RADIATION PRESSURE FROM MASSIVE STAR CLUSTERS AS A LAUNCHING MECHANISM FOR SUPER-GALACTIC WINDS

Star clusters form in dense cores inside GMCs. As shown by Murray et al. (2010), radiation from the young stars will transfer momentum to the surrounding gas through dust grains. The energy from the absorbed photons is then re-emitted isotropically in the infrared; in most galaxies these IR photons simply escape the galaxy.⁶ The bolometric luminosity of a young cluster is carried predominantly by ultraviolet radiation. The dust opacity in the ultraviolet is typically κ ∼ 1000 cm² g⁻¹, so these ultraviolet photons see an absorption optical depth of order unity at a column of N_H ∼ 10²¹ cm⁻². Since the gas in the vicinity of a massive protocluster has a density in excess of 10⁴ cm⁻³, the corresponding length scale is about 10¹⁷ cm, i.e., 0.03 pc. As a result, τ_UV > 1. In the single-scattering regime (where τ_FIR < 1) essentially all photons emitted by the cluster are absorbed once and then leave the system as infrared photons. It follows that the radiation pressure force is

$$\begin{equation} F_{\rm rad} \simeq {L\over c}\,. \end{equation} \tag{ 1 }$$

The accretion of gas and the formation of stars occur until the radiation pressure becomes comparable to the gravitational force acting on the infalling material: F_rad ∼ F_grav. If the accretion is onto a single star, the infall slows down, but does not halt, since in that case the accretion takes place through a disk.

Accretion onto a cluster-forming clump of gas, however, is more subject to disruption. While the accretion onto clumps likely takes place through filaments, on larger scales the filaments draw their gas supply from large regions of the host GMC. By equating the radiation pressure force with that of gravity, Murray et al. (2010) showed that the fraction of gas _G ≡ M_cl/M_G in a GMC that is converted into stars is proportional to the surface density Σ_G of the parent GMC, for GMCs that are optically thin to far-infrared radiation,

$$\begin{equation} \epsilon _{\rm G}={\pi G c\over 2(L/M_*)}\Sigma _{\rm G}, \end{equation} \tag{ 2 }$$

where L/M_* is the light-to-mass ratio of the cluster. This relation shows that the characteristic (or maximum) mass of the star clusters in a galaxy will scale with the surface density Σ_G, and, we will argue, with the surface density Σ_g and the star formation rate of the host galaxy. We will make use of the above relation in Section 3.2.

The star formation process is believed to be rather inefficient. Typical star formation rates in massive (10⁵–10⁶ M_☉) protoclusters are ∼1 M_☉ yr⁻¹, so the time to double the stellar mass is a significant fraction of the dynamical time of the parent GMC and the main-sequence lifetime of a massive star. As long as the mass in stars M_* < _GM_G, the cluster does not strongly affect the GMC. However, once the stellar mass reaches this upper bound, the cluster begins to disrupt the cloud, eventually cutting off its own fuel supply. The disruption takes less than the GMC dynamical time (Murray et al. 2010), so that further star formation will only increase the cluster luminosity by a factor of several at most. The radiation force exerted on the surrounding gas can therefore only be several times larger than the gravitational force. As a result, the ejection velocity with which radiation pressure can expel gas is of order the escape velocity of the system:

$$\begin{equation} v_{\rm ej}\sim v_{{\rm esc}}=\sqrt{2GM_{\rm cl}/r_{\rm cl}}\,. \end{equation} \tag{ 3 }$$

Massive star clusters are compact. For example, Milky Way globular clusters have half-light radii r_h ∼ 2–3 pc, independent of cluster mass, for clusters with masses between 10⁴ and 10⁶ M_☉ (see Figure 1). Observations of young clusters in other galaxies also find r_h ∼ 2 pc independent of luminosity or M_cl (Scheepmaker et al. 2007). Younger clusters of the same mass tend to have even smaller half-light radii, e.g., Larsen (2004) and McCrady & Graham (2007). This is expected if the gas sloughed off by stars as they evolve is also ejected from the cluster; the lack of intercluster gas strongly implies that the latter is the case.

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Clusters with M_cl ≳ 10⁶ M_☉ do show an increase in radius with increasing mass, of the form r_cl ∼ M^3/5_cl (Haşegan et al. 2005; Murray 2009; see Figure 1). Numerically, the velocity of a cluster is given by

$$\begin{equation} v_{\rm ej} \simeq 100 \left(\frac{M_{\rm cl}}{10^6\,M_\odot } \right)^\alpha \,{\rm km \,\, s}^{-1}, \end{equation} \tag{ 4 }$$

where α = 1/2 for M_cl < 10⁶ M_☉ and α = 1/5 for more massive systems. As discussed below, this quantity is crucial in determining whether a newly formed cluster can expel its surrounding material above a galactic disk and beyond.

The star cluster mass function is known to be shallow:

$$\begin{equation} {dN\over dM_{\rm cl}}\propto M_{\rm cl}^{-\beta }, \end{equation} \tag{ 5 }$$

where N is the number of clusters of mass M_cl, with β ≈ 1.8–2.0 (van den Bergh & Lafontaine 1984; Kennicutt et al. 1989). This suggests that a few to a few dozen clusters dominate the luminosity of a star-forming galaxy, and hence certain aspects of the ISM dynamics. Direct counts of the number of clusters that provide half the star formation of the Milky Way (Murray & Rahman 2010) and M82 (McCrady & Graham 2007, see their Figure 8) are consistent with this conclusion.

At early times, i.e., less than a few Myr after a massive star cluster forms, the forces acting on the GMC hosting a massive star cluster include only radiation pressure and pressure from photoionized gas, or from 10⁷ K gas from shocked stellar winds; protostellar jets extend only to ∼1 pc scales, while the first supernovae do not explode for 4 Myr. Nevertheless, observations show that massive star clusters disrupt GMCs both in the Milky Way (Murray & Rahman 2010) and in nearby galaxies, including the Large Magellanic Cloud (LMC; Oey 1996) and the Antennae (Gilbert & Graham 2007).

