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8Mon. Not. R. Astron. Soc.000, 1–6 (2008) Printed 25 October 2018 (MN LATEX style file v2.2)

Stellar dynamical evidence against a cold disc origin for stars in theGalactic Centre

Jorge Cuadra1⋆, Philip J. Armitage1,2, Richard D. Alexander3,11JILA, Campus Box 440, University of Colorado, Boulder, CO 80309, USA2Department of Astrophysical and Planetary Sciences, University of Colorado, Boulder CO 80309, USA3Leiden Observatory, Universiteit Leiden, Niels Bohrweg 2, 2300 RA, Leiden, the Netherlands

Accepted XXX. Received XXX; in original form XXX

ABSTRACTObservations of massive stars within the central parsec of the Galaxy show that, while moststars orbit within a well-defined disc, a significant fraction have large eccentricities and/ orinclinations with respect to the disc plane. Here, we investigate whether this dynamically hotcomponent could have arisen via scattering from an initially cold disc – the expected ini-tial condition if the stars formed from the fragmentation ofan accretion disc. Using N-bodymethods, we evolve a variety of flat, cold, stellar systems, and study the effects of initial disceccentricity, primordial binaries, very massive stars andintermediate mass black holes. Wefind, consistent with previous results, that a circular discdoes not become eccentric enoughunless there is a significant population of undetected 100–1000 M⊙ objects. However, sincefragmentation of an eccentric disc can readily yield eccentric stellar orbits, the strongest con-straints come from inclinations. We show thatnone of our initial conditions yield the observedlarge inclinations, regardless of the initial disc eccentricity or the presence of massive objects.These results imply that the orbits of the young massive stars in the Galactic Centre are largelyprimordial, and that the stars are unlikely to have formed asa dynamically cold disc.

Key words: stellar dynamics – Galaxy: centre – methods: N-body simulations

1 INTRODUCTION

Observations of the Galactic Centre (GC) have revealed a popula-tion of young (∼5 Myr) massive stars orbiting within a fraction ofa parsec of theMBH ≈ 3× 106 M⊙ supermassive black hole Sgr A*.The existence of any young stars so close to the black hole is apuzzle, since the tidal field would prevent star formation from gasclouds with densitiesρ < MBH/[(4π/3)r3]. This critical density is∼ 1010 cm−3 at 0.1 pc, far higher than the densities of normal molec-ular clouds. The dynamics of the stars are also unusual and in-teresting. Although the innermost ‘S-stars’ appear to be consistentwith isotropic orbits, many of the young stars at a distance of theorder of 0.1 pc lie on a single plane (Levin & Beloborodov 2003;Genzel et al. 2003; Paumard et al. 2006; Lu et al. 2006) and rotateclockwise on the sky. The eccentricities of stars in this clockwisedisc are significant. The average value ise ≈ 0.35, with the eccen-tricities of several stars going above 0.5 (Beloborodov et al. 2006;Lu et al. 2006). Moreover, the stars that donot belong to the clock-wise disc (which, by definition, have large inclinations that in manycases exceed 90◦) have even larger eccentricities, perhaps as large

⋆ e-mail: jcuadra@jilau1.colorado.edu

as e ≈ 0.8 for stars on counter-clockwise orbits (Paumard et al.2006)1.

Two classes of theories have been advanced to explain the ori-gin of the disc stars. In the first, the stars formin situ from a mas-sive accretion disc in which the density is large enough to exceedthe tidal limit (Levin & Beloborodov 2003; Milosavljevic &Loeb2004; Nayakshin & Cuadra 2005; Paumard et al. 2006). Accretiondiscs at these radii around black holes are known to be vulnerable tofragmentation and star formation provided only that they can coolon a dynamical timescale (Paczynski 1978; Kolykhalov & Sunyaev1980; Shlosman & Begelman 1989; Gammie 2001; Rice et al.2003; Goodman 2003). The second model posits that the starsformed outside the GC region but migrated in due to dynamicalfriction (e.g. Gerhard 2001; Kim & Morris 2003; Kim et al. 2004;Gurkan & Rasio 2005; Berukoff & Hansen 2006). Understandingthe predicted dynamics of these models in more detail may assistin discriminating between them.

Fragmentation of a thin accretion disc yields, almost by con-struction, starts with the basic disc-like dynamics observed in the

1 Note that there is some dispute as to whether there is evidence for a sec-ond, counter-clockwise disc of stars. This question is not important for ourpurposes. We rely on observations only insofar as they demonstrate thatthere is (at least) one disc, and that some stars have orbits that do not coin-cide with any suggested disc plane. These aspects are uncontested.

