Carbon Chain Molecules toward Embedded Low-mass Protostars^∗

The complete spectrum toward one of our sources, B1-a, with carbon chain identifications is shown in Figure 1. The spectrum contains a number of carbon chain molecules that have been detected toward other high- or low-mass protostars. We focus on six carbon chains, which are commonly seen in our low-mass protostellar survey: sulfur-bearing chains CCS and CCCS, cyanopolyynes HC₃N and HC₅N, and pure hydrocarbons l-C₃H and C₄H. We also include CS and its isotopologues in our analysis to investigate the sulfur carbon chain chemistry. The frequencies, line strengths, and upper-state energies of the observed lines are summarized in Table 2. Line characteristics come from the JPL⁴ and CDMS⁵ catalogs. We note that catalogs sometime differ in their consideration of nuclear spin in the S values and partition functions. In our case, this affects the J = 2–1 transition of C³³S, whose CDMS S value is fourfold that found in some other catalogs, e.g., the SLAIM catalog. Care, therefore, has to be taken to use matching S values and partition functions; in this particular case, we took both from the CDMS catalog. A molecule is considered to be detected if (1) we observed at least one line with a 5σ detection or two lines with 3σ detections; (2) it is free of confusion from other common interstellar or YSO molecular lines; and (3) upper limits of non-detected lines are consistent with predicted populations from detected lines. The treatment of upper limits is discussed in further detail in Section 3.2. In our survey, overlapping lines were typically not a problem, since even the most line-rich sources in our sample are line-poor relative to hot cores. Specifically, our most line-dense sources, such as B1-a, have typical spectral line densities of ∼6–8 lines per GHz, while hot cores can present 10 s or even 100 s of lines per GHz and factors of a few greater line widths (Tercero et al. 2010; Jørgensen et al. 2016; Bonfand et al. 2017). Thus, provided that there are no competing line identifications, a single line is sufficient to claim a detection. All detected lines have upper excitation energies <150 K, consistent with the expected <100 K temperatures of low-mass protostars.

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Table 2.  Carbon Chain Molecular Lines Observed toward Our Protostellar Sample

Molecule	Transition	ν (MHz)	Sa	Eu (K)
C34S	J = 2–1	96412.950	2.00	6.25
C33S	J = 2–1	97172.064	7.98	6.31
CS	J = 2–1	97980.953	2.00	7.05
CCS	JN = 87–76	93870.107	7.97	19.89
	JN = 78–67	99866.521	6.87	28.14
	JN = 89–78	113410.186	7.89	33.58
CCCS	J = 17–16	98268.518	17.00	42.45
	J = 19–18	109828.293	19.00	52.71
HC3N	J = 11–10	100076.392	11.00	28.82
	J = 12–11	109173.634	12.00	34.06
HC5N	J = 35–34	93188.471	34.99	80.50
	J = 36–35	95850.714	35.99	85.10
	J = 37–36	98512.933	36.99	89.83
	J = 41–40	109161.555	40.99	110.02
	J = 42–41	111823.643	41.99	115.39
	J = 43–42	114485.704	42.99	120.88
l-C3H	${J}_{F,l=e}={\left(\tfrac{9}{2}\right)}_{5}-{\left(\tfrac{7}{2}\right)}_{4}$	97995.166	4.89	12.54
	${J}_{F,l=e}={\left(\tfrac{9}{2}\right)}_{4}-{\left(\tfrac{7}{2}\right)}_{3}$	97995.913	3.89	12.54
	${J}_{F,l=f}={\left(\tfrac{9}{2}\right)}_{5}-{\left(\tfrac{7}{2}\right)}_{4}$	98011.611	4.87	12.54
	${J}_{F,l=f}={\left(\tfrac{9}{2}\right)}_{4}-{\left(\tfrac{7}{2}\right)}_{3}$	98012.524	3.87	12.54
C4Hb	NF = 1011–910	95150.397	10.23	25.11
	NF = 109–98	95188.946	8.36	25.13
	NF = 1212–1111	114182.512	11.17	35.62
	NF = 1211–1110	114221.040	10.23	35.64

Notes. Rest frequency, line strength, and upper-state energy are taken from the JPL and CDMS catalogs.

^aLine strength. ^bC₄H transitions were reported in Graninger et al. (2016).

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We detect CS, CCS, CCCS, HC₃N, HC₅N, l-C₃H, and C₄H in 16, 14, 6, 12, 5, 13, and 14 sources, respectively, which correspond to detection percentages of 100%, 88%, 38%, 75%, 31%, 81%, and 88%. We also report additional tentative detections of CCS and CCCS each in one source and HC₅N in two sources, based on single 3σ–4σ line detections.

In Figure 2, we zoom in on the frequency ranges containing CCS lines to visualize line shapes in different sources and the diversity of line strengths across the sample. Spectral windows containing lines of the remaining molecules are presented in the figure set of Figure 2, which is available in the electronic edition of the journal. Since we have a velocity resolution of ∼0.55 km s⁻¹ and a typical line width of ∼1.15 km s⁻¹, many lines are barely resolved, but most lines are consistent with Gaussian shapes. A few sources display small wings, most notably SVS 4–5, which may be due to low-velocity molecular outflows. We see no evidence for jets in any of the sources. Wings were visually identified and then manually excluded by restricting the Gaussian fitting range and hence are not included when determining the molecular abundances below.

