PASJ: Publ. Astron. Soc. Japan 55, 1015–1023, 2003 October 25
c 2003. Astronomical Society of Japan.
Hydrogen Abundance in the Tachocline Layer of the Sun
Masao TAKATA and Hiromoto S HIBAHASHI
Department of Astronomy, School of Science, The University of Tokyo, Bunkyo-ku, Tokyo 113-0033
[email protected], [email protected]
(Received 2003 May 1; accepted 2003 July 22)
Abstract
Sound speed inversions of the Sun show that the profile of the relative difference between the Sun and the
standard solar model has a sharp peak around r/R = 0.65, which is the location of the tachocline layer found by
rotation inversions. It has been suggested that this sharp peak would be due to the difference, between the Sun
and the model, in the hydrogen abundance in the tachocline layer, possibly caused by the weak-mixing process.
In this paper, we quantitatively discuss the hydrogen abundance in the tachocline layer based on a seismic solar
model, which was constructed using the sound speed and density profiles as well as the depth of the convection zone
obtained by helioseismology. One of the important characteristics of the seismic solar model is that it gives us a
hydrogen profile as a part of the solution. We find that the hydrogen abundance of the seismic solar model decreases
more mildly than that of the standard solar models constructed by incorporating the diffusion process. This feature
hardly depends on the profile of the heavy elements as well as the uncertainties in the opacity and the equation of
state.
Key words: Sun: abundances — Sun: helioseismology — Sun: interior — Sun: oscillations
1. Introduction
The solar interior is diagnosed using the solar oscillations
data. We can infer the sound speed profile in the Sun and
deduce the internal rotation angular velocity. Many projects,
such as GONG (Harvey et al. 1996) and SOHO (Fleck et al.
1995), are now going on for helioseismology. The long-time
observation of the solar oscillations by these projects provides
us with very accurate eigenfrequencies of the Sun. The relative
errors in the frequency of the best observed modes are almost
as small as 10−6 (cf. Schou et al. 1998). The relative errors
in the sound-speed profile determined by inverting these eigenfrequencies are less than 0.1% in the range 0.15 ≤ r/R ≤ 0.9
(cf. Gough et al. 1996; Kosovichev et al. 1997). Such highly
accurate results of helioseismology enable us to discuss the
detailed features of the solar structure.
These sound-speed inversions are generally consistent with
each other. One common feature among these inversions is the
fact that there is a sharp bump at r/R ≈ 0.65, just beneath
the base of the convection zone (r/R = 0.713; ChristensenDalsgaard et al. 1991), if we plot the relative differences in the
squared sound-speed profile between the Sun and the standard
solar model as a function of the radius. It is argued (cf.
Gough et al. 1996) that this bump is due to the differences in
the chemical compositions between the Sun and the standard
solar model, which are probably caused by an overestimation
of the diffusion effect in the standard solar model. These
differences in the chemical compositions directly affect the
mean molecular weight, which changes the sound speed. In
fact, in the standard solar model hydrogen (helium) decreases
(increases) rapidly inward just below the base of the convection zone as a result of macroscopic diffusion (cf. ChristensenDalsgaard et al. 1996; Bahcall et al. 1998). However, if we
think critically enough, the above argument can be considered
to be too simplified, because such a local change in the
chemical abundance generally influences not only the sound
speed at the same place, but also other quantities, even at
other places through the stellar structure equations. It is not
clear at all whether such an influence on the sound-speed and
density profiles is consistent with the seismic inversions or
not. Therefore, we think it important to examine the hydrogen
profile of the Sun, which is calculated consistently with the
sound-speed and density inversions based on the stellar structure equations.
The rotation inversions by different groups also have the
following common characteristics (cf. Thompson et al. 1996;
Schou et al. 1998): the angular velocity of the solar internal
rotation is nearly independent of the radius for the fixed latitude
in the convective envelope and the equatorial part rotates faster
than the polar region as we observe at the solar surface; in
the upper part of the radiative core (0.5 ≤ r/R ≤ 0.7), the
rotation rate is almost constant; in other words, the solar
core rotates almost rigidly. The transition from nearly radiusindependent rotation in the convective envelope to the nearly
rigid rotation in the radiative core occurs in a very thin layer
around 0.7 R , which is called the tachocline layer (Spiegel,
Zahn 1992). It is known that the tachocline is confined in such a
thin layer that the rotation inversion cannot resolve it (cf. Antia
et al. 1998). Instead, Elliott and Gough (1999) have estimated
its thickness to be a few percent using the sound-speed inversions. Considering the fact that the base of the convection
zone probably determines the upper limit of the tachocline
layer, we tentatively regard in the following discussions that
the tachocline layer ranges from 0.68 R to 0.713 R , though
our main conclusions deduced below are not sensitive to the
exact location.
