Warped accretion discs and the long periods in X-ray binaries

A(MNLTEXstylefilev1.4)

WarpedaccretiondiscsandthelongperiodsinX-ray

binaries

RalphA.M.J.Wijers1,2andJ.E.Pringle1

ofAstronomy,MadingleyRoad,CambridgeCB30HA,UK

ofPhysicsandAstronomy,SUNYatStonyBrook,StonyBrook,NY11794-3800,USA

E-mail:rwijers@astro.sunysb.eduandjep@ast.cam.ac.uk

2Dept.1Institute

arXiv:astro-ph/9811056v1 3 Nov 1998submittedtoMNRAS,17-Dec-97,revisedsubmit2-Nov-98

2R.A.M.J.WijersandJ.E.Pringle

&Holt1983)and4U1820−30(176d;Priedhorsky&Ter-rell1984a).Othersaresimplylessregular,suchasCenX-3(120–160d;Priedhorsky&Terrell1983).Inthislattercate-gorynumerousobjectshaverecentlybeenaddedwithwell-sampledlightcurvesobtainedwiththeRossiX-rayTim-ingExplorer(RXTE).CygX-2(78d;Wijnands,Kuulkers&Smale1996)andX2127+119(37d;Corbet,Peele&Smith1997)haveperiodsthatseemfairlystable,buttheampli-tudeandlightcurveshapearevariablesotheperiodicitycannotalwaysbeseenequallywell.SMCX-1,ontheotherhand,hasafairlystableamplitudebutaperiodthatslowlydecreasesfromover60dtounder50dinthefirst600daysofRXTEdata(Wojdowskietal.1997).

Moreover,thereappearstobesomeevidenceofdiscsprecessingprogradely.First,therehasbeenadetectionofan11.2-dayperiodinthelightcurveofCygX-2(Holtetal.1976).Thisisthebeatperiodbetweenthe78-daylongpe-riodandthe8.9-dayorbitalperiod,butonlyifthediscpre-cessesprogradely.Notethatthiscannotbeduetotidallyforcedprecession.InX1916−053,theopticalperiodis0.9%longerthantheX-rayperiod,andifthebeatperiodof3.8daysbetweenthetwoisinterpretedastheperiodofdiscpre-cession,itwouldmosteasilybeforaprogradelyprecessingdisc.

ItwasnotedbyPetterson(1977)andIping&Petterson(1990)thatasufficientlystrongilluminationfromthecentrecouldcauseadisctomaintainawarped,tiltedshape.Whileprobablyqualitativelycorrect,theirresultssufferfromtheuseofanincorrectequationofdiscevolution(seePapaloizou&Pringle1983).Anaccretiondiscisindeedunstabletotiltingandwarpingduetoradiationreactionforceswhentheluminosityofitscentralsourceexceedsacriticalvalue(Pringle1996,1997).WeexamineheretheconsequencesofthisinstabilityforthebehaviourofdiscsinX-raybinaries.Weshowthattheinstabilityprovidesamechanismforsus-tainingatilteddiscandmakingitprecesswithaperiodthatagreeswellwiththeobservedlongperiodsinX-raybi-nariessuchasthefamous35-dayperiodofHerX-1.Theyalsoprovidethepossibilityofbothprogradeandretrogradeprecession.Therearealsonon-steadilyprecessingsolutionswithtime-varyingtilt.WeexploretheseinSection3afterdiscussingthenumericalsolutionmethodtothebasicequa-tion,whichisderivedinSection2.ThenweapplytheresultstosomeoftheknownlongperiodsinX-raybinariesandtosomesimilarsystemsthatdonothaveobservedlongperi-ods(Section4).Finally,wediscusssomeimplicationsandlimitationsofourfindings(Section5)andsummariseourconclusions(Section6).

2

EQUATIONOFMOTIONFORANIRRADIATEDDISCINABINARY

TheaccretiondiscisassumedtobethinandKeplerianandisdividedupintoannulithatinteractwitheachotherviaviscousforces,whichhastheadvantagethatonlytheevo-lutiononthelongerviscoustimescaleneedstobefollowed.Therelevantequationofmotionreads∂L

∂

∂

RΣ

∂R∂R∂R2ν2R|L|

∂󰀃2

󰀃−

3ℓ

󰀃dG

2πR

GMR.

ν1istheviscosityassociatedwiththe(R,φ)shear,whereastheviscosityassociatedwiththe(R,z)shearthatdampsthemisalignmentbetweenneighbouringannuliisν2.TheterminvolvingGistheeffectofthetorqueinducedbyirradiationfromthecentralobject:dG

gφ

6πcR

|a−x|3

ΣRdRdφ.(3)

Thetorqueontheannulusisthensimplyobtainedbyin-tegratingtheelementaltorquecontributionsalongit.HereweexpandtheforceinpowersofR/aandonlyretainthelowestnon-vanishingorderofthetorque:

󰀉

sin2βsinψcos(γ−ψ)x×dFc=

3π

a3

󰀊

−sinsin2β2cosβsinψ2(cos(γ−γ−ψ)

ψ)

󰀂

(4)

Notethattheonlyforcetakenintoaccountisthegravityofthecompanion.SinceourframeiscentredonM(butnotrotating)andthusrevolvesaroundthecentreofmassofthebinary,thereisacentrifugalforceaswell.Itisnothardtoshowthatitcontributesnonettorqueonanannulus.ComparisonwiththeexpressionforLleadstotheexpressionfortheprecessionfrequency:Ωp≡ΩpepΩp

=

−

3Ωa3

(5)

󰀃

c0000RAS,MNRAS000,000–000cosψep

=

sinβcos(γ−ψ)

󰀊

sinψ0

󰀂

.

