ńņš. 66 |

ā¢ the photon luminosity is L ,

ā¢ the sound-speed proļ¬le c(r) is that obtained from helioseismology,

ā¢ the density proļ¬le Ļ(r) is that obtained from helioseismology,

ā¢ the model is in hydrostatic equilibrium,

ā¢ the model is in thermal balance,

ā¢ the updated microphysics is adopted concerning the nuclear reactions,

opacity, and the equation of state, and

Solar structure and the neutrino problem 241

ā¢ the envelope is chemically homogeneous, and Z/X there matches the spec-

troscopically determined value.

The basic equations for constructing a model with the above assumptions

are formally identical with those used in constructing evolutionary models.

The only diļ¬erence is that the independent variable used in the evolution

calculation is m which is a Lagrangian coordinate while the Eulerian coor-

dinate r is adopted in the present case:

dm/dr = 4Ļr 2 Ļ, (16.6)

dP/dr = ā’GmĻ/r 2 , (16.7)

dLr /dr = 4Ļr 2 ĻĪµ, (16.8)

ā’3ĪŗĻLr /(16ĻacT 3 r 2 ) if radiative

dT /dr = (16.9)

(1 ā’ 1/Ī“2 ) T d ln P/dr if convective

with the boundary conditions m = 0 and Lr = 0 at r = 0 and m = M

and Lr = L at r = R , where the symbols have their usual meanings (P :

pressure, T : temperature, m: mass inside the radius r, Lr : luminosity at the

radius r, Īµ: rate of nuclear energy generation, Īŗ: opacity, a: Stefan constant,

c: the speed of light, G: gravitational constant, Ī“2 : adiabatic exponent).

In addition to these basic equations, we need the equation of state, the

equations for the opacity and for the nuclear reaction rates,

Ļ = Ļ(P, T, Xi ), (16.10)

Īŗ = Īŗ(P, T, Xi ), (16.11)

Īµ = Īµ(P, T, Xi ), (16.12)

which link the thermal quantities and the chemical abundances.

It may be instructive to remember that, in constructing evolutionary mod-

els, the chemical composition proļ¬les are given at each time step by following

their temporal evolution

ā‚Xi (t, m)/ā‚t = (ā‚Xi /ā‚t)nuclear + (ā‚Xi /ā‚t)diļ¬usion (16.13)

assuming chemical homogeneity at zero age, ā‚Xi (t = 0, m)/ā‚m = 0, and

then the basic equations can be solved as a closed system. A suspected

instability and the resultant mixing may introduce an additional term in

the right-hand side of equation (16.13), but such a term is ignored in the

standard evolution scenario.

Shibahashi

242

If we distinguish only hydrogen and helium separately as X and Y , respec-

tively, and treat all other elements collectively as heavy elements, Z, then

the sound-speed and the density can be regarded as functions of the chem-

ical composition, X and Z, and any two other thermodynamical quantities

such as P and T :

c = c(P, T, X, Z) (16.14)

Ļ = Ļ(P, T, X, Z). (16.15)

It should be noted here that, as demonstrated in Section 16.2, the sound-

speed proļ¬le and the density proļ¬le in the solar interior have already been

determined from helioseismology. Hence equations (16.14) and (16.15) in-

versely relate the hydrogen abundance, X, and the heavy element abun-

dance, Z, at a given r to the pressure, the temperature, the sound-speed,

and the density; X = X(P, T, cinv , Ļinv ) and Z = Z(P, T, cinv , Ļinv ), where

cinv (r) and Ļinv (r) denote the seismically determined sound-speed and den-

sity proļ¬les, respectively. The opacity and the nuclear reaction rates are,

in turn, given in terms of (P, T, cinv , Ļinv ) by equations (16.11) and (16.12),

respectively. Thus all the variables appearing in the right-hand side of equa-

tions (16.6) ā“ (16.9) can be expressed in terms of the variables in the left-

hand side, and hence these equations are solvable. Note that in this way

we obtain directly a model of the present-day sun. Note also that we do

not need to make assumptions about the chemical composition proļ¬les in

the sun, but obtain the X, Y , and Z proļ¬les as a part of the solutions of

equations (16.6) ā“ (16.9).

