Mark Krumholz

Class 14: The IMF: TheoryASTR 4008 / 8008, Semester 2, 2020

Outline• General considerations

• The IMF tail• Competitive accretion• Turbulent fragmentation

• The IMF peak• Setting the peak from the galactic scale• Setting the peak from small scales

Theory of the IMFGeneral considerations

• The IMF consists of two main parts: a broad plateau centred at ~0.2-0.3 M⊙, and a power law tail at high mass

• These features may vary slightly with environment, but, if so, by a surprisingly small amount — e.g., the slope may be ~0.2 flatter in the densest star clusters in the local universe, or the mass to light ratio in ellipticals may be a factor of ~2 higher than in spirals

• A successful theoretical model must explain both parts of the IMF, and also why they vary so little in response to many order of magnitude changes in density, redshift, galactic environment, etc.

Limits of isothermal modelsPart I

• Basic equations for a magnetised, self-gravitating, isothermal fluid:

• Define characteristic length L, velocity V, mean density 𝜌0, mean magnetic field B0, make change of variables:

<latexit sha1_base64="eG8t44Q2MoRL4Yh5rCmkgCrG4m4=">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</latexit>

x = x0L t = t0(L/V ) ⇢ = r⇢0 B = bB v = uV � = G⇢0L2

<latexit sha1_base64="RtdqBIzfRQ9SvlGQRq8oNXI7KQk=">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</latexit>

@⇢

@t= �r · (⇢v)

@

@t(⇢v) = �r · (⇢vv)� c2sr⇢+

1

4⇡(r⇥B)⇥B� ⇢r�

@B

@t= �r⇥ (B⇥ v)

r2� = 4⇡G⇢

Class exercise: write down the equations of motion with this change of variable. What

dimensionless ratios appear in your equations?

Limits of isothermal modelsPart II

• Non-dimensional equations are:

• The dimensionless quantities appearing in these equations are:

• The evolution is entirely determined by these quantities + the initial conditions

<latexit sha1_base64="aWF+n3cHDJzzZYcH5TmZfYZA+m0=">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</latexit>

@r

@t0= �r0 · (ru)

@

@t0(ru) = �r0 · (ruu)� 1

M2r0r +

1

M2A

(r0 ⇥ b)⇥ b� 1

↵virrr0�

@b

@t0= �r0 ⇥ (b⇥ u)

r02 = 4⇡r

Derivative with respect to x’

<latexit sha1_base64="n1xmYmlR/roMSJwhId7X21o+LmQ=">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</latexit>

M = V/cs MA = V/VA = Vp4⇡⇢0/B0 ↵vir = V 2/G⇢0L

2

Limits of isothermal modelsPart III

• Suppose a star forms in an isothermal cloud, with the material going into the star coming from a volume bounded by some initial surface S at time t = 0. The mass of the star is

• Now consider an identical cloud that has just been rescaled by a factor y, with

• One can verify by direct substitution that the rescaled cloud has exactly the same values of , so the evolution is identical — same star forms

• However, the resulting star has mass , so the mass of the star is not the same — it can be changed arbitrarily by the choice of y

<latexit sha1_base64="nMIhIZkvWVYAY8jvo4kJXW/dFGo=">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</latexit>

M⇤ =

Z

S⇢ d3x = ⇢0L

3

Z

Sr d3x0

<latexit sha1_base64="f6mo82gMix6e0skFmj+97it2YVw=">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</latexit>

L0 = yL ⇢00 = ⇢0/y2 B0

0 = B0/y V 0 = V

<latexit sha1_base64="HQcsHoqmN9E9jCw/GREJ/Y9K5lI=">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</latexit>

M,MA,↵vir<latexit sha1_base64="DLnQUDEFNmscYSUgUeD0mKK8q7k=">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</latexit>

M 0⇤ =

⇢00L03

⇢0L3M⇤ = yM⇤

Limits of isothermal modelsImplications

• The conclusion to draw is that isothermal gas does not have a mass scale — the characteristic masses of any stars it forms are not set by the physics of the cloud, but by the initial conditions

• Thus isothermal gas is capable of producing stars with a scale free (powerlaw) mass distribution, but if it produces a distribution that has a scale (like the IMF), that scale has to be set by the initial conditions

• This realisation motivates separate consideration of the IMF tail (which can potentially arise in purely isothermal gas) from the IMF peak (which must depend either on the initial conditions in GMCs, or on deviations from isothermal behaviour)

Numerical confirmation5076 D. Guszejnov et al.

