Neutral Carbon Emission in Luminous Infrared Galaxies: The [C i] Lines as Total Molecular Gas Tracers^∗

Figure 1 shows the correlations between ${L}_{\mathrm{CO}(1-0)}^{\prime }$ and ${L}_{[{\rm{C}}\,{\rm{I}}](1-0)}^{\prime }$ (panel (a)), and ${L}_{[{\rm{C}}\,{\rm{I}}](2-1)}^{\prime }$ (panel (b)), which are the luminosities of CO(1–0), [C i] (1–0), and [C i] (2–1), respectively. The filled black circles are the 23 galaxies with both [C i] (1–0) and [C i] (2–1) detected. The red open circles represent the other galaxies that have [C i] (2–1) detections (panel (b)) and [C i] (1–0) upper limits (panel (a)). These plots show that CO(1–0) is well correlated with both of the [C i] lines, with the corresponding correlation coefficients of 0.81 and 0.85 in panels (a) and (b), respectively. Unweighted least-squares linear fits, using a geometrical mean functional relationship (Isobe et al. 1990), to these 23 sources with both [C i] detections, give:

$$\begin{eqnarray}\mathrm{log}{L}_{\mathrm{CO}(1-0)}^{\prime } & = & (-0.47\pm 0.76)\\ & & +(1.12\pm 0.09)\ \mathrm{log}{L}_{[{\rm{C}}\,{\rm{I}}](1-0)}^{\prime },\end{eqnarray} \tag{ 1 }$$

and

$$\begin{eqnarray}\mathrm{log}{L}_{\mathrm{CO}(1-0)}^{\prime } & = & (0.56\pm 1.55)\\ & & +(1.07\pm 0.19)\ \mathrm{log}{L}_{[{\rm{C}}\,{\rm{I}}](2-1)}^{\prime },\end{eqnarray} \tag{ 2 }$$

with vertical scatters of 0.18 and 0.21 dex, respectively. The fitted trends are shown in Figure 1 as dashed black lines. Furthermore, we obtain the following ${L}_{\mathrm{CO}(1-0)}^{\prime }$−${L}_{[{\rm{C}}\,{\rm{I}}](2-1)}^{\prime }$ relation for all of the 71 galaxies from a geometrical mean fitting:

$$\begin{eqnarray}\mathrm{log}{L}_{\mathrm{CO}(1-0)}^{\prime } & = & (1.46\pm 0.69)\\ & & +(0.97\pm 0.08)\ \mathrm{log}{L}_{[{\rm{C}}\,{\rm{I}}](2-1)}^{\prime },\end{eqnarray} \tag{ 3 }$$

with a scatter of 0.19 dex. The fitted relation is over-plotted in Figure 1(b) with a solid black line. We also obtain the following ${L}_{\mathrm{CO}(1-0)}^{\prime }$−${L}_{[{\rm{C}}\,{\rm{I}}](1-0)}^{\prime }$ correlation (the solid black line in Figure 1(a)) using a Bayesian regression method (Kelly 2007) that also takes into consideration all the upper limits of [C i] (1–0):

$$\begin{eqnarray}\mathrm{log}{L}_{\mathrm{CO}(1-0)}^{\prime } & = & (-0.19\pm 1.26)\\ & & +(1.09\pm 0.15)\ \mathrm{log}{L}_{[{\rm{C}}\,{\rm{I}}](1-0)}^{\prime },\end{eqnarray} \tag{ 4 }$$

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Equations (1)–(4) suggest that the CO(1–0) emission likely has a linear correlation with the [C i] emission. For the [C i] (1–0) and [C i] (2–1) lines, the fitted slopes only have marginal differences, and are consistent with each other within 1σ. The nearly linear relations between ${L}_{\mathrm{CO}(1-0)}^{\prime }$ and ${L}_{[{\rm{C}}\,{\rm{I}}]}^{\prime }$ indicate that the CO(1–0) and [C i] emissions might arise from similar regions within galaxies.

Considering that the low-J CO emission is a commonly used tracer of the total molecular gas, the [C i] (1–0) and [C i] (2–1) lines can thus be a new avenue by which to determine the total molecular gas, at least in (U)LIRGs. This might be particularly useful for measuring the total molecular gas mass in high-z galaxies, since their CO(1–0) lines are difficult to observe using ground-based facilities, whereas the [C i] lines from distant sources become accessible for ground mm/submm telescopes.

