The fate of supernova remnants near quiescent supermassive black holes

2.2 MB·English

by  Alex Rimoldi

Mon.Not.R.Astron.Soc.000,1–20(2014) Printed14January2015 (MNLATEXstylefilev2.2)
The fate of supernova remnants near quiescent supermassive
black holes
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1 A. Rimoldi,1⋆ E. M. Rossi,1 T. Piran2 and S. Portegies Zwart1
0
1LeidenObservatory,LeidenUniversity,POBox9513Leiden,NL-2300RA,TheNetherlands
2
2RacahInstituteofPhysics,TheHebrewUniversity,Jerusalem91904,Israel

n
a
J
Accepted2014December9.Received2014November27;inoriginalform2014October10
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1

ABSTRACT
]
E There is mounting observationalevidence that most galactic nuclei host both supermassive
H black holes (SMBHs) and young populationsof stars. With an abundance of massive stars,
core-collapse supernovae are expected in SMBH spheres of influence. We develop a novel
.
h numerical method, based on the Kompaneets approximation, to trace supernova remnant
p (SNR)evolutioninthesehostileenvironments,whereradialgasgradientsandSMBHtidesare
- present.WetracetheadiabaticevolutionoftheSNRshockuntil50%oftheremnantiseither
o
intheradiativephaseorissloweddownbelowtheSMBHKeplerianvelocityandissheared
r
t apart. In this way, we obtain shapes and lifetimes of SNRs as a function of the explosion
s
a distancefromtheSMBH,thegasdensityprofileandtheSMBHmass.Asanapplication,we
[ focushereexclusivelyon quiescentSMBHs, becausetheir lightmay nothamperdetections

ofSNRsandbecausewecantakeadvantageoftheunsurpasseddetailedobservationsofour
1
GalacticCentre.Assumingthatpropertiessuchasgasandstellarcontentscaleappropriately
v
withtheSMBHmass,westudySNRevolutionaroundotherquiescentSMBHs.Wefindthat,
9
1 for SMBH masses over∼ 107 M⊙, tidaldisruptionof SNRs can occurat less than 104 yr,
8 leadingtoashortenedX-rayemittingadiabaticphase,andtonoradiativephase.Ontheother
2 hand,onlymodestdisruptionisexpectedinourGalacticCentreforSNRsintheirX-raystage.
0 ThisisinaccordancewithestimatesofthelifetimeoftheSgrAEastSNR,whichleadsusto
. expectonesupernovaper104yrinthesphereofinfluenceofSgrA*.
1
0
Key words: accretion, accretion discs — black hole physics — hydrodynamics— shock
5
waves—ISM:supernovaremnants—galaxies:nuclei
1
:
v
i
X
r 1 INTRODUCTION tionenergyiscarriedawaybyradiation(Ichimaru1977;Reesetal.
a 1982;Narayan&Yi1994).
There is compelling evidence for a supermassive black hole
In addition to the ubiquity of SMBHs, young stellar
(SMBH)withamassof4.3×106M⊙inthenucleusoftheMilky populations and appreciable star formation rates are com-
Way, associated withtheSgr A*radio source. Thestrongest evi-
mon in many quiescent galactic nuclei (Sarzietal. 2005;
dencecomesfromtheanalysisoforbitsoftheso-called‘S-stars’
Walcheretal.2006;Schrubaetal.2011;Kennicutt&Evans2012;
verynearthiscompactobject,suchasthatofthestarS2withape-
Neumayer&Walcher2012).1Thisisseenmostclearlyintheabun-
riodofonly16yrandpericentreof 102au(Scho¨deletal.2002,
∼ danceofearly-typestarsinthecentralparsecoftheMilkyWay(see
2003; Ghezetal. 2003; Eisenhaueretal. 2005; Ghezetal. 2008;
Doetal.2013a,b;Luetal.2013,forsomerecentreviews).More-
Gillessenetal.2009).
over, it appears that star formation in the Galactic Centre region
Most other massive galaxies contain SMBHs (Marleauetal.
hasbeenapersistentprocessthathasincreasedoverthepast108yr
2013), some with masses as high as 1010 M⊙ (McConnelletal. (Figeretal.2004;Figer2009;Pfuhletal.2011).Overthattime,an
2011). The observed fraction of active nuclei is no more than a
estimated&3 105M⊙ofstarshaveformedwithin2.5pcofthe
few per cent at low redshifts (Schawinskietal. 2010), and most ×
SMBH(Blumetal.2003;Pfuhletal.2011).
galactic nuclei house very sub-Eddington SMBHs, like Sgr A*
Continuousstarformationingalacticnucleiwillregularlyre-
(Melia&Falcke2001;Alexander2005;Genzeletal.2010).These
plenish the supply of massive stars in these regions. This natu-
SMBHsarebelievedtobesurroundedbyradiativelyinefficientac-
rallyleadstotheexpectationoffrequentcore-collapsesupernovae
cretion flows (RIAFs), where only a small fraction of the accre-
1 Evidence for recent star formation has also been seen around active
galacticnuclei(AGN;forexample,Daviesetal.2007).However,activenu-
⋆ E-mail:[email protected] cleiarenotthesubjectofthisstudy.
2 Rimoldietal.
insuchenvironments.Asanexample,Zubovasetal.(2013)show tribution.WethenspecializeittoaquiescentSMBHenvironment.
that, per 106 M⊙ of stellar mass formed in the Galactic Centre, Section5outlinesthegalacticmodelsusedfortheenvironmentsof
approximately one supernova per 104 yris expected for the past thesupernovasimulations.Section6presentsourresultsforSNR
108yr. shapes and lifetimes. Our concluding remarks are found in Sec-
Onlyonesupernovaremnant(SNR)candidatehasbeeniden- tion7.
tifiedclosetotheSMBHsphereof influence(SOI):anelongated
shell known as Sgr A East, at the end of its adiabatic phase. It
has an estimated age of about 104 yr and appears to be engulf- 2 GASEOUSENVIRONMENTSOFQUIESCENT
ingSgrA*withameanradiusofapproximately5pc(Maedaetal. GALACTICNUCLEI
2002;Herrnstein&Ho2005;Leeetal.2006;Tsuboietal.2009).
Inthissection, weoutline theexpected gasdistributionsnear the
Inaddition,thereareacoupleofobservationsthatindirectlypoint
SMBHinquiescentgalacticnuclei.Thesegasdistributionswillbe
towards supernovae in the SOI. The first is CXOGC J174545.5–
usedastheenvironmentfortheSNRmodelexpositedinSections
285829 (‘The Cannonball’), suspected to be a runaway neutron
3 and 4. We will then proceed to scale the general environment
star associated with the same supernova explosion as Sgr A East
discussedherefortheGalacticCentretootherSMBHsinSection5.
