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Helioseismic constraints on the structure of
the present Sun
V.A.Baturin and S.V.Ajukov
Sternberg Astronomical Institute, Moscow 119899, Russia
1. 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 (Christensen­Dalsgaard 1988) and with six
evolutionary models computed by Christensen­Dalsgaard (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 possibil­
ities of the agreement between such relation and helioseismic determination of
surface helium abundance and entropy is discussed.
2. 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 ff because of the value of ff 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:
s = \Gamma
` @F
@T
'
V;N
=
/
k
X
s
n s
` 5
2 + ln g s
n s \Lambda 3
s
'
+ 4
3 aT 3
! OE
ae
1

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=
p
2úkTm 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.
3. 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 1 by
crosses). To investigate the sensitivity of this Y 0 \Gamma s relation to the internal
physics we have computed models with the opacity locally altered a) in the
energy­generating core and b) in the radiative zone. While opacity is changed
in the core (where luminosity is not constant), the relation Y 0 \Gamma s is preserved.
However, changing of opacity in the radiative zone (between energy­generating
core and convective zone bottom) shifts Y 0 \Gamma 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 \Gamma s plane.
Our calculations were performed with simplified model physics (according
Christensen­Dalsgaard (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,
Christensen­Dalsgaard 1991). To reveal the relation Y 0 \Gamma 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 fi 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 Christensen­Dalsgaard 1988) and accordingly to change
opacities and/or profile of hydrogen abundance in the core.
Acknowledgments. We thank to J.Christensen­Dalsgaard 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.
2

Figure 1. 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 \Gamma s in our models. Labelled squares denote models
from Christensen­Dalsgaard 1992. fi ­ helioseismic determination of
envelope parameters (Vorontsov et al. 1991).
3

References
Christensen­Dalsgaard, J. 1988, Computational Procedures for GONG Solar
Model Project.
Christensen­Dalsgaard, J. 1991, in Challenges to Theories of the Structure of
Moderate­mass Stars, pp.11­36, ed. D.O.Gough and J.Toomre (Lecture
Notes in Physics, 388, Springer, Heidelberg, 1991).
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