Astron. Astrophys. 328, 349-360 (1997)

2. The treatment of convection

2.1. Synthetic colours from convective model atmospheres

The 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 uvby colours based on the Kurucz (1979a) model fluxes. They discussed the accuracy of these colours and compared them to the observed colours of the stars in the Hauck & Mermilliod (1975) catalogue. The colours were found to agree well with the observations, except for late-A and early-F stars. Theoretical colours were in error around 7500 K, and colours were discrepant for 0.050, first too small and then too large. The colours were not expected to agree for 6000 K due to the lack of molecular opacity.

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 uvby colours and adjusted them until they agreed better with colours of stars with known and (e.g. Philip & Relyea 1979; Moon & Dworetsky 1985; see also Smalley 1996). While this approach does give good agreement with fundamental or standard stars, it masks any physical problems with the models.

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 uvby colours could be brought closer to the observed colours, but not enough to entirely remove the discrepancies found by Relyea & Kurucz (1978). For ATLAS9 as published by Kurucz (1993) further modifications were included, the "horizontally averaged opacity" and "approximate overshooting". The original formulation of the "approximate overshooting" lead to several discontinuities in the colour indices obtained from the models (e.g. North et al. 1994). Castelli (1996) presented a discussion on convection in ATLAS and re-defined the "approximate overshooting" so as to remove the discontinuities. A detailed description of mixing-length convection and the modifications used in ATLAS is given by Castelli et al. (1997).

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 model

In 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 less convective flux than MLT for an inefficient convective region, the temperature gradient has to be closer to the purely radiative gradient in the top layers of the convection zone for a larger range of .

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 T - relation is entirely radiative. For the early A-type stars there is a region around the zone of hydrogen ionization that is convectively unstable according to the Schwarzschild criterion, but convective transport remains so inefficient that the resulting temperature gradient cannot be distinguished from the radiative one. In the case of the MLT, minor deviations from the radiative gradient can be observed beginning around 8500 K for = 4. This convection zone is still entirely contained in the stellar atmosphere. For 7500 K the convection zone finally extends below the atmosphere (normal, solar like convection as opposed to "plume convection" for higher ; see Kurucz 1996). Examples and illustrations for the MLT case can be found in Kurucz (1996), as well as in Castelli et al. (1997). For lower surface gravity these transitions occur at lower . For the case of the CM model the last two transition regions occur for about 1000 K less than in the MLT case, but otherwise they have very similar properties (see below). On the other hand, the extent of the overall convectively unstable region of the HR diagram remains unchanged when changing from MLT to the CM model.

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 interactive computation of single ATLAS9 model atmospheres, for background computation of a grid of models. Automatic convergence to a prescribed value of flux constancy (typically less than 3% for the deepest layers) and a zero flux derivative with depth (typically less than 10% for the uppermost layers) is achieved by comparing the output information of ATLAS9 with the criteria just mentioned. The relatively large maximum error for the flux was used in the transition region from "normal" convection to "plume like" convection where the models tend to switch between a radiative and a convective solution for the bottom layer. Only a small neighbourhood of layers at the bottom is contaminated by this effect. Nevertheless, "long-time persistent errors" are created that may slow down convergence considerably. Similar holds for the flux derivative error of the top layer which also affects only a few nearby layers. However, the latter phenomenon is not related to convection. As both the very top and bottom layers mainly affect their local regions and as they either do not contribute to the observable flux (deepest layers with ) or cannot affect it any more (at ), this is a rather safe choice for general model atmosphere grid computations. If the criteria are not fulfilled after a maximum of 200 iterations (which might happen in the transition region from "plume" to "normal" convection if a simple grey model atmosphere is taken as a starting point for the temperature iterations), a notification is generated for the user and the model has to be converged interactively (as an alternative to still more iterations, a different starting model might be chosen or the temperature correction may be changed manually or a different algorithm could be used, if available).

Grids of CM uvby colours were calculated to include the whole range from 5500 K to 8500 K (spaced by 250 K) and from 2.0 to 5.0 (spaced by 0.25) for solar scaled metallicities ranging from -1.0 to 1.0 (spaced 0.5). A microturbulence of 2 km s^-1 was used for all grid models. For 8500 K the models are either totally radiative or have essentially radiative temperature gradients (similar to the case of Vega). Already for a = 8500 K and = 4.0, the temperature differences are less than 20 K for all layers, with resulting differences in colour indices which are an order of magnitude smaller than the typical errors assigned to synthetic colours as a function of physical parameters or vice versa (see Sect. 3). Hence, they can be smoothly completed by the original Kurucz (1993b) models. Fig. 1 shows the CM [ ] and [ ] grids for solar-composition models. The actual numerical values are given in Table 1. The full grids for solar and other metallicities ([-1.0], [-0.5], [0.0], [+0.5], [+1.0]) are available from the authors or by anonymous ftp at the Centre de Données de Strasbourg (CDS), following the instructions given in A&A 280, E1-E2 (1993).

Table 1. The CM uvby colours for solar metallicity models, normalized as described in Sect. 1.

Fig. 1a and b. The [ ] and [ ] grids for solar-composition CM models

In this paper we compare how the different treatments of convection affect the uvby colours for models with 8500 K. We discuss the colours calculated from three grids of solar-composition Kurucz (1993) ATLAS9 models:

1. 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).
2. Standard ATLAS9 models using mixing-length theory, but without convective overshooting. These will be referred to as MLT noOV models.
3. 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

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