EXTERNALLY HEATED PROTOSTELLAR CORES IN THE OPHIUCHUS STAR-FORMING REGION^*

Low-mass protostars form from the collapse of dense molecular clouds. In their youngest stages, such protostars are deeply embedded in an envelope of dust and gas. Investigations of molecular emission lines of this envelope can be used to study both chemical and physical characteristics of the protostar. The temperature of the large-scale envelope is generally low (∼10–15 K; e.g., Bergin & Tafalla 2007), but parts of the envelope can be heated by irradiation from the protostar itself or by external sources.

Molecular emission lines of H₂CO with the same J_u quantum number can be used as a probe of the kinetic temperature for cool ($T\lesssim 50$ K) gas at densities ≳10⁵ cm⁻³ (Mangum & Wootten 1993). Through an unbiased survey of H₂CO toward embedded sources in the Corona Australis (CrA) star-forming region, it was possible to characterize the temperature and external irradiation of such protostars, identifying the Herbig Be star R CrA as the dominant source of irradiation in CrA, heating nearby envelopes to temperatures ∼40 K, but also influencing the temperature on scales of a few 10,000 au (Lindberg et al. 2015). The same molecular transitions were used by Lindberg & Jørgensen (2012) to produce an interferometric map of the physical characteristics in the R CrA cloud (which hosts a handful of embedded low-mass protostars). The rotational temperatures of H₂CO were found to exceed 40 K across the whole cloud (∼6000 au), consistent with Herschel/PACS observations of the extended dust temperature (Lindberg et al. 2014). In particular, strong H₂CO and CH₃OH emission with high temperatures and column densities was detected toward two long (∼5000 au) ridges north and south of the low-mass young stellar objects (YSOs), whereas only faint H₂CO emission was detected toward the YSOs themselves. On the other hand, most of the fainter c-C₃H₂ emission was detected from the region between the two ridges. The signal-to-noise ratio (S/N) of the observations was too low to establish c-C₃H₂ rotational temperatures, but in single-dish APEX observations the rotational temperature of c-C₃H₂ was found to be consistently low (∼10 K) toward the embedded protostars in CrA (Lindberg et al. 2015).

The Ophiuchus star-forming region, at a distance of only 125 pc (de Geus et al. 1989; Wilking et al. 2008), is an excellent laboratory to test the knowledge gained from the CrA survey on a region with a greater number of potential heating sources, but also greater separation between the low-mass sources, allowing for a distinction between effects from internal and external heating. The population of deeply embedded protostars was surveyed by Jørgensen et al. (2008) and Enoch et al. (2009) using Spitzer and JCMT/SCUBA continuum observations, and in combination, the two surveys identified 38 Class 0 and Class I protostars.

Kamegai et al. (2003) found large-scale elevated [C i] excitation temperatures in the ρ Oph A cloud using the Mount Fuji submillimeter telescope, and identified the B2 star HD 147889 as the heating source. Liseau et al. (2015), however, concluded that HD 147889 and the B4 star S1 (also known as GSS 35) both influence the temperature of the cloud, but that S1, despite being less luminous, is the more dominant heating source owing to its proximity to the ρ Oph A cloud. Here, we investigate the influence these luminous stars have on the physical properties of the molecular gas in nearby embedded protostars through 218 GHz spectral line observations.

Our observations were executed with the APEX (Atacama Pathfinder Experiment) 12 m telescope (Güsten et al. 2006) in position-switching mode in August and October 2014 and May 2015. The SHeFI APEX-1 receiver (Vassilev et al. 2008) was used to cover the frequency range 216–220 GHz toward a complete sample of the 38 deeply embedded protostellar sources in the Ophiuchus star-forming region. The sample consists of all embedded protostellar sources in the region detected by Jørgensen et al. (2008) and Enoch et al. (2009); see Table 1 and Figures 1–2. The spectral setup was chosen to cover three H₂CO spectral lines at 218 GHz, and also several spectral lines of c-C₃H₂, an abundant tracer of unsaturated hydrocarbon molecules.

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Table 1.  Coordinates and Other Properties of the Observed Sources

