Large-scale Spectroscopic Mapping of the ρ Ophiuchi Molecular Cloud Complex. I. The C₂H-to-N₂H⁺ Ratio as a Signpost of Cloud Characteristics

With a distance of (Loinard et al. 2008), the ρ Ophiuchi molecular cloud complex (hereafter ρ-Oph MCC), also called the Ophiuchus Molecular Cloud (e.g., Pattle et al. 2015), is one of the most active star-forming regions within several hundred parsecs of the Sun (Lada & Lada 2003). At such a close distance, the beam size (at ∼100 GHz) of a 10 m class millimeter radio telescope corresponds to 0.035 pc. The ρ-Oph MCC consists of Lynds 1688 (hereafter L1688), L1689, and a number of filamentary clouds (L1709, L1740, L1744, L1755, and L1765). These filamentary clouds extending from L1688 are known as the streamers or the cobwebs of the ρ-Oph MCC (Nutter et al. 2006). L1688 is an intermediate-mass star-forming region, filling the gap between low-mass star-forming regions, such as Taurus, and more massive star-forming regions, exemplified by the Orion molecular cloud (Padgett et al. 2008). Motte et al. (1998) identified 13 dense condensations in L1688 from their 1.3 mm dust continuum map with an effective beam resolution of (FWHM and all instances hereafter). Johnstone et al. (2004) identified two more objects outside the boundary of previous maps in one 850 μm map with a 40 mJy beam⁻¹ sensitivity and a beam size. Stanke et al. (2006) presented a 1.2 mm dust continuum L1688 map with a ∼24″ resolution and a noise level of the order of 10 mJy beam⁻¹. They cleaned their map down to the noise level by using wavelet analysis and identified 143 sources. A possible explanation as to why the number of sources Stanke et al. (2006) identified is much larger than that identified by Motte et al. (1998) is that the condensations in Motte et al. (1998) are split into multiple sources. For example, Oph-B1 was identified as a dense condensation with four starless clumps (B1-MMS1, B1-MMS2, B1-MSS3, and B1-MMS4) in Motte et al. (1998), while at least six sources (MMS20, MMS23, MMS27, MMS28, MMS66, MMS86, and maybe MMS49) were identified by Stanke et al. (2006). Other 1.2 mm dust sources in Stanke et al. (2006) are isolated sources or faint and extended dust emission regions. L1689 is to the east of L1688, associated with less star formation (Loren et al. 1990). To further examine the similarities and differences between L1688 and L1689, we explore here the utility of C₂H and N₂H⁺ as tracers.

C₂H is a tracer of relatively high density gas with a critical density of cm⁻³ (see Appendix A for details). It can also be used as an efficient tracer of extended gas owing to its lower effective excitation density (see Shirley 2015). The abundances of C₂H decrease from diffuse gas to dense gas, making it a potential probe of chemical evolution. In diffuse gas, CO is dissociated by UV photons, releasing carbon for C₂H formation and thus sustaining a higher C₂H abundance (Li et al. 2012). In dense gas, without CO dissociation, the C₂H abundance decreases. In contrast, N₂H⁺ is a well-known dense gas tracer with a critical density of cm⁻³ (Appendix A).

In this paper, we present the first large-scale (2.5-square-degree) C₂H N = 1–0 and N₂H⁺ J = 1–0 maps, which cover most high-extinction regions with in L1688 and L1689, including clumps in earlier studies (e.g., Di Francesco et al. 2004; André et al. 2007).

In Section 2, we present the C₂H and N₂H⁺ observations, data reduction, and the Infrared Astronomy Satellite (IRAS) dust temperature map. The calculation of the C₂H-to-N₂H⁺ abundance ratio ([C₂H]/[N₂H⁺]) is presented in Section 3. Our chemical model and interpretation are given in Section 4. Comparative studies of C₂H and N₂H⁺ in terms of centroid velocity, line width, and [C₂H]/[N₂H⁺] are in Section 5. Conclusions are presented in Section 6. Details of critical density estimation and column density calculation are given in the appendices.

2.1. The C₂H and N₂H⁺ Data

The ρ-Oph MCC was mapped with the Delingha 13.7 m telescope of Qinghai Station,⁷ Purple Mountain Observatory (PMO), Chinese Academy of Sciences. The observations were carried out in position switching mode in 2012 December and 2013 January, March, and April.

At Delingha, the ρ-Oph MCC stays above 20° elevation with a maximum of 28° for approximately 5 hr per day. To obtain low system temperatures, we only took data for 2 hr around maximum source elevation.

The 3 × 3 Superconductor-Insulator-Superconductor receiver array of the Delingha telescope (Shan et al. 2012; Zuo et al. 2011) can observe 85–115 GHz. The backend consists of 18 fast Fourier transform spectrometers, each with 16,384 channels. The backend operates in two modes, a 200 MHz bandwidth with 12.2 kHz channel width and a 1000 MHz bandwidth with 61.0 kHz channel width.

As a first test observation, we performed on the fly (OTF) mapping of the Oph-A region with a 200 MHz bandwidth (Δv ∼ 0.05 km s⁻¹) to see whether the velocity resolution in the 1000 MHz bandwidth (Δv ∼ 0.20 km s⁻¹) is suitable for fitting the centroid velocity and line width. The test observation covered a 15′ × 15′ square region centered on the position of Oph-A with the highest Two Micron All Sky Survey (2MASS) extinction. Hyperfine structure fitting was done. The data were then smoothed to a 0.20 km s⁻¹ velocity resolution before another round of fitting was performed. A comparison between these two fitting results showed that the hyperfine structure fitting of spectra with strong emission is affected little by this smoothing, and thus the 0.20 km s⁻¹ velocity resolution is sufficient for observing N₂H⁺ in the ρ-Oph MCC and analysis using hyperfine fitting. For example, the original spectrum with a ∼4 K peak at the position 16^h26^m265, −24°24′09″ (J2000) showed a centroid velocity of 3.60 ± 0.01 km s⁻¹, a line width of 0.60 ± 0.03 km s⁻¹, an optical depth of 0.73 ± 0.2, and an excitation temperature of K from hyperfine fitting. After smoothing to 0.2 km s⁻¹, the centroid velocity was 3.60 ± 0.01 km s⁻¹, the line width was 0.60 ± 0.03 km s⁻¹, the optical depth was 0.73 ± 0.25, and the excitation temperature was K. In this example, only the uncertainties became slightly larger. Compared with the crowded N₂H⁺ hyperfine structures, the C₂H spectra have six well-separated hyperfine components. Observing these six components requires a much lower velocity resolution in a wider bandwidth. The 1000 MHz mode provides enough velocity resolution for C₂H hyperfine structure fitting. For spectra with weak emission, however, we performed hyperfine fitting and found that the results were affected by the smoothing. In the following, we only applied the resulting opacity correction to N₂H⁺ spectra with an integrated intensity of 3 K km s⁻¹ or higher and to C₂H spectra above 2 K km s⁻¹. Low opacities were assumed for spectra with weak emission.