(Parketal.2005;Nynkaetal.2013;Zhaoetal.2013).Thesecond
Quiescent SMBHs are surrounded by RIAFs, which are the
istherecentlydiscoveredmagnetarSGRJ1745–2900,estimatedto
environmentsinwhichtheSNRwillevolve.RIAFsarerelatively
bewithin2pcofSgrA*(Degenaaretal.2013;Kenneaetal.2013;
thick, for which the scale-height, H, is comparable to the radial
Reaetal.2013).
distance,R,fromtheSMBH(H/R 1).Themechanismsofen-
Any supernova exploding in the SOI of a quiescent SMBH ≈
ergy transport within the flow vary depending on the model, and
willexpandintoagaseousenvironmentconstitutedmainlybythe
these variations affect the power-law gradient in density near the
SMBH accretion flow, whose gas is supplied by the winds from
SMBH. Advection-dominated accretion flow (ADAF) models as-
massive stars. The density distribution within the flow is there-
sume that much of the energy is contained in the ionic compo-
fore set by both the number and distribution of young stars and
nentofatwo-temperatureplasma.Astheionsaremuchlesseffi-
thehydrodynamicalpropertiesofaradiativelyinefficientaccretion
cient radiators than electrons, energy is advected into the SMBH
regime.Thisinterplaygivesanoveralldensitydistributionthat is
by the ions before it can be lost via radiation (Narayanetal.
abrokenpowerlaw,forwhichthebreakoccurswherethenumber
1995; Narayan&Yi 1995). Additionally, convection-dominated
densityofstellarwindsourcesdropsoff.FortheGalacticCentre,
accretion flow (CDAF) models rely on the transport of energy
thiscorrespondsto 0.4pc(forexample,Quataert2004).
∼ outward via convective motions in the gas (Quataert&Gruzinov
In suchenvironments, weexpect SNRstoevolve differently
2000;Balletal.2001).Finally,theadiabaticinflow–outflowmodel
fromthoseinthetypicallyflatinterstellarmedium,awayfromthe
(ADIOS;Blandford&Begelman1999,2004;Begelman2012)ac-
SMBH. The density gradients have the potential to distort SNRs
counts for windsfromtheflow that expel hot gasbefore itisac-
anddeceleratethemsignificantly.Oncetheexpansionvelocityfalls
creted.
belowtheSMBHvelocityfield,theremnantwillbetidallysheared
For theregion near the SMBH,predicted exponents, ω , of
and eventually torn apart. This can substantially shorten an SNR in
the power law in gas density, ρ, lie in the range of ω = 1/2
lifetimecomparedtothatinaconstant-densityinterstellarenviron- in
to 3/2. The lower and upper limits of ω are derived from the
ment. In turn, this can reduce the expected number of observed in
predictionsoftheCDAF/ADIOSandADAFmodels,respectively.
SNRsingalacticnuclei.
Adrop-offinstellarnumberdensityataradiusR = R fromthe
Sincequiescentaccretionflowsarefedbystellarwinds,which b
SMBH would cause a break in the mass density, ρ, at the same
canbealsopartiallyrecycledtoformnewstarstogether withthe
radius,sinceitisthewindsfromthesestarsthatfeedtheaccretion
gasreleasedbysupernovaexplosions,thescenarioweconsideris
flow.
of a self-regulating environment, where young stars and gas (or,
The best example of a RIAF is that surrounding Sgr A*. It
inotherwords,starformationandaccretionontotheSMBH)are
hasbeenextensivelystudiedtheoreticallyandobservationallyand
intimatelyrelated. Thisholds until a violent event—for example,
willconstituteourprototype.Adensitydistributionfromtheone-
amerger—drivesabundant starsandgasfromlargerscalestothe
dimensional analytic model of wind sources has approximately a
galacticnucleus.ObservationsandmodellingofourGalacticCen-
broken power-law shape with ω = 1 inside the density break
tresupportthispicture.Inparticular,windsfrommassivestarsare in
andω =3outside(Quataert2004).Simulationsofstellarwind
sufficienttoaccountfortheobservedaccretionluminosityandex- out
accretion show comparable density profiles (Cuadraetal. 2006).
ternalgasfeedingisnotrequired(e.g.Quataert2004;Cuadraetal.
Furthermore,thevalueofω =1isconsistentwithGRMHDac-
2006) or observed. Furthermore, there is strong evidence for the in
cretion simulations (for example, McKinneyetal. 2012). Recent
recentstarformationoccurringinsitu(Paumardetal.2006).
observationsusinglongintegrationsinX-raysuggestthatagradi-
In this paper, we determine the morphology and X-ray life-
entofω 1/2mayprovideabetterfittotheinneraccretionflow
timesof SNRs,which, inturn,can beused toconstraintheenvi- in≈
ofSgrA*(Wangetal.2013).
ronmentofSMBHs.WedevelopanumericalmethodtotraceSNR
Wecantherefore,generallydescribetheambientmediumofa
evolutionanddeterminetheirX-raylifetime.Theinfluenceofthe
quiescentSOIwithabrokenpowerlawforthedensityoftheform:
SMBHonSNRswillbeconsideredfirstindirectly,throughitsin-
fluenceonthegaseousenvironment,andthendirectly,throughits R −ωin
ρ R6R
tidalshearoftheejecta. 0 R b
gaseoTuhseenpvapireornmiseonrtgsafnoiuzneddaarsofuonldlowqusi.esSceecntitoSnM2BiHntsr.odSueccetisonth3e ρ(R)=ρ (cid:18) R0(cid:19)−ωout R>R , (1)
b R b
uses analytic methods to qualitatively trace SNR evolution. Sec- (cid:18) b(cid:19)
tion4describesournumericalmethod,whichallowsustofollow forωin 1/2, 1, 3/2 ,ωout =3,usingareferencepointforthe
∈{ }
theevolutionofanSNRinanarbitraryaxiallysymmetricgasdis- densityatR=R awayfromtheSMBH.
0
FateofSNRs near SMBHs 3
Thestrongest observational constraint on thedensity around tributionsingalacticnuclei,Korycansky(1992)—hereafter,K92—
Sgr A* is given by Chandra X-ray measurements at the scale of showed that, with a specific coordinate transformation, a circular
the Bondi radius (R 0.04 pc) of n 130 cm−3 (ρ solution to the Kompaneets equation can be obtained for explo-
0 0 0
2.2 10−22 gcm−3;≈Baganoffetal. 2003≈). The accretion ra≈te sions offset from the origin of a power-law density profile, R−ω
×
closer to the SMBH can be further constrained by Faraday rota- (forω=2).
6
tionmeasurements,thoughtherelativeerrorislarge(Marroneetal. Theearlyejecta-dominatedandlateadiabaticstagesarewell
2007).Indeed,wefindthatfixingthedensityat0.04pcandvary- characterized by the purely analytic solutions for each stage. In
ingω between1/2and3/2producesarangeofdensitiesatsmall between, the solution asymptotically transitions between these
in
radii that fall within the uncertainty in the density inferred from twolimits(thisisknown as‘intermediate-asymptotic’ behaviour;
Faradayrotation.Theradiusforthebreakinstellarnumberdensity Truelove&McKee1999).2 Thelateevolutionoftheremnant,the
andgasdensityintheMilkyWayistakentobeR =0.4pc. radiative stage, occurs when the temperature behind the shock
b
dropstothepointatwhichthereisanappreciablenumberofbound
electrons. Consequently, line cooling becomes effective, the ra-
diative loss of energy is no longer negligible, and the speed of
3 EVOLUTIONOFREMNANTSAROUNDQUIESCENT
the shock will drop at a faster rate. For SNRs in a constant den-
BLACKHOLES:ANALYTICFOUNDATIONS
sityofn 1cm−3,theradiativephasebeginsatapproximately
Here, we outline the physics describing the early stages of SNR 3 104yr≈(Blondinetal.1998).Wedonotmodeltheremnantdur-
×
evolutionthatareofinterestinthiswork.Thetheorydescribedin ingthisphase,butwewillestimatetheonsetofthetransitiontothe
thissectionwillbeusedasthefoundationofageneral numerical radiativestage.
