Simulations of star formation in a gaseous disc around sgr A* - a failed AGN

0.56 MB·English

by  Sergei Nayakshin

Mon.Not.R.Astron.Soc.000,000–000 (0000) Printed5February2008 (MNLATEXstylefilev2.2)
Simulations of star formation in a gaseous disc around
Sgr A∗– a failed Active Galactic Nucleus
Sergei Nayakshin1⋆, Jorge Cuadra2† and Volker Springel2
1 Department of Physics & Astronomy, Universityof Leicester,Leicester, LE1 7RH, UK
2 Max-Planck-Institut fu¨rAstrophysik, Karl-Schwarzschild-Straße 1, 85741 Garching bei Mu¨nchen, Germany
5February2008
7
0
ABSTRACT
0
We numerically model fragmentation of a gravitationally unstable gaseous disc un-
2
der conditions that may be appropriate for the formation of the young massive stars

n observedinthecentralparsecofourGalaxy.Inthisstudy,weadoptasimpleprescrip-
a tion with a locally constant cooling time. We find that, for cooling times just short
J
enough to induce disc fragmentation, stars form with a top-heavy Initial Mass Func-

5 tion (IMF), as observed in the Galactic Centre (GC). For shorter cooling times, the
discfragmentsmuchmorevigorously,leadingtoloweraveragestellarmasses.Thermal
1
feedbackassociatedwithgasaccretionontoprotostarsslowsdowndiscfragmentation,
v
as predicted by some analytical models. We also simulate the fragmentation of a gas
1 ∗
stream on an eccentric orbit in a combined Sgr A plus stellar cusp gravitational po-
4
tential.Thestreamprecesses,self-collidesandformsstarswithatop-heavyIMF.None
1
1 ofourmodels produceslargeenoughco-movinggroupsofstarsthatcouldaccountfor
0 the observed “mini star cluster” IRS13E in the GC. In all of the gravitationally un-
7 stable disc models that we explored, star formation takes place too fast to allow any
0 gasaccretionontothecentralsuper-massiveblackhole.Whilethiscanhelptoexplain
/ the quiescence of ‘failed AGN’ such as Sgr A∗, it poses a challenge for understanding
h
the high gas accretion rates infered for many quasars.
p
-
o Key words: accretion, accretion discs – methods: numerical – Galaxy: centre
r
t
s
a
:
v 1 INTRODUCTION demonstrated that star formation cannot be quenched by
i
X stellar feedback, unless one is prepared to grossly violate
There is no detailed understanding of how super-massive
constraints that we havefrom AGN spectra.
r black holes (SMBHs) gain their mass, except that
a it must be mainly through gaseous disc accretion Goodman (2003) suggested that thefeeding of SMBHs
(Yu& Tremaine, 2002). The standard accretion disc proceedsviadirect accretion of lowangular momentumgas
model (Shakura& Sunyaev,1973) shows that accre- that settles in a small scale disc. Such a disc would be
tion discs must be quite massive and cold at “large” too hot to allow star formation. The size of this “no-star-
(sub-parsec in this context) distances from the SMBH. formation region” is R∼< 0.03 pc, and is weakly dependent
Many theorists pointed out that these discs should on theSMBH mass. However,one realistically expectsthat
be unstable to self-gravity, and thus must form stars an even larger amount of gas settles at larger radii where
or giant planets by gravitational fragmentation (e.g., it could collapse and form stars, or it could accrete on the
Paczyn´ski, 1978Kolykhalov & Sunyaev,1980Shlosman & BegelmaSnM, B19H8.9CTohlelirnef&oreZathhen,fa1t9e99oBfetrhteinse&seLlfo-gdraatvoi,ta1t9i9n9gGdaimscms iies, 2001).
This creates a dilemma for the field, as star forma- still very much interesting.
tion might be a dynamical, very fast process which Current observations provide new im-
may result in a complete transformation of the gas petus for theoretical work on the topic.
into stars. The SMBH would then be starved of fuel. Wehner& Harris (2006Ferrarese et al. (2006) have shown
Goodman (2003Sirko& Goodman (2003) have recently that many galaxies host “extremely compact nuclei”
(ECN) – star clusters located in the very central ∼ 5
parsec of galaxies. Closer to home, in the centre of our
⋆ E-mail:[email protected] Galaxy, about a hundred massive young stars, mostly
† Current address: JILA, University of Colorado, Boulder, CO arranged into two stellar discs of ∼ 0.05–0.5 parsec scale
80309-0440, USA (Levin & Beloborodov, 2003Genzel et al., 2003Paumard et al., 2006),
(cid:13)c 0000RAS
2 S. Nayakshin, J. Cuadra & V. Springel
circle the super-massive black hole named Sgr A∗. These 2 METHODS AND DISC FRAGMENTATION
stars are suspected to have been formed in situ, as ev- TESTS
idenced by the lack of stars outside the ∼ 0.5 parsec
We use the SPH/N-body code GADGET-2
region (Paumard et al., 2006Nayakshin & Sunyaev,2005).
