Astronomy
&
Astrophysics
A&A 402, 701–712 (2003)
DOI: 10.1051/0004-6361:20030252
c ESO 2003
Evolutionary models for cool brown dwarfs and extrasolar giant
planets. The case of HD 209458
I. Baraffe1,2 , G. Chabrier1 , T. S. Barman3 , F. Allard1 , and P. H. Hauschildt4
1
2
3
4
C.R.A.L (UMR 5574 CNRS), École Normale Supérieure, 69364 Lyon Cedex 07, France
e-mail: ibaraffe,chabrier,fallard@ens-lyon.fr
Max-Planck-Institut für Astrophysik, Karl-Schwarzschildstr.1, 85748 Garching, Germany
Department of Physics, Wichita State University, Wichita, KS 67260-0032, USA
e-mail: travis.barman@wichita.edu
Hamburger Sternwarte, Gojenbergsweg 112, 21029 Hamburg, Germany
e-mail: phauschildt@hs.uni-hamburg.de
Received 27 November 2002 / Accepted 18 February 2003
Abstract. We present evolutionary models for cool brown dwarfs and extra-solar giant planets. The models reproduce the main
trends of observed methane dwarfs in near-IR color-magnitude diagrams. We also present evolutionary models for irradiated
planets, coupling for the first time irradiated atmosphere profiles and inner structures. We focus on HD 209458-like systems
and show that irradiation effects can substantially affect the radius of sub-jovian mass giant planets. Irradiation effects, however, cannot alone explain the large observed radius of HD 209458b. Adopting assumptions which optimise irradiation effects
and taking into account the extension of the outer atmospheric layers, we still find ∼20% discrepancy between observed and
theoretical radii. An extra source of energy seems to be required to explain the observed value of the first transit planet.
Key words. planetary systems – stars: brown dwarfs – stars: evolution – stars: individual (HD 209458)
1. Introduction
The past decade was marked by two major discoveries in the
field of stellar and planetary physics: the detections of the first
unambiguous brown dwarf (BD) GL 229B (Oppenheimer et al.
1995) and the first extrasolar giant planet (EGP) 51 Peg b
(Mayor & Queloz 1995). The near-IR spectrum of GL 229B
was found to be dominated by strong methane absorption
bands, looking more similar to Jupiter than to late type-stars.
On the other hand, the surprisingly small orbital separation
between 51 Peg b and its parent star suggests that the planet
should be affected by irradiation and that, given the expected
large surface temperature, its atmospheric properties should resemble more the ones of relatively hot brown dwarfs than the
ones of jovian planets.
Since then, about thirty methane dwarfs (or the so-called
T-dwarfs) have been identified, due mainly to the near-IR surveys 2MASS (Burgasser et al. 1999), SDSS (Strauss et al.
1999) and the VLT (Cuby et al. 1999). The radial velocity technique has now revealed more than 100 EGPs in orbit around
nearby stars (see Hubbard et al. 2002 for a review and references therein), with a large fraction (∼10%–20%) being extremely close (less than 0.06 AU) to their parent star. The
mass of substellar companions detected by radial velocimetry extends well above the deuterium burning minimum mass
Send offprint requests to: I. Baraffe,
e-mail: ibaraffe@ens-lyon.fr
0.012 M⊙ = 12MJ (Saumon et al. 1996; Chabrier et al. 2000a).
This mass is often used as the boundary between planets and
brown dwarfs, more for semantical than physical reasons. On
the other hand isolated objects with planetary masses are now
discovered in young stellar clusters, down to a few MJ , as recently reported by Zapatero et al. (2002) in σ Orionis. These
observations suggest that there is an overlap between the mass
range of the least massive brown dwarfs and of the most massive giant planets. In principle, different formation processes
should distinguish planets from brown dwarfs. However, such
a distinction is difficult to characterize in terms of atmospheric,
structural and cooling properties since both types of objects
have convective interiors with essentially a metallic H/He mixture. The signature of a central rock+ice core, like in solar giant planets, would be the clear identification of a planet. The
presence of a core can affect the radius of a planet, yielding a
smaller planetary radius than in the absence of a core. For 1 MJ ,
the effect is about 5% on the radius for a core mass <0.06 MJ
(see Saumon et al. 1996). The presence of this core can be inferred from the accurate characterization of the gravitational
moments of the object, and such an observation is currently not
feasible for EGPs. In addition, both giant planets and brown
dwarfs have atmospheres dominated by molecular absorption
and the effects of cloud formation. Although frustrating from
an observational point of view, these similarities imply that the
general cooling theory for BDs, involving detailed models of
the atmosphere and inner structures, can be applied to EGPs.
702
I. Baraffe et al.: Evolutionary models for cool brown dwarfs
In terms of cooling properties, this general theory can even
be applied to Jupiter, as emphasized by Hubbard et al. (2002).
Additional observational constraints, as provided by spacecraft
encounters or by direct probes (e.g. Galileo), have lead to refinements of the models (heavy element core, non-standard
chemical composition). As mentioned above, such constraints
are, unfortunately, far from being accessible for EGPs.
Much effort has been devoted to the modeling of substellar
objects during the past decade, improving our understanding of
cool atmospheres (see Allard et al. 1997 for a review), of the
role of dust (Tsuji et al. 1999; Burrows et al. 2000; Ackerman &
Marley 2001; Allard et al. 2001; Marley et al. 2002), of irradiation (Saumon et al. 1996; Seager & Sasselov 1998; Sudarsky
et al. 2000; Barman et al. 2001), and of their inner structure and
evolutionary properties (Burrows et al. 1997; Chabrier et al.
2000b; see Chabrier & Baraffe 2000 for a review). One remaining major challenge in the theory is the description of dynamical processes of grain formation and diffusion necessary to understand the transition between L-dwarfs and T-dwarfs, which
is expected to take place at T eff ∼ 1300 K–1700 K. The former objects are better reproduced by dusty atmosphere models,
whereas the later are better reproduced by dust-free (or partly
dusty) models. The recent observations of L/T dwarfs at the
transition clearly indicate that complex processes take place in
the atmosphere of these objects (see e.g. Burgasser et al. 2002).
Another important challenge is the modeling of irradiation effects, which are expected to affect the spectra of close-in EGPs,
and may also affect their inner structure and cooling properties. The recent discovery of the planet HD 209458b transiting
its parent star (Charbonneau et al. 2000) provides a unique test
to explore such effects, since its mass and radius can be determined with high accuracy from the modeling of the transit
lightcurve. According to the most recent determination (Cody
& Sasselov 2002), the mass and radius of the planet are estimated to be m = 0.69 ± 0.02MJ and R = 1.42+0.10
−0.13 RJ .
