## 2. The treatment of convection## 2.1. Synthetic colours from convective model atmospheresThe Kurucz (1979a) model calculations represented a landmark in the study of stellar atmospheres. These included the opacity for approximately 900 000 atomic lines and provided realistic emergent flux distributions, spectra and colours of O, B, A, F, and G stars. Since molecular line opacity was not included, systematic errors began to appear in models and fluxes for 6000 K (Relyea & Kurucz 1978). Nevertheless, they have been enormously successful and widely used. Relyea & Kurucz (1978) presented theoretical Possible reasons for the discrepancies between 8500 K and 6000 K were discussed by Relyea & Kurucz (1978). They concluded that the probable sources of error included improper representation of opacities and improper treatment of convection. They stated that convection may account for part or even all the discrepancy between the models and the observations. The choice of mixing length, , might be physically suspect and lead to grossly overestimated convective flux. They showed that changing convective flux can easily induce large changes in the colours (see their Fig. 12). In light of the work of Relyea & Kurucz (1978), several
attempts have been made to improve the accuracy of the model colours
by modifying the treatment of convection. It has to be remembered that
the reason for the discrepancies may not be totally due to convection;
missing opacity (atomic and molecular) could well be as important. In
fact, improvements to opacity has been an ongoing project by Kurucz
(1991a). Others have taken the model Kurucz (1979b) introduced an improvement to the treatment of
convection, compared to that used in Kurucz (1979a), by considering
the different amount of energy loss by the "convective elements"
during their life time in an optically thin medium as compared to an
optically thick one where the diffusion approximation to radiative
transfer is assumed. Lester et al. (1982) discussed the modifications
proposed by Deupree (1979) and Deupree & Varner (1980) to include
"horizontally averaged opacity" and a "variable mixing length". The
model Recently, a model of turbulent convection has been proposed (Canuto & Mazzitelli 1991, 1992; Canuto 1996b) to overcome one of the most basic short-comings of MLT, the "one-eddy approximation". Within this approximation it is assumed that one eddy which has a given size as a function of the local mixing length (and which is usually called "bubble" or "convective element") is responsible for all the transportation of energy due to convection. Because of the one-eddy approximation the MLT systematically overestimates the flux for inefficient convection and underestimates it in the efficient case (Canuto 1996b). The new theory suggested first in Canuto & Mazzitelli (1991) adopts a turbulence model which accounts for eddies of various sizes (scales) that interact with each other. If we take the quantity S, the product of Rayleigh and Prandtl number, as a measure of convective efficiency, the new model, known as the CM model, predicts ten times more flux than MLT for the case of efficient convection and only one tenth of MLT's values in the inefficient case. As in an incompressibility model the pressure becomes a function of the velocity field itself, it is no longer an independent variable and one can no longer construct a unit of length of the type . The only remaining length is the geometrical distance to the nearest stable layer, . This choice also leads to a great degree of generality which was recently confirmed by Stothers & Chin (1997). No free parameter (analogue to MLT's parameter ) was necessary to perform their -luminosity calibrations of the red giant branch for stars with masses ranging from 1-20 . In this sense, the CM model has no adjustable free parameters, unlike MLT which can be "adjusted" to fit observations. Despite the loss of a fit parameter the CM model has had considerable success in explaining observations (see Stothers & Chin 1995; and Canuto 1996b for references). However, the model is still a local concept that has been adapted for one dimensional geometry. Thus, the CM model cannot describe the phenomenon of overshooting or the influence of large scale structures on the integral of the radiation field over the stellar disk. This can be done by "large eddy simulations", which are, however, on a different level of numerical complexity and up to now have not included the same sophistication in their treatment of radiative transfer as classical model atmospheres (e.g. Nordlund & Dravins, 1990; Freytag, 1996). ## 2.2. New grids based on the CM modelIn 1995 the CM convection model was implemented in the ATLAS9 code (Kupka 1996a). The model was tested by F. Kupka with various other prescriptions of a local length. As part of the same project the "approximate overshooting" was applied to the CM model, and a correction for convection in optically thin media was investigated. After applying the model atmosphere code with various treatments of convection to several regions throughout the whole lower and central part of the HR diagram, it was decided to use the CM model in its original form for model grid computations, because the differences found were either small or lacked a convincing physical motivation that could be corroborated by experimental tests. More details on these experiments and on the implementation itself will be discussed in Kupka & Canuto (1997). A brief description of results for A and F type stars has already been given by Kupka (1996b). More extensive discussions of the properties of models in various regions of the HR diagram will be presented in Kupka & Canuto (1997), thus only a few remarks will be given here. In the upper part of a stellar atmosphere (with
) the radiative time scale (see Canuto 1996b) is
necessarily very short as (most of) the observed radiation leaves the
star in this region which must hence be an efficient means of energy
transportation. Well below these layers, at ,
the ionization of hydrogen takes place and decreases the efficiency of
radiative transfer. Where the radiative time scale becomes comparable
to that of buoyancy, energy can be transported by means of convection
instead of radiation. In the observable atmosphere layers of A, F, and
G stars we only encounter the case of inefficient convection. As the
CM model predicts If we map the HR diagram onto a -
plane, we may distinguish between four regions
of different atmospheric conditions for convection. For main sequence
stars and a well above 10 000 K the
Continuous manual interaction during the computation of large grids
of models is rather tedious, but software tools may reduce the
necessary amount of work. To facilitate the determination of
and from photometric
observations, a suite of empirical calibrations was assembled by
Rogers (1995). In addition to this toolbox, he provided another set of
tools which unifies the access to and application of software for the
computation of model grids, synthetic fluxes, and synthetic colour
indices. This was achieved by adapting those parts of the Abundance
Analysis Procedure (AAP) tool (see Gelbmann et al. 1997), which allow
Grids of CM
In this paper we compare how the different treatments of convection
affect the - Standard ATLAS9 models using mixing-length theory with approximate convective overshooting, as modified by Castelli (1996). These models, called COLK95 by Castelli et al. (1997), will be referred to as the MLT OV models in this paper (grids provided by F. Castelli).
- Standard ATLAS9 models using mixing-length theory, but without convective overshooting. These will be referred to as MLT noOV models.
- Modified ATLAS9 models using the Canuto & Mazzitelli (1991, 1992) model of turbulent convection. These will be referred to as the CM models.
All the model grids were calculated identically, except for the treatment of convection. For a detailed description of MLT in ATLAS9 and a comparison between MLT OV and MLT noOV models refer to Castelli et al. (1997). In this paper we are primarily concerned with a comparison between MLT and CM treatments of convection. © European Southern Observatory (ESO) 1997 Online publication: March 24, 1998 |