THE ROLE OF TURBULENCE IN STAR FORMATION LAWS AND THRESHOLDS

During the simulations, stellar particles are formed by conversion of gas. These particles are used for the injection of stellar feedback, but are not used in the post processing analysis of the star-formation rate (SFR), which is recalculated from the density of gas (see Section 4).

Details of star formation and the associated stellar feedback are given in Renaud et al. (2013). The values of the star formation efficiency and the star formation threshold density ρ₀ that we have adopted here (see Table 1) are adjusted to match the observed global SFR for local galaxies: ≈1–5 M_☉ yr⁻¹ for the Milky Way (MW; Robitaille & Whitney 2010) and ≈0.4 M_☉ yr⁻¹ for the Large Magellanic Cloud (LMC; e.g., Skibba et al. 2012), on average. SFR of the simulated Small Magellanic Cloud (SMC) is ≈0.5 M_☉ yr⁻¹, on average, which is higher than the observed value (e.g., 0.05 M_☉ yr⁻¹ obtained by Wilke et al. 2004), perhaps because of different structures, but a one-to-one match is not sought.

Table 1. Summary of Model Parameters

	MWPCa	LMC$_{1.0\,{{Z}}_{\odot }}$	LMCPC	LMCSS	SMC$_{0.1\,{{Z}}_{\odot }}$	SMC$_{0.3\,{{Z}}_{\odot }}$	SMC$_{1.0\,{{Z}}_{\odot }}$	SMCPC	SMCSS
EoS or metallicityb (Z☉)	PC	1.0	PC	SS	0.1	0.3	1.0	PC	SS
Box length (kpc)	100	25			30
AMR coarse level	9	8			8
AMR fine level	21	14			15
Maximal resolution (pc)	0.05c	1.5			1.0
DM halo
mass (× 109 M☉)	453.0	8.0			1.2
Number of particles (× 105)	300.0	3.49			5.0
Primordial starsd
mass (× 109 M☉)	46.0	3.1			0.35
Number of particles (× 105)	300.0	5.75			2.15
Gas
mass (× 109 M☉)	5.94	0.54			0.715
∼ AMR cell number (× 106)	240	385	440	450	43	43	43	45	50
Radial profile	Exponential	Exponential			Exponential
Scale radius (kpc)	6	3			1.3
Radial truncation (kpc)	28	6			2.3
Vertical profile	Exponential	Exponential			Exponential
Scale-height (kpc)	0.15	0.15			0.6
Vertical truncation (kpc)	1.5	0.45			1.3
Intergalactic densitye	10−7	10−7			10−3
Star formation
	3%	3%			3%
ρ0 (cm−3)	2 × 103	102			102
Stellar feedback
Photo-ionization	$\checkmark$	⋅⋅⋅			⋅⋅⋅
Radiative pressure	$\checkmark$	⋅⋅⋅			⋅⋅⋅
SNe	Kinetic	Thermic	Kinetic	Kinetic	Thermic	Thermic	Thermic	Kinetic	Kinetic

Notes. ^aSimulations are labeled mnemonically, with their names having the value of the metallicity or EoS parameter in the subscript: MW_PC, LMC$_{1.0\,{{Z}}_{\odot }}$, LMC_PC, LMC_SS, SMC$_{0.1\,{{Z}}_{\odot }}$, SMC$_{0.3\,{{Z}}_{\odot }}$, SMC$_{1.0\,{{Z}}_{\odot }}$, SMC_SS, SMC_PC. ^bMetallicity is a meaningful parameter only when the heating and cooling processes are evaluated, the name of the EoS is given otherwise. ^cThe analysis is performed by extracting the simulation data at the effective resolution of 1.5 pc (see the text for details). ^dStars initially present in simulation. ^eFraction of the density of gas at the edge of the galaxy that is set beyond the truncation of the gas disk.