(Springelet al., 2001Springel, 2005) to simulate the
Several observational facts are consistent with the hy-
dynamics of stars and gas in the (Newtonian) gravitational
pothesis that these stars formed in a massive gaseous disc
(Paumard et al., 2006). Had this star formation event not field of a point mass with Mbh = 3.5×106M⊙. The code
happened, Sgr A∗ could have been accreting gas from the solves for thegas hydrodynamicsvia thesmoothed particle
gaseous disc even now. Sgr A∗ thus failed to realise itself hydrodynamics (SPH) formalism. The hydrodynamic
treatment of the gas includes adiabatic processes and
as an AGN because of the loss of gas to star formation
artificial bulk viscosity to resolve shocks. The stars are
(Nayakshin & Cuadra, 2005).
modelled as sink particles, using the approach developed
A number of important gaps in our understanding by Springel et al. (2005), modified in the ways described
of young stars near Sgr A∗ remain. The counter clock- below.
wise disc appears to host stars on more elliptical orbits, Table 1 lists some of the parameters and results of the
and it is also geometrically thicker. The same disc also simulations presented in this paper. The tests described in
contains a puzzling “mini star cluster”, IRS13E, that this section are those listed as S1–S5 (the “S” stands for
consists of more than a dozen stars and may be bound “standard”).Theunitsoflengthandmassusedinthesimu-
by an Intermediate Mass Black Hole (IMBH) of mass lationsareRU =1.2×1017cm≈0.04pc,whichisequalto1′′
Mbh ∼> 103M⊙ (Paumard et al., 2005Sch¨odelet al., 2005). atthe8.0kpcdistancetotheGC,andMU =3.5×106M⊙,
The IMF of the observed stars must be top- themass of Sgr A*(e.g., Sch¨odelet al., 2002),respectively.
heavy according to several lines of evidence The dimensionless time unit is tU = 1/Ω(RU), or about
(Nayakshin & Sunyaev,2005Nayakshinet al., 2006Paumard et al.6,020y0e6arAsle(xΩanidsedreefitnaeld.,j2u0s0t6b),elow Eqn. 1). The masses of
which is not expected in the most basic model of a frag- SPH particles are typically around 0.01 Solar masses (see
menting disc, as the Jeans mass there is significantly Table 1).
sub-solar (e.g., Levin 2006, Nayakshin 2006, but see also
Larson 2006).
2.1 Gravitational collapse
In this paper, we discuss numerical simulations of star Toomre (1964) showed that a rotating disc is subject to
formation occurringinagaseousdiscaroundSgrA∗.Given gravitational instabilities when the Q-parameter
the numerical challenges in the problem, we foresee that a
Ω2
reliable modelling of all the questions raised by the young Q= (1)
stars near Sgr A∗ will require an extended effort of con- 2πGρ
stantly increasing complexity. In this study we present nu- becomessmallerthanacriticalvalue,whichisclosetounity.
merical experiments with a locally constant cooling time. This form of the equation assumes hydrostatic equilibrium
This allows a convenient comparison with previousanalyti- to relate the disc sound speed to its vertical thickness, H.
calandnumericalworksthatpredictedtheconditionswhen Ω=(GMbh/R3)1/2 istheKeplerianangularfrequency,and
fragmentation should take place. It might also form a basis ρ is the vertically averaged disc density. Gravitational col-
for comparison with future work. lapse thustakes place when thegas density exceeds
Ω2
Within our formalism with a locally constant cooling ρBH ≡ . (2)
2πG
time, we find that (i) circular and eccentric discs alike can
gravitationally fragment and form stars; (ii) the IMF of Ideally, numerical simulations should resolve the gravita-
formed stars is a strong function of cooling time, becoming tionalcollapse downtostellarscales. Inpractice,thisisim-
top-heavyformarginallystar-formingdiscs;(iii)starforma- possiblefornumericalreasons,andinsteadcollisionless“sink
tion feedback is indeed able to slow down disc fragmenta- particles” are commonly introduced (Bate et al., 1995) to
tion,assuggestedbyseveralearlieranalyticalpapers,butit modelcollapsingregionsofveryhighdensity.Toensurethat
is not yetclear if it can alleviate thefueling problem of the collapseiswellunderwaywhenweintroduceasinkparticle,
SMBHs; (iv) our simulations do form some tightly bound we requirethat thegas density exceeds
binary stars but more populous systems (a la IRS13E) do ρcrit=ρ0+AcolρBH , (3)
not survivelong.
whereρ0=5×10−11 gcm−3,andAcolisalargenumber(we
This paper is structured as follows. In Section 2, we tested values from a few to a few thousand). We in general
discussoursimulationmethodologyandthebasicconditions findthat ourresults are not sensitive to theexact valuesof
for disc fragmentation. Section 3 explains our sink particle ρ0 and Acol, provided they are large enough (Section 5.2).
approach to treat star formation in more detail. We then Further details of our sink particle treatment are given in
analyse theevolution after discfragmentation andtheIMF Section 3.1.
oftheformedstarsinSection4.Thesensitivityofourresults
tonumericalresolutionandfeedbackfromstarsisdiscussed
2.2 Gas cooling
inSections5and6,respectively.Wethenexamineelliptical
orbits of agaseous stream in Section 7, and thequestion of Weaccountforradiativecoolingviaasimpleapproachwith
the formation of mini star-clusters in Section 8. Finally, we alocally constantcooling time.Thecooling termin theen-
summarize and conclude in Section 9. ergy equation is
(cid:13)c 0000RAS,MNRAS000,000–000
Star formation in a disc around Sgr A∗ 3
du u
= − , (4)
(cid:16)dt(cid:17)cool tcool(R) 0.600..66
where u is the internal energy of an SPH particle, and R
is the radial location of the SPH particle, i.e. the distance
0.4
00..44
to the SMBH. We parameterize the cooling time as a fixed
fraction of the local dynamical time,
tcool(R)=β×tdyn(R), (5) 0.200..22
c ]
where t = 1/Ω and β is a parameter of the simula- p
dyn 4
0
ttihoenssi.mTuhliastsioimnpmleetmhooddeolloagllyowasssfourchacloocnavlelyniceonntsvtaalnidtactoioolninogf 0.9 [ 0. -0.0--00..00
time models were investigated in detail in previous litera- y -
ture. Gammie (2001) has shown that self-gravitating discs -0.2--00..22
are bound to collapse if the cooling time, t , is shorter
cool
than about 3/Ω (i.e., β < 3). Rice et al. (2005) presented
a range of runs that tested this fragmentation criterion in -0.4--00..44
detail, and found that, for the adiabatic index of γ = 5/3
(asused throughoutthispaper),thediskfragments as long
-0.6
as β ≤6. --00--..0066..66 --00..44 --00..22 --00..00 00..22 00..44 00..66
-0.6 -0.4 -0.2 -0.0 0.2 0.4 0.6
Notethatmostofthesimulationsexploredinthispaper x -1.1 [ 0.04pc ]
are performed for circular initial gas orbits, and also for
relatively small disc (total gas) masses as compared with
that of the SMBH. As will become clear later, this implies 1.11..000 1.11..555 2.22..000 2.22..555
that during the simulations gas particles continue to follow log Σ [g cm-2]
nearlycircularorbits,andthustheircoolingtimeisconstant
aroundtheirorbits.Oneexceptiontothisisatestdonewith Figure1.Surfacedensityofaregioninamarginallystabledisc
eccentric initial gas orbits (Section 7). fromthesimulationS4(β=4.5)attimet=50.Themapiscen-
tred on the (x,y)= (1.1,−0.9) location. Many small and some-
what dense gas clumps can be seen in the figure. None of the
2.3 Fragmentation tests clumpshowevertrulycollapses,andmostdisappearonadynam-
icaltimescale.
