ISOTOPIC RATIOS IN TITAN's METHANE - IOPscience

The Astrophysical Journal, 749:159 (15pp), 2012 April 20 C 2012.

doi:10.1088/0004-637X/749/2/159

The American Astronomical Society. All rights reserved. Printed in the U.S.A.

ISOTOPIC RATIOS IN TITAN’s METHANE: MEASUREMENTS AND MODELING C. A. Nixon1,2 , B. Temelso3 , S. Vinatier4 , N. A. Teanby5 , B. Bézard4 , R. K. Achterberg1,2 , K. E. Mandt6,7 , C. D. Sherrill8 , P. G. J. Irwin9 , D. E. Jennings2 , P. N. Romani2 , A. Coustenis4 , and F. M. Flasar2 1 Department of Astronomy, University of Maryland, College Park, MD 20742, USA Planetary Systems Laboratory, NASA Goddard Space Flight Center, Greenbelt, MD 20771, USA 3 Dean’s Office, College of Arts and Sciences, Department of Chemistry, Bucknell University, Lewisburg, PA 17837, USA 4 LESIA, Observatoire de Paris, CNRS, 5 Place Jules Janssen, 92195 Meudon Cedex, France 5 School of Earth Sciences, University of Bristol, Wills Memorial Building, Queen’s Road, Bristol BS8 1RJ, UK 6 Space Science and Engineering Division, Southwest Research Institute, 6220 Culebra Road, San Antonio, TX 78228, USA 7 Department of Environmental and Civil Engineering, University of Texas at San Antonio, One UTSA Blvd., San Antonio, TX 78249, USA 8 School of Chemistry and Biochemistry, Georgia Institute of Technology, 901 Atlantic Drive NW, Atlanta, GA 30332-0400, USA 9 Atmospheric, Oceanic and Planetary Physics, University of Oxford, Clarendon Laboratory, Parks Road, Oxford OX1 3PU, UK Received 2011 November 4; accepted 2012 February 13; published 2012 April 4 2

ABSTRACT The existence of methane in Titan’s atmosphere (∼ 6% level at the surface) presents a unique enigma, as photochemical models predict that the current inventory will be entirely depleted by photochemistry in a timescale of ∼20 Myr. In this paper, we examine the clues available from isotopic ratios (12 C/13 C and D/H) in Titan’s methane as to the past atmosphere history of this species. We first analyze recent infrared spectra of CH4 collected by the Cassini Composite Infrared Spectrometer, measuring simultaneously for the first time the abundances of all three detected minor isotopologues: 13 CH4 , 12 CH3 D, and 13 CH3 D. From these we compute estimates of 12 C/13 C = 86.5 ± 8.2 and D/H = (1.59 ± 0.33) × 10−4 , in agreement with recent results from the Huygens GCMS and Cassini INMS instruments. We also use the transition state theory to estimate the fractionation that occurs in carbon and hydrogen during a critical reaction that plays a key role in the chemical depletion of Titan’s methane: CH4 + C2 H → CH3 + C2 H2 . Using these new measurements and predictions we proceed to model the time evolution of 12 C/13 C and D/H in Titan’s methane under several prototypical replenishment scenarios. In our Model 1 (no resupply of CH4 ), we find that the present-day 12 C/13 C implies that the CH4 entered the atmosphere 60–1600 Myr ago if methane is depleted by chemistry and photolysis alone, but much more recently—most likely less than 10 Myr ago—if hydrodynamic escape is also occurring. On the other hand, if methane has been continuously supplied at the replenishment rate then the isotopic ratios provide no constraints, and likewise for the case where atmospheric methane is increasing. We conclude by discussing how these findings may be combined with other evidence to constrain the overall history of the atmospheric methane. Key words: molecular processes – planets and satellites: atmospheres – planets and satellites: composition – planets and satellites: formation – planets and satellites: individual (Titan) – radiative transfer

major enigma, recognized from early Titan chemistry models (Yung et al. 1984), as the entire atmospheric methane inventory would hence be depleted on a timescale presently estimated as ∼20 Myr (Krasnopolsky 2009)—much shorter than the age of the solar system. This reasoning leads us to conclude that either (1) Titan’s methane has been continuously resupplied to the atmosphere from a reservoir, possibly released from the interior where it may be stored as clathrate hydrates, or (2) the methane is a recent and transitory component of the atmosphere, leading to a composition and structure that is much different today from the distant past and far future. Intermediate cases between these two end-member scenarios are also possible, such as an outgassing that began 400–1400 Myr ago (Tobie et al. 2006). Distinguishing between a Titan that is internally active today (e.g., Nelson et al. 2009) and one that is not (Moore & Pappalardo 2011) has led to heated scientific debate in the last several years and remains an important—possibly the most important—unsolved problem regarding Titan. There are multiple avenues by which we can attempt to gain insight into this problem: learning more about Titan’s internal structure, e.g., through measurements of the gravity field (Iess et al. 2010), studying the surface geology to look for signature features of endogenic activity (Nelson et al. 2009), by measuring the amount of hydrocarbons present on the surface to constrain

1. INTRODUCTION Methane (CH4 ) is the second most abundant gas in Titan’s atmosphere, with a stratospheric mole fraction of ∼1.5% (Flasar et al. 2005; Niemann et al. 2010), and with nitrogen (N2 , 98.4%) and hydrogen (H2 , 0.1%) accounting for most of the remainder. The most abundant oxygen species, CO, is present at ∼50 ppm (de Kok et al. 2007). In the troposphere, the CH4 abundance rises reaching 5.65% at the surface. Methane was first identified by Kuiper (1944) and plays a complex role in Titan’s atmosphere, being involved in both a complex cascade of organic chemistry, initiated in the upper atmosphere, and also a meteorological cycle in the troposphere. The chemical cycle begins with the dissociation of CH4 by solar UV or charged particle impacts, resulting in the radicals CH3 , CH2 , and CH that combine with other methane fragments to give rise to higherorder hydrocarbons (Vuitton et al. 2007; Lavvas et al. 2008; Krasnopolsky 2009). In this respect Titan resembles the giant planets, where a similar photochemistry occurs in the middle/upper atmosphere. But unlike the giant planets, where the hydrocarbons are recycled back to methane at the high temperatures and pressures of the deep troposphere, Titan’s complex compounds diffuse downward to the troposphere where they condense and rain out on the surface and are permanently removed. This presents a 1

The Astrophysical Journal, 749:159 (15pp), 2012 April 20

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the volume of chemical processing (Lorenz et al. 2008), by further refining chemical models, and so on. Further important clues come from isotopic ratios, especially D/H and 12 C/13 C in Titan’s methane. These may yield insights into the age of the CH4 , as they are expected to change over time (Mandt et al. 2009), provided that (1) we have a sufficiently accurate model of the rate of change, and (2) accurate measurements of the original and present values of the ratios. In this paper, we seek to advance both the measurement and modeling of the two primary methane isotopic ratios. In Section 2, we describe new determinations of D/H and 12 C/13 C by analysis of spectra from the Cassini Composite Infrared Spectrometer (CIRS) instrument (Flasar et al. 2004). In Section 3, we describe theoretical calculations that estimate for the first time the reaction rates for 12 CH4 and 13 CH4 in a key photochemical reaction that produces fractionation. In Section 4, we develop a simple model of the time-dependent fractionation, and finally in Section 5 we discuss the implications of the combined measurements and modeling for the history and future of Titan’s methane.

for detection of very low abundance trace species (Vinatier et al. 2007; Teanby et al. 2007; Nixon et al. 2009a) including isotopic variants (Bézard et al. 2007; Vinatier et al. 2007; Nixon et al. 2008a, 2008b; Jennings et al. 2008; Coustenis et al. 2008; Jolly et al. 2010). Usually, the instrument is pointed so that the arrays are perpendicular to the disk edge, thereby sampling 10 altitudes with each side-by-side array without repositioning. However, CIRS has only five receiver channels for the 10 detectors of each focal plane, so the receiver must alternate between oddnumbered and even-numbered detectors on each array to build up the full picture—sacrificing half the incident radiation. Recently, several observations have been executed in the orthogonal orientation, where the arrays are instead positioned parallel to disk edge, with all 10 detectors of each array sampling only a single altitude. This provides additional S/N advantages because (1) the data from all 10 detectors may later be coadded, if horizontal variations in signal are negligible, and (2) the receivers may be commanded to work in “pair” mode, where the photon counts from detector pairs (1&2, 3&4 . . . 9&10 on each array) are added together, enabling the entirety of the input signal to be used. In this mode, each pair of detectors form a “virtual” detector dimensions 0.273×0.566 mrad (in the vertical and horizontal directions, respectively). Previously, several of these limb-parallel, detector pair mode observations were successfully analyzed to place upper limits on abundances of predicted, but undetected, trace species (Nixon et al. 2010). In the present study, we have analyzed data from the MIRLMPAIR002 observation described in Nixon et al. (2010), which was made during the T55 flyby on 2009 May 22, beginning at 02:26:41 UT with a 4 hr duration. The arrays were targeted at 25◦ S latitude (sampling ± 2◦ during the observation) and mean altitudes of 107 km (FP3, 7.6 mbar) and 247 km (FP4, 0.27 mbar). We use herein only the spectra recorded by FP4, 941 in all, at the highest spectral resolution of CIRS: 0.5 cm−1 FWHM of instrumental line shape, after Hamming apodization. The range was ∼150,000 km at the mid-point of the observation, and the corresponding projected FOV size for a pair of mid-IR detectors was 41 × 85 km.

