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Astronomy & Astrophysicsmanuscript no. 20339 c© ESO 2018July 13, 2018

Hot-spot model for accretion disc variability as random pro cess –II.

Mathematics of the power-spectrum break frequency

Tomas Pechacek,1 Rene W. Goosmann,2 Vladimır Karas,1 Bozena Czerny,3 and Michal Dovciak1

1 Astronomical Institute, Academy of Sciences, Bocnı II 1401, CZ-14131 Prague, Czech Republic2 Observatoire Astronomique de Strasbourg, F-67000 Strasbourg, France3 Copernicus Astronomical Center, Bartycka 18, P-00716 Warsaw, Poland

Received 6 September 2012; accepted 10 June 2013

ABSTRACT

Context. We study some general properties of accretion disc variability in the context of stationary random processes. In particular, we areinterested in mathematical constraints that can be imposedon the functional form of the Fourier power-spectrum density (PSD) that exhibits amultiply broken shape and several local maxima.Aims. We develop a methodology for determining the regions of the model parameter space that can in principle reproduce a PSD shapewith a given number and position of local peaks and breaks of the PSD slope. Given the vast space of possible parameters, itis an importantrequirement that the method is fast in estimating the PSD shape for a given parameter set of the model.Methods. We generated and discuss the theoretical PSD profiles of a shot-noise-type random process with exponentially decaying flares. Thenwe determined conditions under which one, two, or more breaks or local maxima occur in the PSD. We calculated positions ofthese featuresand determined the changing slope of the model PSD. Furthermore, we considered the influence of the modulation by the orbital motion for avariability pattern assumed to result from an orbiting-spot model.Results. We suggest that our general methodology can be useful in for describing non-monotonic PSD profiles (such as the trend seen, ondifferent scales, in exemplary cases of the high-mass X-ray binary Cygnus X-1 and the narrow-line Seyfert galaxy Ark 564). Weadopt a modelwhere these power spectra are reproduced as a superpositionof several Lorentzians with varying amplitudes in the X-ray-band light curve. Ourgeneral approach can help in constraining the model parameters and in determining which parts of the parameter space areaccessible undervarious circumstances.

Key words. Accretion, accretion-discs – Black hole physics – Galaxies: active – X-rays: binaries

1. Introduction

Accretion discs are believed to drive the variability of su-permassive black holes in active galactic nuclei (AGN) and,on much smaller time-scales, also the fluctuating signal fromGalactic black holes in accreting compact binaries (Vio et al.1992; Nowak et al. 1999; McHardy et al. 2006). Observed lightcurves, f ≡ f (t), exhibit irregular, featureless variationsat all frequencies that can be studied (Lawrence et al. 1987;McHardy & Czerny 1987; Gaskell et al. 2006). Nonetheless,accretion is the common agent (Frank et al. 2002).

Simplified models of accretion disc processes do notexplain every aspect of variability of different sources.Phenomenological studies also show a considerable complex-ity of the structure of accretion flows. Accretion discs co-existwith their coronae and possibly winds and jets, which producea highly unsteady contribution, and so the observed signal ofthe source is determined by the mutual interaction of differ-

ent components (Done et al. 2007). In this paper we study X-ray variability features that can be attributed to stochastic flareevents that are connected to the accretion disc. Our aim is toconstrain the resulting variability using a very general scheme.

Random processes provide a suitable framework for amathematical description of the measurements of a physicalquantity that is subject to non-deterministic evolution, for ex-ample due to noise disturbances or because of the intrinsicnature of the underlying mechanism. In our previous paper(Pechacek et al. 2008, Paper I) we studied the theory of ran-dom processes as a tool for describing the fluctuating signalfrom accreting sources, such as AGN and Galactic black holesobserved in X-rays. These objects exhibit a featureless vari-ability on different time-scales, probably originating from anaccretion disc surface and its corona. In particular, we exploreda scenario where the expected signal is generated by an ensem-ble of spots randomly created on the accretion disc surface,

2 T. Pechacek, R. W. Goosmann, V. Karas, B. Czerny, & M. Dovˇciak: Accretion disc variability as random process

orbiting with Keplerian velocity at each corresponding radius(Galeev et al. 1979; Czerny et al. 2004, and references therein).

Various characteristics can be constructed from the lightcurve of an astronomical object to study its variability prop-erties (Feigelson & Babu 1992). The standard approach is toanalyse the light curve in terms of its power spectral density(PSD), which provides the variability power as a function offrequency (Vaughan & Uttley 2008). The histogram of the fluxdistribution measures the time-scale that the source spends at agiven flux level. Furthermore, the root-mean-square (rms) scat-ter can be computed for different sections of the light curve thatcorrespond to different flux levels. This establishes the rms-fluxrelation (Uttley & McHardy 2001).

In this context an important reference to our work is givenby Uttley et al. (2005), where the authors presented a model toexplain the observed linear rms-flux relations that are derivedfrom X-ray observations of accreting black hole systems. Themodel generates light curves from an exponential applied toaGaussian noise, which by construction produces a log-normalflux distribution consistent with the observed X-ray flux his-tograms of the Galactic black hole binary Cyg-X1, as well asof other sources. Uttley et al. (2005) concluded that the obser-vation of a log-normal flux distribution and the linear rms-fluxrelationship imply that the underlying variability process can-not be additive. Indeed, this statement holds for a large familyof shot-noise variability models and it was extended to othersources in different spectral states (Arevalo & Uttley 2006),also in the optical band (Gandhi 2009).

As mentioned above, in this paper we concentrate on char-acteristics of the PSD profile, in particular on the constraintsthat can be derived from the PSD slope and the occurrenceof break frequencies. Because power spectra do not providecomplete information, even a perfect agreement between thepredicted PSD and the data cannot be considered as a defini-tive confirmation of the scenario. We do not study additionalconstraints, such as the rms-flux relation or time lags betweendifferent bands, which certainly have a great potential of distin-guishing between different schemes. Indeed, these characteris-tics provide complementary pieces of information that may befound incompatible with the spot model. In other words, themodel presented here is testable, at least in principle, andit willbe possible to confirm the model or reject it by observations.

Although well-known ambiguities are inherent to Fourierpower spectra, an ongoing debate is held about the shape ofthe PSD and the dominant features that could reveal impor-tant information about the origin of the variability. The broad-band PSD has been widely employed to examine the fluctu-ating X-ray light curves, often showing a tendency towardsflattening at low frequencies (Lawrence & Papadakis 1993;Mushotzky et al. 1993; Uttley et al. 2002). PSDs of interest forus are characterized by a power-law form of the Fourier powerspectrumωS (ω), which decays at the high-frequency part pro-portionally to the first power ofω. A break occurs at lower fre-quencies, where the slope of the power-law changes. The pro-file of the power spectrum can be even more complex, showingmultiple breaks or even local peaks separated by minima of thePSD curve.

