HIGHLY DEPLETED ETHANE AND MILDLY DEPLETED METHANOL IN COMET 21P/GIACOBINI-ZINNER: APPLICATION OF A NEW EMPIRICAL ν₂-BAND MODEL FOR CH₃OH NEAR 50 K

Comet 21P/Giacobini-Zinner (hereafter 21P/GZ) was discovered in 1900 by Michel Giacobini, was re-discovered independently by Ernst Zinner in 1913, and was recovered in all apparitions since its discovery with the exception of the unfavorable ones in 1907, 1920, and 1953. Its Tisserand parameter with respect to Jupiter (T_J = 2.46) classifies 21P/GZ dynamically as a Jupiter Family comet (JFC), the predominant "feeding zone" for which is the scattered Kuiper disk, as predicted by the pre-eminent Nice model (Gomes et al. 2005; Morbidelli et al. 2008). This comet has the distinction of being the first to be visited by a spacecraft—in 1985 September, the International Cometary Explorer (originally the International Sun–Earth Explorer-3) flew through its plasma tail, passing within 8000 km of the nucleus. Here we report production rates for simultaneously observed C₂H₆, CH₃OH, and H₂O in 21P/GZ during its 2005 apparition, using the cross-dispersed high-resolution IR echelle spectrometer (NIRSPEC) at Keck II, applying our latest fluorescence models for H₂O, C₂H₆, and CH₃OH, including a new empirical model for the ν₂ band of methanol.

Observational studies indicate substantial chemical diversity among comets (e.g., see Mumma & Charnley 2011, and references therein). The most comprehensive work reported optical narrow-band photometric observations of radical (photo-dissociation product) abundances spanning some three decades (A'Hearn et al. 1995). This survey established "typical" and "depleted" classes depending on their abundances of C₂ (and C₃) relative to CN and OH. It was argued that this may reflect differences in carbon-chain chemistry (i.e., differing abundances of progenitor ices for C₂ and/or C₃), with inferred carbon-chain depletion being considerably more pronounced in JFCs compared with nearly isotropic comets (i.e., dynamically new or long-period returning comets, from the Oort cloud; see also Fink & Hicks 1996; Schleicher et al. 2007; Fink 2009; Cochran et al. 2012). Comet 21P/GZ is often regarded as the prototypical carbon-chain depleted comet (A'Hearn et al. 1995).

While highly diagnostic, these findings do not necessarily translate to a similar dichotomy in native ice abundances, primarily due to uncertainties in the parentages of fragmentation products (daughter or granddaughter species). Compared with optical observations, parent species have been studied in relatively few JFCs, owing mostly to their typically lower overall gas production relative to nearly isotropic comets. However, in recent years advances in technology have permitted more detailed studies of native ice abundances in JFCs, in both radio (Biver et al. 2002; Crovisier et al. 2009; Biver et al. 2012) and infrared regimes (Weaver et al. 1999; Mumma et al. 2000, 2005, 2011; Dello Russo et al. 2007, 2008, 2009, 2011; DiSanti et al. 2007a; Kobayashi et al. 2007, 2010; Villanueva et al. 2006; Paganini et al. 2012).

Through the use of modern high-resolution spectrometers, the IR offers the unique capability of measuring trace molecules simultaneously with H₂O, the most abundant ice in comet nuclei and the principal driver of cometary activity for heliocentric distances (R_h) less than ∼3 AU. Symmetric hydrocarbons (e.g., C₂H₂, CH₄, C₂H₆, etc.) have no electric dipole moment and so are not observable at radio wavelengths since pure rotational transitions are forbidden, leaving their infrared vibrational bands as the sole means of measuring their abundances in comets through remote sensing.

In this study, we investigate whether the C₂ depletion observed in 21P/GZ carries over to primary (parent) volatiles, in particular to C₂H₆, which we quantify through application of a comprehensive fluorescence model for the ν₇ band near 3.35 μm (Villanueva et al. 2011a). For CH₃OH we present our newly developed empirical fluorescence model, incorporating more than 100 lines in the ν₂ band that are interspersed with C₂H₆ ν₇ emissions. Application of our empirical g-factors for ν₂ together with our recently published quantum model for the ν₃ band near 3.52 μm (Villanueva et al. 2012a) enables us to retrieve a highly precise production rate for methanol in 21P/GZ.

We compare our results for 21P/GZ with those reported from the IR (for C₂H₆ and CH₃OH) and radio (for CH₃OH) during its previous apparition in 1998. We compare our abundances in 21P/GZ with those measured in other JFCs, and also compare measured abundances of C₂H₆ and C₂ (from optical studies) among seven member comets in this dynamical population. We end with a discussion of prospects for measuring native ice abundances in 21P/GZ during its next favorable apparition in 2018.

We obtained spectra of 21P/GZ on UT 2005 June 03.63 using the near-infrared cross-dispersed echelle spectrograph (NIRSPEC; McLean et al. 1998) and 10 m Keck II telescope on Mauna Kea, HI. NIRSPEC delivers simultaneous coverage of six echelle orders (at L-band) at sufficiently high spectral resolving power (λ/Δλ ∼ 24,000, using the 043 wide slit) to measure emission intensities of individual lines or groups of blended lines for a given molecule.

One half-night (second half) was granted to obtain baseline measurements of Comet 9P/Tempel 1 approximately one month prior to the Deep Impact encounter. Comet 21P/GZ was accessible in the east toward the end of the night, permitting us to dedicate the final 40 minutes of available clock time to the study reported here. Its limited availability and modest brightness (m_v = 10–12) meant that only one instrument setting was possible—this was chosen so as to simultaneously encompass C₂H₆ (in order 23), CH₃OH (in orders 22 and 23), and H₂O (in orders 26 and 27; Table 1).

The telescope was nodded in our standard ABBA cadence, with A- and B-beams each containing comet signal and the telescope nodded by ±6 arcsec relative to the slit center for the two beam positions. For our observations, the slit was oriented at position angle 157° (Figure 1(a)), approximately perpendicular to the Sun–comet direction (P.A. = 260°). A total of six ABBA sets were obtained, corresponding to 24 minutes on source. Individual echelle orders were processed as described previously (Villanueva et al. 2011b; Bonev 2005; Mumma et al. 2001a).

