arXiv:astro-ph/0103449v2 29 Mar 2001 hc i eo h eidgp(hyhave (they gap variable period cataclysmic the of below class lie a which are systems Majoris Ursae SU INTRODUCTION 1 o.Nt .Ato.Soc. Astron. R. Not. Mon. 19) ehv eetyue mohdpril hydrody- particle smoothed a Cannizzo used by recently given have We is (1993). review excellent An obser theory. modern ex and with vations accepted compatability the better as prevailed offering has planation, th that disc is model accretion It instability 1979). the (Hoshi disc in states viscosity outburst varying and from unsta- quiescent a between rate dynamically or being transfer 1973), explain envelope (Bath mass its ble to varying to due a proposed secondary : the been outbursts have nova dwarf models two Historically, mechanisms Outburst 1.1 superoutburst. previous occur the they after if longer all or superoutburst with days a 1987), 170 Paradijs triggering van and outbursts Woerd normal der (van Hyi VW in 2018 October 31 Wynn A. Leicest G. of and University Astronomy, Murray and Physics R. of J. Department Truss, Novae R. M. Dwarf In Superoutbursts Of Nature The On us.Teei vdnefrarlxto ieof time relaxation a out- for normal evidence a is by There initiated burst. being superoutburst each the a nor- of independently with than that coexist other, suggested longer not been do percent has superoutbursts It and few mal system. a the of of period period variation curve a orbital superimposed light with a the brightness - superhumps in superoutburst, days reveals a UMa ten SU to during an outburst Also, the more. show extending or These brightness SU of days, superoutbursts. plateau three longer a to undergo two systems last which outbursts UMa normal novae the dwarf to no- by addition in dwarf displayed : from ways behaviour distinct two outburst in their vae in differ which and c 00RAS 0000 000 0–0 00)Pitd3 coe 08(NL (MN 2018 October 31 Printed (0000) 000–000 , lsi aibe-methods:numerical. variables- clysmic ercnie ihtebhvo fUGmnrmtp wr oa,wh novae, dwarf words: type Key Geminorum t U show of we behavior and the made superoutbursts. with are phen outbursts reconciled these sustain normal be or with initiate enha to superoutbursts No required instability. of is tidal secondary tha of the show result from eccentricity t direct fer disc flux curv a and mass is light superhumps the phenomenon simulated the superhump of The of Analysis rate Chamaeleontis. systems. w growth such Z binary the in system a observed in UMa features two-dimensional superoutburst SU the A a the simulate system. of to binary eters used a is of superout code potential a hydrodynamics tidal of simulation full hydrodynamic the detailed rate first the present We ABSTRACT P orb < ∼ crto,aceindss-isaiiis-bnre:ls oa,cat novae, - binaries:close - instabilities - discs accretion accretion, 2 . 7 days 170 hours) 1 r nvriyRa,Lietr E R,UK 7RH, LE1 Leicester, Road, University er, e - - ltosb hthrt(98 hwdta ic nbinaries in discs ratios Sim- that mass (1995). with showed Warner (1988) in Whitehurst found by be ulations can these superoutburst. of a and of summary (Duschl properties brief some both explain and all 1989) (Osa 1989) Livio (Osaki instabilities superout- instabilities every mass-transfer disc the 1985), in involving particular fail Models in without burst. properties, appear necessary which the howeversuperhumps presented, all been explain have few models Several elusive. more nova 2000). dwarf al., of et range (Truss wide characteristics a simulate outburst to model (SPH) namics risi h icae3:1rsnn ihtebnr orbit. binary the ratios with mass resonant with 1 systems : in 3 of where discs are terms occurs disc Only in that the (1991a) in resonance Lubow orbits Lindblad by inner) detail (eccentric instabil- in an disc explained The was superhumps. ity li explained varying neatly periodically a that to curve rise gave disc stres pre- precessing tidal the disc the on slo the that frame a found Whitehurst binary executed retrogradely. the cessed eccentricity in this Thus frame precession. eccen the prograde inertial significantly in the were fixed In was discs tric. that unstable shape these a frame, had binary which discs stable Unlike ubrt sk rpsdta h icrdu a ml so small super- was a radius of disc end the the over that At significantly proposed cycle. varying Osaki superoutburst radius outburst, a disc of re- outer course model the the His the instabilities. on tidal lied and thermal the combined 1998). (Patterson, ar that superhumps variables have cataclysmic to resul of observed theoretical spread This the resonance. matches the closely reach to enough large opeeter fsprubrt a enmuch been has superoutbursts of theory complete A sk 18)epue ueotus oe which model superoutburst a espoused (1989) Osaki A q T E ∼ < tl l v1.4) file style X 1 / 4 − 1 / ol eoetdlyunstable. tidally become could 3 mn.Comparisons omena. h superoutburst- the t ruhteds and disc the hrough a h oe can model the hat mohdparticle smoothed < q us oincorpo- to burst cdms trans- mass nced t h param- the ith 1 c hwno show ich ssosall shows es / 4 − 1 / are 3 ght ses a- ki A w e t - 2 M. R. Truss et al that it could not access the 3 : 1 resonance. Hence when the disc became thermally unstable, a normal dwarf nova 30.50 outburst occurred. As a result of the outburst however, the disc spread radially. With each successive normal outburst 30.40 the disc grew until it encountered the 3 : 1 resonance and be- came tidally as well as thermally unstable. Osaki