THE INITIAL CONDITIONS OF CLUSTERED STAR FORMATION. I. NH₃ OBSERVATIONS OF DENSE CORES IN OPHIUCHUS

Single-dish observations of emission from the NH₃ (J, K) = (1,1) and (2,2) inversion lines, C₂S J_N = 2₁–1₀ and HC₅N J = 9 − 8 in the Ophiuchus Cores were obtained using the 100 m Robert C. Byrd GBT, located near Green Bank, WV, USA. The observations were done in frequency-switching mode, using the GBT K-band (upper) receiver as the front end, and the GBT spectrometer as the back end. This setup allowed the simultaneous observation of all lines in four 50 MHz-wide IFs, each with 8192 spectral channels, giving a frequency resolution of 6.104 kHz, or 0.077 km s⁻¹ at 23.694 GHz.

The data were taken using the GBT's on-the-fly (OTF) mapping mode, using in-band frequency switching with a throw of 4 MHz. In OTF mode, a map is created by having the telescope scan across the target in right ascension (R.A.) at a fixed declination (decl.), or in decl. at a fixed R.A., writing data at a predetermined integration interval. The maps of Oph B1 and B2 were made while scanning only in R.A. at a fixed decl., while for subsequent targets (Oph B3, C, and F) the scanning mode was alternated to avoid artificial striping in the final data cubes. No striping, however, is apparent in the final B1 or B2 images. At the observing frequency of 23 GHz, the telescope beam was approximately 32'' FWHM. Subsequent rows or columns were spaced by 13'' in decl. or R.A. to ensure Nyquist sampling. Scan times were determined to ensure either one or two full maps of the observed region could be made between pointing observations. For all observations, pointing updates were performed on the point source calibrator 1622-254 every 45–60 minutes, with corrections approximately 2''–3''. The average telescope aperture efficiency η_A and main beam efficiency η_mb were 0.59 ± 0.05 and 0.78 ± 0.06 respectively, determined through observations of 3C286 at the start of each shift. The absolute flux accuracy is thus ∼8%. The average elevation of Ophiuchus for all observations was approximately 26°.

System temperatures (T_sys) varied between 48 K and 92 K over the observation dates with an average T_sys ∼ 62 K. Table 2 gives the area mapped in each region and the final rms sensitivity in K (T_MB).

Initial data reduction and calibration were done using the GBTIDL⁶ package. Zenith opacity values for each night were obtained using a local weather model, and the measured main beam efficiency was used to convert the data to units of main beam temperature, T_MB. The two parts of the in-band frequency switched data were aligned and averaged, weighted by the inverse square of their individual T_sys. The data were then converted to AIPS⁷ SDFITS format using the GBT local utility idlToSdfits. In AIPS, the data were combined and gridded using the DBCON and SDGRD procedures. Finally, the data cubes were written to FITS files using FITTP.

Maps of NH₃ (1,1) and (2,2) inversion line emission of the Ophiuchus B1, B2, C, and F Cores were made over two separate observing runs at the ATCA. The ATCA is located near the town of Narrabri in New South Wales, Australia, and consists of six antennas, each 22 m in diameter. Five of the six antennas are movable along the facility's east–west line and small north line, while the sixth antenna is permanently placed along the east–west line at a distance of 6 km from the other antennas. An 8 MHz bandwidth with 1024 channels was used, which provided a spectral resolution of 7.81 kHz (0.1 km s⁻¹ at 23.694 GHz). This configuration enabled in-band frequency switching for the observations. The Oph B Cores were observed over four 9 hr tracks of the array (2004 August 5–8), and the Oph C and F Cores were observed over three 9 hr tracks of the array (2005 May 5–7). Both sets of observations were done with the array in its H168 configuration. This is a compact, hybrid configuration, where three of the movable antennas are placed along the east–west line and two antennas are located on the north spur. Baselines ranged from 61.2 to 184.9 m (∼4.7–14.2 kλ) with five antennas. At 23.7 GHz, these observations provided a primary beam (field of view) of 2'.

Maps were made of the cores using separate pointings spaced by $\Theta _N = (2/\sqrt{3}) \, \lambda \, / \, 2D \sim 1\farcm 1$ at 23.7 GHz for Nyquist sampling on a hexagonal grid. Table 3 gives the number of individual pointings required to cover each Core, the multiple-beam overlap area observed, and the final rms sensitivity achieved toward each Core for both the ATCA and the VLA observations (described below).

Observations cycled through each individual pointing between phase calibrator observations to maximize uv-coverage and minimize phase errors. The phase calibrator, 1622-297, was observed every 20 minutes, and was also used to check pointing every hour. Flux and bandpass calibration observations were performed every shift on the bright continuum sources 1253-055, 1934-638, and 1921-293.

The ATCA data were reduced using the MIRIAD data reduction package (Sault et al. 1995). The data were first flagged to remove target observations unbracketed by phase calibrator measurements, data with poor phase stability or anomalous amplitude measurements. The majority of the data were good, as the weather was stable during the observations. Much of the data from baselines involving the 6 km antenna, however, were flagged due to poor phase stability. The bandpass, gains, and phase calibrations were applied, and the data were then jointly deconvolved. First, the data were transformed from the spatial frequency (u, v)-plane into the image plane using the task INVERT and natural weighting to maximize signal to noise. The data were then deconvolved and restored using the tasks MOSSDI and RESTOR to remove the beam response from the image. MOSSDI uses a Steer–Dewdney–Ito CLEAN algorithm (Steer et al. 1984). The cleaning limit was set at twice the rms noise level in the beam overlap region for each object. Clean boxes were used to avoid cleaning noise in the outer regions with less beam overlap. Applying natural weighting provided a final synthesized beam of ∼8'' × 10'' FWHM.

Maps of NH₃ (1,1) and (2,2) emission were made at the VLA near Socorro, NM, USA over the period of 2007 January 20 through 2007 February 11. Nine observing shifts were allotted to the project in the array's DnC configuration, each 5 hr in duration covering the LST range 14:00–19:00. The DnC configuration is a hybrid of the most compact D configuration with the next most compact C configuration. This setup ensures a more circular beam shape for southern sources like Ophiuchus while retaining the sensitivity of the D configuration.

For these observations, we used a correlator setup with two IFs, each with a bandwidth of 3.125 MHz with 24.414 kHz spectral resolution (0.3 km s⁻¹). While not providing the same high spectral resolution obtained at the ATCA, this setup enabled simultaneous observations of the (1,1) and (2,2) lines, and allowed the main hyperfine component and the two middle satellite components of the (1,1) line to be contained within the band. Mosaic maps were made with Nyquist-spaced (∼10 at 23 GHz) individual pointings. Observations cycled through pointings between phase calibrator observations. Table 3 gives the pointing and rms sensitivity information for each Core.

During these observations, several antennas in the array had been upgraded as part of the Expanded VLA (EVLA) project. Online Doppler tracking at this time was not yet available, and the observations were thus obtained in a fixed frequency mode, with line sky frequencies calculated using the NRAO's online Dopset tool. The observing frequencies were updated frequently, with phase calibrator observations on 1625-254 before and after a frequency change to avoid phase jump problems. Bandpass and absolute flux calibration were done for each shift using observations of 1331+305.

The data were checked, flagged, and calibrated using the NRAO Astronomical Image Processing System (AIPS), following the procedures outlined in the AIPS Cookbook.⁸ In addition, special processing was required to account for differences in bandpass shape and antenna sensitivity between the VLA and EVLA antennas. System temperatures were lacking for EVLA antennas, so the VLA back-end T_sys values were used when reading the data into an AIPS uv-database. A bandpass table was then created from the line data set and applied to a spectrally averaged "channel 0" data set. The normal VLA calibration steps were then followed. The calibrated uv files were then written to FITS format and imported to MIRIAD, where the data were deconvolved and restored. Applying natural weighting and a taper of 8'' × 6'' provided a final synthesized beam of ∼105 × 85 FWHM.

Since none of the antennas in the ATCA and the VLA can act as single radio dishes, there is necessarily an upper limit to the size of structure to which each is sensitive. This upper scale limit is dependent on the shortest spacing between two antennas in the array, and the missing information is thus referred to as the short- or zero-spacing problem. Mosaicking helps to recover short spacing information. For complex sources with emission on many scales, however, combining interferometer data with single-dish observations provides more complete coverage of the u-v plane and thus creates a more accurate representation of the true source emission structure. Ideally, the single dish diameter should be larger than the minimum interferometer baseline to ensure maximal overlap in the uv-plane and determine accurately flux calibration factors between the observations.

Data from each interferometer were first combined separately with the single dish observations. The GBT data were regridded to match the interferometer data in pixel scale and spectral resolution. The data were then converted to units of Jy beam⁻¹ for combination with the interferometer data using the average beam FWHM measured at the GBT during the observations. Combination of the data was done using MIRIAD's IMMERGE task. IMMERGE combines deconvolved interferometer data with single-dish observations by Fourier transforming both data sets and combining them in the Fourier domain, applying tapering functions such that at small spacings (low spatial frequencies) the single-dish data are more highly weighted than the interferometer data, while conversely at high spatial frequencies the interferometer data are weighted more highly. The flux calibration factor between the two data sets was calculated in IMMERGE by specifying the overlapping spatial frequencies between the GBT and the interferometers. We took the overlap region in both cases to be 35–100 m (2.7–7.7 kλ), yielding a flux calibration factor of ∼1.4 between the GBT and ATCA data sets, and ∼1.0 between the GBT and VLA data sets in the NH₃ (1,1) line emission. These factors were then applied to the NH₃ (2,2) data sets. The final resolution of the combined data is the same as that of the interferometer data.

