Double Compton and cyclo-synchrotron in super-Eddington discs, magnetized coronae and jets

25 Pages·2017·8.81 MB·English

MNRAS467,2241–2265(2017) doi:10.1093/mnras/stx227
Double Compton and cyclo-synchrotron in super-Eddington discs,
magnetized coronae and jets
Jonathan C. McKinney,1‹ Jens Chluba,2 Maciek Wielgus,3 Ramesh Narayan4
and Aleksander Sadowski5
1UniversityofMarylandatCollegePark,DepartmentofPhysics,JointSpace-ScienceInstitute,3114PhysicalSciencesComplex,
CollegePark,MD20742,USA
2JodrellBankCentreforAstrophysics,UniversityofManchester,OxfordRoad,ManchesterM139PL,UK
3CopernicusAstronomicalCenterofthePolishAcademyofSciences,ul.Bartycka18,PL00-716Warszawa,Poland D
4Harvard-SmithsonianCenterforAstrophysics,60GardenSt.,Cambridge,MA02134,USA ow
5MITKavliInstituteforAstrophysicsandSpaceResearch,77MassachusettsAve,Cambridge,MA02139,USA nlo
a
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Accepted2017January24.Received2017January5;inoriginalform2016August30 fro
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ABSTRACT ://a
c
BlackholeaccretiondiscsaccretingneartheEddingtonratearedominatedbybremsstrahlung a
d
e
cooling,butabovetheEddingtonrate,thedoubleComptonprocesscandominateinradiation- m
ic
dominatedregions,whilethecyclo-synchrotroncandominateinstronglymagnetizedregions .o
u
likeacoronaorajet.Wepresentanextensiontothegeneralrelativisticradiationmagnetohydro- p
.c
dynamiccodeHARMRADtoaccountforemissionandabsorptionbythermalcyclo-synchrotron, om
double Compton, bremsstrahlung, low-temperature OPAL opacities, as well as Thomson and /mn
Comptonscattering.TheHARMRADcodeandassociatedanalysisandvisualizationcodeshave ras
been made open-source and are publicly available at the github repository website. We ap- /artic
proximatetheradiationfieldasaBose–Einsteindistributionandevolveitusingtheradiation le
number–energy–momentum conservation equations in order to track photon hardening. We -ab
s
perform various simulations to study how these extensions affect the radiative properties of tra
c
magneticallyarresteddiscsaccretingatEddingtontosuper-Eddingtonrates.Wefindthatdou- t/4
bleComptondominatesbremsstrahlunginthediscwithinaradiusofr∼15r (gravitational 67
g /2
radii)athundredtimestheEddingtonaccretionrate,andwithinsmallerradiiatloweraccre- /2
2
4
tion rates. Double Compton and cyclo-synchrotron regulate radiation and gas temperatures 1
/2
inthecorona,whilecyclo-synchrotronregulatestemperaturesinthejet.Interestingly,asthe 9
6
2
accretion rate drops to Eddington, an optically thin corona develops whose gas temperature 4
3
of T ∼ 109K is ∼100 times higher than the disc’s blackbody temperature. Our results show 5 b
y
the importance of double Compton and synchrotron in super-Eddington discs, magnetized g
u
coronaeandjets. es
t o
n
Key words: accretion, accretion discs–black hole physics–MHD–radiation: dynamics– 0
5
galaxies:jets–X-rays:binaries. A
p
While DC has received attention in cosmology to determine dis- ril 2
1 INTRODUCTION tortions to the cosmic microwave background (Chluba & Sun- 019
Blackhole(BH)accretionflowsbecomeradiation-dominatedand yaev 2012; Chluba 2014) and been applied to gamma-ray bursts
geometrically thick once the luminosity L (cid:2) 0.3L , where L (Vurm,Lyubarsky&Piran2013;Be´gue´ &Pe’er2015),ithasre-
Edd Edd
≈1.3×1039[M/(10M(cid:4))]ergs−1istheEddingtonluminosityfor ceived little attention in accretion theory except by Poutanen &
solarmassM(cid:4)andBHmassM.Suchaccretionflowstendtohave Svensson(1996).
temperaturesof109(cid:2)T(cid:2)106K(Abramowiczetal.1988),where Forphotonnumberdensityn ,ionnumberdensityn,electron
γ i
bremsstrahlung(BR,free–free,ep→e(cid:6)p(cid:6)γ)isimportant. massm ,speedoflightc,Boltzmann’sconstantk andtemperature
e b
Double (radiative) Compton (DC, γ e→γ(cid:6)γ e(cid:6)) (Light- T,theratesofeachprocessshowthatDCdominatesfree–freeif
1 1 2
man 1981; Thorne 1981; Pozdnyakov, Sobol & Syunyaev 1983; (cid:2) (cid:3)
Svensson1984)candominatefree–freewhileregulatingtempera- nγ (cid:2)1.3 mec2 5/2 (1)
turessothatpairsremainsub-dominant(seefig.1inThorne1981). n k T
i b
(Thorne1981;Svensson1984),whichhasthecorrectionbySvens-
(cid:3)E-mail:[email protected] son(1984)of1.3insteadofthevalue0.1byThorne(1981).Inthis
