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Astron. Astrophys. 346, 111-124 (1999)

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2. Methodical aspects

2.1. Hydrodynamical models of solar-type surface convection

We have obtained detailed 2-dimensional models of the surface layers of solar-type stars from extensive numerical simulations solving the time-dependent, non-linear equations of hydrodynamics for a stratified compressible fluid. The calculations take into account a realistic equation-of-state (EOS, including the ionization of H and He as well as formation of [FORMULA]-molecules) and use an elaborate scheme to describe multi-dimensional, non-local, frequency-dependent radiative transfer. Similar to classical model atmospheres, the hydrodynamical models are characterized by effective temperature [FORMULA], acceleration of gravity [FORMULA], and chemical composition. They include the photosphere as well as part of the subphotospheric layers, with an open lower boundary , allowing a free flow of gas out of and into the model. A fixed specific entropy [FORMULA] is (asymptotically) assigned to the gas entering the simulation volume from below. The value adopted for [FORMULA] uniquely determines the effective temperature of the hydrodynamical model. For details about the physical assumptions, numerical method and characteristics of the resulting convective flows see Ludwig et al. (1994) and Freytag et al. (1996).

2.2. From the surface to the base of a convection zone

Fig. 1a shows the mean entropy as a function of depth obtained from a hydrodynamical granulation model of the Sun by averaging over horizontal planes and over time. As in this example, our models in general do not extend deep enough to include those layers where the mean stratification of the convection zone becomes adiabatic. While the mean entropy stratification of the hydrodynamical models does not permit a direct determination of the entropy corresponding to the adiabat of the deep convection zone, the spatially resolved entropy profiles contain additional information. Fig. 1b displays the entropy profiles for an arbitrary instant of the sequence from which the mean stratification in Fig. 1a was computed. The granular convection pattern at the surface of solar-type stars is formed by broad hot upflows accompanied by concentrated cool downdrafts. Fig. 1b shows a remarkable entropy plateau in the subsurface layers, indicating that - in contrast to the narrow downdrafts - the gas in the central regions of the broad ascending flows is still thermally isolated from its surroundings. Neither radiative losses nor entrainment by material of low entropy can produce significant deviations from adiabatic expansion until immediately below the radiating surface layers. The height of the entropy plateau is essentially independent of time and corresponds to [FORMULA].

[FIGURE] Fig. 1a and b. Depth dependence of the entropy in the solar surface layers as obtained from hydrodynamical simulations ([FORMULA], [FORMULA], model code L71D07) performed on a 140 (x) by 71 (z) grid with frequency-dependent radiative transfer. The mean entropy (horizontal and temporal average) is shown in panel a , spatially resolved entropy profiles in panel b . Geometrical height zero corresponds to [FORMULA]. Note that the model comprises only the uppermost part of the 200 Mm deep solar convective zone.

We suggest that [FORMULA] may be identified directly with the entropy [FORMULA] of the deep, adiabatic convective layers:


This idea has been put forward by Steffen (1993) and Ludwig et al. (1997) and is based on the qualitative picture of solar-type convection zones proposed by Stein & Nordlund (1989) (hereafter referred to as "SN scenario") which is fundamentally different from MLT assumptions. According to this scenario the downdrafts continue all the way from the surface to the bottom of the convection zone, merging into fewer and stronger currents at successively deeper levels. The flow closes only near the base of the convective envelope. Most of the gas elements starting from the bottom of a deep convection zone overturn into neighbouring downflows before reaching the surface. Only a very small fraction of gas continues to the surface, reaching the layers corresponding to the location of the lower boundary of our hydrodynamical models essentially without entropy losses, following an adiabat almost up to the visible surface. Hence, [FORMULA] obtained from the simulations is the entropy of the warm, ascending gas throughout the convection zone. This, in turn, is very nearly equal to the mean (horizontally averaged) entropy [FORMULA] near the base of the convection zone because (i) the downflows are markedly entropy-deficient only near the surface and become continuously diluted by overturning entropy-neutral gas as they reach greater depths, and (ii) the fractional area occupied by the downdrafts decreases with depth.

We note that for this investigation it is actually irrelevant whether the downflows possess a plume-like character or take place in another form. The essential point is that sinking material does not locally affect the entropy of rising material by draining heat from it. The role of the downflows is reduced to a dynamical one: while sinking downwards they simply displace buoyancy-neutral material and push it upwards.