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The stellar feedback is modeled by photo-ionization together with radiative pressure in the case of the MW simulation (Renaud et al. 2013) and supernova (SN) feedback, implemented as a Sedov blast (Dubois & Teyssier 2008) in all simulations either in a kinetic or thermal scheme. The choice of the SN feedback scheme is determined by treatment of the heating and cooling processes (see Section 2.2)—every time the gas follows an equation of state (EoS), the total energy of SN (10⁵¹ erg) is injected in the kinetic form, since thermal feedback would have no effect.

The cooling and heating processes occurring in the ISM depend on the metallicity. In our models, the gas cooling due to atomic and fine-structure lines and radiation heating from an uniform ultraviolet background are modeled following Courty & Alimi (2004) and Haardt & Madau (1996), respectively. Metallicity is a parameter fixed for each simulation and represents the average metal mass fraction in the galaxy.

Heating and cooling processes can often substantially slow down the simulation. A piecewise polytropic EoS of form T∝ρ^{γ − 1} with the polytropic index γ can be applied instead. We use a pseudo-cooling (PC) EoS (Figure 1), fitting the heating and cooling equilibrium of gas at one-third solar metallicity (Bournaud et al. 2010; Teyssier et al. 2010). In the above definition of the EoS, we neglect the capacity of the gas to shield itself from the surrounding radiation. At densities around 0.1–1 cm⁻³ and temperatures of several hundred K, SS becomes important—the molecular fraction of the gas increases, enabling it to cool to even lower temperatures (Dobbs et al. 2008). This can be modeled by the alternative EoS shown in Figure 1.

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We study models of a spiral galaxy resembling the MW, a disk galaxy similar to the LMC, and an irregular dwarf galaxy comparable to the SMC. We do not try to reproduce fine details for these galaxies, but propose models representing systems with different morphological and physical properties. Each simulation is performed in isolation and without cosmological evolution.

The details of the MW simulation can be found in Renaud et al. (2013). Here, this simulation is analyzed at resolution comparable to the resolution of other galaxy simulations in our sample, which is 1.5 pc, i.e., not at its maximal resolution. The parameters of all simulations are summarized in Table 1.

Simulations are labeled in a way to stress their principal difference, which is related to the EoS or metallicity parameter. Simulations in which the heating and cooling processes are evaluated have the value of the metallicity in subscript. If the EoS is used instead, the subscript indicates the name of the EoS. The most realistic cases are the LMC$_{1.0\,{{Z}}_{\odot }}$ and the SMC$_{0.1\,{{Z}}_{\odot }}$ simulations for the LMC and SMC, respectively. The solar metallicity we have adopted in the LMC$_{1.0\,{{Z}}_{\odot }}$ simulation is higher than in the real LMC (1/2 Z_☉; Russell & Dopita 1992; Rolleston et al. 1996), but fairly representative of low-redshift and low-mass disk galaxy that we intend to model. The metallicity of 1/10 Z_☉ that we used in the SMC$_{0.1\,{{Z}}_{\odot }}$ simulation falls in the range of estimated values for the real SMC (1/5–1/20 Z_☉; Russell & Dopita 1992; Rolleston et al. 1999).

Figure 2 displays the surface gas density map of the three galaxies. The MW_PC, a spiral galaxy, shows large variety of substructures—bar and spiral arms on the kiloparsec scale as well as dense clumps on the parsec scale (see Renaud et al. 2013, for details). The LMC$_{1.0\,{{Z}}_{\odot }}$ is also a disk galaxy, but with a much less pronounced structure of spiral arms and more diffuse gas present in the inter-arm regions compared to MW_PC. The SMC$_{0.1\,{{Z}}_{\odot }}$ is an irregular dwarf galaxy. Three major dense clumps can be seen within the irregular structure of the diffuse gas.