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Signals from A and B beams were combined, and spectra were extracted over nine spatial pixels (178, or 1880 km) centered on the peak intensity as determined by the measured molecular emission profiles (Figures 1(b)–(d); also see Table 2, note ''b"). The emission line spectrum was isolated by subtracting the cometary dust continuum multiplied by the modeled atmospheric transmittance function (Figures 3 and 5). Flux calibration was accomplished by obtaining spectra of an IR standard star (HR-5487). Absolute line fluxes were referred to the top-of-the atmosphere by ratioing measured intensities to the monochromatic transmittance at each Doppler-shifted line-center frequency (based on geocentric velocity Δ-dot = −5.42 km s⁻¹). Transmittances were modeled using the Line-By-Line Radiative Transfer Model (Clough et al. 2005) and the latest HITRAN database with updated parameters (Rothman et al. 2010; see Villanueva et al. 2011a for detailed discussion and application). We used the NIRSPEC Slit-viewing CAMera to maintain 21P/GZ in the slit, thereby ensuring that loss of flux (slit spillover) was due primarily to seeing. We applied a measured slit-loss correction factor (denoted as GF in Equation (1) below) to convert our nucleus-centered production rates to global (or "total") production rates (Section 4.1).

Methanol is a molecule that is routinely observed in comets, both at IR and radio wavelengths. It has three fundamental vibrational bands in the 3.3–3.6 μm spectral region, ν₃ (parallel band; symmetric CH₃ stretching mode, centered near 2844 cm⁻¹), ν₂, and ν₉ (predominantly perpendicular bands; asymmetric CH₃ stretching modes centered near 2990 and 2970 cm⁻¹, respectively). (The terms "parallel" and "perpendicular" refer, in the symmetric top approximation, to the orientation of the transition moment associated with the vibrational mode relative to the top axis; Herzberg 1945, p. 414.) Of these three bands, ν₃ has traditionally been targeted for measuring CH₃OH in comets, primarily because it is relatively free of blends with emissions from other species and also because it is spectrally more compact compared with the asymmetric bands, having a single strong, well-delineated Q-branch. The ν₂ and ν₉ bands exhibit several sub-band Q-branches dispersed in frequency and interspersed with P- and R-branch lines (e.g., see Herzberg 1945, Figure 128, p. 425).

In addition to providing an alternate means of quantifying CH₃OH in comets, the ability to interpret ν₂-band emissions has a practical application. Prior to the commissioning of NIRSPEC in 1999, the only available high-resolution IR spectrometer was CSHELL at the NASA-IRTF 3 m telescope (Tokunaga et al. 1990; Greene et al. 1994). This heritage instrument revolutionized the field of gas-phase spectroscopy from 1 to 5 μm, albeit with relatively small spectral coverage per setting (within a single echelle order), for example encompassing only about 7 cm⁻¹ in the "cometary organics" region (∼3.2–3.6 μm). CSHELL is adequate for measuring the ν₃ Q-branch (having a total width of ∼1.5 cm⁻¹), however this required a dedicated, stand-alone setting for CH₃OH, and a second setting for measuring other parent molecules, for example C₂H₆ through its ν₇ band centered near 2985 cm⁻¹. A fluorescence model for the methanol ν₂ band permits prominent emission features of C₂H₆ and CH₃OH to be measured simultaneously; even very modern instruments such as CRIRES (Käufl et al. 2004) at the VLT cannot measure C₂H₆ ν₇ and CH₃OH ν₃ emissions in a single setting.

Although since superseded by NIRSPEC, and more recently by CRIRES/VLT, CSHELL/IRTF retains a unique niche by virtue of its daytime capability and its greater flexibility in scheduling observing time compared with Keck or VLT, further emphasizing the need for quantifying CH₃OH through ν₂ emissions. Finally, regardless of instrument, adequate interpretation of cometary spectra in the region from ∼2970 to 3000 cm⁻¹ requires disentangling emissions arising from different molecules, most notably C₂H₆ and CH₃OH, with lesser contributions from CH₄ and OH prompt emission. A major step involves characterizing CH₃OH in this region, which is dominated by ν₂-band emission. Toward this end, we developed a fluorescence model for this band (Section 4.3.4).

4.1. Methodology for Measuring Production Rates

Global (or total) production rates (Q_tot) for parent molecules were calculated using our well-documented methodology (Villanueva et al. 2011b; Bonev 2005; Mumma et al. 2003; DiSanti et al. 2001; Dello Russo et al. 1998). In applying our formalism, we measure a growth factor (GF), relating Q_tot for a primary volatile to that obtained from signal summed over nine spatial pixels centered on the peak emission intensity (we refer to this as the "nucleus-centered" production rate, Q_nc):

$$\begin{equation} Q_{{\rm tot}} \equiv Q_{{\rm nc}} GF. \end{equation} \tag{ 1 }$$

The GF (>1) corrects for loss of flux, due primarily to atmospheric seeing as explained in Section 2. In our analysis of 21P/GZ, the majority of spectral regions (intervals) we selected for determining our production rates consisted of blends of multiple lines. Within each chosen interval, Q_nc for that interval (denoted as Q_nc,int) is proportional to the measured line flux (F_int, W m⁻²) divided by the product of rest frequency (ν_i, cm⁻¹), fluorescence g-factor (g_1,i, photon s⁻¹ molecule⁻¹, evaluated at R_h = 1 AU), and atmospheric transmittance (T_i) at the Doppler-shifted frequency of each encompassed line, summed over all contributing lines (numbering ''n"):⁴

$$\begin{equation} Q_{{\rm nc,int}} = \frac{{4\pi \Delta ^2 }}{{hc\tau _1 f(x)}}\left[ {\frac{{F_{{\rm int}} }}{{\sum_{i = 1}^n {(v_i g_{1,i} T_i)} }}} \right]. \end{equation} \tag{ 2 }$$

In Equation (2), τ₁ (s) is the photo-dissociation lifetime at R_h = 1 AU, f(x) is the fraction of all molecules in the coma contained in the nucleus-centered beam (assuming native release and spherically symmetric outflow; see the Appendix in Hoban et al. 1991), Δ is expressed in meters, and hc = 1.99 × 10⁻²³ W s cm. The overall Q_nc represents the mean of values from all intervals, weighted by their respective stochastic errors (see Section 4.2). For CH₃OH in order 23, the uncertainty in our equivalent empirical g-factor for each interval is also included in this weighting (Table 4). The factor f(x) includes photo-decay, gas outflow speed (taken to be v_gas = 800 R_h^−0.5 m s⁻¹ ∼ 759 m s⁻¹), and geometrical parameters.

We calculate our global production rates using GF = 1.62 (Table 2), based on the spatial emission profile summed from several bright water lines (Figure 1(b)). Although noisier, the GF for CH₃OH was in agreement with this value, and that for C₂H₆, although somewhat larger, was also consistent (within 1σ uncertainty) with that measured for H₂O. Details are given in Table 2 (in particular, see note ''d").