Tocompareournumericalapproachwithknownresults,we
ran several tests (S1–S5 in Table 1) in a setup reminiscent
of that of Rice et al. (2005), who modelled fragmenta-
disc. For definiteness, we simulate a gaseous disc of mass
tion of proto-stellar discs. One key difference between
marginally self-gravitating proto-stellar and AGN discs Md = 3×104M⊙ ≈ 0.01Mbh extending from rin = 1 to
is the relative disc mass. Whereas the former become rout =4,wherer isthedimensionlessdistancefromSgrA∗.
The total initial number of SPH particles used in runs
self-gravitating for a disc to central object mass ratio of
S1–S5is4×106.Gravitationalsofteningisadaptive,witha
Mdisc/M∗ ∼0.1−0.5(e.g., Rafikov,2005),where M∗ isthe minimumgravitational softeninglengthof 3×10−4 forgas,
mass of the central star, the latter become self-gravitating
and 0.001 for sink particles. The disk is in circular rotation
foramassratioassmallasM /M ≃0.003−0.01 (e.g.,
Gammie, 2001Goodman, 2003dNisacyaksbhhin & Cuadra, 2005Levin, 20a0ro6u).nd a SMBH with mass of Mbh = 3.5×106M⊙. The
disc is extended vertically to a height of H/R = 0.02 in
For marginally self-gravitating discs, the ratio of disc
theinitialconditions,whichrendersitstabletoself-gravity.
height to radius, H/R, is of order of the mass ratio,
As the gas settles into hydrostatic equilibrium, it heats up
H/R∼M /M (Gammie, 2001). The disc viscous time
disc bh
due to compressional heating, and then cools according to
is
equation (4). We used fivedifferent values of β =t Ω, in
R 2 cool
tvisc∼α−1 Ω−1 , (6) theruns, namely β =0.3, 2, 3, 4.5 and 6 (see Table 1).
(cid:16)H(cid:17) The discs in runs with β = 0.3,2 and 3 fragmented
where α ∼ 0.1 − 1 is the effective viscosity parame- and formed stars, as expected based on the results of
ter (Lin & Pringle, 1987Gammie, 2001) for self-gravitating Riceet al. (2005), but our tests with β = 4.5 and 6 did
discs. Thus, for H/R∼< 0.01, the disc viscous time is not. However, the run with β = 4.5 and 6 did fragment in
some 4–6 orders of magnitude longer than the disc dy- thesenseofformingtransienthighdensitygasclumps,some
namical time. This is in fact one of the reasons why ofwhichcanbenotedinFigure1.Themaximumdensityin
the discs become self-gravitating, as they are not able theseclumpsfluctuatedbetweenafewtoafewdozentimes
to heat up quickly enough via viscous energy dissipation ρBH,i.e.theclumpsweredenseenoughtobeself-bound,but
(Shlosman & Begelman, 1989Gammie, 2001). not dense enough for our sink particle criterion, since Acol
Also,sincethediscviscoustimeismanyordersofmag- was set to 30 for these tests (cf. Section 2.1).
nitudelongerthantheorbitaltime,aswellasthetotalsim- Rice et al. (2005) reported fragmentation for values of
ulation time, we expect no radial re-distribution of gas in β as large as 6. This difference in the results is most likely
the simulations, and indeed very little occurs. Our models duetoourdiscsbeingmuchthinnergeometrically.Thehigh
are essentially local, as emphasized by Nayakshin (2006), density structures formed in our simulations, like filaments
andhenceitsufficestosimulateasmall radialregion ofthe and clumps, are much smaller in relative terms than they
(cid:13)c 0000RAS,MNRAS000,000–000
4 S. Nayakshin, J. Cuadra & V. Springel
arein thesimulations ofproto-stellar discs. This isnot sur- ally be treated as having a finite life-time in our simula-
prisingasthemostunstablewavelengthfortheQ∼1discis tions. Furthermore, their sizes are only an order of magni-
of order of the disc vertical scale height H (Toomre, 1964), tudesmallerthanthatofthediscscaleheight,H ∼1015 cm
andourdiscsaremuchthinner.Also,thehighdensitystruc- (Nayakshin& Cuadra, 2005).Therefore,thefirstcoreshave
tures are much more numerous in AGN discs. As analyti- a decent chance to merge with one another (Levin,2006).
callyshownbyLevin (2006),self-gravitatingclumpsinsuch They could also be disrupted by a more massive proto-star
discs are quite likely to collide with each other on a dy- thatpassesby,form adiscandbesubsequentlyaccretedby
namical time scale. If clumps are tightly bound they may theproto-star.