Evolutionary models including crude estimates of the effects of irradiation on planet atmospheres suggest that extrinsic
heating is sufficient to maintain a larger planetary radius compared to an isolated planet. It has thus been argued that irradiation could explain the large radius of HD 209458b (Guillot
et al. 1996; Burrows et al. 2000). More recently, Guillot &
Showman (2002) questioned such results and argue that the
radius of HD 209458b can only be reproduced if the deep atmosphere is much hotter than what can be expected from irradiation effects. However, none of these calculations includes a
consistent treatment between the irradiated atmospheric structure and the interior structure of the planet. Such a consistent
treatment is mandatory to get reliable results since the deep
interior entropy profile, which determines the heat content of
the planet to be radiated away while it cools, is affected by the
modification of the atmospheric temperature profile due to the
incoming external heat flux. The main goal of the present paper is to present the first such consistent calculations. As mentioned above, in the case of non-irradiation, these calculations
apply to the evolution of cool (dust-free like) brown dwarfs,
i.e. T-dwarfs, and extrasolar giant planets far enough from the
parent star for the irradiation effects on the thermal structure
of the planet to be negligible. This is the case of the solar
giant planets, the cooling of which is simply characterized by
the cooling properties of the “isolated” planet plus the heating
contribution from the Sun 4πσR2p T ⊙4 , where Rp is the radius of
the planet and T ⊙ represents the equivalent black body temperature of the converted solar radiation (Hubbard 1977; Guillot
et al. 1995). In Sect. 2 we briefly present the input physics of
non-irradiated models, describing methane dwarfs and isolated
EGPs. Apart from the impinging stellar flux, the same input
physics are used to analyse the effects of irradiation. The effects on the radius and cooling properties of giant planets are
described in Sect. 3 and results are compared to the observed
properties of HD 209458b. Discussion follows in Sect. 4.
2. Non-irradiated models
2.1. Model description
The main input physics involved in the present models are the
same as described in our previous works (Chabrier & Baraffe
1997; Baraffe et al. 1998; Chabrier et al. 2000b). The models
are based on the coupling between interior and non-grey atmosphere structures. The treatment of dust in the atmosphere
is described in detail in Allard et al. (2001), with two limiting
cases of dust treatment. The first case, referred to as “DUSTY”,
takes into account the formation of dust in the equation of state,
and its scattering and absorption in the radiative transfer equation. Such models assume that dust species remain where they
form, according to the chemical equilibrium conditions. The
second case, referred to as “COND”, neglects dust opacity in
the radiative transfer equation. In a previous paper (Chabrier
et al. 2000b), we presented the evolutionary models based on
DUSTY atmosphere models, aimed at describing the evolution and the photometric and spectroscopic properties of early
L-dwarfs. The present paper is devoted to evolutionary models based on the COND approach, which are more appropriate to objects with effective temperatures T eff <
∼ 1300 K, such
as methane dwarfs or EGPs at large orbital separation. These
models apply when all grains have gravitationally settled below the photosphere. A forthcoming paper will be devoted to
models taking into account characteristic diffusion timescales
of different processes affecting the dust stratification (e.g. coagulation, gravitational settling, convection). These models aim
at describing in particular the transition objects between late
L-dwarfs and early T-dwarfs (see Allard 2002).
2.2. Evolution of cool brown dwarfs
A preliminary version of the COND models was presented in
Chabrier et al. (2000b) down to 0.01 M⊙ . In the present paper,
we extend our calculations to T eff = 100 K and m = 0.5 MJ .
The evolution of L and T eff as a function of time for different masses is displayed in Fig. 1. The properties of the COND
models for different ages are given in Tables 1–5. As already
stressed in Chabrier et al. (2000b), the treatment of dust in the
atmospheric models barely affects the evolutionary tracks in
terms of L and T eff as a function of time for a given mass (see
Fig. 2 of Chabrier et al. 2000b). Consequently, although the
COND models are more appropriate to describe the spectral
I. Baraffe et al.: Evolutionary models for cool brown dwarfs
703
LHS102B
SDSS1254
GL86
GL570D
Fig. 1. Evolution of the luminosity L and effective temperature T eff as
a function of time (in yr) for different masses. Substellar objects are
indicated by solid lines and stars by short-dashed lines. The horizontal
long-dashed line indicates the limit (T eff ∼ 1300 K) below which the
COND models are appropriate for the photometric and spectroscopic
description of T-dwarfs and EGPs (see text).
<
and photometrical properties of substellar objects with T eff ∼
1300 K, they provide a good description of their cooling properties even at higher T eff . In other words, it is not necessary
to perform evolutionary calculations with the DUSTY models
above T eff ∼ 1300 K and switch to the COND models below.
An object characteristic of the present models was recently discovered by Zapatero et al. (2002) in σ Orionis: S Ori 70. From
a comparison of its observed spectrum with COND synthetic
spectra (Allard et al. 2001), Zapatero et al. (2002) estimate
an effective temperature T eff ∼ 700–1000 K. If the membership of S Ori 70 to σ Orionis is confirmed, implying an age
<10 Myr, its mass mass should be <
∼5 MJ (see Fig. 1).
Several methane dwarfs have been discovered in the solar neighbourhood, implying older ages and thus larger masses
than the extreme case of S Ori 70. At an age of 108 yr, only objects with masses below the deuterium burning minimum mass
(m ≤ 0.012 M⊙ ) have T eff <
∼ 1300 K, whereas at 5 Gyr, it
is the case for all substellar objects with m <
∼ 0.06 M⊙ (see
Fig. 1 and Tables 1 and 4). Photometric observations and parallax determinations of several L- and T-dwarfs (Els et al. 2001;
Leggett et al. 2002a; Dahn et al. 2002) now allow a comparison with models in observational color-magnitude diagrams
(CMD), providing stringent constraints on theoretical models
(see Figs. 2–4).
Fig. 2. Color – Magnitude diagram (J − K) – MK . Observations are
taken from Leggett (1992) (mostly for M-dwarfs) and Dahn et al.
(2002). Also shown: LHS 102B (Goldman et al. 1999), GL86 (Els
et al. 2001). M-dwarfs are shown by dots, L-dwarfs by filled squares
and T-dwarfs by triangles. DUSTY isochrones (Chabrier et al. 2000b)
are displayed in the upper right part of the figure, for different ages,
as indicated. The COND isochrones are displayed in the left part of
the figure. Some masses (in M⊙ ) and their corresponding T eff are indicated on the 1 Gyr isochrones by open squares (COND) and open
circles (DUSTY). The names of two L/T transition objects and of the
faintest T-dwarf known with parallax are indicated.
As already noticed in Allard et al. (1996) for GL 229B,
models free of atmospheric dust clouds better reproduce the
near-IR photometric and spectral properties of methane dwarfs.
This is illustrated in (J − K) and (K − L′ ) colors in Figs. 2
and 3 respectively, where the COND models reproduce the
main observed trends. In Fig. 2, we note the two transition objects, intermediate between L- and T- dwarfs, with (J − K) ∼ 1
(GL 86B: Els et al. 1999; SDSS 1254-01: Leggett et al. 2002a,
Dahn et al. 2002) and the faintest L-dwarfs (Dahn et al. 2002),
which are not described by either the DUSTY or COND limiting cases, and require a detailed treatment of dust diffusion in
the atmosphere, as mentioned in Sect. 2.1.
The predictions of the COND models provide a general
good agreement with observed near-IR photometry and spectra at wavelength >1 µm (Leggett et al. 2002b; Zapatero et al.
2002). The models show however shortcomings at shorter
704
I. Baraffe et al.: Evolutionary models for cool brown dwarfs
Table 1. COND isochrones for 0.1 Gyr.
m/M⊙
0.0005
0.0010
0.0020
0.0030
0.0040
0.0050
0.0060
0.0070
0.0080
0.0090
0.0100
0.0120
0.0150
0.0200
0.0300
0.0400
0.0500
0.0600
0.0700
0.0720
0.0750
0.0800
0.0900
0.1000
T eff
240.
309.
425.
493.
563.
630.
688.
760.
816.
886.
953.
1335.
1399.
1561.
1979.
2270.
2493.
2648.
2762.
2782.
2809.
2846.
2910.
