THE INITIAL CONDITIONS OF CLUSTERED STAR FORMATION. II. N₂H⁺ OBSERVATIONS OF THE OPHIUCHUS B CORE

We next identify discrete emission "clumps" in the Nobeyama N₂H⁺ 1-0 emission data cube using the three-dimensional version of the automated structure-finding routine clumpfind (Williams et al. 1994). clumpfind uses specified brightness contour intervals to search through the data cube for distinct emission features identified by closed contours in position and velocity. The size of the structures found are determined by including adjacent pixels down to an intensity threshold, or until the outer edges of two separate structures meet. The clump location is defined as the emission peak within the core. The identification of emission clumps in the data allows us to discuss the absolute locations of N₂H⁺ emission within Oph B with respect to the locations of objects identified in continuum emission, as well as emission features in other molecular line tracers. We caution that the identification of N₂H⁺ "clumps" in Oph B does not imply any association with underlying physical structure in the core.

The rms noise level in the data cube is essentially constant (rms ∼0.08 K in T*_A units) over most of the region observed, but emission in Oph B1 extends near the edge of the area of best OTF mapping overlap and consequently has slightly lower sensitivity. We therefore created a signal-to-noise ratio (S/N) data cube by dividing the spectrum at each pixel by the rms noise in the off-line spectral channels. We then ran clumpfind specifying brightness contours of 4× the S/N with an intensity threshold of 3× the data rms. The strongest hyperfine components of the N₂H⁺ 1-0 line overlap significantly over much of the region observed, so we ran the algorithm on the isolated F₁F → F'₁F' = 01 → 12 component only. This strategy allowed the clear separation of multiple velocity components along the line of sight, but reduced the S/N of the detections since the isolated component can be substantially fainter (up to a factor of ∼2.3) than the central hyperfine emission. We therefore performed several checks to ensure identified N₂H⁺ objects were legitimate. We removed any identified clumps with areas less than the Nobeyama beam, clumps with a peak location at the edge of the mapped region, and clumps with reported line widths less than twice the velocity resolution of the observations (0.05 km s⁻¹). These limits culled most of the spurious objects, and we finally removed several additional objects through visual inspection.

We found a total of 17 separate N₂H⁺ clumps in Oph B from the Nobeyama data. Table 2 lists the locations and FWHM in AU of the clumps, as well as the peak temperatures (T*_A) of the isolated and main hyperfine N₂H⁺ components, and the S/N of the isolated hyperfine component. We plot the positions of the clumps, i.e., the positions of peak brightness temperature of the isolated component, in Figure 1(b). The centroid velocities and line widths of the N₂H⁺ clumps are also returned by clumpfind, but we determined these values instead through spectral line fitting (described in further detail in Section 3). The line fitting algorithm provided more accurate results given the hyperfine structure of the N₂H⁺ 1-0 line.

We show in Figure 2 the N₂H⁺ clumps as ellipses overlaid on the emission at the velocity channels that most closely match the clumps' centroid velocities determined through spectral line fitting in Section 3. The ellipse axes are the returned clump FWHMs in R.A. and decl. The 850 μm continuum emission contours are also shown in Figure 2. In many cases, an offset between the N₂H⁺ clumps and continuum emission is clear.

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We first compared the locations of the N₂H⁺ clumps with the peak locations of 850 μm continuum clumps identified by Jørgensen et al. (2008) in the SCUBA data. Figure 3(a) shows the angular separation between N₂H⁺ clumps and the peak location of the nearest 850 μm continuum clump. Few continuum clumps lie within a radius of 18'' in angular distance from an N₂H⁺ clump, suggesting that many of the N₂H⁺ clumps thus identified are not correlated with continuum objects. This result is similar to our findings in Paper I, where a similar analysis of NH₃ emission in Oph B revealed significant offsets between locations of NH₃ clumps and continuum clumps. Interestingly, the mean minimum separation between N₂H⁺ and continuum clumps in Oph B1 (∼20'') is half that in Oph B2 (∼40''), suggesting there is a better correlation between N₂H⁺ and continuum emission in Oph B1. Furthermore, Figure 3(b) shows the angular separation between N₂H⁺ clumps and nearest NH₃ clumps from Paper I, and shows a clear correlation between the two sets of objects. In the majority of cases, N₂H⁺ and NH₃ clumps are coincident within the resolution of the N₂H⁺ data.

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Other searches for submillimeter continuum emission clumps in Oph B using different clump-finding algorithms produce different numbers of objects within each sub-Core. For example, the clump catalog by Simpson et al. (2008) was created from a combination of all 850 μm SCUBA data in Ophiuchus, following the technique described by Nutter & Ward-Thompson (2007), and contains a larger number of objects within the Oph B Core. Comparison of the N₂H⁺ clumps with the Simpson et al. continuum clumps shows a better correspondence than with the Jørgensen et al. catalog, but the median offset between N₂H⁺ and continuum clump locations is still greater than the 18'' FWHM beam of the N₂H⁺ observations. In the following, we restrict our analysis to the Jørgensen et al. catalog as we use the same clump identification technique, but note that offsets are also seen between the N₂H⁺ clumps presented here and other continuum clump catalogs.