agglomerate. For tests with larger values of β, the clumps In order to capture this complicated situation in a nu-
are marginally bound (as their maximum density exceeds merically practical way,weintroducetwokindsofsink par-
ρBH byafactorofafewonly),andanyinteractionmayun- ticles. We call those with mass M ≤ Mcore = 0.1M⊙ “first
bind them.In contrast, for high values of H/R∼M /M, cores”. Once the gas reaches thecritical density (eq. 3), we
disc
as appropriate for proto-stellar discs, fewer clumps form, turn SPH particles into sink particles individually, so the
and the interactions are probably too rare to cause clump initial mass of a first core particle is equal to one SPH par-
destruction. ticle mass. First cores have geometrical sizes of Rcore, and
can merge with one another if they pass within a distance
smaller than 2fmRcore of each other, where fm ≥1 is a pa-
rameter that mimics the effect of gravitational focusing in
3 STAR FORMATION
mergers. We assume for simplicity that the lifetime of the
3.1 Two types of sink particles firstcoresislongerthanthedurationofthesimulations.We
performed a few tests with finite core life times and found
To design a physically reasonable and numerically sound
very similar results; most of the merging of first cores hap-
procedure to deal with the gravitational collapse of cool-
pensveryquicklyaftertheyarecreatedinsideofacollapsing
ing gas, we make use of well known results from the field
gas clump.
ofstarformation. Simulationsofcollapsing gashaloes show
Sink particles with mass M > Mcore are assumed to
that the first quasi-hydrostatic object to form has a char-
have collapsed to stellar densities, thus we refer to them
acteristic size of about 5 AU, and may therefore be called
simply as “stars”. Stars are treated as point mass particles
a “first core” (e.g., Larson, 1969Masunaga et al., 1998). As
and thus cannot merge with one another, but they are al-
this optically thick gas condensation slowly cools, the core
lowedtoaccretethefirstcoresiftheirseparationislessthan
decreases in size whileaccreting mass from theinfalling en-
Rcore.Whenthemassofafirstcoreparticleexceedsthecrit-
velope. When themass of thegas accumulated is of theor-
ical mass Mcore as a result of mergers or gas accretion, we
derof0.05–0.1M⊙,H2dissociationandhydrogenionisation turn it into a star particle.
losses become efficient coolants, and the core can collapse
dynamicallytomuchhigherdensitiesandmuchsmallersizes
(Larson, 1969Silk, 1977).Therelevanttimescaleisoftheor- 3.2 Accretion onto sink particles
der of ∼> 103 years, which is very short compared with free
WeusetheBondi-Hoyleformalismtocalculatetheaccretion
falltimesoftypicalmolecularcloudsinagalaxy.Creationof
rate onto thesink particles,
the first collapsed objects is therefore nearly instantaneous
ina“normal” starforming environment,andtheirsizes are (GM)2
M˙ =4πρ , (7)
sufficientlysmallsothattheycanbeconsideredpointmasses (∆v2+c2)3/2
s
compared with thecloud size of ∼1018 cm.
whereM isthesinkparticlemass,andρ,c and∆v arethe
s
The gas densities in self-gravitating AGN discs are
ambient gas density, sound speed, and the relative velocity
ρ/mp ∼ 108 −1012 cm−3, i.e. many orders of magnitude between the accretor and the gas, respectively. Once the
larger than in galactic star forming regions, but smaller1
accretionrateiscomputed,theactualgasparticlesthatare
than the gas density of a first core, ∼ few ×1013 cm−3. We
swallowed by the sink particles are chosen from among the
assume that the initial stages of collapse should be simi-
sink particle’s neighbors via the stochastic SPH method of
lar to “normal” star formation except that it starts from
Springel et al. (2005).
higherinitialgasdensities.ForSgrA∗,atthedistancewhere
theyoungmassivestarsarelocated(Paumard et al., 2006),
the free-fall time in the collapsing gas clouds, t ∼ 3.3 Maximum accretion rates
ff
1/(Gρ)1/2 ∼< tdyn, is of the order of 100–1000 years. For a
As the gas density in these star forming discs is enormous
more massive AGN, the time scales would be even shorter
compared to “normal” molecular clouds, the Bondi-Hoyle
at comparable distances to the SMBH. The life-times of
formula frequently yields super-Eddington accretion rates
the first cores in this environment are then longer than the
free-fall time, since the core collapse is dominated by the (Nayakshin,2006).AssumingthatgasliberatesGM∗/R∗ of
cooling time scale, which is much longer than the t esti- energy per unit mass, where M∗ and R∗ are the mass and
ff
radius of the accretor, one estimates that the accretion lu-
mated above. This implies that these objects should actu-
minosity is
GM∗M˙
1 CloserintoSMBHs,thetidalsheardensity(ρBH)becomesyet Laccr = R∗ , (8)
larger, and therefore there exists a minimum radius in the disc
insideofwhichstarformationdoesnottakeplace,asproto-stars where M˙ is the accretion rate. Setting this equal to the
wouldbetornapartbytidalforces. Eddington luminosity, LEdd = 4πGM∗mpc/σT = 1.3 ×
(cid:13)c 0000RAS,MNRAS000,000–000
Star formation in a disc around Sgr A∗ 5
1038(M∗/M⊙) erg s−1, we obtain the Eddington accretion right panels, respectively. Star formation spreads to larger
rate: radii as time progresses. In the right panel of Fig. 4, the
M˙ = 4πmpR∗c . (9) innermost region of thedisc isalmost completely cleared of
Edd σ gas bycontinuingaccretion ontoexistingstars (ratherthan
T
further disc fragmentation, see Section 4.2). At the end of
Note that this expression depends only on the size
the simulations that do fragment, i.e. S1–S3, very little gas
of the object, R∗. For first cores, R∗ = Rcore, remains. To characterise the rate at which star formation
which yields a very high accretion rate limit of al-
consumes the gas in the disc, we define the disc half-time,
most a Solar mass per year for Rcore = 1014 t ,asthetimefromthestartofthesimulationtothepoint
cm. For stars, we use the observational results of half
atwhichthegaseousmassinthediscisequaltoonehalfof
Demircan & Kahraman (1991Gorda & Svechnikov(1998):
theinitialmass.Table1listst foreachsimulation.Longer
half
R M 0.969 cooling times allow thedisctosurvivelonger. Nevertheless,
= 1.09 for M <1.52M⊙, (10) evenfortherunS3,wheret ≫t ,thehalf-timeof the
R⊙ (cid:18)M⊙(cid:19) half dyn
disc is only around 12,000 years in physical units.
