The Survey of Water and Ammonia in the Galactic Center (SWAG): Molecular Cloud Evolution in the Central Molecular Zone

Raw data are imported via atlod with options birdie, nocacal, noif, and opcorr to automatically flag resonant instrument modes (birdies), initial array calibration data (cacal), set correct intermediate-frequency mapping behavior, and to apply atmospheric opacity corrections. Some of the array setup data were not detected by the automatic routine and had to be flagged manually with uvflag.

Resonant modes in ATCA's backend system cause additional birdies in channels $n\times 1024+1(n=1,2,\ldots )$ that are not identified during data import. The missing channels are interpolated later when averaging and Doppler-correcting the visibilities to a common resolution of 2 km s⁻¹. To remove additional correlator artifacts and extreme RFI, calibrator data were clipped at 200 Jy and CMZ observations at 10 Jy. Additional flagging of bad data was made by eye in the interactive task blflag. Polarization measurements were not required for the presented analysis, and all crosshands/cross-polarizations (XY and YX for ATCA's linear feeds) were flagged. Antenna ca06 offers high spatial resolution at baselines of >4300 m, but was also flagged due to lack of intermediate $u,v$ points, which drastically down-weights these long baselines. Except for the instrumental birdies we described, little radio frequency interference (RFI) is present at frequencies of ∼23 GHz.

Bandpass amplitudes and phases are calculated on the designated bandpass calibrator PKS 1253-055. The resulting correction is then applied to the gain/phase calibrator PKS 1710-269 and flux density calibrator PKS 1934-638, and complex antenna gains are derived with mfcal.

Calibration solutions were inspected visually to identify remaining problems that were subsequently flagged. Recalibrations were performed until well-behaved solutions were found.

The calibration quality is assessed with Figure 4. The phase calibrator PKS 1710-269 was imaged including all calibration corrections for all observation days and all zoombands and the peak flux density listed. Measured flux densities per zoomband are constant within ∼15%, which is typical for radio observations at this frequency. The origin of this variability cannot be identified without absolute measurements and can be due to changing atmospheric conditions or intrinsic luminosity variation in the QSO PKS 1710-269. Thus, 15% is an upper limit on the flux density variation across observation days, which is similar to the absolute flux uncertainty.

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Spectral lines observed in the same zoomband, however, share a common flux density uncertainty, and thus flux ratios and derived quantities are more accurate. This provides a robust basis for the derivation of the kinetic gas temperature, which is primarily determined from ammonia line ratios.

Over the relatively small spectral range of a zoomband of ∼64 MHz, a linear approximation captures the spectral index of any continuum emission well enough. Line-free channels could not be identified in visibility spectra because the brightness of most of the spectral lines is too low; instead, they are calculated from the line's rest frequency and a window of ±400 channels (corresponding to ∼178–150 km s⁻¹ at 21.0–25.0 GHz), which is large enough to exclude the typical velocity range of molecular gas in the CMZ of ±150 km s⁻¹. Any possible emission outside these line-free windows is weak and does not affect the continuum fit. Fit and subtraction were made using the uvlin task, forcing a first-order polynomial.

Rest frequencies are set according to Table 1, which lists values from splatalogue.²³

Table 1.  For Each Ammonia Transition (1), We Detail the (2) Rest Frequency, (3) Excitation Energy of the Upper Transition State, (4) Natural Weighted Beam Sizes (${b}_{\mathrm{maj}}\times {b}_{\min }$), (5) Beam Position Angle, rms Noise for a Channel Width of 2 km s⁻¹ in the Two Regions of (6) Normal (${\sigma }_{\mathrm{normal}}$) and (7) Elevated Noise (${\sigma }_{\mathrm{high}}$)

Transition	Frequency	${E}_{u}/{k}_{{\rm{B}}}$	Beam Size	PA	${\sigma }_{\mathrm{normal}}$	${\sigma }_{\mathrm{high}}$
	(GHz)	(K)	('')	(°)	(mJy beam−1)
(1)	(2)	(3)	(4)	(5)	(6)	(7)
${\mathrm{NH}}_{3}$ (1, 1)	23.69451	24.4	26.2 × 17.8	893	13.4	19.4
${\mathrm{NH}}_{3}$ (2, 2)	23.72263	65.6	26.2 × 17.8	893	13.1	19.3
${\mathrm{NH}}_{3}$ (3, 3)	23.87013	124.7	26.0 × 17.7	893	13.1	18.6
${\mathrm{NH}}_{3}$ (4, 4)	24.13942	201.7	25.7 × 17.5	893	12.9	16.1
${\mathrm{NH}}_{3}$ (5, 5)	24.53299	296.5	25.3 × 17.2	893	12.0	16.2
${\mathrm{NH}}_{3}$ (6, 6)	25.05603	409.2	24.8 × 16.9	893	13.2	16.2
mean value					13.0	17.6

Note. ${\sigma }_{\mathrm{normal}}$ is relevant for most of the maps beside the two outermost strips (see Figure 2) on each side that are described by ${\sigma }_{\mathrm{high}}$. Given values are an average over 20 (${\sigma }_{\mathrm{normal}}$) and 12 (${\sigma }_{\mathrm{high}}$) measurements in two channels at ±150 km s⁻¹ relative to the rest frequency.

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Merging five channels to obtain a spectral resolution of 2 km s⁻¹ increases the signal-to-noise ratio (S/N) by a factor of $\sqrt{5}\sim 2.2$, while the spectral resolution is still adequate for the expected typical line widths of 5–25 km s⁻¹ in ammonia emission.

Fourier transformation from the visibility to the image domain is performed with the task invert, which includes primary beam correction and combines the visibilities of single pointings into a mosaic. The Briggs weighting parameter is set to +2, which is close to natural weighting. The cell size (pixel size) is set to 5'' in R.A. and decl., which oversamples the minor axis of the synthesized beam (248 × 169, 893 at 25.1 GHz) by a factor >3. At lower frequency, the beam size grows up to 262 × 178 at 23.7 GHz, which results in a variation of linear size of ∼5% between ${\mathrm{NH}}_{3}$ (1, 1) and (6, 6). These beam size are ∼1.7 and ∼1.2 times the naively expected size of ∼15'' × 15'' due to weighting. Table 1 lists the obtained beam sizes for ${\mathrm{NH}}_{3}$ (1, 1) to (6, 6).

Each pointing is imaged over 512 pixel (FOV 427) and then integrated linearly into the mosaic with another weighting function that smooths noise across regions of differing pointing coverage (miriad default method, Sault et al. 1996).

The extended emission of most spectral lines is generally better deconvolved by an algorithm that also uses extended sources for modeling. The only possibility of extended modeling that miriad offers is the maximum entropy method (MEM). Our tests with the mosaic versions of MEM (mosmem) and clean (mossdi) on SWAG data confirmed that more extended emission was reconstructed by MEM over regular clean.

A first run with up to 10 iterations deconvolves the dirty image well enough to construct a mask that prevents treating regions without significant emission. Image restoration is done with restor. A mask is calculated to contain all pixels with S/N above ∼5, which corresponds to 65.0 mJy beam⁻¹ in the semi-cleaned image.

Deconvolution is then repeated with 50 iterations in mosmem only inside the pixels that were set as relevant by the mask. The restored images still contain sidelobe structures, especially around strong sources and in strong lines. The difficulty of obtaining better images with mosmem is described in more detail in Appendix B.

