Synthetic Observations of the Evolution of Starless Cores in a Molecular Cloud Simulation: Comparisons with JCMT Data and Predictions for ALMA
Steve Mairs, Doug Johnstone, Stella S. R. Offner, Scott Schnee
AAccepted by ApJ January 13 th , 2014 Preprint typeset using L A TEX style emulateapj v. 5/2/11
SYNTHETIC OBSERVATIONS OF THE EVOLUTION OF STARLESS CORES IN A MOLECULAR CLOUDSIMULATION: COMPARISONS WITH JCMT DATA AND PREDICTIONS FOR ALMA
Steve Mairs
Department of Physics & Astronomy, University of Victoria, Victoria, BC, V8P 1A1, Canada: [email protected], andNational Research Council Canada, Herzberg Institute of Astrophysics, 5071 West Saanich Rd, Victoria, BC, V9E 2E7, Canada
Doug Johnstone
Joint Astronomy Centre, 660 North A’ohoku Place, University Park, Hilo, HI 96720, USANational Research Council Canada, Herzberg Institute of Astrophysics, 5071 West Saanich Rd, Victoria, BC, V9E 2E7, Canada, andDepartment of Physics & Astronomy, University of Victoria, Victoria, BC, V8P 1A1, Canada
Stella S. R. Offner
Department of Astronomy, 260 Whitney Ave, Yale University, New Haven, CT 06511, USA andScott Schnee
National Radio Astronomy Observatory, 520 Edgemont Road, Charlottesville, VA 22903, USA
Accepted by ApJ January 13 th , 2014 ABSTRACTInterpreting the nature of starless cores has been a prominent goal in star formation for many years.In order to characterise the evolutionary stages of these objects, we perform synthetic observations ofa numerical simulation of a turbulent molecular cloud. We find that nearly all cores that we detect areassociated with filaments and eventually form protostars. We conclude that observed starless coreswhich appear Jeans unstable are only marginally larger than their respective Jeans masses (withina factor of 3). We note single dish observations such as those performed with the JCMT appearto miss significant core structure on small scales due to beam averaging. Finally, we predict thatinterferometric observations with ALMA Cycle 1 will resolve the important small scale structure,which has so far been missed by mm-wavelength observations. INTRODUCTION
The coldest and densest regions of the interstellarmedium are the places in which dust and gas form stars(see, for example, Benson & Myers 1989). Although thetime a star spends on the main sequence as well as itssubsequent evolution is one of the most well-understoodproblems in astrophysics today, the same statement can-not be made of the dense, dusty “cores” which are theprogenitors of these objects. The opaque envelopes thathave not yet formed protostars are of particular interestin that they reside at the intersection between the prop-erties of the surrounding molecular cloud and its nascentsuns.A core, as defined in this work, is an object which hasa relatively small mass and will form at most a few stars.Distinguishing “starless” cores from “protostellar” cores isthe first step to answering several open questions in thefield of star formation. For example, many starless coreshave been measured to have masses several times that oftheir “Jeans mass” (Sadavoy et al. 2010a), the limit atwhich thermal pressure alone can provide adequate sup-port against the self gravity of the object (Jeans 1902).One possibility is that there are poorly understood non-thermal support mechanisms, such as magnetic fields orturbulence, preventing the collapse of these cores. An-other likely possibility, however, is that the cores havebeen misclassified as starless when in fact they are col-lapsing and dim protostars lie hidden within their dusty envelopes. In fact, there is strong evidence from recentinterferometric observations (Schnee et al. 2012a; Enochet al. 2010; Dunham et al. 2011; Pineda et al. 2011) thatmany cores classified as “starless” actually harbour VeryLow Luminosity Objects (VeLLOs, objects with lumi-nosities (cid:46) L (cid:12) ; see Young et al. 2004; Kauffmann et al.2005; Di Francesco et al. 2007; Dunham et al. 2008).Another avenue of study which relies on an accurateclassification of starless and protostellar cores is the effortto link the prestellar Core Mass Function (CMF) to thestellar Initial Mass Function (IMF) (for example, Nut-ter & Ward-Thompson 2007; Enoch et al. 2008; Könyveset al. 2010; Alves et al. 2007). It appears that more mas-sive cores are typically found to harbour protostars (see,for example, Sadavoy et al. 2010b and Ragan et al. 2012).As a result, if protostellar cores have been misclassifiedas starless and if the misclassification is more likely forparticular masses, then any attempt to compare the ob-served starless CMF with the IMF is problematic.Finally, as Kirk et al. (2005) describe, the lifetime ofthe subset of starless cores that will go on to form starscan be determined by comparing the relative number ofstarless cores and protostellar cores. Similarly, the num-ber of protostars relative to more evolved young starsis proportional to the protostellar lifetime (Evans et al.2009).The most common method to classify cores as starlessor protostellar is to identify cores via their dust con-tinuum emission by using catalogues such as the “Sub- a r X i v : . [ a s t r o - ph . S R ] J a n millimetre Common-User Bolometer Array (SCUBA)Legacy Survey” (Di Francesco et al. 2008) and then at-tempt to identify embedded sources using infrared data(e.g. Jørgensen et al. 2007; Sadavoy et al. 2010b) suchas the “Molecular Cores to Planet Forming Disks Cata-logue” (c2d; Evans et al. 2003, Evans et al. 2009). Dueto the high optical depth of these dusty envelopes, how-ever, extinction can obscure and even completely hidedim protostars in the centre of these structures leading toerrors in the core classification. In an attempt to explorethe veracity of the non detections of embedded infraredobjects, recent studies have utilised interferometric spec-troscopy of a variety of protostellar and outflow tracerssuch as CO, SiO (2-1), HCO + , and N H + (Pineda et al.2011; Schnee et al. 2012a,b).In order to gain further insight into the misclassifica-tion of starless cores while investigating the effectivenessof observational techniques, we compare starless and pro-tostellar cores observed in the controlled environment ofa simulated turbulent molecular cloud. We analyse theformation and evolving properties of dense structures inthe same manner as real observations taken with SCUBAat the JCMT with the added benefit of knowing preciselocations and masses of forming protostars. This analysisconcentrates on the observed stability of a given object,as defined by the Jeans mass, near the time in whichcollapsing regions begin to form.This paper is organised as follows: Section 2 describesthe numerical simulation. Section 3 outlines the methodswe use to simulate observations and describes our methodto identify objects and derive their stability. We presentthe bulk properties of the observed objects and their gen-eral evolution including stabilities, densities, and proto-star/envelope relationships as “observed” by SCUBA insection 4. Section 5 gives the results of the simulatedinterferometric observations. In section 6, we discuss theresults. Section 7 presents concluding remarks. SIMULATIONS
