Variation in X_CO Factor in N55 Region

Abstract

The $X_{CO}$ factor is defined as $X_{CO}=N\left(H_2\right)/W_{12CO}$. It is useful for estimating cloud mass. However, there is only limited research on how the $X_{CO}$ factor varies within a single cloud. Employing ¹²$CO(J=\text{1-0})$ and ¹³$CO(J=\text{1-0})$ spectral data, we computed an $X_{CO}$ factor of 3.6 $\times {10}^{20}{\mathrm{cm}}^{-2}$ (K km s⁻¹)⁻¹ for luminous gas of the N55 region. Our analysis revealed a V-shaped correlation between the $X_{CO}$ factor and $H_2$ column densities, while the relationship with excitation temperature exhibited obscurity. This suggests that the CO-to-$H_2$ conversion is not consistent on small scale (∼1 pc). Additionally, we found that star formation activity has little influence on the variability in the $X_{CO}$ factor.

1. Introduction

The most abundant molecule in the universe is molecular hydrogen, H₂. Stars form out of clouds made of $H_2$, and the measurement of molecular hydrogen mass is fundamental to comprehending the star formation process [1,2]. While its spectrum of rotational transitions is not a good tracer of the mass in molecular clouds, due to requiring a high temperature to excite its rotational transitions, the emission of bulk $H_2$ in typical clouds is invisible [3]. Therefore, estimates of $H_2$ distribution need some indirect tracers. The lowest rotational transitions of the second most abundant molecule, ¹²$CO\left(\text{1-0}\right)$, have been considered the best tracers of molecular gas due to their strong line emission and easy observability. Due to these reasons, the relation between CO integrated intensity and $H_2$ column density is frequently used to measure the CO-to-$H_2$ conversion factor, $X_{12CO}$ (hereafter, $X_{CO}$) [4]. This also makes the CO-to-$H_2$ conversion factor, $X_{CO}$, widely used for estimating cloud mass. The so-called X factor is formally defined as

$$X_{CO}=N\left(H_2\right)/W_{12CO}\left[\frac{{\mathrm{cm}}^{-2}}{\mathrm{K}\mathrm{km}{\mathrm{s}}^{-1}}\right],$$

(1)

where $N\left(H_2\right)$ is the $H_2$ column density, and $W_{12CO}$ is the integrated ¹²$CO(J=\text{1-0})$ line intensity [5]. In mass units, Equation (1) can be rewritten as

$${\alpha }_{CO}=M_{mol}/L_{12CO}\left[\frac{M_{⨀}}{\mathrm{K}\mathrm{km}{\mathrm{s}}^{-1}{\mathrm{pc}}^2}\right],$$

(2)

where $M_{mol}$ is the total molecular mass. The relationship between $X_{CO}$ and ${\alpha }_{CO}$ can be converted by a factor of 4.5 $\times {10}^9$ [6]. Various methods have been used to estimate the total gas mass, including those employing virial mass [7,8], dust emission or extinction [9,10,11], $\gamma$-ray emission [12,13,14], ¹³$CO$ [6,15], and [C II] [16,17,18].

Most studies of the $X_{CO}$ factor consider an average value over a cloud and discuss the effects of cloud temperature, density, metallicity, and velocity dispersion on the $X_{CO}$ factor [19,20,21,22,23,24]. Pineda et al. and Luo et al. [19,25] found that the $X_{CO}$ factor in low-density regions without CO emission detections is six times higher than the average value in the Milky Way. Through analysis of positionally stacked spectra, Goldsmith et al. [26] suggested that low-intensity, yet large-area, emissions could significantly contribute to the CO emission in distant regions in our galaxy and in other galaxies. Papadopoulos et al. and Madden et al. [18,27] also demonstrate that the $X_{CO}$ factor is exceptionally high in low-metallicity galaxies due to the effects of CO-dark gas. Ignoring metallicity effects, Maloney et al. [21] predicted $X_{CO}\propto T_K^{-1}$, but Shetty et al. [24] found a weaker $X-T_K$ dependence, $X_{CO}\propto T_K^{-0.5}$, where $T_K$ represents kinetic temperature. Feldmann et al. and Bigiel et al. [20,28] predicted that the $X_{CO}$ factor could vary by orders of magnitude in different environments.

