111 Acta Polytechnica CTU Proceedings 2(1): 111–115, 2015 111 doi: 10.14311/APP.2015.02.0111 Flickering in CVs and Accretion Disc Viscosity R. Baptista1 1Departamento de F́ısica, Universidade Federal de Santa Catarina, Campus Trindade, 88040-900, Florianópolis, Brazil Corresponding author: raybap@gmail.com Abstract I review observational constraints on accretion disc viscosity inferred from changes of disc structure with time and from disc flickering distributions. The radial run of the disc viscosity parameter in four cases are presented and discussed. Keywords: cataclysmic variables - dwarf novae - accretion discs - optical - time-series photometry - individual: V2051 Ophi- uchi, HT Cassiopeia, V4140 Sagitarii, UU Aquarii. 1 Context Accretion discs are cosmic devices where angular mo- mentum and gravitational energy are extracted from matter by an anomalous, still unknown viscosity mech- anism, allowing it to be accreted onto a central star (Frank, King & Raine 2002). Currently, the most promising explanation for the disc viscosity is related to magneto-hydrodynamic (MHD) turbulence in the dif- ferentially rotating disc gas (Balbus & Hawley 1991). From the observational standing point of view, because the properties of steady-state discs are largely indepen- dent of viscosity, one must turn to observations of time- dependent disc behavior in order to obtain quantitative information about disc viscosity. Here we adopt the prescription of Shakura & Sunyaev (1973) for the accre- tion disc viscosity, ν = αss cs H, where αss is the non- dimensional viscosity parameter, cs is the local sound speed and H is the disc scaleheight. Non-magnetic1 Cataclysmic Variables (CVs) are excellent sites for stud- ies of disc viscosity, because of their well constrained bi- nary environment and because their accretion discs are usually the dominant light source at UV and optical wavelengths. In particular, the subclass of Dwarf No- vae (DNs) show recurrent outbursts in which their discs brighten by factors 20-100 as a consequence of mass and angular momentum redistribution on timescales of a few days. DN outbursts are explained in terms of either a thermal-viscous disc-instability (DIM, Lasota 2001) or a mass-transfer instability (MTIM, Bath 1975). DIM predicts matter accumulates in a low viscosity disc (αcool ∼ 10−2) during quiescence, whereas in MTIM the disc viscosity is always high (α ∼ 10−1). Therefore, measuring α of a quiescent disc is key to infer which model is at work in a given DN. 2 Flickering in CVs: A Short Review Flickering are the intrinsic brightness fluctuations on timescales of seconds to dozens of minutes seen in light curves of T Tau stars, mass-exchanging binaries and ac- tive galactic nuclei (Bruch 2000 and references therein). The first step in understanding what causes flickering is to find out where is comes from. Clues for the location of the flickering sources in CVs come from the analysis of eclipsing systems. In U Gem, flickering is stronger at orbital hump maximum and disappears during the eclipse of the bright spot (BS) – where the mass transfer stream hits the outer edge of its accretion disc. These results led to the conclusion that flickering in this DN arises at the stream-disc impact region, because of either unsteady mass inflow (Warner & Nather 1971) or post-shock tur- bulence (Shu 1976). On the other hand, the flickering in HT Cas disappears during eclipse of the central source and recovers well before the egress of its BS (Patter- son 1981), indicating that it originates in the inner disc regions or at the WD-disc boundary layer (BL). Pos- sible explanations for this disc+BL flickering include MHD turbulence + convection (Geertsema & Achter- berg 1992), unsteady WD accretion (Bruch 1992) or events of magnetic reconnection at the disc atmosphere (Kawagushi et al. 2000). A series of extensive optical flickering studies along the 90’s (Bruch 1992, 