# Searching for flickering statistics in T CrB

**ABSTRACT** We analyze $V$-band photometry of the aperiodic variability in T CrB. By applying a simple idea of angular momentum transport in the accretion disc, we have developed a method to simulate the statistical distribution of flare durations with the assumption that the aperiodic variability is produced by turbulent elements in the disc. Both cumulative histograms with Kolmogorov-Smirnov tests, and power density spectra are used to compare the observed data and simulations. The input parameters of the model $R_{\rm in}$ and $\alpha$ are correlated on a certain interval and the most probable values are an inner disc radius of $R_{\rm in} \simeq 4 \times 10^9$ cm and a viscosity of $\alpha \simeq 0.9$. The disc is then weakly truncated. We find that the majority of turbulent events producing flickering activity are concentrated in the inner parts of the accretion disc. Comment: 9 pages, 10 figures, accepted for publication in MNRAS

**0**Bookmarks

**·**

**83**Views

- [Show abstract] [Hide abstract]

**ABSTRACT:**We present a study of the flickering activity in two nova like systems KR Aur and UU Aqr. We applied a statistical model of flickering simulations in accretion discs based on turbulent angular momentum transport between two adjacent rings with an exponential distribution of the turbulence dimension scale. The model is based on a steady state disc model which is satisfied in the case of hot ionized discs of nova like cataclysmic variables. Our model successfully fits the observed power density spectrum of KR Aur with the disc parameter alpha = 0.10 - 0.40 and an inner disc truncation radius in the range R_in = 0.88 - 1.67 x 10^9 cm. The exact values depend on the mass transfer rate in the sense that alpha decreases and R_in increases with mass transfer rate. In any case, the inner disc radius found for KR Aur is considerably smaller than in quiescent dwarf novae, as predicted by the disc instability model. On the other hand, our simulations fail to reproduce the power density spectrum of UU Aqr. A tantalizing explanation involves the possible presence of spiral waves, which are expected in UU Aqr, because of its low mass ratio, but not in KR Aur. In general our model predicts the observed concentration of flickering in the central disc. We explain this by the radial dependence of the angular momentum gradient.Monthly Notices of the Royal Astronomical Society 11/2011; 420(3). · 5.52 Impact Factor - SourceAvailable from: Fabienne Bastien[Show abstract] [Hide abstract]

**ABSTRACT:**We present high cadence (1-10 hr^-1) time-series photometry of the eruptive young variable star V1647 Orionis during its 2003-2004 and 2008-2009 outbursts. The 2003 light curve was obtained mid-outburst at the phase of steepest luminosity increase of the system, during which time the accretion rate of the system was presumably continuing to increase toward its maximum rate. The 2009 light curve was obtained after the system luminosity had plateaued, presumably when the rate of accretion had also plateaued. We detect a 'flicker noise' signature in the power spectrum of the lightcurves, which may suggest that the stellar magnetosphere continued to interact with the accretion disk during each outburst event. Only the 2003 power spectrum, however, evinces a significant signal with a period of 0.13 d. While the 0.13 d period cannot be attributed to the stellar rotation period, we show that it may plausibly be due to short-lived radial oscillations of the star, possibly caused by the surge in the accretion rate.The Astronomical Journal 08/2011; 142(4). · 4.97 Impact Factor - SourceAvailable from: A. Bianchini[Show abstract] [Hide abstract]

**ABSTRACT:**Recently, wavelets and R/S analysis have been used as statistical tools to characterize the optical flickering of cataclysmic variables. Here we present the first comprehensive study of the statistical properties of X-ray flickering of cataclysmic variables in order to link them with physical parameters. We analyzed a sample of 97 X-ray light curves of 75 objects of all classes observed with the XMM-Newton space telescope. By using the wavelets analysis, each light curve has been characterized by two parameters, alpha and Sigma, that describe the energy distribution of flickering on different timescales and the strength at a given timescale, respectively. We also used the R/S analysis to determine the Hurst exponent of each light curve and define their degree of stochastic memory in time. The X-ray flickering is typically composed of long time scale events (1.5 < alpha < 3), with very similar strengths in all the subtypes of cataclysmic variables (-3 < Sigma < -1.5). The X-ray data are distributed in a much smaller area of the alpha-Sigma parameter space with respect to those obtained with optical light curves. The tendency of the optical flickering in magnetic systems to show higher Sigma values than the non-magnetic systems is not encountered in the X-rays. The Hurst exponents estimated for all light curves of the sample are larger than those found in the visible, with a peak at 0.82. In particular, we do not obtain values lower than 0.5. The X-ray flickering presents a persistent memory in time, which seems to be stronger in objects containing magnetic white dwarf primaries. The similarity of the X-ray flickering in objects of different classes together with the predominance of a persistent stochastic behavior can be explained it terms of magnetically-driven accretion processes acting in a considerable fraction of the analyzed objects. Comment: 10 pages, 3 figures, 2 tables. Language revision. Accepted for publication in A&AAstronomy and Astrophysics 05/2010; · 5.08 Impact Factor

Page 1

arXiv:0911.3804v1 [astro-ph.SR] 19 Nov 2009

Mon. Not. R. Astron. Soc. 000, ??–?? (2008) Printed 19 November 2009(MN LATEX style file v2.2)

Searching for flickering statistics in T CrB

A. Dobrotka1⋆, L. Hric2†, J. Casares3‡, T. Shahbaz3‡, I.G. Mart´ ınez-Pais3,4‡ and

T. Mu˜ noz-Darias3,5‡

1Departement of Physics, Institute of Materials Science, Faculty of Materials Science and Technology,