The COordinated Molecular Probe Line Extinction Thermal Emission Survey of Star-forming Regions (Ridge et al. 2006, hereafter COMPLETE) provides the 2MASS NICER extinction map and IRAS dust temperature map of the ρ-Oph MCC. These maps were used for observation region selection. In L1688, six 30′ × 30′ areas were selected, covering most regions with extinction . L1689 was covered by four areas next to the observation region of L1688. Reference positions were chosen from the COMPLETE ρ-Oph MCC ¹²CO and ¹³CO maps. Tables 1 and 2 show the details of observation.

Table 1.  Observation Information of 10 30′ × 30′ OTF Scans

No.	Center R.A.	Center Decl.	Ref. Position	Ref. Position	Scan Rate	Sample Interval	Scan Time	Total Scan Time
	(J2000)	(J2000)	R.A. (J2000)	Decl. (J2000)	(arcsec s−1)	(s)	(minutes)	(minutes × repeats)
01	16h26m16s	−24°20′54″	16h29m29s	−24°29′31″	50	0.3	85	85 ×  4
02	16h28m16s	−24°20′54″	16h30m32s	−24°07′28″	50	0.3	85	85 ×  4
03	16h24m16s	−24°20′54″	16h26m18s	−23°46′10″	50	0.3	85	85 ×  4
04	16h26m16s	−24°50′54″	16h30m32s	−24°07′28″	50	0.3	85	85 ×  4
05	16h28m16s	−24°50′54″	16h30m32s	−24°07′28″	75	0.2	45	60 ×  6
06	16h24m16s	−24°50′54″	16h26m18s	−23°46′10″	75	0.2	45	60 ×  6
07	16h30m16s	−24°30′00″	16h30m16s	−24°18′00″	75	0.2	45	60 ×  6
08	16h32m16s	−24°30′00″	16h32m16s	−24°18′00″	75	0.2	45	60 ×  6
09	16h30m16s	−25°00′00″	16h30m16s	−24°18′00″	75	0.2	45	60 ×  6
10	16h32m16s	−25°00′00″	16h32m16s	−24°18′00″	75	0.2	45	60 ×  6

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Table 2.  Frequency Settings of the Observation

Lines	Rest Frequency	Relative Intensity
	(MHz)
C2H N = 1–0 J = – F = 1–1	87284.38	4.25
C2H N = 1–0 J = – F = 2–1	87317.05	41.67
C2H N = 1–0 J = – F = 1–0	87328.70	20.75
C2H N = 1–0 J = – F = 1–1	87402.10	20.75
C2H N = 1–0 J = – F = 0–1	87407.23	8.33
C2H N = 1–0 J = – F = 1–0	87446.42	4.25
N2H+ J = 1–0 F1 = 1–1 F = 0–1	93171.6086	0.33334
N2H+ J = 1–0 F1 = 1–1 F = 2–1	93171.9054	1.66667
N2H+ J = 1–0 F1 = 1–1 F = 1–1	93172.0403	1.00000
N2H+ J = 1–0 F1 = 2–1 F = 2–1	93173.4675	1.66667
N2H+ J = 1–0 F1 = 2–1 F = 3–1	93173.7643	2.33333
N2H+ J = 1–0 F1 = 2–1 F = 1–1	93173.9546	1.00000
N2H+ J = 1–0 F1 = 0–1 F = 1–1	93176.2527	1.00000

Note. The frequencies and relative intensities of C₂H are from Tucker et al. (1974); the frequencies and relative intensities of N₂H⁺ are from Keto & Rybicki (2010). C₂H is in the lower sideband, and N₂H⁺ is in the upper sideband.

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The original OTF data were regridded with a pixel size of . In the regridding procedure, spectra with extremely bad baselines or high rms values were removed, since these spectra may be caused by hardware errors. A 60% main-beam efficiency was used in the calibration. The data were uploaded to the Millimeter Wave Radio Astronomy Database.^{8 ,9 ,10} The astronomical software package Gildas/CLASS¹¹ was used for further reduction.

Given the relatively low elevation of the observations, the spectral baselines are complex. A fifth-order polynomial was subtracted from each C₂H spectrum. For N₂H⁺ data, a third-order polynomial was used. The OTF data were combined into one file for regridding. In regridding, the velocity resolutions are 0.21 km s⁻¹ for C₂H spectra and 0.20 km s⁻¹ for N₂H⁺ spectra. After regridding, only spectra with rms values less than 0.5 K () were selected. For C₂H spectra, velocity ranges of [−50, −40], [−30, 0], and [10, 50] km s⁻¹ were used for rms calculation. For N₂H⁺ spectra, the velocity ranges used were [−50, −8] and [15, 50] km s⁻¹. Though the rms values of the spectra near map edges were still less than 0.5 K (), the baselines of these spectra were too complex for polynomial fitting. Hence, spectra near map edges were rejected.

The integrated intensity maps of C₂H and N₂H⁺ are shown in Figure 1. For C₂H, to present a uniform integrated intensity map, we selected two strongest hyperfine components that were widely detected. The selected velocity ranges for these two components in spectra were [−39.1, −35.1] and¹² [1.9, 5.0] km s⁻¹. When calculating the N₂H⁺ integrated intensity, we selected three hyperfine components, corresponding to a velocity range of¹³ [2.0, 6.0] km s⁻¹. The rms values of C₂H and N₂H⁺ integrated intensity maps are 0.8 and 0.4 K km s⁻¹, respectively. A 5 × 5 pixel boxcar smoothing was then performed to the integrated intensity maps for drawing contours.

