Disc instability models for X-ray transients: evidence for evaporation and low alpha-viscosity ?

14 Pages·1.1 MB·English

by  Kristen Menou

Mon.Not.R.Astron.Soc.000,000–000 (0000) Printed1February2008 (MNLATEXstylefilev1.4)
Disc instability models for X-ray transients: evidence for
α−
evaporation and low viscosity ?
⋆
Kristen Menou1,2,5 , Jean-Marie Hameury3,5, Jean-Pierre Lasota4,5,2
and Ramesh Narayan1,5
1 Harvard-Smithsonian Centerfor Astrophysics, 60 Garden Street, Cambridge, MA 02138, USA,
[email protected]
2 UMR 8629 du CNRS, D´epartement d’Astrophysique Relativiste et de Cosmologie, Observatoire de Paris, Sectionde Meudon,
0
F-92195 Meudon C´edex, France
0
3 UMR 7550 du CNRS, Observatoire de Strasbourg, 11 rue de l’Universit´e, F-67000 Strasbourg, France, [email protected]
0
4 Institut d’Astrophysique de Paris, 98bis Boulevard Arago, F-75014 Paris, France, [email protected]
2
5 Institute for Theoretical Physics, Universityof California, Santa Barbara, CA 93106-4030, USA

n
a
J
1February2008

2
1

ABSTRACT
1
Weconstructtime-dependentmodelsofaccretiondiscsaroundblackholesandneutron
v
stars.Weinvestigatetheeffectofevaporatingthediscinnerregionsduringquiescence
3
on the predictions of the Disc Instability Model (DIM) for these systems. We do not
0
include irradiation of the disc in the models.
2
1 Removing the inner, most unstable parts of the accretion disc increases the
0 predicted recurrence times. However, DIMs with values of the viscosity parameter
0 αhot ∼ 0.1 and αcold ∼ 0.02 (typical of applications of the DIM to standard dwarf
0 nova outbursts) fail to reproduce the long recurrence times of SXTs (unless we resort
/ to fine-tuning of the parameters) independent of the evaporation strength. We show
h
p that models with evaporation and a smaller value of αcold (∼ 0.005) do reproduce
- the long recurrence times and the accretion rates at the level of the Eddington rate
o observedinoutburst.Thelargedifferencebetweenthevaluesofαhot andαcold,ifcon-
r
firmed when disc irradiation is included, suggests that several viscosity mechanisms
t
s operate in these accretion discs.
a
For some choices of parameters our models predict reflares during the decline
:
v from outburst. They are a physical property of the model and result from a heating
Xi front forming in the wake of an initial cooling front and subsequent, multiple front
reflections. The reflares disappear in low–α models where front reflection can not
r
a occur.
Key words: X-ray: stars – binaries: close – accretion, accretion discs – black hole
physics– novae, cataclysmic variables – instabilities – turbulence – MHD
1 INTRODUCTION X-raynovae.Theiroutburstslastfromseveralweekstosev-
eral months during which X–ray luminosities can approach
Low Mass Black Hole Binaries (LMBHBs) and Low Mass
the Eddington limit. The outbursts are separated by long
NeutronStarsBinaries (LMNSBs) areclose binarysystems
periods of quiescence lasting from one to tens of years (see
in which a Roche-lobe filling, low mass (main sequence or
e.g. Chen,Shrader & Livio 1997; hereafter CSL).
sub-giant) secondary star transfers mass to a compact pri-
SXTsareinsomerespectssimilartodwarfnovae(DN),
mary. All known LMBHBs and several LMNSBs are tran-
whichexhibitoutburstsof4–6magnitudesinoptical.These
sients (see e.g. Tanaka & Lewin 1995, van Paradijs & Mc-
outburststypicallylastdaysandareseparatedbyperiodsof
Clintock 1995, White, Nagase & Parmar 1995); these tran-
quiescenceofafewweeks.DNareasubclassofCataclysmic
sientsystemsareknownassoftX-raytransients(SXTs),or
Variables(CVs;seeWarner1995 forareview),close binary
systems similar to LMBHBs and LMNSBs, except for the
⋆ compact primary which is a white dwarf.
Present address: Princeton University, De-
partment of Astrophysical Sciences, Princeton NJ 08544, USA, Isis widely believed that dwarf novaoutburstsare due
[email protected] toathermal-viscousinstabilityintheaccretiondiscofthese
(cid:13)c 0000RAS
2 K. Menou, J.-M. Hameury, J.-P. Lasota and R. Narayan
systems. For effective temperatures between about 5000 K Hur´e 1995; Hameury, Lasota & Hur´e 1997b). A marginally
and8000K,accretiondiscsinwhichviscosityisdescribedby stable,truncateddisc,asproposedbyWarner,Livio&Tout
the “α−prescription” (Shakura & Sunyaev 1973) are ther- (1996),isequivalenttoagloballystablediscfromthispoint
mally and viscously unstable because of large changes in ofview(Hameuryetal.1997b).Ifthetruncateddiscisglob-
the opacity when hydrogen recombines. The disc instabil- ally stable, and α ∼ 0.02, however, there is not enough
cold
itymodel(DIM;seeCannizzo1993b forareview;Hameury massinthedisctoaccountfortheoutburst’stotalenergy,so
et al. 1998 and Menou, Hameury & Stehle 1999a for the thatenhancedmasstransferisalsorequiredduringtheout-
most recent version of themodel) accounts for a numberof burst.Hameuryetal.(1997b)findthatifthisenhancedmass
general properties of dwarf novae. Similarities between the transferisduetoirradiationofthesecondary,themodelre-
properties of DNand SXTsled tothesuggestion thattheir producesthe shape of theobserved lightcurve.
outbursts have the same origin (van Paradijs & Verbunt The presence of “holes” in the accretion discs of quies-
1984; Cannizzo, Ghosh & Wheeler 1982; Huang & Wheeler cent SXTs seems unavoidable if their outbursts are due to
1989; Mineshige & Wheeler 1989). thethermal-viscous instability (Lasota 1996a,b). In theBH
Several important features of DN remain unexplained case, such holescan only result from theevaporation of the
by the standard version of the DIM, which assumes that accretiondiscsinceblackholesarenotmagnetized.Narayan,
the disc extends down to the white dwarf’s surface, that McClintock&Yi(1996)proposedamodelinwhich,asare-
the mass transfer rate from the secondary star is constant, sult of evaporation, the inner parts of the accretion flow in
and that effects of disc and secondary irradiation can be quiescentBHSXTsform an advection-dominatedaccretion
neglected (see e.g. Lasota & Hameury 1998; Lasota 1999; flow(ADAF;seealso Lasota,Narayan&Yi1996; Narayan,
Warner1998;Smak1999,Hameury.Lasota&Warner1999b Barret & McClintock 1997). The model explains the ob-
for a discussion of these effects). This version cannot ac- served spectra and solves the problem of incompatibility
countforthequiescentaccretionratesontothewhitedwarf between the accretion rates predicted by the DIM and the
(they exceed the model predictions by more than two or- observed flux.