2960.
log L/L⊙
−7.418
−6.957
−6.383
−6.112
−5.880
−5.686
−5.534
−5.365
−5.246
−5.103
−4.978
−4.332
−4.281
−4.110
−3.668
−3.386
−3.167
−3.008
−2.879
−2.856
−2.821
−2.776
−2.689
−2.617
R/R⊙
0.114
0.117
0.120
0.121
0.122
0.122
0.121
0.121
0.120
0.120
0.120
0.129
0.124
0.122
0.126
0.132
0.141
0.150
0.160
0.162
0.166
0.170
0.180
0.189
log g
3.020
3.300
3.580
3.746
3.869
3.965
4.048
4.117
4.180
4.232
4.279
4.297
4.424
4.569
4.715
4.797
4.837
4.863
4.874
4.875
4.875
4.880
4.884
4.887
MV
41.98
32.58
29.69
28.71
28.09
27.65
27.36
27.03
26.77
26.45
26.10
23.53
23.30
22.30
19.96
18.46
17.09
16.08
15.33
15.20
15.01
14.77
14.34
14.02
MR
37.51
28.68
25.62
24.48
23.77
23.25
22.92
22.55
22.28
21.96
21.66
19.44
19.24
18.55
16.80
15.63
14.77
14.12
13.59
13.50
13.36
13.18
12.85
12.58
MI
34.00
25.89
22.79
21.66
20.95
20.44
20.09
19.74
19.49
19.19
18.92
16.79
16.46
16.08
14.48
13.31
12.53
12.01
11.60
11.53
11.42
11.29
11.03
10.82
MJ
28.42
22.43
20.05
18.88
17.95
17.23
16.71
16.16
15.76
15.32
14.94
13.20
13.05
12.60
11.52
10.89
10.43
10.10
9.82
9.77
9.69
9.60
9.40
9.24
MH
26.59
22.38
19.76
18.57
17.71
17.02
16.51
16.01
15.65
15.23
14.86
12.97
12.82
12.34
11.20
10.52
10.02
9.68
9.39
9.34
9.26
9.16
8.96
8.80
MK
37.66
29.11
23.13
20.88
19.35
18.15
17.26
16.38
15.79
15.16
14.69
12.76
12.65
12.17
10.90
10.19
9.71
9.37
9.10
9.05
8.97
8.87
8.68
8.52
M L′
19.57
17.41
15.94
15.21
14.59
14.06
13.67
13.26
12.97
12.63
12.34
10.90
10.83
10.53
9.82
9.39
9.04
8.78
8.55
8.51
8.44
8.36
8.19
8.05
MM
17.64
15.69
14.55
13.93
13.50
13.14
12.83
12.55
12.35
12.13
11.96
11.17
11.15
10.99
10.38
9.84
9.37
9.03
8.75
8.70
8.63
8.53
8.35
8.19
Notes: T eff is in K, the gravity g in cgs. The VRI magnitudes are in the Johnson-Cousins system (Bessell 1990), JHK in the CIT system (Leggett
1992), L′ in the Johnson-Glass system and M in the Johnson system.
Table 2. Same as Table 1 for 0.5 Gyr.
m/M⊙
0.0005
0.0010
0.0020
0.0030
0.0040
0.0050
0.0060
0.0070
0.0080
0.0090
0.0100
0.0120
0.0150
0.0200
0.0300
0.0400
0.0500
0.0600
0.0700
0.0720
0.0750
0.0800
0.0900
0.1000
T eff
141.
203.
272.
322.
370.
409.
449.
488.
525.
564.
599.
759.
791.
936.
1264.
1583.
1875.
2116.
2329.
2369.
2426.
2518.
2680.
2804.
log L/L⊙
−8.415
−7.753
−7.218
−6.913
−6.670
−6.496
−6.340
−6.200
−6.080
−5.963
−5.864
−5.447
−5.404
−5.133
−4.636
−4.255
−3.955
−3.729
−3.534
−3.498
−3.445
−3.356
−3.189
−3.047
R/R⊙
0.105
0.109
0.112
0.113
0.114
0.113
0.113
0.112
0.111
0.110
0.110
0.110
0.107
0.104
0.101
0.100
0.101
0.102
0.106
0.107
0.108
0.111
0.119
0.128
log g
3.097
3.365
3.639
3.805
3.928
4.027
4.112
4.185
4.249
4.307
4.358
4.432
4.557
4.704
4.905
5.040
5.131
5.194
5.233
5.238
5.244
5.248
5.241
5.223
MV
56.30
47.57
37.05
32.02
30.65
29.60
29.16
28.71
28.40
28.14
27.91
27.20
27.11
26.53
24.97
23.11
21.31
19.99
18.84
18.60
18.25
17.65
16.54
15.68
wavelength, with a flux excess around 0.8–0.9 µm, characteristic of the I-bandpass. This problem is illustrated in Fig. 4
in a (I − J) – MJ CMD, where the COND models predict
MR
51.03
42.88
33.00
28.23
26.73
25.57
25.05
24.51
24.14
23.82
23.53
22.73
22.63
22.07
20.77
19.28
17.94
16.91
16.04
15.89
15.66
15.27
14.56
13.96
MI
46.60
38.99
30.06
25.75
24.16
22.94
22.39
21.80
21.41
21.07
20.77
19.91
19.81
19.27
17.95
16.74
15.53
14.50
13.68
13.54
13.33
13.00
12.43
11.98
MJ
37.42
31.61
25.07
22.05
21.01
20.20
19.64
19.10
18.65
18.21
17.80
16.38
16.20
15.33
13.90
12.92
12.21
11.69
11.27
11.19
11.08
10.89
10.55
10.25
MH
33.07
29.15
24.62
22.27
21.06
20.11
19.51
18.91
18.46
18.04
17.66
16.29
16.16
15.34
13.87
12.76
11.95
11.36
10.90
10.81
10.69
10.49
10.12
9.80
MK
51.62
43.23
34.02
29.03
26.45
24.54
23.10
21.68
20.74
19.95
19.22
16.82
16.58
15.37
13.69
12.68
11.79
11.13
10.63
10.54
10.42
10.22
9.85
9.54
M L′
23.09
20.93
18.66
17.52
16.85
16.32
15.92
15.52
15.19
14.88
14.59
13.51
13.41
12.78
11.70
10.95
10.44
10.10
9.81
9.75
9.67
9.53
9.25
9.00
MM
20.59
18.68
16.58
15.53
15.05
14.67
14.36
14.06
13.83
13.63
13.43
12.68
12.60
12.21
11.68
11.33
10.99
10.63
10.26
10.19
10.08
9.89
9.53
9.22
significantly bluer (I − J) colors than observations. As mentioned in Allard et al. (2001), uncertainties in the current
treatment of the far wings of the absorption lines of alkali
I. Baraffe et al.: Evolutionary models for cool brown dwarfs
705
Table 3. Same as Table 1 for 1 Gyr.
m/M⊙
0.0005
0.0010
0.0020
0.0030
0.0040
0.0050
0.0060
0.0070
0.0080
0.0090
0.0100
0.0120
0.0150
0.0200
0.0300
0.0400
0.0500
0.0600
0.0700
0.0720
0.0750
0.0800
0.0900
0.1000
T eff
111.
160.
226.
270.
304.
342.
377.
403.
438.
464.
491.
578.
628.
766.
1009.
1271.
1543.
1801.
2082.
2140.
2234.
2383.
2627.
