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The Astrophysical Journal, 650:470–483, 2006 October 10 # 2006. The American Astronomical Society. All rights reserved. Printed in U.S.A.
DETECTION OF FORMALDEHYDE EMISSION IN COMET C/2002 T7 (LINEAR) AT INFRARED WAVELENGTHS: LINE-BY-LINE VALIDATION OF MODELED FLUORESCENT INTENSITIES M. A. DiSanti,1,2 B. P. Bonev,1,2,3,4 K. Magee-Sauer,2,5 N. Dello Russo,2,6 M. J. Mumma,2,7 D. C. Reuter,1 and G. L. Villanueva1 Received 2006 February 2; accepted 2006 June 7
ABSTRACT Formaldehyde ( H2CO) was observed in comet C/2002 T7 (LINEAR) with spectral resolving power k/k 2.5 ; 104 using the Cryogenic Echelle Spectrometer (CSHELL) at the NASA Infrared Telescope Facility, on UT 2004 May 5, 7, and 9. The observations, which sampled emission in the 1 and 5 rovibrational bands between 3.53 and 3.62 m, represent the first spectrally resolved detection, at infrared wavelengths, of monomeric H2CO spanning a range of rotational energies. A comparison of measured line intensities with an existing fluorescence model permitted extraction of rotational temperatures and production rates. Two complementary approaches were used: (1) a correlation analysis that provided a direct global comparison of the observed cometary emissions with the model and (2) an excitation analysis that provided a robust line-by-line comparison. Our results validate the fluorescence model. The overall correlation coefficient was near or above 0.9 in our two principal grating settings. The excitation analysis provided accurate measures of rotational excitation (rotational temperature) on all three dates, with retrieved values of Trot clustering near 100 K. Through simultaneous measurement of OH prompt emission, which we use as a proxy for H2O, we obtained native production rates and mixing ratios for H2CO. The native production of H2CO varied from day to day, but its abundance relative to H2O, Xnative , remained approximately constant within the errors, which may suggest an overall homogeneous composition of the nucleus. We measured a mean mixing ratio Xnative = (0.79 0.09) ; 102 for the three dates. Subject headingg s: astrochemistry — comets: individual (C/2002 T7 ( LINEAR)) — infrared: solar system Online material: color figures
1. INTRODUCTION
The comet nucleus warms when approaching the Sun, causing native ices to release ‘‘parent volatiles’’ into the cometary coma, where they can be sensed spectroscopically. Since 1996, direct detection of parent volatiles (e.g., H2O, C2H6, HCN, CO, and CH4) through high-resolution [resolving power RP k/k (2–3) ; 104] infrared spectroscopy has become routine (Mumma et al. 2003 and references therein). The first (nearly) complete high-resolution infrared survey was of the moderately bright comet C/1999 H1 (Lee) in 1999. Mumma et al. (2001) analyzed selected frequency ranges from this survey, retrieving production rates for seven parent volatiles (H2O, CH4, C2H2, C2H6, CH3OH, HCN, and CO) in a consistent manner and also identifying daughter species (OH and NH2). Subsequent detailed examination of the survey observations, over the spectral range 2.87– 3.70 m, resulted in more than 500 lines being detected ( Dello Russo et al. 2006). Approximately 80% of these lines were identified, the rest arising from as yet unidentified precursors. Although H2CO emissions were identified in the latter work, those data were not as well suited to a general analysis of its fluorescent spectrum compared with the observations presented here. The abundance (and other properties) of a parent volatile can be established if a comprehensive quantum mechanical fluorescence model for its rovibrational lines exists. It is frequently not possible to sample all quantum states in a given molecular band, owing to factors such as atmospheric extinction, limited instrumental sensitivity, and incomplete spectral coverage. However, application of a quantum band model to the lines measured can provide a rotational temperature (Trot ) that in turn permits calculation of the total abundance for that species. In most cases, the analysis requires development of new (or extension of existing) fluorescence models appropriate to the low temperatures
Knowledge of the composition of native ices in comets (i.e., those contained in the nucleus) provides information pertaining to the formative epoch of our solar system. It allows comparison with interstellar ices, with laboratory-processed analogs, and with models of comet formation. The original structures and compositions of native ices should reflect local conditions (chemistry, temperature, degree of radiation processing) prevalent when and where they formed (Mumma et al. 1993; Irvine et al. 2000; Bockele´e-Morvan et al. 2004). Formaldehyde (H2CO) is ubiquitous in dense interstellar clouds (see, e.g., Turner 1994 and references therein), and its presence is expected in comet nuclei if they contain interstellar material. Cometary H2CO was first detected in 1P/ Halley, based on radio (Snyder et al. 1989), infrared (Combes et al. 1988), and mass spectrometric investigations ( Mitchell et al. 1987; Huebner et al. 1987; Eberhardt et al. 1987; Eberhardt 1999). Because comets (icy planetesimals) must have delivered enormous quantities of prebiotic chemicals to the young Earth, the abundance of H2CO in comets is of keen astrobiological interest. 1 Solar System Exploration Division, Code 693, NASA Goddard Space Flight Center, Greenbelt, MD 20771;
[email protected]. 2 Visiting Astronomer at the Infrared Telescope Facility. 3 Department of Physics, Catholic University of America, 620 Michigan Avenue NE, Washington, DC 20064. 4 Work was begun while a graduate student at the University of Toledo. 5 Department of Physics and Astronomy, Rowan University, 201 Mullica Hill Road, Glassboro, NJ 08028. 6 Applied Physics Laboratory, The Johns Hopkins University, 11100 Johns Hopkins Road, Laurel, MD 20723-6099. 7 Solar System Exploration Division, Code 690, NASA Goddard Space Flight Center, Greenbelt, MD 20771.