However, only a few studies pay attention to variations in the $X_{CO}$ factor within a single cloud (scale from ∼10 to ∼100 pc) [29,30,31]. Pineda et al. [30] showed that $X_{CO}$ is heavily affected by the saturation of the emission above extinction $A_V\sim$4 mag. Sofue et al. and Kohno et al. [29,31] point out that the actual $X_{CO}$ factor varies with the $H_2$ column density or with the CO line intensity in giant molecular clouds (GMCs). Therefore, more studies are needed to research the variation in the $X_{CO}$ factor on a small scale.

Molecular clouds are surrounded by atomic envelopes and a transition region in which the hydrogen is primarily molecular, but the carbon is mostly atomic. These regions are known as photodissociation regions or photon-dominated regions (PDRs). More recently, they have been referred to as dark gas. Using analysis of positionally stacked spectra of the Taurus cloud, Goldsmith et al. [26] suggests a factor of two additional masses in this transition region. We also examine the effects of dark gas on the $X_{CO}$ factor in other galaxies; hence, the N55 region was selected. N55 is located inside Large Magellanic Cloud 4 (LMC), the largest supergiant shell in the LMC. In this paper, we analyze the variation in the $X_{CO}$ factor with $H_2$ column density and excitation temperature. We find that the CO-to-$H_2$ conversion is not universal on a small scale ∼1 pc. Additionally, we observe an uncertain correlation between the $X_{CO}$ factor and excitation temperature. Furthermore, we find that the influence of star formation activity on the variation in the $X_{CO}$ factor is minimal.

2. ALMA Archive Data

We use the Atacama Large Millimeter Array (ALMA) archive data of the N55 region in LMC, which is generated by the Additional Representative Image for Legacy (ARI-L) project [32]. The area of coverage was $4^{\prime }\times 6^{\prime }$ at the center position of (${05}^h{32}^m{15}^s.49,-{66}^o{26}^{\prime }{14}^{″}.00$)(J2000). The synthesized beam for ¹²$CO\left(\text{1-0}\right)$ is approximately $3^{″}.5\times 2^{″}.3$, and the position angle is $80.2^{\circ }$, which corresponds to 0.84 × 0.55 pc². The synthesized beam for ¹³$CO\left(\text{1-0}\right)$ is approximately $3^{″}.8\times 2^{″}.7$, and the position angle is $69.8^{\circ }$, which corresponds to 0.91 × 0.65 pc². In order to compare these data sets pixel by pixel, we resample data with a common resolution. Finally, the pixel size is $0.{49}^{″}$ (∼0.1 pc). The rms $\sigma$ per channel over 0.4 km s⁻¹ is ∼57 mJy beam⁻¹ and 18 mJy beam⁻¹ for ¹²$CO\left(\text{1-0}\right)$ and ¹³$CO\left(\text{1-0}\right)$ (hereafter, ¹²$CO$ and ¹³$CO$), respectively. We set zero values at the emission-free pixels (<3$\sigma$) to suppress the noise effect in our analysis. ¹²$CO$ emits beyond the area where ¹³$CO$ is detectable. Thus, comparing them pixel by pixel indicates that the following content is discussed for regions with detectable ¹³$CO$ (>3$\sigma$). Figure 1 shows ¹²$CO-$integrated intensity map of these regions with detectable ¹³$CO$ emission (>3$\sigma$). Meanwhile, we examine the effects of noise on the $X_{CO}$ factor and find that the error of only noise has little influence. Gruendl et al. and Seale et al. [33,34] identified 16 young stellar objects (YSOs) in the N55 region. The appearance of YSOs indicates ongoing star formation in these positions. The 13 YSOs are marked by red pluses in Figure 1.

