The observable prestellar phase of the IMF
aa r X i v : . [ a s t r o - ph . GA ] O c t Draft version May 29, 2018
Preprint typeset using L A TEX style emulateapj v. 11/10/09
THE OBSERVABLE PRESTELLAR PHASE OF THE IMF
Paolo Padoan
ICREA & ICC, University of Barcelona, Marti i Franqu`es 1, E-08028 Barcelona, Spain; [email protected] ˚Ake Nordlund
Centre for Star and Planet Formation and Niels Bohr Institute, University of Copenhagen, Juliane Maries Vej 30, DK-2100,Copenhagen, Denmark; [email protected]
Draft version May 29, 2018
ABSTRACTThe observed similarities between the mass function of prestellar cores (CMF) and the stellar initialmass function (IMF) have led to the suggestion that the IMF is already largely determined in the gasphase. However, theoretical arguments show that the CMF may differ significantly from the IMF. Inthis Letter, we study the relation between the CMF and the IMF, as predicted by the IMF modelof Padoan and Nordlund. We show that 1) the observed mass of prestellar cores is on average a fewtimes smaller than that of the stellar systems they generate; 2) the CMF rises monotonically withdecreasing mass, with a noticeable change in slope at approximately 3-5 M ⊙ , depending on meandensity; 3) the selection of cores with masses larger than half their Bonnor-Ebert mass yields a CMFapproximately consistent with the system IMF, rescaled in mass by the same factor as our model IMF,and therefore suitable to estimate the local efficiency of star formation, and to study the dependenceof the IMF peak on cloud properties; 4) only one in five pre-brown-dwarf core candidates is a trueprogenitor to a brown dwarf. Subject headings:
ISM: kinematics and dynamics — (MHD) — stars: formation — turbulence INTRODUCTION
Molecular clouds (MCs) undergo a highly non-linearfragmentation process, even prior to the emergence ofyoung stars, or in regions with no apparent star for-mation. The fundamental reason for their fragmenta-tion is the presence of supersonic turbulence that orig-inates at large scales from various sources, such as su-pernovae, spiral-arm shocks, or magneto-rotational in-stability. The formation of dense filaments and coresis the natural evolution of the intersection of ran-domly driven shocks in the turbulent flow. Even with-out self-gravity, turbulence simulations with the samerms Mach numbers as MCs generate density contrastsof many orders of magnitude, with characteristic sizeand density of both filaments and cores as observedin MCs (Arzoumanian et al. 2011). Because all starsare born in MCs, and specifically from dense coresfound predominantly within the densest filaments (e.g.Andr´e et al. 2010; Arzoumanian et al. 2011), the frag-mentation by the turbulence must control the earlyphases of the star formation process. It may also con-trol its global properties, such as the star formation rate(Krumholz & McKee 2005; Padoan & Nordlund 2011)and the mass distribution of stars (Padoan & Nordlund2002; Hennebelle & Chabrier 2008), but other processes(e.g. stellar jets and outflows, disk fragmentation, com-petitive accretion, ambipolar drift) can also affect themass and formation rate of stars (e.g. Adams & Fatuzzo1996; Bonnell et al. 2001).Most (sub-)mm or extinction surveys of star-forming regions have resulted in prestellar core massfunctions (CMFs) interpreted to be consistent withthe stellar IMF (Motte et al. 1998; Testi & Sargent1998; Johnstone et al. 2000, 2001; Motte et al. 2001; Onishi et al. 2002; Johnstone et al. 2006; Stanke et al.2006; Andr´e et al. 2007; Enoch et al. 2007, 2008;Belloche et al. 2011). A few studies have also claimed thedetection of a CMF peak, which has been interpreted asa scaled progenitor of the stellar IMF peak, with the scal-ing mass factor giving the local efficiency of star forma-tion (Alves et al. 2007; Nutter & Ward-Thompson 