Models of Neptune-Mass Exoplanets: Emergent Fluxes and Albedos
David S. Spiegel, Adam Burrows, Laurent Ibgui, Ivan Hubeny, John A. Milsom
aa r X i v : . [ a s t r o - ph . E P ] F e b Submitted to ApJ
Preprint typeset using L A TEX style emulateapj v. 04/20/08
MODELS OF NEPTUNE-MASS EXOPLANETS: EMERGENT FLUXES AND ALBEDOS
David S. Spiegel , Adam Burrows , Laurent Ibgui , Ivan Hubeny , John A. Milsom Department of Astrophysical Sciences, Princeton University, Peyton Hall, Princeton, NJ 08544 Steward Observatory, The University of Arizona, Tucson, AZ 85721 and Department of Physics, The University of Arizona, Tucson, AZ 85721
Submitted to ApJ
ABSTRACTThere are now many known exoplanets with M sin i within a factor of two of Neptune’s,including the transiting planets GJ436b and HAT-P-11b. Planets in this mass-range aredifferent from their more massive cousins in several ways that are relevant to their radiativeproperties and thermal structures. By analogy with Neptune and Uranus, they are likelyto have metal abundances that are an order of magnitude or more greater than those oflarger, more massive planets. This increases their opacity, decreases Rayleigh scattering,and changes their equation of state. Furthermore, their smaller radii mean that fluxes fromthese planets are roughly an order of magnitude lower than those of otherwise identicalgas giant planets. Here, we compute a range of plausible radiative equilibrium models ofGJ436b and HAT-P-11b. In addition, we explore the dependence of generic Neptune-massplanets on a range of physical properties, including their distance from their host stars, theirmetallicity, the spectral type of their stars, the redistribution of heat in their atmospheres,and the possible presence of additional optical opacity in their upper atmospheres. Subject headings: equation of state – line: profiles – planetary systems – radiative transfer– stars: individual GJ436, HAT-P-11 – astrochemistry INTRODUCTION
Although in the early days of exoplanet researchlarge, massive planets (gas giants) were discoveredin disproportionate numbers, lower mass objects, in-cluding super Earths (1 to ∼
10 times the mass of theEarth) and ice-giants, are now being found with in-creasing frequency. The latter population, consist-ing of “Neptune-mass” bodies (those with masseswithin a factor of a few of Neptune’s), currentlynumbers several dozen, and is growing rapidly.Exoplanet transits, first observed by Henry et al.(2000) and Charbonneau et al. (2000), allow precisemeasurements both of planet masses (by breakingthe degeneracy between mass and inclination angle)and of radii. These two crucial pieces of data pro-vide constraints on models of structure, evolution,and bulk composition (Ibgui & Burrows 2009; Bur-rows et al. 2007a, 2000; Guillot 2008, 2005; Guillot& Showman 2002; Baraffe et al. 2005). As a result,the best-studied among the Neptune-mass popula-tion are the transiting planets GJ436b (1.33 timesthe mass of Neptune, and 1.27 times its radius; Cac-eres et al. 2009; Bean & Seifahrt 2008; Bean et al.2008; Torres 2007; Deming et al. 2007a; Maness
Electronic address: [email protected], [email protected], [email protected],[email protected], [email protected] See the catalog at http://exoplanet.eu. et al. 2007) and HAT-P-11b (1 . M Nep , 1 . R Nep ;Bakos et al. 2009; Dittman et al. 2009).Observations of an exoplanet can constrain the-oretical models. The marriage of data with theory,therefore, allows us to learn about the physical con-ditions on these distant worlds. For example, obser-vations during transit can constrain various prop-erties, including both the atmospheric compositionand structure at the terminator (Seager & Sasselov2000; Hubbard et al. 2001; Charbonneau et al. 2002;D´esert et al. 2008; Redfield et al. 2008) and the mo-tion of the atmosphere (Brown 2001; Spiegel et al.2007). This paper, however, focuses on the diagnos-tics of planetary structure that are available fromobservations of a planet’s disk, not of its limb. Thedisks of those planets whose orbits are aligned pre-cisely enough that they transit may be studied inexquisite detail. Half an orbit after transit, theyundergo secondary eclipse, when they pass behindtheir host stars. Immediately before and after sec-ondary eclipse, a planet is in its “full-moon” phase,when its reflected and emitted light are at theirmaximum from our vantage. There have been avariety of ground-based and space-based secondary-eclipse observations of exoplanets in the last severalyears (Richardson et al. 2003; Snellen 2005; Dem-ing et al. 2007a,b; Charbonneau et al. 2008; Knut-son et al. 2008, 2009; Alonso et al. 2009b,a; Sing& L´opez-Morales 2009). In the last decade, severalgroups have produced a progression of theoreticalmodels of the atmospheres of Jupiter-mass exoplan-ets (Sudarsky et al. 2000, 2003; Hubeny et al. 2003;Burrows et al. 2005; Fortney et al. 2006; Burrowset al. 2008a; Fortney et al. 2008; Burrows et al.2008b; Spiegel et al. 2009; Showman et al. 2009),including calculations of both optical albedos andinfrared emergent spectra. These calculations arenow being used in interpretative studies of the sec-ondary eclipse observations. A critical difference between the gas giant andthe ice-giant planets motivates the present investi-gation of theoretical models of lower mass objects.The gas giants of our solar system – Jupiter andSaturn – have metal abundances not more thana few times solar (Matter et al. 2009; Saumon &Guillot 2004). However, Neptune, Uranus, and,presumably, extrasolar planets in the same mass-range, are thought to have bulk metallicities of 30-45 times solar (Figueira et al. 2009; Matter et al.2009; Guillot & Gautier 2007). Increased metal-licity leads to increased opacity. Reduced relativehydrogen abundance might also reduce the impor-tance of Rayleigh scattering in the atmospheres ofNeptune-mass planets.In the last several years, infrared observationsby the
