---
abstract: 'Variations of frequency separations of low-degree p-modes are studied over the solar activity cycle. The separations studied are obtained from the frequencies of low-degree p-modes of the Global Oscillation Network Group (GONG). 10.7 cm radio flux is used as an index of solar activity. Small separations of the p-mode frequencies are considered to be mainly dependent on the conditions in stellar interiors. Thus they could be applied to diagnose the changes in the stellar interior. Our calculation results show that the magnitudes of variations of the mean large separations are less than 1 $\sigma$ over the solar activity cycle. Small separations show different behaviors in the ascending and descending phases of activity. In the ascending phase, variations of the small separations are less than 1 $\sigma$. However, the small separations have systematic shifts during 2004 - 2007. The shifts are roughly 1 $\sigma$ or more. The variations of the ratios of the small to large separations with time are similar to the changes of the small separations. The effects of the changes in the large separations on the ratios are negligible. The variations of the separations may be a consequence of the influence from the surface activity or systematic errors in measurements or some processes taking place in the solar interior.'
author:
- Wuming Yang
- Xiangcun Meng
title: 'Frequency separation variations of the solar low-degree p-modes'
---

Introduction
============

It is well-known that the solar p-mode frequencies vary with the solar cycle [@wood85; @elsw90; @libb90; @elsw94; @jime98; @howe99; @chap07]. It has been shown that the variations of the frequencies are significantly related to magnetic activities near the solar surface [@wood91; @jime04; @tou05]. However, a number of attempts have been made to detect the relation between the frequency shifts and the variation of the solar interior. It has been proposed that the frequency shifts may relate to changes taking place in the base of the convection zone [@jime98; @sere05], or in the deep core [@jime98].

Low-degree p-modes can be used to study the solar core because they can penetrate the deep interior [@dzie97]. However, even these modes are primarily sensitive to the solar envelope structure. Thus frequency separations are proposed for a diagnostic tool. The frequency separations of low-degree p-modes in solar-like stars often proposed for diagnostic purposes are large separations $\Delta_{l}(n)$ and small separations $d_{ll+2}(n)$. The large separations approximate to the characteristic frequency $\nu_{0}$ of a star, defined by $$\Delta _{l}(n)\equiv \nu_{n, l} - \nu_{n-1, l} \simeq \nu_{0},
  \label{eqla}$$ where $$\nu_{0} = (2\int^{R}_{0}dr/c)^{-1},$$ in which $c$ is sound speed at radius $r$ and $R$ is some fiducial radius of the star. The small separations are sensitive to the structure of stellar core [@gou90a], defined by $$\begin{array}{lll}
   d_{l l+2}(n)& \equiv & \nu_{n, l} - \nu_{n-1, l+2}\\
   & =& \frac{\varphi_{l+2}-\varphi_{l}}{\pi}\nu_{0},
\end{array}
   \label{eqsm}$$ where $\varphi_{l}$ is the internal phase shift. For the low-degree p-modes, the $\varphi_{l}$ is mainly dependent on conditions in the stellar core and is almost independent of those outside the core [@rox03]. These frequency separations have been extensively investigated by many authors [@chris84; @ulr86; @ulr88; @gou87; @gou90b; @gou03; @gou90a; @rox03; @aud94; @rox05; @flo05]. Additionally @rox93 defines differences $d_{01}(n)$ as $$\begin{array}{lll}
   d_{01}(n)& \equiv & \frac{-\nu_{n, 1} + 2\nu_{n, 0}- \nu_{n-1,
   1}}{2} \\
  & = & \frac{\varphi_{1}-\varphi_{0}}{\pi}\nu_{0},
\end{array}$$ and @yang07 define differences $\sigma_{l-1l+1}(n)$ of low-degree p-modes as $$\begin{array}{lll}
  \sigma_{l-1 l+1}(n)& \equiv & -\nu_{n, l-1}+2\nu_{n, l} - \nu_{n,
  l+1} \\
  & = & \frac{\varphi_{l+1}+ \varphi_{l-1}-2\varphi_{l}}{\pi}\nu_{0}.
\end{array}$$ The differences $\sigma_{02}(n)$ are similar to the scaled small separations $d_{l l+2}(n)/(2l+3)$ in some cases and are mainly sensitive to conditions in stellar core. The difference $\sigma_{02}$ averaged over $n$ is, however, more sensitive to changes in the central hydrogen abundance of solar-like stars than the scaled small separations [@yang07].

