The CO-to-H$_2$ Conversion Factor of Galactic Giant Molecular Clouds using CO isotopologues: High-resolution $X_{\rm CO}$ maps

We investigated the correlation between intensities of the $^{12}$CO and $^{13}$CO ($J=1$-0) lines toward the Galactic giant molecular clouds (GMCs) W51A, W33, N35-N36 complex, W49A, M17SW, G12.02-00.03, W43, and M16 using the FUGIN (FOREST Unbiased Galactic plane Imaging survey with the Nobeyama 45-m telescope) CO line data. All the GMCs show intensity saturation in the $^{12}$CO line when the brightness temperature of $^{13}$CO is higher than a threshold temperature of about $\sim 5$ K. We obtained high-resolution ($\sim 20"$) distribution maps of the $X_{\rm CO}$ factor ($X_{\rm CO, iso}$) in individual GMCs using correlation diagrams of the CO isotopologues. It is shown that $X_{\rm CO, iso}$ is variable in each GMC within the range of $X_{\rm CO, iso} \sim (0.9 {\rm -} 5) \times 10^{20}$ cm$^{-2}$ (K km s$^{-1})^{-1}$. Despite the variability in the GMCs, the average value among the GMCs is found to be nearly constant at $X_{\rm CO, iso} = (2.17 \pm 0.27) \times 10^{20}$ cm$^{-2}$ (K km s$^{-1})^{-1}$, which is consistent with that from previous studies in the Milky Way.


INTRODUCTION
Giant molecular clouds (GMC) mainly consist of hydrogen molecules (H 2 ) and are the sites of star formation in galaxies (Blitz et al. 2007;Fukui & Kawamura 2010;Dobbs et al. 2014;Chevance et al. 2023).The hydrogen molecule is difficult to observe directly because it has no an electric dipole moment for its homonuclear diatomic nature.Therefore, we often observe carbon monoxide (CO) with rotational transitions at millimeter wavelengths (Heyer & Dame 2015) and convert the intensity to the H 2 column density using the CO-to-H 2 conversion factor (hereafter  CO ), which is given by where 12 CO and  H 2 are the 12 CO  =1-0 integrated intensity and column density of hydrogen molecules, respectively (Bolatto et al. 2013).
Recently, Sofue & Kohno (2020)(hereafter paper I) proposed a new method to estimate the molecular cloud mass considering the variability of the  CO factor from the correlation between 12 CO and 13 CO.Paper I analyzed only two Galactic GMCs of M16 (Sofue 2020b;Nishimura et al. 2021) and W43 (Sofue et al. 2019;Sofue 2021;Kohno et al. 2021), while the spatial distributions of  CO inside the GMCs have not yet been discussed.
This paper is structured as follows: section 2 introduces the FU-GIN (FOREST Unbiased Galactic plane Imaging survey with the Nobeyama 45-m telescope) data; section 3 presents the methods of analysis; in section 4, we demonstrate the results; in section 5, we discuss the variability of the  CO factor, and in section 6, we summarize the results.

DATA
We utilized the 12 CO and 13 CO  =1-0 line data obtained with the Nobeyama 45 m telescope (FUGIN: Umemoto et al. 2017;Torii et al. 2019;Fujita et al. 2023).The rest frequencies of 12 CO and 13 CO  =1-0 are 115.271GHz and 110.201GHz, respectively.The front end was the FOur beam REceiver System on the 45-m Telescope (FOREST: Minamidani et al. 2016;Nakajima et al. 2019), which is the four-beam, side-band separating (2SB), and dual-polarization superconductor-insulator-superconductor (SIS) receiver.The observations were performed in the on-the-fly mapping mode (Sawada et al. 2008).The back-end system was used an FX-type spectrometer named SAM 45 (Kuno et al. 2011;Kamazaki et al. 2012).The half-power beam width (HPBW) of the 45 m telescope is 14 ′′ and 15 ′′ at 115 GHz and 110 GHz, respectively.The data are gridded to 8.5 ′′ and 0.65 km s −1 for the spatial and velocity space.The final 3D cube used in this study has a voxel size of (, , ) = (8.5 ′′ , 8.5 ′′ , 0.65 km s −1 ).The effective resolution convolved with a Bessel × Gaussian function is 20 ′′ and 21 ′′ for 12 CO and 13 CO  =1-0, respectively.We subtracted the baselines using a first-order polynomial function from each spectrum.The baseline ranges are adopted from −200 km s −1 to −50 km s −1 and 200 km s −1 to 350 km s −1 for the first Galactic quadrant with the FUGIN CO survey (Umemoto et al. 2017).The data are calibrated from the antenna temperature ( *  ) to the main-beam temperature ( MB ) by measuring the mainbeam efficiency of 0.43 for 12 CO, 0.45 for 13 CO, and C 18 O using standard calibration sources.Detailed information on the FUGIN project is summarized in the project overview paper by Umemoto et al. (2017).Cube data calibrated to the main-beam temperature are available at the Japanese Virtual Observatory (JVO).We also used the 12 CO ( =3-2) High-Resolution Survey (COHRS) archival data of the Galactic Plane obtained by the James Clark Maxwell Telescope (Dempsey et al. 2013;Park et al. 2023).The cube data are converted  *  to the  MB scale using a main-beam efficiency of 0.61 (Park et al. 2023).The spatial and velocity resolutions are 16.6 ′′ and 0.635 km s −1 , respectively.Comparing the FUGIN CO survey data, we smoothed them to 20 ′′ with the same spatial resolution.The 12 CO  =3-2 data are re-gridded to the same voxel size as the FUGIN data using CASA (CASA Team et al. 2022).Table 1 presents the root-mean-square noise level of each GMC at the  MB scale.

