On the Inference of Thermal Inversions in Hot Jupiter Atmospheres
aa r X i v : . [ a s t r o - ph . E P ] O c t Draft version October 18, 2018
Preprint typeset using L A TEX style emulateapj v. 2/16/10
ON THE INFERENCE OF THERMAL INVERSIONS IN HOT JUPITER ATMOSPHERES.
N. Madhusudhan , & S. Seager , Draft version October 18, 2018
ABSTRACTSeveral studies in the recent past have inferred the existence of thermal inversions in some transit-ing hot Jupiter atmospheres. Given the limited data available, the inference of a thermal inversiondepends critically on the chemical composition assumed for the atmosphere. In this study, we explorethe degeneracies between thermal inversions and molecular abundances in four highly irradiated hotJupiter atmospheres, day-side observations of which were previously reported to be consistent withthermal inversions based on
Spitzer photometry. The four systems are: HD 209458b, HAT-P-7b,TrES-4, and TrES-2. We model the exoplanet atmospheres using a 1-D line-by-line radiative transfercode with parametrized abundances and temperature structure, and with constraints of energy bal-ance and hydrostatic equilibrium. For each system, we explore the model parameter space with ∼ models using a Markov chain Monte Carlo routine. Our results primarily suggest that a thoroughexploration of the model parameter space is necessary to identify thermal inversions in hot Jupiteratmospheres. We find that existing observations of TrES-4 and TrES-2 can both be fit very preciselywith models with and without thermal inversions, and with a wide range in chemical composition. Onthe other hand, observations of HD 209458b and HAT-P-7b are better fit with thermal inversions thanwithout, as has been reported previously. Physically plausible non-inversion models of HD 209458band HAT-P-7b fit the data only at the 1.7 σ observational errors; better fits require substantial en-hancement of methane and depletion of CO, which seems implausible in the very hot atmospheresconsidered here. Secondly, in the sample under consideration here, we do not see a correlation betweenirradiation levels and thermal inversions, given current data. Before JWST becomes available, near-IR observations from ground and with HST, along with existing Spitzer observations, can potentiallyresolve thermal inversions in some systems. Observations with only two channels of
Warm Spitzer photometry and good S/N can likely identify or rule out thermal inversions if the difference betweenthe fluxes in the 3.6 and 4.5 µ m channels is very high. Subject headings: planetary systems — planets and satellites: general — radiative transfer INTRODUCTION
Observations of day-side atmospheres of several hotJupiters have indicated the existence of thermal inver-sions. The inference of thermal inversions are a resultof high S/N
Spitzer observations (Knutson et al. 2008;Knutson et al. 2009; Machalek et al. 2009; O’Donovanet al. 2010; Christiansen et al. 2010) and concomitanttheoretical modeling (Burrows et al. 2007, 2008; Fortneyet al. 2008; Madhusudhan & Seager, 2009). Thermal in-versions have been reported based on flux excesses in cer-tain
Spitzer channels. A natural explanation of channel-specific high flux is to invoke molecular emission features,as opposed to absorption features (Burrows et al. 2007,2008; Fortney et al. 2008). Under the assumption of lo-cal thermodynamic equilibrium (LTE), molecular emis-sion features form only in the presence of a thermal in-version, a region in the atmosphere where temperatureincreases outwards, as in the Earth’s stratosphere.Preceding recent observations, early theoretical workon hot Jupiter atmospheres using self-consistent 1D at-mosphere models predicted the existence of thermalinversions based on absorption due to TiO and VO(Hubeny et al. 2003, Fortney et al. 2006). More re- MIT Kavli Institute for Astrophysics and Space Research,and Department of Earth, Atmospheric, and Planetary Sciences,MIT, Cambridge, MA, 02139 Department of Physics, MIT, Cambridge, MA 02139 Corresponding author: [email protected] cently, Fortney et al. 2008, classified hot Jupiters in twocategories based on the degree of incident irradiation.The class of atmospheres with very high incident flux,dubbed “very hot Jupiters” or “pM” class, were consid-ered favorable to host thermal inversions caused due togaseous TiO and VO absorption at low pressures, and at-mospheres with lower fluxes were predicted to be unlikelyto host inversions owing to condensation of TiO/VO. Theflux boundary for this dichotomy was, somewhat arbi-trarily, chosen to be ∼ ergs/s/cm . However, whileinferences of some recent observations have purported toviolate this hypothesis (e.g. Machalek et al. 2009), oth-ers have found present observations insufficient to testthis hypothesis (O’Donovan et al. 2010; Fressin et al.2010).The theory behind the absorbers causing thermal in-versions in hot Jupiters atmospheres is still under de-bate; discussed in detail in 2.1. Recent theoretical stud-ies suggest that TiO/VO may not be able to totally ac-count for the inferred thermal inversions (Spiegel et al.2009). Other alternatives proposed in recent studies in-clude strong UV/visible absorption due of photochemi-cally produced sulfur compounds (Zahnle et al. 2009),and correlation of thermal inversions with chromosphericactivity of host stars (Knutson et al. 2010). Since thenature of absorbers causing the inversions is not known,models that have been successful in inferring thermal in-versions have either adopted a parametric absorber (Bur-rows et al. 2008), or a parametrized temperature profile(see e.g. Madhusudhan & Seager, 2009).Existing models inferring thermal inversions rest onseveral assumptions and parametrization. Models typ-ically invoke several free parameters to induce a ther-mal inversion in the temperature profile, as suggested bythe data. The free parameters range from the locationand magnitude of an unknown opacity source (e.g. Bur-rows et al. 2007 & 2008) to parametrizing the temper-ature profile itself (e.g. Madhusudhan & Seager, 2009).In addition, traditionally exoplanet atmosphere modelsspan only a limited range in chemical abundances, of-ten assuming thermochemical equilibrium (Barman et al.2005; Seager et al. 2005; Fortney et al. 2006; Burrowset al. 2007 & 2008). While some reported models re-quire thermal inversions to explain the observations, it isnot known if the model parameter space has been thor-oughly exhausted. It remains to be seen whether some ofthe observations can be explained without thermal inver-sions, if we were to relax many of the assumptions andthoroughly explore the parameter space.In the present work, our primary goal is to understandthe extent to which thermal inversions can be robustlyinferred in hot Jupiter atmospheres, with current obser-vations. We choose a test sample of four systems forwhich Spitzer observations in the past have been reportedto be consistent with thermal inversions. We then pur-sue a detailed exploration of the model parameter spaceto see the extent to which the observations can be ex-plained by models with and without thermal inversions.We accomplish this by computing large ensembles of in-version and non-inversion models ( N ∼ ), exploringthe parameter space for best-fitting solutions. For eachsystem, we report quantitatively how well the data canbe fit by models with and without thermal inversions andwith what ranges in atmospheric chemical composition.This approach also reveals the underlying correlationsbetween the different chemical species and between thecomposition and temperature structure. And, since thesystems considered here have different levels of irradia-tion, we also seek to understand if the presence or ab-sence of thermal inversions is correlated with the degreeof irradiation, at the level of current data.We focus on four hot Jupiters for which existing ob-servations were reported to be consistent with thermalinversions in their day-side atmospheres, and for whichphotometric observations of thermal emission are avail-able in four or more Spitzer channels. The planets are:HD 209458b, HAT-P-7b, TrES-4, and TrES-2. Bur-rows et al. (2008) and Knutson et al. (2008) first re-ported a thermal inversion in HD 209458b, based on
