Issue 
A&A
Volume 553, May 2013



Article Number  A58  
Number of page(s)  25  
Section  Interstellar and circumstellar matter  
DOI  https://doi.org/10.1051/00046361/201220354  
Published online  06 May 2013 
Online material
Molecular cores toward Aquila targets without detected NH_{3} emission in the (1, 1) and (2, 2) inversion transitions.
Appendix A: Analysis of NH_{3}(1, 1) and (2, 2) spectra
In this appendix we discuss the methods used for calculating physical parameters from observations of the ammonia NH_{3} (J,K) = (1, 1) and (2, 2) transitions. We give a consistent summary of the basic formulae relevant to the study of cold molecular cores.
The recorded NH_{3} spectra were first converted into the T_{MB} scale and a baseline was removed from each of them. The baseline was typically linear in the range covered by the ammonia hyperfine (hf) structure lines, but occasionally it was quadratic. After that the spectra of the same offset were coadded with weights inversely proportional to the mean variance of the noise per channel, rms^{2}, estimated from the channels without emission.
The NH_{3} spectra were fitted to determine the total optical depth τ_{tot} in the respective inversion transition, the LSR velocity of the line V_{LSR}, the intrinsic fullwidth at half power linewidth Δv for an individual hf component, and the amplitude, (e.g., Ungerechts et al. 1980): (A.1)The optical depth τ(v) at a given radial velocity v is defined by (A.2)In this equation, the sum runs over the n hf components of the inversion transition (n = 18 and 21 for the NH_{3}(1, 1) and (2, 2) inversion transitions), r_{i} is the theoretical relative intensity of the ith hf line and v_{i} is its velocity separation from the fiducial frequency. The values of these parameters are given in, e.g., Kukolich (1967) and Rydbeck et al. (1977).
In Eq.(A.2), the total optical depth τ_{tot} is the maximum optical depth that an unsplit (1, 1) or (2, 2) line would have at the central frequency if the hf levels were populated at the same excitation temperature for the two lines (1, 1) and (2, 2). Assuming that the line profile function has a Gaussian shape with width Δv (FWHM) and taking into account that the statistical weights of the upper and lower levels of an inversion transition are equal, one obtains (A.3)where h and k_{B} are Planck and Boltzmann constants, μ is the dipole moment,  μ  = 1.4719 Debye (JPL catalog^{4}), N(J,K) is the inversion state column density – the sum of the column densities of the upper and lower levels of an inversion doublet, ν is the rest frequency of the inversion transition, and T_{ex} is the excitation temperature that characterizes the NH_{3} population across the (J,K) inversion doublet.
In Eq. (A.1), the amplitude can be expressed in terms of the beam filling factor, η = Ω_{cloud}/Ω_{beam} (solid angle Ω = πθ^{2}/(4ln2), where θ is the angular diameter at FWHP), and the excitation temperature, T_{ex}, as (A.4)where T_{bg} = 2.7 K is the blackbody background radiation temperature, and J_{ν}(T) is defined by (A.5)For optically thin lines (e.g., τ_{11} ≪ 1), Eq. (A.1) degenerates into (A.6)and it is not possible to determine and τ_{11} separately. Here ΔV_{i} = v − V_{LSR} + v_{i}.
For some molecular cores we observe twocomponent NH_{3} profiles. The line parameters for the j = 1 and j = 2 components were determined in this case by fitting the function (A.7)This linear form assumes that two clouds in the beam have the same filling factor η and the same T_{ex} for the lines (1, 1) and (2, 2).
At low spectral resolutions, when the channel spacing Δ_{ch} is near to the linewidth Δv, the derived line parameters are affected by the channel profile φ(v). In this case, the recorded spectrum, T′(v), was considered as a convolution of the true spectrum T(v) and φ(v) (A.8)We approximate φ(v) by a Gaussian function with the width FWHM defined by a particular backend setting.
The calculated synthetic spectrum, which is defined as T_{syn}(v) ≡ T(v) if Δ_{ch} ≪ Δv, or T_{syn}(v) ≡ T′(v) if Δ_{ch} ~ Δv, was fitted to the observed spectrum T_{obs}(v) by minimizing the χ^{2} function (A.9)in the fivedimensional space of the following model parameters: and Δv. Attempting to exclude at the 3σ level all noiseonly channels in between the hf lines, the sum runs over the channels with observed NH_{3} emission from all hf components. The minimum of χ^{2} was computed using the simplex method (e.g., Press et al. 1992).
Equation (A.9) was also used to estimate the formal errors of the model parameters by calculating the covariance matrix at the minimum of χ^{2}. Since the uncertainty in the amplitude scale calibration was ~15–20% (Sect. 2), and there were no noticeable correlations between the sequential channels in our datasets, we did not correct the calculated errors by an additional factor as described, e.g., in Rosolowsky et al. (2008).
Given the estimate of the amplitude , the excitation temperature T_{ex} can be calculated from Eq. (A.4) (A.10)where T_{0} = hν/kB, and J_{bg} = J_{ν}(T_{bg}). At T_{bg} = 2.725 K and NH_{3}inversion frequencies, we have J_{bg} = 2.20 K and T_{0} = 1.14 K.
In this equation the only unknown parameter is the filling factor η. If the source is completely resolved, then η = 1. Otherwise, Eq. (A.10) gives a lower bound on T_{ex} corresponding to η = 1, whereas decreasing η drives T_{ex} toward its upper bound at T_{ex} = T_{kin}, which holds for LTE. For emission lines, the radiation temperature within the line is higher than the 2.7 K background and the excitation may be either radiative or collisional. At gas densities higher than the critical density^{5} and for optically thin lines, the local density of line photons is negligible compared to the background radiation field, and the lower metastable states in NH_{3} are mainly populated via collisions with molecular hydrogen^{6}, i.e., the stimulated emission can be neglected. For low collisional excitation, n_{gas} < n_{cr}, the line radiation controls T_{ex} for both optically thin and thick regimes (Kegel 1976). NH_{3}(1, 1) emission usually arises from regions with n_{gas} ≳ 10^{4} cm^{3} (e.g., Ho & Townes 1983). This means that the regime of low collisional excitation does not dominate. We also did not consider anomalous excitation (maser emission) that leads to T_{ex} ≫ T_{bg} because masing is not observed in our data sample. Thus, using T_{ex} = T_{kin} as a formal upper limit for the excitation temperature is legitimate. With this we can restrict the unknown filling factor for an unresolved source in the interval (A.11)and T_{ex} within the boundaries (A.12)From a separate analysis of the (1, 1) and (2, 2) lines with low noise level, we found that the two amplitudes and are equal within the observational errors, which implies that T_{ex}(1, 1) ≈ T_{ex}(2, 2). Albeit radiative transfer calculations show that the excitation temperatures T_{ex}(1, 1) and T_{ex}(2, 2) may differ by about 20% (Stutzki & Winnewisser 1985), assuming that they are equivalent seems to be sufficient in our case.
