The correlation between WISE 12 µm emission and molecular gas tracers on sub-kpc scales in nearby star-forming galaxies

The CO data are mainly from two large programs: CO Multi-line Imaging of Nearby Galaxies (COMING; Sorai et al., 2019) that simultaneously presents ${}^{12}\rm CO$, ${}^{13}\rm CO$ and $\rm C^{18}O$ J=1-0 maps observed with 45 m telescope of the Nobeyama Radio Observatory (NRO45M), and the EMPIRE survey and follow-up programs (Cormier et al., 2018) that provide ${}^{12}\rm CO(1-0)$ and ${}^{13}\rm CO(1-0)$ observations with IRAM 30-m telescope. Then we combine the two survey with the data of these three CO isotopologues in Tan et al. (2011); Nakajima et al. (2018), and the our observations (introduced in Appendix A) using the Purple Mountain Observatory (PMO) 13.7-m millimeter telescope located in Delingha, China. For the galaxies without ${}^{13}\rm CO(1-0)$ observations, we also use ${}^{12}\rm CO(1-0)$ data from the Nobeyama CO mapping survey (Kuno et al., 2007) to complement. In total, we summarize 22 galaxies in ${}^{13}\rm CO(1-0)$ analysis, and 24 galaxies in ${}^{12}\rm CO(1-0)$ sample as listed in Table 1.

For the CO datacubes, we reproject them with $1/\sqrt{2}\times$ beam size to avoid over-sampling, and then extract the spectra for each new position. Based on these spectra that are converted in main-beam temperature ($T_{\rm mb}$) scales, after determining the velocity range of the line emission (from ${}^{12}\rm CO$ spectrum), we measure the new velocity-integrated CO line intensity in the velocity range, $I_{\rm CO}\equiv\int{T_{\rm mb}}\;dv$, and the new rms noise over the full spectrum except the emission line and edge channels. Then we estimate the uncertainty of the integrated intensity using the formula in Gao (1996):

where $f\equiv\Delta v_{\rm FWZI}/\delta v$, $\Delta v_{\rm FWZI}$ is the full width at zero intensity (FWZI) of the emission feature, $\delta v$ is the velocity channel width, and $W$ is the entire velocity coverage of the spectrum. We treat the positions with a signal to noise ratio $S/N\geq$ 3 as detection, and give an (valuable) upper limit of 3 times the uncertainty for the other ones (tentative detection $S/N\geq$ 1.5). Finally, we derived the luminosities of ${}^{12}\rm CO$, ${}^{13}\rm CO$ and $\rm C^{18}O$, and the molecular gas column density from the velocity-integrated CO line intensity $I_{\rm CO}$ using an empirical $X_{\rm CO}=2\times 10^{20}$ cm^-2 (K km s${}^{-1})^{-1}$ (Nishiyama et al., 2001) corresponding to the galactic conversion factor $\alpha_{\rm CO}=3.2$ M_⊙(K km s${}^{-1}{\rm pc^{2}})^{-1}$.

Assuming that the molecular cloud is under local thermal equilibrium (LTE) conditions and ${}^{13}\rm CO$ has the same excitation temperature as ${}^{12}\rm CO$, for the pixels with both ${}^{12}\rm CO$ and ${}^{13}\rm CO$ detections, we can calculate the ${}^{13}\rm CO$ optical depth averaged in the beam from

where $T_{\rm mb}$ is equivalent to the ${T^{*}_{\rm R}}$ (radiation temperature) in the notation of Kutner & Ulich (1981). Then we use the definition of $N{\rm(H_{2})(^{13}CO)}$ in Wilson et al. (2009), which assume the ${}^{13}\rm CO$ abundance ${\rm[^{13}CO]/[H_{2}]}$ is $8\times\ 10^{-5}/60$ (Frerking et al., 1982), and equate the column densities derived from both ${}^{12}\rm CO$ and ${}^{13}\rm CO$, we can estimate the kinetic temperature of the gas, $T_{\rm K}$.