Moreover, the ratios $d_{l l+2}/\Delta_{l}$ are considered to be essentially independent of the structure of the outer layers of a star and determined by interior structures [@ulr86; @rox03; @rox05; @flo05]. The ratios $d_{01}/\Delta_{0}$ and $\sigma_{02}/\Delta_{0}$ also mainly rely on the interior structures [@rox03; @yang07]. Thus changes in stellar interior could be indicated by the variations in these separation ratios.

However frequency variations with solar activity are relative to the angular ($l$) degree, the azimuthal ($m$) degree, and the order $n$. The $m$-dependence is attributed to the solar near-surface activity [@jime04; @tou05] and could be removed by using the frequency centroids which incorporate all the $m$ components of a mode [@chap05]. For the low-degree p-modes, centroid shifts can be regarded as being $l$-independent [@chap05]. Thus separations of frequency centroids of low-degree p-modes should be insensitive to effects of surface activity. Using the p-modes of the Birmingham Solar-Oscillations Network (BiSON) collecting data using a Sun-as-a-star technique, @chap05 studied the impact of the solar activity cycle on the frequency separation ratios and found that the ratios change with the shifting level of global solar activity. @chap05 pointed out that some $m$ components are suppressed in the Sun-as-a-star observations and hence the ratios are expected to be affected by the surface activity. In this paper, we investigate the variations of frequency separations of low-degree p-modes as solar activity cycle proceeds using data collected by the resolved-Sun observations. In Section 2 we present our data and results. Then discussion and conclusion are represented in Section 3.

Data and results
================

In this work, we used the solar p-mode frequencies of the GONG [@harv96] to study the separations. The GONG was planned for measurements of medium and high spherical degrees, but low-degree p-modes can be measured too. @gavr99 showed that the low-l p-modes of the GONG are in good agreement with that of the Michelson Doppler Imager (MDI) and the Global Oscillations at Low Frequency (GOLF). The raw data of GONG are measured in a time series of 36 days labeled a GONG month. A 108-day time series is constructed by concatenating 3 consecutive GONG-month time series. GONG Month Mode Frequencies (GMFs) were then estimated from the 108-day spectra [@ande90; @hill98; @komm99; @howe99]. In the observation of GONG, the solar disk is imaged onto many pixels. The obtained images can then be decomposed into their constituent spherical harmonics. For each degree $l$ and overtone $n$, this imaging strategy provides access to all $2l+1$ components with each being tagged by an azimuthal degree $m$ [@chap04].

The frequencies, $\nu_{n,l,m}$, extracted from resolved-Sun observations are usually represented by $$\nu_{n,l,m}=\nu^{c}_{n,l}+\sum^{2l}_{j=1}a_{j}(n,l)l\mathcal{P}^{j}_{l}(m),
  \label{num}$$ where $\mathcal{P}^{j}_{l}(m)$ are polynomials related to Clebsch-Gordan coefficients [@ritz91; @chap04] and $\nu^{c}_{n,l}$ is the so-called central frequency of the multiplet (i.e. frequency centroid). Using the polynomials $\mathcal{P}^{j}_{l}(m)$ given by @chap04 and equation (\[num\]), one can get the central frequency of low-degree p-modes of the resolved-Sun observations, $$\nu^{c}_{n,l}=\sum^{l}_{m=-l}\frac{\nu_{n,l,m}}{2l+1}.
  \label{centf}$$ The central frequencies are sensitive to the spherically symmetric component of the internal structure of a star and are usually used as inputs to inversions of the internal structure [@chap04; @chap05].

In order to study the variations of frequency separations with the solar activity cycle, we used the GMFs observed between 1996 and 2007. Centroids $\nu_{n,l}$ and individual frequencies $\nu_{n,l,m}$ of the GMFs were published by the GONG team[^1]. Some centroids $\nu_{n,l}$ published are inconsecutive in the order $n$, especially for the modes with $l$ = 1, but the frequencies $\nu_{n,l,m}$ are consecutive. In order to obtain separations, we used the averages $\nu^{c}_{n,l}$ of all $2l$ + 1 components of each mode instead of the centroids $\nu_{n,l}$. In figure \[fig1\], we compared the averages obtained using equation (\[centf\]) with the available centroids estimated by the GONG team. The differences between the frequencies obtained using equation (\[centf\]) and the available GONG centroids are almost much less than the errors of the GONG centroids in the range of 2400 $\mu$Hz $< \nu <$ 3500 $\mu$Hz **except a few modes, whose differences are close to the errors of the GONG centroids**. In this range, the averages obtained using equation (\[centf\]) are good consistent with the centroids estimated by the GONG team. All the following frequency separations are obtained from the averages.