2. MEASUREMENTS OF D/H AND 12 C/13 C USING CASSINI CIRS The infrared signatures of four distinct methane isotopologues have been detected by Cassini CIRS in the range 1100–1400 cm−1 (9–7 μm). The ν4 band of 12 CH4 at 1305 cm−1 is very strong due to the large methane abundance and commonly used to measure stratospheric temperatures by assuming an abundance for the gas: often that measured from direct sampling by the Huygens GCMS instrument (Niemann et al. 2010). The ν6 band of CH3 D at 1156 cm−1 is also strong and has been used since the Voyager era (1980) to measure the D/H ratio in methane by comparison to the 12 CH4 band (see Section 10.5.5 of Strobel et al. 2010). Several measurements −4 have been made by CIRS: (1.17+0.23 Coustenis et al. −0.28 ) × 10 +0.15 (2007); (1.32−0.11 ) × 10−4 Bézard et al. (2007); (1.58+0.16 −0.16 ) × −4 13 10 Abbas et al. (2010). In addition, two C variants have been detected, the ν4 of 13 CH4 at 1296 cm−1 (Nixon et al. 2008a; 12 C/13 C = 77 ± 3) and the ν6 of the doubly substituted species 13 CH3 D at 1148 cm−1 (Bézard et al. 2007; 12 C/13 C = 82+27 −18 ). In the remainder of this section, we describe the acquisition of CIRS spectra covering all four of these bands simultaneously and radiative transfer modeling of the observations leading to a new measurement of the D/H and 12 C/13 C in Titan’s methane.

2.2. Data Analysis and Results After co-adding all 941 FP4 spectra, two spectral regions were isolated for modeling. For the deuterated species, a small region from 1140 to 1160 cm−1 was selected containing only the sharpest (Q-branch) emissions of 12 CH3 D and 13 CH3 D, to minimize problems with fitting the underlying baseline, due to haze absorption and the effects of a weak propane band (the ν7 band of propane centered on 1158 cm−1 ; see Figure 6(d) of Nixon et al. 2009a) for which no line data currently exist to enable modeling. For the non-deuterated species, a wider range was used, 1260–1350 cm−1 , to include the P-Q-R structure of both 12 CH4 and 13 CH4 , giving the best possible S/N. The radiative transfer calculation and fitting was accomplished using the NEMESIS software (Irwin et al. 2008), which predicts the emerging spectrum based on a one-dimensional vertical atmospheric model and iteratively adjusts this model to fit the spectrum using a minimization technique. The modeling was accomplished in a nearly identical way to that described in Nixon et al. (2009a). The model atmosphere included the major gases 12 CH4 , N2 , and H2 using the vertical abundances of Niemann et al. (2010); the minor species C2 H2 , C2 H4 , C2 H6 , and C3 H8 with profiles as described in Nixon et al. (2009a); plus the methane isotopologues 13 CH4 , 12 CH3 D, and 13 CH3 D with profiles initially set to scaled versions of the 12 CH4 at terrestrial

2.1. Observations The CIRS instrument consists of dual Fourier Transform spectrometers (mid-IR and far-IR) that share a common telescope, foreoptics, scan mechanism, and other devices (Kunde et al. 1996; Flasar et al. 2004). The mid-IR spectrometer is a conventional Michelson arrangement, with two parallel 1 × 10 arrays of photodetectors to sense the radiation. These arrays are known as focal plane 3 (FP3, 600–1100 cm−1 , 17–9 μm) and focal plane 4 (FP4, 1100–1400 cm−1 , 9–7 μm). Individual detectors in the arrays are square with a 0.273 mrad field of view (FOV), while the spectral resolution is variable (depending on commanded path difference of the movable mirror) from 0.5 to 15.5 cm−1 . In limb-sounding mode the arrays are pointed above the rim of the body’s solid surface (Nixon et al. 2009b), thereby sensing a much longer path length through the atmosphere that is especially suitable for increasing the signal-to-noise ratio (S/N) 2


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Figure 1. (a) and (b) CIRS spectra of methane isotopologues (black) and model fits. The red line shows all gases present in the model, other colors show the effect of removal of minor isotopologues from the model. (c) and (d) The spectral residual (data-model difference) when minor isotopologues are not included in the model to show their signature in the data.

isotopic abundances. Gas line data used were as described in Nixon et al. (2009a), except for C2 H6 where we used the newly available list for the ν6 and ν8 bands by Di Lauro et al. (2012). Haze spectral opacity used was that of Vinatier et al. (2010), although we also experimented with gray haze and Khare et al. (1984) haze spectral properties and found negligible effect on our results. The variable parameters were vertical temperature profile (information governed by 12 CH4 at fixed profile abundances), and scaling factors (one parameter for all levels) applied to the vertical haze profile, and the profiles of C2 H2 , C2 H4 , C2 H6 , C3 H8 , 13 CH4 , 12 CH3 D, and 13 CH3 D. We tried retrieving in two methods (1) temperature first, then gas and haze abundances, or (2) temperature and gas/haze scale factors simultaneously. A slightly better fit resulted from method (2), which we retained for our final results. Both spectral regions were simultaneously fitted. Note that the finite footprint of the detectors, which convolves the limb radiation, was explicitly accounted for using eight weighted rays spaced 10 km apart, as described in Nixon et al. (2009b). Also, we determined that a small, sub-gridscale correction to the wavenumber scale provided in the CIRS database was required of +0.08 cm−1 , found by oversampling both model and data spectra to a grid of 0.01 cm−1 , and subsequent cross-correlation to obtain the optimum wavenumber shift. This correction is necessary due to slight drifts in the frequency of the reference laser that controls the interferogram sampling. Figure 1 shows the spectral data (upper plots, black) and final model fit (upper plots, red). The lower plots show the

Figure 2. Schematic showing how two estimates of each isotopic ratio (D/H and 12 C/13 C) are obtained from the abundances of the four methane isotopologues.

spectral signature of the isotopologues, calculated by removing the minor CH4 species from the model once the final fitting parameters were calculated, and then computing a synthetic spectrum using these retrieved parameters. The abundances of the three minor methane isotopologues (at 247 km) are found from the fitting (and using 12 CH4 = (1.48 ± 0.09) × 10−2 above the cold trap from Niemann et al. 2010), while the error on the retrieved abundances is computed by NEMESIS according to the formalism given in Equation (22) of Irwin et al. (2008). Figure 2 shows how four isotopic ratio estimates can be derived from the abundances of the four methane isotopologues. The ratios and their errors are given in Table 1, along with the errorweighted mean 12 C/13 C and D/H. Table 2 and Figure 3 compare these results to other recent estimates of D/H and 12 C/13 C, considering only measurements 3


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Figure 3. Recent measurements of Titan’s D/H (a) and 12 C/13 C (b) as detailed in Table 2, including this work. The limits on the weighted mean of the measurements are indicated by the gray regions, and the dashed lines show the terrestrial inorganic standard reference values (SMOW and VPDB).

and also the ground-based telescopes (error bars overlap). Our measured value of (1.59 ± 0.33) × 10−4 is fully consistent with the terrestrial value (1.56 × 10−4 ), while exceeding by one order of magnitude the value measured for Saturn ((1.6 ± 0.2) × 10−5 ; Fletcher et al. 2009). Also, our 12 C/13 C (86.5 ± 8.2) is in agreement with the recent re-measurements by Cassini INMS (Mandt et al. 2012) and Huygens GCMS (Niemann et al. 2010)—revised upward from earlier values due to improvements in the calibration and modeling—and also compatible the earlier CIRS measurements by Bézard et al. (2007) based on the pair (12 CH3 D, 13 CH3 D). With regard to the other previous CIRS measurement of 12 C/13 C, based on 12 CH4 and 13 CH4 : 12 C/13 C = 76.6 ± 2.7 (Nixon et al. 2008a), our present finding is marginally

Table 1 Measured Isotopic Ratios from Cassini CIRS Limb Sounding

12 CH 4 13 CH D 3

13 CH 4

12 CH D 3

86.5 ± 8.2 (1.59 ± 0.44) × 10−4

(1.59 ± 0.33) × 10−4 86.4 ± 28.2

Notes. Error-weighted mean 0.33) × 10−4 .