Spitzer Space Telescope , in conjunction withtheoretical models by our group and others, havesuggested that several giant planets have thermalinversions, wherein, above a relative minimum, theatmospheric temperature increases with altitude.Hubeny et al. (2003) suggested that, if there is anadditional source of upper atmosphere optical opac-ity, incident stellar irradiation could lead to pre-cisely this type of atmospheric structure. The plan-ets whose emergent spectra have suggested ther-mal inversions include HD 209458b (Burrows et al.2007b; Knutson et al. 2008), HD 149026b (Fortneyet al. 2006; Burrows et al. 2008a), TrES-4 (Knut-son et al. 2009), XO-1b (Machalek et al. 2008),XO-2b (Machalek et al. 2009), XO-3b (McCulloughet al. 2008), and, perhaps, υ Andromeda b (Bur-rows et al. 2008a). Various authors have suggestedthat titanium oxide (TiO), as a strong optical ab-sorber, might provide the extra opacity that isneeded to produce the inferred inversions (Fortneyet al. 2008; Showman et al. 2009). However, Spiegelet al. (2009) argue that, without extremely vigor-ous macroscopic mixing, a heavy molecule such asTiO will settle to a level that is too deep for it tocontribute to an inversion. Furthermore, Spiegelet al. (2009) show that several planets for whichinversions have been inferred ought to have cold-trap regions deeper in their atmospheres that arecool enough for titanium to condense and rain out,therefore requiring even more vigorous mixing ifTiO is to survive up to the millibar levels where Comparable calculations have been made in the contextof Super-Earth planets (Miller-Ricci et al. 2009; Kalteneggeret al. 2009). it would be needed. Zahnle et al. (2009) suggestthat sulfur photochemistry provides another avenuefor achieving the additional upper-atmosphere op-tical opacity that is needed to produce inversions.The ultimate cause of inversions remains unknown,and might differ from one planet to another. Nev-ertheless, sources of extra optical opacity might berelated to metal abundance, and so it is reasonableto wonder whether some highly irradiated Neptune-mass planets might have thermal inversions.In order to survey the range of plausible struc-tures of Neptune-mass exoplanets, we compute avariety of model atmospheres. We explore howthe emergent infrared fluxes and optical albedos ofGJ436b and HAT-P-11b depend both on the redis-tribution of heat in their atmospheres and on thepossible presence of an extra source of atmosphericopacity (following the treatment in Burrows et al.2008a and Spiegel et al. 2009). We also examinehow “ice-giant” atmospheres, and their optical andinfrared fluxes, depend on their distances from theirhost stars, their atmospheric metal abundances, andthe spectral types of their hosts. By “ice-giant,”we are simply referring to a planet with approx-imately Neptune’s mass and radius; whether theatmosphere is bounded below by ices is irrelevantto our analysis. Our models are of planets withhydrogen/helium-dominated atmospheres that aremetal-enriched relative to solar by a large factor.We note that some planets currently classified as“Super-Earths” might share this structure.The remainder of this paper is structured as fol-lows: In §
2, we present our numerical techniques forcomputing the atmospheres of “ice-giant” planets.In §
3, we describe the different models that we ran.Section 4 contains the results of our calculations.Finally, in §
5, we summarize our findings. NUMERICAL METHODS
We calculate radiative equilibrium models of ir-radiated planetary atmospheres models. As in ourother recent studies, we use the code
COOLTLUSTY (Hubeny et al. 2003; Sudarsky et al. 2003; Burrowset al. 2006, 2008a; Spiegel et al. 2009). This codeis an variant of the code
TLUSTY (Hubeny 1988;Hubeny & Lanz 1995), with atomic and molecu-lar opacities appropriate to the cooler environmentsof planetary atmospheres and brown dwarfs (Sharp& Burrows 2007; Burrows & Sharp 1999; Burrowset al. 2001, augmented by Burrows et al. 2002 andBurrows et al. 2005). Irradiation in planetary atmo-sphere models is incorporated using Kurucz modelstellar spectra, interpolated to the temperaturesand surface gravities appropriate to our study (Ku-rucz 1979, 1994, 2005).In order to examine the effect of metallicity on The analysis of Showman et al. 2009 appears to agreewith this conclusion. planet atmospheres, we compute some models withsolar abundance of metals, and others with 30 × solarabundance. For both metallicities, we derive thechemical equilibrium abundances (as functions oftemperature and density) of several hundred atomicand molecular species (Sharp & Burrows 2007). Us-ing chemical equilibrium abundances, we calculatethe