(cid:7)C 2017TheAuthors
PublishedbyOxfordUniversityPressonbehalfoftheRoyalAstronomicalSociety
2242 J.C.McKinneyetal.
paper,wefindDCdominatesfree–freeforthemeanopacitywhen Thesephysicaleffectsareincludedwithinourgeneralrelativis-
T >7×107Kρ2/11, (2) tic radiative magnetohydrodynamics (GRRMHD) code HARMRAD
(McKinneyetal.2014)thatusestheM1closure.Similargeneral
whereρismassdensityincgsunits.ForBHX-raybinariesaccreting relativistic(GR;Fragileetal.2012;Takahashietal.2013;Sa¸dowski
at 10 times the Eddington rate, one has ρ ≈ 10−3gcm−3 giving etal.2014;Fragile,Olejar&Anninos2014;McKinneyetal.2014;
equalityatT≈2×107K,whilesimulationsshowthatT∼5×107– Ryan,Dolence&Gammie2015;Takahashietal.2016)andnon-GR
108K(Sadowski&Narayan2015a)indicatingthatDCdominates (Jiang,Stone&Davis2014b)schemeshavebeendeveloped.These
free–freeevenifnotyetincluded.ForsupermassiveBHsrelevant codeshavebeenusedtostudyBHX-raybinariesandtheirdiscs,
fortidaldisruptionevents(TDEs)accretingattheEddingtonrate, coronaeandjets(Jiang,Stone&Davis2014a;McKinney,Dai&
onehasρ∼10−9gcm−3,givingequalityatT≈1.4×106K,which Avara2015;Sadowski&Narayan2015a),growthof10solar-mass
isreachedwithintensofgravitationalradii. BH seeds (McKinney et al. 2014), low-luminosity active galactic
Still, it remains uncertain that which opacity dominates the nuclei(AGN;Sadowskietal.2017),high-luminosityAGN(Jiang,
mean flow behaviour. The Novikov–Thorne solution (Novikov & Davis&Stone2016),AGNfeedback(Sa¸dowskietal.2016a),TDEs D
o
Thorne1973)orslimdiscsolution(Abramowiczetal.1988)could (Sadowski & Narayan 2015b; Sa¸dowski et al. 2016b), ultralumi- w
n
beusedasaguidetohowimportantDCis,butwefindthatthetem- nous X-ray sources (ULXs; Narayan et al. 2016), massive stellar lo
a
peraturesensitivitymakestheconclusionsnotrobust–especially envelopes (Jiang et al. 2015), neutron star atmospheres (Wielgus d
e
aansdthmeasglinmetidcisficelsdosl.uAtiolsno,isunmliokdeififreede–bfyrewe,inDdCs,ivseerntitciarellystrduecpteunre- etaInl.S2e0c1t6io)n,e2t,cw. eoutlineourequationsofmotion;inSection3,we d from
dentupontheradiationtemperature fornon-relativisticelectrons, presentourGRRMHDsimulations,andinSection4,wesummarize h
so distinct gas and radiation temperatures are important to treat. ourresults.Fortheappendices:inAppendixA,wediscussouruse ttp
s
This motivates using simulations to consider both opacities with oftheBEdistribution,inAppendixB,wediscusshowwecompute ://a
c
separategasandradiationtemperatures. themeanopacities,inAppendixC,wediscussfree–freeandrelated a
d
In addition, cyclo-synchrotron (hereafter, synchrotron) is ex- low-temperatureopacities,inAppendixD,wediscusssynchrotron em
pected to be an important source of opacity or emission for opacities,andinAppendixE,wediscussComptonscatteringand ic
.o
low-luminosity accretion flows or in regions where non-thermal DCopacities. u
p
electrons are present. In super-Eddington accretion flows, one .c
o
expects synchrotron to be unimportant in the disc. However, m
2 EQUATIONS OF MOTION /m
the atmosphere above the disc (i.e. corona) and jet can con- n
tain a strong magnetic field at low densities where synchrotron Theconservationlawsare ras
cMancKdionmneiyna2t0e11(D).iInMaadtdteitoi,onC,emloatgtine&ticFalalbyiaanrre1s9te9d7;acUcrzedteionnskdyisc&s (ρuμ);μ =0, (3) /article
(MADs)aremuchmorestronglymagnetizedthanweaklymagne- (cid:4) (cid:5) -a
b
tizeddiscsthatareusuallyconsidered(Igumenshchev,Narayan& Tνμ ;μ =Gν, (4) stra
Abramowicz2003;Narayan,Igumenshchev&Abramowicz2003; c
Tchekhovskoy, Narayan & McKinney 2011). Magnetically sup- (cid:4)Rνμ(cid:5);μ =−Gν, (5) t/467
portedatmospherescanhardenthespectrumbyproducingamore /2
extended vertical disc (Blaes et al. 2006), and synchrotron can whereρ isthegasdensityinthecomovingfluidframe,uμ isthe /22
gas four-velocity as measured in the ‘lab frame’ and Tμ is the 4
provide an abundant source of low-energy soft photons that can ν 1
magnetohydrodynamics(MHD)stress–energytensorinthisframe, /2
undergoinverseComptonbeforebeingabsorbed.Sothemagnetic 9
(cid:2) (cid:3) 6
field strength can regulate photon hardening in disc atmospheres 1 24
andmagnetizedjets. Tνμ=(ρ+ug+pg+b2)uμuν+ pg+ 2b2 δνμ−bμbν. (6) 35 b
In this paper, we consider the competition between free– y
free, bound-free bound–bound opacities from OPAL (Iglesias & Rνμisthestress–energytensorofradiation,Gνistheradiativefour- gu
Rfloowgesr.sW1e99a6l)s,oDcCon,saidnedrssyenpcahrraotetroanbsionrpMtioAnDa-tnydpeemBiHssiaocncrmeteioann fpogracree,dreesscpreibcitnivgeltyh,ethinetienrtaecrtnioalnebneetrwgyeeanndgpasreasnsudrreaodfiathtieonga,suignatnhde est on
opacitiestofocusonemissionratesintheenergyandmomentum comovingframeandbμisthemagneticfieldfour-vector(Gammie, 05
equations,ratherthantheRosselandmeanthatonlyappliesinthe McKinney & To´th 2003). The magnetic pressure is pb = b2/2 Ap
diffusionlimitforthemomentumequation.Theabsorptionmean inourHeaviside–Lorentzunits.Theidealinductionequationand ril 2
handles the effect of irradiation and how the absorption of radia- entropyequationsarealsoused(McKinneyetal.2014).Weassume 01
tionisaffectedbythephotondistributionbeingdifferentthanthat collisions keep gas and electron temperatures similar (Sadowski 9
given by Planck at the local temperature (Sadowski et al. 2017). etal.2017).
Such effects lead to a significant correction to the mean opaci-
tiesandcanqualitativelychangetheoutcome(Hubeny,Burrows&
2.1 Radiativefour-force
Sudarsky2003).Additionally,weevolvethephotonnumbersepa-
ratelyfromthephotonenergy(Sadowski&Narayan2015a),with We use a covariant formalism for computing the interaction due
separatenumberandenergyopacities,inordertotrackphotonhard- to absorption, emission and scattering (McKinney et al. 2014;
eningduetoinverseComptonization(IC).Thephotondistribution Sa¸dowski et al. 2014) via a four-force between the gas and radi-
isassumedtobeaBose–Einstein(BE)distribution,butwithalow- ationof
(cid:4) (cid:5)
energytransitiontoPlanckwhenabsorptionisfasterthaninverse Gμ=−(κ +κ)Rμνu − κRαβu u +λ uμ, (7)
Compton.UnlikeaPlanckdistribution,aBEdistributioncanhandle a s ν s α β
howthephotondistributionevolvestowardsaWiendistributionin whereκ istheenergyabsorptionmeanopacityofanyemissionin
a
ascattering-dominatedregime. thefluidframe(inwhichionsandelectronsareisotropic)inunits
MNRAS467,2241–2265(2017)
DCandsynchrotroninsuper-Eddington flows 2243
ofinverselength,λisthefluid-frametotalenergydensitylossrate of opacity and how the temperature of radiation is evolved, and
before absorption (which includes changes in energy while con- weinvestigatehowtheopacityandradiationtemperatureevolution
servingphotonnumber)andκ isthefluid-frameenergyscattering affecttheaccretionflowbehaviourandproperties.
s
meanopacity.Self-absorptionistreateddirectlybyhavingseparate
totalabsorptionandtotalemission.
3.1 Diagnostics
2.2 Photonnumberevolution ForvariousquantitiesR,weconsidertime-averages([R]),spatial
t
If the radiation is Planckian, then the chemical potential μ = 0 averagesandtheirspatialdistributions.Diagnosticsarecomputed
fromsnapshotsproducedevery∼4r /cforr ≡GM/c2withgrav-
andtheradiationtemperatureisderivedfromonlythefluid-frame g g
radiation energy density via T = (E/a)1/4, where E = Rμνu u , itationalconstantG.
