THE STAR FORMATION LAWS OF EDDINGTON-LIMITED STAR-FORMING DISKS

Galaxies can experience a wide variety of star formation phenomena, from the relative calm of star-forming regions beaded along a spiral arm, to the fury of a nuclear starburst burning through a trillion Suns. Yet, remarkably, the rate of star formation in such wildly disparate environments can be simply related to the density of gas in the star-forming region, $\dot{\Sigma }_{\ast } \propto \Sigma _{\mathrm{gas}}^N$, where $\dot{\Sigma }_{\ast }$ and Σ_gas are the star formation rate (SFR) and total gas (i.e., atomic plus molecular) surface densities. This relation, known as the Kennicutt–Schmidt (K-S) law, has been observed to hold over nearly four decades in Σ_gas with N ≈ 1.4 (e.g., Schimdt 1959; Kennicutt 1998a, 1998b; Yao et al. 2003; Kennicutt & Evans 2012), and is ultimately related to the efficiency of star formation and its connection to both local and global timescales (e.g., Elmegreen 1997, 2002; Silk 1997; Kennicutt 1998b; Krumholz & McKee 2005; Krumholz et al. 2009). The super-linear slope indicates that the star formation efficiency (i.e., the ratio of stellar mass formed to the gas mass in the star-forming region) increases at high densities. Measuring and understanding the K-S law over a wide range of star-forming environments is crucial to elucidating comprehensive theories of star and galaxy formation (see discussion by Kennicutt & Evans 2012).

Most of the star formation in the universe occurred long ago at z > 1 when the majority of massive galaxies were being assembled (e.g., Hopkins & Beacom 2006; Pérez-González et al. 2008). With the advent of new sensitive millimeter arrays and detectors, the last decade has seen a significant advance in the study of the K-S law at high redshifts (e.g., Daddi et al. 2010b; Genzel et al. 2010; Ivison et al. 2011; Freundlich et al. 2013; see Solomon & Vanden Bout 2005 and Carilli & Walter 2013 for reviews). Interestingly, these galaxies, which have both much larger SFRs and gas densities than local objects, seem to follow a similar K-S slope, but are vertically offset from the local relation (Daddi et al. 2010b; Genzel et al. 2010), perhaps implying a fundamental difference in the stellar initial mass function (IMF) in the two regimes. However, many of these results rely on the assumption that the troublesome conversion factor α_CO = Σ_gas/I_CO needed to convert from the observed CO velocity-integrated intensity, I_CO, to Σ_gas is bimodal with one value (α_CO ≈ 4) for galaxies with Milky-Way-like SFRs, and another value (α_CO ≈ 0.8) for galaxies with much larger SFRs (i.e., high-z galaxies). When Narayanan et al. (2012b) recently recomputed the K-S law with a value of α_CO that varied continuously with I_CO, the relation was no longer bifurcated between the high- and low-z galaxies and could be fit over a wide range in Σ_gas with N ≈ 2 (see also Ostriker & Shetty 2011).

Alternatively, it has been argued (Elmegreen 1997; Silk 1997; Daddi et al. 2010b; Genzel et al. 2010) that a more fundamental star formation law is $\dot{\Sigma }_{\ast } \propto (\Sigma _{\mathrm{gas}}\Omega)^{n}$, where Ω is the orbital angular frequency in the star-forming region, usually estimated at the observed outer radius of a galaxy. Both high- and low-z galaxies seem to follow this Elmegreen–Silk (E-S) relation with n ≈ 1, independent of the exact assumptions on α_CO (Daddi et al. 2010b; Genzel et al. 2010; although Narayanan et al. (2012b) found that the scatter in the E-S relation was reduced by using the continuously varying description of α_CO). In this formulation, a larger star-forming efficiency is found in rapidly star-forming galaxies because it occurs in more compact regions with shorter dynamical times.

Analysis and interpretation of the star formation laws at high redshift are hampered by gaps in the theory of star formation in galaxies at these redshifts. The interstellar medium of these galaxies are expected to be predominately molecular (e.g., Solomon et al. 1997; Tacconi et al. 2010; Daddi et al. 2010a; Magdis et al. 2012) have gas fractions of several tens of percent (e.g., Geach et al. 2011; Swinbank et al. 2012; Narayanan et al. 2012a) with significant turbulence and large pressures (e.g., Downes & Solomon 1998; Genzel et al. 2008; Wisnioski et al. 2011). Simulating this environment from first principles is computationally prohibitive, so analytical models have been developed that encompass many aspects of the physics in a well-defined region of parameter space (e.g., Thompson et al. 2005; Ostriker & Shetty 2011). In addition, as eluded to above, there may be severe uncertainties in relating the observed quantities (usually, infrared (IR) luminosity and CO intensity) to the physical parameters $\dot{\Sigma }_{\ast }$ and Σ_gas (e.g., Solomon & Vanden Bout 2005; Feldmann et al. 2012; Narayanan et al. 2012b). Insofar as the IR luminosity captures the bolometric output of embedded star-forming regions, there are well known calibrations between the observed IR luminosity and SFR (e.g., Kennicutt 1998b; Rieke et al. 2009; Calzetti et al. 2010; Murphy et al. 2011). More problematic is how to convert from the line intensity of a molecule that comprises only a tiny fraction of the interstellar gas to an estimate of the entire gas mass. Although the CO lines are most commonly used due to its high abundance, it has been noted that because of their relatively low critical densities and high optical depth, they are likely very poor tracers of high density gas (e.g., Krumholz & Thompson 2007; Narayanan et al. 2008). Indeed, K-S laws derived from a high density tracer such as HCN have produced a simple linear relation between $\dot{\Sigma }_{\ast }$ and Σ_gas for both high- and low-z galaxies (e.g., Gao & Solomon 2004a, 2004b), although the conversion factor α_HCN is just as uncertain as the one for CO (see the discussion by García-Burillo et al. 2012). Taken together, the combined uncertainties in both the existing theoretical framework and the observational conversions severely restrict the use of the K-S or E-S laws.

