THE DISRUPTION OF GIANT MOLECULAR CLOUDS BY RADIATION PRESSURE & THE EFFICIENCY OF STAR FORMATION IN GALAXIES

A key thesis of this work is that stellar feedback is crucial for understanding the low observed values of the star formation efficiency in galaxies. In contrast, Krumholz & McKee (2005) present an explanation of the Kennicutt-Schmidt law (Equation (1)) that does not invoke an explicit form of feedback. Their argument is that turbulent motions prevent the collapse of the bulk of the gas in a GMC (or in other bound objects); only if the density is above a critical density, which depends on the Mach number of the flow, do stars actually form. The fraction of gas in a turbulent flow that lies above this critical density is small, leading to the low observed star formation efficiency per dynamical time.

We find this argument to be compelling, as far as it goes. As long as turbulence is maintained, only a small fraction of gas will collapse into stars per dynamical time. However, the assumption of a constant level of turbulence is essential to the Krumholz & McKee (2005) argument. A key, and yet unanswered, question is thus what maintains the turbulence? If the turbulence in a GMC is not maintained, then the GMC will contract, leading to an increase in the mean density and a decrease in the dynamical time. Indeed, simulations find that turbulence decays on ∼1 crossing time (Mac Low 1999; Ostriker et al. 2001), so that a continued source of energy is needed to maintain the turbulent support of the cloud. It is possible, in principle, that gravitational contraction of a GMC can sustain the turbulence, maintaining approximate virial equilibrium and a slow contraction of the cloud. We argue, however, that an independent internal source of turbulence, provided by stars, is crucial to maintaining the slow rate of star formation.

As an example, we apply this argument to Arp 220. The ISM of Arp 220 has a turbulent Mach number ${\cal M}\approx 100$. The fraction of a GMC (or any bit of molecular gas) that is sufficiently dense to be converted into stars in a free-fall time is then ≃0.013–0.05 for GMC's with a virial parameter α_vir = 0.1–1 (see Figure 3 of Krumholz & McKee 2005).⁷ According to this argument, a GMC will convert half its gas into stars in 10–40 free-fall times, reasonably consistent with Equation (1) for any α_vir. However, this assumes that the cloud does not collapse and reduce its free-fall time. In reality, if turbulence can only maintain α_vir ∼ 0.1, a cloud is likely to collapse, leading to a rapid increase in density and a decrease in the free-fall time. If the star formation rate per free-fall time remains roughly constant, then the actual star formation rate will increase rapidly with time, and most of the gas in the cloud will be turned into stars in roughly one initial (large-scale) free-fall time. Thus, the model of Krumholz & McKee (2005) for the low star formation efficiency in galaxies relies critically on maintaining sufficient levels of turbulence so that α_vir ∼ 1. On larger scales—above the characteristic GMC size—the equivalent argument is that the galactic disk must have Toomre Q ∼ 1, as we discuss below.

The remainder of this paper is organized as follows. In Section 2, we collect a number of relevant astrophysical results used in our modeling. In Section 3, we describe a simple one-dimensional model for the disruption of GMCs which includes the effects of H ii gas pressure, protostellar jets, radiation pressure, gas pressure associated with shocked stellar winds, and wind shock generated cosmic rays (many of the details of how we model these forces are given in Appendix A). In Section 4, we present the results of our numerical modeling of GMC disruption in star-forming galaxies. To explore the wide range of conditions seen in galaxies across the Kennicutt-Schmidt law, we consider models for GMCs in the Milky Way, M82, Arp 220, and the z ∼ 2 galaxy Q2346-BX 482. In Section 5, we discuss the implications of our results, the origin of turbulence in galaxies, and explain physically why radiation pressure is the only source of stellar feedback in principle capable of disrupting GMCs across the huge dynamic range in ISM conditions from normal galaxies to the densest starbursts.

Quirk (1972) showed that normal galaxies have gas disks with Q ≈ 1. Kennicutt (1989) refined this to the statement that within the star-forming part of normal galactic disks, 1/4 ≲ Q ≲ 0.6. At large radii, he found Q>1 and a lack of star formation. More recently, Leroy et al. (2008) studied star formation in detail in 23 nearby galaxies. They found that if they accounted for only the gas surface density, as done above, their disks were stable, with Q ≈ 3–4; using the total (gas plus stars) surface density resulted in Q ≈ 2 with a slight variation in radius. Unlike Kennicutt (1989), they find that star formation occurs at large radii, beyond Kennicutt's Q>1 radius, albeit at reduced rates.

These studies were restricted to normal galaxies and employed a fixed sound speed as the estimate for the random velocity. However, there is evidence that disks with turbulent velocity v_T ≫  c_s also satisfy Q ∼ 1 when v_T is used in evaluating Q (e.g., Thompson et al. 2005; and our discussion of the starbursts M82, Arp 220, and Q2346-BX 482 in Section 4). Motivated by these observations and by theoretical considerations, we will assume that all star-forming galaxies have Q ≈ 1.

We assume that galactic disks initially fragment on the disk scale height H ≈ (v_T/v_c)r. The fragments will form gravitationally bound structures with a mass given by the Toomre mass, M_T ≃ πH²Σ_g. Near the location of the Sun, the gas surface density Σ_g ≈ 2 × 10⁻³ g cm⁻² and H ∼ 300 pc, giving M_T ≈ 2 × 10⁶ M_☉.

This scenario is consistent with observations of GMCs in our Galaxy; in the Milky Way, half the gas is in molecular form in giant molecular clouds with a characteristic mass of order 5 × 10⁵ M_☉ (Solomon et al. 1987), but with a rather wide range of masses. The number of clouds N(m) of mass m is given by

$$\begin{equation} {dN\over dm}=N_0\left({m_0\over m}\right)^{-\alpha _G}, \end{equation} \tag{ 2 }$$

with an exponent α_G ∼ 1.8 (Solomon et al. 1987) or 1.6 (Williams & McKee 1997), so that most of the mass is in the largest clouds. As a cautionary note, we note that Engargiola et al. (2003) find α_G = 2.6 ± 0.3 in M33, which suggests that lower mass clouds contribute a significant fraction of the total mass. In the Milky Way, the largest GMCs have masses of order ∼3 × 10⁶ M_☉ (Solomon et al. 1987), roughly consistent with the Toomre mass. In the Milky Way, and possibly in other galaxies, molecular clouds are surrounded by atomic gas with a similar or slightly smaller mass.

The clouds appear to be somewhat centrally concentrated. We will often employ a Larson-law density distribution,

$$\begin{equation} \rho (r)\sim r^{-1}, \end{equation} \tag{ 3 }$$

where r is the distance from the center of the GMC. We also explored isothermal models ρ(r) ∼ 1/r²; we find that such clouds are slightly easier to disrupt than the less centrally concentrated Larson-law clouds in the optically thick limit.

2.2.1. But are there Molecular Clouds in ULIRGs?

Most of the gas in Milky Way GMCs is diffuse (n ≲ 3 × 10² cm⁻³), but a fraction of order 10% is in the form of dense gas clumps, with sizes around 1 pc (Lada & Lada 2003) and masses from a few tens to a few thousand solar masses. The clumps have a mass distribution similar to that for clouds (Equation (2)), with an exponent α_c ≈ 1.7 (Lada & Lada 2003).