We downloaded the 2 deg $\times$ 2 deg assembled WISE Atlas tiles (image and uncertainty files) for each source from the NASA/IPAC Infrared Science Archive (WISE Team, 2020). Based on them, we produced the background map using the SExtractor package (Bertin & Arnouts, 1996). After masking some saturated pixels (mainly for M82 and NGC1068) and subtracting the background, we smoothed the flux images using the convolution kernels provided by Aniano et al. (2011) to mach the beam size of CO or other gas traces. Combining the smoothed flux images and uncertainty images, we can get the instrumental uncertainty, and add it with the zero-point uncertainty in quadrature to get the total uncertainty of 12 µm flux in corresponding pixel. The recovered core of saturation were not used due to the large uncertainty (20% to 30%) as measured by Jarrett et al. (2019), which lead to, in this work, missing W3 data corresponding to the CO data of NGC1068 provided by Nakajima et al. (2018). We can convert the maps in original unit (digital numbers, DN) to the luminosity maps in units of $L_{\odot}$, following the formula:

which is computed using the W3 zero-point magnitude MAGZP = 18.0 mag, the isophotal flux density $S_{0}$ = 31.674 Jy, and the bandwidth $\Delta\nu=1.1327\times 10^{13}$ Hz (Jarrett et al., 2011). Then scale the value by a factor of 1.133 $\times$ (beam size/pixel size)² to get the 12 µm luminosity L_12µm correspond to gas traces L_gas. Some equations and parameters in calculating the monochromatic luminosity and uncertainty at WISE 12 µm band are described in detail in Chown et al. (2021).

The dense gas data are from the MALATANG survey, which aims to construct a largest and deepest HCN(4-3) and HCO⁺(4-3) maps with FWHM angular resolution about $14^{\prime\prime}$ in a flux-limited sample of nearby IR-bright galaxies. Tan et al. (2018) present the first results of six galaxies, NGC253, NGC1068, IC342, M82, M83, and NGC6946, that were mapped in their central $2^{\prime}\times 2^{\prime}$ regions using a 3 $\times$ 3 jiggle mode (i.e. rapidly step the secondary mirror to observe a grid-pattern of 9 points) with grid spacing of 10 arcsec. Our analysis about dense gas tracers is limited to the maps of these six galaxies.

After producing the pipeline-processed spectra of HCN(4-3) and HCO⁺(4-3) using STARLINK software package (Currie et al., 2014) and the ORAC-DR pipeline (Jenness et al., 2015), we flag some sub-scans with unstable baselines or abnormal noise levels as bad quality using the CLASS package (Pety, 2005), and then we can get average spectrum for each position. We align the ancillary CO data (Kuno et al., 2007; Wilson et al., 2012) with the datacube to determine the velocity range over which CO emission line is significant, and then fit and subtract a first-order baseline using the rest channels. Based on the final spectra, we also use CO-emitting velocity ranges to measure the integrated-intensity of the dense gas tracers. Then we calculate the luminosities of dense gas tracers for all detected positions following the formula in Solomon et al. (1997):

One notable point is that this work include all pixels with $S/N\geq$ 3 even in the outer disks (i.e. $\geq 1^{\prime}$), which (13 pixels) are not shown in the table provided by Tan et al. (2018).

Figure 1 shows the correlation of the ¹²CO(1-0) luminosities, $L_{\rm{}^{12}CO(1-0)}$ and mid-infrared luminosities measured from WISE images in the 12 µm band at small scales of $\sim$ 0.2-1.3 kpc for the 24 MALATANG galaxies (indicated with different symbols and colors). For some galaxies with available ${}^{12}\rm CO$ data from more than one reference (as shown in Table 1), we just use data from the one that can provide most pixel to compute the pixel measurements of log$L_{\rm{}^{12}CO}$(y-axis). Distributions of all detections are shown as a gray-scale background in both panels. The average uncertainty of ${\rm{}^{12}CO(1-0)}$ luminosities measured using Eq. 1 in all pixels is about 0.045 dex, and the one of 12 µm is 0.021 dex (which includes both the photometric uncertainty and the uncertainty of the magnitude zero-point measured by Jarrett et al., 2011), displayed as the error bar in the lower right corner.

