A&A 377, 354-360 (2001)
Dpto. Fisica Aplicada I, Escuela Superior de Ingenieros, Universidad del Pais Vasco, Alameda Urquijo s/n, 48013 Bilbao, Spain
Received 12 February 2001 / Accepted 16 July 2001
We present an assessment of the most plausible dynamical regimes operating in the atmospheres of giant extrasolar planets (EGP) and cold ("methane'') brown dwarfs from the available data on a selected group of objects. The most important parameters controlling the atmospheric circulation are the rotation angular velocity and the energy balance between the internal heat source and the star's insolation. The first parameter can be reasonably constrained for some of these objects by theoretical arguments. The second is constrained by the observations. Assuming a hydrogen composition, we discuss possible scenarios for the first order atmospheric motions in terms of characteristic geophysical fluid dynamic numbers and representative time constants. The analysis is applied to the family of extrasolar giant planets classified recently by Sudarsky et al. (2000) according to their effective temperature and Bond albedo. For completeness we extend this study to cold ("methane'') brown dwarfs. Three main dynamical regimes emerge from this analysis: (A) Close EGP ("hot jupiters'') with spin-orbit locked (slowly rotating) planets, have their atmospheres mainly under the star's radiative control. Super-rotating atmospheric motions between the heated and cooled hemispheres can be expected. (B) Atmospheres with their dynamics controlled by both the internal and external energy sources, with Coriolis forces producing zonal motions (Jupiter like objects). (C) Cold brown dwarfs, with motions controlled by the internal heat source (thermally driven turbulent convection) producing intense vertical velocities that dominate the motion field.
Key words: planets and satellites: general - stars: low mass, brown dwarfs - planetary systems
The application of the similarity theory to the atmospheres of giant extrasolar planets and cold brown dwarfs is the subject of this paper. This represents the simplest attempt to understand what kind of motions we can expect in their tropospheres from a comparative point of view. In addition we extend in this paper the work of Golitsyn (1979, 1984) including new parameters relevant to understanding the bulk atmospheric dynamics of a giant planet. Instead, to cover a broad casuistic of the possible ranges of values of the similarity parameters, we apply the theory to a selected group of bodies that we consider representative of the extrasolar planets known at present. The natural classification scheme based on effective temperatures (or basically distances) introduced by Sudarsky et al. (2000) for their albedo study is also adopted here. Application of this theory to other bodies (discovered or to be discovered in the future) is straightforward.
We have selected the objects to be studied from the updated catalog by
Schneider (2001) using the separation from the star as
a guide. For objects at similar distances, we have used as
selection criteria their different masses and ages, and
among them, those that have the best determined properties. For
example, among the "hot jupiters'',
Boo B has a
much larger mass, is strongly irradiated and probably has a higher
internal heat source than 51 Peg B and HD 209458 B. A similar criteria applies also to
more distant planets, for example
And D and
47 UMa B. Table 1 list the basic data for the
|51 Peg B||0.052||4.231||37-85||1.05||1.3||6-10|
|HD 209458B||0.046||3.52433||86.7 0.25||1.1||1.9||5-6|
|v And C||0.82||241.2||60||1.28||3.4||3.3-5.5|
|v And D||2.56||1308.5||155||1.28||3.4|
|47 UMa B||2.09||1084||44 2421||1.05||1.7||6.9-8|
|Eri B||3.3||2502||46 17||0.85||0.3||<1|
|(in )||(in )||(m s-2)||(gr cm-3)||(hr)||(days)|
|51 Peg B||0.49-0.8||1.2-1.4||6-14||0.2-0.6||4-7||4.23 (syn)|
|Boo B||6-8||1.2||121||3.5||2||3.31 (syn)|
|HD 209458B||0.66||1.35||8||0.30||5.3||3.52 (syn)|
|v And C||2.4||1.06||55||2.5||2.1|
|v And D||10.1 4.7||1.1||94-230||4-15||0.8-1.6|
|47 UMa B||3.7 3.31.1||1.1||56-155||2.5-7.3||1.2-2.1|
|Eri B||1.2 0.3||1.1||28||1.1||3|
All objects have the minimum mass measured from the radial velocity curves. Their values in Jovian masses () are: (51 Peg B), 4.14 ( Boo B), 2.11 ( And C), 4.29 ( And D), 2.6 (47 UMa B), 1.2 ( Eridani B).
