Modelling Accretion Disk and Stellar Wind Interactions: the Case of Sgr A*

1.3 MB·

by  I. M. Christie

Mon.Not.R.Astron.Soc.000,000–000(0000) Printed11April2016 (MNLATEXstylefilev2.2)
Modelling Accretion Disk and Stellar Wind Interactions: the Case of
Sgr A(cid:63)
I. M. Christie1(cid:63), M. Petropoulou1†, P. Mimica2‡, D. Giannios1
1DepartmentofPhysicsandAstronomy,PurdueUniversity,525NorthwesternAvenue,WestLafayette,IN47907,USA
6 2Departamentd’AstronomiaiAstrof´ısicaUniversitatdeVale`ncia
1
0
2 11April2016

r
p
A ABSTRACT
SgrA*isanidealtargettostudylow-luminosityaccretingsystems.Ithasbeenrecentlypro-
8
posed that properties of the accretion flow around Sgr A* can be probed through its inter-

actionswiththestellarwindofnearbymassivestarsbelongingtotheS-cluster.Whenastar
]
E intercepts the accretion disk, the ram and thermal pressures of the disk terminate the stel-
H larwindleadingtotheformationofabowshockstructure.Here,asemi-analyticalmodelis
constructedwhichdescribesthegeometryoftheterminationshockformedinthewind.With
.
h theemploymentofnumericalhydrodynamicsimulations,thismodelisbothverifiedandex-
p tended to a region prone to Kelvin-Helmholtz instabilities. Because the characteristic wind
- and stellar velocities are in ∼ 108 cm s−1 range, the shocked wind may produce detectable
o
X-raysviathermalbremsstrahlungemission.Theapplicationofthismodeltothepericenter
r
t passageofS2,thebrightestmemberoftheS-cluster,showsthattheshockedwindproduces
s
a roughlyamonthlongX-rayflarewithapeakluminosityof L ≈ 4×1033 ergs−1 forastel-
[ larmass-lossrate,disknumberdensity,andthermalpressurestrengthof M˙ =10−7M yr−1,
n =105cm−3,andα=0.1,respectively.Thispeakluminosityiscomparablwetothequi(cid:12)escent
2 d
X-rayemissiondetectedfromSgrA*andiswithinthedetectioncapabilitiesofcurrentX-ray
v
2 observatories.Itsdetectioncouldconstrainthedensityandthicknessofthediskatadistance
3 of∼3000gravitationalradiifromthesupermassiveblackhole.
4
Key words: The Galaxy: centre – Physical Data and Processes: hydrodynamics, radiation
7
mechanisms;thermal,accretion
0
.
1
0
6
1 1 INTRODUCTION Galactic center is a group of early, type-B stars known as the S-
: cluster.Thesestarshavebeenthetargetofextensiveobservational
v Stellarwindbowshocksformduetothesupersonictransitofout- effortinanattempttoidentifytheirevolutionarystageandorigin
i flowinggasfromastarwhileinteractingwithanambientmedium.
X (Eisenhaueretal.2005;Ghezetal.2005;Gillessenetal.2009b).
The formation of such structures has been studied extensively,
r TheorbitsofmanyS-starsarelikelytointerceptandinteractwith
a ranging from colliding winds in a binary system (Stevens et al. the accretion disk surrounding Sgr A*. The brightest member of
1992;Cantoetal.1996),tointeractionsbetweenthestellarwind
the S-cluster, S2, is a massive, main-sequence star with a mass-
of a single star with an ambient medium (Canto et al. 1996; loss rate of M˙ (cid:54) 3×10−7M yr−1 (Martins et al. 2008) and is
Wilkin 1996, henceforth W96), to Modelling heliospheric struc- w (cid:12)
characterizedbyitstightorbitwithapericenterdistanceof∼3000
tures around the heliopause (Drake et al. 2015; Washimi et al.
R , where R is the gravitational radius (Gillessen et al. 2009a).
2015), as well as non-thermal particle acceleration at the shock g g
Interactionsbetweenthediskgasandthestellarwindwillcausea
front(Guoetal.2014).Studyingthepost-shockemissionprovides
bow-shockstructuretoform,eventuallyengulfingthestar.Given
ameanstoprobedifferentpropertiesofboththestellarwindand
thatthecharacteristicvelocitiesofthewindanddiskfluidareafew
theambientmedium(e.g.density,velocity,temperature).
108 cm s−1, the shocked fluid is expected to radiate in the X-ray
Thecompactradiosource,SgrA*,marksthelocationofthe
band.
supermassiveblackholeatourGalacticcenter.Accretionontothe
blackholeisbelievedtopowertheradio,IR,andX-rayemission The use of S2 as a probe of accretion disk properties (e.g
produced around Sgr A* (Genzel et al. 2010). Located near the density and temperature) of Sgr A* was first proposed by Gian-
nios & Sironi 2013, hereafter GS13. By comparing the ratio of
the expansion timescale to the bremsstrahlung cooling timescale
oftheshockedstellarwindwhileadoptingasphericallysymmetric
(cid:63) E-mail:[email protected]
† EinsteinFellow model,theywereabletoestimatetheradiated,thermal,X-raylu-
‡ Email:[email protected] minosity.Thepeakemissionwaspredictedtotakeplacearoundthe
(cid:13)c 0000RAS
2 Christieetal.
pericenterpassageofthestar,wheretherampressurefromthedisk theshockeddiskmaterial,andtheterminationshockofthestellar
becomesmaximal,withtheX-rayluminositypotentiallyexceeding windcoincide.
thequiescentemissionfromSgrA*.Amoredetailedcalculationof In the rest frame of the star, the stellar wind is assumed to
theradiatedpowerisrequiredasthenextpericenterpassageofS2 be isotropic with a mass-loss rate of M˙ and constant velocity
w
isexpectedin2018. v =eˆv ,withtheunitvectordenotingthesphericalradialdirec-
w r w
In this study, an expansion is made upon the previous work tion.Thediskfluidhasaconstantmassdensityρ and,intherest
d
ofGS13byconstructingatwo-dimensional,semi-analyticalmodel frameofthestar,moveswithaspeedv =−V ,whereV istheve-
d (cid:63) (cid:63)
fortheshockedstellarwindregion.Todoso,axisymmetryisas- locityofthestar.Letthez-axisbethesymmetryaxisofthesystem
sumedwhilefollowingasimilaranalysisofmomentumsupported andthemotionofthestarasitprecessesthroughthedisktobein
bow shocks presented by W96 which emphasizes the conserved theeˆ direction.Intherestframeofthestar,thegasfromthedisk
z
momentum flux within the shocked region. With the inclusion of approaches the star from the −eˆ direction. The standoff distance
z
thermal pressure from the disk, the shape of the shock surface is oftheterminationshockisdeterminedbybalancingthepressures
thenderivedasafunctionofα,i.e.theratioofthethermalpres- frombothsidesoftheshockedregion.Therampressureofthestel-
sure to the ram pressure of the disk. Here, the thermal pressure larwinddecreaseswithincreasingradiusRasP = M˙ v /4πR2.
w w w
ofthediskisexpectedtobenon-negligibleduetothediskbeing Thediskpressureisthesumofthethermalandrampressures,the
radiatively inefficient and partially pressure supported. Quantities latterbeingaresultoftherelativemotionofthediskandthestar.