Then we use the IDL script LinMix ¹¹1Available from the NASA IDL Astronomy User’s Library https://idlastro.gsfc.nasa.gov/ftp/pro/math/linmix_err.pro (Kelly, 2007) which is a Bayesian linear regression taking into account upper limits and uncertainties in both the x- and y-axes. Fitting $L_{\rm{}^{12}CO}$ in ${\rm K\;km\;s}^{-1}\;{\rm pc^{2}}$ versus $L_{12\micron}$ in $L_{\odot}$ yields the best-fit result

We can see the best fitting of the entire MALATANG sample is almost the same as the result (the green line) of global galaxies performed by Gao et al. (2019), and show slightly larger total/observed scatter ($\sigma_{\rm tot}=0.27$) but smaller intrinsic scatter ($\sigma_{\rm int}=0.07$). So based on the MALATANG sample, we can extend this relation to low-luminosity end (down to $10^{5}\;[{\rm K\;km\;s}^{-1}\;{\rm pc^{2}}]$ and $10^{5}$ L_⊙ in both $L_{\rm{}^{12}CO(1-0)}$ and $L_{12\micron}$), though the relation may be are various from galaxy to galaxy as indicated by Chown et al. (2021).

Analogously, we plot the similar relation between ${}^{13}\rm CO$ and 12 µm luminosities based on $\sim$ 1000 pixels (with ${}^{13}\rm CO$ detections or valuable upper limits) in the left panel of Figure 2, and get the best-fitting linear relation

For comparison, we also show the relation of ${}^{12}\rm CO$ based on the same pixels in the right panel. The points with same colour and symbol in these two panel belong same galaxy as in Figure 1, and the large filled symbols in Figure 2 are 6 galaxies observed in ${}^{12}\rm CO$ and ${}^{13}\rm CO$ simultaneously by us with PMO and provided by Nakajima et al. (2018). We can see there are strong positive correlations in both panels, though a certain fraction of ${}^{13}\rm CO$ data are upper limits. But the total and intrinsic scatter ($\sigma_{\rm tot}=0.21$ and $\sigma_{\rm int}=0.05$) of ${}^{13}\rm CO$ vs. 12 µm relation is larger than the one of ${}^{12}\rm CO$ ($\sigma_{\rm tot}=0.18$ and $\sigma_{\rm int}=0.02$), meanwhile the Spearman’s correlation coefficient ($r$) also is a little smaller (0.90 compared to 0.94). The relations between ${}^{13}\rm CO$ and 12 µm are more remarkably varied in different galaxies (discussed in Sec 4.2). These results suggest the 12 µm luminosities are more tightly correlated with the ${}^{12}\rm CO$ luminosities than with ${}^{13}\rm CO$ luminosities over the same set of pixels. In Sec 3.3, we will try to figure out the parameter that could contribute the scatter of the relation to confirm the application of these estimator.

Just using ${}^{13}\rm CO$ pixel sample to perform fitting in the right panel of Figure 2, besides with decreased scatter and increased correlation coefficient, we can find the new best-fitted ${}^{12}\rm CO$ vs. 12 µm relation yields a little higher ${}^{12}\rm CO$ luminosity and a shallower slope than the global one (the green line). This change can be simply and naturally caused by selection effect. There are two kinds of effect: the missing position with undetected/faint ${}^{13}\rm CO$ emission are more likely to have lower ${}^{12}\rm CO$ luminosity, which lead to the relation be higher, and the missing fraction could increase with reduced 12 µm luminosity (correlated with CO luminosity), which make the slope lower and difference larger and more significant towards the low end. In the discussion that follows (Sec 4.2), we will further analyse the effect of sample selection.

