Progress in Modeling Very Low Mass Stars, Brown Dwarfs, and Planetary Mass Objects
F. Allard, D. Homeier, B. Freytag, W. Schaffenberger, A. S. Rajpurohit
aa r X i v : . [ a s t r o - ph . S R ] F e b Mem. S.A.It. Vol. 75, 282 c (cid:13) SAIt 2008
Memorie della
Progress in Modeling Very Low Mass Stars,Brown Dwarfs, and Planetary Mass Objects
F. Allard, D. Homeier, B. Freytag, W. Scha ff enberger, & A. S. Rajpurohit CRAL, UMR 5574, CNRS, Universit´e de Lyon, ´Ecole Normale Sup´erieure de Lyon,46 All´ee d’Italie, F-69364 Lyon Cedex 07, France, e-mail: [email protected]
Abstract.
We review recent advancements in modeling the stellar to substellar transition.The revised molecular opacities, solar oxygen abundances and cloud models allow to re-produce the photometric and spectroscopic properties of this transition to a degree neverachieved before, but problems remain in the important M-L transition characteristic of thee ff ective temperature range of characterizable exoplanets. We discuss of the validity of theseclassical models. We also present new preliminary global Radiation HydroDynamical Mdwarfs simulations. Key words.
Stars: atmospheres – M dwarfs – Brown Dwarfs – Extrasolar Planets
1. Introduction
The spectral transition from very low massstars (VLMs) to the latest type brown dwarfsis remarkable by the magnitude of the trans-formation of the spectral features over a smallchange in e ff ective temperature. It is character-ized by i) the condensation onto seeds of strongopacity-bearing molecules such as CaH, TiOand VO which govern the entire visual to near-infrared part (0 . − . µ m) of the spectral en-ergy distribution (hereafter SED); ii) a ”veil-ing” by Rayleigh and Mie scattering of sub-micron to micron-sized aerosols; iii) a weak-ening of the infrared water vapor bands due tooxygen condensation and to the greenhouse (orblanketing e ff ect) caused by silicate dust in theline forming regions; iv) methane and ammo-nia band formation in T and Y dwarfs; and fi-nally v) water vapor condensation in Y dwarfs( T e ff ≤
500 K). Condensation begins to occur
Send o ff print requests to : F. Allard in M dwarfs with T e ff < µ mK I alkali doublets which form out to as muchas 2000 Å from the line center (Allard et al.2007b,a). The SED of those dwarfs is there-fore dominated by molecular opacities and res-onance atomic transitions under pressure ( ≈ ff erential rota-tion and clouds should be distributed in bandsaround their surface much as is shown forJupiter. Burgasser et al. (2002) have suggestedthat brown dwarfs in the L-T transition are af-fected by cloud cover disruption. And indeed,several objects show even large-scale photo-metric variability (Artigau et al. 2009, Radiganet al. 2012) — on the order of 5% to even10-30% in the best studied case. Buenzli et al. llard et al.: Modeling the stellar-substellar transition 283 (2012) find periodic variability both in near-IRand mid-IR for a T6.5 brown dwarf in simul-taneous observations conducted with HST andSpitzer. The phase of the variability varies con-siderably between wavelengths, suggesting acomplex atmospheric structure. Recent large-scale surveys of brown dwarf variability withSpitzer (PI Metchev) have revealed mid-IRvariability on order of a few percent in > ff and spectral types. And the ubiquityof cloud structures in L3-T8 dwarfs stronglysuggests that these may persist into the cooler( > T8) objects.The models developed for VLMs andbrown dwarfs are a unique tool for the char-acterization of imaged exoplanets, if they canexplain the stellar-substellar transition. Andglobal circulation models subjected to cloudformation in presence of rotation are neces-sary to explain the observed weathering phe-nomena. Recently, Allard et al. (2012a,b) andRajpurohit et al. (2012) have published thepreliminary results of a new model atmospheregrid computed with the
PHOENIX atmospherecode accounting for cloud formation and mix-ing from Radiation HydroDynamical (RHD)simulations (Freytag et al. 2010). In this pa-per, we present the latest evolution in modelingtheir SED and their observed photometric vari-ability and present new prospectives for thisfield of research.
