WHICH PHASE OF THE INTERSTELLAR MEDIUM CORRELATES WITH THE STAR FORMATION RATE?

Consider a spherical cloud of mean volume density n_H H nuclei cm⁻³ and mean column density N_H H nuclei cm⁻², mixed with dust with an absorption cross section per H nucleus σ_d to photons near 1000 Å. The mean absorption optical depth to UV photons is τ = N_Hσ_d, and the corresponding mean visual extinction at optical wavelengths is A_V. The outer parts of the cloud are exposed to the ultraviolet interstellar radiation field (ISRF) of the galaxy, and this keeps the hydrogen and carbon in the outer parts of the cloud predominantly in the form of H i and C ii. If the column density is sufficiently large, dust and the small population of hydrogen molecules in the predominantly atomic layer will absorb the ISRF, allowing a transition from H i to H₂ and C ii to CO as the main repositories of hydrogen and carbon. We assume that the gas is neutral, and that there are very few ionizing photons present.

In a real interstellar cloud, even if the density were uniform, the temperature and chemical composition would not be. Surface layers exposed to direct UV radiation would be warmer than the shielded interior, and the hydrogen there would be primarily H i rather than H₂. Thus, there is no single cloud temperature or chemical composition we can compute, and determining the full temperature and chemical profile even when the density profile is given in advance requires solving a photodissociation region model in which one simultaneously determines both the temperature and the chemical state, including line radiative transfer throughout the cloud. While there are many examples of such models in the literature, they are too computationally expensive to allow a wide search of parameter space, and they still rely on arbitrarily chosen density distributions. Instead, our goal is to estimate a single characteristic temperature for a cloud, coupled to a simple description of its chemistry. To this end we will describe a cloud's chemical composition with single numbers giving the mass fraction in a given chemical phase, and we will approximate the cloud interior as a uniform region of volume density n_H and mean column density N_H, consisting of gas at temperature T_g uniformly mixed with dust of temperature T_d. For the purposes of computing the temperature, we will take the chemical composition to be uniform, with the hydrogen all as H i or H₂, and the carbon all as C ii or CO, despite the fact that every H₂ cloud has an outer H i shell, and every CO cloud has an outer C ii shell.

To compute the H i to H₂ transition in our model cloud, we rely on the models of Krumholz et al. (2008, 2009c) and McKee & Krumholz (2010, collectively the KMT model hereafter). In these models the fraction of the hydrogen mass in the H₂-dominated region is given by

$$\begin{equation} f_{\rm H_2} \approx 1 - \left(\frac{3}{4}\right) \frac{s}{1+0.25s}, \end{equation} \tag{ 1 }$$

where

$$\begin{eqnarray} s & = & \frac{\ln (1+0.6\chi +0.01\chi ^2)}{0.6\tau }, \end{eqnarray} \tag{ 2 }$$

$$\begin{eqnarray} \chi & = & 71 \frac{G_0^{\prime }}{n_{\rm H}/\mbox{cm}^{-3}}, \end{eqnarray} \tag{ 3 }$$

and G'₀ is the strength of the ISRF normalized to its value in the solar vicinity (which corresponds to a free-space H₂ dissociation rate of 5.4 × 10⁻¹¹ s⁻¹). Note that, unlike some of the other approximate expressions derived by KMT, this one applies independent of whether the gas is warm or cold neutral medium. It should also be noted that this model assumes chemical equilibrium, which is not necessarily the case for H₂ (Pelupessy & Papadopoulos 2009; Mac Low & Glover 2010). However, Krumholz & Gnedin (2011) perform a detailed comparison between the equilibrium approximation and a set of galaxy simulations including full time-dependent chemistry and radiation. They find that the approximation is very accurate at metallicities of Z' ≳ 0.01, where Z' is the metallicity normalized to the Milky Way value, and we can therefore use it safely. We note that Equation (1) also agrees extremely well with observations both of local galaxies (Krumholz et al. 2009c) and of damped Lyα systems at high redshift (Krumholz et al. 2009b). We therefore conclude that, on the galactic scales relevant to this work (as opposed to the isolated periodic boxes considered by Mac Low & Glover 2010) this approximation is valid.

The C ii to CO transition is predominantly governed by dust extinction, and thus is sensitive primarily to the cloud optical depth. At high optical depths wherever the hydrogen is H₂ the carbon is also CO, while at low optical depths there can be significant regions of H₂ where the carbon is primarily C ii, because the H₂ self-shields while the CO cannot (Glover & Mac Low 2011; Wolfire et al. 2010). Recent semi-analytic work indicates that the ratio of mass where the carbon is CO to total cloud mass is (Wolfire et al. 2010)

$$\begin{equation} f_{\rm CO} = f_{\rm H_2} e^{-4.0 \left(0.53 - 0.045\ln \frac{G_0^{\prime }}{n_{\rm H}/{\rm cm}^{-3}}-0.097\ln Z^{\prime }\right)/A_V}. \end{equation} \tag{ 4 }$$

Alternatively, an empirical fit to numerical simulations for the same quantity (Glover & Mac Low 2011) gives very similar results.

