On the Appearance of Thresholds in the Dynamical Model of Star Formation
aa r X i v : . [ a s t r o - ph . GA ] J a n On the Appearance of Thresholds in the Dynamical Model of StarFormation
Bruce G. Elmegreen
IBM T. J. Watson Research Center, 1101 Kitchawan Road, Yorktown Heights, New York10598 USA [email protected]
ABSTRACT
The Kennicutt-Schmidt (KS) relationship between the surface density of thestar formation rate (SFR) and the gas surface density has three distinct powerlaws that may result from one model in which gas collapses at a fixed fractionof the dynamical rate. The power law slope is 1 when the observed gas has acharacteristic density for detection, 1.5 for total gas when the thickness is aboutconstant as in the main disks of galaxies, and 2 for total gas when the thickness isregulated by self-gravity and the velocity dispersion is about constant, as in theouter parts of spirals, dwarf irregulars, and giant molecular clouds. The observedscaling of the star formation efficiency (SFR per unit CO) with the dense gasfraction (HCN/CO) is derived from the KS relationship when one tracer (HCN)is on the linear part and the other (CO) is on the 1.5 part. Observations ofa threshold density or column density with a constant SFR per unit gas massabove the threshold are proposed to be selection effects, as are observations ofstar formation in only the dense parts of clouds. The model allows a derivationof all three KS relations using the probability distribution function of densitywith no thresholds for star formation. Failed galaxies and systems with sub-KSSFRs are predicted to have gas that is dominated by an equilibrium warm phasewhere the thermal Jeans length exceeds the Toomre length. A squared relationis predicted for molecular gas-dominated young galaxies.
Subject headings: stars: formation — ISM: molecules — Galaxy: local interstellarmatter — galaxies: ISM — galaxies: star formation
1. Introduction
A correlation between surface density of star formation, Σ
SFR , and surface density ofgas, Σ gas , is observed in local galaxies on scales larger than several hundred parsecs (e.g., 2 –Kennicutt et al. 2007) and it is observed for whole galaxies over a wide range of redshifts(Daddi et al. 2010; Genzel et al. 2010, see review in Kennicutt & Evans 2012). This cor-relation is typically a power law in the main parts of spiral galaxy disks with a fall-off inthe outer parts (Kennicutt 1989) that is attributed to a decreasing relative abundance ofcool and molecular gas (e.g., Krumholz 2013) with star formation following the molecules(Schruba et al. 2011). The value of the power law slope is consistently around unity for COin normal galaxies (Wong & Blitz 2002; Bigiel et al. 2008; Leroy et al. 2008) and for densegas tracers like HCN (Gao & Solomon 2004; Wu et al. 2005) and around 1.4 for total gas inthe main parts of galaxy disks (Kennicutt 1998). It is steeper in the outer regions of spi-ral galaxies (Bigiel et al. 2010) and in dwarf irregular galaxies (Roychowdhury et al. 2009;Bolatto et al. 2011; Elmegreen & Hunter 2015) with a slope of around 2.The origin of these correlations has been addressed analytically in many previous studies(e.g., Krumholz & McKee 2005; Krumholz et al. 2009, 2012; Krumholz 2013; Ostriker et al.2010; Hennebelle & Chabrier 2011; Padoan & Nordlund 2011; Renaud et al. 2012; Padoan et al.2014) and also shown to follow from numerical simulations (e.g., Li et al. 2005; Padoan et al.2012; Federrath & Klessen 2012; Kim et al. 2013; Kim & Ostriker 2015; Hu et al. 2016; Murante et al.2015; Semenov et al. 2016; Hopkins, et al. 2017). The basic ingredients are gaseous self-gravity, turbulence and cooling, with cloud geometry and feedback determining the gas scaleheight and limiting the fraction of gas that gets into stars.A recent development is the inference that star formation not only occurs in dense gas,as long believed, but that the rate of star formation per unit gas mass is independent of thevolume density of this gas (e.g., Evans et al. 2014). This inference follows from the linear cor-relation between Σ
SFR and the surface density of dense gas tracers like HCN (Gao & Solomon2004; Wu et al. 2005), and from a linear correlation between the total star formation rate(SFR) in molecular clouds and the mass of gas above some extinction threshold, typically ∼ ∼
2. Multiple KS Relationships2.1. Three Power Laws
Most studies of the relationship between Σ
SFR and the gas surface density have concen-trated on the main disks of spiral galaxies, where a difference in slope is observed for totalgas and molecular gas (Kennicutt 1998, Bigiel et al. 2008, see review in Kennicutt & Evans2012). In the outer parts of spirals and in dIrrs, a third and steeper slope appears. Thetransition to this steeper slope occurs where two things happen simultaneously: the averagegas column density becomes less than the value needed to shield a molecular cloud (e.g., ∼ M ⊙ pc − ; see Krumholz et al. 2009), and the gas mass begins to dominate the stel-lar mass, leading to a flare in the total gas thickness as the surface density decreases withan approximately constant velocity dispersion (Olling 1996; Levine et al. 2006). These twothings leave an ambiguity in what drives the steeper slope: is it the sudden lack of molecules(Krumholz 2013) or the sudden drop of midplane density in the flare (Barnes et al. 2012;Elmegreen 2015)? 5 –Here we take the viewpoint that a single dynamical law produces the three observed KSrelationships regardless of the pre-existence of molecules with the difference between themthe result of primarily two things: the degree of self-gravity in the gas, which determines therelation between the density and the column density, and the ratio of the average density tothe characteristic density for radiation of the tracer used to observe it.To distinguish between these three KS relationships, we group them according to theirpower laws:(KS-1a) Σ SFR on galactic and sub-galactic scales increases approximately linearly with thesurface density of molecular tracers, such as CO in normal spirals at low density or HCN athigh density (Wong & Blitz 2002; Gao & Solomon 2004; Wu et al. 2005; Bigiel et al. 2008;Leroy et al. 2008; Heiderman et al. 2010; Bigiel et al. 2011; Wang et al. 2011; Schruba et al.2011; Bolatto et al. 2011; Leroy et al. 2013; Zhang et al. 2014; Liu et al. 2015; Chen et al.2017);(KS-1b) Σ
SFR for local molecular clouds scales approximately linearly with the dense gassurface density determined from extinction or FIR emission (Vutisalchavakul et al. 2014);(KS-1c) The total star formation rate in a molecular cloud scales about linearly with the massof dense gas (Wu et al. 2010; Lada et al. 2010, 2012; Evans et al. 2014; Vutisalchavakul et al.2016; Shimajiri et al. 2017);(KS-1.5) the star formation rate surface density, Σ
SFR , scales with the total gas surfacedensity, atomic plus molecular, to a power of approximately 1 . SFR scales approximately with the square of the total gas surface density in theouter regions of spiral galaxies and in dwarf irregular galaxies (Roychowdhury et al. 2009;Bigiel et al. 2010; Bolatto et al. 2011; Elmegreen & Hunter 2015; Elmegreen 2015, hereafterPaper I). Teich et al. 2016 also found a slope of 2 for low-mass galaxies, but only on largescales where stochastic variations were smallest;(KS-2b) Σ
SFR scales with the gas surface density in individual molecular clouds (not just thedense gas) to a power that is approximately 2 (Heiderman et al. 2010; Gutermuth et al. 2011;Harvey et al. 2013; Lada et al. 2013; Evans et al. 2014; Willis et al. 2015; Nguyen-Luong et al.2016; Retes-Romero et al. 2017; Lada et al. 2017).There is an offset from the KS-1.5 relationship for some ULIRGs (Greve et al. 2005;Daddi et al. 2010; Genzel et al. 2010; Garcia-Burillo et al. 2012), but that is not viewedhere as physically distinct from KS-1.5 but as a manifestation of either higher densities from 6 –galactic-scale shocks (Juneau et al. 2009; Renaud et al. 2014; Kepley et al. 2016), a changingmolecular conversion factor (Narayanan et al. 2012), or large corrections to the apparent gasmass (Scoville, et al. 2016).These relationships follow from the same physical law if we consider that gas can beobserved with different tracers and that it can have different relationships between the lineof sight depth and the surface density. This law is the commonly assumed dynamical model,which is three-dimensional and written in terms of density, ρ , as ρ SFR = ǫ ff ρ/t ff , (1)where t ff = (32 Gρ/ [3 π ]) − / (2)is the free fall time and ǫ ff is an approximately constant efficiency per unit free fall time (e.g.,Larson 1969; Madore 1977; Elmegreen 1991, 2002; Krumholz & McKee 2005). In the presentmodel, there is no term like the molecular fraction, f H2 , in Krumholz et al. (2012) becausemolecules, molecular clouds, HCN regions etc., are all viewed as incidental and not causal,so they can be ignored in the equation for star formation, which is primarily a dynamicalprocess. Observations of ǫ ff for individual star-forming regions suggest a range of valuesconsistent with time-variability (Lee et al. 2016); numerical simulations get a range for ǫ ff too (Semenov et al. 2016).A summary of analytical models that reproduce these relationships follows. Some arenewly derived and all follow from equation (1) without thresholds. KS-1.5 is the standard relation for star formation in galaxies (Buat et al. 1989; Kennicutt1989, 1998) so we begin with that here. It follows from equation (1) if the observed diskregion has an approximately constant scale height, H , as observed for CO in the Milky Way(Heyer & Dame 2015). Then ρ = Σ / H andΣ SFR = ǫ ff (16 G/ [3 πH ]) / Σ / . (3)With typical H = 100 pc and ǫ ff = 0 .
