Super Star Clusters in the Central Starburst of NGC 4945

We identify candidate star clusters via pointlike sources of emission in the 93 GHz continuum image. Sources are found using PyBDSF (Mohan & Rafferty 2015) in the following way. Islands are defined as contiguous pixels (of nine pixels or more) above a threshold of seven times the global rms value of σ ≈ 0.017 mJy beam⁻¹. Within each island, multiple Gaussians may be fit, each with a peak amplitude greater than the peak threshold of 10 times the global rms. We chose this peak threshold to ensure that significant emission can also be identified in the continuum images made from individual spectral windows. The number of Gaussians is determined from the number of distinct peaks of emission higher than the peak threshold that have a negative gradient in all eight evaluated directions. Starting with the brightest peak, Gaussians are fit and cleaned (i.e., subtracted). A source is identified with a Gaussian as long as subtracting its fit does not increase the island rms.

Applying this algorithm to our 93 GHz image yielded 50 Gaussian sources. We remove five sources that fall outside of the star-forming region. We also remove three sources that appeared blended, with an offset <012 from another source. Finally, we remove 13 sources that do not have a flux density above 10 times the global rms after extracting the 93 GHz continuum flux density through aperture photometry (see Section 3.2). As a result, we analyze 29 sources as candidate star clusters. In Figure 3, we show the location of each source with the apertures used for flux extraction. The sources match well with what we would identify by eye.

For each source, we extract the continuum flux density at 2.3, 93, and 350 GHz through aperture photometry. Before extracting the continuum flux at 2.3 GHz, we convolve the image to the common resolution of 012. We extract the flux density at the location of the peak source within an aperture diameter of 024. Then we subtract the extended background continuum that is local to the source by taking the median flux density within an annulus of inner diameter 024 and outer diameter 030; using the median suppresses the influence of nearby peaks and the bright surrounding filamentary features. The flux density of each source at each frequency is listed in Table 1. When the extracted flux density within an aperture is less than three times the global rms noise (in the 2.3 and 350 GHz images), we assign a 3σ upper limit to that flux measurement.

In Figure 4, we plot the ratio of the flux densities extracted at 350 and 93 GHz (S₃₅₀/S₉₃) against the ratio of the flux densities extracted at 2.3 and 93 GHz (S_2.3/S₉₃). Synchrotron-dominated sources, which fall to the bottom right of the plot, separate from the free–free- (and dust-) dominated sources, which lie in the middle of the plot. One exception is the AGN (source 18), which, due to self-absorption at frequencies greater than 23 GHz, is bright at 93 GHz but not at 2.3 GHz and therefore has the lowest S_2.3/S₉₃ ratio.

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From the continuum measurements, we construct simple SEDs for each source. These SEDs are used for illustrative purposes and do not affect the analysis in this paper. Examples of the SEDs of three sources are included in Figure 5. We show an example of a free–free-dominated source (source 22), which represents the majority of sources, as well as a dust- (source 12) and a synchrotron- (source 14) dominated source. Of the sources with extracted emission of >3σ at 2.3 GHz, nine also have the free–free absorption of their synchrotron spectrum modeled. We plot this information whenever possible. When a source identified by Lenc & Tingay (2009) lies within 006 (half the beam FWHM) of the 93 GHz source, we associate the low-frequency modeling with the 93 GHz source. We take the model fit by Lenc & Tingay (2009) and normalize it to the 2.3 GHz flux that we extract; as an example, see the solid purple curve in the middle panel of Figure 5. The SEDs of all sources are shown in Figure 13 in Appendix C.

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The flux density of optically thin, free–free emission at millimeter wavelengths (see Appendix A.2; Draine 2011) arises as

$$\begin{eqnarray}\begin{array}{rcl}{S}_{\mathrm{ff}} & = & (2.08\,\mathrm{mJy})\left(\displaystyle \frac{{n}_{e}{n}_{+}V}{5\times {10}^{8}\,{\mathrm{cm}}^{-6}\,{\mathrm{pc}}^{3}}\right)\\ & & \times \ {\left(\displaystyle \frac{{T}_{e}}{{10}^{4}{\rm{K}}}\right)}^{-0.32}{\left(\displaystyle \frac{\nu }{100\mathrm{GHz}}\right)}^{-0.12}{\left(\displaystyle \frac{D}{3.8\mathrm{Mpc}}\right)}^{-2},\end{array}\end{eqnarray} \tag{ 1 }$$

where EM_C = n_en₊V is the volumetric emission measure of the ionized gas, D is the distance to the source, and T_e is the electron temperature of the medium. Properties of massive star clusters (i.e., ionizing photon rate) can thus be derived through an accurate measurement of the free–free flux density and an inference of the volumetric emission measure. We will determine a free–free fraction, f_ff, and let S_ff = f_ffS₉₃.

In this section, we focus on determining the portion of free–free emission that is present in the candidate star clusters at 93 GHz. To do this, we need to estimate and remove contributions from synchrotron emission and dust continuum. We determine an in-band spectral index across the 15 GHz bandwidth of the ALMA Band 3 observations. Using the spectral index,¹² we constrain the free–free fraction, as well as the fractional contributions of synchrotron and dust (see Figure 4 and Table 1). We found the in-band index to give stronger constraints and, for 2.3 GHz, be more reliable than extrapolating (assuming indices of −0.8, −1.5) because of the two decades' difference in frequency coupled with the large optical depths already present at 2.3 GHz.

3.3.1. Band 3 Spectral Index

3.3.2. Decomposing the Fractional Contributions of Emission Type

From the in-band spectral index fit, we estimate the fractional contribution of each emission mechanism—free–free, synchrotron, and (thermal) dust—to the 93 GHz continuum. To do this, we simulate how mixtures of synchrotron, free–free, and dust emission could combine to create the observed in-band index. We assume fixed spectral indices for each component and adjust their fractions to reproduce the observations (see below).

We consider that a source is dominated by two types of emission (a caveat we discuss in detail in Section 6): free–free and dust or free–free and synchrotron. With synthetic data points, we first set the flux density at 93 GHz, S₉₃, to a fixed value and vary the contributions of dust and free–free continua, such that S₉₃ = S_d + S_ff. For each model, we determine the continuum flux at each frequency across the Band 3 frequency coverage as the sum of the two components. We assume that the frequency dependence of the free–free component is α_ff = −0.12 and the frequency dependence of the dust component is α_d = 4.0. We fit the (noiseless) continuum of the synthetic data across the Band 3 frequency coverage with a power law, determining the slope as α₉₃, the in-band index. We express the results in terms of free–free and dust fractions, rather than absolute flux. In this respect, we explore free–free fractions to the 93 GHz continuum ranging from f_ff = (0.001, 0.999) in steps of 0.001. We let the results of this process constrain our free–free fraction when the in-band index measured in the actual (observed) data is α₉₃ ≥ −0.12.

Next, we repeat the exercise, but we let synchrotron and free–free dominate the contribution to the continuum at 93 GHz. We take the frequency dependence of the free–free component as α_ff = −0.12 across the Band 3 frequency coverage, and we set the synchrotron component to α_syn = −1.5. We let the results of this process constrain the free–free fraction of the candidate star clusters when the fit to their in-band index is α₉₃ ≤ −0.12.

Our choices for the spectral indices of the three emission types are motivated as follows. In letting α_ff = −0.12, we assume that the free–free emission is optically thin (e.g., see Appendix A.2). We do not expect significant free–free opacity at 93 GHz given the somewhat-evolved age of the candidate clusters in the starburst (see Section 5.2). In letting α_d = 4.0, we assume the dust emission is optically thin with a wavelength-dependent emissivity so that τ ∝ λ⁻² (e.g., see Draine 2011). The low dust optical depths estimated in Section 5.6 imply a dust spectral index steeper than 2, though the exact value might not be 4, e.g., if our assumed emissivity power-law index is not applicable. The generally faint dust emission indicates that our assumptions about dust do not have a large effect on our results. For the synchrotron frequency dependence, we assume α_syn = −1.5. This value is consistent with the best-fit slope of α_syn = −1.4 found by Bendo et al. (2016) averaged over the central 30'' of NGC 4945. Furthermore, the median slope of synchrotron-dominated sources modeled at 2.3–23 GHz is −1.11 (Lenc & Tingay 2009), indicating that even at 23 GHz, the synchrotron spectra already show losses due to aging, i.e., are steeper than a canonical initial injection of α_syn ≈ −0.8. While our assumed value of α_syn = −1.5 is well motivated, on average, variations from source to source are likely present.