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2 J. Cuadra et al

GC. Within such a model those stellar orbits that do not belongto any well-defined disc have to be generated after the disc frag-ments, perhaps via scattering. There is, however, only justenoughtime for significant dynamical evolution to occur. Alexander et al.(2007) found that even with a top-heavy mass function it washard to attain the large eccentricities observed. This problem canbe circumvented if the disc fragments while it is still eccentric(Nayakshin et al. 2007; Alexander et al. 2008), but it is still diffi-cult to explain the high inclinations. Here, we investigatewhetherstellar dynamical evolution fromany plausible cold disc initial con-ditions (i.e. those for which the local velocity dispersionis smallcompared to the orbital speed) can explain the observed stars. Weconsider both circular and eccentric initial conditions, and allowfor the possibility of a high primordial binary population and addi-tional scattering from unobserved massive objects (short-lived mas-sive stars, or intermediate mass black holes).

2 NUMERICAL METHOD

We model the stellar dynamics of a disc of stars using themercury6code, which is optimized to study systems dominated by a centralmass (Chambers 1999). In order to treat close encounters reliablywe adopt the optional Bulirsch–Stoer integrator, in its version forconservative systems.

We simulate stellar discs extending radially from 0.05 to0.15 pc. 2 The discs are composed of 100 stars randomly dis-tributed following a surface density profileΣ(r) ∝ r−1, roughlyas observed in this radial range, with the mass of each star equalto 25 M⊙ = 8.3 × 10−6MBH. In most of our models the stars areinitially rotating in circular Keplerian co-planar orbitsaround thecentral object, with small random perturbations such that the thick-ness of the disc ish/r = 0.01. The exact value of this perturbationis not important, as the evolution of the disc is initially rapid. Weintegrate the systems for a time roughly equivalent to 2700 orbits atr = 0.1 pc, almost 5 Myr. SinceN is rather small, we run three sim-ulations with each set-up to monitor the variance in the outcome.

Our model system includes the number and masses of theob-served population of stars in the GC. Although observational ev-idence favours a top-heavy mass function (Nayakshin & Sunyaev2005; Paumard et al. 2006) there could still be undetected lower-mass stars in the same region. Low mass stars, however, al-ways act todamp the dispersion of the observed massive stars(Alexander et al. 2007), so in ignoring them our models yieldanupper limit to the rms eccentricities and inclinations thatresult fromscattering.

3 SIMULATIONS

3.1 Single stars

To establish a baseline for subsequent comparisons, we firstmodelthe evolution of a disc of single stars similar to that consideredpreviously (Alexander et al. 2007). We also use this simple sys-tem to test the numerical convergence of the integrator, by run-ning three sets of models with different integrator tolerances. Eachset includes three independent realisations of the initialconditions.

2 Young stars are observed further away, but their density decreases rapidlyand their scattering time-scales become longer with radius, so we do notexpect the omission of stars atr > 0.15 pc to alter our results.

Figure 1. Eccentricity (upper curves) and inclination (lower curves) rmsvalues from the nine realisations of discs of single stars. There is a smallbias toward larger dispersion in the runs with less stringent accuracy re-quirements. For comparison, a curvee ∝ t1/4 is also shown.

Figure 1 shows the evolution of the rms values of the eccentric-ity and inclination for each simulation. We find that convergenceis obtained for an accuracy parameter specified byaccuracy =2 × 10−13, and that results accurate to the 10% level, adequate forour purposes, can be obtained withaccuracy = 10−12. We use thelatter value for most of our subsequent runs.

As expected theoretically, the eccentricity and inclination rmsgrow roughly ast1/4, with a proportionalityerms ≈ 2irms (e.g.,Lissauer 1993), wherei is measured in radians. To compare quan-titatively with previous work, we fit the average eccentricity rmsfrom all nine simulations with the analytical solution,

dσ∗dt=

G2N∗M2∗ lnΛ∗

C1r∆rtorbσ3∗

, (1)

whereσ∗ is the velocity dispersion of a stellar disc ofN∗ starseach of massM∗, orbiting at radiusr with radial extension∆r. Thequantity lnΛ∗ is the Coulomb logarithm,torb the orbital period atr,andC1 a geometrical constant of order unity (e.g., Alexander et al.2007). We find a value ofC1 = 1.65, in good agreement with the1.85 that Alexander et al. (2007) found from their high resolutionruns. This is a very small difference in practice, as the eccentricityrms value scales only asC−1/4

1 . In agreement with prior work, weconclude that an initially circular disc set up to match the numbersand masses of observed stars in the GC attains an rms eccentric-ity of around 0.2 within a few million years. This is too smalltoexplain the larger eccentrities observed in the GC population.