The most extreme star-forming regions are those with Σ_gas ≳ 10⁴ M_☉ pc⁻². In this situation the opacity of the dense and dusty gas is so large that the ISM becomes optically thick in the IR, radiation pressure dominates over the turbulent pressure, and the disk is said to be Eddington-limited. Expected to occur in the cores (i.e., central few hundred parsecs) of rapidly star-forming disks, these intense star-forming regions are difficult to investigate observationally, although evidence for such high values of Σ_gas have been recently inferred at the centers of some z < 1 post-starburst galaxies (Diamond-Stanic et al. 2012) and star-forming systems (García-Burillo et al. 2012). Their location also implies that these Eddington-limited star-forming disks may act as a "bridge" between the black hole environment and the host galaxy, ferrying fuel through the disk toward the central black hole and potentially driving nuclear activity (Thompson et al. 2005; Ballantyne 2008). Thus, predictions of the observational signatures of Eddington-limited star-forming regions are needed to guide future observational studies of the centers of both active and inactive galaxies at all redshifts. Ostriker & Shetty (2011) analytically investigated star formation in environments with 10² ≲ Σ_gas ≲ 10⁴ M_☉ pc⁻² when vertical pressure support is provided by supernova driven turbulence, and predicted that the K-S law should have N = 2 (see also Thompson et al. 2005), in decent agreement with the high-z observations if one assumes that α_CO is inversely correlated with I_CO. In the Eddington-limited case the local K-S law is expected to flatten to N = 1 (Thompson et al. 2005; Ostriker & Shetty 2011), but there have been no predictions for the E-S law or work on how clearly these laws will translate for an observation that encompasses an entire >100 pc star-forming disk and contains a significantly variable SFR and Σ_gas. In this paper we use the Thompson et al. (2005) model of an Eddington-limited disk to self-consistently calculate the radial structure of over 100 nuclear star-forming disks, and, through the use of CO radiative transfer calculations, we investigate the relationship between observable (L_IR and I_CO) and physical ($\dot{\Sigma }_{\ast }$ and Σ_gas) variables in these most extreme star-forming environments.

The next section describes the Eddington-limited star-forming disk model, the calculation of our model database, and the CO radiative transfer method. Both the K-S and E-S star formation laws predicted by these models are presented in Section 3. The observational star formation law (the IR-CO relationship) that is predicted by these models and its relationship to the physical K-S and E-S laws is described in Section 4. A discussion and concluding remarks are presented in Section 5. Throughout this paper the total infrared luminosity (L_TIR) is defined as the 3–1100 μm luminosity (Murphy et al. 2011), and helium is not accounted for in Σ_gas or α_CO. Unless otherwise specified all CO intensities or luminosities refer to the J = 1–0 rotational transition.

2.1. The Eddington-limited Star-forming Disk Model

The Eddington-limited starburst disk model utilized here was developed by Thompson et al. (2005), and a short summary of the relevant details is presented below (see also Thompson 2009).

Star formation is modeled simply as an energy source at a location r within a one-dimensional single phase medium that rotates with angular frequency Ω in the potential of a galactic bulge (modeled as an isothermal sphere) with dispersion σ and central black hole of mass M_BH (these two quantities are assumed to be related by M_BH = 2 × 10⁸(σ/200 km s⁻¹)⁴ M_☉; e.g., Tremaine et al. 2002). Gas, radiation, and turbulent pressure driven by supernova all combine to provide the vertical support against gravity:

$$\begin{equation} p_{\mathrm{gas}}+\epsilon \dot{\Sigma }_{\ast }c\left({\tau _{V} \over 2}+\xi \right)=\rho h^2 \Omega ^2, \end{equation} \tag{ 1 }$$

where p_gas = ρkT/m_p, and ρh²Ω² is the total pressure required for hydrostatic balance in the bulge potential (with h denoting the gas scale height and ρ indicating the gas density of the star-forming disk; the self-gravity of the disk is neglected here). Thompson et al. (2005) show that in the optically thick limit the radiation pressure is $\epsilon \dot{\Sigma }_{\ast }c\tau _{V}/2$ where is the IMF-dependent efficiency between SFR and luminosity (i.e., $L=\epsilon \dot{M}_{\ast }c^2$, where $\dot{M}_{\ast }$ is the local SFR) and τ_V = κΣ_gas/2 is the vertical optical depth in the IR (κ(T, ρ) is the Rosseland mean opacity of the dusty gas at temperature T and density ρ). The turbulent pressure driven by supernova can also be written as proportional to the local SFR density, $\epsilon \xi \dot{\Sigma }_{\ast }c$, where ξ ≈ 1 (Thompson et al. 2005). From above, a rms speed c_s can be defined using the total pressure as c_s = hΩ, where it is understood that gas, radiation and turbulent pressure all contribute to this rms speed. As described below, the models studied in this paper all have a radially-averaged τ_V > 1 and are thus dominated by radiation pressure for almost all radii.

