3D Model Atmospheres of Red Giant Stars
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3D Model Atmospheres of Red Giant Stars
Hans-G¨unter Ludwig and Matthias Steffen
Abstract
We provide a brief overview of the modelling of the atmospheres of redgiant stars with the 3D radiation-hydrodynamics code
CO5BOLD . We emphasizeaspects where 3D modelling provides additional insight beyond standard hydrostatic1D models, and comment on present modelling challenges.
Modelling of three-dimensional (3D) atmospheres of cool stars is an active fieldof development (e.g. Nagendra et al., 2009), and particularly 3D models of atmo-spheres of red giant (RG) stars are just on the verge of becoming available for ap-plication to astrophysical problems. In an early application, Kuˇcinskas et al. (2005)used a 3D RG model to estimate color corrections due to thermal inhomogeneities;Collet et al. (2007) considered a set of eight giant models to investigate the impacton line formation and abundances. More recently, Freytag & H¨ofner (2008) devel-oped model atmospheres of AGB stars and their winds, Dupret et al. (2009) derivedthe energy input to solar-like oscillations in giants from 3D models, Ram´ırez et al.(2010) studied convective line-shifts in the metal-poor RG HD 122563 and com-pared them to a 3D model, Chiavassa et al. (2011) applied global 3D models toassess effects of photometric and related astrometric variability, and Pasquini et al.(2011) took recourse to 3D dwarf and RG models to correct for convective blueshiftsin high-precision, spectroscopic radial velocity measurements. While fairly exhaus-tive, the list of examples is still quite short, but illustrates already the variety of
H.-G. LudwigZentrum f¨ur Astronomie der Universit¨at Heidelberg, Landessternwarte, K¨onigstuhl 12, D-69117Heidelberg, Germany, e-mail:
M. SteffenAstrophysikalisches Institut Potsdam, An der Sternwarte 16, D-14482 Potsdam, Germany, e-mail: [email protected] possible applications of 3D RG models. At the moment efforts are under way tocover the Hertzsprung-Russell diagram with 3D model atmospheres including starsin the red-giant branch (Ludwig et al., 2009a; Trampedach & Stein, 2011).In the following we are going to focus on aspects related to abundances from3D models, and the theoretical calibration of the mixing-length parameter a MLT from 3D model atmospheres. Here, we are not so much presenting results as ratherpointing out problems which are still lingering. We finally add some commentsabout predictions of the photometric micro-variability which are of interest in thecontext of high-precision photometry missions like CoRoT.
CO5BOLD
Our 3D model atmospheres were calculated with the radiation-hydrodynamics code
CO5BOLD (Freytag et al., 2002; Wedemeyer et al., 2004; Freytag et al., 2011). Thecode solves the time-dependent equations of compressible hydrodynamics coupledto radiative transfer in a constant gravity field in a Cartesian computational domainwhich is representative of a volume located at the stellar surface. The equation ofstate takes into consideration the ionization of hydrogen and helium, as well as theformation of H molecules according to Saha-Boltzmann statistics. Relevant ther-modynamic quantities – in particular gas pressure and temperature – are tabulatedas a function of gas density and internal energy. The multi-group opacities usedby CO5BOLD are based on monochromatic opacities stemming from the MARCSstellar atmosphere package (Gustafsson et al., 2008) provided as function of gaspressure and temperature with high wavelength resolution. The opacities have beencalculated assuming solar elemental abundances according to Grevesse & Sauval(1998), with the exception of CNO for which values close to the recommendationof Asplund (2005) are adopted (specifically, A(C)=8.41, A(N)=7.80, A(O)=8.67).The metal abundances were scaled according to overall metallicity of the model as-suming an enhancement of the a -elements by +0.4 dex at metallicities [ M / H ] < − × ×
150 to 160 × ×
