Star Formation and Herbig Ae / Be Stars

Kaitlin Kratter University of Arizona

The Missing Link, Santiago, Chile, April 2014 AB & SU Aurigae, Herbig, 1960 1960ApJS....4..337H

Deﬁning Herbig Ae/Be Stars

• A or B spectral type • IR excess • Emission lines • Nebulosity

Herbig, 1960 1960ApJS....4..337H

Deﬁning Herbig Ae/Be Stars

• A or B spectral type • IR excess • Emission lines • Nebulosity table for theorists

Herbig, 1960 1960ApJS....4..337H

Deﬁning Herbig Ae/Be Stars

• A or B spectral type • IR excess • Emission lines • Nebulosity table for theorists

table for observers

Herbig, 1960 1960ApJS....4..337H

Deﬁning an object class, modestly

Herbig 1960 Theoretical deﬁnition of “Intermediate” Mass

Palla & Stahler 2005 • Upper limit: 8 • evolution: SN, winds no “PMS” • formation: no PMS, importance of radiation pressure, formation of HII region born radiative • Lower limit: 1.5-3 • evolution: signiﬁcant mass loss • formation: multiplicity, radiative PMS Herbig Ae/Be’s don’t quite match up.. FIGURE 2. Revised PMS models for low- and intermediate-mass stars with protostellar initial conditions. Selected isochrones are also shown. (From [23])

found by Hayashi, Iben and others can be summarized as follows: • Protostars move from the forbidden zone to the border for quasi-static equilibrium: yes, but the HR diagram is much more reduced, especially for high luminosity stars. • PMS stars descend along the vertical Hayashi tracks: yes & not, since not all of them have an initial fully convective phase. Stars with mass > 3M begin the PMS phase as fully radiative objects. ∼ ⊙ • PMS stars join radiative tracks: yes, all stars go through a radiative phase, apart from stars less massive than 0.4 M that remain fully convective. ∼ ⊙ • PMS stars reach the Main Sequence: indeed, all of them do, but massive ones (M > 10 M ) are born on the main sequence. The main reason is that the Kelvin- Helmoltz∗ ∼ time⊙ scale is a sensitive function of mass and decreases quite rapidly for more massive objects, becoming shorter than the accretion time scale. Thus, such massive stars have no PMS phase at all and appear directly on the main sequence once they are optically revealed. These important properties can be appreciated in the HR diagram shown in Figure 2. A full account of the observational and theoretical properties of young stars and tests on the HR diagram can be found in [18] [19].

SOME OBSERVATIONAL TESTS

As indicated before, the distribution of stars in the color-magnitude diagram of the cluster NGC 2264 provided the ﬁrst evidence of the existence of a population of young stars in the contraction phase. Similar results were obtained by Merle Walker on the brighter members of other clusters, such as NGC 6530, IC 5146, and NGC 6611. Interestingly, the original measurements of NGC 2264 by Walker have been recently

26 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.196.209.100 On: Fri, 04 Apr 2014 21:35:21 Intermediate Mass Stars are in the Transition Zone

• Luminosities: transition zone in stellar structure on the PMS (or the end of the PMS)

• Accretion rates onto cores: transition zone between non-turbulent and turbulent initial conditions

• Ionization: transition zone between irrelevance and HII region dominated feedback

• Disks and Planets: the last (but easier to image) place for planets?

Does transition imply a break in formation process? Do we need more than one mode of star formation: CMF vs IMF

2.2. IMF Universality in the Milky Way 2.2.1. Uncertainties in the IMF Determination From Re- solved Populations PossibleThe basic observational evidence method tofor derive a the single IMF con- sists of three steps. First, observers measure the luminosity function (LF) of a completemode? sample of stars that lie within some defined volume. Next, the LF is converted into a present day mass function (PDMF) using a mass-magnitude ings as a function of scale). This would allow observations log[ M / MOt ] relationship.-2 Finally, the0 PDMF is2 corrected4 for the star6 for- and theory to compare the entire dynamic range of turbulent mation history, stellar evolution, galactic structure, cluster fragmentation using all the information in the field rather dynamical evolution and binarity to obtain the individual- 0 than focusing on some, ultimately arbitrary, definition of a star IMF. None of these steps is straightforward,Hennebelle & and many “core” and a “cloud.” biases and systematic uncertainties Chabrier may betheory introduced during the process.‘last crossing’ A detailed discussion of these limitations at In summary, these theories provide a natural link be- -2 (predicted CMF) tween the CMF, GMC mass function, and star cluster mass low masses and in the substellar regime is given by Jeffries Fig. 1.— IMF functional forms proposed by various au- (2012) and Luhman (2012). function for a wide range of systems. This link derives thors from fits to Galactic stellar data. With the exception -4 When obtaining the LF, defining a complete sample of from the inherently scale free physics of turbulent fluctua- ‘frst crossing’ of the Salpeter slope, the curves are normalized such that log[ dN/dlogM ] stars for a given volume can be challenging. Studies of field (predicted tions plus gravity.the integral The over most mass robust is unity. features When of comparing these models with ob- stars from photometric surveys, either wide and relatively GMC MF) are the Salpeter-likeservational slope data, the (↵ normalization2.2 2.4 is), set which by the emerges total number shallow-6 (e.g. Bochanski et al., 2010) or narrow but deep ⇠ (e.g. SchultheisChabrier et (mass al., x3) 2006), are magnitudeMilky Way limited GMCs and genericallyof from objects scale-free as shown processes, in Figure 2. and the log-normal- Kroupa (mass x3) like turnover, which emerges from the central limit theo- need to be corrected for the Malmquist bias. Indeed, when distances-4 are estimated-2 from0 photometric2 or spectroscopic4 6 rem. The location of the turnover, in any model where tur- log[ M / M ] analyticallyOffner different, et is toal represent 2014 the full mass range of parallax, the volume limited samplesonic inferred will be pol- bulent density fluctuations are important, is closely tied to the IMF as a series of power-laws, such as in Kroupa (2001, luted by more distant bright stars due to observational un- the sonic length,2002). which Above is0 the.2 M scale,, theR multiple-power-laws, below which ther- segments ⇠ Fig.certainties 3.— Core and mass intrinsic spectrum dispersion and GMC in the mass color-magnitude spectrum cal- mal or magneticand log-normal support with dominates a power-law over tail turbulence. at high masses Be- agreeculatedrelation. by ( ThisHopkins effect, can2012b) be fairly as a significant function and of change stellar the mass very well as shown by Figure 1. However, the form of the low this scale density fluctuations become rapidly damped andslope sonic of mass. the luminosity The grey function dotted by curve more than shows 10%. the Stud- CMF IMF at the low mass end is still relatively uncertain and sub- and cannot produce new cores. As a result, the location of inferredies of by nearby (Hennebelle stars, for and which Chabrier the distance, 2008), is determined while the the peak, andject the to ongoing rapidity debate of the (e.g., CMFThies turnover and Kroupa below2007, it, 2008; from trigonometric parallax, are not affected by this bias, Kroupa et al. 2013). The philosophical difference betweenlines indicate structures defined by the last (grey dashed) depend dramatically on the gas thermodynamics (Larson, but they are affected by the Lutz and Kelker (1973) bias, in these two forms reduces to whether one believes that star-and first crossing (dot-dashed) of the self-gravitating bar- 2005; Jappsen et al., 2005). Unlike what is often assumed, which the averaged measured parallax is larger than the true formation is a continuous process or whether there are dis-rier.parallax, The Chabrier if the parallax(2005) uncertainties and Kroupa exceed(2001) 10%. fits These to the however, the dependence on thermodynamics displayed by tinct physical processes that dominate in certain regimes,observedissues IMF reduce are the shown studies by within the grey 20 pc dashed to a completeness and light-grey turbulent modelssuch as and at the simulations stellar/sub-stellar with driven boundary. turbulence is dot-dashedlevel of lines,80% respectively. (Reid et al., 2007), Observed which Milky results Way in small GMCs ⇠ not the same asOther a Jeans-mass proposed functional dependence forms (see includeHennebelle a truncated ex- number statistics, especially at low masses. Cluster stud- ↵ are indicated by the solid grey line with circles. ponential where dN M (1 exp[( M/M ) ])dM and Chabrier 2009). The difference/ between the scalingp of ies do not have these problems since a complete sample can (de Marchi and3 Paresce1/2 , 2001;3/2 Parravano et al., 2011), the the Jeans mass, MJ cs/(⇢ G ), and the sonic mass, in principle be obtained from the complete cluster census. 2 Pareto-Levy' family of stable distributions, which includes However, secure membership cannot be assessed from pho- Ms vturb Rs/G, can be dramatic when extrapolating to ' the Gaussian (Cartwright and Whitworth, 2012), and a log-andtometry the formation alone and of proper cores. motion The measurements most important as well assump- as extreme regionsnormal such joined as with starbursts power laws where at both the low turbulent and high ve- masses tionsspectroscopic in these simulations follow-up are are theoften explicit necessary. treatment In addition, of self- locity, vturb(Maschberger, is large (Hopkins, 2013). It, 2013c). is usually not Consequently, possible to discrim- a gravitythe remaining and turbulent contamination driving, since by field these stars two may processes be large if lie better understandinginate between of these the role forms of on thermodynamics the basis of observations in ex- due cluster properties such as age and proper motion are similar at the very heart of the analytic models described above. treme environmentsto significant is needed. uncertainties. We note that Figure 1 shows to the field. Other processesonly the proposed such as magnetic IMF functions fields, and feedback not the error from associ-Other physicalFor young processes star forming such regions, as magnetic differential fields extinction and the protostars, andated intermittency with the derivations of turbulence of these fits may from also the be data. im- In allequationmay be of a state significant are also problem. very important This is usually but we taken temporar- into cases shown, the authors fit the galactic field IMF over a re- portant and are not always considered in these models (see ily ignoreaccount these by limiting for the the purpose sample of up comparing to a given A theV . analytic How- 5 and thestricted PPVI reviewmass range by andKrumholz then extrapolated et al.). their For functional exam- fitmodelsever, to imposing simulations. a limit Amay more exclude technical the more but central severe regions, issue is § to the whole mass range. Observers and theoreticians nowthe definitionwhich are often of the more object extincted, mass used from the to compute determination the mass of ple, supersonicuse these turbulence functional is intermittent forms without and any spatially error bars cor-to eval- related, so both velocity and density structures can exhibit spectrum.the LF. First, If the stellar the exact spatial definition distribution of isa core not the can same be at impor- all uate how well their data (or models) are consistent within masses and, in particular, if mass segregation is present, ex- coherency andtheir non-Gaussian uncertainties. features, including significant tant, and second, “sink” particles are often used to mimic cluding high AV sources may introduce a strong bias. Such deviations from log-normal density distributions (Feder- protostarsan effect or can cores be tested that accrete by simulating from the a fake surrounding cluster as in gas. rath et al., 2010; Hopkins, 2013b). In addition, the mod- GivenParker that et these al. (2012). sinks cannot self-consistently model all the els above generally assume a steady-state system, where in relevantEven physical with no processes, extinction, mass they segregation are usually must intermediate be taken fact time-dependent effects may be critical (see 5). Cer- betweeninto account a core becauseand a star, the although cluster LF when may be feedback different inis theprop- § tainly these effects are necessary to consider when trans- erly accounted for, they may more closely resemble a star lating the CMF models into predictions of star formation (see 5.2). Another crucial issue is numerical resolution. 3 § rates (Clark et al. 2007; Federrath and Klessen 2012; PPVI Broadly speaking, the IMF covers about four orders of mag- review by Padoan et al.). Preliminary attempts to incor- nitude in mass, which corresponds to five to seven orders of porate time-dependence into the models (Hennebelle and magnitude in spatial scales assuming a typical mass-size re- 1 2 Chabrier, 2011, 2013; Hopkins, 2013a) are worth explor- lation MCORE L (e.g., Larson 1981). Since most nu- / ing in more detail in future work. merical simulations span less than five orders of magnitude, the mass spectrum they compute is necessarily limited. 3.3. The CMF Inferred from Numerical Simulations The CMF has been determined in driven turbulence A variety of numerical simulations have been performed simulations without self-gravity, where prestellar cores are to study the origin of the CMF and the IMF (see 5 for a dis- identified as regions that would have been gravitationally § cussion of cluster simulations). Here, we consider a sub-set bound if gravity had been included. The exact cores iden- of numerical simulations that focus on modeling turbulence tified depend on the types of support considered. Padoan et al. (2007) performed both hydrodynamical and magne-