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Tarball of figure set images

Integrated line intensities were determined by fitting single Gaussian profiles to each spectral feature (excluding line wings) and are listed in Appendix B. Features were fit assuming a zero-order polynomial baseline with a fixed central line frequency. We treated unresolved multiplets arising from the same species as a single line by combining degeneracies and intrinsic line intensities in the analysis below. In addition to the uncertainty in the line fit, we adopt a 10% calibration uncertainty.

In cases of non-detections, 3σ line intensity upper limits were calculated as

$$\begin{eqnarray}&&\sigma =\mathrm{rms}\times \displaystyle \frac{\mathrm{FWHM}}{\sqrt{{{\rm{n}}}_{\mathrm{ch}}}}.\end{eqnarray} \tag{ 1 }$$

In this case, we take the rms from a 40 km s⁻¹ spectral window containing the transition. The FWHM was taken to be the same as that of other lines of the same molecule, if such lines exist, or of closely related molecules in the same source. Here, the number of channels, n_ch, across the FWHM was ∼FWHM/0.6 km s⁻¹ channel⁻¹. The FWHMs for all detected lines (≥3σ) are listed in Table 13 in the Appendix A. For molecules with multiple line detections spread over a range of energies, we used rotational diagrams to calculate rotational temperatures and column densities (Goldsmith & Langer 1999). Upper level populations for each line are given by

$$\begin{eqnarray}&&\displaystyle \frac{{N}_{u}}{{g}_{u}}=\displaystyle \frac{3{k}_{{\rm{B}}}\int {T}_{\mathrm{mb}}{dV}}{8{\pi }^{3}\nu {\mu }^{2}S},\end{eqnarray} \tag{ 2 }$$

where g_u is the statistical weight of level u, k_B is the Boltzmann constant, ν is the transition frequency, μ is the permanent dipole moment, S is the intrinsic line strength, and N_u is the column density in the upper level u. If we assume optically thin lines and local thermodynamic equilibrium (LTE), each molecule's total column density, N_tot, and rotational temperature, T_rot, in each source can be calculated from

$$\begin{eqnarray}&&\displaystyle \frac{{N}_{u}}{{g}_{u}}=\displaystyle \frac{{N}_{\mathrm{tot}}}{Q({T}_{\mathrm{rot}})}\exp \left(-\displaystyle \frac{{E}_{u}}{{T}_{\mathrm{rot}}}\right),\end{eqnarray} \tag{ 3 }$$

where Q(T_rot) is the rotational partition function and E_u is the energy of the upper level u (Goldsmith & Langer 1999). We also assume that all molecules contained within the beam can be characterized by a single temperature (e.g., de Gregorio-Monsalvo et al. 2006; Bisschop et al. 2007; Sakai et al. 2008, 2009b; Öberg et al. 2014; Fayolle et al. 2015; Bergner et al. 2017).

We do not include effects related to absorption against the CMB or optical depths (e.g., Suzuki et al. 1992; Takano et al. 1998). At the frequencies and rotational temperatures we are considering, we find that the inclusion of the CMB contribution only leads to a <10% difference in the J(T) term, as defined in Equation (3) in Takano et al. (1998), which is small compared to other sources of error. The optical depths for all species (expect CS), based on our derived column densities and line widths, are small with τ ≪ 1.

Rotational diagrams for CCS are shown in Figure 3 and diagrams for all detected molecules are presented in Appendix B. We determine rotational excitation temperatures for all sources and species with at least two line detections for a given molecule. The median rotational temperatures for CCS, CCCS, HC₃N, and C₄H are 9 K, 16 K, 13 K, and 10 K, respectively. These molecules all have relatively tight distributions of rotational temperatures, as indicated by their narrow quartile ranges. No rotational diagrams could be constructed for C³⁴S and l-C₃H. For HC₅N, only two sources had a sufficient number of detections to calculate rotational temperatures and of these only one source (SVS 4–5) had strong enough lines to obtain a well-constrained rotational temperature of 42 K.

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Based on the results summarized in Table 3, there is a spread in species median temperatures across molecules of 9–16 K. There does not appear to be any significant trend between the rotational temperature of the sources and their line-richness. Specifically, the line-rich sources B1-a and IRAS 23238+7401, moderately line-dense source B5 IRS1, and line-poor source HH 300 all have approximately the same rotational temperature. For the sulfur-bearing chains, temperature increases with molecular size; the median CCCS rotational temperature is 80% higher than the CCS median. This trend may be explained by either a chemical effect, i.e., overproduction of longer sulfur-bearing carbon chains at higher temperatures, or more likely, an excitation effect, as is seen in dark clouds where longer carbon chains are less efficiently radiatively cooled (e.g., Bell et al. 1998).