Slovak University of Technology in Bratislava, J´ ana Bottu 25, 91724 Trnava, The Slovak Republic

2Astronomical Institute of Slovak Academie of Sciences, 05960 Tatransk´ a Lomnica, The Slovak Republic

3Instituto de Astrof´ ısica de Canarias, La Laguna, Tenerife, Spain

4Departamento de Astrofsica, Universidad de La Laguna, Tenerife, Spain

5INAF - Osservatorio Astronomico di Brera, Via E. Bianchi 46, I-23807, Merate (LC), Italy

Accepted ???. Received ???; in original form 19 November 2009

ABSTRACT

We analyze V -band photometry of the aperiodic variability in T CrB. By applying a

simple idea of angular momentum transport in the accretion disc, we have developed a

method to simulate the statistical distribution of flare durations with the assumption

that the aperiodic variability is produced by turbulent elements in the disc. Both

cumulative histograms with Kolmogorov-Smirnov tests, and power density spectra

are used to compare the observed data and simulations. The input parameters of the

model Rinand α are correlated on a certain interval and the most probable values are

an inner disc radius of Rin≃ 4 × 109cm and a viscosity of α ≃ 0.9. The disc is then

weakly truncated. We find that the majority of turbulent events producing flickering

activity are concentrated in the inner parts of the accretion disc.

Key words: stars: novae: cataclysmic variables - stars: individual: T CrB - accretion,

accretion discs

1INTRODUCTION

Cataclysmic variables (CVs) are interacting binaries with a

white dwarf as the primary component and a late type main

sequence star as the secondary star which fills its Roche-

lobe (see Warner 1995 for review). Consequently mass is

transferred to the white dwarf through the inner Lagrangian

point (L1) and an accretion disc is formed (not in the case of

magnetic polars). The accretion disc is the site where several

known physical mechanisms manifest themselves in the form

of short-term stochastic variations in brightness, generally

known as flickering. These oscillations are one of the most

characteristic observational properties of accreting systems

with typical amplitudes ranging from a few dozen of milli-

magnitudes up to more than one magnitude on timescales

from seconds to minutes. Symbiotic stars are related systems

to CVs where the secondary star is an evolved red giant

where the mass loss can be due to either the wind from the

giant star or the Roche-lobe secondary star, as in ”classical”

CVs. Thus accretion discs can be formed with the associated

stochastic variations.

The first systematic study of mechanisms generating the

⋆E-mail: andrej.dobrotka@stuba.sk

† E-mail: hric@ta3.sk

‡ E-mail: jcv,tsh,igm,tmd@iac.es

flickering was performed by Bruch (1992), where he com-

pared theoretically estimated energies and timescales for in-

dividual mechanisms, with observationally derived values for

a selection of CVs and related symbiotic stars. The author

proposed four possible mechanisms responsible for the ob-

served variations in brightness: (1) unstable mass transfer

from the L1point and interaction with the disc edge, (2) dis-

sipation of magnetic loops, (3) turbulences in the accretion

disc and (4) unstable mass accretion onto the white dwarf.

Apparently it was not enough to restrict the interpretation

of the flickering in the whole family of CVs to only these

four mechanisms. For example, in the case of a truncated

accretion disc (intermediate polars) or in the case of the to-

tal absence of a disc (polars) the source of such variations is

known to be the accretion stream threaded by the magnetic

field and its subsequent interaction with the surface of the

white dwarf – accretion shock.

As mentioned earlier, Bruch (1992) performed a sys-

tematic study of flickering activity in a sample of CVs and

symbiotic systems. The promising models which do not con-

tradict the empirical results, are unstable mass accretion

onto the central accretor or turbulence in the inner accre-

tion disc. Bruch (1996) studied the variations of photomet-

ric data as a function of orbital phase in ZCam. The source

of the flickering was localized in the vicinity of the central

accretor, but the outer disc edge and its interaction with

Page 2

2A. Dobrotka et al.

Table 1. The observing log. T is the duration of the observation

in hours and N is the number of data points in the lightcurve.

n Date HJD-2450000

(start)

T

[h]

N

1

2

3

4

5

6

7

8

9

08.03.1996

09.03.1996

19.04.1996

20.04.1996

22.04.1996

23.04.1996

03.07.1996

11.12.1996

15.01.1997

150.51153

151.44714

192.35582

193.38731

195.48539

196.34682

267.37947

428.61449

463.54001

1.94

1.80

3.85

3.74

2.30

2.79

1.29

1.80

3.87

120

119

240

240

150

179

89

120

219

the gas stream from the L1 point cannot be neglected (hot-

spot). Bruch (2000) studied 4 eclipsing systems: HTCas,

V2051Oph, UXUMa and IPPeg. In these systems, similar

to ZCam, the flickering source is located in the central part

of the disc or the hot-spot. For V2051Oph the region of the

flickering was confirmed by Baptista and Bortoletto (2004)

using eclipse mapping. The hot-spot is the source of low fre-

quency flickering and the central disc of the high frequency

flickering. Finally, TCrB case was studied by Zamanov and

Bruch (1998). The U photometric data showed fast vari-

ability located near the central white dwarf. Following these

observational results it is clear that the dominant source of

flickering variability in CVs and TCrB is the central part

of the disc or the vicinity of the white dwarf. A promis-

ing scenario to explain this phenomenon is the existence of

magneto-hydrodynamic turbulence transporting the angular

momentum outwards (Balbus and Hawley 1998). However

why only the central part of the disc is the source of the

observed turbulence?