10 and excitation conditions, finite resolution, superposition of emission from disparate regions along the line of sight, or i) Not all cores are ‘prestellar’. Here we show the emerging IMF that could arise optical-thickness (concealment of regions along the line of if the low-mass cores in the CMF are transient ‘fluff’. sight). In addition, spectra only probe the radial component of the velocity. These uncertainties aside, the line-widths CMF + + measured from observations of N2H , NH3, and HCO towards dense cores in Perseus and Ophiuchus suggest that the internal velocities, while generally sub-sonic, can con- number number emerging IMF tribute a significant amount of energy (Andre´ et al., 2007; Johnstone et al., 2010; Schnee et al., 2012). Consequently, 0. 1 10 some of the cores identified purely on the basis of dust emis- mass [M ] ⦿ sion or extinction may actually be unbound, transient ob- ii) Core growth is not self-similar. Here we show the emerging IMF that could arise if, say, only the low-mass cores in the CMF are still accreting. jects. Enoch et al. (2008) demonstrated that this is likely to affect only the lowest mass-bins of their CMF, and then only by a small amount, but they note that the statistics are still small. CMF emerging IMF Similar constraints pertain to estimates of the magnetic

number number field in cores. Field estimates are obtained using either the Zeeman effect, which only probes the line-of-sight compo- 0. 1 10 nent, or the Chandrasekhar-Fermi conjecture, which can be mass [M⦿] used to estimate the transverse component. Crutcher et al. iii) Varying star formation efficiency (SFE). Here we show the emerging IMF that (2009) presented statistical arguments to suggest that mag- Beware IMF / CMF could arise if the high-mass cores in the CMF have a lower SFE than their low-mass siblings. netic fields are not able to support prestellar cores against connection gravity. However, observing magnetic field strengths in and excitation conditions, finite resolution, superposition of cores is challenging (see PPVI chapter by Li et al. for more discussion). emission from disparate regions along the lineCMF of sight, or

i) Not all cores are ‘prestellar’. Here we show the emerging IMF that could arise optical-thickness (concealment number ofemerging regions IMF along the line of 4.1.3. Extracting Cores from Column Density Maps if the low-mass cores in the CMF are transient ‘fluff’. sight). In addition, spectra only probe the radial component of the velocity. These uncertainties aside, the line-widths Any evaluation of the CMF inherently depends on the CMF 0. + 1 10 + procedure used to identify cores and map their boundaries. measured from observations of N2Hmass, NH[M⦿]3, and HCO Different algorithms (e.g., GAUSSCLUMPS, Stutzki and towardsiv) Fragmentation dense is not cores self-similar. in Perseus Here we show and the Ophiuchus emerging IMF that suggest could that thearise internalif the cores in velocities, the CMF fragment while based on generally the number of sub-sonic,initial Jeans masses can con- Guesten 1990; CLUMPFIND, Williams et al. 1994; dendro- number number emerging IMF they contain. grams, Rosolowsky et al. 2008b), even when applied to the tribute a significant amount of energyemerging (IMFAndre´ et al., 2007; Johnstone et al., 2010; Schnee et al., 2012). Consequently, same observations, do not always identify the same cores, 0. 1 10 some of the cores identified purely on the basis of dust emis- and when they do, they sometimes assign widely different mass [M⦿] masses. Interestingly, different methods for extracting cores sion or extinction may actually be unbound, transient ob- ii) Core growth is not self-similar. Here we show the emerging IMF that could number usually find similar CMFs even though there may be a poor jects. Enoch et al. (2008) demonstrated that thisCMF is likely arise if, say, only the low-mass cores in the CMF are still accreting. correspondence between individual cores (for example, the to affect only the lowest mass-bins of their CMF, and then CMFs derived for Ophiuchus by Motte et al. 1998 and by only by a small amount, but they note0. that1 the10 statistics are mass [M⦿] Johnstone et al. 2000). A similar problem arises in the still small. CMF emerging analysis of simulations (Smith et al., 2008). Pineda et al. v) VaryingSimilar embedded constraints phase timescale. pertain Here we to show estimates the emerging of IMF the that magnetic IMF could arise if the low-mass cores in the CMF finish before the high-mass cores. (2009) have shown that the number and properties of cores number number field in cores. Field estimates are obtained using either the extracted often depend critically on the values of algorith- Zeeman effect, which only probes the line-of-sight compo- mic parameters. Therefore, the very existence of the cores 0. 1 10 nent, or the Chandrasekhar-Fermi conjecture, whichCMF can be that contribute to an observed CMF should be viewed with mass [M⦿] used to estimate the transverse component. Crutcher et al. caution, particularly at the low-mass end where the sample iii) Varying star formation efficiency (SFE). Here we show the emerging IMF that (2009) presented statistical arguments number emerging to IMF suggest that mag- may be incomplete (Andre´ et al., 2010). could arise if the high-mass cores in the CMF have a lower SFE than their low-mass siblings. netic fields are not able to support prestellar cores against gravity. However, observing magnetic0. 1 field10 strengths in 4.2. Phenomenology of Core Growth, Collapse and Offner et al 2014 cores is challenging (see PPVI chaptermass by [MLi⦿] et al. for more Fragmentation discussion). CMF In the theory of turbulent fragmentation, cores form in Fig. 4.— At the beginning of section 4, we discuss the con-

number number layers assembled at the convergence of large-scale ﬂows or emerging IMF 4.1.3. Extracting Cores from Column Density Maps ditions that are necessary for the CMF to map to the IMF. in shells swept up by expanding nebulae such as HII re- TheseAny schematics evaluation illustrate of the CMF what inherently would happen depends if each on of the gions, stellar wind bubbles and supernova remnants. Nu- 0. 1 10 theseprocedure conditions used towere identify to be violated. cores and map their boundaries. merical simulations indicate that these cores are delivered mass [M⦿] Different algorithms (e.g., GAUSSCLUMPS, Stutzki and by a complex interplay between shocks, thermal instabil- iv) Fragmentation is not self-similar. Here we show the emerging IMF that could arise if the cores in the CMF fragment based on the number of initial Jeans masses Guesten 1990; CLUMPFIND, Williams et al. 1994; dendro- they contain. grams, Rosolowsky et al. 2008b), even when applied to the emerging IMF same observations, do not always identify the same cores,12 and when they do, they sometimes assign widely different masses. Interestingly, different methods for extracting cores

number number CMF usually ﬁnd similar CMFs even though there may be a poor correspondence between individual cores (for example, the 0. 1 10 CMFs derived for Ophiuchus by Motte et al. 1998 and by mass [M⦿] Johnstone et al. 2000). A similar problem arises in the

v) Varying embedded phase timescale. Here we show the emerging IMF that analysis of simulations (Smith et al., 2008). Pineda et al. could arise if the low-mass cores in the CMF ﬁnish before the high-mass cores. (2009) have shown that the number and properties of cores extracted often depend critically on the values of algorith- mic parameters. Therefore, the very existence of the cores CMF that contribute to an observed CMF should be viewed with caution, particularly at the low-mass end where the sample number number emerging IMF may be incomplete (Andre´ et al., 2010).

0. 1 10 4.2. Phenomenology of Core Growth, Collapse and mass [M⦿] Fragmentation In the theory of turbulent fragmentation, cores form in Fig. 4.— At the beginning of section 4, we discuss the con- layers assembled at the convergence of large-scale ﬂows or ditions that are necessary for the CMF to map to the IMF. in shells swept up by expanding nebulae such as HII re- These schematics illustrate what would happen if each of gions, stellar wind bubbles and supernova remnants. Nu- these conditions were to be violated. merical simulations indicate that these cores are delivered by a complex interplay between shocks, thermal instabil-

12 Multiplicity is Mass Dependent

30 RAGHAVAN ET AL. Vol. 190

Figure 12. Multiplicity statistics by spectral type. The thin solid lines represent stars and brown dwarfs beyond the spectral range of this study, and their sources Figure 13. Period distribution for the 259 confirmed companions. The data are listed in the text. For the FGK stars studied here, the thick dashed lines show are plotted by the companion detection method. Unresolved companions our observed multiplicity fractions, i.e., the percentage of stars with confirmed such as proper-motion accelerations are identified by horizontal line shading, stellar or brown dwarfRaghavan companions, for spectral 2010 types F6–G2 and G2–K3. The spectroscopic binaries by positively sloped lines, visual binaries by negatively thick solid lines show the incompleteness-adjusted fraction for the entire F6–K3 sloped lines, companions found by both spectroscopic and visual techniques by sample. The uncertainties of the multiplicity fractions are estimated by bootstrap crosshatching, and CPM pairs by vertical lines. The semimajor axes shown in analysis as explained in Section 5.2. AU at the top correspond to the periods on the x-axis for a system with a mass sum of 1.5 M , the average value for all the pairs. The dashed curve shows aGaussianfittothedistribution,withapeakatlog⊙ P 5.03 and standard publications, when available. Otherwise, they are estimated = deviation of σlog P 2.28. using mass ratios for double-lined spectroscopic binaries, or = from multi-color photometry from catalogs, or using the ∆mag are not available from spectroscopic or visual orbits, we esti- measures in the WDS along with the primary’s spectral type. mate them as follows. For CPM companions with separation Metallicity and chromospheric activity estimates of the primary measurements, we estimate semimajor axes using the statistical are adopted for all components of the system. relation log a log ρ +0.13 from DM91, where a is the ′′ = ′′ 5.3.2. Multiplicity by Spectral Type and Color angular semimajor axis and ρ is the projected angular separation, both in arcseconds. This, along with mass estimates as de- Figure 12 shows the multiplicity fraction for stars and brown scribed in Section 5.3.1 and Newton’s generalization of Kepler’s dwarfs. Most O-type stars seem to form in binary or multiple Third Law yields the period. For the remaining few unresolved systems, with an estimated lower limit of 75% in clusters and pairs, we assume periods of 30–200 years for radial-velocity associations having companions (Mason et al. 1998a, 2009). variables and 10–25 years for proper-motion accelerations. The Studies of OB-associations also show that over 70% of B and period distribution follows a roughly log-normal Gaussian pro- Atypestarshavecompanions(Shatsky&Tokovinin2002; file with a mean of log P 5.03 and σlog P 2.28, where Kobulnicky & Fryer 2007; Kouwenhoven et al. 2007). In sharp P is in days. This average= period is equivalent= to 293 years, contrast, M-dwarfs have companions in significantly fewer somewhat larger than Pluto’s orbital period around the Sun. The numbers, with estimates ranging from 11% for companions median of the period distribution is 252 years, similar to the 14–825 AU away (Reid & Gizis 1997)to34%–42%(Henry Gaussian peak. This compares with corrected mean and me- &McCarthy1990;Fischer&Marcy1992). Finally, estimates dian values of 180 years from DM91. The larger value of the for the lowest mass stars and brown dwarfs suggest that only current survey is a result of more robust companion informa- 10%–30% have companions (Burgasser et al. 2003;Siegleretal. tion for wide CPM companions. The similarity of the overall 2005;Allenetal.2007;Maxtedetal.2008;Joergens2008). profile with the incompleteness-corrected DM91 plot suggests Our results for F6–K3 stars are consistent with this overall that most companions they estimated as missed have now been trend, as seen by the thick solid lines for the incompleteness- found. The shading in the figure shows the expected trend—the corrected fraction. Moreover, the thick dashed lines for two shortest period systems are spectroscopic, followed by com- subsamples of our study show that this overall trend is present bined spectroscopic/visual orbits, then by visual binaries, and even within the range of solar-type stars. Of the blue subsample finally by CPM pairs. The robust overlap between the various (0.5 B V 0.625, F6–G2, N 131), 50% 4% techniques in all but the longest period bins underscores the ! − ! = ± have companions, compared with only 41% 3% for the red absence of significant detection gaps in companion space and ± subsample (0.625

