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Draft version April 29, 2010Preprint typeset using LATEX style emulateapj v. 2/16/10

THE DAYSIDE ATMOSPHERE OF HOT-NEPTUNE GJ 436B

N. Madhusudhan1,2 & S. Seager2,3

Draft version April 29, 2010

ABSTRACT

We present a detailed analysis of the day-side atmosphere of the hot Neptune GJ 436b, based onrecent Spitzer observations. We report statistical constraints on the thermal and chemical properties ofthe planet atmosphere, study correlations between the various molecular species, and discuss scenariosof equilibrium and non-equilibrium chemistry in GJ 436b. We model the planet atmosphere witha 1-D line-by-line radiative transfer code with parametrized molecular abundances and temperaturestructure. We explore the model parameter space with 106 models, using a Markov chain Monte Carloscheme. Our results encompass previous findings, indicating a paucity of methane, an over-abundanceof CO and CO2, and a slight under-abundance of H2O, as compared to equilibrium chemistry with solarmetallicity. The concentrations of the species are highly correlated. Our best-fit constraints requirea CH4 mixing ratio ∼ 10−7 − 10−6, with CO ≥ 10−3, CO2 ∼ 10−6 − 10−4, and H2O ≤ 10−4; higherCH4 would require much higher CO and CO2. Using calculations of equilibrium and non-equilibriumchemistry, we find that the observed high CO abundance can be explained with a combination ofhigh metallicity (30 × solar) and eddy mixing (with Kzz ∼ 106), whereas the low CH4 abundance canpotentially be explained by photochemistry. Our constraints rule out a day-side thermal inversion inGJ 436b. We emphasize that the constraints reported in this work depend crucially on the observationsin the two Spitzer channels at 3.6 µm and 4.5 µm. Future observations with warm Spitzer and JWSTwill be extremely important to improve upon the present constraints on the abundances of carbonspecies in the dayside atmosphere of GJ 436b.

Subject headings: planetary systems — planets and satellites: general — planets and satellites: indi-vidual (GJ 436b) — radiative transfer

1. INTRODUCTION

The last decade in exoplanetary science has demon-strated our capability in detecting and characterizing at-mospheres of transiting extrasolar giant planets. Begin-ning with the first detection of sodium in the atmosphereof HD 209458b in transit (Charbonneau et al. 2002),and the first detections of dayside thermal emission fromhot Jupiters TrES-1 and HD 209458b (Charbonneau etal. 2005; Deming et al. 2005), atmospheric observationsof extrasolar giant planets today are a norm, albeit stillvery challenging (e.g. Knutson et al. 2008; Charbonneauet al. 2008; Swain et al. 2008; Grillmair et al. 2008;Desert et al. 2009; Swain et al. 2009). The intensityof observational efforts have been matched with equallychallenging accomplishments in theoretical modeling ofexoplanet atmospheres, and data interpretation (Seager& Sasselov, 2000; Seager et al. 2005; Burrows et al. 2006& 2008; Fortney et al. 2006; Barman et al. 2007; Tinettiet al. 2007; Showman et al. 2009; Madhusudhan & Sea-ger, 2009). Several inferences of gaseous H2O, CH4, COand CO2, and thermal inversions have subsequently beenmade in hot Jupiter atmospheres (Burrows et al. 2007;Tinetti et al. 2007; Barman et al. 2008; Fortney et al.2008; Grillmair et al. 2008; Swain et al. 2008; Mad-husudhan & Seager, 2009; Swain et al. 2009a & 2009b).A new era in exoplanetary science has now dawned.

Latest observations are leading to discovery and char-

1 Corresponding author:nmadhu@mit.edu2 MIT Kavli Institute for Astrophysics and Space Research,

and Department of Earth, Atmospheric, and Planetary Sciences,MIT, Cambridge, MA, 02139

3 Department of Physics, MIT, Cambridge, MA 02139

acterization of transiting exoplanets much less massivethan the archetypal hot Jupiters, namely extrasolar Nep-tunes and super-Earths. Five such low-mass transitingplanets are known to date, including three hot Neptunesand two super-Earths, opening a new regime in atmo-spheric modeling and data interpretation (Butler et al.2004; Maness et al. 2007; Bakos et al. 2010; Leger etal. 2009; Charbonneau et al. 2009; Borucki et al. 2010).However, atmospheric observations have been reportedfor only one of these planets so far, GJ 436b (Deming etal. 2007, Pont et al. 2009, Stevenson et al. 2010).The first transiting hot Neptune known, GJ 436b, with

a mass of 22.6 M⊕ and radius of 4.2 R⊕, orbits an MDwarf at an orbital separation of 0.03 AU (Butler et al.2004; Gillon et al. 2007; Maness et al. 2007). The aver-age density of the planet is 1.5 g/cc (Torres et al. 2008),i.e. close to the density of water ice, and similar to thebulk density of Neptune. At atmospheric temperatureswell above∼ 700 K, the density originally hinted of a hot-ice interior (Gillon et al. 2007). Refined radius estimatesand more detailed studies of the possible bulk composi-tion of GJ 436b indicate that an additional layer of H/Heatmosphere would be needed to account for the observedradius (Figueira et al. 2009; Rogers & Seager. 2009).The atmosphere of GJ 436b has been a subject of sub-

stantial interest in the recent past. Deming et al. (2007)and Demory et al. (2007) reported independent detec-tions of thermal emission from the dayside of GJ 436bin the Spitzer 8 µm IRAC channel. Although no mean-ingful inferences about the molecular compositions canbe drawn from a single data-point, model fits to the 8µm flux contrast have favored the interpretation of inef-

2

ficient day-night energy redistribution for GJ 436b (De-mory et al. 2007; Spiegel et al. 2010, unless there are ad-ditional unknown optical absorbers in the atmosphere).The models used in Demory et al. 2007 and Spiegel et al.2010 had assumed equilibrium chemistry. Given the lowtemperatures of GJ 436b (Teq ∼ 750 - 800 K ), comparedto hot Jupiters, equilibrium chemistry suggests that theplanet atmosphere must be abundant in methane andwater vapor, and be scarce in carbon monoxide (e.g Bur-rows and Sharp, 2009).Recent observations have suggested distinct depar-

tures from predictions of equilibrium chemistry models.Pont et al. (2009) reported a transmission spectrum ofGJ 436b obtained in the 1.1-1.9 µm bandpass using theHST NICMOS instrument, but found no significant fea-ture in the 1.4 µm water band; the spectrum was flatat the 2-σ uncertainties. More recently, Stevenson et al.(2010) reported planet-star flux contrasts of the day-sideatmosphere of GJ 436b in six channels of Spitzer broad-band photometry, and inferred a deficiency of methane inthe atmosphere of GJ 436b, using models based on Mad-husudhan & Seager (2009) and the present work. Eventhough our models do not assume equilibrium chemistry,the inferred methane mixing ratio of ∼ 10−7 in a hydro-gen dominated atmosphere at ∼ 700K signals a surpris-ing new regime in atmospheric chemistry for extrasolarplanets.In this work, we report detailed statistical constraints

