Central molecular zones in galaxies: 12CO-to-13CO ratios, carbon budget, and X factors

F. P. Israel

doi:10.1051/0004-6361/201834198

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Issue		A&A Volume 635, March 2020


Article Number		A131
Number of page(s)		33
Section		Extragalactic astronomy
DOI		https://doi.org/10.1051/0004-6361/201834198
Published online		20 March 2020

A&A 635, A131 (2020)

Central molecular zones in galaxies: ¹²CO-to-¹³CO ratios, carbon budget, and X factors^⋆

F. P. Israel

Sterrewacht Leiden, PO Box 9513, 2300 Leiden, The Netherlands
e-mail: israel@strw.leidenuniv.nl

Received: 5 September 2018
Accepted: 28 October 2019

Abstract

We present ground-based measurements of 126 nearby galaxy centers in ¹²CO and 92 in ¹³CO in various low-J transitions. More than 60 galaxies were measured in at least four lines. The average relative intensities of the first four J ¹²CO transitions are 1.00:0.92:0.70:0.57. In the first three J transitions, the average ¹²CO-to-¹³CO intensity ratios are 13.0, 11.6, and 12.8, with individual values in any transition ranging from 5 to 25. The sizes of central CO concentrations are well defined in maps, but poorly determined by multi-aperture photometry. On average, the J = 1−0 ¹²CO fluxes increase linearly with the size of the observing beam. CO emission covers only a quarter of the HI galaxy disks. Using radiative transfer models (RADEX), we derived model gas parameters. The assumed carbon elemental abundances and carbon gas depletion onto dust are the main causes of uncertainty. The new CO data and published [CI] and [CII] data imply that CO, C°, and C⁺ each represent about one-third of the gas-phase carbon in the molecular interstellar medium. The mean beam-averaged molecular hydrogen column density is N(H₂) = (1.5 ± 0.2)×10²¹ cm⁻². Galaxy center CO-to-H₂ conversion factors are typically ten times lower than the “standard” Milky Way X_° disk value, with a mean X(CO) = (1.9 ± 0.2)×10¹⁹ cm⁻²/K km s⁻¹ and a dispersion 1.7. The corresponding [CI]-H₂ factor is five times higher than X(CO), with X[CI] = (9 ± 2)×10¹⁹ cm⁻²/K km s⁻¹. No unique conversion factor can be determined for [CII]. The low molecular gas content of galaxy centers relative to their CO intensities is explained in roughly equal parts by high central gas-phase carbon abundances, elevated gas temperatures, and large gas velocity dispersions relative to the corresponding values in galaxy disks.

Key words: galaxies: general / galaxies: abundances / galaxies: ISM / galaxies: statistics / galaxies: starburst / ISM: molecules

^⋆

Table A.1. is only available at the CDS via anonymous ftp to cdsarc.u-strasbg.fr (130.79.128.5) or via http://cdsarc.u-strasbg.fr/viz-bin/cat/J/A+A/635/A131

© ESO 2020

1. Introduction

The aim of this paper is to determine the carbon budget and the amount of molecular hydrogen in the centers of nearby galaxies as accurately as possible, based on extensive new observations and current chemical and radiative transfer models. The bright inner disks of late-type galaxies contain massive concentrations of circumnuclear molecular hydrogen gas. These reservoirs feed central black holes, outflows, and bursts of star formation. Before their crucial role in inner galaxy evolution can be understood and evaluated, the physical characteristics of the gas must be determined. Cool and quiescent molecular hydrogen (H₂) gas is difficult to detect, and studies of the molecular interstellar medium (ISM) in galaxies rely on the observation of tracers such as continuum emission from thermal dust or line emission from the CO molecule. CO is one of the most common molecules in the ISM after H₂, even though its relative abundance is only about 10⁻⁵. It has become the instrument of choice in the investigation of the molecular ISM because it is comparatively easy to detect and traces molecular gas already at low densities and temperatures.

Following the first detections in the mid-1970s, numerous galaxies have been observed in various transitions of CO and its isotopologue ¹³CO. Substantial surveys have been conducted in the J = 1−0 transition of ¹²CO (e.g., Stark et al. 1987; Braine et al. 1993a; Sage 1993; Young et al. 1995; Elfhag et al. 1996; Nishiyama & Nakai 2001; Sauty et al. 2003; Albrecht et al. 2007; Kuno et al. 2007). These surveys sample the nucleus and sometimes also a limited number of disk positions. Extensive surveys in higher ¹²CO transitions are fewer in number and usually only sample the nucleus (J = 2−1: Braine et al. 1993a; Albrecht et al. 2007; J = 3−2: Mauersberger et al. 1999; Yao et al. 2001; Mao et al. 2010). The survey by Dumke et al. (2001) and especially the James Clerk Maxwell Telescope (JCMT) legacy survey of nearby galaxies (NGLS: Wilson et al. 2012; Mok et al. 2016) are exceptional because they provide maps of almost 100 galaxies in the J = 3−2 transition, many of them in the Virgo cluster. Specific surveys of Virgo cluster galaxies have also been published by Stark et al. (1986, J = 1−0), Kenney & Young (1988, J = 1−0), and Hafok & Stutzki (2003, J = 2−1 and J = 3−2).

The ¹²CO lines in the survey are optically thick and cannot be used to measure molecular gas column densities or masses. Even the analysis of a whole ladder of multiple ¹²CO transitions either fails to break the degeneracy between H₂ density, kinetic temperature, and CO column density and leaves the mass an undetermined quantity, or samples only a small fraction of the total gas content in the higher J transitions. Consequently, most molecular gas masses quoted in the literature are critically dependent on an assumed value for the relation between velocity-integrated CO line intensity and H₂ column density, X_CO = N(H₂)/I(CO). Unfortunately, this so-called X-factor does not follow from basic physical considerations. Instead, its empirically estimated value is rather sensitive to assumptions made in the process, and it varies depending on author and method. The most reliable method uses gamma-ray observations to trace hydrogen nuclei, and a useful overview of X values thus obtained can be found in Table E.1 of Remy et al. (2017). The empirically determined X values implicitly include both H₂ gas mixed with CO and H₂ gas that contains no or very little CO (“CO-dark gas”). There is some confusion in the literature as different X values have been referred to as the “standard” CO-to-H₂ conversion factor. In this paper, we define X_°(CO) = 2 × 10²⁰ cm⁻²/K km s⁻¹ (corresponding to 4.3 M_⊙ pc⁻² when it also includes a helium contribution) as the standard factor to convert CO intensity into H₂ column density.

In one form or another, the “standard” factor is frequently applied to other galaxies, often without caveats of any sort. These are essential, however, as the effects of metallicity, irradiation, and excitation may cause X to vary by large factors in different environments such as are found in low-metallicity dwarf galaxies, galaxy centers, luminous star-forming galaxies, molecular outflows, and high-redshift galaxies, as was already explained in the pioneering papers by Maloney & Black (1988) and Maloney (1990). Even the X-factor of our own Galactic center region has been known to be very different since Blitz et al. (1985) discussed the remarkably low ratio of gamma-ray to CO intensities in the central few hundred parsecs and suggested that it is caused by H₂/¹²CO abundances that are an order of magnitude below those in the rest of the disk. These low X values were since confirmed, for instance, by Sodroski et al. (1995; X = 0.22 X_°), Dahmen et al. (1998; X = (0.06−0.33) X_°), and Oka et al. (1998; X = 0.12 X_°).

Conversion factors much lower than the standard Milky Way disk factor have also been ascribed to the central regions of other galaxies. Stacey et al. (1991) used a comparison of [CII] and ¹²CO intensities to suggest a factor of three or more below X_°. Solomon et al. (1997) and Downes & Solomon (1998) argued that in ultra-luminous galaxies the X-factor had to be well below standard for the gas mass to avoid exceeding the dynamical mass, and adopted a value five times lower based on dust mass considerations.

Dust emission is relatively easy to measure but not so easy to interpret. Because the nature of the emitting dust grains is poorly known, uncertainties in interstellar dust composition, dielectric properties, size distributions, and dust-to-gas ratios cannot be avoided, and each of these properties may also change with environment. It is not entirely obvious how the measured intensity of infrared continuum emission should be translated into dust column density, let alone gas column density. These uncertainties allowed authors to err on the side of caution and estimate only moderately low values X ∼ 0.5 X_° (M 82, Smith et al. 1991; M 51, Nakai & Kuno 1995; NGC 7469, Davies et al. 2004), although substantially lower values X ∼ 0.1−0.2 X_° (NGC 3079, Braine et al. 1997; NGC 7469, Papadopoulos & Allen 2000; NGC 4258, Ogle et al. 2014) have also been suggested. Such rather low values were also inferred from the local thermal equilibrium (LTE) analysis of optically thin but weak C¹⁸O isotopologue emission (NGC 1068, Papadopoulos & Seaquist 1999; NGC 6000, Martín et al. 2010).

The potentially problematical use of dust continuum emission for determining the properties of molecular gas is thus not preferred when actual molecular line measurements are available. Both observations and models have increasingly allowed the detailed analysis of CO line intensities using the more sophisticated non-LTE large velocity gradient (LVG) radiative transfer codes. An essential step toward reliable molecular gas mass determinations consists of reducing or breaking the crippling temperature-density degeneracies that plague the analysis of ¹²CO measurements. This is accomplished by including measurements of CO isotopologues with lower optical depth. However, even the strongest of these (¹³CO) is a relatively weak emitter. Consequently, only the brightest galaxies have been analyzed in this way (M 82, Weisz et al. 2001; NGC 4945 and the Circinus galaxy, Curran et al. 2001; Hitschfeld et al. 2008, Zhang et al. 2014; VV 114, Sliwa et al. 2013). These all yield values of X = 0.1−0.2 X_°.

Extensive ¹³CO surveys of external galaxies have so far been lacking in any transition. The survey presented in this paper is therefore a significant addition to the existing CO database on nearby galaxies. The newly determined multi-transition ¹²CO-to-¹³CO isotopologue ratios allow us to determine more accurately the CO gas column densities and their relation to the much more abundant H₂ gas, including the values of X in a large number of galaxy central regions. The analysis of the ¹²CO and ¹³CO spectral lines is of particular importance in the interpretation of the Herschel Space Observatory (2009–2013) observations of galaxies in the two submillimeter [CI] lines and the far-infrared [CII] line (Israel 2015; Kamenetzky et al. 2016; Fernández-Ontiveros et al. 2016; Lu et al. 2017; Croxall et al. 2017; Díaz-Santos et al. 2017; Herrera-Camus et al. 2018). With it, we will place significant constraints on the relation between molecular and atomic carbon and determine the carbon budget in the observed galaxy centers.