Given such a strong (and almost linear) correlation, we also fitted the ${L}_{\mathrm{CO}(1-0)}^{\prime }$−${L}_{[{\rm{C}}\,{\rm{I}}]}^{\prime }$ relations with a fixed slope of 1 in order to reduce any systemic uncertainties caused by the sample itself (e.g., sample size, dynamic range, etc.), resulting in:

$$\begin{eqnarray}&&\mathrm{log}{L}_{\mathrm{CO}(1-0)}^{\prime }=(0.65\pm 0.02)+\mathrm{log}{L}_{[{\rm{C}}\,{\rm{I}}](1-0)}^{\prime },\end{eqnarray} \tag{ 5 }$$

with a scatter of 0.17 dex, and

$$\begin{eqnarray}&&\mathrm{log}{L}_{\mathrm{CO}(1-0)}^{\prime }=(1.19\pm 0.01)+\mathrm{log}{L}_{[{\rm{C}}\,{\rm{I}}](2-1)}^{\prime },\end{eqnarray} \tag{ 6 }$$

with a scatter of 0.19 dex. The fitted relations are also plotted in Figure 1 with dashed–dotted blue lines. If we adopt the CO conversion factor α_CO ≡ M(H₂)_CO/${L}_{\mathrm{CO}(1-0)}^{\prime }$ = $0.8\,{M}_{\odot }\,{({\rm{K}}\mathrm{km}{{\rm{s}}}^{-1}{\mathrm{pc}}^{2})}^{-1}$ (Downes & Solomon 1998), we can obtain the [C i] conversion factors ${\alpha }_{[{\rm{C}}{\rm{I}}],\mathrm{CO}}$, i.e., ${\alpha }_{[{\rm{C}}{\rm{I}}](1-0),\mathrm{CO}}\,=3.6\pm 0.2\,{M}_{\odot }$ ${({\rm{K}}\mathrm{km}{{\rm{s}}}^{-1}{\mathrm{pc}}^{2})}^{-1}$ and ${\alpha }_{[{\rm{C}}{\rm{I}}](2-1),\mathrm{CO}}=12.5\,\pm 0.3\,{M}_{\odot }$ ${({\rm{K}}\mathrm{km}{{\rm{s}}}^{-1}{\mathrm{pc}}^{2})}^{-1}$, respectively. Here we only considered the fitted errors of the intercepts. A more practical choice is to take the scatters in the fitted relations as the final uncertainties, e.g., a factor of ∼1.5.

The galaxy with the largest deviation in Figure 1(b) is NGC 6240, in which the gas heating is likely dominated by shocks (Meijerink et al. 2012). As shown in Figure 2(a), NGC 6240 has the highest excitation temperature in the sample. Therefore, the temperature influence on the level populations of C i in NGC 6240 is likely the highest in this sample.

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In Figure 2, we show the luminosity ratios of ${L}_{[{\rm{C}}\,{\rm{I}}](1-0)}^{\prime }$/${L}_{\mathrm{CO}(1-0)}^{\prime }$ and ${L}_{[{\rm{C}}\,{\rm{I}}](2-1)}^{\prime }$/${L}_{\mathrm{CO}(1-0)}^{\prime }$, as well as the [C i] line ratio ${L}_{[{\rm{C}}\,{\rm{I}}](2-1)}^{\prime }/{L}_{[{\rm{C}}\,{\rm{I}}](1-0)}^{\prime }$ (hereafter ${R}_{[{\rm{C}}{\rm{I}}]}$) as a function of the IRAS $60\,\mu {\rm{m}}/100\,\mu {\rm{m}}$ color (${f}_{60}/{f}_{100}$) for our sample. We check possible correlations in these plots using the cenreg function in the NADA package within the R¹² statistical software environment. As labeled in panels (a) and (b) of Figures 2, the likelihood-r coefficients are 0.11 and 0.33, respectively, with the possibilities of having no correlation at 0.36 and $5.4\times {10}^{-3}$, respectively. Therefore, we conclude that there is no correlation between ${L}_{[{\rm{C}}\,{\rm{I}}](1-0)}^{\prime }$/${L}_{\mathrm{CO}(1-0)}^{\prime }$ and ${f}_{60}/{f}_{100}$ (Figure 2(a)), whereas ${L}_{[{\rm{C}}\,{\rm{I}}](2-1)}^{\prime }$/${L}_{\mathrm{CO}(1-0)}^{\prime }$ shows a weak correlation with ${f}_{60}/{f}_{100}$ (Figure 2(b)), with a monotone increasing trend. Figure 2(c) shows that ${R}_{[{\rm{C}}{\rm{I}}]}$ is modestly correlated with ${f}_{60}/{f}_{100}$.

