ABSTRACT
We use simulations of isolated galaxies with a few parsec resolution to explore the connection between the small-scale star formation rate (SFR)–gas density relation and the induced large-scale correlation between the SFR surface density and the surface density of the molecular gas (the Kennicutt–Schmidt relation). We find that, in the simulations, a power-law small-scale "star formation law" directly translates into an identical power-law Kennicutt–Schmidt relation. If this conclusion holds in the reality as well, it implies that the observed approximately linear Kennicutt–Schmidt relation must reflect the approximately linear small-scale "star formation law."
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1. INTRODUCTION
By now it is reasonably well established that on large scales (many hundreds of parsecs) the star formation rate (SFR) surface density correlates strongly and approximately linearly with the surface density of the CO-emitting gas (and, under the assumption of the constant XCO factor, with the molecular gas; an area under scrutiny but beyond the scope of this Letter). This relation, commonly called Kennicutt–Schmidt, holds both in the local universe (Leroy et al. 2008; Bigiel et al. 2008, 2011; Bolatto et al. 2011; Leroy et al. 2012, 2013) and at intermediate cosmic redshifts (Genzel et al. 2010; Daddi et al. 2010; Tacconi et al. 2013).
The origin of the large-scale linear Kennicutt–Schmidt relation is still not fully understood. Stars form on small scales, hence the large-scale behavior is the result of averaging of the intrinsic small-scale star formation "law." Such averaging can proceed in two distinct regimes: the large-scale linear behavior may be "emergent," i.e., irrespectively how star formation proceeds on small scales, large-scale averaging results in a linear law. For example, in cosmology it is well-known that on sufficiently large scales the galaxy bias is linear even if on small scales galaxy formation is a highly nonlinear process. If the same property holds for the density distribution in the interstellar medium (ISM), it would provide a simple and elegant explanation to why the Kennicutt–Schmidt relation is (nearly) linear on large scales.
Alternatively, the "memory" of small-scale conditions can be preserved, in which case the linearity of the large-scale Kennicutt–Schmidt relation would imply the linear relation between the rate of star formation and gas density on small scales.
The relationship between SFR and gas density on small scales is understood even less. For example, Lada et al. (2010) found that on the scale of individual star-forming cores (≲ 1 pc) the SFR–density relation is also linear, but only if the density is above the threshold of 700 M☉ pc−3. On the other hand, Krumholz & Tan (2007) argued that a "constant efficiency per free-fall time" () is, instead, a better interpretation of several observational measurements.
This apparent inconsistency could arise from mixing diverse spatial scales, so it is worth reviewing how the hierarchy of spatial scales affects the SFR–density relation.
Throughout this Letter we try to be careful with our terminology. We reserve the term "Kennicutt–Schmidt relation" for the relationship between the surface densities of stars and (molecular) gas, and only on large scales (since surface densities are only meaningfully defined on scales much larger than the disk scale height). When we compare the volumetric SFR density with gas density, we use the term "SFR–density relation."
2. SFR–DENSITY RELATION ON VARIOUS SCALES
It is useful to think about relationship between SFR and gas densities (on any scale) in terms of the time-scale for gas consumption, the so-called depletion time τSF,
The above equation is, however, an incorrect way of considering star formation, because the density is only defined on a particular scale—we need to explicitly consider the range of spatial scales that is relevant for our problem.
Let us take some spatial scale L. One can imagine a region of interest (be it the whole universe or a single galaxy) divided into boxes of size L. If we average gas densities on scale L, they become meaningfully defined. Thus, the above equation could be replaced with
but even that is not general enough. SFR is not a deterministic function of gas density, but dependent on other physical parameters (previous history, dynamic state of the gas, etc.); it needs to be treated as a stochastic process, described by a probability distribution of having a particular value of the instantaneous SFR density for a given gas density at scale L, .
For any form of p, however, it is possible to define the average instantaneous SFR density at a given gas density,
Notation here becomes complex, so we use angular brackets 〈〉L to indicate averaging over a given spatial scale (a single box of size L), and the over bar to mark the average over all boxes with the same 〈ρ〉L; to sharpen the terminology, we also call the latter average "the expectation value of the instantaneous SFR surface density on scale L" (EVISFRD). With this notation and terminology, we can finally relate the EVISFRD on spatial scale L to the gas density on the same scale,
(or, equivalently, when considering surface densities). In that case depletion time becomes the function of other gas properties on scale L,
In other words, we need to explicitly consider the star formation relation as (at least) a two-dimensional relation on the plane (L, 〈ρ〉L).