Through this method of decomposition, the approximate relation between the in-band index and the free–free fraction is

$$\begin{eqnarray}{f}_{\mathrm{ff}}=\left\{\begin{array}{cc}0.72\,{\alpha }_{93}+1.09, & -1.5\leqslant {\alpha }_{93}\leqslant -0.12\\ -0.24\,{\alpha }_{93}+0.97, & 4.0\geqslant {\alpha }_{93}\geqslant -0.12\end{array}\right.,\end{eqnarray} \tag{ 2 }$$

for which α₉₃ ≤ −0.12, the synchrotron fraction is found to be f_syn = 1 − f_ff and we set f_d = 0, and for which α₉₃ ≥ −0.12 the dust fraction is found to be f_d = 1 − f_ff and f_syn = 0. This relation is depicted in the bottom panel of Figure 4 using the values that have been determined for each source.

3.3.3. Estimated Free–Free, Synchrotron, and Dust Fractions to the 93 GHz Continuum

Figure 7 also shows integrated spectra derived from intermediate-resolution (07) data. We use these data as a tracer of the total ionizing photons of the starburst region. We expect that the intermediate-resolution data include emission from both discrete, pointlike sources and diffuse emission from any smooth component.

The total integrated emission in the intermediate-configuration data is about three times larger than the integrated emission in the extended-configuration data. Spectra representing the total integrated line flux are shown in Figure 7. In Figure 8, we show the integrated intensity map of H40α emission from the intermediate-configuration (07) data. In Table 3, we include the best-fit line profiles.

We also compare the line profiles of H40α and H42α in the intermediate-configuration (07) data; see Table 4 and Figures 9 and 10. We find that the integrated line emission of H42α is enhanced compared with H40α, reaching a factor of 2 greater when integrated over the entire starburst region. Yet we see good agreement between the two lines at the scale of individual cluster candidates. Spectral lines (possibly arising from c-C₃H₂) likely contaminate the H42α line flux in broad, typically spatially unresolved line profiles.

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4.1.1. Total Emission from the Starburst Region

4.1.2. H42α Contamination

The sources identified through PyBDSF in the 93 GHz continuum image are fit with two-dimensional Gaussians. The average of the major and minor (convolved) FWHM is listed in Table 6 as the FWHM size of the source in units of parsecs. The Gaussian fits are all consistent with circular profiles within the errors. We find FWHM sizes of (1.4–4.0) pc, consistent with the typical sizes of young massive star clusters (Ryon et al. 2017; Leroy et al. 2018). However, the lower end may reflect the resolution limit of our beam, with an FWHM size of 2.2 pc. The uncertainties we report reflect the errors of the Gaussian fit.

Table 6.  Physical Properties of Candidate Star Clusters

Source	FWHM	$\mathrm{log}({{EM}}_{{\rm{C}}})$ a	$\mathrm{log}({{EM}}_{{\rm{L}}})$ a	$\mathrm{log}({Q}_{0})$ a	$\mathrm{log}({M}_{\star })$ a	$\mathrm{log}({M}_{\mathrm{gas}})$ a	$\mathrm{log}({{\rm{\Sigma }}}_{\mathrm{Tot}})$ a	$\mathrm{log}({t}_{\mathrm{ff}})$ a
	(pc)	(cm−6 pc3)	(cm−6 pc3)	(s−1)	(M⊙)	(M⊙)	(M⊙ pc−2)	(yr)
1	3.0 ± 0.1	8.1	8.6	51.2	5.1	<4.5	4.0	4.9
2	2.6 ± 0.1	8.2	⋯	51.3	5.2	4.8	4.3	4.7
3	3.1 ± 0.1	8.2	8.4	51.2	5.2	4.8	4.1	4.9
4	2.5 ± 0.1	7.7	8.5	50.8	4.7	4.6	4.0	4.9
6	2.9 ± 0.1	8.0	8.4	51.1	5.0	<4.5	3.8	5.0
7	2.7 ± 0.1	8.1	8.4	51.2	5.1	4.6	4.2	4.8
8	2.4 ± 0.1	8.5	8.7	51.5	5.5	4.5	4.5	4.6
9	2.4 ± 0.1	8.1	⋯	51.2	5.1	<4.3	4.1	4.8
10	2.9 ± 0.1	8.2	⋯	51.2	5.1	<4.5	4.0	4.9
11	1.4 ± 0.1	8.0	⋯	51.1	5.0	4.0	4.6	4.5
12	2.7 ± 0.1	8.2	8.7	51.3	5.2	5.0	4.4	4.7
13	2.5 ± 0.1	8.5	8.7	51.6	5.5	4.7	4.6	4.6
14	2.2 ± 0.1	8.1	8.9	51.2	5.1	4.4	4.3	4.7
15	2.5 ± 0.1	8.0	8.4	51.0	5.0	<4.4	4.0	4.9
16	2.3 ± 0.1	8.1	⋯	51.2	5.1	4.7	4.3	4.7
17	3.1 ± 0.1	8.7	⋯	51.7	5.7	5.2	4.6	4.6
19	2.4 ± 0.1	7.4	⋯	50.5	4.4	<4.3	3.5	5.1
20	3.7 ± 0.1	8.4	⋯	51.4	5.3	5.1	4.2	4.9
21	3.1 ± 0.1	7.3	8.8	50.4	4.3	<4.6	3.1	5.4
22	2.4 ± 0.1	8.7	8.7	51.8	5.7	4.9	4.8	4.5
23	2.6 ± 0.1	8.2	⋯	51.3	5.2	4.4	4.3	4.7
24	3.1 ± 0.1	7.6	⋯	50.6	4.6	<4.6	3.4	5.2
25	3.9 ± 0.2	8.2	8.5	51.3	5.2	5.0	4.0	5.0
26	3.4 ± 0.1	8.3	8.5	51.3	5.3	4.9	4.2	4.9
27	2.2 ± 0.1	8.2	8.4	51.3	5.2	<4.3	4.3	4.7
28	4.0 ± 0.1	8.0	⋯	51.1	5.0	<4.8	3.6	5.1
29	2.9 ± 0.1	8.4	⋯	51.5	5.4	4.6	4.3	4.7

Note. Source 5 is not included because its free–free fraction is f_ff < 0.01; source 18 is not included because it is the AGN core. The FWHM is the source size as best fit from a Gaussian (to flux that has not been deconvolved); the errors reflect the fit of the Gaussian. Here EM_C is the free–free emission measure derived from the continuum as in Equation (1), and EM_L is the hydrogen free–free emission measure derived from effective H41α as in Equation (3); we note EM_C = (1 + y)EM_L. Here Q₀ is the ionizing photon rate derived from EM_C as in Equation (7), and M_⋆ is the stellar mass derived from Q₀ as in Equation (8).

^aThe error on these quantities is ∼0.4 dex.

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Based on high-resolution imaging of embedded clusters in the nucleus of the Milky Way and NGC 253, some of these clusters might break apart at higher resolution (Ginsburg et al. 2018, Levy et al., in preparation). If they follow the same pattern seen in these other galaxies, each source would have one or two main components with potentially several associated fainter components.

Throughout our analysis, we assume that the candidate star clusters formed in an instantaneous burst of star formation roughly 5 Myr ago (with a likely uncertainty of ∼1 Myr). This (approximately uniform) age is supported through the coincident detection of radio recombination lines (RRLs) and supernova remnants, previous analyses of the global population of the burst (e.g., Marconi et al. 2000; Spoon et al. 2000), and an orbital timescale of ≈3 Myr for the starburst region. We elaborate on this supporting evidence below.