200 points for the hydrodynamical grid. The decision about the resolution primarilyhinges on the effective temperature of the model, hotter models usually require ahigher resolution. The wavelength dependence of the radiation field is representedby 5 multi-group bins in the case of solar metallicity, and 6 bins at sub-solar metal-licities, following the procedures laid out by Nordlund (1982); Ludwig (1992);Ludwig et al. (1994); V¨ogler et al. (2004). For test purposes we have calculated afew models with more bins. Since it is of relevance for the discussion later, weemphasize that all opacity sources – including scattering opacities – are treated astrue absorption. The sorting into wavelength groups is done applying thresholds inlogarithmic Rosseland optical depth { + ¥ , . , − . , − . , − . , − ¥ } for the 5-bin, Page: 2 job: ludwig_roma macro: svmult.cls date/time:27-Oct-2018/18:58
D Model Atmospheres of Red Giant Stars 3 and { + ¥ , . , . , − . , − . , − . , − ¥ } for the 6-bin schemes. In all but one bin aswitching between Rosseland and Planck averages is performed at a band-averagedRosseland optical depth of 0.35; in the bin gathering the largest line opacities, theRosseland mean opacity is used throughout. The decisions about number of bins,and sorting thresholds are motivated by comparing radiative fluxes and heating ratesobtained by the binned opacities in comparison to the case of high wavelength res-olution. LHD
Due to still present limitations in the realism (e.g. by the limited wavelength resolu-tion) of 3D model atmospheres it is often advantageous to work differentially, andexpress 3D effects relative to a 1D comparison structure. To this end we developeda 1D stellar atmosphere code called
LHD which employs the same opacities andequation-of-state as the 3D code
CO5BOLD . The convective energy transport ismodelled in the framework of mixing-length theory as described in Mihalas (1978).The resulting 1D stratifications are in hydrostatic and radiative-convective equilib-rium. See Caffau et al. (2007) for more details on our approach of deriving abun-dance corrections.
Collet et al. (2007) presented 3D-1D abundance corrections for RG models at effec-tive temperatures of around 5000 K, log g = .
2, and metallicities ranging from solarto [ M / H ] = −
3, using the 3D code of Nordlund & Stein (Stein & Nordlund, 1998).Two similar studies were presented by Dobrovolskas et al. (2010) and Ivanauskas et al.(2010) who used
CO5BOLD and
LHD models at about T eff = g = .
5, with metallicities down to [ M / H ] = − CO5BOLD -based abundance corrections are usually noticeably smaller in magni-tude, in particular at the lowest metallicities. Obviously, this is an unfortunate situ-ation, and one would like to see a higher degree of consistency among results fromdifferent 3D codes.In a recent paper, Collet et al. (2011) suggested the treatment of scattering in thesimulations as the reason for the discrepant abundance corrections for RGs at lowmetallicity. The main scattering process is Rayleigh scattering by neutral hydrogen.This is perhaps the simplest case of scattering and can be modelled as coherentisotropic scattering in the continuum. Collet and collaborators implemented a propertreatment of this kind of scattering in 3D. They also put forward an approximate
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Hans-G¨unter Ludwig and Matthias Steffen treatment of scattering by simply leaving out scattering contributions in the binnedopacities in the optically thin regions. They showed that this approximate treatmentprovides results in close agreement with the exact treatment. They further performeda comparison with the case where scattering is treated as true absorption – as isthe case in the
CO5BOLD models. Their models of 2007 used the approximatetreatment of scattering. The models show a sensitive dependence of the resultingtemperature stratification on the treatment of scattering. In their [ M / H ] = − t Ross = − d3t50g25mm30n01, d3t50g25mm30n02 −5 −4 −3 −2 −1 0log t Ross T [ K ] true absorption CO5BOLDtrue absorption 1D LHDapprox. scattering CO5BOLDapprox. scattering 1D LHD Fig. 1
Comparison of the mean temperature structures of two different 3D
CO5BOLD hydrody-namical model atmospheres (solid) and associated 1D LHD models (dashed), computed with adifferent treatment of radiative transfer. In the first case (dark [red] curves), the continuum scatter-ing opacity is treated as true absorption opacity, while in the second case (light [green] curves), thecontinuum scattering opacity is ignored in the optically thin layers. For the 3D models, averagingwas performed on surfaces of constant Rosseland optical depth and over 70 equidistant snapshotscovering a total of 140000 s.