The occurrence of a break frequency has been discussedby various authors (e.g. Nowak et al. 1999; Markowitz et al.2003) for its obvious relevance for understanding observedPSDs and interpreting the underlying mechanism that causesthe slope change. In this context, McHardy et al. (2005) haveexamined the case of the Seyfert 1 galaxy MCG–6-30-15with RXTE and XMM-Newton data. Very accurate fits sug-gest a multi-Lorentzian structure instead of a simple power-law.Furthermore, McHardy et al. (2007), combining the XMM-Newton, RXTE and ASCA observations, discovered multi-ple Lorentzian components in the X-ray timing propertiesextending over almost seven decades of frequency (10−8

.

f . 10−2 Hz) of the narrow-line Seyfert 1 galaxy Ark 564.These authors concluded that the evidence points to two dis-crete, localized regions as the origin of most of the vari-ability. Mushotzky et al. (2011) discussed the AGN opticallight curves monitoring by Kepler. They found, rather sur-prisingly, that the PSD exhibits power-law slopes of−2.6 to−3.3, i.e., significantly steeper than typically seen in the X-rays. Furthermore, within the category of stellar-mass blackholes, Pottschmidt et al. (2003) analysed Cygnus X-1 PSDfrom about three years of RXTE monitoring. These authorsconcluded that the changing form of the power spectrum canbe interpreted as a combination of Lorentzians with graduallyevolving amplitudes.

In this paper, motivated by this and similar observationalevidence, we embark on a mathematical study of PSD that canbe fitted either with a broken power-law or a double-bendingpower law, with the variability powerω S (ω) dropping at lowand high frequencies, respectively. We consider a simplified(but still non-trivial) description of the intrinsic variability (ran-dom flares with idealized light curve profiles of the individualevents), which allows us to develop an analytical approach tothe resulting observable PSD profile. In particular, we deter-mine conditions under which one, two, or more breaks or localmaxima occur in the PSD. We calculate positions of these fea-tures and determine the changing slope of the model PSD.

The paper is organised as follows. In section 2 we dis-cuss the PSDs generated by a shot-noise-like random processwith exponentially decaying flares. We developed a general ap-proach to describe the morphology of the power spectra anddemonstrate the results on a simple, analytically solvableex-ample of a bi-Lorentzian PSD. In section 4 we consider theinfluence of the modulation by the orbital motion on the PSDs.We summarise our conclusions and give perspectives for theapplication of this fast, analytical modelling scheme in sec-tion 5.

2. Exponential flares

It is well known that the power spectra of stationary randomprocesses must be flat at zero (S (ω) ∝ ω0) and integrableover all frequencies (i.e. decaying faster thanω−1 for largeω). Moreover, it was demonstrated for the multi-flare model(Pechacek et al. 2008) that the high-frequency limit of the PSDis a power-law,S (ω) ∝ ωγ, with the slopeγ < −1 determinedby the intrinsic shape of the flare-emission profile. The lowfrequency limit depends mainly on the statistics of the flare-

T. Pechacek, R. W. Goosmann, V. Karas, B. Czerny, & M. Dovciak: Accretion disc variability as random process 3

generating process. The form of the intermediate part of thepower spectrum is influenced by the interplay of all of theseeffects. The simplest possible power spectra generated by theflare model are approximately of the broken power-law form.

It is natural to assume that the power spectrum break-frequency is associated with some intrinsic time-scale of theprocess. This heuristic approach is supported by the basic scal-ing properties of the Fourier transform: rescaling the processin temporal domain by a factora shifts the break frequencyby a−1. The power-law PSDs are invariant under this rescaling,and the break frequency is the only feature of the power spec-tra whose position changes. However, we will see that for tworeasons it is difficult to go beyond this trivial observation.

Rigorous definition of the break frequency can be a prob-lem. Even for the simplest acceptable feature-less power spec-tra such as a Lorentzian PSD, which are power-laws to highaccuracy for most of frequencies, the change of slope occurssmoothly over a long interval. It is therefore not obvious whichfrequency should be defined as the break frequency, and evenif we decide on some particular definition, various problemsofinterpretation may emerge.

In this introductory section, we constrain our discussion tomutually independent exponentially decaying flares with emis-sivity profiles I(t, τ) = I0(τ) exp(−t/τ) θ(t), whereθ(t) is theHeaviside function andτ is the lifetime of individual flares.Exponentials are well suited for our investigations because oftheir simple, feature-less power spectra. Moreover, thereis atime-scaleτ naturally and uniquely associated with each flare.This allows us to relate the break frequencies to the ensembleaverages over the range ofτ.

To simplify the problem even more, we do not take intoaccount the relativistic effects in the rest of this section. Thepower spectrum of a random shot-noise-like process consistingof mutually independent exponentially decaying flares is givenby

S (ω) = λ∫

τ2

1+ τ2ω2I20(τ) p(τ) dτ, (1)

whereλ is the mean rate of flares andp(τ) is the probabilitydistribution ofτ. In the case of identical emissivity profiles, i.e.with pL(τ) = δ(τ−τ0), the power spectrum is a pure Lorentzian,

S L(ω) =λ I2

0 (τ0) τ201+ τ20ω

2. (2)

Even this simple model can reproduce a wide variety of powerspectra. In the context of the variability of the X-ray sourcesthe model was introduced by Lehto (1989).

It is traditional to plot the PSDs inω versusωS (ω) graphs.It can be easily shown that the functionωS L(ω) reaches itsmaximum atωM = τ

−10 , as one can intuitively expect. In the

general case the position of this maximum depends on the ac-tual shape of the functionsp(τ) andI0(τ). Since the power spec-trum is a nonlinear functional of the signal, the peak frequencydiffers from both E [τ]−1 and E

[

τ−1]

, where E [.] denotes theaveraging operator.

To demonstrate this nonlinear behaviour we will next studya process that consists of exponentials with only two different

time-scales, i.e. a process withp(τ) = Aδ(τ− τ1)+ (1−A)δ(τ−τ2), whereA is some constant from the interval〈0, 1〉 andτ1 >τ2 . The resulting power spectrum is

S 2L(ω) =λ A I2

0(τ1) τ211+ τ21ω

2+λ (1− A) I2

0(τ2) τ221+ τ22ω

2. (3)

As long as we are interested only in the process times-cales,the PSD normalization is irrelevant. Without loss of generalitywe can rescale the power spectrum both in normalization andin frequency domain as

S r(ω) = [S 2L(0)]−1S 2L(τ−11 ω) =

α

1+ ω2+

1− α1+ K2ω2

. (4)

Both K andα are from the interval〈0, 1〉. The peak frequencyωM is given by the equation

ddω

[ω S r(ω)]∣

∣

∣

∣

∣

ω=ωM

= 0, (5)

which leads to the bicubical equation

− K2[

(1− α) + αK2]

x3 +[

(1− α) − K2(2− αK2)]

x2

+[

(1− α)(2− K2) + α(2K2 − 1)]

x + 1 = 0, (6)

wherex = ω2M .

Equation (6) can have either one or three positive real roots.For most of the combinations of parametersα andK the rootis unique and the corresponding power spectrum has a singleglobal maximum. Three roots do occur only for parameterswithin a small subset of the configuration spaceCM . The cor-responding power spectra have two local maxima separated bya local minimum. The boundary of setCM is denoted by thered line in figure 1. Note that after rescaling the intrinsic time-scales of the two exponentials are 1 andK. The mean time-scale and the mean frequency of the process are then given by

E [τ] = α + (1− α)K, (7)

E[

τ−1]

= α + (1− α)K−1. (8)

However, the break time-scaleτM = 1/ωM depends nonlinearlyon bothα andK. The comparison of these three quantities isplotted in the figure 2. We can see that the displacement of theactual peak position and the two linearly averaged time-scalescan be very significant.

