DISRUPTION OF MOLECULAR CLOUDS BY EXPANSION OF DUSTY H II REGIONS

We consider a spherical neutral cloud with density distribution

$$\begin{eqnarray}n(r)=\left\{\begin{array}{ll}{n}_{{\rm{c}}}, & \mathrm{for}\;r/{r}_{{\rm{c}}}\ \lt \ 1,\\ {n}_{{\rm{c}}}{(r/{r}_{{\rm{c}}})}^{-{k}_{\rho }}, & \mathrm{for}\;r/{r}_{{\rm{c}}}\geqslant 1,\end{array}\right.\end{eqnarray} \tag{ 9 }$$

where n_c and r_c are the density and radius of a flat core, respectively, and ${k}_{\rho }$ is an index of a surrounding power-law envelope.⁴ A star cluster with total mass ${M}_{*}$, total luminosity L, and total ionizing power ${Q}_{{\rm{i}}}$ is born instantly at the cloud center. Copious energetic photons emitted by the cluster begin to ionize the surrounding medium, causing the ionization front to advance to the initial Strömgren radius, ${r}_{{\rm{IF,0}}}$, within a few recombination timescales (~10³ yr for $n\sim {10}^{2}\;{{\rm{cm}}}^{-3}$). The overpressured, ionized gas creates a shock front that moves radially outward, sweeping up the ambient neutral medium into a dense shell. We assume that the H ii region remains in internal quasi-static equilibrium throughout its dynamical expansion, which is reasonable since the sound-crossing time over the ionized region is sufficiently small compared to the expansion timescale. We further assume that all of the swept-up gas resides in a thin shell of mass ${M}_{{\rm{sh}}}$ located at ${r}_{{\rm{sh}}}$. Since we are interested in the evolution of the shell well after the formation phase, we may take ${r}_{{\rm{c}}}\ll {r}_{{\rm{IF,0}}}$ in describing shell expansion in a power-law envelope.

The momentum equation for the shell is

$$\begin{eqnarray}&&\displaystyle \frac{d({M}_{{\rm{sh}}}{v}_{{\rm{sh}}})}{{dt}}={F}_{{\rm{tot}}}={F}_{{\rm{out}}}-{F}_{{\rm{in}}},\end{eqnarray} \tag{ 10 }$$

where ${v}_{{\rm{sh}}}={{dr}}_{{\rm{sh}}}/{dt}$ is the expansion velocity of the shell, and ${F}_{{\rm{out}}}$ and ${F}_{{\rm{in}}}$ denote the outward and inward forces acting on the shell, respectively. The inward force is due to the cluster gravity and shell self-gravity, given by

$$\begin{eqnarray}&&{F}_{{\rm{in}}}=\displaystyle \frac{{{GM}}_{{\rm{sh}}}({M}_{*}+{M}_{{\rm{sh}}}/2)}{{r}_{{\rm{sh}}}^{2}},\end{eqnarray} \tag{ 11 }$$

so that the cluster gravity and gaseous self-gravity depend on ${r}_{{\rm{sh}}}$ as ${r}_{{\rm{sh}}}^{1-{k}_{\rho }}$ and ${r}_{{\rm{sh}}}^{4-2{k}_{\rho }}$, respectively. The radiation and thermal pressures give rise to the total outward force

$$\begin{eqnarray}&&{F}_{{\rm{out}}}=\displaystyle \frac{{L}_{{\rm{n}}}}{c}{e}^{-{\tau }_{{\rm{d,IF}}}}+8\pi \;{k}_{{\rm{B}}}{T}_{{\rm{i}}}{n}_{{\rm{edge}}}{r}_{{\rm{sh}}}^{2}.\end{eqnarray} \tag{ 12 }$$

Note that ${F}_{{\rm{out}}}$ is the contact force acting on the surface immediately outside the ionization front. Only nonionizing photons reach the neutral shell and exert the outward force, if the shell is optically thick to them.⁵

In Equation (10), we neglect the reaction force on the shell exerted by evaporation flows away from the ionization front because it is not significant for embedded H ii regions. The thrust term would be important in driving expansion of classical, blister-type H ii regions for which ionization fronts are kept D-critical (Matzner 2002; Krumholz et al. 2006; KM09). We also ignore the slowdown of the shell caused by turbulent ram pressure of the external, neutral medium (e.g., Tremblin et al. 2014; Geen et al. 2015a), which is difficult to model within our spherically symmetric, one-dimensional model.

Figure 3 plots ${F}_{{\rm{out}}}$ and the respective contributions of radiative and thermal pressures against the shell radius ${r}_{{\rm{sh}}}$, taken to equal ${r}_{{\rm{IF}}}$, as the black, blue, and red solid lines, respectively, utilizing the Dr11 solutions with $\beta =1.5$ and $\gamma =11.1$. Apparently, the thermal term dominates the radiative term by nonionizing photons over the whole range shown in Figure 3. This should not be interpreted as an indication that radiation is unimportant for the shell expansion. Rather, the effect of the radiation force is indirectly communicated to the shell by increasing the ionized gas density ${n}_{{\rm{edge}}}$ above ${n}_{{\rm{rms}}}$, with the limiting ratio ${n}_{{\rm{edge}}}$/${n}_{{\rm{rms}}}$ given by the factor in Equation (8) when ${Q}_{{\rm{i}}}{n}_{{\rm{rms}}}\to \infty$.

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The behavior of the outward forces for varying ${r}_{{\rm{IF}}}$ can be readily understood in comparison with the usual approximation adopted by previous authors (e.g., Harper-Clark & Murray 2009; KM09; MQT10). Assuming that all the photons from the source are absorbed by the shell, that the H ii region has uniform interior density ${n}_{{\rm{rms}}}$, and that infrared photons reradiated by dust freely escape the system, the effective outward forces on the shell from direct radiation and gas pressure are taken as

$$\begin{eqnarray}&&{F}_{{\rm{rad,eff}}}=L/c\end{eqnarray} \tag{ 13 }$$

and

$$\begin{eqnarray}&&{F}_{{\rm{thm,eff}}}=8\pi \;{k}_{{\rm{B}}}{T}_{{\rm{i}}}{n}_{{\rm{rms}}}{r}_{{\rm{sh}}}^{2},\end{eqnarray} \tag{ 14 }$$

respectively, where ${n}_{{\rm{rms}}}$ is given in terms of ${Q}_{{\rm{i}}}$, ${f}_{{\rm{ion}}}$, and ${r}_{{\rm{sh}}}={r}_{{\rm{IF}}}$ using Equation (1). These, together with the total effective force ${F}_{{\rm{out,eff}}}={F}_{{\rm{rad,eff}}}+{F}_{{\rm{thm,eff}}}$, are plotted as dotted lines in Figure 3. Evidently, ${F}_{{\rm{out,eff}}}$ agrees well (within 20%) with ${F}_{{\rm{out}}}$ from Equation (12). Comparison of the thermal term in Equation (12) with Equation (14) shows that the enhanced thermal force for small radius is due to increased ${n}_{{\rm{edge}}}$/${n}_{{\rm{rms}}}$, which is caused by radiation pressure in the Dr11 solutions. At sufficiently small radius, the total outward force is in fact equal to ${F}_{{\rm{rad,eff}}}$.⁶

Since ${F}_{{\rm{thm,eff}}}\propto {n}_{{\rm{rms}}}{r}_{{\rm{sh}}}^{2}\propto {f}_{{\rm{ion}}}^{1/2}{r}_{{\rm{sh}}}^{1/2}$, while ${F}_{{\rm{rad,eff}}}$ remains constant, one can write

$$\begin{eqnarray}&&{F}_{{\rm{out,eff}}}=\displaystyle \frac{L}{c}\left[1+{\left(\displaystyle \frac{{f}_{{\rm{ion}}}}{{f}_{{\rm{ion,ch}}}}\displaystyle \frac{{r}_{{\rm{sh}}}}{{r}_{{\rm{ch}}}}\right)}^{1/2}\right],\end{eqnarray} \tag{ 15 }$$

with the characteristic radius ${r}_{{\rm{ch}}}$ at which ${F}_{{\rm{rad,eff}}}={F}_{{\rm{therm,eff}}}$ defined by (KM09)

$$\begin{eqnarray}{r}_{{\rm{ch}}} & \equiv & {\left(\displaystyle \frac{(1+\beta ){\sigma }_{{\rm{d}}}}{\gamma }\right)}^{2}\displaystyle \frac{{Q}_{{\rm{i}}}}{12\pi {\alpha }_{{\rm{B}}}{f}_{{\rm{ion,ch}}}}\\ & & \to 5.3\times {10}^{-2}\;{\rm{pc}}\;{Q}_{{\rm{i,49}}},\end{eqnarray} \tag{ 16 }$$

where ${f}_{{\rm{ion,ch}}}\equiv {f}_{{\rm{ion}}}({r}_{{\rm{ch}}})$ is 0.32 for our fiducial parameters $\beta =1.5$ and $\gamma =11.1$.⁷ Expansion is driven predominantly by radiation in the regime with ${r}_{{\rm{sh}}}/{r}_{{\rm{ch}}}\lt 1$, while ionized gas pressure is more important for ${r}_{{\rm{sh}}}/{r}_{{\rm{ch}}}\gt 1$. Thus, the relative importance of radiation pressure within an H ii region can be assessed by the ratio

$$\begin{eqnarray}\displaystyle \frac{{r}_{{\rm{IF}}}}{{r}_{{\rm{ch}}}} & = & {f}_{{\rm{ion,ch}}}{\left[\displaystyle \frac{\gamma {(36\pi {\alpha }_{{\rm{B}}})}^{1/3}}{(1+\beta ){\sigma }_{{\rm{d}}}}\right]}^{2}{\left(\displaystyle \frac{{f}_{{\rm{ion}}}}{{Q}_{{\rm{i}}}^{2}{n}_{{\rm{rms}}}^{2}}\right)}^{1/3}.\\ & & \to {\left(\displaystyle \frac{{f}_{{\rm{ion}}}}{0.32}\right)}^{1/3}{\left(\displaystyle \frac{{Q}_{{\rm{i,49}}}{n}_{{\rm{rms}}}}{2.55\times {10}^{4}{{\rm{cm}}}^{-3}}\right)}^{-2/3},\end{eqnarray} \tag{ 17 }$$

shown as a black dotted line in Figure 2 as a function of ${Q}_{{\rm{i,49}}}{n}_{{\rm{rms}}}$. Thus, shell expansion is dominated by radiation pressure when ${Q}_{{\rm{i,49}}}{n}_{{\rm{rms}}}\gtrsim {10}^{4}\;{\mathrm{cm}}^{-3}$.