We further compared the locations of the N₂H⁺ clumps thus determined with N₂H⁺ clumps identified by André et al. (2007) in N₂H⁺ 1-0 emission observed with the IRAM 30 m telescope (∼26'' half-power beamwidth). We find good agreement with André et al. for most of the objects despite their use of a spatial filtering scheme rather than the clumpfind algorithm to identify clumps.

We show in Figures 9(a) and (b), respectively, the fitted values of v_LSR and Δv of N₂H⁺ 1-0 across Oph B. Relatively little variation in line-of-sight velocity is seen across Oph B1 and B2, with no evidence of a discontinuity in velocity separating the two sub-Cores. A gradient of increasing v_LSR from west to east is seen across Oph B, with the largest v_LSR values found in the eastern half of B2, to a maximum v_LSR = 4.44 km s⁻¹. In Oph B3, the mean v_LSR = 3.06 km s⁻¹, significantly lower than in B1 (〈v_LSR〉 = 3.97 km s⁻¹) or B2 (〈v_LSR〉 = 4.05 km s⁻¹). The rms variation in v_LSR across all Oph B is only 0.23 km s⁻¹, which is dominated by the spread in v_LSR in B2 (rms = 0.20 km s⁻¹, compared with 0.12 km s⁻¹ and 0.14 km s⁻¹ in B1 and B3, respectively).

We find greater variation between Oph B1 and B2 in N₂H⁺ 1-0 line width than in v_LSR. Oph B1 is generally characterized by narrow lines, with a mean Δv = 0.42 km s⁻¹ and an rms variation σ_Δv of only 0.08 km s⁻¹. Oph B2 contains regions of both small and large line widths, with a larger mean Δv = 0.68 km s⁻¹ and σ_Δv = 0.21 km s⁻¹ than found in B1. Both B1 and B2 have similar minimum line widths (Δv = 0.21 km s⁻¹ in B1 and Δv = 0.30 km s⁻¹ in B2). Line widths in Oph B3 are extremely narrow, with a mean Δv = 0.27 km s⁻¹, a minimum Δv = 0.21 km s⁻¹ and an rms variation of only 0.04 km s⁻¹.

Two locations in B2, both near Class I protostars, show intriguing gradients in line-of-sight velocity and line width, which we show more closely in Figure 10. Figure 10(a) shows the first, at a location coincident with the millimeter continuum object B2-MM8, which is found at the thermal dust emission peak in B2 and is the most massive clump identified in Oph B (M ∼ 1 M_☉; Motte et al.; Johnstone et al. 2000b). As noted previously, MM8 is associated with neither an N₂H⁺ clump nor a maximum of integrated N₂H⁺ 1-0 intensity. The nearby Class I protostar, Elias 32, may be associated with MM8, but is also offset from the clump peak by ∼15'' to the southeast. Across the ∼30'' width of the clump (∼3600 AU at 120 pc), v_LSR varies from ∼3.8 km s⁻¹ in the west to ∼4.2 km s⁻¹ in the east. We find no local decrease in line width as might be expected for a starless clump (Di Francesco et al. 2004). Instead, Figure 10(b) shows how at this location Δv increases from east to west, from Δv ∼ 0.5 km s⁻¹ to Δv ∼ 0.9 km s⁻¹. The resulting velocity gradient, given the ∼3600 AU diameter of the clump, is ∼23 km s⁻¹ pc⁻¹. The variation in line width in particular appears centered on the nearby protostar. Alternatively, we noted in Paper I a region of wide NH₃ line width located between Oph B2 and B3, which we suggested was due to the overlap of the two Cores with different v_LSR along the line of sight. In this case, the wide line widths seen near MM8 also may be due to this overlap.

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Figures 10(c) and (d) plot v_LSR and Δv at the second intriguing gradient, near the Class I protostar Elias 33 in Oph B2. Elias 33 is associated with the millimeter object B2-MM10 (Motte et al. 1998, also 162729-24274 in Jørgensen et al. 2008). Again, no N₂H⁺ clump was found coincident with the protostar. Both v_LSR and Δv show strong variations at the protostellar location. An abrupt change in v_LSR from ∼4.1 km s⁻¹ to ∼3.7 km s⁻¹ is seen in over a single 18'' beamwidth (∼2200 AU) from west to east. The resulting velocity gradient is ∼38 km s⁻¹ pc⁻¹. A narrow ridge of large line width, Δv ≳ 1.0 km s⁻¹, is found with an elongation perpendicular to the v_LSR gradient. On either side, line widths are narrow (Δv ≲ 0.6 km s⁻¹).

Given the N₂H⁺ 1-0 line characteristics Δv, τ_tot, and T_ex determined through HFS fitting, and again assuming LTE, we can calculate the volume density, n_ex, and the N₂H⁺ column density, N(N₂H⁺), in Oph B.