To combine all three data sets, the ATCA and VLA data were imaged together using the INVERT task, applying natural weighting and taper as described for the VLA imaging. The interferometer data were then cleaned and combined with the GBT data as described above. The overlap region was taken to be 35–100 m, yielding a flux calibration factor of ∼1.3. By convolving the combined data to match the 32'' resolution of the GBT data, we estimate the total flux of the combined image recovers nearly all (∼98%) the flux in the single-dish image. These data were used to identify structures in the NH₃ data cubes using an automated structure finding algorithm (see Section 3). The data were also tapered to a slightly lower resolution of 15'' FWHM to provide higher signal-to-noise ratios (S/Ns) for a multiple component hyperfine line fitting routine.

We first discuss the NH₃ (1,1) intensity in the combined data sets, and compare the distribution of NH₃ emission and 850 μm continuum emission in the Oph Cores as shown in Figure 1. Figures 2(a), 3(a), and 4(a) show the combined NH₃ (1,1) line emission at ∼106 × 85 FWHM resolution. The emission has been integrated over the central hyperfine components in the Oph B, C, and F Cores, respectively, with a clip of ∼2× the map rms noise level. (Since the outer edge of the combined maps have higher rms noise levels, integrating only over, i.e., the "main component" of NH₃ reduced the amount of signal included in the noisy outer regions.) The respective 850 μm emission for each core at ∼15'' FWHM resolution, i.e., lower than the resolution of the combined NH₃ data, is shown in Figures 2(b), 3(b), and 4(b). We also show locations of submillimeter clumps identified with the 2D version of clumpfind (Jorgensen et al. 2008). In Figure 2(b), we additionally label the ∼1 M_☉ continuum object B2-MM8 (Motte et al. 1998).

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Figures 2, 3, and 4 also show locations and labels of "cold" young stellar objects (YSOs; based on bolometric temperatures derived from fitting their spectral energy distributions) detected and classified through Spitzer infrared observations (Enoch et al. 2008) overlaid on the submillimeter and NH₃ emission. The objects plotted were all identified as Class I protostars (no Class 0 protostars have been associated with Oph B, C, or F). Oph B2 is associated with three YSOs. Of these, two are previously known (IRS45/GY273 and IRS47/GY279). Based on association with a continuum emission object (Enoch et al. 2008) and observed infrared colors (Jorgensen et al. 2008), either three or two protostars in Oph B2 are embedded in gas and dust. One additional Class I protostar is located between B1 and B2, and is only associated with diffuse NH₃ emission. Oph B1 and B3 appear starless. No embedded protostars are associated with Oph C by Enoch et al. (2008), but a "Candidate YSO" is identified by Jorgensen et al. (2008) 30'' south of the core continuum peak (R.A. 16:26:59.1, decl. −24:35:03). Based on the different classifications by the two papers, the significant offset from the continuum emission peak, and the lack of any clear influence on the gas in our data, we will discuss Oph C assuming it is not associated with a deeply embedded object. Four protostars are associated with Oph F, all of which have been previously identified (see Figure 4 for object names). Three are likely embedded in the Core.

Peaks of NH₃ integrated intensity can be used to surmise the presence of "objects," but such identifications can ignore any differences in velocity between adjacent dense gas. We therefore used the 3D version of the automated structure-finding routine clumpfind (Williams et al. 1994) to identify distinct NH₃ emission objects in the combined NH₃ (1,1) data cube, which we will henceforth call "NH₃ clumps." clumpfind uses specified brightness contour intervals to search through the data cube for distinct objects identified by closed contours. The size of the identified clumps are determined by including adjacent pixels down to an intensity threshold, or until the outer edges of two separate clumps meet. The clump location is defined as that of the emission peak. clumpfind was used only on the main emission component, where multiple hyperfine components are sufficiently blended to present effectively a single line given the 0.3 km s⁻¹ spectral resolution and typically wide line widths found. The standard interval of 2 times the data rms between contours, with a slightly larger threshold value worked well to separate distinct emission components in all clumps. A lower limit of 1 K with intervals of 0.4 K, ∼2 times the cube rms, identified separate emission peaks sufficiently for Oph B and C. Oph F was sufficiently fit using a lower limit of 1.2 K and intervals of 0.65 K, as the rms of the combined data was slightly higher. Even so, some identified clumps appeared by eye to be noise spikes at the map edges, and these were culled from the final list, as well as any clumps which contained fewer pixels than in the synthesized beam. Table 4 lists the locations, FWHMs, effective radii, and peak brightness temperatures for clumpfind-identified clumps in Oph B, C, and F, and the clump locations are overlaid on Figures 2–4. The centroid velocities and line widths of the NH₃ clumps were determined through spectral line fitting (described further in Section 4), which provided more accurate measures of v_LSR and Δv given the hyperfine structure of the NH₃ lines.

In Oph B, Figure 2 shows that the large-scale structure of the Core is similar in both line and continuum emission, but some significant differences are also apparent. For example, the integrated line emission displays a less pronounced division between the B1 and B2 Cores seen in the continuum, and indeed shows significant filamentary structure in the region connecting B1 and B2 where little to no continuum emission is observed. Similarly, strong line emission is present in the southeastern edge of B2 where the continuum map shows relatively little emission. Additionally, while B2 is the stronger continuum emitter, B1 is significantly brighter in integrated line emission. Strong NH₃ emission in the northern half of B1 extends beyond the continuum contours, and becomes significantly offset from the bulk of the continuum emission to the east. The NH₃ (1,1) observations of the Oph B Core region also reveal a narrow-line emission peak north of the western edge of B2, which is coincident with a DCO⁺ object, Oph B3 (Loren et al. 1990). B3 is just visible at the lowest contour in the NH₃ integrated intensity map but is not visible in the continuum map.

Furthermore, although the peaks of continuum emission and line emission are often located in the same vicinity in B1 and B2, the brightest continuum peaks and the integrated line emission maxima are typically noncoincident. Overall, the mean separation between an NH₃ integrated intensity peak and the nearest 850 μm continuum peak in B1 and B2 is ∼22'' (2600 AU), or ∼2 times the NH₃ FWHM resolution. Fifteen clumpfind-identified NH₃ clumps are identified in the combined Oph B map, with an average minimum separation between NH₃ clumps of 47'' (5600 AU), or 40'' (4800 AU), if Oph B3 is not included. The mean minimum distance between NH₃ clumps and submillimeter continuum clumps is 44'' (5300 AU), or ∼4 times the NH₃ FWHM resolution. Only five of the fifteen NH₃ clumps are located within 30'' (3600 AU) of a submillimeter continuum clump.

No protostars in Oph B are found at positions of NH₃ (1,1) integrated intensity maxima nor are they associated with identified NH₃ clumps. One protostar, IRS47/GY279, is located south of the NH₃ clump we identify as B2-A6. The offset between the NH₃ clump peak and the protostar, however, is ∼30'', or approximately 3 times the angular resolution of the combined NH₃ data. A second protostar, IRS45/GY273, is coincident with a submillimeter continuum peak but has little associated NH₃ emission. Either of these protostars may be the source of an east–west aligned outflow recently proposed by Kamazaki et al. (2003) from CO observations of B2. The third, previously unidentified protostar, seen west of NH₃ clump B2-A4, is located within a narrow (∼30'') NH₃ integrated intensity minimum between the B2 Core and the filament connecting Oph B1 and B2.

Figure 3 shows that Oph C has more similar overall structure when traced by the continuum and integrated NH₃ (1,1) line emission than Oph B. Extended emission in Oph C is elongated along a southeast-northwest axis and contains a single integrated intensity peak. A thin (∼30'') filament of faint emission extends to the north of the central peak. The submillimeter continuum emission is largely coincident with the integrated NH₃ contours, but the continuum emission peak is offset to the NH₃ integrated intensity peak by ∼20''. clumpfind separates the central NH₃ emission into two clumps, C-A1 and C-A3, and finds a third object, C-A2, at the tip of the northern emission extension. C-A1 and C-A3 are found to the northwest and southeast (30'' offset and 15'' offset, respectively) of the centers of both the NH₃ integrated intensity and of the continuum emission. Continuum emission also extends in the direction of the faint NH₃ northern extension, but there is no secondary peak present.

Like Oph C (but unlike Oph B), Figure 4 shows that Oph F also has very similar structure when traced by either the submillimeter continuum emission or integrated NH₃ intensity. Unlike both Oph B and C, the clumpfind-identified NH₃ clumps are coincident with the integrated intensity peaks. F-A3 is nearly coincident (within a beam FWHM) with a submillimeter continuum clump and is additionally coincident with an embedded protostar, IRS43/GY265. F-A2 is associated with a second embedded protostar (CRBR65) and continuum emission, but not an identified continuum clump. A thin filament (∼15'') extends to the northwest and the third NH₃ clump, F-A1, which is also coincident with extended continuum emission but no identified clump. A third embedded protostar (IRS44/GY259) in the northeast is coincident with a continuum peak, but has no associated NH₃ emission. A fourth protostar, in the south-west, may be coincident with some unresolved NH₃ emission, but is located in a section of the map with larger rms values and consequently the small integrated intensity peak seen at that location may be simply noise.