The star-forming disk is assumed to be always just unstable to gravitational instabilities; that is, Toomre's Q parameter is set equal to one. Material with gas fraction f_gas is fed onto to the disk at a radius r_out from the black hole, and, through a hypothesized global torque (provided by, e.g., a spiral instability or a bar; Goodman 2003; Englmaier & Shlosman 2004; Maciejewski 2004), can slowly accrete towards the center with a radial velocity equal to a fixed fraction m of the local rms speed (Goodman 2003), i.e., $\dot{M}=2\pi r \Sigma _{\mathrm{gas}} mc_s$. At any radius r, Thompson et al. (2005) showed that $\dot{M}$ and f_gas can be related through

$$\begin{equation} f_{\mathrm{gas}} = \left({2^{3/2} \dot{M} G \over Qm\Omega ^3 r^3} \right)^{1/2}\!. \end{equation} \tag{ 2 }$$

As the gas moves through the disk, it is also producing stars at a rate of $\dot{\Sigma }_{\ast }$, which, in the optically thick limit, is determined by requiring radiation pressure in the IR to support the disk (Equation (1)) as well as Q = 1:

$$\begin{equation} \dot{\Sigma }_{\ast } = {\sqrt{2} f_{\mathrm{gas}} Q \sigma ^2 \over \epsilon \kappa c r}, \end{equation} \tag{ 3 }$$

where r is assumed to be large enough that the black hole potential is negligible. The star formation slowly consumes the gas as r decreases, reducing both $\dot{M}$ and f_gas, i.e.,

$$\begin{equation} \dot{M}=\dot{M}_{\mathrm{out}}-\int _{r_{\mathrm{out}}}^r 2\pi r^{\prime } \dot{\Sigma }_{\ast } dr^{\prime }, \end{equation} \tag{ 4 }$$

where $\dot{M}_{\mathrm{out}}$ is the mass accretion onto the disk at r_out. However, the Q = 1 assumption causes a rising density in the inner regions of the disk that substantially increases τ_V so that even a moderately increasing $\dot{\Sigma }_{\ast }$ can still support the disk through radiation pressure. The solution at radius r is determined by finding the minimum central gas temperature T and associated κ to ensure vertical pressure balance, either through the sum of gas and turbulent pressure, or, if there is enough star formation, radiation pressure. Eventually, either the gas available for star formation becomes too small to maintain Q = 1, or energy release through gas accretion pushes Q > 1, and the calculation ceases.

Substituting Equation (2) into Equation (3) results in predictions for both the E-S and K-S laws in the disks:

$$\begin{equation} \dot{\Sigma }_{\ast }=\left({2^{3/2} \pi Q G \over \epsilon ^2 c^2} {c_s \over \kappa ^2} \Sigma _{\mathrm{gas}} \Omega \right)^{1/2} \end{equation} \tag{ 5 }$$

and, using Q to eliminate Ω (see Thompson et al. 2005, their Equation (3)),

$$\begin{equation} \dot{\Sigma }_{\ast }=\left({2^{1/2} \pi Q G \over \epsilon c \kappa }\right)\Sigma _{\mathrm{gas}}. \end{equation} \tag{ 6 }$$

These equations have to be viewed carefully because, as seen in Equation (4), Σ_gas is not an independent variable; its value at some radius r will depend on $\dot{\Sigma }_{\ast }$ at larger radii. As seen in Section 3, this property may affect the predicted slopes of the star formation laws. In sum, radiation pressure supported star-forming disks predict a K-S relation with N ≈ 1, and an E-S law with n ≈ 0.5. Of course, both of these predictions rely on the assumptions that Q = 1 and the disk is optically thick in the IR and therefore radiation pressure supported.

In order to determine how these laws will manifest in observational surveys of several galaxies, the full disk models need to be calculated over a wide range of parameters. There are five input parameters for each model: the black hole mass M_BH (which determines σ), the angular momentum parameter m, the outer radius r_out, the gas fraction at the outer radius f_gas, and a dust-to-gas multiplicative factor for the opacity to account for the enhanced metallicity observed in the centers of galaxies (e.g., Muñoz-Mateos et al. 2009). As in Thompson et al. (2005), the Semenov et al. (2003) calculation of the Rosseland mean opacity for dusty interstellar gas is used for all models. Calculations are performed for each permutation of log (M_BH/M_☉) = 7, 7.5, 8, 8.5, m = 0.0075, 0.01, 0.025, 0.05, 0.075, 0.1, 0.2, r_out = 50, 100, 150, 200, 250 pc, f_gas = 0.1, 0.5, 0.9 and dust-to-gas ratios equal to 1, 5, and 10 times the local ISM value, resulting in a suite of 1260 models. While the models are limited to r_out ⩽ 250 pc, the results on the star-formation laws do not depend on this size (as is seen for the observed K-S and E-S laws), and are valid for any Eddington-limited star-forming region. The range of input gas fractions is consistent with the observed estimates (e.g., Tacconi et al. 2010; Magdis et al. 2012) that show an increase to ∼50% at z ∼ 2 (the f_gas input to the model drops steadily through the disk as the star formation uses up the available gas; Equations (2) and (4)). We employ a standard Kroupa/Salpeter IMF with α_IMF = −2.35 (Salpeter 1955; Kroupa & Weidner 2003) between 1 and 100 M_☉ and α_IMF = −1.3 between 0.1 and 1 M_☉ (where the number of stars with masses between M and M + dM is proportional to $M^{\alpha _{\mathrm{IMF}}}$). This IMF is then input into a Starburst99 stellar synthesis model (Leitherer et al. 1999) to compute the bolometric luminosity of a stellar population with a constant SFR and with the given IMF as a function of time. The value of $\epsilon =L/\dot{M}c^2$ is then measured at an age of 10⁸ yr, yielding = 7.1 × 10⁻⁴ (very similar to the value of 6.2 × 10⁻⁴ used by Ostriker & Shetty 2011).¹