Table 3.  Carbon Chain Rotational Temperatures in K for the 16 Sources

Source	CCS	CCCS	HC3N	HC5N	C4Ha
L1448 IRS1	9 [2]	16 [6]	13 [1]	28 [6]	10 [2]
IRAS 03235+3004	9 [1]	16 [6]	14 [3]	28 [6]	10 [3]
IRAS 03245+3002	12 [2]	18 [4]	13 [1]	28 [6]	10 [4]
L1455 SMM1	9 [1]	37 [11]*	13 [1]	28 [6]	11 [7]
L1455 IRS3	9 [1]	16 [6]	13 [1]	28 [6]	10 [2]
IRAS 03254+3050	9 [2]	16 [6]	13 [1]	28 [6]	10 [1.6]
IRAS 03271+3013	9 [2]	16 [6]	13 [1]	28 [6]	10 [3]
B1-a	10 [1]	16 [6]	12 [2]	28 [6]	9.6 [1.7]
B1-c	7 [1]	10 [6]	13 [1]	28 [6]	10 [2]
B5 IRS1	11 [1]	16 [6]	11 [2]	28 [6]	9.5 [1.0]
L1489 IRS	9 [2]	16 [6]	13 [1]	28 [6]	10 [2]
IRAS 04108+2803	9 [2]	16 [6]	13 [1]	28 [6]	10 [2]
HH 300	7 [1]	16 [6]	13 [1]	28 [6]	10 [2]
SVS 4–5	10 [1]	12 [5]	14 [3]	42 [9]*	15 [3]
L1014 IRS	6 [1]	16 [6]	13 [1]	28 [6]	6.9 [1.0]
IRAS 23238+7401	8 [1]	16 [6]	13 [1]	28 [6]	10 [1]
Species Median	${9.0}_{7.5}^{10.0}$	${16.0}_{12.0}^{18.0}$	${13.0}_{11.8}^{14.0}$	⋯	${10.0}_{10.0}^{10.0}$

Notes. Uncertainties at the 1σ level are reported in brackets. T_rot values in italics are assumed rotational temperatures and are based on median sample temperature. Asterisks (*) indicate unexpectedly high temperatures for CCCS and HC₅N in L1455 SMM1 and SVS 4–5, respectively; additional data are needed to confirm these temperatures.

^aC₄H data are taken from Graninger et al. (2016) with new C₄H detection from source IRAS 03254+3050.

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We calculate column densities without any assumptions about beam dilution, since carbon chains can be produced at a range of temperatures and the emission may therefore fill the telescope beam (see Section 3.2 for further discussion). All column densities are reported in Table 4. For species and sources with well-constrained rotational temperatures, we use Equations (2) and (3) to calculate column density. In calculating column densities for CCS, CCCS, HC₃N, and C₄H in sources with missing rotational temperatures, we adopt the sample median rotational temperature. As noted above, the narrow temperature quartile range for each species implies that using the sample median should not introduce any major errors. For l-C₃H, the column densities were calculated using the median C₄H rotational temperature reported by Graninger et al. (2016) and for C³⁴S, the median CCS rotational temperature was used. These choices were motivated by the fact that l-C₃H/C₄H and C³⁴S/CCS belong to the same families of unsaturated carbon chains and should exhibit similar chemistries. The case of HC₅N is special as we have only one temperature determination. Since this temperature belongs to the warmest source in the best constrained carbon chain molecule, C₄H, it does not seem appropriate to simply adopt it in the sample. Nor does it seem appropriate to adopt the HC₃N or C₄H median temperatures, however, since in SVS 4–5 the HC₅N temperature is three and four times higher, respectively. We instead opt to scale the SVS 4–5 HC₅N temperature with the ratio of the SVS 4–5/sample median temperature for C₄H. The resulting temperature of 28 K is used to calculate HC₅N column densities and upper limits for all sources in the sample. The resulting estimates should be considered order of magnitude constraints. If instead we adopt a HC₅N rotational temperature of 9 K, the lowest among the molecules, typical HC₅N column densities increase by two orders of magnitude, while the distribution of column densities is relatively unchanged. The choice of excitation temperature thus affects any conclusions on the relative abundances of HC₅N with respect to other molecules, but not conclusions based on correlations.