3.2 Primordial binaries

A small number of binary stars have been identified in the Galac-tic Centre (Martins et al. 2006; Peeples et al. 2007; Rafelski et al.2007), and, given our current knowledge of disc fragmentation, itis possible that the primordial binary fraction resulting from a frag-menting disc could be significant. Encounters between starsandhard binaries tend to shrink the binary orbit, releasing energy that

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Dynamics rebut cold-disc origin of GC stars 3

Figure 2. Eccentricity rms values from the nine different simulations ofdiscs of binary stars. The tighter the binaries are initially, the more disper-sion the system acquires.

can heat the system (Heggie 1975; Perets et al. 2008)3. To studywhether binary heating could be significant in the GC we consideran idealised model in which the initial binary fraction is 100%. Wereplace the disc of 100 single stars with one made up of 50 equalmass binaries, whose centres of mass are initially in circular orbitsabout the black hole. For the fiducial case the binary orientationslie in the disc plane. In each simulation all binaries have the sameinitial value of semi-major axis, which we vary between 10−3 pcand 10−4 pc. A binary with semi-major axis 10−4 pc is hard (andhence will shrink and give energy back to the system) if the stellarvelocity dispersionσ∗ <∼ 20 km s−1, which corresponds toe ∼ 0.1.For numerical reasons we are only able to track binaries as theyshrink down to a minimum size which we set at 3× 10−6 pc. If anystars get closer than this limit, the code merges them into a singleobject conserving energy and momentum.

Figure 2 shows the evolution of the rms value of the eccentric-ity for the binary runs4. When the binaries’ initial separations are10−3 pc they are only marginally bound and most of them quicklyget disrupted. As a result the system behaves as if it were made upof single stars from the start, and the resulting eccentricities andinclinations are essentially identical to those discussedin the pre-ceeding section. For initially tighter binaries, we see a trend wherecloser binaries produce more eccentric orbits. The plots also getnoisier, as the instantaneous Keplerian orbital elements of a fewstars are strongly perturbed during close encounters. The changesare relatively modest, and even a disc made up entirely of thetight-est binaries yields a final eccentricity that is only∼ 25% largerthan the single star case. The rms value of the inclination isagainroughly half of the eccentricity. Initial binary orientations that arenot aligned with the disc plane yield modestly smaller inclinationsand eccentricities than the aligned case. We interpret thisas im-plying a reduced cross-section for binary–binary encounters in the

3 Supernovae can also heat the disc, as the surviving companion keeps thevelocity it had in the binary. Perets et al. (2008), however,showed that evenwith 100 km s−1kicks no significant inclinations are reached.4 The average here is defined not over stars, but rather over ‘objects’ thatcould be either bound binaries or single stars. It is very hard to determinewhether the observed stars are actually singles or binaries, so the value weare plotting is what can be compared to current observations.

Figure 3. Eccentricity (upper curves) and inclination (lower curves) rmsvalues from the eccentric disc runs. The eccentricity rms isroughly constantand the inclinations remain small in the more realistic case(see Section 3.3).

non-aligned case. We also experimented with decreasing themin-imum distance below which stars are assumed to merge. A reduc-tion in this scale by a factor of three resulted in slightly smallereccentricities, but we were unable to demonstrate convergence byevolving systems with even smaller merger radii due to numericaldifficulties. Accordingly, we do not attach physical significance tothis difference. In reality, the dynamics of any binaries that shrinkto such small scales (which are less than an AU) would likely beaffected by the presence of circumstellar discs left over from theirformation. We cannot model such effects, but it is hard to see howthey could increase the overall stellar disc eccentricity.

3.3 Eccentric flat discs

If the stars originated from the fragmentation of a self-gravitatingdisc, they did not necessarily form after the disc became circular. Itis possible that gas settled in a plane around the black hole and frag-mented before it had time to circularise. In this case, the stars wouldinitially have inherited the eccentricity of the gas (Nayakshin et al.2007; Alexander et al. 2008).