In this paper we focus on the analysis of turbulent trans-

port of angular momentum in the disc. We have decided to

study TCrB due to its rich phenomenology of rapid variabil-

ity (i.e. flickering activity) and because of it’s disc properties

are suitable for our model. In Sec. 2 we present our observa-

tions. Our treatment of turbulence in the disc as the source

of flickering is explained in Section 3 and the simulations

are described in Section 4. Finally the results are analyzed

in Sec. 5 and discussed in Sec. 6.

2 OBSERVATIONS

The observations of TCrB were taken with the 60 cm

Cassegrain telescope at the Skalnat´ e Pleso observatory of the

Astronomical institute of Slovak Academy of Sciences. We

used a single channel photoelectric photometer with UBV R

Johnson filters. We continuously cycled the UBV R filters

with 10s integrations per filter and 1s dead time for each

filter change, which results in a time resolution of 45s per

filter. The data were first used in the long-term lightcurve

analysis by Hric et al. (1998).

Figure 1 shows two example runs taken in the Johnson

V filter. The individual lightcurves consist of rapid aperiodic

variability with durations of ∼ 102−103seconds. We have in

total ∼ 24 hours of fast photometry divided into 9 observing

runs. Tab. 1 summarizes all our observations.

0.350.4 0.45

9.85

9.8

9.75

9.7

0.550.6 0.65 0.7

10.2

10.15

10.1

10.05

Figure 1. Two examples of our V -band lightcurves. The upper

panel shows data taken on the 23 April 1998 and the lower panel

shows datas taken on 15 January 1997

3 SOURCE OF APERIODIC VARIABILITY

TCrB is a non magnetic system, therefore the source of the

aperiodic variability must be attributed to one of the four

mechanisms proposed by Bruch (1992). Using integration

times of 10-15 minutes, Zamanov et al. (2005) discovered

variability in the Hα emission line which has a character-

istic timescale of ∼ 1 hour. Using the velocity information

the authors localize the variability at ∼ 20 − 30 R⊙. This

is consistent with the disc radius and the gas stream from

the L1 point can be the source of this variations. Our data

also exhibits variability but on timescales shorter than ∼ 25

minutes and therefore must have a different origin. Further-

more Zamanov and Bruch (1998) and Zamanov et al. (2004)

claim that flickering in TCrB with the same timescale as

ours arises from the vicinity of the white dwarf. Therefore,

we conclude that unstable mass transfer from the L1 point

is not likely to be the source of our observed fast aperi-

odic variability. In our paper we focus on the transport of

angular momentum in the disc. This can be produced by

magneto rotational instabilities (Balbus and Hawley 1998)

and we propose that the source of the aperiodic variability

is associated with magnetohydrodynamic turbulence.

4SIMULATIONS

4.1The idea of a turbulent element

Our idea is based on a simple approximation of mass transfer

in the disc. A turbulent eddy in the disc can be represented

by a blob of matter with mass m, Keplerian tangential ve-

locity at distance r

vt(r) = r

?GM1

r3

?1/2

(1)

and dimension scale x. M1 is the white dwarf mass and G

is the Newtonian gravitational constant. The largest possi-

Page 3

Searching for flickering statistics in T CrB3

ble dimension scale x of such a turbulent element at radius r

from the central object is the scale height of the disc H(r) at

this radius. The blob (which has dimension x and a spher-

ical shape approximation) penetrates inwards or outwards

(a blob with excess of angular momentum moves outwards,

whereas a blob with deficit moves inwards) with a distance

proportional to x and a radial viscous velocity vr(r). The

mass of the eddy mt is therefore the product of the volume

and the density ρ(r) ≃ Σ(r)/H(r), where Σ(r) is the surface

density at distance r i.e. Σ(r) =?

mt(r) =4

3πx3ρ(r).

Changing the blob’s position from r to r + x or r − x also

changes it’s Keplerian velocity vt and angular momentum

which is defined by

ρ(r)dz. Hence,

(2)

L = r × mvt. (3)

Thus by summing up all the turbulent elements we obtain

the global transport of angular momentum in the accretion

disc. Every radial penetration of a turbulent element pro-

duces energy (by “friction”) which is liberated in the form of

an observed flare. Therefore, it is expected that the timescale

t of the penetration is equal to the observed flare timescale.

4.2 Statistical distribution of the eddy dimension

scale

It is assumed that the number of eddies decreases with their

dimension scale x. Therefore, we can approximate the dis-

tribution function of eddy sizes f(x) by an exponential func-

tion (in analogy to the stochastic movement of particles in

Brownian motion)

f(x) = A expBx. (4)

where B < 0. The limits of the distribution are f(0) = 1 and

f(H) = 0.01, where H is the scale height of the disc. The

minimum of the distribution is not 0 because the exponential

function (4) cannot reach 0 and thus we obtain

f(x) = exp

?ln f(H)

H

x

?

. (5)

Consequently, the distribution of flare timescales t will be

given by f(t) with t ∼ x/vr(r).

4.3Application of the distribution function

We can define two adjacent rings in a disc, with radius r

and r + ∆r where ∆r is the ring radial thickness. The ring

then has a mass

m(r) = 2πr∆rΣ(r) (6)

and

m(r + ∆r) = 2π(r + ∆r)∆rΣ(r + ∆r). (7)

The difference in angular momentum between these two

rings is ∆L(r) = L(r + ∆r) − L(r), where the angular mo-

mentum of the adjacent rings is

L(r) = 2πr3Σ(r)∆r

?GM1

r3

?1/2

(8)

and

L(r + ∆r) = 2π(r + ∆r)3Σ(r + ∆r)∆r

?

