THE STAR FORMATION EFFICIENCY IN NEARBY GALAXIES: MEASURING WHERE GAS FORMS STARS EFFECTIVELY

The most common formulation of the star formation law is a power law relating gas and star formation (surface) densities following Schmidt (1959, 1963). Kennicutt (1989, 1998a) calibrated this law in its observable (surface density) form. Averaging over the star-forming disks of spiral and starburst galaxies, he found

$$\begin{equation} \Sigma _{\rm {SFR}} \propto \Sigma _{\rm {gas}}^{1.4}, \end{equation} \tag{ 1 }$$

often referred to as the “Kennicutt–Schmidt law.”

The exponent in Equation (1), n ≈ 1.5, can be approximately explained by arguing that stars form with a characteristic timescale equal to the free-fall time in the gas disk, which in turn inversely depends on the square root of the gas volume density, τ_ff ∝ ρ^−0.5_gas (e.g., Madore 1977). For a fixed scale height, ρ_gas ∝ Σ_gas and Σ_SFR ∝ Σ^1.5_gas. Thus, the first star formation law that we consider is

$$\begin{equation} {\rm SFE} \propto \Sigma _{\rm gas}^{0.5}, \end{equation} \tag{ 2 }$$

which is approximately Equation (1).

If the scale height is not fixed, but instead set by hydrostatic equilibrium in the disk, then

$$\begin{equation} \tau _{\rm ff} \propto \frac{1}{\sqrt{\rho _{\rm mp, gas}}} \propto \frac{\sigma _{\rm g}}{\Sigma _{\rm gas} \sqrt{1 + \frac{\Sigma _*}{\Sigma _{\rm gas}} \frac{\sigma _{\rm g}}{\sigma _{*,z}}}}, \end{equation} \tag{ 3 }$$

where σ_g and σ_*,z are the (vertical) velocity dispersions of gas and stars, respectively, Σ_gas and Σ_* are the surface densities of the same, and ρ_mp,gas is the midplane gas density. Equation (3) combines the expression for midplane density from Krumholz & McKee (2005, their Equation 34) and midplane gas pressure from Elmegreen (1989, his Equation 11, used to calculate ϕ_P). The second star formation law that we consider is

$$\begin{equation} {\rm SFE} \propto \tau _{\rm ff}^{-1} \propto \frac{\Sigma _{\rm gas}}{\sigma _{\rm g}} \left(1 + \frac{\Sigma _*}{\Sigma _{\rm gas}} \frac{\sigma _{\rm g}}{\sigma _{*,z}}\right)^{0.5}, \end{equation} \tag{ 4 }$$

which incorporates variations in the scale height and thus gas volume density with a changing potential well.

It is also common to equate the timescale for star formation and the orbital timescale (e.g., Silk 1997; Elmegreen 1997). Kennicutt (1998a) and Wong & Blitz (2002) found that such a formulation performs as well as Equation (1). In this case,

$$\begin{equation} {\rm SFE} \propto \tau _{\rm orb}^{-1} = \frac{\Omega }{2 \pi } = \frac{v(r_{\rm gal})}{2\pi r_{\rm gal}}, \end{equation} \tag{ 5 }$$

where v(r_gal) is the rotational velocity at a galactocentric radius r_gal and Ω is the corresponding angular velocity.

Tan (2000) suggested that the rate of collisions between gravitationally bound clouds sets the timescale for star formation so that

$$\begin{equation} {\rm SFE} \propto \tau _{\rm orb}^{-1} Q_{\rm gas}^{-1} (1 - 0.7 \beta), \end{equation} \tag{ 6 }$$

where Q_gas, defined below, measures gravitational instability in the disk and β = dlog v(r_gal)/dlog r_gal is the logarithmic derivative of the rotation curve. The dependence on β reflects the importance of galactic shear in setting the frequency of cloud–cloud collisions. In the limit β = 0 (a flat rotation curve), this prescription reduces to essentially Equation (5); for β = 1 (solid body rotation), the SFE is depressed by the absence of shear.

If the SFE of an individual GMC depends on its intrinsic properties and if these properties are not themselves strong functions of environment or formation mechanism, then we expect a fixed SFR per unit molecular gas, SFE (H₂). Krumholz & McKee (2005) posited such a case, arguing that the SFE of a GMC depends on the free-fall time in the cloud, itself only a weak function of cloud mass in the Milky Way (Solomon et al. 1987). Bigiel et al. (2008) found support for this idea. Studying the same data used here, they derived a linear relationship between $\Sigma _{{\rm H}_2}$ and Σ_SFR on scales of 750 pc.

SFE (H₂) is likely to appear constant if the scaling relations and mass spectrum (i.e., the intrinsic properties) of GMCs are approximately universal, the gas pressure is low enough that GMCs are largely decoupled from the rest of the ISM, individual resolution elements contain at least a few GMCs, and the properties of a cloud regulate its ability to form stars (Section 5.1 and Bigiel et al. 2008). This is the fifth star formation law that we consider, that star formation in spiral galaxies occurs mostly in GMCs and that once such clouds are formed, they have approximately uniform properties so that

$$\begin{equation} {\rm SFE} ({\rm H}_2) = {\rm constant}, \end{equation} \tag{ 7 }$$

which we can convert to the SFE of the total gas given $R_{\rm mol} = \Sigma _{{\rm H}_2} / \Sigma _{{\rm H}\,\mathsc{i}}$, the ratio of H₂ to H i gas. Then,

$$\begin{equation} {\rm SFE} = {\rm SFE} ({\rm H}_2) \frac{R_{\rm mol}}{R_{\rm mol}+1}, \end{equation} \tag{ 8 }$$

or if we measure only Σ_H i (as is the case in dwarfs), then SFE(HI) = SFE(H₂)R_mol.

The balance between GMC/H₂ formation and destruction will set $R_{\rm mol} = \Sigma _{{\rm H}_2}/\Sigma _{{\rm H}\,\mathsc{i}}$. If GMCs with a fixed lifetime form over a free-fall time or orbital time, then R_mol ∝ τ⁻¹_ff or R_mol ∝ τ⁻¹_orb (Section 5.4), which we have noted in Table 1. Combined with Equation (8), an expression for R_mol predicts the SFE.

Wong & Blitz (2002), Blitz & Rosolowsky (2004), and Blitz & Rosolowsky (2006) explicitly considered R_mol. Following Elmegreen (1989) and Elmegreen & Parravano (1994), they identified pressure as the critical quantity that sets the ability of the ISM to form H₂. They showed that the midplane hydrostatic gas pressure, P_h, correlates with this ratio in the inner parts of spiral galaxies.

Pressure, which is directly proportional to the gas volume density, should affect both the rate of H₂ formation/destruction and the likelihood of a gravitationally unstable overdensity condensing out of a turbulent ISM (Elmegreen 1989; Elmegreen & Parravano 1994). Elmegreen (1989) gives the following expression for P_h:

$$\begin{equation} P_{\rm h} \approx \frac{\pi }{2} G \Sigma _{\rm gas} \left(\Sigma _{\rm gas} + \frac{\sigma _{\rm g}}{\sigma _{*,z}} \Sigma _* \right), \end{equation} \tag{ 9 }$$

and Elmegreen (1993) predicted that the fraction of gas in the molecular phase depends on both P_h and the interstellar radiation field, j, via R_mol ∝ P^2.2j⁻¹. If $\Sigma _{\rm SFR} \propto \Sigma _{{\rm H}_2}$ and we make the simple assumption that j ∝ Σ_SFR, then Elmegreen (1993) predicts

$$\begin{equation} R_{\rm mol} \propto P_{h}^{1.2}\quad {\rm or}\quad \Sigma _{{\rm H}_2} = \Sigma _{{\rm H}\,\mathsc{i}} P_{h}^{1.2}, \end{equation} \tag{ 10 }$$

which combines with Equation (8) to predict the SFE.

Wong & Blitz (2002) and Blitz & Rosolowsky (2006) found observational support for Equation (10). Using a modified Equation (9) appropriate where Σ_* ≳ Σ_gas, Blitz & Rosolowsky (2006) fitted a power law of the form

$$\begin{equation} R_{\rm mol} = \frac{\Sigma _{\rm {H2}}}{\Sigma _{\rm {HI}}} = \left(\frac{P_{h}}{P_0} \right)^{\alpha }, \end{equation} \tag{ 11 }$$

finding P₀ = 4.3 × 10⁴ cm⁻³ K, the observed pressure where the ISM is equal parts of H i and H₂, and a best-fit exponent α = 0.92. Wong & Blitz (2002) found α = 0.8. Robertson & Kravtsov (2008) recently found support from simulations for α ∼ 0.9.