Inthispaper,weshallmaketheapproximationthattheorbitalperiodofthebinaryissmallenoughcomparedtotheprecessionperiodthatwemayaveragetheprecessionoverthebinaryorbitalperiod(i.e.overtheangleψ).Averagingandcross-productarenotinterchangeable,ofcourse,sowehavetoaveragethetorque(Eq.(4))andthenfindanewexpressionforΩp.Theresultis:

Ωavp=Ωpeav

p=Ωpcosβez

(6)

(SeealsoKatzetal1982,Papaloizou&Pringle1983).Notethatthefactorcosβisnotpresentinstudiesthatassumetheinclinationtobesmall.

Fornumericalstudieswecasttheequationindimen-sionlessform.Thedimensionlessvariablesarer=R/R0,σ=Σ/Σ0andτ=t/t0,withR0,t0andΣ0valuestobedeterminedlater.ThenwecanalsosetL=L/L0=L/Σ0

√3∂τ

=

∂r󰀈

r1/2

σr

1/2

+1∂r

󰀈

n1

ηr󰀁

n12+1

1

󰀆

∂r

󰀃󰀇L

󰀋

∂r󰀃󰀃

∂ℓ󰀌

L

󰀋

∂r

+F⋆

gφ

󰀄󰀅

2

R2

0

6πcΣ0R3

0Ω0

1

∂t

=

∂R2

+

whereΓ=L⋆/12πc󰀄

3ν2

∂R

+iΩpW,(9)

ΣR2Ω.ThelocalstabilitythenfollowsbylookingforsolutionsoftheformW=W0expi(σt+kR),withkR≫1.SubstitutinginEq.(9)thisimplies

σ=i[−Γk+

1

4R

.(10)ThecomplexpartofσisidenticaltothatofPringle,soneithertheadditionofforcedprecessionnorretentionofthegradienttermsaffectsthelocallinearstabilityofthedisc.Thus,asbefore,growingmodesoccuronlyforΓ>0and0󰀃

c0000RAS,MNRAS000,000–000WarpedaccretiondiscsinX-raybinaries

3

morerapidlytowardstheoutsideandthustendstodestroytheshapeofthegrowingmode.

3NUMERICALEXPERIMENTS3.1

NumericalMethod

ThenumericalmethodemployedisthatdescribedinPringle

(1997),withchangesmadeinordertomodelaccretiondiscsinbinarysystemsratherthaninAGN.Themostimportantchangeswhichneedtobemadearewithregardtotidaltruncationofthediscattheouteredge,withregardtotheinputofmassatsomeradiusotherthantheouteredge,andwithregardtothetidallyforcedprecessionoftheouterdiscelements.

Weworkinitiallywith40gridpointsbetweenrin=1androut=40,anduseanequallyspacedratherthanalogarithmicgrid(Section3.2).Thiswaschosensothattheouteredgeofthegridwouldbeproperlyresolved,buthasthedisadvantagethattheinnerdiscregionsarenotwellresolved.However,becauseofthis,thecomputationtimerequiredforeachrunisrelativelyshort,andthuswewereabletoexplorealargevolumeofparameterspace.Wereportbelow(Section3.3)onafewrunscarriedoutwithamoreextensivegrid.Thenumberofazimuthalzoneswasnormally68.Increasingthisvalueto120,whichwedidinafewkeyruns,hadnoeffectontheresults.

WeachieveazerotorqueinnerboundaryconditioninthemannerdescribedinPringle(1997)byremovingmassoverthreezonescentredonr=3.Weaddmaterialatacon-stantrateover3zonescentredonradiusradd.Thematerialisaddedtothediscwithspecificangularmomentumvec-torcorrespondingtothematerialbeingaddedintheorbitalplane,andmagnitudeappropriatefortheradiusradd.

Inordertotakeaccountofthetidaltruncationofthediscatradiusrtide(Theforcedprecessioncausedbythetidalpotentialfieldishandledasfollows.Eachannulusisprecessedateachtimestepaboutthevector(0,0,1)atarateωp=ωp0r3/2.Thuswehaveignoredthefactorofcosβpresentintheforcedpre-cessionrate(Equation1.6).Thiscouldbesimplyincluded,butshouldbenegligibleforthekindofsolutionswearelook-ingfor.Thetimestepislimitedtoensurethatanyannulusisnotprecessedbymorethan2π/10inanyonetimestep.Inpracticethetimestepconsiderationsforstabilityofthenumericalmethodensurethattheprecessionrateisnotthedominantfactorinlimitingthetimestep.