2784.
log L/L⊙
−8.851
−8.185
−7.560
−7.244
−7.031
−6.831
−6.664
−6.556
−6.417
−6.325
−6.235
−5.955
−5.835
−5.514
−5.071
−4.696
−4.374
−4.106
−3.829
−3.772
−3.679
−3.527
−3.268
−3.083
R/R⊙
0.102
0.106
0.109
0.111
0.111
0.110
0.110
0.109
0.108
0.107
0.107
0.106
0.103
0.100
0.096
0.093
0.092
0.092
0.094
0.095
0.098
0.102
0.113
0.125
log g
3.115
3.386
3.662
3.827
3.950
4.051
4.134
4.208
4.272
4.331
4.383
4.467
4.587
4.736
4.948
5.099
5.211
5.291
5.333
5.336
5.334
5.323
5.285
5.246
MV
60.75
54.15
44.39
37.64
32.62
31.58
30.53
29.77
29.37
29.06
28.74
28.09
27.86
27.31
26.40
25.19
23.73
22.13
20.44
20.12
19.59
18.67
16.98
15.86
MR
55.23
49.10
39.91
33.60
28.93
27.79
26.63
25.79
25.31
24.94
24.55
23.75
23.47
22.85
21.96
20.99
19.81
18.59
17.31
17.05
16.63
15.96
14.86
14.09
MI
50.50
44.69
36.34
30.73
26.60
25.36
24.12
23.26
22.73
22.33
21.89
21.01
20.72
20.05
19.15
18.13
17.15
16.10
14.87
14.61
14.21
13.61
12.67
12.09
MJ
40.19
35.58
29.28
25.29
22.49
21.71
20.96
20.41
19.93
19.59
19.23
18.15
17.70
16.56
15.10
14.04
13.21
12.56
11.93
11.80
11.60
11.26
10.72
10.33
MH
35.07
32.06
27.80
24.99
22.91
21.98
21.07
20.43
19.89
19.49
19.07
18.03
17.62
16.55
15.16
14.04
13.12
12.36
11.62
11.48
11.25
10.89
10.30
9.89
MK
55.87
49.17
40.31
34.51
30.25
28.28
26.39
25.04
23.77
22.85
21.90
19.86
19.05
17.19
15.14
13.90
13.04
12.27
11.43
11.27
11.02
10.63
10.03
9.62
M L′
24.15
22.40
20.18
18.84
17.91
17.38
16.87
16.54
16.18
15.93
15.65
14.88
14.56
13.72
12.67
11.88
11.25
10.77
10.32
10.23
10.08
9.84
9.40
9.07
MM
21.49
19.95
17.94
16.66
15.73
15.36
15.01
14.76
14.50
14.31
14.11
13.60
13.38
12.80
12.15
11.80
11.53
11.26
10.85
10.75
10.58
10.27
9.72
9.31
R/R⊙
0.105
0.105
0.105
0.105
0.104
0.103
0.103
0.102
0.101
0.100
0.098
0.095
0.090
0.085
0.082
0.079
0.081
0.083
0.089
0.099
0.113
0.125
log g
3.698
3.868
3.994
4.095
4.180
4.254
4.318
4.376
4.429
4.519
4.634
4.786
5.011
5.179
5.313
5.418
5.466
5.453
5.411
5.353
5.289
5.247
MV
60.05
55.32
50.68
46.50
42.71
39.29
36.31
33.73
33.05
31.58
30.20
29.24
28.17
27.58
27.09
26.44
24.33
23.11
21.03
19.11
17.02
15.85
MR
54.63
50.15
45.80
41.92
38.43
35.29
32.57
30.22
29.47
27.86
26.35
25.17
23.83
23.15
22.63
22.03
20.36
19.36
17.80
16.28
14.88
14.09
MI
49.68
45.63
41.76
38.30
35.18
32.37
29.93
27.82
27.06
25.46
24.00
22.66
21.17
20.40
19.82
19.20
17.60
16.75
15.34
13.89
12.70
12.09
MJ
38.16
34.96
32.14
29.83
27.80
26.02
24.50
23.21
22.74
21.77
20.85
19.89
18.34
17.04
15.94
15.01
13.52
12.97
12.20
11.44
10.73
10.33
MH
34.52
32.46
30.52
28.84
27.34
26.00
24.85
23.85
23.30
22.17
21.14
19.92
18.31
17.11
16.05
15.13
13.50
12.85
11.92
11.07
10.31
9.89
MK
53.62
49.21
45.15
41.71
38.67
35.96
33.65
31.64
30.49
28.16
26.01
23.31
20.04
17.99
16.38
15.15
13.44
12.80
11.78
10.82
10.04
9.62
M L′
23.33
22.22
21.28
20.48
19.78
19.16
18.64
18.18
17.91
17.41
16.96
16.18
15.04
14.17
13.38
12.73
11.60
11.16
10.54
9.97
9.42
9.07
MM
20.80
19.81
18.92
18.16
17.49
16.88
16.36
15.91
15.71
15.32
14.97
14.45
13.71
13.13
12.63
12.27
11.77
11.54
11.06
10.43
9.73
9.31
Table 4. Same as Table 1 for 5 Gyr.
m/M⊙
0.0020
0.0030
0.0040
0.0050
0.0060
0.0070
0.0080
0.0090
0.0100
0.0120
0.0150
0.0200
0.0300
0.0400
0.0500
0.0600
0.0700
0.0720
0.0750
0.0800
0.0900
0.1000
T eff
129.
162.
193.
220.
244.
265.
284.
301.
322.
361.
399.
473.
610.
760.
931.
1120.
1524.
1712.
2006.
2320.
2622.
2785.
log L/L⊙
−8.570
−8.166
−7.867
−7.644
−7.469
−7.328
−7.217
−7.124
−7.015
−6.823
−6.671
−6.401
−6.008
−5.670
−5.353
−5.058
−4.504
−4.278
−3.942
−3.603
−3.275
−3.083
elements (Na, K) at such pressures may be responsible for this
discrepancy. No theory, however, exists to date for an accurate description of broadening of atomic lines by collisions
with H2 and He. Attempts to improve current treatments are
under progress (Burrows & Volobuyev 2002).
The correct trend of colors and spectral properties predicted
by the present models at wavelength >1 µm, where most of the
flux is emitted for the concerned range of T eff , comfort us however with their reliability to describe extremely cool objects.
706
I. Baraffe et al.: Evolutionary models for cool brown dwarfs
Table 5. Same as Table 1 for 10 Gyr.
m/M⊙
0.0030
0.0040
0.0050
0.0060
0.0070
0.0080
0.0090
0.0100
0.0120
0.0150
0.0200
0.0300
0.0400
0.0500
0.0600
0.0700
0.0720
0.0750
0.0800
0.0900
0.1000
T eff
125.
149.
172.
193.
213.
232.
249.
265.
293.
330.
389.
504.
634.
776.
941.
1289.
1556.
1997.
2322.
2624.
