HOW TO FIND YOUNG MASSIVE CLUSTER PROGENITORS

What are the initial conditions that lead to bound stellar cluster formation, e.g., YMCs like NGC 3603 and R136? We must consider the differences between low-mass and high-mass star-forming clouds. In the solar neighborhood, the local SFE, defined as the final mass in stars formed in a cloud divided by the initial mass of gas, of the low-mass star forming regions is reported to be 5% or less (Evans et al. 2009). Numerical experiments (e.g., Lada et al. 1984; Geyer & Burkert 2001; Goodwin & Bastian 2006) demonstrate that the formation of gravitationally bound clusters requires that the local SFE ∼30%. Observations show similar results of higher SFE for massive star-forming regions (e.g., Adams 2000; Nürnberger et al. 2002) and hint that some of these systems could evolve into open clusters. The initial conditions that enable the formation of YMCs may be responsible for the high SFE observed in these bound stellar clusters. From the gas clump phase to the stellar clusters the gas is somehow efficiently converted to stars. Assuming that the SFE of 30% is critical for forming a YMC, then the MPC mass should be greater than 3 × 10⁴ M_☉.

Star formation in a clump ends when its gas is dispersed by the feedback energy injected from its newborn stars. Gas can be dispersed when the outward pressure generated by feedback exceeds the inward pressure of the overburden of gas in the cluster gravitational potential well (Bally 2011). The ratio of the local escape velocity divided by the sound speed for photo-ionized gas, $\Omega = v_{{\rm esc}} / c_{\rm \mathsc{ii}}$, plays a crucial role in determining if a clump forms a bound cluster or not. Feedback from accreting low-mass protostars is dominated by bipolar outflows (Bally 2011). As stars reach several solar masses, their non-ionizing near-UV radiation photo-heats surrounding cloud surfaces, raising the sound speed to ∼ 1 to 5 km s⁻¹. Stars greater than 10 M_☉ (early-B and O stars) ionize their surroundings, raising the sound speed up to ∼ 10 km s⁻¹. Although protostellar outflows, stellar winds, and radiation pressure of ever-increasing strength can also raise the effective sound speed by generating internal motions, only the effects of ionizing radiation will be considered here. See Section 2.2 for details.

If Ω < 1, the gas can be dispersed in a few crossing times from the star-forming clump, bringing star formation to a halt with a low-stellar density. On the other hand, when Ω > 1 the gas will remain bound and can continue to form new stars or accrete onto existing ones until further increases in stellar luminosity or mechanical energy injection raise the effective sound speed to a value greater than the escape speed. As the stellar mass increases, energy released by the forming embedded cluster grows (Miesch & Bally 1994). In such a cluster, photo-ionized plasma will remain gravitationally bound by the cluster potential and recombinations in the ionized medium will tend to shield denser neutral clumps, allowing star formation to proceed to high efficiency. This model works under the assumption that the entire gas clump is instantly ionized, which is a worst case scenario for a clump to remain bound. This means that even if there is a large number of OB stars present in a clump, its gas will unlikely be fully ionized and our model would still hold. We discuss the details of ionization and its effect on pressure balance and ongoing star formation below.

Feedback, which plays a major role in the self-regulation of star formation can be subdivided into two forms: mechanical feedback consisting of protostellar outflow, stellar winds, and supernova explosions, and radiative feedback which can be subdivided into radiation pressure, non-ionizing FUV heating, and ionizing EUV heating. In massive cluster forming environments, prior to the explosion of the most massive star, stellar winds will dominate mechanical feedback and all three radiative mechanism can be important (Bally 2011).

