Effects of near-surface changes

I now assume that δFl in equation (5.81) is localized near the stellar surface, in the sense that

φ_r[ξ](r)'0, φ_h[ξ](r)'0 for R−r > , (5.87) for some small . For modes extending substantially more deeply than the region of the perturbation,i.e., withR−r_t , the eigenfunctions are nearly independent ofl at fixed frequency in that region (see also Figure 5.8 and the associated discussion). Hence I_nl

4Thesemodel changesshould not be confused with the Lagrangian perturbations associated with the oscillations, introduced in Chapters 3 and 4.

5As noted by Basu & Christensen-Dalsgaard (1997) one may in addition include a contribution from the intrinsic difference (δΓ1)int between the model and the true Γ1, at fixed thermodynamic conditions and composition.

114 CHAPTER 5. PROPERTIES OF SOLAR AND STELLAR OSCILLATIONS.

depends little on l at fixed ω. To get a more convenient representation of this property, I introduce

Qnl = E_nl

E0(ω_nl) , (5.88)

whereE_l(ω) is obtained by interpolating to ω inE_nl at fixed l. Then

Q_nlδω_nl (5.89)

is independent of l, at fixed ω, for modes such thatR−rt . Conversely, ifQ_nlδω_nl is independent ofl at fixed ω for a given set of modes, thenδFl is probably largely localized outsider = max(r_t) over the set of modes considered.

Figure 5.22: The inertia ratio Q_nl, defined in equation (5.88), against fre-quencyν, for f and p modes in a normal solar model. Each curve corresponds to a given degreel, selected values of which are indicated.

Q_nl has been plotted in Figure 5.22, for selected values ofl. Its variation withlis largely determined by the change in the penetration depth. Modes with higher degree penetrate less deeply and hence have a smaller inertia at given surface displacement. As a consequence of this their frequencies are more susceptible to changes in the model.

An important example concerns the uncertainties in the physics of the model and the oscillations in the near-surface region, discussed in Section 5.1.2 and indicated schematically in Figure 5.1. These are confined to a very thin region, and hence, according to the discussion following equation (5.89), we may expect that their effects on the frequencies, when scaled withQ_nl, are largely independent of degree at a given frequency. As discussed

5.5. THE FUNCTIONAL ANALYSIS OF ADIABATIC OSCILLATIONS 115 below, and in more detail in Sections 7.7 and 9.2, this can be used to eliminate the effects of the uncertainties in the analysis of observed frequencies.

Figure 5.23: Relative sound-speed differenceδ_rc/cbetween a model with mod-ified surface opacity and a normal model, in the sense (modmod-ified model) – (normal model).

To illustrate this behaviour, we may analyze model changes localized very near the solar surface. Specifically, I consider a solar model where the opacity has been artificially increased by a factor of 2 at temperatures below about 10⁵K; this is compared with a normal model of the present Sun. In both models the mixing-length parameter and composition have been adjusted so as to obtain a model with solar radius and luminosity; the effect of this is that the interior of the model is virtually the same. The opacity has no effect in the bulk of the convection zone where energy transport is totally dominated by convection;

hence the change in the model is largely confined to the atmosphere and the uppermost parts of the convection zone. Figure 5.23 shows the sound-speed difference δrc² between the modified and the normal model.

The unscaled differences between frequencies of the two models are shown in Figure 5.24a. It is evident that the differences show a very systematic increase in magnitude with increasing degree. As shown by Figure 5.24b this is entirely suppressed by scaling the differences by Q_nl. Indeed, the differences now seem to decrease with increasing l.

This is predominantly due to the fact that at the largest values of l the modes can no longer be assumed to be independent, near the surface, of degree at fixed frequency (for an asymptotic description of this behaviour, see for example Gough & Vorontsov 1995). A small additional contribution comes from the fact that the modes only penetrate partly, to a degree-dependent extent, into the region where the sound speed has been modified.

