ASSESSING RADIATION PRESSURE AS A FEEDBACK MECHANISM IN STAR-FORMING GALAXIES

The coupling between radiation and gas in star-forming environments is complex primarily because the flux-mean opacity κ_F in Equation (1) has a full range of more than 3 dex, depending on whether the spectral energy distribution of the system considered is dominated by UV or FIR light. However, there are two distinct regimes: optically thick to UV but thin to the re-radiated FIR and optically thick to FIR. We call these the "single-scattering" and "optically thick" limits, respectively.

2.1.1. Single-scattering Limit

Regions in the single-scattering limit are optically thick to the UV but optically thin to the FIR (τ_FIR ∼ κ_FIRΣ_g/2). This limit applies over a wide range in surface density:

$$\begin{equation} \Sigma _{\rm g} \lesssim 5000\,{M_{\odot } \; \rm pc^{-2}} \; \kappa _{2}^{-1} \, f_{\rm dg, \, 150}^{-1}, \end{equation} \tag{ 3 }$$

where κ_FIR = κ₂f_dg, 150 is the Rosseland-mean dust opacity with κ₂ = κ/(2 cm² g^-1) (see Section 2.1.2) and f_dg, 150 = f_dg × 150 is the dust-to-gas ratio. In the single-scattering limit, UV photons are absorbed once and then re-radiated as FIR photons, which free-stream out of the medium.⁵ Since the column-averaged flux-mean optical depth in this limit is always equal to unity, the flux-mean opacity for the single-scattering limit is ∼2/Σ_g. The Eddington flux is then

$$\begin{equation} F_{\rm Edd}^{\rm s} \sim 10^{8} \; {L_{\odot } \; \rm kpc^{-2}} \; \left(\frac{\Sigma _{\rm g}}{10 \, {M_{\odot }\;\rm pc^{-2}}} \right)^2 \, f_{\rm gas}^{-1}, \end{equation} \tag{ 4 }$$

where f_gas = Σ_g/Σ_tot is the gas fraction and Σ_tot ≡ Σ_g + 0.1Σ_⋆ (Wong & Blitz 2002). The wide range of column densities over which this limit is applicable implies that the average medium of most star-forming galaxies, some starbursts, and the GMCs that constitute them is single scattering.

2.1.2. Optically Thick Limit

Dense starbursts and GMCs can reach the high gas surface densities $\Sigma _{\rm g} \gtrsim 5000\,{M_{\odot } \; \rm pc^{-2}} \; \kappa _{2}^{-1} \, f_{\rm dg, \, 150}^{-1}$ necessary to become optically thick to FIR photons (τ_FIR ≳ 1). In this case, P_rad ∼ τ_FIRF/c, and κ_FIR depends on temperature (Bell & Lin 1994; Semenov et al. 2003). The functional form of κ_FIR naturally leads to two regimes: "warm" ($T < 200 \; \rm K$) and "hot" (200 K < T < T_sub, where $T_{\rm sub} \sim 1500 \rm \; K$ is the dust sublimation temperature). For typical numbers, the central temperature of a massive, compact star cluster is

$$\begin{eqnarray} &&T^4 \sim \tau T_{\rm eff}^4 \sim \frac{\kappa _{\rm FIR} \Sigma }{2} \frac{F}{\sigma _{\rm SB}} \sim \frac{\kappa _{\rm FIR} M_{\rm g}}{8 \pi R^2} \frac{M_\star \Psi }{4 \pi R^2 \sigma _{\rm SB}} \nonumber \\&&\ms\ms T \sim 290 \; {\rm K} \; \kappa _{10}^{1/4} \; \Psi _{3000}^{1/4} \; M_{\rm g,\,6}^{1/4} \; M_{\star,\,5}^{1/4} \; R_{\rm pc}^{-1}, \end{eqnarray} \tag{ 5 }$$

where T_eff is the effective temperature, $\kappa _{10}=\kappa /(10 \; \rm cm^2 \; g^{-1})$, $\Psi _{3000}=3000 \; \rm erg \; s^{-1} \; g^{-1}$ is the light-to-mass ratio of a zero-age main-sequence (ZAMS) stellar population, $M_{\rm g,\,6}=M_{\rm g}/(10^6 \, M_\odot \rm)$, and $M_{\star,\,5}=M_{\star }/(10^5 \, M_\odot \rm)$.

Warm Starbursts. For $T < 200 \; \rm K$, the Rosseland-mean opacity increases as κ_FIR(T) ≈ κ_oT², where κ_o ≈ 2 × 10⁻⁴ cm² g^-1 K⁻² f_dg, 150. In this case,

$$\begin{equation} F_{\rm Edd} \approx \left(\frac{3\pi Gc\sigma _{\rm SB}}{\kappa _o^2 \,f_{\rm dg, \, 150}^2 \,f_{\rm gas} }\right)^{1/2} \hspace{-8.5359pt} \sim 10^{13} \; {L_{\odot } \, \rm kpc^{-2}} \, f_{\rm gas}^{-1/2} \, f_{\rm dg, \, 150}^{-1}. \end{equation} \tag{ 6 }$$

Remarkably, the flux necessary to support the medium is independent of Σ (TQM).

