Star Formation Efficiencies at Giant Molecular Cloud Scales in the Molecular Disk of the Elliptical Galaxy NGC 5128 (Centaurus A)

We observed the CO (1−0) line toward a pointed mosaic covering most of the dust lane of Cen A (see Figure 1). The observations include 12 m, 7 m, and Total Power (TP) arrays, and thus the maps capture information from large to small spatial scales. The mosaic's size was defined as 5' × 14, with a position angle of P.A. = 120°. This was done with 46 pointings in the 12 m array and 19 pointings in the 7 m array. Nyquist sampling was used for the separation of the different pointings. The half-power beamwidth (HPBW) is 506 and 868 at 115 GHz for the 12 m and 7 m antennas, respectively. The TP array raster map was a rectangular field of 405'' × 189''.

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The ALMA CO data sets were obtained during 2014 and 2015 as part of program 2013.1.00803.S (P.I. D. Espada). The CO (1−0) data set consisted of 10 executions: two for the 12 m arrays (one extended and one compact configuration data set), two for the 7 m array, and six TP array data sets. The extended and compact 12 m array configurations had longest baseline lengths of 650.3 m and 348.5 m, respectively. The execution IDs, observation dates, time on source, and calibrators used are given in Table 1. The total time on source for extended 12 m, compact 12 m, 7 m, and TP arrays was 27.8, 13.9, 57.2, and 116.2 minutes, respectively.

The spectral setup was designed to observe the CO (1−0) line (ν_rest = 115.271 GHz). The spectral window in the upper sideband where the line was centered had a bandwidth of ∼2450 km s⁻¹ (937.5 MHz) and a velocity resolution of 1.2 km s⁻¹ (488.281 kHz). Additional spectral windows were placed at lower frequencies (center sky frequencies 112.914, 100.999, and 102.699 GHz) using bandwidths of 1875 MHz and resolutions of 31.25 MHz.

Data calibration and imaging were performed using the Common Astronomy Software Applications (CASA; McMullin et al. 2007). We used the packages containing the calibration files provided by the ALMA project. The data sets were delivered using different CASA versions, so we opted to perform the data reduction under version 5.1.1 using the scriptForPI.py script provided by ALMA.

The calibration scheme for all the data sets was standard. We carried out phase calibration (using water vapor radiometer data), system temperature calibration, as well as bandpass, phase, amplitude, and absolute flux calibration using the calibrators indicated in Table 1. We calibrated each of the interferometric data sets independently and concatenated them after performing line-free continuum subtraction. We checked that visibilities from different elements (extended and compact 12 m, 7 m, TP arrays) had correct weights after calibration. These weights were calculated inside CASA by taking into account the system temperature (T_sys) variance, integration time, and antenna diameters.

The single-dish data were calibrated in units of antenna temperature (${T}_{a}^{* }$) in kelvin with frequent observations of blank sky (i.e., OFF position). Further residual baseline corrections were done using line-free channels. A scaling factor of 41 Jy K⁻¹ between ${T}_{a}^{* }$ and flux density (kelvin to Jy beam^–1 factor) as obtained by the observatory using a bright quasar for band 3 was applied to the data to convert them to units of Jy beam⁻¹.

The CO (1−0) interferometric 12 and 7 m data were combined and imaged using the TCLEAN task with Briggs weighting and robust parameter equal to 0.5. In this step CASA 5.4 was used, as inaccuracies in the primary beam pattern have been recently found in earlier CASA versions. The spatial resolution of the final images is 139 × 105 (24.8 × 18.9 pc), and P.A. = 62°. The resulting data cubes were then combined in CASA with the single-dish data cubes using the feathering technique. We generated CO data cubes ranging from V_LSRK = 242 to 820 km s⁻¹ (kinematic local standard of rest velocity) and with 2.0 km s⁻¹ resolution. The combined CO (1−0) data cube is characterized by a typical noise level of 10 mJy beam⁻¹ for a channel width of 2 km s⁻¹.

Finally, we produced moment maps. We first smoothed the CO (1−0) data cubes to 20 × 20 pixels (the size of a pixel is 02 × 02) in the spatial coordinates (Gaussian kernel) and 3 pixels (boxcar) in velocity. Then, we obtained masks for regions above a 1σ threshold in the CO (1−0) smoothed data cube, which were then applied to the original data cube after visual inspection. Note that at the edges of the mosaic there are regions with lower sensitivity, but we exclude them from our analysis. Also, we flagged the absorption-line information (e.g., Espada et al. 2010) toward the continuum emission in our analysis. Figure 2 shows the good agreement between the ALMA single-dish CO (1−0) profile and that obtained from the masked combined CO (1−0) data cube.

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We compiled Spitzer 3.6 μm (emission mostly from the old stellar population), 8 μm (polycyclic aromatic hydrocarbons [PAHs], plus warm dust emission), and 24 μm (warm dust) data from the Spitzer Heritage Archive. The 3.6 μm and 8 μm data were obtained with the IRAC instrument (InfraRed Array Camera; Fazio et al. 2004), and the 24 μm data were obtained with the MIPS instrument (Multiband Imaging Photometer for Spitzer; Rieke et al. 2004) on board the Spitzer Space Telescope (Werner et al. 2004).