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The weather and the relatively low elevation are the main reasons for the variation of rms values. When the observation region is close to its maximum elevation and the weather is good, a typical system temperature is between 160 and 170 K, which increases to 200 K or higher under bad weather or low elevation. The variation of the system temperature caused an approximately ±0.1 K km s⁻¹ variation of the rms values of the smoothed C₂H and N₂H⁺ spectra in different regions. The actual sky coverage of each 30′ × 30′ OTF map is a little larger. Around the edges with overlapped scans, we averaged both data sets to lower the rms values.

2.2. IRAS Temperature Map from COMPLETE

3.1. Integrated Intensity Maps of C₂H and N₂H⁺

3.2. Column Densities and C₂H-to-N₂H⁺ Ratios

Some positions in L1688 and L1689 have weak C₂H or N₂H⁺ emission. The integrated intensities at these positions are significant, but the spectra themselves are too weak for hyperfine fitting, e.g., the C₂H spectrum in L1689NW (see Table 3). We assumed optically thin emission and followed the recipe below (see, e.g., Li 2002).

Table 3.  Mean Chemical Ages

Density NH	L1688 Mean Age	L1689 Mean Age
(cm−3)	(yr)	(yr)

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The column density of an upper-level population of an optically thin transition is

where is the integrated intensity consisting of all six hyperfine components of the C₂H spectra or all seven hyperfine components of the N₂H⁺ spectra. In addition, k is the Boltzmann constant, h is the Planck constant, ν is the frequency, and c is the speed of light. The total column density is

where F_u, , and F_b are the level correction factor, the opacity correction factor, and the correction factor for the background, respectively. In this calculation, we only calculated the C₂H column densities at positions with a C₂H integrated intensity of 1.5 K km s⁻¹ or higher. The threshold for N₂H⁺ column density calculation was 2 K km s⁻¹. The details can be found in Appendix B.

We applied the opacity correction to N₂H⁺ spectra with integrated intensities over [−8, 15] km s⁻¹ greater than 3.0 K km s⁻¹ and optical depths between 0.5 and 5.0. If the fitted optical depth was below 0.5 or above 5.0, however, noise may have made the result unreliable by skewing the peak temperature ratios between hyperfine components, depending on the signal-to-noise ratios of the components. Ignoring an optical depth of 0.5 will lead to an underestimation of 27%. A similar opacity correction was also done for C₂H spectra but for the spectra with integrated intensity greater than 1.0 K km s⁻¹.

In our C₂H and N₂H⁺ column density calculations, using IRAS dust temperatures as excitation temperatures of C₂H and N₂H⁺ can cause large uncertainty. For example, the original IRAS temperatures have an pixel size and a 43 spatial resolution (Schnee et al. 2005). This spatial resolution is approximately four times lower than our molecular line data. The IRAS temperatures come from far-infrared fluxes and may overestimate temperatures in compact, dense cores. Using the Herschel Space Observatory 160 and 250 μm dust continuum maps and James Clerk Maxwell Telescope 450 and 850 μm dust continuum maps, Pattle et al. (2015) derived the mean line-of-sight dust temperatures of sources in the ρ-Oph MCC by fitting the spectral energy distribution with a modified blackbody distribution. These temperatures are lower than the IRAS temperatures. Lower temperatures result in lower C₂H and N₂H⁺ column densities. For example, in the location of the source SM1 (see Motte et al. 1998; 16^h26^m2736, −24°23′528, J2000), the C₂H and N₂H⁺ integrated intensities are 12.3 and 12.0 K km s⁻¹, respectively. Using the IRAS temperature (37 K), the column densities of C₂H and N₂H⁺ are 6.8 × 10¹³ cm⁻² and 3.1 × 10¹² cm⁻², respectively. These two column densities are both 4.6 times higher than those estimated with the temperature (17.2 ± 0.6 K) in the same location in Pattle et al. (2015).

The ratio between the column densities, however, is impervious to the temperature uncertainty for cloud conditions relevant here. We therefore use [C₂H]/[N₂H⁺] to explore the chemical evolution and physical properties of ρ-Oph MCC. [C₂H]/[N₂H⁺] represents both the column density ratio of C₂H and N₂H⁺ and the abundance ratio of C₂H and N₂H⁺. Figure 2 shows an example of [C₂H]/[N₂H⁺] changing with different temperatures when the integrated intensities of C₂H or N₂H⁺ are all assumed to be 5 K km s⁻¹. The [C₂H]/[N₂H⁺] values under different temperatures in Figure 2 come from the column density and abundance ratio calculations that are mentioned above and in Appendix A. For T_ex > 7 K, the [C₂H]/[N₂H⁺] increases only slightly with increasing temperature. Other than Figure 2, with different assumed C₂H or N₂H⁺ integrated intensity, we also calculated the [C₂H]/[N₂H⁺] values in different temperatures. The weak dependence of the abundance ratio on temperatures remains valid. In our observation region, all the sources in Pattle et al. (2015) have temperatures of 8.2 K (8.2 ± 0.3 K for 88N SMM 1) or higher. The [C₂H]/[N₂H⁺] is thus a robust probe of cloud conditions.

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Figure 3 shows the distribution of the measured abundance ratio in L1688 and L1689, which varies between 5 and 110. The differences in the distribution of the abundance ratio will be mentioned in Section 5.

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To interpret our data, we modeled [C₂H]/[N₂H⁺] evolution with both the pure gas and gas-grain chemical networks.

The simulation was performed using Nautilus, a gas-grain chemical code (Hersant et al. 2009), along with the gas-grain chemical reaction network from Hincelin et al. (2011, 2013). An electronic version of the network is available in the KInetic Database for Astrochemistry database¹⁷  (Wakelam et al. 2012). Low metal abundances, as in Semenov et al. (2010), had been assumed for the initial gas phase abundances. Initially, all species were in the gas phase. The cosmic-ray ionization rate was set to s⁻¹. The size of dust grains was assumed to be the same, with a radius of 0.1 μm. Both pure gas phase and gas-grain network simulations were carried out for comparative studies. Four different temperatures (10, 20, 30, and 40 K) and four different densities (5 × 10³, 2 × 10⁴, 1 × 10⁵, and 1 × 10⁶ cm⁻³) were applied in our simulations.