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DETECTION OF H2CO IN COMET C/2002 T7 (20–150 K ) typical of cometary comae. Line-by-line fluorescence models need to be formulated from high-resolution laboratory spectra, synthesized at the desired rotational temperature, and convolved to the instrumental resolution. Interpretation of infrared emission from cometary formaldehyde was approached in this manner nearly a decade before the development of astronomical spectrometers capable of providing the needed high-resolution infrared spectra. A fluorescence model for H2CO with a rotational temperature of 300 K was used to synthesize formaldehyde emission in the infrared spectrum obtained during the Vega 1 flyby of Halley, but the model did not reproduce the shape of the observed emission band (Combes et al. 1988). It was already recognized that rotational temperatures in cometary comae would likely be much lower than 300 K, and indeed high-resolution spectra of H2O in Halley showed a rotational temperature of only about 40 K (Mumma et al. 1986). For this reason, a general line-by-line fluorescence model for the 1 and 5 bands of H2CO was developed for any desired Trot (Reuter et al. 1989). Using this model, Mumma & Reuter (1989) obtained a satisfactory fit to the band shape observed by IKS, the Vega infrared spectrometer, for an H2CO rotational temperature (50 K) in acceptable agreement with values obtained for H2O, and they also applied the model to ground-based spectra of comets Halley and Wilson. However, these early (infrared) observations featured very low spectral resolving power ( RP of order 102), and only general band shapes and areas could be discerned. Since Halley, H2CO has been detected in more than a dozen comets searched at radio wavelengths (Biver et al. 2002). Subsequent ground-based infrared searches (at RP 103) in three comets, in which radio H2CO emission was detected, provided only upper limits (typically 1% relative to water; Reuter et al. 1992). More sensitive infrared searches required the advent of high-resolution astronomical spectrometers (RP > 2 ; 104). Modern infrared echelle spectrometers have small (subarcsecond ) pixels, providing high spatial resolution that favors the detection of nuclear (i.e., native) sources. Although at least in some comets H2CO originates substantially or even predominantly from a source distributed in the coma ( Eberhardt et al. 1987; Biver et al. 1997; Eberhardt 1999; Cottin et al. 2004), infrared detections of (largely native) H2CO were achieved in several long-period (Oort cloud ) comets (e.g., Lee), and also in the Jupiter-family comet 9P/ Tempel 1 ( Mumma et al. 2005). Production rates of native H2CO will be included in a paper on the chemistry of oxidized carbon in comets. In this paper, we present the first quantitative line-by-line analysis of cometary formaldehyde at infrared wavelengths. For this first application of the fluorescence model to high-resolution spectra, we address our observations of comet C/2002 T7 ( LINEAR) only, as these are the highest signal-to-noise ratio data on formaldehyde for any comet in our database. There are two primary objectives: (1) to present unambiguous identification of formaldehyde at infrared wavelengths through multiple rovibrational emission lines sampling a wide range of excitation, and (2) to present (for the first time) a detailed test and validation of the fluorescence model presented in Reuter et al. (1989). These objectives are accomplished by a direct comparison with highresolution spectra targeting H2CO in C/2002 T7, an Oort cloud comet with high emission line–to–continuum contrast. In applying the model, we also present a recently developed formalism for measuring rotational temperatures and native production rates. Abundance ratios for H2CO are obtained by comparing its production rate with that for water, measured simultaneously through OH prompt emission (see x 4). Our combined approach
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establishes infrared fluorescence as a robust means for quantifying H2CO in comets. 2. OBSERVATIONS AND DATA REDUCTION Comet C/2002 T7 was observed with the Cryogenic Echelle Spectrometer (CSHELL) at the NASA Infrared Telescope Facility (IRTF) on Mauna Kea, Hawaii. This instrument incorporates a 256 ; 256 pixel InSb array detector with small (0B2 square) pixels having sensitivity from 1 to 5.5 m (Tokunaga et al. 1990; Greene et al. 1993). CSHELL has a variety of (3000 long) slits; for our observations of C/2002 T7, we used the slit having a width of 100 . This provided sufficiently high spectral resolving power ( RP 2.5 ; 104) to isolate individual (or groups of closely spaced ) H2CO lines. The telescope was nodded along the slit in an A-B-B-A sequence with A and B beams placed equidistant from the slit midpoint and separated by 1500 ; thus the comet was present in both beams. A given echelle setting encompassed only 6 cm1, so multiple settings were required to adequately sample H2CO emission (Table 1). Four settings were chosen to ensure adequate sampling of formaldehyde. Two of these, H2CO_ B and H2CO_ A, include Q-branch emissions from the 1 and 5 bands, respectively. Another ( H2CO_ D) encompasses lines of the 1 P-branch that have the highest predicted intensities for rotational temperatures near 100 K. Because the rotational excitation was not known a priori, a fourth setting (H2CO_ E) was included to allow for temperatures well above 100 K. The spatial-spectral frames were processed using algorithms specifically tailored to our comet observations (for details, see Magee-Sauer et al. 1999; Dello Russo et al. 2000; Magee-Sauer et al. 2002; Appendix 2 of Bonev 2005). Following initial processing, background thermal continuum and sky-line emission were removed by subtracting B frames from A frames. The steps are illustrated in Figure 2 of DiSanti et al. (2001). The observations were conducted almost exclusively during daylight, so the CCD guide camera in CSHELL was not available for use. Instead, the comet was imaged through a 3000 square aperture immediately following each A-B-B-A sequence, which occurred approximately every 5 minutes. A difference image of the comet (Fig. 1a) was used to verify adequate positioning of the comet between A-B-B-A spectral sequences. For each A-B spectral difference frame, a Gaussian function was fitted to the spatial profile of continuum emission in both A and B beams. All spectral frames with the comet in the A-beam position were shifted to a common row, as were the B frames, creating spatially registered, stacked frames with the comet in A- and B-beam positions. Differencing these revealed cometary molecular emission in each beam ( Figs. 1b, 1c). Spatially combining signal in the two beams (Figs. 1d, 1e) improved signal-tonoise ratio and cancellation of thermal background and sky emissions. Absolute flux calibrations were achieved in the usual fashion (see, e.g., DiSanti et al. 2001). 3. DETECTION OF H2CO IN C/2002 T7 (LINEAR) We extracted comet spectra by summing spatially over 15 rows (300 ) centered on the nucleus (100 corresponded to approximately 460, 410, and 350 km on May 5, May 7, and May 9, respectively). These spectra (Figs. 1f, 1g, upper trace) show the cometary continuum and superposed molecular emission as observed through the terrestrial atmosphere; thus, they are attenuated (i.e., multiplied ) by the atmospheric transmittance function and convolved with the instrument function. Absolute wavelength
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TABLE 1 Log of Formaldehyde Observations in C/2002 T7 Using CSHELL
UT
a
2004 May 5: 16:14–16:38...................... 18:11–18:42 ...................... 20:31–20:35...................... 2004 May 7: 15:48–15:58...................... 16:11–16:35 ...................... 17:45–18:30...................... 2004 May 9: 16:17–16:21...................... 16:28–16:32...................... 20:15–20:20...................... a b c d e f g h
Rhb (AU )
c (AU )
˙d ( km s1)
Setting ID
Tinte (minutes)
cf (cm1)
ming (cm1)
maxg (cm1)
0.673 0.674 0.675
0.635 0.632 0.629
66.04 65.83 65.55
H2CO_ B H2CO_ D H2CO_ A
16 16 4
2783.68 2768.70 2832.93
2780.07 2765.19 2828.85
2786.10 2771.01 2835.74
0.692 0.692 0.692
0.561 0.560 0.558
64.55 64.48 64.26
H2CO_ B H2CO_ D H2CO_ E
7 16 16
2783.74 2768.70 2755.16
2780.14 2765.20 2751.77
2786.17 2771.04 2757.40
0.712 0.712 0.714
0.487 0.487 0.482
61.72 61.70 61.05
H2CO_ B H2CO_ Ah H2CO_ Ah
4 4 4
2783.71 2832.73 ...