30 RAGHAVAN ET AL. Vol. 190

Binaries are the norm for A and B stars

• Competitive Accretion

• Turbulent Core Fragmentation Two Star Formation Models

• Competitive Accretion

• Turbulent Core Fragmentation

Two “camps” are converging. Elements of both cartoons are likely correct. Padoan 1995, McKee & Tan 2004 Turbulent Core Accretion

Rturb lower mass

rtherm higher mass

R core turb

• Scaling up from A to O changes rtherm contribution of thermal core

• This causes change in accretion rate with core mass Bonnell et al 1997 Competitive Accretion initial cores

differential growth Do we need more than one formation model theoretically?

• Competitive Accretion: No. All stars start the same, and intermediate mass ones grow more, but not as much as massive ones

• Turbulent Core Model: No. There exists a spectrum of density perturbations that seed stars of diﬀerent masses 8 M.R. Bate . 5 4 r 25 . aT = ect, one E ↵ =0to1 ↵ t/t Comparing two numerical models that create intermediate mass stars ,deﬁnedimplicitlyby r T Column density (left) and density-weighted mean tem-

Figure 2. The global evolution of column density in the radiation hydrodynamical calculation. Shocks lead to the dissipation of the turbulent energy that initially supports the cloud, allowing parts of the cloud to collapse. Star formation begins at t =0.727 t↵ in a collapsing dense core. By t =1.10t↵ the cloud has produced six main sub-clusters, and by the end of the calculation these sub-clusters started to merge and dissolve. Each panel is 0.6 pc (123,500 AU) 20 K. These ﬂashes are associated with brief increases in across. Time is given in units of the initial free-fall time, t =1.90 105 yr. The panels show the logarithm of column density, N, through the cloud, with ↵ ⇥ Bate 2012:2 CA Krumholz et al 2012: TC < the scale covering 1.4 < log N<1.0 with N measured in g cm . White dots represent the stars and brown dwarfs.

c 0000 RAS, MNRAS 000, 000–000 There are a few interesting points to take from these Fig. 2.— numerical differences perature (right) inare run indicated SmNW. in Thein the steps times right of of column,the 0.25. each running positions pair In of from the of starthe column images particles, mass density with of plot, theis the white size star. the circles of radiation In the indicate We show circle the temperature this indicating right rather column,radiation than the the temperatures temperature gas are temperature shown except nearly because in the the equal hot gasand everywhere ambient and medium in in outside the the material clouding ejected cloud, in the run radiation by SmNW, temperature protostellarthis provides outﬂows contribution. a convenient in means run to ﬁlter TuW. Us- plots. One involves the morphologyIn of the the turbulent heated runs, regions. distribution even at looks late more timesgas the like temperature a surrounded series byquite of a near islands to large of thecan medium heated background temperature. discern that something In isthat e like either are individual at heated protostellar or cores bybut the these star areature short-lived or and small, starthe generally keeping system accretion the luminosity embedded temper-simulation that volume are above 10 large K enough for to short heat periods. the entire CA TC ↵ t

turbulent spectrum 4 2 erence k k ↵ erent in ↵ 19 0 evaluated 18 1.5 erences be- density 1.2 10 r ⇢ 8.6 10 r ↵ ⇥ G ⇥ 32 Feedback radiation radiation,/ winds ⇡ 3

13 p 7.5 M = ↵ t RESULTS erent initial conditions, it is im- ↵ ner et al. 2009). 3. 7 ↵ 20%. ⇠ erent in the smooth and turbulent runs, ↵ Overall Evolution and Morphology , based on one-dimensional calculations of the 3.1. M , since this is the dynamical time appropriate to the The second subtlety is that, as in Paper I, we only Note that in Paper I we instead used the volume weighted In runs TuNW and TuW there are sometimes brief increases in erent cases. Moreover, in the runs with turbulent M i Before examining the results of our simulations, we In Figures 2, 3, and 4 we show the large-scale column 6 7 ↵ 6 05 . ⇢ TuNW and TuW.drodynamic Finally, we and note gravitationalnecessarily that, boundary di while conditionsthe the are great hy- majority ofruns the occurs star formation incomputational in subregions the volume, much turbulent and smallerconditions thus than have the the minimal periodic entire shak impact. boundary boundary conditions on We the alsoand radiation in TuW impose runs in TuNW Mar- ordertional volume to (c.f. let O radiation escape the computa- first mention twoto subtleties the in remaindercomparing the of runs this analysis with di discussion. thatportant apply to First, since normalizetween we the the are runs times reflectand so the not underlying that simply physicaldi di that behavior, theinitial dynamical conditions, time is thetees di strong that initial thestructures turbulence that majority guaran- are of significantlyaveraged denser the density. than mass the Givennatural volume- is these approach, considerations, compressed which the into wein most adopt, units is ofat to the a measure free-fall time times densityh equal tobulk the of initial the mass-weightedtage matter. density that This itcomparison, approach is since also the an has mostof observation the would the natural advan- detect mass, basissity the and for at bulk would observational which be thisfollows mass sensitive whenever resides. to we the Fordefined refer this in typical to reason, this den- manner. times, in1. what We we report normalize thisregard to quantity in stars Table as collapsed0 objects once theirmass mass exceeds at which second(Masunaga collapse & to Inutsuka stellar 2000).to densities merge We occurs with allow oneWe smaller another therefore objects and restrict with ourthis more analysis mass. massive to stars. objects larger than density and density-weightedin temperature for distributions runsall SmNW, three TuNW, we andlence observe TuW, the creates respectively. same ancollapse general overdense In and trend: region, form which stars. theamentary, then and The turbu- the collapsing begins stars structures to are areparticularly born fil- along at the the filaments,The and nodes temperature where is initially thespots small, filaments around but intersect. individual asally stars stars spread form appear, over hot time. and these gradu- mean density toever, compute because the the initial free-fallbetween volume- density and time field mean mass-weighted for isfor density very run free-fall this smooth, times SmNW; run the how- is di only the overall background temperature level visible in some snapshots, 8 M.R. Bate

Competitive Accretion: Source for Herbig Ae/Be Stellar and multiple star properties 3133

Table 2. The numbers of single and multiple systems for dif- •resultsLow using radiationmass hydrodynamics end only and a(consistent barotopic equation of with ferent primary mass ranges at the end of the radiation hydrody- state are very similar. The main barotropic calculation gives mul- namical calculation. tiplicitiescluster that are somewhatmass) higher for primary masses >0.2 M than those given by the radiation hydrodynamical calculation, but⊙ Mass range (M )SingleBinaryTripleQuadruple ⊙ the results are consistent within the statistical uncertainties. M<0.03 7 0 0 0 It is important to note that the surveys with which we are compar- 0.03 M<0.07 20 0 0 0 ing the multiplicities are primarily of field stars rather than young ≤ 0.07 M<0.10 8 3 0 0 •stars.Tend This is necessary to be because in the surveys center of young stars of either the do not potential ≤ 0.10 M<0.20 17 7 1 0 sample a large range of separations and mass ratios, or the statistics ≤ 0.20 M<0.50 21 9 2 2 well ≤ are too poor. However, there may be considerable evolution of the 0.50 M<0.80 5 2 0 1 ≤ multiplicities between the age of the stars when the calculations 0.80 M<1.22 1 1 0 5 ≤ were stopped ( 10 yr) and a field population. This question of M>1.24 6 1 4 the subsequent∼ evolution of the clusters produced by hydrodynam- All masses 84 28 5 7 •icalGood simulations agreement was recently tackled bywith Moeckel field & Bate multiplicity (2010) Stellar and multiple star properties 3127 who took the end point of the main barotropic calculation of Bate (2009a)and and IMF evolved it to an age of 107 yr using an N-body code excludes those that are components of triples or quadruples. The under a variety of assumptions regarding the dispersal of the molec- Downloaded from number of triples excludes those that are members of quadruples. ular cloud. Moeckel & Bate found that the multiplicity distribution However, higher-order systems are ignored (e.g. a quintuple system evolved very little during dispersal of the molecular cloud and was may consist of a triple and a binary in orbit around each other, but surprisingly robust to different assumptions regarding gas dispersal. this would be counted as one binary and one triple). We choose Even under the assumption of no gas removal at 107 yr, although quadruple systems as a convenient point to stop as it is likely that the multiplicities were found to have decreased slightly compared

most higher order systems will not be stable in the long term and with those at the end of the hydrodynamical calculation, they were http://mnras.oxfordjournals.org/ would decay if the cluster was evolved for many millions of years. still formally consistent. They concluded that when star forma- The numbers of single and multiple stars produced by the radiation tion occurs in a clustered environment, the multiple systems that hydrodynamical calculation are given in Table 2 following these are produced are quite robust against dynamical disruption during definitions. In Table 3, we give the properties of the 40 multiple continued evolution. Therefore, we do not expect the multiplicities systems. presented in Fig. 17 to evolve significantly as the stars evolve into In the left-hand panel of Fig. 17, we compare the multiplicity afieldpopulation.

