Imaging Optically-Thin Hot Spots Near the Black Hole Horizon of Sgr A at Radio and Near-Inf

A(MNLTEXstylefilev2.2)

ImagingOptically-ThinHotSpotsNeartheBlackHole

HorizonofSgrA*atRadioandNearInfraredWavelengths

AveryE.Broderick⋆&AbrahamLoeb†

InstituteforTheoryandComputation,Harvard-SmithsonianCenterforAstrophysics,60GardenSt.,MS51,Cambridge,MA02138,USA

arXiv:astro-ph/0509237v2 14 Sep 20055February2008

ABSTRACT

Submilli-arcsecondastrometryandimagingoftheblackholeSgrA*attheGalacticcentremaybecomepossibleinthenearfutureatinfraredandsub-millimetrewave-lengths.Motivatedbyobservationsofshort-terminfraredandX-rayvariabilityofSgrA*,inapreviouspaperwecomputedtheexpectedimagesandlightcurves,includingpolarization,associatedwithancompactemissionregionorbitingthecentralblackhole.Weextendthiswork,usingamorerealistichot-spotmodelandincludingtheef-fectsofopacityintheunderlyingaccretionflow.Wefindthatatinfraredwavelengthsthequalitativefeaturesidentifiedbyourearlierworkarepresent,namelyitispossibletoextracttheblackholemassandspinfromspotimagesandlightcurvesoftheob-servedfluxandpolarization.Atradiowavelengths,diskopacityproducessignificantdeparturesfromtheinfraredbehaviour,buttherearestillgenericsignaturesoftheblackholeproperties.Detailedcomparisonoftheseresultswithfuturedatacanbeusedtotestgeneralrelativityandtoimproveexistingmodelsfortheaccretionflowintheimmediatevicinityoftheblackhole.

Keywords:blackholephysics,Galaxy:centre,infrared:general,submillimetre,techniques:interferometric,polarization

1INTRODUCTION

Testinggeneralrelativityinaregimewherespacetimecur-vatureislargeremainsoneoftheprimarygoalsofobser-vationalastronomy.Blackholesprovideanaturaltargetinwhichtofocustheseefforts.Nonetheless,duetotheirextremelycompactnature,anunambiguoussignatureofstronggravityhasbeenelusivesofar.

Anumberofresearchershavesuggestedthatimaginganoptically-thinbackgroundaccretionflowcouldprovideadirecttestofstronggravity(see,e.g.,Falckeetal.2000;Takahashi2005,2004;Broderick&Loeb2005a).Theangu-larscaleoftheblackholeintheGalacticcentre(identifiedwiththeradiosourceSgrA*)is5-10µas,twiceaslargeasthenuclearblackholeinM87,andsubstantiallylargerthanallothercandidates.Asaresult,SgrA*isthemostpromisingcandidateforhighresolutionimaging.WithinthenextdecadeitisexpectedthataVeryLongBaselineArray(VLBA)ofsub-millimetretelescopeswillexist,providing20µasresolutionimagingcapabilities(Doeleman&Bower2004;Miyoshietal.2004).Thus,theoreticaleffortstocom-putetheimagesofrealisticaccretionflowsiswarranted.In

⋆

E-mail:abroderick@cfa.harvard.edu†E-mail:aloeb@cfa.harvard.educ0000RAS󰀄

Broderick&Loeb(2005a)itwasshownthatforatypical

accretionmodel,opacitywillbesignificantatradiowave-lengths,substantiallyalteringtheshapeandvisibilityoftheblackhole“shadow”castbytheblackhole’sphotoncapturecross-section.

Multi-wavelengthpolarizationobservationsofaccret-ingblackholeshavealsobeenproposedasamethodofprobingtheblackholevicinity(Broderick&Loeb2005a;Connorsetal.1980;Laoretal.1990).Foroptically-thinbutgeometrically-thickaccretionflows,asexpectedinlowlu-minositygalacticnucleisuchasourGalacticcentre,itwasshownbyBroderick&Loeb(2005a)thatcomparisonsofthepolarizationnearandabovetheopticallythick/thintransi-tionfrequencyisindicativeoftheblackholespin.

However,observationsofnear-infrared(NIR)andX-rayflaringofSgrA*haveimpliedthattheinnerregionsoftheaccretionflowarenonuniform(Baganoffetal.2001;Genzeletal.2003;Goldwurmetal.2003;Eckartetal.2004;Ghezetal.2004).Thetimescaleofthevariability,∼10min,iscomparabletotheperiodoftheinnermoststablecircularorbit(ISCO),andthussuggestiveofanor-bitinghotspot.Therefore,itislikelythatmodelingofSgrA*imageswillrequiretheinclusionofvariability.InBroderick&Loeb(2005b)itwasshownthattheimagesandlightcurvesassociatedwithahotspotcouldbeused

2AveryE.Broderick&AbrahamLoeb

tomeasurethemassandspinoftheblackhole.ThereareadditionalplanstousethePhaseReferencedImagingandAstrometry(PRIMA)instrument,currentlyunderconstruc-tion,attheVeryLargeTelescope(VLT)toobtainsub-milli-arcsecondastrometryoftheseflares(Paumardetal.2005).