The IRAC 3.6 and 8 μm maps used in this paper were presented in Quillen et al. (2006a, 2006b). The angular resolutions (FWHM) are 22 and 24, respectively. The MIPS data are post-BCD (higher-level products) processed using the MIPS Data Analysis Tool (Gordon et al. 2005). The FWHM of the MIPS point-spread function (PSF) at 24 μm is 60. To remove the backgrounds in all the maps, we first excluded the sources in each image by masking them and then used a sigma-clipping technique to robustly estimate the background level that had to be subtracted.

We also removed the old stellar emission contribution as traced by the 3.6 μm data from the 8.0 μm image, assuming that fluxes at 3.6 μm trace stellar emission only. To do this, we followed the recipe L_ν (PAH) = L_ν (8 μm)–0.232 × L_ν (3.6 μm) (Helou et al. 2004; Boquien et al. 2010).

The FUV (1350–1750 Å) emission from the young stars of Cen A is investigated through the GALEX (Martin et al. 2005) image, delivered by Gil de Paz et al. (2007), where a description of the reduction procedure is given. We corrected for the Galactic extinction using the conversion factor A_FUV =7.9 E(B − V) (Gil de Paz et al. 2007) and the Schlafly & Finkbeiner (2011) recalibration of the Schlegel et al. (1998) infrared-based dust map, assuming an R_V = 3.1 Fitzpatrick (1999) reddening law. The image was provided with the background already subtracted, and no further treatment has been applied.

We are able to resolve for the first time the molecular component along the dust lane of Cen A into tens-of-parsec-scale molecular clouds thanks to ALMA's high resolution, sensitivity, and high dynamic range. Figures 3 and 4 show the CO (1−0) integrated intensity map, velocity field, and velocity dispersion map of the molecular disk of Cen A.

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Figure 5 presents the IRAC 8 μm over the CO (1−0) contours for comparison. The peculiar morphology seen in the 8 μm emission (also in the 24 μm emission) is usually referred to as the "parallelogram structure" (∼3 kpc in size), which can be partially reproduced by a model of a warped and thin disk seen in projection (Quillen et al. 2006b). This model can match partly the distribution of CO emission, but there are some differences, such as the lack of emission on the NE and SW sides of that parallelogram, as well as the curvature of CO (1−0) emission in the form of spiral arms, which are not as clear in the 8 μm map (Espada et al. 2009, 2012). It is not clear within the warped and thin disk model how the gas of the CND is maintained, since an assumption to reproduce the observed morphologies is the existence of a gap of cold gas and dust from r = 0.2 to 0.8 kpc (Quillen et al. 2006b; Espada et al. 2009).

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The CO emission is distributed mainly in three distinct regions: the large-scale disk, the spiral arm features, and the CND. The distribution and kinematics of the CND and spiral arm regions within the inner 1 kpc are in agreement with previous Submillimeter Array (SMA) CO (2–1) observations (Espada et al. 2009). The extension of the spiral arms beyond the 1 kpc scale, reported in Espada et al. (2012) using SMA CO (2–1) mosaic observations, is also in agreement, although their field of view was smaller, and the image was less sensitive and characterized by a poorer dynamic range than the one presented here. Besides the main CO components previously observed, the ALMA CO (1−0) map reveals a much shallower and extended molecular disk relative to the previous SMA data and is composed of multiple filamentary structures covering the dust lane.

The velocity field of Cen A (Figure 4, top panel) shows that the outer disk component follows the kinematics of the parallelogram disk (i.e., receding side on the NW, and approaching on the SE). However, the different filaments in the outermost regions are kinematically distinct to those in the parallelogram structure. Other peculiar components appear at ∼30'' (∼500 pc) to the N and S of the CND, which is probably gas entrained by the jet or associated with the shell-like structure reported by Quillen et al. (2006a). The majority of the clouds in the molecular disk outskirts have velocity dispersions of about ∼5 km s⁻¹ (Figure 4, bottom), except along the arms and parallelogram structure, where dispersions are ∼20 km s⁻¹, and are largest toward the CND. The large velocity dispersions found in the CND are compatible with the results presented in Espada et al. (2017) using the CO (3–2) transition, but as explained there, we note that the velocity dispersions do not exceed ∼40 km s⁻¹. Large velocity dispersions above 40 km s⁻¹, in the CND or elsewhere, are usually composed of multiple lines along the line of sight.

