\Omega_{0}^{2}}$ (16)

which determines the thickness of the torus.

Equation (15) describes the equipotential function in the vicinity of $r_{0}$
in terms of the test particle epicyclic frequencies and is congruent with the
formula derived in Newtonian theory (see equation (9) in Blaes et al (2007)).
In the slender torus limit $\beta\rightarrow 0$, the torus reduces to an
infinitesimally slender ring at the pressure maximum $r_{0}$. In a Newtonian
$1/r$ potential, $\omega_{r0}=\omega_{\theta 0}=\Omega_{0}$ and the slender
torus cross-section has a circular shape. In the general case, however,
$\omega_{r0}\neq\omega_{\theta 0}$ and the isobaric surfaces are ellipses with
semi-axes being in the ratio of the two epicyclic frequencies.

##  3 Perturbation equation

We consider small linear perturbations around the axisymmetric and stationary
torus equilibrium with azimuthal and time dependence in the form
$\propto\exp[\rmi(m\phi-\omega t)]$. The differential equation governing such
perturbations for constant specific angular momentum tori was in Newtonian
theory derived by Papaloizou & Pringle (1984) where it was expressed in terms
of a scalar variable $W$. Abramowicz et al (2006) recently derived the general
relativistic form of the Papaloizou-Pringle equation in terms of

$W=-\frac{\delta p}{A\rho(\omega-m\Omega)},$ (17)

which is related to the Eulerian perturbation in the four-velocity as

$\delta
u_{\mu}=\frac{\rmi\rho}{p+e}\partial_{\mu}W,\qquad\mu\in\\{r,\theta\\}.$ (18)

The relativistic Papaloizou-Pringle equation writes

$\displaystyle\frac{1}{(-g)^{1/2}}\left\\{\partial_{\mu}\left[(-g)^{1/2}g^{\mu\nu}f^{n}\partial_{\nu}W\right]\right\\}$
$\displaystyle-$ $\displaystyle\left(m^{2}g^{\phi\phi}-2m\omega
g^{t\phi}+\omega^{2}g^{tt}\right)f^{n}W$ (19)
$\displaystyle=-\frac{2n\mathcal{A}(\bar{\omega}-m\bar{\Omega})^{2}}{\beta^{2}r^{2}_{0}}f^{n-1}W$

where $\\{\mu,\nu\\}\in\\{r,\theta\\}$, $\mathcal{A}\equiv A^{2}/A^{2}_{0}$,
$\bar{\Omega}\equiv\Omega/\Omega_{0}$, $\bar{\omega}\equiv\omega/\Omega_{0}$
and $g$ denotes the determinant of the metric.

Following Blaes et al (2007) we write (19) as

$\hat{L}W=-2n\mathcal{A}\left(\bar{\omega}-m\bar{\Omega}\right)^{2}W,$ (20)

where $\hat{L}$ is a linear operator given by

$\displaystyle\hat{L}$ $\displaystyle=$
$\displaystyle[\frac{1}{\sqrt{-g}}\partial_{\mu}(\sqrt{-g})g^{\mu\nu}f\partial_{\nu}+\partial_{\mu}(g^{\mu\nu})f\partial_{\nu}+g^{\mu\nu}n\partial_{\mu}f\partial_{\nu}+g^{\mu\nu}f\partial^{2}_{\nu}$
(21) $\displaystyle-\left(m^{2}g^{\phi\phi}-2m\omega
g^{t\phi}+\omega^{2}g^{tt}\right)f]\beta^{2}r^{2}_{0}.$

##  4 Expanding the relativistic Papaloizou-Pringle equation about the slender
torus limit

We now use a perturbation method to derive the expressions for eigenfunctions
and eigenfrequencies of the epicyclic modes for thicker tori ($\beta>0$).

We start by transforming all variables to local coordinates, $\bar{x}=x/\beta$
and $\bar{y}=y/\beta$, measured from the equilibrium point. Then we expand
$\bar{\omega}$, $W$, $\mathcal{A}$, $\bar{\Omega}$, $f$ and $\hat{L}$ into a
power series in $\beta$ by writing

$Q=Q^{(0)}+Q^{(1)}\beta+Q^{(2)}\beta^{2}+\ldots\\\
\,\,\textnormal{where}\,\,\,\,Q\in\\{\bar{\omega},W,\mathcal{A},\bar{\Omega},f,\hat{L}\\}.$
(22)

Substituting all variables (22) into the perturbation equation (20) and
comparing terms of the same order in $\beta$ we obtain formulas for the
respective corrections to eigenfunctions and eigenfrequencies of the desired
modes.

###  4.1 Slender torus limit

In the slender torus limit ($\beta\rightarrow 0$) the relativistic Papaloizou-
Pringle equation (19) reduces to

$\displaystyle
f^{(0)}\left(\frac{\partial^{2}W^{(0)}}{\partial\bar{x}^{2}}+\frac{\partial^{2}W^{(0)}}{\partial\bar{y}^{2}}\right)+n\left(\frac{\partial
f^{(0)}}{\partial\bar{x}}\frac{\partial
W^{(0)}}{\partial\bar{x}}+\frac{\partial
f^{(0)}}{\partial\bar{y}}\frac{\partial W^{(0)}}{\partial\bar{y}}\right)=$
$\displaystyle-2n\mathcal{A}^{(0)}(\bar{\omega}^{(0)}-m\bar{\Omega}^{(0)})^{2}W^{(0)}.$
(23)

It represents the zeroth-order of (20) and may be written in operator form,

$\hat{L}^{(0)}W^{(0)}=-2n\mathcal{A}^{(0)}\sigma^{2}W^{(0)},$ (24)

where $\sigma\equiv\bar{\omega}^{(0)}-m\bar{\Omega}^{(0)}$ denotes the zeroth-
order eigenfrequency in the corotating frame scaled with the orbital velocity
$\Omega_{0}$. The encountered zeroth-order expansion terms are

$\displaystyle\mathcal{A}^{(0)}$ $\displaystyle=$ $\displaystyle 1,$ (25)
$\displaystyle\bar{\Omega}^{(0)}$ $\displaystyle=$ $\displaystyle 1,$ (26)
$\displaystyle f^{(0)}$ $\displaystyle=$ $\displaystyle
1-\bar{\omega}_{r0}^{2}\bar{x}^{2}-\bar{\omega}_{\theta 0}^{2}\bar{y}^{2},$
(27) $\displaystyle\hat{L}^{(0)}$ $\displaystyle=$ $\displaystyle
f^{(0)}\frac{\partial^{2}}{\partial\bar{x}^{2}}+f^{(0)}\frac{\partial^{2}}{\partial\bar{y}^{2}}+n\frac{\partial
f^{(0)}}{\partial\bar{x}}\frac{\partial}{\partial\bar{x}}+n\frac{\partial
f^{(0)}}{\partial\bar{y}}\frac{\partial}{\partial\bar{y}}.$ (28)

Equations (23) and (24) are of the same form as the Newtonian slender torus
limit of the Papaloizou-Pringle equation (Abramowicz et al, 2006; Blaes et al,
2007). They represent an eigenvalue equation for $\sigma$ with $\hat{L}^{(0)}$
being a self-adjoint operator with respect to the inner product

$\left\langle
W_{a}^{(0)}|W_{b}^{(0)}\right\rangle=\int\int\left(f^{(0)}\right)^{n-1}W_{a}^{(0)\ast}W_{b}^{(0)}d\bar{x}d\bar{y}=\delta_{ab},$
(29)

where the integrals are taken over the slender torus cross-section where
$f^{(0)}\geq 0$. This implies that the eigenvalues $\sigma$ are real numbers
and the zeroth-order eigenfunctions $W^{(0)}$ form a complete orthonormal set.
Therefore, any function defined over the torus cross-section may be expanded
in terms of $W^{(0)}$.

The explicit expressions for eigenfunctions and eigenfrequencies of the
complete set of normal modes of a constant specific angular momentum slender
torus in Newtonian potential, where $\bar{\omega}_{r0}=\bar{\omega}_{\theta
0}=1$, were given in Blaes (1985). Recently, Blaes, Arras and Fragile (2006)
derived the eigenfunctions and eigenfrequencies of the lowest-order modes of a
general relativistic slender torus, where
$\bar{\omega}_{r0}\neq\bar{\omega}_{\theta 0}$, for arbitrary specific angular
momentum distribution. The eigenfunctions and eigenfrequencies of the simplest
modes of the relativistic torus with constant specific angular momentum which
are relevant to our calculations are specified in table 1.