HereweimproveuponthecalculationsofBroderick&Loeb(2005b)bymakinguseofamorerealistichotspotmodelconsistingofalocalizedpopulationofnon-thermalelectrons(whichareproduced,forexample,byamagneticreconnectionflare).Inaddition,weincludetheeffectsofdiskopacityforatypicalradiativelyinefficientaccretionflowmodel.Theopacitysystematicallydecreasesthemagnificationandpolarizationfraction,smoothesoutthepolarizationanglelightcurve,circularizesthemotionofthecentroid,andsymmetrizesthetimeaveragedspotimages.

Section2presentsasummaryofthecomputationalmethodsusedandthemodelsfortheaccretiondiskandhotspot.Sections3and5containthelightcurvesandcentroidpathsforNIRandsub-millimetrefrequencies,respectively.Theeffectsofopacityarehighlightedinsection4.Finally,section6summarizesourconclusions.

Inwhatfollows,themetricsignatureistakentobe(−+++),andgeometrisedunitesareused(G=c=1).

2COMPUTATIONALMETHODS2.1

RayTracing&RadiativeTransfer

ThemethodbywhichthelightrayswereproducedandtheradiativetransferperformedisdiscussedatlengthinBroderick&Blandford(2003)(forwhichtracingnullgeodesicsisalimitingcase)andBroderick&Blandford(2004)(seealsoBroderick2005),respectively.Assuch,onlyabriefsummaryispresentedhere.

Nullgeodesicsareconstructedbyintegratingtheequa-tions

dxµ

dλ=−f(r)

󰀇1∂xµ󰀈,kα

(1)

wherethepartialdifferentiationistakenholdingthecovari-antcomponentsofthewavefour-vector,kµ,constant.Thefunctionf(r)isarbitrary(correspondingtothefreedomin-herentintheaffineparameterization,λ)andchosentobe

f(r)=r

2

󰀆

r

,(2)

(whererhisthehorizonradius)inordertoregularizetheaffineparameternearthehorizon.Broderick&Blandford(2003)haveexplicitlydemonstratedthatthisproceduredoesproducethenullgeodesics.

PolarizedradiativetransferincurvedspacetimeismosteasilyperformedbyintegratingtheBoltzmannequation(Lindquist1966;Broderick2005).Inthiscase,itisthephotondistributionfunctionNν∝Iν/ν3thatisevolved.Inthecaseofpolarizedradiativetransfer,itispossibletodefinecovariantanaloguesoftheStokesparameters,

Nν=(Nν,NνQ,NνU,NνV

)(Broderick&Blandford2004).In

03×1071.7×10118×104−2.90.53×1071.4×10115×104−2.80.998

1×1071.5×10111×105

−2.8

dλ

=¯jν

−α¯νNν,(3)

where¯jν

andα¯νareemissivityandabsorptioncoefficients(toseehowthesearerelatedtotheirstandarddefinitions,seeBroderick&Blandford2004;Lindquist1966).Thetwosynchrotronemissioncomponentsconsideredhereinvolvepopulationsofelectronswithathermal(Yuanetal.2003)andpower-law(Jones&O’Dell1977)distributions.Becauseweareprimarilyconcernedwithemissionatsub-millimetreorshorterwavelengths,Faradayrotationandconversionareunlikelytobeimportant,andarethusneglected.2.2

DiskModels

WeusethesamethreeaccretionflowmodelsasdescribedinBroderick&Loeb(2005b).MotivatedbyYuanetal.(2003),whoshowedthatthevertically-averagedelectrondensityandtemperatureareapproximatelypower-lawsinradius,wewritethethermalelectrondensity,ne,tempera-tureTe,andthenon-thermalelectrondensitynnthas

n=n0

e

󰀄ρenn0

󰀅−0.84

nth=nth

󰀄M(4)

ρ2R2

S

󰀂

(5)

󰀄

c0000RAS,MNRAS000,000–000ImagingSpotsinBlackHoleAccretionFlows3

Figure1.Thespectralfluxforthenon-rotating(a=0,solid),moderatelyrotating(a=0.5,long-dashed),andmaximallyrotat-ing(a=0.998,short-dashed)SgrA*diskmodelsasviewedfrom45◦degreesabovetheequatorialplane.Thethicklinescorrespondtothequiescentdiskemissionandthethinlinescorrespondtothepeakflarefluxforaspotlocatedatanorbitalradiusof6M.ThedatapointsaretakenfromthecompilationbyYuanetal.(2004).

µ

where∆rµ≡rµ−rSisthedisplacementfromthespot

µ0

centre(locatedatrS),uµSisthespotfour-velocity,nSisthespotcentralnumberdensity,andRSisameasureofthespotradius.SothatourresultsmaybeeasilycomparedwithBroderick&Loeb(2005b),weusedRS=1.5M.ThespotcentralnumberdensitywaschosentoroughlyreproducetheobservedpeakNIRflareflux.AssuggestedbycoincidentNIR–X-rayobservations,thehotspotspectralindexwasassumedtobe1.3(Eckartetal.2004),andtheminimumLorentzfactorwasassumedtobe100.Suchasituationmayarise,e.g.,inthecaseofreconnectioneventsimilartoaso-larflare.Thespectrumofthediskandourcanonicalflaremodel(atanorbitalradiusof6M,viewedfrom45◦abovetheequatorialplane)areshowninFigure1.Below100GHzthediskphotosphereextendsbeyondthehotspot,quench-ingitscontributiontotheoverallflux.Above100GHzthehotspotbecomesincreasinglyvisible,reachingamaximumnear500GHz.