Cloud	R.A.	Decl.	Our rms	Tbol	Lbol	${\alpha }_{\mathrm{IR}}$	Other Common
Source	(J2000.0)	(J2000.0)	(mK (km s−1)−1)a	(K)	(${L}_{\odot }$)		Identifiers
ρ Oph A
GDS J162625.6	16:26:25.62	−24:24:28.9	15.0	$\lt 72$	>0.039	1.65	⋯
GSS 26	16:26:10.33	−24:20:54.8	26.6	920	0.3	−0.46	⋯
GSS 30	16:26:21.42	−24:23:06.4	13.9	150	8.7	1.46	Elias 21
GSS 39	16:26:45.03	−24:23:07.7	24.6	970	0.23	−0.64	Elias 27
[GY92] 30	16:26:25.49	−24:23:01.6	12.0	135	0.10	0.87	⋯
[GY92] 91	16:26:40.47	−24:27:14.5	16.9	$\lt 370$	>0.065	0.45	CRBR42
ISO-Oph 21	16:26:17.23	−24:23:45.4	13.7	490	0.083	0.69	CRBR12
J162614.6	16:26:14.62	−24:25:08.4	14.7	⋯	⋯	⋯
VLA 1623	16:26:26.42	−24:24:30.0	16.2	57	0.41	1.65	⋯
ρ Oph B
ISO-Oph 124	16:27:17.57	−24:28:56.3	15.1	68	2.6	0.25	⋯
J162728	16:27:28.45	−24:27:21.0	13.3	310	0.48	−0.03	IRS45, [GY92] 273
Oph-emb 5	16:27:21.83	−24:27:27.6	13.1	180	0.019	−0.05	⋯
VSSG 17	16:27:30.18	−24:27:43.4	15.2	530	0.93	−0.12	Elias 33, IRS47, [GY92] 279
ρ Oph C
WL 2	16:26:48.49	−24:28:38.9	22.6	430	0.12	0.02	[GY92] 128
WL 12	16:26:44.19	−24:34:48.3	30.5	290	1.1	2.49	ISO-Oph 65
WL 22	16:26:59.17	−24:34:58.8	14.2	110	1.5	1.99	ISO-Oph 90
ρ Oph E
[GY92] 197	16:27:05.25	−24:36:29.8	20.5	110	0.15	1.27	LFAM26
WL 15	16:27:09.43	−24:37:18.8	27.3	260	18	1.69	Elias 29, [GY92] 214
WL 16	16:27:02.34	−24:37:27.2	23.9	320	4.8	1.53	[GY92] 182
WL 17	16:27:06.78	−24:38:15.0	26.6	310	0.60	0.61	[GY92] 205
ρ Oph F
IRAS 16244-2432	16:27:28.03	−24:39:33.5	26.3	110	15	2.29	IRS44, [GY92] 269
IRAS 16246-2436	16:27:39.83	−24:43:15.1	13.6	570	0.71	−0.15	IRS51, [GY92] 315
ISO-Oph 132	16:27:21.47	−24:41:43.1	25.0	600	1.2	−0.03	[GY92] 252
ISO-Oph 137	16:27:24.61	−24:41:03.4	25.9	191	0.33	1.01	CRBR85
ISO-Oph 161	16:27:37.25	−24:42:38.0	25.9	450	0.13	0.13	[GY92] 301
YLW 15	16:27:26.94	−24:40:50.8	24.3	160	3.8	1.17	IRS 43, [GY92] 265
L1689N
IRAS 16293-2422	16:32:22.56	−24:28:31.8	12.5	47	16	5.03	⋯
L1689S
ISO-Oph 200	16:31:43.75	−24:55:24.6	24.9	500	0.28	0.23	⋯
ISO-Oph 202	16:31:52.06	−24:57:26.0	26.2	$\lt 120$	>0.0093	0.82	⋯
ISO-Oph 203	16:31:52.45	−24:55:36.2	10.9	240	0.13	1.07	⋯
ISO-Oph 209	16:32:01.00	−24:56:42.0	26.4	130	4.0	1.39	L1689S1, IRS67
L1709
GWAYL 4	16:31:35.65	−24:01:29.3	24.1	300	1.4	0.14	IRAS 16285-2355
Oph-emb 4	16:31:36.80	−24:04:20.1	27.8	77	0.18	−0.27	⋯
Solitary sources
Elias 28	16:26:58.44	−24:45:31.9	24.1	860	1.2	−0.86	SR 24N
J162624	16:26:24.07	−24:16:13.5	25.0	980	1.9	−0.71	Elias 24
J1633.92442	16:33:55.61	−24:42:05.0	23.4	1500	0.17	−1.22	⋯
MMS126	16:28:21.61	−24:36:23.4	14.9	41	0.29	1.23	IRAS 16253-2429
Oph-emb 18	16:28:57.85	−24:40:54.9	19.7	300	0.03	0.67	⋯

Note. Spectral properties from Evans et al. (2009), except for the values of [GY92] 30, VLA 1623, and ISO-Oph 137, which are from Enoch et al. (2009).

^aMeasured spectral RMS noise when averaging to 1 km/s channels.

A machine-readable version of the table is available.