Figure 1. Surface density maps from a simulation of a 2 × 105 M⊙ GMC that includes isothermal turbulence and MHD (M2e5 R30 , see Table 1), at about8 per cent star formation efficiency. The colour scale is logarithmic and the circles represent sink particles (stars) that form in high-density regions wherefragmentation can no longer be resolved, their size increasing with mass. This simulation resolves a dynamic range from ∼50 pc to ∼30 au.

3 R ESULTS

We carried out a suite of simulations in the αturb,0–M–µ parameterspace at various resolutions, up to M0/"m = 2 × 108 (see Table 1 fordetails and Fig. 1 for a demonstration of the dynamic range). Thisis the highest mass resolution yet achieved in any 3D simulation ofresolved star cluster formation.

Once the simulation begins, we find that the clouds quickly developa filamentary structure similar to observations (Andre et al. 2010) thatcollapses and forms stars (see Fig. 2). Fig. 3 shows that all our cloudsturn roughly 10 per cent of their gas into stars in a freefall time. Atlow Mach numbers (M < 10), we find a rough trend of SFE ∝ t2

(consistent with the results of Lee, Chang & Murray 2015 whosimulated a M = 9 cloud), while for all highly supersonic clouds(M > 10) the relation becomes steeper, consistent with SFE ∝ t3.This does not necessarily contradict the theory of Murray & Chang(2015), who derived M⋆ ∝ t2 for a single star accreting in a turbulentmedium – our star formation history is the sum of many individualstellar accretion histories.

3.1 Sink mass distribution (IMF)

Fig. 4 shows that varying the initial conditions (in this case thevirial parameter αturb,0 and Mach number M) significantly changesthe mass distribution of sink particles. At high masses, the sinkdistribution is consistent with a dN/dlog M ∝ M−1 power law, similarto the observed IMF (Salpeter 1955; Offner et al. 2014). Meanwhile,at low masses, the distribution becomes shallower, consistent withdN/dlog M ∼ const. This is significantly shallower than the low-mass end of the observed IMF (dN/dlog M ∼ M0.7 in the Kroupa2002 form), leading to an excess of brown dwarfs, which shouldonly make up ∼ 30 per cent of the stellar population (Andersenet al. 2006). Meanwhile, the turnover from the high-mass power-

law behaviour shows that the sink mass distribution does have amass scale inherited from initial conditions. For simplicity, we adoptthe mass-weighted median mass of sinks M50 as the characteristicmass scale of sinks in our subsequent analysis (similar to Krumholz,Klein & McKee 2012), as it roughly corresponds to this turnovermass (see Fig. 4). This characteristic mass M50 monotonicallyincreases as more gas is turned into stars (see Figs 5 and A2 forvalues).

3.2 Effects of turbulent driving and boundary conditions (Boxversus Sphere)

While the global parameters of the initial conditions (αturb,0, M, M0)affect the mass spectrum of sink particles, we find no significantdifference between Sphere and Box runs (see Fig. 5), despite thedifference in initial cloud shape, turbulent driving, density, andmagnetic fields.34 The insensitivity of the sink mass spectrum tothe specifics of the initial conditions is similar to the findings of Bate(2009b), Liptai et al. (2017), and Lee & Hennebelle (2018a).