FortheGalacticcentre,theorbitaltimescaleneartheISCOarebetween5minand30minfororbitsaroundmax-imallyandnon-rotatingblackholes,respectively.Thevia-bilityofahot-spotmodelfortheflaringinSgrA*requiresthatthesynchrotroncoolingtimescalebelargeincompari-sontotheorbitalperiod.Sincetheemissionatafrequencyν󰀃isdominatedbyelectronswithLorentzfactorγe=

103GHz

󰀅−1/2󰀇B

4AveryE.Broderick&AbrahamLoeb

Figure3.Thebackgroundsubtractedtotalandpolarizedflux(bottomandmiddlepanels,respectively)normalizedbytheaver-agebackgroundsubtractedtotalfluxintheoptically-thinregimeasfunctionsoftimeforaspotorbitat6MaroundaSchwarzschildblackholeviewedfrom0◦(solid),22.5◦(dash-dot),45◦(long-dash)and67.5◦(short-dash)abovetheequatorialplane.Thepo-larizationangle,ψ,isshowninthetoppanel.Forablackholemassof4×106M⊙(asinSgrA*),thetimeunitisM=20s.

shouldbenotedthatthepolarizationvariabilitywillbedif-ferentfromthatfoundinBroderick&Loeb(2005b)owingtothedifferenceinthepolarizedemissionmodels(theoneutilizedherebeingthemorephysicallymotivated).Nonethe-less,thegenericfeaturesofaprimaryminimumcausedbythedevelopmentofthesecondaryhot-spotimage(immedi-atelyprecedingmaximumunpolarizedmagnification)andaweakersecondaryminimumcausedbythedevelopmentofatertiaryimage(followingthemaximumunpolarizedmagni-ficationby60M)exist.Inthiscasethepolarizationangleispunctuatedbyrapidrotations(∼90◦)attheprimarymin-imum,anduneventfulotherwise,rotatingasexpectedforatoroidalfield.

Characteristically,thepeakpolarizedfluxfollowsthepeakmagnification.Inthecanonicalcaseofanspotorbitalradiusof6Mviewedfrom45◦abovetheorbitalplane,thiscorrespondstoadelayofapproximately5mininthecontextoftheGalacticcentre.

Again,variationsinviewinginclination,ϑ,producessimilareffectsasthosereportedinBroderick&Loeb(2005b),theprimarydistinctionbeingthetotalmagnifi-cation.Therearestructuralchangesinthepolarizedflux,indicativeoftheprominenceoftheprimaryandsecondaryhot-spotimages.UnlikeBroderick&Loeb(2005b),intheedge-onview(ϑ=0◦),stronggravitationallensingproducesapolarizationcomponentorthogonaltodominantpolariza-tion,leadingtoarapidrotationinpolarizationanglepreced-ingpeakmagnificationbyapproximatelyaquarter-phase.

ThemanifestationofblackholespininthelightcurvesisshowninFigure4.AsinBroderick&Loeb(2005b),thedifferencesamongthemagnificationlightcurvesissmallincomparisontothoseassociatedwithvariationsoftheorbital

Figure4.Thebackgroundsubtractedtotalandpolarizedflux(bottomandmiddlepanels,respectively)normalizedbytheaver-agebackgroundsubtractedtotalfluxintheoptically-thinregimeasfunctionsoftimeforaspotorbitat6MaroundaKerrblackholewitha=0(solid),a=0.5(long-dash)anda=0.998(short-dash)viewedfrom45◦abovetheequatorialplane.Thepolariza-tionangle,ψ,isshowninthetoppanel.Thetimeaxisissetsothatasingleorbitalperiodofthea=0caseisshown.Forablackholemassof4×106M⊙(asinSgrA*),thetimeunitisM=20s.

parameters.Whilethesecondaryminimumofthepolariza-tiondoesvarysignificantlywithspin,inthetotalfluxthisisdifficulttoobserve.