Table 4.  Carbon Chain Column Densities in cm⁻² for the 16 Sources

Source	C34S	CCS	CCCS	HC3N	HC5N	l-C3H	C4Ha
L1448 IRS1	>8.7  × 1010,b	<8.5 × 1010	<1.0 × 1011	<6.8 × 1010	<1.9 × 1011	<8.3 × 1010	<7.3 × 1012
IRAS 03235+3004	6.0 [2.9] × 1011	2.2 [0.7] × 1012	2.0 [4.3] × 1011	3.7 [1.1] × 1012	<2.7 × 1011	5.5 [2.9] × 1011	3.8 [1.5] × 1013
IRAS 03245+3002	1.4 [0.7] × 1012	3.3 [1.0] × 1012	7.0 [2.5] × 1011	4.0 [2.1] × 1012	7.3 [10.8] × 1011	1.7 [0.9] × 1011	1.6 [0.7] × 1013
L1455 SMM1	1.4 [0.7] × 1012	4.3 [1.4] × 1012	1.8 [0.1] × 1011	2.7 [1.4] × 1012	<2.5 × 1011	3.5 [1.9] × 1011	1.8 [0.5] × 1013
L1455 IRS3	4.2 [2.0] × 1011	1.4 [0.2] × 1012	<1.1 × 1011	7.3 [3.8] × 1011	<2.6 × 1011	2.5 [1.4] × 1011	1.1 [0.3] × 1013
IRAS 03254+3050	7.6 [4.0] × 1010	2.3 [1.8] × 1011	<1.5 × 1011	<9.3 × 1010	3.9 [4.1] × 1011	1.9 [1.1] × 1011	1.4 [0.9] × 1013
IRAS 03271+3013	1.1 [0.6] × 1011	3.0 [2.4] × 1011	<1.1 × 1011	1.9 [1.0] × 1012	<2.8 × 1011	4.3 [2.3] × 1011	1.4 [0.5] × 1013
B1-a	3.6 [1.7] × 1012	5.8 [1.7] × 1012	6.1 [4.9] × 1011	4.1 [1.2] × 1012	2.5 [2.6] × 1011	3.9 [2.1] × 1011	2.5 [0.6] × 1013
B1-c	1.3 [0.6] × 1012	6.2 [2.9] × 1012	1.3 [4.9] × 1011	4.7 [2.4] × 1012	1.2 [1.6] × 1012	4.3 [2.3] × 1011	2.7 [0.2] × 1013
B5 IRS1	8.8 [4.3] × 1011	1.7 [0.5] × 1012	<1.8 × 1011	3.2 [1.0] × 1012	1.0 [1.3] × 1012	4.0 [2.3] × 1011	3.6 [0.5] × 1013
L1489 IRS	2.5 [1.3] × 1011	5.4 [5.2] × 1011	<1.1 × 1011	3.8 [3.9] × 1011	<4.3 × 1011	<1.3 × 1011	<3.2 × 1012
IRAS 04108+2803	4.1 [2.0] × 1011	6.0 [4.7] × 1011	<1.5 × 1011	<6.0 × 1010	<2.4 × 1011	1.9 [1.1] × 1011	6.4 [2.4] × 1012
HH 300	4.4 [2.2] × 1011	1.2 [0.8] × 1012	<1.7 × 1011	<1.4 × 1011	<3.4 × 1011	<1.4 × 1011	1.0 [3.2] × 1013
SVS 4–5	4.9 [2.4] × 1012	3.5 [1.3] × 1012	1.4 [1.7] × 1012	1.0 [0.3] × 1013	2.6 [1.2] × 1011	4.1 [2.2] × 1011	2.2 [0.4] × 1013
L1014 IRS	2.9 [1.4] × 1011	2.1 [0.8] × 1012	<1.1 × 1011	4.9 [2.5] × 1011	<1.7 × 1011	2.0 [1.1] × 1011	3.2 [0.8] × 1013
IRAS 23238+7401	7.5 [3.7] × 1011	1.7 [0.2] × 1012	1.5 [3.3] × 1011	3.6 [1.8] × 1012	3.7 [3.7] × 1011	2.1 [1.2] × 1011	1.4 [2.2] × 1013
Species Median	6.013.53.5 × 1011	1.73.40.90 × 1012	2.01.76.6 × 1011	${3.4}_{1.6}^{4.0}\times {10}^{12}$	${3.9}_{3.2}^{8.7}\times {10}^{11}$	${3.5}_{2.0}^{4.1}\times {10}^{11}$	${1.7}_{1.4}^{2.7}\times {10}^{13}$

Notes. Uncertainties at the 1σ level are reported in brackets and tentative detections are shown in italics. Species median column densities with quartiles are reported using only calculated column densities (i.e., no upper or lower limits). While we list the column density data for L1448 IRS1, we exclude this source in all subsequent analysis.

^aC₄H data are taken from Graninger et al. (2016) with new detections from source IRAS 03254+3050. ^bDue to significant self-absorption in this source, we are only able to constrain a lower limit.

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Since several protostars such as IRAS 03254+3050 and B1-c displayed moderate self-absorption in CS lines, we used the isotopologue ratios (CS/C³⁴S and C³⁴S/C³³S) to check the optical depth for each protostar. The integrated intensity ratios of CS/C³⁴S are between 5.0 and 20.5 and are all lower than the expected isotopic ratio of ∼22 (Vocke 1999), albeit marginally so for sources IRAS 03271 and L1448 IRS1. Therefore, we conclude that CS is optically thick in almost all sources. Next, we compared C³⁴S to C³³S and found ratios ranging from 4.9 to 9.3 in 15/16 sources, which, within uncertainties, are all consistent with the expected isotopic ratio of ∼5.5 (Vocke 1999) and indicate that both lines are optically thin. The one exception is L1448 IRS1, where the ratio is 1.5. We used the C³⁴S line to calculate column densities for all sources with the caveat that we are certainly underestimating the column density for L1448 IRS1 (however, this source is not used for further analysis). Using Equation (3) from Takano et al. (1998), we find the highest value of optical depth for the C³⁴S line among our samples (excluding L1448 IRS1) was τ = 0.072 for source B1-a.

Across the sample, a few sources stand out for their especially rich or poor carbon chain inventories, or because they are enhanced in or lack specific molecules. The sources SVS 4–5 and B1-a have large carbon chain column densities compared to the other sources. All seven molecules are detected toward these sources and the derived column densities are often an order of magnitude in excess of the sample medians. In contrast, L1448 IRS1 and L1489 IRS have zero and two carbon chain detections, respectively. While HH 300 shows enhancement in sulfur-bearing molecules C³⁴S and CCS as well as carbon chain radical C₄H, no cyanopolyynes are observed toward this source. L1489 IRS also displays an enhancement in C³⁴S and CCS along with HC₃N, but lacks detections of the carbon chains l-C₃H and C₄H.

Since only upper or lower limits could be constrained for all molecules in L1448 IRS1, we have excluded this source in all subsequent analyses. It is possible that its lack of carbon chains could be related to its high mass. Carbon chain deficiencies have been previously observed in massive YSOs (Tercero et al. 2010; Pagani et al. 2017) and massive clumps in infrared dark clouds (Sakai et al. 2008). In both cases, these deficiencies have been attributed to more advanced stages of chemical evolution compared to their lower-mass counterparts. It is unclear, however, whether this scenario can also explain the low abundances of unsaturated organics in L1448 IRS1 compared to the sample median.