γ μ ν
whichcandifferfromthegastemperatureT orelectrontemperature
g
T . D
e o
A more general photon distribution, like our choice of a BE w
3.1.1 Fluxes n
distribution(seeAppendixA),canbeconsideredbysimultaneously lo
a
feovrolrvaidnigatitohneinsuomtrobpericdienntshiteyforafmpheowtoitnhsfionutrh-veerlaodciiatytiounμr)frvaimae(nr, Ttrhoemsatgrnesesticen(eErMgy)tteenrmsosr:Tνμ includesbothmatter(MA)andelec- ded fro
a(wnnhrduernμr˙e);tμ=he=n˙flunb˙iy,d-Lforarmenetznuinmvabreirandceen,siwtyitohfphotonsisn=nr(−uμru(μ8)) TTMEMAμνμν ==(bρ2u+μuuνg++ppbgδ)νuμμ−uνb+μbpν,gδνμ, (13) m https://a
r c
n˙ =−(κ c)n+λ , (9) Here,ugistheinternalenergydensityandpg=((cid:11)−1)ugistheideal ade
na n gaspressurewithadiabaticindex(cid:11).Thecontravariantfluid-frame m
foranumberabsorptionmeanopacityκna andtotalnumberemis- magneticfour-fieldisgivenbybμ,whichisrelatedtothelab-frame ic.o
seiqounatriaotneaλpnpbroexfoimreataebssothrpetiKono.mTphainseepthsoetoqnuantiuomnbmerosctonacsecruvraattieolyn ttihorneet-efineslodrvainadbμδμ=isBtνhheμνK/urot,nwechkeerredheμνlta=fuunμcutνio+n.δTνμhiesmaapgronjeetcic- up.co
ν m
whenthermalComptonization(TC)isincludedandwhengasand energydensity(ub)andpressure(pb)areub=pb=bμbμ/2=b2/2. /m
radiationtemperaturesaresimilar(Sadowski&Narayan2015a). Thetotalpressureisptot=pg+pb,andplasmaβ≡pg/pb. nra
Thegasrest-massflux,specificenergyfluxandspecificangular s
/a
2W.3e oObptaaicnitfiietssatondservaetreasl different energy-weighted and number- mM˙om=e(cid:6)(cid:6)(cid:6)(cid:6)n(cid:7)tumρuflrudxAaθφre(cid:6)(cid:6)(cid:6)(cid:6),,respectively.givenby (14) rticle-abs
weighted opacities (computed as in Appendix B), including OPAL (cid:8) trac
κmopaena,acenfift)iκeiansn,fAosyrpnsp)oeilnnadrAiaxpbCpuen,nsdydaninxccheDrso,(tearnonendrg(Ceynomemregpaytnomnκeas,acenfaftκatena,rdsiynnnguamn(sdbcenartutmmerebinaengr e≡ [ME˙˙]t =−(cid:8) (Ttr+[M˙R]trt)dAθφ, (15) t/467/2/2
energymeanκs,TCenergyexchangerateλc,andDCenergymean j ≡ J˙ = (Tφr+Rφr)dAθφ, (16) 241
opacityκa,dcandnumbermeanopacityκan,dc)inAppendixE. [M˙]t [M˙]t /29
Then,theenergymeanenergyabsorptionopacityis √ 6
wheredA = −gdx(2)dx(3)formetricdeterminantg=Det(g ) 24
κa=κa,eff+κa,syn+κa,dc, (10) anduniforθmφ codecoordinateswithspacingdx(1),dx(2) anddx(3)μiνn 35 b
andthenumbermeannumberabsorptionopacityis theradial-like,θ-likeandφ-likedirections. y g
Thenetflowefficiencyisgivenby ue
κanF=romκant,hefefs+eoκpana,csyintie+s,κtahne,dce.nergyandnumberemissionrates(1b1e)- η= E˙[M−˙]M˙ . (17) st on 05
fore any absorption are obtained in Appendix B5, giving energy t A
p
emissionrateλeandnumberemissionrateλn,suchthat Pthoesistyivsetevmalautessocmorereraspdoiunsd.Ftooranaejextt,rawchtiiocnhowfepdoesfiitniveeaesntehregyrefgrioomn ril 20
λ=λe+λc, (12) withinwhichb2/ρ>1,thejetefficiencyisηjandincludesallnon- 19
radiativetermsinE˙.Weoftenmeasurethejetefficiencyatr=10r
andλ includesallnumberemission. g
n
Wedonotincludethermalpairsrelevantwhenk T (cid:2)m c2,as andcallthattheinnerjetefficiencyηj,in.Wemeasurethisatsmall
b e e
radii before much of the jet energy is lost to the surroundings at
onlythejetcanbethathotandonerequirespairproductionthere
larger radii, while we measure it off the horizon so that much of
(see,e.g.appendixBinMcKinney&Uzdensky2012).
themassisoutgoinginsidethejet.Theefficiencymeasuredatthe
horizonoverallanglesisgivenbyη .
H
3 SIMULATIONS Themagneticfluxisgiveninanormalizedformas
(cid:8)
Our GRRMHD simulations evolve accretion flows around BHs 0.5|Br|dA
with BH mass of M = 10M(cid:4). All models have identical initial ϒH≈0.7 (cid:9)[M˙] θφ, (18)
conditions, except some have different initial density values due t
to an overall rescaling of the density in order to vary the final which accounts for Br being in Heaviside–Lorentz units (Gam-
massaccretionrateM˙.Fordifferentmodels,wevarythechoices mie1999;Pennaetal.2010).
MNRAS467,2241–2265(2017)
2244 J.C.McKinneyetal.
3.1.2 Inflowequilibrium Thefluid-frameradiationtemperatureis
Inflow equilibrium is defined as when the flow is in a complete Tˆ ≈ E , (27)
quasi-steadystateandtheaccretionfluxesareconstant(apartfrom γ nE¯
0
noise)versusradiusandtime.Theinflowequilibriumtime-scaleis
(cid:7) (cid:2) (cid:3) with E¯0≈2.701kB, which is within 10 per cent of the full BE
rie −1 formulagivenbyequation(A5)thatweactuallyuse.Thelab-frame
t =N dr , (19)
ie [(cid:10)v(cid:11) ] radiationtemperatureis
ri r ρ t
where (cid:10)v(cid:11) is the ρ weighted radial velocity, and N ∼ 3 inflow −Rt
r ρ T ≈ t . (28)
timesfromaninflowequilibriumradiusofr=riedowntori=rH γ nrutrE¯0
atthehorizon.