GM1

(r + ∆r)3

?1/2

.(9)

This difference is produced by discrete blobs of diameter x

and distribution f(x), moving stochastically in radial direc-

tions between the two adjacent rings. The angular momen-

tum of an individual blob of size x located at radius r is

then given by eqs. 1 – 3, i.e.

Lt(r,x) =4

3πx3ρ(r)r2?GM1

r3

?1/2

(10)

whereas for a blob at distance r+∆r, the angular momentum

is given by

Lt(r+∆r,x) =4

3πx3ρ(r+∆r)(r+∆r)2

?

GM1

(r + ∆r)3

?1/2

.(11)

Finally the difference in angular momentum is ∆Lt(r,x) =

Lt(r + ∆r,x) − Lt(r,x).

Until now the distribution function has a maximum

f(0) = 1. This would be valid for the whole disc only if

∆L(r) and H(r) were the same (for all r). This is of course

not true because of the different disc scale height H(r) of

each ring which determines the largest dimension scale of the

turbulent element. Consequently f(x) is then also a function

of r, i.e. f(r,x). The angular momentum difference is differ-

ent for every pair of rings, not only because of H(r), but

also because of vt(r). Therefore, f(r,x) must be multiplied

by some scale parameter k = k(r) which depends on radial

position of the pair of rings. This parameter is needed for the

final construction of the flare duration histogram; for every t

we use k(r)f(r,x/vr(r)) between every pair of rings starting

with the inner disc radius and ending with the outer disc ra-

dius (the step ∆r must be chosen). Of course, the condition

t ? H(r)/vr(r) must be fulfilled, i.e. the flare duration can-

not be longer than that allowed by the biggest turbulence

dimension scale.

The scaling parameter k(r) can be obtained through

the following simple argument. We can compare ∆L(r) with

the sum of ∆Lt(r,x) between one pair of rings at distance r.

Summing up angular momentum of all the eddies is equiv-

alent to integrating ∆Lt(r,x) with the distribution f(r,x)

(equation 5) with the maximum value at 1. We thus obtain

∆l(r) =

H(r)

?

0

f(r,x)∆Lt(r,x)dx. (12)

where this summed angular momentum difference should be

equal to the global angular momentum difference ∆L(r).

The scaling factor k(r) is then given by

k(r)∆l(r) = ∆L(r) (13)

which gives

k(r) =∆L(r)

∆l(r).(14)

Finally, we obtain the distribution of relative duration by

summing up all durations t for each ring, i.e.

ζ(t) =

n

?

i=1

k(ri)f(ri,t).(15)

where n is the number of imaginary rings (derived from Rin,

Rout and ∆r) and ri are the ring diameters.

Page 4

4 A. Dobrotka et al.

4.4 The model and free parameters

For the accretion disc we use the standard Shakura and Sun-

yaev (1973) model (see e.g. Frank et al. 1992). The disc scale

height, surface density, volume density and radial viscous

velocity are then given by the equations

H = 1.7 108α−1/10˙ M3/20

acc,16M−3/8

1,s

r9/8

10f3/5[cm] (16)

Σ = 5.2α−4/5˙ M7/10

acc,16M1/4

1,sr−3/4

10

f14/5[g cm−2](17)

ρ = 3.1 10−8α−7/10˙M11/20

acc,16M5/8

1,sr−15/8

10

f11/5[g cm−3](18)

vr = 2.7 104α4/5˙ M3/10

acc,16M−1/4

1,s

r−1/4

10

f−14/5[cm s−1] (19)

where

f =

?

1 −

?R1

r

?1/2?1/4

(20)

in terms of r10 = r/(1010cm),M1,s = M1/(1M⊙) and

˙ Macc,16 = ˙ Macc/(1016g s−1). In our modelling the free pa-

rameters are the mass of the central star M1, the viscosity

parameter α, the inner disc radius Rin, the outer disc radius

Rout, and the mass transfer rate from the secondary star

which equals the mass accretion rate ˙ Macc through the disc

for a steady state disc. The radius of the white dwarf R1 is

approximated by the formula given in Nauenberg (1972).

5ANALYSIS

5.1Input parameters

The inner disc radius is a free (searched) parameter and

for the starting value we use the radius of the white dwarf

calculated from Nauenberg (1972). The outer disc radius

is first approximated by ∼ 0.5 of the primary Roche lobe

radius RL1 calculated from Paczy´ nski (1971)

RL1 = 0.462 a

?

M1

M1+ M2

?1/3

[cm], (21)

where a is the binary orbital separation and M1 and M2 are

the primary and secondary masses respectively. With the or-

bital period of 227.5 days (Kraft 1958, Paczy´ nski 1965) this

gives a disc radius of up to ∼ 1012cm. Using the Shakura and

Shunyaev (1973) model, for the disc annuli with diameter

1012cm we obtain an effective temperature of ∼ 1000K and

for a disc diameter of 1011cm of ∼ 5000K (see e.g. Frank et

al. 1992). For effective temperatures less than 5000K (an-

nuli with radius larger than 1011cm) the contribution to

the optical emission is not significant and can be neglected.

Therefore, we can adopt as Rout=1011cm in our analysis for

the V filter data.

