VERY LARGE ARRAY OBSERVATIONS OF AMMONIA IN INFRARED-DARK CLOUDS. I. COLUMN DENSITY AND TEMPERATURE STRUCTURE

We acquired single-dish observations of NH₃ (J, K) = (1,1) and (2,2) inversion lines using the Robert C. Byrd GBT from 2006 September 6 to 15. The rest frequencies of the (1,1) and (2,2) lines are 23.6944955 and 23.7226336 GHz (Ho & Townes 1983), respectively. At these frequencies, the beam FWHM was approximately 32''. These upper K-band observations were made in frequency-switching mode. The GBT spectrometer back end was configured to simultaneously observe the two transitions in separate spectral windows, using ∼32,768 spectral channels in each 50 MHz wide band. The spectral resolution for the (1,1) line was 0.03 km s⁻¹ or 1.6 kHz.

We mapped the regions using a single-pointing grid with Nyquist sampling, the parameters of which are summarized in Table 1. Integration times were 2–3 minutes per point, depending on the strength of the line, elevation of the source, and thus the time required to obtain sufficient signal to noise. Pointing corrections were done toward various calibrators depending on the time of observation (1733−1304, 1814−1853, 1822−0938, 1831−0949, and 1851+0035) every 45–75 minutes, weather depending, which resulted in corrections typically of a few arcseconds. System temperatures were between 60 and 100 K, and the elevation of the sources ranged between 20 and 40 throughout the observations.

Data were reduced and calibrated using GBTIDL.⁴ The frequency-switched observations were reduced with the getfs routine, which retrieves, calibrates, and plots the spectrum. We then applied hanning smoothing and then boxcar smoothing over 50 channel bins. Five of the spectral components of the NH₃ signature were always spectrally resolved, and we performed Gaussian fitting on each hyperfine component. The data were then put into a FITS data cube using a homemade IDL-based script.

Observations of the NH₃ (1,1) and (2,2) inversion transitions of the sample were undertaken in the compact D configuration of the VLA over the course of three tracks in 2007 April. At this time, the array consisted of a hybrid of VLA and the updated EVLA receivers, and Doppler tracking was unavailable. The observations were conducted in fixed-frequency mode. Bandpass and calibration were done with observations of 1331+305 and 2253+161. Phase calibration was done with observations of 1833 − 210, 1832 − 105, and 1851+005 every ∼12–15 minutes, and pointing was also performed on these sources every ∼ hour. The data were flagged and calibrated using NRAO Astronomical Image Processing System (AIPS⁵) according to the method described in the AIPS cookbook, taking into account corrections for the absence of Doppler tracking.

These K-band observations were made using a four IF correlator backend configured with 3.125 MHz bandwidth containing 64 channels, yielding a spectral resolution of 48.8 kHz or 0.6 km s⁻¹. This setup was selected to ensure detection of relatively weak lines in IRDCs, both the (1,1) and (2,2) simultaneously, with enough spectral resolution to resolve the lines. Unlike the GBT setup, this configuration only fits the central three of the five main components of the ammonia signature (see Figure 1). We discuss the effects of this in Section 3. Table 1 gives the target, pointing, and sensitivity information for each IRDC.

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The detectable size of structures has an upper limit when using an interferometer, which is set by the shortest spacing between two antennas in the array. Any flux on scales larger than this upper limit will be missing from the final image, a phenomenon known as the short-spacing problem. Because IRDCs are complex structures, with emission on all scales, it is necessary to recover any large scale emission by combining the interferometer data with single-dish observations at the same frequency. The 100 m GBT offers the perfect complement to our D array observations with the VLA.

Following Friesen et al. (2009), the GBT data were first scaled and regridded to match the resolution of the VLA data. Using the MIRIAD task IMMERGE, the deconvolved interferometer images were combined in the Fourier domain with the GBT images. IMMERGE uses a tapering function which weights short spacings higher than long spacings. Using an annulus ranging from 30 to 70 m, where the high- and low-resolution images overlap, we ensure accurate determination of the flux calibration factor. The combined data sets were binned in velocity to 0.6 km s⁻¹, which reduced the level of random errors. This method was applied to the (1,1) and (2,2) data sets independently, yielding images of the same spatial and velocity resolution as the VLA image. Table 1 gives the final beam dimensions, as well as other source parameters, for each object.