monochromatic opacities that are used in theradiative transfer calculations. Furthermore, we re-compute the equation of state tables for the chemi-cal abundances derived for 30 × solar metallicity.To treat the redistribution of incident stellar fluxin a planet’s atmosphere, we use the P n formal-ism described in Burrows et al. (2006) and Burrowset al. (2008a). In this formalism, in lieu of calcu-lating the full three dimensional general circulation,the proportion of day-side incident flux that is trans-ported to and reradiated from the night-side is pa-rameterized as P n , which plausibly ranges between0, corresponding to all stellar flux being instanta-neously reradiated, and 0.5, corresponding to thenight side receiving approximately the same amountof stellar energy as the dayside (as a result of advec-tive heat redistribution). The redistribution takesplace between 0.003 bars and 0.6 bars.For each model planet, we calculate thetemperature-pressure profile, the planet-star fluxratio of emergent infrared radiation, the opticalalbedo, and the temperature and pressure, as func-tions of wavelength, of the τ λ = 2 / However, at wavelengths shorter than 1 µ m, thealbedos we report may be understood as being pre-dominantly measures of reflectance and scatteringproperties of the day-side atmospheres. PLANET MODELS
The known exoplanets in Neptune’s mass rangeare found in a variety of configurations. Table 1summarizes their diversity. Orbital separationsrange from 0.02 AU to 2.7 AU, and stellar typesfrom M through G are represented. The diversityof observed planet-star systems motivates our inves-tigation of a comparably diverse model set.Our goal in this investigation is twofold: ( i ) toexamine the diagnostics of the known transiting ex-trasolar “ice-giants” (GJ436b and HAT-P-11b); and( ii ) to examine more generally the types of radiativeand thermal structures that might obtain in plausi-ble exo-Neptunian atmospheres. To these ends, we In fact, this happens in the mid-infrared, where planetsare strong thermal emitters. compute 29 model atmospheres, listed in Table 2. , Six of the models of GJ436b are namedT35a03Z01P0k0G through T35a03Z01P5k0G (cor-responding to a GJ436b-like planet around a GJ436-like star, with P n ranging between 0.0 and 0.5). Two additional models are T35a03Z01P3k2G (cor-responding to GJ436b with P n = 0 . κ = 0 . g − in the upper atmosphere, where κ is the same as κ e from Burrows et al. 2008a) and T35a03Z30P0k0G(corresponding to GJ436b with P n = 0 . × solar metallicity). As in previous work, κ is gray over the range 3 × -7 × Hz.The models of HAT-P-11b are T48a05Z01P0k0Hthrough T48a05Z01P5k0H (HAT-P-11b with P n ranging from 0.0 to 0.5) and T48a05Z01P3k2H (ex-tra upper atmosphere absorber of opacity κ =0 . g − ). The additional optical opacity inmodels T35a03Z01P3k2G and T48a05Z01P3k2H isconfined to the pressure range from ∼ a ∼ < . × solar metallicity, at thesame distances from the same star). Models test-ing the effect of stellar type are T35a03Z01P1k0Gthrough T62a03Z01P1k0G. These models place aGJ436b-size planet at 0.03 AU from, respectively,GJ436 ( T = 3500 K, log g = 4 . T = 4200 K, log g = 4 . T =4780 K, log g = 4 . T = 5200 K,log g = 4 . T = 5778 K,log g = 4 . T = 6200 K,log g = 4 . There are 31 rows in Table 2, of which two, the 26th and30thre duplicates. ∼ ∼ burrows/. In this labeling convention, the symbols have the follow-ing meanings: “T”, for temperature, is followed by 0.01 timesthe star’s effective temperature; “a”, for semimajor axis, isfollowed by 100 times the orbital semimajor axis in AU; “Z”,for metallicity, is followed by the ratio with respect to solarabundance of metals; “P” is followed by 10 times P n ; “k” isfollowed by 10 times κ (in cm g − ); the final letter (“G” or“H”) indicates whether the radius and surface gravity of themodel planet match those of GJ436b or HAT-P-11b. RESULTS
For each model planet, we present two observ-able variables and three that are not observable.The observables are the emergent infrared flux, pre-sented as a planet-star flux ratio, and the opticalalbedo. The three variables that are unobservable,but are nonetheless useful probes of our models,are the following: the temperature-pressure profile(temperature as a function of depth in the atmo-sphere); the temperature, as a function of wave-length, of the τ λ = 2 / GJ436b and HAT-P-11b
The most salient difference between GJ436band HAT-P-11b is that the latter orbits a hotter,brighter star. The temperature-pressure profiles ofmodels of them, portrayed in Fig. 1, reflect this:HAT-P-11b (bottom panel of Fig. 1) is consistentlyhotter at a given pressure. For both profiles, the re-distribution of day-side irradiation to the night sidereduces the dayside’s temperature. As P n increases,the thermal profile becomes cooler at a given pres-sure in the range where the redistribution takesplace (0.003 bars to 0.6 bars in these models). Thiseffect is more pronounced in GJ436b than in HAT-P-11b.On the other hand, the effect of an additionaloptical absorber (green curves, κ = 0 . g − ) ismore pronounced in HAT-P-11b’s profile – preciselybecause its star is hotter and, therefore, brighter inthe optical. In GJ436b’s model, an additional opti-cal absorber at high altitude increases upper atmo-sphere temperatures by a modest ∼