Consider a fiducial reference clump of M_gas = 2 × 10⁴ M_☉ and M_stellar  =   ∼ 1 × 10⁴ M_☉ (SFE =30%), a total luminosity L_f, Lyman continuum photon luminosity Q_f, and a total stellar wind mass loss rate $\dot{M}_f$ at V_f (wind terminal velocity) which corresponds to 50 O7 stars (30 M_☉) located at the center of the cluster. The assumed quantities for the fiducial cluster are shown in the denominators of the equations below (see Martins et al. 2005; Donati et al. 2002 and references therein). Through 10⁴ Monte Carlo iterations the ratios of L, Q, and $\dot{M}$ between the stellar population of the fiducial cluster and Salpeter-like cluster (8 M_☉ < M_star < 110 M_☉) are shown to be L_f/L_Salpeter = 14.0, Q_f/Q_Salpeter = 2.2, and $\dot{M}_f/\dot{M}_{{\rm Salpeter}} = 8.9$. These calculations were done using Murray & Rahman (2010) for Q and Crowther (2000) for $\dot{M}$. The gravitational radius for this cluster $r_G = 2 {\it G M} / c_{\rm \mathsc{ii}}^2$ = 2.13 pc and α_B ≈ 2.6 × 10⁻¹³ cm³ s⁻¹ is the case-B recombination rate for H at a temperature of 10⁴ K.

Below, we consider the radiation pressure, stellar wind ram pressure, and the internal pressure of a uniform density ${\rm H\,\mathsc{ii}}$ region. These pressures are evaluated at D = 2 pc from the center of the cluster, a distance close to the gravitational radius for the reference cluster.

• 1.  
  Stellar wind:

  $$\begin{eqnarray} P_{w} &=& \rho (r) V^2 = \dot{M} V / {4 \pi D^2}\nonumber\\ &=& 1.32 \times 10^{-10}\, \hbox{dynes\ cm}^{-2}\, \left[\frac{\dot{M}_f}{5 \times 10^{-6}\, M_{\odot }\ {\rm yr}^{-1}} \right] \nonumber\\ &&\times \left[\frac{V_f}{2 \times 10^3\ \mathrm{km\ s}^{-1}} \right] \left[\frac{D_f}{\hbox{2\, pc}} \right]^{-2}. \end{eqnarray} \tag{ 1 }$$
• 2.  
  Radiation pressure on an optically thick surface:

  $$\begin{eqnarray} P_{{\rm rad}} & = L / 4 \pi c D^2\nonumber\\ & = 2.68 \times 10^{-9}\, \hbox{dynes\ cm}^{-2}\, \left[\frac{L_f}{10^7\, L{_{\odot }}} \right] \left[\frac{D_f}{\rm{2\, pc}} \right]^{-2}\!.\quad \end{eqnarray} \tag{ 2 }$$
• 3.  
  Thermal pressure of a uniform density ${\rm H\,\mathsc{ii}}$ region with Strömgren radius equal to D = 2 pc:

  $$\begin{eqnarray} P_{\rm{H\,\mathsc{ii}}} &=& \mu m_H c_{\rm \mathsc{ii}}^2 (3 Q / 4 \pi \alpha _B)^{1/2} D^{-3/2}\nonumber\\ &=& 5.40 \times 10^{-9}\, \hbox{dynes\ cm}^{-2}\, \left[\frac{Q_f}{10^{51}\ \hbox{photons\ s}^{-1}}\right]^{1/2} \nonumber\\ &&\times\left[\frac{D_f}{\hbox{2\, pc}} \right]^{-3/2}. \end{eqnarray} \tag{ 3 }$$

These pressures are listed in order of increasing significance for the fiducial reference cluster. It is important to note that the equations express the feedback pressures as power laws of distance from the center of the fiducial cluster. The pressure from Equation (3) tapers off the slowest among the three pressures, which bolsters the dominance of photo-ionization at large distances. A similar result was found in Krumholz & Matzner (2009), but Murray et al. (2010) conclude that radiation pressure is the dominant mechanism in dispersing gas under different scaling assumptions. If radiation does play a role in dispersing the gas it could mass overload the shell around the H ii region and force some gas to escape, which may imply that a YMC's SFE is unlikely to ever be 100%. However, the ionized gas pressure will not unbind the system and could create large optically thin bubbles, allowing much of the radiation pressure to escape without interacting with any gas particles.