The white dwarf mass in TCrB is 1.37 ± 0.13 M⊙

(Stanishev et al. 2004). Using UV observations the authors

also derive a mass accretion rate onto the central accre-

tor of

˙ Min = 2.5 × 1017g s−1during the low-state and

˙ Min = 1.9 × 1018g s−1during the high-state. Accord-

ing to Fig. 4 in Stanishev et al. (2004), all our observa-

tions were taken during high-state and therefore, we assume

˙ Min = 1.9 × 1018g s−1. According to the disk instability

model (see Lasota 2001 for review) this value is higher than

the critical mass accretion rate for the cold disc (Fig. 2)

for disc radius less than 1011cm, but lower than the critical

mass transfer rate for a hot disc for larger radius. Stanishev

et al. (2004) also proposed that the disc is hot and optically

thick up to a disc radius of ∼ 1R⊙ ∼ 7 × 1010cm. But if

there is an intersection of the mass accretion rate with the

critical values, the disc is unstable and should exhibit dwarf

novae activity and cannot be in a steady state. This is not

observed and all the luminosity variations on timescale of

days to weeks can be explained by the activity of the red

giant atmosphere (Zamanov and Bruch 1998).

Irradiation by the central hot source can stabilize the

disc on the hot branch (see Lasota 2001 for review) to larger

disc radii as allowed by the critical mass accretion rate. The

white dwarf in the TCrB system can have a temperature of

about 105K (see e.g. Hack et al. 1993) which is very high

compared to the typical white dwarf temperature in classical

dwarf novae. Hameury et al. (1999) studied the influence

of the white dwarf irradiation into dwarf novae cycle. The

irradiation temperature is given by

T4

irr= (1 − β)T4

1

1

π

?arcsinρ − ρ(1 − ρ2)1/2?

(22)

where ρ = R1/r, T1 is the white dwarf temperature and

(1−β) is the fraction of the incident flux which is absorbed

in optically thick regions, thermalized and reemited as pho-

tospheric radiation (see e.g. Friedjung 1985, Smak 1989).

Taking the absorbed radiation to be of about ∼ 10% and

a white dwarf radius of 0.2 × 109cm we get temperature

of 7.700K for a distance of 1012cm. This temperature lies

in the region of hydrogen ionisation. Therefore permanent

hot state is possible up to a radius of 1012cm in the case

of ∼ 90% reflection of incident radiation from the central

massive and hot white dwarf.

Finally, Zamanov and Bruch (1998) has claimed that

˙ M2 =

˙ Min in TCrB, and the disc can be assumed to be

in a hot steady state. We thus assume that the material is

hot, ionized and optically thick and the conditions for the

standard Shakura and Shunyaev (1973) model are satisfied.

Therefore, we adopt ˙ Macc = 1.9 × 1018g s−1for the whole

disc.

However, if the disc is irradiated as mentioned earlier,

larger disc radius may have a non-negligible contribution to

the V -band brightness. We therefore use two values (1011

and 1012cm) for the outer disc radius and study the differ-

ences.

5.2Histogram

We have analyzed our data in two different ways, firstly us-

ing cummulative histograms and secondly using power den-

sity spectra. We use the V -band data so that we can compare

both methods directly. In this section we compute the cu-

mulative histogram of the flare duration t from our observed

data. We define the flare duration as the distance between

two adjacent minima of a flare. For a flare to be selected it

must be sampled by a minimum number of points. Because

we smooth the data using a 3-point box-car function, hence

this limit must be larger than 3. However, if we set this limit

to be too large, short flares will be missed. After a detailed

investigation, we decided to take a minimum of 3 points ei-

ther side of the flare peak. Therefore, the shortest flare will

contain 7 points and, because the time resolution is ∼45s,

Page 5

Searching for flickering statistics in T CrB5

9 101112

10

12

14

16

18

20

22

Figure 2. Critical mass accretion rates for the accretion disc in

TCrB as a function of the distance from the central star r.

500 10001500

0

0.2

0.4

0.6

0.8

1

Figure 3.

flare duration, together with 6 simulated histograms for different

α values; from bottom to top – 0.001, 0.01, 0.1, 0.5, 1 and 10.

The simulated histograms have been computed for the following

parameters: M1 = 1.37 M⊙,

˙ Macc = 1.9 × 1018g s−1, Rout =

1011cm and Rin= 3.5 × 109cm.

Normalized cumulative histogram of the observed

they will last for ∼315s. In Fig. 3 we present a cumulative

histogram of the flares, from 300s onwards.

In our simulations we need to choose a time step ∆t to

calculate the final distribution function of duration t. Using

∆t = 200s and a minimum flare duration of t = 300s we

obtain a synthetic cumulative histogram which can be com-

pared with the observed data using a Kolmogorov-Smirnov

test (KS test). This test searches for the maximal difference

D between two cumulative histograms;

log(alpha)

log(Rin) [cm]

-3 -2.5-2 -1.5 -1-0.5 0 0.5 1

8.4

8.6

8.8

9

9.2

9.4

9.6

9.8

10

10.2

10.4

Figure 4. Contour plots of KS test for KS values 0.3, 0.2 and 0.1

(from left to right).

D = max|Oi(t) − Si(t)|(23)

where Oi(t) is the observed histogram and Si(t) is the sim-

ulated distribution function/histogram.

Some examples of simulated histograms are shown in

Fig. 3. The simulations were computed for the parameter α

in the interval 0.001 – 10. Fig. 4 shows the results of the

KS test in the α–Rin plane.It is clear that this method is

not sensitive enough to constrain the two free parameters.

The best KS test value 0.1 is too wide and so there are no

closed contours. Reducing the KS value on the Fig. 3 down

to ∼ 0.0001 does not show more contours features and so

this method is clearly is not suitable for our study.