60 K. In HAT-P-11b’s model, however, an extra optical absorberincreases the temperature at ∼ − bars by ∼
300 K,and creates a thermal inversion of about 100 K.In the figure for GJ436b, we have included oneadditional model (T35a03Z01P0k0G), with P n =0 . × solar opacity. The profile of this modellooks qualitatively different from those of the othermodels, because it does not quite extend to themarginally-stable convection zone. However, theRosseland mean optical depth at the base of thismodel is ∼
80, and the net flux is constant with alti-tude, so the unmodeled portions of the atmospheredo not affect the emergent spectrum.Let us consider our four wavelength-dependentmodel diagnostics, presented in Fig. 2 (GJ436b) andFig. 3 (HAT-P-11b). We have smoothed the curvesin each plot to a spectral resolution of 100 for easein viewing. In the flux-ratio plot of Fig. 2 (upperleft panel), we include the 8- µ m measurement thatDeming et al. (2007a) obtained with the Spitzer
In-fraRed Array Camera (IRAC).The models’ emergent infrared fluxes show var- ious spectral features. The opacity database con-tains significant CH cross section at 3.3 µ m and7.8 µ m, H O and CO absorption at 4.5 µ m, andH O at 6 µ m and 10 µ m (Sharp & Burrows 2007).The emergent flux is insensitive to P n at somewavelengths, and shows a moderate-to-strong de-pendence on P n at others (stronger in the modelsof GJ436b than in those of HAT-P-11b). The effectof increasing P n is to reduce the emergent flux in“emission” features, specifically at ∼ µ m, ∼ µ m, ∼ µ m, and ∼ µ m.The optical albedo (upper right panels of bothfigures) does not exhibit a strong dependence on P n .At wavelengths shorter than 0.5 µ m, the albedo isnonnegligible, due to Rayleigh scattering. At longerwavelengths, it tends to be ∼ < κ = 0 . g − start to increase longward of ∼ µ m. Albedos ofHAT-P-11b models are somewhat higher at thesewavelengths. Note, for both models, the dip in op-tical albedo at ∼ µ m, which is due to a waterabsorption feature (Sharp & Burrows 2007).Although the temperature and pressure of the τ λ = 2 / P n on thesevariables. When P n = 0 .
0, the temperature variessignificantly with wavelength, as narrow windowsof low opacity allow flux from the deeper, warmerportions of the atmosphere to escape. As P n in-creases, the location of the photosphere becomesmore nearly constant with wavelength. The pres-sure of the photosphere does not vary much with P n , except near 5 µ m. As seen in the bottom rightpanel, most of the infrared spectrum originates atpressures between 10 − and 0.1 bars. For these pres-sures, as seen in the upper left panel, the day-sideenergy sink for P n = 0 . P n ’s) decreases the T gradient so that for P n = 0 . κ =0 . g − ) all produce lower planet flux at8 µ m than Deming et al. (2007a) measured, modelT35a03Z01P3k2G ( κ = 0 . g − ) has an IRAC-bandpass-integrated 8- µ m flux that is more than20% higher than that of T35a03Z01P3k0G (the P n = 0 . Furthermore, model T35a03Z30P0k0G,with zero redistribution and 30 × solar opacity, hasan integrated 8- µ m flux that is 7.5% higher thanthat of the analogous model with solar opacity.Still, its 8- µ m flux is 12% lower than bottom of A model with κ = 0 . g − (not shown) passes rightthrough the center of the error bars. the 1- σ range on the Deming et al. (2007a) mea-surement. In both models T35a03Z01P3k2G andT48a05Z01P3k2H (GJ436b and HAT-P-11b with κ = 0 . g − ), an extra optical absorber causesthe upper atmosphere to be warmer and the planetflux and the photospheric temperature to showfewer spectra features. Despite the warmer up-per atmosphere temperatures and generally warmerphotospheres, however, the additional optical opac-ity causes the the albedo to be much lower in theoptical ( ∼ < Consider again the flux-ratio comparisons forGJ436b (upper left panels of Fig. 2). All 8 mod-els have approximately the same flux at 4.6 µ m,but flux varies significantly at 4.0 µ m. The ratioof the planet’s flux at 4.0 µ m to that at 4.6 µ mvaries from ∼ ∼ ∼ ∼ µ m to flux at 4.6 µ m provides a con-straint on the redistribution of heat. This ra-tio varies from ∼ ∼ µ m, extraupper atmosphere opacity (T35a03Z01P3k2G) in-creases the emergent flux to ∼
20% above what itwould be without the extra absorber. We note thatthe difference between the 30 × solar opacity modeland those with solar opacity appears greater whenone considers the full spectral shape than when in-tegrated over the 8- µ m IRAC band.The models of HAT-P-11b exhibit a qualitativelyanalogous dependence on P n , although the ratiosare quantitatively somewhat different because ofthe differences between the two stars and the twoorbital separations. But, again, T48a05Z01P0k0Hhas significantly greater flux at 4.0 µ m (by a factorof ∼
2) and at 10 µ m (by a factor of ∼ µ m,T48a05Z01P3k2H is approximately twice as brightas the models without an extra absorber, and is alsomuch brighter (by ∼ µ m. Distance
The temperature of an irradiated stellar com-panion’s photosphere drops off rapidly with increas- At sufficiently high values of κ and with sufficient irra-diation, this trend would reverse, and higher κ would lead toa larger apparent albedo. ing orbital separation, and its emergent flux de-creases correspondingly. For separations greaterthan ∼ κ = 0 . g − and P n = 0 .
