<figure><img src="image_1.png" /></figure>

# Thermal Physics of the Inner Coma: ALMA Studies of the Methanol Distribution and Excitation in Comet C/2012 K1 (PanSTARRS)

M. A. Cordiner¹,², N. Biver³, J. Crovisier³, D. Bockelée-Morvan³, M. J. Mumma¹, S. B. Charnley¹, G. Villanueva¹, L. Paganini¹,², D. C. Lis⁴, S. N. Milam¹, A. J. Remijan⁵, I. M. Coulson⁶, Y.-J. Kuan⁷,⁸, and J. Boissier⁹

¹Goddard Center for Astrobiology, NASA Goddard Space Flight Center, 8800 Greenbelt Road, Greenbelt, MD 20771, USA; martin.cordiner@nasa.gov

²Department of Physics, Catholic University of America, Washington, DC 20064, USA

³LEISA, Observatoire de Paris, CNRS, UPMC, Université Paris-Diderot, 5 place Jules Janssen, F-92195 Meudon, France

⁴LERMA, Observatoire de Paris, PSL Research University, CNRS, Sorbonne Universités, UPMC Univ. Paris 06, F-75014, Paris, France

⁵National Radio Astronomy Observatory, Charlottesville, VA 22903, USA

⁶Joint Astronomy Centre, Hilo, HI 96720, USA

⁷National Taiwan Normal University, Taipei 116, Taiwan, ROC

⁸Institute of Astronomy and Astrophysics, Academia Sinica, Taipei 106, Taiwan, ROC

⁹IRAM, 300 Rue de la Piscine, F-38406 Saint Martin d'Hères, France

Received 2016 September 16; revised 2016 November 28; accepted 2016 December 7; published 2017 March 16

## Abstract

We present spatially and spectrally resolved observations of CH₃OH emission from comet C/2012 K1 (PanSTARRS), using the Atacama Large Millimeter/submillimeter Array on 2014 June 28–29. Two-dimensional maps of the line-of-sight average rotational temperature ($T_{rot}$) were derived, covering spatial scales 0.′′3–1.′′8 (corresponding to sky-projected distances $\rho \sim 500–2500$ km). The CH₃OH column density distributions are consistent with isotropic, uniform outflow from the nucleus, with no evidence for extended sources of CH₃OH in the coma. The $T_{rot}(\rho)$ radial profiles show a significant drop within a few thousand kilometers of the nucleus, falling from about 60 to 20 K between $\rho = 0$ and 2500 km on June 28, whereas on June 29, $T_{rot}$ fell from about 120 to 40 K between $\rho = 0$ km and 1000 km. The observed $T_{rot}$ behavior is interpreted primarily as a result of variations in the coma kinetic temperature due to adiabatic cooling of the outflowing gas, as well as radiative cooling of the CH₃OH rotational levels. Our excitation model shows that radiative cooling is more important for the $J = 7–6$ transitions (at 338 GHz) than for the $K = 3–2$ transitions (at 252 GHz), resulting in a strongly sub-thermal distribution of levels in the $J = 7–6$ band at $\rho \gtrsim 1000$ km. For both bands, the observed temperature drop with distance is less steep than predicted by standard coma theoretical models, which suggests the presence of a significant source of heating in addition to the photolytic heat sources usually considered.

**Key words:** Comets: individual (C/2012 K1, PanSTARRS) – molecular processes – techniques: imaging spectroscopy – techniques: interferometric

## 1. Introduction

Comets are considered fossils of the early solar system—frozen relics containing ice, dust, and debris from the protoplanetary accretion disk. Having existed in a relatively quiescent state since their formation (e.g., Davidsson et al. 2016), cometary compositions can provide unique information on the thermal and chemical characteristics of the early solar system. Most of our knowledge of cometary compositions comes from remote (ground-based) observations of their gaseous atmospheres/comae (Cochran et al. 2015), for which the (typically relatively low) angular resolution and incomplete spatial coverage limits the amount of information that can be obtained. A particular problem is the lack of understanding regarding the physical and chemical structure of the near-nucleus coma at distances less than a few thousand kilometers from the comet’s surface.

With the advent of the Atacama Large Millimeter/submillimeter Array (ALMA), high-sensitivity, high angular-resolution, millimeter-wave interferometry of typical, moderately bright comets has become possible. ALMA’s unique capabilities allow us to probe the physical and chemical structure of the innermost regions of the coma in unprecedented detail, leading to new insights into the properties of the coma and the nucleus. The first cometary observations using ALMA were reported by Cordiner et al. (2014), who measured the distributions of HCN, HNC, and

H₂CO in comets C/2012 F6 (Lemmon) and C/2012 S1 (ISON). They demonstrated unequivocally that HNC and H₂CO are released in the coma (as a result of photolytic and/or thermal processes), whereas HCN originates from (or very near to) the nucleus.

For simplicity, a constant coma kinetic temperature is commonly assumed during analysis of microwave and sub-mm cometary observations. However, theoretical and observational studies increasingly show that temperatures can vary substantially over short distance scales in the coma. Strong variations in temperature within a few hundred kilometers of the nucleus are predicted by coma hydrodynamic/Monte Carlo simulations (e.g., Körösmezey et al. 1987; Combi et al. 1999a, 1999b; Rodgers & Charnley 2002; Tenishev et al. 2008), but these models are largely untested due to a lack of comparative observational data. Early observational reports of inner coma temperature variations were based on long-slit infrared spectroscopy of HCN and CO in the unusually active comet C/1995 O1 (Hale-Bopp) (Magee-Sauer et al. 1999; DiSanti et al. 2001). In a far-IR study of H₂O and HDO emission from comet C/2009 P1 (Garradd) by Bockelée-Morvan et al. (2012), a strongly variable temperature law was suggested—reaching a minimum at (4–20) × 10³ km from the nucleus and rising to 150 K in the outer coma. Advances in ground-based infrared instrumentation and data-analysis techniques permitted Bonev et al. (2007, 2008, 2013, 2014) to detect significant variations in the H₂O rotational excitation
temperature over distance scales ~20–1500 km in four comets, which led to the first detailed (quantitative) comparisons with theory for comet 73P/Schwassmann–Wachmann 3 (Fougere et al. 2012).

To date, a general lack of information concerning spatial variations in the excitation of cometary molecules impedes the accuracy of important results in the cometary literature, hindering the derivation of accurate cometary mixing ratios, as well as introducing errors into the parent scale lengths of distributed sources. Thus, detailed measurements of coma temperatures are required in order to confirm and expand upon the previous observational findings, and thus stimulate revision and refinement of theoretical models.