5.3 Power Density Spectrum

Our method of flare identification can be very subjective

because different timescales are superimposed. Therefore we

decided to analyze our data in a another way by calculat-

ing the power density spectra (PDS) of our observations

and the synthetic data. We applied the Lomb-Scargle al-

gorithm (Scargle 1982) which is adequate to deal with un-

evenly spaced data. Fig. 5 show PDS of our data in UV BR

Johnson filters. The PDS of the B and V -band lightcurves

are very similar. Whereas the PDS for the other filters show

some fluctuations which can be caused by the larger scat-

ter/noise in U color and the lower amplitude of the vari-

ability in the R-band. Therefore, we decided to focus on the

V -band data, which is also adequate after the outer disc

estimate in Sec. 5.1.

Fig. 7 shows examples of PDS for different viscosity

and outer disc radius. The value for the outer disc radius

of interest are 1011and 1012cm. Differences in Rout only

start to be visible for high value of α. The two Rout cases

are indistinguishable for α = 0.5 and 2.5 (used as upper

limit of viscosity in the following PDS study). Only for α =

5.0 can the two radii be distinguished. We therefore adopt

this viscosity value as our sensitivity limit. for the outer

disc radius in our interval of interest. According to the disc

instability model (see e.g. Lasota 2001 for review) accretion

discs typically have α < 1.0, hence our interval of interest

is lower than the sensitivity limit. Therefore, this parameter

has no influence in our simulation, and so we fixed it to

Rout = 1011cm.

Page 6

6 A. Dobrotka et al.

-1

0

1

2

-1

0

1

2

-4 -3.5-3 -2.5 -2

-2

-1

0

1

2

Figure 5. The power density spectrum of our observed data in

different Johnson filters compared to the V -band data.

For a given pair of the input parameters Rin and α we

produced 100 synthetic runs with the flare duration statis-

tics calculated using the method described in Sec. 4. The

flares has a triangular shape and the time sampling of the

runs were sampled the same as the observed runs (a few ex-

amples are depicted in Fig. 10). We then calculated a mean

PDS. Fig. 6 shows six examples of the simulated PDS for

α = 0.001 − 10. We subsequently calculated the difference

in χ2between the observed and simulated PDS as follows

χ2=

N

?

i=1

(oi− ci)2

ei

, (24)

where oi and ci are the observed and calculated values re-

spectively, ei is the error of the observed PDS value and N

is the number of bins in the PDS (we used 15). We then

computed a 2-D χ2grid fits of the inner disc radius and vis-

cosity parameter (Fig. 8) for the same system parameters as

for the KS test. This method is clearly more suitable for our

study as it is more sensitive to the change in input param-

eters and the wide plateau seen the KS test is not present.

On the contrary the contour-features are well localized and

show that the input parameters α and Rin are correlated at

some interval.

A minimum χ2of 282 is found at Rin ≃ 3.9×109cm and

α ≃ 0.9. Fig. 8 shows the 90% probability (1.6-σ) confidence

levels in the α−Rin plane after rescaling so that the reduced

χ2is 1. The parameters show a linear trend, an indication

that the two are correlated. The correlation is constrained

to the interval Rin ≃ 1.1 − 4.3 × 109cm for all acceptables

α values for a disc in hot branch, i.e. 0.1 – 1.0.

6 DISCUSSION

The 90% confidence level contours of the PDS fits demon-

strates that the inner disc radius and the viscosity parameter

are correlated in the interval Rin ≃ 1.1 − 4.3 × 109cm for

-3.5-3 -2.5 -2

-2

-1

0

1

2

Figure 6. The mean power density spectrum of the observed

runs (the solid points with error bars) compared with 6 ex-

amples of simulated power density spectra (solid line) for α =

0.001,0.01,0.1,0.5,1 and 10 (from bottom to top).

-2

-1

0

1

2

-2

-1

0

1

-2

-1

0

1

-3.5-3 -2.5 -2

-2

-1

0

1

Figure 7. Examples of simulated power density spectra for dif-

ferent viscosity and outer disc radii. The inner disc radius has

been fixed to Rin= 3.5 × 109cm.

all aceptable α values for a disc in hot branch, i.e. 0.1 – 1.0.

Our best solution Rin ≃ 3.9×109cm and α ≃ 0.9 is in agree-

ment with the inner disc radius obtained by Skopal (2005)

through modeling of UV/optical/IR spectra of TCrB; i.e.

0.05 − 0.10R⊙ ≃ 3.5 − 7.0 × 109cm. This is larger than

the estimated white dwarf radius and hence the disc seems

to be truncated. Furthermore, the best model viscosity pa-

rameter is in agreement with expectations for a disc in the

permanent hot branch, i.e. 0.1 < α < 1.0.

Page 7

Searching for flickering statistics in T CrB7

log(alpha)

log(Rin) [cm]

-1.2-1 -0.8-0.6 -0.4-0.2 0 0.2

9

9.2

9.4

9.6

9.8

Figure 8. The χ2contours of the power density spectra at the

90% confidence (1.6-σ) level.