0. Solar opacity models are representedby solid curves, and 30 × solar opacity models withdashed curves; the differences between these arediscussed in § ∼ ∼ µ m because of the con-tribution from the Wien tails of their thermal emis-sion. Metallicity
It was initially surprising to us that modelswith higher metallicity around solar-type stars, withtheir concomitant changes in both opacity and equa-tion of state, nevertheless have extremely similarspectral profiles to their lower metallicity cousins(solid and dashed lines in top left panel of Fig. 5).The temperature-pressure profiles (top panel ofFig. 4) of 30 × solar opacity models are quite dif-ferent from those of solar opacity models; increas-ing metallicity appears to translate the T - P profilesup to higher altitudes (lower pressures). But thephotospheric temperature (bottom panel of Fig. 5)remains essentially unchanged with higher opacity.How is this possible? The photosphere pressureplot (lower right panel of Fig. 5) offers the explana-tion. Because, viewed from the outside, the opticaldepth increases so much more quickly with depthinto atmosphere for the high metallicity models,their τ λ = 2 / × solar metallicitymodels are about half that of the solar metallic-ity models. Furthermore, at the long wavelengthend, a deep absorption feature due to water arisesat ∼ µ m in the high metallicity models that isnot present in the low metallicity ones. Since thetwo characteristics that are significantly differentat high metallicity (the T - P profile and the photo-spheric pressure) are not observable, these compara-tively modest changes to albedo might represent ourbest chance to constrain the atmospheric metallic-ity through whole-disk observations (though transitspectrum observations could provide a complemen-tary constraint). Note, however, that this water fea-ture is the same one seen in solar metallicity modelsof GJ436b and HAT-P-11b in Figs. 2 and 3, and,therefore, might be a signature of high metallicityonly in the context of a modeling effort that includesthe appropriate stellar irradiance spectrum.For a solar-type irradiation spectrum, the opac-ity (solar or 30 × solar) of model planet’s atmo-sphere has a minor influence on the emergent spec-trum. However, the opacity has a stronger influ-ence when the irradiation spectrum is from a coolM-dwarf (as shown in Fig.2, where the irradiationis from GJ436). Stellar Type
How do a Neptune-mass planet’s structure andappearance depend on its host star? ModelsT35a03Z01P0k0G through T62a03Z01P0k0G ad-dress this question. These models are of planets thesize of GJ436b, separated 0.029 AU from stars rang-ing from cool (3500 K) to moderately hot (6200 K).The most prominent differences between modelsaround cooler and dimmer stars and those aroundhotter stars is that, unsurprisingly, planets aroundhotter stars are hotter and bluer. The bottom panelof Fig. 4 and the bottom left panel of Fig. 6 indi-cate how more massive, brighter stars lead to hotterplanets at a given distance.For a planetary system that is of order a Gyror older, a Neptune-mass planet has lost almostall of its heat of formation (Burrows et al. 2000,2003; Baraffe et al. 2005). As a result, for planetsat separations ≤ / (4 π ) times the solid angle subtended bythe planet from the vantage of the star (Ω p / π = πR p / πa ). This leads to the somewhat counter-intuitive flux-ratios plot in the upper left corner ofFig. 6: in the mid-infrared (at wavelengths greaterthan ∼ µ m), a planet around a cool star is signifi-cantly brighter relative to its star than one arounda hot star. Of course, this is because the same totalrelative flux is distributed predominantly at longerwavelengths for a cooler planet and at shorter wave-lengths for a hotter planet.Interestingly, at wavelengths less than ∼ µ m,the photospheric pressures are not dramatically dif-ferent for the six models, despite very different T - P profiles. Finally, similar to the trend seen inalbedo versus distance seen in § SUMMARY AND CONCLUSIONS
We have presented a series of theoretical 1-D models of the radiative and thermal structuresof Neptune-mass, “ice-giant” exoplanets, includingmodels at opacities corresponding both to solarmetallicity and to 30 × solar. To produce the latter,we computed equilibrium chemical abundances ofhundreds of molecular and atomic species, and werecomputed the equation of state with the highermean molecular weight. We produced models ofthe transiting planets GJ436b and HAT-P-11b, andmodels of generic planets in a range of plausibleconditions.In our investigation of GJ436b and HAT-P-11b,we find the following: • The ratio of planet flux at 4.0 µ m or at 10 µ mto that at 4.6 µ m might place a constrainton the amount of redistribution of heat fromthe day to the night side: more redistribu-tion leads