1 Introduction

Understanding the observed solar variability is one of the major goals of solar physics. Because the frequency shifts of solar p-modes are known to be very sensitive to the solar activity cycle, the analysis of helioseismic data has been used to track those physical processes which underly the origin of the cyclic changes observed at the solar surface. Helioseismology based on low-degree p-modes is necessary to look for potential structure or dynamic changes in the deep interior.

The first report of frequency shifts of the low-degree p-modes was given by Woodard & Noyes (1985). Using ACRIM data, they found that the few observed $\ell = 0$ and 1 modes presented a change in the central frequency of $0.42 \pm 0.14~\mu$Hz in average during the declining phase of cycle 21 (1980-1984). These results were confirmed by Fossat et al. (1987) by comparison with observations made at the south pole, and by Pallé et al. (1989) using a long set of data from the Mark-I instrument at Observatorio del Teide covering the full cycle 21 (1977-1988).

Subsequently, Régulo et al. (1994) used Doppler observations collected from the maximum of cycle 21 to the falling phase of cycle 22 (1980-1993), obtained with Mark-I instrument, to calculate monthly frequency shifts. They showed that, for all low degree acoustic modes, there is an important frequency shift of $0.52 \pm 0.02~\mu$Hz correlated with solar activity. In addition, the amplitude of these variations is different when $\ell = 1$, 3 and $\ell = 0$, 2 are considered separately. The odd modes present, on average, a change of $0.58 \pm 0.06~\mu$Hz; the even ones show a full shift of only $0.33 \pm 0.06~\mu$Hz.

Two other important properties of the low-degree changes have also been pointed out recently. Anguera Gubau et al. (1992) observed the frequency dependence of the frequency shifts for the low-degree p-modes, (later confirmed by Chaplin et al. 1998) in agreement with the earlier results of Woodard et al. (1991) for intermediate degrees. Jiménez-Reyes et al. (1998) found that the frequency shifts, when plotted against an activity index, show a hysteresis behavior rather than a simple linear correlation. This result was interpreted as part of structural changes associated with the solar activity which are taking place in the Sun. This must be confirmed by more observations but the interpretation of these later results as being partly due to structural changes in the interior associated with the solar activity has been found to be a complex problem. Recently, Moreno-Insertis & Solanki (2000) have studied in detail the signature left on the low-degree p-mode frequencies by the surface solar magnetic activity. Whether these changes are taking place only close to the surface or not is not completely clear and one of the requirements to address this question is to get precise and reliable measures of the low degree mode parameters for a long period of time.

The spectrophotomer Mark-I, has been collecting solar observations for almost two complete solar cycles. The available database for low-degree p-modes, probably the longest in duration and the most stable, is used in the present work to analyze the signature of the solar cycle in the mode parameters and to parameterize the observed frequency shifts as a function of various classical solar indices.

In the following section, the essential steps of the data reduction leading to the yearly spectra are presented. In general, the frequency shift between a time t_i and a time $t_{\rm o}$ taken as reference, can be written as a function of the frequency and the degree, i.e. $\delta\nu(t-t_{\rm o},\nu,\ell)$. In the following the reference time is 1986 which corresponds to a minimum of solar activity. In Sect. 3, the integrated frequency shift ${\Delta\nu}^i \equiv {<}\delta\nu(t_i-t_{\rm o},\nu,\ell){>}$is analyzed where the brackett indicates an average for all observed low degrees and frequencies between 2.5 and 3.7 mHz. Two techniques are proposed to measure the frequency shift from the cross-correlation function between power spectra of time-series created at different solar activity level. In addition to the frequency shift, the second technique allows us to study the time variation of the total velocity power (TVP) which are presented in Sect. 4. Then, in Sect. 5, we focus on the study of the frequency dependence of the frequency shifts. Again, two different techniques are used. The first one consist in simply cutting the spectra in band of 135 $\mu$Hz before computing the cross-correlation functions. The second one is a new procedure developed here and called simultaneous fitting: all the yearly spectrum are fitted at the same time assuming that the time dependence of the mode frequencies can be described as a linear function of the radio flux at 10.7 cm F₁₀ⁱ, taken as solar activity index i.e.:

$$\delta\nu(t_i-t_{\rm o},\nu,\ell)=\delta\nu(\nu,\ell)(F_{10}^i-F_{10}^{\rm o}),$$ (1)

where $F_{10}^{\rm o}$ represents the radio flux at the 1986 solar minimum. In addition, we define $\delta\nu(\nu)$ as the frequency shift per radio flux unit averaged over $\ell$. We note that if we had used shorter time-series and a magnetic index instead of the radio flux to parameterize the time dependence of the frequency shift, a more complicated formulation would probably have been needed in order to take into account the hysteresis behaviour found when magnetic indices are plotted versus the frequency shift during the cycle (Jiménez-Reyes et al. 1998).

In order to check the $\ell$-dependence of the frequency shift, we use the fact that pairs of low-degree p-modes with the same parity have "almost'' the same frequency and, that they are equally spaced in frequency. This allows us to provide in Sects. 3, 4, 5 not only the $\ell$-averaged quantities defined above ($\Delta\nu$, TVP, $\delta\nu(\nu)$) but also the quantities related to even and odd modes separately i.e. respectively: $\Delta\nu_{0,2}$, TVP_0,2, $\delta\nu(\nu)_{0,2}$ and $\Delta\nu_{1,3}$, TVP_1,3, $\delta\nu(\nu)_{1,3}$. Finally, in Sect. 6 the $\ell$-dependence analysis of the frequency shift is completed by comparing the results with those obtained at higher degrees ( $\ell = 1,99$) using the LOWL database.