In the absence of the inward gravitational forces (${F}_{{\rm{in}}}=0$) and in the limit of negligible core radius, it is straightforward to show that Equations (10) and (15) yield the similarity solutions

$$\begin{eqnarray}&&{n}_{{\rm{mean}}}{r}_{{\rm{sh}}}^{4}=\displaystyle \frac{4-{k}_{\rho }}{2}\displaystyle \frac{L}{c}\displaystyle \frac{3}{4\pi {\mu }_{{\rm{H}}}}{t}^{2},\end{eqnarray} \tag{ 18 }$$

in the limit of ${r}_{{\rm{sh}}}/{r}_{{\rm{ch}}}\ll 1$, and

$$\begin{eqnarray}&&{n}_{{\rm{mean}}}{r}_{{\rm{sh}}}^{7/2}=\displaystyle \frac{3}{2}\displaystyle \frac{\;{k}_{{\rm{B}}}{T}_{{\rm{i}}}}{{\mu }_{{\rm{H}}}}{\left(\displaystyle \frac{3{f}_{{\rm{ion}}}{Q}_{{\rm{i}}}}{4\pi {\alpha }_{{\rm{B}}}}\right)}^{1/2}\displaystyle \frac{{(7-2{k}_{\rho })}^{2}}{9-2{k}_{\rho }}{t}^{2},\end{eqnarray} \tag{ 19 }$$

in the limit of ${r}_{{\rm{sh}}}/{r}_{{\rm{ch}}}\gg 1$, where ${f}_{{\rm{ion}}}$ is regarded as a constant, ${\mu }_{{\rm{H}}}=2.34\times {10}^{-24}\;{\rm{g}}$ is the mean atomic mass per hydrogen, and ${n}_{{\rm{mean}}}=3{n}_{{\rm{c}}}{({r}_{{\rm{sh}}}/{r}_{{\rm{c}}})}^{-{k}_{\rho }}/(3-{k}_{\rho })$ is the mean number density interior to ${r}_{{\rm{sh}}}$ (e.g., Krumholz et al. 2006; KM09). Therefore, the shell radius and velocity vary with time as ${r}_{{\rm{sh}}}\propto {t}^{2/(4-{k}_{\rho })}$ and ${v}_{{\rm{sh}}}\propto {t}^{({k}_{\rho }-2)/(4-{k}_{\rho })}$ in the radiation-pressure-driven limit, and ${r}_{{\rm{sh}}}\propto {t}^{4/(7-2{k}_{\rho })}$ and ${v}_{{\rm{sh}}}\propto {t}^{(2{k}_{\rho }-3)/(7-2{k}_{\rho })}$ in the gas-pressure-driven limit. KM09 presented an analytic approximation valid in both limits by combining Equations (18) and (19). Note that for ${k}_{\rho }=3/2$, the velocity approaches a constant at sufficiently late time.

When a cloud has too steep a density gradient, the shell mass becomes smaller than the ionized gas mass within the ionization front. In this case, the thin-shell approximation we adopt would no longer be valid. For instance, Franco et al. (1990) found that for a thermally driven H iiregion formed in a cloud with ${k}_{\rho }=3/2$, the shock front moves twice as fast as the ionized sound speed, without significant mass accumulation in the shocked shell. For $3/2\lt {k}_{\rho }\lt 3$, an H ii region becomes "density bounded" and develops "champagne" flows rather than forming a shell (see also self-similar solutions by Shu et al. 2002). To be consistent with our thin-shell approximation, therefore, we consider clouds only with ${k}_{\rho }\lt 1.5$ in the following analysis for shell expansion.

When stellar gravity is included, we can relate ${M}_{*}$ to ${Q}_{{\rm{i}}}$ through ${M}_{*}={Q}_{{\rm{i}}}/{\rm{\Xi }}$, where ${\rm{\Xi }}$ is the conversion factor representing the ionizing photon output per unit stellar mass. The corresponding light-to-mass ratio is ${\rm{\Psi }}=L/{M}_{*}=(1+\beta )\langle h\nu {\rangle }_{{\rm{i}}}{\rm{\Xi }}$. Obviously, these quantities depend on ${M}_{*}$ and vary from cluster to cluster, especially for low-mass ones owing to stochastic fluctuations in the stellar population. In Appendix A, we perform Monte Carlo simulations for the spectra of coeval populations using the SLUG code (Krumholz et al. 2015) to find spectral properties as functions of ${M}_{*}$. Equations (33) and (34) provide the fits to the resulting median values of ${\rm{\Psi }}$ and ${\rm{\Xi }}$. While ${\rm{\Psi }}$ and ${\rm{\Xi }}$ saturate to constant values for ${M}_{*}\gtrsim {10}^{4}\;{M}_{\odot }$, they decrease sharply as ${M}_{*}$ decreases below $\sim {10}^{3}\;{M}_{\odot }$, owing to a rapid decrease in the number of O-type stars in the realizations of low-mass clusters. In this work, we use Equation (33) for conversion between ${M}_{*}$ and ${Q}_{{\rm{i}}}$, while fixing to $\beta =1.5$ and $\langle h\nu {\rangle }_{{\rm{i}}}=18\;{\rm{eV}}$: we have checked that varying β and $\langle h\nu {\rangle }_{{\rm{i}}}$ does not affect our results much.

To integrate Equation (10), we first choose a set of values for β, γ, ${Q}_{{\rm{i}}}$, ${n}_{{\rm{rms,0}}}$, and ${k}_{\rho }$. We then specify ${f}_{{\rm{ion}}}$ from the Dr11 solutions and ${r}_{{\rm{IF,0}}}$ from Equation (1). At t = 0, a shell with a zero velocity is located at ${r}_{{\rm{sh}}}(0)={r}_{{\rm{IF,0}}}$. The outside density profile is taken as $n(r)={n}_{{\rm{rms,0}}}{(r/{r}_{{\rm{IF,0}}})}^{-{k}_{\rho }}$ for $r\geqslant {r}_{{\rm{sh}}}$, and the shell mass at any radius is given by ${M}_{{\rm{sh}}}=4\pi {\mu }_{{\rm{H}}}{n}_{{\rm{rms,0}}}{r}_{{\rm{sh}}}^{3-{k}_{\rho }}{r}_{{\rm{IF,0}}}^{{k}_{\rho }}/(3-{k}_{\rho })$.

As illustrative examples, we fix $\beta =1.5$, $\gamma =11.1$, and ${n}_{{\rm{rms,0}}}={10}^{4}\;{{\rm{cm}}}^{-3}$ and vary ${k}_{\rho }$ and ${Q}_{{\rm{i}}}$. Figure 4 plots as solid lines the temporal behavior of the shell radius (upper panels) and velocity (lower panels) for ${Q}_{{\rm{i,49}}}$ = 10 (left) and ${Q}_{{\rm{i,49}}}={10}^{3}$ (right). The solutions without gravity are compared as dotted lines. The cases with ${k}_{\rho }=0$, 1, and 1.4, plotted in red, black, and green, respectively, show that the shell expansion is faster in an environment with a steeper density profile, because the shell mass increases more slowly for larger ${k}_{\rho }$. The horizontal dashed lines mark the characteristic radii, ${r}_{{\rm{ch}}}=0.53$ and $53\;{\rm{pc}}$ for ${Q}_{{\rm{i,49}}}=10$ and 10³, respectively, inside (outside) of which shell expansion is driven primarily by radiation (gas) pressure. The star symbols mark the radius where ${F}_{{\rm{tot}}}=0$, beyond which expansion is slowed down by the shell self-gravity significantly.

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The solutions converge to the asymptotic power-law solutions discussed in Section 3.2 (Equations (18) and (19)), although strong stellar gravity in the case with ${Q}_{{\rm{i,49}}}={10}^{3}$ makes the shell expansion deviate from the nongravitating solutions at early time. Self-gravity of the swept-up shell becomes important in the late stage, eventually halting the expansion. The maximum shell velocity is only mildly supersonic with respect to the ionized gas owing to the rapid increase in the shell mass. Note that when ${Q}_{{\rm{i,49}}}=10$, the driving force changes from radiation to gas pressure at small ${r}_{{\rm{sh}}}$, while shell expansion is always dominated by radiation pressure when ${Q}_{{\rm{i,49}}}={10}^{3}$.

So far we have used a very simple model to study dynamical expansion of a spherical shell surrounding an H ii region. In order to check how reliable our results are, we run direct numerical simulations using the Athena code in spherical geometry (Stone et al. 2008), as described in Appendix B. As an initial state, we consider a source of radiation at the center of a radially stratified cloud with ${k}_{\rho }=1$. To handle the transfer of radiation from the source, we adopt a ray-tracing technique explained in Mellema et al. (2006) and Krumholz et al. (2007). While gas inside the H ii region is evolved self-consistently, we ensure that the outer envelope unaffected by radiation maintains its initial hydrostatic equilibrium.