Theblackbodyassumptiongivesafluid-frametemperatureof
Viscous theory gives a GR α-viscosity estimate for v of
r (cid:2) (cid:3)
vHv/isRc ∼and−GQRα(cHo/rrRe)c2t|ivornotQ|f(oPrargoeta&tioTnhaolrvneel1o9ci7t4y;vProetn,ndaisectathl.ic2k0n1e0s)s, TˆBB= aErad 1/4, (29) Dow
sowecandefineaneffectiveαviscosityas n
forradiationconstanta ,whileinthelabframe,theassumption lo
v rad a
αeff ≡ v r/α. (20) ofPlanckgives ded
Allourmvisocdelshaveαeff∼1inthequasi-steadystate,asexpected T =(cid:2)−Rtt(cid:3)1/4. (30) from
forMADs(McKinney,Tchekhovskoy&Blandford2012). BB arad http
3.1.3 Opticaldepthandradiativequantities Fromthese,onecancomputethelab-framephotonhardeningfactor s://a
f = Tγ , (31) cad
Thesca(cid:7)tteringopticaldepthiscomputedas col TBB em
ic
τ ≈ κdl, (21) andthefluid-framehardeningfactor .o
sca s u
p
whilet(cid:7)he(cid:9)effectiveopticaldepthforabsorptioniscomputedas fˆcol= TˆTˆBγB. (32) .com/m
τ ≈ 3κ (κ +κ )dl. (22) n
eff a s a ra
s
F≈o1rt−he1r/a(duita)2l(daisrevcatliiodna,tdlla=rge−rfaγddiir),,fθγ=≈0u,ta(1nd−th(ev/inct)ecgorsaθl)i,s(fvro/cm) 3.1.4 Numericaldiagnostics /article
anouterradiusofr0=4000rg(chosentobeinsidetheoutergrid’s Themagneto-rotationalinstability(MRI)isalinearinstabilitywith -ab
radialboundary,butstillfaroutsidethescatteringphotosphere)tor fastestgrowingwavelengthof stra
toobtainτr(r).Fortheangulardirection,dl=fγrdθ,θ =π/2,and λ ≈2π|vx,A|, (33) ct/4
tτhe(θin).teNgortaeltihsaftrtohmepehaocthosppohlearreasxinisthtoewpaorldasrjtehtereegqiuoantoarretostroobntgailny x,MRI |(cid:18)rot| √ 67/2
deθterminedbythenumericaldensityfloorsassumed. for x = θ, φ, where |vx,A|= bxbx/(cid:19) is the x-directed Alfve´n /22
To scale M˙c2 or a luminosity L, one can use the Eddington speed,(cid:19)≡b2+ρ+ug+pgandr(cid:18)rot=vrot.(cid:18)rot,vAareseparately 41/2
luminosity angle-volume-averagedateachr,t. 96
TheMRIisresolvedforgridcellsperwavelength(equation33), 2
fLoErddTh=om4πsoGκnesMelcec≈tro1n.3sc×att1e0r4in6g10o8MpMac(cid:4)ityeκrgess.−O1,necanalsocho(2o3se) Qx,MRI ≡ λx(cid:20),MxRI, (34) 435 by gue
ti(onNanolovarickmcoravelit&zioenTM˙ehfofibrycnieMe˙n1Ec9dyd7f3=o)r((tc1ho/emηNNmTo)ovLniklEyod,dv/a–cTfi2h,xowerdhneeηrNtehTηin=NTd0iiss.c1thsioeslnuuostiemodn-, (cid:20)aovfθer≈Qagxri,nMdgxRI(i2s≥)(ddo6θn,/edfaxos(r2w))xitahn=Sdd,(cid:20)MθφR,I≈,φex,rcsewipnhtθevdrxex,A(3/(cid:20))(cid:20)(drxφ≈a/ndddxx((|31(cid:18))))(.rdotrV|/,odwluxhm(i1c)e)h-, st on 05 A
bLuratTdwh=eeri−andc(cid:7)ilautdidveAetθhlφueRmstrpi,ninosdietypeisndceonmcpeu).tedas (24) tad2r=i0esc(0e,s,xewapchlealiprloaetutmeirnloymadoθesdtl,eeMaφlsd-1yvh3o-asvtlthueaamtQtee,iθ-sa,aMlqvluReoIiruta∼ergtmeh4dio0ndbaaeennlfdsdohrihseaavcsfoeoQnQrsmφtθa,i,MnnMgRtRIaIQl∼∼lax6,l1Mo0anR0nIg.datAnhsdoet pril 2019
probablyhassomewhatunderdevelopedMRIturbulence).
whichismeasuredjustbeyondthescatteringphotospheretogive
TheMRIsuppressionfactorcorrespondstothenumberofMRI
the quantity we call L . The radiative efficiency is η =
rad,o rad,o wavelengthsacrossthefulldisc:
L /[M˙]. From any cumulative luminosity L(θ), we can com-
rad,o t
putetheisotropicequivalentluminosity S ≡ 2r(H/R). (35)
∂ L(θ) d,MRI λθ,MRI
L (θ)= θ (25)
iso −∂θcosθ Wavelengths λ<0.5λθ,MRI are stable, so the linear MRI is sup-
pressedforS <1/2whennounstablewavelengthsfitwithin
andthecorrespondingbeamingfactor d,MRI
thefulldisc(Balbus&Hawley1998;Pessah&Psaltis2005).S
d,MRI
b= Liso, (26) usesaveragingweightw=(b2ρ)1/2,conditionβ>1andexcludes
L regions where density floors are activated. When computing the
whereL=L ascomputedoverallangles. averagedS ,v and|(cid:18) |areseparatelyθ,φ-volume-averaged
rad d,MRI A rot
MNRAS467,2241–2265(2017)
DCandsynchrotroninsuper-Eddington flows 2245
within ±0.2r for each t, r. The S ∼0.5 at t = 0, while in at 128 × 64 × 32 in order to resolve the lowest order m modes
d,MRI
quasi-steadystate,S ∼0.1.Sothefieldstrengthhasincreased toallowforaccretioninMADs.ForMADs,suchlowerresolution
d,MRI
considerably due to magnetic flux accumulation. All models are modelshaveshownreasonableconvergence(McKinneyetal.2012;
MADwithS <1/2outtor∼30r . McKinneyetal.2014).Thegridaspectratioisroughly1:1:1for
d,MRI g
Theflowstructurecanalsobestudiedbycomputingthecorre- radii5r ≤r≤30r forourmoderateresolutionmodels.
g g
lationlength-scaleandthencomputinghowmanygridcellscover ModelsM4andM12areverylow-resolution(128×64×16)
eachself-correlatedpieceofturbulence.Wefollowourpriorworks testmodelsthatotherwisematchmodelsM3andM11,respectively.
(McKinneyetal.2012;McKinneyetal.2014)andcomputethis. Thesemodelsandothersatdifferentresolutionsexhibitallofthe
One would desire to have at a minimum six grid cells per cor- sameradiativepropertieswewilldiscuss,whichshowsthatresolu-
relation length-scale, since otherwise uncorrelated parts of turbu- tionisunlikelyadominantfactorincontrollingourresults.Wedo
lence are not independently resolved by our piece-wise parabolic notdiscussmodelsM4andM12further.
