Physical conditions of the molecular gas in metal-poor galaxies

TL;DR

This study investigates the physical conditions of molecular gas in metal-poor galaxies, revealing unique properties such as low temperatures, high densities, and unusual isotopic ratios, through new CO and atomic carbon observations.

Contribution

It provides the first detailed physical modeling of molecular gas in a low-metallicity galaxy, highlighting unexpected chemical abundance ratios and physical conditions.

Findings

NGC 1140 shows extreme CO line ratios (~2).

Molecular gas is cool (Tkin<=20 K) and dense (nH2 >= 10^6 cm^-3).

Unusually low [12CO]/[13CO] ratio (~8-20) in NGC 1140.

Abstract

Studying the molecular component of the interstellar medium in metal-poor galaxies has been challenging because of the faintness of carbon monoxide emission, the most common proxy of H2. Here we present new detections of molecular gas at low metallicities, and assess the physical conditions in the gas through various CO transitions for 8 galaxies. For one, NGC 1140 (Z/Zsun ~ 0.3), two detections of 13CO isotopologues and atomic carbon, [CI](1-0), and an upper limit for HCN(1-0) are also reported. After correcting to a common beam size, we compared 12CO(2-1)/12CO(1-0) (R21) and 12CO(3-2)/12CO(1-0) (R31) line ratios of our sample with galaxies from the literature and find that only NGC 1140 shows extreme values (R21 ~ R31 ~ 2). Fitting physical models to the 12CO and 13CO emission in NGC 1140 suggests that the molecular gas is cool (kinetic temperature Tkin<=20 K), dense (H2 volume density nH2 >= $10^6$ cm$^{-3}$), with moderate CO column density (NCO ~ $10^{16}$ cm$^{-2}$) and low filling factor. Surprisingly, the [12CO]/[13CO] abundance ratio in NGC 1140 is very low (~ 8-20), lower even than the value of 24 found in the Galactic Center. The young age of the starburst in NGC 1140 precludes 13C enrichment from evolved intermediate-mass stars; instead we attribute the low ratio to charge-exchange reactions and fractionation, because of the enhanced efficiency of these processes in cool gas at moderate column densities. Fitting physical models to 12CO and [CI](1-0) emission in NGC 1140 gives an unusually low [12CO]/[12C] abundance ratio, suggesting that in this galaxy atomic carbon is at least 10 times more abundant than 12CO.

Tables

Table 1. Table 1: Parameters for observed galaxies

Name	Pointed position (J2000)		Redshift	Distance	Distance	Sizeb	Beam FWHM
	RA	Dec.		(Mpc)	methoda		12CO(1–0)	12CO(2–1)	12CO(3–2)
CGCG 007$-$025	09:44:01.9	$-$00:38:32.0	0.00483	24.5	CMB	27″ $\times$ 16$\stackrel{\prime \prime }{.}$2	21$\stackrel{\prime \prime }{.}$4	10$\stackrel{\prime \prime }{.}$7	$-$
							2.54 kpc	1.27 kpc	$-$
II Zw 40	05:55:42.6	03:23:31.5	0.00263	11.7	CMB	33$\stackrel{\prime \prime }{.}$6 $\times$ 13$\stackrel{\prime \prime }{.}$2	$-$	27$\stackrel{\prime \prime }{.}$1	18$\stackrel{\prime \prime }{.}$1
							$-$	1.537 kpc	1.027 kpc
Mrk 996	01:27:35.5	$-$06:19:36.0	0.00541	18.1	CMB	36″ $\times$ 30″	21$\stackrel{\prime \prime }{.}$5	10$\stackrel{\prime \prime }{.}$7	18$\stackrel{\prime \prime }{.}$1
							1.88 kpc	0.94 kpc	1.59 kpc
NGC 1140	02:54:33.6	$-$10:01:40.0	0.00501	19.7	TF	102″ $\times$ 54″	21$\stackrel{\prime \prime }{.}$4	10$\stackrel{\prime \prime }{.}$7	18$\stackrel{\prime \prime }{.}$1
							2.04 kpc	1.02 kpc	1.73 kpc
NGC 1156	02:59:42.2	25:14:14.0	0.00125	8.1	Stars	198″ $\times$ 150″	21$\stackrel{\prime \prime }{.}$4	10$\stackrel{\prime \prime }{.}$7	$-$
							0.84 kpc	0.42 kpc
NGC 3353 (Haro 3)	10:45:22.40	55:57:37.0	0.00315	18.1	TF	72″ $\times$ 49$\stackrel{\prime \prime }{.}$8	21$\stackrel{\prime \prime }{.}$4	10$\stackrel{\prime \prime }{.}$7	$-$
							1.88 kpc	0.94 kpc
NGC 7077	21:29:59.6	02:24:51.0	0.00384	17.2	TF	48″ $\times$ 42″	21$\stackrel{\prime \prime }{.}$4	10$\stackrel{\prime \prime }{.}$7	18$\stackrel{\prime \prime }{.}$1
							1.78 kpc	0.89 kpc	1.51 kpc
SBS 0335$-$052	03:37:44.0	$-$05:02:40.0	0.01352	53.6	CMB	13$\stackrel{\prime \prime }{.}$8 $\times$ 12$\stackrel{\prime \prime }{.}$0	$-$	27$\stackrel{\prime \prime }{.}$4	18$\stackrel{\prime \prime }{.}$3
							$-$	7.12 kpc	4.76 kpc
UM 448	11:42:12.4	00:20:03.0	0.01856	81.2	CMB	24″ $\times$ 24″	21$\stackrel{\prime \prime }{.}$7	10$\stackrel{\prime \prime }{.}$9	18$\stackrel{\prime \prime }{.}$4
							8.54 kpc	4.29 kpc	7.244,kpc
UM 462	11:52:37.2	$-$02:28:10.0	0.00353	19.5	CMB	36″ $\times$ 30″	21$\stackrel{\prime \prime }{.}$4	10$\stackrel{\prime \prime }{.}$7	18$\stackrel{\prime \prime }{.}$1
							2.02 kpc	1.01 kpc	1.71 kpc

Table 2. Table 2: CO emission line parameters for CGCG 007 − - 025, Mrk 996, NGC 1156, and NGC 3353 a

Name	Systemic	Transition	Component	Beam	Channel	Trms	Tpeak	Offsetb	FWHM	$I_{CO}$	$I_{CO}^{\mathrm{cor}}$ c	$S_{CO}^{\mathrm{cor}}$ c	Log($L_{\mathrm{cor}}^{\prime }$) c
	velocity			size	width	(mK)	(mK)	( km s-1)	( km s-1)	(K km s-1)	(K km s-1)	(Jy  km s-1)	(K km s-1 pc2)
	( km s-1)			(″)	( km s-1)
CGCG 007-025	1449	12CO(1–0)	Total	21.4	15.7	1.3	2.3	-34	129	0.314 ( 0.092)	0.314 ( 0.092)	1.60	6.368 ( 0.126)
CGCG 007-025		12CO(2–1)	Total	10.7	10.5	3.9	$-$	$-$	$-$	0.31	$-$	$-$	5.83
CGCG 007-025		12CO(2–1)	Total	10.7	15.7	3.4	$-$	$-$	$-$	0.31	$-$	$-$	5.95
Mrk 996	1622	12CO(1–0)	1	21.5	5.2	1.8	6.0	-1	28	0.189 ( 0.034)	0.182 ( 0.033)	0.97	5.890 ( 0.077)
Mrk 996		12CO(1–0)	2	21.5	5.2	1.8	4.7	39	10	0.054 ( 0.019)	0.052 ( 0.019)	0.28	5.347 ( 0.154)
Mrk 996		12CO(1–0)	Total	21.5	5.2	1.8	6.0	0	40	0.243 ( 0.039)	0.235 ( 0.037)	1.25	5.999 ( 0.069)
Mrk 996		12CO(2–1)	Total	10.7	5.2	2.5	3.8	-12	32	0.291 ( 0.047)	0.128 ( 0.020)	2.69	5.729 ( 0.070)
Mrk 996		12CO(2–1)	1	27.2	10.2	2.1	6.0	-12	123	0.563 ( 0.131)	0.782 ( 0.182)	15.34	6.485 ( 0.101)
Mrk 996		12CO(2–1)	2	27.2	10.2	2.1	4.0	72	36	0.113 ( 0.076)	0.156 ( 0.106)	3.07	5.786 ( 0.295)
Mrk 996		12CO(2–1)	Total	27.2	10.2	2.1	6.0	0	84	0.675 ( 0.151)	0.938 ( 0.210)	18.40	6.564 ( 0.097)
Mrk 996		12CO(3–2)	Total	18.1	13.6	1.7	$-$	$-$	$-$	0.21	$-$	$-$	5.74
NGC 1156	375	12CO(1–0)	1	21.4	10.4	7.1	12.2	-78	71	0.933 ( 0.406)	0.929 ( 0.404)	5.00	5.904 ( 0.189)
NGC 1156		12CO(1–0)	2	21.4	10.4	7.1	10.0	25	85	0.916 ( 0.433)	0.912 ( 0.431)	4.91	5.896 ( 0.205)
NGC 1156		12CO(1–0)	Total	21.4	10.4	7.1	12.2	0	103	1.849 ( 0.593)	1.841 ( 0.590)	9.91	6.201 ( 0.139)
NGC 1156		12CO(2–1)	Total	10.7	10.4	4.8	9.4	-5	67	0.725 ( 0.206)	0.668 ( 0.190)	14.19	5.755 ( 0.123)
NGC 3353 [0″, 0″]	944	12CO(1–0)	1	21.4	5.2	5.5	27.6	-31	60	1.824 ( 0.177)	1.771 ( 0.172)	9.50	6.880 ( 0.042)
NGC 3353 [0″, 0″]		12CO(1–0)	2	21.4	5.2	5.5	19.7	21	16	0.339 ( 0.112)	0.329 ( 0.109)	1.76	6.149 ( 0.144)
NGC 3353 [0″, 0″]		12CO(1–0)	Total	21.4	5.2	5.5	27.6	0	52	2.163 ( 0.210)	2.100 ( 0.204)	11.26	6.954 ( 0.042)
NGC 3353 [0″, 0″]		13CO(1–0)	Total	22.4	10.9	2.0	$-$	$-$	$-$	2.10	$-$	$-$	5.95
NGC 3353 [0″, 0″]		12CO(2–1)	1	10.7	5.2	7.9	31.7	-33	42	2.611 ( 0.232)	1.430 ( 0.127)	30.27	6.781 ( 0.039)
NGC 3353 [0″, 0″]		12CO(2–1)	2	10.7	5.2	7.9	19.5	15	22	0.814 ( 0.175)	0.446 ( 0.096)	9.43	6.275 ( 0.093)
NGC 3353 [0″, 0″]		12CO(2–1)	Total	10.7	5.2	7.9	31.7	0	48	3.424 ( 0.290)	1.876 ( 0.159)	39.70	6.899 ( 0.037)
NGC 3353 [0″, $+$10″]	944	12CO(1–0)	1	21.4	5.2	4.5	33.0	-35	39	1.412 ( 0.123)	1.371 ( 0.119)	7.35	6.769 ( 0.038)
NGC 3353 [0″, $+$10″]		12CO(1–0)	2	21.4	5.2	4.5	11.8	19	36	0.468 ( 0.123)	0.454 ( 0.119)	2.43	6.289 ( 0.114)
NGC 3353 [0″, $+$10″]		12CO(1–0)	Total	21.4	5.2	4.5	33.0	0	54	1.880 ( 0.174)	1.825 ( 0.168)	9.79	6.893 ( 0.040)
NGC 3353 [0″, $+$10″]		12CO(2–1)	1	10.7	5.2	7.9	16.2	-34	52	1.635 ( 0.207)	0.896 ( 0.114)	18.96	6.578 ( 0.055)
NGC 3353 [0″, $+$10″]		12CO(2–1)	2	10.7	5.2	7.9	4.3	36	12	0.103 ( 0.079)	0.056 ( 0.044)	1.19	5.377 ( 0.336)
NGC 3353 [0″, $+$10″]		12CO(2–1)	Total	10.7	5.2	7.9	16.2	0	70	1.738 ( 0.221)	0.952 ( 0.121)	20.15	6.605 ( 0.055)
NGC 3353 [$-$6″, $-$9″]	944	12CO(1–0)	1	21.4	5.2	4.6	6.7	-56	29	0.217 ( 0.037)	0.211 ( 0.036)	1.13	5.955 ( 0.075)
NGC 3353 [$-$6″, $-$9″]		12CO(1–0)	2	21.4	5.2	4.6	13.6	-16	28	0.407 ( 0.037)	0.395 ( 0.036)	2.12	6.229 ( 0.040)
NGC 3353 [$-$6″, $-$9″]		12CO(1–0)	3	21.4	5.2	4.6	20.0	22	35	0.758 ( 0.037)	0.736 ( 0.036)	3.95	6.499 ( 0.021)
NGC 3353 [$-$6″, $-$9″]		12CO(1–0)	4	21.4	5.2	4.6	8.8	98	33	0.319 ( 0.037)	0.310 ( 0.036)	1.66	6.123 ( 0.051)
NGC 3353 [$-$6″, $-$9″]		12CO(1–0)	Total	21.4	5.2	4.6	20.0	0	78	1.701 ( 0.075)	1.652 ( 0.073)	8.86	6.850 ( 0.019)
NGC 3353 [$-$6″, $-$9″]		12CO(2–1)	1	10.7	5.2	6.1	5.9	-2	36	0.405 ( 0.190)	0.222 ( 0.104)	4.70	5.972 ( 0.204)
NGC 3353 [$-$6″, $-$9″]		12CO(2–1)	2	10.7	5.2	6.1	13.8	23	12	0.332 ( 0.115)	0.182 ( 0.063)	3.85	5.886 ( 0.151)
NGC 3353 [$-$6″, $-$9″]		12CO(2–1)	Total	10.7	5.2	6.1	13.8	0	25	0.737 ( 0.223)	0.404 ( 0.122)	8.55	6.232 ( 0.131)