4.2. Measurement Uncertainties

The errors associated with all reported parameters (production rates, rotational temperatures, and the ortho–para ratio for H₂O) account for sources of uncertainty including but not limited to stochastic (i.e., photon) noise, for example, uncertainties in fluorescence g-factors of modeled lines, telluric transmittance corrections, and the assumption of a single T_rot over the field of view. This is reflected in scatter about the mean of each measured parameter (e.g., production rate) that cannot be accounted for by photon noise alone. We quantify this scatter in terms of a "standard error" for each distribution. In general, and in our analysis of 21P/GZ, the standard error dominates the stochastic error, as exemplified in Figures 2(d) (for H₂O), 3(c) (for C₂H₆), 3(d) (for CH₃OH), and 5(c), 5(d) (for CH₃OH). In Table 4, we tabulate line fluxes and production rates from application of our ν₂ empirical model for CH₃OH (see Section 4.3.4.2), and in Tables 5–7 from our studies of H₂O, C₂H₆, and CH₃OH ν₃ emission, respectively. For detailed discussions regarding our treatment of errors, see Bonev (2005, Chapter 4), Dello Russo et al. (2006, Section 4.2), Bonev et al. (2007, p. L100), Villanueva et al. (2011b; Section 2.3), and Bonev et al. (2012, Section 3).

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4.3. Results for Individual molecules in 21P/GZ

4.3.1. H₂O

We measured 29 water lines contained within 24 spectral intervals in orders 26 and 27, of which 18 are ortho (parallel nuclear spins of the two H-atoms) and 11 are para (anti-parallel spins; see Figure 2 and Table 5). These sample a large range in rotational energy (>300 cm⁻¹; Figures 2(b)–(d)), allowing the rotational temperature to be determined with high precision (T_rot = 51 ± 3 K). We incorporated all but one ortho line (see below) and used our improved fluorescence model for H₂O (Villanueva et al. 2012b). Analysis of relative ortho- and para-line intensities resulted in a measure of the ortho-to-para ratio (OPR = 2.99 ± 0.23) consistent with equilibrated spins, thereby implying a spin temperature (T_spin) exceeding 50 K (e.g., see Bonev et al. 2007). Our method for decoupling T_rot and T_spin provides a highly precise (and accurate) measure of the global production rate for H₂O in 21P/GZ (Q(H₂O) = 3.885 ± 0.074 × 10²⁸ molecules s⁻¹); see Bonev et al. (2012) for detailed discussion and application of our methodology.

In our rotational analysis, we omitted the strong ortho line at rest frequency 3394.08 cm⁻¹ (200,2₂₁ – 100,3₃₀; labeled ''15" in Figure 2; also see Table 5). This line appears to be affected by one or more bad pixels; its predicted (modeled) intensity is considerably larger than that measured in 21P/GZ, causing its calculated production rate to fall well below the mean Q(H₂O) as determined from the full suite of water lines in orders 26 and 27 (by ∼4.9 times its stochastic error), as shown in Figures 2(b) and (d). Rotational temperature and OPR were highly correlated in our data, and including this line made it difficult to obtain unambiguous values for either parameter, and also increased the standard error associated with the distribution by a factor of nearly two. Fortunately, however, this line has a very minimal effect on our retrieved H₂O production rate—its inclusion decreased Q(H₂O) (and hence increased our retrieved abundance ratios for C₂H₆ and CH₃OH) by only ∼2%.

4.3.2. OH Prompt Emission in Order 23

Our observations encompassed four strong lines of OH in order 23. These lines arise through "prompt emission" (denoted OH*) in the v = 1–0 band, radiating within milliseconds from vibrationally and rotationally excited states following their production from photo-dissociation of H₂O molecules in the coma (Bonev et al. 2006 and references therein). The OH* P12.5 2⁻ line is blended with C₂H₆ ^rQ₃, and also with the strong A^{+→ −} (K = 1→ 0) Q-branch of the CH₃OH ν₂ band (plus a few weaker ν₂ lines, Table 4). We estimated an effective g-factor for the P12.5 2⁻ line through comparison with OH* lines from adjacent multiplets. We also estimated its g-factor empirically from NIRSPEC observations of Comet C/1999 S4 (LINEAR) in 2000 July. This comet was very highly depleted in virtually all primary volatiles, including C₂H₆ and CH₃OH (Mumma et al. 2001b), leaving the OH* lines as the only discernable emissions in excess of the continuum in order 23. These two independent approaches led to g-factors for the P12.5 2⁻ line in the ratio 1.19 (the one based on LINEAR S4 being larger), exemplifying the uncertainties associated with obtaining accurate g-factors for these lines when comparing comets observed at different times, in particular during different phases of the solar cycle.

We verified that including this spectral interval (number ''5" in Figures 3 and 4) does not alter our conclusions regarding the abundances of C₂H₆ and CH₃OH in 21P/GZ. Because we encompass many emissions from both ethane and methanol, the spectral confusion from OH* (which affects only one feature) has a minor influence on their production rates. Including this blend lowers Q(C₂H₆) by only ∼2% (Section 4.3.3) and Q(CH₃OH) by ∼10% (approximately 1σ uncertainty in the production rate; see Section 4.3.4.2, and Table 2, note ''f"). Because of this spectral confusion, together with uncertainty in the equivalent g-factor for the P12.5 2⁻ line, we exclude interval ''5" from order 23 in determining our most reliable values for Q(C₂H₆) and Q(CH₃OH).

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4.3.3. C₂H₆

Order 23 is dominated by emissions from ethane (ν₇ band) and methanol (primarily the ν₂ band). One motivation for our study of 21P/GZ is to investigate (and, in the case of C₂H₆, to confirm) whether the depletion of C₂ observed in this comet (A'Hearn et al. 1995; Fink 2009) carries over to ethane. Depleted C₂H₆ was reported from CSHELL observations of 21P/GZ during the 1998 apparition (Weaver et al. 1999; Mumma et al. 2000). This might be expected if acetylene (C₂H₂) is the dominant source of C₂, as the most efficient means of producing C₂H₆ is H-atom addition to C₂H₂ on the surfaces of interstellar icy grains at low temperatures (Mumma & Charnley 2011; also see Section 5.1.3).

We obtained production rates from the intensities of seven Q-branches of the C₂H₆ ν₇ band (^rQ₃, ^rQ₂, ^rQ₁, ^rQ₀, ^pQ₁, ^pQ₂, ^pQ₃) and identify these by numbers 1–7, respectively, in Figure 3(a) (also see Table 6). Our production rates were based on a full quantum model for the ν₇ band of C₂H₆ that includes P-, Q-, and R-branch intensities as well as contributions from the ν₇+ν₄−ν₄ combination band (Villanueva et al. 2011a). Because each Q-branch encompasses many lines spanning a range of rotational energies, and because there is considerable overlap in the energies sampled among these Q-branches, the resulting production rate is relatively insensitive to T_rot. For this reason, we adopt T_rot = 50 K for C₂H₆, consistent with (to well within 1σ) those we measured for H₂O and for CH₃OH (based on analysis of the ν₃ band; see Section 4.3.4.3 and Figure 5).