It might also be noted that the frequency differences are very small at low frequency.

This is related to the fact that low-frequency modes have very small amplitudes in the surface region (cf. Figure 5.9). However, a proper understanding of this feature requires a more careful analysis (see Christensen-Dalsgaard & Thompson 1997).

116 CHAPTER 5. PROPERTIES OF SOLAR AND STELLAR OSCILLATIONS.

Figure 5.24: Frequency differences between a model with modified surface opacity and a normal model, in the sense (modified model) – (normal model).

Panel (a) shows the raw differences. In panel (b) the differences have been scaled by Q_nl, in accordance with equation (5.89). Points correspond-ing to a given degree have been connected, accordcorrespond-ing to the followcorrespond-ing line styles: l = 0,1,2,3,4,5,10,20,30 (solid); l = 40,50,70,100 (short-dashed);

l = 150,200,300,400 (long-dashed); l = 600,700,800,900,1000,1100 (dot-dashed).

A rather similar behaviour is obtained for the observed frequencies, as shown in Figure 5.25. The unscaled differences (panel a) show a strong dependence on l, which is largely (but not completely) suppressed by scaling. This strongly suggests that most of the errors in the model are located very near the surface. I note that this is precisely the region where, according to the arguments in Section 5.1.2, many of the uncertainties in the physics of the model and the oscillations are located. Thus it is not surprising that we obtain an effect on the frequencies of this nature. As discussed in Section 9.2 it is possible to suppress the effect, precisely because of its systematic dependence on frequency. However, I note also in

5.5. THE FUNCTIONAL ANALYSIS OF ADIABATIC OSCILLATIONS 117

Figure 5.25: Frequency differences between observed data and frequencies of a solar model, in the sense (observations) – (model). The observed frequencies were obtained from the MDI experiment on the SOHO spacecraft (Rhodeset al. 1997, 1998), whereas the model frequencies are for Model S of Christensen-Dalsgaardet al. (1996). Crosses indicate modes with l≤200, diamonds are modes with 200< l ≤ 500 and triangles are modes with 500 < l. Panel (a) shows the raw differences. In panel (b) the differences have been scaled byQ_nl, in accordance with equation (5.89). Here ridges corresponding to low radial ordersnare evident; the lowest values have been indicated in the figure.

Figure 5.25b that there is a significant dependence of the differences on degree; this must be associated with errors in the model in deeper layers. I return to the origin of these differences in Section 7.7.2.

For the highest-degree modes the figure shows a clear ‘pealing-off’ into ridges for each value of the radial order n. Here the upper turning point is so close to the surface that the assumptions of nearly vertical propagation, and degree-independent eigenfunctions, break down. A similar behaviour was noted in Figure 5.24 for differences between model

118 CHAPTER 5. PROPERTIES OF SOLAR AND STELLAR OSCILLATIONS.

frequencies. Also, I note that the f modes (with n = 0) represent a special case. As discussed in Section 7.5.1, these essentially have the character of surface gravity waves, with frequencies given approximately by equation (3.84); in particular, computed frequencies of normal solar models are essentially independent of the details of the structure of the model (see also Gough 1993; Chitre, Christensen-Dalsgaard & Thompson 1998). Thus the error in the f-mode frequencies evident in Figure 5.25b must arise from effects other than differences between the hydrostatic structure of the Sun and the model. A likely cause are dynamical interactions between the modes and the turbulent motion in the solar convection zone (Murawski, Duvall & Kosovichev 1998; M¸edrek, Murawski & Roberts, 1999). However, the details of such mechanisms, and in particular their effects on the p-mode frequencies, are still rather uncertain.