First, we obtained the CO luminosities, given by

$$\begin{eqnarray}&&{L}_{\mathrm{CO}(1-0)}=({c}^{2}/2{k}_{{\rm{B}}}){S}_{\mathrm{CO}(1-0)}\,{\nu }_{\mathrm{obs}}^{-2}\,{D}_{{\rm{L}}}^{2},\end{eqnarray} \tag{ 1 }$$

where c is the light speed, k_B is the Boltzmann constant, ${S}_{\mathrm{CO}}$ is the integrated CO line flux in Jy km s⁻¹, ${\nu }_{\mathrm{obs}}$ is the observed rest frequency in GHz, 115.271 GHz, and ${D}_{{\rm{L}}}$ is the luminosity distance to the source in Mpc (e.g., Solomon & Vanden Bout 2005). It yields

$$\begin{eqnarray}&&{L}_{\mathrm{CO}(1-0)}=2445\times {S}_{\mathrm{CO}(1-0)}\,{D}_{{\rm{L}}}^{2}({\rm{K}}\,\mathrm{km}\,{{\rm{s}}}^{-1}\,{\mathrm{pc}}^{2}),\end{eqnarray} \tag{ 2 }$$

and for the case of Cen A we obtain

$$\begin{eqnarray}&&{L}_{\mathrm{CO}(1-0)}=3.47\times {10}^{8}\,{\rm{K}}\,\mathrm{km}\,{{\rm{s}}}^{-1}\,{\mathrm{pc}}^{2}.\end{eqnarray} \tag{ 3 }$$

The agreement of this luminosity from the combined data set with that of the ALMA CO (1−0) single-dish data assures that we recover all the flux, as shown also by the profiles in Figure 2. We also note that single-dish data are very important to achieve this. Only 47% of the total flux in the ALMA CO (1−0) single-dish data is recovered by the 7 m array data.

The luminosity estimates using single-dish maps from the literature based on CO (1−0) data (SEST) were found to be 1.73 × 10⁸ K km s⁻¹ pc² (corrected by the different distance convention, from 3 to 3.8 Mpc; Eckart et al. 1990; Rydbeck et al. 1993), lower by about a factor of two than ours. This is partly caused by the larger area covered by our map and to extended emission. A comparison of the integrated flux in our combined map and the pointings along the parallelogram structure in Israel et al. (2014), for a common region encompassing 116'' × 45'' along a P.A. of 125°, agrees to within 10% (around 4500 Jy km s⁻¹).

We used as conversion factors between the CO integrated intensity and H₂ column density, ${X}_{\mathrm{CO}}={N}_{{{\rm{H}}}_{2}}/{I}_{\mathrm{CO}}$, X_CO = 2 × 10²⁰ cm⁻² (K km s⁻¹)⁻¹ for the outer disk and X_CO = 5 × 10²⁰ cm⁻² (K km s⁻¹)⁻¹ for the CND. The factor for the external regions of the disk mass was obtained in Paper II using the virial method (i.e., the comparison of the CO luminosity and virial masses of identified GMCs in the Cen A molecular disk) using the same CO (1−0) data that are presented here. This factor matches the recommended factor for the disk of the Milky Way (e.g., Dame et al. 2001) and other nearby disk galaxies (Bolatto et al. 2013). As for the CND, the larger factor we assume is motivated by the value obtained by Israel et al. (2014) using global single-dish measurements of the CO spectral line energy distribution toward the CND and large velocity gradient (LVG) analysis, X_CO = 4 × 10²⁰ cm⁻² (K km s⁻¹)⁻¹. In Paper II, we also confirm a larger factor using the virial method, X_CO = (5 ± 2) × 10²⁰ cm⁻² (K km s⁻¹)⁻¹, in agreement within the uncertainties. In contrast to this result, it should be noted that in the circumnuclear regions of starburst galaxies and mergers, as well as in the Galactic center, lower factors are usually found down to a full order of magnitude (e.g., Bolatto et al. 2013). Larger values than the Milky Way disk X_CO factor are usually found in low-metallicity dwarf galaxies (e.g., Bolatto et al. 2013; Miura et al. 2018). The larger factor is probably caused by high-excitation conditions coupled with CO-dark H₂ gas, but it is not likely due to lower metallicities (Paper II).

Finally, we calculated the molecular gas masses as in Bolatto et al. (2013):

$$\begin{eqnarray}&&{M}_{\mathrm{mol}}[{M}_{\odot }]=4.3\times {X}_{2}\,{L}_{\mathrm{CO}(1-0)},\end{eqnarray} \tag{ 4 }$$

where the factor 1.36 for elements other than hydrogen (Cox 2000) is taken into account, and X₂ is the X_CO factor normalized to 2 × 10²⁰ cm⁻² (K km s⁻¹)⁻¹.

The total molecular gas mass in the observed field is M_mol ≃ 1.6 × 10⁹ M_⊙. The smaller CO luminosity found in the literature as discussed above is reflected in a lower reported molecular gas mass (i.e., M_mol = 0.7 × 10⁹ M_⊙, corrected by different distances and X_CO factors, and including the 1.36 factor to account for He), but they are consistent when considering the smaller area covered (i.e., the parallelogram structure). Our CO (1−0) map is sensitive to more extended gas that is beyond the parallelogram structure. In Table 2 we show the derived molecular gas masses of different regions along the disk (i.e., CND, spiral arms, outer disk, etc.).

Table 2.  Molecular Gas Masses and SFEs in Different Regions of the Disk of Centaurus A

Regiona	Mmol (108 M⊙)b	Mean SFE (Gyr−1)	Median SFE (Gyr−1)	stddev SFE (Gyr−1)
CND	1.65	0.32	0.26	0.24
Arms	2.72	0.55	0.49	0.21
Parallelogram	5.99	1.02	0.88	0.52
Outer disk	5.65	1.29	1.05	0.93

Notes.