downstreamfromtheterminationshock(e.g.density,temperature, TherampressureofthediskisP = ρ v2 whilethethermalpres-
d d d
pressure)maythenbedeterminedbyusingtheRankine-Hugoniot sureisdefinedasamultipleoftherampressure,P =αP .Values
th d
conditions. The semi-analytic formalism adopted for the descrip- of α within the range 0 (cid:54) α (cid:54) 1 are considered2, with α = 0
tionofthesystemallowsforamoreaccuratecomputationofthe representing a cold disk as described by W96. By matching the
thermal bremsstrahlung radiated power produced by the shocked pressures,thestandoffdistanceis
stellarwind. (cid:115)
Since the semi-analytical model is built upon a series of as- R = M˙wvw , (1)
sumptions,thevalidityoftheseresultsistestedthroughdetailedhy- 0 4π(1+α)ρ v2
d d
drodynamicsimulations.Thepresentedmodelisfoundtoprovidea
which sets the characteristic length scale of the system. The dis-
gooddescriptionoftheshocksurfaceandofthethermodynamical
tanceR oftheterminationshockintheoppositedirectionofthe
propertiesinthedownstreamregionoftheforwardshockuptoan 1
standoff point is determined by balancing the ram pressure from
angleθ ≈ π/2,whereθismeasuredwithrespecttothesymmetry
theunshockedwindwiththepressureoftheshockedwind,thelat-
axis.Atlargeranglesandatlargerdistancesinthetailofthebow
shock, the semi-analytical model is not able to describe the mix- ter estimated to be the thermal p√ressure of the disk. This may be
writtenintermsofR asR =R 1+α−1.
ingoftheshockeddiskandstellarwindfluidscausedbyKelvin- 0 1 0
Themassandtotalmomentumfluxtraversinganannulusof
Helmholtzinstabilities.Theroleofthelatterontheradiativeoutput
theshockedregionare2πtimesthefollowing
ofthemixedfluidsis,therefore,investigatedsolelythroughnumer-
icalhydrodynamicsimulations.Thesesimulationsarealsousedto Φ = Rσv sinθ, (2)
m t
investigate,morerealistically,thetime-dependentproblem:namely Φ = Rσv2sinθ (3)
t t
asituationwherethediskdensitychangesonatimescalesimilarto
thetimeittakesforthestartocompleteitspericenterpassage. whereσandvtarethemasssurfacedensityandthetangentialspeed
Thispaperisstructuredasfollows.InSec.2,themathematical ofthefluidwithintheshockedregion,respectively.Inasteady-state
formulationofthesystemofcollidingfluidsispresentedalongwith situation,themassfluxmovingwithinacrosssectionalringofthe
thesemi-analyticalresults.InSec.3,adiscussionisgivenonthe shellisequaltothesumofthemassfluximpartedfromthewind
employmentofnumericalhydrodynamicsimulationsusedinorder onthesolidangleoccupiedbytheshockedregionoftheforward
totestthevalidityofthemodelaswellastodeterminethethermal shellplusthemassfluxfromthediskhittingthecircularareaofthe
radiatedpowerproducedbytheshockedwindregions.InSec.4, projectedcrosssectionoftheshell
thismodelisappliedtotheaccretiondiskofSgrA*andspecificS- M˙ (1−cosθ) 1
starswhileputtinganemphasisonS2.ByModellingthepericenter Φm= w 4π + 2R2ρdvdsin2θ. (4)
transitofS2asitprecessesthroughtheaccretiondiskofSgrA*,
The flow of shocked gases within the shell depends on the
wewillmakepredictionsofpossible,observationalsignatures.A
incomingmomentumfluxes.Letusconsiderasmallwedgeofcon-
discussionandsummaryoftheresultsarethengiveninSec.5and
stantwidth∆φintheazimuthaldirectionaboutthesymmetryaxis.
Sec.6respectively.
Therateatwhichvectormomentumisbeingimpartedonthewedge
oftheshellbythestellarwindis
2 THEMODEL (cid:90) θ
Φ (θ)∆φ = ∆φ ρ R2v v ·eˆ sinθ(cid:48)dθ(cid:48)=
w w w w r
Inordertoderivetheshapeoftheshockedstellarwindregion,it 0
is assumed that the wind-disk system has reached a steady-state1 = ∆φM˙wvw (cid:104)(θ−sinθcosθ)eˆ +sin2θeˆ (cid:105) (5)
andthattheshockedwindregioncanbedescribedasathinshell 8π (cid:36) z
(i.e.,H << RwhereH andRaretheshell’swidthandradialdis- whereρ = M˙ /(4πR2v )wasusedand(cid:36) = Rsinθ denotesthe
w w w
tancefromthestar).Withinthethin-shellapproximation,thecon-
tact discontinuity, which separates the shocked stellar wind from
2 Forα (cid:62) 0.6,ashockdoesnotforminthedisk,sincethemotionofthe
starthroughthehotfluidofthediskbecomessubsonic;ifγ = 5/3isthe
√
1 ThesteadystateassumptionisrelaxedinSec.4wheretime-dependent adiabaticindexofthediskfluid,thencs/vd = γα>1forα>1/γ.The
diskpropertiesaresimulatednumerically. stellarwind,however,isalwaysterminatedthroughastrongshock.
(cid:13)c 0000RAS,MNRAS000,000–000
Wind-DiskInteractions:SgrA* 3
Figure1.Athree-dimensionalviewoftheterminationshocksurface.Fromlefttorightα=10−5,0.05,and0.1.Asphereoffixedsize,centeredattheposition
ofthestar,isoverplottedinordertodemonstratethedifferentscalesamongthestructures.
cylindricalradialcoordinate.Theintegrationwasperformedona θ,atransformationofcoordinatesismadefrom((cid:36),z)to(r,θ)by
spherical surface with unit element dA = eˆR2sinθdθ, which is usingdz/d(cid:36)=Φ /Φ .Thisleadsto
r t,z t,(cid:36)
valid as long as the surface integral of the wedge region is inde-
r(cid:48)cosθ−rsinθ (1−r2)sin2θ
pendentoftheshell’sshape,R(θ).Thelatteristruethankstothe = , (10)
r(cid:48)sinθ+rcosθ (θ−sinθcosθ)(1−λr2)
factthatthesystemismomentumconserving,asfirstmentionedin
W96. wherer(cid:48)≡dr/dθ.Arearrangementofeq.(10)yieldsthefollowing
Therateatwhichvectormomentumisbeingimpartedonthe differentialequationforr(θ)
samewedgebytheaccretiondiskis (cid:104) (cid:105)
rsinθ θ(1−λr2)+r2(λ−1)sinθcosθ
Φ (θ)∆φ=∆φ(cid:90) θ(cid:2)−P I+ρ v v (cid:3)·eˆ R2sinθ(cid:48)dθ(cid:48)= r(cid:48)= (1−λr2)(θ−sinθcosθ)cosθ−sin3θ(1−r2) (11)
d th d d d r
0
R2ρ v2 (cid:104) (cid:105) which can be solved numerically for different values of λ with
−∆φ d d α(θ−sinθcosθ)eˆ +(1+α)sin2θeˆ (6) r(0)=1 as an initial condition. By setting λ = 0, the solution of
2 (cid:36) z √
theequationabovereducestor(θ)=cscθ 3(1−θcotθ),asgiven
whereIistheidentitytensor.Ifthethermalpressureofthediskis byW96.
neglected,i.e.α=0,theresultofW96isrecovered. Figure 1 shows a three-dimensional view of the termination
The rate at which the total vector momentum is being de- shock surface obtained after solving eq. (11) for α = 10−5, 0.05
positedisthereforethesumoftheindividualcontributions,namely and0.1,undertheassumptionofaxisymmetryintheeˆ direction.