monotonicity-preserving scheme that needs six grid cells to re- Therest-massandinternalenergydensitiesaredriventozeronear
solve a structure well. All our models have ≈12 grid cells per theBHwithinthejetandneartheaxis,soweusenumericalceilings D
verticalandradialcorrelationlengthfordensityandmagneticfield ofb2/ρ =300,b2/ug=1011andug/ρ =1010.Thevalueofb2/ρ ow
strength,whileourmoderateresolutionmodelshave≈6cellsinthe isatthecode’srobustnesslimitforthechosenresolution,whilethe nlo
a
φ-direction per correlation length-scale. Our survey models have valueofb2/u ischosentoensurethatanartificialtemperaturefloor d
g e
othnaltyth≈e3sucervllesyinmtohdeeφls-sdhiroeucltdiobnepceorncsoidrerereladtinoontlreensgotlhv.edThenisomugehantos isnAotteinartrlyodtuimceeds.inallsimulations,weramp-upthechemicalpo- d from
demonstratefullyresolvedturbulence,buttheyarestillsufficiently tential factor from Planck towards its desired target. This avoids h
interesting to identify what physical effects (being switch on/off) difficult-to-resolveopacitychangesduetosynchrotronwithinthe ttp
s
could be important in fully resolved simulations. In addition, we first100rg/cintime.Also,thesharpchangesinsynchrotronopac- ://a
comparesomesurveymodelsagainstmoderateresolutionversions ityforhighφ (equationD5)nearPlanck(i.e.exp(−ξ)=1with ca
d
toconfirmthesurveymodelsarereasonable. dimensionlesschemicalpotentialξ =0)aredifficulttohandle.For em
synchrotrononly,weenforceexp(−ξ)<=0.99inordertosmooth ic
.o
outtheseopacitychanges(Seaton1993). u
3.2 Initialconditions p
.c
o
TheinitialdiscisKeplerianwitharest-massdensitythatisGaussian m
in angle with a height-to-radius ratio of H/R ≈ 0.2 and radially 3.4 Models /mn
followsapowerlawofρ∝r−0.6.Thesolutionnearandinsidethe Our goal is to consider the physical effect of various choices for ras
ithneneISr-CmOos,tthsetasbolleuctiiorcnuilsatraoprebrietd(ItSoCaOsm)iaslnleortdaennesqituyi(liρbr→iumρ,(rs/o1n5e)a7r, t‘hsuerrvaedyi’atrivuenstraonvsefreraotpiamceitipese.riWodesoifmaublaotuets1ev0e0r0a0lrlog/wc-,reaslololuwtiinogn /article
within r = 15r ) and a smaller thickness (H/R → 0.2(r/10)0.5, thesimulationstoreachasingleinflowtimeouttor∼50r and -a
g g b
withinr=10rg–baseduponalow-resolutionsimulation).Thetotal inflow equilibrium out to r ∼ 20rg. We check these survey runs stra
ionfteHr/nRal≈encesr/gvyKdfeonrssiotyunudtotspiseeedstcims≈ate√d(cid:11)frtootmPtovt/eρrtiwcailthe(cid:11)qutoitli≈bri4u/m3 wreistohluatcioonuprulenmstoodseereaitferreessoolluuttiioonnprluanyss,aanddomwiencaotenosirdseurbe-vdeonmlionwaenrt ct/46
andKeplerianspeedvK≈(r/rg)/((r/rg)3/2+a/M).Thetotalideal rolecomparedtotheopacityeffects. 7/2
pressurePtot =((cid:11)tot −1)utot israndomlyperturbedby10percent Table 1 shows the physical choices made for each model, in- /22
tsoimsueeladtitohne.MThReI.dTishcehdaisscangaastmhaossp(cid:11)hgeasre=w5it/h3,ρas=us1e0d−5d(urr/irngg)−th1.e1 cplhuodtionngntuhmebBeHrdsepnisni,tychisoeicveoslvfeodr,ohpoawcitthieesr,awdihaetitohnertetmheperaradtiuarteioins 41/296
and gas internal energy density ug = 10−6(r/rg)−5/2. The disc’s computedandwhetherthechemicalpotentialisvaried. 24
3
radiationenergydensityandfluxaresetbyflux-limiteddiffusion The ‘OPAL’ opacity refers to the opacity κeff given by equa- 5 b
and are having the same temperature for both gas and radiation tion (C16). This is physically equivalent to the OPAL-based opac- y g
(McKinneyetal.2014). ity used in Jiang et al. (2016), except ours is more approximate, u
e
The initial magnetic field is large scale and poloidal. For but we also account for radiation temperatures being different st o
r<300rg,thecoordinatebasisφ-componentofthevectorpotential fromgastemperatures.‘DC’referstodoubleCompton,whichno n 0
is simulations have yet accounted for. ‘Syn’ refers to synchrotron, 5
A
Aφ =MAX(rν1040−0.02,0)(sinθ)1+h, (36) w(Fhraicghileha&sMbeeeienra2c0c0o9u;nRteydanfoertailn.2s0u1b5-E;Sdaddinogwtosknieatccarl.et2i0o1n7c).aAselsl pril 2
wmiothnoνp=ola0r.7u5sinangdAhφ==4M.FAoXrr(r≥0ν1r004=0−3000.0rg2,,t0h)e(sfiienlθd)t1r+ahn(sr0it/iro).nTshtoe mpuoteddelsnuwmitbherphmoetoannonpuamcbiteyrneovtolyuettioanccionuclnutdede afosreipnarsaimteluylactioomns- 019
field is normalized with ∼1 MRI wavelength per half-height H (exceptonlysynchrotronnumberopacityinSadowskietal.2017).
givingaratioofaveragegas+radiationpressuretoaveragemagnetic Thesimplifiedfluid-frameradiationtemperaturegivenbyEˆ/(nˆE¯ )
0
pressureofβ≈40forr<100r . with E¯ ≈2.701k is the dominant factor in equation (A5)
g 0 B
(Sadowski & Narayan 2015a). We consider both Planck and BE
photondistributions.
3.3 Numericalsetup
AllmodelsincludeTCexceptmodelM6inordertostudyhow
The numerical grid mapping equations and boundary conditions turningthatoffaffectsgastemperatures.ModelM6wouldbelike
usedhereareidenticaltothatgiveninMcKinneyetal.(2012)and many existing radiative transfer simulations (Fragile et al. 2012;
McKinneyetal.(2014).Thegridfocusesonthediscatsmallradii Fragileetal.2014;Jiangetal.2014b;Takahashietal.2016)except
andonthejetatlargeradii. those by Fragile & Meier (2009), Kawashima et al. (2009) and
Models M14h and M15h are moderate resolution models at Jiangetal.(2014a)andourselves(Sa¸dowskietal.2014;McKinney
N ×N ×N =128×64×64,whiletherestaresurveymodels etal.2014).
r θ φ
MNRAS467,2241–2265(2017)
2246 J.C.McKinneyetal.
Table1. Spin,massaccretionrate,opacityandtemperaturechoices.