ders of magnitude), and for the very long recurrence times Esin, McClintock & Narayan (1997; see also Narayan
ofWZSge–typesystems(Lasota1996a,b).Theobservedde- 1996; Esin et al. 1998) extendedtheideaof two-component
lay between therise in optical and in EUV (e.g. in SS Cyg; accretionflowsastepfurther.Theyshowedthatthevarious
Mauche 1996) was also long considered as a problem, but spectral states of black hole X-ray binaries can be modeled
Smak(1998) recentlyarguedthatitcanbenaturallyrepro- asasequenceofphysicalstateswithvaryingaccretionrates
ducedbytheDIMifappropriateouterboundaryconditions and sizes of the two components of the accretion flow. Ac-
for the disc are used. cording to this model, in outburst, the disc extends down
Similar difficulties, but more pronounced, appear for to the black hole, while the inner disc is gradually replaced
SXTs, since the standard DIM fails to reproduce a num- by an ADAF as a system goes to lower luminosity levels.
berofessentialpropertiesofthesesystems.Inparticular,an A gradual evaporation of the inner disc during the decline
earlyattempttoexplainthepropertiesofBHSXToutbursts is also qualitatively consistent with the accretion geometry
with the DIM (Mineshige & Wheeler 1989) was unable to inferred by Z˙ycki, Done & Smith (1999) from theX-ray re-
reproduce the observed long recurrence times (comparable processing properties of several BH X-raytransients.
tothoseofWZSgesystems)andthehighaccretionratesre- Hameuryetal. (1997a) usedthetwocomponent (inner
quiredinoutburst.Inaddition,theweakX-rayfluxdetected ADAF + outer disc) model to describe the quiescent state
fromquiescentBHSXTs,whichwasmistakenlythoughtby and the rise to outburst of the BH SXT GRO J1655-40.
some to be an argument in favour of the DIM versus the The model predictions agree with existing constraints on
competing mass transfer instability model, is many orders the spectrum of the system in quiescence, the observed 6
of magnitude larger than any flux predicted by the DIM daydelaybetweentherisetooutburstoftheopticalandX-
(Lasota 1996a). ray fluxes, and the rise-times of these two fluxes. Recently,
In the DN case, theproblems can been solved, at least Esin,Lasota&Hynes(1999)showedthatthewhole,unusual
in part, if a “hole” is present in the inner regions of the outburstcanbedescribedintheframeworkoftheDIMthus
disc.The“hole” can eitherbecaused byevaporation of the validating Hameury et al. (1997a) results.
disc (Meyer & Meyer-Hofmeister 1994), or due to the ex- In this paper, we consider the role of disc evaporation
tended magnetospheric cavity of the WD (Livio & Pringle on the predictions of the DIM for SXT outbursts. We shall
1992; Lasota, Kuulkers& Charles 1999). A model in which refer to a DIM that includes evaporation as a Truncated–
a “hole” is included can account for both the EUV delay Disc Instability Model (hereafter TDIM). Cannizzo (1998)
andthequiescentX-rayluminosity(iftheholeisfilledwith alreadyconstructedTDIMsoftheBHSXTA0620-00witha
a tenuous, X–ray emitting gas; Meyer & Meyer-Hofmeister weak evaporation and the prescription α=50(H/R)3/2 for
1994; Hameury,Lasota & Dubus1999a). the viscosity parameter. Cannizzo’s simulations show that
The case of WZ Sge requires additional modifications models without evaporation lead to unreasonably short re-
of thestandard DIM. Thevery long recurrencetime of this currence times, and that evaporation acts to increase the
system, and the large amount of mass accreted during the predictedrecurrencetimestovaluesofseveraltensofyears,
outburst, can be reproduced either by using a value of the asrequired.Thisisexpected,sinceevaporationremovesthe
viscosity parameter α several orders of magnitude lower inner, most unstable parts of the accretion disc. However,
cold
thaninotherDN(Smak1993;Osaki1995),orbymakingthe the outburst rise time predicted by Cannizzo’s model is an
“hole” sufficiently large for the remaining disc to be glob- orderofmagnitudelongerthantheobservedone.Thisisdue
ally stable. In the latter case, the outbursts would have to to the use of the prescription α=50(H/R)3/2 according to
betriggeredbyenhancedmasstransfer(Lasota,Hameury& Lasota & Hameury (1998).
(cid:13)c 0000RAS,MNRAS000,000–000
Disc instability models for X-ray transients: evidence for evaporation and low α−viscosity ? 3
There are two additional reasons to go beyond Can- theobservedvariabilityis‘ready-to-interpret’(freeofnoise,
nizzo’s calculations. First, his calculations do not allow the systematic errors, etc..).
disc outer radius to vary with time. Smak (1984, 1998; see Most of the properties of BH and NS SXTs mentioned
alsoHameuryetal.1998)showedthatincludingthiseffectis in this section are taken from the compilation of CSL. We
crucialfor reliable DIMpredictionsofDNoutbursts,andit identify several important timescales for X-ray transients:
islikelytobeimportantforSXTmodelsaswell.Second,in the recurrence time trec between two successive outbursts,
themodelsof Cannizzo(1998), theeffect ofincludingevap- the e-folding rise timescale τrise and the total rise time trise
oration and using a very small value of α in the quiescent of an outburst (which can be wavelength-dependent), the
discarenotclearlyseparated.Inparticular,inhiswork,itis e-foldingdecaytimescaleτ (whichiswavelengthdepen-
decay
unclear if evaporation alone can be held responsible for the dent), and the total duration of the outburst t . In our
dur
longrecurrencetimes,ifvaluesofαsimilartothoseinferred view, a model which fails to reproduce all these character-
from applications of theDIMto DN (typically α ∼10−1 istic times is not fully satisfactory. None of the previous
hot
andα ∼0.02)areusedforthediscsofSXTs.Inourview, models of SXTs passed this test (e.g. Mineshige & Wheeler
cold
this is an important question because the evidence for disc 1989; Cannizzo 1998).
