Astronomy&Astrophysicsmanuscriptno.hysteresis3 (cid:13)cESO2017 January3,2017 Hystereses in dwarf nova outbursts and low-mass X-ray binaries J.-M.Hameury1,J.-P.Lasota2,3,C.Knigge4,andE.G.Körding5 1 UniversitédeStrasbourg,CNRS,ObservatoireAstronomiquedeStrasbourg,UMR7550,67000Strasbourg,France e-mail:[email protected] 2 Institut d’Astrophysique de Paris, CNRS et Sorbonne Universités, UPMC Paris 06, UMR 7095, 98bis Bd Arago, 75014 Paris, France 3 NicolausCopernicusAstronomicalCenter,PolishAcademyofSciences,Bartycka18,00-716Warsaw,Poland 4 SchoolofPhysicsandAstronomy,UniversityofSouthampton,SouthamptonSO171BJ,UK 5 DepartmentofAstrophysics/IMAPP,RadboudUniversity,POBox9010,NL-6500GLNijmegen,theNetherlands 7 1 0 2 ABSTRACT n Context.Thediscinstabilitymodel(DIM)successfullyexplainswhymanyaccretingcompactbinarysystemsexhibitoutbursts,during a whichtheirluminosityincreasesbyordersofmagnitude.TheDIMcorrectlypredictswhichsystemsshouldbetransientandworks J regardlessofwhethertheaccretorisablackhole,aneutronstarorawhitedwarf.However,ithasbeenknownforsometimethat 2 theoutburstsofX-raybinaries(whichcontainneutron-starorblack-holeaccretors)exhibithysteresisintheX-rayhardness-intensity diagram(HID).Morerecently,ithasbeenshownthattheoutburstsofaccretingwhitedwarfsalsoshowhysteresis,butinadiagram ] combiningoptical,EUVandX-rayfluxes. E Aims.Weexamineherethenatureofthehysteresisobservedincataclysmicvariablesandlow-massX-raybinaries. H Methods.WeusetheHameuryetal.(1998)codeformodellingdwarfnovaoutbursts,andconstructthehardnessintensitydiagram . aspredictedbythediscinstabilitymodel. h Results.WeshowexplicitlythatthestandardDIM–modifiedonlytoaccountfordisctruncation–canexplainthehysteresisobserved p inaccretingwhitedwarfs,butcannotexplainthatobservedinX-raybinaries. - o Conclusions.Thespectralevidencefortheexistenceofdifferentaccretionregimes/components(disc,corona,jets,etc.)shouldbe r basedonlyonwavebandsthatarespecifictotheinnermostpartsofthediscs,i.e.EUVandX-rays,whichisadifficulttaskbecauseof t interstellarabsorption.Theexistingdata,however,indicatethatanEUV/X-rayhysteresisispresentinSSCyg. s a Keywords. accretion,accretiondiscs–Stars:dwarfnovae–X-rays:binaries–instabilities [ 1 v 1. Introduction our, with of course the objective of validating the model, but 2 also of putting constraints on the viscosity which still remains 7 Dwarf novae (DNs) are a subclass of cataclysmic variables poorly understood, despite spectacular progresses with the re- 4 (CVs) which undergorecurrent outbursts usually lasting a few alization that angular momentum transport is most probably 0 0days. Outbursts are separated by quiescent intervals of a few theresultofthemagnetorotationalinstability(Balbus&Hawley .weeks (see e.g. Warner 1995, for an encyclopaedic review of 1991). In the DIM, viscosity is still modelled according to the 1CVs). Itisnowwidelyacceptedthatthese outburstsare there- Shakura&Sunyaev(1973)αparametrization.Whereasαisrea- 0 sultofa thermal-viscousinstabilityofthe accretiondisc which sonablywellconstrainedwhenthediscishot,therearestilllarge 7 arises when hydrogenbecomes partially ionized and the opac- uncertainties when the disc is in quiescence. A new DIM ver- 1 :ities depend sensitively on temperature (see Lasota 2001, for a sionusingviscosityanddiscverticalstructuresobtainedthrough vreview of the model). A stability analysis shows that when the MRI simulation hasbeen recentlydevelopedby Colemanetal. Xilocal mass transfer rate M˙ is in the range [M˙c−rit,M˙c+rit] the disc (2016)butalsointhiscasethecolddiscphysicsiscausingprob- is thermallyand viscouslyunstable;forlower valuesof M˙, the lems. r adiscinstableonacoldbranch,andforhighervaluesof M˙, the Colour variations should be one of the most obvious ob- discisalsostable,butonahotbranch. servational features that models should aim at reproducing. It In quiescence, mass accumulates in the disc which slowly hasbeen knownfor long thatdwarf novaefollow a loopin the heatsup,thelocalmasstransferrateincreasesuntilitreachesat U − B, B−V colour plane (e.g. Bailey 1980, for the cases of somepointinthediscthecritical M˙− forwhichtheinstability SS Cyg and VW Hyi). This fact seems to have been