Note that our formulae are expressed and graphs are la-belled in arbitrary (dimensionless) units. When the underlyingmechanism is interpreted in terms of flaring events in a blackhole accretion disc, the meaning of these units can be identi-fied with geometrical units. Corresponding quantities in physi-cal units can be obtained by a simple conversion:

Mphys

Mphys⊙

=M

1.477× 105cm(9)

for the physical mass in grams, and

f phys=ωphys

2π=

cω2π= (4.771× 109cm · s−1)ω (10)

4 T. Pechacek, R. W. Goosmann, V. Karas, B. Czerny, & M. Dovˇciak: Accretion disc variability as random process

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

α

K

CM

CI1

CI0

CIq c)

0.8

0.9

1

0 0.1 0.2

a)

b)

100

100 101 102

ωS

(ω)

ω

a)

b)

c)

Fig. 1. Left panel: Diagram describing the morphology of the PSD profiles according to eq. (4) for all possible values of theparametersα andK. Different curves represent the boundaries of subsets of the parameters for which the PSD exhibits a double-feature. The red curve encloses setCM of the doubly peaked power spectra. The other curves correspond to the sets of powerspectra with a nontrivial structure of inflection pointsCI0 (blue),CI1 (magenta) andCIq (green). Right panel: Three examplesof the PSD profiles. A double-peaked power spectrum (red) with parametersα = 0.94 andK = 0.01 (i.e., the values withinsetCM , as denoted by the red point in the left panel), doubly-broken power spectrum with a single local maximum (magenta)corresponding toα = 0.85 andK = 0.08 (the magenta point within setCI1 and outside ofCM) and a power spectrum with singlemaximum and break (green) withα = 0.5 andK = 0.5 (the green point outside ofCIq).

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

τ M

K

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

τ M

K

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

τ M

K

Fig. 2. Left: Comparison of the time-scalesτM (red) and E [τ] (magenta). The lines are calculated forα from the set0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8 , avoiding setCM . Middle: As in the left panel, but forα = 0.9. For low K this corre-sponds to double-peaked power spectra. The red curves correspond to 1 over the peak frequency, the green is for the minimum.The blue curve denotes the time-scale 1/E [1/τ]. Right: Comparison of the time-scalesτM (red) and 1/E [1/τ] (blue).

for the frequency in Hertz. Frequencies scale with the centralmass proportionally toM−1; their numerical values are there-fore converted in the physical units by multiplying by the factor

c2πM

= (3.231× 104)

(

MM⊙

)−1

[Hz]. (11)

For the black hole in Cyg X-1 the current mass estimate(Orosz et al. 2011) readsMphys/M⊙ = 14.8± 1.

From figs. 1–2 we learn that two peaks occur within onlya very narrow parameter range. It is even more difficult to pro-

duce two peaks of similar height in theω S (ω) plot. The con-dition is that two small numbers,K and 1− α have to be equal.For example, in the hard state of Cyg X-1 the two most promi-nent Lorentzians are separated by about two decades in fre-quency, and they are of identical height, which means that insuch a stateK ≈ 0.01 and 1− α = 0.01. Hence, the two quan-tities must be strongly correlated. Even if the two Lorentziansarise in different regions, the oscillations have to be physicallylinked. This mathematical statement is thus important and hasto be taken into account in any attempt to explain the nature

T. Pechacek, R. W. Goosmann, V. Karas, B. Czerny, & M. Dovciak: Accretion disc variability as random process 5

of the two mechanisms. Naturally, one has to bear in mind thatall our conclusions are based on a simplified scenario; a morecomplicated (astrophysically realistic) model will very likelycontain more parameters, which will bring additional degreesof freedom.

We have demonstrated that the power spectraS r(ω) aredouble-peaked only for pairs ofα andK within a small area ofthe parameter space. However, this does not mean that single-peaked power spectra are lacking any internal structure. Unlikethe local maximum, it is not obvious how to formalize the pres-ence of the internal structure into equations. The simplestcon-tinuation of the ideas above is to calculate inflection points ofωS (ω),

d2

dω2[ωS r(ω)]

∣

∣

∣

∣

∣

∣

ω=ωI1

= 0. (12)

Apparently, equation (12) has also either one or three positivereal solutions.

We have been investigating the local maxima ofωS (ω), be-cause the power spectraS (ω) of a shot-noise-like processes arepeaked at zero and because frequency times power versus fre-quency plots are traditionally used to analyse of astronomicalsignals. There is, however, no good a priori reason to prefertheinflection points ofωS (ω) over inflection points ofS (ω) itself,

d2

dω2S r(ω)

∣

∣

∣

∣

∣

∣

ω=ωI0

= 0. (13)

The regionsCI0 andCI1 denote areas of the parameter spacethat exhibit more than one inflection point ofS r(ω), respec-tively ωS r(ω). The critical points at their boundaries are con-nected by blue, and by magenta curve in the figure 1. Thesetwo areas clearly do not coincide, because the multiplicationof the power spectra byω can in some cases remove or createnew inflection points. Inspired by the properties of the log-logplotting of power spectra, we can define the local power-lawindex of the PSDq(ω) and associate the internal structure ofthe power spectra with the presence of inflection points,

q(ω) =d ln(S r(ω))

d ln(ω)=ω

S r(ω)dS r(ω)

dω, (14)

d2

dω2q(ω)

∣

∣

∣

∣

∣

∣

ω=ωIq

= 0. (15)

The part of the parameter space corresponding to power spectrawith more than one inflection point ofq(ω) are within setCIq

enclosed by the green curve in fig. 1, which shows the positionsof the inflection points of the three different types for a selectedset ofαs and a whole range ofK.

2.1. A more general distribution of flare profiles

We now investigate a more general case with an arbitrary prob-ability distributionp(τ). First we observe that the normalizationfunction I0(τ) has a similar influence on the resulting powerspectrum asp(τ) itself. Therefore, we can without loss of gen-erality study the power spectrum

S (ω) =∫

11+ τ2ω2

p(τ)dτ, (16)

wherep(τ) = λτ2I20(τ)p(τ).

Without any loss of generality we can setλ = [S (0)]−1.This rescaling does not change the overall shape of the PSDand it normalises ˜p(τ) to unity. DifferentiatingωS (ω) from eq.(16) leads to

ddω

[ωS (ω)] =∫

1− τ2ω2

(

1+ τ2ω2)2

p(τ)dτ = E

1− τ2ω2

(

1+ τ2ω2)2

. (17)

For the peak frequency we find due to linearity of the averagingoperator E[.],

E[

(

1+ τ2ω2M

)−2]

= ω2ME

[

τ2(

1+ τ2ω2M

)−2]

. (18)

Now we substitutexM for ω2M , define a functiong(x) as

g(x) =E

[

(

1+ τ2x)−2

]

E[

τ2(

1+ τ2x)−2

] , (19)

and observe that the position of the local extreme is determinedby the intersection ofg(x) with x:

g(xM) = xM . (20)

It follows from the Jensen inequality that the derivative ofg is always non-negative (see the appendix). Therefore, theposition of xM is constrained to the interval〈g(0), g(∞)〉.Straightforwardly,g(0) = E

[

τ2]−1

. To calculate the upper limit

we observe that 1/(1+ τ2x) ∝ τ−2 for τ2 ≫ x−1. Assuming thatthere is a non-zero lowest time-scaleτmin > 0 so thatp(τ) = 0for t < τmin, we can replace the Lorentzian in (19) byτ−2 andput1 g(∞) = E

[

τ−4]

/E[

τ−2]

. For the break time-scale we fi-nally find the inequality√

E[τ2] ≥ τM ≥√

E[τ−2]/E[τ−4]. (21)

Figure 4 illustrates this again in terms of the power spectrum(4). Althoughg(x) is necessarily a non-decreasing function, theintersectiong(x) = x needs not to be unique. The inequality(21) then holds for all local maxima and minima of the PSD.