on the atmospheric properties of GJ 436b, and explorechannels of equilibrium and non-equilibrium chemistrythat might explain the observed chemical abundances.We first estimate the atmospheric chemical compositionand temperature structure at different levels of fit to thedata, using a 1-D line-by-line radiative transfer model forexoplanet atmospheres (Madhusudhan & Seager, 2009).Our constraints result from exploring the model param-eter space with ∼ 106 models, optimized using a Markovchain Monte Carlo optimization scheme. We then usedetailed calculations of equilibrium and non-equilibriumchemistry, and with different metallicities, to explain theobserved constraints on the various molecular species.Our results indicate that a high planet metallicity com-bined with non-equilibrium chemistry could potentiallyexplain the molecular abundances required by the obser-vations. We also place important constraints on otherobservable parameters like the day-night energy redistri-bution in GJ 436b, which has consequences for the circu-lation patterns and potential interior energy sources inthis planet.In what follows, we first give a brief overview of model

independent interpretation of Spitzer observations in sec-tion 2. We then describe our atmosphere model, the pa-rameter space optimization scheme, and the models forequilibrium and non-equilibrium chemistry we use in thiswork, in section 3. We present the results of our analysisin section 4. Finally, in section 5, we present a summaryof our work and discuss consequences for future observa-tions and theoretical models.

2. OVERVIEW OF SPITZER DATA INTERPRETATION

The presence of molecular species in a planetary atmo-sphere can, in principle, be derived from observations inthe six channels of Spitzer photometry, at 3.6 µm, 4.5 µm,5.8 µm, 8 µm, 16 µm, and 24 µm. Figure 1 shows a listing

Fig. 1.— Molecular absorption features in Spitzer photometricbandpasses. The red dotted lines at the top show the six Spitzerbandpasses. The black lines show the extent of absorption fea-tures due to the corresponding molecules. The gray curve showsa hypothetical model spectrum of GJ 436b based on equilibriumchemistry, and the black filled circles show the corresponding in-tegrated points in the Spitzer channels. The red filled circles witherror bars show the observations of GJ 436b reported by Stevensonet al. (2010).

of the key molecules in the Spitzer channels, along witha synthetic spectrum. Despite having only 6 broad-bandphotometric measurements, identification of molecules ispossible because most molecules have strong features inonly one or two of Spitzer’s photometric bands. CH4 hasstrong features almost exclusively in the 3.6 µm and 8 µmchannels. CO has a strong feature in the 4.5 µm channel,also contributing to the 5.8 µm channel. And, CO2 hasstrong features in the 4.5 µm and 16 µm channels. Al-though H2O has several spectral features in most of theSpitzer channels, the stronger and/or unique features liein the 3.6 µm, 5.8 µm and 24 µm channels.The fact that there are strong spectral features of one

or more of the major species in each of the six Spitzerchannels, high S/N observations can yield remarkableconstraints on the molecular concentrations. One wayto look at this is that, under the assumption that H2,CH4, CO and CO2, are the dominant spectroscopicallyactive molecules in the IR, there are four molecules andsix data points. The temperature gradient contributesan additional free parameter, thus still leading to theprospect of placing statistically meaningful constraintson the four molecular species. Furthermore, the samemolecular species provide constraints on the C/H andO/H ratios in the atmosphere, which serve as probes ofthe overall metallicity.As an example of constraining molecular abundances

from the Spitzer data, consider a planet atmosphere withtemperature decreasing with altitude and a low planetflux in the 3.6 µm channel. This low flux suggests an ab-sorption band due to a high concentration of methane.To confirm a high methane abundance (since water va-por also absorbs weakly at 4.6 µm), the flux should alsobe low in the 8 µm channel where methane also absorbsstrongly. In contrast, high planet fluxes in the 3.6 and 8µm channels indicates a low absorption due to methane,and hence a paucity of methane in the planet atmo-sphere.

Dayside atmosphere of GJ 436b 3

As a second example of how only 6 photometric pointscan constrain molecular abundances we consider CO andCO2. Low fluxes in the 4.5 µm channel must indicate at-mospheric absorption due to CO and/or CO2. At 4.5 µm,CO2 has stronger than the CO, so that a small concen-tration of CO2 can produce an absorption feature com-parable to that from a relatively large concentration ofCO. The degeneracy between CO and CO2 contributionsin a 4.5 µm measurement can be broken by an observa-tion in the 16 µm channel, where only CO2 contributes.As a third example, constraints on H2O are based onfluxes in the 5.8 µm and 24 µm channels. An importantnote concerns the presence of a thermal inversion in theatmosphere, where temperature increases with altitude.In such a case, the molecular features would be emissionfeatures instead of absorption features, thus reversing thelogic of inferences described above (Madhusudhan & Sea-ger, 2010).The recent observations of the day-side atmosphere

of GJ 436b by Stevenson et al. (2010) represent aquintessential example of the above inferences. Their ob-servations indicate extremely high fluxes in the 3.6 µmchannel and extremely low fluxes in the 4.5 µm channel,causing a brightness temperature differential of ∼ 450K. Thus, following the arguments described above, theobservations point towards an extremely low methaneabundance and high CO and/or CO2 abundances, as-suming there is no thermal inversion. This identificationof low CH4 and high CO and/or CO2 was in fact thecentral result of Stevenson et al. (2010), based not onthe qualitative description here but using a much moresophisticated atmospheric modeling procedure.

3. MODEL

Our goal is to determine the best fitting interpretationfor observations of the day-side atmosphere of GJ 436b.We first fit the data with a large ensemble of 1-D daysideatmosphere models of GJ 436b, and determine regions ofthe parameter space that fit the data best. Our resultsyield best-fit constraints on the molecular abundancesand temperature structure. We then use some of thebest-fit temperature profiles, along with independent cal-culations of equilibrium and non-equilibrium chemistryto see if we can explain the observed constraints on themolecular abundances.

3.1. Radiative Transfer Model

In order to fit the observations with model spectra,we use the 1-D exoplanet atmosphere model developedin Madhusudhan & Seager (2009). Our model consistsof a line-by-line radiative transfer calculation, with con-straints of hydrostatic equilibrium and global energy bal-ance, and coupled to a parametric pressure-temperature(P-T) structure and parametric molecular abundances.This modeling approach allows one to compute large en-sembles of models, and efficiently explore the parameterspace of molecular abundances and temperature struc-ture.The major difference of our model from traditional 1-D

atmosphere models is in the treatment of energy balance.Our model requires energy balance at the top of the at-mosphere, instead of an iterative scheme to ensure layer-by-layer radiative (or radiative + convective) equilibriumwhich is assumed in conventional models (e.g. Seager et

Fig. 2.— Pressure-Temperature (P-T) profiles of GJ 436b.Thepurple, red, orange, green and blue profiles correspond to modelswhich fit the observations to within ξ2 of 1.0, 2.0, 3.0, 4.0 and 5.0,respectively.

al. 2005, Burrows et al. 2006, Fortney et al. 2006). Fora given set of model parameters, we require that the netenergy output at the top of the atmosphere is less thanor equal to the net energy input due to the incident stel-lar flux; a deficit indicates energy redistributed to thenight-side. Models where the emergent flux exceeds theincident flux are discarded (see Madhusudhan & Seager,2009).In this work, we consider well mixed atmospheres, i.e.

uniform mixing ratio of each molecular species over theentire atmosphere. The present approach allows us tosample a wider range of compositions independent of anyassumptions of equilibrium chemistry. The molecularspecies in our models include molecular hydrogen (H2),water vapor (H2O), carbon monoxide (CO), carbon diox-ide (CO2), and methane (CH4). Our H2O, CH4, and COmolecular line data are from Freedman et al. 2008, andreferences therein. Our CO2 data are from Freedman(personal communication) and Rothman et al. (2005).And, we obtain the H2-H2 collision-induced opacitiesfrom Borysow et al. (1997), and Borysow (2002).