Note that studies simulating dense, centrally concentrated cloudsfound that the final sink masses depend on the initial condition(Girichidis et al. 2011). These initial conditions, however, are quitedifferent from what is observed in GMCs. Furthermore, Girichidiset al. (2011) simulated isothermal turbulence without magnetic fields,

3It should be noted that while the exact magnitude of magnetic support onlarge scales appears to be irrelevant, having finite (non-zero) magnetic fieldsis crucial because, in the limit of no magnetic fields, clouds undergo aninfinite fragmentation cascade, see Section 3.4 and Guszejnov et al. (2018b)for details.4Note that we use αturb,0 based on equation (12) similar to other studies inthe literature. For a periodic box, this can significantly differ from the valueαturb from equation (2) (Federrath & Klessen 2012).

MNRAS 496, 5072–5088 (2020)

Dow

nloaded from https://academ

ic.oup.com/m

nras/article/496/4/5072/5866504 by R.G

. Menzies Library, B

uilding #2, Australian N

ational University, M

ark Krum

holz on 17 Septem

ber 2020

Can MHD set the mass scale of stars? 5081

Figure 10. Mass PDF of sink particles at 5 per cent SFE for the M2e5 R30run (M0 = 2 × 105 M⊙, αturb, 0 = 2, M = 29, see Table 1) at various massresolutions (M0/"m). Note that unlike Fig. 4, here we plot the full rangeof sink particle masses. We shaded the region where non-isothermal effectsare expected to suppress the formation of new sinks (Bate 2009a; Offneret al. 2009; Lee & Hennebelle 2018b) and mark the brown dwarf regime(M < 0.08 M⊙, dashed line). While the high mass end (that contains mostof the mass) is insensitive to resolution (see Fig. 8), a resolution sensitivepeak forms near the resolution limit for M0/"m → ∞, leading to an excessof brown dwarfs.

scale both the base and maximum AMR resolution levels to achieveconvergence.

Note that while it only contains a small fraction of the total IMFmass in these simulations, the low-mass end of the IMF is clearlynot converged and depends strongly on resolution in our simulations.Plotting the full IMF as a function of resolution in Fig. 10 we seethat the ‘brown dwarf excess’ predicted by ideal MHD physics alonebecomes more severe as our resolution increases. So we emphasizethat our conclusions about M50 and resolution-independence applyonly to the relatively large masses containing most of the mass inthe IMFs here. Note that it is unclear if this would still be trueat much higher mass resolutions (M0/"m ∼ 1010), but probingthat regime is prohibitively expensive with our current code. Wefind that this large number of very low mass sinks originate fromdense regions around massive stars. Note that in these regions ourassumption of isothermality is expected to break down, preventingfurther fragmentation in the gas and the formation of this ‘browndwarf excess’ (for discussion see Section 4.3.1). Furthermore, wefind that this region of the IMF is sensitive to the details of ourangular momentum return algorithm, but the conclusions of our studyis not.

4 D ISCUSSION

4.1 Comparison with other simulation studies

There have been several studies in recent years that investigatedthe sink particle mass spectrum in simulations including MHDturbulence and gravity. In Table 2, we apply our fitting functions fromequations (17) and (21) to the initial conditions of their simulations

and compare them with the mass-weighted median and maximumsink mass in their reported IMFs. Haugbølle et al. (2018), Leeet al. (2019), and Federrath et al. (2017) all used a simulationset-up essentially identical to our Box simulation suite, simulatingisothermal MHD with gravity and sink particles with the RAMSES,ORION2, and FLASH codes, respectively. Compared to ours, thesestudies have subtle differences in the details of turbulence driving,but our results suggest these are unlikely to strongly affect the IMF(Fig. 5).

First, we compare with Haugbølle et al. (2018). Most of thesesimulations included a prescription to model protostellar outflows,by having sink particles accrete only half of the inflowing mass anddelete the rest, so we compare with the IMF from their acc testrun that does have this prescription (their fig. 14). We find that ourpredicted M50 = 7.5 M⊙ and M⋆, max = 19 are quite close to theirvalues of 4.2 M⊙ and 17 M⊙, both <2σ compatible if we estimateerrors by bootstrapping their mass distribution and taking the RMSerror of our fit. We find even better agreement with the values in Leeet al. (2019).

