3 Frequency integrated frequency shifts

Once the power spectrum of each time-series has been calculated, the integrated frequency shifts $\Delta\nu^i$ are determined. This is done by computing the cross-correlation $\rho^i(\nu_{j})$ of each power spectrum i with the power spectrum of the time-series covering 1986 taken as reference. As we show in Sect. 5, the dependence of the frequency shift for low and intermediate-degree p-modes appears to be essentially null below 2 mHz, whereas at high frequency (above 3.7 mHz) the frequency shift is expected to drop quickly. The analysis of the integrated changes is limited to the modes between 2.5 and 3.7 mHz. Considering a larger interval could bias the results by including the strong high frequency variations that are less accurately determined and have possibly different physical explanation (Goldreich et al. 1991).

In order to calculate the position of the cross-correlation main peak, two different methods have been used. The first was used in previous analyses (Régulo et al. 1994; Jiménez-Reyes et al. 1998) and takes the maximum of a second-order polynomial fitted to the logarithm of the cross-correlation function in an interval $\pm\sigma$ around the main peak (where $\sigma$ is the second-order moment). This cross-correlation is calculated starting at an appropriate lag for which the function is symmetric, and which is obtained by calculating the third-order moment. This procedure directly provides a value of the mean frequency shift between the $\ell\leq3$ p-modes and the corresponding values at solar activity minimum, the chosen reference.

The second method, introduced here, is based on the shape of the cross-correlation function. Assuming that each oscillation mode can be modeled by a damped harmonic oscillator, each peak in the spectra has a Lorentzian profile and the correlation function has also a Lorentzian profile. In order to improve the determination of the frequency shift, the model also take into account the presence of sidebands with amplitude $\beta_{\vert 1\vert}$ located at $\pm$11.57 $\mu$Hz. Thus, our model of the cross-correlation function between the spectrum i and the reference spectrum can be written as:

$$M^i_\rho(\nu_j)= \!\!\!\!\sum_{k=-1}^{1} \! \beta_{\vert k\v... ...2)^2}{(\nu_j-\Delta\nu^i+k_D \cdot k)^2+ (\Gamma^i/2)^2} +B^i,$$ (2)

where the parameters to be fitted are:

• Aⁱ, the amplitude of the central peak;
• $\Gamma^i$, the linewidth of the Lorentzian profile;
• $\Delta\nu^i$, the average frequency shift in the chosen frequency interval;
• Bⁱ, the constant background level;
• $\beta_{\vert k\vert}$, the ratio of the sidebands to the central peak ($\beta~=~1$ for k = 0);
• $k_D=11.57\mu$Hz is the constant separation of the sidebands and it is the only fixed parameter in the fit.

The best estimation, in the least square sense, of the parameters related to each time-series i, is obtained by minimizing the following quantities:

$$\chi^{2}_i = \sum_{j=1}^{N}{\mid \rho^i(\nu_{j}) - M^i_\rho(\nu_{j}) \mid^{2}},$$ (3)

where N is the total number of frequency bins in an interval of $\pm$20 $\mu$Hz around the main peak of the cross-correlation function. We applied a Levenberg-Marquard method (Press et al. 1992) but any other minimization routine may be used. We note that the first technique is more objective in the sense that it does not require a physical hypothesis or modeling of the p-mode excitation and damping mechanisms. It uses only information contained in the spectrum while the second method assumes symmetric Lorentzian profiles. There is some evidence that the shape of the peaks in the power spectra are slightly asymmetric (Thiery et al. 2000). In order to check the efficiency of both methods and the influence of the duty cycle in the final result, we analyzed the frequency shifts in periods of 36 days for which the duty cycle varies. The differences between the two methods remain in general within the error bars. The only significant differences are found for time-series with very small duty cycle (around $10\%$) but, for the yearly time-series analysed here, the duty cycle is moderately high and quite stable (around $25\%$) from year to year. Regarding to the observed asymmetry of the p-modes, they are not thought to be important for this analysis. The second method takes into account the known distance from the sidebands to the main peak ($\pm$11.57 $\mu$Hz) which improve the results. Moreover this method allows to study not only the frequency shift but also the TVP by providing amplitudes and widths of the cross-correlation functions (see Sect. 4). Therefore hereafter only the second method will be considered but, for completeness, Fig. 1 shows the marginal differences between the frequency shifts obtained by applying both methods to the yearly time-series.

Figure 1: Differences between the integrated frequency shifts as inferred from the cross-correlation functions by the two methods explained in the text i.e. (1) fit by a polynomial close to the maximum and (2) fit by a Lorentzian profile including side bands. The differences remain small and not significant.

Figure 2: Time variation of the integrated frequency shift for low-degree p-modes is plotted in the upper panel (black dots), where the radio flux at 10.7 cm is also shown (full line). In the sub plot, the results for the even and odd degrees separately are presented. The averages of these, are plotted in the main panel (squares). The lower figure shows the good linear correlation between the frequency shift and the radio flux.