methodtosolvetheproblem,outlinedinSection4.Atthispoint, Inthepresentwork,wemodelSNRsoverthefirsttwo(ejecta-
wedonotdirectlytakeintoaccount thegravitationalforceofthe dominated and adiabatically expanding) stages of evolution in a
SMBH, but instead just the gaseous environment. The gravity of rangeofgalacticnuclearenvironments.Theevolutionbeginswith
theSMBHcanbeignoredwhentheexpansionvelocityoftheSNR a spherically expanding shock, and therefore we do not consider
ismuchlargerthantheKeplerianvelocityaroundtheSMBH.For anyintrinsicasymmetriesinthesupernovaexplosionitself.Collec-
example,aroundSgrA*,atavelocityof104 kms−1 gravitycan tively,anypossibleintrinsicasymmetriesinSNRsarenotexpected
beignoredforradiilargerthan 10−4 pc.Thegravitationalfield to be in a preferential direction, and so they should not bias the
∼
oftheSMBHwillbeaccounted forlater,whenweconsider tidal generalizedresultspresentedhere.
effectsontheexpanding remnant, whichareimportantonlyonce The overall geometry of this analysis is laid out in Fig. 1,
theremnanthassloweddownsignificantly. whichindicatesthemaincoordinates, distancescalesanddensity
A supernova explosion drives a strong shock into the sur- distributions.TheexplosionpointisatadistanceR=a,measured
roundinggasatapproximatelytheradialvelocityoftheejectedde- fromtheSMBH(theoriginofourcoordinatesystem).Theshock
bris.Typically,itisassumedthatasignificantamountoftheejecta frontextendstoradialdistancesR′,measuredfromtheexplosion
iscontainedwithinashelljustbehindtheshockfront(forexample, point.Eachpointalongtheshockisatanangleψ,measuredfrom
Koo&McKee1990).Asitexpands, theshocksweepsupfurther theaxisof symmetry about theexplosion point. Theinitialangle
massfromthesurroundingmedium.Bymomentumconservation, made with the axis of symmetry of each point on the shock, at
thecombinedmassofthefractionofejectabehindtheshockfront t 0,isdenotedψ0.
→
(M )plustheswept-upgas(M )must decelerate. Thedecelera-
ej s
tion is considered to be appreciable when the swept-up mass be-
comescomparable tothatof thedebris, andthereforethisejecta- 3.1 Endoftheejecta-dominatedstage
dominatedphaseholdsforM M .
s ej Inordertoestimatewheretheshockfrontkinematicsappreciably
≪
Thesubsequentadiabaticexpansionoftheshockfrontismod-
deviatefromtheejecta-dominatedsolution,weintegratetheback-
elledwiththeassumptionthatlossesofenergyinternaltotherem-
ground density field along spherical volume elements swept out
nant are negligible. For this decelerating regime, the Rankine–
bytheexpanding remnant. Thisprovidesanestimateofthemass
Hugoniot strong-shock jump conditions can yield exact similar-
sweptupfromtheenvironment,M .Theejecta-dominatedsolution
s
ity (length scale-independent) solutions for the kinematics of the
istakentoendwhenM isequaltosomespecifiedportionofthe
s
shock front. The evolution is determined by its energy, E, and
ejectamass,Mej.Weuseacanonicalvalueof1M⊙ forthisfrac-
the ambient density, ρ (McKee&Truelove 1995). In all of this
tionofejectamass. Thedistancefromtheexplosion point(along
work,weuseacanonical valueof1051 ergfortheexplosionen- the coordinate R′) at which this occurs is denoted the ‘decelera-
ergy. In auniform ambient medium, theadiabatic stage isclassi-
tion length’, L, here (it also known as the ‘Sedov Length’ in the
callymodelledusingthesphericallysymmetricSedov–Taylor so-
standardtreatmentofSNRsinauniformρ).
lution(Taylor 1950;Sedov1959).Thishasself-similarformsfor
Sinceourdensityprofilesarenotuniform,differentdirections
thesphericalradiusandspeedoftheSNRofR′ (E/ρ)1/5t2/5
ofexpandingejectawillsweepupmassatdifferentrates.Ingen-
∝
andv (E/ρ)1/5t−3/5,respectively,whereR′ismeasuredfrom eral,wemustconsiderasolutionforLthatdependsonψ ,theini-
∝ 0
theexplosionsite. tialangleofeachsurfaceelementoftheshockwithrespecttothe
FollowingtheinitialworkbySedovandTaylor,Kompaneets axisofsymmetry(seeFig.1).Wethereforedeterminethevalueof
(1960)developed anon-linear equationfromthejumpconditions L(ψ )correspondingtosmallsurfaceelementsoftheshockfront.
0
that allows self-similar solutions for the shock front evolution WhentheexplosionoccursclosetotheSMBH,thesolutionisex-
in certain density stratifications. The original work by Kompa- pectedtoconvergetothatofanintegraloverasphere,duetothe
neetsconsideredanatmospherewithexponentialstratification,but
manyothersolutionshavesincebeenobtained(seethereviewby
Bisnovatyi-Kogan&Silich1995,aswellasBannikovaetal.2012 2 For an illustration of this transition, see fig. 2 of Truelove&McKee
andthereferencestherein).Ofparticularrelevancetothegasdis- (1999).
4 Rimoldietal.
shocksegment
R
R′
ρ(R)=ρ0(cid:16)RR0(cid:17)−ωin
ψ
θ
O S
a
R=Rb
ρ(R)=ρb(cid:16)RRb(cid:17)−ωout
Figure1.Basicgeometryoftheproblem.ThesupernovaoccursatapointS,adistanceR=aawayfromtheSMBH,whichislocatedattheorigin,O.The
shockfrontextendstodistancesmeasuredradiallyfromtheexplosionpointSbythecoordinateR′.Theanglemadebyapointontheshock,measuredfrom
theθ=0axisabouttheexplosionpoint,isdenotedψ.Eachpointontheshockhasaninitialangleψ(t 0) ψ0.Theentiredensitydistributionρ(R)can
→ ≡
becharacterizedby:thechoiceoftheinnergradientωin(definingthedensitywithintheshadedcircle),theoutergradientωout,thereferencedensityρ0(ata
referenceradiusR0),andabreakatRbbetweenthegradientsωinandωout.
sphericalsymmetryofthebackgrounddensity.3Therefore,asaref- Theassumptionsandmainequationsofthisprescriptionwillalso
erence,wealsofindtheradiusLofthespherewhosevolumeen- beusedinourfullnumericaltreatmentforarbitrarydensityprofiles
closesM M . (Section4).Weshallgiveheretheanalyticsolutionsforω=1and
s ej
≈
TheexplosionoccursatadistanceR=afromtheorigin.For 3.Thesesolutionswillbeusedtovalidateournumericaltreatment
asinglepower-lawstratification,weusetheexplosionpointforthe (Section4).Theyalsogiveanindicationoftheshockbehaviourin
referencedensity,ρ =ρ(a) ρ ,suchthat abrokenpower-lawdensityprofile,whenitexpandsfullyinterior
0 a
≡
orfullyexteriortothedensitybreak.
R −ω
ρ(R)=ρ . (2) The Kompaneets approximation involves setting the post-
a(cid:18)a(cid:19) shock4 pressure, P′, tobe uniform throughout the shock volume
We consider a small surface element of the SNR at an angle ψ andequal to(somefraction, λ,of) themean interior energyden-
0
overaninfinitesimalsolidangle.Inasinglepower-lawstratification sity.ForanarbitraryvolumeV,
withtheformofequation(2),thelengthL(ψ )canbeestimated
0 (γ 1)λE
fromthemassintegratedthroughR′atagivenangleψ : P′= − , (4)
0 V
ρ aω L(ψ0)R−ωR′2dR′=M . (3) wherein theKompaneets approximation proper isto take λ to be
a ej
Z0 constant. Theratioof specificheatsistaken tobe γ = 5/3both
NotethatweareintegratingoverthecoordinateR′thatextendsra- internalandexternaltotheshock.