Kennicutt (1989, 1998a) and Martin & Kennicutt (2001) argued that star formation is only widespread where the gas disk is unstable against large-scale collapse. Following Toomre (1964), the condition for instability in a thin gas disk is

$$\begin{equation} Q_{\rm gas} = \frac{\sigma _g \kappa }{\pi G \Sigma _{\rm gas}} < 1. \end{equation} \tag{ 12 }$$

where σ_g is the gas velocity dispersion, G is the gravitational constant, and κ is the epicyclic frequency, calculated via

$$\begin{equation} \kappa = 1.41 \frac{v(r_{\rm gal})}{r_{\rm gal}} \sqrt{1+\beta }, \end{equation} \tag{ 13 }$$

where β = dlog v(r_gal)/dlog r_gal.

Martin & Kennicutt (2001) found that H ii regions are common where Σ_gas exceeds a critical surface density derived following Equation (12),

$$\begin{equation} \Sigma _{{\rm crit,}Q} = \alpha _Q \frac{\sigma _g \kappa }{\pi G}. \end{equation} \tag{ 14 }$$

In regions where Σ_gas is above this threshold, gas is unstable against large-scale collapse, which leads to star formation. Below the threshold, Coriolis forces counteract the self-gravity of the gas and suppress cloud/star formation. The factor α_Q is an empirical calibration, the observed average value of 1/Q_gas at the star formation threshold. For an ideal thin gas disk, α_Q = 1. At the edge of star-forming disks, Kennicutt (1989) found α_Q = 0.63 and Martin & Kennicutt (2001) found α_Q = 0.69 (Q_gas ∼ 1.5).

Kennicutt (1989) and Martin & Kennicutt (2001) mentioned the influence of stars as a possible cause for Q_gas > 1 at the star formation threshold. Hunter et al. (1998a) presented an in-depth discussion of how several factors influence α_Q, for example stars and viscosity lower it, while the thickness of the gas disk raises it. Kim & Ostriker (2001, 2007) argued, based on simulations, that the observed threshold corresponds to the onset of nonlinear, nonaxisymmetric instabilities. Schaye (2004) and de Blok & Walter (2006) suggested a different explanation: α_Q ≠ 1, because σ_g has been systematically mishandled; they pointed out that σ_g measured from 21 cm emission will overestimate the true velocity dispersion of gas in a cold phase.

The stellar potential well may substantially affect the stability of the gas disk. Rafikov (2001) extended work by Jog & Solomon (1984) to provide a straightforward way to calculate the instability of a gas disk in the presence of a collisionless stellar disk. Rafikov defined

$$\begin{equation} Q_{\rm stars} = \frac{\sigma _{*,r} \kappa }{\pi G \Sigma _*}, \end{equation} \tag{ 15 }$$

where σ_*,r is the (radial) velocity dispersion of stars and Σ_* is the stellar mass surface density. The condition for instability in the gas disk is

$$\begin{equation} \frac{1}{Q_{\rm stars + gas}} = \frac{2}{Q_{\rm stars}} \frac{q}{1+q^2} + \frac{2}{Q_{\rm gas}} R \frac{q}{1 + q^2 R^2} > 1, \end{equation} \tag{ 16 }$$

where q = kσ_*,r/κ, with k the wavenumber of the instability being considered, and R = σ_g/σ_*,r. The minimum value of Q_stars+gas indicates whether the gas disk is unstable to large-scale collapse. In our sample, typical values of q correspond to wavelengths λ = 2π/k ≈ 1–5 kpc at maximum instability.

Hunter et al. (1998a) and Blitz & Rosolowsky (2004) observed strong correlations between star and GMC formation and the distribution of stars, consistent with stellar gravity playing a key role in star formation. Yang et al. (2007) recently showed that Q_stars+gas does an excellent job of predicting the location of star formation in the Large Magellanic Cloud (LMC), and Boissier et al. (2003) showed that including stars improves the correspondence between Q and star formation in disk galaxies. Li et al. (2005, 2006) found the same results from numerical simulations of disk galaxies, that is, stability against large-scale collapse critically depends on the stellar potential well, with star formation where Q_stars+gas ≲ 1.6.

Motivated by the failure of the Toomre Q_gas threshold in dwarf irregular galaxies, Hunter et al. (1998a) suggested that collecting the material for cloud formation may be easier than implied by Q_gas, for example, through the aid of magnetic fields (see also Kim & Ostriker 2001). They hypothesized that the destructive influence of galactic shear may instead limit where GMCs can form and describe a threshold that depends on the ability of clouds to form in the time allowed by shear.

This threshold is based on the local shear rate, described by Oort's A constant:

$$\begin{equation} A = - 0.5 r_{\rm gal} \frac{d\Omega }{dr_{\rm gal}}. \end{equation} \tag{ 17 }$$

Substituting Ω = v(r_gal)/r_gal,

$$\begin{equation} A = 0.5 \left(\frac{v(r_{\rm gal})}{r_{\rm gal}} - \frac{dv(r_{\rm gal})}{dr_{\rm gal}} \right) = 0.5 \frac{v(r_{\rm gal})}{r_{\rm gal}} (1 - \beta). \end{equation} \tag{ 18 }$$

Then the threshold has the form

$$\begin{equation} \Sigma _{\rm crit,A} = \frac{\alpha _A \sigma _g A}{\pi G}. \end{equation} \tag{ 19 }$$

Hunter et al. (1998a) suggested α_A = 2.5, but this normalization for Σ_crit,A is relatively uncertain. The value chosen by Hunter et al. (1998a) corresponds to perturbations growing by a factor of ∼100 during the time allowed by shear, which roughly matches both the surface density contrast between Σ_H i and a GMC and the condition Q_gas ≲ 1 where dv(r_gal)/dr_gal = 0.

The practical advantage of shear over Q_gas is that shear is low in dwarf galaxies and the inner disks of spiral galaxies (β = 1 for solid body rotation), both locales where widespread star formation is observed. In the outer disks of spiral galaxies—where star formation cutoffs are observed—rotation curves tend to be flat (β = 0) so that Σ_crit,A and Σ_crit,Q reduce to the same form.

The very long time needed to assemble a massive GMC from coagulation of smaller clouds suggests that most GMCs in galaxy disks form “top down” (e.g., McKee & Ostriker 2007). However, this does not necessarily require the whole gas disk to be unstable. Where cold H i is abundant, the lower velocity dispersion associated with this phase may render the ISM locally unstable (Schaye 2004), leading to the formation of GMCs and stars.

Therefore, instead of large-scale gravitational instability or cloud destruction by shear, the ability to form a cold neutral medium (McKee & Ostriker 1977; Wolfire et al. 2003) may regulate GMC formation. Schaye (2004) argued, based on modeling, that near the cutoffs observed by Martin & Kennicutt (2001), gas becomes mostly cold H i and H₂, σ_g drops accordingly, and Q becomes less than 1 in the cold gas. In a similar vein, Elmegreen & Parravano (1994) suggested that the SFE in the outer parts of galaxies drops because the pressure becomes too low to allow a cold phase to form even given perturbations, for example, from supernova shocks. Braun (1997) found support for this idea using 21 cm observations; he associated networks of high surface brightness filaments with cold H i and showed that these filaments are pervasive across the star-forming disk, but become less common at large radii (though work on THINGS by A. Usero et al. 2008, in preparation, calls this result into question).

Schaye (2004) modeled the ISM to estimate where the average temperature drops to ≈500 K, the molecular fraction reaches ≈10⁻³, and Q_gas ≈ 1, which are good indicators that cold H i is common and H₂ formation is efficient. These all occur where Σ_gas exceeds

$$\begin{equation} \Sigma _{\rm S04} \approx \frac{6.1}{{M}_{\odot }\;{\rm pc}^{-2}}\;f_{\rm g}^{0.3} \left(\frac{Z}{0.1 Z_{\odot }} \right)^{-0.3} \left(\frac{I}{10^6\;\rm {cm}^{-2}\;{\rm s}^{-1} }\right)^{0.23}, \end{equation} \tag{ 20 }$$

where f_g ≈ Σ_gas/(Σ_gas + Σ_*) is the fraction of mass in gas (we assume a two-component disk), Z is the metallicity of the ISM, and I is the flux of ionizing photons. Σ_S04 also depends on the ratio of thermal to turbulent pressure and higher-order terms not shown here. Schaye (2004) selected fiducial values to match those expected in outer galaxy disks, but concluded that the influence of Z, f_g, and the radiation field is relatively small. Most reasonable values yield Σ_S04 ≈ 3–10 M_☉ pc⁻².

Schaye (2004) argued that a simple column density threshold may work as well as dynamical thresholds. This agrees with the observation by, for example, Skillman (1987) and de Blok & Walter (2006) that a simple H i column density threshold does a good job of predicting the location of star formation in dwarf irregulars. This threshold, Σ_H i ≈ 10 M_☉ pc⁻², also corresponds to the surface density above which H i is observed to saturate (Martin & Kennicutt 2001; Wong & Blitz 2002; Bigiel et al. 2008), that is, gas in excess of this surface density in spiral galaxies is in the molecular phase.