As we discuss in Section 3.3, dust does not significantly contribute to 93 GHz emission (with a median fraction of f_d = 0), but synchrotron emission does through supernova remnants. Supernova explosions begin from ∼3 Myr in the lifetime of a cluster and cease at about ∼40 Myr, when the most massive stars have died out; this puts the loosest bounds on the age of the candidate clusters we observe. The coincident detection of supernova remnants in one-third (6/15) of our recombination line–detected sources implies that the burst is likely not at the earliest stage of the supernova phase. However, the ionizing photon rate changes dramatically over 3–10 Myr, dropping by about two orders of magnitude (Leitherer et al. 1999). As a result, clusters are significantly harder to detect in radio recombination lines or free–free continuum emission after ∼5 Myr.

Properties of star-forming activity in the central starburst have been estimated by combining far-infrared (FIR) and optical/IR tracers. Marconi et al. (2000) discerned an age of 6 Myr and mass of 4 × 10⁷ M_⊙ by using Paα and Brγ to trace the energy distribution of the photon output of the population. However, the dust extinction was underestimated, complicated by the uncertainty in the AGN contribution. The MIR observations of line ratios with the Infrared Space Observatory (ISO; Kessler et al. 1996) further constrained this scenario. Spoon et al. (2000) estimated an extinction of A_V = 36 ^+ 18₋₁₁, determined that the AGN is not dominating the ionizing radiation field, and found that the star-forming population is consistent with a burst of age ≥5 Myr.

As a sanity check on whether a synchronized burst might be expected, we calculate the orbital timescale associated with the burst region. Taking the rotation velocity of ∼170 km s⁻¹ from the integrated spectrum and the radius of ∼80 pc associated with region T1, we estimate an orbital timescale of ∼3 Myr. If we take this as roughly the timescale for the nuclear disk to react to changing conditions, a burst shutting off or turning on in an ∼5 Myr timescale is reasonable.

The ratio of the integrated recombination line flux (Equation (3)) to the free–free continuum flux density (Equation (1)) allows the electron temperature to be determined. Dependencies on the distance, emission measure, and (possible) beam-filling effects cancel out under the assumption that the two tracers arise in the same volume of gas. We show in Appendix A.3.1 that, when taking the ratio of the integrated line to continuum, R_LC, and solving for the temperature, T_e, we arrive at

$$\begin{eqnarray}\begin{array}{rcl}{T}_{e} & = & {10}^{4}\,{\rm{K}}\left[{b}_{{\mathsf{n}}+{\mathsf{1}}}{\left(1+y\right)}^{-1}{\left(\displaystyle \frac{{R}_{\mathrm{LC}}}{31.31\mathrm{km}{{\rm{s}}}^{-1}}\right)}^{-1}\right.\\ & & \times \ {\left.{\left(\displaystyle \frac{\nu }{100\mathrm{GHz}}\right)}^{1.12}\right]}^{0.85},\end{array}\end{eqnarray} \tag{ 4 }$$

where ${b}_{{\mathsf{n}}+{\mathsf{1}}}$ is the non-LTE departure coefficient, and y is the abundance ratio of ionized helium to hydrogen number density, y = n_He+/n_p, which we fix as y = 0.10 (de Pree et al. 1996; E.A.C. Mills et al. 2020, in preparation).

Table 5 lists the temperatures we derive in the region. We focus on the five sources with bright (peak S/N > 4.7σ) and well-fit recombination line emission. Most of these sources have higher free–free fractions than the median. To derive the temperatures, we reevaluate the continuum (fraction of) free–free emission at a resolution of 02, since the free–free fraction may change with resolution. Therefore, we convolve the Band 3 continuum images to 02 resolution. We extract the continuum from the full-bandwidth image through aperture photometry using an aperture diameter of 04. In order to exactly match the processing of the spectral line data, we do not subtract background continuum emission within an outer annulus. Then, by extracting the continuum in each spectral window (using the same aperture diameters just described), we fit for the in-band spectral index. We use the procedure described in Section 3.3 to constrain the free–free fraction from the spectral index fit.

With the free–free fraction and measured fluxes, we plug the line-to-continuum ratio into Equation (4) and take ${b}_{{\mathsf{n}}}=0.73$ (Storey & Hummer 1995) to arrive at the temperature. The departure coefficient at ${\mathsf{n}}=41$ is loosely (<15% variation) dependent on the temperature. We iterate (once) on the input ${b}_{{\mathsf{n}}}$ and output temperature. Here ${b}_{{\mathsf{n}}}=0.73$ is the modeled value for this temperature and typical densities of n_e = (10³–10⁴) cm⁻³ of ionized gas surrounding young massive stars and consistent with the ionized gas densities we derive in Section 5.4.

The uncertainties in the electron temperatures we derive in Table 5 are dominated by the uncertainties in the free–free fraction. We take the mean and standard deviation values of T_e = (6000 ± 400) K as a representative electron temperature of the ionized plasma in the candidate star clusters. This temperature is consistent with the temperature derived from a lower-resolution analysis of NGC 4945 at 23 × 26 resolution, which finds T_e = (5400 ± 600) K (Bendo et al. 2016).

Our estimated temperature implies a thermal line width of (16 ± 4) km s⁻¹ (Brocklehurst & Seaton 1972). Given that this is smaller than our observed line widths, nonthermal motions from bulk velocities (such as turbulence, inflow, or outflow) must contribute to broadening the spectral line profiles.

The electron temperature of free–free plasma surrounding massive stars is related to the metallicity of the plasma, as the metals contribute to gas cooling. Shaver et al. (1983) established a relation,

$$\begin{eqnarray}&&12+{\mathrm{log}}_{10}({\rm{O}}/{\rm{H}})=(9.82\pm 0.02)-(1.49\pm 0.11)\displaystyle \frac{{T}_{e}}{{10}^{4}\,{\rm{K}}},\end{eqnarray} \tag{ 5 }$$

with the temperatures and metallicities derived with (auroral) collisionally excited lines at optical wavelengths. Furthermore, they showed that these temperatures are consistent with electron temperatures derived from radio recombination lines. We find a representative O/H metallicity of 12 + log₁₀(O/ H) =8.9 ± 0.1. This value is in approximate agreement (within 2σ) with the average metallicity and standard deviation of $12+{\mathrm{log}}_{10}({\rm{O}}/{\rm{H}})=8.5\pm 0.1$ (Stanghellini et al. 2015) determined in 15 star-forming regions in the galactic plane of NGC 4945 (and which is consistent with no radial gradient) using strong line abundance ratios of oxygen, sulfur, and nitrogen spectral lines.

We determine the volumetric emission measure of gas ionized in candidate star clusters using Equations (1) and (3) together with the mean temperature derived in Section 5.3. In Table 6, we list the results for each candidate star cluster. Emission measures that we determine from the free–free continuum are in the range ${\mathrm{log}}_{10}({{EM}}_{{\rm{C}}}/{\mathrm{cm}}^{-6}\,{\mathrm{pc}}^{3})$ ∼ 7.3–8.7, with a median value of 8.4. We also calculate the volumetric emission measure of ionized hydrogen as determined by the effective H41α recombination line when applicable, noting that EM_C = (1 + y)EM_L. The line emission measures are in the range ${\mathrm{log}}_{10}({{EM}}_{{\rm{L}}}/{\mathrm{cm}}^{-6}\,{\mathrm{pc}}^{3})$ ∼ 8.4–8.9, with a median value of 8.5. The uncertainty in the emission measures is ∼0.4 dex and is dominated by the errors of the free–free fraction.

Next, we solve for the electron density. We use the emission measure determined from the free–free continuum, assume n_e = n₊, and consider a spherical volume with r = FWHM_size/2. We arrive at densities in the range ${\mathrm{log}}_{10}({n}_{e}/{\mathrm{cm}}^{-3})$ = 3.1–3.9 with a median value of 3.5.

