Methane, Carbon Monoxide, and Ammonia in Brown Dwarfs and Self-Luminous Giant Planets
DDraft version September 3, 2018
Preprint typeset using L A TEX style emulateapj v. 05/12/14
METHANE, CARBON MONOXIDE, AND AMMONIA IN BROWN DWARFS AND SELF-LUMINOUS GIANTPLANETS
Kevin J. Zahnle
NASA Ames Research Center, MS-245-3, Moffett Field, CA 94035; [email protected]
Mark S. Marley
NASA Ames Research Center, MS-245-3, Moffett Field, CA 94035; [email protected]
Draft version September 3, 2018
ABSTRACTWe address disequilibrum abundances of some simple molecules in the atmospheres of solar compo-sition brown dwarfs and self-luminous extrasolar giant planets using a kinetics-based 1D atmosphericchemistry model. Our approach is to use the full kinetics model to survey the parameter space witheffective temperatures between 500 K and 1100 K. In all of these worlds equilibrium chemistry favorsCH over CO in the parts of the atmosphere that can be seen from Earth, but in most disequilibriumfavors CO. The small surface gravity of a planet strongly discriminates against CH when compared toan otherwise comparable brown dwarf. If vertical mixing is like Jupiter’s, the transition from methaneto CO occurs at 500 K in a planet. Sluggish vertical mixing can raise this to 600 K; but clouds ormore vigorous vertical mixing could lower this to 400 K. The comparable thresholds in brown dwarfsare 1100 ±
100 K. Ammonia is also sensitive to gravity, but unlike CH /CO, the NH /N ratio isinsensitive to mixing, which makes NH a potential proxy for gravity. HCN may become interestingin high gravity brown dwarfs with very strong vertical mixing. Detailed analysis of the CO-CH reaction network reveals that the bottleneck to CO hydrogenation goes through methanol, in partialagreement with previous work. Simple, easy to use quenching relations are derived by fitting to thecomplete chemistry of the full ensemble of models. These relations are valid for determining CO,CH , NH , HCN, and CO abundances in the range of self-luminous worlds we have studied but maynot apply if atmospheres are strongly heated at high altitudes by processes not considered here (e.g.,wave breaking). INTRODUCTION
Disequilibrium chemistry has been known in Jupiter’satmosphere for several decades (Prinn and Barshay 1977;B´ezard et al. 2002) and has been expected and sus-pected in brown dwarf atmospheres from the time of theirdiscovery (Fegley and Lodders 1996; Noll et al. 1997;Saumon et al. 2000). The most famous disequilibriumis an overabundance of CO relative to CH . This oc-curs in Jupiter and brown dwarfs when CO is dredgedup from deep, hot layers of the atmosphere more quicklythan chemical reactions with ambient hydrogen can con-vert it to CH . It is to be expected that similar pro-cesses take place in young, self-luminous extrasolar giantplanets. But the apparent paucity of methane in theatmospheres of planets with effective temperatures com-parable to those of methane-rich T-type brown dwarfswas underpredicted and has been met with surprise.Among field brown dwarfs methane appears—bydefinition—at the L to T-type transition where theirnear-infrared colors turn to the blue at effective tempera-tures near 1200 K (Kirkpatrick 2005). The first directlyimaged planets, however, were found to have effectivetemperatures below 1200 K and yet their near-infraredspectra were devoid of signs of methane in K band (e.g.,Barman et al. 2011a). The best example is HR8799c:despite an effective temperature near 1100 K, in bothlow and high resolution spectra—particularly the highresolution spectrum taken by Konopacky et al. (2013)—methane has gone missing. Surface gravity is the defining difference between ex-trasolar giant planets and brown dwarfs, and to dateis the only proven difference, although there are greathopes for metallicity. Field brown dwarfs (hereafter BDs)have high surface gravities ( g of order 10 cm/s ) andtherefore very compressed scale heights. Extrasolar gi-ant planets (EGPs) have modest surface gravities ( g oforder 10 cm/s ) and extended scale heights (e.g., Bur-rows et al. 1997; Saumon & Marley 2008) . A BD and aself-luminous EGP of the same composition and the sameeffective temperature will have similar optical depths as afunction of temperature, at least in the absence of clouds;i.e., the function T ( τ ) is roughly the same. However, be-cause p ∝ τ g , the BD has a much higher pressure ata given optical depth than does the EGP, and a BD ismuch cooler than the EGP at a given pressure. Becauselower temperatures and higher pressure favor CH in itsstruggle with CO, it has been pointed out that CH willbe more easily seen and CO less easily seen at lower grav-ity in BDs (Hubeny and Burrows 2007). Hence Barmanet al. (2011a) suggested that the even lower gravity ofHR8799b might be why no methane is seen in it. Weagree. We will verify that the dependence on g is strongand leads to qualitatively different outcomes for BDs andself-luminous EGPs, a result implicit in previous workbut far from fully appreciated. We do not intend to wade into the nomenclature battles here.For our purposes companion objects below ∼
13 M J are planets. a r X i v : . [ a s t r o - ph . E P ] O c t Zahnle & MarleyPrevious studies of carbon speciation in BDs and EGPshave mostly focused on a few particular objects (Saumonet al. 2006; Geballe et al. 2009; Barman et al. 2011a,b;Moses et al. 2011; Line et al. 2011), or on hot highlyirradiated EGPs (Moses et al. 2011; Visscher 2012), orused one of several quench approximations culled fromthe literature (Fegley and Lodders 1996; Saumon et al.2000; Lodders and Fegley 2002; Hubeny and Burrows2007), or various combinations of the above (Cooper andShowman 2006; Visscher and Moses 2011; Moses et al.2013a,b). The basic idea is that CH and CO will oftenbe seen in disequilibrium because the chemical reactionsthat would enforce equilibrium don’t have time enoughto take place while the gas is cool (Prinn and Barshay1977). The disequilibrium composition that results isdescribed as “frozen-in” or “quenched.”Quenching has been widely used to quantify discus-sions of CO-CH and N -NH disequilibria in a widerange of astrophysical problems, dating at least back toPrinn and Barshay (1977)’s study of CO in Jupiter. Ear-lier discussions of quenching can be found with respect tothe N -O -NO system that is important in thunderbolts(Chameides et al. 1979), meteor entry and rocket r¨eentry(Park and Menees 1978), and explosions in Earth’s at-mosphere (Zel’dovich and Raizer 1967). In the quenchapproximation, the disequilibrium composition that onecan observe is approximated by the equilibrium composi-tion when the relevant chemical reaction time scale t chem equals the relevant cooling timescale, which if due to mix-ing can be written t mix (Prinn and Barshay 1977). Thereare many different prescriptions for defining t chem and t mix that we will discuss below. The history of quenchschemes for jovian planets, exoplanets, and brown dwarfshas been comprehensively recounted in a series of recentpapers by Moses and colleagues (Moses et al. 2010, 2011;Visscher and Moses 2011).Here we do something different. We use a 1D chemi-cal kinetics code coupled to p - T profiles from a detailed1D atmospheric structure code to compute a galaxy ofchemical compositions in a wide range of possible browndwarfs and cooling EGPs. The code explicitly includesreverses of all reactions so that, in the absence of atmo-spheric physics, the chemical composition would relaxto equilibrium at every height. Our strategy is to findthe apparent quench points in all the models and ana-lyze these for their systematic properties. We then finetune our results by using them in quench approximations.Our objective is to describe the emergent properties ofthe chemical network as a whole. Our strategy differsfrom previous work that seeks to determine the one keyrate-limiting step in a network of reactions, which is thentreated as the effective reaction rate for the network asa whole.We limit the study to self-luminous cooling worlds forwhich insolation is not (yet) thermally important. Thisincludes brown dwarfs, free-floating planets, and youngdirectly-imaged planets. This category includes most ofthe exoplanets for which good data can be obtained inthe present or in the near future. What this limitationmeans for carbon speciation is that we are concerned onlywith the conversion of CO to CH . Unlike Line et al.