The temperatures of gas and dust are set by the condition of thermal equilibrium, following a method that combines elements from Goldsmith (2001) and Lesaffre et al. (2005). We consider heating of the gas by the grain photoelectric effect at a rate per H nucleus Γ_pe, heating by cosmic rays at a rate Γ_cr, and cooling via atomic and molecular line emission at a rate Λ_line. Dust is heated via an external radiation field at a rate Γ_dust, and cools via thermal emission at a rate Λ_dust. Finally, energy flows from the dust to the gas at a rate Ψ_gd, which may be negative if the gas is hotter than the dust. The condition for thermal balance is therefore that the temperatures T_g and T_d simultaneously satisfy the equations

$$\begin{eqnarray} \Gamma _{\rm pe} + \Gamma _{\rm cr} - \Lambda _{\rm line} + \Psi _{\rm gd} & = & 0 \end{eqnarray} \tag{ 5 }$$

$$\begin{eqnarray} \Gamma _{\rm dust} - \Lambda _{\rm dust} - \Psi _{\rm gd} & = & 0. \end{eqnarray} \tag{ 6 }$$

Note that we do not include photoionization heating. While this is important at low column densities (e.g., Schaye 2004), the lowest column density clouds we will consider in this work have optical depths of ∼10³ to ionizing photons, and thus photoionization heating is unimportant for them except very close to their surfaces.

To solve this equation we must compute the temperature dependence of each heating and cooling term. We assume that the clouds in question are optically thin to far-infrared radiation, so we need not consider trapping of the dust radiation field. Thus, the dust cooling rate is

$$\begin{equation} \Lambda _{\rm dust} = \kappa (T_d) \mu _{\rm H} c a T_d^4, \end{equation} \tag{ 7 }$$

where c is the speed of light, a is the radiation constant, κ(T_d) is the temperature-dependent dust specific opacity, and μ_H is the mean mass per H nucleus. Dust heating is more complex, as mentioned above, since the external interstellar ultraviolet field dominates at cloud edges and the local re-radiated infrared field dominates at cloud centers. Since we are mostly interested in what happens when clouds are shielded by at least a few magnitudes of extinction in the ultraviolet, we choose to adopt the IR-dominated case. In practice this choice makes very little difference, since the dust–gas coupling is a very minor effect at the densities where we focus our attention. We characterize the IR radiation field by an effective temperature T_rad, so

$$\begin{equation} \Gamma _{\rm dust} = \kappa (T_{\rm rad}) \mu _{\rm H} c a T_{\rm rad}^4. \end{equation} \tag{ 8 }$$

Finally, for the dust–gas energy exchange term we adopt the approximate exchange rate (Goldsmith 2001)

$$\begin{equation} \Psi _{\rm gd} = \alpha _{\rm gd} n_{\rm H} T_g^{1/2} (T_d - T_g), \end{equation} \tag{ 9 }$$

where α_gd is a coupling constant.

For the gas, the cosmic ray heating rate is

$$\begin{equation} \Gamma _{\rm cr} = \zeta q_{\rm CR}\mbox{ s}^{-1}, \end{equation} \tag{ 10 }$$

where ζ is the cosmic ray primary ionization rate and q_CR is the thermal energy increase per primary cosmic ray ionization. For grain photoelectric heating, we must choose a suitable mean heating rate, since extinction through the cloud causes the heating rate to drop substantially as we move toward the interior. For simplicity, and since we are concerned more with cloud interiors than surfaces, we consider the heating rate to be attenuated by half the mean extinction of the cloud. With this approximation, we have

$$\begin{equation} \Gamma _{\rm pe} = 4.0\times 10^{-26} G_0^{\prime } e^{-N_{\rm H} \sigma _{d}/2}\mbox{ erg s}^{-1}, \end{equation} \tag{ 11 }$$

where σ_d is the dust cross section per H nucleus to the UV photons that dominate grain photoelectric heating.

The final term in the thermal balance Equations (5) and (6) is Λ_line, the line cooling rate. In each case we consider only a single coolant species: C ii or CO. This is a reasonable approximation because, at the low temperatures where star formation occurs, one of these two will dominate. To compute the cooling rate we must compute the populations of the various levels, and we do so following the method of Krumholz & Thompson (2007). Let f_i be the fraction of a given species, C ii or CO, in the ith state, and let E_i be the energy of that state. Transitions between states i and j occur due to spontaneous emission, at a rate given by the Einstein coefficient A_ij (which is zero if E_i ⩽ E_j), and due to collisions with rate coefficient $\gamma _{ij,\rm S}$, where S is the species of the collision partner. Note that the collision rate coefficients depend on the gas temperature. We do not consider stimulated emission or absorption due to an external radiation field, since neither is significant for the cooling lines of C ii or CO.