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We note that five of the sources in the sample (GSS 26, GSS 39, Elias 28, J162624, and J1633.92442) should be considered Class II sources by both the ${T}_{\mathrm{bol}}\gt 650$ K and the ${\alpha }_{\mathrm{IR}}\lt -0.3$ criteria (Evans et al. 2009), but are still included in the sample of embedded protostars of Jørgensen et al. (2008) due to their proximity ($\lt 15^{\prime\prime}$) to a SCUBA core. We detect no line emission toward one of these sources (J1633.92442) and only C¹⁸O line emission toward two (GSS 39 and Elias 28), and these sources are thus likely not deeply embedded. GSS 26 and J162624, however, show H₂CO line emission, indicating the presence of dense molecular gas. Finally, toward the two Class I sources WL16 and Oph-emb 4, we only detect emission from C¹⁸O.

To remove quasi-sinusoidal baselines present in the APEX-1 spectra (see Vassilev et al. 2008), we used our running-mean script described in Appendix A of Lindberg et al. (2015). The spectra were thereafter reduced with the X-Spec package,⁵ which was used to average scans, perform Gaussian fits, and calculate line intensities. Throughout the paper we use the T_mb temperature scale, assuming a main beam efficiency ${\eta }_{\mathrm{mb}}=0.75$.

The spectral lines that were used in the analysis are listed in Table 2. These lines were not detected toward all sources, and in addition to these lines, several other lines (of e.g., C₂D, DCN, C¹⁸O, C³³S, and HNCO) were detected in some of the sources, but will not be discussed in this paper. All observed line parameters are listed in Table 4 in Appendix A.

Table 2.  Spectral Lines Used in the Analysis

Molecule	Quantum	Frequency	Type	${E}_{{\rm{u}}}/k$
	Numbers	(GHz)		(K)
DCO+	$3\to 2$	216.11258	⋯	20.7
c-C3H2	${3}_{03}\to {2}_{02}$	216.27876	ortho	19.5
c-C3H2	${6}_{06}\to {5}_{15}$	217.82215	para	38.6
c-C3H2	${6}_{16}\to {5}_{05}$	217.82215	ortho	38.6
c-C3H2	${5}_{14}\to {4}_{23}$	217.94005	ortho	35.4
c-C3H2	${5}_{24}\to {4}_{13}$	218.16046	para	35.4
H2CO	${3}_{03}\to {2}_{02}$	218.22219	para	21.0
CH3OH	${4}_{2}\to {3}_{1}$	218.44006	E	45.5
H2CO	${3}_{22}\to {2}_{21}$	218.47563	para	68.1
c-C3H2	${7}_{16}\to {7}_{07}$	218.73273	ortho	61.2
c-C3H2	${7}_{26}\to {7}_{17}$	218.73273	para	61.2
H2CO	${3}_{12}\to {2}_{11}$	218.76007	para	68.1
SO	${6}_{5}\to {5}_{4}$	219.94944	⋯	35.0

Note. Rest frequencies from the CDMS (Müller et al. 2001) and JPL (Pickett et al. 1998) molecular spectroscopy databases.

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To enable comparison of the line profiles, Figure 3 shows the normalized spectra for the strongest DCO⁺ (216.113 GHz), c-C₃H₂ (217.822 GHz), H₂CO (218.222 GHz), and SO (219.949 GHz) lines in each source where they have been detected. We note that for some sources, the line profiles of different molecular species are dramatically discrepant. In several cases, a double peak is seen in some spectral lines. The CH₃OH line (not plotted) is generally well-correlated with the H₂CO line shape, while C₂D (not plotted) correlates well with c-C₃H₂, although the C₂D line doublet makes analysis of the line profile non-trivial. Of the plotted lines, we find that the line profiles often separate the molecular species into two groups: H₂CO and SO on one hand, and c-C₃H₂ and DCO⁺ on the other. Lindberg et al. (2015) found that H₂CO and other saturated organic species as well as SO and other sulfur-bearing species had significantly higher rotational temperatures than c-C₃H₂ and other unsaturated hydrocarbons in the APEX observations of the embedded protostar R CrA IRS7B. As in CrA, the H₂CO and SO line profiles are generally wider than the c-C₃H₂ line profiles. This is also similar to the six molecular cores studied by Buckle et al. (2006), where CH₃OH and SO were found to be spatially correlated, but separated from the unsaturated hydrocarbon HC₃N.

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For completeness, the corresponding spectra for the Corona Australis survey are shown in Figure 8 of Appendix B.

3.1. Secondary Components

For all sources where at least two H₂CO or c-C₃H₂ spectral lines were detected, the respective rotational temperatures (assuming LTE) have been calculated. We assume that the lines are optically thin, which is valid at the observed column densities (Lindberg et al. 2015). For sources with only one detected line of a certain species, an upper limit of the rotational temperature was calculated using the upper limits on the non-detected line(s). The results are presented in Table 3.