In the slender torus limit the two epicyclic modes correspond to $i=1$ and
$i=2$ from table 1. The corresponding eigenfunctions take the form

$W_{r}^{(0)}\equiv
W_{1}^{(0)}=a_{1}\bar{x}e^{\rmi(m\phi-\omega_{1}^{(0)}t)}\quad\textnormal{and}\quad
W_{\theta}^{(0)}\equiv
W_{2}^{(0)}=a_{2}\bar{y}e^{\rmi(m\phi-\omega_{2}^{(0)}t)}.$ (30)

They describe the global, incompressible modes in which the entire torus moves
in purely radial ($W_{r}^{(0)}$) or purely vertical ($W_{\theta}^{(0)}$)
direction at frequencies which are, in the corotating frame, consistent with
the epicyclic frequencies of free test particles,
$\omega_{r0}=\sigma_{1}\Omega_{0}=\omega_{1}^{(0)}-m\Omega^{(0)}$ and
$\omega_{\theta 0}=\sigma_{2}\Omega_{0}=\omega_{2}^{(0)}-m\Omega^{(0)}$.

###  4.2 Non-slender torus

####  4.2.1 First-order corrections

Expanding the Papaloizou-Pringle equation (20) to first-order in $\beta$ we
arrive at 111We use the index $i$ to label the modes of interest. It takes
values $i=1$ for the radial or $i=2$ for the vertical epicyclic mode.

$\displaystyle\hat{L}^{(0)}W_{i}^{(1)}+\hat{L}^{(1)}W_{i}^{(0)}$
$\displaystyle=$ $\displaystyle
2n(2m\sigma_{i}\bar{\Omega}^{(1)}-\sigma_{i}^{2}\mathcal{A}^{(1)}-2\sigma_{i}\bar{\omega}_{i}^{(1)})W_{i}^{(0)}$
(31) $\displaystyle-2n\sigma_{i}^{2}W_{i}^{(1)}.$

The perturbed basis of eigenfunctions $W_{i}^{(1)}$ may now be expressed in
terms of the orthonormal zeroth-order basis as

$W_{i}^{(1)}=\sum_{j}{b_{ij}W_{j}^{(0)}}.$ (32)

The $b_{ij}$ coefficients may be determined by taking the inner product of
(31) with a zeroth-order eigenfunction $W_{k}^{(0)}$. If the subscript $k$
refers to a different mode from the one we are interested in (i.e., $k\neq i$)
we find

$b_{ij}=\frac{\left\langle
W_{j}^{(0)}|\hat{L}^{(1)}-4nm\sigma_{i}\bar{\Omega}^{(1)}+2n\sigma_{i}^{2}\mathcal{A}^{(1)}|W_{i}^{(0)}\right\rangle}{2n(\sigma_{j}^{2}-\sigma_{i}^{2})}.$
(33)

Whereas if $k$ refers to either of the two epicyclic modes ($k=i$) we obtain
the formula for the first-order correction to the radial ($i=1$) or vertical
($i=2$) eigenfrequency,

$\bar{\omega}_{i}^{(1)}=-\frac{\left\langle
W_{i}^{(0)}|\hat{L}^{(1)}-4nm\sigma_{i}\bar{\Omega}^{(1)}+2n\sigma_{i}^{2}\mathcal{A}^{(1)}|W_{i}^{(0)}\right\rangle}{4n\sigma_{i}}.$
(34)

The first-order expansion terms of $\mathcal{A}$, $\bar{\Omega}$, $f$ and
$\hat{L}$ have a ($\bar{x},\bar{y}$) dependence in the form

$\displaystyle\mathcal{A}^{(1)}$ $\displaystyle=\mathcal{A}_{11}\bar{x},$ (35)
$\displaystyle\bar{\Omega}^{(1)}$ $\displaystyle=\Omega_{11}\bar{x},$ (36)
$\displaystyle f^{(1)}$
$\displaystyle=f_{11}\bar{x}^{3}+f_{12}\bar{x}\bar{y}^{2},$ (37)
$\displaystyle\hat{L}^{(1)}$
$\displaystyle=\left(L_{101}\bar{x}+L_{102}\bar{x}^{3}+L_{103}\bar{x}\bar{y}^{2}\right)\partial_{\bar{x}}^{2}$
(38)
$\displaystyle+\left(L_{104}\bar{x}+L_{105}\bar{x}^{3}+L_{106}\bar{x}\bar{y}^{2}\right)\partial_{\bar{y}}^{2}$
$\displaystyle+\left(L_{107}+L_{108}\bar{x}^{2}+L_{109}\bar{y}^{2}\right)\partial_{\bar{x}}+L_{110}\bar{x}\bar{y}\partial_{\bar{y}},$

where the explicit forms of the coefficients
$\mathcal{A}_{11},\Omega_{11},f_{11},f_{12}$ and $L_{101}-L_{110}$ are
specified in the appendix.

Table 1:  Eigenfunctions of the lowest-order modes of a general slender torus
with constant specific angular momentum (Blaes, Arras and Fragile, 2006). The
normalisation constants $a_{0}-a_{5}$ are given in table 3 and the
coefficients $w_{41}$, $w_{42}$, $w_{51}$, $w_{52}$ are specified in the
appendix.

  * $i$ |  $W_{i}^{(0)}$
---|---
0 |  $a_{0}$
1 |  $a_{1}\bar{x}$
2 |  $a_{2}\bar{y}$
3 |  $a_{3}\bar{x}\bar{y}$
4 |  $a_{4}\left(1+w_{41}\bar{x}^{2}+w_{42}\bar{y}^{2}\right)$
5 |  $a_{5}\left(1+w_{51}\bar{x}^{2}+w_{52}\bar{y}^{2}\right)$

Substituting (35)-(38) into (33) and considering the eigenmodes derived in
Blaes, Arras and Fragile (2006) we find for the radial mode ($i=1$) three non-
zero coefficients that correspond to the modes i = 0, 4 and 5 given in table
1. The resulting eigenfunction $W_{r}$ of the radial epicyclic mode for a
slightly non-slender torus in first-order accuracy reads

$W_{r}=a_{1}\bar{x}+\beta\left(C_{0}+C_{1}\bar{x}^{2}+C_{2}\bar{y}^{2}\right)+\mathcal{O}(\beta^{2}),$
(39)

where $C_{0}=a_{0}b_{10}+a_{4}b_{14}+a_{5}b_{15}$,
$C_{1}=a_{4}b_{14}w_{41}+a_{5}b_{15}w_{51}$ and
$C_{2}=a_{4}b_{14}w_{42}+a_{5}b_{15}w_{52}$. The normalisation constants
$a_{0}$, $a_{4}$, $a_{5}$ are listed in table 3 and the eigenfunction-related
terms $w_{41}$, $w_{42}$, $w_{51}$, $w_{52}$ are specified in the appendix.
The coefficients $b_{10}$, $b_{14}$, $b_{15}$, given by (33), take the
following forms

$\displaystyle b_{10}$ $\displaystyle=$
$\displaystyle-\frac{a_{0}a_{1}\pi}{4n^{2}(n+1)(\sigma_{0}^{2}-\sigma_{1}^{2})\bar{\omega}_{\theta
0}^{3}\bar{\omega}_{r0}^{3}}\left\\{L_{109}\bar{\omega}_{r0}^{2}+\left[L_{108}+2n(m-\bar{\omega}_{r0})\times\right.\right.$
(40)
$\displaystyle\left.\left.\times\left\\{\mathcal{A}_{11}(m-\bar{\omega}_{r0})-2m\Omega_{11}\right\\}+2L_{107}(n+1)\bar{\omega}_{r0}^{2}\right]\bar{\omega}_{\theta
0}^{2}\right\\},$ $\displaystyle b_{1q}$ $\displaystyle=$
$\displaystyle\frac{a_{1}a_{q}\pi}{8n^{2}(n+1)(n+2)(\sigma_{1}^{2}-\sigma_{q}^{2})\bar{\omega}_{\theta
0}^{5}\bar{\omega}_{r0}^{5}}\left[\\{L_{108}+2n\mathcal{A}_{11}(m-\bar{\omega}_{r0})^{2}\right.$
(41)
$\displaystyle\left.-4m^{2}n\Omega_{11}+4mn\Omega_{11}\bar{\omega}_{r0}\\}\times\\{[2(n+2)\bar{\omega}_{r0}^{2}+3W_{q1}]\bar{\omega}_{\theta
0}^{2}\right.$
$\displaystyle\left.+\bar{\omega}_{r0}^{2}W_{q2}\\}\bar{\omega}_{\theta
0}^{2}+L_{109}\\{[2(n+2)\bar{\omega}_{r0}^{2}+W_{q1}]\bar{\omega}_{\theta
0}^{2}+3\bar{\omega}_{r0}^{2}W_{q2}\\}\bar{\omega}_{r0}^{2}\right.$
$\displaystyle\left.+2L_{107}(n+2)\bar{\omega}_{\theta
0}^{2}\bar{\omega}_{r0}^{2}\\{[2(n+1)\bar{\omega}_{r0}^{2}+W_{q1}]\bar{\omega}_{\theta
0}^{2}+\bar{\omega}_{r0}^{2}W_{q2}\\}\right],$

where $q=4$ or $q=5$ in case of $b_{14}$ or $b_{15}$, respectively, and
$\sigma_{0}$, $\sigma_{1}$, $\sigma_{4}$, $\sigma_{5}$ are specified in table
2.

Table 2:  Eigenfrequencies of the eigenmodes given in table 1.