0.6035
R M One notices that some of the stars are followed and/or
= 1.29 for M >1.52M⊙. (11)
R⊙ (cid:18)M⊙(cid:19) preceeded by low column depth regions. These are caused
by angular momentum transfer from the stars to the sur-
For R = R⊙, this yields an Eddington accretion rate limit
of ∼5×10−4M⊙ year−1. rounding gas (Goldreich & Tremaine, 1980). If stars are
massive enough, they could open up gaps in accretion
discs (Syeret al., 1991Nayakshin,2004) in the way plan-
ets open gaps in proto-planetary discs. We do not ob-
4 DISC EVOLUTION AFTER
serve such “clean” gaps around stars in our simulations.
FRAGMENTATION
We believe the main reason for this is the large num-
Weusethetestswithcoolingparameterβ=2and3,briefly ber of stars that we have here. The interactions be-
described in Section 2.3, to point out some general trends tween the gravitational potentials of neighboring stars de-
seeninoursimulations.Theserunswerecontinuedfarlonger stroy the resonant charachter of the gas-star interactions
than was needed to form the first gravitationally bound (Goldreich & Tremaine, 1980).Inaddition,self-interactions
clumps. Sink particles were introduced as described in Sec- betweenstarsforcestarsonslightlyellipticalandslightlyin-
tions2.3and3.1.ThecollapseparameterA wassetto30 clinedorbits(Nayakshin& Cuadra, 2005),whichalsomakes
col
(see equation 3), and the gravitational focusing parameter gap opening harder.
is f =3.
m
Figure 2 shows a column density map of a snapshot
made at dimensionless time t=75 for run S2 (β =2). The
4.1 Evolution of disc vertical thickness
left panel of the figure shows the whole disc, whereas the
rightpanelzoomsinonasmallerregioncentredonx=1.8. The disc starts off as a razor-thin gaseous disc and evolves
Red asterisks show stars more massive that 3M⊙. We do intoathickerstellar disc.This behaviouristobeexpected.
notshowlighterstarsandfirstcoresforclarityinthefigure. Initially,gascooling issufficientlystrong,andthedisctem-
Sincegasdynamicalandcoolingtimesareshorterforsmaller peratureisregulated tomaintain ageometrical thicknessof
radii, high density regions develop faster at smaller radii. order of H/R ≃ Mdisc/Mbh, which is equivalent to having
Thesequenceofeventsisasfollows.Theinnermostedge Q ≈ 1 (e.g., Gammie, 2001). When a good fraction of the
ofthediscformscollapsedhaloesinwhichsinkparticlesare gas isturnedintostars, however,therate at which thedisc
introduced.Thesesinkparticlesgrowbymergerswithother can cool decreases. At the same time, a stellar disc, con-
sinkparticlesandgasaccretion.Withtime,thegaseousdisc sidered on its own, would thicken with time since stellar
is depleted while thestellar disc grows. orbitalenergy istransferred intostellar random motionsby
Looking at larger R in Fig. 2, we observe an interme- thenormal N-body relaxation processes.
diate epoch in star formation, where bound high density In the intermediate state, when both gas and stars are
clumps have formed but there are no stars more massive present in important fractions, disc evolution is not triv-
than 3M⊙ yet. This is because some of these clumps have ial.Intherunsdescribedinthissection,starstransfertheir
notyetsatisfiedthefragmentationcriterion,andotherssat- random energy to the gas via Chandrasekhar’s dynamical
isfied it only recently, thus containing only first cores and friction process (Nayakshin,2006). The disc swells as time
low-massstars.Finally,atthelargest radii,weseedensefil- progresses, as can be seen in the two edge-on views of the
aments(spiralstructure)butnowelldefinedboundclumps. disc from simulation S2 shown in Fig. 5 . The gaseous disc
Figure 3 is identical to Fig. 2, but shows test S3 (i.e. can thus become thicker than it would have been on its
β = 3 instead of β = 2). Comparing the two runs, it is own.However,dependingonconditions(i.e.coolingparam-
apparent that fragmentation of the more gradually cooling eter β, initial total disc mass, mass spectrum of stars), the
disc(S3)isnaturallymuchslowerthaninthetestS2.There two discs can be coupled or decoupled. In the latter case
are fewer stars in thedisc, and thereare fewer high density the stellar disc has a larger geometrical thickness than the
clumps. As we shall see later, within the given model, this gaseousdisc.Insuchacasetherateatwhichstarsheatthe
allowsthestarstogrowtolargermasses,resultinginamore discisnottriviallycalculated. Thisisespecially soifstellar
top-heavymass function. radiation, winds and supernovae are taken into account, as
Figure 4 shows the column density maps for the run stellarenergyreleaseabovethediscismuchlesseffectivein
S3 at two later times, t = 175 and t = 275, in the left and disc heating than it is inside thedisc.
