Prospects of using simulations to study the photospheres of brown dwarfs
Convection in Astrophysics
Proceedings IAU Symposium No. 239, 2007
F. Kupka, I.W. Roxburgh & K.L. Chan, eds.
c© 2007 International Astronomical Union
DOI: 00.0000/X000000000000000X
Prospects of using simulations to study the
photospheres of brown dwarfs
Hans-Günter Ludwig1
1CIFIST, GEPI, Observatoire de Paris-Meudon, 92195 Meudon, France
email: Hans.Ludwig@obspm.fr
Abstract. We discuss prospects of using multi-dimensional time-dependent simulations to study
the atmospheres of brown dwarfs and extrasolar giant planets, including the processes of convec-
tion, radiation, dust formation, and rotation. We argue that reasonably realistic simulations are
feasible, however, separated into two classes of local and global models. Numerical challenges
are related to potentially large dynamic ranges, and the treatment of scattering of radiation in
multi-D geometries.
Keywords. hydrodynamics, convection, radiative transfer, methods: numerical, stars: atmo-
spheres, stars: low-mass, brown dwarfs
1. Introduction
The increasing number of brown dwarfs and extrasolar planets of spectral class L and
later discovered by infrared surveys and radial velocity searches has spawned a great
deal of interest in the atmospheric physics of these objects. Their atmospheres are sub-
stantially cooler than, e.g., the solar atmosphere, allowing the formation of molecules,
or even liquid and solid condensates – in astronomical parlance usually referred to as
“dust”. Convection is a ubiquitous phenomenon in these atmospheres shaping their ther-
mal structure and the distribution of chemical species. Hydrodynamical simulations of
solar and stellar granulation have become an increasingly powerful and handy instru-
ment for studying the interplay between gas flows and radiation. In this paper we discuss
the prospects of developing similar multi-dimensional and time-dependent simulations
of very cool atmospheres. The most important additional process – in view of previous
developments for hotter atmospheres – that one needs to tackle is the dust formation
coupled to the hydrodynamic transport processes and radiative transfer.
In the following, we shall take a slightly broader point of view than just considering
brown dwarf (hereafter BD) atmospheres and include also the atmospheres of extrasolar
giant planets (hereafter EGPs) in the discussion since their atmospheric dynamics is
controlled by similar processes as brown dwarfs atmospheres.
As we shall later see, computational limitations do not allow to address the problem
of the atmospheric dynamics as a whole, i.e., studying the global atmospheric circulation
together with convective flows taking place at relatively small spatial scales given by the
the pressure scale height at the stellar surface. Consequently modelers have addressed
the problem of the local and global circulation separately. Since many BDs and EGPs are
rapid rotators rotation constitutes another important process which one needs to take
into consideration for the global circulation.
We would like to emphasize that we are discussing the perspectives for multi-D time-
dependent simulations. Considerable insight has already been gained by the develop-
ment of one-dimensional, hydrostatic, time independent model atmospheres (hereafter
http://arxiv.org/abs/0704.1296v1
2 H.-G. Ludwig
“standard” models) for L- and T-type objects (e.g., Burrows et al. 2006, Tsuji 2002,
Allard et al. 2001, Ackerman & Marley 2001), and we draw from this work. Simulations
will augment our understanding of BD and EGP atmospheres by adding information
about the detailed cloud meteorology on the local scale of convective cells, as well as on
the scale of the global wind circulation pattern. This includes further characterization of
the effects of irradiation in close-in EGPs. The simulation of the atmospheric dynamics
might also add to our knowledge about local dynamo action in substellar objects, and
acoustic activity contributing to the heating of chromospheres.
2. Micro-physical input
In order to perform realistic simulations micro-physical input data must be available –
radiative opacities, equation-of-state (EOS), and a kinetic model describing the formation
of dust grains. The requirements are similar to those for standard models, and conse-
quently in simulation work one can usually take recourse to the descriptions developed
for 1D models – largely on the same level of sophistication. In all three before-mentioned
areas substantial progress has been made over the last decade, spawned by the discov-
ery of the first brown dwarf and EGP in 1995. In particular, since the early work of
Rossow 1978 kinetic models describing the nucleation, growth, and evaporation of dust
grains under conditions characteristic of brown dwarf atmospheres have been developed,
see Helling et al. 2004 and references therein. Hence, the present input data allow to
set-up simulations on a sufficiently realistic level.
3. Time scales: convection, radiation, dust, rotation, & numerics
To obtain insight into potential challenges one faces in simulations of the dynamics of
brown dwarf and EGP atmospheres it is illuminating to take a look at the characteristic
time scales of the governing physical processes. Figure 1 depicts these time scales in
a representative brown dwarf model atmosphere at Teff=1800K and log g=5.0 of solar
chemical composition. The model comprises the stellar photosphere and the uppermost
layers of the convective stellar envelope. Since it is expected that the cloud decks are
located in vicinity of the boundary of the convective envelope (in this model located at
a geometrical height of 23 km) it also contains the layers in which the dust harboring
layers are expected. We emphasize that the model structure is taken from an experimental
hydrodynamical simulation in which dust formation was not taken into account. Since
here we are interested in order of magnitude estimates only this is not a critical issue.