12 C/13 C

= 86.5 ± 8.2 and D/H = (1.59 ±

of the more abundant species (i.e., CH4 and H2 ). Our D/H from this work is indistinguishable from the value measured by Abbas et al. (2010), and somewhat greater though still compatible with the other measurements by CIRS and GCMS,

Table 2 Recent Measurements of D/H and 12 C/13 C in Bulk Titan Gases Ratio

Species

Measurement

D/H

H2 CH4 CH4

(1.35 ± 0.30) × 10−4 (1.25 ± 0.25) × 10−4 −4 (1.17+0.23 −0.28 ) × 10

CH4 CH4 CH4 CH4 Mean 12 C/13 C

CH4 CH4 CH4 CH4 Mean

−4 (1.32+0.15 −0.11 ) × 10 (1.58 ± 0.16) × 10−4 (1.13 ± 0.25) × 10−4 (1.59 ± 0.33) × 10−4 −4 (1.36+0.07 −0.08 ) × 10 −4 (1.56 × 10 )

82+27 −18

91.1 ± 1.4 88.5 ± 1.4 86.5 ± 8.2 89.7 ± 1.0 89.4

Instrument/Type

Altitude (km)

Reference

Huygens GCMS IRFT TEXES/IR Cassini CIRS/IR

120–0 0–450 95–290

Niemann et al. (2010) Penteado et al. (2005) Coustenis et al. (2007)

Cassini CIRS/IR Cassini CIRS/IR KPNO/IRFT/Near-IR Cassini CIRS/IR

75–260 95–290 0–160 225–275

Bézard et al. (2007) Abbas et al. (2010) de Bergh et al. (2012) This work Earth (SMOW)a

Cassini CIRS/IR Huygens GCMS Cassini INMS Cassini CIRS/IR

75–260 127–0 surfaceb 225–275

Bézard et al. (2007) Niemann et al. (2010) Mandt et al. (2012) This work Earth (VPDB)b

Notes. a Terrestrial Standard Mean Ocean Water. b Terrestrial Standard: Vienna Pee Dee Belemnite.

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compatible by virtue of overlapping error bars. However, the former conclusion that 12 C/13 C in methane is enriched with respect to the terrestrial/giant planet values, and those of heavier hydrocarbons on Titan (ethane and acetylene), is no longer secure. We believe that the change in the central value from ∼77 to ∼87 is due to different systematic errors in the nadir-sounding retrievals (Nixon et al. 2008a) and the limb-sounding retrievals (present work), with limb-sounding having the advantage that the weighting functions for each species are sharply peaked at the same (tangent) altitude. There may be some additional systematic error due to the non-overlapping line approximation made in the correlated-k radiative transfer calculation. Finally, the small error bar of the previous work was predicated on combination of purely random errors, allowing for infinite error reduction, without regard to error floors imposed by systematic errors (e.g., absolute transition strengths in line atlases). From these considerations, we recommend that the present result (86.5 ± 8.2) which has a more conservative error and is also in agreement with the recent GCMS and INMS re-analysis is considered to supersede the previous result.

Table 3 Laboratory Measurements of Methane KIE during Hydrogen Abstraction Reaction

Temperature (K)

qD

q13

CH4 + Cl CH4 + Cl

296 175

1.46a 2.05a

1.062a 1.081a

CH4 + OH CH4 + OH

296 175

1.29b 1.44b

1.004c

CH4 + C2 H CH4 + C2 H

296 175

1.17d 1.21d

CH4 + CN CH4 + CN

296 175

1.22e 1.24e

Notes. a Tyler et al. (2000). b Gierczak et al. (1997). c Sellevåg et al. (2006). d Opansky & Leone (1996). e Yang (1993).

(photolysis plus chemical loss) and is the single largest source of methane loss (Wilson & Atreya 2009) overall. Taking into account the possible isotopic varieties of methane, the possible reactions are (see Figure 4)

3. QUANTIFYING THE H AND C FRACTIONATION IN THE REACTION CH4 + C2 H Methane is irreversibly lost from Titan’s atmosphere by a number of processes, including escape (hydrodynamic or sputtering)—in which the entire molecule is lost—and destruction of methane in situ by photolysis and chemistry. In the model of Wilson & Atreya (2009), escape accounts for 21% of methane loss, while photolysis accounts for 18% and chemistry for the largest part: 61%. The three primary isotopologues of methane are 12 CH4 , 13 CH4 , and CH3 D. As the bond energies are slightly different in each sub-species, there is a potential for fractionation to occur whenever methane is destroyed by either photolysis or chemistry. Fractionation by photolysis occurs due to slight shifts in the absorption cross-section of the isotopologues relative to 12 CH4 . For example, the cross-section of CH3 D is shifted 0.9 nm blueward relative to 12 CH4 (Nair et al. 2005), resulting in a D/H fractionation factor of 0.995 at Mars. 13 CH4 on the other hand has a zero-point energy much more similar to 12 CH4 , resulting in a predicted cross-section shift of only 0.04 nm, and therefore a negligible fractionation due to photolysis. In contrast, chemical fractionation factors can be large and significant, and have been studied extensively due to applications for the Earth’s atmosphere. The importance of these for Titan’s methane chemistry and an estimation of their magnitude for both D/H fractionation and 12 C/13 C fractionation are discussed in this section.

C2 H + 12 CH3 D → C2 H2 + 12 CH2 D [k1a] or → C2 HD + 12 CH3 [k1b] C2 H + 12 CH4 → C2 H2 + 12 CH3 [k12] C2 H + 13 CH4 → C2 H2 + 13 CH3 [k13]

R1A R1B R12 R13.

We may define the Kinetic Isotope Effect (KIE) ratios for this reaction as follows: q13 = k12 /k13 and qD = k12 /(k1a + k1b ). These fractions are > 1 if destruction of 12 CH4 proceeds faster than the alternative (the so-called normal KIE) and < 1 if the rare isotopologue is depleted faster. However, due to the difference in ground state zero-point vibrational energies (ZPEs) for the different methane isotopologues, the reaction rates k should differ by some degree and so q = 1.0. The ZPE shifts for CH3 D and 13 CH4 have been estimated by Nair et al. (2005) to be −599 ± 1 and −25.5 ± 0.5 cm−1 , respectively, relative to 12 CH4 (9834 cm−1 ), therefore the reaction rate for 13 CH4 should more closely resemble that of 12 CH4 than will CH3 D, and the KIE q should be closer to unity. The KIEs for methane in a variety of hydrogen abstraction reactions have been extensively measured for terrestrial atmospheric applications, focusing on CH4 + Cl and CH4 + OH, but CH4 + CN has also been investigated (see Table 3). These show that hydrogen KIE is typically tens of percent (qD ∼ 1.1–2.0) while carbon KIE is a few percent (q13 ∼ 1.01–1.08). The KIE for CH4 + C2 H has been studied only once at cold temperatures, and only for the larger hydrogen effect (Opansky & Leone 1996)—although a recent paper by Matsugi et al. (2011) added data points at warm temperatures (>290 K). The Opansky & Leone (1996) work, undertaken for application to Titan, measured the reaction rates of both CH4 and CD4 with C2 H down to 190 K. We have followed the proposal of Lunine et al. (1999) that an effective reaction rate for CH3 D can be assigned by adding

3.1. Fractionation by Atmospheric Chemistry Once the breakup of methane is initiated by photolysis, a series of reactions (e.g., Yung et al. 1984) leads to the formation of acetylene (C2 H2 ), which is itself easily photolyzed to ethynyl (C2 H). Once this radical has been produced, it quickly attacks methane removing an H-atom (hydrogen abstraction) resulting in C2 H2 and CH3 (see reaction series S5 of Wilson & Atreya 2009). However, this reaction is catalytic in the sense that C2 H is recycled many times from C2 H2 , and because the photolysis of C2 H2 can proceed at longer wavelengths than the photolysis of CH4 , this process effectively photosensitizes methane to longer-wavelength photolysis at altitudes below 700 km. This rapid reaction accounts for ∼46% of total methane destruction

kCH3 D = (0.25 × kCD4 ) + (0.75 × kCH4 ).

(1)

The KIE for hydrogen in this reaction is then qD = kCH4 /kCH3 D and is given in Table 3. 5


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Figure 4. Diagram of the several isotopic variants of the reaction CH4 + C2 H → CH3 + C2 H2 .

However, the carbon KIE for the reaction CH4 + C2 H is also of great interest because the ratio 12 C/13 C in methane may potentially be more informative about the history of Titan’s methane than D/H. This is because we have a larger degree of certainty in the starting value of 12 C/13 C—which varies very little throughout the solar system—compared to D/H, which varies widely. In the absence of laboratory measurements of q13 , we have therefore attempted to constrain the value through ab initio calculations, as described in the following subsection.