Ammonia has long been used as a reliable tracer of high-density gas in studies of low-mass cores (Jijina et al. 1999; Rosolowsky et al. 2008; Friesen et al. 2009) and high-mass cores (Harju et al. 1993; Molinari et al. 1996; Wu et al. 2006), where other molecules, such as CO, are depleted (Bergin & Langer 1997). The initial studies of ammonia in IRDCs (Pillai et al. 2006; Wang et al. 2008) show that ammonia is also a good tracer in these environments that have the potential to form clusters containing massive stars. The ammonia column density can be found through analysis of the (1,1) hyperfine structure, the derivation of which can be found in Appendix A.1.

In Figure 2, we show ammonia column density contours plotted over the 8 μm Spitzer image for each IRDC. The resolution element is shown in the lower-left corner of each panel. In Figure 3, we plot the column density of NH₃ as a function of H₂ column density for each IRDC. The H₂ column density is calculated from the absorption in the Spitzer IRAC 8 μm image⁶ (see Ragan et al. 2009). For these analyses, the H₂ column density map was convolved to match the VLA resolution for each IRDC, thereby lowering the inferred column density (see Vasyunina et al. 2009). In Figures 3, we do not plot data from regions where the NH₃ line opacity exceeds 5, as it is not used in our abundance fits (see below).

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In most cases we find a one-to-one relationship between these two quantities. The slope of the linear relation is an estimate of the ammonia abundance, $\chi _{\rm NH_3}$. Our fits only include regions in the cloud that have low line optical depths (τ_m(1, 1) ≲ 5). Individual abundance values are given in Table 2, except G009.28−0.15 for which no linear fit was possible in this case, although most points (in black) show a one-to-one relation similar to the other IRDCs. We investigated where the apparently anomalous emission originated and found that "high" values (in green) are confined to a southern region and the "low" values are computed for the northern tip. Apart from being contiguous, these data points are not otherwise distinct from the rest of the cloud (i.e., not an optical depth effect) or the rest of the sample; therefore we do not discard them for the sake of the desired fit. Our restriction on the line opacity most heavily affects the data fit for G023.37−0.29, which excludes all of the central region where the NH₃ (1,1) lines are saturated. Ragan et al. (2009) also report the highest values of τ₈ which suggests that N(H₂) in the central part of G023.37−0.29 is also an underestimate. Thus, the abundance here is calculated for the outer edges of the cloud where both tracers are least hampered by optical depth effects.

The IRDCs G005.85−0.22, G009.86−0.04, G024.05−0.22, G034.74−0.12, and the outer regions of G023.37−0.29 show good correlation between the column densities of H₂ and ammonia. The average of these five IRDC abundances is 8.1 × 10⁻⁷, plotted in red on each panel. IRDCs with no evidence of embedded star formation activity show the tightest relation between ammonia and H₂ column densities.

Our abundance value is higher than 4 × 10⁻⁸ found by Pillai et al. (2006) in a single-dish study of ammonia in IRDCs. Though we report similar H₂ column densities, we probe higher NH₃ column densities, namely because our study has effective beams at least five times smaller than single-dish studies. In a single-dish study, the compact emission is beam diluted which would result in lower NH₃ intensity measurements than what we find here with the VLA.

The temperature in cold molecular gas is attainable with knowledge of the optical depth of the line and the ratio of NH₃ (1,1) and (2,2) central line intensities. The derivation of the kinetic temperature is given in Appendix A.2.

This sample contains objects both with and without signatures of star formation. In Figure 4, we present the kinetic temperature maps in all six IRDCs with contours showing the intensity of NH₃ (1,1) and symbols indicating the locations of young stars. The Class II sources (the most evolved YSOs displayed) are the most widely distributed population and do not follow the dense gas traced by ammonia. As such, they are unlikely to have a strong impact on the gas compared to the embedded protostars. The temperature is only calculated for positions with detections three times the rms noise. Since the (2,2) emission is typically much weaker than the (1,1) emission, the temperature map boundaries are limited by the (2,2) signal to noise. Kinetic temperatures range from 8 to 13 K. In Figure 5, we present the kinetic temperature as a function of the molecular hydrogen column density (N(H₂)) derived from 8 μm absorption⁷ (Ragan et al. 2009). Table 2 lists the average temperature associated with each IRDC.