The root mean square (rms) noise is not constant within and across the data cubes, but increases slightly with decreasing frequency because of the proximity to an atmospheric water line that increases the telescope's system temperature. At a level of ∼13.0 mJy beam⁻¹, the difference between the channels at −150 km s⁻¹ and +150 km s⁻¹ relative to rest frequency is 0.1–0.6 mJy beam⁻¹ for the ammonia lines listed in Table 1. Thus, a common value is calculated as the mean of minimum and maximum noise measurements. The spatial variation of noise across pointings is negligible except for the two outermost strips of the data considered in this work (10/11 and 30/31, see Figure 2), which show increased noise levels in the 2014/2015 data that are due to bad weather. Hence, two values ${\sigma }_{\mathrm{normal}}$ and ${\sigma }_{\mathrm{high}}$ are needed to describe the noise properties in the inner map and the ${l}^{+}/{l}^{-}$ edges, respectively. Table 1 lists these noise levels for the six metastable ammonia lines. Any further mention of noise refers to ${\sigma }_{\mathrm{normal}}$ as only two major molecular clouds (Sgr B2 and Sgr C) are located in the region of elevated noise, and they are strong sources with high S/N.

The outer edges of all data cubes exhibit a $\sqrt{3}$ higher noise than the rest of the maps because of non-overlapping pointings. These edges are blanked from the edge inward by ∼50'' and ∼100'' (∼one-third and two-thirds of the primary beam, respectively) as the smaller radius of 50'' proved to be enough to exclude any visible higher noise edges in ammonia distribution maps, whereas spectral fitting is affected by increased noise to ∼90'' from the edge.

Data cubes were collapsed along the spectral axes with the task moment in miriad to obtain moment maps. Map edges are blanked with the 50'' edge mask and noise is excluded at levels of 3σ and 5σ for intensity maps (peak intensity; moment 0, integrated intensity) and velocity maps (moment 1, intensity-weighted mean velocity; moment 2, velocity dispersion), respectively.

Position–velocity cuts through the restored datacube edge masked at 100'' were calculated with the task impv in the CASA software package²⁴ (McMullin et al. 2007) along Galactic longitude, averaging over all emission in Galactic latitude.

A Boltzmann plot (Appendix C.1, cf. Goldsmith & Langer 1999), typical for the ammonia emission in the GC, is shown in Figure 9. The observed shapes do not follow a single linear relation, but become more shallow around ${\mathrm{NH}}_{3}$ (3, 3) to ${\mathrm{NH}}_{3}$ (4, 4), which can be due to the nonlinear T_kin–T_rot conversion or be indicative of a multiple temperature medium, as is already known in the literature (e.g., Hüttemeister et al. 1993; Mills & Morris 2013). Errors in derived column density are typically small (ΔN_u/N_u on the order of a few percent) but larger (a few tens of percent) at the edges of clouds because of the lower S/N, which allows the derivation of accurate temperatures of typically 5%–10% relative error (T_rot) and 10%–25% after conversion into kinetic temperature. Additionally, the ∼15% flux density error (Figure 4) contributes to the column density errors.

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The ${\mathrm{NH}}_{3}$ (3, 3) column density N_l (Figure 33) traces much of the ring-like gas stream by construction of the mask applied before hyperfine structure fitting. A peak of ${N}_{33}=7.7\times {10}^{15}$ cm⁻² is reached in Sgr B2. Massive clouds like the Brick, G+0.10–0.08, and the 20 km s⁻¹ cloud are also found to have high column densities (${N}_{33}\gtrsim 1.5\times {10}^{15}$ cm⁻²), while ${\mathrm{NH}}_{3}$ (3, 3) is not largely excited and detected at column densities $\lesssim 1.5\times {10}^{15}$ cm⁻². The ${\mathrm{NH}}_{3}$ (1, 1) column density reaches a maximum of $1.2\times {10}^{16}$ cm⁻² in Sgr B2, where ${\mathrm{NH}}_{3}$ (6, 6) is still detected at up to $2.6\times {10}^{15}$ cm⁻².

The total ammonia column density as derived from Equation (22) (Figure 10) shows a similar picture as the ${\mathrm{NH}}_{3}$ (3, 3) column density. The highest column densities are reached in Sgr B2 ($6.8\times {10}^{16}$ cm⁻²), the Brick ($3.9\times {10}^{16}$ cm⁻²), and the 20 km s⁻¹ cloud ($3.0\times {10}^{16}$ cm⁻²), whereas most smaller clouds along the "ring" have ${N}_{\mathrm{tot}}\lesssim 1.0\times {10}^{16}$ cm⁻². The clouds at $l\sim 0\buildrel{\circ}\over{.} 1,b\sim 0\buildrel{\circ}\over{.} 2$ show the lowest detected column densities ($1.0\times {10}^{15}\lesssim {N}_{\mathrm{tot}}\ {\mathrm{cm}}^{-2}\lesssim 2.5\times {10}^{15}$) of clouds that do not fall below the fitting thresholds. Other clouds of this size typically exhibit column densities that are higher by a factor of 2–3. A small cloud of high column density is located at l = −040, b = −022 with ${N}_{\mathrm{tot}}$ up to $2.6\times {10}^{16}$ cm⁻² (${N}_{33}=6.4\times {10}^{15}$ cm⁻²) and surrounded by gas of ${N}_{\mathrm{tot}}\,\sim {10}^{16}$ cm⁻². This cloud is assumed to lie in the foreground (Longmore et al. 2013b) and is discussed to be influenced by an intermediate mass black hole (Oka et al. 2016) because of its unusually high CO velocity dispersion.

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The ammonia distribution as traced by the NH₃ column density map closely follows 870 μm dust emission obtained by ATLASGAL (Schuller et al. 2009), as can be seen in Figure 10 (bottom). The correlation is most pronounced for gas at higher column density than $\sim 1\times {10}^{16}$ cm⁻². Lower column density gas is mostly found in regions of weak dust emission, but the presence of dust does not imply the detection of ammonia at ${N}_{\mathrm{tot}}\gtrsim 2\times {10}^{15}$ cm⁻². Almost all (dense) ammonia gas is accompanied by dust emission. Notable exceptions from the overall good correlation are found in Sgr B2 and Sgr A*. Sgr B2 (N) and Sgr B2 (M) are detected in absorption in ammonia, whereas the dust emission peaks at these locations. All of Sgr B2 that is detected in emission, however, does correlate very well with the 870 μm contours. At Sgr A*, the 870 μm emission locally peaks, but the ammonia column density is $\ll 1\times {10}^{15}$ cm⁻² as this region falls below the selection thresholds for hyperfine structure fitting. The emission from Sgr A* at submm is contributed by optically thick synchrotron emission rather than thermal dust emission, and no correlation is observed.

The fit-derived line-of-sight velocity of ${\mathrm{NH}}_{3}$ (3, 3), shown in Figure 11, generally agrees well with the moment-1 map (Figure 7(c)). Owing to the constraints for successful fitting (Section 4.1), the mapped area is considerably smaller. Significant differences between the two maps are not detected, which confirms that "successful" fits are indeed correct and potential "bad" fits were excluded.