Table 4 Computed q13 Using UCCSD(T)/CBS + DBOC + TST Temperature

3.2. Estimation of H and C KIE Fractionation by Transition State Theory (TST) The success of ab initio calculations of chemical kinetics in hydrogen abstraction reactions, including kinetic isotope effects, has previously been demonstrated (Sellevåg et al. 2006; Temelso et al. 2006; Vandeputte et al. 2007). Here we apply highlevel ab initio methods and conventional transition state theory (TST; Eyring 1935) coupled with several one-dimensional tunneling corrections of successively increasing complexity to estimate the reaction rates (R12, R13). The reaction barrier and thermodynamics were determined using the UCCSD(T) (unrestricted coupled-cluster singles, doubles, and perturbative triples) level of theory employing correlation consistent basis sets extrapolated to the complete basis set (CBS) limit. ZPE and thermal corrections were included using UCCSD(T) with the ccpVTZ (correlation-consistent polarized valence triple-ζ ) basis set while a first-order correction to the adiabatic approximation was included using the diagonal Born-Oppenheimer Correction (DBOC) at the UCCSD/cc-pVTZ level of theory. The rate enhancement due to tunneling was determined by calculating the transmission coefficient through three different one-dimensional reaction potentials. The Wigner correction (Wigner 1932) models the reaction barrier as an untruncated parabola of a certain width. The Skodje–Truhlar approach (Skodje & Truhlar 1981) considers the reaction barrier as a truncated parabola of a defined width and height while the Eckart method (Eckart 1930) fits the zero-point-corrected energy of the reactants, transition state, and products to an asymmetrical Eckart function. A detailed description is given in the Appendix.

Tunneling Type

Mean

(K)

None

Wigner

S.-T.

Eckart

90.00 120.00 150.00 175.00 180.00 210.00 242.00 273.15 298.15 313.15 343.15 359.00

1.013 1.014 1.016 1.016 1.016 1.016 1.016 1.017 1.017 1.017 1.017 1.017

1.025 1.022 1.021 1.020 1.020 1.019 1.019 1.018 1.018 1.018 1.018 1.018

1.039 1.027 1.023 1.021 1.021 1.020 1.019 1.019 1.018 1.018 1.018 1.018

1.025 1.021 1.020 1.019 1.019 1.018 1.018 1.018 1.018 1.018 1.018 1.018

Standard Deviation

1.025 1.021 1.020 1.019 1.019 1.018 1.018 1.018 1.018 1.018 1.018 1.018

0.010 0.005 0.003 0.002 0.002 0.002 0.001 0.001 0.001 0.001 0.001 0.001

Our first objective was to attempt to replicate the hydrogen KIE for the reaction C2 H + CD4 → C2 HD + CD3 as measured by Opansky & Leone (1996), and which can be used to infer R1A + R1B. The results of our calculations are shown in Figure 5 using the various tunneling corrections, and generally show good agreement with the laboratory data. This validation was extremely important, as it showed that the model was working well and therefore we should feel confident to extrapolate our scheme to the unmeasured abstraction of hydrogen from 13 CH4 . Using the method of (Lunine et al. 1999; Equation (1)) we find qD = 1.148 ± 0.001 at 298 K and qD = 1.212 ± 0.002 at 175 K, from the mean and standard deviation of our TST calculations. We therefore proceeded to use TST to compute the rate for R13 and also the q13 , which is plotted in Figure 6, and values listed in Table 4. As expected, the KIE for carbon is much less than that for hydrogen, close to 2% across the range of calculated temperatures. Using the standard deviation between the curves as the error, we find q13 1.018 ± 0.001 at 298 K and 1.019 ± 0.002 at 175 K. In the next section, we will show the application of these qD and q13 estimates to modeling the evolution of Titan’s atmosphere. 6


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Figure 5. Hydrogen kinetic isotope effect for the reaction CD4 + C2 H → CD3 + C2 HD calculated using UCCSD(T)/CBS + DBOC(UCCSD/cc-pVTZ) + TST, with Wigner(W), Skodje–Truhlar(ST), and Eckart(E) tunneling corrections. Experimental work conducted by Opansky & Leone (1996) and Matsugi et al. (2011).

Here we consider three idealized cases for isotopic evolution of methane: in the first, methane is not resupplied to the atmosphere, and therefore an initial atmospheric reservoir is constantly being diminished by various depletion processes, even as the isotopic ratios evolve. This model resembles that of Cordier et al. (2008), but with additional fractionation processes. The second possibility considered is that methane is being replenished at a rate approximately equal to its destruction rate, presumably from some interior source, although an external flux would also be possible if it was constant on a timescale of less than a few 107 yr—the approximate timescale for loss of all atmospheric methane. The third possibility is that atmospheric methane is increasing steadily at the present day: this case is examined only semi-quantitatively, emphasizing the range of allowable isotopic values. Note that these three scenarios do not cover all the intermediate possibilities, for example, where there is some resupply of methane, but insufficient to match loss. Also, we make simplifying assumptions that the solar UV flux, aerosol, and chemical atmospheric composition have been constant over Titan’s history—to allow us to use column depletion rates from photochemical models that aim to match Titan’s atmosphere today. The effect of varying such fundamentals on our conclusions would likely be significant, and investigation of this parameter space would be a worthwhile topic for future study. Nevertheless, the idealized cases presented herein provide insight into the categories of allowable histories for Titan’s methane and are therefore valuable, especially as the available constraints are scarce.

TST

1.026

12 13 KIE q13 ( CH4/ CH4)

TST+W 1.024

TST+ST TST+E

1.022 1.020 1.018 1.016 1.014

150

200

250

300

350

Temperature (K)

Figure 6. Carbon kinetic isotope effect for the reaction 12/13 CH4 + C2 H → 12/13 CH + C H calculated using UCCSD(T)/CBS + DBOC(UCCSD/cc3 2 2 pVTZ) + TST, with Wigner(W), Skodje–Truhlar(ST), and Eckart(E) tunneling corrections.

4. MODELING EVOLUTION OF METHANE ISOTOPES ON TITAN Now having estimates of the various fractionation effects for hydrogen and carbon in methane, we can begin to numerically investigate how the D/H and 12 C/13 C may change over time and whether this allows us to make any deductions regarding the origin and history of Titan’s methane. The possibility of KIE fractionation in CH4 /CH3 D on Titan was first examined by Pinto et al. (1986), and subsequently by Lunine et al. (1999) who included nitrogen fractionation processes as well in an attempt to constrain the atmospheric evolution. More recently, Cordier et al. (2008) updated the methane D/H model by including the additional effect of fractionation by hydrodynamic escape due to the difference in mass between 12 CH4 (mass 16) and CH3 D (mass 17). Titan’s 13 C evolution has been modeled, beginning with Wong et al. (2002) who focused on exchange between CO and CH4 , and latterly by Mandt et al. (2009) who made an isotope evolutionary model that tracked three separate stable isotopic ratios (D/H, 12 C/13 C, 14 N/15 N) over time.

4.1. Model 1: No Methane Resupply In the case of no methane resupply, we will expand on the model of Cordier et al. (2008), considering possible fractionation that could occur during the five methane depletion/creation mechanisms identified by Wilson & Atreya (2009). These five processes are included in a generalization of the two-process model of Cordier et al. (2008), which we will use for both D/H evolution, but also 12 C/13 C evolution. We replace their Equation 7


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Table 5 Budget for Methane Depletion in Titan’s Atmosphere Type

F (109 ) (cm−2 s−1 )

q13

qD

F (109 )/q13 (cm−2 s−1 )

F (109 )/qD (cm−2 s−1 )

Direct photolysis Photosensitized (C2 H) Other chemical CH4 production Total:

Model 1A: L08 chemical loss and no escape 2.53a 1.000b 1.006b 7.97a 1.019c 1.212d 1.000e 1.000e 2.93a −1.00a 1.000e 1.000e 12.43 1.012f 1.128f

2.530 7.821 2.930 −1.000 12.281

2.515 6.576 2.930 −1.000 11.021


Model 1B: WA09 chemical loss and no escape 2.50g 1.000b 1.006b 5.00g 1.019c 1.212d 3.30g 1.000e 1.000e −1.00g 1.000e 1.000e 9.80 1.010f 1.100f

2.500 4.907 3.300 −1.000 9.707

2.485 4.125 3.300 −1.000 8.911


Model 1C: K09 chemical loss and no escape 1.000b 1.006b 2.71h 2.00h 1.019c 1.212d 5.69h 1.000e 1.000e −1.01h 1.000e 1.000e 9.39 1.004f 1.041f

2.710 1.963 5.690 −1.010 9.353

2.694 1.650 5.690 −1.010 9.024

Escape Total (Model 1A+ESC):

Model 1D: L08 and hydrodynamic escape 1.35a 1.359i 1.359i 13.78 1.038f 1.147f

0.993 13.275

0.993 12.014

Escape Total (Model 1B+ESC):

Model 1E: WA09 and hydrodynamic escape 2.90g 1.359i 1.359i f 12.70 1.073 1.150f

2.134 11.841

2.134 11.044

Escape Total (Model 1C+ESC):

Model 1F: K09 and hydrodynamic escape 3.48h 1.359i 1.359i 12.87 1.080f 1.111f

2.561 11.913

2.561 11.585

Notes. a Lavvas et al. (2008). b Nair et al. (2005). c This work, Section 3.2. d Opansky & Leone (1996). e Unknown: set to unity. f Weighted means rather than totals: 1/q eff ≡ Σ(Fi /qi )/Σ(Fi ) g Wilson & Atreya (2009). h Krasnopolsky (2009). i From exobase 12 C/13 C: see Mandt et al. (2012).

