Dissecting Nearby Galaxies with piXedfit. II. Spatially Resolved Scaling Relations among Stars, Dust, and Gas

We use three kinds of data sets in this work: broadband imaging data that ranges from FUV to FIR, H i 21 cm line emission maps, and CO intensity maps. We collect archival broadband imaging data of 24 bands from various surveys, including the Galaxy Evolution Explorer (GALEX; Martin et al. 2005), the Sloan Digital Sky Survey (SDSS; York et al. 2000), the Two Micron All Sky Survey (2MASS; Skrutskie et al. 2006), the Wide-field Infrared Survey Explorer (WISE; Wright et al. 2010), the Spitzer Space Telescope (Gallagher et al. 2003), and the Herschel Space Observatory (Pilbratt et al. 2010). In order to obtain the surface densities of gas, including both the neutral and molecular components, we collect the H i 21 cm and CO (J = 2 → 1) emission line maps from The H i Nearby Galaxy Survey (THINGS ⁸ ; Walter et al. 2008) and the Heterodyne Receiver Array CO Line Extragalactic Survey (HERACLES ⁹ ; Leroy et al. 2009), respectively. The basic information for the data sets is given in Table 1. More detailed descriptions of the data sets are given in Paper I (Sections 2.1 and 2.2 therein).

We use the same sample of galaxies as in Paper I, which consists of ten galaxies: NGC 0628 (M74), NGC 3184, NGC 3351 (M95), NGC 3627 (M66), NGC 4254 (M99), NGC 4579 (M58), NGC 4736 (M94), NGC 5055 (M63), NGC 5194 (M51a), and NGC 5457 (M101). We refer the reader to Paper I for the description on the selection criteria for these galaxies and their basic characteristics (Table 2 therein), including the Hubble type, adopted distance, nuclear activity, integrated SFR, M_*, dust mass (M_dust), neutral hydrogen mass (M_HI), molecular hydrogen mass (${M}_{{{\rm{H}}}_{2}}$), half-mass–radius (R_e ), and elliptical isophote fitting results that are characterized by the ellipticity (e) and position angle (PA). The R_e represents the radius along the elliptical semimajor axis that covers half of the total M_*. Overall, our sample are spiral galaxies with relatively face-on configuration (e < 0.6 and b/a > 0.4) and have global properties that span relatively wide ranges: M_* (∼ 10^9.7–10^11.2 M_⊙), SFR (∼ 0.2–12.9 M_⊙yr⁻¹), M_dust (∼ 10^7.2–10^8.7 M_⊙), ${M}_{{{\rm{H}}}_{2}}$ (∼ 10^8.8–10^10.7 M_⊙), and M_HI (∼ 10^8.6–10^9.6 M_⊙). Since our sample are face-on, we do not correct the surface density quantities measured in our analysis for the inclination of the galaxies. Among the sample galaxies, two galaxies (NGC 4254 and NGC 4579) do not have H i data because they are not covered in the THINGS survey. We still include these galaxies in the analysis of this paper because their H₂ information could still provide meaningful information for achieving the goals of this paper.

A detailed description on the methods used in the analysis of our data sets is given in Paper I (Section 3.1 therein). Here we only summarize the key steps of our data analysis, which are all carried out with piXedfit. Basically, we combine all the FUV–FIR broadband imaging data along with the H i and CO maps, which are processed through point-spread function (PSF) matching, spatial registration, and reprojection. As a consequence, all the data are brought to the spatial resolution and sampling of the SPIRE 350 μm band. Based on the processed maps, we define the galaxy's region ¹⁰ with the segmentation maps generated using the SExtractor (Bertin & Arnouts 1996). After that, we calculate the fluxes and their uncertainties (in all bands) of all pixels within the region. Finally, we perform pixel binning to achieve a minimum signal-to-noise ratio (S/N) of 5 (in all bands) for the spatially resolved SEDs.

The H i and CO maps are also processed through the same steps as above. Once we have the H i and CO maps with the same spatial resolution as those of the broadband imaging data, we calculate Σ_HI and ${{\rm{\Sigma }}}_{{{\rm{H}}}_{2}}$ of pixels with the assumption of a constant CO-to-H₂ conversion factor of ${\alpha }_{\mathrm{CO}}=4.35\times {10}^{6}{M}_{\odot }\,{\mathrm{kpc}}^{-2}\,{({\rm{K}}\ \mathrm{km}\ {{\rm{s}}}^{-1})}^{-1}$ (e.g., Bolatto et al. 2013). Such an assumption is made in most of the studies of the resolved scaling relations of galaxies in the literature (e.g., Bigiel et al. 2008; Casasola et al. 2015; Lin et al. 2019; Morselli et al. 2020; Ellison et al. 2021). To investigate the effects of a nonconstant α_CO on the resolved scaling relations derived in this study, in Appendix, we will apply a metallicity-dependent α_CO and discuss the changes in the scaling relations.

Once the pixel binning is done, we perform SED fitting for each bin to derive the spatially resolved properties of the stellar population and dust, again using piXedfit. The SED of the composite stellar population is calculated using the Flexible Stellar Population Synthesis (FSPS ¹¹ ; Conroy et al. 2009) code. The SED modeling in FSPS includes emission from the stars, nebulae, dust, and a dusty torus surrounding an active galactic nucleus (AGN). It implements the energy balance principle (i.e., the energy absorbed by dust in the UV to near-IR is equal to the energy reradiated in the IR). The dust emission is based on the Draine & Li (2007) templates (see Leja et al. 2017), while the modeling of the AGN dusty torus emission uses the Nenkova et al. (2008a, 2008b) CLUMPY templates (see Leja et al. 2018).

In this work, we adopt the following settings in the SED modeling: an initial mass function of Chabrier (2003), Padova isochrones (Girardi et al. 2000; Marigo & Girardi 2007; Marigo et al. 2008), MILES stellar spectral library (Sánchez-Blázquez et al. 2006; Falcón-Barroso et al. 2011), a parametric star formation history in the form of a double power-law function, and the two-component dust attenuation law by Charlot & Fall (2000). For the fitting process, we use the Markov Chain Monte Carlo method. We only include the AGN component for fitting the SED of the spatial bin in the center of a galaxy that is either known to host an AGN or unclassified (see Table 2 in Paper I). The list of free parameters in the SED modeling and fitting, along with the adopted priors, is given in Paper I (Table 4 therein).

As mentioned above, the SED fitting is applied to individual spatial bins in the galaxies, so we obtain inferred parameters at the spatial bin scales. For parameters that scale with the flux (M_*, SFR, and M_dust), we further divide the values associated with a spatial bin into its member pixels in proportion to the flux in a certain band (K_s , 24 μm, and 100 μm bands for M_*, SFR, and M_dust, respectively). Specifically, we weigh the pixel value with w = f_p,i/f_b,i, where f_p,i and f_b,i are the fluxes of the pixel and the parent bin, respectively. Such further division into the pixels is useful in comparing the spatially resolved Σ_*, Σ_SFR, and Σ_dust with Σ_HI and ${{\rm{\Sigma }}}_{{{\rm{H}}}_{2}}$, which are already in pixel basis.