Spitzer photometry in five channels reported by Knut-son et al. (2008) and Deming et al. (2005). Madhusud-han & Seager (2009) confirmed the thermal inversion inHD 209458b, for model fits within the 1.4 σ observationaluncertainties. Additional observations of HD 209458bare available in the near-IR, obtained with HST
NIC-MOS, but were explained by models with and withoutinversions alike (Swain et al. 2009). Knutson et al.(2009) reported a thermal inversion in TrES-4b based onfive-channel
Spitzer photometry using models based onBurrows et al. (2008). O’Donovan et al. (2010) reportedobservations of TrES-2 in four
Spitzer
IRAC Channels,which were explained by models with and without ther- mal inversions; they preferred the inversion model whichseemed more favorable amongst the set of models ex-plored in that study. Croll et al. (2010) reported aground based detection of TrES-2 in the Ks band, andnoted that their observation along with the four
Spitzer observations could be explained equally well by mod-els with and without thermal inversions. More recently,however, Spiegel & Burrows (2010) reported that mod-els assuming radiative and chemical equilibrium cannotexplain the
Spitzer observations of TrES-2 without invok-ing thermal inversions. And, finally, Christiansen et al.(2010) reported a thermal inversion in HAT-P-7b, basedon four
Spitzer
IRAC channels, along with an observa-tion in the visible by the
Kepler Space Telescope (Boruckiet al. 2009). While the Kepler point was not necessar-ily decisive in constraining the thermal inversion itself,it allowed constraints on the albedo, day-night redistri-bution and the TiO/VO abundance. Spiegel & Burrows(2010) confirmed the presence of a thermal inversion inHAT-P-7b using the observations of Christiansen et al.(2010).The observations of TrES-2 and HAT-P-7b notedabove were first interpreted using the exoplanet at-mosphere model developed in Madhusudhan & Seager(2009), which is also used in the present work. In boththose studies, i.e of O’Donovan et al. (2010) and Chris-tiansen et al. (2010), we had reported a representativeset of models which explained the data. In the presentstudy, we report a more exhaustive exploration of themodel parameter space for these systems using a newparameter space exploration scheme described in Sec-tion 3.2.We discuss the theoretical and observational basisof thermal inversions in §
2. We explain the modelframework in §
3, along with the parameter explorationmethod and the selection of systems for our study. In §
4, we present our results, followed by a summary anddiscussion in § THERMAL INVERSIONS IN HOT JUPITERATMOSPHERES
Several arguments have been proposed in the literaturejustifying the existence of thermal inversions in some hotJupiter atmospheres. Compelling
Spitzer
IRAC observa-tions of some hot Jupiters suggest that anomalies in theform of planet-star flux excess in some channels cannotbe explained without invoking thermal inversions (Bur-rows et al. 2007; Knutson et al. 2008; Madhusudhan& Seager, 2009; Swain et al. 2009). However, it is notknown whether such an inference is an outcome of themodel input choices used to infer the observations. Onthe other hand, independent of observations, several the-oretical arguments support the existence of thermal in-versions in some hot Jupiter atmospheres (Hubeny et al.2003; Burrows et al. 2007; Fortney et al. 2008).In this section, we explore the arguments in favor ofthermal inversions, and motivate the framework underwhich they can be tested. We begin with the theoreticalmotivation for thermal inversions, followed by argumentsleading to inference of thermal inversions from observa-tions. We then pose the question of whether the obser-vations can be explained without thermal inversions ifsome of the model constraints are relaxed. Finally, weset up the framework in which the requirement of ther-hermal inversions in hot Jupiter atmospheres 3mal inversions can be robustly tested.
Theoretical Basis for Thermal Inversions
Thermal inversions are a natural consequence of visi-ble/UV absorption of incident star light high in the atmo-sphere. For an isolated planetary atmosphere in hydro-static equilibrium, and no local energy sources, the atmo-spheric temperature decreases with pressure (i.e with in-creasing distance from center); the atmosphere is heatedfrom below, and cools monotonically outwards. How-ever, in planetary atmospheres irradiated by the hoststar, strong optical/UV absorbers in the higher layers ofthe atmosphere can intercept part of the incident starlight. Such local deposition of energy results in a zone inthe planetary atmosphere where temperature increasesoutward, i.e a “thermal inversion”. Most solar systemplanets have thermal inversions in their atmospheres. InEarth’s atmosphere, for example, a thermal inversion iscaused by ozone (O ), which is a strong absorber in theUV (Chamberlain, 1978). And, in Jupiter’s atmosphere,a thermal inversion is caused by strong absorption in thevisible by haze resulting from methane photochemistry. Fig. 1.—
Illustration of P - T profiles. The red and blue curvesshow P - T profiles with and without a thermal inversion, respec-tively. The molecular features due to each of these profiles areshown in Figure 2. The thermal inversion causing absorbers of solar sys-tem planet atmospheres do not survive the temperaturesof hot Jupiters. Nevertheless, it has been proposed that thermal inversions could be formed in the atmospheresof very hot Jupiters due to strong absorption of incidentstellar radiation in the visible by gaseous TiO and VO(Hubeny et al. 2003; Burrows et al. 2007; Fortney etal. 2008). However, Spiegel et al. (2009) suggested thatat mbar pressures where thermal inversions are requiredto explain the observations, TiO and VO may not beabundant in the required amounts. TiO, being a heavyspecies, requires a substantial amount of vertical mixingto keep it aloft; the K zz required is 10 − cm /s,for 0.1 - 10 µ m condensate size, if a cold trap is present.Since neither the particle sizes nor K zz are known, it isuncertain if TiO might be able to explain thermal inver-sions. And, VO is unlikely to help owing to its lower (so-lar) abundance and lower visible absorption compared toTiO (Spiegel et al. 2009). Zahnle et al. (2009) reportedthat photochemically produced Sulfur compounds, HSand S , could have high UV and visible opacities at thetemperatures relevant to hot jupiter atmospheres, mak-ing them potential candidates for causing thermal in-versions in hot jupiters. More recently, Knutson et al.(2010) suggested that the presence of a thermal inver-sion could be inversely correlated with the activity levelof the host star, UV flux from the more active stars po-tentially destroying inversion-causing absorbers. Despitethe continuing debate on the inversion-causing absorbersin hot jupiter atmospheres, thermal inversions have beeninvoked typically by either adopting a parametric opac-ity source of unknown nature (Burrows et al. 2008), orby parametrizing the temperature profile (see e.g. Mad-husudhan & Seager, 2009). Review of Observational Inference of ThermalInversions
The inferences of thermal inversions are motivated byobservations of dayside atmospheres of transiting hotJupiters using
Spitzer photometry. Observations of somehot Jupiters indicate excess emission in some
Spitzer channels over others. For example, in the four IRACobservations of HD 209458b reported by Knutson etal.(2008), there is indication of excess emission in the4.5 µ m and 5.8 µ m channels. The observations showa markedly higher planet-star flux ratio in the 4.5 µ mchannel compared to the neighboring 3.6 µ m channel.And, the flux ratio in the 5.8 µ m channel is higher thanthat in the adjacent channels at 4.5 µ m and 8 µ m.The thermal emission spectrum of a planet is influ-enced by a combination of the atmospheric temperaturestructure and molecular absorption. Let us consider ahypothetical hot Jupiter atmosphere with the stellar andplanetary properties of HD 209458b, and consisting ofgaseous H O, CO, CH , and CO , at nominal mixing ra-tios, close to assumption of chemical equilibrium at solarabundances. In this particular case, we assume that themolecular species are all well mixed in the atmosphere,i.e. uniform volume mixing ratios over the whole atmo-sphere. Figure 2 shows the spectral features of each of themolecules in such an atmosphere, for P - T profiles withand without a thermal inversion (shown in Figure 1).The assumed mixing ratios of the molecules are: H O =10 − , CO = 10 − , CH = 10 − , CO = 5 × − . As isdemonstrated in Figure 2, the atmosphere with a ther-mal inversion gives rise to molecular emission features,whereas the one without a thermal inversion has absorp- Fig. 2.—