(3) (the Rayleigh distillation relation) with −1 i (Fi /qi ) log E(t) log R(t) = 1 − i Fi 1 −1 = 1− log E(t), qeff

explicitly write out the time dependence of E and R in each case. Within this general model, we allow six sub-models for different assumptions about the fractionation processes: (1A) no escape, chemical loss rates from Lavvas et al. (2008; hereafter L08); (1B) as in 1A, but using chemical loss rates from Wilson & Atreya (2009; hereafter WA09); (1C) as in 1A, but using instead chemical loss rates from Krasnopolsky (2009; hereafter K09). (1D), (1E), and (1F) are identical to 1A, 1B, and 1C, respectively, but with additional fractionation by hydrodynamic escape (Yelle et al. 2008; Strobel et al. 2010). In this paper, we do not compute models with sputtering loss of methane, which depends on methane density and is therefore not so easily included in the Rayleigh formula. In any case, the present day sputtering loss rate of 2.8 × 107 cm−2 s−1 (De La Haye et al. 2007) is much smaller than the rates due to photochemistry and possible hydrodynamic loss 109 –1010 cm−2 s−1 , therefore adds only a small perturbation to the results obtained with these assumptions. Fractionation due to sputtering is discussed further in a companion paper (Mandt et al. 2012). Table 5 gives the values for the Fi and qi used in each model. The Fi for each photochemical model were in general obtained

(2) (3)

where E(t) is the enrichment of the heavy isotope in methane (E13 (t) = f13 (t)/f13 (0) and ED (t) = fD (t)/fD (0)); f13 (t) and fD (t) are the (heavy/light) methane mixing ratios (13 CH4 /12 CH4 and 12 CH3 D/12 CH4 ) at time t; R(t) is the reservoir ratio, the ratio of initial methane (12 CH4 total methane) in the atmosphere to the amount of methane today; qi is the fractionation due to the ith process, defined such that q > 1 in the normal case where the lighter isotope is lost faster (inverse definition to Cordier et al. 2008); and Fi is the rate of methane destruction due to the ith process. Also note that we have defined 1/qeff as the single value that replaces the mean of (1/qi ) weighted by the importance of each process (Fi ): 1/qeff = i (Fi /qi )/ i Fi which simplifies both the formula and the concept. For convenience, we will hereafter cease to 8


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from the respective papers, although for L08 the requisite numbers were not given in the paper, but kindly supplied by P. Lavvas upon request. We note that there is significant variation in the values for Fi between the three models (see Figure 7), especially in the proportion of chemical depletion attributed to the reaction of CH4 with C2 H. In particular, only 26% of the chemical loss in K09 is attributed to the reaction of ethynyl with methane—the particular chemical reaction for which we have a measured KIE—compared to 60% in WA09 and 73% in L08, which therefore produces a very substantial change in the inferred fractionation times, as discussed later. In addition, all three photochemical models (K08, WA09, and K09) use 0.3 as the quantum yield for C2 H formation following acetylene photolysis, while recent laboratory measurements (Läuter et al. 2002; Kovács et al. 2010) indicate a yield closer to unity. Thus, all these models may significantly underestimate the C2 H production rate on Titan and similarly the importance of the subsequent C2 H + CH4 loss process. We do not consider the time evolution of the solar UV flux, which is expected to decrease over time as discussed in Mandt et al. (2012). This introduces an error (about 6% decrease in flux over 200 Myr) that affects the chemical loss rate by a similar amount. This error becomes significant when the KIE is very small (qeff 1) and/or E becomes significantly larger than unity, and is discussed further in Section 5. All six models use the same qi values. The q from photolysis follows estimates given by Nair et al. (2005), while q due to the KIE of hydrogen abstraction in methane by ethynyl (or “photosensitized” destruction, in the parlance of Wilson & Atreya 2009) was estimated earlier in this paper for both hydrogen and carbon KIE. The q for other chemical processes is unknown, and set to unity, as is that for chemical production of methane. This model deficiency due to unknown KIE magnitudes causes qeff to be underestimated, and so the enrichment timescales are overestimated by some undetermined amount and should be considered upper limits in this sense. In models 1D, 1E, and 1F we add an additional term for methane loss and fractionation by hydrodynamic escape (Fi ): 1.35, 2.90, and 3.48 (×109 cm−2 s−1 ) in L08, WA09, and K09, respectively, and is a significant fraction of the total methane loss. This loss is also assumed constant over time. For q we use the ratio (12 C/13 C)exobase /(12 C/13 C)surface derived from a new analysis of the INMS data set by Mandt et al. (2012) and is set identically for both heavy isotopologues CH3 D and 13 CH4 . Table 5 also gives both the total F from all processes, calculated separately for the six sub-models, and the qeff in each case. To actually proceed with obtaining predictions from the model, we need to impose some constraints. Our starting point will be to presume that we have knowledge of the enrichment in 13 C at the present day, i.e., E13 , from which we then have sufficient information ((q13 )eff ) to calculate R using Equation (3). We will justify this assumption, by arguing that 12 C/13 C has proved to be surprisingly constant throughout the solar system. Outer solar system planetary values have been derived for the atmospheres of Jupiter (92.6+4.6 −4.1 ; Niemann et al. 1998) and Saturn (91.8+8.4 ; Fletcher et al. 2009), in good agreement with −7.8 12 13 a mean C/ C derived for eight comets (90 ± 4; Hutsemékers et al. 2005) and even with the terrestrial inorganic standard value (Pee Dee Belemnite, 89.4; Werner & Brand 2001). In this work, we use the error-weighted mean of the Jovian, Saturnian, and cometary ratios: 91.3+2.8 −2.7 as the range of the initial 12 C/13 C. Similarly, we derived a mean ratio for present-day Titan from the observations: 89.7 ± 1.0 (Table 2). Combining

Table 6 Calculated Parameters for Methane Evolution Estimate

E13 a

Mean (low) Mean (cen) Mean (high)

1.000 1.017 1.051


ED

Mb (GT)

Model 1A: L08 no escape 1.0 0 4.1 56 64.1 1142

1.000 1.173 1.603

0 368731 7516647

1.000 1.017 1.051

Model 1B: WA09 no escape 1.0 0 5.9 112 186.6 4259

1.000 1.174 1.607

0 581518 2.21 × 107


1.000 1.017 1.051

Model 1C: K09 no escape 1.0 0 69.7 1646 2.75 × 105 6.59 × 106

1.000 1.180 1.629

0 8185862 3.28 × 1010


1.000 1.017 1.051

Model 1D: L08 + escape 1.0 0 1.6 9 3.9 47

1.000 1.061 1.190

0 69521 343415


1.000 1.017 1.051

Model 1E: WA09 + escape 1.0 0 1.3 5 2.1 19

1.000 1.033 1.101

0 33696 129322


1.000 1.017 1.051

Model 1F: K09 + escape 1.0 0 1.3 4 2.0 16

1.000 1.023 1.069

0 30320 113473

R

t (Myr)

Notes. a Mean (low) enrichment limited to 1.00 to prevent lightening of carbon isotopic ratio, which is not possible in this model. b Mass of carbon from methane produced during priori depletion history.

these gives us a range for the present-day enrichment in E13 of 1.017+0.034 −0.032 , or effectively 1.000 < E13 < 1.051 as we do not have any processes in our model that increase the carbon isotopic ratio (i.e., cause E to decrease). The alternative option, beginning with a presumed D/H value, is much riskier, since D/H varies by over an order of magnitude from the giant planets to terrestrial planets and comets. Therefore, we will compute the ED afterward based on timescales derived from E13 . The assumed range of E13 values are shown in Column 2 of Table 6. The corresponding R, given in Column 3, gives us the amount of methane implied in the original methane atmosphere in units of the current methane amount (see also Figure 8). We can then apply Equation (5) of Cordier et al. (2008), with slight modification, to compute the time for this change (depletion) to occur: MCH4 (R − 1) t= . (4) mCH4 i Fi In this formula MCH4 is the present column mass of CH4 = 189.1 g cm−2 , calculated by us by direct quadrature of our atmosphere model; mCH4 is the mass of a CH4 molecule = (16.04/NA ) g, where NA = 6.022 × 1023 ; and Fi has been previously defined. (R − 1) is used rather than R, so that the present inventory is not included in the calculation. The results are given in Column 4 of Table 6. We then use R to compute ED using Equation (3) and replacing the (q13 )eff with (qD )eff . The estimate of the deuterium enrichment can be used to compute the original D/H in Titan’s atmosphere, if a present day-value is assumed, e.g., from Table 2. We note that the maximum computed enrichment in D/H is 9


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Figure 7. Pie charts showing the relative importance of the several methane-loss processes in the six model cases considered.