To explain both of the bump in the sound-speed profile
and the tachocline layer of the rotation profile, the following
1016
M. Takata and H. Shibahashi
picture has been proposed: the strong shear in the tachocline
layer causes a mixing process and homogenizes the chemical
composition profile there (cf. Gough 1997). In fact, Brun
et al. (1999) have demonstrated that a specific mixing process
greatly helps to reduce the bump of the difference in the
sound-speed profile between the Sun and the evolutionary
solar model. This can be regarded as a forward problem
approach. On the other hand, what we take in this paper is the
inverse problem approach. We see how the hydrogen profile is
constrained by the solar oscillation frequencies, without specifying the processes that change the chemical composition in the
tachocline layer of the Sun during its evolution, but with taking
account of various uncertainties such as those in the opacity
and the equation of state. Determining the hydrogen and/or
helium profile based on the solar oscillation frequencies has
been discussed by several people (cf. Kosovichev 1996, 1997;
Takata, Shibahashi 1998a; Antia, Chitre 1998). However, few
of them have explicitly discussed the effect of uncertainties in
the input physics on the hydrogen abundance in the tachocline
layer.
In this paper, we restrict ourselves to the chemical composition profile, particularly the hydrogen profile, in the tachocline
layer of the Sun, which is determined consistently with the
observed solar oscillation data. We discuss the possibility that
the bump in the relative sound-speed profile could be explained
by other reasons than the mixing process, such as errors in our
understanding of the microphysics, or an incorrect hypotheses
about the evolution of the Sun. Though the structure of the
tachocline probably depends on the latitude, we discuss only
its spherically symmetric component.
In section 2, we briefly summarize the way to determine
the chemical composition profile of the Sun based on the
observed solar oscillation frequencies. Then, in section 3,
we calculate the hydrogen profile in the tachocline layer of
the Sun, while accepting the latest microphysics data and the
hypotheses assumed by the standard evolutionary solar models
(except for the one about diffusion). We also examine the effect
of errors in the input physics and the assumed hypotheses.
In section 4, we interpret the hydrogen abundance in the
tachocline layer obtained in section 3 based on an analytic
calculation. Discussions are given in section 5. We finally
summarize the results in section 6.
2.
Seismic Solar Model and the Chemical Composition
Profile of the Sun
When we determine the sound-speed profile and the
density profile of the Sun from the solar oscillation frequencies, we assume only the mass and momentum conservation equations as well as the adiabatic variations of thermodynamical quantities during the course of oscillations. This
procedure is sometimes called primary inversion (Gough,
Kosovichev 1988). On the other hand, we need the energy
equation and the energy transfer equation, in addition to microphysics including the opacity, the equation of state, and the
nuclear reaction rates, to determine other quantities of the
stellar structure, such as the temperature, luminosity, chemical
compositions, etc. In fact, we have constructed a solar
model by assuming thermal equilibrium and radiative heat
[Vol. 55,
transport in the core (Takata, Shibahashi 1998a; Watanabe,
Shibahashi 2001). We call this kind of solar model a seismic
solar model.
The advantages of the seismic solar model over the conventional standard solar models are summarized as follows:
1. The seismic solar model is consistent with the observed
solar oscillation data, while the standard solar models are
not.
2. The seismic solar model gives the profiles of chemical
compositions as a part of the solutions, while the
chemical composition profiles are given as inputs at each
stage of the evolution in the case of the standard solar
models.
The equations governing the seismic solar model are almost
common with the standard solar models: the mass conservation
equation,
dMr
= 4π r 2 ρ,
dr
the equation of the hydrostatic equilibrium,
dP
GMr ρ
=−
,
dr
r2
the energy equation,
dLr
= 4π r 2 ρ,
dr
the equation of the radiative transfer,
(1)
(2)
(3)
3κρLr
dT
=−
,
(4)
dr
64π σ r 2 T 3
and the auxiliary equations for the opacity (κ), the nuclear
reaction rates (), and the thermodynamical quantities (the
equation of state), where the symbols have their usual
meanings. One exception is that we are free from any specific
formulation of the processes that change the chemical compositions, other than the nuclear reactions, in constructing the
seismic solar model. The other exceptions are the sound-speed
and density profiles, which are determined by the primary
inversion and given as input in advance when the differential
equations are solved:
c = c (r)
(5)
and
ρ = ρ (r).
(6)
In fact, these profiles can be alternatives to the two independent
chemical composition profiles, which are input at each time
step to solve differential equations (1)–(4) when we construct
the evolutionary solar models. By assuming the sound-speed
and density profiles, the seismic solar model can be regarded
as a snapshot model of the present-day Sun. We may set the
outer boundary conditions of the seismic solar model at the
base of the convection zone, which is found to be located
at r/R = 0.713 by analyzing the sound-speed profile of the
Sun (Christensen-Dalsgaard et al. 1991). One outer boundary
condition is that the luminosity is equal to the solar surface
luminosity,
Lr = L
at r = rconv ,
(7)
No. 5]
Hydrogen Abundance in the Tachocline Layer of the Sun
1017
The default seismic model is constructed by adopting the
following parameters and input physics: we set rf = 0.6R and
Zc = 0.022; the nuclear reaction rates are those by Adelberger
et al. (1998); the OPAL opacity (Iglesias, Rogers 1996) and
the OPAL equation of state (Rogers et al. 1996) are used for
consistency. All of the seismic solar models discussed in the
next section adopt the same parameters and input physics,
unless otherwise specified. We note that this default model is
equivalent with model (E) in table 4 of Takata and Shibahashi
(1998b), which is found to be most consistent with the soundspeed and density inversions based on the SOHO/MDI data
(Scherrer et al. 1995) by Basu (1998).
