1_Albedo and Reflection Spectra of Extrasolar Giant Planets_(Sudarsky_etal_2000)

THE ASTROPHYSICAL JOURNAL, 538 :885È903, 2000 August 12000. The American Astronomical Society. All rights reserved. Printed in U.S.A.(

ALBEDO AND REFLECTION SPECTRA OF EXTRASOLAR GIANT PLANETS

DAVID SUDARSKY,1 ADAM BURROWS,1 AND PHILIP PINTO1Received 1999 October 27 ; accepted 2000 March 13

ABSTRACTWe generate theoretical albedo and reÑection spectra for a full range of extrasolar giant planet (EGP)

models, from Jovian to 51 Pegasi class objects. Our albedo modeling utilizes the latest atomic andmolecular cross sections, Mie theory treatment of scattering and absorption by condensates, a variety ofparticle size distributions, and an extension of the Feautrier technique, which allows for a general treat-ment of the scattering phase function. We Ðnd that, because of qualitative similarities in the composi-tions and spectra of objects within each of Ðve broad e†ective temperature ranges, it is natural toestablish Ðve representative EGP albedo classes. At low e†ective temperatures K) is a class of(Teff [ 150““ Jovian ÏÏ objects (class I) with tropospheric ammonia clouds. Somewhat warmer class II, or ““ watercloud,ÏÏ EGPs are primarily a†ected by condensed Gaseous methane absorption features are preva-H2O.lent in both classes. In the absence of nonequilibrium condensates in the upper atmosphere, and withsufficient condensation, class II objects are expected to have the highest visible albedos of any class.H2OWhen the upper atmosphere of an EGP is too hot for to condense, radiation generally penetratesH2Omore deeply. In these objects, designated class III or ““ clear ÏÏ because of a lack of condensation in theupper atmosphere, absorption lines of the alkali metals, sodium and potassium, lower the albedo signiÐ-cantly throughout the visible. Furthermore, the near-infrared albedo is negligible, primarily because ofstrong and molecular absorption and collision-induced absorption (CIA) by molecules. InCH4 H2O H2those EGPs with exceedingly small orbital distance (““ roasters ÏÏ) and 900 K (class IV), aK [Teff [ 1500tropospheric silicate layer is expected to exist. In all but the hottest K) or lowest gravity(Teff Z 1500roasters, the e†ect of this silicate layer is likely to be insigniÐcant because of the very strong absorptionby sodium and potassium atoms above the layer. The resonance lines of sodium and potassium areexpected to be salient features in the reÑection spectra of these EGPs. In the absence of nonequilibriumcondensates, we Ðnd, in contrast to previous studies, that these class IV roasters likely have the lowestvisible and Bond albedos of any class, rivaling the lowest albedos of our solar system. For the smallfraction of roasters with K and/or low surface gravity cm s~2 ; class V), the silicateTeff Z 1500 ([103layer is located very high in the atmosphere, reÑecting much of the incident radiation before it can reachthe absorbing alkali metals and molecular species. Hence, the class V roasters have much higher albedosthan those of class IV. In addition, for class V objects, UV irradiation may result in signiÐcant alkalimetal ionization, thereby further weakening the alkali metal absorption lines. We derive Bond albedos

and estimates for the full set of known EGPs. A broad range in both values is found, with(AB) Teff Teffranging from D150 to nearly 1600 K, and from D0.02 to 0.8. We Ðnd that variations in particle sizeA

Bdistributions and condensation fraction can have large quantitative, or even qualitative, e†ects on albedospectra. In general, less condensation, larger particle sizes, and wider size distributions result in loweralbedos. We explore the e†ects of nonequilibrium condensed products of photolysis above or withinprincipal cloud decks. As in Jupiter, such species can lower the UV/blue albedo substantially, even ifpresent in relatively small mixing ratios.Subject heading : molecular processes È planetary systems È scattering

1. INTRODUCTION

Since the discovery of the extrasolar giant planet (EGP),51 Pegasi b, in 1995 (Mayor & Queloz 1995), an explosionof similar discoveries has followed. To date, there are D30known planets orbiting nearby stars, which have collec-tively initiated the new Ðeld of extrasolar giant planetresearch.

While to date most detections have been via Dopplerspectroscopy, other promising methods, both ground-basedand space-based, are in development. These include (but arenot limited to) astrometric techniques (Horner et al. 1998),nulling interferometry (Hinz et al. 1998), coronographicimaging (Nakajima 1994), and spectral deconvolution(Charbonneau, Jha, & Noyes 1998). Furthermore, plannedspace instrumentation such as the Next Generation SpaceTelescope (NGST ) and Space Interferometry Mission (SIM)

1 Department of Astronomy and Steward Observatory, University ofArizona, Tucson, AZ 85721.

may prove to be useful for the detection and character-ization of EGP systems.

With the current push for new instruments and tech-niques, we expect that some of these new EGPs will soon bedirectly detected. One group (Cameron et al. 1999) hasclaimed a detection in reÑected light of the ““ roaster ÏÏ q Boob, while another group (Charbonneau et al. 1999) has notclaimed a detection but quoted an upper limit to thatobjectÏs albedo that is in conÑict with Cameron et al. Ourtheoretical models of EGP albedos are motivated by andcan help guide attempts to detect EGPs directly in reÑec-tion by identifying their characteristic spectral features andby illuminating the systematics.

The theoretical study of EGP albedos and reÑectionspectra is still largely in its infancy. Marley et al. (1999) haveexplored a range of stratosphere-free EGP geometric andBond albedos using temperature-pressure proÐles of EGPsin isolation (i.e., no stellar insolation), while Goukenleuqueet al. (1999) modeled 51 Peg in radiative equilibrium, and

885

886 SUDARSKY, BURROWS, & PINTO Vol. 538

Seager & Sasselov (1998) explored radiative-convectivemodels of EGPs under strong stellar insolation. In thepresent study, our purpose is to provide a broader set ofmodels than previous work, and to establish a generalunderstanding of the albedo and reÑection spectra of EGPsover the full range of e†ective temperatures Rather(Teff).than attempting to model these spectra in a fully consistentway for the almost endless combinations of EGP masses,ages, orbital distances, elemental abundances, and stellarspectral types, we concentrate on representative composi-tion classes based loosely on The ““ Jovian ÏÏ class ITeff.objects K) are characterized by the presence of(Teff [ 150ammonia clouds. (Note that the term ““ Jovian ÏÏ is used herefor convenience, not to imply that this entire class of objectswill be identical to Jupiter.) In somewhat warmer objects

K), ammonia is in its gaseous state, but the(Teff D 250upper troposphere contains condensed These objectsH2O.are designated class II, or ““ water cloud ÏÏ EGPs. Class III,or ““ clear ÏÏ EGPs, are so named because they are too hot

K) for signiÐcant condensation and so are(Teff Z 350 H2Onot expected to contain any principal condensates, thoughthey are not necessarily completely cloud free. The hotterEGPs (900 K; class IV) include thoseK [ Teff [ 1500objects with very small orbital distances (““ roasters ÏÏ) orthose at large distances that are massive and young enoughto have similar e†ective temperatures. In either case, thetroposphere of such an EGP is expected to contain signiÐ-cant abundances of neutral sodium and potassium gases, aswell as a silicate cloud layer. The hottest (Teff Z 1500)and/or least massive cm s~2) has a silicate layer(g [ 103located so high in the atmosphere that much of the incom-ing radiation is shielded from alkali metal and molecularabsorption.

We use a planar asymmetric Feautrier method in con-junction with temperature-pressure (T -P) proÐles, equi-librium gas abundances (assuming Anders & Grevesse 1989elemental abundances), and simple cloud models to accountfor condensed species. The T -P proÐles of isolated EGPs, aswell as proÐles that are nearly isothermal in the outer atmo-sphere, are utilized. This allows us to bracket the e†ects ofvarious T -P proÐles on the resulting EGP albedo spectra.Like Marley et al. (1999), we generate model albedo andreÑection spectra and Bond albedos, assuming a variety ofcentral star spectral types. Similarly, Rayleigh scattering,Raman scattering, Mie extinction due to condensates, andmolecular absorption by a host of species are treated. Inaddition to our broader range of compositions and thanTeffin Marley et al., we treat the important absorption e†ects ofthe alkali metals, include a larger number of relevant con-densates (including some nonequilibrium products typicalof photolysis), and produce a synthetic albedo spectrum ofJupiter that is in reasonable agreement with JupiterÏs actualalbedo spectrum (Karkoschka 1994) from the soft UV tothe near-infrared.

Doppler spectroscopy favors the detection of massivecompanions at small orbital distances, and indeed EGPswith very small orbital radii have been found. q Boo b(Butler et al. 1997), 51 Peg b (Mayor & Queloz 1995), t Andb (Butler et al. 1997), HD 75289b (Mayor et al. 1999), HD187123b (Butler et al. 1998), HD 217107b (Fischer et al.1999), and HD 209458b (Charbonneau et al. 2000) all haveorbital distances of less than 0.1 AU and masses (actually

ranging from D0.4 to 3.4 Jupiter masses. UnderMp

sin i)stellar insolation, the elevated temperatures of EGPs

depend mostly on the level of stellar insolation, rather thanon their masses and ages, which would largely determinetheir in isolation. Using simple radiative equilibriumTeffarguments where is the incident stellar(Teff P Finc1@4, FincÑux), most of the EGPs within 0.1 AU are likely to havevery high (D800 K to over 1600 K). is only weaklyTeff Teffdependent on the Bond albedo for a large range of low-to-moderate albedos, varying only D20% as the Bond albedovaries from 0.01 to 0.6.