somehow crit develops.Theinstabilitypropagatesthroughoutthediscviatwo forgotten,perhapsbecause the precise modellingof the optical heating frontsmovinginwardsand outwards;the outer heating coloursoftheaccretiondiscisnoteasy,asitrequiresinprinci- fronteventuallyreachestheouteredgeofthediscwhichisthen plecalculationsofdetailedspectra,andalsobecauseitcouldbe almostinasteadystatewithamassaccretionrateontothewhite accountedfor as well by the DIM and by the mass transfer in- dwarflargerthanthemasstransferratefromthesecondary.The stability model (Bath&Pringle 1981). A similar phenomenon discemptiesuntilacoolingfrontstartspropagatingattheouter is also seen when comparing the visible and EUV/FUV light edgeofthediscanbringsthesystembacktoquiescence. curves, and is usually described in terms of the so-called UV Confronting the predictions of the disc instability model delay: the UV rise is delayed by approximately 0.5 day in a (DIM) with the observationshas been a long standing endeav- system such as SS Cyg (Cannizzoetal. 1986); it was initially Articlenumber,page1of7 A&Aproofs:manuscriptno.hysteresis3 timebetweenoutbursts(yearsordecadesinsteadofmonths)and longer outburst durations (months instead of a few days). The compactnessoftheaccretingobjectisalsoresponsibleformost oftheemissioninSXTstopeakintheX-raydomaininsteadof the UV/EUV. In LMXBs, hard and soft X-rays originate from theinnermostpartoftheaccretionflow,whereasincataclysmic variables, X-rays are emitted at the very vicinity of the white dwarf, but the optical light is emitted by the whole disc. The observedhysteresisthereforerelatesemissionfromthesamere- gion (or from very nearby regions) in LMXBs, whereas this is notatallthecaseinCVs. In this paper,we reassess the multi-wavelengthtime evolu- tion of dwarf novae as predicted by the disc instability model, with a particular emphasis on SS Cyg, for which we have the most comprehensiveset of observations,and for which the pa- rameters (distance, orbital parameters) are reasonably well de- Fig.1.Hardness-intensitydiagramsforablackhole,aneutronstarand termined, and compare our results with observations. We also a cataclysmic variables. Here, L and L are is the X-ray and EUV providegraphsshowingwhere,accordingtotheDIM,themajor- PL D luminosities respectively. Data is taken from Maitra&Bailyn (2004) ityoftheemittedlightoriginates,asafunctionbothoftimeand andWheatleyetal.(2003).FigurereprintedfromKördingetal.(2008) wavelength.WeshowthattheDIMnaturallyaccountsfortheob- bypermissionofScience. served hysteresis when correlatingX-ray and optical emission, whereas it cannot explain the hysteresis observed in LMXBs when plotting the X-ray flux versus the X-ray hardness. We thoughtthattheimportanceofthedelaywasagoodindicatorof show,however,thattheexistingdataforSS Cygsuggestthata the place where the instability initially develops (Smak 1998): hysteresiscouldalsoexistwhenconsideringtheX-rayplusEUV theso-calledinside-outoutburstsforwhichtheinstabilitydevel- flow,presumablyrepresentativeoftheaccretionrateintheinner opsclosetotheinnerdiscedgeshouldexhibitshorterUVdelays disc, versusthe X-ray / EUV hardnessratio. This hysteresis, if thantheoutside-inoutburstsforwhichtheinstabilitysetsinfur- confirmed,cannotbe explainedby the DIM alone; it would be theraway.Schreiberetal.(2003)showedthatthesituationisin similartotheoneobservedinLMXBsandcouldpointtosimilar fact more complex, that disc truncation plays a major role and mechanisms. that,contrarytoexpectations,thedelayforoutside-inoutbursts isusuallyshorterthanforinside-outoutbursts. Kördingetal. (2008) showed the temporal evolution of the optical flux vs. the EUV/X-ray ratio in SS Cyg: in quiescence, mostofthefluxisemittedintheX-rayband;duringanoutburst, 2. Themodel the optical flux rises first; close to the outburst maximum, the X-ray flux vanishes while the EUV flux reaches its maximum, We use here the version of the DIM as described in Hameuryetal. (1998) in which heating by the tidal torque andduringdeclinetheopticalfluxdecreaseswhereastherelative and stream impact have been incorporated (Buat-Ménardetal. contributionofX-raysincreasesuptoitsquiescentvalue.While thiswas notdiscussedin thepaper,the datashowsa hysteresis 2001).Wealsotakeintoaccountirradiationoftheaccretiondisc intheoptical/hardnessplot,verysimilartotheoneobservedin bythecompactobject,whichimpactstheoutburstcyclebymod- ifying the vertical disc structure and thus the S-curve, as well LMXBs;Fig.1reproducestheirFig.1.Kördingetal.