However,theprimarydistinctionbetweendifferentspinsislikelytobethevariationsintheradiusoftheISCO,andthusthetypicalperiods.Figure5showsthemagnifica-tionandpolarizationlightcurvesforaspotviewedfrom45◦abovetheequatorialplanelocatedattheprogradeISCOforvariousdimensionlessspinparameters(normalizedbyM),namely6M,4.233Mand1.237Mfora=0,0.5and0.998,respectively.Becausetheorbitaltimescalesvarybynearlyanorderofmagnitudebetweena=0anda=0.998,theseareplottedasfunctionsoforbitalphase.Rapidrota-tionsubstantiallyreducesthemaximumpolarizedflux,pri-marilyduetotheenhancedgravitationallensingassociatedwiththecompactnessoftheorbit.However,asmentionedinBroderick&Loeb(2005b),andalludedtoabove,themostsignificantdiscriminatorislikelytobetheeventtimescale.3.2

CentroidPaths

ThePRIMAinstrumentattheVLTisexpectedtoprovidesub-milli-arcsecondastrometry,thusenablinghighresolu-tionmeasurementsofthetheinfraredimagecentroidforSgrA*andinturnconstraintheaccretionflowandblackholeparameters.Becausethelocationoftheimagecentroidswillbedominatedbythebrightestfeatures,subtractingthebackgroundaccretionflowemissionisnecessarytoisolatethelocationofthehotspotduringaflare.However,intrin-sicvariabilityinthesourceand/orthepresenceofmultiplehotspotsmayintroducesubstantialsystematicerrorsinto

󰀄

c0000RAS,MNRAS000,000–000ImagingSpotsinBlackHoleAccretionFlows5

Figure5.Thebackgroundsubtractedtotalandpolarizedflux(bottomandmiddlepanels,respectively)normalizedbytheaver-agebackgroundsubtractedtotalfluxintheoptically-thinregimeasfunctionsofphaseforspotorbitsattheprogradeISCOaroundaKerrblackholewitha=0(solid),a=0.5(long-dash)anda=0.998(short-dash)viewedfrom45◦abovetheequatorialplane.Thepolarizationangle,ψ,isshowninthetoppanel.Thetimeaxisissetsothatasingleorbitalperiodofthea=0caseisshown.Forablackholemassof4×106M⊙(asinSgrA*),thetimeunitisM=20s.

thisprocess.Nonetheless,inordertoclearlyremovetheun-certaintyassociatedwiththeaccretiondiskmodel,incom-putingallofthecentroidpositionsbelowwehaveusedthebackgroundsubtractedimages.Thesemaybegeneratedusingthecentroidsofthetotalemissionifthequiescentcentroidpositionandintensityareknown:

X(t)−FBGF(t)−FBG

,

(7)

X(t)andwhere

F(t)arethelocationofthetotalimagecentroidandthetotalobservedflux,and

6AveryE.Broderick&AbrahamLoeb

Figure8.Thepathsofthebackgroundsubtractedintensitycen-troidintheoptically-thinregimeforcircularspotorbitswithra-dius6MintheequatorialplanearoundaKerrblackholeviewedfrom45◦abovetheorbitalplanefora=0(solid),a=0.5(long-dash)anda=0.998(short-dash).Forreference,acircleinclinedat45◦isalsoshownbythedottedline.AxesarelabeledinunitsofM(correspondingtoanangularscaleofroughly5µasforSgrA*).

Figure9.Thepathsofthebackgroundsubtractedintensitycen-troidintheoptically-thinregimeforcircularspotorbitsaroundaKerrblackholeviewedfrom45◦abovetheorbitalplaneattheprogradeISCOfora=0(solid),a=0.5(long-dash)anda=0.998(short-dash).Forreference,acircleinclinedat45◦isalsoshownbythedottedlinesforspin.AxesarelabeledinunitsofM(correspondingtoanangularscaleofroughly5µasforSgrA*).

eachblackholespin,againviewedfrom45◦abovetheorbitalplane.Herethecentroidpathathighspin,wheretheISCOisclosetothehorizon,issubstantiallyoffset.However,inthecaseofmoderatespin,thegenericdependenceuponradiusseeninFigure6isdominant.

4GENERICEFFECTSOFOPACITY

WhileintheNIRthediskopacitytosynchrotronself-absorptionisnegligible,thisisnotthecaseatsub-millimeterwavelengths,atwhichtheaccretionflowisonlybeginningtobeopticallythin.AsdiscussedinBroderick&Loeb(2005a),theaccretionflowopacityisnotsymmetricbetweenthetwosidesofthedisk,andisenhancedbytheDopplereffectontheapproachingsideofthedisk.Asaresult,thelightcurvesandcentroidpathspresentedintheprevioussectionfortheNIRcanbesubstantiallymodifiedforfrequenciesneartheopticallythick/thintransition.Forthepurposeofhighlight-ingthegenericeffectsofopacity,andtheutilityofhigh-resolutionmulti-wavelengthflareobservations,wepresentnextacomparisonofthelightcurvesandcentroidpathsforNIRandradiofrequencies.4.1

TimeAveragedImages

Theeffectsofasymmetricopacityarereadilyapparentintheorbit-averagedimages,showninFigure10forthecanonicalhot-spotmodelforthethreediskmodelsdiscussedinSec-tion2.2,asobservedat230GHz,350GHzandintheNIR.IntheNIRboththedirectandsecondaryimagesareclearlyvisible,correspondingtothelargerandsmallerrings,respec-tively.Lessvisiblearethetertiaryimages,whichproducetheverythinringdirectlyinsidethatassociatedwiththesec-ondaryimages.Inaddition,asexpectedfromtheDopplershiftandrelativisticbeaming,theapproachingportionofthehot-spotorbit(leftside)appearsconsiderablybrighterthantherecedingportionofthespotorbit(rightside).Atradiofrequenciesthesecondaryimageissubstantiallylessvisibleandthebrightnesscontrastbetweentheapproachingandrecedingportionsofthehot-spotorbitdecreaseswithdecreasingfrequency.Thisisadirectresultoftheincreasedopacityintheapproachingportionofthedisk(alsoontheleftside).AsmentionedinBroderick&Loeb(2005b),inthelimitofathermalspectrumtheorbit-averagedimagesaresymmetric,andthusitisnotsurprisingthatincreasedopacityresultsinmoresymmetricimages.