Our prediction for M⋆,max matches the results of the HighResIsosimulation in Federrath et al. (2017), but for those initial conditionswe predict M50 = 11 M⊙, much greater than their M50 = 1.9 M⊙.This simulation produced 23 objects of mass > 1 M⊙, so while thesampling of the IMF is certainly sparse, the numbers are not so smallthat we can readily attribute a factor of ∼5 discrepancy to statisticalvariations. One difference between our respective calculations is thatthey used a mixture of compressive and solenoidal driving, versusthe purely solenoidal driving used in our BOX simulations. Howevergiven the robustness of our results to the details of turbulent forcing,this is unlikely to strongly affect the result either. We are left with noclear explanation for the discrepancy.

Wurster et al. (2019) simulated a 50 M⊙ dense clump akin toour Sphere suite, with both ideal and non-ideal smoothed-particleradiation MHD; we compare with their µ = 5, ideal MHD model,but note that they found that the IMF is not strongly affected by µ ornon-ideal MHD effects. Our predictions of M50 ∼M⋆,max ∼1 M⊙agrees very well with their results. As such, while it has been shownthat accounting for full radiation transfer is important for suppressingbrown dwarf formation (Bate 2009a; Offner et al. 2009), isothermalMHD may be a sufficient approximation to predict M50 and M⋆,max.

Finally, we compare with Padoan et al. (2019), who ran a 250 pcBox-type set-up containing 1.9 × 106 M⊙, but with turbulence drivenby supernova explosions. We derive approximate RMS M andαturb,0 values of 66 and 4.7, respectively, from the energy statisticsgiven in Padoan et al. (2016), however we emphasize that theseare rough values because (1) their ISM is not isothermal but rathermultiphase, (2) the energetics are highly variable, and (3) the resultsin Padoan et al. (2019) are from a different, higher-resolutionsimulation with the same physical parameters. Nevertheless wepredict M50 = 36 M⊙, within a factor of 2 of their value of ∼20 M⊙.They attribute this overprediction of the IMF turnover to a lack ofnumerical resolution, but our results suggest that they are actuallyclose to the ‘converged’ value. Rather, we believe other, importantprocesses that shape the IMF were neglected, as we will argue furtherin this section.

In summary, we find that our simulations predict M50 and M⋆,max invery good agreement with the predictions of other codes running sim-ilar problems, with the exception perhaps of the FLASH simulationsin Federrath et al. (2017). Whether this represents any meaningfuldifference in code behaviours, or sensitivity to prescriptions, canultimately only be answered by a controlled code comparison study(e.g. Federrath et al. 2010b). Overall the good agreement between the

MNRAS 496, 5072–5088 (2020)

Dow

nloaded from https://academ

ic.oup.com/m

nras/article/496/4/5072/5866504 by R.G

. Menzies Library, B

uilding #2, Australian N

ational University, M

ark Krum

holz on 17 Septem

ber 2020

Guszejnov+ 2020

The power law tailBasic approaches

• Two natural candidate scale-free processes:• Accretion of gas by stars that form near the IMF peak — “competitive

accretion”• Collapse of a distribution of structures with different masses that are

created by turbulent flow — “turbulent fragmentation”

• Reality is almost certainly somewhere between these two extremes, but they provide useful ways of thinking about the problem

The competitive accretion modelKey papers: Bonnell+ (1997), Bonnell+ (2001), Bate+ (2005), Bate (2009)

• Consider a population of “seed” stars formed by fragmentation, all with similar initial masses, that gain mass by accreting additional gas

• Suppose accretion rate depends on current mass as dm/dt ~ m𝜂; for example, for a point mass accreting gas from a uniform, infinite medium, Bondi & Hoyle showed that this holds with 𝜂 = 2

• Mass some time later is

• Suppose all stars start at mass m0, but have different starting accretion rates (e.g. because they are in gas of different density) or accrete for different amounts of time (e.g. because some form earlier) — what is resulting IMF?