2. M dwarfs
Because oxygen compounds dominate theopacities in the SED of VLMs, their syn-thetic spectra and colors respond sensibly tothe abundance of oxygen assumed. Allard et al.(2012a,b) compare models using di ff erent solarabundance values (Grevesse et al. 1993, 2007,Asplund et al. 2009) and found an improvedagreement with constraints (Casagrande et al.2008a) for the solar abundances obtained us-ing RHD simulations by Asplund et al. Inthis paper, we present models based on theCa ff au et al. (2011) solar abundances. Takingthe Grevesse et al. (1993) solar abundance val- ues as reference, the Grevesse et al. (2007)results show a reduced oxygen abundance by-39%, while Asplund et al. obtain a reduc-tion by -34%. These values poses problem forthe interpretation of solar astero-se¨ısmologicalresults (Basu & Antia 2008, Antia & Basu2011). More recently, Ca ff au et al. (2011) usedthe CO5BOLD code to obtain a more conser-vative reduction of oxygen by -22%. The lat-ter value still allows an acceptable representa-tion of the VLMs while preserving the astero-se¨ısmological solar results. Higher spectralresolution and up-to-date opacities also con-tributed to improve VLMs models comparedto previous versions (Hauschildt et al. 1999,Allard et al. 2001). These models rely on listsof molecular transition determined ab initio.See the review by Homeier et al. elsewhere inthis journal for the opacities used in the BT-Settl models presented in this paper.The comparison of the BT-Settl PHOENIX models based on the Ca ff au et al. (2011) solarabundances to the low resolution spectra andto the Casagrande et al. (2008b) temperaturescale is shown in Figs. 1 and 2. Fig. 1 showsan unprecedented agreement with spectral typethrough the M dwarf spectral sequence of themodels. One can see in Fig. 2 that the new BT-Settl models lie slightly to the blue of the BT-Dusty models by Allard et al. (2012) based onthe Asplund et al. (2009) solar abundances, butlargely to the red of the AMES-Cond / Dustymodels by Allard et al. (2001) and the BT-NextGen models Allard et al. (2012) based onthe Grevesse et al. (1993) solar abundances.The even lower oxygen abundance values ofGrevesse et al. (2007) cause the MARCS mod-els by Gustafsson et al. (2008) to lie to theright of the diagram. The NextGen models byHauschildt et al. (1999) also lie to the right ofthe diagram due, among others, to the missingand incomplete molecular opacities.The BT-Settl models are computed solv-ing the radiative transfer in spherical symmetryand the convective transfer using the MixingLength Theory (B¨ohm-Vitense 1958, see alsoLudwig et al. 2002 for the exact formalismused in
PHOENIX ) using a mixing length asderived by the RHD convection simulations(Ludwig et al. 2002, 2006, Freytag et al. 2012)
84 Allard et al.: Modeling the stellar-substellar transition
Fig. 1.
Fig. 1 of the article by Rajpurohit et al. (in prep.). The optical to red SED of M dwarfs from M0 toM9 observed with the NTT at a spectral resolution of 10.4 Å are compared to the the best fitting (chi-squareminimization) BT-Settl synthetic spectra (dotted lines), assuming a solar composition according to Ca ff au etal. (2011). The models displayed have a surface gravity of long = Σ + < – X Σ + system ofMgH by Skory et al. (2003) and the opacities are totally missing for the CaOH band near 5500 Å. Note thatchromospheric emission fils the Na I D transitions in the latest-type M dwarfs displayed here, and that theM9.5 dwarf has a flatten optical spectrum due to dust scattering. Telluric features near 7600 Å have beenignored from the chi-square minimization. and a radius as determined by interior mod-els (Bara ff e et al. 1998, 2003, Chabrier et al.2000) as a function of the atmospheric param-eters ( T e ff , surface gravity, and composition). PHOENIX use the classical approach consist-ing in neglecting the magnetic field, convec-tive and / or rotational motions and other multi-dimensional aspects of the problem, and as-suming that the averaged properties of starscan be approximated by modeling their prop-erties radially (uni-dimensionally) and stati-cally. Neglecting motions in modeling the pho-tospheres of VLMs, brown dwarfs, and planets is acceptable since the convective velocity fluc-tuation e ff ects on line broadening are hiddenby the strong van der Waals broadening andthe important molecular line overlapping pre-vailing in these atmospheres. But this is not thecase of the impact of the velocity fields on thecloud formation and wind processes (see sec-tion 5 below).