We show the resolved star formation scaling relations in Figure 1. The panels from left to right show the rSFMS, rKS, and rMGMS relations, respectively. In each panel, the gray points represent the pixels that are excluded due to the thresholds set above. The data points in the scaling relations are color-coded based on the radial distance of the pixels from the galactic center measured along the elliptical semimajor axis. As we can see, pixels of different radial distances are mixed across the locus of the three relations, in contrast to the naive expectation (based on the radial profiles) that pixels closer to the galactic center would tend to occupy the high surface density locus. This trend is likely caused by the diversity in the global properties of the sample galaxies, especially the M_* (which spans a range of ∼1.5 dex), molecular gas mass (a range of ∼1.8 dex), and SFR (a range of ∼1.8 dex), which define the overall level of the spatially resolved properties in individual galaxies.

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The contours show the number density distributions of the pixels. For a quantitative analysis of the scaling relations, we fit each relation with a linear function of the form

$$\begin{eqnarray}&&\mathrm{log}(Y)=\alpha \mathrm{log}(X)+\beta ,\end{eqnarray} \tag{ 1 }$$

where Y and X are the physical quantities (Σ_*, Σ_SFR, or ${{\rm{\Sigma }}}_{{{\rm{H}}}_{2}}$) on the x- and y-axes of a scaling relation, respectively. The coefficients α and β are the slope and zero-point, respectively. The fitting is performed using the orthogonal distance regression (ODR) method, which considers deviation (i.e., dispersion) along both x- and y-axes, in contrast to the ordinary least square method, which only considers the deviation along the y-axis. Moreover, we calculate the Spearman rank-order correlation coefficient (ρ) in order to evaluate the strength of the correlation. The best-fit linear functions are shown in the figure as the black dashed lines. The slope, zero-point, scatter (σ), and ρ of these scaling relations are summarized in Table 2. The uncertainties of α and β are obtained using the bootstrap resampling method. Throughout this paper, we use these quantities, especially σ and ρ, as an indicator for the significance of a scaling relation. Given the uncertainties of the physical quantities involved in the scaling relations, the observed scatter of the relations is always larger than the intrinsic one. Because of the complexity, we do not attempt to correct the observed scatter for the measurement errors, which include a careful accounting of systematic errors. However, we also use ρ in addition to σ for evaluating the significance of a scaling relation, which provides an independent measure.

Our results show that the rSFMS relation has the largest scatter (0.31 dex) among the three, while the rKS and rMGMS relations are similarly tight, with a scatter of 0.19 dex. The ρ values of the rKS and rMGMS relations are also similarly high (0.76), further indicating the significance of these relations. This result may suggest that the rKS and rMGMS relations are independent of each other, while the rSFMS relation exists as a consequence of these two relations. A similar trend (i.e., the larger scatter of the rSFMS compared to the rKS and rMGMS) is also observed by previous studies (e.g., Lin et al. 2019; Enia et al. 2020; Morselli et al. 2020; Ellison et al. 2021; Baker et al. 2022). Lin et al. (2019) reported scatters of 0.25, 0.19, and 0.20 dex for the rSFMS, rKS, and rMGMS relations, respectively. They analyzed 14 star-forming galaxies from the Atacama Large Millimeter/submillimeter Array (ALMA) Mapping nearby Galaxies at Apache Point Observatory (MaNGA) QUEnching and STar Formation survey (ALMaQUEST; Lin et al. 2020). Using the same data sets but with a larger sample (28 galaxies) that includes the green-valley galaxies, Ellison et al. (2021) obtained the scatters to be 0.28, 0.22, and 0.21 dex for the rSFMS, rKS, and rMGMS relations, respectively. These results are consistent with ours in that the rKS and rMGMS relations are more significant than the rSFMS.

The rMGMS relation is less studied compared to the other two relations. An early study by Shi et al. (2011) has found a correlation between Σ_gas and Σ_* on global (i.e., the whole galaxy) scales. Wong et al. (2013) is the first work that demonstrated the relationship between ${{\rm{\Sigma }}}_{{{\rm{H}}}_{2}}$ and Σ_* on subgalactic scales. Recently, this relation has been studied in greater details, and has been found to be coupled with the rSFMS and rKS relations in the context of star formation on kiloparsec scales (Lin et al. 2019; Morselli et al. 2020; Ellison et al. 2021; Pessa et al. 2021).

Previous studies in the literature have reported a significant diversity in the slope of the rSFMS relation, which ranges from ∼0.6 to 1.4 (Sánchez et al. 2013; Cano-Díaz et al. 2016; Abdurro'uf & Akiyama 2017, 2018; Hsieh et al. 2017; Lin et al. 2019; Enia et al. 2020; Ellison et al. 2021; Pessa et al. 2021). Such a variation can be caused by several factors, including the differences in galaxy sample selection, the methods used to measure the SFR and M_*, and the way the relations are fit. Because of these complications, we refrain from comparing the slopes of the rSFMS relation (as well as the other relations) derived here with those in the literature. Although the slope is sensitive to the above factors, it has important physical implications. The linear slope (α ∼ 1) of the ensemble rSFMS obtained here indicates a broadly constant rate of stellar mass growth (by in situ star formation) on average over the subgalactic regions of the sample galaxies. On the other hand, the superlinear slope of the rKS and the sublinear slope of the rMGMS suggest that, on average, the star formation efficiency (hereafter SFE) in ($\mathrm{SFE}\equiv {{\rm{\Sigma }}}_{\mathrm{SFR}}/{{\rm{\Sigma }}}_{{{\rm{H}}}_{2}}$) tends to increase with increasing ${{\rm{\Sigma }}}_{{{\rm{H}}}_{2}}$ and that the H₂ mass-to-stellar mass ratio (${f}_{{{\rm{H}}}_{2}}\equiv {{\rm{\Sigma }}}_{{{\rm{H}}}_{2}}/{{\rm{\Sigma }}}_{* }$) tends to decrease with increasing Σ_*, respectively.

Previous studies have investigated global scaling relations among the dust mass, gas mass, stellar mass, and SFR (e.g., da Cunha et al. 2010; Santini et al. 2014; Orellana et al. 2017; Casasola et al. 2020). On the other hand, only a small number of studies have focused on such relations on kiloparsec scales (e.g., Foyle et al. 2012; Roman-Duval et al. 2014). By using our data set, we examine the correlations between Σ_dust and the stellar population properties and the gas mass on kiloparsec scales to study the relationship between dust and star formation processes.

