Issue 
A&A
Volume 567, July 2014



Article Number  A78  
Number of page(s)  18  
Section  Interstellar and circumstellar matter  
DOI  https://doi.org/10.1051/00046361/201423732  
Published online  14 July 2014 
Online material
Appendix A: Ammonia spectra toward SS1, SS2, and L1251C and derived physical parameters
The ammonia spectra were analyzed in the same way as in Paper I. The radial velocity V_{c} ( ≡V_{LSR}), the linewidth Δv (FWHP), the optical depths τ_{11} and τ_{22}, the integrated ammonia emission ^{∫}T_{MB}dv, and the kinetic temperature T_{kin}, are well determined physical parameters, whereas the excitation temperature T_{ex}, the ammonia column density N(NH_{3}), and the gas density n_{H2} are less certain since they depend on the beam filling factor η, which is not known for unresolved cores (see Appendix A in Paper I).
The 1σ errors of the model parameters were estimated from the diagonal elements of the covariance matrix calculated for the minimum of χ^{2}. The error in V_{c} was also estimated independently by the Δχ^{2} method (e.g., Press et al. 1992) to control both results. When the two estimates differed, the larger error was adopted. An independent control gives an analytical estimate of the uncertainty of the Gaussian line center by Landman et al. (1982): (A.1)where Δ_{ch} and Δv are the channel width and the linewidth, respectively, and the parameter rms is the root mean square noise level.
The results of our analysis are presented in Tables 1, A.1, and A.2. In Tables A.1 and A.2, Cols. 11 and 12 list the total optical depth τ_{tot} (see Eq. (A.4)) which is the maximum optical depth that an unsplit (J,K) = (1,1) or (2, 2) rotational line would have at the central frequency if the hfs levels were populated with the same excitation temperature for the two lines (1, 1) and (2, 2). Column 13 lists the total column density N_{tot}(NH_{3}) defined in Eq. (A.5).
For high signaltonoise ratio (S/N) data (rms ~ 0.05 K per 0.039 km s^{1} channel and T_{MB}~ 2 K) and a narrow line (Δv~ 1 km s^{1}), Eq. (A.1) gives σ_{V} ~ 0.003 km s^{1}, which is in line with the precision estimated from the covariance matrix. However, the accuracy of the line position centering is probably a few times lower taking into account irregular shifts () of the radial velocities V_{c} measured in our Effelsberg spectra (Levshakov et al. 2013b).
The errors of Δv depend on the S/N ratio and vary from ~ 0.005 km s^{1} for strong ammonia lines (S/N ≳ 30) to ~ 0.03 km s^{1} if S/N ≲ 10. Since the uncertainty in the amplitude scale calibration was ~ 20% (Sect. 2), the same order of magnitude errors are obtained for T_{MB}, T_{rot}, T_{kin}, τ_{11}, and τ_{22}. For T_{ex}, N_{tot}(NH_{3}), and n_{H2}, we estimated lower bounds corresponding to the filling factor η = 1 (Paper I).
Examples of the NH_{3}(1, 1) and (2, 2) spectra observed toward SS1, SS2A, SS2B, SS2C, and L1251C are shown in Figs. A.1−A.5. The measured parameters at the positions with both detected inversion transitions (1, 1) and (2, 2) are listed in Tables A.1 and A.2.
Fig. A.1
Ammonia NH_{3}(1, 1) and (2, 2) spectra toward the molecular core SS1. The channel spacing is 0.039 km s^{1}; the spectral resolution (full width at half peak, FWHP) is 0.045 km s^{1}. The red curves show the fit of a onecomponent Gaussian model to the original data. 

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Fig. A.2
Inverse ammonia NH_{3}(1, 1) spectra (i.e., −1 × T_{MB}, where T_{MB} is the originally measured main beam brightness temperature) toward the second serendipitously detected molecular core SS2C to show its signals, as obtained from offpositions, in emission. The “absorption” feature seen in the blue wing of the central NH_{3} line at offsets (0′′,0′′), (− 40′′,0′′), and (40′′,0′′) is the Doppler shifted ammonia emission line detected at offsets (320′′,440′′), (280′′,440′′), and (360′′,440′′) toward the source SS2B. The channel spacing is 0.039 km s^{1}; the spectral resolution FWHP = 0.045 km s^{1}. The red curves show the fit of a onecomponent Gaussian model to the original data. 

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Fig. A.3
Ammonia NH_{3}(1, 1) and (2, 2) spectra toward the molecular core SS2A. The channel spacing is 0.039 km s^{1}; the spectral resolution FWHP = 0.045 km s^{1}. The red curves show the fit of a one or twocomponent Gaussian model (the latter is marked by two vertical ticks at the position of the central NH_{3} line) to the original data. 

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Fig. A.4
Same as Fig. A.3 but for the serendipitously detected starless molecular core SS2B. 

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Fig. A.5
Ammonia NH_{3}(1, 1) and (2, 2) spectra toward the source L1251C. The channel spacing is 0.039 km s^{1}; the spectral resolution FWHP = 0.045 km s^{1}. The red curves show the fit of a onecomponent Gaussian model to the original data. The model parameters are listed in Table A.2. 

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The physical parameters, such as the total optical depth of an inversion line, τ_{tot}, the radial velocity, V_{c}, the linewidth, Δv, and the amplitude, , were estimated by fitting a Gaussian model, T_{syn}(v), to the observed spectrum, T_{obs}(v), by means of a χ^{2}minimization procedure: (A.2)where (A.3)and the optical depth τ(v) at a given radial velocity v is (A.4)Here, n is the number of magnetic hfs components of the inversion transition (n = 18 and 21 for the (J,K) = (1, 1) and (2, 2) inversion transitions), r_{i} is the relative intensity of the ith hfs line, and v_{i} is its velocity separation from the fiducial frequency (these parameters are taken from Kukolich 1967; and Rydbeck et al. 1977).
The total NH_{3} column density is defined by (A.5)where the relative population of all metastable levels of both orthoNH_{3} (K = 3), which is not observable, and paraNH_{3} (K = 1,2) is assumed to be governed by the rotational temperature T_{rot} of the system at thermal equilibrium (Winnewisser et al. 1979).
If both inversion lines NH_{3}(1, 1) and (2, 2) are detected, we can estimate the kinetic temperature T_{kin} (Eq. (A.15) in Paper I) and find the thermal contribution to the observed linewidth, v_{obs} =Δv/. The thermal velocity is defined by (A.6)where k_{B} is the Boltzmann constant, and m is the mass of a particle. For NH_{3}, it is (km s^{1}).
If the contribution of the thermal v_{th} and nonthermal (turbulent) v_{turb} components to the linewidth are independent of one another, then (A.7)and the nonthermal velocity dispersion σ_{turb} along the line of sight is . This value can be compared with the thermal sound speed (A.8)where P_{th} is the thermal pressure and ρ the gas density. For an isothermal gas (), (km s^{1}).
The Mach number is defined locally as (A.9)The relative errors δ_{Δv} in Δv and δ_{T} in T_{kin} propagate into the relative error of the Mach number, δ_{ℳ}, as (A.10)where δ_{T} ~ 0.2 and δ_{Δv} ~ 0.01−0.05 (see Table 1).
The value of the linewidth Δv and the core radius R can be used to estimate the virial mass (e.g., Lemme et al. 1996): (A.11)where Δv is in km s^{1}, R in pc, and M_{vir} in solar masses M_{⊙}.
© ESO, 2014
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