2007;Rathborne et al. 2009; Andr´e et al. 2010; K¨onyves et al.2010). However, the peak of the CMF is quite close tothe estimated completeness limits of the surveys, whichdepends also on the core shape and selection method (e.g.Kainulainen et al. 2009; Pineda et al. 2009).Simple theoretical considerations show that a one-to-one relation between the mass of observed prestellar coresand that of stars is unlikely. Prestellar cores must growfor some time before they can collapse and form stars.When a core is observed it is unlikely to have just reachedits final mass. Furthermore, the largest observable massof a core may be even lower than the full prestellar massinvolved. Once a core mass grows beyond its Bonnor-Ebert (BE) mass (Bonnor 1956; Ebert 1957), the corerapidly collapses within about a free-fall time, but theaccretion flow that was assembling that core is likely tostill be active, and to continue to bring additional mass tothe new protostar. The prestellar core as such is gone,but the prestellar mass feeding the new protostar maykeep coming. Note that this process is essentially iner-tial , and therefore distinctly different from competitive accretion, which assumes the flow is caused by gravita-tion forces.Clark et al. (2007) have stressed a timescale problemin relating the CMF to the stellar IMF: if cores of dif-ferent masses are assumed to all contain one BE mass,small cores must be denser and free-fall more rapidly Padoan and Nordlundthan larger ones. If the CMF is stationary in time,the corresponding stellar IMF should then be steeperthan the CMF. However, the observations do not show astrong correlation between core mass and density, so thistimescale issue may not be the main problem in deriv-ing the IMF from the CMF. In this Letter, we study therelation between the CMF and the stellar IMF based onour IMF model (Padoan & Nordlund 2002). RANDOM REALIZATIONS OF THE IMF MODEL
The IMF model by Padoan & Nordlund (2002) (PN02hereafter) predicts the mass distribution of gravitation-ally unstable cores generated by a turbulent flow. Themass of a core is the total mass, m accr , the turbulentflow would assemble locally, irrespective of when thecore should collapse and cease to appear as prestellar.In PN02, the mass distribution of all cores (unstable ornot) is a power law with slope x = 3 / (4 − b ), where b isthe slope of the velocity power spectrum of the turbu-lent flow, E ( k ) ∼ k − b . The Salpeter slope, x = 1 .
35, isthus recovered if b = 1 .
78, consistent with power spectraderived from the largest simulations of supersonic andsuper-Alfv´enic turbulence and from observations (e.g.Padoan et al. 2006, 2009). Unstable cores are then se-lected as those more massive than their BE mass, as-suming all cores have the same temperature, while theirexternal density (the local postshock density) follows theLog-Normal pdf of supersonic turbulent flows. Most ofthe massive cores contain more than one BE mass, so theIMF above approximately one solar mass is predicted tobe a power law with slope close to Salpeter’s. Smallercores are usually less massive than their BE mass, so theIMF is expected to peak at a fraction of a solar mass,and to decline towards smaller masses.This assumes that the actual stellar mass is m ⋆ = ǫ m accr , and the local efficiency, ǫ , is approximately in-dependent of mass. In observational studies, the localefficiency, ǫ core , is instead defined as the ratio of the re-sulting stellar mass, m ⋆ , and the current mass of prestel-lar cores, m , so ǫ core = m ⋆ /m . It follows that the ratioof the total core mass predicted by the model and thecurrent core mass can be expressed as the ratio betweenour theoretical local efficiency and the observational one, m accr /m = ǫ core /ǫ , and thus ǫ core /ǫ ≥ m accr , following the predictedpower law distribution with slope x = 1 .