Our results are shown in Figure 2. We fit the relations with a linear function using the ODR method and summarize the results in Table 3. Among the relations, we observe a tight relationship between the surface densities of dust and total gas, with a small scatter of 0.10 dex and a high ρ value of 0.85. This relation is the tightest among the ensemble scaling relations discussed in this paper. The slope of this relation, which is close to unity (σ = 0.96 ± 0.01), indicates a nearly constant dust-to-gas mass ratio. This agrees with the broadly flat dust-to-gas mass ratio radial profile of the sample galaxies that we analyzed in Paper I (Figure 15 therein). It is important to note that these results are sensitive to the assumption for α_CO. The above trends are obtained with the assumption of a constant α_CO. We have shown in Paper I that the dust-to-gas mass ratio profiles deviate from flat and increase toward the galactic center when a metallicity-dependent α_CO is assumed. Since the gas-phase metallicity profiles are increasing toward the galactic center in all the sample galaxies (Pilyugin et al. 2014; Berg et al. 2020), the dust-to-gas mass ratio tends to increase with the increasing gas-phase metallicity. Given the dependency of the dust-to-gas mass ratio on the metallicity, we analyze the effect of the metallicity-dependent α_CO assumption on the Σ_dust–Σ_gas relation in Appendix. Overall, we find that the scatter of this relation is slightly larger (but still considerably tight) when we assume the metallicity-dependent α_CO (see Table 4).

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Table 3. Results of the Fitting of Linear Function to the Resolved Dust Scaling Relations

Correlation	ρ	α	β	σ
Σdust–ΣHI	0.53	0.48 ± 0.01	4.17 ± 0.07	0.15
Σdust–${{\rm{\Sigma }}}_{{{\rm{H}}}_{2}}$	0.76	1.48 ± 0.01	−1.02 ± 0.07	0.14
Σdust–Σgas	0.85	0.96 ± 0.01	2.06 ± 0.04	0.10
Σdust–ΣSFR	0.62	2.14 ± 0.03	−14.07 ± 0.19	0.20
Σdust–Σ*	0.58	2.05 ± 0.04	−3.22 ± 0.21	0.20
Σdust–sSFR	0.06	28.21 ± 13.01	−164.48 ± 71.08	0.27

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Table 4. Comparison of the Linear Function Fits between the Scaling Relations Derived with the Constant and Metallicity-dependent α_CO

	αCO Assumption
Correlation	Constant αCO				Metallicity-dependent αCO
	ρ	α	β	σ	ρ	α	β	σ
rKS	0.75	1.25 ± 0.01	−11.15 ± 0.10	0.18	0.78	1.77 ± 0.03	−15.43 ± 0.19	0.20
rMGMS	0.83	0.95 ± 0.01	−0.51 ± 0.06	0.14	0.71	0.66 ± 0.01	2.16 ± 0.07	0.21
Σdust–${{\rm{\Sigma }}}_{{{\rm{H}}}_{2}}$	0.76	1.45 ± 0.01	−0.91 ± 0.08	0.13	0.79	0.96 ± 0.02	2.17 ± 0.09	0.18
Σdust–Σgas	0.85	0.93 ± 0.01	2.24 ± 0.05	0.09	0.83	0.71 ± 0.01	3.68 ± 0.07	0.14

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When we separate the gas into the atomic and molecular components, the dust surface density is more tightly correlated with the molecular gas compared to the atomic one. This trend agrees with the picture in which dust plays roles in catalyzing the formation of molecular gas (e.g., Gould & Salpeter 1963; Hollenbach & Salpeter 1971; Cazaux & Tielens 2004; Yamasawa et al. 2011). The strong correlation between dust and H₂ has also been observed at even smaller scales (10–50 pc) by Roman-Duval et al. (2014), who studied the dust-to-gas mass ratio in the Magellanic Clouds. They found a higher dust-to-gas mass ratio in the dense ISM environment dominated by molecular gas. Other studies of nearby galaxies, by Foyle et al. (2012) and Hughes et al. (2014), have also observed tight relations of Σ_dust–Σ_gas and Σ_dust–${{\rm{\Sigma }}}_{{{\rm{H}}}_{2}}$, but a loose relation between the Σ_dust and Σ_HI, in good agreement with our results.

On global scales, the total dust mass is also found to be correlated more strongly with the total gas mass than with either M_HI or ${M}_{{{\rm{H}}}_{2}}$ (e.g., Corbelli et al. 2012; Orellana et al. 2017; Casasola et al. 2020), in agreement with the trend we observe on kiloparsec scales. However, there is a disagreement among previous studies in terms of the relative significance between the M_dust–M_HI and M_dust–${M}_{{{\rm{H}}}_{2}}$ relations. While Corbelli et al. (2012) and Orellana et al. (2017) found that M_dust tends to correlate more strongly with ${M}_{{{\rm{H}}}_{2}}$ than with M_HI (in agreement with the trend in subgalactic scales that we observe), Casasola et al. (2020) found the opposite trend. Another interesting result by Bertemes et al. (2018), who performed cross-calibration of CO- and dust-based H₂, implies that dust traces not only molecular hydrogen but also part of the atomic hydrogen in the inner molecular-dominated region, which is consistent with our spatially resolved Σ_dust–${{\rm{\Sigma }}}_{{{\rm{H}}}_{2}}$–Σ_HI relations that mostly emerge from the regions within the H₂-dominated regime.

The second row in Figure 2 shows the scaling relations of Σ_dust with Σ_*, Σ_SFR, and sSFR. While Σ_dust is strongly correlated with Σ_SFR and Σ_* (as corroborated by the small σ and high ρ values), it appears to have no correlation with the sSFR on spatially resolved scales. The lack of the correlation with the sSFR may be a consequence of the linear slope (α ∼ 1) of the rSFMS relation, which suggests a broadly constant sSFR.

A previous study on the global properties of galaxies by Santini et al. (2014) suggested that the M_dust–M_* relation is likely to be a direct consequence of the SFR–M_dust and SFMS relations. They further suggest that the relationship between SFR and M_dust is a natural consequence of the KS and M_dust–${M}_{{{\rm{H}}}_{2}}$ relations. However, our results on kiloparsec scales do not necessarily support their arguments. We find that the Σ_dust–Σ_* and Σ_dust–Σ_SFR relations are similarly tight (i.e., significant, as indicated by the small scatter and high ρ value), and both relations are tighter than the rSFMS. Therefore, it is unlikely that the relation between the Σ_dust and Σ_* is originated from the rSFMS.

The color-coding in Figure 2 reveals a clear radial trend in the majority of the scaling relations. The radial gradient in the Σ_dust–Σ_HI appears to be perpendicular to the best-fit relation, indicating that the radial trend is mainly responsible for the scatter in this relation. This trend suggests an increasing H i-to-dust mass ratio with radius (i.e., having a positive gradient). The radial trends in Σ_dust–${{\rm{\Sigma }}}_{{{\rm{H}}}_{2}}$, Σ_dust–Σ_gas, and Σ_dust–Σ_SFR show increasing surface densities with decreasing radius, as expected from the radial profiles of these quantities (see Figure 8 in Paper I). However, we do not see a clear radial trend in the Σ_dust–Σ_* and Σ_dust–sSFR relations.