(2008)ar- guedthattheEUV/X-rayratioisameasureoftheopticaldepth astheemittedspectrabecauseofreprocessing.Forthespectral modelling,wefollowtheproceduredescribedinSchreiberetal. oftheinnerpartsoftheaccretionflow.Theysuggested,thatsim- (2003),withthedifferencethatthediscspectrumisobtainedby ilartoXRBsCVsshouldemitthemostprominentradioemission attheinitialriseandduringthetransitionfromopticallythinto summing blackbodiesinstead of Kurucz (1979) spectra. Given the small differences found in Schreiberetal. (2003) between opticallythickinneraccretionflow(inCVs,itispresumablythe these two different approximations, and given the fact that the boundarylagerthatchangesopticaldepth;inLMXBs,thetran- sitionopticallythin/opticallythickwouldrather–certainlyfor spectrum of an accretion disc is not very well represented lo- callybyaKuruczspectrum,thisisasimpleandfairassumption. LMXBs containingblack holes – takes place in the disc). This A detailed calculation as done in Idanetal. (2010) for time- is the phase where the authorsdetected the source in the radio bands. dependentaccretion discs would be CPU intensive and clearly outsideofthescopeofthispaper. The interpretation of the similarity between the HIDs for LMXBs and CVs is, however, not as straightforward as it Inadditiontothefluxemittedbytheaccretiondisc,wetake might seem. Low-mass X-ray binaries (LMXBs) are also sub- intoaccountthecontributionoftheboundarylayer,whichisas- jecttothesameinstabilityasdwarfnovae(King&Ritter1998; sumed to emit hard X-ray when optically thin; we assume that Dubusetal.2001)whichproducessoftX-raytransients(SXTs) thishappenswhentheaccretionrateisbelow2 1016gs−1.This (seee.g.Tanaka&Shibazaki1996;Chenetal.1997;Yan&Yu valueistwiceaslargeastheoneusedinSchreiberetal.(2003) 2015;Tetarenkoetal.2016,forreviewoftheobservations).The to account for the smaller white dwarf mass and hence larger differencesbetweenSXTsandDNsareduetothenatureofthe emittingsurfaceweassumehere(seebelow).Inordertoobtaina compactobject,whichresultsinstrongirradiationoftheaccre- smoothtransition,weassumethattheX-rayfluxiscutbyanex- tion disc during SXT outbursts, together with larger accretion ponentialfactorexp(−M˙ /2 1016g s−1).Whenopticallythick, acc discsdetobothalargercompanionmassandlongerorbitalpe- thespectrumisthatofablackbody,andweassumethattheemit- riodsinLMXBs.Theseaccountforthemuchlongerrecurrence tingareais f 4πR2 ,with f givenby(Patterson&Raymond em wd em Articlenumber,page2of7 J.-M.Hameury etal.:Hysteresesindwarfnovaoutburstsandlow-massX-raybinaries Table1.BinaryparametersforSSCyg Parameter Value References P (hr) 6.603 1 orb M (M ) 1.00 2(seetext) wd ⊙ M /M 0.67 2 2 wd R (cm) 5.17×1010cm tid R (cm) 5.5×108cm wd Distance(pc) 114 3 References. (1) Ritter&Kolb (2003); (2) Bitneretal. (2007); (3) Miller-Jonesetal.(2013) 1985;Schreiberetal.2003): M˙ 0.28 f =10−3 acc (1) em 1016gs−1! Fig. 2. Light curve obtained using parameters given in Table 1. The where R is the white dwarf radius, and M˙ is the accretion solidlinecorrespondstotheunilluminatedcase;thedashedlinetothe wd acc maximallyirradiatedcase.SnapshotsinFigs.3–5aretakenatphases rateontothewhitedwarf. labelledR,M,D,Q. Wealsoincludethecontributionofthewhitedwarf,assumed to be a blackbodywith a constant temperature, and that of the fractionof the hotspotluminositywhichis notincludedin the discmodel;weassumethatthehotspotradiatesasablackbody withatemperatureof10,000K.Weassumethatonehalfofthe stream impactenergyis emitted by the hot spot. We finally in- cludethecontributionfromthesecondarystar,whichisthesum oftwoblackbodieswithdifferenteffectivetemperaturesbutthe sameemittingarea:theunilluminatedbacksideofthesecondary, plus the contribution of the illuminated side of the secondary whichisassumedtohaveatemperatureT givenby: 2 R 2 T4 =T4+ wd T4 f +T4 (2) 2 ∗ (cid:18) a (cid:19) (cid:16) BL em wd(cid:17) where T , T and T are the secondary, boundary layer and ∗ BL wd whitedwarftemperaturesrespectively,andaistheorbitalsepa- ration.IrradiationbythediscisnotincludedinEq.