35. We thengenerate a random value of external density for each core,according to the Log-Normal gas density distribution ofthe turbulent flow, converted to mass fraction, p (˜ ρ ) d ˜ ρ ∝ exp (cid:20) − (ln˜ ρ + σ / σ (cid:21) d ˜ ρ, (1)where ˜ ρ is the core external density in units of the cloudmean density, ˜ ρ = ρ/ρ , and the standard deviation, σ ,of the logarithmic density field is σ = ln " (cid:18) M S , (cid:19) (1 + β − ) − , (2)where M S , is the rms sonic Mach number, M S , = v /c S , , with v the three-dimensional rms velocity and c S , the isothermal sound speed corresponding to themean temperature T , and β is a characteristic ratioof gas to magnetic pressure in the postshock gas (seePadoan & Nordlund 2011, eqs. 16, 17, 18, 27 and 28).The corresponding standard deviation of the linear den-sity field, σ ˜ ρ , is given by σ ˜ ρ = (1 + β − ) − / M S , / . (3)Finally, we associate a random age to each core, assum-ing for simplicity that the star formation rate (SFR) isuniform over time and independent of core mass.In order to relate the PN02 model to observations ofprestellar cores (cores without a detectable embeddedprotostar), we also need to account for the time evolutionof cores, and define when, during its growth, a core wouldbe observed as prestellar. In PN02, cores are assumed tobe chunks of dense postshock filaments (or sheets), withsize equal to the postshock thickness, λ , and mass m accr = (4 / πρ ( λ/ , (4)where ρ is the postshock density. Assuming that a com-pression from the scale ℓ has a shock velocity that followsthe second order velocity structure function of the tur-bulent flow, v ∼ ℓ a (with a = ( b − / t accr = ℓ/v = t ( ℓ/L ) − a , (5)where t is the cloud crossing time, defined as the ratioof cloud size, L , and rms velocity, v , t = L /v . As-suming that the postshock thickness, λ , grows linearlywith time, we can then model the mass evolution of aprestellar core, up to the time t = t accr , when the totalcore mass, m accr , is reached, with the simple law m ( t ) /m accr = ( t/t accr ) , (6)which shows that cores must spend a significant fractionof their lifetime at a mass significantly lower than theirfinal mass, and therefore the relation between the massesof prestellar cores and stars cannot be trivial.In order to express t accr and m as a function of m accr ,instead of ℓ , we can relate m accr and ℓ using equa-tion (4) with the shock jump conditions, ρ = ρ M A ,ℓ and λ = ℓ/ M A ,ℓ , assuming that the shock Alfv´enic Machnumber scales like the shock velocity, M A ,ℓ ∼ v ∼ ℓ a ,and with the effective rms Alfv´enic Mach number ex-pression of Padoan & Nordlund (2011) (equations (13)and (14)), M A , = σ ρ . We then obtain the followingexpression for t accr expressed as a function of m accr t accr = t σ − a − a ˜ ρ (cid:18) m accr m (cid:19) − a (3 − a ) . (7)We assume that cores that do not reach their BE massare seen only during their formation time, t accr , whilethose growing past their BE mass continue to be observedas prestellar for one free-fall time, reaching a maximumprestellar mass, m max , given by m max m accr = (cid:18) t BE + t ff t accr (cid:19) , (8) Fig. 1.—
Ratio of the total core mass and the current coremass, m . Dots are for cores that will never collapse into stars,plus symbols for cores that form stars and can reach m accr whileprestellar, m f = m accr < m max , and diamond symbols for thosethat form stars but do not reach the mass m accr while prestellar, m f = m max < m accr . where t BE is the time when cores reach their BE mass, t BE = t accr (cid:18) m accr m BE (cid:19) − / , (9)and the BE mass (Bonnor 1956; Ebert 1957) is m BE = 1 . c / ( G / ρ / ) , (10)where c S is the isothermal sound speed in the cores, cor-responding to the mean core temperature T , ρ the post-shock density (assumed to be the external density of theBE sphere), G the gravitational constant, and t ff is thefree-fall time, t ff = (3 π/ (32 Gρ )) / . The value of m max may be larger than m accr (if t BE + t ff > t accr ), and there-fore the final (observable) mass of a prestellar core assuch is m f = min[ m accr , m max ] . (11) RESULTS
We show results for a model with characteristic molec-ular cloud (MC) parameters, T = 10 K, T = 7 K(the core mean temperature), L = 10 pc, ρ = 2 × − g/cm , M S , = 25, β = 0 .
4. The total MC massis then m = 1 . × M ⊙ , and its three-dimensionalrms velocity v = 4 . t = L /v = 1 . t = t , assumingthat prestellar cores are detected above a minimum sur-face density of N det = 10 cm − , corresponding to theHerschel 5- σ detection limit due to cirrus noise in Aquila(Andr´e et al. 2010; K¨onyves et al. 2010). We generatea random distribution of total core masses, m accr , withprobability following a power law with slope x = 1 . ⊙ . The total mass inthe cores is 0 . m , while the total mass of those thatcollapse into stars gives a final star formation efficiencyof SFE f =0.05. At the time we study the core popula-tion, t = t , the total mass in stars (defined as all thecores that have reached their total mass, m accr ) is suchthat SFE( t ) = 0 .
02, which is a reasonable value for MCs
Fig. 2.—
Histograms of the ratio m accr /m shown in Figure 1.The solid line is the histogram for all cores with mass m > . ⊙ ,yielding a mean value of h m accr /m i = 5 .