By virtue of its large abundance in comets (typically on the order of a few percent, with respect to H₂O), and its strong rotational bands throughout the mm/sub-mm, methanol (CH₃OH) is an ideal molecule for mapping the coma temperature distribution. Here, we present results exploiting the high resolution and sensitivity of ALMA, to provide new information on the distribution and excitation of CH₃OH in the inner coma of the Oort-cloud comet C/2012 K1 (Pan-STARRS). We interpret our observations using non-LTE radiative transfer models.

## 2. ALMA CH₃OH Observations

ALMA observations of C/2012 K1 (PanSTARRS) were obtained pre-perihelion, on 2014 June 28 19:07-20:05 and 2014 June 29 17:37-18:26, while the comet was at a heliocentric distance $r_H = 1.42–1.43$ au and geocentric distance $\Delta = 1.96–1.97$ au (the comet reached perihelion at $r_H = 1.05$ au on 2014 August 28). The CH₃OH K = 3 – 2 rotational band near 251.9 GHz (in ALMA receiver band 6) was observed on June 28, and the J = 7 – 6 band near 338.5 GHz (in band 7) was observed on June 29. Atmospheric conditions were excellent throughout (with zenith PWV <0.4 mm). Thirty 12 m antennas, with baseline lengths 20–650 m, resulted in an angular resolution of $0.′′80 × 0.′′43$ at 252 GHz and $0.′′71 × 0.′′33$ at 338 GHz. These beam dimensions correspond to $1100 × 610$ km and $1000 × 470$ km, respectively, at the distance of the comet; the measured RMS noise levels per channel were 1.8 and 3.0 mJy bm⁻¹, respectively. The correlator was configured to simultaneously observe as many strong CH₃OH lines as possible (spanning different upper-state energy levels) in a single 976 MHz spectral window, using a spectral resolution of 488 MHz (0.58 km s⁻¹ in band 6 and 0.43 km s⁻¹ in band 7).

The observing sequence on each date consisted of an interleaved series of scans of the science target and a continuum source (quasar) for phase calibration, alternating between seven minutes of integration on the comet and 30 s on the phase calibrator, for a total of 46 minutes on-source at 252 GHz and 38 minutes on-source at 338 GHz. The comet was tracked, and the position of the array phase center updated in real time, using the latest JPL Horizons orbital solution (sampled at 15 s intervals and interpolated using a fourth-order polynomial). The data were flagged, calibrated, and cleaned using standard CASA routines (e.g., Cordiner et al. 2014), with Ganymede as the flux calibrator. Imaging was performed using a grid size of 768 × 768 pixels, with 0.′′05 pixel scale. The resulting data cubes were corrected for the response of the ALMA primary beam, and then transformed from celestial coordinates to sky-projected distances with respect to the center

<figure><img src="image_2.png" /><figcaption>**Figure 1.** Production rates for OH in comet C/2012 K1 (PanSTARRS) as a function of heliocentric distance, obtained using the Nançay radio telescope. The values shown are pre-perihelion, and cover the period 2014 June 14 to 2014 July 17. Dashed horizontal line shows the mean value of $9.0 × 10^{28}$ s⁻¹. Vertical dotted line corresponds to the epoch of the ALMA observations.</figcaption></figure>

of the comet. Spectral fluxes per beam were subsequently converted to the Rayleigh–Jeans brightness temperature scale ($T_B$) for further analysis.

## 3. Nançay OH Observations

Production rates for the dominant volatile H₂O are required for the measurement of CH₃OH mixing ratios and the derivation of the collisional excitation rates. The strengths of the 18 cm OH lines (at 1667 and 1665 MHz) in comet C/2012 K1 (PanSTARRS) were monitored using the Nançay radio telescope during the period 2014 April 03 to 2014 September 27. The observational procedure and data analysis were described previously by Crovisier et al. (2002), and an expansion velocity of 0.70 km s⁻¹ was retrieved following the analysis method of Tseng et al. (2007). The resulting OH production rates ($Q$(OH), for the period 2014 June 14 to 2014 July 17) are shown as a function of heliocentric distance (pre-perihelion) in Figure 1, including error bars due to thermal noise. The OH line-inversion parameter was large for these observations, therefore errors on the OH production rate due to excitation uncertainties are expected to be negligible. There is no clear trend in $Q$(OH) versus $r_H$ around the time of our ALMA observations, so we take the mean value of $9.0 × 10^{28}$ s⁻¹ (with a standard deviation of $1.7 × 10^{28}$ s⁻¹). Assuming H₂O is the sole parent of OH, and adopting a branching ratio of 0.86 for the H₂O + hν → OH + H photolysis channel (Huebner et al. 1992), we obtain a water production rate of $(1.05 ± 0.20) × 10^{29}$ s⁻¹.