  * $i$ | $\sigma_{i}^{2}$
---|---
0 | 0
1 | $\bar{\omega}_{r0}^{2}$
2 | $\bar{\omega}_{\theta 0}^{2}$
3 | $\bar{\omega}_{r0}^{2}+\bar{\omega}_{\theta 0}^{2}$
4 | $\left\\{(2n+1)(\bar{\omega}_{r0}^{2}+\bar{\omega}_{\theta 0}^{2})-[4n(n+1)(\bar{\omega}_{\theta 0}^{2}-\bar{\omega}_{r0}^{2})^{2}+(\bar{\omega}_{r0}^{2}+\bar{\omega}_{\theta 0}^{2})^{2}]^{1/2}\right\\}/(2n)$
5 | $\left\\{(2n+1)(\bar{\omega}_{r0}^{2}+\bar{\omega}_{\theta 0}^{2})+[4n(n+1)(\bar{\omega}_{\theta 0}^{2}-\bar{\omega}_{r0}^{2})^{2}+(\bar{\omega}_{r0}^{2}+\bar{\omega}_{\theta 0}^{2})^{2}]^{1/2}\right\\}/(2n)$

Similarly, for the vertical epicyclic mode ($i=2$) we find one non-zero
coefficient corresponding to $j=3$ and receive the final expression for the
eigenfunction $W_{\theta}$ of the vertical epicyclic mode,

$W_{\theta}=a_{2}\bar{y}+\beta C_{3}\bar{x}\bar{y}+\mathcal{O}(\beta^{2})$
(42)

with $C_{3}=a_{3}b_{23}$. Again, $a_{3}$ is the normalisation constant (see
table 3) and the $b_{23}$ coefficient is given by

$\displaystyle b_{23}$ $\displaystyle=$
$\displaystyle\frac{a_{2}a_{3}\pi}{8n^{2}(n+2)(n+1)(\sigma_{2}^{2}-\sigma_{3}^{2})\bar{\omega}_{\theta
0}^{3}\bar{\omega}_{r0}^{3}}\left\\{L_{110}-2n(m-\bar{\omega}_{\theta
0})\times\right.$ (43)
$\displaystyle\left.\times\left[\mathcal{A}_{11}(\bar{\omega}_{\theta
0}-m)+2m\Omega_{11}\right]\right\\}$

with $\sigma_{2}$, $\sigma_{3}$ listed in table 2.

Table 3:  Normalisation constants of the eigenmodes in table 1 (Blaes et al,
2007). They are calculated such that the eigenfunctions are normalised in the
inner product (29).

  * $i$ | $a_{i}$
---|---
0 | $\left({n\bar{\omega}_{r0}\bar{\omega}_{\theta 0}/\pi}\right)^{1/2}$
1 | $a_{0}[2(n+1)\bar{\omega}_{r0}^{2}]^{1/2}$
2 | $a_{0}[2(n+1)\bar{\omega}_{\theta 0}^{2}]^{1/2}$
3 | $a_{0}[4(n+1)(n+2)\bar{\omega}_{r0}^{2}\bar{\omega}_{\theta 0}^{2}]^{1/2}$
4 | $a_{0}\left\\{{(n+2)[\sigma_{4}^{2}-(\bar{\omega}_{\theta 0}^{2}+\bar{\omega}_{r0}^{2})]/\left[2n\sigma_{4}^{2}-(2n+1)(\bar{\omega}_{\theta 0}^{2}+\bar{\omega}_{r0}^{2})\right]}\right\\}^{1/2}$
5 | $a_{0}\left\\{{(n+2)[\sigma_{5}^{2}-(\bar{\omega}_{\theta 0}^{2}+\bar{\omega}_{r0}^{2})]/\left[2n\sigma_{5}^{2}-(2n+1)(\bar{\omega}_{\theta 0}^{2}+\bar{\omega}_{r0}^{2})\right]}\right\\}^{1/2}$

Then, using (35)-(38), we find the terms

$\displaystyle\mathcal{A}^{(1)}|W_{1}^{(0)}\rangle=a_{1}\mathcal{A}_{11}\bar{x}^{2},\qquad\qquad$
$\displaystyle\mathcal{A}^{(1)}|W_{2}^{(0)}\rangle=a_{2}\mathcal{A}_{11}\bar{x}\bar{y},$
$\displaystyle\bar{\Omega}^{(1)}|W_{1}^{(0)}\rangle=a_{1}\Omega_{11}\bar{x}^{2},\qquad\qquad$
$\displaystyle\bar{\Omega}^{(1)}|W_{2}^{(0)}\rangle=a_{2}\Omega_{11}\bar{x}\bar{y},$
$\displaystyle\hat{L}^{(1)}|W_{1}^{(0)}\rangle=a_{1}(L_{107}+L_{108}\bar{x}^{2}+L_{109}\bar{y}^{2}),\qquad\qquad$
$\displaystyle\hat{L}^{(1)}|W_{2}^{(0)}\rangle=a_{2}L_{110}\bar{x}\bar{y}.$
(44)

Substituting (44) into the formula (34) for $\omega_{i}^{(1)}$, we obtain in
the inner product for both modes $i=1$ and $i=2$ odd functions of $\bar{x}$
and $\bar{y}$, such that the integration over the elliptical torus cross-
section yields zero. Therefore, to find the relevant pressure corrections to
the radial and vertical mode frequencies, one needs to extend the expansion to
second-order in torus thickness.

####  4.2.2 Second-order corrections

The perturbation equation (20) expanded to second-order in $\beta$ reads

$\displaystyle\hat{L}^{(0)}W_{i}^{(2)}+\hat{L}^{(1)}W_{i}^{(1)}+\hat{L}^{(2)}W_{i}^{(0)}$
$\displaystyle=$
$\displaystyle-2n\left\\{\sigma_{i}^{2}W_{i}^{(2)}+(\sigma_{i}^{2}\mathcal{A}^{(1)}\right.$
(45)
$\displaystyle\left.-2m\sigma_{i}\bar{\Omega}^{(1)})W_{i}^{(1)}+\left[\sigma_{i}^{2}\mathcal{A}^{(2)}\right.\right.$
$\displaystyle\left.\left.+m^{2}\left(\bar{\Omega}^{(1)}\right)^{2}-2m\sigma_{i}\bar{\Omega}^{(2)}+2\sigma_{i}\bar{\omega}_{i}^{(2)}\right.\right.$
$\displaystyle\left.\left.-2m\sigma_{i}\mathcal{A}^{(1)}\bar{\Omega}^{(1)}\right]W_{i}^{(0)}\right\\}.$

Analogous to the first-order case one may write

$W_{i}^{(2)}=\sum_{j}{c_{ij}W_{j}^{(0)}}$ (46)

and take the inner product of (45) with a zeroth-order eigenfunction
$W_{k}^{(0)}$. For a subscript $k$ referring to the mode of interest ($k=i$)
we obtain the formula for the second-order correction,

$\displaystyle\bar{\omega}_{i}^{(2)}$ $\displaystyle=$
$\displaystyle-\frac{1}{4n\sigma_{i}}\left[\left\langle
W_{i}^{(0)}|\hat{L}^{(2)}+2nm^{2}\left(\bar{\Omega}^{(1)}\right)^{2}-4nm\sigma_{i}\bar{\Omega}^{(2)}\right.\right.$
(47)
$\displaystyle\left.\left.+2n\sigma_{i}^{2}\mathcal{A}^{(2)}-4nm\sigma_{i}\mathcal{A}^{(1)}\bar{\Omega}^{(1)}|W_{i}^{(0)}\right\rangle\right.$
$\displaystyle\left.+\sum_{j}b_{ij}\left\langle
W_{i}^{(0)}|\hat{L}^{(1)}+2n\sigma_{i}^{2}\mathcal{A}^{(1)}-4nm\sigma_{i}\bar{\Omega}^{(1)}|W_{j}^{(0)}\right\rangle\right].$

The second-order terms of $\mathcal{A}$, $\bar{\Omega}$, $f$ and the $\hat{L}$
operator take now the form

$\displaystyle\mathcal{A}^{(2)}$ $\displaystyle=$
$\displaystyle\mathcal{A}_{21}\bar{x}^{2}+\mathcal{A}_{22}\bar{y}^{2},$ (48)
$\displaystyle\bar{\Omega}^{(2)}$ $\displaystyle=$
$\displaystyle\Omega_{21}\bar{x}^{2}+\Omega_{22}\bar{y}^{2},$ (49)
$\displaystyle f^{(2)}$ $\displaystyle=$ $\displaystyle
f_{21}\bar{x}^{4}+f_{22}\bar{x}^{2}\bar{y}^{2}+f_{23}\bar{y}^{4},$ (50)
$\displaystyle\hat{L}^{(2)}$ $\displaystyle=$
$\displaystyle\left\\{L_{201}\bar{x}^{4}+L_{202}\bar{x}^{2}+L_{203}\bar{x}^{2}\bar{y}^{2}+L_{204}\bar{y}^{2}+L_{205}\bar{y}^{4}\right\\}\partial_{\bar{x}}^{2}$
(51)
$\displaystyle+\left\\{L_{206}\bar{x}^{4}+L_{207}\bar{x}^{2}+L_{208}\bar{x}^{2}\bar{y}^{2}+L_{204}\bar{y}^{2}+L_{209}\bar{y}^{4}\right\\}\partial_{\bar{y}}^{2}$
$\displaystyle+\left\\{L_{210}\bar{x}^{3}+L_{211}\bar{x}+L_{212}\bar{x}\bar{y}^{2}\right\\}\partial_{\bar{x}}$
$\displaystyle+\left\\{L_{213}\bar{x}^{2}\bar{y}+L_{214}\bar{y}+L_{215}\bar{y}^{3}\right\\}\partial_{\bar{y}}$
$\displaystyle+\left\\{L_{216}+L_{217}\bar{x}^{2}+L_{218}\bar{y}^{2}\right\\},$

with the coefficients $\mathcal{A}_{21}$, $\mathcal{A}_{22}$, $\Omega_{21}$,
$\Omega_{22}$, $f_{21}$, $f_{22}$, $f_{23}$ and $L_{201}$ \- $L_{218}$ again
specified in the appendix.