<latexit sha1_base64="QLX4agl0Xye9KFdHhDQk0vo+Jnw=">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</latexit>

m(t) = m0

⇢[1� (⌘ � 1)⌧ ]1/(1�⌘), ⌘ 6= 1exp(⌧), ⌘ = 1

, where ⌧ = tm0/m0

Starting massStarting accretion rateDimensionless time parameter

Competitive accretionPredicted IMF

• Varying initial accretion rate or time corresponds to varying 𝜏

• For a distribution of 𝜏 values dn / d𝜏, resulting mass distribution is

• Thus accretion converts an initial 𝛿-distribution in mass (all stars at mass m0) into a power law distribution

• BH accretion (𝜂 = 2) is a bit too shallow compared to observed IMF, but in a crowded environment likely to have somewhat larger 𝜂 due to tidal truncation of feeding zones — plausible explanation for IMF slope

<latexit sha1_base64="C9pXYbpnBi8nmVdzGcgHL4LGVKY=">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</latexit>

dn

dm=

dn/d⌧

dm/d⌧=

✓dn

dt

◆m�⌘

Turbulent fragmentationKey papers: Padoan & Nordlund (2002), Hennebelle & Chabrier (2008), Hopkins (2012)

• Basic idea: consider density field smoothed at some size scale 𝓁, and imagine drawing iso-density contours at density 𝜌

• Define mass m enclosed by each contour — typically most numerous contours will be as small as possible on the smoothing scale, so m ~ 𝜌𝓁3

• What is the distribution of m values? Can estimate from the density PDF: total mass dM of objects with density from 𝜌 to 𝜌 + d𝜌 is dM ~ 𝜌 p(𝜌) d𝜌

• Thus number of contours of mass m must follow<latexit sha1_base64="kS6+rWI+iejMIKyl/WA8O5VZ3C0=">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</latexit>

dn

dm=

dM

m⇠ 1

`3

Zp(⇢) d⇢

Turbulent fragmentationThe barrier function

• Not all of the identified contours will be gravitationally bound, so not all will make stars; IMF will come from distribution of bound contour masses

• Contour of mass m, size scale 𝓁 is bound if Gm2 / 𝓁 ≳ m𝜎(𝓁)2

• Can express this condition in terms of a minimum density: 𝜌 ≳ 𝜎(𝓁)2 / G𝓁2

• Using LWS to get 𝜎(𝓁) = cs (𝓁 / 𝓁s)1/2: 𝜌 ≳ cs2 / G𝓁𝓁s

• This sets lower limit to integral over PDF; integrate over all smoothing scales 𝓁, and, given model for smoothed PDF, calculate resulting IMF; result is a power law

The IMF peakBasic approaches

• Mass scale of IMF peak must be set by two possible mechanisms:• “Initial conditions”, i.e., properties of molecular clouds at the Galactic scale• Deviations from isothermal behaviour caused by additional physical

processes

• Numerous papers published using both approaches, but consensus seems to be moving toward deviations from isothermality, for reasons that will become clear as we go through the arguments

The peak from initial conditionsKey papers: Bate+ (2005), Hennebelle & Chabrier (2008), Hopkins (2012)

• Simplest hypothesis is that peak mass set by Jeans mass in molecular cloud at largest scale: Mpeak ~ cs3 / (G3𝜌)1/2

• Problem: mean density 𝜌 in molecular clouds varies by a lot from galaxy to galaxy, and even within galaxies• Example: mean GMC density near Sun is n ~ 100 cm−3, but near MW centre

n ~ 104 cm−3 • Predicted change in Mpeak by factor of ~10 — ruled out by observations

• Can be partly compensated by change in cs, but requires considerable fine-tuning to explain the small amount of observed variation in IMF peak mass

The peak from initial conditionsRevised version

• More refined hypothesis: peak mass set by sound speed and sonic length, Mpeak ~ cs2𝓁s / G