Figure 5 plots as solid lines the radial density distributions at a few epochs for Model A (with ${Q}_{{\rm{i,49}}}=1$ and $n({r}_{0})={10}^{3}\;{{\rm{cm}}}^{-3};$ top), Model B (with ${Q}_{{\rm{i,49}}}={10}^{2}$ and $n({r}_{0})={10}^{4}\;{{\rm{cm}}}^{-3};$ middle), and Model C (with ${Q}_{{\rm{i,49}}}={10}^{3}$ and $n({r}_{0})={10}^{4}\;{{\rm{cm}}}^{-3};$ bottom), all with ${k}_{\rho }=1$ and ${r}_{0}=1\;{\rm{pc}}$. In Model A, the effect of radiation on shell expansion is almost negligible since ${Q}_{{\rm{i,49}}}{n}_{{\rm{rms}}}\lt {10}^{4}\;{{\rm{cm}}}^{-3}$. Its overall expansion is in good agreement with the classical picture of thermally driven expansion (e.g., Spitzer 1978). As soon as the ionization front reaches the initial Strömgren radius ($\sim 0.3\;{\rm{pc}}$), an isothermal shock wave forms and propagates outward. At the same time, rarefaction waves excited at the ionization front propagate radially inward (e.g., Arthur et al. 2011), gradually turning into acoustic disturbances that travel back and forth between the ionization front and the center. In this model, the density profile in the ionized region is nearly flat.

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On the other hand, Model C is dominated by radiation. Gas at small r is pushed out supersonically to create a central cavity together with an ionized shell within t~10³ yr.⁸ After the shell formation, the gas in the ionized region relaxes into a quasi-static equilibrium state within a sound-crossing time across the H ii region. In Model B, the shell expands mostly as a result of radiation pressure until it reaches ${r}_{{\rm{sh}}}=6.3\;{\rm{pc}}$ at $t=0.82\;{\rm{Myr}}$, after which the driving force switches to ionized gas pressure.

Figure 5 also plots as red dashed lines the corresponding Dr11 quasi-static solutions inside the ionization front, which are in good agreement with the results of the time-dependent simulations. This not only confirms that the static solutions of Dr11 are applicable even for expanding H ii regions, but also validates our ray-tracing treatment of radiation in the numerical simulations.

Figure 6 plots the temporal changes (solid lines) of the shell radius from the simulations in comparison with the solutions (dashed lines) of Equation (10). The cases with and without gravity are plotted in blue and red, respectively. Small differences at early time between the results from the different approaches are caused mainly by the assumption of a vanishing shell velocity at t = 0 in solving Equation (10), while the shell in simulations has a nonzero velocity when it first forms. Nevertheless, semianalytic and numerical solutions agree with each other within ~5% after $0.05\;{\rm{Myr}}$, suggesting that our semianalytic model of shell expansion is quite reliable. Although the inclusion of gravity does not make significant changes in the internal structure of the ionized region, it is self-gravity that makes the swept-up shell decelerate and eventually stall at ${r}_{{\rm{sh}}}=51\;{\rm{pc}}$ when $t=23\;{\rm{Myr}}$ in Model A. Gravity becomes more significant for larger ${Q}_{{\rm{i}}}{n}_{{\rm{rms}}}$, stopping the shell expansion at $t\approx 3$–$4\;{\rm{Myr}}$ in Models B and C, despite correspondingly stronger outward forces.

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Let us consider an isolated spherical cloud with mass ${M}_{{\rm{cl}}}$, radius ${R}_{{\rm{cl}}}$, and mean surface density ${{\rm{\Sigma }}}_{{\rm{cl}}}\equiv {M}_{{\rm{cl}}}/(\pi {R}_{{\rm{cl}}}^{2})$. We assume that the cloud is gravitationally bound with the one-dimensional turbulent velocity dispersion $\sigma \ ={({\alpha }_{{\rm{vir}}}{{GM}}_{{\rm{cl}}}/5{R}_{{\rm{cl}}})}^{1/2}$, where ${\alpha }_{{\rm{vir}}}$ is the usual virial parameter of order unity (e.g., Bertoldi & McKee 1992). At t = 0, the cloud forms a star cluster with mass ${M}_{*}=\varepsilon {M}_{{\rm{cl}}}$ instantaneously at its center. The remaining gas mass $(1-\varepsilon ){M}_{{\rm{cl}}}$ is distributed according to Equation (9) with ${r}_{{\rm{c}}}=0.1{R}_{{\rm{cl}}}$ within the cloud radius ${R}_{{\rm{cl}}};$ we have checked that the choice of the core radius has little influence on the resulting ${\varepsilon }_{{\rm{min}}}$ as long as it is sufficiently small. The cluster emits ionizing photons at a rate ${Q}_{{\rm{i}}}={\rm{\Xi }}{M}_{*}$, producing a shell at ${r}_{{\rm{sh}}}={r}_{{\rm{IF,0}}}$, which we determine by substituting ${n}_{{\rm{rms}}}(\lt {r}_{{\rm{IF,0}}})$ for ${n}_{{\rm{rms}}}$ in Equation (1).

To determine ${\varepsilon }_{{\rm{min}}}$, we first take a trial value of $0\lt \varepsilon \lt 1$ and integrate Equation (10) over time until the shell reaches the cloud boundary. At ${r}_{{\rm{sh}}}={R}_{{\rm{cl}}}$, we check whether the shell meets one of the following four disruption criteria: (1) ${v}_{{\rm{sh}}}({R}_{{\rm{cl}}})={v}_{{\rm{bind}}}\equiv {[(1+\varepsilon ){{GM}}_{{\rm{cl}}}/{R}_{{\rm{cl}}}]}^{1/2}$, corresponding to a vanishing total (kinetic plus gravitational) energy of the shell (see Matzner 2002; Krumholz et al. 2006);⁹  (2) ${v}_{{\rm{sh}}}({R}_{{\rm{cl}}})=\sigma$, corresponding to a stalling of the shell expansion by turbulent pressure inside the cloud (e.g., Matzner 2002; KM09; Fall et al. 2010); (3) ${v}_{{\rm{sh}}}({R}_{{\rm{cl}}})$ equal to a large-scale turbulent ISM velocity dispersion, ${v}_{{\rm{turb}}}\sim 7\;{\rm{km}}\;{{\rm{s}}}^{-1}$ (Heiles & Troland 2003), corresponding to merging of the shell with the diffuse ISM; and (4) ${F}_{{\rm{tot}}}({R}_{{\rm{cl}}})=0$, corresponding to a balance between gravity and outward forces (e.g., Ostriker & Shetty 2011; MQT10). If none of these conditions are satisfied or if ${r}_{{\rm{sh}}}$ is unable to expand to ${R}_{{\rm{cl}}}$, we return to the first step and repeat the calculations by changing $\varepsilon$.

We take $\beta =1.5$, $\gamma =11.1$, ${\alpha }_{{\rm{vir}}}=1$, and ${k}_{\rho }=1$ for our fiducial parameters. Figure 7 plots as solid lines ${\varepsilon }_{{\rm{min}}}$ (a) as a function of ${{\rm{\Sigma }}}_{{\rm{cl}}}$ for clouds with ${M}_{{\rm{cl}}}={10}^{5}\;{M}_{\odot }$ and (b) as a function of ${M}_{{\rm{cl}}}$ for clouds with ${{\rm{\Sigma }}}_{{\rm{cl}}}={10}^{3}\;{M}_{\odot }\;{{\rm{pc}}}^{-2}$, for the four disruption criteria given above. As expected, clouds with smaller surface density are more easily destroyed by feedback, while more compact clouds need to have ${\varepsilon }_{{\rm{min}}}$ increasingly closer to unity for disruption. The minimum efficiency depends more sensitively on ${{\rm{\Sigma }}}_{{\rm{cl}}}$ than ${M}_{{\rm{cl}}}$. In the case of the condition ${v}_{{\rm{sh}}}({R}_{{\rm{cl}}})={v}_{{\rm{bind}}}$, for instance, ${\varepsilon }_{{\rm{min}}}$ rises rapidly from 0.02 to 0.66 as ${{\rm{\Sigma }}}_{{\rm{cl}}}$ increases from 10² to ${10}^{3}\;{M}_{\odot }\;{{\rm{pc}}}^{-2}$ for fixed ${M}_{{\rm{cl}}}={10}^{5}\;{M}_{\odot }$, whereas it changes less than a factor of two over ${10}^{2}\;{M}_{\odot }\leqslant {M}_{{\rm{cl}}}\leqslant {10}^{6}\;{M}_{\odot }$ for fixed ${{\rm{\Sigma }}}_{{\rm{cl}}}={10}^{3}\;{M}_{\odot }\;{{\rm{pc}}}^{-2}$. Furthermore, a criterion based on a higher shell velocity at the cloud boundary results in higher ${\varepsilon }_{{\rm{min}}}$.