fraction of the stars and brown dwarfs as a function of stellarFigure mass 2. The globalIn evolution detail, of we column find density the following. in the radiation hydrodynamical calculation. Shocks lead to the dissipation of the turbulent energy that obtained from the radiation hydrodynamical calculation with obser-initially supports the cloud, allowing parts of the cloud to collapse. Star formation begins at t =0.727 t↵ in a collapsing dense core. By t =1.10t↵ the vations. The results from a variety of observational surveys (seecloud the has produced sixSolar-type main sub-clusters, stars: and byDuquennoy the end of the calculation & Mayor these sub-clusters (1991) found started to mergean ob- and dissolve. Each panel is 0.6 pc (123,500 AU) across. Time is given in units of the initial free-fall time, t =1.90 105 yr. The panels show the logarithm of column density, at University of Arizona on April 4, 2014 N, through the cloud, with ↵ ⇥ figure caption) are plotted using black open boxes with associatedthe scale coveringserved1.4 < log multiplicityN<1.0 with fractionN measured of in gmf cm 2. White0.58 dots represent0.1. the However, stars and brown the dwarfs. SFE too high error bars and/or upper/lower limits. The data point from the survey recent larger survey carried out by Raghavan= ± et al. (2010) revised Figure 7. Histograms giving the IMF of the 590 stars and brown dwarfs at t 1.20tff fromof theDuquennoy main barotropic & Mayor (1991) calculation is plotted of using Bate dashed(2009a) lines (left), for the and the this downwards to 0.44 0.02 and they concluded that the higher c 0000 RAS, MNRAS 000, 000–000 = 183 objects formed at the same time in the radiation hydrodynamical calculation (right). Theerror double bars hatched since this histograms survey has are been used recently to denote superseded those byobjec thatts that value obtained by Duquennoy± & Mayor was due to them overesti- have stopped accreting, while those objects that are still accreting are plotted using single hatching.of Raghavan The et radiational. (2010). hydrodynamical The results from the calculation radiation hydrody- produces far mating their incompleteness correction. The radiation hydrodynam- namical simulation have been plotted in two ways. First, using the ical calculationDownloaded from gives a multiplicity fraction of 0.42 0.08 over the fewer brown dwarfs and low-mass stars and more stars with masses !1.5M and is in good agreement with the Chabrier (2005) fit to the observed IMF for ± individual objects. Two other parametrizations of the IMF are also plotted: Salpeter⊙ (1955) andnumbers Kroupa given (2001). in Table 2 we compute the multiplicity in six mass mass range 0.5–1.2 M which is in good agreement with the result ranges: low-mass brown dwarfs (masses <0.03 M ), VLM objects of Raghavan et al. (2010).⊙ excluding the low-mass brown dwarfs (masses⊙ 0.03–0.10 M ), M-dwarfs: Fischer & Marcy (1992) found an observed multi- observational constraints, but the statistical uncertainty is large. accretionlow-mass and ‘ejection’. M-dwarfs (masses There 0.10–0.20 was no M significant), high-mass dependence M-dwarfs⊙ of plicity fraction of 0.42 0.09. In the mass range 0.1–0.5 M we (masses 0.20–0.50 M ), K-dwarfs and⊙ solar-type stars (masses obtain 0.36 0.05. Fischer± & Marcy’s sample contains stars⊙ with Here we obtain a ratio of N(0.08–1.0)/N(0.03–0.08) 117/31 the mean accretion rate of an object on its final mass. Rather, there ± = ≈ 0.50–1.20 M ), and intermediate-mass⊙ stars (masses >1.2M ). masses between 0.1 and 0.57 solar masses, but the vast majority 3.8. Eight of the 31 low-mass objects and 84 of the 117 stars was a roughlyThe filled linear blue⊙ correlation squares give the between multiplicity an fractionsobject’s in final these mass mass⊙ and have masseshttp://mnras.oxfordjournals.org/ in the range 0.2–0.5 M whereas in the simulation were still accreting when the calculation was stopped, so there the time betweenranges, while its the formation surrounding and blue the hatched termination regions give of its the accretion. range almost half of the low-mass stars have⊙ masses less than 0.2 M .In is some uncertainty in this figure due to unknown future evolution. Furthermore,in stellar the masses accretion over which on to the an fraction object is was calculated usually and terminated the 1σ the 0.2–0.5 M mass range we obtain 0.38 0.06. All these values⊙ ⊙ ± But the value is well within observational uncertainties. For the by a dynamical(68 per cent) interaction uncertainty between on the multiplicity the object fraction. and another In addition, system. a are consistent with the statistical uncertainties. thick solid line gives the continuous multiplicity fraction computed VLM objects: there has been much interest in the multiplicity main barotropic calculation of Bate (2009a), this ratio was much Note thatusing such a boxcar an interaction average of does the results not necessarilyfrom the radiation require hydrody- that the of VLM objects in recent years (Mart´ın et al. 2000, 2003; Bouy lower: 212/146 1.45 at t 1.20t and 408/326 1.25 at object isnamical ejected simulation. from the The cluster. width of Many the boxcar times average this is is the one case,order but et al. 2003, 2006; Burgasser et al. 2003, 2006; Close et al. 2003, ≈ = ff = t 1.50tff . moving anof magnitude object into in stellar a lower mass. density part of the cloud (e.g. out of 2007; Gizis et al. 2003; Pinfield et al. 2003; Siegler et al. 2003, 2005; =Below 0.03 M , the IMF is very poorly constrained, both obser- its natal core)The or radiation substantially hydrodynamical increasing calculation the object’s clearly speed produces without a Luhman 2004; Maxted & Jeffries 2005; Kraus, White & Hillenbrand ⊙ multiplicity fraction that strongly increases with increasing primary 2005, 2006; Basri & Reiners 2006; Reid et al. 2006; Ahmic et al. vationally and from the calculation presented here. The radiation it becomingmass. unbound Furthermore, can the also values dramatically in each mass reduce range are its in accretion agreement rate 2007; Allen et al. 2007; Artigau et al. 2007; Konopacky et al. 2007; hydrodynamical calculation produced six objects with masses in [cf. the Bondi–Hoylewith observation. accretion In the right-hand formula panelM˙ of Fig.ρ/ 17,(c2 we providev2)3/2,where the Law, Hodgkin & Mackay 2008; Maxted et al. 2008; Reid et al. 2008; ∝ s + at University of Arizona on April 4, 2014 this range, with a minimum mass of 17.6 MJ.Alloftheseobjects v is the velocityequivalent of quantities the object obtained relative from to the the main gas]. barotropic Thus, calculation Bate & Bon- Burgasser, Dhital & West 2009; Luhman et al. 2009a; Radigan et al. had stopped accreting when the calculation was stopped. This is nell foundof that Bate objects (2009a) at formed the same with time very as the low end masses of the radiation (a few hy- Jupiter 2009; Faherty et al. 2011). For a recent review, see Burgasser et al. drodynamical calculation. Those readers who wish to examine the (2007). Over the entire mass range of 0.018–0.10 M ,wefinda very different to the main barotropic calculation. At the same time masses)multiplicity and accreted at the to end higher of the massesbarotropic until calculation their can accretion find this was very low multiplicity of just 0.08 0.05, although this⊙ is twice the ± (1.20tff ), the barotropic calculation had produced 217 brown dwarfs terminated,in Bate usually, (2009a). by The a dynamical barotropic calculations encounter. also This produce combination a mul- value found from the main barotropic calculation of Bate (2009a). (40 still accreting) with masses less than 30 MJ with a minimum of competitivetiplicity accretion that is a strongly and stochastic increasing function dynamical of mass. interactions In fact, the pro- However, the multiplicity drops rapidly with decreasing primary mass of only 5.6 MJ. Even discounting objects that were still ac- duced theC mass distributions, and Bate & Bonnell (2005) presented ⃝ 2011 The Author, MNRAS 419, 3115–3146 C creting, the inclusion of radiative feedback has cut the production a simpleMonthly semi-analytic Notices of the model Royal Astronomicalwhich could Society describe⃝ 2011 RAS the numerical of these VLM brown dwarfs by a factor of 30 and significantly in- results in which the characteristic stellar mass was given by the creased the minimum mass. It is interesting≈ to note that the minimum product of the typical accretion rate and the typical time between an mass is substantially higher than the opacity limit for fragmenta- object forming and having a dynamical interaction that terminated tion (Low & Lynden-Bell 1976; Rees 1976; Silk 1977a,b; Boyd & its accretion. Bate (2009a) found the IMF in their larger barotropic Whitworth 2005). This is because the opacity limit provides a min- calculations also originated in this manner. They found the mean imum mass, but generally objects will accrete from their surround- accretion rate of a low-mass star did not depend on its final mass, but ings to greater masses. Perhaps more importantly, the minimum that objects that accreted for longer ended up with greater masses mass is also greater than the estimated masses of some objects ob- and that protostellar accretion was usually terminated by dynam- served in star-forming regions (Zapatero Osorio et al. 2000, 2002; ical interactions. Here we analyse the radiation hydrodynamical Kirkpatrick et al. 2001, 2006; Lodieu et al. 2008; Luhman et al. calculation using the same methods. 2008, 2009b; Bihain et al. 2009; Burgess et al. 2009; Weights et al. In Fig. 8, we plot the final mass of an object versus the time 2009; Quanz et al. 2010). While an exact cut-off is difficult to at which it formed (i.e. the time of insertion of a sink particle). It determine from either numerical simulations or observations, the is clear that the most massive stars at the end of the calculation results of the radiation hydrodynamical calculation presented here were some of the first to begin forming. During the calculation, do imply that brown dwarfs with masses !15 MJ should be very as other lower-mass stars have formed and some have had their rare. accretion terminated, these stars have continued to grow to higher and higher masses. Maschberger et al. (2010) have argued that such aclusterformationprocessnaturallyproducesarelationbetween 3.2.1 The origin of the initial mass function cluster mass and maximum stellar mass similar to that which is Bate & Bonnell (2005) analysed two barotropic star cluster for- observed (Weidner & Kroupa 2006; Weidner, Kroupa & Bonnell mation simulations that began with clouds of different densities to 2010), although others argue that the observations are also consistent determine the origin of the IMF in those calculations (see also Bate with random sampling from a universal IMF (Lamb et al. 2010; 2005). They found that the IMF resulted from competition between Fumagalli, da Silva & Krumholz 2011).

