On solar p-mode frequency shifts caused by near-surface model changes

Transcription

1 Mon. Not. R. Astron. Soc. 284, (1997) On solar p-mode frequency shifts caused by near-surface model changes J. Christensen-Dalsgaard 1,2 and M. J. Thompson 3 1 Teoretisk Astrofysik Center, Danmarks Grundforskningsfond, nstitut for Fysik og Astronomi, Aarhus Universitet, DK-8000 Aarhus C, Denmark 2nstitut for Fysik og Astronomi, Aarhus Universitet, DK-8000 Aarhus C, Denmark 3 Astronomy Unit, School of Mathematical Sciences, Queen Mary and Westfield College, Mile End Road, London E1 4NS Accepted 1996 August 2. Received 1996 July 29; in original form 1996 March 7 1 NTRODUCTON ABSTRACT The effect on p-mode frequencies of changing the near-surface structure of solar models is investigated. As is well known, changes confined to the near-surface region have little effect on the low-frequency p modes: this is as one would expect from a simple asymptotic description which shows that these modes have upper turning points located well beneath the photosphere. However, some examples of structural changes show that, if the changes are viewed at fixed fractional radius (an Eulerian description), the small frequency shifts at low frequency come about through near-cancellation of different contributions which are individually much larger than the resultant shifts themselves; the reason is that even socalled near-surface changes extend substantially below the upper turning points of these modes. We demonstrate that the corresponding changes at fixed fractional mass (a Lagrangian description) are confined much closer to the surface, so that the small frequency shifts come about in a natural way. Key words: Sun: interior - Sun: oscillations. There is good reason to study in detail how solar p-mode frequencies are affected by changes in the structure of the Sun in the photosphere and the layers immediately below. All observed modes are sensitive to this near-surface region, to a greater or lesser extent; indeed, it is evident from comparing observed frequencies with those of solar models that uncertainties there contribute substantially to the errors in the theoretical frequencies (e.g. Christensen Dalsgaard, Dappen & Lebreton 1988). On the one hand, this discrepancy should allow us to use the observed frequencies to improve our models of the near-surface region. On the other hand, the presence of errors in the surface layers might be a source of error in inversions for the structure and rotation of the deeper interior. A further reason for wishing to understand the connection between near-surface structure and p-mode frequencies is that the observed changes in solar frequencies with time appear to be caused predominantly by changes in the near-surface layers (Libbrecht & Woodard 1990; cf. Balmforth, Gough & Merryfield 1996). The p modes are standing waves formed in an acoustic cavity between a lower turning point, at radius r t, and an upper turning point, at radius R t The lower turning point occurs where downward-propagating waves get refracted back up towards the surface by the increasing adiabatic sound speed c(r). The upper turning point, which for many modes is in the near-surface layers, occurs when the vertical scale of the waves becomes comparable to the scale on which the vertical stratification is changing: this occurs where the frequency w of the wave is approximately equal to the acoustic cut-off frequency wc(r) given by (1.1) where H(r) is the density scaleheight (Deubner & Gough 1984). Mode frequencies are determined principally by conditions in this acoustic cavity, for their energy density is largest in that region, while outside it decays away exponentially. The acoustic cut-off frequency increases from the deep interior, where it is small, towards the surface, so that the upper turning point is located deeper for a low-frequency mode than for a mode with higher frequency. For example, for one typical solar model a lowdegree 1-mHz mode has its upper turning point at r = 0.987R, R being the photospheric radius, whereas the corresponding 4-mHz mode has the upper turning point at r = 0.999R (e.g. Christensen Dalsgaard & Perez Hernandez 1992). Thus one expects from this argument that, if the near-surface layers are modified in some way, the frequency of the low-frequency mode should be less affected than the frequency of the higher-frequency mode. This is precisely what is found in model calculations (e.g. Christensen-Dalsgaard 1986, 1990; see also below). Yet a more careful inspection of model changes indicates that this explanation may be naive. Such changes are conventionally evaluated at fixed r; it is found that, even if the physical cause of the modifications is located in and immediately below the atmosphere, their effects on the structure of the model typically extend well into the acoustic cavities of modes of even quite low frequency. Thus 1997 RAS