As described in detail by Thompson et al. (2005), roughly two different types of star-forming disks result from this theoretical description. In about one-third of the models the gas maintains a large optical depth over the entire disk, resulting in the temperature in the inner parsec of the disk exceeding the dust sublimation temperature and causing a significant burst of star formation in order to maintain Q = 1. These parsec-scale bursts could puff up the disk, and, if they do not use up all the gas, may both fuel and obscure the central black hole. The observational properties of these disks have been studied elsewhere (Ballantyne 2008; Armour & Ballantyne 2012), and, as they are not directly related to the study of general star-forming galaxies, the description of their star formation laws are deferred to a later paper. The remaining models reach the maximum SFR, SFR_max, at r_out with the SFR declining toward smaller r. Many of these disks are actually gas-pressure dominated or do not extend very far before $\dot{\Sigma }_{\ast }$ becomes too small to maintain Q = 1. As our interest here is studying radiation pressure dominated star-forming disks that can extend tens to hundreds of parsecs, we selected only models with a radially averaged τ_V > 1 and an inner radius <0.04r_out for further investigation (including less extended disks only increases the scatter of the derived relations). This selection criteria yields 132 model star-forming disks. Figure 1 shows several of the properties of one of these Eddington-limited star-forming disks. The plot clearly shows the implications of fixing Q = 1: the increasing Σ_gas keeps τ_V large enough so that radiation pressure dominates over nearly 200 pc. This fact, in turn, keeps the disk very warm with temperatures T > 100 K and $\dot{\Sigma }_{\ast } \sim 10^3$ M_☉ yr⁻¹ kpc⁻².

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Figure 1 illustrates that the quantities that enter into the star formation laws (i.e., SFR, $\dot{\Sigma }_{\ast }$ and Σ_gas) can vary dramatically over the size of the star-forming disk. As these variations will be unresolved at most redshifts, we consider both the radial average and the mass-weighted average of these variables when constructing the predicted K-S and E-S laws. For example, the mass-weighted average of the SFR along the disk, 〈SFR〉_mw, is calculated as follows:

$$\begin{equation} \langle \mathrm{SFR}\rangle _{{\rm mw}} = {{\displaystyle {\sum } (2\pi r \Delta r \Sigma _{\mathrm{gas}}(r)/f_{\mathrm{gas}}(r))\mathrm{SFR}(r)\ \Delta r} \over {\displaystyle {\sum } (2\pi r \Delta r \Sigma _{\mathrm{gas}}(r)/f_{\mathrm{gas}}(r))\ \Delta r}}, \end{equation} \tag{ 7 }$$

where Σ_gas/f_gas is the total dynamical mass surface density. The simple radial average, 〈SFR〉, is computed as in Equation (7) but with the (2πrΔrΣ_gas/f_gas) factors omitted.

Finally, it is important to bear in mind the limitations associated with this model including the lack of any clumpiness which is found by hydrodynamical simulations of turbulence supported disks (e.g., Shetty & Ostriker 2012). In addition, as these disks are Eddington-limited they may be subject to Rayleigh–Taylor instabilities (Krumholz & Thompson 2012), and produce outflows and feedback matter and energy into the galaxy (Murray et al. 2005; Andrews & Thompson 2011). The effects of outflows on the structure of these disks is not considered here. Feedback from an accreting black hole is also not treated, so comparisons between the predictions and data should be limited to sources without active nuclei. It is expected that the impact of these limitations will be mostly offset by considering radially averaged quantities, a wide range of model parameters, and by comparing to constraints derived from observations of many dozens of individual galaxies.

2.2. Infrared Luminosity and SFR

The SFR is one of the most important observational properties of a galaxy as it gives crucial information on the galaxy's gas content and evolutionary state. Estimating the SFR typically relies on utilizing one of a number of relationships between SFR and the luminosity of the galaxy in a specific wavelength or energy range (Kennicutt & Evans 2012). As most of the starlight produced in high-z star-forming galaxies is absorbed by dust and re-emitted in the IR, the SFR–L_TIR relationship determined by Kennicutt (1998b) and Murphy et al. (2011) is one of the most frequently used SFR estimators, as well as serving as the calibration for SFR estimates at single IR wavelengths (Rieke et al. 2009; Calzetti et al. 2010). As described by Kennicutt (1998b) and Murphy et al. (2011), the SFR-L_TIR relationship was calculated using Starburst99 models assuming that 100% of the bolometric luminosity of a stellar population with a constant SFR is captured and emitted in the IR. This scenario is also fulfilled by the optically thick star-forming disk models with 〈τ_V〉 > 1, as τ_V is the vertical optical depth in the IR. Thus, as the SFR is known throughout each disk, it is interesting to determine how accurately the Murphy et al.-derived SFRs describe the star-forming properties of the model disks.

Since these disks are largely optically thick at all radii, the effective temperature T_eff of one side of the disk surface at radius r is related to half of the total flux produced by star formation: $\sigma _{\mathrm{SB}}T_{\mathrm{eff}}^4 = (1/2)\epsilon \dot{\Sigma }_{\ast }c^2$, where σ_SB is the Stefan–Boltzmann constant (Thompson et al. 2005). Assuming that each annulus of the disk radiates as a blackbody and is viewed face-on, the IR spectral energy distribution (SED) of each model is calculated as

$$\begin{equation} \lambda L_{\lambda }={2 \pi h c^2 \over \lambda ^4} \int ^{r_{{\rm out}}}_{r_{{\rm in}}} {2 \pi r dr \over \exp [hc/\lambda k_{\mathrm{B}} T_{\mathrm{eff}}(r)]-1}, \end{equation} \tag{ 8 }$$

where r_in is the inner radius for each model, h is Planck's constant, and k_B is Boltzmann's constant. The L_TIR for each model is then computed by integrating the SED from 3–1100 μm. This luminosity can then be compared to various calculations of the disk SFR, such as 〈SFR〉, SFR_max, and 〈SFR〉_mw.