The distributions of column densities and fractional abundances reveal variations in chemistry among sources and between different molecules. Figure 4(a) shows a histogram of each molecule's column density across the sample. The column densities of C³⁴S, CCS, and HC₃N span two orders of magnitude or more. By comparison, the l-C₃H and C₄H distributions are narrow. Table 5 lists lower and upper quartiles for the column density of each molecule. For CCS and HC₃N, lower and upper quartile values differ by a factor of ∼3–4, while l-C₃H and C₄H quartiles differ by a factor of ∼2. Due to the large fraction of non-detections for CCCS and HC₅N, we are not able to reliably compare the distributions or quartile ranges of these species.

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Table 5.  Median Column Densities and Fractional Abundances

	Median Column Densities		Median Abundances
	(1012 cm−2)		$({10}^{-12})$
	Detections Only	Survival Analysis	Survival Analysis
C34S	${0.60}_{0.35}^{1.35}$	${0.60}_{0.35}^{1.35}$	${1.38}_{1.17}^{3.48}$
CCS	${1.7}_{0.90}^{3.4}$	${1.7}_{0.90}^{3.4}$	${4.08}_{2.01}^{6.95}$
CCCS	${0.20}_{0.17}^{0.66}$	${0.12}_{0.0079}^{0.19}$	${0.16}_{0.037}^{0.54}$
HC3N	${3.4}_{1.6}^{4.0}$	${2.3}_{0.32}^{3.8}$	${4.06}_{2.18}^{8.00}$
HC5N	${0.39}_{0.32}^{0.87}$	${0.25}_{0.10}^{0.39}$	${0.41}_{0.28}^{0.72}$
l-C3H	${0.35}_{0.20}^{0.41}$	${0.23}_{0.18}^{0.40}$	${0.60}_{0.20}^{1.61}$
C4H	${17.0}_{14.0}^{26.5}$	${15.0}_{10.7}^{25.5}$	${46.0}_{16.2}^{73.2}$

Note. Lower and upper quartiles are shown to the right of the medians.

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To compare our observational results with theoretical predictions for low-mass YSO chemical abundances, we need to calculate fractional abundances with respect to atomic hydrogen. Thus, we first need to determine the hydrogen column density that is contained within the beam. To obtain an order-of-magnitude estimate (more precise estimates require detailed models of each individual source, which is beyond the scope of this study), we use existing constraints on envelope masses toward the majority of our sources and assume a spherically symmetric physical model with power-law density profile:

$$\begin{eqnarray}&&{n}_{{\rm{H}}}(r)={n}_{{{\rm{M}}}_{\mathrm{env}}}{\left(\displaystyle \frac{r}{1000\mathrm{au}}\right)}^{-\alpha },\end{eqnarray} \tag{ 4 }$$

where ${n}_{{{\rm{M}}}_{\mathrm{env}}}$ is a normalizing constant based on the protostellar envelope mass, which we have estimates of for 11 sources (see Table 1). We assume that all of the envelope material is molecular hydrogen. From the median values derived from radiative transfer modeling of low-mass protostars by Jørgensen et al. (2002), α = 1.5, which is also the expected value of a free-falling core. To determine the total number of H nuclei within the beam for a typical protostar with radial density profile n_H(r) and emitting area A(r), we integrate radius from minimum (r_min) to maximum (r_max):

$$\begin{eqnarray}&&{\eta }_{{\rm{H}}}={\int }_{{r}_{\min }}^{{r}_{\max }}A(r){n}_{{\rm{H}}}(r){dr}.\end{eqnarray} \tag{ 5 }$$

For our sources, we assume r_min = 1 au and a maximum extent of r_max = 20,000 au, which is consistent with the values adopted in Jørgensen et al. (2002). We can express A(r), in terms of the cylindrical line of sight, as

$$\begin{eqnarray}A(r)=\left\{\begin{array}{ll}4\pi {r}^{2}, & r\leqslant {r}_{b}\\ 2\pi [{r}_{b}^{2}+{(r-\sqrt{{r}^{2}-{r}_{b}^{2}})}^{2}], & r\gt {r}_{b}.\end{array}\right.\end{eqnarray} \tag{ 6 }$$

Equivalently, the emitting surface, when the radius is less than the beam radius, is that of the surface area of a sphere at that particular radius. For radii larger than the beam radius, the emitting surface corresponds to that of spherical caps in front of and behind a sphere with radius r_b. Each cap has a fixed base radius equal to the beam radius r_b and a height that is allowed to vary based on radius. This methodology is adapted from Bergner et al. (2017).

Using this method, the beam is determined to contain ≈50% and ≈46% of the protostellar envelope mass for Perseus and Taurus sources, respectively. To test the effect of selecting different r_max values, beam fractions were calculated for r_max = 10,000 au. For Perseus sources, there was a difference of about 8% in beam fraction and about 5% for Taurus sources.

Then, the beam-averaged hydrogen column density is given by

$$\begin{eqnarray}&&{N}_{{\rm{H}}}=\displaystyle \frac{{\eta }_{{\rm{H}}}}{\pi {r}_{b}^{2}},\end{eqnarray} \tag{ 7 }$$

where r_b is the beam radius. We adopt a 12'' beam radius based on the median observing beam size (see Section 2) and a distance of 250 pc to Perseus (Schlafly et al. 2014) and 140 pc to Taurus (Torres et al. 2007).