Twoadditionalassumptionsinthetreatmentarethatthedirec-
diallyfromtheexplosionpoint,butthatthedensityvariesradially
tionsofthelocalvelocityvectorsalongtheshockfrontarenormal
withthecoordinateRasmeasuredfromtheSMBH.Forintegrals
totheshockfront,andthatthemagnitudeofthevelocityisdeter-
overabrokenpower-lawdensity,thedensitybreakaddscomplica-
minedbytakingthepost-shockpressuretobeequaltothatofthe
tionstotheintegralsanalogous toequation(3).Thesolutionsare
discussedfurtherinAppendixA. rampressureoftheenvironment(ρvs2,whereρisthedensityofthe
unshockedgas)atthatpoint(K92):
Thesemethodsforestimatingthedecelerationlengthprovide
ameansfortestingthelevelofasymmetryanddistancescalesinthe
(γ2 1)λE
einjeScetac-tdioonm6in.1a,tewdhsetraegweeofsheovwoluretisounl,tsa.ndwillbefurtherdiscussed vs(R,t)=s 2ρ(R−)V(t) . (5)
FollowingthecoordinatetransformationofK92,the‘time’is
parametrizedbyy(whichactuallyhasadimensionoflength)via
3.2 Decelerationintheadiabaticstage
WeusetheKompaneets(1960)approximationalongsidethecoor- (γ2 1)λE
dy= − dt, (6)
dinatetransformationidentifiedbyK92tofollowtheadiabaticde- s 2ρ0V(y)
celerationoftheshockfrontinasinglepower-lawdensityprofile.
as well as the dimensionless parameter x = 2 ω y/(2a)
| − | ≡
3 Thethree-dimensionalvolumeintegrals(ofanoffsetsphere)overasin-
gulardensityconvergefortheshallowpowerlawsusedhere:1/26ωin6 4 For thermodynamic variables, we use primes (′) to indicate the post-
3/2forρ R−ωin. shockvalues(thevaluesbehindtheshockfront).
∝
FateofSNRs near SMBHs 5
y/y .Theparameterxis,therefore,equaltoyscaledwithrespect Withthiseffectivemassterm,theSNRleavestheejecta-dominated
c
toacritical value y , whichiswhen the shock either reaches the phasearoundthepointatwhichthemasssweptupfromtheenvi-
c
origin(ω=1)or‘blowsout’toinfinity(ω=3).Therefore,x(like ronmentiscomparabletotheinitialmassbehindtheshock.
y)canbeconsideredtorepresentthe‘time’inthistransformation.
Theconstantλ 1isgivenbythedifferenceinpressurebehindthe
≈ 3.4 Transitiontotheradiativestage
shockfrontrelativetotheaveragepressureinternaltotheremnant,
andinapower-lawprofileis(Shapiro1979) Astheshockslows,thelateevolutionofatypicalSNRismarked
by an increase in radiative losses. Although we will not model
(17 4ω)/9
λ= − . (7) this stage, we intend to check the time-scales over which SNRs
1 (9 2ω)−(17−4ω)/12−3ω
− − willreachthisstageinquiescentnuclei(iftheysurvivesufficiently
Inanambientdensitywithasinglepower-lawformofequa- long).
tion(2),theK92transformationgivesaself-similarsolutiontothe Typically,coolingfunctions showamarkedincreaseinther-
Kompaneetsequation(seeequations510and11ofK92)foranex- malradiationoncethegastemperaturedropsto 106 K(forex-
∼
plosionatR=a(seeFig.1): ample, Schureetal. 2009). Thisoccurs due tothe formation of a
sufficientnumberofelectronsboundtoionstoallowforeffective
R 2α R α
2 cos(αθ) x2+1=0, (8) linecooling.Onceregionsofgasbehindtheshockdroptothistem-
a − a −
(cid:18) (cid:19) (cid:18) (cid:19) perature,thedecelerationoftheSNRbecomesmorepronounced.
for the polar coordinates R and θ, where α (2 ω)/2. This Bycalculatingthetemperaturebehindtheshockwecandetermine
can be identified as a circular solution for a≡given−x in the two the time at which parts of the remnant begin to cool more effec-
variables(R/a)α andαθ.Analyticsolutionsforthevolume,time tively.
andvelocityinω=1and3densitiesarepresentedinAppendixB. Itispossibletodeterminethetemperatureoftheshockedgas
Theequationsdescribingtheshockfrontcanalternativelybe viatheidealgaslaw,
parametrizedbyψ0,theinitialangleofapointontheshockwith k ρ′T′
respecttotheaxisofsymmetry.Thesubsequentequationsofmo- P′= B (12)
m µ
u
tionforagivenψ describethepathsofflowlinesintheshockin
0
termsofthepolarcoordinatesmeasuredfromtheSMBH(K92): (where, again, we denote post-shock values with primes, kB is
Boltzmann’s constant, m is the atomic mass unit and µ is the
u
R=a 1+2xcosψ +x2 1/(2α), (9) meanmolecularmass),aswellasthejumpconditionsforthe(post-
0
shock)densityandpressure,
θ= 1(cid:0) arctan xsinψ(cid:1)0 . (10)
|α| (cid:18)1+xcosψ0(cid:19) ρ′=ργ+1 and P′= 2ρvs2 . (13)
γ 1 γ+1
Thisflowline-basedtreatmentisausefulcontextforthenumerical
−
approachtotheshockevolutionpresentedinSection4.1,andthese Forγ =5/3,
equationswillbeusedtocomparewiththenumericalresults.
2(γ 1)m µ 3m µ
T′= − u v2= u v2, (14)
(γ+1)2kB s 16kB s
3.3 Intermediate-asymptotictransition
andthepost-shocktemperatureisfoundtobeT′ 106Kforv
s
The Kompaneets solution for the velocity diverges for x 0, 300kms−1.Therefore,ifwemonitoreachpoin≈talongtheshoc≈k
→
given that the volume V(x) 0. In this limit, the energy den- forthetimeatwhichthevelocitydropsbelowthisvalue,wemay
→
sityand,therefore,alsothevelocity,tendtoinfinity.Thesolution estimatethetimeatwhichradiativeprocessesbecomesignificant.
is,however,notintendedtodescribetheinitialevolutionarystage As the cooling function is also dependent on ρ, for a given
of the remnant. In order for the numerical treatment to correctly temperaturetherateofcoolingisalsoexpectedtobeamplifiedin
followtheSNRevolution,wemustaccountfortheinitialcoasting regionsofpost-shockmaterialwithhigherdensity.However,bythe
stage.Afullanalyticjoiningoftheintermediate-asymptoticsolu- timethatSNRsareradiative,theyhavesurvivedtheexpansionpast
tions between the ejecta-dominated and adiabatic stages is com- theSMBHandenteredintothemoreuniformdensitybeyondthe
plex, even for an ω = 0 ambient medium (see, for example, SOI,suchthattheambientdensityissimilaracrossallpointsofthe
Truelove&McKee1999). shock. At this stage, the SNRs are reasonably symmetric around
Asamodelforthisintermediatebehaviour,weemployanef- theSMBHandthevelocityissimilaracrossalloftheshockfront,
fective density (mass) term to the solution that gives a transition sothatmostoftheSNRreachestheradiativestageatsimilartimes.
betweentheexpectedsolutions.Thedensityofthemediumismod- IftheSNRsurvivesexpansionpasttheSMBH,thislateevolution
ifiedto: islargelyuninfluencedbythedetailsofanyearlyinteractionsnear
theSMBH.6.