Our simulations are purely chemistry models without dynamic evolution. This is sufficient for examining the cloud chemistry as long as the timescale for density variation is much longer than the chemical relaxation time, which is a reasonable assumption for the densities concerned here.

We compared the pure gas model and the gas-grain model. Figure 4 shows the C₂H abundance, N₂H⁺ abundance, and [C₂H]/[N₂H⁺] as functions of time in a pure gas phase model and a gas-grain model with cm⁻³ and cm⁻³, respectively. The temperature in both models is fixed at 10 K. The most pronounced difference between these two models shows up in the abundance of C₂H. It remains constant or rises slightly in gas-grain models, whereas in pure gas models, it plummets after approximately 10⁵–10⁶ yr. The major destruction route of C₂H is reaction with O to form CO, while its major production route is the dissociative recombination of C₂H and C₂H. Since only neutral species are allowed to accrete on grain surfaces in our gas-grain reaction network simulation, the abundances of C₂H and C₂H in the gas-grain model are not all that different from those in the pure gas model. O can be frozen on grain surfaces as H₂O and be largely depleted from the gas phase in later stages of the gas-grain model. C₂H experiences a lower destruction rate in gas-grain networks. Thus, the C₂H abundance in late stages is sensitive to the existence of dust, mainly due to depletion of O onto grain surfaces. For N₂H⁺, its major production route is the reaction

while its major destruction route is reaction with CO (e.g., Turner 1995; Vasyunina et al. 2012). The depletion temperatures of both N₂ and CO are similar and lower than 20 K. Such temperatures lead to the result that the trend of the change of N₂H⁺ abundance is little affected by adding the dust into the model. Thus, the N₂H⁺ abundance keeps rising in both the pure gas and gas-grain models, until t ∼ 10⁶ yr. After t ∼ 10⁶ yr, the decreasing of N₂H⁺ abundance is the result of the evolution of CO, N₂, and . Comparing the models with and without dust, the [C₂H]/[N₂H⁺] values are insensitive to the existence of dust, also until t ∼ 10⁶ yr. We use the gas-grain model in subsequent analyses. It is also worth noting that there is no significant difference seen in the evolution of C₂H abundances or N₂H⁺ abundances between two densities that differ by a factor of 5.

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We ran the model with different extinctions and UV field strengths to see whether the variations of these two parameters affect the chemical evolution. In the model, both the extinction and the UV field affect the photodissociation rate coefficients (PRCs), which are proportional to . The factor γ is a value around 2, depending on the balance between molecular photon absorption and differential dust absorption, A_V is the extinction, G₀ = 1.6 × 10⁻³ erg s⁻¹ cm⁻² is the unit of UV field, and F₀ is the scaling factor of the UV field. We ran the chemical model with two sets of parameter grids: (1) F₀ = 1, density cm⁻³, temperature T = 30 K, and extinctions A_V = 1, 5, 8, and 20 mag; and (2) F₀ = 0.01, 100, 1000, and 10,000, density cm⁻³, temperature T = 30 K, and extinction A_V =  10 mag.

Different parts of molecular clouds may be exposed to different external far-UV (FUV) radiation fields, and different positions in molecular clouds have different extinctions. These variations produce different chemical products in different parts of a molecular cloud. The PRCs for cosmic-ray-induced secondary photons, however, are independent of A_V. For visual extinction A_V = 10 mag, the PRCs due to secondary photons are more than three orders of magnitude larger than those of the external FUV radiation in our reaction network (Hincelin et al. 2011, 2013). Thus, the variation of A_V and the scaling factor do not affect the chemical evolution much in our models if A_V is larger than 10 mag. In our map, the extinction values in the place with C₂H or N₂H⁺ emissions are relatively high (e.g., 10 mag). We expect that the variation of extinction and external FUV radiation fields has little effect on the chemical evolution of dense molecular clouds.

The extinctions at almost all the positions of the N₂H⁺ clumps fulfill A_V ≥ 10 mag. Figure 5 shows the evolutionary trends of the abundances of C₂H or N₂H⁺ and their abundance ratio under different extinctions. We see that the C₂H abundance, the N₂H⁺ abundance, and the [C₂H]/[N₂H⁺] all hardly differ for A_V ≥ 8 mag, and the C₂H abundance is more easily affected by the different extinctions. The evolution of [C₂H]/[N₂H⁺] with time differs substantially when A_V = 1 mag. Therefore, we ignored differences in extinctions and used 10 mag as the basis for further model simulations.

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Figure 6 shows the evolutionary trends of the abundances of C₂H or N₂H⁺ and the abundance ratio under different external UV fields. With extinction 10 mag, four models with F₀ values of 0.01, 100, 1000, and 10,000 are shown. From the early evolutionary time of 10³ yr to ∼5 × 10⁵ yr for C₂H or from 10³ yr to ∼2 × 10⁶ yr for N₂H⁺, the model calculation results were little affected by the different F₀ values from 0.01 to 10,000. From the later evolutionary time of ∼5 × 10⁵ yr to 10⁷ yr for C₂H or from ∼2 × 10⁶ yr to 10⁷ yr for N₂H⁺, the C₂H and N₂H⁺ abundances only slightly increase by factors of ∼10 and 5, respectively. As seen from Figure 5, when extinctions are high enough, both C₂H and N₂H⁺ abundances are barely affected by the external UV field. For a cloud with extinction A_V = 10 mag, however, the chemical evolution will be affected by the external UV field when F₀ reaches 1000. Liseau et al. (1999) estimated the UV field in the ρ-Oph MCC by sampling 33 positions, including Oph-A, Oph-B1, Oph-B2, Oph-C1, and Oph-D. The F₀ values (F in Liseau et al. 1999) in these 33 positions are all below 150. Therefore, we ignored the differences in external UV field and used F₀ = 1 for further simulations.