3. Global RHD M dwarf simulation
VLMs and brown dwarfs are fully convec-tive, and their convection zone extends into llard et al.: Modeling the stellar-substellar transition 285
J−K s T e ff [ K ] MARCSAtlas9DRIFT PHOENIXUCM Tc=1700KUCM Case CBurrows ClearBurrows CloudyNextGenAMES-CondAMES-DustyBT-NextGenBT-CondBT-DustyBT-Settl
Fig. 2.
Estimated T e ff and metallicity (decreasing from lighter to darker tones) for M dwarfs byCasagrande et al. (2008a) on the left, and brown dwarfs by Golimowski et al. (2004) and Vrba et al.(2004) on the right are compared to the NextGen isochrones for 5 Gyrs Bara ff e et al. (1998) using modelatmospheres by various authors: MARCS by Gustafsson et al. (2008), ATLAS9 by Castelli & Kurucz(2004), DRIFT-PHOENIX by Helling et al. (2008), UCM by Tsuji (2002), Clear / Cloudy by Burrows et al.(2006), NextGen by Hauschildt et al. (1999), AMES-Cond / Dusty by Allard et al. (2001), the BT-Cond / Dusty / NextGen models by Allard et al. (2012) and the current BT-Settl models. Some curves labeledin the legend can only be seen in the more extended Fig. 4. their atmosphere up to optical depths of even10 − (Allard 1990, 1997). Convection is e ffi -cient in M dwarfs and their atmosphere, ex-cept in the case of 1 Myr-old dwarfs whichdissociate H , little sensitive to the choice ofmixing length (Allard et al. 1997). A mixinglength value between α = l / H p = . − . T e ff .The magnetic breaking known to operatein low mass stars is not or less e ffi ciently op- erating in fully convective M dwarfs ( < M3)and brown dwarfs. Brown dwarfs can there-fore rotate with equatorial speeds as large as 60km / sec or periods of typically 1.5 hour. Somehave even been observed with periods nearlyas low as their breakup velocity (1 hour). Incomparison, planets of our solar system havea larger rotation period, such as 10 hours forJupiter, and 24 hours for the Earth. This rapidrotation (as well as their magnetic field) causesa suppression of the interior convection ef-ficiency leading to a slowed down contrac-tion during their evolution, and to larger radiithen predicted by classical evolution models(Chabrier et al. 2007).
86 Allard et al.: Modeling the stellar-substellar transition −0.5 0.0 0.5 1.0 1.5 2.0 2.5
J−K s T e ff [ K ] MARCSAtlas9DRIFT PHOENIXUCM T =1700KUCM Case CBurrows ClearBurrows CloudyNextGenAMES-CondAMES-DustyBT-NextGenBT-CondBT-DustyBT-Settl
Fig. 4.
Same as Fig. 2 but extending into the brown dwarf regime for an age of 3 Gyrs. The region below2900 K is dominated by dust formation. The dust free models occupy the blue part of the diagram and onlyat best explain T dwarf colors, while the Dusty and DRIFT models explain at best L dwarfs, becoming onlyredder with decreasing T e ff . The BT-Settl, Cloudy and UCM T crit = T e ff ≤
500 K).
The CO5BOLD code solves the coupledequations of compressible hydrodynamics andnon-local radiative energy transport on a carte-sian grid with a time-explicit scheme. It can beused in a local setup (with constant downwardgravity) to model small patches of the stel-lar surface or in a global “star-in-a-box” setup(with central gravitational potential) to modelentire stars. The “star-in-a-box” setup has beenused by Ste ff en & Freytag (2007) to computeglobal RHD simulations of the scaled-downSun in the presence of rotation, described byCoriolis and centrifugal forces. In the cur-rent code version, angular-momentum conser-vation and the isotropy of the flows at smallMach numbers have been further improved.This setup has been used, see Fig. 3, to com- pute a sequence of radially scaled-down toymodels of M dwarfs with various rotation pe-riods to determine the change in mixing lengththat corresponds to the suppression of convec-tion by rotation (Scha ff enberger & Freytag, inprep.). It is found that at realistic Coriolis num-bers, rotation is indeed able to reduce the con-vective flux (predominantly towards the equa-tor) to contribute significantly to the increasein radius as suggested from analysing classicalmodels.