The observed weak correlations involving the [C i] (2–1) line might be due to the relatively higher energy required for exciting [C i] (2–1), compared to that of [C i] (1–0) and CO(1–0). [C i] (2–1) is sensitive to high gas temperatures, which also correlates to ${f}_{60}/{f}_{100}$ as an indicator of the intensity of the ambient UV field (thus the gas temperature, e.g., Abel et al. 2009). As shown in next subsection, the excitation temperatures of our sample galaxies are typically in the range of 20–30 K, which is very close to the excitation energy of [C i] (1–0) (24 K) but significantly lower than the excitation energy of [C i] (2–1) (63 K). Therefore, the [C i] (2–1) is more sensitive to temperature.

The nearly linear luminosity correlations between the [C i] lines and CO(1–0) suggest that either [C i] line can substitute CO(1–0) as a tracer of the total molecular gas mass. In this subsection, we further calculate the total molecular gas mass directly from the observed [C i] line fluxes to derive a carbon conversion factor ${\alpha }_{[{\rm{C}}{\rm{I}}]}$, and check whether ${\alpha }_{[{\rm{C}}{\rm{I}}]}$ agrees with ${\alpha }_{[{\rm{C}}{\rm{I}}],\mathrm{CO}}$. Here we also include those seven galaxies that are detected in both [C i] lines but lack CO(1–0) data.

To derive the total molecular gas mass, we adopt the equation given by Papadopoulos et al. (2004) assuming optically thin [C i] emission:

$$\begin{eqnarray}&&M{({{\rm{H}}}_{2})}_{[{\rm{C}}{\rm{I}}]}=\displaystyle \frac{4\pi {m}_{{{\rm{H}}}_{2}}}{{{hcA}}_{\mathrm{ul}}{X}_{[{\rm{C}}{\rm{I}}]}}\left(\displaystyle \frac{{D}_{{\rm{L}}}^{2}}{1+z}\right){Q}_{\mathrm{ul}}^{-1}{I}_{[{\rm{C}}{\rm{I}}]},\end{eqnarray} \tag{ 7 }$$

where ${Q}_{\mathrm{ul}}={N}_{{\rm{u}}}/{N}_{[{\rm{C}}{\rm{I}}]}$ is the excitation factor that depends on the gas temperature (${T}_{\mathrm{kin}}$), density (n) (Papadopoulos et al. 2004), and the radiation field, A_ul the Einstein coefficient, with ${A}_{10}=7.93\times {10}^{-8}\,{{\rm{s}}}^{-1}$ and ${A}_{21}=2.68\times {10}^{-7}\,{{\rm{s}}}^{-1}$, ${X}_{[{\rm{C}}{\rm{I}}]}$ the abundance ratio of C i to ${{\rm{H}}}_{2}$, and here we adopt $3\times {10}^{-5}$ (Weiß et al. 2003), z the redshift, and ${D}_{{\rm{L}}}$ the luminosity distance.

The line ratio ${R}_{[{\rm{C}}{\rm{I}}]}$ can be used to estimate the excitation temperature by adopting the equation ${T}_{\mathrm{ex}}=38.8\,{\rm{K}}\,/\mathrm{ln}[2.11/{R}_{[{\rm{C}}{\rm{I}}]}]$ (Stutzki et al. 1997). For those 30 galaxies having both [C i] lines detected, their T_ex are given in Table 2. The table shows that the excitation temperatures of most sources in our sample are in the range of 20–30 K. To calculate ${Q}_{\mathrm{ul}}$, we used Equations (A21) and (A22) in Papadopoulos et al. (2004) to avoid assuming local thermodynamic equilibrium (LTE). These equations are derived assuming an optically thin case and using the weak radiation field approximation, i.e., ${T}_{\mathrm{kin}}\gg {T}_{\mathrm{CMB}}\sim 2.7$ K, where ${T}_{\mathrm{CMB}}$ is the temperature of the cosmic microwave background. We also simply set T_ex as ${T}_{\mathrm{kin}}$, since it is difficult to have an accurate ${T}_{\mathrm{kin}}$ and ${T}_{\mathrm{ex}}$ might not deviate much from ${T}_{\mathrm{kin}}$ for our sample galaxies. Then, at each ${T}_{\mathrm{ex}}$, we computed ${Q}_{\mathrm{ul}}$ for the four gas densities of 10³, 3 × 10³, 5 × 10³, and ${10}^{4}\,{\mathrm{cm}}^{-3}$ according to Papadopoulos & Greve (2004), and adopted the averaged value as the final ${Q}_{\mathrm{ul}}$. However, we find that ${Q}_{10}/{Q}_{10}^{(\mathrm{LTE})}\sim 0.94$ and ${Q}_{21}/{Q}_{21}^{(\mathrm{LTE})}\sim 0.76$, where ${Q}_{\mathrm{ul}}^{(\mathrm{LTE})}$ was calculated in LTE using the same assumed ${T}_{\mathrm{kin}}$, indicating that our adopted physical parameters are very close to an LTE condition. The derived masses of carbon and ${{\rm{H}}}_{2}$ are also listed in Table 2.