2 528 J. Christensen-Dalsgaard and M. J. Thompson one might expect such modes to be substantially affected by the modifications, unlike the asymptotic intuition and the computational results. Here we show that this behaviour can be understood quite simply by considering instead model changes evaluated at fixed interior mass m. n Section 2 we discuss some properties of two specific model changes and the resulting frequency shifts. Section 3 considers the general properties of near-surface model changes, whereas Section 4 sets forth the machinery for relating the frequency changes to the model changes and applies it to the examples considered in Section 2, and shows how the results can be understood in terms of the asymptotic properties of the oscillations. Finally, Section 5 contains a brief discussion. 2 EXAMPLES OF NEAR-SURFACE PERTURBATONS We have considered two different examples of mode\ changes the physical origin of which is localized to the superficial layers of the star: a change in the low-temperature opacity; and a modification of the treatment of the superadiabatic gradient in the uppermost parts of the convection zone. Some aspects of similar model changes, and their effects on the frequencies, were discussed by Perez Hernandez & Christensen-Dalsgaard (1994). Although these examples are far from exhausting the possible uncertainties near the top of stellar models, they may at least be representative of some of the known problems. n particular, recent proposed alternative models for convection result in qualitatively similar modifications (e.g. Canuto & Mazzitelli 1991; see also Monteiro, Christensen-Dalsgaard & Thompson 1996); analogous results are also obtained from averaged hydrodynarnical models of near-surface convection (e.g. Rosenthal et al. 1995). 2.1 Characterization of the models Except for the specific modifications, the physics of the models was essentially the same as for the normal model of Christensen Dalsgaard, Proffitt & Thompson (1993); in particular, we used the CEFF equation of state (Christensen-Dalsgaard & Dappen 1992) and the OPAL opacities (Rogers & glesias 1992; glesias, Rogers & Wilson 1992). All models considered were static models of the present Sun and had the same total mass M. The hydrogen abundance X(q) as a function of mass fraction q = mlm (m being the mass interior to the given point) was obtained by scaling the abundance profile in the normally evolved non-diffusive model of Christensen-Dalsgaard et al. (1993). The abundance scaling factor X and a parameter characterizing the convective efficacy were adjusted so as to obtain a model of solar luminosity and photospheric radius (cf. Christensen-Dalsgaard & Thompson 1991). Unless otherwise noted, convective transport was treated using the mixing-length formulation of Bohm-Vitense (1958), characterized by the mixing-length parameter O!c; penetration beyond the convectively unstable region was not taken into account. We note that the effect of adjusting o!c (or more generally the parameter characterizing convection) is essentially to fix the specific entropy S in the deeper almost adiabatically stratified parts of the convection zone where S is very nearly constant: o!c determines the change in S between its value at the top of the convection zone, which is largely fixed by the conditions in the atmosphere, and the value Sa in the deep convection zone. As a result of the neglect of penetration, S is continuous at the base of the convection zone, thus relating Sa to conditions in the radiative interior. f the -adiative interior is virtually unchanged by a certain modification to the model, so therefore is Sa' and hence the change in the convective parameter Table 1. Properties of solar models. Model X-!Xc dblr Tb(K) Ph (gcm- 3 ) x x x x x x Some properties of the static solar models used in this paper. The quantities Tb and Ph are the temperature and density at the bottom of the convective envelope; tit, is the depth of the convective envelope;!xc is the mixing-length parameter; and X is the factor by which the hydrogen profile of an evolutionary model had to be scaled to give a static model with the correct luminosity. Relative to Modell, Model 2 has higher atmospheric opacity, and Model 3 has a smaller but broader superadiabatic gradient. must be such as to compensate for any possible change in the atmospheric value of s. We characterize the effects of the modifications to the physics of the models both in terms of changes at fixed radial distance r (in the following called Eulerian changes) and in terms of changes at fixed mass m (in the following referred to as Lagrangian changes). We denote Eulerian changes by 8 r and Lagrangian changes by 8 m As discussed in Section 4, the effects on the frequencies can be characterized by the changes in pressure p, density p and the adiabatic exponent r 1 = (a lnp/a lnp). (the derivative being at fixed specific entropy s), or in quantities derived from these: of special relevance to acoustic modes are the adiabatic sound speed c, given by 2 = rp, (2.1) p and the quantity r 1 v =-. (2.2) c We note in passing that v is closely related to Lamb's (1909) acoustical cut-off frequency for an isothermal atmosphere under constant gravity (cf. equation 4.12). Some relevant properties of the reference model (in the following Modell), as well as of the modified models, are given in Table 1. For all models the abundance scalefactor is slightly larger than unity; this increase in the hydrogen abundance is required to compensate for the fact that the static models lack the release of gravitational energy which contributes to the luminosity of normally evolving models. To illustrate the location and extent of the superadiabatic region and the ionization zones, Fig. 1 shows V - Vad (where ' ' '1 ' , , : './\.~ Figure 1. Properties of the reference Modell. The solid line shows r 1> and the dotted line V - V ad -