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The line widths of the fitted map (Figure 12) and the moment-2 map (Figure 7(d)) qualitatively agree about regions of higher or lower line width, but show very different quantitative results. After correcting for the conversion between FWHM (fitted map) and velocity dispersion σ (moment map), a typical factor of ∼1.5–2 (coresponding to differences of <10 km s⁻¹) remains, but extended regions all across the CMZ display factors of 3–5 (differences of 20 km s⁻¹ up to >50 km s⁻¹). The larger line widths are derived from the moment map in all cases, meaning the image moment analysis overestimates the line width with respect to line fitting. This is partly due to unaccounted blending of hyperfine structure and mostly caused by multiple line-of-sight components in the moment map. Henshaw et al. (2016b) find similar differences between these two line width estimates in the CMZ based on HNCO data. Depending on the number of detected emission components along the line of sight, their median difference varies between 3.4 km s⁻¹ (single component) and 18.8 km s⁻¹ when four components are present. The maps derived from hyperfine structure fitting are thus considered to be more reliable than the simple moment analysis when investigating the kinematics, although fitting discards all but the strongest component along the line of sight, introducing a bias toward strong sources.

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The distribution of fitted line widths for all ammonia transitions is shown in Figure 13 and shows a typical line width of 8–16 km s⁻¹ consistently for all six observed ammonia lines. The peak of the distribution is located at 8–10 km s⁻¹, and only ${\mathrm{NH}}_{3}$ (3, 3) has its peak at a slightly higher velocity of 10–12 km s⁻¹.

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The opacity map of ${\mathrm{NH}}_{3}$ (3, 3) (Figure 14) shows that a significant number of fitted pixels of the six ammonia species exhibit opacities τ ≳ 1, where the often used assumption of optically thin emission does not hold anymore. This is the case not only in the very high column density clouds (like Sgr B2 and the Brick), but also in other dust ridge clouds and the inner regions of smaller clouds at negative longitudes and the far side of the "ring" (${l}^{+},{b}^{-}$). A weak correlation between high column density (Figure 33) and high opacity (Figure 14) is found in the Brick, G+0.10 + 0.00, the 20 and 50 km s⁻¹ clouds, and the foreground cloud G-0.40-0.20. Other clouds are optically thin at low column density or do show high opacity despite having low column density (${N}_{u,33}\lt 2.5\times {10}^{15}$ cm⁻², e.g., G0.50+0.05 or G-0.60-0.09), which is indicative of a smaller filling fraction of clumpy gas. High opacities, especially in very small regions, must be taken with care as the edge of a region corresponds to a low S/N (≳3σ, see Section 4.1 for details), and thus random noise peaks may be mistaken as hyperfine structure satellite components by the fitting routine. Because of the low flux of these pixels, the derived column density is only marginally affected, but opacity, which depends on the flux ratio of hyperfine structure components, can be artificially elevated. This opacity distribution is consistent with Hüttemeister et al. (1993), who observed ${\mathrm{NH}}_{3}$ (1, 1), (2, 2), (4, 4), and (5, 5) in selected clouds in the GC. In the (1, 1) line, they find "strong" sources at ${\tau }_{11}\sim 4$ and an average ${\bar{\tau }}_{11}=2.3$ for "weak" sources, which matches our observations of ${\bar{\tau }}_{11}=2.0$ (median 1.2) and ${\tau }_{11}\gg 1$ in clouds when considering that SWAG also observed the less dense ISM in between clouds. We can also confirm the trend to lower opacity in Hüttemeister et al. (1993) for higher-J lines as the median opacity decreases to the lower fitting limit (τ = 0.1) for ${\mathrm{NH}}_{3}$ (4, 4) and higher.

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The ammonia gas temperature as traced by kinetic temperature T₂₄, which is derived from ${\mathrm{NH}}_{3}$ (2, 2) and ${\mathrm{NH}}_{3}$ (4, 4), is shown in Figure 15. The typical temperature range observed in the GC is 50–120 K, only Sgr B2 is considerably hotter with ∼200 K.

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A histogram of temperature measures ${T}_{12}$, ${T}_{24}$, ${T}_{45}$, and ${T}_{36}$ is shown in Figure 16 to illustrate the temperature distribution of the CMZ. ${T}_{12}$ cannot reliably trace temperatures ≳50–60 K (see excitation energies listed in Table 1), and thus its distribution peaks at ∼35 K, tracing cooler molecular gas. The other three temperature measures of higher lines are reliable to at least ∼200 K before the rotational-kinetic conversion starts to introduce large uncertainties, and they are therefore easier to interpret. Their distributions are similar in peak temperature and spread, with ${T}_{24}$ yielding higher values by ∼15 K. Regarding the spatial distribution, most of the gas is found at intermediate temperatures at 60–100 K and in a few hot regions T > 150 K.

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The detection of two temperature components in the molecular gas is consistent with Hüttemeister et al. (1993), who found gas at rotational temperatures ${T}_{\mathrm{rot}}\sim 25$ K and ${T}_{\mathrm{rot}}\sim 200$ K. Our cold (${T}_{\mathrm{kin}}=25\mbox{--}50$ K) component is thus compatible with Hüttemeister et al. (1993) considering the rotational-kinetic temperature conversion. However, the temperature of the warm components differ significantly by a factor of ∼2–2.5 in rotational temperature. Ao et al. (2013) and Ginsburg et al. (2016) derived kinetic temperatures of dense molecular gas over large areas of the CMZ based on APEX observations of H₂CO (formaldehyde). They find the gas to be warm at 50 K to >100 K (Ao et al. 2013) and ∼60 K to >100 K (Ginsburg et al. 2016), which is consistent with our detection of a warm gas component. Regarding the spatial distribution, the temperatures known in the literature match our results: Sgr B2 is hot at ∼200 K (this work), >150 K (Ginsburg et al. 2016); the 20 km s⁻¹ and 50 km s⁻¹ cloud and the Brick show temperatures of 80–100 K (this work, Güsten et al. 1981; Mauersberger et al. 1986; Ao et al. 2013; Ginsburg et al. 2016).

We estimate the fraction of cold (∼35 K) and warm (∼80 K) gas of the total detected ammonia column density by fitting the Boltzmann plot (Figure 9) with two components. This allows the calculation of the column density of cold and warm ammonia independently and hence the relative abundances. Over the whole CMZ, we find the mean column density fractions to be ∼55% and ∼45% for the cold and warm component, respectively. In the 20 and 50 km s⁻¹ clouds, the cold component dominates at 50%–70% (mean: ∼68%) the relative contribution to the observed ammonia column density. In Sgr B2, a larger fraction of the column density is in the warm component, which results in both components being almost equally strong (∼50% cold and warm). Mean values of ∼60% of the cold gas fraction are found in the dust ridge and ∼53% in clouds at negative longitudes (excluding the 20 km s⁻¹ and 50 km s⁻¹ clouds). The "Three Little Pigs" (G+0.05–0.07 "sticks," G+0.10–0.08 "stones," and G+0.15–0.09 "straw") are a group of clouds of similar global properties but very different substructure. They are also striking in the cold gas fraction ${f}_{\mathrm{cold}}$ as G+0.10–0.08 ("sticks") shows the highest ${f}_{\mathrm{cold}}$ over an extended region at ∼81%, whereas its neighbor G+0.05–0.07 ("stones"; ${f}_{\mathrm{cold}}\sim 65 \%$) seems inconspicuous and much closer to the CMZ mean. G+0.15–0.09, "straw," does not show up as its ammonia emission was to low to be hyperfine structure fitted. These results show that the typical gas temperature is not the same over the whole CMZ, but is either ∼35 K or ∼80 K depending on the considered region.