It has been suggested that these bubbles are inflated by shocked winds from O stars (Castor et al. 1975; Weaver et al. 1977; Bisnovatyi-Kogan & Silich 1995). However, observations of real bubbles do not support this contention. In particular, observations consistently reveal that the radii of bubbles are a factor of ∼5 smaller than the theory predicts (Dorland et al. 1986; Dorland & Montmerle 1987; Oey 1996; Rauw et al. 2002; Dunne et al. 2003; Smith et al. 2005). The theory is premised on the fact that the pressure inside the bubble is set by energy conservation (the shocked gas radiates only a small fraction of its energy in a dynamical time), so that

$$\begin{equation} P\approx {5\over 11}{L_{{\rm wind}}\,t\over 4\pi R^3/3}. \end{equation} \tag{ 6 }$$

If the bubble radii are a factor of five smaller than the theory predicts, but the wind energy is trapped inside the bubble, the pressure is ∼100 times higher than predicted. However, the pressure can be measured directly from the free–free radio emission produced by the photoionized gas seen in the same bubbles; it is several orders of magnitude below that predicted by Equation (6). We conclude that the hot gas from shocked stellar winds is either not confined or that it loses energy; in either case the low measured pressures show that the hot gas is not inflating the bubbles.

This is consistent with observations of H ii regions in the Milky Way, which have shown that the hot gas pressure is equal to that of the associated H ii (10⁴ K) gas (McKee et al. 1984; Harper-Clark & Murray 2009; Murray et al. 2010), or even lower in 30 Doradus in the LMC (Lopez et al. 2011). The H ii gas pressure is consistent with that predicted from the observed rate of ionizing photons Q produced by the cluster stars, and with the bubble radius, assuming a filling factor of order unity. It follows that the hot gas is dynamically irrelevant.

Why do shocked stellar winds not affect the bubble dynamics? Examination of actively star-forming regions in the Milky Way shows that the bubble walls are far from uniform; multiple partial shells, clumps, and "elephant trunks" (pillars pointing toward the center of the bubble) are common (Murray & Rahman 2010). This is not unexpected, given the turbulent nature of the ISM; the gas near the cluster is clumpy, even before any star formation begins. The fact that the bubble walls do not actually isolate the bubble interior from the region outside the walls explains both the low luminosity of diffuse X-ray emission inside the bubbles and the dynamical irrelevance of the hot gas (Harper-Clark & Murray 2009). However, it is worth stressing that there are obvious (fragmentary) shell structures surrounding the bubbles in all of the star-forming regions examined by Rahman & Murray (2010), and that the bubble walls are moving radially outward at 10–20 km  s⁻¹.

As outlined in Section 2.1, once the radiation pressure of a newly formed star cluster balances the gravitational force acting on the surrounding gas, the star formation rate slows. The luminosity gradually increases, driving gas out of the cluster. As the system evolves, a bubble in the ISM forms around the cluster.

Since radiation pressure supplies the bulk of the momentum absorbed by the gas, the clumpy nature of the ISM will not prevent the formation of a bubble: any gas in the vicinity will absorb momentum from the radiation field. Some very dense clumps will not be appreciably accelerated, but the bulk of the gas in a GMC is in a relatively diffuse state (n ∼ 100 cm⁻³) and will be pushed outward, piling up in multiple partial shells.

Over time the bubble expands, sweeping up more gas, and the swept-up mass M_sh(r) increases with increasing radius. The evolution of the optical depth depends on the surface density through the swept-up gas, Σ_sh ≡ M_sh(r)/4πr²; for a Larson-like GMC density profile ρ(r)∝r⁻¹, Σ_sh(r) is constant. When the bubble, or some part of it, emerges from the GMC, near the surface of the gas disk, the growth rate of M_sh(r) slows and eventually halts. After that, the column of the clouds will decrease with increasing radius.

Murray et al. (2010) studied the behavior of such a bubble in a GMC centered at the disk midplane for bubble radii smaller than the scale height H of the gas disk. Their model included forces due to protostellar jets, shocked stellar winds, H ii gas pressure, turbulent pressure, gravity, and radiation pressure. The jets exert an outward force, but are relevant only on small scales ∼1 pc; the shocked stellar winds may also be confined, briefly, on these scales; we will refer to the combination of the two as the small-scale force, F_sm. These authors found that, on larger scales, the radiation pressure and the gravity dominate the dynamics when the star cluster has M_cl ≳ 3 × 10⁴ M_☉.

Since the largest GMCs have radii only a factor of two or three smaller than the disk scale height, bubbles capable of disrupting their parent GMCs will often break out of the disk. During the expansion, some clouds will be accelerated in the plane of the gas disk, while others will be accelerated vertically, out of the plane of the disk. In this paper, we are interested in the latter, which we will assume cover a fraction $C_\Omega$ of the sky as seen from the center of the cluster. Before any supernovae erupt in the driving cluster and before the clouds rise above the disk and become exposed to radiation or any supernova-driven hot outflows from other stars or clusters in the disk, their equation of motion is given by

$$\begin{eqnarray} \frac{{d} P_{\rm sh}}{{d} t}&\simeq & F_{\rm rad} - F_{\rm grav}\nonumber \\ &\simeq & C_\Omega \left(\frac{L_{\rm cl}(t)}{c}-\frac{G M_{\rm sh}(r)M_{\rm sh}(r)}{2r^2} - {GM_{\rm sh}M_{\rm cl}\over r^2}\right), \end{eqnarray} \tag{ 7 }$$

where P_sh is the momentum of the swept-up gas, and L_cl is the luminosity of the cluster. For simplicity, we have neglected the dynamical pressure of the ISM overlying the cloud, as it is much smaller than the internal pressure of the GMC (Murray et al. 2010). We argue that the emergence of the clumps from the gas disk is the crucial step in driving a superwind. We refer to fragments that manage to rise above the gas disk as clouds. As we will show below, radiative driving, acting alone, can drive clouds above the disk scale height in a few Myr, i.e., before the explosion of any supernovae from the parent cluster.

Here we provide an estimate of the critical gas and star formation rate surface densities required to launch a galactic wind, starting from the assumption that GMC disruption is the limiting step. To do so, we first estimate the mass and radius of the largest GMCs in a galaxy with a specified circular velocity v_c, disk radius R_d, and gas surface density Σ_g. The surface density Σ_G of the GMC then allows us to estimate the stellar mass of the largest star cluster in the GMC. Given the mass–radius relation for star clusters described in Section 2, we can then compare the velocity at which the cluster expels gas to the galaxy circular velocity v_c.