The volume density is calculated following Caselli et al. (2002a) using two-level statistical equilibrium

$$\begin{equation} \frac{n_{\rm ex}}{n^\prime _{{\rm cr}}} = \frac{T_{\rm ex} - T_{{\rm bg}}}{(h\nu /k) (1 - T_{\rm ex}/T_{k})}, \end{equation} \tag{ 1 }$$

where the critical density, corrected for trapping, of the N₂H⁺ 1-0 line is given by

$$\begin{equation} n^\prime _{{\rm cr}} = n_{{\rm cr}} \frac{1 - \exp (-\tau _{{{{\rm hf}}}})}{\tau _{{{{\rm hf}}}}}. \end{equation} \tag{ 2 }$$

The opacity τ_hf is the opacity of a typical hyperfine component. For N₂H⁺ 1-0, we take τ_hf = τ/9. We assume that the N₂H⁺ kinetic temperature is equal to the T_K derived in Paper I from NH₃ observations. This assumption introduces some uncertainty in the resulting n_ex on the order of ∼10%–50% if a difference in T_K exists of ∼2–5 K between gas traced by NH₃ and N₂H⁺ emission (a temperature difference >5 K is unlikely given the mean T_K = 15 K for Oph B determined in Paper I and the factor ∼10–100 difference in the expected gas densities traced by NH₃ and N₂H⁺). Mean, standard deviation, minimum, and maximum densities for Oph B1, B2, and B3 are given in Table 5. For Oph B as a whole, 〈n_ex〉 = 2.4 × 10⁵ cm⁻³, with an rms variation of 2.5 × 10⁵ cm⁻³. Separately, the mean n_ex for Oph B1 and B2 (2.0 × 10⁵ cm⁻³ and 3.1 × 10⁵ cm⁻³, respectively) are lower by factors of ∼2 and ∼3.5 than the mean volume densities calculated by Motte et al. (1998) from 1.3 mm continuum observations. We find only moderate variation in n_ex across the Core, and consequently the densities determined for the N₂H⁺ clumps are similar to the mean Core values. This result is discrepant with the n(H₂) ∼ 10⁷ cm⁻³ densities for the small-scale continuum objects in Oph B, determined by Motte et al., but agrees within a factor of ∼2 with the average 850 μm clump densities (which we calculated from reported clump masses and radii) from Jørgensen et al. (2008).

We follow Di Francesco et al. (2004) and derive the total N₂H⁺ column density N(N₂H⁺) using

$$\begin{eqnarray} N_{\rm tot} &=& \frac{3h}{8\pi ^3}\frac{1}{\mu ^2} \frac{\sqrt{\pi }}{2\sqrt{\ln 2}}\frac{1}{(J+1)\exp (-E_J/kT_{\rm ex})} \biggl (\frac{kT_{\rm ex}}{hB} + \frac{1}{3}\biggr) \nonumber \\ && \times \frac{\Delta v \tau }{1 - \exp (-h\nu /kT_{\rm ex})}, \end{eqnarray} \tag{ 3 }$$

where the lower rotational level number J = 0, the dipole moment of the N₂H⁺ molecular ion μ_e = 3.4 D (Green et al. 1974), the N₂H⁺ rotational constant B = 46.586702 GHz (Caselli et al. 1995), and the rotational level energy above ground of the lower level E_J = J(J + 1)hB. Here, the rotational partition function Q_rot has been approximated by its integral form for a linear molecular ion, Q_rot = kT_ex/hB + 1/3. Note that this calculation is equivalent to that presented by Caselli et al. (2002b). Figure 9(c) shows the N(N₂H⁺) distribution in Oph B.

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

$$\begin{equation} N(\mbox{H}_2) = S_\nu / [ \Omega _m \mu m_{{{\rm H}}} \kappa _\nu B_\nu (T_d)], \end{equation} \tag{ 4 }$$

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_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. As in Paper I, we take κ_ν = 0.018 cm² g⁻¹, following Shirley et al. (2000), using the OH5 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 incorporates a gas-to-dust mass ratio of 100. The 15'' resolution continuum data were convolved to a final angular resolution of 18'' and regridded to match the Nobeyama N₂H⁺ data.

We assume that the dust temperature T_d per pixel is equal to the kinetic temperature T_K determined in Paper I for NH₃ in Oph B, where we found 〈T_K〉 = 15 K. This assumption is expected to be good at the densities probed by the NH₃ data (n ≳ 10⁴ cm⁻³), when thermal coupling between the gas and dust by collisions is expected to begin (Goldsmith & Langer 1978). Our analysis in Paper I, however, suggested that NH₃ observations may not trace the densest gas in Oph B. As we expect the denser gas to be colder in the absence of a heating source, this assumption may systematically overestimate T_d in the Core. The derived N(H₂) values would then be systematically low. There is additionally a ∼20% uncertainty in the continuum flux values, and estimates of κ_ν can 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). Emission on scales greater than 130'' is suppressed by convolving the original continuum map with a σ = 130'' continuum beam, and subtracting the convolved map from the original data. While the flux density measurements of bright sources are likely accurate, the column densities measured at core edges may be underestimates. Following Paper I, 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 N(H₂) and N(N₂H⁺), we next calculate per pixel the fractional abundance of N₂H⁺ relative to H₂, X(N₂H⁺), in Oph B. In Figure 9(d), we show the distribution of X(N₂H⁺) in Oph B. The mean, rms, minimum and maximum values of the derived N₂H⁺ column density and fractional abundance are given in Table 5, while specific values for individual N₂H⁺ clumps are given in Table 4. The N(H₂) and consequently the X(N₂H⁺) uncertainties given in Table 4 include the ∼20% uncertainty in the submillimeter continuum flux values only; uncertainties in T_d and κ_ν are not taken into account.