evaporation, even indirect, is stronger than for any specific We focus our discussion on a sample of SXTs with
α−prescription. clearly identified primaries (see Garcia et al. 1998): theBH
In § 2, we recall some important observational proper- systemshavefirmdynamicallowerlimitsonthemassofthe
tiesofBHandNSSXTs,laterusedtoconstrainourmodels. primary,and theNS SXTsshowed typeI X-ray bursts. We
In § 3, we first describe the numerical truncated disc insta- also focus on systems which have exhibited standard Fast
bility model used. We then show that, in the limit where RiseExponential Decay (FRED) lightcurves. This excludes
illumination canbeneglected,TDIMswithvaluesofαcon- the long period systems GRO J1655-40 and V 404 Cyg
sidered as standard for DN discs are unable to reproduce and reduces the sample to four BH SXTs (A0620-00, GS
the long recurrence times of SXTs, no matter how strong 2000+25,GRS1124-683=NovaMus91andGROJ0422+32
theevaporationis.In§4,wepresentotherplausiblemodels – “best examples of FREDs” according to CSL) and two
of SXTs. We find that a TDIM with a small value of α NS SXTs (Cen X-4 and Aql X-1; here we chose only their
cold
(∼ 5×10−3) and strong evaporation does reproduce the outbursts of the FRED type : it is not clear how the stan-
long recurrence times and the high maximum values of the dard DIM could account for the other types of lightcurves
accretion ratein thediscsof SXTs.Wealso presentmodels observed in SXTs).
in which the disc of SXTs are globally stable and the out-
burstsaretriggered byaslowvariationofthemasstransfer
2.1.1 Recurrence times
rate of the secondary. In § 5, we discuss important conse-
quences,limitationsandpossibleextensionsofthiswork.In Aql X-1 experiences an outburst every year approximately,
§ 6, we summarize ourmain results. while therecurrence time of Cen X-4 is ∼10 years. A0620-
00 experienced two outbursts separated by 58 years (note
that noX-raydataareavailable for thepresumed first out-
burst).MostBHSXTsdidnotexperienceasecondoutburst
2 MAIN OBSERVATIONAL sincetheirdiscoverybyX-raysatellites,sothatthereisgood
CHARACTERISTICS OF SXTS evidencefor therecurrencetimes of BHSXTsgenerally ex-
ceeding20−30years.Asmentionedabove,SXTrecurrence
BH and NS X-ray Transients show a rather complex be-
timesaresimilar tothoseofWZSge-typeDN.Forcompar-
haviour(see,e.g.,Tanaka&Shibazaki1996andCSLforre-
views).Inthissection,weidentifyseveralimportantproper- ison, the recurrence time of SS Cyg is trec ∼ 50 days (e.g.
Cannizzo & Mattei 1992) and the recurrence time of other
tiesofthesesystemswhichmaybeconsideredascharacter-
UGemtypeDNisgenerallyoftheorderofonemonth(e.g.
isticoftheirclassandthereforemustbeexplainedbyasuc-
Szkody& Mattei 1984).
cessfulmodelofSXToutbursts.Wewillalsooften compare
SXTs with DN, since the same underlying thermal-viscous
instabilityissupposedtoberesponsiblefortheoutburstsin 2.1.2 Rise times
thetwo classes of systems.
As a guide, we use the observed properties of SS Cyg, ThecharacteristicX-rayrisetimescales(τrise)ofFRED-type
SXTsvary from 0.3 to2 days(for theperiod of fastest flux
Aql X-1 and A0620-00 as indicative of the general proper-
ties of U Gem-type DN, NS SXTs and BH SXTs respec- increase–CSL).Thetotalrisetimetrise from quiescenceto
outburstpeakisdifficulttoestimatebecauseitisinstrument
tively,keepinginmindthatsignificantdifferencesexisteven
dependent,and because the coverage of the rising phase is,
between two outbursts in a same system (e.g. at least four
typesof outburstsin SS Cyg; Warner1995). for obvious reasons, rather poor. For SXTs, trise is ∼ 5 to
10 days in X–rays, or perhaps longer. For comparison, the
totalopticalrisetimeinSSCygistrise∼3days(fora‘type
B’ outburst - see below). The optical rise time of several U
2.1 Timing properties
Gem-type DN in the sample of Szkody & Mattei (1984) is
The timescales of evolution of an unsteady disc are simple trise∼2 days.
and robust predictions of the DIM, which provide powerful In several DN, there is evidence for at least two types
testsofthemodelwhencomparedtotheobservations.This ofoutbursts.‘TypeA’outburstscorrespond toFREDsand
is because the time evolution of the disc is due to the exis- are usually interpreted as ‘outside-in’ outbursts, i.e. out-
tence of a well defined limit cycle in the DIM, and because burststriggeredintheouterregionsofthedisc(Smak1984).
(cid:13)c 0000RAS,MNRAS000,000–000
4 K. Menou, J.-M. Hameury, J.-P. Lasota and R. Narayan
Thisinterpretationseems,inafewcases,consistentwiththe M˙ =1.39×1018 M1 g s−1, (1)
observed delay between the rise to outburst of optical and Edd (cid:18)M⊙(cid:19)
EUV(orUV)fluxes(Warner1995).‘TypeB’outburstshave
therateatwhichanaccretionflowaroundacentralobjectof
a more symmetric shape with a slower rise phase, and are
usuallyinterpretedas‘inside-out’outbursts(Smak1984;see mass M1 reaches the Eddington luminosity for a standard
10% radiative efficiency. The Eddington luminosity, how-
also Menou et al. 1999a).
ever,seems tobethemaximumluminosity for SXTs.Com-
In the case of SXTs, thereis no observational evidence
bined with outburst durations this implies that the mass
for outside-in outbursts. The delay between the rise of the
accreted during outbursts by the central body is ∼ a few
opticalandX-rayfluxes(observedonlyduringanon-FRED
1024 g (roughly the same as accreted onto the white dwarf
outburstoftheBHSXTGROJ1655-40)isnotanindication
duringWZSgeeruptions).Thisconstraintmustbesatisfied
of an outside-in outburst if the disc is truncated. Indeed,
bySXT models as well.
Hameuryetal.(1997a) obtaincorrectrisetimesandtheX-
Observations of broadened H emission lines provide
ray/opticaldelayforaninside-outoutburst.Inthiscasethe α
‘inside’ corresponds totheinneredgeof thetruncated disc, an upper limit on the value of the disc inner radius Rin
in quiescence for several BH SXTs (e.g. Orosz et al. 1994;
far from thecentral object.
Narayan et al. 1996; Menou, Narayan & Lasota 1999b). In
most systems, the constraints indicate a value of Rin less
2.1.3 Decay times than104toafew104RS,whereRs =2.95×105(M1/M⊙)cm
istheSchwarzschildradiusoftheblackholeprimaryofmass
TheX-ray decay timescales τdec of SXTswith FREDlight- M1. The limits provide a useful constraint on the strength
curves are typically 25 - 40 days (CSL). The optical decay of evaporation to be included in the TDIMs of SXTs (see
timescales of BH SXTs are ∼ 50 to 200 days, while the below).
opticaldecaytimescalesofNSSXTsare∼10to30days(but Forsomesystems,otherconstraintsonRin existatvar-
theavailable information is poor). Wenote that long decay ious stages of the outburst from the modeling of X-ray re-
timescalesarealsoobservedinWZSge–typeDN(Kuulkers, flection properties (Z˙ycki et al. 1998), as well as spectral
Howell & van Paradijs 1996; Kuulkers1998). fits which involve Rin as a parameter (e.g. Sobczak et al.