## 4. Results

### 4.1. Spectra and Maps

Fourteen individual emission lines of the K = 3 – 2 band were detected on 28 June, and 16 lines of the J = 7 – 6 band were detected on 29 June. The complete list of detected CH₃OH transitions and their upper-state energies is given in Table 1, and the observed spectra are shown in Figure 2. These spectra were extracted by averaging the data over a circular aperture 1.′′5 in diameter, centered on the peak of the CH₃OH
<table><caption>**Table 1**
CH₃OH Lines Detected in C/2012 K1 (PanSTARRS)</caption><thead><tr><th>ALMA Band</th><th>Transition</th><th>Freq. (MHz)</th><th>E<sub>u</sub> (K)</th><th>W<sub>p</sub> (K km s<sup>-1</sup>)</th></tr></thead><tbody><tr><td rowspan="15">6</td><td>10<sub>3</sub> - 10<sub>2</sub> A<sup>-+</sup></td><td>251164</td><td>177.5</td><td>0.14 (0.10)</td></tr><tr><td>9<sub>3</sub> - 9<sub>2</sub> A<sup>-+</sup></td><td>251360</td><td>154.3</td><td>0.54 (0.10)</td></tr><tr><td>8<sub>3</sub> - 8<sub>2</sub> A<sup>-+</sup></td><td>251517</td><td>133.4</td><td>0.71 (0.10)</td></tr><tr><td>7<sub>3</sub> - 7<sub>2</sub> A<sup>-+</sup></td><td>251642</td><td>114.8</td><td>0.62 (0.10)</td></tr><tr><td>6<sub>3</sub> - 6<sub>2</sub> A<sup>-+</sup></td><td>251738</td><td>98.5</td><td>0.47 (0.10)</td></tr><tr><td>5<sub>3</sub> - 5<sub>2</sub> A<sup>-+</sup></td><td>251812</td><td>84.6</td><td>1.07 (0.10)</td></tr><tr><td>4<sub>3</sub> - 4<sub>2</sub> A<sup>-+</sup></td><td>251867</td><td>73.0</td><td>0.60 (0.10)</td></tr><tr><td>5<sub>3</sub> - 5<sub>2</sub> A<sup>--</sup></td><td>251891</td><td>84.6</td><td>0.78 (0.10)</td></tr><tr><td>6<sub>3</sub> - 6<sub>2</sub> A<sup>--</sup></td><td>251896</td><td>98.5</td><td>1.04 (0.10)</td></tr><tr><td>4<sub>3</sub> - 4<sub>2</sub> A<sup>--</sup></td><td>251900</td><td>73.0</td><td>0.56 (0.10)</td></tr><tr><td>3<sub>3</sub> - 3<sub>2</sub> A<sup>-+</sup></td><td>251906</td><td>63.7</td><td>0.44 (0.10)</td></tr><tr><td>3<sub>3</sub> - 3<sub>2</sub> A<sup>--</sup></td><td>251917</td><td>63.7</td><td>0.38 (0.10)</td></tr><tr><td>7<sub>3</sub> - 7<sub>2</sub> A<sup>-+</sup></td><td>251924</td><td>114.8</td><td>0.84 (0.10)</td></tr><tr><td>8<sub>3</sub> - 8<sub>2</sub> A<sup>--</sup></td><td>251985</td><td>133.4</td><td>0.68 (0.10)</td></tr><tr><td rowspan="15">7</td><td>7<sub>0</sub> - 6<sub>0</sub> E</td><td>338124</td><td>78.1</td><td>0.65 (0.11)</td></tr><tr><td>7<sub>-1</sub> - 6<sub>-1</sub> E</td><td>338345</td><td>70.6</td><td>1.08 (0.11)</td></tr><tr><td>7<sub>0</sub> - 6<sub>0</sub> A<sup>+</sup></td><td>338409</td><td>65.0</td><td>1.22 (0.11)</td></tr><tr><td>7<sub>-4</sub> - 6<sub>-4</sub> E</td><td>338504</td><td>152.9</td><td>0.50 (0.11)</td></tr><tr><td>7<sub>-4</sub> - 6<sub>-4</sub> A<sup>-</sup></td><td>338513</td><td>145.3</td><td>1.46 (0.11)</td></tr><tr><td>7<sub>-4</sub> - 6<sub>-4</sub> A<sup>+</sup></td><td>338513</td><td>145.3</td><td>B</td></tr><tr><td>7<sub>-2</sub> - 6<sub>-2</sub> A<sup>-</sup></td><td>338513</td><td>102.7</td><td>B</td></tr><tr><td>7<sub>-4</sub> - 6<sub>-4</sub> E</td><td>338530</td><td>161.0</td><td>0.32 (0.11)</td></tr><tr><td>7<sub>-3</sub> - 6<sub>-3</sub> A<sup>+</sup></td><td>338541</td><td>114.8</td><td>1.28 (0.14)</td></tr><tr><td>7<sub>-3</sub> - 6<sub>-3</sub> A<sup>-</sup></td><td>338543</td><td>114.8</td><td>B</td></tr><tr><td>7<sub>-3</sub> - 6<sub>-3</sub> E</td><td>338560</td><td>127.7</td><td>0.39 (0.12)</td></tr><tr><td>7<sub>-3</sub> - 6<sub>-3</sub> E</td><td>338583</td><td>112.7</td><td>0.30 (0.12)</td></tr><tr><td>7<sub>-1</sub> - 6<sub>-1</sub> E</td><td>338615</td><td>86.1</td><td>0.83 (0.12)</td></tr><tr><td>7<sub>-2</sub> - 6<sub>-2</sub> A<sup>+</sup></td><td>338640</td><td>102.7</td><td>0.64 (0.12)</td></tr><tr><td>7<sub>-2</sub> - 6<sub>-2</sub> E</td><td>338722</td><td>87.3</td><td>1.63 (0.14)</td></tr><tr><td>7<sub>-2</sub> - 6<sub>-2</sub> E</td><td>338723</td><td>90.9</td><td>B</td></tr></tbody></table>

**Note.** Integrated line intensities $W_p$ are given for the CH₃OH column density peak, with $1\sigma$ statistical errors in parentheses. Here, B denotes transitions that are blended with the line above.

emission. Spectrally integrated maps of the CH₃OH emission (shown in Figure 3) were obtained by integrating over the spectral ranges covered by the lines in Table 1. The CH₃OH emission peaks were approximately spatially coincident on both dates, offset by 1′′2 south-west from the ALMA phase center, which is within the typical range of uncertainty for the position of the nucleus using optically derived cometary ephemerides. No continuum emission was detected, and a $3\sigma$ upper limit of 0.3 mJy was obtained for the average continuum flux (in band 7).

The CH₃OH maps show a compact distribution with a strong central peak. The flux falls rapidly with distance from the center, which is consistent with the combined effects of (isotropic) outflow expansion and photodissociation due to solar irradiation. Due to the lack of ALMA baselines shorter than 20 m, the largest angular scale detectable in our observations is ≈7′′4 (or ≈10⁴ km), which results in filtering out of coma structures larger than this size. Based on a simple Haser parent model for K1/PanSTARRS (see Section 4.5), spatial filtering is expected to have a negligible impact on the peak CH₃OH flux (less than a few percent), whereas only ~75% of the predicted flux per beam may be recovered at 2000 km (1′′4) from the nucleus.