(cid:13)c 0000RAS,MNRAS000,000–000
6 S. Nayakshin, J. Cuadra & V. Springel
444 1.011..00
222 0.500..55
c ] c ]
p p
0.04 000 0.04 0.000..00
y [ y [
-2--22 -0.5--00..55
-4 -1.0
--44--44 --22 00 22 44 --11--..1100..00 --00..55 00..00 00..55 11..00
-4 -2 0 2 4 -1.0 -0.5 0.0 0.5 1.0
x [ 0.04pc ] x -1.8 [ 0.04pc ]
00..00 00..55 11..00 11..55 22..00 00 11 22 33
0.0 0.5 1.0 1.5 2.0 0 1 2 3
log Σ [g cm-2] log Σ [g cm-2]
Figure2.Snapshotofthedisccolumndensityattimet=75fortestS2(β=2).Thegasisrotatingclockwiseinthisandalltheother
figuresinthispaper.Thelefthandpanelshowsthefullsimulationdomain,whereastherighthandonezoomsinonaregionofthedisc
centredatx=1.8.Starswithmassesgreater than3M⊙ areplottedasredasterisks.
444 1.011..00
222 0.500..55
c ] c ]
p p
0.04 000 0.04 0.000..00
y [ y [
-2--22 -0.5--00..55
-4 -1.0
--44--44 --22 00 22 44 --11--..1100..00 --00..55 00..00 00..55 11..00
-4 -2 0 2 4 -1.0 -0.5 0.0 0.5 1.0
x [ 0.04pc ] x -1.8 [ 0.04pc ]
00..00 00..55 11..00 11..55 22..00 22..55 00..55 11..00 11..55 22..00 22..55
0.0 0.5 1.0 1.5 2.0 2.5 0.5 1.0 1.5 2.0 2.5
log Σ [g cm-2] log Σ [g cm-2]
Figure3.SameasFig.2butfortestS3(β=3),i.e.foralongercoolingtime.Notethesmallernumberofstarsandhighdensityclumps
inthistestcomparedwithFig.2,wherethecoolingtimeisshorter.
(cid:13)c 0000RAS,MNRAS000,000–000
Star formation in a disc around Sgr A∗ 7
4 4
44 44
2 2
22 22
c ] c ]
p p
0.04 000 0.04 000
y [ y [
-2 -2
--22 --22
-4 -4
--44--44 --22 00 22 44 --44--44 --22 00 22 44
-4 -2 0 2 4 -4 -2 0 2 4
x [ 0.04pc ] x [ 0.04pc ]
00..00 00..55 11..00 11..55 22..00 --00..55 00..00 00..55 11..00 11..55 22..00
0.0 0.5 1.0 1.5 2.0 -0.5 0.0 0.5 1.0 1.5 2.0
log Σ [g cm-2] log Σ [g cm-2]
Figure4.SnapshotsofthediscfortestS3attimest=175(leftpanel)andt=275(rightpanel).TogetherwiththeleftpanelofFig.3,
thesesnapshotstracethegradualdepletionofthegaseous discandthebuild-upofthestellardisc.
0.300..33 0.300..33
0.2 0.2
00..22 00..22
0.1 0.1
00..11 00..11
c] c]
p p
0.04 -0.0--00..00 0.04 -0.0--00..00
z [ z [
-0.1 -0.1
--00..11 --00..11
-0.2 -0.2
--00..22 --00..22
-0.3 -0.3
--00--..3300..33 --00..22 --00..11 --00..00 00..11 00..22 00..33 --00--..0033..33 --00..22 --00..11 --00..00 00..11 00..22 00..33
-0.3 -0.2 -0.1 -0.0 0.1 0.2 0.3 -0.3 -0.2 -0.1 -0.0 0.1 0.2 0.3
x [ 0.04pc ] x [ 0.04pc ]
11..55 22..00 22..55 33..00 33..55 --11..00 --00..55 00..00 00..55 11..00 11..55
1.5 2.0 2.5 3.0 3.5 -1.0 -0.5 0.0 0.5 1.0 1.5
log Σ [g cm-2] log Σ [g cm-2]
Figure5.EvolutionofthegaseousandstellardiscthicknessesinthetestS2.Theimageiscentredatx=1.3.Theleftpanelisplotted
attimet=50,when∼20%ofgaswasturnedintostars,whereastherightoneisfort=175,atwhichpointonly∼15%ofgasremains
in the disc. The disc thickens with time as the remaining gas is not able to provide sufficient damping and cooling of stellar N-body
heating.
(cid:13)c 0000RAS,MNRAS000,000–000
8 S. Nayakshin, J. Cuadra & V. Springel
1.0000
-1ar] 0.1000
e 1000
y
M sun
n rate [ 0.0100 M ]sun
ntatio 0.0010 M) [ 100
e (
m N
ag M
Fr 0.0001
10
0 5.0•103 1.0•104 1.5•104
time, years
1
Figure6.Discfragmentationrateinthetestswithβ=0.3(S1, 0.1 1.0 10.0 100.0
thick solid),β=2(S2, dashed), and β =3(S3, thin solid).The M [ M ]
sun
shorterthecoolingtime,thefasterdiscfragmentationproceeds.