In the escape probability formalism we approximate that each spontaneously emitted photon produced by an atom transiting from level i to level j has a probability β_ij of escaping; photons that do not escape are re-absorbed on the spot within the cloud, yielding no net change in the level populations. With these definitions and approximations, in statistical equilibrium the level populations are given implicitly by

$$\begin{eqnarray} &&\hspace{-2.6pc}f_i \sum _j (\beta _{ij} A_{ij} + n_{\rm H\,{\mathsc i}} \gamma _{ij,\rm H\,{\mathsc i}} + n_{\rm He} \gamma _{ij,\rm He} \nonumber \\ && + n_{\rm p\hbox{-}H_2}\gamma _{ij,\rm p\hbox{-}H_2} + n_{\rm o\hbox{-}H_2} \gamma _{ij,\rm o\hbox{-}H_2})\nonumber \\ & =& \sum _i (\beta _{ij} A_{ij} + n_{\rm H\,{\mathsc i}} \gamma _{ij,\rm H\,{\mathsc i}} + n_{\rm He} \gamma _{ij,\rm He} \nonumber \\&& + n_{\rm p\hbox{-}H_2}\gamma _{ij,\rm p\hbox{-}H_2} + n_{\rm o\hbox{-}H_2} \gamma _{ij,\rm o\hbox{-}H_2}) f_i, \end{eqnarray} \tag{ 12 }$$

where the sums run over all quantum states and $n_{\rm H\,{\mathsc i}}$, $n_{\rm p\hbox{-}H_2}$, $n_{\rm o\hbox{-}H_2}$, and n_He are the number densities of H i, para-H₂, ortho-H₂, and He, respectively. The left-hand side of this equation represents the rate of transitions out of state i to all other states j, while the right-hand side represents the rate of transitions into state i from all other states j. The escape probabilities themselves depend on the level population. If we let X be the abundance of the species in question relative to H nuclei and σ be the one-dimensional gas velocity dispersion, then we have (Krumholz & Thompson 2007)

$$\begin{equation} \beta _{ij} \approx \frac{1}{1+0.5\tau _{ij}}, \end{equation} \tag{ 13 }$$

where

$$\begin{equation} \tau _{ij} = \frac{g_i}{g_j} \frac{3A_{ij} \lambda _{ij}^3}{16(2\pi)^{3/2} \sigma } X N_{\rm H} f_j \left(1 - \frac{f_i g_j}{f_j g_i}\right), \end{equation} \tag{ 14 }$$

g_i and g_j are the statistical weights of levels i and j, and λ_ij is the wavelength of a photon associated with the transition from level i to level j.

Equations (12)–(14) constitute a complete system of algebraic equations for the level populations f_i. Given the solution to these equations, the line cooling rate per H nucleus is then given by

$$\begin{equation} \Lambda _{\rm line} = \sum _{i,j} \beta _{ij} A_{ij} f_i h \nu _{ij} / n_{\rm H}, \end{equation} \tag{ 15 }$$

where ν_ij is the frequency of a photon emitted in a transition from state i to state j.

We have now written down a complete set of equations to determine the gas and radiation temperatures. To solve them we use a double-iteration method. We first select trial values of T_d and T_g, and then we compute all the heating and cooling rates Γ, Λ, and Ψ. In order to compute Λ_line, we must solve for the equilibrium level populations f_i by solving Equations (12)–(14). We do so by fixing T_g (and thus all the collision rate coefficients) and applying Broyden's method (Press et al. 1992). Once we have determined Λ_line, we check if the thermal balance Equations (5) and (6) are satisfied to within a specified tolerance. If not, we iteratively update T_d and T_g using Newton's method to update T_d and T_g until Equations (5) and (6) are satisfied to the required tolerance.

We can use our result that star formation correlates with H₂ to predict an approximate correlation between SFRs and the masses of all gas, H₂ gas, and CO gas as a function of galaxy metallicity and surface density. Consider a portion of a galaxy with a mean surface density Σ_g (averaged over the ∼kpc scales accessible to current observations). Within this region a fraction $f_{\rm H_2}$ of the gas, given by Equation (1), is in the form of star-forming H₂ clouds, and within these a smaller fraction f_CO of the gas, given by Equation (4), has most of its carbon in CO molecules.

We evaluate $f_{\rm H_2}$ following the methods outlined in Krumholz et al. (2009c, 2009d). First, we must adopt a clumping factor c to scale from the surface densities of individual atomic molecular complexes on ∼100 pc scales to the ∼kpc scales accessible to current observations. This is necessary because the mean surface density averaged over a kpc-scale region that we observe is lower than the surface densities of the ∼100 pc sized giant molecular clouds, but it is the latter rather than the former that determines the atomic to molecular ratio in these clouds. Based on a combination of theoretical arguments and fits to observation, Krumholz et al. (2009c) adopt c = 5, and we do so here, so τ = 5Σ_gσ_d/μ_H. Second, we adopt a characteristic value of G'₀/n = 0.044(1 + 3.1Z'^0.365)/4.1 expected for a two-phase atomic medium (Krumholz et al. 2009c, 2009d). (For a given value of G'₀, this corresponds to assuming that clouds on Figures 1–3 form a horizontal sequence at a fixed n value that depends on G'₀ and Z'.) To evaluate f_CO and the SFR, we note that star-forming H₂ clouds appear to develop column densities N_H ∼ 7.5 × 10²¹ cm⁻² (∼85 M_☉ pc⁻²) independent of galactic environment (Heyer et al. 2009; Bolatto et al. 2008; Krumholz et al. 2006). Since the CO-dominated regions are the inner parts of these clouds, this implies that the column density and A_V that enter the calculation of the CO fraction should not be the mean galactic surface density or extinction, but instead the fixed A_V of star-forming H₂ clouds. We adopt this value for A_V in Equation (4).