• Can rewrite in terms of dimensionless parameters using LWS plus definition of virial parameter, 𝜎(L) = cs (L / 𝓁s)1/2 and 𝛼vir ~ 𝜎2L / GM → Mpeak ~ cs4 / 𝛼virG𝛴2

• Advantage of this approach: predicts constant peak mass in GMCs of fixed surface density and virial parameter, so predicts no variation in Milky Way

• However, still has problems with external galaxies where GMCs can have (much) higher 𝛴, but no big changes in peak mass are observed

The IMF peak from non-isothermalityKey papers: Larson (2005), Krumholz+ (2006, 2011, 2012), Bate (2009, 2012)

• Jeans mass depends on density and temperature as MJ ~ (T3/𝜌)1/2, so as long as T ~ constant, Jeans mass decrease without limit as gas compresses

• However, if T starts to increase as 𝜌 decreases, fragmentation can be slowed or halted (depending on how strongly T rises with 𝜌)

• Thus any physical process that makes T start increasing at some characteristic density 𝜌 can “freeze in” a mass scale at the value of MJ evaluated at the (𝜌, T) where the temperature increase starts

The Astrophysical Journal, 754:71 (18pp), 2012 July 20 Krumholz, Klein, & McKee

Figure 8. Same as Figure 7, but showing differential rather than cumulative mass distributions. The histogram value in each bin shows the total fraction of all stellarmass (for the top panels) or the total fraction of the number of stars (for the bottom panels) falling within that bin. Thick colored lines indicate the simulation result,and gray lines indicate the results of drawing an equal mass stellar population from the Da Rio et al. (2012) IMF. For the gray histogram, the histogram values givethe median result, and the vertical lines indicate the range from the 10th to the 90th percentile. We omit the Chabrier (2005) IMF here to reduce clutter.(A color version of this figure is available in the online journal.)

10

The Astrophysical Journal, 754:71 (18pp), 2012 July 20 Krumholz, Klein, & McKee

Figure 8. Same as Figure 7, but showing differential rather than cumulative mass distributions. The histogram value in each bin shows the total fraction of all stellarmass (for the top panels) or the total fraction of the number of stars (for the bottom panels) falling within that bin. Thick colored lines indicate the simulation result,and gray lines indicate the results of drawing an equal mass stellar population from the Da Rio et al. (2012) IMF. For the gray histogram, the histogram values givethe median result, and the vertical lines indicate the range from the 10th to the 90th percentile. We omit the Chabrier (2005) IMF here to reduce clutter.(A color version of this figure is available in the online journal.)

10

Origin of non-isothermalityStellar radiation

• Main source of deviation from isothermal behaviour is radiation from forming stars

• Once a small star forms, accretion onto it releases energy, which comes out as radiation, and heats the surrounding gas

• Simulations including this effect reproduce observed IMF well

The Astrophysical Journal, 754:71 (18pp), 2012 July 20 Krumholz, Klein, & McKee

Figure 10. Phase diagrams of the three runs at different times. The three columns correspond to runs SmNW, TuNW, and TuW, as indicated. The three rows correspondto times t/tff = 0.75, 1.0, and 1.25. In each panel, the color indicates the gas mass in a given bin of density and temperature; bins are 0.025 dex wide in both ρ andT. The color scale is normalized so that the bin containing the largest amount of mass is 1.0. The long-dashed line indicates the locus in density and temperature atwhich the code inserts sink particles. The short-dashed lines indicate the locus in density and temperature where the Bonnor–Ebert mass is 0.01 M⊙, 0.1 M⊙, and1 M⊙ as indicated. Note that gas in the winds is run TuW is heated to ∼104 K, well above the temperature range shown here, but there is relatively little mass at thesetemperatures.(A color version of this figure is available in the online journal.)