In this paper, we analyse a series of snapshots from ahydrodynamic simulation of a turbulent molecular cloudthat is forming stars. This simulation was previouslypresented in Offner et al. (2013) (simulation Rm6), inwhich it was used to study the chemical distribution inmolecular clouds. We briefly summarise the numericalprocedure and parameters below.The simulation was performed with the orion adap-tive mesh refinement (AMR) code (Truelove et al. 1998;Klein 1999), and it includes large-scale driven turbu-lence, self-gravity, and sink particles (Krumholz et al.2004). The simulation was first driven for two cross-ing times without gravity and then evolved for a globalfree-fall time with gravity. Sink particles were insertedat the finest AMR level when the local density violatedthe Truelove criterion for J = 0 . (Truelove et al. 1997).This corresponds to a mass density of 4.6 x 10 − g cm − ( n H = 1 . cm − ). Throughout this work, the terms“sink particle” and “protostar” will be used interchange-ably when discussing the simulation. The basegrid is cells and the run has 4 AMR levels.The bulk properties of the simulation were chosen torepresent a typical Galactic low-mass, star-forming re-gion. The simulation domain has a length of 2 pc andcontains ∼ M (cid:12) , which corresponds to an average number density of n H = 1300 cm − . The Mach num-ber, M =6.6, was set so that the simulated cloud isapproximately virialised and satisfies the linewidth-sizerelation (e.g., McKee & Ostriker 2007).The simulation was run for one global free-fall time of0.95 Myr. At the final time, the cloud contains 88 pro-tostars and has a star formation rate per free-fall timeof 0.18. Since the simulation does not include magneticfields or stellar feedback, the sink particles represent anupper limit on the true star formation (Offner et al. 2009;Commerçon et al. 2011; Hansen et al. 2012). Since thecollapse has not been followed down to the sizes of indi-vidual protostars, and feedback such as outflows is notincluded, the sink particles likely over-estimate the stellarmass by a factor of ∼ (Matzner & McKee 2000; Enochet al. 2007; Alves et al. 2007). The most massive sinkparticle formed throughout the simulation is 8.5 M (cid:12) . SYNTHETIC OBSERVATION METHODS
Single Dish Synthetic Observations
The simulation was placed at a distance of 250 pc torepresent the Perseus molecular cloud as this region hasbeen well-studied by c2d (Evans et al. 2003, 2009) andother surveys (Kirk et al. 2006; Hatchell et al. 2005; Sa-davoy et al. 2010b). At this distance, 2 pc correspondsto 1650 (cid:48)(cid:48) . We analyse column density maps integratedalong each of the x, y, and z directions. Each integratedimage was gridded to 512 x 512 square pixels 3.22 (cid:48)(cid:48) on aside.For the optically thin case, as we have here, it is simpleto convert from column density, N, to flux, S ν : S ν = N κ ν B ν , where B ν is the Planck function and κ ν isthe opacity calculated at frequency ν . The flux can thenbe related to the core mass via Equation 2 in Sadavoyet al. (2010a) (modified for a typical core temperature of10 K): S Jy = 0 . (cid:18) M c M (cid:12) (cid:19) (cid:18) d
250 pc (cid:19) − (cid:18) κ .
01 cm g − (cid:19) × exp (cid:104) . (cid:16)
10 K T d (cid:17)(cid:105) − . − − . (1)Here, S represents the flux received at 345 GHz (850 µ m), M c is the core mass, d = 250 pc is the distance tothe source, κ is the opacity at 345 GHz (see below),and T d = 10 K is the isothermal dust temperature.We adopt an opacity value appropriate for a dusty pro-tostellar core at 230 GHz, κ = 0 . cm g − (Os-senkopf & Henning 1994), in accordance with previousobservations (e.g. Schnee et al. 2012a). This value as-sumes MRN grains with a thin ice mantle for a core witha density of 10 cm − which is typical in nearby starforming regions (see Johnstone et al. 2000, Schnee et al.2012a, Sadavoy et al. 2010b).By assuming a spectral index, β , where κ ν ∝ ν β , onecan extrapolate to other frequency values. For β = 2 . ,which we adopt here, κ = 0.0202 cm g − .The flux maps are smoothed to 20 (cid:48)(cid:48) to compare againstthe smoothed SCUBA catalogues . Then, to furthermatch the observations, we remove large-scale structureby smoothing the same images by 120 (cid:48)(cid:48) and subtractingthis smoothed map from the former 20 (cid:48)(cid:48) maps (see alsoKirk et al. 2006). In total, we analyse 68 simulation out-puts distributed between t = 0 − t ff . For each of theseoutputs, we consider each projection separately. Notethat the mass of each core changes depending on the pro-jection (see Section 3.2 below) but, if detected multipletimes, each detection will be considered as an individualcore. The sink particle masses remain unchanged overeach projection. Core Definition
To extract the bulk properties of the core populationand analyse their time evolution, we use the automatedroutine CLFIND2D (Williams et al. 1994).The lowest flux level which defines the boundary of anobserved core is 0.09 Jy/beam set by comparison withthe observations. This threshold is defined by the opacityand, assuming the material is isothermal, it is equivalentto the mass depth to which we are sensitive (see Equation1). Thus, the choice in our opacity value sets our scalingbetween the observations and the simulation.Each core’s radius is then compared to the full widthat half maximum (FWHM) of the smoothed 850 µ mSCUBA observations, 20 (cid:48)(cid:48) . If the core is smaller thanthis, it is deemed spurious and removed from the analy-sis (since it might be noise). For the non-spurious cores,we convert the measured flux into a mass by invertingEquation 1.Once the mass is attained, an object’s stability canbe analysed using simple assumptions. We estimatethe Jeans mass, M J , c , of an identified core by applyingthe simple scaling relation presented by Sadavoy et al.(2010a), M J , c = 1 . (cid:18) T d