01, this becomesΣ
SFR M ⊙ pc − Myr − = 8 . × − (cid:18) Σ gas M ⊙ pc − (cid:19) . . (4)This result was shown in Paper I to agree in both slope and intercept with the observationsin Kennicutt & Evans (2012), which are for average star formation rates in the main disks 7 –of spiral galaxies (the average rate per unit area in a galaxy is a good reflection of the localrate; Elmegreen 2007a).The approximately constant H requires a separate model with a more complete theoryof interstellar processes (e.g., Kim & Ostriker 2015). For example, the squared velocitydispersion of the gas, σ , should be proportional to the total mass surface density in the gaslayer because H = σ / ( πG Σ tot ) is then constant. Such a relation may be a consequence ofthe KS-1.5 relation if we consider that the energy density decay rate per unit area in thegas, 0 . gas σ /H (for dissipation in a crossing time H/σ ; Mac Low et al. 1998; Stone et al.1998), is proportional to the product of the gas surface density and the SFR, ∝ Σ gas Σ SFR .Then Σ gas σ /H = πG Σ gas Σ tot σ ∝ Σ gas Σ SFR ∝ Σ . so that σ ∝ Σ . (Σ gas / Σ tot ) and H ∼ (Σ gas / Σ tot ) . This ratio for H is about constant in the main disks of spiral galaxies whereboth stars and gas have the same exponential scale lengths. KS-1a follows for a constant effective density for emission, ρ mol (Evans 1999; Shirley2015; Jim´enez-Donaire et al. 2017; Leroy et al. 2017a), that is much larger than the averageinterstellar density, ρ gas . Then equation (1) converts toΣ SFR = ǫ ff Σ mol /t ff , mol (5)for constant t ff , mol equal to the free fall time at the characteristic density for emission bythe molecule (which could be CO, HCN, or some other tracer of a particular phase of gas).Here we have set the fraction of the interstellar medium in the molecular phase equal to thefraction of the time spent as molecules, based on the local dynamical time (Paper I). f mol = ρ − . ρ − . + ρ − . ∼ (cid:18) ρ gas ρ mol (cid:19) . (6)(the approximation is for partially molecular regions, ρ gas << ρ mol ). This expression assumesthat both the cloud formation time before star formation and the cloud break-up time afterstar formation are proportional to the local dynamical times, in agreement with numericalsimulations of a supernova-agitated interstellar medium (Padoan et al. 2016a). ThenΣ mol = f mol Σ gas (7)and equation (5) follows from equation (1). 8 – To evaluate equation (5) in case KS-1a, we consider that CO appears in local clouds atabout 1.5 magnitude of visual extinction (Pineda et al. 2008), which corresponds to ∼ M ⊙ pc − of column density. For a typical large cloud near this extinction threshold with a sizeof ∼
30 pc, the 3D density is ∼
17 H cm − and for this density t ff , mol = 8 . ǫ = 0 .
01 again, equation (5) becomesΣ
SFR M ⊙ pc − Myr − = 1 . × − (cid:18) Σ mol , CO M ⊙ pc − (cid:19) (8)giving a consumption time of 0 .
80 Gyr. This molecular consumption time is too shortby a factor of ∼ ǫ ff /t ff should be lower by this factor. For example,Leroy et al. (2017b) find ǫ ff ∼ .
003 on 40 pc scales in M51 and discuss how ǫ ff is oftenobserved to be lower than 0.01. Murray (2011) suggest ǫ ff ∼ .
006 on average in the MilkyWay.Leroy et al. (2017b) considered star formation relationships for CO observations at 40pc resolution in M51. They found that the molecular depletion time, which is Σ mol / Σ SFR = t ff , mol /ǫ ff in our notation, is approximately constant instead of their expected Σ − . for dy-namical star formation at fixed ǫ ff . However, this depletion time should be constant if theaverage density for observations of CO is constant, as above, because that gives the linearmolecular relation, KS-1a. The stated expectation was that the density used for the dynam-ical time would be proportional to the average density in the 40 pc region, but that is notthe case if the average CO density in the resolution element is less than the characteristicdensity for CO emission. The average density comes from the summed mass of the COclouds in the 40 pc region, but each cloud could have about the same characteristic densityfor CO emission and the same t ff , mol . The best correlation they found was with the virialparameter, 5 Rσ / ( GM ) for R = 40 pc and mass M inside the region. They determined thatthe depletion time scales with the virial parameter to a power of ∼ .
9. At the same time,Leroy et al. (2017b) found that ǫ ff is nearly independent of the virial parameter. These tworesults imply, for the dynamical model, that the average density per molecular cloud, whichoccurs inside t ff , mol , depends on the average virial parameter measured on the scale of 40 pc.In the case of a dense molecular tracer, like HCN or HCO + , the characteristic densityof observation is ∼ × cm − in equation (5), giving t ff , mol = 0 .
19 Myr and with ǫ = 0 . SFR M ⊙ pc − Myr − = 0 . (cid:18) Σ mol , HCN M ⊙ pc − (cid:19) (9)The average observed coefficient is slightly lower than 0.052, i.e., more like 0.02 (Sect. 6), 9 –so ǫ ff is proportionally lower or the characteristic density for emission is slightly higher. Theresult is close enough to the observation to support the general model, given the uncertaintiesin density, star formation rate, and dense mass, plus the approximate nature of the modelitself.At high interstellar density (which usually corresponds to high Σ gas ), ρ gas & ρ mol and f mol ∼
1, in which case equation (3) applies with Σ mol ∼ Σ gas . Thus molecular emission hasa 1.5 power law at high ρ gas (if H is still about constant) and a linear law at interstellardensities below the effective density for emission where f mol < . ± .
10 forCO emission in ULIRGS and high redshift galaxies where the density is large ( ρ gas & ρ mol for CO), and they got a slope closer to unity, 1 . ± .
05, for the dense gas tracer HCN inthe same galaxies, presumably because ρ gas . ρ mol and f mol < An important correlation appears for average interstellar densities that are between thecharacteristic densities for CO and HCN observations (or any other low and high densitytracers). Above the CO density, CO tracks the total interstellar density fairly well because f mol , CO ∼ SFR ∝ Σ . for a constant thickness galaxy, as mentioned above.Below the HCN density, f mol , HCN < SFR ∝ Σ HCN . Thus the ratio of HCN to CO, which is viewed as the “densegas fraction,” increases with the star formation rate, f dense = Σ HCN Σ CO ∝ Σ / ∝ Σ / . (10)Similarly, the “star formation efficiency”, measured as the ratio Σ SFR / Σ CO , should scalelinearly with f dense : SFE = Σ SFR Σ CO ≈ Σ CO3 / Σ CO = Σ / ≈ Σ / ≈ f dense (11)These correlations are consistent with observations of star-forming galaxies and ULIRGs inUsero et al. (2015) and elsewhere. Similarly, Gao & Solomon (2004) observed L HCN /L CO ∝ L . which is similar to our prediction of a power of ∼ . H , and they observe L IR /L CO ∝ ( L HCN /L CO ) . , which is similar to our predictedpower of 1. Sections 4 and 6 return to discuss star formation in dense gas, including sublinearrelations between SFR and HCN which are not considered above. 10 – KS-2 follows when the line-of-sight thickness of the region is in pressure equilibriumwith gas self-gravity for the observed gas column density. This should be the case in thegas-dominated parts of galaxy disks. We set the disk scale height H = σ /πG Σ and thenderive (Paper I) Σ
SFR = (4 / √ ǫ ff G Σ /σ. (12)With a constant velocity dispersion σ = 6 km s − as typically observed in dwarf irregularsand outer spiral disks, and for ǫ ff = 0 . SFR M ⊙ pc − Myr − = 1 . × − (cid:18) Σ gas M ⊙ pc − (cid:19) . (13)This relation was shown in Paper I to agree with observations of the outer parts of spiraldisks and dwarf irregular galaxies. The main reason for the steepening of the slope is theincrease in scale height with radius, i.e., the disk flare in a galaxy. That increase drops themidplane gas density faster than the surface density so the dynamical rate at the midplanedensity drops more quickly too. A disk flare was also present in the Krumholz (2013) modelalthough not mentioned explicitly.These regions of low surface brightness are also where the metallicity tends to be low(Rosales-Ortega et al. 2012; Bresolin & Kennicutt 2015), but the drop in Σ SFR is probablynot from an inability to make H on dust. This is because the star formation relation in thisregime is the same for a wide range in metallicities, i.e., comparing outer spiral disks wherethe metallicity is slightly below solar to dwarf irregular galaxies, where the metallicity is ∼
10% solar (Roychowdhury et al. 2015; Jameson et al. 2016). The squared dependence ofΣ
SFR on Σ gas is also not from a drop in molecular fraction with decreasing density becausethe density dependence of the molecular fraction for conventional theory (Krumholz et al.2009) is much steeper than the observed decrease in Σ
SFR with gas density in galaxies(Elmegreen & Hunter 2015).Ostriker & Shetty (2011) derived a squared KS relation on a galactic scale by assum-ing that the interstellar pressure is proportional to Σ
SFR through momentum injected bysupernovae, and that this pressure is also proportional to Σ as in an equilibrium galaxydisk. The application of supernova regulation in outer spiral disks and dIrr galaxies is notclear though, considering the very low star formation rate and pressure there. For example,in an exponential disk, the surface density of supernovae decreases as exp( − R/R D ) for starformation scale length R D , and the midplane gas density decreases as ρ ∝ exp( − R/R D )considering the outer-disk flare (i.e., ρ = ( π/ G Σ /σ for an equilibrium disk of pure gas 11 –with a near-constant velocity dispersion σ and the same exponential for gas surface density,Σ). Considering that the radius at which a supernova remnant merges with the ambientmedium scales as ρ − / (Cioffi et al. 1988), it follows that the volume filling factor of rem-nants, which is this radius cubed multiplied by the space density of supernovae, decreaseswith galactocentric radius as f SNR ∼ e − R/R D e R/ R D ∼ e − . R/R D . (14)Thus, outer galaxy disks should have relatively sparse stirring by supernovae. A flatterdecrease than this for the total gas surface density, e.g., from an extended HI disk, makesthis conclusion even stronger. KS-2b may follow from the same relationship as KS-2a if it is applied to the interiors ofself-gravitating clouds or to whole self-gravitating clouds. We assume a power-law internaldensity profile ρ ( r ) = ρ edge ( r edge /r ) α from some small core radius, r core to the edge radius r edge where the density is ρ edge (an internal profile that explicitly includes ρ core is in equation(30)). This gives a radius-dependent mass M ( r ) = 4 π − α ρ edge r α edge r − α (15)and surface density Σ( r ) = M ( r ) / ( πr ). As an approximation, we take the one-dimensionalvelocity dispersion σ ( r ) from the virial theorem,3 Z r σ ( r ) ρ ( r )4 πr dr = Z r ( GM [ r ] /r ) ρ ( r )4 πr dr, (16)which gives σ ( r ) = 4 πG − α ) ρ edge r α edge r − α = GM ( r ) / (3 r ) . (17)The internal surface density for the SFR then follows from equation (1), which is also afunction of radius, Σ SFR ( r ) = 1 πr Z r (cid:18) ǫ ff ρ ( r ) t ff ( r ) (cid:19) πr dr. (18)For a singular isothermal sphere, α = 2, the SFR surface density has a logarithmic divergencenear the center of the cloud, which requires the use of a core radius,Σ SFR ( r ) = p / ǫ ff (cid:18) G Σ( r ) σ ( r ) (cid:19) ln ( r/r core ) . (19) 12 –For other α <
2, Σ
SFR ( r ) = p / (cid:18) (3 − α ) . − . α (cid:19) ǫ ff (cid:18) G Σ( r ) σ ( r ) (cid:19) . (20)We assume for comparison with observations that ln( r/r core ) ∼ (3 − α ) . / (3 − . α ) ∼ . α = 2 if r/r core = 11 . α = 1 .