Our best values are derived from the minimal value of

χ2. The 90% contours levels suggest a large uncertainty in

parameters which are correlated. The determination of one

of the two parameters requires an independent measurement

of the other. Taking the inner disc radius from Skopal (2005)

to lie in the interval of 3.5 < Rin < 4.3×109cm then gives a

viscosity interval of 0.7 < α < 1.0. King et al. (2007) report,

for fully ionized geometrically thin accretion discs, that the

observational evidence suggests α ∼ 0.1 − 0.4. In particu-

lar, the simulations for SSCyg (Schreiber et al. 2003) yield

a viscosity value of about 0.1 − 0.2, whereas Schreiber and

G¨ ansicke (2002) obtain α = 0.5 for SSCyg in the ionized hot

state. Our derived viscosity value of 0.7−1.0 is considerably

higher than previous values. However, Schreiber et al. (2003)

used a maximum mass transfer rate of ∼ 1×1017g s−1in his

simulations. This is one order of magnitude lower than mass

accretion rate in TCrB. The high mass transfer rate through

the disc together with the irradiation by the central source

may explain the higher temperature and viscosity than in

standard dwarf nova system. Furthermore the lower viscos-

ity values were determined from the long-term light curve

behaviour where the considered timescales are typical for

the largest disc radii, whereas our constraints apply mainly

at the inner disc radius. This gives an idea that the viscos-

ity parameter α can also be a function of the distance r. In

the standard disc instability model (see e.g. Lasota 2001 for

review) the main difference between the viscous parameter

in the cold and hot branch is due to the ionisation of hy-

drogen. Once the material is in the hot state, the ionization

state of hydrogen does not change with increasing tempera-

ture and the viscosity parameter is supposed to be constant

in the whole disc. Intensive studies of the disc behaviour

with constant α were performed in the case of dwarf novae

and low mass X-ray binaries. But the typical disc radii of

these binaries are of the order of 1010−1011cm. The case of

symbiotic systems, such as TCrB is different because of the

large binary separation and the possible outer disc radii up

to 1012cm. But we showed that our simulations are not sen-

sible to such large disc radii in the acceptable α regime. The

outer disc radius used for TCrB is 1011cm, and hence is in

agreement with the radius of low mass X-ray binaries mod-

elled by Lasota (2001) with constant α values. Therefore we

conclude that radial variations of the viscous parameter is

0

5

10

15

6

7

8

9

9 10 11

5

6

7

8

9

Figure 9. A simulation of the scaling parameter k, disc scale

height H and radial velocity vr for M1= 1.37 M⊙, ˙ Macc = 1.9×

1018g s−1and α = 0.5.

not required. An alternative explanation for the different α

results could be a different model other than the Shakura

and Shunyaev (1973) model for the large outer discs in sym-

biotic systems. An application of our simulations to dwarf

novae with disc radii of 1010cm can test whether the α differ-

ences remain or disappear. This case is problematic because

dwarf novae discs are not in a steady-state. Another pos-

sibility is to study dwarf novae in outburst when the disc

is fully ionised and is in a quasi steady-state (see e.g. La-

sota 2001) and in a permanent hot state except for in the

lightcurve minima e.g. in VYScl systems (see e.g. Warner

1995).

Another interesting question is the location of the flick-

ering events in the inner disc. This can be understood from

the behavior of the scaling parameter k(r) (Fig. 9). For very

small r values, in the vicinity of the central star, k(r) rises

very steeply because of the small disc scale height (Fig. 9)

and hence small turbulent eddies. Furthermore, k(r) de-

creases with increasing r and the disc scale height, and thus

less events are needed to transport the angular momentum.

This strongly suggests that the inner parts of the disc are

the source of the fast flickering, which is in agreement with

observations (see e.g. Bruch 1992, Zamanov & Bruch 1998,

Bruch 1996, Bruch 2000, Baptista & Bortoletto 2004, Za-

manov 2004). A similar behavior is deduced from the sen-

sitivity of our modeling to the input parameters Rin and

Rout, which is very high for Rin ∼ 109−1010cm but not for

Rout ∼ 1011− 1012cm. This suggests that the majority of

turbulent events are located in the central part of the disc

up to radial distance ∼ 1010cm.

The inner disc radius and the viscosity is correlated

only for low values of r comparing to the disc radius. This

can be explained because the viscosity must increase with

higher Rin in order to get the same χ2residual. This means

that by truncating the central part of the disc, we are re-

moving many fast events which should be compensated by

Page 8

8 A. Dobrotka et al.

the increase of the radial velocity of slower events at larger

radii. The radial velocity vr ∼ α4/5strongly decreases at

very small r (see Fig. 9). An alternative explanation relies

on the total number of events in the central disc. The scaling

parameter k gives an estimate of the number of events and

this is very large at small r. Hence by truncating the central

part of the disc we are removing many events. Whereas at

larger r the decrease of k and vr is flatter and the correlation

between Rin and α is not as strong as for small values of r.

Fig. 10 shows three examples of simulated lightcurves

computed for different viscosities. It is obvious that high fre-

quency fluctuations increase with α, whereas low-frequency

fluctuations decrease with α. This disappearance of large

events is visible in our PDS (Fig. 6). The low viscosity PDS

has a strong peak near frequency 10−3Hz which is caused by

the quasi-periodic oscillations present in the simulated data.

Such variations has a timescale of ∼ 0.01 day clearly visible

on Fig. 10 for α = 0.01 case. The timescale of the data is too

short to distinguish whether the fluctuations are stochastic

or quasi-periodic. For higher viscosities, large events disap-

pear and so does the evidence for quasi-periodic signals in

the simulated data. Hence the peak at 10−3Hz dissapear.