In both the Milky Way (Elmegreen & Efremov 1997; van den Bergh & Lafontaine 1984) and in nearby galaxies (McKee & Williams 1997; Kennicutt et al. 1989; Murray & Rahman 2010;), the number of stellar clusters of mass m_* is given by

$$\begin{equation} {dN_{*\rm cl}\over dm_{*\rm cl}}=N_{*\rm cl,0}\left({m_{*\rm cl,0}\over m_{*\rm cl}}\right)^{\alpha _{\rm cl}} \end{equation} \tag{ 4 }$$

with α_cl ≈ 1.8. In other words, most stars form in massive clusters; in the Milky Way, at least, these clusters are made from gas in massive gas clumps, inside of massive GMCs.

Star clusters are observed to have sizes ranging from r_cl ≈ 0.1 pc (for M_cl ≈ 10 M_☉) to r_cl ≈ 10 pc (M_cl ≈ 10⁸ M_☉). There are hints that clusters with masses M_cl ≲ 10⁴ M_☉ have a mass–radius relation of the form

$$\begin{equation} r_{\rm cl}\approx 2\left({M_{\rm cl}\over m_0}\right)^\beta \,{\rm pc } \end{equation} \tag{ 5 }$$

with m₀ = 10⁴ M_☉ and β ≈ 0.4 (Lada & Lada 2003), but it is entirely possible that this is a selection effect. Intermediate mass clusters, those with 10⁴ M_☉ ≲ M_cl ≲ 3 × 10⁶ M_☉ have r_cl ≃ 2 pc independent of mass (β = 0), albeit with substantial scatter. This characteristic size is seen for young (≲30 Myr) and old (≳30 Myr) clusters in M51 with masses in the range 10³–10⁶ M_☉ (Scheepmaker et al. 2007), for superstar clusters in M82 with M_cl = 10⁵–4 × 10⁶ M_☉ (McCrady et al. 2003; McCrady & Graham 2007), and in globular clusters with masses ∼10⁵–10⁶ M_☉ (Harris 1996). Finally, high-mass clusters with M_cl ≳ 10⁶ M_☉ have β = 0.6 and m₀ = 10⁶ M_☉ in Equation (5) (Walcher et al. 2005; Evstigneeva et al. 2007; Barmby et al. 2007; Rejkuba et al. 2007; Murray 2009). When using the radii of stellar clusters in our GMC models, we will be guided by these observed mass–radius relations.

Recent work has revealed that the Milky Way harbors a number of very young massive star clusters, with masses ranging up to 10⁵ M_☉ (Figer et al. 1999; Brandner et al. 2008; Figer et al. 2006; Murray & Rahman 2010). For example, consider the case of G298.4-0.3 in the Carina arm. Murray & Rahman (2010) show that the free–free flux emerging from this region implies a total ionizing flux Q = 7.7 × 10⁵¹ s⁻¹ and suggest that ∼60% of this comes from a single massive cluster residing in the prominent ∼50 pc bubble revealed by Spitzer GLIMPSE images. Most of the remaining flux comes from two clusters associated with the two giant H ii regions G298.2-0.3 and G298.9-0.4; both sources appear to lie in the rim of the bubble seen in the GLIMPSE images. Grabelsky et al. (1988) find two massive GMCs in this region, numbers 24 and 26, both with M_GMC ≈ 3 × 10⁶ M_☉ and R_GMC ≈ 100 pc. Their radial velocities are 22 km  s⁻¹ and 24 km  s⁻¹, in good agreement with the range of radio recombination line, i.e., H ii region, radial velocities in this direction, which range from +16 km  s⁻¹ to 30.3 km  s⁻¹, with a mean ∼+23 km  s⁻¹.

Accordingly, our model for G298.4-0.3 consists of a GMC with R_GMC = 150 pc and M_GMC = 3 × 10⁶ M_☉. In the spirit of our simplified one-dimensional modeling, we lump all of the star clusters together into a central star cluster with L ≈ 7 × 10⁷L_☉ and initial cluster radius r_cl = 1.5 pc. Half the Galactic star formation takes place in 17 complexes, with a minimum Q = 3 × 10⁵¹ (Murray & Rahman 2010), so G298.4-0.3 is representative of star-forming clusters in the Milky Way.

The astute reader may note that the density of the GMC associated with G298.4-0.3 is lower than that of typical GMCs. The large star clusters (including G298) mentioned in the first paragraph of this section are all associated with very massive GMCs, in fact, with the most massive GMCs in the Milky Way. The surface densities of Milky Way GMCs are roughly constant (e.g., Solomon et al. 1987). It follows that the volume densities of massive GMCs are lower than the volume densities of typical (less massive) GMCs.

However, there do exist fairly massive GMCs with smaller radii, for example the GMC associated with the W49 star formation region. Simon et al. (2001) find a GMC with a radius of 22 pc and a CO luminosity derived mass of ≈5 × 10⁵ M_☉, and a viral mass ≈10⁶ M_☉. Using a mass 7.5 × 10⁵ M_☉, the mean density of the W49 GMC is n ≈ 700 cm⁻³. The ionizing flux associated with W49 is Q ≈ 3 × 10⁵¹ s⁻¹ when corrected for dust absorption (Murray & Rahman 2010); this is about half the ionizing flux of G298.4-0.3. The corresponding luminosity is L = 2.5 × 10⁴¹ erg  s⁻¹ and the corresponding stellar mass M_cl = 4 × 10⁴ M_☉. We present models of both G298.4-0.3 and W49.

The top two panels of Figure 1 show the radius of the shell surrounding the central cluster in G298.4-0.3 as a function of time (upper left panel) and the shell velocity as a function of shell radius (upper right panel). The shell starts at our putative initial radius for the cluster of ∼1.5 pc, and reaches r ∼ 80 pc at about 6.5 Myr (dashed line), at which point the star cluster luminosity has dropped by a factor of 3. The most massive stars begin to explode after about 3.6 Myr (vertical dotted line), while the last O stars explode after about 1.3 × 10⁷ yr. The solid vertical line marks the dynamical time R/v_c for the Milky Way at R = 8 kpc. Note that the dynamical time for the GMC is somewhat shorter, ∼6 Myr.

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The lower left panel of Figure 1 shows the forces acting on the shell as a function of shell radius in this model; recall that in our thin shell approximation all outward forces are contact forces, so they only act at the radius of the shell. Since the velocity only varies by a factor ∼2 for t ≳ 300, 000 yr, the plot can also be used to infer (roughly) the force acting on the shell as a function of time. Note that the radiation and H ii gas pressures drop after 3.6 Myr, when the bubble has r ∼ 80 pc, because the luminosity and ionizing flux of the cluster drop rapidly. Despite this, the bubble continues to expand at very nearly the same rate. The evolution after 3.8 Myr, i.e., radii larger than ∼80 pc, may underestimate the rate of expansion somewhat; the bubble may expand slightly more rapidly after several megayears due to energy input by SNe. We say the rate of expansion may be underestimated since the hot gas from the SNe is likely to escape the bubble as easily as the hot gas from shocked winds apparently does. In addition to SNe, other unmodeled effects also become important at late times and large radii. For example, because the inner radius of the shell exceeds the outer radius of the initial GMC, the surface density of the gas decreases to A_V ≈ 1, so that ionizing photons from the Galactic radiation field can penetrate and ionize the shell.

We halt the integration in our model when the radius of the expanding bubble exceeds the Hill radius r_Hill of the GMC, i.e., when the tidal shear from the Galaxy exceeds the self-gravity of the GMC: r_Hill ≈ (M_GMC/2M(r))^1/3a, where M(r) = v²_ca/G is the enclosed dynamical mass of the galaxy at the galactocentric radius a of the GMC. After this time, the remaining molecular gas will be dispersed (although not necessarily converted to atomic gas).