φ
M˙ v (cid:104) (cid:105) Toillustratethedifferentscalesamongthethreecases,asphereof
Φt(θ)= 8wπw eˆ(cid:36)(θ−sinθcosθ)(1−λr2)+eˆz(1−r2)sin2θ (7) fixedsizeisoverplottedandcenteredatthepositionofthestar.The
radiusofthespheredoesnotrepresenttheactualradiusofthestar.
whereλisdefinedas
Theincreasingthermalpressureofthesurroundingmediumdrives
λ≡ α (8) R0 closertowardsthestarwhileconfiningtheshockintoaclosed,
1+α oval-shapedsurface.Figure2showstheradialdependenceofthe
andristhedimensionlessradialdistancedefinedas terminationshockobtainedbysolvingeq.(11)forvariousαvalues
up to an angle of π/2. The reasons for choosing the particular θ
R(θ)
r(θ)≡ . (9) range will become apparent in Sec. 3. The distance of the shock
R
0 increasesalreadyby10%withrespecttothestandoffvalueatθ (cid:39)
The inclusion of the thermal pressure in the disk results in a de- π/5−π/4,forallαvalues.However,onlyforα>0.11dotheeffects
creaseofthecylindricalradialcomponentofthetotalmomentum ofthedisk’sthermalpressureontheshapeoftheshocksurfaceat
flux imparted on the shell. This affects the shape of the shell in θ=π/2becomeevident.
suchawaythat,ifαisnon-negligible,theterminationshockwill
enclosethestarandwillnolongerextendouttoinfinity,asfound
byW96forthecaseofacolddisk. 2.2 Theshockedstellarwind
The properties of the shocked, stellar wind, e.g. temperature and
numberdensity,arecrucialforthecalculationoftheradiatedther-
2.1 Theshapeoftheterminationshock
malbremsstrahlungpower.Thesepropertiesaredeterminedbyus-
ThekeypointstressedbyW96isthattheshapeoftheshellisnot ingtheRankine-Hugoniotjumpconditionsacrosstheshockfront
determinedbythelocalbalanceofdifferentpressuresactingonit, withtheassumptionthattheshapeoftheterminationshockisad-
butbytheconditionthatthetotalvectormomentumflowsinthedi- equatelydescribedbyr(θ)andthatthestellarwindisanidealgas
rectiontangentialtotheshockedregion(i.e.thenaturaldirection). withaspecificheatratioγ=5/3.
To determine the shape of the termination shock as a function of Theapplicationofthejumpconditionsrequirestheknowledge
(cid:13)c 0000RAS,MNRAS000,000–000
4 Christieetal.
1.8 writtenas
α=0.005 α=0.3
2
α=0.05 α=0.5 Psw(θ) = γ+1ρwv2w⊥(θ) (19)
α=0.1
(γ−1) m
1.6 T (θ) = pv2 (θ), (20)
sw (γ+1)2 k w⊥
B
wherek andm aretheBoltzmannconstantandthemassofthe
B p
proton,respectively.Fromeq.(20)wealsoretrievethewell-known
θ) 1.4 result that the average thermal energy of the post-shock region,
r( (3/2)k T , is a fraction of the wind kinetic energy, (3/32)m v2,
B sw p w
whereweusedγ = 5/3.Thetemperatureoftheshockedwindat
θ = 0 is independent of the disk thermal pressure, i.e. T (0) =
sw
1.2 (γ−1)m v2/(γ+1)2k .Thiscanbeunderstoodasfollows:atthe
p w B
shockfront,P ∝ (1−λ)−1 isdeterminedbybalancingthepres-
sw
sureswhileitwasshownthatρ (0)alsoscalesas(1−λ)−1.From
sw
the ideal gas equation of state, it directly follows that T (0) ∝
sw
1.0 P (0)/ρ (0)∝const.
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 sw sw
ThethicknessoftheshockedregionHcanbeestimatedusing
θ(rad)
therelationH(θ) ≈ σ(θ)/ρ (θ),whereσisthemasssurfaceden-
sw
Figure2.PlotoftheterminationshockdistanceR,normalizedtoR0,asa sityoftheshockedwind.Thisisdefinedasσ=Φ2m/RsinθΦtand
functionofθfortheαvaluesmarkedontheplot.Thesevaluescorrespond isgivenby
toλvaluesof0.005,0.05,0.1,0.23,and0.35.
g2(θ)
σ(θ)=σ λ . (21)
02r(θ)(1−λ)sinθf (θ)
λ
ofthenormalandtangentialunitvectorsalongtheshocksurface.
whereσ ≡R ρ and
ThelatterisobtainedbyknowingthatΦ flowstangentialtosurface 0 0 d
t
v
Φ g (θ)=2 d(1−cosθ)+r2sin2θ(1−λ). (22)
nˆt≡ |Φt| =A(cid:36)(θ)eˆ(cid:36)+Az(θ)eˆz, (12) λ vw
t
Bycombiningeqs.(17)and(21)thethicknessisfoundtobe
where
A(cid:36)(θ) = (θ−sinθcfoλ(sθθ))(1−λr2) (13) HR((θθ)) = 12γγ+−11(cid:32)vvwd(cid:33)2 fλ(gθ2λ)(sθi)nθ (23)
sin2θ(1−r2) Figure3summarizesthesemi-analyticalresultsontheproper-
A (θ) = , (14)
z f (θ) tiesoftheshockedwind:theangulardependenceofthedensityra-
λ
tioρ /ρ ,thetemperatureT andthethicknessHoftheshocked
and f isdefinedas sw d sw
λ stellarwindshelluptoπ/2.Afewthingsthatareworthcomment-
f (θ)=(cid:104)(θ−sinθcosθ)2(1−λr2)2+sin4θ(1−r2)2(cid:105)1/2. (15) ingfollow:
λ
(i) Ahigherthermalpressureinthediskcausestheshockfront
The normal unit vector outward to the termination shock is then
to displace closer towards the star (see Figs. 1 and 2). Both the
determinedfromeq.(12)andiswrittenas
increaseofλandthedecreaseinr(θ)pointtowardsahigherdensity
nˆ⊥=−Az(θ)eˆ(cid:36)+A(cid:36)(θ)eˆz. (16) rTahtiioseofffetchtebsehcoocmkeesdpsrtoelmlairnewnitnfdortoλt≈heαd>isk0,.1ρρ.sdw,forallanglesθ.
Allquantitiesoftheshockedstellarwind,i.e.inthedownstreamof
(ii) Foragivenλ,theratioρ /ρ becomesmaximumatθ=0,
theshock,willbenotedwiththesubscript‘sw’.Themassdensity sw d
i.e.attheshockfront.Althoughρ isadecreasingfunctionofθ,at
ρ oftheshockedstellarwindisgivenby sw
sw θ (cid:39)π/2,itisonlyafactorof∼1.5lowerthanitsmaximumvalue
γ+1(cid:32)v (cid:33)2 1 (seealsoeq.(A1)inAppendixA).Thus,theshockedstellarwind
ρsw(θ)=ρdγ−1 vd (1−λ)r2(θ). (17) uptothatangleisexpectedtohaveanon-negligiblecontributionto
w
theradiatedpower.