(2)sincethe disc luminosity is emitted perpendicular to the disc plane and only a small fraction of it can contribute to the heating of the secondary. The unilluminated temperature is taken to be 4000 Fig.3.Energyspectraemittedbytheaccretiondiscattimesindicated K.Asforthedisc,onecouldhaveassumedthesecondaryspec- in Fig. 2. The black curve corresponds to the rise to maximum (R), trumtobegivenbyKurucz(1979);again,giventheapproxima- theredonetotheoutburstmaximum(M),thegreenonetodecay(D), andtheblue one toquiescence (Q).Thesolid linecorresponds tothe tionsmadehere,inparticularthefactthatthetemperatureofthe unilluminated case; the dashed line to the maximally irradiated case. irradiatedsideofthesecondaryisconstantoverthewholehemi- Note that contributions from other components (white dwarf surface, sphere instead of decreasing from the equator to the pole, this boundarylayer,hotspotandsecondarystarhavenotbeenincludedhere. refinementisnotappropriate,andwethereforeuseablackbody spectrum. We use the orbital parameters given in Table 1. We have fitfortheUV delayinSS Cyg(Schreiberetal. 2003).We take chosen M1 = 1 M⊙, which does not correspondto the fiducial thetruncationradiustobe: valuegivenbyBitneretal.(2007)of0.81M andissignificantly ⊙ asmparilmlerartyhamnatshseopfr0ev.8io1uMsly⊙,utsheeddvisaclueexotefn1s.i2onMis⊙sbmecaalluasendwtihthe rin =5.21×108µ340/7M1−1/7 10M1˙6agccs−1!−2/7 cm, (3) peakluminositycannotreachtheobservedvalue. M = 1.0M 1 ⊙ correspondsto theupperrangeofallowedvalues[0.62−1.00] which correspondsto the case where truncation is caused by a M⊙ providedbyBitneretal.(2007).Thevalueoftheradiopar- dipolarmagneticfieldwithmomentµ30 inunitsof1030 Gcm3, allaxdeterminedbyMiller-Jonesetal.(2013)alsorequiresthat andM istheprimarymassinsolarunits. 1 theprimarymasstobeclosetothemassoftheSun. Wehaveconsideredtwoextremecasesfortheirradiationof the disc: the case where irradiation is neglected, and the case whereitistreatedasinHameuryetal.(2000),i.e.theirradiation 3. Resultsforaccretingwhitedwarfs fluximpingingontothediscisgivenby: We first consider the case where the disc is truncated by e.g. a magnetic field and does not extend all the way down to the GM M˙ 31 σT4 = wd acc [arcsinρ−ρ(1−ρ2)1/2] (4) whitedwarfsurfaceduringquiescence,whichprovidesthebest ill 8πR π wd Articlenumber,page3of7 A&Aproofs:manuscriptno.hysteresis3 heating variesas r−3, irradiation will always overcomeviscous heatingat largedistance if the disc is largeenough.This is the case in LMXBs; in CVs, using the same value for C does not leadtosignificanteffects,becauseofthesmallerdiscextentand of the smaller accretion efficiency. Our numerical simulations haveshownthatthisisindeedthecase. The resultinglightcurveis presentedin Fig. 2. Itshowsan alternationof long and shortoutbursts,as observedin SS Cyg. Theirradiatedandunirradiatedcasesareverysimilar;whenir- radiationistakeninto account,themajoroutburstslastslightly longer, the quiescence level is reduced because the disc mass is smaller at the end of an outburst, and the recurrence time is slightlyincreased.WeshowinFig.3thediscspectraobtainedat 4differentepochs,representativeoftherisetomaximum,max- imum,decayandquiescence.Contributionsfromothercompo- nentsofthesystem(whitedwarf,secondary,boundarylayer,hot spot)havenotbeenincludedhere;exceptfortheboundarylayer, theircontributionissignificantonlyduringquiescence.Irradia- Fig. 4. Radial distribution of the UV flux during rise, maximum and decay. Thecolour coding isthesameasinFig.5. Thesolidlinecor- tion has little effect on these spectra; the main differences are responds to the unilluminated case; the dashed line to the maximally thattheyextendtoslightlylowerwavelengthsandthefluxesare irradiatedcase.TheUVfluxinquiescenceisvanishinglysmallandis slightly higher in the irradiated case, despite the accretion rate notshownhere. beingalmostidentical,exceptinquiescence.Figs.4and5show thecontributionofeachringinthedisctotheUV(1250Å)and optical (5500 Å) fluxes. The position of the maximum in r2F ν indicates