A main aim of this work is to figure out the origin of the correlation between 12 µm and CO (molecular gas) emission. It is presumed that the correlation is like the KS relation, in other words, the 12 µm emission (maybe dominated by 11.3 µm PAH bands) is an accurate, quantitative measure of SFR (Xie & Ho, 2019; Cluver et al., 2017), so the correlation can be simply understood as the formation of stars from molecular gas. If this conjecture is right, the spatially resolved relation between the masses of dense molecular clumps (traced by HCN(4-3) and/or HCO⁺(4-3) luminosity) and stars formed (traced by 12µm luminosity) should be more linear and tight than the correlation with total molecular mass (traced by CO luminosity).

In Figure 3, we show the ${L^{{}^{\prime}}_{\rm dense}}$-${L_{\mbox{12\micron}}}$ relations, which are based on all HCN(4-3) or HCO⁺(4-3) detected regions/pixels in MALATANG maps (different galaxy is indicated using different symbol and colour). The rightest panel shows the 12 µm vs. ${}^{12}\rm CO(1-0)$ relation over the same set of HCN(4-3) detected pixels. In this panel, we derive the ${}^{12}\rm CO(1-0)$ luminosity of M82 from COMING survey (Sorai et al., 2019), and from the spatially resolved spectra provided by Kuno et al. (2007) for other galaxies, which are observed using NRO45M with similar angular resolution.

Then we can see the ${L^{{}^{\prime}}_{\rm dense}}$-${L_{\mbox{12\micron}}}$ relations is nonlinear with larger scatter of 0.29 and 0.33 dex, compared with the $L_{\rm{}^{12}CO(1-0)}$-$L_{\mbox{12\micron}}$ relation. We also compute the best linear least-squares fitting (logarithmic) and intrinsic scatter in the three scaling relations, with parameters shown in Table 2. The parameters also illustrate that $L_{\rm{}^{12}CO}$ is more tightly correlated with $L_{\mbox{12\micron}}$, with a very small intrinsic scatter of 0.06 dex, than dense gas (0.09 or 0.11 dex). Meanwhile, we also do the fits with ${{\rm log}L_{\mbox{12\micron}}}$ on the y-axis, the total (and intrinsic) scatters are 0.53, 0.50 and 0.34 (0.29, 0.25 and 0.12) dex for HCN(4-3), HCO⁺(4-3) and ${}^{12}\rm CO$, which are much larger than the ones of correlations shown in Figure 3. Based on these results, we conclude that the relation between 12 µm and ${}^{12}\rm CO(1-0)$ is unlikely to be due to star-formation law, though this sample is small.

We also notice there is one AGN (NGC1068) in these six nearby galaxies, whose data points are indicated as the red crosses in the panels. If we remove this galaxy, the slope of $L_{\rm{}^{12}CO(1-0)}$-$L_{\mbox{12\micron}}$ will be much steeper (about 0.9) and in general consistent with the one in Figure 1, while the slope and correlation coefficients of ${L^{{}^{\prime}}_{\rm HCN(4-3)}}$-${L_{\mbox{12\micron}}}$ and ${L^{{}^{\prime}}_{\rm HCO^{+}(4-3)}}$-${L_{\mbox{12\micron}}}$ relations become much lower and keep constant. So we speculate the relation of $L_{\rm{}^{12}CO(1-0)}$-$L_{\mbox{12\micron}}$ could be further complicated by the possible effects of AGN, which modify the expected bias in such small sample (detail discussion about AGN in Sec 4.3).

Based in these analysis, we conclude that the WISE image can be used to derived molecular gas mass more directly and simply than using some kinds of star formation law, so the estimated molecular gas mass can be used widely, e.g. in studying the star formation process.