2786.
log L/L⊙
−8.629
−8.325
−8.087
−7.888
−7.724
−7.584
−7.469
−7.368
−7.204
−7.016
−6.759
−6.358
−6.004
−5.695
−5.393
−4.832
−4.472
−3.954
−3.602
−3.274
−3.082
R/R⊙
0.104
0.104
0.103
0.102
0.102
0.101
0.100
0.099
0.098
0.096
0.093
0.088
0.083
0.079
0.076
0.078
0.081
0.089
0.099
0.113
0.125
log g
3.879
4.006
4.109
4.195
4.270
4.335
4.393
4.445
4.536
4.650
4.802
5.029
5.200
5.338
5.450
5.503
5.481
5.415
5.353
5.289
5.246
MV
61.46
58.00
54.62
51.29
48.17
45.19
42.49
39.95
35.23
32.81
30.75
28.98
28.18
27.64
27.20
25.69
24.17
21.10
19.10
17.01
15.85
MR
55.89
52.62
49.46
46.38
43.50
40.77
38.30
35.99
31.68
29.25
26.99
24.87
23.85
23.21
22.74
21.45
20.22
17.85
16.27
14.88
14.09
MI
50.79
47.86
45.05
42.30
39.72
37.28
35.07
33.00
29.12
26.80
24.64
22.35
21.19
20.47
19.93
18.60
17.48
15.39
13.89
12.69
12.08
MJ
38.29
36.01
34.06
32.23
30.55
29.00
27.62
26.34
24.04
22.70
21.28
19.72
18.33
17.10
16.04
14.37
13.44
12.23
11.43
10.73
10.32
MH
35.04
33.56
32.18
30.85
29.62
28.46
27.41
26.44
24.64
23.33
21.73
19.68
18.33
17.18
16.18
14.43
13.41
11.95
11.06
10.31
9.88
MK
54.28
51.11
48.26
45.53
43.01
40.66
38.55
36.59
33.06
30.46
27.02
22.59
19.99
18.07
16.55
14.36
13.36
11.81
10.82
10.04
9.62
M L′
23.36
22.59
21.92
21.29
20.71
20.18
19.70
19.26
18.48
17.93
17.21
16.04
15.03
14.23
13.50
12.27
11.55
10.56
9.97
9.41
9.07
MM
20.89
20.19
19.57
18.97
18.41
17.89
17.42
16.98
16.20
15.76
15.19
14.37
13.75
13.21
12.73
12.08
11.75
11.08
10.43
9.73
9.30
3. Irradiated models
3.1. Effect on atmosphere structure
As mentioned in the introduction, a non-negligible fraction
of EGPs orbit close to their parent star and their thermal
and mechanical structure is affected by irradiation effects.
Therefore, a general theory of cool substellar objects must take
these effects into account. Recently Barman et al. (2001) have
modeled irradiated atmospheres by including the impinging radiation field in the solution of the radiative transfer equation.
As shown by these authors, for a given intrinsic luminosity,
non-irradiated planets have very different temperature structures than irradiated planets. Thus, substituting non-irradiated
atmospheric structures with T eff = T eq (see definition below,
Eq. (8)) for irradiated structures, as done up to now in the literature, yields incorrect inner boundary conditions for evolutionary calculations (see e.g. Fig. 13 of Barman et al. 2001). Given
the present lack of an accurate treatment of atmospheric dust
diffusion, the calculations were performed only for the DUSTY
and COND limit cases, respectively. The results emphasize the
strong dependence of the emergent spectrum and atmospheric
structure on the presence or absence of dust. In the absence of
dust, the impinging flux can penetrate in deeper layers of the
planet atmosphere, affecting more drastically the inner structure of the planet than in the dusty case.
Except for a possible detection of sodium absorption in
the atmosphere of HD 209458b (Charbonneau et al. 2002),
no constraints on the atmospheric composition of EGPs are
available at the present time. The only strong observational
constraint available for irradiated models is the transit planet
HD 209458b. The determination of its mass and radius provides a stringent test to irradiated atmosphere calculations and
to the resulting structure and evolution. We thus apply our calculations of irradiated EGPs to HD 209458-like systems.
Fig. 3. Color – Magnitude diagram (K − L′ ) – MK . K is in
the MKO-NIR system. Observations are from Leggett et al. (2002a).
Symbols are the same as in Fig. 2. A DUSTY isochrone of 1 Gyr
(Chabrier et al. 2000b) is indicated by the long-dashed line. The
COND isochrones are displayed for 0.1 Gyr (dash), 1 Gyr (solid)
and 10 Gyr (dash-dot). Some masses (in M⊙ ) and their corresponding T eff are indicated on the 1 Gyr COND isochrone by open squares.
I. Baraffe et al.: Evolutionary models for cool brown dwarfs
707
directions µ (µ = cos θ, where θ is the angle of incidence).
Assuming there is no extra source or sink of energy (e.g. no
horizontal energy transfer), energy conservation implies that
all the incident energy coming in must go out. Therefore, in
the case of irradiation, the in-coming flux from the parent star
cancels out the extra out-going, absorbed and reradiated flux
due to the heating of the upper layers of the planet atmosphere
(see Fig. 5). The in-coming flux at the surface is Fin = −Finc
4
,
and the out-going flux at the surface is Fout = Finc + σT eff
4
4
where σT eff defines the intrinsic, unperturbed flux σT eff of the
initial, non-irradiated atmosphere structure. Energy conservation thus implies:
4
.
Fnet = Fout + Fin = σT eff
(2)
The non-irradiation case (Finc = 0) corresponds to the usual
4
.
condition Fnet = Fout = σT eff
Our atmosphere models, irradiated or not, are thus characterized by the parameters T eff and g. Of course, the same net
flux Fnet corresponds to two different atmospheric structures,
in the non-irradiated and irradiated case, because of the extra
energy source Fin , 0 in the latter case (see Fig. 5). Given
the above definitions, the net flux characterizes the intrinsic
luminosity, i.e. the rate of energy released by the planet as it
contracts and cools down:
Z
dS
2
4
−T
Lint = 4πRp σT eff =
dm.
(3)
dt
Fig. 4. Color – Magnitude diagram (I − J) – MJ . Observations are
from Leggett (1992) and Dahn et al. (2002). Symbols and curves are
the same as in Fig. 3.
We have computed a grid of irradiated atmosphere models based on the COND input physics described in Sect. 2, as
in Barman et al. (2001). Although more appropriate for EGPs
with T eff <
∼ 1300 K, the COND models maximise the effect
of irradiation on the inner atmosphere structure and thus on
the evolution of EPGs (Barman et al. 2001). The grid covers a
wide range of T eff from 40 K to 100 K, in steps of 20 K, and
from 100 K to 2800 K, in steps of 100 K. It covers a range
of surface gravities from log g = 2.5 to log g = 4.5, in steps
of 0.5 dex. We adopt the characteristics of HD 209458, assuming for the primary an effective temperature T eff ⋆ = 6000 K,
a radius R⋆ = 1.18 R⊙ (Mazeh et al. 2000; Cody & Sasselov
2002) and an orbital separation a = 0.046 AU (Charbonneau
et al. 2000). As in Barman et al. (2001), we make the simplifying assumptions that the impinging radiation field is isotropic
and the incident flux Finc is redistributed only over the dayside, i.e.
1 R⋆ 2
Finc =
F⋆ ,
(1)
2 a
where F⋆ is the total flux from the primary (see discussion
in Sect. 4).
Before proceeding any further, we briefly re-specify definitions of fluxes (see e.g. Brett & Smith 1993), since use of various terminologies leads to confusion. In all cases, the integrated
net flux Fnet , obtained from the solution of the transfer equation,
is the intensity integrated over both in-coming and out-going
This quantity determines the cooling properties of the planet
for a given set of outer boundary conditions provided by the
atmospheric profile (see Sect. 3.2 below). We stress that, in the
case of irradiation, T eff does not characterize the total flux emitted by the planet, which is given by:
1 R⋆ 2
4
4
+ Finc = σT eff
+
F⋆ .