to a smaller flux ratio. The ratio ofplanet flux at 7 µ m to that at 4.6 µ m mightplace a constraint on the possible presence ofan extra optical absorber in the high atmo-sphere. • An extra optical absorber, if present in eitherplanet, would have a more dramatic effect onboth the T - P profile and the emergent spec-trum of HAT-P-11b than of GJ436b, becauseHAT-P-11 is brighter in the optical. • Models of GJ436b without an extra ab-sorber are significantly dimmer than the Dem-ing et al. (2007a) measurement, while themodel with an extra absorber of opacity κ = 0 . g − is closer to being consis-tent with the measurement, though still shyof it. • Models of these two planets have moderateoptical albedos ( ∼ • Higher metallicity implies higher opacity for agiven column mass of atmosphere. As a result,photospheres of higher metallicity models areat lower pressures. • Nonetheless, when the irradiation spectrum isfrom a solar-type star, higher metallicity mod-els appear to differ little from solar metallic-ity models in observable variables. The mostnoticeable difference, among close-in planetsaround solar-type stars, is the water absorp-tion feature at ∼ µ m that shows up inhigher metallicity models. When the irradia-tion spectrum is from a cool M-dwarf (GJ436),the metallicity of the planet’s atmosphere hasa larger influence on the emergent spectrum. • Neptune-mass planets around cooler starshave larger flux ratios in the mid-infrared thanones at the same distance around hotter stars. (This conclusion is germane to Jupiter-massplanets, as well.)Close-in Neptune-mass planets present an ob-servational challenge because, with radii a factorof ∼ Spitzer , and additional ones are planned. ForGJ436b, 3.6 and 4.5 µ m IRAC observations willhelp to quantify both the amount of redistributionof heat from the day to the night sides and the pos-sible presence of an extra absorber in the high atmo-sphere. Future observations with new facilities willprovide constraints on models of other planets inthis mass range, and will sharpen our understand-ing of this class of objects.We thank Jason Nordhaus, Nikole Lewis, andAdam Showman for helpful discussions. Wethank our anonymous referee for a number ofhelpful comments that improved the manuscript.This study was supported in part by NASAgrant NNX07AG80G. We also acknowledge sup-port through JPL/Spitzer Agreements 1328092,1348668, and 1312647. REFERENCESAlonso, R., Alapini, A., Aigrain, S., Auvergne, M., Baglin,A., Barbieri, M., Barge, P., Bonomo, A. S., Borde, P.,Bouchy, F., Chaintreuil, S., De la Reza, R., Deeg, H. 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TABLE 1Characteristics of ∼ Neptune-Mass Exoplanets
Planet Mass Radius log g Semi-major Axis Stellar Type References( M Nepa ) ( R Nepb ) (cgs) (AU)CoRoT-7b 0.65 0.43 3.59 0.02 K0 V Fressin et al. 2009GJ436b 1.33 1.04 3.15 0.03 M2.5 Torres 2007HAT-P-11b 1.50 3.05 1.22 0.05 K4 Bakos et al. 2009Dittman et al. 2009HD 181433b 0.44 - - 0.08 K3 IV Bouchy et al. 2009HD 285968b 0.49 - - 0.07 M2.5 V Forveille et al. 2009HD 40307d 0.53 - - 0.13 K2.5 V Mayor et al. 2009HD 7924b 0.54 - - 0.06 K0 V Howard et al. 2009HD 69830b 0.61 - - 0.08 K0 V Lovis et al. 2006HD 160691c 0.62 - - 0.09 G3 IV-V Pepe et al. 200755 Cnc e 0.63 - - 0.04 G8 V Poveda & Lara 2008GJ 674b 0.69 - - 0.04 M2.5 Bonfils et al. 2007HD 69830c 0.70 - - 0.19 K0 V Lovis et al. 2006OGLE-05-169L b 0.74 - - ∼ a The mass of Neptune is M Nep = 1 . × g = 0 . M Jup , where M Jup = 1 . × g. b The radius of Neptune is R Nep = 2 . × cm = 0 . R Jup , where R Jup = 7 . × cm. TABLE 2Exoplanet Models a Model Name Stellar Model a Metallicity P n R p log g κ (AU) ( Z ⊙ ) ( R Nep ) (cgs) (cm g − )T35a03Z01P0k0G GJ436 ( T = 3500 K; log g = 4 .
80) 0.03 1 0.0 1.04 3.145 0.0T35a03Z01P1k0G GJ436 ( T = 3500 K; log g = 4 .
80) 0.03 1 0.1 1.04 3.145 0.0T35a03Z01P2k0G GJ436 ( T = 3500 K; log g = 4 .
80) 0.03 1 0.2 1.04 3.145 0.0T35a03Z01P3k0G GJ436 ( T = 3500 K; log g = 4 .
80) 0.03 1 0.3 1.04 3.145 0.0T35a03Z01P4k0G GJ436 ( T = 3500 K; log g = 4 .
80) 0.03 1 0.4 1.04 3.145 0.0T35a03Z01P5k0G GJ436 ( T = 3500 K; log g = 4 .
80) 0.03 1 0.5 1.04 3.145 0.0T48a05Z01P0k0H HAT-P-11 ( T = 4780 K; log g = 4 .
60) 0.05 1 0.0 1.22 3.052 0.0T48a05Z01P1k0H HAT-P-11 ( T = 4780 K; log g = 4 .
60) 0.05 1 0.1 1.22 3.052 0.0T48a05Z01P2k0H HAT-P-11 ( T = 4780 K; log g = 4 .
60) 0.05 1 0.2 1.22 3.052 0.0T48a05Z01P3k0H HAT-P-11 ( T = 4780 K; log g = 4 .
60) 0.05 1 0.3 1.22 3.052 0.0T48a05Z01P4k0H HAT-P-11 ( T = 4780 K; log g = 4 .
60) 0.05 1 0.4 1.22 3.052 0.0T48a05Z01P5k0H HAT-P-11 ( T = 4780 K; log g = 4 .
60) 0.05 1 0.5 1.22 3.052 0.0T35a03Z01P3k2G GJ436 ( T = 3500 K; log g = 4 .
80) 0.03 1 0.3 1.04 3.145 0.2T48a05Z01P3k2H HAT-P-11 ( T = 4780 K; log g = 4 .
60) 0.05 1 0.3 1.22 3.052 0.2T35a03Z30P0k0G GJ436 ( T = 3500 K; log g = 4 .