5, which are two cases considered also in Section 3. ThenΣ
SFR ( r ) ≈ . ǫ ff G Σ( r ) /σ ( r ) . (21)These results were written to resemble equation (12) in form.Table 2 in Gutermuth et al. (2011) gave fits for the Σ SFR − Σ gas relation for 8 localmolecular clouds. Averaging their coefficients in the log and averaging their powers of Σ gas ,we get from their observationsΣ SFR M ⊙ pc − Myr − ≈ . × − (cid:18) Σ gas M ⊙ pc − (cid:19) . (22)This observation agrees with equation (21) at Σ gas = 100 M ⊙ pc − and σ = 1 km s − if ǫ ff = 0 .
09. This efficiency inside molecular clouds is factor of ∼ SFR M ⊙ pc − Myr − ≈ . × − (cid:18) Σ gas M ⊙ pc − (cid:19) . , (23)which is a factor of ∼ ǫ ff = 0 . σ ∼ − . These studies determine the local star formation ratesfrom counts of young stars, which involve assumptions about stellar ages and masses, andthere could also be stochastic effects for low counts. Lada et al. (2013) note that the relationis steeper for Orion B, where the slope is 3.3. The real relation should scale with Σ /σ ,however, and σ may vary with Σ gas (see below).Gutermuth et al. (2011) considered a reason for their molecular cloud relation that issomewhat like ours, deriving a KS-2b relation based on counting the areal density of thermalJeans mass objects in a thin self-gravitating cloud. Equation (21) is more general and showsdirectly the connection between the molecular cloud relation and the galactic relation forgas-dominated regions (KS-2a). Parmentier & Pfalzner (2013) derived approximately thesame squared star formation law for molecular clouds as in equation (21) using the area 13 –integral over a collapse-model density profile (see also Parmentier et al. 2011). They appliedit to the formation of star clusters.Other observations of the KS relation on molecular cloud scales are in Heiderman et al.(2010), Harvey et al. (2013), Lada et al. (2013), Willis et al. (2015), Heyer et al. (2016),Nguyen-Luong et al. (2016), Retes-Romero et al. (2017) and Lada et al. (2017), who allfound a power index of Σ gas close to 2 (or sometimes larger) and a similar factor of ∼ Spitzer legacy program (Evans et al. 2003; Dunham et al. 2013). They plottedthe KS relation for these clouds with 3D density instead of column density, using the samethickness for the star formation rate and the gas, so it is essentially the same as plottingsurface densities. They got a slope of 2 . ± .
07 and suggested that this slope is inconsistentwith the dynamical star formation model. That inconsistency is only in comparison to KS-1.5, however, with its slope of 1.5 for main galaxy disks.Equation (21) should also apply to whole molecular clouds if the radius is taken to be thecloud radius so the average surface density and cloud dispersion are used. For the Evans et al.(2014) data with tabulated values of Σ
SFR , Σ gas and σ , we derive a proportionality constant ǫ ff = 0 .
016 for equation (21). Another survey of star-forming complexes was made for theLarge Magellanic Cloud (Ochsendorf et al. 2017). The average value of Σ
SFR / (cid:0) Σ /σ (cid:1) forthat implies ǫ ff = 0 .
04. These values are reasonably consistent with other values of ǫ ff discussed in this paper, considering the difficulty in defining the local star formation rate ina molecular cloud.Heyer et al. (2016) saw no KS relationship for young stellar objects in Milky Way molec-ular clouds because the range in Σ gas was too small (their figure 10a), but they compared itonly with the galactic ∼ . SFR and Σ gas /t ff , however, andthis is the basic model assumed in equation (21).Wu et al. (2010) compared the total star formation rates in molecular clouds, dM star /dt ,with the local rates in the dynamical model, ρ . , and got a decreasing relationship in theirFigure 35 which they claimed was inconsistent with an expected positive correlation in thedynamical model. However, they should have compared the total rate with the productof the cloud volume times the local rate, which would have introduced an additional term(Σ gas /ρ gas ) for cloud volume to be multiplied by the local rate ρ . . The result would havebeen proportional to Σ ρ − . as in their observed decreasing relation, considering that their 14 –Σ gas had a narrow range. Thus their result is also consistent with molecular cloud evolutionon a self-gravitating timescale. Their explanation for the decreasing relationship is that cloudmass and therefore IR luminosity is correlated inversely with density, as found by Larson(1981), but the KS relationship should be between IR luminosity per unit volume and gasdensity (or IR luminosity per unit area and gas surface density).The small-scale mechanism of star formation inside molecular clouds is not addressedby the simple dynamical model of equation (21). Molecular clouds appear to be composed ofnumerous filaments (Andr´e et al. 2010; Molinari et al. 2010) and it might be that collisionsbetween these filaments trigger local star formation (Myers 2009; Schneider et al. 2012).Parmentier (2017) considered the cloud KS law with filamentary extensions to large radius.The density PDF for filamentary structure has been considered by Myers (2015). If mutualgravity is involved, causing the filaments to collide with each other, then the dynamicalmodel should still apply because it states only that the rate of star formation on a largescale, i.e., averaged over many filaments, is proportional to the rate of mutual gravitationalattraction. Krumholz et al. (2012) suggested a two-regime model for KS-1.5 using the line-of-sightintegrated form of equation (1) where Σ
SFR is proportional to Σ gas /t ff . In one regime, thedensity used for t ff was the 3D cloud density of a Jeans-mass cloud with a fixed surfacedensity, Σ GMC , and in another regime, the density for t ff was the average disk value whenToomre Q = 1. The value of t ff used for the relationship was the smaller of these two. Whilethis method gave a good fit to the data, we consider that clouds with a fixed Σ GMC are moreor less star-forming depending on the local interstellar pressure (see equation 24 below), andthat Q ∼ constant is not a dependable criterion for interstellar properties (Sect. 7). Still,the utility of a KS law written explicitly in terms of Σ gas /t ff rather than Σ gas alone, or onewritten in terms of Σ /σ for self-gravitating regions, is that important dynamical processescan be considered when t ff or σ are included in addition to the total available gas for starformation.KS-1.5 and KS-1 break down on small scales because star formation and cloud evolutionare time dependent, and observations on small scales no longer see the average values thatare used in the simple theory reviewed here (Schruba et al. 2010; Kruijssen & Longmore2014).KS-1.5, KS-1, and KS-2a also contain another, hidden, relation that is independent of 15 –star formation and that is the simultaneous radial decrease of both Σ SFR and Σ gas from theexponential profile of a galaxy disk (Bolatto et al. 2017). This dependence stretches out therelation to cover a large range in both quantities. Azimuthal variations from spiral arms alsocontain an independent relation that stretches out the parameter range. Spiral arms collectand disperse molecular gas and its associated star formation without much of a change inthe star formation rate per unit molecular mass (Ragan et al. 2016; Schinnerer et al. 2017).These additional dependencies can be important in some cases. For example, azimuthalvariations of Σ gas in dwarf irregular galaxies may not have associated variations in the diskthickness and then KS-1.5 or KS-1a would apply to those variations, while KS-2a still appliesin the radial direction as the thickness increases.The dense gas relations KS-1b and KS-1c do not follow from the dynamical model wherecloud evolution is always proportional to the gravitational collapse rate. We suspect KS-1band KS-1c may be artifacts of observational selection, as discussed in Sections 4, 5 and 6.
