W. J. Chaplin, 1P T. Appourchaux, 2 Y. Elsworth, 1 G. R. Isaak 1 and R. New 3

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1 Mon. Not. R. Astron. Soc. 324, (2001) The phenomenology of solar-cycle-induced acoustic eigenfrequency variations: a comparative and complementary analysis of GONG, BiSON and VIRGO/LOI data W. J. Chaplin, 1P T. Appourchaux, 2 Y. Elsworth, 1 G. R. Isaak 1 and R. New 3 1 School of Physics and Astronomy, University of Birmingham, Edgbaston, Birmingham B15 2TT 2 ESA Space Science Department, Solar System Division, ESTEC, 2200 AG Noordwijk, The Netherlands 3 School of Science & Mathematics, Sheffield Hallam University, Sheffield S1 1WB Accepted 2001 January 10. Received 2001 January 10; in original form 2000 September 4 ABSTRACT We use high-quality helioseismic data collected by three different observational programmes during the declining phase of activity cycle 22, and a substantial portion of the rising phase of the current cycle (23), to study the phenomenological nature of the cycle-induced (centroid) eigenfrequency variations. We have analysed the frequency dependence of the shifts by fitting a power law of the form dn /ðn Þ a /E to the data (where the E are the mode inertias, and a is the power-law index to be extracted). Previous studies have suggested that a relation with a ¼ 0 provides an adequate description of the shifts up to n < 3500 mhz. However, here we show that while nevertheless describing the shifts well up to,2500 mhz, the linear scaling breaks down conspicuously at higher frequencies. Above this threshold, the shifts follow a power-law dependence with a, 2. Our analyses (for 1600 # n # 4000 mhzþ make use of observations made by the groundbased GONG over the angular degree range 4 # l # 150; the ground-based BiSON over 0 # l # 2; and the VIRGO/LOI instrument on board the ESA/NASA SOHO satellite over 0 # l # 8. We show that GONG shifts averaged over different ranges in l, together with the BiSON and LOI data averaged over their full quoted ranges, all scale at fixed frequency with the normalized mode inertia ratio Q. This is to be expected if the solar-cycle perturbation affecting the modes is confined in the surface layers; the excellent agreement also reflects favourably on the external consistency of the different observations. Key words: Sun: activity Sun: oscillations. 1 INTRODUCTION: CONTEXT OF THIS STUDY Observations of variations in the eigenfrequencies of the acoustic vibrations of the Sun that are correlated with the solar activity cycle are now well documented, for example, at low and intermediate angular degrees (Woodard & Noyes 1985; Elsworth et al. 1990, 1994; Anguera Gubau et al. 1992; Regulo et al. 1994; Chaplin et al. 1998; Jimenez-Reyes et al. 1998; Libbrecht & Woodard 1990, 1991; Woodard et al. 1991; Bachmann & Brown 1993; Rhodes Jr et al. 1993; Howe, Komm & Hill 1999; Bhatnager, Jain & Tripathy 1999; Dziembowski et al. 2000). Here, we perform the first proper, detailed comparative (and complementary) analysis of eigenfrequency shifts derived from independent observations of the modes. To do this, we have made use of data collected by the P wjc@bison.ph.bham.ac.uk ground-based Global Oscillations Network Group (GONG) and Birmingham Solar-Oscillations Network (BiSON), and the Luminosity Oscillations Imager (LOI) which is part of the VIRGO instrument package on board the ESA/NASA SOHO spacecraft. In particular, we concern ourselves with a thorough analysis of the dependence of the magnitude of the shifts both as a function of frequency and angular degree on the inertial properties of the modes. Libbrecht & Woodard (1990) were the first to demonstrate that the shift observed for a mode of given radial order and angular degree (n and l ) scales approximately linearly with the reciprocal of the inertia associated with the mode. These general findings were confirmed by Chaplin et al. (1998), who used BiSON data to reveal a similar dependence for the first time at low l, and Howe et al. (1999), who analysed GONG data up to l ¼ 150, but used oy a restricted temporal subset of their data to study the frequency dependence of the shifts. A common feature of these studies is the conclusion that the linear scaling holds fairly well up q 2001 RAS