^aRegions as indicated in Figure 7. Note that in the CND the inner r < 86 pc is not included owing to contamination in the maps by the AGN. ^bMolecular gas mass assuming ${X}_{\mathrm{CO}}={N}_{{{\rm{H}}}_{2}}/{I}_{\mathrm{CO}}$ = 2 × 10²⁰ cm⁻² (K km s⁻¹)⁻¹. For the CND we use X_CO = 5 × 10²⁰ cm⁻² (K km s⁻¹)⁻¹ (Paper II; see also Israel et al. 2014).

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One of the goals of the present work is to disentangle how the SF activities may vary in the different environments of this galaxy, from the CND close to the AGN to the outermost parts of the molecular disk. We need to reach a high spatial resolution to investigate the KS law locally in Cen A. Approaching the excellent spatial resolution (∼18 pc) of the ALMA maps for the SFR is not possible, but the use of the 8 μm instead of 24 μm emission to trace the SFR mitigates this issue by a factor of 2 (Maragkoudakis et al. 2017), and so we chose the 8 μm band for our analysis because of its higher angular resolution (24, or 43 pc) compared to that of the 24 μm data (60, or 108 pc). Moreover, Elson et al. (2019) showed that the 8 μm PAH emission is a robust SFR tracer down to physical spatial resolutions of 49 pc.

PAH emission represents generally more than 70% of the 8 μm emission (Calzetti 2011) and dominates by more than a factor of 100 above the warm dust emission and about a factor of 5 above the stellar continuum (Li & Draine 2001; Dale et al. 2005; Young et al. 2014a). As a consequence, PAH emission (and in particular from the Spitzer 8 μm band) has been used in the literature to estimate SFR in galaxies (Calzetti et al. 2005; Wu et al. 2005; Pérez-González et al. 2006; Zhu et al. 2008; Pancoast et al. 2010; Calzetti 2011; Kennicutt & Evans 2012; Young et al. 2014a; Cluver et al. 2017; Kokusho et al. 2017; Maragkoudakis et al. 2017; Hall et al. 2018; Elson et al. 2019; Mahajan et al. 2019). Nevertheless, there may be variations from galaxy to galaxy, due to metallicity or different beam filling factors (Calzetti et al. 2005; Engelbracht et al. 2005; Pérez-González et al. 2006; Young et al. 2014a), so it is necessary to assess the validity of the 8 μm PAH emission as a reliable SF tracer in Cen A.

To do so, we compared the 8 μm PAH emission of Cen A to the more robust and widely used 24 μm emission (Calzetti et al. 2005; Alonso-Herrero et al. 2006; Pérez-González et al. 2006). In Figure 5 we show the 8 μm map (with CO (1−0) contours for comparison). We checked the spatial pixel-to-pixel agreement between 8 μm PAH and 24 μm fluxes (${F}_{8\mu {\rm{m}}\mathrm{PAH}}$ and ${F}_{24\mu {\rm{m}}}$, respectively, in units of MJy sr⁻¹) in Figure 6. The 8 μm PAH map was convolved to the FWHM of the 24 μm map (6''). We note that in this comparison only data points outside the inner 10'' are shown because of the difficulty in subtracting the bright central AGN component in the 24 μm map, which is worsened by the complex shape of the point-spread function. Also, we note that we clipped some data points at the low noise level end. We find a very tight (with a correlation coefficient of 0.98) linear correlation, which can be expressed as

$$\begin{eqnarray}&&\mathrm{log}{F}_{\nu }(8\,\mu {\rm{m}}\,\mathrm{PAH})=1.01\times \mathrm{log}{F}_{\nu }(24\,\mu {\rm{m}})+0.71.\end{eqnarray} \tag{ 5 }$$

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In Cen A, the linear coefficient against the 24 μm emission and the very tight correlation over more than two orders of magnitude ensures that the 8 μm PAH emission is a reliable SFR tracer that, hence, could be linked to the hot dust reprocessing photons from star-forming regions (as traced by the 24 μm emission). Crocker et al. (2011) found that, in a sample of 12 elliptical and lenticular galaxies, the 8 μm PAH and the 24 μm emissions yield similar SFR estimates. Therefore, the 8 μm PAH emission will be used to study the spatially resolved KS law in Section 5. In the next subsection we will estimate the global SFR in Cen A from a variety of standard SFR tracers.

A large number of luminosity-to-SFR calibrations can be found in the literature, and these conversions can vary by up to ∼50% and create artificial offsets between different works (Kennicutt & Evans 2012), so it is necessary to use homogenized calibrations. In this section, we use the homogenized SF calibrations compiled by Calzetti (2013) (from works by Calzetti et al. 2005, 2007; Kennicutt et al. 2007, 2009; Hao et al. 2011; Liu et al. 2011). All the calibrations make the implicit assumption that the stellar initial mass function (IMF) is constant across all environments and given by the double-power law expression with slope −1.3 in the range 0.1–0.5 M_⊙ and slope −2.3 in the range 0.5–120 M_⊙ (Kroupa 2001) unless noted otherwise.