10 K (cid:19) (cid:18) R c .
07 pc (cid:19) M (cid:12) . (2)Here, T d = 10 K is the (assumed) isothermal dust tem-perature and R c is the radius of the core.To determine the stability of each core, we comparethe mass attained from inverting Equation 1 to the Jeansmass calculated by Equation 2. If M c ≥ M J , c , the ob-ject is defined as “super-Jeans” an unstable configurationwhich should show signs of gravitational collapse if ther-mal pressure alone were counteracting the force of grav-ity. M c < M J , c represents a stable, “sub-Jeans” objectwhich would not be expected to collapse since the ther-mal pressure within the assumed spherical object wouldbe more than enough to balance the gravitational forces.Once we determined the stability parameter using onlythe envelope mass, we correlated the positions of proto-stars (sink particles) with the CLFIND2D objects. If aprotostar lies within 75 % of the circular radius (see be-low) of the centre of a core, we define the core to beprotostellar. Therefore, a comparison can be made, forexample, between the cores that are observed to be stable(cores without protostars) and the cores with evidence ofcollapse (due to the fact that they have embedded proto- The JCMT beam is 15 (cid:48)(cid:48) but the SCUBA observations to whichwe are comparing were smoothed to 20 (cid:48)(cid:48) (Di Francesco et al. 2008). stars). We tracked the growth of protostellar and starlesscore masses through time along with density, stability,and position.To visually display the cores on the flux maps, the im-ages include circles and squares corresponding to the sizeof each core at the location of each of the core centroids(see Figure 1). Circles represent cores that are Jeans un-stable (M ≥ M J , c ) and squares denote cores with massesless than the calculated Jeans mass. Plus signs symbol-ise the location of protostars. We constructed movies bystitching together images of sequential outputs (labeledat the top of each frame, see Figure 1).Since protostellar masses cannot be directly measured,adding the protostellar mass to the core mass does notmake observational sense but allows us to track the sta-bility of an object. When more than one protostar isassociated with a given core, the protostellar masses aresimply added. Without sufficient feedback, however, theavailable material will continue to accrete onto a givenprotostar. Without outflows, protostellar masses maybe overestimated by up to a factor of ∼ (Matzner &McKee 2000; Enoch et al. 2007; Alves et al. 2007).Once a protostar is formed the inner envelope will beheated. Observationally this makes it harder to con-vert from observed 850 micron flux to mass and thusmakes Jeans stability investigations difficult for proto-stellar cores. For the simulations investigated here, how-ever, the lack of included heating makes the mass deter-mination and stability analysis straightforward.Note that the mass, size, and density scales of objectswe extract are dependent on the large-scale structure inthe image. To investigate, we performed structure identi-fication for maps that had not undergone the 120 (cid:48)(cid:48) scaleremoval. In these cases we find large reservoirs of mass,which are strongly associated with filamentary structure(see Section 4.1), surrounding each core. These extendedzones are approximately twice the size of cores identi-fied when the large-scale structure is removed. It is use-ful to consider that it has been previously determinedthat CLFIND2D works reasonably well when a field issparsely populated with discrete objects, but strugglesto sensibly pick out important structures when the fieldis “crowded” (see, for example, Pineda et al. 2009). Interferometric Synthetic Observations
The Common Astronomy Software Applications(CASA) package is used to simulate 100 GHz AtacamaLarge Millimetre Array (ALMA) Cycle 1 interferometricobservations of several individual cores in order to com-pare with the single dish results. We choose a five second“snapshot” integration time to match real observations;90 seconds in total is required to create a mosaic for eachcore.Beginning with the simulated column density maps, weagain generate flux maps as described above. For a givencore, we use a square 2 arcminutes on a side centred onthe object’s coordinates as the input sky model (the sim-ulated flux map). Then, we construct the UV visibilitiesfor the most compact arrangement of antennas using theCASA package simobserve . We employ simanalyze to ∼ smairs/research/starless/movies/movies.html Fourier transform these visibilities into the image plane.In this study, the simulation has similar densities andtemperatures to the Perseus cloud. Therefore, the decli-nation of the observation is set to a reasonable approx-imation of the cloud’s position: J2000 +30d00m00. Weuse a hexagonal stitching pattern for each mosaic.All observations include thermal noise. In CASA, arobust atmospheric profile exists for the ALMA site in-cluding the altitude, ground pressure, relative humidity,sky brightness temperature, and receiver temperatures.In simobserve , the user need only define the precipitablewater vapour (pwv) which had a chosen value of 1.262mm. With these assumptions, the noise for the ALMACycle 1 image is ∼ . mJy/beam. Since these simulatedobservations are only meant to note whether the sourcewas detected, the maps are not “cleaned”; we simply con-volve them with the point spread function of the instru-ment to produce a “dirty” map for analysis. This will notsignificantly alter the mass of the objects detected, how-ever, a cleaned map would reveal more structure over apotentially larger area.Simulated observations using the Submillimetre Ar-ray (SMA) and the Combined Array for Research inMillimeter-wave Astronomy (CARMA) were also per-formed, but we did not detect any cores in the resultingmaps. RESULTS FROM SINGLE DISH “OBSERVATIONS”
Over the course of one free-fall time, we traced sev-eral core properties both qualitatively and quantitatively.The following sections describe our main results.
Filamentary Structure
The idea that filaments play a role in in the formationof dense cores has been entertained for several decades(see, for example, Schneider & Elmegreen 1979). Thegrowth of instabilities in filaments and subsequent frag-mentation have been numerically analysed in both thelinear (e.g. Inutsuka & Miyama 1992) and nonlinearregimes (e.g. Inutsuka & Miyama 1997). Robust sim-ulations of isothermal, self gravitating cylinders have re-vealed several properties of filaments such as characteris-tic temperatures, external pressures, densities, and radiiwhich are consistent with Herschel observations (Fischera& Martin 2012b,a). The Herschel observations show thatfilaments are present throughout many star forming re-gions (André et al. 2010). Additionally, Hacar et al.(2013) performed observations of C O(1-0), N H + (1-0), and SO( J N = 3 − ) in the Taurus star formingregion with the 14 m FCRAO telescope (and supple-mented the data with APEX 870 µ m and IRAM 30 m1200 µ m) where they found that cores appear to formin a two step process. First, velocity-coherent filamentsform. Then, these large structures fragment into cores.Kirk et al. (2013) and Friesen et al. (2013) have found ev-idence of filamentary accretion flows in the Serpens starforming region using a variety of spectral lines observedwith the ATNF Mopra 22 m telescope and the K-BandFocal Plane Array at the Robert C. Byrd Green BankTelescope, respectively. Hennemann et al. (2012) sug-gest that gravitationally unstable filaments are the driv-ing factor for star formation in the Cygnus X region witha specific emphasis on the DR21 ridge. Furthermore,Myers (2009) shows that young, embedded star clusters are associated with multiple filaments. It has also beenshown that filamentary geometry is ideal for the growthof small-scale perturbations that lead to large scale col-lapse in a preferential dimension (the length of the fil-ament) and that filaments containing only a few Jeansmasses can easily fragment (for a thorough description,see Pon et al. 2011).Of course, some detections of filaments may be at-tributed to multiple beam diluted objects so careful in-vestigation of properties such as velocity coherence fromspectral fitting are necessary to characterise a structurewith certainty. As suggested by the authors above andreferences therein, a typical filament within a molecu-lar cloud has a width of ∼ .