We matched (see Section 3.2) five of the candidate star clusters that have recombination line emission detected—sources 1, 6, 14, 21, and 27—with the 2.3 GHz objects of Lenc & Tingay (2009), which have the free–free optical depth modeled through their low-frequency turnovers. Although the 2.3 GHz objects have nonthermal indices, it is their radio emission that is opaque to free–free plasma. Using the optical depths derived in Lenc & Tingay (2009) and our fiducial electron temperature, we solve for the density through the relation $\tau \approx 3.28\times {10}^{-7}{\left(\tfrac{{T}_{e}}{{10}^{4}{\rm{K}}}\right)}^{-1.35}{\left(\tfrac{\nu }{\mathrm{GHz}}\right)}^{-2.1}\left(\tfrac{{{EM}}_{{\ell }}}{{\mathrm{cm}}^{-6}\,\mathrm{pc}}\right)$ (Condon & Ransom 2016), where EM_ℓ = n_en₊ℓ and for which a spherical region the path length ℓ translates as ${\ell }=\tfrac{3}{4}r$. We find densities in the range ${\mathrm{log}}_{10}({n}_{e}/{\mathrm{cm}}^{-3})$ = 3.3–3.6. This agrees well with the values we separately derive.

We convert the ionized gas density and source sizes to an ionized gas mass through

$$\begin{eqnarray}&&{M}_{+}=1.36\,{m}_{{\rm{H}}}\,{n}_{+}\displaystyle \frac{4}{3}\pi {r}^{3},\end{eqnarray} \tag{ 6 }$$

where we have assumed a 1.36 contribution of helium by mass, and we let r = FWHM_size/2. The ionized gas masses of the candidate star clusters are in the range ${\mathrm{log}}_{10}({M}_{+}$/M_⊙) =2.7–3.5 with a median value of 3.1. The ionized gas mass is a small fraction (≲1%) of the stellar mass (see Section 5.5).

We estimate the number of ionizing photons needed per second to maintain the total free–free emitting content (see Table 6). From the emission measure of ionized gas and the temperature-dependent recombination coefficient for case B recombination, the rate of ionizing photons (see Appendix A.3.2) with E > 13.6 eV is

$$\begin{eqnarray}\begin{array}{rcl}{Q}_{0} & = & \left(3.8\times {10}^{51}\,{{\rm{s}}}^{-1}\right)\left(\displaystyle \frac{{n}_{e}{n}_{+}V}{5\times {10}^{8}\,{\mathrm{cm}}^{-6}\,{\mathrm{pc}}^{3}}\right)\\ & & \times \ {\left(\displaystyle \frac{{T}_{e}}{{10}^{4}{\rm{K}}}\right)}^{-0.83},\end{array}\end{eqnarray} \tag{ 7 }$$

where EM_C = n_en₊V is the volumetric emission measure of the total ionized gas we take from the continuum-derived emission measure, and T_e is the electron temperature of the ionized gas. Our candidate star clusters have ionizing photon rates in the range ${\mathrm{log}}_{10}({Q}_{0}/{{\rm{s}}}^{-1})$ ∼ 50.4–51.8. The sum of the ionizing photon rate over all candidate massive star clusters is 5.3 × 10⁵² s⁻¹. In the top panel of Figure 11, the ionizing photon rates of the candidate clusters are plotted as complementary cumulative fractions.

We use Starburst99 calculations (Leitherer et al. 1999) to infer the stellar mass from the ionizing photon output of a 5 Myr old stellar population via

$$\begin{eqnarray}&&{M}_{\star }\approx \displaystyle \frac{{Q}_{0}}{4.7\times {10}^{45}}{{\rm{M}}}_{\odot }.\end{eqnarray} \tag{ 8 }$$

We arrive at this value by simulating a single 10⁶ M_⊙ stellar population with the initial mass function (IMF) of Kroupa (2001), a maximum stellar mass of 100 M_⊙, and the default stellar evolution tracks and tuning parameters. Then we divide the ionizing photon output at 5 Myr by the initial mass of the stellar population. We note that this is a rough approximation that has not accounted for the amount of ionizing photons absorbed by dust, mass ejected from the system, and/or enhanced emission from stellar binaries.

Our candidate star clusters have stellar masses in the range ${\mathrm{log}}_{10}({M}_{\star }/{{\rm{M}}}_{\odot })$ ∼ 4.7–6.1 (see Table 6) with a median of 5.5. The error on the mass estimate is ∼0.4 dex. The sum of the stellar masses of the candidate star clusters is ≈1.1 × 10⁷ M_⊙. In the bottom panel of Figure 11, the estimated stellar masses of the candidate clusters are plotted as cumulative fractions.

We estimate the mass of gas associated with each candidate star cluster (see Table 6) from dust emission at 350 GHz. We determine the dust optical depth by comparing the measured intensity with that expected from an estimate of the true dust temperature. Assuming a mass absorption coefficient, we convert the optical depth to a dust column density. We arrive at a gas mass by multiplying the dust column density by the measured source size and an assumed dust-to-gas mass ratio (DGR).

We assume a dust temperature of T_dust = 130 K, as has been determined for the gas kinetic temperature in the forming super star clusters in NGC 253 (Gorski et al. 2017). This is an approximation, though the uncertainty is linear. Then we convert the 350 GHz flux density into an intensity (I₃₅₀) and solve for the optical depth through I₃₅₀ ≈ τ₃₅₀B_ν(T_dust), where B_ν(T_dust) is the Planck function evaluated at 350 GHz. We measure optical depths in the range τ₃₅₀ ∼ 0.02–0.10 with a median value of τ₃₅₀ ∼ 0.04, justifying our optically thin assumption. We note that the 3σ upper limit of the sources that have not been detected at 350 GHz corresponds to τ₃₅₀ < 0.02.

Next, we convert the optical depth to a dust column density using an assumed mass absorption coefficient (κ). We adopt κ = 1.9 cm² g⁻¹, which should be appropriate for ν ∼ 350 GHz and dust mixed with gas at a density of ∼10⁵–10⁶ cm⁻³ (Ossenkopf & Henning 1994), but we do note the large (factor of 2) uncertainties on this value. Finally, we combine the dust surface density with an adopted DGR of 1 to 100, approximately the Milky Way value and similar to the value found for starburst galaxies by Wilson et al. (2008). Our estimate for the gas surface density is determined with

$$\begin{eqnarray}&&{{\rm{\Sigma }}}_{\mathrm{gas}}=\displaystyle \frac{{\tau }_{350}}{\kappa \,\mathrm{DGR}}.\end{eqnarray} \tag{ 9 }$$

We determine the gas mass by multiplying the gas surface density by the two-dimensional area of the source size, M_gas = A Σ_gas.

The gas masses we estimate are included in Table 6. We find values in the range of ${\mathrm{log}}_{10}({M}_{\mathrm{gas}}$/M_⊙) = 4.4–5.1 with a median value of 4.7. Upper limits for the sources that have not been detected in 350 GHz emission are included in the table.

We estimate the current total mass of each candidate star cluster as the sum of the gas and star masses, where M_Tot = M_gas + M_⋆. The total mass is dominated by the stellar mass, as we find low gas mass fractions of M_gas/ M_Tot =0.04–0.22 and a median of 0.13. When calculating the total mass of the 10 sources that are not detected at 350 GHz, we do not include the lower limit of the dust mass; we only consider the stellar mass.

We express the total mass of each source in terms of a surface density (in Table 6). This is calculated within the FWHM of the region; we thereby divide the total mass by 2 and divide by the two-dimensional area of the FWHM. The values are in the range $\mathrm{log}({{\rm{\Sigma }}}_{\mathrm{Tot}})=3.1$–4.8 M_⊙ pc⁻², or, alternatively, Σ_Tot = 0.3–13 g cm⁻².

Using the total mass, we estimate the gravitational freefall time of the clusters, ${t}_{\mathrm{ff}}={\left(\tfrac{\pi {r}^{3}}{8{{GM}}_{\mathrm{Tot}}}\right)}^{1/2}$. The values we derive are included in Table 6 and in the range ${\mathrm{log}}_{10}({t}_{\mathrm{ff}}$/yr) = 4.3–5.2 with a median of 4.6. This is the gravitational freefall time that would be experienced by gas with no support if all of the cluster mass were gas. The fact that the gas mass fractions are low and that the age appears longer than the freefall time adds support to the idea that these clusters have mostly already formed.

Summing all sources, we find a total mass within the candidate clusters of M_Tot ≈ 1.5 × 10⁷ M_⊙.

Both the free–free fractions at 93 GHz and the age of the burst could be better constrained using future observations, and this would result in more accurate estimates of almost all properties of the cluster candidates.

The first source of uncertainty that we discuss is the estimate of the free–free flux, by which we estimate a free–free fraction to the 93 GHz continuum.