Given the column density N_X of a species, the species specific fractional abundance is simply given by the ratio

$$\begin{eqnarray}&&{f}_{X}=\displaystyle \frac{{N}_{X}}{{N}_{{\rm{H}}}}.\end{eqnarray} \tag{ 8 }$$

Fractional abundance distributions are shown in Figure 4(b). Since envelope masses are not available for SVS 4–5, IRAS 23238, L1014 IRS, HH 300, and IRAS 04108, these sources are omitted from the comparison. The spread in carbon chain abundances is considerable, demonstrating that the column density spread (as seen in Figure 4(a)) is not simply an effect caused by different source sizes. Relative to the column densities, the abundance distributions broaden for l-C₃H and C₄H, tighten for HC₃N, and remain roughly the same for C³⁴S and CCS. This tightening of the HC₃N distribution is particularly prominent, which in terms of column density, had previously spanned two orders of magnitude but in fractional abundance only spans a little over one order of magnitude.

The calculated fractional abundances are naturally quite uncertain, since in reality envelope sizes and radial profiles differ substantially between sources (e.g., Jørgensen et al. 2002). We note that all sources but one for which this analysis is possible (i.e., envelopes masses are known) are located in Perseus, which implies that we do not suffer from errors related to different relative distances within the sample. Still, better constraints on both envelope and carbon chain radial profiles are required for meaningful data–model comparisons beyond order-of-magnitude checks.

We calculate median column densities and abundances using both detections only and detections with upper limits. If median abundances are determined by using only detections, we are omitting information supplied by the upper limits and risk overestimating median occurrences. Hence, we used survival analysis in the form of the Kaplan–Meier (KM) estimator with left censorship (Feigelson & Nelson 1985). All detections and non-detections were ordered, and then the values of positive detections were used to define intervals for the total range of values. Whenever an upper limit was contained in an interval, it was treated as being the lower limiting value of that interval. Equivalently, every positive detection was weighted by the number of upper limits between it and the next largest positive detection.

To account for the discrete nature of the survival function, we calculated median abundances via linearly interpolating between the values above and below where the cumulative density function (CDF) equaled 0.5. But, if the lowest 50% of values consist of solely upper limits, the first positive detection will happen after the CDF has already exceeded 0.5 and it will be impossible to compute medians. In our sample, this applies to CCCS and HC₅N. To circumvent this limitation, we assigned the lowest value in our sample a "detection" status, irrespective of its true nature as a detection or a non-detection and then calculated medians via the KM estimator (e.g., Feigelson & Nelson 1985; Bergner et al. 2017). While this method may artificially inflate estimates for median values, we believe this procedure still provides a more realistic estimate of the median than simply using detections alone. As can be seen in Figure 5 and Table 5, there is a significant difference between medians determined using only detections and those calculated with survival analysis, which reveals the importance of incorporating the constraints provided by non-detections. Note that we include tentative detections for CCS, CCCS, and HC₅N to estimate their median column densities. The inclusion of tentative detections does not have a large effect on the median column densities (∼10%) but increases the HC₅N median fractional abundance by ∼50%.

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Median column densities span three orders of magnitude. For the sulfur-bearing chains, each additional carbon atom leads to approximately an order of magnitude reduction in the median column density. Similarly, the addition of two C atoms from HC₃N to HC₅N leads to a reduction of one order of magnitude. For the carbon chain radicals, however, the addition of one C atom from l-C₃H to C₄H leads to a two order of magnitude increase in column density.

Median fractional abundances calculated using survival analysis are shown in Figure 5(b). The median fractional abundances also span three orders of magnitude and exhibit the same trends as found for the median column densities. The distribution of abundances, reflected in the error bars spanning the lower and upper quartiles, is tighter for the small sulfur-bearing molecules C³⁴S and CCS than for both the nitrogen-bearing chains and the carbon chain radicals.

To assess chemical relationships between carbon chains, we searched for correlations between different species. Both chemically related molecules (e.g., if one forms from another) or those that depend similarly on an underlying physical parameter (e.g., envelope mass, temperature) are expected to exhibit strong positive correlations. When comparing column densities, some correlation is expected regardless of chemical relationships, as most molecules increase with an increasing total column density along a sight line. In fact, we see a positive correlation between column densities and envelope masses for all molecules, except CCCS (not shown). We therefore focus our correlation study on the derived abundances.

We calculate the Spearman's rank correlation coefficient for each carbon chain with one another, as shown in Figure 6 and list the results in Table 6. Unlike Pearson's correlation, which characterizes linear relationships, Spearman's correlation assesses monotonic relationships or, alternatively, the rank order between two variables. Spearman's correlation was chosen since we do not have any prior expectation of linear correlation and because Spearman's correlation is less sensitive to outlier values than Pearson's. Only detections were used for calculating the correlations with the implication that for some species, the relatively low number of mutual detections implies that our determined correlation coefficients are not robust to the inclusion/exclusion of a few low- or high-valued data points. We also note that, if we instead consider Pearson's correlations, all identified correlations are still found to be significant.

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Correlations are found at the 90% confidence level between similar families of carbon chains (e.g., CS/CCS, l-C₃H/C₄H) and between HC₃N and the carbon chains l-C₃H and C₄H. This latter correlation could be explained by a closely related production chemistry of l-C₃H and C₄H and that of HC₃N in low-mass protostars, as discussed in Section 4.1. This result is independent of whether we only consider Perseus sources or also include the one Taurus source for which we could calculate abundances. However, the correlation could also be due to a mutual dependence on an unobserved underlying variable. Without a clear mechanism or a multivariable correlation study, these two explanations cannot be distinguished.