In this formalism, the large-scale Kennicutt–Schmidt relation can be expressed as
with substantial intrinsic (i.e., exceeding the formal observational error) scatter and the actual value for τSF noticeably different at z = 0 (τSF ≈ 2 Gyr; Leroy et al. 2008; Bigiel et al. 2008, 2011; Bolatto et al. 2011; Leroy et al. 2012, 2013) and high redshift (τSF ≈ 0.7 Gyr; Genzel et al. 2010; Daddi et al. 2010; Tacconi et al. 2013). In this Letter we are going to overlook these important details, since we are only interested in the broad-brush effect of varying spatial scale on the SFR–density relation. Hence, for our purposes it is sufficient to consider τSF as constant on large scales.
Lada et al. (2010) observations can now be simply expressed as
with ρmin = 700 M☉ pc−3. For this equation to be consistent with the Kennicutt–Schmidt relation on large (say, 500 pc) scale,
exactly 1% of the molecular gas must sit above the small-scale density threshold ρmin—after all,
In the language of depletion time the "constant efficiency per free-fall time" ansatz becomes
or, in a more familiar form,
This ansatz is consistent with the observational constraints presented by Krumholz & Tan (2007). However, those constraints sample not only various densities, but also various spatial scales. Indeed, the lowest densities nH ≳ 10 cm−3 in Krumholz & Tan (2007) are sampled by CO measurements, which typically include whole molecular clouds on scales L ∼ 10–30 pc; densities around nH ∼ 103 cm−3 are observed in Infrared Dark Clouds, which have sizes 1–3 pc; finally, the highest densities of nH ∼ 105 cm−3 are only present in HCN-emitting star-forming cores with sizes around 0.1–0.3 pc or below. Hence, the observational constraints used by Krumholz & Tan (2007) all fall along a particular track L2 × 〈ρ〉L ∼ (1–10) × 103 cm−3 pc2 in the two-dimensional plane (L, 〈ρ〉L). In other words, observational constraints that support the "constant efficiency per free-fall time" () equally well support the "constant efficiency per unit scale" (τSF∝L).
In fact, the following two models for the depletion time:
- "linear"
- "3/2"
with , cannot be distinguished at present without additional observational constraints. The parameter L0 in both cases must be in the range of a few hundred parsecs (for example, the scale height of the gaseous disk) to be consistent with the large-scale Kennicutt–Schmidt relation. In the Milky Way galaxy ρ0 is such that the Lada et al. (2010) result is matched (ρmin(L) ≈ 700 M☉ pc−3 at L ∼ 1 pc), but in other galaxies it may be different (for example, being proportional to the density of the atomic-to-molecular transition).
Even if observations are not yet conclusive, we can at the very least explore the scale dependence of the SFR–density relation in high resolution numerical simulations. In fact, since we are only concerned with the effect of changing the spatial scale L on the relation, we do not even need dynamic simulations—all that is required is a snapshot with a realistic gas distribution down to sufficiently small (few parsec) scales.
3. SIMULATIONS
In order to test such a conjecture, we use three adaptive mesh refinement simulations of isolated galaxies. For the first two simulations, the galaxy model is based on observations of M83 and has a grand design bar and spiral excited by the gravitational potential from particles which rotate at a fixed pattern speed. The simulations were run using the three-dimensional hydrodynamical code, Enzo (The Enzo Collaboration et al. 2013), with radiative cooling down to 300 K, and did not include active star formation or feedback. To prevent unresolved collapse, a pressure floor was introduced when the Jeans length covered less than four cell widths. The limiting resolution (smallest cell size) for the high resolution run, M83HighRes, was 1.5 pc, and for the lower resolution realization, M83LowRes, it was 6 pc. Further analysis of these simulations will be found in Fujimoto et al. (2014).
The third simulation is of a Milky-Way-sized galaxy (MWLowRes) without an imposed grand design. The model initial conditions are presented in Tasker (2011); its spatial resolution is similar to the M83LowRes run, but the MWLowRes run includes star formation and supernovae feedback in the form of a thermal energy injection.
A summary of the three simulations is given in Table 1.