At the other end of the scale, several objects with moretraditional orbital distances of AU have been dis-Z1covered. These EGPs include 16 Cyg Bb (Cochran et al.1997), 47 UMa b (Butler & Marcy 1996), t And d (Marcy,Butler, & Fischer 1999), Gl 614b (Marcy, Cochran, &Mayor 1998), HR 5568b, HR 810b, and HD 210277b(Marcy et al. 1998) and have ranging from D0.75M

psin i

to 5 At these larger orbital distances, EGPs receiveMJ.much less stellar radiation and, therefore, have lower TeffK). Still, many other EGPs, such as 70 Vir b (Butler([300& Marcy 1996), Gl 86 Ab (Queloz et al. 2000), and HD114762b (Latham et al. 1989), have orbital distancesbetween 0.1 and 1 AU and between 0.7 and 10M

psin i MJ.Over the full set of currently known EGPs, spectral classes

of the central stars range from F7 V to M4 V.The albedo of an object is simply the fraction of light that

the object reÑects. However, there are several di†erent typesof albedos. The geometric albedo refers to the reÑectivity ofthe object at full phase ('\ 0, where ' represents theobjectÏs phase angle) relative to that by a perfect Lambertdisk of the same radius under the same incident Ñux. Sinceplanets are essentially spheres, the factor projecting a unitsurface onto a disk orthogonal to the line of sight is givenby cos / sin h, where / is the objectÏs longitude (deÐned tobe in the observer-planet-star plane) and h is its polar angle(n/2-latitude). The geometric albedo is given by integratingover the illuminated hemisphere :

Ag\ 1

nIinc

PÕ/~n@2

n@2 Ph/0

nI(/, h, '\ 0) cos / sin hd) , (1)

where is the incident speciÐc intensity, is the inci-Iinc nIincdent Ñux, and I(/, h, '\ 0) is the emergent intensity. Moregenerally, I\ I(/, h, but at full phase all inci-' ; /0, h0),dent angles are equal to the emergent ones. The(/0, h0)geometric albedo is usually given as a function of wave-length, although it is sometimes averaged over a wavelengthinterval and stated as a single number.

The spherical albedo, refers to the fraction of incidentAs,

light reÑected by a sphere at all angles. Usually stated as afunction of wavelength, it is obtained by integrating thereÑected Ñux over all phase angles. The Ñux (F(')) as afunction of phase angle (') is given by the more generalform of equation (1). Assuming unit radius (Chamberlain &Hunten 1987),

F(') \PÕ/'~n@2

n@2 Ph/0

nI(/, h, ' ; /0, h0) cos / sin hd) .

(2)

The spherical albedo is obtained by integrating F(') overall solid angles :

As\ 1

nIinc

P4n

F(')d)\ 2Iinc

P0

nF(') sin 'd' . (3)

No. 2, 2000 EXTRASOLAR GIANT PLANETS 887

Note that the spherical and geometric albedos are relatedby whereA

s\ A

gq,

q \ 2F('\ 0)

P0

nF(') sin 'd' (4)

is known as the phase integral.The Bond albedo, is the ratio of the total reÑected andA

B,

total incident powers. It is obtained by weighting the spher-ical albedo by the spectrum of the illuminating source andintegrating over all wavelengths :

AB\ /0= A

s,j Iinc,j dj/0= Iinc,j dj

, (5)

where the j subscript signiÐes that the incident intensityvaries with wavelength.

Spherical, geometric, and Bond albedos of objects arestrong functions of their compositions. Within the solarsystem, they vary substantially with wavelength, and fromobject to object. At short wavelengths, gaseous atmospherescan have high albedos because of Rayleigh scattering, andlow albedos at longer wavelengths because of molecularrovibrational absorption. Icy condensates, whether theyreside on a surface or are present in an upper atmosphere,are highly reÑective and increase the albedo. Other conden-sates, such as silicates or nonequilibrium products of pho-tolysis, can lower the albedo substantially over a broadwavelength region.

Some of the lowest albedos seen in the solar system arethose of asteroids containing large amounts of carbon-aceous material. Many have Bond albedos of less than 0.03(Lebofsky, Jones, & Herbert 1989). The Bond albedo of theEarth is 0.30 (Stephens, Campbell, & Vonder Haar 1981),and that of the Moon is 0.11 (Buratti, Hillier, & Wang1996). Jupiter and Saturn have somewhat higher Bondalbedos, both near 0.35 (Conrath, Hanel, & Samuelson1989).

In ° 2, we describe our approach to modeling EGPs.Section 3 describes our radiative transfer method, ° 4 con-tains a discussion of molecular absorption and scattering,and ° 5 describes the properties of and our treatment of therelevant condensates in EGP atmospheres. In ° 6, wediscuss the application of our methods to Jupiter, ° 7 con-tains our EGP model albedo and reÑection spectra results,as well as and Bond albedo estimates for currentlyTeffknown EGPs, and ° 8 describes the e†ects of varying keyparameters of the models. We summarize our results in ° 9.

2. EXTRASOLAR GIANT PLANET MODELS

Depending upon their proximity to their central stars aswell as their masses and ages, EGP e†ective temperatureslikely span a large range, from below 100 K to well over1600 K, with highly varying temperature-pressure-composi-tion proÐles. However, an EGPÏs outer atmospheric com-position, rather than its speciÐc temperature-pressureproÐle, is of primary importance in modeling albedos andreÑection spectra. With our composition classes, weencompass the range of behaviors of EGP albedos andreÑection spectra. We do not model emission spectra, nordo our models account for object-speciÐc details, such aselemental abundance di†erences or cloud patchiness. EGPsare surely at least as rich and varied as the planets of oursolar system, but simple modeling reveals many interestingsystematics.

2.1. Temperature-Pressure ProÐlesIdeally, temperature (T )-pressure (P) proÐles are com-

puted directly via radiative equilibrium models of EGPsunder stellar insolation. A move toward such models forvery strong stellar insolation has been made by Seager &Sasselov (1998) and Goukenleuque et al. (1999), while forlower temperature objects Marley et al. (1999) utilize T -PproÐles of isolated EGPs. The main e†ect of stellar inso-lation on the T -P proÐle of an EGP is to make the outeratmosphere more nearly isothermal. Studies of strongstellar insolation conclude that a stratosphere does not existin the high-temperature roasters (Seager & Sasselov 1998 ;Goukenleuque et al. 1999). However, it is not completelyclear what might occur in the upper atmosphere if ultraviol-et photochemical processes are fully modeled. Under solarinsolation, Jupiter and Saturn do exhibit stratospheres, andwe suspect that the class I EGPs are likely to have strato-spheres as well. In an albedo spectrum, the existence of astratosphere is made manifest mainly by the scattering andabsorption e†ects of nonequilibrium aerosols that residethere. Additionally, photochemical processes in the strato-sphere may be the origin of ““ chromophores,ÏÏ nonequilib-rium solids that settle near or in the upper cloud layers andare largely responsible for the coloration of Jupiter.

To bracket the range of albedos under stellar insolation,we use two sets of pressure-temperature proÐles. The Ðrst isa subset of proÐles for theoretical isolated objects (Marleyet al. 1999 ; Marley 1998 ; Burrows et al. 1997) with Teff B130 K (representing an isolated class I EGP), 250 K (classII), 600 K (class III), and 1200 K (class IV). We estimate thatthese representative isolated T -P proÐles are valid forsurface gravities between D3 ] 103 to 3 ] 104 cm s~2. Aset of modiÐed proÐles is obtained by altering these isolatedproÐles to simulate a stellar insolated T -P proÐle by usingthe model results of Seager & Sasselov (1998) as a guide. Toapproximate the T -P proÐles of the very hottest close-inobjects (class V), we scale the 1200 K proÐle up to 1700 K.We stress that these modiÐed proÐles are very approximate,but along with the isolated T -P proÐles, they bracket abroad range of possible EGP T -P proÐles.

Figure 1 shows both the isolated and modiÐed T -P pro-Ðles for classes I through IV, as well as our modiÐed class VproÐle. Also shown are condensation curves, which indicatethe highest temperatures and pressures at which species cancondense. Cloud bases are located approximately where theproÐles intersect the condensation curves (dotted curves).Class I (Jovian) objects contain both ammonia and deeperwater cloud layers, while water is likely the only principalcondensate present in the tropospheres of class II objects.(As shown in Fig. 1, a thin ammonia haze layer mightappear very high in the atmosphere for an isolated class IIT -P proÐle.) The class III T -P proÐle does not cross anyprincipal condensation curves in the upper atmosphere,regardless of the level of stellar insolation. Finally, the classIV and V roasters contain a silicate cloud deck and a deeperiron cloud deck throughout the full range of possible T -PproÐles, though their cloud depths di†er considerably.

2.2. Determination of Gaseous AbundancesUsing the analytic formulae in Burrows & Sharp (1999),

we calculate gaseous mixing ratios of the main compoundsof carbon, oxygen, and nitrogen CO,(CH4, H2O, NH3, N2)over the full range of temperatures and pressures in themodel EGP atmospheres. and He mixing ratios are setH2


FIG. 1.ÈTemperature-pressure proÐles for each of the EGP classes ofthis study. Both isolated and modiÐed (more isothermal) proÐles areshown for classes I through IV, as well as a modiÐed proÐle for class V.Also plotted are the condensation curves for some principal condensates,as well as the and abundance equilibrium curves.NH3/N2 CH4/CO

according to Anders & Grevesse (1989) solar abundances,and the mixing ratio is set in accordance with theH2SAnders & Grevesse abundance of sulfur (D3 ] 10~5). Theabundances of the alkali metals (Na, K, Rb, Cs), importantin the class III through class V EGPs, are calculatednumerically using the formalism of Burrows & Sharp(1999).

Overall, the e†ect of di†erences in the T -P proÐle ongaseous mixing ratios tends to be greatest for the class IVobjects because of the temperature and pressure depen-dences of neutral alkali metal mixing ratios and the fact thatthe T -P proÐles are in the vicinity of the andCH4/CO

equilibrium curves. From the standpoint ofNH3/N2gaseous abundances, the T -P proÐles have little e†ect onthe albedos of cooler EGPs.