3. Data Analysis

According to the radiative transfer equation, the brightness temperature ($T_B$) is expressed in terms of the excitation temperature, $T_{ex}$, and optical depth, $\tau$, as

$$T_B=T_0(\frac{1}{e^{T_0/T_{ex}}-1}-\frac{1}{e^{T_0/T_{bg}}-1})(1-e^{-\tau })\left[\mathrm{K}\right],$$

(3)

where $T_{bg}$ = 2.725 K and $T_0=h\nu /k$ are the blackbody temperature of the cosmic background radiation and the Planck temperature, respectively.

We assumed that ¹²$CO$ is optically thick; then, $1-e^{-\tau }$ tend to 1. We can rewrite Equation (3) as

$$T_B\approx T_0(\frac{1}{e^{T_0/T_{ex}}-1}-\frac{1}{e^{T_0/T_{bg}}-1})\left[\mathrm{K}\right].$$

(4)

By equivalent transformation of Equation (4), the excitation temperature is written as

$$T_{ex}\approx 5.53194/ln(1+\frac{5.53194}{T_{max}{(}^{12}CO)+0.8632})\left[\mathrm{K}\right],$$

(5)

where 5.53194 K = $h\upsilon {(}^{12}CO)/k$, and $T_{max}{(}^{12}CO)$ is the main beam brightness temperature at the peak of ¹²$CO$ emission. The excitation temperature of our sample ranges from 5 to 42 K. Assuming optically thick causes higher values of our excitation temperature.

Assuming that the emission is in local thermodynamic equilibrium (LTE), the column density of ¹³$CO$ molecules is given by Sofue et al. [31]

$$N{(}^{13}CO)\approx 3\times {10}^{14}\frac{\tau }{1-e^{-\tau }}\frac{1}{1-e^{-5.28864/T_{ex}}}{I}_{{}^{13}CO}\left[{\mathrm{cm}}^{-2}\right],$$

(6)

where $I_{{}^{13}CO}$ is the ¹³$CO$ velocity-integrated intensity. Then, we calculate $H_2$ column density using

$$N\left(H_2\right)\approx Y{(}^{13}CO)N{(}^{13}CO)\left[{\mathrm{cm}}^{-2}\right],$$

(7)

where $Y{(}^{13}CO)$ is the abundance ratio of $H_2$ to ¹³$CO$. We adopt abundance ratios of 50 for [¹²$CO$/¹³$CO$] and 1.6 × 10⁻⁵ for [¹²$CO$/$H_2$] [35]. Hence, we obtain the abundance ratio of $Y{(}^{13}CO)=3.125\times {10}^6$ and $H_2$ column density of

$$N\left(H_2\right)\approx 9.375\times {10}^{20}\frac{\tau }{1-e^{-\tau }}\frac{1}{1-e^{-5.28864/T_{ex}}}{I}_{{}^{13}CO}\left[{\mathrm{cm}}^{-2}\right].$$

(8)

4. Results and Discussion

4.1. $X_{CO}$ Factor

Fukui et al. [36] obtained an $X_{CO}$ factor of 7 × 10²⁰${\mathrm{cm}}^{-2}{\left(\mathrm{K}\mathrm{km}{\mathrm{s}}^{-1}\right)}^{-1}$ for the LMC molecular clouds. Naslim et al. [2] gave a similar value of 6.5 × 10²⁰${\mathrm{cm}}^{-2}$ (K km s⁻¹)⁻¹ for N55 clumps in the LMC. However, Hughes et al. [37] reported a value of 4 × 10²⁰${\mathrm{cm}}^{-2}$ (K km s⁻¹)⁻¹. So, we try to calculate the factor by Equation (1). We may consider that ¹³$CO$ is optically thin and is a more natural tracer of the true column density. Using Equation (3) and assuming that the excitation temperature of ¹³$CO$ is equal to that of ¹²$CO$, the optical depth can be calculated by

$$\tau {(}^{13}CO)\approx -ln(1-\frac{T_{max}{(}^{13}CO)/5.28864}{{(e^{5.28864/T_{ex}}-1)}^{-1}-0.167667}),$$

(9)

where $T_{max}{(}^{13}CO)$ is the main beam brightness temperature at the peak of ¹³$CO$ emission. We calculate optical depth and excitation temperature in each cell (grid) of the channel maps at the line-center velocity. The calculated optical depth $\tau {(}^{13}CO)$ ranges from 0.03 to 0.58, with a mean value of 0.09, which suggests that ¹³$CO$ is nearly optically thin. The optically thin line is a more natural tracer of the true column density. So, we can calculate cloud mass using optically thin tracers.