M
ρ ρ(R)+ ej, (11)
eff ≡ V
where the additional effective term counters the divergent be-
4 EVOLUTIONOFREMNANTSAROUNDQUIESCENT
haviourofthevelocityatsmallvolumes.Thishasthedesiredprop-
BLACKHOLES:NUMERICALTREATMENT
ertythatwhenthevolumeislargeρ ρ(R)inthestandardKom-
→
paneets approximation, while at small volumes the second term PurelyanalyticsolutionsfortheshockfrontevolutionviatheKom-
dominates to provide the initial coasting phase of the remnant. paneetsequationarenotfeasibleformanydensityconfigurations.
5 Notethat there aretwosignerrors intheexponents ofequation 11in 6 Foradetailedconsiderationoftheradiativetransitioninpower-lawme-
K92. dia,seePetruk(2005).
6 Rimoldietal.
weneedtheenergydensitywithintheshockedvolume(whichisas-
localshockproperties
sumedtobeaconstantwithinthisvolume)aswellasthelocalmass
(position,velocitymagnitudeand
direction,ejectamassfraction) densityintheenvironment(seeequations5and11.7Thisstageof
evolutionisadiabatic,andsogivenaninitialexplosionenergywe
thereforecalculatetheenergydensityusingtheinstantaneousvol-
umeenvelopedbytheshockfront.
flowline flowline flowline
(object) (object) (object) Inanaxisymmetricarrangementofgasdensityandexplosion
point, calculation of the volume is simplified by the geometrical
symmetry; it isdetermined byasolid of rotationof the areaof a
globalshock
properties ambient accretion two-dimensionalsliceaboutthisaxis.Anyarbitraryorderedsetof
ca(usnrhvdoeci,nkvtoefrlruonnmatle sh(oocbkjefcrto)nt m(oebdjeiucmt) dSmisMcoBdaenHld pthoeisnetstw(xoi-,dyimi)encsainonsaplecciofoyrdthinealtoesc,atthioenvooflutmhee,sbhyoctkhefrsoencot.nGditvheen-
energydensity)
orem of Pappus, is equal to the product of the area of the non-
intersectingpolygondefinedbythesecoordinatesandthedistance
travelled by its centroid under rotation about the symmetry axis
evolver
(Kern&Bland 1948). Using the fact that the components of the
centroid,C =(C , C ),ofapolygonaregivenby
x y
Figure2.Thebasicnumericalscheme,asdescribedinSection4.1.Shaded 1 n−1
boxesshowthebasictypesofobjects inthenumericalconstruction. The Cξ = 6A (ξi+ξi+1)(xiyi+1−xi+1yi) (15)
choiceofthenumberofflowlinesdeterminestheresolution,andonlyafew i=1
X
areshownhereschematically.Theflowlinestracklocalphysicalproperties
forξ x,y ,thevolumeoftheSNRcanbedeterminedfrom
oftheshock(velocityandejectamassfraction).Collectively,theydefinethe ∈{ }
locationoftheshockfront,withglobalphysicalpropertiessuchasitsen- πn−1
ergydensity,determinedbythevolume.Alongwiththeenvironment(most V = (y +y )(x y x y ). (16)
3 i i+1 i i+1− i+1 i
importantly,thebackgroundmassdensity),theglobalshockpropertiesde- i=1
X
terminetheevolutionofalltheindividualflowlinesinthesubsequenttime
Withthemagnitudeofthevelocityknown,eachpointonthe
step.
shockevolvesbydeterminingtheunitvector forthevelocitythat
is perpendicular to its neighbouring points. The position is then
linearlytranslatedoverasmalltimestepusingthevelocityvector.
Therefore, we developed a numerical method that solves for the
We assign an ejecta mass element to each flowline. Fig. 3
evolution of a shock front using the physical assumptions of the
showsaschematicofthisimplementation.Duetorotationalsym-
KompaneetsapproximationdescribedinSection3.
The primary assumptions that must be encompassed by the metry,eachpointontheshockatt 0representsaringsegment
→
oftheSNRinthreedimensions.Weassignathicknesstoeachof
method culminate in constraints on the velocity. Namely, the di-
theseringsegmentsbasedonthespacingbetweentheflowlinesin
rection of the velocity of any point must be perpendicular to the
theinitialsphericalstate.Thefractionofejectamassrepresentedby
shockfront,andthemagnitudeofthevelocitymustbedetermined
theflowlineisthentheratiooftheareaofthiszoneofthesphere
bytheenergydensitybehindtheshockandlocalambientdensity
tothetotalsurfaceareaofthesphere.
asprescribedinequation(5).
Duetoasymmetryinthebackgrounddensityandthepresence
ofastrongdensitycontrastneartheorigin,segmentsoftheshock
maycollidewithoneanother.Thisshockfrontself-interactioncan
4.1 Generalprescription
lead to collisions in which kinetic energy is converted into inter-
Thenumericaltreatmentfollowsanapproachbywhichtheshockis nal energy. Since it cannot be easily radiated away, we expect a
describedbytheevolutionofflowlinesthroughthebackgroundgas. transient acceleration outward of the heated gas, after which the
Theflowlinesarethepathsfollowedbytracer ‘particles’(points) fluidwillreturntothedynamicsimposedbytheglobalexpansion.
distributed along the shock front, analogous to the analytic treat- Thenumericaltreatmentoftheseself-interactionsisoutlinedinAp-
mentwiththeψ0parameterofequations(9)and(10). pendixC.Thistreatmentresultsinthedeletionofsomeflowlines,
Fig.2showsaschematicoftheapproach.Theinitial(spheri- accountingforthemodificationoftheflowinthisregion.
cal)stateoftheshockisbrokendownintoflowlinescharacterized
bytheirangleψ .Duringtheevolution,thenumberofflowlinesis
0
dynamic.Tokeepareasonableresolutionoftheshockfront,new 4.2 Comparisonwithanalyticsolutionsforsinglepower-law
flowlines can be inserted, withmean properties of adjacent flow- profiles
lines, if the distance between two points on the shock is over a
Fig. 4 shows the morphology of an SNR,running into a circum-
definedthreshold.Foroursimulations,athresholdoftheorderof
SMBHenvironmentwithdensitypower-lawgradientω = 1(left-
0.05pchasprovensufficienttodescribeasmoothshockfrontevo-
handpanel)andω = 3(right-handpanel).There,wecomparethe
lution on Milky Way-like scales. Flowlines may also be deleted
analyticprescriptiondescribedinSection3.2andAppendixBwith
inregionswherepartsoftheshockfrontarecolliding.Theback-
ournumericalmethod.Thenumericalsolutionsarefoundtomatch
groundgasprescribestheevolutionoftheshock,butthebehaviour
ofthepost-shockgasisnottracked,andthusthebackground gas
canbetreatedasbeingindependentoftheshock. 7 Forsimplicity,theratioλ 1ofthepost-shockpressuretomeaninterior
Thekinematicsoftheshockfront aredetermined bytheve- energydensityissettobeex≈actlyunityinequation(5).Astheshockveloc-
locityvectorsateachflowline.Todeterminethemagnitudeofthe ityisproportionalto√λ,theeffectofthisissmallcomparedtootherlimi-
velocity,weusethejumpconditionsacrosstheshock,andforthose tationsinherentintheKompaneetsapproximationdiscussedinSection4.3.