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The mass-weighted density probability distribution function (PDF) of the gas for the MW_PC, LMC$_{1.0\,{{Z}}_{\odot }}$, and SMC$_{0.1\,{{Z}}_{\odot }}$ is shown in Figure 3. The MW_PC's PDF has a log-normal shape, followed by a power-law tail at high densities (ρ ≳ 1000 cm⁻³) probed thanks to the high resolution reached in this simulation. Similarly, the LMC$_{1.0\,{{Z}}_{\odot }}$'s PDF can be approximated by a log-normal functional form in the density range from 10⁻² to 10² cm⁻³ with an excess of dense gas with respect to a log-normal fit above density of about 100 cm⁻³. Truncation, possibly due to the resolution limit, is visible at a density of ∼2 × 10⁴ cm⁻³. The PDF of the SMC$_{0.1\,{{Z}}_{\odot }}$ is rather irregular, with two components—one at low densities (∼10⁻¹ cm⁻³) and the other at high densities (∼2 × 10⁴ cm⁻³). Such irregular PDF reveals the density contrast between diffuse gas and several high-density clumps.

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The shape of the density PDF is determined by global properties of galaxies and physical processes of the ISM. As suggested by Robertson & Kravtsov (2008), the density PDF can vary from galaxy to galaxy and that of a multiphase ISM can be constructed by summing several log-normal PDFs, each representing approximately an isothermal gas phase. Similarly, Dib & Burkert (2005) showed that the PDF of a bi-stable two-phase medium evolves into a bimodal form with a power-law tail at the high-density end in the presence of self gravity (see also Elmegreen 2011; Renaud et al. 2013). However, in most cases, a single, wider log-normal functional form is a reasonably good approximation of the PDF of disk galaxies up to ≳ 10⁴ cm⁻³ (see, e.g., Bournaud et al. 2011; Tasker & Bryan 2008; Tasker & Tan 2009).

Note that the SMC$_{0.1\,{{Z}}_{\odot }}$, which has a lower metallicity than the LMC$_{1.0\,{{Z}}_{\odot }}$, is able to reach the highest densities. Metallicity is important for cooling—more metallic gas is more efficient at cooling the gas and should allow it to reach higher densities. However, we do not observe such a trend. This could indicate that factors other than thermal may be key in setting the gas distribution.

Another possible explanation could be a mismatch between the choices of threshold density ρ₀ for star formation and the metallicity in the LMC$_{1.0\,{{Z}}_{\odot }}$. If ρ₀ is chosen to be low, stars will form in an intermediate density medium, i.e., without the need of gravitational collapse of a cloud into a dense region. Furthermore, stellar feedback helps the destruction of the densest clumps, which produces more intermediate-density gas and further prevents the gravitational contraction leading to high densities. The maximum density of the ISM is thus lower than with a high ρ₀, and the resulting PDF does not yield the classical high-density power-law tail. However, in the case of the LMC$_{1.0\,{{Z}}_{\odot }}$, the transition of the gas from high (>10³–10⁴ cm⁻³) to intermediate densities (10–10² cm⁻³, just below the actual ρ₀) due to the feedback would lead to a substantial reduction in the SFR (because of the ρ_SFR∝ρ^3/2 used in our model, the SFR is dominated by high-density gas). Consequently, feedback itself would be substantially reduced.

Another more likely explanation is that the SMC$_{0.1\,{{Z}}_{\odot }}$ contains a much higher gas fraction compared to the LMC$_{1.0\,{{Z}}_{\odot }}$ (see Table 1), leading to a much lower value of the Toomre parameter ($\mathrm{Q} \propto \Sigma _{\mathrm{gas}}^{-1}$), which allows the SMC$_{0.1\,{{Z}}_{\odot }}$ to reach higher densities than in the LMC$_{1.0\,{{Z}}_{\odot }}$.