2.3. Cloud ModelingOur treatment of clouds in our Ðducial EGP models

assumes that the gaseous form of a condensable species iscompletely depleted above the cloud deck and that thespecies settles within the cloud layer in its condensed form.The base of the cloud resides where the T -P proÐle of theEGP meets the condensation curve of the given species, andthe cloud top is simply set at one pressure scale heightabove the base. Not all of the given condensable within thecloud is in condensed form. Rather, at the base of the cloud,the gaseous form is assumed to be at the saturation vaporpressure. For a given abundance of a condensable, if weassume that the portion of the condensable that exceeds thesaturation vapor pressure is entirely in condensed form, wewill refer to this as ““ full condensation.ÏÏ However, as inJupiterÏs outer atmosphere (see ° 6), it is possible that thecondensation fraction will be smaller. Hence, we retain thecondensation fraction as a parameter. Furthermore, theparticle size distributions in EGP atmospheres are impossi-ble to ascertain at this point, so particle size remains a freeparameter as well.

The standard model for Jupiter lends some support toour prescription for clouds. The base of JupiterÏs ammoniacloud deck resides approximately where its T -P proÐlemeets the condensation curve (D0.7 bar) and theNH3

cloud tops extend roughly 1 pressure scale height, to D0.3bar (West, Strobel, & Tomasko 1986 ; Griffith et al. 1992).Although present, gas is largely depleted above theNH3cloud layer.

In the case of silicate condensation, where the condensateand gas molecules are not identical (unlike andNH3 H2O),the condensate abundance is estimated by the Anders &Grevesse abundance of the limiting species. We use enstatite

though a number of other silicates with di†ering(MgSiO3 ;optical properties are certainly present), for which the limit-ing element is silicon. For the full condensation limit, it isassumed that the entire mass of silicon above the pressure ofthe cloud base settles into within the cloud.MgSiO3

3. RADIATIVE TRANSFER METHOD

Because of the forward scattering from condensates inEGP atmospheres, an appropriate radiative transfermethod must allow for a forward-backward asymmetricscattering phase function. Although the conventional Feau-trier method (e.g., Mihalas 1978) does not allow for such anasymmetry, a straightforward extension of this techniquecan be derived by separating the source function intoupward- and downward-propagating rays (Mihalas 1980 ;Milkey, Shine, & Mihalas 1975).

At Ðrst thought, it may seem inappropriate to use aplanar transfer code in the modeling of albedos and reÑec-tion spectra from spherical objects. However, it is fairlystraightforward to derive the equivalence between uniformradiation from one direction onto a unit sphere anduniform radiation from 2n sr onto a plane with unit area.Hence, provided that we set the incident intensity to beuniform in angle, the spherical albedo is the ratio of theoutward and incident Ñuxes.

The fundamental transfer equation is

kLI(k)Lq

\ I(k) [ S(k) , (6)

where the source function is given by

S(k) \ 12

pP~1

1R(k, k@)I(k@)dk@ (7)

and the thermal term is neglected in this albedo study. R(k,k@) is the azimuth-independent angular redistribution func-tion (azimuthal symmetry is assumed) and p is the single-scattering albedo, where is the scatteringp \ pscat/pext, pscatcross section and is the extinction cross section.pextSeparated into upward (I`) and downward (I~) com-ponents, the transfer equation becomes

kLI`(k)

Lq\ I`(k) [ S`(k) (8)

and

[kLI~(k)

Lq\ I~(k) [ S~(k) , (9)

where the source functions are given by

S`(k) \ 12

pP0

1[R(k, k@)I`(k@) ] R(k,[ k@)I~(k@)]dk@

(10)


and

S~(k)\ 12

pP0

1[R([k, k@)I`(k@)] R([k,[ k@)I~(k@)]dk@

(11)

for the I` and I~ equations, respectively.Forming symmetric and antisymmetric averages, and

using the Feautrier variables, and v\u \ 12(I`] I~)equations (8) and (9) are rewritten as12(I` [ I~),

kLv(k)Lq

\ u(k)[ 12

[S`(k)] S~(k)] (12)

and

kLu(k)Lq

\ v(k)[ 12

[S`(k)[ S~(k)] . (13)

Since R(k, k@) depends only upon the angle between k andk@, the following symmetries exist :

R(k, k@)\ R([k,[ k@) , (14)

R([k, k@)\ R(k,[ k@) . (15)

With the deÐnitions, R`(k, k@)\ R(k, k@)] R([k, k@)and R~(k, k@)\ R(k, k@)[ R([k, k@), equations (12) and(13) become

kLvLq

\ u [ 12

pP0

1R`(k, k@)u(k@)dk@ (16)

and

kLuLq

\ v[ 12

pP0

1R~(k, k@)v(k@)dk@ . (17)

This system of Ðrst-order equations is discretized fornumerical computation by replacing the derivatives withdi†erence quotients and by substituting Gaussian quadra-ture sums for the integrals. The principal equations thenbecome

kivd,i [ v

d~1,i*q

d\ u

d,i [12

p ;j

ujR`(k

i, k

j)u

d,j (18)

and

kiud`1,i[ u

d,i*q

d] 1/2

\ vd,i[

12

p ;j

ujR~(k

i, k

j)v

d,j , (19)

where d signiÐes a given depth zone (d \ 1, . . . , D), and i andj signify angular bins (i, j\ 1, . . . , N). equals*q

d`1@2 qd`1and equals To achieve[ q

d, *q

d12(*q

d`1@2 ] *qd~1@2).numerical stability, *q is staggered by one-half zone in

equation (19) relative to equation (18). The are theujGaussian weights.

The upper boundary conditions are given by the rela-tions, andu1,i[ v1,i\ I

i~

kiu2,i[ u1,i

*q1@2\ u1,i[ I

i~[ 1

2p ;

ju

j

] [u1,j [ Ij~]R~(k

i, k

j) , (20)

where and signify the incident intensity as a functionIi~ I

j~

of angle at the surface. We set I~ to unity at all angles sinceonly the ratio of the outward and inward Ñuxes determines

the spherical albedo. The lower boundary conditions aregiven by andu

D,i] vD,i \ I

i`

kiuD,i[ u

D~1,i*q

D~1@2\ I

i` [ u

D,i [12

p ;j

uj

] [Ij`[ u

D,j]R~(ki, k

j) , (21)

where and signify the outward-traveling intensity atIi` I

j`

the base of the atmosphere (set to zero in this study).The system of equations can be represented by angle

matrices and column vectors and(Ad, B

d, C

d, . . .) (u

d¿d)

such that equations (18) and (19) can be written as

Ad¿d~1 ] B

dud] C

d¿d\ 0 (22)

and

Dd

ud] E

d¿d] F

d¿d`1\ 0 . (23)

Given D depth zones and N angles, the system results in ablock matrix containing [2] D]2 submatrices, each oforder N. Implementing the boundary conditions describedabove, this system is solved directly via LU decompositionand substitution.

Our atmosphere models utilize 100 optical depth zoneswith logarithmic sizing near the surface, where higherresolution is essential, and a continuous transition to linearzoning at depth. Sixteen polar angular bins per hemisphereare used.

3.1. Quantitative Comparison for Uniform AtmospheresIn order to test our asymmetric Feautrier code, we

compare our resulting spherical albedos for uniform atmo-sphere models with those derived employing both MonteCarlo and analytic techniques. Van de Hulst (1974) deriveda solution for the spherical albedo of a planet covered witha semiinÐnite homogeneous cloud layer. Given a single-scattering albedo of p and a scattering asym-(\pscat/pext)metry factor of g \ Scos hT (the average cosine of thescattering angle), van de HulstÏs expression for the sphericalalbedo is

AsB

(1[ 0.139s)(1[ s)1 ] 1.170s

, (24)

where

s \A 1 [ p1 [ pg

B1@2. (25)

Figure 2 shows the spherical albedo of a homogeneous,semiinÐnite atmosphere as a function of scattering asym-metry factor and single scattering albedo. Along with vande HulstÏs semianalytic curves are our asymmetric Feautrierand Monte Carlo model results using a Henyey-Greensteinscattering phase function,

p(h) \ 1 [ g2(1] g2[ 2g cos h)3@2 . (26)

For nearly all values of g and p, the agreement is verygood, di†ering by less than 1%. There are slightly largervariations when both g and p approach unity because of theÐnite number of angles and depth zones used in our numeri-cal models, but in actual planetary or EGP atmospheresthis corner of parameter space is rarely realized.

Real planetary atmospheres are usually highly stratiÐedand the optical depth is a strong function of wavelength.

0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1


FIG. 2.ÈComparison of our asymmetric Feautrier code results toMonte Carlo and analytic solutions for deep, homogeneous atmospheres.The spherical albedo is plotted as a function of the single-scattering albedo

and the average value of the cosine of the scattering angle(\pscat/pext)(g \ Scos hT).

Given the atmospheric temperature-pressure-compositionproÐle, an appropriate conversion to optical depth isrequired. Assuming hydrostatic equilibrium and using anideal gas equation of state, this conversion is

dq\ pext(P)gk(P)

dP , (27)

where is the e†ective extinction cross section per parti-pextcle at depth P, k is the mean molecular weight, and g is thesurface gravity (assumed constant because the depth of thee†ective atmosphere is a very small fraction of the planetÏsradius).

4. ATOMIC AND MOLECULAR SCATTERING AND

ABSORPTION

The gases present in EGP atmospheres are many(Burrows & Sharp 1999). However, only some of them havethe requisite abundances and cross sections at the tem-peratures and pressures of upper EGP atmospheres to havesigniÐcant spectral e†ects in the visible and near-infrared.These species include CO, andH2, CH4, H2O, NH3, H2S.Additionally, Na and K are important absorbers in classIII, IV, and V EGPs.