Model a/M M˙H Opacities Radiationnumberdensity Radiationtemperature Chemicalpotential
M˙Edd
M1 0.8 140 OPAL+Syn+DC Evolved E/(nE¯0) 1
M2 0.8 140 OPAL Evolved E/(nE¯0) 1
M3 0.8 80 OPAL Evolved Bose–Einstein Bose–Einstein
M5 0.8 120 OPAL PlanckatTˆγ PlanckatTˆγ 1
M6 0 5.9 OPAL(noTC) PlanckatTˆγ PlanckatTˆγ 1
M7 0 4.8 OPAL Evolved E/(nE¯0) 1
M8 0 5 OPAL+Syn+DC Evolved E/(nE¯0) 1
M9 0.8 50 OPAL+Syn+DC Evolved Bose–Einstein Bose–Einstein
M10 0.8 27 OPAL+DC Evolved Bose–Einstein Bose–Einstein
M11 0.8 36 OPAL+Syn+DC Evolved Bose–Einstein Bose–Einstein D
M13 0.8 1.2 OPAL+Syn+DC Evolved Bose–Einstein Bose–Einstein ow
M14 0.8 3.5 OPAL+Syn+DC Evolved Bose–Einstein Bose–Einstein nlo
M14h 0.8 2.4 OPAL+Syn+DC Evolved Bose–Einstein Bose–Einstein ad
M15 0.8 14 OPAL+Syn+DC Evolved Bose–Einstein Bose–Einstein ed
M15h 0.8 31 OPAL+Syn+DC Evolved Bose–Einstein Bose–Einstein fro
m
h
ModelsM9,M11,M12,M13,M14,M14h,M15andM15hare ttp
s
used to explore how varying M˙ affects the results while using ://a
c
our full opacity physics, with M9 and M11 not behaving much a
d
differentlyduetohavingsimilarM˙. e
m
Throughoutourdiscussionofmodelresults,wefocusonspecific ic
.o
models,oftenshowingmoredetailsformodelM15becauseithas u
p
anintermediatemassaccretionrateanddemonstratespropertiesof .c
o
allourmodels. m
/m
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3.5 Initialandfinalstate s/a
Fig.1showstheinitialandfinalstateoftheaccretionflowformodel rticle
M15,whichistypicalofallmodelswithrotatingBHs.Theinitial -a
b
mdiascgnthetaitcifiseilnditsiatrlulyctaurGeaiusslsairagnedsicsaclewainthdHpo/lRoi≈dal0,.a2nadntdhraeapdoswtehre- strac
lawradialbehaviour.TherotatingBHanddischavelaunchedajet t/46
7
thatstartsoutmagneticallydominatedbutconvertsitsenergyinto /2
kineticenergyatlargeradii. /22
4
Table 2 shows results for dynamical quantities (like fluxes and 1
/2
efficiencies) for all our models as time-averaged from 4000rg/c 96
till the end of the simulation. This table can be used to compare 24
resultsformodelswithdifferentopacityphysicsanddifferentM˙. 35
b
The models vary in mass accretion rate through the horizon with y
M˙/M˙Edd=1–140,andtheradiativeluminosityisgivenbyLrad,o. gue
Otherquantitiesaremeasuredonthehorizon(e.g.ηH),ataninner st o
radius of 10rg for inner (‘i’) quantities, or at large radii for outer n 0
(‘o’)quantities. 5
A
surTehdeaetffithceiehnocryizηoHn,iswthhiechtoitsalcoefnfisctaienntctyoowfitthhiens3y0stpemercaesnmt oeua-t pril 2
tothescatteringphotospherewhereouterquantitiesaremeasured. 01
9
Thetotalandradiativeefficiencyatlargeradii(η )arecom-
RAD,o
parable to the NT standard thin disc efficiencies (η ). The total
NT
non-radiativejetefficiencyη istheefficiencyatr=10r inthe
j,in g
jet with b2/ρ > 1. This tracks each model’s total efficiency, but
thejetenergyisprogressivelylosttothesurroundingmaterialthat Figure1. ModelM15,showingtheinitialcondition(toppanel)andfinal
heatsupandradiates(McKinneyetal.2015),leadingtorelatively state(bottompanel).Theinitialdiscisshownasrest-massdensity(scaled
lowgasjetefficienciesbyr=1000rg.Thetotalradiativeefficiency by an Eddington density value inferred from M˙Edd, rg, and c, shown in
is therefore dependent upon the mass-loading physics, which is colourwithlegend)threadedbymagneticfieldlines(greenlines).Thefinal
dischasbecomemuchhigherdensityneartheBHandtherotatingBH,and
controlledpartiallybynumericalfloorinjectioninoursimulations.
dischavelaunchedamagneticallydominatedjetduetotheaccumulationof
Fornon-rotatingBHmodels,thejetefficiencyislowat1percent,
magneticfluxintotheMADstate.TheBHspin-drivenjetemergesandhas
exceptforthemodelwithoutTCwithjetefficiencyat4percentdue aboundarythatcanbeseenwhereb2/ρ=1(redline).Atlargeradiiatlate
tothethermalenergycontentofthejet(thisalsoleadstoaslightly times,thejethasbecomekineticallydominatedwithLorentzfactorγ ∼2
higherradiativeefficiency). byr=500rg,sothatb2/ρ<1there.
MNRAS467,2241–2265(2017)
DCandsynchrotroninsuper-Eddington flows 2247
Table2. Accretionrates,luminosities,efficiencies(percent)andmagnetic Thenormalizedmagneticfluxonthehorizonϒ ∼10forrel-
H
fluxes. ativelythickdiscsathighersuper-Eddingtonrates,whilelowerM˙
leadtodowntoϒ ∼4asseeninthinMADsimulations(Avara,
H
Model M˙M˙EHdd LLrEadd,do ηH ηj,in ηoRAD ηNT ϒH MshcoKwinunpeyto&ϒRe∼yno2l0dsan2d01h6a)v.eAqfueiwtemhiogdhelesffi(Mci9en,cMie1s0aapnpdarMen1t1ly)
H
M1 140 67 54.7 30.3 5.68 12.2 9.8
duetoradiativesuppressionofthemagneticRayleigh–Taylormodes
M2 140 110 58.3 40.7 9.76 12.2 9 duetoopacityeffectsattheirintermediateM˙/M˙ ∼30–50,but
M3 80 81 114 84.6 12.3 12.2 13 Edd
one suspects that higher resolutions would show no such effect
M5 120 81 59.2 36.3 8.39 12.2 9.5
(which M15h approaches and does show more moderate ϒ and
M6 5.9 6 9.83 4.24 5.86 5.72 3.9 H
M7 4.8 2.4 7.98 0.926 2.84 5.72 3.9 efficiencies).
M8 5 1.9 8.39 0.979 2.21 5.72 5 Fig.2showsasnapshotfromthesimulationandshowsvarious
M9 50 240 124 71.6 59.4 12.2 14 fluxesandefficienciesversustime,whoseconstancyindicatesthat
M10 27 160 267 184 71.8 12.2 20 theflowhasreachedaquasi-steadystateinwhichthetotalefficiency D
M11 36 290 185 110 95.5 12.2 18 isηH∼30percent,almostthreetimestheNTthindiscefficiency, ow
MM1134 13..25 12..18 1283..57 118..42 91.07.97 1122..22 53..16 ManAdDthestaratedioautitvteoeafbfiocuietnrc∼yi5s0ηrradw,oith∼ev5ipdeernctemnat.gTnehteicdRisacyilseiignh–a nload
MMM111455hh 213.144 211.185 332021...398 111471...237 11452.5..677 111222...222 647...771 TflauyxloϒrHin≈sta7b,icliotimespairnabthleewy–itxhpngloann-er.adTihaetivdeimdiesncssiowniltehssHm/Rag≈ne0t.i3c ed from
(McKinneyetal.2012).Theeffectivephotospherereachescloseto h
thedisc,exceptwherethejethasrelativelyhighdensities. ttp
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Figure2. EvolvedsnapshotofmodelM15hatt≈5000rg/cshowinglogofrest-massdensity(scaledbyanEddingtondensityvalueinferredfromM˙Edd,rg,
andc,shownincolourwithlegendonright)inboththez–xplaneaty=0(topleft-handpanel)andy–xplaneatz=0(topright-handpanel).Blacklines
tracefieldlines.Inthetopleft-handpanel,thethickredlinecorrespondstowhereb2/ρ =1andthepurplelinecorrespondstotheeffectivephotosphere
(computedradiallyinward).Thebottompanelhasthreesub-panels.Thetopsub-panelshowsM˙ throughtheBH(M˙H)andradiativeluminosity(Lrad,o).The
middlesub-panelshowsthemagneticfluxpassingthroughthehorizon(ϒH).Thebottomsub-panelshowsthetotalefficiency(ηH),innerjetefficiency(ηj,in),
andradiativeefficiency(ηrad,o).Horizontalsolidlinesofthesamecoloursshowtheaveragesovertheaveragingperiod.Forsuper-Eddingtonaccretionat
M˙/M˙Edd≈31,thetotalBHefficiencyismoderateatηH≈31percentwithradiativeefficiencyofηrad,o≈5percent.