What are the expected physical geometries of MPCs? Using both the YMC progenitor model and observations of YMCs, we derive the maximum sizes and mass of the MPCs. In the bound ${\rm H\,\mathsc{ii}}$ gas clump model, the key requirement is $v_{{\rm esc}}>c_{\rm \mathsc{ii}}$ to form a YMC. By setting the potential and kinetic variables to −GMmr⁻¹ and ${1}/{2} m c_{\rm \mathsc{ii}}^2$, we can solve for r. We denote this radius as r_Ω, which is a function of the clump mass, $r_{\Omega }(M_{{\rm clump}}) = 2 {\it G M}_{{\rm clump}} c_{\rm \mathsc{ii}}^{-2}$. For MPC candidates with masses of 3 × 10⁴–3 × 10⁶ M_☉, r_Ω ranges between 2.6 and 258.1 pc (see Table 1 for details). The r_Ω values for >10⁶ M_☉ clumps are large and are very unlikely to form a >10⁶ M_☉ gravitationally bound star cluster as the free-fall time of the system could exceed when supernovae could begin and disrupt the system. Hence, an upper size limit for these objects is needed.

Table 1. Predicted Proto-cluster Properties

		Galactic	d ⩽ 20 kpc
Mass	log Pcen/k	rΩ	rvir	θ20 kpc	F500	F850	F1100
(M☉)	(K m−2)	(pc)	(pc)	"	(Jy)	(Jy)	(Jy)
3 × 104	13.15	2.6	5.1	53.62	30.78	4.17	1.81
3 × 105	13.81	25.8	11.0	226.89	17.19	2.33	1.01
3 × 106	14.47	258.1	23.8	490.91	36.72	4.98	2.16
		Extragalactic	d ⩽ 40 Mpc	θALMA = 05
Mass	log Pcen/k	rΩ	rvir	θ40 Mpc	F450	F850	F1200
(M☉)	(K m−2)	(pc)	(pc)	"	(mJy)	(mJy)	(mJy)
3 × 107	15.13	...	51.2	0.53	77.7	7.8	2.3
3 × 108	15.80	...	110.3	1.14	167.5	16.7	5.1
3 × 109	16.47	...	237.7	2.45	360.6	36.6	11.0

Notes. Top half: the predicted physical properties of the YMC progenitors, which are observable throughout the Milky Way (< = 20 Kpc) with the Hi-GAL, ATLASGAL, and BOLOCAM Galactic plane surveys. Bottom half: the predicted physical properties of the YMC progenitors, which are observable up to 40 Mpc with ALMA assuming an angular resolution of 05. The values for r_Ω are not shown for the extragalactic sources since r_vir ≪ r_Ω in all cases and hence such objects are immediately considered as PMC candidates. All fluxes are derived using r_Ω for the ⩽8.4 × 10⁴ M_☉ and r_vir for >8.4 × 10⁴ M_☉. Note that the fluxes for the extragalactic clumps are calculated using the rest wavelengths.