We assume that galactic disks initially fragment on the disk scale height H ≃ (v_T/v_c)R_d (given by hydrostatic equilibrium), where v_T is the larger of the turbulent velocity or the sound speed of the gas. The fragments will form gravitationally bound structures with a mass given by the Toomre mass,

$$\begin{equation} M_{\rm G}=\pi H^2\Sigma _g\,. \end{equation} \tag{ 8 }$$

The value of v_T required to estimate H can be obtained by assuming that the disk is marginally gravitationally stable, so that the Toomre (1964) Q is of order unity:

$$\begin{equation} Q={v_cv_{\rm T}\over \pi G R_{\rm d}\Sigma _g}\sim 1. \end{equation} \tag{ 9 }$$

It follows that H = πGQ(R_d/v_c)²Σ_g. The mass of a large GMC is then

$$\begin{equation} M_{\rm G}= \pi ^3G^2Q^2\left({R_{\rm d}\over v_c}\right)^4\Sigma _g^3. \end{equation} \tag{ 10 }$$

The star formation efficiency in a GMC, i.e., the limit beyond which the GMC is disrupted by radiation pressure (see Equation (2)) can be expressed as

$$\begin{equation} \epsilon _{\rm G} = {\pi G c\over 2(L/M_*)} \Sigma _{\rm G}={\pi G c\over 2(L/M_*)} \phi ^2 \Sigma _g, \end{equation} \tag{ 11 }$$

where ϕ ≡ H/r_G ≈ 2–3, and the light-to-mass ratio (L/M_*) is 3000 cm² s⁻³ for a Chabrier (2005) initial mass function. Using a Salpeter (1955) initial mass function would reduce L/M_* by a factor ∼2. The characteristic cluster mass is then

$$\begin{equation} M_{\rm cl}=\epsilon _{\rm G}\,M_g = {\pi ^4G^3c\over 2(L/M_*)} Q^2\phi ^2\left({R_{\rm d}\over v_c}\right)^4\Sigma _g^4. \end{equation} \tag{ 12 }$$

The velocity v_ej at which the cluster ejects gas, as given by Equation (3), reads

$$\begin{equation} v_{\rm ej}\simeq v_{{\rm esc}}={ {\pi ^2G^2c^{1/2}\over (2\,L/M_*)^{1/2}} {Q \phi \over \sqrt{r}_{\rm cl}} \left({R_{\rm d}\,\Sigma _g\over v_c}\right)^2 }. \end{equation} \tag{ 13 }$$

If this velocity is comparable to or larger than v_c, then a galactic scale wind will result. We define the velocity ratio

$$\begin{equation} \zeta \equiv {v_{\rm ej}\over v_c} ={\pi ^2G^2c^{1/2}\over (2\,L/M_*)^{1/2}}\,{Q \phi \over \sqrt{r}_{\rm cl}}\,{R_{\rm d}^2\,\Sigma _g^2\over v_c^3}. \end{equation} \tag{ 14 }$$

For ζ ≳ 1, we expect the largest star clusters in a galaxy to launch galactic scale winds. For a specified disk radius and circular velocity, this defines a critical gas surface density in order to launch a galactic wind.

3.2.1. Critical Star Formation Rate

We now use Equation (14), in conjunction with the Kennicutt (1998) star formation law

$$\begin{equation} \dot{\Sigma }_* = \eta \left({v_c\over R_{\rm d}}\right)\Sigma _g, \end{equation} \tag{ 15 }$$

with η = 0.017, to find the critical surface density of star formation required to launch a super-galactic wind. We find

$$\begin{equation} \dot{\Sigma }^{{\rm crit}}_*={ (2\,L/M_*)^{1/4}\over \pi G c^{1/4}} \;{\eta \,\zeta \over \sqrt{Q\phi }}\; {r_{\rm cl}^{1/4}v_c^{5/2}\over R_{\rm d}^2}. \end{equation} \tag{ 16 }$$

Scaling to an L_* galaxy,

$$\begin{equation} \dot{\Sigma }^{{\rm crit}}_*\approx 0.06 \left({v_c\over 200\,{\rm km \,\, s}^{-1}}\right)^{5/2} \left({5\;{\rm kpc }\over R_{\rm d}}\right)^2 \,M_\odot \;{\rm yr }^{-1}\;{\rm kpc }^{-1}. \end{equation} \tag{ 17 }$$

From observations of galaxies with and without superwinds, Heckman (2002) gives $\dot{\Sigma }^{{\rm crit}}_*\approx 0.1\,M_\odot \,{\rm yr }^{-1}\;{\rm kpc }^{-2}$. Finally, we can express our radiatively motivated threshold in terms of critical star formation rate

$$\begin{equation} \dot{M}_*^{{\rm crit}}\approx 5\left(v_c\over 200\,{\rm km \,\, s}^{-1}\right)^{5/2}\,M_\odot \,{\rm yr }^{-1}. \end{equation} \tag{ 18 }$$

We have shown that radiation from a single massive star cluster can eject clouds from the disk of a galaxy. However, neither radiation pressure nor ram pressure from supernova-driven hot winds arising from a single isolated cluster can drive the clouds to tens of kiloparsecs; doing so requires the collective effect of all the clusters in the galaxy, as we now show.

4.1.1. Radiative Force

Once the cloud emerges from the galactic disk, it is subject to radiation pressure from stars in the disk. The observed surface brightness of star-forming disks follows distributions that range from nearly constant for R < R_d, where R is the distance from the galactic center, to exponential, i.e., $\Sigma _{{\rm UV}}\approx \exp ^{-R/R_{\rm d}}$ (Martin & Kennicutt 2001; Azzollini et al. 2009).