Of course, is the most abundant species, followed byH2helium. The dominant carbon-bearing molecule is a func-tion of both temperature and pressure. Chemical equi-librium modeling (Burrows & Sharp 1999 ; Fegley &Lodders 1996) shows that, at solar metallicity, willCH4dominate over CO in most EGP atmospheres. At high tem-peratures, the CO abundance overtakes that of CH4(D1100 K at 1 bar ; D1400 K at 10 bars). There is a similartransition for the nitrogen-bearing molecules : domi-NH3nates at low temperatures, but it is overtaken by atN2higher temperatures (D700 K at 1 bar ; D900 K at 10 bars).Some species condense into solids at low temperatures,thereby depleting the gaseous phase. In particular, NH3condenses below 150È200 K (depending on pressure), asdoes below 250È300 K.H2OAt visible and near-infrared wavelengths, molecularabsorption is due to rovibrational transitions, so molecular

opacity is a very strong function of wavelength. Even whenno permanent dipole moment exists, such as with the H2molecule, the high gas pressures in EGP atmospheres caninduce temporary dipole moments via collisions. Thiscollision-induced absorption (CIA) is responsible for broad

(and absorption bands in Jupiter and SaturnH2-H2 H2-He)(Zheng & Borysow 1995 ; Trafton 1967).

The temperature- and pressure-dependent gaseous opa-cities are obtained from a variety of sourcesÈa com-bination of theoretical and experimental data as referencedin Burrows et al. (1997). Additionally, for this study the CH4opacity was extended continuously into the visible wave-length region using the data of Strong et al. (1993) and amethane absorption spectrum from Karkoschka (1994).These two data sets were then extrapolated in temperatureand pressure by scaling with existing temperature andpressure-dependent near-infrared data (Burrows et al.CH41997, and references therein).

Many prominent molecular absorption features may beseen in EGP albedo and reÑection spectra. At relatively lowtemperatures, broad and CIA bands peak atH2-H2 H2-HeD0.8, 1.2, and 2.4 km. At higher temperatures and pres-sures, the CIA cross sections become larger at all wave-lengths. CIA is especially important in cloud-free gaseousobjects, where incident radiation is absorbed deeper in theatmosphere. absorption bands shortward of 2.5 kmNH3occur at D1.5, 2.0, and 2.3 km. (Note that our databasedoes not contain the visible bands of ammonia.) H2Oabsorption occurs at D0.6, 0.65, 0.7, 0.73, 0.82, 0.91, 0.94,1.13, 1.4, 1.86, and 2.6 km. A large number of featuresCH4appear in the visible and near-infrared. Some of the moreprominent ones occur at D0.54, 0.62, 0.67, 0.7, 0.73, 0.79,0.84, 0.86, 0.89, 0.99, 1.15, 1.4, 1.7, and 2.3 km. CO absorp-tion bands occur at D1.2, 1.6, and 2.3 km and featuresH2Smay be found at D0.55, 0.58, 0.63, 0.67, 0.73, 0.88, 1.12, 1.6,and 1.95 km. Of course, depending upon mixing ratios andcross sections, only some of these features will appear in agiven EGP albedo spectrum.

Strong pressure-broadened lines of neutral sodium andpotassium are expected to dominate the visible albedos ofclass III and class IV EGPs. The most prominent absorp-tion lines of sodium occur at 3303, 5890, and 5896 whileA� ,those of potassium occur at 4044, 7665, and 7699 A� .

Atomic and molecular scattering includes conservativeRayleigh scattering as well as nonconservative Raman scat-tering. In the case of Rayleigh scattering, cross sections arederived from polarizabilities, which are in turn derived fromrefractive indices. Since the refractive indices are readilyavailable at 5893 (Weast 1983), the Rayleigh cross sec-A�tions are derived at this wavelength via

pRay\83

nk4An [ 1

2nL 0

B2, (28)

where k is the wavenumber at this wavelength (2n/j ^ 106,621 cm~1) and is LoschmidtÏs number. Assuming thatL 0the refractive indices are not strong functions of wavelength,we simply extrapolate these cross sections as j~4.

Raman scattering by involves the shift of continuumH2photons to longer or shorter wavelengths as they scatter o†exciting or deexciting rotational and vibrational tran-H2,sitions. Raman scattering is not coherent in frequency, so a

rigorous treatment is not possible with our transfer code.Instead, we adopt the approximate method introduced by


Pollack et al. (1986) and used by Marley et al. (1999) in theiralbedo study. At a given wavelength, the single scatteringalbedo within a particular depth zone is approximated by

p \pRay ] pscat@ ] ( fj*/fj)pRampRay] pext@ ] pRam

, (29)

where denotes the spectrum of incident radiation (thefjspectrum of an EGPÏs central star), jp~1\ j~1] *j~1,where *j is the wavelength of the vibrational fundamen-H2tal (*j~1\ 4161 cm~1), is the Raman cross section,pRamand and are the e†ective condensate scattering andpscat@ pext@extinction cross sections, respectively. Raman scatteringmay be signiÐcant in deep gaseous planetary atmospheres,where it can lower the UV/blue albedo by up to D15%(Cochran & Trafton 1978). However, our models show thatin higher temperature EGP atmospheres, alkali metalabsorption can dominate over this wavelength region, whilein cooler EGP atmospheres, condensates largely dominate.Over our full set of EGP models, we Ðnd that Raman scat-tering is relatively insigniÐcant.

5. MIE THEORY AND OPTICAL PROPERTIES OF

CONDENSATES

Condensed species in EGP atmospheres range fromammonia ice in low temperature objects to silicate grains athigh temperatures. Some of the condensates relevant toEGP atmospheres include K),NH3 ([150È200 NH4SH

K), K), low-abundance sulÐdes and([200 H2O ([250È300chlorides K), silicates such as([700È1100 MgSiO3K), and iron or iron-rich compounds([1600È1800

K). Additionally, photochemical processes in([1900È2300the upper atmosphere can produce nonequilibrium conden-sates. Stratospheric hazes may be composed of polyacety-lene (Bar-Nun, Kleinfeld, & Ganor 1988) and otheraerosols. Chromophores, those nonequilibrium species thatcause the coloration of Jupiter and Saturn, might include P4(Noy, Podolak, & Bar-Nun 1981) or organic species similarto Titan tholin (Khare & Sagan 1984).

Condensates can have drastic e†ects on EGP reÑectionspectra, increasing the albedo at most wavelengths, butsometimes depressing the albedo in the UV/blue. Of course,those condensates that are higher in the atmosphere willgenerally have a greater e†ect than those that reside moredeeply. The presence and location of a particular condensedspecies is determined largely by an objectÏs T -P proÐle andby the tendency of the condensate to settle (because of rain)at a depth in the atmosphere near the region where the T -PproÐle crosses the condensation curve. Hence, a given low-temperature K) atmosphere might consist of an(Teff [ 150ammonia cloud deck high in the troposphere and a watercloud deck somewhat deeper, with purely gaseous regionsabove, beneath, and between the clouds. Similarly, a high-temperature K) atmosphere might consist of a(Teff D 1200tropospheric silicate cloud deck above a deeper iron clouddeck. Depending upon the amount of condensate in theupper cloud and the wavelength region, the presence ofdeeper clouds may or may not have any e†ect on the albedoand reÑection spectrum.

The scattering and absorption of electromagnetic radi-ation by condensed species in planetary atmospheres is avery complex problem. The extinction properties of ices,grains, and droplets of various sizes, shapes, and composi-tions cannot be described accurately by simple means. Most

often, these properties are approximated by Mie theory,which describes the solution of MaxwellÏs equations insideand outside a homogeneous sphere with a given complexrefractive index.

We use a full Mie theory approach that utilizes the for-malism of van de Hulst (1957) and Deirmendjian (1969), andresults in scattering and extinction cross sections as well asa scattering asymmetry factor, g \ Scos hT, given thecomplex index of refraction and particle radius (a). Largerparticles require an increasing number of terms in an inÐn-ite series to describe these parameters accurately, and sothey require more computing time. However, while thecross sections and scattering asymmetry factors of small- tomoderately sized particles vary substantially(2na/j [ 75)with wavelength, these variations are greatly reduced forlarger spheres. For these larger particles, we use an asymp-totic form of the Mie equations outlined fully by Irvine(1965). Interpolation between the full Mie theory results andthese asymptotic limits yields the parameters for large par-ticles. However, inherent assumptions in the asymptoticform of the Mie equations render them inadequate for thecomputation of the scattering cross sections in the weak-absorption limit in which case we use the(nimag[ 10~3),geometric optics approximation (Bohren & Hu†man 1983),

Qsca \ 2 [ 83

nimagnreal

[nreal3 [ (nreal2 [ 1)3@2]x , (30)

where is the usual scattering coefficient (the ratio of theQscascattering cross section to the geometric cross section), x isthe size parameter (\2na/j), is the real index of refrac-nrealtion, and is the imaginary component of the refractivenimagindex.

The principal condensates to which we have applied Mietheory include ice, ice, and (enstatite),NH3 H2O MgSiO3where the optical properties, namely, the complex indices ofrefraction, were obtained from Martonchik, Orton, &Appleby (1984), Warren (1984), and Dorschner et al. (1995),respectively. The complex refractive indices of wereNH3interpolated in the 0.7È1.4 km wavelength region, becauseof the lack of data there.

Each of these condensates has absorption features, as ismade evident by the behavior of the imaginary index ofrefraction (Fig. 3). Shortward of 2.5 km, ice absorptionNH3occurs at D1.55, 1.65, 2.0, and 2.25 km. ice producesH2O

FIG. 3.ÈImaginary refractive indices of the principal condensates usedin this study.


FIG. 4.ÈImaginary refractive indices of stratospheric haze and tropo-spheric chromophore candidates. Tholin and provide a great deal ofP4absorption in the UV/blue.

broader features at D1.5 and 2.0 km. Enstatite is mostlyfeatureless below 2.5 km, except shortward of D0.35 km.