About the correlations between $L_{\rm gas}$ and $L_{\mbox{12\micron}}$ shown above, the scatter and some systematic deviations are significant different. We hereby examine the scatters in these relations as a function of the (spatially resolved) physical conditions of molecular gas.

We use the best-fitting relations (Eq. 5 and Eq. 6) between $L_{{}^{12}\rm CO}$ or $L_{{}^{13}\rm CO}$ and $L_{12\micron}$ to predict CO luminosity from W3 luminosity, then we compute residuals about these fits, e.g. for the ${}^{12}\rm CO$-12 $\mu$m relation,

and likewise for ${}^{13}\rm CO$ residuals. Based on more than one thousand of pixels with both ${}^{12}\rm CO$ and ${}^{13}\rm CO$ detections, we plot the residuals as a function of ${}^{13}\rm CO$ optical depth and gas kinetic temperature in the left two panels of Figure 4 and Figure 5. In each panel, to show the trend, we divide the galaxies into four subsamples (with the same number of galaxies in each) according to the parameter considered, and compute the median and scatter for the galaxies in different bins, which is plotted as the large red triangles and error bars. Then we can see the ${}^{12}\rm CO$ luminosity residuals show nearly no dependence on the optical depth or gas kinetic temperature with the Spearman’s correlation coefficient $r$ of -0.1 and 0.12 respectively, while the residuals of $L_{{}^{13}\rm CO}$ is enhanced when the optical depth increase and gas kinetic temperature decrease ($r\sim$ 0.6 and -0.6). These can lend support to a guess that the relation between WISE 12µm band (PAH) and ${}^{12}\rm CO$ is more physically fundamental. And the relation between 12µm and ${}^{13}\rm CO$ emission, most if not all, is the consequence of relation between 12µm and ${}^{12}\rm CO$ with the effect of optical depth and temperature.

In Figure 4, we use the linear fit shown in Figure 1 to predict the ${}^{12}\rm CO$ luminosity, because it’s based on a more complete sample of pixel and more consistent with the global one provided in Gao et al. (2019), compared with the one in the right panel of Figure 2. Then we also check the correlations between the ${}^{12}\rm CO$ luminosity residuals and the spatially resolved parameters using the second fitting, and find no correlation (${\lvert}r{\rvert}\leq 0.1$), which support the different dependence between ${\Delta}$ log($L_{{}^{12}\rm CO})$ and ${\Delta}$ log($L_{{}^{13}\rm CO})$ on these parameters is real instead of sample selection bias.

We find ${\Delta}$ log($L_{{}^{12}\rm CO})$ tend to be negative in the region with very low molecular gas mass surface density shown in the right panel of Figure 4, which also explain a lot of upper-limits are not above the best-fittng line in Figure 1. The Spearman’s correlation coefficient ($r$) is 0.34 for all pixels, which is mainly contributed by the regions with $\Sigma_{\rm mol}<10{\rm M_{\odot}pc^{-2}}$, and this trend disappears when we just include the pixels with $\Sigma_{\rm mol}>20{\rm M_{\odot}pc^{-2}}$ ($r$ = 0.1 for 2623 detected pixels)).

But, we can’t provide the exact range of $\Sigma_{\rm mol}$ value where the ${}^{12}\rm CO$ luminosity would be significantly underestimate using 12 µm band, because the CO-to-H₂ conversion factor ($\alpha_{\rm CO}$ or $X_{\rm CO}$) can be largely different from the value we assumed in Section 2.1, which is dependent on the physical conditions in the molecular clouds, $X_{\rm CO}\propto n^{0.5}T_{\rm K}^{-1}$, even ignoring metallicity effects and assuming CO emission from the gravitationally bound and virialized cloud cores (Maloney & Black, 1988; Bolatto et al., 2013). Non-correlation between ${}^{12}\rm CO$ luminosity residual on mass surface density at the mid-to-high end is consistent with independence of ${}^{13}\rm CO$ luminosity residual shown in the right panel of Figure 5, because the regions with ${}^{13}\rm CO$ measurements are more likely with high molecular gas mass surface density.