AlsoshowninFigure10isthedegreeandorientationofthepolarizedflux.Inallcasesthisisdominatedbytheapproachingportionoftheorbit.IntheNIR,asecondpo-larizationcomponentispresentduetothesecondaryim-age.Thisissubstantiallysuppressedatradiowavelengths,andthusthestrongswingsinpolarizationanglearenotex-pectedintheradio.Finally,thedegreeofpolarizationisalsoexpectedtodecreaseasaresultofthedecreaseinthebrightnessasymmetrymentionedbefore.4.2

LightCurves

Figures11–13shownthemagnificationandpolarizedfluxlightcurvesintheradioandNIRforahot-spotorbitingana=0,0.5and0.998blackhole,respectively.Theprimary

󰀄

c0000RAS,MNRAS000,000–000ImagingSpotsinBlackHoleAccretionFlows7

Figure10.Comparisonoftheorbit-averaged,disk-subtractedimagesofaspotfortworadiofrequencies(atwhichopacityisimportant)andintheinfrared(atwhichthediskistransparent).Forreference,thephoton-capturecrosssectionforaSchwarzschildblackholeisshownbythedashedwhiteline.

distinctionbetweenthedifferentobservingwavelengthsisinthedegreeofpolarization,decreasingbyroughlyanorderofmagnitudefromtheNIRto230GHz.Inaddition,asex-pectedfromtheimages,thepolarizationanglelightcurveisprogressivelysmoothedastheobservingwavelengthin-creases.Itshouldalsobenotedthatthesecondaryminimuminthepolarization,alsoresultingfromthedevelopmentoftertiaryimages,issignificantlylessvisibleintheradio.

4.3CentroidPaths

ThecentroidpathsfordifferentobservingfrequenciesarecomparedinFigures14–16.Astheobservingfrequencyde-creasesthecentroidpathbecomeslargerandmoresymmet-ric.Asaresult,theorbitalparametersaremoreeasilydeter-minedatradiowavelengths.Thisimpliesthatsimultaneoushigh-resolutionradioandNIRobservationsofthecentroidarecapableofidentifyingblackholespin.

c0000RAS,MNRAS000,000–000󰀄

8AveryE.Broderick&AbrahamLoeb

Figure11.Thebackgroundsubtractedtotalandpolarizedflux(bottomandmiddlepanels,respectively)normalizedbytheav-eragebackgroundsubtractedtotalfluxasfunctionsoftimeforaspotorbitat6MaroundaSchwarzschildblackholeviewedfrom45◦abovetheequatorialplaneasobservedintheinfrared(solid),350GHz(long-dash)and230GHz(short-dash).Thepolarizationangle,ψ,isshowninthetoppanel.Thetimeaxisissetsothatasingleorbitalperiodofthea=0caseisshown.Forablackholemassof4×106M⊙(asinSgrA*),thetimeunitisM=20s.

Figure12.Thebackgroundsubtractedtotalandpolarizedflux(bottomandmiddlepanels,respectively)normalizedbytheav-eragebackgroundsubtractedtotalfluxasfunctionsoftimeforaspotorbitat6MaroundaKerrblackhole(a=0.5)viewedfrom45◦abovetheequatorialplaneasobservedintheinfrared(solid),350GHz(long-dash)and230GHz(short-dash).Thepolarizationangle,ψ,isshowninthetoppanel.Thetimeaxisissetsothatasingleorbitalperiodofthea=0caseisshown.Forablackholemassof4×106M⊙(asinSgrA*),thetimeunitisM=20s.

Figure13.Thebackgroundsubtractedtotalandpolarizedflux(bottomandmiddlepanels,respectively)normalizedbytheav-eragebackgroundsubtractedtotalfluxasfunctionsoftimeforaspotorbitat6MaroundaKerrblackhole(a=0.998)viewedfrom45◦abovetheequatorialplaneasobservedintheinfrared(solid),350GHz(long-dash)and230GHz(short-dash).Thepo-larizationangle,ψ,isshowninthetoppanel.Thetimeaxisissetsothatasingleorbitalperiodofthea=0caseisshown.Forablackholemassof4×106M⊙(asinSgrA*),thetimeunitisM=20s.

Figure14.ThepathsofthebackgroundsubtractedintensitycentroidforcircularspotorbitsaroundaSchwarzschildblackholeviewedfrom45◦abovetheorbitalplaneat6Masobservedintheinfrared(solid),350GHz(long-dash)and230GHz(short-dash).Forreference,acircleinclinedat45◦isalsoshownbythedottedlinesforeachradius.AxesarelabeledinunitsofM(correspond-ingtoanangularscaleofroughly5µasforSgrA*).