4R.A.M.J.WijersandJ.E.Pringle

Weassumeaviscosityoftheform:

ν1=ν10r3/4,

(11)

whichischosentoapproximatetheviscositydependenceinsuchdiscs(seeeq.15).Wetakeη=1.Thediscissetupinitiallywithasurfacedensityappropriatetoasteadydiscwiththisviscosityandwiththeappropriateboundaryconditions(viz.Σ=0atr=1,vR=0atr=rtide,andadimensionlessmassinputrateofunityatr=radd).Thediscisgivenaninitialwarpoftheformofaprogradespiralwhichissuchthatβincreasesfromzeroatr=1toβ=0.1atr=40,withγincreasingbyanangleofoneradianoverthesamedistance.3.2

NumericalResults

Wehaveinvestigatedthenon-linearbehaviourofirradiateddiscsinbinarysystems,takingshadowingfullyintoaccount.Ouraimhasbeentodiscoverwhatregionsofparameterspace(ifany)giverisetodiscswithafinite,and(reasonably)steadytiltangleofthekindwhichmightberelevanttoex-planationsofsystemslikeHerX-1/HZHer.Thusourinitialaimwastosearchforthekindof‘mode-like’behaviourdis-cussedbyMaloney&Begelman(1997).Wefound,however,thatthetime-dependenceofbinaryaccretiondiscsgivesrisetoaricherandmorevariedbehaviourthancanbedescribedbysimple‘modes’.3.2.1

radd=10,rtide=30,ωp0=0

WithaneyeontheparametersofthebinarysystemsweareinterestedinasshowninTable3,wechooseradd/rtide=0.33.Weinitiallysettheforcedprecessionratetozero,andinvestigatewhatvaluesofthediscluminosity(F⋆)giverisetorelevantbehaviour.InFigure1weshowthebehaviouroftheinclinationoftheouterdiscedgeasafunctionofdimensionlesstimeforvariousvaluesofF⋆.WefindthatforF⋆toinstability≤0.04theanddisctheirradiationdiscflattensisnotintostrongtheorbitalenoughplane.

toleadForF⋆=0.045thedisctiltsettlesdowntoasteadyfunctionalform,withβsmallattheinside,andβ′positiveatmostradii,reachingavalueofβ=0.275(correspondingtoanangleof16◦)attheoutside(i.e.atr=rtide).Thedisciswarpedintheshapeofaprogradespiral,asistobeexpectedforadiscunstabletoself-irradiation(Pringle1996).AtthesametimethewholediscprecessessteadilyinaretrogradedirectionwithadimensionlessperiodofPp=58.Thedirectionofprecessionisretrogradebecauseoftheshapeofthedisc.Thesignofβ′determineswhichfaceofthediscisilluminatedbythecentralsource.Apositiveβ′leadstoretrogradeprecession(Pringle1996).WenotethatthiscontrastswiththeAGNdiscsdiscussedinPringle1997.Theretheouterboundaryconditionensuredthattheouteredgeofthediscremainedintheinitialdiscplane.Theniftheinnerdiscistilted,muchofthediscislikelytohavenegativeβ′,andthereforeprecessesinaprogradefashion.Inthebinarycasetheedgeofthediscisfreeinthesensethatnotorqueactstheretochangethetiltofthedisc.

SimilarbehaviouroccurswhenF⋆=0.05,althoughinthiscaseβattheoutsidesettlesdowntothelargervalueofβ=0.44(correspondingtoanangleof25◦)andtoamorerapidretrogradeprecessionratewithaperiodPp=48.

Figure1.Thebehaviourofthediscinclinationatr=26,justinsidertide,fordifferentvaluesofF⋆andusingthesmallgrid(Section3.2).WithincreasingF⋆,thediscfirstisstable,thengrowstoafiniteandconstanttiltthatisgreaterforhighervaluesofF⋆.TheoscillationsatstillhigherF⋆areanartefactofthepoorlyresolvedinnerdisc.

Whentheeffectofradiationisfurtherincreased,thediscbehaviourisnolongersteady.ThebehaviourofβattheouterradiusisshowninFigure1.Thebehaviourisintheformofalimitcycle.WenoteherethatforsimplicitywehaveassumedthatF⋆isconstantthroughouteachrun.However,inreality,sincetheradiationilluminatingthedisccomesdirectlyfromtheaccretionrateatthedisccentre,F⋆islikelytovarywithtimewhenthediscdisplayssuchnon-steadybehaviour.Thustheactualdiscbehaviourislikelytobeyetmorecomplicatedthanwefindhere.3.2.2

radd=10,rtide=30,F⋆=0.05

Inthissectionwedescribetheeffectonthesteadyprecessingsolutionsofaddingretrogradeforcedprecessionoftheforminducedbytidaltorquesfromacompanion.Themagnitudeoftheforcedprecessionisdescribedbytheparameterωp0(eq.20).Withzeroforcedprecession,ωp0=0,wehaveseen(Section3.2.1)thatthediscsettlesdownintheshapeofaprogradespiral,withβanincreasingfunctionofr,reachingβ=0.44attheouteredge,andprecessingsteadilyinaretrogradedirectionwithperiodPp=48.

Whenasmallamountofretrogradeforcedprecessionisadded,thediscsettlesdowntoasimilarbehaviourasforωp0=0,butwiththefinalvalueofβ(rtide)slightlyincreased,andtheprecessionratealsoincreased.Thuswefindthatforωp0=−0.0002,β(rtide)=0.54andPp=40;forωp0=−0.0005,β(rtide)=0.63andPp=35,andforωp0=−0.001,β(rtide)=0.56andPp=29.However,astheforcedprecessionrateisincreasedfurtherthediscinstabilityisremoved.Thus,forωp0=−0.002,wefindthatβ(rtide)tendstozero,andthediscsettlesdownintotheorbitalplane.Wesuggestthatthisbehaviourcomesaboutbecause

󰀃

c0000RAS,MNRAS000,000–000inordertobeunstablethediscmusttaketheformofaprogradespiral(Pringle1996).Howeverstrongretrogradeforcedprecession,whichisdifferentialinthesensethattheprecessionrateincreaseswithradius(here∝r3/2),tendstounwindtheprogradespiralandthusactstopreventtheinstabilityfromoccurring.