Hot starbursts. Intense, compact starbursts may have central temperatures greater than 200 K. The corresponding opacity is roughly constant with temperature: κ_FIR(T) ≈ 5–10 cm² g^-1 f_dg, 150 for temperatures 200 K ≲ T ≲ T_sub. For typical numbers,

$$\begin{equation} F_{\rm Edd}^{\rm thick} \sim 10^{15} \; {L_{\odot }\;\rm kpc^{-2}} \; \left(\frac{\Sigma _{\rm g}}{10^6 \, {M_{\odot }\;\rm pc^{-2}}} \right) \; f_{\rm gas}^{-1/2} \, f_{\rm dg, \, 150}^{-1}. \end{equation} \tag{ 7 }$$

The high surface densities necessary to enter this regime may only be attained in the parsec-scale star formation thought to attend the fueling of bright active galactic nuclei (AGNs; Sirko & Goodman 2003; TQM; Levin 2007).

In order to gauge the importance of GMC evolution and intermittency, we adopt the simple picture presented by MQT that marginally stable (Q ≈ 1) disks fragment into sub-units on the gas disk scale height (h) to form GMCs. An individual star cluster is born, reaches the critical Eddington luminosity threshold, and then expels the overlying gas. Importantly, the timescale for collapse and expansion of the GMC is the disk dynamical timescale, t_dyn, which can be much longer than the characteristic timescale for the stellar population to decrease in total luminosity, the main-sequence lifetime of massive stars, t_MS ∼ 4 × 10⁶ yr. In this picture, a low-density star-forming galaxy with radius r should have ∼(r/h)² sub-units, but only a small fraction ξ (the "intermittency factor") should be bright at any one time. If each subregion reaches the Eddington luminosity for a time t_MS and is then dark, and then if a large number of subregions are averaged, one expects

$$\begin{equation} \xi \equiv \frac{N_{\rm on}}{N_{\rm tot}} \sim \frac{L_{\rm obs}}{L_{\rm Edd}} \sim \frac{t_{\rm MS}}{2t_{\rm dyn}+t_{\rm MS}}, \end{equation} \tag{ 8 }$$

where

$$\begin{eqnarray} t_{\rm dyn} &\sim & \left(\frac{3 \pi (2h)}{32 G \Sigma _{\rm tot}}\right)^{1/2} \nonumber \\ \hspace{-8.5359pt}&\sim & 3.5\times 10^7\,{\rm yr}\,\,\,h_{100}^{1/2}f_{\rm gas}^{1/2} \left(\frac{10\,\,{M_\odot \rm \,\,pc^{-2}}}{\Sigma _{\rm g}}\right)^{1/2}, \end{eqnarray} \tag{ 9 }$$

$h_{100} = h/(100 \; \rm pc)$, N_on is the number of sub-units that are "on," and N_tot is the total number of sub-units. For example, the normal star-forming galaxy M51 has an observed bolometric luminosity (L_obs = 0.2L_Edd) that is a factor of ∼4 larger than its intermittency-corrected Eddington luminosity (L^int_Edd = 0.05L_Edd). Although the approximation that the stellar population is bright for a time t_MS and then dark is crude, the parameter ξ gives us a way to judge the importance of intermittency in normal star-forming galaxies.

Note that for higher densities, t_dyn decreases and ξ → 1 at a critical surface density

$$\begin{equation} \Sigma _{\rm crit} \sim \frac{3 \pi (2h)}{32 G (0.5t_{\rm MS})^2} \sim 3 \times 10^3 \, {M_\odot \rm \; pc^{-2}} \; h_{100}, \end{equation} \tag{ 10 }$$

which corresponds with a critical midplane pressure P_crit ∼ 5 × 10⁻⁸ erg cm^-3 h²₁₀₀ (see Equation (15)). For Σ_tot > Σ_crit, massive stars live longer than the time required to disrupt the parent sub-unit (t_MS > t_dyn). MQT argue that in this regime the massive stars continue to drive turbulence in the gas and maintain hydrostatic equilibrium in a statistical sense until t_MS, when the process then repeats until gas exhaustion.

Several additional elements of GMC evolution are important in judging whether or not the star formation of galaxies is regulated by radiation pressure on dust. First, the GMCs collapse from regions of size h², with total mass Σ_gπh², and to a size R_GMC = h/ϕ. This implies that the surface density of individual GMCs is Σ_GMC ∼ ϕ²Σ_g, where ϕ can be ∼2–5 in the Galaxy (MQT). For example, taking _GMC = M_⋆/M_GMC (∝Σ_g in the single-scattering limit), Ψ₃₀₀₀, and assuming κ_FIR ∼ κ_oT² (appropriate for $\rm T \lesssim 200 \, K$), one finds that the required gas surface density for a GMC to be optically thick to the FIR is $\Sigma _{\rm GMC}^{\tau _{\rm FIR}=1}\sim 7000\epsilon _{\rm GMC,\,0.1}^{-1/3}$ M_☉ pc⁻². This GMC gas surface density would correspond to an average gas surface density for the galaxy that is ϕ² times smaller (Σ_g ∼ 350–1800 M_☉ pc⁻²). Thus, the medium surrounding the central star cluster may be optically thick to the FIR even if the average gas surface density of the galaxy is less than the naive estimate given in Section 2.1.2.

