Weather Variability Of Close-in Extrasolar Giant Planets

a r X i v :a s t r o -p h /0210499v 2 12 M a r 2003

Draft version February 2,2008

Preprint typeset using L A T E X style emulateapj v.04/03/99

“WEATHER”VARIABILITY OF CLOSE-IN EXTRASOLAR GIANT PLANETS

Kristen Menou a,1,2,James Y-K.Cho b,c ,Sara Seager c &Bradley M.S.Hansen d

a Princeton

University,Department of Astrophysical Sciences,Princeton,NJ 08544,USA

b Spectral Sciences Inc.,99South Bedford St.,#7,Burlington,MA 01803,USA

c Carnegie Institution of Washington,Dept.of Terrestrial Magnetism,5241Broa

d Branch Rd.

NW,

Washington,DC 20015,USA

d Division of Astronomy,8971Math Sciences,UCLA,Los Angeles,CA 90095,USA

Draft version February 2,2008

ABSTRACT

Shallow-water numerical simulations show that the atmospheric circulation of the close-in extrasolar giant planet (EGP)HD 209458b is characterized by moving circumpolar vortices and few bands/jets (in contrast with ～10bands/jets and absence of polar vortices on cloud-top Jupiter and Saturn).The large spatial scales of moving circulation structures on HD 209458b may generate detectable variability of the planet’s atmospheric signatures.In this Letter ,we generalize these results to other close-in EGPs,by noting that shallow-water dynamics is essentially speci?ed by the values of the Rossby (R o )and Burger (B u )dimensionless numbers.The range of likely values of R o (～10?2–10)and B u (～1–200)for the atmospheric ?ow of known close-in EGPs indicates that their circulation should be qualitatively similar to that of HD 209458b.This results mostly from the slow rotation of these tidally-synchronized planets.Subject headings:planetary systems –planets and satellites:general –stars:atmospheres –turbulence

1.INTRODUCTION

The focus of extrasolar planet research has broadened to now include the characterization of their physical prop-erties,as shown by the recent sodium detection in the atmosphere of HD 209458b (Charbonneau et al.2002).Atmospheric circulation is expected to play a key role in determining a number of observational characteristics of EGPs,including their albedo and transmission spectrum (see,e.g.,Seager &Sasselov 1998,2000;Sudarsky et al.2000;Brown 2001).This is especially true for close-in EGPs,which are thought to be tidally-locked to their par-ent star and irradiated on one side only:circulation will be essential in redistributing heat from the day to the night side on these planets,thus determining to a large extent how they will appear to the distant observer (Cho et al.2002a,b;Showman &Guillot 2002).

Recently,we have presented a set of detailed shallow-water numerical simulations of the atmospheric ?ow on HD 209458b (Cho et al.2002a,b),currently the only EGP with known mass and radius from the transit light curves and radial velocity measurements (Charbonneau et al.2000;Henry et al.2000;Mazeh et al.2000;Jha et al.2000;Brown et al.2001).These simulations suggest that,contrary to the simple day/night (hot/cold)picture,the circulation on this planet is characterized by two mov-ing circumpolar vortices and a small number of latitudi-nal bands/jets.The vortices act as dynamically distinct thermal spots whose motion around the poles generates variability as seen by an observer interested in quantities integrated over the planetary disk (or circumference).It is possible to determine the general features of the circulation pattern expected within the framework of shallow-water dynamics by specifying the two dimension-

less numbers –Rossby (R o )and Burger (B u )–for the atmospheric ?ow.In this Letter ,we estimate a range of likely R o and B u values for known close-in EGPs and con-clude that their atmospheric circulation pattern should be qualitatively similar to that of HD 209458b.In §2,we re-call how the atmospheric ?ow pattern can be characterized by the knowledge of R o and B u .In §3,we describe the sample of close-in EGPs selected for our study and how we estimate likely values for various global planetary param-eters entering into the de?nition of R o and B u .Finally,our results and conclusions are presented in §4.

