DIFFERENT STAR FORMATION LAWS FOR DISKS VERSUS STARBURSTS AT LOW AND HIGH REDSHIFTS

Exploring the relation between the gas content and star formation rate (SFR) of galaxies is crucial to understanding galaxy formation and evolution. This is required to understand the nature of SF, the parameters that regulate it, and its possible dependence on local and global galaxy properties (e.g., Silk 1997; Elmegreen 2002; Krumholz & Thompson 2007; McKee & Ostriker 2007). In addition, this information is a critical ingredient of theoretical models of galaxy formation, either based on semianalytical realizations or on numerical simulations (e.g., Guiderdoni et al. 1998; Somerville et al. 2001; Monaco et al. 2007; Ocvirk et al. 2008; Dekel et al. 2009; Gnedin et al. 2009; Croton et al. 2006). In fact, the physics associated with the conversion of gas into stars inside galaxies is overwhelmingly complicated, so that theoretical models generally resort to scaling laws that are calibrated using observations of nearby galaxies.

Schmidt (1959) first suggested the existence of a power-law relation between surface densities of SFR and gas masses. Kennicutt (1998, hereafter K98) presented a calibration of the Schmidt law with a slope of 1.4 in log space, which has since been the most widely used in the community. K98 fits the local populations of spiral galaxies and IR-luminous galaxies (LIRGs/ULIRGs), with the gas mass including both neutral (H i) and molecular (H₂) hydrogen for spirals, and molecular gas only for (U)LIRGs (as their H i content is likely negligible). The molecular gas component is routinely estimated using its most luminous tracer, carbon monoxide (CO), that is generally optically thick. This requires an empirical derivation of the conversion factor to derive molecular gas masses from CO luminosities (α_CO = M_gas/L'_CO). K98 used the Galactic value for all objects in his sample.

Downes & Solomon (1998) showed that α_CO is smaller for local (U)LIRGs than for spirals, by about a factor of 6—see a detailed discussion and additional references in the review by Solomon & Vanden Bout (2005). Bouché et al. (2007) applied such different conversion factors for spirals and (U)LIRGs, resulting in a steeper relation with a slope of about 1.7. Observations of high-redshift submillimeter selected galaxies (SMGs) can also be fit by the same relation, following Tacconi et al. (2008) who argued that a ULIRG-like α_CO is appropriate for SMGs. Integrated quantities, L'_CO and IR luminosities (L_IR), also appear to follow a correlation with a slope of 1.7 (e.g., Solomon & Vanden Bout 2005; Greve et al. 2005). In the following discussion, we use a Chabrier (2003) initial mass function (IMF) and a standard Wilkinson Microwave Anisotropy Probe cosmology.

In this Letter, we explore the validity of the SF law further by including new observations of CO emission in distant near-IR selected galaxies at z = 0.5 and z = 1.5 (Daddi et al. 2008, 2010—hereafter D10; F. Salmi et al. 2010, in preparation). These allow us to study more typical high-redshift galaxies with SFRs much larger than those of local spirals but less extreme than those of distant SMGs. The sample of six CO-detected z = 1.5 normal (BzK-selected) galaxies is presented in D10. We also use CO detections of three near-IR selected disk galaxies at z = 0.5. A detailed discussion of the z = 0.5 data set will be presented elsewhere (F. Salmi et al. 2010, in preparation). For comparison, we also show measurements for normal CO-detected galaxies at z = 1–2.3 from Tacconi et al. (2010), although we do not use these in our analysis. These new observations are placed in context with the literature data for ULIRGs, SMGs, and local samples of disk galaxies.

In order to investigate the location of these populations of normal high-z galaxies in the gas mass versus SFR plane, either for the integrated properties or for the surface densities, a crucial ingredient is, again, the α_CO conversion factor. Comparing the dynamical and stellar mass estimates, D10 derive a high α_CO = 3.6 ± 0.8 M_☉ (K km s⁻¹ pc²)⁻¹ for the BzK galaxies⁸, quite similar to that for local spirals (α_CO = 4.6). This is not unexpected, given the evidence that the z ∼ 1.5 near-IR selected galaxies appear to be high-redshift analogs of local disks with enhanced gas content (see, e.g., discussions in Daddi et al. 2008, 2010; Dannerbauer et al. 2009; Tacconi et al. 2010, and later in this Letter). In the following, we adopt this value of α_CO = 3.6 for the z = 0.5–2.5 normal galaxies⁹ and the "consensus" value for the other populations (α_CO = 4.6 for local spirals, α_CO = 0.8 for local (U)LIRGs and distant SMGs/QSOs), and explore the consequences for the relation between gas masses and IR luminosities/SFRs.