2780.10 2828.79 ...
2786.19 2835.52 ...
Date and universal time interval over which comet C/2002 T7 was observed at each echelle grating setting. Heliocentric distance. Geocentric distance. Geocentric radial velocity. Total on-source integration time. Central wavenumber of grating setting, in observer’s rest frame. Frequency range encompassed, in cometary rest frame. ˙ = 61.40 were used in the ensuing analysis. These two A-B-B-A sequences for H2CO_ A were combined, and Rh = 0.713, = 0.485, and
calibration and determination of column burdens of absorbers in the terrestrial atmosphere (and hence of atmospheric opacity at each absorbing frequency) were achieved using the Spectrum Synthesis Program (SSP; Kunde & Maguire 1974), which accesses the HITRAN molecular database (Rothman et al. 1992). The optimized SSP model was convolved to the spectral resolution of the comet data and scaled to the cometary continuum level (Figs. 1f, 1g, dashed line). Subtracting the scaled synthetic continuum from each comet spectrum isolated the observed cometary molecular emissions (Figs. 1f, 1g, lower trace). Aside from the two lines of OH prompt emission (indicated by ‘‘OH’’ in Fig. 1f ), essentially all residual emission shown in Figure 1 is attributable to H2CO. 4. OH PROMPT EMISSION SAMPLED SIMULTANEOUSLY WITH H2CO The two lines labeled OH (Fig. 1f ) belong to the P-branch of the (1–0) band of hydroxyl and correspond to very high rotational excitation (J 0 = 16.5; Maillard et al. 1976). The origin of this prompt emission is photodissociation of H2O leaving the daughter product (OH ) in vibrationally and rotationally excited states ( Mumma 1982; Crovisier 1989; Bockele´e-Morvan & Crovisier 1989; Mumma et al. 2001). Brooke et al. (1996) noted OH emission features near 3.0 m (J 0 = 5.5) in comet C/1996 B2 (Hyakutake) that were sharply peaked on the nucleus. In stark contrast to the relatively extended spatial signature of fluorescent OH emission, these states decay to lower vibrational levels within milliseconds after the dissociation event and therefore faithfully trace the spatial distribution of their parent (H2O) molecules in the coma. This was later demonstrated by comparing simultaneous measures of H2O and OH in several comets (Bonev et al. 2004, 2006; Bonev 2005). Quantitatively, infrared OH prompt emission has been utilized as a proxy for H2O production in comets. Mumma et al. (2001) postulated that OH is a convenient and relatively temperatureinsensitive means of obtaining water production rates and presented (albeit without error estimates) an equivalent g-factor
(emission efficiency, expressed in OH photons s1 [ H2O molecule]1) for the combined multiplet near 3046 cm1 in comet Lee. Bonev et al. (2004, 2006) and Bonev (2005) presented lineby-line equivalent g-factors (including uncertainties) from simultaneous observations of water and hydroxyl, for lines of OH spanning a range in rotational quantum number, thereby allowing derivation of H2O production rates from measured OH line intensities. The simultaneous detection of OH and H2CO has two important implications for our study of C/2002 T7: (1) The parentlike spatial profile of OH prompt emission permits us to quantify slit losses associated with release of a parent volatile. This is required to calculate the native production rate of H2CO based on the signal contained in a nucleus-centered extract (see x 7). (2) The OH lines allow a simultaneous measure of H2O production in C/2002 T7 and, thus, a robust measure of the abundance ratio of native H2CO relative to H2O. 5. FLUORESCENCE MODEL FOR H2CO 5.1. Calculating g-Factors for Fundamental Band Transitions The main goal of this paper is to conduct, for the first time, a line-by-line comparison of an existing model for H2CO fluorescence and high-resolution (RP 2.5 ; 104) comet spectra. The detailed model, in which the rotational temperature (Trot ) is a free parameter, was presented in Reuter et al. (1989). In that work, line-by-line g-factors for the 1 band (symmetric CH2 stretching, centered at 2781 cm1) and the 5 band (asymmetric CH2 stretching, centered at 2834 cm1) were presented at several values of Trot. For convenience, in this section we summarize the steps involved when calculating g-factors for any desired value of Trot . As usual for asymmetric-top molecules, rotational levels are specified by the three quantum numbers J, Ka, and Kc (Herzberg 1945, p. 44). J represents the total angular momentum, while Ka and Kc represent the limiting-case prolate and oblate quantum numbers, respectively. All (electric dipole–allowed) transitions with
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Fig. 1.—(a) Difference image showing comet C/2002 T7 (LINEAR) though a 3000 ; 3000 (open) aperture, in the A-beam (white) and B-beam (black) positions, separated by 1500 (corresponding to 7300, 6100, and 5300 km at the comet on May 5, May 7, and May 9, respectively). Also shown is a superposed schematic of the 100 wide CSHELL slit. Relative to its placement in imaging mode, the spectral signal is displaced by 500 (25 rows) on the array to avoid contamination by residual afterglow from imaging a bright source. For this reason, in imaging mode the slit extends beyond the array as shown. (b, c) Difference of spectral frames in the H2CO_ B and H2CO_ D settings, respectively, after spatially registering A and B beams separately. Continuum and superposed molecular emissions are evident in each beam. (d, e) Respective cropped difference frames, after spatially combining the beams. ( f, g) Upper trace: Extracted spectra (solid line) representing the comet signal contained in a 100 ; 300 (490 ; 1470 km) aperture centered on the row containing peak continuum intensity, with optimized atmospheric transmittance function (dashed line). Lower trace: Observed comet residuals (solid line), showing the emission intensity in excess of the continuum. Also shown (dotted lines) is the 1 stochastic noise envelope (this also applies to subsequent figures). Essentially all residual emission in these settings, other than the two lines of prompt OH, is attributable to formaldehyde. [See the electronic edition of the Journal for a color version of this figure.]