Table 3. Table 3: CO emission line parameters for NGC 7077, UM 448 and UM 462 a

Name	Systemic	Transition	Component	Beam	Channel	Trms	Tpeak	Offsetb	FWHM	$I_{CO}$	$I_{CO}^{\mathrm{cor}}$ c	$S_{CO}^{\mathrm{cor}}$ c	Log($L_{\mathrm{cor}}^{\prime }$) c
	velocity			size	width	(mK)	(mK)	( km s-1)	( km s-1)	(K km s-1)	(K km s-1)	(Jy  km s-1)	(K km s-1 pc2)
	( km s-1)			(″)	( km s-1)
NGC 7077	1152	12CO(1–0)	1	21.4	5.2	1.7	11.6	-5	21	0.265 ( 0.034)	0.257 ( 0.033)	1.38	5.996 ( 0.055)
NGC 7077		12CO(1–0)	2	21.4	5.2	1.7	6.5	36	31	0.224 ( 0.040)	0.217 ( 0.039)	1.16	5.923 ( 0.077)
NGC 7077		12CO(1–0)	Total	21.4	5.2	1.7	11.6	0	41	0.489 ( 0.052)	0.474 ( 0.050)	2.54	6.262 ( 0.046)
NGC 7077		12CO(2–1)	1	10.7	5.2	3.9	9.6	-9	29	0.578 ( 0.086)	0.298 ( 0.044)	6.30	6.055 ( 0.064)
NGC 7077		12CO(2–1)	2	10.7	5.2	3.9	10.4	37	23	0.485 ( 0.084)	0.250 ( 0.043)	5.28	5.979 ( 0.075)
NGC 7077		12CO(2–1)	Total	10.7	5.2	3.9	10.4	0	46	1.063 ( 0.120)	0.548 ( 0.062)	11.58	6.320 ( 0.049)
NGC 7077		12CO(3–2)	1	18.1	6.8	2.8	12.1	-1	16	0.259 ( 0.057)	0.208 ( 0.046)	9.12	5.864 ( 0.096)
NGC 7077		12CO(3–2)	2	18.1	6.8	2.8	4.2	37	50	0.277 ( 0.092)	0.223 ( 0.074)	9.78	5.894 ( 0.143)
NGC 7077		12CO(3–2)	Total	18.1	6.8	2.8	12.1	0	38	0.536 ( 0.085)	0.431 ( 0.068)	18.89	6.180 ( 0.069)
UM 448	5564	12CO(1–0)	1	21.7	5.3	1.6	4.8	-115	46	0.106 ( 0.034)	0.104 ( 0.033)	0.54	6.934 ( 0.138)
UM 448		12CO(1–0)	2	21.7	5.3	1.6	6.1	-59	61	0.398 ( 0.140)	0.392 ( 0.137)	2.04	7.509 ( 0.152)
UM 448		12CO(1–0)	3	21.7	5.3	1.6	10.9	0	46	0.547 ( 0.125)	0.539 ( 0.123)	2.80	7.647 ( 0.099)
UM 448		12CO(1–0)	Total	21.7	5.3	1.6	10.9	0	115	1.051 ( 0.190)	1.035 ( 0.187)	5.38	7.931 ( 0.079)
UM 448		12CO(2–1)	1	10.9	5.3	4.0	3.8	-145	71	0.212 ( 0.093)	0.105 ( 0.046)	2.16	6.932 ( 0.190)
UM 448		12CO(2–1)	2	10.9	5.3	4.0	3.7	-95	52	0.407 ( 0.118)	0.202 ( 0.058)	4.15	7.216 ( 0.126)
UM 448		12CO(2–1)	3	10.9	5.3	4.0	12.3	-15	71	1.871 ( 0.099)	0.930 ( 0.049)	19.10	7.879 ( 0.023)
UM 448		12CO(2–1)	Total	10.9	5.3	4.0	12.3	0	130	2.490 ( 0.179)	1.237 ( 0.089)	25.41	8.003 ( 0.031)
UM 448		12CO(2–1)	1	27.6	10.4	2.1	8.9	-47	22	0.153 ( 0.081)	0.208 ( 0.111)	3.98	7.198 ( 0.232)
UM 448		12CO(2–1)	2	27.6	10.4	2.1	6.7	-45	118	0.614 ( 0.212)	0.839 ( 0.290)	16.03	7.803 ( 0.150)
UM 448		12CO(2–1)	3	27.6	10.4	2.1	12.1	34	82	0.776 ( 0.191)	1.059 ( 0.261)	20.25	7.904 ( 0.107)
UM 448		12CO(2–1)	Total	27.6	10.4	2.1	12.1	0	81	1.543 ( 0.297)	2.106 ( 0.406)	40.27	8.202 ( 0.084)
UM 448		12CO(3–2)	1	18.4	13.8	1.6	3.9	-70	22	0.111 ( 0.042)	0.090 ( 0.034)	3.82	6.827 ( 0.165)
UM 448		12CO(3–2)	2	18.4	13.8	1.6	4.2	-27	23	0.126 ( 0.069)	0.101 ( 0.056)	4.32	6.881 ( 0.238)
UM 448		12CO(3–2)	3	18.4	13.8	1.6	6.7	33	72	0.637 ( 0.049)	0.513 ( 0.039)	21.88	7.585 ( 0.033)
UM 448		12CO(3–2)	Total	18.4	13.8	1.6	6.7	0	103	0.875 ( 0.094)	0.705 ( 0.076)	30.02	7.723 ( 0.047)
UM 462	1057	12CO(1–0)	Total	21.4	15.7	2.5	6.7	-8	123	0.872 ( 0.157)	0.872 ( 0.157)	4.43	6.613 ( 0.078)
UM 462		12CO(2–1)	Total	10.7	15.7	4.8	$-$	$-$	$-$	0.87	$-$	$-$	5.72
UM 462		12CO(3–2)	Total	18.1	13.6	1.6	$-$	$-$	$-$	0.87	$-$	$-$	5.78

Table 4. Table 4: Emission line parameters for NGC 1140 a

Name	Transition	Component	Beam	Channel	Trms	Tpeak	Offsetb	FWHM	$I_{CO}$	$I_{CO}^{\mathrm{cor}}$ c	$S_{CO}^{\mathrm{cor}}$ c	Log($L_{\mathrm{cor}}^{\prime }$) c
			size	width	(mK)	(mK)	( km s-1)	( km s-1)	(K km s-1)	(K km s-1)	(Jy  km s-1)	(K km s-1 pc2)
			(″)	( km s-1)
NGC 1140	12CO(1–0)	1	21.4	5.2	1.6	9.5	-21	20	0.205 ( 0.016)	0.200 ( 0.015)	1.07	6.004 ( 0.033)
NGC 1140	12CO(1–0)	2	21.4	5.2	1.6	7.8	0	24	0.201 ( 0.016)	0.196 ( 0.015)	1.05	5.996 ( 0.034)
NGC 1140	12CO(1–0)	3	21.4	5.2	1.6	4.5	35	25	0.123 ( 0.016)	0.120 ( 0.015)	0.64	5.782 ( 0.055)
NGC 1140	12CO(1–0)	4	21.4	5.2	1.6	5.4	64	8	0.049 ( 0.016)	0.048 ( 0.015)	0.26	5.386 ( 0.138)
NGC 1140	12CO(1–0)	Total	21.4	5.2	1.6	9.5	0	85	0.578 ( 0.031)	0.564 ( 0.031)	3.02	6.455 ( 0.024)
NGC 1140	12CO(2–1)	1	10.7	5.2	5.0	6.0	-71	20	0.212 ( 0.047)	0.126 ( 0.028)	2.65	5.797 ( 0.096)
NGC 1140	12CO(2–1)	2	10.7	5.2	5.0	17.0	-17	37	1.124 ( 0.047)	0.668 ( 0.028)	14.10	6.522 ( 0.018)
NGC 1140	12CO(2–1)	3	10.7	5.2	5.0	8.5	10	25	0.389 ( 0.047)	0.231 ( 0.028)	4.88	6.061 ( 0.052)
NGC 1140	12CO(2–1)	4	10.7	5.2	5.0	6.4	38	23	0.260 ( 0.047)	0.155 ( 0.028)	3.26	5.886 ( 0.078)
NGC 1140	12CO(2–1)	Total	10.7	5.2	5.0	17.0	0	109	1.985 ( 0.093)	1.180 ( 0.056)	24.89	6.769 ( 0.020)
NGC 1140	12CO(3–2)	1	18.1	6.8	3.6	20.0	-30	27	0.686 ( 0.182)	0.577 ( 0.153)	25.24	6.423 ( 0.115)
NGC 1140	12CO(3–2)	2	18.1	6.8	3.6	11.0	0	22	0.308 ( 0.171)	0.259 ( 0.144)	11.33	6.075 ( 0.241)
NGC 1140	12CO(3–2)	3	18.1	6.8	3.6	9.9	47	13	0.162 ( 0.044)	0.136 ( 0.037)	5.96	5.796 ( 0.117)
NGC 1140	12CO(3–2)	Total	18.1	6.8	3.6	20.0	0	77	1.155 ( 0.252)	0.972 ( 0.212)	42.52	6.650 ( 0.095)
NGC 1140	12CO(4–3)	1	13.6	10.2	7.5	18.2	-37	23	0.645 ( 0.168)	0.440 ( 0.114)	33.11	6.291 ( 0.113)
NGC 1140	12CO(4–3)	2	13.6	10.2	7.5	9.2	-1	14	0.209 ( 0.132)	0.142 ( 0.090)	10.71	5.801 ( 0.274)
NGC 1140	12CO(4–3)	Total	13.6	10.2	7.5	18.2	0	38	0.854 ( 0.213)	0.582 ( 0.146)	43.82	6.413 ( 0.109)
NGC 1140	13CO(1–0)	Total	22.4	10.9	0.6	2.1	2	23	0.052 ( 0.016)	0.053 ( 0.016)	0.26	5.423 ( 0.131)
NGC 1140	13CO(2–1)	1	11.2	5.5	0.9	1.3	-21	41	0.093 ( 0.020)	0.057 ( 0.012)	1.09	5.450 ( 0.093)
NGC 1140	13CO(2–1)	2	11.2	5.5	0.9	0.8	74	58	0.079 ( 0.021)	0.048 ( 0.013)	0.93	5.381 ( 0.117)
NGC 1140	13CO(2–1)	Total	11.2	5.5	0.9	1.3	0	95	0.172 ( 0.029)	0.105 ( 0.018)	2.02	5.718 ( 0.073)
NGC 1140	C18O(1$-$0)	Total	22.5	11.0	0.5	$-$	$-$	$-$	0.10	$-$	$-$	5.40
NGC 1140	HCN(1$-$0)	Total	21.8	12.8	0.5	$-$	$-$	$-$	0.10	$-$	$-$	5.44
NGC 1140	CN(1$-$0)	Total	21.8	10.6	1.2	$-$	$-$	$-$	0.10	$-$	$-$	5.75
NGC 1140	CS(2$-$1)	Total	25.2	12.3	0.6	$-$	$-$	$-$	0.10	$-$	$-$	5.64
NGC 1140	[Ci](1–0)	Total	12.7	9.6	5.7	6.8	5	51	0.563 ( 0.188)	0.369 ( 0.123)	31.63	6.215 ( 0.145)

Table 5. Table 5: 3 σ 𝜎 \sigma upper limits of undetected emission lines for II Zw 40 and SBS 0335 − - 052 a

Name	Transition	Component	Beam	Channel	Trms	$I_{CO}$
			size (″)	width ( km s-1)	(mK)	(K km s-1)
II Zw 40	12CO(2–1)	Total	27.1	10.2	2.5	0.23
II Zw 40	12CO(3–2)	Total	18.1	13.6	1.9	0.23
SBS0335-052	12CO(2–1)	Total	27.4	10.3	2.2	0.20
SBS0335-052	12CO(3–2)	Total	18.3	13.7	2.7	0.33

Table 6. Table 6: Best-fit RADEX parameters for NGC 1140 a

$\Delta V$	${\chi }^2$	$\tau$						$T_{\mathrm{kin}}$	Log($n_{\mathrm{H2}}$)	Log($N_{12CO}$)	12CO/13CO
( km s-1)		12CO(1–0)	12CO(2–1)	12CO(3–2)	12CO(4–3)	13CO(1–0)	13CO(2–1)	(K)	(cm-3)	(cm-2)
(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)	(10)	(11)	(12)
Fit to intensities with exponential beam correction
10	0.94	0.02	0.06	0.06	0.04	0.002	0.006	17.3	7.4	15.6	8
$-$	$0.94-1.93$	$0.02-0.04$	$0.05-0.10$	$0.06-0.10$	$0.04-0.06$	$0.002-0.004$	$0.005-0.010$	$16.1-18.7$	$4.9-8.0$	$15.6-15.8$	$8-10$
20	0.93	0.02	0.05	0.05	0.03	0.002	0.005	16.7	7.5	15.8	8
$-$	$0.93-1.93$	$0.01-0.05$	$0.03-0.13$	$0.03-0.13$	$0.02-0.08$	$0.001-0.005$	$0.003-0.012$	$16.1-19.4$	$4.9-8.0$	$15.6-16.2$	$8-10$
Fit to intensities with Gaussian beam correctionb
10	0.95	0.02	0.06	0.06	0.04	0.002	0.005	18.0	7.1	15.6	10
$-$	$0.95-1.95$	$0.01-0.06$	$0.05-0.15$	$0.06-0.16$	$0.04-0.10$	$0.001-0.005$	$0.004-0.014$	$16.1-26.3$	$4.3-8.0$	$15.6-16.0$	$8-10$
20	0.85	0.01	0.03	0.03	0.02	0.001	0.003	17.3	5.9	15.6	10
$-$	$0.85-1.85$	$0.01-0.05$	$0.03-0.13$	$0.03-0.13$	$0.02-0.08$	$0.001-0.005$	$0.002-0.012$	$16.1-24.4$	$4.5-8.0$	$15.6-16.2$	$8-10$
Fit to “Set 1” line ratiosc with exponential beam correction
10	0.39	0.09	0.23	0.24	0.14	0.009	0.022	17.3	7.9	16.2	10
$-$	$0.39-1.39$	$0.00-0.24$	$0.00-0.72$	$0.00-1.01$	$0.00-0.75$	$0.000-0.022$	$0.000-0.068$	$15.5-44.5$	$4.0-8.0$	$14.0-16.8$	$8-10$
20	0.38	0.07	0.18	0.19	0.11	0.007	0.017	17.3	7.5	16.4	10
$-$	$0.38-1.37$	$0.00-0.23$	$0.00-0.63$	$0.00-0.85$	$0.00-0.58$	$0.000-0.021$	$0.000-0.060$	$15.5-41.3$	$4.0-8.0$	$14.0-17.0$	$8-14$
Fit to “Set 1” line ratiosc with Gaussian beam correctiond
10	0.44	0.09	0.23	0.24	0.14	0.009	0.022	17.3	7.1	16.2	10
$-$	$0.44-1.44$	$0.00-0.33$	$0.00-0.85$	$0.00-1.03$	$0.00-0.75$	$0.000-0.030$	$0.000-0.080$	$15.5-44.5$	$4.0-8.0$	$14.0-16.8$	$8-10$
20	0.43	0.12	0.29	0.30	0.18	0.011	0.028	17.3	6.9	16.6	10
$-$	$0.43-1.43$	$0.00-0.29$	$0.00-0.74$	$0.00-0.85$	$0.00-0.59$	$0.000-0.027$	$0.000-0.070$	$15.5-44.5$	$4.0-8.0$	$14.0-17.0$	$8-14$
Fit to “other” sets of line ratiosc with exponential beam correction
10	0.85	0.06	0.15	0.15	0.09	0.005	0.014	17.3	7.8	16.0	10
$-$	$0.85-1.84$	$0.00-0.27$	$0.00-0.91$	$0.00-1.40$	$0.00-1.20$	$0.000-0.016$	$0.000-0.056$	$16.1-44.5$	$4.0-8.0$	$14.0-17.0$	$8-20$
20	0.83	0.05	0.12	0.12	0.07	0.004	0.011	17.3	8.0	16.2	10
$-$	$0.83-1.83$	$0.00-0.24$	$0.00-0.64$	$0.00-0.78$	$0.00-0.56$	$0.000-0.014$	$0.000-0.038$	$16.1-27.3$	$4.3-8.0$	$14.0-17.0$	$8-20$
$\Delta V$	${\chi }^2$	$\tau$						$T_{\mathrm{kin}}$	Log($n_{\mathrm{H2}}$)	Log($N_{12CO}$)	12CO/C
( km s-1)		12CO(1–0)	12CO(2–1)	12CO(3–2)	12CO(4–3)	[Ci](1–0)		(K)	(cm-3)	(cm-2)
Fit to intensities with exponential beam correction
10	0.55	0.02	0.06	0.06	0.04	0.05		17.3	7.1	15.6	0.03
$-$	$0.55-1.54$	$0.02-0.04$	$0.06-0.10$	$0.06-0.10$	$0.03-0.06$	$0.047-0.077$		$16.1-18.7$	$5.1-8.0$	$15.6-15.8$	$0.03-0.03$
20	0.66	0.02	0.05	0.05	0.03	0.04		17.3	6.2	15.8	0.03
$-$	$0.66-1.65$	$0.01-0.05$	$0.03-0.13$	$0.03-0.13$	$0.02-0.08$	$0.023-0.097$		$16.1-18.7$	$4.9-8.0$	$15.6-16.2$	$0.03-0.03$
Fit to intensities with Gaussian beam correctione
10	0.75	0.04	0.10	0.10	0.06	0.08		16.7	7.4	15.8	0.03
$-$	$0.75-1.75$	$0.00-0.04$	$0.07-0.10$	$0.09-0.11$	$0.06-0.07$	$0.015-0.077$		$16.1-41.3$	$4.0-8.0$	$15.8-15.8$	$0.03-0.10$
20	0.66	0.02	0.05	0.05	0.03	0.04		16.7	7.5	15.8	0.03
$-$	$0.66-1.66$	$0.02-0.02$	$0.04-0.05$	$0.05-0.05$	$0.03-0.03$	$0.037-0.039$		$16.1-18.7$	$4.9-8.0$	$15.8-15.8$	$0.03-0.03$
Fit to [Ci]/12CO ratios with exponential beam correction
10	0.03	0.19	0.67	0.93	0.70	0.30		26.3	4.5	16.8	0.06
$-$	$0.03-1.03$	$0.00-0.94$	$0.00-2.86$	$0.00-4.32$	$0.00-4.25$	$0.000-1.927$		$16.1-44.5$	$4.0-8.0$	$14.0-17.6$	$0.03-0.16$
20	0.03	0.10	0.68	1.14	0.94	0.20		38.3	4.1	17.2	0.10
$-$	$0.03-1.03$	$0.00-0.97$	$0.00-2.89$	$0.00-3.88$	$0.00-3.64$	$0.000-1.546$		$16.1-44.5$	$4.0-8.0$	$14.0-17.8$	$0.03-0.16$
Fit to [Ci]/12CO ratios with Gaussian beam correctionf
10	0.01	0.34	1.09	1.43	1.08	0.48		24.4	4.6	17.0	0.05
$-$	$0.01-1.01$	$0.00-1.19$	$0.00-3.88$	$0.00-6.06$	$0.00-6.40$	$0.000-1.927$		$15.5-44.5$	$4.0-8.0$	$14.0-17.8$	$0.03-0.25$
20	0.01	0.24	1.10	1.70	1.46	0.34		32.9	4.2	17.4	0.10
$-$	$0.01-1.01$	$0.00-1.22$	$0.00-3.93$	$0.00-6.09$	$0.00-6.58$	$0.000-1.537$		$15.5-44.5$	$4.0-8.0$	$14.0-18.2$	$0.03-0.25$

