Resolved Star Formation Efficiency in the Antennae Galaxies

SSCs are the most massive and dense type of stellar cluster, and represent one of the most extreme modes of star formation in the universe (O'Connell et al. 1994). First identified as bright blue knots of star formation (e.g., Arp & Sandage 1985; Melnick et al. 1985), these objects are now generally considered to be the precursors to globular clusters (Meurer et al. 1995). However, various lines of evidence suggest that perhaps as few as ∼1% of SSCs survive to become globular clusters (Fall & Zhang 2001; Fall et al. 2005; Fall 2006). A number of factors play a role in determining whether a specific SSC will survive to become a globular cluster, including stellar feedback, tidal shocks, two-body relaxation, and tidal truncation (Gnedin & Ostriker 1997). One of the first criteria that a given SSC must meet to (eventually) evolve into a globular cluster is that it must be gravitationally bound. Moreover, even if an SSC is originally bound at the time of its formation, if a significant amount of the binding mass is in the form of gas, the cluster is likely to become unbound when the gas is removed through stellar feedback. Thus, SFE also has an important role in the formation and survival of globular clusters.

The impact of SFE on the evolution and long-term survival of star clusters has been studied by a number of authors for a distribution of conditions. For example, early work by Hills (1980) based on the virial theorem suggested that some clusters can lose as little as 10% of their mass (SFE = 90%) and still become unbound after the gas was lost from the system. The general requirement for high SFEs is supported by Geyer & Burkert (2001), who suggest that the formation of bound clusters (e.g., globular clusters) requires extremely high SFEs of ∼50%.

More recent work using simulations has been able to study the impact of SFE on cluster evolution using more sophisticated physical processes. For example, using a grid of simulations, Baumgardt & Kroupa (2007) found that star clusters can remain bound with SFEs as low as 10% provided that gas is removed from the clusters over long timescales. If, however, gas is removed instantaneously, clusters with SFE < 33% did not survive. Subsequent work by Pelupessy & Portegies Zwart (2012) used simulations of clusters with 1000 stars to explore their dynamical evolution under different scenarios. Their simulations allow for clusters with SFEs as low as 5% to survive, but as with Baumgardt & Kroupa (2007), cluster survival is more likely if the gas removal does not take place during violent quasi-instantaneous episodes.

Observationally determining the SFE for globular clusters presents numerous challenges. The first issue is that with current observing capabilities, it is not possible to observe the formation of globular clusters ≳10 Gyr in the past. Instead, we turn to relatively nearby objects that are generally considered to be analogous to adolescent globular clusters—super star clusters. The second issue is that determining the SFE requires knowing both the total stellar mass that forms in a given molecular cloud, and the total mass of the molecular cloud. However, these two properties are largely mutually exclusive—once stars have formed, they start disrupting the cloud, and if there is still molecular material available, additional stars could form. In fact, whether molecular material can remain or be reintroduced after the first generation of stars is being investigated by a number of studies. For example, whether various types of "polluters" (e.g., AGB stars, fast rotating massive stars, and massive stars in binary systems D'Ercole et al. 2008; Renzini 2008; Decressin et al. 2009; de Mink et al. 2009; Conroy & Spergel 2011; Krause et al. 2013) was explored by Cabrera-Ziri et al. (2015) in three young (50–200 Myr) molecular clouds. They found no evidence of CO(3–2) emission in these YMCs, placing strong constraints on the various "polluter" scenarios.

This problem can be at least partially mitigated if a statistically significant sample of SSCs with known ages is available. In this case, the relative fraction of stellar and gas mass can be tracked through a cluster's evolution. In this case, the ratio ${M}_{\mathrm{stars}}/({M}_{\mathrm{gas}}+{M}_{\mathrm{stars}})$ becomes a function of time, and here we adopt the term "instantaneous mass ratio" (IMR). The IMR is only equal to the actual SFE at a time when stars have ceased forming and the molecular cloud has not been disrupted. One can debate whether such a time is overly idealized or actually exists—whether stars have truly stopped forming, or whether the molecular cloud is not significantly disrupted until star formation has ceased. In any case, one would predict a trend in which the youngest SSCs would be associated with the lowest IMR values, and that as the molecular cloud is destroyed the IMR should approach unity.