Figure 1 shows that the central cluster in G298.4-0.3 should disrupt its natal GMC. What force is responsible for this disruption? At the current radius, ∼55 pc, the radiation force and the gas pressure force are within a factor of 2 of each other, and will remain so until most of the O stars explode; the force from protostellar jets is substantially smaller. However, at early times, the jet force was as much as a factor of 2 larger than the radiation pressure force, and the gas pressure force was negligible.

Finally, the lower right panel of Figure 1 plots the momentum of the shell as a function of time. At the current radius of the bubble in G298.4-0.3, r ∼ 55 pc, the radiation has deposited about twice the momentum supplied by the H ii gas pressure. The stellar jets are not active at this time, but over the time they were active (corresponding to radii below ∼15 pc) they deposited a momentum comparable to that of the radiation pressure (at those early times). In these models, a combination of protostellar jets and radiation pressure disrupts the natal cluster, while a combination of gas and radiation pressure disrupts the GMC. The shell velocity at late times is of the order of the turbulent velocity seen in the ISM of the Galaxy (upper right panel), demonstrating that, even in the absence of supernovae, massive star formation can generate turbulent motions on large (50 pc or larger) scales comparable to those observed.

Figure 2 shows the results for W49; again, we find that the GMC will be disrupted by the stars that power the observed free–free radio emission.

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We argue below that the fraction _GMC is proportional to Σ_GMC based on a simple force balance. The GMC in W49 has a significantly higher surface density Σ_GMC ≈ 0.1 g cm⁻² compared to ≈0.01 g cm⁻² for G298.4-0.3. The observed _GMC for the two objects are different, although not by a factor of 10.

Because star formation is a complex process, it would be surprising if it were not somewhat stochastic, so that, for example, the star formation efficiency varied from GMC to GMC even when R_GMC and M_GMC are held fixed. Our estimate of the efficiency is then more in the nature of a lower bound.

We have run models in which we artificially lowered the stellar mass (and hence luminosity) in our models for G298.4-0.3 and W49. We can reduce the stellar mass by a factor of ∼6 and still see the GMC in G298.4-0.3 disrupted. Lowering the mass further results in a failed bubble, i.e., the GMC is not disrupted.

In contrast, we find that if we use a lower efficiency for W49 by more than 40%, our models predict that the GMC will not be disrupted.

We note that W49 is younger than G298.4-0.3, based on the size of the observed bubble in each object. Star formation will continue as the bubble in W49 expands, so that the final _GMC will be larger than that currently observed. However, it seems unlikely that the stellar mass will increase by more than a factor of 2, so that the final stellar mass will remain within a factor ∼2–3 of the critical mass required to disrupt the GMC.

M82 is one of the nearest (D = 3.6 Mpc; Freedman et al. 1994) starburst galaxies, with an infrared luminosity L_IR = 5.8 × 10¹⁰L_☉ (Sanders et al. 2003). The galaxy is small compared to the Milky Way, with a circular velocity v_c ≈ 110 km  s⁻¹ (Young & Scoville 1984), and a CO inferred gas mass 2 × 10⁸ M_☉ inside r = 350 pc (Weiß et al. 2001) (adjusted to our assumed distance), yielding a gas surface density Σ_g ≈ 0.1 g cm⁻² and a gas fraction f_g ≈ 0.2. The metallicity is 1.2–2.0 times solar (Smith et al. 2006).

The radius and mass of the most massive star clusters in M82 are well established; there are about 200 clusters with M>10⁴ M_☉ (Melo et al. 2005) and about ∼20 well-studied superstar clusters (M_cl>10⁵ M_☉). With 1 arcsec corresponding to a spatial scale of 17.5 pc, a number of superstar clusters are resolved by Hubble Space Telescope (HST; Smith & Gallagher 2001; McCrady et al. 2003; McCrady & Graham 2007). Typical half-light-projected radii for these massive objects are ∼008 or 1.4 pc. McCrady et al. (2003) list 20 such clusters. The total mass of the 15 clusters for which they measure viral masses is ∼1.4 × 10⁷ M_☉. Their largest cluster, "L," is a monster, with a mass of 4 × 10⁶ M_☉ and a half-light radius of 1.5 pc; more typical masses are ∼5 × 10⁵ M_☉. A rough fit of the form (Equation (4)) gives α_cl ≈ 1.9 (McCrady & Graham 2007).

The masses and radii of the GMCs in M82 are also known; the distribution is well fitted by Equation (2), with α_G ≈ 1.5 ± 0.1, and a maximum mass of M_GMC ≈ 3 × 10⁶ M_☉ (Keto et al. 2005); the radii of these 3 × 10⁶ M_☉ GMCs are R_GMC ∼ 30 pc (E. Keto 2008, private communication). The Toomre mass is ≈7 × 10⁶ M_☉. Both are comparable to the mass of the two largest superstar clusters given by McCrady & Graham (2007). Either there were more massive GMCs in M82 in the past, or _GMC ≈ 1 for the GMCs out of which these two clusters formed.

Tables 1 and 2 summarize our assumed galaxy, GMC, and star cluster properties in M82. These are all motivated by, and reasonably consistent with, the observations summarized above. Our results for the disruption of GMCs are summarized in Figure 3. The top two panels show the shell radius as a function of time and the velocity as a function of radius. The main-sequence lifetime of a 120 M_☉ star (the dashed vertical line) is comparable to the disk dynamical time (R_d/v_c, the solid vertical line). The GMC is disrupted (reaches the Hill radius) about one disk dynamical time after the cluster forms. The velocity of the shell reaches higher values than those found in our Milky Way model because of the much larger cluster masses in M82, combined with the fact that the star cluster radii in the two galaxies are nearly the same. Initially, the shell velocity is comparable to the escape velocity from the cluster. The velocity begins to slow once the swept-up mass is similar to the mass in the cluster (at r ∼ 4 pc).

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What is responsible for disrupting the GMC in our M82 model? The lower panels of Figure 3 show that the cluster gas on small scales is expelled by a combination of protostellar jets and radiation pressure, while the overlying GMC is disrupted primarily by radiation pressure. The contribution from the gas pressure of the H ii region is negligible over most of the evolution: the lower right panel shows that the H ii gas pressure contribution to the momentum is ∼10% of that contributed by radiation when the shell radius reaches the original size of the GMC.

Q2346-BX 482 is a redshift z = 2.26 disk galaxy with a disk radius of R_d ≈ 7 kpc and a gas mass, as estimated from inverting the Schmidt Law in Kennicutt (1998), of M_g ≈ 3 × 10¹⁰ M_☉ (Genzel et al. 2008). We interpret the clumps in rapidly star-forming redshift z ∼ 2 galaxies as Toomre-mass GMCs, with radii R_GMC ≈ 1 kpc, and we model the giant clump in BX 482, as one of the most extreme examples of this phenomenon. With turbulent and circular velocities of v_T ≈ 55 km  s⁻¹ and v_c ≈ 235 km  s⁻¹, respectively, we infer a disk scale height of H ≈ 1.6 kpc, and using ϕ_G = 3 a GMC size of order 500 pc. Our model uses the observed GMC radius, around 1 kpc (R. Genzel 2008, private communication).