Table 7. Table 7: Flux corrections to a common beam size a

Galaxy	$r_s$	Transition	Original beam	Common beam	Aperture	$T_{\mathrm{mb}}$	$I_{CO}$
	(arcsec)		(arcsec)	(arcsec)	correction	correction	corrected
					$A{P}_{\mathrm{cor}}$		(K km s-1)
Mrk 0996	3.6	12CO(1–0)	21.4	22.0	1.014	0.964	0.235
Mrk 0996	3.6	12CO(2–1) (IRAM)	10.7	22.0	1.843	0.438	0.128
Mrk 0996	3.6	12CO(2–1) (APEX)	27.2	22.0	0.908	1.389	0.938
Mrk 0996	3.6	12CO(3–2)	18.1	22.0	1.13	0.77	0.161
NGC 1140	6.4	12CO(1–0)	21.4	22.0	1.027	0.976	0.564
NGC 1140	6.4	12CO(2–1)	10.7	22.0	2.502	0.595	1.180
NGC 1140	6.4	12CO(3–2)	18.1	22.0	1.238	0.841	0.972
NGC 1140	6.4	12CO(4–3)	13.6	22.0	1.784	0.682	0.582
NGC 1140	6.4	13CO(1–0)	22.4	22.0	0.980	1.019	0.053
NGC 1140	6.4	13CO(2–1)	11.2	22.0	2.342	0.609	0.105
NGC 1140	6.4	[Ci](1–0)	12.7	22.0	1.952	0.655	0.369
NGC 1156	44.9	12CO(1–0)	21.4	22.0	1.055	0.995	1.841
NGC 1156	44.9	12CO(2–1)	10.7	22.0	3.903	0.921	0.668
NGC 3353	5.4	12CO(1–0)	21.4	22.0	1.025	0.971	2.100
NGC 3353	5.4	12CO(2–1)	10.7	22.0	2.314	0.548	1.876
NGC 7077	4.8	12CO(1–0)	21.4	22.0	1.022	0.969	0.474
NGC 7077	4.8	12CO(2–1)	10.7	22.0	2.175	0.516	0.548
NGC 7077	4.8	12CO(3–2)	18.1	22.0	1.186	0.804	0.431
UM 448	4.4	12CO(1–0)	21.4	22.0	1.009	0.985	1.035
UM 448	4.4	12CO(2–1) (IRAM)	10.7	22.0	2.036	0.497	1.237
UM 448	4.4	12CO(2–1) (APEX)	27.6	22.0	0.869	1.365	2.106
UM 448	4.4	12CO(3–2)	18.1	22.0	1.154	0.806	0.705

Table 8. Table 8: Flux corrections to a common beam size b

Galaxy	Transition	Original beam	Common beam	Aperture	$T_{\mathrm{mb}}$	$I_{CO}$
		(arcsec)	(arcsec)	correction	correction	corrected
				$A{P}_{\mathrm{cor}}$		(K km s-1)
NGC 1140	12CO(1–0)	21.4	22.0	1.022	0.972	0.562
NGC 1140	12CO(2–1)	10.7	22.0	2.406	0.572	1.135
NGC 1140	12CO(3–2)	18.1	22.0	1.204	0.818	0.945
NGC 1140	12CO(4–3)	13.6	22.0	1.705	0.652	0.556
NGC 1140	13CO(1–0)	22.4	22.0	0.983	1.023	0.053
NGC 1140	13CO(2–1)	11.2	22.0	2.246	0.584	0.100
NGC 1140	[Ci](1–0)	12.7	22.0	1.865	0.626	0.352

Equations

$$N_u=\frac{8\pi k{\nu }^{2}W}{h{c}^3{A}_{ul}}\left(\frac{\Delta {\Omega }_a}{\Delta {\Omega }_s}\right)\left(\frac{\tau }{1-e^{-\tau }}\right),$$

$$N_u^{\mathrm{thin}}=\frac{N}{Z}{g}_u{e}^{-E_u/kT},$$

$$\ln \frac{N_u^{\mathrm{thin}}}{g_u}+\ln C_{\tau }=\ln N-\ln Z-\frac{E_u}{kT},$$

$${\chi }^2=\sum _{i=1}^6{\left(\frac{F{I}_i^{\mathrm{mod}}-I_i^{\mathrm{obs}}}{\Delta I_i^{\mathrm{obs}}}\right)}^2$$

$${\chi }^2=\sum _{j=1}^5{\left(\frac{R_j^{\mathrm{mod}}-R_j^{\mathrm{obs}}}{\Delta R_j^{\mathrm{obs}}}\right)}^2$$

$${}^{13}{\mathrm{C}}^{+}+{}^{12}\mathrm{CO}\leftrightharpoons {{}^{12}\mathrm{C}}^{+}+{}^{13}\mathrm{CO}+\Delta E(=35\mathrm{K}).$$

$$S_{\nu }=A_0{\int }_0^{2\pi }{\int }_0^{{\theta }_b/2}\exp (-r/r_s)rdrd\varphi$$

$$S_{\nu }=2\pi A_0\left\{r_s^2\left[1-\exp \left(-\frac{{\theta }_b}{2r_s}\right)\right]-\frac{{\theta }_b{r}_s}{2}\exp \left(-\frac{{\theta }_b}{2r_s}\right)\right\}$$

$$D(x,0)=A_0{e}^{-x^2/{\sigma }_b^2}{\int }_0^{\infty }{\int }_0^{2\pi }{e}^{-\frac{r_0}{r_s}}{e}^{\left(-\frac{r_0^2-2x{r}_0\cos \varphi }{{\sigma }_b^2}\right)}{r}_{0}d{r}_{0}d\varphi$$

$$D(x,0)=2\pi A_0{e}^{-x^2/{\sigma }_b^2}{\int }_0^{\infty }{e}^{-\frac{r_0}{r_s}}{e}^{-\frac{r_0^2}{{\sigma }_b^2}}{I}_0\left(\frac{2x{r}_0}{{\sigma }_b^2}\right)r_{0}d{r}_0$$

$$T_{\mathrm{mb}}(22)=T_{\mathrm{mb}}(\mathrm{orig})A{P}_{\mathrm{cor}}\frac{\Omega (\mathrm{orig})}{\Omega (22)}$$

$$S_{\nu }=A_0{\int }_0^{2\pi }{\int }_0^{{\theta }_b/2}\exp \left(\frac{-r}{2{\sigma }_s^2}\right)rdrd\varphi$$

$$S_{\nu }=2\pi A_0{\sigma }_s^2\left[1-\exp \left(-\frac{{\theta }_b^2}{8{\sigma }_s^2}\right)\right]$$

$$D(x,y)=A_0{\int }_{-\infty }^{\infty }{\int }_{-\infty }^{\infty }{e}^{-\frac{-(x_0^2+y_0^2)}{{\sigma }_s^2}}{e}^{-\frac{-(x-x_0{)}^2-(y-y_0{)}^2}{{\sigma }_b^2}}d{x}_{0}d{y}_0$$

$$D(r)=A_0^{\prime }\frac{{\sigma }_s^2}{({\sigma }_b^2+{\sigma }_s^2)}\exp \left[\frac{-r^2}{({\sigma }_b^2+{\sigma }_s^2)}\right]$$

$$S_{\nu }$$

Full text

††offprints: L. K. Hunt

1 Introduction

Physical conditions in the molecular component of the interstellar medium (ISM) of metal-poor galaxies are difficult to measure. The main obstacle is the faintness at low metallicities of the common proxy of H2, carbon monoxide emission (Sage et al. 1992; Taylor et al. 1998; Gondhalekar et al. 1998; Barone et al. 2000; Leroy et al. 2005; Buyle et al. 2006; Leroy et al. 2007; Schruba et al. 2012; Cormier et al. 2014). While the proximity of Local Group galaxies facilitates studies of molecular gas (e.g., Cohen et al. 1988; Rubio et al. 1993; Fukui et al. 1999; Israel et al. 2003; Bolatto et al. 2008; Pineda et al. 2012; Elmegreen et al. 2013; Paron et al. 2014; Shi et al. 2015; Rubio et al. 2015; Paron et al. 2016; Shi et al. 2016; Schruba et al. 2017), beyond the Local Group, CO detections in low-metallicity galaxies are arduous (e.g., Schruba et al. 2012; Cormier et al. 2014), and few in number. In a previous paper (Hunt et al. 2015, hereafter Paper I), we reported 12CO(1–0) detections in 8 metal-poor galaxies outside the Local Group with metallicities ranging from 12$+$log(O/H) $\sim$7.7 to 8.4, or equivalently 0.1 Z*⊙* to 0.5 Z*⊙*111We assume the Asplund et al. (2009) Solar abundance calibration of 12$+$log(O/H) = 8.69.. This more than doubles the previous number of such detections, and sets the stage for a more detailed study of the physical conditions in molecular gas in a metal-poor ISM.

In this paper, we present observations of 12CO(2–1) and 12CO(3–2) for some of the galaxies in Paper I, and for one of them, NGC 1140, also 12CO(4–3), 13CO(1–0), 13CO(2–1), and Ci detections, as well as upper limits for other molecular transitions including HCN(1-0). In Sect. 2, we briefly discuss the sample, and in Sect. 3, describe the observations and data reduction. Section 4 presents our approach for correcting the observed temperatures and fluxes for the different beam sizes. Sect. 5 analyzes the observed line ratios from an empirical point of view, comparing them with other metal-rich samples, and Sect. 6 discusses dense-gas tracers. Physical models of the line emission of NGC 1140 are presented in Sect. 7, in order to infer H2 volume density, $n_{\rm H2}$, kinetic temperature, $T_{\rm kin}$, CO column density, $N_{CO}$, and the 12CO/13CO as well as the [Ci]/12CO abundance ratios. More limited models are computed for the other galaxies with fewer multiple transitions. We discuss the results in Sect. 8, and give our conclusions in Sect. 9.

2 The targets

A complete description of source selection and the individual galaxies is given in Paper I. Here we briefly outline the main selection criteria and general characteristics of our targets.

The majority of the galaxies in the observing sample already had H2 detections, either of the ro-vibrational transitions at 2 $\mu$m or the rotational transitions in the mid-infrared with Spitzer/IRS (Hunt et al. 2010). Additional galaxies were included with previous CO observations lacking a clear detection. Stellar masses, estimated from 3.6 $\mu$m luminosities after the subtraction of nebular emission, range from $\sim 10^{8}$ to $10^{10}$ M*⊙* (UM 448 is more massive than this, and is the clear result of an interaction with another galaxy, see Paper I for more details). The relatively high specific star-formation rates (ratio of SFR and stellar mass, SFR/Mstar = sSFR) of the sample make them starburst galaxies (sSFR $\ga 10^{-10}$ yr*-1*). They are also gas rich, with gas-mass fractions (relative to total baryonic mass) as high as $\sim$0.8.

3 The observations

We have observed 12CO(1–0) and 12CO(2–1) in our sample galaxies over a three-year period from 2008 to 2010 with the IRAM 30-m telescope (Pico Veleta, Spain). The measurements of 12CO(3–2) and for a few galaxies also of 12CO(2–1) were acquired with the APEX 12-m telescope (Chile)222This publication is based on data acquired with the Atacama Pathfinder Experiment (APEX). APEX is a collaboration between the Max-Planck-Institut fur Radioastronomie, the European Southern Observatory, and the Onsala Space Observatory.. NGC 1140 was singled out for a more intense observing campaign in which we observed 13CO(1–0), HCN(1–0), other 3-mm transitions, and 13CO(2–1) at the 30m; 12CO(4–3) and Ci measurements were obtained with APEX. The 12CO(1–0) observations of this sample were presented in Paper I. Table 1 reports selected observational parameters for the sample galaxies. Our single-dish observations sample fairly large regions within each galaxy, ranging from 420 pc for the smallest beam in the closest galaxy (12CO(2–1), NGC 1156) to 8.5 kpc for the largest beam in the most distant one (12CO(1–0), UM 448).

3.1 IRAM

We observed eight galaxies at the 30m in 12CO(1–0) and 12CO(2–1), first with the older ABCD receivers (proposals 036-08, 227-09), then with the Eight Mixer Receiver (EMIR, proposal 097-10) using the Wideband Line Multiple Autocorrelator (WILMA) backend. Earlier spectra were acquired at an intrinsic resolution of 4 MHz, while later ones using EMIR at 2 MHz. For NGC 1140, the EMIR E090 receiver was tuned to an intermediate frequency, $\sim$113 GHz, in order to cover other 3 mm transitions simultaneously together with 12CO(1–0) in the lower-inner sideband in two polarizations. We adopted wobbler switching with a throw of 90″, and used the standard intensity calibration with two absorbers at different temperatures. Pointing was checked every hour or two on nearby planets or bright quasars, and focus tests were performed every 4 hrs during the night and every 3 hrs during the day. Typical pointing errors were $\la$2″, and never exceeded $\sim$3″.

With the EMIR setup for NGC 1140, we were able to also observe additional transitions, including 13CO(1–0), 13CO(2–1), C18O(1–0), and the dense-gas tracers HCN(1–0), CN(1–0), and CS(2–1).