The relation between Hi-GAL dust temperature (Molinari et al. 2011) and ammonia gas temperature is shown in Figure 17 for ${T}_{24}$. Further tracers are shown in Appendix F. ${T}_{24}$ is offset well above the one-to-one relation by a factor of ∼3 to >5. A linear best fit, calculated in bins of 1 K for $18\,{\rm{K}}\leqslant {T}_{\mathrm{dust}}\leqslant 30$ K, yields a gradient of ${{dT}}_{24}/{{dT}}_{\mathrm{dust}}=1.8$, and fitting without binning results in no correlation (gradient ∼0.0) because of cutoff effects of the limited temperature reliability range. This result implies that in the CMZ, the same heating mechanism is likely to be responsible for both dust and gas heating although their temperatures are not coupled to the same value. This hint toward a common heating source contradicts the prediction by Clark et al. (2013), who explain the temperature structure of the Brick (G0.253+0.016) by independent heating of gas and dust by cosmic rays and an interstellar radiation field, respectively. The observed correlation is compatible with heating by turbulence on large scales and compression on cloud scales (cf. Section 5), as was also suggested by Ginsburg et al. (2016). The fact that dust acts as a coolant in the CMZ has been known in the literature for a long time (e.g., Güsten et al. 1981; Molinari et al. 2011; Ginsburg et al. 2016) and can be confirmed by our new data. Similar results are obtained for warm molecular gas traced by T₄₅ and T₃₆ (${{dT}}_{45}/{{dT}}_{\mathrm{dust}}=2.4$, ${{dT}}_{36}/{{dT}}_{\mathrm{dust}}=1.4$). The cold molecular gas component (T₁₂) only weakly correlates with dust temperatures (${{dT}}_{12}/{{dT}}_{\mathrm{dust}}=0.5$), which implies a temperature component that is not reflected by dust measurements.

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A simple estimation of the beam filling factor ${\eta }_{f}$ can be made based on the comparison of T_kin and brightness temperature T_b. For ${\eta }_{f}=1$, both temperatures must be equal, but if the beam is partly covered by emission of temperature T_kin and partly by weak (i.e., cold) background, the observed T_b is lower. A first-order estimate is given by ${\eta }_{f}={T}_{b}/{T}_{\mathrm{kin}}$ under the assumptions of ${\eta }_{f}$ being identical for the two lines ${\mathrm{NH}}_{3}$ (i, i) and ${\mathrm{NH}}_{3}$ (j, j) that define temperature ${T}_{{ij}}$. This furthermore assumes high optical depth and the line excitation temperatures being approximately given by ${T}_{\mathrm{kin}}$. Typical values of dense clouds (the 20 km s⁻¹ cloud, the Brick, cloud d, Sgr B2) are in the range 0.10–0.15, Sgr C and less dense clouds are covered at ${\eta }_{f}=0.05\mbox{--}0.10$ (e.g., the "Three Little Pigs" G0.05–0.07, G0.10–0.08, and G0.15–0.09). G-0.40–0.25 sticks out from all other clouds at negative longitudes with ${\eta }_{f}\sim 0.10$, which is the third highest ${\eta }_{f}$ observed after Sgr B2 and the 20 km s⁻¹ cloud. All other clouds show beam filling factors below 0.05. The average size of cloud substructure (clump size) for ${\eta }_{f}=0.10$ (0.05 0.15) is then 0.27 pc (0.19 pc, 0.33 pc). This is consistent with recent interferometer measurements with the SMA and ALMA (Kauffmann et al. 2013, 2017a; Rathborne et al. 2014b; Lu et al. 2017).

Figure 18 shows the relation between line width and temperature for ${T}_{24}$ and the corresponding line ${\mathrm{NH}}_{3}$ (2, 2). As before, temperatures scatter between 50 K and the sensitivity cutoff at 200 K for line widths of ∼5 km s⁻¹ to 30–40 km s⁻¹ with no obvious correlation. Temperature medians in bins of 2 km s⁻¹ width, however, follow a linear relation at ${dT}/d({\rm{\Delta }}v)=1.2$. The increase in line width due to thermal broadening scales sub-linearly as ${\rm{\Delta }}v\propto \sqrt{T}$ and is about one order of magnitude (${\rm{\Delta }}{v}_{\mathrm{thermal}}\sim 2$ km s⁻¹) weaker than the observed linear relation. Assuming that the line width is not contaminated by blending of clouds along the line of sight, this implies that on a statistical basis, the more turbulent gas is warmer but large variations between clouds do exist, as has also previously been detected by Immer et al. (2016) and Ginsburg et al. (2016).

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The top panel of Figure 20 shows the kinetic ammonia temperature ${T}_{24}$ as a function of time. Median kinetic temperatures of ∼55 K to ∼135 K are detected with typical error margins of [−11 K, +15 K]. Four sequences of consistently increasing temperature are found in the time intervals [−1.9 Myr, −1.1 Myr], [−0.3 Myr, 0.05 Myr], [1.5 Myr, 1.8 Myr], and [1.8 Myr, 2.3 Myr]. Positive linear temperature gradients of 51 and 56 K Myr⁻¹ are derived in the dust ridge and around the near-side pericenter, whereas steeper gradients of 114 K Myr⁻¹ at pericenter far and 97 K Myr⁻¹ at pericenter near are fitted. Owing to the short sampled time range and the few temperature measurements, the sequence at pericenter far must be taken with caution. The error intervals are large because of the chosen conservative error estimation (Section 5.2), which causes the heating rate errors derived from χ²-fitting to also show a substantial error of typically ∼100%, which is likely overestimated given the consistent trend. The mean heating rate of the three individual measurements (${T}_{24}$, ${T}_{45}$, ${T}_{36}$) thus has a relative error of approximately 58%.

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In the dust ridge (Brick to Sgr B2, −1.9 < t [Myr] < −1.1), the covered time range (0.8 Myr) and statistics (∼9200 measurements) yield reliable heating rates of 47.1 K Myr⁻¹ to 57.9 K Myr⁻¹ (mean of 52.0 K Myr⁻¹) in the three partly independent temperature measures ${T}_{24}$, ${T}_{45}$, and ${T}_{36}$ (Table. 2). The absolute increase is from 75 K to 110 K (${T}_{24}$), 60 K to 105 K (${T}_{45}$), and 60 K to 100 K (${T}_{36}$) or ∼40 K on average.

Table 2.  Ammonia Kinetic Temperature Gradients (Heating Rates) of the Four Recovered Temporal Sequences (Figure 20) and of the Orbital Phase without and with Adjusting the Zero-point (Figure 21)

Time Range	$\tfrac{{T}_{24}}{{dt}}$	$\tfrac{{T}_{45}}{{dt}}$	$\tfrac{{T}_{36}}{{dt}}$	Mean
	[K Myr−1 ]	[K Myr−1 ]	[K Myr−1 ]	[K Myr−1 ]
$-1.90\lt t\,[\mathrm{Myr}]\lt -1.10$	51.0	47.1	57.9	52.0
(dust ridge)
$-0.30\lt t\,[\mathrm{Myr}]\lt 0.00$	113.8	77.3	67.7	86.3
(pericenter far)
1.50 < t [Myr] < 1.75	96.9	122.8	84.8	101.5
(Sgr C)
1.80 < t [Myr] < 2.30	56.0	57.2	41.6	51.6
(pericenter near)
combined (phase)	14.6	14.6	17.7	15.6
combined (phase, adjusted)	47.7	58.4	46.3	50.8

Note. Errors are ∼100% for individual heating rates and ∼58% for the mean.