First, we use a mixed indicator combining the FUV flux from massive stars, as well as its dust-obscured counterpart (Calzetti 2013):

$$\begin{eqnarray}&&\begin{array}{l}\qquad {\mathrm{SFR}}_{\mathrm{FUV}+24\mu {\rm{m}}}\ [{M}_{\odot }\,{\mathrm{yr}}^{-1}]\\ \,=\,4.6\cdot {10}^{-44}\times [L(\mathrm{FUV})+3.89\cdot L(24\,\mu {\rm{m}})],\end{array}\end{eqnarray} \tag{ 6 }$$

where L is the luminosity in units of erg s⁻¹ for a wavelength (or range) λ. We obtained L(FUV) = 4.32 × 10⁴² erg s⁻¹ and L(24 μm) = 5.19 × 10⁴² erg s⁻¹. This leads to ${\mathrm{SFR}}_{\mathrm{FUV}+24\mu {\rm{m}}}\,=1.13\pm 0.09\,{M}_{\odot }\,{\mathrm{yr}}^{-1}$ for Cen A. Second, we use the monochromatic calibration by Rieke et al. (2009) referenced to the same Kroupa (2001) IMF to be consistent with the other calibrations by Calzetti (2013),

$$\begin{eqnarray}&&{\mathrm{SFR}}_{24\mu {\rm{m}}}\ [{M}_{\odot }\,{\mathrm{yr}}^{-1}]=2.04\times {10}^{-43}L(24\,\mu {\rm{m}})\ (\mathrm{erg}\,{{\rm{s}}}^{-1}),\end{eqnarray} \tag{ 7 }$$

and it yields ${\mathrm{SFR}}_{24\mu {\rm{m}}}=1.06\pm 0.11\,{M}_{\odot }\,{\mathrm{yr}}^{-1}$. Another standard monochromatic calibration, based on the 70 μm emission (Equation (22) in Calzetti et al. 2010), reads

$$\begin{eqnarray}&&{\mathrm{SFR}}_{70\mu {\rm{m}}}\ [{M}_{\odot }\,{\mathrm{yr}}^{-1}]=5.9\times {10}^{-44}L(70\,\mu {\rm{m}})\ (\mathrm{erg}\,{{\rm{s}}}^{-1}).\end{eqnarray} \tag{ 8 }$$

We obtain L(70 μm) = 1.95 × 10⁴³ erg s⁻¹ (from the 70 μm flux in Bendo et al. 2012), and it yields ${\mathrm{SFR}}_{70\mu {\rm{m}}}=1.15\,\pm 0.23\,{M}_{\odot }\,{\mathrm{yr}}^{-1}$. Finally, we estimate the total IR (TIR) flux (in W m⁻²) in the range 3–1000 μm, F(TIR), by using the following relation (Dale & Helou 2002; Verley et al. 2010):

$$\begin{eqnarray}&&\begin{array}{l}F(\mathrm{TIR})={10}^{-14}\times [19.5\ {F}_{\nu }(24\,\mu {\rm{m}})+3.3\ {F}_{\nu }(70\,\mu {\rm{m}})\\ \,\,+\,2.6\ {F}_{\nu }(160\,\mu {\rm{m}})],\end{array}\end{eqnarray} \tag{ 9 }$$

where ${F}_{\nu }(24\,\mu {\rm{m}})$, ${F}_{\nu }(70\,\mu {\rm{m}})$, and ${F}_{\nu }(160\,\mu {\rm{m}})$ are the MIPS flux densities in Jy for Cen A as listed in Table 3 of Bendo et al. (2012). This TIR flux turns into a luminosity of $L(\mathrm{TIR})\,=1.29\pm 0.16\times {10}^{10}\,{L}_{\odot }$. We can then derive the SFR based on dust-processed stellar light from the TIR luminosity using the following calibration (Calzetti 2013), where the assumptions are a mass range of the stars in the IMF spanning 0.1–100 M_⊙ and the SF remaining constant during 100 Myr:

$$\begin{eqnarray}&&{\mathrm{SFR}}_{\mathrm{TIR}}\ [{M}_{\odot }\,{\mathrm{yr}}^{-1}]=2.8\times {10}^{-44}\,L(\mathrm{TIR})\ (\mathrm{erg}\,{{\rm{s}}}^{-1}),\end{eqnarray} \tag{ 10 }$$

which leads to SFR_TIR = 1.38 ± 0.17 M_⊙ yr⁻¹ for Cen A. All these estimates of global SFR lead to a consistent result for an SFR in Cen A of about 1 M_⊙ yr⁻¹.