Using Equation (9) and assuming that the excitation temperature of ¹³$CO$ is equal to that of ¹²$CO$, we can obtain the optical depth of ¹³$CO$. Then, we derive the $H_2$ column density by Equation (8). Finally, using Equation (1), we obtain a CO-to-$H_2$ conversion factor of 3.6 × 10²⁰${\mathrm{cm}}^{-2}$ (K km s⁻¹)⁻¹ for the entire region, which is similar to the value given by Hughes et al. [37]. It is interesting that there is a discrepancy between the two different ways (virial mass and LTE methods) for the $X_{CO}$ factor. The result of the LTE method is smaller than the result of the virial mass method by a factor of 2. The discrepancy is similar to the result of Goldsmith et al. [26]. Using a column-density-dependent model for the CO fractional abundance, Goldsmith et al. [26] derive a mass more than twice as large as would be obtained using a canonical fixed fractional abundance of ¹³$CO$. The gas mass from the virial mass method includes all media (CO luminous and dark gases), but the $X_{CO}$ factor method only includes CO luminous gas. Our results suggest that the dark gas mass is close to the luminous gas mass.

The ¹²$CO$ emission traces the column density of molecular gas over a narrow dynamic range. It saturates at moderate column densities, as shown in a study by Kennicutt et al. [3]. We present plots of the calculated $H_2$ column densities using the $X_{CO}$ and LTE methods in Figure 2. The $H_2$ column densities are calculated using $X_{CO}=6.5\times {10}^{20}{\mathrm{cm}}^{-2}$ (K km s⁻¹)⁻¹ shown in Figure 2a, which shows almost no saturation in high-density regions. However, the $H_2$ column densities exhibit saturated values in high-density regions when using $X_{CO}=3.6\times {10}^{20}{\mathrm{cm}}^{-2}$ (K km s⁻¹)⁻¹ shown in Figure 2b. Meanwhile, Sofue et al. and Pineda et al. [30,31] have reported saturated values of ¹²$CO$ in high-density regions of clouds. Therefore, $X_{CO}=3.6\times {10}^{20}{\mathrm{cm}}^{-2}$ (K km s⁻¹)⁻¹ may be applicable for the N55 clumps.

We also examine the correlation between the mean column density $N\left(H_2\right)$ and the mean integrated ¹²$CO$ intensity $W_{12CO}$ for each region in Figure 3. A least-squares linear fit is then performed, yielding a best-fit slope of $X_{CO}=(3.4\pm 0.1)\times {10}^{20}{\mathrm{cm}}^{-2}$ (K km s⁻¹)⁻¹, consistent with the value reported by Hughes et al. [37]. The slight discrepancy between the $X_{CO}$ factor of the entire region, and each small region may be caused by different scales. We use Ramsey’s reset test to find whether the relationship is non-linear. The F-test is statistically significant (p-value $=0.016$), suggesting omitted variable bias. In other words, the non-linear relationship in Figure 3 exists. The typical scale of these single small regions is 1 pc. The non-linear relationship suggests that the fixed $X_{CO}$ factor is unbefitting on a scale of ∼1 pc.

4.2. Variability in $X_{CO}$ Factor

Regardless of the $X_{CO}$ factor, Figure 2 shows the non-linear growth of the curve compared to the LTE method. We use Ramsey’s reset test to find whether the relationship is non-linear. The F-test is statistically significant (p-value ≪ 0.001), suggesting that there is omitted variable bias. In other words, the non-linear relationship in Figure 2 exists. This non-linear correlation implies that the $X_{CO}$ factor is not constant at the pixel scale (∼0.1 pc), and different $X_{CO}$ factors are required. In Figure 4, plots of the distribution of the $X_{CO}$ factor are shown. The $X_{CO}$ factor varies within the range of $X_{CO}$∼(1.3–20.6) × ${10}^{20}{\mathrm{cm}}^{-2}$ (K km s⁻¹)⁻¹.