the region of the disc which is the main contributor to the flux, provided that F (r) varies on scales of order of r, ν whichisobviouslynotthecasefortheoutermostregionsofthe disc, where heating is dominatedby dissipation of tidal forces. Despite the largevalue of r2F atthe outer edge,these regions ν contributelessthan1%tothetotalUVfluxthroughoutthewhole cycle,andto10%oftheopticalfluxatmaximumbut60%dur- ingdecay.Ascanbeseen,inthehotstate,theUV-domainr2F ν curve is rather flat over most of the disc, and all regions con- tribute more or less evenly to the UV flux. This contrasts with the opticalwavelengthswhere most of the flux originatesfrom the outermostring in the hot state. It can also be seen that dif- ferencesbetweentheirradiatedandnon-irradiatedcasesexistin opticalduring decay. They appear in regionsof the disc which havereturnedtothecoldstate,anddonotcontributemuchtothe opticallightanyway.Differencesarealsointhecontributionsof the inner disc to the optical light duringquiescence, which are Fig.5.RadialdistributionoftheopticalfluxattimesshowninFig.2. alsosmall;theseareconsistentwiththereducedM˙ intheirra- acc Theblack curvecorresponds totherisetomaximum (R),theredone diatedcase. totheoutburstmaximum(M),thegreenonetodecay(D),andtheblue onetoquiescence(Q).Thesolidlinecorrespondstotheunilluminated These resultscould in principlebe comparedwith observa- case;thedashedlinetothemaximallyirradiatedcase. tionsofdwarfnovaeusingeclipsemappingtechniques(seee.g. Baptista2016,andreferencestherein).TheexaminationofFig. 3inBaptista&Catalán(2001)showingtheradialdistributionof whereρ = R /r,andristheradialcoordinateinthedisc.This theintensityofthedwarfnovaEXDrashows,however,anum- wd berofdifferenceswiththepicturepresentedhere.Inparticular, assumes that the energy released by accretion onto the white in quiescence,mostof the opticallightis emittedby the inner- dwarf is emitted isotropically by the white dwarf surface. It is mostpartoftheaccretiondisc,withasignificant,thoughsmaller, a fair approximation when the disc is truncated. It clearly is contributionfromanouterringwhichisnotaxisymmetric.One an overestimatewhen an optically thick boundarylayer forms, should keep in mind that, because this technique is limited to since only a fraction of this energy will be thermalised by the highinclinationsystems, some artefactsmay be present(Smak wholewhitedwarfsurface.Notethatirradiationbyahotwhite dwarf decreases as r−3; it is quite significant in the inner parts 1998).Irradiationoftheinnerdisc bythewhite dwarfcanalso besignificantduringquiescence,andhasnotbeenincludedhere. of the disc because the white dwarf extends very much above thediscsurface,buttheouterpartsofthediscarenotmuchaf- Figure5showsthatthediscluminosityattimeRduringthe fected by irradiation. This contrasts with the LMXB case, for rise phase is much larger than the luminosity at time D during which this effect does not exist. In LMXBs, irradiation is ob- decay;yet,theaccretionratesattheinnerdiscedgearecompa- servedtoberesponsibleformostoftheopticallightduringout- rable in both cases. The reason for this is that during rise, the bursts or in steady systems. In this case irradiationcan vary as accretionrateattheinneredgeofthediscincreasesonaviscous r−2,butwitharelativelysmallcoefficient(Dubusetal.1999,as- time, while the heating front propagates outwards on a much sume that σT4 = CL /4πr2 with C ∼ 5 × 10−3). As viscous shorterthermaltime.Ontheotherhand,duringthedecayphase, ill x Articlenumber,page4of7 J.-M.Hameury etal.:Hysteresesindwarfnovaoutburstsandlow-massX-raybinaries Fig.7.Hardnessmass-accretion-rateontothecompactobjectdiagram. Fig.6.HardnessintensitydiagramfromtheDIM.Severaloutburstscy- AsinFig.6,several cyclesareplottedonthisdiagram.Suchafigure clesarerepresentedhere,andeachpointisseparatedbyapproximately also represents the predictions of the DIM alone for low-mass X-ray 3hr.Theunirradiatedcaseisshowninblack,theirradiatedcaseinred. binaries. the cooling front propagates inwards at a slower speed than a Thismeansthattheexponentialcut-offoftheX-rayluminosity heatingfront(Menouetal.1999).Foragivendiscopticallumi- isprobablytoosharp. nosity,andhenceagivenpositionofthetransitionfront,theac- cretionrateshouldbesmallerduringrisethanduringthedecay. In order for this comparison to be meaningful, the spectral It is therefore not surprising that an hysteresis such as the one model has to be as realistic as possible. Modelling the contri- seenbyKördingetal.