To study the growth of inclinations in this scenario, we ranseveral simulations with initially eccentric stellar discs. The starswere set up in orbits withe = 0.5, with a surface densityΣ(a) ∝ a−1

between 0.05 and 0.15 pc. We considered both the case wherethe angle of pericentre of the eccentric orbits was initially con-stant (so all the ellipses are nested), and the case where this anglewas randomly distributed (in this case the in-plane velocity disper-sion is initially large). The aligned case is the expected outcomeimmediately after the fragmentation of an eccentric gaseous disc,but theMcusp ∼ 105 M⊙ worth of old stars between 0.05–0.15 pc(Genzel et al. 2003) will lead to differential precession and ran-domly oriented orbits on a time-scale∼ (MBH/Mcusp)torb. This time-scale is very short compared to the current age of the stellarsystem,so the more realistic set up for our simulationswithout the cusp po-tential are those with randomly chosen pericentre angles.

Figure 3 shows the results for these simulations, with the bluecurves depicting the case of randomly aligned orbits. The typicaleccentricity remains fixed ate ≈ 0.5, while the inclination reachesa typical value of 0.1, just as in the circular case (Sect. 3.1). Theruns starting with aligned orbits (red curves) have a largerspreadin eccentricities (although stillerms ≈ 0.5), with approximately

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4 J. Cuadra et al

Figure 4. Eccentricity (upper curves) and inclination (lower curves) rmsvalues from the six different simulations of discs of stars that included asignificant number of very massive stars. The very massive stars, however,are not included in this analysis, as they are not observed and are supposedto have already died. The typical eccentricity reaches an interesting value,but the inclinations are still too low to match the GC observations.

10 of the innermost stars acquiring large eccentricities and incli-nations as a result of the interaction with the eccentric potentialof the rest of the stars. This seems to be a resonant relaxation ef-fect (Rauch & Tremaine 1996), a mechanism already invoked byHopman & Alexander (2006) and Levin (2007) to explain the higheccentricities and inclinations of the innermost ‘S-stars’. The frac-tion of the stars that get highi orbits is still too small to match theGC population, and, as explained above, we expect the backgroundstellar potential to substantially weaken this resonant process.

3.4 Influence of Massive Stars

So far we have concentrated on the stars that are currently observed.However, it is possible that many more massive stars were bornbut have already exploded as supernovae. To take into account thiseffect, we ran a set of simulations where – in addition to the 10025- M⊙ stars previously considered – we include an extra numberof 100 M⊙ stars. These stars are included throughout the simulationbut are not taken into account in the final analysis, so in practice itis as if these very massive stars disappeared a very short time ago.As shown by Alexander et al. (2007), the more massive stars willsink toward the mid-plane, while the lower mass stars will acquirehigher eccentricities and inclinations.

We ran several simulations with either 50 or 100 of the100 M⊙ stars. Figure 4 shows the eccentricity and inclination rmsfor the 25 M⊙ stars. The eccentricities now reach a valuee ≈ 0.4,comparable to that observed. The inclinations, however, are stilltoo low to explain the non-disc population in the Galactic centre.There is not much difference between the cases with 50 and 100very massive stars, and, although we cannot provide a strictlimit, itis hard to imagine that even more massive stars were formed intheGC. It does not seem likely that a significant non-disc populationcan be formed via scattering off such objects.

To illustrate the failure of the model more clearly, Figure 5shows the orbital parameters and angular momentum direction ofeach 25 M⊙ star from a run with 100 extra 100 M⊙ stars. Even inthis case only a handful of stars seem to be outside of the disc, and

Figure 5. Aitoff projection of the angular momenta of the 25- M⊙ stars ina simulation that also included 100 100- M⊙ stars. The angle definition ischosen such that the disc initially has the same orientationas the observedclockwise disc (cf., Lu et al. 2006, Fig. 5). No star reaches the bottom halfof the plot, where they would be observed rotating counter-clockwise. Theinset shows the distribution of semi-major axes and eccentricities.

none of them appears to be rotating in the opposite direction. Theeccentricities, however, are comparable to the observed values.

3.5 Intermediate Mass Black Holes

Finally, we investigated the possibility that scattering from a hypo-thetical population of intermediate mass black holes mightheat thedisc stars. Portegies Zwart et al. (2006) argued that core collapsein GC star clusters could yield a significant population of such ob-jects, with masses around 1000M⊙. Adopting their numbers, therecould be∼50 intermediate black holes within 10 pc of the GC, andsome would likely have orbits that intrude upon the observedstellardisc.