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Global production rates for the seven Q-branches were calculated using their integrated line fluxes and applying the growth factor measured for H₂O (GF = 1.62); these are indicated in Figure 3(c). However, because the g-factor for the blended P12.5 2⁻ OH* line is uncertain (Section 4.3.2), ^rQ₃ (labeled "1" in Figures 3(a) and (c)) is excluded from our quantitative analysis of C₂H₆. The global production rate is then the mean of the remaining six points, weighted by their individual stochastic uncertainties (denoted by the error bars in Figure 3(c)). We obtained a global production rate [Q(C₂H₆) = 5.27 ± 0.90 × 10²⁵ s⁻¹] that translates to an abundance ratio of (0.136 ± 0.023) × 10⁻² relative to H₂O (Tables 2 and 3). This is well below the current "normal" abundance among comets (∼0.6 × 10⁻²; Bockelée-Morvan et al. 2004; DiSanti & Mumma 2008; Mumma & Charnley 2011), confirming highly depleted C₂H₆ in 21P/GZ, in analogy with its observed C₂ depletion (A'Hearn et al. 1995; Fink 2009; see also Table 3).

Table 3. Volatile Abundances in 21P/G-Z and Other Jupiter Family Comets

Comet	Yeara	C2H6a	CH3OHa	Log C2b	C2/OHc	C2H6c
		H2O	H2O	OH	(%)	C2
21P/GZ	1998	0.22 ± 0.13d
	1998	<0.05–0.08e	0.9–1.4e	−3.41(1)	.056
	1998		1.6 ± 0.4f	−3.14(2)
	2005	0.136 ± 0.023g	1.22 ± 0.11g			2.7
9P/Tempel 1	2005	0.23 ± 0.04h	1.4 ± 0.2h	−2.90(1)	.097	2.6
		0.55 ± 0.09i	1.1 ± 0.2h	−3.17(2)
73P/SW3-B	2006	<0.3j		−3.29(2)	.051	2.7
73P/SW3-B		0.14 ± 0.01k	0.25 ± 0.04k
73P/SW3-C		0.15 ± 0.04j	<0.38j
73P/SW3-C		0.11 ± 0.01k	0.21 ± 0.04k
17P/Holmes	2007	1.78 ± 0.26l	3.2 ± 0.6l	−2.42(3)	0.38	5.2
6P/d'Arrest	2008	0.26 ± 0.06m	1.99 ± 0.42m	−2.37(1)	0.28	1.0
				−2.90(2)
103P/Hartley 2	2010	0.75 ± 0.03n	1.87 ± 0.13n	−2.36(1)	0.35	2.3
		0.65 ± 0.09o	1.72 ± 0.11o	−2.59(2)
10P/Tempel 2	1999		1.8 ± 0.2p
	2010	0.39 ± 0.04q	1.58 ± 0.23q	−2.70(1)	0.20	2.1
				−2.71(2)

Notes. ^aYear observed (excluding optical observations; see note ''b") and abundances of ethane and methanol with respect to H₂O = 100 (all uncertainties represent 1σ, and upper limits are 3σ). Previously published values for Q(CH₃OH) from IR observations using the ν₃ Q-branch intensity are inter-normalized to g(Q-br) = 1.5 × 10⁻⁵ photons s⁻¹ molecule⁻¹ (Villanueva et al. 2012a). ^bAbundances of C₂ are taken from the following papers: (1) A'Hearn et al. (1995), (2) Fink (2009), and (3) Schleicher (2009). With the exception of 17P, the optical observations refer to apparitions different from those listed under "year"; e.g., years of observation for Fink et al.: 21P, 1985; 9P, 1994; 73P, 1990 and 1995; 6P, 1995; 103P, 1997 and 1998; 10P, 1988. A'Hearn et al. (1995) classify comets having −1.22 < log(C₂/CN) < −0.21, and −4.13 < log(C₂/OH) < −2.98 as "carbon-chain depleted", and those having −0.09 < log(C₂/CN) < 0.29 and −2.90 < log(C₂/OH) < −2.10 as "typical". The values for 17P refer to the mean of 2007 November 01.16, November 20.12, and 2008 January 01.23, during the decline from its spectacular outburst in late 2007 October. Its rapidly evolving coma complicates comparison of abundances for primary (parent) and product species, and in this sense 17P represents a special case. ^cFor comets with multiple entries for log(C₂/OH), the quantity C₂/OH refers to their mean value. In calculating C₂H₆/C₂, the relation Q(H₂O) = 1.1 × Q(OH) is assumed. The entry for 21P is based on the value for C₂H₆/H₂O reported here (0.136 × 10⁻²). The entry for 9P is for the pre-impact abundance ratio C₂H₆/H₂O (note ''h"). The entry for 73P uses the weighted mean value from the three measurements of C₂H₆/H₂O having 1σ errors (∼0.14 × 10⁻²). The entry for 103P uses the weighted mean value from the two measurements listed for C₂H₆/H₂O. ^dMumma et al. (2000). ^eWeaver et al. (1999). Their value for Q(CH₃OH) was reported as a 3σ upper limit. It was based on the entire ν₃ band using g(ν₃) = 1.5 × 10⁻⁴ s⁻¹, therefore no re-scaling of production rate from the published value is required. ^fBiver et al. (2002). ^gThis work. The value listed for C₂H₆ is based on six ν₇ Q-branches; i.e., excluding ^rQ₃—interval/point ''1" in Figures 3(a) and (c) (see related discussion in Section 4.3.3). Including ^rQ₃ modifies its abundance very little, to (0.133 ± 0.021) × 10⁻². The value listed for CH₃OH is based on signal in 36 spectral intervals in order 22 (Figure 5) and 17 spectral intervals in order 23 (i.e., with interval/point ''5" in Figures 3(b) and 3(d) omitted). Including interval ''5" from order 23 results in a mean CH₃OH abundance of (1.12 ± 0.11) × 10⁻². ^hMumma et al. (2005). The first row corresponds to pre-impact, the second row to the DI ejecta. ⁱDiSanti et al. (2007b). ^jVillanueva et al. (2006). ^kDello Russo et al. (2007) and Kobayashi et al. (2007) reported consistent values for fragment B, based on independent measurements. ^lDello Russo et al. (2008). ^mDello Russo et al. (2009). ⁿMumma et al. (2011), as revised by Villanueva et al. (2012a). ^oDello Russo et al. (2011), as revised by Villanueva et al. (2012a). ^pBiver et al. (2012). ^qPaganini et al. (2012).