At surface densities in excess of τ_FIR = 1 for the GMCs, the models of MQT rely on the fact that the medium is in fact optically thick to FIR radiation. This is in sharp contrast to the work of KM09, where they argue that the effective optical depth is always ∼1 because instabilities allow the radiation to leak out of otherwise optically thick media. MQT argue that the effective optical depth must be much larger than unity in the GMCs of dense starbursts and ULIRGs for radiation pressure to be effective as a feedback mechanism. In addition, they show that the effective momentum coupling between the radiation and the gas can exceed the naive estimate based on a disk-averaged τ_FIR by a factor of a few in systems as dense as the putative GMCs in Arp 220 because of the time dependence of the GMC disruption process (see Section 4.3).

We show the total IR luminosity L_IR as a function of molecular line luminosity⁶ L'_CO (Figure 1) and L'_HCN (Figure 2) for our sample of star-forming galaxies.⁷ L_IR is known to trace the total light from massive stars (e.g., Kennicutt 1998),⁸ whereas L'_CO and L'_HCN provide a measure of the total gas mass and dense gas mass, respectively. Under the assumption that the total gravitational potential is dominated by the gas on the physical scales where the stars are forming, the Eddington luminosity is related to L'_CO by

$$\begin{equation} L_{\rm Edd} = \frac{4\pi G c}{\kappa } X_{\rm CO} \, L_{\rm CO}^\prime, \end{equation} \tag{ 11 }$$

where X_CO is the L'_CO-to-$M_{\rm H_2}$ conversion factor and κ is the appropriate flux-mean or Rosseland-mean opacity (either single-scattering or optically thick; see Sections 2.1.1 and 2.1.2). Although counterintuitive, the single-scattering Eddington luminosity lies below the optically thick Eddington limit because the dust opacity is column-averaged and highly non-gray (see Section 2.1). We adopt X^MW_CO = 4.4 M_☉(K km s⁻¹ pc²)⁻¹ for normal galaxies (including a correction factor of 1.36 to account for He; Strong & Mattox 1996; Dame et al. 2001), and X^ULIRG_CO = 0.8 M_☉(K km s⁻¹ pc²)⁻¹ for galaxies with L_IR ⩾ 10¹¹ L_☉ (as appropriate for starbursts and ULIRGs; e.g., Downes & Solomon 1998). Similarly, if the majority of the IR luminosity comes from regions where the dense molecular gas dominates the potential, then L_Edd is related to L'_HCN by

$$\begin{equation} L_{\rm Edd} = \frac{4\pi G c}{\kappa _{\rm FIR}} X_{\rm HCN} \, L_{\rm HCN}^\prime, \end{equation} \tag{ 12 }$$

where we explicitly write κ = κ_FIR because the critical density for HCN emitting gas is large enough that these regions should always be optically thick (Section 2.1.2).⁹ In Equation (12), we take an L'_HCN -to-$M_{\rm H_2}^{\rm dense}$ conversion factor of $X_{\rm HCN}=3 \,M_{\odot }\,\rm (K \;km \;s^{-1} \;pc^2)^{-1}$, but we caution that X_HCN is uncertain to a factor of ∼  3 (Gao & Solomon 2004a, 2004b; see Section 4.5).

In Figure 1, the lines indicate the Eddington limit for various limiting cases. The lines in the left panel show the single-scattering Eddington limit (solid line) and the single-scattering Eddington limit accounting for intermittency (dot-dashed line) assuming h = 100 pc and r = 10 kpc (Equation (11)). The shaded region in the right panel is the optically thick Eddington limit (Equation (11) with κ_FIR = 5–$10 \; \rm cm^2 \; g^{-1}$) and the dashed line shows the optically thick Eddington limit for an enhanced opacity (κ_FIR = 30 cm² g^-1 f_dg,50; where f_dg,50 = f_dg × 50) due to an assumed higher dust-to-gas ratio in dense star-forming environments. We plot the single-scattering (left panel) and optically thick (right panel) Eddington limits separately for clarity, but the data are the same in both panels. The different symbols indicate various rotational transitions of CO: solid circles (J = 1–0), crosses (J = 2–1), squares (J = 3–2), and triangles (J = 4–3).

Note that no galaxies exceed the optically thick Eddington limit and most galaxies are neither significantly super- or sub-Eddington with respect to the single-scattering Eddington limit. We caution that the applicable Eddington limit for any individual galaxy cannot be determined in this plot due to the lack of surface density measurements, which dictate the optical depth to the FIR and the relevant Eddington limit. For example, high surface density star-forming regions can be optically thick at $L_{\rm CO}^\prime \lesssim 10^9 \rm \; K \; km \; s^{-1} \; pc^{2}$ and lie to the left of the single-scattering Eddington limit (solid line in left panel) but below the optically thick Eddington limit (shaded region in right panel). For the single-scattering limit, our assumption of r = 10 kpc is accurate to a factor of ∼5 for most of the sample, but the single-scattering opacity scales as r⁻², so it is only accurate to a factor of ∼25. Some compact starbursts have radii much smaller than our assumed radius, so they are optically thick, even at low L'_CO and this explains why they exceed the single-scattering limit in the left panel but are below the optically thick limit in the right panel. In addition, note that the optically thick Eddington limit is a hard upper bound to a galaxy's IR luminosity, which suggests that radiation pressure feedback may set the maximum SFR of a galaxy. In the Eddington-limited model, the scatter in the L_IR–L'_CO relation may be due to variations in h, X_CO (see Section 4.5), the dust-to-gas ratio/metallicity (see Section 4.4), the effective radii, and the depth of the stellar potential.¹⁰

