Äîêóìåíò âçÿò èç êýøà ïîèñêîâîé ìàøèíû. Àäðåñ îðèãèíàëüíîãî äîêóìåíòà : http://crydee.sai.msu.ru/ftproot/pub/local/science/publications/hague.94/hague94p.ps.gz Äàòà èçìåíåíèÿ: Tue Jul 30 00:16:38 1996 Äàòà èíäåêñèðîâàíèÿ: Mon Dec 24 11:11:47 2007 Êîäèðîâêà: Ïîèñêîâûå ñëîâà: massive stars

STRUCTURE OF THE SUN'S CORE:
EVOLUTIONAL AND SEISMOLOGICAL CONSTRAINTS
V . A . Baturin
Queen Mary and Westfield College, Mile End Road, London E1 4NS, UK
Sternberg State Astronomical Institute Moscow State University Moscow 119899, Universitetsky
pr.13, Russia (permanent address)
S. V. Ajukov
Sternberg State Astronomical Institute Moscow State University Moscow 119899, Universitetsky
pr.13, Russia
Introduction
A variety of solar models (namely, models of stars having solar values of mass, radius,
luminosity) is a subject of our study. The hydrogen profile in the star's interior is usually
determined by course of the evolution, but we also consider models with arbitrary hydrogen
profiles or diffusion profiles also. Applying additional restrictions on the model which follow
from recent helioseismic results we consider the question of existence of the models. These
restrictions are: 1) calibration of the surface helium abundance and the specific entropy in the
convection zone (Vorontsov et al. 1991; Baturin, Vorontsov 1994; there are few results on the
helium calibration but we used cited ones because they contain simultaneous determination of
the helium and the entropy); 2) estimation of the convection zone depth (Christensen­Dalsgaard
et al., 1991); 3) sound speed profile inferred from oscillation frequencies (Vorontsov, Shibahashi
1992). These results are basically independent of a specific solar model. Hereafter we roughly
divide a regular solar model into three zones: 1) the outer convection zone (the envelope); the
chemical composition and specific entropy are constant there; 2) the radiative zone; the chemical
composition is constant there when diffusion isn't taken into account; the structure of this area
depends mainly on the opacity; 3) the energy­generating core where the thermonuclear reactions
are significant; there is definitely variable hydrogen/helium profile. Commonly speaking we
study the possibility of fitting some cores and convection zones while the fit conditions are
determined by the radiative zone. The boundaries between these areas are rather approximate,
of course. We set the core's upper boundary at 10 million K assuming that thermonuclear
reactions are not significant outside this area and helium abundance is nearly constant there in
the standard solar model.
As already mentioned we study possibility to fit envelopes (having fixed helium content Y
and entropy S) and the cores (with arbitrary hydrogen profiles and opacities). We used the
relations between Y and S from one side and opacities from the other (Baturin, Ajukov 1994).
Simplifying, this relation means impossibility to change S while modifications of opacities and
hydrogen profile X(m/M) are limited in the core, or, in another words, the connection between
S and the opacity in the radiative zone. At the same time increase of core opacities leads t o
raising of Y in the standard models (Fig. 1; vector K core
corresponding core opacity changes).
Additional result concerns changes of opacity near t he convection zone bo ttom: it affects

significantly the convection zone depth H b
but the rest of the model is still the same.
Analysis of the standard solar models with different opacity tables reveals t hat (Y, S) in these
models do not correspond to helioseismological estimate (Fig. 1). We assume that it is caused
by errors in the opacity or/and chemical composition profile. Thus the main question is--does it
possible or not to construct a solar model with helioseismically consistent parameters? And if
so, what's the changes we have to imply?
Estimate of opacity near the bottom of the convection zone
Fig. 2 shows the relation between H b
and S/R g
for the different values of Y (solid lines). This
figure is computed for IR91 (Iglesias, Rogers 1991) opacities. The fact that the grayed areas do
not overlay indicates the inconsistency between the (H b
, Y, S) set of values and IR91 opacity.
Provided (Y, S, H b
) are determined we can state that IR91 opacities must be significantly
lowered (by 30--40%) at the temperatures about 2.3·10 6 K. But this conclusion is correct while
convection is treated using Schwarzchild criterion and magnetic field and dynamical effects can
be neglected. In con t rast, we might state t hat Y , H b
and opacities indicate the error in the entropy
calibration (the appropriate value may be S/R g
= 21.3 mol/g).
Solar models with ``helioseismologically correct'' envelopes
Next we consider models with helioseismological values of (Y, S, H b
). These values obliously
fix the stucture of the envelope (i.e. convection zone). The question is: do such models exist?
First we can try to apply the opacity corrections to IR91 standard model (see Fig. 1): the only
problem is to get the right Y = 0.25 saving fixed S. This model is possible with decrease of the
opacity or/and the hydrogen content in the core. The necessary opacity correction is about 60%
that is rather large but can be accepted with additional mechanisms of heat transfer.
Now we use the sound speed profile obtained from the oscillation frequencies in Vorontsov,
Shibahashi 1992. When the function c 2 (r) is given the solar structure problem reduced to second
order system without radiative transfer law but requiring equation of state to be known (MHD
equation of state (Dappen et al. 1988) is used). Two questions arise: 1) is there any core to fit
this envelope; 2) what's the corrections necessary if 1) is present. To construct such model is
possible; however, the changes necessary are rather strange: decrease of opacity in the core by
80% and lowering the hydrogen content in the centre by 0.2 (up to X c
= 0.15).
Diffusion models
Let's now consider a some expansion of ``solar model'': the models with possible gradient of
the chemical composition in the radiative zone. The so­called `` diffusion '' models (e.g.,
Christensen­Dalsgaard et al., 1993) belong to this class. The helium content is variable in the
r adiative zone due to gravitational settling. Of course, all abundances are constant in the
convection zone where the mixing is present. Detailed calculations (Christensen­Dalsgaard et
al., 1993) revealed tha t t hese models have nearly constant hydrogen abundance X in the radiative
zone excluding rather thin layer just below the bottom of the convection zone where X rapidly
falls. In our investigations we've used models with piece­constant beyond of the core X(r) profile:
X = 0.73 in the convection zone, X = 0.70 in the radiative zone. Such model provides some
approximation of a ``detailed'' diffusion model from Christensen­Dalsgaard, et al. 1993 (see
Fig. 1). It is notable the increase of the entropy due to implementing of diffusion. The core and