Figure 1 is equivalent to Figure 13 in D10, after replacing L'_CO with M_H2. The right panel shows the ratio of L_IR to M_H2 plotted versus L_IR. The implied gas consumption timescales (τ_gas = M_H2/SFR; right panel of Figure 1) are 0.3–0.8 Gyr for the BzK galaxies,¹⁰ about 2–3 times that for spirals, and over 1 order of magnitude smaller for local (U)LIRGs and distant SMGs. In a simple picture, this finding can be interpreted in terms of two major SF modes: a long-lasting mode appropriate for disks, that holds for both local spirals and distant BzK galaxies, and a rapid starburst mode appropriate for ULIRGs, local starbursts like M82 or the nucleus of NGC 253, and distant SMGs/QSOs. For the disk galaxies we formally fit

$$\begin{equation} {\rm log}\, L_{\rm IR}/L_\odot = 1.31\times {\rm log}\ M_{\rm H2}/M_\odot -2.09, \end{equation} \tag{ 1 }$$

with an error on the slope of 0.09 and a scatter of 0.22 dex. Combining ULIRGs and SMGs we find that they define a trend with a similar slope, but with about 10 times higher L_IR at fixed M_H2.

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A similar picture applies to the surface densities (Figure 2). We here use the original K98 measurements for local spirals and (U)LIRGs, but apply our choice of α_CO and a Chabrier (2003) IMF. For consistency with the K98 relation, we measure Σ_gas adding H i and H₂ for spirals, and H₂ for IR-luminous galaxies, in Figures 2 and 3. The results would not change if we had used H₂ only for all galaxies. Values for SMGs are taken from Bouché et al. (2007). For the BzK galaxies, we derive gas and SFR surface densities using the UV rest-frame (SFR) sizes. These are consistent with the CO sizes (D10) but are measured at a higher signal-to-noise ratio. Again, we find that the populations are split in this diagram and are not well fit by a single sequence. Our fit to the local spirals and the BzK galaxies is virtually identical to the original K98 relation:

$$\begin{eqnarray} &&\hspace{-5.2pc}{\rm log}\, \Sigma _{{\rm SFR}}/[M_\odot \;{\rm yr}^{-1} \;{\rm kpc}^{-2}] \nonumber \\ &=& 1.42\times {\rm log}\, \Sigma _{\rm gas}/[M_\odot \;{\rm pc}^{-2}]-3.83. \end{eqnarray} \tag{ 2 }$$

The slope of 1.42 is slightly larger than that of Equation (1), with an uncertainty of 0.05. The scatter along the relation is 0.33 dex. Local (U)LIRG and SMGs/QSOs are consistent with a relation having a similar slope and normalization higher by 0.9 dex, and a scatter of 0.39 dex.

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Despite their high SFR ≳100 M_☉ yr⁻¹ and Σ_SFR ≳ 1 M_☉ yr⁻¹ kpc⁻², BzK galaxies are not starbursts, as their SFR can be sustained over timescales comparable to those of local spiral disks. On the other hand, M82 and the nucleus of NGC 253 are prototypical starbursts, although they only reach an SFR of a few M_☉ yr⁻¹. Following Figures 1 and 2, and given the ∼1 dex displacement of the disk and starburst sequences, a starburst may be quantitatively defined as a galaxy with L_IR (or Σ_SFR) exceeding the value derived from Equation (1) (or Equation (2)) by more than 0.5 dex.

The situation changes substantially when introducing the dynamical timescale (τ_dyn) into the picture (Silk 1997; Elmegreen 2002; Krumholz et al. 2009; Kennicutt 1998). In Figure 3, we compare Σ_gas/τ_dyn to Σ_SFR. Measurements for spirals and (U)LIRGs are from K98, where τ_dyn is defined to be the rotation timescale at the galaxies' outer radius (although Krumholz et al. 2009 use the free-fall time). For the near-IR/optically selected z = 0.5–2.3 galaxies, we evaluate similar quantities at the half-light radius. Extrapolating the measurements to the outer radius would not affect our results substantially. Quite strikingly, the location of normal high-z galaxies is hardly distinguishable from that of local (U)LIRGs and SMGs. All observations are well described by the following relation:

$$\begin{eqnarray} &&\hspace{-2.2pc} {\rm log}\, \Sigma _{{\rm SFR}}/[M_\odot \,{\rm yr}^{-1} \,{\rm kpc}^{-2}] \nonumber \\[3pt] &&=1.14\times {\rm log}\, \Sigma _{\rm gas}/\tau _{\rm dyn}/[M_\odot \;{\rm yr}^{-1} \;{\rm kpc}^{-2}]-0.62, \end{eqnarray} \tag{ 3 }$$

with a slope error of 0.03 and a scatter of 0.44 dex. The remarkable difference with respect to Figures 1 and 2 is due to the fact that the normal high-z disk galaxies have much longer dynamical timescales (given their large sizes) than local (U)LIRGs.