J < 16 are included in the model, representing states with rotational energies up to 600 cm1 (or Trot Elow /k 900 K, where Elow represents rotational energy in the ground vibrational state) above the ground state. The model is expected to be highly accurate for representative temperatures in the cometary coma (Trot 150 K or less). In fundamental transitions, vibrational excitation to an upper rovibrational state ( j 0 ) is accomplished by solar infrared pumping from a rotational level (i 00 ) in the ground vibrational state ( Fig. 2). This is followed by radiative decay from j 0 back to a rotational level (k 00 ) in the ground vibrational state (not necessarily the rotational level i 00 from which the pump occurs). Fluorescent g-factors (gj 0 k 00, photons s1 molecule1) for emission from j 0 to k 00 are calculated in a three-step process: (1) For a specified Trot ,
ground-state rotational populations are calculated assuming a Boltzmann population distribution. This is characterized by a rotational temperature that can be constrained by observations and is generally valid in the collisionally dominated inner coma (Weaver & Mumma 1984; Xie & Mumma 1992). (2) Solar pumping rates into j 0 from all relevant (i.e., electric dipole–allowed ) rotational levels i 00 are summed. (3) Each summed pump is multiplied by the downward rotational branching ratio to state k 00 . In this fashion, the g-factors (at heliocentric distance Rh = 1 AU ) are calculated as gj 0 k 00 ¼ Sun
X i00
Bi 0 0 j 0 popi 0 0
A 0 00 P jk i 0 0 Aj 0 i 00
;
ð1Þ
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Di SANTI ET AL. ture as higher energy rotational levels become populated. The asymmetric ( 5) band encompasses a considerably broader range in frequency, and its (relatively more spread out) Q-branch becomes stronger at lower temperatures as a larger fraction of the total band intensity becomes concentrated in the lowest-J transitions. 5.2. Model Parameters
Fig. 2.—Schematic diagram depicting the processes involved in resonance fluorescence. Formaldehyde molecules are pumped (upward-pointing arrows) from multiple rotational states (i 00 ) in the ground vibrational state to rotational state j 0 in an excited vibrational state, with subsequent emission of a photon through radiative de-excitation (downward-pointing arrow) to rotational state k 00 in the ground state. [See the electronic edition of the Journal for a color version of this figure.]
where popi 0 0 is the fractional ( Boltzmann) population of state i 00 and Sun is the solar radiation flux density [ photons cm2 s1 (cm1)1, assumed to be independent of pumping frequency] at Rh = 1 AU. The Einstein A- and B-coefficients are calculated in the standard manner: Bi 00 j 0 ¼
Si 00 j 0 (T ) ; popi 0 0 (T )
Aj0k 00 ¼
wk 0 0 8c 2 Bi 0 0 j 0 ; wj 0
ð2Þ
where wk 00 and wj 0 (2 Jk 0 0 þ 1 and 2 Jj 0 þ 1) are the statistical weights of rotational states k 00 and j 0 , respectively. Because not all relevant 1 and 5 transitions have been measured in the laboratory, Reuter et al. (1989) determined the line frequencies () and strengths (Si 00 j 0 ) from a fit of an accurate Hamiltonian to the available subset of experimental frequencies and strengths ( Brown et al. 1979). An implicit assumption in this model is that the comet absorption lines are optically thin. The relative populations at two rotational temperatures, T1 and T2, are related by popi 0 0 (T1 ) ¼ popi 0 0 (T2 )
T2 T1
3=2
xEi 0 0 (T2 T1 ) exp ; T1 T2
ð3Þ
where x hc/k 1.44 cm K and Ei 0 0 (cm1) is the rotational energy of level i 00 . Once the popi 00 are established, line g-factors can be calculated for any desired temperature based on gline at a given T (e.g., 100 K ). Stick spectra ( RP 3 ; 105) of the 1 ( Fig. 3a) and 5 ( Fig. 3b) bands reveal stark differences in the distribution of their line positions and intensities. The symmetric ( 1) band has relatively well defined P-, Q-, and R-branches, with very closely spaced (in frequency) lines in the Q-branch ortho manifold for J > 2. Reuter et al. (1989) predict the Q-branch intensity near 2781 cm1 to increase with increasing tempera-
Like H2O, modeled formaldehyde g-factors vary with two parameters: the rotational temperature of the ground vibrational state and the ortho-to-para ratio (OPR) of the H2CO molecule. For H2CO, the dependence on OPR is a secondary effect because the lowest ortho level lies only 10.5 cm1 above the lowest para level (the ground state); the corresponding separation for H2O is 23.0 cm1. Consequently, in the case of H2CO, nonequilibrium values (OPR < 3.0) are expected only for spin temperatures below 15 K (see, e.g., Crovisier 1998). Such low values of Tspin have not been found for H2O (Dello Russo et al. 2005; Bonev 2005) or for other species ( Kawakita et al. 2003, 2005) in comets. The observed relative intensities of H2CO emission lines in C/2002 T7 are reproduced well with Trot as the sole free parameter in the model (i.e., with g-factors based on OPR = 3). 