FateofSNRs near SMBHs 7
3.0
2.5
2.0
a
/ 1.5
y
R
1.0
0.5
0.0
1 0 1 2 3 0 1 2 3 4
−
Rx/a Rx/a
Figure4.Numericalresults(blue)comparedwithanalyticsolutions(red,dashed)forthelocationsoftheshockfrontindensityprofileswithω=1(left-hand
panel)andω=3(right-handpanel).Theinitial(spherical)stateforthenumericalsolutionsisshowningreen.Unitsaregivenasratiosofdistance(Rx,Ry)
toexplosiondistanceafromthedensitysingularityat(0,0)(theSMBHinourmodel).Theresultscanbewritteninparametricformintermsofx,which
increaseswithtimetuptoacriticalvalueofx=1;seeSection3.2aswellastheexpressionsfort(x)inAppendixB.Thesolutionsforω=1arefoundup
tox = 1,whileforω = 3theyaregivenuptox = 0.8duetothedivergenceofsolutionsasx 1inthislattercase.Thetrailingpartoftheshock(the
→
partdirectlytowardstheSMBH)forω =1solutionreachesRx/a=0atx=1,whileleadingpoint(directlyawayfromtheSMBH)reachesRx/a=4.
Thetrailingpartoftheω=3solutionasymptoticallyapproachesadistanceofRx/a=1/4asx 1,whileinthesamelimittheleadingpartoftheshock
→
followsRx/a .
→∞
y
i−1 power-lawmediumwithω =1andω =3.Thisfigureshows
in out
i the distance and velocity evolution of selected sample points on
theshockfront.Wefollowtheportionoftheshockthatpropagates
towardstheSMBH(bluelines),awayfromtheSMBH(redlines)
i+1
and at an initial angle of ψ = π/2 (green line). The numerical
0
radiusandvelocityareseentotransitionfromthecoasting(radius
R′ t,velocityv = const.)phasetoformssimilartothoseseen
∝
inthepureKompaneetssolutions.
As expected, the evolution of the trailing part of the shock,
x
SN asitgetsclosertotheSMBH,approachestheanalyticsolutionfor
a pure ω = 1 medium. Likewise, the leading part of the shock
asymptotes to the pure ω = 3 analytic solution, as it expands
away from the SMBH. The green line shows how the evolution
of a flowline that emerges at 90◦ from the θ = 0 axis has, in-
stead,anintermediatebehaviour,whichisinfluencedbytheover-
allbrokenpower-lawdensity.Thefigurealsoshowsforreference
theSedov–Taylorsolution(blackdashedline,Taylor1950;Sedov
1959)foranexplosioninauniformambientmedium(R′ t2/5
h andv t−3/5). ∝
∝
Figure3.Aschematicoftheinitial,sphericalstateoftheSNR,withini-
tialpositionsfortheflowlines(filledcircles)aroundthepointofexplosion
(opencirclelabelled‘SN’).Flowlinesinthepositive-y portionofthex–y
4.3 Caveatsandlimitationsofthemodel
planedefineathree-dimensional shockfrontbyrotationabouttheaxisof
symmetry.Rotatingeachflowlineaboutthisaxisproduces aring(shown Formorecomplexbackgrounddensityconfigurations,suchasone
fortheithflowlineasathickline).Themidpointsbetweentheithflowline withmanylargedensitycontraststhattriggerself-interactionsand
anditsneighbours definethelimitsofthezoneofthesphereassignedto turbulence,onemayconsideratreatmentofself-interactingshocks
thatflowline(thinlines).ForasphereofradiusR′,theareaofthiszoneis
thatismorein-depthandsophisticatedthanthatpresentedinAp-
proportionaltotheheightofthezone,h,sinceitssurfaceareais2πR′h.
pendix C. The increase in velocity of any small self-intersecting
Thefractionoftotalejectamassassignedtotheflowlineisthentheratioof
region is expected to be a brief transient phenomenon; therefore,
thisareatothetotalareaofthesphere.
we do not presently apply any boost in velocity when merging
flowlines,insteadonlyaccountingforthenetdirectionoftheflow
theanalyticformverywell.Thereisveryslightdeviationbetween that results from two colliding parts of the shock. The reason is
the two methods, more noticeably in the ω = 3 case, which is that, in all our simulations, the portion of the shock front which
duetothefactthatthenumericalmethodrequiresasmallspherical undertakes self-interaction is limited, and therefore the treatment
initialstep.Theanalyticsolutionisclosertoasphereatsmalltimes of theseregions have asmall effect on theoverall volume evolu-
intheω=1solutionsothereisalmostnodiscerniblediscrepancy. tion.Obviously,ifoneconsidersamorecomplexgeometrywhere
Fig.5showsacomparisonbetweenournumerical(solidlines) self-interactiondominatestheevolvingvolume,fullhydrodynami-
andtheanalytic(dashedlines,arbitraryscaling)resultsinabroken calsimulationsaretheonlyreliabletoolofinvestigation.
8 Rimoldietal.
104
1]
pc] s−
[ m
R′ k
[
v
1 103
10 102 103 10 102 103
t[yr] t[yr]
Figure5.Exampleofradiusandvelocityevolutionforaremnantinabroken-power-lawmedium(solidlines).Theexplosionoccursat3pc(nearthedensity
breakbetweenωin =1andωout =3)arounda5×108M⊙SMBHusingthescalingdescribedinSection5.Solidlinesareplottedfromsnapshotsofthe
evolutionoftheremnantinthenumericaltreatment,whereblue(lowermost)curvesareforthetrailingflowline(towardstheSMBH)andred(uppermost)
curvesarefortheleadingone(awayfromtheSMBH).Thegreencurveshowsthebehaviourofaflowlineinthenumericaltreatmentthatemergesat90◦from
theθ = 0axis(ψ0 = π/2).Someanalyticresults(witharbitraryscaling)aregivenasdashedlinesforcomparison.BlackistheformoftheSedov-Taylor
(uniformmedium,ω=0)solutions.ThereddashedcurveshowsthesolutionforthepointontheshocktravellingdirectlyawayfromtheSMBHforashock
inapurelyω = 3medium.ThebluedashedcurveshowsthesolutionforthepointontheshocktravellingdirectlytowardstheSMBHinapurelyω = 1
medium.
The Kompaneets approximation itself has some drawbacks, ofPeebles(1972),weuse
in that it generally predicts too large a velocity, and therefore
size, for the shock once it accelerates (Koo&McKee 1990; RSOI≡ GσM2•, (17)
Matzner&McKee 1999). In the context of the present problem,
thisismorepronouncedintheouterdensityregionwithasteeper, for a black hole of mass M•, where σ is the velocity disper-
R−3,gradient.IfmuchoftheshockisintheR−3region,theover- sion of stars about the SMBH. We use this parameter not only
to define the outer edge for the range of explosion distances
estimationofvelocitiesandsizeswillthereforebegreater.
considered, but also to rescale Milky Way properties to galac-
tic nuclei with different M•. To obtain an expression for the
SOI which depends only on M•, we use the well-known (‘M•–
σ’) relation between black hole mass and velocity dispersion
5 GALACTICNUCLEIMODEL (Ferrarese&Merritt2000;Gebhardtetal.2000).Usingtheobser-
InourquiescentSOIs,withnoappreciableinflowofgaseousmate- vationallydeterminedGebhardtetal.(2000)result,8M• = 1.2
×
rialfromfurtherout,thegasdensitydistributionisthatofanRIAF. 108 σ/ 200kms−1 15/4M⊙,weobtain
Thedistributionofearly-typestars(theonlypopulationofinterest
here)isdictatedonlybylocalandcurrentconditions,notbearing R (cid:0) (cid:0)3 M•(cid:1)(cid:1) 7/15 pc, (18)
imprints of the long term history of theassembly of the nucleus. SOI≈ (cid:18)4.3×106M⊙(cid:19)
ThesefactswillallowustorescalefeaturesofourGalacticCentre where here, and hereafter, we rescale equations for the Galactic
(observationallyconstrainedbecauseofitsproximity)toquiescent Centre black hole mass. For what follows, a useful parameter to
nucleiwithdifferentSMBHmasses.