3 " 0.1 a) "... n,.'o... ~ J.:.... ;,.:.:::::.:::::::.:... "" " "" Near-surface perturbations to solar p-mode frequencies """TTTTTTTTTT"l'TMrTTrTTTTTTTTTTTTT"l'TMrTTr~ 0.01 b) ~:::::::::.:.::::..:::::.~~.:.. ~ , " \ \ \ \ ' t!.u..l.u..llljllu..l.u..llljllu..l.u..llljllu....lljo...l.u Figure 2. Eulerian differences between Model 2, with increased atmospheric opacity, and Modell. The following quantities are shown: 0, n c 2 (solid); 0, np (short-dashed); 0, np (long-dashed); 0, nr 1 (triple-dot-dashed); 0, n v (dotted). Panel (b) shows the same quantities as panel (a) but on a larger ordinate scale. The inset, which shows the whole range of radius, demonstrates that tbe structural differences are indeed minuscule except in the outer part of the model. V' = dlnt/dlnp and V'ad is its adiabatic value) and r 1 in the reference model. The behaviour of V' - V' ad shows that the region of significant superadiabaticity is confined to the outer 10-4 R of the convection zone. The variation of r 1 reflects the ionization zones of hydrogen and helium, ionization of hydrogen producing the strong decrease in r 1 near the surface ' ' ' 0.0 ~;;;::':;::::'::::::':::':":':::':":':':::":':':::":':':::::.::::::.:::::':'::::::':~"'" ----'3-0.1 ~ ~ a) :- -"-\0-----::; 1 1 1,, b) Figure 3. Panel (a) shows Eulerian differences between Model 2, witb increased atmospheric opacity, and Modell in tbe very near-surface layers. Panel (b) shows Lagrangian differences between the two models. The lines correspond in variables and style to those shown in Fig ncrease in the atmospheric opacity We first consider in Model 2 the effects of an increase in the opacity K in the outermost parts of the model. Specifically, in computing Model 2 we added to 10glO K a function of temperature which was 0.3 at the temperature characterizing the atmosphere and upper part of the convection zone of the model, decreasing smoothly to zero at higher temperatures so that the opacity was unmodified in the radiative interior of the model. As shown in Table 1 this had very little effect on the abundance sca1efactor or on properties at the base of the convection zone, confirming that the changes are confined to the outer parts of the model. On the other hand, a very considerable change in (Xc is required to maintain the radius of the model. This is related to the large changes in pressure and density in the atmosphere of the model. ndeed, for simple atmospheric models it may be shown that the product pk is roughly constant (e.g. Schwarzschild 1958); thus the opacity increase causes a decrease in atmospheric pressure by almost a factor of 2. Since the temperature is essentially fixed at the effective temperature of the Sun, the specific entropy in the atmosphere is decreased; this is compensated by the increase in (Xc, making convection more efficient and decreasing the superadiabatic gradient required for convective transport and hence the drop in entropy between the atmosphere and the deep convection zone. Details of changes in several model variables are illustrated in Figs 2 and 3. Fig. 2 shows the Eulerian changes in the outer 5 per cent of the model, on two different scales, as well as the changes throughout the model on a strongly expanded scale. The very large differences in pressure and density in the outer parts of the model are evident. They decrease with increasing depth but remain at a substantial level through the outer 10 per cent of the radius of the model. The change in c is smaller and largely confined below the photosphere: in the atmosphere, the temperature T, mean molecular weight p. and r 1 do not change much, and thus neither does c. There is a sharp feature in Or n c 2 in the superadiabatic region, followed by a gradual decrease towards greater depth. Also, there are fairly substantial changes in r 1 in the ionization zones of hydrogen and helium. Model changes in the outer 0.5 per cent by radius are illustrated in Fig. 3. The Lagrangian model changes (panel b) show a behaviour strikingly different from the Eulerian changes (panel a). The Lagrangian differences are strongly confined to the atmosphere

4 530 J. Christensen-Dalsgaard and M. J. Thompson and the superadiabatic part of the convection zone, with only a small and rapidly decreasing tail extending into the deeper parts of the model. The strong localization suggests that the Lagrangian changes provide a physically more meaningful description of the effects of near-surface modifications, particularly insofar as the corresponding frequency changes are concerned. We shall show in Section 4 that this is indeed the case. t is worth remarking that in this example the Eulerian changes in sound speed and adiabatic exponent are smaller in the superficial layers than are the corresponding Lagrangian differences. The relation between radius and temperature, and hence to a large extent sound speed and adiabatic exponent, is roughly determined by the optical depth; since K]J and hence Kp are approximately fixed, so is optical depth. n contrast, the change in p causes substantial changes in the mass scale and hence leads to the large Lagrangian differences (see also the discussion in Section 4). When analysing the frequency changes resulting from the changes in the model, account must be taken of the fact that highdegree modes penetrate less deeply than do low-degree modes, and therefore involve a smaller fraction of the mass of the model. As a result they are more strongly affected by changes in the outer parts of the star. This effect can be eliminated by considering scaled frequency differences Qn/OWnl; here En/ Qnl '" Eo(wn/), (2.3) where En/ is the inertia of the mode of radial order n and degree, normalized with the square of the photospheric amplitude, and Eo(w) is obtained by interpolating the values of Enl for radial (i.e. = 0) modes to the frequency w. t was argued by Christensen Dalsgaard (1986) (see also Christensen-Dalsgaard & Berthomieu 1991) that for model changes confined near the solar surface the QnlOWnl are largely functions of frequency alone; physically, this may be understood from the fact that within the region of modification the modes propagate essentially vertically, the horizontal wavelength being much longer than the vertical wavelength, so that the modes behave as radial modes. This property is confirmed by the scaled frequency differences between Models 1 and 2, shown in Fig. 4. Only at degrees higher than 200 do the scaled differences show significant dependence on. At such high degree the departure from vertical propagation becomes significant (for an asymptotic description of this behaviour, see for example Gough & Vorontsov 1995). This is the dominant contribution to the -dependence seen in Fig. 4. There is also a small contribution from the fact that for such high-degree modes the lower turning point is located so close to the surface that the modes do not fully sample the modified region, leading to a dependence of the frequency change on the depth of penetration and hence on the degree. Also, it should be noticed that the differences are very small at low frequency. 