Correlations of abundances with physical properties may also yield insight into the underlying chemistry, e.g., if the carbon chemistry depends on protostellar heating, a positive correlation with protostellar luminosity. We checked for correlations between bolometric luminosity and carbon chain abundance and found moderate to strong negative correlations for all species. The luminosity correlations are presented in Figure 7. Both l-C₃H and C₄H as well as HC₃N stood out as being particularly well anti-correlated with bolometric luminosity. While this finding suggests that more luminous and evolved protostars have envelopes exhibiting lower carbon chain abundances, we find that most of this anti-correlation might be explained by a positive correlation ($r=0.69$) between hydrogen column density and bolometric luminosity.

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The observed carbon chain abundances provide constraints on the chemistry that could produce them. We note that such a chemistry must be general enough to always produce some carbon chains, as we detect them in almost all sources. The formation and/or destruction chemistry must also be sensitive to some aspects of the protostellar environment and/or evolutionary stage since we see order-of-magnitude variations across the sample. As shown in the right panel of Figure 5, the decreasing column density with size in the cyanopolyynes and sulfur-bearing chains is consistent with bottom-up chemistry, as is the increasing column density from l-C₃H to C₄H, since these molecules form via different pathways (see the subsequent discussion). Finally, the lack of correlations between the pure hydrocarbon chains and the sulfur-bearing molecules suggest that there are multiple, fairly independent carbon chain chemistries that affect the final carbon chain composition in a protostellar envelope.

Within the pure hydrocarbon chain molecular family, we find that the relative abundances of l-C₃H and C₄H are consistent with model expectations (Hassel et al. 2011) and previous observations (Gratier et al. 2016). The fact that C_nH with even-numbered n tends to be much more abundant than species with odd n is well known and is related to the molecular structure and reactivity of these species (e.g., Cernicharo et al. 1987; Kawaguchi et al. 1991; Guelin et al. 1997; Sakai et al. 2010). Based on our observations, we find that this trend is also present in the relatively warm and dense regions of gaseous envelopes surrounding protostars. C₄H is expected to form efficiently at low temperatures (Chastaing et al. 1998) through two different pathways involving C_2nH₂ with C₂ (Canosa et al. 2007) and C₂H (Agúndez et al. 2017), respectively. The production of unsaturated carbon chains with an odd number of carbon atoms also occurs efficiently at low temperatures and is due to reactions of neutral carbon atoms and CH radicals with C₂H₂ (Chastaing et al. 2001; Loison & Bergeat 2009). In particular, l-C₃H can form by the neutral–neutral reaction ${\rm{C}}+{{\rm{C}}}_{2}{{\rm{H}}}_{2}\,\to \,{{\rm{C}}}_{3}{\rm{H}}+{\rm{H}}$ (Kaiser et al. 1997, 1999). Sakai et al. (2008) also suggested a lukewarm pathway involving methane and C⁺. The strong correlation between l-C₃H and C₄H is consistent with their shared origin in C₂H₂. It would be interesting to obtain more sensitive data and explore whether this correlation continues to be as strong when considering larger hydrocarbon chains.

For the cyanopolyynes, we also find that the relative abundances of HC₃N and HC₅N are consistent with model predictions and previous observations (Hassel et al. 2011; Gratier et al. 2016). Cyanopolyynes can form efficiently at low temperatures (Seki et al. 1996) and without activation barriers (Fukuzawa & Osamura 1997; Choi et al. 2004) via a neutral–neutral reaction of CN with C_2nH₂. Additional formation pathways involve the growth of cyanopolyyne carbon chains by C₂H₂⁺ + HC_2n+1N and reactions between hydrocarbon ions and nitrogen atoms followed by electron recombination reactions (see Figure 2 in Taniguchi et al. 2016). Based on observational results, Taniguchi et al. (2016) suggest that the hydrocarbon ion and N atoms reactions are the dominant pathway for forming HC₅N, while the neutral–neutral reaction is likely the most important for HC₃N (Takano et al. 1998). The strong correlation we observe between HC₃N and the hydrocarbons is consistent with their shared origin in C₂H₂ and a neutral–neutral production pathway and is compatible with the above model.

The decreasing trend in median column density from n = 1–3 seen in the C_nS molecules is consistent with previous observations and model expectations (Hassel et al. 2011; Gratier et al. 2016). Since the ionization potential of the sulfur atom is lower than that of the carbon atom, most of the sulfur atoms are ionized by the interstellar UV radiation in regions where there is only partial ionization of carbon atoms (Suzuki et al. 1992). Since S⁺ does not interact with lower vibrational states (v ≤ 1) of H₂ in cold clouds (Prasad & Huntress 1982; Zanchet et al. 2013), the S⁺ ion is expected to be abundant in the gas phase in regions that produce carbon chains (Suzuki et al. 1992). Reactions of hydrocarbons such as CH, C₂H, and l-C₃H with S⁺ are major routes to produce CS, CCS, and CCCS, respectively (see Figure 11 in Suzuki et al. 1992). Numerous neutral–neutral reactions have also been identified (e.g., Yamada et al. 2002; Sakai et al. 2007; Agúndez & Wakelam 2013) and may significantly contribute to the production of CCS and CCCS. In particular, Sakai et al. (2007) found that the reaction between CH and CS is the most probable neutral–neutral pathway to produce CCS, while the formation of CCCS via the neutral–neutral reaction C₂ + H₂CS →CCCS + H₂ has been suggested by Yamada et al. (2002).

The strong mutual correlations between CS and CCS are consistent with both the proposed ion–molecule and neutral–neutral production routes. However, as our survey did not detect CCCS in the majority of sources, higher-resolution data of sulfur-bearing chains and the pure hydrocarbons could help clarify the relative contributions of the proposed formation pathways of sulfur-containing carbon chains, especially larger (n ≥ 3) chains, and to determine if neutral–neutral or ion–molecule reactions are more dominant in protostellar envelopes. Specifically, an observed strong correlation between CCCS and C₃H would support formation via ion–molecule reactions, while the absence of such a correlation would imply that neutral–neutral reactions are more important for CCCS production.