3.2. Parameter Space Exploration for Model Fits

We use the Markov chain Monte Carlo (MCMC)method to explore the model parameter space. TheMCMC method is a Bayesian parameter estimation algo-rithm which allows the calculation of posterior probabil-ity distributions of the model parameters conditional to agiven set of observations, and prior probabilities (see e.g.

4

Fig. 3.— Constraints on the atmospheric properties of GJ 436b. The contours show surfaces of minimum ξ2 in the space of atmosphericcomposition and temperature structure. The purple, red, orange, green and blue, correspond to minimum ξ2 of 1.0, 2.0, 3.0, 4.0 and 5.0,respectively. The gray and black regions correspond to models which have methane mixing ratios greater than 10−7, along with differentconditions on the compositions of the remaining molecules, and allowing fits within ξ2 = 3. For the gray surfaces, CO2 ≤ 10−3, H2O≥ 10−5, and CO≤ 10−1. Our most plausible interpretation is represented by the black surfaces, CO2 ≤ 10−4, H2O ≥ 10−4, and CO≤ 10−2

(see section 4.1.1). Mixing ratios are shown as ratios by number density.

Tegmark et al. 2004; Ford et al. 2005). In this work,our goal is not parameter estimation – the number ofmodel parameters (N = 10) exceeds the number of avail-able observations (Nobs = 6) and rendering the problemunder-constrained . However, the MCMC method al-lows an efficient means of exploring the parameter spacein search of regions which provide the best fits to the ob-servations. We, therefore, use the MCMC method witha Metropolis-Hastings scheme within the Gibbs sampler,

for fine sampling of the model parameter space . And,we report error contours in the space of the molecularcompositions and temperature structure. Our statistic ofchoice is ξ2, defined as χ2/Nobs (Madhusudhan & Seager,2009). In this metric, a ξ2 = 1 indicates models fittingthe observations within the 1-σ observational uncertaini-ties, on average. Similarly, ξ2 of 2 and 3 indicate fits atthe 1.41-σ (i.e

√2) and 1.73 -σ error bars, respectively.

Our model described in § 3.1 above has ten free pa-

Dayside atmosphere of GJ 436b 5

Fig. 4.— Observations and model spectra for day-side emission from GJ 436b. The black filled circles with error bars show the Spitzerobservations in the six photometric channels, from Stevenson et al. 2010. The blue dashed lines shows a planet blackbody spectra at 800K. The green and red curves show two model spectra, and the colored circles show the corresponding channel integrated model points.The green model is a best-fit model spectrum (see section 4.1.1, with molecular mixing ratios of H2O = 10−4 , CO = 7 × 10−3, CH4

= 10−6, and CO2 = 6× 10−6. The red model has a composition close to solar abundance, with H2O = 2× 10−3 , CO = 10−5, and CH4

= 7 × 10−4; it also contains CO2 = 10−6, which is allowed by photochemistry (e.g. Zahnle et al. 2009). Both models have the samepressure-temperature profile, shown in the inset.

rameters (Madhusudhan & Seager, 2009). Six parame-ters concern the P -T profile: T0, P1, P2, P3, α1, and α2 .And, four parameters correspond to the uniform molecu-lar mixing ratios: fH2O, fCO, fCH4

, and fCO2. We define

the mixing ratio of a molecule as the number fractionwith respect to molecular hydrogen.We define some physically motivated boundaries in the

parameter space explored by the Markov chain. Werestrict all the molecular mixing ratios to a range of[10−10, 0.1]. We also impose the constraint of globalenergy balance by restricting η to [0.0,1.0], where, η =(1 − A)(1 − fr) is the ratio of emergent flux output onthe day-side to incident stellar flux input on the day-side, weighted appropriately (Madhusudhan & Seager,2009). Here, A is the Bond Albedo and fr is the day-night energy redistribution. The “fit” parameters forthe MCMC are T0, log(P1), log(P2), log(P3), α1, α2,log(fH2O), log(fCO), log(fCH4

), and log(fCO2). We con-

sider uniform priors in all the parameters. For each sys-tem under consideration, we run one chain of 106 links,which takes ∼ 22 hours on a single processor.

3.3. Chemistry Model

After we obtain the constraints on the molecular abun-dances from model fits to data, we investigate processesof atmospheric chemistry that could explain the required

abundances. Observations of giant planet atmospheresin the solar system and those of brown dwarfs have re-vealed a complex interplay between equilibrium and non-equilibrium chemical processes. At high pressures deepin a planetary atmosphere, molecules react fast enoughthat all species are in thermochemical equilibrium. Asthe pressures decrease with increasing altitude, thermo-chemical reaction rates decrease, allowing for compet-ing non-equilibrium processes with shorter timescales toshift the involved species out of equilibrium. We com-pute the atmospheric compositions in equilibrium usingthe equilibrium chemistry code adapted from Seager etal. (2005).

3.3.1. Equilibrium chemistry

Our objective here is to determine whether the ob-served constraints on the molecular mixing ratios areconsistent with chemical equilibrium. To this end, wecalculate the equilibrium compositions of the species us-ing the equilibrium chemistry code developed in Seageret al. (2005) and Miller-Ricci et al. (2009). We calculatethe gas phase molecular mixing ratios for 172 molecules,resulting from abundances of 23 atomic species, by min-imizing the net Gibbs free energy of the system. Themulti-dimensional Newton-Raphson method described inWhite et al. (1958) was used for the minimization.

6

We adopt polynomial fits for the free energies of themolecules, based on Sharp & Huebner (1990). We as-sume a hydrogen dominated atmosphere for GJ 436b,and we compute equilibrium concentrations of all thespecies at varying metallicites (see section 4.1.1), overa grid in pressure and temperature.At the temperatures of GJ 436b, the most abundant

and spectroscopically active molecules in the Spitzerbandpasses are H2O, CH4, CO and CO2. Ammonia(NH3) is also abundant, but does not contain major fea-tures in the Spitzer channels. At high temperatures, e.g.& 1300 K at 1 bar pressure, like those in hot Jupiteratmospheres, CO is predicted to be the dominant car-bon and oxygen bearing species. At lower temperatures,on the other hand, CH4 is the dominant carbon bearingspecies in equilibrium. Water vapor is a major carrier ofoxygen in either regime. The specific amounts of each ofthese species also depend strongly on the assumed metal-licity, and pressure. Finally, the amount of carbon diox-ide in equilibrium is a very strong function of metallicity.At the temperatures of GJ 436b, solar metallicity yieldsa CO2 mixing ratio up to 10−7, whereas amounts as highas 10−4 can be obtained for 30 × solar metallicity.