In order to constrain the impact of SFE on cluster survival through observations, an optimal target will be relatively nearby (for both sensitivity and resolution limitations), and also host a statistical sample of star clusters. As the closest and youngest example in Toomre's 1977 sequence of prototypical mergers (Toomre 1977), the Antennae galaxies (NGC 4038/39) have been a target for multiwavelength observations on the internal structure of galaxy mergers. The IRAS satellite found that virtually all extragalactic luminous infrared sources were mergers (Sanders et al. 1988), making them a prime laboratory for studying early stages of star formation. With an infrared luminosity of ${L}_{\mathrm{IR}}\sim {10}^{11}\,{L}_{\odot }$, the Antennae galaxy system is barely classified as an LIRG, but the vast molecular gas reservoir ($\gt {10}^{10}\,{M}_{\odot }$, Gao et al. 2001; Wilson et al. 2003) suggests that the Antennae may evolve into a superluminous infrared galaxy as the merger continues.

With a molecular gas reservoir comparable to that of an early-stage ULIRG, the Antennae has an abundance of the ingredients needed for star formation. HST observations have identified >10,000 stellar clusters in the overlap region alone (Whitmore et al. 2010). The close distance to the Antennae galaxies (d ∼ 22 Mpc, Schweizer et al. 2008) yields better linear resolution than that available for ULIRGs more common at higher redshifts. As a nearby snapshot in the evolution of a gas-rich merger, the Antennae provides a crucial opportunity to better understand when, and how efficiently, stars form in a merger environment and the initiation of starburst galaxies in exquisite detail.

The organizational structure of this paper is as follows. In Section 2, we describe the submillimeter and optical observations utilized in this study and methods used for calculating the stellar and molecular masses. In Section 3, we present the process of determining the IMR(t) through measurements centered on stellar clusters. In Section 4, we present the algorithm and method for determining the IMR(t) through measurements centered on molecular gas. In Section 5, we report the total mass information and IMRs for both measurement schemes. In Section 6, we discuss the implications of the results in the context of understanding early stages of star formation. Finally, in Section 7, we summarize our findings.

We utilize previously attained ALMA CO(3–2) Cycle 0 data of the Antennae overlap region (Figure 1). These data were first presented in Whitmore et al. (2014), and detailed information on the configuration and calibration of such data is given in that work. In summary, a 13 point mosaic was obtained with the "extended" configuration (∼400 m maximum) covering the overlap region between the two interacting galaxies. These data did not contain short or zero spacings; however, the size scales of the structures of interest here are sufficiently compact (≲100 pc ≈ 1''), and this is not a concern.

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The resulting CO (3–2) data cube has an angular resolution FWHM of 056 × 043, equivalent to (59 × 45 pc) at the distance of the Antennae, ∼22 Mpc. The rms of this data cube was determined to be 3.3 mJy beam⁻¹ using line-free channels. These observations enabled the high resolution cospatial study of the mass of molecular gas as compared to the mass of stars and star clusters (i.e., the SFE).

We retrieved Hubble Space Telescope images of the Antennae galaxies (NGC 4038/4039) using the Advanced Camera for Surveys (ACS) from the Hubble Legacy Archive. The original observations were published in Whitmore et al. (2010) and include multiband photometry in the following optical broadband filters: F435W, F550M, and F814W. All optical observations were made with the Wide Field Camera on the ACS and exposure times total 2192 s, 2544 s, and 1680 s, respectively. The observations were taken with a scale of 0049 pixel⁻¹ and yield a field of view of ∼202'' × 202''. The registration of the HST optical images was done following the methods of Brogan et al. (2010). Catalogs of clusters and their optical properties in the Antennae were constructed from the $F550M$ image using DAOFIND in IRAF and were presented in Whitmore et al. (2010). For our purposes, we use the "concentration index" CI of each source in the catalog to select for clusters and remove any bright stars from our sample. This is defined in Whitmore et al. (2010) as the difference between magnitudes measured within a 1 and a 3 pixel radius with more concentrated objects having lower values of CI. As in Whitmore et al. (2010), we consider sources with $\mathrm{CI}\gt 1.52$ to be clusters and include them in our analysis.

Our prescription for the IMR includes a term representing the stellar mass of a given region, which therefore necessitates determining the mass-to-light ratio, $M/{L}_{814}$. We attempted several methods of determining the mass-to-light ratio of spatial bins produced by the Weighted Voronoi Tessellation (WVT; see Section 4) before settling on the procedure described later in this section. These approaches aimed to characterize the mass-to-light ratio in the Antennae overlap region as a function of age using (1) model predictions and (2) an empirical relationship.