3.2.3radd=10,rtide=30,F⋆=0.09

HereweinvestigatetheeffectofaddingforcedretrogradeprecessiononthesolutionsdiscoveredinSection3.2.1whichtaketheformoflimitcycles.Wethereforeinvestigatedso-lutionswithF⋆=0.09,andwithωp0=−0.001,−0.002,−seems0.003toandhave−0little.004.effectIncreasingonthetheamplitudeforcedprecessionofthelimitratecy-cle,untilatthevalueof−0.004theinstabilityisquenchedaltogether,andthediscsettlesintotheorbitalplane.How-evertheperiodofthelimitcycleisaffected,anddecreaseswithincreasingωp0,fromavalueof200forωp0=0toavalueofabout120forωp0=−0.003.Whiletheselimit-cyclesolutionsarepartlyanartefactofthesmallgrid,thedamp-ingofeventhesehigh-luminositysolutionsillustratesthehowpowerfulforcedprecessionisinsuppressingdisctilts.

3.2.4radd=20,rtide=30,ωp0=0

Oneeffectofaddingmaterialatradd=10intheprevioussectionswastohelppinthedisctowardstheorbitalplaneatthatradius.Thishelpedtocontrolthebehaviourofthediscwithinthatradius(seealsoSection3.3),andtoallowtheouterregionsofthedisc(afactorofthreeinradius)toevolvefreelywithregardtotilt.Sincetheself-irradiationwarpinginstabilityactsmorestronglyatlargerradii,itwasalwaystheouterregionsofthediscwhichrespondedmosttotheradiationfluxfromthecentralobject.Inthissectionwedescribetheeffectofchangingtheradiusatwhichmassisaddedtothediscfromradd=10toradd=20.Thishastwomajoreffects.First,addingmatterclosertotheoutsidehastheeffectoftendingtopinthediscintotheorbitalplaneattheoutside,whileallowingtheinnerdiscregionstotilt(seealsoSection3.3).Second,asteadydiscinsidetheradiusraddhasνΣconstantatradiiinsideradd,whereasasteadydiscwithavR=0outerboundaryconditionhasνΣ∝r−1/2forradiioutsideradd.Thusforagivenaccretionrate,sinceνistypicallyanincreasingfunctionofbothΣandr,theouterpartsofthediscbecomelessmassiveasraddisdecreased.Thusbyincreasingraddfrom10to20,weexpecttheradiationinstabilitytorequirelargervaluesofF⋆.

ForvaluesofF⋆toself-irradiation.For≤values0.15,weoffindF⋆thatthediscisstableofthediscturnovercompletelyinthe≥manner0.25thediscussedinnerpartsfortheAGNdiscsinPringle1997.ForF⋆=0.2thediscdoestendtoafairlysteadyconfiguration.Howevertheshapeofthediscnowdiffersfromthosediscussedaboveinthesensethatthedisctiltβislargestattheinside,anddecreaseswithradius(i.e.β′ispredominantlynegative;Fig.2).Moreovertheinnerandouterpartsofthediscbehaveinquitedifferentandindependentmanners.TheinclinationatallradiiinthediscoscillatesmoreorlessinphasewithaperiodofaboutPinc=10.Theinnermostradiioscillatebetweenβ=0.53and0.57,andtheoutermostradiioscillatebetweenβ=0.06

󰀃

c0000RAS,MNRAS000,000–000WarpedaccretiondiscsinX-raybinaries

5

Figure2.Comparisonofthevariationofthedisctiltβwith

radiusformassadditionwellinsidetheouterradius(radd=10,solidcurve)andmassadditionnearertheouteredge(radd=20,dashedcurves).Theformer,havingβ′>0inmostofthedisc,precessesretrogradely,whereasthelatterprecessesprogradely.Notethatwhenradd=20thediscisnotafixed-shapeobjectprecessingatasinglerate(seetext).Thetwocurvesspanap-proximatelytherangeofshapesitdisplays.

and0.12.Atradiusr=24,theoscillationinβvariesbe-tween0and0.14,anditisthisradiuswhichappearstosep-aratetheinnerandouterpartsofthedisc.Theinnerpartofthedisc(roughlythoseradiir<24)hasaninclinationwhichdecreaseswithradius,andwhichprecessesinaprogradedi-rectionwithaperiodofaboutPin=12.Theouterpartsofthedisc(roughlyradiiintherange24inc=Pin+Pout).3.2.5

radd=20,rtide=30,F⋆=0.2

Wenowinvestigatetheeffectofaddingretrogradeforced

precessiontotheF⋆=0.2discwhichwasdescribedintheprevioussection.Asthemagnitudeofωp0increasesfromzero,thebehaviourofthediscstaysinitiallythesame,exceptthattheinclinationdecreases,theinner(pro-grade)precessionperiod,Pin,increases,andtheouterpre-cessionperiod,Pout,decreases.Thus,incomparisonwithβ(rin)=0.55,Pin=12andPout=50forωp0=0,wefindthatforωp0=−0.0015,β(rin)=0.46,Pin=14,Pout=20;forωp0=−0.004,β(rin)=0.32,Pin=20,Pout=11;for

