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Bull. Astron. Soc. India (1996) 24,
Solar models with helioseismologically correct sound speed profile
S. V. Ajukov
Sternberg Astronomical Institute, Moscow 119899, Russia
V. A. Baturin
Astronomy Unit, Queen Mary and Westfield College, London E1 4NS, UK
Sternberg Astronomical Institute, Moscow 119899, Russia (permanent address)
Abstract. The problem of computation of the model of the present Sun with
given sound speed profile in the solar radiative zone is considered. It is shown
that the chemical composition is still a free parameter but the entropy of the
adiabatic part of the convection zone is determined by this procedure. Pa­
rameters of the models are compared with the helioseismic calibration of solar
envelope and the disagreement is revealed. A study of the seismic properties
of the models' cores indicates that the best model is the one closest to the
standard solar model.
Key words: solar interior, helioseismology
1. Introduction
We are investigating the possibility of computing of the helioseismologically consistent
model of the Sun. The final goal would be the model those frequencies are indentical
to the observed ones. However, this way is plagued with many difficulties, mainly due
to two facts: 1) frequencies depend on lots of factors, like outermost layers properties;
2) frequencies are integrands over wide parts of the Sun. To avoid these difficulties we
used special derivatives instead of an oscillation spectrum: the parameters of the solar
envelope: Y (surface helium abundance) and S (specific entropy of the adiabatic part
of the convection zone) (Baturin, Vorontsov 1995), convection zone depth (Christensen­
Dalsgaard et al. 1991) and sound speed profile (Vorontsov, Shibahashi 1991). Then we
try to obtain the model of the Sun which is consistent with them.
To be able to get a model which is different from the standard one, we are broaden­
ing the class of solar models by introducing an assumption that modern opacity tables

S. V. Ajukov, V. A. Baturin
(namely, Livermore opacities (Iglesias, Rogers 1991)) can be modified in a reasonable
manner. The opacity is probably the most poorly known component of solar model
physics, so this assumption wouldn't look unnatural.
The possibility to obtain a solar model with the seismic envelope parameters (S, Y )
and convection zone depth was studied by (Baturin, Ajukov 1995a,b). It is shown that
such model can be computed, but required opacity changes are very large and have non­
uniform shape (see Baturin, Ajukov (1995b)). Furthermore, this `best' model has a sound
speed profile inconsistent with seismic one (Vorontsov, Shibahashi 1991); see Fig. 1.
Figure 1. Sound speed profiles in two solar models -- standard model using Livermore opacities (Iglesias,
Rogers 1991) and the model M1 from (Baturin, Ajukov 1995b) which has opacity adjusted to obtain
right values of solar envelope parameters. Zero (c 0 ) is the sound speed from (Vorontsov, Shibahashi
1991), derived from oscillation frequencies.
As you can see from Fig. 1, the standard solar model has sound speed profile pretty close
to helioseismic one.
2. Solar models with given sound speed profile
Having failed to obtain solar model using seismic envelope parameters (Baturin, Voro­
ntsov, 1995), one can try to use the sound speed profile instead. The sound speed profile
isn't available everywhere in the Sun: it cannot be obtained with good accuracy in the
solar core mainly due to restricted range of acoustic spectrum. On the other hand, all
solar models have nearly the same sound speed profile in the most part of the convective
zone due to adiabatic stratification. Hence we only considered fitting the sound speed in

Solar models with helioseismologically correct sound speed profile
the radiative zone of the Sun, i.e. from r=R = 0:3 to the lower boundary of the convection
zone (approximately r=R = 0:7).
Desirable sound speed profile can be achieved by choosing opacity value according to
the sound speed gradient:
d ln c 2 = ln P = 1 \Gamma 1=\Gamma 1 \Gamma Q
Ÿ
r ad \Gamma
3L
16acúGM r
P
T 4
Ÿ
–
(1)
(assuming \Gamma 1 = const).We assume that no diffusion takes place, so helium abundance
in the convective zone is the same as in the radiative zone. However we still have one
free parameter to adjust while fixing sound speed profile: a surface helium abundance
Y . We've computed a set of models with different Y s, (Baturin, Ajukov 1995b); opacity
corrections in these models are plotted on Fig. 2.
Figure 2. Opacity corrections in the models with different surface helium abundancies Y . All models
have helioseismic sound speed profile in the radiative zone.
Correction is minimal for the model with Y ú 0:28. Most of the latest helioseismic he­
lium abundance estimations agree at Y = 0:25 (Y = 0:2505 \Sigma 0:004 (Baturin, Vorontsov
1995), Y = 0:252 \Sigma 0:003 (Antia, Basu 1994), Y ú 0:25 (Christensen­Dalsgaard, P'erez
Hern'andez 1991), Y = 0:268 \Sigma 0:01 (D¨appen et. al. 1991), Y = 0:234 \Sigma 0:003 (Dziem­
bowski et al. 1995)). Model with Y = 0:25 requires very large opacity correction in the
core. The situation indicates that structure of the core is close to one from the standard
model, but there must be gradient of helium abundance in the radiation zone, to achieve
the low Y in the convection zone. This may point to diffusion­like effects.

S. V. Ajukov, V. A. Baturin
We also have computed a small spacing of low­l frequencies which is determined
mainly by the solar core structure. Comparison with observational data values (Elsworth
et al. 1994) indicates that the best model again has high helium abundance in the core
(Y = 0:28), just like the standard solar model. (see Table 1).
Table 1. Oscillation properties of model's cores.
l = 0; 2 l = 1; 3
ffi ffi 0 dffi=dn ffi ffi 0 dffi=dn
Y = 0:24 8:570 8:657 \Gamma0:2950 15:335 15:341 \Gamma0:4215
Y = 0:25 8:463 8:533 \Gamma0:2943 15:199 15:212 \Gamma0:4295
Y = 0:26 8:157 8:233 \Gamma0:2928 14:799 14:830 \Gamma0:4313
Y = 0:28 9:157 9:210 \Gamma0:3034 16:077 16:063 \Gamma0:4281
Standard model 9:201 9:266 \Gamma0:3061 16:150 16:137 \Gamma0:4302
Elsworth et al. 8:960 9:040 \Gamma0:3044 15:770 15:747 \Gamma0:4435
3. Conclusions
1) It is shown that models with the helioseismic sound speed profile exhibit a relation
between the specific entropy of the convection zone and opacity in radiative zone. This
relation is parametrized by the surface helium abundance.
2) Convective zone parameters in the model with the seismic sound speed profile are
inconsistent with values obtained in phase­shift helioseismic study (Baturin, Vorontsov
1995). However, the standard solar model has sound speed profile rather close to the
helioseismic one.
3) A set of models with seismic sound speed profile is computed. Seismic properties
of the model's cores again indicate that the best solar model has opacities close to recent
Livermore ones.
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