Illustration of molecular features. Each panel shows molecular line features due to a single molecule in a hypothetical atmosphere.The stellar and planetary properties are assumed to be those of the HD 209458b system. The blue curves show the spectral features of anatmosphere without a thermal inversion, and red curves show features of an atmosphere with a thermal inversion. The corresponding P - T profiles are shown in Figure 1. The mixing ratios, with respect to H , in the four panels are assumed to be: H O = 10 − , CO = 10 − ,CH = 10 − , CO = 5 × − . The continuum in each spectrum is of the black-body of the bottom of the atmosphere with features dueto H -H collision induced opacities. The green curves show the six Spitzer band passes at 3.6, 4.5, 5.8, 8, 16, and 24 µ m. tion features.The excess emission in some Spitzer channels over oth-ers can be qualitatively explained by a thermal inversionalong with some key molecular features. As can be seenfrom Figure 2, H O has several spectral features in the3.6 µ m, 5.8 µ m, 8 µ m, 16 µ m, and 24 µ m Spitzer chan-nels. CH has strong features almost exclusively in the3.6 µ m and 8 µ m channels. CO has a strong feature inthe 4.5 µ m channel, also contributing to the 5.8 µ m chan-nel. And, CO has strong features in the 4.5 µ m and 16 µ m channels. The 4.5 µ m feature of CO is degeneratewith the contribution of CO in the same channel. Thehigh fluxes in the 4.5 µ m and 5.8 µ m channels, as seenin HD 209458b for example, can therefore be explainedsimply by having strong emission due to CO and CO ,and only moderate emission due to CH and H O (Mad-husudhan & Seager, 2009). And, since emission featurescan form only due to a thermal inversion, the observa-tions can be interpreted as suggesting the presence of athermal inversion along with CO and/or CO .The molecular species required to explain the obser-vations are physically plausible. In chemical equilibrium(Burrows & Sharp, 1999), CO occurs naturally in thetemperature range of HD 209458b (see P - T profiles inBurrows et al. 2008, Madhusudhan & Seager, 2009, Swain et al. 2009). And, CO mixing ratios up to ∼ − are allowed by equilibrium chemistry and/or photochem-istry (Liang et al. 2003; Zahnle et al. 2009). Moregenerally, hot Jupiters for which thermal inversions arepredicted, are characterized by their very hot day-sideatmospheres (Fortney et al. 2008), suggesting that dom-inant contribution to the emergent spectra is expectedfrom CO (Burrows & Sharp, 1999).Care must be exercised while exploring the space of at-mospheric composition in order to fit the data. Since ahigh flux in the 4.5 µ m channel can be explained by COand/or CO , it is possible for a fitting model to infer thelack of CO in the atmosphere, by allowing an implausiblyhigh CO . Such a proposition would likely be implausiblegiven the hot temperatures where CO is expected to bethe dominant form of carbon. The degeneracy betweenCO and CO can be broken by fitting the model to the16 µ m Spitzer
IRS photometry, where available. The 16 µ m channel has dominant contribution due to a strongfeature of CO alone. For example, an observation ofHD 209458b in the 16 µ m channel by Deming (personalcommunication, 2009) was used in Madhusudhan & Sea-ger (2009) to place simultaneous constraints on CO andCO . The same observation is also used in the presentwork.hermal inversions in hot Jupiter atmospheres 5 Alternate Qualitative Interpretation
Is there any conceivable scenario, independent of ex-isting models in which the observations can be explainedwithout thermal inversions? A few qualitative alterna-tives seem feasible. Let us re-consider the situation ex-plained in § µ m and 5.8 µ m IRAC channels are higher than those in the 3.6 µ mand 8 µ m channels, respectively. Let us now investigateif we can explain the same observations with a planet at-mosphere which has no thermal inversion (the blue modelin Figure 1 and Figure 2). In this case, instead of con-sidering the emission features in the 4.5 µ m and 5.8 µ mchannels, one can consider the absorption features in the3.6 µ m and 8 µ m channels, to explain the same flux dif-ferential between adjacent channels. The fitting modelwould then require strong absorption features in the 3.6 µ m and 8 µ m channels, and weaker absorption in the4.5 µ m and 5.8 µ m channels. One conceivable solution isprovided by methane (CH ) which has strong absorptionfeatures only in the 3.6 µ m and 8 µ m channels. So, inprinciple, a high contribution due to CH , over CO orCO , could provide the required absorption signatures.Although H O also has features in the 8 µ m channel, ahigh abundance of it may not be desirable since it wouldlead to absorption in the 5.8 µ m channel. In presentingthis solution, we have exploited the degeneracy betweenthe presence or absence of a thermal inversion and com-plementary molecular features.Despite our simple qualitative explanation of anon-thermal-inversion fit to observations like those ofHD 209458b, whether or not a model without inversionfits the data, and is physically plausible, is subject tofurther tests. Firstly, CH has stronger absorption inthe 8 µ m channel than in the 3.6 µ m channel. In otherwords, for the same molecular composition of CH , theabsorption in the 8 µ m channel with respect to the 5.8 µ m channel can be deeper than the absorption in the 3.6 µ m channel with respect to the 4.5 µ m channel. This iscontrary to what is required by the IRAC observationsof HD 209458b (Knutson et al. 2008).A fitting model without a thermal inversion must alsosatisfy the constraint of energy balance. Explaining thehigh flux in the 4.5 µ m channel with a non-thermal in-version model means that the black-body continuum ofthe spectrum must be at the level of the 4.5 µ m point orhigher. While such a high emergent flux balances the in-cident stellar flux remains to be verified. Therefore, it isnot certain that a non-inversion configuration which fitsthe data necessarily satisfies the fundamental constraintof energy balance. This latter point could, in principle,be obviated by P - T profiles steep enough to produce deepspectral features in most parts of the spectrum except inthe 4.5 µ m and 5.8 µ m channels. Finally, even if themodel fits and maintains energy balance, it is not clearhow such extremely hot atmospheres can be dominatedby CH over CO. Future theoretical work might explainthis possibility. A Test for Thermal Inversions
A rigorous inference of a thermal inversions from agiven set of observations would involve running a largepopulation of models thoroughly exploring the parame-ter space in search of non-inversion models that fit the data. Not finding a statistically significant fit in such anexploration would constitute strong evidence in favor ofthermal inversions from the data set in question. In arecent work, we demonstrated the capability of running ∼ Spitzer broadband photometry.Such a capability was possible because of the efficientparametrization of the model temperature structure andmolecular abundances.In this study, we combine the model developed in Mad-husudhan & Seager (2009) with an efficient parameterspace exploration procedure to test the requirement ofthermal inversions for a select sample of hot Jupiters. Weconsider four hot Jupiters at different levels of irradiationfor which observations have been known to be consistentwith thermal inversions in their respective atmospheres,and for which
Spitzer photometry is available in four ormore channels. We run ∼ models, with and withoutthermal inversions, for each planet under consideration,and report goodness-of-fit contours in the space of atmo-spheric composition and temperature structure. In whatfollows, we describe our model set-up, the optimizationalgorithm, and the systems considered in this study. MODEL AND METHOD
Model