3. Power law Probability Distribution Functions and their Role in the KSRelation3.1. Integrals over the Convolution PDF Function
The previous section showed how observations of star formation in total gas or in COand HCN molecules follow from equation (1) for main and outer galaxy disks, dIrrs, andindividual GMCs. Another way to derive the SFR has been to use an integral over theinterstellar density PDF above some threshold density, assuming the PDF is a log-normal(Elmegreen 2002; Kravtsov et al. 2003; Krumholz & McKee 2005). This use of a thresholddensity assumes that all of the gas above the threshold is involved with the star formationprocess and is therefore strongly self-gravitating. This cannot be the case for a fixed thresh-old, however. What matters is the virial parameter, 5 Rσ /GM , for a particular interstellarcloud of radius R , velocity dispersion σ and mass M (Padoan et al. 2012). This parameterhas to be less than about unity for strong self-gravity. That condition is the same as re-quiring a threshold (minimum) column density that depends only on the ambient pressure P : Σ Threshold = (cid:18) PπG (cid:19) . . (24)(given that the pressure in a self-gravitating cloud is πG Σ / ? ). Interstellar gas that has some density structure inaddition to local turbulence should have a PDF equal to the convolution of that structurewith the log-normal from local turbulence (Elmegreen 2011). This implies that gravitatinggas with power-law density gradients in dense cores and filaments should have a power-law PDF, as simulated (Klessen 2000; V´azquez-Semadeni et al 2008; Kritsuk et al. 2011;Federrath & Klessen 2013; Pan et al. 2016) and observed with dust extinction (Froebrich & Rowles2010; Kainulainen et al. 2011), dust emission (Schneider et al. 2013; Lombardi et al. 2015;Schneider et al. 2015a,c) and molecular line emission (Schneider et al. 2016). Schneider et al.(2015b) found a power law characteristic of collapse up to A V ∼
100 mag and then a flat-ter power law beyond, which they supposed was from some termination of the collapse.Schneider et al. (2012) determined the PDF of the Rosette molecular cloud and suggested ithad an extension to higher density because of compression from the nebula.In a steady-state, collapse-like motions have a local velocity, v ( r ), proportional to( GM ( r ) /r ) . for mass M ( r ) inside radius r , and these motions produce a density gradi-ent toward the collapse center, ρ ( r ), that makes the inflow flux, 4 πr v ( r ) ρ ( r ), approximatelyconstant. The solution to these equations is ρ ∝ r − . A singular isothermal sphere in virialequilibrium also has this density profile, while a collapsing envelope onto a core can havea ρ ( r ) ∝ r − / profile (Shu 1977). Observations of these density gradients in dense coresmapped by sub-mm wave dust emission were in Mueller et al. (2002).In general, if ρ ∝ r − α then the PDF slope is − /α (Kritsuk et al. 2011; Elmegreen 2011).This result may be derived for ρ ∝ r − α from the expression where density and radius corre-spond one-to-one, P PDF , ( ρ ) dρ = P ( r ) dr , which gives (e.g., see also Federrath & Klessen2013; Schneider et al. 2013; Girichidis et al. 2014) P PDF , ( ρ ) = P ( r ) / ( dρ/dr ) . (25) 17 –Considering that the probability of radius r is P ( r ) = 4 πr and that dρ/dr ∝ − r − α − , thisgives P PDF , ∝ r α +3 ∝ ρ − − /α for equal intervals of ρ and ρ − /α for equal intervals of ln ρ .In the same way, the power in the PDF for surface density may be obtained from theone-to-one relation P PDF , (Σ) d Σ = P ( b ) db for impact parameter b . With Σ ∝ b − β and P ∝ b , the result is P PDF , ∝ b β +2 ∝ Σ − − /β for equal intervals of Σ and Σ − /β for equalintervals of ln Σ. For a spherical cloud, β = α −
1, so if α = 2, then β = 1 and P PDF , ∝ Σ − for equal ln Σ intervals. For α = 3 / P PDF , ∝ Σ − for equal ln Σ intervals.In a more general situation (Elmegreen 2011), the total 3D PDF may be approximatedby the convolution of the average radial density profile, ρ ave ( r ), and the local PDF repre-senting fluctuations around this average, P PDF , local : P PDF , ( ρ ) = Z ρ ave max ρ ave min P PDF , local ( ρ | ρ ave ) P ave ( ρ ave ) dρ ave . (26) P PDF , local ( ρ | ρ ave ) is the conditional probability distribution function for density ρ , given theaverage ρ ave . We assume a local PDF from supersonic turbulence: P ′ PDF , local ( ρ ) = (2 πD ) − / e − . ( ln( ρ/ρ pk ) /D ) , (27)where P ′ indicates the function is written per unit logarithm of the argument. The peakand average densities are related by ρ pk = ρ ave e − . D , (28)and the Mach number M and log-normal width D are related approximately by (Padoan, Nordlund & Jones1997) D = ln(1 + 0 . M ) . (29)The average density profile is assumed to be a cored power law, ρ ave ( r ) = ρ edge r α edge + r α core r α + r α core . (30)Then equation (26) becomes (Elmegreen 2011) P ′ PDF , ( y ) = 3 C α (2 π ) . Z / C exp − ln ( yze . D )2 D ! ( z C − (3 − α ) /α D ( C − /α dz, (31)where y = ρ/ρ edge is the local density including turbulent fluctuations, normalized to thevalue at the cloud edge, z = ρ edge /ρ ave ( r ) is the inverse of the average density, and C is thedegree of central concentration, C = ρ ave ( r = 0) ρ edge . (32) 18 –In the dynamical model of star formation, essentially all of the gas evolves toward higherdensity when it is not being pushed back by stellar pressures or sheared out by galacticrotation. The result is a delayed or resistive collapse, i.e., one filled with obstacles, butstill a progression toward higher densities at some fraction of the dynamical rate. Such amodel implies that the 3D star formation rate, ρ SFR , is proportional to the integral of thedensity-dependent collapse rate over the entire 3D density PDF: ρ SFR = ǫ ff Z ∞ P PDF , ( ρ ) ( ρ/t ff ) dρ. (33)There is no lower limit to the density in this expression because the low-density gas con-tributes very little to the integral. In practice, this lower limit is around ρ edge and the valueof that is approximately the average midplane density. Lower-density gas tends to be warmor hot-phase and unable to join the cooler gas that is condensing from self-gravity. Fornumerical integrations in this sub-section, we take the lower limit of the integral equal to0 . ρ edge and the upper limit 10 ρ edge .The KS relation uses the SFR surface density, for which we should integrate ρ SFR overthe line of sight through the galaxy. For the moment we write this asΣ
SFR = 2 Hρ SFR . (34)The total gas column density isΣ gas = 2 H Z ∞ P PDF , ( ρ ) ρdρ (35)and the molecular column density isΣ mol = 2 H Z ∞ ρ mol P PDF , ( ρ ) ρdρ, (36)where we use a fixed lower limit to the density in the region of the interstellar medium wherethe molecules appear. This is considered to be an effective minimum density for emission(Glover & Clark 2012; Shirley 2015; Jim´enez-Donaire et al. 2017; Leroy et al. 2017a) and istaken to have a universal value independent of star formation rate and interstellar structure.In Section 4 we also consider a local column density threshold for the appearance of molecules.Figure 1 plots Σ SFR versus Σ gas as a blue curve and Σ
SFR versus Σ mol as a red curveassuming α = 2. These quantities come from the PDF integrals without conversion tophysical units and are shown to illustrate the slopes. Each curve is a sequence of increasing ρ edge , which tracks the density variation in the interstellar medium. We assume the scale 19 –height H is constant for these curves. The effect of increasing ρ edge is shown in Figure 2,which plots the 3D density PDFs from equation (31) assuming a core-to-edge density contrast C = 10 . Each curve uses a different ρ edge , increasing from 7 . × − to 3 . × in equation(31) as the curves move to the right. The vertical line in Figure 2 is the fixed value of ρ mol .As mentioned above, both Σ SFR and Σ gas come from integrals under the whole curve, butΣ mol comes only from the integral of the part of the curve to the right of the vertical line.The assumption of a constant scale height in Figure 1 makes the total-gas relation (bluecurve) have a power of 1.5 like the 3D relation because Σ gas is proportional to ρ edge . Themolecular gas relation is different with a slope of unity. This difference is because of theconstant density, ρ mol , used as a lower limit in the PDF integral for the red curve. At highsurface density, the molecular slope also becomes 1.5 because most of the disk has a densityabove the molecular excitation density. In terms of the PDFs, this means that most of thePDF has shifted to the right of the ρ mol in Figure 2 so the total integral over the PDF from ρ = 0 to ρ = ∞ is about the same as the integral above ρ mol .This shift from a linear law to a 1.5 law is consistent with observation of the KS lawwhich show a steepening slope from ∼ ∼ . CO ofstarburst galaxies in Gao & Solomon (2004) produced a steep molecular relation too. A steeprelation was observed for HCN at high star formation rates in ULIRGS (Graci´a-Carpio et al.2008). Kepley et al (2014) observed a high star formation rate per unit HCN luminosity inthe center of the starburst M82. The decrease in the molecular depletion time with decreasingradius in galaxies found by Utomo et al. (2017) may be the same effect.The cyan curve in Figure 1 shows Σ SFR versus f mol Σ gas , which is the molecular part ofthe gas determined with fraction f mol from timing (Paper I). Using the PDF, this fractionbecomes the ratio of average times in the molecular and total-gas phases, f mol = R ∞ ρ mol P PDF , ( Gρ ) − . dρ/ R ∞ ρ mol P PDF , dρ R ∞ P PDF , ( Gρ ) − . dρ/ R ∞ P PDF , dρ ; (37)this is the quantity used in the figure. The cyan curve is parallel to the red curve through-out Figure 1, meaning that the molecular fraction obtained from integrating the PDF overdensities exceeding ρ mol is the same as the fraction of the time that the total gas spends inthe molecular phase.Semenov et al. (2017) obtain the linear molecular law in a numerical simulation becauseopacity provides a lower limit to the densities of molecular regions and feedback from starformation provides an upper limit. With the resulting molecular density in a narrow range, 20 –equations (36) and (37) apply and t ff , mol is nearly constant. Similarly, numerical simulationsby Padoan et al. (2016b) of molecular cloud formation in a turbulent medium find best agree-ment with observations when the molecular fraction increases suddenly at a characteristicdensity, effectively giving most of the molecular gas approximately that density.In gas-dominated regions of a galaxy, such as the outer parts of spirals and most of dwarfirregulars where σ is also nearly constant, the gas scale height is determined by self-gravity, H = σ /πG Σ gas , as mentioned above. If ρ edge is approximately the mid-plane density, then H = σ/ (2 πGρ edge ) . , which means that at the edge of the self-gravitating structures, ρ edge ∼ πG Σ σ . (38)Figure 3 shows the KS relations in this case, using equation (38) for ρ edge in equation (32)for C , along with equation (31) for the PDF. Now Σ SFR ∝ Σ as in equation (12), but Σ SFR is still proportional to the first power of Σ mol except in the high-Σ gas , molecular-dominatedregions, where the squared-law appears for molecules too.Such a squared-law for molecules has not been observed in galaxies yet because usuallythe molecular-dominated interstellar regions are stellar-dominated in mass. Only individualmolecular clouds have shown a squared molecular KS law so far, as discussed in Section2.5. There may be applications of the squared molecular law in high-redshift galaxies whichmight have both gas dominating the stars and molecules dominating the gas (for a reviewof the high-redshift KS relation, see Tacconi et al. 2013).
