Sub-kiloparsec scaling relations between hot gas, dense gas and star formation rate in five nearby star-forming galaxies

For M82, IC 342 and NGC 253, the data of HCN $J=4\rightarrow 3$ and HCO${}^{+}\ J=4\rightarrow 3$ are from Tan et al. (2018). Because M83 and NGC 6946 were both observed by MALATANG I and MALATANG II survey from September 2015 to October 2021, we processed the MALATANG I and II data together. The observations of $J=4\rightarrow 3$ lines of HCN and HCO⁺ were obtained with the 16-receptor array receiver Heterodyne Array Receiver Program (HARP; Buckle et al., 2009). The full width at half maximum (FWHM) beamwidth of each receptor at 350 GHz is about $14^{\prime\prime}$ using a 3$\times$3 jiggle mode with grid spacing of $10^{\prime\prime}$ (see Figure 1). All the five sample galaxies were mapped in the central $2\arcmin\times 2\arcmin$ region. We measured the uncertainty in the absolute flux calibration using standard line calibrators and it was estimated to be about 10%.

We reduced the raw data of M83 and NGC 6946 using the Starlink (Currie et al., 2014) software package ORAC-DR (Jenness et al., 2015) to obtain spectral data. It should be noted that MALATANG I observations used the position-switch (PSSW) mode for these two galaxies, which may have caused unstable baselines and high level noise. To enhance data quality, the beam-switch(BSW) mode was used during MALATANG II observations to obtain a steady baseline with low noise. After processing and inspecting the pipeline, we converted the spectra to the GILDAS/CLASS ¹¹1http://www.iram.fr/IRAMFR/GILDAS/ format for further analysis. The spectral intensity units were converted from antenna temperature $T^{\ast}_{\rm A}$ to main beam temperature $T_{\rm mb}$ adopting a main beam efficiency of $\eta_{\rm mb}=0.64$ ($T_{\rm mb}\equiv T^{\ast}_{\rm A}/\eta_{\rm mb}$). We adopted the same criteria of Tan et al. (2018) (SNR $\geqslant$ 3) to identify detections of the emission lines. The uncertainty ($\sigma$) on the integrated line intensities was

where $T_{\rm rms}$ is the rms of the spectrum for a spectral velocity resolution of $\Delta v_{\rm res}$, $\Delta v_{\rm line}$ is the velocity range of the emission line, and $\Delta v_{\rm base}$ is the velocity width used to fit the baseline. For positions without significant detections, the 3$\sigma$ upper limits of the line integrated intensities are estimated. We derived the line luminosities $L^{\prime}_{\rm dense}$ for each position following Solomon et al. (1997):

where $S\Delta v$ is the velocity-integrated flux density, $\nu_{\rm obs}$ is the observed line frequency, and $D_{\rm L}$ is the luminosity distance. For JCMT telescope at $\thicksim$ 350 GHz, we adopted the conversion factor of $S/T_{\rm mb}=15.6/\eta_{\rm mb}=24.4\ {\rm Jy\ K^{-1}}$. The luminosities where the $J=4\rightarrow 3$ lines of HCN and HCO⁺ $\geqslant$ 3$\sigma$ detection are listed in Table 2. The more detailed description of MALATANG survey and reduction strategy are given in Zhang et al. (in prep., also in Tan et al., 2018).

All X-ray data are obtained from the $Chandra$ Data Archive, with a full list summarized in Table 1. We first used the Chandra$\_$repro task to reprocess the data with CIAO v.4.13 and CALDB v.4.9.6. Figure 1 shows the R-band and the X-ray maps based on the $Chandra$ data for the five galaxies. To examine the diffuse emission from the hot interstellar medium, we used the wavelet-based source detection algorithm wavdetect (Freeman et al., 2002) to search for discrete sources on the scales of 1, $\sqrt{2}$, 2, 2$\sqrt{2}$, 4, and 8 pixels ($0^{\prime\prime}.492$/pixel) and in the soft (0.5 ${-}$ 2.0 keV), hard (2.0 ${-}$ 8.0 keV) and total (0.5 ${-}$ 8.0 keV) energy bands. For most point sources covered by our JCMT observations of the five galaxies, excluding the aperture of 90% energy-encircled fraction of the point spread function (PSF) was sufficient. For a few bright point sources, we manually increased the size of the elliptical mask to remove visible ring-like features due to the PSF wing to minimize the contamination from the PSF spillover.

The diffuse X-ray emission spectra of the regions matching the FWHM $\thicksim$ 14$\arcsec$ of JCMT beam were extracted using the CIAO task specextract, and the background were chosen from the source free regions within the same field. We combined the multiple observed spectra of each position by the combine$\_$spectra task with appropriate grouping. The XSPEC v.12.11.0 (Arnaud, 1996) software was used to fit the spectra. The models included an absorbed thermal plasma (APEC; Smith et al., 2001), and when necessary, Gaussian emission-line components. The best-fit diffuse X-ray luminosities corrected for both Galactic and intrinsic absorption were shown in Table 2.

