% Our poster

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\title{Helioseismic constraints on the structure of \break the present Sun}
\author{V.A.Baturin and S.V.Ajukov}
\affil{Sternberg Astronomical Institute, Moscow 119899, Russia}

%               INTRODUCTION

\section{Introduction}
Given the chemical composition (hydrogen abundance mainly),  the mass  and
internal physics, the structure of the star is  determined  uniquely.
The commonly  used  method  for  the  choice  of  the hydrogen abundance
 profile in  the  model  of  the  present  Sun  is evolutionary calculations. 
But the existing uncertainties in physics  and  evolution  give us an idea
to consider a more extended set of possible solar models. We  consider
models with solar values of radius, luminosity and mass, but with an
arbitrary profiles of hydrogen abundance in the core.  Of  course,  
to get up to solar values, profile  $X(m/M)$ must satisfy some (nonstrict)
restrictions, which are determined by fitting.

In  this  poster  we
propose a hypothesis that within fixed assumptions about internal physics
(opacity, thermodynamics, nuclear reactions) a strict relation between the
surface helium abundance and the entropy of the adiabatic part of the
convective zone exists for all solar models.  This hypothesis is verified
with models under simplified physics (Christensen-Dalsgaard 1988) and
with six evolutionary models computed by Christensen-Dalsgaard (1992).
Such relation mainly depends on the functions of the
opacities in radiative zone between the lower boundary of convection and
the region of nuclear reactions.  The possibilities of the agreement between
such relation and helioseismic determination of surface helium abundance
and entropy is discussed.

%               SOLAR MODELS WITH FIXED ENVELOPE

\section{Solar models with fixed envelope}
Let's  consider solar models assuming that chemical composition (hydrogen
and helium abundance  $Y_0$) in convective envelope is fixed.  To obtain
solar values  of  radius $R$ and luminosity $L$ of the model
we vary the hydrogen
profile $X(m/M)$ in  the core and the convective envelope entropy $s$.
 We use the specific  entropy  $s$  in  the adiabatic  part  of  the
convective zone as a parameter of the solar envelope instead of convection
parameter  $\alpha$ because of the  value  of  $\alpha$  depends  on
structure of the outermost layers (in particularly, opacities).  The
specific entropy $s$ is also related with convection zone depth but
opacities affect this connection too.  To compute the specific entropy we
use the expression for free energy from Mihalas et al. (1988), but
electron degeneracy was omitted:
\begin{displaymath}
s=-\left( {\partial F \over \partial T} \right)_{V,N} =
\left( k \sum_s n_s \left( { 5 \over 2 } + \ln {g_s \over n_s
\Lambda^3_s} \right)+ { 4 \over 3 }a T^3 \right) \biggr / \rho
\end{displaymath}
where sum is taken over all particles, $g_s=2$ for electrons and $g_s$ is a
partition function for atoms and ions.  $\Lambda_s$ denotes thermal
wavelength: $\Lambda_s =  h / \sqrt{2 \pi k T m_s}$.

As a result, fitted value of $s$ was revealed to be the same for these  models,
and all our attempts to  construct  solar  model  with  another
entropy in convective envelope, but  constant  $Y_0$ in envelope
were a failure.  In other words, there are
many models with different $X(m/M)$ profiles in the interior, but the same
$Y_0$  in envelope, and all of them have the same  $s$.

%               RELATION...

\section{Relation between  helium  and entropy in envelope}
We  also  have  computed  solar models  with various
helium abundance  $Y_0$  in  the envelope.   Fitting  to solar  $R$   and
$L$ is possible again,  but  these  models  have  other  entropy  in   the
convective zone (and accordingly other convective  zone  depth $d_b$).
However, for given $Y_0$ only one $s$ is  allowed, i.e.  there  is  a
relation between surface helium abundance $Y_0$ and convective  zone
entropy $s$ in models with various $X(m/M)$ profiles in the interior
(as it plotted on Figure~\ref{fig-1} by crosses).
\begin{figure}
\vspace{3in}
%\epsffile{}
\caption{The specific entropy $s/R$ ($R$ - gas constant) and the surface helium
abundance in the solar models.  The crosses connected by lines represent
relation $Y_0-s$ in our models.  Labelled squares denote models from
Christensen-Dalsgaard 1992.  \sun\/ - helioseismic determination
of envelope parameters (Vorontsov et al. 1991).} \label{fig-1}
\end{figure}
     To investigate the sensitivity of  this  $Y_0-s$  relation to  the
internal physics we  have  computed  models  with  the opacity locally
altered a) in the energy-generating core and b) in the  radiative zone.
While opacity is changed
in the core (where luminosity is  not  constant), the relation $Y_0-s$ is
preserved. However, changing of
opacity  in  the  radiative zone (between energy-generating core  and
convective  zone bottom) shifts $Y_0-s$ curve as a whole.  Thus opacity in
the radiative zone is one of the factors determining  the  position of
the  curve  on $Y_0-s$ plane.

     Our calculations were performed with simplified  model physics
(according Christensen-Dalsgaard (1988)) and the helium abundance and
entropy of our evolutionary
model do not correspond commonly obtained in recent evolutionary
calculations (Kim et al. 1991, Sackmann et al. 1990, Courtaud et al. 1990,
Christensen-Dalsgaard 1991).  To reveal the relation $Y_0-s$ for other
models we have calculated specific entropy for 8 models computed by
J.Christensen-Dalsgaard (1992).  We find a similar relation for three
models with different age (22, 6, 21) and for three models with
 corrected opacity  in the core
(7, 27, 28).  Model 4 differs from model 6 due to another opacities and
model 14 from model 4 because of including Coulomb interactions.  But all
these models have too high entropy comparing with helioseismic determination
by Vorontsov et al. (1991)
(marked by \sun\/ on figure).  Preliminary investigation indicates
that to reach low entropy and required helium contents it is necessary to
increase opacities in radiative zone (by factor 1.5 or 2) in comparison with
used by us opacities (formulae from Christensen-Dalsgaard 1988)
and accordingly to change opacities and/or profile of hydrogen abundance
in the core.

\acknowledgments
     We thank  to J.Christensen-Dalsgaard  for  stimulating
discussion and making your models available to us. We are also grateful to
W.Dappen for help with preparation of this poster to print.

\begin{references}

\reference Christensen-Dalsgaard, J. 1988, {\it Computational
Procedures  for GONG Solar Model Project}.
\reference Christensen-Dalsgaard, J. 1991, in {\it Challenges to
Theories  of the  Structure of Moderate-mass Stars\/}, pp.11-36,
ed. D.O.Gough and J.Toomre (Lecture Notes in Physics, {\bf 388},
Springer, Heidelberg, 1991).
\reference Christensen-Dalsgaard, J. 1992,  {\it  Geophys.  Astrophys.
Fluid Dynamics}, in  press.
\reference Courtaud, D., Damamme, G., Genot, E.,  Vuillemin, M., and
Turck-Chieze, S. 1990, {\it Solar Phys.}, {\bf 128}, 49-60.
\reference Kim, Y.-C., Demarque, P., and Guenther D.B. 1991,  \apj,
{\bf 377}, 407-412.
\reference Mihalas, D., D\"appen, W., and Hummer, D.G. 1988, \apj,
{\bf 331}, 815-825.
\reference Sackmann, I.-J., Boothroyd, A.I., and Fowler W. 1990, \apj,
{\bf 360}, 727-736.
\reference Vorontsov, S.V., Baturin, V.A., and Pamyatnykh, A.A. 1991,
{\it  Nature}, {\bf 349},  49-51.
\end{references}

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