Fig. 12.3. Intrinsic di¬erence between γ1 obtained from an inversion of helioseis-

mological data (Basu et al. 1999), and γ1 of 4 MHD models (M1-4: ¬lled points)

and 4 OPAL models (M5-8: empty points), respectively. All results are in the sense

“sun “ model” (for more detail, see Basu et al. 1999).

the di¬erence in the equation of state (Basu & Christensen-Dalsgaard 1997).

The error bars shown in Figure 12.3 are based on combined errors of the

inversion method and observational errors.

First, Figure 12.3 reveals the accuracy of present-day helioseismology,

which has attained almost 10’4 for γ1 . Second, Figure 12.3 reveals the

in¬‚uence of the equation of state. The method is by an analysis of the

di¬erence between the solar values obtained from inversions and the ones

computed in reference models (standard solar models). Since in Figure 12.3

two sets of reference models (MHD and OPAL) are compared with the real

solar data, whereas all other model ingredients are kept the same, the ¬gure

Solar constraints on the equation of state 185

not only re¬‚ects the di¬erence between the solar models and the sun but

also the di¬erence between the MHD and OPAL models themselves.

Figure 12.3 should not be over-interpreted, however, because present un-

certainties in the inversion of the upper layers of the sun (e.g., turbulent

pressure, magnetic ¬elds, nonlocal thermodynamic e¬ect due to radiation,

uncertainties in the chemical composition) preclude so far a de¬nitive inter-

pretation, and further clarifying work is in progress. In the slightly deeper

regions (below a depth of about 3% of the solar radius) the ¬ndings of the

study (Basu et al. 1999) are more reliable, and they con¬rm the ¬ndings of

Figure 12.1, that is, overall OPAL is a better equation of state than MHD.

However, should the results in the top 2% of the sun remain in favor of

MHD, they would demonstrate the signi¬cance of the di¬erent implementa-

tions of many-body interactions in the two formalisms. In principle, since

density decreases in the upper part, OPAL, by its nature of a systematic

expansion, inevitably becomes itself more accurate; but MHD might, by its

heuristic approach (and by luck!), have incorporated even ¬ner, higher-order

e¬ects. Since helioseismology gives localized information, it is natural that

the various equations of state have their preferred regions in the sun. One

should, however, resist the temptation to produce a “combined” solar equa-

tion of state, with di¬erent pieces for di¬erent parts of the sun. Such a hybrid

solution is fraught with danger, because patching equations of state together

can introduce spurious e¬ects (D¨ppen et al. 1993). It is better to seek an

a

improvement of individual equations of state, such as MHD and OPAL, in

parallel and independently, guided by the progress of helioseismology.

12.2.4 Heavy elements

In solar modeling, an adequate treatment of the heavy elements and their

excited states is important. The treatment is subject to the stringent re-

quirements of helioseismology. In a recent analysis (Gong et al. 2001a), the

contribution of various heavy elements in a set of thermodynamic quantities

was examined.

To show speci¬cally the contribution of each individual heavy element,

Gong et al. (2001a) have calculated solar models with particular mixtures.

In each case, hydrogen and helium abundances were ¬xed with mass frac-

tions X = 0.70 and Y = 0.28; the remaining 2% heavy-element contribution

was topped o¬ by only one element, carbon, nitrogen, oxygen and neon,

respectively. Gong et al. (2001a) then compared these models with a pure

hydrogen-helium mixture (such a mixture is obtained by replacing the two

per cent reserved for heavy elements with additional helium). The expecta-

D¨ppen

a

186

’3

x 10

5

b A B

4

3

2

1

0

1

∆γ

’1

’2

’3

’4

’5

’6

3.5 4 4.5 5 5.5 6 6.5 7

log T

Fig. 12.4. Di¬erence in thermodynamic quantities between the H-He-C mixture

and the H-He mixture for various equations of state. Di¬erences are in the sense

[X(Model)H’He’C - X(Model)H’He ]. Line styles are: Dashed line: MHD; Thick

Solid line: OPAL; Dashed-dotted line: CEFF; Dotted line: SIREFF; Thin Solid

line: MHDGS . See Gong et al. (2001a) for more details.

tion is that the biggest deviations between these special models, and the one

with the complete heavy-element mixture, will result (i) from the change in

the total number of particles per unit volume and (ii) from the di¬erent

ionization potentials of the respective elements. Because the solar plasma is

only slightly non-ideal, the leading pressure term is still given by the ideal-

gas equation pV = N kB T , with p standing for pressure, N the number of

particles, and kB the Boltzmann constant. Because of their higher mass,

the number of heavy-element atoms is obviously smaller than that of helium

atoms representing the same mass fraction. However, against this reduction

in total number there is an o¬set due to the ionization of their larger number

of electrons which becomes stronger at higher temperatures. The resulting

Solar constraints on the equation of state 187

net change of the total number of particles is therefore a combination of

these two e¬ects. Details can be found in Gong et al. (2001a).

A further comparison done by Gong et al. (2001a) had the purpose to dis-

entangle even further the contribution of the mere presence of each heavy ele-

ment from the more subtle in¬‚uence of di¬erent physical formalisms. In Fig-

ure 12.4 various H-He-C models (with mass fractions 0.70:0.28:0.02) are com-

pared with the analogous H-He models (0.70:0.30) for a set of di¬erent equa-

tions of state, respectively. In addition to OPAL and MHD, three other equa-

tions of state were considered. First, MHDGS , which is MHD but with parti-

tion functions truncated to ground states; second CEFF, which is the Eggle-

ton, Faulkner & Flannery (EFF) (1973) formalism with an added Coulomb

term in the Debye-H¨ckel approximation (Christensen-Dalsgaard & D¨ppen

u a

1992); third, SIREFF, another EFF sibling (Guzik & Swenson 1997), which

includes hydrogen molecules and mimics OPAL in some respects.

Comparing H-He-C with H-He models allows one to eliminate most of the

di¬erence due to the treatment of hydrogen and helium in each individual

equation of state. The comparison therefore isolates the behavior of carbon

in each of our equations of state. (The slight roughness of the OPAL curve

in Figure 12.4 is due to the fact that we have almost reached the limit

interpolation accuracy of the available OPAL tables.)

From Figure 12.4, which shows ∆γ1 graph between H-He-C and H-He, it

follows that γ1 reveals the biggest di¬erences between the various equations

of state in the temperature range of 4.5 < log T < 5.5 (named region “A” in

the ¬gure). Common features can be identi¬ed however, when log T > 5.5

and log T < 4.5.

The bump in ∆γ1 in the region “B” of Figure 12.4 is totally independent

of the equation of state used. Gong et al. (2001a) showed that the bump

feature is likely due to the ionization of carbon at that temperature in gen-

eral, and importantly, quite independent of details in the equation of state.