Asρsw ∝ ρd,changesinthedensityofthediskfluidwilldirectly (iii) Thetemperatureoftheshockedwind,whichisdetermined
affectthethermalbremsstrahlungpowerfromtheshockedstellar by the normal component of the stellar wind velocity, has a very
wind (see Sec. 3.3.3). The density of the shocked wind depends weakdependenceonθ.Forθ = 0thetemperatureisindependent
alsoonthethermalpressureofthedisk,notonlyinadirectway fromthethermalpressureofthedisk(seediscussionaftereq.(20)).
throughthe(1−λ)−1term,butalsoindirectlythroughtheshapeof Itcanalsobeseenthatthetemperaturegradientintheθdirection
the shock surface r(θ) (see also Fig. 2). In particular, the density issmoothedoutasthethermalpressureofthediskincreases.For
shouldincreaseforahigherλvalueforallanglesθ,whileatthe typicalvaluesofthewindvelocity,wefindthetemperaturetobein
standoff-distance,i.e.forθ=0,itscalesasρsw(0)∝(1−λ)−1. the∼107Krange(∼keVband).
The normal components of the velocities across the shock (iv) Thewidthoftheshockedregion,whichisboundedbythe
frontarerelatedasvsw⊥=ρwvw⊥/ρsw,where contactdiscontinuityandthestellarwindshockfront,increasesfor
v (θ)≡v ·nˆ =v (−sinθA (θ)+cosθA (θ)). (18) higherλvalues,sincetheshockfrontmovesclosertothestar(see
w⊥ w ⊥ w z (cid:36) alsopoint(i)).Forλ (cid:54) 0.1,wefindthat H/R (cid:28) 1forallangles
The thermal pressure and temperature of the shocked region are uptoπ/2,whichjustifiestheuseofthethin-shellapproximation.
(cid:13)c 0000RAS,MNRAS000,000–000
Wind-DiskInteractions:SgrA* 5
1.2
7
0.55
6
0.50 α=0.005 α=0.3
5 1.1
α=0.05 α=0.5
0.45
4 K) α=0.1
2ρ/(ρu)swd 23 α=0.005 α=0.3 72(θ)/v(×10ww,8 01..90 α=0.005 α=0.3 H(θ)/R(θ) 000...334050
Ts α=0.05 α=0.5
α=0.05 α=0.5
0.8 α=0.1
0.25
α=0.1
0.7
0.0 0.5 1.0 1.5 0.0 0.5 1.0 1.5 0.0 0.5 1.0 1.5
θ(rad) θ(rad) θ(rad)
Figure3.Plotofthedensityratioρsw/ρdu2(leftpanel),thetemperatureTsw(middlepanel),andthethicknessHinunitsofRoftheshockedstellarwindshell
(rightpanel)asafunctionoftheangleθfortheαvaluesmarkedontheplot.Here,u=vd/vwandvw,8=vw/(108cms−1).
Thisbeginstobreakdownforthecaseofaveryhotdisk,where
Table1.Valuesofthefixedparametersusedinthenumericalsimulations.
H/R∼0.35−0.55.
Parameter Symbol Referencevalue
3 NUMERICALSIMULATIONS Stellarwindvelocity(cms−1) vw 108
Mass-lossrate(M(cid:12)yr−1) M˙w 10−7
To test the validity of the assumptions used to derive the semi- Windnumberdensity(cm−3) nw 480×104
analytical expressions presented in Sec. 2, numerical hydrody- Diskvelocitya (cms−1) vd 8×108
namicsimulationswereemployed.Here,wereportonthenumeri- Windinjectionradius(cm) Rinj 2.5×1013
Ratioofthermaltoramdiskpressureb α 0.005,0.1,0.20.3,0.4,0.5
calsetupusedfordescribingtheaccretiondisk-stellarwindinter-
actions(i.e.,initialandboundaryconditions)andpresentthetech- aMeasuredintherestframeofthestar.
nicaldetailsofthenumericalcode.Acylindricalcoordinatesystem bTherespectiveMachnumberofthediskfluidwithrespecttotheshockforthelisted
was used in all simulations. Assuming axisymmetry, the simula- αvaluesare11,2.4,1.7,1.4,1.2,and1.1respectively.
tions were performed in two dimensions and in the rest frame of
thestar,whichwasassumedtobelocatedatadistancez∗alongthe withtheparticularsetup.Alternatively,onecouldinitiallyfillthe
symmetryz-axis. gridofthesimulationboxwiththestellarwindmedium,assuming
ar−2 dependenceofitspropertieswithrespecttothecenterofthe
star.Itwasverifiedthatneithertheresultsnorthedurationofthe
3.1 Numericalsetup
adjustmentperiodwouldchangesignificantlyinthiscase.
Simulations were carried out for λ = 0.005,0.1,0.17,0.23,0.29, (iv) Reflection boundary conditions at the z−axis (inner radial
and 0.35 (equivalently for α = 0.005,0.1,0.2,0.3,0.4, and 0.5). boundary)andoutflowboundaryconditionsatboth(radialandax-
The parameter values used as initial conditions in the numerical ial) outer boundaries were imposed. The outer boundaries had to
simulations are summarized in Table 1. All simulations were al- beplacedsufficientlyfarawayfromthestartoavoidthepropaga-
lowedtoreachasteadystateafteraninitialperiodofadjustment tionofnumericalartifactsproducedatthoseboundariestowardsthe
(fordetails,seeSection3.2).Inordertominimizethedurationof wind-disk interaction region. Typically, the outer radial and axial
theadjustmentperiod,thesimulationswereoptimizedasdescribed boundarieswere,respectively,locatedat1.2×1015cmand2.4×1015
below: cm.
(i) Aradialstellarwindwascontinuouslyinjectedintothegrid
fromasphericalsurfaceofradiusR centeredatz .Forawindve-
inj ∗ 3.2 Technicaldetails
locityv ,thewindnumberdensityattheinjectionpointwasdeter-
w
minedbyn = M˙ /(4πR2 v m ).Thewindthermalpressurewas High-resolutionsimulationsofthewind-diskinteractionwereper-
w w inj w p
chosentobeasmallfractionofthewind’srampressure,namely formedusingtherelativistichydrodynamicscodemrgenesis(Mim-
P =α P =α n m v2 withα =10−3or,P =10−8n m c2. icaetal.2009).Auniformnumericalgridwithatypicalspacing
w,th w w w w p w w w,th w p
SuchlowvaluesofP werenecessaryforavoidinganyunwanted ∼ 6.67 × 1011 cm was used. In terms of numerical resolution,
w,th
windacceleration. this corresponds to a resolution of 1800 points in the radial and
(ii) Auniformdiskmediumwascontinuouslyinjectedfromthe 3600pointsintheaxialdirection.mrgenesisusesathirdorderto-
z = 0 boundary with velocity v , number density n and thermal talvariationdiminishingRunge-Kuttascheme(Shu&Osher1988)
d d
pressureP =αn m v2. forthetimeintegrationandthepiecewise-parabolicmethod(PPM;
th d p d
(iii) Initially, the grid of the simulation box was filled with a Colella & Woodward 1984) for the spatial interpolation. The in-
uniformdiskmediummovingalongthezdirectionandhavingthe tercellfluxesarecomputedwiththeMarquinafluxformula(Donat
samepropertiesastheseofthefluidthatwasbeinginjectedatthe et al. 1998). The fluids are assumed to obey the ideal gas equa-
boundaryz = 0.Thedurationoftheadjustmentperiodisreduced tionofstatewiththeadiabaticindex5/3.Thecodeisparallelized