󰀄

c0000RAS,MNRAS000,000–000ImagingSpotsinBlackHoleAccretionFlows9

Figure15.ThepathsofthebackgroundsubtractedintensitycentroidforcircularspotorbitsaroundaKerrblackhole(a=0.5)viewedfrom45◦abovetheorbitalplaneat6Masobservedintheinfrared(solid),350GHz(long-dash)and230GHz(short-dash).Forreference,acircleinclinedat45◦isalsoshownbythedottedlinesforeachradius.AxesarelabeledinunitsofM(correspondingtoanangularscaleofroughly5µasforSgrA*).

Figure16.ThepathsofthebackgroundsubtractedintensitycentroidforcircularspotorbitsaroundaKerrblackhole(a=0.998)viewedfrom45◦abovetheorbitalplaneat6Masobservedintheinfrared(solid),350GHz(long-dash)and230GHz(short-dash).Forreference,acircleinclinedat45◦isalsoshownbythedottedlinesforeachradius.AxesarelabeledinunitsofM(correspondingtoanangularscaleofroughly5µasforSgrA*).

󰀄

c0000RAS,MNRAS000,000–000Figure17.Thebackgroundsubtractedtotalandpolarizedflux(bottomandmiddlepanels,respectively)normalizedbytheav-eragebackgroundsubtractedtotalfluxasfunctionsoftimeat350GHzforspotorbitsat6M(solid),8M(long-dash)and10M(short-dash)aroundaSchwarzschildblackhole,viewedfrom45◦abovetheequatorialplane.Thepolarizationangle,ψ,isshowninthetoppanel.Thetimeaxisissetsothatasingleorbitalperiodofthe10Mcaseisshown.Forablackholemassof4×106M⊙(asinSgrA*),thetimeunitisM=20s.

5RADIO

Anticipatinghigh-resolutionsub-millimetreobservationsoftheGalacticcentre,wepresentthemagnificationandpo-larizedfluxlightcurvesandimagecentroidsatradiofre-quencies.Asdiscussedintheprevioussection,thesemaybeexpectedtobequantitativelydifferentfromthosepresentedinSection3.

5.1LightCurves

Figure17showsthatat350GHztheabsorptionhaslittleef-fectuponthemagnificationlightcurvesformoderateorbitalinclinations(comparetoFigure2).However,thepeakpolar-izedfluxisreducedbymorethan70%andobservedearlierthanintheNIR.Whilethepolarizationanglecontinuestoexhibitarapidrotationassociatedwiththedevelopmentofasecondaryimage,itissignificantlysmoothed.

ThisisespeciallyapparentinFigure18,whichshowsthelightcurvesforanumberofviewinginclinations,ϑ,(comparetoFigure3).Inthecasewhereϑ=0◦,theemer-genceofanorthogonalpolarizationissuppressed,leadingtoaconstantpolarizationangle.Forsmallϑ,thepeakmagnifi-cationisalsomarginallyreduced,asmaybeexpectedfromthelargercolumndensityforsuchviewingangles.Never-theless,theflaretime-scaleandpeakmagnificationarestillindicativeoftheorbitalradiusandinclination.Furthermore,thepresenceofaflareinthepolarizedfluxisgeneric.

Figures19and20showthemagnificationandpolar-izedfluxlightcurvesforvariousblackholespinsasseenat350GHzand230GHz,respectively(comparetoFigure

10AveryE.Broderick&AbrahamLoeb

Figure18.Thebackgroundsubtractedtotalandpolarizedflux(bottomandmiddlepanels,respectively)normalizedbytheav-eragebackgroundsubtractedtotalfluxasfunctionsoftimeat350GHzforaspotorbitat6MaroundaSchwarzschildblackholeviewedfrom0◦(solid),22.5◦(dash-dot),45◦(long-dash)and67.5◦(short-dash)abovetheequatorialplane.Thepolariza-tionangle,ψ,isshowninthetoppanel.Forablackholemassof4×106M⊙(asinSgrA*),thetimeunitisM=20s.

Figure19.Thebackgroundsubtractedtotalandpolarizedflux(bottomandmiddlepanels,respectively)normalizedbytheav-eragebackgroundsubtractedtotalfluxasfunctionsoftimeat350GHzforaspotorbitat6MaroundaKerrblackholewitha=0(solid),a=0.5(long-dash)anda=0.998(short-dash)viewedfrom45◦abovetheequatorialplane.Thepolarizationan-gle,ψ,isshowninthetoppanel.Thetimeaxisissetsothatasingleorbitalperiodofthea=0caseisshown.Forablackholemassof4×106M⊙(asinSgrA*),thetimeunitisM=20s.

Figure20.Thebackgroundsubtractedtotalandpolarizedflux(bottomandmiddlepanels,respectively)normalizedbytheav-eragebackgroundsubtractedtotalfluxasfunctionsoftimeat230GHzforaspotorbitat6MaroundaKerrblackholewitha=0(solid),a=0.5(long-dash)anda=0.998(short-dash)viewedfrom45◦abovetheequatorialplane.Thepolarizationan-gle,ψ,isshowninthetoppanel.Thetimeaxisissetsothatasingleorbitalperiodofthea=0caseisshown.Forablackholemassof4×106M⊙(asinSgrA*),thetimeunitisM=20s.