We can test if this holds also for integrated quantities by dividing the gas masses in Figure 1 by the average (outer radius) dynamical timescale in each population. Spirals and (U)LIRGs (whose τ_dyn does not depend on luminosity) have average values of τ_dyn = 370 Myr and τ_dyn = 45 Myr, respectively (K98). This can be compared to τ_dyn = 33 Myr for SMGs (Tacconi et al. 2006; Bouché et al. 2007). For the QSOs, we use the SMG value. Assuming a flat rotation curve for BzKs, we get an average τ_dyn = 330 Myr at the outer radius, about three times longer than at the half-light radius, given that for an exponential profile 90% of the mass is enclosed within ∼3 half-light radii. A similar value is found for our z = 0.5 disk galaxies and the z = 1–2.3 objects from Tacconi et al. (2010). Despite this simple approach, Figure 4 shows a remarkably tight trend:

$$\begin{equation} \log \mbox{SFR}/[M_\odot \;{\rm yr}^{-1}] = 1.42\times {\rm log} (M_{\rm H2}/\tau _{\rm dyn})/[M_\odot \;{\rm yr}^{-1}] -0.86, \end{equation} \tag{ 4 }$$

with an error in slope of 0.05 and a scatter of 0.25 dex. Figure 4 suggests that roughly 10%–50% of the gas is consumed during each outer disk rotation for local spirals, and some 30%–100% for BzK, z = 0.5 galaxies, and local (U)LIRGs. Some SMGs/QSOs might even consume their gas in less than one rotation.

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The bimodal behavior in Figures 1 and 2 obviously depends on our assumptions for α_CO. Even though we have assumed the most accurate values that are available in the literature, one should keep in mind that this factor is still relatively poorly constrained. A possible variation of α_CO, e.g., as a function of galaxy properties, may change the simple picture outlined here. However, based on our current understanding, this is unlikely to greatly affect the overall conclusions. Working only with observed quantities, the BzK galaxies are already offset from local ULIRGs in the L'_CO/L_IR diagram by a factor of 3 (D10).

In summary, the observations of near-IR selected galaxies at z = 1.5 do not fit a single relation of M_gas versus L_IR, neither in terms of integrated quantities nor in terms of surface densities (Figures 1 and 2). There appears to be two main sequences, one for disks and one for starbursts, with the latter having 10 times higher L_IR at fixed M_gas (either integrated, or for surface densities). However, all populations define a single sequence when dividing the gas masses or surface densities by the respective dynamical timescales.

Figures 1 and 2 cannot be interpreted in terms of a single average trend for all populations with a large intrinsic scatter. Significant systematic shifts are observed between the BzK and z = 0.5–2.3 normal galaxies and other highly star-forming galaxies, such as local (U)LIRGs and SMGs. It is worth recalling here that major differences exist in the properties of local disk galaxies versus local (U)LIRGs, and the same is observed (at least on average) between distant near-IR selected BzK galaxies and SMGs (see D10, in particular Section 7.4, for a detailed discussion). Overall, this dichotomy in physical properties of spirals/(U)LIRGs and BzKs/SMGs makes it less surprising that the two classes define separate sequences in the M_gas/L_IR plane and correspond to two distinct SF modes.

Previously it was argued that above 100 M_☉ pc⁻² the gas conversion factor would drop by a large factor (see, e.g., the discussion and references in Bouché et al. 2007; Tacconi et al. 2008). This does not seem to be the case for BzK galaxies (D10) that exhibit SF at 10 times higher gas and SFR surface densities than are seen locally. One could thus wonder how the same SF mode can be maintained at such high gas surface densities. It appears that the velocity dispersion is high in these systems (e.g., Förster Schreiber et al. 2009), probably due to high turbulence. The implication would be that BzKs have considerably thicker disks than those of local spirals. A higher gas-filling factor in BzK galaxies might also apply.

We now discuss how the observed differences in the M_gas versus L_IR plane could be interpreted. We note that the dichotomy could be altered in the M_gas versus SFR plane if stars were formed with different IMFs in disks and starbursts. For example, if the L_IR to SFR conversion factor was lower for (U)LIRGs and SMGs by a factor of 10, then if we were to plot SFR instead of L_IR in Figures 1 and 2 we would basically end up with a single SF sequence. It has been claimed that the IMF in bursts could be top heavy (e.g., Baugh et al. 2005; Elbaz et al. 1995). While the general consensus is that the appropriate IMF for local spirals is bottom light as in Chabrier (2003) or Kroupa (2002), very little is known observationally for the case of local (U)LIRGs and even less for high-redshift galaxies. In principle, the IMF might explain the bimodal behavior at least in part.