C ⃝ 2011 The Author, MNRAS 419, 3115–3146 C Monthly Notices of the Royal Astronomical Society ⃝ 2011 RAS 13

cores, but instead tightly-packed agglomerations of many smaller cores. For the final low mass core (shown in the second column of Figure 12) the point at which the star forms is a slight overdensity in the middle of a filament, and there is no centrally-concentrated object at all. Thus massive cores and low mass cores have clearly distinct properties. However, we also find that these di↵erences are completely indistinguishable in the smeared images, indicating that it is not possible to distinguish true high mass cores from agglomerations of low mass ones with the resolution available in pre-ALMA telescopes, at least for objects at the kpc distances typical of massive star- Turbulent12 Core with winds, radiation forming regions. This⇠ conclusion is consistent with that of O↵ner et al. (2012). • Fig.Feedback 13.— reducesAccretion SFE rate (not versus enough) stellar mass for the four most It is important to note that the di↵erences between massive stars present at the end of run TuW. Colored unbroken high and low mass stars is not simply a function of for- lines• Massive indicate cores the don’t measured all fragment simulation accretion rates, while the dashed black line is the prediction of the McKee & Tan (2003) mation time. It is certainly true that the most massive model (Equation 15). The simulation accretion rates have been stars at the end of the simulation preferentially began smoothed• Sub fragmentation over 500 yr timescalesnot visible to even reduce scatter. when occurring (see Schnee et al forming early.13 However, their greater masses are far less 2009, Offner et al 2012) a reflection of this than it is of their di↵erent forma- the Lagrangian elements making up a core change with tion environments. The four massive stars grow at time- 4 1 time is irrelevant to the question ofcores, whether, but instead as tightly-packed a mas- agglomerationsaveraged of many rates of 3.6 4.6 10 M yr , compared 5 ⇥1 sive star forms, it sits at the center ofsmaller a gravitationally cores. For the final low massto core 1 (shown.7 in8. the8 10 M yr for the low mass stars. At second column of Figure 12) the point at which the star⇥ bound Eulerian structure. Figure 11forms shows is a slightthat overdensity it does. in the middlethe ofaccretion a filament, rates typical of the low mass stars, it would We can make the link between theand massive there is no cores centrally-concentrated and require object at all. Thus10t↵ for one of them to grow to the 10 M massive cores and low mass cores have clearly distinct⇠ ⇠ the stars they form more quantitativeproperties. by comparing However, we also to find thattypical these di↵ oferences the massive stars. The massive stars are not the massive core evolution model of McKeeare completely & Tan indistinguishable (2002, in thesimply smeared those images, that form first; they are those that form sur- indicating that it is not possible to distinguish true high 2003) and Tan & McKee (2004). Thismass cores model from predicts agglomerations of lowrounded mass ones by with coherent, bound, non-subfragmented struc- that the accretion rate onto a star asthe resolution a function available of in its pre-ALMAtures telescopes, that at least provide high accretion rates. This is somewhat for objects at the kpc distances typical of massive star- massFig. should 11.— Images be of the initial cores that produced the fourforming most massive regions. stars This⇠ in simulation conclusion TuW. is consistentsimilar In each column, with to the that the upper competitive image accretion model in that mas- shows the column density distribution, centered on the 0.05 M protostars that will grow to be massive stars. The lower image shows McKee &the Tan same column density distribution, smeared with a⇠ 1700of AU O↵ Gaussianner et al. beam. (2012). In the upper panels we indicate the mass of the core Fig. 13.— Accretion rate versus stellar mass for the3/ four4 most 1/2 sive stars’ preferred locations at the centers of collapsing 2004(defined as the projected mass within a radius of 0.01 pc, as indicatedIt is important by the dashed to circles) note that and the the time di↵ aterences which the between snapshot is taken. massive stars present at the end of runm TuW. Colored unbroken m In the lower panels we3 indicate thef core mass that would3/4 be inferred from the beam-smeared1 image and the final mass of the resulting star. lines indicate the measured simulation accretion⇤ rates, while the high⇤ and low mass stars is not simplyregions a function that of for- provides their high accretion rates. How- m˙ Note=1 that.2 the10 second and fourth columns are nearly⌃cl identical because two of theM finalyr massive , stars both form in the same core. dashed black⇤ line is the⇥ prediction of30 theM McKee & Tan (2003) mmationf time. It is certainly true thatever, the most it massive di↵ers from competitive accretion in that these model (Equation 15). The simulation✓ accretion rates◆ have been ✓ stars⇤ at◆ the end of the simulationKrumholz preferentially et al began2012 smoothed over 500 yr timescales to reduce scatter. forming early. However,(15) their greatercores masses are are far lessnon-subgfragmented and have masses at the a reflection of this than it is of their di↵erent forma- the Lagrangianwhere melementsis making the star’s up a core instantaneous change with tion mass, environments.m f Theis four its massivesame stars grow order at time- of magnitude as the final stars, 10 M , ⇤ ⇤ 4 1 ⇠ timefinal is irrelevant mass, to the⌃ questioncl is the of whether, surface as a mas- densityaveraged of the rates molecular of 3.6 4.6 10 Mandyr therefore, compared intermediate between that of the entire 5 ⇥1 sive star forms, it sits at the center of a gravitationally to 1.7 8.8 10 M yr for the low mass stars. At 3 clump from which it forms, and we have used⇥ McKee & star cluster, 10 M and the thermal Jeans mass, 1 bound Eulerian structure. Figure 11 shows that it does. the accretion rates typical of the low mass stars, it would⇠ ⇠ WeTan’s can make fiducial the link between parameter the massive choices, cores and withrequire the exception10t↵ for one that of them to growM to the. In10 theM competitive accretion model such structures the stars they form more quantitative by comparing to typical of⇠ the massive stars. The massive stars⇠ are not the massivewe have core evolution increased model of the McKee accretion & Tan (2002, ratesimply by athose factor that form of first; 2.6 they are thoseshould that form be sur- absent, because everything fragments down to 2003)to and include Tan & McKee subsonic (2004). contraction, This model predicts followingrounded Tan by coherent, & McKee bound, non-subfragmentedthe thermal struc- Jeans mass (Bonnell et al. 2004; Bate & Bon- that the accretion rate onto a star as a function of its tures that provide high accretion rates. This is somewhat mass(2004). should be To evaluate this equation andsimilar compare to the competitive it to our accretion modelnell in2005) that mas- and some objects subsequently grow to larger simulations, we3/4 take m f1/2 10 M , sincesive stars’ this preferred is roughly locations at the centersmasses of collapsing by Bondi-Hoyle accretion. There are no 10 3 m f 3/4 m ⇤ 1 regions that provides their high accretion rates. How- m˙ =1.2 10 ⇤ ⌃cl ⇤ ⇡M yr , ⇠ ⇤ the⇥ mass30 ofM our fourm mostf massive starsever, it di at↵ers the from end competitive of accretionM inobjects that these that do not subfragment in competitive ac- ✓ ◆ ✓ ⇤ ◆ the simulation. For ⌃cl, we note(15) that,cores in are the non-subgfragmented simulation, and havecretion. masses at the where m is the star’s instantaneous mass, m f is its same order of magnitude as the final stars, 10 M , the⇤ core is better-defined than⇤ the clump, so we adopt Finally,⇠ we note that both the high mass and the low final mass, ⌃cl is the surface density of the molecular and therefore intermediate between that of the entire 3 clumpMcKee from which & it forms, Tan’s and result we have used with McKee⌃cl &replacedstar cluster, by ⌃10coreM .Inand the thermalmass Jeans cores mass, are1 above the column density threshold ⌃ > 1 Tan’s fiducial parameter choices, with the exception that M . In the competitive⇠ accretion model such structures2 ⇠ we havetheir increased fiducial the accretion model rate these by a factor agree of 2.6 to withinshould be a absent,factor because1. everything2, g fragments cm downfor massive to star formation posited analytically by to include subsonic contraction, following Tan & McKee the thermal Jeans mass' (Bonnell et al. 2004; Bate & Bon- so thisFig. 12.— doesSame not as Figure significantly 11, but for the four astars↵ect closest the to the accretion median of the final rate. mass distribution.Krumholz & McKee (2008) and confirmed numerically by (2004). To evaluate this equation and compare it to our nell 2005) and some objects subsequently grow to larger simulations,As shown we take m inf Figure10 M , since 11, this our is roughly cores havemasses masses by Bondi-Hoyle of order accretion. ThereKrumholz are no et10 al. (2010). This means that both the high the massfragmented. of our four most There⇤ ⇡ massive are at most stars one at theor two end density of max-M objectsceived that extensive do not subfragment investigation in competitiveby Bonnell⇠ et ac- al. (2004) 10 imaM in eachin one, radii not manyof order density 0maxima..01 pc, These which struc- correspondsand Smith et al. to (2009a,b),mass among and others. low It may mass well cores have the potential to form mas- the simulation. For ⌃cl, we note2 that, in the simulation, cretion. the core⌃cl istures better-defined=6 look.6 much g cm than like the the. turbulent clump, With so cores we these adopt posited parameter in theFinally,be we that choices, note particular that both in fluid the high elementssive mass that stars; and are the present lowindeed, in the in one of the four cases shown in Figure McKee & Tan (2003) theory for massive star forma- cores at the time shown in Figure 11 do not accrete onto McKee & Tan’s result with ⌃cl replaced by ⌃core.In mass cores are above the column density threshold ⌃ > 1 Figuretion. When 13 smeared we plot on a theresolution accretion of 1700 AU, rate distinct as2 athe function final star and of are instead12, accreted the low by other mass stars star or is in fact forming in a core that puts their fiducial model these agree to within a factor 1.2, g cm for massive star formation posited analytically by so thisstellar doescentrally-condensed not mass significantly for thestructures a↵ect four the remainaccretion most visible' rate. massive for threeKrumholz starstorn & McKee at o↵ theby (2008) turbulent end and confirmed motions,most while numerically of fluid its elements by mass not into a single high mass star. That does As shownof theof in the Figure simulation, four 11, massive our cores stars, whose have indicating masses cores that of order these are objects shownKrumholz inpresent et al.Figure (2010). in the core 11,This at means the timenot that shown both appear the are high eventually to be ac- the case for the other three low mass 10 M inwould radii be of orderdetectable 0.01 as pc, massive which corresponds cores in an observation. to mass andcreted low mass into cores the have final the star.potential Indeed,to McKee form mas- & Tan (2003) It is important2 to understand that our analysis says predicted in their analytic model that turbulent cores ⌃cl =6and.6 g compare cm . With to these the parameter McKee choices, & Tan in sive prediction. stars; indeed, in As one of the the four casesstars shown shown in Figure in the Figure, however. Thus a high column Figureplot 13nothing we shows, plot about the accretion the the Lagrangian simulation rate as trajectories a function accretion of of the fluid12, rates the lowshould mass agree star over is quite the in fact course forming of theirdensity in a lives core interact that is puts clearly with a sur- a necessary but not a sucient condi- stellar masselements for the that four eventually most massive coalesce stars to at form the end the massivemost of itsrounding mass into gas a mass single comparable high mass star. to that That which does eventually of thewell simulation,stars with in our whose the simulations, cores analytic are a shown topic predictions. that in Figure has previously 11, not re- appearends to be up the in their case for central the other stars.tion three However, for low massive mass the fact that star formation. A high column density and compareIn contrast, to the McKee the & Tan cores prediction. that As give the risestars to shown low in mass the Figure, cores however. Thusallows a high radiative column heating to suppress the growth of grav- plot shows, the simulation accretion rates agree quite density is clearly a necessary but not a sucient condi- well with(Figure the analytic 12) predictions. are quite noticeably di↵erenttion for from massive the star formation. high A highitational column density instabilities that would lead to fragmentation In contrast,mass ones. the cores In that three give rise of to the low four mass cores cases (theallows first, radiative third, heating and to suppress theand growth prevent of grav- a massive star from forming. However, if (Figure 12) are quite noticeably di↵erent from the high itational instabilities that would lead to fragmentation massfourth ones. In three columns of the four in cases the (the first, Figure) third, and theyand are prevent also a centrally- massive star from forming.the turbulent However, if density field present before a star begins fourthcondensed columns in the lumps Figure) they of gas. are also However, centrally- the unlike turbulent the density massive field present beforeradiating a star begins is already highly non-linearly fragmented, as condensed lumps of gas. However, unlike the massive radiating is already highly non-linearly fragmented, as corescores they are they highly are sub-fragmented highly and sub-fragmented show many is the and case show for several many of the low massis cores the shown case in for several of the low mass cores shown in densitydensity maxima. maxima. Clearly these objects Clearly are not these single objectsFigure 12, are radiative not heating single will not undoFigure this fragmen- 12, radiative heating will not undo this fragmen- 12