Figure 8. Calculated relation between initial atmospheric methane reservoir (present atmosphere = 1) and enrichment of 13 C for six chemistry/escape models with no methane resupply. Dashed lines show the values in Table 6.

∼1.65, much less than required to enrich a Saturnian D/H ((1.6 ± 0.2) × 10−5 ; Fletcher et al. 2009) to any realistic Titan value, which is typically one order of magnitude greater (see Table 2). Finally, we have computed the mass of carbon that has been processed during the methane depletion from M = Matm (R − 1) × (12.00/16.04), where Matm is the mass of methane in the present atmosphere, calculated by us to be 1.576 × 105 GT (gigatons). The implications of the D/H and carbon mass ranges are discussed further in Section 5. 4.2. Model 2: Steady State Methane

destruction by all mechanisms. In the absence of resupply all Titan’s methane is predicted to vanish in the next ∼20 Myr, therefore this model avoids the problem posed by our observing a rather unique and unsustainable point in Titan’s history. It is possible to show that in this case, a simple relation holds between the isotopic ratio in the atmosphere and the isotopic ratio in the supply flux as follows. We consider two isotopes of methane: “light” (L, i.e., 12 CH4 ) and “heavy” (H, i.e., either 13 CH4 or 12 CH3 D), with atmospheric molecular populations NL (t) and NH (t), and total loss rates from all mechanisms lL (t) and lH (t). The general time-evolution equations for the populations of each molecular species are

In the second scenario, we assume that methane is being replenished to the atmosphere at a rate equivalent to its net

dNL f0 φ(t) = − lL (t)NL (t) + dt 1 + f0 10

(5)


dNH 1 = − lH (t)NH (t) + φ(t), dt 1 + f0

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Model 1). In this case, the supply of methane occurs faster than depletion, and therefore any effect of isotopic enrichment is washed out by the greater supply of fresh methane. In this model, the isotopic ratios are maintained arbitrarily close to the supply values, and must exceed (for 13 C, or be lesser than, for D) the steady state isotopic ratios: (12 CH4 /13 CH4 )supply /(q13 )eff < (12 CH4 /13 CH4 )atm < (12 CH4 / 13 CH4 )supply and (CH3 D/CH4 )supply < (CH3 D/CH4 )atm < (qD )eff (CH3 D/CH4 )supply .

(6)

where φ(t) is the total methane supply flux (e.g., from an internal, or perhaps external source) with constant isotopic ratio f0 . Therefore when supply balances loss, the relation φ(t) = lL (t)NL (t) + lH (t)NH (t) must hold. If in addition, the loss processes are unchanging with time (or changing on long timescales compared to equilibration of molecular species) and the supply constant then nothing perturbs the populations of both isotopologues and dNL /dt = dNH /dt = 0. Therefore, lL NL =

f0 φ(t) 1 + f0

(7)

lH NH =

1 φ(t). 1 + f0

(8)

5. DISCUSSION In the previous section we considered three idealized scenarios for methane longevity in Titan’s atmosphere: either no resupply of methane, in which case the present inventory will be entirely depleted in a few 10s of Myr (Model 1); replenishment matching the destruction rate (Model 2); or resupply exceeding the destruction rate (Model 3). Distinguishing between these scenarios is not possible based on the currently available data regarding methane isotopes alone: all models are potentially compatible with the zero or small enrichment seen in 13 CH4 , while the original D/H value is too uncertain to provide any constraint. Therefore, we can only proceed by drawing in evidence from other sources to examine the feasibility of each scenario.

Dividing these last equations yields lH NL = f0 ⇒ f = f0 /qeff , NH lL

(9)

as qeff is just the total effective fractionating factor we met earlier. In other words, the (light/heavy) isotopic ratio ((D/H)−1 or 12 C/13 C) stabilizes under steady resupply conditions at a value equal to the isotopic ratio in the source flux, divided by the effective (weighted) fractionation factor, also assumed constant. If, however, the loss conditions change over time (e.g., changing solar flux), then the supply must also change to maintain balance. Assuming the 12 C/13 C of supply is the outer solar system value we estimated earlier (f0 = 91.3) we can use the six (q13 )eff derived for Model 1 (see Table 5) and Equation (9) to derive six steady state predictions for 12 C/13 C today: 90.2 (2A: L08, no escape); 90.4 (2B: WA09, no escape); 90.9 (2C: K09, no escape); 88.0 (2D: L08, no escape); 85.1 (2E: WA09, escape), and 84.5 (2F: K09, escape). The values for the first three (non-escape) models are very compatible with our estimated Titan mean value (89.7 ± 1.0) while the values for the escape models are compatible with individual measurements in Table 2 other than GCMS. Similarly, (CH3 D/CH4 )atm = (qD )eff (CH3 D/ CH4 )supply , so if the present day value is close to terrestrial, then the source must be some ∼4%–15% lower in D/H (depending on model) to allow for subsequent enrichment. In this model scenario, it is not possible to put any constraint on the amount of time for which methane has been resident in the atmosphere because once a steady state has been reached there is no evolution over time. Note that this steady state condition has previously been advocated by Jennings et al. (2009), who measured a zero-enrichment (terrestrial) value for 12 C/13 C in ethane and contrasted it with the enrichment seen in 13 CH4 . They suggested that a carbon KIE with q13 1.1 could produce a stable enhancement in methane’s 12 C/13 C consistent with the (preliminary) GCMS value of 82.3 ± 1.0 (Niemann et al. 2005). Since this work, both the enhancement in the present-day Titan 13 CH4 and the q13 have been substantially revised closer to unity, allowing the scenario to remain self-consistent.

5.1. Model 1: No Resupply of Methane This model would be strongly favored if, for example, Titan turned out to have no present endogenic activity, as argued by Moore & Pappalardo (2011). Contrariwise, this creates the Copernican problem mentioned previously that we are observing Titan during a unique period in its history, when the final remnants of a (possibly much larger) methane inventory are finally vanishing. Using the probabilistic version of the Copernican argument advocated by Gott (1993) for continuous events, there is a 50% probability that Titan had no more than twice the present methane present in the atmosphere, and 90% probability that the initial inventory was no more than 10× the present amount, prior to the beginning of the current depletion cycle. This initial input may have come from a phase of episodic outgassing as envisioned by Tobie et al. (2006), with the most recent event occurring 400–1400 Myr ago. The results from our Model 1 indicate a very large range of possible enrichment times based on the measured present and assumed initial values for 12 C/13 C. Using the mean values for enrichment, we find timescales of 4–9 Myr if hydrodynamic escape is occurring, increasing to 56, 112, or 1646 Myr based on the no-escape photochemistry of L08, WA09, and K09, respectively. Using maximum possible enrichments increases these times dramatically: up to 47 Myr for hydrodynamic escape, and 1.1, 4.3, and 6600 Gyr for the chemistry of L08, WA09, and K09—the last of which is clearly implausible. Note that while the models of L08 and WA09 agree more closely on the chemical depletion rates compared to K09, the models of WA09 and K09 agree more closely on the escape rate than does L08. This reinforces the earlier observation that our model is extremely sensitive to input parameters, and better agreement between the photochemical models for Fi due to C2 H + CH4 in particular would be helpful. A drawback exists in our simple model, namely, the lack of time dependence in the solar flux, and hence destruction rate. Calculations using a time-dependent model of methane depletion (Mandt et al. 2012) indicate much shorter timescales:

4.3. Model 3: Increasing Methane The third possibility is that atmospheric methane is in a non-steady state, but increasing, rather than decreasing (as in 11