Considering the complexity of equation (31), a simpler derivation could make the resultsmore intuitive. Here we consider just the power-law part of the PDF to illustrate the variousslopes of the KS relations and the effect of widespread density gradients of the type ρ ∝ r − α .As mentioned above, this gradient translates to a power-law column density PDF with aslope of − − / ( α −
1) when plotted in linear intervals of column density, and − / ( α − − − /α and − /α , respectively. We now use this power law PDF to derive the various KS slopesshown in Figures 1 and 3.To be specific, we write the normalized 3D PDF for α = 2 as P , PDF dρ = 1 . ρ . ρ − . dρ (39) 21 –between ρ = ρ edge and ρ = ρ max >> ρ edge ; the average density out to the edge is ¯ ρ ( r edge ) =3 ρ edge . We define the scale height based on the average quantities, indicated by a bar overthe symbol: H = σ (2 πG ¯ ρ ) − . = σ (cid:0) πG ¯Σ gas (cid:1) − where ¯Σ gas = 2 H ¯ ρ . ThenΣ SFR = 2 Hǫ ff Z ρ max ρ edge P , PDF ( ρ ) ( ρ/t ff ) dρ = 3 Hǫ ff (32 G/ [3 π ]) . ρ . Z ρ max ρ edge ρ − dρ (40)= (4 G/ [9 πH ]) . ǫ ff ¯Σ . ln ( ρ max /ρ edge ) (constant H)= (2 / ǫ ff (cid:0) G ¯Σ /σ (cid:1) ln ( ρ max /ρ edge ) (constant σ ) . The first result is for a region in a galaxy where the scale height is constant, and the secondresult is for a region where the velocity dispersion is constant. These expressions have thesame forms as equations (3) and (12), respectively, with additional weak dependencies on ρ edge .The average surface density comes from an integral as in equation (35),¯Σ gas = 2 H Z ρ max ρ edge P PDF , ( ρ ) ρdρ = 3 σρ edge (2 πG ) . × Z ρ max ρ edge ρ − . dρ = 6 σρ . (2 πG ) . (41)The average molecular surface density is:¯Σ mol = 2 H Z ρ max ρ mol P PDF , ( ρ ) ρdρ = 3 σρ edge (2 πG ) . × Z ρ max ρ mol ρ − . dρ (42)= (cid:18) Hρ mol (cid:19) . ¯Σ . (constant H)= (cid:18) πG σ ρ mol (cid:19) . ¯Σ (constant σ ) . The Σ gas dependencies for molecular column density are the same as the Σ gas dependenciesfor Σ
SFR , so the two scale linearly with each other, as discussed in Section 3.1.The molecular fraction also follows from this simple model using the fraction of the timespent in the molecular phase (i.e., at ρ > ρ mol ), from equation (37), f mol = R ρ max ρ mol P PDF , ( Gρ ) − . dρ/ R ρ max ρ mol P PDF , dρ R ρ max ρ edge P PDF , ( Gρ ) − . dρ/ R ρ max ρ edge P PDF , dρ = (cid:18) ρ edge ρ mol (cid:19) . . (43)Then Σ mol = f mol Σ gas from equations (41) and (42), and Σ SFR = ǫ ff f mol Σ gas /t ff , mol to withina factor of 2 / ln( ρ max /ρ edge ). This slight inaccuracy for the simple model is the differencebetween integrating over ρ . in equation (40) and taking the product of the integral over ρ from equation (41) and the ρ part of the dynamical rate, ρ . . 22 –The various star formation relations are shown graphically in Figure 4. The total gasrelation is plotted with two blue lines. It has a slope of 1.5 where the scale height is aboutconstant, which tends to be in the star-dominated regions at high Σ tot and high Σ mol . Because H ∼ constant, σ is expected to decrease with increasing galacto-centric radius approximatelyas an exponential with a scale length that is twice the disk scale length for the stars. The gas-dominated regions are shown by a blue line at low Σ gas with a slope of 2. In a spiral galaxy,the transition from star-dominated to gas-dominated occurs in the outer disk, so a singleradial profile should show both KS-1.5 and KS-2a if it goes far enough. In a dwarf irregulargalaxies, only the gas-dominated part might be present and then the total relation is KS-2awith a slope of ∼
2, as found by Elmegreen & Hunter (2015). The molecular star formationrelation is shown in Figure 4 by a red line. This has a slope of 1 at low-to-moderate Σ mol because of the selection effect to pick regions defined by a characteristic density, ρ mol , whichmakes t ff = t ff , mol constant when the molecular fraction is low (e.g., Krumholz & Thompson2007).On a log-log plot like Figure 4, the molecular fraction at a particular star formation rateis the difference between the logs of the molecular and total surface densities, representedby the horizontal distance between the blue and red lines. In the star-dominated regionsat low-to-moderate Σ mol , the molecular fraction scales with the square root of the totalgas surface density. In the gas-dominated regions, it scales with the first power. Thesescalings are evident directly from the figure and may also be derived from equation (43):if H = constant in the first case, then ρ edge ∝ Σ gas and f mol ∝ Σ . (as also recognized byHeiderman et al. 2010; Krumholz et al. 2012); if H = σ /πG Σ gas in the second case, then ρ edge ∝ Σ , and f mol ∝ Σ gas .The power law expression for the PDF is simple enough to allow us to see a problemwith a SFR based on equation (5) if the power in the radial profile of density, α , is not equalto 2, as assumed above. For a more general case with 3D PDF power γ = 3 /α on a log-logplot, the normalized PDF is P PDF , dρ = γρ γ edge ρ − − γ dρ . Then the integrals in equations(40)-(42) become Σ SFR = (cid:18) G ( γ − πH ( γ − . γ (cid:19) . ǫ ff ¯Σ . (constant H) (44)= (cid:18) γ − γ − . γ (cid:19) . ǫ ff (cid:0) G ¯Σ /σ (cid:1) (constant σ ) . ¯Σ gas = (cid:18) γγ − (cid:19) . (cid:18) σ (2 πG ) . (cid:19) ρ . (45)¯Σ mol = (cid:18) γ − γHρ mol (cid:19) γ − ¯Σ γ gas (constant H) (46) 23 –= (cid:18) ( γ − πG γσ ρ mol (cid:19) γ − ¯Σ γ − (constant σ ) . Now the Σ gas dependencies for molecular column density and Σ
SFR are not the same if α = 2, so the SFR does not scale linearly with molecules. The above equations suggestΣ SFR ∝ Σ . /γ mol , which is sub-linear for shallow cloud profiles, α <
2, and the corresponding γ > .
5. Note that this γ is the slope of the 3D density PDF for log intervals of density,which is not observed directly. The slope of the PDF for surface density, which is directlyobserved, is related to this γ by γ = 2 γ − γ . (47)Thus γ > . γ > α < γ > α and γ (eg. 44). The sublinearmolecular relation implies that at high gas surface density, an increasing fraction of theobserved molecules are in the form of diffuse clouds, i.e., not strongly self-gravitating, andtherefore not contributing to the SFR at the full dynamical rate. Also, it means that at lowgas surface density, there is star formation in a phase of gas that is not revealed by thatparticular molecular emission, in dark molecular gas or atomic gas. We discuss the sublinearKS relation more in the next section.