2 Solar-cycle variations: GONG, BiSON and VIRGO/LOI 911 to mode frequencies of n < 3500 mhz; however, none of studies presented the phenomenological dependence in any great detail. This we do here, making use of large ranges of the available data to enable a precise study of the frequency dependence of the eigenfrequency shifts to be made. Furthermore, we are able to demonstrate external consistency between the shifts extracted from the analysis of the three data sets by showing that the inertia scaling successfully removes the dependence in l over the analysed range of 1600 # n # 4000 mhz. 2 DATA GONG makes resolved observations of the visible solar disc from six ground-based observatories. Here, we use Doppler velocity data collected over the period 1995 May through 1999 October. The data were processed through the GONG pipeline (Hill et al. 1996), producing power spectra of duration 108 d. The peaks in these spectra were then fitted to symmetric Lorentzians to yield estimates of the mode frequencies up to l ¼ 150. In spite of the fact that there is indisputable evidence that the modal profiles are asymmetric in frequency, the use of a Lorentzian limit model is not of great concern here since we are interested in the analysis of the time variation of the frequencies (of fractional size,1 part in 10 4 ), not the absolute values themselves. Furthermore, an analysis of BiSON data (Chaplin et al. 1999) has failed to uncover any solarcycle dependence in the asymmetries, at the level of precision of these observations. Table 1. shifts extracted from the analysis of GONG data over the angular degree range 4 # l # 25. The presented shifts have been scaled to the expected full swing between the minimum and maximum of the activity cycle, under the assumption that (allowing for the resolution afforded by the length of each time series) the 10.7-cm radio flux changes by 150 RF units. GONG 4 # l # ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ BiSON also consists of six ground-based stations. Observations are made of the unresolved solar disc which provide data that are dominated by the Doppler variations arising from the low-l modes. We have analysed BiSON data collected over the period 1991 January through 1999 November by dividing the complete data set into two different time-bases, these being 108 and 216 d. Since an individual BiSON data set is made up of substantially fewer modes than those for GONG and LOI (i.e. low-l data, and small range in l ) the associated shift uncertainties will be appreciably larger. Even though they cover an earlier phase of cycle 22 than the GONG and LOI data, we have therefore decided to make use of this more extensive set in order to minimize the uncertainties on the final results. We have satisfied ourselves that similar shifts are returned by dividing the full data set into two parts. While there may be a change in behaviour on the falling and rising phases of the cycle, the uncertainties in these data are such that we are unable to uncover any clear, significant differences. In order to extract time-dependent estimates of the modal eigenfrequencies for 0 # l # 3, we fitted the resonant peaks in the power spectrum of each time series to a suitable model which reflects the asymmetric nature of those peaks [the polynomial asymmetric formalism of Nigam & Kosovichev (1998); see Chaplin et al. (1999) for more details]. LOI makes low-spatial-resolution intensity observations in the continuum at 500 nm. The data set which spans the period 1996 March through 2000 April has been divided into 136-d segments. Complex Fourier (frequency) spectra were fitted to yield estimates of the modal parameters for 1 # l # 8 (see Appourchaux, Gizon & Rabello-Soares 1998 for a full discussion of the technique), while power spectra of the full-disc proxy were fitted to extract the radial ðl ¼ 0Þ mode frequencies. We again used the model of Nigam & Kosovichev to describe each modal component. Table 2. shifts extracted from the analysis of GONG data over the angular degree range 30 # l # 90. GONG 30 # l # ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ 0.026