## 4.2. Rotational Diagrams

Detailed spectral modeling is required to interpret the observed multi-line CH₃OH data. We begin by using the method outlined by Bockelée-Morvan et al. (1994) to construct rotational excitation diagrams to obtain the internal (rotational) temperature of the CH₃OH molecules ($T_{rot}$), averaged along the line of sight. Figure 4 shows rotational diagrams for June 28 (left) and June 29 (right), obtained from spectra extracted at the CH₃OH column density peak. The integrated line intensities (along with their statistical $1\sigma$ errors) for this position are given in Table 1. The gradient of the rotational diagram is equal to $-1/T_{rot}$, and the intercept is equal to $\ln(N/Q)$, where Q is the partition function and N is the column density. Errors on $T_{rot}$ and N were obtained from the errors on the gradient and intercept, resulting in values of $T_{rot} = 63.0 \pm 6.8$ K, $N = (2.8 \pm 0.7) \times 10^{14}$ cm⁻² on June 28, and $T_{rot} = 119 \pm 23$ K, $N = (2.7 \pm 1.0) \times 10^{14}$ cm⁻² on June 29. No significant differences were detected between the abundances of the A and E nuclear spin states of CH₃OH, consistent with an equilibrium (high-temperature) spin distribution —as observed in comet Hale-Bopp by Pardanaud et al. (2007). Thus, we assumed equal abundances of A and E CH₃OH in our analysis from here on.

Unfortunately, the rotational diagram method suffers from several shortcomings: blended lines with differing upper-state energies ($E_u$), and lines with signal-to-noise $<1$, cannot be included in the diagram, which limits the accuracy of the results. Furthermore, the full frequency extent of each line is not known a priori, which hinders the accuracy of the individual line measurements. In determining the integration widths for weak lines, a balance has to be found between including the full line flux and excluding noisy data in the line wings, which inevitably results in some reduction in the overall signal-to-noise for each line (for our data, we used integration limits of $\pm 1.5$ km s⁻¹ for all lines). In order to obtain the most reliable temperatures and column densities, spectral line modeling is employed to maximize the available information in our ALMA data.

## 4.3. Line Modeling

Using a modified version of the spectral line modeling algorithm previously applied to interstellar rotational emission lines by Cordiner et al. (2013), the spectra at each point in our CH₃OH maps were extracted and fitted (using a nonlinear least-squares algorithm) to determine $T_{rot}$ and N as a function of the spatial coordinate. The technique works by fitting Gaussian optical depth curves to each emission line (as a function of frequency), parameterized by $T_{rot}$, N and the line FWHM. The observed CH₃OH line-intensity ratios encode the rotational temperature, and the overall intensity ($T_B$) scaling encodes the column density, so the best fit to each spectrum can be found for a unique pair of [$T_{rot}$, N] values, as determined by the minimum of the sum-of-squares of the residuals between the observation and model (or $\chi^2$ value). Error estimates were obtained using a Monte Carlo noise-resampling technique, whereby for each spectrum, 300 synthetic, Gaussian, random noise spectra were generated (with standard deviation equal to the rms noise of the observed spectrum). These were then added one at a time to the (noise-free) best-fitting model spectrum. The same least-squares fitting procedure was then repeated for each of the 300 synthetic data sets. The $1\sigma$ errors
<figure><img src="image_3.png" /><figcaption>**Figure 2.** CH₃OH spectra observed on 2014 June 28 of the *K* = 3 − 2 band (left), and on 2014 June 29 of the *J* = 7 − 6 band (right). Ticks indicate the detected CH₃OH lines. These spectra were averaged over a 1′′-5-diameter circle (≈2100 km) centered on the emission peak.</figcaption></figure>

on *T*<sub>rot</sub>, *N* were obtained from the ±68% percentiles of the resulting set of fit parameters.

This procedure assumes a Gaussian shape for the emission
lines, which can be a poor approximation for cometary lines
observed at high spectral resolution—indeed, many of our
observed CH₃OH line profiles show slight asymmetries
(consistent with asymmetric outgassing). To avoid problems
with the fitting due to such non-Gaussianity, each extracted
spectrum was first convolved with a Gaussian broadening
kernel of FWHM 3 km s⁻¹. This is sufficiently broad, with
respect to the FWHM of the observed lines (≈0.9–1.2 km s⁻¹),
that it effectively results in the smoothing out of their profiles,
producing a line shape practically indistinguishable from
Gaussian, and independent of the specific outflow geometry
of the comet. This smoothing also has the benefit that any small
variations in the line FWHM and Doppler shift of the gas over
the ALMA field of view (≈±0.25 km s⁻¹) can be neglected,
allowing these values to be held fixed during the fitting to
further improve the accuracy of the results, which is
particularly useful for the noisier data toward the edge of the
field of view.

Examples of the line-modeling results are shown in the lower
panels of Figure 4, obtained using spectra extracted at the
CH₃OH column-density peak position. The quality of these fits
is very good (with reduced χ² values in the range 1.0–1.3), and
the *T*<sub>rot</sub> and *N* values derived using this method
(*T*<sub>rot</sub> = 62.3 ± 5.3 K, *N* = (2.8 ± 0.1) × 10¹⁴ cm⁻² on June
28 and *T*<sub>rot</sub> = 116 ± 18 K, *N* = (2.7 ± 0.4) × 10¹⁴ cm⁻² on
June 29) are in good agreement with those obtained using the
rotational diagram method. Due to improved utilization of the
noisier data, this line-modeling technique results in signifi-
cantly improved accuracy of the derived parameters, and is
therefore adopted as the preferred method of analysis from here
on. The uncertainties derived from Monte Carlo noise
replication are also expected to be more reliable, because they
implicitly account for correlations between the errors on *T*<sub>rot</sub>
and *N*, whereas in the rotational diagram analysis, the (partially
correlated) errors on the gradient and intercept cannot be easily
disentangled.

4.4. *Temperature and Column Density Maps*

Spectra were extracted pixel-by-pixel from the CH3OH data
cubes within an area of 2" × 2" (40 × 40 pixels) centered on

the integrated emission peak. Using the line-fitting method
described in Section 4.3, temperatures and column densities
were obtained for each pixel to create the maps shown in
Figure 5. These maps only show the derived T<sub>rot</sub> and N values
with uncertainties of less than 50%—values with larger errors
have been masked (and are shown in white). The mean errors
on these T<sub>rot</sub> and N maps are ±14 K and ±1.8 × 10<sup>13</sup> cm<sup>-2</sup> for
June 28, ±15 K, and ±1.7 × 10<sup>13</sup> cm<sup>-2</sup> for June 29. Naturally,
a higher CH<sub>3</sub>OH column density leads to a stronger spectrum,
and hence lower uncertainties on T<sub>rot</sub> and N. Therefore, the
regions closer to the centers of the maps are more reliable. The
maps have been positioned with their origins at the respective
column density peaks, which are marked with white crosses.

