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PARAMETERS OF THE SOLAR CONVECTION ZONE IN
EVOLUTIONARY AND SEISMIC MODELS
VLADIMIR A. BATURIN
Queen Mary & Westfield College, Mile End Road,
London E1 4NS, UK
AND
SERGEY V. AYUKOV
Sternberg Astronomical Institute, Moscow 119899, Russia
Abstract. Three alternative approaches in evaluating of the entropy of the
adiabatic part of the solar convection zone are compared.
Key words: solar physics, equation of state
1. Entropy as a parameter of solar structure
The specific entropy (s) and the helium abundance (Y ), chief parameters
of the solar convection zone, are still poorly determined. They are principal
characteristics of the equation of state (EOS hereafter), whereas others
apparent to be significant EOS's features can be rarely characterized in a
simple way (e.g. nonideality deviations).
We focus on the specific entropy, which is ''implicitly'' presented in a
wide range of solar studies. Thus the evolutionary calibration of a convec­
tion parameter appears to be anything else but calibration of the entropy.
Entropy defines (T \Gammaae) profile of the adiabatic convection zone, but together
with surface conditions it accurately determines the mechanical structure
(ae(r), m(r)) too. A direct way to calibrate the entropy is adequate modelling
of the superadiabatic convection and finding the jump of the entropy be­
tween the atmosphere and adiabatic layers. On the other hand, one should
equalize the entropy jump before compare of different convection theories.

VLADIMIR A. BATURIN AND SERGEY V. AYUKOV
Figure 1. Filled circles correspond to the standard solar model from Baturin and Ayukov,
1995 (labels are opacity tables used). Filled diamonds are for models of others authors.
Dashed line connects the crosses is Y (s)­dependence of models with helioseismic sound
of speed. Two empty circles with error bars represent results of phase­shift calibrations
of the helium ionization zone (with type of EOS as labels). Filled square with a vertical
error bar is the result of the convective calibration by Ludwig et al., 1996. Dot­dashed
lines show density at the radius r = 0:8 in models.
An arbitrary constant in the thermodynamic definition of entropy is
naturally eliminated in the statistical descriptions of contemporary EOS's.
The numerical approximation of absolute entropy in MHD EOS needed for
comparative studies is given by Baturin and Ayukov, 1996 (Hereafter MHD
EOS is collectively for Mihalas, Hummer, D¨appen).
2. Determination of the entropy in a solar model
Classic procedure of ''evolutionary calibration'' is in matching of a convec­
tive parameter and the initial helium abundance to get the correct radius
and luminosity of a model of the present Sun. As a result one can get the
helium content (and such estimation is widely accepted in astrophysics)
and, at the same time, specify the entropy of the adiabatic convection zone
as a conjugated to helium parameter. On Fig. 1, where coordinates are Y env
and s env , the parameters of several standard models (Baturin and Ayukov,
1995) with different opacity are plotted as filled circles.
But we intend to extend the method of ''model calibration'' of the en­
tropy to nonevolutionary models of the Sun. Indeed entropy calibration is
essentially connected with mechanical relations. The condition of hydro­

Parameters of the solar convection zone in evolutionary and seismic models
static equilibrium is enough to get a mechanical structure (i.e. P (r); ae(r))
from the profile of u = P=ae = c 2 =\Gamma 1
(Dziembowski et al., 1990). Moreover,
the necessary information can be restricted further. Due to mass concentra­
tion in solar­type model, the profile u(r) only below considered point is suf­
ficient to get (P; ae) r0 in the lower part of the convective zone. Furthermore,
Baturin and Ayukov, 1996 showed that u(r) in the very core of the model
(r ! 0:3) does not affect convection zone entropy. So to define mechanical
parameters we need u(r) only in the radiative zone (0:3 ! r ! r CZ ), which
is available from helioseismic inversion of c 2 (r), assuming \Gamma 1 is close to 5=3.
But knowledge of (P; ae) is equivalent to some Y (s)­dependance, because Y
is not mechanical variable (and it does not appear in this consideration),
but it affects an absolute value of entropy. As a result, any model with some
given (seismically inverted one, for example) sound speed in the radiative
zone will lay on the Y (s)­dependence. This dependence represents also the
line of the equial density at a fixed temperature/radius, see Fig. 1.
3. Calibration of entropy on the helium ionization zone
Apart from the model calibration of entropy, a study of the helium ion­
ization zone can be used as an alternative approach. The peculiarity of a
sound speed profile gives a trace in the oscillation spectrum and can be cal­
ibrated with the phase­shift of a frequency­dependent function (Vorontsov
et al., 1992, Baturin, Vorontsov, 1995, Perez Hernandez and Christensen­
Dalsgaard, 1994, Basu and Antia, 1995), or within helioseismic inversion,
based on variational principle (Dziembowski et al., 1995, Kosovichev, 1995).
We refer mainly on our results of the phase­shift calibration due to two
reasons. First, the variational calibrations are not unconditional in Y \Gamma s
space, because only models with ''good'' sound speed are considered, and
they always belong to specific Y (s)­dependence. Second, the value of the
calibrated entropy is rarely available from other authors. Our present re­
sults were obtained with a technique very close to described by Baturin,
Vorontsov, 1995, except two improvements: we analyze the differential sig­
nal and scale the atmospheric opacity individually in every model.
Because the helium zone calibration is dealing with the thermodynamic
peculiarity, it is very sensitive to EOS. We used two modern EOS's -- MHD
and OPAL (OPAL EOS described by Rogers et al., 1996) and compared
with other's results (see Table 1 and Fig. 1). We should point out that
neither MHD nor OPAL EOS's do not supply us with an exact descrip­
tion of the helium ionization zone (although OPAL EOS has advantages
in the description of deep layers of the convection zone) -- the residuals for
calibrated models still exceed a level of noise. So the differences between
Y ­calibrations with both EOS's (mentioned also by Kosovichev, 1995) and

VLADIMIR A. BATURIN AND SERGEY V. AYUKOV
TABLE 1. Helioseismic calibration of the helium zone
Author(s) MHD EOS OPAL EOS
Our results, 1996 Y = 0:25 \Sigma 0:008 Y = 0:23 \Sigma 0:008
S=Rg = 21:05 \Sigma 0:08 S=Rg = 21:8 \Sigma 0:1
Kosovichev, 1995 Y = 0:232 \Sigma 0:006 Y = 0:253 \Sigma 0:006
Basu and Antia, 1995 Y = 0:246 Y = 0:249
Dziembowski et al., 1995 Y = 0:244 \Sigma 0:003 Y = 0:2505
Perez Hernandez,
Christensen­Dalsgaard, 1994 Y = 0:242 \Sigma 0:003
the disagreement with the model calibration do not look surprising.
4. Entropy from hydrodynamic simulation
Ludwig et al., 1996 presented the result of the most direct calibration of
the entropy -- from 2D numerical hydrodynamic calculation of convection.
Results for model with Y = 0:24 are also plotted on Fig. 1. Note that
their point is rather close to the Y (s)­dependence for the models with
helioseismic sound speed.
5. Conclusions
The model calibration of the entropy appears to be most certain. The direct
helium ionization calibration is perhaps too sensitive to any EOS's errors,
so the discrepancy between approaches can indicate EOS's errors.
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