In a given beam, the effective Mach number $\mathcal {M}$ is defined as

$$\begin{equation} \mathcal {M} = \frac{\sigma _{v}}{\sqrt{3}} \frac{1}{{c_s}}, \end{equation} \tag{ 1 }$$

where σ_v and c_s are the mass-weighted velocity dispersion and the mass-weighted sound speed with respect to the beam, respectively, calculated as follows:

$$\begin{equation} \sigma _{v} = \sqrt{\frac{\sum _i m_i v_i^2}{\sum _i m_i} - \left(\frac{\sum _i m_i v_i}{\sum _i m_i}\right)^2}, \end{equation} \tag{ 2 }$$

and

$$\begin{equation} {c_s} = \sqrt{\frac{\sum _i m_i T_i \gamma \frac{k_{\mathrm{B}}}{m_{\mathrm{H}}} }{\sum _i m_i}}. \end{equation} \tag{ 3 }$$

Summations are done over all AMR cells in the analyzed beam and the index i refers to cell related quantities. T_i, m_i, and v_i are the cell temperature, gas mass, and speed, respectively. γ = 5/3 is the adiabatic index for monoatomic gas, m_H is the mass of the hydrogen atom, and k_B is the Boltzmann constant.

An alternative to the above "beam-based" average could be to compute the mass-weighted $\mathcal {M}$ with a cell velocity dispersion itself calculated with respect to its closest cells, but we find that this does not lead to a significant difference in the results.

Temperature in the beam is computed as a mass-weighted average:

$$\begin{equation} T = \frac{\sum _i m_i T_i}{\sum _i m_i}. \end{equation} \tag{ 4 }$$

To estimate the actual thickness of the star-forming regions within each beam, we apply a Gaussian fit to a one-dimensional (1D) projection of the gas density along one of the mid-plane axes. The thickness is then defined as the FWHM of the resulting fit. Although the estimation method of the thickness parameter is simplistic, the obtained values are in good agreement with visual inspection of density maps of individual star-forming regions.

Note that in our analysis, we do not use the SFR computed directly in the simulation. The main reason is that the conversion of gas into stars is modeled as a stochastic process leading to the discretization of the Σ_SFR values, which make the analysis difficult by introducing more noise.

The SFR of a beam is estimated from the gas content of each cell by

$$\begin{equation} \rho _{\star } = \epsilon \frac{\rho }{t_{\mathrm{ff}}} \propto \epsilon \rho ^{3/2} \quad {\rm for}\ \rho > \rho _{0}, \end{equation} \tag{ 5 }$$

where ρ_⋆ is the local SFR density, ρ is the density of gas in the cell, is the star formation efficiency per free-fall time $t_{\mathrm{ff}} = \sqrt{3 \pi /(32 G \rho)}$, and ρ₀ is the star-formation threshold.

Σ_SFR is then given by

$$\begin{equation} \Sigma _{\mathrm{SFR}} = \frac{\sum _i {\rho _{\star }}_i V_i}{S}, \end{equation} \tag{ 6 }$$

where ρ_⋆i and V_i are the cell SFR density and volume, respectively, and S is the surface of the beam. Similarly, Σ_gas is

$$\begin{equation} \Sigma _{{\rm gas}} = \frac{\sum _i \rho _i V_i}{S}, \end{equation} \tag{ 7 }$$

with ρ_i representing the cell gas density.

4.2.1. Beam Size Effects

Our choice of the beam size is related to the adopted analytical formalism that is tightly linked to the turbulence-driven structure of the ISM. Supersonically turbulent isothermal gas is found to be well described by a log-normal PDF. However, once these hypotheses about the state of gas are relaxed, strictly log-normal PDF is not recovered. The PDFs of the density field in our sample of galaxies are close to, but not exactly, log-normal functional forms when all scales are considered (see Section 3.3). Individual beams should be large enough to be representative samples of star-forming regions at different evolutionary stages. In addition, the choice of the beam size is somehow linked to the turbulence and its cascade from large scales where the turbulence is injected, down to the small scales where the energy dissipation overcomes its transfer. In order to capture the turbulent cascade, the size of the beam should not be too large compared to the injection scale,⁷ nor too small compared to the dissipation scale. In the former case, the simulation would capture processes other than turbulence and in the latter case, turbulence would be already dissipated.