Figure 7. Mass function of stars (“IMF”) formed in the simu-
lations with β = 0.3 (S1, solid), β = 2 (S2, dotted), and β = 3
4.2 Fragmentation quenching
(S3,dashed).Thedot-dashedpower-lawinthisandthefollowing
We expect that, following an increase in disc effective tem- plotscorrespondstotheSalpeter IMF.
peratureandgeometricalthicknesswithtime,thevertically
averaged density of the disc must drop. The disc may then M2 1/2.Itisclearfromboththefigureandthetable,that
evolveintoanonself-gravitating stateasQincreasesabove ∗
unity (see equation 1). Further disc fragmentation should (cid:10)the lo(cid:11)nger the cooling time, the more top-heavy (or, equiv-
alently,bottom-light) is theresulting IMF.Thisoutcome is
cease. This effect can be noticed in the right hand panel of
not surprising. As we saw in Section 4.2, fragmentation is
Fig. 2. Near the inner edge of the disc, there are stars but
fastest for the smallest values of the cooling parameter β.
no high density clumps or filaments implying that the disc
Further, fragmentation stalls in our models when the stars
is no longer fragmenting in that region.
heatupthediscaboveQ=1(seeFig.6).Atthispoint,disc
Figure6showstheradius-integratedfragmentationrate
fragmentationstopsbutaccretionontostarscontinues.The
of the disc in the simulations with β = 0.3 (run S1, thick
average mass of a star reached by the time the gas supply
solid curve), β =2 (S2, dashed) and β =3 (S3, thin solid).
is exhausted is roughly inversely proportional to the num-
The fragmentation rate is defined as the total mass of first
berofstarsatthetimethediscfragmentation stalls. Aswe
corescreatedperunittime.Notsurprisingly,theshorterthe
find many more stars in the tests where fragmentation is
cooling time (smaller β), the more rapid is disc fragmenta-
quickest, these simulations will also have the least massive
tion. The more vigorous disc fragmentation explains why
IMF.
there are more stars and dense bound gas clumps in Fig. 2
The shape of stellar IMF formed in our simu-
than in Fig. 3 at the same time (t=75).
lations is very different from the IMF in the Solar
One can also see that the expectation of a decrease of
neighbourhood, which is close to the power-law func-
thefragmentationratewithtimeisborneout.Atthepeaks
tion of Salpeter (1955). Several authors argued that
of the respective curves, most of the disc mass is still in
this might be expected based on theoretical grounds
gaseous form. Therefore, the decline in the fragmentation
(Morris, 1993Nayakshin,2006Larson, 2006Levin,2006).
rate with time is indeed mostly a consequence of a change
Theproportion of massive stars in thestellar population of
in the disc state (higher Q-parameter) rather than due to
thesediscsmightbeveryimportantforthephysicsofthese
thedisc runningout of gas.
discs, e.g., theirsurvival chances against star formation.
4.3 IMF of the formed stars
5 SENSITIVITY OF RESULTS TO
Itisinterestingtocomparethedistributionfunctionsofstel-
PARAMETER CHANGES
lar masses from our simulations. The end state of our sim-
ulations, i.e., when the majority of gas is turned into stars, Duetonumericallimitations,oursimulationsdonotinclude
correspondstothe“initialmassfunction”(IMF)ofastellar anumberofimportantphysicalprocesses.Inparticular,ra-
population.Thereforeweusethisnametorefertoourmass diation transfer and chemical processes taking place during
distributions. Figure 7 shows the IMF of stars formed in collapse of a protostar are dealt with merely via prescrip-
the three simulations S1–S3. Table 1 lists the first two mo- tions.Theseprescriptionsemployparameterswhichwehave
ments of the distribution, i.e., the average mass, hM∗i, and tried toset toas physically realistic valuesas possible. It is
(cid:13)c 0000RAS,MNRAS000,000–000
Star formation in a disc around Sgr A∗ 9
importanttounderstandtowhichdegreeourresultsdepend
on the choices we made for the values of these parameters.
Herewedescribeaselectionoftestsaimedtowardsthisgoal.
Thesetests,labelledE1–E8inTable1,weredoneforslightly
different parameters than runsS1–S5. In particular, thein- 1000
nerandouterradiiwereset torin=1androut =2,respec-
tively,and thegravitational focusing parameter was chosen
tobef =1. The radiativecooling parameter β was set to
m ]n
3 for all of thesetests. Msu 100
) [
M
(
N
5.1 Maximum accretion rate M

We calculate the accretion rate using the Bondi-Hoyle for- 10
malism, with therestriction thatit isnotallowed toexceed
theEddingtonlimit(equations7and9).Inprinciple,excess
angularmomentuminthegasflowaroundproto-starscould
decrease theaccretion rate below our estimates. To explore
1
possible changes in the results, simulations E1 and E3 dif-
fered only by the maximum accretion rate allowed, with 1 0.1 1.0 10.0 100.0
and 0.1 times the Eddington limit, respectively. M [M ]
sun
The resulting IMFs of these two runs are shown in
Fig. 8, along with the IMF for test S3, which was already
Figure 8. IMF from the simulations with two different maxi-
presented in Fig 7. Comparing the solid curve (simulation
mumallowedaccretion rates(seeSection 5.1),i.e.,0.1timesthe
E1)withthedashedcurve(S3),weseethatchangesinrout, Eddingtonrate(E3inTable1;dottedcurve)andtheEddington
AcolandfmdidinfluencetheIMF,makingitlesstop-heavy. rate (E1; solid). These curves are compared with the IMF from
Themaineffectstemsfromthereductioninthevalueofthe simulationS3(dashed).
gravitational focusing parameter, f . Fewer mergers imply
m
that the finalaverage stellar mass is smaller.
Comparing the runsE1 and E3, onenotices that while
5.2 Critical collapse density
the IMFs of the two tests are similar at the high mass end,
aroundthepeakin thecurves,thereisalarger discrepancy Our fragmentation criterium is controlled by two parame-
at the low mass end. The largest difference in the IMFs ters, ρ0 and Acol (equation 3). The default values for these
occursataroundthecriticalmass,Mcrit =0.1M⊙.Thedif- parameters are ρ0 = 5×10−11 g cm−3, and Acol = 60.
ferences are nevertheless not as large as one could expect Thesevalueswere usedin runE3. Thesimulations E2, E3a
givenanorderofmagnitudechangeinthemaximumaccre- and E4 are completely analogous to the run E3 except for
tionrate.Weinterprettheseresultsasfollows.Astarisfirst the two parameters controlling the fragmentation. In E2,
bornasalowmassstarinoursimulations,anditthengains the fragmentation density threshold is drastically relaxed,
mass by either accretion or mergers with the first cores. If with ρ0 = 5×10−12 g cm−3 and Acol = 9. In run E3a,
the maximum possible accretion rate onto the star is high, ρ0 =5×10−12 g cm−3, but Acol =60, as in E3. Finally, in
it“cleansout”itsimmediatesurroundingsofgas,andgains therunE4weusedρ0=5×10−11 gcm−3,andAcol =300.