In Section 1, we argued that a critical observation for theories of galactic-scale star formation is that the SFE declines in the outer parts of spiral galaxies. Figure 1 shows this via plots of SFE against galactocentric radius (normalized to r₂₅) in our two subsamples.

In spiral galaxies (top row), the SFE is nearly constant where the ISM is mostly H₂ (magenta), which agrees with our observation of a linear relationship between $\Sigma _{{\rm H}_2}$ and Σ_SFR in Bigiel et al. (2008). Typically, the ISM is equal parts of H i and H₂ at r_gal = 0.43 ± 0.18r₂₅ (Section 5.2). Outside this transition, the SFE decreases steadily with increasing radius. This decline continues to r_gal ≳ r₂₅, the limit of our data. This is similar, though not identical, to the observation by Kennicutt (1989) and Martin & Kennicutt (2001) that star formation is not widespread beyond a certain radius.

The SFE in spirals can be reasonably described in two ways. First, a constant SFE in the inner parts of galaxies followed by a break at 0.4r₂₅ (slightly inside the transition to a mostly-H i ISM):

$$\begin{equation} {\rm SFE} = \left\lbrace \begin{array}{@{}ll} 4.3 \times 10^{-10} & r_{\rm gal} < 0.4 r_{25} \\ \dsty 2.2 \times 10^{-9} \exp \left(\frac{-r_{\rm gal}}{0.25 r_{25}}\right) & r_{\rm gal} > 0.4 r_{25} \\ \end{array} \right. {\rm yr}^{-1}. \end{equation} \tag{ 21 }$$

Alternatively, we can adopt Equation (8), appropriate for a fixed SFE (H₂), and derive the best-fit exponential relating R_mol to r_gal:

$$\begin{eqnarray} {\rm SFE} &=& 5.25 \times 10^{-10} \frac{R_{\rm mol}}{R_{\rm mol} + 1}\; {\rm yr}^{-1}, \nonumber\\&&[-12pt]\\ R_{\rm mol} &=& 10.6 \exp (-r_{\rm gal} / 0.21 r_{25}),\nonumber \end{eqnarray} \tag{ 22 }$$

which appears as a thick dashed line in the upper panels of Figure 1. The two fits reproduce the observed SFE with similar accuracy; the scatter about each is ≈0.26 dex, slightly better than a factor of 2.

In dwarf galaxies (lower panels), we observe a steady decline in the SFE with increasing radius for all r_gal, approximately described by

$$\begin{equation} {\rm SFE} = 1.45 \times 10^{-9} \exp (-r_{\rm gal} / 0.25 r_{25})\;{\rm yr}^{-1} \end{equation} \tag{ 23 }$$

with ∼0.4 dex scatter about the fit, that is, a factor of 2–3.

In dwarfs, we take $\Sigma _{\rm gas} \approx \Sigma _{{\rm H}\,\mathsc{i}}$, so that ${\rm SFE}= \Sigma _{\rm SFR} / \Sigma _{{\rm H}\,\mathsc{i}}$. For comparison with Equation (22), however, we rewrite Equation (23) assuming that SFE (H₂) = 5.25 × 10⁻¹⁰ yr⁻¹, the value measured in spirals. In terms of R_mol, Equation (23) becomes

$$\begin{eqnarray} {\rm SFE} &=& \frac{\Sigma _{\rm SFR}}{\Sigma _{\rm H\,{\mathsc{i}}}} = 5.25 \times 10^{-10}\;R_{\rm mol}\;{\rm yr}^{-1},\nonumber\\&&[-12pt]\\ \nonumber R_{\rm mol} &=& 2.76 \exp (-r_{\rm gal} / 0.25 r_{25}). \end{eqnarray} \tag{ 24 }$$

The outer parts of dwarfs, r_gal ≳ 0.4r₂₅, appear similar to the outer disks of spiral galaxies in Figure 1. Surprisingly, however, we find the SFE to be higher in the central parts of dwarf galaxies than in the molecular gas of spirals. A higher SFE in dwarf galaxies is quite unexpected. Their lower metallicities, more intense radiation fields, and weaker potential wells should make gas less efficient at forming stars. A simple explanation for the high observed SFE is the presence of a significant amount of H₂. Figure 1 assumes that $\Sigma _{\rm gas} \approx \Sigma _{{\rm H}\,\mathsc{i}}$ in dwarfs. If we miss a significant amount of H₂ along a LOS, we will overestimate the SFE because we underestimate Σ_gas. We quantify the possibility of substantial H₂ in dwarfs in Section 5.3, but the magnitude of the effect can be directly read from Equation (24). At r_gal = 0, if dwarf galaxies have the same SFE (H₂) as spirals, R_mol ≈ 2.76, that is, $\Sigma _{{\rm H}_2} \approx 2.76 \Sigma _{{\rm H}\,\mathsc{i}}$.

Galactocentric radius is probably not intrinsically important to a local process like star formation, but Figure 1 suggests that local conditions covariant with radius have a large effect on the ability of gas to form stars. The radius, r₂₅, that we use to normalize the x-axis is defined by an optical isophote and thus measures stellar light. Therefore, r₂₅ is closely linked to the stellar distribution.

Figure 2 shows this link directly. We plot the stellar scale length, l_*, measured via an exponential fit to the 3.6 μm profile as a function of r₂₅ for our spiral subsample. We see that r₂₅ = (4.6 ± 0.8)l_* and that we could have equivalently normalized the x-axis in Figure 1 by l_*. We may then suspect that the stellar surface density, Σ_*, underlies the well-defined relation between SFE and r_gal observed in Figure 1.

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In Figure 3, we explore this connection by plotting SFE as a function of Σ_*. In both spiral and dwarf galaxies, we see a nearly linear relationship between SFE and Σ_* where the ISM is H i-dominated (blue points).

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A basic result of THINGS is that over the optical disk of most star forming galaxies, the H i surface density varies remarkably little (Appendices E and F and Walter et al. 2008). Inspecting our atlas, one sees that $\Sigma _{{\rm H}\,\mathsc{i}} \approx 6$ M_☉ pc⁻² (within a factor of 2) over a huge range of local conditions, including most of the optical disk in most galaxies. Because Σ_gas is nearly constant in the H i-dominated (blue) regime, SFE ∝ Σ_* approximately defines a line of fixed specific SFR (SSFR), that is, SFR per unit stellar mass.

The inverse of the SSFR is the stellar assembly time, τ_* = Σ_*/Σ_SFR. This is the time required for the present SFR to build up the observed stellar disk. In our spiral subsample, the mean log₁₀τ_* ≈ 10.5 ± 0.3, that is, 3.2 × 10¹⁰ years or slightly more than 2 Hubble times. Dwarf galaxies have shorter assembly times, log₁₀τ_* ≈ 10.2 ± 0.3 years, about a Hubble time (dashed lines in Figure 3 show these values using average values of Σ_gas for each subsample). Taking these numbers at face value, dwarfs are forming stars at about their time-average rate, while spirals are presently forming stars at just under half of their average rate.

We only observe SFE ∝Σ_* where the ISM is mostly H i. Where the ISM is mostly H₂ in spirals galaxies, we observe a constant SFE at a range of Σ_*; similar to the constancy as a function of r_gal observed in the inner parts of spirals (Figure 1). The transition between these two regimes occurs at Σ_* = 81 ± 25 M_☉ pc⁻² (Section 5.2) in spirals. In dwarfs, LOSs with Σ_* above this transition value exhibit systematically high SFE, lending further, albeit indirect, support to the idea that these points correspond to unmeasured H₂.

Figures 1 and 3 show that where the ISM is mostly H₂, the SFR per unit of gas (SFE) is nearly constant and that where the ISM is mostly H i, the SFR per unit of stellar mass (SSFR) is nearly constant. Together these observations suggest that H₂, stars, and star formation have similar structure with all three embedded in a relatively flat distribution of H i. Figure 4 shows that the scale lengths of these three distributions are, in fact, comparable. The SFR (black) and CO (gray) scale lengths of spiral galaxies are both roughly equal to the stellar scale length:

$$\begin{equation} l_{\rm CO} = (0.9 \pm 0.2) l_*\quad {\rm and}\quad l_{\rm SFR} = (1 \pm 0.2) l_*. \end{equation} \tag{ 25 }$$

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Regan et al. (2001) also found that l_CO ≈ l_* comparing K-band maps to BIMA SONG and Young et al. (1995) found l_CO ≈ 0.2r₂₅, which is almost identical to our l_CO ∼ 0.9l_* and (4.6 ± 0.8)l_* = r₂₅.

This link between Σ_* and the SFE is somewhat surprising because it is common to view Σ_SFR, and thus the SFE, as set largely by Σ_gas alone over much of the disk of a galaxy (following, e.g., Kennicutt 1998a). In Figure 5, we show this last slice through SFR-stars-gas parameter space, plotting SFE as a function of Σ_gas.