3.3.2. Non-equilibrium chemistry

Turbulent vertical motions can mix up molecularspecies over regions of the atmosphere, where thetimescales of vertical transport are shorter than thetimescales governing chemical equilibrium between therelevant species. Such mixing shifts molecular speciesaway from their equilibrium partitions. “eddy” diffu-sion” or “eddy mixing” has been known in the context ofatmospheres of solar system giant planets (e.g. Prinn &Bashay, 1976), and brown dwarfs (Griffith & Yelle, 1999;Saumon et al. 2002; Hubeny & Burrows, 2007). Addi-tionally, Cooper & Showman (2006) studied eddy mixingin hot Jupiter atmospheres. Eddy mixing offers a viableexplanation to the excess carbon monoxide discovered inhydrogen dominated atmospheres (see e.g Yung & De-more, 1999). The primary reaction governing the rela-tive abundances of CH4 and CO in equilibrium is givenby:

CO + 3H2 ⇋ CH4 +H2O. (1)

This reaction favors CO at high temperatures and CH4

at the low temperatures of GJ 436b. Thus, based on equi-librium chemistry, it is expected for CO to be dominantin hotter lower regions of the atmosphere (T & 1200K),and CH4 to be dominant at higher altitudes where tem-peratures are lower (T . 1000K). However, atmosphericspectra of solar system planets and cooler brown dwarfssuggest significant amounts of CO in the upper layers ofthe atmosphere (e.g. Griffith and Yelle, 1999; Yung andDemore, 2000; Saumon et al. 2004). This is achievedby eddy mixing which vertically transport CO from thelower regions of an atmosphere to the upper regions.Eddy mixing dominates when the mixing time scale

(τmix) is shorter than the chemical time scale (τchem) ofCO in the forward reaction in (1). The forward reactionin (1) in fact proceeds via multiple steps, and Yung etal. (1998) suggested the rate determining step in thereaction chain to be:

H + H2CO+M → CH3O+M (2)

The life time of CO is given by:

τchem ∼ [CO]

d[CO]/dt=

[CO]

kf [H][H2CO], (3)

where, kf is the rate constant for the forward reaction.kf is not known directly, but can be estimated fromthe reverse reaction rate constant (kr), which is knownfrom laboratory experiments, and the equilibrium con-stant (Keq) for the reaction (see Griffith & Yelle, 1999,for a detailed discussion). In the present study, we usethe following estimate of kf , based on Line et al. (2010):

kf = 5.77× 10−12T−1.2e(3327/T ) (4)

Then, τchem can be calculated from (3), using equilibriumconcentrations of the CO, H and H2CO.The mixing time (τmix) in radiative regions of the at-

mosphere is determined by the eddy diffusion coefficient(Kzz) and the atmospheric scale height (L), as

τmix ∼ L2

Kzz(5)

As is evident from the above discussion, both τmix andτchem vary with height in the atmosphere; although, τmix

varies to a lesser extent than τchem. But, while τchem in-creases towards higher levels in the atmosphere (i.e withdecreasing pressure), τmix increases in the opposite direc-tion. The pressure (p0) at which τmix = τchem is calledthe “quench” level. Above this pressure, i.e in deeper lay-ers of the atmosphere, the species are in chemical equi-librium, and below this pressure the concentration of thespecies (CO in this case) is fixed, or “quenched”, at theequilibrium value at p0. This yields a uniform mixingratio profile for the species for pressures below p0.Thus, a higher p0 implies that CO can be dredged up

from deeper levels in the atmosphere. And, since in equi-librium CO concentration increases with pressure, a highp0 implies higher concentration of CO in the upper layersof the atmosphere. It can be shown that, for a given τchemprofile, p0 increases monotonically with Kzz, thus corre-lating a high CO concentration in the upper atmospherewith a high Kzz (see for e.g. Griffith & Yelle, 1999).In the discussion here, we have assumed a temperatureprofile that increases monotonically with pressure. Thedependence of observed CO on Kzz deviates from thismonotonic behavior for more complicated temperaturestructures, for example in the presence of thermal inver-sions or partial isotherms, as will be shown in section 4.2.In this work, we explore the CO mixing ratios resulting

from different combinations of Kzz and metallicities. Weexplore values of Kzz between 102 - 1010, and metallic-ities of solar – 30 × solar. Our choices of metallicitiesare motivated by the constraints on CO and CO2 abun-dances which indicate high C/H and O/H ratios apriori.And, our range in Kzz encompasses values found in solarsystem planets and brown dwarfs. For comparison, plan-etary atmospheres in the solar system have Kzz rangingbetween 105 − 109 cm2/s, and Kzz for brown dwarf at-mospheres can be as low as 102 − 104 cm2/s (Prinn andBashay, 1977; Yung and Demore, 1999; Saumon et al.2003).

Dayside atmosphere of GJ 436b 7

Apart from eddy mixing, photochemistry can play avery important role in irradiated atmospheres. A de-tailed photochemical calculation is beyond the scope ofthe present work. Nevertheless, we make a preliminaryattempt to address whether photochemistry can causethe observed abundances in GJ 436b, based on photo-chemical models reported in the literature (e.g. Zahnleet al. 2010). Given a best-fit temperature profile ofGJ 436b, we obtain the mixing ratio profiles from the lit-erature in the same temperature range, and investigateunder what conditions the observed low methane abun-dance of GJ 436b can be explained (see section 4.2.2).

4. RESULTS

In this section, we present constraints on the atmo-spheric properties of the dayside atmosphere of GJ 436b,as placed by the six channel Spitzer photometry. Wefirst report constraints on the molecular abundances,and discuss the correlations between the various species.We then discuss the physical plausibility of the solu-tions, and present calculations of equilibrium and non-equilibrium chemistry attempting to explain the ob-served constraints. Finally, we present constraints on theatmospheric temperature structure and day-night energyredistribution. Our constraints follow from a detailedexploration of the model parameter space with ∼ 106

models, using the procedures described in section 3.2.Our modeling approach allows placing statistically ro-

bust model constraints from the data. The pressure-temperature (P -T ) profiles explored by our models areshown in Figure 2, color coded by their degree of fit todata. The goodness-of-fit contours in the space of atmo-spheric composition are shown in Figure 3. As mentionedin section 3.2, our statistic for the goodness-of-fit is givenby (Madhusudhan & Seager, 2009):

ξ2 =1

Nobs

Nobs∑

i=1

(

fi,model − fi,obsσi,obs

)2

, (6)

where, fi,model is the planet-star flux contrast of themodel in each channel, and fi,obs and σi,obs are the ob-served flux contrast and the 1-σ uncertainty, in thatchannel. Nobs is the number of observations. Here,Nobs = 6, corresponding to the six channels of Spitzerphotometry. In this metric, a ξ2 = 1 indicates modelsfitting the observations within the 1-σ observational un-certainities, on average. Similarly, ξ2 of 2 and 3 indicatefits at the 1.41-σ (i.e

√2) and 1.73 -σ error bars, respec-

tively.The constraints depend on the ξ2 surface one chooses

for interpretation, apart from any conditions of physi-cal plausibility one would like to impose on the models.We first report constraints at the ξ2 = 1 and ξ2 = 2levels, and with only the barest assumptions of physicalplausibility. We then discuss additional constraints thatresult from considering some nominal conditions of phys-ical plausibility, from equilibrium and non-equilibriumchemistry.