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There is a well-measured constraint on YMCs that we can apply to predict the upper limit radii for the MPCs. Recent high resolution, multi-epoch spectral studies of YMCs like NGC 3603, Arches, Westerlund 1, and R136 have shown these systems to be in or close to virial equilibrium at ages of ∼1 Myr (Rochau et al. 2010; Clarkson et al. 2011; Cottaar et al. 2012; Hénault-Brunet et al. 2012). This implies that the YMCs must have gone through at least one full crossing time before their presently observed age. Using the crossing time equation from Portegies Zwart et al. (2010), t_cross = (GMr⁻³_vir)^−1/2, and fixing the crossing time to 1 Myr we can solve for r_vir = (GMt²_cross)^1/3, which is only dependent on the mass. For the same clump mass range mentioned for r_Ω above, r_vir spans from 5.1 to 23.8 pc (see Table 1 for details). Note that the r_Ω and r_vir values are the upper limit radii for the MPCs. This is an important aspect to keep in mind as there is no evidence for YMCs to have a proportionality between mass and radius (Larsen 2004; Bastian et al. 2012). r_Ω and r_vir have similar radii between 10⁴ M_☉ and 10⁵ M_☉ intersecting at 8.4 × 10⁴ M_☉, but at >10⁶ M_☉ r_Ω ≫ r_vir. Extragalactic predictions regarding more massive MPCs (3 × 10⁷–3 × 10⁹ M_☉) are provided in Table 1.

Figure 1 shows the upper limit radii relative to clump masses for r_Ω and r_vir. We include a shaded region marking where MPC candidates reside. Infrared dark clouds and clumps reported in Rathborne et al. (2006) and Walsh et al. (2011) are included to show where they lay on the plot relative to r_Ω and r_vir. Five gas clumps fit within the MPC criterion: G0.253+0.016 (Longmore et al. 2012), an MPC candidate in the Antennae galaxy (Herrera et al. 2012), and three Galactic clumps reported in Ginsburg et al. (2012) which are discussed in further detail in their paper.

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We took a collection of Galactic YMCs summarized in Portegies Zwart et al. (2010) and estimated their progenitor masses by multiplying their current stellar mass by a factor of 3 to account for SFE∼30%. Eight of the twelve estimated YMC progenitors are within the virial and Ω radii limits. The other four are found not only close to the upper limit of the r_Ω line, but appear to follow the line. If we assume that the SFE ≳ 30% then the estimated YMC progenitors will move above r_Ω line.

From the derived physical properties of the MPCs discussed, we predict the MPC's integrated fluxes in the HiGAL, ATLASGAL, and BGPS surveys in Table 1 using mass to flux conversions discussed in Kauffmann et al. (2008) assuming T = 20 K. If T = 40 K, then the flux will increase by a factor of ∼2. Between the wavelengths of 500 μm and 1100 μm, the emission from the bright MPCs and the interstellar medium are optically thin throughout the Galactic plane. We predict that the surveys should be sensitive to all of the MPCs in the Milky Way. Tackenberg et al. (2012) independently came to similar conclusions for ATLASGAL. Moreover, the sources for the mass clumps ⩾3 × 10⁴ M_☉ are resolvable up to 20 kpc. We calculated the clump central pressures from Johnstone et al. (2000). These values are provided in Table 1. Note that HiGAL has advantages over ATLASGAL and BGPS, since it has no spatial filtering and has the highest sensitivity among these surveys.

Ginsburg et al. (2012) used the predicted properties to develop a selection criteria from the BGPS catalog and discovered three MPC candidates. The masses of the objects can be determined from the dust emission (M_dust), but comparing the dust masses to virial mass, M_vir, is important in determining whether an MPC candidate is gravitationally bound or not. Following similar treatment from Longmore et al. (2012), one can use ΔV from molecular line tracers (e.g., HCO + and N₂H +) to approximate a clump's virial mass, M_vir∝r_clumpΔV². If M_dust ∼ M_vir, then we can assume that the clump is consistent with being gravitationally bound. Estimating the mass of a clump from continuum emission requires a known distance to the objects using V_LSR, which line surveys can provide (e.g., Walsh et al. 2011; Schlingman et al. 2011). The combination of these line and continuum Galactic plane surveys will allow us to obtain a near complete census of MPCs in the Milky Way.

In Table 1, we show our predictions for the most massive MPCs (>10⁷ M_☉) in the Local Universe regarding ALMA's observing capabilities. The given fluxes are calculated for the MPCs at a distance of 40 Mpc and θ ∼05. These objects, if near the upper limit of r_vir, could be resolvable at these distances.