80) 0.03 30 0.0 1.04 3.145 0.0T58a03Z01P0k0G G2 V ( T = 5778 K; log g = 4 .
44) 0.03 1 0.0 1.04 3.145 0.0T58a05Z01P0k0G G2 V ( T = 5778 K; log g = 4 .
44) 0.05 1 0.0 1.04 3.145 0.0T58a10Z01P0k0G G2 V ( T = 5778 K; log g = 4 .
44) 0.10 1 0.0 1.04 3.145 0.0T58a25Z01P0k0G G2 V ( T = 5778 K; log g = 4 .
44) 0.25 1 0.0 1.04 3.145 0.0T58a50Z01P0k0G G2 V ( T = 5778 K; log g = 4 .
44) 0.50 1 0.0 1.04 3.145 0.0T58a03Z30P0k0G G2 V ( T = 5778 K; log g = 4 .
44) 0.03 30 0.0 1.04 3.145 0.0T58a05Z30P0k0G G2 V ( T = 5778 K; log g = 4 .
44) 0.05 30 0.0 1.04 3.145 0.0T58a10Z30P0k0G G2 V ( T = 5778 K; log g = 4 .
44) 0.10 30 0.0 1.04 3.145 0.0T58a25Z30P0k0G G2 V ( T = 5778 K; log g = 4 .
44) 0.25 30 0.0 1.04 3.145 0.0T58a50Z30P0k0G G2 V ( T = 5778 K; log g = 4 .
44) 0.50 30 0.0 1.04 3.145 0.0T35a03Z01P0k0G b GJ436 ( T = 3500 K; log g = 4 .
80) 0.03 1 0.0 1.04 3.145 0.0T42a03Z01P0k0G M0 V ( T = 4200 K; log g = 4 .
70) 0.03 1 0.0 1.04 3.145 0.0T48a03Z01P0k0G HAT-P-11 ( T = 4780 K; log g = 4 .
60) 0.03 1 0.0 1.04 3.145 0.0T52a03Z01P0k0G K0 V ( T = 5200 K; log g = 4 .
50) 0.03 1 0.0 1.04 3.145 0.0T58a03Z01P0k0G c G2 V ( T = 5778 K; log g = 4 .
44) 0.03 1 0.0 1.04 3.145 0.0T62a03Z01P0k0G F9 V ( T = 6200 K; log g = 4 .
25) 0.03 1 0.0 1.04 3.145 0.0
This table contains 31 rows describing the 29 models we computed. Two rows are repeated. a The model name is a shorthand for the model characteristics, as follows: “T” is for temperature, followed by the temperaturein Kelvin divided by 100; “a” is for semimajor axis, followed by 10 times the orbital separation in AU; “Z” is for metallicity,followed by the multiple of solar abundance; “P” is for P n , followed by 10 × P n ; “k” is for κ , followed by 10 × κ (in cm g − ).The final letter is either “G” or “H”: if “G”, the planet’s radius and surface gravity are those of GJ436b; if “H”, they are thoseof HAT-P-11b. b This is a repeat of the first row. c This is a repeat of the 15th row.
400 600 800 1000 1200 1400 1600 1800 2000 220010 −8 −6 −4 −2 P n = 0.5P n = 0.4P n = 0.3P n = 0.2P n = 0.1P n = 0.0P n = 0.3; κ = 0.2P n = 0.0; 30x solar T (K) P ( ba r s ) T−P ProfilesGJ436b Models
600 800 1000 1200 1400 1600 1800 2000 220010 −8 −6 −4 −2 P n = 0.5P n = 0.4P n = 0.3P n = 0.2P n = 0.1P n = 0.0P n = 0.3; κ = 0.2 T (K) P ( ba r s ) T−P ProfilesHAT−P−11b Models
Fig. 1.—
Model temperature-pressure profiles of GJ436b and HAT-P-11 b.
Top:
Models of GJ436b, with the redistributionparameter P n varying from 0 (no redistribution) to 0.5 (complete redistribution of heat to the night side). A model is shownwith added upper atmosphere opacity κ = 0 . g − (here, κ is the same as the κ e from Burrows et al. 2008a). An additionalmodel shows the influence of 30 × solar atmospheric opacity. Bottom:
Same, for HAT-P-11 b, except without the 30 × solarmodel. P n = 0.5P n = 0.4P n = 0.3P n = 0.2P n = 0.1P n = 0.0P n = 0.3; κ = 0.2P n = 0.0; 30x solar λ ( µ m) F P / F * Planet−Star Flux RatiosGJ436b Models
Deming et al. (2007) P n = 0.5P n = 0.4P n = 0.3P n = 0.2P n = 0.1P n = 0.0P n = 0.3; κ = 0.2P n = 0.0; 30x solar λ ( µ m) A l bedo Albedos: GJ436b Models P n = 0.5P n = 0.4P n = 0.3P n = 0.2P n = 0.1P n = 0.0P n = 0.3; κ = 0.2P n = 0.0; 30x solar λ ( µ m) T ( K ) Temperature of τ λ =2/3 PhotosphereGJ436b Models −6 −5 −4 −3 −2 −1 P n = 0.5P n = 0.4P n = 0.3P n = 0.2P n = 0.1P n = 0.0 P n = 0.3; κ = 0.2P n = 0.0; 30x solar λ ( µ m) P ( ba r s ) Pressure of τ λ =2/3 PhotosphereGJ436b Models Fig. 2.—
Additional characteristics of models of GJ436b depicted in the top panel of Fig. 1.