Inserting all first and second-order expansion terms into (47) we gain a
fairly long expression for $\bar{\omega}_{1}^{(2)}$. Along with (22) it leads
to the resulting formula for the eigenfrequency $\bar{\omega}_{r}$ of the
radial mode. In order to keep the overview we write the expression in the
following form

$\displaystyle\bar{\omega}_{r}$ $\displaystyle=$
$\displaystyle\bar{\omega}_{r0}+m-\frac{\beta^{2}}{4n\sigma_{1}}\sum_{l=1}^{4}P_{l}+\mathcal{O}(\beta^{3})$
(52)

where

$\displaystyle P_{1}$ $\displaystyle\equiv$ $\displaystyle\langle
W_{1}^{(0)}|\hat{L}^{(2)}+2nm^{2}\left(\bar{\Omega}^{(1)}\right)^{2}-4nm\sigma_{1}\bar{\Omega}^{(2)}+2n\sigma_{1}^{2}\mathcal{A}^{(2)}$
$\displaystyle-4nm\sigma_{1}\mathcal{A}^{(1)}\bar{\Omega}^{(1)}|W_{1}^{(0)}\rangle$
$\displaystyle=$
$\displaystyle\frac{a_{1}^{2}\pi}{4n(n+1)(n+2)\bar{\omega}_{\theta
0}^{3}\bar{\omega}_{r0}^{5}}\left\\{\left(L_{212}+L_{218}\right)\bar{\omega}_{r0}^{2}\right.$
(53)
$\displaystyle\left.+\left[3\left(L_{210}+L_{217}+2m^{2}n\Omega_{11}^{2}\right)\right.\right.$
$\displaystyle\left.\left.+2\left(L_{211}+L_{216}\right)\left(n+2\right)\bar{\omega}_{r0}^{2}\right]\bar{\omega}_{\theta
0}^{2}\right.$
$\displaystyle\left.-4mn\left[3(\mathcal{A}_{11}\Omega_{11}+\Omega_{21})\bar{\omega}_{\theta
0}^{2}+\Omega_{22}\bar{\omega}_{r0}^{2}\right]\bar{\omega}_{r0}\right.$
$\displaystyle\left.+2n\left(3\mathcal{A}_{21}\bar{\omega}_{\theta
0}^{2}+\mathcal{A}_{22}\bar{\omega}_{r0}^{2}\right)\bar{\omega}_{r0}^{2}\right\\},$

$\displaystyle P_{2}\,$ $\displaystyle\equiv$ $\displaystyle\,b_{10}\,\langle
W_{1}^{(0)}|\hat{L}^{(1)}+2n\sigma_{1}^{2}\mathcal{A}^{(1)}-4nm\sigma_{1}^{2}\bar{\Omega}^{(1)}|W_{0}^{(0)}\rangle$
(54) $\displaystyle=$
$\displaystyle\frac{a_{0}a_{1}b_{10}\pi\bar{\omega}_{r0}\left(\mathcal{A}_{11}\bar{\omega}_{r0}-2m\Omega_{11}\right)}{(n+1)\bar{\omega}_{\theta
0}\bar{\omega}_{r0}^{3}},$

$\displaystyle P_{q-1}\,$ $\displaystyle\equiv$
$\displaystyle\,b_{1q}\,\langle
W_{1}^{(0)}|\hat{L}^{(1)}+2n\sigma_{1}^{2}\mathcal{A}^{(1)}-4nm\sigma_{1}^{2}\bar{\Omega}^{(1)}|W_{q}^{(0)}\rangle$
(55) $\displaystyle=$ $\displaystyle
b_{1q}\frac{a_{1}a_{q}\pi}{2n(n+1)(n+2)\bar{\omega}_{\theta
0}^{3}\bar{\omega}_{r0}^{5}(\bar{\omega}_{\theta
0}^{2}-\bar{\omega}_{r0}^{2})}\left\\{-2(n+1)\bar{\omega}_{r0}^{2}\bar{\omega}_{\theta
0}^{2}\times\right.$ $\displaystyle\left.\times\left[3\bar{\omega}_{\theta
0}^{2}(L_{102}-L_{105}+L_{108})+2(n+2)\bar{\omega}_{\theta
0}^{2}\bar{\omega}_{r0}^{2}(L_{101}-L_{104}\right.\right.$
$\displaystyle\left.\left.+L_{107})+\bar{\omega}_{r0}^{2}(L_{103}-L_{106}+L_{109}-L_{110})\right]+n\sigma_{q}^{2}\left\\{\bar{\omega}_{r0}^{4}\left[L_{103}\right.\right.\right.$
$\displaystyle\left.\left.\left.+L_{109}+2(2+n)\bar{\omega}_{\theta
0}^{2}(L_{101}+L_{107})\right]-\bar{\omega}_{r0}^{2}\bar{\omega}_{\theta
0}^{2}\left[L_{106}+L_{110}\right.\right.\right.$
$\displaystyle\left.\left.\left.+2(n+2)\bar{\omega}_{\theta
0}^{2}L_{104}\right]+3\bar{\omega}_{r0}^{2}\bar{\omega}_{\theta
0}^{2}(L_{102}+L_{108})-3\bar{\omega}_{\theta 0}^{4}L_{105}\right\\}+\right.$
$\displaystyle\left.n(\bar{\omega}_{\theta
0}^{2}-\bar{\omega}_{r0}^{2})\bar{\omega}_{r0}(\mathcal{A}_{11}\bar{\omega}_{r0}-2m\Omega_{11})\left[2(n+2)\bar{\omega}_{r0}^{2}\bar{\omega}_{\theta
0}^{2}\right.\right.$ $\displaystyle\left.\left.+3\bar{\omega}_{\theta
0}^{2}w_{q1}+\bar{\omega}_{r0}^{2}w_{q2}\right]\right\\}.$

Here $q=4$ or $q=5$ in case of $P_{3}$ or $P_{4}$, respectively.

For the vertical epicyclic mode we find a much shorter expression,

$\displaystyle\bar{\omega}_{\theta}$ $\displaystyle=$
$\displaystyle\bar{\omega}_{\theta
0}+m-\beta^{2}\frac{a_{2}^{2}\pi}{128n^{4}(n+1)^{2}(n+2)^{2}\left(\sigma_{2}^{3}-\sigma_{2}\sigma_{3}^{2}\right)\omega_{\theta}^{8}\omega_{r}^{6}}\times$
(56) $\displaystyle\times a_{3}^{2}\pi\left[L_{110}+2n\bar{\omega}_{\theta
0}\left(\mathcal{A}_{11}\bar{\omega}_{\theta
0}-2m\Omega_{11}\right)\right]\left\\{3L_{109}\bar{\omega}_{r0}^{2}+\bar{\omega}_{\theta
0}^{2}\left[L_{108}\right.\right.$
$\displaystyle\left.\left.+L_{110}+2L_{107}(n+2)\bar{\omega}_{r0}^{2}+2n\bar{\omega}_{\theta
0}(\mathcal{A}_{11}\bar{\omega}_{\theta 0}-2m\Omega_{11})\right]\right\\}$
$\displaystyle-8n^{2}(n+1)(n+2)\bar{\omega}_{\theta
0}^{3}\bar{\omega}_{r0}^{3}\left(\sigma_{2}^{2}-\sigma_{3}^{2}\right)\left\\{3\left(L_{215}+L_{218}\right)\bar{\omega}_{r0}^{2}+\bar{\omega}_{\theta
0}^{2}\left[L_{213}\right.\right.$
$\displaystyle\left.\left.+L_{217}+2m^{2}n\Omega_{11}^{2}+2\left(L_{214}+L_{216}\right)(n+2)\bar{\omega}_{r0}^{2}\right]-4mn\left[\left(\mathcal{A}_{11}\Omega_{11}\right.\right.\right.$
$\displaystyle\left.\left.\left.+\Omega_{21}\right)\bar{\omega}_{\theta
0}^{2}+3\Omega_{22}\bar{\omega}_{r0}^{2}\right]\bar{\omega}_{\theta
0}+2n\left(\mathcal{A}_{21}\bar{\omega}_{\theta
0}^{2}+3\mathcal{A}_{22}\bar{\omega}_{r0}^{2}\right)\bar{\omega}_{\theta
0}^{2}\right\\}$ $\displaystyle+\mathcal{O}(\beta^{3}).$

##  5 The properties of epicyclic modes for non-slender tori

The equations (39), (42), (52) and (56) represent the epicyclic mode
eigenfunctions and eigenfrequencies of non-slender tori as functions of
$r_{0}$, $a$, $m$, $n$ and $\beta$. In this section we use these formulas to
illustrate the behaviour for the axisymmetric ($m=0$) and lowest-order non-
axisymmetric ($m=\pm 1$) epicyclic modes for variable torus thickness and
varying black hole spin value. We take the polytropic index $n=3$ that refers
to a radiation-pressure dominated torus 222Note that varying $n$ makes no
relevant difference to the results..