8 Recent studies imply that the M•–σ relation is steeper than this,
and σ may have an exponent closer to 5 (for example, Morabito&Dai
5.1 Characteristicradii 2012). Although there is still some ambiguity in the value of this ex-
ponent, we tested the effect of a very steep relationship M• = 1.2
WeconsiderSNexplosionswithintheSOIofanSMBH.Theirfate 108(cid:0)σ/(cid:0)200kms−1(cid:1)(cid:1)5.3M⊙ motivated by Morabito&Dai (2012×).
canbeinfluencedbyboththeSMBHgravityanditsgaseousenvi-
Evenwiththislargeexponent,wefindthatourmainresults,thetime-scales
ronment.Correspondingly, therearecharacteristicradiiinthenu-
inSection6.3,aregenerallyonlyincreasedbyafactorof2(whilethescal-
cleusassociatedwiththeseproperties.ThefirstisthatoftheSOI: ingofradiibythespheresofinfluencealsoincreasesbyatmostafactorof
therangeouttowhichthegravityoftheSMBHdominatesoverthat 2).Astheoverallconsequenceissmall,wedonotpresentadditionalresults
ofthegravitationalpotentialofthebulge.Followingthedefinition forasteeperM•–σrelationinthiswork.
FateofSNRs near SMBHs 9
rescaletheMilkyWaypropertiesistheratioζ M•7/15,between theSOI.Sincethetotalmassof starsintheSOIscaleswithM•,
∝
theRSOIofagenericM•andthatofSgrA*. then so too will the number of massive stars and, therefore, the
The closer a supernova explodes to the SMBH, the stronger accreted mass: M˙ M•. Asabove, M˙crit M•, as well, and
the tidal forces, which may become high enough to disturb and thereforetheratioM∝˙/M˙crit isconstantover∝M•.Inotherwords,
eventuallydisrupttheremnantinadynamical time.Thishappens forourphysicallymotivatedpictureofquiescentnuclei,theSMBH
when the velocity of the shock front becomes comparable to the is accreting at the same fraction of Eddington as Sgr A* (M˙
Keplerianvelocity,v ,associatedtotheSMBHgravityfield.We 10−5M˙ ).9 ≈
K Edd
do not model distortions due to tidal effects but we account for GiventhisaccretionrateofM˙/M˙ 10−3,wecanesti-
crit
≈
thetidaldisruptionoftheremnantwhenwequantifyits‘lifetime’ matethedensityatR0fornucleiwithadifferentM•.Todosowe
(seeSection6.3).Tothisendwetestforwhetherv <v todetect usethecontinuityequationfortheflow,
s K
partsoftheshockthathavedeceleratedenoughtobeshearedbythe
M˙ 4πR2ρ(R )v (R ), (21)
SMBH.Wethereforeintroduceanother characteristicradius—the ≈ 0 0 K 0
innermostradiusfortheexistenceofSNRs,Rsh,whichislimited wherethescaleheightH RandtheradialvelocityvR vK.
by SMBH shearing. This minimal shearing radius is the point at Thereferencedensity≈forour general galacticnucle≈i models
whichvKiscomparabletotheinitialSNRejectavelocity,vinit: istherefore
Rsh ≡ Gvi2Mnit• n0 =n(R0)≈130 4.3 M10•6M⊙ 1/2ζ−3/2cm−3. (22)
(cid:18) × (cid:19)
=1.9×10−4(cid:18)4.3×M10•6M⊙(cid:19)(cid:16)104vkinmits−1(cid:17)−2 pc dAinffeerxeanmtipnlneeorfgroaudriednetnssiistygimveondeinl fFoirg.M6,•w=her1e0t7hMee⊙ffeacntdonthrtheee
=900 104vkinmits−1 −2Rg, (19) ddeennts.ityprofilesof fixingthescalingreferencepoint atR0 isevi-
(cid:16) (cid:17)
whereR isthegravitationalradiusoftheSMBH.
g
SupernovaethatoccurwithinR arecompletelysheared,as
sh
5.3 Massivestardistributions
the velocity of the ejecta in all directions is less than the Keple-
rianvelocityaroundtheSMBH.Note,however,thatSNRscanbe Given a physical number density n∗(R) of stars at a distance R
shearedalsoatlargerradiiastheejectaslowsdown,andmayreach from the centre of mass, the projection on to the celestial sphere
thelocalKeplerianvelocityataradiuslargerthanRsh.Comparing gives,asafunctionoftheprojectedradiusRpr,asurfacedensity
theshockvelocitytotheKeplerianvelocityiseffectivelyequivalent of stars Σ∗(Rpr). Observationally, the latter quantity is typically
tocomparingtherampressurewiththeambientbaryonicpressure, given.Assumingasphericallysymmetricspatialdistribution,itis
Pgas, since vK cs Pgas/ρ (where cs is the sound speed possibletoreversetheprojectiontoinferthesphericalnumberden-
≈ ∝
in the external medium). Additionally, we note that for the same sity,
p
reason we can ignore the shearing of remnants during the initial
emxoptlioosnioonffthoerapr>ogeRnsihto,rwsetacrsa.nalsoneglectthe(Keplerian)orbital n∗(R)= −π1ZR∞ dΣd∗R(Rprpr) Rdp2Rrp−rR2, (23)
Finally,forthegasmodels,thereferenceradiusforthedensity,
providedthatthephysicalnumbperdensityn∗(R)fallsoffatlarge
R0,andthelocationofthebreakinthegasdensitypowerlawRb RatarategreaterthanR−1.Forpower-lawdistributions,thisgives
(asexplained inFig.1)arescaledinourmodel bytheSOI,such acorrespondence oftheobservedradialdependence, R−Γ,tothe
thatR0 ≡ζR0,MWandRb ≡ζRb,MW. physicaldependence,R−γ,viatherelationshipγ Γ+p1r.
∼
FortheMilkyWay,thereisevidencefortwodifferentpower-
law distributions in the old and young stellar populations of the
5.2 Gasmodels
GalacticCentre.Aparticularcuriosityisanapparentdepletionof
Asmentionedpreviously,weexpectthatquiescentSMBHsaresur- late-type(K, M) giantsin theinner 0.5 pc(Doetal. 2009).This
roundedbyRIAFs,similartothatwhichissuggestedintheGalac- leadstoamuchshallower(possiblyinverted)innerpower-lawfor
ticCentre.Thisimpliesthatallflowshavesimilardensitygradients, the late-type distribution compared to that of the early-type (O,
whichissetbythephysicalprocesseswhichcharacterizethisaccre- B) stars. Recent analyses estimate the radial dependence of the
tionregime.Additionally,theiraccretionratemustbemodestand, early-type stars in the Milky Way nuclear star cluster to be ap-
inparticular,lowerthanthecriticalvalueforadvection-dominated proximately R−1 inside the power-law break and R−3.5 outside
pr pr
accretionofM˙ =α2M˙ ,or (Buchholzetal.2009;Doetal.2013a),correspondingtovaluesof
crit Edd
γof2and4.5,respectively.