Top left:
Planet-star fluxratios. Varying P n has little influence on the planet’s flux at 8 µ m, but an additional upper atmosphere absorber results in asignificantly better match of the model to the data. 30 × solar atmospheric opacity also helps bridge the gap between the solar-opacity models and the observed data. The black square is the Spitzer
IRAC-4 measurement of Deming et al. (2007a); verticalbars indicate 1- σ uncertainty; the horizontal bar indicates the full width at 10% maximum of the IRAC-4 band’s transmissionfunction. Large X’s indicate integrated planet flux divided by integrated stellar flux over this band. Top right:
Albedos. Thealbedo is lower for lower P n , and much lower for nonzero κ . Bottom left:
Temperature of the τ λ = 2 / Bottom right:
Pressure of the τ λ = 2 / P n = 0.5P n = 0.4P n = 0.3P n = 0.2P n = 0.1P n = 0.0P n = 0.3; κ = 0.2 λ ( µ m) F P / F * Planet−Star Flux RatiosHAT−P−11b Models P n = 0.5P n = 0.4P n = 0.3P n = 0.2P n = 0.1P n = 0.0P n = 0.3; κ = 0.2 λ ( µ m) A l bedo Albedos: HAT−P−11b Models P n = 0.5P n = 0.4P n = 0.3P n = 0.2P n = 0.1P n = 0.0P n = 0.3; κ = 0.2 λ ( µ m) T ( K ) Temperature of τ λ =2/3 PhotosphereHAT−P−11b Models −5 −4 −3 −2 −1 P n = 0.5P n = 0.4P n = 0.3P n = 0.2P n = 0.1P n = 0.0 P n = 0.3; κ = 0.2 λ ( µ m) P ( ba r s ) Pressure of τ λ =2/3 PhotosphereHAT−P−11b Models Fig. 3.—
Additional characteristics of models of HAT-P-11b depicted in the bottom panel of Fig. 1. Completely analogousto Fig. 2. −8 −6 −4 −2 T (K) P ( ba r s ) T−P Profiles, "Neptune" Models
Solar opacity: solid30x solar opacity: dashed(G2V)
500 1000 1500 2000 2500 300010 −8 −6 −4 −2 GJ436: T = 3500 K; log (g) = 4.8 T = 4200 K; log (g) = 4.7 HAT−P−11: T = 4780 K; log (g) = 4.6 T = 5200 K; log (g) = 4.5 T = 5778 K; log (g) = 4.44T = 6200 K; log (g) = 4.25 T (K) P ( ba r s ) T−P Profiles forDifferent Host Stars (GJ436b at 0.029 AU)
Fig. 4.—
Model temperature-pressure profiles of GJ436b-like planet. In all models, the redistribution parameter P n = 0. Top:
Models are computed at a range of distances from a Sun-like star, at solar and at 30-times solar metallicity. More distantmodels are cooler, and more metal-rich models are warmer at a given pressure.
Bottom:
Models are computed for a range ofhost stars, from GJ436 (a cool M-dwarf) through an F9 star (6200 K). λ ( µ m) F P / F * Planet−Star Flux Ratios, "Neptune" Models
Solar opacity: solid30x solar opacity: dashed (G2V)
Solar opacity: solid
Albedos, "Neptune" Models A l bedo λ ( µ m) 0.5 0.6 0.7 0.8 0.9 1
30x solar opacity: dashed (G2V) λ ( µ m) λ ( µ m) T ( K ) Temperature of τ λ =2/3 Photosphere"Neptune" Models (G2V) −6 −5 −4 −3 −2 −1 λ ( µ m) P ( ba r s ) Pressure of τ λ =2/3 Photosphere"Neptune" Models (G2V) Fig. 5.—
Additional characteristics of models for different orbital distances and metallicities depicted in the top panel ofFig. 4. Analogous to Fig. 2.
Top left:
Planet-star flux ratios for a GJ436b-like model planet at a variety of distances froma G2V star, at solar and at 30-times solar metallicity. Changes in metallicity have only a very modest influence on modelspectra.
Top right:
Optical albedos for these same models. At high metallicity, generally the cooler, more distant models(dashed curves) have larger optical albedo. At solar metallicity, the albedo varies less smoothly with semimajor axis.
Bottomleft:
Temperature of the τ λ = 2 / Bottom right:
Pressure of the τ λ = 2 / GJ436: T = 3500 K; log (g) = 4.8 T = 4200 K; log (g) = 4.7 HAT−P−11: T = 4780 K; log (g) = 4.6 T = 5200 K; log (g) = 4.5 T = 5778 K; log (g) = 4.44T = 6200 K; log (g) = 4.25 λ ( µ m) F P / F * Planet−Star Flux RatiosDifferent Host Stars (GJ436b at 0.029 AU)
GJ436: T = 3500 K; log (g) = 4.8 T = 4200 K; log (g) = 4.7 HAT−P−11: T = 4780 K; log (g) = 4.6 T = 5200 K; log (g) = 4.5 T = 5778 K; log (g) = 4.44T = 6200 K; log (g) = 4.25 λ ( µ m) A l bedo AlbedosDifferent Host Stars (GJ436b at 0.029 AU)