3 912 W. J. Chaplin et al. Table 3. shifts extracted from the analysis of GONG data over the angular degree range 100 # l # 150. Table 4. shifts extracted from the analysis of the LOI and BiSON data over the angular degree ranges 0 # l # 8 and 0 # l # 2, respectively. GONG 100 # l # ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ LOI 0 # l # ^ ^ ^ ^ ^ BiSON 0 # l # ^ ^ ^ ^ ^ ^ ^ ANALYSIS For each instrument data set (the three were treated separately) we selected a subset of spectra that were made from data collected at low levels of solar activity. To do this, we used the 10.7-cm radio flux averaged over the duration of each time series as our proxy of the level of activity. The fitted frequencies from these data were then averaged at each (n, l ) to yield mean estimates of the eigenfrequencies at low activity. These then served as a reference against which to compute the shifts (residuals) for all spectra. We then carried out a linear regression of the frequency residuals in adjacent segments in frequency space with the 10.7-cm radio flux (averaged over the duration of the time series of each spectrum). The modes that contributed to the regression in each frequency band came from chosen ranges in l; we detail the ranges selected below. From the extracted gradient of the regression, we could then infer from the variation of the radio flux the total mean change in mode frequency from activity minimum to maximum over the considered bandwidth (i.e. over those modes used in the regression). At each stage of the analysis, we made use of the rejection algorithms developed by Chaplin et al. (1998) to treat outliers in the data. 4 RESULTS AND DISCUSSION The upper right-hand panel of Fig. 1 displays the extracted mean shifts as a function of frequency over the ranges: 4 # l # 25, 30 # l # 90 and 100 # l # 150 for GONG; 0 # l # 2 for BiSON; and 0 # l # 8 for LOI. The shifts are those inferred between the activity minimum and maximum of the solar cycle. As expected, the curves reveal a clear frequency and angular-degree dependence. Furthermore, the form of the plotted data is that expected from a time-dependent perturbation acting on the modes in the near-surface layers. These data are presented in Tables 1 4. Modes over the angular degree range analysed here propagate essentially radially in the vicinity of the upper turning point (UTP), the absolute location of which shows negligible variation with l. Since their properties are therefore largely l invariant at the surface, one would expect a similar response in the modes (at a given frequency) to any highly localized structural changes. Here, by localized, we mean those constrained over a range in radius that is significantly smaller than the depth of penetration of the modes considered. The remaining l dependence in the shifts reflects the associated modal inertia. We use the definition for the normalized inertia, E, given by Christensen-Dalsgaard & Berthomieu (1991): ð ð Rs E ¼ M 21 ( j 2 r dv ¼ 4pM 21 ( j 2 rr 2 dr ¼ M ; ð1þ M ( V where j is the displacement associated with the mode, suitably normalized at the photosphere, and the integration is made over the volume V of the Sun, the mass of which is M (. M is therefore the mass associated with the mode, and the time-averaged kinetic energy of the oscillation satisfies 1=2M v 2 ¼ 1=2E M ( v 2 ; ð2þ such that v 2 is the mean (both spatially, over the solar surface, and temporally) of the squared total velocity of the mode. Note that the solar model is normalized to unity radius where T ¼ T eff ; this height lies some,50 km above the radial datum where optical depth reaches unity at a wavelength of 500 nm (the surface reference often chosen for observational data). In order to explain the variation of E across the mode set, we consider the angular degree and frequency dependence in turn. The l dependence is best understood if we are guided by a physical interpretation of the inertia, i.e. as some measure of the interior mass affected by any given mode. As noted above, the location of the upper turning point can be regarded as being independent of l over the range of modes considered here. However, the lower turning point shows a marked dependence in l: acoustic waves that form modes of lower l penetrate to a greater depth before they are refracted back toward the surface. At fixed n, a decreased l therefore leads to an increased E, i.e. a greater 0