We show the resolved star formation scaling relations of individual galaxies in Figure 3. In each panel, the scaling relation of a galaxy is obtained by calculating the median of the distribution of a quantity on the y-axis for each bin of the quantity on the x-axis with a bin width of 0.2 dex. In deriving the running median, we discard the bins that contain less than 10 pixels. The scaling relations of different galaxies are shown by lines and shaded areas, color-coded by their distances from the global SFMS ridge line (ΔSFMS), as defined in Paper I (Figure 1 therein). The shaded area around a running median represents the range given by the 16th and 84th percentiles. The black dashed lines are the best fits to the ensemble scaling relations as shown in Figure 1.

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All three relations exhibit variations from galaxy to galaxy in terms of the normalization, slope, and shape. The rSFMS relation shows the most significant variations among the three, while the rKS relation has the least variations. The small variation in the slope of the rKS relation may imply a universal star formation law, while the variation in the normalization of this relation reflects the diversity in the local SFEf among the galaxies.

The scatter of the relations in individual galaxies is significantly smaller than the scatter of the ensemble relation, indicating that the scatter in the ensemble relation is mainly driven by the galaxy-to-galaxy variation (i.e., differences in slope and normalization). Considering the radial profiles of the Σ_*, Σ_SFR, and ${{\rm{\Sigma }}}_{{{\rm{H}}}_{2}}$ (Paper I, Figure 8 therein), which decrease with increasing galactocentric distances, the pixels from the bottom left to the top right sides of each relation (of individual galaxies) tend to have decreasing radii.

As can be seen from the left panel of Figure 3, the rSFMS relation shows a saturation of Σ_SFR at high Σ_* for most of the galaxies. Such a flattening has also been observed by previous studies (e.g., Abdurro'uf & Akiyama 2017, 2018; Ellison et al. 2021), and is mainly caused by the central quiescent bulge component. A similar flattening trend is also found in the rMGMS relation for most of the galaxies. This flattening trend is likely a consequence of the suppression of the molecular gas-to-stellar mass ratio in the central regions of the galaxies, as we observed in Paper I (Figure 12 therein). A previous study by Ellison et al. (2021) also observed a similar saturation of ${{\rm{\Sigma }}}_{{{\rm{H}}}_{2}}$ in high Σ_* regions of the rMGMS relation.

The color-coding clearly shows that the galaxies with a higher global SFMS normalization (i.e., higher global sSFR) tend to have a higher normalization in the three resolved star formation scaling relations. This trend suggests that the normalization of these scaling relations, which represents the overall levels of the sSFR, SFE, and H₂ fraction on local kiloparsec scales within galaxies, is likely correlated with some global processes that determine the global sSFR. Based on these results, we can also see that the diversity in the global sSFR contributes to the scatter of the resolved star formation scaling relations formed with the ensemble of all pixels (see Section 3.1).

Next, we investigate the galaxy-to-galaxy variations of the resolved dust scaling relations. The dust scaling relations of individual galaxies are shown in Figure 4. We only analyze four relations (Σ_dust–${{\rm{\Sigma }}}_{{{\rm{H}}}_{2}}$, Σ_dust–Σ_gas, Σ_dust–Σ_SFR, and Σ_dust–Σ_*) as the other two relations (Σ_dust–Σ_HI and Σ_dust–sSFR) are weak (see Figure 2). As can be seen from the figure, the Σ_dust–${{\rm{\Sigma }}}_{{{\rm{H}}}_{2}}$ and Σ_dust–Σ_gas relations of individual galaxies are located in the same tight locus, which indicates that this relation holds similarly among our sample galaxies. We note that our sample is limited to the relatively massive spiral galaxies and thus may not be representative of the general population of galaxies. Although our sample covers a wide range of global sSFR, it does not cover a sufficiently wide range of gas-phase metallicity. Referring to Moustakas et al. (2010), our sample galaxies have a global $12+\mathrm{log}({\rm{O}}/{\rm{H}})$ that ranges from 8.31 to 8.68, if the Pilyugin & Thuan (2005) calibration is adopted, or from 8.99 to 9.22, if the calibration of Kobulnicky & Kewley (2004) is applied. Our sample only covers the high-mass plateau in the mass–metallicity relation of the SINGS galaxies as analyzed in Moustakas et al. (2010, Figure 11 therein). One may expect that the Σ_dust–Σ_gas relation should be influenced by the gas-phase metallicity because of the known relationship between the dust-to-gas mass ratio and the gas-phase metallicity (e.g., Rémy-Ruyer et al. 2014; Chiang et al. 2018). The influence of metallicity could come from the dependency of α_CO on the metallicity. To check this, in Appendix, we recalculate ${{\rm{\Sigma }}}_{{{\rm{H}}}_{2}}$ of the subgalactic regions with a metallicity-dependent α_CO, and find that the above two scaling relations still hold with slightly larger scatter and shallower slope.

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In the second row of Figure 4, we see that the Σ_dust–Σ_SFR and Σ_dust–Σ_* relations exhibit a more significant variation in normalization than the other two relations. Similar to the case of the resolved star formation scaling relations (Section 4.1), these resolved dust scaling relations in individual galaxies are tighter than the corresponding ensemble scaling relations (see Section 3.2), and it is the galaxy-to-galaxy variations in normalization of these relations that mainly contributes to the scatter of the ensemble scaling relations. As can be seen from the color-coding, the galaxies that are more actively forming stars tend to have higher (lower) normalization in the Σ_dust–Σ_SFR (Σ_dust–Σ_*) relations.

We start by quantifying the offset of each scaling relation in individual galaxies with respect to the median profile of the corresponding ensemble scaling relation that is constructed from all subgalactic regions of the sample galaxies (see Section 3). To do this, we calculate the distance along the y-axis between the median profile of the scaling relation in each galaxy and that of the ensemble relation. Then we define the offset of each relation as the median of these distances. The uncertainty of the offset is defined as the range given by the 16th and 84th percentiles. We compare the offsets of the resolved star formation scaling relations (ΔrSFMS, ΔrKS, and ΔrMGMS) in Figure 5. In each panel, different galaxies are indicated with different symbols and colors that represent the distance from the global SFMS (as we have done in Section 4).

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We find significant positive correlations among the normalization offsets of the scaling relations: ΔrKS–ΔrSFMS, ΔrMGMS–ΔrSFMS, and ΔrMGMS–ΔrKS. ΔrSFMS is strongly correlated with ΔrKS with a high ρ value of 0.96 and a small scatter of 0.07 dex, indicating that the galaxies with higher rKS normalization tend to have higher normalization of the rSFMS relation. This means that the galaxies with a higher level of resolved SFE tend to have a higher level of resolved sSFR and vice versa. ΔrSFMS is also strongly correlated with ΔrMGMS, although it is less significant compared to the ΔrKS–ΔrSFMS correlation. The σ and ρ values of the ΔrMGMS–ΔrSFMS relation are 0.12 dex and 0.85, respectively. This correlation indicates that the rate of stellar mass growth (by star formation) is also strongly influenced by the molecular gas mass fraction, such that a higher molecular gas fraction leads to a higher rate of stellar mass growth. These results are reminiscent of trends seen globally (Saintonge & Catinella 2022, and reference therein).