(2011), Visscher and Moses (2011), and Visscher (2012),we do not address the kinetic inhibition against oxidizing CH to CO. The latter is an issue in strongly irradiatedplanets that are warm at high altitudes where, if verticalmixing is fast and the temperature not too hot, CH andits photochemical products will be overabundant (Lineet al. 2010; Miller-Ricci Kempton et al. 2011; Morley etal. 2013). Methane oxidation could be an issue for BDsand cooling EGPs if the higher parts of their atmospheresare strongly heated by wave-breaking processes not takeninto account in our radiative-convective model, but we donot further address this possibility here.Quenching in the nitrogen system has also been thesubject of many studies over the years (Abelson 1966;Chameides and Walker 1981; Prinn and Fegley 1987; Fe-gley and Lodders 1994, 1996; Saumon et al. 2006; Moseset al. 2010, 2011; Line et al. 2011). As with methane,the underabundance of ammonia in brown dwarfs coldenough to favor it has been attributed to disequilibriumchemistry (Saumon et al. 2006). The visibility of am-monia has been made the distinguishing characteristic ofthe Y dwarf, the newest, coldest, and possibly last mem-ber of the stellar spectral sequence (Cushing et al. 2011).The NH -N -HCN system differs from CO and CH ininteresting ways that lead to significantly different be-havior.Finally, for completeness, we address quenching ofCO , a gas that can be relatively easy to observe froma space-based observatory and, because its abundance issensitive to metallicity, can be relatively telling. CO is not always thought of as a species subject to quench-ing (but see Prinn and Fegley (1987)). In the H -rich,UV-poor worlds that are the subject of this study, CO quenching does take place. OVERVIEW OF CO AND CH Figure 1 illustrates the most important chemical path-ways between CO and CH in a warm H -rich atmo-sphere. The path from CO to CH climbs over threeenergy barriers, the first between CO and formaldehyde(H CO), the second between formaldehyde and methanol(CH OH), and the third between methanol and methane.The three barriers can be thought of as reducing theC ≡ O triple bond to a double bond, reducing the C=Odouble bond to a single bond, and splitting C from Oentirely.In photochemistry the first barrier is unimportant be-cause atomic hydrogen is generated in abundances thatvastly exceed equilibrium (Liang et al. 2003). Kineticbarriers to adding H to CO to make HCO or addingH to HCO to make H CO are insignificant. The thirdbarrier is unimportant in photochemistry because inci-dent UV radiation readily splits CH OH into CH andOH, which is quickly followed by adding photochemicalH to CH to make CH . In photochemistry the energyto overcome the first and third barriers comes from UVphotons.By contrast, the middle barrier is not easily overcomeby photochemistry. Formaldehyde is readily photolyzedbut the products are either CO or HCO; i.e., regress thatleaves the C=O bond unbroken. Meanwhile successive3-body additions of photochemical H to H CO to makeCH OH face considerable kinetic barriers, save at hightemperatures where CH is not favored. The historicfocus of the planetary literature was therefore on reac-lanetary Disequilibrium Chemistry 3 Fig. 1.—
Major chemical pathways linking CO and CH in an H -rich atmosphere. Reactions from left to right arewith H or H. Key intermediate molecules are formaldehyde(H CO) and methanol (CH OH). Other intermediates (HCO,H COH, CH O, CH ) are short-lived free radicals. Wherethe species are plotted gives a rough indication of the ener-getics. Energy barriers correspond to breaking C-O bonds:from triple to double, from double to single, and from singleto freedom. Relative magnitudes of reaction rates are indi-cated by arrow thickness for conditions near the quench pointin a particular model (red star on Fig 4) that gives roughlyequal amounts of CO and CH . At higher temperatures andlower pressures the plot as a whole would tilt to the left, withcarbon pooling in CO. At lower temperatures and higher pres-sures the plot as a whole tilts to the right and carbon poolsin CH . The energy barriers pose greater obstacles when thegas is colder. tions that go from formaldehyde to methanol. Prinn andBarshay (1977) suggested H + H CO → CH + OH, anambitious reaction that jumps two hurdles at once, asthe rate-limiting step, while Yung et al (1988) suggestedthat H + H CO + M → CH O + M (where M represents athird body) would be the bottleneck. Both schemes havebeen widely used, for Jupiter (B´ezard et al. 2002), EGPs(Cooper and Showman 2006; Line et al. 2011), and BDs(Hubeny and Burrows 2007; Barman et al. 2011a).Without the photochemical assist, any of the threebarriers could potentially block the reaction. Recently,Moses et al. (2011) concluded that the rate-limitingreaction is the one that breaks the C-O single bond.They report that the bottleneck is between methanoland methane, CH OH + M → CH + OH + M whenmethane is abundant, or goes through a free radicalH + H COH → CH + OH when methane is scarce. Wewill concur with the former but not the latter. THE PHOTOCHEMICAL MODEL
Standard 1D codes simulate atmospheric chemistry bycomputing the gains and losses of chemical species at dif-ferent altitudes while accounting for vertical transport.Vertical transport is parameterized as a diffusive pro-cess with an “eddy diffusion coefficient,” denoted K zz [cm /s]. Volume mixing ratios f i of species i are ob-tained by solving continuity N ∂f i ∂t = P i − L i N f i − ∂φ i ∂z (1)and force (flux) φ i = b ia f i (cid:16) m a gkT − m i gkT (cid:17) − ( b ia + K zz N ) ∂f i ∂z (2)equations. In these equations N is the total number den-sity (cm − ); P i − L i N f i represent chemical production and loss terms, respectively; φ i is the upward flux; b ia ,the binary diffusion coefficient between i and the back-ground atmosphere a , describes true molecular diffusion;and m a and m i are the molecular masses of a and i . Weimplement molecular diffusion of a heavier gas throughH by setting b ia = 6 × ( T / . cm − s − —appropriate for CO—for all the heavy species. This isa reasonable approximation for present purposes, as ourconcern is with quenching at altitudes well below the ho-mopause. The physical meaning of b ia is the ratio of therelative thermal velocities of the two species to their mu-tual collision cross section. In the present circumstances,the relative thermal velocity is effectively that of H , sothe only important source of variation in b ia stems fromthe different diameters of the molecules. The code hasbeen used by Zahnle et al. (2009) to address hot Jupitersand by Miller-Ricci Kempton et al. (2011) and Morleyet al. (2013) to address H -rich hot Neptunes and super-Earths.Temperature and pressure profiles are imported fromthe output of detailed radiative-convective models thatare described elsewhere (Saumon & Marley 2008). Themodels are cloudless and of solar metallicity. Addingcloud opacity would make the p - T profiles hotter andless favorable to CH . The cloudless model is somethingof a best case for methane.Treating vertical transport as diffusion is a necessaryevil in a 1-D code. We will regard K zz as a mildly con-strained free parameter. In free convection, K zz has beenestimated from mixing length theory (Gierasch & Con-rath 1985), K zz = 13 H (cid:18) R gas F conv µρC p (cid:19) / , (3)where R gas is the universal gas constant, F conv the con-vective heat flux, µ the mean molecular weight (dimen-sionless), ρ the density, and C p the heat capacity of thegas. The mixing length is approximated by the scaleheight. An upper bound is obtained from Eq 3 by equat-ing F conv with σT , K zz < . × (cid:18) T eff (cid:19) / (cid:18) g (cid:19) cm / s . (4)These large values of K zz might be regarded with someskepticism. If applied to Earth’s troposphere, Eq 3 pre-dicts K zz > cm /s, which is two orders of magnitudetoo high. Even in the present context, F conv can be muchsmaller than σT (or even fall to zero in places), becausea considerable fraction of energy transport is by radia-tion. Depending on how well one tolerates theory, onemight expect K zz to be of the order of 10 -10 cm/s inthe convecting zones of self-luminous EGPs, and 100 × smaller for BDs with g = 10 cm/s .It is likely that K zz would be much smaller in stratifiedgas above the convecting regions. On Earth, K zz dropsby two orders of magnitude at the tropopause. Hubenyand Burrows (2007) reduce K zz by several orders of mag-nitude in the stratosphere. Simulations by Freytag et al.