(cid:13)c 0000RAS,MNRAS000,000–000
6 Christieetal.
usingMPI3.AllsimulationswereperformedontheTirantsuper-
computer4.Inallcases,thesimulationtimewassetto∼104hours,
whilethetypicaladjustmentperiodlasted∼ 103 hours.Snapshots
ofthegridstateweresavedevery(cid:39) 20hours,allowingustofol-
lowthetemporalevolutionofthewind-diskinteraction.Eachsnap-
shotcontainedthedensity,pressureandvelocitymaps(fordensity
maps,seeFig.4)thatwereusedtocomputethefree-freeemissivity
(e.g.Fig.7).
3.3 Comparison
Wefirstperformedaqualitativecomparisonbetweentheresultsof
the semi-analytical model and the numerical simulations regard-
ingtheshapeoftheshocksurfaceandproceededwithaquantita-
tivecomparisonofthepropertiesoftheshockedstellarwind,such
astemperatureanddensityangularprofiles.Forthispurpose,after
verifyingthatasteadystatehasbeenreached,thesimulationsnap-
shotsfordifferentλvalueswerepost-processedinordertoobtain
thefollowing:(i)thecharacteristicdistancesR andR ,(ii)thetem-
0 1
peratureanddensityprofilesoftheshockedstellarwindshell,and
(iii) the thermal bremsstrahlung power emitted from the shocked
stellarwind.
3.3.1 Geometricalcharacteristics
Figure4presentstwo-dimensionaldensitymapsasobtainedfrom
the simulations for α = 0.005,0.1 and 0.3. In the first two cases
the star is positioned at (x ,z ) = (0,5), while in the third case,
(cid:63) (cid:63)
itslocationis(0,4).Thedensityvalues(inunitsof104 cm−3)can
be read from the colour tables included in each panel. The semi-
analyticalresultsforRandR+H asafunctionoftheangleθare
overplotted with thick yellow lines. The results for θ (cid:54) π/2 are
shownwithsolidlines,whereasdottedlinesareusedforlargeran-
gles. Inspection of Fig. 4 shows that the derived curves straddle
alongtheinterfacesfairlywelluptoanangleofπ/2.Atlargeran-
gles, the results start to diverge from the numerical ones; hence,
thereasoningfortheθplottingrangeinthefiguresofSec.2.Fur-
thermore,thenumericalresultsverifytheassumptionofahomo-
geneouspost-shockregion,atleastuptoπ/2.Themainreasonfor
thedeviationbetweenthesemi-analyticalandnumericalresultsis
thegrowthofKelvin-Helmholtzinstabilitiesthatcannotbetreated
within the adopted analytical framework. These take place at the
interface separating the shocked disk fluid and the shocked stel-
lar wind (i.e. the contact discontinuity). The mixing of the fluids
spreads over a larger volume as the thermal pressure of the disk
increases,whileitleadstomoreturbulentflows.
Regardless,thesemi-analyticalmodelcapturestheroughfea-
tures of the shocked wind shell. The only appreciable difference
betweenthenumericsandanalyticsforθ (cid:46) π/2isidentifiedclose
tothesymmetryaxis(seebottompanelsinFig.4)andisanumer-
ical artifact caused by the reflection boundary along the symme-
tryaxis.Aswilldescribebelow,thisdeviationisimportantwhen
comparing geometrical characteristics, but for the comparison of
volume-integratedquantities,suchasthethermalradiatedpower,it
doesnothaveasignificanteffect.Nevertheless,thisdeviationtends
todiminishastheresolutionincreases.
Figure4.Densitymapsasobtainedfromthehydrodynamicsimulationsfor
α=0.005,α=0.1,andα=0.3.Thesemi-analyticalresultsfortheposition
3 MessagePassingInterface,http://www.mpi-forum.org. oftheterminationshockR(θ)andthecontactdiscontinuityR(θ)+H(θ)are
4 More information (in Spanish): http://www.uv.es/siuv/cas/ shownasthickgreenlinesforθ (cid:54) π/2(solidlines)andθ > π/2(dotted
zcalculo/res/descripcion.wikiattheUniversityofValencia. lines).Theactualvalueofthedensity(inunitsof104cm−3)canberead
fromthecolorbarafterraisingthelistedvaluestothepowerof4/3.
(cid:13)c 0000RAS,MNRAS000,000–000
Wind-DiskInteractions:SgrA* 7
5500
×:α=0.1 2200 ×:α=0.1
⊙:α=0.3 ⊙:α=0.3 ×
×
4400
1155
×
%Error 3300 × × Error ⊙ ×
% 1100 ⊙
d ⊙ ×
ρ/ρsw 2200 × ⊙ ⊙ ⊙ Tsw ⊙×
× ⊙
× 55 ×
1100 ⊙ ⊙ × ×
⊙ × ⊙
× ⊙
⊙ ⊙ ⊙
00 00
00..00 00..55 11..00 11..55 00..00 00..55 11..00 11..55
θ(rad) θ(rad)
Figure5.Percentageerrorofthedensityratio(leftpanel)andtemperatureoftheshockedstellarwind(rightpanel)obtainedfromthesimulationsascompared
withthetheoretical(i.e.semi-analytical)expressionsfordifferentθvalues.Theshadedareadenotestheangularregionwherenumericalartifactsrelatedtothe
inneraxialboundaryarenon-negligible,thusnotallowingforapropercomparisonwiththesemi-analyticalresults.
Thestandoff-distance,R0,marksthestartingpointofthean- Table 2. Distance of the termination shock at θ = 0 and θ = π as de-
alyticalmodelandisthebestestimateforthelocationoftheter- rivedfromthesimulationandthesemi-analyticalmodel.Forα = 0.005,
minationshockatθ=0.Similarly,acharacteristicdistanceR1can themodelassumptionrequiredforthedeterminationofR1 breaksdown
bedefinedatθ≈π(seeSec.2).Botharefoundbymatchingpres- (fordetails,seetext).