4 Solar-cycle variations: GONG, BiSON and VIRGO/LOI 913 volume of the interior is associated with the motions generated by the mode. An explanation of the frequency dependence is less obvious, and requires us to recognize that the modes are evanescent at the photosphere. At fixed l there is a marked increase observed in E as the mode frequency, n, decreases. This is because lower-frequency modes become evanescent (at their UTP) at a greater depth in the atmosphere than their higher-frequency counterparts. The displacement at the solar surface associated with the mode is therefore somewhat reduced, and the inertia must increase to satisfy equation (2). The larger the inertia, the less sensitive the mode is to Figure 1. The phenomenology of solar p-mode shifts, as extracted from the analysis of GONG, BiSON and LOI data. The plotted shifts are those inferred between the minimum and maximum of the solar activity cycle. Upper left-hand panel, the raw, extracted shifts, averaged over the indicated ranges in l, asa function of frequency. Upper right-hand panel, frequency shifts multiplied by the inertia ratio, Q. Middle left-hand panel, the inertia-ratio modified shifts plotted as a function of the inverse fractional mode inertia (normalized to the inertia of a notional radial mode with frequency 3000 mhz). The dotted line is a linear fit to the 30 # l # 90 GONG shifts, but oy for those modes with a fractional inverse inertia e 21 # 0:6; the continuation of the plotted fit at higher e 21 is meant to serve as an eye guide. Middle right-hand panel, the changing value of the reduced chi-squared statistic for linear fits to different numbers of points from the 30 # l # 90 GONG set. The returned statistic is plotted as a function of the largest value of the independent variable e 21 used in the fit. Lower lefthand panel, the residuals, computed with respect to the linear fit indicated in the middle left-hand panel, from the five l-averaged sets, now plotted as a function of frequency. Lower right-hand panel, the product of the eigenfrequency shift and the fractional mode inertia, e, plotted as a function of frequency for GONG data averaged over 30 # l # 90. The broken curves show the results of fitting the plotted data to the power-law model described by equation (4), below and above 2500 mhz. The higher-frequency dependence is described by a power law with index a n.2500 ¼ 1:91 ^ 0:03; at low frequencies a n#2500 ¼ 0:22 ^ 0:23 provides the best fit to the data.

5 914 W. J. Chaplin et al. near-surface perturbations of a given size. If the assumption of a surface-confined perturbation is valid, we may render the shifts essentially l independent by multiplying each by the inertia ratio, Q (Christensen-Dalsgaard & Berthomieu 1991), given by Q ¼ E = Eðn Þ: Here, Ē(n ) corresponds to the normalized inertia that a radial mode would have at the frequency of the mode identified in the numerator. The multiplication of each (n, l ) datum by Q therefore re-normalizes the shift to its expected radial equivalent. We plot the re-normalized shifts in the upper right-hand panel of Fig. 1, having made use of E data derived from model S of Christensen- Dalsgaard (see Christensen-Dalsgaard et al. 1996). The calculated inertias are normalized at the photosphere (at T ¼ T eff Þ where the perturbation responsible for the shifts is assumed to reside. The spread present between the raw averages in the left-hand panel is seen to contract noticeably. This confirms our prejudice regarding the restricted, near-surface location of the source, and also reflects favourably on the external consistency between the results obtained from the three observational sources. With the l dependence therefore largely removed, we turn to an analysis of the frequency dependence. In order to quantify the dependence, we describe the shifts according to the relation dn ¼ k E ðn Þ a : The precise nature of the frequency dependence, i.e. the value of a in equation (4), depends upon the manner in which the background state in the solar atmosphere is perturbed. Modifications