(2008)isfoundindwarfnovaeoutbursts. bution of the inner and outer disc and of the boundarylayer is Figure 6 shows the hardness vs intensity diagram expected relativelysimple,in contrastwith the LMXBcase (see below). foradwarfnovasuchasSSCygaspredictedbytheDIM.Here, Itshouldbestressedthatthehysteresiswouldappearinanycase thex-axisistheratiobetweentheX-rayandtheX-ray-plus-EUV whenconsideringtheopticalluminosityandanyquantitylinked (definedasλsmallerthan130Å)luminosities.Severaloutbursts to themass accretionrate at theinneredge ofthe disc, andthe arerepresentedonthisdiagram,anditisworthnotingthatboth conclusionthattheDIMisresponsiblefortheobservedhystere- the long and short outbursts follow the same track, despite the sisisinfactverysolidanddonotdependonthedetailedspectral fact that the accreted mass is different in each case (10% vs. modellingoftheinnerflow. 40% of the total disc mass is accreted during short and long Figure7showsthesamediagramasFig.6,butwehereplot outbursts respectively). Our simulations have shown that short the mass accretion rate onto the white dwarf instead of the op- outburstswouldproducedifferenttracksonsuchadiagramonly ticalluminosity.Ascanbeseen,thehysteresishasdisappeared iftheheatingfrontwerenotabletoreachtheouteredgeofthe and there is a univocalrelation between M˙ and the hardness disc and the peak luminosity was much smaller than for long acc ratio. This is merely due to the fact that the innermost part of outbursts – which is not the case for SS Cyg. We have found thesystemwhichemitstheX-raysandtheEUVhaveverysmall that this happens when e.g. the primary mass is large and the characteristictime-scalesandarethereforeinaquasisteadystate truncationradiusissmall.Theirradiatedandunirradiatedcases situation,withalocalmasstransferratedeterminedbytheouter areverysimilar;irradiationslightlyreducestheamplitudeofthe hysteresis,becausesomeopticalflux,proportionalto M˙ (and parts of the disc which have a much longer viscous time. The acc preciseshapeofthedependencebetween M˙ andthehardness hencedependingonlyontheratioL /(L +L ))isadded.But acc X X euv ratiodependsonthevariousassumptionsmadefortheboundary because irradiation is negligible in the outer parts of the disc layerand is thereforequiteuncertain;but the veryexistenceof whichcontributetomostoftheopticalflux,thiseffectissmall. thehysteresisinthehardness–opticalluminositydiagramisnot This figure can be directly compared with Fig. 1 from affectedbytheseassumptions. Kördingetal. (2008). It is interesting to note that the general characteristicsofourmodel-generateddiagramsarerathersim- For the sake of completeness, we also calculated the hard- ilar to the observational diagram. The maximum optical lumi- ness intensity diagram we obtain when we do not assume that nosityis,asexpected,closetotheobservedvalue.Thetransition the disc is truncated in quiescence, all other parameters being X-ray/EUVontheupperbranchoccursatabouttherightoptical identical.Thelight-curveswefindarerelativelydifferenttothe luminosity,whichmeansthatthecritical M˙ forthetransition truncatedcase:we obtainonlyonetypeofoutbursts,similar to acc betweenopticallythin/opticallythickX-rayemissionisnotvery theshortonesinFig.2,separatedby47days,andthepeakopti- differentfromthevalueusedhere(21016 g/s).Aslightlylarger calluminosityissmaller.Yetthehardnessintensitydiagramwe value,wouldhavegivenanevenbetteragreement.Notealsothat getin both casesare almostidentical.This shouldnotcomeas thetransitionisquitesharp.Theopticalluminosityonthelower a surprise, since this diagram is largely determined by the vis- branchisalsoquitecomparabletotheobservedone;note,how- coustimeintheouterpartsofthediscwhichisresponsiblefor ever, that the transition on the lower branch is smoother than thedelaybetweentheriseintheopticalandtheincreaseofthe that on the upper branch, but not quite as smooth as observed. accretionrateontothewhitedwarf. Articlenumber,page5of7 A&Aproofs:manuscriptno.hysteresis3 4. ThecaseofLMXBs In LMXBs, the DIM providesa naturalexplanationfor the oc- currenceof SXT outbursts,butthe DIM focusis essentially on theouterregionsofthedisc,wheretheeffectivetemperatureis of order of a few thousands degrees, and where the instability occurs. Moreover, standard geometrically thin, optically thick Shakura-Sunyaevaccretiondiscs are unable to producethe ob- served hard X-rays observed in LMXBs. The formation of an advection dominated flow when the mass transfer rate is low (ADAF, Esinetal. 