2. Observations and data handling

2.1. SEST 15 m observations

With the 15 m Swedish-ESO Submillimetre Telescope (SEST) at La Silla (Chile)¹ we conducted seven observing runs between May 1988 and January 1992, and another three runs between 1999 and 2003. Observations in the first period were mostly in the J = 1−0 ¹²CO transition, with some J = 1−0 ¹³CO observations of the brightest galaxies. In the second period we obtained additional J = 2−1 ¹²CO and J = 1−0 ¹³CO observations simultaneously. The SEST full width at half-maximum (FWHM) beam sizes were 45″ at 115 GHz (J = 1−0 ¹²CO) and 23″ at 230 GHz (J = 2−1 ¹²CO). All observations were made in a double beam-switching mode with a throw of 12′. Using the CLASS package, we binned the spectra to resolutions of 10–30 km s⁻¹ after which third-order baselines were subtracted if the spectral coverage allowed it; otherwise, only a linear baseline was fit. A sample of the SEST observations is shown in Fig. 1. Line parameters were determined by fitting with one or two Gaussians as required by the shape of the profile. In the ¹³CO profiles, we set the fitting range to be the same as determined in the ¹²CO profiles with higher signal-to-noise ratios (S/N). Intensities were reduced to main-beam brightness temperatures T_mb = $T_{A}^{*} / η_{mb}$ $T_{A}^{*}/\eta_{\mathrm{mb}}$ , using main-beam efficiencies at 115 GHz η_mb(115) = 0.66 until October 1988, 0.74 until June 1990, 0.75 until October 1990, and 0.70 thereafter (L.E.B. Johansson, private communication), and η_mb(230) = 0.50 for the whole period. The resulting velocity-integrated line intensities are listed in Tables 2 and 3.

Fig. 1.

Sample of SEST J = 1−0 CO observations of galaxy centers, showing ¹²CO (histogram) and superposed ¹³CO (continuous lines) profiles; the intensities of the latter have been multiplied by a factor 5. Intensities are in $T_{A}^{*}$ $T_{\mathrm{A}}^{*}$ (K). Velocities are V(LSR) in km s⁻¹. Galaxies are identified at the top.

Table 1.

Galaxy sample.

Table 2.

Galaxy center J = 1−0 line intensities.

Table 3.

Galaxy center J = 2−1 line intensities.

2.2. IRAM 30 m observations

Using the IRAM 30 m telescope on Pico de Veleta (Granada, Spain)², we conducted four observing runs between December 2004 and July 2006, simultaneously observing the J = 1−0 and J = 2−1 transitions of both ¹²CO and ¹³CO with the facility 3 mm and 1.3 mm SIS receivers coupled to 4 MHz backends. All observations were made in beam-switching mode with a throw of 4′. The FWHM beam sizes were 22″ at 110/115 GHz and 11″ at 220/230 GHz. The diameter of the IRAM telescope is twice that of the JCMT (and the SEST) so that J = 1−0 (IRAM) and J = 2−1 (JCMT) observations are beam-matched, as are the J = 2−1 (IRAM) and J = 4−3 (JCMT) observations. A sample of the IRAM observations in the J = 2−1 transition is shown in Fig. 2. The profile analysis was similar to that described for the SEST. Intensities were converted into main-beam brightness temperatures using main-beam efficiencies η_mb of 0.79/0.80 at 110/115 GHz and 0.59/0.57 at 220/230 GHz. The resulting velocity-integrated line intensities are listed in Tables 2 and 3.

Fig. 2.

Sample of IRAM J = 2−1 CO observations of galaxy centers, showing ¹²CO (histogram) and superposed ¹³CO (continuous lines) profiles; the intensities of the latter have been multiplied by a factor 5. Intensities are in $T_{A}^{*}$ $T_{\mathrm{A}}^{*}$ (K). Velocities are V(LSR) in km s⁻¹. Galaxies are identified at the top. IRAM J = 1−0 profiles (not shown) are similar, with better S/N.

2.3. JCMT 15 m observations

The observations with the 15 m JCMT on Mauna Kea (Hawaii)³ were obtained at various periods between 1988 and 2005. When the JCMT changed from PI-scheduling to queue-scheduling in the late 1990s, most of the survey measurements were made in back-up service mode. In both the J = 2−1 and the J = 3−2 transitions, ¹²CO and ¹³CO observations were made closely together in time. The JCMT FWHM beam-sizes were 22″ at 220/230 GHz and 14″ at 330/345 GHz. All observations were made in a beam-switching mode with a throw of 3′. We have discarded almost all early observations, preferring to use those obtained after 1992 with more sensitive receivers and the more sophisticated Dutch Autocorrelator System (DAS) back-end. We included data extracted from the CADC/JCMT archives on galaxies relevant to our purpose that had been observed by other observers (e.g., Devereux et al. 1994; Papadopoulos & Allen 2000; Zhu et al. 2003; Petitpas & Wilson 2003).

We reduced the JCMT observations using the SPECX package, and subtracted baselines up to order three, depending on source line-width. We determined integrated intensities by summing channel intensities over the full range of emission. In the ¹³CO profiles, we set this range to be the same as determined in higher S/N ¹²CO profiles. Antenna temperatures were converted into main-beam brightness temperatures with efficiencies η_mb(230) = 0.70 and η_mb(345) = 0.63. The velocity-integrated line intensities are listed in Tables 3 and 4. Available J = 4−3 ¹²CO observations, many of which were discussed in earlier papers (Israel 2009b, and references therein) were re-reduced and the results are listed in Table 5.

Table 4.

Galaxy center J = 3−2 line intensities.

Table 5.

Galaxy center J = 4−3 line intensities.

For almost half of the sample, small maps of the J = 3−2 and J = 2−1 ¹²CO emission from the central region were obtained in addition to the central profiles. Maps and profiles of more than 16 galaxies have already been published (Israel 2009a,b, and references therein). A sample of the new JCMT J = 3−2 profiles is shown in Fig. 3. All JCMT J = 3−2 ¹²CO maps not included in our previous papers are shown in Fig. 4.

Fig. 3.

Sample of JCMT J = 3−2 CO observations of galaxy centers, showing ¹²CO (histogram) and superposed ¹³CO (continuous lines) profiles; the intensities of the latter have been multiplied by a factor 5. Intensities are in $T_{A}^{*}$ $T_{\mathrm{A}}^{*}$ (K). Velocities are V(LSR) in km s⁻¹. Galaxies are identified at the top. JCMT J = 2−1 profiles (not shown) are similar, with better S/N.

Fig. 4.

JCMT ¹²CO(3-2) 1′×1′ galaxy center maps. Linear contours ∫T_mbdV (K km s⁻¹) are superposed on grayscales $\int T_{A}^{*}$ $\smallint T_{A}^{*}$ dV (K km s⁻¹). Galaxy names, the values of the lowest white contour, and the contour step are as follows: Row 1: NGC 628 (2, 0.5), NGC 695 (20, 4), NGC 972 (24, 6), NGC 1667 (7.5, 1.5); Row 2: NGC 1808 (150, 30), NGC 2273 (12, 2), NGC 2559 (36,6), NGC 2623 (24,4); Row 3: NGC 2903 (48, 8), NGC 3175 (15, 5), NGC 3227 (48, 8), NGC 3256 (48, 8); Row 4: NGC 3310 (12, 3), NGC 3627 (40, 8), NGC 3982 (14, 2), NGC 4051 (32, 8); Row 5: NGC 4258 (30, 6), NGC 4303 (20, 4), NGC 4321 (32, 8), NGC 4388 (9, 1.5); Row 6: NGC 4735 (25, 5), NGC 5713 (20, 5), NGC 5775 (12, 3), NGC 7674 (7.5, 1.5).

2.4. Observational error

We usually integrated until the peak S/N in individual 10–20 km s⁻¹ channels exceeded a value of 5–10. Especially for ¹³CO line measurements, this required long integration times, sometimes up to several hours. The JCMT B-band receiver system had a relatively high system temperature spike around 330 GHz, resulting in a decreased sensitivity for the J = 3−2 ¹³CO line. The higher profile noise level and the limited bandwidth of 920 MHz (800 km s⁻¹) caused additional uncertainties in the line parameters that could only partly be alleviated using longer integration times. In the SEST 1999–2003 and all IRAM observing runs, the need to obtain a good detection of the ¹³CO line automatically provided very high S/N for simultaneously observed stronger ¹²CO lines.

From repeated observations, and from comparison with published measurements by others (summarized in Appendix A), we find the uncertainty in individual intensities obtained with the SEST in 1988–1992 to be about 30%, and those obtained in 1999–2003 to be about 20%. Depending on profile width, galaxies with intensities above 40–70 K km s⁻¹ have somewhat lower uncertainties, whereas galaxies with intensities below 10 K km s⁻¹ have larger uncertainties of up to 50%. The IRAM profiles in particular were obtained with wide velocity coverage and well-defined baselines, which is especially important for observations of galaxy center profiles with large velocity widths. They have relatively high S/N and are generally superior to those obtained in earlier measurements as well as to our own SEST and JCMT data. The uncertainty in the intensities observed with IRAM is ∼10% for ¹²CO, and 20−25% for ¹³CO. Again from repeated observations, individual intensities measured with the JCMT have an uncertainty of 15−20%, except for those of J = 3−2 ¹³CO, where uncertainties range from 20% for bright narrow lines to 50% for weak broad lines. However, because the ¹²CO and ¹³CO intensities were measured (almost) simultaneously, the uncertainty in their ratio is lower, typically 10−20% for the J = 1−0 and J = 2−1 transitions and 15−25% for the J = 3−2 transition. A comparison of the ¹²CO-to-¹³CO ratios determined in this paper and published in the literature may be found in Appendix B.

3. Results

Tables 2 through 5 list all directly observed ¹²CO intensities measured with the SEST (S), the JCMT (J), and the IRAM 30 m (I) telescope, with the resolution in arcseconds indicated in the headers. For comparison purposes, we also listed additional intensities at lower resolutions determined by the convolution of JCMT ¹²CO maps such as those shown in Fig. 4. In a few cases we have included published measurements obtained by others with the same telescopes; these are identified in the footnotes.

With Tables 2 through 4 we have constructed transition line ratios in matched beams. Individual ratios have typical errors of 25% to 30%. The histograms in Fig. 5 show the distributions of the transition line ratios. These are clearly peaked, and their width reflects in roughly equal parts the measurement error and the intrinsic variation. The average (1–0):(2–1):(3–2):(4–3) ¹²CO line intensities relate to one another as (1.09 ± 0.04):(1.00):(0.76 ± 0.05):(0.62 ± 0.05). As a practical application, the quantities 1.1 × I_CO(2–1) or 1.4 × I_CO(3–2) can thus be used to estimate the central I_CO(1-0) intensities in gas-rich spiral galaxies when these are needed but not measured. Oka et al. 2012 found the identical (3–2):(1–0) ratio for the central region of the Milky Way. The central (2–1):(1–0) ratio of 0.9 exceeds the value 0.7 used by Sandstrom et al. 2013 for galaxy disks. The bottom diagram in Fig. 5 shows (3–2):(2–1) ratios as a function of the parent galaxy FIR luminosity, ranging from log L(FIR) = 9 for normal galaxies over log L(FIR) = 10 for star-burst galaxies to log L(FIR) = 11 for luminous infrared galaxies (LIRGs). It does not reveal a clear dependence on galaxy class, nor does any of the other transition line ratios.

Fig. 5.