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As in Figures 1 and 3, we observe two distinct regimes. In spirals, where Σ_gas > 14 ± 6 M_☉ pc⁻² (Section 5.2), the ISM is mostly H₂ and we observe a fixed SFE. This Σ_gas, shown by a vertical dotted line, corresponds approximately to both the N(H) ∼ 10²¹ cm⁻² star formation threshold noted by Skillman (1987) and the saturation value for H i observed by, for example, Martin & Kennicutt (2001) and Wong & Blitz (2002) (and seen strikingly in THINGS at Σ_gas = 12 M_☉ pc⁻² by Bigiel et al. 2008, who quote Σ_gas = 9 M_☉ pc⁻² but do not include helium).

In contrast to r_gal and Σ_*, Σ_gas does not exhibit a clear correlation with the SFE where the ISM is mostly H i. Instead, over the narrow range Σ_gas ≈ 5–10 M_☉ pc⁻², the SFE varies from ∼3 × 10⁻¹¹ to 10⁻⁹ yr⁻¹. We see little evidence that Σ_H i plays a central role regulating the SFE in either spirals or dwarfs. Rather, the most striking observation in Figure 5 is that Σ_H i exhibits a narrow range of values over the optical disk and is therefore itself likely subject to some kind of regulation.

The possibility of a missed reservoir of molecular gas in dwarfs is again evident from the lower panels in Figure 5. A subset of data has SFE higher than that observed for H₂ in spirals and just to the left of the H i saturation value. If H₂ were added to these points, they would move down (as the SFE decreases) and to the right (as Σ_gas increases), potentially yielding a data distribution similar to that we observe in spirals.

A dashed line in Figure 5 illustrates SFE ∝Σ^0.5_gas, expected if the SFE is proportional to the free-fall time in a fixed scale height gas disk (similar to the Kennicutt–Schmidt law; Kennicutt 1998a). The normalization matches the H₂-dominated parts of spirals and roughly bisects the range of SFE observed for dwarfs, but large areas have much lower SFE than one would predict from this relation. Adjusting the normalization can move the line up or down but cannot reproduce the distribution of data observed in Figure 5.

The culprit here is the small dynamic range in Σ_H i. Because Σ_H i does not vary much across the disk, while the SFE does, the free-fall time in a fixed scale height disk, or any other weak dependence of SFE on Σ_gas alone, cannot reproduce variations in the SFE where the ISM is mostly H i. A quantity other than Σ_gas must play an important role at radii as low as ∼0.5r₂₅ (a fact already recognized by Kennicutt 1989, among others).

In Section 4.1, we saw that where the ISM is mostly H i, the SFE correlates better with Σ_* than with Σ_gas. This might be expected if the stellar potential well plays a central role in setting the volume density of the gas, ρ_gas, because Σ_* varies much more strongly with radius than Σ_H i. In Section 2, we presented two predictions relating SFE to ρ_gas: that the timescale over which GMCs form depends on the τ_ff, the free-fall time in a gas disk with a scale height set by hydrostatic equilibrium,⁹ and that the ratio $R_{\rm mol} = \Sigma _{{\rm H}_2}/\Sigma _{{\rm H}\,\mathsc{i}}$ primarily depends on midplane gas pressure, P_h.

Under our assumption of a fixed σ_gas, P_h ∝ ρ_gas and both predictions can be written as a power law relating SFE or R_mol to P_h. In Figure 6, we plot SFE as a function of ρ_gas and P_h (top and bottom x-axis), estimated from hydrostatic equilibrium (Equation 9).

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Where the ISM is mostly H₂ (magenta points) in spirals (top row), we observe no clear relationship between P_h and SFE, further evidence that SFE (H₂) is largely decoupled from global conditions in the ISM in our data.

Where the ISM is mostly H i (blue points) in dwarf galaxies and the outer parts of spirals, the SFE correlates with P_h. P_h predicts the SFE notably better than Σ_gas in this regime, supporting the idea that the volume density of gas (at least H i) is more relevant to star formation than surface density. Wong & Blitz (2002) and Blitz & Rosolowsky (2006) observed a continuous relationship between R_mol and P_h, mostly where $\Sigma _{{\rm H}_2} \gtrsim \Sigma _{{\rm H}\,\mathsc{i}}$. Figure 6 suggests that such a relationship extends well into the regime where H i dominates the ISM.

The solid line in Figure 6 illustrates the case of 1% of the gas formed into stars per τ_ff(SFE ∝ ρ^0.5_gas), a typical value at the H i-to-H₂ transition in spirals (Section 5.2). Adjusting the normalization slightly, such a line can intersect both the high and low ends of the observed SFE in spirals, but predicts variations in SFE (H₂) that we do not observe and is too shallow to describe dwarf galaxies.

The dash-dotted line shows R_mol ∝ τ⁻¹_ff ∝ P^0.5_h, expected for GMC formation over a free-fall time. In dwarf galaxies, where we take $\Sigma _{\rm gas} = \Sigma _{{\rm H}\,\mathsc{i}}$, this is equivalent to SFE∝τ⁻¹_ff. This description can describe spirals at high and intermediate P_h, but is too shallow to capture the drop in SFE at large radii in spirals and across dwarf galaxies. If τ_ff is the characteristic timescale for GMC formation, effects other than just an increasing timescale must suppress cloud formation in these regimes.

A dashed line shows the steeper dependence, R_mol ∝ P^1.2_h, expected for low R_mol based on modeling by Elmegreen (1993). This may be a reasonable description of both spiral and dwarf galaxies (note that at high SFE, P_h may be underestimated in dwarf galaxies because we fail to account for H₂). We explore how P_h relates to R_mol more in Section 5.

The orbital timescale, τ_orb, varies strongly with radius, and Kennicutt (1998a) found τ_orb to be a good predictor of disk-averaged SFE. In Figure 7, we plot SFE as a function of τ_orb in our sample.

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The solid line shows 6% of the gas converted to stars per τ_orb and is a reasonable match to spirals near the H i-to-H₂ transition (vertical dotted line). This value agrees with the range of efficiencies found by Wong & Blitz (2002) and with Kennicutt (1998a), who found ≈7% of gas converted to stars per τ_orb averaged over galaxy disks (converted to our adopted initial mass function, IMF). Like Wong & Blitz (2002), we do not observe a clear correlation between SFE and τ_orb where the ISM is mostly H₂.

Where the ISM is mostly H i (blue points), the SFE clearly anticorrelates with τ_orb in both spiral and dwarf galaxies. However, we do not observe a constant efficiency per τ_orb. In both subsamples, SFE drops faster than τ_orb increases, so that data at large radii (longer τ_orb, lower SFE) show lower efficiency per τ_orb than those from inner galaxies. Although τ_orb correlates with the SFE, the drop in τ_orb is not enough on its own to explain the drop in SFE.

We reach the same conclusion if we posit that τ_orb is the relevant timescale for GMC formation, so that R_mol ∝ τ⁻¹_orb. The dashed lines in Figure 7 show this relation combined with a fixed SFE (H₂) and normalized to R_mol = 1 at τ_orb = (1.8 ± 0.4) × 10⁸ yr, which we observe at the H i-to-H₂ transition in spirals (Section 5.2). This dependence is even shallower than SFE ∝τ⁻¹_orb and cannot reproduce the SFE in both inner and outer disks by itself. If τ_orb is the relevant timescale for cloud formation, then the fraction of gas that actively forms stars must substantially vary between the middle and the edge of the optical disk.

Tan (2000) suggested that cloud–cloud collisions regulate the SFE. The characteristic timescale for such collisions is τ_orb modified by the effects of galactic shear. In Figure 7, we saw that the SFE of molecular gas is not a strong function of τ_orb. Therefore, in Figure 8, we plot the SFE as a function of β, the logarithmic derivative of the rotation curve (we plot SFE against Q_gas, the other component of this timescale in Section 4.3.1). This isolates the effect of differential rotation; β = 0 for a flat rotation curve and β = 1 for solid body rotation (no shear).

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Figure 8 shows a simple relationship between β and SFE in spirals: β>0 is associated with high SFE. High β occurs almost exclusively at low radius (where the rotation curve rises steeply) and in these regions, the ISM is mostly H₂ with accordingly high SFE. However, the outer disks of spirals have β ∼ 0 and a wide range of SFE. Beyond the basic relationship, it is unclear that β has utility predicting the SFE. In particular, we see no clear relationship between SFE and β where the ISM is mostly H₂ (magenta points). If collisions between bound clouds regulate the SFE, we would expect an anticorrelation between β and SFE because cloud collisions are more frequent in the presence of greater shear.