The plot of the $X_{CO}$ factor and $H_2$ column densities shows a V-shaped behavior, with the turning point occurring at around ${10}^{21}{\mathrm{cm}}^{-2}$. The V-shaped behavior suggests a higher $X_{CO}$ factor is needed at low $H_2$ density regions, as observed in [19,25]. At higher $H_2$ density regions, the $X_{CO}$ factor increases with increasing column density. This can lead to underestimated or overestimated $H_2$ column densities if a fixed $X_{CO}$ factor, usually derived from the mean value of the column density in individual molecular clouds or regions, is used. The V shape is due to the functional property of the “Q” function, as emphasized in reference [29,31]. Hence, the V shape is a general law of the transfer equation as a function of $T_{ex}$ but does not express clouds.

We find that the distribution of $X_{CO}$ factors for clumps with and without YSOs shows no significant differences in Figure 4, implying that star formation has little influence on determining the $X_{CO}$ factor. Similar conclusions were drawn by Hughes et al. [37], where the difference in $X_{CO}$ factors between young GMCs and other GMCs was only marginally significant. Additionally, Naslim et al. [2] found that molecular cores associated with YSOs generally exhibit larger linewidths and masses. The similarity in the distributions of $X_{CO}$ factors between clumps with and without YSOs also suggests that the $X_{CO}$ factor is insensitive to the velocity structure, which agrees with the results of Shetty et al. [24].

In Figure 5, we plot the $X_{CO}$ factor as a function of excitation temperature for the entire region. We find that the $X_{CO}$ factor decreases with increasing excitation temperature and exhibits a lower limit ranging from $1.3\times {10}^{20}{\mathrm{cm}}^{-2}$ (K km s⁻¹)⁻¹ to $6\times {10}^{20}{\mathrm{cm}}^{-2}$ (K km s⁻¹)⁻¹. However, this lower limit shows significant dispersion, with some pixels even showing a lack of convergence. To further investigate, we examine the correlation between the $X_{CO}$ factor and excitation temperature for each clump. In Figure 6, we display the distribution of the $X_{CO}$ factor for two clumps for contrast, while additional plots are provided in the Supplementary Materials. We find that these clumps show lower dispersion and different properties. one clump shows a V-shaped distribution of the $X_{CO}$ factor, increasing with excitation temperature when $T_{ex}>10$ K, while the other clump shows that the $X_{CO}$ factor remains almost constant. Similarly, in the Supplementary Materials, some clumps show an increase in the $X_{CO}$ factor with increasing excitation temperature, whereas others show a constant $X_{CO}$ factor, suggesting an obscure correlation between the $X_{CO}$ factor and excitation temperature. Additionally, the commonly observed lower transitions of CO are easily thermalized ($T_{ex}=T_K$). If we consider that the kinetic temperature is equal to the excitation temperature, this also suggests an obscure correlation between the $X_{CO}$ factor and kinetic temperature. This is consistent with Kohno et al. [29] that there is no clear correlation between the $X_{CO}$ factor and the ¹²$CO(J=\text{3-2}/\text{1-0})$ intensity ratio, which depends on the kinetic temperature.

5. Conclusions

Using ALMA spectral data, we computed an $X_{CO}$ factor of $3.6\times {10}^{20}{\mathrm{cm}}^{-2}$ (K km s⁻¹)⁻¹ for the N55 region in the LMC. Furthermore, we investigated the variation in the $X_{CO}$ factor with $H_2$ column density and excitation temperature. The correlation between the $X_{CO}$ factor and $H_2$ column densities reveals a V-shaped trend, while the relationship between the $X_{CO}$ factor and excitation temperature exhibits obscurity. These findings suggest that the CO-to-$H_2$ conversion is not consistent on a small scale (∼1 pc). Additionally, star formation activity appears to have minimal influence on the variation in the $X_{CO}$ factor.