1999). However, these two methods are model dependent
and somewhat controversial, because the relevant spectral
featurescanbemodeleddifferently(M.Nowak,privatecom-
2.1.4 Duration times
munication). Wechoose to ignore them here.
The reported duration times for the FRED–type outbursts
of BH SXTs range from 170 to 260 days in X-rays (and
even longer in optical), while t ∼ 70 days in X-rays for
dur 3 “STANDARD” TRUNCATED DISC
FRED–type outbursts of NS SXTs. The typical duration
INSTABILITY MODELS OF SXTS
time of UGem–type DNin thesample of Szkody &Mattei
(1984) is t ∼10 days(optical). In this section, we investigate a “standard” version of the
dur
One has to remember, however, that the reported du- TDIM for SXTs. As mentioned above, by “standard”, we
rations for SXTs depend on the detection threshold of the meanamodelwithvaluesoftheviscosityparameter αthat
instrument and that the coverage of some outbursts is in- are usually considered as standard for DN discs: αhot ∼0.1
complete; sometimes the reported duration is only a lower andαcold ∼0.02.Non-standardmodels,inparticularmodels
limitonthetrueduration.Insomecases,thedecaytimescale using significantly smaller values of αcold, are considered in
τ maybemoreappropriatethanthedurationtimet for §4.
dec dec
comparison of the model predictions with theobservations.
3.1 Numerical specifications
2.2 Other important observational constraints 3.1.1 General
Anestimateofthemaximumluminosityreachedinoutburst We use the numerical code described in Hameury et al.
byaSXTdependsonthedistancetothesystemand,when (1998) to simulate the evolution of discs around NSs and
compared to the Eddington luminosity, on the mass of the BHs. The equations of conservation of mass, angular mo-
primary. According to Garcia et al. (1998), the NS SXTs mentum and energy for a thin Keplerian disc are solved on
considered here approximately reached the Eddington lu- anadaptivegridwhichresolvesnarrowstructuresinthedisc
minosity at outburst peak, as did some of the BH SXTs. (transition fronts in particular). A fully implicit numerical
According to CSL, the maximum luminosities reached are schemeisusedfortimeevolution,toavoidanyCourant-type
closer to 10% of the Eddington luminosity, but their val- limiting condition on the timestep of integration.
ues could be underestimated. For example, CSL obtain for A grid of disc vertical structures, which give the local
NovaMuscae 1991 apeak luminosity (0.4 –10 keV)of 0.08 cooling rate of the disc as a function of its surface density,
inEddingtonunits,whereasaccordingtoEsinetal.(1997), centraltemperatureandtheverticalgravityisprecalculated
whousedtheentireavailablespectralinformation,thepeak before running the disc evolution. The effect of changing
luminosity in thissystem was almost exactly Eddington.In the mass of the central object (from a NS SXT to a BH
our view, this means that models of SXTs have to be able SXT) enters the time-dependent calculations both through
to reach the Eddington accretion rate. The Eddington ac- the radial conservation equations solved, and through the
cretion rate is defined here as grid of vertical structures with different gravities.
(cid:13)c 0000RAS,MNRAS000,000–000
Disc instability models for X-ray transients: evidence for evaporation and low α−viscosity ? 5
Theouterradiusofthediscisallowedtovarywithtime et al. 1999b), whereas typical SXT mass–transfer rates are
undertheactionofthetorqueappliedbythesecondarystar much lower: ∼ 1015−1016 g s−1 (van Paradijs 1996; CSL;
on the outermost regions of the disc. Including this proper Menou et al. 1999b). In addition, for mass transfer rates
boundary condition (as opposed to a rigid boundary condi- ≤ 1017 g s−1, the stable discs would be truncated at radii
tion with a fixed value of the outer radius) has been found ≥ 3×104 R , which is inconsistent with the upper limits
s
todrasticallyaffectthepredictionsoftheDIMforDNdiscs derived from H emission lines (§ 2.2).
α
(seeHameuryetal.1998fordetails).Theeffectofincluding Tocircumventthesedifficulties,andensurethatdiscsin
thisproperboundaryconditionfordiscsofSXTsisdiscussed ourmodelsexperienceoutburstsevenatlowermasstransfer
in §3.2.1. rates, M˙evap(R) must be less than M˙cAriDtAF(R). We use the
following evaporation law:
3.1.2 Formation of a hole in the disc 0.08M˙
The formation of a hole in the discs of SXTs is proba- M˙evap(R)= R 1/4+EEddR 2. (3)
Rs 800Rs
bly determined by the physics of the evaporation of gas (cid:0) (cid:1) (cid:0) (cid:1)
where E is a constant ensuring that a maximally truncated
from the disc, possibly into an ADAF. This physics is
discisstillunstable.ThepowerlawsandscalingsinEq.(3)
poorlyunderstood.Severalmechanismshavebeenproposed are such that the profile of M˙evap(R) is initially slowly de-
intheliteraturetoexplainthetransition(see,e.g.,Meyer&
creasingwithradius,andthendecreasesmuchmoresteeply
Meyer-Hofmeister1994;Liu,Meyer,Meyer-Hofmeister1997; beyond a radius ∼ 800R ×E−1/2. In all the models with
Narayan & Yi 1995; Honma 1996; Shaviv,Wickramasinghe s
evaporationdescribedbelow,wetakeE =20(exceptinsec-
& Wehrse 1999), but none of these models has reached a
tion 4.1, in which E = 30). This corresponds to a situation
sufficient level of detail to allow robust quantitativepredic-
wheretheevaporationrateduringquiescenceissignificantly
tions.
smallerthanthemasstransferratefromthesecondary,thus
Disc evaporation has already been included in the cal-
ensuring that thedisc is not marginally unstable.
