5 Frequency dependence of the frequency shifts

In the previous sections, the variations of the frequency-integrated velocity power and frequency shifts have been studied. However, the time variation of the frequency shift is expected to be different at different frequencies. Previous works (e.g. Libbrecht & Woodard 1990; Anguera Gubau et al. 1992; Chaplin et al. 1998) have shown that the modes at high frequency become more sensitive to the solar cycle. The quality of our time-series and a new method to fit all the spectra at once motivates us to determine the frequency-dependence of the frequency shift for low degrees.

The first method used is related again to the fact that pairs of low-degree acoustic modes are equally spaced in the spectrum. Each spectra is divided in regions of 135 $\mu$Hz containing a set of modes $\ell = 0$, 1, 2 and 3. Then, every region is cross-correlated with the corresponding region of the reference spectrum (corresponding to the solar activity minimum of 1986), and the method explained above (Sect. 3) is used to calculate the frequency shift. Finally, the frequency shift of each region is fitted as a linear function of the integrated radio flux at 10.7 cm.

In the second method proposed here, we try to fit together all the spectra at once. We have established that the central frequencies of the solar acoustic modes vary during the solar cycle and there is a strong linear correlation with any of the solar indices. So, in order to improve the statistics we can fit all the spectra together introducing the frequency shift as a new parameter. Here, as well as in the first method, the radio flux was chosen because, according to Table 1, it leads to the best linear correlation coefficients for the central frequency variations but, as quoted before, this choice is not crucial as all indices present similar correlations. We emphasize that, while the first method is faster, the second provides us not only individual frequency shifts but also valuable mode parameters, i.e. mode resonant frequencies corrected for the solar-cycle effects.

As the structure of the power spectrum is complicated by the presence of the $11.57~\mu$Hz sidebands, modes close in frequency must be fitted simultaneously in order to maintain the stability in the fits. Therefore adjacent $(n,\ell)$ and $(n - 1,\ell + 2)$ peaks are fitted together. The multiplet structure induced by the rotation and the temporal sidebands for each mode of the pair are also included in the model. If we label by p the 60 $\mu$Hz wide part of the spectrum including the adjacent $(n,\ell)$ and $(n - 1,\ell + 2)$ peaks and $\nu_p$ the mean frequency of the pair, the model for this part of the spectrum of each time-series i can be expressed as:

 

$$M_p^i(\nu,\vec{a})= \sum_{k=-1}^{1} \beta_{\vert k\vert} \left[\s... ...mma_{p}/2)^2}{ (\nu-\nu_{(n-1)(\ell+2)m}^{ik})^2 +(\Gamma_p/2)^2} \right]+ B_p,$$ (5)

with:

$$\nu_{n\ell m}^{ik} = \nu_{n\ell} + \delta\nu(\nu_{p})_{\ell,... ...(F^i_{10} - F_{10}^{\rm o}) + m \cdot s_{n\ell} + k_D \cdot k,$$ (6)

where:

• $\nu_{n \ell}$, is the central resonance frequency of the mode (n,l) at solar minimum;
• $\delta\nu(\nu_p)_{\ell,\ell+2}$, is the searched frequency shift per solar index unit, assumed to be the same for all the components of the pair;
• $\nu_p=(\nu_{n\ell}+\nu_{n-1,\ell+2})/2$ is the center of the $60~\mu$Hz wide interval of the spectrum fitted;
• $s_{n \ell}$, is the synodic rotational splitting for a multiplet taken to be constant for all of them and equal to 400 nHz;
• Fⁱ₁₀, is the averaged radio flux for the given series ( $F_{10}^{\rm o}$ being the value at the solar cycle minimum);
• B_p, is a constant background level at the fitted frequency interval;
• $A_{n \ell}$, is the power at the resonance for sectoral components $m = \ell$
• $\alpha_{\mid m \mid}^\ell$, is a given ratio $A_{n \ell m}$/ $A_{n \ell}$ constrained to take the theoretical value, for an instrument without spatial resolution, calculated by Christensen-Dalsgaard (1989) for the special case of resonant scattering observation using potassium line;
• $\Gamma_p$, is the linewidth of the components of the multiplet assumed to be the same for all components of the two multiplets; and
• $\beta_{\vert k\vert}$, is the ratio of the power of the sidebands (|k|=1) to the central peak (|k|=0). These are directly related to the duty cycle and can be estimated empirically. The duty cycles of the yearly time-series remaining close to their average value of $25\%$, this allows us to fix the amplitude of the sidebands to $\beta_{\vert 1\vert} = 0.5$ for $\beta_{0} = 1$.