The mass of the central star cluster is set so that the luminosity matches that observed, roughly L ≈ 4 × 10¹¹L_☉ (R. Genzel, private communication). We take the radius of the star cluster to be ∼7 pc (see Section 2.4). In reality, there will likely be many star clusters, with a distribution given by Equation (4), and with a spread in age of several to ten megayears. We assume solar metallicity, consistent with the observations. Finally, the gas mass of the clump is not known, but we assume it is roughly the Toomre mass, ∼10⁹ M_☉. This is consistent with the mass of ionized gas for the observed luminosity of the clump, at the observed size R_GMC ≈ 1 kpc, and for a stellar population less than 4 Myr old (Equations (A9) and (A11)). The clump is seen to emit Hα, with an estimated A_v ≈ 2, suggesting that the mass cannot be much larger than 10⁹ M_☉. If A_v is larger, then both the luminosity and potentially the GMC mass could be larger.

Figure 4 shows the results of our model for the giant GMC in Q2346-BX 482. The right panel shows that radiation pressure dominates the evolution of the GMC at nearly all times. The GMC is disrupted in about 15 Myr, half the disk dynamical timescale. The shell velocity, shown in the middle panel, is ∼60–80 km  s⁻¹ when the radius is ∼1 kpc, in reasonable agreement with the observed velocity dispersion of the galaxy.

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Our conclusion that radiation pressure is disrupting the massive clump in BX482 is directly supported by observations, independent of the specific assumptions in our model: the self-gravity of the clump is

$$\begin{equation} F_{\rm grav}={{\it GMM}\over 2r^2}=3\times 10^{34} \left({M\over 10^9\, M_\odot }\right)^2 \left({1\,{\rm kpc }\over R_{\rm GMC}}\right)^2\,{\rm dynes }, \end{equation} \tag{ 10 }$$

where we have scaled M to the Toomre mass. This can be compared directly with the radiation pressure force,

$$\begin{equation} F_{\rm rad}={L\over c}=5\times 10^{34} \left({L\over 4\times 10^{11}L_\odot }\right)\,{\rm dynes }. \end{equation} \tag{ 11 }$$

The clump should thus be expanding.

Arp 220, at ∼77 Mpc, is the prototypical ULIRG in the local universe. The gas mass of each of the r_d ≈ 100 pc star-forming disks in Arp 220 is 10⁹ M_☉, the circular velocity v_c ≈ 300 km  s⁻¹, v_T ≈ 80 km  s⁻¹, and the disk scale height H = (v_T/v_c)r_d ≈ 23 pc (Downes & Solomon 1998; Sakamoto et al. 1999). The mean surface density is Σ_g ≈ 7 g cm⁻², about 100 times larger than that of M82 and more than a thousand times higher than in the Milky Way. We estimate that Arp 220 has GMC masses of ≈5 × 10⁷ M_☉, R_GMC ≈ 5 pc and a turbulent velocity in each GMC of ∼170 km  s⁻¹, about twice that measured for the disk as a whole. Although the metallicity is uncertain, we take a fiducial metallicity of 3 times solar; this increase in metallicity is important because it increases the dust optical depth and hence the overall importance of radiation pressure.

Wilson et al. (2006) found ∼40 young superstar clusters in and around Arp 220; they estimate masses for about a dozen, with a number having M_cl ∼ (2–4) × 10⁶ M_☉; the largest has M_cl ≈ 10⁷ M_☉. Given the huge extinction toward the twin disks, this is likely to be a rather conservative lower limit on the mass of the most massive cluster in the system. The clusters are unresolved in the HST images (which have a resolution of order 15 pc at the distance of Arp 220), except possibly their brightest cluster, with a half-light radius r_cl ≈ 20 pc. Wilson et al. (2006) do not obtain either a velocity dispersion or a half-light radius for their clusters, so they cannot calculate a dynamical mass. Rather, they use a Salpeter (1955) IMF and Bruzual & Charlot (1993) stellar synthesis models combined with their photometry.

We find that for M_cl = 1.4 × 10⁷ M_☉ (L = 3 × 10¹⁰L_☉), even a Toomre-mass GMC (4 × 10⁷ M_☉) in Arp 220 would be disrupted (see Figure 5). The disruption of the GMC occurs on the dynamical time of the disk, well before any supernovae explode in the GMC's central star clusters.

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Our estimated GMC mass in Arp 220 (5 × 10⁷ M_☉) is a factor of 10 higher than the largest GMC mass seen in M82; this is a result of the much larger surface density in Arp 220 compared to M82. In contrast, the star cluster masses found so far in Arp 220 are only a factor 2–3 times larger than the masses of the largest clusters observed in M82, the latter being around (2–4) × 10⁶ M_☉ (McCrady & Graham 2007). Given that our predicted GMC star formation efficiency is not that different in the two cases (Table 2), we expect that more massive clusters are lurking in Arp 220.

The right panel of Figure 5 shows the forces acting on the shell of swept-up mass in our model of Arp 220. As in Figures 1 and 3, the force due to protostellar jets is initially similar to that due to radiation pressure. This situation lasts only while the shell accelerates from the initial clump radius of about 4 pc, until the shell reaches a little less than 6 pc. For the rest of the evolution radiation pressure provides the only significant outward force. The outward force supplied by ionized gas is completely negligible; the short dash line in the figure is the gas pressure multiplied by 100 (Figure 5, lower left panel). Both the hot gas and cosmic rays produced by shocked stellar winds are dynamically unimportant, even though we have assumed complete trapping of the shocked stellar wind material (an assumption that fails in the Milky Way; Harper-Clark & Murray 2009).

The middle panel of Figure 5 shows that radiation pressure will stir the ISM of Arp 220 to ∼50 km  s⁻¹, somewhat less than the escape velocity from the cluster and similar to the velocity dispersion seen in CO observations.

Using four examples that cover conditions ranging from Milky Way-like spirals to the densest starbursts (see Tables 1 and 2), we have explored the physical processes that can disrupt GMCs, one of the basic building blocks of star formation. We find that radiation pressure produced by the absorption and scattering of starlight by dust grains can contribute significantly to disrupting GMCs in nearly all types of galaxies. By contrast, protostellar jets are important only at early times during GMC disruption, while the thermal gas pressure in H ii regions is important for GMC dispersal in spiral galaxies like the Milky Way, but not in more luminous starbursts. For the Milky Way and M82, where the observations are particularly detailed, our results demonstrate that observed massive star clusters have precisely the luminosities and structural properties required to disrupt Toomre-mass GMCs via radiation pressure.

The results presented here support Thompson et al.'s (2005) model of radiation pressure supported star-forming galaxies. In that paper, we focused on the large-scale properties of star formation in galaxies and the fueling of massive black holes in galactic nuclei. Here we have extended that model by taking into account the fact that star formation is not smooth and homogeneous; rather, most stars form in massive star clusters inside massive GMCs (Section 2). Our conclusion that Toomre-mass GMCs can be disrupted by radiation pressure is qualitatively and quantitatively similar to Thompson et al.'s conclusion that radiation pressure can regulate star formation in galactic disks to have Toomre's Q ≈ 1.

It is useful to consider simple scaling arguments in order to understand why, over the range of surface densities probed by the observed Schmidt law (10⁻³  g  cm^-2 ≲ Σ_g ≲ 10  g cm⁻²), radiation pressure is the most viable mechanism for GMC disruption. To rough approximation, the self-gravity of the gas in a GMC is

$$\begin{equation} F_{\rm sh}=\phi _G^2{GM_TM_T\over 2H^2}\propto \frac{M_T^2}{H^2}\propto M_T\Sigma _g, \end{equation} \tag{ 12 }$$

which varies by a more than a factor of ∼10⁶ from normal galaxies to starbursts.