The discrepancies between NH₃ and submillimeter continuum emission in Oph B are in contrast to earlier findings of extremely high spatial correlation between the two gas tracers in isolated, low-mass starless clumps. For example, Tafalla et al. (2002) found that both the millimetre continuum and integrated NH₃ (1,1) and (2,2) line intensity were compact and centrally concentrated in a survey of five starless clumps, including L1544 in the Taurus molecular cloud. In these clumps, the integrated intensity maxima of both transitions are approximately coincident (within the 40'' angular resolution of the NH₃ observations) with the continuum emission peaks. This same coincidence between NH₃ and millimeter continuum was found in B68 (Lai et al. 2003). When observed at higher angular resolution, NH₃ emission in L1544 remained coincident with the continuum but the line integrated intensity peak was offset by ∼20''. This offset was explained, however, as being due to the NH₃ emission becoming optically thick (Crapsi et al. 2007).

It is also possible that different methods of identifying structure in molecular gas (such as the gaussclump method of Stutzki & Guesten 1990, or using dendrograms as in Rosolowsky et al. 2008b, for example) would create a different "clump" list than presented here. We have additionally compared our results in Oph B with the locations of millimeter objects identified using multiwavelet analysis by Motte et al. (1998), and find a smaller yet still significant mean minimum distance of 27'' between NH₃ clumps and millimeter objects. Given the severe positional offsets in some locations between the NH₃ emission and submillimeter continuum, it is unlikely that different structure-finding methods would provide substantially different results.

The GBT observations of C₂S (2₁–1₀) emission only resulted in single, localized detections in Oph B1 and Oph C, while only Oph C had a single, localized detection in HC₅N (9–8). The C₂S emission in B1 was confined to a single peak at its southern tip. The single-dish spectra of all observed species at the C₂S peak locations in B1 and C are presented in Figure 5 (note that in Figure 5, the NH₃ line is so narrow in Oph C that we are detecting the hyperfine structure of the (1,1) line). The integrated intensity GBT maps of all molecules observed in Oph C are shown in Figure 6. Within the ∼30'' resolution limits of the GBT data, the NH₃, C₂S and HC₅N spectral line integrated intensity peaks overlap with the local 850 μm continuum emission peak in Oph C. We fit the spectra of the two C₂S and single HC₅N detections with single Gaussians to determine their respective v_LSR, line width Δv, and peak intensity of the line in T_MB units. We additionally fit a 2D Gaussian to the integrated intensity maps to determine the FWHMs of the emitting regions. We find a beam-deconvolved FWHM = 9200 AU and 5800 AU for the C₂S and HC₅N emission, respectively, in Oph C. The C₂S emission in Oph B1 is elongated, with a FWHM = 9400 AU in R.A. but only 3700 AU in decl. for an effective FWHM = 5100 AU. The results of the Gaussian fitting are listed in Table 5. See Section 4.4.2 for further analysis of C₂S and HC₅N.

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The metastable J = K rotational states of the symmetric-top NH₃ molecule are split into inversion doublets due to the ability of the N atom to quantum tunnel through the hydrogen atom plane. Quadrupole and nuclear hyperfine effects further split these inversion transitions, resulting in hyperfine structure (HFS) of the (J, K) = (1,1) transition, for example, containing 18 separate components. This hyperfine structure allows the direct determination of the optical depth of the line through the relative peaks of the components. Additionally, since transitions between K-ladders are forbidden radiatively, the rotational temperature describing the relative populations of two rotational states, such as the (1,1) and (2,2) transitions, can be used to determine directly the kinetic gas temperature (Ho & Townes 1983).

For a given (J, K) inversion transition of NH₃ in local thermodynamic equilibrium (LTE), the observed brightness temperature T*_A as a function of frequency ν can be written as

$$\begin{eqnarray} T_{A,\nu }^*(J,K) &=& \eta _{\rm MB} \, \Phi \, (J(T_{\rm ex}(J,K)) - J(T_{bg}))\nonumber\\ &&\times (1 - \exp (-\tau _\nu (J,K))) \end{eqnarray} \tag{ 1 }$$

assuming the excitation conditions of all hyperfine components are equal and constant (i.e., T_ex,ν = T_ex). Here, η_MB is the main beam efficiency, Φ is the beam filling factor of the emitting source, T_ex is the line excitation temperature, T_bg = 2.73 K is the temperature of the cosmic microwave background, and J(T) = (hν/k)[exp(hν/kT) − 1]⁻¹. The line opacity as a function of frequency, τ_ν, is given by

$$\begin{equation} \tau _\nu = \tau _0 \sum _{j=1}^{N} a_j \exp \biggl (-4\ln 2\,\biggl (\frac{\nu -\nu _0 - \nu _j}{\Delta \nu }\biggr)^2\biggr) \end{equation} \tag{ 2 }$$

where N is the total number of hyperfine components of the (J, K) transition (N = 18 for the (1,1) transition and N = 21 for the (2,2) transition). For a given jth hyperfine component, a_j is the emitted line fraction and ν_j is the expected emission frequency. The observed frequency of the brightest line component is given by ν₀, with a FWHM Δν. Values of a_j and ν_j were taken from Kukolich (1967). Here, we assume Φ = 1. If the observed emission does not entirely fill the beam, the determined T_ex will be a lower limit. In regions where the emission is very optically thin (τ < <1) there is a degeneracy between τ and T_ex and solving for the parameters independently becomes impossible. We restricted our analysis to regions where the NH₃ (1,1) intensity in the central component is greater than 2 K, which corresponds roughly in our data to an S/N of 8 − 10 in the main component and ∼2–3 in the satellite components. With this restriction, we also find τ ≳ 0.5 throughout the regions discussed.

To improve the S/N of the data and to match the resolution of the 850 μm continuum data, we first convolved the combined data to a final FWHM of 15'' (from 106 × 85), and then binned the convolved data to 15'' × 15'' pixels. Assuming Gaussian profiles, the 18 components of the NH₃ (1,1) emission line were fit simultaneously using a chi-square reduction routine custom written in idl. The returned fits provide estimates of the local standard of rest (LSR) line centroid velocity (v_lsr), the observed line FWHM (Δv_obs), the opacity of the line summed over the 18 components (τ), and [J(T_ex) − J(T_bg)]. The satellite components of the (2,2) line are not visible above the rms noise of our data. These data were consequently fit (again in idl) with a single Gaussian component.

The line widths determined by the hyperfine structure fitting routine are artificially broadened by the velocity resolution (0.3 km s⁻¹) of the observations. To remove this effect, we subtract in quadrature the resolution width, Δv_res, from the observed line width, Δv_obs, such that $\Delta v_{\rm line} = \sqrt{\Delta v_{\rm obs}^2 - \Delta v_{\rm res}^2}$. In the following, we simply use Δv = Δv_line for clarity. The limitations of the moderately poor velocity resolution are discussed further in Appendix B, but do not significantly impact our analysis. In regions where lines are intrinsically narrow, such as Oph C and parts of Oph F, the derived line widths may be overestimated by up to ∼20%–35%.

The uncertainties reported in the returned parameters are those determined by the fitting routine, and do not take the calibration uncertainty of ∼8% into account. The calibration uncertainty affects neither the derived parameters that are dependent on ratios of line intensities, such as the opacity and kinetic temperature T_K, nor the uncertainties returned for v_LSR or Δv. The excitation temperature, however, as well as the column densities and fractional NH₃ abundances discussed below (see Section 4.4) are dependent on the amplitude of the line emission, and are thus affected by the absolute calibration uncertainty.

Table 6 lists the mean, rms, minimum, and maximum values of v_lsr, Δv, τ, and T_ex found in each of the Cores using the above restrictions for the combined NH₃ line emission. In the following sections, we describe in detail the results of the line fitting and examine the resulting line centroid velocities and widths, as well as T_ex and τ. In addition, we use the fit parameters to calculate the gas kinetic temperature (T_K), nonthermal line widths (σ_NT), NH₃ column density (N(NH₃)), gas density (n(H₂)) and NH₃ abundance (X(NH₃)) across all the Cores, as described further in Sections 4.2, 4.3, and 4.4. Table 7 summarizes the mean, rms, minimum, and maximum values of T_K, σ_NT, N(NH₃), n(H₂), and X(NH₃) for each core. For each NH₃ clump, Table 8 summarizes the mean, rms, minimum, and maximum values of all determined parameters (means were obtained by uniformly weighting each pixel).

Figures 7(a), 8(a), and 9(a) show maps of v_LSR of the fitted NH₃ (1,1) line in Oph B, C, and F respectively from the combined, smoothed and regridded data. These maps reveal that although variations of v_LSR are seen within the Cores, they are not that kinematically distinct from each other. For example, only 0.35 km s⁻¹ (≲ the mean Δv) separates the average line-of-sight velocity in Oph B from Oph F. This result agrees with the one-dimensional velocity dispersion of ∼0.36 km s⁻¹ found by André et al. (2007) through N₂H⁺ observations of the Oph Cores.