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4.3.4. CH₃OH

4.3.4.1. CH₃OH ν₂ Emission in Order 23: A New Empirical Fluorescence Model

We developed an empirical model that includes 157 individual lines of the ν₂ band, with assignments and frequencies adopted from jet-cooled laboratory spectra (Xu et al. 1997) and lower state energies taken from Mekhtiev et al. (1999, Section 3; also see note ''b" of Table 4). We incorporated approximately 70% of these (109 lines) to quantify CH₃OH in order 23 (see Figures 3 and 4, and related discussion below).

Table 4. CH₃OH ν₂ empirical model results applied to 21P/Giacobini-Zinner on UT 2005 June 03

Int	νmax − νmina		Rot IDb	Rest νb		Elowb	Flinec	Td	g-face	Qf
	(cm−1)			tabulated	wtd mean	(cm−1)	(10−20 W m−2)		(10−7 s−1)	(1025 s−1)
1	3002.18 − 3001.70		A+ (0→0) P3	3002.154		9.68
			A+ (1→0) P4	3002.121		16.13
			E(1→2) R6	3002.054		49.35
			E(1→0) R2	3001.975		13.96
			A+→ − (3→3) Qg	3001.940		∼50.00
			E(−1→ −2) R5	3001.905		42.21
			E(−3→ −3) R6	3001.799		11.73
			A− (1→1) R1	3001.775		14.90
			A+ (0→1) R2	3001.777		77.47
			A+ (1→1) R1	3001.735		11.70
					3001.954	25.65	7.77 ± 2.65	0.935	15.72 ± 1.91	111.0 ± 37.9(40.4)
2	3001.21 − 3000.97		E(−1→0)Qg	3001.080		38.50
			E(0→0)R3	3001.042		18.80
					3001.079	37.65	7.40 ± 1.97	0.937	32.69 ± 1.51	50.73 ± 13.49(13.82)
3	3000.01 − 2999.77		A+ (2→1)P4	3000.002		26.10
			A+/−(2→2)R5	2999.912		50.41
			E(−1→ −1)P4'	2999.894		50.41
			E(−1→ −1)P4"	2999.836		78.34
			E(0→ −1)Qg	2999.83		20.32
					2999.854	24.31	5.89 ± 1.91	0.942	23.36 ± 1.36	56.19 ± 18.20(18.61)
4	2997.31 − 2997.07		E(1→2)R3	2997.250		25.14
			A+ (0→1)P7	2997.161		45.16
			E(1→0)Qg	2997.14		32.51
			E(0→1)R4	2997.130		30.76
			E(−2→−2)P3	2997.116		27.68
					2997.139	33.02	13.18 ± 1.86	0.973	37.21 ± 1.30	76.59 ± 10.82(11.49)
5h	2996.95 − 2996.42		A+→ −(1→0)Q2	2996.870		43.69
			A+→ −(1→0)Q1	2996.847		11.73
			A+→ −(1→0)Q3	2996.811		14.99
			A+→ −(1→0)Q4	2996.763		19.87
			E(−1→ −1)P6"	2996.715		26.38
			A+→ −(1→0)Q5	2996.702		37.75
			A+/− → +/−(2→2)R3	2996.674		34.52
			E(0 → −1)P2'	2996.639		35.89
			A+→ −(1→0)Q6	2996.626		35.89
			E(−1→ −1)P6'	2996.626		44.28
			E(0 → −1)P2"	2996.611		8.72
			E(2→2)P5	2996.542		8.72
			A+→ −(1→0)Q7	2996.537		37.75
			E(1→1)R2	2996.470		55.67
			A+→ −(0→1)Q8	2996.431		39.67
					2996.675	32.01	7.59 ± 2.61	0.938	85.88 ± 1.98	19.82 ± 7.08(7.13)
6	2992.44 − 2992.08		A− (1→2)R5	2992.389		50.41
			E(0→1)R1	2992.324		16.24
			E(1→0)P3	2992.294		18.80
			E(−1→ −2)Qg	2992.23		55.46
			A+ (1→1)P4	2992.153		26.10
			A+ (0→1)P3	2992.131		19.70
					2992.224	45.85	5.35 ± 3.18	0.776	45.83 ± 2.67	31.68 ± 18.81(18.94)
7	2990.89 − 2990.53		E(1→2)Qg	2990.76		48.69
			A− (1→2)R4	2990.693		42.34
			E (1→0)P4	2990.684		25.25
			A+ (1→1)P5	2990.580		26.10
			A+ (0→1)P4	2990.575		34.10
					2990.715	42.30	4.86 ± 2.34	0.951	36.60 ± 1.67	29.43 ± 14.19(14.29)
8	2986.14 − 2985.65		A+ (0→1)P7	2986.017		54.89
			E (−2→ −3)R5	2985.956		25.14
			E(1→2)P3	2985.956		67.79
			E(0→1)P2'	2985.887		19.47
			E(1→0)P7	2985.859		19.47
			E(0→1)P2"	2985.859		54.27
			E (−1→ −2)P4	2985.708		34.14
					2985.907	37.69	1.91 ± 2.66	0.956	23.36 ± 1.90	18.10 ± 25.19(25.24)
9	2984.62 − 2984.33		A+ (0→1)P8	2984.524		67.68
			E(−2→ −3)R4	2984.382		42.35
			E(1→2)P4	2984.342		31.60
					2984.403	52.40	1.27 ± 2.57	0.819	13.61 ± 1.76	24.15 ± 48.88(49.33)
10	2983.14 − 2982.90		A−− > +(1→2)Q10	2983.131		96.13
			A−− > +(1→2)Q9	2983.010		98.82
			A+ (0→1)P9	2983.044		82.07
			A−− > +(1→2)Q8	2982.898		84.30
					2983.0197	92.44	7.23 ± 1.97	0.958	17.91 ± 1.43	89.16 ± 24.32(25.54)
11	2982.86 − 2982.48		A+→ − (1→2)Q7	2982.797		53.27
			E(−2→ −3)R3	2982.797		71.39
			E(1→2)P5	2982.726		39.67
			A+→ − (1→2)Q6	2982.705		60.10
			E(0→1)P4	2982.635		81.64
			E(1→0)P9	2982.635		30.76
			A+→ − (1→2)Q5	2982.624		50.41
			A+→ − (1→2)Q4	2982.555		51.89
			E(−1→ −2)P6'	2982.573		42.35
			E(−1→ −2)P6"	2982.487		35.89
			A+→ − (1→2)Q3	2982.501		51.89
					2982.647	49.90	3.72 ± 2.68	0.838	50.57 ± 1.89	18.55 ± 13.39(13.43)
12	2981.99 − 2981.74		E(1→1)P6	2981.941		48.51
			A+→ − (2→3)Qg	2981.86		81.24
			A−→ + (2→3)Qg	2981.83		76.24
					2981.848	78.03	6.55 ± 1.97	0.958	39.17 ± 1.40	36.89 ± 11.14(11.30)
13	2981.21 − 2980.76		E(1→2)P6	2981.106		49.35
			E(0→1)P5	2981.012		38.83
			E(−1→ −2)Q*q	2981.04		30.00
			E(−1→ −2)P7	2980.924		63.19
			A−→ + (3→4)Qg	2980.825		115.1
			A+→ − (3→4)Qg	2980.793		103.8
					2980.918	69.87	8.71 ± 3.84	0.733	44.96 ± 4.91	71.10 ± 31.33(32.38)
14	2979.56 − 2979.15		E(0→ −1)P12	2979.549		129.6
			E(1→2)P7	2979.479		60.65
			E(0→1)P6	2979.380		48.51
			E(−1→ −2)P8	2979.332		76.10
			A+ (1→2)P2	2979.181		31.05
			A− (1→2)P2	2979.204		31.05
					2979.352	54.16	8.80 ± 3.61	0.538	24.33 ± 3.06	142.4 ± 58.34(61.25)
15	2977.16 − 2976.92		E(1→1)P9	2977.044		87.24
			A− (2→3)P3	2976.965		44.29
			A+ (2→3)P3	2976.966		44.29
					2976.973	46.42	5.94 ± 2.09	0.932	11.36 ± 1.58	118.7 ± 41.78(45.13)
16	2976.25 − 2975.30		E(−1→ −2)P10	2976.192		88.09
			E(0→1)P8	2976.043		72.72
			A− (1→2)P4	2976.047		42.34
			A+ (1→2)P4	2975.909		42.35
			A+ (2→2)P9	2975.750		98.82
			A− (2→2)P9	2975.735		98.80
			E(2→3)P7	2975.656		78.34
			A− (2→3)P4	2975.356		50.75
			A+ (2→3)P4	2975.357		50.75
					2975.732	57.00	16.69 ± 3.91	0.928	36.29 ± 2.81	105.0 ± 24.61(26.20)
17	2974.35 − 2974.14		E (0→1)P9	2974.282		87.24
			A+ (1→2)P4	2974.260		50.41
			A+ (3→4)P4	2974.240		71.98
			A− (3→4)P4	2974.239		71.98
					2974.251	67.48	−0.720 ± 2.03	0.912	16.63 ± 1.95	−10.01 ± 28.30(28.33)
18	2973.89 − 2973.64		A− (2→3)P5	2973.748		58.81
			A+ (2→3)P5	2973.751		58.81
					2973.750	58.81	4.69 ± 2.37	0.863	8.70 ± 1.70	132.6 ± 67.07(72.06)