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Figure 7 shows the abundances of C₂H and N₂H⁺ and [C₂H]/[N₂H⁺] as functions of time for different temperatures and densities in the gas-grain model. Here, we find that the C₂H abundance depends strongly on temperature and density. These dependencies can be explained from the formation and destruction routes of C₂H. The abundance of plays a crucial role in the production of C₂H. Protonated methane is produced by radiative association of C⁺ with H₂ and hydration addition reactions. All carbon atoms are in the form of in the beginning of the model and are gradually converted to species such as CO or protonated methane. Thus, the abundance of C₂H increases in the beginning when abundances of methane or species such as CO and OH also increase. Atomic O accretes on grain surfaces and reacts with other species, and thus is depleted from the gas phase. At 10 K, when most gas phase O atoms are depleted after approximately 2 × 10⁶ yr, C₂H can hardly be destroyed and the abundance of C₂H increases again. Indeed, our simulation results bear out this trend. O starts to desorb from grain surfaces at around 15 K. As temperature increases, it becomes increasingly difficult to freeze atomic O on grain surfaces. Hence, C₂H abundance decreases with increasing temperature at late stages. Density also affects C₂H abundance. The dependence of C₂H abundance on density is due to both O and C⁺ depleting more quickly from the gas phase when density increases. The desorption of O at around 15 K is also the reason that the 20 K models differ so dramatically from those at lower and higher temperatures.

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The situation is different for the changes of N₂H⁺ abundances. The major production and destruction routes of N₂H⁺ are related to N₂ and CO, respectively. The desorption energies are 1000 K for N₂ and 1150 K for CO. Both N₂ and CO can be depleted from the gas phase when the temperature is lower than 20 K. The relative abundance of these two species determines the abundance of N₂H⁺. The N₂H⁺ abundance is affected little by temperature and volume density variations.

Combining the evolutions of both species, the [C₂H]/[N₂H⁺] decreases with time between 10² and 10⁶ yr for a volume density of 5.0 × 10³ cm⁻³ and from the beginning to approximately 2.0 × 10⁴ yr for a volume density of 1.0 × 10⁶ cm⁻³. This evolutionary trend is little affected by temperature for T_k < 40 K. At high density (n_H > 10⁵ cm⁻³), the time it takes for the abundance ratio to drop at least one order of magnitude becomes less than the typical dynamical time. For example, a 0.1 pc clump at the distance of ρ-Oph MCC spans about 29, implying a turbulence crossing time of ∼10⁵ yr for a characteristic 1 km s⁻¹ turbulence velocity.

Concerning the observed [C₂H]/[N₂H⁺] values, the trends for all three modeled densities are consistently decreasing with time. The possible volume densities in the positions with the observed [C₂H]/[N₂H⁺] values are 1.0 × 10⁵ cm⁻³ or higher (see, e.g., Motte et al. 1998; Young et al. 2006). In such conditions, the [C₂H]/[N₂H⁺] values are more sensitive to the volume densities than to the chemical ages, as we describe below.

5.1. [CH]/[NH as a Tracer of Chemical Evolution and Volume Density

According to our chemical model, the C₂H abundance decreases while the N₂H⁺ abundance increases with time (see Figure 7). Therefore, [CH]/[NH can be used as a tracer of chemical evolution. Figure 8 shows a comparison between the observed [CH]/[NH values in L1688 or L1689 and models of different volume density and temperature. The mean estimated chemical ages of the observed [CH]/[NH values in L1688 (yellow squares in Figure 8) or L1689 (purple triangles in Figure 8) are listed in Table 3. Linear interpolation is used to include our observed [CH]/[NH values with the modeled data. The details of the interpolation can be found in Appendix C. Though the IRAS temperatures could also bring uncertainties here, as shown in Figure 8, the uncertainties brought about by temperature are insignificant.

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Figure 8 shows that both chemical ages and volume densities affect the [CH]/[NH values. In the first and second panels from the left of Figure 8, the chemical ages of the observed [CH]/[NH are both around 10⁵ yr. In such a volume density range, the estimated chemical ages seem to be affected more by the [C₂H]/[N₂H⁺] values than by different volume densities. From left to right, in the third and fourth panels in Figure 8, for these same observed [CH]/[NH values, the estimated ages decrease to ∼5 × 10⁴ yr and ∼1 × 10⁴ yr, respectively. For the same [CH]/[NH values and in high volume density gas, Figure 8 shows that higher volume densities result in younger estimated ages of the same observed abundance ratios, and both the [C₂H]/[N₂H⁺] values and volume densities affect the chemical ages at similar levels. In addition, the relationship between the time/chemical age and [C₂H]/[N₂H⁺] steepens when the volume density increases. The mean ages in Table 3 corresponding to the observed [CH]/[NH values in L1688 or L1689 decrease more when the volume density increases to higher volume density ranges. For example, when the volume density increases 20 times from 5.0 × 10³ cm⁻³ to 1 × 10⁵ cm⁻³, the derived chemical mean age corresponding to the observed abundance values in L1688 decreases from 1.4 × 10⁵ yr to one-third of this value. When the volume density increases 10 times further from 10⁵ cm⁻³, the derived chemical mean age corresponding to the observed abundance values in L1688 decreases from 4.4 × 10⁴ yr to one-fifth of that value.

Figure 9 in another way shows the relationship between [C₂H]/[N₂H⁺], the chemical age, and the volume density. The top panel shows, using models, how [C₂H]/[N₂H⁺] is affected by chemical age. When the volume density is low, e.g., 5 × 10³ cm⁻³, the abundance ratios at different ages look distinct in the panel. When volume density is high, e.g., 10⁶ cm⁻³, the abundance ratios at different ages seem to be indistinct in the panel. Note that the axis for chemical age (Y-axis) is linear to reflect the chemical age decreasing when the volume density is increasing; thus, the total range variation at high density is hard to recognize in such a panel, although they may still be significant. The bottom panel shows how [C₂H]/[N₂H⁺] is affected by volume density. When the volume density is 10⁵ cm⁻³ or lower, chemical age changes of one order of magnitude higher or lower result in [C₂H]/[N₂H⁺] changes of only ∼1.5 orders of magnitude lower or higher, respectively. When the volume density is 10⁶ cm⁻³, however, [C₂H]/[N₂H⁺] drops roughly seven orders of magnitude lower when the chemical age only increases one order of magnitude higher. It appears that 10⁵ cm⁻³ is roughly the critical value between the regimes where [C₂H]/[N₂H⁺] is sensitive to volume density or not.