In figure 1 the line labeled “C-F-L” computed as the sound crossing time over a pres-
sure scale height depicts the upper limit of the time step which is allowed in an explicit
hydrodynamical scheme due to the Courant-Friedrichs-Levy stability criterion. Depend-
ing on the actual resolution of the numerical grid this number may be one to two orders of
magnitude smaller than indicated. The line labeled “convection” depicts the modulus of
the Brunt-Vaiäsälä period providing a measure of the time scale on which the convective
flow evolves. Two lines labeled “radiation” indicate the time scale on which radiation
changes the thermal energy content of the gas. The dashed-dotted line is computed using
Rosseland mean opacities which give the correct behavior in the optical thick layers,
the solid line is based on Planck mean opacities which give a better representation in
the optically thin layers. For the rotational period depicted by the dashed line labeled
“rotation” we took a representative value close to one day. For three different dust grain
diameters of 1, 10, and 100µm we plotted the sedimentation (“rain out”) time scale
taking as drifting time over a pressure scale height. The drift velocities were taken from
Prospects for simulations of brown dwarf photospheres 3
0 20 40 60 80
Geometrical height [km]
 2  1  0 −1 −2 −3
radiation
convection
rotation
100µm
rain out
C−F−L
Figure 1. Characteristic time scales of various processes in a brown dwarf atmosphere of
Teff=1800K as a function of geometrical height. The tick marks close to the abscissa indicate
the (log Rosseland) optical depth. For details see text.
the work of Woitke & Helling 2003. We did not depict the formation time scale of the
dust grains in the figure: for grains of 100µm diameter it is of the same order as the sed-
imentation time scale. Consequently, it is unlikely that larger grains can stay in brown
dwarf atmospheres. The formation time scale becomes rapidly shorter for smaller grains
so that they can be considered being essentially formed in quasi-static phase-equilibrium
(see also Helling 2005.
Computing resources available today typically allow to simulate a dynamical range in
time of 104. . . 105, and 102. . . 103 per dimension (for 3D models) in space. From figure 1
we conclude that it should be feasible to include convection, radiative transfer effects, and
dust formation in a simulation of a BD/EGP atmosphere. The simultaneous inclusion of
rotation is beyond reach, in particular if one takes into consideration that one would like
to simulate many rotational periods to obtain a statistically relaxed state. However, the
substantial difference between the time scales on which rotation and convection operate
moreover indicates that rotation is dynamically not relevant for the surface granulation
pattern in BD/EGPs.
A rather strong modeling limitation comes about by the large spatial scale separation
between the typical size of a convective cell and the global scale of a BD or EGP of
about 104, at best reduced to 103 for the case of young, low mass EGPs. Hence, typical
BD/EGP conditions are hardly within reach with 3D models, and the steep increase of the
computational cost with spatial resolution (for explicite numerical schemes with (∆x)4)
makes it likely that this situation prevails during the nearer future. We expect that 3D
simulations will for some time be either tailored to simulate the global meteorology, or
will be restricted to local models simulating the convective flow in detail.
Figure 2 illustrates the kinematics of the flow in a local BD simulation analogous to
figure 1. The horizontal root-mean-square of the vertical velocity component is depicted
by the diamond symbols. The key-point to note is that the convective motions proper
are largely confined to the convectively unstable layers. The velocities in the convectively
stable layers with log τ < 0 are almost exclusively related to sound waves. As essentially
oscillatory motions they are ineffective for mixing so that they provide little updraft
to keep dust grains aloft in the atmosphere. The green line is illustrating an estimate
4 H.-G. Ludwig
0 20 40 60 80
Geometrical height [km]
100.0
1000.0
locit
RMS vertical hydro velocity
ix  h
ctive
 2  1  0 −1 −2 −3
Figure 2. Characteristic velocities in brown dwarf atmosphere analogous to figure 1.
of the effective mixing velocity provided by the convective motions. The decline of the
amplitude is rather steep – in the test model with a scale height of about 1/3 of the local
pressure scale height.
Comparing the mixing velocities with typical grain sedimentation velocities indicates
that the kinematics could support cloud decks in the convective zone and a thin adjacent
overshooting layer at its top boundary. Fitting observed spectra Burrows et al. 2006 find
a preference for a rather large grain size of ≈ 100µm in BD atmospheres. This would
make the grain sedimentation velocities comparable to convective velocities which is
numerically uncritical. More demanding would be small grain sizes. The distribution of
small grains would hinge on the capability of a numerical code to deal with large velocity
ranges. Any non-physical diffusivity in a code can artificially extend the region over which
clouds of small grains could exist.