Our model atmosphere includes a 1D parametric P - T profile coupled with line-by-line radiative transfer, hydro-static equilibrium, and the requirement of energy balanceat the top of the atmosphere (Madhusudhan & Seager,2009). We consider 100 atmospheric layers in the pres-sure range between 10 − −
100 bar. The key aspect ofour model is the parametrization of the P - T profile andthe chemical composition, which allows us to run largeensembles of models, exploring the parameter space, ina computationally efficient manner.The major difference of our model from traditional at-mosphere 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) equilibriumas is done in conventional models. We note that therequirement of layer-by-layer radiative equilibrium in a1-D model is not strictly physical since complex hydrody-namics flows in highly irradiated hot Jupiter atmospherescan alter the temperature structure away from radiativeequilibrium (Showman et al. 2009). The global energybalance, e.g., at the top of the atmosphere, however, is astrict requirement. For a given set of model parameters,we require that the net energy output at the top of theatmosphere is less than or equal to the net energy inputdue to the incident stellar flux; a deficit indicates energyredistributed to the night-side. Models where the emer-gent flux is greater than the incident flux are discarded(see Madhusudhan & Seager, 2009). By running a largenumber of ( ∼ ) models in the parameter space, anddiscarding those that did not satisfy energy balance, wewere left with a population of models that satisfied en-ergy balance.We parameterize the chemical abundances of themolecular species by considering deviations over chem-ical equilibrium (explained at length in Madhusudhanand Seager, 2009). For each molecule under considera-tion, we compute its mixing ratio in a layer of the atmo-sphere by multiplying a parametric factor to the mixingratio that would be expected under thermochemical equi-librium with solar abundances (TE ⊙ ). The parametricfactor for a given molecule is constant over the entire at-mosphere, i.e the mixing ratio profile of the molecule overthe entire atmosphere would be shifted relative to thatobtained from TE ⊙ by the constant factor. We use thistreatment for H O, CO, and CH . For CO , we perturbover a uniform mixing ratio of 10 − (1 ppmv or 10 − isjust a reference; it is the approximate mixing ratio of CO expected from TE ⊙ and/or photochemistry, for 5 × solarmetallicity and T ∼ K ; Zahnle et al. 2009, Lianget al. 2003). Thus corresponding to the four prominentmolecules, we have four parameters: f H O , f CO , f CH ,and f CO . We reiterate that f X is not the absolute mix-ing ratio of “X”. It is the ratio between the mixing ratioof “X” and the mixing ratio under TE ⊙ ; except for CO ,for which it is with respect to 10 − . Our models alsoinclude NH , fixed at the TE ⊙ value, and TiO and VOat solar abundances. Additionally, we include H -H col-lison induced cross-sections, which are a source of con-tinuum opacity. Our H O, CH , CO and NH molecularline data are from Freedman et al. (2008), and refer-ences therein. Our CO data are from Freedman (per-sonal communication) and Rothman et al. (2005). And,we obtain the H -H collision-induced opacities from Bo-rysow et al. (1997), and Borysow (2002). We use a Ku-rucz model for the stellar spectrum (Castelli & Kurucz,2004).In the current work, we have made one significantchange to our approach in Madhusudhan & Seager(2009). Previously we had run models on a predeter-mined grid, chosen based on some model independentarguments outlined in that work. While we were ableto run a large population (tens of millions) of models inthat approach, the grid resolution was still coarse andsampled evenly over evidently unnecessary regions of theparameter space. In this approach, we run the modelsusing a more efficient parameter space exploration pro-cedure, allowing us to sample the desired error surfacesat much higher resolution. Parameter Space Exploration
Our primary requirement in this work is to be ableto explore the model parameter space at fine resolution.Even a single plausible model without thermal inversions,in a million models, would still be evidence against therequirement of thermal inversion by a given set of obser-vations.We use the Markov chain Monte Carlo (MCMC)method to explore the parameter space of models withoutthermal inversions. The MCMC method is a Bayesianparameter estimation algorithm which allows the calcu-lation of posterior probability distributions of the modelparameters conditional to a given set of observations. Anextensive body of literature exists on the applicationsof MCMC for parameter estimation (Gilks et al. 1998;Tegmark et al. 2004; Ford et al. 2005). The MCMC method allows an efficient means of exploring the param-eter space in search of a global solution, with very finesampling in the allowed range of parameter values. Inthis work, however, the observations are always less thanthe number of parameters, i.e there is no unique solu-tion. However, it is still possible to explore the parameterspace and find contours in the error surface of some mea-sure of fit. We, therefore, use the MCMC method witha Metropolis-Hastings scheme within the Gibbs sampler,for fine sampling of the model parameter space.Our model described in § P - T profile: T , P ,P , P , α , and α (Madhusudhan & Seager, 2009). And,four parameters correspond to the departures of molecu-lar abundances from the reference abundances describedin § f H O , f CO , f CH , and f CO .We define some physically motivated boundaries in theparameter space explored by the Markov chain. Weimpose the constraint of global energy balance by re-stricting η to [0.0,1.0], where, η = (1 − A )(1 − f r ) isthe ratio of emergent flux output on the day-side to in-cident stellar flux input on the day-side, weighted ap-propriately (Madhusudhan & Seager, 2009). Here, A is the Bond Albedo and f r is the day-night energy re-distribution. And, we impose some nominal bound-aries on the temperatures and departures from equilib-rium chemistry. We explore a wide range of deviationsfrom chemical equilibrium (Burrows & Sharp, 1999),empirically selected so as to be general enough. Formodels without thermal inversions, we set the bound-aries as − < log( f H O ) < − < log( f CO ) < − < log( f CH ) < − < log( f CO ) <
4. The lim-its are similar for models with thermal inversions, exceptthe lower boundaries for log( f H O ) and log( f CO ), whichare set at -4 and -5, respectively. We report all thosemodels which have the overall elemental C/H and O/Habundances within the broad range of (10 − − ) × solar (Anders & Grevesse, 1989; Burrows & Sharp, 1999;but c.f. Asplund & Grevesse, 2005; Allende-Prieto et al.2002). For the temperature structure, the constraintof no thermal inversion is imposed by requiring that P ≥ P . The “fit” parameters for the MCMC are T ,log(P ), log(P ), log(P ), α , α , log( f H O ), log( f CO ),log( f CH ), and log( f CO ). We consider uniform priors inall the parameters. For each system under consideration,we run one chain of 10 links for models with thermal in-version and one for models without thermal inversion.Our parametric P − T profile provides a simple means todemarcate between inversion and non-inversion models.The condition for the P - T profile to have no thermal in-version is P ≤ P ≤ P . And, that to have a thermalinversion is P ≤ P ≤ P . Quantitative Measure of Fit
Central to our analysis is the definition of what con-stitutes a “fit” to the data. We can only report to whatextent a model fits the data relative to the “observa-tional” uncertainties, i.e within the 1 σ error bars, or 1.5 σ errors, and so on. Given that the number of broadbandobservations are typically less than the number of modelparameters, we cannot report a formal fit with confidencelevels. Nevertheless, we evaluate our models based on the ξ statistic, defined as χ /N obs (Madhusudhan & Seager,hermal inversions in hot Jupiter atmospheres 7 Fig. 3.—
Pressure-Temperatures ( P - T ) profiles for the four systems. Each panel shows the P - T profiles with no thermal inversionsthat explain the observations at different levels of fit. Best-fitting profiles with thermal inversions for each of these systems have beenreported in the literature referred in the text. For HD 209458b, the profiles in magenta correspond to models that fit the observations with2 < ξ ≤ .