Table 3.  Measured Rotational Temperatures, Column Densities, Median Line Widths, and LSR Velocities

	H2CO				c-C3H2
Cloud	Trot	N	${\rm{\Delta }}v$	vLSR	Trot	Nrot	${\rm{\Delta }}v$	vLSR
Source	(K)	(1012 cm−2)	(km s−1)	(km s−1)	(K)	(1012 cm−2)	(km s−1)	(km s−1)
ρ Oph A
GDS J162625.6 (main)	$25.8\pm 1.0$	$18\pm 3$	0.8	3.8	$12.4\pm 0.8$	$2.5\pm 0.4$	0.8	3.8
GDS J162625.6 (sec.)	$36.7\pm 2.1$	$11\pm 1$	1.4	2.6	⋯	⋯	⋯	⋯
GSS 26	$39.9\pm 8.7$	$4.3\pm 1.4$	1.1	3.4	⋯	⋯	⋯	⋯
GSS 30	$35.4\pm 1.2$	$19\pm 2$	1.3	3.3	$14.7\pm 3.7$	$0.46\pm 0.26$	0.9	3.5
GSS 39	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯
[GY92] 30	$36.2\pm 0.4$	$79\pm 8$	1.2	3.4	$9.9\pm 0.3$	$4.0\pm 0.3$	0.4	3.4
[GY92] 91	$26.7\pm 2.0$	$12\pm 2$	0.6	3.2	⋯	⋯	⋯	⋯
ISO-Oph 21 (main)	$23.2\pm 1.2$	$15\pm 3$	0.9	2.8	⋯	⋯	⋯	⋯
ISO-Oph 21 (sec.)	$48.0\pm 4.7$	$4.4\pm 0.6$	1.2	3.9	⋯	⋯	⋯	⋯
J162614.6 (main)	$29.0\pm 0.5$	$47\pm 6$	0.9	2.7	⋯	⋯	⋯	⋯
J162614.6 (sec.)	$49.2\pm 2.2$	$11\pm 1$	1.5	4.3	⋯	⋯	⋯	⋯
VLA 1623 (main)	$30.6\pm 1.0$	$23\pm 3$	0.8	3.8	$11.2\pm 0.5$	$4.2\pm 0.5$	0.8	3.8
VLA 1623 (sec.)	$28.3\pm 2.1$	$12\pm 2$	1.4	2.5	⋯	⋯	⋯	⋯
ρ Oph B
ISO-Oph 124 (main)	$17.0\pm 1.2$	$23\pm 5$	1.1	4.3	⋯	⋯	⋯	⋯
ISO-Oph 124 (sec.)	$19.1\pm 1.7$	$9.2\pm 2.4$	0.5	3.1	⋯	⋯	⋯	⋯
J162728 (main)	$20.0\pm 1.1$	$21\pm 4$	0.6	4.2	$10.2\pm 0.2$	$4.1\pm 0.3$	0.4	4.3
J162728 (sec.)	$16.2\pm 1.5$	$13\pm 4$	0.5	3.4	$6.9\pm 0.5$	$1.9\pm 0.5$	0.5	3.7
Oph-emb 5	$15.5\pm 0.7$	$55\pm 12$	0.8	3.7	$8.7\pm 1.0$	$1.0\pm 0.4$	0.5	3.9
VSSG 17 (main)	$23.7\pm 1.8$	$12\pm 2$	0.8	4.4	$10.9\pm 0.4$	$3.2\pm 0.4$	0.5	4.3
VSSG 17 (sec.)	$18.7\pm 2.9$	$6.2\pm 2.4$	0.5	3.3	⋯	⋯	⋯	⋯
ρ Oph C
WL 2	$\lt 120$	⋯	0.5	3.7	⋯	⋯	⋯	⋯
WL 12	$\lt 215$	⋯	0.9	4.0	⋯	⋯	⋯	⋯
WL 22	$\lt 28.8$	⋯	0.8	3.8	$8.2\pm 0.5$	$2.4\pm 0.5$	0.4	3.8
ρ Oph E
[GY92] 197	$\lt 61.6$	⋯	0.9	4.8	⋯	⋯	⋯	⋯
WL 15	$\lt 109$	⋯	2.2	4.5	⋯	⋯	⋯	⋯
WL 16	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯
WL 17	$\lt 292$	⋯	0.9	4.6	⋯	⋯	⋯	⋯
ρ Oph F
IRAS 16244-2432	$\lt 66.6$	⋯	1.9	3.7	⋯	⋯	⋯	⋯
IRAS 16246-2436	$\lt 28.5$	⋯	0.6	3.7	$\lt 7.5$	⋯	0.3	3.7
ISO-Oph 132	$\lt 138$	⋯	0.3	4.0	⋯	⋯	⋯	⋯
ISO-Oph 137	$\lt 34.3$	⋯	1.3	4.4	$6.7\pm 0.8$	$2.5\pm 1.0$	0.2	4.0
ISO-Oph 161	$\lt 43.6$	⋯	0.6	3.8	$\lt 7.9$	⋯	0.4	3.7
YLW 15	$\lt 35.4$	⋯	2.4	3.9	$7.8\pm 0.7$	$2.6\pm 0.8$	0.4	4.1
L1689N
IRAS 16293-2422 (main)	$69.4\pm 0.9$	$47\pm 3$	2.7	4.8	$14.4\pm 0.3$	$6.5\pm 0.3$	1.3	4.6
IRAS 16293-2422 (sec.)	$124\pm 3$	$32\pm 1$	1.7	3.2	$16.5\pm 1.5$	$2.4\pm 0.5$	1.7	3.1
L1689S
ISO-Oph 200	$\lt 35.2$	⋯	0.8	4.7	⋯	⋯	⋯	⋯
ISO-Oph 202	$\lt 46.4$	⋯	0.8	4.6	⋯	⋯	⋯	⋯
ISO-Oph 203 (main)	$23.4\pm 1.9$	$5.9\pm 1.2$	0.5	4.5	$9.3\pm 0.4$	$1.9\pm 0.3$	0.4	4.4
ISO-Oph 203 (sec.)	$22.3\pm 2.7$	$4.0\pm 1.1$	0.4	4.9	⋯	⋯	⋯	⋯
ISO-Oph 209	$\lt 62.3$	⋯	1.3	4.2	⋯	⋯	1.3	2.9
L1709
GWAYL 4	$\lt 54.7$	⋯	0.5	2.8	⋯	⋯	1.3	3.0
Oph-emb 4	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯
Solitary sources
Elias 28	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯
J162624	$\lt 68.6$	⋯	0.8	3.3	⋯	⋯	⋯	⋯
J1633.92442	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯
MMS126	$\lt 22.4$	⋯	0.4	4.1	$8.6\pm 0.3$	$4.4\pm 0.5$	0.4	4.1
Oph-emb 18	⋯	⋯	0.2	3.5	⋯	⋯	⋯	⋯