1996, 2000) strengthened the idea that there are two different sources of flickering in CVs: in objects with strong anisotropic emission from the BS the BS-stream flickering component dominates, whereas the disc-BL flickering component prevails in objects where emission is mostly from the accretion disc. 1White dwarf (WD) surface magnetic fields Bwd < 10 5 G. 111 http://dx.doi.org/10.14311/APP.2015.02.0111 R. Baptista The power density spectra (PDS) of flickering sources show a characteristic power-law behaviour at higher frequencies (∝ f−n), with indexes ranging n = 1 − 2, and a flat slope below a cut-off frequency, fc (Bruch 1992). The values of n and fc vary from one object to the other and for the same object at different brightness levels (Fig. 1). Figure 1: Optical (B-band) power density spectra for V2051 Oph (top) and HT Cas (bottom) at two different brightness levels in their quiescent states (differences between bright, intermediate, and faint are 1-2 orders of magnitude lower than differences between outburst and quiescence). Dashed lines show the best-fit power- law in each case; the corresponding power-law index n is depicted in each panel. 3 Estimating the αss Parameter 3.1 Time changes in disc structure By measuring the (viscous) timescale tv = r/vr ' r2/ν ' r2/(αss cs H) with which the disc responds to changes in mass input rate (Frank et al. 2002), one might infer a spatially-averaged disc viscosity parame- ter αss, αss ' r cs tv [ H r ]−1 (1) where vr is the viscous radial drift speed. Application of this time-lapse mapping technique to accretion discs of DN in outburst lead to αss ' 10−1 (e.g., Baptista & Catalán 2001). 3.2 Flickering relative amplitude Geertsema & Achterberg (1992) investigated the effects of MHD turbulence in an accretion disc. They found that the convection of turbulent eddies lead to large fluctuations in the energy dissipation rate per unit area at the disc surface D(r), which could be a source of flickering in CVs and X-ray binaries. Encouragingly, the PDS of these fluctuations resemble those of flick- ering sources, with a power-law dependency of similar index range and the flat slope at low-frequencies. Per- haps more important, their model gives a direct relation between the energy dissipation rate fluctuations and the disc viscosity parameter, providing an interesting obser- vational way to infer the local accretion disc viscosity – by measuring the relative amplitude of the energy dis- sipation rate/flickering. In this model, the number of turbulent eddies that contribute to the local fluctuation is, N(r) = 4 π r H ( H L )2 , (2) where L is the size of the largest turbulent eddies. The local rms value of the fluctuations σD(r) in the average energy dissipation rate per unit area 〈D(r)〉 is given by, σD(r) ' 2.5〈D(r)〉/ √ N(r) , (3) while the disc viscosity parameter can be written as, αss = 3 νt 2 cs H ' 0.9 ( L H )2 , (4) where νt is the local disc viscosity. If the disc-related flickering is caused by MHD turbulence, it is possible to infer αss from the relative flickering amplitude, σD/〈D〉, αss ' 0.23 [ r 50 H ][ σD(r) 0.05〈D(r)〉 ]2 . (5) In the thin disc limit (H � r), disc regions with flick- ering amplitudes of a few percent already lead to large local αss values (≥ 10−1). In this scenario, CVs with highly viscous accretion discs are expected to show sig- nificant disc flickering component. How can we measure σD/〈D〉? From a large, uni- form ensemble of light curves of a given CV it is possible to separate the steady-light component, low- and high- frequency flickering amplitudes as a function of binary phase, to derive corresponding maps of surface bright- ness distributions from their eclipse shapes and, there- after, to compute the radial run of the relative ampli- tude of the disc flickering component (flickering map- ping, see Baptista & Bortoletto 2004). 