6. COMPARISON WITH COMET SPECTRA The combined fluorescence model (i.e., both the 1 and 5 bands) at 100 K is shown in Figure 3c, along with the four sampled spectral regions (Table 1). Two of these regions, ‘‘B’’ and ‘‘D’’ (corresponding to the H2CO_ B and H2CO_ D settings, respectively, in Table 1), provided particularly diagnostic comparisons of modeled line-by-line g-factors with observed comet residuals (Figs. 4 and 5). Atmospheric opacity reduces the observed emission intensities compared with their values at the top of the atmosphere, so we multiplied the modeled g-factor for each line by the fully resolved atmospheric transmittance at its Dopplershifted line center frequency. We next summed contributions over the range offrequencies included within each spectral channel. We then used a sliding kernel to convolve this binned model to the resolution of the observed comet residuals. We explored kernels ranging from triangular to trapezoidal; a seven-element hybrid (quasi-trapezoidal ) kernel provided the best match to the observed comet residuals. We refer to this as the ‘‘convolved model.’’ Our methodology for extracting the rotational temperature involves two complementary (though not equivalent) approaches, which we refer to as ‘‘correlation’’ and ‘‘excitation’’ analyses. For each setting, spectral regions are selected for which the convolved model (when scaled to the comet residuals) remains statistically significant over a range of values in Trot. We increase the rotational temperature in small (1 K ) steps over a broad range of values (e.g., 30–200 K ) that span the expected Trot of the inner coma, and we compare the model with the observed comet residuals at each step. 6.1. Correlation Analysis This approach permits the degree of agreement between the convolved model and the observed comet residuals to be visualized. The correlation coefficient, R, between these is calculated at each step in rotational temperature, and the plot of R versus Trot is referred to as a ‘‘temperature correlogram.’’ The maximum value of R corresponds to the most probable rotational temperature, and the sharpness of this maximum indicates how well Trot can be constrained. We prefer the test for maximum correlation over other statistical approaches (e.g., 2 minimization) because
Fig. 3.—Stick spectra of modeled H2CO (k/k = 3 ; 105), showing calculated line g-factors for (a) the 1 band and (b) the 5 band at several temperatures. In (c), we show the model at 100 K with both bands combined, illustrating the spectral coverage of our four H2CO settings (with ‘‘A’’ denoting H2CO_ A, etc.; Table 1). [See the electronic edition of the Journal for a color version of this figure.]
Fig. 4.—(a) Observed comet residuals in the H2CO_ B setting, compared with modeled line-by-line g-factors at 100 K multiplied by the atmospheric transmittance at each Doppler-shifted line center frequency. (b–d ) Expansions of the frequency scale over regions of high line density. The number above each modeled line represents its lower state energy, and ‘‘(2)’’ indicates that there are two lines at that particular frequency. Below each modeled line, ‘‘O’’ and ‘‘P’’ pertain to ortho and para lines, respectively, and ‘‘5’’ indicates lines of the 5 band (all other lines are in the 1 band). (e) Observed comet residuals with the convolved fluorescence model superposed. [See the electronic edition of the Journal for a color version of this figure.]
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Fig. 5.—Similar to Fig. 4, but for the H2CO_ D setting. [See the electronic edition of the Journal for a color version of this figure.]
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ture near rest frequency ( 0) 2767.6 cm1 over that predicted by the model, both on May 5 ( Fig. 6b) and (especially) on May 7 (Fig. 6d ). Also, excess emission is apparent near 0 = 2768.0 cm1, particularly on May 5. Accordingly, these spectral regions are omitted from our analysis. The correlation analysis is useful because the correlation coefficient reflects the overall agreement between model and data; however, this global statistical approach does not provide an easy means by which to quantify uncertainties in Trot . Also, all points are given equal weight, whereas one might expect stronger lines to be given relatively higher weight because they have higher signal-to-noise ratio. Moreover, this approach does not incorporate information pertaining to the spread in rotational energy sampled by the observed transitions. Our excitation analysis addresses these issues and provides more robust estimates for the rotational temperature, and also a direct measure of its uncertainty. 6.2. Excitation Analysis
Fig. 6.—Correlation analysis: (a) May 5 H2CO_ B; (b) May 5 H2CO_ D; (c) May 7 H2CO_ B; (d ) May 7 H2CO_ D; (e) May 9 H2CO_ B. Left, correlation coefficient vs. rotational temperature, based on comparing observed comet residuals and convolved model within selected spectral intervals (these are shown in the plots to the right as horizontal lines below each residual spectrum); right, observed comet residuals and convolved model at the most probable Trot , which corresponds to the peak in the correlogram. [See the electronic edition of the Journal for a color version of this figure.]