radiative zone in the diffusion model are very similar to those of the standard model---a l l
differences are localized in the envelope and since changes of Y and S is a property of envelope
itself. So we could say that the core of the standard model can be fit with several envelopes
belonging some dependence S(Y). On the other hand, the entropy of the diffusion model is
somewhat higher than helioseismological value but the latter contradicts to the value necessary
to obtain right H b
(see above paragraph). This also means that if S/R g
assumed to be 21.3 mol/
g then one do not need any opacity corrections near the convection zone boundary and minimal
opacity changes to obtain helioseismological c 2 (r) profile (about 5% only).
However the entropy of diffusion model still does not correspond to helioseismological value,
and this discrepancy could only be corrected with very large (or even inconsistent) opacity
changes: large decrease near the bottom of the convection zone (to get correct value of H b
) plus
large increase in the radiative zone (to get a fit with the core) that causes c 2 (r) to be wrong (see
Fig. 3,4).
Conclusion
The specific entropy in the adiabatic part of the convection zone of the solar models is in a
disagreement with a helioseismological estimate: the latter is too low. Large and compicated
opacity corrections (for I R91 opacity t ables) are needed to construct solar model with t he proper
entropy. The best nonstandard model is p robably a model including molecular diffusion effects:
it has helium abundance lowered in the convection zone by 0.03.
Acknowledgments
We thank to J. Christensen­Dalsgaard, C.R.Profitt and M.J.Thompson for the data about
diffusion model. We also thank to S.V.Vorontsov and H.Shibahashi for the sound speed profile
inferred from oscillation frequencies. We are grateful to W . D.ppen for providing us with MHD
equation of state tables.
References
Baturin, V.A., Ajukov, S.V. 1994, Astr. Zh., submitted.
Baturin, V.A., Vorontsov S.V. 1994, Proceedings of GONG'94 Annual meeting.
Christensen­Dalsgaard, J. 1982, M.N.R.A.S., 199, 735.
Christensen­Dalsgaard, J., Gough, D.O., Thompson, M.J. 1991, Astrophys. J., 378, 413.
Christensen­Dalsgaard, J., Profitt, C.R., Thompson, M.J. 1993, Astrophys. J., 403, L75.
D.ppen, W., et al. 1988, Astrophys. J., 332, 261---270.
Iglesias, C.A., Rogers, F.J. 1991, Astrophys. J., 371, 408---417.
Vorontsov, S.V., Baturin, V.A., Pamyatnykh, A.A. 1991, Nature, 349, 49.
Vorontsov, S. V., Shibahashi, H. 1992, Publ. Astron. Soc. Japan, 43, 739---753.

Fig. 1. Solar models on the (Y, S) plane.
Asterisks indicate solar models computed by authors; ¤ is the helioseismic estimate; ( are two
models from Christensen­Da lsgaard et al. 1993--- st andard and d iffusion models. The
abbreviations of opacity tables are:
CS70 -- Cox A.N., Stewart J.N. 1970, Astrophys. J. Suppl., 19, 243.
CT76 -- Cox A.N., Tabor J. E. 1976, Astrophys. J. Suppl., 31, 271.
BU88 -- Bahcall J.N., Ulrich R. K., 1988, Rev. Mod. Phys., 60, 297.
WKM -- Weiss A., Keady J.J., Magee N.M. Jr., 1990, Atomic Data and Nuclear Data Tables, 45, 209.
IR91 -- Iglesias C. A., Rogers F. J., 1991, Astrophys. J. 371, 408.
Two vectors K core and K rad indicate directions in which the standard model moves when
decreasing opacity in the co re and radiative zone, respectively. The amplitude of opacity
modification is about 50%.
R g
is the gas constant.

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Fig. 2. The relation between convection zone depth and entropy in the solar envelope.
Dashed lines---helioseismologically allowed ranges for H b
and S/R g
according Christensen­
Dalsgaard, Gough, Thompson 1992 and Vorontsov, Baturin, Pamyatnykh 1991. Solid lines
represent H b
(S) dependence in envelopes with IR91 opacities, and two central lines correspond
to Y = 0.25±0.05 as proposed in Baturin, Vorontsov 1994. You can note that the three values
( H b
, Y, S) are in mutual disagreement when IR91 opacities are used.

Fig. 3. Sound speed profiles in the solar models.
The three lines represent the relative differences between the sound speed profiles of the three
models and a model from Christensen­Dalsgaard 1982. Solid line--as obtained in Vorontsov,
Shibahashi 1992; sho rt­d ashed line--diffusion model from Ch ristensen­Dalsgaard, Profit t,
Thompson 1993; long­dashed line--model compu ted by au thors with helioseismologically correct
H b
, S, Y. Note that the diffusion model gives rather good sound speed profile comparing with
observations.

Fig. 4. Opacity correction needed to obtain correct sound speed profile.
If S=21.3 mol/g is assumed instead of 21.0 mol/g the opacity correction necessary is much smaller .
This may indicate an error in the entropy estimate in Baturin, Vorontsov 1994.