The CH₃OH column density maps show a single, dominant
peak, as expected for spherically symmetric outgassing from a
compact nucleus. The shapes of the column density peaks are
strongly influenced by the elliptical telescope beam, which
hinders interpretation of the true structure of the innermost
coma. On both dates, the column density maps show a
significant extension toward the top left, which is in addition to
the dominant outflow structure. The direction of this extension
coincides approximately with the direction of the comet's
orbital trail, perhaps implying some CH₃OH release from
trailing material, although the presence of a directional CH₃OH
jet/vent cannot be ruled out.

Interestingly, on June 29 the *T*<sub>rot</sub> distribution shows a strong central peak around the nominal position of the nucleus (falling from 116 ± 18 K to ≈50 ± 10 K within about 500 km), whereas on June 28 the main source of CH<sub>3</sub>OH appears to be located within a temperature trough, which forms part of a much broader region of high temperatures that extends down the whole map in a roughly north-south direction. The central temperature on June 28 was 62 ± 5 K and the two flanking temperature maxima have *T*<sub>rot</sub> ≈ 85 ± 14 K on the left and ≈73 ± 12 K on the right. However, the statistical significance of these peaks relative to the central temperature is low (~1–2σ), so a monotonically decreasing (or flat) temperature as a function of distance from the nucleus in this region cannot be ruled out. Similarly, although some of the temperature structure toward the bottom left of the central peak on June 29 could be real, the uncertainties preclude a robust interpretation of the other features in these maps.
<figure><img src="image_4.png" /><figcaption>**Figure 3.** Spectrally integrated ALMA flux maps of CH₃OH in comet C/2012 K1 (PanSTARRS), observed on 2014 June 28 at 252 GHz (left) and 2014 June 29 at 338 GHz (right), integrated over all detected transitions. Coordinate axes are aligned with the R.A.–decl. grid, with celestial north toward the top. White crosses indicate the emission peaks, which are employed as the origin of the coordinate axes. Contours are plotted at 3σ intervals, where σ is the rms noise in each map. The FWHM of the (Gaussian) restoring beams are shown lower left. Direction of the Sun (S) and orbital trail (T) with respect to the comet are indicated lower right, along with the illumination phase angle of 30°.</figcaption></figure>

## 4.5. Radial Profiles

To help average out any clumpiness or small-scale coma structures, and provide a clearer picture of the dominant physical and chemical processes as a function of distance from the nucleus, we perform azimuthal averaging of the ALMA data. Such azimuthally averaged data also benefit from a significantly improved signal-to-noise ratio, particularly at large radii, allowing us to probe the conditions further from the nucleus than in the 2D maps.

Taking the CH₃OH column density peak as the center, our data were binned into a series of 0.′′1-wide annuli, and the average flux per spectral channel was taken in each annulus, resulting in azimuthally averaged spectra ($\bar{S}_{\nu}$) as a function of sky-projected radius $ρ$. These $\bar{S}_{\nu}(ρ)$ were subject to the same fitting procedure outlined in Section 4.3 to derive radial [$T_{rot}$, $N$] profiles, shown in Figure 6.

The 1D radial temperature profiles show a general trend for falling $T_{rot}$ as a function of distance. Similar to the 2D maps, on June 28 $T_{rot}$ shows a possible (1σ confidence) increase from 62 ± 5 K to 68 ± 6 K between $ρ = 0–500$ km, followed by a relatively slow, steady decrease to 22 ± 9 K at $ρ = 2500$ km. Conversely, on June 29 the temperature decreased sharply from 116 ± 18 K to 42 ± 4 K between $ρ = 0–1000$ km. In addition to starting at least 30 K hotter near the nucleus, overall, the CH₃OH rotational temperatures remained at least 20 K higher, throughout the observed coma, on the 29th than the 28th, implying the presence of a substantial additional coma heat source on the 29th (22.5 hr later).

The possible temperature increase at distances $ρ > 1500$ km on June 29, identified by Cordiner et al. (2016), is less clear in our new results; the large error bars on our $T_{rot}$ profile at $ρ > 2000$ km indicate that this result may have been at least partly due to an underestimate of the errors by Cordiner et al. (2016), who used a rotational diagram method rather than spectral line fitting. Furthermore, the azimuthal profiles in the present study were taken around the CH₃OH column density

peak, whereas those of Cordiner et al. (2016) were taken around the integrated emission line peak; these positions were coincident on June 28, but on June 29 the column density peak is offset northeast by 0.′′16, leading to differences in the radial temperature profiles. The elliptical shape of the telescope beam introduces some additional uncertainty into our radial temperature and column density profiles, which would be particularly problematic for any asymmetric coma features that happen to line up with the long axis of the beam. The (long-axis) spatial resolution HWHM of 550 km on June 28 and 500 km on June 29 should thus be considered as upper limits for the possible radial error margins on these azimuthally averaged profiles.

On both dates, the azimuthally averaged column densities decreased smoothly with distance from the peak value of about 2.8 × 10¹⁴ cm⁻², with a shape closely consistent with uniform, spherically symmetric outflow. This is confirmed by comparison of the observed $N(ρ)$ profiles in Figure 6 with profiles obtained from a Haser-type spherical outflow model, in which CH₃OH is assumed to flow isotropically from the nucleus at a constant velocity of 0.5 km s⁻¹ (consistent with the average CH₃OH line Doppler FWHM of ≈1.0 km s⁻¹), and is photodissociated at a rate of 10⁻⁵ s⁻¹ (Huebner et al. 1992). To match the observational point spread functions, the model column density maps were run through the ALMA simulator in CASA with array configuration, imaging, and cleaning parameters matching those of our observations. Good fits to the observed $N(ρ)$ profiles were obtained for CH₃OH production rates of 1.9 × 10²⁷ s⁻¹ on June 28, and 1.4 × 10²⁷ s⁻¹ on June 29. Using the H₂O production rate of (1.05 ± 0.20) × 10²⁹ s⁻¹ (Section 3), these results imply that CH₃OH was sublimated directly from the nucleus of comet K1/PanSTARRS, with mixing ratios of 1.1%–2.2%.