it tests the relative intensities of H2CO lines and the range of permissible values for R is always the same (1 R þ1). ( We note that, by definition, the maximum value of R corresponds to the minimum 2.) The limits correspond to complete correlation (þ1) and complete anticorrelation (1), with no correlation corresponding to R = 0. The resulting correlograms suggest that the setting containing the 1 Q-branch (H2CO_ B; Figs. 6a, 6c, 6e) should constrain the low end of Trot better than the high end. The other setting ( H2CO_ D; Figs. 6b, 6d ) is centered near the peak line intensity in the 1 P-branch (for Trot near 100 K; see Fig. 3c) and should constrain Trot better overall (especially on the high end). The results in Figure 6 provide a test of the H2CO fluorescence model, as well as an approximate measure of Trot (100 K ). This comparison represents an efficient means by which to identify emission features that are not fitted well (or not fitted at all ) by the model at any value of Trot . For example, in the H2CO_ D setting, the comet residuals show excess intensity in the fea-
Modeled formaldehyde g-factors vary with rotational temperature. After multiplying each line g-factor by the fully resolved transmittance at its Doppler-shifted line center frequency, we express g in watts per molecule to permit direct comparison with the observed line flux (i.e., with Fline , W m2). (Note that the ratio Fline /g is proportional to the column abundance of H2CO molecules; see, e.g., eq. [1] in DiSanti et al. 2006.) Because the line g-factors are temperature dependent, so is the ratio Fline /g. The basis of our excitation analysis involves examining this ratio for lines that sample a range of rotational energies. At the correct rotational temperature, Fline /g is independent of rotational energy (see Dello Russo et al. [2004, 2005] and Bonev [2005] for applications to H2O). In our excitation analysis (Fig. 7), we perform a linear leastsquares weighted fit to Fline /g versus Elow at each temperature step. The quantity Fline represents the observed wattage per square meter in the comet residuals, summed over each range of points (each frequency interval ) indicated in Figure 6 (right), and g represents the (transmittance-multiplied ) model intensity similarly summed. Elow is the mean lower state energy for all lines encompassed within each frequency interval, weighted by each corresponding line g-factor (times transmittance). For our H2CO_ B and H2CO_ D observations, we take the temperature at which the best-fit slope equals zero to be the optimal value, Trot (opt) (this corresponds to the zero-slope intercept in the leftmost panels of Fig. 7). For evaluating uncertainties in rotational temperature, we adopt the approach applied to analysis of H2O nonresonant fluorescent emission (Dello Russo et al. 2005; Bonev 2005). We use the larger of the ‘‘stochastic’’ error or the ‘‘standard’’ (i.e., ‘‘variance’’) error in the slope when evaluating the uncertainty (Trot). The stochastic error pertains solely to the signal-to-noise ratio of the data (the error bars on the points), while the standard error is a measure of the dispersion of individual values about the mean (weighted by the inverse square of each individual stochastic error). Both errors depend on the spread in rotational energy sampled. This dependence reflects the fact the rotational distribution is better constrained when a larger range in rotational energy is included in the analysis. The remaining panels in Figure 7 correspond to Trot (opt) and Trot (opt) (Trot). For Trot < Trot (opt), g-factors for lower energy lines are overestimated, while g-factors for higher energy lines are underestimated, and hence the least-squares slope is positive. Conversely, for Trot > Trot (opt), the opposite is true, and the slope is negative. It is relatively easy to assess small changes in Trot using this approach.
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Fig. 7.—Excitation analysis. From left to right, linear least-squares slope (and its 1 uncertainty; dotted lines) vs. Trot and fits for Trot(opt) (Trot), for Trot(opt), and for Trot(opt) þ (Trot). The optimal value of Trot corresponds to the zero-slope intercept, and (Trot) is determined as indicated: (a) May 5 H2CO_ B; (b) May 5 H2CO_ D; (c) May 7 H2CO_ B; (d) May 7 H2CO_ D; (e) May 9 H2CO_ B. In all cases the linear fits are weighted by the stochastic errors (error bars) on each point. [See the electronic edition of the Journal for a color version of this figure.]
6.3. Search for Additional Spectral Features Nearly all intensity (in excess of the continuum) contained in the H2CO_ B and H2CO_ D settings is accounted for by either OH or H2CO emission. Subtracting the ( best-fit) convolved model from the comet residuals (Fig. 8) can reveal additional intensity not accounted for by the fluorescence model. Visual inspection of the resulting weighted mean spectrum for H2CO_ B reveals no apparent excess (above the 1 noise envelope) other than the two lines of OH prompt emission. In the H2CO_ D setting, the most obvious feature is near 0 = 2767.6 cm1. It is very strong on May 7, yet it is only marginally present on May 5. The excess near 0 = 2768.0 cm1 is apparent on May 5 but is virtually absent (at the 1 noise level ) on May 7. We mention these and other nearby marginal (largely because of limited signalto-noise ratio) features that may or may not be causally related only as points of reference for future IR studies of H2CO in comets.
6.4. Additional Settings The settings H2CO_ B and H2CO_ D permitted a robust lineby-line comparison with the fluorescence model. We used two additional settings to characterize formaldehyde emission more completely in C/2002 T7. The setting H2CO_ A ( Fig. 9a) targeted the 5 Q-branch region, while H2CO_ E ( Fig. 9b) sampled high-J lines in the 1 P-branch. At low rotational temperature, line g-factors in H2CO_ A are expected to become very strong (see Fig. 3b). The H2CO_ E setting samples lines with high rotational quantum number (up to J = 13), thereby permitting an accurate measure of Trot for population distributions well above 100 K. Inspection of Figure 9a shows considerable emission in excess of that predicted by the model. This could be expected a priori, given that the H2CO_ A setting also encompasses emission in the P-branch of the CH3OH 3 band and CH3OH was
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Fig. 8.—Residuals with convolved model [at Trot(opt) from our excitation analysis, Fig. 7], and their difference (residuals minus model) for (a–c) the H2CO_ B setting on three dates and (e–f ) the H2CO_ D setting on two dates. The respective bottom panels [(d) for H2CO_ B, and (g)–(h) for H2CO_ D] represent the mean differences over all dates, weighted according to the respective 1 values for each date [(g) and (h) show the same data; the scale in (h) provides a direct comparison with flux densities in (d)]. Note that each flux density has been multiplied by the appropriate growth factor (‘‘qscale’’ in Table 2), compared with that shown in previous figures. [See the electronic edition of the Journal for a color version of this figure.]
relatively abundant in C/2002 T7 (approximately 4% relative to H2O; Crovisier et al. 2005). Based on modeled line intensities, we expect this setting to be of great value in comets having a higher H2CO/CH3OH abundance ratio (and lower rotational temperature) than that observed in C/2002 T7. The comparison between model and comet residuals in the H2CO_ E setting (Fig. 9b) is more perplexing, because very little agreement is seen. This may suggest the presence of an additional emitting species, or of H2CO emissions not included in the model (in which no lines with J > 15 are included ). Alternatively, it may reflect the relaxation of high-J levels in the ground vibrational state (v = 0).