0. The dashed line inFigure 2 is the histogram including only core masses above halftheir BE mass, m > m BE /
2, yielding a mean value of h m accr /m i =2 . (e.g. Evans et al. 2009). The core formation efficiency at t = t is CFE( t )=0.01, also consistent with observations(e.g. Enoch et al. 2007).Figure 1 shows the ratio between the total core massand its current mass at time t = t , m accr /m = ǫ core /ǫ .The plot shows that prestellar cores may have a masssignificantly smaller than the mass of the stellar systemthey will form, assuming reasonable values of ǫ , whichmeans that ǫ core >
1, contrary to the usual assumptionof observational studies. Observed prestellar cores withmasses between 0.1 and 1.0 M ⊙ , for example, may be ontheir way to form stellar systems with masses 10 timeslarger.The relation between current core mass and totalcore mass is quantified by the histograms of the ratio m accr /m , shown in Figure 2. The solid line is the his-togram for all prestellar cores with mass m > . ⊙ ,showing a broad distribution, with a mean value of h m accr /m i = 5 .
0. Cores that will never grow above theirBE mass to form stars ( m accr < m BE ) are shown asdots in Figure 1. These cores are not included in thehistogram, but they certainly contaminate observationalsamples.The dashed line in Figure 2 is the histogram includingonly core masses above half their BE mass, m > m BE / h m accr /m i =2 .
6. The ratio of the peak of the stellar IMF from ourmodel (see Figure 4 below) and the peak of the multiplesystem IMF of Chabrier (2005) is approximately 2.1, cor-responding to ǫ ≈ .
48. With this value of the theoreticallocal efficiency of star formation, the average ratio be-tween the actual mass of stellar systems, m ⋆ , and the cur-rent mass of cores, m , would be ǫ core = ǫ h m accr /m i = 2 . m > . ⊙ , and ǫ core ≈ . m > m BE / t = t . It shows that a significant frac-tion of the cores with m > . ⊙ may be found tohave a mass larger than their BE mass. It also showsthat the selection of cores with mass m > m BE / Fig. 3.—
Ratio of current core mass to BE mass, plotted versusthe current core mass. The solid line shows the ratio between thenumber of cores that will never collapse and form stellar systemsand that of true prestellar cores, after selecting cores with m >m BE / lapse into stars, while still providing a large enough coresubsample to allow a meaningful estimation of the CMF.This strategy is already being implemented in the anal-ysis of observational surveys (e.g. K¨onyves et al. 2010).The solid line in Figure 3 gives the ratio between thenumber of non-prestellar cores and that of true prestel-lar cores (computed from five different realizations of thesame model), as a function of core mass, for cores withmass m > m BE /
2. Around 1 M ⊙ , only 16% of thesecores are not prestellar, while that fraction grows to 35%around 0.1 M ⊙ .Smaller cores that may be progenitors of brown dwarfscan barely exceed their BE mass (towards the end of theirlifetime). Only one in five of the selected pre-brown-dwarf core candidates with mass m BE / < m < .
15 M ⊙ (assuming ǫ ≈ .
48) is a true pre-brown-dwarf core. Halfof them are growing progenitors of higher mass stars,and one third are non-prestellar cores. Distinguishingtrue pre-brown-dwarf cores or non-prestellar cores in ob-servational surveys is a difficult task that may requirethe characterization of such cores using synthetic obser-vations of turbulence simulations.The CMF at the time t = t is shown in Figure 4by the black histogram (and by the continuous red linefor the average of five different realizations of the samemodel). The CMF seems to change its slope at two massvalues, m ≈ . ⊙ (approximately the peak of themodel IMF) and m ≈ ⊙ . For masses above m , theslope is steeper than x = 1 .
35, it becomes shallower thanthat for masses below m , and it steepens again below m , but still remaining a bit shallower than x = 1 . m is presumably very dif-ficult to derive from observations, because core samplesare generally incomplete at such low masses, dependingon the survey sensitivity, confusion, and method of coreselection.We have verified that the values of m and m scaleapproximately as ρ / ( m remains close to the peakof the model IMF), while they are approximately thesame in models where equation (6) is modified to assume m ( t ) ∼ t , and in models where cores are assumed to re-main prestellar for only a fraction f of t ff after they reach Fig. 4.—
Black histogram: CMF at time t = t . Blue histogram:CMF for a prestellar core subsample with masses m ( t ) > m BE / t = 2 t . Dashed lines: Chabrier (2005) system IMFs,shifted in mass by a factor of 2.1. their BE mass. However, with decreasing values of f , theslope of the CMF for masses in the range m < m < m increases, becoming approximately the same as the slopefor m < m when f ≈ .