4).Typically,thereislittledifferenceamongthemagnifica-tionlightcurves.However,inbothcasesthepolarizedfluxclearlydiscriminatesbetweenhighandmoderate/lowblackholespins,varyingbyroughlyafactoroftwoinbothcases.Thisislikelyduetothelowerthermalelectrondensityper-missibleinthehighspinmodelemployed(seeTable1).

Asintheoptically-thinregimedescribedinSection3.1,theprimarydifferencebetweenhighandlowspinblackholesislikelytobetheflaringtimescales.SimilarlytoFigure5,Figure21showsthemagnificationandpolarizedfluxlightcurvesasfunctionsoforbitalphase.AsintheNIR,thepo-larizationismoresensitivetotheblackholespinthanthedegreeofmagnification.Theopacityassociatedwiththehot-spotorbitingthemaximallyrotatingblackholeissignif-icantlylargerdueprimarilytotherapiddiskvelocityatthisposition.Asaresult,thismodelwassignificantlymoreab-sorbedthantheothers.Combinedwithstronggravitationallensing,thiseffectproducesavariationinthefluxthatisunpolarizedandsmallincomparisonwiththequiescentdiskemission.5.2

CentroidPaths

Next,weconsiderthepathsofthebackgroundsubtractedim-agecentroids.WhileintheopticallythinregimedescribedinSection3.2,thediskandhot-spotemissionarecompletelydisentangled,inthiscasethediskopacitycansubstantiallymodifytheimagecentroids.Inthefrequencyregimeoftheopticallythick/thintransition,opacityismostsignificantontheapproachingsideoftheaccretionflow,nearwherethehot-spotisthebrightest.Thishastheeffectofreducingthe

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c0000RAS,MNRAS000,000–000ImagingSpotsinBlackHoleAccretionFlows11

Figure21.Thebackgroundsubtractedtotalandpolarizedflux(bottomandmiddlepanels,respectively)normalizedbytheav-eragebackgroundsubtractedtotalfluxasfunctionsofphaseat350GHzforspotorbitsattheprogradeISCOaroundaKerrblackholewitha=0(solid),a=0.5(long-dash)anda=0.998(short-dash)viewedfrom45◦abovetheequatorialplane.Thepo-larizationangle,ψ,isshowninthetoppanel.Thetimeaxisissetsothatasingleorbitalperiodofthea=0caseisshown.Forablackholemassof4×106M⊙(asinSgrA*),thetimeunitisM=20s.

brightnesscontrastbetweentheapproachingandrecedingportionsofthehot-spotorbit.Sincethebrightestregionsdominatethelocationoftheimagecentroid,thishassignif-icantimplicationsforthecentroidpaths.

Figure22showsthecentroidpathsforhot-spotsorbit-ingatthreeradii,viewedfrom45◦abovetheorbitalplane.ThisissimilartoFigure12inBroderick&Loeb(2005b),primarilyduetothereducedbrightnesscontrastmentionedabove.TheorbitsarenoticeablylargerthanthoseinFigure6,thoughthemajoraxisdistanceissimilar.

VariousviewinginclinationsareshowninFigure23(cf.,Figure7).AsinFigure22,thecentroidpathsarelargerthanthosefoundintheNIR.Forlowϑ,thissubstantiallymodifiestheshapeofthecentroidpath.Nonetheless,thenearestregionoftheorbit(bottom)isclosetotheunlensedorbitpositions(shownbythedottedlines),andthusthesemi-minoraxisisstilldiagnosticoftheorbitalinclination.

Figures24and25showthecentroidpathsforvariousblackholespinsasviewedat350GHzand230GHz,respec-tively(cf.,Figure8).InallcasesthepathsarelargerthanintheNIRand,unliketheNIR,arelargerthanexpectedintheabsenceofgravitationalandopacityeffects(shownbythedottedline).Inallcases,thesemi-minoraxisagainappearstobeagoodmeasureoftheorbitalinclination.

ThecentroidpathsfororbitsattheISCOofthethreeblackholespinsweconsideredareshowninFigure26.AsintheNIR,theinnerorbitsaresubstantiallymodifiedbystronglensing,andinthiscase,opacity.However,asmen-tionedpreviously,thefluxvariationassociatedwiththe

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c0000RAS,MNRAS000,000–000Figure22.Thepathsofthebackgroundsubtractedinten-sitycentroidat350GHzforcircularspotorbitsaroundaSchwarzschildblackholeviewedfrom45◦abovetheorbitalplanewithradii6M(solid),8M(long-dash)and10M(shortdash).Forreference,acircleinclinedat45◦isalsoshownbythedottedlinesforeachradius.AxesarelabeledinunitsofM(correspondingtoanangularscaleofroughly5µasforSgrA*).