112 Flickering in CVs and Accretion Disc Viscosity Figure 2: The radial run of the disc viscosity parameter inferred from disc flickering relative amplitude distri- butions, for three strong flicker DNs and for the nova-like variable UU Aqr. Dots with solid line show average αss values, while dashed lines indicate their 1-σ limits. Horizontal dashed lines depict the typical αcool = 10 −2 value expected for quiescent DN discs according to DIM. 4 Results and Discussion Baptista et al. (2011) analyzed an extensive data set of optical light curves of HT Cas. Their observations frame a 2 d transition from a low state (largely reduced mass transfer rate) back to quiescence, allowing the applica- tion of both time-lapse and eclipse mapping techniques to estimate αss and to compare the independently de- rived results. They find that, in the low state, the gas stream hits the disc at the circularization radius rcirc, and the accretion disc has its smallest possible size. The disc fast viscous response to the onset of mass transfer, increasing its brightness and expanding its outer radius at a speed v ' +0.4 kms−1, implies αss ' 0.3 − 0.5. The newly added disc gas reaches the WD at disc centre soon after mass transfer recovery (∼ 1 d), also implying a large disc viscosity parameter, αss ' 0.5. Flickering mapping reveal a minor BS-stream flickering compo- nent in the outer disc, and a main disc flickering compo- nent the amplitude of which rises sharply towards disc centre, leading to a radial dependency αss(r) ∝ r−2, and to large values αss > 10 −1 for r < rcirc (Fig. 2) – in agreement with the time-lapse results. A similar analysis was performed for the DN V4140 Sgr (Baptista et al. 2012). Standard eclipse mapping in quiescence indicate that the steady-light is dominated by emission from an extended disc with neg- ligible contribution from the WD, suggesting that effi- cient accretion through a high-viscosity disc is taking place. Flickering maps show an asymmetric source at disc rim (BS-stream flickering) and an extended cen- tral source (disc flickering) several times larger in ra- dius than the WD at disc centre. Unless the thin disc approximation breaks down, the relative amplitude of the disc flickering leads to large αss’s in the inner disc regions (' 0.15 − 0.3), which decrease with increasing radius. Flickering mapping of the DN V2051 Oph reveals that the low-frequency flickering arises mainly in an overflowing gas stream and is associated with the mass transfer process. The high-frequency flickering origi- nates in the accretion disc and has a relative amplitude of a few percent, independent of disc radius and bright- ness level, leading to large αss ' 0.1 − 0.2 at all disc radii (Baptista & Bortoletto 2004). In UU Aqr, optical flickering arises mainly in tidally- induced spiral shocks in its outer disc (Baptista & Bor- toletto 2008). Assuming that the turbulent disc model applies, its disc viscosity parameter increases outwards and reaches αss ∼ 0.5 at the position of the shocks, sug- 113 R. Baptista gesting that they might be an effective source of angular momentum removal of disc gas. Since αss increases by two order of magnitude with increasing radius, it is not surprizing that Dobrotka et al. (2012) were not able to reproduce the observed PDS of UU Aqr with a turbu- lent disc model of constant αss. 5 Summary and Future Steps The picture that emerges from flickering mapping ex- periments of three DNs and of the nova-like vari- able UU Aqr underscores earlier results, indicating that there are mainly two sources of flickering in CVs: the stream-disc impact region in the outer disc and a tur- bulent inner accretion disc, the relative importance of which varies from system to system. In combination with an MHD turbulent disc model, flickering mapping provides a powerful probe of accre- tion disc viscosity by allowing one to derive the local magnitude and the radial run of αss from the distribu- tion of the disc