For simplicity, we will work with a constant surface brightness out to R_d. If the height above the disk (which we approximate as r − H, where r is the distance from the center of the star cluster and H is the scale height of the gas disk) is smaller than R_d, the disk flux seen by the cloud, which quickly dominates that provided by the natal cluster, is roughly constant. The radiation force then becomes

$$\begin{equation} F_{\rm rad}= \left\lbrace \begin{array}{@{}ll}F_{\rm rad,cl} + C_\Omega \dsty\frac{L_{\rm gal}}{c}\,\left(\dsty\frac{r-H}{R_{\rm d}} \right)^2 & \mbox{if } H<r<R_{\rm d}\\ F_{\rm rad,cl} + C_\Omega \dsty\frac{L_{\rm gal}}{c}\,\left(\dsty\frac{R_{\rm d}-H}{r}\right)^2 & \mbox{if } r>R_{\rm d},\\ \end{array} \right. \end{equation} \tag{ 20 }$$

where r is the distance from the center of the cluster, and F_{rad, cl} is the radiation force due to the birth cluster of the cloud, given by the first term on the second line of Equation (7). This expression slightly underestimates the effective luminosity of the disk, but the scale height of the disk is assumed to be smaller than the disk radius, and the entire model is a rather crude approximation in any case. This expression also ensures that when r > R_d, the flux seen by the cloud drops as 1/r². We have assumed that the cloud covers the same fraction of the sky as seen from its parent star cluster ($C_\Omega$) as seen from the disk center; this approximation is fairly good by the time r ≈ R_d.

At early times, the clouds are optically thick to UV radiation, so the radiation force does not depend on the opacity κ (due primarily to dust) of the gas. As a cloud moves outward, its column density will decrease, and the cloud will become optically thin, a point we return to below.

The radiation force also depends on the size of the cloud, which expands perpendicular to the direction of acceleration due to its finite temperature. The rate of expansion perpendicular to the direction of acceleration will depend on a number of factors, primarily the pressure of any hot component in the disk proper, as we now discuss.

The clouds are overpressured compared to the ISM as a whole, since they are subject to the intense radiation from the central star cluster. If this pressure is large compared to the pressure of the hot gas component, which is the case while the cloud is in the gas disk, and possibly for heights of order the disk radius, the cloud will effectively expand into a vacuum; in this case the size of the cloud in the direction perpendicular to its acceleration will be given by l_⊥ = l_{⊥, 0} + c_s(t − t₀), where l_⊥ is the radius of the cloud in the direction perpendicular to the acceleration vector, and c_s refers to the sound speed of the cloud.

If, on the other hand, the pressure of the hot gas in the disk is large, the cloud volume will vary with height above the disk, tracking the pressure in the hot gas. The latter will be in rough hydrostatic equilibrium, with a scale height much larger than that of the molecular gas disk, so the size of the cloud will not vary until it reaches a height of order the scale height of the hot gas. At that point the hot gas density will decrease as 1/r², as will the density in the cloud. In most of the numerical work we present here, we assume the cloud is overpressured compared to any hot gas; this is the case in the Milky Way, but the situation may be different in starburst galaxies.

If the cloud is at a distance less than R_d above the disk, any expansion in the perpendicular radius will give rise to an increase in the amount of radiation impinging on the cloud. For r > R_d, the 1/r² radiative flux decrease will tend to cancel the effect of the increased area of the cloud, and the amount of light impinging on the cloud will increase less rapidly, or not at all.

The expansion of the cloud is also responsible for a third change in the radiative force; as noted below Equation (20), the cloud eventually becomes optically thin to the continuum emission from the disk. The fraction of the incident radiation that the cloud absorbs then drops like 1/r², tracking the decrease in the column density and hence optical depth τ, which we account for in our numerical models. If r > R_d, the decrease in incident flux tends to cancel the geometric growth of the cloud, and the radiative driving, which is proportional to τ, decreases as r increases.

4.1.2. Ram Pressure from Supernovae

The hot gas from isolated supernovae is likely to be trapped in the disk, so that it does not affect clouds above the disk. However, supernovae in large clusters will have a different fate. As we will demonstrate, the ISM above and below large clusters will be expelled from the disk, opening up a pathway for the hot supernova gas to escape. We focus on the large-scale behavior of this hot gas, since the bulk of the cold gas is expelled from the vicinity of the cluster long before any stars in the cluster explode. We assume that the starburst lasts much longer than the lifetime of a very massive star (which we take to be 4 Myr), so that we can average over many cluster lifetimes and use a mean supernova rate.

Supernovae produce hot gas capable of driving a wind. The ram-pressure-derived force on a cloud of cross-section πl²_⊥ with a drag coefficient C_D is

$$\begin{equation} F_{\rm ram} = C_{\rm D}\pi l_\perp ^2 \rho _h (v_h-v_{\rm cold})^2\,, \end{equation} \tag{ 21 }$$

where v_cold is the velocity of the cold gas cloud. The kinetic luminosity of the hot wind is given by

$$\begin{equation} {1\over 2}\dot{M}_h v_h^2 = \epsilon L_{\rm SN}, \end{equation} \tag{ 22 }$$

where is the fraction of the supernova luminosity that is thermalized to produce a hot phase. It is generally found to range from 0.01 to 0.2 (Theis et al. 1992; Cole et al. 1994; Padoan et al. 1997; Thornton et al. 1998). However, Strickland & Heckman (2009) find > 0.3, possibly reaching = 1, in the local starburst M82.

Following Strickland & Heckman (2009), we introduce the mass loading parameter β

$$\begin{equation} {\dot{M}_{\rm h}}= \beta \dot{M}_{{\rm SN+SW}}=0.2\beta {\dot{M}_*}, \end{equation} \tag{ 23 }$$

where, by definition, β > 1. The factor 0.2 in the second equality corresponds to a Chabrier (2005) initial mass function. For very high values of the mass loading β, the hot gas becomes radiative, cooling on a dynamical time (Silich et al. 2004; Strickland & Heckman 2009). This effectively limits the range of mass loading, 1 ⩽ β ≲ 17.

