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Appendix A: Analysis of NH₃(1, 1) and (2, 2) spectra

In this appendix we discuss the methods used for calculating physical parameters from observations of the ammonia NH₃ (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 NH₃ 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 co-added with weights inversely proportional to the mean variance of the noise per channel, rms^-2, estimated from the channels without emission.

The NH₃ 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 full-width 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 NH₃(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 catalog⁴), 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 NH₃ 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 = πθ²/(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 black-body background radiation temperature, and J_ν(T) is defined by (A.5)For optically thin lines (e.g., τ₁₁ ≪ 1), Eq. (A.1) degenerates into (A.6)and it is not possible to determine and τ₁₁ separately. Here ΔV_i = v − V_LSR + v_i.

For some molecular cores we observe two-component NH₃ 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 back-end 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 χ² function (A.9)in the five-dimensional space of the following model parameters: and Δv. Attempting to exclude at the 3σ level all noise-only channels in between the hf lines, the sum runs over the channels with observed NH₃ emission from all hf components. The minimum of χ² 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 χ². 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 T₀ = hν/kB, and J_bg = J_ν(T_bg). At T_bg = 2.725 K and NH₃-inversion frequencies, we have J_bg = 2.20 K and T₀ = 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 density⁵ 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 NH₃ are mainly populated via collisions with molecular hydrogen⁶, 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). NH₃(1, 1) emission usually arises from regions with n_gas ≳ 10⁴ 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 NH₃ 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 H₂ 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 K-ladders of the (J,K) levels are forbidden and because the unknown filling factor η is canceled in the line intensity ratio of the NH₃ (1, 1) and (2, 2) transitions, the estimate of the rotational temperature is more reliable than the excitation temperature.

For a two-level system with a lower kinetic temperature than the energy gap between the (1, 1) and (2, 2) states (ΔE₁₂ = 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 beam-averaged 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 ζ₁₁ = 1.3850 × 10¹³, ζ₂₂ = 1.0375 × 10¹³, and Δv is in km s^-1 (for details, see Ungerechts et al. 1980). As mentioned above, , and for both transitions T₀ = 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 N₁₁ if η ∈  [η_min,1]: (A.18)The uncertainty interval transforms directly into the estimate of the total NH₃ column density: (A.19)which assumes that the relative population of all metastable levels of both ortho-NH₃ (K = 3), which is not observable, and para-NH₃ (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 A-coefficient and C is the rate coefficient for collisional de-excitation. For a typical kinetic temperature in the dense molecular cores of ~10–20 K, the collision coefficient is ~4 × 10^-7  s^-1   [n(H₂)/10⁴  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(H₂) 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(H₂)_min, at η = 1.

Given a fractional NH₃ 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 density⁷. 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 =  [NH₃] / [H₂] 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(NH₃)_min at d_max = θ_mD 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(NH₃), and n(H₂). 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 NH₃ 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 NH₃(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 τ₁₁, the gas density n_H2, and the total ammonia column density N(H₂) 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 single-component Gaussian model to the original data. The residuals between the observed and model spectra are shown in black.
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	Fig. B.2 Same as Fig. B.1 but for the source Do279P6.
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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 single-component Gaussian model to the NH3(1, 1) original data and the upper limit on NH3(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 high-resolution spectra (channel spacing 0.015 km s-1), and the other panels low-resolution spectra (channel spacing 0.077 km s-1). Two NH3 components are marked by ticks and labeled by letters A and B in the upper panels.
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	Fig. B.5 Same as Fig. B.1 but for the source Do321P2.
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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.7 Same as Fig. B.6 but for the source Do279P7.
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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 single-component (two upper panels) and a double-component (three lower panels) Gaussian model to the original NH3 data. Two components of the NH3 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 NH3(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 NH3(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 NH3(1, 1) and (2, 2). The corresponding numerical values are listed in Table B.4.
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© ESO, 2013