Distribution of the J = 2−1/J = 1−0, the J = 3−2/J = 2−1, and the J = 4−3/J = 2−1 ¹²CO intensities. Bottom: J = 3−2 intensities relative to the J = 2−1 ¹²CO intensity as a function of galaxy total FIR luminosity.

An essential part of this survey is the measurement of ¹²CO-to-¹³CO isotopologue ratios in the J = 1−0, J = 2−1, and J = 3−2 transitions; high atmospheric opacities render the J = 4−3 ¹³CO line practically unobservable from the ground. In galaxy disks and centers, the observed ¹²CO lines are optically thick (τ > 1), but in the observed three lowest transitions, the ¹³CO lines have optical depths (well) below unity. This is important because including lines with low optical depth reduces the degeneracy that severely limits the analysis of the optically thick ¹²CO lines. The measured ¹³CO fluxes are listed in Tables 2–4, and Figs. 1 through 3 show that the ¹³CO and ¹²CO line profiles are very similar in width and shape. The single but frequently occurring difference is a dip in the central ¹³CO profile at the systemic velocity where the ¹²CO profile shows a flat top. This dip suggests an optical depth decrease in the nuclear line of sight that is consistent with a lack of material (an unresolved “hole”) in the very galaxy center.

Taking into account the errors, the isotopologue ratios in the lower two transitions do not depend on the aperture size. We therefore averaged whenever possible the isotopologue ratios in the 45″ and 22″ and the 22″ and 11″ apertures. The resulting distributions in the lower three transitions are shown in Fig. 6. The ¹²CO-to-¹³CO ratios peak around R = 10 in the J = 2−1 transition and well above that in the other two transitions. The isotopologue ratios in the three transitions are clearly related to one another. In all three transitions, most isotopologue ratios occur between R = 8 and R = 16. Only a few galaxies have R < 8, which is characteristic of the relatively high optical depths of dense star-forming molecular clouds in the spiral arm disk of the Milky Way.

Fig. 6.

Top: distribution of the J = 1−0, the J = 2−1, and the J = 3−2 isotopologue ratios. The histogram fraction representing luminous galaxies (log L_FIR/L_⊙ ≥ 10) is filled. The remainder represent the normal galaxies (log L_FIR/L_⊙ < 10) in the sample. Bottom: J = 2−1 and J = 3−2 isotopologue ratio as a function of the J = 1−0 ratio.

4. CO maps and radial extent

4.1. Global CO flux and central fraction

The literature provides J = 1−0 ¹²CO observations at various resolutions for about 50 sample galaxies that are accessible from the northern hemisphere, and CO fluxes of entire galaxies have been published by Stark et al. (1987), Sage (1993), Young et al. (1995), and Chung et al. (2009). These are summarized in Appendix A, and examples of the multi-aperture photometry diagrams that can be constructed from them are shown in Fig. 7. In this section CO intensities are expressed as line fluxes (Jy km s⁻¹) per beam in order to emphasize their increase as larger areas are covered.

Fig. 7.

J = 1−0 ¹²CO multi-aperture photometry of galaxies observed with different telescopes. The points on the vertical axis refer to the integrated CO line flux of the entire galaxy. In each panel, their average is marked by a horizontal line. References to the measurements used in these diagrams and in the photometry analysis are given in Appendix A.

We determined slopes α_CO (defined by F ∝ θ^α) describing the increase of flux F with beam-width θ. In extended sources much larger than the sampling beams, the measured flux increases with the beam surface area so that α = 2. Point-like sources much smaller than the sampling beams have identical fluxes in all beams so that α_CO = 0⁴. The observed CO emission does not represent either extreme, as Figs. 7 and 8 illustrate. The average slope is close to unity, α_CO = 0.96 ± 0.06, with a standard deviation of 0.42 (see Col. 2 in Table 6 and Fig. 9) and is independent of galaxy distance. Assuming that this sample is representative, we conclude that CO fluxes of gas-rich spiral galaxies can be extrapolated from one beam to another with a modest uncertainty of about 30% by taking the linear beam width ratio.

Fig. 8.

Left: slope α derived from J = 1−0 ¹²CO multi-aperture photometry as a function of galaxy distance. Completely unresolved galaxies have α = 0, and fully resolved galaxies have a constant CO surface brightness with α = 2. The solid line marks the mean value of the sample, the two dashed lines mark half-widths of the distribution. Right: fraction of the total J = 1−0 CO flux of the sample galaxies contained within a beam of FWHM 22″ as a function of galaxy distance.

Table 6.

Spatial distribution of CO emission.

The galaxy CO extent (d_CO) equals the angular size at which the extrapolated CO flux in Fig. 7, for instance, reaches the total CO flux taken from the literature. From internal consistency, we find that the average error in the total fluxes is 26% (cf. Appendix A), which dominates the error in the global size. We list the extrapolated global sizes in Col. 3 of Table 6, both as an angular size in arcminutes and as a fraction of the optical galaxy size D₂₅ (taken from Col. 5 in Table 1). The distribution in Fig. 9 is distinctly peaked at 0.35 D₂₅ but the average value is slightly higher at (0.44 ± 0.03) D₂₅. As the average HI disk radius is 1.7 D₂₅ (cf. van der Kruit & Freeman 2011), the extent of CO-emitting gas is typically only 25% that of HI: in late-type galaxies, the molecular gas is much more concentrated than the atomic gas.

Fig. 9.

Distributions of the sample galaxies as a function of (left) slope α, marking the change in measured J = 1−0 CO flux as a function of increasing observing beam size, (center) the fraction f₂₂ of the extrapolated total galaxy CO flux detected in a 22″ beam, and (right) the extrapolated galaxy CO size as a fraction of the optical size (D₂₅) (see text).

4.2. Inner galaxy CO concentrations

For more than half of the sample galaxies, small maps of the central CO emission at the relatively high resolution of 14″ are provided by JCMT J = 3−2 ¹²CO observations. These include the 24 galaxies shown in Fig. 4, the 16 galaxies published in earlier papers (Israel et al. 1995; Israel & Baas 2006, 1999, 2001, 2003; Israel 2009a,b), and some 20 more by the authors identified in the notes to Table 6. The resolution of these maps is sufficient to separate the emission of a central compact source from the extended disk emission discussed in the previous section. We determined both the projected solid angle subtended by the central compact sources and their radial extent (FWHM along the major axis). We have at least partial information for 73 galaxies in Table 6. Ten of these do not have a central CO peak, but a central CO minimum instead (e.g., NGC 628 in the upper left corner in Fig. 4). In 6 galaxies, the central CO peak is unresolved. Except for NGC 3310, all are very distant galaxies, at distances of 60 Mpc or more. The observed central peak solid angle (Ω_CO) of 52 galaxies is listed in Col. 4 of Table 6. We corrected the central peak FWHM radius (R_CO) observed in 57 galaxies for finite resolution by (Gaussian) deconvolution. Column 5 lists the resulting angular radii as well as the corresponding linear radii using the distances from Table 1. The distribution of the linear radii is shown in Fig. 10. As also shown in Fig. 4, in most of the sample galaxies, a significant amount of molecular gas is concentrated within a kiloparsec from the nucleus (mean radius of 400 pc). Another group of CO peak radii 2 ≤ R_CO ≤ 4.5 kpc represents galaxies with more extended inner disk features such as “rings” (e.g., NGC 1068 and NGC 1097) or bars (e.g., NGC 1365). All galaxies with a central CO minimum, absent in the first group, are present in the second group with bright CO emission. Sakamoto et al. (1999) obtained a similar result for 20 nearby spiral galaxies, many of which are also included in our sample. Their average “local” scale length r_e = 0.53 kpc and average “global” scale length R_e = 2.6 kpc closely correspond to the first two peaks in Fig. 10. The occurrence of compact circumnuclear molecular gas is probably more frequent than suggested by Fig. 10 because galaxies with distances beyond 15–20 Mpc are imaged with relatively limited linear resolution, making it hard to separate compact circumnuclear and extended inner disk emission.

Fig. 10.

Histogram of the intrinsic (beam-deconvolved) radii of the central concentrations in galaxy CO maps. Three characteristic radii are distinguished (see text).

4.3. CO size and beam-dependent intensity ratio

In the absence of maps, beam-dependent intensity ratios are sometimes used to estimate sizes. The map-derived solid angles in Table 6 can be used to determine the reliability of (effective) source sizes recovered from the ratio of line intensities in different apertures. In Fig. 11 we show the ratio of the ¹²CO(3–2) intensities in 14″ and 22″ beams (Table 4) as a function of the measured solid angle (Table 6). In each case, both intensities were derived from the same map data-set. The observed points roughly follow the dashed line that marks the expected relation for circularly symmetric isolated compact peaks. The observational scatter is increased by the non-circularity of the peaks (points above the dashed line) and by the presence of extended emission especially in case of barely resolved peaks (points below the dashed line).

Fig. 11.

Intensity ratio of J = 3−2 ¹²CO emission in beams of 22″ and 14″ as a function of the effective surface area of the central CO concentration taken from Table 6. Very extended emission has a ratio of unity, and fully unresolved (point-like) sources have a ratio of 2.25. The vertical line corresponds to the surface area of a 14″ beam. The dashed curve indicates the relation expected for circular Gaussian sources without contamination by more extended emission.

Figure 11 suggests that the central peak diameters estimated from homogeneous beam intensity ratios have errors of up to ∼40% that are mostly caused by unknown emission structure. In reality, the errors are larger because this method is used precisely when no map is available. In this case, the combination of heterogeneous data results in additional scatter. In Fig. 12 we compare 22″-to-11″ beam intensity ratios from unrelated J = 2−1 ¹²CO JCMT and IRAM measurements (center panel) with 45″-to-22″ (J = 2−1, left panel) and 22-to-″/14″ (J = 3−2, right panel) beam intensity ratios extracted from the same JCMT map data-sets. The dispersion of the ratios from the heterogeneous data in the center panel is twice that of the homogeneous ratios based on the same map data, which is especially clear for the J = 3−2 ratios in the rightmost panel. The additional errors in the heterogeneous intensity beam ratios increase the errors in the derived source size to 70% or more. As long as the actual morphology of the emission remains unknown, more sophisticated treatments of the problem (e.g., Yamashita et al. 2017) do not significantly change these uncertainties. With such uncertainties, the multi-aperture method is only useful when no great accuracy is required.

Fig. 12.

Left: histogram of J = 2−1 CO intensity ratios in beams of 45″ and 22″. Center: same for J = 2−1 in 22″ and 11″. Right: same for J = 3−2 CO in 22″ and 14″ beams.

5. CO radiative transfer modeling

The various transitions in our survey have been measured at different resolutions, but a meaningful comparison requires intensities at the same resolution. These are provided by the data measured directly (J = 1−0, J = 2−1) or indirectly (J = 3−2, J = 4−3) at a resolution of 22″. Table 7 give all ratios in that aperture for all galaxies with at least two measured line ratios. Galaxies with a determination of the ¹²CO-to-¹³CO in the J = 1−0 transition only are separately listed in Table 8. The ¹²CO transition ratios in Cols. 3 through 5 of Table 7 have typical errors of 30%. The isotopologue ratios in Cols. 6 through 8 were determined by fitting each ¹³CO to its corresponding ¹²CO profile rather than by a division of the ¹²CO and ¹³CO intensities in Tables 2 through 4. By comparing the two methods, we find that the isotopologue ratios listed here indeed have typical uncertainties close to those suggested in Sect. 2.4. In Sect. 3 we noted that the ¹²CO-to-¹³CO isotopologue ratio is effectively independent of aperture in the J = 1−0 and J = 2−1 transitions (cf. Appendix B), and we have assumed that this is also true for the J = 3−2 ratios measured in 14″ apertures. Complementary values taken from the literature are identified by footnotes.