4. Cloud formation in
PHOENIX
To describe the cloud formation in
PHOENIX
Allard et al. (2003) took the approach of us-ing a cloud model drawn from an extensive llard et al.: Modeling the stellar-substellar transition 287
Fig. 3.
Global RHD simulation of an M dwarfswith solar composition. Obtained parameters are T e ff = .
44 K, log g = .
38, and a radius of0.003 R ⊙ . The surface pressure is 68227.6 g / cm s ,the surface density is 4.52146e-07 g / cm , the cen-tral temperature is 28889.7 K, the central pres-sure is 1.67214e +
09 g / cm s , the central density is6.92703e-4 g / cm. The model covers 10 pressurescale heights from the center to the surface. The sim-ulations use grey PHOENIX opacities in the outer lay-ers merged with OPAL data for the interior layers.The rotation period of the simulation is 1 hour andthe simulation is shown equator-on in the referenceframe of rotation. One can notice a slight oblatenessof the model at this exaggerated velocity for an Mdwarf. study of cloud formation in planetary atmo-spheres (Rossow 1978) which compares layer-by-layer the timescales of the main processes(mixing, sedimentation, condensation, coales-cence and coagulation). The mixing timescalesare taken from RHD simulations (Freytag et al.2010, see section 5 below for details). For theAllard et al. (2012a,b) pre-release of the BT-Settl models, the cloud model was improvedby a dynamical determination of the super-saturation — the ratio of the saturation va-por pressure to the compound local gas pres-sure ( P g ( t ) / P sv / ). In the current pre-release thecloud model is further improved by the imple-mentation of a grain-size-dependent forwardscattering, and by accounting for nucleationbased on cosmic rays studies (Tanaka 2005). This latter change allows the cloud model tolimit its refractory element depletion and formrelatively more dust grains in higher atmo-spheric layers.The cloud model is solved from the inner-most to the outermost atmospheric layer, de-pleting the gas composition due to sedimenta-tion gradually from the bottom to the top of theatmosphere. This model obtains both the den-sity distribution of grains and the grain averagesize per layer, and includes 55 types of grains.We do not assume a seed composition since ourcloud model modifies the equilibrium chem-istry iteratively to the nucleation rate limit ateach atmospheric layers. The cloud compo-sition in the photospheric layers varies withspectral type from zirconium oxide (ZrO ) andrefractory ceramics (perovskite and corundum;CaTiO , Al O ) in late-M dwarfs, to silicates(enstatite, forsterite, etc.) in early-L dwarfs,and to salts (CsCl, RbCl, NaCl) and ices (H O,NH , NH SH) in late-T and Y dwarfs. Thismethod is coherent with the observed weak-ening and vanishing of TiO and VO molecu-lar bands (via CaTiO , TiO , and VO grains)from the optical spectra of late M and L dwarfs,revealing CrH and FeH bands otherwise hid-den by the molecular pseudo-continuum, andthe resonance doublets of alkali transitionswhich are only condensing onto salt grains inlate-T dwarfs. We solve the Mie equation forspherical grains using complex refraction in-dex of materials as a function of wavelengthcompiled by the Astrophysikalisches Institutund Universit¨ats-Sternwarte in Jena.The results are shown in Fig. 4 wherethe new BT-Settl model atmospheres colorsare interpolated onto the published theoreticalisochrones (Bara ff e et al. 2003). New interiorand evolution models consistent with the BT-Settl model atmospheres are in development.
5. Cloud Formation in CO5BOLD
Cloud formation is included in the CO5BOLDsimulations of VLMs and brown dwarfs(Freytag et al. 2010) by assuming a mass den-sity of forsterite dust “monomers” initially setto its maximum abundance at solar metallic-ity, i.e. assuming no settling. Evaporation is ac-
88 Allard et al.: Modeling the stellar-substellar transition mt22g50mm00n11 time=21785sec
100 200 300x [km]100200300 y [ k m ] mt26g50mm00n03 time=3585sec
100 200 300 400x [km]100200300400 y [ k m ] mt22g50mm00n08 time=4505sec
100 200 300x [km]100200300 y [ k m ] mt20g50mm00n09 time=7945sec
100 200 300x [km]100200300 y [ k m ] mt18g50mm00n11 time=7955sec
100 200 300x [km]100200300 y [ k m ] mt15g50mm00n06 time=13920sec
50 100 150 200 250 300x [km]50100150200250300 y [ k m ] Fig. 5.