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Figure 8 plots the contours of ${\varepsilon }_{{\rm{min}}}$ on the ${M}_{{\rm{cl}}}$–${{\rm{\Sigma }}}_{{\rm{cl}}}$ plane for the various criteria. The solid curves draw the results for the variable light-to-mass ratio (see Equations (33) and (34)), while the case of constant ${\rm{\Psi }}=943\;{L}_{\odot }\;{M}_{\odot }^{-1}$ and ${\rm{\Xi }}\ =5.05\times {10}^{46}\;{{\rm{s}}}^{-1}\;{M}_{\odot }^{-1}$ are shown as dotted curves. The difference between the solid and dotted contours is caused by an incomplete sampling of the initial mass function (IMF), which lowers ${\rm{\Psi }}$ for low-mass clusters and thus yields higher ${\varepsilon }_{{\rm{min}}}$ for ${M}_{*}\lesssim {10}^{3}\;{M}_{\odot }$. For the criterion ${v}_{{\rm{sh}}}({R}_{{\rm{cl}}})={v}_{{\rm{bind}}}$, typical GMCs with ${10}^{4}\;{M}_{\odot }\lesssim {M}_{{\rm{cl}}}\lesssim {10}^{6}\;{M}_{\odot }$ and ${{\rm{\Sigma }}}_{{\rm{cl}}}\sim {10}^{2}\;{M}_{\odot }\;{{\rm{pc}}}^{-2}$ in the Milky Way could be destroyed by a handful of O stars (${Q}_{{\rm{i,49}}}\lesssim 10$) with efficiency of $\lesssim 10\%$. On the other hand, denser and more compact cluster-forming clouds or clumps with ${{\rm{\Sigma }}}_{{\rm{cl}}}\gtrsim {10}^{3}\;{M}_{\odot }\;{{\rm{pc}}}^{-2}$ require a star formation efficiency larger than ~50% for disruption.

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In our model, the gas density is proportional to $(1-\varepsilon )$ and the ionizing rate ${Q}_{{\rm{i}}}$ is proportional to $\varepsilon$, so that the size of the H ii region becomes increasingly larger for higher $\varepsilon$. It is thus possible to have ${r}_{{\rm{IF,0}}}\geqslant {R}_{{\rm{cl}}}$, indicating that the whole cloud is completely photoionized from the beginning. The upper right part above the thick blue line in Figure 8(a) corresponds to these clouds destroyed by photoionization without involving shell expansion. This can be seen more quantitatively by considering the critical efficiency ${\varepsilon }_{{\rm{S}}}$ that makes the initial Strömgren radius equal to ${R}_{{\rm{cl}}}$, for which Equation (1) and

$$\begin{eqnarray}&&{n}_{{\rm{rms}}}(\lt {R}_{{\rm{cl}}})\approx (1-\varepsilon ){\left(1+\displaystyle \frac{{k}_{\rho }^{2}}{9-6{k}_{\rho }}\right)}^{1/2}\displaystyle \frac{3{\pi }^{1/2}}{4{\mu }_{{\rm{H}}}}\displaystyle \frac{{{\rm{\Sigma }}}_{{\rm{cl}}}^{3/2}}{{M}_{{\rm{cl}}}^{1/2}},\\ &&\mathrm{for}\;\;{r}_{{\rm{c}}}\ll {R}_{{\rm{cl}}}\end{eqnarray} \tag{ 20 }$$

yield

$$\begin{eqnarray}&&{\varepsilon }_{{\rm{S}}}={ \mathcal C }-\sqrt{{{ \mathcal C }}^{2}-1},\end{eqnarray} \tag{ 21 }$$

where

$$\begin{eqnarray}&&{ \mathcal C }=1+\displaystyle \frac{{f}_{{\rm{ion}}}{\rm{\Xi }}{\mu }_{{\rm{H}}}^{2}}{2{\pi }^{1/2}{\alpha }_{{\rm{B}}}}{M}_{{\rm{cl}}}^{1/2}{{\rm{\Sigma }}}_{{\rm{cl}}}^{-3/2},\;\;\;\mathrm{for}\;\;{k}_{\rho }=1.\end{eqnarray} \tag{ 22 }$$

The red dashed contours in Figure 8(a) plot ${\varepsilon }_{{\rm{S}}}$, demonstrating that the photodestruction boundary corresponds to the loci of ${\varepsilon }_{{\rm{min}}}={\varepsilon }_{{\rm{S}}}$. For the other criteria, ${\varepsilon }_{{\rm{min}}}\lt {\varepsilon }_{{\rm{S}}}$ for the ranges of ${M}_{{\rm{cl}}}$ and ${{\rm{\Sigma }}}_{{\rm{cl}}}$ shown in Figure 8.

The region bounded by a thick red line (solid and dotted lines for varying and fixed ${\rm{\Psi }}$ and ${\rm{\Xi }}$, respectively) in the upper left corner of Figures 8(b) and (d) corresponds to clouds with ${F}_{{\rm{tot}}}({r}_{{\rm{IF,0}}})\lt 0$ from the beginning or ${v}_{{\rm{sh}}}=0$ somewhere before reaching the cloud boundary, indicating that a shell undergoes infall rather than expansion due to strong gravity. In order for the outward force to overcome gravity and drive expansion all the way to the cloud surface, clouds in these regions must have star formation efficiency higher than required by the respective criterion. To describe these "trapped" H ii regions correctly, one needs to consider accretion flows and gas rotation in a flattened geometry, as in Keto (2002, 2003, 2007), which is beyond the scope of the present paper.

Once we calculate the minimum efficiency required for cloud disruption, we are positioned to derive various properties of H ii regions and their host clouds. Figure 9 plots contours of ${Q}_{{\rm{i}}}$ (black) and the mean gas number density in the cloud (red) corresponding to ${\varepsilon }_{{\rm{min}}}$ for the various disruption criteria. Clearly, larger ${Q}_{{\rm{i}}}$ is necessary to destroy more massive and compact clouds. The shaded regions shown in Figures 8 and 9 that run roughly diagonally from the upper left to lower right corners correspond to H ii regions that make a transition from the radiation-dominated to thermally dominated regime in the course of the expansion before disruption (${r}_{{\rm{IF,0}}}\lt {r}_{{\rm{ch}}}\lt {R}_{{\rm{cl}}}$).¹⁰ For clouds with parameters above (below) the shaded area, radiation (thermal) pressure plays the dominant role throughout the expansion. As a demarcation line between thermal- and radiation-pressure-dominated H ii regions, Fall et al. (2010) suggested ${{\rm{\Sigma }}}_{{\rm{cl}}}/(1\;{\rm{g}}\;{{\rm{cm}}}^{-2})=0.15\times {[{M}_{{\rm{cl}}}/({10}^{4}{M}_{\odot })]}^{-1}$ by taking constant ${\varepsilon }_{{\rm{min}}}=0.5$. This is plotted as a blue dashed line in Figure 8(b). As will be shown below, the condition of ${F}_{{\rm{rad,eff}}}={F}_{{\rm{thm,eff}}}$ at ${R}_{{\rm{cl}}}={r}_{{\rm{ch}}}$ results in ${{\rm{\Sigma }}}_{{\rm{cl}}}\propto {\varepsilon }^{-2}{M}_{{\rm{cl}}}^{-1}$ in our model. Therefore, since ${\varepsilon }_{{\rm{min}}}$ is a function of ${{\rm{\Sigma }}}_{{\rm{cl}}}$ and ${M}_{{\rm{cl}}}$ instead of a fixed value as assumed by Fall et al. (2010), the shaded demarcation differs from their proposal.

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We also calculate the time $t({R}_{{\rm{cl}}})$ for the shell to move from ${r}_{{\rm{sh}}}={r}_{{\rm{IF,0}}}$ to ${r}_{{\rm{sh}}}={R}_{{\rm{cl}}}$, plotted in Figure 10 as red contours. Note that $t({R}_{{\rm{cl}}})$ is comparable to or smaller than the gas free-fall time ${t}_{{\rm{ff}}}={(1-{\varepsilon }_{{\rm{min}}})}^{-1/2}{(3{\pi }^{1/2}/8G)}^{1/2}{M}_{{\rm{cl}}}^{1/4}{{\rm{\Sigma }}}_{{\rm{cl}}}^{-3/4}$ shown as gray contours. This is naturally expected for virialized clouds with ${v}_{{\rm{sh}}}({R}_{{\rm{cl}}})=\sigma$ or ${v}_{{\rm{sh}}}({R}_{{\rm{cl}}})={v}_{{\rm{bind}}}$ since the shell velocity approximately follows a power law in time as ${v}_{{\rm{sh}}}({r}_{{\rm{sh}}})\approx {v}_{{\rm{sh}}}({R}_{{\rm{cl}}}){(t/t({R}_{{\rm{cl}}}))}^{\alpha -1}$ before gravity takes over (see Figure 4), with an exponent $2/3\lt \alpha \lt 4/5$ for ${k}_{\rho }=1$, yielding $t({R}_{{\rm{cl}}})=\alpha {R}_{{\rm{cl}}}/{v}_{{\rm{sh}}}({R}_{{\rm{cl}}})\sim {R}_{{\rm{cl}}}/\sigma$. This suggests that cloud disruption by an expanding H ii region is rapid, occurring roughly over the free-fall timescale, provided that the star formation efficiency is larger than the minimum value. For the condition of ${v}_{{\rm{sh}}}({R}_{{\rm{cl}}})={v}_{{\rm{turb}}}$, the slopes of the $t({R}_{{\rm{cl}}})$-contours are different since $t({R}_{{\rm{cl}}})\propto {R}_{{\rm{cl}}}$. Nevertheless, the shell expansion takes less than the free-fall time, except for clouds with very large ${{\rm{\Sigma }}}_{{\rm{cl}}}$ or very large ${M}_{{\rm{cl}}}$.