Figure 2 plots the Murphy et al. (2011) calibration (log (SFR/M_☉ yr⁻¹) = log (L_TIR/erg s⁻¹) − 43.41) as the solid line, SFR_max versus L_TIR (open triangles), and 〈SFR〉 against L_TIR (solid points). The SFR_max, 〈SFR〉 and L_TIR are derived directly from the models as described above. Evidently, the Murphy et al. (2011) calorimetric relation provides a coarse approximation to the maximum SFR of each disk. This is a natural consequence of the assumption of a single SFR that underlies the original computation of the SFR–L_TIR relationship. The maximum SFR in the disk provides the largest contribution to the luminosity, so the estimated SFR from the L_TIR relationships will be closest to the maximum rate. However, as shown in Figure 2, a better (with ∼10 times smaller scatter) SFR–L_TIR relationship is found with the 〈SFR〉 which is, on average, 2.2  ±  0.3 smaller than the values predicted by the Murphy et al. (2011) calibration at the same luminosity. A least-squares fit to the 〈SFR〉–L_TIR relationship (dashed line) yields log (〈SFR〉/M_☉ yr⁻¹) = (0.991 ± 0.008)log (L_TIR/L_☉) − (10.07  ±  0.09). Thus, the 〈SFR〉 is used throughout the paper to quantify the SFR for any particular model.

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2.3. Computing the CO Line Emission

3.1. The Kennicutt–Schmidt Relation

Turning now to the star formation laws predicted by the Eddington-limited star-forming disks, Figure 3 plots the K-S law in two ways: 〈$\dot{\Sigma }_{\ast }$〉 versus 〈Σ_gas〉 (left panel) and 〈$\dot{\Sigma }_{\ast }$〉 against 〈Σ_gas〉_mw (right panel). The theoretical expectation is for $\dot{\Sigma }_{\ast } \propto \Sigma _{\mathrm{gas}}/\kappa$ (Equation (6)), and, indeed, we find that the K-S laws, calculated from these radial averages, have slopes close to unity and are separated vertically by the dust-to-gas enhancement factor (that increases κ). The scatter in the K-S relations is smallest when plotted against 〈Σ_gas〉 because the same spatial averaging is being performed on both quantities. However, due to the dependence of Σ_gas on $\dot{\Sigma }_{\ast }$ (see Equation (4) and surrounding discussion), the slopes of the K-S laws in the left-hand panel can drift away from the expectation of N = 1. This effect is enhanced for larger dust-to-gas enhancement factors because a larger κ reduces $\dot{\Sigma }_{\ast }$ and thus Σ_gas does not decrease as fast with radius. The spatial averaging therefore results in a larger Σ_gas which then flattens the slope of the relations. By plotting the quantities with different spatial averaging, the right-hand panel breaks the dependence of Σ_gas on $\dot{\Sigma }_{\ast }$ and shows that indeed the K-S laws follow the theoretical expectation with N ≈ 1.

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A random selection of galaxies will likely exhibit a wide range of dust-to-gas ratios that may be difficult to observationally separate. The dotted lines in Figure 3 plot the K-S relations found when including all the model points, regardless of the dust-to-gas factor. The resulting slopes are N = 1.2 and 1.3 for the 〈Σ_gas〉 and 〈Σ_gas〉_mw panels, respectively. These values are very similar to the observed K-S laws (e.g., Kennicutt & Evans 2012), and indicate that the common practice of measuring the K-S relation by compiling a large collection of heterogeneous data may be hiding important clues of the physics of star-forming disks. Future investigations of the star formation laws may benefit from the analysis of a smaller sample of galaxies with well-known properties.

3.2. The Elmegreen–Silk Relation

The E-S law predicted from the radiation pressure dominated star-forming disk models are shown in Figure 4, with the same symbols and line styles as the previous K-S law figure. Following current observational practice, the orbital frequency is set to the inverse of the dynamical timescale at r_out, i.e., τ_dyn = 1/Ω(r_out). The theoretical expectation for the E-S law is $\dot{\Sigma }_{\ast }$∝(Σ_gasΩ/κ²)^0.5 (Equation (5)), which is in good agreement with the relations seen in Figure 4. There is a flattening of the slope with the dust-to-gas enhancement factor that is reduced in the 〈Σ_gas〉_mw/τ_dyn plot. The explanation for this effect is the same as with the K-S relations discussed above. However, it is clear from a comparison of Figures 3 and 4 that the E-S law has significantly more scatter than the K-S relations (about a three-times-larger rms in the 〈Σ_gas〉 plots; ∼50% larger rms in the 〈Σ_gas〉_mw panels). This increased scatter is likely due to dividing a radial average by the orbital time at one radius, as well as a contribution due to variations in the rms speed in each model (Equation (5)). It is notable that even when all the models are fit with a power law, irrespective of the dust-to-gas factor, the measured slopes (n = 0.70 and 0.73) are still significantly sub-linear. Thus, measurements of a sub-linear E-S-law in high-z star-forming galaxies may provide compelling evidence for a high-density Eddington limited environment.

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The previous section described the theoretical expectations for the star formation laws governing the radiation pressure supported disks. To test the theory, these laws must be observationally inferred from data of a sample of galaxies, which, at high-z, are typically limited to photometric measurements such as an IR luminosity and a molecular line intensity. The challenge is then to turn these measured quantities into accurate estimates of $\dot{\Sigma }_{\ast }$ and Σ_gas for each galaxy. In this section, we make use of our predicted IR luminosities and I_CO for each disk model² to test how well the observationally derived star formation laws recover the theoretical laws presented in Section 3.