2.3 Change in the superadiabatic gradient To illustrate the effects of modifications to the superadiabatic region of the convection zone, we use the expression for the superadiabatic gradient V - Vad proposed by Christensen-Dalsgaard (1986; see also Christensen-Dalsgaard & Perez Hernandez 1992). t is characterized by a parameter f3e which determines the extent of the substantially superadiabatic region and a parameter lie which determines the maximum value of V - Vad and is used to calibrate the model. We choose f3e = 3, as did Christensen-Dalsgaard (1986) and Christensen-Dalsgaard & Perez Hernandez (1992): this results in a superadiabatic region that is broader, but -N -5 == ::t -i! ~ '0 i! C)' -10 o lint (JLHz) Figure 4. Frequency differences between Model 2, with increased atmospheric opacity, and Model 1; the differences have been scaled by the norma1ized mode inertia (cf. equation 2.3). Modes with the same degree have been connected with continuous lines for l:s; 150, dashed lines for = 200,300,400,500, and dot-dashed lines for = 600,700,800,900, 1000 and with a smaller superadiabatic gradient, than for the reference model. As shown by the quantities given in Table 1, the resulting Model 3 is very similar to the reference model at the base of the convection zone. Eulerian and Lagrangian changes in the outer parts of the model are illustrated in Figs 5 and Fig. 6. t is evident (Fig. 6) that the atmosphere is unaffected by the modification to the superadiabatic gradient. Except for the atmosphere and the superadiabatic region, the behaviour of the Eulerian differences is quite similar to those obtained for an increase in the atmospheric opacity, although of the opposite sign. n particular, the changes extend to a significant depth within the convection zone. Once again the Lagrangian differences are much more tightly confined to the superficial layers. However, it is interesting to note the differences between the near-surface behaviour in our two examples (Figs 3 and 6). n the present case the Eulerian and Lagrangian differences in sound speed and adiabatic exponent are very similar in the superadiabatic region, whereas previously the corresponding Eulerian differences were very small. Also, the detailed behaviour of the Lagrangian differences is very different in the two cases. Fig. 7 shows scaled frequency differences between Models 3 and 1. As before, they depend little on except at high degree, and are small at low frequency. 3 ANALYSS OF NEAR-SURFACE MODEL CHANGES n order to understand the properties of the model changes, we

5 0.3 '1 '1 ' ,,, -0.1 a) ", b) ,,, Near-surface perturbations to solar p-mode frequencies " " " / _..., " '."",."" ~ ----= ~::~:::::::::::::::::::::.:::::::: ", '"... :...: ',-0.02~~~~~~~~~~~~~~g Figure 5. Eulerian differences between Model 3, with modified superadiabatic gradient, and ModelL The same line styles are used as in Fig. 2. assume in this section that the modifications are sufficiently small for a linear approximation to be valid. By considering the linearized equations of stellar structure it is in fact possible to explain the general features of the results obtained in the previous section. Throughout we consider only models of fixed radius and surface luminosity. We first note that there are straightforward connections between the Eulerian and Lagrangian changes: df 0",/ = orf + omr dr ' (3.1a) /~.\ /..'. 0.1 / '''''.1./.....', 0.0 ~... ~d:~-:::...,:: - 't--~ -0.1 b) Figure 6. Panel (a) shows Eulerian differences between Model 3, with modified superadiabatic gradient, and Modell in the very near-surface layers. Panel (b) shows Lagrangian differences between the two models. The lines correspond in variables and style to those shown in Fig. 2. df Or! = 0",/ + orm dm ' (3.1b) for any model quantity f; here orm is the Eulerian change in mass, and omr is the Lagrangian shift in radius. Also, since obviously Orr = omm = 0, we have, e.g., from equation (3.1a) that dm orm = -omr dr = -47f"z p omr. (3.2) From the equation for mass, which we write as dr 1 = ,." 8 N ::t -i! 6 ~ '0 i! 0' o (3.3) no ", (p.hz) Figure 7. Frequency differences between Model 3, with modified superadiabatic gradient, and Modell; the differences have been scaled by the normalized mode inertia (cf. equation 2.3). Modes with the same degree have been connected using the same styles as in Fig. 4.