In estimating the free–free fractions, we enforce that only two contributions of emission type, free–free and synchrotron or free–free and dust, compose the flux at 93 GHz. If we consider three emission types, we would obtain nonunique solutions in estimating the fractions of emission from the spectral index. Assuming two types is the best that we have assessed with the currently available data. Small dust opacities and a majority of negative slopes measured at 93 GHz independently indicate that dust does not substantially contribute in the majority of sources. Therefore, assuming two dominant components to the emission at 93 GHz appears to be valid in the majority of sources. In order to decompose the emission at 93 GHz through SED fitting, additional observations at intermediate frequencies (within 2.3 and 350 GHz) of comparable resolution are needed.

To estimate the free–free fraction, we also assume fixed energy distribution slopes for emission from free–free, synchrotron, and dust. However, variations from source to source may be expected. If we rederive the free–free fractions, letting α_d = 2 without changing the free–free and synchrotron indices, we find that the median dust fraction of sources remains at f_d = 0. This is another indication that the spectral index assumed for dust does not have a considerable impact on the majority of sources. The range of reasonable values to consider for the frequency dependence of nonthermal emission is less constrained, where single-injection indices of α_syn = −0.5 to −0.8 can be expected all the way up to the dramatic exponential cutoff¹³ in a single-injection scenario where the highest-energy electrons (at the high-frequency end) completely depopulate after a characteristic energy-loss time (e.g., Klein et al. 2018). As we note in Section 3.3, our assumed index of α_syn − 1.5 agrees well with the measured value of −1.4 at lower resolution (Bendo et al. 2016). If we rederive the free–free fractions assuming the canonical value of α_syn = −0.8 determined at 10 GHz (Niklas et al. 1997), the median free–free fraction of sources would be f_ff = 0.28, and the synchrotron fraction would be f_syn = 0.72.

On average, the fractional contributions to the 93 GHz emission that we obtain assuming indices of α_ff = −0.12, α_syn = −1.5, and α_d = 4 appear to be reasonable and consistent. Our median free–free fraction of f_ff = 0.62 with a median absolute deviation of 0.29 agrees within the errors with the free–free fraction derived by Bendo et al. (2016) of f_ff = 0.84 ± 0.10 at 86 GHz. Our median free–free fraction is also consistent with that derived for NGC 253, f_ff =0.70 ± 0.10, at our same frequency but averaged over 30'' (Bendo et al. 2015). Similarly, pilot survey results of the MUSTANG Galactic Plane Survey at 90 GHz and with parsec-scale resolution indicate that >80% of the emission in (candidate) clusters is composed of synchotron or free–free emission (Ginsburg et al. 2020).

We can also perform another self-consistent check on the free–free emission at 93 GHz by using the recombination line emission and assuming the candidate star clusters have uniform temperatures. A constant temperature implies a constant integrated recombination line–to–continuum ratio (see Equation (4)). Letting R_LC ≈ 35 km s⁻¹ (for a temperature of T_e = 6000 K), we plug in our measured line and continuum fluxes and solve for f_ff. A median free–free fraction of f_ff = 0.85 with a median absolute deviation of 0.33 is found when considering all sources with detected recombination line emission. In comparison, the relation assumed for the in-band spectral index, also at 02 resolution, results in a median value of f_ff = 0.66 and a median absolute deviation of 0.24. These values are in reasonable agreement given the uncertainties in the two methods.

Overall, the assumptions made in decomposing the emission at 93 GHz, especially regarding synchrotron emissions, likely affect individual clusters at the ∼30% level. This is not enough to bias our overall results, but follow-up observations at other frequencies would be extremely helpful.

The second potentially major source of uncertainty is the assumed age of the burst. We discuss in Section 5.2 how we arrive at an adopted age of the clusters of ∼5 Myr. The assumed age has a large impact on the stellar mass inferred from the ionizing photon rate. The ionizing photon output changes substantially (by a factor of 40 from a zero-age main sequence to an age of 5 Myr) as the most massive stars explode as supernovae. An uncertainty of ∼1 Myr about an age of 5 Myr of a star cluster results in an uncertainty in the inferred stellar mass by a factor of 4. This is roughly included in the 0.4 dex uncertainty, though it would represent a systematic offset.

The estimated properties (mass, size, age; Table 6) that we derive for these candidate star clusters meet the criteria for young massive clusters (e.g., Portegies Zwart et al. 2010), and these sources can be considered super star clusters (SSCs). Star clusters forming in high gas surface density environments may be able to acquire significant amounts of mass before feedback effects (likely radiation pressure) can disrupt and/or disperse the cluster (e.g., Adamo & Bastian 2016). Super star clusters with stellar masses of M_⋆ ≳ 10⁵ M_⊙ typically have high star formation efficiencies (ε > 0.5) and therefore remain bound.

With the mass surface densities that we estimate and the age of the burst equaling many multiples of the freefall times, these star clusters will likely remain bound, at least initially. They are being born into a violent environment, and clusters often still experience significant mortality after forming. Estimates of the virial mass, escape velocity, and momentum driving determined through molecular line observations will help quantity their mass and initial dynamical state.

In Figure 11, we plot the cluster stellar-mass function of candidate SSCs in NGC 4945 and compare it with the cluster mass distributions in additional galaxies with SSC populations.

A power-law fit to the cluster mass distribution of 22 sources down to M_⋆ = 2.4 × 10⁵ M_⊙ in NGC 4945 results in a slope of β = −1.8 ± 0.4. Note that the fit to the data results in β = −1.76 ± 0.07, but given the uncertainty in our mass estimates (0.4 dex), we adopt the more conservative figure. We include the SSCs of NGC 253 with stellar-mass properties determined on similar spatial scales (∼2 pc resolution) and through H40α recombination lines for a zero-age main-sequence population (Leroy et al. 2018; E.A.C. Mills et al. 2020, in preparation). A power-law fit to the cluster mass distribution of NGC 253, including 12 star clusters down to M_⋆ = 7.9 × 10⁴ M_⊙, results in a slope of β = −1.6 ± 0.3. We estimated the completeness limits by eye. The turnoff from power-law distributions likely includes nonphysical effects resulting from the depth/sensitivity of the observations, as well as source confusion due to the high inclination in which we view the starburst regions (and that appears to be higher in NGC 253).

We also include recent results from a homogeneous analysis by Mok et al. (2020) of the cluster mass distributions of young (τ < 10 Myr) massive clusters in six galaxies. As representative examples, we include the best-fit power laws from three systems: the Large Magellanic Cloud (LMC), M51, and the Antennae System. A main result from Mok et al. (2020) is that the cluster mass functions across the six galaxies are consistent with slopes of β = −2.0 ± 0.3. Our measurements for NGC 4945 and NGC 253 are consistent with these findings.

As we discuss in Section 6.6, we estimate the total stellar mass in the burst to be 8.5 × 10⁷ M_⊙. Extending the best fit of our cluster mass function down to a cluster mass of ∼2 × 10⁴ M_⊙ would account for the additional mass and, correspondingly, the additional ionizing photon luminosity. As we discuss in Section 6.4, if these lower-mass clusters were present, they would need to be located in a more extended region than the super star clusters. Therefore, the cluster mass distribution may not reach down to 2 × 10⁴ M_⊙ in the region where the SSCs are located but perhaps ∼4 × 10⁴ M_⊙ if a third of the additional recombination line emission results from diffuse ionized gas.

Total ionizing photon rate of the starburst. We measure an H40α recombination line flux of (6.4 ± 1.0) Jy km s⁻¹ integrated within the T2 aperture (see Figure 8) in the intermediate-configuration (07) data. Assuming the temperature of T_e = (6000 ± 400) K from Section 5.3, the total ionized content, which accounts for the full star formation rate of the starburst, yields an ionizing photon rate of Q_T2 = (3.9 ± 0.3) × 10⁵³ s⁻¹.