Using the relations given above, and assuming that v_cold ≪ v_h, the ram pressure force can be expressed as

$$\begin{equation} F_{\rm ram}\simeq \left({\pi l_\perp ^2\over 4\pi r^2}\right)0.2\,C_{\rm D}\,{\dot{M}_*}\,\times \,\sqrt{\epsilon \,\beta }\sqrt{10L_{{\rm SN}}\over {\dot{M}_*}}. \end{equation} \tag{ 24 }$$

The radiation pressure on the same cloud is

$$\begin{equation} F_{{\rm rad}}= \left({\pi l_\perp ^2\over 4\pi r^2}\right)\,\times \,\rm {min(\tau _{UV},1)}\,\it {L\over c}. \end{equation} \tag{ 25 }$$

The ratio of the two forces is

$$\begin{equation} {F_{\rm ram}\over F_{\rm rad}} \approx {2\,C_{\rm D} \sqrt{\epsilon \,\beta }}\,\times \,\frac{1}{ \rm {min}(1,\tau _{UV})}. \end{equation} \tag{ 26 }$$

For typical values C_D ≈ 0.5, ≈ 0.3, β = 4, and τ_UV = 1 in the optically thick regime, this ratio is ∼1. At large distances, of order 10 kpc, when the clouds are optically thin, the ram pressure force will start to dominate.

4.1.3. Gravitational Force

While in the disk, the cloud takes part in the rotational motion of the galaxy, so that the force exerted by the galaxy on the GMC is canceled by the centrifugal force due to rotation. However, as it rises above the disk, the cloud will lose the centrifugal support it enjoyed while pursuing its circular path around the galaxy. We take this into account by using

$$\begin{equation} g_{{\rm grav}}= {v_c^2\over R_{\rm d}}\left(r\over R_{\rm d}\right) \end{equation} \tag{ 27 }$$

if r < R_d. This assumes that the GMC lies at a distance ∼R_d from the galactic center.

We model the overall gravitational effect of the galaxy and its dark matter distribution by using a singular isothermal sphere model for the galactic halo:

$$\begin{equation} g_{{\rm grav}}={v_c^2 \over r}. \end{equation} \tag{ 28 }$$

This accurately reflects observed rotation curves of spiral galaxies out to r ≈ 50 kpc, e.g., Casertano & van Gorkom (1991). It probably overestimates the force of gravity acting on the cloud at large radii (approaching the virial radius of the galaxy), where lensing measurements suggest that a Navarro–Frenk–White profile is a better fit, e.g., Mandelbaum et al. (2006).

Given the forces introduced above, we now solve the one-dimensional equation of motion and present results aimed at characterizing the fate of the clouds.

In Figure 2, we consider the case of an M82-like galaxy, i.e., with a circular velocity v_c = 110 km  s⁻¹ and a luminosity L = 5 × 10¹⁰L_☉. For such a galaxy, our idealized model (Equation (10)) predicts a characteristic cluster mass of about 10⁶ M_☉, in agreement with observations (McCrady & Graham 2007). As indicated in Section 3.2, the ζ parameter of such a system is greater than one, suggesting that the star cluster can drive a cloud of gas above the plane. Our numerical calculation confirms the analytic result. We now describe the detailed behavior of the cloud as a function of time and radius. The bottom panel of the figure presents the amplitude of the relevant forces as a function of radius.

• 1.  
  Small-scale force (F_sm, cyan dot-dashed line): it represents the momentum per unit time deposited by protostellar jets, outflows, and possibly shocked stellar winds on small scales. The details of this force are described in Murray et al. (2010). Numerous observations of expanding bubbles in the Milky Way (Murray & Rahman 2010) and in nearby galaxies (Whitmore et al. 2007; Gilbert & Graham 2007) demonstrate that such outflows start before any supernovae occur. The amplitude of this force quickly decreases, on a timescale of about 0.1 Myr, and effectively extends only to ∼1–3 pc, consistent with observed protostellar jets.
• 2.  
  Gravity (red solid line): at early times, the gravity is dominated by that exerted by the star cluster on the nascent bubble; for r < R_GMC this force drops as 1/r². At larger radii, H ≲ r ≲ R_d, the mass in the cloud is constant, but the cloud is losing the rotational support it enjoyed in the disk, so the effective gravitational force increases. At still larger radii the force is dominated by the interaction between the cloud and the combined gravity of the stars and dark matter in the galaxy proper, so that g(r) = v²_c/r.
• 3.  
  Radiation pressure (blue solid line): initially, the radiation pressure force originates only from the parent star cluster, and the cloud is optically thick to far infrared photons. As a result, this force initially scales as 1/r², but the expansion of the cloud gradually reduces the far-IR optical depth below unity, and the net radiation pressure force becomes roughly constant. The radiation pressure from the parent cluster lasts only for a period of about 8 Myr, i.e., the time for the cluster luminosity to decrease by a factor of two. If the cloud does not reach an altitude greater than the height of the disk during that time, it will fall back toward the midplane of the galaxy. If the cloud succeeds in rising above the disk, it then feels the radiation pressure from the neighboring clusters, which increases until r = R_d. Above this height, the radiative flux decreases as 1/r². The fraction of the luminosity impinging on the cloud is given by the area of the cloud divided by r². On those scales, the size of the cloud increases roughly linearly with time, but the radius increases more rapidly. As a result, the momentum deposition decreases. Finally, due to its continuing expansion, the cloud becomes optically thin at about half a kpc, so F_rad drops as 1/r² beyond this point.
• 4.  
  Ram pressure (green dot-dashed line): we have already noted that observations of the Milky Way and the LMC show that stellar winds are not dynamically important, so the ram pressure force is gradually ramped up when the cool gas emerges from the disk, around 60 pc in Figure 2. It is comparable to the radiation pressure, with the exact ratio of the two forces varying with and β, as in Equation (26). The ratio changes when the cloud becomes optically thin to UV radiation, around r ≃ 0.5 kpc. At large radii, the ram pressure is the dominant force, causing the cloud to accelerate slowly beyond r ∼ 50 kpc. Because the area of the cloud continues to increase with increasing r, but not as rapidly as r², the ram pressure force decreases with increasing r.

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From an observational point of view, we note the following results: clouds escape the disk in less than 3 Myr, before any supernovae explode in the cluster. They reach velocities of several hundred kilometers per second at a radius of 10 kpc, after a time ∼10 Myr. The clouds reach a distance of 50 kpc after ∼100 Myr.

4.2.1. Star-forming Galaxy: Varying Cluster Mass

As shown in Equation (14), for a given galaxy the main parameter defining the fate of gas clouds ejected by radiation pressure is the cluster mass—only sufficiently massive clusters can transfer enough momentum to eject a cloud of gas. To demonstrate this result numerically, we use the M82-like galaxy parameters described above, and solve the equation of motion for a range of cluster masses, from 10⁵ M_☉ up to 10⁸ M_☉. The results are presented in Figure 3. We can see that, in a halo of mass M_h ≈ 10¹¹ M_☉, clusters with M = 10⁵ M_☉ will launch galactic fountains rather than galactic winds. More massive clusters launch winds to larger distances from the center of the galaxy, with clusters above M = 10⁷ M_☉ expelling gas from the halo.