15 pc (120 (cid:48)(cid:48) at the distance of the Perseus star form-ing region) from the analysis of the simulation presentedhere, however, to focus on the cores themselves and nottheir parent structures.With this in mind, the convolved simulation hosts coreswhich are forming almost exclusively along what appearto be dense filamentary structures. As the simulationevolves in time, the objects appear to travel along thesestriations, fragmenting and coalescing into a variety ofmorphologies (see movies).Figure 1 shows images of four of the sixty-eight snap-shots. The top left panel represents an early time inwhich no protostars have formed; the bottom right panelrepresents one free-fall time, t ff . The filaments stand outclearly and their qualitative association with the identi-fied cores is obvious. Bulk Properties of the Ensemble
In this section we present the mass and density dis-tribution of the identified cores, where we analyse coresidentified from three orthogonal views. Figure 2 showsthe distributions of core masses and densities at differ-ent times throughout the simulation. We calculate thedensity of each core by assuming the mass is uniformlydistributed over the object as if it were a sphere with aradius determined by its projected area.The top row of Figure 2 presents the dataset in whichno protostellar masses were included in the mass deter-mination in order to compare directly with observations.We note that the range in core masses and densitiesfound in the simulation are consistent with real obser-vations of Perseus (Sadavoy et al. 2010b; Enoch et al.2008). The bottom row illustrates the core mass distribu-tion including the protostellar masses in order to analysethe “true” stability of a given core. As time progresses,the core masses increase as gas accretes and collapsesto higher densities. This is more evident in the bottomrow than the top. New cores are identified throughoutthe simulation and therefore an approximately constantlow mass population of objects is present throughout thelater stages of the simulation.At the end of the simulation when protostars are nottaken into account in the mass estimate, the mediancore mass and the mean core mass are both found tobe .
89 M (cid:12) . Including the mass of protostars yieldsnearly equivalent median and mean masses of approxi-mately 1.9 M (cid:12) . The median and mean number densi-ties of the dataset without protostellar masses are both n H = 1 . cm − . The dataset including protostellarmasses has a median number density of n H = 1 . Fig. 1.—
Four snapshots ranging from “early” times to “late” times (t = 0.15 t ff to t = t ff , see labels). Protostellar masses have beenincluded in the stability calculations. The Y-dimension integrated images are shown. Circles represent unstable cores, squares show thelocations of stable cores, and plus signs display the locations of protostar formation sites. Three cores that we study in greater detail arehighlighted in the top right panel. cm − and a mean number density of n H = 2 . cm − .The density of a core can be compared with three ref-erence values: the average density of the box, n , thedensity of a typical shocked region, n s , and the “modaldensity”, n c . n = 1 . cm − . This is an order of magni-tude lower than the density of any identified object.To estimate the compressed density, we consider a 1D,isothermal shock at the typical density: n s = M n =1 . cm − (see section 6.1 for more details). Thisvalue is shown by the vertical line in Figure 2, which is close to the lowest densities of the cores.The “modal density” is an empirically derived densitynoted by the vertical dashed line in Figure 2. Whether ornot the masses of the sink particles are taken into accountin the analysis, there appears to be a peak in the densitydistribution at a number density of n H ∼ . cm .Note that the tail of the distribution in Figure 2 is quitedifferent depending upon whether we include the proto-stellar masses in the estimates, especially at later times. Core Stability − )0.010.020.030.040.050.0 N u m be r o f C o r e s Sink Masses Not Included -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0Log Core Mass (Log M ⊙ )0.010.020.030.040.050.0 N u m be r o f C o r e s Sink Masses Not Included − )0.010.020.030.040.050.0 N u m be r o f C o r e s Sink Masses Included -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0Log Core Mass (Log M ⊙ )0.010.020.030.040.050.0 N u m be r o f C o r e s Sink Masses Included
Fig. 2.—
The core density (left column) and mass (right column) distributions at three different times: 0.5 t ff (solid line), 0.8 t ff (dashedline), and t ff (dotted line) including all three projections. The protostellar masses are not included in the top row (to emulate realobservations of cores with hidden protostars); they are included in the bottom row. The solid vertical line in the density plots showsan estimate of the typical density of shocked regions. The dashed vertical line highlights a peak in the density distribution, the “modaldensity”. Over time, cores become visible because gravity pro-duces densities which exceed the threshold of observabil-ity. Thus, more cores are identified as the simulation pro-ceeds (see Figure 3). At t = 0.5 t ff there are 78 identifiedcores over all three projections. 80 cores are identifiedby t = 0.8 t ff , and the simulation ends with 86 identifiedcores at t = t ff .As expected, the simulation begins with few identifiedcores, then mass accumulates and observationally sta-ble starless cores begin to form in what appears to be abottom-up fashion. As the simulation proceeds, many ofthese cores become unstable and begin to form protostars(see right hand panel of Figure 3).Figure 4 shows the number of cores in different stabilitystates throughout the observed portion of the simulationfor cases in which the dust envelope alone is taken into consideration (left panel) and when protostars are addedto the flux maps (right panel). Together, the two panelsshow the evolution of cores defined as sub-Jeans (obser-vationally stable), super-Jeans (observationally unstableto collapse), protostellar (contain a sink particle), andstarless (contain no sink particles).It is important to note that when the protostellarmasses are not taken into account, the true masses ofthe identified objects are underestimated and there is apopulation of cores that are deemed “stable” even thoughthey are collapsing and forming protostars. This suggeststhat, when the embedded protostar is unobservable, theJeans stability argument used by Sadavoy et al. (2010a)may not be sufficient to identify super-Jeans cores.Although we find super-Jeans cores without any proto-stars inside, it appears these objects are only marginallyunstable. All but one of these cores have masses whichare less than a factor of two greater than their respec-tive Jeans masses (see Figure 5). As noted by Sadavoyet al. (2010a), starless cores which satisfy M ≥ . M J , c are those which are deemed “unusual relative to the pro-tostellar cores”. We find no such cores throughout thissimulation.When protostellar masses are included, the observedcore stability changes significantly. We see that prac-tically all of the previously sub-Jeans cores which hadprotostars are now classified as super-Jeans as expected. Evolution