We now further investigate properties of the functiong(x).Eq. (20) states that the extrema of the PSD are fixed pointsof g(x). We can define then-th iteration ofg(x) by recurrentrelations,

g(n)(x) ≡ g(g(n−1)(x)), g(1)(x) ≡ g(x). (22)

It trivially follows from eq. (20) that the local extremaxM arealso fixed points ofg(n)(x) for all values ofn. Assuming dis-tributions p(τ) with finite momentsg(0) andg(∞), the func-tion g(x) maps the real half-line〈0,∞〉 into the finite interval〈g(0), g(∞)〉. From the monotonicity ofg, it then follows thatgmaps the latter interval within itself.

By repeated applications ofg we obtain an infinite se-quence of nested inequalities,

g(0) ≤ g(n)(0) ≤ g(n+1)(0), (23)

g(n+1)(∞) ≤ g(n)(∞) ≤ g(∞). (24)

1 We denoteg(∞) as an abbreviation for a graphically less compactbut formally more correct expression, lim

x→∞g(x).

6 T. Pechacek, R. W. Goosmann, V. Karas, B. Czerny, & M. Dovˇciak: Accretion disc variability as random process

0

0.5

1

1.5

2

0 0.2 0.4 0.6 0.8 1

τ I0

K

0

0.1

0.2

0.3

0.4

0.5

0.6

0 0.2 0.4 0.6 0.8 1

τ I1

K

0

0.5

1

1.5

2

2.5

3

0 0.2 0.4 0.6 0.8 1

τ Iq

K

Fig. 3. Time-scalesτI0 (left), τI1 (middle) andτIq (right) calculated for the PSD given by eq. (4). We assumed the values ofαfrom the set0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, andK from the interval〈0, 1〉.

10-1

100

101

102

103

104

105

10-2 10-1 100 101 102 103 104 105 106

g(x)

x

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

τ M

K

Fig. 4. Left: Functiong(x) (eq. (19)) for the power spectrum (4), calculated forα = 0.95, andK = 0.01 (red) andK = 0.2 (blue),respectively. The green line represents the identity function g(x) = x. Right: The corresponding plot of time-scales, similar to theright panel of fig.2. The light-blue lines correspond to the limits from the inequality (21).

Because of the Bolzano-Weierstrass theorem, the followinglimits exist and the same inequality holds for them as well,

g(∞)(0) = limn→∞

g(n)(0), (25)

g(∞)(∞) = limn→∞

g(n)(∞), (26)

g(∞)(0) ≤ xM ≤ g(∞)(∞). (27)

Bothg(∞)(0) andg(∞)(0) are fixed points ofg(x) and correspondto the lowest and highest local maxima ofωS (ω), respectively.If they coincide, the power spectrum has a single peak.

The inflection points of the power spectrum can be investi-gated in a similar way. Defining a function

h(x) =E

[

τ2(

1+ τ2x)−3

]

E[

τ4(

1+ τ2x)−3

] , (28)

it can be easily proven that

d2

dω2S (ω)

∣

∣

∣

∣

∣

∣

ω=√

xI0

= 0 : 3xI0 = h(xI0), (29)

d2

dω2[ωS (ω)]

∣

∣

∣

∣

∣

∣

ω=√

xI1

= 0 : xI1 = 3h(xI1). (30)

As we show in the appendix,h(x) is also a non-decreasingfunction. Therefore,h(x) has properties similar tog(x), withh(0) = E[τ2]/E[τ4], h(∞) = E[τ−4]/E[τ−2] and with a similarstructure of nested subintervals,

〈h(n)(0), h(n)(∞)〉 ⊇ 〈h(n+1)(0), h(n+1)(∞)〉. (31)

There are several applications of the previous relations.Clearly, finding the fixed points ofg andh numerically is ashard as finding the extrema and inflection points ofS (ω) di-rectly. But by calculatingg(0), h(0), g(∞) andh(∞), we canroughly estimate the position of each feature. Every applica-tion of these functions requires calculating two expected valuesof some functions ofτ. It was demonstrated in the case of twoLorentzians that these limits are sometimes too wide. In thiscase we can always determine a tighter constraint by applyingg andh, respectively, on the result at the cost of calculating twoadditional expected values per each iteration. Moreover, we can

T. Pechacek, R. W. Goosmann, V. Karas, B. Czerny, & M. Dovciak: Accretion disc variability as random process 7

useg andh to find a constraint on the numbers of local extremaand inflection points.

In the simple case of two Lorentzians we were able to cal-culate directly for which sub-sets of the parameter space ofα

and K is the PSD doubly-peaked and doubly-broken. Let uscall these setsCM , CI0, CI1 andCIq. The parameter space ofthe general case with an arbitrary probability function ˜p(τ) isinfinite-dimensional. In analogy with the two-dimensionalcaseof S r(ω) we can define a subsets of the space of all probabilitydistribution functionsCM , such that the power spectrum (16)has multiple local extremaxM if and only if p(τ) ∈ CM.

Testing if a particular function ˜p(τ) is contained inCM

means calculating the power spectrum directly. However, usingthe functiong(x), we can construct approximation setsCN

M andCS

M representing the necessary and sufficient conditions on ˜p(τ)to produce a multiply peaked power spectrum,CS

M ⊆ CM ⊆ CNM .

Usingg andh, the approximating sets can be defined by an in-equality, or a set of inequalities between some moments ofτ.We now present two examples of such constructions and applythem to the example of the PSD (4).

2.2. Examples of the construction procedure

We takex+ andx− such thatx+ < xM < x− for all fixed pointsg(xM) = xM . By construction,g(x+) > x+ and g(x−) < x−.If there is more than one fixed pointxM , the functionsg(x) =g(x)− x changes its sign in the interval〈x+, x−〉more than once.We can pickL valuesx j from the interval such thatx+ < x1 <

. . . < xL < x−. If the sequence ofsg(x j) changes sign morethan once, the fixed point is not unique. Figure 5 shows theapplication of the test on the power spectrum (4) forx+ = g(0),x− = g(∞) and x j placed uniformly over the whole interval.The results are plotted forL equal to 4, 5, 9 and 50. Note thatthis is not the only possible choice of thex. All x+ = g(n)(0) andx− = g(n)(∞) are admissible, and likewise,x j can be chosenin some irregular or even adaptive way by using firstL stepsof some numerical root-finding method (e.g. the regula falsimethod). What matters is the final complexity of the test. Itproduces a set ofL inequalities that must be satisfied by ˜p(τ) tobe inCS

M .For setCN

M we can use a different approach. The well-known Banach fixed-point theorem says that if the functiong isa contracting mapping, i.e. if there exists a number 0≤ λ < 1such that|g(x) − g(y)| < λ|x − y| for all admissiblex andy,then the fixed-pointxM is unique. Becauseg(x) is a continuousand non-decreasing function, it follows from the mean valuetheorem that it is a contracting mapping if the maximum of itsderivative is lower than one,g′(x) < 1. Defining a new functionℓ(x) as