3.
Fig. 1. Parametrized heavy element profile used to construct the
seismic solar model. We introduce two parameters, rf and Zc . The
default value of the position of the base of the convection zone rconv is
set to 0.713 R as determined by Christensen-Dalsgaard et al. (1991)
because there is no energy generation in the convective
envelope. As the other outer boundary condition, the temperature gradient is required to match the adiabatic value,
∇ad = ∇rad ≡
3κLr P
64π σ GMr T 4
at r = rconv ,
(8)
which means that the neutral stability against convection
holds there. We also demand that the ratio of the heavy
element abundance to the hydrogen abundance be equal to the
photospheric value of Z/X = 0.0245 (Grevesse, Noels 1993),
because the matter in the convection zone is fully mixed on the
order of one month.
Since it is practically difficult to determine both of
the hydrogen and heavy-element abundance simultaneously
from given sound-speed and density profiles (cf. Takata,
Shibahashi 2001; Antia, Chitre 2002), we alternatively
constrain the density and hydrogen profiles by given soundspeed and heavy element profiles. Only if the thus-determined
density profile is consistent with the helioseismic inversion,
can the model be regarded as a seismic solar model. We assume
that the heavy element profile is constant in the central region
and a linear function of the radius in the outer layer. This is the
same treatment as that which we used in previous work (Takata,
Shibahashi 1998b, 1999). Such a profile is parametrized by
two parameters (see figure 1). One is the radius rf under which
the heavy element abundance is constant. The other is Zc , the
central heavy element abundance. As explained above, these
two parameters are adjusted so as to give a density profile
consistent with that given by helioseismology. We restrict
ourselves to adopt only those parameters that give a consistent density profile with helioseismology at the 2-σ error level.
It has been found that rf between 0.3 and 0.65 and Zc between
0.020 and 0.023 are allowed, though rf outside of the above
range has not been tried (Takata, Shibahashi 1998b). The heavy
element abundance at the base of the convection zone, on the
other hand, is determined so that the condition of Z/X = 0.0245
is satisfied.
Hydrogen Profile of the Seismic Solar Model in the
Tachocline Layer
There are some uncertainties in the input parameters or
microphysics which are necessary to construct a seismic solar
model and hence to determine the hydrogen profile of the Sun.
Among them, we pick here the following four factors: the
heavy element profile, the equation of state, the opacity, and
the depth of the convection zone. The hydrogen profile in
the tachocline layer of the Sun may be insensitive to the other
factors. For example, the cross sections of the nuclear reactions
affect the hydrogen abundance only in the core of the Sun
because the nuclear reaction is active only in the central region
of the Sun. On the other hand, the statistical errors, which
originate from those in the sound-speed profile determined by
the solar oscillation frequencies, are too small to be comparable
to the above systematic uncertainties. Actually their magnitude
is as small as the line width of the curves shown in figures 2, 4,
and 5, which are explained below.
In figure 2, we see the hydrogen profile determined for
five different heavy element profiles, which are displayed in
figure 3, as well as those of the standard solar models. Note
that the profiles of the default model, explained at the end
of the previous section, are drawn neither in figure 2 nor in
figure 3. Because the range of the parameter Zc is constrained
to be between 0.020 and 0.023, both the sound-speed profile
and the density profile of the seismic solar model are consistent with the inversion results. Figure 2 tells us that the effect
of the other parameter rf on the structure of the seismic solar
model is smaller than that of Zc . For all of the parameters
shown in figure 2, the hydrogen abundance of the seismic solar
model is higher than that of the standard solar models in the
range 0.66 ≤ r/R ≤ 0.713. Among the five Z profiles shown
in figure 3, the two profiles with label ‘rf = 0.65 Zc = 0.020’
and label ‘rf = 0.60 Zc = 0.023’ have the similar gradient in the
tachocline layer to that of the standard solar models, while the
curve with label ‘rf = 0.65 Zc = 0.023’ has a steeper gradient.
The gradient of the remaining profiles, one labeled ‘rf = 0.60
Zc = 0.020’ and the other labeled ‘rf = 0.30 Zc = 0.020’, is
milder than that of the standard solar models. In section 4,
we discuss the reason for such a dependence of the hydrogen
profile on the heavy element profile, as shown in figures 2
and 3.