Since an IR luminosity is correlated with the SFR (Section 2.2) and the molecular line intensity is connected in some manner to the total gas content of galaxy, a common zeroth-order measurement of the K-S law is made by plotting the observed IR luminosity against a molecular line luminosity or intensity (Figure 5). The CO J = 1–0 transition is used as the molecular line tracer in this plot because it is one of the most commonly measured lines; however, the densities and temperatures of these disks are large enough that all the low J CO lines are fully thermalized and thus have approximately equal luminosities (see Armour & Ballantyne 2012, for example CO SLEDs from the radiation pressure dominated disks). As a result, neither the slope nor the scatter in the L_TIR–I_CO relationship depend significantly on the CO transition.

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Interestingly, the slope of the L_TIR–I_CO relationship (1.1 ± 0.1) is very similar to the one predicted from the theoretical K-S law (N = 1.2 or 1.3 when including all models; Figure 3). However, this is purely a coincidence because in these warm, dense and well-mixed disks both the dust and molecular gas emitters are thermalized, optically thick and represent the same amount of disk mass. Thus, a quasi-linear relationship between the L_TIR (indicating the dust mass) and I_CO (indicating the gas mass) is expected in this scenario (Krumholz & Thompson 2007). We conclude that L_TIR–I_CO plots are poor estimates of the star formation laws of radiation pressure dominated disks.

The next step is to use the observables to estimate the physical quantities that enter into the star formation laws, beginning with $\dot{\Sigma }_{\ast }$. This quantity is reasonably straightforward to calculate, as one can simply translate the L_TIR into an SFR through one of the well known calibrations and then divide by the observed area of the disk to obtain the star formation surface density.

Figure 6 illustrates how well that procedure recovers the actual $\langle \dot{\Sigma }_{\ast }\rangle$ of the model star-forming disks by plotting $\langle \dot{\Sigma }_{\ast }\rangle$ versus the values obtained from L_TIR, the relationship from Section 2.2, and the model inner and outer radii. The "observed" $\dot{\Sigma }_{\ast }$ is highly correlated with $\langle \dot{\Sigma }_{\ast }\rangle$, and the least-squares fit to the data (dashed line) has a slope of 1.04 ± 0.02, but is ≈2 times smaller than the one-to-one relationship. The dotted line is the least-squares fit to the model data if the Murphy et al. (2011) L_TIR–SFR relationship is used and results in a $\dot{\Sigma }_{\ast }$ that is approximately correct at low $\langle \dot{\Sigma }_{\ast }\rangle$, but becomes progressively too small at large $\langle \dot{\Sigma }_{\ast }\rangle$. These offsets from the actual $\langle \dot{\Sigma }_{\ast }\rangle$ is a result of approximating the average $\dot{\Sigma }_{\ast }$ with (〈SFR〉)/area or (SFR_max/area), but are relatively minor. Most importantly, there is a good linear correlation between $\langle \dot{\Sigma }_{\ast }\rangle$ and the observational estimate, which indicates that the translation from L_TIR results in a reasonable estimate of $\langle \dot{\Sigma }_{\ast }\rangle$.

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As described in Section 1 the conversion factor α_CO between I_CO and Σ_gas is not well determined except for a small number of galactic star-forming regions and some local ULIRGs (e.g., Downes & Solomon 1998; Solomon & Vanden Bout 2005; Narayanan et al. 2012b). In our case, since we have both I_CO and 〈Σ_gas〉_mw for every model, we can calculate an unique value of α_CO for each star-forming disk (calculating α_CO with 〈Σ_gas〉 instead of 〈Σ_gas〉_mw results in more scatter, so we focus on those computed with the mass-weighted average). Figure 7 plots the relationship between the resulting α_CO and I_CO and indeed finds a reasonable anti-correlation (R = −0.64, where R is the linear correlation coefficient) with a slope of −0.56  ±  0.06. The median value of α_CO is 3.56. As the CO intensity is correlated with the SFR (Figure 5), this plot also indicates that α_CO is anti-correlated with the SFR. The slope of the α_CO–I_CO anti-correlation is steeper than the one of −0.32 found by Narayanan et al. (2012b) from hydrodynamical simulations of galaxies. Similarly, the amplitude of the dotted line in Figure 7 is ≳ 2 times larger than the Narayanan et al. (2012b) relation. As the physical scenario and calculation of the CO intensity performed by Narayanan et al. (2012b) differs significantly from our setup the similarity in the results is striking. The relatively high gas temperatures found in the Eddington-limited star forming disks will tend to drive α_CO lower (Magnelli et al. 2012; Narayanan et al. 2012b), which may explain the steeper slope in Figure 7. The larger amplitude may be caused by the lack of clumpiness in our star-forming medium which, when combined with the optically-thick nature of the CO emission, would result in the CO line intensity being a worse tracer of the total gas mass than in a clumpy medium where the CO intensity is directly related to a large number of smaller clouds. However, Papadopoulos et al. (2012) recently argued that the centers of ULIRGs (which may be radiation pressure dominated; Thompson et al. 2005) are unlikely to be comprised of individual molecular clouds and that the small values of α_CO inferred from these environments may be erroneously low, in agreement with the results presented in Figure 7.

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It is apparent from Figure 7 that there is no single value of α_CO for a given I_CO; indeed, the scatter in α_CO at one I_CO can reach a factor of 10. Using the best-fit α_CO–I_CO relation will then propagate this large scatter through to the estimated Σ_gas (as seen in Figure 8). The values of Σ_gas derived from α_CO are only weakly correlated (R = 0.55) with the 〈Σ_gas〉_mw calculated directly from the models. This poor relationship between I_CO and Σ_gas will substantially complicate any attempt to recover the star formation laws from the observational quantities.