For the first approach, we generated Starburst99 models for the I-band mass-to-light ratio, $M/{L}_{I}$, i.e., we generated models for M_V and V − I and combined them in such a way to result in M_I (Leitherer et al. 1999). These models were generated for total cluster masses of ${10}^{4}\mbox{--}{10}^{7}\,{{\rm{M}}}_{\odot }$, various metallicity models, Z from 0.008–0.04, Salpeter and Kroupa IMFs, and upper mass limits on the IMF from 20 to 100 M_⊙. From these models, we generated functions of M/L_I with respect to age. The reduced chi-squared statistic of the models was calculated with respect to the relationship of cluster M/L_I with age derived from Whitmore et al. (2010). In addition to chi-square values >100, adopting this approach requires assuming an extinction value a priori, and given the range of extinction values (particularly for clusters with ages ≲10 Myr), we found that this method was not robust.

We then attempted to derive a strictly empirical relationship between the observed value of cluster M/L_I from Whitmore et al. (2010) as a function of age. This was done by calculating the median of cluster M/L_I for each logarithmic age bin as described below.

From Whitmore et al. (2010), we gathered the following parameters for each cluster in the overlap region: mass (M_⊙), age (τ), apparent magnitude in the I-band (m_I), and the best estimate of reddening ($E(B-V)$). The I-band apparent magnitudes were then corrected for extinction using the procedure described in Section 4.2 and converted to absolute magnitudes assuming a distance to the Antennae galaxies of d = 22 Mpc. We then calculated the mass-to-light ratio to be the ratio of the mass to the corrected I-band luminosity for a given cluster. Figure 2 shows the relationship of this ratio versus age of the cluster. Here we opt to use the I-band luminosity as the primary value for "light" as opposed to converting to a luminosity in solar luminosity units L_⊙, which requires additional assumptions in the conversion to bolometric luminosity.

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As with the model-based approach, the scatter in the data at the youngest ages (the clusters we are most interested in) spans three decades in the mass-to-light ratio, and any ensemble approach improperly estimates M/L_I for the majority of the young clusters. For this reason, we viewed the ensemble median M/L_I as a function of age to be too general for the wide variety of clusters and environments. Rather, we assigned to each region the mass-to-light ratio of the enclosed star cluster. In the case where there are multiple clusters in a region, we assigned the median of those to that particular region. These mass-to-light ratios are shown in relation to positions in the Antennae overlap region in Figure 3.

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The assumptions made in this process regarding noncluster light in a region follow those discussed in Section 3. Namely, we assume that the diffuse intercluster light in the aperture between the cluster and the larger aperture boundary has approximately the same mass-to-light ratio as the central cluster, i.e., they are both dominated by the same population. In a given WVT bin that contains at least one star cluster, the mean fraction of light than can be attributed to the cluster is 11%, the distribution of this fraction for all WVT bins with a star cluster can be seen in Figure 4.

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The total CO(3–2) emission in a given aperture was measured from the moment 0 (integrated intensity) map of the ALMA Cycle-0 observations (see Section 2.1). The α_CO conversion factor assumes a rotational transition of CO(J = 1−0). Since our measured emission of CO is the J = 3−2 rotational transition, we must first scale our luminosity of this upper transition to that expected for the ground state. We adopt CO(J = 1−0) ∼2 × CO(J = 3−2), consistent with the median value found by Wilson et al. (2008) for a sample of LIRGs and with the values derived by Ueda et al. (2012) found for giant complexes in the Antennae.

We assume a Milky Way metallicity for the Antennae galaxies. For this assumption, the CO-to-H₂ conversion factor is 4.3 M_⊙ (K km s⁻¹ pc²)⁻¹ (Bolatto et al. 2013). The units of the α_CO conversion factor necessitate a CO luminosity L_CO, yet simply summing the flux density in a region yields a measurement in units K km s⁻¹ pixel². We adopted the following to calculate the corresponding CO luminosity of a region. The pixel scale was taken from the header of the CO(J = 3−2) moment-0 map as Δ = 2.22 × 10⁻⁵ degree, corresponding to 008 pixels. Assuming a distance to the Antennae of d = 22 Mpc (Schweizer et al. 2008), we determined the physical scale corresponding to a pixel, 8.5 pc pixel⁻¹. We then converted the summed CO(J = 3−2) emission in a region with units K km s⁻¹ pixel² to a CO luminosity by multiplying this total by the conversion factor 72.6 pc² pixel⁻². With the CO luminosity determined, we arrive at the following conversion from CO(J = 3−2) to a molecular gas mass in solar units:

$$\begin{eqnarray}&&{M}_{\mathrm{mol}}=2\,{\alpha }_{\mathrm{CO}}\,{L}_{\mathrm{CO}}.\end{eqnarray} \tag{ 1 }$$

The WVT binning algorithm was developed for use in sparse X-ray data. It is advantageous over the traditional Voronoi tessellations by introducing an additional parameter, the scale length δ_i, to be associated with each bin. This scale length works to stretch or compress the metric inside each bin, which are now defined to minimize the quantity $| {x}_{k}-{z}_{i}| /{\delta }_{i}$. The adaptability of the metric allows for the boundaries of each bin to have curvature that approximately follows a constant gradient of the CO emission intensity. This allows an objective, but adaptive, pixelation of the CO integrated intensity map into bins within which the IMR can be derived.