Figure23.Thepathsofthebackgroundsubtractedinten-sitycentroidat350GHzforcircularspotorbitsarounda

Schwarzschildblackholewithradius6Mviewedfrom0◦(solid),22.5◦(dash-dot),45◦(long-dash)and67.5◦(short-dash)abovetheorbitalplane.Forreference,acirclesinclinedateachangleareshownbythedottedlines.AxesarelabeledinunitsofM(correspondingtoanangularscaleofroughly5µasforSgrA*).

12AveryE.Broderick&AbrahamLoeb

Figure24.Thepathsofthebackgroundsubtractedintensitycentroidat350GHzforcircularspotorbitswithradius6MintheequatorialplanearoundaKerrblackholeviewedfrom45◦abovetheorbitalplanefora=0(solid),a=0.5(long-dash)anda=0.998(short-dash).Forreference,acircleinclinedat45◦isalsoshownbythedottedline.AxesarelabeledinunitsofM(correspondingtoanangularscaleofroughly5µasforSgrA*).Figure25.Thepathsofthebackgroundsubtractedintensitycentroidat230GHzforcircularspotorbitswithradius6MintheequatorialplanearoundaKerrblackholeviewedfrom45◦abovetheorbitalplanefora=0(solid),a=0.5(long-dash)anda=0.998(short-dash).Forreference,acircleinclinedat45◦isalsoshownbythedottedline.AxesarelabeledinunitsofM(correspondingtoanangularscaleofroughly5µasforSgrA*).

Figure26.Thepathsofthebackgroundsubtractedintensitycentroidat350GHzforcircularspotorbitsaroundaKerrblackholeviewedfrom45◦abovetheorbitalplaneattheprogradeISCOfora=0(solid),a=0.5(long-dash)anda=0.998(short-dash).Forreference,acircleinclinedat45◦isalsoshownbythedottedlinesforspin.AxesarelabeledinunitsofM(correspond-ingtoanangularscaleofroughly5µasforSgrA*).

high-spincaseissmall,andthusitisnotclearthatthecentroidpathwillbemeasurableinthiscase.

6CONCLUSIONS

ThephotontrappingradiusoftheblackholeinthecenteroftheMilky-Waygalaxyhasthelargestapparentsizesamongallknownblackholes,occupyingananglularscaleoftensofmicro-arcsecondsonthesky.Thisscalewillbewithinreachofforthcomingobservatories,suchasasub-millimetreVLBA(forimaging),orthePRIMAinstrumentontheVLT(formonitoringshiftsintheinfraredimagecentroidduringflares).Imaginghotspotsintheimmediatevicinityofanac-cretingblackholeprovidesanewmethodfortestinggeneralrelativityandmeasuringtheblackholemassandspin.

Theimageandcentroidshiftsofacompactpopulationofnon-thermalelectronsmaybeusedtoconstrainthemassandspinofthecentralblackhole.Atradiowavelengthssuchaspotwouldpersistformanyorbitallyperiods,ultimatelybeingshearedintoaring,whileintheNIRsuchaspotwouldcoolrapidly,survivingapproximatelyasingleorbit.

Atthesub-millimetrefrequenciescurrentlyproposedfortheVLBAimagingoftheGalacticcentre,theopacityoftheunderlyingsteadyaccretionflowhasasignificanteffectupontheimagesandcentroidmotions.Nonetheless,itre-mainspossibletoextracttheblackholemassandspinfromthese.Furthermore,combinedwiththeNIRdata(atwhichtheaccretionflowopacityisnegligble)thiswillprovideamethodbywhichtheaccretionflowmaybeisolated.

WhilepolarizationissignificantinboththeradioandNIRregimes,theenhancedopacityatradiowavelengthsreducesthepolarizationsignaturesofstronglensingthat

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c0000RAS,MNRAS000,000–000ImagingSpotsinBlackHoleAccretionFlows

aremostnoticeableforthepolarizationangleandpolarizedfluxintheNIR.Immediatelypriortomaximummagnifi-cation,rapidchangesintheorientationofthepolarizationtakeplaceintheNIR.Thesearesystematicallysmoothedintheradio,thoughthepolarizationdoesrotateasexpectedforatoroidalmagneticfield.

Wehaveusedsimplemodelsfortheunderlyingac-cretionflowandthehotspot.Futureworkcanimproveuponourresultsusingthree-dimensional,fullyrelativistic,magneto-hydrodynamicsimulationsoftheaccretionflow,inwhichhotspots(orflares)arisenaturallyasmagneticre-connectioneventsand/oratshocks.

13

”ESOAstrophysicsSymposia”,”Scientificprospectsforvltiinthegalacticcentre:Gettingtotheschwarzschildradius”inpress

TakahashiR.,2004,ApJ,611,996

TakahashiR.,2005,Publ.Astron.Soc.Japan,57,273YuanF.,QuataertE.,NarayanR.,2003,ApJ,598,301YuanF.,QuataertE.,NarayanR.,2004,ApJ,606,894

AThispaperhasbeentypesetfromaTEX/LTEXfileprepared

bytheauthor.

ACKNOWLEDGMENTS

ThisworkwassupportedinpartbyNASAgrantNAG5-13292andNNG05GH54G(forA.L.).A.E.B.gratefullyac-knowledgesthesupportofanITCFellowshipfromHarvardCollegeObservatory.

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