Table Of ContentAstronomy&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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