In this work, we also utilize the CO $J=1\rightarrow 0$ data from the Nobeyama CO Atlas of Nearby Spiral Galaxies (Sorai et al., 2000; Kuno et al., 2007), and infrared archival imaging data from Spitzer (MIPS 24 $\mu$m) and Herschel (PACS 70$\mu$m, 100$\mu$m, and 160$\mu$m). All ancillary data were convolved to $14^{\prime\prime}$ resolution and regridded to the same pixel scale (see Tan et al. 2018 for more details). These processed infrared data allowed us to calculate the total infrared Luminosity $L_{\rm IR}$ (8-1000 $\mu$m) by combining the 24 $\mu$m, 70 $\mu$m, 100$\mu$m, and 160 $\mu$m luminosities (Galametz et al., 2013).

Figure 2 shows the relationships between hot gas and dense gas on a sub-kpc scale ($\thicksim$ 230 pc) in the central region (within 2 kpc) of our five sample galaxies, covering a range of log$L_{\rm 0.5-2\,keV}^{\rm gas}$ $\thicksim$ 5 dex and a range of log$L^{\prime}_{\rm dense}$ $\thicksim$ 2 dex. We adopt the Bayesian method code from the public IDL routine LINMIX_ERR (Kelly, 2007) for linear regression. In the code, the parameters were estimated by posterior median, and the error was adopted as the median absolute deviation of the posterior distribution. The density distributions of the fitted slopes are presented as insets in Figure 2. The best regression fits (not including the 3$\sigma$ upper limits) with uncertainties are listed below:

The solid lines in Figure 2 (top panels) represent the fits for HCN $J=4\rightarrow 3$ and HCO${}^{+}\ J=4\rightarrow 3$, with Spearman correlation coefficients of $r_{\rm s}$=0.84 and $r_{\rm s}$=0.90, respectively. The corresponding $p$-values are $2.7\times 10^{-10}$ and $2.0\times 10^{-14}$. The results indicate that there is a strong connection between dense gas and hot gas in the nuclear regions of galaxies. This is consistent with our expectation because both dense gas and hot gas are closely interrelated to the star formation process (Gao & Solomon, 2004a, b; Wu et al., 2005; Mineo et al., 2012b; Zhang et al., 2014; Usero et al., 2015; Tan et al., 2018).

Jiang et al. (2020) found that the HCN $J=4\rightarrow 3$ and HCO${}^{+}\ J=4\rightarrow 3$ emissions were concentrated at the inner $\thicksim$ 2 kpc of NGC 253, with a rapid drop until $\thicksim$ 0.5 kpc. Pietsch et al. (2000) found a similar concentration result that a third of diffuse soft X-ray emission of NGC 253 is in the nuclear area (1.1 kpc), which accounts for about 25% of the total X-ray luminosity. Considering our Equation (3) or Equation (4), these imply that the spatial distribution of hot gas in the central regions of galaxies may follow a similar trend to that of molecular gas, characterized by a sharp decrease at small radii and a slower decline further out.

Among the five sample galaxies, the HCN line luminosity of M82 is significantly lower than that of HCO⁺, with a mean HCN/HCO⁺ ratio of $\thicksim$ 0.3. We consider that this situation could be attributed to the low abundance of nitrogen in the sub-solar metallicity environment and/or the relatively low gas density condition of M82 (Origlia et al., 2004; Nagao et al., 2011). Recently, studies found the deficiency of nitrogen-bearing molecules was associated with the low metallicity in Large Magellanic Cloud and IC 10 (Nishimura et al., 2016a, b). Meanwhile, for low-metallicity Local Group galaxies, Braine et al. (2017) also observed weak HCN emission line compared to the HCO⁺ emission. On the other hand, the lack of high density molecular gas (Jackson et al., 1995) may also lead to the weak HCN emission, as HCO⁺ is more easily collisionally excited to $J=4\rightarrow 3$ ($n_{\rm crit}\thicksim 1.3{\times}{10^{6}}{\rm cm}^{-3}$) than HCN ($n_{\rm crit}\thicksim 5.6{\times}{10^{6}}{\rm cm}^{-3}$ for $J=4\rightarrow 3$) (Schöier et al., 2005).