The sensitivity of the hydrogen profile to the depth of the
convection zone is shown in figure 4. The error in the depth of
the convection zone is estimated to be as small as ±0.003 R
1018
M. Takata and H. Shibahashi
Fig. 2. Hydrogen profiles in the tachocline layer of the seismic solar
model for various profiles of heavy elements. While the abscissa is
the fractional radius, the ordinate means the hydrogen abundance, X,
subtracted by its value in the convection zone, Xconv . The meanings
of parameters rf and Zc are shown in figure 1. Also depicted are the
hydrogen profiles of the two standard solar models, one with label
‘BP98’ by Bahcall et al. (1998) and the other with label ‘Model S’
by Christensen-Dalsgaard et al. (1996).
Fig. 3. Profiles of the heavy element abundance of the seismic solar
model and the standard solar models near the base of the convection
zone at r ≈ 0.71 R . Each profile corresponds to the curve in figure 2
drawn in the same color with the same label.
(Christensen-Dalsgaard et al. 1991). We find that, if the
position of the base of the convection zone is modified from
0.713 to 0.710, the profile of the hydrogen abundance relative
to its surface value (X − Xconv ) shifts downward with essentially no change in its shape between 0.6 R and 0.68 R . We
should pay attention to the fact that the hydrogen abundance of
the standard solar models still decreases inward more rapidly
than that of the seismic solar models between 0.66 R and
0.71 R .
It is found that, even if we change the input microphysics,
the equation of state, and the opacity, the hydrogen profile in
the tachocline layer is hardly influenced (see figure 5). Even if
the errors in the opacity table is as large as 10%, the hydrogen
abundance in the tachocline layer is modified by less than 1%,
as we can judge from the lines labeled ‘κ ± 10%’ in figure 5.
[Vol. 55,
Fig. 4. Hydrogen profiles in the tachocline layer of the seismic solar
model for various values of the depth of the convection zone. The
meanings of the abscissa and the ordinate are the same as those in
figure 2. Note that rconv is the radius of the base of the convection
zone. The profile with label ‘rconv = 0.713 Rsun ’ corresponds to the
default model, which is explained in the main text. As in figure 2, the
hydrogen profiles of the two standard solar models are also depicted.
Fig. 5. Hydrogen profiles in the tachocline layer of the seismic solar
model for various input physics. The meanings of the abscissa and the
ordinate are the same as those in figure 2. The line with label ‘OPAL
EOS, OPAL OPACITY 95’ is the profile of the default model, which is
calculated with the OPAL equation of state (Rogers et al. 1996) and the
OPAL opacity (Iglesias, Rogers 1996). Instead of the OPAL equation
of state, that of the ideal gas is assumed in calculation of the line with
label ‘Ideal Gas EOS’. The opacity is modified artificially by ±10%
when we obtain lines labeled ‘κ + 10%’ and ‘κ − 10%’, respectively.
As in figures 2 and 4, the hydrogen profiles of the two standard solar
models are also plotted.
As for the equation of state, we regard that its errors could
be estimated by the difference between the up-to-date table
(OPAL equation of state, Rogers et al. 1996) and the ideal gas
case. We calculated the hydrogen profiles for both of these
equations of state. The results are drawn by the lines labeled
‘Ideal Gas EOS’ and ‘OPAL EOS, OPAL OPACITY95’ in
figure 5. It is clear that the difference between these two
profiles is less than 1.2% in the interval [0.6 R , 0.713 R ].
This is consistent with the fact that, for a given density, a
temperature, and chemical compositions, which correspond to
No. 5]
Hydrogen Abundance in the Tachocline Layer of the Sun
the condition of the tachocline layer of the Sun, the pressure
given by the OPAL equation of state differs from that given by
the equation of state of the ideal gas by about 1%. Figure 5
tells us that the hydrogen abundance in the tachocline layer is
more sensitive to the errors in the equation of state than those in
the opacity because the effects of a ±10% modification of the
opacity are as small as those of the 1% errors in the equation
of state. We discuss this point in section 4.
4. Analytical Treatment of the Hydrogen Profile in the
Tachocline Layer
The reason why the Z profile considerably influences the
hydrogen abundances, as shown in figure 2, can be explained
as follows: since the opacity is most sensitive to the heavy
element abundance, a larger heavy element abundance causes
a steeper temperature gradient in the radiative zone, which
results in a higher temperature because the temperature at the
base of the convection zone is constrained by the boundary
conditions of the seismic solar model; because the (squared)
sound speed, which is essentially proportional to the temperature and the inverse of the mean molecular weight, is fixed by
the inversions, such a temperature increase must be compensated by an increase in the mean molecular weight, which
is accomplished by a decrease in the hydrogen abundance.
Comparing the hydrogen profiles in figure 2 with the corresponding Z profiles in figure 3, we actually find that the higher
is the heavy element abundance, the lower is the hydrogen
abundance.