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The short sequence at the far-side pericenter passage (−0.30 < t [Myr] < 0.00) is affected by low number statistics and has heating rate errors larger than 100%. Any conclusion for this time range should thus be treated merely as a qualitative but not as a quantitative result.

Between the western apocenter of the orbit and the pericenter on the near side, two independent sequences of increasing temperature are found, separated by a discontinuity at 1.8 Myr, where the median temperature sharply drops from ∼120 K to ∼85 K. This is again seen at a very similar level in ${T}_{24}$, ${T}_{45}$ and ${T}_{36}$. Heating rates differ between the sequence at Sgr C and pericenter, the former being consistently higher at ∼102 K Myr⁻¹ than the latter (∼52 K Myr⁻¹, mean heating rates).

An overview of derived heating rates in the four sequences is given in Table 2.

The median column densities show no evidence of time dependence (Figure 20, middle top, Appendix G) in the dust ridge. No consistent trend is apparent as column densities increase strongly in the Brick and Sgr B2 above a flat or, at most, very weakly rising base level. These clouds are already known to be denser than other dust ridge clouds and thus are expected to be more massive outliers to any potential underlying relation (Lis & Carlstrom 1994; Lis et al. 1994; Molinari et al. 2011; Longmore et al. 2012; Pillai et al. 2015). Given that the column densities in between the Brick and Sgr B2 show no rise and also drop to the same level after Sgr B2, we conclude that there is no evolutionary column density sequence in the dust ridge.

The clouds around near-side pericenter passage show weak evidence for decreasing column densities where the median column density drops for J = 1, 2, 3 but can be considered constant within the scatter for J = 4, 5, and 6. This relation is likely also influenced by two individual objects, the 20 km s⁻¹ and 50 km s⁻¹ clouds, although density bumps are not as prominent as in Sgr B2.

At far-side pericenter, medians scatter by up to ∼1 dex and no conclusion can be drawn. The column densities in the Sgr C region are constant for all six ammonia tracers, with very low scatter compared to the rest of the covered time ranges.

Qualitatively and quantitatively very similar results are derived for opacity due to the correlation of opacity and column density, and this is hence not discussed further.

The evolution of the median line width (FWHM) is similar to that of the column density, but shows an even less consistent behavior, as can be seen in Figure 20 (middle bottom) for ${\mathrm{NH}}_{3}$ (3, 3) (J = 1, ..., 6 in Appendix G). In the dust ridge, line widths vary by a factor of ∼2.5 with peaks in the Brick and Sgr B2. There is no sign that these could be just outliers to an underlying trend as line widths are at similar levels of 6–10 km s⁻¹ before the Brick, in between Brick and Sgr B2, and after Sgr B2. Any potential time dependency of the line width thus cannot be stronger than a few km s⁻¹ Myr⁻¹ to be hidden within the observed scatter. Results are very similar for the other ammonia lines (J = 1, 2, 4, 5, 6) at slightly varying absolute level, as could already be seen in the line width histogram (Figure 13).

In the other time ranges that show temperature evolution, no sign of consistent line width trend can be found. As for column density, the 20 and 50 km s⁻¹ clouds are weak outliers toward higher values.

The increase in line width at the position of large and massive clouds can be caused by scaling relations ("Larson laws," Larson 1981), be an effect of hyperfine structure fitting, or an effect of projection. CMZ clouds are known to follow a steep scaling relation of line width σ with cloud radius R: $\sigma \propto {R}^{0.6\mbox{--}0.7}$ (Shetty et al. 2012; Kruijssen & Longmore 2013; Kauffmann et al. 2017b), and thus line widths are expected to increase by a factor of ∼1.5 between the small clouds b/c and larger clouds like the Brick and cloud e/f (Immer et al. 2012). The observed variation in line width of a factor ∼2.5 is thus only partly due to variations in cloud size and other sources of scatter must be present, such as unresolved blending of multiple emission components along the line of sight, as shown in Figure 36(d)). Spectra with two close components with separations of a few to a few tens of km s⁻¹ are more likely to be present in larger clouds and thus cause the same signature as the scaling relation does, increasing the overall scatter. Projection effects may also increase the apparent line width through pile-up along the line of sight at the tangent points of the orbit. This can play a role in Sgr B2, but is not expected to affect the other clouds (cf. Figure 19).

Qualitatively, increasing temperatures during pericenter passage can occur by converting the gravitational energy released by cloud compression or collapse into heat through a turbulence cascade if cooling is not efficient enough to keep the cloud at a constant temperature (e.g., Ginsburg et al. 2016). The effect on the line width of such a process is not immediately clear and likely needs detailed simulations. In general, turbulence should increase the observed line width, which would result in a rising sequence of line width following the rise in temperature. However, as the energy makes its way through the cascade to smaller scales R, the dissipation rate $\epsilon \propto \sigma /R$ also increases, allowing for little or even no change in the effective line width σ. If the turbulent line width does increase, it will still be difficult to detect because of variations among clouds (Section 5.2.3).

Radiative cooling generally is an efficient process (e.g., Juvela et al. 2001), and it is not obvious why it could be so inefficient for temperatures to rise by >50 K Myr⁻¹. As shown by Figure 14, a significant fraction of clouds in the CMZ are optically thick for line emission at ∼25 GHz, which can trap cooling photons inside clouds and diminish cooling efficiency. Ammonia is not a dominant cooling line, but other lines that are more important for cooling (CO, C⁺, and, at least in the Brick, neutral oxygen; Clark et al. 2013) can be expected to be even more optically thick than NH₃, and the same argument applies. If true, the hint toward decreasing column density might also be an effect of opacity because in optically thick media, the observed emission originates from an optical depth of τ ∼ 1, which is the envelope rather than the core of the cloud. The outer layers of a cloud moving through the pericenter along the stream orbit are sheared and can even become unbound, while the inner cloud is pushed into self-gravity and collapses, as shown observationally by Rathborne et al. (2014a), Federrath et al. (2016), and modeled by Kruijssen (2017) and Kruijssen et al. (2017, in preparation). Hence, a trend of decreasing column density can be observed.