We define the ratios of HCN $J=4\rightarrow 3$ / CO $J=1\rightarrow 0$ and HCO${}^{+}\ J=4\rightarrow 3$ / CO $J=1\rightarrow 0$ as proxies for the dense-gas fraction $f_{\rm dense}$ (Jiang et al., 2020). The bottom panels of Figure 2 show the diffuse X-ray luminosity as a function of $f_{\rm dense}$. A Spearman test yields a correlation coefficient of $r_{\rm s}$ = 0.84 with a $p$-value of $3.4\times 10^{-8}$ for HCN $J=4\rightarrow 3$, and $r_{\rm s}$ = 0.79 with a $p$-value of $4.2\times 10^{-9}$ for HCO${}^{+}\ J=4\rightarrow 3$, suggesting a significant correlation between hot gas and dense-gas fraction. However, we ignore the $f_{\rm dense}$ of M82 for HCN $J=4\rightarrow 3$ when analyzing the correlation between them, because the ratios of the weak HCN emission lines (discussed above) bring a huge dispersion and even lead to non-correlation.

Based on the calibrations of Kennicutt (1998) and Murphy et al. (2011), we use the total IR luminosity to calculate the SFR:

To investigate the correlation between hot gas and SFR on sub-kiloparsec scales, we analyze all JCMT beam matched positions (see Figure 1) in the central $50\arcsec\times 50\arcsec$ regions of the five galaxies. The $L_{\rm 0.5-2\,keV}^{\rm gas}$ $-$ SFR relation is shown in Figure 3, fitted using the Bayesian method code (see Sect. 3.1) which yields the following results:

with a Spearman rank correlation coefficient of 0.88 and a $p$-value of $1.4\times 10^{-34}$. This super-linear relation bears a larger dispersion of rms = 0.69 dex than the dispersions (typically 0.3-0.4 dex) observed in the global-galaxy relations (Mineo et al., 2012a, b, 2014; Lehmer et al., 2016). Kouroumpatzakis et al. (2020) suggested that the scatter in their $L_{\rm X}-{\rm SFR}$ relation at sub-galactic scale ($\geq$ 1$\times$1 ${\rm kpc}^{2}$) could be the difference of the stellar population, while the local variations of stellar population and the associated X-ray emission could be smeared out when measured for the entire galaxy. For this study at higher resolution, the intrinsic scatter is more prominent, as more factors can have a large impact on our results in such a small area.

In the seminal work by Mineo et al. (2012b), a global linear relation is found between the luminosity of unresolved diffuse emission and the SFR. For comparison, the relation, $L_{bol}$/SFR $\thicksim$ 1.5$\times$10⁴⁰ erg/s per $M_{\odot}$/yr, between the hot gas intrinsic bolometric luminosity and the SFR of them are shown in Figure 3. We also add the intrinsic bolometric luminosities of the five galaxies in Figure 3 (black circles), and the detailed values are presented in Table 1. The intrinsic bolometric luminosities are derived from the 0.5 $-$ 2 keV luminosities using a bolometric correction factor of 2, as suggested by Mineo et al. (2012b). It is clear that the global luminosities of the five galaxies are consistent with the linear relation, but there is a significant difference in our relations at low SFR. This difference implies that the hot gas environment may have dramatic variations from galactic center to the outskirts of the galaxy, thereby resulting in different relationships at different spatial scales. We consider there may be a large amount of hot gas concentrating in a small region of galaxies.

The contribution of several different types of unresolved compact sources (faint LMXBs, coronally active binaries (ABs), and cataclysmic variables (CVs)) to the total X-ray luminosity is relatively small and usually safely ignored in previous global studies. However, these unresolved stellar sources could become an important factor on small physical scales, especially for early type galaxies, where X-ray emission could be dominated by stellar sources (Revnivtsev et al., 2007a). Therefore, in order to determine the contribution of unresolved LMXBs and collective emission of ABs and CVs to the 0.5$-$2 keV band emission, we use K-band luminosity to estimate the X-ray luminosity from these components based on the calibrations of Boroson et al. (2011):

where $L_{\mathrm{K\odot}}$ is in solar luminosity. We find that, on average, the sum of the contribution of these faint compact souces accounts for about 20% of the total soft X-ray luminosity, and the low contribution does not significantly change the $L_{\rm 0.5-2\,keV}^{\rm gas}$ $-$ SFR relation. Therefore, we conclude that in the 0.5$-$2 keV band, the source-subtracted resolved emission is a reasonable estimate of the hot gas emission. Furthermore, we also find the contribution of ABs and CVs on sub-kiloparsec scales indeed dominates the unresolved X-ray emission (about twice as much as LMXBs), consistent with the results of M32 and the Galactic bulge reported by Revnivtsev et al. (2007a, b, 2009).

Last but not least, it is worth mentioning that all derived luminosities in Table 2 are determined in 2D. Projection effect is unavoidable in such measurements and may contribute to the scatters in Figure 2 and 3. Although the inclination of galaxies in our samples ranges, the X-ray emission and the associated dense gas/star formation activity are physically linked, as we are working with the inner disk regions. In addition, as shown in Figure 3, the data appear to separate into two groups with a flatter slope but different intercepts. This may be due to the limited number of measurements. We will explore with future observations to examine these differences. If confirmed, it implies that the high SFR regions have a proportionally higher production efficiency for X-ray hot gas.