suresoneithersideoftheirrespectiveinterface.Thesevalueswere
determinedbyfirstcreatingapressuremap,similartothedensity α λ R0(in1014cm) R1(in1014cm)
mapsinFig.4,ofasnapshottakenafterthesimulationhasreached
model simulations model simulations
asteady-state.Thefirstnon-negligiblejumpsinthepressurefound
at z < z and z > z represent the locations of R and R , re- 0.005 0.005 0.68 0.60 9.68 2.52
(cid:63) (cid:63) 0 1
spectively.TheresultsaresummarizedinTable2.Overall,agood 0.1 0.10 0.65 0.54 2.06 1.56
agreement is found: the distances determined by the simulations 0.2 0.17 0.63 0.52 1.54 1.20
scalewithαaspredictedbytheanalytics(e.g.seeeq.(1)),while 0.3 0.23 0.60 0.51 1.25 0.95
0.4 0.29 0.58 0.50 1.08 0.82
the fractional errors are ∼ 20% for all cases with α > 0.1. For
0.5 0.35 0.55 0.48 0.93 0.77
colderdisks(α (cid:28) 0.1),theoriginalassumption,thatthepressure
of the post-shock wind is of the same order as the disk pressure,
breaksdown,thuspreventingusfromobtainingareasonablevalue changes;thepressureate.g.θ = π/2canvarybyasmuchasone
forR1(seealsoTable2).Inthespecificexampleofα=0.005,the orderofmagnitudewhichcorrespondstosmallvariationsinR1.
numericalsimulationsshowthatP (cid:29)P .
sw th
Thequantitativedifferencesfoundbetweentheanalyticaland 3.3.2 Temperatureanddensityprofiles
numerical values of R are most likely caused by numerical arti-
0
Thedensityandtemperatureoftheshockedstellarwindregionas
factsattheinneraxialboundaryofthesimulation.Byfurtherex-
aminingthen andv oftheunshockedwindclosetoθ=0,within afunctionofθaredescribedbytheanalyticalexpressions(20)and
w w
(17) presented in Sec. 2. In order to compare the derived results
theregionboundedbytheshockfrontandtheinjectionradius,it
wasfoundthatn istwentypercentlowerthanthespecifiedinitial with those from the simulations, density and temperature values
w
conditionforα = 0.005whileitisonlytenpercentlowerforthe wereextractedfromtheirrespectivetwo-dimensionalmapsfordif-
remainingαvalues. ferent θ values while moving along a curve that falls within the
shell.Basedontheassumptionthatthesequantitiesareradiallyin-
The deviations in R , displayed in Table 2, are much larger dependent(refertothesimulationsnapshotsinFig.4foravisual
1
than those in R . Moreover, the distance R oscillates during the confirmation),anychoiceofcurvethatoverlapswiththeshocked
0 1
simulationaboutameanvaluebutneverreachesastationaryvalue, shellisacceptable.Here,acurvewaschosenwitharadialdistance
incontrasttoR .BothresultsarerelatedtothegrowthofKelvin- ofr(θ)+ H(θ) fromthestar.
0 4R0
Helmholtzinstabilitieswithinasignificantfluidvolumebehindthe Themainobjectiveinthecomparisonofthetemperatureand
star (see Fig. 4). On the one hand, the instabilities cause waves density profiles is to determine the deviation between the semi-
to propagate towards the axis (behind the star) upon which they analyticalandnumericalresults.Forthisprocedure,thepercentage
are reflected. The reflected waves, in turn, make the shape of the error was computed while adopting the semi-analytical results as
shock surface far beyond the star to oscillate. On the other hand, the exact values. The results of the comparison for α = 0.1 and
the shocked disk medium behind the star is very complex due 0.3aredisplayedinFig.5.Ineachplot,anglesclosetozero,rep-
to Kelvin-Helmholtz instabilities, thus leading to large pressure resentedbytheshadedregioninthefigure,were excludedinthe
(cid:13)c 0000RAS,MNRAS000,000–000
8 Christieetal.
comparisonforreasonsexplainedintheprevioussection.There- λ
sultsforthetemperature(rightpanelinFig.5)areingoodagree- 0.0480.091 0.13 0.17 0.20 0.23 0.26 0.29 0.31 0.33 0.35 0.38
5
ment with the semi-analytical model and fall within 15% of the
predictedvalues.Asfortheangularprofileofthedensityratio(left ⋄
panelinFig.5),agoodagreementisfoundforbothα = 0.3and ⋄ ⋄ ⋄
0.1withmosterrorsfallingwithin35%ofthetheoreticalvalues. ⋄ ⋄
However,thereisaslightdeviationfoundintheα = 0.1casefor
largeangles.Thisdeviationisreasonableforanglesclosetoπ/2,
)
1
forthisisthepointatwhichthemodeltendstobreakdown. - 1
s
g
r
e
2
3.3.3 Thermalbremsstrahlungluminosity 30 0.50 ×
1 ×
× × ×
Adisskth,ethsetesllhaorcwkeinddgiassshboecgkinesdtboycthoeolinvtiearatchteiormnawlitbhretmhessatcrcarheltuionng (hell × ×
Ls
emission.Thetotalradiatedthermalluminosityfromtheshocked
shellisgivenby
L =2π(cid:90) π/2sinθdθ(cid:90) Rmax(θd)R(cid:48)R(cid:48)2d2W(θ), (24) 0.10 PureHydrogen
shell dtdV
0 R(θ) SolarMetallicity
whereR (θ)≡R(θ)+H(θ),d2W/dtdVisthefree-freeemissivity,
max 0.05
andtheintegrationisperformeduptoθ=π/2wheretheagreement 0.0 0.1 0.2 0.3 0.4 0.5 0.6
α
betweenthesemi-analyticalmodelandthesimulationsisgood.The
aboveexpressionalsoimpliesthatthereisnogradientofthetem-
Figure6.Logarithmicplotofthethermalbremsstrahlungluminosityemit-
peratureanddensityintheradialdirectionwithintheshell,namely tedfromtheshockedstellarwindshellasfunctionofα.Theresultsobtained
that the shocked fluid in the shell is homogeneous. This is veri- assuminggaswithsolarmetallicityandpurehydrogenareplottedwithred
fiedbythesimulationresults(seeFig.4).Forthecaseofawind and black lines, respectively. The results from simulations for various α
composedofpurehydrogen,analyticalexpressionsforthefree-free valuesareoverplottedwithsymbols.
emissivityareavailable(seee.g.eq.(5.15b)inRybicki&Lightman
(1986)). In the more realistic scenario of a stellar wind with so-
larmetallicity,appropriatetablesforthefree-freecoolingfunction
wereused(Sutherland&Dopita1993). mentbetweentheresultsofthesemi-analyticalmodelandthesim-
Beforeperforminganycomputationoftheradiatedpowerpro- ulationsforbothgascompositionsconsidered.Theoutcomeofthe
ducedbytheshockedwind,itisimportanttoremoveanycontribu- comparisonwasnottrivial,giventhattheradiatedpowerisaconvo-
tions from the unshocked wind and disk. The emission from the lutionofthegastemperature,densityandemittingvolume.When
unshockeddisk,inparticular,isexpectedtobenon-negligiblefor solarmetallicityistakenintoaccount,anincreaseoftheradiated
largeαvaluesandisconsideredtobeabackgroundemissionfor powerisfound,forallαvalues,comparedtothecaseofpurehy-
thepurposesofthepresentstudy.Toremovethese‘unwanted’re- drogen.Theeffectbecomesmoreprominentforhotterdisksandis
gionsfromthesimulationsnapshots,certaincutsinthetemperature ofimportancefortheapplicationtoSgrA*,aswillbeshownnext
mapsweremadebysettingthetemperatureofspecificdatapoints inSec.4.
tozero.Throughouteverysimulation,itwasfoundthattheshocked Forfluidscomposedofpurehydrogen,inparticular,thegood
wind never reaches temperatures below 106 K. Cutting any data agreementbetweentheresultsallowsustousethesemi-analytical
pointwithatemperaturebelowthisvalueallowedfortheremoval modelinordertoderivescalingsofthebremsstrahlungluminosity
of unwanted contributions from the cool, unshocked wind. A cut onparameters,suchasthewindvelocityanddiskthermalpressure.
is also made for regions with temperatures above 108.5 K, as this Thebolometricbremsstrahlungpoweroftheshellmaybewritten
is the maximum temperature value included in the cooling factor as
tables(Sutherland&Dopita1993).