The mean N₂H⁺ column density 〈N(N₂H⁺)〉 = 5.5 × 10¹² cm⁻² in Oph B. As seen in Table 5, N₂H⁺ column densities in Oph B2 are slightly greater, on average, than in Oph B1. B1, however, contains the largest N₂H⁺ column density peak, with a maximum N(N₂H⁺) = 1.7 × 10¹³ cm⁻² found near (but not at the peak position of) the N₂H⁺ clump B1-N4. The smallest average column density is found toward B3, and the maximum value in B3 is ∼2–4 times less than peak values found in both B1 and B2 (N(N₂H⁺) = 1.1 × 10¹³ cm⁻² toward B2-N9; similar high values are also found toward B2-N2). With the exception of the B1 N(N₂H⁺) peak, observed peaks in N(N₂H⁺) are only factors of ∼2 above the mean values.

In Figure 9(c), we show the locations of N₂H⁺ clumps identified in Section 3.1.1 over the N(N₂H⁺) distribution in Oph B. The peak emission locations of N₂H⁺ clumps are often found near, but are not perfectly coincident with, local N(N₂H⁺) maxima. Clumps identified in continuum emission have varying correspondence with N(N₂H⁺) maxima. In Oph B1, two of three 850 μm continuum clumps are coincident with the two largest N(N₂H⁺) maxima, while in Oph B2 only two of five continuum objects is co-located with an N(N₂H⁺) maximum. The embedded protostars in B2 are not associated with N(N₂H⁺) maxima.

The mean fractional N₂H⁺ abundance for Oph B, 〈X(N₂H⁺)〉 = 8.0 × 10⁻¹⁰, is greater by a factor of ∼2–4 than previous measurements in isolated starless cores (Caselli et al. 2002a; Tafalla et al. 2002, 2006; Keto et al. 2004). As seen in Table 5, the mean N₂H⁺ abundance in Oph B1 is greater by a factor of ∼2 than in Oph B2, although the maximum values are similar. The greatest N₂H⁺ abundances are found toward the sub-Core edges and the connecting filament. Conversely, the smallest X(N₂H⁺) values are located at the centers of B1 and B2. Little small-scale variation is found in the sub-Core interiors. We discuss further in Section 5.2 the general trend in X(N₂H⁺) with H₂ column density.

At the peak locations of N₂H⁺ clumps in B2, the mean X(N₂H⁺) = 3.2 × 10⁻¹⁰ is slightly greater than typical values found in isolated starless cores. N₂H⁺ clumps, however, are not located at clear X(N₂H⁺) maxima at the 18'' resolution of the single-dish data, as shown in Figure 9(d) where we plot the N₂H⁺ clump locations over the X(N₂H⁺) distribution in Oph B. In Oph B2, two of five continuum clumps are found toward N₂H⁺ abundance minima, and no clear abundance minima or maxima are associated with the remaining three. In Oph B1, two of three continuum clumps are co-located with X(N₂H⁺) minima, while a good abundance measurement was not found at the position of the third.

We list in Table 6 the physical parameters derived from N₂H⁺ 1-0 emission toward 850 μm continuum clump locations (Jørgensen et al. 2008) and embedded Class I protostars (Enoch et al. 2009). Columns are the same as in Table 4. In Table 7, we list the mean values of each physical parameter toward N₂H⁺ clumps, continuum clumps, and protostars. Note that we were only able to solve for all parameters listed toward a single protostar, and discuss below only those mean parameters where we obtained values from three or more objects.

We find no difference in the mean v_LSR between N₂H⁺ clumps, continuum clumps and protostars. There is a clear reduction in Δv toward N₂H⁺ clumps relative to continuum clumps (mean Δv = 0.49 km s⁻¹and 0.68 km s⁻¹, respectively), but excluding one continuum clump (162712-24290) reduces the mean Δv to 0.58 km s⁻¹. Protostars are associated with wider mean Δv, but the larger mean is also driven by large Δv toward a single object. The differences between N₂H⁺ and continuum clumps are marginally significant given the variance of Δv across the entire Core is only 0.08 km s⁻¹ in Oph B1 and 0.21 km s⁻¹ in Oph B2. We find similar N₂H⁺ line opacities and excitation temperatures toward continuum and N₂H⁺ clumps. Continuum clumps are associated with larger non-thermal motions, with a mean σ_NT/c_s = 1.24 which is 1.5 times that associated with N₂H⁺ clumps (mean σ_NT/c_s = 0.86). N₂H⁺ column densities are similar toward N₂H⁺ and continuum clumps (within ∼25%), but N₂H⁺ clumps are associated with smaller N(H₂) and hence greater X(N₂H⁺) by a factor of > 2. In fact, the mean N₂H⁺ abundance toward continuum clumps is less than X(N₂H⁺) averaged over the entire Oph B Core. Derived volume densities are the same for N₂H⁺ and continuum clumps.