3D RHD simulations using
CO5BOLD (Freytag et al. 2010) of a small box of – from top to rightto bottom right – 2600K, 2200K, 2000K, 1800K and 1500K atmospheres of log g = ff ects on the surface convectioncells and the atmopheric wave pattern.llard et al.: Modeling the stellar-substellar transition 289 counted for by applying a rate determined fromthe forsterite saturation vapor curve. Graingrowth by condensation and sedimentation areaccounted for using the formulations describedby Rossow (1978). The grains are assumed tohave sizes up to 1 µ m and the correspond-ing geometric cross-section is added to theopacity bins on-the-fly. Fig. 5 shows local 3D”box-in-a-star” type RHD simulations usingCO5BOLD and the same cloud model, wherethe e ff ects of cloud formation (restricted toforsterite formation) and Coriolis e ff ects (in thetop left case) are investigated. Here, the cen-trifugal force has been neglected since over thebox size the centrifugal force would simplytranslate into a reduced gravity of the simula-tion. The Coriolis force is exaggerated to ex-plore the maximum possible e ff ect of rotationlocally at the surface.Global simulations of quasi-hydrostatic M,L and T dwarfs that can resolve the convec-tion and cloud formation processes, must besignificantly scaled down ( ≈
20 so far) in ra-dius compared to the real object. Conclusionsfrom these RHD simulations must thereforebe wisely drawn from both types of local andglobal simulations to learn about the cloud sur-face coverage of across the M-L-T transition.The preliminary conclusions are that the ro-tation does not a ff ect remarkably the convec-tive shape and size of convective surface cells,the related gravity waves and the cloud for-mation process at the surface. We know alsofrom several global circulation simulations ofrotating planets (periods around 3 days fortidally locked for Jupiter around solar typestars) that strong winds and currents are gener-ated on larger scales at the surface. These largescale features are believed to be responsiblefor the distribution of clouds on the surface ofJupiter. The higher rotation velocities of browndwarfs can therefore similarly and easily causecomparable patterns at their surface. However,it is not yet clear how patchy cloud cover-age and variability can occur. Correspondingglobal simulations are being developed.
6. Summary and Prospectives
We showed that M dwarfs can be modeled ad-equately using up-to-date opacities and the re-vised solar abundances by Ca ff au et al. (2011)which preserve the agreement with the resultsfrom solar astero-se¨ısmology (Antia & Basu2011). We also showed that VLMs and browndwarfs allow to study the process of cloud for-mation in a relatively simple context where ir-radiation and ground interaction e ff ects impor-tant for planets are not present. This allows toidentify the fundamental mechanism of cloudformation and the determination of the velocityfield by RHD simulations (Freytag et al. 2010).We have compared the behavior of the re-cently published model atmospheres from var-ious authors across the M-L-T spectral tran-sition from M dwarfs through L type and Ttype brown dwarfs and confronted them to con-straints. If the onset of dust formation is oc-curring below T e ff = ff ects of dust cloud forma-tion impact strongly ( J − K s < .