3.2 APEX

The 12CO(3–2) observations of six of our targets were performed with the Swedish Heterodyne Facility Instrument (SHeFI) at the 12-m Atacama Pathfinder Experiment telescope (APEX). The observations were acquired over three observing periods from 2008 to 2010 as part of the proposals 082.B-0795A, 085.F-9320A, and 087.F-9316A. 12CO(2–1) measurements were also obtained for some sources, as reported in Tables 2, 3, and 5. We used the “standard” receiver$+$backend configurations, APEX-1 (HET230$+$FTTS1) at 230 GHz and APEX-2 (HET345$+$FTTS1) at 345 GHz, with an intrinsic resolution of 0.12 MHz at both frequencies. The sources were small enough to be observed in one pointing with wobbler switching (in ON/OFF mode); pointing was checked every 2 hrs, and calibrations were performed every 10 minutes. Pointing accuracy was typically $\la$2″. System temperatures for the 12CO(3–2) (12CO(2–1)) observations ranged from $\sim$180 K to 670 K ($\sim$160 K to 270 K) with precipitable water vapor from 0.2 mm to 1.1 mm (1.2 mm to 3 mm).

Only NGC 1140 was observed in the 12CO(4–3) and Ci transitions (Period 82), following the same observing protocols as for the lower frequencies but with APEX-3 (HET460$+$FTTS1) at 460 GHz (intrinsic resolution 0.98 MHz). System temperatures for these higher frequency observations ranged from $\sim$380 K to 740 K ($\sim$370 K to 550 K) for Ci (12CO(4–3)). Water vapor columns ranged from 0.2 mm to 0.6 mm for [Ci], and from 0.3 mm to 0.4 mm for 12CO(4–3).

3.3 Data reduction

We adopted the GILDAS/CLASS data reduction package (http://www.iram.fr/IRAMFR/GILDAS) to obtain averaged spectra for all transitions. To remove the baseline from each spectrum, a polynomial was fitted to the line-free regions of each scan (defined a priori) and subtracted; the scans were thereafter Hanning smoothed and averaged, and a constant baseline subtracted. We measured the peak intensities, central velocities, full width half-maximum (FWHM) and velocity-integrated fluxes of the detected lines by fitting Gaussian profiles to the data. In most cases, there is significant velocity structure in the emission, so we fit the line profiles to multiple Gaussians. In these cases we use the sum of the integrated line intensities from the multiple Gaussian fits in the analysis.

Antenna temperatures ($T_{\rm A}^{*}$) have been converted to main-beam brightness temperatures ($T_{\rm mb}$) by dividing the antenna temperatures by $\eta\,\equiv\,B_{\rm eff}/F_{\rm eff}$, where $B_{\rm eff}$ and $F_{\rm eff}$ are the beam and forward hemisphere efficiencies, respectively (i.e., $T_{\rm mb}$ = $T_{\rm A}^{*}$/$\eta$). To convert the measured 12CO(1–0) antenna temperatures, [$T_{\rm mb}$ (K)] to fluxes [$S$ (Jy)], we used the standard conversion factors for both telescopes. The 12CO(1–0), 12CO(2–1), and 12CO(3–2) line profiles for the galaxies in which CO was detected are shown in Fig. 1, together with the Gaussian fits obtained using the GILDAS/CLASS package as described above and reported in Tables 2, 3, and 4. The upper limits for II Zw 40 and SBS 0335$-$052 are given in Table 5. Limits for non-detections are 3$\sigma$, taking the width of three contiguous velocity channels and the rms noise in line-free regions of the spectrum.

For the total flux in each transition, we have included the different velocity components in each galaxy into a global, spectrally-integrated sum. While some of the spectra show slightly different profiles in the different transitions, the signal-to-noise (S/N) is insufficient to allow an analysis of the separate components. Although a few of the individual components may be too narrow to be significant (because their width is smaller than three velocity channels, e.g., Mrk 996, NGC 3353, NGC 1140, see Figs. 1, 2), we have checked that the total intensities are robust to the details of the fitting by repeating the fits with different numbers of components. Some of the measured values in the tables differ slightly from those given in Paper I; this is because we have redone the data reduction for all transitions in order to render the results as consistent as possible. The main difference is the newer version of CLASS used here that seems to give integrated fluxes which can differ by a few percent (but within the error limits) with respect to those in Paper I (performed with an older version of CLASS).

In Paper I, for the galaxies with the largest apparent sizes, to estimate total 12CO(1–0) flux beyond even the largest beam we applied aperture corrections. Here they are not adopted because we prefer to correct the observations to a common beam size, rather than also extrapolating to an uncertain total flux.

3.4 Comparison with previous observations

A detailed comparison of our 12CO(1–0) measured intensities relative to previous work is given in Paper I. Here we compare only the higher-$J$ CO transitions with published literature values. Sage et al. (1992) observed II Zw 40, NGC 3353 (Haro 3), NGC 7077, and UM 462 in 12CO(2–1), and securely detected only the first three. Like the 12CO(1–0) measurements, their results from the IRAM 30m for 12CO(2–1) in NGC 3353 are in good agreement with the line parameters deduced from our central pointing. For NGC 7077, our 12CO(2–1) flux of 1.06 K km s*-1* is $\ga$50% larger than the value of 0.68 K km s*-1* reported by Sage et al. (1992); given the similar uncertainties, it is difficult to understand the discrepancy, especially since our 12CO(1–0) flux is smaller than theirs by $\sim$30%. Our 12CO(2–1) non-detection for UM 462 is consistent with the non-detection by Sage et al. (1992), while our upper limit is smaller. Instead, our non-detection with APEX of 12CO(2–1) in II Zw 40 is inconsistent with Sage et al. (1992); at the flux level found by them, we should have detected this galaxy, as our 3$\sigma$ upper limit is several times smaller.

NGC 1140 and II Zw 40 were also observed previously in 12CO(2–1) with the IRAM 30m by Albrecht et al. (2004). Although the 2–1/1–0 line ratios for NGC 1140 are similar, they find $\sim$40-60% larger intensities in both lines. The line intensity they report for the IRAM 30-m spectrum of II Zw 40 (UGCA 116) is larger than our APEX upper limit, although less than half of the 30m detection reported by Sage et al. (1992). The peak 12CO(3–2) temperatures observed with ALMA in the main knots in II Zw 40 by Kepley et al. (2016) exceed those of our upper limits; but beam dilution is probably playing an important role since the ALMA beam is $\sim$0$\aas@@fstack{\prime\prime}$5 compared to the APEX beam of $\sim$18″.

One of our targets, NGC 7077, was observed by Mao et al. (2010). The ratio of 12CO(3–2) to 12CO(1–0) we find for this galaxy of 1.1 is in excellent agreement with their study.

In conclusion, there is no clear systematic difference between our results and previous work. Nevertheless, the formal uncertainties given in Tables 2–5 should probably be considered as lower limits to the true uncertainties.

4 Correction to a common beam size

The main difficulty in the interpretation of line ratios from single-dish observations is the correction for different beam sizes. The physical models we discuss in Sect. 7 rely on brightness temperatures $T_{\rm B}$ within a common beam size, while we observe main-beam brightness temperatures, $T_{\rm mb}$, within different beams. To correct for this, we have pursued two independent approaches:

1. (1)

used maps of the cool dust emission ($\sim$160 $\mu$m) to scale the different transitions to a common beam size of 22″, with the underlying assumption that the molecular emission is distributed like the cool dust. The PACS 160 $\mu$m maps are particularly well suited to such an endeavor because their intrinsic FWHM is $\sim$11″, similar to the smallest beam size in our observations (12CO(2–1), 10$\aas@@fstack{\prime\prime}$7). 2. (2)

assumed that the molecular emission follows a Gaussian distribution with a FWHM size to be determined by letting it vary in the fit to the line intensities. This can be done in a robust way only for NGC 1140 where we have a sufficient number of line intensities to constrain the fit.

Observations at high spatial resolution ($\la 2$ pc) of metal-poor galaxies in the Local Group suggest that the CO clouds are more concentrated than the dust emission (e.g., Rubio et al. 2015; Schruba et al. 2017); if this were true also on the spatial scales probed by our observations ($\ga$500 times larger), then our assumption that the gas follows the dust would be incorrect. Thus, analyzing beam correction factors under the assumption of a Gaussian distribution enables an independent assessment of the validity of the inferred beam corrections.

Appendix A describes the formulation of the correction to a common beam size assuming that the molecular emission follows the dust. In brief, we have established that the distribution of the cool dust emission in our targets is well described by an exponential distribution by extracting azimuthally-averaged radial brightness profiles; they are all well fit by an exponential. Moreover, for our targets, 160 $\mu$m emission is longward of the spectral energy distribution peak, thus tracing as well as possible the cool dust component. We have checked that free-free emission is unimportant at 160 $\mu$m by inferring its amplitude from the observed SFR (Paper I); for all galaxies for which we have dust images, the expected free-free component is $\la$ 0.1% of the observed emission.

We performed aperture photometry to obtain growth curves, in order to approximate the cumulative flux distribution we expect for the molecular gas in the different beams. Finally, we have followed an analytical approach to fit the growth curves, in order to derive the corrections to the expected flux in different apertures. Table 7 in Appendix A gives the multiplicative correction factors, and the corrected velocity-integrated $T_{\rm mb}$ values that we will use throughout the paper unless otherwise noted; corrected intensities are also given in the last three columns in Tables 2, 3, and 4. While high-$J$ CO emission may be more compact than the gas traced by the lower-$J$ lines, the different radial distributions are expected to be a second-order effect; our proposed correction provides the bulk of the “first-order” correction.

Appendix B discusses the form of the growth curves for a Gaussian source distribution. The main difficulty with applying the Gaussian assumption is the unknown source size; while the source extent can be measured from the analysis of the dust emission, there is no similar way to assess the size of a Gaussian distribution. Thus, when we estimate the physical conditions in the molecular gas by fitting the line intensities (for NGC 1140, the only source for which this is possible), we use the fitting procedure itself to define the source size. The expectation is that because of the similarity of some of the beams in our observations (e.g., 12CO(2–1) with IRAM, 12CO(4–3) with APEX), the fit will be sensitive to the beam correction. This turns out to be the case as will be described in Sect. 7.3.

Mrk 996 shows a markedly broader 12CO(2–1) profile in the larger (27″) APEX beam, relative to the IRAM one (11″). This could be an illustration of the spatial extension of the 12CO(2–1) emitting gas, because of the broader velocity range in the bigger beam; on the other hand, the APEX spectrum is rather noisy, so its reliability is uncertain. Nevertheless, the optical diameter of Mrk 996 is 36″ so it is conceivable that there is more CO emission in the larger APEX beam. UM 448 was also observed in 12CO(2–1) with both the 30m and APEX, but in this galaxy the shape and breadth of the two profiles are more similar, possibly because the galaxy is smaller, 24″ in diameter.

5 Empirical line ratios

Before describing our models of the molecular gas, here we discuss three commonly used diagnostic ratios, 12CO(2–1)/12CO(1–0), 12CO(3–2)/12CO(1–0), and Ci/12CO(1–0). These ratios are calculated using velocity-integrated main-beam temperature, $\int T_{\rm mb}\ d\varv$ (see Tables 2, 3, and 4).

5.1 CO(2–1)/CO(1–0)

The 12CO(2–1) transition traces slightly warmer (upper level equivalent temperature $\sim$17 K) and denser (critical density $\sim 10^{4}$ cm*-3*) gas than the $1-0$ transition. In most galaxies, CO emitting gas is optically thick and the observed CO(2–1)/CO(1–0) ratios are $\la$ 1 (e.g., Braine & Combes 1992). Figure 3 shows the distribution of the 12CO(2–1)/12CO(1–0) line temperature ratios ($R_{21}$) of our low-metallicity targets together with $R_{21}$ ratios (corrected for beam dilution) of spiral galaxies from Braine & Combes (1992). In Fig. 3, our data are corrected to a common beam size using the exponential approach described in Sect. 4 and in Appendix A.

Even after correction for beam dilution, half of our targets have $R_{21}$ $\ga 2$, higher than most of the normal spirals studied by Braine & Combes (1992). Braine & Combes (1992) found an extremely high ratio for another sub-Solar metallicity starburst, NGC 3310 with $R_{21}$ $\sim 2.6$; however Zhu et al. (2009) were unable to confirm this, finding a somewhat lower ratio ($R_{21}$ $\sim 1.5$), after convolving the observations to a common beam size through their maps.

Although it is tempting to claim that the high $R_{21}$ observed in some of these dwarf galaxies is due to optically thin gas, the $R_{21}$ values are somewhat uncertain because of the beam correction. First, the ratios are dependent on the kind of correction applied. The corrections are sensitive both to the source distribution (exponential versus Gaussian) and to the source size (exponential scale length versus Gaussian FWHM); since we have no way of accurately determining a Gaussian FWHM, we cannot assess how much the $R_{21}$ values would change under another assumption for source distribution. Second, if we use the IRAM 12CO(2–1) observations (with a smaller beam) for Mrk 996 and UM 448, we obtain beam-corrected R_{21}$$\sim0.6 for Mrk 996, and R_{21}$$\sim1.2 for UM 448. Thus the high $R_{21}$ values are not altogether robust. The only galaxy in our sample for which we are fairly certain that the true $R_{21}$ value is $\ga$2 is NGC 1140. The interpretation of $R_{21}$ for this galaxy will be discussed in Sect. 7.3 where we present physical models for the molecular gas.

5.2 CO(3–2)/CO(1–0)

Figure 4 shows the distribution of the ratio of 12CO(3–2)/12CO(1–0) ($R_{31}$ in $T_{\rm mb}$ units) for the three sources in our sample with 12CO(3–2) detections, compared to the large sample of Mao et al. (2010). Unlike for the 12CO(2–1)/12CO(1–0) ratio observed with the IRAM 30m, the APEX beam is $\sim$18″, similar to the IRAM 30-m beam of $\sim$22″ for the 1–0 line; the Mao et al. (2010) observations were obtained with the Heinrich Hertz Telescope with a 12CO(3–2) beam of $\sim$22″. Thus, we have not applied any correction for beam dilution; the $R_{31}$ values shown in Fig. 4 should be relatively impervious to beam corrections, but rather representative of the beam-averaged physical conditions in the molecular gas.

Of the three galaxies in our sample with 12CO(3–2) detections (Mrk 996 was observed but not detected, R_{31}$$<0.7), NGC 1140 shows the largest ratio ($R_{31}$ $=\,2.00\,\pm\,0.45$; with the exponential beam correction $R_{31}$ $=\,1.72\,\pm\,0.39$), roughly equivalent to that found for NGC 3310 ($R_{31}$ = 1.9$\,\pm\,$0.52) by Mao et al. (2010). Such a ratio exceeds typical values in luminous infrared galaxies (LIRGs) and those expected in the Orion “hot spots” (Papadopoulos et al. 2012). $R_{31}$ has been historically taken as an indicator of H2 density (e.g., Mauersberger et al. 1999), and high $R_{31}$ is possibly indicative of somewhat excited, optically thin, gas, although shocks and cool foreground absorbing layers are also possible causes (e.g., Oka et al. 2007). We shall explore this point further with physical radiative transfer models in Sect. 7.2.

5.3 CI(1–0)/CO(1–0)

Together with CO and Cii (at 158 $\mu$m), the atomic carbon line Ci is an important coolant of the ISM (e.g., Bolatto et al. 1999; Gerin & Phillips 2000; Bolatto et al. 2000a, b, c; Israel & Baas 2002; Papadopoulos et al. 2004; Bayet et al. 2006; Israel 2009). Despite the difficulty of observation from the ground (the ${}^{3}{\rm P}_{1}-^{3}{\rm P}_{0}$ [Ci] transition lies at 609 $\mu$m), the potential of atomic carbon to trace molecular gas was recognized early on. Photo-Dissociation Regions (PDRs) are thought to be larger at low metallicity, because of the decreased attenuation of the UV field through the lower dust content (e.g., Lequeux et al. 1994; Bolatto et al. 1999). The brighter UV field in low $A_{V}$ environments tends to change the structure of PDRs, enlarging the region where atomic and ionized carbon dominate, and diminishing the region where H2 is associated with CO emission (e.g., Bolatto et al. 2013). The spatial distributions of [Ci] and CO gas are similar in regions within the Milky Way (e.g., Kramer et al. 2004; Zhang et al. 2007), the Small Magellanic Cloud (SMC, Requena-Torres et al. 2016), and other galaxies (e.g., Israel & Baas 2003; Krips et al. 2016), so [Ci] seems to be tracing molecular gas. This is especially important in low-metallicity environments or cosmic-ray dominated regions where CO emission is suppressed (e.g., Hopkins et al. 2013; Bisbas et al. 2015, 2017), or in high-redshift galaxies where [Ci] observations are facilitated (e.g., Walter et al. 2011).