Table 7.

Line intensity ratios normalized to 22″ aperture^(a).

Table 8.

Galaxies with J = 1−0 isotopologue ratio only.

From the previous section, we determined that this normalized beam covers between 3% and 11% of the total CO surface area of the sample galaxies. When we restrict the sample to galaxies with distances between 10 Mpc and 40 Mpc, we obtain the same result. The fraction f₂₂ of all CO flux contained in an aperture of 22″ (Col. 2 of Table 7) is much higher, on average 26%. As expected, Fig. 8 shows that the individual values increase with increasing distance D. The distribution of individual f₂₂ values is also shown in Fig. 9.

We have modeled the data in Table 7 with the statistical equilibrium radiative transfer code RADEX (van der Tak et al. 2007). It provides model line intensities as a function of three input parameters per molecular gas phase: gas kinetic temperature T_k, molecular hydrogen density n_H₂, and the CO column density per unit velocity N(CO)/dV. Each combination of physical parameters uniquely determines a set of line intensities and ratios. The opposite is not true because the same line ratio may result from different combinations of physical input parameters. Reverse tracing is therefore not a unique process. Nevertheless, by comparing for each galaxy as many observed line ratios as possible to extensive grids of precalculated model line ratios, we may constrain and identify the physical parameters that best describe the actual conditions.

The large linear beam sizes that apply to galaxy center observations encompass molecular gas clouds at distinctly different temperatures and densities, which require more than one model gas phase to produce acceptable fits to the observations (see, e.g., Israel & Baas 1999; Papadopoulos & Seaquist 1999; Israel 2009a,b). Good model fits are easily obtained for data sets containing only ¹²CO observations, but the high degree of degeneracy between H₂ temperature and density renders such excellent fits non-unique and not very useful. Not even long ¹²CO ladders (such as those extending up to J = 13–12 obtained with Herschel-SPIRE) provide significant constraints (e.g., see Meijerink et al. 2013). Fortunately, the degeneracy can be broken by measuring lines with low optical depth such as ¹³CO in addition to the mostly optically thick ¹²CO lines at the cost, however, of more physical parameters to be determined. Such combinations of ¹²CO with related species yield constraints that although still not unique, are much tighter than those based on ¹²CO alone.

We have modeled our data under the assumption that the emission is dominated by two distinct model gas phases. This is an important and necessary improvement over models assuming homogeneous single-phase gas that tend to provide a poor fit to the data when overconstrained. With only two gas phases, however, an ambiguity in temperature and density remains. It would be more realistic to model the gas with a smoothly changing temperature and density over a range of phases. This is, however, unfeasible even with the present relatively extensive data set because including more gas-phase components rapidly increases the number of unconstrained free parameters, which renders the result less rather than more realistic.

To simplify matters, we assumed that both gas phases have the same fixed isotopologue abundance, and we considered abundances of 40 and 80, respectively. In nearby galaxies such as NGC 253, NGC 4945, M 82, IC 342, and NGC 1068 the lower value seems appropriate (Henkel et al. 1994, 1998, 2014; Bayet et al. 2004; Giannetti et al. 2015; Tang et al. 2019). The higher value may be more appropriate to (some of) the more distant very luminous galaxies such as NGC 5135, NGC 6240, NGC 7469, and Mrk 231 (e.g., Henkel et al. 2014; Sliwa et al. 2014; Tunnard et al. 2015; see also Israel 2015).

For any particular set of line ratios, the RADEX model-fit line intensity, column density gradient, spatial density, and temperature in the two phases do not vary independently. The beam-averaged CO column density is sensitive only to the combined effect of these variations, and its resulting dispersion of about 30% is much lower than the uncertainty in each of the individual constituent model parameters, as illustrated in Table C.1.

The fraction of gas-phase carbon contained in CO is a function of the actual total carbon column densities N_C. We determined for each phase the fractional CO abundance [CO]/[C] as well as the total beam-averaged carbon column densities N_C using the chemical models presented by van Dishoeck & Black (1988) and updated by Visser et al. (2009). The detailed results of the two-phase modeling are given in Table C.2, where we present for each galaxy the model solution closest to the observations, regardless of the other possible model ratios within the observational error.

These results were combined to derive the beam-averaged fractional CO abundance [CO]/[C] and the beam-averaged total carbon column density N_C (Cols. 2 and 3) in Table D.1 for both of the assumed isotopologue abunandances. The beam-averaged CO column density N_C is the given by the producte: $N_{CO} = N_{C} \times \frac{[CO]}{[C]}$ $N_{\mathrm{CO}} = N_{\mathrm{C}}\,\times\, \frac{\mathrm{[CO]}}{\mathrm{[C]}}$ . Out of 72 galaxies, 64 (90%) are successfully modeled with a [¹²CO]/[¹³CO] abundance of 40, and 28 galaxies (39%) even require this abundance for successful modeling. Only 8 galaxies (11%) need to be modeled with a high isotopological ratio of 80 instead. Half of the galaxy sample can be modeled with either ratio, but in most cases, the lower ratio of 40 provides better fits. Four galaxies (NGC 1614, NGC 4293, NGC 4527, and NGC 5236) have poor fits at either abundance.

6. Gas-phase carbon budget

6.1. Carbon monoxide fraction

The distribution of the fractional CO abundances is shown in Fig. 13 for the two isotopological abundances 40 (left) and 80 (center), with average values f_CO = 0.28 and f_CO = 0.38 and standard deviations of 0.18 and 0.17, respectively. We constructed the combined distribution (right) by averaging the results for the galaxies that could be fit at either ratio and by taking the single result for the galaxies that could not. The combined distribution has an average value f_CO = 0.33. The standard deviation 0.16 exceeds the uncertainty in the individual values and represents an intrinsic spread of the ratios. The final adopted beam-averaged fractional CO abundances and gas-phase total carbon column densities are summarized in Table 9 (Cols. 2 and 3).

Fig. 13.

Distribution of the fraction of all carbon contained in CO. Left: results for an isotopological ratio of 40. Center: results for an isotopological ratio of 80; for comparison, the distribution for the ratio of 40 is shown as well (unshaded). Right: most probable distribution derived from both data sets (see text). Typically, one-third of the gas-phase carbon is in CO and two-thirds is in atomic or ionic form.

Table 9.

Physical properties of 22″ central regions.

In Table 10 we have selected the data for all galaxies for which [CI] and [CII] measurements are also available. On average, molecular carbon represents only one-third of all gas-phase carbon in the observed galaxies. The remainder is atomic carbon either in neutral (C°) or in ionized form (C⁺). The [CI] and [CII] line fluxes that are needed to further investigate this are found in the literature.

Table 10.

Gas-phase carbon fractions.

6.2. Neutral atomic carbon fraction

We took central [CI] line data from the compilations by Israel (2015), Kamenetzky et al. (2016), and Lu et al. (2017). Most of these were fluxes obtained with the SPIRE instrument onboard the ESA Herschel Space Observatory⁵ in a 35″ aperture. We expressed them as integrated main-beam brightness temperatures in units of K km s⁻¹, reduced to our “standard” beam by assuming identical [CI] and ¹²CO distributions and filling factors and using the multi-aperture CO data in Tables 2 and 3 to estimate the 35″ → 22″ beam conversion factors (typically between 1.1 and 2.2). The resulting [CI] line intensities are given in Table 10.

We cannot derive two-phase atomic carbon column densities in the same way as the carbon monoxide column densities without additional assumptions because only two [CI] transitions are available for analysis. Fortunately, the [CI] intensities scale quite well with the observed ¹²CO intensities. We are uncertain of the cause, but it is reasonable to expect that the [CI] emission either results from photodissociation of the CO clouds in the beam or from material that is left over in the formation of these CO clouds. In either case, neutral carbon and carbon monoxide are closely related and associated with the same H₂ gas. We therefore used the H₂ densities, kinetic temperatures, and relative filling factors from the CO analysis (Table C.2) as RADEX input to determine model [CI] intensities. From these, we derived beam-averaged column densities of [CI] in the same way as those of CO. This procedure is less critical for [CI] than for CO. With energy levels of 24 and 39 K and a critical density of 10³ cm⁻³, the [CI] emission is thermalized and close to being optically thin, roughly proportional to the C° column, and only weakly dependent on temperature and density (Schilke et al. 1993; Stutzki et al. 1997). We calculated neutral carbon column densities separately for the J = 1−0 and the J = 2−1 transitions and for the two isotopologue abundances. As expected, the average column densities are identical for the two [CI] transitions, and use of the parameters of the ¹²CO/¹³CO = 40 case yields column densities lower than those for the ¹²CO/¹³CO = 80 case by a factor of 0.6 (standard deviation 35%), reflecting the corresponding decrease in average optical depth. The final [CI] column densities in Table 10 are the averages of the independent determinations, as are the fractional [C°]/[C] abundances; their uncertainty is about 40%. The original SPIRE measurements are quite accurate, therefore most of this uncertainty must be due to assumptions in the analysis.

Our previous single gas-phase modeling of [CI] and CO intensities of galaxy centers (Israel 2015) suggested a significantly higher CO-to-C^o ratio. However, the two studies measured different quantities. In the earlier study we used line ratios of the mid-level J transitions of ¹²CO, representing the more highly excited gas. We did not scale these results with line intensity, and the derived column densities were not corrected for the differentiating effects of beam filling factor and cloud velocity width. They sampled purely local conditions rather than the global conditions derived here.

The average gas-phase neutral atomic carbon fraction is 0.31. When we leave out the very high fractions derived for NGC 1365 (0.93) and NGC 5135 (0.86), the average drops to 0.26. In general, the [CI] fraction is somewhat below the CO fraction. The derived [CI] fraction exceeds that of CO in less than 20% of all cases

6.3. Ionized atomic carbon fraction

Ionized carbon [CII] line measurements useful for our purpose have been carried out with the PACS instrument (Poglitsch et al. 2010) onboard the ESA Herschel Space Observatory at a resolution of 11″ in square pixels of 9.4″ × 9.4″ size. We used the compilations published by Fernández-Ontiveros et al. (2016), Croxall et al. (2017), Díaz-Santos et al. (2017), and Herrera-Camus et al. (2018). We interpolated intensities in the central nine PACS pixels (28.2″ × 28.2″) and in the single central pixel to those expected in an intermediate 22″ aperture. We also extrapolated central PACS pixel intensities to those expected in a 22″ aperture assuming the emission to be point-like, with consistent results.