On the other hand, we have shown that the ratio of L_IR to M_gas correlates inversely with the dynamical timescales of the systems. The systematic differences disappear once the gas masses are divided by τ_dyn, given that (U)LIRGs and SMGs have much shorter τ_dyn than disk galaxies at 0 < z < 2.3. This is an important result because, for the first time, we show that global galaxy properties (that determine τ_dyn) are related to the regulation of a local (punctual) process like SF. The dynamical time is expected to scale with the gas volume density (ρ) as τ_dyn ∝ ρ^−0.5 (e.g., Silk 1997). This suggests that the different L_IR/M_gas ratios are not driven by the IMF (a difference in the IMF would break the single relations found using τ_dyn) but by the fact that in some galaxies the gas can reach high volume densities and the systems can have short dynamical timescales so that the fuel is consumed more rapidly and the resulting SFR per unit of gas mass is higher. This is consistent with the existence of a unique and approximately linear correlation between L_IR and HCN luminosity for spirals and local (U)LIRGs (Gao & Solomon 2004; Juneau et al. 2009) and QSOs (with the exception of the highest redshift objects; Riechers et al. 2007; Gao et al. 2007). The HCN luminosity is a measure of dense gas, requiring H₂ volume densities >10⁴ cm⁻³, while CO can be thermally excited for densities ≲10³ cm⁻³. Based on the HCN/L_IR relation, the bimodal trends of L_IR to M_gas in disks versus starbursts can be interpreted in terms of a similarly bimodal behavior for the dense gas fractions, with roughly 10 times higher fraction of dense gas in the starbursts compared to disks at fixed L_IR (but also with about twice higher dense gas fractions in BzKs versus local spirals). All the observations might thus be explained by a genuine increase of SFR efficiency in some galaxy classes, probably due to the concentration of the gas at high volume densities.

Major mergers or other kinds of instabilities appear to be the most natural explanation for the increased efficiencies in the starbursts (although not all mergers will necessarily produce this effect; di Matteo et al. 2008). This is not a new scenario. A higher SF efficiency in merger-driven starbursts than in rotating disks has often been implemented in semianalytical models of galaxy formation (e.g., Guiderdoni et al. 1998; Somerville et al. 2001), although this has been usually interpreted as the natural outcome of a single SF law with an exponent >1, as long as mergers make gas lose angular momentum and concentrate in the galaxy center. However, in such implementations the occurrence of very high SFRs in gas-rich disks is neglected; they thus have a bimodality in surface density, not in the SF law. Our analysis suggests that the original K98 calibration can account for the properties of disk galaxies at low and high redshifts but would underestimate the SF efficiency of starbursts by a factor of 10. An SF law with a higher exponent of 1.7 (Bouché et al. 2007) would in turn overestimate SF efficiency of gas-rich disks by a similar amount. An implementation of such a double SF law would surely influence predictions from semianalytical models of galaxy formation.

The difference in α_CO for disks and starbursts helps to hide what we are interpreting here as large differences in the SF efficiency, expressed in terms of L_IR/M_gas, by reducing the observed differences in L_IR/L'_CO. There seems to be a conspiracy at work such that the particular physical conditions that lead to high L_IR/M_gas in starbursts also determine variations in α_CO that obscure observationally the differences in SF efficiency. We emphasize thus that the distinction of starburst (or merging versus non merging) systems is important for interpreting CO observations, although this may be difficult on a case by case basis. We caution that blindly applying the same conversion factor to all high-z observations can lead to confusion. Also, care must be taken that SFR/IR luminosities are accurately derived. The estimates for BzK galaxies here are based on the cross-comparison of three independent SFR indicators that agree within each other very well, and overall on the global assessment by Daddi et al. (2007a, 2007b) on SFR measurements of near-IR selected galaxies at z ∼ 2. On the other hand, purely radio selected or mid-IR selected populations are likely to produce a mixed bag of merging systems and disk galaxies and can be affected by AGNs.

We thank Adam Leroy for discussions and help with Figure 2 and Padelis Papadopoulos for discussions. We acknowledge the funding support of the ERC-StG grant UPGAL-240039, ANR-07-BLAN-0228, and ANR-08-JCJC-0008. D.R. acknowledges NASA Hubble Fellowship grant HST-HF-51235.01 awarded by the STScI, operated by AURA for NASA, contract NAS-5-26555.