In the bottom panel of Figure 5, we see that ΔrMGMS and ΔrKS are positively correlated, although the significance level is not as high as the other two correlations, as indicated by the larger scatter (σ = 0.13 dex) and lower ρ value (0.81) in comparison to the other two correlations. This indicates that the SFE is positively correlated with the molecular gas mass fraction, such that the galaxies with more molecular gas mass supplies tend to have a higher SFE. A similar comparison of the normalization offsets of the rSFMS, rKS, and rMGMS relations by Ellison et al. (2021) found strong positive correlations in the ΔrKS–ΔrSFMS and ΔMGMS–ΔrSFMS relations, which agree with our result. However, they found a very weak correlation between the ΔrMGMS and ΔrKS, in contradiction with our result. This disagreement can be caused by several factors, including the sample selection and methods for deriving SFRs and stellar masses. Nevertheless, it is worth noting that we also find the weakest correlation in the ΔrMGMS–ΔrKS correlation than in the other two.

Basically, we can think of ΔrSFMS, ΔrKS, and ΔrMGMS of individual galaxies as measures of the overall levels of the spatially resolved sSFR, SFE, and H₂ mass fraction (${f}_{{{\rm{H}}}_{2}}$) in galaxies, respectively. The latter three quantities are related via

$$\begin{eqnarray}&&\mathrm{sSFR}=\mathrm{SFE}\times {f}_{{{\rm{H}}}_{2}}.\end{eqnarray} \tag{ 2 }$$

Although ΔrSFMS, ΔrKS, and ΔrMGMS tell about the normalization of the resolved scaling relations, they are actually global-scale quantities. To investigate the correlations among equivalent quantities of those normalizations on kiloparsec scales, we plot the spatially resolved sSFR–SFE–${f}_{{{\rm{H}}}_{2}}$ correlations in Figure 6. In the left and middle panels, we see that there is a tendency of the subgalactic regions of galaxies with different ΔSFMS (i.e., global sSFR) to be separated diagonally, in contrast to the resolved star formation and dust scaling relations, where the regions associated with the star-forming and quiescent galaxies tend to be separated vertically. This trend indicates that the global sSFR, SFE, and ${f}_{{{\rm{H}}}_{2}}$ are preserved on kiloparsec scales. However, there seems to be no correlation between SFE and ${f}_{{{\rm{H}}}_{2}}$ in individual galaxies, with SFE staying broadly constant over an order of magnitude change in the H₂ mass fraction. This is in contrast to the trend we see in the equivalent relation on global scales (ΔrMGMS–ΔrKS), where we see a significant positive correlation. This trend may indicate that the SFE is roughly constant in the disk, and it is likely governed by some processes that act on global scales.

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Next, we investigate possible correlations between the resolved star formation and dust scaling relations, to examine how dust is associated with star formation activities on kiloparsec scales. For the dust scaling relations, we only consider the Σ_dust–Σ_* and Σ_dust–Σ_SFR relations because they exhibit stronger galaxy-to-galaxy variations than the Σ_dust–${{\rm{\Sigma }}}_{{{\rm{H}}}_{2}}$ and Σ_dust–Σ_gas relations, while the other two relations (Σ_dust–Σ_HI and Σ_dust–sSFR) are weak (see Figure 2). In Figure 7, we show correlations of the normalization offsets between the resolved star formation scaling relations and the above two dust scaling relations. It can be seen that the normalization offset of the Σ_dust–Σ_SFR relation is strongly correlated (σ < 0.15 dex and ρ > 0.7) with the offsets of the rSFMS and rKS relations, but only weakly correlated with the offset of the rMGMS relation (ρ = 0.39). On the other hand, the normalization offset of the Σ_dust–Σ_* relation has a strong negative correlation with the offset of the rMGMS relation, but has a relatively weak negative correlation with the offsets of the rSFMS and rKS relations.

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The ΔΣ_dust–Σ_SFR versus ΔrSFMS and ΔΣ_dust–Σ_SFR versus ΔrKS correlations indicate that the galaxies that have a higher average SFR-to-dust mass ratio over their spatial regions tend to have higher averages of sSFR and SFE. The ΔΣ_dust–Σ_* versus ΔrMGMS relation suggests that the galaxies that have a lower average dust-to-stellar mass ratio tend to have a lower average ${f}_{{{\rm{H}}}_{2}}$. There is also a hint of decreasing sSFR with decreasing dust-to-stellar mass ratio, as indicated by the ΔΣ_dust–Σ_* versus ΔrSFMS relation. Looking at the quantities involved in the above relations, one may argue that the strong relationship is just the consequence of the fact that the two axes share a quantity that is the same. However, this is not always the case. As we have seen in the right panel of Figure 6, although both axes in the ${f}_{{{\rm{H}}}_{2}}$–SFE plane share the same quantity ${{\rm{\Sigma }}}_{{{\rm{H}}}_{2}}$, there is no clear correlation on this plane.

After having examined the connections between the resolved dust and star formation scaling relations, now we investigate the possible connection between Σ_dust–Σ_* and Σ_dust–Σ_SFR relations. We show the comparison between the normalization offsets of the Σ_dust–Σ_* and Σ_dust–Σ_SFR in Figure 8. There appears to be no correlation between ΔΣ_dust–Σ_* and ΔΣ_dust–Σ_SFR, suggesting that these two relations tend to be independent of each other, in the sense that the Σ_dust–Σ_* relation is not the consequence of the Σ_dust–Σ_SFR and rSFMS relations, nor is the Σ_dust–Σ_SFR relation the consequence of the Σ_dust–${{\rm{\Sigma }}}_{{{\rm{H}}}_{2}}$ and rKS relations. This weak correlation between the ΔΣ_dust–Σ_* and ΔΣ_dust–Σ_SFR also supports our earlier argument that involving a same quantity into both axes does not always result in a strong relationship.

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As shown in Section 4.1, the resolved star formation scaling relations exhibit variations from galaxy to galaxy (Figure 3). To investigate the possible causes for the galaxy-to-galaxy variations, we check the correlations between the normalization of the rSFMS, rKS, and rMGMS relations and the global properties of galaxies. Figure 9 shows the normalization offsets of the sample galaxies in the rSFMS, rKS, and rMGMS relations as functions of the global sSFR (first row), M_* (second row), and concentration index (C; third row). The concentration index, which measures the compactness of the stellar distribution in a galaxy, is defined as C ≡ R₉₀/R₅₀, where R₉₀ and R₅₀ are the galactocentric distances (measured along the elliptical semimajor axis) that enclose 90% and 50% of the total stellar mass, respectively. Generally, the galaxies with early-type morphology tend to have higher C than those with late-type morphology (e.g., Shimasaku et al. 2001; Strateva et al. 2001; Nakamura et al. 2003; Park & Choi 2005). Previous studies have also shown that C correlates positively with the bulge-to-total ratio (B/T; e.g., Kim et al. 2016).