METHODS
We derived the H 2 column density per velocity channel using the local thermal equilibrium (LTE) method and the  CO factor, as described in Paper I and Pineda et al. (2008).The brightness temperature ( B ) of the CO line intensity with the excitation temperature ( ex ) and optical depth () is given by where  bg = 2.725 K is the temperature of the cosmic microwave background radiation. 0 = ℎ/ is the Planck temperature with ℎ, , and  being the Planck constant, rest frequency, and Boltzmann constant, respectively.If we assume that the 12 CO line is optically thick, the excitation temperature is given by where  B ( 12 CO) max and  115 0 = 5.53194 correspond to the 12 CO peak intensity and the Planck temperature at the rest frequency of 12 CO  =1-0, respectively.We assume that  ex is equal in the 12 CO and 13 CO line emissions, and express the optical depth as follows: where  B ( 13 CO) max and  110 0 = 5.28864 represent the 13 CO peak intensity and the Planck temperature at the rest frequency of 13 CO  =1-0, respectively (Pineda et al. 2008).According to Wilson et al. (2009), the 13 CO column density is given by where 13 CO is the 13 CO integrated intensity.Then, we converted 13 CO to the H 2 column density using the abundance ratio of H 2 to 13 CO molecules given by Here, 13 CO is adopted as (5.0±2.5)×10 5 (Dickman 1978) following Paper I.
We then calculated the H 2 column density per velocity channel,  The points are plotted of  B ( 13 CO) > 3.The 1 noise level of each GMC is presented in Table 1.
which is defined as the spectral column density, using the  CO and LTE method from the intensity of 12 CO and 13 CO, as described in Paper I. The spectral column densities (SCD) of 12 CO and 13 CO are expressed as and Here, SCD 12X and SCD 13L are the H 2 column densities at the peak velocity channels, assuming a standard (normalization) value of  CO = 2.0 × 10 20 cm −2 (K km s −1 ) −1 (Bolatto et al. 2013) and 13 CO = 5.0 × 10 5 (Dickman 1978), respectively.
Figure 2(a-h) presents scatter plots between SCD 12X and SCD 13L of GMCs presented in Figure1(a-h).Black dashed lines show the linear relation of SCD 12X = SCD 13L .SCD 12X shows saturation at high SCD 13L , and the nonlinear relation of each GMC has different saturation levels.Indeed, previous studies have reported that the 12 CO intensity shows apparent saturation at the high-intensity level of 13 CO from the observation of Galactic molecular clouds (Langer et al. 1989;Pineda et al. 2008;Yoda et al. 2010).Curve fitting is useful for quantitatively evaluating the nonlinear relationship between the 12 CO and 13 CO intensities.Here, we performed curve fitting to the scatter plots between SCD 12X and SCD 13L using the free parameters of  and SCD c (spectral column density coefficient), as in Paper I. SCD 12X is given by Blue curves show the fitting results for individual GMCs.To confirm whether the scatter plots shown in Figure 2 depend on the physical resolutions due to different distances from the GMCs, we smoothed the data of GMCs to the same physical resolution and grid size of ∼ 1 pc and ∼ 0.45 pc.This physical resolution corresponds to a beam size of ∼ 20 ′′ at the distance of W49A, the GMC furthest from us in this study.In Figures A1 and A2(a-h) of the Appendix, we show the obtained peak intensity maps and scatter plots of GMCs for equal physical resolution.The results are almost the same as those in Figure 2. The fitting parameters have the range of  = 0.30-0.54and SCD c =(1.6-4.4)×10 21 cm −2 (km s −1 ) −1 .Thus, we find that the opacity and internal clump sizes in a GMC do not affect the relation of the 12 CO and 13 CO correlation.Our analysis also shows that the saturation level of correlations does not depend on physical spatial resolution and pixel size.