It appears plausible that the differing 3D-1D abundance corrections are a conse-quence of the different thermal structures resulting from the different treatment ofscattering in the
CO5BOLD and Nordlund-Stein class of models. To test this idea,we calculated a RG model with the same atmospheric parameters as before but withthe approximate treatment of scattering as suggested by Collet et al. (2011). Fig-ures 1 and 2 illustrate the outcome. The most striking aspect is that our modelsshow a very much reduced sensitivity to the treatment of scattering in comparisonto the models of Collet and co-workers. The approximate treatment of scattering
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D Model Atmospheres of Red Giant Stars 5 d3t50g25mm30n01, d3t50g25mm30n02 −5 −4 −3 −2 −1 0log t Ross D T r m s [ K ] true absorptionapprox. scattering Fig. 2
Total rms temperature fluctuation D T rms , tot as a function of Rosseland optical depth forthe two 3D models shown in Fig. 1, computed as D T rms , tot = q h T i x , y , t − h T i x , y , t , where h . i x , y , t denotes horizontal averaging over surfaces of constant Rosseland optical depth and over time. Wehave verified that the amplitude of the total temperature fluctuation is completely dominated by thespatial temperature variations: D T rms , tot ≈ D T rms , xy = Dq h T i x , y − h T i x , y E t . leads to a structure which is only 120 K cooler at log t Ross = −
4, in comparisonto ≈
600 K found by Collet et al. (2011). This also carries over to the temperaturefluctuations which are little affected by the treatment of scattering (see Fig. 2). Wealready emphasized the importance of a differential approach, and Fig. 1 also showsthe effects on the associated 1D
LHD models. Temperature differences between the1D and 3D models at given optical depth are changing even less. While we didnot perform spectrum synthesis calculations yet to derive new abundance correc-tions, we consider it as unlikely that the modest changes in the thermal structurecan change our abundance corrections so much that they become consistent withthe values of Collet et al. (2007).The situation remains puzzling. The very different sensitivity to the treatment ofscattering is difficult to explain. We only can hint at the differences in the calcula-tion of the band-averaged opacities in the various codes: Collet and collaboratorsuse intensity-averaged opacities in the optically thin regions, while we use Planck-averages – except for the band collecting the strongest lines where a Rosselandaverage is used throughout. We speculate that these choices, together with the defi-nition of the opacity bins, may have a significant influence on the resulting thermalstructures and their sensitivity to the treatment of scattering.
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Hans-G¨unter Ludwig and Matthias Steffen a MLT and turbulent pressure
It is well known from the theory of stellar structure that convection is generallyan efficient means of transporting energy, and that it establishes a thermal struc-ture close to adiabatic. Only in the vicinity of the boundaries of convective regionsnoticeable deviations from adiabaticity occur. In convective envelopes of late-typestars the upper boundary of the convective envelope – usually located close to oreven in the optically thin layers – constitutes the bottle-neck for the energy trans-port through the stellar envelope assigning a special role to it. Despite its smallgeometrical extent and low mass, it largely determines the properties of the convec-tive envelope as a whole. It is the value of the entropy of the adiabatically stratifiedbulk of the convective s env which is most important from the point of view of stellarstructure since it influences the resulting radius and effective temperature of a stellarmodel. s env is controlled by the efficiency of convective and radiative energy trans-port in the thin, superadiabatically stratified surface layers. 3D model atmospherescan be applied to model this region, and allow to quantify the mutual efficiency ofthe convective and radiative energy transport, and to predict s env . Comparing themodel predictions to standard 1D models based on mixing-length theory (MLT) thevalue of s env can be translated into a corresponding mixing-length parameter a MLT (Trampedach et al., 1999; Ludwig et al., 1999, 2008).In stellar evolution calculations the free mixing-length parameter is usually cal-ibrated against the Sun. However, it is unclear whether mixing-length theory pro-vides a suitable scaling of the convective efficiency at constant a MLT across theHertzsprung-Russell diagram. The depth of the surface convective envelope andthe related a MLT can be constraint by asteroseismology. However, degeneracieswith other parameters often make it difficult to obtain a unique solution (e.g.Goupil et al., 2011). Hence, it would be useful to have an independent estimateavailable which 3D models can provide in principle.In main-sequence models turbulent pressure plays generally only a minor rolebut becomes relatively more important towards lower gravities – and causes trou-ble when one is interested in a well-defined