CO and [CI] emission can only originate in a neutral gas, but [CII] emission can also come from an ionized gas. The fraction of the [CII] emission from the neutral gas f_CII relevant to our analysis is estimated from the Herschel intensities of the [NII] 122 μm (PACS) and 205 μm (SPIRE) lines in the usual way; for a detailed description of this procedure and mapping results on many of the sample galaxies, see Croxall et al. (2017). Typically, 20% of the [CII] emission comes from ionized gas, and the neutral gas fractions of interest to us range from 0.57 to 0.92 (mean 0.81, average 0.75, with a standard deviation 0.21). Table 10 provides normalized and corrected [CII] intensities I[CII] available for all galaxies in which [CI] has also been measured. In three cases, the actual fraction of [CII] emission arising from neutral gas could not be determined; here we inserted the average value, denoted by a colon.

The analysis of ionized carbon is more problematical than that of the neutral carbon in the preceding section. Because the [CII] emission can be more extended and associated with dense hydrogen gas that is not traced by [CI] or CO emission, the CO parameters that we used to guide our [CI] analysis are now of little use. The 158 μm [CII] line is the only strong C⁺ emission line in the far-infrared, and if it is optically thin, the line-of-sight column density N_C+ is related to the line intensity I_[CII] (in K km s⁻¹) by Eq. (1) from Pineda et al. (2013): N_C⁺ = I_[CII] × (3.05 × 10¹⁵ (1 + 0.5(1 + 2840/n) e^91.2/T). The temperatures and densities of the [CII]-emitting gas cannot be determined directly because there are three unknown parameters and only one equation. The equation provides a lower limit N_C⁺ = 4.6 × 10¹⁵ I(CII) to the column density in the high-temperature, high-density limit, but no upper limit. We calculated C⁺ column densities (Col. 6 of Table 10) for a more reasonable temperature T_k = 100 K and density n(H₂) = 3000 cm⁻³, so that N_C⁺ = 1.04 × 10¹⁶ I(CII), doubling the high-T, high-n limit. The corresponding fractional abundances [C⁺]/[C] are listed in Col. 12 of Table 10. They show a large spread; in particular, the values for NGC 2146, M 82, and NGC 4536 are quite high, which might indicate that their [CII] emission comes from gas that is denser and hotter than we have assumed. The derived central C⁺/C fractions and the galaxy FIR luminosities are correlated, with considerable scatter. Because we calculated ionized carbon column densities for fixed temperatures and densities, this is loosely related to a correlation between [CII] and FIR intensities. No other clear pattern seems to emerge from the data in Table 10, consistent with significant variation in the physical conditions of the ISM and morphology even in galaxies that otherwise appear similar.

6.4. Carbon budget in galaxy centers

The combined column density of the three individual carbon gas-phase components listed in Col. 13 in Table 10 is generally close to the total gas-phase column density derived independently of the analysis of the ¹²CO and ¹³CO intensities sample (Table C.2). The average ratio of the two values is 1.06, with a standard deviation of 0.45. The consistency of these results underlines the validity of the analysis that produced them.

The relative amounts of the three gas components vary significantly among the observed galaxies, but the average fractional CO, C°, and C⁺ contributions to the total C are quite similar. The respective contributions in the 33 galaxies are 0.32, 0.31, and 0.43, with standard deviations of 0.13, 0.23, and 0.36, respectively. The three forms of carbon occur in comparable amounts in the gas phase. Slightly less than one-third of all gas-phase carbon is in molecular form, and somewhat less than half of all gas-phase carbon is ionized.

As argued in Sect. 6.1, the beam-averaged carbon monoxide column density is well constrained. In Sect. 6.2, we argued that the [CI] emission arises from the same molecular gas, so that the beam-averaged neutral carbon column densities should likewise be robust. If the actual CO-to-C and C°-to-C ratios were constant across the sample, the standard deviations would represent the measurement error in their determination. More realistically, they serve as upper limits to the uncertainty in the actual individual cases. The ionized carbon column densities are not so robust because they depend more strongly on assumptions. As we showed, there is a firm lower limit corresponding to very hot (T_k > 250 K) and dense (n_H₂ > 10⁵ cm⁻³) gas, so that the column densities in Table 10 can be lowered by a factor of two at most. In contrast, the [CII] line measurements do not define an upper limit. For instance, column densities would be higher by a factor of ten for modest temperatures T_k ≈ 50 K and low densities n_H₂ ≈ 300 cm⁻³. This is very unlikely because it requires the total carbon column densities N_C derived before to be underestimated by factors of five, incompatible with the models used. The corresponding very small CO fractions would also seem incompatible with the high-metallicity environment of galaxy centers as they are more characteristic of low-metallicity objects such as the Magellanic Clouds (Requena-Torres et al. 2016). As it is, modest temperature decreases to 60 K, or equally modest density decreases to 1000 cm⁻³, implying 70% higher [CII] column densities, delineate the limits of what is feasible in view of the various uncertainties associated with Table 10.

7. Hydrogen column density and mass

Total hydrogen column densities N_H would follow directly from the carbon column densities N_C if the gas phase carbon-to-hydrogen abundance were directly known, which is not the case. Instead, we must infer this abundance from our knowledge of (i) the relative oxygen abundance [O]/[H], (ii) the relative carbon abundance [C]/[O], and (iii), the fraction δ_C of all carbon that is in the gas phase rather than locked up in dust grains. Based on the detailed discussion in Appendix D, we adopt for all galaxy centers a metallicity of twice that of the solar neigborhood, identical [C] and [O] abundances and a carbon depletion factor δ_C = 0.5 ± 0.2. This yields a “nominal” (N) gas phase ratio N_H/N_C = (2 ± 1)×10³. In Table D.1 we list the implied beam-averaged total hydrogen column densities N_H for a range of assumptions, as well as the derived molecular hydrogen column densities N_H₂ and overall hydrogen gas masses M_H. The adopted nominal values are summarized in Table 9. The beam-averaged molecular hydrogen column densities N_H₂ (Col. 4) are corrected for the (small) contribition by HI. The total gas masses M_gas (Col. 5) incorporate a 35% contribution by helium. As discussed in Appendix D, the uncertainty in individual values of N_H and M_H, hence also in N_H₂ and M_gas, is a factor of slightly more than two.

In Fig. 14 we plot N_H and M_H as a function of galaxy distance D for both the nominal N and the extrapolated E carbon abundance. The Galactic abundance case G is not shown, but would be represented by points offset from the nominal abundance points by +1.7 in the log. The hydrogen gas column densities peak in the center, causing beam-averaged column column densities to decrease with galaxy distance as ever larger linear areas are covered by the fixed 22″ beam. At the same time, the encompassed hydrogen mass increases with distance when the beam includes ever larger areas of the galaxy disk.

Fig. 14.

Calculated beam-averaged column densities N_H (left) and central gas masses M_H (right) as a function of distance D. Filled circles: Values based on the preferred “nominal” carbon abundance. Open circles: Values bases on extrapolated maximum carbon abundances (see text). All values are based on an assumed isotopological ratio of 40.

Figure 15 shows the distributions of N_H and M_H. We merged the two isotopological abundance data sets and averaged values where appropriate. The column densities have a well-defined peak at about N_H = 1.5 × 10²¹ cm⁻², with a tail to higher values primarily caused by long sight-lines through highly tilted galaxies. The mass distribution shows a wide range of values from 10⁶ M_⊙ to 10⁹ M_⊙ with a broad peak around a few times 10⁷ M_⊙ and a narrow peak around 10⁸ M_⊙. Figure 14 shows that with a few exceptions, the higher masses are all found in galaxies at distances of 10 Mpc or more and mostly refer to the “inner disks” in Fig. 10. The lower masses are found over a wider range of distances, from 4 to 25 Mpc, and thus characterize both “circum-nuclear disks” and low-mass “inner disks”.

Fig. 15.

Distribution of the total hydrogen column densities (left) and total hydrogen masses (right). See text for sample definition and errors.

8. CO as a tracer of H₂

8.1. Conversion factors X(CO), X[CI], and X[CII]

We are now in a position to consider to what extent CO, [CI], or [CII] line intensities trace H₂ molecular gas column densities. In Fig. 16 we plot the N_H₂ column densities from Table 9 as a function of the J = 1−0 CO, [CI], and [CII] line intensities from Tables 2 and 10. In each panel, the dashed line marks the relation between the two quantities with a slope corresponding to the X factor. Because the same molecular hydrogen column densities are plotted as a function of the observed line intensities, the results are not subject to the uncertainties that plague the determination of neutral and ionized carbon column densities discussed in Sects. 6.2 and. 6.3. The distribution of the individual X values is shown in Fig. 17 for each line. Inasmuch as the dispersion exceeds the observational error, the dispersion around the mean X value is an important quantity to judge the relative performance of each of the three lines in predicting the molecular gas column density.

Fig. 16.

Column densities N_H₂ as a function of (left) CO, (center) [CI], and (right) [CII] intensities. Points representing the nearby bright galaxies NGC 253, NGC 3034 (M 82), and NGC 4945 are outside the box limits. Dashed lines are linear regression fits to all data points, including these galaxies. The fits correspond to conversion factors X(CO) = 1.9 × 10¹⁹ cm⁻²/K km s⁻¹ and X[CI] = 9.1 × 10¹⁹ cm⁻²/K km s⁻¹. There is no meaningful fit for I(CII).

The individual I_CO-to-N(H₂) ratios occur in a fairly narrow range, ten times below the commonly assumed Milky Way solar neighborhood conversion factor X_MW = 2.0 × 10²⁰ cm⁻²/K km s⁻¹. A linear regression fit on I_CO and N(H₂) yields a high-quality solution (r² = 0.78) corresponding to a conversion factor that is very close to the average of the individual I_CO-to-N(H₂) ratios: X(CO) = (1.9 ± 0.2)×10¹⁹ cm⁻²/K km s⁻¹. The average X-factor applicable to galaxy centers is quite robust and well defined even when individual CO-to-hydrogen conversion factors are still subject to uncertainties of a factor of two.

A similar regression fit on the much lower intensities of the neutral carbon line (Fig. 16 central panel) yields a five times higher value X[CI]=(9 ± 2)×10¹⁹ cm⁻²/K km s⁻¹. The average of the individual values is almost twice as high, with a large standard deviation. This is caused by the cluster of low-intensity, low-column-density points in the lower left corner of the central panel in Fig. 16, and it suggests that [CI] intensities are useful but less reliable as N(H₂) indicators than CO.

The linear regression fit on all [CII] intensities gives a value (1.9 ± 0.5)×10¹⁹ cm⁻²/K km s⁻¹ with very low significance (r² = 0.07). The average is more than twice as high, (4.4 ± 0.8)×10¹⁹ cm⁻²/K km s⁻¹. These [CII] intensities include a contribution from ionized gas. If corrections were included, the N_H₂/I(CII) slopes would be steeper by about 20%, but the relative distribution of points would suffer little change. In any case, from both Figs. 17 and 16 it follows that the present data do not define a clear-cut single value for X([CII]).

Fig. 17.

From left to right: distributions of the CO-toH₂, [CI]-toH₂, and [CII]-toH₂ conversion factors X(CO), X[CI], and X[CII]. See text for sample definition and errors.