Table 2.  Physical Parameters of the Sample

Galaxy	${R}_{[{\rm{C}}{\rm{I}}]}$	Tex	${M}_{[{\rm{C}}{\rm{I}}]}$ a	$M{({{\rm{H}}}_{2})}_{\mathrm{CO}}$	$M{({{\rm{H}}}_{2})}_{[{\rm{C}}{\rm{I}}](1-0)}$	$M{({{\rm{H}}}_{2})}_{[{\rm{C}}{\rm{I}}](2-1)}$
		(K)	$({10}^{6}\,{M}_{\odot })$	$({10}^{9}\,{M}_{\odot })$	$({10}^{9}\,{M}_{\odot })$	$({10}^{9}\,{M}_{\odot })$
Arp 193	0.54 ± 0.07	28.4 ± 2.9	1.5 ± 0.2	4.3 ± 0.9	8.3 ± 1.1	10.5 ± 2.2
Arp 220	0.35 ± 0.07	21.7 ± 2.3	2.5 ± 0.5	6.6 ± 1.3	14.2 ± 2.6	17.8 ± 4.0
CGCG 049-057	0.25 ± 0.05	18.2 ± 1.6	0.5 ± 0.1	1.0 ± 0.2	2.9 ± 0.6	3.6 ± 0.9
ESO 320-G030	0.34 ± 0.05	21.4 ± 1.8	0.3 ± 0.05	0.8 ± 0.2	1.7 ± 0.3	2.1 ± 0.5
IRASF 18293-3413	0.34 ± 0.03	21.3 ± 1.2	3.5 ± 0.4	9.8 ± 3.0	19.5 ± 2.0	24.4 ± 5.5
Mrk 331	0.38 ± 0.04	22.6 ± 1.5	1.0 ± 0.1	4.5 ± 1.1	5.4 ± 0.6	6.8 ± 1.5
NGC 3256	0.45 ± 0.04	25.3 ± 1.6	1.1 ± 0.1	3.6 ± 1.1	5.9 ± 0.5	7.5 ± 1.6
NGC 5135	0.43 ± 0.04	24.4 ± 1.5	1.2 ± 0.1	2.8 ± 0.4	6.8 ± 0.6	8.5 ± 1.9
NGC 6240	0.81 ± 0.09	40.7 ± 4.8	3.5 ± 0.3	7.5 ± 1.2	19.6 ± 1.8	25.2 ± 4.6
NGC 7469	0.44 ± 0.05	24.9 ± 1.7	1.1 ± 0.1	2.9 ± 0.9	5.9 ± 0.6	7.5 ± 1.6
NGC 7552	0.46 ± 0.05	25.5 ± 1.7	0.3 ± 0.03	0.7 ± 0.2	1.7 ± 0.2	2.2 ± 0.5
NGC 7771	0.31 ± 0.03	20.1 ± 1.2	1.1 ± 0.1	2.7 ± 0.6	5.9 ± 0.7	7.4 ± 1.7
NGC 6286	0.20 ± 0.03	16.5 ± 1.1	2.5 ± 0.4	3.0 ± 0.7	14.0 ± 2.2	17.4 ± 4.3
CGCG 052-037	0.23 ± 0.05	17.7 ± 1.5	1.8 ± 0.3	1.6 ± 0.5	9.8 ± 2.0	12.3 ± 3.0
NGC 0828	0.23 ± 0.05	17.2 ± 1.7	1.7 ± 0.4	4.4 ± 0.7	9.3 ± 2.1	11.6 ± 2.9
NGC 2369	0.33 ± 0.05	20.9 ± 1.7	0.8 ± 0.1	2.5 ± 0.7	4.2 ± 0.7	5.3 ± 1.2
NGC 6701	0.18 ± 0.03	15.6 ± 1.1	1.6 ± 0.3	1.8 ± 0.3	9.1 ± 1.6	11.3 ± 2.8
UGC 02238	0.25 ± 0.06	18.4 ± 1.9	1.4 ± 0.3	3.0 ± 0.6	8.0 ± 1.8	10.0 ± 2.4
VV 340	0.20 ± 0.03	16.5 ± 1.0	5.0 ± 0.8	19.9 ± 6.0	27.9 ± 4.3	34.7 ± 8.5
NGC 2623	0.51 ± 0.09	27.5 ± 3.3	0.6 ± 0.1	2.1 ± 0.4	3.4 ± 0.6	4.3 ± 0.9
MCG+12-02-001	0.54 ± 0.09	28.4 ± 3.6	0.5 ± 0.08	2.2 ± 0.4	2.8 ± 0.5	3.5 ± 0.7
IC 1623	0.35 ± 0.04	21.6 ± 1.4	1.6 ± 0.2	7.8 ± 1.6	9.0 ± 1.0	11.3 ± 2.5
NGC 0232	0.26 ± 0.04	18.5 ± 1.5	2.1 ± 0.4	5.4 ± 1.6	11.7 ± 2.0	14.6 ± 3.4
ESO 264-G036	0.23 ± 0.05	17.5 ± 1.6	2.0 ± 0.4		11.1 ± 2.2	13.8 ± 3.3
IRAS 12116-5615	0.38 ± 0.09	22.6 ± 3.1	1.5 ± 0.3		8.4 ± 2.0	10.6 ± 2.3
UGC 12150	0.21 ± 0.05	16.8 ± 1.7	1.6 ± 0.4		8.6 ± 2.0	10.8 ± 2.6
IRAS 13120-5453	0.42 ± 0.06	24.1 ± 2.1	4.8 ± 0.7		26.9 ± 3.7	33.9 ± 7.2
ESO 173-G015	0.55 ± 0.07	28.9 ± 2.6	0.5 ± 0.06		3.0 ± 0.3	3.9 ± 0.8
IC 4687	0.34 ± 0.05	21.4 ± 1.7	0.7 ± 0.1		4.0 ± 0.6	5.1 ± 1.1
NGC 0034	0.25 ± 0.04	18.1 ± 1.5	0.8 ± 0.1		4.2 ± 0.8	5.2 ± 1.3