6R.A.M.J.WijersandJ.E.Pringle

ωp0=−0.006,β(rin)=0.2,Pin=26,Pout=6;andforωp0=−0.007,β(rin)=0.16,Pin=40andPout=4.5.Inaddition,asthesizeofωp0isincreased,theradiuswhichsep-aratestheinnerdiscandouterdiscbehaviours(theradiusatwhichtheβoscillationpassesthroughzero)movesout-wardsuntilatωp0=−0.006itisattheouteredgetowithintheresolutionofthegrid.Howeverwhenωp0isincreasedfurtherto−0.008thewholediscnowsettlestoaconstantβprofile,whichispositiveatallradii,andhasβ(rin)=0.12,andβdecreasingwithradius.ThediscasawholeprecessesinaprogradedirectionwithperiodPp=22.Theshapeofthediscissuchthatithasaprogradespiralinsideradd,andaretrogradespiraloutsideradd.3.3

Numericalresultsonamoreextensivegrid

Sinceallthenumericalresultsabovewerecomputedonagridwhichhadlimitedresolutionintheinnerregions,wefeltthatitwasnecessarytoexplorethelimitationsofsuchaprocedure.Weusealogarithmicgridconsistingof80gridpointsbetweenRin=0.136andRtide=30.Westilladdmaterialover3gridpointscentredonRadd=10,andweremovematerialover3gridpointscentredonR=0.155.InthismannertheouterpartofthegridbetweenRaddandRtideisreasonablywellresolved(15gridpoints),andtheinnerdiscregionextendsinwardofRaddbyalmosttwoor-dersofmagnitude,sothatthebehaviouroftheinnerpartsofthegridcannowbemodelledmoreaccurately.

Becausethegridisnowmoreextensive,andespeciallybecausethegridnowextendstosmallerradii(wherevis-coustimescalesareshorter)thecomputationalruntimesarenowlongerbyabouttwoordersofmagnitude.Thuswehavelimitedourselvestoafewrepresentativeexamplesforcomparisonwiththeresultsintheprevioussection.WefindthattheinstabilitysetsinatvaluesofF⋆whicharelargerbyaboutafactoroftwo.Thiscomesaboutbecauseinasteadyaccretiondiscwithaninnerboundaryconditioncor-respondingtovanishingsurfacedensity(i.e.zerotorque),itisthequantityνΣ[1−(r/rin)1/2]whichisconstantwithradius,ratherthanjustνΣ.Thustheeffectofmovingtheinnerboundaryinwardsistoincreasethesurfacedensityoftheouterdiscbyaboutafactoroftwo.

SincethetidallyinducedprecessiondoesnotseemtoplayastrongroleforthosediscswhicharerelevanttotheX-raybinariesweareinterestedinhere,wehavetakenωp=0.InFigure3weshowthebehaviourofthediscinclinationatr=26forF⋆=0.09,0.12,0.135,0.15,0.175,0.2and0.3.AscanbeseenthediscwithF⋆=0.09isstable.ThediscwithF⋆=0.12eventuallyprecessesinasteadyfashioninaretrogradedirectionwithaperiodofabout35,andthediscinclinationattheouteredgesettlesdowntoavalueof0.15(i.e.about8.5degrees).ThediscwithF⋆=0.15settlesdowntoasolutioninwhichtheinclinationandoscillateswithaperiodofabout15andsemi-amplitude5percent,aboutasteadyvaluefortheinclinationof0.25(14degrees).Theprecessionperiodis30fortheouterdisc.Theinnerdischasinclinationnearzero,withsmalloscillationsofthesameperiodastheouterdisc(Figure4).Theperiodoftheseos-cillationsisthebeatperiodbetweentheinnerandouterdiscperiods,aswiththesmallgrid.Theircauseissimplythattheregionwheretheouterandinnerdiscjoin(justoutsideradd)triestoadjusttothetiltsofbothsides.Thisitcan-

notdosimultaneously,ofcourse,andthetiltofthiszoneoscillatesbetweennearlyzeroandafinitevalue,dependingontherelativephasebetweentheinnerandouterdiscso-lutions.Thisoscillationisthencommunicatedthroughoutthedisc,withanamplitudethatdecreasesawayfromthecontactzone.

ForgreaterF⋆,theinnerdiscsuddenlygetsasubstan-tialtiltaswell,presumablybecauseittooisnowunstable(thesuddentransitioniscausedbythefactthatthein-nerradiusoftheunstableregiondecreasesastheinversesquareofF⋆,seePringle1997).ThediscswithF⋆=0.2and0.3varychaotically,withthetiltoftheinnerdiscusu-allygreaterthan90degrees,andthatoftheouterdiscless.Theouterdiscstillprecessesretrogradelyonaverage,withroughlythesameperiodasbefore(25–35),butwithirreg-ularitiessuperimposed.Theazimuthoftheinnerdisctiltwandersirregularlywithoutanylong-termtrend.