ℓ(x) =E

[

(1+ τ2x)−3]

E[

τ4(1+ τ2x)−3] , (32)

we can write the derivative ofg,

g′(x) = 2ℓ(x) − h2(x)

(h(x) + x)2. (33)

Similarly tog andh, alsoℓ(x) is a non-decreasing function thatcan be expressed in terms of the functionsg andh,

ℓ(x) = g(x) (h(x) + x) − xh(x). (34)

Because the functiong maps the whole semi-axis〈0,∞〉 into〈g(0), g(∞)〉 it is sufficient to investigate the derivativeg′(x)only on the latter interval. Becauseg, h, and ℓ are non-decreasing functions, the following inequality holds for all x+,x− andy that satisfyg(0) ≤ x+ ≤ y ≤ x− ≤ g(∞),

g′(y) ≤ Dmax(x+, x−) = 2l(x−) − h2(x+)

(h(x+) + x+)2. (35)

Using this inequality, we can take forCNM the set of all dis-

tributions p(τ) for which Dmax(x+, x−) > 1. Figure 5 demon-strates this on the bi-Lorentzian power spectrum. We employedx+ = g(n)(0) andx− = g(n)(∞) with n from 1 to 4. The marginsderived from this criterion can be relatively wide, both becauseDmax(x+, x−) overestimates the maximum ofg′(x), and also be-cause some functionsg(x) can have a single fixed pointxM anda derivativeg′(y) > 1 (y , xM) at the same time. On the otherhand, if Dmax(x+, x−) < 1, g(x) surely exhibits a single fixedpoint.

Unfortunately, the analysis of inflection points of the lo-cal power-law slope cannot be generalized in the same way.However, we note thatq(ω) is related tog by

q(ω) =−2ω2

ω2 + g(ω2). (36)

From here we can immediately find thatq(0) = 0 andq(∞) =−2. Furthermore, from (20) we see thatq(ωM) = −1 and con-versely, by expressingg in terms ofq, we can see that allωwith q(ω) = −1 are the fixed points ofg. This is not surprisingsince in the vicinity of the local extreme (ωM + y)S (ωM + y) =const.+ o(y). It is, however, a good motivation to briefly inves-tigate the behaviour of the functionωγS (ω) in terms of its localextremaωMγ and inflection-pointsωIγ,

ddω

[

ωγS (ω)]

∣

∣

∣

∣

∣

ω=√

xMγ

= 0,d2

dω2

[

ωγS (ω)]

∣

∣

∣

∣

∣

∣

ω=√

xIγ

= 0. (37)

Following the same procedure that led to the equations forxM ,xI0, andxI0, we obtain

γg(xMγ) − (2− γ)xMγ = 0, (38)

Aγx2Iγ + Bγh(xIγ)xIγ +Cγl(xIγ) = 0, (39)

whereAγ = (γ−2)(γ−3), Bγ = 2(γ2−3γ−1) andCγ = γ(γ−1).We see that the solution of the problem is fully determined bythe functionsg andh.

This generalization brings nothing new for the local ex-tremaxMγ. The constraints for the number of fixed pointsxMγ

and methods for approximating their positions can be obtainedfrom the corresponding methods forγ = 1 by replacingg(x)with gγ(x) = γ(2− γ)−1g(x) and the local slope at the extremalpoints isq(ωMγ) = −γ, as expected. However, the problem ofthe inflection points is more complicated with the exceptions of

8 T. Pechacek, R. W. Goosmann, V. Karas, B. Czerny, & M. Dovˇciak: Accretion disc variability as random process

0.85

0.9

0.95

1

0 0.05 0.1 0.15

α

K

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

αK

0.75

0.8

0.85

0.9

0.95

1

0 0.05 0.1 0.15 0.2 0.25

α

K

0

0.2

0.4

0.6

0.8

1

0 0.05 0.1 0.15 0.2 0.25

α

K

Fig. 5. Top left: Boundaries of approximating setsCSM for the power spectrumS r(ω). The functiong(x) was evaluated inL points,

evenly placed in betweeng(0) andg(∞). The nested curves correspond toCSM calculated forL equal to 4, 5, 9, and 49. Top right:

The nested setsCNM calculated from the upper limit ofg′(x). We have takenx+ = g(n)(0) andx− = g(n)(∞) for n from 1 to 4.

Bottom left: The same as in the top left panel for setsCSI1. Bottom right: The same as in the bottom left panel, this timewith a

grid of points that samples the area around point 3h(0) more densely but that does not cover the whole interval〈3h(0), 3h(∞)〉.The values ofL are the same as in the previous examples. The approximating setsCS

I1 cover a broader part of the whole setCI1.However, they omit a small area of highα. This demonstrates that an adaptive grid can perform significantly better than a regularone.

γ = 0 andγ = 1. A possible generalization ofh(x) for γ ∈ 〈0, 1〉is given by

hγ(x) = −Bγ2Aγ

h(x) +

√

B2γ

4A2γ

h2(x) −CγAγℓ(x). (40)

3. Polynomial flares

In all of the previous considerations, we have assumed that theflare profiles are given by decaying exponentials and the distri-bution p(τ) is positive and can be normalized to one. No otherassumptions on ˜p(τ) were imposed. Therefore, the resulting in-

T. Pechacek, R. W. Goosmann, V. Karas, B. Czerny, & M. Dovciak: Accretion disc variability as random process 9

equalities are very general. We now approach the case of flaresof the form

Ik(t|τ) = I0(τ)Pk

( tτ

)

exp(

− tτ

)

θ(t), (41)

wherePk is a polynomial ofk-th order, andI0(τ) is an arbi-trary non-negative and quadratically integrable function. As amotivation we tookI0(τ) = 1/τ, k = 1 andP1(t/τ) = t/τ andcalculated a power spectrum of a mixture of these profiles withthe probability distributionp(τ),

S (ω) =

τmax∫

τmin

S (ω|τ)p(τ)dτ =

τmax∫

τmin

p(τ)(

1+ τ2ω2)2

dτ. (42)

It can be easily proven that the termS (ω|τ) from the previousequation satisfies a relation

S (ω|τ) =(

1+ τ2ω2)−2=

12τ

ddττ2

(

1+ τ2ω2)−1. (43)

Using this relation, we can rewrite equation (42) by integrationby parts,

S (ω) =[

τ

2p(τ)

(

1+ τ2ω2)−1

]τmax

τmin

+12

τmax∫

τmin

p(τ) − τ ddτ p(τ)

1+ τ2ω2dτ. (44)

We observe that the power spectrum can be rewritten in theform

S (ω) =

τmax∫

τmin

p2(τ)1+ τ2ω2

dτ, (45)

p2(τ) =12

[

τp(τ) (δ(τ − τmax) − δ(τ − τmin))

+ p(τ) − τ ddτ

p(τ)]

. (46)

If p(τmin) = 0 andp(τ) − τp′(τ) ≥ 0 for all τ ∈ 〈τmin, τmax〉,p2(τ) is behaving well, can be normalized to unity, and thepower spectrum can be studied in terms of the functionsg andh, as described previously, with the only difference that all av-erages must be calculated usingp2(τ) instead ofp(τ).