In summary, the N₂H⁺ clumps are associated with smaller line widths and subsonic non-thermal motions relative to continuum clumps and protostars, but the difference in mean values is similar to the variance in line widths across the Core. The most significant difference between N₂H⁺ and continuum clumps are found in their N₂H⁺ abundances, with a mean X(N₂H⁺) toward continuum clumps that is lower than X(N₂H⁺) in N₂H⁺ clumps.

The B2-N6 clump is a pocket of narrow N₂H⁺ 1-0 line emission within the larger, more turbulent B2 Core, with a large fraction (≳ 50%) of the emission at the clump peak coming from the small size scales probed by the ATCA observations (see Figure 5). Although we do not consider all N₂H⁺ clumps to trace underlying physical structure in Oph B, the remarkable properties of B2-N6 suggest that further attention be paid to that particular clump. Line widths measured from the interferometer data are a factor of ∼2 less than those recorded from the single-dish data (Δv = 0.21 ± 0.04 km s⁻¹ compared with 0.39 ± 0.01 km s⁻¹). The millimeter continuum object B2-MM15, identified through a spatial filtering technique (Motte et al. 1998), lies offset by ∼12'' from the peak interferometer integrated intensity (see Figure 5). Motte et al. calculate a mass of only 0.17 M_☉ associated with the B2-MM15 clump, but based on its small size (unresolved at 15'' resolution, using an estimate of r ∼ 800 AU) derive an average density of n ∼ 1.2 × 10⁸ cm⁻³. This density is significantly larger than the densities calculated in Section 4.1.2 at 18'' resolution, but higher sensitivity N₂H⁺ 1-0 interferometer observations are needed to probe the clump density on small scales.

We calculated in Paper I a gas kinetic temperature T_K = 13.9 ± 1.1 K for the co-located B2-A7 NH₃ (1,1) clump. We find σ_NT = 0.06 km s⁻¹ for B2-N6, equal to the expected N₂H⁺ thermal dispersion, and σ_NT/c_s = 0.26. A similar compact N₂H⁺ 1-0 clump with nearly thermal line widths was found in Oph A (Oph A-N6, Di Francesco et al. 2004). While both clumps are associated with larger-scale N₂H⁺ emission, these objects are only distinguishable as compact, nearly purely thermal clumps when observed with interferometric resolution and spatial filtering.

Di Francesco et al. proposed Oph A-N6 may be a candidate thermally dominated, critical Bonnor–Ebert sphere embedded within the more turbulent Oph A Core. Such objects have sizes comparable to the cutoff wavelength for MHD waves and are in equilibrium between their self-gravity and internal and external pressures. Myers (1998) called these objects "kernels," and showed that they could exist in cluster-forming cores with FWHM line widths Δv>0.9 km s⁻¹ and column densities N(H₂)>10²² cm⁻³, with sizes comparable to the spacing of protostars in embedded clusters, suggesting that a population of kernels within a dense Core could form a stellar cluster. The physical conditions in both Oph A and Oph B2 fit these requirements. The B2-N6 clump, however, is significantly smaller (0.004 pc compared with a predicted ∼0.03 pc) and less massive (0.17 M_☉ compared with a predicted ∼1 M_☉) than the objects discussed by Myers.

If we calculate the mass of a critical Bonnor–Ebert sphere given the ATCA N₂H⁺ 1-0 line width and a radius of ∼800 AU, where M/R = 2.4σ²/G (Bonnor 1956), we find M_BE = 0.02 M_☉, or approximately 10× smaller than the mass determined by Motte et al. (1998). We find a virial mass M_vir = 0.04 M_☉, assuming B2-N6 is a uniform sphere and M_vir = 5σ²R/G. If the clump mass is accurate, this suggests that while small, B2-N6 may be gravitationally unstable. André et al. (2007) made a tentative detection of infall motions toward the clump through observations of optically thick lines such as CS, H₂CO or HCO⁺. Thus B2-N6 may be at a very early stage of clump formation, and may eventually form an additional low-mass protostar to the three YSOs already associated with dense gas in Oph B2.

We first look at the velocities and non-thermal line widths of the gas traced by N₂H⁺ gas and those determined from NH₃ emission in Paper I. Figure 13(a) shows the distribution of v_LSR derived from N₂H⁺ and NH₃ emission in Oph B. There is no significant difference in the measured line-of-sight velocity between the two dense gas tracers, with the mean v_LSR = 3.98 km s⁻¹ for NH₃ and 4.01 km s⁻¹ for N₂H⁺.

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In Figure 13(b), we show the distribution of σ_NT/c_s as determined with N₂H⁺ or NH₃ using the NH₃-derived T_K values. The non-thermal line widths measured in N₂H⁺ 1-0 are substantially smaller than those measured in NH₃ in Paper I, and additionally the variation in the relative magnitude of the non-thermal motions between Oph B1 and B2 was not found in NH₃ emission. The mean NH₃ σ_NT/c_s = 1.64 in Oph B, and does not vary significantly between B1 (〈σ_NT/c_s〉 = 1.62) and B2 (〈σ_NT/c_s〉 = 1.68).