6 532 J. Christensen-Dalsgaard and M. J. Thompson we have that!g~m~) hence we obtain 1 ~mp --- 4rrpp ~ _ ~ lr ~mp f2dr' umr - 2 r. r r P (3.4) (3.5) n particular, we note that if ~mp is largely confined to a narrow region near the surface, of extent h say, it follows from equation (3.5) that ,,,,, -0.1 ~;rl_~ 1~;pl (~y ~1~;pl (3.6) To obtain an expression for the Lagrangian change ~mp in pressure we use the equation of hydrostatic support in the form dp Gm dm = - 41Tr4 ' where G is the gravitational constant. Hence (3.7) d~mp 4~mr dp dm =--r-dm' (3.8) or ~mp _ ~mpo 1 JP ~mr dp' p-p--p Po-r-, (3.9) where Po is the pressure at a suitable reference level. t is perhaps most natural to take Po = P.. the pressure at m = M. Note that the last term is purely an effect of the sphericity of the model: had we assumed that the model was plane-parallel, r would have been replaced by the constant R in equation (3.7) and ~mp/p would have been given by the first term in equation (3.9). We note from equation (3.6) that the term in ~mr is expected to be small. Also, since p increases rapidly with increasing depth this term decreases rapidly with increasing depth beneath the reference level. From Figs 3 and 6 it is evident that, in the specific examples considered, ~mp/p is in fact very small compared with the other changes, except perhaps in the atmosphere. To close the description, we need a relation between ~mp and ~mp. This can be obtained from the equation of state which we express as a function of p, the specific entropy s and composition. We assume a priori that the changes are essentially confined to the convection zone. Also, we shall assume that the equation of state and the abundance Z of heavy elements have not been modified. As the radiative interior is unchanged, so is the hydrogen abundance required to obtain the correct luminosity. Thus the composition of the convection zone is unchanged, and we have that ~mp "" (ijnp) ~mp + (ijnp\ ~ms. (3.10) P ijnp s p ijlnsh, s Similar relations clearly hold for the Lagrangian changes in any other thermodynamic variable. Now consider a model change that is predominantly localized near the surface. Since the radiative interior is essentially unaffected, so is the value Sa of s in the adiabatic part of the convection zone, for r < ra, p > Pa, say. Here, therefore, ~ms "" 0 in equation (3.10). Also, since R - ra ~R, we have from equation (3.6) that ~mr/rl ~ l~mp/pl The relation between ~mp/p and ~mp/p near r = r a obviously depends on the details of the intrinsic modification to the model. However, it is not unreasonable to assume that they are of the same order of magnitude as would be the case, for example, if the temperature perturbation were smaller than, or comparable to, either; this is certainly true for the modifications Qn~~~~~nn~~nn~~nn~~nq b) i ~ : :1 1: ::E '::'::::',:,,::::'':':: '::':':':::':':':::.:.:'::: :.:':::: ::.:::: :.::::: ~:::::::: :~} \:.: ::::': ~-: ~~~~~~~~~~~~~ww~ Figure 8. Panel (a) shows the Lagrangian differences 6 m n r for Model 2 minus Modell (solid line) and for Model 3 minus Modell (dashed line). Panel (b) shows logarithmic derivatives with respect to radius in the reference Modell. The following quantities are shown: dnc 2 /dnr (solid line); dnpldnr (short-dashed); dnpldnr (long-dashed); dnr1/dn T (triple-dot-dashed); din vldnr (dotted line). considered in Section 2. Under this assumption the term in ~mr/r in equation (3.9) is much smaller than ~mpa/p near r.; thus ~mp/p decreases with increasing depth as P -1, at least in the vicinity of r., and it follows from equation (3.10) that the same is true of ~mp/p (as well as of the relative perturbations of any other thermodynamic quantities, such as c or v). t is this steep decrease with increasing depth which leads to the strong confinement of the Lagrangian perturbations to the superadiabatic region and the atmosphere. t should be noted also that, according to equation (3.5), ~mr/r is essentially constant in this region, at least insofar as effects of sphericity can be ignored. The behaviour of the Eulerian perturbation, for a quantity f whose Lagrangian perturbation is confined to the superadiabatic region, may now be obtained from equation (3.1a) which we write as (3.11) Beneath the superadiabatic region the term in ~"J can be neglected and ~mr is approxinlately constant; hence the behaviour of ~rf/f is essentially determined by the variation of dnf/dn r. t follows from the preceding analysis that the relation between the Lagrangian and Eulerian differences is controlled by the logarithmic derivatives of the corresponding model quantities, and by the modification ~m n r of the radius scale at fixed mass. These quantities are illustrated in Figs 8 and 9. As inferred, the values of ~m n r for the two examples considered (cf. Fig. 8a) are

7 3000 '1' a) '1' '1' Near-surface perturbations to solar p-mode frequencies 533 '1,,,,,,, / 100 b) o '-"'-"'-"'-"'-"'-"'-"'-'",...! ~...-., , ',-300~~~~~~~~~~~~~~~ Figure 9. Logarithmic derivatives in the reference Modell. The same quantities as in Fig. 8(b) are shown, but on different scales. The line styles are the same as in Fig. 8(b). nearly constant in the outer parts of the model and of much smaller magnitude than, e.g., 6 m n p. Comparison of Figs 2(b) and 5(b) with Fig. 9(b) shows clearly that the differences beneath the superadiabatic region are indeed dominated by the derivatives. n particular, this explains the similarity (apart from the sign) between the changes in these two cases. The overall behaviour of the derivatives can be understood by noting that dnp 1 dnp dnr = r 1 dnr ' dine = ~ dnu = ~ (1-~) dnp dnr 2dnr 2 r 1 dnr (3.12) (neglecting in the last relation the dependence of c on r 1), where u!! pp. Furthermore, dnp Gmp Gm dnr = ---;p = -"'iii (3.13) Thus the Eulerian perturbations of p, p, c and u decrease with increasing depth approximately as u- 1 X T-l, compared with the p -1 decrease found for the Lagrangian perturbations. Since p varies approximately as depth to the fourth power (assuming an effective polytropic index of about 3), and temperature varies only as the first power of depth, it follows that the Eulerian perturbations decrease much more slowly with increasing depth than do the Lagrangian perturbations. We note that the Eulerian change in r 1 can be