The intermittency of star formation will likely affect the Eddington limit for CO-emitting gas (dot-dashed line in left panel). L'_CO traces the total molecular gas reservoir including the molecular gas that is not actively participating in star formation, such as GMC envelopes and diffuse intercloud gas. The gas mass relevant for the Eddington limit may be overestimated for galaxies in the single-scattering limit. To account for this, we multiply the Eddington luminosity by the intermittency factor for CO-emitting gas ξ ∼ 0.06 for the Milky Way value of X_CO, h = 100 pc, r = 10 kpc, and $L_{\rm CO}^\prime =10^9 \rm \; K\;km\;s^{-1} \;pc^2$ (see Equation (8) and Section 2.2). The intermittency factor approaches unity when L'_CO ∼ 2 × 10¹¹ K km s^-1 pc² h₁₀₀ r²₁₀/X^MW_CO. Thus, compact star-forming regions, such as the nuclear starbursts of ULIRGs, have ξ ∼ 1 at low L'_CO due to their very small radii (e.g., Downes & Solomon 1998).

Figure 2 shows the L_IR–L'_HCN relation for our sample of star-forming galaxies. The shaded region represents the optically thick Eddington limit (Equation (12) with κ_FIR = 5–$10 \; \rm cm^2 \; g^{-1}$) and the dashed line shows the optically thick Eddington limit for an enhanced opacity (κ_FIR = 30 cm² g^-1 f_dg,50; where f_dg,50 = f_dg × 50), which may result from a higher dust-to-gas ratio in dense star-forming environments. The circles and crosses correspond to the J = 1–0 and the J = 2–1 rotational transitions of HCN, respectively.

The L_IR–L'_HCN relation (Figure 2) is tight and linear over several orders of magnitude, implying that stars form out of dense gas (Gao & Solomon 2004a, 2004b; Wu et al. 2005). The dense gas fraction (L'_HCN/L'_CO) is nearly constant for galaxies with L_IR ≲ 10¹¹L_☉ (Gao & Solomon 2004b), so L'_CO can be used to indirectly trace the dense gas mass $M_{\rm H_2}^{\rm dense}$. However, the dense gas fraction increases dramatically in luminous infrared galaxies (LIRGs) and ULIRGs (L_IR ≳ 10¹¹L_☉), so CO does not trace dense gas mass in these galaxies (Gao & Solomon 2004b). HCN, on the other hand, has a critical density for excitation that is ∼2 orders of magnitude larger than that of CO, so it traces dense, optically thick gas in star-forming GMC cores rather than diffuse GMC envelopes. The dynamical time for HCN-emitting gas is much less than the main-sequence lifetime of the most massive stars, so the intermittency factor for HCN-emitting gas will be approximately unity:

$$\vspace*{-3pt} \begin{equation} t_{\rm dyn}^{\rm HCN} \sim 2 \times 10^5 \; {\rm yr} \; \rho _{\rm crit, \, HCN}^{-1/2} \ll t_{\rm MS} \rightarrow \xi \approx 1, \end{equation} \tag{ 13 }$$

where $\rho _{\rm crit, \, HCN} \sim 10^{-19} \; \rm g \; cm^{-3}$.

If the picture of radiation pressure feedback is correct, then it should determine the L_IR–L'_HCN correlation directly. In fact, both the Eddington limit and the data show a linear relation between L_IR and L'_HCN. The galaxies closely follow but do not exceed the Eddington limit for our preferred value of the Rosseland-mean opacity (κ_FIR = 5–10 cm² g^-1 f_dg,150). If the opacity is higher (κ_FIR = 30 cm² g^-1 f_dg,50), then many galaxies are consistent with Eddington and a number are super-Eddington. For any of the values of the opacity that we assume, the general agreement between L_IR and L'_HCN suggests that radiation pressure may play an important role in regulating star formation (Scoville 2003). However, a number of important factors remain uncertain, which we discuss in Section 4.

The Schmidt law is a tight power-law relation between the surface density of SFR $\dot{\Sigma }_{\star }$ and the gas surface density ($\dot{\Sigma }_{\star } \propto \Sigma _{\rm g}^{1.4}$; Kennicutt 1998). Furthermore, Bigiel et al. (2008) found that the Schmidt law for molecular gas is linear within local star-forming galaxies ($\dot{\Sigma }_{\star } \propto \Sigma _{\rm H_2}^{1.0}$). In the left panel of Figure 3, we plot $\dot{\Sigma }_{\star }$ versus $\Sigma _{\rm H_2}$ for individual apertures of THINGS galaxies (small dots; Bigiel et al. 2008; Helfer et al. 2003; Leroy et al. 2008), THINGS galaxies with $\rm H_2$ detections (open circles), starburst galaxies¹¹ (solid circles), M82 super star clusters (stars; McCrady et al. 2003; McCrady & Graham 2007), and the Galactic center star cluster (diamond; Paumard et al. 2006). We compare the data with the Eddington limit using $\dot{\Sigma }_{\star }$ and $\Sigma _{\rm H_2}$ as proxies for the radiation and gravitational forces,