We assume that each m-component is well represented by a symmetric Lorentzian profile (even though Thiery et al. 2000 have found a slight asymmetry on them) and that the individual m-components are independent (see Foglizzo et al. 1998). B_p, $A_{n \ell}$, $\Gamma_p$, $\nu_{n \ell}$ and $\delta\nu_{n \ell}$ are the parameters to be fitted (vector $\vec{a}$, hereafter). Notice that, in Sect. 4, it has been demonstrated that the TVP depends on the solar activity. This indicates that time variations may also occur for amplitudes and linewidths. Such variations have indeed been found recently from BiSON (Chaplin et al. 2000) for low degree and GONG (Komm et al. 2000) for higher degrees, both showing an increase of the linewidths and a decrease of the amplitudes leading to decrease of the TVP in agreement with our result. We therefore tried to parameterize amplitudes and linewidths as a function of time in the way used for frequencies, but the fits turned out to be very difficult and in many cases did not converge.

At higher frequencies ($\nu>3500$ $\mu$Hz) the peaks get wider, the width being greater than their rotational splittings ( $\Gamma_p \gg s_{n \ell}$). Moreover, the linewidths get so large that they become comparable to, or bigger than, the small frequency separations between the pair $(\ell,n)$, $(\ell + 2,n - 1)$. Therefore the fit is made at frequency intervals (labeled by q hereafter) of 165 $\mu$Hz centered at $\nu_q$ and containing one pair of even modes (n,0)(n - 1,2) labeled hereafter by p=1 and one pair of odd modes (n,1)(n - 1,3) labeled hereafter by p=2. Only one Lorentzian is fitted for even and another one for odd modes. Thus the model becomes:

$$M_q^i(\nu,\vec{a})= \sum_{p=1}^{2} \frac{ A_{pq} (\Gamma_{pq}/2)^2}{(\nu-\nu_{pq}^i)^2+ (\Gamma_{pq}/2)^2} +B_q,$$ (7)

with:

$$\nu_{pq}^i = \nu_{pq} + \delta\nu(\nu_q) \cdot (F^i_{10}-F_{10}^o).$$ (8)

Here the parameters $\vec{a}$ to be fitted are: B_q, A_pq, $\Gamma_{pq}$, $\nu_{pq}$ and $\delta\nu(\nu_q)$.

As Woodard (1984) points out, in the case of observations without spatial resolution, the power spectrum of the solar p-mode oscillations is distributed around the mean Lorentzian profiles with a $\chi^{2}$ probability distribution with two degrees of freedom. Consequently, the power spectrum appears as an erratic function where an abundance of frequency fine structure can be found. For this type of statistics the joint probability function associated to the observed power spectrum $\vec{X}^i=\{X^i(\nu_j)\}_{j=1,N}$ corresponding to the time-series i is given by:

$$f^i(\vec{X}^i) = \prod_{j=1}^N \frac{1}{M^i(\nu_{j},\vec{a})} \exp\left[-\frac{X^i(\nu_{j})}{M^i(\nu_{j},\vec{a})}\right],$$ (9)

where N is the number of frequency bins in the interval considered (i.e. the 60 $\mu$Hz wide interval for $\nu<3.5$ mHz or the 165 $\mu$Hz wide interval for higher frequencies), and Mⁱ is given either by Eq. (5) or Eq. (7) depending also on the frequency domain.

Figure 5: The slope of the frequency shift in its assumed linear dependence with the radio flux is plotted here as a function of the frequency for low-degree p-modes. The results for all modes (triangles) are obtained using the first method described in the text while the separate fits for odd and even degrees are obtained using a simultaneous fit (Eqs. (5), (6)). The solid line is the best fit of the inverse mode mass calculated for the same set of modes. The bottom figure shows the results obtained at high frequency using Eq. (7). The estimation of the frequency shifts corresponding to low degree using the first method as well as the best fit of the inverse mode mass have been again plotted in the bottom figure as reference. RF stands for radio flux units, namely 10-22 J/s/m2/Hz.

The likelihood for the 30 observed spectra is then given by the product of the individual likelihood function for each year. Thus, it can be written as:

$$L(\vec{a})= \prod_{i=1}^{30} {f^i(\vec{X}^i)}.$$ (10)

We then look for the vector $\vec{a}$ that will maximize the likelihood of the observed spectra according to our model. For numerical reasons one minimizes S, defined as the negative logarithm of the likelihood function,

$$S(\vec{a}) = \sum_{i=1}^{30} \sum_{j=1}^{N} \left[{\ln(M^i(... ...,\vec{a}))+ \frac{X^i(\nu_j)}{M^i(\nu_j,\vec{a})}}\right]\cdot$$ (11)

To minimize this expression, we have used a modified Newton method (Press et al. 1992). The initial guesses for the parameters are important to avoid local minima. In that aspect, the frequency shift appears to be the more sensitive parameter and the initial guess was taken from the results shown Table 2 independently of the frequency range fitted. In the case of some parameters (amplitude, noise and linewidth), the natural logarithm of those have been fitted and not the parameters themselves. Doing this, S follows a normal distribution near the minimum and the covariance matrix for the vector $\vec{a}$ can be approximated by the inverse of the Hessian matrix found at the minimum of S. The uncertainties on each fitted parameter are therefore taken as the square roots of the diagonal elements of the inverted Hessian matrix.