Note. The H₂CO column densities were calculated by non-LTE methods assuming n(H₂)=(8.9 ± 1.0) × 10⁵ cm⁻³ (see Section 4). The line widths and LSR velocities are median values. Section 3.1 describes the main and secondary components.

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The three observed H₂CO lines all have the same ${J}_{{\rm{u}}}=3$, and their rotational temperature is thus an excellent proxy for the kinetic temperature, but makes estimates of the column density more uncertain (Mangum & Wootten 1993; Lindberg et al. 2015). We estimate the beam-averaged H₂CO column densities using the non-LTE radiative transfer code RADEX (van der Tak et al. 2007) assuming an H₂ number density of $(8.9\pm 1.0)\times {10}^{5}$ cm⁻³ in all sources. This value was measured from observations of multiple H₂CO ${J}_{{\rm{u}}}=3$ and ${J}_{{\rm{u}}}=5$ lines toward the protostar R CrA IRS7B at a similar beam size using non-LTE methods (RADEX), and is of the same order of magnitude as found in the other protostars in CrA where this value could be measured (Lindberg et al. 2015). All observed H₂CO lines are para lines, and an ortho-to-para ratio is therefore not important for the excitation analysis. However, such a ratio is necessary to calculate the total H₂CO column density, and for this purpose an ortho/para ratio of 1.6 is assumed (see Dickens & Irvine 1999; Jørgensen et al. 2005). For the c-C₃H₂ excitation analysis, an ortho/para ratio of 3 is assumed (see Lucas & Liszt 2000). Non-LTE RADEX models of c-C₃H₂ show that its physical temperature is well-represented by the rotational temperature if $n({{\rm{H}}}_{2})\gtrsim {10}^{5}$ cm⁻³, at temperatures typical for protostellar envelopes. This is further discussed in Appendix C (see also Spezzano et al. 2016).

The H₂CO rotational temperatures in the Ophiuchus star-forming region are generally higher than what is expected in low-mass protostars on the scale of the APEX primary beam (29'', or 3600 au at a distance of 125 pc). Internal heating by low-mass embedded protostars would produce temperatures ∼10–15 K on these scales (Bergin & Tafalla 2007). The H₂CO temperatures observed toward most sources thus require an external radiation field to be explained (see e.g., Jørgensen et al. 2006; Lindberg & Jørgensen 2012).

4.1. IRAS 16293-2422

4.2. External Heating of Sources in $\rho$ Oph A and $\rho$ Oph B

The H₂CO rotational temperatures are somewhat higher toward the sources in the ρ Oph A cloud than in the ρ Oph B cloud. With two exceptions (IRAS 16293-2422 and ISO-Oph 203), all of the sources where the H₂CO temperatures could be measured are located in these two clouds, and it is therefore not possible to judge whether the H₂CO temperatures of these clouds are representative of protostellar envelopes in the whole region.