We can compare this force directly to the radiation pressure force. In the optically thin limit⁸

$$\begin{equation} F_{\rm rad}=\frac{L}{c}=\frac{\Psi M_*}{c}, \end{equation} \tag{ 13 }$$

where Ψ is the light-to-mass ratio in cgs units. Thus,

$$\begin{eqnarray} \frac{F_{\rm sh}}{F_{\rm rad}}&\sim &\frac{\phi _G^2Gc}{2\Psi }\frac{M_T}{H^2} \left(\frac{M_T}{M_*}\right) \nonumber \\ &\sim &1\left(\frac{3000\,\,{\rm cgs}}{\Psi }\right) \left(\frac{0.02}{\epsilon _{\rm GMC}}\right)\left(\frac{\Sigma _g}{2\times 10^{-3}\,{\rm g\,\,cm^{-2}}}\right),\nonumber \\ \end{eqnarray} \tag{ 14 }$$

where we have scaled to values appropriate to the Galaxy. We see that if _GMC increases with gas surface density, as our calculations indicate (e.g., Table 2), then radiation pressure provides a mechanism for GMC disruption in both spiral galaxies like the Milky Way and in somewhat denser and more luminous galaxies.

Another way to express this is to note that the minimum efficiency

$$\begin{equation} \epsilon _{\rm GMC}={\pi Gc\over 2(L/M_*)}\Sigma _{\rm GMC}. \end{equation} \tag{ 15 }$$

For galaxies with sufficiently large surface densities, Σ_g ≳ 0.5 g cm⁻², GMCs will be opaque to the emission by dust in the far-infrared (FIR). This increases the radiation pressure force so that (in the thin shell approximation used here)

$$\begin{equation} F_{\rm rad}=\tau _{\rm rad}\frac{L}{c}\propto M_*\Sigma _g, \end{equation} \tag{ 16 }$$

where τ_rad = κ_FIRΣ_sh/2, κ_FIR is the Rosseland mean opacity of the GMC in the FIR, and Σ_sh is the surface density of material in the shell. Comparing the optically thick radiation pressure force with that due to gravity, we find that

$$\begin{eqnarray} \frac{F_{\rm sh}}{F_{\rm rad}}&\sim &\frac{4 \pi Gc}{\kappa _{\rm FIR}\Psi } \left(\frac{M_T}{M_*}\right)\nonumber \\ &\sim & 1 \left(\frac{3000\,\,{\rm cgs}}{\Psi }\right) \left(\frac{30\,\,{\rm cm^2\,\,g^{-1}}}{\kappa _{\rm FIR}}\right) \left(\frac{0.25}{\epsilon _{\rm GMC}}\right), \end{eqnarray} \tag{ 17 }$$

where we have scaled to a relatively high value for the Rosseland-mean dust opacity (see below). Note that the ratio in Equation (17) does not explicitly depend on stellar/gas mass, because both F_sh and F_rad are ∝M². Using the scalings in Appendix A, it is easy to see that no other stellar feedback process has this property. Indeed, most of the previously suggested support mechanisms scale as F ∝ M^β_* with β ⩽ 1, namely, H ii gas pressure, stellar winds, and pressures associated with shocked stellar winds. For this reason, although many feedback mechanisms are competitive with gravity in normal spirals, the self-gravity of the disk quickly overwhelms the forces due to stellar feedback in starburst galaxies. In contrast to these other feedback processes, radiation pressure in optically thick gas scales as F_rad ∝ M_*M_g, so that it is at least, in principle, possible that radiation pressure can disrupt GMCs even in the densest, most gas-rich environments (e.g., ULIRGs and z ∼ 2 galaxies). Radiation pressure is, to our knowledge, the only stellar feedback process that has this property.

Equation (17) shows that the efficiency of star formation in a GMC at very high densities is sensitive to the metallicity and dust composition, which influence the FIR opacity κ_FIR, and to the stellar IMF, which determines the light-to-mass ratio of the stellar population Ψ. In very dense environments, there are some reasons for suspecting that the IMF may be top heavy (e.g., Murray 2009), as appears to be the case in regions of massive star formation more generally (Krumholz et al. 2007). If this is indeed the case, it would increase Ψ and thus decrease the _GMC required for GMC disruption.⁹ For a relatively normal IMF, however, the star formation efficiency in GMCs must be appreciable at high densities, with _GMC ∼ 0.25 or perhaps even larger.

A key part of our argument for the importance of radiation pressure is the fact that star clusters have very high surface densities, which can trap the FIR radiation produced by dust, thus enhancing the radiation force by a factor of ∼τ_rad in the optically thick limit (Equation (17)). Figure 6 shows the Rosseland mean dust optical depth (τ_rad) through the shell as a function of shell radius r_sh(t) in the M82 and Arp 220 models (compare with Figures 3 and 5). In the M82 model, τ_rad>1 for r ≲ 20 pc, while for Arp 220 τ_rad ≫ 1 at all radii. In our Milky Way models, by contrast, we find that the GMC is essentially always in the optically thin limit, i.e., opaque to the UV but not to the FIR. Figure 6 also shows the effective optical depth in both the M82 and Arp 220 models (dashed lines), which we define as

$$\begin{equation} \tau _{\rm eff}\equiv {P_{\rm rad}\over (Lt/c)}. \end{equation} \tag{ 18 }$$

The effective optical depth quantifies the enhanced coupling of photons emitted by a cluster in the center of a GMC compared to photons originating at the mid-plane of a uniform density disk. In M82, the effective optical depth at the end of the bubble evolution is about equal to the mean optical depth at the mid-plane. By contrast, in our Arp 220 models, the momentum deposited per photon in the bubble shell is a factor of ∼3 larger than would deposited by a photon traversing a uniform density disk. This shows that the effect of the radiation pressure may be 3 times larger than calculated using the mean mid-plane optical depth, as was done in Thompson et al. (2005). This effect increases the importance of radiation pressure in the densest galaxies, where it is needed most, effectively decreasing the normalization in Equation (17). Note also that because most of the star formation—and thus radiation—occurs in a few massive star clusters in the most massive GMCs (Section  2), there is very little "cancellation" due to different radiation sources driving the gas in different directions (as was suggested by Socrates et al. 2008); the distribution of radiation sources in galaxies is not well approximated as infinite and homogeneous (Section  2).

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Given the turbulent and clumpy nature of the ISM in GMCs, one may question whether or not the photon coupling efficiency is as large as τ_rad or τ_eff, since these expressions assume uniform shells of matter. We have, after all, used the argument that GMCs are porous to argue that hot gas from shocked stellar winds escapes rapidly from GMCs in the Galaxy. The optical depth τ_rad is measured from the center of the protocluster outward and is proportional to the column density of overlying gas. The optical depth through a GMC has been observationally measured by a number of authors (e.g., Wong et al. 2008; Goodman et al. 2009) and is consistent with a log-normal distribution. Numerical simulations also find log-normal surface density distributions (Ostriker et al. 2001).¹⁰ Using the convention that $x = ln (N_{\rm H}/\bar{N}_{\rm H})$, Goodman et al. (2009) find a dispersion in the logarithm of the column density of 0.25 < σ < 0.53, and a range of mean logarithmic column densities −0.14 < μ < −0.02, where μ ≡ 〈x〉. The relation between μ and σ and their numerical values agree well with the high Mach number turbulence simulations of Ostriker et al. (2001).

In Appendix B, we estimate the reduction in the effective FIR optical depth, using an expression given in McKee & Tan (2008), for a log-normal distribution of column density. We find that the correction is of order 10% (see Figure 7). We apply this correction in our models for M82 and Arp 220; in the Milky Way and Q2346-BX 482, the optical depth to the FIR is so small that the correction is negligible.