The masses (M) of the EGP are at present bounded by constraining the inclination of the orbital plane i. For 51 Peg B it is constrained from the lack of transit detection (Henry et al. 1997), and from the observed spectral rotational velocity of the star and its chromospheric emission (François et al. 1996; Henry et al. 1997; Gonzalez 1998). Note that there is, however, some uncertainty in the rotation period of the star (Henry et al. 2000b). For Boo B, i is constrained from the upper limit detection of the light reflected off of this giant planet (Charbonneau et al. 1999; Cameron et al. 2000). Transit photometry precisely constraints i for HD 209458 B (Henry et al. 2000a; Charbonneau et al. 2000; Mazeh et al. 2000; Jha et al. 2000; Brown et al. 2001). The mass of Eridani B is constrained from the tilt observed in its dust disk and from the assumption that this object lies in the disk plane (Hatzes et al. 2000). The mass of And D is constrained from Hipparcos astrometry (Mazeh et al. 1999; Han et al. 2000). Using then the retrieved value of i for And D, we estimate the mass of And C assuming it lies in the same orbital plane. The mass of 47 UMa B is constrained from the star's spectroscopy (Gonzalez 1998) and Hipparcos astrometry (Perryman et al. 1996; Han et al. 2000). The mass of Gl 229 B comes from Leggett et al. (1999).
The radius () of the planet is estimated from the evolution models (Burrows et al. 1997, 2000; Guillot 1999) using the age of the star as determined by different methods (see below). The only exception is the planet HD 209458 B for which a direct measurement of the radius has been performed during transit observations (Charbonneau et al. 2000; Henry et al. 2000a; Mazeh et al. 2000; Jha et al. 2000; Brown et al. 2001). A constraint of the radius of Boo B from the initial tentative detection of the reflected spectra has been given by Cameron et al. (1999) but later modified from a more detailed analysis (Cameron et al. 2000). In fact, the non-detection of the reflected light from this planet puts an upper limit to its radius (Charbonneau et al. 1999) that is consistent with the "inflating'' models by Bodenheimer et al. (2001). Their value in Table 2 is given in terms of the Jovian radius (, 1 km). The acceleration of gravity g in the troposphere and the mean density of the object are derived from these data.
The next parameter, rotation period, is critical for atmospheric
dynamics but unfortunately it has not been measured for any of these
bodies. We can only constraint it in the lower limit by the condition
that the body is not broken apart by the centrifugal force
In Table 3 we present the data for the sources of atmospheric heating and their related temperatures.
|(W m-2)||(W m-2)||(K)||()||(W m-2)||(K)|
|51 Peg B||6.6 105||0.03||1.6 105||1300||5 10-10||3||1300|
|Boo B||1.9 106||0.03||4.6 105||1690||10-7||423||1690|
|HD 209458B||1.2 106||0.03-0.55||1-3 105||1350||10-9||3||1300|
|v And C||6886||0.14||1480||402||6 10-9||34||404|
|v And D||1317||0.4-0.8||36-108||184||3 10-8||151||250|
|47 UMa B||542||0.4-0.8||90||195||10-8||55||210|
|Eri B||38||0.45||5||97||8 10-9||40||168|
|G1 229B||0.013||-||0.002||14||6 10-6||39555||900|
and it is used to retrieve the equilibrium temperature (
being the Stephan-Boltzman constant.
As stated above we assume that the atmospheres are composed of molecular hydrogen. Condensed phases of some compounds are surely present but they are not considered in our dynamical study. Latent heat release from phase changes could be important in controlling some meteorological processes, as for instance occurs in the development of localized large-scale convective storms in the water clouds of Jupiter and Saturn (Sanchez-Lavega & Gomez 1996b; Sanchez-Lavega et al. 1991, 1996a). We assume here that they do not play a main role in controlling the global atmospheric dynamics. This is also assumed for the heating effect due to the hydrogen ortho to para conversion because of the high effective temperatures expected in the atmospheres of the planets under study. The mean molecular weight is taken to be gr mol and the specific heat J kg K . The relevant atmospheric parameters for our study are the following.
The "effective pressure'' marks the level where the hydrogen optical depth is 1. It can be derived from the hydrostatic relation using the ideal gas law and the definition of the optical depth in terms of the hydrogen mass absorption coefficient (see Barnet 1990 for details). For a pure hydrogen atmosphere, it is given by
This velocity can be expected to represent an upper limit for the horizontal wind (mass motion) speeds present in the tropospheres. On the other hand, the most intense expected vertical motions should be those related to convection in the troposphere. They can be characterized by means of the mixing length theory (Spiegel 1971; Stone 1976), so the vertical velocities within the convective layer are given by
and taking the mixing - length scale to be the atmospheric scale
height, we can define a local convective time as
|(bar)||(km)||(m s-1)||(m s-1)||(hr)||(hr)|
|51 Peg B||0.55||407||2555||2||150||9 days||10|
|Boo B||2.5||40||950||10||2||0.8 days||25|
|HD 209458B||0.51||470||2525||2||200||9 days||11|
|v And C||0.7||28||463||3||6||51 days||45|
|v And D||1||6||368||4||1||105 days||59|
|47 UMa B||0.7||7.5||330||3||2||197 days||66|
|Eri B||0.3||23||298||3||6||1.7 years||73|
|G1 229B||3.5||4||2100||37||0.05||2 days||10|
We now present a comparative view of the dynamical state of the
atmospheres of these bodies through a "similarity analysis'' using some
basic non-dimensional numbers that characterize the dynamical state of
the atmosphere (see e.g. Golitsyn 1979).