1997)is an optionwhichgotstrongsupport whenrealizingthattheaccretiondischadtobetruncatedforthe DIM to be applicable to SXTs (Dubusetal. 2001). It remains, however, that there is no widely accepted model that can ac- countforthespectraltransitionsofLMXBs,andhenceforthfor providingtheoreticalspectraofaccretiondiscsforgivenphysi- calparameters(seee.g.Salvesenetal.2013;Nixon&Salvesen 2014,forrecentalternativemodels). Inaddition,modellingtheHIDforLMXBsrequires,incon- trast with the CV case, an accurate modellingof the spectra of each ofthe componentsof the flow.Thisis becausethe energy bandsusedtodeterminethehardnessratioareadjacentwhereas intheCVcase,theEUVandX-raybandsarewellseparatedand originatefromwelldefinedregions.Andindeed,spectralmodels ofaccretiondiscsincorporatingtheeffectsoflocaldissipationin the vertical structure of the disc differ significantly from black Fig. 8. Observed HID for SS Cyg when considering the EUV/X-ray bodies(Tao&Blaes2013). domain. Despitethesedifficulties,itis,however,simpletoshowthat the hysteresis in LMXBs is different from the one observed in CVs.Theviscoustimeintheregionsoftheaccretiondiscwhere InFig.8weshowtheHIDobtainedforSSCygwhenplot- X-rayareproducedis tingtheEUVluminosityversusthehardnessdefinedastheratio oftheEUVandX-rayplusEUVluminosities,whichshouldbe t = r2 = 1v2K = 1 c2rs (5) representative of the accretion rate onto the white dwarf. The visc ν α c2 2αc2 r data are taken from Wheatleyetal. (2003). Unfortunately the s s hysteresis hinges on 2 data-points but has a magnitude larger whereνistheviscosity,v istheKeplerviscosityandr is the K s than 0.3 dex. A similar picture can be seen if one plots the X- Schwartzschild radius. For X-rays to be emitted, the effective raysasafunctionofhardness.IfonecombinesboththeX-rays temperaturehasto be largerthan107 K, andhencethe viscous andtheEUVtogetsomethinglikeabolometricluminosity,one timeislessthan: obtainsa hardnessintensitydiagramnotdissimilar to thatseen 1r inXRBs.Aswedonotknowthevariablebolometriccorrection t <105 s s (6) visc α r factors,itisunclearhowtheseplotsrelatetoaproperhardness intensity diagram for the inner accretion flow. But the data is i.e.isalwayslessthanadayorafewdays.Onthetransienttime indicativethatsucha hysteresiscanexist. Ifitwereconfirmed, scale,theinnerpartofthediscwillthereforebeinaquasisteady- such an hysteresis would not be related to the DIM, as shown state situation,with a localmasstransferbeingimposedby the byFig.7,andwouldinfactbringnoconstrainontheinstability externalpartsofthediscwhichevolveonamuchlongerviscous responsible for dwarf nova outbursts; any mechanism account- time.Thisthenrequiresthatadditionalmechanismsareresponsi- ingfortheoutburstofdwarfnovaoutburst(suchase.g.themass blefortheexistenceoftwodifferentaccretionflowsforagiven transferinstabilitymodel,Bath1973)wouldproduceexactlythe X-ray luminosity. Many such mechanismshave been proposed sameHIDbecausetheinnerdiscisquasi-steady.Thishysteresis (see e.g. Begelman&Armitage 2014; Nixon&Salvesen 2014; would,however,bekeyinunderstandingthephysicsofaccretion Cao 2016, for recent examples); they are clearly disconnected intheimmediatevicinityofthewhitedwarf. fromtheDIM. 5. SSCyg:anhysteresisinthe L − L domain? 