or whether all stars are born from cores with masses !1 M⊙,and massive stars subsequently grow from these small seedsby Bondi–Hoyle accretion (Bonnell et al. 1997, 2001a, 2001b,2004, 2006; Bonnell & Bate 2002, 2006; Bate & Bonnell 2005;Smith et al. 2009a, 2009b). A number of authors have also pro-posed hybrid models, in which massive stars form from gravita-tionally bound gas structures, but these structures are assembledand fed from larger scales at the same time as they form mas-sive stars (Peretto et al. 2006; Wang et al. 2010). To addressthis question, we examine the four most massive stars presentat the end of run TuW; these have masses of 10.8, 9.8, 8.8, and8.3 M⊙, respectively, and thus each is large enough that, evenif it were to accrete no further, it would be expected to end itslife as a supernova. For comparison, we also examine the fourstars whose masses are closest to the median mass at the endof the simulation, 0.34 M⊙. For each of these stars, we identifythe point in space and time at which that star first appeared inour simulations, and examine the gas density distribution in itsvicinity.

We show the results in Figure 11 for the high-mass coresand Figure 12 for the low-mass cores. To facilitate comparisonwith observations, in addition to showing the true gas densitydistribution, we show the distribution smeared with a 1700 AUGaussian beam; we choose this size scale because it is approxi-mately the spatial resolution of the highest published resolution

maps of massive cores (e.g., Beuther & Schilke 2004; Bon-temps et al. 2010), though the Atacama Large Millimeter Array(ALMA) will soon produce images at significantly higher res-olution. Figure 11 demonstrates that the massive stars in oursimulation form in distinct, massive overdensities that can beidentified as cores. Their characteristic sizes, determined fromvisual inspection, are roughly 0.01 pc. Comparing the gravita-tional and kinetic energies in this structures shows that they areroughly gravitationally bound and virialized. The flows withinthem are highly supersonic, producing a filamentary morphol-ogy. Nonetheless, these objects are not highly sub-fragmented.There are at most one or two density maxima in each one, notmany density maxima. These structures look much like the tur-bulent cores posited in the McKee & Tan (2003) theory formassive star formation. When smeared on a resolution of 1700AU, distinct centrally condensed structures remain visible forthree of the four massive stars, indicating that these objectswould be detectable as massive cores in an observation.

It is important to understand that our analysis says nothingabout the Lagrangian trajectories of the fluid elements thateventually coalesce to form the massive stars in our simulations,a topic that has previously received extensive investigation byBonnell et al. (2004) and Smith et al. (2009a, 2009b), amongothers. It may well be that particular fluid elements that arepresent in the cores at the time shown in Figure 11 do not accrete

12

The Astrophysical Journal, 754:71 (18pp), 2012 July 20 Krumholz, Klein, & McKee

Figure 10. Phase diagrams of the three runs at different times. The three columns correspond to runs SmNW, TuNW, and TuW, as indicated. The three rows correspondto times t/tff = 0.75, 1.0, and 1.25. In each panel, the color indicates the gas mass in a given bin of density and temperature; bins are 0.025 dex wide in both ρ andT. The color scale is normalized so that the bin containing the largest amount of mass is 1.0. The long-dashed line indicates the locus in density and temperature atwhich the code inserts sink particles. The short-dashed lines indicate the locus in density and temperature where the Bonnor–Ebert mass is 0.01 M⊙, 0.1 M⊙, and1 M⊙ as indicated. Note that gas in the winds is run TuW is heated to ∼104 K, well above the temperature range shown here, but there is relatively little mass at thesetemperatures.(A color version of this figure is available in the online journal.)

or whether all stars are born from cores with masses !1 M⊙,and massive stars subsequently grow from these small seedsby Bondi–Hoyle accretion (Bonnell et al. 1997, 2001a, 2001b,2004, 2006; Bonnell & Bate 2002, 2006; Bate & Bonnell 2005;Smith et al. 2009a, 2009b). A number of authors have also pro-posed hybrid models, in which massive stars form from gravita-tionally bound gas structures, but these structures are assembledand fed from larger scales at the same time as they form mas-sive stars (Peretto et al. 2006; Wang et al. 2010). To addressthis question, we examine the four most massive stars presentat the end of run TuW; these have masses of 10.8, 9.8, 8.8, and8.3 M⊙, respectively, and thus each is large enough that, evenif it were to accrete no further, it would be expected to end itslife as a supernova. For comparison, we also examine the fourstars whose masses are closest to the median mass at the endof the simulation, 0.34 M⊙. For each of these stars, we identifythe point in space and time at which that star first appeared inour simulations, and examine the gas density distribution in itsvicinity.