We have obtained a Ci measurement of one of our targets, NGC 1140, as shown in Fig. 5 where we plot the [Ci]/12CO(1–0) ratio (in energy units) together with similar measurements from the literature (Bolatto et al. 2000a, b, c; Israel & Baas 2002; Bayet et al. 2006). We have applied the correction for beam dilution as described in Sect. 4 and Appendix A. The regression lines are taken from the PDR model by Bolatto et al. (1999, dashed line), and the best fit to their data (Bolatto et al. 2000a). Despite the paucity of data, the trend of enhanced Ci/12CO(1–0) with decreasing O/H is evident.

The expected metallicity-dependent behavior of Ci in PDR environments has been modeled by various groups (e.g., Papadopoulos et al. 2004) who confirm that the abundance of [Ci] dominates that of CO for large regions within a low-metallicity cloud (A_{V}$$\ga2). Glover & Clark (2016) present more recent simulations of Ci emission relative to 12CO(1–0) as a function of metallicity. Our (beam-corrected) Ci/12CO(1–0) velocity-integrated temperature ratio for NGC 1140 of 0.65 (see Table 4) is more than a factor of two higher than their predictions for the highest PDR illumination they model ($G_{0}$ = 100 in units of the interstellar radiation radiation field by Draine 1978), and the disagreement increases for less intense fields.

The [Ci]/12CO(1–0) ratio tends to decrease with increasing PDR density; the observed ratio for NGC 1140 is higher than the predictions of (Solar metallicity) PDR models with volume number densities $n_{0}\sim 10^{4}$ cm*-3* and $G_{0}\sim 10^{4}$ (Hollenbach et al. 1991). The observed Ci/13CO(2–1) for NGC 1140 is a factor of $4-5$ higher than the PDR predictions (for Solar, and twice Solar metallicities) by Meijerink et al. (2007), for volume number densities $n_{0}\ga 10^{4}$ cm*-3* and $G_{0}\sim 10^{5}$. However, they assume [12CO/13CO] = 40; a lower isotopic ratio (see Sect. 7.3) or a lower metallicity would reduce the disagreement. In Sect. 7.3, we explore physical models of the gas in NGC 1140 and derive relative abundances of atomic carbon and CO.

6 Dense-gas tracers

The relatively high critical densities of 12CO(3–2) and HCN(1–0) (n_{\rm crit}$$\sim 10^{4}-10^{5} cm*-3*) make these transitions good tracers of dense gas, which are possibly better correlated with SFR than the lowest-$J$ CO lines (Gao & Solomon 2004; Narayanan et al. 2005, 2008; García-Burillo et al. 2012; Lada et al. 2012; Hopkins et al. 2013; Greve et al. 2014). Given the relatively high sSFRs of our targets (sSFR $\ga 10^{-10}$ yr*-1*, see Paper I), here we examine luminosity correlations of 12CO(3–2) and HCN(1–0) for these metal-poor galaxies and compare them with more massive metal-rich ones.

Figure 7 shows HCN luminosity plotted against FIR luminosity, $L_{\rm FIR}$ (left panel) and CO(1–0) luminosity, $L^{\prime}_{\rm CO(1-0)}$ (right). The only galaxy for which we have HCN(1–0) observations is NGC 1140333We do not apply beam corrections to the upper limit, because of the similarity of the HCN(1–0) and 12CO(1–0) beam sizes.; also shown in Fig. 7 are luminous (and ultraluminous) infrared galaxies from Gao & Solomon (2004); Graciá-Carpio et al. (2008); García-Burillo et al. (2012), together with regions within M 33 (Buchbender et al. 2013) and the Milky Way (MW, Wu et al. 2010); a luminous irregular galaxy, He 2–10, is also plotted (Santangelo et al. 2009). The dashed lines show a robust regression444Here and throughout the paper we use the “robust” regression algorithm, effective for minimizing the effects of outliers, as implemented in R, a free software environment for statistical computing and graphics, https://www.r-project.org/. performed on all galaxies excepting He 2–10 and NGC 1140; the slope of the $L^{\prime}_{\rm HCN(1-0)}$-$L_{\rm FIR}$ correlation is $0.97\,\pm\,0.01$ and for $L^{\prime}_{\rm HCN(1-0)}$-$L^{\prime}_{\rm CO(1-0)}$ $1.16\,\pm\,0.03$ (this last without the MW points, for which there were no CO(1–0) data). The slopes are consistent with previous work (e.g., Gao & Solomon 2004; García-Burillo et al. 2012), and suggest that HCN in irregular galaxies such as He 2–10 and NGC 1140 is somewhat deficient compared to $L_{\rm FIR}$ but within normal limits compared to $L^{\prime}_{\rm CO(1-0)}$.

Both He 2–10 and NGC 1140 contain extremely massive, compact star clusters (e.g., Johnson et al. 2000; de Grijs et al. 2004; Moll et al. 2007) (Super Star Clusters, SSCs), but He 2–10 is approximately Solar metallicity (Kobulnicky et al. 1999) while NGC 1140 is $\sim$0.3 Z*⊙. Feedback from these SSCs would be expected to substantially reduce the fraction of dense gas in these galaxies as traced by HCN(1–0) (Hopkins et al. 2013). Indeed, with L^{\prime}_{\rm CO(1-0)}$$\sim 10^{8} K km s-1* pc2, the $L^{\prime}_{\rm HCN(1-0)}$/$L^{\prime}_{\rm CO(1-0)}$ ratio for He 2–10 is $\sim$0.02, and $\la$0.10 for NGC 1140 (with L^{\prime}_{\rm CO(1-0)}$$\sim 10^{6} K km s*-1* pc2), roughly consistent with the model predictions by Hopkins et al. (2013). More observations of HCN in such extreme galaxies are needed to further explore the survival of dense gas in these galaxies, and the ability of HCN to trace it (see also Sect. 7.3).

Figure 7 plots $L^{\prime}_{\rm CO(3-2)}$ vs. $L_{\rm FIR}$ and $L^{\prime}_{\rm CO(1-0)}$ as in Fig. 7 for HCN(1–0), but here for the four galaxies in our sample which were observed in 12CO(3–2) (Mrk 996, NGC 1140, NGC 7077 UM 448). Like HCN(1–0), 12CO(3–2) is deficient relative to $L_{\rm FIR}$. The slopes of $0.93\,\pm\,0.05$ ($L^{\prime}_{\rm CO(3-2)}$ vs. $L_{\rm FIR}$) and $0.96\,\pm\,0.04$ ($L^{\prime}_{\rm CO(3-2)}$ vs. $L^{\prime}_{\rm CO(1-0)}$) found by fitting the combination of the Mao et al. (2010) and Narayanan et al. (2005) samples are consistent with those found by Mao et al. (2010) and Greve et al. (2014), despite different fitting techniques for the latter; however the $L^{\prime}_{\rm CO(3-2)}$ vs. $L^{\prime}_{\rm CO(1-0)}$ slope is significantly flatter than the one of $\sim$1.5 found by Narayanan et al. (2005) and used to calibrate the Hopkins et al. (2013) simulation results. The deficit of $L^{\prime}_{\rm CO(3-2)}$ relative to $L_{\rm FIR}$ is almost certainly due to metallicity as shown in Paper I; for metal-poor galaxies, we found an offset between the CO emission relative to the SFR in the sense that there is less CO for a given SFR value. This deficiency results from the metallicity dependence of the CO conversion factor to H2 mass. Interestingly, the four galaxies for which we have CO(3–2) measurements are not deficient in 12CO(3–2) relative to 12CO(1–0) in the other galaxies taken from the literature; they show similar CO luminosity ratios despite their low metallicity ($\sim$0.25 Z*⊙*). This could be implying that excitations in these galaxies are similar to those in more metal-rich systems, a point which we explore more fully in the next section.

7 Modeling physical conditions at low metallicity

We now explore via modeling the physical conditions in the molecular gas in our metal-poor targets. Detailed physical modeling is possible only for NGC 1140, for which we have four 12CO and two 13CO detections (see also Sect. 7.3). This is the first time that a low-metallicity galaxy outside the Local Group can be modeled in such detail with CO; for the remaining galaxies we explore what can be learned from the available 12CO detections.

7.1 Local thermodynamic equilibrium

For optically thin thermalized emission in Local Thermodynamic Equilibrium (LTE), antenna temperature is directly proportional to the column density in the upper level of the observed transition. With knowledge of the kinetic temperature, and under the above assumptions, we can estimate the column density of the species. Population (or rotation) diagrams are a useful tool for this (e.g., Goldsmith & Langer 1999), through the analysis of the column density per statistical weight of different energy levels as a function of their energy above ground state. In LTE, the statistically-weighted column densities will be a Boltzmann distribution, so that the slope of a plot in natural logarithmic space will correspond to the inverse of the “rotational” temperature, equal to the kinetic temperature in the case of thermalized emission.

Despite its usefulness, there are several problems with such an approach. First, the assumptions required are not usually met by the physical conditions in molecular clouds; molecular emission in the 12CO lines is usually optically thick and not necessarily thermalized. Second, to be able to correct the emission for the different beam sizes with which the various transitions are observed requires a knowledge of the source distribution and size. Nevertheless, a population diagram analysis sets useful limits on the column density and temperature and is a fruitful starting point for non-LTE considerations.

The column density of the upper energy state can be written as (e.g., Goldsmith & Langer 1999):

[TABLE]

where $W$ is the velocity-integrated main-beam brightness temperature, $A_{ul}$ the Einstein spontaneous emission coefficient, and $\tau$ is the optical depth of the transition. For molecules in LTE, both a line’s excitation temperature and a molecule’s rotation temperature, which determines its relative energy level populations, equal the kinetic temperature ($\equiv\,T$); the population in each level is then given by the Boltzmann distribution:

[TABLE]

where $N^{\rm thin}_{u}$ corresponds to optically-thin emission, ignoring the $\tau$ term in Eqn. (1); $N$ is the total column density, $E_{u}$ is the upper energy level, $g_{u}$ is the statistical weight, and $Z$ is the partition function. Thus, under the assumption of LTE, and with knowledge of $T$, we can obtain the total molecular column density from the column density of any single transition:

[TABLE]

where, following Goldsmith & Langer (1999), we have equated the optical-depth correction factor $\tau/(1-e^{-\tau})$ to $C_{\tau}$, and $N_{u}$ is calculated from the first two factors in Eqn. 1. To evaluate the population diagrams numerically, the transition frequencies, upper energy levels, Einstein spontaneous emission, and statistical weights ($\nu$, $E_{u}$, $A_{ul}$, $g_{u}$) for the 12CO and 13CO molecules were calculated using physical quantities from Schöier et al. (2005) and found to be highly consistent with the data from HITRAN (Rothman et al. 2009).

We first assume that the emission is optically thin, with $\tau\ll 1$ ($C_{\tau}\,=\,1$). A CO population diagram with this assumption for NGC 1140 is shown in Fig. 8, including the two available 13CO transitions. Here, as elsewhere, we have used the velocity-integrated $T_{\rm mb}$ corrected to a 22″ beam size as described in Sect. 4 and Appendix A. By fitting the $\ln(N_{u}/g_{u})$ vs. $E_{u}$ values with a linear regression (shown as a gray dashed line), we were able to infer the total 12CO and 13CO column densities and temperatures: $T_{\rm 12CO}\,=\,18.1\,\pm\,1.0$ K; $\log(N_{\rm 12CO})\,=\,14.7$ cm*-2*; $T_{\rm 13CO}\,=\,15$ K. With only two transitions, the 13CO temperature is ill determined, so we fixed it to $T_{\rm 12CO}$ and obtained $\log(N_{\rm 13CO})\,=\,13.6$ cm*-2*.

We then attempted to correct for optical depth effects, according to Goldsmith & Langer (1999, their Eqn. 27): the largest correction term $C_{\tau}$ is for 12CO(4–3)  with $C_{\tau}\,=\,1.01$, and the largest optical depth $\tau\sim$0.002. If we take these results at face value, and infer the 12CO/13CO abundances by taking the ratio of the total column densities, we obtain a value of 12CO/13CO $\sim 12-13$. To our knowledge this is one of the lowest values ever inferred for a galaxy, but cannot be considered reliable because of the assumption of optically thin gas in the LTE analysis. We will explore this further in Sects. 7.3 and 8.

The population diagrams under the optical-thin assumption for UM 448 and NGC 7077 (the only other galaxies for which we have 12CO(3–2) detections) are shown in Fig. 9. The observations were corrected for beam dilution as described in Sect. 4 and Appendix A. Again, from the best-fit regression, we infer $T$ and total $N_{\rm CO}$ column densities: $T_{\rm 12CO}\,=\,10.8\,\pm\,0.8$ K and $\log(N_{\rm 12CO})\,=\,15.0$ cm*-2* for UM 448; and $T_{\rm 12CO}\,=\,12.3\,\pm\,2.0$ K and $\log(N_{\rm 12CO})\,=\,14.7$ cm*-2* for NGC 7077. As for NGC 1140, we inferred the optical depths for the various transitions, and for both galaxies find that the largest $\tau$ is for the $J=2-1$ transition with $\tau_{\rm CO}$ = 0.001 (NGC 7077) and $\tau_{\rm CO}$ = 0.002 (UM 448).

7.2 Modeling 12CO line ratios with RADEX

To better understand the observed 12CO line ratios in the galaxies in our sample, in particular the high ratio of NGC 1140 compared with other, more metal-rich, galaxies (see Figs. 3, 4), we have plotted the line ratios predicted by RADEX as a function of $T_{\rm kin}$ in Fig. 10 ($R_{21}$ and $R_{31}$ for NGC 1140 have been corrected for beam dilution as described in Sect. 4 and Appendix A). Two sets of column densities $N_{\rm 12CO}$ and volume densities $n_{\rm H2}$ are illustrated in the figure; the solid (purple, upper) line passing through the locus of NGC 1140 corresponds to the approximate best-fit RADEX model described in Sect. 7.3. It is evident from Fig. 10 that the combination of moderately low column density (and thus low $\tau_{\rm CO}$) and high $n_{\rm H2}$ is a necessary ingredient for the high line ratios observed in NGC 1140. Line ratios would be high also in the case of high $N_{\rm 12CO}$ and high $n_{\rm H2}$ (see lower blue solid line), but only at high temperatures; this is the case in NGC 1068 where gas is warm and dense in the circumnuclear disk around the active nucleus (Viti et al. 2014). We discuss below in Sect. 7.3.3 why we consider high temperatures in NGC 1140 to be unlikely. In the case of more moderate $n_{\rm H2}$ (here illustrated as $10^{4}$ cm*-3*), there is no $T_{\rm kin}$ that can sufficiently excite the upper $J$ level.

The remaining galaxies for which we have 12CO(3–2) observations (Mrk 996, NGC 7077, UM 448) are also shown in the right panel of Fig. 10. The $R_{31}$ values of these are less extreme, and can be well modeled with H2 volume densities as low as $10^{4}$ cm*-3*. In these galaxies, the gas does not necessarily have to be cool and optically thin (i.e., low $N_{\rm 12CO}$) in order to achieve the observed $R_{31}$; it can be warmer and have a higher column density than in NGC 1140.