6. Conclusions X EUV The explanation by the DIM of the hysteresis observed by TheDIMnaturallyproducesanhysteresisinadiagramplotting Kördingetal. (2008) does not of course mean that in CVs an the opticalluminosity versusthe X-ray / EUV hardnessor any hysteresissimilartotheoneobservedinLMBXsdoesnotexist, quantity which is directly related to the accretion rate onto the especially that for SS Cyg the white-dwarf jet-formation crite- whitedwarf.Thishysteresisissimplyduetothefactthatittakes rion by Soker&Lasota (2004) is notsatisfied. But such a hys- aviscoustimeforthemasstransferratetovaryattheinnerdisc teresisshouldrelatequantitiescharacterizingthestateofthein- edgewhereastheopticalluminosityvariesonthemuchshorter ner disc. This is, however,difficult to observe, as one needs to thermaltime.ThisisverydifferentfromtheLMXBcase,where measure the accretion rate at the inner edge, while most of the thehysteresisrelatessoftandhardX-rayswhicharebothemit- relatedemissionisintheUV-EUVband,thusheavilyabsorbed, ted in the innermost parts of the disc, which, on the observa- andthebolometriccorrectionsareimportantanduncertain. tionaltime-scalescan then safely be assumed to be in a steady Articlenumber,page6of7 J.-M.Hameury etal.:Hysteresesindwarfnovaoutburstsandlow-massX-raybinaries statesituation,meaningthatforagivenlocalmassaccretionrate, Kurucz,R.1979,ApJS,40,1 twoquasi-stablesolutionscanexist.Inaddition,inCVsthehys- Lasota,J.-P.2001,NewARev.,45,449 teresis relatesemissionfromvariouspartsof the accretiondisc Maitra,D.,&Bailyn,C.D.,ApJ,608,444 MaucheC.W.,2004b,RMxAC,20,174 whereasincaseofLMXBsitmightrelateemissionfromthedisc MaucheC.W.,2004b,ApJ,610,422 andfromahotextendedflownotbeingpartofthedisc. Menou,K.,Hameury,J.M.,&Stehle,R.1999,MNRAS,305,79 The X-ray / EUV hardness-intensity diagram for SS Cyg, Miller-Jones,J.C.A.,Sivakoff,G.R.,Knigge,C.,Körding,E.G.,Templeton,M., which is similar to the classical HID for LMXBs shows clear Waagen,E.O.2013,Science,340,950 Nixon,C.,&Salvesen,G.2014,MNRAS,437,3994 indications that an hysteresis is present, which cannot be ac- Patterson,J.,&Raymond,J.C.1985,ApJ,292,550 countedforbytheDIM,butisindicativeofseveralcomponents Polidan,R.S.,Mauche,C.W.,&Wade,R.A.1990,ApJ,356,211 of the accretion flow being present for a given mass accretion Ritter,H.,&Kolb,U.2003,A&A,404,301 rateintheinnerdisc,andwhoseexistencedependonthehistory Salvesen,G.,Miller,J.M.,Reis,R.C.,&Begelman,M.C.2013,MNRAS,431, ofthesystem.Thishysteresisneeds,however,tobeconfirmed. 3510 Schreiber,M.R.,Hameury,J.M.,Lasota,J.P.2003,A&A,410,239 Itisdifficult,butnotimpossibletotestthemodelswithcur- Shakura,N.I.&Sunyaev,R.A.,A&A,24,337 rentinstrumentation.ThemajordifficultyisaccessingtheEUV Smak,J.1994,AcA,44,265 (CVs)and/orsoftX-rays(XRBs).Therearetwoproblemsthere: Smak,J.1998,AcA,48,677 (i)theyaresoeasilyabsorbedawaybytheISM;(ii)therearefew Soker,N.,&Lasota,J.-P.2004,A&A,422,1039 observatories/instruments capable of observing in this regime. Tanaka,Y.,&Shibazaki,N.1996,ARA&A,34,607 Tetarenko,B.E.,Sivakoff,G.R.,Heinke,C.O.,&Gladstone,J.C.2016,ApJS, Overcomingthe ISM issue – which especiallyaffects the EUV 222,15 – requires observing systems specifically selected to have low Tao,T.,&Blaes,O.2013,ApJ,770,55 NH columns.Such systemsare rare,butthey doexist. Perhaps Warner,B.1995,Cataclysmicvariablestars,Camb.Astrophys.Ser.,28 thebestexampleofsuchasystemisVWHyi,whichhasacol- Wheatley,P.J.,Mauche,C.W.,&Mattei,J.A.2003,MNRAS,345,49 umn of only N = 6 1017 cm−2 (Polidanetal. 1990). There Yan,Z.,&Yu,W.2015,ApJ,805,87 H arealsoothersthatmayinprinciplebeobservable(e.g.SSCyg wassuccessfullyobservedwithEUVE).Overcomingtheinstru- mental issue is harder. EUVE does not exist any more. Of the currentlyavailablespaceobservatories/instruments,theonlyone whichcouldbeusedforCVsinthisregardistheChandra/LETG combination. In fact, this has already been used to study WZ Sge(Mauche2004a)andSSCyg(Mauche2004b).GOALS(the Great Observatories Accretion Legacy Survey) is an observa- tionalcampaignbeingdeveloped,whichisaimedatfollowingan entiredwarfnovaoutburstfromrisetodeclineacrossallwave- lengths,coveringX-rays,EUV,FUV,NUV,optical,NIR,radio. AGOALS-PathfindercampaignwithChandra–aimedatestab- lishingwhethertwocandidatetargets(RXAndandYZCnc)are detectablewiththeLETG–isalreadyapprovedandunderway. Acknowledgements. This work was supported by a National Science Centre, Poland grant 2015/19/B/ST9/01099. JPL was supported by a grant from the FrenchSpaceAgencyCNES. 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