4. The KS Relation with a column density threshold for molecules
Observations suggest that the KS relation for HCN sometimes becomes sublinear withtoo little emission for the SFR at high surface densities (Chen et al. 2015; Bigiel et al. 2016;Gowardhan et al. 2017), and sometimes becomes sublinear with excess HCN for the SFRat low surface densities (Usero et al. 2015; Kauffmann et al. 2017a). Sublinear molecularemission like this was discussed in the previous section as a possible result of a shallowdensity profile inside molecular clouds ( α <
2) in a large-scale survey, where the projectedPDF for the cloud population is steep ( γ > mass and star formation rate.CO observations in local galaxies have also been plotted with a sublinear slope (Shetty et al.2014), with the interpretation that much of the CO is in a diffuse, non-gravitating phase. 24 –Diffuse CO could be responsible for a decrease in the star formation efficiency with increasingmolecular mass for clouds in the Milky Way (Ochsendorf et al. 2017). HCN also contains adiffuse component in the central parts of galaxies, where the pressure is high (Helfer & Blitz1997; Kauffmann et al. 2017b). Molecules that require a certain column density for self-shielding or extinction, such as H or CO (Pineda et al. 2008), should have a sub-linear KSrelation when this column density is less than the threshold for strong self-gravity, whichdepends on pressure (Eq. 24).A column density threshold for molecule visibility has about the same effect as decreasingthe value of ρ mol for an increase in Σ gas . Decreasing ρ mol extends the integral in equation(36) to include a larger part of the PDF in molecular form, increasing Σ mol without changingthe integral in equations (33) and (40) that control Σ SFR .In the present model with average density variations like ρ = ρ edge r /r , we can writea characteristic column density Σ edge = ρ edge r edge . For a self-gravitating medium at pressure P , Σ edge ∼ Σ Threshold for large-scale r edge . Also with this density gradient, a cloud’s mass outto the edge is M edge = 4 πρ edge r . Combining these quantities gives r edge = (cid:18) M edge π Σ edge (cid:19) . . (48)Now consider an effective critical surface density for the appearance of a particular molecule,Σ c . Because in general for this density gradient, Σ( r ) = ρ edge r /r , the critical radius in acloud that has this surface density is r c = ρ edge r Σ c = (cid:18) Σ edge Σ c (cid:19) (cid:18) M edge π Σ edge (cid:19) . . (49)Putting this radius into the ρ ( r ) density law gives the corresponding effective critical density, ρ c = Σ edge r edge r = Σ (cid:18) π Σ edge M edge (cid:19) . . (50)This equation states that a critical column density has a corresponding critical density thatdepends on cloud mass and decreases slowly with increasing pressure (through Eq. 24).Figure 5 shows the KS relation for a hypothetical molecule that has a constant densitythreshold at low Σ gas , and a constant surface density threshold at high Σ gas . The 1.5 slopeis still present for the total gas and for the molecular emission at high surface density, thelinear slope is present for the molecule at low surface density, but now a sub-linear slope ispresent at moderate-to-high surface density where there is a surface density threshold forthe molecule. Two sample cases are indicated by the split in the red and cyan lines. These 25 –curves are solutions to equations (33)–(36) using equation (31) as in figure 1, but now thelower limit to equation (36) is ρ c from equation (50). ρ c is taken to be 100 for the same rangeof ρ edge as in Figure 2, but for ρ edge > ρ c in equation (36) is replaced by ρ c = 100 ρ − . inone case (slope 0.86 line) and 100 ρ − in the other case (slope 0.64 line).These hypothetical examples illustrate how molecular tracers can have a sublinear slopein the KS relation if the molecular gas is in a diffuse, non-self-gravitating state. Such astate tends to occur when the cloud column density is less than the threshold value given byequation (24). If the column density of a molecule exceeds a first threshold for the existenceor appearance of the molecule, but not the second threshold given by self-gravity (Eq. 24),then it should present a sub-linear slope on the KS relation. Another way to say this is thatmolecular clouds of a certain column density (for detection) that are self-gravitating at lowpressure will not be self-gravitating at high pressure.At high column density, i.e., one that is above both the detection and the self-gravitythresholds, the KS relation for total gas mass should be recovered, i.e., with a slope of 1.5 ifthe galaxy thickness is about constant. This is shown in Figure 5 as well. The sub-linear tosuper-linear transition is reminiscent of observations by Leroy et al. (2017b), who find thisfor CO in M51.Mok et al. (2016) found that the molecular fraction, as determined from CO(3-2) andHI emission, for gas in Virgo spiral galaxies is higher than in group galaxies. They also foundthat the molecular emission is high in Virgo compared to the star formation rate. Theseobservations suggest there is an excess of non-star-forming, or diffuse, CO gas in Virgospirals compared to group spirals. According to equations (24) and (50), a high interstellarpressure would do this by lowering the density at which CO is observed, thereby makingproportionally more CO, and by increasing the column density at which clouds become self-gravitating, thereby limiting the fraction of the interstellar medium that forms stars. Highpressure in Virgo cluster spirals is expected because of the high intergalactic pressure fromhot gas and the ram pressure from galaxy motions (Mok et al. 2017).
5. The appearance of threshold densities and column densities for starformation5.1. Observations and Models with Thresholds
In typical regions that have been observed, stars tend to form where the column densityexceeds a certain value corresponding to ∼ . Z/ . − . cm − , andIllustris (Vogelsberger et al. 2014) has a constant threshold, 0.13 cm − . Hu et al. (2016)simulate dIrrs with a constant threshold of 100 cm − . Hopkins, et al. (2017) note that thethreshold value does not matter in a simulation as long as it is high enough to avoid theessential physics of interstellar collapse; this is the same point as in the present paper, wherethere is no physical density threshold separating star-forming gas from non-star-forming gas(see also Saitoh et al. 2008).Analytical derivations of the star formation rate (e.g., Elmegreen 2002; Krumholz & McKee2005; Hennebelle & Chabrier 2011; Padoan & Nordlund 2011) also assume threshold densi-ties to get the efficiency of star formation correctly when integrating over the PDF above thisdensity. For example, Padoan et al. (2017) define a critical density as the external density ofa critical Bonnor-Ebert sphere that fits in the postshock layer of a supersonically turbulentgas. Hennebelle & Chabrier (2011) choose a critical density at which fluctuations smallerthan a fixed fraction of the cloud size can fragment. Various models like this are reviewedin Federrath & Klessen (2012) and Padoan et al. (2014).The dynamical model discussed here does not need a threshold density for star formationbecause it reproduces the large-scale star formation properties of galaxies by assuming theentire interstellar medium is evolving at some fixed fraction of the dynamical rate, withno transition below and above any particular density. The model only has characteristicdensities or surface densities for the detection of certain molecules, but not for the starformation process itself. The same model was applied on the molecular cloud scale byParmentier & Pfalzner (2013) to study the formation of bound clusters. Parmentier (2016)also considered there is no physical density threshold for star formation, but interpret the 27 –appearance of one as the result of combining local sources with a steep KS relation insideindividual clouds and distant sources with a linear relation from poor angular resolution.Burkert & Hartmann (2013) and Lada et al. (2013) explain the appearance of a threshold asthe result of a decreasing cloud area at higher surface density, with no actual threshold-likechange for the KS relation inside molecular clouds.Threshold-free models do not deny that certain densities play an important role in reg-ulating the various stages of star formation. At high density, magnetic fields should decouplefast from the gas (Goodman et al. 1998; Elmegreen 2007b) and turbulent motions becomesubsonic causing turbulent fragmentation to stop (Padoan 1995; Vazquez-Semadeni, et al.2003; Krumholz & McKee 2005). The present model implies two other things instead, thatthe rate limiting step for star formation on a large scale is the free fall time at the lowestdensity, and that stars of a certain young age tend to appear where t ff is about this age.Also, the lack of a bump or leveling off of the density PDF in star forming regions priorto the power law part from collapse implies that there is no bottleneck at some physicalthreshold between no star formation and star formation. Here we show that the appearance of a threshold density for star formation could be amanifestation of the dynamical process itself in the sense that very young objects tend toappear at a density where the dynamical time is comparable to their age. Star formationtracers with very young ages tend to show up at densities where the dynamical time is short.This proposed correlation between stellar age and dynamical time follows from twoconcepts. First, stars drift a distance proportional to their age and relative speed, so youngstars are still near their birth sites, and, second, interstellar gas tends to be hierarchicallyclumped, so if a star forms in a certain dense region, then there is likely to be anotherdense region nearby (e.g., Gouliermis et al. 2017; Grasha et al. 2017). Putting these togethermeans that young stars tend to be found near high gas densities, and slightly older stars,which have drifted further from their birth sites, tend to be near gas that has a more average,i.e., lower, density. Stars sufficiently old will have drifted past many molecular clouds in theirlifetimes and they will have lost any correlation with the density of their birth.The implications of this proposed age-density correlation may be illustrated by the time-dependent collapse of a spherical cloud core, whose radius evolution is given by (Spitzer 1978) d rdt = − GM ( r init ) r . (51) 28 –The solution is r = r init cos β (52)where β + 0 . β = π tt ff (53)is solved iteratively. Here, t is the time when r = r init and t ff is the free fall time at thedensity when the collapse begins. The solution gives the radius as a function of time andtherefore the density as a function of time.If stars form hierarchically (i.e., together) on a dynamical timescale, then the youngeststars will still be near gas where other stars are forming on about the same timescale. Ifyoung stars of a particular type have an average age T star , then these stars are likely to be ina region where the time before other star formation, i.e., at the singularity of the collapse, t − t ff , is comparable to T star . The density at this time t is known from the above solution,so we can plot the probability of seeing this star, which is approximately P star ( ρ ) = T star T star + t ff − t ( ρ ) , (54)as a function of the instantaneous density, ρ ( t ). This plot is shown in figure 6 for 3 stellarages and 4 initial densities of the collapse, 1, 10, 10 , and 10 in cm − of H ; Helium andheavy elements are included in the mass density for t ff . The different initial densities showup as different starting points for the curves in the lower left, but at the low values assumed,they have little effect on ρ ( t ) after the start.The curves show an increasing probability of seeing the stars with increasing density.The time at the half-probability point, in units of the starting t ff , ranges from 0.999 to 0.991for a starting density of 1 H cm − as T star increases from 3 × yrs to 3 × yrs, itranges from 0.997 to 0.972 at a starting density of 10, 0.991 to 0.910 at a starting densityof 100, and 0.972 to 0.716 at a starting density of 1000 H cm − . The stars appear at thevery last few percent of the collapse time after the density has become high, regardless ofwhen the collapse started. Such timing would give the appearance of a threshold densitywhen in fact the collapse is continuous with no physical threshold separating stability fromcollapse. Similar curves result from other collapse solutions (e.g., Huff & Stahler 2006) aslong as there is a singularity in density at a certain time.Although this is an idealized model of singular cloud core collapse, it illustrates thebasic point that in a probabilistic sense, the youngest objects should appear near gas atthe highest densities. They should also appear near each other because of the hierarchicalstructure of the gas, and those that have already formed should be near others that are justabout to form. This correlation between age and density should persist even to larger scales, 29 –as long as the interstellar medium is in a state of resisted collapse where all phases last forsome relatively constant number of dynamical times within the full cycle of cloud formationand dispersal.