MNRAS467,2241–2265(2017)
2248 J.C.McKinneyetal.
Table3. Radiativepropertiesandradiation/jetopeninganglesinradians.
(cid:10) (cid:11)
Model TTˆgγas r=10rg Tˆγ,r=10rg(K) Tγ,r=100rg,rad.beam(K) fˆcro=l10rg,disc fcro=l100rg,rad.beam e−r=ξ10rg,disc e−r=ξ100rg,rad.beam θrr=1000rg θjr=1000rg
M1 0.96 4.2e7 1.7e7 1 1.4 1 1 0.12 0.067
M2 0.93 1.3e8 7.7e8 3.9 55 1 1 0.089 0.0038
M3 1 1.1e8 6e8 3.2 42 0.88 0.36 0.084 0.021
M5 0.97 3.8e7 9.1e6 1 0.77 1 1 0.092 0.025
M6 9.8e-5 1.5e7 3.3e6 1 0.58 1 1 0.6 0.12
M7 0.87 1.4e8 2.5e7 8.9 5.4 1 1 0.6 0.19
M8 0.63 1.9e7 6.7e6 1.1 1.6 1 1 0.6 0.15
M9 0.98 4.1e7 1.5e7 1 1.1 0.97 0.96 0.13 0.16
M10 0.99 4.9e7 2.1e7 1.3 1.6 0.87 0.69 0.13 0.041
M11 0.99 4.3e7 1.5e7 1 1 0.99 0.97 0.098 0.21 D
o
M13 0.12 3e7 9.2e6 4.5 2.6 0.43 0.45 0.28 0.0091 w
n
M14 0.23 1.5e7 2.1e6 1.3 0.48 0.79 0.94 0.26 0.011 lo
a
M14h 0.2 1.3e7 2e6 1.2 0.45 0.76 0.99 0.47 0.055 d
e
MM1155h 00..8957 22..45ee77 77..32ee66 11.1 11..21 00..8969 00..7962 00..1367 00..002987 d fro
m
h
ttp
s
Table3showsresultsforourradiativediagnostics,whichcanbe beamingfactorisuptob=15forrotatingBHmodelsandb=3 ://a
usedtodeducehowtheradiativepropertiesareaffectedbydifferent fornon-rotatingBHmodels(similartothoseseeninSadowski& ca
opacitychoicesanddifferentM˙.Thistableincludesthefluid-frame Narayan 2015b). This corresponds to an enhanced radiative flux de
m
draidscia,tfliounidt-efmrapmeerartaudrieatpioernutenmitpgearsattuermepinertahtuerdeisact,rla=b-f1r0armgeinratdhie- ahtigshpeerciafi/cMv.ieTwhiengMa1ngclleoss,uwreithslhigighhtleyrobveearmesintigmfaotershitghheebreMa˙mainndg ic.ou
hatairodnenteinmgpfearcattourrefˆintihnethraeddiaistico,nlabbe-afrmamaetrha=rde1n0i0nrgg,faflcutiodr-fframine factors(Narayanetal.2016). p.com
theradiationbeamcoaltr=100r ,chemicalpotentialfactorsincolthe /m
disc and radiation beam, radiatgion beam half-opening angle (θ ) 3.6 Magneticandradiativefluxes nra
r s
aotperni=ng1a0n0g0lreg(,θaj)ndatjret=(k1i0n0e0tircg+.eDnitshcalqpuya+netlieticetsroamreagconmetipcu)tehdalafs- Fciiegn.c3y)sahnodwlsabth-ferammaegrnaedtiiactiflonuxflulixnestsre(awmithlineelsec(twroitmhargadnieattiicveefefif-- /article
weightedbyvolumeofthegridcelltimessquareofdensity,while ficiency).Theradiationisbroadlydistributed,buthasanenhanced -a
b
radiation or electromagnetic beam quantities are measured at the beamedregionwhoseopeningangleisabouttwicelargerthanthe s
aperaekpriensethnetaltuiomninoofswithyaptearnuonbistearnvgelres(e∂esθL(a(sθ)w).oTulhdiscodmoeesfnroomtgtihvee eplleocttterodm) ashgonweticthjeet’pseoapkenEiMngalunmglien.oMsiotydeplserwuitnhitzearnogBleHesmpienrg(ninogt tract/4
6
luminosityperunitsolidangle),andinsteadshowswhatopening fromcylindricalradiusneartheISCO(Tchekhovskoy,McKinney 7/2
angleshavethemostefficiencyassociatedwiththem.Thiscanbe & Narayan 2012), instead of the rotating BH models where the /2
2
usefulwhenconsideringhowanambientmediumwouldbeaffected peakpowerperunitangleemergesfromneartheequatorialregion 41
bythejetorradiationandleadtosecondaryemissionduetothat oftheBHhorizon(McKinney&Gammie2004;McKinney2006; /29
6
interaction,likeinTDEaftergloworULXionizationcones. McKinney&Blandford2009). 2
4
ThelowM˙ ∼M˙Eddmodelshavehighergastemperaturesinthe 35
disc,butgastemperaturesareatmostabout10timestheradiation b
temperatureswithTC.OnlymodelM6withoutTCshowsverylow 3.7 Effectiveenergyphotospheres y gu
Tsˆpγe/cTtrguasm.TihnemroaddiealtsiownitbheoaumtDaCndodriisncmhaovdeelhsawrdiethniMn˙g∼wiMt˙hEaddW. ien Fraidgi.i,4thsheotwotsaltheeffeefcfteicvteivpehpohtoostpohspehreerseistsfoarbomvoedtehleMd1is5c.aAntdladrigsec est on
ThephotondistributionstendtobesomewhatWieninthecoronae wind. Sitting inside the total effective photosphere is the free– 05
formodelswithchemicalpotentialevolution.ModelswithoutDC free photosphere, the DC photosphere, and the synchrotron pho- A
p
and synchrotron (like M2, M3 and M7) show significant photon tosphere. The free–free, DC and synchrotron opacities all merge ril 2
hardening,whichbecomesmuchmorelimitedwhenincludingthese withinsomeradius,showingthattheybecomecomparablyimpor- 0
1
opacities.Radiationbeamlab-frametemperaturesarecomparable tant. While free–free and DC are clearly important in metal-free 9
to the disc core, except for models without DC and synchrotron. plasmas,bound-freeandbound–boundcontributionsareimportant
ModelswithoutDCandsynchrotron(e.g.M7)havemuchhigher withsolarabundances,whichleadstoaneffectivephotospherefar
radiation temperatures than otherwise identical models (e.g. M8) beyondthefree–freephotosphere.ForhigherM˙ models,thesedif-
with DC and synchrotron. This shows that DC and synchrotron ferentopacitiesbecomemorecomparableatlargerradiithaninthis
are crucial to include in order to obtain accurate observer-frame lowerM˙ model.Thescatteringphotosphereisatr∼800rginthis
radiationtemperaturesforflowswithM˙ (cid:2)M˙ . model.