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IRDCs G005.85−0.23 and G024.05−0.22 lack any 24 μm point sources coincident with the gas traced by NH₃. We infer that these IRDCs contain no embedded star formation, though there are stars projected in the vicinity (shown in Figure 4). In both cases, the NH₃ intensity peaks at the same position of both the 8 and 24 μm absorption peaks. The average temperature profiles (Figure 5) in these two starless IRDCs are the flattest of the entire sample (〈T_kin〉 varies by ∼1 K in both cases), which is consistent with these objects being least influenced by radiation sources.

The four remaining IRDCs in the sample (G009.28−0.15, G009.86−0.04, G023.37−0.29, and G034.74−0.12) coincide with at least one 24 μm point source, indicating that these objects likely host embedded protostars. The "northern" region of G009.28−0.15 (shown in cyan in Figures 2 and 3), which coincides with two 24 μm point sources, appears to have a slightly elevated temperature compared to the "southern" region (shown in green in Figures 2 and 3), which appears infrared-dark. While these regions are each spatially coherent, the absolute difference in the mean temperature is within the temperature measurement error shown in Figure 5.⁸

There are six 24 μm point sources in G009.86−0.04, five of which are concentrated in the eastern portion of the cloud where it is not possible to calculate the temperature because of very weak (2,2) emission. The kinetic temperature at the integrated intensity peak is 9 K, and the point source to the southwest has an elevated temperature of 13 K. We see the strongest temperature gradient in this source (see Figure 5), with the lowest temperatures tending toward the highest optical depths.

IRDC G023.37−0.29 exhibits the strongest NH₃ emission in the entire sample, and we also estimate the highest abundance in this object. The lines near the center of this object are saturated, indicating that the emission is very optically thick. The temperature then is likely not representative of the conditions near to the 24 μm source near the center of the object, which could explain why we do not see any elevation in kinetic temperature as we do for the other IRDCs in the sample. It is likely that this object would potentially exhibit strong emission in higher J, K lines and, therefore, be at a higher temperature not probable by the present observations.

In the IRDC G034.74−0.12, emission is widespread in both the (1,1) and (2,2) transitions. We find the highest ammonia column densities in this object (∼3 × 10¹⁶ cm⁻²). This map is also the noisiest, and the (2,2) emission map only passed our 3σ detection limit test at the two strong peaks also in (1,1) emission. If the noise requirement for the (2,2) emission is relaxed, the resulting temperature across the source is very uniform, between 8 and 9 K. Four 24 μm point sources are detected on the IRDC, and the IRAS point source 18526+0130 (L_bol ∼ 10⁴ L_☉; Robitaille et al. 2007) is 30'' north of the NH₃ map boundary, corresponding to 0.7 pc at the distance to this IRDC, though no direct influence of the IRAS source is evident in our temperature measurements.

Throughout the studies of star formation regions with molecular emission, ammonia has been one of the best tracers of cold dense gas, showing no compelling evidence of depletion up to very high densities, n > 10⁵ cm⁻³ (Bergin & Langer 1997). The same holds true in IRDCs, though the ammonia line can become optically thick in regions of very high H₂ column density. We find no evidence of depletion of ammonia in IRDCs; thus it can serve as a reliable tracer of kinematics throughout the bulk of the gas which is not yet forming stars. We will, in the following sections, discuss the limitations of the (1,1) and (2,2) lines in probing warm gas in compact regions near embedded protostars.

We note that our estimated NH₃ abundances are high when compared to other cold objects. For example, Tafalla et al. (2004) find central abundances on the order of ∼2 × 10⁻⁸ in a few cold low-mass cores in Taurus, while Harju et al. (1993) find values from (1 to 5) × 10⁻⁸ in a sample of isolated cores in Orion and Cepheus. Interestingly, Tafalla et al. (2004) find evidence for an increase in the NH₃ abundance in the denser gas. They postulate that perhaps greater CO depletion might lead to enhancements of the ammonia abundance via gas phase reactions. Our sources likely have greater densities and perhaps that induces greater formation of ammonia. However, this question needs to be addressed with more complete modeling spanning a wide range of densities.

We find that the temperatures in this sample of IRDCs, despite spanning from quiescent to active phases, are largely uniform in the bulk of the cloud, ranging only from 8 to 13 K. These results are in general agreement with the single-dish measurements of temperature in IRDCs by Pillai et al. (2006), and also the temperatures found in Perseus (Rosolowsky et al. 2008) and Ophiuchus (Friesen et al. 2009).