$$\begin{equation} \dot{\Sigma }_{\star }^{\rm Edd} = \frac{4 \pi G \Sigma _{\rm H_2}}{\epsilon c \kappa }, \end{equation} \tag{ 14 }$$

where ≈ 5 × 10⁻⁴ is the efficiency of the conversion of mass into luminosity during the star formation process assuming a Kroupa (2001) broken power-law initial mass function (IMF) that extends up to $120 \, M_\odot \rm$. For star clusters we assume a light-to-mass ratio appropriate for a ZAMS stellar population ($\Psi =3000 \; \rm erg \; s^{-1} g^{-1}$) and that the stellar mass is a lower limit on the gas mass of the parent GMC. We use the same X_CO values as in Figure 1 and again caution that X_CO is uncertain to a factor of a few and may vary systematically from normal galaxies to starbursts (see Section 4.5).

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For the molecular Schmidt law, we find that the Eddington limit is an upper bound to $\dot{\Sigma }_{\star }$. Most star-forming regions and galaxies follow the Eddington limit (solid line) and are within ∼1.5 dex of it. The Eddington limit accounting for intermittency (dot-dashed line) appears to agree better with the data than the naive single-scattering Eddington limit, suggesting that intermittency may be an important effect (similarly, Papadopoulos & Pelupessy 2010 showed that time dependence might strongly affect the Schmidt law). As the medium becomes optically thick near the critical surface density for intermittency (hatched region; see Section 2.2), the optically thick Eddington limit ($\dot{\Sigma }_{\star }^{\rm Edd} \propto \Sigma _{\rm H_2}/\kappa _{\rm FIR} \propto \Sigma _{\rm H_2}^{1.0}$) provides a firm upper bound to $\dot{\Sigma }_{\star }$ for our preferred value of the dust opacity (κ_FIR = 5–10 cm² g^-1 f_dg,150). If the dust opacity (κ_FIR = 30 cm² g^-1 f_dg,50) is enhanced due to a higher assumed dust-to-gas ratio, then some galaxies reach the optically thick Eddington limit and a few galaxies exceed it.

In the right panel of Figure 3, we plot the radiation pressure from UV and FIR photons versus the midplane pressure. These pressures will balance each other at Eddington (solid line):

$$\begin{equation} P_{\rm rad} \sim (1+\tau _{\rm FIR})\frac{F}{c} \sim P_{\rm mid} = \frac{\pi }{2} G \Sigma _{\rm g} \Sigma _{\rm tot}, \end{equation} \tag{ 15 }$$

where we take Σ_tot ≡ Σ_g + 0.1Σ_⋆ (Wong & Blitz 2002). The P_rad–P_mid plot shows that radiation pressure correlates strongly with midplane pressure over 10 orders of magnitude. The Eddington limit serves as a rough upper limit to P_rad, and most galaxies are within 2 dex of the Eddington limit. We note that some of the THINGS apertures and some dense starbursts reach or exceed the Eddington limit. For galaxies with P_mid < P_crit, the critical midplane pressure (hatched region; see Sections 2.2 and 4.2), we expect that the effects of intermittency are important; however, the intermittency-adjusted Eddington limit (dot-dashed line) underpredicts P_rad for star-forming regions with $P_{\rm mid} \lesssim 10^{-11.5} \rm \; erg \; cm^{-3}$. The intermittency factor may overestimate the importance of intermittency because of the simplifying assumption that subregions are "on" or "off" (see Section 2.2). We also see that galaxies and star-forming regions with 10⁻¹¹ erg cm^-3 ≲ P_mid ≲ P_crit tend to fall significantly below the Eddington limit, possibly because our simple parameterization of X_CO (see Section 4.5) is overestimating $M_{\rm H_2}$ (and P_mid) for these systems. Radiation pressure becomes increasingly more important in the optically thick limit (P_rad ≳ P_crit) as some galaxies and star-forming regions meet and exceed Eddington. As expected from Figures 1 and 2, if we assume a larger dust-to-gas ratio and opacity (κ = 30 cm² g^-1 f_dg,50) potentially appropriate for dusty galaxies, then more of the optically thick starbursts would be super-Eddington.

So far we have evaluated radiation pressure on a galaxy-wide scale; however, the distribution of gas and star formation in galaxies is inhomogeneous. Consequently, the Eddington ratio (Γ = P_rad/P_mid) will likely vary on sub-galactic scales. We use observations from the THINGS survey (Walter et al. 2008; Leroy et al. 2008; Bigiel et al. 2008) to calculate the Eddington ratio as a function of radius in azimuthally averaged radial bins and for semi-resolved (750 pc) apertures. Since the THINGS galaxies are generally in the single-scattering limit (see Equation (3)), we conservatively adopt the radiation pressure to be P^IR_rad = F_IR/c (see Equation (15)). For the midplane pressure given in Equation (15), the corresponding Eddington ratio is Γ_tot = P^IR_rad/(0.5πGΣ_gΣ_tot). Stars and atomic gas may not contribute significantly to the surface density in regions of active star formation, so we also calculate the Eddington ratio assuming that the midplane pressure depends only on the total gas surface density Γ_g = P^IR_rad/(πGΣ²_g) or the molecular gas surface density $\Gamma _{\rm H_2}=P_{\rm rad}^{\rm IR}/(\pi G \Sigma _{\rm H_2}^2)$. Intermittency may be important because the THINGS observations cannot resolve individual star-forming regions. We calculate the Eddington ratio corrected for intermittency (Γ^int_tot = Γ_tot/ξ, see Equation (8)). In Figure 4, we plot Γ_tot (open circles), $\Gamma _{\rm H_2}$ (solid triangles), and Γ^int_tot (open squares) as a function of radius for azimuthally averaged rings. We find that Γ_g is similar to Γ_tot, so we omit Γ_g for clarity.