most of itsmass byaccretion. Ontheotherhand,if theac- Figure9showstheresultingmassfunctionforthestars
cretionrateiscappedatlowervalues,asintherunE3,then formedinthesefourtests.Evidently,asignificantrelaxation
gas gravitationally captured bythestar will first settleinto of the fragmentation criteria resulted in very large changes
a small scale disc around the star. This disc is somewhat in the IMF at its the low mass end. The high mass end is
smaller than the Hill’s radius RH = R(M∗/3MBH)1/3 ≈ not so strongly affected. In fact, the IMF of the tests E3
0.01R for the star of mass M∗ = 10M⊙. The density in and E4 are almost identical, implying that the results be-
that disc will often exceed the critical density for star for- comeindependentofthevalueofthecriticalcollapsedensity
mation,andhencenewfirstcoreswillbeintroducedinclose (equation 3) when it is chosen large enough.
proximity to the star. These cores then mostly and rela-
tively quickly merge with the star. Because the rate of gas
capturefromthelargerdisc(theHillortheBondirate,e.g.,
5.3 First core size
Goodman & Tan, 2004Nayakshin& Cuadra, 2005) is con-
trolled by the total mass inside the Hill radius, the rate Runs E3 and E5 are different only in the size of the first
atwhichthedominantstarinsidetheHillsphereisgrowing cores, Rcore (see Table 1). Figure 10 demonstrates that the
is comparable. The end result (the IMF for massive stars) IMF of the simulations is remarkably similar at the high
is then similar, regardless of whether the captured gas was massend,whereitcontainsalmostallofthestellarmass.At
directly accreted or first turned into fluffy proto-stars (the the low mass end, however, simulation E5 has significantly
first cores) and then accreted. more mass than E3. Clearly, the smaller first core radius
Another difference between the results of the runs E1 Rcore permitsfewermergersandslowergrowthoffirstcores
andE3 isthelongerdisclife-time, thalf,for theformer sim- by accretion. The exact value of Rcore is hence important
ulation for ratherobvious reasons. for thelow-mass end of the IMF.
(cid:13)c 0000RAS,MNRAS000,000–000
10 S. Nayakshin, J. Cuadra & V. Springel
1000 1000
] ]
M sun M sun
[ [
) )
M M
N( 100 N( 100
M M

10 10
0.1 1.0 10.0 100.0 0.1 1.0 10.0 100.0
M [ M ] M [ M ]
sun sun
Figure9.IMFofrunsE2,E3,E3aandE4(bluedash-dotted,red Figure 10. Comparison of the IMFs of runs E3 (solid) and E5
dashed, dotted and solid curves, respectively). The simulations (dotted). Thesimulationshaveadifferent“firstcore”radiuspa-
differbythevalueofthecriticalcollapsedensity(seeSection5.2 rameter, Rcore = 1014 and 3×1013 cm, for E3 and E5, respec-
fordetails). tively.SeetextinSection5.3andTable1fordetails.
5.4 SPH particle mass
ThesimulationsE6–E8hadidenticalparametersettingsbut
used different initial SPH particle numbers to test the sen-
sitivityoftheresultsontheSPHmassresolution.Figure11
showstheIMFfromthesetests.Clearly,thereisasignificant
1000
difference between run E6 (solid; 1 million SPH particles)
and the two others, E7 (dotted; 2 million) and E8 (dashed;
]
4rumnsillwiointh). Tthheerheigisheastfarressmolaultlieornd,ifftheroeungche,bseutgwgeeesntinthgetthwaot M sun
the results are reasonably close to full convergence. Given ) [
other uncertainties of the runs, e.g., the exact value of the M 100
(
N
parameterRcore andthecooling parameterβ,etc.,it seems M
sufficient to have an SPH mass resolution at the level of
the run E7, which has msph = 0.02M⊙. Note that except
for the simulations E6 and E7, all the other runs presented
here used msph =0.01M⊙, as in the test E8. 10
6 IMF AND FEEDBACK
0.1 1.0 10.0 100.0
Nayakshin (2006) suggested that the luminosity of the M [ M ]
sun
young stars which accrete gas from within a star-forming
disc is sufficient to heat up the disc, increasing the
Figure 11. IMF of runs E6–E8. The simulations differ in the
Toomre (1964) Q-parameter of the disc above unity and
initial number of SPH particles used. In particular, Nsph =1, 2
hence making the disc stable to further fragmentation. and4×106 forE6,E7andE8,respectively.
Nayakshin (2006) also suggested that this will lead to a
top-heavy IMF for stars born in such marginally unstable
self-gravitating discs. To test these ideas, we ran two addi- ontothestar(equation8).Thefeedbackluminosity,FLacc,
tional simulations that had thermal accretion feedback in- is spread over the SPH neighbours of the accreting star ac-
cluded.The feedback is implemented in thesame way asin cording to theirweight in the SPH kernel.
Springel et al. (2005), with a parameter 0 ≤ F ≤ 1 intro- Our two test simulations F1 and F2 had feedback pa-
duced to quantify the fraction of the accretion luminosity rameters of 0.01 and 0.5, respectively, and are otherwise
that is fed back as heat into the surrounding gas. The ac- identicaltotherunE3,whichhadF =0.Onenoticesright
cretion luminosity of course depends on the accretion rate awayfromTable1thatthedisclifetimet becomeslonger
half
(cid:13)c 0000RAS,MNRAS000,000–000