(2010) support this expectation. The tradeoff is betweensimplicity (constant K zz ) and realism (a vertical discon-tinuity in K zz ). The latter choice would introduce twomore free parameters, the value of K zz in the strato- Zahnle & Marleysphere (ill constrained) and the altitude of the disconti-nuity (reasonably well constrained, but results could besensitive to this). A two layer model also makes inter-preting the numerical results in terms of a quench ap-proximation less straightforward because there can bemore than one quench point. For this study we chooseto treat K zz as constant with height, but vary it over arange wide enough to encompass all likely values.The chemical system used here comprises 366 for-ward chemical reactions and 32 photolysis reactions of64 chemical species made of H, C, N, O, and S. Themost important missing species is probably methylamine,CH NH . Reaction rates when known are selected fromthe publicly available NIST database (http://kinetics.nist.gov/kinetics). Although many of the important re-actions have been measured in both directions in the lab(e.g., both CH +H → CH +H and CH +H → CH +Hhave been heavily studied), in general one direction ismuch better characterized than the other. Thermody-namic data (enthalpies and entropies) are usually betterknown over a wider range of temperatures. Thus it is bet-ter to complement each specific reaction with its exactreverse, with the forward and reverse rates linked self-consistently by the thermodynamic data of the speciesinvolved. This way the transition from equilibrium tokinetically-controlled abundances is automatic. Equilib-rium is reached when all the important forward and re-verse reactions are fast compared to changes in the statevariables.How this is done is fully explained by Visscher andMoses (2011). What follows is a telegraphic summary.For reactions of the form A + B → C + D with for-ward reaction rate k f , the reverse rate (i.e., the ratefor C + D → A + B) is k r = k f exp ( − ∆ G/RT ), where∆ G , the Gibbs free energy, is obtained from enthalpiesand entropies of the of the reactants and products,∆ G = H A + H B − H C − H D − T ( S A + S B − S C − S D ).For associative reactions of the form A + B → AB,the rate for the reverse reaction (dissociation of AB)is k r = k f ( kT /P ◦ ) exp ( − ∆ G/RT ), where P ◦ = 10 dynes/cm is one atmosphere. Similarly, the associa-tive reverse of a dissociative reaction is given by k r = k f ( P ◦ /kT ) exp ( − ∆ G/RT ).Thermodynamic data as a function of temper-ature are available for atoms and most smallmolecules from NIST in the form of empiricalShomate equation fits for enthalpy and entropy(http://webbook.nist.gov/chemistry/form-ser). Zeropoint data for HS are corrected by Lodders (2004). ForCH OH and C H we use heat capacities as a functionof temperature to derive Shomate equation fits for en-thalpy and entropy. Unfortunately, many of the moreexotic free radicals are not listed in the publicly avail-able NIST databases. For many of these we use estimatesgiven in Burcat and Ruscic (2005)’s widely available grayliterature compilation. For NNH we follow Haworth etal. (2003), and for N H and N H we follow Matuset al. (2006). Inaccurate thermodynamic data for freeradicals are not a problem for equilibrium calculationsof well-characterized abundant species because poorly-characterized free radicals are never abundant. On theother hand, poorly-characterized species do pose a prob-lem in disequilibrium kinetics because net reaction rates depend on the uncertain abundances of free radicals,which are determined by their thermodynamic proper-ties.The lower boundary is set deep enough that all speciesare in thermodynamic equilibrium. We find that pres-sures above 300 bars and temperatures above 2000 Kusually suffice, with the limiting species being N . Theupper boundary condition is zero flux for all species. Weplace the upper boundary at ∼ − bars, as higheraltitudes require additional physics and chemistry (ionchemistry, thermospheric heating) that go beyond thescope of this work. Steady state solutions are found byintegrating the time-dependent chemical and transportequations through time using an overcorrected fully im-plicit backward-difference method. Most models take afew minutes to run to steady state from arbitrary initialconditions on a vintage laptop computer, although someparticular cases can be more challenging.Photolysis significantly affects the composition of at-mospheres at very high altitudes even when the incidentUV flux is small. In our models we have set the incidentUV flux to 0.1% that at Earth. Photolysis plays almostno part at the higher pressures germane to quenching andis not further discussed here. We have not included thereverses of photolysis reactions (radiative attachment,e.g. OH + H → H O + hν ) in detailed balancing. Recentwork has shown that radiative attachment can be im-portant in hydrocarbon growth in planetary atmosphereswhen the resulting molecule is complex enough that ra-diative relaxation can be effective (Vuitton et al. 2012). Two examples
Figure 2 shows how K zz effects chemistry on a tepid( T eff = 600 K), cloud-free, relatively low mass ( g = 10 cm/s ) extrasolar planet of solar composition. Insolationis insignificant. Three different K zz are compared. (i)Strong vertical mixing— K zz = 10 cm /s—suppressesCH and maintains f CO (cid:29) f CH to very high alti-tudes. CO and CH are in chemical equilibrium belowthe quench point at ∼ K zz = 10 cm /s raises the quench point to1.5 bars and 1280 K, which is better for CH . (iii) Weakmixing, K zz = 10 cm /s, raises the quench point to 0.6bars and 1000 K, which is cool enough, barely, to fall intothe CH field, so that f CH > f CO in this planet’s photo-sphere. The effect of molecular diffusion (in which heavymolecules sink through H ) is apparent above 3 × − bars at K zz = 10 cm /s and above 3 × − bars at K zz = 10 cm /s.At a given effective temperature, surface gravity de-termines whether the atmosphere is CO or CH domi-nated. Figure 3 shows CO and CH abundances in threeworlds that differ only in their surface gravities. Theatmospheres are cloud-free and of elementally solar com-position. The models shown here have effective radiat-ing temperatures of 600 K and a Jupiter-like K zz = 10 cm /s. Surface gravities range from a Saturn-like g = 10 cm/s to a brown-dwarf-like g = 10 cm/s . The com-puted CO/CH ratio is sensitive to surface gravity. Thisis because the p - T profile is displaced to higher pres-sures and lower temperatures when gravity is higher.Quenching at higher pressures and lower temperaturesfavors CH over CO. The result is that the 600 K planetlanetary Disequilibrium Chemistry 5 -6 -5 -4 -3 -2 -1 -5 -4 -3 P r e ss u r e [ ba r s ] Volume Mixing Ratio
Temperature [K] COCH Temp
COCH K zz =10 Fig. 2.—
Methane and CO in an exemplary lukewarm EGP.The planet is cloud-free, has an effective radiating temper-ature of 600 K, g = 10 cm/s , and receives insignificantinsolation. The p - T profile of the atmosphere is compared tothe p - T curve (gold dot-dash) corresponding to equilibrium f CH = f CO . Methane is thermodynamically favored when T is to the left of the f CH = f CO curve, CO is favored to theright. Computed CO and CH mixing ratios are shown fromthree 1D kinetics models that differ only in vertical eddy mix-ing: K zz = 10 cm /s (solid), K zz = 10 cm /s (dashed),and K zz = 10 cm /s (dots). Deep in the atmospheres COand CH are in equilibrium. High in the atmospheres CO andCH are unreactive and f CO and f CH would be constant butfor molecular diffusion. The quench points are obvious andwell-defined in all three models. -6 -5 -4 -3 -2 -1 -5 -4 -3 P r e ss u r e [ ba r s ] Volume Mixing Ratio
Temperature [K]
CO CH g=10 g=10 g=10 CO CO CH CH Fig. 3.—
Three models with three different surface gravities.The plot shows p - T profiles and CO and CH abundances.The planets are 600 K, self-luminous, cloudless, of solar com-position, with K zz = 10 cm /s. Surface gravities are g = 10 cm/s (solid), g = 10 cm/s (dashed), and g = 10 cm/s (dots). Higher gravity models are colder at a given pressureand are more favorable to CH . ( g = 10 cm/s ) has abundant CO and relatively littleCH , while the 600 K brown dwarf is methane-rich withonly a trace of CO. The ensemble of models
Solar Metallicity 3X Solar Metallicity g = g = g = T e ff T e ff T e ff zz Vertical Diffusivity K zz Fig. 4.—
An overview of the ensemble of models. Thereare six panels for six different combinations of gravity andmetallicity. Relative CO and CH abundances above thequench point are indicated by the areas of the semicircles,CO black and following, CH green and preceding. In gen-eral, higher gravities, cooler temperatures, and lower metal-licity favor CH vs. CO. Different metallicities can give thesame CO/CH ratio at the same T eff and g with modest andprobably unmeasurable differences in K zz . Blue squares de-note the specific cases in Figures 2 and 3. Red circles denotethe 16 cases used in Fig 7 to illustrate quenching. Orangediamonds denote the two models used in Fig 8 to comparereaction rates. The red star is the guide to Fig 1. In allcases, equilibrium f CH > f CO . We use the 1D chemical kinetics models to explore thephase space defined by 500 < T eff < 30. Warmer tem-peratures excite hydrogen’s vibrational modes, for which C p ≈ . R and R/C p ≈ . 24. Detailed calculations showthat R/C p decreases monotonically from 0 . 29 at 1000 Kto 0 . 25 at 2100 K to 0 . 20 at 3000 K as dissociation (an-other energy sink) becomes important.Equation 7 needs to be compared to contours of equi-librium chemistry. Defining ξ ≡ CH / CO, we can writethe equilibrium (Eqs 5 and 6) in the form ξ · (cid:18) f O − f C ξ (cid:19) = p f Ae B/T (8)where f H ≈ . 