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This correction is based on observations that do not probe, and simulations that do not include, some of the physics relevant to our models (e.g., the Rayleigh–Taylor and photon-bubble instabilities; Blaes & Socrates 2003; Turner et al. 2007; Thompson 2008; Krumholz et al. 2009). Nonetheless, these results provide some support for our argument that the dense ISM will be opaque to FIR radiation, thus significantly enhancing the radiation pressure force in the optically thick limit. In the absence of very large changes in the IMF (Ψ) or the average dust-to-gas ratio (κ_FIR), this enhancement is required for radiation pressure to disrupt GMCs in dense starbursts. For Milky Way-like galaxies, our conclusions are not sensitive to the radiation force in the optically thick limit, since the GMCs/star clusters are not optically thick except on very small scales.

McKee & Tan (2008) also give an expression for the reduction in optical depth when there is a literal hole in the shell wall, i.e., no absorption at all, so that photons stream out freely. If this were to occur in Arp 220 or M82, even for holes that pierce as little as 2% of the shell, our calculations show that essentially all the gas in the natal GMC must be in stars for radiation pressure to disrupt (what little is left of) the GMC.

Is it possible that the bubble walls in very dense regions do not have even such small holes? One observational hint comes from our own Galaxy. Observations of mature H ii regions clearly show breaches in bubble walls. Examples include the very extended free–free emission associated with giant H ii regions, and similarly extended 8 μm emission associated with massive star-forming regions (e.g., Murray & Rahman 2010).

Ultracompact H ii regions, on the other hand, furnish an example of almost completely confined UV and optical radiation fields. This is clear evidence against holes with a cover fraction of more than a few percent, since many of the central stars are luminous enough to power detectable free–free emission if 1% of UV light escapes. De Pree et al. (2000) give examples of objects in W49 with ionizing fluxes Q ∼ 2 × 10⁴⁹ s⁻¹; if 1% of this flux were to free stream out of the confined region, it would produce a free–free luminosity L_ν ∼ 1.5 × 10²¹ erg s⁻¹ Hz⁻¹; at the distance of W49 (11 kpc) this is 10 mJy, easily detected in their maps. The gas densities in their objects reach n ∼ 10⁶ cm⁻³, which makes them similar to our estimates of GMC densities in Arp 220. The surface densities of ultracompact H ii regions are also high, but two orders of magnitude or more below our estimates for Arp 220 GMCs, suggesting that radiation trapping in the latter case may be yet more complete. Clearly, more work is needed to address this question with any certainty.

In all of our calculations, the shell velocity at r ≈ H_d is similar to the turbulent velocity required to maintain Toomre's Q ∼ 1 in the ambient disk, thus staving off self-gravity on the largest scales. This is not a coincidence: disrupting the GMC—which we have explicitly taken to be the Toomre mass—requires a force comparable to that needed to stir up the gas in the disk proper to Q ∼ 1. We have interpreted the shell velocity at large radii as a turbulent velocity because the disruption of the shell by shear, tidal forces, and instabilities (e.g., Rayleigh–Taylor) will inevitably convert this relatively ordered kinetic energy into random fluid motions. More sophisticated models are clearly called for, but it is also clear that such models must include the effects of radiation pressure, particularly in very optically thick systems like Arp 220, but also in models of Milky Way-like galaxies.

A key question we are unable to fully address is how the efficiency of star formation in a GMC (_GMC) relates to the global star formation rate in the galaxy as a whole, as encapsulated by, e.g., the parameter η in the Kennicutt law (Equation (1)) or the observed Schmidt law. We briefly discuss some of the relevant issues here, but leave a more detailed analysis of this problem to future work.

In our models of GMCs in Milky Way-like spirals, the time for a star cluster to disrupt the natal GMC is 1/3 to 1/2 the local dynamical time. When we add to this the time for the GMC to contract to its present size, the total time involved will be ∼2 longer than the local dynamical time of the disk. If the subsequent supernovae do not greatly prolong the process of reincorporating the bulk of the gas back into the disk, which we suspect is correct, the time averaged star formation rate per unit area will be

$$\begin{equation} \dot{\Sigma }_* =\epsilon _{\rm GMC}\Sigma /(2t_{\rm dyn}) = \eta \Sigma /t_{\rm dyn}, \end{equation} \tag{ 19 }$$

with η given roughly by η ≈ _GMC/2 ≈ 0.02, in good agreement with the observations.

The number of clusters capable of disrupting a Toomre-mass GMC, in our feedback model, should be proportional to the number of Toomre masses, ∼(R/H)² ∼ 700. The corresponding number of giant H ii regions observable at any time is (R/H)²t_MS/(2t_dyn) ≈ 30, where t_MS ≈ 3.6 × 10⁶ years is the lifetime of an early O star, which is also the time over which a star cluster will emit a large luminosity of ionizing photons. This estimate is consistent with the fact that half of the free–free emission in the Galaxy is produced by about 17 giant H ii regions (Murray & Rahman 2010).

The GMC star formation efficiency is _GMC = 0.38 for our fiducial Arp 220 model (Table 2); more generally, it is _GMC ∼ 0.25 in the optically thick limit for typical IMFs and dust-to-gas ratios (Equation (17)). This suggests that η is also ≈0.25, higher than implied by the Kennicutt relation (Equation (1)), although reasonably consistent with the conclusions of Bouché et al. (2007). However, this estimate assumes that after the dispersal of the GMC, gas falls back into the disk and forms a new GMC in a single dynamical time; as the left panel of Figure 5 shows, the timescale for cluster disruption is in fact ∼1/10 of the timescale for the cluster luminosity to drop significantly. Thus, the star cluster luminosity can drive motions in the remaining gas ∼v_T ∼ (H/R)v_c, sufficient for hydrostatic equilibrium over many (∼10) dynamical timescales. Because the photon diffusion timescale is rapid compared to the dynamical timescale, the medium cannot be supported stably (Thompson et al. 2005; Thompson 2008). Hydrostatic balance will only be maintained in a statistical sense within a volume ∼4H³ of the star cluster. After the cluster's luminosity decreases on a timescale ∼t_MS, the gas will recollapse to form a new GMC and the process will repeat until gas exhaustion. We suspect that this may reduce η by a factor of ∼t_dyn/t_MS, but clearly more work is needed.

A.1.1. Gravity

We assume that the gravitational force acting on the bubble shell consists of three components, F_grav = F_stars + F_shell + F_disk. The central star cluster exerts a force on the bubble shell given by

$$\begin{equation} F_{\rm stars}=-{GM_*M_{\rm sh}\over r^2}, \end{equation} \tag{ A1 }$$

the shell self-gravity is

$$\begin{equation} F_{\rm shell}=-{GM_{\rm sh}^2\over 2r^2}, \end{equation} \tag{ A2 }$$

while the mass in the galactic disk exerts a force

$$\begin{equation} F_{\rm disk}=-{v_c^2M_{\rm sh}\over R_d}{r\over R_d}. \end{equation} \tag{ A3 }$$

The last force is that exerted on the part of the shell rising vertically away from the disk; gas in the plane of the disk will not feel this force, but we ignore this complication, just as we ignore Coriolis forces.