The assumption that T_K is similar for both N₂H⁺ and NH₃ is reasonable if both molecules are excited in the same material. The good agreements between the locations of NH₃ and N₂H⁺ clumps, illustrated in Figure 3, and the v_LSR, illustrated in Figure 13(a), support this assumption. The substantial offset in the σ_NT/c_s ratio seen in Figure 13(b), however, between NH₃ and N₂H⁺ emission indicates significantly different motions are present in the gas traced by NH₃ than in the gas traced by N₂H⁺. The gas densities traced by N₂H⁺ are ∼1–2 orders of magnitude greater than those traced by NH₃ (∼10⁵ cm⁻³ and ∼10^3–4 cm⁻³, respectively). In fact, if we attempt to derive the gas kinetic temperature by assuming equal non-thermal motions for the N₂H⁺ 1-0 and NH₃ (1,1) emission (effectively assuming N₂H⁺ and NH₃ trace the same material), the resulting average value is a highly unlikely T_K = 190 K for Oph B2. Given the narrow N₂H⁺ lines observed in Oph B1, no physical solution for T_K can be found under the assumption of equal σ_NT.

Starless cores are typically found to be well described by gas temperatures that are either constant or decreasing as a function of increasing density (Di Francesco et al. 2007). Given that N₂H⁺ traces denser gas, we then expect T_K from NH₃ measurements to be biased high in starless cores, e.g., gas traced by N₂H⁺ should be colder than gas traced by NH₃. The mean T_K found in Oph B in Paper I was 15 K. With a lower T_K, the sound speed would be smaller and the returned σ_NT would be larger on average. A gas temperature T_K = 10 K rather than 15 K would increase the average N₂H⁺ σ_NT/c_s ratio in Oph B by ∼20%–25%. This increase, while significant, is not large enough for the mean N₂H⁺ σ_NT/c_s to match the NH₃ results in Oph B. With both a constant temperature or decreasing temperature with density, non-thermal motions in Oph B1 and B3 would remain subsonic, while Oph B2 would still be dominated by transonic non-thermal motions. Alternatively, it is possible that the two Class I protostars in Oph B2 could raise the temperature of the dense gas above that traced by NH₃ (no T_K difference was found between protostars and starless areas in Paper I). In this case, the returned N₂H⁺ σ_NT would be smaller, and would further increase the differences in magnitude of non-thermal motions seen between the NH₃ and N₂H⁺ emission in Oph B1. In B2, a temperature of 20 K would decrease the mean σ_NT/c_s ratio to 1.1. In a study of dust temperatures in the Ophiuchus Cores, Stamatellos et al. (2007) showed, however, that embedded protostars in the Cores will only heat very nearby gas and will not raise the mean temperature of the Cores by more than ∼1–2 K, so an average T_K = 20 K over all Oph B2 is unlikely.

In Figure 14, we show the isolated F₁F → F'₁F' = 01 → 12 component of the N₂H⁺ 1-0 emission line toward the three N₂H⁺ clumps in Oph B1 (B1-N1, B1-N3, and B1-N4) which show double-peaked line profiles, as described in Section 4. Figure 14 also shows a single Gaussian line profile overlaid on each N₂H⁺ spectrum, which represents the v_LSR and Δv of the 18-component Gaussian fit to the NH₃ (1,1) emission at the clump peak, normalized to a peak amplitude of 1 K. We plot the Gaussian fit for clarity since there are no isolated components in the NH₃ (1,1) hyperfine structure. Additionally, spectra of C₂S 2₁ − 1₀ emission line at the peak clump locations are shown. Note that over all Oph B, C₂S emission was only detected toward the southern tip of Oph B1, and thus no significant C₂S emission was found at the peak location of B1-N1. The C₂S emission was observed with the GBT at ∼32'' spatial resolution and 0.08 km s⁻¹ spectral resolution, with the observations and analysis described in Paper I. Arrows show the locations of the fitted v_LSR for all species, including both N₂H⁺ velocity components.

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The clumps which show the double-peaked line profile structure are the clumps with the highest total N₂H⁺ 1-0 line opacities based on a single velocity component fit, with τ = 5, 6, and 9, respectively, for B1-N1, B1-N3, and B1-N4. The F₁F → F'₁F' = 01 → 12 component is expected to have an opacity τ_iso = 1/9 τ, so τ_iso = 0.6, 0.7, and ∼1 for the three clumps, based on a single component fit. If the line is optically thick, however, this fit is likely to underestimate the true opacity given the missing, self-absorbed flux. This is suggestive that the lines are indeed self-absorbed. Since the NH₃ and C₂S emission is optically thin (Paper I), however, if the N₂H⁺ emission was self-absorbed due to high optical depth we would expect the emission peak of both NH₃ and C₂S to be found between the two N₂H⁺ components in v_LSR, which is not the case. In all three clumps, we find the v_LSR of the NH₃ emission more closely matches the v_LSR of the red N₂H⁺ component, while the C₂S emission in two clumps (v_LSR = 3.67 km s⁻¹) more closely matches the v_LSR of the blue N₂H⁺ component (v_LSR = 3.85 km s⁻¹ and 3.62 km s⁻¹ for B1-N3 and B1-N4, respectively). This behavior suggests the N₂H⁺ 1-0 emission is not self-absorbed toward these positions. A similar offset of C₂S emission relative to N₂H⁺ was found by Swift et al. (2006) toward the starless Core L1551, however the case of L1551 the C₂S emission is redshifted with respect to the systemic velocity of the Core rather than blueshifted, as we find in B1.