Table 1. Simulations
Simulation | Resolution | SF/Feedback | Grand Design |
---|---|---|---|
M83HighRes | 1.5 pc | No | Yes |
M83LowRes | 6.0 pc | No | Yes |
MWLowRes | 7.8 pc | Yes | No |
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4. RESULTS
In order to explore the effect of averaging in the simulations, we adopt a simple power-law-with-threshold relation between the EVISFRD and the gas density ρ at the resolution scale of the simulation l. For the sake of brevity, we use just ρ instead of 〈ρ〉l and instead of for the density and the EVISFRD on the smallest resolved scale,
Since we are only interested in the relative changes in with averaging over various scales, the normalization of Equation (4) is not important to us, and we always measure the SFR in arbitrary units.
In the simulations, the molecular gas is not followed explicitly. However, the values for the density threshold we choose are always at or above nH = 103 cm−3, which, for even parsec scales, corresponds to molecular gas. At this value of the threshold the density probability density function (PDF) in the M83HighRes run already deviates strongly from lognormal, having developed the power-law tail induced by self-gravity, in agreement with turbulent ISM simulations (e.g., Kritsuk et al. 2011). Hence, we only allow star formation in self-gravitating gas.
Starting with the simulation resolution scale l, we first assign the SFR on the most refined cells throughout the whole computational domain. Then we simply average both ρ and independently over cells in groups of eight, building up the SFR–density relation sequentially on scales L = 2 × l, then L = 4 × l, etc. At each scale we compare versus 〈ρ〉L; that relationship at two spatial scales is shown in Figure 1 for our fiducial simulation M83HighRes.
The immediately obvious conclusion is that the slope α of the small-scale power-law relation (4) is preserved through the averaging procedure all the way to low densities. At the lowest densities the power-law scaling steepens, reflecting the density threshold, with the characteristic density until which the power-law scaling is maintained decreasing with increasing averaging scale. For sufficiently large averaging scales (many hundreds of parsecs), the power-law scaling extends well beyond threshold densities and even into the regime where the gas is expected to be mostly atomic. In other words, the density structure of the ISM (at least in the simulation) is not like the density distribution of large-scale structure in the universe, on large scales the "bias" does not become linear.
Such a behavior likely indicates that the ISM density distribution is highly correlated in space. This is not particularly surprising, though, since in the real ISM density structures are, indeed, organized along spiral arms (either grand design or flocculent) and star forming sites are strongly clustered in their parent molecular clouds (notice, that the latter statement does not imply that the clustering properties of molecular clouds along spiral arms are similar to clustering properties of stars inside giant molecular clouds (GMCs)—they clearly cluster differently; all that matters is that they cluster strongly in each case, with their respective distributions, while being different from each other, also both being very different from a Gaussian random field).
It is already clear from Figure 1 that the nonlinear small-scale relation should be preserved for surface densities as well (since at high densities/surface densities the relation is scale independent). Just to illustrate that, we show in the top panel of Figure 2 the surface densities of gas and the EVISFRD on the largest averaging scale (400 pc) for nmin = 103 cm−3 (it is apparent from Figure 1 that large-scale behavior is weakly dependent on the density threshold nmin) for three representative simulations. Indeed, the small-scale slope α is also preserved in the relationship between surface densities (at least until very low surface densities, well below observational detection thresholds).
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Standard image High-resolution imageWhile observations indicate the best correlation lies between star formation and CO emission, our model only follows atomic gas. We therefore can only use "star-forming gas" whose density is above our specified threshold, when plotting the Kennicutt-Schmidt (KS) relation. The steepening in the relation due to the threshold is, expectantly, absent in that case, but all the qualitative features of the relation are fully preserved in that case as well.
Figure 2 also demonstrates the robustness of our results to changing numerical resolution and modeled physics: simulations M83HighRes and M83LowRes differ by a factor of four in spatial resolution, but are barely distinguishable above Σgas ≈ 5 M☉ pc−2; simulations M83LowRes and MWLowRes have comparable resolutions, but the first does not include the feedback, while the second one does, nevertheless, the results from them are fully consistent.
5. DISCUSSION
Our main conclusion is that the large-scale Kennicutt–Schmidt relation is not emergent, but, instead, directly inherits the functional form of the small-scale SFR–density relation. In the simulations this conclusion is robust—it holds for runs with and without feedback, and in simulations with varied resolution.