to the speed of sound in a thin layer at the photosphere arising from, for example, changes to the Lorentz force acting on the fluid from a varying magnetic field would be expected to produce a functional dependence with a, 3 (Gough 1990; Libbrecht & Woodard 1990; Goldreich et al. 1991). If the extent of the perturbation were such that it extended well beneath the photosphere, the dependence on frequency would be weakened, i.e. a would decrease. Gough (1990) notes that if changes to the efficacy of convection are the principal agent of change then a, 21. We begin here by recalling that previous similar studies (Libbrecht & Woodard 1990; Chaplin et al. 1998; Howe et al. 1999) have suggested that the magnitudes of the observed shifts are linearly proportional to the inverse mode inertia up to frequencies of <3500 mhz: this corresponds to a functional dependence with a, 0. Libbrecht & Woodard (1990) noted the resulting marked disparity with a third-power dependence, and therefore argued for there being a significant contribution to the observed shifts from the subphotospheric layers. The middle left-hand panel of our Fig. 1 displays the inertia-ratio modified shifts as a function of the inverse fractional mode inertia, which we define according to e 21 ¼½E = Eð3000ÞŠ 21 ; where Ē(3000) is the normalized inertia that a radial mode would have at a frequency of 3000 mhz. A visual inspection of the data reveals that a linear relation between the two variables appears to be appropriate oy at low to moderate values of e 21, while at higher e 21 (i.e. higher frequencies), clear departures from linearity are revealed. Given the precision inherent in these data, it is evident ð3þ ð4þ ð5þ that an a ¼ 0 dependence cannot wholly account for the observed shifts. In order to assess the range over which the linear model is appropriate, we made a series of straight-line fits to different numbers of data; the numbers of data were varied for each fit by omitting one point at a time from the high e 21 end of the set. The middle right-hand panel of Fig. 1 shows the results of a series of fits made to the 30 # l # 90 GONG frequencies (the other GONG sets return similar results). Here, we plot the reduced chi-squared statistic, x 2 n, of each fit as a function of the mean inverse fractional inertia of the highest point used in the fit. When the threshold value is large, data are admitted that are clearly out of line with the linear trend (cf. middle left-hand panel); the linear model is then a poor representation of the data and this is reflected by the high x 2 n values., 0:6, the reduced chi-squared value passes through unity: this occurs at a frequency of,2500 mhz, and we take this to be the threshold above which a linear description can no longer be considered as appropriate. At lower thresholds, the returned x 2 values drop below unity. This could reflect: (i) that the scatter on the data is less than that implied by the formal uncertainties (error bars); or (ii) that the formal uncertainties are too large. The observed values probably arise from a combination of these two effects, although we are inclined to believe the first may dominate since the low-threshold values were generated from analyses of small numbers of data over a part of the spectrum where the shifts are not much greater than the errors themselves. However, the difficulties associated with obtaining accurate estimates of the true uncertainties on the frequencies should also be borne in mind (see, e.g., Toutain & Appourchaux 1994; Chaplin et al. 1998, 1999). We have subtracted each set of l-averaged shifts from the linear fit made to the 30 # l # 90 GONG data, and these residuals are plotted in the lower left-hand panel of Fig. 1. These begin to depart from the zero level at frequencies above,2500 mhz (where At a threshold of e 21 < 0:6Þ. Above this, a dependence on frequency with a, 0is implied by the data. In order to extract an estimate of a, we have recast equation (4) in logarithmic form: e 21 log ½dn E Š¼log k 1 a log n : The fitted gradient of a linear regression of the logarithm of the shift-inertia product against logarithmic frequency is then the power-law index, a, we seek. Table 5 displays