proposed 30.30 but could not conclusively show, that the tidal removal of angular momentum was much more efficient from an ec- 30.20 centric disc. Consequently, when the disc encountered the resonance it would dump a large fraction of its mass upon Viscous dissipation from outer disc 0 5 10 15 20 the white dwarf. A prolonged “super” outburst that left a Time (days) much diminished disc resulted. Osaki’s model could not be carried too far as the efficiency with which tides removed Figure 1. Viscous dissipation from disc radii R > 20Rwd. This is angular momentum from the outer disc could not be deter- representative of the observed visual lightcurve. The scale of the mined with certainty. Osaki, and other workers, were relying dissipation program units is logarithmic and t=0 represents the upon one dimensional numerical models for disc evolution in beginning of the outburst. which the tidal forces had only been included approximately. In this paper we report the results of a smoothed parti- cle hydrodynamics (SPH) simulation of a cataclysmic vari- from azimuthal symmetry in response to the tidal resonance. able binary with mass ratio q = M2/M1 = 0.15, representa- We therefore calculate our trigger condition absolutely lo- tive of the SU UMa system Z Chamaeleontis. As the code cally for every particle in the simulation. We also previ- is fully three dimensional (though for the purposes of this ously used an enhanced viscosity parameter (α = 0.1 in paper the third dimension has been suppressed), tidal forces quiescence and α = 1.0 in outburst) to improve the run- are included exactly. Thus for the first time we are able to time of the code. The desire to examine the growth-rate follow the evolution of an accretion disc over the course of of superhumps on the correct time-scale has motivated us a superoutburst. to use more realistic viscosity parameters here at the ex- pense of simulating many outburst cycles. We use α = 0.01 in quiescence and α = 0.1 in outburst, typical of the val- ues in a dwarf nova disc (see, Shakura & Sunyaev (1973) 2 NUMERICAL METHOD for a discussion of this viscosity parameterisation). We use other physical parameters relevant to observations of Z Cha 15 −1 2.1 The Model - Porb = 0.075 d, M˙ 2 = 2x10 gs (both from Wood et al. 1986) and M1 = 0.84M⊙ (Wade et al. 1988). The disc The model and methods used in this work have already been is built up from a mass stream and undergoes normal out- described in our recent paper (Truss et al., 2000), as applied bursts during the build-up to steady state in exactly the to a system with the properties of SS Cygni (0.6). The SPH same way as our previous simulations (Truss et al. 2000). code directly includes the tidal forces due to the secondary, and incorporates a simple model for the dwarf nova thermal instability. The key to our calculations lies in the implementation 3 RESULTS of a viscosity switch. If the local surface density exceeds a 3.1 Light Curves defined value Σmax then that region is transformed to the high-viscosity outburst state on a time-scale appropriate to We present light curves through the entire superoutburst. the thermal time-scale. There is a corresponding second trig- The curves are constructed by summing the viscous dissipa- ger level, Σmin, which causes the viscosity to be switched tion in different regions of the disc. In this way we are able back down to the quiescent level. In this way we can simulate to compare the results with observed lightcurves in different the limit-cycle behaviour of a dwarf nova with an isothermal wavebands. This approach differs slightly from our previous simulation. Viscosity switches are nothing new in themselves work, in which we attempted to reconstruct several wave- in disc simulations, but in the past they have used trigger bands by treating the disc as a black body. Since the ob- levels which are constant throughout the disc. This is an served optical emission is expected to be dominated by the unrealistic situation. Cannizzo, Shafter and Wheeler (1988) cooler outer regions of the disc and the extreme ultraviolet have calculated the format for the critcal values of surface (EUV) emission is dominated by the very hot inner region, density Σ in a steady-state disc and found an almost lin- we can gain a qualitative understanding of the behaviour ear relationship with disc radius R (Cannizzo, Shafter and of the disc with this simplified approach. A more quanti- Wheeler, 1988, equations 2.1 and 2.2). Therefore, we imple- tative analysis should be left to future work in which full ment a linear Σ−R relationship for the disc in steady-state. thermodynamics is incorporated self-consistently. Figure 1 There are, however, some important refinements that shows the viscous dissipation from the outer parts of the > have been made for this study which were not made previ- disc (R ∼ 20 Rwd). The profile and duration of the outburst ously. In Truss et al. (2000) we incorporated an azimuthally is in good agreement with V-band observations of SU UMa smoothed trigger condition, in which the density condition systems. The initial rise to outburst is fairly rapid, lasting was only tested for annuli in the disc. This is unsatisfac- for two days, and is followed by the characteristic super- tory for work on a system with a more extreme mass ra- outburst plateau. Superhumps begin to appear prominently tio because the outer regions of the disc depart strongly at t = 8 days. The decay of the superhumps in the sim-