4.1. Constraints on Chemical Composition

The constraints on the molecular abundances arestrongly influenced by the correlations between them.The correlations between the molecules result from their

overlapping absorption features in the Spitzer channelsas described in section 2. We present constraints on themixing ratios of methane (CH4), water vapor (H2O), car-bon monoxide (CO), and carbon dioxide (CO2). Becausethe abundances of molecules are correlated, constraintson any molecule have to be discussed with respect toabundances of one or more of the remaining molecules.The constraints on all the molecules and the correla-

tions between them are shown in Figure 3.Methane CH4: Our results indicate a substantial

paucity of methane in the day-side atmosphere of GJ436b. Our results place an absolute upper-limit on themixing ratio of methane to be 3 × 10−6 − 6 × 10−6, forξ2 ranging between 1 – 5, and assuming nothing aboutthe remaining molecules. However, these upper-limitsallow for a wide range of abundances of the remainingmolecules, including some manifestly impractical values.Primarily, the constraints include CO2 abundances ashigh 0.3, implying 30% of a hydrogen dominated at-mosphere to be composed of CO2! Assuming a highmetallicity for the planet atmosphere (about 30 × so-lar), CO2 mixing ratios as high as∼ 10−4 can be attainedat the relevant pressure levels by equilibrium chemistry,as shown in Figure 5, and by photochemistry (see e.g.Zahnle et al. 2009), but not much more.The methane mixing ratio is constrained to values be-

low 10−6, if we impose a plausible limits on the CO2

abundance. A generous upper-limit on the CO2 abun-dance can be assumed to be ∼ 10−3, based on the ar-guments above. Under this assumption, at the ξ2 ≤ 1surface (purple surfaces in Figure 3), the methane mix-ing ratio is constrained to between 10−7− 10−6, for CO2

mixing ratios between 10−7 − 10−3. And, at the ξ2 ≤ 2surface (red surfaces in Figure 3), CH4 = 10−7 − 10−6,for CO2 = 10−8 − 10−3. There is no lower bound on theCH4 abundance; mixing ratios below 10−9 do not havediscernible features at the resolution of the current data.The low methane requirement is enforced primarily by

the hight planet-star flux contrast in the 3.6 µm SpitzerIRAC channel. Inflating the uncertainties in the 3.6 µmchannel does not obviate the low methane requirement.And, the strong correlation of methane with CO2 andarises from the large flux differential between the 3.6 µmand 4.5 µm channels, as has been described in section 2.Methane is also correlated with water vapor which alsohas features in the 3.6 µm channel, albeit to a lesserextent, as shown in Figure 3).The low mixing ratio of methane is a clear indication of

non-equilibrium chemistry in the atmosphere of GJ 436b,as has been suggested in Stevenson et al. (2010). Equi-librium chemistry at the temperatures of GJ 436b causesmethane to be dominant carbon bearing molecule. Atsolar abundances the methane mixing ratio in chemicalequilibrium, for typical temperature profiles of GJ 436b,is predicted to be 7× 10−4, and 2× 10−2 for 30 × solarabundances, as is evident from Figure 5.Water vapor (H2O): Our results place an absolute

upper-limit on the H2O abundance, as shown in Fig-ure 3. The H2O mixing ratio is constrained to < 10−3

and < 3 × 10−3 for ξ2 ≤ 1 and ξ2 ≤ 5, respectively, ifwe make no assumptions of physical plausibility of thesolutions. As in the case of methane, however, the H2Oabundance is also correlated with the CO2 abundance.

8

Fig. 5.— Molecular compositions in chemical equilibrium. Each panel shows contours of molecular mixing ratios, with respect to molecularhydrogen, over the pressure-temperature grid representing the axes. The left panels show compositions for solar metallicity, and the rightpanels (“30×S”) have 30 × solar metallicity. The gray, brown, red, orange, yellow, green, cyan, blue and purple contours correspond tomixing ratios greater than 10−2, 10−3, 10−4, 10−5, 10−6, 10−7, 10−8, 10−9, and 10−10, respectively. The black lines are some best fittingpressure-temperature profiles.

If we restrict CO2 to a generous upper-limit of 10−3, asdescribed for the case of methane, the ξ2 ≤ 1 and ξ2 ≤ 2surfaces constrain the H2O abundance to < 3 × 10−4

and < 10−3, respectively. H2O is also correlated withCO and CH4. Thus the H2O abundance can be furtherconstrained if we assume conditions of physical plausi-bility of all the species simultaneously (discussed belowin section 4.1.1). The correlations of H2O with all theremaining molecules arise from it numerous features inall the Spitzer channels, as described in section 2.

Carbon monoxide (CO): The abundance of CO ishighly correlated with the abundance of CO2. If no as-sumption is made on the CO2 abundance, the observa-tions provide no constraint on the CO mixing ratio, asshown in Figure 3. If the mixing ratio of CO2 is restrictedto the conservative upper-limit of 10−3, the observationsconstrain CO to be ≥ 3 × 10−5 for ξ2 < 1, while stillleaving it unconstrained at ξ2 ∼ 2 and higher.Most importantly, despite the constraints on CO

above, simultaneous conditions of physical plausibility

Dayside atmosphere of GJ 436b 9

on all the molecules indicate that a very high CO abun-dance (≥ 10−3) is essential to explain the observations.This will be discussed in section 4.1.1 below.The strong correlation between CO and CO2 arises

from the fact that both molecules have strong featuresin the 4.5 µm IRAC channel, the CO2 feature beingstronger, as described in section 2. Although not ap-parent in Figure 3, CO is also correlated with H2O andCH4, via their correlations with CO2.Carbon dioxide (CO2): The observations require a

clear presence of CO2 in the atmosphere, as shown inFigure 3. At the ξ2 ≤ 1 surface, the lower limit on CO2

is ∼ 5 × 10−8, which requires for very high CO levels of0.1. At the same error surface, CO ≤ 10−2 requires CO2

≥ 10−7. At the ξ2 ≤ 2 level, CO2 mixing ratios as low10−8 can be attained for CO mixing ratios greater than10−3.The observations alone do not place any upper-limit on

the CO2 abundance. However, the maximum amount ofCO2 possible can also be constrained theoretically, basedon equilibrium and non-equilibrium chemistry. For thebest-fit temperature profiles of GJ 436b shown in Fig-ure 2, and pressures in the 10−3 − 1 bar range, thermo-chemical equilibrium can yield CO2 mixing ratios up to10−7 for solar metallicity and up to 10−4 for ∼ 30 × so-lar metallicities (see Figure 5). And, photochemistry canyield CO2 mixing ratios up to ∼ 10−6 and ∼ 10−4 forsolar and super-solar metallicities, respectively (Liang etal. 2003; Zahnle et al. 2009).