In dwarf galaxies, increasing β mostly corresponds to increasing SFE. This relationship has the sense of the shear threshold proposed by Hunter et al. (1998a) that where rotation curves are nearly solid body low shear allows clouds to form via instabilities aided by magnetic fields (also see Kim & Ostriker 2001). The rotation curves in dwarf galaxies rise more slowly than those in spirals, leading to β>0 over a larger range of radii in dwarf galaxies and limiting β = 0 to the relative outskirts of the galaxy. A positive correlation between β and SFE is opposite the sense expected if cloud collisions are important: at high β, collisions should be less frequent.

Figure 11 shows SFE as a function of Q_gas, Toomre's Q parameter for a thin gas disk; the top left panels in Figures 9 and 10 show Q_gas as a function of radius.

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In each plot, a gray area indicates the theoretical condition for instability. We immediately see that almost no area in our sample is formally unstable. Rather, most LOSs are strikingly stable, Q_gas ∼ 4 is typical inside ∼0.8r₂₅ and Q_gas > 10 is common.

We find no clear evidence of a Q_gas threshold (at any value) that can unambiguously distinguish regions with high SFE from those with low SFE. In spirals, Q_gas ≲ 2.5 appears to be a sufficient, but by no means necessary, condition for high SFE; there are also areas where the ISM is mostly H₂, SFE is quite high, and Q_gas ≳ 10. In dwarfs, Q_gas appears, if anything, anticorrelated with SFE, though this may partially result from incomplete estimates of Σ_gas.

These conclusions appear to contradict the findings by Kennicutt (1989) and Martin & Kennicutt (2001), who found marginally stable gas (Q_gas ∼ 1.5) across the optical disk with a rise in Q_gas corresponding to dropping SFE at large radii. In fact, after correcting for different assumptions, our median Q_gas matches theirs quite well. Both Kennicutt (1989) and Martin & Kennicutt (2001) assumed X_CO = 2.8 × 10²⁰ cm⁻² (K km s⁻¹)⁻¹ and σ_gas = 6 km s⁻¹, while we take X_CO = 2.0 × 10²⁰ cm⁻² (K km s⁻¹)⁻¹ and σ_gas = 11 km s⁻¹, respectively. As a result, we estimate less H₂ and more kinetic support than they did for the same observations. If we match their assumptions, our median Q_gas in spirals and the outer parts of dwarfs agree quite well with their threshold value, though we find the central regions of dwarfs systematically above this value (as did Hunter et al. 1998a). We show this in Figures 9–11 by plotting the Martin & Kennicutt threshold converted to our assumptions (Q_gas ∼ 3.9) as a a dashed line.

The main observational difference between our result and Martin & Kennicutt (2001) is that Q_gas shows much more scatter in our analysis. As a result, a systematic transition from low to high Q_gas near the edge of the optical disk is not a universal feature of our data, though a subset of spiral galaxies does show increasing Q_gas at large radii (Figure 9).

This discrepancy in Q_gas derived from similar data highlights the importance of assumptions. The largest effect comes from σ_gas, which we measure to be ≈11 km s⁻¹ and roughly constant in H i-dominated outer disks (Appendix B). We assume σ_gas to be constant everywhere, an assumption that may break down on small scales and in the molecular ISM. In this case, we expect σ_gas to be locally lower than the average value, lowering Q_gas and making gas less stable. Black dots in the upper right panel of Figure 11 show the effect of changing σ_gas from 11 km s⁻¹ (our value) to 6 km s⁻¹ (the Martin & Kennicutt value) and then to 3 km s⁻¹, the value expected and observed for a cold H i component (e.g., Young et al. 2003; Schaye 2004; de Blok & Walter 2006). If most gas is cold, then Q_gas may easily be ≲1 for this component (if only a small fraction of gas is cold, the situation is less clear).

Stars dominate the baryon mass budget over most of the areas we study and stellar gravity may be expected to affect the stability of the gas disk. In Section 2, we described a straightforward extension of Q_gas to the case of a disk containing gas and stars (Rafikov 2001). In Figure 12, we plot SFE as a function of this parameter, Q_stars+gas, which we plot as a function of radius in the top right panels of Figures 9 and 10.

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The gray region indicates where gas is unstable to axisymmetric collapse. Including stars does not render large areas of our sample unstable, but it does imply that most regions are only marginally stable, Q_stars+gas ∼ 1.6. This in turn suggests that it is not so daunting to induce collapse as one would infer from only Q_gas.

In addition to lower values, Q_stars+gas exhibits a much narrower range of values than Q_gas; mostly areas in both spiral and dwarf galaxies show Q_stars+gas = 1.3–2.5. This may offer support to the idea of self-regulated star formation, but it also means that Q_stars+gas offers little leverage to predict the SFE. High SFE, mostly molecular regions show the same Q_stars+gas as low SFE regions from outer disks (indeed, the highest values we observe come from the central parts of spiral galaxies).

As with Q_gas, our assumptions have a large impact on Q_stars+gas. In addition to σ_gas and X_CO(which affect the calculation via Q_gas), the stellar velocity dispersion, σ_*, and the mass-to-light ratio, ϒ^K_⋆, strongly affect our stability estimate. We assume that σ_* ∝ Σ^0.5_* in order to yield a constant stellar scale height. If we instead fixed σ_*, we would derive Q_stars+gas, increasing steadily with radius. Radial variations in ϒ^K_⋆ may create a similar effect.

Boissier et al. (2003) found results similar to our own when they incorporated stars in their stability analysis; they adopted a lower σ_gas than we did, but also lower X_CO and the effects roughly offset. Yang et al. (2007) recently derived Q_stars+gas across the LMC and found widespread instability that corresponded well with the distribution of star formation. If we match their adopted σ_gas (5 km s⁻¹) and assumptions regarding σ_* (constant at 15 km s⁻¹), we also find widespread instability throughout our dwarf subsample, Q_stars+gas, decreasing with radius; we find a similar result for spirals if we fix σ_* at a typical outer disk value.

Our approach is motivated by observations of disk galaxies (see Appendix B), but direct observations of σ_* at large radii are still sorely needed.

If clouds form efficiently, for example, through the aid of magnetic fields to dissipate angular momentum, then Hunter et al. (1998a) suggest that the time available for a perturbation to grow in the presence of destructive shear may limit where star formation is widespread. Kim & Ostriker (2001) described a similar scenario where magneto-Jeans instabilities can grow in regions with weak shear or strong magnetic fields. In the bottom left panels of Figures 9 and 10, we plot this shear threshold as a function of radius, and in Figure 13, we compare it to the SFE.

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The gray region shows the condition for instabilities to grow into GMCs, Σ_crit,A/Σ_gas < 1. This matches the condition Q_gas < 1 where β = 0, for example in outer disks of spirals. In the inner parts of spirals and in dwarf galaxies, however, Σ_crit,A/Σ_gas is lower than Q_gas, that is, the conditions for star formation are more nearly supercritical (because shear is low in these regions). These areas harbor H₂ or widespread star formation, so supercritical values are expected.

This trend of more supercritical data at lower radii agrees with the steady increase of SFE with decreasing radius that we saw in Figure 1. However, the scatter in Σ_crit,A/Σ_gas < 1 is as large as that in Q_gas (as one would expect from their forms; see Table 1). As a result, a direct plot of SFE against Σ_crit,A/Σ_gas does not yield clear threshold behavior or a strong correlation between Σ_crit,A/Σ_gas and SFE. The strongest conclusion we can draw is that the inner parts of both spiral and dwarf galaxies are marginally stable for the shear threshold (an improvement over Q_gas and Q_stars+gas in these regions).

Even where the ISM is stable against gravitational collapse on large scales, star formation may still proceed if a cold (narrow-line width) phase can form locally and thus induce gravitational instability in a fraction of the gas (recall the effect of lower σ_gas in Figure 11). Schaye (2004) argued that this is the usual path to star formation in the outer parts of galaxies and modeled the critical gas surface density for such a phase to form, Σ_S04. The bottom right panels in Figures 9 and 10 show Σ_S04/Σ_gas as a function of radius, and Figure 14 shows the SFE as a function of this ratio. The gray area in both figures shows where a cold phase can form.

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We calculate Σ_S04 from Equation (20), which depends on I/(10⁶ cm⁻² s⁻¹), the flux of ionizing photons. In outer disks, we assume I = 10⁶ cm⁻² s⁻¹, Schaye's fiducial value, and in inner disks, we take I ∝ Σ_SFR:

$$\begin{equation} I \approx 10^6\; {\rm cm^{-2}\; s^{-1}} \left(\frac{\Sigma _{\rm SFR}}{5 \times 10^4\; {M}_{\odot }\; {\rm yr}^{-1}\; {\rm kpc}^{-2}}\right). \end{equation} \tag{ 26 }$$

The normalization is the average Σ_SFR between 0.8 and 1.0r₂₅ in our spiral subsample.