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From the first row in Figure 9, we observe that the normalization offsets of the three resolved star formation scaling relations are strongly correlated with the global sSFR in such a way that the galaxies with higher sSFR tend to have higher normalization of rSFMS, rKS, and rMGMS relations. Among these three, the sSFR is correlated most strongly with ΔrKS (σ = 0.08 dex and ρ = 0.92), suggesting a strong positive correlation between the global sSFR and the SFE within galaxies. One may expect the sSFR to be more strongly correlated with ΔrSFMS than with ΔrKS because both of these quantities are different expressions of the same physical property. However, our result shows that this is not necessarily the case. The strong positive sSFR–ΔrMGMS correlation suggests that there is a tight relation between the global sSFR and the overall level of the molecular gas fraction in galaxies. The color-coding shows a clear trend of an increasing normalization of the global SFMS relation from the bottom left side to the top right side on these three diagrams. A similar analysis carried out by Ellison et al. (2021), using a larger sample of low-z galaxies, also obtained strong positive correlations between the global sSFR and ΔrSFMS, ΔrKS, and ΔrMGMS, in good agreement with our results.

In the second row of Figure 9, we observe weak correlations between the global M_* and the normalization offsets of the resolved star formation scaling relations. While ΔrMGMS seems to be constant across a wide range of M_* (see the right panel), there is a tendency of decreasing normalization in the rSFMS and rKS relations with increasing M_*, as shown in the left and middle panels. These results are again in good agreement with the finding of Ellison et al. (2021).

The third row of Figure 9 shows the correlations between the concentration index and the normalization offsets. Despite the small size and the limited morphological type (i.e., only spirals) of our sample, we still observe clear correlations between C and the normalization offsets of the scaling relations. The correlations are all significant, with ∣ρ∣ ≳ 0.67. This result suggests that more compact galaxies (i.e., having higher C) tend to have lower levels of resolved sSFR, SFE, and molecular gas fraction. The suppressed H₂ fraction and SFE in galaxies with higher C (i.e., higher B/T) seem to support the scenario in which the star formation suppression is promoted by the existence of the massive bulge component (Martig et al. 2009).

Previous studies have also explored the correlations between galaxy morphology and star formation scaling relations on both global and kiloparsec scales. On global scales, the early-type galaxies tend to reside on the lower envelope or below the SFMS, while the late-type galaxies are mostly located on the upper envelope and the SFMS itself (e.g., Wuyts et al. 2011; González Delgado et al. 2016; Cano-Díaz et al. 2019). For the global ${M}_{{{\rm{H}}}_{2}}$–SFR relation, the B/T tends to increase with decreasing SFR at a given ${M}_{{{\rm{H}}}_{2}}$ (Dou et al. 2021). A correlation between H₂ depletion time and C that is observed by Saintonge et al. (2012) is also consistent with the picture that more compact galaxies tend to have longer H₂ depletion time (i.e., lower SFE). Because the MGMS relation has been studied in detail only recently, its correlations with the global properties are not well understood. However, we can regard the MGMS normalization as the H₂ fraction, which is better studied. In line with this, Saintonge et al. (2011) observed a rather weak negative correlation between the C and the H₂ fraction. On kiloparsec scales, Ellison et al. (2021) found strong negative correlations between the Sérsic index and the normalization offsets of the rSFMS and rKS relations, in agreement with our results. However, they found no correlation between the Sérsic index and ΔrMGMS.

Based on our current understanding of the galaxy formation and evolution, the star formation in galaxies is regulated by gas flow in and out of the galaxies (e.g., Dekel & Birnboim 2006; Bouché et al. 2010; Tacconi et al. 2010; Lilly et al. 2013; Tacchella et al. 2016). The accreted gas from the cosmic web cools down to form atomic hydrogen first, and then condenses to form molecular hydrogen on dust surfaces (e.g., Gould & Salpeter 1963; Hollenbach & Salpeter 1971; Cazaux & Tielens 2004). Star formation can occur when the molecular gas collapses owing to gravitational instability (Kennicutt 1989, 1998b; Martin & Kennicutt 2001). It has been known that H₂ is centrally concentrated, while H i is spatially extended (i.e., more abundance in the outskirt regions) in galaxies (e.g., Young & Scoville 1991; Bigiel et al. 2008; Leroy et al. 2008). As we have shown in Paper I, the radius of transition from the H i-dominated to H₂-dominated regions varies from galaxy to galaxy. Here, we examine whether this transition occurs in a well-defined local condition, especially in terms of the local surface densities of dust, gas, and stars.

Figure 10 shows the relationship between the molecular-to-neutral hydrogen ratio (${{\rm{\Sigma }}}_{{{\rm{H}}}_{2}}/{{\rm{\Sigma }}}_{\mathrm{HI}}$) and the surface densities of the dust (left panel), gas (middle panel), and stars (right panel). To investigate the connections between the molecular-to-neutral hydrogen ratio and star formation processes on kiloparsec scales, we bin the pixels on these diagrams and calculate the average of the perpendicular distances from the best fit of the ensemble rSFMS relation (left panel of Figure 1) of the member pixels in each bin. Then, we color-code the bins based on this distance. We also show the density contours for the pixel distributions on the diagrams. The figure shows that there is an overall strong correlation between the H₂-to-H i ratio and the surface densities of dust, gas, and stars, as indicated by the small scatter (σ ≤ 0.2 dex) and high ρ value (≳0.5).

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From these relations, we try to infer the typical ISM surface densities at the transition between the molecular and atomic hydrogen by collecting pixels that have ${{\rm{\Sigma }}}_{{{\rm{H}}}_{2}}$ within 0.8–1.2Σ_HI (represented by the horizontal bands in Figure 10), and calculate the median values of Σ_dust, Σ_gas, and Σ_*. The typical ISM surface densities at the H i-to-H₂ transition are found to be as follows: $\mathrm{log}({{\rm{\Sigma }}}_{\mathrm{dust}}[{M}_{\odot }\ {\mathrm{kpc}}^{-2}])=5.34\pm 0.02$, $\mathrm{log}({{\rm{\Sigma }}}_{\mathrm{gas}}[{M}_{\odot }\ {\mathrm{kpc}}^{-2}])=7.15\pm 0.02$, and $\mathrm{log}({{\rm{\Sigma }}}_{* }[{M}_{\odot }\ {\mathrm{kpc}}^{-2}])=7.73\pm 0.02$. These critical values are shown by the vertical dashed lines in Figure 10. The uncertainties of the medians are estimated using the bootstrap resampling method, which is relatively conservative. While the uncertainties of the medians are small, the standard deviations are modest, but still considerably tight, with 0.23, 0.17, and 0.27 dex for the surface densities of dust, gas, and stars, respectively. In agreement with this result, a similar study by Leroy et al. (2008) found critical surface densities of the stellar mass and gas at the H i-to-H₂ transition of $\mathrm{log}({{\rm{\Sigma }}}_{* }[{M}_{\odot }\ {\mathrm{kpc}}^{-2}])=7.91$ and $\mathrm{log}({{\rm{\Sigma }}}_{\mathrm{gas}}[{M}_{\odot }\ {\mathrm{kpc}}^{-2}])=7.15$.