2.3. Functional dependence of [FORMULA] on stellar parameters

When considering the MLT picture and the SN scenario it is interesting to ask what stellar properties determine [FORMULA] or - in other words - what coordinates are appropriate to describe its functional dependence in the HRD. We argued that hydrodynamical models of the surface layers are able to provide information about [FORMULA]. The models are characterized by the atmospheric parameters and consequently we describe [FORMULA] in terms of them. Whether the standard atmospheric parameters [FORMULA] and [FORMULA] are the most suitable coordinates is not clear. One might speculate that e.g. the surface opacity is a physically more relevant quantity. Nevertheless, for solar-type stars the conditions at the stellar surface govern the global envelope structure and the standard atmospheric parameters are suitable coordinates to parameterize them. They have the advantage that they are external control parameters and not part of the solution of the problem. For solar-type stars with superadiabatic regions which are thin in comparison to the stellar radius we expect the same qualitative behaviour irrespective of whether we consider the MLT picture or the SN scenario. Global stellar parameters (mass, radius, or age) play only an indirect role for the entropy jump. The situation changes when the size of the granular cells or the thickness of the superadiabatic layer become comparable to the stellar radius. This might happen in giants and one has to account for effects introduced by the global stellar structure, i.e. sphericity effects.

2.4. [FORMULA] from envelope models

Although [FORMULA] allows the construction of the envelope structure with the necessary precision, it is of interest to translate this quantity to an equivalent [FORMULA] since this parameter is conventionally used in stellar structure models. For this purpose we computed grids of standard stellar envelope models (not subject to central boundary conditions) based on MLT, covering the relevant range of effective temperature and gravity and assuming [FORMULA] and solar metallicity for the chemical composition. In these models [FORMULA] was a free parameter. For given stellar parameters ([FORMULA], [FORMULA]) we obtained [FORMULA] as a function of [FORMULA]. By matching [FORMULA] from the RHD models we deduced the corresponding [FORMULA]. Fig. 2 illustrates this procedure for a number of representative models. Each panel shows the entropy profile of the envelope model matching [FORMULA] from a RHD simulation. For comparison two further envelope models are plotted with [FORMULA] varied by [FORMULA] with respect to the matching one, as well as an estimate of the uncertainty of [FORMULA] stemming from the temporal fluctuations of [FORMULA] which we observe during the simulation run. Although at first glance the matching is a straightforward procedure, it is important to take care of a number of fine points in order to ensure a unique and well-defined calibration of [FORMULA]. We discuss these points in the following.

[FIGURE] Fig. 2. Representative examples of entropy profiles (in units of [FORMULA]) as a function of Rosseland optical depth [FORMULA]. Each panel shows the [FORMULA]-averaged entropy of a RHD model (thick solid line) with envelope models (thin solid lines) for the matching [FORMULA] and [FORMULA] for a given [FORMULA] combination. In all models the radiative transfer was treated in the grey approximation. The dotted horizontal lines indicate [FORMULA] of the RHD model and its temporal RMS-fluctuations. Note the significantly different entropy jumps.

Concerning the opacities and EOS we tried to stay as close as possible to the physical description used in the RHD models. Due to this differential procedure, systematic uncertainties are substantially reduced. In the RHD models ATLAS6 opacities according Kurucz (1979) are adopted. These opacities contain no contribution from molecular lines. Since the envelope models reach much deeper than the RHD models we supplemented the ATLAS6 opacities by OPAL opacities for higher temperatures (Rogers & Iglesias 1992). Care was taken that the chemical mixture assumed in both data sets was as similar as possible. In the envelope models we used the RHD EOS throughout. It is a simple EOS taking into account the ionization equilibria of hydrogen and helium according to the Saha-Boltzmann equations.

The RHD EOS has the advantage of great smoothness and computational efficiency. It might appear that this description is too crude, especially for the envelopes of cooler objects where non-ideal effects become more pronounced. However, this simplification is tolerable since the entropy profile, which is the major concern here, is not sensitively dependent on the EOS. The entropy gradient is proportional to the difference between actual and adiabatic temperature gradient which in MLT is related to the convective flux. Hence, the constraint that a nominal total flux has to be transported essentially fixes the entropy gradient irrespective of the EOS employed, while the temperature profile is noticeably affected by the choice of the EOS.