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Finally, Figure 9 presents the K-S and E-S laws of the radiation pressure supported star-forming disks derived solely from the observational quantities L_TIR and I_CO. To guide the analysis and comparison with the theoretical expectations (Figures 3 and 4), the models are separated by their dust-to-gas enhancement factor, although this may be impractical for a real observational sample. Recall that the Σ_gas computed from I_CO is an estimate of 〈Σ_gas〉_mw, so Figure 9 should be compared to the right-hand panels of Figures 3 and 4. As expected, the scatter of the "observed" K-S laws (rms  ≈  0.2) is larger than the ones found from the theoretical relations (rms ≈0.1). More striking are the differences in the correlation coefficient and slopes. The observationally derived K-S laws have slopes of N = 1.3 ± 0.3, 0.8 ± 0.2, and 0.5 ± 0.2 and correlation coefficients of R = 0.6, 0.5, and 0.4 as the dust-to-gas enhancement factor increases. In contrast the theoretically expected slopes are 0.94–1.0 and have correlation coefficients of >0.9. Collecting all the models together yields a K-S slope of N = 1.4 ± 0.2 (with R = 0.5), surprisingly close to the canonical value, but, of course, yielding an entirely misleading view of the underlying physics. Thus, the K-S law governing the Eddington-limited star-forming disks is nearly unrecognizable due to the very poor conversion between I_CO and Σ_gas. Our data indicate that the use of a higher density CO transition (such as CO J = 3–2) would not improve the conversion as this line is thermalized in these very warm disks. A high-J transition from a large density tracing molecule, such as HCN, may provide the best probe of Eddington-limited disks (although this needs to be confirmed by subsequent computations).

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In contrast to the K-S law, the "observational" E-S laws (right panel of Figure 9) have similar correlation coefficients (R ≈ 0.8) and about the same amount of scatter (rms ≈0.2) as the theoretically expected relations (right panel of Figure 4). However, the slopes of the different E-S laws are all n > 0.7, inconsistent with the expected slope of n ≈ 0.5. The slope for the entire ensemble of models is n = 1.2 ± 0.1, which, much like the K-S law, leads to an erroneous description of the correct E-S law. The steep slopes are a result of the uncertain α_CO–I_CO relation. If, for example, a constant α_CO = 3.56, was used to estimate Σ_gas, then the slopes flatten to n = 0.4–0.7 for the individual dust-to-gas factors, and n = 0.8 ± 0.1 when including all the models. These results are consistent with the theoretical E-S law from Section 3.2. Applying this constant α_CO to the observed K-S law results in a slope that is much too flat (N = 0.2–0.6). The slopes of the observed E-S and K-S relations are therefore very sensitive to the assumptions placed on α_CO; however, the low rms and large correlation coefficients found in the observed E-S law are robust to the assumptions on α_CO, indicating that, out of these two relationships, the E-S law may provide the most promising means to investigate SF laws at high redshift.

The purpose of this paper is to (1) elucidate the K-S and E-S laws of a Eddington-limited star-forming disk (including the impact of spatial averaging), and (2) to determine how well the theoretically expected star formation laws can be recovered by the standard conversions from the observables (L_IR and I_CO). To explore the effects of spatial averaging and how these disks might appear to telescopes, 1260 star-forming disk models, spanning a wide range of size and mass scales, were run to produce a bank of test data. From this model suite, 132 of the largest and most optically thick disks were selected to best describe the properties of Eddington-limited starbursts that may be embedded in the centers of the gas-rich star-forming galaxies.

The analytical model of a radiation pressure supported star-forming disk developed by Thompson et al. (2005) predicts a K-S law with N ≈ 1, and an E-S law with n ≈ 0.5, both significantly flatter than the canonical values (N ≈ 1.5 and n ≈ 1). An important aspect of these star formation laws is the dependence on the opacity, or, equivalently, the dustiness of the star-forming gas. Indeed, the star formation process in a radiation supported disk strongly depends on the dust opacity (Thompson et al. 2005), and, when spatially averaged, can even impact the slopes of the observed star formation laws (Figures 3 and 4). Another important effect of the gas opacity is the vertical separation of the star formation laws based on the dust-to-gas enhancement factor (Figures 3 and 4). A heterogeneous sample of Eddington-limited star-forming disks may then appear to have a significantly steeper slope than what is governing the disks and would lead to the wrong conclusion about the underlying star formation relations.

To study how these disks might appear observationally, molecular line radiative transfer calculations were performed and CO line velocity-integrated intensities, I_CO, were derived for all 132 star-forming disk models. Combining these results with the infrared luminosity, L_TIR, calculated from the SED of each model, allowed a quantitative test of deriving the star formation laws from observational data. The calculated L_TIR and I_CO are approximately linearly correlated with each other (albeit with a substantial scatter of rms = 0.4; Figure 5), as might have been expected from the theoretical K-S law. However, in the Eddington-limited star-forming disks, the gas density and temperature are always above the critical density and excitation temperature of the first several CO rotational transitions. These lines are therefore completely thermalized and are tracing the same gas as the infrared emission, resulting in a near-linear relationship (e.g., Krumholz & Thompson 2007; Narayanan et al. 2008). Indeed, the densities and temperatures of these disks are so large that a linear relationship is also expected for the low J transitions of molecules with higher critical densities (such as HCN or HCO⁺), as seen in some ULIRGs (e.g., García-Burillo et al. 2012). While such a linear relationship may indicate that the star-forming region could be radiation-pressure dominated, it does not give any information on the underlying K-S law (Krumholz & Thompson 2007). Observations of high-J transitions from these molecules may be needed to determine the K-S and E-S laws from the observed IR luminosity and molecular line intensity.