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Table 8. Derived Parameters at NH₃ (1,1) Peak Locations

ID	vlsr (km s−1)	Δv (km s−1)	τ	Tk (K)	Tex (K)	σNT (km s−1)	N(H2) (1021 cm−2)	$N(\mbox{NH}_3)_{\rm {tot}}$ (1014 cm−2)	X(NH3) (10−8)	n(H2) (104)
B1-A1	4.01(3)	0.85(7)	1.6(3)	16.3(1.9)	9.4(1)	0.35(3)	⋅⋅⋅	5.0(9)	⋅⋅⋅	2.8(1)
B1-A2	3.84(2)	0.92(5)	1.5(2)	16.4(1.3)	12.1(1)	0.38(2)	14(3)	6.3(9)	6(1)	6.5(2)
B1-A3	3.98(1)	0.62(3)	3.4(2)	13.7(1.0)	9.7(1)	0.25(2)	18(4)	9.1(7)	7(1)	4.2(1)
B1-A4	3.83(2)	0.54(3)	4.2(2)	12.4(0.9)	8.4(1)	0.22(2)	12(2)	9.5(8)	11(2)	3.2(1)
B2-A1	4.18(2)	0.57(5)	1.6(3)	13.3(1.2)	11.6(1)	0.23(2)	⋅⋅⋅	4.9(9)	⋅⋅⋅	12.6(2)
B2-A2	4.13(2)	0.86(5)	0.3(*)	14.1(*)	31.0(*)	0.35(*)	⋅⋅⋅	3.9(*)	⋅⋅⋅	0.0(*)
B2-A3	4.10(3)	0.80(7)	1.3(3)	14.2(1.2)	10.2(1)	0.33(3)	15(3)	4(1)	4.2(8)	4.7(1)
B2-A4	4.00(2)	1.00(6)	1.4(2)	13.5(1.0)	11.6(1)	0.42(3)	32(6)	7(1)	3.2(6)	11.7(8)
B2-A5	3.91(2)	0.79(5)	2.4(2)	13.2(1.0)	9.7(1)	0.32(2)	45(9)	8.6(9)	2.7(5)	4.8(2)
B2-A6	4.32(2)	0.66(4)	1.3(2)	14.5(1.1)	14.1(1)	0.27(2)	33(7)	5.2(9)	2.2(4)	72(26)
B2-A7	4.57(1)	0.33(2)	2.8(2)	13.9(1.1)	11.9(1)	0.12(1)	35(7)	4.8(6)	2.0(4)	11.5(8)
B2-A8	4.35(3)	0.95(8)	0.5(*)	14.6(*)	16.9(1)	0.39(*)	43(8)	3.5(*)	1.2(2)	0.0(*)
B2-A9	⋅⋅⋅	⋅⋅⋅	⋅⋅⋅	⋅⋅⋅	⋅⋅⋅	⋅⋅⋅	⋅⋅⋅	⋅⋅⋅	⋅⋅⋅	⋅⋅⋅
B2-A10	4.22(2)	0.71(6)	1.6(3)	14.3(1.2)	10.3(1)	0.29(3)	35(7)	5.0(9)	2.1(4)	4.8(2)
B3-A1	3.15(2)	0.08(8)	0.7(*)	13.9(*)	13.9(1)	0.0(*)	⋅⋅⋅	2.2(6)	⋅⋅⋅	12.4(*)
C-A1	4.08(1)	0.16(1)	9.3(6)	10.4(0.7)	8.0(2)	0.0(*)	84(17)	7.4(6)	1.3(3)	4.2(2)
C-A2	3.97(2)	0.36(4)	2.7(6)	12.0(1.2)	8.8(2)	0.13(2)	15(3)	4(1)	4.5(9)	4.1(2)
C-A3	3.98(1)	0.29(2)	9.3(5)	10.7(0.8)	8.0(2)	0.10(1)	68(14)	13(1)	2.8(6)	3.9(2)
F-A1	4.82(4)	0.8(1)	1.5(7)	15.3(2.0)	7.4(2)	0.32(5)	31(6)	4(2)	1.7(3)	1.6(1)
F-A2	4.31(2)	0.58(5)	1.9(4)	15.6(1.6)	9.9(2)	0.23(2)	35(7)	4(1)	1.8(4)	3.4(1)
F-A3	4.31(2)	0.43(3)	2.6(4)	16.8(1.8)	9.8(2)	0.16(2)	24(5)	4.1(8)	2.5(5)	3.0(1)

Notes. Uncertainties are given in brackets beside values and show the uncertainty in the last digit of the value, with the exception of the T_K uncertainties which are given in K units. A (*) indicates that the uncertainty in the value at this position were large, even though in most cases the values themselves are reasonable. A good HFS fit was not found at the peak position of NH₃ clump B2-A9.

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In Oph B (see Figure 7(a)), the v_LSR of NH₃ emission has little internal variation, with a mean v_LSR = 3.96 km s⁻¹ and an rms of only 0.24 km s⁻¹. An overall gradient is seen across Oph B1 and B2, with smaller v_LSR values (3.2 km s⁻¹) at the southwest edge of B1 increasing to 4.6 km s⁻¹ at the most eastern part of B2. Correspondingly, B1 has a characteristic velocity somewhat less than the average (3.8kms⁻¹ ± 0.1 km s⁻¹), while the B2 v_LSR is slightly greater (4.1 ±  0.2 km s⁻¹). The filament connecting B1 and B2 is kinematically more similar to B2, but there is no visible discontinuity in line-of-sight velocity of the lines. B3 has the lowest v_LSR in Oph B, with an average velocity of 3.3kms⁻¹ ± 0.2 km s⁻¹, or 0.7 km s⁻¹ less than the average of the group. This large difference, greater than that between the mean v_LSR values of Oph B and F, suggests that B3 may not be at the same physical distance as the rest of Oph B. The change in velocity between B2 and B3 occurs over a small projected distance (∼30'', or ∼3600 AU). There is some indication that B2 and B3 may overlap along the line of sight, as the determined v_LSR values in B2 immediately south of B3 are less than those to the east and west, as might be expected if lower velocity emission is also contributing to the line at that location (as discussed further below, the Δv line widths in this area are larger than the average, as would be expected if unresolved emission from two different velocities is contributing to the observed line).

Oph C (see Figure 8(a)) has a mean v_LSR = 4.01 km s⁻¹ with an rms of only 0.07 km s⁻¹. A small velocity gradient of ∼0.4 km s⁻¹ is evident in Oph C, with a minimum line-of-sight velocity of ∼3.7 km s⁻¹ in the southeast, increasing to a maximum of ∼4.1 km s⁻¹ in the northwest. A similar gradient was noted in N₂H⁺(1–0) observations by André et al. (2007), and may be indicative of rotation. The identification of two NH₃ clumps in the region, however, could also be indicative of two objects with slightly different v_LSR. A slight decrease in v_LSR values is seen in the NH₃ extension to the north, with C-N2 associated with emission at a slightly lower v_LSR than the mean.

Oph F (see Figure 9(a)) has a mean v_LSR = 4.34 km s⁻¹ and an rms of only 0.26 km s⁻¹ over the region containing the bulk of the NH₃ (1,1) emission and two of the three identified NH₃ peaks, despite the presence of four protostars. The filament extending toward the northwestern NH₃ peak gradually increases in v_LSR, but only by ∼0.2 km s⁻¹. With higher sensitivity, single-dish data show that outside this area v_LSR drops to ∼3.7 km s⁻¹. There is clear evidence for two velocity components along the line of sight at the peak position of F-A1, with a secondary component at ∼3.7 km s⁻¹. André et al. (2007) also find two velocity components near F-A1 in N₂H⁺observations. For the brighter component, their N₂H⁺ data agree with our NH₃ data, but for the secondary component they find a higher v_LSR = 4.1 km s⁻¹. Some blue asymmetry is found in the line profiles of F-A2 and F-A3, but this is more likely due to the complicated velocity structure of the core rather than infall motions (comparison with an optically thin tracer at this position is necessary to confirm infall).

In summary, v_LSR varies little across any of the Cores (rms <0.24 km s⁻¹), and additionally varies little between them (maximum mean difference ∼0.38 km s⁻¹). Some small gradients were found in the larger Cores (i.e., on scales larger than the individual NH₃ clumps) which may be indicative of rotation.

Figures 7(b), 8(b), and 9(b) show the line Δv for Oph B, C, and F obtained from the combined, smoothed and regridded data. As stated above, the line widths have been corrected for the resolution of the spectrometer (0.3 km s⁻¹). In the few cases where the returned FWHM from the fits is similar or equal to the resolution, we set the corrected FWHM to the thermal line width (see Appendix A for further discussion).

Line widths range from Δv ≲ 0.1 km s⁻¹ to ∼1.2 km s⁻¹ in both Oph B and F, with a minimum line width of 0.08 km s⁻¹ in B3 and a maximum of 1.37 km s⁻¹ in B2. Line widths in Oph C range from 0.11 km s⁻¹ to 0.70 km s⁻¹. The rms variations Δv and v_LSR in Oph B and C are similar (0.2 km s⁻¹ and 0.1 km s⁻¹, respectively), while in Oph F the rms in Δv is much larger than the rms in v_LSR (i.e., 0.3 km s⁻¹ compared to 0.12 km s⁻¹, respectively).