4. The multi-D story so far
A number of simulations of BD/EGP atmospheres have been already conducted in 2D
and 3D geometry. Here, the problem of the circulation between the day- and night-side of
close-in EGPs (“hot Jupiters”) achieved particular attention. However, to our knowledge
none of the studies has addressed the coupled problem of hydrodynamics, dust formation,
radiation, and rotation, but rather have focused on different parts of the overall problem.
We would like to refer the interested reader to Showman & Guillot 2002, Cho et al. 2003,
Burkert et al. 2005, including the follow-up works of these groups.
5. Serendipity
In the previous sections we summarized expectations about the insights one might gain,
and challenges one might face when trying to construct multi-D models for BD/EGP
atmospheres. We added figure 3 as a reminder that of course the unforeseen results
are the most interesting ones. Figure 3 illustrates a slight but distinct change of the
granulation pattern between the familiar solar granulation and granulation in an M-
dwarf. E.g., similarly and perhaps more drastically the formation of dust in BD/EGP
atmospheres might modulate the convective dynamics in unexpected ways – who knows.
Prospects for simulations of brown dwarf photospheres 5
Figure 3. Grey-scale images of the vertical velocity component of a solar hydrodynamical model
(top row) and a M-dwarf model (bottom row). From left to right, the velocities are depicted
at (Rosseland) optical depth unity as well as one and two pressure scale heights below that
level in the respective models. The absolute image scales are 5.6 × 5.6Mm2 for the solar and
0.25× 0.25Mm2 for the M-dwarf model.
6. Conclusions
Reasonably realistic local or global models of brown dwarf and extrasolar giant planet
atmospheres coupling hydrodynamics, radiation, dust formation, and rotation are numer-
ically in reach at present. However, “unified” models spanning all spatial scales from the
global scale down to scales resolving the flow in individual convective cells are stretch-
ing the computational demands beyond normally available capacities. Hence, we expect
that a separation between local and global models will prevail during the nearer future.
Whether this will turn out to be a severe limitation remains to be seen. While appar-
ently subtle we would like to point to the solar dynamo problem where the still not fully
satisfactory state of affairs might be related to the lack of the inclusion of small enough
scales when modeling the global dynamo action. In BD/EGP atmospheres it is perceiv-
able that the local transport of momentum by convective and acoustic motions might
alter the global flow dynamics – in the simplest case by adding turbulent viscosity.
If the sizes of dust grains in BD/EGP atmospheres turn out to be small, and the
grains consequently exhibit low sedimentation speeds, numerical simulations must have
the ability to accurately represent the large dynamic range between grain and convec-
tive/acoustic velocities. Overly large numerical diffusivities artificially enlarge the height
range over which cloud decks can persist.
Standard model atmospheres are treating the wavelength-dependence of the radiation
field commonly in great detail which is not possible in the more demanding multi-D
geometry of simulation models. An approximate multi-group treatment of the radiative
transfer has been developed for simulations of stellar atmospheres which has also been
6 H.-G. Ludwig
proven to provide reasonable accuracy at acceptable computational cost in cooler (M-
type) atmospheres. We expect that the scheme also works for even cooler atmospheres.
However, one simplification usually made is treating scattering as true absorption. De-
pending on the specific dust grain properties this approximation might need to be re-
placed by a more accurate treatment of scattering. Hence, another challenge a modeler
might face is to device a computationally economic scheme to treat scattering in the
time-dependent multi-D case.
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Discussion
C. Helling: Your wish list implies that no progress has been made in the brown dwarf
modeling. Additionally, I am convinced that we will need to work on both sides: on 1D
models which are fast and applicable, not only on 3D models though they will play an
important role.
Ludwig: My wish list was intended as overall collection of things we would like to
understand about brown dwarf atmospheres. Progress related to the various points has
indeed already been made. Concerning the mutual role of 1D and 3D models, I fully
agree. 3D models should address crucial aspects that are in principle not accessible in
1D. Insight emerging from 3D models should then be transferred to 1D models.
F. Kupka: Considering the complexity of molecular opacity I am actually surprised how
robust the opacity binning seems to be.
Ludwig: Tests have been performed for M-type stars where the effect of many millions
of – primarily molecular – lines is captured quite well. As for brown dwarfs: the dust
opacity has a rather smooth functional dependence on wavelength. Hence, it should be
easy, but scattering is a problem.
I.W. Roxburgh: You said overshooting was small, could you quantify this in terms of
local scale height?
Ludwig: The velocity amplitude declines exponentially with a scale height of about
1/3 of the local pressure scale height. For comparison: in solar models the scale height
of decline is about six times larger. However, keep in mind that the hydrodynamical
model presented here is experimental, in particular does not include any effects of dust
formation.
	Introduction
	Micro-physical input
	Time scales: convection, radiation, dust, rotation, & numerics
	The multi-D story so far
	Serendipity
	Conclusions