The GBT NH₃ (1,1) observations described in Paper I found extensive emission around Oph B, such that it is possible to determine the v_LSR and Δv of NH₃ beyond where continuum emission traced the Oph B Core. We use an intensity threshold of 2 K in the NH₃ (1,1) main component to separate Core and off-Core gas. The off-Core gas surrounding Oph B1 has a mean v_LSR = 3.72 km s⁻¹ with an rms variation of 0.13 km s⁻¹, which is significantly different from the mean B1 Core v_LSR = 3.96 km s⁻¹ and rms variation of 0.15 km s⁻¹. Both the C₂S emission and the blue N₂H⁺ component are thus more kinematically similar to the off-Core NH₃ gas. The critical densities of C₂S 2₁ − 1₀ and NH₃ (1,1) are similar (Suzuki et al. 1992; Rosolowsky et al. 2008). Kinematically, we find σ_NT/c_s = 0.7 for the C₂S emission if we assume the T_K determined from NH₃ emission accurately represents the temperature of the gas traced by C₂S. This result closely matches the N₂H⁺ results in Oph B1, but the non-thermal motions in the gas traced by NH₃ are significantly larger (Section 5.1 and Paper I). It is impossible to match the non-thermal C₂S and NH₃ motions for a given T_K, since the NH₃ (1,1) line σ_NT>σ_obs of the C₂S.

At densities n≳ a few ×10³-10⁴ cm⁻³, chemical models predict C₂S is quickly depleted from the gas phase (timescale t_dep ∼ 10⁵ yr; Millar & Herbst 1990; Bergin & Langer 1997), and accordingly the molecule is observed to be a sensitive tracer of depletion in isolated cores (Lai & Crutcher 2000; Tafalla et al. 2006). The volume density in Oph B1 is greater than that required for C₂S depletion (n ≳ 10⁵–10⁶ cm⁻³ from Section 4.1.2 and continuum observations), so we would not expect to see any C₂S emission unless the gas has only been at high density for t < t_dep. C₂S was only detected toward southern B1. An explanation, therefore, for both the presence of C₂S in southern B1 and for its velocity offset relative to the dense gas is that gas from the ambient molecular cloud is accreting onto B1, reaching a density high enough to excite the C₂S emission line. This material would be chemically "younger" than the rest of the B1 Core gas, such that the C₂S has not had enough time to deplete from the gas phase. A similar result was found in a multi-species study of the starless L1498 and L1517B cores, where Tafalla et al. (2004) conclude that non-spherical contraction of the cores produced asymmetric distributions of CS and CO, where the CS and CO "hot spots" reveal the distribution of recently accreted, less chemically processed dense gas.

A single 850 μm continuum clump (162715-24303; Jørgensen et al. 2008) was identified in southern Oph B1 where we find B1-N3 and B1-N4. The authors found a clump mass M = 0.2 M_☉, which is a factor of ∼2 less than the virial mass M_vir = 0.4 M_☉ we calculate based on the mean line width (of the single velocity component fit) of B1-N3 and B1-N4 (Δv = 0.52 km s⁻¹, or σ = 0.22 km s⁻¹) and the continuum clump radius. This suggests that the clump is currently stable against collapse. If the clump is gaining mass through ongoing accretion from the ambient gas, however, then eventually the non-thermal support may not be large enough to prevent gravitational collapse, leading to the formation of a low-mass protostar.

We show in Figures 13(c) and (d) the distribution of total line opacity τ and excitation temperature T_ex derived from hyperfine line fitting of NH₃ (1,1) and N₂H⁺ 1-0 in Oph B. We find higher total opacities in N₂H⁺ emission (〈τ〉 = 2.5) than in NH₃ emission (〈τ〉 = 1.0) in Oph B. The N₂H⁺ emission is characterized by lower excitation temperatures than those found for NH₃ emission in Paper I, with a mean T_ex = 7.1 K for N₂H⁺ 1-0 compared with T_ex = 9.5 K for NH₃ (1,1).

We next compare the column densities of NH₃ and N₂H⁺ toward the Oph B Core. We only show results for pixels which have well-determined values for both N(NH₃) and N(N₂H⁺). In Paper I, we found a trend of decreasing fractional NH₃ abundance with increasing N(H₂) for both B1 and B2. After convolving the NH₃ data to match the single-dish N₂H⁺ resolution, we find the following linear relationships between log X(NH₃) and log N(H₂):

$$\begin{eqnarray} &&\mbox{B1}{:} \log X(\mbox{NH$_3$}) = (5 \pm 4) - (0.6 \pm 0.2) \log N(\mbox{H$_2$}), \end{eqnarray} \tag{ 6 }$$

$$\begin{eqnarray} &&\mbox{B2}{:} \log X(\mbox{NH$_3$}) = (3.5 \pm 2.2) - (0.5 \pm 0.1) \log N(\mbox{H$_2$}).\quad \end{eqnarray} \tag{ 7 }$$

The slopes found for Oph B1 and B2 are both negative and similar to that found in Equation (5) for the N₂H⁺ fractional abundance as a function of N(H₂). This result is suggestive of a trend of increasing depletion of both NH₃ and N₂H⁺ with increasing N(H₂) in Oph B.