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At intermediate radii ($r \sim 1 \rightarrow \rm several \; kpc$), Γ_tot and $\Gamma _{\rm H_2}$ generally increase from sub-Eddington (Γ ∼ 0.1) to approaching or exceeding the Eddington limit (Γ ∼ 1). Γ_tot reaches a maximum Eddington ratio at r ∼ 5–10 kpc, where it falls off steeply. As the observations near the $\rm H_2$ detection threshold, $\Gamma _{\rm H_2}$ increases rapidly due to a small $\Sigma _{\rm H_2}$ with large error bars (see, e.g., Figure 40 of Leroy et al. 2008), and thus the large value of $\Gamma _{\rm H_2}$ at large r is consistent with Eddington to within the errors on $\Sigma _{\rm H_2}$. For $r > 1 \rm \; kpc$ where Γ_tot < 1, the intermittency factor can boost Γ^int_tot to the Eddington limit, suggesting that intermittency is important.

The Γ < 1 regions in the inner parts of star-forming galaxies present a challenge for radiation pressure regulated star formation. Intermittency cannot account for the low Eddington ratios of these regions. However, a metallicity gradient, as seen in observations of star-forming galaxies (Muñoz-Mateos et al. 2009), increases the Eddington ratio at small radii (see Section 4.4). A metallicity gradient that rises at smaller radii correlates with a decreasing gradient in X_CO (Sodroski et al. 1995; Arimoto et al. 1996) and an increasing gradient in the dust-to-gas ratio (Muñoz-Mateos et al. 2009). We adopt the X_CO gradient given by Equation (10) of Arimoto et al. (1996) for data from the Milky Way, M31, and M51 (log X/X_e = 0.41[r/r_e − 1], where r_e is the effective radius, which we assume to be 7 kpc and X_e is the value of X_CO at the effective radius). To account for the dust-to-gas ratio gradient, we use a power-law interpolation between f_dg = 1/30 at 0.1 kpc and f_dg = 1/150 at 10 kpc, motivated by Figure 15 of Muñoz-Mateos et al. (2009). In addition, collapsing GMCs enhance the surface density by a factor of ϕ² (ϕ = h/R_GMC; see Section 2.2), making some regions optically thick to FIR radiation. In Figure 5, we show the molecular Eddington ratio as a function of radius for NGC 6946 since this galaxy is well below (∼ 2 dex) the Eddington limit at small radii (see Figure 4). After accounting for intermittency, an X_CO gradient, a dust-to-gas ratio gradient, and a surface density enhancement in the GMCs, we find that $\Gamma _{\rm H_2} \sim 1$ for almost all radii in NGC 6946. We find qualitatively similar results for all of the THINGS galaxies shown in Figure 4 assuming that the metallicity, X_CO, and dust-to-gas ratio gradients are similar to those adopted for NGC 6946. Thus, it is at least in principle possible to explain the nominally sub-Eddington inner regions of local star-forming galaxies using a combination of these effects.

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In Figure 6, we plot the distributions of the individual Eddington ratios for the THINGS apertures (750 pc resolution): Γ_tot (dotted line), $\Gamma _{\rm H_2}$ (dashed line), and Γ^int_tot (solid line). The distribution of Γ_tot is peaked around Γ_tot ∼ 0.1, and the majority of apertures are sub-Eddington for Γ_tot. The high Γ_tot tail of the distribution extends above the Eddington limit, with super-Eddington apertures comprising 5% of the total apertures and containing 5% of the total flux. Star-forming regions are unresolved on 750 pc scales, so Γ_tot should be adjusted to account for intermittency (Γ^int_tot). However, most apertures lie above the intermittency-adjusted Eddington limit. As in Figures 1–5, this shift suggests that intermittency is important for radiation pressure supported star formation in normal spirals; however, ξ appears to overestimate the importance of intermittency, possibly due to the simplifying assumption that subregions are either "on" or "off" (see Sections 2.2 and 4.2). The distributions of Γ_tot and Γ_g are similar, so we do not plot Γ_g for clarity.