The above physical picture can be justified with the help of
mathematics. First of all, we show that the outer boundary
conditions of the seismic model constrain all of the four structure variables (the mass, pressure, temperature, and luminosity)
as well as the chemical compositions at the base of the convection zone, whose radial location is expressed as r = rconv in
the following. The luminosity can be simply set to the surface
value (L ), while the mass is obtained by integrating the
density profile, which is given by the helioseismic inversion,
over the volume contained in the sphere of the radius r = rconv
because of the continuity equation. Once we know the mass
and density profiles, we get the pressure by integrating the
equation of the hydrostatic equilibrium from the surface to
r = rconv . Or, the pressure can be roughly calculated from
the density and the sound speed, both of which are given by
the inversion, by setting the adiabatic index to 5/3. There are
three additional conditions available at the base of the convection zone: the equation of state, the condition of the temperature gradient, and the ratio of the heavy element abundance to
the hydrogen abundance, Z/X, which can be measured at the
surface. Therefore, the remaining three variables (the temperature, hydrogen abundance, and heavy element abundance) can
be fixed by these conditions. In table 1, we summarize the
structure variables and the chemical compositions at the base of
the convection zone of the default seismic solar model, which
is described at the end of section 2.
Next, we derive the differential equation that determines the
hydrogen abundance in the tachocline layer. We start from the
equation of radiative transfer,
1019
Table 1. Structure of the default seismic solar model at the base of
the convection zone.
Mass
Density
Pressure
Temperature
Luminosity
Sound speed
Hydrogen mass fraction
Metal mass fraction
0.975 M = 1.94 × 1033 [g]
0.191 [g cm−3 ]
5.73 × 1013 [dyn cm−2 ]
2.20 × 106 [K]
1 L = 3.85 × 1033 [erg s−1 ]
2.24 × 107 [cm s−1 ]
0.735
0.018
3κρ (r)L
dT
=−
,
(9)
dr
64π σ r 2 T 3
in which the meanings of the symbols are as follows: T is
the temperature; κ the opacity; ρ (r) the density determined
by helioseismology; σ the Stefan-Boltzmann constant; and r
the radius. We assume the Kramers-type expression for the
opacity,
κ = κ0 (1 + X)Zρ (r)α T −β ,
(10)
where all of κ0 , α, and β are positive constants, because free–
free transitions are dominant under the condition relevant to the
tachocline layer of the Sun (cf. Kippenhahn, Weigert 1990).
Fitting equation (10) to the OPAL opacity table (Iglesias,
Rogers 1996) by the method of least-squares in the range
of 0.191 g cm−3 ≤ ρ ≤ 0.500 g cm−3 , 2.20 × 106 K ≤ T ≤
3.10 × 106 K, 0.695 ≤ X ≤ 0.740, and 0.018 ≤ Z ≤ 0.022,
we have α = 0.501, β = 3.17, and κ0 = 1.82 × 1023 , whose unit
is defined from equation (10) so that κ has units of g−1 cm2 .
The error in this fit is found to be within 4.8% everywhere
in the above fitting ranges. As for the equation of state, we
slightly correct that of the ideal gas, which is fully ionized. We
assume that the sound speed is related to the temperature and
the chemical compositions as follows:
c (r)2 =
5RT
(1 + ∆) ,
3µ
(11)
where c (r) is the sound speed determined by helioseismology, R the gas constant, ∆ the correction factor, which is
determined by a fit to the realistic equation of state, and µ the
mean molecular weight defined by
−1 −1
5
5
1
3
3
X− Z +
X+
µ=
≈
.
(12)
4
4
4
4
4
Here, we have assumed Z 1. The correction factor ∆ is
actually set to −0.0154 as a result of the fit of equation (11)
to the OPAL equation of state (Rogers et al. 1996) in the
same ranges as those for the opacity fitting. The maximum
error of this fit is 0.43% in the fitting ranges. If we substitute
equation (10) into equation (9) and eliminate the temperature,
T , and the mean molecular weight, µ, using equations (11)
and (12), then we get the following differential equation for
the hydrogen abundance:
β + 5
2 d ln c
dX
3
3
= F Z (X + 1) X +
+
X+
, (13)
dr
5
dr
5
where F is a function of the radius defined by
1020
M. Takata and H. Shibahashi
[Vol. 55,
equation (13) and the hydrogen profile of the seismic solar
model is attributed to the crudeness of the various approximations used to derive equation (13).
We show that equation (15) is quite useful to explain the
following results obtained in section 3:
Fig. 6. Approximations to the hydrogen abundance, X, are plotted as
functions of the fractional radius. The dashed line labeled ‘numerical
integration’ was obtained by a numerical integration of equation (13),
while the dotted line labeled ‘approximation’ was calculated based on
expression (15). For a comparison, the profile of the default seismic
solar model is also drawn in the solid line labeled ‘seismic solar model’.
F≡
β + 4
3κ0 ρ (r)α + 1 L 25R (1 + ∆)
.
64π σ r 2
12c (r)2
(14)
We can integrate equation (13) inward from the base of the
convection zone, r = rconv , where we know the value of X, for a
given Z profile. Thus, we can obtain the hydrogen abundance
in the tachocline layer.
We show in the appendix that the solution of equation (13)
is approximately given by
− β 1+ 5
rconv
3β + 20
,
(15)
F̃ Z dr
−
X ≈ c (r)2 ξ +
5(β
+ 6)
r
where
ξ ≡ c (rconv )
2(β + 5)
−β−5
3β + 20
Xconv +
5(β + 6)
(16)
and
F̃ ≡ (β + 5)c (r)2(β + 5) F
β + 4
3(β + 5)κ0 L 25R (1 + ∆)
=
64π σ
12
ρ (r)α + 1 c (r)2
.