The observed mean heating rates at pericenter passage of ∼52 and ∼86.3 K Myr⁻¹ are high, but cannot be easily compared to theoretic predictions because the complex interplay of dynamic and hydrodynamic processes requires simulations that include radiative transfer. Without these simulations, it is also not known at which time before pericenter passage tidal effects start to become non-negligible and affect temperature. The same applies to the time after pericenter passage, when compression and shear are reduced, which could in principle also lead to cooling, but this is not observed. Detailed simulations can be expected in the near future as the general feasibility was already demonstrated (Clark et al. 2013; Kruijssen 2017). Without simulations, we can estimate the time range of strong tidal influence on the clouds from Figure 20. The indication of increasing temperature in Figure 20 suggests that a temperature change is detectable at least 0.25 Myr before reaching pericenter. A change in temperature might occur earlier than 0.25 Myr before pericenter passage, but unfortunately, our data do not cover that time range, therefore we cannot check if temperatures are flat before heating sets in. At 0 Myr and 2 Myr, the deepest point in the potential is reached at ${R}_{\mathrm{peri}}\sim 59$ pc, while 0.25 Myr corresponds to R ∼ 1.22 ${R}_{\mathrm{peri}}=72$ pc. Apocenter is reached at R ∼ 120 pc, so tidal effects becoming relevant at ∼75–80 pc from the center of the potential is plausible. In the spherically symmetric enclosed mass distribution in the GC (Launhardt et al. 2002), the gravitational potential at ≲0.25 Myr before reaching pericenter²⁹ is symmetric to the respective range after pericenter, i.e., tidal effects are expected to play a role until at least 0.25 Myr after pericenter passage. After pericenter passage, some processes reverse, i.e., tidal pressure is released, which can allow cooling, but cloud collapse is ongoing and early star formation may set in. This complex interplay of physical processes makes it very difficult to define an end point of the strong tidal influence from the observational data. Generally, observations show that the temperatures are consistent with turbulent energy dissipation (Ginsburg et al. 2016), as would be expected from triggered cloud collapse.

This observed sequence in the dust ridge starts about one free-fall time ${\tau }_{\mathrm{ff}}=0.34\,\mathrm{Myr}$ after pericenter passage, whereas the other sequences start before. This means that if cloud collapse was triggered, a cloud core collapses, whereas previously sheared but still bound gas is reaccreted. As the collapse of outer cloud layers proceeds, the release of gravitational energy declines, which implies flattening temperature gradients. However, no obvious decrease in heating rate is found in the dust ridge, meaning that cloud collapse is still ongoing at a similar intensity as at pericenter passage, or alternatively, that a second heating source starts to contribute. In the framework of triggered SF, star formation could be a second source of heating. As an estimate, star formation can set in as early as one free-fall time after the cloud itself started to collapse, which is 0.34 Myr for a typical CMZ cloud, and this coincides with the time since the Brick passed pericenter (K15). Accordingly, early signs of embedded star formation have been found in one region of the Brick (Longmore et al. 2012; Johnston et al. 2014; Rathborne et al. 2014a), i.e., heating by star formation is just about to set in.

These effects increase the temperature if the gas cannot cool efficiently enough, which apparently is the case as gas temperatures are known to be significantly offset from dust temperatures (Figure 17 and e.g., Güsten et al. 1981; Molinari et al. 2011; Ginsburg et al. 2016). Therefore, the increase in temperature cannot simply be attributed to a single of these processes, but to a combination at varying relative strengths, which is not known at the moment for GC conditions and cannot be inferred from our observations. In order to disentangle the contribution of star formation from the kinematic and gravitational effects, a complex hydrodynamic gas simulation including star formation feedback is needed.

Collapsing clouds are expected to show a signature of increasing column density that is not observed, which might be a consequence of shearing outer cloud layers during pericenter passage. It might also be an observational effect of too low resolution to trace the collapse. Collapse occurs quickest on the highest density, lowest virial ratio scales. It is known in the literature that the column density profiles of the Brick and bricklet clouds are centrally concentrated and that the volume density of the Brick increases toward the center (Walker et al. 2015; Rathborne et al. 2016). The collapse will therefore occur quickest at the center, on scales smaller than the SWAG beam, which will wash out potential small-scale effects. In higher resolution observations, Walker et al. (2015) found evidence for increasing gas and (proto-) stellar density along the stream orbit from the Brick to Sgr B2.

As discussed in Section 5.2.1, the sequence of increasing temperature around Sgr C is likely not connected to the sequence at near-side pericenter because of the sharp discontinuity and differing heating rate. The observation of 24 μm emission in this region and the higher temperature compared to the pericenter sequence suggest that these clouds already underwent tidal triggering of collapse at far-side pericenter (see also Kruijssen et al. (2015) regarding the position of Sgr C). This places Sgr C as a successor to the dust ridge sequence in terms of evolution, but not in time. Finding no column density trend in Sgr C matches the picture, as this cloud is supposedly collapsed already and is actively forming stars (Kendrew et al. 2013; Lu et al. 2017), which powers the rise in temperature.

According to the K15 model, the empirically found sequences are directly associated in the sense that the phasse dominated by star formation (dust ridge and Sgr C) follows the phase dominated by tidal compression (near and far pericenter passage). The orbital phase, however, cannot be the only parameter controlling the physical state of gas along the stream, as discussed in Section 5.2.1. It seems likely that the clouds at and after pericenter passage are similar but not identical in properties, i.e., all clouds may follow the same evolution, but the gas in the dust ridge was initially cooler or could cool more efficiently when cloud collapse was triggered. The offset in temperature that is required to match the sequences is small (∼15–20 K) compared to the observed heating rates (∼52 K Myr⁻¹). As we do not know the initial cloud properties, this remains speculation, and we can only state that the observed sequences can be the result of tidally triggered collapse assuming a scatter of initial cloud temperature of ∼20 K. On this note, it should be kept in mind that individual cloud properties as well as collapse triggering are statistically distributed and are not identical for subsequent occurrences. Estimates for the variation of physical parameters from cloud to cloud can be drawn from the analysis of velocity corrugations in the $[{l}^{-}$, ${b}^{-}]$ segment of the stream, which is expected to be a precursor to the dust ridge (Henshaw et al. 2016a). For this analysis, Kruijssen (2017) report standard deviations of initial cloud parameters of 0.19 dex (mass), 0.09 dex (radius), 0.16 dex (density), and 0.08 dex (free-fall timescale) among the potential precursors to the dust ridge only. The standard deviations would naturally be larger when the progenitors to the other temperature sequences identified in this paper were also included. If the variations of initial cloud temperature are at a similar level as these properties, the expected temperature scatter is ≳20 K, larger than the observed offsets of 15–20 K. These numbers show that the observed temperature sequences at and after pericenter passage may be causally related as the temperature deviation is within the expected scatter.

The periodic occurrence of pericenter passages every ∼2.0 Myr suggests that temperatures can also be plotted against orbital phase, which is shown in Figure 21 (top panel). Although four temperature sequences could be defined as a function of absolute time, no global sequence is found. Only the time range 0.5 to 0.9 Myr shows a consistent trend of rising temperatures as the data in this region originate almost exclusively from the dust ridge. The other sequences are lost because differing temperature levels are superposed, which also smears the discontinuity at ${t}_{\mathrm{peri}}=1.75\,\mathrm{Myr}$ that occurs at a phase of ${t}_{\mathrm{last}}=-0.25\,\mathrm{Myr}$. However, fitting a linear relation in the time range −0.25 to 0.90 Myr (sum of the four ranges identified in Section 5.2) still results in a positive heating rate of 14.6 K Myr⁻¹. Almost identical results are obtained for ${T}_{45}$ and ${T}_{36}$ with relative errors of ∼100%, as listed in Table 2. These results are in line with Kauffmann et al. (2017b), who find a trend in density gradients of six CMZ clouds, but no clear trend in star formation activity per unit dense gas as a function of orbital phase.