0) the near-infrared SED of late-M and early L-type at-mospheres for 1300 < T e ff < < T e ff < ,
000 K). In the M dwarfrange, the results appear to favor the BT-Settlbased on the Ca ff au et al. (2011) solar abun-dances versus MARCS and ATLAS 9 modelsbased on other values. In the M dwarfs range,the BT-Settl models show an unprecedented fitquality, even in the V bandpass despite stillmissing or incomplete opacities (in particularfor the electronic bands of CaOH, and missingopacities in the corresponding MgH B-X sys-tem bands), above 2500 K. In the brown dwarf(and planetary) regime, on the other hand, theunified cloud model by Tsuji (2002) succeedsin reproducing the constraints, while the BT-Settl models also show a plausible transition.No models succeed in reproducing perfectlythe M-L transition between 2500 and 2000 Kat this stage. This T e ff range is similar tothat of young (directly observable by imaging)and strongly irradiated planets (Hot Jupiters).However, Bonnefoy et al. (2010, 2013) obtainreasonable parameters and fit quality to the col-
90 Allard et al.: Modeling the stellar-substellar transition ors and SINFONI near-infrared integral fieldspectra of giant planets.Possible further development of the cloudmodel could lie in an improvement of the treat-ment of coagulation and of the grain opacitiesin general (e.g. porosity, non-sphericity, mixedgrain composition rather than adding the con-tribution of pure grain species, etc.).In this paper, we have presented the firstpreliminary global RHD simulations of an Mdwarf in presence of rotation. This simulationis precise enough in the interior while able toresolve the atmospheric convective cells at thesurface. The strong molecular and atomic res-onance line opacities prevailing in M dwarfatmospheres lead to a control of their cool-ing evolution by the atmosphere, such that in-terior and evolution codes must consider theatmospheric structure as their boundary con-dition (Bara ff e et al. 1998). This requires thecomputation of large and fine ( ∆ T e ff =
100 K; ∆ log g ≤ . ffi cult to be achieveby computationally expensive RHD codes likeCO5BOLD. Implicit hydrodynamical simu-lations more adequate to address the inte-rior convection conditions are being developed(Viallet et al. 2011). But these simulations willhave to resolve the atmosphere and treat theradiative transfer and the surface cooling ad-equately. RHD simulations do not cover evo-lution timescales and cannot replace classicalstellar evolution and atmosphere models. In themeanwhile, therefore, classical 1D static inte-rior models are being developed using the newBT-Settl model atmospheres as surface bound-ary conditions, and will be published shortly.Global RHD simulation which account forrotation are needed to resolve the cloud surfacedistribution and induced potential variability.In principle, it is only a small step to go fromthe global RHD simulations of an M dwarfto global simulations of late-type M dwarfs,brown dwarfs, and gas giant planets that ac-count for dust cloud formation and rotation.Such simulations are under development andto be expected for the end of 2013.M dwarfs are magnetically active withpossibly an important magnetic spot coveragedue to their more concentrated magnetic field lines (same magnetic field strengths oversmaller radii) than solar type stars. Magneticspots are due to the local suppression ofconvection by the magnetic field lines emerg-ing at these points of the stellar surface.In the Sun, spots are showing an importantsuppression of the local convection, causingan important cooling (almost half the solar T e ff ). The suppression of convection e ffi ciencyby rotation and magnetic field is also known tolead to a slowed down contraction during theirevolution, and to larger radii then predicted byclassical interior models for M dwarfs withstrong magnetic fields in close-binary systems(Chabrier et al. 2007). Global Magneto RHD(MRHD) simulations which account for rota-tion are needed to identify the mixing lengthto use in classical models to compensate forthis suppression. The non-magnetic globalRHD models presented in this paper alreadyshow that rotation alone can have a significantimpact on the convective e ffi ciency and thestellar radius. MRHD simulations are beingdeveloped also using CO5BOLD (Nutto et al.2012, Freytag et al. 2012,Wedemeyer-B¨ohm etal. 2012). The CO5BOLD MRHD simulationshave not yet been applied to global M dwarfmodels (to investigate the dynamo mechanismand the suppression of convection by magneticfields in the interior, or to simulate magneticspots). Fortunately however, we do not expecte ff ects as important as for solar spots sinceM dwarf model atmospheres have provento be insensitive to the value of the mixinglength (Allard et al. 1997) unless of course ifconvection is completely suppressed at depthover the stellar disk. The BT-Settl model at-mospheres and synthetic spectra can thereforebe used safely to determine the parametersof moderately active VLMs, brown dwarfs,and planetary mass objects. The BT-Settlsynthetic colors and spectra are distributed viathe PHOENIX web simulator . Acknowledgements.
The research leading to theseresults has received funding from the French“Agence Nationale de la Recherche” (ANR),the “Programme National de Physique Stellaire” http: // phoenix.ens-lyon.fr / simulatorllard et al.: Modeling the stellar-substellar transition 291(PNPS) of CNRS (INSU), and the EuropeanResearch Council under the European Community’sSeventh Framework Programme (FP7 / References
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