So we guess the correlation between ${}^{12}\rm CO$ and 12 µm emission should be sustained as long as the molecular gas mass density is not too low, in other words, star formation could take place, where the 12 µm emission is dominated by the PAH formed in molecular clouds. We also examine the correlation between the offset and the dense gas fraction, which is indicated by the luminosity ratios of HCN(4-3) and HCO⁺(4-3) over ${}^{12}\rm CO$. We find that the offset appears to be nearly independent (with correlation coefficient less than 0.2), because there should be a large amount of molecular gas existing if the dense gas can be detected.

All these results support that the ${}^{12}\rm CO$ estimators based on 12 µm emission is usable in the regions with $\Sigma_{\rm mol}$ higher than a threshold.

Our findings show that there is a tight and strong correlation between ${}^{12}\rm CO$ and 12 µm band emission at scales of about 1 kpc or less. By combining with the global and resolved studies presented by Jiang et al. (2015); Gao et al. (2019); Chown et al. (2021), this linear relation holds over 5 orders of magnitude, and is highly applicable to different physical scales, from spatially resolved regions to whole galaxies. But the physical origin of this relation is still debated.

Our comparisons of the correlations between 12 $\mu$m emission and various molecular species may suggests that emission in WISE 12 µm band is mainly from the cold dust (mostly PAHs).

On the one hand, in molecular clouds, PAH molecules and ${}^{12}\rm CO$ are spatially well-mixed, and are excited/destroyed under similar conditions. Dense gas is found deeper inside molecular clouds, while PAHs are found primarily on the surfaces of clouds, leading to a weaker association between 12 $\mu$m emission and dense gas. Considering the ${}^{13}\rm CO$ is mostly optically thin (with median optical depth about 0.09), ${}^{13}\rm CO$ emission originates in denser cores than ${}^{12}\rm CO$ as suggested in Mao et al. (2000), which lead to the larger scatter between ${}^{13}\rm CO$ and 12 $\mu$m emission than that between ${}^{12}\rm CO$ and 12 $\mu$m emission. The slope of ${\Delta}$ log($L_{{}^{13}\rm CO})$-log(T_k) is also consistent with the power-law index (-1.4 ) of ${}^{13}\rm CO$/${}^{12}\rm CO$ intensity ratio as a function of gas kinetic temperature for high densities presented by homogeneous cloud model in Paglione et al. (2001). Similarly we also find the scatter of correlation between $\rm C^{18}O$ and 12 µm is significant larger than that of ${}^{12}\rm CO$ over the same set of pixels (37 pixels in 13 galaxies).

On the other hand, the PAH emission can be detected in wider varieties of astrophysical regions, even in early-type galaxies, where the PAH spectra are dominated by the 11.3 and 12.7 µm band (Kaneda et al., 2005, 2008; Vega et al., 2010). So the PAH emission from the atomic interstellar medium (ISM) not traced by CO can explains why the observed ${}^{12}\rm CO$ -12 µm correlation breaks down in more diffuse gas, which are also shown as the pixels with very low molecular gas mass surface density in Figure 4. This kind of regions or galaxies with few molecular clouds and/or very low molecular gas mass surface density will also be analysed in our next work based on a large sample of nearby early-type galaxy.

Overall, it is plausible our finding support that the 11.3 µm PAH emission is highly correlated with ${}^{12}\rm CO$ emission in both position and flux, which is the main reason why WISE 12 µm band emission can be used to estimate the molecular gas. Future work could confirm this with a study on smaller physical scales to reveal more detail about PAH formation/destruction and emission mechanisms in molecular clouds. In the future, we will do some analysis based on some observations in Milky Way (e.g. Milky Way Imaging Scroll Painting; Su et al., 2019), and focus on 11.3 µm PAH that were obtained with the Spitzer Space Telescope or with the James Webb Space Telescope.