Following Wada & Tomisaka (2005), we also considered the ¹²CO J = 3−2 intensities as a tracer for N(H₂). Their three-dimensional, non-LTE radiative transfer calculations for circum-nuclear molecular gas disks predict that the J = 3−2 line is more useful than the J = 1−0 line as a tracer for N(H₂) and they suggest a conversion factor of ∼2.7 × 10¹⁹ cm⁻²/K km s⁻¹ for this transition. Our linear regression fit yields a value (2.1 ± 0.4)×10¹⁹ cm⁻²/K km s⁻¹. There is considerable scatter around this value and the average comes out higher, at (3.1 ± 0.3)×10¹⁹ cm⁻²/K km s⁻¹. We conclude that the X(CO3–2) value calculated by Wada & Tomisaka (2005) is very close to the actual value following from our work, but that X(CO1–0) is still the better performer, contradictory to their expectations.

In a previous paper, Israel (2015) discussed the suggestion by Papadopoulos et al. (2004) and others that [CI] line emission might provide a tracer of molecular hydrogen at least as good as CO emission. The above discussion, and indeed inspection of the CO, and [CI] panels in Figs. 17 and 16, establishes that the scatter is somewhat greater in the [CI] diagrams and that the X[CI] distribution is less strongly peaked than the X(CO) distribution. This result confirms and expands our earlier conclusion that the J = 1−0 ¹²CO line should be preferred over the [CI] line as a molecular gas tracer if both are available. If only the [CI] line is accessible, as may be the case for redshifted objects, it should be regarded as an acceptable substitute, provided an adequate calibration can be established for the differing environmental conditions.

The scatter in the [CII] diagrams is much greater than that in either the ¹²CO or [CI] counterparts, and our data do not establish a convincing unique value of X[CII]. It follows that [CII] intensities are not a useful tracer of extragalactic molecular gas column densities, and given the minor contribution of HI, are not a useful tracer of total gas either.

8.2. Low X(CO) and H₂ mass in galaxy centers

In the above, we have come to the conclusion that molecular hydrogen column densities and masses in the centers of galaxies are an order of magnitude lower than suggested by the ‘standard’ conversion factor. This conclusion depends to some extent on the correctness of the carbon abundances assumed in the derivation. If, for instance, these were substantially lower, as in the Pilyugin et al. (2014) calibration already mentioned, the distribution of the individual X factors would still be similar to that depicted in Fig. 17, but it would be shifted upward to a mean value X(CO) = 5 × 10¹⁹ cm⁻²/K km s⁻¹. Even then, the conversion factors would still be over four times lower than the local Milky Way disk factor X_°.

Our results fit the historical downward trend of the published values of X in galaxy centers as opposed to galaxy disks. Sandstrom et al. (2013) studied the disks of 26 nearby galaxies, many in common with our survey. They derived a more or less constant factor X = 1.95 × 10²⁰ from ¹²CO(2–1), FIR, and HI emission, but suggested a lower central X. Follow-up J = 1−0 ¹²CO and ¹³CO observations of nine nearby galaxies led Cormier et al. (2018) to a similar result, with a low average center X ∼ 0.15 X_° which, fortuitously, happens to be close to our result. Even lower X factors of about 0.05 X_° are now discussed for the more extreme case of optically thin CO gas outflows from the centers of nearby luminous galaxies (Alatalo et al. 2011; Sakamoto et al. 2014; Oosterloo et al. 2017)

As the low X value of the ISM in the center of our own galaxy is thus revealed to be characteristic of galaxy centers in general, it is instructive to take a closer look at the well-studied Central Molecular Zone (CMZ) in the Milky Way. The density of most of the gas in the CMZ is not very high. High-density molecular tracers other than CO are generally subthermally excited, implying densities of only ∼10⁴ cm⁻³ (Jones et al. 2012). Observations of J = 1−0 ¹²CO and its optically thin isotope C¹⁸O have shown very complex molecular gas distributions characterized by moderate or low optical depths (Dahmen et al. 1998) on large scales. Observations of several ¹²CO and ¹³CO transitions (but not including the J = 1−0 transition) rule out excitation by a single component and instead suggest the superposition of various warm gas phases (Requena-Torres et al. 2012). Requena-Torres and collaborators performed a two-component LVG analysis, analogous to the one in this paper, that suggested a dominant phase with T_kin ≈ 200 K and a density n(H₂) ∼ 3 × 10⁴ cm⁻³ and a minor phase (20-30% by mass) with a higher T_kin ≈ 300-500 K, n(H₂) ∼ 2 × 10⁵ cm⁻³. The lack of the low-J transition biases their analysis to higher densities and temperatures, whereas our coverage of all lower transitions up to J = 3−2 or J = 4−3 samples the lowest temperatures in phase 1, but may underestimate phase 2 temperatures.

The bulk of the CMZ molecular gas has temperatures T_kin = 50–120 K, and the average gas temperature outside the densest clouds is 65 ± 10 K (Ao et al. 2013; Ginsburg et al. 2016). Both the temperature range and the average temperature are very similar to those of our sample of galaxy centers (cf. Table C.2), for which we find a mass-weighted mean temperature T_kin = 55 ± 5 K. The emission-weighted mean gas density of the galaxy centers in our sample, n_H₂ = (1.8 ± 0.6)×10⁴ cm⁻³, is likewise similar to the values obtained for the CMZ.

Requena-Torres et al. (2012) concluded that the gas sampled by them is not organized by self-gravity, is unstable against tidal disruption, and is transient in nature. Velocity dispersions of clouds in the Galactic center region are five times higher than those of clouds in the disk (Miyazaki & Tsuboi 2000). This, the widespread presence of shocked gas emitting in the J = 2−1 SiO line (Hüttemeister et al. 1998), and the high but variable gas temperatures throughout the CMZ noted by Requena-Torres et al. (2012) indicate that the dense gas is dominated by turbulent heating. Neither heating by UV photons nor by cosmic rays can be important on global scales in the CMZ (Ao et al. 2013; Ginsburg et al. 2016), as we have also concluded in the individual cases of NGC 253 and NGC 3690 after detailed modeling of molecular line data (Rosenberg et al. 2014a,b).

The picture that emerges of the molecular gas in the CMZ, and by implication also in the observed galaxy centers, is very different from the picture of the star-forming molecular gas in the disk of the Milky Way and other galaxies. In the CMZ, molecular gas appears to occur mostly in extended diffuse clouds with modest optical depths in CO but with relatively high surface filling factors. This warm gas does not only rotate rapidly around the nucleus, but is also continuously stirred up and rather turbulent, even though the precise mechanisms are not yet clear (see, e.g., the review by Mills 2017).

That CO-to-H₂ conversion factors are much lower for molecular gas in galaxy centers than in galaxy disks can be understood in terms of different physical properties. For instance, Stacey et al. (1991) already suggested that higher excitation temperatures were responsible for the drop in X(CO) implied by their [CII] measurements. Downes & Solomon (1998) speculated that the central CO in luminous galaxies is only moderately opaque and highly turbulent. They also believed it to be subthermally excited, but higher J CO intensities show this to be an oversimplification. More recently, models and simulations have started to elucidate the effects of different environments on empirical quantities such as X. In one example, Bell et al. (2006) used photon-dominated region (PDR) time-dependent chemical models to show that increases in density, cosmic-ray ionization rate, metallicity, and turbulent velocity all act to depress X-values. Bell et al. (2007) explicitly noted that application of their models to M 51 and NGC 6946 yielded results that are very close to those obtained by us and ascribed the low X values primarily to high density and metallicity. In another example, Narayanan et al. (2011) combined smoothed particle hydrodynamics (SPH) simulations of galaxy disks with physical and radiative transfer ISM models and the CO-to-H₂ conversion factor in star-forming and merger galaxy disks. Although their results do not directly apply to galaxy centers, they do draw attention to the significant dependence of X on kinetic temperature and velocity dispersion.

The bulk of the gas in galaxy centers is, like that in the CMZ, carbon rich, moderately dense, warm, and turbulent. With respect to H₂ column density, the higher CO abundance and emissivity increase the CO line intensity, whereas lower CO mean optical depth imply lower H₂ column densities relative to CO intensity. Thus, in galaxy centers, the molecular gas radiates in CO with much enhanced efficiency. The gains thus made by galaxy center molecular gas with respect to disk gas can be quantitatively estimated from the model dependencies X ∝ T^−1/2 and X ∝ σ^−1/2 derived for the CO-to-H₂ conversion factor by Shetty et al. (2011). The thrice higher carbon gas phase abundance leads to a three times lower X. The elevated average kinetic temperature of 55 K lowers X by a further factor of two. Most of the central CO is of modest optical depth, and taking our cue from the five times higher CMZ velocity dispersions, another drop in X by a factor of 2.2 is to be expected. Taking all together, we would expect X ≈ 1.5 × 10¹⁹ cm⁻²/K km s⁻¹ (or 0.075 X_°), which is very close to the average value X ≈ 1.9 × 10¹⁹ cm⁻²/K km s⁻¹ from Section 8.1. This agreement shows that the conditions causing the low value of central X are reasonably well understood. It also shows that no single cause prevails; all three contributing factors are of a similar magnitude, and all three are needed.

9. Conclusions

We determined intensities of galaxy centers in the lowest J transitions of ¹²CO and ¹³CO. Out of a total of 126 galaxies 112, 103, 88, and 24 were measured in the J = 1−0, J = 2−1, J = 3−2, and J = 4−3 transitions of ¹²CO, respectively, as well as 89, 71, and 61 in the J = 1−0, J = 2−1, and J = 3−2 transitions of ¹³CO, respectively. In 15 galaxies, only the J = 1−0 transition was measured and 30 galaxies lack ¹³CO measurements⁶.
Multi-aperture J = 1−0 ¹²CO fluxes from the survey and the literature show that CO luminosities increase roughly linearly with observing beam size. The extrapolated CO-emitting gas extends to ∼40% of the optical size D₂₅ and ∼25% of the HI size. The molecular gas is thus much more concentrated than the atomic gas.
The ¹²CO J = 1−0 to J = 4−3 transition ladder has relative intensities 1.00:0.92:0.70:0.57. The mean isotopologue ¹²CO-to-¹³CO intensity ratios are 13.0 ± 0.7 (J = 1−0), 11.6 ± 0.6 (J = 2−1), 12.8 ± 0.6 (J = 3−2), but individual values may be as low as 5 and as high as 25.
For more than 70 galaxies, physical parameters of a two-phase gas were determined with the use of non-LTE radiative transfer models (RADEX). On average, only one-third of the gas-phase carbon (32 ± 8%) is found to reside in CO. The full gas-phase carbon budget was determined for 45 galaxies, using literature data for neutral and ionized carbon line intensities. The intensities of the [CI] and ¹²CO lines are closely related; on average, neutral carbon C° accounts for somewhat less of the gas-phase carbon (30 ± 4%). Somewhat more than one-third of the gas-phase carbon is available for ionized carbon C⁺. This condition is met if [CII] emission originates in a moderately dense and warm (n ≥ 3000 cm⁻³, T ≥ 100 K) gas.
Averaged over a 22″ beam, mean total hydrogen column densities are N_H = (3.3 ± 0.2)×10²¹ cm⁻² and mean molecular hydrogen column densities are N(H₂) = (1.5 ± 0.2)×10²¹ cm⁻². Total gas masses of central molecular zones up to one kiloparsec radius are typically a few times 10⁷ M_⊙, whereas the total molecular gas masses of the inner disk are typically 10⁸ M_⊙.
The observed J = 1−0 CO intensities and the derived N(H₂) values yield CO-to-H₂ conversion factors with a well-defined mean value X(CO) = (1.9 ± 0.2)×10¹⁹ cm⁻²/K km s⁻¹. This is a factor of ten below the “standard” solar neighborhood Milky Way factor X_MW. The mean [CI]-to-H₂ conversion factor is X[CI]=(9 ± 2)×10¹⁹ cm⁻²/K km s⁻¹. There is no meaningful conversion factor for [CII].
Use of a conversion factor based on J = 3−2 ¹²CO line intensities yields results that are better than those obtained with [CI], but not as good as those derived from the J = 1−0 ¹²CO line.
From comparisons with the well-studied CMZ in the Milky Way galaxy, it appears that the order-of-magnitude decrease of the CO-to-H₂ conversion factor in the central molecular zones of nearby other galaxies with respect to canonical galaxy disk conversion factors is caused in equal parts by the higher gas-phase carbon abundances in galaxy centers, elevated kinetic gas temperatures, and high molecular cloud velocity dispersions.