Notes. ^aRange of rest frequencies encompassed by each spectral interval in Figure 3(b). ^bRotational designation and tabulated rest frequencies are taken from Xu et al. 1997. The asterisk in interval 13 denotes an E-type Q-branch of the ν₉ band; its frequency and energy are only approximate. The heading "wtd mean" for each spectral interval denotes the mean rest frequency weighted by contributing line-by-line empirical g-factors (multiplied by atmospheric transmittance at each Doppler shifted line center frequency, using Δ-dot = −5.42 km s⁻¹), based on the "CH₃OH residuals" in 8P/Tuttle (bottom trace in Figure 4). Equations (A1) and (A2) were used as a guide for modeling intensities of lines having similar rotational designation and symmetry, as discussed in Appendix A. Lower state energies (E_low) were taken from Mekhtiev et al. (1999), in particular see the anonymous ftp link listed in Section III of that paper. ^cMeasured nucleus-centered line flux and 1σ stochastic error for each interval, based on signal contained within an aperture of size 460 x 1880 km at the distance of 21P/GZ. ^dWeighted mean atmospheric transmittance for all encompassed lines within each interval. ^eSummed fluorescence-efficiency g-factor (at R_h = 1 AU) for each interval. Uncertainties in g represent 1σ and are based on the stochastic noise summed within each spectral interval in the 8P/Tuttle order 23 spectrum. ^fGlobal (total) production rate based on the signal contained in each interval. The first uncertainty incorporates the 1σ error in line flux (plus σGF, which adds only a minor contribution). The second uncertainty (in parentheses) includes these errors plus the 1σ uncertainty in empirical g-factor for each interval. ^gBecause the lines comprising these Q-branches are closely spaced in frequency, entries for individual J's are not presented. Instead, the equivalent integrated g-factor and approximate rotational energy for each Q-branch are listed. The approximate spectral dispersion per pixel = 0.041 cm⁻¹ in order 23; correspondingly one resolution element is ∼0.2 cm⁻¹ (5 pixels). The spectral widths of these Q-branches range from approximately 0.04 cm⁻¹ for E(1→2)Q(J) to 0.2 cm⁻¹ for E(0→−1)Q(J) or E(−1→0)Q(J), including transitions having J ⩽ 8. ^hBold-faced entries for interval "5" are shown in italics since we excluded this interval in determining our most reliable Q for CH₃OH in order 23, as discussed in the text.

Download table as:  ASCIITypeset images: 1 2 3 4

Fluorescence efficiencies (g-factors) for our empirical model are based on higher signal-to-noise spectra of Halley Family Comet 8P/Tuttle, which we observed with NIRSPEC on UT 2007 December 22, approximately 2.5 years following our observations of 21P/GZ reported here. Comet Tuttle exhibited a high abundance ratio CH₃OH/C₂H₆ and a rotational temperature near 50 K (Bonev et al. 2008; also see Böhnhardt et al. 2008; Kobayashi et al. 2010), making it an excellent "template" for our study. Figure 4 shows the resulting model compared with the 8P/Tuttle residuals from order 23. After accounting for emissions from other species (specifically C₂H₆, CH₄, and OH*), we judiciously identified 18 separate spectral intervals in which identified CH₃OH ν₂ emissions are isolated. Values for our empirical g-factors were assigned such that the production rate for each interval, based on a spectral extract centered on the nucleus (Q_nc,int; see Equation (2)), agreed with that obtained independently from application of our quantum model (to 8P/Tuttle) for the simultaneously observed ν₃ band in order 22 (Villanueva et al. 2012a).⁵

4.3.4.2. CH₃OH Abundance in 21P/GZ Based on ν₂ Emissions in Order 23

Subtracting modeled C₂H₆ (and OH*) emissions from the (continuum-subtracted) spectrum in Figure 3(a) results in a net emission spectrum dominated by methanol (labeled "CH₃OH residuals" in Figure 3(b)). Application of our empirical model then provides a production rate for each spectral interval (see Table 4), and thereby an additional measure of CH₃OH production. In Figure 3(d) we show the resulting global production rates for CH₃OH in 21P/GZ based on signal contained in each of the 18 spectral intervals in order 23. The mean from all intervals (excluding number ''5") results in Q(CH₃OH) = (4.88 ± 0.71) × 10²⁶ s⁻¹ or an abundance ratio CH₃OH/H₂O = (1.26 ± 0.18) ×10⁻², while including interval ''5" results in CH₃OH/H₂O = (1.03 ± 0.17) × 10⁻², based on CH₃OH in order 23 alone.