Inconclusionwefindthattheresultsobtainedwiththelargergridarefullyconsistentwiththebehaviourfoundwiththecrudergrid.ThustheresultspresentedinSection3.2areexpectedtobereasonablyrepresentative.Themaindifferencesappeartobethatthelargergridhasalargersurfacedensity(forthereasonsexplainedabove)andsogoesunstableforsomewhatlargervaluesofF⋆,thattherangeofF⋆forwhichthediscdisplayssteady,ornearlysteady,precessingtiltedbehaviourissomewhatlarger,andthatthetiltanglesreachedbytheouterdiscaresomewhatsmaller.Inordertocheckthegrid-dependenceoftheselargesimulationswefurtherdoubledthenumberofradialandazimuthalgridpointsinafewkeysimulations;nosignificantchangesintheresultswereobserved.

3.4Precessionperiodsandstability

Inordertoseehowwellsimpleestimateswork,wenowcomparetheprecessionperiodsobtainedinthesimulationswithexpectedvalues.FromEq.(7),oneobtainsaprecessiontimescaleestimate,takinggφ/2πtobeunity:tΓp=

Lr

3/2Fσr.

(13)

∗

Wenowhavetotakethequantitiesfromsimulationstocom-paretheleftandrighthandsidesnumerically.Usingthere-sultsfromSect.3.2.1,wehave,e.g.anumericalperiodof48whenF∗=0.05.Attheouteredgeofthissimulation,r=30.25andσ=2.26×10−3,givingapredictedperiodfromEq.(13)ofPp=47,inverygoodagreementwiththevalueseeninthesimulation.FortherunwithF∗=0.045wehaveσ=2.76×10−3,andhencecomputePp=64,com-paredwiththeactualsimulationvalueof58.Itisclearthattheradiativetimescaleattheouteredgeofthegridpredictsthenumericalprecessionperiodquiteaccurately.Thehighprecisionisdoubtlesssomewhatfortuitous,sinceshadowingwilltendtolengthentheperiodatthegridedgesomewhat(gφ/2π<1),whereasthefactthatthepatternspeedmustbeanaverageoverafiniterangeofradiiinthedisctendsto

󰀃

c0000RAS,MNRAS000,000–000WarpedaccretiondiscsinX-raybinaries7

Figure3.Thebehaviourofthediscinclinationatr=26,justinsidertide,fordifferentvaluesofF⋆.WithincreasingF⋆,thediscfirstisstable,thengrowstoafiniteandconstanttiltthatisgreaterforhighervaluesofF⋆.Theoscillationsaroundthestablelevelareduetothefactthattheinnerandouterdiscprecessinoppositedirectionsoncetheunstableregioninthediscincludesenoughofthediscinsideradd,sothatthesolutionisnolongerstationary.AtevengreaterF⋆thedisctiltsthrough90degreesandthebehaviourbecomeschaotic.ThisistheregimepreviouslydiscussedbyPringle(1997)forAGN.

Figure4.Thebehaviourofthediscinclinationatr=2.5,wellinsideradd,fordifferentvaluesofF⋆.NotethelargertiltsincomparisonwiththeouterdiscandthefactthatatlargeF⋆theinnerdiscismostlycounter-rotating.

c0000RAS,MNRAS000,000–000󰀃

8R.A.M.J.WijersandJ.E.Pringle

shortenit,becausetheradiativetimescalebecomesshorteratsmallerradii.

Thecriticalnumberforstabilityistheratioofradiativetoviscoustimescalesattheouterdiscedge.Inthescaledequations,wehaveγ=

tΓ

F∗

ν20

νΣ(RΩ′)2tothelocalemissionrate

fromeachdiscsurface.SinceatitsinneredgethispartofthediscdoesconnecttoaregimewhereΣfollowsfromtheaccretionrateintheusualwaywecanstillsetthenormal-isationoftheouter-discsurfacedensityfromtheaccretionrate.Thenetresultisthesameasabove(eq.15),butwithanadditionalfactor(R/Radd)−3/20.WedonotconsideranydiscswithRtide/Radd>4,sowehaveignoredthesmall

2

c0000RAS,MNRAS000,000–000󰀃

WarpedaccretiondiscsinX-raybinaries

correctiontotheviscosityinourcalculations.Notethatthesurfacedensityissignificantlyaffected,changingfromΣ∝R−3/4insideRaddtoΣ∝R−11/10outside,sothattheouterdiscbecomessignificantlylighterandmoresuscepti-bletoradiativewarpingwhenmassisinjectedwellwithinRtide.

ν2R

9

1/4˙−3/105/4

=39.4α−4/5η−1M1.4MR11days,(17)−8

4.1.2Discsizeandmassinput

tΓ=

0.1

Inabinarysystemtheaccretionstreamemergingfromthe

3/4˙−3/103/4M1.4MR11−8

󰀅−1

days,(18)

companionatL1initiallyhasanon-circularorbit,butonceitself-intersectsitsettlesinacircularorbitwithradiusRJwhichhasthesamespecificangularmomentumabouttheaccretingobjectastheincomingstream.ThisradiusistabulatedbyFlannery(1975)for0.053RJ

tΩ2π

p=

2

day

󰀇󰀆

1+q

110R.A.M.J.WijersandJ.E.Pringle

Table2.Derivedparametersusingtherelationsinthispaper,assumingα=1,η=1,andǫ=0.1(andMX=1.4M⊙).ThetimescalesinthelastthreecolumnsaretakenatRtide.