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One can deduce an approximate result for the minimum efficiency analytically by utilizing the effective outward force given in Equation (15). Multiplying both sides of Equation (10) by ${M}_{{\rm{sh}}}{v}_{{\rm{sh}}}$ and integrating the resulting equation over time, we obtain

$$\begin{eqnarray}&&\int {{dp}}_{{\rm{sh}}}^{2}=2\int {M}_{{\rm{sh}}}({F}_{{\rm{out,eff}}}-{F}_{{\rm{in}}}){{dr}}_{{\rm{sh}}},\end{eqnarray} \tag{ 23 }$$

where ${p}_{{\rm{sh}}}={M}_{{\rm{sh}}}{v}_{{\rm{sh}}}$ is the shell momentum. Assuming ${r}_{{\rm{c}}}\ll {R}_{{\rm{cl}}}$, ${f}_{{\rm{ion}}}=1$, and ${p}_{{\rm{sh}}}\to 0$ as ${r}_{{\rm{sh}}}\to 0$, Equation (23) yields an expression for the shell velocity at the cloud boundary as

$$\begin{eqnarray}{v}_{{\rm{sh}}}^{2}({R}_{{\rm{cl}}}) & = & {\eta }_{{\rm{rad}}}\displaystyle \frac{{\rm{\Psi }}}{c}\displaystyle \frac{\varepsilon }{1-\varepsilon }{R}_{{\rm{cl}}}+{\eta }_{{\rm{thm}}}{ \mathcal T }\displaystyle \frac{{\varepsilon }^{1/2}}{1-\varepsilon }\displaystyle \frac{{R}_{{\rm{cl}}}^{3/2}}{{M}_{{\rm{cl}}}^{1/2}}\\ & & -[{\eta }_{{\rm{sh}}}(1-\varepsilon )+{\eta }_{*}\varepsilon ]\displaystyle \frac{{{GM}}_{{\rm{cl}}}}{{R}_{{\rm{cl}}}},\end{eqnarray} \tag{ 24 }$$

where ${\eta }_{{\rm{rad}}}=2/(4-{k}_{\rho })$, ${\eta }_{{\rm{thm}}}=4/(9-2{k}_{\rho })$, ${\eta }_{{\rm{sh}}}\;=1/(8-3{k}_{\rho })$, ${\eta }_{*}=2/(5-2{k}_{\rho })$, and ${ \mathcal T }=8\pi \quad {k}_{{\rm{B}}}{T}_{{\rm{i}}}{[(3{f}_{{\rm{ion}}}{\rm{\Xi }})/(4\pi {\alpha }_{{\rm{B}}})]}^{1/2}$. Note that the total force on the shell at ${r}_{{\rm{sh}}}={R}_{{\rm{cl}}}$ is approximately given by

$$\begin{eqnarray}&&{F}_{{\rm{tot}}}\approx \displaystyle \frac{\varepsilon {\rm{\Psi }}{M}_{{\rm{cl}}}}{c}+\displaystyle \frac{{ \mathcal T }{M}_{{\rm{cl}}}^{3/4}{\varepsilon }^{1/2}}{{\pi }^{1/4}{{\rm{\Sigma }}}_{{\rm{cl}}}^{1/4}}-\displaystyle \frac{\pi G{{\rm{\Sigma }}}_{{\rm{cl}}}{M}_{{\rm{cl}}}}{2}(1-{\varepsilon }^{2}).\end{eqnarray} \tag{ 25 }$$

The first and second terms on the right-hand side of Equation (25) represent the radiative and thermal pressure forces, respectively, while the last term comes from the total gravity. The ratio of the first to second term is proportional to $\sim {({\varepsilon }^{2}{{\rm{\Sigma }}}_{{\rm{cl}}}{M}_{{\rm{cl}}})}^{1/4}$, suggesting that the relative role of radiation pressure to thermal pressure depends not only on the cloud mass and size but also on the star formation efficiency. The relation in Equation (24) may be combined with the disruption criterion ${v}_{{\rm{sh}}}({R}_{{\rm{cl}}})={v}_{{\rm{bind}}}$, ${v}_{{\rm{sh}}}({R}_{{\rm{cl}}})=\sigma$, and ${v}_{{\rm{sh}}}({R}_{{\rm{cl}}})={v}_{{\rm{turb}}}$ to obtain estimates for $\varepsilon$. Similarly, Equation (25) may be used with the disruption criterion ${F}_{{\rm{tot}}}({R}_{{\rm{cl}}})=0$. The estimates for the minimum efficiencies are plotted as dotted lines in Figure 7. At high and low ${{\rm{\Sigma }}}_{{\rm{cl}}}$, the analytic estimates are very close to the solutions of the ODE (integrating Equation (10)), although they can depart by up to a factor of two between ${{\rm{\Sigma }}}_{{\rm{cl}}}\;=\;$ 10² and 10³$\;{M}_{\odot }\;{{\rm{pc}}}^{-2}$.

4.2.1. Radiation-pressure-driven Limit

When the expansion is dominated by radiation pressure, we can keep only the first term on the right-hand side of Equation (24) to obtain

$$\begin{eqnarray}&&\displaystyle \frac{{\varepsilon }_{{\rm{min}}}}{1-{\varepsilon }_{{\rm{min}}}^{2}}=\displaystyle \frac{\pi {Gc}}{{\eta }_{{\rm{rad}}}{\rm{\Psi }}}{{\rm{\Sigma }}}_{{\rm{cl}}},\;\;\mathrm{for}\;\;{v}_{{\rm{sh}}}({R}_{{\rm{cl}}})={v}_{{\rm{bind}}},\end{eqnarray} \tag{ 26 }$$

and

$$\begin{eqnarray}&&\displaystyle \frac{{\varepsilon }_{{\rm{min}}}}{1-{\varepsilon }_{{\rm{min}}}}=\displaystyle \frac{\pi {\alpha }_{{\rm{vir}}}{Gc}}{5{\eta }_{{\rm{rad}}}{\rm{\Psi }}}{{\rm{\Sigma }}}_{{\rm{cl}}},\;\;\mathrm{for}\;\;{v}_{{\rm{sh}}}({R}_{{\rm{cl}}})=\sigma .\end{eqnarray} \tag{ 27 }$$

These clearly show that the minimum efficiency increases with the mean surface density, independent of the mass. This is consistent with the results of Fall et al. (2010), although their effective outward force includes the trapping factor accounting for hot stellar winds and dust-reprocessed radiation. Similarly, a criterion based on radiation-gravity force balance at the cloud boundary in Equation (25) yields

$$\begin{eqnarray}&&\displaystyle \frac{{\varepsilon }_{{\rm{min}}}}{1-{\varepsilon }_{{\rm{min}}}^{2}}=\displaystyle \frac{{{\rm{\Sigma }}}_{{\rm{cl}}}}{2{\rm{\Psi }}/(\pi {Gc})},\;\;\mathrm{for}\;\;{F}_{{\rm{tot}}}=0,\end{eqnarray} \tag{ 28 }$$

independent of ${M}_{{\rm{cl}}}$ again (see also MQT10 and Raskutti et al. 2015).

4.2.2. Gas-pressure-driven Limit

For expansions driven primarily by ionized gas pressure, Equations (24) and (25) give

$$\begin{eqnarray}&&\displaystyle \frac{{\varepsilon }_{{\rm{min}}}}{{(1-{\varepsilon }_{{\rm{min}}}^{2})}^{2}}={\left(\displaystyle \frac{{\pi }^{5/4}G}{{\eta }_{{\rm{thm}}}{ \mathcal T }}\right)}^{2}{M}_{{\rm{cl}}}^{1/2}{{\rm{\Sigma }}}_{{\rm{cl}}}^{5/2},\;\;\mathrm{for}\;\;{v}_{{\rm{sh}}}({R}_{{\rm{cl}}})={v}_{{\rm{bind}}},\end{eqnarray} \tag{ 29 }$$

$$\begin{eqnarray}&&\displaystyle \frac{{\varepsilon }_{{\rm{min}}}}{{(1-{\varepsilon }_{{\rm{min}}})}^{2}}={\left(\displaystyle \frac{{\pi }^{5/4}{\alpha }_{{\rm{vir}}}G}{5{\eta }_{{\rm{thm}}}{ \mathcal T }}\right)}^{2}{M}_{{\rm{cl}}}^{1/2}{{\rm{\Sigma }}}_{{\rm{cl}}}^{5/2},\;\;\mathrm{for}\;\;{v}_{{\rm{sh}}}({R}_{{\rm{cl}}})=\sigma ,\end{eqnarray} \tag{ 30 }$$

and

$$\begin{eqnarray}&&\displaystyle \frac{{\varepsilon }_{{\rm{min}}}}{{(1-{\varepsilon }_{{\rm{min}}}^{2})}^{2}}={\left(\displaystyle \frac{{\pi }^{5/4}G}{2{ \mathcal T }}\right)}^{2}{M}_{{\rm{cl}}}^{1/2}{{\rm{\Sigma }}}_{{\rm{cl}}}^{5/2},\;\;\mathrm{for}\;\;{F}_{{\rm{tot}}}=0.\end{eqnarray} \tag{ 31 }$$

Equations (29)–(31) imply ${\varepsilon }_{{\rm{min}}}\propto {M}_{{\rm{cl}}}^{1/2}{{\rm{\Sigma }}}_{{\rm{cl}}}^{5/2}$ for ${\varepsilon }_{{\rm{min}}}\ll 1$.

In Section 4.1, we considered a stratified cloud with ${k}_{\rho }=1$ and assumed that infrared photons emitted by dust freely escape the cloud. Here we relax these two constraints to calculate ${\varepsilon }_{{\rm{min}}}$ in more general situations.