The luminous Herbig Be star S1 is located only 2' ($\sim 15\,000$ au) east of ρ Oph A. It has spectral class B4 (Bouvier & Appenzeller 1992) and luminosity ∼1000–$1600{L}_{\odot }$ (Bontemps et al. 2001; Wilking et al. 2005). About 15' (0.5 pc) west of the cloud lies the even more luminous B2 star HD 147889 (Houk & Smith-Moore 1988), with a luminosity $\sim 4500{L}_{\odot }$ (Greene & Young 1989). In their radiative transfer models, Liseau et al. (2015) use the luminosities of S1 and HD 147889 of $1100{L}_{\odot }$ and $4500{L}_{\odot }$, respectively.

In Figure 4, the rotational temperatures of H₂CO and c-C₃H₂ are plotted as a function of the projected distance to S1. We also calculate how the luminous source S1 influences the dust temperature, and thus indirectly the molecular gas temperature, using a 1D Transphere model (Dullemond et al. 2002). We assume the luminosity of S1 to be $1600{L}_{\odot }$ and a constant cloud density $n({{\rm{H}}}_{2})={10}^{4}$ cm⁻³. The model (dashed line in Figure 4) underpredicts the observed H₂CO temperatures of the sources in ρ Oph A and ρ Oph B somewhat, but the irradiation from HD 147889 and the internal irradiation from the embedded sources are not included in the model, which could account for the difference. A density different from our assumption, or a non-uniform density distribution within the cloud, could also play a role in explaining the discrepancy.

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To compare the contribution from the two heating sources, and to find the dominant source of heating in any given part of the cloud, Figure 5 shows where the heating from both sources are equal (black contour). This plot was made with two important assumptions limiting the level of interpretations to be made, namely that the cloud is uniformly dense ($n({{\rm{H}}}_{2})={10}^{4}$ cm⁻³) and that all sources lie in the same plane of sky.

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4.3. Temperatures of H₂CO and c-C₃H₂

In this section, to evaluate the statistical significance of any correlations, we use Spearman's ρ rank correlation test, which tests the monotonicity of two sets of variables without requiring a linear relation or a normal distribution of the variables. The H₂CO and c-C₃H₂ rotational temperatures and column densities are given in Table 3. The CH₃OH column densities were estimated using the CH₃OH line at 218.440 GHz and the H₂CO rotational temperature as a proxy for the CH₃OH rotational temperature. This assumption is based on the similar origin of these two molecules (hydrogenation of CO on grain mantles; see, e.g., Charnley 1997), interferometric observations showing them to be spatially co-aligned in protostellar envelopes (e.g., Lindberg & Jørgensen 2012), and the similar critical densities of the observed H₂CO and CH₃OH lines (a few $\times \,{10}^{5}$ cm⁻³; Guzmán et al. 2011, 2013). We used RADEX models to find that the CH₃OH line at 218.440 GHz is optically thin and that our method provides reliable values on the CH₃OH column density assuming n ∼ 10⁶ cm⁻³ and $T\lesssim 40$ K (see also Lindberg et al. 2016).

Lindberg et al. (2015) suggested a trend between the H₂CO rotational temperature and the CH₃OH/H₂CO ratio in the embedded protostars of CrA. In Figure 6 we plot the CH₃OH/H₂CO column density ratio as a function of the H₂CO rotational temperature, and find that they have a strong correlation (Spearman's rank test gives $\rho =0.62$ with $p\lt 0.001$). If the data points of the hot corino source IRAS 16293-2422 are disregarded (since the bulk of this emission should originate in the hot inner envelope), the correlation weakens somewhat ($\rho =0.52$ with $p\lt 0.01$). However, if the secondary components are also disregarded due to their uncertain origin, there is no significant correlation between the CH₃OH/H₂CO and the H₂CO rotational temperature. A correlation would be in agreement with laboratory results that, suggest that the grain-surface hydrogenation reactions forming CH₃OH are more efficient at higher temperatures (e.g., Fuchs et al. 2009). The exact mechanism behind CH₃OH evaporation at low temperatures, however, remains unclear (Bertin et al. 2016; Martín-Doménech et al. 2016).