(17)
r2
In equation (16), subscript ‘conv’ means the value at the base
of the convection zone. In figure 6, we compare expression (15) with the numerical integration of equation (13) and
the hydrogen profile of the default seismic solar model. We can
hardly distinguish the approximation given by equation (15)
from the numerical integration of equation (13), since the
difference in these two curves is within 0.1%. On the other
hand, the difference between expression (15) and the profile
of the default seismic solar model is about 0.2% at most over
the whole range shown in figure 6, while the agreement is
even better in the tachocline layer, 0.68 < r/R < 0.713. We
have similar results for other Z profiles shown in figure 3.
Note that the difference between the numerical integration of
×
• Since F̃ , which is defined by equation (17), is positive,
an increase in Z causes the integral in equation (15) to
grow, which results in a decrease in X because of the
negative power, −1/(β + 5).
• An increase in the opacity, which corresponds to that of
κ0 in equation (17), has the same effect as that of Z.
• Following a similar argument, we find that the hydrogen
abundance also decreases in the case where the equation
of state of the ideal gas is adopted. Note that this case
can be simulated by setting ∆ = 0 in equation (17). The
physical explanation for the decrease in hydrogen is as
follows: the equation of state of the ideal gas gives
a larger pressure for a given density, temperature, and
chemical compositions, because neither the Coulomb
correction nor the ionization effect is taken into account;
since we fix both the sound speed and the density, the
pressure is also fixed due to the almost constancy of the
adiabatic index; therefore, to keep the same pressure,
T /µ has to be decreased, which means that either T or
µ−1 must be decreased, at least; it can be shown from the
characteristics of the radiative transfer that an increase
(decrease) in the temperature has to be accompanied by
that of the hydrogen abundance; for example, we cannot
decrease the temperature while keeping µ−1 (hydrogen)
fixed, because the opacity is then increased, which must
result in an increase in the temperature gradient and the
temperature itself; therefore, the decrease in T and that
in µ−1 (hydrogen) must occur simultaneously in this
case. Note that the temperature decrease is much smaller
than the hydrogen decrease because of the large power
index of the temperature in the opacity expression (β) in
addition to that in the expression for the radiation energy
density (4).
• Because of the large power index, β + 4 = 7.17, in
equation (17), the change in the equation of state (∆)
has a larger effect on the hydrogen abundance than that
in the opacity (κ0 ) of the same size.
On the other hand, it is not so simple to explain the results
shown in figure 4 qualitatively based on equation (15) because
not only the integral, but also the constant ξ , is modified if
rconv changes. We note that, if we input the appropriate numerical values, equation (15) still reproduces the results shown in
figure 4 very well.
It is important to note that the Z abundance outside of the
tachocline layer never affects the hydrogen abundance in the
tachocline layer, because the integration of equation (13) starts
at the base of the convection zone and depends only on the Z
profile in the tachocline layer.
The simplified Z profile of the seismic solar model is qualitatively different from those of the standard solar models in
that the former is flat in the central region, while the latter
gradually decrease as the radius increases because of diffusion. One might suspect that the larger hydrogen abundance
No. 5]
Hydrogen Abundance in the Tachocline Layer of the Sun
in the tachocline layer of the seismic solar model than the
standard solar models would be possibly caused by the simplified heavy element profile shown in figure 1. We do not have
to worry about such a suspicion at all because what is important to the hydrogen profile in the tachocline layer is the heavy
element abundance only at the same place. Therefore, even if
the simplified Z profile does not reflect the behavior of the real
Z profile well below the tachocline layer in the Sun, such a
simplification is not essential to the result that hydrogen in the
tachocline layer is more reduced in the standard solar models
than in the seismic solar model.
5. Discussion
We can see from figures 2, 4, and 5 that the Z profile
and the depth of the convection zone influence the hydrogen
profile more significantly than microphysics, such as do the
equation of state and the opacity. As we can understand
from equation (15), such a dependence on the Z profile is not
because of an intrinsic difference in the sensitivity, but simply
because the fractional changes in Z in figure 3 are larger than
those of the opacity and the equation of state in figure 5. Note
that the uncertainties in the Z profile and the depth of the
convection zone can be expected to be reduced by helioseismic
observations in the future.
Though we do not take account of the overshooting explicitly, we may consider its effect in the following way: if
overshooting occurs, the temperature gradient at the base of the
convection zone does not coincide with the adiabatic temperature gradient; both gradients rather agree at a position slightly
displaced inward in the convection zone; therefore we can
effectively take the overshooting into account by decreasing
the depth of the convection zone. In fact, if we adopt rconv =
0.716 R , the hydrogen abundance in the tachocline layer
relative to its surface value becomes greater than that of the
case for rconv = 0.713 R , which is essentially the value of the
standard solar model (see figure 4).