Equation (20) also accounts for variations about Schaye's fiducial metallicity Z = 0.1 Z_☉, typical for the outer disk of a spiral. We lack estimates of Z and so neglect this term but note the sense of the uncertainty. Inner galaxy disks will tend to have higher metallicities, which will lower Σ_S04. We already find Σ_gas > Σ_S04 over most inner disks; therefore, missing Z seems unlikely to seriously bias our results.

Figures 9, 10, and 14 show that we expect a cold phase over most of the disk in both spiral and dwarf galaxies. In our spiral subsample, most data inside r_gal ∼ 0.9r₂₅ meet this criterion. Because most subcritical data come from large radii, we also find that most LOSs with Σ_gas < Σ_S04 exhibit low SFEs or upper limits.

Because most data are supercritical, the Schaye (2004) threshold is of limited utility for predicting the SFE within a galaxy disk. Schaye (2004) did not predict the ratio of H₂ to H i where cold gas formed; he was primarily concerned with the edges of galaxies. Figures 9 and 14 broadly confirm that his proposed threshold matches both the edge of the optical disk and the typical threshold found by Martin & Kennicutt (2001).

This relevance of this comparison to the SFE within the optical disk is that based on the Schaye (2004) model, we expect a widespread narrow-line phase throughout most of our galaxies (Wolfire et al. 2003 obtained a similar result for the Milky Way). This suggests that cold-phase formation followed by collapse may be a common path to star formation and offers a way to form stars in our otherwise stable disks.

In Section 2, we discuss two basic ways that R_mol might be set by environment. First, the timescale to form GMCs may depend on local conditions. If H i and H₂ are in approximate equilibrium, with the entire neutral ISM actively cycling between these two phases, then

$$\begin{equation} R_{\rm mol} = \frac{\Sigma _{{\rm H}_2}}{\Sigma _{{\rm H}\,\mathsc{i}}} \approx \frac{{\rm GMC\ lifetime}}{\tau ({\rm H\,{\mathsc i}}\rightarrow {\rm H}_2)}. \end{equation} \tag{ 30 }$$

For constant GMC lifetimes—perhaps a reasonable extension of fixed SFE (H₂)—R_mol is set by $\tau ({\rm H}\,\mathsc{i}\rightarrow {\rm \hbox{H}_2})$. In Sections 4.2.2 and 4.2.3, we saw that SFE anticorrelates with τ_ff and τ_orb where $\Sigma _{{\rm H}\,\mathsc{i}} > \Sigma _{{\rm H}_2}$. If GMCs form over these timescales, then R_mol ∝ τ⁻¹_ff or R_mol ∝ τ⁻¹_orb.

However, we found that the SFE decreased more steeply than one would expect if these timescales alone dictated R_mol, so that increasing timescale for GMC formation cannot explain all of the decline in R_mol. Figure 17 directly shows this: dotted lines in the bottom two panels illustrate R_mol ∝ τ⁻¹_ff and R_mol ∝ τ⁻¹_orb, respectively. In both cases, the prediction is notably shallower than the data in both the H₂- and H i-dominated regimes.

Of course, the entire ISM may not participate in cloud formation. Star formation thresholds are often invoked to explain the decrease in SFE between inner and outer galaxy disks. The amount of stable, warm H i may depend on environment, with a variable fraction of the disk actively cycling between H i and GMCs. This suggests a straightforward extension of Equation (30):

$$\begin{equation} R_{\rm mol} = \frac{\Sigma _{{\rm H}_2}}{\Sigma _{{\rm H}\,\mathsc{i}}} \approx \frac{{\rm GMC\ lifetime}}{\tau ({\rm H\,\mathsc{i}}\rightarrow {\rm \hbox{H}_2})} \times f_{\rm GMC\, forming}, \end{equation} \tag{ 31 }$$

which again balances GMC formation and destruction but now includes the factor f_{GMC forming} to represent the fact that only a fraction of H i actively cycles between the molecular and atomic ISMs.

We considered three thresholds in which large-scale instabilities dictate f_{GMC forming}—Q_gas, Q_stars+gas, and shear. One would naively expect these thresholds to correspond to f_{GMC forming} ∼ 1 for supercritical gas and f_{GMC forming} ≪ 1 for subcritical gas, yielding a step function in SFE or R_mol. However, we do not observe such relationships between thresholds and SFE (Section 4.3); this agrees with Boissier et al. (2007) who also based their SFR profiles on extinction-corrected FUV maps and found no evidence for sharp star formation cutoffs.

If these instabilities regulate star formation but operate below our resolution, we still expect a correspondence between SFE and the average threshold value, which should indicate what fraction of the ISM is unstable. Despite this expectation, Q_gas shows little correspondence to the SFE and almost all of our sample is stable against axisymmetric collapse. Kim & Ostriker (2001, 2007) discussed Q_gas thresholds for the growth of nonaxisymmetric instabilities, but these are in the range Q_gas ∼ 1–2, still lower than the typical values that we observe (Section 4.3.1). Even independent of the normalization, Q_gas shows little relation to the SFE, particularly in dwarf galaxies (also see Hunter et al. 1998a; Wong & Blitz 2002; Boissier et al. 2003).

Including the effects of stellar gravity reduces stability. Over most of our sample, Q_stars+gas ≲ 2 with a much narrower range than Q_gas (similar improvements were seen by Boissier et al. 2003; Yang et al. 2007). These values are roughly consistent with the conditions for cloud formation found from simulations. Li et al. (2005) found that gas collapses where Q_stars+gas ≲ 1.6 and Kim & Ostriker (2001, 2007) found runaway instabilities where Q_gas ≲ 1.4 (though this is Q_gas and not Q_stars+gas; for a region like the solar neighborhood, Kim & Ostriker 2007 argued that disk thickness, which tends to increase stability, approximately offsets the effect of stars on Q).

Q_stars+gas increases toward the central parts of spirals, so that although the ISM in these regions is usually dominated by H₂, they appear more stable than gas near the H i-to-H₂ transition. Hunter et al. (1998a) and Kim & Ostriker (2001) suggested that because of low shear, instabilities aided by magnetic fields may grow in these regions despite supercritical Q. Comparing to the shear threshold proposed by Hunter et al. (1998a), we find some support for this idea: at ≲0.2r₂₅, many dwarf and spiral galaxies appear unstable or marginally stable. As with Q_gas, however, Σ_crit,A/Σ_gas shows large scatter and no clear ability to predict the SFE.

Thus, we find no clear evidence that disk stability at large scales drives the observed variations in SFE and R_mol. Improved handling of second-order effects (disk thickness, σ_gas, X_CO, σ_*, and ϒ^K_⋆) may change this picture, but comparing our first-order analysis to expectations and simulations, disks appear marginally stable more or less throughout with little correlation between proposed thresholds and SFE.

Timescales and thresholds computed at 400 (dwarfs) and 800 pc (spirals) scales do not offer a simple way to predict R_mol. An alternative view is that physics on smaller scales regulates cloud formation. Comparison with models by Schaye (2004) suggests that a cold phase can form across the entire disk of most of our sample, which agrees with results from Wolfire et al. (2003) modeling our own Galaxy. High-density, narrow-linewidth clouds may easily be unstable or be rendered so by the passage of spiral arms or supernova shocks, even where the ISM as a whole is subcritical. Both Schaye (2004) and de Blok & Walter (2006) emphasized the effect of lower σ_gas on instability, and we have seen that a shift from the observed σ_gas = 11 km s⁻¹ to σ_gas = 3 km s⁻¹ would render most gas disks in our sample unstable or marginally stable (of course, a proper calculation requires estimating the density and fraction of the mass in this phase as well).

A narrow-line component is observed from high-velocity resolution H i observations of nearby irregular galaxies (Young et al. 2003; de Blok & Walter 2006), but an important caveat is the lack of direct evidence of such a component in THINGS. With ∼2.5 or 5 km s⁻¹ velocity resolution, one cannot distinguish a narrow component directly. Therefore, A. Usero et al. (2008, in preparation) followed up work by Braun (1997), who used the peak intensity along each LOS to estimate the maximum contribution from a cold phase and found pervasive networks of high brightness filaments. A. Usero et al. (2008, in preparation) found no clear evidence for a cold phase traced by networks of high brightness filaments, suggesting that a cold phase, if present, is mixed with the warm phase at the THINGS resolution of several times ∼100 pc.

Despite this caveat, our results offer significant circumstantial support that ISM physics below our resolution dictates R_mol: the lack of obvious threshold behavior, marginal stability of our disks, the ability of a cold phase to form, and the continuous variations in SFE and R_mol as a function of radius, Σ_*, and P_h.

In particular, the relationship between R_mol and P_h has been studied before. Following theoretical work by Elmegreen (1993) and Elmegreen & Parravano (1994), Wong & Blitz (2002) and Blitz & Rosolowsky (2006) showed that R_mol and P_h correlate in nearby spiral galaxies (mostly at R_mol > 1) and Robertson & Kravtsov (2008) recently produced a similar relationship from simulations that include cool gas and photodissociation of H₂; they emphasized the importance of the latter to reproduce the observed scaling.