Fig. 11.— Images of the initial cores that produced the four most massive stars in simulation TuW. In each column, the upper image shows the column density distribution, centered on the 0.05 M protostars that will grow to be massive stars. The lower image shows the same column density distribution, smeared with a⇠ 1700 AU Gaussian beam. In the upper panels we indicate the mass of the core (defined as the projected mass within a radius of 0.01 pc, as indicated12 by the dashed circles) and the time at which the snapshot is taken. In the lower panels we indicate the core mass that would be inferred from the beam-smeared image and the final mass of the resulting star. Note that the second and fourth columns are nearly identical becauseLow two of thevs final Intermediate massive stars both form in the / same High core. Mass Stars

“Cores”Fig. 11.— areImages not of di theff initialerent cores in mass, that produced but thein fragmentation four most massive stars properties. in simulation The TuW. distinction In each column, is the upper image Fig. 12.— Same as Figure 11, but for the four stars closest to the medianshows the of the column final density mass distribution. distribution, centered on the 0.05 M protostars that will grow to be massive stars. The lower image shows duethe same to columnstochastic density distribution,ICs, location smeared in withpotential. a⇠ 1700 AU Gaussian beam. In the upper panels we indicate the mass of the core (defined as the projected mass within a radius of 0.01 pc, as indicated by the dashed circles) and the time at which the snapshot is taken. fragmented. There are at most one or two density max- Inceived the lower extensive panels we indicate investigation the core mass by Bonnell that would et be al. inferred (2004) from the beam-smeared image and the final mass of the resulting star. ima in each one, not many density maxima. These struc- Noteand that Smith the second et al. and (2009a,b), fourth columns among are nearly others. identical It may because well two of the final massive stars both form in the same core. tures look much like the turbulent cores posited in the Likebe thatCA: particular massive fluid stars elements need thatpotential. are present Like in TC: the identifiable, long lasting core. McKee & Tan (2003) theory for massive star forma- cores at the time shown in Figure 11 do not accrete onto tion. When smeared on a resolution of 1700 AU, distinct the final star and are instead accreted by other stars or centrally-condensed structures remain visible for three torn o↵ by turbulent motions, while fluid elements not of the four massive stars, indicating that these objects present in the core at the time shown are eventually ac- would be detectable as massive cores in an observation. creted into the final star. Indeed, McKee & Tan (2003) It is important to understand that our analysis says predicted in their analytic model that turbulent cores nothing about the Lagrangian trajectories of the fluid should over the course of their lives interact with a sur- elements that eventually coalesce to form the massive rounding gas mass comparable to that which eventually stars in our simulations, a topic that has previously re- ends up in their central stars. However, the fact that

Fig. 12.— Same as Figure 11, but for the four stars closest to the median of the ﬁnal mass distribution.

fragmented. There are at most one or two density max- ceived extensive investigation by Bonnell et al. (2004) ima in each one, not many density maxima. These struc- and Smith et al. (2009a,b), among others. It may well tures look much like the turbulent cores posited in the be that particular fluid elements that are present in the McKee & Tan (2003) theory for massive star forma- cores at the time shown in Figure 11 do not accrete onto tion. When smeared on a resolution of 1700 AU, distinct the final star and are instead accreted by other stars or centrally-condensed structures remain visible for three torn o↵ by turbulent motions, while fluid elements not of the four massive stars, indicating that these objects present in the core at the time shown are eventually ac- would be detectable as massive cores in an observation. creted into the final star. Indeed, McKee & Tan (2003) It is important to understand that our analysis says predicted in their analytic model that turbulent cores nothing about the Lagrangian trajectories of the fluid should over the course of their lives interact with a sur- elements that eventually coalesce to form the massive rounding gas mass comparable to that which eventually stars in our simulations, a topic that has previously re- ends up in their central stars. However, the fact that UnderstandingIdentifying turbulent Dendrogram fragmentation Leaves Scaled Density (log(Density in g/cm^3 * 10e22)) * 10e22)) g/cm^3 in (log(Density Density Scaled

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FIG.1.—Schematicdiagramofthedendrogramprocess.Theleftpanel shows a one-dimensional emission proﬁle with three distinct local maxima. The dendrogram of the region is shown in blue and repeated in the right panel where the components of the dendrogram are labeled. The left-hand panel indicates three characteristic contour levels through the data. Thresholding at I1 produces a single object whereas thresholding at I2 produces two. The level separating these two regimes is indicated as Icrit .

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4 1 2 6 5 7 6 10 18 ()%&#&)" (K) 8 9 12 21 mb 11 14 19 T 15 4 13 22 16 20 23 17 24 25 26 FIG.3.—Schematicdiagramoftheparametersthatdeterminethedeci- 2 mation of local maxima. The same profile as in Figure 1 is used. The local maximum indicated with the white point would be considered a valid lo- 0 cal maximum if (a) it is the highest point in a window Dmax on either side of it (and an analogous width ∆Vmax in velocity space), (b) the interval be- FIG.2.—Dendrogramsof13CO emission in L1448. The top panel shows tween the maximum and the highest merger level with a valid local maximum the dendrogram of L1448 using the standard algorithm parameters. The bot- ∆T1 > ∆Tmax and if the number of pixels associated with the shaded region tom panel shows the dendrogram after relaxing the conditionsfornoisesup- is larger than Nmin.Thesecriteriarestricttheanalysistothesubsetoflocal pression resulting in more independent leaves in the dendrogram (see the end maxima that are most distinct. of §3.1). However, the basic structure of the dendrogram remains the same; the isosurfaces used in the top plot are a subset of those used in the bottom. By default, these are set to be Dmax =3and∆Vmax =7resolu- Each leaf of the dendrogram is labeled in the top plot and the corresponding ∆ leaf is identified in the bottom figure. Leaves appearing in both dendrogram tion elements, Tmax =4σrms and Nmin =4pixelsforindepen- also have a circle at their tip in the bottom plot. dent pixels. The bottom panel of Figure 2 shows the resulting dendrogram for Dmax =1and∆Vmax =3resolutionelements of the isosurfaces that contain only that maximum is less than and ∆Tmax =0.Figure3isaschematicdiagramillustrating some minimum number of pixels (Nmin,usuallytakentobe the definition of these parameters. Changing ∆Tmax results in 4). Furthermore, we only recognize a significant bifurcation the largest changes in the dendrograms for typical radio line in structure when both local maxima are more than a given data since a larger fraction of the local maxima fail the check interval ∆Tmax above the highest contour level that contains against the contrast than any other noise suppression criterion. both of the maxima, i.e. the level at which the objects merge. The default values represent a compromise between sensitiv- Such a criterion has been used previously in data cube analy- ity to dendrogram structure and algorithm performance. sis (Brunt et al. 2003; Rosolowsky & Blitz 2005): noise fluc- Noise has an additional affect on the dendrogram, namely tuations will typically only produce maxima with characteris- intensity fluctuations can alter the levels at which two isosur- tic height σrms so variations significantly larger than this are faces merge. A positive fluctuation can join two surfaces at nominally real. If this criterion is not fulfilled, we reject the ahigherlevelthanthesurfaceswouldjoinintheabsenceof lower of the two local maxima and consider the emission pro- noise. We have modeled the influence of the noise by com- file to representonlya single object. We note that the resulting paring the merge levels of surfaces in a model cube in the dendrogram using a decimated set of local maxima represents absence of noise to those with noise added. We find, in gen- asetofisosurfacesthatareasubsetoftheisosurfacesthat eral, that the merge levels are uncertain on a scale of ∼ 2σrms would be considered including all local maxima (see Figure with some variation based on algorithm parameters and the 2). precise model used. In addition, there is a bias towards merg- Hence, the initial leaves of the dendrogram are determined ing ∼ 1σrms higher than the surfaces would merge in the ab- by four free parameters: Dmax, ∆Vmax, ∆Tmax and the Nmin. sence of noise. The structure of the tree can only be con- 14

the fraction of stars in multiple systems from the simulation requires some care, as pointed out by Bate (2009a). Many of the stars in our simulation form a bound cluster, and thus many stars are bound to many other stars, often in hierarchical structures consisting of dozens of individual stars; for example a binary and a triple system may orbit one another, and these in turn may have additional stars orbiting them. Such agglomerations would be extremely unlikely to survive dynamically even for the lifetime of a massive star, and would break up if we could continue the simulation further. Binaries form from filaments on these scales, then Thus we follow Bate in defining stellar multiplicity via the following algorithm. We first compute the total evolve energy (gravitational plus kinetic in the center of mass frame) pairwise for each pair of stars in the simulation. Fig. 14.—• BothMultiplicity low and fraction highf as mass a function star of systemform primary We find the most bound system and replace it with a mass mprim in run TuW. The thick red line shows f as a running single point mass, with a mass equal to the sum of the average. Thebinaries light red and boxes interact show f computed at early over times discrete bins in two components, a position located at their center of mprim.Ineachcase,thewidthoftheboxshowstheprimarymass range for that bin, the asterisk shows the mean multiplicity fraction mass, and a momentum equal to the sum of their two for stars in that bin, and the vertical extent of the box shows the momenta. We then continually repeat this process, with statistical• Lower uncertainty mass on that star value, form computed in highly as described in the the exception that we do not create aggregates consist- text. Finally,fragmented black crosses cores indicate start observational out in results, very with the ing of more than four individual stars; should the most horizontal width indicating the mass range for the observations and the verticalhierarchical range showing multiples, the stated but uncertainty. likely decay The two bound system contain five our more stars, we proceed to highest mass observational data points are lower limits, indicated the next most bound pair with fewer than five members by the upward arrows. The data shown are taken from, from left instead.10 We terminate the process once there are no to right,• Higher Basri & Reinersmass (2006)stars and form Allen as (2007) multiples (shown in as a single combined point), Fischer & Marcy (1992), Raghavan et al. more bound pairs consisting of fewer than five individual (2010), Preibischcentrally et al. condensed (1999), and Mason regions et al., (2009); and may the data stars. At the end we are left with a list of star systems, compilationkeep shown a herehigher is the fraction same as that of inbond Bate (2012). some single and some containing up to four individual stars. companions Given this list, we can compute the fraction of multiple systems as a function of primary star mass. For a set of star systems, we define the multiplicity fraction B + T + Q f = , (16) S + B + T + Q where S, B, T , and Q and the numbers of single, binary, triple, and quadruple systems, respectively.11 We choose our sets of star systems in two ways. The first is as a running average; for a primary mass mprim, we compute f considering all systems for which the primary mass is within half a dex of mprim. The second is in discrete Offner, Kratter et al 2010, bins, chosen to roughly match the mass ranges selected Krumholz et al 2012 in observational surveys. We consider primary mass bins in the range 0.05 0.1 M ,0.1 0.2 M ,0.2 0.5 M , Fig. 15.— Semi-major axis versus primary star mass for all the 0.5 0.8 M ,0.8 1.2 M ,1.2 3 M , and >3 M .In binaries in our simulations. For triple and quadruple systems, we addition to the mean value, we compute the statistical plot them only once, showing the properties of the most bound pair 12 of stars. Points are coded by the mass ratio of the system: purple uncertainty in this value for each bin. stars for q<0.1, red squares for q =0.1 0.25, green triangles for q =0.25 0.5, and blue circles for q>0.5. capacity because to omit them would artificially make it impossible for stars near our 0.05 M cut to have companions. tation and prevent the core from forming a small cluster 10 The results are not particularly sensitive to the choice of four of low mass stars rather than a few massive ones. as the maximum size of a system, as long as we stop at some point well short of allowing the entire cluster to be considered a single large star system. 3.6. Stellar Multiples 11 Following Bate (2009a) and Hubber & Whitworth (2005), It is also illuminating to consider the properties of the we measure this quantity rather than either the companion star fraction (B +2T +3Q)/(S + B + T + Q)orthefractionofstarsin stellar multiples that form in run TuW, since producing multiple systems (2B +3T +4Q)/(S +2B +3T +4Q)becauseit the correct multiplicity fraction has been proposed as is more robustly determined observationally. If a new member of test for star formation models in addition to producing amultiplesystemisfound,forexampleleadingtoabinarybeing the correct IMF (e.g. Bonnell et al. 2007). We therefore reclassified as a triple, f does not change, while the companion star 9 fraction and the fraction of stars in multiple systems does. examine the final time slice for this run. Extracting 12 We determine the statistical uncertainty by assuming that there is a true multiplicity fraction ftrue for stars in that bin, and 9 In this section of the paper alone we do not exclude stars that our sample of systems in that bin represent a series of random smaller than 0.05 M from consideration, but we consider them drawings that follow a binomial distribution. From these assump- only as companions to larger stars. We allow them to count in this tions, we can compute the probability distribution for ftrue given Disk Fragmentation becomes alternate binary formation mechanism for more massive stars