the results of fitting Table 5. Results of fitting different sets of eigenfrequency shifts to the power-law model described by equation (4), both below and above 2500 mhz. We re-cast equation (4) in logarithmic form to enable a straightforward linear regression to be made (equation 6) that would allow estimates of the power-law index, a, to be extracted from the fitted gradients. There were an insufficient number of low-frequency fits in the LOI set to allow a to be extracted for n # 2500 mhz. Data set l range a n # 2500 a n.2500 GONG 4 # l # ^ ^ 0.09 GONG 30 # l # ^ ^ 0.03 GONG 100 # l # ^ ^ 0.12 GONG 4 # l # ^ ^ 0.03 LOI 0 # l # ^ 0.40 BiSON 0 # l # ^ ^ 0.69 ð6þ

6 Solar-cycle variations: GONG, BiSON and VIRGO/LOI 915 the different data sets over different ranges in l. For completeness, we have also fitted each set for n # 2500 mhz. There were an insufficient number of low-frequency data in the LOI set to enable us to extract an estimate of a n#2500. The large uncertainty on the corresponding low-frequency BiSON estimate arises from the difficulty in determining the small solar-cycle-induced eigenfrequency shifts present in this part of the spectrum from the restricted numbers of data available at low l. The lower-frequency indices in Table 5 are consistent with an a ¼ 0 dependence, given the associated errors, while for n mhz we find a, 2 over all the different ranges fitted in l. The lower right-hand panel of Fig. 1 shows the product of the eigenfrequency shift and the fractional mode inertia, e, plotted as a function of frequency for GONG data averaged over 30 # l # 90. The broken curves show the results of fitting the plotted data to the power-law model below and above 2500 mhz. The higher-frequency dependence is described by a power-law with index a n,2500 ¼ 1:91 ^ 0:03; at low frequencies a n#2500 ¼ 0:22 ^ 0:23 provides the best fit to the data. The plot shows the clear difference in behaviour at low and high frequencies. There are also indications that at increasingly higher frequencies, the power law may steepen. The division of the data into two parts has been made on a purely artificial basis, i.e. we have applied a heuristic approach based upon quantitative, statistical tests of a simple model, and have not Figure 2. Upper panel, the upper turning point as a function of frequency for radial modes, as determined from model S of Christensen-Dalsgaard. Here, we define the zero-point location as that corresponding to where optical depth reaches unity at a wavelength of 500 nm. Lower panel, fit residuals from the linear model plotted as a function of the upper turning point of the modes. Some points with UTPs more than 700 km beneath the base of the photosphere have been omitted in order increase clarity by restricting the range plotted on the abscissa. been guided by any associated physics. The quality of the data are such that we are unable to gauge any departures from a linear scaling with inverse mode inertia below,2500 mhz; it may be that an analysis of increased numbers of data will uncover some other functional dependence. In spite of this, our imposed breakpoint may perhaps have some physical relevance beyond satisfying the constraints imposed by the tested model and the precision in the observations, and we hope that our analysis may provide additional clues for those addressing the theoretical aspects of the problem. Here, we point out a few facts that may or may not be relevant. The upper panel of Fig. 2 shows the location of the cavity UTP calculated from model S of Christensen-Dalsgaard. This is strongly dependent upon the properties of the model at the top of the convection zone. It can be seen that the depth beneath the solar surface of the cavity UTP increases quite sharply with frequency below <2300 mhz, while above this there exists a much shallower trend. The two regimes delineate approximately those uncovered by our analyses. We further emphasize this point in the lower panel of Fig. 2, where the residuals from the linear fits are plotted as a function of the UTP. They are seen to increase sharply in magnitude when the UTP lies within,200 km or so of the base of the photosphere. Recent analyses indicate that the excitation source