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The distribution of $\Gamma _{\rm H_2}$ is less peaked and shifted to systematically higher values than the distribution of Γ_tot with 10% of these apertures radiating at or above the Eddington limit. This is not surprising (given Figure 4) because radiation pressure will likely be more important in $\rm H_2$-dominated star-forming regions. The detection limit for $\rm H_2$ is higher than that for H i, so the distribution of $\Gamma _{\rm H_2}$ contains fewer apertures than Γ_tot. In addition, the apertures with $\rm H_2$ detections tend to be within 0.4R₂₅ (Bigiel et al. 2008), so they might have increased metallicities and dust-to-gas ratios with depressed X_CO values, which would increase the Eddington ratio (see Figure 5, Sections 4.4 and 4.5). Super-Eddington apertures contain 6% of the total flux in $\rm H_2$ -detected apertures across the whole sample; in NGC 6946, for example, super-Eddington apertures contain 10% of the total flux. The super-Eddington apertures indicate that radiation pressure can be dynamically dominant even when individual star-forming regions remain unresolved, suggesting that radiation pressure may be more important on the scale of GMCs and massive star clusters. Finally, we calculated $\Gamma _{\rm H_2}$ assuming that UV photons contribute to the radiation pressure P_rad = (F_UV + F_IR)/c. The distribution of $\Gamma _{\rm H_2}$ remains nearly the same because star-forming regions have F_IR/F_UV ≫ 1, so we refrain from plotting it in Figure 6.

The gas depletion timescale, the time required to consume a galaxy's gas reservoir at the current SFR, is observed to be ∼2 Gyr in normal spirals (Kennicutt 1998; Leroy et al. 2008). Radiation pressure sets the gas depletion timescale to be

$$\begin{equation} t_{\rm gas} = \frac{M_{\rm g}}{\dot{M}_\star } = \frac{M_{\rm g} \epsilon c^2}{\xi L_{\rm Edd}} = \frac{\epsilon c \kappa }{4 \pi G \xi }. \end{equation} \tag{ 16 }$$

Using typical numbers for a spiral galaxy in the single-scattering limit, the gas depletion timescale is

$$\begin{equation} t_{\rm gas} \sim 2.1 \; {\rm Gyr} \; \Sigma _{30}^{-3/2} \, h_{100}^{1/2}, \end{equation} \tag{ 17 }$$

where $\Sigma _{30}=\Sigma /30 \, M_\odot \rm \; pc^{-2}$ and h₁₀₀ = h/100 pc. This normalization of t_gas is in good agreement with the observed gas depletion timescale, but Equation (17) predicts that the gas depletion timescale should have a strong dependence on Σ_g, in contrast to the observations of Leroy et al. (2008) (see their Figure 15). For completeness, we note that variations in the dust-to-gas ratio will not affect the dependence of t_gas on Σ_g in the single-scattering limit, but uncertainties in X_CO, ξ, and ϕ (see Sections 4.4 and 4.5) might impact the gas depletion timescale.

Hot starbursts and optically thick subregions (see Section 2.1.2) have intermittency factors that approach unity and nearly constant opacities, so the gas depletion time is approximately constant:

$$\begin{equation} t_{\rm gas} \approx 5.7 \; {\rm Myr} \; \kappa _{10}, \end{equation} \tag{ 18 }$$

where $\kappa _{10}=\kappa _{\rm FIR}/10 \; \rm cm^2 \; g^{-1}$. For comparison, Sakamoto et al. (2008) find that the optically thick western nucleus of Arp 220 has a gas depletion time of ${\sim} 6 \;\rm Myr$. The fact that the gas depletion timescale set by radiation pressure is consistent with the observed gas depletion timescale in spirals and ULIRGs is equivalent to the statement of Figures 1–5 that starbursts approach the dust Eddington limit. In addition, we point out that radiation pressure feedback predicts that the specific SFR (SSFR) will be SSFR ∼ 4πGξf_gas/(cκ) for small f_gas.

The intermittency factor ξ (see Section 2.2; MQT) relates the properties of radiation pressure dominated star-forming subregions to the global properties of a galaxy. However, ξ may overestimate the effect of intermittency in some galaxies (see Figures 3, 4, and 6). We expect the determination of ξ to be complicated by uncertainty in the timescale for the central cluster of a sub-unit to be bright ($t_{\rm MS} \sim 4 \; \rm Myr$). We adopt t_MS as the time a cluster will be bright, since the cluster luminosity drops rapidly after the most massive stars in the cluster explode as supernovae. However, a cluster will continue to emit after t_MS. Indeed, models of cluster luminosity indicate that a similar amount of momentum will be transferred to the gas during the time t₀ → t_MS and during the time t_MS → 4t_MS, where the cluster luminosity has dropped by 1 dex after ∼4t_MS (Leitherer et al. 1999). Further uncertainty in ξ is due to ambiguities in calculating the lifetime of a sub-unit (∼2t_dyn + t_MS), especially the disk dynamical time (see Equation (9)). The dynamical time likely varies with galactocentric radius because Σ is a strong function of radius (Leroy et al. 2008). Thus, trying to determine an effective ξ applicable to a galaxy as a whole may be difficult if ξ changes locally. Overall, we expect the uncertainty in ξ to be a factor of a few to several.