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208 and 586 Myr, respectively, for E = 1.017 and E = 1.051 using the chemistry of K09 and no escape (cf. 1.6 and 6600 Gyr above). Therefore, we conclude that using the simple analytical Rayleigh distillation relationship in the presence of timedependent processes can result in large errors, and should only be used for order-of-magnitude estimates when the enrichment times are short compared to the timescales of time-dependent processes. Although many uncertainties exist, it does seem possible to conclude that models that include escape can achieve the inferred present levels of carbon fractionation in short times, no more than one present methane lifetime (∼20 Myr). On the other hand, fractionation by chemistry alone will take much longer, hundreds to thousands of Myr, based on our model results for most likely E13 (56–1646 Myr) and the 586 Myr upper limit derived using the time-dependent model. We note that Hörst et al. (2008) have estimated that ∼300 Myr is required to create the present inventory of atmospheric CO, which requires both the presence of carbon (from methane) and oxygen (O+ , deposited at the top of the atmosphere). If this model is interpreted as a minimum time for the existence of methane in the atmosphere, then models that allow fractionation during hydrodynamic escape (1D–1F) must be excluded. Additional considerations may include the age of the surface, if the crust underwent significant upheaval and resurfacing during a major outgassing event and has subsequently aged only due to cratering and mechanical erosion. Neish & Lorenz (2012) recently estimated the surface age in two slightly different impact models as 200 Myr or 1000 Myr: incompatible once again with our hydrodynamic escape models, which fractionate too rapidly. We therefore arrive at an estimated range for the methane duration of 300–600 Myr, bounded by the CO chemistry and the time-dependent model for methane depletion, and contained inside the surface age range. We note that the probabilistic Copernican argument (Gott 1993), however, implies that there are only 6% and 3% likelihoods respectively of observing the last ∼20 Myr of a 300 or 600 Myr continuous process. This is a quantitative way of saying that it appears unlikely that we are observing a narrow period in Titan’s history when the last methane of a much larger initial inventory is disappearing. Regarding the implied amount of carbon processed during the methane depletion, we find most-likely amounts (in GT) of 370,000 (1A), 580,000 (1B); 8,200,000 (1C); 70,000 (1D); 34,000 (1E); 30,000 (1F). We can infer that this total carbon is ultimately deposited on the surface, although we must correct the latter three totals for carbon escape, arriving at: 63,000 (1D); 26,000 (1E); 22,000 (1F). These may be compared with the surface deposits of carbon estimated by Lorenz et al. (2008): 16,000–160,000 GT (lakes) and 160,000–640,000 GT (dunes). Therefore, considering these six cases where methane is not resupplied to the initial atmosphere, cases 1D–1F with hydrodynamic escape do not produce sufficient methane to explain the minimum surface inventory (176,000 GT). Case 1C also probably produces too much, while 1A and 1B fall inside the nominal range.

features (“cryovolcanic flows”) that could supply methane, trapped originally in Titan’s interior as clathrate hydrates, to the atmosphere (e.g., Fortes et al. 2007; Choukroun et al. 2010). Some early putative volcanic features from Cassini mapping (Sotin et al. 2005; Lopes et al. 2007) have later been disproven, while more recent evidence (e.g., Nelson et al. 2009; Wall et al. 2009) has been disputed (Soderblom et al. 2009; Moore & Pappalardo 2011). A full discussion of the arguments for and against endogenic activity on Titan is beyond the scope of this paper, and changes rapidly due to discoveries made by the ongoing Cassini mission. Suffice it to say that while such phenomena present significant advantages for explaining the present atmosphere, the evidence as yet is insufficient to sustain or disprove the hypothesis. The case of the total surface carbon deposits (176,000– 800,000 GT as discussed in previous section) is potentially more serious, as the steady state scenario processes about 120,000 GT of carbon (the current atmospheric inventory) every 20 Myr, or 6000 GT Myr−1 . Therefore, durations of this state beyond ∼130 Myr produce in excess of 800,000 GT and require explanations for the missing surface carbon, possibly reburied or otherwise incorporated into the crust, as advocated by Wilson & Atreya (2009). In summary, while Model 2 is compatible with available isotopic evidence from Titan’s methane, resolving this scenario with the other existing evidence, especially the surface hydrocarbon deposits, remains to be proven. 5.3. Model 3: Increasing Methane It is unlikely that Titan’s methane is in an exact steady state with supply equaling depletion (Model 2). If methane resupply is ongoing, it is likely sporadic, or else either decreasing (variant of Model 1) or increasing (Model 3) at the present time. Steadily increasing atmospheric methane could occur due to the gradual brightening of the Sun, slowly thawing out Titan’s crust. In this case, the surface would also likely be active and young, and similar evidence for endogenic activity might be required as in Model 2. However, this scenario could help with the issue of carbon buildup at the surface, if methane was less abundant in the past. At the present time, isotopic information from methane can neither confirm not deny such a hypothesis. 6. CONCLUSIONS In this paper, we present new measurements of the D/H and 12 C/13 C in Titan’s methane from Cassini CIRS infrared spectroscopy. We have modeled simultaneously, for the first time, the emissions of all four detected methane isotopologues: 12 CH4 , 13 CH4 , CH3 D, and 13 CH3 D. Following the retrieval of temperature from 12 CH4 , and the abundances of the other three species, we calculate 12 C/13 C = 86.5 ± 8.2 and D/H = (1.59 ± 0.33) × 10−4 . We have also advanced the study of chemical fractionation of isotopes (KIE) in methane by studying the important hydrogen abstraction reaction, CH4 + C2 H → CH3 + C2 H2 , using transition state theory. This has led us to confirm earlier experimental findings for the hydrogen KIE, and to a first estimate for the carbon KIE q13 over a range of temperatures. We then added these fractionation effects to a simple model for isotopic enrichment in Titan’s atmosphere and attempted to constrain possible ages for Titan’s atmospheric methane based on assumptions and measurements for the initial and final 12 C/13 C. In this modeling we assumed that the initial 12 C/13 C equaled that recently measured for Saturn, and use a mean

5.2. Model 2: Steady State Methane As we have previously stated, Model 1 presents a significant problem in that it requires us to be observing a unique period of Titan’s history (the Copernican Problem). Therefore, much attention has been focused on mechanisms for replenishing the methane, in particular a search for endogenic eruptive 12


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received additional support from the Leverhulme Trust. The authors thank Panayotis Lavvas for supplying methane column depletion rates predicted by his photochemical model (Lavvas et al. 2008). Facility: Cassini

12 C/13 C from three Cassini–Huygens instruments. We note, however, that a major source of error in our models at the present time is the lack of knowledge for q13 and qD for many chemical reactions (i.e., all reactions other than the hydrogen abstraction by ethynyl) that contribute significantly to the depletion of methane. Under the assumption that Titan’s methane is not actively being replenished (our Model 1), our model indicates probable prior lifetimes of 56–1646 Myr (depending on chemical model) for Titan’s atmospheric methane, in the absence of escape, or likely within the last 10 Myr if hydrodynamic escape is occurring. If longer than this time, the 12 C/13 C would deviate from the original value (presumably Saturnian) by an amount greater than observed. However, timescales less than several hundreds of Myr appear inconsistent with other age constraints: (1) a most-recent predicted episode of major outgassing of 400 Myr from an interior evolution model (Tobie et al. 2006); (2) an inferred minimum duration for methane in the atmosphere of 300 Myr, based on the production of CO (Hörst et al. 2008); (3) surface age estimates, if interpreted as correlated with the last major eruptive activity releasing significant atmospheric methane, of 200–1000 Myr (Neish & Lorenz 2012). Based on these constraints, we infer that the models that include hydrodynamic escape (1D, 1E, 1F) in particular are forbidden, as they act to fractionate methane’s carbon much too quickly, while other models lacking methane replenishment (1A, 1B, 1C) but without hydrodynamic escape are compatible. If on the other hand methane is being replenished at the present day, at (Model 2) or above (Model 3) the depletion rate, then the methane isotopic measurements are compatible and not constraining. In Model 1 we can estimate the original D/H based on the depletion timescale, in Model 2 we also infer the steady-state enrichment, while in Model 3 we can constrain a range of original D/H values. It is not possible in any of these scenarios to enrich D/H from a low value similar to Saturn (1.6±0.2×10−5 ; Fletcher et al. 2009) to a Titan value some order of magnitude higher (see Table 2). This confirms arguments by Mousis et al. (2002a, 2002b), Cordier et al. (2008), and Mousis et al. (2009) that Titan’s D/H in methane was enriched relative to the giant planets prior to incorporation into Titan, likely by hydrogen exchange between gas (H2 ) and ices (including CH4 ). We also note that a maximum timescale of ∼130 Myr is implied by the amounts of surface carbon detected by Cassini, requiring significant reburial of ethane and other hydrocarbon products in the crust if the atmospheric methane has been produced continuously at or near replacement levels for longer than this time. It is unlikely that isotopic studies, even if much more accurate measurements and models are eventually obtained, will be sufficient alone to definitively distinguish between the possible histories for Titan’s methane, although they can provide important constraints. Rather, a combined case of evidence must be built from isotopic studies along with studies of surface age and morphology, and more general models of the atmospheric origin, chemistry, evolution, and escape to advance our understanding toward finally answering the perplexing riddle posed by the existence and persistence of Titan’s methane.