We made use of the WVT binning algorithm by Diehl & Statler (2006), which is a generalization of the algorithm developed in Cappellari & Copin (2003). The original algorithm builds the tessellation such that pixels are accreted into bins until a target S/N level is reached. This is appropriate for X-ray data where there exists independent Poisson noise in each pixel. Since the noise in interferometric images is correlated in a more complicated manner, this target parameter is ill-suited for our data. We adapted this algorithm such that pixels are accreted to bins until a limiting luminosity level (L_target) is reached. Given a conversion prescription from CO radiation luminosity to total mass, by seeking a target total luminosity we are effectively probing a particular gas mass scale in each tessellation.

Rather than running our adapted WVT binning algorithm over the entire CO moment 0 map of the Antennae overlap region, we masked the CO moment 0 map to only consider significant (5σ) star-forming gas emission. The rms of the CO moment 0 map was calculated with CASA in isolated patches of the map and was found to be σ ∼ 11 K km s⁻¹. We then created a mask of the CO map that only included locations of emission greater than 55 K km s⁻¹. The WVT binning algorithm was then run on the masked CO moment 0 map for the following limiting luminosity levels: 3.6 × 10⁵ K km s⁻¹ pc², 7.3 × 10⁵ K km s⁻¹ pc², and 14.5 × 10⁵ K km s⁻¹ pc². The tessellations of differing luminosity levels probe different size and mass scales. Figure 5 shows snapshots of the WVT binning algorithm at work during the 25%, 50%, and 100% completion stages.

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The WVT bin boundaries were applied to the optical image. With the units in e⁻/s, the flux was summed over each WVT bin. Taking the PHOTFLAM keyword from the image header as 7.07236 × 10⁻²⁰ erg/cm²/Å/e⁻, the e⁻/s per bin were converted to a flux in ergs/cm²/s/Å per bin. This flux was converted to an absolute magnitude assuming an I-band flux for Vega of ${f}_{\mathrm{vega}}=1.134\times {10}^{-9}$ erg/cm²/s and a distance to the Antennae of d = 22 Mpc (Schweizer et al. 2008),

$$\begin{eqnarray}&&{M}_{I}=-2.5\mathrm{log}\left[\displaystyle \frac{({e}^{-}/{\rm{s}})7.07236\times {10}^{-20}}{{f}_{\mathrm{vega}}}\right]-5\mathrm{log}d.\end{eqnarray} \tag{ 3 }$$

For each bin that contained at minimum one cluster, the absolute magnitudes could be corrected for extinction using the best estimate of reddening, $E(B-V)$, from Whitmore et al. (2010). We converted the reddening values to extinctions at 814 nm using ${A}_{V}=3.1E(B-V)$ and the relation between A₈₁₄ and A_V determined by Cardelli et al. (1989):

$$\begin{eqnarray}&&{A}_{814}\approx 0.6\,{A}_{V}.\end{eqnarray} \tag{ 4 }$$

From Figure 3, it is evident that there exist bins without any identified star clusters, particularly in Super Giant Molecular Complexes (SGMCs) 2 and 3. For those bins, the IMR cannot be calculated, as we lack a suitable method for determining the M/L ratio for the stellar component. For bins with multiple clusters, we follow the procedure discussed in Section 2.3 and assume the M/L ratio of the bin can be approximated by the median of the M/L ratios of the enclosed clusters.

Using our best estimate for the mass-to-light ratio in a WVT bin (Section 2.3), the extinction-corrected absolute I-band magnitudes were converted to stellar masses in solar units. The stellar mass was combined with the molecular mass in each WVT bin (calculated using the methods of Section 2.4) following Equation (2) to obtain the IMR in each WVT bin containing at least one stellar cluster.