LMCX-4CenX-3SS433

X1907+097LMCX-3SMCX-1CygX-1HerX-1X2127+119CygX-2gen.LMXBX1916−0534U1626−6711.412.92.14.30.612.92.11.70.640.50.360.070.029.513.50.34.11.19.34.6.53.117.1.20.360.310.200.190.320.190.420.190.320.340.420.440.470.600.681.72.214.5.43.93.39.41.91.16.60.510.190.180.260.260.250.260.290.260.250.250.290.300.330.500.7553.110.890.800.250.110.900.130.180.420.34.10.12.8.112.100.47.21.22.44.14.8.7140.3.31.84.07.413.110.62.62.11.180.17.31.29.8.54.04.9

name

Porb(days)MX(M⊙)Mdonor(M⊙)

˙−8M

Plong

(days)

nameωp,tideγtiderJ

t0(days)

Formostofthesimulations,weuseadiscthatextendsfromrin=1tortide=30.SincetheinnerdiscdoesnotmattermuchforourresultswescaletheparametersforeachsystemsothatRtideinthesystemcorrespondstothetrun-cationradiusinthenumericalgrid.MassisaddedtothediscatrJ=rtideRJ/Rtide(seeTable3).Theappropriatevalueofωp,tidefollowsfromωp,tide

=

2

Ωp(Rtide)Rtide/ν1

enoughforcedprecessiontohavetheirwarpssurvive(Ta-ble3),butnotbymuch;alsoforcedprecessionmaysome-whatshortenretrogradelongperiods(Sect.3.2.2).

Similarly,γtidecanberelatedtotheparametersofarealsystemviaγtide

=

tν2(Rtide)

=

η

󰀆

1day

q

󰀇−2

×

0.1

Notethatγdoesnotdependondetailsoftheviscosity.Thisisbecauseboththesurfacedensityandtheviscoustimescaleareinverselyproportionaltotheviscosity,sotheeffectthatalower-viscositydischasalongerdampingtimefordisctiltsisexactlycompensatedbythefactthatitsmassalsoincreasesthegrowthtimeoftheradiativeinstability.Intable3welistthevaluesofthenumericalparametersthatfollowfromtheserelationsforallsystemsintable1.Wealsolistt0,whichistherealtimethatpassesineachsystemperunitofdimensionlesstimeinthesimulation.

󰀅−1

ηM1.4Rtide,11

1/2−1/2

(21)

c0000RAS,MNRAS000,000–000󰀃

WarpedaccretiondiscsinX-raybinaries11

Figure5.Theobservedlongperiodsasafunctionofmassratio.Thesolidcurveindicatestheexpectedforcedprecessionperiod.

4.3

Comparingforcedandradiativeprecessionwithdata

Letusnowassumethatthediscradiiinrealbinariesaread-equatelyapproximatedbyRtide,andthatthedisctiltisbigenoughthatmassinputoccurspredominantlyatRJ.Thenwecanpredictwhattheforcedandradiativeprecessiontimescalesareforallthesystemsintable3andcomparethemwiththeirobservedlongperiods.

Forforcedprecession,aparticularlysimplerelationen-suesifwedividetheresultbytheorbitalperiod:Pp

3

󰀄

0.87RL1+q

12R.A.M.J.WijersandJ.E.Pringle

Figure7.Theshapeofadiscwithmassinputwellinsidetheouteredge.Itflarestohightiltsneartheouteredgeandprecessesretrogradelyundertheactionofradiationtorques.(runwithsmallgrid,radd=10,rtide=30,F⋆=0.045)

Figure8.Aprojectionofthediscshapeforamildlyinclinedretrogradelyprecessingdiscontotheskyasseenfromthecentralsource.(runwithsmallgrid,radd=10,rtide=30,F⋆=0.045)

source.Forlowinclinations,oneseestwoonstatesofdiffer-entlength(Fig.9),oncewhenthelineofsightpassesunderthedisc(short-on)andoncewhenitpassesoverit(long-on).Bycontrast,aprogradelyprecessingdiscsuchasper-hapsthatofCygX-2hasitshighesttiltinthemiddle.AtypicalexampleisshowninFig.10.Whenmassinputinsuchadiscisnotattheveryoutsideedgetheouterpartcanbegoinginaretrogradedirectionatthesametime,withadifferentperiod(Sect.3.2.4).Underfavourableincli-nationsanobservercouldseeboththeouterandinnerdiscperiodicitiesinthesequenceofX-rayonandoffstates.AnexampleofthisisshowninFig.11.4.4

HerculesX-1

ForthespecificcaseofHerX-1,wherearelativelystablecycleexistsoveralongtime,andtheretrogradeprecessionhasbeenestablishedwellfromobservations,wemaylookat

Figure9.CurvesmarkingX-rayonandoffstatesforthepre-cessingdiscofFig.8asseenfordifferentinclinationsofthebinaryorbittothelineofsight.Forhighinclinationsonlyoneonstateoccursperperiod,butfori=1◦oneobservesanalterationoflong-onandshort-onstates,similarperhapstothemain-highandshort-highstatesofHerX-1(Jones&Forman1976).(runwithsmallgrid,radd=10,rtide=30,F⋆=0.045)