A.1.2. Turbulent Pressure

The pressure in the ISM and in the GMC provides an inward force on the shell. We refer to these pressures as turbulent pressure, although there may be other components such as magnetic or cosmic-ray pressure. We approximate them as

$$\begin{equation} F_{\rm turb}=-4\pi r^2 P_{\rm ISM} -{GM_{\rm GMC}^2\over R_{\rm GMC}^2} \left[1-\left({r\over R_{\rm GMC}}\right)^2\right]. \end{equation} \tag{ A4 }$$

Here we have assumed that the GMC has a Larson-like mass distribution. A similar expression holds for clouds with ρ(r) ∼ 1/r². The pressure of the ISM is given by

$$\begin{equation} P_{\rm ISM}\approx \pi G \Sigma _g^2. \end{equation} \tag{ A5 }$$

A.2.1. Gas Pressure Forces

The large luminosity Q (number per second) of ionizing photons produced by the massive stars in a young star cluster will photoionize and heat gas in the vicinity of the cluster, raising the gas pressure above that in the neutral gas. This hot gas will exert an outward force on the bubble wall, with a magnitude

$$\begin{equation} F_{\rm H\,{\mathsc {ii}}}=4\pi r^2 P_{\rm H\,{\mathsc {ii}}} \end{equation} \tag{ A6 }$$

The gas pressure is taken to be

$$\begin{equation} P_{\rm H\,{\mathsc {ii}}}=nkT, \end{equation} \tag{ A7 }$$

where T = 8000 K (Gardner et al. 1970) and

$$\begin{equation} n=\sqrt{Q\over \alpha _{\rm rec} V}. \end{equation} \tag{ A8 }$$

The volume V = 4πr³/3 and the recombination coefficient α_rec ≈ 4 × 10⁻¹³. In models for clusters in the Milky Way, the shell velocity is subsonic, so the gas pressure is roughly constant throughout the bubble. If there is any hot gas or cosmic rays due to, for example, shocked stellar winds, they will also be roughly isobaric, and in pressure equilibrium with the H ii gas. If the hot gas or cosmic-ray pressure is in excess of the estimate given here, the H ii gas will be confined to a fraction of the bubble volume. However, observations of Carina and massive Milky Way and LMC clusters suggest that the pressure is well approximated by (A7) and (A8) (Harper-Clark & Murray 2009).

For large enough stellar clusters, those with at least one 35 M_☉ star, Q is proportional to L,

$$\begin{equation} {L\over Q}\equiv \xi \approx 8\times 10^{-11}\,{\rm erg }, \end{equation} \tag{ A9 }$$

or about 3.6 Rydbergs (1Ryd ≈ 2.2 × 10⁻¹¹ erg is the energy required to ionize hydrogen). The H ii gas pressure inside the bubble is then

$$\begin{eqnarray} P_{\rm H\,\mathsc {ii}}=&&5.0\times 10^{-10}\nonumber \left({L\over 10^{41}}\right)^{1/2}\left({5\,{\rm pc }\over r}\right)^{3/2}\nonumber\\ &&\times\left({T\over 8000\,{\rm K}}\right)\,{\rm dynes }\,{\rm cm }^{-2}, \end{eqnarray} \tag{ A10 }$$

assuming a unit filling factor for the H ii gas, i.e., ignoring the hot shocked winds.

It is helpful to have an estimate for n in terms of the luminosity. The number density n(L, r) is given by

$$\begin{equation} n(L,r)=\sqrt{3L\over 4\pi \alpha \xi r^3}. \end{equation} \tag{ A11 }$$

If the bubble is breached, so that H ii gas leaks out, the pressure in the bubble will be lower than the estimate given here, but the force experienced by the bubble wall will actually be larger. The reason is that fewer ionizing photons are absorbed in the bubble cavity, leaving more to heat the gas on the interior of the bubble wall. The heated gas escapes away from the wall into a partial vacuum, exerting a force

$$\begin{equation} F=\dot{M} c_{\rm H\,{\mathsc {ii}}} \end{equation} \tag{ A12 }$$

on the wall. The mass loss rate is given by $4\pi r^2 m_p n c_{\rm H\,\mathsc{ii}}$. In our numerical work, we have assumed that the H ii gas leaks out of the bubble, since the rapidly expanding ionized gas at the ionization front will exert a larger force, and we want an upper limit on the efficacy of H ii gas pressure.

In models for ULIRGs like Arp 220, the bubble expansion velocity is larger than the sound speed of 8000 K gas, so the H ii pressure may be twice that in the subsonic case even if no gas escapes from the bubble. However, the pressure associated with H ii gas is negligible in ULIRGs, as we show in the main text.

A.2.2. Forces Associated With Shocked Stellar Winds

A.2.3. The Force Due to Hot, Shocked Stellar Wind Gas: X-ray Constraints

O stars emit high velocity massive winds; in clusters these winds are seen to shock and emit diffuse X-rays at ∼ KeV energies (Seward et al. 1979; Oey 1996; Smith et al. 2005). If the stellar winds are confined to the bubble interior, the associated pressure can be far larger than the ram pressure of the wind (Castor et al. 1975). The force is given by

$$\begin{equation} F_h = 4\pi r^2P_h, \end{equation} \tag{ A13 }$$

where P_h = 2E_h/(3V). The energy equation for the hot gas is

$$\begin{equation} {dE_h\over dt} = L_w - 4\pi r^2 P_h v_{\rm sh} - \Lambda n_h^2 V; \end{equation} \tag{ A14 }$$

recall that V is the volume of a bubble of radius r, and Λ is the cooling function. The wind luminosity L_w = (1/2)η(v_∞/c)L_bol, (η ≈ 0.5 is the fraction of stellar luminosity scattered in the wind) and L_bol is the bolometric luminosity of the cluster.

We follow Castor et al. (1975) to find the number density n_h of hot gas, i.e., we assume that heat conduction (at the Spitzer rate C_SpT^5/2_h) drives a mass flow into the bubble interior (possibly supplemented from cold gas clouds in the interior of the bubble) at a rate

$$\begin{equation} {dM_h\over dt}= {16\pi m_P C_{\rm Sp} T_h^{5/2}r\over 25 k}. \end{equation} \tag{ A15 }$$

Given E_h and the radius of the bubble we find the hot gas pressure; from Equation (A15), we find n_h, and hence the temperature. With n_h and T_h, we can find the cooling rate (the third term on the right-hand side of the energy Equation (A14).

The solar-mass stars in clusters also emit X-rays; in low resolution (non-Chandra) observations, these stars appear as diffuse emission, so care must be taken to distinguish the two. Assuming that one can extract the stellar emission, the diffuse X-ray flux can be used to constrain the pressure of any hot gas component. Harper-Clark & Murray (2009) use this to show that for clusters in the Milky Way and the LMC, the X-ray gas is in approximate pressure equilibrium with the H ii gas, and that the H ii gas has a filling factor near unity (greater than ∼0.1). The implication is that the shocked winds either radiate away the bulk of their energy, or else leak out of the bubble wall. In either case, they do not play any role in the dynamics of the bubble, a conclusion also reached by McKee et al. (1984).

Harper-Clark & Murray (2009) show that observations of ultracompact H ii regions (Rauw et al. 2002; Tsujimoto et al. 2006) reveal a similar story.

If the bubbles that form around star clusters in ULIRGs are also leaky, the shocked wind pressure would be negligible, as in the Milky Way. However, even if we assume that the winds are perfectly trapped, the high ambient pressures in ULIRGs ensure that the shocked gas is not dynamically important.