7.3 RADEX models for NGC 1140

To determine the excitation conditions of the molecular gas in NGC 1140, we have fitted the six 12CO and 13CO detected transitions with the radiative transfer code RADEX developed by van der Tak et al. (2007), using the uniform sphere approximation to calculate the escape probabilities. A one-zone model was adopted, namely the kinetic temperature and H2 volume density were required to be the same for all molecules, implying that all transitions sample the same gas. We calculated RADEX predictions for $\Delta V$ =10 and 20  km s*-1*, covering the following range of parameters sampled in logarithmic steps555Temperature was sampled linearly.:

• •

H2 volume densities $n_{\rm H2}$ = $10^{2}-10^{8}$ cm*-3*;

• •

kinetic temperature $T_{\rm kin}$ = $5-200$ K;

• •

CO column densities $N_{CO}$ = $10^{12}-10^{20}$ cm*-2*.

All parameters were calculated assuming a cosmic microwave background radiation temperature of $T$ = 2.73 K. The 13CO abundance was included in the calculation through a second RADEX grid for the isotope; the column density $N_{13CO}$ was adjusted by sampling a range of abundances (from $N_{12CO}$/$N_{13CO}$ = 1 to 100), and refining the fineness of the grid once the initial best-fit parameters were established. Ultimately, $\ga$400 000 total models in the combined 12CO $+$ 13CO grid were sampled.

The observed emission is generally only a small fraction of integrated line intensity from RADEX (and Large-Velocity Gradient, LVG) models. The fraction of these two quantities, $F\,=\,(\Delta\varv_{\rm obs}T_{\rm obs})/(\Delta V_{\rm model}T_{\rm model})$, is the filling factor. We incorporated $F$ as a free parameter in the models, and fit the grid of RADEX models via a $\chi^{2}$ minimization technique (e.g., Nikolić et al. 2007):

[TABLE]

where $I_{i}^{\rm obs}$ is the velocity-integrated $T_{\rm mb}$ of the $i^{th}$ transition, $I_{i}^{\rm mod}$ the $i^{th}$ model prediction, and $\Delta I_{i}^{\rm obs}$ are the 1$\sigma$ uncertainties given in Table 4.

We also experimented with a minimization technique not including the filling factor $F$, but rather based on line ratios:

[TABLE]

where $R_{j}^{\rm obs}$ is the $j^{th}$ velocity-integrated temperature ratio, $R_{j}^{\rm mod}$ is the corresponding model line ratio, and $\Delta R_{j}^{\rm obs}$ are the 1$\sigma$ uncertainties. These fits do not explicitly require equality of the filling factor $F$ and thus may be more accommodating. We experimented with several sets of line ratios, in order to avoid potentially biasing the fits to any of the individual ratios.

As described in Sect. 4, we explored two different methods for correcting the velocity-integrated $T_{\rm mb}$ (and uncertainties) to a 22″ beam size: (1) analyzing the cool-dust distribution and assuming that the molecular gas is distributed like the dust (see Appendix A); and (2) assuming the molecular gas is distributed in a Gaussian, and letting the Gaussian’s FWHM ($\theta_{s}$) be a free parameter in the fit (see Appendix B). We used both methods to calculate the observed $I_{i}^{\rm obs}$ in Eqns. (4, 5). In the first case based on the fitted exponential function of the dust emission, $I_{i}^{\rm obs}$ were calculated independently of the fitting procedure. Thus for the six intensities (Eqn. 4), we fit five parameters ($n_{\rm H2}$, $T_{\rm kin}$, $N_{CO}$(12CO), [12CO]/[13CO], $F$), and for the five independent line ratios (Eqn. 5), we fit four parameters (same as before, but not including $F$). For the second case of Gaussian beam corrections, $I_{i}^{\rm obs}$ were calculated within the fit procedure; we explored $\theta_{s}$ ranging from 2″ to 40″, and narrowed the step size in successive iterations. Thus for the Gaussian beam corrections, the fitting procedure comprised an additional parameter, $\theta_{s}$, and the minimum $\chi^{2}$ value includes this dependence. All the $\chi^{2}$ values quoted here are not reduced, but rather the total summed $\chi^{2}$.

Table 6 gives the best-fit parameters obtained for NGC 1140. The different fitting methods and even the different beam correction approaches give very similar results, namely cool temperatures, high volume densities, low CO opacities, and low 12CO/13CO abundance ratios. The main difference among the fits is the CO column density, $N_{12CO}$, which for a given $\Delta V$ varies by a factor of $\sim$4. Because the opacity in the RADEX models $\tau\propto N_{\rm co}/\Delta V$, and $\tau$ governs the line ratios and thus must remain relatively constant, for higher $\Delta V$, $N_{12CO}$ is consequently higher within the resolution of our RADEX grid. For higher $N_{12CO}$, the filling factor $F$ is also commensurately lower, which we will discuss in Sect. 7.3.1. In any case, the low values of $\chi^{2}$ imply that a one-zone spherical cloud with uniform physical properties gives a good fit to the molecular gas emission in NGC 1140. The extremely low 12CO/13CO abundance ratio of $\sim 8-20$ is obtained for NGC 1140 even with a non-LTE analysis; we discuss this unexpected result further in Sect. 8.1.

Figure 11 shows the $\chi^{2}$ surface for the Set 1 line ratios666“Set 1” was taken with the ratios relative to the adjacent lower-$J$ line, specifically: [12CO(2–1)/12CO(1–0), 12CO(3–2)/12CO(2–1), 12CO(4–3)/12CO(3–2), 13CO(1–0)/12CO(1–0), 13CO(2–1)/13CO(1–0)]. See Table 6 for more details. with the exponential beam correction as a function of the fitted parameters ($T_{\rm kin}$ vs. $n_{\rm H2}$ in the left panel, and $N_{12CO}$ vs. $\tau$[12CO(1–0)] in the right). In both panels, the error bars correspond to $\chi^{2}\la 1$ at the best-fit value, rather than the full posterior range given in Table 6. It can also be seen from Fig. 11 that the parameters are fairly well constrained by the model; specifically, the volume density $n_{\rm H2}$ cannot be much lower than $\sim 10^{5}$ cm*-3*, and because of the well-known degeneracy between $T_{\rm kin}$ and $n_{\rm H2}$, the temperature also cannot be much greater than $\sim$20 K.

Figure 12 gives the $\chi^{2}$ surface for the Set 1 line ratios using the Gaussian beam correction plotted against the 12CO column density $N_{\rm 12CO}$ and the Gaussian FWHM $\theta_{s}$. The fit is somewhat sensitive to the assumed $\theta_{s}$; for $\theta_{s}\la$15″ and for $\theta_{s}\ga$34″, $\chi^{2}>$ 10, and thus is not plotted. The best-fit Gaussian sizes range from 26″ to 31″, implying that under the assumption of a Gaussian distribution, the macroscopic size of the emission is $\ga$2 kpc. This would imply that the CO emission is more extended than the dust (FWHM($r_{s}$) $\sim$850 pc, see Appendix A), which seems unlikely. In any case, even the largest CO beam is sampling less than half of the galaxy ISM; as shown in Fig. 17, 50% of the dust emission lies within an effective radius (diameter) of $\sim$20″ (40″, 3.8 kpc).

It should be emphasized that small $\chi^{2}$ values are not easy to achieve, even with several trials with many different formulations of beam corrections, in addition to the ones described in Sect. 4 and in the Appendices. The small $\chi^{2}$ values of the fits in Table 6 are telling us that for reasonable assumptions for beam corrections, the single-zone RADEX models can well accommodate the data. Moreover, the physical conditions inferred from the different fitting techniques are mutually consistent, independently of the kind of beam correction. Because of this similarity, in what follows, we will use the exponential beam corrections described in Appendix A; the virtue of these is that they can be measured or estimated for all the galaxies observed, under the assumption that on kpc scales the molecular gas is following the dust.

7.3.1 Filling factor

The mean 12CO line width of the various components in NGC 1140 (see Table 4) is $\sim 21$  km s*-1*; we thus prefer the RADEX fits with $\Delta V$ = 20  km s*-1*. For $\Delta V$ = 20  km s*-1*, fitting the intensities as in Eqn. (4) gives a best-fit filling factor $F$ of $0.10\,\pm\,0.06$. This best-fit $F$ is rather large, but related to the somewhat low $N_{12CO}$ value of $10^{15.8}$ cm*-2*; as discussed below, the fits of the line ratios as in Eqn. (5) give higher $N_{12CO}$ thus consequently lower, possibly more realistic, values of $F$.

For the line-ratio fits, the best-fit RADEX model for NGC 1140 predicts a velocity-integrated brightness temperature $T_{B}($12CO(1–0)$)$ = 21.0 K km s*-1*; from Table 4, we have a beam-corrected velocity-integrated $T_{\rm mb}$ of 0.56 K km s*-1*. Dividing the two values gives a nominal filling factor of $\sim$0.03. Similar filling factors are derived for the other transitions including 13CO, as a consequence of the assumption of a single-zone model. If we compare with the model from the “other” $\Delta V$ = 20  km s*-1* line-ratio fits (see Table 7.3), with lower $N_{12CO}$ and $\tau$ (see Table 6), we obtain a slightly higher value, $F\sim$0.04. Using the main-beam peak temperatures rather than velocity-integrated values (see Table 4), and adjusting to the common 22″ beam size, gives filling factors of $\sim$0.01.

These filling factors of a few percent for the molecular gas in NGC 1140 are similar to those found from high-resolution ALMA maps of 30 Doradus in the Large Magellanic Cloud (LMC, Indebetouw et al. 2013). They are also similar to those obtained by a similar $\chi^{2}$ analysis of non-LTE models of single-dish line ratios in the SMC and the LMC (Nikolić et al. 2007); these last are as high as $\sim$0.14 (for what they consider to be good fits with $\chi^{2}\la 2$), consistent with $F\,=\,0.1$ for NGC 1140 obtained from the intensity fitting.

7.3.2 CO excitation ladder

The modeled CO excitation ladders for NGC 1140 are shown in Fig. 13. As in Fig. 10, the observations have been corrected for beam size as described in Sect. 4 and Appendix A. The line segments show the best-fit RADEX model for both 12CO and 13CO transitions adopting the best fit for the Set 1 line ratios with $\Delta V$ = 20  km s*-1*; the blue area corresponds to the range of ratios allowed by the models (for $\chi^{2}$ = $\chi^{2}({\rm min})+1$). The similar 12CO and 13CO slopes for NGC 1140 (the data points follow parallel lines) suggest that even the 12CO emission may be optically thin, as also implied by the best-fit results in Table 6. The CO gas is in an excited state, more similar to the center of M 82 (Weiß et al. 2005) than to the Milky Way (e.g., Carilli & Walter 2013). This is not a surprising result given the population of SSCs in the nuclear region of NGC 1140.

The low optical depths of even the 12CO lines steepen the excitation ladder at low $J$, steeper than $\nu^{2}$ as would be expected for thermalized, optically thick lines. The shape is an indirect consequence of the normalization to $J$=1–0, due to the increase of $\tau_{\rm CO}$ with increasing $J$ up to the inflection in the excitation. Because 12CO(1–0) has the smallest $\tau$ of the 12CO transitions, with a different normalization, it would drop below the $\nu^{2}$ trend. The results of the RADEX fit (see Table 6) are telling us that the $J=4$ level is sparsely populated because of the decreased $\tau_{\rm CO}$ relative to the lower-$J$ transitions; this produces the flattening in the excitation ladder.

7.3.3 Atomic carbon

We have performed additional RADEX fits to the 12CO and Ci emission, both with intensities as in Eqn. (4) and with ratios (Eqn. 5). Fitted parameters include, as before, ($n_{\rm H2}$, $T_{\rm kin}$, $N_{CO}$(12CO), $F$), but now [12CO]/[12C], rather than [12CO]/[13CO]. Here, again, the implicit assumption is that atomic carbon and CO occupy the same volume, and are uniformly intermixed within an ISM of a single H2 kinetic temperature and volume density; such an assumption is consistent with the findings of Requena-Torres et al. (2016) for Hii regions in the SMC and of Okada et al. (2015) for the LMC. The results of the fits are given in Table 6 and shown graphically in Fig. 14 where the $\chi^{2}$ surfaces (in logarithmic terms) are shown as a function of $n_{\rm H2}$ and $N_{\rm CI(1-0)}$ column density. As for the previous CO analysis, we prefer the $\Delta V$ = 20  km s*-1* fits, roughly the mean component line width observed in NGC 1140.

The intensity fit of the 12CO and [Ci] emission gives parameters very similar to the CO fit, although the inferred $n_{\rm H2}$ for the Ci fit is lower. On the other hand, the fit to the Ci ratios (with 12CO) gives higher $T_{\rm kin}$, lower $n_{\rm H2}$, and 25 times higher $N_{CO}$, higher than any other fit we performed. Folding in the best-fit [12CO]/[12C] abundance ratio, which varies by a factor of $\sim$3 between the two fits, shows that the best-fit column density of atomic carbon $N_{C}$ is also different for both fits: $\sim 10^{17.3}$ cm*-2* in the intensity case and $\sim 10^{18.2}$ cm*-2* for the ratios. Consequently, the filling factors for the two fits also differ. The best-fit $F$ for the [Ci] intensity fits is 0.10 as for (the intensity fitted) CO, but for the [Ci] ratio fits, it is much lower, $F\la$0.01. This is because the higher $N_{CO}$ requires a lower filling factor to accommodate the RADEX predictions with the observations. We found a similar behavior in Sect. 7.3 for the CO ratio fits relative to the intensity ones. The reasons for the disagreement are unclear, although the low values of $\chi^{2}$ for all the [Ci] fits make it unlikely that the model is failing to capture the physical conditions in the clouds. In any case, these values of $F$ are between 2 and 20 times lower than those found for [Ci] emission in the Magellanic Clouds by Pineda et al. (2017); however their observations probe physical scales more than 50 times smaller, so a comparison is not straightforward.

Independently of the disagreement, our RADEX fits for [Ci] all find that the atomic carbon column density must be larger than $N_{CO}$, implying a [12CO]/[12C] abundance of $<$1. For the intensity fit, with lower $N_{CO}$, this requires a very low [12CO]/[12C] of $\sim$0.03, while for the ratio fit with higher $N_{CO}$, [12CO]/[12C]$\sim$0.1. This would indicate that atomic carbon is at least 10 times more abundant than 12CO in NGC 1140. Such a result would be consistent with the theoretical predictions of Bialy & Sternberg (2015) for interstellar ion-molecule gas-phase chemistry in galaxies; their models predict that [12CO]/[12C] abundance ratios decrease with decreasing metallicity and with higher ionization parameters. On the other hand, observational studies of luminous (metal-rich) galaxies found that atomic carbon is less abundant than CO, [12CO]/[12C] $\sim 4-8$ (e.g., Israel 2009; Israel et al. 2015). However, the observed Ci/12CO(4–3) ratio (in flux units) for NGC 1140 ($\sim 0.7$) is more than 4$\sigma$ higher than the mean of the galaxies observed by Israel et al. (2015), so the relatively high abundance of atomic carbon is perhaps not altogether surprising. Although our Ci detection is marginal ($\sim 3\sigma$, see Table 4), to reduce the inferred [12CO]/[12C] ratio to unity or less would require a reduction in the Ci intensity by more than a factor of 10; this seems unlikely given the trends shown in Fig. 5.