Edd
Thehalf-openinganglesinradiansidentifythemaximuminL
iso
withintheradiationbeamorgasjet.Wealsocomputed(notintable)
3.8 Gasoverheatedregions
thebeamingfactor(b=L /L,i.e.isotropicequivalentluminosity
iso
per unit total luminosity) measured at r = 1000r . The electro- Fig.5showsthelab-frameradiationtemperatureandfluid-framegas
g
magnetic jet is beamed by factors up to b = 15 for rotating BH temperatureformodelM15.Theradiationtemperaturereachesupto
modelsanduptob=6fornon-rotatingBHmodels.Theradiation Tγ ∼108K,whilethegastemperaturereachesuptoTgas∼109K
MNRAS467,2241–2265(2017)
DCandsynchrotroninsuper-Eddington flows 2249
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Figure3. ModelM15,withtoppanelshowingtime-φ-averagedmagnetic /2
2
fluxlines(translucentgreylines)withelectromagneticluminosityperunit 4
1
angle(∂θLEM)/(M˙Hc2)(shownincolourwithlegend),withbluelineshow- /29
ingwhereb2/ρ=1,greenlineshowingwhereur=0andpurplelineshow- 6
2
ingtotaleffectivephotosphere.Imageisduplicatedacrossthex=0line. 43
5
Thebottompanelshowssamethingsforlab-frameradiationfluxlinesand b
radiationluminosity.Theelectromagneticenergyfluxactsasafunnel-wall y g
jetbyfollowing(andsittingjustoutside)theboundarywhereb2/ρ =1. u
e
MostoftheelectromagneticenergyisreleasedfromtheBH’sspinenergy. st o
TheMADequatorialregionisquitehotanddynamic,leadingtotheradi- n
ationemergingfromquiteclosetotheBH(andinatime-averagedsense, 05
someradiationappearstoemergefromabovetheBHduetotransientbut A
p
powerfulpolarmagneticfieldsthatjumpbetweentheBHanddisc).Radia- ril 2
tionmovesinwardwithinthediscthataccumulatesmoreradiativeenergy, 0
1
whichsomewhatfollowsthepathofthewindandultimatelybecomesmore 9
Figure4. ModelM15,showingeffectivephotospheresfordifferentpro-
radiallydirectedatlargerdistances.
cesses, including OPAL+DC+synchrotron (purple line), synchrotron (red
line),DC(brownline),free–free(blackline),scattering-only(yellowline)
in the jet region. Given our discussion in the Introduction, this ondifferentsizeregions(top,middleandbottompanels).Thephotospheres
suggests that DC should be important throughout the flow, while arecomputedradiallyinwardfromr=4000rg.Rest-massdensityshown
synchrotron is likely important in the jet region. Note that as M˙ incolour(withlegend),scaledbyanEddingtondensityvalueinferredfrom
drops, the disc becomes thinner, although such MAD type discs iMn˙EFdidg,.r3g.aFnodrcr.(cid:2)Gr5eregn,,DblCuea,nydelflroewe–farnedepbuercpolmeelincoesmapraer,abiflypirmespeonrtt,aanst
are also magnetically compressed by the large-scale poloidal and
processesinthedisc.Theeffectivephotospheresomewhatfollowsthetran-
toroidalfieldsthreadingtheBHanddisc(McKinneyetal.2012). sitionbetweendiscinflowandwindoutflow.Thepolaraxisregionscontains
Fig.6showsthefluid-frameradiationtemperatureperunitgas ahigh-densityportionofthejet(launchedbymassinjectionneartheBH),
temperature.TCactstoregulategastemperaturestowardstheradi- whichleadstohigheropacitiesthere.
ationtemperatureinradiation-dominatedplasmas.Weshowseveral
model’spoloidalplanetemperatureratiosinordertopresenthow
MNRAS467,2241–2265(2017)
2250 J.C.McKinneyetal.
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4
Figure5. ModelM15,showinglab-frameradiationtemperatureinKelvin 1/2
(toppanel)andfluid-framegastemperatureinKelvin(bottompanel).Green, 9
6
blue,yellowandpurplelinesare,ifpresent,asinFig.3.Radiationtemper- 24
aturesarehighintheequatorialandpolarregions,whilegastemperatures 35
arehighinthejetregionduetoinsufficientComptonization. b
y
g
u
e
thiseffectworksathighM˙ ∼100M˙Edd tolowM˙ ∼1M˙Edd.Note st o
thatmodelM14hasalong-livedhemisphericalasymmetry,leading n
0
toscattering photosphere far away onthe upper hemisphere. The 5
fiargauterelyshtiomwes-φfl-uaivde-frraagmede)T.ˆγW/Thiglaes(lnouwmererMa˙tomraonddeldsenhoavmeinaatsolirgshetply- April 2
overheatedgasregionwithinsomeradius,thecorediscregionhas 01
9
atmost10percenthighergastemperaturesthanradiationtemper-
aturesforM˙ ∼10M˙ models.
Edd
ModelM13withM˙ ≈1M˙Eddshowsanopticallythincoronawith Figure 6. Model M1 at high M˙ ∼100M˙Edd (top panel), model M15 at
gastemperaturesabout100timeslargerthanthedisc’sblackbody lowerM˙ ∼10M˙Edd(middlepanel),modelM14atlowM˙ ∼3M˙Edd(next
temperature and about 20 times larger than the disc’s radiation panel)andmodelM13atlowM˙ ∼1M˙Edd(bottompanel),showingfluid-
temperature that is hardened by fˆcol≈4.5 (see related data from frameTˆγ/Tgas.BlacklinehasTˆγ/Tgas=1.Green,blue,yellowandpurple
Fig.9).Thegaspressureisuptoatenthoftheradiationpressure linesare,ifpresent,asinFig.3.ModelsM13andM14haveaphotosphere
andthediscthicknessH/R∼0.1inthismodel. nearthedisc,exceptthejetthatislaunched,whichkeepsthedensityhigh
Table 3 includes a sequence of a/M = 0 models M6, M7 and atlargeradii.AthighorlowM˙,thevalueofTˆγ/Tgasisorderunitydueto
TC.Thediscisevidentlythinnerandcooleratlowermassaccretionrates
M8thathavenoTCandnophotonhardening(M6),haveTCand
ofM˙ ∼10M˙Eddeventhoughtheinflowisstillquitesuper-Eddington.Only
photonhardeningbutwithoutDCorsynchrotron(M7),andhave forthelowestM˙ ∼M˙Edd modelM13,doesTˆγ/Tgas∼0.1inthecentral
both along with all our opacities (M8). This shows that the lack disc,butprogressivelymorecoronalmaterialhashighergastemperatures
ofTCleadstounphysicallyhighgastemperaturessimilartothose asM˙ drops.
MNRAS467,2241–2265(2017)