The "energy balance'' E gives the strength of the internal heat source against the star flux (insolation)
The action of rotation on vertical motions (buoyancy) can be
quantified by a vertical Rossby number
The Golitsyn number Go characterizes the thermal inertia of the
atmosphere or its degree of diabaticity (Golitsyn 1979,
1984; Chamberlain & Hunten 1987),
i.e. the importance of the heating or cooling of the atmosphere by
radiation against dynamics and is given by the ratio
According to the values of the similarity parameters (Table 5),
|51 Peg B||1||10||0.6||0.05|
|v And C||1||>0.05||2||0.04|
|v And D||3.4||>0.02||13||0.02||5 10-2-|
|47 UMa B||2.2||>0.02||6||0.01||3-13||2 10-4|
|Jupiter||1.67||0.4||0.7||7.2 10-4||5.2||5 10-4|
(A) Synchronized close EGP (CEGP or "roasters'' - "Hot Jupiters'', albedo classes IV and V). Their dynamics is controlled by the strong stellar radiation (E = 1, K) and the Coriolis forces have a small influence in shaping the horizontal horizontal motions (Ro >1). The radiative time constant is very short (days) and since Go < 1, the diabatic heat transport by atmospheric motions should have an important role in establishing the thermal state of the atmosphere. One exception is the planet Boo B that has and so its upper troposphere should be dominated by radiative effects alone. It is at deeper tropospheric levels where convection in this object must be very intense (see below).
(B) Intermediate EGP with dynamical properties similar to those of Jupiter (albedo classes I, II and III). The tropospheric dynamics is controlled most probably by both the internal and external heat source (they have moderate values of -9) with the rotation effects being very important in shaping the motions (). The diabatic heat transport should establish small horizontal temperature gradients and the atmosphere should be locally far from radiative equilibrium ( Go << 1).
(C) Cold brown dwarfs, with the dynamics strongly controlled by the internal heat source (E >> 1). Intense vertical motions are expected due to convective heat transport from the interior.
Note that there are many planets where Rov > 1 if one assumes that groups B and C have typically hr, and that the synchronous rotation time characterizes group A. Convective motions should play a more important role in these objects than in Jupiter. It is somewhat surprising to find in particular a high value Ro in two very different objects, Boo B and Gl 229B. The reason is that Boo B is expected to have an intense internal energy source deeper in its atmosphere (at the level where radiation does not penetrate) and thus the vertical motions should dominate the dynamics. Finally, two other ratios permit one to asses the importance of the annual, i.e. seasonal changes, ( ), and the diurnal radiative variations ( ) in the dynamical state of the atmosphere. They are valuable as long as (i.e. when the stellar insolation is important) and when the planet is not synchronized. From the data in Table 5 we can see that this applies to our group B planets, where -13 and -0.0002. This indicates that the tropospheres of these planets would have slow adjustment times to the radiative forcing during the diurnal cycle but could have noticeable adjustments during their annual cycle.
We have presented a first order estimation, through a similarity theory, of the dynamical state of the atmospheres of giant extrasolar planets and cold brown dwarfs. A broad but distinctive dynamical classification scheme emerges from this analysis. Three main families result, each one represented by archetype planets whose properties are at present best measured or constrained. They follow a dynamical classification where E is the main parameter. The other constraint to dynamics is the angular rotation that tends to make the motions two-dimensional or quasi-horizontal as rotation increases.
For group A objects (slowly rotating but strongly irradiated planets), the motions should be driven by the intense winds blowing between the heated and cooled hemispheres. Objects with increasing E but under rapid rotation should develop two dimensional motions in geostrophic balance because of the increasing intensity of Coriolis forces. This is most plausible for group B planets. Based on our experience with Jupiter and Saturn, a system of zonal (East-West) jets varying their intensity with latitude (Limaye 1986; Sanchez-Lavega et al. 2000) could be the main dynamical mode there. Finally, for those cases where E is enormous, as in group C, three-dimensional convective motions driven by the internal energy source should be the main dynamical mode. Even under rapid rotation, the vertical motions are so vigorous in these objects that rotation cannot regulate the visible motion field.
All this tells us that dynamics could play a significant role in establishing the thermal state of the atmosphere of extrasolar planets. Thus, "pure radiative'' equilibrium models, such as those used to infer the cloud vertical structure and albedos, should be handled with care. Future advances in atmospheric circulation studies will require the knowledge of the rotation period of these bodies. In the meantime, the CEGP family, whose rotation state and other properties are well constrained (in particular the planet HD 209458 B), should be the main objective of detailed dynamical modeling (Guillot & Showman 2001).
This work was supported by Spanish MCYT research grant PNAYA2000-0932.