culations of Hameury et al. (1997a) and Hameury et al. This expression for the evaporation law is ad hoc but
(1999a). In these studies, a ‘reasonable’ formula was used
possesses several advantages. First, it is a continuously de-
(close to the one proposed by Meyer & Meyer-Hofmeister
creasing function of radius, which seems quite natural, and
1994) and evaporation was included in the numerical mod-
is appropriate for the numerical implementation of evap-
els as a sink of matter in the mass conservation equation
oration described above. Second, the maximum value of
(seeHalesroe,thweemchoodseelsawsliitghhtelvyadpioffrearteionntaopfpCroaancnhiz.zWoe19a9ss8u)m. e M˙evap(R), as defined in Eq. (3) when R ∼ RS, is consis-
tent with the high value required in the ADAF models of
an ad hoc law for the rate of evaporation as a function of
Esin et al. (1997) to fit the high luminosity states of the
radius,asinHameuryetal.(1997a; 1999a), butinaddition
BHSXTNovaMuscae91.Third,itleadstodisctruncation
weassumethatevaporationactsonlyinaverynarrowregion
radii in quiescence which are smaller than the upper lim-
at the inner edge of the disc. This is a good approximation
its inferred from the broadening of H emission lines. The
if the rate of evaporation of the disc material is a steep α
resulting discs are unsteady and experience limit cycles for
function of radius. reasonable mass transfer rates (∼ 1015 g s−1), as required
To be more specific, the inner radius of the disc, Rin, for theDIM to bethe explanation of SXToutbursts.
varies according to:
Naturally, one could worry that the predictions of the
M˙disc(Rin)=M˙evap(Rin), (2) time-dependentmodelsdependontheevaporationlawused.
This is true about the strength of evaporation, but we find
whereM˙disc(Rin)isthetime-dependentaccretionrateatthe that our conclusions do not crucially depend on the pre-
inneredgeRin of thedisc,and M˙evap(R) istheevaporation cise shape of the evaporation law. We discuss the effects of
lawwithanarbitraryfunctionalform.Thisevaporationlaw
changing the evaporation law on the model predictions in
is chosen so that, in outburst, M˙evap(R) is much less than §5.
the accretion rate in the disc at any radius, and the disc
Notethatinthemodelspresentedhere,thepresenceof
is fully extended. The radius of truncation of the disc in
atenuousregion in theinneraccretion flow, presumablyan
quiescencedependsonthefunctionalform ofM˙evap(R)and ADAF,isonlytaken intoaccount throughdisc evaporation
the profile of M˙ in the disc. We find that this numerical
disc (i.e. as a sink of mass): we neglect any other physical effect
recipeismuchmorestablethanthatusedinHameuryetal.
thatthisinnerregionoftheaccretionflowcouldhaveonthe
(1997a), while it gives very similar results for the evolution
disc evolution. Disc irradiation by the central X-ray source
of Rin and theoutburst cycles in general. duringhighluminositystatesisalsoneglected,althoughitis
Following Hameury et al. (1997a), we could have
apotentially important effect (see §5for a discussion). For
assumed that the disc is evaporated into an ADAF
simplicity,we usethesame evaporation law for BH and NS
which accretes everywhere at the maximum rate allowed, SXTs,except that M˙ andR are scaled with theactual
MM˙˙cAAriDDtAAFF((RR)).,Ranedcentatlkye,hfoorwtehveere,vtahpeoersattiimonatleaswofMM˙˙evAaDp(ARF)(R=) primary mass M1. Edd S
crit crit
havebeen revisedtohighervalues(Esin et al. 1997; Menou
et al. 1999b). Using these recent estimates for the evapora-
3.2 Results
tion law (i.e. a stronger evaporation than, e.g., in Hameury
et al. 1997a) would imply that truncated discs are glob- We present in this section two SXT models, one with pa-
allystable(andstationary)forallmasstransferratesbelow rameters relevant for theBH SXTA0620-00, and theother
∼1017 g s−1 (this is demonstrated by the Fig. 5 of Menou for the NS SXT Aql X-1. In both models, α = 0.1,
hot
(cid:13)c 0000RAS,MNRAS000,000–000
6 K. Menou, J.-M. Hameury, J.-P. Lasota and R. Narayan
Figure 1.EvolutionofthediscmassMdisc,theVbandmagni- Figure 2. Details on the evolution of the V band magnitude of
tudeofthedisc,theaccretionrateatthediscinneredgeM˙in,the the disc, the accretion rate at the disc inner edge M˙in and the
disc outer radius Rout and the disc inner radius Rin in a model discinnerradius Rin duringan outburst of the model presented
withparametersrelevantfortheBHSXTA0620-00(seetextfor inFig.1.Notethemultiplereflaresduringthedecline,asseenin
details).Therecurrencetimeistrec≃5yearsinthismodelwith themodulationsoftheVbandmagnitude, M˙in andRin.
αcold=0.02.
α = 0.02, the disc inner radius is limited during the does not reach the outer disc radius. In theDIM, theinner
cold
evolutiontotheminimumvalueRmin =5×108 cm(see§5 regions of quiescent accretion discs are more subject to in-
for a discussion of this numerical limitation) and Eq. (3), stabilitiesthantheouterones,sothatthetruncationofthe
scaled with M1, is used for the evaporation law. innerdischasastabilizing effect.Thisexplainstheabsence
The model of A0620-00 has the following parameters: of ‘small’ outbursts and the reduced number of reflares in
primary mass M1 = 6M⊙, time-averaged value of the disc models that includeevaporation (see below in § 3.2.6).
outer radius < Rout >= 1.24×1011 cm and mass trans- The panel representing the total mass in Fig. 1 shows
fer rate M˙ = 3×1015 g s−1. The model of Aql X-1 has thatthediscevolutionisperiodicandthereforerelaxed(i.e.
T
thefollowingparameters:primarymassM1 =1.4M⊙,time- does not depend on the arbitrary properties of the initial
averagedvalueofthediscouterradius<Rout >=1.8×1011 discusedatthebeginningofthesimulation).Thisistrueof
cm and mass transfer rate M˙ =2×1016 g s−1. The main every model discussed in this paper. Strictly periodic out-
T
propertiesofthesemodelsaresummarizedinFigures1to5, burstcyclesareproducedinSXTmodels,likeforDN,while
and are discussed in more detail below. observed cycles show, in general, only some regularity.
For comparison, we also show in Fig. 3 and 4 results Inthemodelspresentedhere(aswellasinalltheother
from a model with parameters relevant for A0620-00, but sections),theoutburstsaretriggered intheinnerregionsof
withoutevaporation.Resultsaresimilar,exceptforasignif- the truncated disc (inside-out type). Such outbursts never
icantly shorter recurrence time, a larger numberof reflares, start exactly at the disc inner edge, so that there is always
andtheexistenceoftwotypeofoutbursts.Successivesmall aheatingfronttravelingsomedistanceinwards.Itseffect is
and large outbursts are also predicted in models of dwarf seen as a ‘spike’ in the light-curve in Fig. 4, which lasts for
novae (see e.g. Cannizzo 1993a; Menou et al. 1999a). Dur- less than one minute and is therefore not observable. In a
ing small outbursts, the outward propagating heating front truncated disc, these short-lived fronts are absent, because
(cid:13)c 0000RAS,MNRAS000,000–000
Disc instability models for X-ray transients: evidence for evaporation and low α−viscosity ? 7
Figure 3.Evolutionof thetotal discmassMdisc,visualmagni-
tudeofthediscandmassaccretionrateontothecompactobject
M˙in in a model with the same parameters as the one shown in
Fig.1,butwithoutevaporation.Theinnerdiscradiusisfixedat
5×108cm.Heresuccessivesmallandlargeoutburstsarepresent.