To estimate the impact of the size of the beam, we compare the results in the Σ_gas–Σ_SFR plane obtained by varying the beam width by a factor of 2.5 with respect to the one used in analysis. Figure 4 shows the comparison for the MW_PC simulation. Increased beam size leads to an overall reduction of Σ_gas for the beam, which can be understood as a consequence of decreased volume fraction occupied by the dense gas. On the contrary, smaller beam size allows reaching higher values of Σ_SFR. This comparison suggests that the position of points in the Σ_gas–Σ_SFR plane depends on the considered spatial scale. Schruba et al. (2010) found such dependence in the study of the Local Group spiral galaxy M33. Similarly, Lada et al. (2013) found a more efficient star formation (SF) at scales of molecular clouds, indicating that caution should be used when comparing SF relations involving different spatial scales.

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However, the global behavior of the Σ_gas–Σ_SFR does not seem to be strongly affected by the size of the beam, at least for the range of sizes that we explored.

4.2.2. Parsec and Sub-parsec Physics

We remind that the MW_PC simulation is analyzed at the resolution of 1.5 pc, which is different from its maximal resolution of 0.05 pc. To study the impact of the resolution, we compare the Σ_gas–Σ_SFR relation for these two resolutions in Figure 5. Sub-parsec physics does not influence our results at low and intermediate surface gas densities, but it plays a role in the densest regions, where it leads to higher values of Σ_SFR. The increased resolution leads to the modification of structures mainly at high densities, which translates into higher values of Σ_SFR computed at a fixed 100 pc scale.

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Figure 6 shows the impact of metallicity on the Σ_SFR. The left panel compares two systems with comparable metallicities, 0.1 Z_☉ and 0.3 Z_☉, while in the right panel, two more extreme metallicities are compared, 0.1 Z_☉ and 1.0 Z_☉. In the region below the break, high-metallicity systems tend to have higher Σ_SFR for a fixed value of Σ_gas than systems with lower metallicity.

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The impact of the EoS on the SFR is presented in Figure 7. The SMC$_{0.1\,{{Z}}_{\odot }}$ simulation is compared to that of the SMC_PC, using the EoS of PC, and to that of the SMC_SS with the EoS of SS. The similarity of two contour plots on the left panel shows that the PC EoS is a good approximation to the actual heating and cooling processes even for a slightly lower metallicity in this case (we remind that the PC EoS is derived using the metallicity of one-third Z_☉; see Section 2.2). In the case of the SS EoS, for a given value of Σ_gas, Σ_SFR tends to be higher compared to that of the simulation with metallicity of 0.1 Z_☉.

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We do not assume any metallicity gradient in the gas, nor chemical evolution. We use the model for SS without an implicit metallicity dependence, similar to the work of Dobbs et al. (2008). As shown in Figure 7, the SS EoS leads to higher Σ_SFR for fixed Σ_gas compared to the model of the SMC with metallicity of 0.1 Z_☉.

The existence of the break in the Schmidt–Kennicutt relation in our models does not seem to depend on SS effects. The exact position of this break is, however, sensitive to metallicity—the slope at low Σ_gas, i.e., below the break, is higher in metal-poor galaxies as shown on Figure 6. A similar metallicity-dependent position of the break is present in the theoretical model of Krumholz et al. (2009), including the effect of hydrogen SS, which in turn determines the amount of gas in molecular form. In addition, Dib (2011) explored the metallicity-dependent feedback and found that it can lead to a modification of the position of the break for a given metallicity-dependent molecular gas fraction. It is clear that SS has an impact on the Schmidt–Kennicutt relation (see the right panel of Figure 7), but it does not seem to be the only factor determining the presence of the break.