First, we explore the cases with differing ${k}_{\rho }$. H ii regions in clouds with a larger central density tend to be denser and smaller in size initially. As a result, the radiation pressure becomes more important in the initial expansion phase for larger ${k}_{\rho }$. Figure 11 plots ${\varepsilon }_{{\rm{min}}}$ as a function of ${{\rm{\Sigma }}}_{{\rm{cl}}}$ for ${k}_{\rho }=0$, 1, and 1.4 in the case of the disruption criterion ${v}_{{\rm{sh}}}({R}_{{\rm{cl}}})={v}_{{\rm{bind}}}$. Three cloud masses ${M}_{{\rm{cl}}}={10}^{4},{10}^{5},{10}^{6}\;{M}_{\odot }$ are chosen. Although clouds with steeper density profile have smaller ${\varepsilon }_{{\rm{min}}}$ and are thus more readily disrupted, ${\varepsilon }_{{\rm{min}}}$ does not exhibit strong dependence on ${k}_{\rho }$. In fact, Equation (24) suggests that the shell momentum depends weakly on ${k}_{\rho }$ only through the η coefficients.

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Next, we examine the effect of dust-reprocessed infrared radiation. In a high-column cloud optically thick to infrared radiation, dust-reprocessed radiation can be trapped within the cloud, providing a significant boost to H ii region expansion (e.g., KM09; MQT10; Hopkins et al. 2011; Ostriker & Shetty 2011; Skinner & Ostriker 2015). For the momentum injection rate by the trapped radiation, we adopt the usual simple prescription

$$\begin{eqnarray}&&{F}_{{\rm{rad,IR}}}={\tau }_{{\rm{IR}}}\displaystyle \frac{L}{c},\end{eqnarray} \tag{ 32 }$$

applicable to smooth and spherical clouds. Here ${\tau }_{{\rm{IR}}}\ ={\kappa }_{{\rm{IR}}}({{\rm{\Sigma }}}_{{\rm{sh}}}+{\int }_{{r}_{{\rm{sh}}}}^{{R}_{{\rm{cl}}}}\rho (r){dr})$ is the infrared optical depth through the cloud, with ${\kappa }_{{\rm{IR}}}$ being the Rosseland mean dust opacity (treated as a constant for simplicity).¹¹ Note that we include the contribution to ${\tau }_{{\rm{IR}}}$ from the portion of the cloud outside the shell as well, corresponding to the maximum possible efficiency of feedback by the trapped radiation. As ${r}_{{\rm{sh}}}$ increases, ${\tau }_{{\rm{IR}}}$ experiences a moderate decrease (by a factor of three from ${r}_{{\rm{sh}}}=0$ to ${r}_{{\rm{sh}}}={R}_{{\rm{cl}}}$ for ${k}_{\rho }=0$).

We integrate Equation (10) by adding ${F}_{{\rm{rad,IR}}}$ to ${F}_{{\rm{out}}}$. Figure 12 plots ${\varepsilon }_{{\rm{min}}}$, determined from two disruption criteria, ${v}_{{\rm{sh}}}({R}_{{\rm{cl}}})={v}_{{\rm{bind}}}$ (left panels) and ${v}_{{\rm{sh}}}({R}_{{\rm{cl}}})=\sigma$ (right panels), for several different values of ${\kappa }_{{\rm{IR}}}$, as functions of ${{\rm{\Sigma }}}_{{\rm{cl}}}$ for ${M}_{{\rm{cl}}}={10}^{6}\;{M}_{\odot }$ (upper panels) and as contours in the ${M}_{{\rm{cl}}}$–${{\rm{\Sigma }}}_{{\rm{cl}}}$ plane (lower panels). The other parameters are the same as in the fiducial case. Clearly, ${\kappa }_{{\rm{IR}}}$ tends to reduce ${\varepsilon }_{{\rm{min}}}$ since clouds are more easily disrupted. But, its effect is significant only for sufficiently large ${\kappa }_{{\rm{IR}}}$ and/or sufficiently massive, high-column clouds so as to have ${r}_{{\rm{sh}}}\lesssim {r}_{{\rm{ch}}}$ and ${\tau }_{{\rm{IR}}}\gtrsim 1$. When ${\kappa }_{{\rm{IR}}}\approx 5\;{{\rm{cm}}}^{2}\;{{\rm{g}}}^{-1}$ appropriate for solar-metallicity gas (Semenov et al. 2003), ${\varepsilon }_{{\rm{min}}}$ is insensitive to ${F}_{{\rm{rad,IR}}}$. For larger values of ${\kappa }_{{\rm{IR}}}$, the minimum efficiency is reduced considerably and the photodestruction regime becomes less extended, as Figure 12(c) shows. For clouds with ${\tau }_{{\rm{IR}}}\gg 1$, the trapped radiation dominates the direct radiation (${F}_{{\rm{rad,IR}}}\gt L/c$). In this case, ${\varepsilon }_{{\rm{min}}}$ computed from the criterion ${v}_{{\rm{sh}}}({R}_{{\rm{cl}}})=\sigma$ becomes close to ${\varepsilon }_{{\rm{min,IR}}}={[{\rm{\Psi }}{\kappa }_{{\rm{IR}}}/(2\pi {cG})-1]}^{-1}$, a prediction from the force balance between ${F}_{{\rm{rad,IR}}}$ and gravity (Ostriker & Shetty 2011). The simulations of Skinner & Ostriker (2015) consider the evolution of turbulent clouds in the limit ${\tau }_{{\rm{IR}}}\gt 1$ where reprocessed radiation dominates direct radiation.

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Young massive stars have dramatic effects on the surrounding ISM through ionizing radiation, winds, and supernova explosions. H ii regions created by ionizing radiation from young star clusters embedded in molecular clouds are able to destroy their natal clouds under some circumstances, preventing further star formation in them. In this paper we have used a simple semianalytic model and hydrodynamic simulations to study dynamical expansion of a dusty H ii region around a star cluster and its role in cloud disruption. Our expansion model is one-dimensional, assuming spherical symmetry, and treats the structure of the ionized region using the solution of Dr11. We assume that as the H ii region expands, the swept-up shell formed at the interface between the ionized region and surrounding neutral neutral gas remains very thin. We solve the shell's temporal evolution subject to outward contact forces arising from radiation and thermal pressures and inward gravity from the cluster and the shell.

In our model, radiation pressure affects the shell expansion indirectly through the enhanced thermal pressure at the ionization front, a feature of the Dr11 solutions. The total outward contact force in our detailed model agrees within ~20% with the combination of the effective radiation force ${F}_{{\rm{rad,eff}}}=L/c$ and the effective gas pressure force ${F}_{{\rm{thm,eff}}}=8\pi \;{k}_{{\rm{B}}}{T}_{{\rm{i}}}{r}_{{\rm{sh}}}^{2}{n}_{{\rm{rms}}}$ used in the simplified model KM09 (see Figure 3). Since ${F}_{{\rm{rad,eff}}}$ is constant while ${F}_{{\rm{thm,eff}}}$ depends on the shell radius ${r}_{{\rm{sh}}}$ as ${F}_{{\rm{thm,eff}}}\propto {r}_{{\rm{sh}}}^{1/2}$ (apart from the weak dependence on ${f}_{{\rm{ion}}}$), expansion is driven primarily by radiation pressure when ${r}_{{\rm{sh}}}\lt {r}_{{\rm{ch}}}$ (or equivalently when ${Q}_{{\rm{i,49}}}{n}_{{\rm{rms}}}\gtrsim {10}^{4}\;{{\rm{cm}}}^{-3}$), and by thermal pressure when ${r}_{{\rm{sh}}}\gt {r}_{{\rm{ch}}}$. Here ${r}_{{\rm{ch}}}$ is the characteristic radius where ${F}_{{\rm{rad,eff}}}={F}_{{\rm{thm,eff}}}$ (Equation (16)). We note that, in practice, radiation forces are conveyed to the surrounding neutral shell indirectly, by compressing the ionized gas strongly to the outer portion of the H ii region and increasing the density immediately interior to the shell. The cluster gravity is important in the early phase of shell expansion, especially in the radiation-driven limit, while shell self-gravity eventually halts expansion, after about the free-fall timescale of the whole cloud.

To validate the assumptions used in our expansion model, we also perform direct numerical simulations of expanding spherical H ii regions for sample cases. We find that despite the presence of small-amplitude acoustic disturbances, the radial density structure of the ionized region in our numerical simulations is overall in good agreement with the static equilibrium solutions of Dr11 throughout the shell expansion. This justifies the use of the Dr11 solutions even for expanding H ii regions. The temporal changes in the shell position and velocity in our expansion model also agree with the results of the numerical simulations until gravity completely stops the expansion (Figures 5 and 6).

Using our expansion model, we explore requirements for cloud disruption by an expanding H ii region around a star cluster in diverse star-forming environments, as characterized by the total cloud mass ${M}_{{\rm{cl}}}$ and the mean surface density ${{\rm{\Sigma }}}_{{\rm{cl}}}$. We also allow for power-law internal density profiles in the cloud. As criteria of cloud disruption, we consider the following four conditions: (1) the shell velocity at the cloud boundary ${R}_{{\rm{cl}}}$ is reduced to ${v}_{{\rm{sh}}}({R}_{{\rm{cl}}})={v}_{{\rm{bind}}}={[(1+\varepsilon ){{GM}}_{{\rm{cl}}}/{R}_{{\rm{cl}}}]}^{1/2}$, corresponding to a vanishing total energy; (2) ${v}_{{\rm{sh}}}({R}_{{\rm{cl}}})=\sigma$, the one-dimensional internal velocity dispersion of the cloud; (3) ${v}_{{\rm{sh}}}({R}_{{\rm{cl}}})=7\;{\rm{km}}\;{{\rm{s}}}^{-1}$, the mean turbulent velocity dispersion of the diffuse ISM; and (4) the net force on the shell at the cloud boundary is zero. As a function of ${M}_{{\rm{cl}}}$ and ${{\rm{\Sigma }}}_{{\rm{cl}}}$, we calculate the minimum efficiency of star formation ${\varepsilon }_{{\rm{min}}}$ needed to satisfy each of the four conditions above (Figures 7 and 8). Using the effective outward forces, we also derive analytic expressions for ${\varepsilon }_{{\rm{min}}}$ in the radiation- and gas-pressure-driven limits (Section 4.2).