Our model column density profiles (red dots overlaid on the blue curves in Figure 6) provide an excellent fit to the observations on June 28, and a reasonably good fit on June 29. As with the study of Cordiner et al. (2014), this confirms that the Haser model can be usefully applied to azimuthally
<figure><img src="image_5.png" /><figcaption>**Figure 4.** Top: rotational diagrams for K1/PanSTARRS CH₃OH, constructed using spectra obtained at the column density peaks for the *K* = 3 – 2 (left) and *J* = 7 – 6 transitions (right). Bottom: least-squares model fits to the observed (smoothed) spectra; for clarity, only a portion of the entire spectral region is shown for each date.</figcaption></figure>

averaged ALMA observations of the inner coma. Slight discrepancies may be explained by the presence of clumpy structure, jets, or variations in the outflow velocity. Some acceleration of the coma gas within our ALMA field of view is plausible. Indeed, the derived CH₃OH expansion velocity of 0.5 km s⁻¹ in our ~2000 km field of interest is smaller than the value of 0.7 km s⁻¹ obtained for OH within a ~10⁵ km beam, as expected if the outflow speed increased with cometocentric distance. This possibility may be explored through detailed analysis of the CH₃OH outflow dynamics, in a future study.

## 5. Discussion

In the range $\rho$ = 0–2000 km, our ALMA observations indicate, as expected, a general trend of decreasing CH₃OH rotational temperatures as a function of cometocentric distance in the coma. Interpretation of this result requires an understanding of the meaning of $T_{\text{rot}}$. As explained by Bockelée-Morvan et al. (1994, 2004), the temperature $T_{\text{rot}}$ obtained from the analysis of CH₃OH rotational line ratios is representative of the distribution of internal rotational level populations. Unless the molecules are in local thermodynamic equilibrium (LTE, which occurs in the dense, inner coma through collisional equipartition), $T_{\text{rot}}$ can deviate significantly from the kinetic temperature of the gas ($T_{\text{kin}}$). The relationship between $T_{\text{kin}}$ and $T_{\text{rot}}$ depends on the competing influences of microscopic

collisional and radiative processes. As a result, the prevalence of LTE depends primarily on the coma density, the spontaneous radiative decay rate of the gas, and the flux of external (solar) radiation. Within a few hundred kilometers of the nucleus, where densities are high, the collisional rates between molecules are usually sufficient to maintain LTE. However, as the molecules flow out into the less dense, outer coma regions (at $r_c \gtrsim 1000$ km), collisions become less frequent, allowing the rotational levels to radiatively cool, so that $T_{\text{rot}}$ falls below $T_{\text{kin}}$. Pumping of molecular ro-vibrational levels by solar radiation can also have an important effect on $T_{\text{rot}}$ at large distances (beyond those considered in the present study).

Using the radiative transfer model of Biver et al. (1999, 2015), which computes the CH₃OH level populations that result from collisions with H₂O and electrons, as well as pumping by solar radiation, we calculated the departure of $T_{\text{rot}}$ from $T_{\text{kin}}$ with distance from the nucleus. Using dashed line styles, Figure 7 shows $T_{\text{rot}}$ as a function of projected distance ($\rho$) for the *J* = 7 – 6 and *K* = 3 – 2 bands, under the assumption of constant kinetic temperatures of 60 K (left panel) and 120 K (right panel). The H₂O production rate was taken to be 10²⁹ s⁻¹, with an outflow velocity 0.5 km s⁻¹, and a collisional cross section between H₂O and CH₃OH of 5 × 10⁻¹⁴ cm⁻². In this figure, $T_{\text{rot}}$ (*K* = 3 – 2) and
<figure><img src="image_6.png" /><figcaption>**Figure 5.** Column density maps (top) and rotational temperature maps (bottom) for K1/PanSTARRS CH<sub>3</sub>OH, derived from pixel-by-pixel spectral line modeling. Coordinate axes are aligned as in Figure 3, with celestial north toward the top. Left panels show the results for the *K* = 3 – 2 transitions and right panels are for the *J* = 7 – 6 transitions. Values with an uncertainty of greater than 50% have been masked (shown as white pixels). White crosses (“+”) mark the column density peak, and the hatched ellipses show the instrumental resolution. The 0.05 pixel scale corresponds to 71 km at the distance of the comet.</figcaption></figure>

$T_{\text{rot}}$ ($J = 7 - 6$) both show increasing departures from LTE with increasing distance from the nucleus. However, although the *K* = 3 – 2 band remains relatively close to LTE (with $T_{\text{rot}} > 0.75T_{\text{kin}}$ for $\rho < 2500$ km), for the *J* = 7 – 6 band, $T_{\text{rot}}$ falls much more rapidly with distance, reaching $0.25T_{\text{kin}}$ by $\rho = 2500$ km in the 120 K model. This is due to the larger Einstein A coefficients of the *J* = 7 – 6 transitions, which result in more rapid rotational cooling. Although this trend is qualitatively similar to the observed $T_{\text{rot}}$ pattern for the two CH<sub>3</sub>OH bands in Figure 6, the observed $T_{\text{rot}}$ curves both fall significantly more rapidly than expected with a coma kinetic temperature that remains constant as a function of distance. Sub-thermal excitation is therefore insufficient to fully explain the observed $T_{\text{rot}}$ behavior in comet K1/PanSTARRS, and a variable $T_{\text{kin}}$ ($r$) profile is required.

A variety of different kinetic temperature profiles were tested in our radiative transfer model, with the aim of reproducing the general behavior of the observed $T_{\text{rot}}$ ($r$) profiles. For the *K* = 3 – 2 profile on June 28, a good fit to the observed $T_{\text{rot}}$ ($r$) was obtained using a single-slope $T_{\text{kin}}$ ($r$) profile, starting at 90 K at the nucleus and falling to 35 K at $r_c = 2500$ km (shown

by the solid black line in the left panel of Figure 7). For the *J* = 7 – 6 data from June 29, however, it was more difficult to obtain a good fit to the measured $T_{\text{rot}}$ ($r$) data. Even with a rapidly falling $T_{\text{kin}}$ ($r$) profile in the inner coma, the very steep initial drop in $T_{\text{rot}}$ ($r$) could not be accurately reproduced. This is partly due to line-of-sight averaging, because cooler, more distant parts of the coma can dominate the temperature contribution from a compact, warm inner region (combined with the need to keep the inner coma temperatures physically reasonable). Our best fit to the $T_{\text{rot}}$ (*J* = 7 – 6) profile was obtained using the $T_{\text{kin}}$ ($r$) shown in black in the right panel of Figure 7, beginning at a relatively high temperature of 150 K, falling to 40 K at $r_c = 1000$ km, then rising back to 150 K by $r_c = 2500$ km. Difficulties in modeling the $T_{\text{rot}}$ ($r$) profile on June 29 may be due to coma asymmetries or other physical factors not included in our model. Future attempts to more robustly derive the coma kinetic temperatures from these data may require modeling the coma structure in 3D, as well as exploring the effects of variations in electron density profiles and temperatures, and gas collisional cross-sections.
<figure><img src="image_7.png" /></figure>