Independent one-zone model fitting of the 12CO, 13CO, and Ci emission of NGC 1140 gives similar physical conditions. Specifically, by fitting a vast grid of LVG models (Weiß et al. 2007) to the NGC 1140 data, and considering dust continuum, and molecular and atomic abundances relative to H2, we obtain high densities ($n_{\rm H2}$ $\ga 10^{5}$ cm*-3*), cool temperatures ($T_{\rm kin}$ $\sim$20 K), low [12CO]/[13CO] abundance ratios ($\sim 13$), low [12CO]/[12C] abundance ratios ($\sim 0.13$), and moderate column densities ($N_{CO}$ $\sim 5\times 10^{16}$ cm*-2*), similar to the RADEX results. However, with these models, a “warm” solution with $T_{\rm kin}$ $\sim$52 K is also possible. This solution would result in higher abundance ratios ([12CO]/[13CO] $\sim 36$, [12CO]/[12C] $\sim 0.3$), and higher $N_{CO}$ ($\sim 7\times 10^{17}$ cm*-2*). Nevertheless, in this case there would be an unrealistically low gas-to-dust mass ratio (GDR), $\sim$62; this is lower than even the Milky Way (GDR $\sim$ 100, Draine et al. 2007), which is unlikely considering the low metallicity of NGC 1140. With ${\mathrm{Z}/Z}_{\odot}$ $\sim$0.3 for NGC 1140, and given the linear trend of GDR with metallicity in this abundance range (e.g., Rémy-Ruyer et al. 2014), we would expect a GDR of $\sim$300, not far from the GDR of $\sim$380, inferred from the “cool” solution obtained with the Weiß et al. (2007) LVG models. Thus, the “cool” solution with $T_{\rm kin}$ $\sim$ 20 K seems more likely for NGC 1140.

The “warm” solution with the Weiß et al. (2007) LVG models for 12CO, 13CO, and Ci is not far from the “warm” solution (with $T_{\rm kin}$ $\sim 38$ K) that emerged from the RADEX fitting of Ci and 12CO line ratios (see Table 6). However, our RADEX models for 13CO and 12CO do not allow the higher $T_{\rm kin}$ found with the Weiß et al. (2007) models; we obtain $\chi^{2}>10$ for $T_{\rm kin}$ $>46$ K, and a mean $\chi^{2}$ of $\sim 6-7$ for $40\leq$ $T_{\rm kin}$ $\leq 46$ K, compared to $\chi^{2}$ values of $<1$ for our best fits. Based on the much lower $\chi^{2}$ values for the CO fits with cool $T_{\rm kin}$, we do not consider further the “warm” RADEX fits. Observations of higher-$J$ CO lines and Ci would help settle this potential ambiguity.

Combined with the [12CO]/[13CO] abundance ratio of $\sim 10-12$ found above (see also Sect. 7.1), this would imply a [12C]/[13CO] abundance ratio of $\ga 100-300$, within the range of values found by Israel et al. (2015) for galaxies with low CO/Ci] ratios. However, this may be a specious comparison because of the unusually low [12CO]/[13CO] abundance ratio which, in some sense, compensates for the extremely low [12CO]/[12C] found here for NGC 1140. As already mentioned, we discuss further the low [12CO]/[13CO] abundance ratio found for NGC 1140 in Sect. 8.1.

7.3.4 Modeling HCN with RADEX

We did not detect HCN(1–0) in NGC 1140, even though the RADEX best fit suggests that there is dense gas in this galaxy. Figure 15 shows the predictions of RADEX models for the HCN(1–0)/12CO(1–0) line ratio plotted against H2 volume density, $n_{\rm H2}$ and assuming [CO]/[HCN]$\,=\,10^{5}$. The implicit assumption here, almost certainly incorrect, is that the filling factor of the emitting gas is the same in both transitions. Fig. 15 suggests that the abundance ratio [CO]/[HCN] must be high $\ga 10^{5}$. Relative to the best-fit RADEX model for NGC 1140, [CO]/[HCN]$\,=\,10^{5}$ corresponds to an HCN column density of $\sim 2\times 10^{11}$ cm*-2*, an order of magnitude lower than in diffuse clouds in the Galaxy (Liszt & Lucas 2001).

The upper limit (UL) for the HCN(1–0)/12CO(1–0) line temperature ratio of NGC 1140 is $\sim 0.1$, higher than the detected HCN(1–0)/12CO(1–0) ratio of $\sim$0.05 in the prominent star-formation region N 113 of the LMC (Wang et al. 2009). Thus, even after 18 hr with the IRAM 30m, our observations are not able to constrain the HCN content of NGC 1140. More observations of dense-gas tracers are needed at low metallicity, especially outside the Local Group where conditions can be even more extreme than in the massive star-forming regions in the Magellanic Clouds.

8 Discussion

Unlike some previous work covering a similar range of CO $J$ values (e.g., Meier et al. 2001; Israel 2005; Nikolić et al. 2007), our one-zone non-LTE models give very good fits to the CO and atomic carbon emission in NGC 1140. Part of this success is almost certainly due to the lack of high-$J$ transitions in our data set; we are fitting only the cool, dense clouds in this galaxy, while there may be a substantial contribution of warmer, more diffuse gas. It may also be a consequence of the extreme conditions in this galaxy. None of the six dwarf galaxies observed by Cormier et al. (2014) in 12CO(1–0) and 12CO(2–1) have R_{21}$$>1, independently of the beam size of the observations or the upper limits. The N 113 region in the LMC has R_{21}$$\la1 after beam deconvolution (R_{21}$$\sim1.2 before, see Wang et al. 2009), and Minamidani et al. (2008) found maximum values of R_{31}$$\ga1 in only a handful of the 33 LMC regions observed. These ratios are much higher in NGC 1140 ($R_{21}$ = 2.1, $R_{31}$ = 1.7, see Figs. 3, 4, and Table 7), independently of the beam corrections applied. Because of these high line ratios, and the implied more extreme physical conditions dominating the ISM of this galaxy, one-zone models may be more effective.

The CO excitation in NGC 1140 is starburst-like, similar to M 82, but as mentioned before, this is not particularly surprising given the presence of at least 6 SSCs within its central region (de Grijs et al. 2004; Moll et al. 2007). However, unlike M 82, where the gas is warm (T_{\rm kin}$$\sim 60-200 K) and not particularly dense (n_{\rm H2}$$\sim 10^{3} cm*-3*) (Weiß et al. 2005; Mühle et al. 2007), in NGC 1140 the gas is dense and cool. Given the relatively high excitation, it is not straightforward to understand why the CO has not been photo-dissociated through the intense radiation field, although as discussed below, the dense, small-filling factor clouds may be one part of the explanation. High atomic gas columns may also contribute to self-shielding as discussed below and suggested by Wong et al. (2009) for the LMC.

Finding evidence for cool, dense gas, n_{\rm H2}$$\ga 10^{6} cm*-3* in NGC 1140 was unexpected. However, at least one other low-metallicity galaxy outside the Local Group shows similar properties. In Haro 11, PDR modeling (Cormier et al. 2012; Cormier et al. 2014) also suggests that the molecular gas is dense, with hydrogen densities $n_{H}$ $\sim 10^{5-6}$ cm*-3*. The Cormier et al. (2014) PDR model for Haro 11 implies that CO emission is produced at intermediate depths within the cloud ($3<A_{V}<6$ mag), and for larger $A_{V}$, the gas temperature is $\sim$30 K, somewhat warmer than the (non-LTE) gas temperature $T_{\rm kin}$ of $\sim$17 K we find for NGC 1140.

Another unexpected result is the low optical depth of the 12CO transitions. Our RADEX fits (see Table 6, $\Delta V$ = 20  km s*-1*) suggest that the 12CO and 13CO transitions have a maximum $\tau_{\rm CO}$ $\sim 0.2$ for 12CO(3–2). The best fit of 12CO with [Ci] gives higher optical depths, with $\tau_{\rm CO}$ $\sim 1$ for 12CO(3–2) but lower for the other transitions. Apparently even these low optical depths allow sufficient self-shielding for detectable 12CO and 13CO emission. This could be possible because the clouds are dense and occupy only a small fraction of the macroscopic source size (see Sect. 7.3.1).

It could also be that high column densities are contributing to the shielding of the molecular gas. Hunter et al. (1994a) find a peak Hi column $N_{HI}$ of $2.4\times 10^{21}$ cm*-2* (averaged over a 30″$\times$37″ beam), $\sim$10 times higher than the $N_{\rm H2}$ of $2\times 10^{20}$ cm*-2* estimated from N_{CO}$$\sim 10^{16} cm*-2*, assuming a roughly Solar abundance ratio [H2]/[CO]$\sim 5\times 10^{-5}$ (e.g., Sakamoto 1999). Even the disk-averaged $N_{HI}$ in NGC 1140 of $\sim 10^{21}$ cm*-2* (Fumagalli et al. 2010) is 5 times higher than the H2 column that we would infer from $N_{CO}$. As proposed by Wong et al. (2009), high Hi columns may even be necessary (but not sufficient) for the detection of CO in low-metallicity galaxies. However, the LVG abundance fits using the models of Weiß et al. (2007) indicate that the [CO]/[H2] abundance ratio in NGC 1140 is extremely low: $\sim$ 100 times lower than Solar for the “cool” solution ([CO]/[H2]$\sim 8\times 10^{-7}$). This would imply that the H2 columns could be comparable to, if not exceed, the Hi, thus dominating the column density contribution necessary for self-shielding.

Such a low [CO]/[H2] abundance and cold $T_{\rm kin}$ in NGC 1140 could be the result of a high level of photo-dissociation of CO, leaving only the densest, best shielded cores. This is a similar situation to ammonia (NH3) in the LMC where it is only detected in N 159 W (Ott et al. 2010). In this region, the NH3 is cold with $T_{\rm kin}$ $\sim$16 K, and has very low abundance; NH3 abundance is between 1.5 and 5 orders of magnitude lower than observed in Galactic star-forming regions, but similar to that found in the late-stage starburst M 82 (Weiß et al. 2001). NH3 is particularly sensitive to UV radiation (Sato & Lee 1983), but formaldehyde (H2CO) is less vulnerable to photo-dissociation than NH3, and in N 159 W is warmer, T_{\rm kin}$$\sim30-35 K (Tang et al. 2017). The implication is that H2CO is sampling regions that are more exposed to radiation of young massive stars, while NH3 resides in the most shielded knots of dense gas (Tang et al. 2017). The stronger radiation field in NGC 1140 from the concentration of SSCs would annihilate NH3, and possibly also H2CO, but CO could survive, thus playing the role that NH3 plays in the LMC, tracing the surviving cool pockets of dense gas.

Such a scenario may occur in any specific low-metallicity star-forming region. That it is seen over a $\sim$2 kpc area of a galaxy with six SSCs requires that the evaporation of dense molecular gas occurs roughly simultaneously. Thus, galaxies like NGC 1140 are probably rare. Consistently with observations, the starburst must also be either quite young (all activity is occurring at almost the same time) or the last one of several possibly very short starburst episodes (e.g., de Grijs et al. 2004; Moll et al. 2007). With 6 SSCs, all of them clearly surpassing 30 Doradus in the LMC, this last starburst episode must then be at least (if not the only one) the strongest one. More extensive observations of molecular tracers are necessary to test this scenario, and constrain models of CO/H2 abundance ratios in NGC 1140, and, more generally, in metal-poor starbursts.

Our RADEX fits with 12CO and [Ci] also give an unusually high atomic carbon abundance (see Sect. 7.3.3), C/CO $\ga$ 10. This result is premised on the assumption that [Ci] and CO occupy the same volume in a cloud characterized by a single kinetic temperature and volume density (e.g., Okada et al. 2015; Requena-Torres et al. 2016). Such a high [Ci] abundance relative to CO is more extreme than found in the SMC with the LVG analysis of Requena-Torres et al. (2016); there the clouds tend to be less dense ($n_{\rm H2}$ $\sim 10^{4}$ cm*-3*), and warmer ($T_{\rm kin}$ $\sim 30-50$ K), although of similar CO column density ($N_{CO}$ $\sim 2-4\times 10^{16}$ cm*-2*, see also Nikolić et al. 2007). Spatially-resolved observations of [Ci] and CO in NGC 1140 would help understand whether the high [Ci] abundance we find is dictated by a different spatial distribution relative to CO, unlike what is observed in the Magellanic Clouds.

8.1 13CO charge-exchange reactions and fractionation

Another surprising result from our RADEX analysis of NGC 1140 is the unusually low [12CO]/[13CO] abundance ratio, $\sim 8-20$, lower even than the value of $\sim$24 found in the Galactic Center (Langer & Penzias 1990). Isotopic ratios are generally governed by two competing processes: photo-selective dissociation that would tend to decrease the relative 13CO abundance, and low-temperature carbon isotope exchange reactions that would tend to increase it.

Low [12CO]/[13CO] abundance ratios are typically found in older intermediate-mass stellar populations ($\ga$500 Myr), in which 13C is produced in nuclear reactions in the cores of asymptotic giant branch (AGB) stars, and “dredged up” through convective mixing where it is ejected into the ISM in stellar winds (e.g., Milam et al. 2005, 2009). However, the SSCs in NGC 1140 are quite young, with ages ranging from $\sim$ 5 to 12 Myr, and the overall starburst is younger than $\sim$55 Myr (de Grijs et al. 2004; Moll et al. 2007). These young ages make it unlikely that the isotopic ratio in NGC 1140 has been lowered by stellar reprocessing (e.g., Boothroyd & Sackmann 1999; Pavlenko et al. 2003; Henkel et al. 2014).

Instead, the extremely low 12CO/13CO abundance ratio for NGC 1140 could be explained by charge-exchange reactions, i.e., fractionation (e.g., Watson et al. 1976). Isotopic selective photo-dissociation of CO is effective only in diffuse gas (n_{\rm H2}$$\la 100 cm*-3*); in denser regions with higher column density but moderate extinction (1 mag $\la$ $A_{V}$ $\la$ 3 mag), fractionation reactions become important:

[TABLE]

At intermediate $N_{CO}$ (and $T\la 35$ K), the rightmost (exothermic) reaction dominates, enhancing 13CO and leading to a reduced [12CO]/[13CO] abundance ratio. Thus, if molecular gas in low-metallicity galaxies is dense and cool, but with moderate $N_{CO}$ as in NGC 1140, it is possible that fractionation drives low 12CO/13CO ratios.

The low value of [12CO]/[13CO] we obtain for NGC 1140 is roughly consistent with (although slightly higher than) the trends of CO column density and 13CO fractionation found by Röllig & Ossenkopf (2013) and Szűcs et al. (2014). Fig. 16 shows the trend of 12CO/13CO abundance ratio and 12CO column density given by Szűcs et al. (2014); the range of $N$(12CO) for NGC 1140 as estimated by the best-fit RADEX (and LTE) models is shown as a shaded rectangle. Szűcs et al. (2014) show that there is a “sweet spot” in $N_{CO}$ for maximizing the fractionation process, as long as the temperature barrier of $\sim 35$ K is not exceeded. Interestingly, the best-fit $T_{\rm kin}$ for NGC 1140 is $\sim 16-18$ K (see Fig. 11 and Table 6), consistent with the low temperatures necessary for optimizing 13CO fractionation. In NGC 1140, the low X(12CO)/X(13CO) abundance ratio could be fostered by the moderately low $N_{CO}$ and the low temperature, which in turn are probably related to the high $n_{\rm H2}$ volume density. We are pursuing an observational program to establish whether such a phenomenon is common in metal-poor starbursts.