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The signature of rising temperature could be expected to become stronger by transforming into orbital phase if the four sequences are samples of a single consistent evolution. However, it is highly unlikely that all clouds are triggered to collapse exactly at pericenter, this is expected to occur rather within some time range around pericenter. By allowing the zero-point ${t}_{\mathrm{start}}$ of each sequence to vary by up to ±0.3 Myr, the sequential behavior can be recovered (Figure 21 bottom panel). The heating rate is maximized at a value of ∼48 K Myr⁻¹ for zero-points of ${t}_{\mathrm{start}}=-0.3\,\mathrm{Myr}$ (dust ridge), ${t}_{\mathrm{start}}=+0.2\,\mathrm{Myr}$ (pericenter near) and ${t}_{\mathrm{start}}=+0.2\,\mathrm{Myr}$ (pericenter far). At 0.3 Myr before or after pericenter, the radial distance from the bottom of the GC potential is R = 77 pc (R = 60 pc at pericenter), and thus the clouds are still deep in the potential where tidal forces can be expected to be sufficiently strong to trigger cloud collapse. The fact that heating rates similar to the individual values can be recovered by only adjusting the exact trigger point without taking potential variations of initial cloud conditions or variations of the gravitational potential into account suggests that ${t}_{\mathrm{start}}$ is one of the major sources of scatter of a downstream SF sequence. The stacked and ${t}_{\mathrm{start}}$-adjusted mean heating rates are lower than the per-segment heating rates, implying that another factor still erases part of the trend. This is the vertical (temperature) offset, which is due to slightly different initial conditions, as has been discussed in Section 5.3.3. This observation confirms the warning by Kruijssen (2017) that simple stacking of pericenter passages is not advisable because of scatter. However, problems can be diminished if it is possible to estimate the scatter in ${t}_{\mathrm{start}}$. Additional scatter can result from the local environment (e.g., radiation field), which does not have to be identical for two clouds of the same evolutionary stage at different positions. For instance, pericenters on the near and far side are identified with each other in terms of evolutionary stage despite the spatial separation of ∼120 pc. Hence, influences of the environment can be very different at a given time since the last pericenter passage.

Assuming optically thin emission, column densities can simply be derived from observed intensities with the appropriate scaling factors because every detected photon directly corresponds to one emitting molecule. Following Ott et al. (2005, 2014), Mills & Morris (2013), whose work is based on Hüttemeister et al. (1993, 1995), and Mauersberger et al. (2003), the column density N in state (J, K) depends on the electric dipole moment μ in Debye and on the line brightness temperature T_mb as

$$\begin{eqnarray}&&N(J,K)[{\mathrm{cm}}^{-2}]=1.6698\cdot {10}^{14}\cdot \displaystyle \frac{J(J+1)}{{K}^{2}}\\ &&\qquad \times \,\displaystyle \frac{1}{{\mu }^{2}\,[{D}^{2}]\cdot \nu \,[\mathrm{GHz}]}{\displaystyle \int }_{\mathrm{line}}{T}_{\mathrm{mb}}\ {dv}\,[{\rm{K}}\,\mathrm{km}\,{{\rm{s}}}^{-1}]\end{eqnarray} \tag{ 11 }$$

$$\begin{eqnarray}&&=\displaystyle \frac{7.77\times {10}^{13}}{\nu \,[\mathrm{GHz}]}\displaystyle \frac{J(J+1)}{{K}^{2}}{\int }_{\mathrm{line}}{T}_{\mathrm{mb}}\ {dv}\,[{\rm{K}}\,\mathrm{km}\,{{\rm{s}}}^{-1}].\end{eqnarray} \tag{ 12 }$$

Equation (11) lists the general formula for any symmetric top molecule with dipole moment μ, whereas Equation (12) already includes μ = 1.472 D for ammonia and shows the form given in the literature.

The datacube structure in discrete channels of width ${\rm{\Delta }}{v}_{\mathrm{chan}}$ simplifies the integration in velocity over the spectral line to a summation of spectral channels, which in turn is just the definition of moment-0 flux ${F}_{\mathrm{mom}0}=\int {T}_{\mathrm{mb}}\ {dv}=\sum {T}_{\mathrm{mb}}\ {\rm{\Delta }}{v}_{\mathrm{chan},}$

$$\begin{eqnarray}&&N(J,K)[{\mathrm{cm}}^{-2}]=1.6698\cdot {10}^{14}\displaystyle \frac{J(J+1)}{{K}^{2}}\\ &&\qquad \cdot \,\displaystyle \frac{1}{{\mu }^{2}\,[{{\rm{D}}}^{2}]\cdot \nu \,[\mathrm{GHz}]}\cdot \displaystyle \sum _{\mathrm{line}}{T}_{i}\ {\rm{\Delta }}{v}_{\mathrm{chan}}\,[{\rm{K}}\,\mathrm{km}\,{{\rm{s}}}^{-1}],\end{eqnarray} \tag{ 13 }$$

$$\begin{eqnarray}&&=\displaystyle \frac{7.77\times {10}^{13}}{\nu \,[\mathrm{GHz}]}\displaystyle \frac{J(J+1)}{{K}^{2}}{{\rm{F}}}_{\mathrm{mom}0}\,[{\rm{K}}\,\mathrm{km}\,{{\rm{s}}}^{-1}],\end{eqnarray} \tag{ 14 }$$

By solving the radiative transfer equation, it is possible to derive an expression for the column density without assuming high or low optical depth because an exact solution for the relation of T_ex (${T}_{{JJ}^{\prime} }$ for rotational states) and observed brightness temperature ${T}_{\mathrm{mb}}$ is known (Schilke 1989).

The derivation starts again from the general approach in Equation (10), but then solves the integral over a spectral line of Gaussian shape (peak T_L and FWHM ${\rm{\Delta }}{v}_{\mathrm{int}}$) with excitation temperature as in Equation (9). The integral over a single hyperfine component thus is given by

$$\begin{eqnarray}&&{\rm{A}}=\displaystyle \frac{\sqrt{\pi }}{2\sqrt{\mathrm{ln}2}}\cdot {T}_{L}\cdot {\rm{\Delta }}{v}_{\mathrm{int}}\end{eqnarray} \tag{ 15 }$$

and the velocity integrated temperature becomes

$$\begin{eqnarray}&&\int {T}_{\mathrm{ex}}\ {dv}=\displaystyle \frac{\sqrt{\pi }}{2\sqrt{\mathrm{ln}2}}{\rm{\Delta }}{v}_{\mathrm{int}}\displaystyle \frac{{T}_{\mathrm{ex}}\tau }{{f}_{J}},\end{eqnarray} \tag{ 16 }$$

$$\begin{eqnarray}&&=\displaystyle \frac{\sqrt{\pi }}{2\sqrt{\mathrm{ln}2}}{\rm{\Delta }}{v}_{\mathrm{int}}\displaystyle \frac{{T}_{L}\tau }{{{ff}}_{J}(1-{e}^{-\tau })},\end{eqnarray} \tag{ 17 }$$

T_L denotes the line peak intensity and must not be confused with the integrated line intensity T_mb we used before. ${\rm{\Delta }}{v}_{\mathrm{int}}$ stands for the fitted FWHM of a single hyperfine component.