If the L_TIR–I_CO relation cannot be trusted as a measurement of the star formation laws of Eddington-limited disks, then these quantities must be translated to $\dot{\Sigma }_{\ast }$ and Σ_gas. Figure 6 shows that the use of a L_TIR–SFR conversion and the outer radius of the disk results in a $\langle \dot{\Sigma }_{\ast }\rangle$ that is within a factor of two of the correct value. On the other hand, α_CO, the phenomenological conversion factor between I_CO and Σ_gas, exhibits significant scatter (about an order of magnitude for a given I_CO; Figure 7) and yields a very weak correlation between the estimated Σ_gas and the true 〈Σ_gas〉_mw (Figure 8). This poor conversion from I_CO to Σ_gas causes the resulting K-S laws (Figure 9 (left)) to have only a weak correlation between $\dot{\Sigma }_{\ast }$ and Σ_gas. The slopes of this K-S relation vary strongly with the dustiness of the gas, reaching as low as N = 0.5  ±  0.2 for a dust-to-gas factor of 10. The slope of N = 1.4  ±  0.2 found from including all the models is consistent with the canonical slope, which perfectly illustrates how the observed K-S law can distort and mislead the interpretation of star formation laws.

The situation is improved somewhat when considering the E-S law, calculated by dividing the I_CO-derived Σ_gas by the orbital time at r_out (Figure 9 (right)). In this case, the observed E-S laws retain the large correlation coefficients expected from the theoretical predictions, but, as with the K-S law, the slopes are significantly steeper (n > 0.7) than what the theory predicts (n ≈ 0.5). Similar to the K-S law, the slope found from the ensemble of models (n = 1.2 ± 0.1) is close to the canonical linear slope, leading to yet another erroneous conclusion about the underlying star formation law.

These experiments all point to the same problem: the α_CO conversion factor is too blunt and inaccurate an instrument to study the star formation laws in radiation pressure dominated disks. In many ways this is unsurprising, as the α_CO factor has only been shown to accurately trace Σ_gas in local Galactic molecular clouds up to Σ_gas ≈100 M_☉ pc⁻² (Kennicutt & Evans 2012). The Eddington-limited disks are at such high surface densities that the medium may not be accurately described as a series of molecular clouds, but rather a continuous structure with a near unity molecular fraction. In their study of the CO SLEDs from these disks, Armour & Ballantyne (2012) noted that in many ways the thermalized molecular emission from these disks acts like a stellar spectrum, with a SLED shape that depends on excitation temperature and a normalization that depends on the internal energy generation, i.e., star formation. Therefore, a large, massive disk may produce a similar I_CO as a smaller, less massive disk that has a larger 〈SFR〉 (due to, e.g., a larger opacity). The CO emission is largely immune to the total gas mass in each disk, and therefore the α_CO is a poor tracer of Σ_gas. We do find that α_CO is inversely related to I_CO and 〈SFR〉, in agreement with other authors (Narayanan et al. 2012b), but with an amplitude ≈2 times larger. This may be an effect of the high density and uniform molecular medium of these disks (Papadopoulos et al. 2012), but this result should be checked with more realistic two- or three-dimensional simulations.

Given the difficulties in using α_CO, how can we make progress in studying star formation laws, especially when Σ_gas ≳ 10⁴ M_☉ pc⁻²? Clearly the traditional method of CO rotational lines will not be particularly useful in this regime, so observations should focus on molecular species with higher critical densities, such as HCN. Recent results using HCN observations of ULIRGs that reach into this high-Σ_gas range do give K-S and E-S slopes consistent with the theoretical expectation for Eddington-limited disks, suggesting that radiation pressure dominated regions may be playing a role in these sources (García-Burillo et al. 2012). Unfortunately, the conversion factor of HCN and other higher density tracers is as uncertain as α_CO. Observational campaigns focusing on measuring these conversion factors over as wide a range of Σ_gas as possible will be necessary in order to confidently apply them for ULIRGs and at high redshift where large gas surface densities will be more common. Interestingly, the E-S law appears to be more robust to changes in opacity and maintains a strong correlation even with the use of α_CO (although the slope is still incorrect). This result implies that, at least for the Eddington-limited star-forming disks studied here, measurements of the E-S law will be more useful in determining the underlying star formation relations than the K-S law. Finally, the star formation laws of radiation pressure dominated disks are strongly dependent on the gas opacity, causing a vertical separation in the K-S and E-S plots. Fitting a line through the models without controlling for the differences in the dust-to-gas ratio results in a much steeper slope than the underlying star formation law. Thus, future observational work at these values of Σ_gas would benefit from analyzing samples of galaxies with metallicity or dust-to-gas ratio estimates in order to control for this potentially misleading effect.

In summary, the K-S and E-S star formation laws for radiation pressure supported star-forming disks have flatter slopes (N ≈ 1 and n ≈ 0.5) than what is expected for star formation at Σ_gas ≲ 10⁴ M_☉ pc⁻². These slopes are relatively robust to spatial averaging over the disks, but the strong dependence on the opacity causes the relations to be vertically offset based on the dust-to-gas ratio. Thus, an erroneously large slope for the K-S and E-S laws could result if the opacity dependence is not recognized. Attempting to recover these star formation laws from the predicted L_TIR and I_CO failed because of the poor translation from I_CO to Σ_gas at these high densities and temperatures. The "observed" K-S and E-S laws have slopes that differ significantly from the theoretically expected relations, even when controlling for the gas opacity. Progress in identifying and studying the star formation laws at these values of Σ_gas can be made by moving to a well-calibrated high critical density molecular tracer, selecting a sample of galaxies with metallicity estimates, and focusing on the E-S relation, as its slope seems less affected by variations in metallicity and spatial averaging.

The authors thank D. Narayanan for comments on a draft of the paper. This work was supported in part by NSF award AST 1008067 to D.R.B.