Based on the first criterion, ${v}_{{\rm{sh}}}({R}_{{\rm{cl}}})={v}_{{\rm{bind}}}$ (see Figure 8(a)), GMCs in normal disk galaxies (typically ${10}^{4}\;{M}_{\odot }\ \lesssim {M}_{{\rm{cl}}}\lesssim {10}^{6}\;{M}_{\odot }$ and ${{\rm{\Sigma }}}_{{\rm{cl}}}\quad \sim$ 50–200 $\;{M}_{\odot }\;{{\rm{pc}}}^{-2}$) can be destroyed with ${\varepsilon }_{{\rm{min}}}\lesssim 10\%$. For ${M}_{{\rm{cl}}}\leqslant {10}^{5}\;{M}_{\odot }$ and ${\rm{\Sigma }}\leqslant {10}^{2}\;{M}_{\odot }\;{{\rm{pc}}}^{-2}$, expansion is primarily due to gas pressure, whereas both gas and radiation pressures are important at larger ${M}_{{\rm{cl}}}$ and ${{\rm{\Sigma }}}_{{\rm{cl}}}$. Disruption of cluster-forming clumps (${M}_{{\rm{cl}}}\gtrsim {10}^{3}\;{M}_{\odot }$ and ${{\rm{\Sigma }}}_{{\rm{cl}}}\gtrsim {10}^{3}\;{M}_{\odot }\;{{\rm{pc}}}^{-2}$, from Figure 1 of Tan et al. 2014, p. 149) requires a significantly higher efficiency of ${\varepsilon }_{{\rm{min}}}\gtrsim 50\%$ and is mainly driven by radiation pressure. Massive clouds in starbursts (${M}_{{\rm{cl}}}\gtrsim {10}^{5}\;{M}_{\odot }$ and ${{\rm{\Sigma }}}_{{\rm{cl}}}\gtrsim 1\;{\rm{g}}\;{{\rm{cm}}}^{-2}$) would need to convert most of their gas into stars (${\varepsilon }_{{\rm{min}}}\gtrsim 0.9$) for the H ii region to achieve disruption by direct radiation forces. The minimum efficiency for the criterion ${v}_{{\rm{sh}}}({R}_{{\rm{cl}}})=\sigma$ is only slightly higher than that required from ${F}_{{\rm{tot}}}({R}_{{\rm{cl}}})=0$, implying that the gravity is already taking over when the shell velocity drops to σ, naturally expected for a virialized cloud.

Star clusters with masses below $\sim {10}^{3}\;{M}_{\odot }$ are likely to have fewer OB stars and thus a smaller light-to-mass ratio than the predictions of a fully sampled IMF (see Appendix A). The required ${\varepsilon }_{{\rm{min}}}$ increases accordingly for these low-mass systems. Since the size of H ii regions increases with increasing efficiency, there exists a critical efficiency ${\varepsilon }_{{\rm{S}}}$ (Equation (21)) at which the initial Strömgren radius is equal to the cloud size. Clouds with ${\varepsilon }_{{\rm{min}}}\geqslant {\varepsilon }_{{\rm{S}}}$ are regarded as being disrupted by photoionization rather than shell expansion. Such clouds are located in the upper right corner in the ${M}_{{\rm{cl}}}$–${{\rm{\Sigma }}}_{{\rm{cl}}}$ plane in the case of the criterion ${v}_{{\rm{sh}}}({R}_{{\rm{cl}}})={v}_{{\rm{bind}}}$ (Figure 8(a)). Under the criterion ${v}_{{\rm{sh}}}({R}_{{\rm{cl}}})=\sigma$ or ${F}_{{\rm{tot}}}({R}_{{\rm{cl}}})=0$, shells in the upper left corner are subject to gravitational infall rather than expansion. For clouds disrupted by shell expansion, the disruption time is typically comparable to or smaller than the gas free-fall time (Figure 10), implying that the disruption is rapid once an H ii region is formed.

We also examine the effect of differing cloud density profiles by varying the power-law density index ${k}_{\rho }$. When all other quantities are held fixed, momentum deposition is only slightly larger for clouds with larger ${k}_{\rho }$ (Figure 11). Therefore, disruption is quite insensitive to the degree of density concentration. Finally, we explore the effect of trapped IR radiation using a usual spherical symmetry prescription in which the associated radiation force is taken proportional to the IR optical depth through a whole cloud (see Figure 12). The minimum efficiency is lowered only slightly for a dust opacity of ${\kappa }_{{\rm{IR}}}\approx 5\;{{\rm{cm}}}^{2}\;{{\rm{g}}}^{-1};$ a substantial reduction in ${\varepsilon }_{{\rm{min}}}$ requires ${\kappa }_{{\rm{IR}}}\gtrsim 15\;{{\rm{cm}}}^{2}\;{{\rm{g}}}^{-1}$, appropriate for dust-enriched environments.

In this paper, we have calculated ${\varepsilon }_{{\rm{min}}}$ required for cloud disruption. Our results strongly suggest that GMCs are able to end their lives within a single internal crossing time after the formation of a large H ii region with stellar mass ${\varepsilon }_{{\rm{min}}}{M}_{{\rm{cl}}}$. While this ${\varepsilon }_{{\rm{min}}}$ is a minimum, it would also be a reasonable estimate for the net star formation efficiency (at least in the idealized spherical case) for the following reasons. Consider a cloud with $\varepsilon \lt {\varepsilon }_{{\rm{min}}}$. Since it cannot be destroyed by shell expansion, it will continue to form stars, and $\varepsilon$ will increase. However, once $\varepsilon$ reaches ${\varepsilon }_{{\rm{min}}}$, if the disruption is rapid compared to the free-fall time, the efficiency would not increase much beyond this value.

Similarly to our study, MQT10 obtained ${\varepsilon }_{{\rm{GMC}}}$ for five GMCs in various galactic environments. However, there are several notable differences between their and our expansion models. First, MQT10 made a distinction between the star formation efficiency of the GMC and that of cluster-forming gas at the center, the latter of which was fixed to 50%, assuming that half of the "cluster gas" turns into stars and the other half goes into the initial shell mass. Second, the shell expansion in MQT10 begins at a radius (typically a few parsecs) that follows the observed mass–radius relations of star clusters, whereas our model starts from the initial Strömgren radius. Third, MQT10 included an additional inward force, of order $\sim {{GM}}_{{\rm{cl}}}^{2}/{R}_{{\rm{cl}}}^{2}$, due to turbulent pressure, which is not considered in our models. We omit this term as turbulent pressure is dominated by the largest scales and is difficult to model; swept-up gas may add either inward or outward momentum to the shell. The expanding shells in MQT10 have a higher surface density and stronger inward force initially than in our model, which is partly compensated by the outward forces they include to represent protostellar outflows and direct/dust-reprocessed radiation. Despite these differences, the conclusions of MQT10 that the ${\varepsilon }_{{\rm{GMC}}}$ increases with surface density and that the disruption takes place in about a free-fall time of the cloud are all qualitatively consistent with our results.

Murray (2011) estimated star formation efficiency of Galactic star-forming complexes by matching a WMAP sample of luminous free–free sources to host GMCs with typical mass $\sim {10}^{6}\;{M}_{\odot }$ and typical surface density $\sim {10}^{2}\;{M}_{\odot }\;{{\rm{pc}}}^{-2}$, finding ${\varepsilon }_{{\rm{GMC}}}=0.08$, on average, for the luminosity-limited samples (see his Figure 2). These observed star formation efficiencies are in the same range as our minimum efficiencies for the destruction criterion ${v}_{{\rm{sh}}}({R}_{{\rm{cl}}})={v}_{{\rm{bind}}}$ shown in Figure 8(a), suggesting that these clouds may be in the process of disruption by expanding H ii regions. Indeed, most of these star-forming complexes show evidence for expanding, bubble-like morphologies in infrared and radio recombination lines, in excess of turbulent motions (Rahman & Murray 2010; Lee et al. 2012). Murray (2011) also compared inward and outward forces to find ${F}_{{\rm{out}}}\gt {F}_{{\rm{in}}}$ and ${F}_{{\rm{rad,eff}}}\gt {F}_{{\rm{therm,eff}}}$, suggesting that expansion is driven by radiation in many of their samples. García et al. (2014) presented a catalog of GMCs in the inner southern Galaxy and estimated star formation efficiency of those associated with ultracompact H ii regions. In their study, stellar mass was inferred from far-IR luminosity from the IRAS point-like source catalog assuming that far-IR luminosity traces OB stellar population with an age of less than $100\;{\rm{Myr}}$. They found an average star formation efficiency of 3%. While these observations are roughly consistent with our results, the data need to be interpreted with caution, because of selection bias (Murray 2011), neglect of extended emission (García et al. 2014), difficulty in identifying the boundary of a cloud that is being destroyed, and uncertainties involved in lifetime of stellar tracers. Also, it is important to distinguish between net star formation efficiency $\varepsilon$ and observational estimates of "current" star formation efficiency, the latter of which does not allow for the gas inflow/outflow experienced in the past, as well as future star formation before cloud destruction (e.g., Matzner & McKee 2000; Feldmann & Gnedin 2011).