The dash-dotted line in the bottom left panel of Figure 17 shows R_mol ∝ P^1.2_h, predicted by Elmegreen (1993) from balancing H₂ formation and destruction in a model ISM. This is a reasonable description of dwarf galaxies, where we derive a best-fit power law with index ≈1.05. Spirals show a slightly shallower relation between R_mol and P_h with the best-fit power law index ≈0.80. The thick dashed line in the bottom left panel of Figure 17 shows our best fit to the spiral subsample (both CO/H i and SFR/H i) over the range R_mol = 0.1–10. Table 6 lists this fit along with fits to dwarf galaxies and the results of Wong & Blitz (2002) and Blitz & Rosolowsky (2006).

The entry “spiral subsample (combined)” in Table 6 lists the best-fit power law like Equation (11) for our spiral subsample. This fit has an index α = 0.80 and normalization log₁₀P₀/k_B = 4.23 (this is an ordinary least squares bisector fit over the range 0.1 < R_mol < 10 giving equal weight to each of the red and green points in Figure 16). Formally, the uncertainty in the fit is small because it includes a large number of data points. However, both log₁₀P₀/k_B and α scatter by several tenths when fitted to individual galaxies. This agrees well with α = 0.8 derived by Wong & Blitz (2002) and with α = 0.92 ± 0.07 obtained by Blitz & Rosolowsky (2006), given the uncertainties. Fitting the dwarf subsample in the same manner yields log₁₀P₀/k_B = 4.51, the pressure at the H i-to-H₂ transition. This is 0.2–0.3 dex higher than log₁₀P₀/k_B = 4.23 in spirals, suggesting that at the same pressure (density), GMC/H₂ formation in our dwarf subsample is a factor of ∼2 less efficient than in spirals.

The fits between R_mol and P_h in Table 6 are reasonable descriptions of the data, but do not represent a “smoking gun” regarding the underlying physics; radius, Σ_*, P_h, and τ_orb are all covariant and each could be used to predict R_mol with reasonable accuracy in spirals. Therefore, we close our discussion by noting a set of four scaling relations between R_mol and environment that describe our spiral subsample:

$$\begin{eqnarray} &&R_{\rm mol} = 10.6 \exp (-r_{\rm gal} / 0.21 r_{25}), \\ &&R_{\rm mol} = \Sigma _{*} / 81\;M_\odot \;{\rm pc}^{-2}, \\ &&R_{\rm mol} = (P_{\rm h} / 1.7 \times 10^4\;{\rm cm}^{-3}\;{\rm K}\;k_{\rm B})^{0.8}, \\ &&R_{\rm mol} = (\tau _{\rm orb} / 1.8 \times 10^8 \;{\rm yr})^{-2.0}. \end{eqnarray} \tag{ 32 }$$

These appear as thick dashed lines in Figure 17.

In particular, we stress the relationship between R_mol and Σ_* (also see Figure 3). This has several possible interpretations, the most simple of which is that stars form where they have formed in the past. However, there are physical reasons to think that the relationship may be causal. Considering a similar finding in dwarf irregular galaxies, Hunter et al. (1998a) suggested that stellar feedback may play a critical role in triggering cloud formation. Recently, the importance of the stellar potential well has been highlighted, either to triggering large-scale instabilities (Li et al. 2005, 2006; Yang et al. 2007) or in bringing gas to high densities in order for small-scale physics to operate more effectively (Elmegreen 1993; Elmegreen & Parravano 1994; Wong & Blitz 2002; Blitz & Rosolowsky 2004, 2006).

We use FUV maps obtained by the GALEX satellite (Martin et al. 2005) as part of the GALEX Nearby Galaxies Survey (NGS; Gil de Paz et al. 2007). The GALEX FUV band covers λ = 1350–1750 Å with a resolution of 56 and a 1°.25 diameter FOV. These maps have excellent sensitivity and well-behaved backgrounds over a large FOV. GALEX simultaneously observes in a near-UV (NUV) band (λ = 1750–2750 Å). We use these data to measure UV colors and to identify foreground stars.

We correct the FUV maps for Galactic extinction using the dust map of Schlegel et al. (1998). We subtract a small background, measured away from the galaxy. We identify and remove foreground stars using their UV color: any pixel with a NUV-to-FUV intensity ratio of ≳15 (varying ± 5 from galaxy to galaxy) that is also detected in the NUV map with greater than 5σ significance is blanked. In convolution to our working resolution, blank pixels are replaced with the average of nearby data. We also blank a few regions with obvious artifacts. These include bright stars (e.g., in NGC 3198 and NGC 6946) that are usually beyond the optical radius of the galaxy and NGC 5195, the companion of NGC 5194.

We use maps of 24 μm emission obtained as part of the SINGS Legacy program (Kennicutt et al. 2003), using the Multiband Imaging Photometer for Spitzer (MIPS) instrument (Rieke et al. 2004). Gordon et al. (2005) described the reduction of these scan maps, which have 6″ resolution. The sensitivity and background subtraction are both very good, and it is typical to find 3σ emission at ∼r₂₅ in a spiral. The MIPS point-spread function (PSF) at 24 μm is complex at low levels, but our working resolution of ∼20″ makes this only a minor concern.

NGC 3077, NGC 4214, and NGC 4449 are not part of SINGS. For these galaxies, we use 24 μm (and IRAC) maps from the Spitzer archive. We use the post basic calibrated data produced by the automated Spitzer pipeline.

As with the FUV maps, we subtract a small background from the 24 μm maps, which we measure away from the galaxy. We blank the same set of foreground stars as in the FUV maps. In convolution to our working resolution, these pixels are replaced with the average of nearby data. We also blank the edges of the 24 μm maps perpendicular to the scan, which are noisy (and outside the optical radius) and the same artifacts blanked in the FUV maps.

The SINGS fourth data release includes Hα maps, which we use to compare Σ_SFR derived from Hα, FUV, and 24 μm emission in 13 galaxies. We convert Hα to Σ_SFR following Calzetti et al. (2007) and the SINGS data release documentation. We correct for [N ii] contamination following Calzetti et al. (2005) and Lee (2006) and correct for Galactic extinction using the Schlegel et al. (1998) dust maps. We check the flat-fielding by eye and mask regions or fit backgrounds where necessary.

In conjunction with a direct measurement, it is helpful to have a basic expectation for W_FUV. We calculate this by combining a Galactic extinction law and a typical nebular-to-stellar extinction ratio. In terms of Hα extinction, A_Hα, and FUV extinction, A_FUV, Equations (D2) and (D3) are

$$\begin{eqnarray} {\rm SFR}_{\rm Tot} &=& {\rm SFR}^{\rm unobscured}_{\rm H\alpha } 10^{A_{\rm H\alpha } / 2.5}, \\ \nonumber {\rm SFR}_{\rm Tot} &=& {\rm SFR}_{\rm FUV}^{\rm unobscured} 10^{A_{\rm FUV} / 2.5}. \end{eqnarray} \tag{ D8 }$$

For a Galactic extinction law, A_FUV/A_R ≈ 8.24/2.33 (Cardelli et al. 1989; Wyder et al. 2007). We may also expect that FUV originates from a slightly older and more dispersed population than Hα. If we assume a typical nebular-to-stellar extinction ratio of A_Hα/A_R ≈ 2 (Calzetti et al. 1994; Roussel et al. 2005), then we expect A_FUV/A_Hα ≈ 1.8 (if FUV comes mostly from a very young population coincident with Hα, we instead expect A_FUV/A_Hα ∼ 3.6). Combined with Equations (D3)–(D8), these assumptions yield

$$\begin{eqnarray} &&\frac{1}{f + W_{\rm FUV} - 1} + 1 = \left(\frac{W_{\rm FUV}}{f} + 1\right)^{1/1.8} {\rm where}\nonumber\\ && f = \frac{{\rm SFR}_{\rm FUV}^{\rm unobscured}({\rm FUV})}{{\rm SFR}_{\rm H\alpha }^{\rm embedded} (24\;\mu {\rm m})}, \end{eqnarray} \tag{ D9 }$$

which we may solve for W_FUV given f, the ratio of observed FUV to 24 μm intensities (in SFR units).

For A_Hα = 1.1 mag, a typical value in disk galaxies (Kennicutt 1998b), f ≈ 0.26 and Equation (D9) suggests W_FUV ≈ 1.3. For higher A_Hα, expected for the inner parts of spiral galaxies or arms, f will be lower and we expect lower values of W_FUV, approaching W_FUV = 1 where both FUV and Hα are almost totally absorbed (and SFR_Tot is totally determined from 24 μm emission). For lower A_Hα, for example, expected in dwarf galaxies or the outer parts of spirals, we expect W_FUV to approach the ratio of extinctions, 1.8, in the optically thin case.