Complicated and unknown uncertainties in both the stellar and gas mass calculations make estimating formal errors on the resulting IMR nearly impossible. Despite this, it is simple to calculate the dependencies of the error in IMR on several quantities. This allows for a better understanding of the areas in parameter space for which the reported values of the IMR(t) can be trusted and to what extent. As an example, we illustrate how the uncertainties depend on the relative errors in the gas mass and stellar mass in Figure 6. Throughout the following calculation, we assume a nominal error in the gas mass of 20%, ${\sigma }_{{M}_{\mathrm{gas}}}=0.2\,{M}_{\mathrm{gas}}$. Rewriting the IMR from its traditional notation (${M}_{\mathrm{stars}}/({M}_{\mathrm{gas}}+{M}_{\mathrm{stars}})$) into a function of the gas-to-stellar mass ratio yields:

$$\begin{eqnarray}&&\mathrm{IMR}={\left(1+\displaystyle \frac{{M}_{\mathrm{gas}}}{{M}_{\mathrm{stars}}}\right)}^{-1}.\end{eqnarray} \tag{ 5 }$$

Traditional propagation of error gives the fractional error in the IMR as a function of the IMR itself and the fractional error in stellar mass. Exploring this relationship with varying values of the IMR and the fractional stellar mass error (Figure 6) results in fractional errors in the IMR around ∼20% for much of the parameter space. The error in the IMR is minimized as the ratio approaches unity with errors in the stellar mass of less than ∼10%.

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The Antennae galaxies contain molecular clouds of unusually high mass. Wilson et al. (2000) derived mass values that are 5–10 times more massive than the largest molecular complexes of similar size in the grand-spiral galaxy M51 (Rand & Kulkarni 1990). Wei et al. (2012) measured clouds in the overlap region of the Antennae to be larger than any in the Milky Way or nearby disk galaxies (Solomon et al. 1987; Bolatto et al. 2008; Heyer et al. 2009), causing them to be referred to as SGMCs. Within the SGMCs, individual clouds are also grand in scale, Johnson et al. (2015) found a >5 × 10⁶ M_⊙ cloud with a radius of <24 pc, making it one of the densest clouds known. As some of the most massive molecular complexes, updated molecular mass values for the Antennae SGMCs remain crucial for improved understanding of star formation in intense environments.

As a consistency check with previous work, we determined the mass of each WVT bin in the three SGMCs following Section 2.4 and summed over these values in each of the three SGMCs to derive total gas mass. For the first, second, and third SGMCs, the total gas mass is approximately 7.8 × 10⁸ M_⊙, 6.3 × 10⁸ M_⊙, and 6.6 × 10⁸ M_⊙. These values are within a factor of ∼2 with previous mass estimates of Wilson et al. (2003) for the three SGMCs: 6.3 × 10⁸ M_⊙, 3.9 × 10⁸ M_⊙, and 9.3 × 10⁸ M_⊙ in SGMCs 1, 2, and 3, respectively.

The averaged IMRs are 0.2%, 0.3%, and 0.3% for SGMCs 1, 2, and 3, respectively. While these values are quite low, this is reasonable given the enormity of these molecular clouds and the large age range of the stellar clusters enclosed.

The instantaneous mass ratio $\mathrm{IMR}={M}_{\mathrm{stars}}/({M}_{\mathrm{stars}}+{M}_{\mathrm{gas}})$, was calculated for each aperture. The prescription for determining the M/L ratio of a bin, and therefore the corresponding stellar mass, is described in Section 2.3. The prescription for determining the total gas mass from the CO(3–2) gas emission is described in Section 2.4. These values were then combined to determine the IMR associated with each stellar cluster. We expect that this method will have a bias toward regions with optically visible (unembedded) clusters that are more efficient at expelling gas. Indeed, Cabrera-Ziri et al. (2015) found that for the three most massive clusters in the overlap region (of ages between 50 and 200 Myr), the gas mass is only <9% of the stellar mass.

The resulting IMR values are presented in Figure 7. We see a clear trend for the IMR value to increase with cluster age, as we expected, with some regions reaching a value of unity (100% of the gas has been removed and/or converted to stars) at ages of <10 Myr. For ages ≲2 Myr (before which we do not expect supernovae and winds from Wolf–Rayet stars to have had a significant effect), the IMR values range from close to zero to ∼0.5.

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A significant caveat to this approach is that for a given optical cluster, we do not know whether the CO emission in the aperture is associated with the cluster, or simply along the same line of sight. Therefore, we expect a fraction of older clusters to appear to be associated with molecular material, when in fact they are not. This contamination will result in scatter toward lower IMR values, as discussed in Section 5.3. The centering of measurements on either the product of star formation (stellar clusters), or the gas produces different results on these small size scales. Zhang et al. (2001) found that feedback effects may modify the Schmidt law at scales below 1 kpc. A similar effect of star formation relations breaking down on small-scales was presented by Schruba et al. (2010) in M33 and by Kruijssen & Longmore (2014) for a sample of regions including the solar neighborhood, the Central Molecular Zone of the Milky Way, a "disk galaxy," a "dwarf galaxy", and a starbursting "submillimeter galaxy."