5.2.1. Mach Number

In Figure 8, we show how the Σ_gas–Σ_SFR relation depends on the Mach number, temperature, and velocity dispersion calculated using Equations (1), (4), and (2), respectively. We show the example of the MW_PC, but we obtain qualitatively similar results for all other galaxies. The Mach number dependence for the MW_PC, SMC$_{0.1\,{{Z}}_{\odot }}$, and LMC$_{1.0\,{{Z}}_{\odot }}$ is displayed in Figure 9.

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Two regimes in the star-formation relation are identified. The points located in the region below the break typically have Mach numbers with values below unity. Furthermore, for a given Σ_gas, Σ_SFR increases with increasing Mach number. At high surface densities of gas, Σ_SFR and Σ_gas are found to be correlated. The gas reservoirs that happen to be in this regime of efficient star formation tend to have supersonic velocity dispersions.

Both the temperature and velocity dispersion contribute to the resulting Mach number dependence in the star-formation relation. Despite the variation in temperature, the overall variation in Mach number relies on σ_v. The velocity dispersion of the ISM can be increased by several processes. Among them, Bournaud et al. (2010) found, in simulations similar to those analyzed here, self-gravity to play the dominant role compared to stellar feedback. Therefore, by increasing the velocity dispersion, self-gravity sets the level of turbulence, i.e., the compression of gas and thus the SF. This suggests that the power law part of the Σ_gas–Σ_SFR relation arises from self-gravity at a high Mach number, while this connection is weaker in the break.

5.2.2. Vertical Scale of the Gas

Figure 10 shows the variation of the Σ_gas–Σ_SFR relation with the thickness of the star-forming regions in the SMC$_{0.1\,{{Z}}_{\odot }}$. For a given surface gas density, thicker regions tend to have lower surface star formation density. This relation between Σ_SFR and the thickness parameter at fixed Σ_gas results from Equation (5) relating the volume density of gas with that of SFR.

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Most spatially resolved studies of spiral galaxies find the presence of a power-law Σ_gas–Σ_SFR relation with a break at surface gas densities of the order of a few M_☉ pc⁻² (see Kennicutt & Evans 2012, and references therein). The slope of the power-law relation in the high surface density regime is found to be in the range of 1.2–1.6 when the total (molecular plus atomic) gas surface density is considered.

Less agreement about the power-law slope in observations is reached when the molecular gas surface density is considered solely. Some recent studies (e.g., Eales et al. 2010; Rahman et al. 2011; Leroy et al. 2013) have reported an approximately linear relation between the surface density of the SFR rate and the surface density of molecular gas. Other studies (e.g., Kennicutt et al. 2007; Verley et al. 2010; Liu et al. 2011) have found a much steeper relation, with a slope in the range of 1.2–1.7, similar to that of integrated measurements (Kennicutt 1998). This discrepancy between different results in observations is still debated. A possible interpretation of the sublinear relation was recently proposed by Shetty et al. (2013). They suggest that the CO emission used in the estimation of Σ_gas is not all necessarily associated with SF. Not subtracting such a diffuse component could lead to a slope close to unity.

The distribution of data points from the observations of the SMC (Bolatto et al. 2011) in the Σ_gas–Σ_SFR plane has a similar shape than that of spiral galaxies, but is noticeably shifted toward higher total Σ_gas.

In Figure 11, we show three of our models: the MW_PC, SMC$_{0.1\,{{Z}}_{\odot }}$, and LMC$_{1.0\,{{Z}}_{\odot }}$ in the Σ_gas–Σ_SFR plane. The MW_PC and LMC$_{1.0\,{{Z}}_{\odot }}$ models lie in the loci of observed spiral galaxies (e.g., Kennicutt 1998; Kennicutt et al. 2007; Bigiel et al. 2008). Our SMC$_{0.1\,{{Z}}_{\odot }}$ model has a lower Σ_SFR for a given Σ_gas when compared to both the MW_PC and LMC$_{1.0\,{{Z}}_{\odot }}$ models. The region below the break of our SMC$_{0.1\,{{Z}}_{\odot }}$ model is located at slightly lower Σ_gas than the real SMC, but its displacement with respect to spiral galaxies (the MW and LMC) is well reproduced (Figure 11). In our simulations, Equation (5) sets the slope of the power-law relation with the value of 1.5. A shallower relation, closer to the observed values, might be reached by accounting for a stronger regulation of star formation (e.g., pre-SN stellar feedback; see Renaud et al. 2012), but this slope change has not been demonstrated by simulations yet.