6. On the appearance of a constant star formation rate per unit dense gasmass
The increased likelihood for young stars to appear in dense gas implies that star forma-tion rates correlate best with the mass of dense nearby gas. Such correlations are commonlyfound, and they seem to contradict the dynamical model which also involves density (e.g.,Evans et al. 2014). Here we show that the star formation rate should be weakly dependenton density, as observed, even in the dynamical model if dense gas is defined either by emissionfrom molecules with a fixed characteristic density for emission, or by a high column densityobserved in dust emission or extinction. The first point with dense gas defined by dense-tracing molecules like HCN is essentially the same as that discussed in Sections 2.3 and 3,where the density used for the free fall time is a factor of order unity times the characteristicdensity of the molecule’s emission.The second point follows from equation (50) when a region is defined by a thresholdcolumn density, Σ c . In Section 4 we used this equation to suggest that the KS relationflattens for molecular column densities that exceed a threshold for excitation but not athreshold for self-gravity, given the ambient turbulent pressure. However, this flattening ofthe KS relation is also the main point of the observation that the star formation rate per unitmass is constant, regardless of density. Equation (50) shows that the threshold density forobservation is insensitive to the actual cloud column density or mass when there is a constantvalue, Σ c , used to define the “dense” region. Dense gas surveys that define their selectionto have extinctions exceeding 8 mag or some such value are in this category. The free fallrate in these regions scales with ρ . , which scales with Σ − . from equation (50), which,at the threshold of self-gravity, scales with interstellar pressure, P , as P − . according toequation (24). Thus there is a very weak dependence on density or ambient conditions oncea self-gravitating region is chosen to exceed a certain fixed column density. The appearanceof a fixed star formation rate per unit dense gas mass is thus a selection effect resulting fromthe definition of dense gas.Setting the threshold column density at 8 magnitudes of visual extinction, which corre-sponds to Σ gas , threshold = 160 M ⊙ pc − , we can rewrite equation (24) as P threshold = 8 . × k B (cid:18) Σ gas , threshold M ⊙ pc − (cid:19) . (55) 30 –The corresponding average molecular density for a velocity dispersion of σ threshold = 10km s − is n H = P/ ( µσ ) = 30 cm − for mean molecular weight µ . Evidently, when theaverage interstellar density exceeds the equivalent of 30 H cm − at a 10 km s − velocitydispersion, “dense” regions with more than 8 magnitudes of extinction start to becomediffuse and should lose their ability to form stars. This is another way of saying that thevirial parameter, 5 Rσ / ( GM ) becomes large at the corresponding cloud radius R and mass M . Evidently, at very high pressures, “dense” gas in star forming regions should not bedefined by 8 magnitudes of extinction. Defining dense gas in terms HCN emission or otherdense molecular tracers may still be practical, but then the linear relation follows (KS-1a)until the average interstellar density is higher than the HCN density. Gowardhan et al.(2017) recognized this problem with the standard definition of dense gas and suggested thatdensity contrast rather than absolute density is important.To be more quantitative, we consider the star formation rate per unit dense gas mass,which is conveniently defined in terms of the dense gas depletion time, M dense ( dM stars /dt ) − for dense gas mass M dense and star formation rate dM stars /dt . For surveys with HCN andhigh density molecular tracers, this time has been evaluated to be ∼
56 Myr for galaxiesin Gao & Solomon (2004), ∼
69 Myr for normal spiral galaxies and ∼
37 Myr for ULIRGSin Liu et al. (2015), ∼
83 Myr for local clouds and galaxies combined in Wu et al. (2005)and Retes-Romero et al. (2017), and ∼
60 Myr for local clouds in Heiderman et al. (2010).For surveys at high extinction or in cloud regions with high FIR emission intensities, thedense gas depletion time has been evaluated as ∼
22 Myr (Lada et al. 2012; Shimajiri et al.2017) and ∼
25 Myr (Evans et al. 2014) for local molecular clouds. For a combination oflocal clouds and galaxies using both CO and FIR emission, Vutisalchavakul et al. (2016)got ∼
66 Myr.These depletion time vary between ∼
20 Myr and ∼
80 Myr, depending on the sourcesand the observational techniques. For a representative value of ∼
50 Myr, which means dM stars dt = M dense
50 Myr = ǫ ff M dense /t ff , dense (56)we require t ff , dense /ǫ ff , dense ∼
50 Myr. With a characteristic density for these regions of n dense ∼ × cm − , t dense = 0 .
19 Myr and ǫ ff = 0 . ρ ∝ /r , the mass of a gas concentration increases linearly with sizeand the column density decreases linearly with the inverse of size. This implies that the mass 31 –ratio of high density to low density gas in a cloud scales inversely with the column densitythresholds used to define these gases. If extinction A V = 2 defines the low density part and A V = 8 defines the high density part (e.g., Evans et al. 2014), then f dense = 25%. If densegas is defined by line emission that is sensitive to density rather than column density, thenbecause mass out to some cloud radius scales inversely with the square root of density, f dense should be the square root of the ratio of densities of the low- and high-density tracers, whichmight be √ . ∼ f dense ∼ constant with radius in the Milky Way even thoughthe star formation fraction in this dense gas varies slightly.The dense gas fraction defined by line emission, e.g., L HCN /L CO , can increase substan-tially in ULIRGS and other regions where the average density exceeds the threshold for COemission. The reason for this was given in Section 2.3.3, i.e., the HCN luminosity increaseslinearly with the star formation rate when f mol , HCN <
1, and the CO luminosity increaseswith the 2/3 power of the star formation rate when f mol , CO ∼
1. Then the ratio of HCN toCO luminosity increases with the 1/3 power of the star formation rate, making it seem likeregions of high star formation are different with larger dense gas fractions. In fact it is onlya comparison of f mol < f mol ∼ f mol < t ff < T star , such asthe dense part of a molecular cloud or a starburst galaxy with a high average interstellardensity, the curves in Figure 6 all lie at high probability. Figure 7 shows examples. Now thestarting densities for the T star = 10 yrs curves include higher values, 10 cm − , 10 cm − and 10 cm − , representing surveys of dense-gas regions inside molecular clouds. Also, forobservations of distant galaxies, only older ages for young stellar regions can be discernedbecause individual stars and protostars often cannot be seen by themselves. For example, T star might be as high as 1 Myr for the brightest young stars, or 10 Myr for stars that exciteHII regions. These cases are also in Figure 7. For high interstellar densities or relatively oldstars at modest densities, the probability of seeing a star associated with the gas tends to be 32 –high. This would lead to the impression that stars form primarily where the average densityis high, with lower average densities for older stars. The spatial and temporal resolution ofevents on short timescales is lost when equally young stars are not distinguished.
7. Discussion
A physical model for star formation that is consistent with the above explanation forthe KS relations is discussed here. The main assumption is that gaseous gravity is theprimary driver of star formation on a scale comparable to the scale height. As is well known,interstellar gas in a disk galaxy is stabilized against gravitational collapse by rotation on largescales and pressure on small scales (Safronov 1960; Toomre 1964; Goldreich & Lynden Bell1965). The mass surface density, Σ gas , and epicyclic frequency, κ , determine the rotationallystabilized length in the radial direction, which is the inverse Toomre wavenumber k − =2 πG Σ gas /κ , where self-gravity balances the Coriolis force. Because a region of this sizecontains the mass M T = π Σ gas k − , it has a specific potential energy σ / GM T k T ,which corresponds to a potential velocity dispersion σ = 2 πG Σ gas /κ . For conditions in thesolar neighborhood, Σ gas ∼ M ⊙ pc − and κ = 0 .
037 Myr − , this dispersion is σ = 7 . − , much larger than the sound speed in the cool, diffuse phase of interstellar gas, which isonly σ s = 0 . − at 100 K. The result should be gravitationally-generated motion thatis supersonic, compressive, and dissipative, with a tendency to collapse and form stars.Magnetic forces, galactic shear, and intermittent local expansions from supernovaeand other stellar feedback act to resist this collapse (Kim & Ostriker 2015; Padoan et al.2016a; Pan et al. 2016; Ibanez-Mejia, et al. 2016; Mac Low et al. 2017; Semenov et al. 2017).Kinematic pressures like these stabilize a disk on scales smaller than the Jeans length, k − = σ /πG Σ. For an adiabatic gas, conventional theory states that if the Jeans lengthis larger than the Toomre length, corresponding to the condition Q = 2 k T /k J >
1, thenthere should be stability in the radial direction of a galaxy. Also, if Q eff > Q − ∼ Q − + 2( k azim σQ/κ ) , then both radial and azimuthal motions at wavenumber k azim are sta-bilized (Lau & Bertin 1978; Bertin 1989). However, kinematic resistance at supersonic speeddamps quickly (Stone et al. 1998; Mac Low et al. 1998), so the Jeans length evaluated withsupersonic σ should not be a short-term barrier to collapse on small scales (Elmegreen 2011).In that case, Q is not a measure of absolute stability. For example, in low surface-brightnessregions where Q appears to be high, star formation still seems normal (Elmegreen & Hunter2015). Moreover, a combined Q from gas and stars appears to be dominated by stellar Observations of disk quenching (Martig et al. 2009; Genzel et al. 2014; French et al. 2015) seem to con-
33 –mass and motions (Romeo & Mogotsi 2017), which means that stellar spirals, when they arepresent, do most of the work to regulate Q ∼
2, not gas processes such as star-formation feed-back. Stars form because self-gravity, pressure gradients, magnetic torques, and energy dis-sipation decouple the gas from the stars on scales less than k − . This makes gravitationally-driven contraction of the gas inevitable for a wide range of scales (e.g., Kravtsov et al. 2003;Li et al. 2005; Ballesteros-Paredes et al. 2011; Ibanez-Mejia, et al. 2017).The role of the background stellar disk in the star formation process could be minor aslong as the gas can dissipate thermal and turbulent energy. Gas that forms new stars hasto move through the background stars to come to a high density. The two-fluid instabilitydoes not include this process because it assumes that the two fluids have perturbations withthe same wavelength, which is usually the large scale of spiral arms. Gas moves throughstars because of pressure forces from turbulence and stellar feedback, and it decouples fromstars during the gravitational swing amplifier because of magnetic forces (Elmegreen 1987;Kim & Ostriker 2001). The point of view here is that background stars mostly affect thethickness of the gas layer through gravity, heating, and the generation of turbulence (e.g.,Ostriker et al. 2010) and therefore it affects the conversion between the observed surfacedensity and the spatial density that controls the collapse rate. This background effect isimportant in the main disks of spiral galaxies and possibly elsewhere, but in the outer partsof these disks or in dwarf Irregular galaxies, the gas surface density dominates the stellarsurface density and although starlight continues to heat the gas, a disk flare has a largereffect on the KS relation than the thermal properties (Elmegreen & Hunter 2015). As longas there is a cool phase with a thermal Jean length less than the Toomre length, and aslong as the timescales for thermal, turbulent, and feedback processes are comparable to orshorter than the timescale for gravitational processes, gravity, as a persistent and monotonicforce toward condensation, should control the rate of star formation. Then the precedingderivations of the KS relation should contain most of the relevant physics.