The corresponding SFR is expected on theoretical grounds to be approximately (Krumholz et al. 2009d)

$$\begin{eqnarray} \dot{\Sigma }_* & = & \frac{f_{\rm H_2} \Sigma _g}{2.6\mbox{ Gyr}} \nonumber \\ & & \times \left\lbrace \begin{array}{@{}ll@{}}(\Sigma _g/85\,M_{\odot }\,{\rm pc}^{-2})^{-0.33}, & \Sigma _g < 85 \,M_{\odot }\,{\rm pc}^{-2} \\ (\Sigma _g/85\,M_{\odot }\,{\rm pc}^{-2})^{0.33}, & \Sigma _g > 85 \,M_{\odot }\,{\rm pc}^{-2}. \\ \end{array} \right.\quad \end{eqnarray} \tag{ 17 }$$

In reality, based on the argument we have just made, we should compute the SFR not based directly on the H₂ fraction but instead based on the mass of cold gas. However, we have already seen that the H₂ and cold gas fractions are very similar, and Equation (1) provides a convenient analytic approximation. We therefore use it to estimate both $f_{\rm H_2}$ and the star-forming gas fraction. We have therefore computed, for a portion of a galaxy of total gas surface density Σ_g and metallicity Z', the expected surface densities of H₂, CO, and star formation.

Figure 4 shows our predicted correlation between specific star formation rate (SSFR) with respect to the mass of each gas constituent, $\mbox{SSFR}_{\rm (CO,H_2,total)} = \dot{\Sigma }_* / \Sigma _{\rm (CO,H_2,total)}$, and the surface density of that component at a range of metallicities. We see that at high surface densities and high metallicities the SSFRs for total gas, H₂, and CO are essentially identical, consistent with observations. At lower surface densities or metallicities, though, the SFR per unit total gas falls. Conversely, at lower metallicity the specific star formation with respect to CO rises, reflecting the fact that, at lower metallicity, the mass of gas where the chemical makeup is H₂ plus C ii rises as a fraction of the total H₂ mass. Since star formation follows H₂, SSFR_CO rises as a result. We note that Pelupessy & Papadopoulos (2009) reached a similar conclusion about SSFR_CO based on simulations that used a star formation recipe that assumed roughly constant $\mbox{SSFR}_{\rm H_2}$.

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The most striking feature of Figure 4 is that it predicts very strong variation in SSFR with metallicity for CO and total gas, but not for H₂. In the low-metallicity galaxies where we would like to test this prediction it is difficult to measure either the total gas mass or the H₂ mass, because we lack a reliable H₂ tracer except in a few limited cases (see Section 4.3 for further discussion). We can, however, measure CO luminosities and thereby estimate masses of gas that are CO dominated. As shown in Figure 4, we predict that the SSFR for CO should be much higher in the lower metallicity systems.

To test this prediction, we compile galaxy-integrated CO luminosities, L_CO, and SFRs from the literature. These measurements cover a wide range of metallicity from highly sub-solar to super-solar and include both spirals and dwarfs. The data come from a variety of sources, summarized in Table 2. Whenever they are available, we give preference to L_CO derived from integrating complete single-dish maps of a galaxy, but many of the data still come from sparse sampling. For SFRs, we draw heavily from the recent set of galaxy-integrated measurements by Calzetti et al. (2010); these have the advantage of being computed in a uniform way and include an IR contribution to correct for extinction. Both L_CO and the SFR are luminosity-like quantities, so the ratio of the two is independent of distance. We also draw metallicities for each target from the literature. When available, we give preference to the recent compilation by Moustakas et al. (2010), using the average of their two characteristic metallicities. For targets not studied by Moustakas et al., we use values from the compilations by Marble et al. (2010), Calzetti et al. (2010), and Engelbracht et al. (2008) and a variety of literature sources.