The nonequilibrium condensates to which we haveapplied Mie theory include phosphorus (Noy et al. 1981),tholin (Khare & Sagan 1984), and polyacetylene (Bar-Nunet al. 1988). and tholin are chromophore candidates,P4particularly for the coloration of Jupiter and Saturn,because of their large imaginary indices of refraction in theUV/blue (Fig. 4) and plausibility of production. A some-what yellowish allotrope of phosphorus, was producedP4in the laboratory by Noy et al. (1981) by UV irradiation ofan gaseous mixture. It is believed that this sameH2/PH3process may be responsible for its production in Jupiter.Tholin is a dark-reddish organic solid (composed of over 75compounds) synthesized by Khare & Sagan (1984) by irra-diation of gases in a simulated Titan atmosphere. It isbelieved that a tholin-like solid may be produced similarlyin giant planet atmospheres. Polyacetylenes, polymers of

were investigated by Bar-Nun et al. (1988) and likelyC2H2,are an optically dominant species in the photochemicalstratospheric hazes of giant planets, where hydrocarbonsare abundant (Edgington et al. 1996 ; Noll et al. 1986).

Cloud particle sizes are not easily modeled and are astrong function of the unknown meteorology in EGP atmo-spheres. Inferred particle sizes in solar system giant planetatmospheres can guide EGP models, though they rangewidely from fractions of a micron to tens of microns.

We have investigated various particle size distributions.A commonly used distribution, and the one that we use inour Ðducial models, is

n(a)PA aa0

B6exp

C[6A aa0

BD, (31)

which reproduces the distributions in cumulus water cloudsin EarthÏs atmosphere fairly well if the peak of the distribu-tion is km (Deirmendjian 1964). Stratospherica0D 4aerosolsÈat least those in EarthÏs stratosphereÈcan berepresented by the ““ haze ÏÏ distribution (Deirmendjian1964),

n(a)Paa0

expC[2A aa0

B1@2D. (32)

6. THE ALBEDO OF JUPITER

Jupiter is an important test bed for the theory of albedosbecause full-disk geometric albedo spectra have beenobtained (Karkoschka 1994, 1998) and because space-basedand ground-based studies have provided a fair amount ofinformation concerning JupiterÏs atmosphere. At visible andnear-infrared wavelengths, JupiterÏs upper troposphere andstratosphere shape its albedo spectrum. According to thestandard model, a somewhat heterogeneous cloud deckextends from D0.3 to 0.7 bars in the troposphere (West etal. 1986 ; Griffith et al. 1992). Although the bulk of the clouddeck consists primarily of particles at least 10 km in size, alayer of smaller particles (D0.5È1.0 km) resides near thecloud tops (West et al. 1986 ; Pope et al. 1992). Beneath thisupper cloud deck is a and cloud layer atNH4SH NH3D2È4 bars and an cloud condenses somewhat deeper.H2OAbove the cloud deck, a stratospheric haze resides atNH3pressures near D0.1 bar. It is worth mentioning that theGalileo probe results deviate from this standard model. Onedi†erence relevant to the albedo and reÑection spectra is atropospheric haze inferred from the probe data, likely com-posed primarily of above a somewhat deeperNH3, NH3cloud deck (BanÐeld et al. 1998), but it is not knownwhether the probe entry location is characteristic of theplanet as a whole.

In addition to abundant gaseous species in the upperH2,troposphere include He and with mixing ratios rela-CH4,tive to of 0.156 and D2.1] 10~3, respectively (NiemannH2et al. 1996). Gaseous and are presentNH3, H2O, H2S, PH3in small mixing ratios.It is suggested that the color di†erences of JupiterÏs belts

and zones are largely due to the visibility of chromophoresresiding within the cloud deck (West et al. 1986). NoNH3appreciable altitude di†erences between the belts and zonesare found, although the zones likely contain thicker uppercloud and/or haze layers than the belts (Chanover, Kuehn,& Beebe 1997 ; Smith 1986). JupiterÏs UV/blue albedo isdepressed substantially from what one would expect fromthe increase with frequency of the Rayleigh scattering crosssections, and Raman scattering cannot account for thealbedo in this wavelength region. This depressed UV/bluealbedo likely is produced by the large imaginary refractiveindices of the tropospheric chromophores and, to a lesserdegree, stratospheric aerosols (West et al. 1986).

Because of the large optical depth of JupiterÏs upperammonia cloud deck at visible and near-infrared wave-lengths, a two-cloud model of the atmosphere suffices (West1979 ; Kuehn & Beebe 1993). We model the top of the uppercloud deck (D0.35 bar) with a ““ cloud ÏÏ distribution (see ° 5)peaked at 0.5 km. Deeper in the cloud, from 0.45 to 0.7 bar,a particle distribution peaked at 30 km is used. This sizedistribution is also utilized in the lower cloud, spanning 2È4bars.

In addition to condensation, a small mixing ratio ofNH3a chromophore, either tholin (2] 10~8) or (5 ] 10~9), isP4added to the upper cloud. As inferred from limb darkeningobservations, the condensed chromophore becomes wellmixed in the upper ammonia cloud, and perhaps deeper aswell (West et al. 1986 ; Pope et al. 1992). As per Noy et al.(1981), the peak of the chromophore particle size distribu-tion is set to 0.05 km. However, the nature of the size dis-tribution and whether the chromophore adheres to theammonia ice particles are as yet unclear.


Gaseous abundances are modeled using the GalileoProbe Mass Spectrometer values (Niemann et al. 1996) as aguide. The He, and abundances are taken directlyH2, CH4from the Probe results. However, the tropospheric NH3abundance varies considerably with depth. At D0.4 bar, itsmixing ratio has been found to be D5 ] 10~6 (Griffith et al.1992 ; Kunde et al. 1982), while at D0.7 bar, its mixing ratiois D5 ] 10~5 to 10~4. In an infrared study using VoyagerIRIS data (Gierasch, Conrath, & Magalhaes 1986), it wasfound that only a small fraction (D1%) of the ammonia is incondensed form. Based on our visible albedo modeling,where the smaller particle size distribution dominates, andusing the gaseous mixing ratios above, we Ðnd that aNH3condensation fraction of D5% in the upper cloud isrequired to provide the necessary reÑectance.

We model JupiterÏs stratospheric haze using Deirmend-jianÏs haze particle size distribution (see ° 5) of polyacetylenepeaked at 0.1 kmÈa particle size justiÐed by limb-darkening studies (Rages et al. 1997 ; West 1988 ; Tomasko,Karkoschka, & Martinek 1986). The abundance of inC2H2JupiterÏs stratosphere is D10~8 to 10~7 (Edgington 1998 ;Noll et. al. 1986), though the polymerized abundance is notknown. In this study, the polyacetylene mixing ratio is set to5 ] 10~8, in a haze layer from 0.03 to 0.1 bar.

Figure 5 shows two model geometric albedo spectraalong with the observed full-disk albedo spectrum of Jupiter(Karkoschka 1994). We convert our model spherical albedoto a geometric albedo using an averaged phase integral ofq \ 1.25 (Hanel et al. 1981). The upper model utilizes tholinas the chromophore throughout the upper ammonia clouddeck, while the lower model utilizes P4.Although the general character of JupiterÏs geometricalbedo is reproduced fairly well, many of the methaneabsorption features are modeled too deeply. Furthermore,

the gaseous ammonia features at D0.65 and 0.79 km do notappear in the models because our database does not includeammonia absorption shortward of D1.4 km. Karkoschka(1998) indicates that the absorption feature centered atD0.93 km may be due to ammonia as well. Relying uponMie scattering theory and our choices for chromophoreparticle size distributions, we Ðnd that tholin appears toreproduce the UV/blue region of the albedo better than P4.However, the actual chromophore (s) in JupiterÏs atmo-sphere remains a mystery.

The published Bond albedo of Jupiter is 0.343 (Hanel etal. 1981). Using our models and limited wavelength cover-age (0.3È2.5 km), we estimate a Bond albedo (see ° 7) in the0.42È0.44 rangeÈa fair approximationÈdepending uponwhether or tholin is used as the chromophore.P4Uncertainties in the vertical structure of JupiterÏs atmo-sphere, heterogeneities in JupiterÏs cloud layers, and our useof an averaged phase integral all likely play a role inexplaining the di†erences between observational andmodeled albedo spectra. These details aside, JupiterÏs atmo-sphere remains a useful benchmark for our models of EGPalbedo and reÑection spectra.

7. RESULTS FOR EGPsWe produce Ðducial albedo models for each EGP class

using both isolated and modiÐed temperature-pressure pro-Ðles. We adopt DeirmendjianÏs cloud particle size distribu-tion with a peak at the moderate size of 5 km. Our modelEGP spherical albedos for the full range of e†ective tem-peratures are shown in Figures 6, 7, and 8. For these Ðducialmodels, full condensation is assumed (as described in ° 2.3).

The Jovian class I albedo spectra are determined mainlyby the reÑectivity of condensed and the molecularNH3absorption bands of gaseous Stratospheric and tropo-CH4.

FIG. 5.ÈGeometric albedo spectrum of Jupiter. Our model albedo spectra (thin curves) are compared with the observational full-disk albedo spectrum(thick curve ; Karkoschka 1994). The top model utilizes tholin as a chromophore, while the bottom model uses P4.

FIG. 6.È(a) Spherical albedo of a class I Jovian EGP. Irrespective of the T -P proÐle, an cloud deck resides above an cloud deck. The thin curveNH3 H2Ocorresponds to an isolated T -P proÐle model, while the thick curve signiÐes a modiÐed, more isothermal proÐle model. (b) Spherical albedo of a class II watercloud EGP. A thick cloud deck in the upper troposphere produces a high albedo. Isolated (thin curve) and modiÐed (thick curve) T -P proÐle models areH2Oshown.

FIG. 7.È(a) Spherical albedo of a class III clear EGP. In addition to the isolated (thin curve) and modiÐed (thick curve) T -P proÐle models, the dashedcurve depicts what the albedo would look like in the absence of the alkali metals. (b) Spherical albedo of a class IV roaster. Theoretical albedo spectra ofisolated (thin curve) and modiÐed (thick curve) T -P proÐle class IV models are depicted.