2 10 0.5

0.45

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1 0.3 10 0.25 µ

Final Mass 0.2

0.15

0.1 0 10 0.05

3 4 5 6 10 10 10 10 years Intermediate mass stars might harbor massive, self-gravitating disks at early times. Magnetic disk instabilities should also be more efﬁcient Kratter et al 2008, 2010 7

3.6. Summary of Model also that the mass accretion rate is a fraction ⇧facc of 3 We summarize our model via the flowchart shown in the characteristic rate vK (r) /G. Then, figure 2, which illustrates a simplified version of the f 3 f = K acc . (28) code’s decision tree. At a given time t we know the ⌥in ⇧2 current disk and star mass, and the current angular momentum and mass infall rates as prescribed in 2.1 and (In this expression, negative three powers of ⇧ appear § because the binding mass is ⇧ times smaller for the disk 2.2. We can calculate Rd and d directly, and find the appropriate§ stellar luminosity based on its evolution, cur- than for the core; one of these is compensated by the rent mass, and accretion rate. Using these variables we reduction of the accretion rate.) In 2.1 we adopted the McKee & Tan (2003) model for self-consistently solve for the appropriate temperature, § Q, and disk accretion rate as described in 3.2. With massive star formation due to core collapse of a singular, this information in hand, we determine whether§ the disk turbulent, polytropic sphere in initial equilibrium. Their is stable, locally fragmenting, or forming a binary. If equations (28), (35), and (36) imply 1/2 the disk is stable, we proceed to the next iteration. If 3 k Q < 1, then the disk puts mass into fragments according facc = 0.84(1 0.30k) (29) 1 + H to equation [23]. If µ > 0.5 we consider binary forma- ⇤ 0 ⌅ tion to have occurred, and the net angular momentum within 2%, where 1 + H0 2 represents the support due and disk mass over the critical threshold is placed into to static magnetic fields ⌃(Li & Shu 1996). (Note, their a binary (see 3.5). The simulation is halted at either equation [28] is a fit made by McKee & Tan 2002 to the 5 Myr, or if the§ entire disk turns into fragments. We results of McLaughlin & Pudritz 1997.) choose 5 Myr as a stopping time for two reasons: first, KM06 predicted the turbulent angular momentum of 1/2 the most massive stars in our parameter space are signif- these cores; our parameter fK equals (⇥j⌅j) in their icantly evolved and have exploded by 5 Myr, and so our paper. Their equations (25), (26), and (29) imply stellar evolution models are no longer su⌃cient; and sec- 0.49 (1 k /2)0.42 ond, because many other e⇤ects begin to dominate the fK = 1/2 1/2 , (30) disks appearance at late stages due to gas-dust interac- (k 1) ⌅B tion and photo-evaporation. with excursions upward by about 50% and downward by 4. EXPECTED TRENDS about a factor of three expected around this value; here How do we know if disks fragment?⌅B 2.8 represents the magnetic enhancement of the Before examining the numerical evolution, it is useful turbulen⌃ t pressure. All together, we predict to make a couple analytical predictions for comparison. 1.26 First, can we constrain where disks ought to wander in 1/2 k 0.10 3 k 1 2 the plane of Q and µ? This turns out to depend critically Environment in = Derived3/2 3/2 (1 0.30k) on the dimensionless system accretion rate ⌥ 2 1 + H0 (k 1)⇥ ⇧ ⌅B ⇤ ⌅ 2 M˙ M˙ j3 MG˙ M 0.5 in = in in (26) d 0.02 cs (31) in = 2= 3 µ = ⇧ Q =⇧ ⌥ ⇥ M d⇥(Rcirc) G M 3 cds,disk Md + M ⇤ ⇥⌅G where the evaluation uses 1 + H 2, ⌅ 2.8, and which is the ratio of the mass accreted per radian of disk 0 B k 1.5. ⇧ ⇧ rotation (at the circularization radius Rset by) coreto theand total set by core/ turbulence: circ set by relativeImp⇧ ortan tlypurely, localis diska function of (1 + H ), ⌅ , , and system mass M orbital= M time+ / Mmass. Since thediskactiv e inner in 0 B d d accretionk , ratesbut not thequantity⌥core mass. We therefore expect similar disk has a radius comparabledoubling timet o R ,thermodynamicsthis controls how circ values of to describe all of present-day massive star rapidly the disk gains mass via infall. in formation,⌥at least insofar as these other parameters take The importance of is apparent in the equation gov- in similar values. Suppose, for instance, that the formation erning the evolution of⌥ the disk mass ratio µ: of 10M and 100M stars were both described by the same ⇥. According⇥to equation (27), the di⇤erence in µ µ˙ M˙ in 1 M˙ in = 1 between⌥ these two systems would be controlled entirely µ⇥ M d⇥ µ Md⇥ ⇤ ⌅ by the thermal e⇤ects that cause them to take di⇤erent ⇥(Rcirc) 1 M˙ values of Q. = 1 in . (27) A few additional expectations regarding Q itself can ⇥d µ ⌥ Md⇥ ⇤ ⌅ be gleaned from the analytical work of Matzner & Levin Toomre, 1964, Kratter et al 2008, 2010 Since we consider M˙ /(Md⇥) to be a function of µ and (2005) and KM06: Q, we must know the disk temperature to solve for µ(t). - The Toomre parameter remains higher than unity Regardless, equation (27) shows that larger values of in for low-mass stars ( 2M ) in low-column cores tend to cause the disk mass to increase as a fraction⌥of ⇥ (c,0 1), but falls to unity during accretion for the total mass. We may therefore view in and Q as the ⌅ ⌥ massive stars and for low-mass stars in high-column two parameters that define disk evolution – of which in cores; is imposed externally and Q is determined locally. ⌥ Moreover, in takes characteristic values in broad - A given disk’s Q drops during accretion, reach- classes of accretion⌥ flows, such as the turbulent core mod- ing unity when the disk extends to radii beyond els we employ. Suppose the rotational speed in the pre- 150 AU (in the massive-star case), or to peri- ⇤ collapse region is a fraction fK of the Kepler speed, so ods larger than 460 yr (in the case of an optically 1/2 ⇤ that jin = fK rvK (r) = fK [GrMc(r)] , and suppose thick disk accreting from a low-mass, thermal core). M˙ = in High Mass Stars M d⇥ Low Mass Stars 0.1

0.01 normalized rotation rate Singles multiples / fragments binary twins? 0.005

1.4 2 3 5 25 M˙ G45 normalized accretion rate = 3 cs Kratter et al 2010 Orbital properties are not ﬁxed at birth

•Cluster Evolution •Disk interactions?

•Dynamical Interactions

Kroupa 1995

Kratter et al 2010 Disk-Disk interactions with moving mesh code, AREPO (Springel 2009)

Diego Munoz Disk-Disk interactions

• Need high stellar densities for close enough interactions

• Disk-orbit misalignment aids in orbit change, but outcome is not well aligned

14 Munoz˜ et al.

Munoz, Kratter et al, in prep

Figure 11. Same as Figure 10, but now for the ‘PARA2’, ‘PARA3’, ‘PARA4’ and ‘PARA5’ configurations. As before, the color of the curves is chosen according to the combined disk spin in each configuration proper pericenter is not entirely captured by the simulation. This is the case of ‘PARA4’ and ‘PARA5’, for which the tidal interaction is expected to begin at wider separations than the ones included in our initial conditions (see discussion on the fractional error in orbital angular momentum in Section 2.3). Although it is not possible to say without running simulations with much wider initial separations D, it is possible that some of the notable differences between ‘PARA4’ and ‘PARA5’ with respect to the other orbital tests (for example, that there is no steep jump in eccentricity right before pericenter in Figure 13) could be due to an extremely wide tidal interaction window and that ‘PARA4’ and ‘PARA5’ are simply “incompletete”, that is, their integration should have be- gun at greater separations in order to cover the asymptotic interaction in greater extent. Figure 14 summarizes the energy change during first pericenter passage (i.e., restricted only to the shaded region in Figure 13) ‘PARA1’ for all 35 simulations. Note that the energy change in the Figure 14. Energy change for during first pericenter passage for all sim- and ‘PARA2’ is similar. However, these data include only first ulations. Colors represent the orbital configurations ‘PARA1’ (blue) , passage. As it can be seen from Figure 13, the orbital energy in ‘PARA2’ (green), ‘PARA3’ (orange), ‘PARA4’ (cyan) and ‘PARA5’ the ‘PARA1’ simulations keeps chaining repeatedly as the stars (magenta), while symbols represent the different disk orientations: go through pericenter over and over. On the other hand, In the ‘PARA -1’ or Sz =2(diamond), ‘PARA -2’ or Sz =1.7 (square), ‘PARA2’ examples, repeated pericenter passages are more rare, ‘PARA -3’ or Sz =1(upright triangle), ‘PARA -4’ or Sz =0.3 (circle), ‘PARA -5’ or Sz =0(pentagon), ‘PARA -6’ or Sz = 2 since runaway decay of the binary orbit is not taking place. (sideways triangle) and ‘PARA -7’ or Sz = 1.7 (bowtie).