A key theoretical uncertainty in calculating the Eddington limit for dense starbursts is the effective optical depth (τ_eff) for surface densities where τ_FIR > 1. In order for radiation pressure to be dynamically important in optically thick GMCs, τ_eff must exceed unity. Based on the high Mach number turbulence simulations of Ostriker et al. (2001), MQT conclude that if the ISM is optically thick on average, then the vast majority of sight lines will be optically thick. For comparison, KM09 argue that instabilities, such as Rayleigh–Taylor and photon–bubble instabilities, will reduce the effective optical depth of the dense ISM to ∼1. However, MQT note that both the midplane pressure from gravity P_mid ∼ πGΣ² and optically thick radiation pressure P_rad ∼ τF/c ∝ Σ² scale as Σ², a feature unique to radiation pressure among stellar feedback processes. Thus, if radiation pressure cannot regulate star formation in dense, optically thick gas, then no known stellar feedback process can.

The coupling between radiation and gas directly depends on the dust-to-gas ratio (κ ∝ f_dg). In this analysis, we assume the Galactic value for the dust-to-gas ratio (f_dg = 1/150) and solar metallicity; however, there is strong evidence that f_dg and metallicity change with environment. The dust-to-gas ratio has been shown to correlate with metallicity and radius (Issa et al. 1990; Lisenfeld & Ferrara 1998; Draine et al. 2007; Muñoz-Mateos et al. 2009). Muñoz-Mateos et al. (2009) find that the dust-to-gas ratio can climb to values as high as f_dg ∼ 1/10 in the centers of spiral galaxies. This increase in metallicity and dust-to-gas ratio is necessary for the centers of star-forming spirals to be at Eddington (see Figures 4 and 5 and Section 3.3). The average dust-to-gas ratio of local spirals also varies by a factor of a few (e.g., M51 has a f_dg ∼ 1/75; Draine et al. 2007). Furthermore, the dust-to-gas ratio is observed to be higher in some dense starbursts, such as SMGs (f_dg ∼ 1/50; Kovács et al. 2006; Michałowski et al. 2010) and submillimeter faint ULIRGs (f_dg ∼ 1/20; Casey et al. 2009). Importantly, if we adopt a dust-to-gas ratio potentially appropriate for dusty starbursts (short dashed line in Figures 1, 2, and the left panel of Figure 3), then a substantial fraction of optically thick galaxies would be at or above the Eddington limit (Figure 5 illustrates this for NGC 6946).

The Eddington limit depends strongly on the CO-to-H₂ (X_CO) and the HCN-to-H₂ (X_HCN) conversion factors (see Equations (11) and (12)). These conversion factors are two of the largest sources of observational uncertainty in our calculations because they vary with excitation conditions ($X \propto \sqrt{n_{\rm H_2}}/T_{\rm b}$, where T_b is the brightness temperature) and metallicity. Several lines of evidence suggest that X^MW_CO overestimates $M_{\rm H_2}$ in starburst galaxies. For example, X_CO is a factor of ∼3 lower in M82 (Weiß et al. 2001) and a factor of ∼5 lower in a sample of local ULIRGs (Downes & Solomon 1998). To account for the different values of X_CO in normal spirals and extreme starbursts, we apply the Milky Way X_CO value to galaxies with L_IR < 10¹¹ L_☉ and the ULIRG X_CO value to galaxies with L_IR > 10¹¹ L_☉. Because this prescription is somewhat simplistic, it probably overestimates $M_{\rm H_2}$ in moderate luminosity starbursts (L_IR < 10¹¹ L_☉), such as M82, and in the centers of star-forming spirals (see Figures 4 and 5 and Section 3.3). Additionally, it likely underestimates $M_{\rm H_2}$ in ultra-luminous (L_IR ∼ 10¹² L_☉) high-redshift disk (BzK) galaxies, for which Daddi et al. (2010a) find a value of X_CO that is consistent with the Galactic value. As a result, moderate luminosity starbursts and the centers of star-forming spirals may be closer to Eddington and BzK galaxies might be further below the optically thick Eddington limit than Figure 3 would suggest.

Unfortunately, X_HCN is more uncertain than X_CO because there is no direct calibration of X_HCN from Milky Way GMCs. For normal spirals, Gao & Solomon (2004a, 2004b) find $X_{\rm HCN} \sim 10 \, M_{\odot }\,\rm (K\;km\;s^{-1}\;pc^2)^{-1}$ for virialized cloud cores with $\langle n \rangle = 3\times 10^4 \; \rm cm^{-3}$ and $T_{\rm b}=35 \; \rm K$. They caution that X_HCN could be lower in regions of massive star formation due to significantly higher brightness temperatures $T_{\rm b} \sim {\rm few} \times 10^2 \; \rm K$ (Boonman et al. 2001). In addition, Papadopoulos (2007) finds large variations in excitation conditions in LIRGs, suggesting that using a universal X_HCN conversion factor might be a poor tracer of dense gas mass. Ultra-luminous starbursts exhibit widespread intense massive star formation, so one might expect that X_HCN is lower in more luminous galaxies. For example, Graciá-Carpio et al. (2008) estimate that X_HCN should be ∼4.5 times lower for galaxies at L_FIR ∼ 10¹² L_☉ than at L_FIR ∼ 10¹¹ L_☉. We note that if X_HCN is smaller than the assumed value of $3 \,M_\odot\, \rm (K\;km\;s^{-1}\;pc^2)^{-1}$, then more galaxies will approach or exceed the Eddington limit (see Figure 2). For example, decreasing X_HCN by a factor of ∼2 brings essentially all galaxies in line with the Eddington limit for the nominal value of κ_FIR (∼5–10).