It is not apparent on first sight how to interpret these aver-age values. Therefore, we now study the general case of an ar-bitrary polynomial (42) assuming that the power spectrum canbe reduced to the pure-exponential case. The PSD of a singleprofile (42) can be written as

S (ω|τ) =∣

∣

∣Ik(ω|τ)∣

∣

∣

2=

Qk(τ2ω2)(

1+ τ2ω2)k+1, (47)

whereQk(z) is a k-th order polynomial, different than (but re-lated to)Pk(t). We defineQk by

Qk1

1+ τ2ω2=

Qk(τ2ω2)(

1+ τ2ω2)k+1. (48)

The operator is constructed entirely fromτ, derivatives overτ,and from the coefficients ofPk (see the appendix).

The generalization of equation (44) is

S (ω) =

τmax∫

τmin

p(τ)Qk(τ2ω2)

(

1+ τ2ω2)k+1

dτ =

τmax∫

τmin

p(τ)Qk1

1+ τ2ω2dτ

=

τmax∫

τmin

11+ τ2ω2

Q†k p(τ)dτ =

τmax∫

τmin

pQ(τ)

1+ τ2ω2dτ, (49)

whereQ†k is the Hermitian conjugate ofQk. For pQ(τ) posi-tive over the interval of all admissibleτs, we can calculate thefunctiong(x),

g(x) =EQ

[

(

1+ τ2x)−2

]

EQ

[

τ2(

1+ τ2x)−2

] =

E[

Qk

(

1+ τ2x)−2

]

E[

Qkτ2(

1+ τ2x)−2

] , (50)

where EQ [.] is an averaging operator with respect to the dis-tribution pQ(τ). We note that we do not have to compute thefunction pQ(τ).

As Qk operates withτ only, it also commutes with thederivative inω. Therefore, we can easily calculate the numera-tor and denominator of (50),

Qkτ2

(

1+ τ2ω2)2= − 1

2ωd

dωS (ω|τ), (51)

Qk1

(

1+ τ2ω2)2=

ddωωS (ω) − ω

2d

dωS (ω|τ). (52)

The value ofg(0) is related to the mean moment profiles as

g(0) =E

[

E2I (τ)

]

E[

E2I (τ)W2

I (τ)] , (53)

EI(τ) =

∞∫

0

I(t|τ)dt, (54)

W2I (τ) = E−1

I (τ)

∞∫

0

t2I(t|τ)dt −

E−1I (τ)

∞∫

0

tI(t|τ)dt

2

. (55)

The functionEI(τ) can be interpreted as the total energy emit-ted by the flare andWI(τ) is a measure of the width of theflare in the temporal domain. Equation (55) is formally iden-tical with the prescription for the standard deviation. Forthehigh-frequency limit we find

g(∞) =((k + 1)qk − qk−1) E

[

τ−4]

qkE[

τ−2] , (56)

where ql are the coefficients of the polynomialQk(z) =∑k

l=0 qlzl. The situation ofh(x) is completely analogical. Wecan use the commutation properties ofQk and expressh(ω2) as

h(ω2) =E

[

S ′′(ω|τ) − 3ω

[ωS (ω|τ)]′′]

E[

3ω2 S ′′(ω|τ) − 1

ω2 [ωS (ω|τ)]′′] . (57)

The asymptotic slope of the power spectraS (ω|τ) is q(∞) =−2(l + 1), wherel is the order of the lowest non-zero coef-ficient of the polynomialPk(t). Apparently, such a case can

10 T. Pechacek, R. W. Goosmann, V. Karas, B. Czerny, & M. Dovciak: Accretion disc variability as random process

not be transformed onto the purely exponential case, becauseits asymptotical slope (36) is alwaysq(∞) = −2. However,we can easily repeat the whole analysis for flares of the formI(t, τ) = (tk/τk+1) exp(−t/τ) θ(t), leading to the power spectrum

S k(ω) = E

(k!)2

(

1+ τ2ω2)k+1

. (58)

For everyk we can define the functionsgk andhk,

gk(x) =E

[

(

1+ τ2x)−k−2

]

E[

τ2(

1+ τ2x)−k−2

] , (59)

hk(x) =E

[

τ2(

1+ τ2x)−k−3

]

E[

τ4(

1+ τ2x)−k−3

] . (60)

It is easy to prove by direct calculation that the extrema andinflection points ofωS k(ω) andS k(ω) satisfy the relations

xM =1

2k + 1gk(xM), (61)

xI0 =1

2k + 3hk(xI0), xI1 =

32k + 1

hk(xI1). (62)

Both gk andhk are non-decreasing (see the appendix) and cantherefore be used analogously tog andh. Apparently,gk(0) =g(0) andhk(0) = h(0), for the upper limits we find,

gk(∞) = hk(∞) =E

[

τ−4−2k]

E[

τ−2−2k] . (63)

Finally, the local slope ofS k(ω) is related togk(x) by the for-mula

qk(ω) =−(2k + 2)ω2

gk(ω) + ω2. (64)

3.1. Alternative forms of the function g

In the previous sections we described the properties of thefunction g. By construction,g-values are in every point di-rectly related to the power spectrum and its derivatives; seeequations (36) and (50)–(52). However, in all previous appli-cations we have required only thatg is a non-decreasing func-tion, g(0) andg(∞) are positive finite numbers, and its fixedpointsg(xM) = xM correspond to the local extrema ofωS (ω).Clearly, for givenS (ω), these conditions do not fix the functionuniquely. For instance, every member of the sequenceg(n)(x)has the required properties. It is therefore natural to ask whetherthere is an alternative, easily calculable form ofg.

A simple set of solutions to this problem can be found byadding a zero to equation (17) to obtain

ddω

[ωS (ω)] = E

1− (τ2 + f (ω2|τ) − f (ω2|τ))ω2

(

1+ τ2ω2)2

= 0, (65)

where f (ω2|τ) is a function ofω2 andτ. Properties off (ω2|τ)are specified below. Due to linearity of the averaging operatorwe can define a new functiong[ f ] (x) as

g[ f ] (x) =E

[

(1+ x f (x|τ))(

1+ τ2x)−2

]

E[

(τ2 + f (x|τ)) (1+ τ2x)−2

] . (66)

By construction, the functionsg[ f ] (x) and g(x) have identicalfixed points irrespective of the choice off (x|τ). The limits ofg[ f ] are given by

g[ f ] (0) = E[

τ2 + f (0|τ)]−1, (67)

g[ f ] (∞) =E

[

τ−4(1+ φ(τ))]

E[

τ−2] , (68)

whereφ(τ) is the limit

φ(τ) = limx→∞

x f (x|τ). (69)

Both f (0|τ) andφ(τ) must be finite and chosen such that 0<g[ f ] (0) ≤ g[ f ] (∞) < ∞. Unlike the previous cases, the positivityof the derivative ofg[ f ] does not follow from any theorem andmust be checked separately for every particular choice off .

4. Exponential flares with periodic modulation

In analogy with the previous section, we start the investigationwith the simplest case of identical exponential flares modulatedby sine function,

I(t, τ,Ωs) = I0

(

A + B sin(

Ωst +C)

)

e−t/τ θ(t). (70)

Without loss of generality we can putτ = 1 and rescale thepower spectrum to the form

S rD(ω) =β

1+ ω2+

1− β1+ (ω −Ωs)2

+1− β

1+ (ω + Ωs)2, (71)

whereβ is taken from the interval〈0, 1〉. Figure 6 shows theboundary of set of the doubly peakedω S rD(ω) in the parameterspace and the break time-scalesτM for some choices ofβ andΩs. The structure of the setCM corresponds again to the cuspcatastrophe. However, the structure inflection points, shown inthe figure 7, is more rich.