7. PRODUCTION RATES 7.1. General Approach Our method of calculating production rates (Q, molecules s1) has been described elsewhere in detail (see, e.g., Dello Russo et al. 1998; Magee-Sauer et al. 1999; Dello Russo et al. 2000; DiSanti et al. 2001) and so is only briefly summarized here. For a given molecular species, we step a 100 ; 100 square aperture along the slit and calculate Q at each step according to Q¼
42 Fline : 1 AU f (x) gline;1 AU
ð4Þ
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Fig. 9.—(a) Similar to Figs. 4 and 5, but for the H2CO_ A setting. The upper trace (15-row spectral extract) is included to show the fit of the atmospheric transmittance function. (b) Same as (a), but showing the H2CO_ E setting. All lines in the H2CO_ A setting belong to the 5 band. [See the electronic edition of the Journal for a color version of this figure.]
Here (meters) is the geocentric distance, 1 AU (seconds) is the photodissociation lifetime (evaluated at Rh = 1 AU ), and f (x) is the fraction of molecules of the species encompassed in the beam at each position, incorporating photodecay (assuming constant production and uniform outflow; see the appendix of Hoban et al. 1991 for details). The parameters Fline and gline,1 AU are as defined previously (note that g as calculated in eq. [4] pertains to Rh = 1 AU, through Sun; see eq. [1]). We average Q at corresponding projected distances to either size of the nucleus and refer to the resulting plot of Q versus projected distance as the ‘‘symmetric Q-curve.’’ Owing to slit losses, production rates calculated in this manner invariably increase from the nucleus-centered value to a constant terminal value (within statistical uncertainty) at some distance from the nucleus. We take the terminal Q to represent the true global production rate. The terms ‘‘nucleus-centered’’ and ‘‘terminal’’ (off-nucleus) production rates and the relation between them have been clearly defined and extensively discussed in our previous work ( Dello Russo et al. 1998, 2000; Magee-Sauer et al. 1999; DiSanti et al. 2001; Magee-Sauer et al. 2002; Bonev 2005; Bonev et al. 2006). We define a ‘‘growth factor’’ as the ratio of terminal to nucleuscentered production rates. Once established, this factor can be applied to production rates based on line fluxes within a nucleuscentered extract (these have higher signal-to-noise ratio than do corresponding intensities from extracts taken off the nucleus) to retrieve reliable global production rates. For a native source, the offset distance required to reach the terminal Q is realized fairly quickly, depending on the effective point-spread function ( PSF ) for the particular observation. The PSF is influenced primarily by seeing, plus possible drift of the comet perpendicular to the slit during each A-B-B-A sequence or along the slit during each individual exposure (each A or B). In the presence of extended release in the coma, the terminal value is
reached farther from the nucleus, assuming the scale for such release is large compared with the PSF. 7.2. Native Production Rate of Formaldehyde Our spectral extracts represent nucleus-centered sums over 15 rows. This corresponds to an aperture of 100 spectrally (the slit width) by 300 spatially (1B5 in either direction about the nucleus). To best constrain the native production rate, we combined our nucleus-centered production rate for H2CO with the growth factor for a parent volatile. The nucleus-centered production rate was calculated (eq. [4]) using the mean value of Fline /g at each corresponding best-fit rotational temperature (Fig. 7), and the growth factor was obtained from spatial profiles for OH prompt emission measured simultaneously with H2CO in setting H2CO_ B (see x 4). In addition to native H2CO in C/2002 T7, we must also consider possible distributed sources. Their significance depends on the relative size of the two components and on the spatial scale for distributed release in the coma. Neither of these factors is known a priori. In comet 1P/Halley, a significant extended source was seen by the Giotto spacecraft, several thousand kilometers from the nucleus (Meier et al. 1993). A precursor scale length of 8 ; 103R1:5 h is generally assumed in deriving Q( H2CO) from radio observations (Colom et al. 1992; Biver et al. 1997, 2002), implying a value of 4.5 ; 103 km for our C/2002 T7 observations (Table 1). Our nucleus-centered beam is dominated by molecules lying within 600–700 km of the nucleus (corresponding to 1B5 at 0.6 AU ), and comparison between spatial profiles for H2CO and OH in C/2002 T7 (Fig. 10) suggests a significant ( perhaps dominant) native-source contribution. Therefore, our analysis is expected to provide a good approximation for the production of native H2CO in C/2002 T7. We defer more a detailed investigation of distributed-source contributions to a future paper.
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The production rates of native H2CO obtained from the two settings on a given day are in agreement on May 5 and May 7 (Table 2). While the absolute production rate of H2CO showed some variation, its abundance relative to H2O (Xnative), as well as Q( H2O), remained constant (within error) on all three dates (Table 2). Based on this limited chemical sampling (i.e., of two molecules only), we do not see evidence for chemical heterogeneity within the nucleus. 8. SUMMARY Comet C/2002 T7 ( LINEAR) provided the best opportunity to date to compare an existing fluorescence model for formaldehyde with high-resolution spectra. To accomplish this we used two complementary approaches: (1) a correlation analysis, providing a visual comparison between the observed spectrum and the fluorescence model, and a measure of their global agreement, and (2) an excitation analysis, providing a robust line-by-line comparison incorporating the stochastic noise and spread in rotational energy for the sampled lines. This work validated the model, with global correlation between data and model near or above 0.9 for both H2CO_ B and H2CO_ D settings on all dates. Our line-by-line comparison has resulted in accurate quantitative measures of the rotational excitation and native production of H2CO. Overall, the rotational temperature was constrained better by the H2CO_ D setting, which targeted lines sampling a larger range in rotational energy. Based on observations of H2CO on three dates, we find rotational temperatures clustering around 100 K. From simultaneous measurements of OH prompt emission, which indirectly samples H2O, we measured a mixing ratio (8 ; 103 relative to H2O) that remained constant (within error) over the 4 day interval sampled by our observations. Our analysis has revealed possible shortcomings in the model in the region sampling high-J lines of the 1 band. It is also possible that most of the emission features observed in this region arise from a different emitting species. These possibilities will be pursued in a future study. This work represents an example of
Fig. 10.—Comparison of the spatial distributions of H2CO, OH, and continuum emissions in C/2002 T7 (LINEAR), from the H2CO_ B setting on UT 2004 May 5, scaled to a common mean intensity over the central 300 (between the dashed vertical lines); 100 corresponds to 460 km. The approximate agreement between H2CO and OH suggests a substantial native source contribution to H2CO and justifies our approach to measuring Qnative (x 7.2). [See the electronic edition of the Journal for a color version of this figure.]