The IMR was also calculated for all WVT bins that had an associated stellar cluster (see Section 4). This procedure was completed for WVTs resulting from target luminosities of L_target = 3.6 × 10⁵ K km s⁻¹ pc², 7.3 × 10⁵ K km s⁻¹ pc², and 14.5 × 10⁵ K km s⁻¹ pc². We find very few WVT bins with IMRs greater than ∼10%, with the mean of the distribution being 2.4% (see Figure 8). The cluster with consistently the highest IMR (shown as a black star in Figures 9, 11, and 12) is identified as WS80 and has been studied extensively in the literature (e.g., Whitmore & Zhang 2002). As the size of the WVT bin increases, the fractional increase of stellar light due to both the diffuse stellar component and compact clusters within the WVT bin increases more slowly than the increase in the diffuse CO(3–2) emission. Correspondingly, as the size of the WVT bins (L_target) decreases, the IMR values systematically increase. We find that determining the IMR by centering measurements on molecular gas does not follow the predicted trend as seen in Section 5.2.

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A necessary condition for determining the IMR in a WVT bin is the presence of at least one identified stellar cluster in the projected area of the bin. The age dating of these clusters, done by Whitmore et al. (2010), enables us to compare the relationship of the IMR with time, see Figure 9. The majority of WVT bins contain only one identified cluster, but in the case where there are multiple, the median age of the enclosed clusters is taken. Our original hypothesis was that, as a cluster ages, more of its gas is converted into stars, destroyed, or expelled, and the IMR would approach 100%. We do not observe this trend in the overlap region of the Antennae galaxies for regions selected based on their CO(3–2) emission. Rather, we observe systematically decreasing IMRs for clusters of increasing age. This result is interesting, but may be better explained by an observational bias rather than an astrophysical phenomenon (this bias is explored further in Section 6.1). In particular, because the bins are determined based on their CO(3–2) flux (for which a 5σ detection is required), we expect that this method will be systematically biased toward regions with significant CO emission. By requiring the presence of significant CO emission in the analyzed WVT bins, we bias ourselves exclusively toward clusters with a significant mass of gas. We repeated all procedures using a 2σ and 3σ threshold for the CO emission to determine whether the trend of decreasing IMR with age would change, but no significant change was observed. Given that the clusters are also identified in projection, we also expect there to be contamination from clusters simply along the line of sight, and not physically associated with the molecular gas, which would systematically manifest itself as older clusters appearing to be gas-rich. These combined effects may explain the systematically low IMR values for relatively older clusters in the overlap region of the Antennae galaxies.

As a direct comparison of the two methods, we take an arbitrary cluster that is both one of the "50 Most Massive" clusters (included in the analysis of Section 3) and surrounded by a 5σ detection of CO(3–2) emission (included in the analysis of Section 4). The cluster chosen is the 19th most massive cluster in the entire Antennae system (Figure 10), with an ID of 18848 in Whitmore et al. (2010) and nominal values of the stellar mass and age estimate of 1.2 × 10⁶ M_⊙ and 4.8 Myr, respectively.

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Centering the measurement on the stellar cluster yields an IMR of 23% and 21% for the 25 pc and 50 pc apertures, respectively. In comparison, the WVT bins containing this cluster have IMRs of 34%, 11%, and 10% for increasing L_target, respectively. In order to compare these values, we consider the physical area of the circular apertures and WVT bins. The 25 and 50 pc radii apertures have areas 1960 pc² and 7850 pc², respectively, while the WVT bins have areas of 874 pc², 2040 pc², and 4440 pc², for increasing L_target. The smallest WVT bin has an IMR greater than the apertures. This is reasonable due to its small size, which has a higher fraction of cluster light to diffuse stellar light and minimizes the mass from the gas reservoir that encompasses the cluster. The larger IMR ratios for the apertures centered on the stellar clusters are likely due to the differing focuses of the two methods. While the circular apertures are centered on the cluster, the WVT bins follow the gas emission, which is not symmetrically distributed and leads to larger gas masses and lower IMRs.