2.TURBULENT SHALLOW-W ATER DYNAMICS

Shallow-Water equations describe the motion of a thin,homogeneous layer of hydrostatically-balanced,inviscid ?uid with a free surface,in motion around a rotating planet (Pedlosky 1987,Holton 1992).The ?uid is subject to gravitational and Coriolis forces and obeys the following equations

?v

?t

+v ·?h =?h ?·v ,(2)where v is the horizontal velocity,h is the thickness of the modeled layer,f =2?sin ?is the Coriolis parame-ter,?is the rotation rate of the planet,?is the latitude,g is the gravitational acceleration and k is the unit vec-tor normal to the surface of the planet.In dimensionless form,shallow-water equations become functions only of the Rossby (R o )and Burger (B u )numbers:

R o ≡

U

2“WEATHER”VARIABILITY

B u≡ L D gH/|f|,(4)

where U,L and H are characteristic velocity,length and

layer thickness scales,respectively;L D is the Rossby de-

formation radius.Note that|f|～?at mid-latitudes and

that the planetary radius,R p,is the relevant length scale

when discussing the large-scale atmospheric circulation.

The Rossby number measures the importance of rotation

on the?ow,while the Burger number measures the strati-?cation of the atmosphere via the Brunt-V¨a is¨a l¨a frequency (Holton1992).

The atmospheric structure in bands of gaseous giant planets in our Solar System is well described as emerg-ing from freely-evolving shallow-water turbulence on the sphere(Cho&Polvani1996b).Turbulence in a thin at-mospheric layer is quasi-2D in nature.Contrary to the forward turbulent energy cascade observed in3D geome-try,2D turbulence is characterized by an inverse energy cascade(transfer from small to large scales)and a forward cascade of enstrophy3down to small scales,where it is dissipated by viscous processes.Qualitatively,the inverse cascade is associated with the growth of vortices through continuous mergers.

Turbulence in a thin atmospheric layer is also strongly constrained by the combined e?ect of spherical geometry and rotation(the“β?e?ect”).While the force balance is everywhere the same along a latitude circle,it changes with latitude because of the dependence of the Coriolis term with?.This anisotropy,as measured by the param-eterβ=2?cos?/R p(the latitudinal gradient of f),is strongest at the equator,whereβis maximum..While ?uid motions are free to grow to the largest available scale in the longitudinal direction,their growth is limited in the latitudinal direction by the characteristic Rhines scale, Lβ=π

(2R o)and that the presence of circumpolar vortices is expected for B u～>1/9.Thus,if the values of R o and B u for other close-in EGPs can be estimated,one can get an idea of the type of large-scale atmospheric circulation expected on these planets in the stable,radiative region.In our estimates of R o and B u,we will set U=ˉU,which is the global velocity scale of the atmospheric?ow and is only known for the Solar System giants(Table1),

An important parameter entering the de?nition of both R o and B u is the planetary rotation rate,?.While the value of?is generally unknown for EGPs,a number of close-in EGPs have the advantage of being probably tidally-synchronized to their parent star,so that their ro-tation rate has e?ectively been measured via the orbital period(?=?orb for circular orbits).As we show be-low,the knowledge of?for this sample of close-in EGPs restricts the range of possible values for R o and B u to a small enough region of the parameter space that their atmospheric circulation pattern can be inferred.

3.SAMPLE OF CLOSE-IN EXTRASOLAR GIANT PLANETS Since tidal synchronization occurs faster than orbital circularization,it is possible that some close-in EGPs with substantial eccentricities are nonetheless(pseudo-)synchronized(i.e.synchronized at the periastron orbital frequency).The shallow-water results described in§2were established only in the limit of negligible eccentricity,how-ever.Hence,we must restrict our sample to planets with small eccentricities.4Table1lists all the EGPs selected for our study,plus the four Solar System giants.Parame-ters for the EGPs were collected from the extrasolar planet almanac5and encyclopedia.6

Low-eccentricity EGPs were divided into two groups, based on their orbital distance to the parent star.In the ?rst group,EGPs with semi-major axes a≤0.066AU are most likely tidally-synchronized since all known EGPs with such small values of a are also circularized.The tidal synchronization status of the more distant planets(in the second group)is less clear because several eccentric EGPs with distances of closest approach7as small as0.05AU are also known.It is thus not clear why some EGPs with pe-riastron distances larger than this value would be tidally-circularized while others would not.We note,however, that for values of the tidal parameter Q not too di?erent from that of Jupiter(～105),EGPs in this second group

3The?ow enstrophy is de?ned as1

MENOU ET AL.3

are also expected to be synchronized.We will assume it is indeed the case in our calculations.