Figure \[fig2\] shows fractional changes of the separation ratios in two sets of GMFs, one at a low level and one at a high level of activity. The error bars indicate 1 $\sigma$ errors, which were propagated from uncertainties on the individual frequencies of the GMFs. The departure of the separation ratios from the null level is within 1 $\sigma$ but shows a systematic shift, which is consistent with the prediction of the 108 day spectra of @chap05 (in their figure 1). However, just as the observed shifts of ratios of the BiSON frequencies [@chap05], 'the shifts are clearly not significant for individual estimates of the ratios'. In the interest of temporal variations of separations, for each set of separations with the same degree $l$, we averaged the separations over $n$ in the range of 2400 $\mu$Hz $< \nu_{n,l} <$ 3500 $\mu$Hz, which was implied by symbols $< >$. The frequency range is often used to obtain a mean frequency shift [@elsw94; @jime04].

                                         Ascending phase                                                                                                    
  ---------------------------- ----------------------------------- --------------------- ------- ----------------------------------- ---------------------- -------
           Separation                Slope (nHz/year)$^{a}$             $\chi^{2}$          R          Slope (nHz/year)$^{a}$              $\chi^{2}$          R
           $<d_{02}>$                    - 6.9 $\pm$ 5.1                   0.228          0.177           - 35.4 $\pm$ 7.6                   0.019           0.697
           $<d_{01}>$                     1.1 $\pm$ 5.3                    0.244          0.027           - 76.9 $\pm$ 13.4                  0.058           0.768
        $<\sigma_{02}>$                  - 9.9 $\pm$ 7.1                   0.446          0.184             108 $\pm$ 23                     0.179           0.692
           $<d_{13}>$                    - 8.6 $\pm$ 3.3                   0.092          0.327            43.0 $\pm$ 9.0                    0.029           0.691
         $<\Delta_{0}>$                  11.8 $\pm$ 2.8                    0.072          0.485        - 14.7 $\pm$ 2.3$^{b}$                0.041           0.654
         $<\Delta_{1}>$                   5.2 $\pm$ 2.2                    0.041          0.303        - 11.4 $\pm$ 1.3$^{b}$                0.012           0.771
     $<d_{02}/3\Delta_{0}>$     - (1.9 $\pm$ 1.2)$\times 10^{-5}$   1.3$\times 10^{-6}$   0.205   - (8.8 $\pm$ 1.9)$\times 10^{-5}$   1.2$\times 10^{-7}$    0.698
     $<d_{01}/\Delta_{0}>$       (5.6 $\pm$ 3.9)$\times 10^{-5}$    1.3$\times 10^{-5}$   0.019   - (5.7 $\pm$ 1.0)$\times 10^{-4}$   3.2$\times 10^{-6}$    0.766
   $<\sigma_{02}/\Delta_{0}>$   - (7.5 $\pm$ 5.3)$\times 10^{-6}$   2.5$\times 10^{-5}$   0.188    (8.0 $\pm$ 1.7)$\times 10^{-4}$    9.8 $\times 10^{-6}$   0.694
     $<d_{13}/5\Delta_{1}>$     - (1.4 $\pm$ 0.5)$\times 10^{-5}$   2.0$\times 10^{-7}$   0.348    (6.5 $\pm$ 1.4)$\times 10^{-5}$    6.2 $\times 10^{-8}$   0.699

  : Results of linear fits between separations and time. Left panel for the ascending phase of solar activity, while right panel for the during $\sim$ 2004 to 2007 except for the large separations. \[tab1\]

a\. Not for ratios.

b\. The results were obtained from the data between $\sim$ 2001 and 2007.