All figures that are related to frequency behaviour (1-3, 5-7) display the
frequencies defined as $\nu=\omega/2\pi$ 333As it is commonly used, throughout
the paper we refer to quantities $\omega$ as to ’frequencies’ as well,
although in exact notation they should be called ’angular velocities’.. The
horizontal axis in all these figures starts at the radius of the marginally
stable orbit calculated for the appropriate black hole spin.

###  5.1 Axisymmetric epicyclic modes

In the axisymmetric ($m=0$) case the $\phi$-components vanish from the
equations and the problem becomes symmetric with respect to the rotational
axis ($\theta=0$). The corresponding axisymmetric radial and vertical
epicyclic frequencies are shown in figures 1, 2 and 3. The $\beta=0$ line
always refers to the epicyclic frequency of a test particle, while the
$\beta>0$ lines illustrate the behaviour of the two frequencies when the torus
becomes thicker. As a result of the topology of the equipotential surfaces
there is an upper limit on $\beta$ for a given torus pressure maximum above
which no closed equipotential surfaces and consequently no equilibrium tori
may exist. This limit is incorporated into the figures as a dash-dotted line
that defines the region of ’allowed frequencies’ above it and to its right
(inside the shaded region).

Figure 1:  The axisymmetric radial epicyclic frequency as a function of the
torus pressure maximum $r_{0}$ for tori of various thickness that orbit a
black hole of mass $M=10M_{{}_{\odot}}$. The allowed frequencies corresponding
to equilibrium tori lie inside the shaded region. Left In the Schwarzschild
case $a=0$. Right For a rotating black hole of $a=0.8$.

For any black hole spin both axisymmetric frequencies decrease with increasing
torus thickness. A comparison of the left ($a=0$) and right ($a=0.8$) panels
of figures 1 and 2 and the respective panel of figure 3 ($a=0.999$)
illustrates the influence of the black hole rotation.

The radial frequency qualitatively retains for all spin values the same
profile albeit the torus thickness. The radius where it becomes zero moves,
however, away from the central object. It is interesting to note that for all
values of $a$ the frequency maximum for a fixed $\beta$ is reduced by almost
exactly the same relative amount444This does not apply to the actual thickness
of the torus, since a given $\beta$ implies for different $a$ a different
extent of the torus.. Moreover, for a non-extremely rotating black hole with
$a\lesssim 0.96$, this is more or less true for any location of the pressure
maximum $r_{0}$, while for $a\gtrsim 0.96$ the frequencies at small radii tend
to crowd together, as may be seen in figure 3.

The vertical frequency for a Schwarzschild black hole changes its profile with
increasing torus thickness from a monotonic function in the case of test
particle frequency to a function exhibiting a maximum value (see the left
panel of figure 2). For a Kerr black hole the non-monotonicity is already
present for the test particle frequency (although for low to moderate spin
values the frequency maximum is located at radii inside the marginally stable
orbit) and it remains also for a thicker torus (see the right panels of
figures 2 and 3). At very high spin values ($a\gtrsim 0.96$), the frequency
shape is modified in such a way that the frequency maxima for all tori almost
coincide with the $\beta=0$ curve (figure 3, right panel).

Figure 2:  The same as figure 1, but for the axisymmetric vertical epicyclic
mode.

In order to illustrate the characteristic features of the flow for the modes
of a thicker torus we use equations (18), (39) and (42) to plot the
corresponding poloidal velocity fields. The axisymmetric radial mode shows a
relatively coherent flow with only slight deviations from the radial motion in
the outer regions of the torus (see the left panel in figure 4). The
axisymmetric vertical mode, however, shows a more complex behaviour. As seen
in the right panel of figure 4, it involves vertical motions near the inner
and outer edges of the torus, as well as smaller radial flows in regions close
to the pressure maximum. Its velocity pattern exhibits the features as one of
the lowest-order slender torus modes calculated in Blaes, Arras and Fragile
(2006), the so-called x-mode (see their figure 1). The velocity fields of both
modes keep their characteristics similar to those described above
independently of the black hole spin.

Figure 3:  The same as figure 1, but for the axisymmetric radial (left) and
vertical (right) epicyclic mode in case of a near-extreme Kerr black hole of
$a=0.999$.

Figure 4:  Poloidal flow velocity fields of the axisymmetric radial (left) and
vertical (right) epicyclic mode for a torus of $\beta=0.15$ with pressure
maximum at $r_{0}=8r_{\rm{g}}$ orbiting a Kerr black hole with $a=0.5$.

###  5.2 Non-axisymmetric epicyclic modes

The lowest-order non-axisymmetric ($m=\pm 1$) radial and vertical epicyclic
mode frequencies are shown in figures 5-7. Again, the $\beta=0$ curve in each
case refers to the test particle frequency and the dash-dotted line specifies
the region of allowed frequencies (inside the shaded region). Like above, the
left panels of figures 5 and 6 display the frequencies for tori that orbit a
Schwarzschild black hole ($a=0$), while the corresponding right panels show
the same for a rapidly spinning Kerr black hole ($a=0.8$). Figure 7 then
displays the frequencies for a near-extreme Kerr black hole ($a=0.999$).

Figure 5:  The non-axisymmetric $m=1$ (top ) and $m=-1$ (bottom) radial
epicyclic frequency as a function of $r_{0}$ for different values of $\beta$
for a $M=10M_{{}_{\odot}}$ black hole. Left For a non-rotating black hole.
Right For a spinning black hole of $a=0.8$.

The $m=1$ radial epicyclic mode frequency of a slightly non-slender torus is
given by equation (52). For a test particle ($\beta=0$) it reduces to the sum
of $\Omega_{0}$ and the test particle axisymmetric radial epicyclic frequency
$\omega_{r0}$. For all values of $a$ this frequency decreases with increasing
torus thickness (see the top panels of figure 5 and the top left panel of
figure 7).

The $m=-1$ radial mode frequency, again given by (52), is in the test particle
case represented (in absolute value) by the difference between $\Omega_{0}$
and $\omega_{r0}$. At small radii the frequencies for all spin values increase
with growing torus thickness (see the bottom panels of figure 5 and the bottom
left panel of figure 7). With rising $r_{0}$, the non-slender torus
frequencies start oscillating about the $\beta=0$ frequency and eventually
converge to the test particle profile.

In spherically symmetric spacetimes, the orbital frequency $\Omega_{0}$ and
the axisymmetric vertical epicyclic frequency $\omega_{\theta 0}$ of a test
particle coincide. This is no longer true in case of an axially symmetric
(Kerr) spacetime or the frequencies of a non-slender torus. The non-
axisymmetric $m=1$ vertical epicyclic frequency (given by (56)) for test
particles corresponds to the sum of $\Omega_{0}$ and $\omega_{\theta 0}$. For
a non-slender torus the frequency behaves up to $a\lesssim 0.96$ similarly to
the axisymmetric $\omega_{\theta}$ (compare the top panels of figure 6 to
figure 2). Its form slightly differs only for very high spin values ($a\gtrsim
0.96$) (see the top right panel of figure 7).

The $m=-1$ vertical mode frequency (equation (56)) is for a test particle
given (in absolute value) by the difference between $\Omega_{0}$ and
$\omega_{\theta 0}$. In a Schwarzschild spacetime this difference equals zero
for test particles, but for increased torus thickness the frequency grows
(bottom left panel of figure 6). For a Kerr black hole, the frequency
converges for all spin values to the test particle frequency as the pressure
maximum $r_{0}$ moves away from the black hole. At smaller radii and for
$a\lesssim 0.96$ there is a crossing point with the $\beta=0$ frequency, such
that, in the first interval, rising $\beta$ causes the frequency to increase
in contrast to the second interval where the frequencies show opposite
behaviour (figure 6, bottom right panel). For $a\gtrsim 0.96$ the crossing
point vanishes and rising torus thickness evokes rising frequencies at all
radii (bottom right panel of figure 7).