EXTRASOLAR GIANT PLANETS 895

FIG. 8.ÈSpherical albedo of a class V roaster. A silicate layer high in the atmosphere results in a much higher albedo than a class IV roaster. Noionization is assumed in this model.

spheric nonequilibrium species are not included in theseÐducial models. Their e†ects are explored in ° 8. Becauseboth isolated and nearly isothermal T -P proÐles of EGPswith K cross the condensation curve, theTeff [ 150 NH3details of the T -P proÐles do not have a large impact on theresulting albedos of class I objects. As shown in Figure 6a,the reÑective clouds keep the albedo fairly highNH3throughout most of the visible spectral region. Toward theinfrared, the gaseous absorption cross sections tend tobecome larger, so photons are more likely to be absorbedabove the cloud deck. Hence, at most infrared wavelengths,the albedo is below that in the visible region.

The isolated and modiÐed proÐle class II albedos areshown in Figure 6b. Relative to a class I EGP, a class IIalbedo is even higher in the visible because of very stronglyreÑective clouds in the upper atmosphere. GaseousH2Oabsorption features tend to be shallower because these H2Oclouds form higher in the atmosphere than the cloudsNH3of most class I objects. The intersection of the isolatedproÐle and the condensation curve near 0.01 bars mayNH3result in a thin condensation layer high in the atmo-NH3sphere, but condensation is assumed to be negligibleNH3for this model.

The clear class III does not contain any principal conden-sates in the upper atmosphere (irrespective of the T -PproÐle), although a silicate cloud deck exists deeper, at D50bars. The presence of alkali metals in the troposphere has asubstantial lowering e†ect on the albedo. As per Figure 7a,sodium and potassium absorption lowers the albedo atshort wavelengths, resulting in a spherical albedo belowD0.6 throughout most of the UV/blue spectral region. Inthe red region, lower Rayleigh scattering cross sections andstrong alkali metal absorption result in spherical albedosthat drop below 0.1. In contrast, in the absence of the alkalimetals, the spherical albedo would remain high (Z0.75)throughout most of the visible. In both cases, the near-infrared albedo is essentially negligible, largely because ofabsorption by and CIA. Our modelsCH4, H2O, H2-H2show that, in class III objects, the details of the T -P proÐlewill have only minor e†ects on the albedo. If low-abundance sulÐde or chloride condensates were to exist inthe troposphere, they could appear at pressures as low as afew bars. Based on theoretical abundances (Burrows &Sharp 1999), thick clouds are very unlikely, but it is worthmentioning that even cirrus-like condensation could raisethe albedo in the visible and near-infrared.

In the higher-temperature (900 K) classK [Teff [ 1500IV roasters, the e†ect of the alkali metals is most dramatic.Unlike the class III EGPs, a silicate cloud deck exists atmoderate pressures of D5È10 bars, depending on the detailsof the T -P proÐle. An iron or iron-rich condensate likelyexists below the silicate deck, but it is sufficiently below theopaque silicate cloud that it does not have any e†ect on thevisible and near-infrared albedos. Figure 7b shows thespherical albedo of a class IV EGP. Assuming a fairly““ isothermal ÏÏ T -P proÐle (the modiÐed proÐle) in the upperatmosphere, absorption by sodium and potassium atoms,coupled with rovibrational molecular absorption, results ina surprisingly low albedo throughout virtually the entirevisible and near-infrared wavelength region explored in thisstudy (¹2.5km). The silicate cloud is deep enough that itse†ects are rendered negligible by the absorptive gases aboveit.

Although a class IV model with a modiÐed T -P proÐleresults in an albedo that is signiÐcantly lower than that ofeven a class III model, the albedo of a class IV model withan isolated T -P proÐle is a di†erent story : because theupper atmosphere in such a model is signiÐcantly coolerthan in the modiÐed T -P proÐle case, the equilibrium abun-dances of the alkali metals are lower. Furthermore, the sili-cate cloud deck is expected to be somewhat higher in theatmosphere (Fig. 1) and to have a nonnegligible e†ect on thealbedo in both the visible and near-infrared regions (Fig.7b).

Because of the low ionization potentials of sodium (5.139eV) and potassium (4.341 eV), it is likely that signiÐcant NaII and K II layers exist in the outer atmospheres of class IVEGPs (and perhaps class III EGPs). Nevertheless, assuminga silicate cloud layer at D5-10 bars, simple ionization equi-librium estimates indicate that these layers should not reachthe depths of the silicate layer in class IV EGPs, so substan-tial column depths of Na I and K I should remain to absorbvisible radiation. The full absorption and emission featuresof such ionization layers will be explored in future EGPstudies.

The very hot K) class V roasters have a sili-(Teff Z 1500cate cloud layer that is located much higher in the atmo-sphere relative to the class IV roasters, so alkali metal andmolecular absorption is reduced. Figure 7b illustrates themuch higher albedo expected of the class V objects,assuming the silicate layer is composed predominantly ofenstatite grains. If the sodium and potassium ionization

896 SUDARSKY, BURROWS, & PINTO

layers are substantial in these objects, then the alkaliabsorption lines are much weaker or are converted intoemission lines because of the heating of the upper atmo-sphere. We also note that a roaster of particularly low mass(e.g., HD 209458b) is expected to exhibit a signiÐcantlylarger radius than such an object in isolation (Burrows et al.2000). For such a low surface gravity cm s~2) object,([103the silicate layer will form high in the atmosphere even for

K. Hence, the lower limit to required toTeff \ 1500 Teffrender a roaster a class V EGP is reduced in the case of lowsurface gravity.

Via spectral deconvolution, Charbonneau et al. (1999)have constrained the geometric albedo of the roaster, q Boob, to be below 0.3 at 0.48 km. This limit, which was obtainedwith an assumed phase function and orbital inclination near90 degrees, contrasts with the Ðndings of Cameron et al.(1999), who infer that the albedo is high in this region. Usingour class IV T -P proÐle model K),(Teff [ 1500we Ðnd that the geometric albedo at 0.48 km is only 0.03.However, if in fact q Boo b is a class V EGP K),(Teff Z 1500we derive a geometric albedo of 0.39 at 0.48 km, still smallerthan the assumed Cameron et al. value of 0.55, from whichthey derive a planetary radius as high as 1.8 Jupiter radii.The widely varying albedos of classes IV and V coupledwith the fact that q Boo b appears to have an e†ectivetemperature near the transition region between these classesindicates that the detailed modeling of this EGP will benecessary in order to ascertain its nature.

It is instructive to examine the temperatures and pres-sures to which incident radiation penetrates an EGPÏsatmosphere as a function of wavelength. For each class,Figures 9 and 10 show the pressures and temperatures,respectively, corresponding to one mean free path of anincident photon. In clear atmospheres, these temperaturesand pressures are very strong functions of wavelength,largely mirroring molecular absorption bands and/oratomic absorption lines. Conversely, when thick cloudlayers are present, the wavelength dependence is muchweaker, because of the efficient extinction of radiation by asize distribution of condensed particles.

FIG. 9.Èlog of the pressure (P) at a depth equal to the mean free path ofincident radiation, as a function of wavelength (j) in microns, for each ofthe EGP classes. A size distribution of particles and the assumption of fullcondensation conspire to make the class I and class II curves only weakfunctions of wavelength.

FIG. 10.ÈTemperature (T ) at a depth equal to the mean free path ofincident radiation, as a function of wavelength (j) in microns, for each ofthe EGP classes.

Because of the azimuthal symmetry of our Feautrier tech-nique, we do not compute the phase integrals of EGPs. Intheir absence, the characteristics of the atmosphere at qjD1 are useful for the approximate conversion from sphericalalbedos to geometric albedos. Using the asymmetry factorand single scattering albedo values, the phase integral, isqj,estimated by interpolating within the tables of Dlugach &Yanovitskij (1974) (Marley et al. 1999). Geometric albedosare then obtained using the relation, Esti-A

g,j\As,j/qj.mated geometric albedo spectra are shown in Figure 11.

Our class II geometric albedo compares qualitatively withthat of the ““ quiescent ÏÏ water cloud model of Marley et al.(1999). Given the di†erences in particle size distributions,the Marley et al. albedo tends to fall o† a bit more sharplywith increasing wavelength, while having shallower gaseousabsorption features in the visible. Our class IV models maybe compared with the Marley et al. ““ brown dwarf ÏÏ modelwith silicate (enstatite) clouds, as well as with the 51 Peg bmodel of Goukenleuque et al. (1999). Our inclusion of thealkali metals results in a qualitatively very di†erent, andmuch lower, albedo spectrum than in these previous studies.Our class V model may be compared with the high-temperature model K) of Seager & Sasselov(Teff \ 1580(1998). We Ðnd that, similar to Seager & Sasselov, the pres-ence of silicate (enstatite) grains results in signiÐcant reÑec-tion, but our inclusion of the alkali metals results inprominent absorption lines as well.

We combine a geometric albedo spectrum from eachEGP class with appropriately calibrated stellar spectra(Silva & Cornell 1992) to produce representative EGPreÑection spectra. Figure 12a shows theoretical full-phasereÑection spectra of EGPs from 0.35 to 1.0 km, assuming aG2 V central star, orbital distances of 0.05 AU (class IV), 0.2AU (class III), 1.0 AU (class II), and 5.0 AU (class I), and aplanetary radius of 1 Jupiter radius For a class IV(RJ).roaster, at 0.45 km, the ratio of reÑected and stellar Ñuxesis D5 ] 10~6, while for class I, II, and III EGPs, it isD5 ] 10~9, 10~7, and 10~6, respectively. This ratio at 0.65km drops to D5 ] 10~7 for a class IV object, and isD5 ] 10~9, 10~7, and 3 ] 10~7 for class I, II, and IIIEGPs, respectively. Figure 12b shows theoretical full-phasereÑection spectra of class IV and V roasters, assuming anF7 V central star, orbital distances of 0.1 AU (class IV) and

FIG. 11.È(a) Estimated geometric albedos of class I, II, and III EGPs. A modiÐed T -P proÐle model is used in each case. These conversions from sphericalalbedos are made by approximating the phase integral based on the single scattering albedo and scattering asymmetry factor at an atmospheric depth(qj)equal to the mean free path of incident radiation. (b) Estimated geometric albedos of class IV and V EGPs.