We show the results in Figure 11 for the high-mass coresand Figure 12 for the low-mass cores. To facilitate comparisonwith observations, in addition to showing the true gas densitydistribution, we show the distribution smeared with a 1700 AUGaussian beam; we choose this size scale because it is approxi-mately the spatial resolution of the highest published resolution

maps of massive cores (e.g., Beuther & Schilke 2004; Bon-temps et al. 2010), though the Atacama Large Millimeter Array(ALMA) will soon produce images at significantly higher res-olution. Figure 11 demonstrates that the massive stars in oursimulation form in distinct, massive overdensities that can beidentified as cores. Their characteristic sizes, determined fromvisual inspection, are roughly 0.01 pc. Comparing the gravita-tional and kinetic energies in this structures shows that they areroughly gravitationally bound and virialized. The flows withinthem are highly supersonic, producing a filamentary morphol-ogy. Nonetheless, these objects are not highly sub-fragmented.There are at most one or two density maxima in each one, notmany density maxima. These structures look much like the tur-bulent cores posited in the McKee & Tan (2003) theory formassive star formation. When smeared on a resolution of 1700AU, distinct centrally condensed structures remain visible forthree of the four massive stars, indicating that these objectswould be detectable as massive cores in an observation.

It is important to understand that our analysis says nothingabout the Lagrangian trajectories of the fluid elements thateventually coalesce to form the massive stars in our simulations,a topic that has previously received extensive investigation byBonnell et al. (2004) and Smith et al. (2009a, 2009b), amongothers. It may well be that particular fluid elements that arepresent in the cores at the time shown in Figure 11 do not accrete

12

0.01 M

⊙

0.1 M

⊙

1 M⊙

0.01 M

⊙

0.1 M

⊙

1 M⊙

0.01 M

⊙

0.1 M

⊙

1 M⊙

Krumholz+ 2012

SimulationObserved

Universality of the IMF from stellar radiationPart I

• Consider a protostar with luminosity L; temperature of material a distance R from it will be approximately given by L = 4𝜋 𝜎SB R2 T4

• Jeans mass MJ = [ (kBT / 𝜇mHG)3 / 𝜌]1/2

• Consider spherical regions of increasing radius R around a seed star; as R increases, T and MJ fall and enclosed mass rises

• Hypothesis: characteristic mass where fragmentation suppressed set by condition that enclosed mass ≈ MJ

• Result: <latexit sha1_base64="Cee8j85fSNwmYd9Ot+QjI7pGZjA=">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</latexit>

M =

✓1

36⇡

◆1/10 ✓ kBGµmH

◆6/5 ✓ L

�SB

◆3/10

⇢�1/5

Universality of the IMF from stellar radiationPart II

• Luminosity comes from accretion; we will show later in the course that energy yield from accretion ~constant for all protostars, L ≈ 𝜓 (dM/dt), 𝜓 ≈ 1014 erg/g

• Accretion time must be ~tff, so dM / dt ~ M (G𝜌)1/2

• Substitute in for L and solve for M:

• This is nearly independent of environmental conditions; basic reason: higher density favours fragmentation (lowers MJ) but also higher accretion rate and thus higher temperature (raises MJ), and effects almost perfectly cancel

<latexit sha1_base64="owqpRJcCpkrD9wbhjc9ajaHAud8=">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</latexit>

M =

✓1

36⇡

◆1/7 ✓ kBGµmH

◆12/7 ✓

�SB

◆3/7

⇢�1/14 ⇡ 0.3⇣ n

100 cm�3

⌘�1/14M