 

 

Table 1: Correlation coefficients between different solar indices and the yearly frequency shift. $r_{\rm P}$ is the Pearson linear correlation coefficient, $r_{\rm S}$ the Spearman rank correlation coefficient and $P_{\rm s}$ is the probability of having no correlation.

Index		$\Delta\nu$				$\Delta\nu_{0,2}$				$\Delta\nu_{1,3}$
	$r_{\rm P}$	$r_{\rm S}$	$P_{\rm s}$		$r_{\rm P}$	$r_{\rm S}$	$P_{\rm s}$		$r_{\rm P}$	$r_{\rm S}$	$P_{\rm s}$
$R_{\rm I}$	0.94	0.88	9 $\times$ 10-11		0.76	0.70	1 $\times$ 10-5		0.90	0.85	4 $\times$ 10-9
F10	0.94	0.87	4 $\times$ 10-10		0.77	0.72	7 $\times$ 10-6		0.90	0.83	1 $\times$ 10-8
KPMI	0.90	0.86	8 $\times$ 10-10		0.79	0.71	1 $\times$ 10-5		0.85	0.84	8 $\times$ 10-9
MPSI	0.94	0.88	5 $\times$ 10-10		0.83	0.76	2 $\times$ 10-6		0.88	0.85	1 $\times$ 10-8
TSI	0.89	0.89	3 $\times$ 10-8		0.78	0.78	1 $\times$ 10-5		0.83	0.83	2 $\times$ 10-7
He	0.94	0.88	9 $\times$ 10-11		0.79	0.74	2 $\times$ 10-6		0.89	0.83	2 $\times$ 10-8

As mentioned in the introduction, the asymptotic theory predicts that, for low-degree p-modes, pairs of modes with alternately odd and even degrees are equally spaced in frequency with a separation of about 67 $\mu$Hz i.e. half of the so-called big separation (e.g. Deubner & Gough 1984). The contributions of even ($\ell = 0$, 2) and odd degrees ($\ell = 1$, 3) to the integrated frequency shift can therefore be separated simply by applying a mask to the spectra before using the procedure explained above. It should also be noted that, for full disk observations, the contribution of the two modes of a pair is not the same, due to the geometry of the modes at the surface. In the case of the odd modes, the frequency shifts come essentially from $\ell = 1$ due to the high ratio in sensitivity between $\ell = 3$ and 1 ($\sim$0.1) whereas, in the case of the even modes the sensitivity ratio is close to one and therefore the contributions of $\ell = 0$ and 2 are nearly the same.

Figure 2 illustrates the frequency averaged frequency shift between 2.5 and 3.7 mHz. The results corresponding to $\ell = 0$, 2 and $\ell = 1$, 3 have been plotted in the sub plot at the top-right corner and their average is shown in the main figure. The solid line represents the radio flux averaged over the same periods as the time-series used for this work. Although there are few departures from the general trend which do not agree with the smooth behavior of the solar index, the integrated signal concurs very well with the radio flux, which represents here the behavior of the solar cycle. The average of both contributions ($\ell = 0$, 2 and $\ell = 1$, 3) follows well the integrated signal as expected. The amplitude, measured as the straight difference from peak-to-peak, for all observed modes is $0.45 \pm 0.05$ $\mu$Hz, while in the case of the even and odd degree modes are $0.48 \pm 0.05$and $0.55 \pm 0.07$ $\mu$Hz respectively. Although the difference in amplitude between even and odd degree modes seems to remain, the frequency shift for the even modes is larger than that obtained by Régulo et al. (1994).

 

 

Table 2: Intercept and slope of the frequency shift expressed as a linear function of different solar indices.

Solar		Intercept a	Slope b
Index		(nHz)	(nHz per activity unit*)
$R_{\rm I}$	$\Delta\nu$	18.25 $\pm$ 14.77	2.56 $\pm$ 0.18
	$\Delta\nu_{0,2}$	-48.08 $\pm$ 25.94	1.98 $\pm$ 0.32
	$\Delta\nu_{1,3}$	60.62 $\pm$ 21.51	2.91 $\pm$ 0.26
F10	$\Delta\nu$	-139.07 $\pm$ 24.17	2.69 $\pm$ 0.19
	$\Delta\nu_{0,2}$	-173.73 $\pm$ 41.78	2.11 $\pm$ 0.32
	$\Delta\nu_{1,3}$	-115.88 $\pm$ 36.40	3.03 $\pm$ 0.28
KPMI	$\Delta\nu$	-98.43 $\pm$ 27.42	22.92 $\pm$ 2.04
	$\Delta\nu_{0,2}$	-154.75 $\pm$ 37.20	19.08 $\pm$ 2.78
	$\Delta\nu_{1,3}$	-63.15 $\pm$ 40.42	25.34 $\pm$ 3.02
MPSI	$\Delta\nu$	33.79 $\pm$ 14.04	149.06 $\pm$ 10.46
	$\Delta\nu_{0,2}$	-39.06 $\pm$ 21.10	124.22 $\pm$ 15.59
	$\Delta\nu_{1,3}$	79.81 $\pm$ 22.19	164.63 $\pm$ 16.56
TSI	$\Delta\nu$	-528.06 $\pm$ 51.63	386.71 $\pm$ 37.80
	$\Delta\nu_{0,2}$	-437.13 $\pm$ 65.43	320.07 $\pm$ 47.90
	$\Delta\nu_{1,3}$	-585.31 $\pm$ 71.58	428.67 $\pm$ 52.40
He	$\Delta\nu$	-438.86 $\pm$ 42.45	10.92 $\pm$ 0.73
	$\Delta\nu_{0,2}$	-417.59 $\pm$ 73.56	8.87 $\pm$ 1.26
	$\Delta\nu_{1,3}$	-447.34 $\pm$ 67.87	12.21 $\pm$ 1.17