The extended emission in Oph B is dominated by highly nonthermal motions (mean Δv = 0.83 km s⁻¹), shown in Figure 7(b). Several localized pockets of narrow line width are found embedded within the more turbulent gas. The single Oph B3 clump (B3-A1) and B2-A7 are both characterized by extremely narrow Δv (i.e., 0.08 km s⁻¹ and 0.33 km s⁻¹ respectively). As mentioned above, the line emission broadens from B3 to B2 over only ∼30'' (3600 AU) to Δv ∼ 1.4 km s⁻¹, possibly due to line blending along the line of sight if B2 and B3 overlap in projection at these locations. We find small Δv ∼ 0.5 km s⁻¹ toward the southeastern edge of the mapped region in B1 near NH₃ clumps B1-A3 and B1-A4, but the line width minimum does not coincide with either core. Another region of low Δv ∼ 0.6 km s⁻¹ is coincident with the NH₃ clumps B2-A1 and B2-A2. A final Δv minimum, Δv ∼ 0.4 km s⁻¹ is found between the two eastern protostars in B2. Note that the protostars in Oph B are all associated with smaller Δv than the average value for the Core, but none are coincident with a clear local minimum in line width.

The maximum line width in Oph C, Δv = 0.70 km s⁻¹, is half that found in Oph B. Figure 8(b) shows that most of the emission in C is narrow, with a minimum Δv = 0.11 km s⁻¹, similar to the extremely narrow lines found in B3. These narrowest line widths are found centered on NH₃ clump C-A1 in a band perpendicular to the elongated direction of Oph C and are coincident with the highest velocity emission, and are thus not coincident with the NH₃ integrated intensity maximum nor the continuum emission peak. Curiously, if C-A1 and C-A3 are indeed physically distinct clumps, we would expect the broadest lines between them due to overlap, but instead find narrow lines at this location. The largest line widths are found at the edges of the integrated intensity contours.

Oph F is characterized by moderately wide line emission more similar to that found in Oph B, with a mean Δv = 0.63 km s⁻¹ (see Figure 9(b)). Line widths associated with the central NH₃ clumps F-A2 and F-A3 are smaller than the mean (i.e., 0.6 km s⁻¹ and 0.3 km s⁻¹ respectively), while clear minima in line width (Δv ∼ 0.33 km s⁻¹) are found at the locations of the two central protostars. Line widths along the Core extending to the northwest integrated intensity peak broaden to larger values (∼1.2 km s⁻¹), but at the tip NH₃ clump F-A1 is associated with Δv = 0.4 km s⁻¹ in a single 15'' pixel.

Overall, the observed NH₃ Δv in the Cores are generally large, excepting Oph C. We also find regions of localized narrow line emission, some of which are associated with NH₃ clumps. Line widths near protostars tend to be smaller than the mean values, but only in Oph F are the protostars coincident with clear minima in Δv.

Following Mangum et al. (1992), among others, we use the returned τ, Δv and line brightnesses of the NH₃ (1,1) and (2,2) lines in each pixel to calculate the kinetic temperature of the gas. Details of our calculations can be found in Appendix A. Propagating uncertainties from our hyperfine structure fitting routine gives typical uncertainties in T_K of ∼1 K. The mean, rms, minimum, and maximum kinetic temperatures are given for Oph B, C, and F in Table 7. Figures 7(c), 8(c), and 9(c) show the kinetic temperatures calculated across the Oph Cores.

In Oph B, we find a mean T_K = 15.1 K with an rms variation across the entire Core of only 1.8 K. Most NH₃ emission peaks are associated with lower than average gas temperatures, but only a few are coincident with clear T_K minima. The lowest temperatures in the Core, T_K = 13.7 K and 12.4 K, are found in southern B1 toward B1-A3 and B1-A4, respectively. These low temperatures are found at the same location as the detection of single dish C₂S emission. Gas temperatures appear colder (T_K ∼ 13 K) toward the center of Oph B2, but the minimum temperature, T_K = 12.4 K, is not coincident with an NH₃ clump. Instead, the lowest temperatures are found directly between the NH₃ clumps B2-A4, B2-A5, and B2-A6, and closer to the central submillimeter clump.

Oph C is the coldest of the observed Cores, with a mean T_K = 12.8 K and a similar rms variation (1.6 K) as in Oph B. The central region is effectively at a single low temperature T_K = 10.6 K, with the lowest values found near the emission peaks C-A1 and C-A3 in the northwest and southeast, while the gas temperature of the northern clump, C-A2, is consistent with the average.

Oph F is characterized by the highest temperatures of the observed Cores, with a mean T_K = 16.6 K, slightly warmer than Oph B, and with a large rms variation of 3.2 K. No clear minima in gas temperature are observed near any of the NH₃ clumps, protostars or continuum peaks identified in the region. The protostar associated with NH₃ clump F-A2 is coincident with a temperature maximum in a single 15'' pixel.

The gas temperatures traced by NH₃ emission in the Oph B and F Cores are consistently higher than those found in isolated dense clumps. For example, all five of the starless clumps surveyed by Tafalla et al. (2002) were found to have a constant gas temperature T_K = 10 K determined through an analysis similar to that done here. Two recent studies of NH₃ emission in dense clumps in the Perseus molecular cloud and the less active Pipe Nebula at 32'' resolution also found slightly lower temperatures than those found here, with a median T_K = 11 K in Perseus (Rosolowsky et al. 2008a) and a mean T_K = 13 ± 3 K for ≲1 M_☉ clumps in the Pipe Nebula (Rathborne et al. 2008, we note that one object in the Pipe is warmer than the typical T_K found in Oph B and F). In a sample of NH₃ observations toward Galactic high mass star forming regions, Wu et al. (2006) found a mean T_K = 19 K. The mean kinetic temperatures found in Oph B and F are also slightly greater than the median T_K = 14.7 K found in a survey of NH₃ observations by Jijina et al. (1999), although their analysis showed that the median temperature of dense gas in clusters was significantly higher, T_K = 20.5 K, than in nonclustered environments where the median T_K = 12.4 K. Since L1688 is a clustered star forming environment, it is not unreasonable to expect temperatures higher than those in more isolated regions, given the Jijina et al. results.

Evidence for an extremely cold temperature of 6 K (obtained through observations of H₂D⁺(1₁₀–1₁₁), which likely probes denser gas than NH₃ (1,1) and (2,2)) was recently found for the nearby Oph D core (Harju et al. 2008). In addition, while the mean gas temperature T_K = 12 K in Oph C, temperatures in the highest column density gas drop to 10 K, similar to temperatures found in the studies of isolated clumps described above.

We find little difference in average gas temperatures between NH₃ clumps and NH₃ emission associated with submillimeter continuum emission peaks, and a mean T_K increase of only ∼1 K, i.e., similar to our uncertainty in T_K, in gas temperatures near protostars (but note only five protostars are associated with emission with sufficient S/N to fit the HFS). The Jijina et al. (1999) survey found that NH₃ clumps not associated with identified IRAS sources (presumably protostellar objects) were slightly colder than those with coincident IRAS detections (12.4 K compared with 15.0 K), but the effect was much smaller than temperature differences seen due to association with a cluster.

Given the determined gas temperature T_K, we calculate the expected one-dimensional thermal velocity dispersion σ_T of the gas across the Cores:

$$\begin{equation} \sigma _T = \sqrt{\frac{k_B T_K}{\mu _{\rm{NH_3}} m_{\rm{{H}}}}}. \end{equation} \tag{ 3 }$$

Here, k_B is the Boltzmann constant, $\mu _{{{\rm NH_3}}} = 17.03$ is the molecular weight of NH₃ in atomic units, and m_H is the mass of the hydrogen atom. Similarly, the thermal sound speed c_s of the gas can be calculated using a mean molecular weight μ = 2.33.

The nonthermal velocity dispersion σ_NT is given by

$$\begin{equation} \sigma _{\rm NT} = \sqrt{\sigma ^2_{\rm obs} - \sigma ^2_T} \end{equation} \tag{ 4 }$$

where $\sigma _{\rm obs} = \Delta v / (2 \sqrt{2 \ln {2}})$. The mean, rms, minimum, and maximum values for both σ_NT and the nonthermal to thermal velocity dispersion ratio of the gas, given by σ_NT/c_s, are given for each of Oph B, C, and F in Table 7.

Figures 7(d), 8(d), and 9(d) show the resulting nonthermal to thermal velocity dispersion ratio over the Cores. The mean σ_NT/c_s values show supersonic velocities are present. At the limits of the velocity resolution of our data, the smallest observed line widths are consistent with motions being purely thermal in nature.

We find that the mean σ_NT = 0.35 km s⁻¹ and σ_NT/c_s = 1.5 in Oph B. Across the Core, we find a moderate σ_NT/c_s rms of 0.4. The majority of the gas traced by NH₃ in Oph B is thus dominated by nonthermal, mildly supersonic motions. Several NH₃ clumps are associated with smaller, but still transsonic, nonthermal motions. Thermal motions dominate the observed line widths in only two well defined locations in Oph B which additionally coincide with NH₃ clumps: B2-A7, with σ_NT/c_s = 0.5, and B3-A1, where the observed line width is consistent with purely thermal motions. Otherwise, little difference is seen between the nonthermal line widths of individual clumps and the surrounding gas, with a mean σ_NT/c_s = 1.4 km s⁻¹ for the NH₃ clumps.

In contrast, Oph C has a mean σ_NT = 0.14 km s⁻¹. Consequently, the mean σ_NT/c_s = 0.6 with an rms of only 0.2, showing that thermal motions dominate the observed line widths over much of the Core. The minimum nonthermal line width is found associated with C-A1, which is consistent with purely thermal motions within our velocity resolution limits. On average, nonthermal motions in Oph F are also similar to the expected thermal values, with a mean σ_NT/c_s = 0.9, but with a larger spread around the mean (0.6 rms) and a maximum value (σ_NT/c_s = 2.2) similar to that found in Oph B (σ_NT/c_s = 2.5). The lowest values are found toward F-A2 and F-A3 and the nearby protostars.