In the rightmost panel of Figure 10, the color-coding shows a clear correlation between the scatter of the Σ_*–$({{\rm{\Sigma }}}_{{{\rm{H}}}_{2}}/{{\rm{\Sigma }}}_{\mathrm{HI}})$ relation and the rSFMS normalization. A clear bimodality of the rSFMS normalization is also seen on the Σ_*–(${{\rm{\Sigma }}}_{{{\rm{H}}}_{2}}/{{\rm{\Sigma }}}_{\mathrm{HI}}$) diagram where the majority of the subgalactic regions on the upper envelope of this relation are those that are located on the upper envelope of the rSFMS relation and vice versa. However, we do not see a similar trend in the other two relations, where the quiescent spatial regions (rSFMS normalization ≲ − 0.5 dex) are mostly residing on the uppermost and lowermost envelopes of both relations, while the star-forming spatial regions (rSFMS normalization ≳0.0 dex) are residing around the best-fit line. In contrast to our result, Morselli et al. (2020) found a steadily decreasing H₂-to-H i ratio with increasing rSFMS normalization. Unfortunately, they only analyzed the correlations between the rSFMS normalization and the scatter of the Σ_gas–(${{\rm{\Sigma }}}_{{{\rm{H}}}_{2}}/{{\rm{\Sigma }}}_{\mathrm{HI}}$) relation. Therefore, we can not compare our results on the Σ_dust–(${{\rm{\Sigma }}}_{{{\rm{H}}}_{2}}/{{\rm{\Sigma }}}_{\mathrm{HI}}$) and Σ_*–(${{\rm{\Sigma }}}_{{{\rm{H}}}_{2}}/{{\rm{\Sigma }}}_{\mathrm{HI}}$) relations with their results. This discrepancy can be caused by a possible discrepancy in the SFR estimate (which have a direct effect on the rSFMS normalization): Morselli et al. (2020) derived the SFR using a prescription involving the total UV and IR luminosities, while we have shown in Paper I that our SFR estimates deviate from this kind of prescription mainly because of a significant contribution of old stellar populations to the dust heating.

The small scatter (σ = 0.19 dex) and high ρ (0.78) of the Σ_*–(${{\rm{\Sigma }}}_{{{\rm{H}}}_{2}}/{{\rm{\Sigma }}}_{\mathrm{HI}}$) relation suggest that the surface density of stellar mass plays the strongest role in defining the H₂-to-H i ratio when compared to the other baryonic components. It also sets the color gradient clearly, such that the H₂-to-H i ratio appears to be increasing with increasing rSFMS normalization at a given Σ_*. This may indicate the importance of the gravitational potential of the stars in governing the physical processes in the ISM, which include the condensation of gas to form H₂.

The inverted trend of rSFMS normalization between the regions with low and high surface densities of dust and gas (as shown in the left and middle panels of Figure 10) seems to agree with the picture in which the H₂-to-H i ratio is controlled by the balance between the H₂ formation in the dense environment and H₂ destruction by photodissociation due to the stellar feedback that is expected to dominate in the less dense environment (e.g., Sternberg et al. 2014; Tacconi et al. 2020). Overall, a plausible explanation could be that this inverted rSFMS normalization trend is also influenced by the gravitational potential. In regions with low surface densities, the gravitational potential is weak enough for radiative feedback from the stars, particularly those of young and massive ones, to disrupt and dissociate H₂ in the regions of intense star formations (Sternberg et al. 2014). This makes the H₂ destruction dominate over the H₂ formation in more actively star-forming regions. In high density regions, the gravitational potential is strong such that the stellar feedback is not strong enough to disrupt and dissociate H₂. Because of this, the H₂ destruction in star-forming regions becomes ineffective, and as a result, we see an increasing molecular-to-atomic gas ratio with the increasing rSFMS normalization. This trend is more prominent in the relationship between the H₂-to-H i ratio and Σ_gas, where an opposite rSFMS normalization trend is clear between the regions with surface density higher and lower than Σ_gas ∼ 10^7.8 M_⊙ kpc⁻².

Having established the spatially resolved star formation scaling relations from the panchromatic SED fitting with the energy balance approach, here we compare the relations obtained from our work and those reported by previous studies in the literature, particularly those involving integral field spectroscopy (IFS) observations and determining the spatially resolved SFR from the Hα emission that is corrected for the dust attenuation based on the Balmer decrement. We choose the following studies (and the associated surveys) as references: Cano-Díaz et al. (2016; Calar Alto Legacy Integral Field Area; CALIFA; Sánchez et al. 2012), Medling et al. (2018; Sydney Australian Astronomical Observatory Multi-object IFS; SAMI; Croom et al. 2012), Lin et al. (2019) and Ellison et al. (2021; ALMaQUEST; Lin et al. 2020), and Pessa et al. (2021; Physics at High Angular Resolution in Nearby Galaxies; PHANGS; Leroy et al. 2021; Emsellem et al. 2022). For simplicity, we do not include previous studies that used other methods for the SFR determination. This comparison can also serve as an indirect validation for our SFR estimate.

Figure 11 shows a comparison of the resolved star formation scaling relations between our work and those from the previous studies mentioned above. The relations from our work are represented by the contours and the best-fit lines (shown as the black dashed lines) taken from Figure 1. Among the aforementioned surveys, only ALMaQUEST and PHANGS surveys analyzed all three scaling relations, while the CALIFA and SAMI only examined the rSFMS relation. The ALMaQUEST survey combines the optical IFS data from the MaNGA (Bundy et al. 2015) survey and the CO J = 1 → 0 emission line maps from new observations with ALMA, while the PHANGS survey combines the optical IFS data from the PHANGS-MUSE (Emsellem et al. 2022) survey and the CO J = 2 → 1 emission line maps from the PHANGS-ALMA (Leroy et al. 2021) survey. We include two results from the ALMaQUEST survey because both studies analyzed different sets of sample galaxies: Lin et al. (2019) analyzed only star-forming galaxies, while Ellison et al. (2021) included star-forming and green-valley galaxies in their analysis. For the PHANGS result from Pessa et al. (2021), we only adopt the relations that were derived with a spatial resolution of 1 kpc to match that achieved in our study.