M˙crit =9×10−3(cid:18)4.3×M10•6M⊙(cid:19) M⊙yr−1, (20) lar diIsntrtihbeutniouncslefioorfldoinfgfeerre-lnitvegdalastxayrstyaprees,povsasriibatlieondsueintothdeifsfteerl--
whereM˙ istheEddingtonrate,α= 0.3(Narayan&Yi1995) ingnuclear assembly histories. However, inour pictureof aself-
Edd
andweassumea10%radiationefficiencyforM˙ .Atareference regulatingSOI,theyoungstardistributionsaretakentobethesame
Edd
distanceofR 0.04pc,theaccretionrateisestimatedfrom acrosstherangeofM•,wherethemostrecentstarformationinthis
0,MW
observations and sim≈ulationstobearound M˙ 10−5 M⊙yr−1
(Cuadraetal.2006;Yuan2007).Therefore,Sgr≈A∗isaccretingat
9 It is possible for SMBHs to be accreting at different fractions
∼10−3ofitscriticalrate. M˙/M˙crit < 1andstillbetermed‘quiescent’intheconventional sense.
We extend properties of the Sgr A* accretion flow to other However, the accretion rate given here is the most physically motivated
quiescentnucleiasfollows.Theprimarymaterialforaccretionin valuebasedonscalingofquantitiesbyM•,anddeviationsfromthisvalue
quiescent nuclei originates from the winds from massive stars in arebeyondthescopeofthiswork.
10 Rimoldiet al.
pendrespectivelyontheSMBHaccretionrateandaccretionmode
106
(forexample,CDAFversusADAF).
Considering the values of L in various environments, we
105
obtain an indication of the length- and time-scales over which
SNRs will end their ejecta-dominated stage and start decelerat-
104 ing. As a reference for the crossing time-scale of a nucleus, re-
callthatanSNRthatdoesnotappreciablydeceleratefromitsini-
] 103 tial 104 kms−1 wouldreacharadiusof1pcinapproximately
−3 100∼yr.AninvestigationofLindifferentdirectionsindicateswhich
m
[c 102 SNRswilldeceleratewithinthistime-scale.Italsoprovidesatest
) forthelevelofasymmetryoftheSNRduringthisstageofevolu-
R
( tion.WewilllaterproceedtomodelSNRsthroughthedecelerating
n 10
stage.
Figs 7 and 8 depict two curves describing the deceleration
1
length,L,approximatelytowardsandawayfromtheSMBH.The
angleψ approximatelytowardstheSMBHistakentobe10−3π,
10−1 suchtha0ttheintegratedpaththroughthedensityrunsverycloseto
theSMBH,butdoesnotpassthoughthesingularityattheorigin.
10−2 Thedifferencebetweenthesetwocurvesprovidesameasureofthe
10−3 10−2 10−1 1
asymmetryoftheremnantattheendoftheejecta-dominatedstage.
R[pc]
Forcomparison, inFig.7,athirdcurve(solidline)isshownthat
describesanaveragedecelerationlengthderivedbyanintegralover
asphere.
Figure6.Exampleofagasdensitymodelusedforagalaxywitha107M⊙ WefirstconsideramodelofthebackgroundgasintheMilky
SMBH, showing the number density, n, as a function of radius R from
Way.Fig.7showsLforthreedifferentdensityvalues:theobserva-
theSMBH.Thered(shallowest)linecorrespondstoaninnergradientwith
tionallymotivated‘canonical’densityρ =ρ (innumberdensity,
ωin = 1/2,thegreentoωin = 1andtheblue(steepest)toωin = 3/2. n 130 cm−3, left-hand panel), and0 3 (ccentral panel) and 30
Allmodelshaveagradientoutsidethebreakofωout = 3.Thedensityis c ≈
scaledusingareferencepointR0,seenasthepointofconvergenceofall (right panel) times that value. We consider values other than the
theinnerdensitygradients (inthiscase,R0 = 0.06pc).Abreakinthe canonicaldensity,as,eveninquiescentnucleisuchastheGalactic
densitydistributionislocatedataconstantR =Rbforallchoicesofthe Centre, thereisthe possibilityfor variation in theoverall density
densitygradient(inthiscase,Rb =0.6pc).Theleft-handandright-hand oftheaccretionflow.Forexample,denseraccretionflowscanre-
limitsofthehorizontalaxisaredeterminedbytheshearingradius(equation sultfromsuddenaccretionepisodesfromtidallydisruptedstarsor
19)andSOI(equation17),respectively. clouds,10 ortheycanbeassociatedwithmoreintensestarforma-
tionactivityinthenucleus.Scalingthedensityalsoshowstheeffect
ofunder-ormisestimatingthegasdensityfromtheX-rayemission.
regionisindifferenttothehistoryofthenucleus.Therefore,forour
Asexpectedwithincreasingdensity,thereisanoveralltrend
galacticnucleimodel,weusethesamevaluesforγasthosegiven
towardslower values of L/a.Thereisalsoatrendtowards more
abovefortheearly-typestarsaroundSgrA*.Asbefore,wescale
symmetric remnants with increasing density, since in general the
thebreakinthestellarnumber densitybytheSOIof theSMBH,
ratioL/aisreducedforhigherdensities.
whichdefinesthetransitionradiusbetweenthetwovaluesofγ.
Theinvestigationofdifferentdensityprofiles(seeFig.6)leads
us to conclude that CDAF/ADIOS model, preferred by Galac-
ticCentreobservations (Wangetal.2013), gives, quitegenerally,
6 RESULTS
shorterdecelerationlengths(redlinesinFigures6and7).Theflat-
Weproceedtodescribeourmainresults,basedonthemethodout- terCDAF/ADIOSprofile(smallerω )isdenserinmostoftheSOI
in
linedintheprevioussections. InSection6.1,weexaminetheef- oftheblackhole,thereforereducingL/a.
fectofblackholemassandgasdensityprofileonthedeceleration ForthecanonicalvalueofdensityintheMilkyWay(leftpanel
lengthusingtheprescriptionsof Section3.1.Then,usingthenu- ofFig.7),thedecelerationlengthsareL & 1pc.Consideringthe
mericalmethodofSection4,theoverallSNRmorphologyispre- CDAF model, we remark that, for the canonical density, the ma-
sentedinSection6.2.Finally,inSection6.3,weinvestigatetheX- jorityof the SNRswould decelerate beyond the SMBH location.
rayemittinglifetimesbasedonshearingoftheSNRejectabythe EjectafromastarsuchasS2(markedwithabluedot–dashedline
SMBH.WethenusethislastresulttopredictthemeanSNRlife- inFig.7)isexpectedtoevolvemoresymmetricallythanthatfroma
timesexpectedwithintheSMBHspheresofinfluencefordifferent starfurtherout,inthestellardisc(s)( R ).Alreadywithafactor
b
∼
M•. offewenhancementindensity,SNRsinandbeyondthestellardisc
woulddecelerateappreciablybeforetheyreachSgrA*(seecentral
panel).
6.1 Decelerationlengths
Fig. 8 shows deceleration lengths for each of ω
in
∈
TheSNRbeginstoappreciablydecelerateoncethesweptupmass 1/2, 1, 3/2 forgalacticnucleiwithSMBHmassesof107 M⊙
{ }
becomes comparable to the ejecta mass. This end of the ejecta-
dominatedstagecanbecharacterizedbyadecelerationlengthfrom
theexplosionpoint,L,determinedbythedensityintegralsofSec- 10 A recent example around Sgr A* is the object G2 (for example,
tion3.1,whichvarieswithdirection.Thisdecelerationlengthde- Burkertetal.2012);though,ifacloud,itsmassistoosmalltohaveasig-
pendsonthegasdensityandontheradialdensityprofile.Thesede- nificantimpactontheoveralldensity.