3.5 Comparison to linear theory The tidal interaction between a star+disk system and another stellar flyby was studied in detail by Ostriker (1994). Although that work focused on a simpler system containing only one disk, comparison should be meaningful for our wide-separation simulations. tions, the first five runs of each subset only differ in the value of ✓2) Assuming only one disk changes orientation (in our simula- the angular momentum loss suffered by the victim disk is, in the

c 2014 RAS, MNRAS 000, 1–?? Disk Alignment in Binaries

does this shedA&A light 532, A28 on (2011) formation?

Fig. 4. The distribution in the diﬀerence between spectropolarimetrically predicted disk PA and observed disk PA for the sample presented in Table 3 (blue dashed). This is compared to a random distribution (black short dotted). On the left we show a distribution where polarisation signatures are always orientated perpendicularly to circumstellar disks and on the right we present the distribution for a scenario where the spectropolarimetric signatures can be either perpendicular or parallel to disks (see the text for more detail). Both model distributions have a maximum error of 15◦.Inbothcases,arandomorientationofdiskandpolarisation position angles can be discarded at the 3σ level.

1σ level and is thus consistent with the data. This confirms the if the spectropolarimetric signatures are due to optical pumping earlier finding that spectropolarimetric signatures trace disks and and absorption in a wind, the wind geometry essentially mirrors Wheelwrightdemonstrates that this is not dependentet al on 2011 spectral type and clas- that of the disk, at least in the regions where the polarisation sification (i.e. T Tauri star or HAe/Be star). occurs. To summarise, we show that the spectropolarimetric data This is partly substantiated by recent observations of the presented in Table 3 do appear to trace circumstellar disks. This Hα emission of the Herbig Ae star AB Aur with the CHARA would be expected if the polarisation is due to scattering in these array by Rousselet-Perraut et al. (2010). This object has been disks. However, Harrington & Kuhn (2009)claimmanyofthe proposed to exhibit polarisation due to optical pumping since it HAe/Be stars in our sample have polarisation signatures which displays polarisation across the P Cygni absorption component require optical pumping and absorption in outflows (e.g. AB Aur of its Hα emission (Harrington & Kuhn 2007). Rousselet-Perraut and MWC 480). We show here that regardless of the polarising et al. (2010)resolvedtheHα emitting region around AB Aur and mechanism, spectropolarimetry can be employed to trace the ori- found that it could be modelled as the base of a wind represented entation of circumstellar disks. by a flattened torus encompassing a circumstellar disk. Provided the inclination and the angle between the wind surface and disk 4. Discussion and conclusion mid-plane is low (20◦ and 35◦ in the case of the disk-wind model of Rousselet-Perraut et al. 2010), such a flattened torus might 4.1. On the use of spectropolarimetry to probe circumstellar well appear to have a similar morphology to the disk. disks

The results presented in this paper indicate a direct correla- 4.2. On the alignment between binary systems tion between the spectropolarimetric signatures of pre-main- and their circumprimary disks sequence stars and the orientations of their circumstellar disks. This is significant above the 3σ level and appears independent To investigate the relative alignment of HAe/Be binary systems of the classification of the young stellar objects. Therefore, we and circumstellar disks we have used spectropolarimetry and conclude that spectropolarimetric signatures of young stellar ob- high spatial resolution data to determine the orientation of cir- jects do indeed trace the orientation of their circumstellar disks. cumstellar disks around the primary components of such sys- This is expected in the case of polarisation of stellar and ac- tems. We then combined these disk angles with binary param- cretion shock photons by disks (McLean & Clarke 1979; Vink eters to assess whether HAe/Be circumstellar disks and binary et al. 2002). In the case of polarisation via optical pumping and systems are co-planar. Studies of lower mass T Tauri stars have absorption, the relationship between spectropolarimetric signa- found that the circumstellar disks in T Tauri star binary sys- tures and disks is less clear. tems tend to be aligned, suggesting that such systems may form We note that Kuhn et al. (2011)suggestthatthepolarisa- via fragmentation (see e.g. Wolf et al. 2001; Jensen et al. 2004; tion signature of the Herbig Be star HD 200775 is due to op- Monin et al. 2006). Here we investigate whether this is also the tical pumping and that the signature does trace the orientation case for HAe/Be systems. We note that Maheswar et al. (2002) of an imaged disk. However, the spectropolarimetric signature also compared HAe/Be binary and polarisation angles (although of this object is observed across a double-peaked emission line these were calculated via broadband polarimetry and thus sub- profile, and might therefore be the result of depolarisation af- ject to uncertainties in the correction for the interstellar polari- ter all. Nevertheless, many Herbig Ae/Be stars, several of which sation). These authors find that their data are inconsistent with are in our sample, exhibit polarisation signatures associated with arandomassociationofdiskandbinarypositionangleswitha PCygnilineprofiles,i.e.outflowinggas(seee.g.Harrington significance of 84 per cent. This is not very conclusive and the &Kuhn2009). Here we show that the signatures still appear authors note that they do not account for projection effects and to trace the orientation of circumstellar disks. This implies that, thus the actual correlation may be stronger. We do account for

A28, page 8 of 10 Stellar Structure, Disks

• Early development of radiative zone means that internal luminosity gets more important At accretion midpoint... L 1.5M acc 100 ! Lint ⇡ L 5M acc 1 ! Lint ⇡ ˙ 4 3M⌦ L ⇤ Td = f(⌧) + ✏ 2 8⇡ 4⇡Rd !

inﬂuences planet formation / accretion disk physics for Palla & Stahler 2004, 2005 intermediate mass stars Planet formation and accretion depends on radiation

The Astrophysical Journal,736:144(18pp),2011August1 Gammie 1996 Bai & Stone

4 B /B =0 10 φ z = 10−4 B /B =1.25 α 3 z 10 φ B /B =4 3 φ z 10 Bai & Stone 2011 B /B =∞ −3 Dullemond & Monnierφ z α = 10 2

> 10 β > 2 MRI Permitted < 2010 β 10 < α = 0.01

1 10 1 10 α = 0.1 MRI Prohibited 0 10 0 What is the maximum−4 −3 mass−2 −planet1 0 host? 10 10 10 10 10 10 −2 −1 0 1 2 3 4 α 10 10 10 10 10 10 10 Am Figure 14. Scatter plot of the total turbulent stress α and the plasma β at the saturated state of the MRI turbulence from all our simulations. Simulations Figure 16. Diagnostics of the MRI in the AD regime. At a given Am, MRI is with different field geometries are marked by different symbols and colors as permitted when β ! βmin(Am) (see Equation (26)), and in the MRI permitted region, the stress⟨ α⟩can be inferred given the field strength at the saturated state indicated in the legend, where Bφ /Bz 0andBφ /Bz correspond to pure = = ∞ characterized by β . net vertical and pure net toroidal flux simulations, respectively. Dashed line ⟨ ⟩ shows the fitting curve β 1/2α. (A color version of this figure is available in the online journal.) ⟨ ⟩ = (A color version of this figure is available in the online journal.) provides useful diagnostics on the properties of the MRI in the B /B =0 AD regime in a concise fashion. φ z First, at a given Am, the ultimate strength of the MRI B /B =1.25 3 φ z turbulence (e.g., α and β )dependsonthefieldgeometry 10 B /B =4 ⟨ ⟩ φ z (including the net flux), but there exists a maximum α (or B /B =∞ φ z minimum β )atthemostfavorablefieldgeometry(usually contains both⟨ ⟩ net vertical and toroidal fluxes). One way to think 2

> 10 about it is to start with a weak regular field as we perform our β < simulations. As the system evolves and as the MRI amplifies the field, the corresponding position of the system in the diagram 1 moves downward and until it stops at some β " βmin. 10 Second, MRI can be self-sustained for any⟨ value⟩ of Am even for Am 1. Although we have explored the Am parameter down to Am≪ 0.1, we believe that it can be extended further to 0 = 10 smaller Am because of the following reasons. Linear analysis −1 0 1 2 3 10 10 10 10 10 by Kunz & Balbus (2004)andDesch(2004)showsthatinthe Am presence of both vertical and toroidal field, MRI can grow at an 1 Figure 15. Scatter plot of AD coefficient Am and the plasma β at the saturated appreciable rate (approximately 0.13 Ω− when Bφ/Bz 4) state of the MRI turbulence from all our simulations. Simulations with different even in the limit of Am 0+ provided that the field= is field geometries are marked by different symbols and colors as indicated in the → legend. The dashed curve shows the fitting formula (26)asalowerboundof sufficiently weak. This means that MRI turbulence can always β βmin(Am). be self-sustained. Meanwhile, we find that the linear dispersion ⟨ ⟩ ! (A color version of this figure is available in the online journal.) relation has already approached the small Am asymptote for Am ! 0.3. Therefore, we expect the trend in Figure 15 on βmin to hold to further smaller Am values. both net vertical and toroidal fluxes and the actual βmin may be Third, the boundary between the MRI permitted and pro- somewhat smaller than obtained here. Nevertheless, this regime hibited regions is only suggestive but it does not necessarily is closer to ideal MHD and is less concerning. By combining all imply sharp transitions. Our simulations are restricted by the the available simulations, we obtain a fitting formula for βmin limited box height (H)sincetheyareunstratified.Inreality,as given by one increases the field strength, the transition from sustained

2 2 1/2 MRI turbulence to its suppression involves the effect of vertical 50 8 stratification of gas density in the disks and may be a smooth βmin(Am) + +1 , (26) = Am1.2 Am0.3 process. Before justified by stratified simulations, which is left !" # " # $ for our future work, this result should be taken with some cau- and is indicated in Figure 15.Itasymptotesto1atAm as tion. In particular, when vertical stratification is included, linear 1.2 →∞ one expects, and approaches 50/Am for Am ! 1. analysis by Gammie & Balbus (1994)andSalmeron&Wardle The constraint on βmin at a given Am allows us to identify (2005)foridealandnon-idealMHDhassuggestedtheexis- the regions in the Am– β plane at which MRI can or cannot tence of global modes in the disk even in low β0 and a small operate. In the mean time⟨ ⟩ the correlation between α and β Elsasser number. On the other hand, in the case of Ohmic re- provides the corresponding stress when MRI is permitted.⟨ ⟩ sistivity, the criterion that the Ohmic Elsasser number equaling Combining them together, the main results from this paper are one is the boundary between MRI permitted and suppressed best summarized in Figure 16.MRIpermittedregionsarein regions identified in unstratified simulations (Sano et al. 1998; the upper right with the boundary given by Equation (26). It Fleming et al. 2000;Sano&Stone2002b)doesagreewith

16 Conclusions

• Even theorists agree that one mechanism can mostly explain low, intermediate, and high mass star formation. And there aren’t even two very diﬀerent theories.

• Observations (see next talk) agree with continuity, but very diﬀerent physical systems look the same observationally

• Starting with Herbig Ae/Be’s, conditions are plausible for fragmentation of disks into low or equal mass companions

• Herbig Ae/Be’s can be interesting sites to study planet formation and disks, but may not be totally representative of solar analogs due to stellar structure on PMS