Figure 5. Evolution of the total disc mass Mdisc, the V band
magnitude of the disc, the accretion rate at the disc inner edge
M˙in and the disc inner radius Rin for two outbursts in a model
withparametersrelevantfortheNSSXTAqlX-1.Therecurrence
timeistrec≃1yearinthismodelwithαcold=0.02.Herereflares
areabsent.
theinnermostregionswheretheywouldpropagatenolonger
exist.Heatingfrontsonlymarginallyreachtheouteredgeof
thediscinthemodelwithparametersrelevantforA0620-00.
InthemodelwithparametersrelevantforAqlX-1,thedisc
is so extended that the heating fronts never go beyond one
third of the disc outer radius. This would of course not be
thecaseforshorterperiodsystemssuchasGROJ0422+32.
In the following discussion of the rise times and decay
timespredictedbythemodels,wewillassumeforsimplicity
that the X-ray emission from the system is proportional to
the accretion rate M˙in at the inner edge of the disc. The
real situation is likely to be more complicated, in particu-
larifsomeof theX-raysobservedareemittedbyanADAF
whose luminosity is not simply proportional to M˙ (in that
case,thetimescalescouldbeevenshorterbecausetheradia-
tiveefficiencyincreaseswithM˙).Similarly,theVmagnitude
shown are not supposed to directly represent the V magni-
tudeobservablefromasystembutonlytheintrinsicoptical
luminosity variations of the disc. In observed systems, the
Vbandmagnitudeemission hasalsocontributionsfrom the
secondary, from reprocessing of X-ray photons by the disc
and possibly from an ADAF. We do not attempt to de-
Figure 4.Detailsofoneoftheoutbursts ofFig.3 scribe all these processes here, and we concentrate on the
basic properties of themodels.
(cid:13)c 0000RAS,MNRAS000,000–000
8 K. Menou, J.-M. Hameury, J.-P. Lasota and R. Narayan
3.2.1 Recurrence times data (GRO J1655-40 being the only system in which the
rise in optical was observed; see Orosz et al. 1997). This
AsforDNdiscs,itisimportant toincludeacorrect bound-
timescale is approximately the thermal time at the disc in-
ary condition at the outer edge of the disc of SXTs. We
neredge,since thisis wheremost oftheopticallight comes
findthattherecurrencetimespredictedbymodelsinwhich
fromatthebeginningofanoutburst.Itisnoteasilycompa-
Rout is allowed to vary with time are typically a factor of a rable to the observations, because of light dilution by both
few (say 2-3) times shorter than those predicted if R is
out thesecondary and theADAF.
keptfixed.Indeed,when aheatingfront reachesRout,mass In the model of Aql X-1 shown in Fig. 5, it takes the
is spread over a more extended region if Rout is allowed to disc∼3daystoreachthecompactobject onceanoutburst
vary. Consequently the critical surface density Σmin below is triggered. The fact that this time is shorter than in the
which a cooling front appears in the disc is reached by the
modelofA0620-00canbepartlyattributedtoalessefficient
disc earlier, less mass is accreted during the outburst, and evaporationinthismodel(becauseM˙ andR ,inEq.[3],
overall it takes less time for the disc to reach the state of Edd S
critical surface density Σmax at which the next outburst is dscaaylsefwoirththteheacmcraestsioMn1raotfetahtetphreimdaisrcy)i.nInteralseodgteakteosr∼ea1ch2
triggered(Hameuryetal.1998).Therefore,allexistingmod-
its maximum value. Overall, the total rise time is therefore
oelvseorefsBtiHmaStXeTdsthweitrhecfiuxrerdenvcaeluteimsoefsRtroeuctbhyavaefsaycsttoermoaftiacafellwy ttrhiseea≃va9iladbalyesd,awthaicohnaFgRaiEnDis-tyinperelaigshont-acbulreveasg.reement with
when the heating front was able to reach the disc’s outer
edge. In the present paper, we only consider models with The total rise time in optical is not so short: trise ≃ 9
days. This is because the disc is less strongly truncated
varyingRout. thanin themodelof A0620-00, which resultsin muchmore
TherecurrencetimeinthemodelofA0620-00istrec ∼5 “inside-out/symmetric” outburst shapes (e.g. Smak 1984).
years, i.e. shorter than the known recurrence time by more
In addition, because the disc is large, the inner parts do
than one order of magnitude (§2.1.1). The recurrence time
notcontributeasmuchasinthecaseofA0620-00. Therise
minetnhtewmitohdetlheofkAnoqwlXr-e1cuisrrternecce∼ti1myee.ar,inverygood agree- timescale for the period of fastest flux increase is τrise ≃ 1
days in X-rays, which is consistent with available observa-
tional data. It is τrise≃1.7 hours in optical.
3.2.2 quiescence
During quiescence, all of the accreted mass is processed
3.2.4 Decay and duration times
through the ADAF and is therefore evaporated, so that
the mass accretion rate shown in figs. 1 and 5 is the ac- InthemodelofAqlX-1,thetotaldurationtimeofoutbursts
tual evaporation rate except when the disc radius reaches ist ≃25 days.Thisis short compared totheduration ∼
dur
its minimum value during outbursts. As can be seen, the
70daysreportedintheliteratureforFRED-typeoutbursts.
evaporationrateremainssignificantlysmallerthanthemass Again,wenotethattheoutburstsproducedinthismodelare
transfer rate from the secondary, varying between 0.15 and not really of the FRED-type, but have a rather symmetric
0.60 M˙T in the case of the A0620-00 model, and between “triangle” shape.
0.10 and 0.15 M˙T in the Aql X-1 case. These are close to The decay timescale early in the decline is τ ≃ 3.5
dec
themaximumevaporationrateforwhichthediscisunstable days in X-raysand τ ≃4 days in optical. The timescales
dec
but they donot imply finetuning.
become smaller later on because the light-curves steepen.
Thesevaluesareclearly shorterthantheobservedvaluesof
25-40 daysin X-raysand 10-30 days in optical.