Following the conventional way to treat the atmosphere in stellar structure models we prescribe a certain [FORMULA]-relation representing the temperature run in the optically thin regions. Its significance for the envelope structure emerges from the fact that it determines the level of the entropy minimum in the deeper atmosphere where the stratification becomes convectively unstable (the starting level for the entropy jump). In evolutionary calculations various descriptions of the atmosphere are commonly used, differing in the level of sophistication between simple analytical or empirical [FORMULA]-relations and full-fledged model atmospheres. Here we chose a [FORMULA]-relation that mimics closely the average atmospheric structure of the hydrodynamical models. We reproduce in particular the atmospheric entropy minimum found in the hydrodynamical models. The hydrodynamical models predict an average temperature structure in the deeper photosphere (around [FORMULA]) that closely resembles a stratification in radiative equilibrium. We emphasize that this statement refers to an averaging of the temperature on surfaces of constant optical depth in the RHD models. This averaging procedure preserves the radiative flux properties of the RHD models to a good approximation (see Steffen et al. 1995), and hence appears suitable to characterize their atmospheric structure in the present context since radiative surface cooling plays the dominant role for the convection driving.

In practice, we used an analytical fit to the exact grey [FORMULA]-relation for an atmosphere in radiative equilibrium according Unsöld (1955) in envelope models intended for comparison with RHD models adopting grey radiative transfer. From the small set of RHD models adopting frequency-dependent radiative transfer we derived the average [FORMULA]-relation and constructed an analytical fit to it. Later in this paper we shall discuss the role of grey versus frequency-dependent radiative transport in the RHD models, demonstrating the need for a clean distinction between both types of [FORMULA]-relations.

The influence of the [FORMULA]-relation grows as the entropy jump itself decreases, so it is largest for models at low effective temperature and high gravitational acceleration. In earlier papers on the subject (Ludwig et al. 1996, 1997) we used [FORMULA]-relations which were not selected on a strictly differential basis, leading to systematic differences between the older and the present calibration of [FORMULA]. Fig. 2 illustrates the match of the atmospheric entropy minima. (The range in optical depth over which the RHD models are plotted do not represent the full extension of the RHD models.) Our present procedure gives now an improved match with the largest deviation found for the ([FORMULA]) model. We expect that a further improved fit of the atmospheric stratification of the RHD model would lead to an increase of [FORMULA]. But note that using the [FORMULA] derived from models in this region of the HRD would produce a consistent [FORMULA] in a stellar structure model as long as it is basing on a description of the stellar atmosphere by a [FORMULA]-relation resembling an atmosphere in radiative equilibrium.

As a side-issue the Fig. 2 demonstrates that in RHD models the higher atmospheric layers are systematically cooler than in an atmosphere in radiative equilibrium. This is due to the situation that in a convective atmosphere the temperature is governed by the competition between adiabatic cooling of overshooting material and heating by the radiation field. Quantatively, our grey RHD models overestimate the temperature reduction since they do not include the important heating by spectral lines in the higher layers. However, as we will see in the next section this is of minor importance for the determination of [FORMULA]. The deviations between average RHD stratification and envelope models underline that our calibrated [FORMULA] only fits the asymptotic entropy and the entropy jump. The overall stratification is not well matched by an envelope model with the fitted [FORMULA].

Effects of turbulent pressure are ignored in the envelope models. This is not only done for computational convenience but also reflects our opinion that the recipes provided for its treatment in the framework of MLT overestimate the effects, doing more harm then good when included. In MLT the convective motions are restricted to the unstable layers according to the Schwarzschild criterion. At the upper boundary of a surface convective zone this produces a sharp decline of the convective velocity amplitude and a correspondingly large gradient of the turbulent pressure that alters the hydrostatic structure of a model significantly. In contrast, hydrodynamical models (see Freytag et al. 1996) predict a rather smooth run of the velocity amplitude in that region with a correspondingly smaller gradient of the turbulent pressure. Clearly, the inclusion of turbulent pressure in 1D envelope models requires a proper treatment of overshooting.

Finally, it is important to realize that a unique formulation of MLT does not exist. Rather, different versions of MLT are in use. In the following we refer to the version of MLT originally given by Böhm-Vitense (1958). To further elaborate this point, we explicitly present details of our MLT implementation together with commonly used other formulations in Appendix A.

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© European Southern Observatory (ESO) 1999

Online publication: May 6, 1999