obtained from the changes in, e.g., p and p as 5r r l = (anrl) 5rP + (anrl) 6rP, (3.14) r l anp pp anp p p or an equivalent relation expressed in terms of Lagrangian modifications. Since r 1 is almost constant and equal to i outside the ionization zones of hydrogen and helium, the modification to r is essentially confined to these ionization zones, as was also found in Figs 2 and 5. This result can evidently also be obtained by applying equation (3.11) to r l. n the case when the atmospheric opacity is modified, we remarked that the Eulerian differences in c, r and v were small in the superadiabatic region compared with the corresponding Lagrangian differences (Fig. 3). n those circumstances it is evident from equation (3.11) that the Lagrangian differences in that region largely reflect the logarithmic derivatives: comparison of Fig. 3(b) and Fig. 8(b) shows that this is indeed the case. n contrast, for the modification of the superadiabatic gradient illustrated in Fig. 6, the term in 15 m n r is comparatively small in the superficial layer; as a result, the Eulerian and Lagrangian differences are rather similar. 4 ANALYSS OF THE FREQUENCY CHANGES The differences in frequencies between pairs of models can be related to the differences in their internal structure. Provided that the differences are small, the frequency differences can be considered to be linear functionals of the structural differences. The derivation of these functionals from a variational principle is described by, e.g., Gough & Thompson (1991) and Gough (1993): some details are given in Appendix A n the adiabatic approximation the relevant structural differences are completely described by 5 rp, 6 rp and 6 rr l' Assuming hydrostatic equilibrium allows 5 rp to be expressed in terms of 5 rp and hence the three variables may be reduced to two. Exactly which pair of variables is retained is to some extent a matter of taste: one choice is 6 r c 2 and 6 r P. With this choice of variables one may therefore express the frequen,cy changes as - c'-,p r Knl ( ) 5rP] d _? + pc'- r r. Wnl 0 ~, p 15wnl _ JR [Knl ( ) 6r~ (4.1) Fig. 10 shows K~,p and K;~c'- for an 1 = 20, n = 15 mode with frequency 3080 J.Hz. The global structure of the kernels mimics that of the square of the eigenfunction, the number of peaks in the kernels being approximately equal to the order of the mode. To be more precise, K~,p is proportional to the square of the divergence of displacement eigenfunction (e.g. Gough & Thompson 1991) o a) ot---~" -10 b) Figure 10. Eulerian kernels for the variable pair (c 2,p), fer the 1 = 20, 11 = 15 mode (v = 3080,.uz). Panel (a) shows the kernel with respect to c 2 at constant p; while panel (b) shows the kernel with respect to p at constant c 2

8 534 J. Christensen-Dalsgaard and M. J. Thompson a) /...../ \ t :,\ ~,.:, \ \1 :", 1 " :', \,.~... \.: 1 ~... ~.>.. ::..~.~./;;-;..' '<.~.. ~./... > \; \ ,. " i! ~ /: ~"'/ J, 1.00 Figure 11. Eulerian kernels (a)o,p and (b)x;,c2 in the near-surface region for various 1=20 modes: n = 2, v = 986 jlhz (solid line); n = 7, v = 1889 jlhz (dotted line); n = 15, v = 3080 jlhz (dashed line) 'i! :.. ) ' lin! ('Hz) Figure 12. Contributions tq &"1,,, from ljrcc (dashed line) and ljrplp (dotdashed line) for (a) Model 2 minus Modell and (b) Model 3 minus Modell. The total of the two contributions is shown as a solid curve; the exact frequency differences are shown with plus symbols. Results are for = 20. Fig. 11 shows ~,p and~c2 in the near-surface region, for = 20 at selected frequencies. As well as exhibiting more peaks with increasing n (and hence frequency), the higher-frequency kernels also have maxima closer to the surface. This is as one would expect from asymptotics, since the upper turning point is closer to the surface for higher-frequency modes. t should be noticed, however, that the Eulerian model differences (cf. Figs 2 and 5) extend well into the region where even the 30 [~' 1 25 r- Kr!,p( r) E '" ' Kp.Ar) 40~ OE , 0.997, ' '1 1 a) _ ,,' ' 1 b) Figure 13. Lagrangian kernels (a)~,p and (b) K;,c" in the near-surface region for various 1=20 modes: n = 2, v = Hz (solid line); n = 4, v = 1384 jlhz (dotted line); n = 7, v = Hz (dashed line); n = 10, v = Hz (dot-dashed line); n = 15, v = 3080ll-Hz (tripledot-dashed line). lowest frequency kernels illustrated (approximately 1 mhz) have substantial amplitude. t is to be expected then that even for lowfrequency modes the individual contributions from the two terms on the right-hand side of equation (4.1) will not be negligible. This is confirmed in Fig. 12, where the frequency differences and the individual contributions from lirc2 and lirp are shown for both Models 2 and 3 relative to the reference model. n each case the contributions are both significant. nterestingly, in both cases the frequency differences at low frequencies are smaller than either of the individual contributions, which to a large extent cancel out; as we shall see this cancellation is not fortuitous but a general consequence for near-surface changes. The frequency differences can similarly be related to structural changes measured at fixed fractional mass: the equation corresponding to (4.1) for this Lagrangian case is liwnl = [iq/r) li~~2 + k: c2 (r) limp] dr. (4.2) Wnl 0 ~, P The derivation of such kernels is presented in Appendix A Note that the choice of integration variable (in this case r) is quite independent of whether the differences are evaluated at constant mass or constant radius: the choice is simply one of convenience, and changing the integration variable, say to some other variable z, would merely scale the kernels by dr/dz. Lagrangian kernels corresponding to the Eulerian ones in Fig. 11 are shown in Fig. 13.