of the modes may be located in this thin layer (see, e.g., Nigam & Kosovichev 1998, 1999a,b; Kumar & Basu 1999a,b; Chaplin & Appourchaux 1999; Chaplin et al. 2000). The analysis of Chaplin & Appourchaux (1999) also showed that when the source is considered to be of a particular type (polarity), the asymmetries of low-l modes (in BiSON data) constrain the location of the source to lie outside the resonant cavity for modes with frequencies below <2500 mhz, and within it at frequencies above this. We note, however, that the absolute source location inferred from the type of analysis performed by Chaplin & Appourchaux is sensitive to the form of the cavity acoustic potential adopted (a simple, square-well model in this case). ACKNOWLEDGMENTS This work utilizes data collected by the VIRGO/LOI instrument on board the SOHO satellite, and the ground-based Global Oscillations Network Group (GONG) and Birmingham Solar- Oscillations Network (BiSON). We thank the members of all three teams for allowing us to use these data. SOHO is a project of international cooperation between ESA and NASA. We thank all members of the VIRGO team for allowing us to use these data. The GONG project is managed by the National Solar Observatory, a Division of the National Optical Astronomy Observatories, which is operated by AURA, Inc. under a cooperative agreement with the National Science Foundation. The data were acquired by instruments operated by the Big Bear Solar Observatory, High Altitude Observatory, Learmonth Solar Observatory, Udaipur Solar Observatory, Instituto de Astrofísico de Canarias, and Cerro Tololo Interamerican Observatory. BiSON is funded by the UK Particle Physics and Astronomy Research Council. We are extremely grateful to R. Howe for providing the azimuthally averaged GONG frequencies; we would also like to thank J. Christensen-Dalsgaard for the model S inertias and useful discussions, and T. Sekii for the modified acoustic potential data. REFERENCES Anguera Gubau M., Palle P. L., Perez Hernandez F., Regulo C., Roca Cortes T., 1992, A&A, 255, 363

7 916 W. J. Chaplin et al. Appourchaux T., Gizon L., Rabello-Soares M. C., 1998, A&AS, 132, 107 Bachmann K. T., Brown T. M., 1993, ApJ, 411, L45 Bhatnagar A., Jain K., Tripathy S. C., 1999, ApJ, 521, 885 Chaplin W. J., Elsworth Y., Isaak G. R., Lines R., McLeod C. P., Miller B. A., New R., 1998, MNRAS, 300, 1077 Chaplin W. J., Appourchaux T., 1999, MNRAS, 309, 761 Chaplin W. J., Elsworth Y., Isaak G. R., Miller B. A., New R., 1999, MNRAS, 308, 424 Chaplin W. J., Appourchaux T., Elsworth Y., Isaak G. R., Miller B. A., New R., 2000, MNRAS, 314, 75 Christensen-Dalsgaard J., Berthomieu G., 1991, in Cox A.N., Livingston W.C., Matthews M., eds, Solar Interior and Atmosphere. Univ. Arizona Press, Tucson, p. 401 Christensen-Dalsgaard J. et al., 1996, Sci, 272, 1286 Dziembowski W. A., Goode P. R., Kosovichev A. G., Schou J., 2000, ApJ, 537, 1026 Elsworth Y., Howe R., Isaak G. R., McLeod C. P., New R., 1990, Nat, 345, 322 Elsworth Y., Howe R., Isaak G. R., McLeod C. P., Miller B. A., New R., Speake C. C., Wheeler S. J., 1994, ApJ, 434, 801 Goldreich P., Murray N., Willette G., Kumar P., 1991, ApJ, 320, 752 Gough D. O., 1990, in Osaki Y., Shibahashi H., eds, Progress of Seismology of the Sun and Stars. Springer-Verlag, Berlin, p. 135 Hill F. et al., 1996, Sci, 272, 1292 Howe R., Komm R., Hill F., 1999, ApJ, 524, 1084 Jimenez-Reyes S. J., Regulo C., Palle P. L., Roca Cortes T., 1998, A&A, 329, 1119 Kumar P., Basu S., 1999a, ApJ, 519, 389 Kumar P., Basu S., 1999b, ApJ, 519, 396 Libbrecht K. G., Woodard M. F., 1990, Nat, 345, 779 Libbrecht K. G., Woodard M. F., 1991, Sci, 253, 152 Nigam R., Kosovichev A. G., 1998, ApJ, 505, L51 Nigam R., Kosovichev A. G., 1999a, ApJ, 510, L149 Nigam R., Kosovichev A. G., 1999b, ApJ, 514, L53 Regulo C., Jimenez A., Palle P. L., Hernandez F. P., Cortes T. R., 1994, ApJ, 434, 384 Rhodes E. J., Jr, Cacciani A., Korzennik S. G., Ulrich R. K., 1993, ApJ, 406, 714 Toutain T., Appourchaux T., 1994, A&A, 289, 649 Woodard M. F., Noyes R. W., 1985, Nat, 318, 449 Woodard M. F., Libbrecht K. G., Kuhn J. R., Murray N., 1991, ApJ, 373, L81 This paper has been typeset from a TEX/L A TEX file prepared by the author.