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From our observations, [C₂H]/[N₂H⁺] varies from 5 to 110. In the top panel of Figure 9, two nonvertical dotted lines show the results of the [C₂H]/[N₂H⁺] values of 5 and 110. Motte et al. (1998) evaluated the volume densities of cores in L1688 with their 1.3 mm dust continuum map, finding values between 1.4 × 10⁵ cm⁻³ and 8.0 × 10⁶ cm⁻³. These two volume densities are also shown in the top panel of Figure 9 with two vertical dotted lines. For comparison, Young et al. (2006) evaluated the volume densities of cores with their 1.1 mm dust continuum map, which included both L1688 and L1689. Using a method different from the one used by Motte et al. (1998), they found 48 cores in their map. Among these 48 cores, 41 are in L1688 or L1689, and these have volume densities between 1 × 10⁵ cm⁻³ and 3 × 10⁷ cm⁻³, which do not differ too much from the results of Motte et al. (1998). With such limits of observed [C₂H]/[N₂H⁺] values and volume densities (the region with dotted line boundaries), for the overall values of [C₂H]/[N₂H⁺], they are likely the results from a combination of chemical age and volume density.

In the integrated intensity map (see Figure 1), the C₂H emission is more extended in the relatively low density regions (e.g., the C₂H ring) in L1688 than in L1689. The emission distribution difference can be explained by L1688 having chemically younger gas in relatively less dense regions that are without detectable N₂H⁺ emission. With no N₂H⁺ detection, the values of [C₂H]/[N₂H⁺] indicate that these regions have young chemical ages. In the abundance ratio map (see Figure 3), the observed [C₂H]/[N₂H⁺] values in L1688 tend to be higher than those in L1689. For any given clump, the abundance ratio in the inner region tends to be lower than in the outer region. For the overall distribution of [C₂H]/[N₂H⁺], they are also likely the results of time evolution, accelerated at higher densities.

5.2. Centroid Velocities and Line Width Comparison between C₂H and N₂H⁺

We selected 14 positions of C₂H or N₂H⁺ intensity peaks in our maps and analyzed their spectra. Figure 10 shows spectra from these positions. The locations of these positions are represented by plus signs and crosses in Figure 1. The coordinates, rms values, centroid velocities with errors, and line widths with errors of these spectra are listed in Table 4. We ignored Oph-E, because it is only 5′ south of Oph-C and connected in N₂H⁺ emission. The eight N₂H⁺ peaks in clumps of L1688 (Oph-A through F and H) are all co-located with dust continuum sources (e.g., dust clumps in Motte et al. 1998), i.e., within or less.

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Table 4.  Hyperfine Line Fitting Result

Source	Molecular	R.A.	Decl.	rms	V	Δv	Offset from N2H+ Peak	Comment
		(J2000)	(J2000)	(K)	(km s−1)	(km s−1)	(arcsec and direction)a
Oph-A	N2H+	16h26m284	−24°23′42″	0.27	3.58(0.02)	0.79(0.05)	E	SM1b in Oph-A
	C2H	16h26m282	−24°23′42″	0.26	3.39(0.03)	1.11(0.05)
Oph-B1	N2H+	16h27m124	−24°30′12″	0.26	3.75(0.02)	0.96(0.08)	W	MM3b in Oph-B1
	C2H	16h27m122	−24°30′12″	0.27	3.61(0.05)	1.18(0.20)
Oph-B2	N2H+	16h27m277	−24°27′12″	0.24	4.03(0.01)	1.07(0.03)		MM8b in Oph-B2
	C2H	16h27m275	−24°27′12″	0.27	4.02(0.05)	1.06(0.12)
Oph-C	N2H+	16h27m014	−24°34′42″	0.26	3.72(0.02)	0.56(0.04)	SE	MM6b in Oph-C
	C2H	16h27m012	−24°34′42″	0.28	3.78(0.03)	0.71(0.04)
Oph-D	N2H+	16h28m291	−24°19′12″	0.28	3.47(0.02)c	0.44(0.04)c	SE	MM2b in Oph-D
	C2H	16h28m289	−24°19′12″	0.31	3.49(0.02)	0.71(0.06)
Oph-F	N2H+	16h27m234	−24°40′42″	0.26	4.13(0.03)	0.82(0.08)	W	MM2b in Oph-F
	C2H	16h27m233	−24°40′42″	0.29	4.23(0.04)	1.09(0.12)
Oph-H	N2H+	16h27m585	−24°33′12″	0.30	4.18(0.03)	0.63(0.07)	S	MM1d in Oph-H
	C2H	16h27m584	−24°33′12″	0.29	4.24(0.02)	0.58(0.02)
L1688-C2H-1	C2H	16h25m399	−24°30′42″	0.24	3.56(0.08)	2.08(0.18)
L1688-C2H-2	C2H	16h26m040	−24°42′42″	0.27	3.45(0.10)	1.37(0.10)
L1688-C2H-3	C2H	16h24m492	−24°41′42″	0.35	3.00(0.15)	2.46(0.38)
L1688-C2H-4	C2H	16h25m133	−24°51′42″	0.31	3.28(0.10)	1.59(0.29)
L1689W	N2H+	16h31m391	−24°49′42″	0.31	4.46(0.02)	0.59(0.04)	SW	SMM16e in L1689-West
	C2H	16h31m390	−24°49′42″	0.33	4.57(0.09)	0.79(0.19)
L1689S	N2H+	16h31m571	−24°57′42″	0.27	4.41(0.02)	0.62(0.03)	SW	SMM8e in L1689-South
	C2H	16h31m570	−24°57′42″	0.32	4.47(0.05)	1.20(0.16)
L1689NW	N2H+	16h32m288	−24°28′42″	0.40	3.76(0.02)	0.62(0.04)	S	SMM19e in L1689-NorthWest
	C2H	16h32m286	−24°28′42″	0.37	f	f		Weak C2H emission

Notes. CLASS fitting errors of centroid velocities and line widths are shown in the brackets behind the numbers.