We selected several cores to study in greater detail.These cores are taken from isolated positions in the fluxmaps so other objects and protostars will not signifi-cantly affect the measurements (three are highlighted inFigure 1). We track one of these cores through the entiresimulation and choose specific timesteps for the othersbased on the protostar formation time. Figure 6 showsthe mass and density of three example cores tracked overa number of times.In Figure 6, the solid horizontal line shows the den-sity associated with the shocked material (see Section6.1). The dashed horizontal line highlights the “modaldensity” as defined in Section 4.2. The diagonal line in-dicates the minimum density for a core to be consideredobservationally unstable (Equation 2).It is evident that there are a few discontinuous jumps(annotated by “D’s”) in Figure 6. These are due toCLFIND2D itself and how it defines multiple objects.Although a core is initially isolated, it fragments andcoalesces with its pieces as it evolves. When an extendedobject becomes large enough to exceed the flux thresh-old between two regions, CLFIND2D draws a boundarybetween the regions and labels each as a separate ob-ject. Sometimes, this bifurcation lasts only for a briefperiod of time and the object reassembles into its pre-vious configuration in the next timestep. Of course, asudden decrease in radius becomes a sudden increase indensity and vice versa. Both in the synthetic observationand in actual observations, there are occasionally multi-ple objects and filaments that have been “smeared” intoone identified core due to the 20 (cid:48)(cid:48) smoothing.Both panels in Figure 6 highlight specific regions ofinterest. In the left panel, which is directly comparablewith real observations, stage 1 shows the beginning ofthe core evolution when a protostar has not yet formed.There is a short period (stage 2) in which the mass in-creases but the density remains constant. A brief subdi-vision and merging takes place before the core splits intotwo distinct objects, losing mass and becoming far moredense as it enters stage 3. Another CLFIND2D iden-tified bifurcation and amalgamation takes place whenthe defined flux threshold is briefly achieved before theobject settles into its final evolutionary state (stage 4).From this point until the end of the simulation, the massand the density both increase as in stage 1, but to amuch larger degree. When considering the envelope massalone, the evolution is less monotonic but still exhibitsperiods of collapse.The right panel of Figure 5 shows density versus massincluding the protostellar mass. There are two obviousregions marked with “D’s” in which CLFIND2D sub- divided, then merged this object. Beginning in stage1, there is a very steady mass and density increase asthis object forms a protostar and becomes observation-ally Jeans unstable shortly thereafter. After the earlycollapse, the density begins to level off for a short periodof time. Meanwhile, the mass continues to increase, in-dicating that the radius must also be increasing beforea large upward jump in density. Here again, the corebifurcates causing an abrupt density spike. During thisperiod, the mass is essentially constant for the rest of thesimulation, which suggests the core is simultaneously de-creasing in size.The bottom left panel of Figure 6 shows the evolutionof a core which is recognisably Jeans unstable before aprotostar forms. The right panel shows an isolated corethat we select at random in the simulation to highlightthe diversity in the core population. When only the en-velope masses are taken into account, each core resem-bles the left panel of Figure 6. When protostellar massesare included as shown there is a rapid evolution in thecore properties. Both the mass and the density increasesubstantially over time, such that the cores appear ob-servationally unstable.
Protostar and Envelope Relationship
In this section, we explore the relationship between themass of a protostar and its parent core. Figure 7 showsthe fraction of core mass in sink particles. The pointswhich lie at exactly 1.0 on the ordinate axis are proto-stars which did not have an associated envelope. Theseprotostars lie outside 75 % of their nearest core’s radius(measured from the core centre). Thus, with no appar-ent envelope, the sink mass is one hundred percent ofthe object’s mass. Note that because of the backgroundsubtraction, some of the core mass may be lost in thisanalysis.As expected, many young objects have not formed pro-tostars and lie along the bottom of the plot. As timeprogresses, the sink particles begin to dominate the coremasses quickly. The bottom right quadrant of Figure 7is empty. This means that the more massive cores ob-served in the simulation are dominated by the protostarspresent. Stepping through time, it is clear that this isa rapid process. Once sink particles form, they quicklyaccrete a large amount of mass. Therefore, observationsignoring the embedded protostars could miss a signifi-cant portion of mass. This will lead to errors in stabil-ity classification using non-interferometric observationaltechniques.When each of the timesteps are analysed in a sequentialfashion, the points travel from the bottom left throughan “S” shape to the top right of the plot. Since the totalmass of a given object substantially increases through-out the formation and growth of a protostar, it is clearthat the protostar mass cannot come from the initial ma-terial detected in the envelope alone. One possibility isthat cores extend below our flux limit and are collaps-ing to higher densities such that they enter the observ-able regime. Another likely possibility is the accretion ofmass due to bulk flows in the simulation in conjunctionwith gravity. The obvious filamentary structures in themap are the most likely sources of mass. In fact, massflow along filaments associated with protostars have beenobserved in many star forming regions (e.g. Kirk et al. t ff T o t a l N u m be r o f C o r e s t ff F r a c t i ono f M a ss i n C o r e s EnvelopeEnvelope + ProtostarProtostar
Fig. 3.—
Left:
The total number of cores identified through time during the simulation.
