B. Gelly1 - M. Lazrek1,2,3 - G. Grec3 - A. Ayad 1,4 - F. X. Schmider1 - C. Renaud3 - D. Salabert1 - E. Fossat1
1 - Laboratoire d'Astrophysique, UMR 6525 du CNRS, Université de Nice-Sophia Antipolis, 06108 Nice Cedex 2, France
2 - C.N.R., 52 Charii Omar Ibn El Khattab, BP 8027, Rabat, Morocco
3 - Département Cassini, UMR 6529 du CNRS, Observatoire de la Côte d'Azur, 06304 Nice, France
4 - Laboratoire de Physique des Hautes Énergies et Astrophysique, Université Cadi Ayyad, Marrakech, Morocco
Received 26 March 2002 / Accepted 10 July 2002
With the GOLF instrument onboard the SoHO observatory, 1979 days of full-disc Doppler velocity observations have been compiled into a study of p-mode properties. We develop a multi-step iterative method (MSIM) algorithm to access all p-mode parameters while minimizing any perturbating effect or cross-talk between parameters during their determination. We present frequency and splitting tables, amplitudes, linewidths, line asymmetries, pseudo-modes, and background noise determinations. We have a first look at the changes induced by the transition from the low-activity to the high-activity part of solar cycle 23: we have recorded frequency shifts with a downturn at 3.7 mHz followed by a possible higher upturn, and linewidth changes to a good accuracy. We detect an effect on the noise background at 3 mHz possibly related to an interaction between noise and the modes and connected to the asymmetry of the profiles.
Key words: Sun: oscillations - Sun: helioseismology
The GOLF helioseismology instrument was launched onboard the SoHO spacecraft on December 2, 1995 (Gabriel et al. 1995). After a commissioning phase which ended on April 1, 1996, GOLF has been running continuously, except for an interruption due to the spacecraft loss on June 25, 1998 followed by its recovery on August 8, 1998, and a much shorter period in January 1999. Altogether, the duty cycle of GOLF is close to 95% of possibly the finest data available for the helioseismic investigation. A number of important topics have already been addressed using the GOLF data, including the validation of the solar models against sound-speed inversions (Turck-Chièze 2001; Brun et al. 1999), the solar neutrino production puzzle (Turck-Chièze et al. 2001), the solar internal rotation through splitting measurements (Bertello et al. 2000a), the effect of the asymmetry of the modes on the p-mode parameters, and the effect of such changes in the inversions (Thiery et al. 2000; Basu et al. 2000), and of course the search for g-modes (Gabriel et al. 2002; Grec & Renaud 2002). Most of the GOLF science so far is based upon p-mode studies, from the pioneering paper of Lazrek et al. (1997) to recent attempts in the detection of the very low frequency modes (Bertello et al. 2000b; García et al. 2001b). Solar activity effects have already been reported using GOLF data (Lazrek et al. 2001; Thiery 2000) on shorter time-series. The present work uses 1979 days of observations, and we are confident that some solar activity effects must be present, because the beginning of the mission was at the solar minimum of cycle 22 to 23, and that the current data corresponds to the maximum of cycle 23. This paper focuses on making a running analysis along the duration of this experiment to determine the p-mode parameters to the highest accuracy allowed by the quality and the duration of the selected data. We then check for changes in these parameters, and find out how much of those can be attributed to solar-cycle effects.
|Figure 1: GOLF data calibration overview: each point represents 18 days of data. a) and b) Granulation and supergranulation magnitude versus time. c) Sum of the spectral density of the calibrated signal in the p-mode range (2 to 4 mHz). d) The photon noise in the experiment with the clearly visible yearly modulation. All ordinates are m2 s-2 UpsymnmHz-1.|
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GOLF is an instrument dedicated to the monitoring of the global solar velocity field and has been operated onboard the spatial observatory SoHO (Domingo et al. 1995) since 1996. GOLF is a spectrophotometer using optical resonance of Na atoms. A permanent magnet separates the Zeeman components and allows the creation of 2 narrow bandpasses (0.02 Å) centered on the wings of the Fraunhofer D lines (Gabriel et al. 1995). The rotation of a polarizer is added to alternatively tune the 2 bandpasses.
The motor used to rotate the polarizer during the flight has shown poor reliability and was used only briefly during the P2 period (Table 1). The later observations were made in a back-up photometric mode in a single optical bandpass. During P3, GOLF was tuned on the blue wing of the Na lines. At the start of P4, GOLF was tuned on the red wing of the Na lines and is still working in this configuration. The observations used for this study cover the period from April 1996 to December 2001 (this is an arbitrary choice, more measurements being now available). There are 2 significant gaps in 1998 due to temporary loss of SoHO and a gyro failure.
The relation of the GOLF single-band signal to the average across the solar disc of the solar velocity field has been estimated for the p-modes (Pallé et al. 1999; Renaud et al. 1999). Other velocity components are present and are related to several sources of variation of the monochromatic intensity, to the instrumental noise and to the photon noise.