Ionizing photon rate from candidate super star clusters. The sum of the ionizing photon rate over all massive star clusters is ≈5.3 × 10⁵² s⁻¹. The total integrated line flux of H40α in the extended-configuration (02) data is $\int S\,\mathrm{dv}=(2.1\pm 0.6)$ Jy km s⁻¹ within an aperture (T1) that covers the clusters in the starburst region. With the temperature of T_e = (6000 ± 400) K, the integrated line flux corresponds to an ionizing photon rate of Q_T1 = (1.2 ± 0.4) × 10⁵³ s⁻¹. Given the large uncertainties derived for individual clusters, we consider this value consistent with the sum over individual clusters. Therefore, we conclude that 20%–44% of the total ionizing photons in the starburst can be attributed to the candidate super star clusters identified in NGC 4945.

Low-mass clusters. The intermediate-configuration (07) data are sensitive to emission on larger physical scales, but they also reach deeper sensitivities per unit area. Consequently, emission from a distribution of many compact, low-mass clusters could be traced in these data but not in the extended-configuration data.

We do not find it likely that the deficit of line emission in the extended-configuration data can be attributed primarily to low-mass clusters. Focusing only on the low-resolution (07) observations, the integrated line flux is greater in the larger aperture (T2) than in the aperture covering only the star clusters (T1), indicating that additional line emission originates outside of the area where massive clusters are forming. In order for low-mass clusters to account for the difference, these would need to form in a more extended region than the bright SSCs that we see.

The AGN and the circumnuclear disk. The circumnuclear disk, including the AGN, could be the origin of ionizing photons observed only on larger scales. The strongest recombination line emission is seen at low resolution in this region, yet no significant line detections are obtained toward sources 17, 18, and 20. We test this scenario by extracting spectra from the 07 intermediate-configuration data and the 02 extended-configuration data in seven nonoverlapping apertures of 3'' diameter consecutively distributed along the major axis of the starburst region. The ratios of the line flux in the intermediate- to extended-configuration data are all consistent within the errors; the ratios in the seven regions have a mean and standard deviation of 2.3 ± 0.5. This indicates that a deficit of line emission on the physical scales probed in the high-resolution data is ubiquitous (and not unique to the region surrounding the AGN), and at most ∼20% may originate from the circumnuclear disk and AGN.

Diffuse ionized gas on large scales. Summarizing the information above, we find that up to ∼70% of the total radio recombination line flux may originate on scales larger than ∼100 pc; this emission cannot be directly attributed to the AGN. Ionizing radiation may be escaping the immediate surroundings of massive stars and reaching larger scales. The approximate age of the burst indicates that clusters have had time to shed their natal material, dust is not a significant contribution of their emission, low gas mass fractions have been determined, and the current freefall times of the clusters indicate abundant time to expel gas. Despite these indicators, extinctions may still be considerable, and patchy regions may help to leak ionizing photons. Note that if radiation is escaping from star clusters, this would also imply that the stellar masses (derived from the ionizing photon rates) are underestimated.

However, the missing 70% of line flux in the 02 resolution data should be considered as an upper limit. Given the low signal-to-noise ratio of these lines, artifacts in both the image and spectral domains can impact the properties of the line profiles. For example, incomplete uv coverage on the scales of the emission can result in incorrect deconvolution and thus systematic underestimates. Deeper observations at high resolution (∼01) would allow radio recombination lines to be mapped out with adequate signal-to-noise ratios and obtain sensitivities approaching that of the low-resolution (07) observations.

Although the Seyfert 2 AGN in NGC 4945 is one of the brightest in our sky at X-ray energies, obscuration by, e.g., A_V ≥ 160 (Spoon et al. 2000) has prevented its observation at virtually all other wavelengths (also given the spatial resolutions observed). Our observations at 93 GHz provide another piece of evidence for its existence.

We detect a point source in the 93 GHz image with a deconvolved size determined by PyBDSF of FWHM =0.84 ± 0.01 pc. The flux of 9.7 ± 1.0 mJy that we extract from the source is ∼10% of the total continuum (∼120 mJy) flux in the 012 resolution data. The Band 3 spectral index we derive is α₉₃ = −0.85 ± 0.05, which is consistent with freshly accelerated electrons emitting synchrotron radiation. However, we follow the procedure outlined in Section 3.3, assuming free–free emission may also contribute, and place a limit on the ionizing photon rate. The spectral index at 93 GHz corresponds to f_ff ≥ 0.47 ± 0.03. If we assume a temperature of T_e = 6000 (30,000) K, this would correspond to Q₀ = 1.1 (0.5) × 10⁵² s⁻¹. The low level of escaping ionizing radiation is consistent with previous work from MIR line ratios (Spoon et al. 2000).

We place a limit on the ionizing photons that could be leaking into medium. The bolometric luminosity estimated from the X-ray luminosity is L_bol ≈ 6 × 10⁴³ erg s⁻¹, which we determined using the relation for Seyfert AGNs of L_bol ∼ 20 L_X (Hopkins et al. 2007). Given the bolometric luminosity, we can now estimate the total expected ionizing photon luminosity. We use the standard relation of Elvis et al. (1994), which posits L_bol/L_ion ∼ 3 with an average ionizing photon energy of 113 eV. Therefore, the expected ionizing photon luminosity of the AGN is L_ion ≈ 2 × 10⁴³ erg s⁻¹ ≈5 × 10⁹ L_⊙, while our limit corresponds to ≈2 × 10⁴² erg s⁻¹. This would imply that <10% of the ionizing photon luminosity of the AGN is escaping into the surrounding nuclear starburst and creating ionized gas.

We can convert the total ionizing photon rate of the burst Q_T2 (see Section 6.4), as measured in the T2 region, to a luminosity. Assuming an average ionizing photon energy of $\left\langle h\nu \right\rangle \approx 17\,\mathrm{eV}$, the ionizing luminosity is L_T2 ≈ 1.1 ×10⁴³ erg s⁻¹ = 2.8 × 10⁹ L_⊙. A ratio of bolometric luminosity to ionizing photon luminosity L_bol/L₀ ∼ 14 is predicted for a 5 Myr old population using Starburst99 (Leitherer et al. 1999). Thus, we can expect a bolometric luminosity of 4 × 10¹⁰ L_⊙, which agrees within the errors with the bolometric luminosity derived from FIR observations, L_bol = (2.0 ± 0.2) × 10¹⁰ L_⊙ (Bendo et al. 2016).

We note that, if we had adopted a lifetime for the burst of 4 Myr rather than 5 Myr, Starburst99 calculations would expect a bolometric luminosity of L_bol ≈ 2 × 10¹⁰ L_⊙. Also, this calculation assumes that the source of the additional recombination line emission in the intermediate-configuration data is also characterized by a 5 Myr old stellar population.

The ratio of bolometric luminosity to mass of a 5 Myr old burst is estimated at Ψ ∼ 470 L_⊙/M_⊙. With an L_bol =4 × 10¹⁰ L_⊙, the inferred total stellar mass of the population is ∼8.5 × 10⁷ M_⊙. The total stellar mass in our candidate clusters is ≈1.1 × 10⁷ M_⊙, or ∼13% of the expected total stellar mass.

An outflow of warm ionized gas in NGC 4945 has been modeled as a biconical outflow with a deprojected velocity of v ≈ 525 km s⁻¹ with emission out to 1.8 kpc (Heckman et al. 1990). The outflow has been traced in [N II], [S II], and Hα in multiple analyses (Heckman et al. 1990; Moorwood et al. 1996; Mingozzi et al. 2019). Heckman et al. (1990) estimated a total energy of 2 × 10⁵⁵ erg and momentum flux of 9 × 10³³ dyne = 1400 M_⊙ km s⁻¹ yr⁻¹ (after correcting for the updated distance). These values do not include possible contributions by colder phases, which are likely associated with the NGC 4945 outflow (Bolatto et al., in preparation). While these values are uncertain, they are useful for an order-of-magnitude comparison with the expected properties of the star clusters.