To see this, note that we expect a total number of young (ionizing) clusters in a ULIRG to be

$$\begin{equation} N_{\rm cl}=\left({R\over H}\right)^2 \approx 20. \end{equation} \tag{ A16 }$$

If there are ∼20 clusters powering Arp 220, each must have L_cl ≈ 5 × 10¹⁰L_☉, and a mass M_cl ≈ 2 × 10⁷ M_☉ assuming a normal (Muench et al. 2002) IMF; this cluster luminosity is also that needed to disrupt a GMC in Arp 220, as we showed in Section 4.4. The wind luminosity of a single such cluster is L_w ≈ 7 × 10⁴¹ erg  s⁻¹. We note that L_cl ≈ 5 × 10¹⁰ L_☉ is about 3–4 times the luminosities of known young star clusters in Arp 220 (Wilson et al. 2006).

From Figure 5, the bubble disrupts the GMC (R_GMC ≈ 10 pc) after a time t ∼ 5 × 10¹² s. Assuming no losses, the wind energy accumulated in the bubble is E_h = 2.5 × 10⁵⁴ erg. The pressure is 2E_h/3V = 10⁻⁵ dynes cm⁻², about equal to the ambient pressure in the disk but well below the radiation pressure, τF/c ≈ 10⁻⁴ dynes cm⁻².

The force we have estimated from the hot gas is 1.6 × 10³⁵ dynes; in the right panel of Figure 5 the force from the hot gas is a factor of ∼5 below this estimate. The reason is that the hot gas suffers radiative losses, so that we have overestimated E_h. We note that in our numerical work we have assumed that the hot gas fills the volume of the bubble; if it does not, the cooling rate will be higher than we have calculated.

In addition to producing hot gas, wind shocks will generate cosmic rays. Perhaps a third of the wind energy may be expected to be deposited into cosmic rays. If, as observed in the Milky Way, the hot gas is advected out of the shell, it will probably take the cosmic rays with it, so that both the hot gas and the cosmic rays are not dynamically relevant.

We note that the pressure due to cosmic rays, if perfectly confined, would be 1/6 that of similarly confined shocked winds. As noted in the introduction, the dynamics of bubbles in the Milky Way and the LMC are not consistent with the high pressures associated with trapped hot gas; it follows that cosmic rays are probably also not confined to the interiors of such bubbles.

If the shocked winds in ULIRGs also flow through the swept-up shell, cosmic rays are unlikely to be dynamically important. Even if the cosmic rays are confined, the cosmic-ray pressure is still negligible. The cluster bolometric luminosity is 2 × 10⁴⁴ erg  s⁻¹, so the wind luminosity is 5 × 10⁴¹ erg  s⁻¹. Assuming 1/3 of this is deposited in cosmic rays, the cosmic-ray luminosity is L_cr ≈ 1.6 × 10⁴¹ erg  s⁻¹. The cosmic-ray pressure when the bubble radius is 10 pc is

$$\begin{equation} P_{\rm cr}={L_{\rm cr}t_{\rm dyn}\over V}\approx 7\times 10^{-6} \left({\,{\rm pc }\over R_{\rm GMC}}\right)^3 \,{\rm dynes }\,{\rm cm }^{-2}. \end{equation} \tag{ A17 }$$

This is about a half of the mean dynamical pressure in the galaxy, but only several percent of the dynamical pressure in the bubble. This is still likely to be an overestimate of the cosmic-ray pressure, as cosmic rays will quickly diffuse out of the bubble. The cosmic-ray mean free path in the Milky Way is 0.1 pc < λ_cr < 1 pc (Schlickeiser 2002); scaling to the lower value, the time for a cosmic ray to diffuse out of a GMC in Arp 220 is

$$\begin{equation} t_{\rm cr}={R_{\rm GMC}^2\over D_{\rm cr}}\approx 2\times 10^{11} \left({0.1\,{\rm pc }\over \lambda _{\rm cr}}\right)\,{\rm s }, \end{equation} \tag{ A18 }$$

where D_cr = λ_crc/3 is the cosmic-ray diffusion coefficient. This diffusion time is ∼10–20 times shorter than the bubble dynamical time, so even if the mean free path for cosmic rays in Arp 220 is substantially smaller than in the Milky Way, cosmic rays will diffuse out of the bubbles in the ULIRG. Figure 5 shows that the cosmic-ray pressure is a little less than 1% of the radiation pressure; the lower cosmic-ray pressure is the result of taking P_cr = L_crt_cr/V, which accounts for diffusive losses, instead of Equation (A17). Note that we have optimistically ignored cosmic-ray losses due to inelastic proton–proton collisions, which cool the cosmic-ray population with energies ≳GeV on a timescale t_pp ≈ 5 × 10³(10⁴cm^-3/n) yr, where we have scaled to the average volumetric gas density of Arp 220.

A.3.1. Protostellar Jets

While stars are actively accreting in the protocluster, we assume that they will emit high velocity outflows or jets. Once the cluster disrupts, the accretion halts, and the jets turn off. We assume that this happens over a time given by ϕ_ff times t_ff of the cluster; in addition, we allow for an extra factor of 2 to account for the fact that the accretion disks will not vanish once the cluster gas is dissipated, but instead will accrete onto the star in a disk viscous time:

$$\begin{equation} t_{\rm jet}=2\phi _{\rm ff}t_{\rm ff}. \end{equation} \tag{ A19 }$$

We take ϕ_ff = 1. The jets deposit momentum into the surrounding gas at a rate

$$\begin{equation} F_{\rm jet}=\epsilon _{\rm jet}\dot{M}_{\rm acc} v_{\rm jet}\exp ^{-t/t_{\rm jet}}. \end{equation} \tag{ A20 }$$

The jets expel mass at a fraction _jet ≈ 0.1–0.3 of the mass accretion rate (Matzner & McKee 2000), at a velocity v_jet that depends on the mass of the star. We estimate v_jet by calculating the escape velocity from the surface of a star of mass m, using the radius of a star of that mass found from the main-sequence radius as given by the Padova models; this is a slight over estimate of the escape velocity, since accreting stars are larger than main-sequence stars, but we compensate for this by using the escape velocity rather than the (larger) terminal velocity of a wind or jet. We then average over the IMF to yield v_jet. For a Muench et al. IMF, we find v_jet ≈ 3 × 10⁷ cm s⁻¹, but for top heavy IMFs this can increase to 10⁸ cm s⁻¹. We varied _jet over the range 0.01–0.3 giving an effective velocity in the range 5–90 km s⁻¹, bracketing that found by, e.g., Matzner & McKee (2000) and Nakamura & Li (2007). The results did not depend strongly on this parameter, but because the clusters we consider are so dense, we expect a substantial amount of momentum cancellation (from jets colliding nearly head on) we used low values for _jet, typically 0.03 in the massive cluster models, i.e., not the Milky Way model, presented here.

A.3.2. Radiation Force

The radiation force is given by

$$\begin{equation} F_{\rm rad}=(1+\tau _{\rm rad}){L\over c}, \end{equation} \tag{ A21 }$$

where τ_rad is the Rosseland mean opacity through the shell. The gas in the shell has a temperature that is of order 100 K, so for most clusters in the Milky Way τ_rad < 1. We calculate the optical depth as follows: we assume that the shell has a thickness 1/10 of its radius, and divide it into 100 pieces. We assume that the density is constant through the shell. We calculate T_eff from L and the radius r of the shell. Given T and ρ, we use the opacity tables of Semenov et al. (2003) to find the Rosseland mean opacity in the outermost slice of the shell. Subsequently, we find the optical depth, then integrate the radiative transfer equations inward through the shell. The factor of unity in Equation (A21) accounts for the fact that the temperature of the radiation field is of order 30,000 K at the inside edge of the shell, before the photons have encountered any dust grains; upon striking a dust grain, the photons provide an impulse per unit time given by L/c.