Table 2. Data Compilation

Galaxy	log LCO (K km s−1 pc2)	CO Reference	$\log \dot{M}_*$ (M☉ yr−1)	SFR Reference	12 + log(O/H)	Metallicity Reference
SMC	5.2 ± 0.2	M06	−1.3 ± 0.2	W04	8.0 ± 0.2	D84; MA10
LMC	6.5 ± 0.1	F08	−0.7 ± 0.2	H09	8.3 ± 0.3	D84; MA10
IC10	6.3 ± 0.2	L06	−1.0 ± 0.5	L06	8.2 ± 0.2	L79; L03
M33	7.6 ± 0.1	H04	0.0 ± 0.5	H04	8.3 ± 0.2	R08
IIZw40	6.2 ± 0.3	T98	−0.2 ± 0.3	C10	8.1 ± 0.3	E08; C10
NGC1569	5.1 ± 0.3	T98	−0.6 ± 0.3	C10	8.1 ± 0.2	M97
NGC2537	5.5 ± 0.3	T98	−1.1 ± 0.3	C10	8.4 ± 0.3	MA10
NGC4449	6.8 ± 0.3	Y95	−0.5 ± 0.3	C10	8.3 ± 0.2	M97
NGC5253	5.8 ± 0.3	T98	−0.3 ± 0.3	C10	8.2 ± 0.3	MA10
NGC6822	5.1 ± 0.2	I97	−2.0 ± 0.3	C10	8.4 ± 0.3	MO10
NGC0628	8.3 ± 0.1	L09	−0.2 ± 0.2	C10	8.8 ± 0.3	MO10
NGC0925	7.6 ± 0.1	L09	−0.3 ± 0.2	C10	8.6 ± 0.3	MO10
NGC1482	8.8 ± 0.3	Y95	0.5 ± 0.2	C10	8.6 ± 0.4	MO10
NGC2146	9.3 ± 0.3	Y95	0.9 ± 0.2	C10	8.7 ± 0.3	E08; C10
NGC2403	7.1 ± 0.3	Y95	−0.4 ± 0.2	C10	8.6 ± 0.2	MO10
NGC2841	8.4 ± 0.1	L09	0.1 ± 0.2	C10	9.0 ± 0.3	MO10
NGC2782	9.0 ± 0.3	Y95	0.7 ± 0.2	C10	8.6 ± 0.4	E08; C10
NGC2798	8.8 ± 0.3	Y95	0.5 ± 0.2	C10	8.7 ± 0.3	MO10
NGC2976	7.0 ± 0.1	L09	−1.0 ± 0.2	C10	8.7 ± 0.3	MO10
NGC2903	8.8 ± 0.1	H03	0.3 ± 0.2	C10	8.9 ± 0.3	MA10
NGC3034	8.8 ± 0.3	Y95	0.9 ± 0.2	C10	8.8 ± 0.3	MO10
NGC3077	6.0 ± 0.3	T98	−1.0 ± 0.2	C10	8.6 ± 0.3	MA10
NGC3079	9.4 ± 0.3	Y95	0.5 ± 0.2	C10	8.6 ± 0.4	E08; C10
NGC3184	8.4 ± 0.1	L09	−0.5 ± 0.2	C10	8.9 ± 0.3	MO10
NGC3198	7.9 ± 0.1	L09	−0.0 ± 0.2	C10	8.8 ± 0.3	MO10
NGC3310	8.2 ± 0.3	Y95	0.9 ± 0.2	C10	8.2 ± 0.4	E08; C10
NGC3351	8.2 ± 0.1	L09	−0.2 ± 0.2	C10	8.9 ± 0.3	MO10
NGC3368	8.3 ± 0.3	Y95	−0.4 ± 0.2	C10	9.0 ± 0.3	MA10
NGC3521	8.8 ± 0.1	L09	0.1 ± 0.2	C10	8.8 ± 0.3	MO10
NGC3628	9.2 ± 0.3	Y95	0.3 ± 0.2	C10	9.0 ± 0.3	MA10
NGC3627	9.0 ± 0.1	H03	0.2 ± 0.2	C10	8.7 ± 0.3	MO10
NGC3938	8.5 ± 0.1	H03	−0.1 ± 0.2	C10	8.7 ± 0.3	E08; C10
NGC4194	8.9 ± 0.3	Y95	1.1 ± 0.2	C10	8.7 ± 0.4	E08; C10
NGC4214	6.1 ± 0.1	L09	−1.0 ± 0.2	C10	8.2 ± 0.2	T98
NGC4254	9.9 ± 0.3	Y95	1.3 ± 0.2	C10	8.8 ± 0.3	MO10
NGC4321	9.2 ± 0.1	H03	0.4 ± 0.2	C10	8.8 ± 0.3	MO10
NGC4450	8.9 ± 0.3	Y95	−0.2 ± 0.2	C10	8.9 ± 0.4	C10; MA10
NGC4536	8.6 ± 0.3	Y95	0.3 ± 0.2	C10	8.6 ± 0.4	MO10
NGC4569	8.8 ± 0.1	H03	−0.1 ± 0.2	C10	8.9 ± 0.4	E08; C10
NGC4579	9.0 ± 0.3	Y95	0.2 ± 0.2	C10	9.0 ± 0.4	C10; MA10
NGC4631	8.5 ± 0.3	Y95	0.4 ± 0.2	C10	8.4 ± 0.3	MO10
NGC4725	9.1 ± 0.3	Y95	0.0 ± 0.2	C10	8.7 ± 0.4	MO10
NGC4736	7.9 ± 0.1	L09	−0.4 ± 0.2	C10	8.7 ± 0.4	MO10
NGC4826	8.1 ± 0.1	H03	−0.5 ± 0.2	C10	8.9 ± 0.3	MO10
NGC5033	9.3 ± 0.1	H03	0.1 ± 0.3	K03	8.7 ± 0.3	MO10
NGC5055	8.9 ± 0.1	L09	0.1 ± 0.2	C10	8.9 ± 0.4	MO10
NGC5194	9.2 ± 0.1	H03	0.4 ± 0.2	C10	9.0 ± 0.3	MO10
NGC5236	8.9 ± 0.1	Y95	0.4 ± 0.2	C10	9.0 ± 0.3	MO10
NGC5713	9.1 ± 0.3	Y95	0.6 ± 0.2	C10	8.7 ± 0.4	MO10
NGC5866	8.1 ± 0.3	Y95	−0.6 ± 0.2	C10	8.7 ± 0.4	C10; MA10
NGC5953	9.0 ± 0.3	Y95	0.4 ± 0.2	C10	8.7 ± 0.4	E08; C10
NGC6946	8.8 ± 0.1	L09	0.5 ± 0.2	C10	8.8 ± 0.3	MO10
NGC7331	9.0 ± 0.1	L09	0.4 ± 0.2	C10	8.8 ± 0.3	MO10