The WVT binning algorithm allows us to investigate the dependence of the IMR on various physical parameters. For example, we can explore whether the efficiency of star formation depends on the surface density of the CO gas in molecular clouds. The ratio of the flux in a given WVT bin and the area in units of pc² probes the CO gas density. Since our WVT bins were defined algorithmically from the data itself, we are able to objectively measure the dependence of the IMR as a function of molecular gas surface density. When considering WVT bins that include optical clusters, the IMR does not seem to correlate with the CO surface brightness (Figure 11), which agrees with results of Leroy et al. (2017) showing that the small-scale surface density of clouds has a relatively weak effect on star formation. This is rather unexpected, and may be truly physical or it may indicate that the surface density is not a good probe of the volumetric density of CO gas and we are therefore unable to establish a relationship between IMR and CO gas density. Even when considering WVT bins with only the more massive optical clusters of M > 10⁵ M_⊙, the relationship between IMR and surface brightness remains largely consistent with a random distribution (Figure 11).

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In addition to the relationship between the IMR and CO surface brightness, we can also investigate the dependency of IMR and the 814 nm A₈₁₄ extinction in each of the WVT bins. Higher extinction values may indicate that the cluster is more deeply embedded in the molecular cloud from which its stars are forming. Due to the increase in the amount of available dense gas in this environment, we expect higher IMR for clusters with higher values of extinction. Since the WVT bins were determined independent of extinction, they provide an objective means for investigating whether more deeply embedded clusters indeed correspond to higher IMRs. We see a clear trend across all SGMCs of increasing IMR with extinction, confirming that more deeply embedded clusters are capable of higher rates of SFE (Figure 12). A comparison of the IMR with respect to both age and extinction can be seen in Figure 13, along with a comparison of the IMR to both age and surface brightness.

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Each of the two primary methods we used to estimate the IMR(t) for regions in the Antennae is subject to limitations. With the WVT method, there is power in mathematically defining regions whose areas adjust based on the amount of CO emission at a given point. However, it constrains our analysis to regions where there is significant CO emission, which inherently biases our results toward regions rich in molecular gas, but which may not have associated star formation or the star formation episode may be in its early stages. For these young clusters, it is likely that we are missing optical light and thereby underestimating the stellar mass associated with the WVT bin. This method also selects and highlights slightly older clusters that are either inefficiently producing stars or feedback from massive stars has failed to clear out much of the initial gas from the molecular cloud. These scenarios are expected to be rare and cannot explain the presence of multiple clusters with ages ≳5 Myr with very low IMRs. The more plausible explanation is that these regions are "contaminated" by clusters along a WVT bin's line of sight and are not physically associated with the observed molecular gas. This limitation of the algorithm biases our results toward lower values of the IMR, particularly for older clusters.

Because of these biases toward detecting the population of older clusters that appear to remain in a dense molecular cloud, the results of the evolution of IMR with age by centering our measurement on molecular gas contradict our initial hypothesis. Specifically, we observe a decreasing trend in the relationship between these two quantities.

On the other hand, the trend is reversed for the analysis done by focusing on circular apertures around luminous and massive clusters. This result follows our prediction: the ratio of stellar to total mass increases as stars form and gas is expelled through stellar feedback. Although this result supports our original hypothesis, this method is also subject to biases. Building a sample of clusters by requiring their luminosity and mass to be high will preferentially select a sample of clusters that have been successful at clearing much of their gas reservoir. Whitmore et al. (2010) found that ∼15% of clusters in the entire Antennae system were completely dust-obscured. This ratio may be higher in the overlap region due to the larger gas reservoir. While there are still some of the optically brightest clusters visible within the SGMCs, the majority of this sample resides outside of these giant gas clouds. There are very few older clusters that are surrounded by a significant amount of gas, and it is more likely that older clusters with low IMR are a result of chance superposition or incorrect age estimates. This sways the stellar-to-total mass ratios of this sample to approach unity as age increases.

The ability of star clusters to remain bound and evolve into globular clusters is strongly tied to their SFE. However, different theoretical efforts have not converged on consistent values, but instead have predicted a large range of SFEs necessary for clusters to remain bound. For example, the simplest case without considering environmental effects (e.g., turbulence and noninstantaneous timescales Hills 1980) requires an SFE of at least 90%, while Baumgardt & Kroupa (2007) and Pelupessy & Portegies Zwart (2012) report values as low as 5%–10% provided the gas is removed over long timescales. With only ∼8 clusters presented here with IMR > 10%, our findings suggest that SFEs of several tens of percent (but much less than 90%) are not especially common. Taken at face value, the apparently low SFE with which clusters in the Antennae are forming could contribute significantly to their destruction.