Spatial distributions of 𝑋 CO,iso in the Galactic GMCs
We obtained the  CO factor (hereafter  CO,iso ) of each GMC from the correlation between SCD 12X and SCD 13L . CO,iso at each pixel of the molecular clouds is expressed as follows: where  and SCD c are the fitting parameters obtained in Figures 2  and A2. Figure 3(a-h) shows the  CO,iso maps of each GMC. CO,iso has a variability with the range of (0.9-5.0) × 10 20 [cm −2 (K km s −1 ) −1 ] in a GMC.Comparing Figure 1 and Figure 3, we find an anti-correlation between the 12 CO peak intensity and  CO,iso .Figure 4 shows the scatter plot between  CO,iso and   ( 12 CO) in each GMC. CO,iso monotonically decreases with increasing brightness temperature, except for G012.02-00.03.It also shows a lower limit around  CO ∼ 2 × 10 20 cm −2 (K km s −1 ) −1 above  ( 12 CO) > 20 K.These results may correspond to the saturation of the 12 CO intensity in the dense cores of the GMC.This is consistent with previous studies of Perseus molecular clouds obtained by the correlation between the 12 CO integrated intensity and H I column number density derived from the optical depth of 353 GHz dust emission ( 353 ) by Planck satellite observations (see Figure 12 and 15 in Okamoto et al. 2017).

The variability of 𝑋 CO,iso
Our fitting results show that the correlation of column densities between 12 CO and 13 CO is not universal in Galactic GMCs.Each GMC has its own correlation parameters (Figures 2 and A2).Weiß et al. (2001) reported that the  CO factor for a virialized GMC depends on the kinetic temperature ( kin ) and molecular hydrogen number density ((H 2 )) of the molecular gas.(see also Chapter 2.1 in Bolatto et al. 2013).To reveal the origin of the variations in  CO,iso , we investigated the 12 CO  =3-2/1-0 intensity ratio (hereafter  12 3−2/1−0 ) using the JCMT 12 CO  =3-2 archival data (Dempsey et al. 2013;Park et al. 2023). 12 3−2/1−0 depends on  kin and (H 2 ) in molecular clouds assuming the large velocity gradient model (Goldreich & Kwan 1974).We analyzed the W51, W33, N35, W49A, G12.02-00.03,and W43 GMC because the JCMT data are only covered with || < 0.5 • .Figure 5 shows the scatter plot between  CO,iso and The plots of each GMC do not show a clear correlation.Thus, we suggest that the origin of own correlation parameters and variations of  CO,iso might not be simply density and/or temperature variations in a molecular cloud.In our analysis, we assumed that 13 CO is constant in all GMCs.We point out that the change of the isotope abundance ratio with density might explain the variability of  CO,iso in a GMC.

Radial gradient of the 𝑋 CO factor
In addition, we investigated the radial gradient of the  CO factor in the Galactic disc.The averaged  CO factor has a high value in W49A at Galactocentric distance  =7.5 kpc, while G012.02-00.03has a low value  =2.3 kpc.According to Arimoto et al. (1996), the  CO factor increases with the Galactocentric radius by an exponential function given by log 10 where   = 6.2 kpc is the scale radius of the Galactic disc and  0 is  CO at  =  0 = 8.0 kpc (VERA Collaboration et al. 2020; Reid et al. 2019).This equation can be deformed as follows: Figure 6 shows a plot of  CO,iso obtained in this paper as a function of the Galactocentric distance.It is found that  CO,iso increases with the distance from the Galactic Centre, and the plot may be fitted by an exponential function by Previous studies reported that a value of the  CO factor depends on   the metal abundance in galaxies (e.g., Arimoto et al. 1996;Israel 1997;Leroy et al. 2011;Genzel et al. 2012).We suggest that the galactocentric dependency of  CO,iso is caused by the radial metallicity gradient from the Galactic center to the outer Galaxy in the Milky Way.Our fitting result yields a significantly larger  CO at the Galactic Centre by a factor of 3 compared to that by Arimoto et al. (1996) indicated by the green dashed line.This discrepancy may be caused by no data point within  < 2 kpc in our study as well as by the normalization of the local value to 2 × 10 20 [H 2 cm −2 (K km s −1 ) −1 at 8 kpc in the previous study.Sodroski et al. (1995) reported that the  CO factor within 400 pc in the Galactic Centre region is by a factor 3-10 lower than the Galactic disk.Therefore, the value of  CO in the Galactic Centre is still controversial.The present method applied to the 12 CO and 13 CO  =1-0 line data in the Galactic Centre region would provide a more precise determination of  CO in the innermost Milky Way.A detailed analysis of this point will be presented in a separate paper using the CO survey data of the Galactic Centre (e.g., Oka et al. 1998;Torii et al. 2010;Enokiya et al. 2014;Tokuyama et al. 2019).