The results are summarized in Fig. 5. In the top figure, the triangles represent the frequency shift integrated in bands of 135 $\mu$Hz using the first method. The frequency dependence is quite clear, the frequency shift being close to zero at 2 mHz and then increasing progressively with the frequency to reach approximately 2 nHz/RF (where RF stands for radio flux units, namely 10^-22 J/s/m²/Hz) at the center of the p-mode and a maximum of 4 nHz/RF around 3.6 mHz, where it drops fast. The solid line denotes the inverse mode mass extracted from the solar model of Morel et al. (1997). It has been averaged over the same regions in frequency and what we show represent the best fit to the data. The results corresponding to the simultaneous fits are also shown in the same figure. The frequency shift for the even modes are represented by crosses while black circles represent the odd ones. Both confirm the previous results showing a similar frequency dependence. These results are also in agreement with the analysis carried out by Chaplin et al. (1998) on BiSON data for frequency below 3.6 mHz. The mean $<\delta\nu(\nu)>$ in the 2.5-3.7 mHz range can be estimated by integrating the best fit of the inverse mode mass to our results divided by the length of the frequency range i.e. 1.2 mHz. This leads to a value of 2.66 nHz/RF compatible with the slope b reported in Table 2 for the linear dependance between the integrated frequency shift and radio flux. A comparison of Eqs. (1) and (4) gives the straightforward correspondance between b and the averaged frequency shift per radio flux units. We note however that the two estimates are not equivalent due to the variation of the p-mode energy across the five minute band. The cross-correlation function weight more the peak with higher amplitude located around $\sim$2.9-3.0 mHz while the present integration gives the same weight to all the modes.

At high frequency, the first points corroborate the frequency shift obtained using the first method, where a sharp downturn is found. Then, at higher frequency, a large fluctuation appears. A similar feature was found by Anguera Gubau et al. (1992) and also found for intermediate degree by Libbrecht & Woodard (1990) in the analysis of BBSO data. These authors observed that the sensitivity of the mode frequency shifts show similar sharp downturn located around 3.8 mHz. Chaplin et al. (1998) also reported (in their Fig. 4) a sharp downturn with negative frequency shift of about -10 nHz/RF around 4.3 mHz but they did not find the first downturn found at 3.75 mHz as in our analysis. The better sampling of the Mark-I data set and the fact that this first downturn was obtained with the two different methods we used for low and high frequencies make us confident in this result; moreover, this result is also found by Anguera Gubau et al. (1992).
The frequency dependence of the frequency shift has been addressed from the theoretical point of view in different works. The model developed by Goldreich et al. (1991) suggests that a combination of an increase of the chromospheric temperature and a chromospheric resonance can be responsible for the sharp downturn at high frequency followed by an oscillation while the progressive increase in the five-minute band can be interpreted as an increase of the filling factor of the small scale photospheric magnetic fields. On the other hand, Jain & Roberts (1993, 1996) argue that the presence of a magnetic field in the chromosphere and a combination of temperature and magnetic field strength variations could, qualitatively, explain the observed frequency dependence of the frequency shift at both low and high frequencies. We notice that both models, those of Goldreich et al. (1991) and Jain & Roberts (1996) present a wavelike behavior at high frequency qualitatively similar to the one found in our analysis. The particular model of Goldreich et al. (1991) is even able to reproduce quantitatively this result, including the upturn around 4.5 mHz but no evidence has been found for the required chromospheric resonance (Woodard & Libbrecht 1991) and Kuhn (1998) point out that the change in the photospheric magnetic field strength needed in this model is much higher than the one obtained from recent infrared splitting observations of the quiet region field strengths or MDI magnetograms. Thus, Kuhn (1998) argues that photospheric magnetic fluctuations are unlikely to be responsible for the observed frequency shifts and proposes instead that turbulent pressure and mean solar atmosphere stratification variations resulting from entropy perturbations through the solar cycle may be the dominant process affecting p-mode frequencies. In this model, the associated temperature fluctuations may originate from near the base of the convection zone but in order to explore those possibilities and locate the different possible perturbations one needs to invert the even splitting coefficients (see Dziembowski et al. 2000) and track any fluctuation in the sound speed inversion using long term observations. We intend to extend our work in that direction using the observations of both low and intermediate degrees from LOWL and Mark-I.

Figure 6: The cross shows the normalized frequency shift for $\ell = 1$-99 integrated between 2.5 and 3.7 mHz using LOWL data, and the solid line denotes the best fit of the inverse mode mass for those modes. We have also plotted the results corresponding to very low degree using Mark-I data (Table 2): the dashed horizontal bar is for $\ell = 0$, 1, 2, 3; the triangle for $\ell = 1$, 3 is placed close to $\ell = 1$ because this component gives the main contribution; and the square for $\ell = 0$, 2 has been arbitrarily placed close to $\ell = 2$.