84 is the H mixing ratio, f O ≈ f H O + f CO is the total oxygen mixing ratio, and f C ≈ f CH + f CO isthe total carbon mixing ratio. These are generally verygood approximations for T < f H O . For contours of constant ξ , the lefthand side of Eq 8 is constant. Taking derivatives, thetemperature gradient is (cid:18) ∂T∂p (cid:19) ξ = 2 TB Tp ≈ . T Tp . (9)Thus ( ∂T /∂p ) ξ < ( ∂T /∂p ) S for temperatures lower thanabout 2900 K, by which point several of the assump-tions made in deriving this relation have begun to breakdown. Even with K zz as high as 10 cm /s, quench-ing in the CO-CH system takes place at temperaturesless than 2200 K. Hence we can always presume that COis most abundant and CH least abundant at the lower boundary of our model. It follows that we can deter-mine the quench conditions (temperature T q , pressure p q ) by working up from the lower boundary and findingthe lowest altitude at which the product K test is appre-ciably smaller than the equilibrium constant K CH · CO .In practice we searched for the lowest altitude where K test < . K CH · CO .We also wish to determine a quench timescale t q . Inprinciple, the right way to do this is to determine thetrue chemical reaction timescale from the sum of all rel-evant reaction rates at the quench point. In general thisinformation is surprisingly hard to extract from a model,because the breakdown of equilibrium appears as smalldifferences between much larger forward and reverse re-action rates in many reactions. This approach is practicalonly if one already knows exactly which species and reac-tions are key. Once known, for t q to be useful in a quenchapproximation one then needs a comparably fine-graineddescription of the cooling time implicit in K zz and theparticular chemical reactions that establish t q .Much easier, and more useful, is to define a quenchtimescale for the system as a whole in terms of a relevantmixing time, t q = t mix = L /K zz where L is a character-istic length scale. The simple choice is to set L = H q , thescale height at T q . This definition is arbitrary to withina multiplicative factor because the true mixing length L is not in general equal to H q . But this definition of t q isprecisely what is needed for constructing a new quenchapproximation that assumes t mix = H q /K zz and, as seenin Figure 5, it leads to a very simple description of theoverall behaviour of the full network of chemical reac-tions.Smith (1998) demonstrated that L is often consider-ably shorter than H , mostly because the key reactionsdetermining t q are much more sensitive to temperaturethan to pressure. Smith (1998) developed a five step iter-ative algorithm for determining L and presented it in theform of a “recipe” to be used in quenching calculations,which is especially useful when the system is controlledby a single reaction whose rate is known. Several workershave employed Smith’s recipe. Unfortunately, the rangeof mixing times that results can be very great. For exam-ple, Visscher and Moses (2011) list values for L between0 . H and 0 . H for the CO-CH system, the equivalentof a factor of 50 in t mix and therefore equivalent to a fac-tor of 50 in the inferred value of K zz , itself a parameterof easy virtue. The resulting uncertainty frustrates in-tercomparison between models. Our position is that themost useful approximations are simple approximations.We therefore stick with the old rule L = H to define t mix ,rather than iterate between L and T q . How L is definedis important for comparing to previous work that triesto estimate t chem from particular reaction rates, but it issimply a scaling factor for us.Irrespective of how L is defined, we find that we canfit the quench time t q to a simple functional form thatspans all our models. We start from the expectationthat CO → CH will depend on temperature and pres-sure, perhaps the C/O ratio, and possibly on metallicity.It is reasonable to hope that the net reaction has anArrhenius-like rate t q = A p − b m − c exp ( B/T ) , (10)lanetary Disequilibrium Chemistry 7 t q × p q × p q [ s e c ond s - ba r s ] Quench Temperature 1000/T q t chem × p = 40e bar •sec10 t q X p q [ s e c ond s - ba r s ] Quench Temperature 1000/T q t chem × p = 3 × -6 e bar•sec Fig. 5.— CO-hydrogenation timescales as a function of quench point temperature T q and quench pressure p q for the ensembleof solar composition kinetics models. Quench conditions T q and p q are determined numerically from the full kinetics modelsusing the entire network of chemical reactions. How this is done is explained in the text. The quench timescale used here is amixing timescale, t q = t mix ≡ H q /K zz , where H q is the pressure scale height at the quench point. The relation between the truechemical timescale and t q is arbitrary to a factor L /H q , where L refers to the true mixing length. We do not plot t q itself (forwhich the points are scattered), but rather t q × p q ( left ) and t q × p q ( right ), which resolve into orderly arrays. Left . A simpleexponential provides a good fit to t q × p q for all but the coldest T q . Right . A different simple exponential can be fit to t q × p q for the coldest T q . Lines show fits. where possible pressure and metallicity dependencies arewritten as power laws.Our hope is nicely borne out by Figure 5, which plots t q for the CO-CH reaction for our complete roster of solarmetallicity models. All but the coldest quench points aredescribed by a rather well-defined Arrhenius expressionwith b = 1. The Arrhenius fit (for m = 1) is t (cid:48) q = 3 . × − p − exp (42000 /T ) sec , (11)where p is in bars. The time scale for models with m = 3(not shown) is roughly half of this.The prime is placed on t (cid:48) q because, by using K test < . K eq rather than K test < K eq , we have overestimatedthe quench time. To correct for this—and to verify thata quench scheme works—we compare the predictions ofthe quench scheme with the predictions of the ensembleof models. Fits are shown in Figure 6. These use t q = 1 . × − p − m − . exp (42000 /T ) sec . (12)The methods of Figure 6 and Figure 5 are complemen-tary. A satisfactory fit on Figure 6 is more sensitive tothe A factor (overall rate) and the metallicity (Eq 10),whilst the fit on Figure 5 is more sensitive to the B factor(temperature dependence) and pressure.The half-dozen models that are ill fit by Eq 11 are fitinstead by a second Arrhenius relation t q = 40 p − exp (25000 /T ) sec . (13)where we have given the iterated rate. The models fit byEq 13 are those with very low quench temperatures T q and very little CO. We have no such models with m = 3.The high and low temperature fits combine harmon-ically to give a general description of reaction times in the CO-CH system, t CO = (cid:18) t q + 1 t q (cid:19) − . (14)Equation 14 when used with the timescale t mix = H /K zz works for all of our models for CO and CH and we expect should work generally for compositionsnot too far from solar. Analysis We examine the ensemble of models for quench pointspertinent to the reaction intermediates, to determinewhere in the CO-CH reaction network quenching firsttakes place. We consider the CO-H CO, H CO-CH OH,and CH OH-CH equilibria. We analyzed 16 modelstaken arbitrarily from the ensemble. The chosen ones aremarked on Fig 4 with scarlet circles. We find, to withinour ability to determine this, that the CH OH-CH reac-tion quenches at the same temperature as the CO-CH reaction as a whole. This is shown by panels (c) and(d) of Figure 7. By contrast the CO-H CO and H CO-CH OH reactions stay near equilibrium for temperaturesabove 900 K (Figures 7a and 7b). Hence as a parcel coolsequilibrium first breaks down between CH OH and CH .This is in accord with conclusions reached by Moses etal. (2011) and Visscher and Moses (2011).Figure 8 examines in detail the key reactions that breakthe CO-CH equilibrium. The important reactions formthe methyl radical CH from species such as CH OH inwhich the C-O bond is still intact. Figure 8 illustratestwo cases, one a cool ( T eff = 500 K) brown dwarf with apreference for CH ( f CO = 1 . × − , f CH = 4 . × − ),the other a warmer ( T eff = 1100 K), smaller world that Zahnle & Marley -7 -6 -5 -4 -3 -7 -6 -5 -4 -3 Q uen c h M i x i ng R a t i o f ( C O ) Model Mixing Ratio f(CO) 10 -7 -6 -5 -4 -3 -7 -6 -5 -4 -3 Q uen c h M i x i ng R a t i o f ( CH ) Model Mixing Ratio f(CH ) Fig. 6.