4.3.4.3. Measurement of CH₃OH ν₃ Emission in 21P/GZ: T_rot and Abundance

We identified 36 spectral intervals in order 22 and, applying our new ν₃ quantum band model (Villanueva et al. 2012a), we measured a rotational temperature for CH₃OH (T_rot = 48⁺¹⁰/₋₇ K; Figure 5) consistent with that measured for H₂O in 21P/GZ. Our production rate for CH₃OH from order 22 alone (4.63 ± 0.55 × 10²⁶ s⁻¹) translates to an abundance ratio CH₃OH/H₂O = (1.19 ± 0.14) × 10⁻² (Table 2).

We note that although the residuals in Figure 5(a) are noisy, including this many spectral intervals actually improves our detection of CH₃OH in order 22 over that obtained by considering the Q-branch alone. The Q-branch is clearly detected, however its stochastic uncertainty (8.58 × 10²⁵ s⁻¹; see Table 7) is considerably larger than either stochastic (4.57 × 10²⁵ s⁻¹) or standard (5.53 × 10²⁵ s⁻¹) uncertainties associated with the ensemble of 36 spectral intervals.

Combining CH₃OH from orders 22 and 23 (but excluding interval ''5" in order 23) results in Q_tot(CH₃OH) = (47.5 ± 4.4) × 10²⁵ s⁻¹, or (1.22 ± 0.11) ×10⁻² relative to H₂O. We take this to represent our most reliable measure of CH₃OH in 21P/GZ. Global production rates for each of the 54 spectral intervals included in these two orders are shown graphically in Figure 5(d).

5.1. Comparisons with Other Observations of 21P/GZ

5.2. Abundance Comparisons of C₂H₆ and CH₃OH Among Jupiter Family Comets

5.3. Comparisons with C₂ Abundances from Optical Observations of JFCs

Because of its non-integral orbital period (6.6 years), apparitions of 21P/GZ provide widely varying observational opportunities as viewed from Earth. This is particularly true when comparing those in 2012 and 2018. Near perihelion, the geocentric distance Δ was >1.8 AU in 2012, but Δ will be only ∼0.4 AU in 2018. Moreover, the geocentric velocity in 2018 exceeds 10 km s⁻¹ (and thus is adequate for studying CO and CH₄) when Δ reaches ∼0.48 AU. Assuming gas production rates (and rotational temperatures) similar to those measured during previous apparitions, line intensities in 2018 are expected to be larger by factors of three to four compared with those observed from 2005, and by factors of two to three compared with those from 1998 (for which Δ ∼ 0.9–1.1 AU). Combined with anticipated improvements in instrumentation (e.g., higher spectral resolving power and sensitivity), the 2018 apparition will enable deep searches for trace species in 21P/GZ—for example, detection of C₂H₂ or, if severely depleted, establishing stringent upper limits for its abundance—and will allow the question of potential compositional heterogeneity in the nucleus to be addressed in greater detail.

We used NIRSPEC at Keck II to obtain infrared spectra of 21P/GZ on UT 2005 June 03.6 that allowed simultaneous measurement of H₂O, C₂H₆, and CH₃OH. For H₂O, we obtained a production rate (3.885 ± 0.074 × 10²⁸ molecules s⁻¹) consistent with that reported previously for 21P/GZ from optical, IR, and millimeter-wavelength observations. Our water analysis also provided values for rotational temperature (T_rot = 51 ± 3 K) and spin temperature (T_spin > 50 K, based on OPR = 2.99 ± 0.23).

For C₂H₆ we applied our quantum-mechanical fluorescence model for the ν₇ band that includes contributions from the Q-, P-, and R-branches and also from a combination band. We obtained an abundance ratio C₂H₆/H₂O = 0.136 ± 0.023 × 10⁻², indicating nearly five-fold depletion of ethane in 21P/GZ relative to the current median value in comets (∼0.6 × 10⁻²).

For CH₃OH, we applied our recently published fluorescence model for the ν₃ band to obtain a rotational temperature (T_rot = 48⁺¹⁰/₋₇ K) consistent with the value we measured for H₂O. The first application of our newly developed empirical model for the ν₂ band provided a CH₃OH production rate consistent with that obtained from ν₃. Combining results for both bands (ν₂ and ν₃) resulted in an ∼11σ detection of CH₃OH, and an abundance ratio CH₃OH/H₂O = 1.22 ± 0.11 × 10⁻² in 21P/GZ, somewhat below its current median value (2 × 10⁻²).

Our results therefore confirm strongly depleted C₂H₆ and mildly depleted CH₃OH in 21P/GZ, as reported previously from observations during the 1998 apparition. Together with those results, our study provides a measure of primary volatile abundances from ground-based IR spectroscopy for this JFC over multiple apparitions. It is also a comparator for more extensive studies of 21P/GZ during future apparitions, beginning with that in 2018.

M.A.D. and G.L.V. acknowledge support from NASA's Planetary Atmospheres Program (RTOP 09-PATM-0080; PI: M.A.D., and 08-PATM-0031; PI: G.L.V.). B.P.B. acknowledges support from the NSF Astronomy and Astrophysics Grants Program (AST-1211362; AST−0807939) and NASA's Planetary Atmospheres Program (PATM-NNX12AG60G). We also acknowledge support from NASA's Planetary Astronomy Program for MAD (RTOP 09-PAST09–0034) and for M.J.M. and G.L.V. (RTOPs 344–32-07, 08-PAST08–0033/34), and from NASA's Astrobiology Program through the NASA Astrobiology Institute (RTOP 344–53-51; PI: M.J.M.). We thank an anonymous reviewer for providing comments that improved the manuscript. The data presented herein were obtained at the W. M. Keck Observatory, operated as a scientific partnership among CalTech, UCLA, and NASA. This Observatory was made possible by the generous financial support of the W. M. Keck Foundation. The authors recognize and acknowledge the very significant cultural role and reverence that the summit of Mauna Kea has always had within the indigenous Hawaiian community. We are most fortunate to have the opportunity to conduct observations from this mountain.