Right:
The fraction of the simulation’s totalmass contained within identified cores over all three projections. The dashed line represents the dust envelope mass only; the solid lineshows the dust envelope mass as well as their contained protostar masses, the dash-dot line shows the protostellar structure mass only.Note that the large majority of the mass contained within cores is locked in protostars where it cannot be directly observed. After onefree-fall time, protostellar masses account for 15% of the 600 M (cid:12) box, or, 90 M (cid:12) while core envelopes account for ∼
5% of the mass of thebox, or 30 M (cid:12) . t ff F r a c t i ono f T o t a l C o r e N u m be r Protostellar Mass NotIncluded in Jeans Calculation
Protostellar Super-JeansProtostellar Sub-JeansStarless Super-JeansStarless Sub-Jeans t ff F r a c t i ono f T o t a l C o r e N u m be r Protostellar MassesIncluded in Jeans Calculation
Protostellar Super-JeansProtostellar Sub-JeansStarless Super-JeansStarless Sub-Jeans
Fig. 4.—
Different core stability states.
Left:
Protostellar masses are not included in the analysis,
Right:
Protostellar masses areincluded. M p is the protostellar mass. INTERFEROMETRIC ANALYSIS
In this section, we assess the conditions for which thedetection of an embedded protostar is possible. Employ-ing the high resolution and sensitivity offered by inter-ferometers is the logical next step in characterising thedynamic nature of cores.As described in section 3.3, we performed syntheticALMA Cycle 1, SMA, and CARMA simulated interfero-metric observations. It was found, however, that a ninetysecond observation performed at 100 GHz with ALMACycle 1 is comparable to an eight hour observation takenby SMA at 230 GHz and achieves a far better signal tonoise ratio than an eight hour observation taken with
TABLE 1ALMA Cycle 1 observations performed on three cores.
Core Identifier Radius (AU) Envelope Mass ( M (cid:12) ) Sink Particle Mass ( M (cid:12) ) Envelope Mass M J Density (cm − )1 829.30 0.27 0.619 2.44 1.70 x 10 J,c N u m be r o f C o r e s All Starless Super-Jeans Cores
Fig. 5.—
The masses of the starless super-Jeans core population(the unstable prestellar cores) over all timesteps and projections inthe simulation given in terms of their individual Jeans masses.
CARMA at 100 GHz. Offner et al. (2012) found a simi-lar result. The SMA and CARMA observations producedonly non detections; evidently, to identify any substruc-ture present in faint cores or to even detect the objectswith σ confidence, a greater sensitivity is required. Thisnull result is compatible with recent interferometric ob-servations (see Schnee et al. 2012b).Thus, we focus primarily on ALMA. We analyse eachinterferometric image down to the same flux threshold( σ = 0 . mJy/beam, defined by-eye) for a consistentanalysis. Purely investigating the results of CLFIND2D,we see that in most cases there is no significant buildupof mass before a protostar forms. In two of the cases thecore is identified very briefly after the formation of theprotostar. In one case, the core is identified concurrentlywith the formation of a sink particle and continues togrow in time. In the last case, CLFIND2D does notidentify a core at any time output.Figure 8 shows the observations of the object in thebottom left panel of Figure 6 (labeled “Core 1” in Fig-ure 1) at three different timesteps; two approaching theformation of the first protostar and one at the time thefirst protostar appears. The core begins in an undetectedstate. By the time a protostar forms, a clear detection ispossible.Table 1 shows the properties of the three identifiedobjects at the point of their first detection. In all threecases, the objects already contained sink particles. Cores1 and 2 are the same as in Figure 6 (left and right panels,respectively); note the increase in density in the centreof the core compared to the core average (compare with Section 4.4).CASA simulations such as these provide a strong pre-diction for real interferometric observations. Currently,ALMA Cycle 1 telescope time has been awarded withhighest priority to observe the 3mm continuum emissionfrom all 60 starless cores and 13 protostellar cores inthe Chamaeleon I molecular cloud, as identified in Bel-loche et al. (2011). These observations will be sensitiveto point sources with masses (cid:38) . M (cid:12) , with less than2 minutes of on-source integration time per object (seeTable 1). Once these data are collected, we will be ableto perform robust comparisons between simulated coreproperties and their observed counterparts. DISCUSSION
Shocked Densities and Structure
In this section we consider some simple estimates ofthe role of turbulence and gravity to place our results incontext.The characteristic size scale at which gravity domi-nates over thermal pressure is given by: L J = (cid:115) πc Gρ . (3)where c s = ( k B T /µm H ) is the sound speed, µ = 2 . isthe mean molecular weight, k B = 1 . x − ergK − , Gis the gravitational constant, m H is the mass of hydrogen,T is the isothermal gas temperature, and ρ is the averagemass density in the simulation.In a typical shocked region in the simulation, the den-sity and hence the Jeans length will be higher. Theshocked density of a 1D isothermal shock is given by ρ s = M ρ , where M = M / √ . is the 1Dsimulation Mach number. Plugging this into Equation 3,we can expresses the compressed Jeans length as: L Jm = (cid:115) πc G M ρ (4)or, L Jm = M − L J . (5)Values of T = 10 K and ρ = 5 . − g cm − givea compressed Jeans length of L Jm = 4 . cm or126.42 (cid:48)(cid:48) , assuming a distance of 250 pc. For the 2 pcbox, the low resolution (512 x 512) grid used in this anal-ysis had 3.2 x 3.2 arcsecond pixels. So, a typical shockforces material together on scales of ∼
40 pixels in thelow resolution grid.The Jeans mass, M J , associated with the Jeans lengthis given by Equation 2 replacing R c with the Jeans ra-dius, R J = L J / . Thus, the Jeans mass corresponding tothe average density in the simulation is M J , = 7 . M (cid:12) − . − . − . . . . . . . Mass of Core (Log M ⊙ ) . . . . . . . . . . M ean C o r e D en s i t y ( Log c m − ) − . − . − . . . . . . . Mass of Core (Log M ⊙ ) . . . . . . . . . . M ean C o r e D en s i t y ( Log c m − ) − . − . − . . . . . . . Mass of Core (Log M ⊙ ) . . . . . . . . . . M ean C o r e D en s i t y ( Log c m − ) Core 1 − . − . − . . . . . . . Mass of Core (Log M ⊙ ) . . . . . . . . . . M ean C o r e D en s i t y ( Log c m − ) Core 2
Fig. 6.—
Top panels: One core tracked over all the outputs of the simulation.
Left:
Dataset in which protostellar masses are not included.