— CO (left) and CH (right) mixing ratios predicted by the quenching approximation t chem = t mix (y-axis) plotted againstthe actual asymptotic mixing ratio determined from the full ensemble of 1-D kinetics model (x-axis). A perfect approximationwould adhere to the line. Open red symbols are for m = 1 and solid black symbols are for m = 3 using Eq 12. favors CO ( f CO = 4 . × − , f CH = 1 . × − ). Bothmodels assume solar metallicity ( m = 1). In both casesthe most important reaction is simple thermal decompo-sition of methanol, CH OH + M → CH + OH, althoughthe reaction of methanol with atomic hydrogen is nearlyas fast in the colder, higher gravity case. The reactionH COH+H, which Moses et al. (2011) report as the mostimportant in their CO-rich models, does not appear tobe very important in ours. We do not claim that our sys-tem is better; we merely point out that abundances andreaction rates of furtive species like H COH are highlyuncertain. Comparisons with previous work To put our results in perspective, we compare themto some quench approximations seen in the literature.Published quench approximations begin by identifying aparticular forward reaction as the bottleneck and thencalculate CO loss time scales from the chosen reaction’srate; variety lies in the reactions that are chosen and thereaction rates adopted.We begin with the classic Prinn and Barshay (1977)prescription, used by Hubeny and Burrows (2007) astheir “slow” case. Prinn and Barshay (1977) suggestedH + H CO → CH + OH (15)as a rate-limiting reaction, with reaction rate k f = 2 . × − e − /T cm /s. R15 is a reaction that jumps overtwo energy barriers in Figure 1, which makes it look likeit should be relatively unlikely. Figure 8 shows that R15is modestly important in our system, but we use a newerslower rate estimated by Jasper et al (2007). The timeconstant for CO loss by reaction R15 using Prinn andBarshay (1977)’s rate is approximated by the forwardreaction t − = − ∂ [CO] ∂t = k f [H ] [H CO][CO] sec − (16)where the notation [CO] refers to number density [cm − ]. To evaluate t chem for R15, we put formaldehyde in equi-librium with CO and H , p H CO p H · p CO = K H CO = 3 . × − exp (1420 /T ) , (17)to obtain t chem = 1 . × exp (34780 /T ) p N f sec . (18)The number density N is related to the pressure p inbars by N kT = 10 p . The predictions of Eq 18 are com-pared to results from our complete models in Figure 9.Equation 18 works quite well in a quench approximationto our full model, especially for cases where CH is pre-dicted to be abundant, despite its being based on thewrong reaction with the wrong rate. As the Prinn andBarshay (1977) approximation has been widely used fora very long time, it is valuable to see that it seems towork rather well. When compared to Eq 12, Eq 18 hasa stronger pressure dependence ( p − vs. p − ), a weakertemperature dependence ( T · e /T vs. e /T ), andno dependence on metallicity.More recent discussions of the CO-CH quench approx-imation omit enough details that they can be challengingto reproduce. We attempt to do so here because the com-parisons are illuminating. Yung et al. (1988) suggestedthat the rate-limiting step is the 3-body reactionH + H CO + M → CH O + M . (19)Hubeny and Burrows (2007) use R19 as their “fast” ratewith L = H . Barman et al. (2011a) also use this rate,but following Smith (1998) they set the mixing length L to unspecified values between 10% to 20% of the scaleheight, which effectively slows the “fast” rate by a factorof 25-100 compared to L = H . Cooper and Showman(2006) implemented the CH O channel with a differentreaction rate while following Smith’s “recipe.” We usethe asymptotic high pressure ratelanetary Disequilibrium Chemistry 9 -6 -5 -4 600 800 1000 1200 K = p H C O / p H / p C O Temperature equilibrium CO + H ↔ H CO (a)10 -6 -5 -4 -3 -2 800 1000 1200 1400 1600 1800 2000 K = p CH O H * p H / p CH / p H O Temperature K equilibrium CH OH + H ↔ CH + H O (c) 10 -4 -3 -2 -1 500 1000 1500 2000 K = p CH O H / p H C O / p H Temperature K equilibrium H CO + H ↔ CH OH (b)10 -8 -6 -4 -2 800 1000 1200 1400 1600 1800 2000 K = p CH * p H O / p C O / p H Temperature K equilibrium CO + 3H ↔ CH + H O (d) Fig. 7.— Quenching behavior of the three intermediate reactions (a, b, c) and of the CO ↔ CH system as a whole (d). Resultsare shown from 16 models chosen to provide a representative sample of the ensemble (marked by red circles in Fig. 4). Panels(a,b) show that CO ↔ H CO and H CO ↔ CH OH appear to be equilibrated in all models at T > 900 K. Panel (c) shows thatequilibration of CH OH and CH is more difficult. In detail the quench points for CH OH ↔ CH appear to be the same asthe quench points for CO ↔ CH as shown in panel (d) for the same 16 models. Evidently the bottleneck is between CH OHand CH , in accord with conclusions reached by Moses et al (2011). k ∞ = 8 . × − exp ( − /T ),a very fast rate which we obtained by reversing thefirst order rate for CH O → H + H CO given by Rauket al. (2003). If we presume that H and H CO are bothin equilibrium, we can use the hydrogen equilibrium con-stant K H · H = 1155 e − /T = p H · p H − . to obtain t − = k ∞ [H] [H CO][CO] = k ∞ p . f . (cid:18) kT (cid:19) K H CO K H · H . (20)Evaluated, t chem = 4 . × − T p − . f − . exp (28656 /T ) . (21)Results of using Eq 21 are shown as red diamonds on Fig9 with L = 0 . H , which is comparable to what Barmanet al. (2011a) use (the fit would look worse with L = H ).These stand out with too much CH and too little CO,because R19 is not a true bottleneck.Line et al. (2011) moved the bottleneck to CH O +H → CH OH + H. To compute a rate constant, weuse p CH O = K CH O · p H · p H · p CO, where K CH O = 3 . × − e /T is the relevant equilibrium constant(the large number of significant digits in these equilib-rium constants do not imply accuracy). To completethe equation we also assume that H and H are in equi-librium. We use the reaction rate given by Line et al.(2011), which results in t chem = 1 . × exp (45720 /T ) T N pf sec . (22)Results are shown as green squares on Figure 9 using L = H , as Line et al. (2011) do. Because it predicts toomuch CO and too little CH , t chem given by Eq 22 is tooslow.Based on detailed analysis of the reactions in a com-plete 1D model similar to our own, Moses et al. (2011)concluded that the bottleneck is associated with break-ing the C-O bond. They propose several key channels.One channel is thermal decomposition of methanol,CH OH + M → CH + OH + M . (23)Visscher and Moses (2011) and Moses et al. (2011) state0 Zahnle & Marley H+CH OHH+H COH CH OH + M H +H CO A l t i t ude ( k m ) -3 s -1 )10 H+CH OHH+H COH CH OH + M H +H CO A l t i t ude ( k m ) -3 s -1 )10 Fig. 8.— The four most important reactions for quenching in our model CO-CH system. All four reactions yield CH + OH asproducts. Solid curves show rates of the forward reaction that creates CH , dashed curves show the reverse reaction that destroysCH , and wide dots the difference (the net production of CH ), which is equivalent to net loss of CO. Black dot-dash lines arethe altitudes where CH and CO reach their final quenched values. Left: A warm, strongly-mixed ( T eff , K zz , g = 1100 , , ),model dominated by CO. Disequilibrium reactions are fastest at 210 km (at 1700 K) but full quenching is delayed until 270km (at 1360 K); the distance corresponds to a bit more than a pressure scale height. Right: A cool, moderately-mixed( T eff , K zz , g = 500 , , ) BD dominated by CH . Disequilibrium reactions are fastest at 7 km but quenching is at 9 km (thescale height is 5 km). Q uen c h q ( C O ) − − − − − − Model f(CO)10 − − − − − Q uen c h q ( CH ) − − − − − Model f(CH )10 − − − − − Fig. 9.— Comparison of some quench approximations (y-axes) to the actual CO and CH mixing ratios computed by the full1D model (x-axes). A perfect approximation would adhere to the line. Only results for solar metallicity ( m = 1) are shown.Purple plusses follow the widely applied Prinn and Barshay (1977) model with L = H . Red diamonds use the channel throughCH O suggested by Yung et al. (1988) but using a more recent estimate for the reaction rate and reducing the mixing lengthto L = 0 . H ; this is not the same as Yung et al. (1988)’s algorithm. Green squares implement Line et al. (2011)’s CH Ochannel with L = H . Black triangles implement the methanol decomposition channel favored for low f CO by Moses et al. (2011)and Visscher and Moses (2011). These are computed with mixing length L = 0 . H . Blue circles represent our attempt toimplement the Moses et al. (2011) H COH channel with L = 0 . H . that this is more important when CH is abundant. An- other channel, which they state is more important whenlanetary Disequilibrium Chemistry 11CO (cid:29) CH , isCH OH + H → CH + OH . (24)The rate of R24 depends on the uncertain thermody-namic properties of CH OH. In the high pressure limit,methanol decomposition R23 is a first order reaction, ∂ [CH OH] ∂t = − k ∞ [CH OH] . (25)Jasper et al (2007) estimate k ∞ from theory k ∞ = 6 . × (300 /T ) . exp ( − /T ) sec − . The methanol equilibrium constant K CH OH = p CH OH p CO · p H = 1 . × − e /T relates [CH OH] to [CO]. The reaction timescale is then t chem = (cid:0) k ∞ p H K CH OH (cid:1) − = 4 . × − T . exp (33573 /T ) p − f − sec . (26)Equation 26 has a weaker dependence on T , a strongerdependence on p than we find for the ensemble, and nodependence on metallicity.Figure 9 illustrates quenching using Eq 26 as blacktriangles. For the comparison we take L = 0 . H . Thisapproximates what Moses et al. (2011) may be using.Equation 26 does well, especially in cases where CO (cid:28) CH at quenching, which is the regime for which Moseset al. (2011) report that Eq 26 applies. Equation 26 doesbetter if L > . H . The match is best for CO with L = 0 . H , and better for CH with L ≈ . H .To reconstruct a simple form for the H COH chan-nel, we need to estimate the equilibrium abundance ofH COH. We use p H COH = K H COH · p H · p H · p COwith K H COH = 1 . × − e /T . We assume that Hand H are in equilibrium. Jasper et al. (2007) list a fastrate, k = 