Notes. C10 = Calzetti et al. (2010); D84 = Dufour (1984); E08 = Engelbracht et al. (2008); F08 = Fukui et al. (2008); H03 = Helfer et al. (2003); H04 = Heyer et al. (2004); H09 = Harris & Zaritsky (2009); I97 = Israel (1997); K03 = Kennicutt et al. (2003); L79 = Lequeux et al. (1979); L03 = Lee et al. (2003); L06 = Leroy et al. (2006); L09 = Leroy et al. (2009); M97 = Martin (1997); M06 = Mizuno et al. (2006); MA10 = Marble et al. (2010); MO10 = Moustakas et al. (2010); R08 = Rosolowsky & Simon (2008); T98 = Taylor et al. (1998); W04 = Wilke et al. (2004); Y95 = Young et al. (1995).

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To convert observed CO (1 − 0) luminosities to masses of gas traced by CO, we adopt a conversion factor 2 × 10²⁰ H₂ molecules cm⁻²/(Kkms⁻¹)⁻¹ (Abdo et al. 2010b; Blitz et al. 2007; Draine et al. 2007; Heyer et al. 2009), so the mass is

$$\begin{equation} M_{\rm CO} = 4\times 10^{20} \mu _{\rm H} \left(\frac{L_{\rm CO}}{\mbox{K km s}^{-1}\mbox{ cm}^2}\right), \end{equation} \tag{ 18 }$$

where μ_H is the mean mass per H nucleus. This is equivalent to

$$\begin{equation} \frac{M_{\rm CO}}{M_{\odot }} = \alpha _{\rm CO} \left(\frac{L_{\rm CO}}{\mbox{K km s}^{-1}\mbox{ pc}^2}\right), \end{equation} \tag{ 19 }$$

with α_CO = 4.4. Thus, we effectively assume a Milky Way conversion factor for CO-emitting gas. It is important to note that our X factor represents the conversion from CO luminosity to mass of gas where the carbon is predominantly CO, and not the conversion from CO luminosity to total mass of gas where the hydrogen is H₂. These concepts are often not clearly distinguished in the literature. The approximately constant conversion factor derived from virial mass measurements of extragalactic clouds (Blitz et al. 2007; Bolatto et al. 2008; which include some but not all of the H₂ that is not associated with CO) motivates this assumption, though there may still be changes in α_CO of CO emitting at the factor of two level across the range of metallicities that we study. The sense of these would be to increase α_CO and to decrease the SFR-to-M_CO ratio, moving points down in Figure 5. Regardless of systematic effects, the y-axis in Figure 5 should be very close to the ratio of ionizing photon rate to CO luminosity.

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To convert measured oxygen abundances to metallicities relative to Milky Way, we adopt (Caffau et al. 2008)

$$\begin{equation} \log Z^{\prime } = [12 + \log ({\rm O}/{\rm H})] - 8.76. \end{equation} \tag{ 20 }$$

As emphasized by Kewley & Ellison (2008) and Moustakas et al. (2010), the adopted calibration has a large influence on the metallicities derived from measurements of strong optical lines. Therefore, the overall normalization of the metallicities in Figure 5 must be considered uncertain by at least ∼0.1–0.2 dex, though the internal ordering is likely to be relatively robust.

A final complication is that most of the literature data we have gathered consist of galaxy-integrated values, rather than local values. We therefore do not have surface densities, and we must adopt characteristic values in order to compute the relationship between total gas, molecular gas, and CO luminosity. Fortunately, the results for the SSFR for CO are not particularly sensitive to this assumption, as shown below.

We plot the literature data against our predictions in Figure 5. As the plot shows, there is a clear correlation between SSFR_CO and metallicity that agrees within the uncertainties with what one expects if star formation follows H₂ rather than total gas or CO. Though the data are sparse with significant scatter, the difference appears to be more than two orders of magnitude in the SFR-to-CO ratio over about an order of magnitude in metallicity. The shape of this trend agrees well with our theoretical predictions. Given the uncertainties in estimating several of the parameters shown here, we believe that this constitutes a reasonable first check on the model. Future improvements to the measured dust-to-gas ratios, improved estimates of α_CO from CO-emitting gas, and observations of CO from larger samples of low-metallicity galaxies will improve the accuracy of the observed trend and allow more stringent tests.

The elevated value of SSFR_CO in dwarf galaxies and the outer parts of spirals relative to the inner parts of spirals has been pointed out before (Young et al. 1996; Leroy et al. 2006, 2007b; Gardan et al. 2007), but it was unclear if this was due to change in the star formation process or a change in the CO to H₂ ratio combined with star formation following H₂ rather than CO. Pelupessy & Papadopoulos (2009) argued that the elevated value of SSFR_CO could be explained if the latter were true, but they simply assumed that star formation was correlated with H₂, rather than explaining the correlation from first principles. Here, we have provided the missing explanation, and Figure 5 demonstrates that a model based on it can quantitatively reproduce the observations.