calibration of the mixing-length pa-rameter. Figure 3 shows the average temperature profile of a 3D red giant model( T eff ≈ g =1.0, [ M / H ] =0.0) in comparison to standard 1D model atmo-spheres of the same atmospheric parameters. While the turbulent pressure P turb isnaturally included in the 3D models, it is modelled in a ad-hoc fashion in 1D mod-els, assuming a parameterisation P turb = f turb r v , where f turb is a free parameter oforder unity, r the mass density and v c the convective velocity according MLT.Figure 3 shows that it is essentially impossible to reproduce the mean thermalprofile of the 3D model with any of the 1D models – irrespective of the choiceof a MLT and f turb . The failure is related to the local nature of MLT confining theaction of the turbulent pressure gradients strictly to the convectively unstable re-gions. While formally one can still match s env of the 3D model by a 1D profilewith suitably chosen a MLT and/or f turb such a match becomes physically little mo-tivated, and is unlikely to provide a robust scaling with changing atmospheric pa-rameters. An improved 1D convection description including non-local effects like Page: 6 job: ludwig_roma macro: svmult.cls date/time:27-Oct-2018/18:58
D Model Atmospheres of Red Giant Stars 7 −4 −2 0 2 4 6 8log t ross S pe c i f i c en t r op y [ e r g g − K − ] env Fig. 3
Entropy-optical depth profiles (horizontal and temporal average) of a 3D red giant model(thick solid line) in comparison to 1D stellar atmosphere models of different a MLT leaving out (thinsolid lines) or including (dashed and dashed-dotted lines) turbulent pressure. The lines are labeledwith values a MLT / f turb (details see text). The horizontal dashed line indicates the value of s env aspredicted by the 3D model. overshooting is clearly desirable to handle this situation. Empirical calibrations of a MLT using giants are likely to suffer from ambiguities related to the way turbulentpressure is treated in the 1D models. One may take the result as an indication thattaking recourse to 1D models is not warranted, and one may give up the benefitsof a differential approach by relating 3D to 1D structures. Alternatively, one maytake the absolute entropy of the convective envelope (perhaps translated to equiv-alent pressure-temperature pairs) as predicted by the 3D model as constraint to bematched in 1D stellar structure models.
High-precision photometry of satellite missions (foremost M
OST , CoRoT,
Kepler )allow the detection of stellar variability associated with the random changes of thegranulation pattern on the surfaces of late-tape stars – by asteroseismologists usu-ally referred to as “granular background noise”. 3D model atmospheres represent thegranulation pattern in detail and allow to predict the power spectrum of the variabil-ity signal (Trampedach et al., 1998; Svensson & Ludwig, 2005; Ludwig, 2006). De-spite this possibility, no comprehensive theoretical study has been conducted so far.One of the reasons is that long time series need to be calculated to collect sufficientstatistics, which is computationally demanding. The F-dwarf HD 49933 – a promi-nent CoRoT-target – is an exception for which Ludwig et al. (2009b) performedan analysis. However, the growing body of observational data in particular for giantstars should motivate further efforts in this direction. Recently, Kjeldsen & Bedding(2011) suggested a new scaling relation for the amplitude of solar-like oscillations,and also discuss the scaling of the granulation background signal. It would be in-
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Hans-G¨unter Ludwig and Matthias Steffen teresting to see whether 3D model atmospheres can lend further support to the sug-gested relations.To illustrate the feasibility, we show in Fig. 4 a rough comparison of the photo-metric variability between the RG HD 181907 (HR 7349) and predictions from two3D models. The plot focuses on the frequency region where the granulation-relatedsignal is expected. CoRoT acquired a high-quality time series for HD 181907;Carrier et al. (2010) give atmospheric parameters 4780 ± . ± . − . ± .
10 ( T eff /log g / [ M / H ] ). The two 3D models have atmospheric parameters 4500/2.5/0.0and 5000/2.5/0.0, bracketing the star in effective temperature, as well as havingcomparable surface gravity and metallicity. Although no dedicated modelling wasperformed, the spectra appear quite similar. n [mHz]10 −1 P n [ pp m / m H z ] Fig. 4
Comparison of power spectra of photometric variability. Black line: HD 181907 as observedby CoRoT. The predicted power spectra from two 3D models are depicted with light [green ]( T eff = T eff =
3D model atmospheres of cool stars, including red giants, have reached a level ofrealism which allows a direct confrontation with observations. In some areas theyallow to make predictions beyond the capabilities of classical 1D models. However,as we have seen there still exist modelling challenges, and last but not least quitesome work is still necessary to fully exploit the potential of such models.
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D Model Atmospheres of Red Giant Stars 9
References