2 . × − cm /s, for the reaction of H andH COH. Assembling the parts, we obtain t chem = 2 . × exp (38000 /T ) N pf − sec . (27)Results are shown as blue circles on Figure 9. Equation27 has a similar T dependence to what we find for thesystem as a whole, but the overall rate is about 3 ordersof magnitude slower using L = 0 . H , or about ten timesslower using L = H .Disagreement between our model and Moses et al overthe H COH channel is also apparent in Figure 8. TheH COH radical plays a much more modest role in our1-D code than it does in Moses et al. (2011)’s. The sameis true for CH O. Why this should be so is likely relatedto differing guesses of H COH’s and CH O’s ill-knownthermodynamic properties.There are some more subtle differences between ourmodel and previous quench models that should be noted.One is that we find t chem ∝ p − by fitting to the en-semble, whilst quench models that attempt to identifythe one key forward reaction tend to get t chem ∝ p − .This difference is the reason why quench prescriptions inFigure 9 that do well with f CO (cid:28) f CH miss the mark somewhat with f CH (cid:28) f CO . Figure 6 that uses t q ∝ p − shows fine agreement at both ends of the scale.Another difference is that we see a well-defined metal-licity dependence when fitting to the ensemble. Themetallicity dependence probably stems from the reversereaction, which as noted is usually ignored. As equilib-rium breaks down, the reaction slows in both directions,more quickly in the reverse direction than in the forwarddirection. This behavior is shown clearly in Figure 8,where forward, reverse, and net reaction rates are plot-ted. It is the net reaction, the difference between theforward and reverse reactions, that defines the retreatfrom equilibrium, not the forward reaction alone. Forexample, the time scale for the methanol decompositionchannel is t − = − ∂ [CO] ∂t = k f [CH OH][CO] − k r [CH ] [OH][CO] . (28)The reverse reaction is quadratic in metallicity, whilst[CO] is quadratic in metallicity only for f CO (cid:28) f CH ,which suggests that it is through the reverse reaction thatmetallicity enters t CO . The same consideration appliesfor all the important reactions in Figure 8, as the reversereaction in every case is between CH and OH. QUENCHING IN THE N -NH SYSTEM The approach we used to search for quenching of COand CH does not work well for N and NH . Indeed,Figure 10—the analog to Fig 4 for N and NH —showsno sign that quenching plays any role in the N -NH reaction, although doubtless quenching occurs. Solar Metallicity 3X Solar Metallicity T e ff g = T e ff g = T e ff g = zz Vertical Diffusivity K zz Fig. 10.— Nitrogen. The six panels are as in Fig. 4. Relativeabundances of NH (red, east) and N (blue, west) are indi-cated by area. HCN abundances are multiplied by 5 to makethem visible at lower g and superposed in gold on N . Highergravities and cooler temperatures favor NH . The NH /N ratio is insensitive to K zz and only weakly sensitive to m ,which suggests that NH can be a proxy for g . HCN faresbest at high g and high K zz . The HCN/NH ratio is sensi-tive to K zz and only weakly sensitive to metallicity. The twoboxed models are ilustrated in Figure 11. Two things get in the way. Of less importance, a thirdspecies—HCN—becomes non-negligible at high temper-atures. HCN also quenches at high temperature, which2 Zahnle & Marleyleaves the nitrogen system with three possibly distinctquench points: NH -N , NH -HCN, and N -HCN. Al-though HCN is never very abundant in equilibrium, itis often abundant enough that its decomposition can in-crease the NH abundance by more than 10% after theNH -N reaction has quenched. When confused with thesecond, greater obstacle, HCN can make it difficult topinpoint where quenching occurs.The greater obstacle is that curves of constant NH /N are nearly parallel to the adiabat at the pressures andtemperatures where quenching occurs. This is illustatedin Figure 11. Consequently the NH /N ratio in a par-cel remains close to equilibrium well after quenching hastaken place, which makes the NH /N ratio indifferentto quenching (Saumon et al. 2006). To show this, writethe N -NH equilibrium in the form K NH · N = 5 . × − exp (13207 /T ) = p NH p N · p H . (29)Define ξ ≡ NH / N . If we approximate the total mixingratio of N by f N ≈ f NH + 2 f N (including HCN makesthis very complicated), we can write ξ (cid:18) f N ξ (cid:19) = p f Ae B/T . (30)With context-obvious substitutions, the temperaturegradient for contours of constant ξ is (cid:18) dTdp (cid:19) ξ = 2 TB Tp ≈ . T Tp . (31)The temperature gradient in Eq 31 is parallel to the adi-abat (Eq 7) at 1750 K, and nearly parallel for 1500 Fig. 11.— Two examples of quenching in nitrogen. The particular models are indicated by boxes on Figure 10. Volume mixingratios of N , NH , and HCN are shown for two different K zz pertinent to brown dwarfs. The relative insensitivity of NH andN to K zz is a consequence of contours of constant NH / N being nearly parallel to the adiabat p ( T ). This is illustrated hereby showing three relevant contours of constant ξ ≡ NH / N . Both ξ and p ( T ) are plotted against the temperature axis (top).In the full models ξ = 0 . 28 and ξ = 0 . 24 for K zz = 10 cm /s and K zz = 10 cm /s, respectively. Q uen c h q ( NH ) − − − − Model f(NH )10 − − − − Q uen c h q ( NH ) − − − − Model f(NH )10 − − − − Fig. 12.— Ammonia predicted by the quenching approximation q (NH ) (y-axes) plotted against the actual asymptotic mixingratio f (NH ) determined from the ensemble of 1-D kinetics models (x-axes). A perfect approximation would adhere to the line.Open red symbols are for m = 1 and solid black symbols are for m = 3. Left. Using Eq 32. Right. Open red diamonds m = 1 andfilled black circles m = 3 are based on Lodders and Fegley (2002); open green triangles and purple crosses are based on Line etal. (2011). bottom and find the altitude where the HCN abundancestops decreasing.We looked for quench points defined against Eq 35, Eq 36, and a third equilibrium with CO and NH , K HCN · CO = p CO · p NH p HCN · p H O = 0 . /T ) . (37)4 Zahnle & MarleyQuenching with N ( K HCN · N ) gives an indifferent fit toan Arrhenius relation, as might be expected given theinsensitivity of Eq 36 to f HCN and the high thermal sta-bility of N . The other two equilibria both give plausibleArrhenius-like fits, although far from perfect (Figure 13).A direct fit to the chemical time scale derived from theequilibrium K HCN · CH for the full ensemble of models is t (cid:48) q1 = 1 . × − p − m − . exp (37000 /T ) sec , (38)whilst the corresponding fit to K HCN · CO is t (cid:48) q2 = 1 . × − p − m − exp (34500 /T ) sec . (39)At quench temperatures greater than 1600 K, the pres-sure dependence is better described by t (cid:48) q ∝ p − . , andthe T dependence is stronger with an Arrhenius B -factorof order 46000 K. The stronger temperature dependencefor high T q suggests that reactions with N with its highactivation energy are becoming important. The higher T q cases correspond to higher f HCN .As noted above, the quench approximation is not verysensitive to the details of t q , and that is the case here aswell. Computed quenched abundances using t q = t mix = H /K zz provide a good approximation to the HCN mix-ing ratios computed by the full model with t HCN = 1 . × − p − m − . exp (36000 /T ) sec . (40)This expression seems to work well for all cases we haveconsidered (Fig 14).Fegley and Lodders (1996) treat HCN destruction ascontrolled by direct reaction of HCN with H to makeNH and CH . The corresponding chemical time scale is t chem = 9 . × exp (70456 /T ) N f sec . (41)This has a very steep temperature dependence. Quenchapproximations using Eq 41 with L = H are shown inFigure 14. This approximation predicts much more HCN(a higher quench temperature) than we find in our 1-Dmodels, a result consistent with the steep temperaturedependence of Eq 41.Moses et al. (2010) presume that HCN destruction iscontrolled by reaction of H with the H CN radical. Thisis a much faster reaction. To convert their discussion intoa reaction time requires defining the H CN equilibriumabundance. We write p H CN = K H CN · p HCN · p H with K H CN = 1 . × − e /T . This is likely not the sameas what Moses et al. (2010) use. Atomic and molecularhydrogen are also assumed to be in equilibrium. Otherpertinent information is given in Moses et al. (2010). Thereaction time scale that results is t chem = 8 . × exp (23358 /T ) T . N p . f . sec . (42)Figure 14 shows that Eq 42 used in a quench approxima-tion predicts much less HCN than we compute in our 1Dmodels. This means that reactions destroying HCN areoccurring at relatively low temperatures. This fits withthe relatively weak temperature dependence of t chem inEq 42.The comparison of models may be frustrated in partby a hole in our model. We did not include methylamine (CH NH ), which Moses et al. (2010) argue plays thesame role in hydrogenation of HCN at low temperaturesand high pressures that methanol plays for CO. Theirscheme