APPENDIX NUMERICAL APPROACH TO ESTIMATION OF KIE FRACTIONS In order to accurately calculate KIEs, we used (1) ab initio methods to determine the reaction barrier and energy, (2) conventional transition state theory to determine the reaction rate, and (3) one-dimensional tunneling corrections to incorporate the role of quantum tunneling on the reaction rates. A.1. Calculating the Reaction Barriers and Energies Modeling hydrogen abstraction barriers involving radicals is challenging because of spin contamination in many ab initio wavefunction methods (Andrews et al. 1991), and large selfinteraction errors in density functional methods (Zhang & Yang 1998), as demonstrated for the case of ethynyl (Temelso et al. 2006) and other radical species (Chuang et al. 2000; Vandeputte et al. 2007). A proven way to overcome spin contamination is to use the robust CCSD(T) (Raghavachari et al. 1989) method with a large basis set. Here, we fully optimized the geometry of the stationary points (reactants, transition state, and products) and determined their harmonic vibrational frequencies using the unrestricted CCSD(T) with cc-pVTZ basis set. We utilized the analytic gradient and Hessian code implemented in the CFOUR package (Stanton et al. 2010). Employing the cc-pVXZ (X = T, Q, 5, . . . , ∞) (Dunning 1989; Kendall et al. 1992) basis sets has the great advantage that the UCCSD(T)/cc-pVXZ electronic energies systematically converge to their infinite (complete) basis set (CBS) limit with increasing cardinal number X. The UCCSD(T)/cc-pVXZ energy is decomposed into its reference Hartree–Fock energy (EXHF ) which converges exponentially to the HF CBS limit HF (ECBS ) and the correlation energy (EXC ) which converges as HF −3 X to the E CCBS . We extrapolated EXHF (X = T, Q, 5) to the ECBS by fitting to Equation (A1) (Feller 1993) and the EXC (X = T, Q) using Equation (A2) (Helgaker et al. 1997): HF −CX E HF CBS = E X + Be

E CCBS =

X3 E CX − (X + 1)3 E CX+1 X3 − (X + 1)3

C E CBS = E HF CBS + E CBS .

(A1)

(A2) (A3)

The extrapolated E CBS (A3) was used to calculate benchmark quality reaction barrier and energy within the framework of the Born-Oppenheimer (BO) approximation. Table 7 shows the convergence of the UCCSD(T) reaction barrier and energy toward the CBS. A first-order correction to the BO energy was included using the DBOC using UCCSD/cc-pVTZ at the UCCSD(T)/cc-pVTZ optimized geometry. DBOC is a firstorder adiabatic correction to the BO approximation, and instead of assuming that nuclei are infinitely heavy, it takes into account the finite mass of the nuclei (Handy et al. 1986). While it is a small correction (Valeev & Sherrill 2003), its mass dependence is important in calculating reaction rates and KIEs. See Table 8.

C.A.N., R.K.A., D.E.J., P.N.R., and F.M.F. were supported by the NASA Cassini Mission during the period in which this work was conducted. N.A.T. and P.G.J.I. received support for their portion of this work from the UK STFC, and N.A.T. 13


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constant. κ(T ) is unity in the classical (no tunneling) case; however, in reactions involving light atoms like hydrogens, quantum mechanical tunneling through the reaction barrier can increase the transmission coefficient, and hence enhance the reaction rates. The effect of tunneling is typically most pronounced at low temperatures where it can be the dominant reaction mechanism.

Table 7 Convergence of the Reaction Barrier(ΔE ‡ ) and Energy(ΔE) to the CBS (in kcal mol−1 )

UCCSD(T)/cc-pVTZ UCCSD(T)/cc-pVQZ UCCSD(T)/CBS

ΔE ‡

ΔE

1.697 1.480 1.288

−27.046 −27.334 −27.602

A.3. Correcting for Quantum Mechanical Tunneling

Table 8 Computed Thermochemical Reaction Profile for R2 and R3 12 CH +C H 4 2

13 CH +C H 4 2

CD4 +C2 H

235.358

216.919

1.288 1.320 0.563 0.595 2.136 3.194

1.288 1.293 0.924 0.930 2.612 3.694

Ignoring all tunneling pathways but the one dimension along the reaction coordinate, we have estimated the tunneling transmission coefficient, κ(T ). The simplest way to account for the effect of quantum mechanical tunneling on reaction rates is to use the Wigner approximation (Wigner 1932). It assumes that the reaction proceeds along a parabolic potential whose width is defined by the imaginary frequency (ν˜ ‡ ) of the transition state. The Wigner transmission coefficient κw (T ) is

Imaginary mode ||˜ν ‡ ||(cm−1 )

238.954 Reaction barriera,b,c

ΔE ‡ b ΔE ‡ [+DBOC] ΔE ‡ (0 K) ΔE ‡ (0 K)[+DBOC] ΔG‡ (175 K)[+DBOC] ΔG‡ (298.15 K)[+DBOC]

1.288 1.319 0.564 0.596 2.130 3.184

κw (T ) = 1 +

Reaction energya,b,c ΔE b

ΔE[+DBOC] ΔE(0 K) ΔE(0 K)[+DBOC] ΔG(175 K)[+DBOC] ΔG(298.15 K)[+DBOC]

−27.602 −27.622 −29.552 −29.572 −29.600 −29.624

−27.602 −27.623 −29.528 −29.549 −29.578 −29.606

1 24

hcν˜ ‡ kb T

2 ,

(A5)

where ν˜ ‡ is the magnitude of the imaginary frequency along the reaction coordinate in wavenumbers and c is the speed of light. This simplification is valid only when kB T hc˜ν ‡ . Using our UCCSD(T)/cc-pVTZ ν˜ ‡ of 239 and 235 cm−1 for the 12 CH4 –C2 H and 13 CH4 –C2 H transition states, we can see that the Wigner tunneling correction will work well only for T 340 K. The Skodje–Truhlar (Skodje & Truhlar 1981) tunneling correction is a more robust approximation that is valid over a large temperature range because it factors in not only the width of the barrier but also its height. The Skodje–Truhlar transmission coefficient for an exothermic reaction with α β like ours is given by

−27.602 −27.629 −28.732 −28.760 −28.801 −28.894

Notes. a In kcal mol−1 . b UCCSD(T)/CBS electronic energy. c [+DBOC] included at the UCCSD/cc-pVTZ level; (0 K) implies the inclusion of ZPVE corrections using UCCSD(T)/cc-pVTZ vibrational frequencies; (175 K) and (298.15) include thermal corrections on top of the DBOC and ZPVE corrections.

κST (T ) =

A.2. Determining Reaction Rates Using Transition State Theory

β βπ/α ‡ − e[(β−α)ΔE (0)] , sin(βπ/α) α − β

(A6)

2π where α = h˜ , β = kb1T , and ΔE ‡ (0) is the zero-point energy ν‡ corrected barrier. Unlike the Wigner and Skodje–Truhlar corrections which assume a parabolic form for the reaction potential, the Eckart tunneling method (Eckart 1930) fits the zero-point-corrected energies of the reactants, transition state, and products to an asymmetric Eckart function. It models the reaction potential more realistically and it has analytical solutions. For a full description of the Eckart tunneling correction, the reader is encouraged to see Johnston & Heicklen (1962) or Vandeputte et al. (2007). A summary of the q13 KIE using the different tunneling approximations is shown in Table 4. The three zero-curvature tunneling methods described above assume tunneling occurs in one dimension along the reaction path. There exist other more complicated ways of calculating the reaction rate (such as variational transition state theory) and accounting for multi-dimensional tunneling. Truhlar et al. (1996) provide a good summary of these approaches.

The above scheme provides very accurate electronic energies for the reactants, transition state, and products and one can easily calculate the classical barrier (ΔE ‡ ) to the hydrogen abstraction from methane and its isotopologues by ethynyl radical. To get further thermodynamic information, we need to calculate the partition function for the reactants and transition state and employ TST (Eyring 1935) to get reaction rates. First, the zero-point vibrational energy correction (ΔZPVE), is estimated simply as one-half of the sum of the harmonic vibrational frequencies and added to the classical barrier to give ΔE ‡ (0). We obtain finite temperature corrections to the enthalpy ΔH (T ) ΔS(T ) using the vibrational, rotational, and translational partition functions in conjunction with the harmonic oscillator, rigid rotator, and particle-in-a-box models (Mcquarrie 2000; Cramer 2004). Adding these thermodynamic corrections to the classical barrier gives the enthalpy (ΔH ‡ ) and Gibbs free energy (ΔG‡ ) of activation. Given the these quantities, one can use TST to get the reaction rate [k(T )]: kb T −ΔG‡ kb T −ΔH ‡ +T ΔS ‡ = κ(T ) e RT e RT . (A4) k(T ) = κ(T ) h h

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