Comparison with the virial methods
Finally, we calculated the  CO factor from the virial mass of each GMC compared with our estimation.According to Solomon et al. (1987), the virial mass of GMCs is given by where  is the effective radius of a GMC in parsec and   is the 1D velocity dispersion in km s −1 , assuming a density profile of () ∝  −1 .We estimated the radius of a GMC with  = √︁ /, where S is the cloud area within 30% levels of the 12 CO peak integrated intensity.The velocity dispersion is adopted as the averaged value of each pixel in a GMC.Then, the  CO factor using virial methods is expressed as where  H 2 ∼ 2.8 is the mean molecular weight per hydrogen molecule (Appendix A.1 in Kauffmann et al. 2008),  H = 1.67 × 10 −24 g is proton mass and  CO is the 12 CO total luminosity.Table 3 shows the results of the  CO factor in each GMC obtained by the virial methods.The ratio of averaged  CO taken from the virial mass and CO isotopologues is 1.80 ± 0.52.

SUMMARY
The conclusions of this paper are summarized as follows: (i) We studied the correlation between 12 CO and 13 CO intensities toward the Galactic GMCs W51A, W33, N35-N36 complex, W49A, M17SW, G12.02-00.03,W43, and M16 using the FUGIN CO survey data taken with the Nobeyama 45 m telescope.
(ii) All the GMCs show intensity saturation of the 12 CO line in regions with high brightness of 13 CO.
(iii) We also present high-resolution  CO,iso maps made from the correlations of the CO isotopologues, which revealed local variability (1) GMC names (2) Cloud radius defined by the %30 levels of the peak integrated intensity (3) Mean velocity dispersion in a molecular cloud (4) Virial mass (5) 12 CO luminosity (6) The  CO factor derived by the virial mass.(7) Ratio of  CO,iso to  CO,vir .*The averaged value is calculated from the ratio of averaged  CO,iso and  CO,vir .
of  CO,iso in each GMC.We also show that the  CO,iso monotonically decreases with increasing 12 CO brightness temperature.
(iv) The averaged value of all GMCs is calculated to be  CO,iso = (2.17 ± 0.27) × 10 20 cm −2 (K km s −1 ) −1 , which is consistent within the margin of error with the reported values in the previous review.
(v) We show that  CO,iso increases with the Galactocentric distance in accordance with the previous works while suggesting a larger value for the Galactic Centre by a factor of 3 compared to that by Arimoto et al. (1996).
(vi) The averaged value of the  CO factor taken from the virial mass and CO isotopologues of GMCs is consistent within the error by a factor of 2.  MNRAS 000, 000-000 (2024)

Figure 2 .Figure 3 .
Figure 2. Scatter plots between SCD 13L and SCD 12X of (a) W51A, (b) W33, (c) the N35-N36 complex, (d) W49A, (e) M17SW, (f) G012.02-00.03,(g) W43, and (h) M16.Red points show the averaged values of each bin, and the error bars are standard deviations of SCD 12X .Blue curves indicate the fitting results of scatter plots.The black dashed lines indicate the linear relation of SCD 12X =SCD 13L .The color bars show the number of data points in a bin.( data /bin)

Figure 4 .Figure 5 .
Figure 4. Scatter plots between  CO,iso and  B ( 12 CO) in (a) W51A, (b) W33, (c) the N35-N36 complex, (d) W49A, (e) M17SW, (f) G012,02-00.03,(g) W43, and (h) M16.Black points show the averaged values of each bin, and the error bars are standard deviations of  CO,iso .The color bars show the number of data points in a bin.( data /bin) GMC names (2) The fitting parameter of the non-linear relation.(3) The coefficient of the spectrum column density (SCD C ). (4) The mean value of  CO,iso in a GMC.(5) Correlation coefficient.The errors of averaged value are adopted as the standard variation in all GMCs.

Figure 6 .
Figure 6.The  CO,iso gradient from the distance of the Galactic Center.The blue line shows the fitting result adopted by the exponential function.The green dashed line indicates the relation from Arimoto et al. (1996).

Figure A1 .
Figure A1.Same as Figure 1, but the data was smoothed to the same physical resolution of ∼ 1 pc.

Figure A2 .
Figure A2.Same as Figure 2, but the data was smoothed to the same physical resolution of ∼ 1 pc.

Table 1 .
Properties of the Galactic massive star forming regions

Table 2 .
Fitting results and < CO,iso > taken from the correlation of SCD 12X and SCD 13L .