When we constructed a seismic solar model, we assumed
that the effect of the magnetic field is negligibly small, though
it is believed that the tachocline layer is the seat of the solar
dynamo. In a case where we consider the influence of the
magnetic field, we have to redetermine the sound-speed and
density profiles, which are the inputs to the seismic solar
model, from the solar oscillation frequencies, because the
equation of hydrostatic equilibrium, which is assumed in the
inversion process, is modified if a magnetic field exists. It is,
however, easily shown that even a magnetic field as strong as
106 G corresponds to a magnetic pressure on the order of only
0.1% of the thermal pressure in the tachocline layer, which
means that even such a strong magnetic field hardly affects the
sound speed and the density. Hence, we can safely neglect
the effect of the magnetic field on the structure of the seismic
solar model. In fact, Chou and Serebryanskiy (2002) claim
that they have detected a signature of the solar cycle variation
at the base of the convection zone, which roughly corresponds
to a magnetic field on the order of 105 G at the solar maximum,
whereas Basu and Antia (2003) cannot find any considerable
temporal variation in the tachocline structure between 1995 and
2002.
1021
In spite of the fact that the opacity is usually thought to
be the main source of uncertainties in the evolutionary solar
models (cf. Tripathy, Christensen-Dalsgaard 1998), our experiments shown in figure 5 suggest that the small errors in the
opacity are not sufficient to explain the sound-speed difference between the evolutionary solar models and the Sun in the
tachocline layer. Of course, we cannot exclude the possibility
that the sound-speed difference in other parts of the Sun can be
understood by such a small change in the opacity.
One might suspect that the apparent higher hydrogen
abundance in the tachocline layer of the present seismic solar
models would actually be caused by a smoothed hydrogen
profile as a result of the limited resolution of the sound-speed
and density inversions. Such a resolution problem is actually
not serious at all in the current arguments, as we explain in
the following: the sound-speed and density profiles used in the
present work were obtained by connecting the local averages,
which are the outputs of the optimally localized averaging
(OLA) method of the helioseismic inversion (Basu 1998);
the bump at the tachocline layer in the sound-speed profile
obtained with the OLA method must be milder than the real
one because of the limited resolution of the inversion; then,
the true difference in the hydrogen abundance in the tachocline
layer between the Sun and the standard solar models should
be regarded as being larger than that shown in the present
work; hence, our conclusion that the hydrogen abundance in
the tachocline layer is higher in the Sun than in the standard
solar models is unlikely to be influenced by the limited resolution of the sound-speed and density inversions.
All of figures 2, 4, and 5 clearly show that the hydrogen
profiles of the seismic solar model in the tachocline layer are
different from those of the evolutionary solar models regarding
two points. One of them is that the hydrogen abundance of
the seismic solar model decreases inward more mildly than
that of the evolutionary solar model in the tachocline layer.
The other point is that the shape of the profiles is convex in
the seismic models, while it is concave in the evolutionary
models. Figure 7 shows the relative difference in the squared
sound speed, the temperature, and the hydrogen abundance
between the default seismic solar model and the standard solar
model by Christensen-Dalsgaard et al. (1996). It is clear that
the bump in the sound-speed difference is mostly explained
by the difference in the hydrogen abundance. Note that both
of the evolutionary solar models by Bahcall et al. (1998) and
Christensen-Dalsgaard et al. (1996) take account of only the
diffusion process as a possible mechanism which changes the
chemical abundance in the tachocline layer. This implies that
it seems difficult to explain the convex profile only by the
diffusion process. On the other hand, if mixing is the most
dominant process that changes the chemical composition in
the tachocline layer, then the hydrogen abundance must be
homogenized, and we can expect a flat hydrogen profile there.
Therefore, the hydrogen profile in the tachocline layer of the
seismic solar model strongly suggests that there is competition
among some processes, such as diffusion, which inhomogenizes the hydrogen abundance, and mixing, which tends to
give a constant hydrogen profile. We note that the mass
loss process could be the alternative to the mixing process
to explain the bump in the sound-speed difference, as Gough
1022
M. Takata and H. Shibahashi
[Vol. 55,
2
/dr change rapidly in
3. Both of the functions F and d lnc
the tachocline layer.
From the first point, we expect that a good approximation
would be obtained by subtracting a small constant from the
factor X + 1 and adding another constant to the factor X +
3/5 so that both factors become identical. The second point
suggests that the magnitude of such a constant shift in the factor
X + 3/5 must be smaller than that of the factor X + 1. We
therefore denote the former by 2 and the latter by −γ , where
is a small positive number and γ is a positive constant on the
order of 1. Introducing a new dependent variable, y, by
3
+ 2,
5
we can rewrite equation (13) as follows:
δ/
dy
= R F Zy δ + 1 1 + γ y − 1 − 2 y −
dη
2
d ln c
y 1 − 2 y − ,
+
dx
where
r − rconv
η≡
,
R
r
,
x≡
R
y ≡ X +
Fig. 7. Plotted are the relative differences in the squared sound speed
(c2 ), the temperature (T ), and the hydrogen abundance (X) between
the default seismic solar model and the standard solar model (model S)
by Christensen-Dalsgaard et al. (1996). The abscissa is the fractional
radius.
et al. (1996) discuss. What is important is that the hydrogen
profile of the seismic solar model reflects that of the presentday Sun independently of any specific formulations of diffusion, mixing, and mass-loss.