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We now consider the case of a quiescent galaxy: the Milky Way. We take v_c = 220 km  s⁻¹, $\,{\cal M }_g=2\times 10^9\,M_\odot$, and R_d = 8 kpc. The results are shown in Figure 4. As a result of the Toomre criterion (see Equation (10)), typical star clusters are expected to have M_cl = 10⁴–10⁵ M_☉. The trajectories of clouds expelled by such star clusters are shown by the solid lines. Those of hypothetically more massive clusters are represented by dashed lines.

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As can be seen, a galaxy like the Milky Way is able to launch galactic fountains to heights of order a kiloparsec. However, even clusters with M_* = 10⁵ M_☉, similar to the most massive clusters in the Milky Way, do not drive large-scale outflows. This behavior is in agreement with the scaling relation presented in Equation (14). It reflects the fact that in such a galaxy, the initial kick resulting from radiation pressure is not strong enough to trigger a large-scale outflow.

We now model the evolution of clouds of various masses in a typical z = 2 starburst galaxy. We parameterize such a system with v_c = 100 km  s⁻¹, $\,{\cal M }_g=2\times 10^{10}\,M_\odot$, R_d = 5 kpc, and L = 4 × 10¹¹ L_☉. We consider the case of a galaxy for which most of the luminosity comes out in the optical, for example a Lyman break galaxy. Given Equation (10), the star cluster mass expected for such a galaxy is about 3 × 10⁸ M_☉. This stellar mass will be distributed in a number of clumps, reducing the self-gravity of each clump. Such compact, massive systems have been observed in z = 2–3 galaxies, e.g., Jones et al. (2010).

Individual GMCs are seen to host multiple star clusters. In the Milky Way or M82, the free-fall time of the most massive GMCs is ∼5–10 Myr, e.g., Grabelsky et al. (1988), similar to the luminosity half-life of a cluster, so that the spread in ages is similar to the cluster lifetime. In contrast, the massive clumps seen in high-redshift galaxies have free-fall times significantly longer than the lifetime of an individual star cluster. There is likely to be a spread in the ages of the individual clumps of order the free-fall time in the host GMC. To take this effect into account, we increase the timescale over which the radiation pressure from the (simulated, single) parent cluster is on by a factor of several. In our calculations we use a factor of four, but any larger factor will produce similar results.

Here, as an illustration, we do not include the ram pressure contribution from supernovae and consider only radiation pressure. We find that, for such a galaxy, radiation pressure can expel the entire cloud with a typical velocity v ≃ 300 km  s⁻¹; see Figure 5. We also consider the case of lower mass clouds: these can easily reach velocities of order 1000 km  s⁻¹ and might be related to the tails of blueshifted self-absorption observed in star-forming galaxies (Weiner et al. 2009; Steidel et al. 2010).

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If a wind is launched from a cluster, the column density through the wind will be given roughly by

$$\begin{equation} N_{\rm H}(r) \approx N_g \left({H\over r}\right)^\beta, \end{equation} \tag{ 29 }$$

where N_g ≡ Σ_g/m_p and β ≈ 1–2. The power-law index β depends on the details of the model, in particular on the initial size of the cloud, how v(r) varies, and on whether the cloud is pressure confined by hot gas. The initial column density is simply that of the disk (when the GMC has expanded to the size of the disk scale height). As already noted, the gas will be in photoionization equilibrium, and illuminated by thousands of O stars, so its temperature will be of order 10⁴ K. Since the cloud is being accelerated, the width of the cloud in the direction of the acceleration will be

$$\begin{equation} l_{||} \approx {c_s^2\over g}, \end{equation} \tag{ 30 }$$

where the acceleration g ≈ v²/r. Since v ≈ v_cl ≈ v_c, we have

$$\begin{equation} l_{||} \approx \left({c_s\over v_c}\right)^2 r\approx 8\times 10^{-3}r \left({T\over 10^4\,{\rm K }}\right) \left({100\,{\rm km \,\, s}^{-1}\over v_c}\right)^2. \end{equation} \tag{ 31 }$$

Combining Equations (29) and (31), the number density is

$$\begin{eqnarray} n(r)&\approx &{N_g\over r} \left({H\over r}\right)^\beta \left({v_c\over c_s}\right)^2\hfill \nonumber \\ &\approx & 250\left({N_g\over 10^{23}\,{\rm cm }^{-2}}\right) \left({H\over 50\,{\rm pc }}\right)^2 \left({1\;{\rm kpc }\over r}\right)^3\,{\rm cm }^{-3},\qquad \end{eqnarray} \tag{ 32 }$$

where we have taken β = 2 and v_c = 100 km  s⁻¹. For a star formation rate of 5 M_☉ yr⁻¹, this corresponds to an ionization parameter

$$\begin{equation} U\equiv {Q\over 4\pi r^2 n c}\approx 10^{-3} \left({Q\over 10^{54}\,{\rm s }^{-1}}\right) \left({r\over 1\;{\rm kpc }}\right), \end{equation} \tag{ 33 }$$

where Q is the number of ionizing photons emitted by the galaxy per second; we have scaled to a value appropriate to the assumed star formation rate. If β < 2, the r dependence is weaker, e.g., β = 1 implies U = const. We have run the photoionization code CLOUDY (Ferland et al. 1998) to determine the ionization state of the gas. The bulk of magnesium is in the form of Mg ii for U = 10⁻³, while a fraction of order 0.1 of sodium is neutral. At very large r the background UV flux will dominate that from the galaxy, and the ionization fractions of both ions will drop, even for the case β = 1.