Notes. Column 2 is the line ratio of [C i] (2–1) to [C i] (1–0); Column 3 is the excitation temperature derived with the [C i] lines; Column 4 is the carbon mass calculated from [C i] (1–0); Column 5 is the ${{\rm{H}}}_{2}$ gas mass calculated from the CO(1–0) line; Columns 6 and 7 are the ${{\rm{H}}}_{2}$ gas mass calculated from the [C i] (1–0) and [C i] (2–1) lines.

^aThe carbon mass ${M}_{[{\rm{C}}{\rm{I}}]}=6\times M({{\rm{H}}}_{2}){X}_{[{\rm{C}}{\rm{I}}]}$, is calculated using the [C i] (1–0) line.

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In the top panels of Figure 3, we plot the resulting $M({{\rm{H}}}_{2})$ as a function of ${L}_{[{\rm{C}}\,{\rm{I}}]}^{\prime }$. The red and olive lines in each plot show the two cases with $(n,{T}_{\mathrm{kin}})=({10}^{3}\,{\mathrm{cm}}^{-3}$, $20\,{\rm{K}}$) and (${10}^{4}\,{\mathrm{cm}}^{-3}$, $30\,{\rm{K}}$), respectively. The dashed black lines are unweighted least-squares fits to these data points, which give:

$$\begin{eqnarray}\mathrm{log}M{({{\rm{H}}}_{2})}_{[{\rm{C}}{\rm{I}}](1-0)} & = & (0.63\pm 0.19)\\ & & +(1.02\pm 0.02)\ \mathrm{log}{L}_{[{\rm{C}}\,{\rm{I}}](1-0)}^{\prime },\end{eqnarray} \tag{ 8 }$$