25; the best-fit non-inversion model has ξ = 2.04. For HAT-P-7b, the red profiles correspond to models fitting the observationsto within 1 . < ξ ≤
2; the best-fit model had ξ = 1 .
65. The brown profiles for both HD 209458b and HAT-P-7b are 30 profiles thatfit best, shown for illustration. For TrES-4 and TrES-2 the red profiles correspond to models that fit within 1 ≤ ξ ≤
2, and the purpleprofiles fit to within ξ <
1; only 100 randomly chosen profiles from each category are shown, for clarity.
Fig. 4.—
Departures from chemical equilibrium of CH and CO. The dots indicate the regions in the space of CO and CH mixing ratiosexplored by the MCMC chain (see Section 3.2); each dot is a model realization. All models with C/H and O/H within (10 − − ) × solar are shown. The boundaries in the composition space are described in section 3.2. For each planet, f CO and f CH are the departuresin the mixing ratios of CO and CH from those corresponding to thermochemical equilibrium with solar abundances (TE ⊙ ) for the sametemperature structure (see Section 3.1). For example, f CO = 1 implies a CO concentration that is at TE ⊙ . The left (right) panel for eachsystem shows constraints on models without (with) thermal inversions. Non-inversion and inversion models are labelled with (NI) and (I),respectively. The purple, orange, green and black colors correspond to ξ ≤
1, 1 ≤ ξ ≤
2, 2 ≤ ξ ≤
3, and 3 ≤ ξ ≤
4, respectively. Theblue dots for non-inversion models in HD 209458b and HAT-P-7b correspond to 2 . < ξ ≤ .
25; the best fitting model for HD 209458bhas a ξ = 2 .
04, and that for HAT-P-7b has ξ = 1 . hermal inversions in hot Jupiter atmospheres 9 Fig. 5.—
Departures from chemical equilibrium of H O and CO . For each planet, f H O is the departure in H O mixing ratio from thatcorresponding to TE ⊙ (see Figure 4), and f CO is the departure in CO mixing ratio from a constant value of 10 − (see Section 3.1). Thedescription of panels and colors is identical to that in Figure 4. All models with C/H and O/H within (10 − − ) × solar are shown.The boundaries in the composition space are described in section 3.2. ξ = 1 N obs N obs X i =1 (cid:18) f i,model − f i,obs σ i,obs (cid:19) , (1)where, f i,model and f i,obs are the model and observed fluxratios, respectively, and σ i,obs is the 1 σ measurementuncertainty. N obs is the number of observations.For each system, we report the best value of ξ wefind with a non-inversion model. We also present therange in parameter space which fit the observations atdifferent levels of ξ , for example, ξ ≤ ξ ≤
2, orhigher, as applicable. In this framework, ξ ≤ σ errorbars on average. And, a ξ ≤ √ . σ of the error bars on average, and so on. RESULTS
In this section, we report the constraints on the chemi-cal compositions of the day-side atmospheres of the plan-ets in our study. For each planet, we present the rangeof composition, temperature structure, and day-night en-ergy redistribution required by the best-fit models withno thermal inversions. We also present the constraintson the composition and day-night energy redistributionof models with thermal inversions for each system.
HD 209458b
We consider planet-star flux contrasts of HD 209458bin six channels of
Spitzer broadband photometry. Thedata include four IRAC observations reported by Knut-son et al. (2008), and observations in the 16 µ m IRSchannel and the 24 µ m MIPS channel, by Deming (per-sonal communication) and Deming et al.(2005), respec-tively. We focus on these observations which were re-ported in the literature as suggestive of a thermal inver-sion in HD 209458b, especially the four IRAC observa-tions.Our results indicate that HD 209458b is a likely can-didate to host a thermal inversion in its day-side atmo-sphere. However, whether or not HD 209458b actuallyhas a thermal inversion depends on the level of fit, andthe physical plausibility of the fitting models one is will-ing to consider. Figure 3 shows populations of pressure-temperature ( P - T ) profiles with no thermal inversionswhich fit the observations at different levels of ξ . Thecorresponding constraints on the atmospheric composi-tion are shown in Table 1, and the constraints for inver-sion models are shown in Table 2. Figure 4 and Figure 5show departures of molecular species from TE ⊙ , for mod-els of HD 209458b with and without thermal inversions.The observations require a thermal inversion in the at-mosphere of HD 209458b at the ξ = 2 level. The bestfitting model with no thermal inversion has a ξ of 2.04,implying a fit at 1.43 σ (i.e √ .
04) of the observations, onaverage. And, even at this level of fit, the models showsubstantial departures from thermochemical equilibriumassuming solar abundances (TE ⊙ ). Figure 4 shows thedepartures in the mixing ratios of CO and CH from TE ⊙ at different levels of fit. At the 2 . < ξ < .
25 surface(shown in blue dots for the non-inversion case), it can beseen that non-inversion models of HD 209458b require adepletion of CO of at least 10 − times TE ⊙ (the depar-tures shown in Figure 4 are in fraction with respect to TE ⊙ ). This low a mixing ratio of CO can, in principle, beachieved by having similarly low abundances of C and Orelative to solar. However, the simultaneous requirementof an overabundance of CH , is hard to explain. Thus,non-inversion models fitting the observations at the 1.5 σ errors seem physically implausible.The observations can be explained by physically plau-sible non-inversion models at the ξ ∼ σ observational errors. As shown in Fig-ure 4, the ξ = 3 level (region with green dots) allows fornon-inversion models which have CO within a factor of ∼
10 from TE ⊙ . Such small a factor can potentially beexplained either by just having different C/H and O/Habundances or due to non-equilibrium processes (Cooper& Showman, 2006; Zahnle et al. 2009; Line et al. 2010;Madhusudhan & Seager, 2010).The observations can be fit to within ξ = 1 by modelswith thermal inversions, as has been demonstrated previ-ously by Madhusudhan & Seager (2009). The molecularmixing ratios for inversion models as constrained by theobservations are shown in Table 2. Figures 4 and 5 showthe deviations from TE ⊙ required by the inversion mod-els. We find that the best-fitting models, within ξ = 1,allow compositions with CO and CH deviant from TE ⊙ by a factor of ∼
10 and higher. However, if we considerthe ξ = 2 surface, the observations can be fit with in-version models containing close to TE values of CO andCH , with C/H and O/H abundances only slightly en-hanced over solar. One potential problem in the inversionscenario, however, is the requirement of low H O by thebest-fit models. The low observed flux in the 3 µ m and24 µ m channels requires low H O, which is contrary tothe high H O requirement imposed by the high observedflux in the 5.8 µ m channel. This problem has been pre-viously discussed in the literature (Deming et al. 2005,Seager et al. 2005, Madhusudhan & Seager, 2009), andis a subject for future studies.Two best-fit model spectra, with and without a ther-mal inversion, for HD 209458b are shown in Figure 6,along with the P - T profiles. The corresponding at-mospheric composition and day-night redistribution areshown in Table 3. As can be seen from Figure 6, thedominant source of error for the non-inversion modelcomes from the high flux in the 5.8 µ m IRAC obser-vation. While most of the observations can be fit at the ∼ σ error bars, the 5.8 µ m is fit only at ∼ . σ . Thedominant source of opacity in this channel comes fromH O. In the non-inversion scenario, a low water contentcould potentially explain the lack of absorption in thischannel but will over predict the flux in the 24 µ m chan-nel. HAT-P-7b
HAT-P-7b is one of the hottest transiting hot Jupitersknown. Being the hottest of our sample of planets, itis also the most expected hot Jupiter in our sample tohost a thermal inversion. We use the day-side observa-tions of HAT-P-7b reported in four channels of
Spitzer
IRAC photometry by Christiansen et al. (2010). Theconstraints on the molecular abundances, and departuresfrom equilibrium, for the inversion and non-inversionmodels are shown in Table 1, Table 2, Figure 4, andFigure 5.Our results confirm previous findings that the inver-hermal inversions in hot Jupiter atmospheres 11
TABLE 1Constraints on the atmospheric properties for models without thermal inversions
HD 209458b a HAT-P-7b a TrES-4 TrES-22 < ξ ≤ .