Figure \[fig3\] shows the temporal variation of mean separations $<d_{ll+2}>$, $<\sigma_{02}>$, $<d_{01}>$, $<\Delta_{0}>$ and $<\Delta_{1}>$. Sunspot numbers and 10.7 cm radio flux [^2] (F10), which are always used as an index of the solar activity, are also plotted as a function of time. The activity indices have been averaged over the same time interval covered by the GONG date. **The error bars show 1 $\sigma$ errors, which come from the formal errors on the frequencies. When averaging these separations from each spectrum, the correlations of neighbouring separations were considered**. The lines in this figure represent the results of linear least-squares fits. The fits in the descending phase of solar activity were divided into two periods in accordance with the variations of separations: one from $\sim$ 2001 to 2004 and one from $\sim$ 2004 to 2007. Variations in the latter period are more obvious and systematic. The mean large separations $<\Delta_{0}>$ and $<\Delta_{1}>$ increase/decrease in the ascending/descending phase of solar activity and reach a maximum in 2001. However, the magnitude of the shifts of the large separations is about 0.07 $\mu$Hz, which is less than 1 $\sigma$. Variations of small separations with time are more complicated than those of the large separations. In the ascending phase of solar activity, the separations $<d_{ll+2}>$, $<\sigma_{02}>$ and $<d_{01}>$ almost have no systematic shifts; the magnitudes of the shifts of these separations, obtained from their regression equations, are also less than 1 $\sigma$. In the descending phase of solar activity, variations of these separations seem to be disorderly. Between 2001 and 2002, the separations $<\sigma_{02}>$ and $<d_{13}>$ increase suddenly. They decrease firstly and then increase obviously between $\sim$ 2001 and 2007. However, the magnitudes of their shifts are roughly 1 $\sigma$ or more. The separation $<d_{01}>$ has an obvious and systematic decrease between $\sim$ 2004 and 2007. The magnitude of the variation is about 1.5 $\sigma$. The separation $<d_{02}>$ decreases slightly between $\sim$ 2001 and 2007, but the magnitude is less than 1 $\sigma$. The results of linear fits: slope, residual sum of squares $\chi^{2}$, and correlation coefficient $R$ of regression equations, are summarized in Table \[tab1\].

                      Ascending phase                                                             
  ----------------- ------------------- ------------ ------- ----------------------- ------------ -------
     Separation       Slope (nHz/SFU)    $\chi^{2}$     R        Slope (nHz/SFU)      $\chi^{2}$     R
     $<d_{02}>$      - 0.34 $\pm$ 0.19     0.223      0.235      2.13 $\pm$ 0.69        0.026      0.542
     $<d_{01}>$      0.024 $\pm$ 0.197     0.244      0.016       5.6 $\pm$ 1.1         0.068      0.722
   $<\sigma_{02}>$   - 0.33 $\pm$ 0.27     0.449      0.162      - 8.4 $\pm$ 1.8        0.176      0.699
     $<d_{13}>$      - 0.25 $\pm$ 0.13     0.096      0.254      - 3.2 $\pm$ 0.8        0.032      0.657
   $<\Delta_{0}>$     0.33 $\pm$ 0.11      0.081      0.361   0.57 $\pm$ 0.09$^{a}$     0.041      0.648
   $<\Delta_{1}>$     0.22 $\pm$ 0.08      0.040      0.329   0.45 $\pm$ 0.05$^{a}$     0.012      0.791

  : Results of linear fitting between separations and 10.7 cm radio flux. \[tab2\]

a\. The results were obtained from the data between $\sim$ 2001 and 2007.

In order to further understand the variations of separations with the solar activity, Figure \[fig4\] shows separations plotted against F10 in the ascending phase of solar activity. Figure \[fig4\] shows that the separations almost not vary with activity. The results of linear least-squares fits between the separations and F10 are summarized in the left panel of Table \[tab2\]. The magnitudes of variations of these separations are only about 0.04 $\mu$Hz, which is less than 1 $\sigma$. Moreover, Figure \[fig5\] shows that the changes of the large separations with the F10 in the descending phase of the solar activity. The shifts of the large separations are also less than 1 $\sigma$. Figure \[fig5\] also shows the variations of small separations with the F10 during $\sim$ 2004 - 2007. In this period of time, the changes of these separations with the F10 are obvious and systematic. However, the magnitudes of variations of these separations are around 1 $\sigma$ or more. The results of linear fits between the separations and F10 are shown in the right panel of Table \[tab2\].