(4)
Fout = σT eff
2 a
Note that Fout is the important quantity for observers, since it
characterises the total radiation of the planet, including both
thermal and reflected parts of the flux. However, we do not focus on this quantity, since a forthcoming paper will be devoted
to spectral properties of irradiated planets (Barman et al. 2003,
in preparation).
For the sake of comparison with non-irradiated atmosphere
profiles (see Barman et al. 2001), we also define the quan4
emitted
tity T therm which characterises the thermal flux σT therm
by the irradiated fraction of the planet (in the present case, the
day side only). This quantity reads:
4
4
= σT eff
+ (1 − A)Finc ,
σT therm
(5)
where A is the Bond albedo. According to the definitions above:
4
+ Frefl ,
Fout = σT therm
(6)
where
Frefl = AFinc
(7)
is the reflected part of the incident flux.
Within the conditions of the present calculations (Eq. (1),
a = 0.046, T eff ⋆ = 6000 K, R⋆ = 1.18 R⊙ ), our Bond albedo
708
I. Baraffe et al.: Evolutionary models for cool brown dwarfs
is close to 0.1 for the coolest models (T eff ∼ 100 K). A final
quantity, often used in the literature, is the equilibrium temperature, T eq , which characterizes the planet’s luminosity after
having exhausted all its internal heat content (see e.g. Guillot
et al. 1996; Saumon et al. 1996):
4
=
T eq
(1−A)
σ F inc
=
(1−A)
2σ
( Ra⋆ )2 F⋆
(8)
4
→ T therm
when T eff → 0.
4
Note that given our definition of Finc , T eq
defined by Eq. (8)
differs by a factor 2 from the definition usually used in the literature, because of the redistribution only over the day side.
Note also that T eq and T therm differ significantly at young ages,
when the intrinsic flux of the planet is not negligible.
The effect of irradiation on atmosphere structures is illustrated in Fig. 5 for different values of the effective temperature T eff . As already stressed in Barman et al. (2001), an irradiated structure characterised by T therm can differ significantly
from a non-irradiated structure at the same effective temperature T eff = T therm . This point (see also Seager & Sasselov 1998;
Guillot & Showman 2002) emphasizes the fact that adopting
outer boundary conditions, for evolutionary calculations, from
atmospheric profiles of nonirradiated models with T eff = T therm ,
or T eff = T eq (as e.g. Burrows et al. 2000), is incorrect and yields
erroneous evolutionary properties for irradiated objects.
3.2. Effect on evolution
The main effect of irradiation on convective atmospheres and
its consequences on evolution is well known (see Hubbard
1977; Brett & Smith 1993; Guillot et al. 1996; Hubbard et al.
2002). The heating of the outer layers by the incident flux reduces the temperature gradient between these layers and the
interior. They become radiative and the top of the convective zone is displaced to larger depths compared to the
non-irradiated case, as clearly illustrated in Fig. 5. The inner
atmosphere structure is hotter at a given pressure than the nonirradiated atmosphere model of same T eff (see Fig. 5). In order
to match the same inner entropy, or the same values of P and T ,
characteristic of the boundary layer between the interior structure and the irradiated atmosphere structure, characterized by
a given T eff and log g, one would need a nonirradiated atmosphere model with higher T eff , i.e. a larger heat loss. Therefore,
for a given planet heat content, i.e. internal entropy, the heat
loss is reduced in the case of irradiation and the planet maintains a higher entropy for a longer time. Since for a given mass,
the interior (P, T ) profile and thus the entropy fix the radius,
the irradiated planet has a larger radius than the nonirradiated
counterpart at a given time, starting from the same initial configuration. In other terms, gravitational contraction, which is
the dominant source of energy of the planet, proceeds more
slowly with irradiation than without it.
Our calculations proceed as for our low-mass star or brown
dwarf calculations, by coupling the interior and atmosphere
profile at a deep enough optical depth, which defines unequivocally the fundamental properties of the object, m, R, T eff , L
along its evolution t (Chabrier & Baraffe 1997). The boundary condition between inner and atmosphere structure is fixed
Fig. 5. Effect of irradiation on atmosphere profiles, T (K) versus
P (dyn/cm2 ), characterized by a surface gravity log g = 3.0, T eff =
1000 K (upper panel) and T eff = 100 K (lower panel). Dashed lines correspond to nonirradiated structures. Solid lines are irradiated models
at a separation a = 0.046 AU from a primary with T eff⋆ = 6000 K. The
corresponding equilibrium temperature is T eq ∼ 1630 K. The squares
on the curve refer to optical depth, defined at λ = 1.2 µm, τstd = 1 and
the circles to τstd = 100. The triangles indicate the top of the convective
zone.
at τstd = 100, which corresponds to a range of pressure P =
0.1–200 bar for the whole range of atmosphere models1 . The
irradiated atmosphere models are integrated down to an optical
depth τstd = 100 for T eff ≥ 1000 K and τstd = 105 for T eff <
1000 K. In both cases, this is deep enough to reach the top
of the convective zone and to provide a good spatial resolution of these layers, even for the coolest models (see Fig. 5).
In any case, the incident flux Finc drops to zero at τstd ≪ 50,
well above the deepest layers of the atmosphere models. Note
that for the coolest atmospheric structures, convection does not
reach the layers corresponding to τstd = 100 (see Fig. 5). In that
case, the radiative gradient in the interior is calculated with the
Rosseland means of the same atmospheric opacities, for a consistent treatment between the interior and atmosphere thermal
structures.
We have calculated the evolution of planets covering a
range of mass from 0.5 MJ to 10 MJ with and without irradiation. The evolution of the radius as a function of time is
shown in Fig. 6 for irradiated and nonirradiated EGPs of 1 MJ
and 10 MJ . As expected, the less massive the planet, the larger
1
τstd is defined at λ = 1.2 µm.
I. Baraffe et al.: Evolutionary models for cool brown dwarfs
709
HD209458b
Fig. 6. Radius (in RJ ) versus age (in yr) for EGP masses of 1 MJ
(lower panel) and 10 MJ (upper panel). The dashed lines are nonirradiated models. The solid lines are irradiated models at a separation
a = 0.046 AU from the parent star with T eff⋆ = 6000 K. The dotted
lines are nonirradiated models from Burrows et al. (1997).
the effect of irradiation, for a given incident flux. At 1 Gyr,
the 0.5 MJ EGP has a 14% larger radius than its nonirradiated counterpart, whereas for the 1 MJ (resp. 10 MJ ), R is only
10% larger (resp. 7%). We also compare our COND models
(the nonirradiated models) to the Burrows et al. (1997, hereafter B97) nonirradiated models. Significant differences appear
at young ages (<1 Gyr), due certainly to different initial conditions (see Baraffe et al. 2002). For ages > 1 Gyr and m >
∼ 5 MJ ,
the differences between the B97 models and ours are of the
order of the irradiation effects. This reflects the different input physics, mainly in the dust treatment and molecular opacities and illustrates the present uncertainties in the models.
For m <
∼ 5 MJ , however, irradiation effects become larger than
the differences between the B97 and our models.
The specific case of HD 209458b, with a mass mp =
0.69 MJ , is illustrated in Fig. 7. The intrinsic luminosity
and corresponding effective temperature in the irradiated case
(solid lines) are compared to the non-irradiated case (dashed
lines). Starting from the same initial configuration in both
cases, the heat loss is reduced at early ages in the case of irradiation, as expected. Consequently, the irradiated model evolves
at larger entropy and radius than its non-irradiated counterpart.