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In Figure 7, we investigate the correlation between the rotational temperatures and column densities of H₂CO, CH₃OH, and c-C₃H₂. The CH₃OH column density was estimated assuming T_rot(CH₃OH) = T_rot(H₂CO). We find no correlation between the H₂CO rotational temperature and column density ($\rho =0.14$, $p\approx 0.44$), but the CH₃OH column density is moderately correlated with the H₂CO temperature ($\rho =0.58$, $p\approx 0.001$), which indicates that CH₃OH production is enhanced by high temperatures or radiation fields (grain-mantle evaporation and/or enhancement of hydrogenation reactions on grain surfaces; cf. Öberg et al. 2009). If  the hot corino source IRAS 16293-2422 is disregarded, the correlation is still significant ($\rho =0.48$, $p\lt 0.02$). For c-C₃H₂ there is no significant correlation between its rotational temperature and column density (not plotted; $\rho =0.19$, $p\approx 0.41$), which would suggest that its formation is not temperature dependent. We find a very strong correlation between the H₂CO and CH₃OH column densities ($\rho =0.82$, $p\lt {10}^{-6}$), indicating that these molecules share a common origin (e.g., grain-mantle hydrogenation of CO). To exclude the possibility of this correlation being biased by the fact that we use the H₂CO temperature to find the CH₃OH column density, we also performed this calculation assuming a fixed T_rot(CH₃OH) = 10 K for all sources, and still find a strong correlation between N(H₂CO) and N(CH₃OH). Finally, the c-C₃H₂ and H₂CO column densities show a barely significant correlation ($\rho =0.52$, $p\approx 0.04$).

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We find that the H₂CO temperature in protostellar cores depends strongly on the distance to external sources of strong irradiation, whereas the c-C₃H₂ temperature is insensitive to such irradiation (see Figure 4). This implies that the H₂CO traces the outer, externally irradiated layers of the protostellar cores, whereas the emission from c-C₃H₂ originates primarily in the inner, more shielded region. However, in the hot corino source IRAS 16293-2422, the majority of the c-C₃H₂ gas appears to mainly trace an intermediate region, shielded from any external irradiation, but sufficiently distant from the central protostar to remain relatively cool.

We also find that the column densities of the two molecules H₂CO and CH₃OH are closely related, suggesting that they share a common origin. Interferometric observations would be necessary to confirm that they spatially co-exist, and also to study the nature of the secondary components found toward many of the studied sources.

The chemical implications of external heating are still not well understood. A comparative study using interferometric observations of similar protostars within the same star-forming region but with different levels of external irradiation could be used to investigate the origin of the differences in chemical composition and distribution. We suggest future line surveys searching for complex organic molecules and hydrocarbon species, and interferometric mapping of a few key species in the most luminous sources of ρ Oph A and ρ Oph B.

This research was supported by NASA's Emerging Worlds Program and by an appointment to the NASA Postdoctoral Program at the NASA Goddard Space Flight Center to J.E.L., administered by the Universities Space Research Association through a contract with NASA. We thank the anonymous referee for insightful comments and suggestions, which helped us improve the manuscript.

Table 4 lists the detected lines in all of the observed sources. The errors on integrated intensities are $1\sigma$ rms errors. In a few of the sources, we found that considerable C¹⁸O emission in the off position made the C¹⁸O line data unreliable for those sources. Those are indicated by "Off-em." in their integrated intensity column. We do not expect any of the other lines, which are much fainter (in particular in the diffuse gas where the off positions were measured), to be affected by off-position emission.

Table 4.  Measured Line Parameters

Source	Molecule	Transition	vLSR	${\rm{\Delta }}v$	$\int {T}_{\mathrm{mb}}{dv}$
Rest freq.
(MHz)			(km s−1)	(km s−1)	(K km s−1)
GDS J162625.6 (main)
216112.58	DCO+	J = 3–2	3.8	0.8	3.84 ± 0.03
216278.76	c-C3H2	${3}_{03}$–${2}_{21}$	3.8	0.8	0.28 ± 0.02
216373.32	C2D	N = 3–$2,J=7/2$–5/2a	4.0	1.3	0.48 ± 0.03
216428.32	C2D	N = 3–$2,J=5/2$–3/2b	3.6	1.0	0.28 ± 0.02
217238.54	DCN	J = 3–2	3.8	1.1	0.58 ± 0.03
217822.15	c-C3H2	${6}_{06}$–${5}_{15}$, ${6}_{16}$–${5}_{05}$	3.8	0.8	0.38 ± 0.03
217940.05	c-C3H2	${5}_{14}$–${4}_{23}$	3.8	0.7	0.22 ± 0.02
218160.46	c-C3H2	${5}_{24}$–${4}_{13}$	3.9	1.3	0.09 ± 0.02
218222.19	H2CO	${3}_{03}$–${2}_{02}$	3.9	0.8	1.87 ± 0.02
218475.63	H2CO	${3}_{22}$–${2}_{21}$	3.8	0.7	0.17 ± 0.02
218760.07	H2CO	${3}_{21}$–${2}_{20}$	3.8	0.9	0.16 ± 0.02
219560.35	C18O	J = 2–1	3.9	0.8	10.65 ± 0.02
219949.44	SO	${6}_{5}$–${5}_{4}$	4.0	0.7	1.45 ± 0.02

Notes. Rest frequencies from the CDMS (Müller et al. 2001) and JPL (Pickett et al. 1998) molecular spectroscopy databases. A portion is shown here for guidance regarding its form and content.