The depth of the base of the convection zone, rconv , is well determined

from helioseismology. Takata & Shibahashi (1998) shifted the outer bound-

ary from r = R to r = rconv and required that the radiative tempera-

ture gradient matches the adiabatic temperature gradient there. The outer

boundary conditions are then Lr = L and āad = 3ĪŗLr P/(16ĻacGmT 4 ) at

r = rconv . This cutting oļ¬ of the convective outer 30% of the sun has little

eļ¬ect on the solar interior where the neutrinos are generated. By setting

the outer boundary at the base of the convection zone and treating only the

radiative core, we do not need to worry about the treatment of convection.

It should be noted that the sound-speed proļ¬le is more accurately de-

termined helioseismically than the density proļ¬le, as seen in Figure 16.1.

Indeed, in early 1990s, attention was paid to the sound-speed proļ¬le rather

than to the density proļ¬le. In those days, Shibahashi (1993) outlined a

recipe for constructing a solar model with the constraint of the seismically

determined sound-speed proļ¬le and with the assumption of a homogeneous

Solar structure and the neutrino problem 243

Z proļ¬le and simpliļ¬ed microphysics. Takata & Shibahashi (1998) polished

this recipe and succeeded in constructing a realistic solar model by impos-

ing a constraint of the seismically determined sound-speed proļ¬le cinv (r) and

with the assumption of a homogeneous Z proļ¬le in advance.

16.7 Seismic solar model and the neutrino ļ¬‚ux estimate

In practice it is fairly hard to determine the Z proļ¬le directly as outlined

in the previous section, since the dependence of the equation of state on Z

is weak. Alternatively, Watanabe & Shibahashi (2002) constructed a series

of solar models with various Z proļ¬les by imposing the constraint of the

seismically determined sound-speed proļ¬le, and searched among them for

the model for which the density proļ¬le ļ¬ts best with the seismically deter-

mined density proļ¬le Ļinv (r). If Z(r) is given, X is represented in terms

of (P, T, cinv , Z) by equation (16.14), and in turn the density, the opacity

and the nuclear reaction rates are given as well in terms of (P, T, cinv , Z) by

equations (16.10) ā“ (16.12), then equations (16.6) ā“ (16.9) can be solved.

The problem is changed to ļ¬nding, among various seismic solar models, the

rconv

[Ļinv (r) ā’ Ļmodel (r)]2 /ĻĻ dr, where ĻĻ is

model which minimizes F ā” 2

0

the standard error of the helioseismically determined density. By carrying

out numerical experiments reproducing the solar model of Bahcall et al.

(2001) with a constraint of the modelā™s true sound-speed proļ¬le, Watan-

abe & Shibahashi (2002) demonstrated that the modelā™s Z proļ¬le could be

determined no better than to within 0.001. In applying the recipe for the

realistic seismic data they restricted themselves to trying stepwise Z pro-

ļ¬les, varying the height with a step of 0.001; the step width ār/R was 0.1

except for the innermost step which was taken to be 0.2.

Figure 16.2 shows the X, Y , and Z proļ¬les determined in this way. The

thick solid lines correspond to the best ļ¬t model, and the error bars in the

X and Y proļ¬les are the 1-Ļ uncertainty level due to all the uncertainties in

microphysics and the errors in the inverted proļ¬les of the sound speed and

of the density. The 1-Ļ uncertainty level in the Z proļ¬le is shown by dotted

lines. For comparison, the evolutionary model by Bahcall et al. (2001) is

also shown with the dashed lines in each panel. As seen in the panel (c)

of Figure 16.2, the uncertainty in the Z proļ¬le is quite large. But the very

important point is that even the Z proļ¬le can also be deduced from helioseis-

mology. It should be noted here that the main source of the error is not the

error in the helioseismically determined proļ¬le of the sound speed nor that

of the density, but the nuclear cross-section of the pp-reaction (S11 -factor).

Shibahashi

244

SeiSM01 ā“1Ļ

+ SeiSM01

(a) (b) (c)

1Ļ

BPB01

0.026

0.700 0.600 BPB01

0.024

0.600 0.500

0.022

X Y Z

0.500 0.400 0.020

ńņš. 66 |