¹

The Swedish-ESO Submm Telescope (SEST) was operated jointly by the European Southern Observatory (ESO) and the Swedish Science Research Council (NFR) from 1987 until 2003.

²

IRAM is supported by INSU/CNRS (France), MPG (Germany), and IGN (Spain). The IRAM observations in this paper have benefited from research funding by the European Community Sixth Framework Programme under RadioNet R113CT 2003 5058187.

³

Between 1987 and 2015, the (JCMT) was operated by the Joint Astronomy Centre on behalf of the Particle Physics and Astronomy Research Council of the United Kingdom, the Netherlands Organization for Scientific Research (until 2013), and the National Research Council of Canada.

⁴

When we express CO intensities in temperature (K km s⁻¹) instead of flux units (Jy km s⁻¹), we have α′ = α−2.

⁵

Herschel was an ESA space observatory with science instruments provided by European-led Principal Investigator consortia and with important participation by NASA.

⁶

The raw JCMT data are publicly available from the CADC JCMT Science Archive at https://www.cadc-ccda.hia-iha.nrc-cnrc.gc.ca/en/jcmt/. Neither SEST nor IRAM data are archived, but the processed galaxy data from this paper can be downloaded in CLASS format from ftp://ftp.strw.leidenuniv.nl/pub/israel/data/galaxyfiles

⁷

Some studies (cf. Pilyugin et al. 2012, 2014) derive abundances that are systematically lower by a factor ∼2.5. See the discussion by Peimbert et al. (2017).

Acknowledgments

Some of the SEST and IRAM observations and most of the JCMT observations were made in service mode. With gratitude I acknowledge my indebtedness to the many colleagues and facility staff, above all the JCMT operators, who over many years helped to collect the large and unique data base described in this paper. It is fitting to commemorate the generous advice and assistance of Lars E.B. Johansson (SEST) and Fred Baas (JCMT), both of whom sadly died before they could see the results. I also thank Thijs van der Hulst for supplying his unpublished SEST observations of NGC 613, NGC 1097, and NGC 1365 as well as Rodrigo Herrera-Camus for sharing his machine-readable [CII] data in advance of publication. Critical comments by the referee, Jonathan Braine, led to a substantial improvement of the paper. We made extensive use of the JCMT archive part of the facilities of the Canadian Astronomy Data Centre operated by the National Research Council of Canada with the support of the Canadian Space Agency.

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Appendix A: CO survey results and literature data

A.1. J = 1−0 fluxes versus observing beam size

Table A.1 (only available at CDS) collects J = 1−0 ¹²CO data from this paper and from the literature as a function of observing beam size. Column 2 identifies the telescope used. The beam size is listed in Col. 3; “total” indicates the extrapolated line flux of the whole galaxy taken from the reference cited. Column 4 gives the line flux in units of Jy km s⁻¹. The factors required to convert flux density into temperature are given in a footnote. In all cases where two or more fluxes (including the new measurements from this paper) are available at the same resolution, their average (marked “ave”) is also given in Col. 4. The IRAM fluxes presented in this paper are extracted from profiles with generally better S/Ns and better baselines than those taken from the literature. For the same galaxy and the same aperture, fluxes given in the literature can differ by as much as a factor of two. Given this spread, we did not attempt to identify discrepant fluxes or to eliminate them from the compilation.

The average ratios of the fluxes for beam sizes of 45″ and 22″ measured by us to those collected from the literature are given in Table A.3. They are close to unity and have modest standard deviations. This shows that the results of the present survey and those of previous work are consistent. The data in Table A.1 have been used to construct Fig. 7.

A.2. Comparison CO(2–1) and CO(3–2) survey results

The literature provides far fewer data for the J = 2−1 and J = 3−2 CO transitions. Braine et al. (1993a) used the IRAM telescope with early receivers and backends to measure a large number of galaxies in the J = 2−1 transition, of which 17 are in common with our survey. They convolved small maps to match the J = 2−1 intensities to the J = 1−0 beam. These are compared to the JCMT J = 2−1 measurements in Table A.2. Our results are somewhat higher on average, but the dispersion (standard deviation) is significant. The other large extragalactic IRAM J = 2−1 survey by Albrecht et al. (2007) unfortunately has few objects in common with our survey, as is the case for the JCMT J = 3−2 survey by Yao et al. (2001). Of more interest are the J = 3−2 CO surveys conducted with the 10m Heinrich Hertz Submillimeter Telescope (HHSMT) with a beam size of 22″ (Mauersberger et al. 1999; Dumke et al. 2001; Mao et al. 2010), which have several galaxies in common with our survey. Data from the latter two surveys are also summarized in Table A.2 together with JCMT intensities from maps convolved to 22″. We disregarded the Mauersberger et al. (1999) results because they are superseded by the Mao et al. (2010) survey and suffer from serious calibration and pointing issues. To a lesser extent, these also plague the later two surveys, as inspection of Cols. 6 and 7 shows. This issue is discussed in more detail by Mao et al. (2010). It is therefore not surprising that the ratios of the HHSMT to the JCMT intensities vary either way by factors of up to 2.5. Notwithstanding the relatively large dispersions, the average intensities are quite consistent.

Table A.2.

Galaxy center line intensities from JCMT, IRAM, and HHSMT.

Table A.3.

Statistics of the line intensity comparisons.

Appendix B: Observed and literature isotopologue ratios

Large-scale extragalactic ¹³CO hence ¹²CO/¹³CO isotopologue surveys are lacking in all J transitions. There are, however, a significant number of J = 1−0 ¹³CO small-scale surveys and individual measurements from which isotopologue ratios can be constructed. In Table B.1 we collect these ratios for the J = 1−0 transition as could be found in the literature for the galaxies in the present sample. For Cols. 3 through 10, we determined the ratio of each value to the corresponding value in Col. 2 (i.e., the ratio determined by the measurements in this paper). At the bottom of the table we present for each column the average of these ratios as well as its standard deviation. In all columns the average is close to unity, implying that for resolutions between 22″ and 55″, the isotopologue ratio in the J = 1−0 transition does not vary significantly with beam width. In Table B.2 we summarize the much more sparse data for the J = 2−1 transition, as well as the ratios of the value determined in this paper to the literature value, their average, and their standard deviation. With one exception (J = 1−0 AROKP), the standard deviations are all between 0.22 and 0.31.

Table B.1.

Detailed comparison of J = 1−0 ¹²CO/¹³CO.

Table B.2.

Literature comparison of J = 2−1 ¹²CO/¹³CO

Table B.3.

J = 1−0 ¹²CO/¹³CO comparison statistics.

These standard deviations represent the combined error of the isotopologue ratio derived from the literature and our value. In both transitions, our data are derived from simultaneous dual-frequency measurements, which compared to the literature data have well-defined baselines and relatively high S/Ns so that their contribution to the listed standard deviations is substantially below that of the literature values.

Appendix C: CO radiative transfer modeling

Our modeling assumes that the measured intensities are described by two distinct model gas phases. As noted in Section 5, this is an oversimplification, but two phases is the most that is allowed by the present data without introducing major additional assumptions. We preferred to fit the more diagnostic and more accurate ¹²CO-to-¹³CO ratios, and we assumed that both phases have the same isotopological abundance. The modeling was accomplished by searching a grid of line intensity ratios resulting from the superposition of two model gas clouds with kinetic temperature T_k between 10 K and 150 K, densities n_H₂ between 10² cm⁻³ and 10⁵ cm⁻³, and CO velocity gradients N(CO)/dV between 6 × 10¹⁵ cm⁻²/km s⁻¹ and 1 × 10¹⁸ cm⁻²/km s⁻¹ for ratios matching the observed set, with the relative weights of the two as a free parameter. This parameter space contains the full range of physical conditions, from translucent gas to dense clouds, that the ¹²CO and ¹³CO transitions included in this paper are expected to distinguish. Because we consider only the lower J ¹²CO and ¹³CO transitions, our models lack sensitivity to molecular gas at the very high densities and temperatures that are sampled by the higher CO transitions or by molecular species such as HCN or HCO⁺. Measurements of ¹²CO and ¹³CO at higher J would provide more information on very high-pressure molecular gas but would not add much to the present model results on the cooler and less dense lower-pressure gas of which the bulk of the molecular ISM consists.

Residual degeneracies occur in the majority of galaxies modeled and the observed ratios can usually be fit with a range of comparable solutions. Thus, the model solutions obtained are not unique but instead only constrain values to limited regions of parameter space. This is illustrated in Table C.1 where we show the search results for a few galaxies. Although each of the solutions is acceptable, individual fit parameters sometimes vary considerably. However, these variations are not independent and the final beam-averaged column densities resulting from the combination of the two phases normalized by the observed CO(2–1) line intensity is much less variable. The highest and lowest value differ by a factor of two or less, and dispersions are typically 30% or less.

Table C.1.

Two-phase fitting: example

In Table C.2 we list the model solution that was closest to the observations, even when other model ratios were only marginally different and within the observational error; for examples, again see Table C.1. We rejected solutions in which the denser gas component is also hotter than the more tenuous component because we consider the large pressure imbalances implied by such solutions physically implausible, certainly on the observed kiloparsec scales. Table C.2 gives the model H₂ gas volume densities n_H₂ and kinetic temperatures T_k for each of the two phases as well as their relative contributions f₁:f₂ = f₁₂ to the observed J = 2−1 ¹²CO intensity. Because of the residual degeneracy in the two-phase modeling, kinetic temperatures and gas densities are uncertain by factors of two and three, respectively. Moreover, because our analysis is limited to the lower three ¹²CO and ¹³CO transitions (only 14 galaxies were also observed in J = 4−3 ¹²CO), we cannot meaningfully distinguish temperatures above 100 K or densities above 10⁴ cm⁻³, even though our analysis may formally do so. The results in Table C.2 must therefore be taken as representative rather than individually accurate.