The column densities, densities, sizes, and ionization parameters just given are those of individual cloud in the outflow from a single cluster. Observations of expanding gas around clusters in the Milky Way make it clear that while shell-like structures do form, they are not very regular, nor is all the material at the same radius from the cluster. Physically, it is clear that higher column density gas will accelerate less rapidly than lower column density gas, leading to clouds at a range of radii. Further, it is likely that material at the fringes of the clouds will be ablated, as is seen in star-forming regions in the Milky Way. When seen in the large, the outflows from a cluster will appear quasi-continuous; if the flows from multiple clusters cannot be resolved, the outflow from the galaxy will appear even more continuous.

We note that the outflow will not be seen as such until the continuum (dust) optical depth drops below unity; only then may absorption lines be seen against the cluster or galactic light. The corresponding column is N_H ≈ 10²¹ cm⁻², depending on the wavelength of the absorption line used to trace the outflow.

While the continuum optical depth drops below unity at fairly small radii, the Mg ii resonance transition at 2900 Å is optically thick out to radii as large as 100 kpc. The dynamical time to 100 kpc is ∼100 Myr. Hence, we expect blueshifted absorption to be seen in post-starburst galaxies as well as in active starbursts.

Individual GMCs and their cluster progeny occupy a small fraction of the disk area, so individual outflows have a small disk covering factor

$$\begin{equation} C_{{\rm fd}}\equiv {{a}^2(v)\over R_{\rm d}^2}. \end{equation} \tag{ 34 }$$

This covering factor is an increasing function of cloud velocity v for small distances r, since initially both v(r) and a(r) are increasing functions of r. When the clumps enter the large-scale hot outflow, their radii may decrease as they come into pressure equilibrium, but after that they will resume their expansion. However, the summed cover factor of all the clouds will be a significant fraction of unity.

The ratio of disk scale height to disk radius is related to the star formation rate of the galaxy. For high star formation rates, it can reach (H/R_d)² ≈ 0.1–0.3 but is smaller for most starburst galaxies. However, clusters above and below the disk midplane can drive gas from either side of the disk (and possibly from both sides, as just noted), so that sightlines from most directions will encounter the outflow from one or more clusters.

Murray et al. (2010) show that the fraction of gas in a GMC that ends up in stars is given roughly by

$$\begin{equation} \epsilon _{\rm G}\approx {\Sigma _{\rm G}\over 1+\Sigma _{\rm G}/0.25\,{\rm g }\,{\rm cm }^{-2}}\,, \end{equation} \tag{ 35 }$$

interpolating between their optically thin and optically thick expressions. The balance of the GMC is either returned to the ISM or driven out of the galaxy as a wind. We have argued in this paper that for systems above the threshold given in Section 3.2.1, one-quarter to one-half of the GMC is launched as a wind. The ratio of mass launched out of the disk to mass locked up in stars is very roughly

$$\begin{equation} {\dot{M}_w\over \dot{M}_*}\approx {0.25\,{\rm g }\,{\rm cm }^{-2}+\Sigma _{\rm G}\over \Sigma _{\rm G}}. \end{equation} \tag{ 36 }$$

This predicts a very large ratio of mass lost in a wind compared to stellar mass in low-Σ_G systems. Since we require that the galaxy be above the threshold star formation rate, this high mass-loss ratio applies to galaxies with low circular velocities.

In addition to this local criterion, there is a global limit to the momentum-driven mass-loss rate, namely that the rate of momentum carried out by the wind is no larger than the rate of momentum supplied by starlight and ram pressure. This limit is

$$\begin{eqnarray} \hspace{-8pc}\dot{M}_w &=& \left(1+{F_{\rm ram}\over F_{\rm rad}}\right) L/c v_\infty \end{eqnarray} \tag{ 37 }$$

$$\begin{eqnarray} &=& \left(1+{F_{\rm ram}\over F_{\rm rad}}\right) \left[ \left({L\over M_{{\rm cl}}}\right) {\tau _{{\rm ms}}\over c^2 } \right] \left({c\over v_\infty }\right) \dot{M}_*. \end{eqnarray} \tag{ 38 }$$

The maximum wind mass-loss rate per star formation rate for such a momentum-driven outflow is

$$\begin{equation} {\dot{M}_w \over \dot{M}_*} \approx 0.4\left(1+{F_{\rm ram}\over F_{\rm rad}}\right) \left({v_\infty \over 300\,{\rm km \,\, s}^{-1}}\right)^{-1}. \end{equation} \tag{ 39 }$$

These expressions assume that the bulk of the star formation occurs in clusters above the critical mass necessary to launch galactic winds. As noted in Section 2.2, the mass function of star clusters is shallow so that this criterion will be satisfied for galaxies with sufficiently high star formation rates.

6.1.1. Ram Pressure from Supernovae and Cloud Survival

There is little doubt that hot gas from supernovae pushes cool gas out of galaxies. However, it has been argued here and elsewhere that supernovae do not push cool gas out of GMCs. There is good evidence in the Milky Way that GMCs are disrupted well before any supernovae explode in them (Harper-Clark & Murray 2009; Murray & Rahman 2010).

The crucial point, however, is that cool gas entrained in a hot flow on GMC scales will not survive at a low temperature long enough to escape the galaxy. As noted in the introduction, clouds caught up in a hot outflow are compressed on a cloud crossing time, t_cc = χ^1/2a/v_b. For a cloud with T = 1000 K at a distance r ≈ 10 pc from its natal cluster, we find

$$\begin{equation} t_{{\rm cc}}\approx 6\times 10^{12} \left({{a}\over 1\,{\rm pc }}\right)\,{\rm s }, \end{equation} \tag{ 40 }$$

where we have scaled to the temperature T_h = 5 × 10⁷ K of the hot flow seen in M82 (Strickland & Heckman 2009).

We have scaled to a cloud size of one parsec. Star clusters have radii of this order (see Figure 1, for example), as do the clumps of gas associated them, e.g., Simon et al. (2001) or Kauffmann et al. (2010); we remind the reader that in fact there is a distribution of clump sizes and masses. These clumps are the raw material of both the swept-up clumps and the clouds that are launched out of the galaxy.