The probable presence of near-surface errors in the model must be taken into account when relating differences between observed and computed frequencies to the corresponding differences between the structure of the star and the model. Specifically, consider the determination of corrections δ_rc² and δ_rρ to a stellar model, from the differences δω_nl = ω_nl^(obs)−ω_nl between observed frequenciesω_nl^(obs) and model frequenciesω_nl: here we expect differences between the star and the model both in the internal structure, characterized by (c², ρ), and in the near-surface layers. In addition, the observed frequencies are unavoidably affected by errors. From the analysis above we therefore expect that equation (5.84) must be replaced by

δω_nl ω_nl =

Z _R

0

"

K_c^nl2,ρ(r)δ_rc²

c² (r) +K_ρ,c^nl2(r)δ_rρ ρ (r)

# dr

+Q⁻_nl¹G(ω_nl) +_nl , (5.90)

where the functionG(ω) accounts for the effect of the near-surface uncertainties, and_nlare the relative observational errors in the frequencies. The term inG must then be determined as part of the analysis of the frequency differences, or suppressed by means of suitable filtering of the data. As discussed above, for high-degree modes the scaled frequency effects of the near-surface problems can no longer be regarded as purely a function of frequency; a representation which takes this into account has been developed by Di Mauroet al. (2002).

Chapter 6

Numerical techniques

The differential equations (4.61), (4.62) and (4.64), in combination with boundary con-ditions such as equations (4.65) – (4.68), constitute a two point boundary value problem.

Non-trivial solutions to the problem can be obtained only at selected values of the frequency ω, which is therefore an eigenvalue of the problem. Problems of this nature are extremely common in theoretical physics, and hence there exists a variety of techniques for solving them. Nevertheless, the computation of solar adiabatic oscillations possesses special fea-tures, which merit discussion. In particular, we typically need to determine a large number of frequencies very accurately, to match the volume and precision of the observed data.

Specific numerical techniques are discussed in considerable detail by, for example, Unno et al. (1989) and Cox (1980). Here I concentrate more on general properties of the solution method. The choice of techniques and examples is unavoidably biased by my personal experience, but should at least give an impression of what can be achieved, and how to achieve it.

I note that a self-contained package for computing stellar adiabatic oscillations, with documentation and further notes on the numerical techniques, is available on the WWW athttp://astro.phys.au.dk/∼jcd/adipack.n/.

6.1 Difference equations

The numerical problem can be formulated generally as that of solving dy_i

dx = XI

j=1

a_ij(x)y_j(x), for i= 1, . . . , I , (6.1) with suitable boundary conditions at x =x₁ and x₂, say. Here the order I of the system is four for the full nonradial case, and two for radial oscillations or nonradial oscillations in the Cowling approximation.

To handle these equations numerically, I introduce a mesh x₁ = x⁽¹⁾ < x⁽²⁾ < . . . <

x^(Nme) = x2 in x, where Nme is the total number of mesh points. Similarly I introduce y_i⁽ⁿ⁾ ≡y_i(x⁽ⁿ⁾), and a⁽ⁿ⁾_ij ≡a_ij(x⁽ⁿ⁾). A commonly used, very simple representation of the differential equations is in terms of second-order centred differences, where the differential

119

120 CHAPTER 6. NUMERICAL TECHNIQUES equations are replaced by the difference equations

y_i⁽ⁿ⁺¹⁾−y_i⁽ⁿ⁾

These equations allow the solution at x = x⁽ⁿ⁺¹⁾ to be determined from the solution at x=x⁽ⁿ⁾.

More elaborate and accurate difference schemes (e.g. Presset al. 1986; Cash & Moore 1980), can be set up which allow the rapid variation in high-order eigenfunctions to be represented with adequate accuracy on a relatively modest number of meshpoints. Alter-natively one may approximate the differential equations on each mesh interval (x⁽ⁿ⁾, x⁽ⁿ⁺¹⁾) by a set of equations with constant coefficients, given by

dη_i⁽ⁿ⁾ analytically on the mesh intervals, and the complete solution is obtained by continuous matching at the mesh points. This technique clearly permits the computation of modes of arbitrarily high order. I have considered its use only for systems of order two, i.e., for radial oscillations or non-radial oscillations in the Cowling approximation.