6, and approximately equal to1.35 when f ≤ . t = 2 t , when all coreshave reached their total mass, m accr , and have alreadycollapsed into stars (with SFE f = 0 . ⊙ ). This shift inmass corresponds to ǫ = 0 . m > m BE /
2. Thisis illustrated in Figure 4 by the CMF of such coresubsample (blue histogram), which is fit by a Chabrier(2005) system IMF with masses shifted by a factor of2.1 (dashed line), except for the largest masses, wherethe slope is steeper than Salpeter’s. This CMF peaksat the same mass as the model system IMF, and thusmay be used to estimate the value of ǫ (because in thiscase ǫ core ≈ ǫ , based on the peaks alone). Its peak couldalso be used to study possible variations of ǫ with cloudproperties. CONCLUSIONS
We have computed the observable CMF predicted bythe PN02 IMF model, assuming characteristic MC pa-rameters that are shown to yield a Chabrier (2005) sys-tem IMF with ǫ ≈ .
48. Our main results are the fol-lowing: 1) The observed mass of prestellar cores is onaverage a few times smaller than that of the stellar sys-tems they generate, so ǫ core >
1. 2) The CMF risesmonotonically with decreasing mass, and shows a no-ticeable change in slope at approximately 3-5 M ⊙ , de-pending on mean density; 3) the selection of cores withmasses m > m BE / ǫ , and itspossible dependence on cloud properties; 4) Only one infive pre-brown-dwarf core candidates is a true progenitorto a brown dwarf.We have not discussed the proto stellar CMF, nor thesystem IMF at time t = t (when relatively massive pro-tostars are still growing in mass). The definition of theprotostellar phase faces the difficulty of accounting forthe rate of mass transfer between the core and the proto-star (McKee & Offner 2010). However, we have verifiedthat with a loose definition of protostars as all the coresthat have reached their final prestellar mass, m max , butnot yet their total mass, m accr (core ages between t BE + t ff and t accr ), the protostellar CMF is significantly shallowerthan the prestellar CMF, in agreement with the obser-vations (e.g. Hatchell & Fuller 2008; Enoch et al. 2008),and the current system IMF has a high-mass slope a bitsteeper than Salpeter’s (not observationally tested yet,to our knowledge), due to the fact that more massivecores remain longer in the protostellar phase than lowermass ones.Upcoming starless core samples from the HerschelGould belt survey (Andr´e et al. 2010) will allow com-parisons with our model predictions. In this Letter wehave only discussed the results of a very large core sampleand a specific set of cloud parameters, without simulat-ing observational uncertainties and incompleteness, andthus setting aside a detailed comparison with observedCMFs for a separate work.We thank Alyssa Goodman for reading the manuscriptand providing comments, and the anonymous referee foruseful comments and corrections. PP is supported by theSpanish MICINN grant AYA2010-16833 and by the FP7-PEOPLE-2010-RG grant PIRG07-GA-2010-261359. Thework of ˚AN is supported by the Danish National Re-search Foundation, through its establishment of the Cen-tre for Star and Planet Formation.), the protostellar CMF is significantly shallowerthan the prestellar CMF, in agreement with the obser-vations (e.g. Hatchell & Fuller 2008; Enoch et al. 2008),and the current system IMF has a high-mass slope a bitsteeper than Salpeter’s (not observationally tested yet,to our knowledge), due to the fact that more massivecores remain longer in the protostellar phase than lowermass ones.Upcoming starless core samples from the HerschelGould belt survey (Andr´e et al. 2010) will allow com-parisons with our model predictions. In this Letter wehave only discussed the results of a very large core sampleand a specific set of cloud parameters, without simulat-ing observational uncertainties and incompleteness, andthus setting aside a detailed comparison with observedCMFs for a separate work.We thank Alyssa Goodman for reading the manuscriptand providing comments, and the anonymous referee foruseful comments and corrections. PP is supported by theSpanish MICINN grant AYA2010-16833 and by the FP7-PEOPLE-2010-RG grant PIRG07-GA-2010-261359. Thework of ˚AN is supported by the Danish National Re-search Foundation, through its establishment of the Cen-tre for Star and Planet Formation.