The SFRs we have obtained are also in agreement with previous measurements in the literature obtained from IRAS data. The total infrared luminosity was estimated as $L(\mathrm{TIR})$ ≈ 2 $L(\mathrm{FIR})=1.12\times {10}^{10}\,{L}_{\odot }$ (Marston & Dickens 1988; Voss & Gilfanov 2006) (corrected to our distance convention). Without the contribution of the AGN as estimated by Voss & Gilfanov (2006), then $L(\mathrm{TIR})\approx 9.4\,\times \,{10}^{9}\,{L}_{\odot }$. The total SFR is estimated to be $\approx 1\,{M}_{\odot }\,{\mathrm{yr}}^{-1}$ (Colbert et al. 2004; Voss & Gilfanov 2006). Colbert et al. (2004) estimated SFR = 1 M_⊙ yr⁻¹ (converted to our distance convention) from a combination of $L(\mathrm{FIR})$ and UV luminosity, L(UV), and a conversion factor to convert from $L(\mathrm{FIR}+\mathrm{UV})$ to SFR of $5.7\times {10}^{-44}$. However, the contribution of the AGN was not subtracted. Voss & Gilfanov (2006) obtained SFR = 1.6 M_⊙ yr⁻¹ (also converted to our distance convention) from $L(\mathrm{TIR})$ and after subtraction of the AGN contribution. In this estimate, Voss & Gilfanov (2006) used the SFR calibration of Kennicutt (1998), SFR_TIR (${M}_{\odot }\,{\mathrm{yr}}^{-1})\,=4.5\times {10}^{-44}L(\mathrm{TIR})(\mathrm{erg}\,{{\rm{s}}}^{-1}$). To be able to compare this with our analysis, we must convert it to a Kroupa IMF, and therefore we divided it by a factor of 1.59 (Bigiel et al. 2008). Finally, we obtain SFR ≃ 1 M_⊙ yr⁻¹, which is consistent with all the previously mentioned estimates.

As the 8 μm PAH emission is so closely tied to the 24 μm emission (see Figure 6), we can provide a calibration of the global SFR in Cen A from the 8 μm PAH emission to reproduce the global SFR obtained from the 24 μm emission (i.e., 1.06 M_⊙ yr⁻¹):

$$\begin{eqnarray}&&\begin{array}{l}{\mathrm{SFR}}_{8\mu {\rm{m}}\mathrm{PAH}}\ [{M}_{\odot }\,{\mathrm{yr}}^{-1}]\\ \qquad =\,6.14\times {10}^{-44}L(8\,\mu {\rm{m}}\,\mathrm{PAH})\ (\mathrm{erg}\,{{\rm{s}}}^{-1}).\end{array}\end{eqnarray} \tag{ 11 }$$

We present in Figure 7 the spatially resolved KS SF law plot (i.e., Σ_SFR vs. Σ_mol) along the dust lane of Centaurus A. The scatter is large, but by visual inspection it is clear that there are two main clusters of data points, one cluster at high Σ_mol and Σ_SFR, and another centered at Σ_mol and Σ_SFR about 1 dex smaller.

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We used a machine learning technique to separate the data set in the $\mathrm{log}{{\rm{\Sigma }}}_{\mathrm{SFR}}-\mathrm{log}{{\rm{\Sigma }}}_{\mathrm{mol}}$ parameter space (see Figure 7) into clusters. We first remove the 19 points corresponding to the AGN to focus only on separating the low and high components in this parameter space. We use a Gaussian mixture model (GMM), based on an iterative expectation–maximization algorithm, to extract a mixture of multidimensional Gaussian probability distributions that best model our data set (VanderPlas 2016). To separate our data set into two components, random initializations are used, and we tested several covariance options (diagonal, spherical, tied, and full covariance matrices). A Bayesian Information Criterion (BIC) selects the best model, which is the one obtained with the full covariance matrix, allowing each cluster to be modeled by an ellipse with arbitrary orientation.

The results of the separation are shown in Figure 7. This machine learning segregation is able to effectively separate the inner from the outer parts of the galaxy. The highest Σ_SFR–Σ_mol cluster corresponds to the CND, arms, and parallelogram regions, while the lowest cluster defines the outer disk region. We also use a color/symbol scheme in Figure 7 (bottom) to separate the data points from the different regions (i.e., CND, arms, parallelogram, and outer disk). Although the separation has a physical meaning regarding regions with high or low SFR/gas surface densities in the CND, arms, and outer disk, note that in the parallelogram structure regions the surface densities may artificially be higher because of the projection effect.

The slope of the orthogonal distance regression fit to the data points characterizing the SF law is N = 0.63 (intercept is −2.4). We note, however, that the scatter is large, there is a trend of decreasing slope for regions at inner radii (for the CND N = 0.28 and intercept −1.6), and also there might be further uncertainties in the fits due to projection effects, so this result should be taken with caution.

In this section we study the SFE, defined as the SFR per unit of molecular gas mass, Σ_SFR/Σ_mol. The depletion time is then defined as the inverse of SFE, τ = 1/SFE. We find a total molecular gas mass of 1.6 × 10⁹ M_⊙ and a global SFR of ∼1 M_⊙ using different recipes, so the SFE is 0.6 Gyr⁻¹ (or depletion time τ = 1.5 Gyr), similar to that in spiral galaxies. For reference in the resolved KS plot in Figure 7 (top panel), we indicate constant SFE lines, at SFEs 0.1, 1, 10, and 100 Gyr⁻¹ (or equivalently depletion times τ = 0.01, 0.1, 1, and 10 Gyr).

Figure 8 displays the map of the SFE. The SFEs across the disk vary by at least two orders of magnitude from pixel to pixel. We confirm with this plot that the SFEs along the arms and the CND are lower by a factor of 2–4 compared to the remaining area and that higher SFEs are found preferentially in the outskirts of the disk (blue data points in Figure 7, bottom panel).