flickering relative amplitude. The large αss values found for the three quiescent DNs are critical for the outburst mechanism dispute. They are at odds with DIM and indicate that the out- bursts of a group of DN (strong flickers) are powered by mass-transfer instabilities. Two questions related to this result seem to demand further theory development. The first one is why (and how) mass transfer from the donor star in CVs is unstable? The second one is what is responsible for the difference between the low-viscosity discs of quiescent DIM-driven dwarf nova and the high- viscosity discs of their MTIM-driven counterparts? A good first step towards solving this problem might be asking the related question of what is the influence of the WD magnetic field on the radial distribution of αss (e.g., see Bisikalo 2014)? An important test of the MHD turbulent disc model still to be done is to perform a flickering mapping exper- iment on a bona-fide DIM-driven DN to check whether it has a low-viscosity accretion disc (as predicted by DIM) and if αss is independent of radius (as generally assumed by DIM). Acknowledgement This work is supported by CNPq/Brazil grant 302.443/2008-8. References [1] Baptista, R., Catalán, M.S.: 2001, MNRAS, 324, 599 doi:10.1046/j.1365-8711.2001.04320.x [2] Baptista, R., Bortoletto, A.: 2004, AJ, 128, 411 [3] Baptista, R., Bortoletto, A.: 2008, ApJ, 676, 1240 doi:10.1086/528706 [4] Baptista, R. et al.: 2011, in The Physics of Accreting Binaries, Universal Academic Press (arXiv 1105.1382) [5] Baptista, R., Borges, B., Oliveira, A.: 2012, IAU Symposium 285, R.E.M. Griffin, R.J Hanish & R. Seaman (eds.), Cambridge Univ. Press, 278 [6] Balbus, S.A., Hawley, J.F.: 1991, ApJ, 376, 214 doi:10.1086/170270 [7] Bath, G.T.: 1975, MNRAS, 171, 311 doi:10.1093/mnras/171.2.311 [8] Bisikalo, D.: 2014, this proceedings [9] Bobinger, A. et al.: 1997, A&A, 327, 1023 [10] Bruch, A.: 1992. A&A, 266, 217 [11] Bruch, A.: 1996. A&A, 312, 97 [12] Bruch, A.: 2000. A&A, 359, 998 [13] Dobrotka, A., Mineshige, S., Casares, J.: 2012, MNRAS, 420, 2467 doi:10.1111/j.1365-2966.2011.20210.x [14] Frank, J., King, A., Raine, D.: 2002, Accretion Power in Astrophysics, Cambridge Univ. Press doi:10.1017/CBO9781139164245 [15] Geertsema, G.T., Achterberg, A.: 1992, A&A, 255, 427 [16] Kawagushi, T., et al.: 2000, PASJ, 52, L1 [17] Lasota, J.P.: 2001, New Astronomy Review, 45, 449 doi:10.1016/S1387-6473(01)00112-9 [18] Patterson, J.: 1981, ApJS, 45, 517 doi:10.1086/190723 [19] Shakura, N.I., Sunyaev, R.A.: 1973, A&A, 24, 337 [20] Shu, F.H.: 1976, in Structure and Evolution of Close Binary Systems, P. Eggleton, S. Mitton & J. Whelan (eds.), Dordrecht, 253 [21] Warner, B., Nather, R.E.: 1971, MNRAS, 152, 219 doi:10.1093/mnras/152.2.219 DISCUSSION JOHN CANNIZZO: What is the theoretical expres- sion used in determining αss from the speed of the cool- ing front? RAYMUNDO BAPTISTA: Inferences of αss from time-lapse mapping of outbursting DNs are generally 114 http://dx.doi.org/10.1046/j.1365-8711.2001.04320.x http://dx.doi.org/10.1086/528706 http://dx.doi.org/10.1086/170270 http://dx.doi.org/10.1093/mnras/171.2.311 http://dx.doi.org/10.1111/j.1365-2966.2011.20210.x http://dx.doi.org/10.1017/CBO9781139164245 http://dx.doi.org/10.1016/S1387-6473(01)00112-9 http://dx.doi.org/10.1086/190723 http://dx.doi.org/10.1093/mnras/152.2.219 Flickering in CVs and Accretion Disc Viscosity based on measurements of the speed of the heating wave during rise, instead of the cooling wave. [NOTE ADDED: Bobinger et al. (1997) adopted the expression αcool = vcool/cs to estimate the disc viscosity parame- ter from their measured speed of the cooling wave vcool. This expression lead to α values smaller than derived from Eq. (1) by a factor (H/r).] 115 Context Flickering in CVs: A Short Review Estimating the ss Parameter Time changes in disc structure Flickering relative amplitude Results and Discussion Summary and Future Steps