Table C.2.

RADEX model results.

Notwithstanding this reservation, successful fits at either abundance generally suggest modest kinetic temperatures well below 100 K, typically between 20 K and 30 K for the densest phase, usually with densities above 3000 cm⁻³. The less dense phase exhibits greater variety in temperature and density. In most galaxies, two-thirds or more of the J = 2−1 ¹²CO emission come from a single gas phase, often at relatively low kinetic temperatures of 10-30 K. Overall, no more than one-third of the detected J = 2−1 ¹²CO emission is contributed by high-density (n > 3000 cm⁻³) gas. About one-third of the observed galaxies can be modeled with only a difference in density between the two phases, and another one-third with only a difference in temperature. The variation in molecular gas suggested by our modeling even in apparently similar galaxies will at least partly reflect the rapidly changing state of the molecular gas in galactic centers as a function of inflow and outflow rates, as well star formation and AGN activity. Each of these processes may significantly influence the balance of central gas phases in any particular galaxy on timescales much shorter than the Hubble time.

Appendix D: Hydrogen amount

D.1. H column density and mass

Total hydrogen column densities N_H would follow directly from the carbon column densities N_C if the gas phase carbon-to-hydrogen abundance were directly known, which is not the case. Instead, we must infer this abundance from our knowledge of (i) the relative oxygen abundance [O]/[H], (ii) the relative carbon abundance [C]/[O], and (iii), the fraction δ_C of all carbon that is in the gas phase rather than locked up in dust grains.

Optical spectroscopy of disk HII] regions has yielded oxygen abundances that are expressed as metallicities 12+log(O/H) for various galaxies. The results for 34 galaxies from our sample are summarized in Col. 4 of Table D.1. They are taken from the compilations of extragalactic HII region abundances by Vila-Costas & Edmunds (1992) and later publications based on similar methods (references given in the table)⁷. Unfortunately, individual measurements of HII regions in galaxy centers may suffer errors of up to factors of two. Frequently, no suitable HII regions occur in the very center of a galaxy. In these cases, central metallicities are deduced from a linear extrapolation of metallicity gradients to zero radius, but the non-negligible dispersion of individual abundances causes relatively large errors in the slopes of the metallicity gradient. In addition, there is some evidence that these gradients flatten at small radii so that linear extrapolations to zero radius overestimate the central metallicity. The data in Table D.1 define a mean zero-radius abundance 12+log(O/H) = 9.2 ± 0.2, which is three times the solar neighborhood metallicity. However, in view of the uncertainties involved, we consider an intermediate metallicity twice solar to be more reasonable. This implies an elemental ratio [O]/[H] = 10⁻³, which we consider uncertain by a factor of one and a half.

Table D.1.

Physical parameters of molecular gas in galaxy centers.

The relation of carbon to oxygen, the carbon abundance [C]/[O], has been investigated at solar metallicity and below by various authors (e.g., Kobulnicky & Skillman 1998; Garnett et al. 1999, 2004; Esteban et al. 2014; Berg et al. 2016). From solar metallicity downward, [C]/[O] drops linearly proportional to [O]/[H] to a metallicity of about a tenth solar and then flattens. The few available data suggest equal [C]/[H] and [O]/[H] abundances at metallicities just above solar. It is unlikely that the trend observed at subsolar abundances can be extrapolated much farther as this would quickly lead to unrealistically high carbon fractions. Accordingly, we assume equal carbon and oxygen abundances at supersolar metallicities, so that [C]/[H] ≈10⁻³.

A final source of uncertainty is the carbon-depletion factor δ_C. In galaxy disks, as much as two-thirds of all carbon may be tied up in dust particles, rendering it unavailable for the gas phase (see Jenkins 2009). However, turbulence and shocks may cause substantial dust grain erosion in galaxy centers, leading to a higher carbon gas fraction. It is thus reasonable to adopt a depletion factor δ_C = 0.5 ± 0.2. Taken together, these three considerations C suggest that the best result is obtained with an intermediate “nominal” (N) phase ratio N_H/N_C = (2 ± 1)×10³.

In Table D.1 we present the beam-averaged total hydrogen column densities N_H based on this nominal gas-phase carbon ratio (columns headed “N”), which is taken to be the same for all galaxies unless otherwise noted. We present results for isotopologue ratios of 40 and 80, respectively, denoted by superscript. For comparison, we also list column densities (“E”) assuming the individual extrapolated high gas-phase carbon abundance from Col. 5 to apply, as well as column densities (“G”) assuming for all galaxies the same low solar neighborhood gas-phase carbon abundance. The “E” and “G” columns represent the extreme lower and upper limits.

Only the extrapolated case E total hydrogen column densities N_H in Table D.1 are based on individually determined carbon abundances; they have a dispersion of 0.09, corresponding to a factor 1.25. The case N as well as case G values are based on a fixed value for the whole sample. We assume that these are characterized by the same dispersion as case E. The mean values of the beam-averaged case N column densities for isotopological abundances of 40 and 80 differ by a factor of 1.6, thus introducing an error of 27% in the combined data set. Taking into account the uncertainty in the beam-averaged carbon column densities themselves (see Sect. 6.1), we find a combined uncertainty in each derived set of hydrogen column densities slightly over a factor of two.

The final uncertainty lies, however, in our choice of the assumed metallicity-dependent carbon abundance. We chose a nominal value (N) for N_H that is on average a factor of four higher than the extrapolated value (E) and is by definition a factor of three lower than the Galactic value (G) in Col. 3. This is a collective rather than an individual uncertainty, however.

D.2. H₂ column density

The column density of molecular hydrogen N(H₂) follows from that of total hydrogen N_H after subtraction of the neutral hydrogen contribution N(HI). In Col. 9 of Table D.1 we list the HI column densities from the literature at resolutions similar to those of the normalized CO beam used in this paper; almost all were originally obtained with either the Westerbork Synthesis Radio Telescope (NL) or the Very Large Array (USA). Most of the HI column densities are relatively low. The few high values originate in strongly tilted galaxies where the long lines of sight include gas at large radii. Maps show that the distributions of CO and HI in galaxies are anticorrelated: CO usually peaks in the center where HI maps frequently exhibit clear central holes. For galaxies with b/a ≥ 0.6, we took the actual N(HI) values from Col. 9, and for tilted galaxies with b/a < 0.6 as well as those where no HI data were found, we set N(HI) to 0.5 × 10²¹ cm⁻², the average for the galaxies with b/a ≥ 0.6.

Beam-averaged molecular hydrogen column densities assuming nominal abundances are listed in Table D.1 for the two isotopological abundances. The values in Cols. 10 and 13 are corrected for the contribution of HI. We did not separately list uncorrected column densities as these are just the values in Col. 8 divided by two. For most galaxies, the HI contribution is minor, typically 15%. In a recent study, Gerin & Liszt (2017) reached an almost identical conclusion for the inner Milky Way (R < 1.5 kpc) using a completely different line of reasoning.

For carbon abundances as low as those of the Solar Neighbourhood (case G), the HI contribution is in fact negligible (typically ≤5%). However, if the carbon abundances were as high as suggested by the full extrapolation of the abundance gradients (case E), the total hydrogen column densities would be reduced to the levels of HI. This would leave no room for molecular hydrogen, implying improbably high CO emissivities as well as improbably low gas-to-dust ratios. The high case E carbon abundances are therefore ruled out unless almost all carbon is in dust and very little in the gas phase. This is not expected in (dynamically) active environments such as galaxy centers.

The errors in the derived N(H₂) values are almost identical to those discussed in the preceding section: about a factor of 2.5 for for the nominal (N) case and slightly less for the low abundance (G) case. This level of uncertainty is similar to the systematic uncertainty represented by the difference between the two cases.

Finally, we derived the CO-to-H₂ conversion factors X = N(H₂)/I_CO(1−0) for the nominal carbon abundance with and without HI subtraction for each of the two isotopologue abundances considered. The real but relatively small effects of HI correction and isotopologue abundance are illustrated by Cols. 12 and 13 in Table D.1.

All Tables

Table 1.

Galaxy sample.

	Fig. 1. Sample of SEST J = 1−0 CO observations of galaxy centers, showing ¹²CO (histogram) and superposed ¹³CO (continuous lines) profiles; the intensities of the latter have been multiplied by a factor 5. Intensities are in $T_{A}^{}$ $T_{\mathrm{A}}^{}$ (K). Velocities are V(LSR) in km s⁻¹. Galaxies are identified at the top.
In the text

	Fig. 5. Distribution of the J = 2−1/J = 1−0, the J = 3−2/J = 2−1, and the J = 4−3/J = 2−1 ¹²CO intensities. Bottom: J = 3−2 intensities relative to the J = 2−1 ¹²CO intensity as a function of galaxy total FIR luminosity.
In the text

	Fig. 6. Top: distribution of the J = 1−0, the J = 2−1, and the J = 3−2 isotopologue ratios. The histogram fraction representing luminous galaxies (log L_FIR/L_⊙ ≥ 10) is filled. The remainder represent the normal galaxies (log L_FIR/L_⊙ < 10) in the sample. Bottom: J = 2−1 and J = 3−2 isotopologue ratio as a function of the J = 1−0 ratio.
In the text

	Fig. 7. J = 1−0 ¹²CO multi-aperture photometry of galaxies observed with different telescopes. The points on the vertical axis refer to the integrated CO line flux of the entire galaxy. In each panel, their average is marked by a horizontal line. References to the measurements used in these diagrams and in the photometry analysis are given in Appendix A.
In the text

	Fig. 9. Distributions of the sample galaxies as a function of (left) slope α, marking the change in measured J = 1−0 CO flux as a function of increasing observing beam size, (center) the fraction f₂₂ of the extrapolated total galaxy CO flux detected in a 22″ beam, and (right) the extrapolated galaxy CO size as a fraction of the optical size (D₂₅) (see text).
In the text

	Fig. 10. Histogram of the intrinsic (beam-deconvolved) radii of the central concentrations in galaxy CO maps. Three characteristic radii are distinguished (see text).
In the text

	Fig. 12. Left: histogram of J = 2−1 CO intensity ratios in beams of 45″ and 22″. Center: same for J = 2−1 in 22″ and 11″. Right: same for J = 3−2 CO in 22″ and 14″ beams.
In the text

	Fig. 14. Calculated beam-averaged column densities N_H (left) and central gas masses M_H (right) as a function of distance D. Filled circles: Values based on the preferred “nominal” carbon abundance. Open circles: Values bases on extrapolated maximum carbon abundances (see text). All values are based on an assumed isotopological ratio of 40.
In the text

	Fig. 15. Distribution of the total hydrogen column densities (left) and total hydrogen masses (right). See text for sample definition and errors.
In the text

	Fig. 17. From left to right: distributions of the CO-toH₂, [CI]-toH₂, and [CII]-toH₂ conversion factors X(CO), X[CI], and X[CII]. See text for sample definition and errors.
In the text