That conclusion can be formalized further if one considers explicitly the spatial averaging of Equation (4) as we move from the small-scale l to some large-scale L. Spatial averaging inside each large box L can be represented by an integral over a local density PDF,
where is the location of the center of a given box i, and the integral is taken over the volume V = L3. Then for the EVISFRD we obtain (Equation (1))
where is the average density PDF in our region of interest for all boxes with the mean density 〈ρ〉L.
For our model ansatz (Equation (4)) we can, thus, obtain
where flow is the relative contribution to the integral from densities below the threshold. We can now consider several special cases as illustrative examples.
Example 1. For α = 1, for any form of , we have
i.e., the small-scale linear SFR-density relation is preserved at large scales for the total gas Kennicutt–Schmidt relation in environments where flow is small—for example, when at progressively higher average densities 〈ρ〉L the PDF shifts toward progressively higher small-scale densities (so that flow decreases with increasing 〈ρ〉L). At low enough densities, where flow is not negligible, the Kennicutt–Schmidt relation becomes steeper than linear, as was first shown by Kravtsov (2003) and is also visible from Figure 2 (see also Feldmann et al. 2011, 2012). The Kennicutt–Schmidt relation for the star-forming gas then simply becomes mathematical identity (up to a constant proportionality factor) in all environments and for any PDF (for example, that may explain the linear dependence of the SFR on HCN luminosity).
Example 2. If the PDF is self-similar, , as, for example, is often found in the simulations of the turbulent ISM (McKee & Ostriker 2007; Kritsuk et al. 2011; Padoan & Nordlund 2011; Kritsuk et al. 2013), then for any α
and, again the small-scale SFR-density relation directly propagates into the total gas Kennicutt–Schmidt relation in the limit flow ≪ 1.
It is worth noting that since we consider only the instantaneous KS relation, we do not comment upon the effect of feedback processes that may alter the fraction of gas available for future star formation. Feedback has been proved to affect the gas distribution on time-scales of several tens Myr, causing self-regulation in the form of weak dependence of the Kennicutt–Schmidt relation on the star formation efficiency (Schaye et al. 2010; Agertz et al. 2011, 2013; Hopkins et al. 2011, 2013), but this impacts the PDF from Equation (5) which is independent of our results. Such a shift will affect the amplitude of the KS relation, but does not affect whether the slope is preserved over large scales.
In this Letter we deliberately measured star formation in arbitrary units, to sidetrack any effect of varying efficiency and the whole question of the amplitude of the Kennicutt–Schmidt relation. Instead, we entirely focus on the slope of the relation, which is indeed preserved on large scales in other simulations as well. For example, both Agertz et al. (2013) and Hopkins et al. (2013) used a "3/2" model for the small-scale SFR–density relation, and found a similar slope of the Kennicutt–Schmidt relation on large scales, in full agreement with our results.
Hence, irrespective of the strength or a specific implementation of the feedback model, in cosmological and galactic scale simulations the slope of the small-scale SFR–density relation is preserved in the large-scale Kennicutt–Schmidt relation.
Is this conclusion robust in real life? Barring unexpected problems with simulations, one should never forget that, while the observations measure the instantaneous value of the molecular gas density, any large-scale measure of SFR always returns a time averaged value—over a time scale of about 20 Myr for UV observations and about 4–5 Myr for Hα measurements. Hence, the observational determinations of the large-scale Kennicutt–Schmidt relation always compare apples and oranges, as was convincingly shown by, for example, Schruba et al. (2010).
The mismatch between the measurements of the instantaneous gas density and of the time-averaged SFR may, potentially, cause the emergence of the large-scale linear Kennicutt–Schmidt relation from a non-linear small-scale SFR-density relation (to the best of our knowledge, that has not been demonstrated, though, so we only mention this as a possibility). If that was the case, however, it would be really bad news for the observers, implying that all of the existing large-scale studies of the Kennicutt–Schmidt relation are, in some sense, misleading, as they obscure the true non-linear relation between the instantaneous gas density and the SFR.
Irrespectively of the proper interpretation of our results, it is clear that studying the linear small-scale relation between the SFR and molecular gas density is as important as exploring the "constant efficiency per free-fall time" ansatz.
We are grateful to Andrey Kravtsov, Diederik Kruijssen, and the anonymous referee for the constructive criticism that significantly improved the original manuscript.