Unmodulated PSDs of the form (16) always satisfy the con-ditionsS ′′(ω) > 0 for ω → ∞ andS ′′(0) = −2E[τ2] < 0, andhence they have an odd number of inflection points. However,the power spectrum (71) can be convex both at zero and inthe limit of high frequencies and therefore it can have an evennumber of inflection points. These two cases are separated bya curveβcrit(Ωs) given by the conditionS ′′rD(0) = 0. By directcalculation we find

βcrit(Ωs) = 23Ω2

s − 1

Ω6s + 3Ω4

s + 9Ω2s − 1

. (72)

4.1. General periodic modulation

We assumed that the exponentially decaying flares are at thesame time orbiting in the equatorial plane of an accretion discand that the observed signal is periodically modulated by theDoppler effect and abberation. It has been shown (Paper I) thatthe resulting power spectrum is described by the formula

S (ω) =∞∑

n=−∞

Rout∫

Rin

τmax∫

τmin

λI20(τ) τ2 |cn(r)|2

1+ τ2 (ω − nΩ(r))2p(τ, r) dτdr, (73)

T. Pechacek, R. W. Goosmann, V. Karas, B. Czerny, & M. Dovciak: Accretion disc variability as random process 11

0

0.2

0.4

0.6

0.8

1

0 2 4 6 8 10

β

ΩS

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 2 4 6 8 10

τ M

ΩS

Fig. 6. Left: Boundary of setCM of doubly-peaked power spectraωS rD(ω). Right: The time-scalesτM calculated forβ from theset0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9 and forΩs from the interval〈0, 10〉. The blue line corresponds toΩ−1

s . Again, thered lines denote theτM of the peaks, and the green lines denote the local minimum.

0

0.2

0.4

0.6

0.8

1

0 2 4 6 8 10

β

ΩS

0

0.2

0.4

0.6

0.8

1

0 2 4 6 8 10

β

ΩS

0

0.2

0.4

0.6

0.8

1

0 2 4 6 8 10

β

ΩS

Fig. 7. Left panel: The boundary of the setCI0 (red) for the PSD (71). Parameters inside of this area leads to PSD with a morethan one inflection point. Unlike the power spectrum (4)S rD(ω) can have a global maximum atω ≈ Ωs. In some,CI1 (middle)andCIq (right) of the power spectrum (71). The common feature of in the three panels is the critical curve (72).

whereΩ(r) is the Keplerian orbital frequency,cn(r) are theFourier coefficients of the periodic modulation, andp(τ, r) isthe probability density of finding a flare with the lifetimeτ atthe radiusr. The formula (73) is far too complicated for cal-culations. Apparently, the factorI2

0(τ) τ2 can be again absorbedinto the probability distribution. Integration overr and summa-tion overn can be transformed into a single integral by appro-priate change of the variables,r → Ω(r)/n. Finally, we canrewrite the equation (73) as

S (ω) =

∞∫

−∞

τmax∫

τmin

1

1+ τ2 (ω −Ω)2p(τ,Ω) dτdΩ

= E

[

1

1+ τ2 (ω −Ω)2

]

. (74)

The function ˜p(τ,Ω) contains contributions from all harmonicsof the orbital frequenciesΩ(r) and is by construction symmet-rical in the second variable ˜p(τ,Ω) = p(τ,−Ω). The distribution

p(τ,Ω) = δ(τ − 1)[

βδ(Ω)

+ (1− β)(δ(Ω −Ωs) + δ(Ω + Ωs))]

(75)

leads to the PSD (71).

We now repeat the procedures which, applied to equation(16), led to the definition of the functionsg andh. We beginwith the local extrema ofωS (ω). By differentiating (74), wefind

ddω

[ωS (ω)] = E

1+ τ2Ω2 − τ2ω2

(

1+ τ2 (ω −Ω)2)2

. (76)

The term 1+ τ2Ω2 is always positive. Therefore, we can inanalogy to eq. (19) define the functiongΩ(x) as

gΩ(ω2) =E

[

(1+ τ2Ω2)(

1+ τ2(ω −Ω)2)−2

]

E[

τ2(

1+ τ2(ω −Ω)2)−2

] , (77)

and observe that the position of the local extremaω2M = xM is

again given by the fixed point

gΩ(xM) = xM . (78)

The functiongΩ is positive and bothgΩ(0) andgΩ(∞) are finite.Unfortunately, in a general case,gΩ(x) is not a non-decreasingfunction. To be able to use the techniques developed in section

12 T. Pechacek, R. W. Goosmann, V. Karas, B. Czerny, & M. Dovciak: Accretion disc variability as random process

1

10

1 10

ω∞

ω0

ωM

ωM 0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

α

K

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

α

K

Fig. 8. Left panel: Graph ofω0 =√

g(0) versusω∞ =√

g(∞) calculated for the PSD from eq. (4). The deformed grid of contourlines corresponds to curves ofα = const andK = const in the parameter space of the model, whereω∞ ≥ ω0 by construction.It follows that the image of the entire parameter space lies above the green diagonal line ofω∞ = ω0. Middle panel: The upper-left rectangular sector of acceptable pairs ofω0 andω∞ from the left panel is projected back onto the (α,K) parameter space.Assuming the first-order boundaries ofω0 =

√

g(0) andω∞ =√

g(∞), the pairs ofα vs.K outside of the filled (red) area lead tothe power spectrum, which cannot have any local extrema atωM . These parameter values are therefore ruled out. Right panel:Analogical to the middle panel, but employing tighter (second-order) boundariesω0 =

√

g(2)(0) andω∞ =√

g(2)(∞); in this case,the parameter constraints are more stringent (the filled area is smaller).

2 we have to define a perturbed version ofgΩ(x) analogical to(66) as

g[ f ]Ω(ω2) =E

[

(1+ τ2Ω2 + ω2 f (ω2|τ,Ω))L2(ω|τ,Ω)]

E[

(τ2 + f (ω2|τ,Ω))L2(ω|τ,Ω)] , (79)

where L(ω|τ,Ω) =(

1+ τ2(ω −Ω)2)−1

is the elementary

Lorentzian term. The perturbationf (ω2|τ,Ω) has to be finiteatω = 0 and to decay faster thanω−1 to ensure the finitenessof g[ f ]Ω(0) andg[ f ]Ω(∞). Again, the positivity of the derivativeof g[ f ]Ω has to be tested for every particular choicef . For thepower spectrum (71) it can be shown that

f (ω2|τ,Ω) =Ω3

s + 1

1+ ω2(80)

leads to a non-decreasing and positive functiong[ f ]Ω(ω2).A general calculation of the points of inflection is even

more problematic. One can proceed in the following way.Analogically to equations (39) and (40), the functionhΩ(ω) isgiven by the solution of the following equations:

3E[

τ4L3(ωI0|τ,Ω)]

ω3I0 − 6E

[

τ2ΩL3(ωI1|τ,Ω)]

ωI0

+E[

τ2(3τ2Ω2 − 1)L3(ωI0|τ,Ω)]

= 0, (81)

E[

τ4L3(ωI1|τ,Ω)]

ω3I1 − 3E

[

τ2(1+ τ2Ω2)L3(ωI1|τ,Ω)]

ωI1

+2E[

τ2(1+ τ2Ω2)L3(ωI1|τ,Ω)]

= 0. (82)

The positivity of the derivative ofhΩ has to be ensured by theperturbation procedure.

