5. Temporal behaviour

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Author: Dr. Marek A. Abramowicz, Physics Department, Göteborg University, Sweden and N. Copernicus Astronomical Center, PAN, Warsaw, Poland
Author: Miss Odele Straub, N. Copernicus Astronomical Center PAN, Warsaw, Poland

One of the key features observed in accretion discs is the strong and often chaotic time variability in the X-ray spectrum. Timing measurements (e.g. by the Rossi X-ray Timing satellite RXTE or the X-ray Multi-Mirror Mission XMM-Newton) have produced power density spectra of the fluctuations in the light flux of binaries and showed low and high frequency peaks in the PSD, known as quasi-periodic oscillations (QPOs). They were first noticed in dwarf novae, i.e., erupting cataclysmic variables, as incoherent pulses with time scales of 30-170s along with coherent periodic oscillations with a period of 20s, the so called dwarf nova oscillations (DNOs).

These flux variations occur at all sorts of time scales and some of them are supposedly caused by an ensemble of waves and oscillation modes in the innermost region of the accretion disc. In general, oscillations are the result of restoring forces acting on perturbations. For instance, if one perturbs a fluid element radially inwards, it conserves its own angular momentum and will be rotating too slow for its new location. Centripetal forces consequently push it outwards again. These kind of inertial oscillations are the epicyclic oscillations. The field of studying the temporal behaviour of discs by means of oscillations is called discoseismology

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Thin discs

One can then express the Eulerian perturbations of all physical quantities through a single function $\delta W \propto \delta p/\rho$ which satisfies a second-order partial differential equation. Since the accretion disc is considered to be stationary and axisymmetric, the angular and time dependences are factored out as $\delta W = W(r,z)e^{i(m\phi - \sigma t)}$ , where the eigenfrequency $\sigma(r,z) = \omega - m\Omega$ and $m$ is the azimuthal wave number. It is assumed that the variation of oscillation modes in radial direction is much stronger than in vertical direction. The resulting two (separated) partial differential equations for the functional amplitude $W(r,z) = W_r(r) W_y(r,y)$ are given by

$\frac{d^2W_r}{dr^2}-\frac{1}{(\omega^2-\omega_r^2)}[\frac{d}{dr}(\omega^2-\omega_r^2)]\frac{dW_r}{dr} +\alpha^2(\omega^2-\omega_r^2)(1-\frac{\Psi}{\tilde{\omega}^2})W_r = 0$ and

$(1-y^2)\frac{d^2W_y}{dy^2}-2gy\frac{dW_y}{dy}+2g\tilde{\omega}^2[1-(1-\frac{\Psi}{\tilde{\omega}^2})(1-y^2)]W_y = 0$ .

The radial eigenfunction, $W_r$ , varies fast with $r$ and the vertical eigenfunction, $W_z$ , varies slowly with $r$ . The radial and vertical epicyclic frequencies are given by $\omega_r(r)$ and $\omega_{\theta}(r)$ , respectively, $\tilde{\omega}(r) = \omega(r)/\omega_{\theta}(r)$ and $\alpha(r) = \frac{dt}{d\tau}\frac{\sqrt{g_{rr}}}{c_s(r,0)}$ , using coefficients of the Kerr metric in Boyer-Lindquist coordinates. $\Psi$ is the eigenvalue of the (WKB) separation function. The radial boundary conditions depend on the type of mode and its capture zone (see below). Oscillations in accretion discs are studied by means of $\Psi(r,\sigma)$ with the angular mode number $m$ , the vertical and radial mode numbers (number of nodes in the corresponding eigenfunction) $j$ and $n$ , respectively.


Figure 1: Schematic picture showing trapped axisymmetric g-modes with n=1 in the region between $r_1$ and $r_2$ , below the radial epicyclic mode $\kappa(r)=\omega_r(r)$ . p-modes only exist above the Keplerian frequency $\Omega_K$ . (Figure credits: Kato et al., 1998)		Figure 2: Schematic picture showing the dependence of three characteristic disc frequencies on the spin $a$ of the black hole. (Figure credits: Wagoner, 1999)

Classification:

A mode oscillates in the radial range outside the inner disc, $r > r_i$ , where

$(\omega^2 - \omega_r^2) (1 - \frac{\Psi}{\tilde{\omega}^2}) > 0$

The two points $r=r_{\pm}(m, a, \sigma)$ refer to the location where the Lindblad resonances occur.

p-modes are inertial acoustic modes defined by $\Psi < \tilde{\omega}^2$ and are trapped where $\omega^2 > \omega_r^2$ in two zones between the inner ( $r_i$ ) and outer ( $r_o$ ) radius of the disc. The inner p-modes are trapped between the inner disc edge and the radial epicyclic frequency at $r_i < r < r_-$ where the gas already is accreted rapidly. The outer p-modes occur at $r_+ < r < r_o$ . The latter, shown in Figure 1, produce the stronger luminosity modulation. In the corotating frame these modes appear at frequencies slightly higher that the radial epicyclic frequency. Pressure is their main restoring force.