Figures 4 and 5 also contain clear predictions for observations. At present it is extremely difficult to measure either total gas masses or H₂ masses in galaxies with low metallicity, because CO breaks down as a tracer of H₂ in these environments, and because the H₂ itself does not emit. Observations are available for only a few galaxies based on using proxies other than CO for the H₂ (e.g., Leroy et al. 2007a). However, future dust observations with facilities such as Herschel and ALMA will make it easier to obtain H₂ masses, and thus total gas masses, for more galaxies. Our work contains a clear prediction: for these galaxies, the SSFR with respect to H₂ mass should be essentially independent of metallicity, while the SSFR for the total gas mass will behave in the opposite sense as for CO: low-metallicity galaxies will have lower SSFR_total, even as they have higher SSFR_CO.

With resolved observations an additional test becomes possible. Figure 4 shows that SSFR_total is essentially flat at high surface densities, but turns down sharply at surface densities below a metallicity-dependent value. Such a drop is seen below Σ_total ≲ 10 M_☉ pc⁻² in observations of solar metallicity galaxies (Bigiel et al. 2008). Models in which the SFR depends on global gravitational instability or similar phenomena that do not care about gas cooling or chemistry predict that the surface density at which the total gas SSFR drops should not vary with metallicity (e.g., Li et al. 2006). In contrast, our work here suggests that it should scale roughly inversely with metallicity; Schaye (2004), using a thermally based model that anticipates some of this work, makes a similar prediction (his Equation (25)), although the metallicity dependence in his model is weaker than in ours. Resolved measurements of the total gas surface density in nearby galaxies should be able to test which prediction is correct: does SSFR_total always change sharply at ∼10 M_☉ pc⁻², or is that value metallicity dependent? Preliminary results indicate appear consistent with a dependence on metallicity (Fumagalli et al. 2010), but a more systematic survey is needed.

For the dust-gas thermal exchange rate coefficient α_dg, we adopt a standard value of 3.8 × 10⁻³⁴ erg cm³ K^−3/2 in H₂-dominated gas in the Milky Way (Goldsmith 2001), and we assume that this will also hold in galaxies of similar metallicity. We must extrapolate this both to lower metallicity galaxies and to predominantly H i gas. For the former, we assume that the total surface area of grains is proportional to the metal abundance. For the latter, we must account for both the change in the number of particles and the masses of the individual particles, which alters their speed. We assume that the accommodation coefficient is equal for H, H₂, and He. (More accurate approximations are possible if the grain size distribution is known (e.g., Hollenbach & McKee 1979), but given the uncertainties in how size distributions vary from galaxy to galaxy and the relative lack of importance of this effect, we omit this complication.) With this assumption, the dust–gas energy exchange rate provided by gaseous members of species X is proportional to n_X/m^1/2_X, where n_X is the number density of members of that species and m_X is the mass of that species. Thus, in regions of atomic and molecular hydrogen, respectively, we have

$$\begin{eqnarray} \alpha _{\rm dg, \rm H\,{\mathsc i}} & \propto & \frac{n_{\rm H\,{\mathsc i}}}{m_{\rm H}^{1/2}} + \frac{n_{\rm He}}{m_{\rm He}^{1/2}} \end{eqnarray} \tag{ A1 }$$

$$\begin{eqnarray} \alpha _{\rm dg, H_2} & \propto & \frac{n_{\rm H_2}}{m_{\rm H_2}^{1/2}} + \frac{n_{\rm He}}{m_{\rm He}^{1/2}}, \end{eqnarray} \tag{ A2 }$$

where the constants of proportionality are the same in each case. In a predominantly atomic region $n_{\rm H\,{\mathsc i}} = n_{\rm H}$, while in a predominantly molecular region $n_{\rm H_2} = n_{\rm H}/2$; in both cases n_He ≈ n_H/10. Similarly, m_He = 4m_H and $m_{\rm H_2} = 2 m_{\rm H}$. Combining the dependence on metallicity with that on chemical phase, we arrive at our final expressions for α_dg:

$$\begin{eqnarray} \alpha _{\rm dg, H_2} & = & 3.8\times 10^{-33} Z^{\prime }\mbox{ erg cm}^3\mbox{ K}^{-3/2} \end{eqnarray} \tag{ A3 }$$

$$\begin{eqnarray} \alpha _{\rm dg, \rm H\,{\mathsc i}} & = & 1.0\times 10^{-33} Z^{\prime } \mbox{ erg cm}^3\mbox{ K}^{-3/2}. \end{eqnarray} \tag{ A4 }$$

For the radiation temperature, T_rad, as discussed above we must select a single value to characterize the re-radiated infrared field within a cloud. We adopt T_rad = 8 K for this, near the minimum temperature seen in ammonia observations of Galactic cold clouds (Jijina et al. 1999). Both this choice and our choice of α_dg have very little impact on our results due to the weak dust-gas coupling in the density range with which we are concerned.