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Overall, our scaling relations are consistent with those from the previous studies mentioned above. All the scaling relations seem to occupy the similar loci. The slight deviation on slope can be caused in part by the difference in sample selection.

As pointed out in Section 1, the scaling relations among physical properties of galaxies have provided valuable insights into our current understanding of galaxy formation and evolution. Some scaling relations are intuitively understandable, such as the KS relation, which reflects the star formation process from the molecular gas in the ISM, but other relations are more difficult to understand. The empirical scaling relations have provided valuable constraints for the cosmological simulations of galaxy formation (e.g., Sparre et al. 2015; Tacchella et al. 2016; Taylor & Kobayashi 2016; Donnari et al. 2019; Trayford & Schaye 2019). More fundamental relations like the KS can be used for fine-tuning or calibrating the subgrid physics in the cosmological simulations, while other more high level relations (e.g., the SFMS and stellar mass–metallicity) can serve as a standard reference against which the simulation results will be compared (i.e., tested). These mutually beneficial interactions between observations and simulations allow us to gain deeper insights into the physical mechanisms that drive the physical processes in galaxy evolution (e.g., Tacchella et al. 2016; Matthee & Schaye 2019; Trayford & Schaye 2019).

There have been several attempts to reproduce the global SFMS relation from cosmological simulations (e.g., Tacchella et al. 2016; Donnari et al. 2019; Matthee & Schaye 2019). Tacchella et al. (2016) used the VELA cosmological simulation (Ceverino et al. 2014; Zolotov et al. 2015) to study how star-forming galaxies evolve on the global SFMS plane across cosmic time. Despite the tightness (∼0.3 dex) of the bulk distribution of galaxies on the SFMS plane, they found that star-forming galaxies oscillate across the SFMS locus during their evolution. This confinement of galaxies in the tight sequence is regulated by the processes of gas compaction (e.g., by mergers, counterrotating streams, and violent disk instabilities), depletion due to the star formation, and gas outflows. Similar attempts have also been made on local kiloparsec scales within galaxies (e.g., Trayford & Schaye 2019; Nelson et al. 2021). By using the EAGLE cosmological zoom-in hydrodynamical simulations (Crain et al. 2015; Schaye et al. 2015), Trayford & Schaye (2019) studied the evolution of the rSFMS and Σ_*–gas-phase metallicity relations. They found that the rSFMS relation started with a steep slope at high redshift (z ∼ 2) and became shallower with time, a behavior attributed to the inside-out quenching process in galaxies. The mass–metallicity relation also tends to become flattened (at the high surface density regions) with cosmic time. They found that the AGN feedback plays a significant role in the inside-out quenching and the flattening of those two relations.

The significant galaxy-to-galaxy variations of the resolved scaling relations presented in Section 4 indicate that those relations do not hold universally true for all galaxies. Figure 9 further shows that the global sSFR and concentration index correlate strongly with the normalization of the resolved star formation scaling relations. The trend with the global sSFR indicates that the relative normalization of the global SFMS among the galaxies is preserved on kiloparsec scales. On the other hand, the internal morphological factor, in particular the bulge fraction, also plays an important role in the quenching of star formation in galaxies, as has been shown here and in previous studies (e.g., Martig et al. 2009; Genzel et al. 2014).

Recently, some papers have discussed which of the resolved star formation scaling relations is more fundamental than the others (e.g., Lin et al. 2019; Morselli et al. 2020; Ellison et al. 2021; Pessa et al. 2021). Considering the scatters in those relations, our results suggest that the rKS and rMGMS relations are likely independent of each other and are more fundamental from which the rSFMS relation emerges, in agreement with the results from previous studies (e.g., Lin et al. 2019; Ellison et al. 2021). The lack of correlation between the rKS and rMGMS relations is further supported by the lack of correlation between the SFE and ${f}_{{{\rm{H}}}_{2}}$ on kiloparsec scales as shown in the right panel of Figure 6. Though, we see a strong positive correlation between the normalization of the rKS and rMGMS relations for individual galaxies in Figure 5. This may indicate that the SFE and H₂ fraction are correlated on global scales, but they are somehow uncorrelated on kiloparsec scales. The constant SFE over more than an order of magnitude variation in the H₂ fraction on kiloparsec scales (the right panel of Figure 6) suggests that the SFE is likely governed by physical processes that act globally. In line with this result, previous studies have shown that the SFE (and H₂ depletion time) is correlated with other global physical properties, including the sSFR, M_*, concentration index, and dynamical time, which is the timescale for the formation of stars per galactic orbital time due to the gravitational instability of the cold gas in the galactic disk (e.g., Silk 1997; Kennicutt 1998a; Genzel et al. 2010; Saintonge et al. 2011; Huang & Kauffmann 2014; Tacconi et al. 2018).

We have also analyzed the sSFR–SFE and ${f}_{{{\rm{H}}}_{2}}$–sSFR relations on kiloparsec scales (left and middle panels of Figure 6). Despite the tightness of these relations, we still find galaxy-to-galaxy variations that appear to separate the spatial regions of different galaxies diagonally, in contrast to the vertical (i.e., normalization) variations we see in the other relations. It is also clear that the galaxy-to-galaxy variations in the above two relations are mainly driven by the global sSFR. The sSFR is a key parameter that describes the rate of the stellar mass growth due to star formation, and it is thought to be primarily driven by the accretion history of dark matter halo, as discussed in previous studies (e.g., Lilly et al. 2013; Peng & Maiolino 2014; Dekel & Mandelker 2014; Dou et al. 2021). It has also been suggested that the sSFR and SFE are key fundamental parameters in galaxies (e.g., Huang & Kauffmann 2014; Dou et al. 2021). Both of these parameters can be considered as the primary parameters for the evolution of the stellar populations and gas in galaxies (Peng et al. 2010; Tacconi et al. 2020).

While the resolved star formation scaling relations have been studied extensively in the last decade, the resolved scaling relations between the dust surface density and other physical parameters are still unexplored. We find tight correlations between the surface densities of dust and gas (in both molecular and total gas), which applies universally to all galaxies in our sample, as indicated by the lack of the galaxy-to-galaxy variation in these relations (see Figure 4). Despite the variation in global properties (e.g., sSFR, SFE, M_*, and C), the subgalactic regions in the galaxies follow these tight relations. The tightness and the universality of the Σ_dust–${{\rm{\Sigma }}}_{{{\rm{H}}}_{2}}$ relation support the view that dust plays important roles in catalyzing the formation of molecular gas (e.g., Hollenbach & McKee 1979; Yamasawa et al. 2011). Moreover, as demonstrated in Figure 7, the galaxies with higher levels of resolved dust-to-stellar mass ratios tend to have higher levels of resolved sSFR, SFE, and H₂ fraction. This is consistent with the picture that dust also plays an important role in promoting the star formation (e.g., by shielding the gas from the ISRF, which can cool the gas; Hollenbach & Tielens 1999; Krumholz et al. 2009; Yamasawa et al. 2011; Glover & Clark 2012).