Poloidal velocity fields of the non-axisymmetric modes are displayed in figure
8. The $m=1$ radial mode velocity field exhibits radial flows originating in
the central regions of the torus and pointing outwards in opposite directions
(top left panel). The $m=-1$ radial mode velocity field has a similar
character, except that the radial flows point inwards (bottom left panel). The
poloidal flow of the $m=1$ vertical mode has analogous features to that in the
axisymmetric case (compare the top right panel of figure 8 to the right panel
of figure 4), while for the $m=-1$ vertical mode the flow is, apart from small
variations, mainly vertical (figure 8, bottom right panel). Once again, the
shape of the velocity fields described here remains preserved for all values
of the black hole spin.

Figure 6:  The same as figure 5, but for the vertical epicyclic mode.

Figure 7:  The non-axisymmetric $m=1$ (top) and $m=-1$ (bottom) epicyclic mode
frequencies in case of a near-extreme Kerr black hole of $a=0.999$. Left The
radial epicyclic mode. Right The vertical epicyclic mode.

Figure 8:  Poloidal velocity fields of the non-axisymmetric epicyclic modes
for the same torus as in figure 4. Top The $m=1$ radial (left) and vertical
(right) epicyclic mode. Bottom The $m=-1$ radial (left) and vertical (right)
epicyclic mode.

Blaes et al (2007) presented the properties of the axisymmetric and non-
axisymmetric radial epicyclic mode frequencies for non-slender tori in a
spherical pseudo-Newtonian potential, likewise based on calculations accurate
to second-order with respect to the torus thickness. Comparing their figures
2, 6, 7 to our figures 1, 2, 3, 5 and 7, one may distinguish the behaviour of
frequencies calculated in a pseudo-Newtonian potential from those calculated
in a fully-relativistic Schwarzschild and Kerr geometry. Our results show that
the radial and vertical mode frequencies in a rotating, axially symmetric
(Kerr) spacetime follow the same trend as those calculated in a non-rotating,
spherically symmetric potential, except that the vertical frequency has in a
rotating spacetime a non-monotonic profile already in the case of test
particles. This non-monotonicity feature is for thicker tori present in both,
axially and spherically symmetric, potentials. Generally, only extremely high
black hole spin values ($a\gtrsim 0.96$) cause slight variations in the
frequency behaviour.

In order to compare the poloidal velocity fields of the axisymmetric modes in
Kerr geometry (our figure 4) to those derived in the pseudo-Newtonian
potential see figure 5 of Blaes et al (2007).

##  6 Conclusions

We have assumed a pressure-supported torus of small radial extent in a Kerr
spacetime that performs epicyclic oscillations and studied the pressure
effects on the epicyclic modes properties, i.e., how the modes eigenfunctions
and eigenfrequencies of a torus differ from those of a free test particle. For
this purpose we calculated the relevant pressure corrections to the
axisymmetric and lowest-order non-axisymmetric epicyclic mode eigenfunctions
and eigenfrequencies within first- (eigenfunctions) and second-order
(eigenfrequencies) accuracy in torus thickness.

In the limit of an infinitely slender torus, the radial and vertical epicyclic
oscillations occur as global oscillations that correspond to purely radial and
vertical displacements of the whole torus at epicyclic frequencies of free
test particles orbiting at the position of the torus pressure maximum
(Abramowicz et al, 2006; Blaes, Arras and Fragile, 2006). Several numerical
studies (e.g. Montero et al, 2004; Rubio-Herrera and Lee, 2005a, b; Šrámková
et al, 2007) have shown that when the torus gets thicker, its (axisymmetric)
oscillations occur at lower frequencies. This has been confirmed recently by
analytic pseudo-Newtonian calculations (Blaes et al, 2007), where pressure
corrections to epicyclic modes of a small-size, constant specific angular
momentum torus in the Paczyński and Wiita (1980) potential were derived. Using
the same approach, but within the framework of general relativity, we extended
their results into the rotating Kerr geometry. For both axisymmetric and the
radial non-axisymmetric oscillations explored in Blaes et al (2007), our
calculations qualitatively confirm the trends as carried out there for a
spherically symmetric potential. As expected (and also demonstrated in Blaes
et al (2007)), the epicyclic mode eigenfunctions of thicker tori no longer
describe a purely radial or vertical displacement since there appear some
deviations of the flow in the poloidal velocity fields for both modes.

The configuration considered here is represented by the idealised model of a
non-selfgravitating, non-accreting, non-magnetised torus with constant
specific angular momentum555Uniform specific angular momentum distribution
throughout the torus is (aside from simplicity reasons) assumed because the
relativistic Papaloizou-Pringle equation for non-constant distributions does
not describe a self-adjoint eigenvalue problem (see equation (26) in Blaes,
Arras and Fragile (2006)), and we do not have the appropriate complete
orthonormal set of eigenfunctions that is necessary to apply the perturbation
method (see also discussion in Blaes et al (2007))., studied within a purely
hydrodynamical regime. These tori are widely known to be dynamically unstable
under global, non-axisymmetric perturbations (Papaloizou & Pringle, 1984).
However, this instability can be suppressed when accretion through the inner
edge of the torus takes place (Blaes, 1987; Blaes and Hawley, 1988). A
possible interpretation is that torus-like structures can be formed within the
innermost regions of a standard, nearly-Keplerian accretion disc where several
physical processes may give rise to pressure gradients that in turn form tori.
Such torus-like accretion flows seem to appear in three-dimensional global MHD
simulations (De Villiers and Hawley et al, 2003; Machida et al, 2006).

Whether purely hydrodynamic modes may exist in the presence of magnetic fields
that surely play a significant role in the physics of accretion flows is an
issue still to be investigated more deeply. A few studies in that context have
been carried out, e.g. by Montero et al (2007) who explored the oscillation
properties of relativistic tori comprising a toroidal magnetic field, and
reported similar results as obtained in previous investigations of non-
magnetised torus oscillations. Their initial set-up, however, did not allow
for development of the magneto-rotational turbulence (MRI). One of the first
attempts to investigate the effects of MRI on the properties of hydrodynamic
oscillation modes via relativistic MHD simulations was done by Fragile (2005).

In general, studying oscillations of black hole accretion discs can improve
our understanding of the origin of the observed X-ray variability. In
particular HF QPOs that are detected in the X-ray light curves of several
X-ray binaries are often attributed to disc oscillations. The results
discussed here can be applied to models that directly involve epicyclic
oscillations. Assuming a particular oscillation model, the identification of
oscillation frequencies with the frequencies of observed QPOs can provide a
precise determination of the mass or spin of the black hole (e.g. Wagoner et
al, 2001; Abramowicz and Kluźniak, 2001; Kato and Fukue, 2006). Black hole
spin estimates for several microquasars have been carried out, based on the
resonance model considering epicyclic oscillations in a thin disc that occur
at frequencies of free test particles (Török et al, 2005). Applying pressure
corrections to epicyclic frequencies, our results should be taken into account
to obtain (more) accurate estimations (Blaes et al, 2007).

This work was supported by the Polish Ministry of Science grant N203 009
31/1466 (OS) and the Czech grant LC06014 (ES). The authors would like to thank
Marek Abramowicz for initiating this work and his invaluable support, as well
as Włodek Kluźniak and Omer Blaes for their kind advice and encouragement. We
also thank Pavel Bakala and Gabriel Török for many useful discussions and
technical help. Then we would like to acknowledge the hospitality of the
Silesian University in Opava and NORDITA in Copenhagen and Stockholm where
most of the work was carried out.

## Appendix

Here we display the coefficients introduced to abbreviate the analytic terms
we derive and use in section 4.