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In Columns (3)–(5) of Table 2 we show the mean, median, and standard deviation of the SFEs obtained from the pixels within the different regions. While the mean SFE in the CND and arms is 0.3–0.5 Gyr⁻¹, in the outer disk it is 1.3 Gyr⁻¹. Also the standard deviation of the SFEs increases with radius, which likely implies variable physical conditions of molecular gas and SF activities in the outskirts. This trend is contrary to what is observed in the central regions of other AGNs, as will be discussed in Section 6.2.

Cen A is a rare case of a giant elliptical galaxy with a large amount of molecular gas recently accreted and with ongoing SF. In agreement with Young et al. (2009) for other early-type galaxies, the bulk of its mid-IR emission can be traced to SF. However, the SF law seems remarkably different from the trends observed for other early-type galaxies. Thanks to its recent merger with a gas-rich galaxy, the recent SF history varied substantially compared to other objects of its class. The global SFE (∼0.6 Gyr⁻¹) is compatible within the uncertainties to the average of local star-forming galaxies, SFE ∼ 1 Gyr⁻¹ (e.g., Tacconi et al. 2018), found across different galaxy environments (Lisenfeld et al. 2011; Martinez-Badenes et al. 2012).

In the literature it is usually argued that SFEs in early-type galaxies are either equal to or lower than the standard relation for late-type galaxies. The derived SFRs and molecular gas surface densities (measured globally) place E/S0 galaxies overlapping the range spanned by disk galaxies, although with a larger scatter (Shapiro et al. 2010; Wei et al. 2010). Crocker et al. (2011) find that although 12 E/S0s in the SAURON project lie closely around a constant SFE, as in spiral galaxies, there are some hints of lower SFEs. In fact, it was reported that molecular gas-rich early-type galaxies in the ATLAS3D project have a median SFE of ∼0.4–0.09 Gyr⁻¹ (e.g., Davis et al. 2014, 2015), considerably lower than the typical SFEs of ∼1 Gyr⁻¹ in star-forming galaxies. It has been suggested that the SF in early types is suppressed owing to the stability of the molecular disks against gravitational fragmentation using the Toomre Q test (Kohno et al. 2002; Davis et al. 2014; Boizelle et al. 2017). This result is supported by hydrodynamic models of Martig et al. (2013), where stars form two to five times less efficiently in early-type galaxies than in spirals. Davis et al. (2015) argue that dynamical effects induced by the minor merger might be responsible for the low SFE values. Recent studies using ALMA observations of some of the most extreme cases in terms of low SFE bulge-dominated galaxies with dust lanes confirmed that SFE ranges from 0.002 to 0.04 Gyr⁻¹ (van de Voort et al. 2018). However, Kokusho et al. (2017) find that most of the ATLAS3D local early-type galaxies follow the same SF law as local later type star-forming galaxies. Contrary to recent results suggesting slightly lower SFEs, they claim that this is directly related to the somewhat lower SFRs derived in those works. They note, however, that there is evidence that early-type galaxies whose cold gas has an external origin (kinematically misaligned gas with respect to stars) have more varied SFEs, which may suggest a more bursty and variable recent SF history. This might be the case for Cen A.

A parameter that is crucial to understanding the different claims regarding the SFE in early-type galaxies is the stellar mass. In Cen A the stellar mass is M_⋆ = 10¹¹ M_⊙ (Kormendy & Ho 2013). This value is within the range of the early-type galaxy sample with recent minor mergers studied by Davis et al. (2015). About 80% of their low-mass E/S0s (M_⋆ < 4 × 10¹⁰ M_⊙) present high SFEs (above the KS law) and are usually star-forming (i.e., blue sequence) objects (Wei et al. 2010). The fraction of E/S0s that are star-forming increases with decreasing stellar mass (Kannappan et al. 2009). Indeed, Cen A's stellar mass is close to the "shutdown mass" (i.e., the mass at which blue sequence E/S0s first emerge, and above which nearly all galaxies are old and red) for blue sequence early-type objects, M_s ∼ (1–2) × 10¹¹ M_⊙, which only a minority (2%) of E/S0s exceed (Kannappan et al. 2009). Hence, in summary Cen A is at the massive end where it is rare to find a star-forming elliptical. It is therefore likely that it is mostly by accretion events that such an object can exist.

Next, we put the case of Cen A in the context of previous works where the SFEs were found to be either equal to or lower than the standard values of spiral galaxies. Because Cen A recently accreted material, it would match the selection criteria of galaxies in the samples of Davis et al. (2015) and van de Voort et al. (2018). Indeed, they focus on a sample of bulge-dominated galaxies with large dust lanes and with signs of a recent minor merger. In these works it is argued that the SFE is suppressed as a result of dynamical effects. How can we reconcile the apparent contradiction of Cen A presenting a nearly standard SFE similar to star-forming galaxies? This can be due to different impact parameters of the merger, merged galaxy properties (such as mass and gas fraction), and the dynamical relaxation time. At any rate, the case of Cen A shows that the SFE is not always suppressed in early-type galaxies after gas-rich minor mergers. ALMA observations of the elliptical and radio jet galaxy NGC 3557 have recently shown that its molecular disk within the inner hundreds of parsecs presents a global SFE that may be compatible with the standard relation in spiral star-forming galaxies too (Vila-Vilaro et al. 2019).