25 1 . < ξ ≤ ξ ≤ ξ ≤ ξ ≤ ξ ≤ O b − − × − − − × − − − × − − − .
15 8 × − − .
16 10 − − . − − − × − − − − − .
06 10 − − .
07 10 − − .
04 10 − − . × − − × − × − − .
05 10 − − .
07 10 − − .
07 10 − − .
07 10 − − . − − × − × − − × − × − − .
02 10 − − .
03 10 − − − − − × − C/O 0 . − .
82 15 − × . − × − − × × − − × × − − × η c . − . . − . . − . . − . . − . . − . a For HD 209458b and HAT-P-7b, the best-fit non-inversion model has ξ of 2.04 and 1.65, respectively, and hence the reported ranges of ξ .See text for details. b The molecular mixing ratios are quoted as ratios by number with respect to molecular hydrogen. c η = (1 − f r )(1 − A ), where, f r is the day-night redistribution, and A is the bond albedo. (1- η ) gives the maximum day-night redistributionallowed by the model, i.e assuming zero albedo. TABLE 2Constraints on the atmospheric properties for models with thermal inversions
HD 209458b HAT-P-7b TrES-4 a TrES-2 a ξ ≤ ξ ≤ ξ ≤ ξ ≤ ξ ≤ ξ ≤ O b × − − × − × − − − − − .
16 3 × − − .
16 5 × − − .
13 5 × − − . × − − .
06 3 × − − .
07 3 × − − .
07 10 − − .
07 4 × − − .
07 10 − − . × − − .
06 10 − − .
06 10 − − .
07 10 − − .
07 10 − − .
07 10 − − . − − × − − − × − − − × − − − .
01 10 − − .
02 10 − − . −
146 1 −
105 5 × − −
53 5 × − −
207 4 × − − × × − − η c . − .
60 0 . − .
84 0 . − .
83 0 . − .
85 0 . − . . − . a For TrES-2 and TrES-4, the constraints at the ξ = 2 level are almost identical to those at the ξ = 1 level, and hence we do not reportthem here. b The molecular mixing ratios are quoted as ratios by number with respect to molecular hydrogen. c η = (1 − f r )(1 − A ), where, f r is the day-night redistribution, and A is the bond albedo. (1- η ) gives the maximum day-night redistributionallowed by the model, i.e assuming zero albedo. sions models fit the observations of HAT-P-7b betterthan models without thermal inversions. As shown inFigure 4, inversion models can fit the observations towithin the 1- σ errors (i.e. ξ ≤
1) for a wide range ofmethane and CO concentrations, including those closeto TE ⊙ values. On the other hand, the best fitting non-inversion model has a ξ = 1 .
65, indicating a fit at 1.3- σ errors. Even then, the best fitting non-inversion mod-els of HAT-P-7b shown in Figure 4 (blue dots) requiremethane abundances that are over five orders of magni-tude greater than the TE ⊙ values, which are seeminglyimplausible.The observations of HAT-P-7b can plausibly be ex-plained without a thermal inversion at the ξ = 3 level,i.e fits at 1.7- σ errors (green dots). At this level, mod-els with methane and CO concentrations only marginallydeviant from TE ⊙ values can explain the data. However,such a degree of fit may not be statistically representativeof the true nature of the planet atmosphere. A sampleof non-inversion temperature profiles at different levelsof fit is shown in Figure 3. The P - T profiles at ξ ≤ P - T profiles below ξ = 2.Two model spectra, corresponding to models with andwithout a thermal inversion, are shown in Figure 6, alongwith the corresponding temperature profiles. The modelparameters for each case are shown in Table 3. As shownin Figure 6, the non-inversion model is unable to fit the low flux at 3.6 µ m and the high flux at 8 µ m, simul-taneously. The reason for this behavior is because bothchannels have absorption features due to methane. A low(high) methane concentration in a non-inversion modelcauses high (low) observed flux in both the channels, con-trary to what is observed. On the other hand, the inver-sion model easily explains all the observations. The lowflux in the 3.6 micron channel is explained by the tem-perature decreasing outward in the lower layers, and thehigh flux at 8 microns is explained by the high tempera-tures, due to the thermal inversion, in the higher layerswhere the contribution to the 8 µ m channel peaks.The day-night redistribution is well-constrained by thedata. At the ξ = 2 level, the constraint on η =(1 − A )(1 − f r ) is 0.7 - 1.0, for non-inversion models.This range allows for a maximum redistribution of 0.26,assuming zero albedo, and hence implies relatively ineffi-cient advection of energy to the night side. This findingis consistent with the low redistribution of an inversionmodel reported in Christiansen et al. (2010), and thefinding of low redistribution in the visible light curve of Kepler . The inversion models on the other hand allowfor a wider range of η = 0 . − .
85 at the ξ ≤ µ m) obtained by the Kepler space2
Fig. 6.—
Sample model spectra for each system. Two model spectra, corresponding to models with and without a thermal inversion, arepresented for each system. The models represent a balance between degree of fit and physical plausibility. The corresponding P - T profilesare shown in the insets. The atmospheric compositions and day-night redistribution corresponding to each model are shown in Table 3. Ineach panel, the black circles with error bars (and the upper-limit for TrES-4) are the available Spitzer observations (see text for details).The red and green circles are the channel integrated model spectra in the six
Spitzer photometric channels, corresponding to the red andgreen curves, respectively. telescope (Borucki et al. 2009). However, modeling thevisible flux in the
Kepler bandpass introduces anotherfive free parameters in terms of the dominant opacitysources in the visible - the atmospheric concentrationsof TiO, VO, Na, K, and a prescription for scattering.These five parameters, which are largely decoupled fromthe opacity sources in the IR, allow a large degree of flex-ibility in fitting the one
Kepler observation. By exploringa preliminary range of values for the five parameters, wedo find that some of our best-fit non-inversion models forthe
Spitzer data are also able to fit the
Kepler point. Inan earlier work, we found that the
Kepler point could alsobe fit with models with thermal inversions (Christiansenet al, 2010).