We show in Figure 15 the NH₃ column density, N(NH₃), versus the N₂H⁺ column density, N(N₂H⁺), in Oph B1 and B2, and also the ratio of N(NH₃) to N(N₂H⁺) as a function of N(H₂), calculated in Section 4.1.2 (we do not show results in B3 due to the small number of pixels with both well-determined NH₃ and N₂H⁺ column densities, but note mean values below). Note that the column density ratio, N(NH₃)/N(N₂H⁺), is equivalent to the ratio of fractional abundances, X(NH₃)/X(N₂H⁺). Figure 15(a) shows a significant difference in the N(NH₃)/N(N₂H⁺) ratio between Oph B1 and B2. We find the mean and the standard deviation of this ratio are each twice as large in B1 (N(NH₃)/N(N₂H⁺) = 135, σ = 52) as in B2 (N(NH₃)/N(N₂H⁺) = 65, σ = 31) and B3 (N(NH₃)/N(N₂H⁺) = 80, σ = 56). In Oph B1 and B2, Figure 15(a) illustrates that this difference is largely driven by lower N(N₂H⁺) values toward B1, while the spread in the N(NH₃)/N(N₂H⁺) appears due to a wider spread of N(NH₃) values in B1. We find no variation in the N(NH₃)/N(N₂H⁺) ratio as a function of N(H₂).

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In the clustered star-forming region IRAS 20293+3952, Palau et al. (2007) found strong NH₃ and N₂H⁺ differentiation associated with the presence or lack of embedded YSOs. Low N(NH₃)/N(N₂H⁺) ∼ 50 values were found near an embedded YSO cluster, while higher N(NH₃)/N(N₂H⁺) ∼ 300 were found toward cores with no associated YSOs. In a study of two dense cores, each containing a starless main body and a YSO offset from the core center, Hotzel et al. (2004) find a factor of ∼2 smaller X(NH₃)/X(N₂H⁺) values toward the starless cores compared with Palau et al., with abundance ratios (∼140–190, toward the starless gas, and ∼60–90 toward the YSOs) similar to those found in this study. The relative abundance variation found by Hotzel et al. is driven by a varying NH₃ abundance while X(N₂H⁺) remains constant over the cores. Although we find both a decrease in N₂H⁺ column density as well as an increase in NH₃ column density drives the greater fractional NH₃ abundance toward Oph B1 (see Figure 15), both X(N₂H⁺) and X(NH₃) are greater in B1, by factors of ∼2 and ∼4, respectively, compared with B2 given the lower H₂ column densities found in B1 in Section 4.1.2. In a survey of 60 low-mass cloud cores, Caselli et al. (2002a) find correlations between N(NH₃) and N(N₂H⁺) in starless cores, but the relative column density values do not vary significantly between starless and protostellar objects.

Based on this work and the studies described above, it appears that the relative fractional abundance of NH₃ to N₂H⁺ remains larger toward starless cores than toward protostellar cores by a factor of ∼2–6 in both isolated and clustered star-forming regions. We find a 1σ positive trend in the relative NH₃ to N₂H⁺ abundance in Oph B2 with increasing distance to an embedded protostar, suggestive of a direct impact by the protostars on the relative abundances. In their work on X-ray emission from young protostars in Ophiuchus, Casanova et al. (1995) show that protostellar X-ray emission is potentially significant enough to increase the ionization fraction in nearby dense gas, potentially affecting the local gas chemistry. In this region, the complicated line structures found in both species near the protostars make accurate column density measurements difficult, however, with resulting large uncertainties.

In their models of collapsing prestellar cores, Aikawa et al. (2005) found that NH₃ can be enhanced relative to N₂H⁺ in core centers at high densities (n = 3 × 10⁵–3 × 10⁶ cm⁻³) due to dissociative recombination reactions of N₂H⁺ and e⁻ to form NH and N. NH then reacts with H⁺₃ and H₂ to form NH₃. The N₂H⁺ recombination reaction is dominant only where CO is depleted (if CO is abundant, N₂H⁺ is destroyed mainly through proton transfer to CO), and occurs when the abundance ratio of CO to electrons, n(CO)/n_e ≲ 10³. When a collapsing core reaches higher central densities (n ∼ 10⁷ cm⁻³), however, Aikawa et al. predict that the N(NH₃) will begin to decrease toward the core center due to depletion, while N(N₂H⁺) continues to increase, leading to a higher fractional N₂H⁺ abundance relative to NH₃ at later times. This prediction is in agreement with X(NH₃)/X(N₂H⁺) results since B2 is more evolved than B1, having already formed protostars, but seems inconsistent with the common slope we find for the decrease in X(NH₃) and X(N₂H⁺) with N(H₂) in both Cores. Detailed modeling of the physical and chemical structure of the (non-spherical and clumpy) Cores is beyond the scope of this study, but would help to constrain the central density and abundance structure as a function of Core radius needed for a direct comparison with the Aikawa et al. models.