###  A.1 The first-order terms

The coefficients that appear in the first-order expansion terms of $A$,
$\bar{\Omega}$, $f$, $\hat{L}$ in subsection 4.2.1 read

$\displaystyle\mathcal{A}_{11}$ $\displaystyle=$
$\displaystyle-2L_{101}\frac{r_{0}^{1/2}(a^{2}-2ar_{0}^{1/2}+r_{0}^{2})}{(r_{0}-a^{2})(r_{0}^{3/2}+a)},$
(57) $\displaystyle\Omega_{11}$ $\displaystyle=$ $\displaystyle-
L_{101}\frac{r_{0}^{2}(2a-3r_{0}^{1/2}+r_{0}^{3/2})}{(r_{0}-a^{2})(r_{0}^{3/2}+a)},$
(58) $\displaystyle f_{11}$ $\displaystyle=$
$\displaystyle\frac{2}{r_{0}^{4}}\left[\frac{r_{0}^{2}}{a^{2}+(r_{0}-2)r_{0}}\right]^{1/2}\left\\{-a^{4}+2a^{3}r_{0}^{1/2}-4a^{2}(r_{0}-1)r_{0}\right.$
(59)
$\displaystyle\left.+2ar_{0}^{3/2}(5r_{0}-6)+r_{0}^{2}[8+(r_{0}-8)r_{0}]\right\\},$
$f_{12}=\frac{2}{r_{0}^{4}}\left[\frac{r_{0}^{2}}{a^{2}-2r_{0}+r_{0}^{2}}\right]^{1/2}\left(3a^{2}-4ar_{0}^{1/2}+r_{0}^{2}\right)\left(a^{2}-2r_{0}+r_{0}^{2}\right),$
(60) $\displaystyle L_{101}$ $\displaystyle=$
$\displaystyle\frac{2(r_{0}-a^{2})}{r_{0}^{2}}\left[\frac{r_{0}^{2}}{a^{2}+r_{0}(r_{0}-2)}\right]^{1/2},$
(61) $\displaystyle L_{102}$ $\displaystyle=$ $\displaystyle-
L_{101}\bar{\omega}_{r0}^{2}+f_{11},$ (62) $\displaystyle L_{103}$
$\displaystyle=$ $\displaystyle-L_{101}\bar{\omega}_{\theta 0}^{2}+f_{12},$
(63) $\displaystyle L_{104}$ $\displaystyle=$
$\displaystyle-2\left[\frac{r_{0}^{2}}{a^{2}+r_{0}(r_{0}-2)}\right]^{-1/2},$
(64) $\displaystyle L_{105}$ $\displaystyle=$ $\displaystyle-
L_{104}\bar{\omega}_{r0}^{2}+f_{11},$ (65) $\displaystyle L_{106}$
$\displaystyle=$ $\displaystyle-L_{104}\bar{\omega}_{\theta 0}^{2}+f_{12},$
(66) $\displaystyle L_{107}$ $\displaystyle=$
$\displaystyle\frac{r_{0}(r_{0}-1)}{(r_{0}-a^{2})}L_{101},$ (67)
$\displaystyle L_{108}$ $\displaystyle=$
$\displaystyle-(2nL_{101}+L_{107})\bar{\omega}_{r0}^{2}+3nf_{11},$ (68)
$\displaystyle L_{109}$ $\displaystyle=$ $\displaystyle-
L_{107}\bar{\omega}_{\theta 0}^{2}+nf_{12},$ (69) $\displaystyle L_{110}$
$\displaystyle=$ $\displaystyle 2n(f_{12}-L_{104}\bar{\omega}_{\theta
0}^{2}).$ (70)

###  A.2 The second-order terms

The coefficients in the second-order terms of $A$, $\bar{\Omega}$, $f$,
$\hat{L}$ in subsection 4.2.2 take the following forms

$\displaystyle\mathcal{A}_{21}$ $\displaystyle=$
$\displaystyle\frac{1}{\left[2a+\left(r_{0}-3\right)r_{0}^{1/2}\right]\left[a^{2}+\left(r_{0}-2\right)r_{0}\right]\left(a+r_{0}^{3/2}\right)^{2}\,r_{0}^{3/2}}\times$
$\displaystyle\times\left[5a^{6}+2a^{5}\left(3r_{0}-2\right)r_{0}^{1/2}+a^{4}\left[\left(r_{0}+10\right)r_{0}-64\right]r_{0}\right.$
(71)
$\displaystyle\left.+4a^{3}\left[\left(6r_{0}-23\right)r_{0}+40\right]r_{0}^{3/2}+a^{2}\left\\{\left[\left(6r_{0}-67\right)r_{0}+100\right]r_{0}\right.\right.$
$\displaystyle\left.\left.-108\right\\}r_{0}^{2}+2a\left(r_{0}+16\right)r_{0}^{9/2}+\left(5r_{0}-16\right)r_{0}^{6}\right],$
$\displaystyle\mathcal{A}_{22}$ $\displaystyle=$
$\displaystyle-\frac{5a^{3}+a^{2}(r_{0}-8)r_{0}^{1/2}+3ar_{0}^{2}-r_{0}^{7/2}}{[2a+(r_{0}-3)r_{0}^{1/2}](a+r_{0}^{3/2})r_{0}^{3/2}},$
(72) $\displaystyle\Omega_{21}$ $\displaystyle=$
$\displaystyle\frac{a+(r_{0}-2)r_{0}^{1/2}}{[a^{2}+(r_{0}-2)r_{0}](a+r_{0}^{3/2})^{2}}\left[a^{3}-a^{2}(r_{0}+6)r_{0}^{1/2}\right.$
(73) $\displaystyle\left.+3a(3r_{0}+2)r_{0}+3(r_{0}-4)r_{0}^{5/2}\right],$
$\Omega_{22}=1-\frac{2a}{a+r_{0}^{3/2}},$ (74) $\displaystyle f_{21}$
$\displaystyle=$
$\displaystyle\frac{1}{2\left[2a+(r_{0}-3)r_{0}^{1/2}\right]r_{0}^{4}\left[a^{2}+(r_{0}-2)r_{0}\right]}\times\left[8a^{7}\right.$
(75)
$\displaystyle\left.+4a^{5}r_{0}(3r_{0}+8)+a^{6}r_{0}^{1/2}(4r_{0}-37)+a^{4}r_{0}^{3/2}\left[5r_{0}(2r_{0}-17)+74\right]\right.$
$\displaystyle\left.+16a^{3}r_{0}^{2}\left[r_{0}(3r_{0}-1)-5\right]+4ar_{0}^{3}\left\\{r_{0}[(124-21r_{0})r_{0}-192]+88\right\\}\right.$
$\displaystyle\left.+a^{2}r_{0}^{5/2}\left\\{r_{0}\left[r_{0}(32r_{0}-307)+500\right]-188\right\\}\right.$
$\displaystyle\left.+r_{0}^{7/2}\left(r_{0}\left\\{r_{0}\left[(77-6r_{0})r_{0}-286\right]+380\right\\}-168\right)\right],$
$\displaystyle f_{22}$ $\displaystyle=$
$\displaystyle-\frac{1}{r_{0}^{4}(2a-3r_{0}^{1/2}+r_{0}^{3/2})}\times\left(24a^{5}-85a^{4}r_{0}^{1/2}+72a^{3}r_{0}\right.$
(76)
$\displaystyle\left.+34a^{2}r_{0}^{3/2}+12a^{4}r_{0}^{3/2}-48ar_{0}^{2}+2a^{3}r_{0}^{2}-69a^{2}r_{0}^{5/2}+52ar_{0}^{3}\right.$
$\displaystyle\left.+12r_{0}^{7/2}+11a^{2}r_{0}^{7/2}-6ar_{0}^{4}-14r_{0}^{9/2}+3r_{0}^{11/2}\right),$
$\displaystyle f_{23}$ $\displaystyle=$
$\displaystyle\frac{1}{6r_{0}^{4}\left(2a-3r_{0}^{1/2}+r_{0}^{3/2}\right)}\times\left(24a^{5}-63a^{4}r_{0}^{1/2}+24a^{3}r_{0}+24a^{2}r_{0}^{3/2}\right.$
(77)
$\displaystyle\left.+12a^{4}r_{0}^{3/2}-10a^{2}r_{0}^{5/2}-24ar_{0}^{3}+8ar_{0}^{4}+9r_{0}^{9/2}-4r_{0}^{11/2}\right),$
$\displaystyle L_{201}$ $\displaystyle=$ $\displaystyle
f_{21}+L_{101}f_{11}-L_{202}\bar{\omega}_{r0}^{2},$ (78) $\displaystyle
L_{202}$ $\displaystyle=$ $\displaystyle\frac{3a^{2}-2r_{0}}{r_{0}^{2}},$ (79)
$\displaystyle L_{203}$ $\displaystyle=$ $\displaystyle
f_{22}+L_{101}f_{12}-L_{202}\bar{\omega}_{\theta
0}^{2}-L_{204}\bar{\omega}_{r0}^{2},$ (80) $\displaystyle L_{204}$
$\displaystyle=$ $\displaystyle-\frac{a^{2}}{r_{0}^{2}},$ (81) $\displaystyle
L_{205}$ $\displaystyle=$ $\displaystyle f_{23}-L_{204}\bar{\omega}_{\theta
0}^{2},$ (82) $\displaystyle\vskip 12.0pt plus 4.0pt minus 4.0ptL_{206}$
$\displaystyle=$ $\displaystyle
f_{21}+(L_{101}-L_{107})f_{11}-L_{207}\bar{\omega}_{r0}^{2},$ (83)
$\displaystyle L_{207}$ $\displaystyle=$
$\displaystyle\frac{3[a^{2}+(r_{0}-2)r_{0}]}{r_{0}^{2}},$ (84) $\displaystyle
L_{208}$ $\displaystyle=$ $\displaystyle
f_{22}+(L_{101}-L_{107})f_{12}-L_{207}\bar{\omega}_{\theta
0}^{2}-L_{204}\bar{\omega}_{r0}^{2},$ (85) $\displaystyle L_{209}$
$\displaystyle=$ $\displaystyle f_{23}-L_{204}\bar{\omega}_{\theta 0}^{2},$
(86) $\displaystyle\vskip 12.0pt plus 4.0pt minus 4.0ptL_{210}$
$\displaystyle=$ $\displaystyle
L_{101}\left(3n+\frac{r_{0}^{2}-r_{0}}{r_{0}-a^{2}}\right)f_{11}-\left[\frac{6na^{2}-2r_{0}(r_{0}+2n-2)}{r_{0}^{2}}\right]\bar{\omega}_{r0}^{2}$
(87) $\displaystyle+4nf_{21},$ $\displaystyle L_{211}$ $\displaystyle=$
$\displaystyle-2+\frac{4}{r_{0}},$ (88) $\displaystyle L_{212}$
$\displaystyle=$ $\displaystyle
2nf_{22}+L_{101}\left(n+\frac{r_{0}^{2}-r_{0}}{r_{0}-a^{2}}\right)f_{12}-L_{211}\bar{\omega}_{\theta
0}^{2}-2nL_{204}\bar{\omega}_{r0}^{2},$ (89) $\displaystyle\vskip 12.0pt plus
4.0pt minus 4.0ptL_{213}$ $\displaystyle=$ $\displaystyle
2n\left[f_{22}+(L_{101}-L_{107})f_{12}-L_{207}\bar{\omega}_{\theta
0}^{2}\right]+\bar{\omega}_{r0}^{2},$ (90) $\displaystyle L_{214}$
$\displaystyle=$ $\displaystyle-1,$ (91) $\displaystyle L_{215}$
$\displaystyle=$ $\displaystyle 4nf_{23}-(2nL_{204}-1)\bar{\omega}_{\theta
0}^{2},$ (92) $\displaystyle\vskip 12.0pt plus 4.0pt minus 4.0ptL_{216}$
$\displaystyle=$
$\displaystyle\frac{r_{0}}{a^{2}+r_{0}(r_{0}-2)}\left\\{[r_{0}^{3}+a^{2}(r_{0}+2)]\left(\omega_{i}^{(0)}\Omega_{0}\right)^{2}\right.$
(94)
$\displaystyle\left.-{m^{2}(r_{0}-2)-4ma\omega_{i}^{(0)}\Omega_{0}}\right\\},$
$\displaystyle L_{217}$ $\displaystyle=$ $\displaystyle-
L_{216}\bar{\omega}_{r0}^{2},$ (95) $\displaystyle L_{218}$ $\displaystyle=$
$\displaystyle-L_{216}\bar{\omega}_{\theta 0}^{2}.$ (96)