Figure10.Theshapeofadiscwithmassinputclosetotheouteredge.Itshighesttiltoccursattheinsideandthebulkofitprecessesprogradelyundertheactionofradiationtorques.(runwithsmallgrid,radd=20,rtide=30,F⋆=0.2)

thenumbersinsomewhatgreaterdetail.Usingγcrit=0.1andEq.(17),wecanestablishamaximumpossibleradiativeprecessiontimescalePp,max

=2πγcrittν2

=

82.5α−4/5η−1days,

(23)

󰀃

c0000RAS,MNRAS000,000–000Figure11.CurvesmarkingX-rayonandoffstatesforthepre-cessingdiscofFig.10asseenfordifferentinclinationsofthebi-naryorbittothelineofsight.Forhighinclinationsonlyonepe-riodicityisvisible,butforloweronesthebeatbetweentheouterdiscandinnerdiscperiodsisclearlyvisible.(runwithsmallgrid,radd=20,rtide=30,F⋆=0.2)

wherewehaveusedtheparametersofHerX-1fromourtables.ThestablelongperiodindicatesthatHerX-1mustbewithin30%ofthestabilityboundary(Sect.3),sosupposingweequatethisperiodtotheactuallyobservedvalueof35d,thisgivesα4/5η=2.36.

(24)

Ifwethenuseη=1/(2α2)(KumarandPringle1985,Ogilvie1998)wefindα=0.27,whichisnotanunreasonablevalue(seeSection5).Notethatourresultsimplyquitegenerallythatthemeasurementofaprecessionperiodconstitutesacrudemeasurementofthediscviscosity.

Ratherthanassumingnear-criticality,wemayalsojustcomputePΓ=2πtΓfromEq.(18)togetP.3α−4/5

0.1

Γ=17󰀃

c0000RAS,MNRAS000,000–000WarpedaccretiondiscsinX-raybinaries13

Ω

≪

H

5q

Ω

=

a

Takingthesystem󰀅3

.(27)

parametersforHerX-1inTables2and3wefindthatneartheedgeofthedisc,R=Rtide,|Ω−κ|

3

R.(28)

tide

Bycomparisonwefindthat󰀅H

Rtide

󰀅1/8

.(29)

14R.A.M.J.WijersandJ.E.Pringle

Thusclosetothediscedgeweexpectwavepropagationtobemarginallypossiblebythiscriterion,andtoimprovesignificantlyatsmallerradii.

However,forwavepropagationwealsorequirethatthediscviscositybesufficientlysmall.AdiscwarpwaveinaKepleriandiscpropagatesadistanceoforderH/α,beforeitisdissipatedbyviscosity.ThusforawavetobeabletopropagateoveraradialdistanceR,werequireα≪

H

R

ForthevalueofH/Rappropriatehere(eq.29),thiscorre-spondstoα=

˙−9/46M−45/920.45M−81.4

󰀅1.5

.(31)

󰀄

R

R

˙−1/11M−5/22≪0.01M−81.4

󰀄

R

otherstudies,thisnumericalstudytakesaccountoftheim-portanteffectsofself-shadowingofthedisc.(iii)Thequanti-tativeagreementbetweentheobservedlongperiodsofX-raybinariesandtheresultsofoursimulationsisgood,whereasforcedprecessionduetothecompanion’sgravitationalpullonthediscfaresratherpoorlywhencomparedquantita-tivelywiththedata.Also,radiativeprecessioncanbepro-gradeaswellasretrograde,unlikeforcedprecession.ThereistentativeevidencethatthediscsinCygX-2andpossiblyX1916−053doprecessprogradely.Toputitsuccinctly,wehaveshownthatiftheradiativeinstabilityofPringle(1996)givesrisetothedisctiltinthesesystems,thenitalsoauto-maticallygivesrisetodiscprecessionatapproximatelytheobservedrate.Moreover,wehavealsoshownthatiftidallyinducedprecessionbecomesdominant,thentheinstabilityislikelytobestabilised,andthedisctoremainintheorbitalplane.

Inadditiontosteadilyprecessingdiscs,radiativewarpsathighluminositycanalsobenon-stationary:theirtiltan-glevariesperiodicallywithtimeinoursimulations,andinrealisticcaseswithfeedbackbetweencentralsourceluminos-ityandaccretionratewouldprobablyexhibitnon-periodicbehaviouraswell.ThismaybeapplicabletomanyofthelongX-rayperiodsobservedinnature,whicharenotverystableinamplitudeand/orperiod.

Oneparticularfeatureofhigh-luminositysystemsisthattheinnerdiscmaytiltthroughmorethan90degrees,andthusrotatecountertothenormaldirection.Whenitencountersthemagetosphereofaneutronstaritwillthenprovideastrongspin-downtorque,possiblyexplainingthetorquereversalsseeninsystemssuchas4U1626−67.OnewouldexpecttheX-raysourcetobebehindthediscmuchofthetimewhenthewarpissostrong,butthestronglywarpeddiscshavemuchlowersurfacedensities,sotheycouldbe(partly)transparenttoXrays(vanKerkwijketal.1998).

AcknowledgementsRAMJWgratefullyacknowledgessupportfromtheRoyalSocietythroughaURFgrant.WealsothankM.BegelmanandP.Maloneyforusefuldiscus-sions,andforapreprintoftheirpaper.

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