Ammonia studies of temperature structure in massive star-forming complexes have focused mainly on Orion. Wiseman & Ho (1998) map the filamentary region surrounding the Orion-KL massive star-forming core complex. One source within this complex is IRc2, which drives a large bipolar outflow. The surrounding molecular ridge, OMC-1, is comprised of several finger-like filaments in which they find the kinetic temperature ranges between 15 and 30 K. In the central core around Orion-KL, they find temperatures near 80 K, and near the edges of the filaments the temperature is elevated to ∼40 K. The heating is detectable up to 0.3 pc away from the central core, and is attributed to radiation and winds from Orion-KL sources and shock interactions near the core where the highest temperatures are found. If such luminous sources were present in IRDCs, we would see greater temperature variation. The low, unvarying temperature of the bulk of the IRDC gas indicates that IRDCs are in an earlier evolutionary stage than Orion.

Li et al. (2003) select quiescent cores far away from the Orion-KL region, a site of active massive star formation. The authors observe a similar temperature trend as the one presented here: the lowest temperatures are found in the cloud interiors and warmer gas is near the edges. They attribute this to external heating from the elevated radiation field due to a nearby active star formation region in the Trapezium, despite the large amount of extinction surrounding the region probed by ammonia. These quiescent Orion cores can be most favorably compared to the IRDCs G005.85−0.23 and G024.05−0.22 in our sample. They show no 24 μm point sources coincident with the dense gas. We see very little heating in these objects, and by comparing the colored region (where there is an NH₃ (1,1) and (2,2) detection) and the contours (NH₃ (1,1) intensity), the region where temperature calculation is possible is already heavily embedded in dense gas, and therefore shielded from the interstellar radiation field.

One high-resolution follow-up study of IRDC G028.34+0.06 (Wang et al. 2008) showed a significant rise in rotation temperature at the sites of 24 μm point sources. In this case, the more evolved, warmer region contains a well-characterized high-mass protostellar object and IRAS source (L ∼ 10³L_☉) and has a peak rotation temperature of 30 K, much higher than detected in any IRDC in this study. In the single-dish study of this object, however, the rotation temperatures across the IRDC were found to be between 13 and 16 K. Thus, Wang et al. (2008) demonstrate that, without the high resolution of an interferometer, these compact features are diluted by the emission from the cold, quiescent bulk of the IRDC.

Our sample of six IRDCs was selected to be at the earliest evolutionary stage, in regions devoid of maser emission, radio emission, and IRAS sources (Ragan et al. 2006). Thus, the lower average kinetic temperature (∼11 K) is consistent with this selection, and this sample of IRDCs are at an earlier stage in their evolution compared to G028.34+0.06. Furthermore, we have no instances of such elevated temperatures as that seen in Wang et al. (2008), indicating that nowhere in these IRDCs is there massive star formation at such an advanced stage, if it exists at all.

The absence of temperature structure does not preclude the presence of star formation. Indeed there exists a surface population of young (low-mass) stars in these IRDCs (Ragan et al. 2009); however, these appear to have a minimal impact on the gas traced with these observations. In the cases where we detect embedded star formation, the effect on the temperature is localized and does not influence the widespread dense gas in IRDCs.⁹

The temperatures determined in this study can place constraints on how fragmentation proceeds in IRDCs. The traditional characteristic fragmentation scale (which assumes isothermality, spherical symmetry, and uniform density) is given by the Jeans length:

$$\begin{equation} \lambda _{J} = \sqrt{\frac{\pi c_s^2}{G \rho }}, \end{equation} \tag{ 1 }$$

where c_s is the isothermal sound speed (0.18 km s⁻¹ for 10 K gas). For an average density of 10⁵ cm⁻³ (ρ ∼ 4 × 10⁻¹⁹ g cm⁻³), the Jeans length is 0.07 pc, which is below the effective physical resolution of our VLA observations (0.1–0.2 pc). The corresponding Jeans mass (M_J = ρλ³_J) is about 2 M_☉. While the detectable star formation activity within these IRDCs is limited to a few objects, in the cases where there are multiple objects of Class I or earlier (G009.28−0.15, G009.86−0.04, and G034.74−0.12), the average separation between protostars is 0.4 pc, within a factor of five of the Jeans length scale. This result is similar to what Teixeira et al. (2006) found in a filamentary star formation region (within a factor of three of λ_J), which is roughly consistent with thermal fragmentation. Given that IRDCs are at distances of several kiloparsec, hampering our sensitivity to faint stars, our scale is an upper limit.