In about 1 Gyr, objects like Cen A may look more similar to other objects of its class (i.e., giant elliptical galaxies without much molecular gas and with low SFEs). A fast evolutionary sequence may be playing a role, which would explain why only a minority of massive early-type galaxies are in the blue sequence even though they are in high galaxy density environments where accretion events are relatively frequent. The molecular gas will be exhausted preferentially in the external regions. The SFE is higher in the outskirts by a factor of four than in the central regions of Cen A, so it will be consumed faster. In addition, the gas will be either driven to the central regions or destroyed because of a lack of self-shielding to massive SF combined with the hot environmental conditions within the elliptical galaxy (i.e., a hot ISM thermally radiating at X-rays; Henkel & Wiklind 1997). Dense gas that is stable against collapse owing to inflowing motions, shear, and/or excess of shocks/turbulence will remain, preferentially in compact components or filamentary structures within the stellar component, which is the result observed in other massive elliptical galaxies.

For the amount of molecular gas in the CND, the SFR is comparatively smaller than that found in other regions. The mean SFE in the CND is ∼0.3 Gyr⁻¹ at 40 pc scales, four times smaller than in the outer regions (see Section 5.2). Using photon-dominated region (PDR) modeling of the neutral and ionized gas from the far-infrared lines, Israel et al. (2017) claimed that because the radiation field in the CND is weaker by an order of magnitude than in the extended disk, no significant SF in the CND is required. The radiation field strength could be explained just by the radiation coming from the stellar component of the extended disk. We do find that the SFR in the CND is small but not negligible. We note that this SFR is a lower limit because we have excluded pixels close to the AGN that may be contaminated by it, but the SFE we quote is representative for the outer (r = 100–200 pc) regions of the CND.

A source of uncertainty in the deduced low SFE of the CND is the X_CO factor used to calculate the molecular gas mass. For the CND we used a conversion factor X_CO = 5 × 10²⁰ cm⁻² (K km s⁻¹)⁻¹, while for the rest of the disk we applied 2 × 10²⁰ cm⁻² (K km s⁻¹)⁻¹. However, a different X_CO in the central regions is supported by an analysis in Israel et al. (2014) using LVG modeling on CO spectral line energy distribution and in Paper II using the virial method. These are independent methods, so this result is likely robust. Moreover, in Paper II we found a gradual increase of X_CO toward inner radii. We note that X_CO is driven by metallicity in the low-metallicity regime, but in the case of Cen A metallicity across the molecular disk (including the CND) is ∼0.75 Z_⊙ and relatively constant.

The SFEs are also relatively constant at 1–2 Gyr⁻¹ along galactocentric radii on kiloparsec scales (Leroy et al. 2008, 2013; Muraoka et al. 2019). However, Leroy et al. (2013) systematically found higher SFEs for a given X_CO in the inner kiloparsec of a sample of 30 nearby disk galaxies that show nuclear molecular gas concentrations, including AGN and nuclear starbursts, but not galaxy-wide starbursts. Our finding of lower SFEs in the CND compared to the outer regions of Cen A contradicts this. Since X_CO is typically smaller by a factor of 2 below the galaxy mean on average close to their nuclei (Sandstrom et al. 2013), and in some cases 5–10 times below the standard galaxy disk value, the resulting SFEs are higher than when assuming a constant X_CO.

A direct comparison of SFEs between different objects requires the use of similar spatial scales. Using ALMA CO (2–1) maps of 14 disk galaxies, Utomo et al. (2018) found that on 120 pc scales SFEs range from 0.43 to 1.65 Gyr⁻¹, similar to the range we see for the different regions in Cen A, from SFE = 0.3–0.5 Gyr⁻¹ in the CND/arms region to SFE = 1.3 Gyr⁻¹ in the outer disk. Casasola et al. (2015) studied the central regions (from 20 to 200 pc) of a sample of four galaxies possessing LLAGNs, i.e., Seyfert and LINER nuclei, and found that they were characterized by higher SFEs than the standard value for disk galaxies (adopting a common X_CO factor of $2.2\times {10}^{20}$ cm⁻² (K km s⁻¹)⁻¹), with a behavior that is in between galaxy-wide starburst systems and more quiescent galaxies. The SFR and gas surface densities of the central regions of these objects are comparable to those in the CND of Cen A, in the ranges of ∼0.01–1 M_⊙ yr⁻¹ kpc⁻² and ∼10²–10³ M_⊙ pc⁻², respectively. Possible causes for the lower SFEs in the CND of Cen A are stronger shear motions and shocks. Also, the more continuous period of AGN feedback in the case of Cen A than in other AGNs may be responsible for the observed differences. However, we also note that other late-type AGN galaxies with low SFEs toward the circumnuclear regions also exist, such as in the inner spiral regions of NGC 1068 (Tsai et al. 2012). In our Galactic center the current SFR per unit mass of dense gas was also found to be an order of magnitude smaller than in the disk (Longmore et al. 2013).