FIG. 12.È(a) Full-phase EGP reÑection spectra, assuming a G2 V central star, a planetary radius equal to that of Jupiter, and orbital distances of 0.05 AU(class IV), 0.2 AU (class III), 1.0 AU (class II), and 5.0 AU (class I). These reÑection spectra are obtained by combining the geometric albedo of each EGP classwith a G2 V stellar spectrum (Silva & Cornell 1992) and a Kurucz (1979) theoretical spectrum longward of 0.9 km. (b) Full-phase EGP reÑection spectra,assuming an F7 V central star, a planetary radius equal to that of Jupiter, and orbital distances of 0.1 AU (class IV) and 0.04 AU (class V).


0.04 AU (class V), and 1 At 0.45 km, the reÑected toRJ.stellar Ñux ratios are D10~6 (class IV) and 5 ] 10~5 (classV). At 0.65 km, these ratios are D10~7 (class IV) and5 ] 10~5 (class V). For larger planetary radii and di†erentorbital distances, these ratios should be scaled accordingly.

In the reÑection spectrum of a class IV object (Fig. 12),absorption by the resonance lines of sodium (5890 A� /5896

and potassium (7665 is extreme. These linesA� ) A� /7699 A� )are also very signiÐcant, though substantially weaker, inclass III objects. Methane absorption bands shortward of 1km, especially those at D0.73, 0.86, and 0.89 km, are quiteprominent in class I and III objects. These bands are alsoclearly present in class II objects, but with sufficient watercondensation high in the troposphere, the bands are not asprominent as in class I or class III EGPs. Although present,methane absorption is even weaker in class IV EGPs, whereCO is the dominant carbon-bearing molecule. At the highe†ective temperature of a class V object K), the(Teff Z 1500methane abundance is completely overwhelmed by that of

CO, and we expect that no strong methane bands will beseen in reÑection.

Bond albedos for EGPs are obtained using equation (5).Our lower and upper wavelength limits of integration are0.3 and 2.5 km, respectively, rather than formally from 0 toinÐnity. Hence, our derivations are estimates of actual Bondalbedos, accurate to D10%È15%, depending on the centralstellar spectral type and the uncertainties in the EGP spher-ical albedos shortward of 0.3 km and longward of 2.5 km.The Bond albedos for our Ðducial modiÐed T -P proÐlemodels and for isolated T -P proÐle models are shown inTables 1A, 1B, 2A, and 2B. Assuming full condensation ofprincipal condensates and no nonequilibrium species, theBond albedos of class I and II objects are high. Over thespectral range, M4 V to A8 V, the peak of the stellar energyÑux ranges from D0.9 to 0.4 km. Class I EGP Bond albedosrange from D0.4 to 0.65, while those of class II EGPs reachnearly 0.9. These albedos tend to be signiÐcantly lowerwhen smaller condensation fractions and nonequilibrium

TABLE 1A

ESTIMATED BOND ALBEDOS OF EGPs

Star EGP class Fiducial ““ Isolated ÏÏ 10% Condensation 1% Condensation

A8 V . . . . . . I 0.63 0.64 0.62 0.59II 0.88 0.88 0.79 0.47III 0.17 0.13 . . . . . .IV 0.04 0.21 . . . . . .V 0.57 . . . . . . . . .

F7 V . . . . . . I 0.59 0.61 0.57 0.51II 0.84 0.83 0.74 0.40III 0.14 0.10 . . . . . .IV 0.03 0.18 . . . . . .V 0.56 . . . . . . . . .

G2 V . . . . . . I 0.57 0.59 0.55 0.47II 0.81 0.81 0.71 0.37III 0.12 0.09 . . . . . .IV 0.03 0.16 . . . . . .V 0.55 . . . . . . . . .

NOTES.ÈBond albedos of EGPs, using modiÐed T -P proÐle models (Ðducial) with full condensation,isolated T -P proÐle models (““ isolated ÏÏ) with full condensation, modiÐed T -P proÐle models with 10%condensation, and modiÐed T -P proÐle models with 1% condensation. Because class III and class IVBond albedos are not signiÐcantly a†ected by condensates, the columns referring to fractional conden-sation models are left blank. The existence of class V is a result of very strong stellar insolation, soisolated T -P proÐle models are not calculated. In all cases, nonequilibrium condensates are ignored.

TABLE 1B

ESTIMATED BOND ALBEDOS OF EGPS

Star EGP Class Fiducial ““ Isolated ÏÏ 10% Condensation 1% Condensation

G7 V . . . . . . I 0.55 0.58 0.52 0.44II 0.79 0.79 0.69 0.34III 0.10 0.07 . . . . . .IV 0.02 0.15 . . . . . .V 0.55 . . . . . . . . .

K4 V . . . . . . I 0.48 0.52 0.44 0.33II 0.70 0.70 0.60 0.25III 0.05 0.04 . . . . . .IV \0.01 0.11 . . . . . .V 0.53 . . . . . . . . .

M4 V . . . . . . I 0.38 0.43 0.33 0.16II 0.56 0.55 0.47 0.16III 0.01 \0.01 . . . . . .IV \0.01 0.08 . . . . . .V 0.51 . . . . . . . . .


TABLE 2A

CLASS II EGPS (““WATER CLOUD ÏÏ)

Mp

sin i Condensation TeffObject Star (MJ) a (AU) (%) A

B(K)

Gl 876b M4 V 1.9 D0.2 Full 0.56 18010 0.47 1821 0.16 199

HR 5568b K4 V 0.75 D1.0 Full . . . . . .10 . . . . . .1 0.25 160

HD 210277b G7 V 1.28 D1.15 Full 0.79 17710 0.69 1941 0.34 232

HR 810b G0 V 2.0 D1.2 Full 0.82 19210 0.72 2131 0.38 254

NOTES.ÈClass II EGPs and their central stars, masses, and orbital distances, alongwith estimated Bond albedos and e†ective temperatures, assuming various conden-sation fractions. Nonequilibrium condensates are ignored. Internal luminosities areestimated using the evolutionary models of Burrows et al. 1997 and assuming an age of5 Gyr, except for HD 210277b (8 Gyr), 47 UMa b (7 Gyr), and t And d (3 Gyr).Absence of albedo and entries indicates that, for the expected EGP Bond albedoTeffassuming the given condensation fraction, is low enough such that condensedTeffammonia, rather than water, should reside in the upper troposphere. Hence, thiscombination of parameters should not result in a class II EGP.

condensates are considered. For example, the Bond albedoof our Jupiter model about a G2 V central star is in the0.42È0.44 range, depending upon whether or tholin isP4used as the chromophoreÈsomewhat higher than JupiterÏsactual Bond albedo of 0.343 (Hanel et al. 1981).

In contrast, Bond albedos of class III and IV EGPs arevery low. Those of class III objects vary from D0.01 to 0.2over the spectral range, M4 V to A8 V. Class IV EGPsreÑect the smallest fraction of incident radiation, with Bondalbedos ranging from below 0.01 up to only 0.04, assumingour modiÐed T -P proÐle model and no nonequilibriumcondensates. These Bond albedos are signiÐcantly lowerthan those of Marley et al. (1999) because we include thee†ects of the alkali metals. For example, assuming a G2 Vcentral star, our Ðducial class III model yields a Bondalbedo of 0.12, while those of Marley et al. are in the 0.31È

0.33 range (cloud-free 500 K models), and our Bond albedofor a class IV EGP is only 0.03, while those of Marley et al.are in the 0.30È0.44 range (cloud-free and cloudy 1000 Kmodels). The Bond albedos of the very hot class V objectsare much higher than those of class III or IV, ranging fromD0.51 to 0.57 over the spectral range, M4 V to A8 V.

Estimated Bond albedos and e†ective temperatures ofknown EGPs are shown in Tables 2A, 2B, 3, and 4. Theequilibrium temperature of an irradiated object is

Teq \C(1[ A

B)L

*16npa2

D1@4(33)

(Saumon et al. 1996), where is the stellar luminosity, p isL*the Stefan-Boltzmann constant, and a is the orbital distance

of the planet. For massive and young EGPs with sufficiently

TABLE 2B

CLASS II EGPS (““WATER CLOUD ÏÏ)

Mp

sin i Condensation TeffObject Star (MJ) a (AU) (%) A

B(K)

16 Cyg B b . . . . . . G2.5 V 1.66 1.7 Full 0.81 15810 0.71 1701 0.37 198

47 UMa b . . . . . . . G0 V 2.4 2.1 Full 0.82 16010 0.72 1721 0.38 199

t And d . . . . . . . . . F7 V 4.61 2.50 Full 0.84 22810 0.74 2331 0.40 247

Gl 614b . . . . . . . . . K0 V 3.3 2.5 Full 0.75 16810 0.65 1701 0.30 177

55 Cnc c . . . . . . . . . G8 V D5 3.8 Full 0.78 19810 0.68 1991 0.33 201


TABLE 3

CLASS III EGPS (““ CLEAR ÏÏ)

Mp

sin i TeffObject Star (MJ) a (AU) A

B(K)

HD 130322b . . . . . . K0 V 1.08 0.08 0.07 81055 Cnc b . . . . . . . . . . G8 V 0.84 0.11 0.10 690Gl 86 Ab . . . . . . . . . . K1 V 4.9 0.11 0.07 660HD 195019b . . . . . . G3 V 3.4 0.14 0.12 720HD 199263b . . . . . . K2 V 0.76 0.15 0.07 540o Cr Bb . . . . . . . . . . . G0 V 1.13 0.23 0.13 670HR 7875b . . . . . . . . . F8 V 0.69 D0.25 0.14 650HD 168443b . . . . . . G8 IV 5.04 0.277 0.10 620HD 114762b . . . . . . F9 V D10 0.38 0.13 51070 Vir b . . . . . . . . . . . G4 V 6.9 0.45 0.11 380t And c . . . . . . . . . . . F7 V 2.11 0.83 0.14 370

NOTES.ÈClass III EGPs and their central stars, masses, and orbitaldistances, along with estimated Bond albedos and e†ective temperatures.Nonequilibrium condensates are ignored.