During the first Myr of evolution, both evolutionary sequences
contract with increasing central density and temperature, the
non-irradiated model being denser. The latter becomes partially
Fig. 7. Effect of irradiation on the evolution of a planet with mp =
0.69 MJ at a separation a = 0.046 AU from its parent star with T eff =
6000 K. The panels from top to bottom display respectively the intrinsic luminosity Lint , the effective temperature (in K) and the radius
versus time (in yr). The solid curves correspond to the irradiated case
and the dashed curves to the nonirradiated counterpart. We recall that
in the case of irradiation, T eff and Lint do not characterize the total flux
emitted by the planet. The position of HD 209458b in the lower panel
is from Cody & Sasselov (2002).
degenerate earlier, its contraction slows down and its heat loss
becomes smaller than in the irradiated case (at log t ∼ 6.2 yr).
The situation reverses at log t ∼ 7.4 yr when the effect of partial degeneracy becomes important in the irradiated sequence.
The age of HD 209458 is about 4–7 Gyr, according to Cody
& Sasselov (2002). At 5 Gyr, the irradiated sequence displayed
in Fig. 7 predicts a radius R = 1.09 RJ , 26% smaller than the
observed value. Without including irradiation effects, the radius
is >30% than the observationally determined one. Note that the
nonirradiated sequence stops at T eff = 100 K, corresponding to
an age of ∼2 Gyr and a radius R = 1 RJ . In the following section, we analyse the possible reasons for such a discrepancy.
4. Discussion
4.1. Uncertainties of irradiated
atmosphere/evolutionary models
The question rises whether uncertainties of current models can
explain the mismatch of HD 209458b predicted versus observed radius, and whether irradiation effects can still provide
the solution to the problem. We first note that our choice of
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I. Baraffe et al.: Evolutionary models for cool brown dwarfs
parameters for the irradiated atmosphere calculations certainly
overestimates the effects of irradiation (see Sect. 3). We assume
redistribution of the incident flux over the day-side of the planet
only, without taking into account varying angles of incidence
of the impinging flux. As shown in Brett & Smith (1993), the
effect of irradiation on the atmosphere will decrease with increasing angles of incidence. Note that adopting the maximum
case, i.e. no redistribution at all of Finc , affects significantly the
outer atmospheric profile, but only slightly the inner profile.
We did not consider horizontal energy flows, which may exist
in a real planet receiving a varying amount of incident flux over
its surface. But our assumption of isotropic incident flux, with
the maximum amount of flux allowed, should provide an upper
limit for the expected effects of irradiation on the evolution of
a planet.
Although present atmosphere models still have some shortcomings, due either to incomplete molecular opacities or to
dust treatment (see Sect. 2.2), the inner profiles of the irradiated models must be changed and heated drastically to provide
the effect required on evolutionary models to match the radius
of HD 209458b. Figure 7 indeed shows that the contraction of
the irradiated sequence proceeds too rapidly after the first Myr
of evolution to maintain a radius close to the observed value. At
an age t1 = 1 Myr, the model has a radius R1 = 2 RJ and an intrinsic luminosity log L1 /L⊙ = −5.12, corresponding to T eff =
670 K. This corresponds to a characteristic Kelvin-Helmholtz
timescale τKH = Gm2p /(R1 L1 ) ∼ 10 Myr. In fact, the model
has already reached a radius of R10 = 1.48 RJ, about the radius of HD 209458b, after only 10 Myr (see Fig. 7). In order to
slow down the planet contraction to reach the observed value
at ∼5 Gyr, the inner profiles of the present irradiated atmosphere models, for a given T eff , should be drastically modified.
Such a major modification seems unlikely, for a fixed incident
flux. Note that assuming a constant flux from the parent star
during the whole planet evolution overestimates the total incident flux received by the planet over ∼5 Gyr. For a parent star
mass ∼1.06 M⊙ (Cody & Sasselov 2002), most of the star evolution after the first 1 Myr proceeds at a luminosity L smaller
than its value at 5 Gyr.
Finally, possible uncertainties due to abundance effects,
such as non-solar metallicity and/or helium abundance are not
expected to affect significantly the present results. To estimate
such uncertainties, we have computed a grid of irradiated atmosphere models for an over-solar metallicity [M/H] = +0.3
and the corresponding evolutionary sequence for a mass mp =
0.69 MJ . After a few Gyr evolution, the radius of the later sequence is essentially the same as for the solar metallicity case.
Given the optimization of irradiation effects provided by our
assumptions, and the huge effect required on the inner profile
to reduce the mismatch between observed and predicted radii,
we do not expect uncertainties in the present models to be the
source of the discrepancy.
4.2. Observed versus theoretical radius
The definition of the radius in low mass stars, brown dwarfs
or isolated giant planets is usually not a matter of confusion,
given the negligible extension of their atmosphere compared to
the total radius of the object. The picture could be different in
the case of irradiated atmospheres, where extension effects due
to the large heating of the upper layers may not be negligible
(Seager & Sasselov 2000; Hubbard et al. 2001).
As discussed in Baschek et al. (1991), the condition of
compactness for a photosphere in hydrostatic equilibrium is
HP /r ≪ 1, whit HP the pressure scale height and r the radial
distance to the center. This condition is known to be perfectly
fulfilled in (non-irradiated) low mass objects (see Chabrier &
Baraffe 1997), where the extension of the photosphere is usually less than 1% the total size of the object. Thus the radius can unambiguously be defined, quoting Baschek et al.
(1991), as the distance of the atmosphere to the object center. In all our previous work, we fix the boundary condition
between atmosphere and inner structure at τstd = 100, knowing that R(τstd = 100) is essentially the same as R(τstd ∼ 1) (see
Chabrier & Baraffe 1997). As already mentioned, we define τstd
at 1.2 µm, which corresponds to the peak of the flux emitted by
cool (non-irradiated) objects. Usually, the region where τstd ∼ 1
is close to the region where τRosseland ∼ 1 (for the present irradiated models as well). Above this region, the atmosphere contains little mass and contributes negligibly to the luminosity.
The evolutionary calculations for irradiated models presented
in Sect. 3.2 determine also the radius at τstd = 100. Within the
present assumptions of irradiation, the atmospheric extension
between τstd = 100 and τstd ∼ 1 represents only 1–2% of the
total radius. The presently calculated theoretical radius is thus
essentially equivalent to a photospheric radius at 1.2 µm.
However, the observed radius of HD 209458b (e.g. the one
estimated by Cody & Sasselov 2002) is based on the analysis
of optical light curves. It corresponds to a region of the atmosphere where the optical depth is near unity at 0.6 µm, which is
near τstd = 10−2 in our atmosphere models. Therefore, the radius predicted by the evolutionary models is not equivalent to
the observed radius. If the atmospheric extension and the opacity of the atmosphere are large at the observed wavelengths, the
measured radius could be different from the radius predicted by
the evolutionary calculation. However, based on our irradiated
atmosphere models with the gravity predicted by the evolution
and the age of HD 209458b (i.e. log g = 3.2), the atmospheric
extension between τstd = 100 and τstd = 10−2 (where the optical depth is close to unity at 0.6 µm) is very small, namely
0.05 RJ , compared to the overall radius (Barman et al. 2003,
in preparation). Adding this value to the radius predicted by
the models at 5 Gyr yields an optical-depth radius at 0.6 µm
of ∼1.14 RJ , still 22% less than the observed value. For younger
planets or planets undergoing stronger irradiation effects, with
much lower gravities, the extension will be more important and
should be taken into account for a consistent comparison between theoretical and observed radii, as already stressed by
Seager & Sasselov (2000) and Hubbard et al. (2001).