**Figure 6.** Azimuthally averaged CH₃OH rotational temperatures (upper panels) and column densities (lower panels), taken around the column density peaks labeled in Figure 5. The *K* = 3 – 2 transitions are shown (left) and the *J* = 7 – 6 transitions (right). Simulated (Haser-type) isotropic outflow model column density profiles are overlaid with red points, corresponding to CH₃OH production rates of 1.9 × 10²⁷ s⁻¹ on June 28 and 1.4 × 10²⁷ s⁻¹ on June 29. Dashed lines in the upper panels indicate the rotational temperature profiles obtained from radiative transfer modeling, using the kinetic temperature distributions shown in Figure 7 (solid black lines).

<figure><img src="image_8.png" /></figure>

Compared with previous generations of chemical/hydro-
dynamic coma models (see Rodgers et al. 2004 for a review),
the temperature trends observed in K1/PanSTARRS differ
from past theoretical expectations. In those models, the neutral
gas temperature falls rapidly as it travels outward from the
nucleus, starting at ~100 K and reaching a minimum of ~10 K
by 100 km, as a result of adiabatic cooling. Around a
cometocentric distance r_c ~ 100 km, the onset of photoioniza-
tion due to solar irradiation results in the production of a
population of fast (hot) electrons, ions, and neutrals. Collisions
with these hot photoproducts heats the parent gases. However,
the strong heating trends apparent in these conventional multi-
fluid hydrodynamic models—which predict temperatures
above a few hundred kelvins for cometocentric distances
>1000 km—primarily reflects the fact that the neutral gas
temperature is obtained by calculating the weighted average
temperature over all coma neutrals, including the fast-moving
photoproducts (primarily O, H, H₂, and OH from the photolysis
of H₂O). Because the kinetic temperatures of these photo-
products can reach thousands of kelvins in the outer coma, their
presence creates a strong bias in the average temperature
of neutrals, which is then no longer properly representative of
the temperature of parent gases. To account for this problem,
the temperatures of the parents and photoproducts must be
treated separately, as in the direct simulation Monte Carlo
(DSMC) models of, e.g., Tenishev et al. (2008) and Fougere
et al. (2012). These models show a clear separation in the
kinetic temperatures of parents and photoproducts, and confirm
the expectation that parent species are dominated by adiabatic
cooling for distances <~10⁵ km.

However, standard DSMC models still predict a very rapid
drop with distance in the temperature of parent molecules that is
inconsistent with our ALMA CH3OH observations. For example,
in the model for comet 67P/Churyumov-Gerasimenko by
Tenishev et al. (2008), the H2O temperature falls by an order of
magnitude within 100 km of the nucleus, whereas on June 28,
Tkin(r) in K1/PanSTARRS fell by only a few percent over this
range. To resolve this discrepancy, an additional source of coma
heating is required in the models. Fougere et al. (2012) considered
the sublimation of (dirty) ice grains, which can significantly raise

the H₂O rotational temperature in the region r_c ≲ 1000 km,
provided their mass represents a significant fraction of the total
gas-phase H₂O production. Such sublimation heating could also
raise the CH₃OH rotational temperature, either through injection
of a source of heated CH₃OH into the coma, or by collisions with
the heated H₂O molecules. Other possible heating sources that
may be considered in future models include suprathermal
electrons and ions that could be produced through interaction of
the coma with the solar wind, UV, and X-rays.

The temperature trend in K1/PanSTARRS is compatible
with the results of other observational studies. The steadily
decreasing T<sub>kin</sub> behavior found on June 28 is comparable to the
linear temperature slope deduced for the inner few thousand
kilometers of comet P1/Garrad (Bockelée-Morvan et al. 2012).
Further, on June 29 the falling kinetic temperature in the inner
coma, which rises back to ~150 K at larger radii, is
qualitatively similar to that study. Our observations on both
dates are also broadly consistent with the general trend for
decreasing H<sub>2</sub>O rotational temperatures with distance over
scales ρ ~ 10–1000 km in comets C/2004 Q2 (Machholz) and
73P-B/Schwassmann–Wachmann 3 (Bonev et al. 2007, 2008).
Comets 103P/Hartley 2 and C/2012 S1 (ISON) showed a
more complex temperature behavior. At r<sub>H</sub> = 0.53 au, comet
ISON’s T<sub>rot</sub> fell from ~120 K to ~85 K within 1000 km of the
nucleus, whereas evidence for a double-peaked temperature
structure was observed at r<sub>H</sub> = 0.35 au, rising to maxima at
cometocentric distances ~500–1000 km (Bonev et al. 2014).
For 103P/Hartley 2, T<sub>rot</sub> was observed to decrease with
distance on the sunward side of the nucleus, whereas evidence
was found for a significant increase in T<sub>rot</sub> between
ρ = 0–75 km (Bonev et al. 2013) on the anti-sunward side
(as projected on the sky plane at a phase angle of 54°). This T<sub>rot</sub>
behavior is analogous to the possible double-peaked temper-
ature structure identified in K1/Panstarrs on 28 June (Figure 5).
As discussed by Bonev et al. (2014), it may be that such
unexpected temperature peaks in the coma can arise as a result
of photolytic or sublimation heating, but additional theoretical
studies will be required to confirm this hypothesis.