The second temperature, which describes the NH_{3} population, is the rotational temperature T_{rot}, which characterizes the population of energy levels with different (J,K). As mentioned above, the population of the lower metastable inversion doublets is determined by collisions with H_{2} and, thus, is regulated by the kinetic temperature T_{kin}. The population ratio between the (1, 1) and (2, 2) states is defined as (A.13)Assuming that both transitions trace the same volume of gas and their linewidths Δv are equal, one finds from Eqs. (A.3) and (A.13) that (A.14)We note that since radiative transitions between the different Kladders of the (J,K) levels are forbidden and because the unknown filling factor η is canceled in the line intensity ratio of the NH_{3} (1, 1) and (2, 2) transitions, the estimate of the rotational temperature is more reliable than the excitation temperature.
For a twolevel system with a lower kinetic temperature than the energy gap between the (1, 1) and (2, 2) states (ΔE_{12} = 41.5 K), the rotational temperature can be related to the kinetic temperature through detailed balance arguments (e.g., Ragan et al. 2011): (A.15)The beamaveraged column density N_{JK} (in cm^{2}) can be calculated from Eqs. (A.3) and (A.4) using the estimated values of the amplitude , the total optical depth τ_{JK}, and the linewidth Δv: (A.16)where ζ_{11} = 1.3850 × 10^{13}, ζ_{22} = 1.0375 × 10^{13}, and Δv is in km s^{1} (for details, see Ungerechts et al. 1980). As mentioned above, , and for both transitions T_{0} = 1.14 K and J_{bg} = 2.20 K. Substituting these numerical values in Eq. (A.16), one obtains (A.17)This gives us lower and upper boundaries of N_{11} if η ∈ [η_{min},1]: (A.18)The uncertainty interval transforms directly into the estimate of the total NH_{3} column density: (A.19)which assumes that the relative population of all metastable levels of both orthoNH_{3} (K = 3), which is not observable, and paraNH_{3} (K = 1,2) is governed by the rotational temperature of the system at thermal equilibrium (Winnewisser et al. 1979). Substituting and from (A.18) into (A.19), one finds: (A.20)The detailed balance calculations also provide a relation between the gas density, gas kinetic temperature, and the excitation temperature (e.g., Ho & Townes 1983): (A.21)where A is the Einstein Acoefficient and C is the rate coefficient for collisional deexcitation. For a typical kinetic temperature in the dense molecular cores of ~10–20 K, the collision coefficient is ~4 × 10^{7} s^{1} [n(H_{2})/10^{4} cm^{3}]. The value of A for the inversion transition (1, 1) is 1.67 × 10^{7} s^{1}. However, the gas density calculated using Eq. (A.21) may be significantly underestimated if the beam is not filled uniformly. Moreover, for T_{ex} = T_{kin}, Eq. (A.21) is invalid and, hence, n(H_{2}) needs to be calculated by different methods (see, e.g., Hildebrand 1983; Pandian et al. 2012). We used (A.21) just to set a lower bound on the gas density, n(H_{2})_{min}, at η = 1.
Given a fractional NH_{3} abundance, an upper bound on the gas density may be estimated from the deduced N(NH assuming that the ammonia emission traces the real distribution of the gas density^{7}. If the source is unresolved, its diameter d and the beam filling factor are related as (A.22)where θ_{m} is the beam angular diameters (FWHP), and D the distance of the source. The highest gas density is obtained at the smallest diameter, : (A.23)where X = [NH_{3}] / [H_{2}] is a given abundance ratio. In this equation the unknown distance D can be found from the requirement that both values of the lowest gas densities calculated at η = 1 from Eq. (A.21) and from N(NH_{3})_{min} at d_{max} = θ_{m}D be equal. This gives (A.24)For the unresolved source its axis – major, θ_{a}, or minor, θ_{b} – is less than θ_{m}, η < 1, and only limiting values can be obtained for T_{ex}, N_{JK}, N(NH_{3}), and n(H_{2}). If the source is resolved, i.e., θ_{b} > θ_{m}, then η = 1 and these physical parameters are directly defined. In the latter case we calculated the deconvolved values of θ_{a} and θ_{b} and their geometrical mean (A.25)which is used as a formal estimate of the source angular diameter.
A similar procedure was applied to calculate the virial mass of the ammonia core through the linewidth, Δv. If the observed value Δv is of order of the spectral resolution Δ_{sp} (FWHM), then only an upper limit on M_{vir} is defined. Otherwise, if Δv> Δ_{sp}, then the deconvolved value of the linewidth, , and the core radius r give the virial mass (e.g., Lemme et al. 1996) (A.26)where is in km s^{1}, r in pc, and M_{vir} in solar masses M_{⊙}.
To conclude, we note that if T_{kin} varies slowly within an ammonia clump, the NH_{3} column density can be deduced from the (1, 1) line alone assuming the same average T_{kin} in the core as in the outskirts of a cloud. This allows us to extend the radial gas distribution to zones with unobservable (2, 2) emission (Morgan et al. 2013).
Appendix B: Ammonia spectra toward the Aquila rift cloud complex and derived physical parameters
The observed spectra of the NH_{3}(1, 1) and (2, 2) transitions detected at the peak positions of ammonia emission toward each core are shown in Figs. B.1–B.9. The measured physical parameters are listed in Tables B.1–B.4. The distributions of the physical parameters for the three most abundant cores are presented in Figs. B.10–B.12. They demonstrate variations of the kinetic temperature T_{kin}, the excitation temperature T_{ex}, the linewidth Δv (FWHM), the total optical depth τ_{11}, the gas density n_{H2}, and the total ammonia column density N(H_{2}) across the mapped area of the core.
Fig. B.1
Ammonia (1, 1) and (2, 2) spectra (blue) toward the source Ka01. The channel spacing is 0.015 km s^{1}, the spectral resolution FWHM = 0.024 km s^{1}. The red curves show the fit of a singlecomponent Gaussian model to the original data. The residuals between the observed and model spectra are shown in black. 

