All Figures

$\begin{figure} \par\includegraphics[width=8cm,clip]{9720fig1}\end{figure}$ Figure 1: Illustrating the correspondence between the ignition moments of the elementary events and the resulting light curve. The model is fully determined by a set of points in ( $t-\tau$ ) plane, representing the pairs of ignition time t versus the time constant $\tau$ of each event (panel a)), and the elementary light curves with the profile $I(t,\tau)=\left(t/\tau\right)^2 \exp\left(-t/\tau\right)\theta(t)$ (panel b)). The total light curve (panel c)) is represented as a sum of the individual contributions.
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In the text

$\begin{figure} \par\includegraphics[width=8.7cm,clip]{9720fig2}\end{figure}$ Figure 2: Distinguishing between the Chinese process a) and the Hawkes processes b). In both panels, points represent ignition times of the spots. Each sequence starts with a spontaneously generated parent spot (open circles) and it continues with subsequent secondary ones (filled circles). Arrows symbolise the parent-daughter relation. The difference between the two scenarios is described in Sects. 4.1 and 4.2, respectively. Within the schematic level of this graph, the pulse avalanche process (see Sect. 4.3) belongs also to case b).
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In the text

$\begin{figure} \par\includegraphics[width=14.5cm,clip]{9720fig3.eps}\end{figure}$ Figure 3: Power spectra from the spot model in which the birth and duration of spots are governed by the market Hawkes process with the exponential infectivity (76). Values of $\alpha$ are given on top of the plots. Frequency is given in geometrical units (see the text for its scaling to physical units). In each frame, the four curves correspond to different values of $\nu$ ; from bottom to top $\nu =0.1$ , 0.5, 0.9, and 0.99. The mean number of spontaneous spots has been set to $\lambda =(2\pi )^{-1}$ . The relative normalisation of these curves scales as the mean number of all spots, i.e. proportionally to $1/(1-\nu )$ . Left: the case when all profiles I(t) are identical; $\tau =1$ , i.e. $\zeta (\tau )=\delta (\tau -1)$ . Right: the case when the life-times of spots are distributed uniformly, i.e. according to $\zeta (\tau )=1/(\tau _{\rm max}-\tau _{\rm min})$ , between $\tau _{\rm min}=0.01$ and $\tau _{\rm max}=1$ .
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In the text

$\begin{figure} \par\includegraphics[width=14.5cm,clip]{9720fig4.eps}\end{figure}$ Figure 4: Graphs of the model PSD as in the previous figure, but for the case of Hawkes process with the power law infectivity (80). Parameter values $\beta$ are given on top of each panel; values of $\nu$ are as in Fig. 3. All spots have the same (exponential) profile, as in the previous figure. For $\beta~{\protect \ga}~1.5$ we notice the new profile resembles a power law that develops in between the two peaks. However, this situation is rather rare within the parameter space of our models. By parameter tuning - e.g., by setting $\nu \rightarrow 1$ , which means enhancing the contribution of avalanches while suppressing the importance of spontaneous parent spots - the extent of the flat part of the PSD profile (like the one seen for $\beta =1.5$ ) can be stretched farther towards small frequencies.
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In the text

$\begin{figure} \par\includegraphics[width=16.5cm,clip]{9720fig5.eps}\end{figure}$ Figure 5: Decomposition of the total PSD curve, $S_{\!\rm t}(\omega)$ , into a product of two terms which are responsible for the two peaks in the final profile (see Eq. (83)). Left: this case corresponds to $\alpha =10$ , $\nu =0.5$ curve shown in the bottom-left panel of Fig. 3. Right: this case corresponds to $\beta =1.9$ and $\nu =0.99$ curve in the middle-right panel of Fig. 4. Similar behaviour of the PSD profile is typical in our model, although in some cases only one peak dominates the spectrum whereas the other one is suppressed.
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In the text

$\begin{figure} \par\includegraphics[width=14.5cm,clip]{9720fig6.eps}\end{figure}$ Figure 6: The pulse avalanche model for the uniform probability density $\zeta (\tau )$ , constrained by limits $\tau _{\rm min}=0.01$ and $\tau _{\rm max}=1$ . Left: the PSD calculated from the formulae (33) and (86). Right: the ratio between the PSD of the pulse avalanche process to the Hawkes process; in the latter the exponential infectivity $\mu (t)$ is related to the corresponding infectivity of the pulse avalanche process by $\mu(t)=\mu(t~\vert~\bar{\tau})$ .
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In the text

$\begin{figure} \par\includegraphics[width=16.5cm,clip]{9720fig7.eps}\end{figure}$ Figure 7: Upper panels: power spectra from the Poisson-driven spot model, calculated for a thin accretion disc extending between radii r=6 M and r=100 M (in geometrical units), for three inclinations $\theta _{\rm o}$ . The purpose of this plot is to demonstrate a general agreement between the PSD calculated from the model light curve and from the analytical formula. The wiggly (red) curve is a result of direct numerical simulation, including the relativistic effects. The smooth (blue) curve is the analytical result, derived from Eq. (14) by specifying the probability density function $\zeta (\tau )\propto 1/\tau$ . The vertical (magenta) lines represent the Keplerian orbital frequencies, $\Omega (r)$ , at the inner and the outer edges of the disc. Lower panels: power spectra from the Hawkes process with the exponential infectivity ( $\alpha =7$ , $\nu =0.8$ ). The analytical curve was calculated by using formula (33) and assuming the same probability density function $\zeta (\tau )\propto 1/\tau$ . In all panels we set $\tau _{\rm min}=300$ , $\tau _{\rm max}=5000$ .
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In the text

$\begin{figure} \par\includegraphics[width=16.7cm,clip]{9720fig8.eps}\end{figure}$ Figure 8: The analytical curves of the PSD profiles are plotted, with relativistic effects included. In order to allow comparison with previous figures, we selected appropriate combinations of the model parameters: the infectivity $\alpha$ , the mean number of secondary spots $\nu$ , and the spot distribution are the same as in Fig. 3. The inclination $\theta _{\rm o}$ and the spot distribution as in Fig. 7. The vertical lines again indicate the range of orbital frequencies corresponding to the assumed range of radii, 6 M<r<100 M, where the spots are distributed. The analytical form is quick to evaluate, hence it is convenient to obtain the PSD form for variety of different situations.
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In the text

$\begin{figure} \par\includegraphics[width=16cm,clip]{9720fig9.eps}\end{figure}$ Figure 9: The ratio $R_{\rm {}S}(\omega )$ of PSD computed with and without relativistic effects. The difference is most visible around intermediate frequencies; Keplerian orbital frequencies at the outer/inner edges of the spot radial distribution are again indicated by vertical lines. The arrangement of panels and the assumed parameter values are the same as in the previous Fig. 8. In particular, the vertical lines indicate the minimum and the maximum orbital frequencies corresponding to the outer and the inner edges of the spot distribution.
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In the text