8. Conclusions
Galaxies form stars because feedback and other energy sources are unable to resistself-gravity when gas motions are supersonic and highly dissipative. The resulting collapseis not rapid, however, because that would mix the young stellar populations and erasewidespread correlations between positions and ages, which suggest turbulent compression is tradict these statements, but perhaps quenching is from other effects, such as a high supernova rate comparedto the average collapse rate, or high shear (Davis et al. 2014).
34 –involved. The collapse is more likely resistive, with a rate proportional to the gravity ratebut significantly slower. In this case, the star formation rate per unit volume is related tothe gas density in a fairly simple way (Eq. 1), and the density structure in the gas may alsobe written simply using a power-law PDF.With these two assumptions, the various relationships between the star formation rateand the amount of gas can be explained by a combination of resistive collapse and a selectioneffect for gas observations. These relationships were divided into three types: (KS-1.5)Σ
SFR ∝ Σ . for main galaxy disks with constant thickness; (KS-1a) Σ SFR ∝ Σ mol forthe molecular part of the gas when the average density is less than the characteristic densityfor observation of the molecule (and KS-1.5 again when the average density exceeds thischaracteristic density); (KS-1b) Σ SFR ∝ Σ gas , dense and (KS-1c) dM star /dt ∝ M dense for densegas when the selection of this gas involves an observational threshold for either density orcolumn density; (KS-2a) Σ SFR ∝ Σ for galaxies or parts of galaxies where gaseousself-gravity determines the disk thickness and the velocity dispersion is about constant (e.g.,outer parts of spirals and dIrrs); and (KS-2b) Σ SFR ∝ Σ inside molecular clouds wherethe size is also determined by gaseous self-gravity.Numerous observations were collected to illustrate these four regimes, and the assumedmodel for star formation was shown to agree with the observations fairly well if the efficiencyper unit free fall time is always around one percent. The model ignores the details of collapseand feedback, but the general agreement with these observations lends support to the basicpremise that star formation is a pervasive dynamical process on all relevant scales.The star formation relationships were also reproduced in general form using a proba-bility distribution function for interstellar gas that results from turbulence convolved withself-gravitational density gradients. Using these PDFs, the molecular component can bedetermined either from integration above the characteristic density for observation of themolecule or from averaging over the time spent in the molecular phase. These two methodsare equivalent for the dynamical model (Paper I).A column density threshold for the detection of certain molecules flattens the KS rela-tions by increasing the proportion of diffuse (non-gravitating) gas as the average interstellarsurface density increases. Such flattening has been observed for the HCN KS relationshipat both low and high pressure in different situations. Cloud selection at a minimum columndensity (e.g., using a lower limit to the extinction or far-infrared intensity) leads to a corre-sponding minimum density that varies only weakly with true cloud column density or mass,or with ambient pressure in the case of self-gravitating clouds. Then the characteristic freefall time is also nearly invariant for that region and the star formation rate per unit densegas will be about constant. Such near-constancy will be observed even if the actual free fall 35 –rate scales with the square root of local density.The appearance of a threshold density for star formation was attributed to the likelyassociation between new-born stars and nearby gas with a free fall time comparable to thestar’s age. The association between young stars and high densities gets even stronger as theambient density increases or the ages of the observable stars decreases. There is no physicalthreshold for star formation in this model, only an apparent one resulting from the timing ofcollapse. When combined with the nearly invariant collapse time in gas selected by densityor column density, the observation of a universal star formation rate per unit dense gasresults. Such a universal specific rate has no physical basis, however, and is even contraryto the observation that stars form in self-gravitating clouds, considering that the degree ofself-gravity follows from a balance between column density and ambient pressure.We conclude that star formation can be a continuous and dynamical process, with noactivity thresholds in density or column density, and no special physics or universal rates indense gas. The observation of star formation is fraught with selection effects, however, andthis gives the KS relation many different forms.The dynamical model gives some insight into what might be different in ultra diffusegalaxies (van Dokkum et al. 2015), damped Lyman- α galaxies (Rafelski et al 2016), andother seemingly failed systems where star formation rates are extremely low for the amountof gas present (e.g., Filho et al. 2016; Lisenfeld et al. 2016). If the basic ingredient for thedynamical model is supersonic turbulence generated by pervasive gravitational instabilities,then either these systems are stable, perhaps because they are thick or relatively fast-rotating,or their gas motions are subsonic, perhaps because there is no cool neutral phase. Becausemetallicity effects do not seem to matter for the cessation of star formation (Rafelski et al2016), or for the star-formation rates in general in the KS relation (Section 2.4), H for-mation is apparently not involved, nor is it a necessary precursor to star formation. Lackof a cool neutral phase occurs when the pressure is very low for the ambient radiationfield (Elmegreen & Parravano 1994; Wolfire et al. 2003; Schaye 2004; Kanekar et al. 2011;Krumholz 2013). This would seem to be a natural state for galaxies at very low surfacedensities, i.e., low pressures, where there is faint background radiation from cosmologicalsources and an early generation of stars.I am grateful to Dr. Ralf Klessen for discussions at an early stage of this research andto the referee for a careful reading of the manuscript. 36 – REFERENCES
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This preprint was prepared with the AAS L A TEX macros v5.2.
46 – –2 –1 S tot , S mol , f mol S gas S S F R f mol S gas S mol S tot slope = 1.5slope = 1 slope = 1.5 Fig. 1.— Solutions to equations (31), (33)- (37) when the average density profile inside eachcloud has a slope α = 2. Two parts of the KS law in main galaxy disks are shown: (1)the 1.5 slope (blue) for all phases of gas at high gas surface density when the interstellarmedium is mostly molecular and the same 1.5 slope also for any gas surface density whenplotted versus total gas; (2) the linear slope for molecules (red) at low-to-intermediate surfacedensities when the characteristic density for molecular emission is larger than the averageinterstellar density. The red curve uses equation (36) directly and the cyan curve calculatesthe molecular fraction from timing considerations, equation (37). The offset between the redline and the cyan line is non-physical, it is the difference between integrating the PDF over ρ . and integrating it over ρ with a separate multiplication by ρ . . The first method is themost physically relevant and the second illustrates the importance of dynamical evolution inthe molecular medium. 47 – –3 –2 –1 –7 –6 –5 –4 –3 –2 –1 r / r edge P P D F ( r ) –3 –2 –1 –7 –6 –5 –4 –3 –2 –1 r P P D F ( r ) Fig. 2.— (left) The density PDF from equation (31) for a density contrast C = 10 , a Machnumber 2, and α = 2, normalized to unit area. (right) A sequence of PDFs with increasing ρ edge illustrating how the interstellar PDF moves through a constant threshold for molecularemission (red line) as Σ gas increases. This sequence corresponds to the points used to plotthe curves in Figure 1. 48 – –2 –1 S tot , S mol , f mol S gas S S F R f mol S gas S mol S tot slope = 2slope = 1 Fig. 3.— Solutions to equations (31)-(37) as in Figure 1 but for a pure-gas disk where thescale height varies inversely with the gas surface density. Now the KS relation for total gas(blue) and high-density molecular gas (blue/red) has a slope of 2, although the moleculargas (red) still has a slope of 1 at low surface density. α = 2 is assumed. 49 – –4 –3 –2 –1 –3 –2 –1 S tot , S mol S S F R S mol S tot slope = 1.5slope = 2slope = 1 f mol a S gas0.5 f mol a S gas f mol Fig. 4.— Schematic KS relation showing star formation rate surface density versus totalgas (blue) and molecular gas (red) as calculated from equations (39) – (43), which assume apower-law PDF. The slope is 1.5 for all phases at high gas surface density, 1.5 for total gasat intermediate surface density, 1 for molecular gas at intermediate-to-low surface density,and 2 for total gas at low surface density. All of these relationships follow from one three-dimensional star formation law, equation (1), but they are viewed with different radialvariations of galaxy thickness and different selection effects. The molecular fraction is thehorizontal distance between the molecular and the total-gas KS curves in this logarithmicplot. 50 – –2 –1 S tot , S mol , f mol S gas S S F R f mol S gas S mol S tot slope = 1.5slope = 1 slope = 1.50.86 0.64 Fig. 5.— Solutions to equations (33)–(36) using equation (1) as in Figure 1, with the lowerlimit to equation (36) taken to be the critical density ρ c from equation (50). This solutionillustrates the flattening of the KS relation for molecules that have a threshold column densityfor formation or emission (which is at Σ ∼ . cm –3 ) P ( s t a r) P star ( r )= T star /(T star +t ff –t[ r ])Pressureless Collapse T star (yrs)=3x10 Fig. 6.— Solutions to equations (51)–(54) for the probability of observing stars with ayoung age, T star , near a region of a cloud with the H density indicated on the abscissa. Theprobability increases rapidly at the density where the free fall time is comparable to theage of the star. Different curves for the same T star are for different starting densities in thecollapsing cloud. 52 – cm –3 ) P ( s t a r) T star (yrs)=10 T star (yrs)=10 Starting densities =1, 10, 10 , 10 , 10 , 10 Fig. 7.— Same as Figure 6 but for a wider range of starting densities in the lower curvesand for larger stellar ages in the upper curves. This figure shows that stars slightly olderthan the free fall time in a gas selected for observation will usually appear to be associatedwith that gas, giving the impression for the youngest observable stars that they tend to formwhere the averageaverage