L ∝M˙3/2n1/2v−3/2v (0.3+0.47α1.41). (26)
Thetotalbremsstrahlungpowerproducedbyvariousregions shell w d w d
withinthesimulationiscomputedforthecasesofpurehydrogen Theappropriatescalingisdeterminedbyfirstnotingthatthefree-
andsolarmetallicitiesbyconvertingthegeneralformofthedouble freeemissivity,givenineq.(24),scalesasT1/2 andn2 (seealso
sw sw
integralineq.(24)intoadouble,discretesum,namely eq. (5.14a) in Rybicki & Lightman (1986)), while the total lumi-
(cid:88)(cid:88) dW(x,z) nosityscalesasR3.Byuseofeqs.(1),(17),and(20)onemayac-
L=2π x ∆z∆x (25) curatelyobtainthe0 appropriatescalingsfor M˙ ,n ,v ,andv .To
dtdV w d w d
z x obtaintheappropriatescalingforα(i.e.lasttermintheright-hand
where ∆x and ∆z are the spacing between each data point in the sideofeq.(26)),anon-linearfitwasperformedontheblackpoints
xandzdirections,respectively.Thetotalradiatedpowerproduced inFig.6.
bytheshellisdeterminedbyeq.(25)whilelimitingtheangleof Similarmethodscanbeusedtodeterminetheappropriatepa-
integrationuptoθ=π/2.TheresultsarepresentedinFig.6where rameterscalingsforastellarwindcompositionofsolarmetallici-
thesimulationmarkersarerepresentativeofthesixαvaluesgiven ties.Forthiscase,theluminosityhasthesamescalingforM˙ and
w
atthebeginningofSec.3.Theresultsobtainedassuminggaswith n asineq.(26)whilethedependenceonαis(1.83+2.2α1.18).The
d
solarmetallicityandpurehydrogenareplottedwithredandblack αscalingismadebyperforminganon-linearfittotheredpointsin
lines, respectively. Inspection of the figure shows a good agree- Fig.6.
(cid:13)c 0000RAS,MNRAS000,000–000
Wind-DiskInteractions:SgrA* 9
25
)
-1 20
s
g
r
e 15
2
3
0
1
10
×
(
L
5 α=0.1 α=0.3 α=0.5
α=0.2 α=0.4
0
0 5 10 15 20 25 30
z(×1014cm)
Figure8.Cumulativebremsstrahlungluminosity,L((cid:54)z),calculatedassum-
ingsolarmetallicity,fortheαvaluesmarkedontheplot.Theverticalline
indicatethepositionofthestar,z∗=5×1014cm,inthesimulationruns.
TheresultslistedinTable3alsoraisethequestionofwhether
thesizeofthesimulationboxissufficientlylargeenoughtocapture
Figure 7. Free-free emissivity map as obtained from the simulation for
alltheradiatedpowerproducedbythemixedfluids.Thisissuecan
α = 0.3, assuming gas composed of pure hydrogen. The star is located
at(x(cid:63),z(cid:63))=(0,4). b(cid:82)ezdazd(cid:48)dLr(ezs(cid:48))s.edThbiysicsalpcruelsaetnintegdthineFciugm.u8lafotirvethleumfivineoαsitvyalLue((cid:54)slizs)te≡d
atthebeginningofSec.3.1whileassumingsolarmetallicity.The
verticallineindicatesthepositionofthestar,z = 5×1014 cm,in
Table3.Totalluminosityproducedfromsimulationsincludingcontribu- ∗
thesimulationruns.Inbothcases,itwasfoundthatthecumulative
tionsfromthelongtailofmixedfluidsbehindthestar.
luminositybeginstoconvergetoafinalvalueatz ≈ 2.5z ≈ 12×
∗
1014cm.However,thegradientalongthesymmetryaxis,i.e.dL/dz,
PureHydrogen SolarMetallicity
ismuchsteeperinhotterdiskcases.Theincreaseinαalsocauses
α λ L(×1032ergs−1) L(×1033ergs−1) anincreaseinthetotalluminosityofthesimulationsnapshots;this
isfoundtoscaleasα0.56.
0.005 0.005 0.736 0.511
0.1 0.1 2.008 1.168
0.2 0.17 3.017 1.541
0.3 0.23 3.752 1.759
0.4 0.29 5.292 2.424 4 APPLICATIONTOSGRA*
0.5 0.35 6.203 2.815
Sgr A*, the compact radio source associated with the supermas-
siveblackholeinourGalacticcenter,spendsmostofitstimeina
low-luminosity,X-rayemissionstatewithanabsorptioncorrected
3.4 TheroleofKelvin-Helmholtzinstabilitiesonthe
luminosity of ≈ 2.4×1033 erg s−1 (Baganoff et al. 2003; Genzel
bremsstrahlungluminosity
et al. 2010). However, X-ray flares with durations ranging from
Aresultthatcouldnotbepredictedfromthesemi-analyticalmodel minutestohoursandluminositieswellabovethequiescentemis-
presentedinSec.2istheefficientmixingofthediskandwindflu- sionofSgrA*arefrequentlyobserved(e.g.Baganoffetal.2001;
idsduetoKelvin-Helmholtzinstabilities.Althoughthedensityof Barrie`reetal.2014).Thephysicalcauseoftheserapidflaresisnot
thegasisnotashighasintheshockedshellregion,theoccupied yetunderstood.Someofthehypotheticalscenariosusedtomodel
volume by the mixed fluids is considerably larger. This is illus- theseoccurrencesareanepisodicoutflowtriggeredbymagneticre-
trated in Fig. 7 where a large scale, two-dimensional map of the connection(Yuanetal.2009),enhancedmassaccretion(Tagger&
free-freeemissivity,inthecaseofpurehydrogen,isplottedfora Melia2006),orapossiblehotspotintheaccretionflow(Dodds-
hotdiskwithα = 0.3.Althoughthedetailsoftheturbulentstruc- Edenetal.2010).
turesarenotvisible,Fig.7clearlyshowsthattheemissivityfrom Month-longX-rayflaresareexpectedfromthepericenterpas-
themixedfluidsatdistancesof14×1014 cmawayfromthestar sage of certain S-stars as they precess through the accretion disk
is non-negligible. To better assess the role of Kelvin-Helmholtz ofSgrA*(Giannios&Sironi2013).Thepresentworkwillbeori-
instabilitiesonthebremsstrahlungluminosity,theradiatedpower entedtowardsS2,whoseupcomingpericenterpassagewilloccurin
wascomputedbyuseofeq.(25)foralargervolumewithinthesim- 2018.Thisearly,B-dwarfstarischaracterizedbyamass-lossrate
ulation.Thisvolumeincludedtheshockedshellregionforθ>π/2 of M˙ (cid:54) 3×10−7M yr−1,stellarwindvelocitiesofv = 108 cm
w (cid:12) w
andtheregionsextendingbeyondR ,whichstretchesanywherebe- s−1,andapericenterorbitalvelocityofv = 8×108 cms−1.Dur-
1 d
tween7×1014−2×1015cmabovez .Theresultsaresummarizedin ingthetransitofthestarthroughthedisk,thestarwillencounter
∗
Table3anddemonstratethatthemixingregionproducesasignifi- sectionsofthediskwithdifferentdensities.Inthefollowingsec-
cantcontributiontotheluminosityoftheentiresimulation.Thisis tion,thischangeindensityismimickedbyemployingtheuseof
ofthesameorderorevenlargerthanthatproducedbytheshocked a time-dependent density simulation, without adopting a specific
shellascomparedwithFig.6. modelfortheaccretiondisk.