In reality, the material in IRDCs is much more clumped and filamentary than a uniform sphere. An alternative quantification of such a cloud is the critical fragmentation mass per unit length scale in a filament (e.g., Inutsuka & Miyama 1992; André et al. 2010), which gives 15 M_☉ pc⁻¹ for 10 K gas (not taking into consideration any magnetic field support). The mass per unit length in these IRDC filaments is typically a few hundred solar masses per parsec, well in excess of the critical value. This would indicate that these objects are undergoing collapse and actively forming stars. The dynamical state of this IRDC sample will be analyzed in detail in a forthcoming publication (S. E. Ragan et al. 2011, submitted).

In Ragan et al. (2009), we discussed the overall mass distribution of IRDCs in comparison to the Orion Molecular Cloud, which is the local example of massive and clustered star formation. From our absorption study we can estimate a gas mass surface density for IRDCs, Σ_IRDC, between 1000 and 5000 M_☉ pc⁻², while in Orion the mass surface density is Σ_OMC ∼ 30 M_☉ pc⁻². Thus, IRDCs are not distant Orions, but rather something fundamentally different. Here, we examine what role the environment of IRDCs might play in supporting such a large concentration of material where the prevalence of star formation appears relatively low.

Jackson et al. (2008) have examined the spatial distribution of IRDCs which were previously identified with the molecular ring that contains over 70% of the galactic molecular mass. They argue that the distribution is best fitted by a model of the Milky Way with two major spiral arms emerging from a central bar. The first quadrant IRDCs lie preferentially in the Scutum–Centarus arm. This is in agreement with other results probing the dust structure in the galaxy (e.g., Drimmel & Spergel 2001; Benjamin et al. 2005).

Theoretical models of gas dynamics in barred galaxies suggest that gas flow produces arms of gas/dust density enhancement that extend from the leading side of the bar (Athanassoula 1992). These enhancements are the result of shocks and other dynamical processes which result in the formation of molecular clouds from the atomic medium (Combes & Gerin 1985; Bergin et al. 2004). The IRDC surface density corresponds to an average density of $\langle n_{\rm H_2}\rangle \sim 10^5$ cm⁻³. Given a typical uniformly low temperature of ∼10 K, this corresponds to a internal thermal pressure of P_int/k ∼ few × 10⁶ cm⁻³ K, well in excess of that in Orion, where the gas temperature is typically 20 K, or the general interstellar medium (ISM), where P_ISM/k = 2240 cm⁻³ K (Jenkins & Tripp 2001). These high pressures require a strong dynamical interaction to gather ISM gas (cf. Bergin et al. 2004), enabling gravitational collapse to commence and fragmentation to proceed. This interaction may be related to gas flow induced by the central Galactic bar.

We calculate the NH₃ (1,1) column density following Rohlfs & Wilson (2004), taking into account both parity states of the (1,1) energy level (Friesen et al. 2009):

$$\begin{equation} N(1{,}1) = \frac{8\pi \nu _0^2}{c^2} \frac{1}{A(1{,}1)} \frac{\frac{g_1}{g_2} + e^{-h\nu _0/(kT_{{\rm ex}})}}{1 - e^{-h\nu _0/(kT_{{\rm ex}})}} \int \tau (\nu) d\nu, \end{equation} \tag{ A1 }$$

where g₁ and g₂ are the statistical weights for the upper and lower states of the NH₃ (1,1) transition (equal in the case of our two-level approximation), A(1, 1) is the Einstein spontaneous emission coefficient, 1.68 × 10⁻⁷s⁻¹, $\int \tau (\nu) d\nu = \sqrt{2\pi } \sigma _{\nu } \nu _0 \tau _1 / c$ with ν₀ the rest frequency, τ₁ is the total line opacity, and σ_ν is the dispersion (Rosolowsky et al. 2008). T_ex is the gas excitation temperature, which can be found by fitting the observed spectrum, T_{a, ν}, for a given frequency

$$\begin{eqnarray} T_{a,\nu }^{\*}(J,K) &=& \eta _{{\rm MB}} \Phi (J(T_{{\rm ex}}(J,K)) - J(T_{{\rm bg}}))\nonumber\\ && \times (1 - {\rm exp}(-\tau _{\nu }(J,K))), \end{eqnarray} \tag{ A2 }$$