8.2 The H2 mass for NGC 1140 revealed with atomic carbon

In Paper I, we derived the metallicity dependence for the CO-to-H2 conversion factor, $\alpha_{\rm CO}$ $\propto(Z/Z_{\odot})^{-2}$, which, at the metallicity of NGC 1140, is roughly equivalent to the exponential variation with abundance found by Wolfire et al. (2010). Here we compare the molecular mass for NGC 1140 using $\alpha_{\rm CO}$ with the molecular mass estimated from [Ci] luminosity according to Glover & Clark (2016). After correcting to a 22″ beam size as in Table 4 (see also Table 7), $L^{\prime}({\rm CO})$ = $2.9\times 10^{6}$ K km s*-1* pc2. Using $\alpha_{\rm CO}$ =  3.2 M*⊙* (K km s*-1* pc${}^{2})^{-1}$ (not including helium, Saintonge et al. 2011), and scaling with $(Z/Z_{\odot})^{-2}$ (see Paper I) for NGC 1140 (${\mathrm{Z}/Z}_{\odot}$ = 0.31; 12$+$log(O/H) = 8.18, and assuming 12$+$log(O/H)⊙ = 8.69 from Asplund et al. 2009), we find a total molecular gas mass of $\sim 9.3\times 10^{7}$ M*⊙*.

Glover & Clark (2016) use their hydrodynamical simulations of Ci emission to estimate H2 masses, and provide a [Ci] luminosity-to-H2 conversion factor. At the highest $G_{0}$ modeled by Glover & Clark (2016), they obtain a mean value $X_{CI}\,=\,8.8\times 10^{20}$ cm*-2* (K km s*-1*)-1 for $Z$ $\sim$ Z*⊙* and an approximately linear increase with decreasing metallicity. For NGC 1140 the Ci luminosity $L^{\prime}$(Ci) = $1.6\times 10^{6}$ K km s*-1* pc2 (from Table 4, including the aperture correction for a 22″ beam). Converting the Glover & Clark (2016) $X_{CI}$ to equivalent $\alpha_{\rm CO}$ units, we would find $\alpha_{CI}$ = 53.2 M*⊙* (K km s*-1* pc2)-1 for 0.31 ${\mathrm{Z}/Z}_{\odot}$, and a resulting molecular mass MH2 of $\sim 8.7\times 10^{7}$ M*⊙*. This value is within 10% of what we infer with the quadratic metallicity scaling of $\alpha_{\rm CO}$, despite the larger observed Ci/12CO(1–0) ratio (see Sect. 5.3) relative to their models777In these calculations, unlike Paper I, we have not included the factor of 1.64 to correct to total flux in NGC 1140; these values are corrected to 22″ as in Tables 4 and 7. These values of molecular gas mass for NGC 1140 are within 15% of those estimated using the models of Weiß et al. (2007).

9 Summary and conclusions

We have presented 12CO observations of ten galaxies, with detections for eight. With metallicities ranging from 12$+$log(O/H) $\sim$7.7 to 8.4 (0.1 Z*⊙* to 0.5 Z*⊙*), at and below the abundance of the SMC, this is the largest sample of metal-poor galaxies with CO detections so far obtained outside the Local Group. For one of the galaxies, NGC 1140, we report additional 13CO, Ci, and HCN(1–0) measurements. Our main conclusions are the following:

• •

After correcting for differences in beam sizes, NGC 1140 shows high velocity-integrated temperature ratios of 12CO(2–1)/12CO(1–0) ($R_{21}$ = 2.1 $\pm$ 0.06) and 12CO(3–2)/12CO(1–0) ($R_{31}$ = 2.0 $\pm$ 0.45, or $R_{31}$ = 1.7 $\pm$ 0.39 with the correction to 22″ beams), while the other galaxies have less extreme ratios (see Figs. 3, 4).

• •

A comparison of dense-gas tracers such as 12CO(3–2) (of the four observed metal-poor galaxies) and HCN(1–0) (upper limit for NGC 1140) with data in the literature shows a deficit in 12CO(3–2) relative to SFR (see Fig. 7), similar to the 12CO(1–0) deficit found in Paper I. However, gas excitation at low metallicity measured by comparing 12CO(3–2) to 12CO(1–0) luminosities seems similar to that of more metal-rich systems. HCN(1–0) was observed but not detected in NGC 1140. The lower limit of HCN(1–0) for NGC 1140 may be due to a combination of stellar feedback (e.g., Hopkins et al. 2013) and an extremely high [CO]/[HCN] abundance ratio ($\ga 10^{5}$) probably because of the low metallicity (see Fig. 15).

• •

Fitting LTE models to the 12CO(3–2), 12CO(2–1), and 12CO(1–0) emission in NGC 1140, NGC 7077, and UM 448 suggests that CO column densities are moderate and temperatures are low. These fits also suggest low optical depths, but the lack of higher-$J$ lines for NGC 7077 and UM 448 prevents confirming this as a general result with radiative-transfer models.

• •

Fitting physical models (RADEX) of the 12CO and 13CO emission in NGC 1140 suggests that the molecular gas is cool ($T_{\rm kin}$ $\la$ 20 K), dense ($n_{\rm H2}$ $\ga 10^{6}$ cm*-3*), with moderate CO column density ($N_{CO}$ $\sim\,10^{16}$ cm*-2*) and low filling factor ($F\,\sim 0.01-0.1$). 12CO optical depths are fairly low ($\tau_{\rm CO}$ $\la$0.2), and we speculate that the CO survives photo-dissociation because of the high Hi column density in this galaxy. The molecular excitation in NGC 1140 is starburst-like, similar to M 82 (see Fig. 13).

• •

The [12CO]/[13CO] abundance ratio in NGC 1140 inferred from the fit is very low, $\sim 8-20$, lower even than the value of $\sim 24$ found in the Galactic center (Langer & Penzias 1990). Because the starburst in NGC 1140 is quite young, it is difficult to interpret this low ratio as due to stellar reprocessing by older intermediate-mass populations (e.g., Milam et al. 2005, 2009). Instead, we attribute it to enhanced CO fractionation, from the combination of moderate column densities and cool gas (see Fig. 16), as predicted by the chemical models of Röllig & Ossenkopf (2013) and Szűcs et al. (2014).

• •

Fitting RADEX models of the 12CO and Ci emission in NGC 1140 gives physical conditions similar to the 12CO$+$13CO fits, although there is a possibility for warmer ($T_{\rm kin}$ $\sim$ 38 K), less dense ($n_{\rm H2}$ $\sim\,10^{4}$ cm*-2*) gas at higher column densities. Independent fitting results for NGC 1140 using the models of Weiß et al. (2007) give similar results. Both our RADEX fits and the LVG fits suggest that atomic carbon is at least 10 times more abundant than 12CO in NGC 1140.

For the first time outside the Local Group, the six CO transitions measured for NGC 1140, together with Ci, have enabled an analysis of physical conditions in the molecular gas of a low-metallicity galaxy with at least 6 SSCs. The unusual [12CO]/[13CO] and [12CO]/[12C] abundance ratios, and the cool, dense gas at moderate column densities, may be the consequence of the SSCs in this galaxy and their feedback effect on the ISM. Future work will attempt to put this speculation on a more quantitative footing, both with models of ISM chemistry and with more observations of the molecular ISM in low-metallicity starbursts.

Acknowledgements.

We acknowledge the anonymous referee for her/his useful comments and suggestions. LKH is grateful to Carlo Giovanardi for interesting discussions and mathematical insights for the beam corrections. SGB acknowledges economic support from grants ESP2015-68964-P and AYA2016-76682-C3-2-P. We warmly thank the IRAM staff, both in Granada and at Pico Veleta, for their capable management of the logistics and the telescope/receiver operations. We are also indebted to the APEX service observing team for their dedication in the challenging high-frequency observations obtained for NGC 1140. We gratefully acknowledge the International Space Science Institute (Bern) for hospitality during the conception of this paper. Heavy use was made of the NASA/IPAC Extragalactic Database (NED).

Appendix A Exponential beam size correction

Following Liu et al. (2017), we have corrected the velocity-integrated $T_{\rm mb}$ values to a common beam size according to the distribution of cool dust, thus explicitly assuming that the molecular gas and the dust have the same distribution. In order to apply such a correction also to the two galaxies (Mrk 0996, NGC 7077) for which this was not possible, we have adopted an analytical approach; such an approach also tends to average out photometric uncertainties and possibly improves the reliability of the correction.

PACS 160 $\mu$m images are available from the Herschel archive for NGC 1140, NGC 3353, and UM 448, but for NGC 1156, we had to use the MIPS 160 $\mu$m image from Spitzer. In all cases, the 160 $\mu$m PACS fluxes reported by Rémy-Ruyer et al. (2013) are beyond the peak of the dust emission, implying that in these dwarf galaxies the 160 $\mu$m is tracing cool dust. We obtained surface brightness profiles by performing an azimuthal extraction centered on the brightness peak, and growth curves by measuring photometric flux in increasingly large circular apertures. The 160 $\mu$m surface brightness profiles are shown in the left panels and the growth curves in the right panels of Figs. 17, 18, 19, and 20. The dust distribution is clearly exponential (rather than Gaussian) in all cases examined. For the two remaining galaxies, Mrk 996 and NGC 7077, MIPS 160 $\mu$m images are available, but the galaxies are not resolved (even at 24 or 70 $\mu$m where we obtained clear profiles of a diffraction-limited Airy ring).

Taking the source distribution to be exponential, with folding length $r_{s}$, the integrated flux density within a beam with FWHM $\theta_{b}$ corresponds to the following integral (assuming azimuthal symmetry) in polar coordinates:

[TABLE]

where $A_{0}$ is the brightness at the origin of the exponential, and $r$ corresponds to the angular distance from the origin. Eqn. 6 can be solved analytically to obtain:

[TABLE]

The source FWHM $\theta_{s}$ is related to the exponential folding length by $\theta_{s}=2r_{s}\ln(2)$, obtained by calculating the diameter (i.e., twice the radius) at which the exponential distribution is equal to $A_{0}/2$.

However, Eqn. (6) is formally incorrect because of the need to convolve the pure exponential with the Gaussian beam when the exponential scale length $r_{s}$ is small relative to the beam radius $\theta_{b}/2$. This convolution is given by the following, where we have taken advantage of the azimuthal symmetry and calculate the convolution along a single radial ray here given by $x$:

[TABLE]

where $\sigma_{b}$ corresponds to the Gaussian $\sigma$ assuming that the beam is Gaussian with FWHM $\theta_{b}$ ($\theta_{b}\,=\,2\,\sigma_{b}\sqrt{2\,\ln\,2}$).

Equation (8) can be integrated by separating the variables ($r_{0}$, $\phi$). After integrating in $\phi$, we obtain:

[TABLE]

where $I_{0}$ is a modified Bessel function of the first kind. Eqn. (9) can then be integrated numerically to obtain the surface brightness profile of the convolved exponential, as shown as the lower (red) dashed curves in Figs. 17, 18, 19, and 20.

These figures show that for the four galaxies for which we were able to fit surface-brightness profiles and growth curves the pure exponential ($2\pi\,A_{0}\,\exp(-r/r_{s})$) is equivalent to the integral given in Eqn. 9 as long as the scale length $r_{s}$ is larger than $\theta_{b}/2$. We thus fitted the growth curves with Eqn. (7) in order to obtain the beam corrections. This is advantageous from a numerical point of view because otherwise it would be necessary to fit the growth curve to the integral of Eqn. (9) which is in itself an integral.

To correct the fluxes (e.g., $S_{\nu}$ in Jy), we scaled the observed dust fluxes to a common beam size of 22″ by comparing the growth-curve integrals in the different beam sizes using the best-fit of the analytical formula in Eqn. (7) to obtain $r_{s}$ and $A_{0}$. We thus derive a multiplicative factor $AP_{\rm cor}$ as given in Col. (6) of Table 7: $S_{22}\,=\,S_{\rm orig}\,AP_{\rm cor}$. We then apply the correction to velocity-integrated $T_{\rm mb}$ values by noting that $S/\Omega\,=\,2\,k\,\nu^{2}\,/c^{2}$ $T_{\rm mb}$, where $\Omega$ is the beam solid angle, $k$ is the Boltzmann constant, $\nu$ is the observed frequency, and $c$ is the speed of light. Thus:

[TABLE]

where “orig” refers to the original values observed in the different beams. The flux and $T_{\rm mb}$ corrections necessary to convert our observations to a common beam size of 22″ are given in Table 7, together with the corrected velocity-integrated $T_{\rm mb}$ values that we use throughout the paper when we are examining line ratios (12CO(2–1)/12CO(1–0)) or fitting physical models. Corrected values are also reported in Tables 2, 3, and 4 in the main text.

For Mrk 996 and NGC 7077, the only available cool dust images are from MIPS and, as mentioned above, the galaxies are not resolved. Thus, for these we assumed that the disk scale length $r_{s}$ is equal to 0.2 $R_{\rm opt}$ (optical radius from NED), consistently with Young et al. (1995); Leroy et al. (2009); Kuno et al. (2007) who all found that CO exponential scale lengths are, in the mean, equal to 1/5 the optical radius. This is also roughly consistent with the other four galaxies for which the mean $r_{s}$/$R_{\rm opt}$ given by the best fits of the growth curves is 0.27$\pm$0.16.

Appendix B Gaussian beam size correction

To ensure that our results do not depend on the specific formulation of the correction for beam dilution, here we investigate a different assumption for the distribution of the CO emission, namely a Gaussian. For a Gaussian source distribution with a FWHM $\theta_{s}$ (corresponding to a Gaussian $\sigma_{s}\,=\,\theta_{s}/[2\,\sqrt{2\,\ln(2)}]$), the integrated flux density within a beam with FWHM $\theta_{b}$ is given by the following integral (assuming azimuthal symmetry) in polar coordinates:

[TABLE]

where $A_{0}$ is the normalization constant, and $r$ corresponds to the angular distance from the origin. Eqn. 11 can be solved analytically to obtain:

[TABLE]

However, like Eqn. (6), Eqn. (11) is formally incorrect because of the need to convolve the source Gaussian distribution with the Gaussian beam. This convolution is given by the following, where we have taken advantage of the azimuthal symmetry and calculate the convolution in Cartesian coordinates:

[TABLE]

where $\sigma_{b}$ corresponds to the Gaussian $\sigma$ assuming that the beam is Gaussian with FWHM $\theta_{b}$ ($=\,2\,\sigma_{b}\sqrt{2\,\ln\,2}$).

It is straightforward to integrate Eqn. (13) analytically to obtain:

[TABLE]

where $A_{0}$ has been substituted by the inclusion of the normalization constant of a two-dimensional Gaussian with integral unity, $A_{0}\,=\,A^{\prime}_{0}/(\pi\,\sigma_{b}^{2})$; the independent Cartesian coordinates $x$ and $y$ have now been substituted with $r\,=\sqrt{x^{2}+y^{2}}$. Unlike Eqn. (9) for the exponential source distribution, Eqn. (14) can be easily integrated to a circular aperture radius $\theta_{b}/2$ to obtain the aperture correction:

[TABLE]

For the case of a Gaussian source distribution, to correct the fluxes (e.g., $S_{\nu}$ in Jy) to a common beam size, we scaled the observed CO fluxes by comparing the growth-curve integrals in the different beam sizes using the analytical formula in Eqn. (15). Figure 21 shows the aperture and the $T_{\rm mb}$ corrections for different transitions in the case of a Gaussian source distribution. Also shown are the exponentially-derived beam corrections for NGC 1140 (see Appendix A).

In the best-fit procedure described in the main text, we sample source FWHM $\theta_{s}$ ranging from 2″ to 40″ with steps of 1″, and for each of these derive a multiplicative factor $AP_{\rm cor}$ $S_{22}\,=\,S_{\rm orig}\,AP_{\rm cor}$ [see Eqn. (10)]. We then apply this correction to velocity-integrated $T_{\rm mb}$ values, and consider the best fit as the one with the lowest $\chi^{2}$ value. Corrected values for NGC 1140 are given in Table 8.

It is interesting that the Gaussian beam corrections found by the $\chi^{2}$ minimization technique are the same as the independently-derived exponential beam corrections. It seems unlikely that this is a fortuitous result, but rather that it is confirming the validity of either approach to beam corrections.