9 ;} ') '<l a) ", (p.hz) b) "",.?,,, ", (/LHz) Figure 14. Contributions to 6wlw from 6 mc c (dashed line) and 6 mplp (dotdashed line) for (a) Model 2 minus Model 1 and (b) Model 3 minus Modell. The total of the two contributions is shown as a solid curve; the exact frequency differences are shown with plus symbols. Results are for = 20. The contributions from the individual terms on the right-hand side of equation (4.2) are shown in Fig. 14, for both our pairs of models. Here no near-cancellation between contributions is required to produce the small frequency shift at small frequencies. Notice that this difference in the behaviour between the Lagrangian and the previously considered Eulerian formulations is not principally a result of any difference between the Lagrangian and Eulerian kernels; in fact the kernels are very similar. Rather, the effect arises because the Lagrangian structural differences are confined very close to the surface: the low-frequency kernels are small in this region and so the small frequency difference comes out naturally. Similar results are found with other choices of pairs of structure variables. As a second example of such a choice we take (c, v). Fig. 15 shows some Eulerian kernels K~v, K.:~c; the corresponding Lagrangian kernels K~!v, K~~c are shown in Fig. 16. Strikingly, the Eulerian and Lagrangian kernels for changes in c at constant v are dominated in the outer part of the Sun by a smoothly varying behaviour, whereas the corresponding kernels for changes in v reflect the relatively rapid spatial variation of the eigenfunction. t is precisely this separation that originally motivated the choice of this pair of variables for studying the helium ionization zone and nearsurface structure (Christensen-Dalsgaard & Perez Hernandez 1992). The Eulerian and Lagrangian contributions to the frequency Near-surface perturbations to solar p-mode frequencies Kc,v o a) Kv,c 0 A'VV~ b) Figure 16. Lagrangian kernels for the variable pair (c, v), for the = 20, n = 15 mode (v = 3.08 mhz). Panel (a) shows the kernel with respectto c at constant v; while panel (b) shows the kernel with respect to vat constant c. differences are illustrated in Fig. 17. While the Eulerian case again shows some contribution from each structure variable, the contribution from the Lagrangian perturbation omc at constant v is essentially zero. The reason for this striking behaviour can be understood from the leading-order behaviour of the Lagrangian kernels, derived by Goldreich et al. (1991) and summarized in Appendix B. t follows from this that K:.c=w - -2(lM 2 )-1 2(d~)2 2 0 ~ dm 41Tr dr cp, (4.3) while k:!v = O. Further insight into the behaviour of the frequency changes can be obtained from asymptotic analyses. By JWKB theory, J R' kr(r) dr = (n + e)1t, (4.4) r, where kr is the radial component of the wavenumber. The constant e accounts for the behaviour near the turning points r = rl and r = R For our purposes it is adequate to take e,. = (w2 ~ w~ _ ~) 112 (4.5) 0.003r""1, ~...,..-~-1'1'"...,..-..,...-.., ~"'"1' a) ;: ') '<l Or---~ a) b) w-~~~~~~~~ Figure 15. Eulerian kernels for the variable pair (c, v), for the = 20, n = 15 mode (v = 3.08mHz). Panel (a) shows the kernel withrespectto c at constant v; while panel (b) shows the kernel with respect to vat constant c. Figure 17. Contributions to 6wlw for Model 3 minus Modell (a) from 6rC2lC2 and fjrvlv and (b) from 6mC2lC2 and fjmvlv. n each case, the contribution from the sound-speed differences is indicated with the dashed curve, and the contribution from the v differences with the dotdashed curve; their total is shown with the solid curve, and the exact frequency differences with the plus symbols. Results are for = 20.

10 536 J. Christensen-Dalsgaard and M. J. Thompson (cf. Deubner & Gough 1984) where kh E Jl(l + 1)/r is the horizontal wavenumber. Formally perturbing equation (4.4), assuming that e is unaffected by the perturbations, gives r [(a~w) 8: + (aa~j 8~e + (~) 8 r W e ] dr = o. (4.6) alnwe We (Note that kr vanishes at the endpoints, so there is no contribution from perturbing them.) We now consider a mode oflow frequency, such that it can be assumed that within its acoustic cavity the Lagrangian perturbation of any quantity f vanishes; here, therefore, M = _ dlnf 8 r f dr m (4.7) from equation (3.1a). n particular let this be true for f.. e and for f.. w e Now dk,. = ( akr ) dine + (~) dlnwe (4.8) * alne * aln~ *. n this approximation, the variation of kh with radius, which arises from the spherical geometry, has been neglected: this is consistent with the derivation of the dispersion relation (4.5) (cf. Deubner & Gough 1984). Furthermore, it was argued in Section 3 that, if effects of sphericity can be ignored, 8 mr may be considered to be constant. Expressing 8 r c/e and 8 rw elw e in equation (4.6) by the corresponding logarithmic derivatives, in accordance with equation (4.7), and using equation (4.8), we therefore obtain 8WJ R t ( ak, ) R -;;; rt alnw dr = 8mr[krlrtt = O. (4.9) Thus, for modes of sufficiently low frequency that the region with non-zero 8 me and 8 mw e lies entirely above their asymptotic upper turning point, the frequency shift is essentially zero, as indeed we have found to be the case numerically. Of course for higherfrequency modes, for which the Lagrangian perturbations to e and We would not generally vanish in the acoustic cavity, equation (3.1a) should be used instead of (4.7): this would give rise to additional terms in equation (4.9), and so the frequency perturbations would not necessarily vanish, again in accordance with the numerical results. nstead of expressing the perturbed equation (4.4) in terms of Eulerian perturbations, it is instructive to consider Lagrangian perturbations directly. For simplicity we consider radial modes, for which kh = 0 and rt = O. Perturbing equation (4.4) then yields 8(1:' krdr) 8(C 41T~2p krdm) C 8m(41T~p kr) dm 0, where M t '" m(r t ). Now equation (4.5) with kh = 0 gives 1 (W~)ll2 1 rp k, = W 1 - w2 rpv. (4.10) (4.11) For simplicity we approximate We with its value for an isothermal atmosphere: (4.12) (Lamb 1909). Then equation (4.10) implies that 8w JR t (1 _ W~) -1/2 * W 0 w 2 e JRt (1- W~)1/2 [8mV + 8m(?p) o w2 V rp + We 2 ( 1 _ We 2)-18mWe] dr w 2 w 2 We e t 1- We J R ( 2)-112 8mvdr o w 2 V e + JRt (1 _ W~) 1/2 [8m(?P) o w 2 rp _ 2 We 2 ( 1 _ We 2)-18mr] dr, w2 w2 re (4.13) using 8 mglg = -28 mrlr [cf. equation (4.3) of Balmforth et al. 1996]. Thus, in Lagrangian terms, the change in frequency is described in terms of perturbations to v, p and r. n cases when the Lagrangian perturbations to radius and pressure are negligible in the acoustic cavity, the frequency differences will therefore be given essentially by the difference in valone, as we found above (cf. Fig. 17 and equation 4.3). 