We compare our carbon chain abundances against those found in dense cold clouds to address questions of chemical inheritance. Due to its relative closeness (∼140 pc) and rich molecular complexity, TMC-1 has been extensively observed to study the chemical processes taking place in dark clouds (e.g., Ohishi & Kaifu 1998; Kaifu et al. 2004; Smith et al. 2004; McCarthy et al. 2006; Marcelino et al. 2007). Figure 8 shows our carbon chain observations compared to the abundances found in TMC-1. We note that while the majority of our sources are in Perseus, which is ∼55% further away than TMC-1, we have a beam size that is ∼60% smaller than the smallest beams used in the original observations by Kaifu et al. (2004) and therefore the two sets of observations probe similar physical scales.

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We find that all our sources are underabundant in all species, often by several orders of magnitude, relative to the TMC-1 abundances. HC₅N is the most depleted (∼1 × 10⁴), followed by CCCS (∼8 × 10³) and l-C₃H (∼4 × 10³), while smaller decreases were seen for HC₃N (∼3 × 10³), CCS (∼2 × 10³), and C₄H (∼2 × 10²).

The different depletion factors suggest a differentiation in carbon chain composition in the protostellar phase. Figure 9 directly compares two column density ratios, CCS/HC₃N and CCS/HC₅N, in cold clouds and protostellar envelopes using a different set of cloud observations. The cloud data include all dark cloud core sources in Suzuki et al. (1992) for which ratios or ratio limits of CCS, HC₃N, and HC₅N could be derived. Figure 9 shows that while there is overlap between the cloud and protostellar samples, a large portion of the protostellar sources are overabundant in CCS or underabundant in HC₃N and HC₅N compared to cloud sources. These two samples are not observed in a homogeneous way and it is therefore difficult to quantify this difference, but it warrants further study to determine whether it is a general effect. An additional caveat is related to the fact that cyanopolyyne formation chemistry tends to proceed more slowly, causing nitrogen-bearing chains to be less abundant in the earlier phases of cold clouds (e.g., Bergin et al. 1997), while sulfur-bearing chains are more quickly formed (e.g., Bergin & Langer 1997; Aikawa et al. 2001, 2003, 2005). Thus, a cold cloud survey such as the one conducted by Suzuki et al. (1992) could either be biased toward higher ratios of CCS/HC_nN column density (since they were likely observing these systems when cyanopolyynes were still not yet all formed) or toward lower ratios if CCS has already begun freezing out.

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We next compare our beam-averaged carbon chain abundances with observed abundances in WCCC sources to assess whether any of our sources show signs of WCCC chemistry. The left panel of Figure 10 shows a comparison of our observations with known WCCC sources taken from Hassel et al. (2008, 2011). The fractional abundances for B228 are based on observational data from the Nobeyama 45 m Telescope (Sakai et al. 2009a), which has a typical beam size of 20'' at the observed frequencies. Since B228 is at a distance of 155 pc (Lombardi et al. 2008), these observations are likely probing smaller spatial scales than our IRAM 30 m observations. The fractional abundances for L1527, which is at a distance of 137 pc (Torres et al. 2007), are based on heterogeneous observations with beam sizes ranging from ∼17''–37'' (see details within Sakai et al. 2008, 2009a, 2009b). The data from species CCS and C₄H represent the same spatial scales as our IRAM 30 m observations, while those for HC₃N, HC₅N, and l-C₃H are likely probing somewhat smaller (∼1.5–2.5×) spatial scales.

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The medians for all molecules in our sample are underabundant relative to the WCCC observations by about two orders of magnitude, although there is significant scatter in carbon chain abundances among molecular families and individual protostars. L1455 IRS3 possesses the highest abundances for most molecules but no sources in our sample reach the elevated abundances observed in WCCCs, even when taking into account the factors of ∼2–5 arising from possible differences in beam dilution.

Since we have adopted a different approach for determining abundances than Hassel et al. (2008, 2011), it is possible that abundances have not been derived consistently between our sample and theirs. To mitigate this concern, the right panel of Figure 10 shows the same comparison but with column densities instead of fractional abundances. Our sources are still underabundant compared to WCCC sources, with the possible exception of CCS. We note that the abundance pattern is quite similar between our sample and the WCCC sources, but it is unclear whether this similarity is informative, since the pattern was also quite similar to what is observed in TMC-1.

Finally, we compare our sources with protostellar chemical models developed to explain the observed carbon chain abundances in WCCC sources. The protostellar model expectations shown in Figure 11 are those taken from the latest and most comprehensive warm-up models of Hassel et al. (2011). The model fractional abundance for CCCS was provided by G. E. Hassel (2018, private communication). The majority of our sources have abundances lower than the cold cloud model predictions (T = 10 K), and no sources in our sample are consistent with a temperature of 30 K, which is that of WCCC sources, or the hot corino warm-up model expectations (T = 100, 200 K). However, the abundance patterns look similar between the model and our sample. This could indicate that the observed carbon chains are formed in situ but less efficiently than in the models or, alternatively, that these carbon chains were simply swept up as remnants from the cold cloud stages. To definitively distinguish between inheritance and WCCC would require spatially resolved observations that can determine whether there are enhancements around the CH₄ desorption front or not.

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