The nonthermal NH₃ line widths we measure are similar to those recently found for N₂H⁺(1-0) emission in the Cores at larger angular resolution (∼26''), where the mean σ_NT/c_s = 1.6 ± 0.3 in Oph B, 0.9 ± 0.2 in Oph C, 1.5 ± 0.8 in Oph F (André et al. 2007), and are less than those found in DCO⁺ emission (σ_NT/c_s ∼ 2 in B2, ∼1–1.5 in B1, B3, C, and F (Loren et al. 1990).

4.4.1. NH₃

Given Δv, τ, and T_ex from the NH₃ (1,1) line fitting results, we calculate the column density of the upper level of the NH₃ (1,1) inversion transition. We then calculate the NH₃ partition function to determine the total column density of NH₃ following a revised calculation based on Rosolowsky et al. (2008a). Relevant equations are given in Appendix A.

We also calculate the H₂ column density, N(H₂), per pixel in the cores from 850 μm continuum data using

$$\begin{equation} N(\mbox{H}_2) = S_\nu / [ \Omega _m \mu m_{\rm{{H}}} \kappa _\nu B_\nu (T_d)], \end{equation} \tag{ 5 }$$

where S_ν is the 850 μm flux density, Ω_m is the main-beam solid angle, μ = 2.33 is the mean molecular weight, $m_{\rm{{H}}}$ is the mass of hydrogen, κ_ν is the dust opacity per unit mass at 850 μm, and B_ν(T_d) is the Planck function at the dust temperature, T_d. We take κ_ν = 0.018 cm² g⁻¹, following Shirley et al. (2000), using the dust model from Ossenkopf & Henning (1994) which describes grains that have coagulated for 10⁵ years at a density of 10⁶ cm⁻³ with accreted ice mantles and incorporating a gas-to-dust mass ratio of 100. The 15'' resolution continuum data were regridded to 15'' pixels to match the combined NH₃ observations. The dust temperature T_d per pixel was assumed to be equal to the gas temperature T_k derived from HFS line fitting of the combined NH₃ observations. This assumption is expected to be good at the densities probed by our NH₃ data (n ≳ 10⁴cm⁻³), when thermal coupling between the gas and dust by collisions is expected to begin (Goldsmith & Langer 1978). If these temperatures are systematically high, however, then the derived N(H₂) values are systematically low. There is a ∼20% uncertainty in the continuum flux values, and estimates of κ_ν can additionally vary by ∼3 (Shirley et al. 2000). Our derived H₂ column densities consequently have uncertainties of factors of a few.

Due to the chopping technique used in the submillimeter observations to remove the bright submillimeter sky, any large-scale cloud emission is necessarily removed. As a result, the image reconstruction technique produces negative features around strong emission sources, such as Oph B (Johnstone et al. 2000a). While the flux density measurements of bright sources are likely accurate, emission at the core edges underestimates the true column. We thus limit our analysis to pixels where S_ν ⩾ 0.1 Jy beam⁻¹, though the rms noise level of the continuum map is ∼0.03 Jy beam⁻¹. For a dust temperature T_d = 15 K, this flux level corresponds to N(H₂) ∼ 6 × 10²¹ cm⁻².

Using the calculated H₂ and NH₃ column densities, we have calculated per pixel the fractional abundance of NH₃ relative to H₂, X(NH₃) = N(NH₃)/N(H₂) for each Core. The results of these calculations are shown in Figures 7, 8, and 9, which show the H₂ column density derived from submillimeter continuum data, N(H₂) and the fractional NH₃ abundance, X(NH₃). The mean, rms, minimum, and maximum of the derived column density and fractional abundance in each Core are given in Table 7, while specific values for identified NH₃ clumps are listed in Table 8. The N(H₂) and consequently the X(NH₃) uncertainties given in Table 8 include the ∼20% uncertainty in the submillimeter continuum flux values only; uncertainty in κ_ν is not taken into account.

Oph B has a mean NH₃ column density of 4.6 × 10¹⁴ cm⁻², with the highest N(NH₃) values (maximum N(NH₃) = 12 × 10¹⁴ cm⁻²) found in B1. Two peaks in NH₃ column density are found in B1 which correspond closely with the integrated intensity maxima, but are offset from the B1 NH₃ clumps. In B2, the column density also generally follows the integrated intensity contours, with lower values overall than in B1. The highest NH₃ column in B2 of N(NH₃) = 8.5 × 10¹⁴ cm⁻² is found toward B2-A5. The highest opacity in B2, τ = 4.7, was found associated with B2-A7, but the NH₃ column density at this location is similar to the core average. Small column density increases are seen at other NH₃ intensity peak locations. The NH₃ extension connecting B1 and B2 is characterized by similar NH₃ column densities to those at the edges of the core with no obvious N(NH₃) maxima.

The discrepancy between the bright NH₃ and faint submillimeter continuum emission in B1 indicates a high NH₃ fractional abundance relative to B2, with fractional abundances a factor of ≳2–3 higher than the typical values of X(NH₃) ∼ 10⁻⁸ within B2. This is shown in Figure 7(f). Abundance minima are seen in B2, most notably toward B2-A6 and nearby protostars. Despite the higher NH₃ column densities in B1 and B2, the lack of submillimeter emission in the NH₃ extension connecting the two regions suggests this connecting material has a higher fractional NH₃ abundance. Prominent negative features in the continuum data in this extension preclude a quantitative X(NH₃) estimate.

The mean and maximum NH₃ column densities in Oph C are similar to those found in Oph B (〈N(NH₃)〉 = 5.3 × 10¹⁴ cm⁻² and N(NH₃) = 13 × 10¹⁴ cm⁻², respectively). Oph C contains two N(NH₃) maxima. One is coincident with C-A3 and the second is offset to the west by ∼15'' from C-A1. C-A3 is correspondingly associated with a maximum in NH₃ fractional abundance (X(NH₃) = 2.8 × 10⁻⁸), but C-A1 is coincident with an elongated minimum X(NH₃) ∼ 1 × 10⁻⁹ that extends along the same axis perpendicular to the long axis of the core where the smallest line widths were found. The highest NH₃ abundances X(NH₃) ∼ 3 × 10⁻⁸, are found in the northern extension. The mean abundance in Oph C, X(NH₃) = 1.7 × 10⁻⁸, is slightly more than half the Oph B average.

In Oph F, the mean N(NH₃) = 2.6 × 10¹⁴ cm⁻² is less than that found in Oph B and C by a factor of ∼2. The maximum N(NH₃) = 6.6 × 10¹⁴ cm⁻² is also significantly less than the maxima in either B or C, and is found in the emission extending to the northwest from the two central NH₃ clumps and protostars. The fractional NH₃ abundances are also low compared with B and C, with a mean X(NH₃) = 1.8 × 10⁻⁸ and a maximum X(NH₃) = 3.1 × 10⁻⁸ found near but not coincident with F-A3.

Studies of isolated starless clumps have determined a wide range of NH₃ abundance values for these objects. While in some cases different methods have been used to determine H₂ column density values than that performed here, typical observed fractional abundance values in cold, dense regions tend to be on the order of a few ×10⁻⁹ to a few ×10⁻⁸ (Tafalla et al. 2006; Crapsi et al. 2007; Ohishi et al. 1992; Larsson et al. 2003; Hotzel et al. 2001). Abundances as low as X(NH₃) = 7 × 10⁻¹⁰ and 8.5 × 10⁻¹⁰ have been proposed for B68 (Di Francesco et al. 2002) and Oph A (Liseau et al. 2003), respectively. The values found here agree well with previous studies. The wide variations of X(NH₃) in the same general environment suggests dramatic differences in the chemical states of the Cores in L1688 (see Section 5).

4.4.2. C₂S and HC₅N

Similarly, we can calculate the abundance of C₂S and HC₅N from respective emission detected in the single-dish data, where

$$\begin{equation} N = \frac{8 \pi k \nu _0}{h c^2} \frac{g1}{g2} \frac{1}{A_{ul}} \sqrt{2 \pi } \sigma _v [J(T_{\rm ex}) - J(T_{bg})] \tau _{ul} \end{equation} \tag{ 6 }$$

is the column density of the upper state of the observed transition (Rosolowsky et al. 2008a). The values for A_ul, g₁, and g₂ were taken from Pickett et al. (1998) for each transition. Assuming the transitions are optically thin, the observed temperature of the line T_MB = [J(T_ex) − J(T_bg)]τ_ul. The partition function $Z = \sum _i g_i \exp \left(\frac{-E_i}{kT_K}\right)$ was then used to calculate the total column density of each species as for NH₃, with g_i and E_i values taken from Pickett et al. (1998). The column densities thus derived are given in Table 5. The molecular column densities derived for C₂S in B1 and C (N(C₂S) ∼ 10¹²–10¹³ cm⁻²) are similar to results in young starless cores (Tafalla et al. 2006; Rosolowsky et al. 2008a; Lai et al. 2003). The N(HC₅N) results agree with previous measurements in the Taurus molecular cloud (Codella et al. 1997; Benson & Myers 1983) and the Pipe Nebula (Rathborne et al. 2008). We calculate molecular abundances as above and find X(C₂S) = 3.1 × 10⁻¹⁰ and 1.5 × 10⁻¹⁰ at the C₂S emission peaks in Oph B1 and C, respectively. We further find an abundance X(HC₅N) = 4.6 × 10⁻¹¹ at the HC₅N emission peak in Oph C.