We measure W_FUV directly from observations by comparing FUV and 24 μm emission to various estimates of SFR_Tot. We perform these tests in the 13 galaxies with SINGS Hα data. Over a common set of LOSs where we estimate Hα, FUV, and 24 μm to all be complete, we estimate Σ_SFR and W_FUV (from Equation (D7)) in five ways, as follows.

• 1.  
  Combining Hα+24 μm using Equation (D2) (Calzetti et al. 2007).
• 2.  
  From 24 μm emission alone, using the (nonlinear) relation found by Calzetti et al. (2007, their Equation 8).
• 3.  
  From Hα alone, taking A_Hα = 1.1 mag, a typical extinction averaged over disk galaxies, though not necessarily a good approximation for each LOS (Kennicutt 1998b).
• 4.  
  From Hα emission, estimating A_Hα from Σ_H i and $\Sigma _{{\rm H}_2}$ following Wong & Blitz (2002). We assume a Galactic dust-to-gas ratio and treat dust associated with H i as a foreground screen obscuring Hα while treating dust associated with H₂ as evenly mixed with Hα emission.
• 5.  
  From FUV emission, estimating A_FUV for every LOS by applying the relationship between FUV-to-NUV color and A_FUV measured for nearby galaxies by Boissier et al. (2007).

In principal, the first method is superior to the others because Calzetti et al. (2007) directly calibrated it against Paα, and because it incorporates both Hα and IR emission, offering direct tracers of both ionizing photons and dust-absorbed UV light. The other four methods offer checks on SFR_Tot that are variously independent of 24 μm, FUV, or Hα emission, allowing us to estimate the plausible range of both W_FUV and the uncertainty in Σ_SFR.

Figure 22 shows the results of these calculations. In the top panel, we plot the normalized distribution of W_FUV for each estimate of Σ_SFR. The bottom left panel shows how each distribution of W_FUV depends on the FUV-to-24 μm ratio, f (Equation D9). The bottom right panel shows how W_FUV varies with normalized galactocentric radius.

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The median W_FUV derived in various ways spans a range from ∼1.0–1.8. The two 24 μm-based methods (blue and gray) both yield W_FUV ∼ 1.3 with relatively narrow distributions. Using FUV and UV colors yields the highest expected W_FUV, ∼1.8; estimating A_Hα from gas yields the lowest W_FUV, peaked near ∼1.0, though the distribution is very wide. This range of values reasonably agrees with our extrapolation (seen as a dash-dotted curve in the top right panel), which also lead us to expect a typical W_FUV of 1.3 and a reasonable range of 1.0–1.8.

The bottom panels show that while individual methods to estimate W_FUV do exhibit significant systematics (particularly at very high and low f), simply fixing W_FUV = 1.3 is a reasonable description of most data (the dashed lines in the center panel bracket ∼80% of the measured f). W_FUV does not have to be constant. Indeed, we expect it to vary with f based on simple assumptions and very basic arguments. However, a constant W_FUV is consistent with the data and is also the simplest, most conservative approach. Therefore, this is how we proceed: taking W_FUV = 1.3^+0.5_−0.3, Equation (D3) becomes

$$\begin{eqnarray} {\rm SFR}_{\rm Tot} &=& 0.68 \times 10^{-28} L_{\nu }({\rm FUV}) + 2.14^{+0.82}_{-0.49}\nonumber\\ &&\times 10^{-42} L ({\rm 24}\;\mu {\rm m}). \end{eqnarray} \tag{ D10 }$$

We convert Equation (D10) from luminosity to intensity units using ν_24 μm = 1.25 × 10¹³ Hz, 1 MJy =10⁻¹⁷ erg s⁻¹ Hz⁻¹ cm⁻², and L_ν = 4πAI_ν, where A is the physical area subtended by the patch of sky being considered. This yields Equation (D1),

$$\begin{eqnarray} \Sigma _{\rm SFR} &=& 8.1 \times 10^{-2} I_{\rm FUV}+3.2^{+1.2}_{-0.7}\times 10^{-3} I_{24}, \end{eqnarray} \tag{ D11 }$$

with I_FUV and I₂₄ in units of MJy ster⁻¹ and Σ_SFR in units of M_☉ kpc⁻² yr⁻¹.

If we adopt the naive tack of treating all approaches as equal, the aggregate data in Figure 23 yield a median ratio Σ_SFR(other)/Σ_SFR(FUV + 24) ≈ 1.05 with ≈0.22 dex (i.e., ∼65%) 1σ scatter. The dominant sources of this scatter are the choice of “other” Σ_SFR and galaxy-to-galaxy variations. Once a galaxy and methodology are chosen, the data tend to follow a fairly well-defined and often nearly linear relation. For comparison, the 24 μm part of the Hα+24 μm calibration has ≈20%–30% uncertainty (Calzetti et al. 2007; Kennicutt et al. 2007), considering only star-forming peaks. In light of the wider range of star formation histories and geometries encountered working pixel by pixel or averaging over whole rings, the estimate of ∼65% seems quite reasonable. Comparing our integrated SFRs (Figure 23, bottom right) with those estimated by 11HUGS (Lee 2006) and SINGS (Kennicutt et al. 2003) bears out this estimate; we match these estimates with a similar scatter. Another view of this comparison may be seen in Appendix F, where we present radial profiles of Σ_SFR based on Hα in the same plots as our FUV+24 μm profiles.

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Despite the overall good agreement between our Σ_SFR and other estimates, Figures 22 and 23 do show systematic differences among tracers. We note several of these before moving on.

• 1.  
  Using only 24 μm emission (gray) yields a low estimate of Σ_SFR for the two low metallicity galaxies in our comparison sample: Holmberg II and IC 2574. Dust is known to be deficient in these galaxies (Walter et al. 2007), which is likely to lead to a breakdown in the fit between 24 μm emission and Paα. This effect, already recognized by Calzetti et al. (2007), highlights the importance of including a non-IR component in a SFR tracer.
• 2.  
  Estimating A_Hα from gas (red; Wong & Blitz 2002) yields very high Σ_SFR (and high W_FUV) in the inner parts of galaxies. This underscores the complexity of the geometry and timescale effects at play; it is extremely challenging to reverse engineer the true luminosity of a heavily obscured source knowing only the amount of nearby interstellar matter. These high values are almost certainly overestimates; stellar feedback, turbulence, or simply favorable geometry likely always allow at least some light from deeply-embedded H ii regions to escape.
• 3.  
  Particularly at low Σ_SFR, inferring A_FUV from UV colors (purple) yields higher embedded SFR than using 24 μm emission (and this method appears to completely fail in NGC 6946, the horizontal row of points). A possible explanation is that where Σ_SFR is relatively low, the UV originates from a somewhat older (and thus redder) population (e.g., Calzetti et al. 2005); the FUV–NUV color relation depends on the recent star formation history (e.g., differing between starbursts and more quiescent galaxies; Boissier et al. 2007; Salim et al. 2007).This discrepancy (and the close association between our SFR tracer and stellar mass seen in the main text) argues for a comparison among metallicity, stellar populations, and mid-IR emission that is beyond the scope of this paper. We restrict ourselves to a first-order check: we compare the ratio of 24 μm-to-3.6 μm and FUV-to-3.6 μm emission in our sample with those in elliptical galaxies, which should be good indicators of how much an old population contributes to 24 μm or FUV emission. Very approximately, in ellipticals I₂₄/I_3.6 ∼ 0.1 (Temi et al. 2005; Dale et al. 2007; Johnson et al. 2007), with ∼0.03 expected from stellar emission alone (Helou et al. 2004), while I_FUV/I_3.6 ∼ 2–4 × 10⁻³ (Dale et al. 2007; Johnson et al. 2007, taking the oldest bin from the latter). We measure I₂₄/I_3.6 and I_FUV/I_3.6 for each ring in our sample galaxies and compare these to the elliptical colors. In both cases, only ∼5% of individual tilted rings have ratios lower than those seen in elliptical galaxies and the mean color is ∼10 times that found in elliptical galaxies, though the ratio I_FUV/I_3.6 shows large scatter due to the effects of extinction. Both the 24 μm and FUV bands do appear to be dominated by a young stellar population almost everywhere in our sample. Discrepancies among various tracers thus likely seem to arise from the different geometries and age sensitivities of FUV (τ ∼ 100 Myr), Hα (τ ∼ 10 Myr), and 24 μm (likely intermediate) emission.

Finally, we emphasize that uncertainties estimated via these comparisons mainly reflect the ability to accurately infer the total UV light or ionizing photon production from young stars. They do not include uncertainty in the IMF, ionizing photon production rate (e.g., at low metallicity), or any of the other factors involved in converting an ionizing photon count or FUV intensity into a SFR.