where η_MB is the main beam efficiency (0.81), Φ is the filling factor (assumed here to be 1), J(T) is the Rayleigh–Jeans temperature, and T_bg is the background temperature, 2.73 K. Finally, the total column density of ammonia is determined via use of the partition function, Z:

$$\begin{equation} Z = \displaystyle \sum \limits _{J} (2J+1) S(J) {\rm exp} \left[ \frac{-h[BJ(J+1)+(C-B)J^2]}{kT_k} \right], \end{equation} \tag{ A3 }$$

where B and C are the rotational constants of NH₃, 298,192 and 186,695 MHz, respectively,¹⁰ and S(J) accounts for the nuclear spin statistics of the system. The total ammonia column density is then

$$\begin{equation} N({\rm NH}_3) = N(1{,}1) \times \frac{Z}{Z_1} \end{equation} \tag{ A4 }$$

in which the Z in the numerator is summed over J = 0, 1, and 2, and Z₁ in the denominator is evaluated only at J = 1. At the low temperatures of these IRDCs, we assume that these metastable states represent a good approximation to the column density of ammonia.

The rotation temperature derived from NH₃ inversion lines depends on the optical depth and the ratio of the main lines in the (J, K) = (1,1) and (2,2) signatures. In turn, the optical depth of the NH₃ (1,1) main line (τ_m (1,1)) can be measured by comparing the ratio of the main line intensity to the satellite line intensity, as described in Ho & Townes (1983):

$$\begin{equation} \frac{\Delta T_a^{\*}(J,K,m)}{\Delta T_a^{\*}(J,K,s)} = \frac{1 - e^{-\tau (J,K,m)}}{1 - e^{-a\tau (J,K,m)}}, \end{equation} \tag{ A5 }$$

where the left-hand side is the ratio of the intensities of the main and satellite lines in the NH₃ (1,1) signature, and a is the ratio of the intensities compared with the main line intensity. With measurements of $\Delta T_a^{\*}$ for the main line and the two inner satellite lines (for which a = 0.28), one can solve for τ numerically. The total optical depth of the (1,1) line, τ₁, is twice the main line optical depth (Li et al. 2003). The rotational temperature (T_Rot) characterizes the distribution of molecules in these two energy levels, which are separated by 41.5 K in energy, denoted as T₀:

$$\begin{eqnarray} T_{\rm Rot} &=& - T_0 \div {\rm ln} \bigg[ \frac{-0.282}{\tau _m (1{,}1)} {\rm ln} \bigg[ 1 - \frac{\Delta T_a^{\*}(2{,}2{,}m)}{\Delta T_a^{\*}(1{,}1{,}m)}\bigg.\bigg.\nonumber\\ && \bigg.\bigg. \times (1 - e^{-\tau _m(1{,}1)}) \bigg] \bigg]. \end{eqnarray} \tag{ A6 }$$

Since there are no radiative transitions between the (2,2) and (1,1) levels, their population ratio, represented by T_Rot, probes the importance of collisions, and thus can be used to estimate T_k. This assumption is robust in environments such as pre-stellar cores and IRDCs, where temperatures are typically 20 K and lower (Walmsley & Ungerechts 1983). At such temperatures, only the (1,1), (2,2), and (2,1) states are important; thus Swift et al. (2005) showed that the relationship between the kinetic and rotational temperatures is as follows:

$$\begin{equation} T_{\rm Rot} = T_k \left[ 1 + \frac{T_k}{T_0} {\rm ln} \left(1 + 0.6 e^{-15.7/T_k} \right) \right]^{-1}. \end{equation} \tag{ A7 }$$

Tafalla et al. (2004) perform Monte Carlo models including the six lowest levels of para-NH₃, varying the temperature between 5 and 20 K and self-consistently calculating the (1,1), (2,2), and (2,1) level populations. They find that the approximation

$$\begin{equation} T_k = \frac{T_{\rm Rot}}{1 - \frac{T_{\rm Rot}}{T_0} {\rm ln} [ 1 + 1.1 e^{-15.7/T_{\rm Rot}} ]} \end{equation} \tag{ A8 }$$

is accurate to 5% or better in the 5–20 K temperature range. Recent work by Maret et al. (2009) shows that this calibration of the ammonia thermometer holds with this precision taking into account the most up-to-date collision rates in the literature. Thus, the two-level approximation for the ammonia thermometer described above is robust.