6.
Summary and Conclusions
In this paper, we have discussed the hydrogen profile in the
tachocline layer of the Sun. The point is that the hydrogen
profile of the Sun is given by the seismic solar model, which
is consistent with the sound-speed profile, the density profile,
and the depth of the convection zone determined by helioseismology.
Even if we take account of the uncertainties in the input
physics and parameters of the seismic solar model, we always
find that the hydrogen abundance just beneath the base of the
convection zone decreases inward more mildly than that of the
standard solar models, and that the sign of the curvature of the
seismic profile is negative, whereas that of the evolutionary
profiles is positive. Such a milder change in the hydrogen
abundance is compatible with the speculation that some mixing
process in addition to the diffusion process operates in the
tachocline layer.
γ =
2
− 2,
5
(A1)
(A2)
(A3)
(A4)
(A5)
and
δ
= β + 5.
(A6)
Here, we have assumed that δ is also a positive constant on the
order of 1. Note that is not exactly defined in advance, but
will be set later to some appropriate value. We also note that the
2
functions R F and d ln c
/dx do not change very rapidly as
functions of η, while they do as functions of x = r/R (the third
point at the beginning of this section). This kind of coordinate
stretching is typical in the boundary layer problems (cf. Nayfeh
1973).
Regarding as a small parameter and substituting the expansion
y = y0 + y1 + 2 y2 + O( 3 )
(A7)
We thank T. Sekii and A. Birch for the helpful comments.
This research was partially supported by the Grant-in-Aid for
Scientific Research of the Japanese Ministry of Education,
Culture, Sports, Science and Technology (No. 12047208).
into equation (A2), we have the following perturbation
equations:
Appendix. Derivation of the Approximate Expression for
the Hydrogen Abundance in the Tachocline Layer
and
We solve equation (13) by a perturbative approach. There
are three points to note about this equation before we start the
analysis:
1. If X + 1 is replaced by X + 3/5, or if X + 3/5 is replaced
by X + 1, we can have the exact analytical solution.
2. The power index β + 5 = 8.17 is a large number.
2
d ln c
dy0
= R F Zy0δ + 1 +
y0
dη
dx
(A8)
2 d ln c
dy1
= (δ + 1)R F Zy0δ +
y1
dη
dx
+ (γ − δ)R F Zy0δ + 1
(A9)
for the zeroth and first orders in , respectively. Taking y1 = 0 at
the base of the convection zone as the boundary condition for
equation (A9), we can entirely eliminate the first-order term,
y1 , from expansion (A7), provided that
No. 5]
Hydrogen Abundance in the Tachocline Layer of the Sun
γ = δ.
(A10)
We use this condition to determine as well as γ and δ. Using
equations (A5), (A6), and (A10), we have
2
=
= 0.21
(A11)
5(β + 6)
and
γ = δ = (β + 5)
2
= 1.7,
5(β + 6)
(A12)
where the numerical values are evaluated by using β = 3.17.
Note that these values are consistent with the three assumptions: 1, γ ∼ 1, and δ ∼ 1. Since the first-order term
vanishes, the solution of equation (A8), which is given by
2δ/
2/ c (rconv )
y0 = c (r)
y0 (rconv )δ
−1/δ
0
2δ/ +δ
R F (η )Z(η )c r (η )
dη
,(A13)
η(r)
has an error on the order of only 2 . Substituting this into
equation (A1) and utilizing equations (A3), (A11), and (A12),
we find
− β 1+ 5
rconv
2
1 + O( 3 )
F̃ Z dr
X = c (r) ξ +
1023
where ξ and F̃ are defined by equations (16) and (17), respectively. Note that the error in expression (A14) is not on
the order of 2 , but 3 because of the power index in
equation (A1).
Accepting γ = δ and y1 ≡ 0, we have the equation of the
second-order perturbation, y2 , as follows:
2 d ln c
dy2
δ
= (δ + 1)R F Zy0 +
(A15)
y2 − Hy0 ,
dη
dx
where
2
d ln c
1
H ≡ δ 2 R F Zy0δ +
.
(A16)
2
dx
If we evaluate function H by using the inverted sound-speed
and density profiles, it turns out that H is a quantity on the
order of . Therefore, the second term on the right-hand-side
of equation (A15) can be actually transferred to the equation of
the third-order perturbation. As a result, we can also neglect y2
because of the boundary condition that y2 = 0 at η = 0. Hence,
the error in equation (A14) can be essentially regarded as being
on the order of 4 . This explains the remarkable agreement
shown in figure 6 between equation (A14) and the numerical
solution of equation (13).
r
−
3β + 20
,
5(β + 6)
(A14)
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