^aBlend of $F=9/2$–7/2, $F=7/2$–5/2, and $F=5/2$–3/2. ^bBlend of $F=7/2$–5/2, $F=5/2$–3/2, and $F=3/2$–1/2. ^cBlend of $F=13/2$–11/2 and $F=15/2$–13/2. ^d ${F}_{1}=2$–$2,F=2$–1. ^e ${F}_{1}=2$–$1,F=2$–1. ^f ${F}_{1}=2$–$1,F=3$–2. ^g ${F}_{1}=3$–$2,F=2$–1.

Only a portion of this table is shown here to demonstrate its form and content. A machine-readable version of the full table is available.

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In sources with two components, these are listed as separate sources. LSR velocities and line widths are from Gaussian fits. For the sources with two components, the integrated intensities are from Gaussian fits. In all of the other sources, they are integrated over the whole spectral line.

For completeness, Figure 8 shows the spectral profiles of the sources in the CrA survey of embedded protostars (Lindberg et al. 2015) by plotting the normalized spectra for the strongest DCO⁺ (216.113 GHz), c-C₃H₂ (217.822 GHz), H₂CO (218.222 GHz), and SO (219.949 GHz) lines in each source where they have been detected (cf. Figure 3).

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In our observations, the rotational temperatures of c-C₃H₂ were generally found to be lower than those of H₂CO. To verify that this reflects a difference in physical temperatures and is not just a non-LTE or optical depth effect, we used the radiative transfer code RADEX (van der Tak et al. 2007) to study line ratios within a range of physical parameters. A similar investigation of the H₂CO rotational temperature was performed by Mangum & Wootten (1993), showing that a rotational temperature calculated from H₂CO transitions with the same J_u will provide a good measure of the physical temperature.

We here use RADEX to calculate non-LTE line ratios of four of the five c-C₃H₂ lines covered in our observations (see Table 2; the 218.733 GHz line is not included since it is only detected toward one source). One of these lines is a blend between an ortho and a para line, two lines are ortho lines, and the final one is a para line. Excitation analysis involving a combination of ortho and para lines will therefore depend on the assumed ortho-to-para ratio. For c-C₃H₂, we use a ratio of 3 (Lucas & Liszt 2000). We use the collisional rates of Chandra & Kegel (2000) retrieved from the LAMDA database (Schöier et al. 2005).

Since three of the four lines have roughly the same upper level energy (${E}_{{\rm{u}}}=35$–39 K), we only show ratios between the 216.279 GHz line (${E}_{{\rm{u}}}=19.5$ K) and the three remaining lines. We perform the non-LTE calculations assuming three different column densities (total ortho+para c-C₃H₂ column densities): $N={10}^{12}$ cm⁻², $N={10}^{13}$ cm⁻², and $N={10}^{14}$ cm⁻². Our observations show that the c-C₃H₂ column densities typically are $\lt {10}^{13}$ cm⁻², so the $N={10}^{14}$ cm⁻² case is unlikely to be applicable to this work. A model with $N={10}^{11}$ cm⁻² (not plotted) has results identical to the $N={10}^{12}$ cm⁻² model. The results are shown in Figure 9.

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We find that the 216.279 GHz/217.822 GHz non-LTE line ratio is within ∼5 K of the LTE solution given ${T}_{\mathrm{kin}}\lesssim 20$ K for all densities and column densities. The other two ratios are worse off in predicting the temperature, in particular for low H₂ densities. At ${T}_{\mathrm{kin}}\lesssim 20$ K and $n({{\rm{H}}}_{2})\gtrsim 5\,\times \,{10}^{5}$ cm⁻³, the temperature is, however, only underpredicted (or overpredicted) by $\lesssim 5$ K. Densities are expected to exceed 5 × 10⁵ cm⁻³ within the inner 1800 au of a majority of embedded protostars (e.g., Jørgensen et al. 2002), which is the radius of the APEX beam projected to the distance of the Ophiuchus cloud (125 pc).

We compare the rotational (LTE) temperatures calculated when using only the 216.279 GHz and the 217.822 GHz lines with those calculated from using all of the detected lines for all sources in the sample, and find that in no source is the difference greater than 2 K. The rotational temperatures of c-C₃H₂ presented in Table 3 use all observed c-C₃H₂ lines and might thus be underestimated, but calculating rotational temperatures from RADEX intensities shows that this difference is $\lesssim 5$ K. We therefore conclude that the physical temperature of the c-C₃H₂ gas is significantly lower than that of the H₂CO gas in the externally irradiated sources in the sample.