Given the determined excitation and kinetic temperatures, T_ex and T_K, and assuming the metastable states can be approximated as a two level system, we have calculated the gas density n(H₂) from the NH₃ (1,1) transition following Ho & Townes (1983). We list the mean, rms variation, and range of densities found for each Core in Table 7. Note that this density is effectively a mean density along the line of sight. In general, we find n(H₂)∼ a few ×10⁴ cm⁻³ in all three Cores, with only moderate variation and no clear spatial correspondence with NH₃ or continuum clumps. The largest n(H₂) values (n(H₂) ∼ 8 × 10⁵ cm⁻³) were found in Oph B2 toward the central continuum clump B2-MM8 (labeled in Figure 2(b)). While these values agree with n(H₂) estimates based on NH₃ emission in other regions, they are an order of magnitude lower than estimates of Ophiuchus clump densities derived from dust continuum emission studies at similar spatial resolutions (Motte et al. 1998; Johnstone et al. 2000b).

5.1.1. Correlation Between NH₃ clumps, NH₃ Integrated Intensity, and Dust Clumps

5.1.2. Comparison of NH₃ Clumps, Submillimeter Clumps and Protostars

5.1.3. σ_NT/c_s in Individual NH₃ Clumps

We next look at the nonthermal line widths in the individual NH₃ clumps in all three Cores. In general, Jijina et al. (1999) found that nonthermal NH₃ line widths in clustered environments are larger than those found in isolated regions. In Figure 10, we plot Δv_NT versus Δv_T for the NH₃ clumps in each Core. We omit NH₃ clumps B3-A1 and C-A1 where the corrected nonthermal line width is effectively zero. We also show the best fit lines found by Jijina et al. to the relationship between thermal and nonthermal line widths in clustered and in isolated regions. Most of the Oph B clumps lie above the Δv_NT–Δv_T trend for isolated clumps and near the trend for the clumps in clustered regions. The good agreement is somewhat surprising given that the majority of the objects in the Jijina et al. sample were observed with ∼4–8 times poorer angular resolution, while those observed with high angular resolution are high mass star forming regions ∼3–7.5 kpc distant and thus have very low linear resolution (excepting Orion B, at a distance of 420 pc). Even at the small spatial scales probed by our observations, Oph B is characterized by wide line widths that follow the relationship found for larger objects in clustered environments. In comparison, the NH₃ clumps in Oph C lie well below the clustered Δv_NT–Δv_T trend, with Δv_NT < Δv_T, and also below the Δv_NT–Δv_T trend seen for objects not associated with a cluster. Two of the three Oph F clumps also fall significantly below the isolated object trend, while the third is more turbulent.

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5.2.1. Trends with N(H₂)

In Figure 11, we plot the distribution of T_K, σ_NT/c_s, N(NH₃) and X(NH₃) with N(H₂) (calculated in Section 4.4 assuming T_d = T_K) in Oph B, C, and F. We additionally analyze Oph B1 and B2 separately to examine any potential differences between the two. In Figure 11(a), we find that T_K values in Oph C are nearly universally lower than those in the other filaments, and show a tendency to decrease with increasing H₂ column density. As described previously, the other Cores are warmer but also do not show a significant trend with N(H₂). We show in Figure 11(b) that Oph B1 and B2 are both consistent with having a constant, mildly supersonic ratio of nonthermal to thermal line widths over all N(H₂). Oph C line widths are generally dominated by thermal motions, and σ_NT/c_s decreases significantly at N(H₂) ≳ 4 × 10²² cm⁻². It is interesting to note that there are few data points at these column densities in the other cores, and no observed decrease in σ_NT/c_s. Figure 11(c) and (d) show that in all cores N(NH₃) tends to increase with N(H₂), following the general trend that the NH₃ emission follows the continuum emission in the cores. The differences between NH₃ and continuum clumps are due to small-scale differences in maxima. In addition, the fractional NH₃ abundance, X(NH₃), tends to decrease with increasing N(H₂), although we note that Oph B1 and F have relatively few data points. Figure 11(c) also illustrates the high N(NH₃) values in Oph B1 relative to N(H₂) compared with values in Oph B2, C, and F.

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If NH₃ is depleting at high densities, as the variation of X(NH₃) with N(H₂) suggests, NH₃ may not trace well the T_K in the densest and likely coldest regions. Hence, our assumption of T_d = T_K may not be valid toward the highest columns and we may be underestimating T_d and N(H₂) by factors of ∼2. For example, Stamatellos et al. (2007) predicted that T_d in the centers of the Oph cores could be as low as ∼7 K. Furthermore, X(NH₃) (∝N(H₂), see Figure 11) may be overestimated by similar factors. High-resolution multiwavelength continuum observations and models are needed to obtain independent assessments of T_d throughout the Oph cores.

5.2.2. σ_NT/c_s in Oph B

Jijina et al. (1999) compiled numerous observations of NH₃ in dense gas and found that while most starless clumps are characterized by largely thermal motions, a large fraction of clumps in clusters have σ_NT>σ_T, and that the identification of an NH₃ clump as part of a cluster has a larger impact on the observed line widths than association with a protostar. These findings are in agreement with our results in Oph B. The line widths in Oph B, with a mean Δv = 0.89 km s⁻¹ and σ_NT/c_s = 1.5, are significantly wider and more dominated by nonthermal motions than those found in isolated cores. While B2 is associated with at least two embedded protostars and B1 appears starless, when studied separately (shown in Figure 12), both cores have similar σ_NT/c_s distributions.

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If the nonthermal component is caused by turbulent motions in the gas, then it is interesting to consider the turbulence source. In the following, we consider the source of wide lines in this region as being due to "primordial" (i.e., undamped) turbulence, protostar-driven turbulence, bulk motions, or biased sampling.

Firstly, Oph B may have nonthermal motions throughout all the core that are inherited from the surrounding cloud and that have not yet been damped. Since Oph B is associated with a few embedded protostars, it is likely that parts of the Core, at the very least, have been at high density for over a free-fall time t_ff (∼3 × 10⁵ years for n(H₂) ≈ 10⁴ cm⁻³, ∼n_cr for the NH₃ (1,1) and (2,2) transitions). Since the dissipation timescale for turbulence is ∼t_ff (Mac Low & Klessen 2004), it is unlikely that the nonthermal motions from the parent cloud have been retained, if the embedded protostars are indeed physically connected to the Core. Determining the relative velocities of the YSOs compared with the Core v_LSR would help to address this question.

Secondly, the embedded protostars in Oph B may be adding turbulence to the Core through energy input associated with mass loss. One outflow has been found in CO emission in the region associated with one of the two protostars near the peak NH₃ integrated intensity in B2 (IRS45/GY273 and IRS47/GY279, labeled in Figure 2), with a blue lobe toward the west and a red lobe toward the southeast (Kamazaki et al. 2003). We do not, however, see localized regions of wide line widths associated with these or any protostars in Oph B. In fact, protostars are associated with Δv minima. Additionally, wide line widths are found in B1, where there are no embedded protostars and no known outflows. This suggests that the large nonthermal motions across the Core are not driven by embedded YSOs.

Thirdly, the wide line widths seen in the NH₃ emission may be indicative of global infall in the Oph B Core. Given the mean NH₃ line width in Oph B2, we calculate a virial mass M_vir ∼ 8 M_☉, assuming a density distribution which varies as ρ ∝ r⁻². Given mass estimates of the Cores by Motte et al. (1998; which are uncertain to factors ∼2) and accounting for the different Ophiuchus distance used by the authors (160 pc compared with our preferred value of 120 pc), we find M/M_vir ∼ 5 in Oph B2. Since the mean NH₃ (1,1) opacity τ = 1.6 over Oph B, the average individual hyperfine component is optically thin, and we would not expect to see the asymmetrically blue line profiles found in collapsing cores in optically thick line tracers. André et al. (2007) find these spectroscopic signatures of infall motions in B2, with a clear blue infall profile toward our B2-A10 NH₃ clump, and profiles suggestive of infall toward B2-A7 and other continuum objects in its eastern half. No evidence of infall motions in the tracers used were found toward central B2, but the lines they used (CS, H₂CO and HCO⁺) may suffer from depletion at the high densities and low temperatures found at this location, masking any infall signature. Oph B1, however, is also characterized by wide NH₃ line widths and has M/M_vir ∼ 1. Additionally, spectroscopic infall signatures were found in Oph C toward our C-A3 NH₃ clump by André et al., where we find extremely narrow NH₃ line widths (similarly, B2-A7 contains the narrowest lines observed in B2). It does not appear, then, that infall motions are the sole contribution to the wide lines we find in Oph B.