c 0000 RAS, MNRAS 000, 000–000 Superoutbursts In Dwarf Novae 3

31.2

31.0

30.8

Viscous dissipation 30.6

30.4 0 5 10 15 20 Time (days)

Figure 2. Viscous dissipation from R < 10Rwd. There is a no- ticeable rebrightening around t = 10 days. Figure 5. Surface density evolution of the disc.The plots corre-

7000 spond to times t=5.0 days (solid line),t=10.3 days (dot-dashed line) and t=18.1 days (stars). The straight lines are the critical 6000 surface density conditions Σmax and Σmin from the S-curve disc instability model. a is the binary separation 5000

4000 ible. In quiescence, the disc is not in contact with the 3:1 3000 resonance; consequently it is circular. The innermost part of No.of particles accreted the disc remains in the high state. This is in agreement with 2000 our previous work on SS Cygni. The outburst progresses just 1000 0 5 10 15 20 as for a normal outburst, but during this time the disc radius Time (days) increases and the edge of the disc encounters the resonance. We stress that this resonance is not available to systems with Figure 3. Mass accretion rate onto the primary. less extreme mass ratios. The disc rapidly becomes eccen- tric and virtually the entire disc is transformed to the high ulation is slower than observations have suggested - they viscosity outburst state. It is interesting to note the bright- persist (albeit with decreasing amplitude) during and just ness of the hot-spot in the lower-left hand panel of fig. 4. after the decline from supermaximum. Buat-M´enard et al. During supermaximum, we find that the brightness of the (2000) included the effects of stream-impact heating in a hotspot varies according to the orientation of the eccentric one-dimensional model, and showed that this causes the disc. This is not due to a varying mass transfer rate from functional form of the critical surface density to rapidly de- the secondary - this remains constant at all times during crease at the disc edge. We believe that this would cause the the simulation - it is purely a result of the geometry of the outer disc to drain faster than in our simulation and drive stream impact region. The existence of such an effect in the the edge away from the resonance more quickly, suppressing nova-like V348 Pup has been reported by Rolfe et al. (2000). the superhumps sooner. The decay from supermaximum is Figure 5 shows the evolution of surface density with disc ra- < slower than the rise, lasting for 5 days. The total duration dius. Initially, the inner regions (R ∼ 0.25a) go into outburst of the simulated superoutburst is around 18 days. Recent (solid line), but it should be noted that the large reservoir of EUVE observations of OY Carinae, which has a similar mass mass at larger radii is very close to the upper critical thresh- ratio to Z Cha, have shown that in addition to the expected old. Consequently, it is only a short time before nearly the delay between the optical and EUV rise, the EUV emis- whole disc is in outburst, and at supermaximum the large sion shows a rebrightening during superoutburst which is reservoir of mass is seen to move to smaller radii (diamonds). not seen in the V-band (Mauche et al., 2000). We are able At the end of the superoutburst, a large fraction of the disc ∼ ∼ − to look for this effect in the simulation by consideration of has been accreted (stars); 60%, compared with 5 10% the viscous dissipation coming from the hot, inner part of for a normal outburst. < the disc (R ∼ 10 Rwd), which is shown in Fig. 2 and the accretion rate onto the white dwarf (a good measure of the 3.3 Modal Analysis and Superhumps X-Ray lightcurve - Fig. 3). The rebrightening can clearly be seen in both figures, and is caused by the arrival of addi- We compare the response of the disc to the tidal field of tional material from the outer disc which has been tidally the secondary star with the analysis of Lubow (1991a). The forced into the high state. tidal potential can be decomposed into a set of functions ∞ φ(r,θ,t)= X φm(r)cos[m(θ − Ωorbt)], (1) 3.2 Disc Response m=0 It is particularly instructive to construct a visualisation of where each mode m generates a sinusoidal response in the the disc at various times through the superoutburst cycle. disc with argument (kθ − lΩorbt). The (k,l) = (1,0) mode Figure 4 shows dissipation maps of the disc in four states. represents the disc eccentricity, which grows exponentially The hot-spot and spiral structure of the disc are clearly vis- at a rate