4.1.1. Plausibility of the abundance constraints

The constraints reported above assume nothing withregards to the physical plausibility of the models, exceptthe conservative limit of CO2 ≤ 10−3 used for sake of ar-gument. However, reasonable theoretical constraints canbe placed over the observed constraints based on well es-tablished arguments of equilibrium and non-equilibriumchemistry (see section3.3). The mixing ratios of CH4,H2O, CO and CO2 under chemical equilibrium are shownin Figure 5, for a range of temperatures and pressurespertinent to GJ 436b, along with some best-fit P -T pro-files. A detailed calculation of non-equilibrium chem-istry, via eddy mixing, is presented in section 4.2, alongwith a discussion on allowed mixing ratios due to photo-chemistry.We find that stringent constraints on the molecular

abundances required by the data can be placed even withmodest assumptions of atmospheric chemistry. As dis-cussed in section above, CO2 mixing ratios up to 10−4

are allowed by equilibrium and non-equilibrium chem-istry. H2O is a major carrier of oxygen in the desiredtemperature range. The H2O mixing ratio is expectedto be ≥ 10−4 and ≥ 10−3, for solar and 30× solar abun-dances, respectively. And, while there is no lower limiton the CO abundance, the CO upper-limit is fixed bythe metallicity; i.e. with the assumption that all oxygenis in CO. Finally, the upper-limit on methane is fixed bythe carbon abundance. At the temperatures of GJ 436b,methane is supposed to be highly abundant, based onequilibrium chemistry, as shown in Figure 5. However,methane can be depleted to some extent due to photo-chemistry (Zahnle et al. 2009), which depends criticallyon the metallicity and temperature structure.The constraints due to the considerations of physi-

cal plausibility are shown in Figure 3. The gray sur-face shows regions assuming conservative boundaries ofξ2 ≤ 3, CH4 ≥ 10−7, CO2 ≤ 10−3, H2O ≥ 10−5, andCO≤ 10−1. And, the black surfaces show a subset of thegray surface with ξ2 ≤ 3, CH4 ≥ 10−7, CO2 ≤ 10−4,H2O ≥ 10−4, and CO≤ 10−2. The black contours repre-sent our most likely interpretation of the atmosphere ofGJ 436b.Our most plausible explanation for the atmosphere of

GJ 436b is the existence of a high metallicity, along withnon-equilibrium chemistry. The black surfaces in Fig-ure 3 show that a CO abundance ≥ 10−3 is required tohave a maximum possible H2O abundance of 10−4 anda methane abundance above 10−7. This surface allowsrather plausible values of CO2, between 10−6 − 10−4.While the CO2 abundances can be completely explainedbased on equilibrium chemistry with high metallicityalone (see Figure 5), the high CO abundance requiresa high metallicity alone with non-equilibrium chemistry.As shown in Figure 5), very high CO abundances can ex-ist at the bottom of the atmosphere, for high metallicity,however the cooler upper layers of the atmosphere havemuch lower CO abundance. Non-equilibrium chemistryin the form of eddy mixing can transport the high COto the upper-layers of the atmosphere, as discussed insection 4.2 below. On the other hand the low CH4 abun-dance could potentially be cause by photochemistry. Abest-fit model consistent with our interpretation aboveis shown in Figure 4.

4.2. Explanations for Non-equilibrium Abundances

Our best-fit constrains on the chemical compositionof GJ 436b require substantial deviations from equilib-rium chemistry with solar metallicity. While equilibriumchemistry with high metallicities can yield the high COand/or CO2 abundances required, it exacerbates the re-quirement of low methane, and relatively low H2O. Ofparamount importance to fitting the observations is thesimultaneous requirement of low methane and high COand/or CO2.Here, we explore sources of non-equilibrium chemistry

in search of potential explanations to the observed con-straints. Non-equilibrium processes have been known toinfluence chemical compositions of several planetary andbrown dwarf atmospheres. Vertical eddy diffusion hasbeen used extensively to explain the observed CO over-abundances in hydrogen rich atmospheres (see e.g. Yungand Demore, 1999; Griffith & Yelle, 1999). And, photo-chemical processes have been advocated in hot Jupiteratmospheres to cause depletion of methane and forma-tion of hydrocarbon soots (Zahnle et al. 2010).

4.2.1. Eddy mixing as an explanation for high CO

The high observed mixing ratio of CO in GJ 436b canbe achieved by eddy mixing with a high metallicity. Asexplained in 3.3.2, eddy mixing transports CO from thedeeper layers of the atmosphere to the upper layers ofthe atmosphere where CO is less abundant. The mostrelevant quantity is the quench pressure (p0), which de-notes the pressure level in the atmosphere from whichCO could be transported upwards. Figure 6 shows thedependence of the quench level (p0) on Kzz, assuminga best-fit P-T profile of GJ 436b. Higher values of Kzz

10

Fig. 6.— Vertical eddy diffusion in GJ 436b. The two panels show the dependence of the ”quench” pressure (p0) and the correspondingCO mixing ratio (qCO), on the eddy diffusion constant, Kzz. “S”, 10 × S, and 30 × S refer to solar, 10 times solar and 30 times solarmetallicities.

lead to mixing of species from deeper levels of the atmo-sphere, i.e higher p0. It can be seen from the figure thatKzz values between 106 - 107 can cause mixing from aquench level of ∼ 1bar.The mixing ratio of CO in the upper atmosphere de-

pends on Kzz and the metallicity. Figure 6 also shows therange of CO mixing ratios that are attainable with differ-ent values of Kzz and metallicities. For solar metallici-ties, a maximum CO mixing ratio of ∼ 10−4 is attainablefor Kzz values between ∼ 107 − 108. For a higher metal-licity of 30 × solar, however, CO as high as 10−4 can beattained with a rather low Kzz of ∼ 103 cm2/s, and at-tains a maximum of ∼ 10−2 for Kzz between ∼ 106−108

cm2/s. For comparison, planetary atmospheres in thesolar system have Kzz ranging between 105− 109 cm2/s,and that for brown dwarf atmospheres can be as low as102− 104 cm2/s (Prinn and Bashay, 1977; Yung and De-more, 1999; Saumon et al. 2003).While eddy mixing can account for the required high

CO mixing ratios, it cannot account for the other speciesin entirety. At the quench levels of ∼ 1 bar, the equi-librium mixing ratio of CO2 is 10−7 and 10−4, for so-lar and 30 × solar metallicity. A CO2 mixing ratio of10−4 can potentially meet the constraints, if the methaneconcentration were at the 10−7 level. However, at thequench pressures of ∼ 1 bar, equilibrium chemistry yieldsmethane mixing ratios of ∼ 10−4 and ∼ 10−3 for solarand 30 × solar metallicity. Therefore, the paucity ofmethane as required by the observations still remainsunexplained.