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The existence of the break in the Σ_gas–Σ_SFR relation is, in our models, equivalent of having a non-zero value of the volume density threshold in the local three-dimensional star-formation relation. Setting no threshold leads to a power-law relation without a break.

Figure 12 shows that the value of the density threshold ρ₀ that we have used in our analysis has an impact on the Σ_gas–Σ_SFR relation. Changing the value of ρ₀ changes the slope at low Σ_gas in the Σ_gas–Σ_SFR relation (Figure 12). This could suggest that the transition from the inefficient to the power-law regime could be due to the density threshold ρ₀ we imposed by hand in the star-formation law (see Equation (5)). However, we have checked that the deviation from the power-law regime occurs at $\mathcal {M} \approx$ 1, independently of ρ₀. In addition, in Figure 9, we have shown that beams located in the break tend to have $\mathcal {M}$ below unity, while regions at high Σ_gas are mostly supersonic.

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To better understand the behavior of the ISM in our simulations, we show in Figure 13 the Mach number as a function of average volume density of gas (computed as the mass-weighted average density of the gas in each beam) <ρ > in the beam for the MW_PC. The Mach number varies with the average density as $\mathcal {M} \propto \left<\rho \right>^{0.5}$, similarly to the two-phase turbulent flow studied by Audit & Hennebelle (2010). Although caution should be used when doing such a comparison (temperature and velocity dispersion vary with density differently in both models), the onset of the supersonic regime, i.e., the transition from an inefficient regime to a power law, happens at densities of ≈10 cm⁻³(see also Audit & Hennebelle 2010, their Figures 4 and 9).

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Other interpretations of the observed break are possible. The Σ_gas–Σ_SFR relation could be an effect of the galactic radial distance with low Σ_gas at large radius and high Σ_gas at small radius, as found by Kennicutt et al. (2007) and Bigiel et al. (2008). The break could then be explained as a consequence of the drop in the average volume density in the outer regions of galaxies as proposed by Barnes et al. (2012). However, Figure 14 shows no such radial dependence for Σ_gas nor Σ_SFR. Star-forming regions in outer parts of a galaxy can exhibit both star formation regimes. A possible explanation why we do not see any radial dependence in our simulations may be a missing metallicity gradient. The outer regions of our simulated galaxies have the same metallicity than the innermost regions, thus the metallicity is probably too high at the edge of disk and allows for efficient cooling and, consequently, efficient star formation while it may lie in the break regime otherwise.

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When azimuthally averaged, Σ_gas and Σ_SFR both decline steadily as a function of radius in many galaxies, despite different morphologies (see Bigiel et al. 2008 for a sample of nearby spiral galaxies and Leroy et al. 2009 for CO intensity radial profiles for the same sample). Figure 15 shows Σ_gas and Σ_SFR as functions of radius for the LMC$_{1.0\,{{Z}}_{\odot }}$. Both radial profiles decline with galactic radius as in observed spiral galaxies.