To explain the variation in 12 µm-CO relations, we explore the source of variation and offset from the main relations from galaxy to galaxy, beside the study presented above about the spatially resolved physical conditions.

For that, we first do the similar fitting as Figure 2 for all galaxies with a minimum 12 measured pixels (detections or valuable upper limits), to calculate the Spearman rank correlation coefficient $r$, slope $k$ and intercept $b$ between log$L_{\rm CO}$ and log$L_{12\micron}$ for each galaxy. Figure 6 shows the histograms of these fitting parameters for different isotopologues.

We discuss the effect of sample selection on the best-fit relations (logarithmic) between 12 µm and $\rm CO(1-0)$ luminosity. In the left panel of Figure 6, we can find about the 12 µm vs. ${}^{12}\rm CO$ relation, most galaxies are with a very high Spearman rank correlation coefficient regardless of how to select the pixels used in fitting ( 0.85 $\pm$ 0.15 and 0.90 $\pm$ 0.19 for all ${}^{12}\rm CO$ measured pixels and ${}^{13}\rm CO$ measured pixels). Then we check the details of 7 galaxies with low value ($r\leq 0.8$), and find the reasons for poor correlations are few CO-detected pixels in such galaxies, or small parameter range in the pixels that are detected (e.g. the $L_{12\micron}$ region covered by the pixels in most such galaxies is less than 1 dex). So it is encouraging to see that the strong correlations between 12 µm and ${}^{12}\rm CO$ luminosity exist in most galaxies with enough statistical data.

In contrast, the distribution of Spearman rank correlation coefficient between log$L_{12\micron}$ and log$L_{{}^{13}\rm CO(1-0)}$ luminosity are much lower (0.69 $\pm$ 0.2). Similarly, the scatter of slope ($k$) and intercept ($c$) ($\sigma_{k}=0.34$ and $\sigma_{c}=2.6$) is significantly larger than the one between 12 µm and ${}^{12}\rm CO$ ($\sigma_{k}=0.15$ and $\sigma_{c}=1.1$ dex for ${}^{13}\rm CO$ measured pixels) as expected.

Meanwhile we find: due to the effect of measurement uncertainty, the slope of best-fitting between 12 µm and ${}^{12}\rm CO$ luminosity tend to be smaller, and the intercept is higher, for galaxies without enough statistical CO pixels, which is similar as the different distribution of $k$ and $c$ fitted using ${}^{12}\rm CO$ and ${}^{13}\rm CO$ pixel samples. The selection effect combining the AGN contribution may explain the small slope in the right panel of Figure 3. And we could speculate that the intrinsic 12 µm-${}^{12}\rm CO$ relation (without selection bias) should be nearly linear (with a slope closer to unity).

To assess the applicability of the best-fitting linear relation in different galaxies, we compute the median of the residuals about them for each galaxy. In the panel (d) of Figure 6, we show the histograms of median ${\Delta}$ log($L_{\rm CO})$: the histogram with grey lines is computed using Eq. 5 based on all ${}^{12}\rm CO$ detected pixels in 24 galaxies, and the red and blue one is computed using Eq. 6 and Eq. 5 respectively, based on all ${}^{13}\rm CO$ detected pixels in 21 galaxies. Overall, both estimators can work well for most galaxies, though the scatter is large in the distribution of ${\Delta}$ log($L_{{}^{13}\rm CO})$. Based on the grey histogram, we can see the most of median ${}^{12}\rm CO$ offset is between -0.3 to -0.3, except only 3 galaxies with ${\Delta}$ log($L_{\rm{}^{12}CO}/[{\rm K\;km\;s}^{-1}\;{\rm pc^{2}}])<-0.3$. Two of these three outliers are AGNs: NGC1068 and NGC4736 (-0.49 dex and -0.42 dex),about which we will discuss in Sec 4.3). So the estimator (Eq. 5) can predict reliable $L_{\rm{}^{12}CO}$ for all MALATANG galaxies, even for the incomplete ${}^{13}\rm CO$ pixel sample (the blue histogram).