Gas reaching 1.8 kpc and moving at 525 km s⁻¹ would have been launched ∼3 Myr ago assuming a constant velocity, which is within the time frame of the burst. An ∼3 Myr timescale and a velocity of 525 km s⁻¹ would put 8 × 10⁶ M_⊙ of warm ionized gas mass into the outflow. Using Starburst99 (Leitherer et al. 1999), an estimated mechanical luminosity of ∼3 × 10⁴¹ erg s⁻¹ from the clusters, fairly constant over the 5 Myr age, equates to an injected energy of ≈6 × 10⁵⁵ erg. It would require 30% of the expected mechanical energy output of the clusters to drive the ionized outflow. From simulations, the total momentum supplied to the ISM per supernova is expected to be 2.8 × 10⁵ M_⊙ km s⁻¹ (Kim & Ostriker 2015, and references therein), with stellar winds contributing an additional 50% to the momentum. Starburst99 predicts the supernova rate to be fairly constant at ∼0.009 yr⁻¹ for these clusters. From these values, we estimate that the momentum supplied to the ISM by the candidate star clusters would be ∼3800 M_⊙ km s⁻¹ yr⁻¹. It would require about 40% of the expected momentum output of the clusters to drive the ionized outflow. Therefore, neither the energetics nor the inferred momentum of the outflow prevent it from being driven solely by the star clusters in NGC 4945, although there are very considerable uncertainties associated with this calculation.

The nearly edge-on galaxy NGC 253 is located nearby (3.5 ± 0.2 Mpc; Rekola et al. 2005) with similar properties as NGC 4945. It hosts a central starburst spanning ∼200 pc with young, massive clusters (Leroy et al. 2018; E.A.C. Mills et al. 2020, in preparation). A major difference between these two galaxies is that NGC 4945 shows unambiguous signatures of harboring an active supermassive black hole. Since NGC 4945 is just the second analysis we have undertaken with millimeter-wavelength ALMA observations on scales that resolve the clusters, it is relevant to compare the properties of their super star clusters.

Analyses of the cluster population of NGC 253 at ∼2 pc resolution with ALMA (Leroy et al. 2018; E.A.C. Mills et al. 2020, in preparation) revealed 18 sources as (proto-)super star clusters that are in the process of forming or close to a zero-age main sequence (∼1 Myr; see also Rico-Villas et al. 2020). Overall, the properties of the clusters are strikingly similar to those in NGC 4945; they have a median stellar mass of 1.4 × 10⁵ M_⊙ and FWHM sizes of 2.5–4 pc. The star clusters of NGC 253 boast slightly higher ionizing photon luminosities and larger gas fractions M_gas/M_tot ∼ 0.5, reflecting a slightly younger age. The slopes of the cluster mass function we derive are also consistent (see Section 6.3), just offset by a factor of ∼3.5 in mass. As in NGC 4945, at least 30% of the nuclear starburst of NGC 253 originates in a clustered mode of star formation.

While we have not estimated some of the dynamical properties of NGC 4945, the presence of broad recombination line emission in four sources in NGC 253 indicates the star clusters are operating under similar processes. In NGC 253, the sources are young enough that feedback has not managed to unbind a large fraction of the gas from the clusters. The slightly higher total mass surface densities and smaller freefall times in NGC 4945 indicate that its clusters might be surviving a young disruptive stage.

The concerted feedback of the young (<10 Myr) forming SSCs identified in NGC 253 (Leroy et al. 2018) is likely not responsible for the starburst-driven outflow, as they are expected to impart momentum that is a factor of 10–100 lower than the outflow momentum estimated in CO (Bolatto et al. 2013; Krieger et al. 2019). Some (global) event may be initiating the active formation of star clusters. On the other hand, the star clusters in NGC 4945 could potentially influence the outflow of warm ionized gas; some event appears to be inhibiting the formation of new star clusters.

Considering Kirchoff's law of thermodynamics with no net change in intensity through the medium, the emission of a radio recombination line in LTE, ${S}_{{\mathsf{n}}}(\mathrm{LTE})$, is related to ${\kappa }_{{\mathsf{n}}}$, the fractional absorption per unit path length ℓ, for small line optical depths by

$$\begin{eqnarray}&&{S}_{{\mathsf{n}}}(\mathrm{LTE})\approx {\kappa }_{{\mathsf{n}}}\,{\ell }\,{B}_{\nu }({T}_{e})\,{\rm{\Omega }}\end{eqnarray} \tag{ A1 }$$

for a transition to the final principal quantum number ${\mathsf{n}}$, where B_ν is the Planck function and Ω is the solid angle on the sky. For small optical depths of the line and the free–free continuum τ_c, the non-LTE line flux density is, by definition,

$$\begin{eqnarray}&&{S}_{{\mathsf{n}}}\approx {S}_{{\mathsf{n}}}(\mathrm{LTE}){b}_{{\mathsf{n}}+{\mathsf{1}}}\left(1+\displaystyle \frac{{\tau }_{C}}{2}\beta \right).\end{eqnarray} \tag{ A2 }$$

Here ${b}_{{\mathsf{n}}+{\mathsf{1}}}$ is the departure coefficient, which is defined as ${b}_{{\mathsf{n}}}=n/{n}_{\mathrm{LTE}}$, the ratio of the actual number density of atoms with an electron in level ${\mathsf{n}}$ to the number that would be there if the population were in LTE at the temperature of the ionized gas, such that for LTE, ${b}_{{\mathsf{n}}}=1$. Here β is the departure coefficient that accounts for stimulated emission. The second term in parentheses is usually negligible at millimeter wavelengths, such that (1 + 0.5τ_cβ) ≈ 1. In Equation (A5) of Section A.2, the absorption coefficient of the free–free continuum is given, and the optical depth can be derived to show that the typical parameters (e.g., EM_l = 10⁷ cm⁻⁶ pc and T_e = 10⁴ K) of ionized regions around massive stars result in τ_c ≲ 3 × 10⁻⁶ at 100 GHz. Typical values of β for ${\mathsf{n}}\gtrsim 40$ are $\left|\beta \right|\lesssim 40$ (e.g., Storey & Hummer 1995).

In its expanded form, the LTE absorption coefficient of a radio recombination line can be expressed in terms of the Saha–Boltzmann distribution, oscillator strength, and line frequency as

$$\begin{eqnarray}&&\begin{array}{l}{\kappa }_{{\mathsf{n}}}\approx {n}_{e}{n}_{+}\displaystyle \frac{2}{\sqrt{\pi }}\displaystyle \frac{{e}^{2}}{{m}_{e}}{\left(\displaystyle \frac{{h}^{2}}{2{m}_{e}{k}_{B}}\right)}^{3/2}\displaystyle \frac{h}{{k}_{B}}\,{T}_{e}^{-2.5}{R}_{H}{Z}^{2}{\rm{\Delta }}{\mathsf{n}}\\ \quad \ \times \ \left(1-\displaystyle \frac{3}{2}\displaystyle \frac{{\rm{\Delta }}{\mathsf{n}}}{{\mathsf{n}}}\right)\exp \left(\displaystyle \frac{{\chi }_{{\mathsf{n}}}}{{k}_{B}{T}_{e}}\right)M({\rm{\Delta }}{\mathsf{n}})\left(1+\displaystyle \frac{3}{2}\displaystyle \frac{{\rm{\Delta }}{\mathsf{n}}}{{\mathsf{n}}}\right){\phi }_{\nu },\end{array}\end{eqnarray} \tag{ A3 }$$

where n_e and n₊ are the number densities of electrons and ions, respectively; R_H is the Rydberg constant for hydrogen; Z is the effective nuclear charge; ${\rm{\Delta }}{\mathsf{n}}$ is the change in energy levels of the given transition; ${\chi }_{{\mathsf{n}}}$ is the energy required to ionize the atom from state ${\mathsf{n}}$ but $\exp \left({\chi }_{{\mathsf{n}}}/{k}_{B}{T}_{e}\right)$ is small (<1.02) for ${\mathsf{n}}$ ≥ 40 and typical ionized gas temperatures; $M({\rm{\Delta }}{\mathsf{n}}=1,2)=0.190775,0.026332$ is an approximation factor for the oscillator strength; and ϕ_ν is the line profile (normalization; in SI units, Hz⁻¹) such that ${\int }_{-\infty }^{\infty }{\phi }_{\nu }{\rm{d}}\nu =1$.