TABLE 4

CLASS IV EGPS (ROASTERS)

Mp

sin i aObject Star (MJ) (AU) A

BTeff(K)

HD 187123b . . . G3 V 0.52 0.0415 0.03 146051 Peg b . . . . . . . G2.5 V 0.45 0.05 0.03 1240t And b . . . . . . . . F7 V 0.71 0.059 0.03 1430HD 217107b . . . G7 V 1.28 0.07 0.02 1030

NOTES.ÈClass IV EGPs and their central stars, masses, and orbitaldistances, along with estimated Bond albedos and e†ective temperatures.Nonequilibrium condensates are ignored.

large orbital distances, because of their signiÐcantTeff [ Teqinternal energies. We estimate the e†ective temperatures ofsuch objects simply by adding the stellar-insolated andinternal contributions to the luminosity, and noting that

The internal contribution is deÐned to beL \ 4nRp2 pT eff4 .

the luminosity of an isolated object of the given mass andage, and it is found using the evolutionary models ofBurrows et al. (1997).

Given the list of over two dozen known EGPs, it is pos-sible that none is cold enough to be a class I (Jovian) object(HR 5568b is an ambiguous case). Classes II, III, and IV arewell-represented (Tables 2A, 2B, 3, and 4), while class Vlikely includes HD 209458b, and perhaps q Boo b and/orHD 75289b.

8. PARAMETER STUDIES

In EGP atmospheres, variations in condensation frac-tions and particle size distributions, as well as the possiblepresence of stratospheric and tropospheric nonequilibriumspecies, can have large e†ects on the spherical and Bondalbedos. First, we consider the e†ects of lowering the con-densation fraction to 10% and 1%. Figures 13a and 13bshow the substantial changes in class I (Jovian) and class II(water cloud) EGPs. The class II case best illustrates thesystematic e†ects, since only an cloud deck exists.H2O(Recall that the class I model contains an ammonia clouddeck above a water cloud deck.) The condensation fractionhas a substantial e†ect on the spherical and geometricalbedos. Less condensation clearly results in lower albedos,especially in the red/near-infrared, where gaseous opacitiesare strong (Marley et al. 1999). Note that the e†ects of the

FIG. 13.È(a) Dependence of the spherical albedo of a class I EGP on condensation fraction. Full condensation (thick curve), 10% condensation (thincurve), and 1% condensation (dashed curve) models are shown. (b) Dependence of the spherical albedo of a class II EGP on condensation fraction. Fullcondensation (thick curve), 10% condensation (thin curve), and 1% condensation (dashed curve) models are shown.


alkali metals, deep in the atmosphere, are apparent in theUV/blue albedo of the class II, 1% condensation model. Asshown in Tables 1A and 1B, the Bond albedos of these 1%condensation models are signiÐcantly lower than those oftheir full condensation counterparts, particularly for theclass II EGPs.

Cloud particle size distributions in EGPs are not known.As alluded to in ° 5, for a given condensate abundance, thenet extinction by condensates (almost pure scattering for

ice) is smaller when particle sizes are larger. This isH2Oshown explicitly in Figure 14a, comparing spherical albedosfor Deirmendjian ice cloud distributions with sizeH2Opeaks of 0.5, 5 (Ðducial), and 50 km. The qualitative e†ect ofincreasing the peak size is similar to the e†ect of reducingthe condensation fraction. Widening the distribution hassimilar consequences because the largest particles squanderthe condensate, reducing the number density of smallerscattering particles.

Nonequilibrium species in the upper atmospheres ofEGPs may be produced by UV-induced processes. Whileboth gaseous and condensed species are likely to be produc-ed, the condensates will generally have greater e†ects on thealbedos and reÑection spectra. As in the atmosphere ofJupiter, stratospheric hazes and tropospheric chromo-phores, or impurities within or above the principal cloudlayers, can lower the albedo spectra in the UV/blue rangeand can also modify their character at other wavelengths. Inaddition to their compositions, the size distributions ofthese nonequilibrium species play a role. Figure 14b showsthe e†ect of including a representative upper tropospherichaze of tholin (with mixing ratio of 10~8) on the spherical

albedo of a class I EGP. In analogy with our Jupiter model,the size distribution is peaked at 0.05 km. Although theabundances and size distributions of such particles in EGPsare unknown, we present this model as an indication of thequalitative e†ect that this type of haze would have on thealbedo. The associated class I Bond albedo, assuming a G2V central star, decreases from 0.57 to 0.48.

We represent the optically dominant aerosol withinstratospheric hazes by polyacetylene, although other pos-sibilities certainly exist. Our models show that the e†ect ofpolyacetylene on the albedo is minor, lowering the UV/bluealbedo no more than a few percent, assuming a mixing ratioas large as 10~7. We stress that the actual compositions,abundances, and the size distributions of nonequilibriumspecies in EGPs are unknown, and that the quantitativee†ects on EGP albedos may or may not be signiÐcant.

9. CONCLUSIONS

The classiÐcation of EGPs into Ðve composition classes,related to is instructive, since the albedos of objectsTeff,within each of these classes exhibit similar features andvalues. The principal condensate in class I Jovian EGPs

K) is while in class II water cloud EGPs it(Teff [ 150 NH3,is ice. Gaseous molecular absorption features, espe-H2Ocially those of methane, are exhibited throughout class Iand II albedo spectra. Assuming adequate levels of conden-sation, class II EGPs are the most highly reÑective of anyclass. For lower condensation fractions, the albedos of bothclasses fall o† more quickly with increasing wavelengthrelative to full condensation modelsÈespecially the class IIobjects. Even a small mixing ratio of a nonequilibrium tro-

FIG. 14.È(a) Dependence of the spherical albedo on the particle size distribution. Class II EGP models with Deirmendjian cloud distributions peaked at0.5 (thin curve), 5 (Ðducial ; thick curve), and 50 km (dashed curve) are shown. Note that the alkali metals are not included in these model albedo spectra. (b)E†ect of the presence of an upper tropospheric tholin haze (with mixing ratio of 10~8) on a class I EGP spherical albedo. The albedo-lowering e†ect isgreatest in the UV/blue region of the spectrum.


pospheric condensate within or above a cloud deck candepress the UV/blue albedo and reÑection spectrum signiÐ-cantly.

In class III clear EGPs K), little condensation(Teff Z 350is likely, and so albedos are determined almost entirely byatomic and molecular absorption and Rayleigh scattering.Radiation generally penetrates more deeply into theseatmospheres, to pressures and temperatures where sodiumand potassium absorption and collision-inducedH2-H2absorption (CIA) become substantial. Throughout most ofthe visible spectral region, the albedo decreases withincreasing wavelength. In the near-infrared, CIA, andH2O,

conspire to keep the albedo very low.CH4In the upper atmospheres of the high-temperature (900K) class IV roasters, the equilibrium abun-K [Teff [ 1500

dances of the alkali metals are higher than in the class IIIEGPs, so the absorption lines of sodium and potassium areexpected to lower the albedo more dramatically. A silicatecloud exists at moderate depths (D5È10 bars), but the largeabsorption cross sections of the sodium and potassiumgases above it likely preclude the cloud from having a sig-niÐcant e†ect on the albedo. Like class III EGPs, the near-infrared albedo is expected to remain close to zero in theabsence of nonequilibrium condensates.

The hottest K) and/or lowest gravity(Teff Z 1500 (g[103 cm s~2) roasters (class V) have a silicate layer locatedmuch higher in the atmosphere relative to the class IV roas-ters. This layer is expected to reÑect much of the incidentradiation before it is absorbed by neutral sodium and pot-assium and molecular species. Hence, class V EGPs havemuch higher albedos than those of class IV. Furthermore, ifsubstantial sodium and potassium ionization layers exist,the neutral alkali metal absorption lines could be muchweaker than those shown in these class V models.

While stratospheres generally are not anticipated in hightemperature EGPs (Seager & Sasselov 1998 ; Goukenleuqueet al. 1999), it is possible that more detailed modeling willshow that they do exist. The presence of a stratospherewould give rise to visible and infrared emission features nototherwise seen. Furthermore, the presence of nonequilib-rium solids due to photochemistry may decrease the albedoin the UV/blue, but increase it somewhat in the red/near-infrared because even largely absorbing condensates aremore reÑective than gaseous molecular species in this spec-tral region.

Di†erences in particle size distributions of the principalcondensates can have large quantitative, or even qualitativee†ects on the resulting albedo spectra. In general, less con-densation, larger particle sizes, and wider size distributionsresult in lower albedos.

Despite many uncertainties in the atmospheric details ofEGPs, our set of model albedo spectra serves as a usefulguide to the prominent features and systematics over a fullrange of EGP e†ective temperatures, from D100 to 1700 K.Full radiative equilibrium modeling of a given EGP at aspeciÐc orbital distance from its central star (of given spec-tral type), and of speciÐc mass, age, and composition isnecessary for a detailed understanding of an object.However, as observational EGP spectra become available,our set of model albedo spectra o†ers a means by which aquick understanding of their general character is possible,and by which some major atmospheric constituents, bothgaseous and condensed, may be inferred.

We thank Mark Marley, Sara Seager, Bill Hubbard, Jon-athan Lunine, Christopher Sharp, and Don Hu†man for avariety of useful contributions. This work was supportedunder NASA grants NAG5-7073 and NAG5-7499.

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1_Albedo and Reflection Spectra of Extrasolar Giant Planets_(Sudarsky_etal_2000)