TrES-4
TrES-4 is the second hottest planet in our sample, afterHAT-P-7b, and is highly favored to host a thermal inver-sion, on theoretical grounds (Fortney et al. 2008). Weuse the five
Spitzer photometric observations of Knutsonet al. (2009), which have been previously reported asevidence for a thermal inversion in TrES-4.We find that existing observations of TrES-4 can be ex- plained almost equally well by models with and withoutthermal inversions, contrary to previous findings. Ourfits for each case are better than ξ = 1, i.e within the1- σ error bars. The non-inversion P - T profiles fittingthe data at the ξ ≤ ξ ≤ ξ ≤
1, for a wide rangeof CO and CH abundances, including the values at TE ⊙ ,represented by f CO = 1 and f CH = 1. However, we findthat the best-fitting models (within ξ ≤
1) without in-versions require a high concentration of CO , & − , forTE ⊙ concentrations of the remaining molecules like H O(e.g. Figure 5). This requirement arises from the non-detection of planet flux observed in the 16 µ m channel.Since the dominant contribution to this channel comesfrom the CO feature at 15 µ m, a non-detection of fluxindicates substantial absorption due to CO and hencethe high CO requirement. At the high temperaturesof TrES-4, a CO concentration of 10 − is not feasiblevia equilibrium chemistry at solar abundances. However,CO concentrations as high as ∼ − are, in principle,feasible for a high metallicity, over 30 × solar, (Zahnle ethermal inversions in hot Jupiter atmospheres 13 TABLE 3Chemical compositions and day-night redistribution of sample spectra a H O CO CH CO C/O η ξ HD 209458bI 6 × − (0.006) 3 × − (50) 4 × − (0.001) 6 × − (0.006) 1.00 0.38 1.60NI 4 × − (0.04) 8 × − (0.01) 9 × − (0.8) 5 × − (5 × − ) 0.36 0.56 2.75HAT-P-7bI 7 × − (7) 6 × − (0.8) 8 × − (0.1) 1 × − (10) 0.07 0.59 0.89NI 2 × − (0.02) 5 × − (0.63) 2 × − (0.01) 8 × − (80) 0.85 0.63 3.0TrES-4I 8 × − (8) 4 × − (0.6) 5 × − (1) 2 × − (0.02) 0.05 0.40 1.76NI 5 × − (5) 4 × − (0.6) 2 × − (0.2) 2 × − (23) 0.09 0.41 1.80TrES-2I 2 × − (2) 1 × − (2) 1 × − (0.3) 1 × − (0.01) 0.44 0.72 0.54NI 1 × − (1) 4 × − (0.6) 2 × − (0.7) 2 × − (0.02) 0.23 0.46 0.98 a The two rows for each system correspond to the two model spectra for each system presented in Figure 6. “I” and “NI”correspond to the models with and without a thermal inversion, respectively, in Figure 6. For each molecule, the mixing ratioaveraged over all the layers is reported, along with the deviation from TE ⊙ shown in parentheses. al. 2009; Madhusudhan & Seager, 2010).Non-inversion model fits at the ξ ≤ ⊙ (Table 3 shows the composition).Also shown for reference is a model with a thermal in-version. The constraints on the molecular abundancesare shown in Table 1 and Table 2, for models withoutand with thermal inversions, respectively. We concludethat the observations place almost no constraints on themolecular concentrations in the atmosphere of TrES-4, ineither scenario. While the wide range in allowed chemi-cal compositions is seemingly implausible, it does allowfor a significant population of non-inversion models thatare physically plausible (see 5.2). We find this evidenceenough to conclude that there is no sure sign of a thermalinversion in TrES-4, given current data. TrES-2
Our results show that the observations of TrES-2 canbe explained to a high degree of fit by models both withand without thermal inversions. This general conclusionis similar to that of O’Donovan et al. (2010), where wefirst reported that the data could be fit by models withand without thermal inversions. In that study our non-inversion models fitting the data required a CO abun-dance that was lower than TE ⊙ by about two orders ofmagnitude, whereas the best-fit inversion models allowedTE ⊙ composition. However, with the new parameter ex-ploration routine, in the present study we have been ableto explore regions of parameter space well beyond whatwe could pursue in O’Donovan et al. (2010). While ourpresent results for inversion models agree with our pre-vious findings, our results for non-inversion models gobeyond our findings in O’Donovan et al. (2010).The best-fit solutions with no thermal inversions in thepresent study span a wide range in chemical composition,including that of TE ⊙ . Our results show that the P - T profile is mostly unconstrained by the data, resulting ina large region of parameter space that can explain theobservations even at the ξ = 1 level. Figure 3 shows thenon-inversion P - T profiles corresponding to ξ ≤ ξ ≤ ⊙ ). The composition in each scenario ispractically unconstrained even at the ξ ≤ × − - 4 × even at the ξ = 1 level, for the non-inversion models,for example. The large range in C/O is a consequence ofallowing the molecular mixing ratios to vary arbitrarily.Nevertheless, it does indicate that best-fit solutions withno thermal inversions can be found at the ξ ≤ § ξ = 1 level.Based on the existing Spitzer
IRAC observations,therefore, our results show that a thermal inversion can-not be inferred in the day-side atmosphere of TrES-2.The weak constraints on the atmosphere of TrES-2 isevident from the data. The flux ratio in the 4.5 µ mIRAC channel is noticeably higher than that in the 3.6 µ m channel, hinting at a possible emission feature dueto a thermal inversion. However, the flux ratio in the 5.8 µ m channel is noticeably lower than that in the 8 µ mchannel allowing for H O absorption, and hence the lackof a thermal inversion. Thus, while a model with thermalinversion can explain the data, a thermal inversion is notrequired by existing IRAC observations of TrES-2. Twobest-fit models for TrES-2, with and without a thermalinversion, are shown in Figure 6, and the correspondingcompositions are shown in Table 3. The models for boththe cases shown here span a rather plausible range ofchemical compositions. SUMMARY AND DISCUSSION
We have investigated the question of whether thermalinversions can be robustly inferred from existing
Spitzer
Spitzer observations at four or more wavelengths, and are highlyirradiated, so that they are theoretically favored to hostthermal inversions (Hubeny et al. 2003; Fortney et al.2008): HD 209458b, HAT-P-7b, TrES-4, and TrES-2.Furthermore, the observations considered have also beenpreviously reported to be consistent with thermal inver-sions in the corresponding systems, albeit less robustly.In this work, we addressed to what level of statisticalsignificance and physical plausibility thermal inversionscan be inferred in each of these systems.Our primary finding is that a detailed exploration ofthe model parameter space is necessary to make robustinferences of thermal inversions in exoplanetary atmo-spheres. We find that the observations of TrES-4 andTrES-2 can be explained by models with and withoutthermal inversions, and with physically plausible chem-ical compositions, at the ξ ≤ ξ ≤ mixing ratio ( ≥ − ), whichmight be an indication of enhanced metallicity in TrES-4(Zahnle et al. 2009; Madhusudhan & Seager, 2010).For HD 209458b and HAT-P-7b, we find that the ob-servations cannot be explained without thermal inver-sions to within a ξ <
3, i.e to within 1 . σ observationaluncertainties, for any plausible composition. Any betterfit would require substantial enhancements in methaneand depletion of CO, which is implausible at the hightemperatures in the systems considered. Our inference ofthermal inversions in HD 209458b and HAT-P-7b is con-sistent with previous findings of Burrows et al. (2008),Knutson et al. (2008), Christiansen et al. (2010), andSpiegel & Burrows (2010). Our results show that a de-tailed exploration of the model parameter space and anaccurate assessment of the observational errors is essen-tial to robustly infer thermal inversions based on existingphotometric observations. For example, if one considersthe 2- σ error bars on the data, thermal inversions maynot be required even for compositions in chemical equi-librium. Thermal Inversions or Not?