g-modes are inertial gravity modes defined by $\Psi > \tilde{\omega}^2$ . They are trapped where $\omega^2 < \omega_r^2$ in the zone $r_- < r < r_+$ given by the radial dependence of $\omega_r$ , i.e., g-modes are gravitationally captured in the cavity of the radial epicyclic frequency and are thus the most robust among the modes. Since this is the region of the temperature maximum of the disc, g-modes are expected to be observed best. In the corotating frame these modes appear at low frequencies. Gravity is their main restoring force. - In Figure 1 $r_1=r_-$ and $r_2=r_+$ and in figure 2 the spin dependency is plotted for $m=0$ .

c-modes are corrugation modes defined by $\Psi = \tilde{\omega}^2$ . They are non-radial ( $m=1$ ) and vertically incompressible modes that appear near the inner disc edge and precess slowly around the rotational axis. The c-modes are controlled by the radial dependence of the vertical epicyclic frequency. In the corotating frame these modes appear at highest frequencies.

All modes have frequencies $\propto 1/M$ . Upon the introduction of a small viscosity ( $\nu \propto \alpha, \,\, \alpha << 1$ ) most modes grow at a dynamical timescale such that the disc should become unstable.

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Thick discs

Geometrically thick accretion discs, i.e. accretion tori, always allow axisymmetric, incompressible modes corresponding to global oscillations of the entire torus at radial ( $\sigma=\omega_r$ ) and vertical ( $\sigma=\omega_{\theta}$ ) epicyclic frequencies. Other possible modes, provided $m=0$ , are basically acoustic (p-), surface gravity (g-) and internal inertial (c-) modes and can be found by solving the relativisic Papaloizou-Pringle equation

$\frac{1}{(-g)^{1/2}}\left\{\partial_{\mu}\left[(-g)^{1/2}g^{\mu\nu}f^n\partial_{\nu}W\right]\right\} - \left(m^2g^{\phi\phi} - 2m\omega g^{t\phi} + \omega^2 g^{tt}\right)f^n W = -\frac{2nA(\bar{\omega}-m\bar{\Omega})^2}{\beta^2 r^2_0}f^{n-1}W$

together with the boundary condition that the Lagrangian perturbation in pressure at the unperturbed surface ( $f=0$ ) vanishes:

$\Delta p = (\delta p + \xi^{\alpha}\nabla_{\alpha}p) = 0$

Classification:

$\times$ -modes are surface gravity modes (k=2) derived from an eigenfunction $W = a xy$ , for some constant $a$ , which is odd in $x$ and $y$ and results in two modes.

$\bar{\sigma}_0^2 = \frac{1}{2}\{\omega_r^2+\omega_{\theta}^2 \pm [(\omega_r^2+\omega_{\theta}^2)^2+4\kappa_0^2\omega_{\theta}^2]^{1/2}\},$

where $\bar{\sigma}_0^2=\sigma_0/\Omega_0$ taken at the location $r_0$ of the pressure maximum in the torus centre (hence the index 0). $\kappa_0^2 = \frac{\mathcal{E}_0^2}{l_0 A_0^2}(\frac{g^{tt}_{,r}-l_0 g^{t\phi}_{,r}}{g_{rr}}\frac{dl}{dr})$ is the squared frequency of the inertial oscillation in the fluid due to an angular momentum gradient. For constant angular momentum distribution this term vanishes. The positive square root of which gives the x-mode that is a surface gravity mode (Figure 8). The negative square root gives a purely incompressible inertial (c-) mode whose poloidal velocity field represents a circulation around the pressure maximum.

breathing- and $+$ -modes are derived from an eigenfunction $W = a + bx + cy$ , for some constants $a,b,c$ . The resulting eigenfrequencies are (for $b=c=0$ ) the zero corotation frequency mode as well as

$\bar{\sigma}_0^2 = \frac{1}{2n}\{(2n+1)(\omega_r^2+\omega_{\theta}^2)-(n+1)\kappa_0^2 \pm [ ((2n+1)(\omega_{\theta}^2-\omega_r^2)^2 + (n+1)\kappa_0^2)^2 + 4(\omega_r^2 - \kappa_0^2) \omega_{\theta}^2 ]^{1/2} \},$

- breathing - modes have frequencies corresponding to the upper sign in the above equation. The torus cross section contracts and expands (Figure 6). Breathing modes are comparable to acoustic modes (k=0, j=1) in the incompressible Newtonian limit for $l=const.$ , while in the Keplerian limit the mode frequency becomes that of a vertical acoustic wave.

- $+$ -modes have frequencies corresponding to the lower sign. In the incompressible $n\rightarrow 0$ limit they are comparable to (k=2) gravity modes.

Figure 3: Non-oscillating torus	Figure 4: Radial epicyclic oscillations	Figure 5: Vertical epicyclic oscillations
Figure 6: $\times$ -mode oscillations	Figure 7: breathing mode oscillations	Figure 8: $+$ -mode oscillations

To non-axisymmetric oscillation modes, however, accretion tori are dynamically unstable. The instability, discovered by Papaloizou & Pringle (1984), affects all torus configurations, and most violently tori with a constant angular momentum distribution.

Whether or not hydrodynamical oscillation modes may survive such global instabilities or the presence of a weak magnetic field (MRI turbulence), is subject of current, numerical investigations.

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5. Temporal behaviour

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