To study the effect of these black holes, we set up simula-tions in which the stellar disc was initialised as describedin Sec-tion 3.1. We then addedN intermediate mass black holes, with or-bits appropriate for an isotropic cluster with number density pro-file n(r) ∝ r−7/4 (Bahcall & Wolf 1976) (i.e. drawn from an energydistribution f (E) ∝ E1/4 with random directions). We consider amodel with N = 5 1000-M⊙ inside 1 pc (similar to the numberexpected from Portegies Zwart et al. 2006 analysis), together withadditional cases that assumeN = 10 500-M⊙ andN = 20 500-M⊙intermediate mass black holes.

The resultingerms curves are shown in Figure 6. There is alot of dispersion between the different realisations, since the smallnumbers of black holes involved mean that in some cases none (orvery few) of the massive objects intersect the stellar disc.However,it is clear that the typical effect is rather similar to the case of mas-sive stars discussed previously. Quite large eccentricities are readilyproduced, but the inclinations (irms ≈ erms/2, not shown in the plotfor clarity) remain too small to explain the observed distribution ofstars in the Galactic Centre5.

5 Note that although Kozai (1962) resonancewould allow for the gener-ation of high inclinations, the rapid precession of stellarorbits due to thestellar cusp is likely to destroy this resonance.

c© 2008 RAS, MNRAS000, 1–6

Dynamics rebut cold-disc origin of GC stars 5

Figure 6. Eccentricity rms values from the nine different simulations ofdiscs of single stars perturbed by IMBHs of different masses. The results aresimilar to those of the models with extra massive stars in thedisc (Fig. 4).

4 CONCLUSIONS

In this paper we have studied the dynamical evolution of initiallyflat cold stellar discs consisting of the massive stars observed todayin the GC. We find that these systems do not evolve to match theobserved stellar population – which includes significant numbers ofstars that are not clearly identified with a well-defined discplane –even when scattering from plausible (and perhaps not so plausible)additional perturbers is included. It is clear that the mostimportantobservational constraints on theoretical models comes from mea-surements of the orbital inclinations. Although a flat circular discdoes not become eccentric enough in 5 Myr to match the observa-tions, it is possible to create quite eccentric systems either by start-ing from eccentric gas discs, or by perturbing circular discs withadditional massive objects (short-lived stars, or intermediate massblack holes). However, in no cases that we have considered isit pos-sible to generate enough very high inclination or counter-rotatingstars within the available time. No physical process we havecon-sidered breaks the basic relationerms ≈ 2irms between the meaneccentricity and inclination and this, coupled with the slow rate ofheating at late times (typically∝ t1/4), precludes formation of asignificant non-disc population.

The simplest disc scenario for forming the sub-pc stars in theGC appeals to the fragmentation of a flat gaseous disc into a pop-ulation of massive stars. Both the tendency of self-gravitating ac-cretion discs at these radii to fragment, and the subsequentforma-tion of a top-heavy mass function, receive at least qualifiedsupportfrom theory (Shlosman & Begelman 1989; Levin & Beloborodov2003; Nayakshin et al. 2007). However, our results suggest that itis very unlikely that subsequent evolution of a stellar system formedin this manner would yield any substantial population of starsoutside of the original disc plane or planes. Possible alternativesare that some or all of the GC stars migrated inward from largerradii (though this model has its own challenges, e.g., Kim etal.2004; Nayakshin & Sunyaev 2005); that a non-spherical compo-nent in the background stellar population, or a second stellar disc(Nayakshin et al. 2006), influences the dynamics of the main stel-lar disc in ways we have not considered; or that in situ star for-mation occurs in a very complex geometry. In the latter case thepresent orbits of stars are expected to trace, to a first approxima-tion, the angular momentum distribution of the star forminggas.

Ideas that would yield dynamically hot initial conditions –such asvery strongly warped discs (Nayakshin 2005) or clouds that col-lapse as they engulf the black hole (Yusef-Zadeh & Wardle 2008),perhaps forming two coeval discs (Genzel et al. 2003) – have beenproposed, though whether such scenarios can be analysed in thesame framework as the simpler disc models is dubious.

It is to be hoped that better observational data on the stellarorbits, which should confirm or refute the existence of additionalclustering in the distribution of orbital vectors, will afford someinsight into which models are viable.

ACKNOWLEDGEMENTS

We thank Clovis Hopman, Mark Morris, and the anonymous ref-eree for useful comments. This research was supported by NASAunder grants NNG04GL01G, NNX07AH08G and NNG05GI92G,and by the NSF under grant AST 0407040. RDA acknowledgessupport from the Netherlands Organisation for Scientific Research(NWO) through VIDI grants 639.042.404 and 639.042.607.

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