###  A.3 Eigenfunctions-related coefficients

The coefficients $w_{41}$, $w_{42}$, $w_{51}$, $w_{52}$ introduced in table 1
have the form

$\displaystyle w_{41}=-\frac{\bar{\omega}_{r0}^{2}(2\bar{\omega}_{\theta
0}^{2}+2n\bar{\omega}_{\theta 0}^{2}-n\sigma_{4}^{2})}{\bar{\omega}_{\theta
0}^{2}-\bar{\omega}_{r0}^{2}},$ (97) $\displaystyle
w_{42}=\frac{\bar{\omega}_{\theta
0}^{2}(2\bar{\omega}_{r0}^{2}+2n\bar{\omega}_{r0}^{2}-n\sigma_{4}^{2})}{\bar{\omega}_{\theta
0}^{2}-\bar{\omega}_{r0}^{2}},$ (98) $\displaystyle
w_{51}=-\frac{\bar{\omega}_{r0}^{2}(2\bar{\omega}_{\theta
0}^{2}+2n\bar{\omega}_{\theta 0}^{2}-n\sigma_{5}^{2})}{\bar{\omega}_{\theta
0}^{2}-\bar{\omega}_{r0}^{2}},$ (99) $\displaystyle
w_{52}=\frac{\bar{\omega}_{\theta
0}^{2}(2\bar{\omega}_{r0}^{2}+2n\bar{\omega}_{r0}^{2}-n\sigma_{5}^{2})}{\bar{\omega}_{\theta
0}^{2}-\bar{\omega}_{r0}^{2}}.$ (100)

## References

## References

  * Abramowicz and Kluźniak (2001) Abramowicz M A and Kluźniak W 2001 Astron. Astrophys. 374 L19
  * Abramowicz et al (2006) Abramowicz M A, Blaes O M, Horák J, Kluźniak W and Rebusco P 2006 Class. Quantum Grav.23 1689
  * Aliev and Galtsov (1981) Aliev A N and Galtsov D V 1981 Gen. Rel. Grav.  13 899
  * Blaes (1985) Blaes O M 1985 Mon. Not. R. Astron. Soc. 216 553
  * Blaes (1987) Blaes O M 1987 Mon. Not. R. Astron. Soc. 227 975
  * Blaes and Hawley (1988) Blaes O M and Hawley J F 1988 Astrophys. J.  326 277
  * Blaes, Arras and Fragile (2006) Blaes O M, Arras P and Fragile P C 2006 Mon. Not. R. Astron. Soc. 369 1235
  * Blaes et al (2007) Blaes O M, Šrámková E, Abramowicz M A, Kluźniak W and Torkelsson U 2007 ApJ 665 642
  * De Villiers and Hawley et al (2003) De Villiers J P and Hawley J F 2003 Astrophys. J.  665 642
  * Fragile (2005) Fragile P C 2005 eprint arXiv:astro-ph/0503305
  * Fragile et al (2008) Fragile P C, Lindner C C, Anninos P, Salmonson J D 2008 eprint  arXiv:0809.3819
  * Kato et al (1998) Kato S, Fukue J and Mineshige S 1998 Kyoto Univ. press Kyoto 1998
  * Kato (2001a) Kato S 2001a Publ. Astron. Soc. Japan 53 1
  * Kato (2001b) Kato S 2001b Publ. Astron. Soc. Japan 53 37
  * Kato (2003) Kato S 2003 Publ. Astron. Soc. Japan 55 801
  * Kato (2004) Kato S 2004 Publ. Astron. Soc. Japan 56 905
  * Kato and Fukue (2006) Kato S and Fukue J 2006 Publ. Astron. Soc. Japan 58 909
  * van der Klis (2004) van der Klis M 2004 eprint arXiv:astro-ph/0410551
  * Kluźniak and Abramowicz (2000) Kluźniak W and Abramowicz M A 2000 submitted to Phys. Rev. Lett.  eprint arXiv:astro-ph/0105057
  * Kojima (1986) Kojima Y 1986 Prog. Theor. Phys. 75 L1464
  * Machida et al (2006) Machida M, Nakamura K E and Matsumoto R 2006 Publ. Astron. Soc. Japan 58 193
  * McClintock and Remillard (2003) McClintock J E and Remillard R A 2003 eprint arXiv:astro-ph/0306213
  * Montero et al (2007) Montero P J, Zanotti O, Font J A and Rezzolla L 2007 Mon. Not. R. Astron. Soc. 378 1101
  * Montero et al (2004) Montero P J, Rezzolla L, Yoshida S 2004 Mon. Not. R. Astron. Soc. 354 1040
  * Nowak and Lehr (1998) Nowak M A and Lehr D E 1998 Theory of black hole accretion disks ed M A Abramowicz, G Björnsson and J E Pringle (Cambridge University Press) p 233
  * Paczyński and Wiita (1980) Paczyński B and Wiita P J 1980 Astron. Astrophys. 88 23
  * Papaloizou & Pringle (1984) Papaloizou J C B and Pringle J E 1984 Mon. Not. R. Astron. Soc. 208 721
  * Proga & Begelman (2003) Proga D and Begelman 2003 Astrophys. J. 583 69
  * Rezzolla et al (2003a) Rezzolla L, Yoshida S and Zanotti O 2003a Mon. Not. R. Astron. Soc. 344 978
  * Rezzolla et al (2003b) Rezzolla L, Yoshida S, Maccarone T J and Zanotti O 2003b Mon. Not. R. Astron. Soc. 344 37
  * Rubio-Herrera and Lee (2005a) Rubio-Herrera E and Lee W H 2005a Mon. Not. R. Astron. Soc. 357 L31
  * Rubio-Herrera and Lee (2005b) Rubio-Herrera E and Lee W H 2005b Mon. Not. R. Astron. Soc. 362 789
  * Šrámková et al (2007) Šrámková E, Torkelsson U and Abramowicz M A 2007 Astron. Astrophys. 467 641
  * Stella and Vietri (1998) Stella L and Vietri M 1998 Astrophys. J. Lett.  492 L59
  * Török et al (2005) Török G, Abramowicz M A, Kluźniak W and Stuchlík Z 2005 Astron. Astrophys.  436 1
  * Török and Stuchlík (2005) Török G and Stuchlík Z 2005 Astron. Astrophys.  437 775
  * Wagoner (1999) Wagoner R V 1999 Phys. Rev. 311 259
  * Wagoner et al (2001) Wagoner R V, Silbergleit A S and Ortega-Rodríguez M 2001 Astrophys. J. Lett.  559 L25
  * Zanotti et al (2003) Zanotti O, Rezzolla L and Font J A 2003 Mon. Not. R. Astron. Soc. 341 832