3.2.3 Rise times
InthemodelofA0620-00,thetotaldurationtimeofan
In the model of A0620-00, once an outburst is triggered, it outburst is tdur ≃55 days. This is quite short compared to
takesthedisc∼9daystoreachthecompactobject(thetime the typical duration times for BH SXTs with FRED-type
required to go from 5×108 cm to ∼ 107 cm, not included light-curves, ∼ 170−260 days in X-rays and even more in
in our simulations, is small because of the short timescales optical. Note that tdur would be shorter if reflares did not
in the inner disc, so that we neglect it). It also takes ∼ 14 occur (see below for a discussion of reflares).
days for the accretion rate at the disc inner edge to reach Estimating the decay timescales is complicated by the
its maximum value.Overall, thetotal rise time in X-raysis presence of reflares in this model. For completeness, we es-
therefore trise ∼ 5 days, which is in reasonable agreement timate two types of τdec. The overall τdec corresponds to
with the available information on FRED-type light-curves. the decay timescale for the global light-curve, ignoring the
The total rise time in optical is much shorter because the “fluctuations” induced by the reflares. It is τ ≃ 20 days
dec
outbursts are triggered in a truncated disc, i.e. effectively in X-rays and τdec ≃ 22 days in optical. The actual decay
in the outer regions of the accretion flow. Our results are timescales, in between two successive reflares, are shorter:
similar to those of Hameury et al. (1997a) who addressed τ ≃ 9 days in X-rays and τ ≃ 11 days in optical. We
dec dec
in more details the issue of time delays between the rise in find it difficult to compare these timescales with observed
optical and X-ray. values,sinceobservedlight-curvesdonotresemblethelight-
Therisetimescalefortheperiodoffastest fluxincrease curvesshown in Fig. 2 (see § 5 for a discussion)
is τrise ≃ 4 days in X-rays. This is somewhat longer than One should keep in mind that we did not take into ac-
theobservedrisetimescalesof0.3-2days(§2.1.2).Forcom- countouterdiscX–rayirradiationwhichwillincreasesignif-
pleteness,wealsogivetherisetimescaleinoptical, τrise≃1 icantly both the duration and the decay timescale (Dubus
hour, for which there is very little available observational 1999a,b).
(cid:13)c 0000RAS,MNRAS000,000–000
Disc instability models for X-ray transients: evidence for evaporation and low α−viscosity ? 9
3.2.5 Outburst amplitude models: reflares are a generic property of the DIM and the
TDIM for some range of masses and viscosity parameters.
In the model of A0620-00, the maximum accretion rate
One can rather easily see how the presence or absence
reached at the inner edge of the disc is substantially sub–
of reflares depends on the mass of the accreting object. As
Eddington (∼ 0.015M˙ ; Fig. 2). This is far too small. In
Edd noticed by Menou et al. (1999a) the surface density at the
the model of Aql X-1, on the other hand, the Eddington
cooling front is
limit is reached without difficulty (Fig. 5).
The total amount of mass accreted during an outburst Σ(Rfront)=Σmin(Rfront), (4)
is also quite small in the model of A0620-00. This is not
where Σmin(R) is the ‘minimum’ critical surface density in
surprisingbecausetherecurrencetimeismorethan10times
the disc. Vishniac (1997) emphasized that the cold, outer
shorter than observed, and not much mass is accumulated
regions of the disc behind the cooling front are essen-
in quiescence. On the other hand, the total accreted mass
tiallyfrozenduringthecoolingfrontpropagation.Therefore,
is comparable to what is deduced from the observations in
Σ(R) behind the front is constant with time and equal to
the case of Aql X-1. Again, this is not a surprise since the
K(M1,α)Σmin, where K depends on the viscosity parame-
recurrence time is about right.
terαandthemassoftheaccretingobjectM1.Accordingto
the disc models of Menou et al. (1999a), for a central mass
M1 = 1.2 M⊙, K ≈ 4, while for M1 = 7 M⊙, K ≈ 6−7.
The ratio of the ‘maximum’ to ‘minimum’ surface densities
3.2.6 Reflares
is (Hameury et al. 1998)
Ablestorcikciunrgrefnecaetuorferoeffltahreesmdoudreinlsgdtehsecrdibeceldinheeroef iasnthoeutpbousrssit- Σmax(αcold =0.02) ≈6.4, (5)
Σmin(αhot =0.1)
(Fig. [2]). These reflares correspond to multiple reflections
ofcoolingandheatingfrontsduringtheirpropagationinthe sothatforM1 =1.2M⊙ thepost-cooling-frontsurfaceden-
disc. For the specified set of parameters, we find that they sityKΣmin <Σmax,andnoreflaresareexpected.ForM1 =
are present in models with a BH primary, but absent when 7M⊙,thepost-cooling-frontsurfacedensityKΣmin ∼>Σmax
and it is not surprising that reflares are present.
theprimaryisaNS.Thenumberofreflaresisreducedwhen
Toexplain thedependenceofK on themass of theac-
thediscistruncatedbecauseincreasingtheinnerdiscradius
cretingobject, onecan usethecooling front modelof Vish-
gradually removes the regions of the disc where reflections
niac&Wheeler(1996).Althoughsomeofthepredictionsof
occur.
thismodelwerefoundtobesomewhatinaccuratebyMenou
ReflaresareinfactanaturaloutcomeoftheDIM.They
etal.(1999a),weexpectthescalingsderivedheretobevalid.
have been observed in many simulations over the years, in
In thismodel thefront speed can be written as:
particular in models of DN discs in which a same value
αhot = αcold was used (e.g. Smak 1984). In that case, a H 0.7
v ∝ . (6)
cooling front develops,propagates oversomefraction ofthe front R
(cid:16) (cid:17)
discradialextent,isreflectedintoaheatingfrontwhichgoes
From mass conservation the post-cooling front density can
backall thewayuptothediscouteredge.This isrepeated
bewritten as (see e.g. Vishniac 1997)
indefinitely and leads to small amplitude variations of the
dbyisculsuinmginaoshitigyh.eTrhvisalfuaieluorfeαofftohretmheodheolthiaosnbizeeedndovisecrctohmane Σpf ∼Σminvrv(ffrroonntt), (7)
for the cold neutral disc, which results in large amplitude where v (front) is the speed of matter at the cooling front.
r
outburstsas those observed. Using Vishniac & Wheeler (1996) model, one obtains
Thefundamentalreasonwhyareflareoccursisbecause
Σ
the surface density just behind a cooling front reaches the pf ∝M0.35. (8)
critical value Σmax sometime during its propagation. This Σmin 1
happens typically when the front is located at Rfront ∼< This scaling, K ∝ M10.35, is in good agreement with the
1010cminoursimulations.Anoutwardpropagatingheating numerical results of Menou et al. (1999a).
frontthenappearswhichdestroystheinnercoolingfrontas
mass starts being accreted efficiently in the cooling region.