Calculating all numerical factors and logically rearranging the parts to match Equation (11)–(14) results in

$$\begin{eqnarray}N(J,K) & = & 1.77745\cdot {10}^{14}\,{\mathrm{cm}}^{-2}\cdot \displaystyle \frac{J(J+1)}{{K}^{2}}\\ & & \times \displaystyle \frac{{\rm{\Delta }}{v}_{\mathrm{int}}\,[\mathrm{km}\,{{\rm{s}}}^{-1}]}{{(\mu [{\rm{D}}])}^{2}\ \nu \,[\mathrm{GHz}]}\displaystyle \frac{\tau \ {T}_{L}\,[{\rm{K}}]}{{f}_{J}(1-{e}^{-\tau })}.\end{eqnarray} \tag{ 18 }$$

The pre-factor of $1.77745\cdot {10}^{14}$ applies when the beam filling factor is set to unity, line width ${\rm{\Delta }}{v}_{\mathrm{int}}$ is given in km s⁻¹, dipole moment μ in Debye, frequency of the inversion line in GHz and line peak intensity in Kelvin. f_J is a factor to correct the fitted opacity τ for its line profile and is basically the ratio of flux in the central hyperfine component (see Schilke 1989, for details). Table 3 lists f_J for ${\mathrm{NH}}_{3}$ (1, 1) to (6, 6) for five hyperfine components per line, whereof the central ("component 0'') enters Equation (18).

Table 3.  Ammonia Hyperfine Components as Calculated from Tabulated Data in Townes & Schawlow (1975) and splatalogue.net

Line	Component
	−2	−1	0	1	2
${\mathrm{NH}}_{3}$ (1, 1)	−19.48	−7.46	0	7.59	19.61	[km s−1]
	0.111	0.139	0.500	0.139	0.111
${\mathrm{NH}}_{3}$ (2, 2)	−25.91	−16.30	0	16.43	26.03	[km s−1]
	0.052	0.054	0.789	0.054	0.052
${\mathrm{NH}}_{3}$ (3, 3)	−29.14	−21.10	0	21.10	29.14	[km s−1]
	0.027	0.029	0.888	0.029	0.027
${\mathrm{NH}}_{3}$ (4, 4)	−30.43	−24.22	0	24.22	30.43	[km s−1]
	0.016	0.016	0.935	0.016	0.016
${\mathrm{NH}}_{3}$ (5, 5)	−31.41	−25.91	0	25.91	31.41	[km s−1]
	0.011	0.011	0.956	0.011	0.011
${\mathrm{NH}}_{3}$ (6, 6)	−31.47	−26.92	0	26.92	31.47	[km s−1]
	0.008	0.008	0.969	0.008	0.008

Note. For each ammonia species, the top row lists spectral offset in km s⁻¹, the bottom row shows the fractional line strength.

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Figure 33 shows the column density distribution of the upper state ${N}_{u}$ of ${\mathrm{NH}}_{3}$ (3, 3) as an example.

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Note that Equations (18) and (12) and Figure 33 give the column density of ammonia in the lower transition state only. The total column density of the state (J, K) must also take the molecules in the upper transition state into account. In thermal equilibrium, the column densities of both states are related by a Boltzmann factor

$$\begin{eqnarray}&&N(J,K)={N}_{l}(J,K)\left(1+\exp \left(-\displaystyle \frac{h\nu }{{k}_{{\rm{B}}}T}\right)\right)\end{eqnarray} \tag{ 19 }$$

$$\begin{eqnarray}&&\simeq 2\times {N}_{l}(J,K)\qquad \qquad \mathrm{for}\ T\gg \displaystyle \frac{h\nu }{{k}_{{\rm{B}}}}.\end{eqnarray} \tag{ 20 }$$

In the ground state (J = 0), the inversion states are degenerate and no transition can be observed, which means that the column density N₀₀ cannot be derived directly but needs to be extrapolated from low metastable states. As the slope in a typical Boltzmann plot (excitation state-scaled column density versus excitation temperature) of ammonia emission of the GC starts to flatten around J = 4–6, the best estimate for N₀₀ is derived from the slope between measurements at J = 1 and J = 2, i.e., rotational temperature T₁₂. If the ortho/para ammonia abundance ratio is not known, the simplest approach is to assume occupation by thermal equilibrium of states J = 0, 1, 2 at temperature ${T}_{12}$, hence to extrapolate N₀₀ according to Equation (21) (Ungerechts et al. 1986). The factor 23.2 K is given by the energy difference of ${\mathrm{NH}}_{3}$ (0, 0) and ${\mathrm{NH}}_{3}$ (1, 1),

$$\begin{eqnarray}&&{N}_{00}=\displaystyle \frac{1}{3}\exp \left(\displaystyle \frac{23.2\,{\rm{K}}}{{T}_{\mathrm{kin},12}}\right){N}_{11}.\end{eqnarray} \tag{ 21 }$$

The "total" column density of ammonia is then derived by summing the six directly measured column densities ($N={N}_{l}+{N}_{u}$) for states J = 1, ..., 6 and the extrapolated J = 0 column density. In thermal equilibrium, higher states are increasingly less populated and can thus be approximated to contribute negligibly to the total column density for low temperatures. Equation (22) uses these assumptions, i.e., does not contain the contribution of ammonia in rotation states higher than J = 6 and consequently must be understood as a lower limit. How close this limit is to the real ammonia column density can roughly be estimated by the ratio ${N}_{00}/{N}_{66}$, which in SWAG typically exceeds 10 with only few very hot clouds such as Sgr B2 that contain relevant amounts of ammonia in higher excitation states, as was also found by Mills & Morris (2013). The assumption of thermal equilibrium, which is required to extrapolate the total column density, does not hold in all CMZ clouds, whereas the lower limit is supported by direct measurements and thus is more robust.

The total column density of ammonia (defined to include emission up to J = 6) is thus given by

$$\begin{eqnarray}{N}_{\mathrm{tot}} & = & \left[\displaystyle \frac{1}{3}\exp \left(\displaystyle \frac{23.2\,{\rm{K}}}{{T}_{12,\mathrm{kin}}}\right)+1\right]{N}_{11}\\ & & +{N}_{22}+{N}_{33}+{N}_{44}+{N}_{55}+{N}_{66}.\end{eqnarray} \tag{ 22 }$$

As described in Appendix C, temperatures can also be calculated assuming optically thin emission, as is often done (in extra-galactic contexts), with the advantage of being able to derive temperatures over a larger area. From the fitted opacities (Figure 14), it is already known that this assumption is not satisfied in a significant portion of the map. The practical differences for derived temperatures are shown in Figure 34, which plots the ratio ${T}_{\mathrm{thin}}/{T}_{\tau }$ of the kinetic temperature T₂₄ derived under the assumption of optically thin emission and considering fitted opacities. The differences are minor in large areas of the map, but the temperature can be overestimated by ${T}_{\mathrm{thin}}$ in dense regions by factors of up to 1.6 in Sgr B2. Temperatures in less dense clouds and at the edges of the fitted area are typically underestimated by factors of ≲2. This can also be seen in a histogram of the map (Figure 35) as an increase in ratios >1.0 over what is expected from statistical, normally distributed errors (Gaussian fit). In an optically thick molecular cloud, not all emission can be detected because of self-absorption and scattering, as would be the case in an optically thin cloud. The derived column densities thus increase when opacity is considered. The temperature is derived from the inverse slope of the column density as a function of molecular energy level, which results in lower temperatures for two conditions: when the opacity is considered in calculating the column density, and when the opacity decreases for higher states, which typically is the case. Given that the opacity typically decreases for higher states, the effect on column density is stronger for lower J transitions. As the temperature is derived from the inverse slope of the column density as a function of molecular energy level, considering the opacity results in lower temperatures.

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