The behavior of the coma temperature as a function of time
can provide more information on the nature of the coma
<figure><img src="image_9.png" /><figcaption>**Figure 7.** Dashed green and purple lines show simulated CH₃OH rotational temperatures as a function of distance from the nucleus (using the excitation model of Biver et al. 2015), for the *K* = 3 – 2 and *J* = 7 – 6 transitions respectively, assuming a constant coma kinetic temperature of *T*<sub>kin</sub> = 60 K (left panel) and 120 K (right panel). Whereas the *K* = 3 – 2 levels remain quite close to the kinetic temperature, the *J* = 7 – 6 levels quickly become very sub-thermal due to faster radiative cooling. The solid black lines show our best-fitting *T*<sub>kin</sub>(*r*) profiles on June 28 (left) and June 29 (right).</figcaption></figure>

heating processes, and is possible because of the 22.5 hr separation between our *K* = 3 – 2 and *J* = 7 – 6 CH₃OH observations. Although our excitation model shows that *T*<sub>rot</sub>(*J* = 7 – 6) may still be somewhat sub-thermal in the inner coma, the *J* = 7 – 6 and *K* = 3 – 2 bands are both expected to be close to the coma kinetic temperature at around *r*<sub>c</sub> = 0. The implied dramatic increase in coma kinetic temperature between June 28th and 29th is therefore surprising (given the expected relative constancy of solar radiation input), and is likely to have been caused by an increase in the heating rate of the inner coma—for instance, an increase in the supply of hot electrons or sublimating icy grains. If the electron temperature dropped below the threshold for dissociative electron impacts with H₂O, this could also result in a sudden increase in the coma heating rate, as a larger fraction of the electron energy can then be converted into kinetic energy (as discussed by Bodewits et al. 2016). Interactions with the solar wind could also lead to short-timescale variability in the coma energy balance.

Large variations in the CH3OH rotational temperature were observed by Drahus et al. (2012) over timescales of several hours, in comet 103P/Hartley 2. These variations were explained as due, in part, to the theorized correlation between water production rate and coma heating efficiency (see Bockelée-Morvan & Crovisier 1987; Combi et al. 2004). However, the relative constancy of the CH3OH production rate during our observations appears to rule out this possibility in K1/PanSTARRS, and the full explanation for such strong, transient variations in coma heating requires further investigation.

Several review studies have drawn comparison between interstellar, protostellar, and cometary ice abundances (e.g., Ehrenfreund & Charnley 2000; Mumma & Charnley 2011; Boogert et al. 2015). To date, every coma species detected in the radio has also been found in interstellar clouds, and the dominant cometary ice constituents (CO, CO₂ and CH₃OH) show abundances with respect to H₂O that are generally within the range of values observed in protostellar environments. Bockelée-Morvan et al. (1994) identified the similarity between cometary and interstellar CH₃OH/H₂O ice ratios, and

our values of 1.1%–2.2% in C/2012 K1 (PanSTARRS), compared with 1%–30% in low-mass protostars (Mumma & Charnley 2011), confirm this result. Recent detailed models for disk gas and ice chemistry confirm the plausibility of a close chemical relationship between cometary and protoplanetary material, e.g., Drozdovskaya et al. (2016) predict CH₃OH/H₂O ~1%–4% in the mid-plane ices for low-mass protoplanetary disks. The CH₃OH abundance in K1/PanSTARRS is also consistent with the detection of this molecule for the first time in a protoplanetary disk by Walsh et al. (2016), who obtained a gas-phase CH₃OH/H₂O ratio ~1%–5% in the disk surrounding the low-mass TW Hya system (at a distance of 54 pc). The fact that CH₃OH appears to be depleted in comets and protoplanetary disks, compared with the median protostellar abundance of 6% in the nearby Galaxy (Boogert et al. 2015), implies that significant processing of interstellar/protostellar envelope material occurs during or after its passage to the accretion disk, thus confirming the importance of cometary ices as a record for the physical and chemical processes occurring during the formation of the solar system.

**6. Conclusion**

Using ALMA observations of C/2012 K1 (PanSTARRS), the first instantaneous spatial/spectral maps of CH3OH rotational emission have been obtained in a cometary coma. Through rotational excitation analysis, 2D spatial maps of the CH3OH column density and rotational temperatures, averaged along the line of sight, have been derived, revealing new information about the physics and chemistry of the coma on scales of 500–5000 km. We find that the *T*<sub>rot</sub>(*J* = 7 – 6) and *T*<sub>rot</sub>(*K* = 3 – 2) radial profiles both exhibit a relatively rapid drop with distance, which cannot be explained purely through sub-thermal excitation, and must therefore be due to coma kinetic temperatures falling with distance from the nucleus. The observed temperature behavior is more consistent with the DSMC model of Fougere et al. (2012) (that includes coma heating from sublimating dirty ice grains), than the behavior seen in standard multi-fluid models, highlighting a need for continued research into coma heating (and cooling) mechanisms.
The CH₃OH radial column density profile is in good agreement with spherically symmetric, uniform outflow from the nucleus. Accordingly, no evidence is found for significant production of CH₃OH in the coma, either from icy grain sublimation or photochemistry. The CH₃OH mixing ratios of 1.1%-2.2% in K1/PanSTARRS are consistent with previous observations of comets at infrared and radio wavelengths. Combined with the observation of CH₃OH outgassing directly from the nucleus, our results confirm the utility of radio interferometric observations as a probe for the abundances of complex organic molecules in cometary ice. Our CH₃OH abundance adds to the evidence confirming a close chemical similarity between protostellar/protoplanetary and cometary ices.

The derivation of accurate coma kinetic temperatures (using CH₃OH or other molecules with a high density of rotational lines in the mm/sub-mm such as H₂CO or CH₃CN), combined with detailed theoretical modeling, is necessary to provide new constraints on the coma physics and further elucidate the gas heating and cooling mechanisms. Continued ALMA observations (including observations at higher angular resolution and higher sensitivity), will therefore play a crucial role in improving our knowledge of coma energetics, as well as leading to higher accuracy in cometary molecular production rates, parent scale lengths, and gas outflow velocities. Improved accuracy and statistics in measurements of cometary ice abundances are key requirements for ongoing studies on the origin and evolution of icy materials in planetary systems.

This work was supported by NASA's Planetary Atmospheres and Planetary Astronomy Programs, and by the National Science Foundation under Grant No. AST-1616306. It makes use of the following ALMA data: ADS/JAO. ALMA#2013.1.01061.S. ALMA is a partnership of ESO (representing its member states), NSF (USA), and NINS (Japan), together with NRC (Canada), and NSC and ASIAA (Taiwan), in cooperation with the Republic of Chile. The Joint ALMA Observatory is operated by ESO, AUI/NRAO, and NAOJ. The National Radio Astronomy Observatory is a facility of the National Science Foundation, operated under cooperative agreement by Associated Universities, Inc. The Nançay Radio Observatory is the Unité Scientifique de Nançay of the Observatoire de Paris, associated as USR No. B704 to the CNRS. The Nançay Observatory also gratefully acknowledges the financial support of the Conseil Régional of the Région Centre in France. We acknowledge the advice of Dr. Boncho Bonev on the measurement of spatial variability in coma rotational temperatures.

Facility: Atacama Large Millimeter/submillimeter Array.

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