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Fig. B.3
Same as Fig. B.1 but for the source Do279P8 and the channel spacing 0.077 km s^{1} (FWHM = 0.123 km s^{1}). The red curves show the fit of a singlecomponent Gaussian model to the NH_{3}(1, 1) original data and the upper limit on NH_{3}(2, 2). 

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Fig. B.4
Same as Fig. B.1 but for the source Do279P12. The ammonia spectra have a double structure. The upper panels show the highresolution spectra (channel spacing 0.015 km s^{1}), and the other panels lowresolution spectra (channel spacing 0.077 km s^{1}). Two NH_{3} components are marked by ticks and labeled by letters A and B in the upper panels. 

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Fig. B.6
Same as Fig. B.1 but for the source Ka05 and the channel spacing 0.038 km s^{1} (FWHM = 0.044 km s^{1}). 

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Fig. B.8
Same as Fig. B.1 but for the source SS3. The channel spacing is 0.77 km s^{1} and 0.038 km s^{1} in the two upper and three lower panels. The corresponding spectral resolutions are 0.895 km s^{1} and 0.044 km s^{1} (FWHM). The red curves show the fit of a singlecomponent (two upper panels) and a doublecomponent (three lower panels) Gaussian model to the original NH_{3} data. Two components of the NH_{3} emission are marked by ticks and labeled by letters A and B. 

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Fig. B.9
Same as Fig. B.3 but for the sources Do243P2, Do279P18, Do279P13, and Do321P1. 

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Fig. B.10
Spatial distributions of the physical parameters measured in Do279P6 from the ammonia inversion lines NH_{3}(1, 1) and (2, 2). The corresponding numerical values are listed in Table B.2. 

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Fig. B.11
Spatial distributions of the physical parameters measured in Do279P12 from the ammonia inversion lines NH_{3}(1, 1) and (2, 2). The corresponding numerical values are listed in Table B.3. 

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Fig. B.12
Spatial distributions of the physical parameters measured in SS3 from the ammonia inversion lines NH_{3}(1, 1) and (2, 2). The corresponding numerical values are listed in Table B.4. 

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Observed parameters of the NH_{3}(1, 1) and (2, 2) lines and calculated model parameters for Kawamura 01 and 05, Dobashi 279 P7 and P8, and Dobashi 321 P2.
© ESO, 2013
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