Right:
Dataset in which protostellar masses are included. Bottom Panels: Densities of two individual cores which form protostars trackedover a subset of the outputs of the simulation. The left panel shows the object labeled “Core 1” in Figure 1 and the right panel shows “Core2”. Protostellar masses have not been taken into account for either of these latter cores. Circles indicate when the core does not contain aprotostar within its boundaries. Plus signs indicate at least one protostar exists within the core boundaries. Points lying above the soliddiagonal line are defined to be observationally unstable using Equations 1 and 2; points lying below are classified as observationally stable.The solid horizontal line shows the fiducial shock density (see Section 6.1). The dashed horizontal line shows the empirically derived “modaldensity”. “D” represents a discontinuous feature introduced by CLFIND2D. and the Jeans mass corresponding to shocked densitiesis M J , m = 2 . M (cid:12) . The average CLFIND2D core massis much closer to the shocked Jeans mass than the Jeansmass corresponding to the average density of the box.This is, of course, a very simple estimate. The ac-tual processes at play are more complicated and caninvolve a number of oblique shocks and shears. Notethat the “modal density” is approximately a factor of 5greater than the shocked density calculated here; thiscorresponds to a higher characteristic velocity than thatassociated with M .To gain a better understanding of the true densitiesattained in the centre of an individual object, we anal- ysed one of the cores tracked in Sections 4.3 and 4.4 inmore depth at a higher resolution (level 2 in the AMRsimulation). The core we selected was the first objectto achieve a super-Jeans state in the Y-dimension fluxmaps.Figure 9 shows number density maps for Core 1 atfour different timesteps centred on the formation of thefirst sink. The outer contours show the expected num-ber density achieved using the M coefficient ( n H =1 . cm − ). The inner contours show the “modaldensity” ( n H = 1 . cm − ); approximately a factorof 5 larger. Clearly, both these values are significant intracing the structure present in the map.1 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0Total Core Mass (Log M ⊙ )0.00.20.40.60.81.0 F r a c t i ono f T o t a l C o r e M a ss i n S i n k P a r t i c l e s Fig. 7.—
The fraction of a total core’s mass (protostar andenvelope) found in the protostars contained within the object’sboundaries plotted against total core mass for all objects in allthree projections observed at three timesteps. The Y dimensionintegrated images only are shown here for clarity. Red represents50 % of the box free-fall time, green represents 80 % of the box free-fall time, and blue represents one free-fall time. The solid verticalline is drawn at the Jeans mass corresponding to the shocked den-sity (see section 6.1). The dashed vertical line highlights the Jeansmass associated with the empirically derived “modal density”. Thesolid horizontal line simply shows the 50 % mark (i.e. where thecollapsing regions dominate the core mass). Evidently, much higher densities are achieved thanthose expected by the simple argument presented above.Throughout the observed frames, both before and afterthe protostar has formed within the core, gravity quicklycreates regions of high density on scales much smallerthan the beam. Thus, important structure will remainunobserved without deep interferometric maps.Analysing the simulation itself, a typical number den-sity value within Figure 9’s inner contour was found tobe n H = 5 x 10 cm − by averaging over the projection.The highest inferred average density of a region exceeds n H = 10 cm − (see Table 1). This typical density cor-responds to a dynamical collapse time of approximately5% of the total box free-fall time. Near the beginningof the simulation, this is close to the time resolution inwhich observations were performed. Therefore, for densi-ties significantly greater than 5 x 10 cm − , the collapsewill not be resolved temporally with the chosen, observedsimulation snapshots in the earlier stages of core evolu-tion. Single Dish Results
The CLFIND2D objects span a range of masses andmorphologies. The objects are initially detected whenthey are starless; quickly, a subset undergoes mass ac-cretion, gravitational collapse, and protostar formation.In fact, the majority of the cores detected go on to formprotostars. This indicates that the objects observed arenot transient, but “real” star formation sites. In the ear-liest timesteps, before gravity has had a chance to signif-icantly affect localised regions, there are very few coresidentified. As time progresses, the number of observa- tionally defined super-Jeans cores as well as the numberof protostars increases.It is interesting to note that the majority of the massin cores accretes onto the protostars and the protostarmass rapidly dominates the mass budget. In fact, in thissimulation the most massive envelope including at leastone protostar was approximately 10 M (cid:12) . The cumu-lative mass of the protostars present within the largestcore, however, totalled more than M (cid:12) . Recall, how-ever, that without removing the 120 (cid:48)(cid:48) scale structure fromthe flux maps, we find these cores reside within muchlarger mass envelopes (Section 3.2). These extended re-gions predominately trace the filamentary structure inthe simulation and act as reservoirs for the smaller scalecores.When the mass of the protostars is not included (toresemble observations of cores wherein the dense centralobject is unobservable) there exists a small populationof cores forming stars which are deemed stable from anobservational perspective. When the protostellar mass isadded, however, these cores are found to be unstable tocollapse as expected.Observationally detecting collapse proves to be quitedifficult. For example, if the protostar is deeply embed-ded and undetectable, its mass cannot be accounted forin the gravitational analysis. The problem of detectingembedded protostars is generally expected to be moresevere when the protostars are small and dim. Conse-quently, the results presented here are most applicableto the transition between the starless and protostellarstage. In many cases, the core, the protostar, or bothmay be too faint to detect in the first place; and detec-tion is especially difficult at early times (Schnee et al.2012b; Pineda et al. 2011; Dunham et al. 2008; Bourkeet al. 2006; Young et al. 2004).Smith et al. (2012) show that the problem of detect-ing protostars extends to observations of molecular lines.They performed radiative transfer calculations for coresembedded in filaments in a turbulent hydrodynamic sim-ulation. They find that in over 50% of viewing angles,there is no “blue asymmetry”: a classic sign of materialinfall in an isolated spherical core. In a continuation ofthis work, Smith et al. (2013) highlight the need for highresolution observations with ALMA in order to test howline profiles and results change with beam size.Note that a few significantly super-Jeans starless ob-jects (core mass > . M J,c ) were identified by Sadavoyet al. (2010a), little evidence of objects fitting this clas-sification is found in this simulation.Of course, the detection and analysis of cores relieson many assumptions. We adopted typical values forthe dust properties (Johnstone et al. 2000; Kirk et al.2006; Schnee et al. 2012a; Sadavoy et al. 2010b). Weneglected internal heating due to protostars. In actualobservations, the warmed dust grains cause an increasein flux which can easily be misconstrued as a core witha larger mass.One can clearly see a low mass core population whichis maintained throughout the simulation in Figure 2. Itappears that once a core is identified, it continues to col-lapse and gain mass. Most of this mass is accreted ontoembedded protostars. A sufficient amount of the massflow, however, replenishes the dust envelope, leaving itdetectable.2
Fig. 8.—
ALMA Cycle 1 simulated observations of Core 1. The left column shows three original, simulated, images at different timesteps.The right column shows the interferometric observations of these same three timesteps. The top and middle rows show times 0.20 t ff and0.24 t ff . The third time shows the object at 0.29 t ff , just after a protostar has formed. The large circle on the bottom right hand panelrepresents the effective 20 (cid:48)(cid:48) smoothed beam in the single dish analysis. The smaller circle shows the 3.2 (cid:48)(cid:48)