5. Discussion and Conclusions

We have described a general formalism that can be used tocharacterize the overall form of a power spectrum, namely, we

determined the constraints on the functional form of the PSDshape that follow from the occurrence of local peaks. In par-ticular, our approach allowed us to distinguish between PSDsthat have the form of multiple power-law profiles with dif-ferent numbers of local maxima. Our methodology is usefulin the context of non-monotonic PSD profiles. In fact, Figs.1 and 2 demonstrate that the occurrence of separate peaksof the PSD can put tight constraints on the model. The cor-rect interpretation of these strict constraints still remains to befound. Naturally, a possible line of interpretation suggests thatthe spot/flare scenario is not generic and needs some kind offine-tuning of the model parameters to reproduce observations.However, this calls for a more thorough investigation; in thepresent paper we only explored a highly idealized scheme inwhich strong assumptions were imposed. More complicatedmodels (for instance involving avalanches) also exhibit a morevaried behaviour.

A potential application of our approach is illustrated byFigure 8, where we construct a mesh of two-dimensionalcontour-lines in (α,K) parameter space of the model.Assuming that the exemplary power spectrum has a local max-imum (or minimum) at frequencyωM , it follows directly fromeq. (27) that the three frequencies must satisfy the relationω0 ≤ ωM ≤ ω∞. Only those pairs of parametersα andK whoseimages are within the upper-left sector (Fig. 8, left panel), asdescribed by the mentioned inequality, can produce a powerspectrum with the required feature atω = ωM .

An example for this is Cyg X-1, which has been describedas a superposition of several Lorentzians in the X-ray band lightcurve (Pottschmidt et al. 2003). These authors have demon-strated that the change of the timing parameters of the source ismainly caused by a strong decrease in the amplitude of one ofthe Lorentzian components present in the PSD. This character-

T. Pechacek, R. W. Goosmann, V. Karas, B. Czerny, & M. Dovciak: Accretion disc variability as random process 13

istic is closely connected with the normalization of the modelcomponents. During the state transition of the source one oftheLorentzians becomes suppressed relative to the other. It wassuggested that this behaviour is associated with the accretiondisc corona, which is believed to be responsible for the hard-state spectrum. The evolution of the PSD form can be studiedin terms of our method, which allowed us to discuss the entireclass of multiple-power-law PSDs within a uniform systematicscheme. Although it is a versatile approach, its practical usecan be illustrated in a very simple way.

The hard-state X-ray spectrum is very likely related withthe presence of soft photons from the accretion disc, which areCompton up-scattered in a hot electron gas. The state transi-tions are then caused by the disappearance of the Comptonizingmedium. Such a change in the coronal configuration could ex-plain the change of the mutual normalization of the Lorentziancomponents, namely, the disappearance or recurrence of localpeaks in the PSD. In the idea of coronal flares modulating theaccretion disc variability of the source, which seems to be rel-evant for Cyg X-1, the position of the source in Fig. 8 willconstrain possible parameter values of the model.

Finally, we have checked that a certain proportionality be-tween the light curve rms and the flux does exist in our modelas well. This is an interesting fact by itself, however, it isnotclear at present whether the slope and the scatter of the rms-flux relation are in reasonable agreement with the actual data,and how generic the model is. Furthermore detailed investiga-tions are needed to clarify whether the spot model could satisfythe constraints arising beyond the PSD profiles. Moreover, itremains to be seen if our model requires some kind of spe-cial fine-tuning of the parameters (such as the inclination an-gle), which would make this explanation less likely (work inprogress).

Acknowledgements. The research leading to these results has re-ceived funding from the Czech Science Foundation and DeutscheForschungsgemeinschaft collaboration project (VK, GACR-DFG 13-00070J). We also acknowledge the Polish grant NN 203 380136 (BC)and the French GdR PCHE (RG). Part of the work was supported bythe European Union Seventh Framework Programme under the grantagreement No. 312789 (MD, BC, RG). The Astronomical Institutehas been operated under the program RVO:67985815 in the CzechRepublic (TP).

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Appendix A: Monotonicity of the functions g and h

We prove that the derivatives ofgk(x) and hk(x) are non-negative. We define an operator Fx,k[.] by its action on an ar-bitrary function f (τ) as

Fx,k[ f (τ)] = Nx,k E

f (τ)(

1+ τ2ω2)k+3

, (A.1)

whereNx,k is a normalization constant ensuring that Fx,k[1] =1 for all k and x. For a fixed value ofx it acts as anmean value operator with a probability distributionpx(τ) =

p(τ)/(

1+ τ2ω2)−k−3

. Using this operator, we can express thederivatives as

dgk(x)dx

= (k + 2)Fx,k[τ4] −

(

Fx,k[τ2])2

(

Fx,k[τ2(1+ τ2x)])2, (A.2)

dhk(x)dx

= (k + 3)Fx,k+1[τ6]Fx,k+1[τ2] −

(

Fx,k+1[τ4])2

(

Fx,k+1[τ4(1+ τ2x)])2

. (A.3)

14 T. Pechacek, R. W. Goosmann, V. Karas, B. Czerny, & M. Dovciak: Accretion disc variability as random process

The denominators in both equations are always non-negative.Furthermore, non-negativity of the numerator of (A.2) followsfrom the Jenssen theorem. The numerator of (A.3) is a deter-minant of a matrix of the following quadratic form:(

ab

)T (

Fx,k+1[τ6] Fx,k+1[τ4]Fx,k+1[τ4] Fx,k+1[τ2]

) (

ab

)

= Fx,k+1[(aτ3 + bτ)2] ≥ 0. (A.4)

Because the form is positively (semi-)definite, the determinantmust be a non-negative function.

Appendix B: Operator associated with thepolynomial Q

We give an explicit formula for the operatorQk. We define op-eratorsN andD as

N =1τ

ddτ, D = τ3

ddτ. (B.1)

It can be proven by direct calculation that

Nl 11+ τ2ω2

=l!(−2)lω2l

(

1+ τ2ω2)l+1, (B.2)

Dl τ2

1+ τ2ω2=

l!(2)lω2lτ2l+1

(

1+ τ2ω2)l+1. (B.3)

From these relations it follows, for arbitraryl andm,

(−1)l

(l + m)!2l+mτ−2m−1 Dm τ2l+2 Nl 1

1+ τ2ω2

=

(

τ2ω2)l

(

1+ τ2ω2)l+m+1

. (B.4)

Assuming thatQk(z) =∑k

l=0 ql zl, we can define the operatorQk as

Qk =1

k!2k

(

1τ

)2k+1 k∑

l=0

(−1)l ql τl Dk−l τ2l+2 Nl. (B.5)

The polynomialsQk andPk are mutually related by the formula

Qk(τ2ω2) = (1+ τ2ω2)k+1

∣

∣

∣

∣

∣

∣

∣

k∑

l=0

l! pl (1+ iτω)−l−1

∣

∣

∣

∣

∣

∣

∣

2

, (B.6)

wherepl are the coefficients ofPk.