In Figure \[fig6\], we show the temporal variations of separation ratios $<d_{02}/3\Delta_{0}>$, $<d_{13}/5\Delta_{1}>$, $<d_{01}/\Delta_{0}>$ and $<\sigma_{02}/\Delta_{0}>$. The results of linear fits between the ratios and time are presented in Table \[tab1\]. The changes of the ratios with time are similar to those of the corresponding small separations. In the ascending phase of solar activity, shifts of the ratios are less than 1 $\sigma$. In the descending phase of activity, from $\sim$ 2004 to 2007, the changes of the ratios with time are systematic. However, the magnitudes of the changes are still roughly 1 $\sigma$ or more. Since the fractional changes of the large separations are much less than those of small separations during the activity, the contribution to the changes in the ratios from the variations of the large separations is much less than that from the small separations. The effects of the changes in the large separations on the changes in the ratios are almost negligible. The changes in the ratios with time are mainly determined by the changes in the small separations.

The magnitudes of temporal variations for some individual separations are very large but should not be significant. In the ascending phase of the solar activity, the magnitudes of variations of separations, obtained from regression equations, are about 0.04 $\mu$Hz, which is less than 1 $\sigma$. During $\sim$ 2004 - 2007, the changes in separations are systematic. However, the magnitudes of the changes are roughly 1 $\sigma$ or more. Using BiSON data, @chap05 show that variations of the separation ratios between low and high activity is at a marginal level of significance. Moreover, using simple averages, from the gradient of mean frequency shift of the MDI given by @jime04 in their Table 1, one can get a shift gradient in the centroids would be 1.61 and 2.02 nHz/SFU for $l$ = 0 and 2 respectively. For a 120 SFU increase in solar activity, the change in separation $<d_{02}>$ would then be about 0.05 $\mu$Hz, which is consistent with our result.

Discussion and conclusion 
==========================

The characteristic frequency $\nu_{0}$, i.e. the large separation, is an integral property of the entire star. Because the solar sound speed is a decreasing function of the solar radius, the contribution to $\nu_{0}$ from the outer layers of the Sun is larger than that from the solar inner regions [@gou90a]. Thus the large separation should be more sensitive to the changes in the outer layers than those in the inner regions and could be affected by the near surface activity. Moreover small separations are more sensitive to the conditions in stellar interior than those in outer layers. Thus the large separations should be more sensitive to the activity than the small separations. However, in fact, the magnitudes of variations of the large separations are about 0.07 $\mu$Hz during the activity, which is less than 1 $\sigma$. The magnitude of variations of $<d_{02}>$ is also less than 1 $\sigma$ over the activity cycle. However, other small separations show a systematic shift during $\sim$ 2004 - 2007, and the magnitudes of their variations are roughly 1 $\sigma$ or more.

Small separations $<d_{ll+2}>$, $<\sigma_{02}>$ and $<d_{01}>$ show a sudden change between 2001 and 2002. The fractional changes shown in Figure \[fig2\] of ratios in two GMFs (one in 1996 and one in 2001) are consistent with the prediction of @chap05. However, if we use the data of 2002 instead of the data of 2001, the results would be different. The small separations show different behaviors in the ascending and descending phases of the cycle. In the ascending phase, variations of the small separations are less than 1 $\sigma$. In the descending phase, the small separations show a systematic shift during 2004 - 2007. And the magnitudes of the changes in $<d_{13}>$, $<\sigma_{02}>$ and $<d_{01}>$ are roughly 1 $\sigma$ or more. It is not clear whether the variations are caused by the systematic errors in measurements.

The large separations increase slightly with increasing of the solar activity. But the magnitudes of temporal variations of the mean large separations of the low-degree p-modes are only about 0.07 $\mu$Hz over the activity cycle, which is less than 1 $\sigma$. Small separations $<d_{ll+2}>$, $<\sigma_{02}>$ and $<d_{01}>$ show different behaviors in the ascending and descending phases of the cycle. In the ascending phase, variations of the small separations are less than 1 $\sigma$. However, the small separations show a systematic shift during 2004 - 2007. The magnitudes of the changes in $<d_{01}>$, $<\sigma_{02}>$ and $<d_{13}>$ in this period of time are about 1 $\sigma$ or more. The variations of the ratios of the small to large separations with time are similar to the changes of the small separations. Changes in the large separations have a negligible impact on the variations in the ratios. The variations of the separations may be a consequence of the influence from the surface activity or systematic errors in measurements or some processes taking place in the solar interior. For further work, the data over cycle 22/23 would be required.