Our initial hypothesis that the IMR(t) value should exhibit a trend in which IMR values increase with age and approach unity appears to be approximately borne out (as shown in Figure 7), which is based on localized emission in the vicinity of optical SSCs. However, this hypothesis is generally inconsistent with the results from the second method based on the intensity of the CO emission (as shown in Figure 9). These apparently contradictory results can be explained by both line-of-sight contamination and the Antennae system not being a steadily evolving post-starburst, but rather having a significant CO reservoir that is actively forming star clusters. In a system crowded with both newly formed and mature star clusters, any arbitrary and sufficiently large aperture is likely to contain a mixed population, which makes it challenging to disentangle the gas associated with any particular cluster of a given age.

For these reasons, we assume that the IMR is better determined through the method that localizes emission associated with optical clusters. If we assume that the IMR is roughly equivalent to the SFE for clusters with sufficiently young ages (e.g., IMR $(t\leqslant {10}^{6.5})$), we can assess the fraction of clusters that could remain bound. Based on the analysis discussed in Section 5.2, only 2/5 of the youngest massive clusters (t ≤ 10^6.5 years, mass ≳7 × 10⁵ M_⊙) optically visible clusters considered here appear to have IMRs ≳0.5. If SFEs as low as ∼30% could still produce a bound cluster, the relative fraction of these massive clusters that could survive increases to 3/5. Although we recognize small sample statistics may be at play, this fraction does not appear to increase until the SFE threshold is lowered to 5%, which may only result in bound clusters under very specific physical scenarios (see Section 1.1). However, we note that this method inherently assumes that the gas within the 25 pc or 50 pc radius around each cluster is associated with that cluster. For the youngest clusters, the IMR values for the 25 and 50 pc radii are very similar, suggesting that the gas is highly concentrated on the clusters. However, if the gas associated with the clusters has been overestimated, the corresponding IMR value will be higher—suggesting a higher fraction of potentially bound clusters.

If instead we look to the IMR(t) values determined based on the CO emission in Section 5.3, only a single region appears to have IMR (t <10^6.5) ≳ 0.3, suggesting that very few of the regions at large in this system will produce bound clusters. Thus, on the surface, these results suggest that only a few of the massive clusters that recently formed in the Antennae have the chance to evolve into globular clusters, especially considering the number of other physical processes that will affect their survival after the first few million years (e.g., Gnedin & Ostriker 1997; Hollyhead et al. 2015). This is consistent with other studies (e.g., Whitmore 2004; Fall et al. 2005; Whitmore et al. 2007) that find that only a few percent of young star clusters survive to become old globular clusters.

The timescales on which molecular material is destroyed or expelled from a massive star cluster is important for determining whether these massive clusters could potentially undergo a "second generation" of star formation, as has become an outstanding issue in globular cluster evolution in recent years. Observations of globular clusters have unambiguously shown evidence for multiple chemical populations; however, the origin of these populations remains unclear (e.g., N. Bastian et al. 2018, in preparation; Cabrera-Ziri et al. 2015).

From Section 5.2, we can estimate the timescales on which the CO gas is destroyed or expelled from the cluster environments. Figure 7 shows a clear trend indicating that the relative fraction of CO associated with clusters decreases with age, as expected. By ages of ∼10^6.7 years, some of the clusters appear to have already cleared the molecular material from within radii of 50 pc (where Whitmore et al. 2014 found that the lifetime of GMCs in the region is ≈10⁷ years). By ages of ∼10^7.5 years, this is true for the majority of clusters. This agrees well with the clearing timescales of 2 and 7 Myr for gas that is associated and unassociated with the stellar cluster, respectively (K. Grasha et al. 2018, in preparation). Moreover, we note that for clusters older than ∼10^6.7 years, the IMR values derived from the 25 and 50 pc radii are more discrepant (see Figure 7), potentially indicating that gas is no longer as centrally concentrated on the clusters by this time. We hypothesize that the clusters with ages >10^7.5 years that appear to have molecular material still associated with them may well simply be contaminated by CO emission along the line of sight.

Determination of the IMR through centering the measurement on gas or clusters is subject to the additional issue that we are likely missing stellar light from the youngest clusters, which may not be visible at optical wavelengths. This additional effect is more strongly felt by the WVT binning method as the resulting bins are systematically more enshrouded in obscuring material than the clusters assembled by having a high luminosity and/or mass. In both methods (but particularly for the one based on CO emission), we are very likely underestimating the stellar mass of the youngest clusters (and older enshrouded ones). This effect extends to the IMR, which is also likely to be slightly underestimated at the youngest ages. Properly accounting for stellar mass of young clusters could be done with high spatial resolution thermal radio observations as an extinction-free tracer of recent star formation. Although Neff & Ulvestad (2000) present VLA images of the Antennae at 6 and 4 cm of resolutions ∼1'' and 26, respectively, these observations likely suffer from too much nonthermal synchrotron contamination.