and

$$\begin{eqnarray}\mathrm{log}M{({{\rm{H}}}_{2})}_{[{\rm{C}}{\rm{I}}](2-1)} & = & (1.78\pm 1.00)\\ & & +(0.96\pm 0.13)\ \mathrm{log}{L}_{[{\rm{C}}\,{\rm{I}}](2-1)}^{\prime },\end{eqnarray} \tag{ 9 }$$

with scatters of 0.04 dex and 0.24 dex, respectively. We also fit these models using a fixed slope of 1, and obtained:

$$\begin{eqnarray}&&\mathrm{log}M{({{\rm{H}}}_{2})}_{[{\rm{C}}{\rm{I}}](1-0)}=(0.88\pm 0.01)+\mathrm{log}{L}_{[{\rm{C}}\,{\rm{I}}](1-0)}^{\prime },\end{eqnarray} \tag{ 10 }$$

and

$$\begin{eqnarray}&&\mathrm{log}M{({{\rm{H}}}_{2})}_{[{\rm{C}}{\rm{I}}](2-1)}=(1.44\pm 0.02)+\mathrm{log}{L}_{[{\rm{C}}\,{\rm{I}}](2-1)}^{\prime },\end{eqnarray} \tag{ 11 }$$

with scatters of 0.05 and 0.24 dex. The fitted results are plotted in Figure 3 as dashed–dotted blue lines.

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As shown in Figures 3(a) and (b), the derived model of $M{({{\rm{H}}}_{2})}_{[{\rm{C}}{\rm{I}}](1-0)}$ from ${L}_{[{\rm{C}}\,{\rm{I}}](1-0)}^{\prime }$ shows a smaller dispersion than that of $M{({{\rm{H}}}_{2})}_{[{\rm{C}}{\rm{I}}](2-1)}$ from ${L}_{[{\rm{C}}\,{\rm{I}}](2-1)}^{\prime }$. This is because ${Q}_{21}$ is much more sensitive to temperature and density than ${Q}_{10}$, which is illustrated clearly in Figures 3(c) and (d). Hence, the [C i] (1–0) emission is a better total molecular gas mass tracer compared to the [C i] (2–1) line in the theoretical aspect, if we do not have constrains on the gas temperature and density. However, in practice the variation of the C i abundance (see below) and/or violation of the assumption of optically thin [C i] emission may wash out this effect and result in similar scatter for both lines, as indicated in Figure 1.

We calculated the conversion factors using Equations (10) and (11) and obtained ${\alpha }_{[{\rm{C}}{\rm{I}}](1-0)}=7.6\pm 0.2$ ${({\rm{K}}\mathrm{km}{{\rm{s}}}^{-1}{\mathrm{pc}}^{2})}^{-1}$ and ${\alpha }_{[{\rm{C}}{\rm{I}}](2-1)}=27.5\pm 1.3$ ${({\rm{K}}\mathrm{km}{{\rm{s}}}^{-1}{\mathrm{pc}}^{2})}^{-1}$. These values are about 2 times larger than those (see Section 3.1) derived from the [C i]−CO relations with ${\alpha }_{\mathrm{CO}}=0.8\,{({\rm{K}}\mathrm{km}{{\rm{s}}}^{-1}{\mathrm{pc}}^{2})}^{-1}$. Several reasons could cause these discrepancies: (1) the adopted ${\alpha }_{\mathrm{CO}}$ is smaller than its real value; (2) our adopted ${X}_{[{\rm{C}}{\rm{I}}]}$ is too high; and/or (3) the density we adopted to calculate ${Q}_{\mathrm{ul}}$ is too low. Indeed, Papadopoulos et al. (2012) and Scoville et al. (2016) found that even in (U)LIRGs ${\alpha }_{\mathrm{CO}}$ can be as high as the Galactic value. Regarding ${X}_{[{\rm{C}}{\rm{I}}]}$, several studies on local and high-redshift star-forming environments have shown that it is in the range of $(2\mbox{--}16)\times {10}^{-5}$ (Pety et al. 2004; Weiß et al. 2005; Walter et al. 2011). Furthermore, ${Q}_{\mathrm{ul}}$ varies by a factor of 1–2 when the density changes from 10³ to 10⁴ cm⁻³ within our temperature range. Therefore, all of these uncertainties give our estimators accuracies of a factor of 2–3.