Numerous simulations of interactions of hot density gas with cool (T ⩽ 10⁴ K) gas demonstrate that the clouds are rapidly destroyed, i.e., in ∼(3–5)t_cc, in the simulations of Klein et al. (1994) and Poludnenko et al. (2002). These early simulations employed an adiabatic equation of state, so that the temperature of the cool cloud increased as it was compressed. Simulations which include radiative cooling find that clouds live longer, ∼10t_cc, reaching velocities of several hundred kilometers per second (Mellema et al. 2002; Cooper et al. 2009; Pittard et al. 2010). For example, Cooper et al. (2009) employ high-resolution three-dimensional radiative models, finding that a substantial fraction of the gas in a cloud is lost, via Kelvin–Helmholtz and Rayleigh–Taylor instabilities, to the hot flow by the time the cloud has traveled 75 pc (about a million years), consistent with the simple analytic estimate of ∼10t_cc.

Laboratory experiments also show that dense clumps of matter subject to high-velocity, low-density flows are destroyed after a few tens of cloud crossing times (Hansen et al. 2007).

The short lifetimes of such ram-pressure-driven clouds was stressed by Marcolini et al. (2005), who noted another form of destruction, that of thermal conduction. They find that conductive heating suppresses the Kelvin–Helmholtz and Rayleigh–Taylor instabilities that disrupt the clouds in non-conductive simulations. However, the conduction also causes rapid mass loss from the clouds, leading to lifetimes ∼1–10 Myr. As in the simulations of Cooper et al. (2008), the clouds are essentially completely disrupted by the time they reach r ≈ 1 kpc, even in the most favorable simulations.

Observations of the solar wind have shown that the conductivity is similar to the classical Spitzer value, e.g., Salem et al. (2003), modified to account for saturation at the free-streaming heat flux. This, combined with the rough agreement between the analytically estimated cloud lifetimes against conduction and those found by the numerical simulations of Marcolini et al. (2005), strongly suggest that ram-pressure-driven cold gas clouds have lifetimes of order ∼1 Myr, simply due to heat conduction, independent of the effects of Kelvin–Helmholtz instabilities.

To summarize, the destruction time t_dest ≲ 20t_cc of cold clouds entrained in a hot outflow is given by

$$\begin{equation} {t_{\rm dest}}\lesssim 4 \left({T_h\over 5\times 10^7\,{\rm K }}\right) \left({T_c\over 10^3\,{\rm K }}\right)^{-1} \left({a\over 1\,{\rm pc }}\right) {\rm Myr}{.} \end{equation} \tag{ 41 }$$

Assuming a cloud velocity of 300 km  s⁻¹, the cloud is destroyed after traveling a distance

$$\begin{equation} R_{\rm destruction}\lesssim 1\;{\rm kpc }\left({a\over 1\,{\rm pc }}\right). \end{equation} \tag{ 42 }$$

This is a rather generous estimate, since it assumes the upper limit for estimates of the cloud destruction time, and assumes that the clouds are accelerated instantaneously. The sizes of starburst disks range from R_d ≈ 0.3–1 kpc or larger, implying that the hot gas density does not begin to drop until r = 0.3–1 kpc.

We conclude that cold clouds entrained in hot flows on GMC scales do not survive beyond distances of ∼1 kpc.

Radiatively driven cold gas clouds escape their natal clusters well before any supernovae explode, so they are not subject to the ill effects of hot gas, at least until they reach the surface of their galactic disk. If the hot gas fills the volume above the disk surface, the cold clouds will be subject to both destruction by Kelvin–Helmholtz instabilities and conductive evaporation, but at a much lower rate than the clouds simulated above. The slower evaporation results from the fact that the conductivity is saturated; in that case the mass-loss rate $\dot{M}_{\rm cond}\sim n_h^{5/8}$, where n_h is the number density of the hot gas. The longer Kelvin–Helmholtz times follow from the larger length scale a of the clouds, which grows as the clouds move away from the launching cluster.

This leads to a clear observational distinction between the two types of driving. Small clouds driven by hot gas have lifetimes of order 1 Myr or shorter in most simulations; in the most optimistic case the lifetime may reach ∼3 Myr. The cold clouds survive to r ≲ 1 kpc from the disk (assuming a rather generous instantaneous v = 300 km  s⁻¹).

In contrast, radiatively driven clouds are not subject to such rapid initial mass loss. They will travel for ∼4 Myr, reaching distances 300 pc before the hot gas from the first cluster supernovae explode, and about twice that before the bulk of the supernovae in the cluster explode. The cold clouds will be traveling at ∼100 km  s⁻¹ or more when the hot gas reaches them, and they will have sizes of order ∼50 pc, assuming only that they expand at their sound speed (initially the clouds may expand more rapidly, as the radiation field diverges). As a result, radiatively driven clouds will survive to reach r ∼ 50 kpc or more. Cool outflows extending to r ≫ 1 kpc from the host disk are a clear signature of a radiatively driven outflow.

A second test that can distinguish between radiative driving and supernova driving is even more direct. Individual star clusters are known to drive winds. If radiative driving is important, winds will be seen emanating from clusters younger than 4 Myr. Since no stars have lifetimes shorter than this, supernova driving cannot account for any winds seen emerging from such young clusters.

6.1.2. Shocked Stellar Winds

6.1.3. Post-shock Cooling in Multiple Star Cluster Driven Winds

6.1.4. Other Hybrid Models

We have neglected any hot gas which might reside in the galaxy halo, and which could potentially halt the outflow of cold gas clouds within the halo. This drag (or destruction) is relevant for any proposed launching mechanism.

If the halo contained the cosmic abundance of baryons, M_b ≈ 0.173M_h (Dunkley et al. 2009), the mean column of hot gas would be N_H ∼ 3 × 10¹⁹ cm⁻². Since this is much larger than the cloud column density at the virial radius, the hot gas would either stop or shred the outgoing clouds. However, the hot gas content of cluster and group size halos has been measured to be smaller than the cosmic abundance, e.g., Dai et al. (2010). These authors measure a hot gas fraction f_g that ranges from f_g ≈ 0.08 ± 0.02 for halo masses M_h = 6 × 10¹⁴ M_☉ down to f_g = 0.013 ± 0.005 for M_h = 2.6 × 10¹³ M_☉. Written as a ratio to the cosmic baryon abundance, this is f_g/f_c = 0.46–0.075. It is believed that the hot gas fraction in individual galaxies (M_h ≲ 10¹³ M_☉) is yet smaller (Bregman 2007). If we take f_g/f_c = 0.1, then cold clouds reach r ∼ 50–100 kpc before they run into their own column of hot halo gas.