Whether or not the observations considered in thiswork can be explained without a thermal inversion de-pends on what level of fit, and physical plausibility ofmodels, one is willing to consider. If we do not considerthe physical plausibility of the best-fit models, the ob-servations of all the four hot Jupiters can be explainedwithout thermal inversions to within the 1.5 σ error bars,i.e ξ ≤ .
25. On the other hand, if we consider onlymodels fitting within the 1 σ error bars, and/or enforcearguments of physical plausibility (see § ∼ σ error bars by non-inversion models (e.g., Figure 6).A large contribution to the ξ comes predominantly fromthe 5.8 µ m point which is fit only at greater than 2 σ .Thus, for these models it is only the 5.8 µ m IRAC pointwhich guides any inference we make about thermal inver-sions. Similarly, for the model spectra shown for HAT-P-7b, the dominant contribution to ξ comes from the8 µ m IRAC point. Therefore, any inference of thermalinversions can be highly sensitive to the reported obser-vational uncertainties in a single channel, which varieson a case by case basis.At the level of current observations, our results poten-tially deviate from theory. Given that the atmospheresof TrES-4 and TrES-2 can be explained by models with-out thermal inversions, it is possible that these systemsdo not host thermal inversions. If that happens to bethe case, the results are in contrast to theoretical predic-tions, as both TrES-4 and TrES-2 have higher levels ofincident star flux as compared to HD 209458b (Fortneyet al. 2008), and hence are more likely to host thermalinversions. Nevertheless, since the observations for thesesystems are consistent with models both with and with-out thermal inversions, the only conclusion is that it istoo early to claim thermal inversions in these systems,contrary to some previous studies. Plausibility of Models
An important point concerns the physical plausibil-ity of non-inversion models fitting the observations. Asshown in Figure 4, the best-fit non-inversion models forHD 209458b and HAT-P-7b require substantial enhance-ment of CH as compared to CO. However, as explainedin § enhancement at theexpense of CO in a very-hot atmosphere.Our best-fit models explore an unrestricted range ofatmospheric compositions. In trying to conduct an unbi-ased exploration of the parameter space, we have allowedfor all the molecules to vary over a large range of values,that might be seemingly unphysical. For example, inTrES-2 the ξ = 1 limits extend to mixing ratios as highas 0.1 for H O, CO and CH . Such high mixing ratios in-dicate extreme metallicities that are too high to be plau-hermal inversions in hot Jupiter atmospheres 15sible, although not impossible. For reference, solar abun-dances have number fractions of C and O at 3 . × − and 7 . × − , respectively (Anders & Grevesse, 1989;used in equilibrium chemistry calculations of Burrows &Sharp, 1999; but c.f. Allende Prieto et al., 2002; As-plund et al., 2005, for recent values which are lower bya factor of ∼ , which are also manifestly unphysi-cal. For instance, the solar C/O ratio is ∼ .
5. However,the best-fitting models, with and/or without inversions,for all the systems do allow C/O ratios in the plausiblerange of 0.1 - 1.Finally, we have not explored the realm of drasticallyinhomogeneous models - those models where the mixingratio of a molecule could be very different in different lay-ers of the atmosphere. It is understandable that photo-chemistry and vertical mixing can deviate molecular mix-ing ratios away from equilibrium (e.g. Line et al. 2010;Madhusudhan & Seager, 2010). We expect that our pre-scription for molecular species, which is parametrized asdeviations from chemical equilibrium, spans the space ofpossible deviations. However, we have not consideredarbitrarily populating different layers of the atmospherewith different species. Adhoc filling of the layers withspecific molecules might allow a large, albeit unphysical,degree of freedom in fitting the observations.
Future Observations to Resolve the Degeneracy
In this work, we have addressed the apparent degen-eracy between atmospheric composition and thermal in-versions in hot Jupiter atmospheres. The large numberof model parameters allows the freedom to fit the lim-ited observations of some atmospheres in any scenario, i.ewith or without inversions; although, at different levels offit. Future developments in observations and theory areneeded to break the apparent degeneracies. Theoreticalefforts are needed to put limits of physical plausibility onthe atmospheric composition and temperature structure.Such limits, for example, might exclude many of the non-inversion models that fit the observations considered inthis work.New observations are important for better constraintson thermal inversions. In the near future, multiple obser-vations of thermal emission from transiting hot Jupitersin the near-IR, from ground and with
HST , along withexisting
Spitzer data can help constrain models to a goodextent, on a case-by-case basis. For instance, the twomodel spectra of HAT-P-7b in Figure 6 show only modestdifferences in the
Spitzer bandpasses. However, the spec-tra are markedly different in the near-IR, especially inthe continua between the molecular features, which canpotentially be observed from ground, e.g., in the J , H ,and K bands. Additional constraints can be also placedby observations within the molecular features, which are possible with space-based observations (e.g. with HST ),except for molecules like methane where ground-basedobservations might also be feasible. Near-IR observa-tions have been reported for a few systems to date, notparticularly constraining thermal inversions (e.g. Swainet al. 2009; Croll et al. 2010). However, multi-bandnear-IR photometry and/or spectroscopy can prove tobe a rich resource for targeted searches for thermal in-versions in exoplanetary atmospheres. Targets can be se-lected based on constraints from already existing
Spitzer observations, irradiation levels, and sensitivity of a giveninstrument to the planet-star flux contrasts. The severalnear-IR bandpasses mentioned above are currently ripefor this purpose. In the long run, high resolution spectrawith the
James Webb Space Telescope will have the sen-sitivity to conclusively identify the presence of thermalinversions based on spectrally resolved emission features.An important point concerns the opportunity to ob-serve in the two IRAC channels on
Warm Spitzer . Whileit is true that robust inferences of thermal inversionscould not be made even with four
Spitzer points in sev-eral known cases, a large difference between the 3.6 µ mand 4.5 µ m channels can place stringent constraints onthe existence of thermal inversions. A very large fluxexcess in the 4.5 µ m channel over the 3.6 µ m channelis highly indicative of a thermal inversion; although, italso depends on the irradiation level of the planet whichgoverns the blackbody continuum. A large excess in the3.6 µ m channel, on the other hand, is almost a sure signof no thermal inversion, as in the cases of HD 189733b(Madhusudhan & Seager, 2009) and GJ 436b (Stevensonet al. 2010; Madhusudhan & Seager, 2010). Thus, obser-vations of hot Jupiters with Warm Spitzer would likelybe able to identify the extreme cases of systems with orwithout thermal inversions.Tremendous progress has been made in the last decadein our understanding of exoplanetary atmospheres. Atthe same time, recent and current observations allow usa chance to revisit previous interpretations made withlimited observations. It is now upon us to evaluate allthe theoretical options and observational uncertainties soas to determine a framework in which to interpret obser-vations. Judicious target-selection and efficient planningof future observations, from ground and from space, willbe critical to characterizing the atmospheres of the grow-ing number of transiting exoplanets.We thank Heather Knutson and Jonathan Fortney forhelpful discussions. This work is based on published ob-servations made with the
Spitzer Space Telescope , whichis operated by the Jet Propulsion Laboratory, CaliforniaInstitute of Technology under a contract with NASA.Support for this work was provided by NASA throughan award issued by JPL/Caltech.
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