Accordingly, for the H2CO_ B setting Qnative = QH2 CO (100 ; 300 ) ; QOH (terminal )/QOH (100 ; 300 ). The H2CO_ D setting does not contain emission from prompt OH, so the growth factor for it is obtained by comparing Q-curves for the continuum in the two settings. This assumes that the true volumetric distribution of dust about the nucleus remained constant in the interval between the H2CO_ B and H2CO_ D observations. Because the H2CO_ B setting provides a measure of the growth factor for a native volatile, we calculate mixing ratios for native H2CO relative to water (Xnative) from these observations alone. We measure a value of Xnative = (0.79 0.09) ; 102 for the mean mixing ratio from May 5, May 7, and May 9 (Table 2).
TABLE 2 Rotational Temperatures and Native Production Rates of Formaldehyde in C/2002 T7
UT Date 2004 May 5.............................. 2004 May 7.............................. 2004 May 9..............................
Setting IDa
Trot Corr.b ( K)
Trot Boltz.c ( K)
hFline /gi
B D B D B
110 106 97 97 121
102þ10 8 96þ9 8 102þ32 16 108þ6 6 94þ27 14
1.88 0.09 1.24 0.07 1.67 0.08 2.33 0.07 1.65 0.06
d
f (x)e (1 ; 300 ) 00
0.2429 0.2139 0.1812
qscalef
Qnativeg
Q ( H2O)h
Xnativei
1.356 0.158 1.964 0.247 2.823 0.351 1.844 0.242 2.487 0.445
2.65 0.32 2.50 0.34 4.33 0.59 3.93 0.53 3.33 0.62
3.55 0.70
0.75 0.12
4.62 1.11
0.94 0.20
4.37 1.13
0.76 0.15
Grating setting from Table 1: ( B) H2CO_ B, ( D) H2CO_ D. Optimum rotational temperature based on correlation analysis. c Optimum rotational temperature based on Boltzmann excitation analysis, with 1 uncertainty. d Weighted mean ratio of observed line flux to line g-factor times monochromatic transmittance (107 molecules m2) at the optimum value of Trot , based on our excitation analysis ( Fig. 7). This ratio is proportional to the column abundance of H2CO, and the uncertainty listed represents the larger of stochastic and standard error. e Fraction of all (native) H2CO molecules included in a 100 ; 300 box centered on the nucleus, based on a photodissociation lifetime of 1 = 4500 s for formaldehyde at Rh = 1 AU ( Huebner et al. 1992). f Scaling factor for converting the production rate in a 100 ; 300 box to the terminal Q. For the H2CO_ B setting, qscale was based on simultaneously observed OH prompt emission (OH); for the H2CO_ D setting, qscale was approximated by assuming a similar spatial distribution of continuum emission in the two settings on each particular date (see text). g Production rate of native formaldehyde (1027 molecules s1): Qnative = 42 /½1 f (x)(Fline /g)(1;3) ; qscale. The error incorporates the tabulated uncertainty in hFline /gi and also the uncertainty in qscale. The production rates for H2CO and H2O are proportional to the gas outflow speed, vg (assumed to be 0.8 km s1); however, their ratio, X (note i), is independent of vg . h Water production rate (1029 molecules s1) obtained from the OH emission lines P(17.5) 1+ and 1 (at rest frequencies 0 of 2784.188 and 2785.8529 cm1), having g1 AU of 0.88 ; 108 and 2.68 ; 108 photons s1 ( H2O molecule)1 respectively, as determined from comet C/2004 Q2 ( Machholz) observations in 2005 January ( Bonev 2005; Bonev et al. 2006). The error incorporates stochastic noise and uncertainties in individual line g-factors, and also the uncertainty in qscale. Q(H2O) values (1029 molecules s1) determined separately from the 1+ and 1 lines, respectively, are 5.27 1.14 and 3.36 0.50 ( May 5), 5.79 1.95 and 4.52 0.79 ( May 7), and 4.99 1.78 and 4.30 0.94 ( May 9). i Abundance of native formaldehyde relative to H2O ( percent). Because qscale is common to both H2CO and OH in the H2CO_ B setting, the uncertainty in X does not include the error in qscale. The weighted 3 day mean for the mixing ratio of native H2CO was (0.79 0.09) ; 102. a
b
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using comets as ‘‘cosmic laboratories’’ for testing predicted fluorescent intensities of spectral lines. The ability to accurately measure H2CO in comets represents a critical piece in the overall puzzle of their chemical taxonomy. Along with the chemically related species CO and CH3OH, both of which are measured routinely in comets at infrared wavelengths (and elsewhere), knowledge of native H2CO abundances allows a more complete assessment of the efficiency of H-atom addition to CO ice on the surfaces of interstellar grains prior to their incorporation into the nucleus. The present work represents a step toward completing this puzzle.
We thank an anonymous reviewer whose comments improved the manuscript. This research is supported by the NASA Planetary Atmospheres Program ( RTOP 344-33-55 to M. A. D. and
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NNG05GA64G to N. D. R.), the NASA Planetary Astronomy Program (RTOP 344-32-98 to M. A. D. and RTOP 344-32-07 to M. J. M.), and the NASA Astrobiology Program ( RTOP 344-5351 to M. J. M.). K. M.-S. acknowledges support from the National Science Foundation Research in Undergraduate Institutions program (0407052). G. L. V. acknowledges support under the NPP-NRC Resident Research Associateship Program. We also thank the staff of the IRTF, especially B. Golisch and D. Griep, whose expertise allowed us to successfully obtain these highquality data under difficult daylight observing conditions. The NASA IRTF is operated by the University of Hawaii under cooperative agreement NCC 5-538 with the NASA Planetary Astronomy Program. The authors wish to 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.
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