Precise measurements of solar oscillation frequencies provide data for ac-

curate inversions for the sound speed in the solar interior. Except in the

very outer layers, the strati¬cation of the convection zone is almost adia-

batic and the Reynolds stresses are negligible. The sound-speed pro¬le is

governed principally by the speci¬c entropy, the chemical composition and

the equation of state, and it is therefore essentially independent of the un-

certainties in the radiative opacities. The inversions thus reveal, via minute

variations in the adiabatic exponent of the solar plasma, physical processes

that have just a small in¬‚uence on the equation of state.

Although a simple ideal-gas model of the plasma of the solar interior

179

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180

was adequate before helioseismology, helioseismic equation-of-state analyses

require more sophisticated physical models. The need to go beyond the

ideal-gas approximation for helioseismic applications had been recognized in

the early 1980s (e.g. Berthomieu et al. 1980, Ulrich 1982, Noels et al. 1984).

With the better data available towards the end of the 1980s, a clearer picture

began to emerge. Christensen-Dalsgaard et al. (1988) demonstrated that

it was essential to include the leading Coulomb correction. The Coulomb

correction is due to the sum of all pair interactions between charged particles

(electrons, nuclei and compound ions); together they lead to a screening of

the Coulomb potential and to a negative pressure correction with respect to

the ideal-gas value. The leading-order Debye-H¨ckel (DH) theory is a good

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approximation for solar conditions.

Helioseismic equation-of-state studies typically use solar models based on

sophisticated new equations of state, in particular, the ones underlying the

two ongoing major opacity recomputation e¬orts. One of these e¬orts is the

international Opacity Project (OP; see the books by Seaton 1995, Berrington

1997); it contains the so-called Mihalas-Hummer-D¨ppen equation of state

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(Hummer & Mihalas 1988; Mihalas, D¨ppen & Hummer 1988; D¨ppen et

a a

al. 1988; hereinafter MHD) and it deals with heuristic concepts about the

modi¬cation of atoms and ions in a plasma. The other e¬ort is being pursued

at Lawrence Livermore National Laboratory by the OPAL group (Iglesias &

Rogers 1996; Rogers, Swenson & Iglesias 1996); its equation of state is based

on a detailed systematic method to include density e¬ects in a plasma.

Speci¬c reviews address the helioseismic determination of the equation

of state (e.g., Christensen-Dalsgaard & D¨ppen 1992; Baturin et al. 2000).

a

The article by Christensen-Dalsgaard et al. (2000) contains a signi¬cant part

dedicated to the helioseismic equation-of-state diagnosis, too. Finally, there

is an article (D¨ppen & Guzik 2000), which contains a practical compilation

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of available stellar equations of state (and opacities), including their detailed

speci¬cations, and information about how to obtain them.

Although approximate asymptotic techniques (see Christensen-Dalsgaard

et al. 1985; Gough 1993) exist to invert solar oscillation frequencies for the

internal sound speed, for an accurate analysis of the observations, a fully-

¬‚edged, non-asymptotic numerical treatment of the oscillations is manda-

tory (see Gough et al. 1996). Figure 12.1 is a typical result of such a numer-

ical inversion (Basu & Christensen-Dalsgaard 1997). It shows the relative

di¬erence (in the sense sun “ model) between the squared sound speed ob-

tained from inversion of oscillation data and that of a two standard solar

models. The two solar models used are identical in all respects except for

their equation of state, MHD (circles) and OPAL (triangles), respectively.

Solar constraints on the equation of state 181

Fig. 12.1. Di¬erence between squared sound speed from inversion and two solar

models. Figure by S. Basu.

For the present purpose, we can consider inversion results such as Figure 12.1

as the data of helioseismology, disregarding the procedure through which

they were actually obtained from solar oscillation frequencies. It follows

from Figure 12.1 that in most parts of the sun the OPAL equation of state

is a better approximation to reality than MHD, but OPAL needs to be im-

proved as well. In the outermost layers of the sun, however, the general

trend might be reversed (Section 12.2.3).

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182

Fig. 12.2. Di¬erence between γ1 (here denoted “1 ) of a solar model and observation,

for models with nonrelativistic electrons (top) and relativistic electrons (bottom).

Figures by J. Elliot.

12.2 Equation of state issues

12.2.1 Coulomb correction

As mentioned above, the most important result from earlier helioseismic

equation-of-state analyses was that it is essential to add the leading Coulomb

correction (Debye-H¨ckel term) to ideal-gas thermodynamics (Christensen-

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Dalsgaard et al. 1988, 1996). The relative Coulomb pressure correction

peaks in the outer part of the convection zone (about “8 per cent) and

in the solar core (about “1 per cent). Both MHD and OPAL contain the

Debye-H¨ckel correction; the good agreement seen in Figure 12.1 would be

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Solar constraints on the equation of state 183

an order of magnitude less without it (Christensen-Dalsgaard et al. 1996).

The discrepancy between theory and present observations is clearly much

smaller than the Debye-H¨ckel correction.

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12.2.2 Relativistic electrons

The strong constraints from helioseismology revealed the in¬‚uence of rela-

tivistic electrons. The original versions of MHD and OPAL had not included

relativistic electrons (although both did include degeneracy). In a recent he-

lioseismic inversion for the adiabatic gradient γ1 = (‚ ln p/‚ ln ρ)s (s being

speci¬c entropy), Elliot and Kosovichev (1998) found a discrepancy between,

on the one hand, the observed structure of the sun, and, on the other hand,

models using the OPAL or MHD equation of state.

The top panel of Figure 12.2 shows this discrepancy for MHD. The rel-

evant deviation occurs in the central 30% parts of the sun. (A corre-

sponding ¬gure for OPAL would look essentially the same.) A relativistic

treatment of the degenerate electrons in the solar model (bottom panel) re-

moves the discrepancy nicely. As a result, both MHD (Gong et al. 2001b;

Gong et al. 2001c) and OPAL (Rogers, private communication) were since

upgraded to include relativistic electrons.

12.2.3 E¬ect of excited states in hydrogen and helium

Another e¬ect beyond the Debye-H¨ckel correction is the signature of the

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internal partition functions. Nayfonov and D¨ppen (1998) discovered a “wig-

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gle” in the thermodynamic quantities, located in the hydrogen and helium

ionization zones. This e¬ect, due to excited states, has probably already

been observed in the sun, because new observations (Basu et al. 1999) sug-

gest that in the top 2% of the solar radius, MHD models can give a more

accurate match with the data than OPAL models. Since it turns out that

in this region, the discrepancy between MHD and OPAL is essentially re-

¬‚ected by the aforementioned wiggle (Nayfonov & D¨ppen 1998), the result

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of the inversion (Basu et al. 1999) could mean a validation of an MHD-like

treatment (Hummer & Mihalas 1988) of exited states.

The main result of Basu et al. (1999) is shown in Figure 12.3. It is

the result from an inversion of observed solar oscillation frequencies for the

intrinsic γ1 di¬erence between the sun and a solar model. The intrinsic

di¬erence is that part of the γ1 di¬erence which is due to the di¬erence in

the equation of state itself; there is a further component to the γ1 di¬erence

caused by the change to the structure of the solar model resulting from

D¨ppen

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184