is plausible but almost entirely hypothetical be-cause it passes through several free radicals that mustexist but about which little else is known. Our omissionof a CH NH channel implies that Eq 40 overestimatesHCN, especially in cool worlds. Another issue under-mining comparison between models is that we have notimplemented Moses et al. (2010)’s full quench scheme:Moses et al. (2010) require that HCN quench with re-spect to already quenched abundances of CH and NH .But at the temperatures at which HCN might actuallybe abundant enough to be detectable in EGPs and BDs,our prescription should work and is easy to use. CO In principle CO is also subject to quenching (Prinnand Fegley 1987), with caveats. First, because CO quenches at a lower temperature than CO, it quencheswith respect to the disequilibrium (quenched) abundanceof CO. Second, in practice, CO can be much enhancedby photochemistry if the world in question is subjectto significant stellar irradiation. Under such conditionsquenching is a poor guide. But for solitary brown dwarfsand planets in wide orbits it should do fine.The approach is similar to that used for CO and CH above. Equilibrium between CO , CO, H , and H O canbe approximated by K CO = p CO · p H O p CO · p H = 18 . (cid:16) − /T − (932 /T ) (cid:17) . (43)The equilibrium product Eq 43 is evaluated using thequenched values of p CO and p H O, beginning at thealtitude where CO and CH quench, and then extend-ing to all higher altitudes. CO quenching is pinnedat the altitude where the equilibrium defined by Eq 43breaks down. Figure 15 shows the results of doing sofor the ensemble of models. Figure 15 is noisy becausethe deviations from equilibrium are modest and can goin either direction. It is interesting that the quenchingtimescale t q varies inversely with the square root of thequench pressure p q , and that unlike the CO-CH systemthere is no discernible dependence on metallicity. Thetemperature dependence in the Arrhenius-like relation t q · p . q ∝ exp (38000 /T ) is similar to the other cases wehave looked at involving CO, which is also notable.Figure 15 shows that CO abundances in the ensembleof models are well approximated by a quench model pro-vided that the appropriate disequilibrium CO and H Omixing ratios are used. A chemical reaction timescalethat works well for CO quenching is t CO = 1 . × − p − . exp (38000 /T ) sec (44)where p is in bars. The results shown in Fig 15 are ratherinsensitive to the Arrhenius A factor in t CO , which per-haps is to be expected given the weak temperature de-pendence of the equilibrium constant Eq 43 compared tothe very strong temperature dependence of Eq 44. DETECTABILITY Whether or not a molecule can be detected dependson the abundance and opacity of the species in questionlanetary Disequilibrium Chemistry 15 Q uen c h t i m e t q · p [ ba r · s e c ] Quench Temperature 1000/T q Q uen c h t i m e t q · p . [ ba r . · s e c ] Quench Temperature 1000/T q Fig. 13.— HCN quenching time scales determined from the ensemble of models with respect to NH and CH (red m = 1 andblack m = 3) and CO (green m = 1 and violet m = 3). Symbol areas are proportional to f HCN . Left. The time scale plotted is t q ∝ p − q . Right. High T q results fit better to the Arrhenius form with time scale t q ∝ p − . q . Q uen c h f ( HCN ) − − − − − − HCN Mixing Ratio f(HCN)10 − − − Q uen c h q ( HCN ) − − − − − − − − HCN Mixing Ratio f(HCN)10 − − − − − Fig. 14.— HCN mixing ratios predicted by various quenching approximations (y-axes) plotted against the actual asymptoticHCN mixing ratio as computed by the full ensemble of 1-D kinetics models (x-axes). A perfect approximation would adhere tothe line. Left. Quench predictions using Eq 40. Open red symbols are for m = 1 and solid black symbols are for m = 3. Right. Quench approximations using Eq 41 with L = H (open green circles, m = 1; filled black circles, m = 3) and using Eq 42 with L = 0 . H (open red diamonds, m = 1; filled blue circles, m = 3). and on the opacities of other molecules and clouds. Forreference, Figure 16 shows absorption cross sections at650 K and 1 bar pressure for H O, NH , CH , HCN,and CO. These can be compared to illustrative columndensities shown in Figure 17. The latter are integratedupward from the 1, 0.1, and 0.01 bar pressure level, typ-ical near-IR photospheric pressures for planets or browndwarfs with gravities of 10 , 10 , and 10 cm s − , respec-tively. For CO, CH , CO , and HCN we plot column densities for only one value of K zz for each ( g, T eff ) pair.We arbitrarily select K zz = 10 /g , a high value of K zz but consistent with Eq 3, for the illustration. For N andNH we plot column densities for all K zz to emphasizehow little these depend on mixing.While only a complete model spectrum can definitivelypredict the visibility of each molecule given a set of as-sumptions, we can use Figure 16 together with Figure 17to make some generalizations. We defer the task of prop-6 Zahnle & Marley Q uen c h T i m e , t q · p . [ ba r . · s e c ] Quench Temperature, 1000/T q Q uen c h A pp r o x i m a t i on f ( C O ) − − − − − − Full Model Computation f(CO )10 − − − Fig. 15.— Left. CO quenching time scales determined from the ensemble of models. Here quenching refers to disequilibriumbetween CO and CO and H O; CO and H O are themselves not in equilibrium because they have already quenched. The timescale plotted is t q × p . q . Right. f CO predicted by the quenching approximation (y-axis) plotted against the actual asymptoticmixing ratio determined from the full ensemble of 1-D kinetics model (x-axis). A perfect approximation would adhere to theline. Open red symbols are for m = 1 and solid black symbols are for m = 3. erly including our new chemical network into a completemodel atmosphere to the future. CH H ONH HCNCO C r o ss s e c t i on [ c m ] − − − − − − − Wavelength [microns]1.0 1.5 2.0 2.5 3.0 3.5 4.0 Fig. 16.— Absorption cross sections of molecules of interesthere, computed at 650 K and 1 bar pressure, for wavelengthsbetween 1 and 4 microns. UCL opacities are provided by theUniversity College London and the Hitemp opacities are fromthe HITRAN opacity database (Rothman et al. 2003). The absorption cross section per molecule for the COband head at 2 . µ m is about 10 − cm per molecule.We know from model comparisons to data (e.g., Cushinget al. 2008) that by T eff = 1100 K at g = 10 this bandhead produces only a slight spectral feature. ConsultingFigure 17 and scaling appropriately we expect, all elsebeing equal, that CO would remain detectable down to T eff ∼ 900 K at g = 10 and 500 K at g = 10 . Thistemperature is considerably cooler than expected by theconventional wisdom.We likewise can consider the appearance of the 2 . µ mmethane band which by definition first appears at theT0 spectral type at T eff ∼ g = 10 planetuntil T eff ∼ 650 K. The directly imaged companion GJ504 b (Kuzuhara et al. 2013; Janson et al. 2013) providesa test of this reasoning. The mass and effective tempera-ture of GJ 504 b have been estimated at ∼ J and 600K. Its gravity should be ∼ − . By the reasoningabove methane would be expected and indeed has beendetected (Janson et al. 2013).Methane is far more detectable in the 3 . µ m funda-mental band where the absorption cross section is overan order of magnitude larger. CH has been detectedhere as early as spectral type L5 (Noll et al. 2000) infield dwarfs. Likewise spectra over this wavelength rangewould better help to constrain the arrival of methane atlow gravity in the directly imaged planets. Indeed Ske-mer et al. (2013) report a possible detection of this samefeature in HR 8799 c, suggesting a trace abundance inthe upper atmosphere.Another test of these ideas may be provided by field Tor Y dwarfs with unusual colors not predicted by models.Faint, red objects in particular would be good candidatesfor the low gravity, methane-poor objects we predict. Forexample the cool Y dwarf WISE 1828+2650 (Cushing etal. 2011), is both faint and unusually red in J − H color(Beichman et al. 2014). While Cushing et al. did detectmethane in this object, chemical equilibrium models doa very poor job fitting the available photometry (Beich-man et al. 2014). Near-IR spectra should be obtainedof all such color outliers to search for methane spectralfeatures.We note that some searches for planetary companionsto young stars (e.g., Liu et al. 2010) as well as surveysfor ‘planetary-mass’ T-dwarfs in young stellar clusters(e.g., Parker & Tinney 2013) have employed differential-methane band imaging. In this technique (Tinney et al.lanetary Disequilibrium Chemistry 17 CO CH CO C o l u m n D en s i t y [ c m - ] T eff 500 600 700 900 1100 N NH HCN C o l u m n D en s i t y [ c m - ] T eff 500 600 700 900 1100 Fig. 17.