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Another alternative explanation for the existence of the break is that it corresponds to the transition from atomic to molecular hydrogen (Krumholz et al. 2009). According to this scenario, the transition from atomic to molecular hydrogen and the subsequent star formation depend on local conditions that vary with galactic radius, e.g., metallicity, gas pressure, and shear.⁸ Bigiel et al. (2008) found such radial dependence in the sample of nearby galaxies in agreement with the findings of Wong & Blitz (2002) and the threshold interpretations of, e.g., Kennicutt (1989), Martin & Kennicutt (2001), and Leroy et al. (2008). Similar results are reproduced in some simulations, e.g., Halle & Combes (2013), who find that molecular gas is a better tracer of star formation than atomic gas and plays an important role in the low surface density regions of galaxies by allowing for more efficient star formation. However, our models that do not include chemodynamics are able to reproduce the observed break at low Σ_gas. Therefore, this seems to indicate that the presence of molecules is not a necessary condition to trigger the process of star formation. However, we acknowledge numerous observational evidences showing that molecules are involved at a later stage of the SF process.

To summarize, we consider two representative beams having the same Σ_gas, but different Σ_SFR (Figure 16). These beams have similar average volume densities <ρ >, which can be several orders of magnitude smaller than ρ₀. However, the beam that happens to have the highest Σ_SFR always has the highest Mach number, as previously suggested by Figure 9. We have argued above that the density threshold ρ₀, the thickness of the star-forming regions and the molecules do not have impact on the transition from the regime of inefficient star formation to the efficient power-law regime. The role of the artificial threshold ρ₀ imposed in the simulations is to set a frontier between the diffuse non-star-forming gas and the star-forming component, but not to tune the efficiency of star formation per se. Therefore at a given Σ_gas, this efficiency depends mostly on the level of turbulence ($\mathcal {M}$), i.e., the compression of the ISM.

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Renaud et al. (2012) proposed that the break indeed corresponds to the onset of supersonic turbulence that, by generating shocks, triggers gravitational instabilities leading to star formation. In Figures 17 and 18, we compare simulations of the MW_PC and SMC$_{0.1\,{{Z}}_{\odot }}$ with the analytical model of Renaud et al. (2012). In this model, the relation between Σ_gas and Σ_SFR depends on three parameters: the Mach number $\mathcal {M}$, the scale height h, and the star formation volume density threshold ρ₀. We do not compare each star-forming region in the simulation with the model, but we are rather interested in what values these parameters should take to obtain upper and lower limits for simulated data. The break is in the subsonic regime (measured values of $\mathcal {M}$ are below unity), which corresponds to the regime where the analytical model deviates from its asymptotic behavior (at high Σ_gas). In this regime the scale heights of the beams set the efficiency of star formation spanning the range given by the model and quantitatively in accordance with the values measured in the simulations (Figure 10). In the analytical model, the power-law regime can be reached even with the Mach number below unity (red curve). However, our simulations do not probe this area of the Σ_gas–Σ_SFR plane. The data points in the power-law regime are exclusively supersonic and can only be described by a model with the Mach number above unity (black curve).

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In Figure 6, we have shown that the exact position of the break in the Σ_gas–Σ_SFR plane depends on metallicity. A comparison of different metallicities in otherwise identical systems shows that the slope at low Σ_gas has a greater value in metal-poor galaxies. Figure 3 suggests that metallicity is not the only factor determining the gas density distribution in our simulations. A similar lack of direct dependence of the fraction of dense gas on metallicity is found when simulations of the SMC with different metallicities are compared (not shown here). Thus the slope at low Σ_gas cannot be explained by the presence of a higher fraction of dense gas in systems with higher metallicity compared to systems with lower metallicity. However, metallicity has an impact on star formation, even though indirect. Metallicity directly influences the temperature of the gas—the higher the metallicity, the more efficient the cooling and therefore the temperature, which in turn impacts the Mach number. In Figure 19, we show Mach numbers for the SMC$_{1.0\,{{Z}}_{\odot }}$ and SMC$_{0.1\,{{Z}}_{\odot }}$. Higher values of Mach number are reached in galaxies with higher metallicity.

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This work does not include metallicity-dependent SS and feedback. Accounting for those, Dib (2011) showed that both the fraction of gas in molecular form and the efficiency of star formation per unit time depend on metallicity. This leads to the metallicity-dependent Σ_gas–Σ_SFR relation at any Σ_gas.