3.2.7 Consequences
Later on, this heating front is reflected when the surface
densityjustbehindthefrontreachesthecriticalvalueΣmin. Ourmodels confirm theresults of Cannizzo (1998) showing
A cooling front developswhich shutsoff theoutward trans- thatastrongerevaporationincreasestrecbecauseitremoves
portofangularmomentumattheoriginoftheheatingfront the inner, most unstable parts of the disc. The recurrence
propagation.Asequenceofheatingandcoolingfrontreflec- times are also increased if the mass transfer rate M˙ in
T
tions corresponds to the multiple reflares observed in the the model is reduced, since more time is then needed to
simulations. accumulatemassinthediscuptothecriticalsurfacedensity
What was probably not previously realized is that the Σmaxatwhichthenextoutburstistriggered.Anexploration
occurrence of reflares depends not only on the values of α of the parameter space of the models shows that it is not
but also on themass of the centralobject, as shown byour possible, however, to reproduce recurrence times of ten to
simulations. We note that reflares were also found in the tens of years in TDIMs with ‘standard’ values of α. This is
simulations of the decline phase of BH SXT outbursts of becausetoobtaintheselongrecurrencetimes,theincreasein
Cannizzo, Chen & Livio (1995). Cannizzo et al. interpreted thestrengthofevaporationorthereductionofM˙ required
T
thereflaresasnumericalartifacts.Thisisnotthecaseinour lead to globally stable discs.
(cid:13)c 0000RAS,MNRAS000,000–000
10 K. Menou, J.-M. Hameury, J.-P. Lasota and R. Narayan
This result on short recurrence times is confirmed by
an analytical estimate of the recurrence times of SXTs.
Anoutburststartswhenthesurfacedensityreachesthe
critical value Σmax. In the case of outside-in (type A) out-
bursts,thishappenswhenenoughmasshasaccumulatedat
the disc outer edge. If the accumulation time at the outer
edgeislongerthantheviscoustime(asitisthecaseforlow
accretion rates), matter transferred from thesecondary dif-
fuses inwards and accumulates at shorter radii, giving rise
to an inside-out (type B) outburst. Since, for a given set
of parameters, the recurrence time of a type A outburst is
shorter than the corresponding time for a type B outburst
(tA < tB), we consider here only type B outbursts (in any
case the only typeobtained in our models).
FollowingSmak(1993),therecurrencetimeofatypeB
outburst is
∂lnΣ −1
tB≈ , (9)
∂t
(cid:16) (cid:17)
which is the characteristic growth-time of a surface density
contrast. From the standard diffusion equation describing
theevolution of Σ, oneobtains therelation between tB and
νΣ, whereν is thekinematicviscosity coefficient. The rela-
tion between νΣ and the effectivetemperature T is
eff
Figure 6.Recurrence times predicted bymodels ofNSand BH
SXTs (assuming parameters relevant to Aql X-1 and A0620-00, σSBTe4ff = 98GRM31ξνΣ, (10)
respectively) for various values of the disc fixed inner radius
Rin (αhot = 0.1, αcold = 0.02). The dotted, dashed and solid whereσSB istheStefan-Boltzmannconstant,Gisthegrav-
linescorrespondtomasstransferratesM˙T =1016,3×1016 and itationalconstant,Risthedistancefrom thecentralobject
1017 gs−1, respectively. For both NSand BHSXTs, truncating ofmassM1,andξ≈3−5isaconstantfactorthataccounts
morethediscresultsingloballystablediscs(trec→∞)insteadof for the non-stationarity of the quiescent disc (Idan et al .
trec≥10−50yearsasexpected. Modelswitharbitrarylongtrec 1999). Using the values of Σmax calculated by Hameury et
can be constructed by fine tuning a marginally stable disc (e.g.
al. (1998) and assuming a constant effective temperature,
thetwoupperpointsintheBHSXTpanel),butthesemodelsdo
which is a good approximation in quiescence (Smak 1984;
nothaveanyphysicalmeaning(seetextfordetails).
Hameury et al. 1998), one gets
To demonstrate this, we computed a series of models tB≈3(cid:16)3ξ(cid:17)(cid:18)MM⊙1(cid:19)0.62(cid:16)101R0 cm(cid:17)0.14×
of NS and BH SXTs with fixed values of Rin. The vari- α −0.83 T −4
ous values of Rin are equivalent to various strengths of disc cold eff yr. (11)
0.02 3000 K
evaporation.Forexample,amodelwithparametersrelevant (cid:16) (cid:17) (cid:16) (cid:17)
for A0620-00, with a fixed value of Rin = 6×109 cm and Thischaracteristic timetB isexpectedtobeanupperlimit
M˙T =1016gs−1hasarecurrencetimetrec ≃3years.Thisis to the recurrence time (e.g. Teff may be larger than 3000
slightly smaller than therecurrence time in the model with K). Again, this rough estimate is much less than tens of
evaporation shown in Fig. 1, in which Rin ≃ 6×109 cm years. It also suggests that one way to obtain long enough
before the onset of an outburst, but M˙T is slightly smaller. recurrencetimesistoreducethevalueofαcold.Interestingly,
Thistypeofcomparisonshowsthatthemaineffectofevap- we expect the maximum accretion rate reached by the disc
oration on the outburst cycles is to affect the radial extent during outburst to increase as well if αcold is reduced. This
of the disc in quiescence, which in turn determines trec. isbecausemoremasswillbepresentinthediscatthetime
Figure6showstherecurrencetimespredictedforthree an outburst is triggered (Σmax ∝ α−co0ld.83; e.g. Hameury et
mass transfer rates (M˙ = 1016, 3×1016 and 1017 g s−1) al. 1998). Models of SXTs with smaller values of αcold are
and several values of the fixed inner radius Rin. At a given considered in the following section.
M˙ ,thesametrendisobserved:therecurrencetimeinitially
T
increases rather slowly with Rin, but then the disc rapidly
becomes globally stable at some critical value of Rin. For 4 “NON-STANDARD” TRUNCATED DISC
reasonable values of M˙T, this critical value of Rin corre- MODELS OF SXTS
sponds to recurrence times well below ten to tens of years.
It is possible to adjust the inner disc radius so that the re- Inthefollowing,wepresenttwonon-standardtruncateddisc
currence time will bearbitrary long (see e.g. thetwo upper models of SXTs. The first model is a non-standard TDIM
points in the lower panel of Fig. 6) but the resulting disc because the value of the viscosity parameter α used is
cold
is marginally stable. Keeping it in this state for the entire significantly smaller than the standard value ∼ 0.02. The
quiescence period requires a mass-transfer rate M˙ strictly second model explores the possibility that quiescent, trun-
T
constant, which is clearly unrealistic. cateddiscsin SXTsareglobally stable.Thismodel isnota
(cid:13)c 0000RAS,MNRAS000,000–000