In addition, the observed PDS in Fig. 6 are scattered

which produce a high χ2. This behavior tells us something

about the observational demands. Our observed PDS is com-

puted from 9 runs, whereas the simulated PDS is computed

from about 100 synthetic runs. Smoother observed PDS

(made from a higher number of observational data sets) are

required for the analysis to get lower χ2and then make the

method more sensitive to the input parameters. The mean

duration of our data is ∼ 2.6 hours. Longer observations

could remove the quasi-periodic signals at low viscosities

and reduce the scatter in the observed and simulated PDS

at low frequencies (∼ 10−4Hz). Using of a single filter can

improve the data resolution to ∼ 10 seconds and hence the

quality of the PDS at higher frequencies (∼ 10−2Hz). Also

the observational gaps needed to observe a comparison star

(clearly visible on Figs. 1 and 10) affect the PDS. Therefore

CCD observations with fast readout speeds are preferred.

Our best model yields a parameters in reasonable agree-

ment with other works. We therefore conclude that our sim-

ple idea of angular momentum transport through turbulent

eddies is sufficient to explain the observed flickering statis-

tics. No other mechanisms are then required to generate the

observed flickering activity.

7 SUMMARY

We have analyzed V -band photometry of the aperiodic vari-

ability of TCrB during the high state, taken using data with

a time resolution of ∼ 45s. We developed a method to sim-

ulate flickering activity based on a simple idea of angular

momentum transport through turbulent processes in the ac-

cretion disc. We simulated artificial data sets and compared

them with the observations using cumulative histogram with

Kolmogorov-Smirnov tests and power density spectra. The

best model yielded Rin ≃ 4 × 109cm and α ≃ 0.9 but with

large uncertainties. However, using the results from litera-

ture for the inner disck radius, we derived the viscosity to

lie in the interval 0.7 < α < 1.0. The disc is weaklu trun-

cated and we provide an explanation for the concentration

1

0

-1

1

0

-1

1

0

-1

0.55 0.60.65 0.7

10.18

10.16

10.14

10.12

10.1

10.08

Figure 10. Three examples of the synthetic data compared to

an observed lightcurve. The simulated runs are computed for M1

= 1.37 M⊙, ˙ Macc= 1.9×1018g s−1and Rin= 3.5×109cm and

three different viscosities α = 0.01, 0.50 and 10.0.

of flickering events in the central part of the accretion disc.

The correlation between the parameters Rin and α is ex-

plained by the coupling between the number of fast events

and the radial velocity, i.e. the removal of fast events pro-

duced by increasing Rin is compensated by an increase in

vr and hence α. Our turbulent scenario of angular momen-

tum transport successfully explains the observed flickering

statistics without any other mechanism required.

ACKNOWLEDGMENT

AD and LH acknowledge the Slovak Academy of Sciences

Grant No. 2/7011/7. AD also acknowledge HPC Europa

grants HPC04MXW87 and HPC0477ZZL for supercomput-

ing training and V.Antonuccio-Delogu from INAF Catania

for computer providing. JC, TS, IM and TMD acknowledge

support from the Spanish Ministry of Science and Technol-

ogy under the grant AYA200766887. Partially funded by

the Spanish MEC under the Consolider-Ingenio 2010 Pro-

gram grant CSD2006-00070: “First Science with the GTC”

(http://www.iac.es/consolider-ingenio-gtc/).

REFERENCES

Balbus, S.A., Hawley, J.F., 1998, RvMP 70, 1

Baptista, R., Bortletto, A., 2004, AJ 128, 411

Bruch, A., 1992, A&A 266, 237

Bruch, A., 1996, A&A 312, 97

Bruch, A., 2000, A&A 359, 998

Frank, J., King, A.R., Raine, D.J., 1992, Accretion Power

in Astrophysics, Cambridge University Press, Cambridge

Friedjung, M., 1985, A&A 146, 366

Hack, M., La Dous, C., Jordan, S.D., Thomas, R.N., Gold-

berg, L., Pecker, J.-C., 1993, Cataclysmic Variables and

Page 9

Searching for flickering statistics in T CrB9

Related Objects, Monograph Series on Nonthermal Phe-

nomena in Stellar Atmospheres - NASA SP, Paris: Centre

National de la Recherche Scientifique, Washington, D.C.,

NASA

Hameury, J.-M., Lasota, J.-P., Dubus, G., 1999, MNRAS

303, 39

Hric, L., Petr´ ık, K., Urban, Z., Niarchos, P., Anupama,

G.C., 1998, A&A 339, 449

King, A.R., Pringle, J.E., Livio, M., 2007, MNRAS 376,

1740

Kraft, R.P., 1958, ApJ 127, 625

Lasota, J.-P., 2001, NewAR 45, 449

Nauenberg, M., 1972, ApJ 175, 417

Paczy´ nski, B., 1965, AcA 15, 197

Paczy´ nski, B., 1971, AcA 21, 417

Scargle, J.D., 1982, ApJ 263, 835

Schreiber, M.R., G¨ ansicke, B.T., 2002, A&A 382, 124

Schreiber, M.R., Hameury, J.-M., Lasota, J.-P., 2003, A&A

410, 239

Shakura, N.I., Sunyaev, R.A., 1973, A&A 24, 337

Skopal, A., 2005, A&A 440, 995

Smak, J., 1989, AcA 39, 317

Stanishev, V., Zamanov, R., Tomov, N., Marziani, P., 2004,

A&A 415, 609

Warner, B., 1995, Cataclysmic Variable Stars, Cambridge

University Press, Cambridge

Zamanov, R.K., Bruch, A., 1998, A&A 338, 988

Zamanov, R., Bode, M.F., Stanishev, V., Mart´ ı, J., 2004,

MNRAS 350, 1477

Zamanov, R., Gomboc, A., Bode, M.F., Porter, J.M., To-

mov, N.A., 2005, PASP 117, 268