— A sequence of models that illustrates how column densities of important carbon-bearing species change withtemperature T eff and gravity g . K zz is restricted to 10 /g for CO, CH , CO , HCN. All K zz are plotted for N and NH .Columns are integrated above p = 10 − · g bars. Small symbols represent g = 1000, mid-sized symbols g = 10 , and big symbols g = 10 . Left. CO (black circles), CH (green diamonds), CO (blue squares). Right. N (blue diamonds), NH (red circles),HCN (gold squares). . − µ m CH band and one which probesthe entire H band. When the two images are differenced,methane-bearing objects stand out as they are dark inthe CH filter. Our conclusions here suggest that suchtechniques must be used with caution as methane maysimply not yet be present in planetary-mass objects evenat effective temperatures below 1000 K.Similar arguments can be made for the appearance ofother spectral features of interest. We expect that NH will appear at effective temperatures about 200 K coolerin planets compared to field brown dwarfs. HCN, with across section of 10 − cm per molecule at 1 . µ m is un-likely to be detectable in most objects as the computedcolumn abundances are less than 10 cm − . HCN’sprospects are poor at 3.0 microns despite a high cross-section unless the C/O ratio is higher than solar andwater’s abundance reduced. Water’s cross section is100-fold smaller, but with solar abundances its column(1 − × cm − ) is 1000 times what HCN can reach atits best. Carbon dioxide has an absorption cross section10 − cm per molecule at 4 . µ m, and with abundancesapproaching 10 cm − we expect it to be detectable ataround 900 to 1100 K in field brown dwarfs. Indeed theAKARI space telescope discovered CO features in sev-eral late L and early T dwarfs (Yamamura et al. 2010).Judging by Figure 17, we would predict CO to be de-tectable to 500 K and cooler in the lowest mass planets. CONCLUSIONS We use a reasonably complete 1-D chemical kineticscode to survey the parameter space that encompasses at-mospheres of cool brown dwarfs and warm young extra-solar giant planets. Our model contains only gas phasechemistry of small molecules containing H, C, N, O, andS. We use realistic p - T profiles for cloudless atmosphereswith effective temperatures between 500 and 1100 K andsurface gravities between 10 cm/s to 10 cm/s . Verti- cal transport is described by an eddy diffusivity K zz thatwe vary over a wide range. Our objective is to describecarbon and nitrogen speciation, especially at lower (plan-etary) surface gravities. Overviews of what we found arepresented in Figure 4 for carbon and Figure 10 for nitro-gen.We find that carbon in cloudless brown dwarfs is pre-dominantly in the form of methane at 900 K for g = 10 cm/s . The small surface gravity of planets strongly dis-criminates against CH when compared to an otherwisecomparable brown dwarf. If vertical mixing is compara-ble to Jupiter’s, methane first predominates over CO inplanets cooler than 500 K. Sluggish vertical mixing canraise the transition to 600 K; clouds or more vigorousvertical mixing could lower it to 400 K.The detectability of specific molecular features in aspectrum depends on the strength of the molecular ab-sorption cross sections as well as the gaseous abundance.Nevertheless a natural prediction of our model is thatthere will be cool planets with no methane observedin the H or K spectral bands. The refractory behav-ior of CO in low gravity objects is likely at least par-tially responsible for the lack of cool planets found bythe NICI survey, which relied upon the methane absorp-tion H band to identify planets (Liu et al. 2010).Ammonia is also sensitive to gravity, but unlikemethane and CO, ammonia is insensitive to mixing,which makes it a proxy for gravity. We did not ex-plore temperatures low enough to determine the tran-sition from N to ammonia in planets, but it is nearlyas abundant as N at 500 K in brown dwarfs, which isbroadly consistent with the observed properties of theY dwarfs (Cushing et al. 2011) for which NH is seenin H band. On the other hand, ammonia persists as anabundant minor species to rather high temperatures andthis is consistent with it being readily detected in mid-IRspectra of T-dwarfs (Cushing et al. 2006). HCN might8 Zahnle & Marleybecome interesting in high gravity brown dwarfs if verti-cal mixing is very vigorous.When expressed in terms of quenching parameters,nearly all our results for CO and CH can be reducedto a simple equation in Arrhenius form that is easy touse but not easy to interpret. From a strictly practicalperspective, what we find is close to what one gets us-ing Prinn and Barshay (1977)’s algorithm in its originalform (a fortuitous accident of using too high a rate forthe wrong reaction). The apparent simplicity is some-what surprising given the complexity of the chemistryinvolved and the many different ways the system hasbeen described in the literature. Nor is our result quitewhat we expected. First, we find that the timescale t CO for hydrogenation of CO is shorter for higher metallic-ity. Because the forward reactions that destroy CO haveno metallicity dependence, we infer that metallicity de-pendence enters through the reverse reaction (oxidationof CH ). By definition the reverse reaction is as fast asthe forward reaction while in equilibrium, but it falls offmore quickly than the forward reaction as the tempera-ture drops. The important forward reactions are linearin metallicity whilst the important reverse reactions arequadratic in metallicity.Second, we find that t CO is inversely proportional topressure. Analyses based on isolating a limiting reactionpredict a stronger dependence on pressure, typically p − .The weaker p − pressure dependence is expected for areaction involving CO itself, such asH + CO → HCO + H . (45)It is interesting that R45 actually does have the highest∆ G value in the CO hydrogenation sequence, and there-fore the highest energy barrier, and that the magnitudeof ∆ G of R45 is what we infer in Eq 14. But it is alsoquite clear from Figure 7 that equilibrium breaks downbetween CH OH and CH well before if breaks down be-tween CO and CH OH, as was found by Moses et al.(2011). Reactions involving CH OH imply a p − pres-sure dependence, because the equilibrium abundance ofCH OH with respect to CO goes as the square of the H pressure. The system behaves—both in the 40,000 K en-ergy barrier and the p − pressure dependence— as if theoverall rate depends on the entry reaction R45 ratherthan on the rate of R23 in which equilibrium actuallybreaks down. The apparent paradoxes might be resolvedthrough a more complete analysis of the reverse reac-tions, or they may result from systematic biases stem-ming from our treating the mixing length as equal to ascale height, rather than as the complex function of manyvariables that it probably is (Smith 1998).For nitrogen and ammonia we were unable to recoveran emergent Arrhenius-like behavior from the full sys-tem. On the other hand, it was easy to devise a goodquench approximation, and we also found that some pub-lished quench approximations (e.g., Fegley and Lodders1994) work very well. For HCN, we met with mixed success. We did find emergent Arrhenius-like behav-iors, and we developed a new quench approximation thatworks quite well for our models. On the other handthere does not yet appear to be a consensus on whatthe quench chemistry of HCN actually is. We found awide scatter of different outcomes when comparing dif-ferent published quench schemes. We think our quenchapproximation should work well for warm objects whereHCN is predicted to be relatively abundant but, becausewe did not consider hypothetical hydrogenation channelsthrough methylamine (Moses et al. 2010), our modelsmay overestimate f HCN , especially in cooler objects.There are many effects that our models do not address.None of these effects have much sway over our quenchapproximations, but they have much to do with whatmight actually be present in a real atmosphere.Clouds are likely the most important. Adding infraredopacity to an atmosphere is effectively equivalent to re-ducing the gravity. This is because opacity raises the p - T profile (i.e., the adiabat) to a lower pressure for a giveneffective temperature. A cloudy atmosphere favors COand N over CH and NH , other things equal. Increas-ing metallicity also increases opacity to the detriment ofmethane and ammonia.Another issue for real worlds is that K zz may be muchsmaller in higher altitude radiative regions of the atmo-sphere than it is in the convecting region. The radiativestratosphere is the part of the atmosphere where K zz the modeling parameter is most suspect. For N andNH pronounced vertical structure in K zz will not mat-ter much. For CO and CH , sharply lower values of K zz in the stratosphere might be important, because if K zz is low enough, the atmosphere can have a second quenchpoint in the stratosphere. Such behavior no problem to afull kinetics model, but it can make implementing quenchapproximations more complicated.Photochemistry induced by irradiation from a nearbystar can deplete NH and increase CO and HCN. Light-ning and/or impact shocks in cool NH - and CH -richatmospheres can generate CO and HCN (Chameides andWalker 1981). 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