In the same vein, Hubbard et al. (2001) estimate a radius of 94 430 km (1.32 RJ ) at a pressure of 1 bar, based
on a detailed analysis of physical effects influencing the observed light curve of HD 209458b. At 5 Gyr, our models
predict a radius at 1 bar of ∼1.1 RJ , 18% smaller than the
I. Baraffe et al.: Evolutionary models for cool brown dwarfs
HD209458b
Fig. 8. Effect of extra source of energy dissipation on the evolution of a
planet with mass mp = 0.69 MJ . The solid and long-dashed lines correspond to the irradiated and non-irradiated sequences respectively, with
no extra source of energy dissipation. The other curves correspond to
irradiated sequences with a total amount of additional energy Lextra
(in erg s−1 ) deposited in layers between the surface and the mass shell
mp − ∆m, as indicated in figure (see text).
711
Showman & Guillot (2002) suggested that downward transport
of kinetic energy produced by atmospheric circulation could
be dissipated in the planet interior, leading to a substantial
deposit of energy. Within the present input physics and treatment of irradiation, we can estimate the amount of energy required to reach the radius of HD 209458b. As in Guillot &
Showman (2002), we arbitrarily add an extra term of energy
generation ǫ̇extra in the energy equation at different depths. We
have explored several
cases displayed in Fig. 8. We add a total
R
amount Lextra = ǫ̇extra dm in a region of mass ∆m enclosed between the surface and an arbitrary depth at mass shell m1 (i.e.
∆m = mp − m1 ). Various tests indicate that an amount of energy
Lextra ∼ 1027 − 5 × 1027 erg s−1 dissipated along the internal adiabat yields a radius within the error bars of the observed value
(see Fig. 8). As expected, the larger the fraction of Lextra deposited in the convective layers, the more important the effect.
Note that the case displayed in Fig. 8 with Lextra ∼ 1027 erg s−1
dissipated all over the star (∆m/mp = 1, dashed curve) is equivalent to depositing the same amount of energy only at the very
center. Our quantitative estimates are in general agreement with
Guillot & Showman (2002). Such an amount of energy represents more than 100 times the intrinsic luminosity Lint of the
planet, which is ∼1025 erg s−1 at 1 Gyr and ∼2 × 1024 erg s−1
at 5 Gyr (see Fig. 7). However, it represents only ∼1% of the
incoming luminosity, Linc = 2πR2p Finc ∼ 1029 erg s−1 , which
largely dominates the planet energetic balance. Thus, an alternative possibility is the release of an external source of energy
caused by the incident radiation (see e.g. Showman & Guillot
2002). As illustrated in Fig. 8, however, the extra source of energy must be dissipated at the top of the internal adiabat, i.e. at a
much deeper level than the penetration of the incident photons
(≫τ = 1).
5. Conclusion
Hubbard et al. (2001) estimate. Such a discrepancy is consistent
with the afore-mentioned mismatch for the radius at 0.6 µm.
4.3. Other sources of energy deposit
If irradiation effects alone do not explain the large observed
radius of HD 209458b, other sources of energy must be invoked. Tidal interactions between the star and the planet
can provide a source of energy associated to the synchronization and/or circularization of the planet orbit, dissipated
within the planet (Lubow et al. 1997; Rieutord & Zahn 1997;
Bodenheimer et al. 2001). However, as discussed recently by
Guillot & Showman (2002) and Showman & Guillot (2002),
these processes are efficient only during the early stages of the
planet evolution. Estimates based on the current understanding of such processes yield typical circularisation timescale
τcirc ∼ 108 yr (Bodenheimer et al. 2001) and synchronisation timescale τsyn < 108 yr (Lubow et al. 1997; Rieutord &
Zahn 1997). Such an energy source seems thus unlikely to slow
down the long term evolution of the planet, unless a second
planet orbiting HD 209458a is present. Such a detection has
been suggested very recently in the literature (Bodenheimer
et al. 2003) but remains to be confirmed unambiguously.
We have presented calculations describing the evolution of
cool brown dwarfs and extra-solar gaseous planets. The present
models reproduce the main trends of observed methane-dwarfs
in near-IR color-magnitude diagrams (J − K, K − L). Problems
still remain at wavelengths <1 µm, with a flux excess predicted
in the I-bandpass. The treatment of atomic line broadening in
such dense objects may be the source of the present discrepancy. The models fail to reproduce the coolest L-dwarfs and a
detailed treatment of dust diffusion/sedimentation is required
for a correct description of the transition region between Land T- dwarfs. Work is in progress in this direction.
We have included the effects of irradiation, coupling irradiated atmosphere profile and inner structure, and providing consistent evolutionary models for irradiated planets. The effect
of irradiation are shown to modify significantly the mechanical (mass-radius) and thermal evolution of irradiated EGPs.
However, a significant discrepancy (26%) remains between the
theoretical and observed radii of the transit planet HD 209458b.
We have explored possible uncertainties inherent in the models
to explain such a discrepancy. Although solving these uncertainties may modify the outer structure of the models (extension of the atmosphere, albedo) and perhaps slightly reduce the
discrepancy, none of the uncertainties is likely to modify
712
I. Baraffe et al.: Evolutionary models for cool brown dwarfs
significantly the inner entropy profile of the models, which determines the radius of the planet. Indeed, a drastic modification
on the inner thermal structure is required to bring the theoretical radius in agreement with the observed one.
In summary, we do not expect irradiation effects alone to
explain the large observed radius of HD 209458b. In the same
vein, tidal interactions will affect only the early stages of evolution of the planet but will probably be dissipated too rapidly
to affect the long term contraction of the object. Other sources
of energy, representing about 100 times the intrinsic luminosity of the planet, seem to be required to explain the observed
radius. The first extra-solar planet transit thus remains a challenge for theory. Detection of other transits is now crucial to
conclude whether HD 209458 is a peculiar system, whether a
second planetary companion is confirmed or not, or whether we
are missing something in the current understanding of close-in
giant planets.
Note: Isochrones for t ≥ 1 Myr of the COND models
(from 0.5 MJ to 0.1 M⊙ ) are available at:
http://www.ens-lyon.fr/˜ibaraffe/COND03 models
Acknowledgements. We are very grateful to H. Harris and S. Leggett
for providing data under ascii files and to Doug Lin for mentioning the possible detection of a second planet. We thank our
anonymous referee for valuable comments. I.B thanks the MaxPlanck-Institut für Astrophysik in Garching for hospitality during
elaboration of part of this work. This research was supported in part
by the LTSA grant NAG 5-3435, the NASA EPSCor grant to Wichita
State University, NSF grants AST-9720704 and AST-0086246, NASA
grants NAG5-8425, NAG5-9222, as well as NASA/JPL grant 961582
to the University of Georgia. This work was supported in part by the
Pôle Scientifique de Modélisation Numérique at ENS-Lyon. Some
of the calculations presented in this paper were performed on the
IBM pSeries 690 of the Norddeutscher Verbund für Hoch- und
Höchstleistungsrechnen (HLRN), on the IBM SP “Blue Horizon” of
the San Diego Supercomputer Center (SDSC), with support from the
National Science Foundation, on the IBM SP and the Cray T3E of the
NERSC with support from the DoE, and using the computer facilities
at Centre d’Études Nucléaires de Grenoble, CINES and IDRIS. We
thank all these institutions for a generous allocation of computer time.
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