Lastly, rather than acting as a tracer of the densest gas in Oph B, the NH₃ emission may be dominated in this high density environment by emission from the more turbulent outer envelope of the Core. Modeling of the expected emission given a constant abundance of NH₃ would be necessary to determine accurately whether depletion is a factor in Oph B (note that we see evidence for decreasing abundance of NH₃ with N(H₂); see Figure 11(d)). Alternatively, observations of species which are excited at higher densities than 10⁴ cm⁻³, such as N₂H⁺ and deuterated species such as N₂D⁺ and H₂D⁺, may better probe the dense clump gas. Moderate resolution (∼26'') resolution observations of N₂H⁺(1–0) (André et al. 2007) find narrower line emission in small-scale N₂H⁺ condensations (which we call "clumps") in Oph B1 and the filament connecting B1 and B2 (σ_NT = 0.15 ± 0.04 km s⁻¹, or σ_NT/c_s ∼ 0.65 for T_K = 15 K), but nonthermal line widths in Oph B2 condensations remain transsonic (σ_NT/c_s ∼ 1.1 < 2 on average). In an upcoming paper, we present H₂D⁺ 1₁₀–1₁₁ observations in B2, showing that nonthermal line widths of gas at densities of ∼10⁶ cm⁻³ (approximately the critical density of H₂D⁺, depending on the collisional cross section used) in the core are also transsonic, σ_NT ∼ 0.25 km s⁻¹, or σ_NT/c_s ∼ 1.4 at 10 K (or ∼1.1 at 15 K; R. K. Friesen et al. 2009, in preparation). Regardless of whether or not the line widths seen in NH₃ emission are tracing the highest density gas, the mean nonthermal motions of gas in B2 are greater than typically found in isolated clumps.

5.2.3. σ_NT/c_s in Oph C

5.2.4. Oph B3

Comparisons of the mean values of many parameters found in each of the Cores (Tables 6 and 7) show significant differences between Oph B and C. In particular, Oph C is characterized by significantly narrower line widths and lower kinetic temperatures. This distinction is further illustrated in Figures 11 and 12. An object like Oph C (and perhaps Oph B3) resembles isolated clumps in the close correspondence between the NH₃ line map and the dust continuum map, and in the evidence that the peak of the intensity map coincides with local minima in both T_K and Δv. Such a core would fit well the idea that clustered star formation is just a spatially concentrated version of isolated star formation, with smaller, denser star forming clumps packed closer together than in isolated regions. The Oph B Core shows a remarkably different behavior. Oph B contains a larger number of NH₃ clumps than Oph C or F, and there is much less correspondence between the locations of the NH₃ and continuum clumps. The NH₃ clumps show modest contrast with interclump gas in their intensity, and little contrast in their velocity or their line width. The greater amount of fragmentation of Oph B than found in C or F may be related to the relatively higher levels of turbulence in the Core, which are significantly greater than typically found in isolated regions. Thus, dense NH₃ gas in Oph B does not resemble the dense gas in regions of isolated star formation, and this raises the issue whether this presents a different "initial condition" for clustered star formation.

Since the metastable states across K-ladders are coupled only by collisions, and if we neglect the upper (J ≠ K) nonmetastable states, then the populations of molecules in the metastable states can be described by the Boltzmann equation.

$$\begin{equation} \frac{n(J^{\prime },K^{\prime })}{n(J,K)} = \frac{g(J^{\prime },K^{\prime })}{g(J,K)} \exp \biggl (-\frac{\Delta E(J^{\prime },K^{\prime };J,K)}{T_{\rm rot}(J^{\prime },K^{\prime };J,K)}\biggr) \end{equation} \tag{ A1 }$$

where T_rot(J', K'; J, K) is the rotational temperature relating the populations in the (J, K) and (J', K') states, and ΔE(J', K'; J, K) is the energy difference between the two states. For (J', K') = (2, 2) and (J, K) = (1, 1), ΔE(2, 2; 1, 1)/k = −41.5 K. The state statistical weights, g(J, K) and g(J', K'), are equal to 3 and 5 respectively for the NH₃ (1,1) and (2,2) states (Ho & Townes 1983). Assuming the molecular cloud is homogenous such that n(J', K')/n(J, K) = N(J', K')/N(J, K), we can solve for T_rot(2, 2; 1, 1) using

$$\begin{eqnarray} T_{\rm rot}^{21} &=& T_{\rm rot}(2,2;1,1) = -41.5 \biggl [ \ln \biggl (-\frac{0.283}{\tau (1,1,m)}\frac{\Delta \nu (2,2)}{\Delta \nu (1,1)}\nonumber\\ &\times&\! \ln \biggl [1 - \frac{\Delta T_A (2,2,m)}{\Delta T_A (1,1,m)}\biggl (1 - \exp (-\tau (1,1,m))\biggr)\biggr ]\biggr)\biggr ]^{-1}\nonumber\\ \end{eqnarray} \tag{ A2 }$$

where ΔT_A(1, 1, m) and ΔT_A(2, 2, m) are the antenna temperatures of the main component of the (1,1) and (2,2) transitions, respectively, τ(1, 1, m) is the opacity of the (1,1) line summed over the main component only, and we assume that the line widths Δν(1, 1) = Δν(2, 2). We then calculate the gas kinetic temperature T_K from the rotational temperature following the updated result given by Tafalla et al. (2002) in their Appendix B:

$$\begin{equation} T_K = \frac{T_{\rm rot}^{21}}{1 - \frac{T_{\rm rot}^{21}}{42} \ln \big[1 + 1.1\,\mbox{exp}\big({-}16/T_{\rm rot}^{21}\big)\big]}. \end{equation} \tag{ A3 }$$

We calculate the column density of the (1,1) transition following Equation (13) of Rosolowsky et al. (2008a):

$$\begin{eqnarray} N(1,1) &=& \frac{8\pi \nu _0^2}{c^2}\frac{g_1}{g_2}\frac{1}{A(1,1)}\nonumber \\ &&\times \frac{{1+\exp(-h\nu _0}/kT_{\rm ex})}{1\!-\exp(-h\nu_0/kT_{\rm ex})}\int{\tau (\nu)d\nu}, \end{eqnarray} \tag{ A4 }$$

where we have revised the equation from that presented by (Rosolowsky et al. 2008a) to include both parity states of the NH₃ (1,1) level in the derived NH₃ (1,1) column density. This revision results in greater NH₃ (1,1) column densities (and consequently greater total NH₃ column densities) than those determined through the original equation by a factor which depends to a small degree on the excitation temperature T_ex, and is ∼2 for the regions studied. The statistical weights for the upper and lower states of the (1,1) inversion transition, g₁ and g₂ are equal. The Einstein A coefficient A(1, 1) = 1.68 × 10⁻⁷ s⁻¹ (Pickett et al. 1998), and $\int {\tau (\nu)d\nu } = \sqrt{2\pi } \sigma _v / (c \nu _0) \tau _{\rm tot}$. The total NH₃ column density N(NH₃) can then be determined by calculating the value of the partition function Z of the metastable states:

$$\begin{equation} Z = \sum _J (2J+1) S(J) \exp {\frac{-h[B J (J+1)+(C-B)J^2]}{kT_{\rm {rot}}}}, \end{equation} \tag{ A5 }$$

where the total NH₃ column is then N(1, 1) × Z/Z(1, 1). The values of the rotational constants B and C are 298117 MHz and 186726 MHz, respectively (Pickett et al. 1998). The function S(J) accounts for the extra statistical weight of the ortho- over para-NH₃ states, with Z = 2 for J = 3, 6, 9, ... and Z = 1 for all other J.

Figure 13 plots the returned Δv against that of the model spectrum for 0.1 km s⁻¹ and 0.3 km s⁻¹ resolution for a range of line widths. For both resolutions, the returned line width is greater than the actual for small values, with the 0.3 km s⁻¹ resolution showing a significantly greater effect. The trends, however, follow that expected for addition in quadrature of the true line width and the resolution, i.e. $\Delta v_{\rm obs} = \sqrt{\Delta v_{\rm line} + \Delta v_{\rm res}}$ with some additional offset at small values and larger scatter due to poor resolution. For the mean values found in this study, the corrected line widths consequently reflect the true line widths, but at small widths the corrected values still overestimate the true value. For this reason, we cannot accurately state the true widths of lines where returned (uncorrected) values Δv ≲ 0.35 km s⁻¹, but do show that these uncorrected line widths are consistent with purely or nearly thermal values.

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Figure 14 plots the returned opacity τ against that of the model spectrum for 0.1 km s⁻¹ and 0.3 km s⁻¹ resolution over a range of line widths. The relatively low spectral resolution of our observations has a significant impact on the returned opacity, with our fitting routine significantly underestimating the opacity by greater than 20% for line widths ≲0.7 km s⁻¹. In addition, the scatter in the returned opacities is large at small line widths, and this affects the uncertainty in the calculated NH₃ column density, N(NH₃), in Section 4.3. Since N(NH₃) depends linearly on the opacity (see Equation (A4)), the uncertainty in the opacity dominates the uncertainty in the column density at small line widths. Since the column density also depends linearly on line width, however, and the returned line width at small values is greater than the true value by a similar relative amount, the returned column density is more accurate than the uncertainties suggest.

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The results from the above tests show that at small NH₃ line widths, the returned line widths and opacities can vary significantly from the true values. Regardless, the returned kinetic temperatures from our NH₃ line fitting are robust, as shown in Figure 15 (for T_K = 15 K). Even at small line widths (Δv ≲ 0.3 km s⁻¹), the kinetic temperatures derived from the fits are accurate within uncertainties of the true T_K, with relatively little scatter. At the lowest temperatures found in the Oph cores, T_K = 10 K, the results are similar with slightly more scatter at the lowest line widths.

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