4.2.2. Potential photochemical depletion of methane

The low methane abundance required from the best fitsto the data is not explained by non-equilibrium chemistry

due to eddy-mixing alone. Values only as low as 10−4

can be reached, whereas the required value is < 10−7.A natural solution is photochemistry, which at high al-titudes depletes methane by photodissociation, reactionwith OH radicals, and polymerization.We argue, using previous photochemistry studies to-

gether with the derived metallicities and quench levelsfrom this work, that photochemical depletion is a plausi-ble explanation of the low methane abundance. Severalrecent studies have demonstrated that methane can bedepleted in hot Jupiter atmospheres due to photochem-istry (e.g. Zahnle et al. 2009, Line et al. 2010). Of rel-evance here, Zahnle et al. (2009) showed that methanemixing ratios as low as 10−7 could be attained at the1 bar level, for an isothermal temperature structure of1200 K, a metallicity of [M/H] = 0.7, and a diffusioncoefficient of 106 cm2/s. In their example, the methaneconcentration was high (∼ 10−3) at the lower boundary,at 100 bar, where equilibrium chemistry was assumed,but was rapidly depleted moving up in the atmosphere,reaching 10−7 at the 1 bar level. This methane mixingratio at the 1-bar level, the same pressure as the quenchlevel required for the CO mixing ratio, agrees with themethane mixing ratio derived from the data.The above scenario is a plausible explanation for the

low methane abundance in GJ 436b. Many of the bestfitting temperature profiles have temperatures greaterthan ∼ 1200 K for pressures greater than ∼1 bar (seee.g. Figure 2). Figure 7, shows a sample best-fit P-Tprofile (panel A), which has an isothermal temperaturestructure of 1200 K, for pressures above 1 bar, which issimilar to the cases presented in Zahnle et al. (2009). Ifit is true that the methane concentration in the lower at-mosphere (between ∼ 1 - 100 bar) is driven primarily by

Dayside atmosphere of GJ 436b 11

Fig. 7.— Non-equilibrium chemistry in GJ 436b. Panel A shows the pressure-temperature (P -T ) profile of a best-fit model. Panel Bshows the times scales of eddy mixing (τmix) and of CO–CH4 equilibrium chemistry (τchem). Panel C shows the potential influence ofphotochemistry coupled with eddy mixing, in the form of mixing ratios of the four prominent species. The solid lines show the compositionsfrom equilibrium chemistry, which continue as dotted lines above the quench level, and fixed to the values at the quench level. The dashedlines show the mixing ratios from Zahnle et al. (2009), which yield methane mixing ratios below 10−6 above the quench level. The y-axisin all the panels is pressure in bars. The common black dashed line in each panel corresponds to the quench pressure level.

the lower boundary condition, then the methane mixingratios at the 1 bar level for GJ 436b, can be as low as10−7, for metallicities of ∼ 5 × solar, and an eddy diffu-sion coefficient of ∼ 106, as in Zahnle et al (2009). PanelB shows the times scales of eddy mixing and chemical re-action for the CO-CH4 reaction. For a Kzz of 106 – 107,we find that quenching occurs at the p0 ∼ 1 bar level.Panel C of Figure 7, shows mixing ratio profiles from oureddy mixing calculation (solid lines for P ¿ p0 and dottedlines for P ¡ p0), and mixing ratio profiles from Zahnleet al. (2009) (shown in dashed lines), from their 1200 Kand high metallicity models. For pressures below p0, weassume that the atmosphere is well mixed in all speciesin our calculation. The constant vertical profiles for CO,CO2 and H2O are also found by Zahnle et al. 2009, usinga full photochemical model. We note that our profilescannot be compared directly to those of Zahnle et al.(2009), as the two models are of different metallicitiesand, very likely, different stellar irradiation. The pro-files from Zahnle et al. (2009) are shown here primarilyto illustrate that methane mixing ratios below 10−6 arefeasible at the 1 bar level due to photochemistry. At evenhigher altitudes, further depleting of methane would beexpected due to more vigorous photochemistry.The observed spectrum is primarily sensitive to the

methane mixing ratios in only a limited altitude regionin the atmosphere. We emphasize that the methane con-centration at the 1 bar level is of particular importanceto the Spitzer observation at 3.6 µm. The contributionfunction in the IRAC 3.6 µm channel peaks at a pres-sure of 1 bar (Spiegel et al. 2010; Stevenson et al. 2010).This is the primary reason why the high flux in the 3.6µm channel is an important probe for the methane abun-

dance.

4.3. Temperature Structure and Day-Night EnergyRedistribution

The dayside atmosphere of GJ 436b is the coolest ex-oplanet atmosphere observed to date, at an equilibriumtemperature of 770 K1. The observed brightness tem-peratures in the six Spitzer channels range from the 3-σupper limit of 700 K in the 4.5 µm IRAC channel toabout 1100 K in the 3.6 µm channel (Stevenson et al.2010). With the exception of the 3.6 µm IRAC chan-nel, observations in all the remaining five channels areconsistent with a black-body planet spectrum at 750 K± 100 K. However, the 3.6 µm observation of ∼ 1100 Kbrightness temperature is a major exception, requiring atemperature differential of ∼ 400 K between the 3.6 µmand 4.5 µm channels, implying a very steep temperaturegradient in the atmosphere.The dayside pressure-temperature (P-T) profiles of GJ

436b constrained by the observations are shown in Fig-ure 2. The best fitting P-T profiles (in purple) have tem-perature gradients in excess of ∼ 400 K per decade inpressure, required primarily by the large temperature dif-ferential observed between the 3.6 µm and 4.5 µm chan-nels. The large temperature gradients for P& 0.05barcause most of the contributions to the emergent flux toarise in the lower layers of the atmosphere. Furthermore,the observations distinctly rule out the presence of a sig-nificant thermal inversion in the dayside atmosphere ofGJ 436b. We find that we cannot fit the observationswith an inversion model for any chemical composition(although very small inversions which might cause only

1 assuming zero albedo and uniform energy distribution

12

weak observable features cannot be ruled out by presentdata). A significant thermal inversion in this atmospherewould have caused the brightness temperatures in the4.5, 5.8, and 8 µm channels to be markedly higher thanthe 3.6 µm channel, much higher than what the currentdata indicate. Finally, as has been known for all hotJupiters, the observations indicate that the dayside at-mosphere of GJ 436b is mostly radiative, with the radia-tive zones of some of the best fitting P-T profiles extend-ing to pressures above ∼ 10 bar. The isotherms at thehigh pressure ends of the P-T profiles are suggestive ofthe radiative diffusion approximation in the high opticaldepth limit (see e.g. Madhusudhan & Seager, 2009), andare also found in other self-consistent models reported inthe literature (e.g. Spiegel et al. 2010).The large brightness temperature observed in the 3.6

µm channel is also indicative of low day-night energy re-distribution in GJ 436b. The 3.6 µm channel probeslevels deep in the atmosphere (around pressures of 1bar), and hence a high brightness temperature indicatesa high blackbody continuum. Thus, most of the incidentstar flux is reradiated on the day side in the form of aT & 1100K blackbody. The bottom-right panel of fig-ure 3 shows the ξ2 contours in the η = (1 − A)(1 − fr)and Teff . η = (1 − A)(1 − fr) is obtained from energybalance, where A is the bond albedo and fr is fractionof energy redistributed to the night side (Madhusudhan& Seager, 2009). The best fitting models favor a maxi-mum day-night redistribution (fr) of 0.25, i.e for A = 0.An albedo of, say 0.1, further restricts the distributionto 0.2. Our results support similar conclusions arrivedat by previous works (Deming et al. 2007; Spiegel et al.2010). Our conclusion of a low day-night redistributionon this planet assumes special significance for potentialfuture observations of thermal phase curves of GJ 436b.Thermal phase curves in the 3.6 µm and 4.5 µm IRACchannels, feasible with warm Spitzer, should show clearmodel-independent evidence of a high day-night temper-ature contrast, according to our present results. A find-ing on the contrary, i.e finding efficient redistribution inthe phase curves, can imply the possibility of a substan-tial interior energy source in GJ 436b.