For our empirical model of the CH₃OH ν₂ band, we adopt notation according to the G₆ molecular symmetry group, as this takes into account the large amplitude torsional tunneling between the three equivalent CH₃ orientations relative to OH (Xu et al. 1997, 2008). Under the G₆ group representation, the rotational structure is described using three quantum numbers (J, K, l), where J (⩾ 0) is the usual total angular momentum, K is its projection along the symmetry a-axis of the molecule (such that 0 ⩽ K ⩽ J), and l is related to the parity of the states involved in the transition. The G₆ symmetry group uses the representations A₁, A₂, E₁ and E₂, where subscript "1" is associated with positive parity of a ro-vibrational state and "2" with negative parity, thereby leading to the representations A⁺, A⁻, E⁺, and E⁻. Following Mekhtiev et al. (1999), we differentiate between E⁺ to E⁻ states by associating the latter with negative K values. Using the standard convention, a single-prime denotes the upper vibrational state (v = ν₂) and a double-prime denotes the lower (ground) vibrational state (v = 0). Then, for example (see Table 4), the designation A⁺ (1→0)P4 corresponds to J' = 3, K' = 1, l' = 1, and J'' = 4, K'' = 0, l'' = 1, while E (−1→−2)R5 corresponds to J' = 6, K' = −1, l' = 2, and J'' = 5, K'' = −2, l'' = 2.

To establish spectral intervals for determining the CH₃OH production rate using our empirical model, we first identified regions in the order 23 spectrum of Comet 8P/Tuttle having residual intensity after accounting for emissions due to C₂H₆, OH, and CH₄ (see Figure 4, in particular the bottom trace labeled "CH₃OH residuals"). We next incorporated line assignments and frequencies from the compilation in Xu et al. (1997) based on supersonic (jet) spectra of CH₃OH measured at low temperature (∼17 K). We assigned fluorescence g-factors for lines contained in each spectral interval such that their sum reproduced the nucleus-centered production rate (Q_nc,int in Equation (2)) retrieved for 8P/Tuttle (=29.0 × 10²⁵ molecules s⁻¹), based on application of our recently published ν₃ quantum band model to its molecular emissions in order 22 (the "revised" Q_tot for 8P/Tuttle, which includes GF = 1.5 as discussed in Bonev et al. 2008, is presented in Table 4 of Villanueva et al. 2012a; also see Section 4.3.4.1 in this 21P/GZ paper).

Rigorous calculation of line g-factors requires a full quantum-mechanical fluorescence model for the ν₂ band, in which solar pumps from all (dipole-allowed) contributing levels in the ground vibrational state (v = 0) into each upper level (in v = ν₂) are summed (with potential cascades from higher ro-vibrational levels included). Each particular line g-factor is then given by this summed pump rate multiplied by the rotational branching ratio for the line in question. However, such a treatment is beyond the scope of the present paper.

Instead, in our empirical treatment, we preformed checks on lines having similar rotational designation (J, K) and symmetry (A, E). This approach provided more consistent g-factors, and also provided insights as to which spectral regions not to include in our measurements. To accomplish this, we examined the intensities of spectral lines, which for a given transition can be expressed as a proportionality with respect to line parameters and rotational temperature:

$$\begin{eqnarray} S_{{\rm line}} &\propto & \;\frac{\nu }{{\nu _0 }}L_{{\rm HL}} [1 - \exp (- hc\nu /kT_{{\rm rot}})]\,[(2J + 1)\nonumber\\ &&\times \exp (- hcE/kT_{{\rm rot}})]. \end{eqnarray} \tag{ A1 }$$

Here, and in the formulae below, ν is the line frequency (cm⁻¹), ν₀ is the band frequency (2999 cm⁻¹), J, K, and E (cm⁻¹) are respectively lower state rotational quantum numbers and energy (denoted by "E_low" in Table 4, and taken from Mekhtiev et al. 1999), T_rot is rotational temperature, and L_HL is the Hönl–London factor. For a perpendicular band such as ν₂, the intensity of which is dominated by lines having ''A" symmetry in b-type sub-bands (for which ΔK = ±1; e.g., see Xu et al. 1997), L_HL is given by

$$\begin{eqnarray} L_{{\rm HL}} &=& \;\frac{{(J\, - \,1\; \mp \;K)(J\; \mp \;K)}}{{J(2J\; + \;1)}}\nonumber\\ &&\qquad\qquad\quad { {\rm for}\ \Delta J = } - { 1 (P - {\rm branch\; lines}),}\nonumber\\ L_{{\rm HL}} &=& \;\frac{{(J\, + \,1\; \pm \;K)(J\; \mp \;K)}}{{J(J\; + \;1)}}\nonumber\\ &&\qquad\qquad\quad\;\;\;\, {{\rm for}\ \Delta J = 0 (Q - {\rm branch\; lines}),}\nonumber\\ L_{{\rm HL}} &=& \;\frac{{(J + \,2\; \pm \;K)(J\; + 1 \pm \;K)}}{{(J\; + \;1)(2J\; + \;1)}}\nonumber\\ &&\qquad\qquad\quad\;\; {{\rm for}\ \Delta J = + 1 (R - {\rm branch\; lines}),}\qquad \end{eqnarray} \tag{ A2 }$$

where the upper sign in each equation refers to ΔK (=K_upper → K_lower) = +1, and the lower sign refers to ΔK = −1 (Herzberg 1945, p. 426).

An example is provided by P-branch lines in the A(ΔK = 1 → 2) sub-band. In the Comet 8P/Tuttle "CH₃OH residual" spectrum (Figure 4), the intensity contained in the left-most (highest frequency) peak in interval 16 (near 2976.0 cm⁻¹) is dominated by A^−/+→^−/+ P4 components. The corresponding P3 components at ∼2977.6 cm⁻¹ coincide approximately with a feature having strong residual intensity (additionally, a second feature near 2977.4 cm⁻¹ has lower intensity but no viable corresponding ν₂ line). Using Equations (A1) and (A2) (and assuming T_rot = 50 K) suggests an intensity ratio P3/P4 = 1.07, and this is reflected in the model shown in Figure 4. (We note that the modeled "P3" feature appears weaker than the feature in interval 16; this results from its relatively poorer transmittance (∼50%) compared with that of the "P4" feature (∼95%).] However, to account for the intensity in the CH₃OH residuals near 2977.6 cm⁻¹ requires an unrealistically large ratio (∼1.8), and we therefore exclude this region from our analysis. Similar reasoning pertains to other excluded spectral regions (marked ''x" in Figure 4), and this approach will be used as our empirical model is improved through extension to other comets in our database encompassing a range of rotational temperatures. Table 4 provides a summary of our empirical model as applied to Comet 21P/Giacobini-Zinner.

Tables 5–7 present fluxes and production rates for spectral intervals included in our analysis of 21P/GZ.