Our calculations depend on the absorption cross section per H nucleus σ_d to UV photons, and the visual extinction per unit hydrogen column A_V/N_H. Note that σ_d reflects only absorption, while A_V includes scattering as well. These quantities depend on the extinction curve and vary between dense and diffuse environments even within a single galaxy at a single metallicity, and thus their exact values are uncertain by a factor of two. As a guide we examine the values given by the models of Weingartner & Draine (2001) for the Milky Way, Large Magellanic Cloud, and Small Magellanic Cloud.⁵ Once we scale the LMC and SMC models to the same total dust-to-gas ratio as the Milky Way models, we obtain values of σ_d/10⁻²¹cm⁻² = 0.7, 1.7, 2.0, 1.8 and (A_V/N_H)/10⁻²²magcm⁻² = 4.5, 5.1, 3.6, 2.8 for Weingartner & Draine's models for Milky Way sight lines with R_V = 5.5, Milky Way sight lines with R_V = 3.1, the average LMC sight line, and a sight line through the SMC bar, respectively. We therefore adopt intermediate values σ_d = 1.0 × 10⁻²¹Z' cm² and A_V/N_H = 4.0 × 10⁻²²Z' mag cm². With these fiducial choices, the UV absorption is larger than the visual extinction by a factor of 2.5/1.08 = 2.31 (where the factor of 1.08 accounts for the conversion from magnitudes to true dimensionless units).

The FUV radiation field, parameterized by G'₀, affects both gas temperature and chemistry. There is no unique value of G'₀ that characterizes all gas clouds across all galaxy types, and even within a single star-forming cloud the ambient UV field increases with time as stars form. As a fiducial choice we adopt G'₀ = 1, the solar neighborhood value. In regions where star formation is ongoing the mean is likely larger, but we are interested in the initiation of star formation, so it seems prudent to select a value typical of regions before the onset of star formation.

To test our sensitivity to this choice, we have recomputed our model grids with values of G'₀ = 0.1 and G'₀ = 10, and re-plotted Figures 1–3 of the main text in Figure 6. Not surprisingly, the H i to H₂, C ii to CO transitions, and warm to cold gas transitions all move to higher density and A_V for higher G'₀, and to lower density and A_V for lower G'₀. Nonetheless, over a factor of 100 range in G'₀, we will retain the excellent correlation between the H i to H₂ transition and the drop in temperature from hundreds of K to ∼10 K, along with the concomitant drop in the Bonnor–Ebert mass from several 1000 M_☉ or more to a few M_☉. At all three values of G'₀, contours of constant $f_{\rm H_2}$ align closely with contours of constant Bonnor–Ebert mass. In the lower two panels, note that contours of constant M_BE remain largely vertical, indicating that the Bonnor–Ebert mass drops systematically as the H₂ fraction increases. Thus our conclusions are robust against large variations in G'₀.

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The cosmic ray heating rate is uncertain in two ways. First, the energy yield per primary cosmic ray ionization q_CR is 6.5 eV in predominantly neutral H i (Dalgarno & McCray 1972; Wolfire et al. 1995). In H₂ the yield is uncertain. Both dissociative recombination of H₂ and excitation of its rotational and vibrational levels followed by collisional de-excitation provide extra channels for energy transfer from primary cosmic ray electrons to thermal motion, but the importance of these processes is likely density dependent. Values given in the literature range from nearly the same energy yield as in H i up to 20 eV per primary ionization (Glassgold & Langer 1973; Dalgarno et al. 1999). We follow Wolfire et al. (2010) in adopting an intermediate value q_CR = 12.25 eV, but this should be regarded as uncertain by a factor of two.

A similar uncertainty affects the cosmic ray ionization rate. For Milky Way like galaxies we adopt ζ = 2 × 10⁻¹⁷ s⁻¹ (Wolfire et al. 2010), but the primary cosmic ray ionization rate even in the Milky Way is substantially uncertain and may be higher than this value (Neufeld et al. 2010). The cosmic ray intensity also varies with the SFR in galaxies (Abdo et al. 2010a). Thus in low-metallicity galaxies, which also tend to have low SFRs, the cosmic ray ionization rate is likely lower than the Milky Way value. The scaling is extremely uncertain; for lack of a better alternative, we simply take the cosmic ray ionization rate to scale with the metallicity.

To test our sensitivity to our choice of cosmic ray ionization rate, we have recomputed our temperature grid with cosmic ray ionization rates that are a factor of three larger and a factor of three smaller than our fiducial choice, thereby exploring a ∼1 decade range in cosmic ray heating rate. We plot the results in Figure 7. Again, we see that the results are robust, in the sense that contours of constant Bonnor–Ebert mass remain closely aligned with contours of constant H₂ fraction as we vary the cosmic ray ionization rate.

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We do caution, however, that this breaks down if we select a cosmic ray ionization rate that is extremely high (more than 10–30 times our fiducial one). For such high cosmic ray ionization rates, cosmic ray heating becomes more important than grain photoelectric heating even in unshielded, low A_V gas. In this case the temperature and Bonnor–Ebert mass become uncorrelated with A_V and vary with density alone. However, such high cosmic ray ionization rates are inconsistent with observations in the Milky Way.