5. DISCUSSION AND SUMMARY

We presented a detailed analysis of the day-side atmo-sphere of GJ 436b. Our results show that a high metal-licity along with non-equilibrium chemistry are requiredto explain the observations. We also studied the cor-relations between the various molecular species, and re-ported detailed constraints on the metallicity, chemicalprocesses, and day-night energy circulation. Althoughour results come from observations in six channels ofSpitzer photometry, some channels are more importantthan others. Here, we discuss the relative importanceof the different Sptizer channels to our conclusions. Wealso discuss some potential alternate interpretations ofthe data, followed by a summary of our results.

5.1. Sensitivity of Results to Spitzer Observations

The constraints reported in this work depend criticallyon the two Spitzer IRAC observations at 3.6 µm and 4.5µm. The high planet-star flux contrast in the 3.6 µmchannel is responsible for the constraints of low methane

abundance and low energy circulation. The low flux con-trast in the 4.5 µm channel is responsible for the require-ment of high CO and/or CO2 in GJ 436b. While theobservation in the 3.6 µm channel was reported to beof the higher S/N of all channels, the 4.5 µm channelwas a non-detection Stevenson et al (2010). Neverthe-less, future observations in both these channels would beextremely important to confirm that these fluxes actu-ally represent the steady state atmosphere in GJ 436b.The observations in the remaining four channels (5.8 µm- 24 µm) are much less constraining, although still veryuseful. For instance, the moderate flux observed in the 8µm channel, which has a strong methane feature, is im-portant to the conclusion that the very high flux in the3.6 µm channel could not have been due to a thermalinversion causing methane emission.Our constraints on the molecular abundances are fairly

robust with respect to the observational uncertainties.The high flux in the 3.6 µm channel cannot be explainedby equilibrium chemistry. A model with equilibriumchemistry and solar or 30 × solar metallicities predictsplanet-star flux contrasts that are lower than the ob-served value by over 4 -σ (also in agreement with mod-els of Demory et al. 2007; Spiegel et al. 2010). Onthe other hand, the contrasts predicted in the 4.5 µmchannel based on equilibrium chemistry alone would behigher than the observed non-detection by over 3-σ (alsosee Spiegel et al. 20010). We have been conservative inour analysis by allowing our best fits to the 4.5 µm pointto lie within the 3 − σ ± 1 − σ upper-limits. Had weconsidered this point to be a strict non-detection at 1-σ,our results would predict even higher CO and/or CO2.Finally, the large uncertainties in the observed fluxes inthe 16µm and 24 µm channels provide only fiducial con-straints on the temperature structure and the H2O andCO2 abundances.

5.2. Alternate Interpretations

The high planet-star flux contrast observed in the 3.6µm IRAC channel is central to most of the constraintsreported in this work. Our inferences could partly berestricted by the specific choices that are inherent to ourmodels. For example, our models do not include cloudsor hazes. Although scattering from hazes has been sug-gested to be potentially relevant in the optical and near-IR (e.g. Sing et al. 2009), a high contribution at thelonger wavelengths of the 3.6 µm channel is unlikely. Wehave also assumed the planet atmosphere to be in localthermodynamic equilibrium (LTE). Swain et al. (2010)reported a potential signature due to non-LTE methaneflorescent emission in HD 189733b observed in the rangeof the 3.6 µm channel. However, Stevenson et al. (2010)found that a florescence features of the same strength asreported in Swain et al. (2010) could not explain theobserved flux in the 3.6 µm channel for GJ 436b. Theday-side atmosphere of GJ 436b might be variable, ashas been reported previously for hot Jupiters (Grillmairet al. 2008; Madhusudhan & Seager, 2009). However,for variability to explain the observed flux in the 3.6 µmchannel, the temperature at the ∼ 1 bar level in the at-mosphere of GJ 436b has to exhibit fluctuations greaterthan 400 K, between subsequent observations of Steven-son et al. (2010). These and other alternate explanationsare worth exploring with more data at different epochs.

Dayside atmosphere of GJ 436b 13

5.3. Summary

We have presented constraints on the chemical com-position and temperature structure of the hot NeptuneGJ 436b, based on recent data. Our statistical con-straints at the 1-σ fit level require that the planet havemethane mixing ratio below ∼ 10−7, for CO2 mixing ra-tio below 10−5 , CO & 10−3, and H2O . 10−4. At thetemperatures of GJ 436b, equilibrium chemistry with so-lar abundances predicts abundances of methane, CO andCO2 to be ∼ 7× 10−4, . 10−5 and . 10−7, respectively.This observed scarcity of methane and the over abun-dance of CO and CO2 point towards unambiguous non-equilibrium chemistry in this atmosphere, as has beenpreviously reported by us in Stevenson et al. (2010).The observed constraints can be explained with a high

metallicity in the atmosphere, along with processes ofnon-equilibrium chemistry. We find that the high COand CO2 abundances can be explained by a combinationof high metallicity (30 × solar) and eddy mixing in theatmosphere of GJ 436b. We estimate the eddy mixingcoefficient (Kzz) to be between 106 − 108.Neither high metallicity alone, nor high metallicity

with eddy diffusion, can explain the required methaneabundance. We find that the paucity of methane canpotentially be explained by photochemistry, combinedwith a high metallicity and physically plausible amountsof eddy mixing. Future detailed calculations of photo-chemistry in GJ 436b are needed to confirm this plausibleexplanation of the low-methane requirement.Our results also constrain the day-side temperature

structure and the day-night energy redistribution in theatmosphere of GJ 436b. A temperature inversion isruled out by the current observations; although, small

inversions which are not observable at the resolutionof the current photometric data cannot be conclusivelyruled out. The observations also suggest inefficient day-night energy redistribution (fr) in GJ 436b, requiring(1 − AB)(1 − fr) = η ≥ 0.7, at the 1-σ fit. Thus, themaximum fr allowed by the data at the 1-σ fits is 0.3for zero bond albedo (AB), and 0.23 for AB = 0.1. Fu-ture observations of thermal phase curves in the availablewarm Spitzer channels will be instrumental in validatingthe low redistribution requirement. A finding on the con-trary might indicate a substantial interior energy source.We emphasize that the constraints reported in this workdepend primarily on the two Spitzer channels (3.6 µmand 4.5 µm). Thus, future observations in these channelswill be extremely important in confirming the presentand previous results on the atmosphere of GJ 436b.The atmosphere of the hot-Neptune GJ 436b presents

new challenges and opportunities for detailed modeling ofexoplanet atmospheres. As low-mass transiting planetscontinue to be discovered, unexpected findings are likelyto continue. The next generation of models will help upunravel those mysteries, and help put our solar systemin perspective.

NM thanks Adam Showman, James Kasting, JonathanFortney, Evan Hubeny, and David Spiegel, for helpfuldiscussions. We thank Richard Freedman for providingmolecular line lists, especially with helpful informationon CO2 opacities. We thank Larry Rothman for accessto the HITEMP database. We thank the MIT KavliInstitute for facilitating access to the computer clusterused for this work. Support for this work was providedby NASA through an award issued by JPL/Caltech.

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