Whether dust-reprocessed radiation can be effective in dispersing star-forming clouds or not has been actively debated (e.g., KM09; MQT10; Skinner & Ostriker 2015). Because the mean free path of infrared photons can often be comparable to the system size, the nonlocal nature of radiation makes it difficult to obtain the solution of the radiative transfer equation. It is only in recent years that the usage of the simplified prescription for ${F}_{{\rm{rad,IR}}}$ has been tested by numerical simulations. For example, Krumholz & Thompson (2012, 2013) used a flux-limited diffusion scheme to investigate matter–radiation interaction in a radiation-supported dusty atmosphere. They found that the photon-trapping efficiency can be greatly reduced by radiation-induced Rayleigh–Taylor instabilities that provide channels for photons to escape, resulting in ananticorrelation between matter and radiation. Using a more advanced (variable Eddington tensor) algorithm, Davis et al. (2014) revisited these calculations and found a reduced anticorrelation, corresponding to stronger matter–radiation coupling. More recently, Skinner & Ostriker (2015) adopted the M1 closure relation to run simulations of turbulent GMC disruption by reprocessed radiation feedback, self-consistently including self-gravitating collapse to produce sources of radiation. This work showed that the usual trapping factor based on the dust optical depth overestimates the radiation momentum deposition rate by a factor of ∼4–5, in part owing to matter–radiation anticorrelation, and in part owing to the cancellation of radiation forces where sources are distributed rather than centrally concentrated. Skinner & Ostriker (2015) also showed that reprocessed radiation is able to limit collapse only when the opacity is large, ${\kappa }_{{\rm{IR}}}\gt 15\;{{\rm{cm}}}^{2}\;{{\rm{g}}}^{-1}$. Here we also find that reprocessing only significantly affects the minimum efficiency when ${\kappa }_{{\rm{IR}}}$ is large.

While in this paper we exclusively focus on the effects of H ii region expansion, diverse feedback mechanisms with different degrees of importance are believed to operate in star-forming environments (MQT10; Fall et al. 2010; Krumholz et al. 2014, p. 243; Matzner & Jumper 2015). For example, outflows and jets from protostars feed turbulent motions within cluster-forming clumps, prevent global collapse, and reduce star formation efficiency therein (e.g., Matzner & McKee 2000; Matzner 2007; Wang et al. 2010). They are a dominant source of momentum injection before massive stars form, but are unlikely to be capable of destroying the intermediate-mass and massive clumps (e.g., Matzner 2002; Fall et al. 2010; Nakamura & Li 2014). Supernova explosions are regarded as the most powerful feedback mechanism for driving turbulence in the ISM (Joung & Mac Low 2006), providing vertical pressure support against gravity and regulating the star formation rates in galactic disks (e.g., Ostriker & Shetty 2011; Kim et al. 2013). Supernovae may also significantly impact or destroy molecular clouds if they are not dispersed by other feedback processes during the lifetime of massive cluster stars (e.g., Hennebelle & Iffrig 2014; Geen et al. 2015b; Walch & Naab 2015). Indeed, observations of mixed-morphology supernova remnants suggest that this is often the case (e.g., Rho & Petre 1998).

Traditionally, thermal pressure of hot gas created by shocked stellar winds is thought to dominate expansion of ionized bubbles (Castor et al. 1975; Weaver et al. 1977; Koo & McKee 1992). The pressure of hot gas trapped within the H ii region would push the inner boundary of the H ii region outward, indirectly doing mechanical work on the swept-up shell. A semianalytic spherical model by Martínez-González et al. (2014) suggests that shocked wind pressure is more important than radiation pressure in driving H ii region expansion. However, observed X-ray luminosity is lower than the theoretical prediction of hot confined gas, casting doubt on the effectiveness of stellar winds in controlling the dynamics of gas around star clusters (Harper-Clark & Murray 2009; Yeh & Matzner 2012; Rosen et al. 2014). Estimates of wind energy lost via various mechanisms suggest that leakage of hot gas through holes in the shell and/or turbulent mixing at hot–cold interfaces can explain the observed low luminosity (Rosen et al. 2014). The approximate calculation by KM09 shows that a shocked wind brings only a modest increase in effective outward force on a porous shell, with a wind trapping factor of order unity. The star formation efficiencies we present here are the minimum for disruption solely by dynamical expansion of an H ii region, but other feedback processes could reduce ${\varepsilon }_{{\rm{min}}}$.

We now comment on caveats of the present study in various aspects. First, we have assumed that the ionizing luminosity remains constant during the shell expansion. For most clouds, this is acceptable as the expansion time $t({R}_{{\rm{cl}}})$ is shorter than $3.8\;{\rm{Myr}}$, a typical main-sequence lifetime of ionizing stars (McKee & Williams 1997; Krumholz et al. 2006). For clouds located in the lower right corner in the ${M}_{{\rm{cl}}}$–${{\rm{\Sigma }}}_{{\rm{cl}}}$ plane (Figure 10), however, the ionizing output may experience a significant drop before cloud disruption. With a decrease in the outward force, the shell expansion would be slowed down and possibly fall back owing to gravity, implying that higher ${\varepsilon }_{{\rm{min}}}$ is required than in our models. Second, while we assume spherical symmetry, expanding shells may be subject to various nonradial instabilities such as ionization front instability (Vandervoort 1962; Kim & Kim 2014), Rayleigh–Taylor instability, and Vishniac instability (Garcia-Segura & Franco 1996; Whalen & Norman 2008) in the early phase, and gravitational instability assisted by external pressure in the late phase (Wünsch et al. 2010; Iwasaki et al. 2011; Kim et al. 2012). At the nonlinear stage, the shell may break up into pieces and create holes through which photons and ionized gas can leak out, reducing feedback efficiency. Moreover, any shell that forms in an inhomogeneous cloud would itself be inhomogeneous, such that acceleration would be nonuniform (see below).

Third, our model considers a situation where the shell expansion is driven solely by a single embedded H ii region, as in Fall et al. (2010) and MQT10. In reality, however, star formation in a molecular cloud may be distributed spatially with a population of subclusters (McKee & Williams 1997). This results in expanding H ii regions of various sizes interacting with each other. In the early stage of expansion, the momentum injection by individual subclusters may not simply add up to the total due to cancellation; this effect is evident in the simulations of Skinner & Ostriker (2015). But once a main shell created by the most luminous H ii region expands to a volume large compared to that of most ionizing sources (assuming some degree of subcluster concentration within the cloud), the effective radiation source is the same as for a single central cluster.

More importantly, we have ignored blister-type H ii regions that can effectively vent ionized gas through low-density regions. Since real turbulent clouds have a lognormal density distribution characterized by many clumps and holes, even an initially fully embedded H ii region is likely to transform into blister-type. The analytic model of Matzner (2002), in which all H ii regions are taken to be blister-type, showed that photoevaporative mass loss from small subclusters (dominated by those around turnover in the cluster luminosity function; McKee & Williams 1997) alone can limit star formation efficiency of galactic GMCs to below ~10%, with the photodestruction timescale decreasing as a function of the cloud mass. When both mass-loss mechanisms are considered, dynamical disruption by shell expansion is more frequent than photodestruction alone, but significant photoevaporative mass loss also occurs prior to disruption if clouds survive for several free-fall times (Krumholz et al. 2006). However, these analyses of photoevaporation did not allow for the reduced surface area that gas confined in dense filaments presents to radiation, which tends to reduce photoevaporation. Indeed, Dale et al. (2012, 2013a) found that less than 10% of the gas is photoionized in their simulations. The reduced effective area of clumpy clouds also reduces radiation forces (Raskutti et al. 2015; Thompson & Krumholz 2016). The relative importance of the mass-loss mechanisms in inhomogeneous clouds is difficult to assess and would require numerical simulations (but see Matzner & Jumper 2015).

In inhomogeneous turbulent clouds, gas dispersal would take place in a gradual, rather than impulsive, fashion. Both analytic and numerical investigations of radiation-only feedback models show that broad distributions of surface density and/or radiation flux cause locally low-column, super-Eddington gas parcels to be ejected from the system early on, while higher surface density parcels are ejected only later, when the total luminosity of stars has increased (e.g., Raskutti et al. 2015; Thompson & Krumholz 2016). For example, the blue shaded region in Figure 11 shows the range of net star formation efficiency of turbulent clouds with width of lognormal surface density distribution in the range $0\lt {\sigma }_{\mathrm{ln}{\rm{\Sigma }}}\lt 1$, predicted from the analytic model of Raskutti et al. (2015) (see their Equation (21)), which considers just radiation pressure on dust from nonionizing radiation. Stellar mass grows until there is little gas above the critical surface density, making $\varepsilon$ increase beyond what is expected from the uniform case (lower bound). Note also that the inclusion of gas pressure in our model is responsible for the dramatic difference in efficiency for low-${{\rm{\Sigma }}}_{{\rm{cl}}}$ clouds.

Because the current models consider only uniform spherical shells of gas rather than a broadened surface density probability distribution function, we underestimate the luminosity and hence ${\varepsilon }_{{\rm{min}}}$ required to eject the denser clumps in a cloud via direct radiation forces. However, the direct photoevaporation enabled by star formation close enough to the cloud periphery would also tend to lower ${\varepsilon }_{{\rm{min}}}$. Additional star formation can also in principle be triggered in shocked shells by the collect and collapse process (e.g., Elmegreen & Lada 1977; Hosokawa & Inutsuka 2006; Dale et al. 2007; Iwasaki et al. 2011; Dale et al. 2013b). To address these complex issues, it is necessary to perform three-dimensional RHD simulations of star cluster formation in a turbulent cloud with both ionizing and nonionizing radiation.