Radial velocity surveys only measure M p sin i,which is a lower limit to the planet’s mass,M p,given the unknown orbital inclination,i.For randomly oriented systems,the distribution of cos i is uniform.We adopt the value of M p corresponding to sin i=0.5for our?ducial estimate of R o and B u,and we allow sin i to vary from0.1to1when estimating the range of likely values for R o and B u.8

For a given mass,M p,the radius of an isolated planet is estimated from the mass–radius relation for sub-stellar objects of Chabrier&Bara?e(2000),supplemented at the low mass end by a constant density law that empirically ?ts values for the Solar System giants.To account for the slower cooling under strong stellar irradiation,we also al-low the radius to be up to50%larger than the value for an isolated planet,in agreement with published cooling EGP models(Burrows et al.2000;Guillot&Showman 2002).Our results depend only weakly on the planetary radius,as long as R p～R Jup(as expected for all masses of interest).The gravitational acceleration is derived as g=GM p/R2p,where G is the gravitational constant.

For the mean layer thickness,H,we adopt the atmo-spheric pressure scale-height,H atm≡R T atm/g,where R is the perfect gas constant.The global radiative equilib-rium temperature of the planet is T atm=T?(R?/2a)1/2(1?A b)1/4,which is a function of the parent star luminosity (L?∝T4?R2?),the planet’s semi-major axis a,and Bond albedo A b.We adopt A b=0.5for all our numerical esti-mates;our results only weakly depend on the value of A b unless it approaches unity.The stellar luminosity is de-rived from the mass through the simple mass-luminosity relation L?=(M?/M⊙)3.6L⊙.

The last two parameters needed to determine R o and B u are the planetary rotation rate?and the global kinetic en-ergy scaleˉU.We assume that?=?orb(as determined by radial velocity surveys)in all cases.We allowˉU to vary from50m s?1,the smallest observed value for giant planets in the Solar System(Jupiter),to1000m s?1,a rather large value for which the typical wind speeds in the atmosphere of hot,close-in EGPs approaches the sound speed.A valueˉU=400m s?1is adopted for our?ducial estimate of R o and B u.

4.RESULTS

Estimated values for R o and B u are given in Table1for Solar System giants and close-in EGPs.The values listed for close-in EGPs correspond to the range of min./max. values found given the various assumptions detailed in§3. Fiducial estimates are also reported in?gure1,where solid dots correspond to group1EGPs(safe tidal synchroniza-tion assumption)and open circles to group2EGPs(tidal synchronization assumption less safe).HD209458b is indicated as a star.

It is clear from?gure1that close-in EGPs,as a group, occupy a di?erent region of the R o–B u parameter space than Solar System giants.In particular,they systemat-ically have a Burger number B u>1/9(even when ac-counting for the large range of allowed values;Table1), which indicates that the presence of circumpolar vortices is expected in the radiative region of close-in EGPs within the framework of shallow-water dynamics.The larger val-ues of R o also indicate that generally few bands/jets are expected on these planets(the uncertainty onˉU strongly a?ects this number;see Table1),thus allowing the forma-tion of larger“great spots”(which could also contribute to the variability;Cho et al.2002b).The near alignment of all the points representing close-in EGPs in?gure1shows that the dominant parameter determining their position in this diagram is their rotation rate(R o∝??1;B u∝??2). The small values of R o and B u for Solar System giants re?ect their relatively fast rotation rates.