5 CONCLUSONS The near-surface region, i.e., the solar atmosphere and the substantially superadiabatic part of the convection zone, introduces substantial uncertainties into current calculations of solar p-mode oscillation frequencies, both in terms of the structure of the underlying hydrostatic equilibrium model and in terms of the dynamics and energetics of the oscillations. We have shown that, if models are compared at fixed interior mass m, the resulting Lagrangian differences are confined essentially to that region of the Sun which is directly affected by the modifications to the physics of the model. t then follows immediately that the resulting frequency changes are very small at low frequency, because the eigenfunctions are evanescent in the immediate sub-photospheric layers. n contrast, Eulerian model differences, at fixed radius r, extend to substantial depths, as a result of the change in linear scale induced by the modification; here the small frequency differences at low frequencies follow from a near-cancellation between two substantial terms of opposite sign. These properties of the model and frequency differences have been verified by numerical calculations involving two different types of near-surface modifications. n addition, they follow from an asymptotic analysis of the oscillations. We have also seen that the frequency differences are naturally expressible in terms of the pair of variables (e, v), where v'" r le, with the Lagrangian difference in v dominating the frequency shift (see Fig. 17b and equation 4.13). This behaviour is evident also from the corresponding numerical kernels. t is interesting therefore to compare the frequency differences arising from modifying the opacity or the superadiabatic gradient, and to relate those to the Lagrangian differences in v (Fig. 18). Compared with the case of changing the surface opacity, modifying the superadiabatic gradient produces a difference 8 m n v that extends more deeply, down to a radius of about 0.999R. Consequently the frequency differences become substantial at lower frequencies, being appreciably different from zero at about 3000 ~Hz. On the other hand, 8 m n V in this case is negligible in the atmosphere above the superadiabatic layer,

11 z ;:>E '0 0.00, a) ;: ') ~ b) " lin! (p.hz) Figure 18. (a) Lagrangian differences in v = r11e between Model 2 and Modell (solid line) and Modell and Model 3 (dashed line; note the change of sign relative to Fig. 6). (b) Corresponding relative frequency differences. and so the frequency differences tum over; in contrast, the frequency differences caused by the atmospheric opacity modification exhibit a monotonic trend up to 5000 jlhz. Our results have implications for inverse analyses of observed frequencies of solar oscillation, aimed at determining corrections to models of solar structure. The inversion is generally carried out in terms of Eulerian differences, the effects of the near-surface errors being eliminated by representing them as suitably scaled slowly varying functions of frequency (e.g. Basu et al. 1996, and references therein). t follows from our analysis that such suppression will eliminate from the solution of the inverse problem that part which is expressible as in equation (4.7), for some 8 m,. Thus we might hope that the non-local nature of the Eulerian changes will not corrupt the results of the inversion, beyond the immediate region of the errors in the model, after the suppression of their dominant effect on the frequencies. However, we note that inversion in terms of Lagrangian differences should offer a clean separation between the uncertain physics of the near-surface region and changes at even slightly greater depth. This could provide a valuable alternative to the inversions in terms of Eulerian differences, particularly in investigations of the thermodynamical state of solar matter in the hydrogen and helium ionization zones where separation of the nearsurface effects represents a substantial problem. ACKNOWLEDGMENTS We are grateful to F. Perez Hernandez for inspiring the investigation, and to Y. Elsworth for comments on an earlier version of the paper which substantially improved the presentation. This work was supported by the Danish National Research Foundation through its establishment of the Theoretical Astrophysics Center, Near-surface perturbations to solar p-mode frequencies 537 by the Danish Natural Science Research Council and by grant GR/ K94133 from the UK Particle Physics and Astronomy Research Council. The work was conceived and begun at the nstitute for Theoretical Physics, University of California at Santa Barbara, and hence supported in part by the US National Science Foundation under grant PHY , supplemented by funds from the National Aeronautics and Space Administration. REFERENCES Balmforth N. J., Gough D.O., Merryfield W.., 1996, MNRAS, 278, 437 Basu S., Christensen-Dalsgaard., Perez Hernandez F., Thompson M.., 1996,MNRAS,280,651 BOhm-Vitense E., 1958, Z. Astrophys., 46, 108 Canuto V. M., Mazzitelli., 1991, ApJ, 370, 295 Christensen-Dalsgaard J., 1986, in Gough D.O., ed., Seismology of the Sun and the Distant Stars. Reidel, Dordrecht, p.23 Christensen-Dalsgaard J., 1990, in Berthomieu G., Cribier M., eds, Proc. AU Colloq. 121, nside the Sun. Kluwer, Dordrecht, p.305 Christensen-Dalsgaard J., Berthomieu G., 1991, in Cox A N., Livingston W. C., Matthews M., eds, Solar nterior and Atmosphere. 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Princeton University Press, Princeton, NJ APPENDX A: NOTE ON THE DERVATON OF SOME EULERAN AND LAGRANGAN KERNELS A Eulerian kernels The derivation of kernels for 'Eulerian perturbations to (c 2, p) was presented by Gough & Thompson (1991). n brief, the adiabatic oscillation frequencies satisfy a variational principle (e.g. Ledoux & Walraven 1958). By perturbing this variational principle, and retaining only terms that are linear in perturbation quantities, the frequency differences between two models (or a model and the Sun)