The AGN-dominated (Seyfert) galaxies show a larger scatter in global $L_{\rm CO(1-0)}$–$L_{\mbox{12\micron}}$ relation (Gao et al., 2019), where PAHs could survive in the highly ionized medium and show surprising abundant features (e.g. Roche et al., 1991; Tommasin et al., 2010). On the one hand, PAH features at 6.2, 7.7, and 8.6 µm are often substantially suppressed in AGN compared to star-forming galaxies (e.g. Valiante et al., 2007; Sajina et al., 2008; Diamond-Stanic & Rieke, 2010), which is interpreted as the selective destruction of small PAHs molecules by the hard radiation field of AGNs (Smith et al., 2007). On the other hand, the 11.3 µm PAH features can be excited by photons or the radiation field from AGNs itself or circumnuclear star formation (Jensen et al., 2017). Li (2020) speculates that the 11.3/7.7 ratios observed in a number of Seyferts could be produced by catacondensed PAHs with an open, irregular structure (be able to have more H atoms on a per C atom basis). And all these features may appear weaker, due to increased dilution from the AGN continuum (Alonso-Herrero et al., 2014) or IR (especially in short wavelengths ) emission from dust heated by the AGN (Shipley et al., 2013).

There are a considerable amount ($\geq$ 50, and about 1000 in total) of measured ${}^{12}\rm CO$ pixels for 5 out of 8 AGNs in MALATANG sample, which can be used to determine whether and how AGN affect the 12 µm vs. ${}^{12}\rm CO$ relation. There are only 3 AGNs in ${}^{13}\rm CO$ sample, and no one with the number of detected pixels $\geq$ 30, so we didn’t provide the analysis on them.

In the Figure 6, we can see that all AGNs also show high correlation coefficient between 12 µm and ${}^{12}\rm CO$ emission, the best fit slope and intercept of AGNs tend to locate at the two end, and the ${}^{12}\rm CO$ estimator (Eq. 5) still work well basically in the AGN sample. It seems these AGNs would not have significant effect on the derived primary $L_{\rm CO(1-0)}-L_{\mathrm{12\mu m}}$ relationship shown in Figure 1. A linear least-squares fit to the data points excluding the two AGN outliers (NGC1068 and NGC4736) or all five AGNs yields:

which is closer to the relation of global galaxies (Gao et al., 2019).

In the two AGN hosts with significant ${}^{12}\rm CO$ residuals, the pixels in these galaxies are well-fit with a linear relation that is approximately parallel to the best-fit relation in Eq. 5, which covers an extensive radius from the center: 90^′′ ($\sim$6 $r_{K,e}$, corresponding to a physical scale of $\sim$ 4.4 kpc) for NGC1068, 60^′′ (about 1.8 $r_{K,e}$, corresponding to $\sim$ 1.3 kpc) for NGC4736, and beyond those regions the uncertainty becomes larger due to few detected pixels. The near-infrared $K_{s}$ band half-light "effective" radii $r_{K,e}$ is taken from the 2MASS Large Galaxy Atlas (Jarrett et al., 2003; Skrutskie et al., 2003). The larger 12 µm luminosities at fixed CO luminosity in AGN hosts compared to galaxies without AGN may be explained by dust-reprocessed emission peaking between rest frame 15 and 60 µm with respect to star-formation dominated galaxies as shown in Mullaney et al. (2011); Salvestrini et al. (2022), and/or by a prevalence of dust grains that are heated by old stars in weak AGN (predominantly associated with massive, early-type galaxies), which may be contributed by circumstellar dust from AGB stars (Donoso et al., 2012; Villaume et al., 2015). So the different behaviors in 12 µm-${}^{12}\rm CO$ relation may provide new constraints on the physical processes behind how AGN affect the surrounding ISM and subsequent galaxy evolution.