(cid:13)c 0000RAS,MNRAS000,000–000
10 Christieetal.
4.1 MimickingthepericenterpassageofS2
40
PureHydrogen
As shown in Sec. 3, a significant fraction of the bolometric
SolarMetallicity
bremsstrahlungemissionoriginatesfromalongtailofmixedflu- nd(T)
ids.Fortheastrophysicalapplicationweareinterestedin,itisnot
knownaprioriiftheS2pericenterpassagelastslongenoughasto
30
allow for the full mixing of the fluids to distances far behind the
star.Ift andt denotethetwocharacteristictimescalesofthe
per mix
)
pericenter passage and mixing, the maximum bremsstrahlung lu- -1
s
minosityisexpectedforthecasetper (cid:29)tmix.Toaddressthisissue, rg
time-dependentsimulationsarerequired. e 20
2
Inwhatfollows,thepericenterpassageofthestarismimicked 30
1
byvaryingthediskdensityusedasaninitialconditioninthesim- ×
(
ulation. The temporal dependence of the disk number density is L
modeledas
10
9
n˜d(T)=1+ cosh(cid:16)T−25T0(cid:17) (27)
3T0
where T = 7.5 × 105 s and n˜ = n /(104cm−3). The density
smoothly0increases from its referdence vdalue at T = 0, reaches a 0
maximumvalueofn˜ = 10at∼ 5200hr,anddecreasestoitsini- 100 150 200 250 300 350
d
tialvalueonwards.Thedensitypeaksonatimescaleof∼3T ∼1 T(days)
0
month,whichiscomparabletothetimeittakesS2tocompleteits
Figure9.Lightcurvesproducedfromtimedependentdensitysimulation
pericenterpassage.Duetothevariablediskdensity,thesurfaceof
forthecaseα = 0.1.Thecolourcodedcurvescorrespondtothedifferent
the termination shock oscillates by factor of 2 in distance, lead-
stellarwindcompositionsasmarkedontheplot.
ing also to a variable bremsstrahlung emission. Interestingly, the
regionofmixedfluids,describedinSec.3.4,wasquicklyformed
beforethepeakdensitywasreached,whileitwasfoundtoextend
ferencecanbeexplainedbythefollowing.Thepreviousworkof
toafairlylargedistancebeyondthestar.Theinclusionofthistail
GS13assumedthatacircularterminationshockformedinthewind
region to the total luminosity of the system is, therefore, a valid
at a radial distance of R and that the shocked fluid radiates on
0
assumption.
anexpansiontimescalewhiletraversingadistanceofR afterthe
0
Trackingtheevolutionoftheluminosityiscompletedbycal-
terminationshock.Thisisequivalenttohavinganemittingregion
culating the thermal bremsstrahlung emission from several snap- withavolumeV ∼4πR3.Thisworkhasshownthatthethicknessis
shots, as described in Sec. 3.3.3. The results are summarized in insteadH ∼ R /4mean0ingthatthevolumeoftheemittingregion
Fig.9wherethetwolightcurvesareplottedforthedifferentwind isV ∼2πR2H0∼0.5πR3.Itisaninterestingcoincidencethatwhen
compositions under consideration. For comparison, the temporal effectsofth0elargeback0regionareincludedintothetotalluminos-
variation of the density is overplotted (green line). The dashed,
ityofthesystem,theluminositiesproducedbythisworkandGS13
blue line denotes the X-ray luminosity corresponding to the qui-
areofthesameorder.
escent emission of Sgr A*. Any model prediction falling in the
In Sec. 3.3.3 a scaling relation of the shell’s bremsstrahlung
blue-shadedregionistoolowtobedistinguishedfromthequies- power on the model parameters M˙ ,n and α was derived (see
w d
cent emission. For the case of solar metallicity, it was found that
eq. (26)). The derivation of a similar scaling relation for the
amonth-longflareabovethequiescentemissionisexpected,with
bremsstrahlungpowerfromthewholeemittingvolume(i.e.,shell
apeakreachingL ≈ 4×1033ergs−1.Thisisaratherconservative
plustail)isnotstraightforward.Themainreasonisthatthevolume
value, given that a higher luminosity is expected for hotter disks
ofthetailregioncouldnotbedeterminedanalytically.Aslongas
withα>0.1(seealsoFig.8).
thetailofmixedfluidscanbeapproximatedbyacylindricalvolume
Figure9showsthatthereisatime-lagbetweenthedensityand withcrosssectionalareaA ∝ R2 andlength∝ R ,nolargedevia-
photontemporalprofiles.Thepeakofbothlightcurvesappearsata 0 0
tionsfromthescalingsineq.(26)areexpected.Thishasbeenveri-
latertimethanthepeakofthedensityprofile.AsstatedinSec.3.4, fiedbyrunningvarioussteady-statesimulationsfordifferentvalues
thebackareaofmixedfluidsthatispronetoKelvin-Helmholtzin- of M˙ whilekeepingallotherparametersfixed.Itwasdiscovered
w
stabilities,contributesalargeportiontothetotalluminosityofeach thatthetotalluminosityscalesasL∝M˙1.3whichverifiesourprevi-
w
simulationsnapshot.Thetime-lagbetweenthepeakluminosityand
ousstatementthatthevolumeoftheemittingregionroughlyscales
thepeakdensityreflectsthetimerequiredforthetailregiontofeel asR3.Bytrackingtheevolutionofthetotalbremsstrahlungpower
theeffectsofthedensitychanges. 0
andthediskdensityduringthetime-dependentsimulation,wefind
L ∝ n0.512.Thedependenceondensityisthereforesimilartothat
d
of eq. (26). We note that the scaling relation was determined by
performingalinearfittoaplotoflog(L/L )vs.log(n /n ).
5 DISCUSSION 33 d d,4
Althoughnospecificmodelwasadoptedfortheaccretiondisk
Thismodelcanbeconsideredanextensionoftheworkperformed duringthiswork,itbehoovesustobrieflydiscussafewofthecur-
by GS13. Upon comparing the two models, similarities can be rentproposedmodels.ThroughGRMHDsimulations,theprofileof
found in the temperature and density of the shocked shell. How- thenumberdensitywasshowntoscaleasR−1 withtypicalvalues
ever, even with these similar values, the two works differ in the ataradialdistance∼ 3000R being∼ 104 cm−3 (Sadowskietal.
g
amountofbremsstrahlungpowerproducedbytheshell.Thisdif- 2013;Tchekhovskoy&McKinney2012;McKinneyetal.2012).If
(cid:13)c 0000RAS,MNRAS000,000–000