100 GHz synthesised ALMA beam.
This distribution of mass within an evolving proto-stellar core can be best illustrated by Figures 2 and7. The top right hand panel of Figure 2, which takesinto account only what is observable in the submillimetreregime, shows the total remains approximately constantover time. If the mass of protostar is included, the to-tal mass within the core increases over time. Even whenthe embedded protostars become quite massive, the coremasses remain comparatively low. When only consider-ing the material in the envelope, the density increasesonly slightly over time. In general, when the protostellarmasses are added, the peak of the mass and density dis-tributions stay approximately constant while more highermass objects are observed. This is also shown in Figure 7where the protostars dominate the mass by a significant factor.These distributions were also analysed assuming eachprotostar was a factor of 3 less massive in order to com-pensate for the overestimate inherent in the simulation(see Section 2). We found a less accentuated but still sig-nificant increase in higher mass objects as expected. Amodified version of Figure 7 did not undergo any substan-tial changes. The protostar masses still greatly dominatethe envelope masses.
Interferometric Results
To truly determine the nature of a given core, one needsto look more closely at the internal structure. The largeJCMT beamsize “washes out” the more compact struc-3
Fig. 9.—
Density map for one individual core observed at a resolution of 1.6 (cid:48)(cid:48) /pixel. The top row shows the core before it forms aprotostar (Left to right: 0.15 t ff and 0.24 t ff ); the bottom row shows two timesteps shortly after a protostar forms (Left to right: 0.34 t ff and 0.41 t ff ). The plus signs indicate the locations of the protostar. The outer contour indicates a density of . cm − ; the inner: . cm − (see text). ture, causing the object to appear less dense. In orderto investigate the details of protostar formation at thescales, sensitivity, image fidelity, and resolution neces-sary, an interferometer such as ALMA is required.The bottom right panel of Figure 8 shows the differencebetween the JCMT and the ALMA beams for a simulatedALMA Cycle 1 observation (100 GHz). It is clear thatthe JCMT beam blends much smaller structures that can be detected with an interferometer.Comparing the Table 1 values with the same cores asin the single dish data, we highlight several interestingpoints. Beginning with Core 1, the mass inferred usinginterferometry is an order of magnitude less than foundby the JCMT. This is to be expected as the envelope ob-served by ALMA is much smaller than that observed bythe JCMT. The reduction in mass indicates that there is structure present that is on larger angular scales thatcannot be recovered by ALMA’s 12m array in its mostcompact configuration at 100 GHz. The object was de-tected concurrently with the snapshot in which the pro-tostar formed.Core 2 appears more unstable when synthetically ob-served by ALMA. Removing the noise from the observa-tions and redefining the clump boundaries at a lower fluxthreshold which is discernible by-eye, the object is foundto be up to 1.5 times more massive. This indicates thatthe signal to noise ratio significantly influences the coreclassification and stability calculation. The object wasdetected shortly after the sink particle first appeared.The third isolated core in which a detection was madewith ALMA is similar to the second. The core mass toJeans mass ratio indicates more instability in the case4of ALMA. Probing the central densities of these cores isimportant for understanding the single dish data.Our interferometric analysis of core stability also as-sumes isothermal dust and spherical cores. It is clear,however, that the morphologies of these objects could in-dicate collapse along preferred axes (see Pon et al. 2011for a thorough description of collapse modes). This typeof analysis is vital to perform at small scales near thecentre of identified objects in order to truly observe howstars are forming. CONCLUSION
In this study, we performed synthetic single dish andinterferometric observations of a simulated star formingregion. Assuming the gas and dust were optically thin,we inferred masses and densities by assuming the objectswere spheres. We calculated the gravitational stabilityusing the density of each core and correlated protostarswith the objects. The single dish analysis was performedwith and without including protostellar masses; the for-mer was to emulate real observations while the latter wasto compute the core’s “true” stability. We investigatedthe relationship between the protostar and its parent en-velope in terms of their mass and we considered the sig-nificance of the core densities with respect to turbulenceand gravity.There are several key results:1. Our analysis is consistent with various observa-tions. Namely, the masses and densities of the sim-ulated cores we detect are very similar to “real”cores in Perseus (see Section 4.2). We find manymore sub-Jeans cores than super-Jeans, in termsof their mass and size, which is consistent with Sa-davoy et al. (2010b). The fact that we do not detectsubstructure with simulated CARMA observationsis consistent with Schnee et al. (2012b).2. Nearly all cores that we detect eventually form pro-tostars. This suggests that observed cores detectedin this manner (assuming the physical conditions and distance of Perseus) are probably “real”; thatis, they will likely go on to form protostars in thefuture (see Section 6.2). The mass of the observedenvelope, however, does not appear to be a goodtracer of the eventual protostellar mass. This hasimplications for comparing the core mass functionwith the stellar initial mass function (see section4.5). Note, however, that the simulation does notinclude magnetic fields, which could provide addi-tional support that may inhibit a core from collaps-ing.3. Nearly all cores we identify are associated with fil-aments. This is consistent with the ubiquity of fil-aments recently observed by Herschel and suggeststhat if observations had better resolution and sensi-tivity they would also see a similar correspondencebetween cores and filaments (see Section 4.1).4. Single dish observations such as those with theJCMT as well as previous-generation interferom-eters appear to miss significant core structure onsmall scales due to flux averaging. Interferometricobservations with ALMA are necessary to recoverthis information (see Sections 5 and 6.3). ACKNOWLEDGEMENTS
Doug Johnstone is supported by the National ResearchCouncil of Canada and by a Natural Sciences and Engi-neering Research Council of Canada (NSERC) DiscoveryGrant. Support for this work was provided by NASAthough Hubble Fellowship grant
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