Acknowledgments {#acknowledgments .unnumbered}
===============

This work utilizes data obtained by the Global Oscillation Network Group (GONG) program, managed by the National Solar Observatory, 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 Astrofisica de Canarias, and Cerro Tololo Interamerican Observatory.

Anderson, E. R., Duvall, Jr., Jefferies, S. M. 1990, ApJ, 364, 699 Audard, N., Provost, J. 1994, A&A, 282, 73 Chaplin, W. J., Appourchaux, T., Elsworth, Y., Isaak, G. R., Miller, B. A., New, R. 2004. A&A, 424, 713 Chaplin, W. J., Elsworth, Y., Miller, B. A., New, R., Verner, G. A. 2005, ApJ, 635, L105 Chaplin, W. J., Elsworth, Y., Miller, B. A., Verner, G. A., New, R. 2007, ApJ, 659, 1749 Christensen-Dalsgaard, J. 1984, in Mangeney A., Praderie, F., eds, Space Research Prospects in Stellar Activity and Variability, Paris Observatory Press, Paris, P11 Dziembowski, W. A., Goode, P. 1997, A&A, 317, 919 Elsworth, Y., Howe, R., Isaak, G. R., McLeod, C. P., New, R. 1990, Nature, 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 Gough, D. O. 1987, Nature, 326, 257 Gough, D. O. 1990, in Progress of Seismology of the Sun and Stars, Proc. Oji International Seminar Hakone (Japan: Springer Verlag), Lect. Notes Phys., 367, 283 Gough, D. O. 2003, AP&SS, 284, 165 Gough, D. O., Novotny, E. 1990, So. phys., 128, 143 Gavryusev, V., Gavryuseva, E. 1999, in: Magnetic Fields and Solar Processes, Wilson, A., ed., ESA Publication Division, ESA SP-448, p99 Harvey, J. W., Hill, F., Hubbard, R. et al.  1996, Science, 272, 1284 Hill, F., Anderson, E., Howe, R., Jefferies, S. M., Komm, R., Toner, C. 1998, in Dynamics of the Interior of the Sun and Sunlike Stars, ed. S. G. Korzennik & A. Wilson (ESA SP-418; Noordwijk: ESA), 1998, 231 Howe, R., Komm, R., Hill, F. 1999, ApJ, 524, 1084 Jiménez-Reyes, S. J., Régulo, C., Pallé, P. L., Roca Cortés, T. 1998, A&A, 329, 1119 Jiménez-Reyes, S. J., García, R. A., Chaplin, W. J., Korzennik, S. G. 2004, ApJ, 610, L65 Komm, R. W., Gu, Y., Hill, F., Stark, P. B., Fodor, I. K. 1999, ApJ, 519, 407 Libbrecht, K. G., Woodard, M. F. 1990, Nature, 345, 779 Otí Floranes, H., Christensen-Dalsgaard, J., Thompson, M. J. 2005, MNRAS, 356, 671 Ritzwoller, M. H., Lavely, E. M. 1991, ApJ, 369, 557 Roxburgh, I. W. 1993, in PRISMA, Report of Phase A Study, ed. T. Approurchaux, et al., ESA 93, 31 Roxburgh, I. W. 2005, A&A, 434, 665 Roxburgh, I. W., Vorontsov, S. V. 2003, A&A, 411, 215 Serebryanskiy, A., Chou, D. Y. 2005, ApJ, 633, 1187 Toutain, T., Kosovichev, A. G. 2005, ApJ, 622, 1314 Ulrich, R. K. 1986, ApJL, 306, L37 Ulrich, R. K. 1988, in IAU Symp. 123, Advances in Helio- and Asteroseismology, ed. J. Christensen-Dalsgaard, S. Frandsen (Dordrecht: Kluwer), 299 Woodard, M. F., Noyes, R. W. 1985, Nature, 318, 449 Woodard, M. F., Libbrecht, K. G.. Kuhn, J. R., Murray, N. 1991, ApJ, 373, L81 Yang, W. M., Bi, S. L. 2007, A&A, 472, 571

[^1]: ftp://gong2.nso.edu/MFS/

[^2]: www.ngdc.noaa.gov/stp/SOLAR/getdata.html

{'source': 'arXiv_out/0911'}