^aThe numbers in this column are the angular distances between the positions of the dust sources and the peaks of the N₂H⁺ emission in the integrated intensity map. ^b1.3 mm dust continuum sources in Motte et al. (1998). ^cThe Gildas/CLASS reports that it is an optimistic fitting. ^d850 μm dust continuum sources in Johnstone et al. (2004). ^e850 μm dust continuum sources in Nutter et al. (2006). ^fThe line is too weak to fit the hyperfine structures.

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L1688-C₂H-1, L1688-C₂H-2, L1688-C₂H-3, and L1688-C₂H-4 are located in the molecular ring Oph-RingSW. Their line widths are larger than those at other C₂H peak positions in L1688 or L1689. The centroid velocities of L1688-C₂H-3 and L1688-C₂H-4 are smaller than those of any other C₂H spectra at the selected positions. In these four positions, no N₂H⁺ emission is detected. With small centroid velocities and large line widths, the C₂H spectra at these four positions indicate that the molecular ring is not a part of the denser regions of L1688. In L1689NW, the C₂H line is detectable but weak. For the other nine selected sources with both C₂H and N₂H⁺ spectra, the centroid velocity differences between C₂H and N₂H⁺ spectra are smaller than 0.19 km s⁻¹, while the velocity resolutions of C₂H and N₂H⁺ spectra are 0.21 and 0.20 km s⁻¹, respectively. In these nine positions, differences in line widths between C₂H and N₂H⁺ spectra are smaller than 0.27 km s⁻¹, except for those of Oph-A (0.32 km s⁻¹) and L1689S (0.58 km s⁻¹). Such similar centroid velocities and line widths indicate that C₂H N = 1–0 and N₂H⁺ J = 1–0 trace similar gas in our observation region. Although the differences in both the centroid velocities and the line widths are similar to the channel width, from the fitting errors, the line widths of C₂H are significantly larger than those of N₂H⁺ in almost all cases, except in the positions of Oph-B2 and Oph-H. This behavior confirms our understanding that C₂H traces more extended and relatively more diffuse gas than N₂H⁺ despite their transitions having similar critical densities.

We have mapped 2.5 square degrees of the ρ-Oph MCC in the emission of the C₂H N = 1–0 and N₂H⁺ J = 1–0 transitions. The observations are the first large-scale C₂H and N₂H⁺ maps of the ρ-Oph MCC. Our main results are as follows:

• (1)   
  Four N₂H⁺ clumps and one C₂H ring are identified and named here as Oph-J, Oph-K, Oph-L, Oph-M, and Oph-RingSW, respectively. Oph-M and the molecular ring Oph-RingSW are identified for the first time. Eight N₂H⁺ peaks in clumps of L1688 are all co-located with dust peaks within one beam (60″).
• (2)  
  The observed C₂H-to-N₂H⁺ ratio, [C₂H]/[N₂H⁺], varies from 5 to 110 and is little affected by the uncertainties of derived temperatures. [C₂H]/[N₂H⁺] in L1688 tends to be higher than that in L1689. For any given clump, [C₂H]/[N₂H⁺] in the inner region tends to be lower than in the outer region.
• (3)   
  We modeled C₂H and N₂H⁺ abundances with 1D chemical models. The main difference between pure gas and gas-grain models is the dramatic drop in C₂H abundance in later stages of pure gas models due to destruction by reactions with CO, which will be depleted with the presence of dust.
• (4)  
  The diverging trends of abundances of C₂H and N₂H⁺ in our gas-grain models make [C₂H]/[N₂H⁺] a signpost of cloud conditions. For clouds with densities n_H ≤ 10⁵ cm⁻³, [C₂H]/[N₂H⁺] drops with time by ∼2 orders of magnitude. In the gas with a density of n_H ≥ 10⁵ cm⁻³ or higher, the [C₂H]/[N₂H⁺] becomes more sensitive to density.
• (5)   
  The observed [C₂H]/[N₂H⁺] values are the results of time evolution, accelerated at higher densities. For the relatively low density regions in L1688 where only C₂H emission was detected, the gas should be chemically younger.

This work is supported by National Basic Research Program of China (973 program) grant Nos. 2012CB821800 and 2015CB857101, National Science Foundation of China grant no. 11373038, and CAS International Partnership Program, grant No. 114A11KYSB20160008. We thank the anonymous referee for his or her help. This paper was improved so much with the comments and suggestions. The observation was made with the Delingha 13.7 m telescope of the Qinghai Station of Purple Mountain Observatory (see footnote 6), Chinese Academy of Sciences. We appreciate all the staff members of the observatory for their help during the observation. The telescope and the millimeter wave radio astronomy database are supported by the millimeter and submillimeter wave laboratory of PMO. D.L. acknowledges the sponsorship provided by the Guizhou Normal University and support from the CAS Interdisciplinary Innovation Team. L.Q. is supported by the FAST FELLOWSHIP. The FAST FELLOWSHIP is supported by Special Funding for Advanced Users, budgeted and administrated by Center for Astronomical Mega-Science, Chinese Academy of Sciences (CAMS). We thank Ugo Hincelin for allowing us to use the Nautilus codes. We thank Dr. Juan Li for helping us with the C₂H hyperfine fittings.

The critical density is the density of molecular hydrogen at which the collisional de-excitation rate of the target molecules by hydrogen equals the spontaneous de-excitation rate of the target molecule, i.e.,

We define σ as the collisional cross section, v as the velocity of hydrogen molecules relative to target molecules, and A as the Einstein A coefficient (see next subsection). In addition, the relative velocity of hydrogen molecules can be estimated as

With these definitions, the critical density can be written as

Values of the collisional cross section can be found in BASECOL¹⁸ (Dubernet et al. 2013), which is part of the “Virtual Atomic and Molecular Data Center” (VAMDC). The spectroscopic data of BASECOL come from the CDMS (see footnote 17; the Cologne Database for Molecular Spectroscopy; Müller et al. 2001, 2005) and JPL databases¹⁹ (Pickett et al. 1998). The original data of the C₂H and N₂H⁺ collisional cross sections are from Spielfiedel et al. (2012) and Daniel et al. (2005), respectively. Tables 5 and 6 show the calculation results of critical densities n_cr for C₂H N = 1–0 and N₂H⁺ J = 1–0 hyperfine lines, respectively. The critical density of N₂H⁺ J = 1–0 can also be found in the literature, for example, cm⁻³ (Tanaka et al. 2013).