^*  Units: nHz; nHz/(10^-22 J/s/m²/Hz), nHz G^-1, nHz G^-1, nHz W^-1m², nHz mÅ^-1 respectively.

Since the time-series created are one year long, the time variation of the integrated frequency shift analyzed here informs only on long-term changes. A recent analysis covering different time scales from one to seven months using 9 years of BiSON data can be found in Chaplin et al. (2001). Average values of the following solar activity indices have been computed over the same one year periods than the frequency shifts in order to obtain the corresponding correlation coefficients:

• the International Sunspot Number, $R_{\rm I}$;
• the integrated radio flux at 10.7 cm, F₁₀;
  (both obtained from the Solar Geophysical Data);
• the Kitt Peak magnetic index (KPMI) extracted from the Kitt Peak full disk magnetograms (Harvey 1984);
• the Mount Wilson Magnetic Plage Strength Index (Ulrich 1991), MPSI;
• the Total Solar Irradiance, TSI (Fröhlich & Lean 1998) and;
• the equivalent width of HeI 10830 Å averaged over the whole solar disk using data from Kitt Peak observatory.

The Pearson correlation coefficient $r_{\rm P}$, which is a measure of the strength of the linear relationship between two indices, and the Spearman rank correlation coefficient, $r_{\rm S}$ which provides a measure of the correlation between the ranks of two indices during the chosen period, are shown in Table 1 for the frequency shift. The correlation between the different solar activity indices themselves have been investigated in details by Bachmann & White (1994).

In addition, the probability $P_{\rm s}$ of having null correlation between the ranks of any of the solar indices and the frequency shift is indicated. We note that the database of MPSI and TSI indices, covers only the first 28 time-series. The correlation analysis for these two indices was therefore made with two fewer points than for the others.

The integrated signal shows very high correlation with the various solar indices, whereas the frequency shift corresponding to $\ell = 1$, 3 and, more significantly, $\ell = 0$, 2 are slightly lower. The general trend of the frequency shift corresponding to the even and odd degree modes separately is quite clear but, in addition to the difference in amplitude already mentioned, the frequency shifts measured for the even modes seem to be sensitive to the solar cycle later than the odd ones and the resulting difference in phase between the two curves is probably at the origin of the lower correlation coefficients found for $\ell = 0$, 2. As demonstrated by Moreno-Insertis & Solanki (2000) different modes would respond differently at different phases of the cycle depending on the positions of the activity (i.e. sunspots) on the disk. Aside of the long term differences, there are also fluctuations at shorter time scales which are different for the two data sets.

Because of the excellent linear correlation coefficients found, the frequency shifts were fitted as a linear function of the different solar indices I by:

$$\Delta\nu^i=a + b \cdot I^i.$$ (4)

In Table 2 we report the intercepts and the slopes obtained for all the solar indices considered, and an example is given for the radio flux in the lower part of Fig. 2. The slopes can be compared with the results shown in Table 1 of Régulo et al. (1994) also obtained for low-degree p-modes. The three solar indices in common to both works present similar slopes; the differences are less than 3 times our error bars. Notice that, more or less, all activity indices used here produce similar values of the correlation coefficient with the frequency shifts leading to the conclusion than none is much better, for this purpose, than others.

The numbers also agree with those obtained recently by Jain et al. (2000) for intermediate-degrees ($\ell = 20$-100) using GONG data. However, intermediate degree modes are confined closer to the surface and one may expect them to be more sensitive to the activity changes and therefore the slope to be larger for those modes than for low-degree modes. This is indeed what we found analyzing LOWL data (Jiménez-Reyes et al. 2001) but, as we shall see, this depends also strongly on the range of frequencies considered to calculate the average values.

The interpretation of the different behavior found for $\ell = 1$, 3 and $\ell = 0$, 2 is not straightforward. In order to understand better the underlying physics, one may instead look at the velocity power variations and at the frequency and $\ell$-dependences. This is considered in the following section.

Figure 3: Time variation of the TVP changes in percent compare to its minimum value (i.e. $100\cdot({\it TVP}^i-{\it TVP}_{\min})/{\it TVP}_{\min}$) for low-degree p-modes integrated between 2.5 and 3.7 mHz. In the sub-panel the TVP corresponding to $\ell = 1$, 3 and $\ell = 0$, 2 are also shown. The radio flux at 10.7 cm calculated for the same periods is plotted as a full line in the upper plot and its linear correlation with the frequency shift is shown in the lower plot.