Bringing the above equations together, using the Rayleigh–Jeans approximation of the Planck function, and integrating over the line profile, we arrive at the expression for the integrated line flux density,

$$\begin{eqnarray}&&\begin{array}{l}\displaystyle \int {S}_{{\mathsf{n}}}\,\mathrm{dv}=\left(65.13\,\mathrm{mJy}\,\mathrm{km}\,{{\rm{s}}}^{-1}\right){b}_{{\mathsf{n}}+{\mathsf{1}}}\left(\displaystyle \frac{{n}_{e}{n}_{p}V}{5\times {10}^{8}\,{\mathrm{cm}}^{-6}\,{\mathrm{pc}}^{3}}\right)\\ \quad \ \times \ {\left(\displaystyle \frac{D}{3.8\mathrm{Mpc}}\right)}^{-2}{\left(\displaystyle \frac{{T}_{e}}{{10}^{4}{\rm{K}}}\right)}^{-1.5}\left(\displaystyle \frac{\nu }{100\,\mathrm{GHz}}\right),\end{array}\end{eqnarray} \tag{ A4 }$$

where we take n₊ = n_p, Z = 1, and ${\rm{\Delta }}{\mathsf{n}}=1$. For convenience and self-consistency, we express the emission measure in terms of source volume, V, and the solid angle in terms of distance to the source, D. Let ${\rm{\Omega }}=\tfrac{\pi {r}^{2}}{{D}^{2}}$, where r is the radius of the region, and if we let ${\ell }=\tfrac{4}{3}r$, the volumetric emission measure is given as EM_V = n_en_pV, where $V=\tfrac{4}{3}\pi {r}^{3}$.

The absorption coefficient for free–free continuum (Oster 1961) in the Rayleigh–Jeans limit is, as a function of frequency,

$$\begin{eqnarray}&&{\kappa }_{{\rm{c}}}=\displaystyle \frac{{n}_{e}{n}_{+}}{{\nu }^{2}}\displaystyle \frac{8{Z}^{2}{e}^{6}}{3\sqrt{3}{m}_{e}^{3}c}{\left(\displaystyle \frac{\pi }{2}\right)}^{1/2}{\left(\displaystyle \frac{{m}_{e}}{{k}_{B}{T}_{e}}\right)}^{3/2}{g}_{\mathrm{ff}},\end{eqnarray} \tag{ A5 }$$

where the gaunt free–free factor is (Draine 2011)

$$\begin{eqnarray}&&{g}_{\mathrm{ff}}\approx 13.91{\left(Z\displaystyle \frac{\nu }{\mathrm{Hz}}\right)}^{-0.118}{\left(\displaystyle \frac{{T}_{e}}{{\rm{K}}}\right)}^{0.177},\end{eqnarray} \tag{ A6 }$$

valid for ν_p ≪ ν ≪ kT_e/h, where the plasma frequency is ν_p = 8.98(n_e/cm⁻³)^1/2 kHz, and to within 10% when $1.4\,\times {10}^{-4}\lt Z\nu {T}_{e}^{-1.5}\lt 0.25$.

The continuum intensity for an optically thin medium is

$$\begin{eqnarray}&&{S}_{{\rm{c}}}\approx {B}_{\nu }\,{\kappa }_{{\rm{c}}}\,{\ell }\,{\rm{\Omega }}.\end{eqnarray} \tag{ A7 }$$

As we did for the line intensity, we can also express the above relation in terms of distance D and the volumetric emission measure of the region EM_V = n_en_pV, where $V=\tfrac{4}{3}\pi {r}^{3}$,

$$\begin{eqnarray}&&\begin{array}{l}{S}_{c}=(2.080\,\mathrm{mJy})\,{Z}^{1.882}\left(\displaystyle \frac{{n}_{e}{n}_{+}V}{5\times {10}^{8}\,{\mathrm{cm}}^{-6}\,{\mathrm{pc}}^{3}}\right)\\ \quad \ \times \ {\left(\displaystyle \frac{D}{3.8\mathrm{Mpc}}\right)}^{-2}{\left(\displaystyle \frac{{T}_{e}}{{10}^{4}{\rm{K}}}\right)}^{-0.323}{\left(\displaystyle \frac{\nu }{100\mathrm{GHz}}\right)}^{-0.118},\end{array}\end{eqnarray} \tag{ A8 }$$

evaluated in the Rayleigh–Jeans limit.

A.3.1. Line-to-Continuum Ratio and Temperature

Using the previous expressions, the ratio between the RRL and continuum flux density is

$$\begin{eqnarray}&&\begin{array}{l}\displaystyle \frac{\displaystyle \int {S}_{{\mathsf{n}}}\,\mathrm{dv}}{{S}_{{\rm{c}}}}=\left(31.31\,\mathrm{km}\,{{\rm{s}}}^{-1}\right){b}_{{\mathsf{n}}+{\mathsf{1}}}{\left(\displaystyle \frac{{T}_{e}}{{10}^{4}{\rm{K}}}\right)}^{-1.177}\\ \quad \ \times \ {\left(1+y\right)}^{-1}{\left(\displaystyle \frac{\nu }{100\mathrm{GHz}}\right)}^{1.118},\end{array}\end{eqnarray} \tag{ A9 }$$

where $y={n}_{{\mathrm{He}}^{+}}/{n}_{p}$ is the ratio of singly ionized helium to hydrogen by number and ${n}_{{\mathrm{He}}^{+}}$ is the singly ionized helium number density. If we assume the emission region is composed of only hydrogen and singly ionized helium, then

$$\begin{eqnarray}&&\displaystyle \frac{{n}_{p}}{{n}_{+}}=\displaystyle \frac{{n}_{p}}{{n}_{p}+{n}_{{\mathrm{He}}^{+}}}={\left(1+\displaystyle \frac{{n}_{{\mathrm{He}}^{+}}}{{n}_{p}}\right)}^{-1}={\left(1+y\right)}^{-1}.\end{eqnarray} \tag{ A10 }$$

We can rearrange the integrated RRL line-to-continuum ratio and solve for the electron temperature T_e of the emission region,

$$\begin{eqnarray}&&\begin{array}{l}{T}_{e}={10}^{4}\,{\rm{K}}\left[{b}_{{\mathsf{n}}+{\mathsf{1}}}{\left(1+y\right)}^{-1}\right.\\ \quad \ \times \ {\left.{\left(\displaystyle \frac{{R}_{\mathrm{lc}}}{31.31\mathrm{km}{{\rm{s}}}^{-1}}\right)}^{-1}{\left(\displaystyle \frac{\nu }{100\mathrm{GHz}}\right)}^{1.118}\right]}^{0.85},\end{array}\end{eqnarray} \tag{ A11 }$$

where we denote the integrated RRL line-to-continuum ratio as ${R}_{\mathrm{lc}}=\tfrac{\int {S}_{{\mathsf{n}}}\mathrm{dv}}{{S}_{{\rm{c}}}}$.

A.3.2. Ionizing Photon Rate

The rate of ionizing photons (E > 13.6 eV) is given by

$$\begin{eqnarray}&&{Q}_{0}={n}_{e}{n}_{+}V{\alpha }_{B},\end{eqnarray} \tag{ A12 }$$

where α_B is the case B recombination coefficient (Draine 2011),

$$\begin{eqnarray}&&{\alpha }_{B}=2.59\times {10}^{-13}\,{\mathrm{cm}}^{3}\,{{\rm{s}}}^{-1}{\left(\displaystyle \frac{{T}_{e}}{{10}^{4}{\rm{K}}}\right)}^{-0.833\mbox{--}0.034\mathrm{ln}({T}_{e}/{10}^{4}{\rm{K}})},\end{eqnarray} \tag{ A13 }$$

which is valid for 3000 K < T_e < 30,000 K. Thus, we have

$$\begin{eqnarray}&&\begin{array}{l}{Q}_{0}=\left(3.805\times {10}^{51}\,{{\rm{s}}}^{-1}\right)\left(\displaystyle \frac{{n}_{e}{n}_{+}V}{5\times {10}^{8}\,{\mathrm{cm}}^{-6}\,{\mathrm{pc}}^{3}}\right)\\ \quad \ \times \ {\left(\displaystyle \frac{{T}_{e}}{{10}^{4}{\rm{K}}}\right)}^{-0.833\mbox{--}0.034\mathrm{ln}({T}_{e}/{10}^{4}{\rm{K}})}.\end{array}\end{eqnarray} \tag{ A14 }$$