Although we argued in favor of variable atmospheric sig-natures for close-in EGPs,it is important to note that models do not yet quantitatively predict how much vari-ability is expected.In Cho et al.(2002a,b),we emphasized that the combination ofˉU(unknown)and the amplitude of day-night heating(parametrized in adiabatic simula-tions)determines the contrast of the thermal spots asso-ciated with circumpolar vortices.In the future,diabatic shallow-water models will allow a self-consistent determi-nation of the day-night forcing.More sophisticated mod-els,combined with detailed radiative transfer and chem-istry descriptions(Seager&Sasselov1998;2000;Seager et al.2000),will allow us to make quantitative predic-tions regarding the level of variability expected for various atmospheric signatures.

ACKNOWLEDGMENTS

Support for this work was provided by NASA through Chandra Fellowship grant PF9-10006awarded by the Smithsonian Astrophysical Observatory for NASA under contract NAS8-39073.

REFERENCES

Brown,T.M.2001,ApJ,553,1006

Brown,T.M.,Charbonneau, D.,Gilliland,R.L.,Noyes,R.W.& Burrows,A.2001,ApJ,552,699

Burrows,A.et al.2000,ApJ,534,L97

Chabrier,G.&Bara?e,I.2000,ARA&A,38,337 Charbonneau,D.,Brown,T.M.,Latham,D.W.&Mayor,M.2000, ApJ,529,L45

Charbonneau,D.,Brown,T.M.,Noyes,R.W.&Gilliland,R.L.2002, ApJ,568,377

Cho,J.Y-K.,Menou,K.,Hansen,B.M.S.&Seager,S.2002a,ApJ Lett.,submitted

Cho,J.Y-K.,Menou,K.,Hansen,B.M.S.&Seager,S.2002b,in preparation

Cho,J.Y-K.&Polvani,L.M.1996a,Phys.Fluids,8,1531Cho,J.Y-K.&Polvani,L.M.1996b,Science,273,335

Guillot,T.&Showman,A.P.2002,A&A,385,156

Henry,G.W.,Marcy,G.W.,Butler,R.P.&Vogt,S.S.2000,ApJ, 529,L41

Holton,J.R.1992,‘Introduction to Dynamic Meteorology’( Academic Press,San Diego)

Jha,S.et al.2000,ApJ,540,L45

Mazeh,T.et al.2000,ApJ,532,L55

Pedlosky,J.1987,‘Geophysical Fluid Dynamics’(2nd ed.,Springer-Verlag,New york)

Rhines,P.B.1975,J.Fluid Mech.,69,417

Seager,S.&Sasselov,D.D.,1998,ApJ,502,L157

Seager,S.&Sasselov,D.D.,2000,ApJ,537,916

Seager,S.,Whitney,B.A.,&Sasselov,D.D.2000,ApJ,540,504

8Note that for low values of sin i,some EGPs in Table1have M p>13M Jup and are thus brown dwarfs.We expect shallow-water results to be applicable even in that limit.

4“WEATHER”VARIABILITY

Showman,A.P.&Guillot,T.2002,A&A,385,166Sudarsky,D.,Burrows,A.,&Pinto,P.2000,ApJ,538,885

MENOU ET AL.5

Table1

GLOBAL PLANETARY PARAMETERS

Planet M?P orb a e M p R p g?HˉU R o B u (M⊙)(days)(AU)(M Jup)(m)(m s?2)(s?1)(m)(m s?1)

(1)(2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)

Group2

Group1

NOTES:(1)In order of decreasing orbital period(2)Stellar mass(3)Orbital period(4)Semi-major axis(5)Eccentricity(6) Planet mass(7)Planet radius(8)Surface gravity(9)Rotation rate(10)Atmospheric scale-height(11)Global atmospheric velocity scale(12)Rossby number(13)Burger number

6“WEATHER”VARIABILITY

Fig.1.—Location of Solar System and close-in extrasolar giant planets in the Rossby-Burger space.The assumption of tidal synchronization for extrasolar planets represented by solid circles is the safest(group1;see Table1).HD209458b is indicated by a star.A representative range of possible values around the?ducial estimates for extrasolar giant planets(each individual solid or open circle)is shown as an errorbar (see Table1for details).Formation of circumpolar vortices is expected in the region to the right of the vertical dotted line(B u～>1/9).A larger number of bands is expected for smaller values of R o(see text for details).