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Figure 4. Viscous dissipation from the disc. Top left : quiescence (t=0 days); top right : rise to outburst before contact with the resonance (t=3 days); bottom left : superoutburst (t=13 days); bottom right : after superoutburst (t=23 days). The colour scaling is logarithmic 7 −1 −2 9 −1 −2 from 10 ergs cm to 10 ergs cm . The white central portion of the disc (R < 4Rwd) is not modelled in the simulation.

30.52 0.70 (1,0) 30.5 0.60 30.48

0.50 30.46 30.44 0.40 30.42 0.30 30.4 Mode Strength 0.20 30.38

0.10 Dissipation from outer disc 30.36 (2,3) 0.00 30.34 0 5 10 15 20 30.32 Time (days) 11.7 11.8 11.9 12 12.1 12.2 Time (days)

Figure 6. Relative mode strengths. The edge of the disc en- Figure 7. Superhumps in lightcurves from the outer disc. counters the 3:1 resonance around t = 5 days, producing a rapid growth in the eccentric mode (1,0). The (2,3) mode represents a travelling spiral wave launched at the resonance. which compares favourably with the analytical prediction for this system of ≃ 2 2 Σ(rres) λ 0.01Ωorb. λ = 2.08 × 2πΩorbq rres . (2) Mdisc Let us turn now to the superhumps themselves. The su- Σ(r) is the surface density of the disc at radius r, Mdisc is the perhump period is, as expected from observations, slightly mass of the disc and rres is the radius of the Lindblad reso- longer than the orbital period, and gradually decreasing as nance. This resonance couples with any existing eccentricity the outburst progresses. A Fast Fourier Transform of the to launch a two-armed travelling spiral wave with (k,l) = dissipation data in the range where the eccentricity growth (2,3). The response of these two modes in our simulation is a maximum (9.5 - 12 days) yields the period is shown in Fig. 6. It is immediately clear that, as Lubow Psh = 1.039 ± 0.008Porb . predicted, the growth of eccentricity is proportional to the strength of the (2,3) mode. For times less than t=11 days in However, between 12 and 18.5 days, when the superhumps the simulation, the growth rate of the (1,0) mode was are fully developed, we obtain

λ = 0.0075 ± 0.0002Ωorb Psh = 1.029 ± 0.003Porb .

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Long-term observations of dwarf novae are still needed across a variety of wavebands in order to improve our un- derstanding of these objects. EUV and X-Ray observations will reveal much about the physics of the boundary layer between the disc and the surface of the white dwarf, while eclipse mapping and d¨oppler tomography should probe the structure of the disc and the rˆole of the spiral waves. We also point out that observations of the growth rate of superhumps can be used to gain estimates for mass ratio, through referral to Lubow’s modal analysis that we have reprised here. It is very important to concentrate on two and three di- mensional simulations in the future, as the full tidal field is Figure 8. Surface density plot of the disc showing the spiral wave inherent in these models. The next step should be to perform launched on contact with the resonance. a 3D calculation which includes fully self-consistent thermo- dynamics. One can envisage a hybrid of a 2D isothermal These results are in excellent agreement with the observa- code such as this one with a traditional 1D thermodynam- tions collated by Warner & O’Dononghue (1988), who found ics code. This is a mammoth task in terms of computational the superhump period to decrease according to requirements, but all the components of such a model al- ready exist. Psh = (1.0389 − 0.0007T )Porb where T is the time after the beginning of the outburst in days. Fig. 7 shows a 15 hour portion of the lightcurve from ACKNOWLEDGMENTS the same region of the disc as defined in fig. 1. We also find Research in theoretical astrophysics at Leicester is sup- a strong superhump signal produced from the inner part of ported by a PPARC rolling grant. The simulations were the disc. This emission results from the spiral shock wave performed using GRAND, a high-performance computing reaching all the way down into this region, and the effect facility based at Leicester and funded by PPARC. MRT ac- can be seen with striking clarity in the map of the disc in knowledges a PPARC studentship and the support of the fig. 8. William Edwards Educational Charity.

4 DISCUSSION REFERENCES

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