Table 5.  C₂H N = 1–0 Critical Densities

Transition	Rest Frequency
	(MHz)	(Unit)	(Unit)	(Unit)	(Unit)
C2H N = 1–0 J = – F = 1–1	87,284.156
C2H N = 1–0 J = – F = 2–1	87,316.925
C2H N = 1–0 J = – F = 1–0	87,328.624
C2H N = 1–0 J = – F = 1–1	87,402.004
C2H N = 1–0 J = – F = 0–1	87,407.165
C2H N = 1–0 J = – F = 1–0	87,446.512

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Table 6.  N₂H⁺ J = 1–0 Critical Densities

Transition	Rest Frequency
	(MHz)	(Unit)	(Unit)	(Unit)	(Unit)
N2H+ J = 1–0 F1 = 1–1 F = 0–1	93,171.621
N2H+ J = 1–0 F1 = 1–1 F = 2–1	93,171.917
N2H+ J = 1–0 F1 = 1–1 F = 1–1	93,172.053
N2H+ J = 1–0 F1 = 2–1 F = 2–1	93,173.480
N2H+ J = 1–0 F1 = 2–1 F = 3–1	93,173.777	No Data	No Data	No Data	No Data
N2H+ J = 1–0 F1 = 2–1 F = 1–1	93,173.967
N2H+ J = 1–0 F1 = 0–1 F = 1–1	93,176.265

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A simple solution to the radiative transfer equation is

where the source temperature T_A, the excitation temperature T_x, and the background temperature T_bg are all modified Planck functions.

The relation between the optical depth at the frequency ν and the molecular column density in the upper level, N_u, is

Here, A_ul is the spontaneous radiative decay coefficient, c is the speed of light, h and k are the Planck and Boltzmann constants, respectively. Combining Equations (7) and (8), the N_u⁰, which is the column density for the molecule in the upper level, can be calculated in the Rayleigh–Jeans approximation and optically thin limits as

The total column density can be obtained after some correction (Li 2002; Qian et al. 2012),

where F_u, , and F_b are the level correction factor, the correction factor for opacity, and the correction for the background. F_u can be written as

where Z is the partition function. The correction factor F_RJ corrects for the difference between Planck’s law and the Rayleigh–Jeans limit. This difference is caused by the Taylor series expansion of in the low column density and low-temperature assumptions.

The correction factor F_RJ is

The opacity correction and background radiation correction are analyzed. The correction factor accounting for opacity is (Li 2002)

and the correction factor for background radiation is (Li 2002)

For an optically thin spectrum, . For the correction for spectra with optical depth between 0.5 and 5, we use

Combining Equations (9), (12), (13) (or 15), and (14), the corrected column density of the molecular in the upper level, N_u, is

In our C₂H and N₂H⁺ column density calculation, we assume that the C₂H and N₂H⁺ spectra are all optically thin, the gas is in LTE, and . Temperatures from the IRAS temperature map in COMPLETE are used as T_x, which is the excitation temperature. is the integrated intensity.

Table 7 shows the parameters in Equation (16). The A_ul for C₂H comes from Tucker et al. (1974), and the A_ul for N₂H⁺ comes from Turner (1974). B_e values come from JPL Molecular Spectroscopy (Pickett et al. 1998).

Table 7.  Parameters Used for Calculating Column Density

Molecular	Frequency	Aul	Be
	(GHz)	(s−1)	(MHz)
C2H	87.3486		43,674.534
N2H+	93.1738		46,586.867
Constant	k (J/K)	h (J.S)	c (m s−1)

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The total column density N_tot can be calculated by

where

Here, as an approximation, we use

where the errors will be less than 1%.

These equations are for N₂H⁺. Though the transition of C₂H is N = 1–0, where N is the total angular momentum exclusive of spin, the same calculation is also suitable for C₂H. One similar calculation was presented in Tucker et al. (1974) for the first detection of C₂H in a number of Galactic sources.

The chemical model was run with four temperatures (T₁ = 10 K, T₂ = 20 K, T₃ = 30 K, and T₄ = 40 K) and four volume densities (n₁ = 5.0 × 10³ cm⁻³, n₂ = 5.0 × 10³ cm⁻³, n₃ = 5.0 × 10³ cm⁻³, and n₄ = 5.0 × 10³ cm⁻³). For one temperature and one volume density, we obtained a set of [C₂H]/[N₂H⁺] values (namely, [C₂H]/[N₂H⁺], with x = 1, 2, 3, or 4 the index of the temperature, y = 1, 2, 3, or 4 the index of the volume density, and z the index in the set) and corresponding chemical ages (namely, , with x, y, and z having the same meanings as those in [C₂H]/[N₂H⁺]) from the chemical model. With the four temperatures and four volume densities, we obtained 16 sets of [C₂H]/[N₂H⁺] values and corresponding chemical ages. For an observed [C₂H]/[N₂H⁺] value, namely, [C₂H]/[N₂H⁺]_oph, under the given temperature and volume density, its chemical age can be calculated by linear interpolation as

where K₁ is

Here, is the obtained chemical age, [C₂H]/[N₂H⁺] and [C₂H]/[N₂H⁺] are the abundance values that are the two values closest to the observed [C₂H]/[N₂H⁺]_oph and

and and are the chemical ages corresponding to [C₂H]/[N₂H⁺] and [C₂H]/[N₂H⁺], respectively. For the 16 sets, we have 16 values of in which x = 1, 2, 3, or 4 and y = 1, 2, 3, or 4. If the IRAS dust temperature of [C₂H]/[N₂H⁺]_oph is T_oph, the estimated chemical age of [C₂H]/[N₂H⁺]_oph under one volume density is

where K₂ is

Here, t_oph is the determined chemical age under the given volume density, and are two temperatures that are closest to T_oph and

and and are two chemical ages that come from Equation (20) under the same volume density and temperatures of and , respectively.