|begin||Feb. 18, 1996||Apr. 10, 1996||Oct. 15, 1998|
|end||Mar. 31, 1996||Jun. 24, 1998||Dec. 31, 2001|
|photometry||differential||blue wing||red wing|
The data acquisition involves 2 photometric chains, experiencing different performances and data coverage. In order to remain in a well defined condition for the signal to noise ratio, we use only the channel having higher efficiency, referred to as PM2. The conversion of the GOLF measurement to the variation of the disc-averaged solar velocity depends on a calibration procedure. A method is proposed in Ulrich et al. (2000). The basis are found in the instrumental laboratory calibrations, together with additional studies of the solar D lines. Nevertheless one of the purposes of our work is to study the dynamic properties of the solar atmosphere, which could be interdependent with the line profile used in this complex calculation. We decided not to follow this calibration procedure in order to avoid any model fitting and to check the dependency of the results on several critical steps of the data reduction. Our computation of the Doppler shift is made in 2 major steps, allowing the conversion of the raw photometric signal I(t) to a "velocity'' signal related to the disc-averaged surface velocity:
The first step is to normalize intensities to a reference photometric condition as the measurements include components related to the distance from the Sun, the radial component of the orbital velocity, and the decay of the measured light due to the aging of the optical components. SoHO moves around the Sun on a halo orbit circling the Lagrange point L1, that is a perturbated elliptic solar orbit. The parameters of this trajectory are computed to great accuracy. The solar intensity is first computed for a constant distance to the Sun. The instrumental drifts give also proportional photometric factors. A second order modulation is then detected at low frequency and can originate either from changes of the magnetic pattern or from changes in the average over the solar disc of the velocity field. This modulation is clearly related to the solar rotation (Grec et al. 1999). The effects of such a modulation on the p-mode signal are proportional only if, due to a change in the magnetic pattern, there is a change of the global monochromatic solar flux: we consider this as the most likely situation to happen. Both effects can be removed simultaneously from the signal: by using a low-pass filter with a cutoff at 23 UpsymnmHz, we can extract the photometric "carrier'' <I(t)>.
The second step is to normalize to a reference sensitivity for the small velocity changes. The seasonal changes of the orbital velocity give an additive component of <I(t)>, so far a perturbating term in the photometric analysis. i is the intensity collected at the solar wavelength. As depends on the orbital Doppler shift, the line slope is related to the solar lines and varies accordingly to the seasonal change in the orbital velocity V(t). This results in a variable sensitivity . Those 2 effects can be regarded as a linear departure from linearity and the slope is simply , where is a constant number.
The observational function
for the p-mode study is then given
in Eq. (1), in which the constant factors
depend on the optical bandpass selected.
We describe in Sect. 3.1 the method used to study the p-mode properties. As a by-product, we have computed 4 basic parameters from 18-day independent time-series. The key parameter of the calibration method is the result obtained for the integrated value of the power spectrum in the 5 min spectral range. The coefficients and are adjusted to the following conditions:
Our model of the spectral profiles takes into account all the identified components in the low degree solar velocity spectrum:
|Figure 2: Visual flowchart of the current MSIM procedure.|
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Doing so, we aim to account for every spectral feature of the power spectrum, and such that the residuals, if any, could be tracked to either some un-modeled physical phenomenon, or to some numerical problem. The flowchart of our fitting procedure is summarized in Fig. 2. It is an iterative algorithm that never attempts to solve all the unknown parameters at once but find them step by step and gradually includes the fresh estimates in the target parameters vector. For example we start by the estimation of all the central frequencies, while amplitudes linewidths and splittings remain set to their input table values; then, using these "fresh'' frequencies kept constant until the next iteration, we estimate the amplitudes and the linewidths. Finally, setting these 3 parameters, we estimate the splitting as a single number for a given multiplet. This process is iterated 4 times at most. It must start initialized to some realistic p-mode parameter table, and stops whenever convergence in the parameters is achieved but usually 4 iterations are sufficient. The convexity of function to minimize is quite different for parameters such as the frequency, where it is rather simple, and the splitting where it is usually very tricky (Fierry-Fraillon 1999). Thus, we think that this iterative approach is efficient in avoiding the cross-talk arising from the simultaneous fit of parameters. At each stage, the minimization of the cost function is done using a maximum likelihood criterion based on a spectral probability density function (PDF). Techniques consisting of fitting the entire spectrum at once (modes and background, "WSF'') have already been successfully implemented (Roca Cortés et al. 1998). This requires the minimization of a vector of at least 250 parameters just for mode amplitudes, frequencies and linewidths, possibly much more if one wants to go into asymmetry and splitting. It was tested successfully on small time-series but would be extremely time-consuming in our case. Our procedure, to be referred as Multi-Step Iterative Method (MSIM) is distinct from the so-called WSF method because:
Finally, we have been making this analysis at 3 typical time scales using a running window:
The solar noise model we used is derived from the classical "Harvey
model'', except that instead of a 4-component model of the solar
noise which includes noise from the active regions, the supergranulation,
the mesogranulation and the granulation, we use only the two most
relevant contributions, which, in our case, are the granulation (in
the 600-1200 UpsymnmHz range) and the supergranulation (in the 20-600 UpsymnmHz range). Active regions are at very low frequency and
their effects are filtered out in the calibration process (Eq. (1))
so we do not consider them. We also found that using a mesogranulation
component does not improve the result of the model for our purpose.
Finally our noise model is:
|Figure 3: Global background noise fit, using 2 components for the solar noise and a constant level for the photon noise (not represented here). The top curve is the sum of the 3 components and is the reference background level used in the p-mode fit. The extra noise component discussed in Sect. 3.3 is visible as the superimposed "bump'' peaking at 3000 UpsymnmHz. The bump curve here is a model of the average of the determination of the component over all our subseries.|
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Concerning the estimation of photon noise level, the 40 s sampling time used in this analysis gives a cutoff frequency of 12.5 mHz. Although the solar acoustic cutoff is at about 5.4 mHz, we know that the range 6 to 10 mHz does contain so-called HIPs (or pseudo-modes, García et al. 1998), and some p-mode wings, so that the photon noise cannot be safely estimated there. Finally, our photon noise is computed as a constant level in the 10 to 12.5 mHz range.
In determining p-mode parameters using our former model for the background noise, we noticed that an extra component was needed to ensure a correct determination of amplitudes and linewidths. This extra component has a specific frequency behavior which is shown in Fig. 3. An error in the background determination results in a truncation in the mode amplitude which is also fairly well reflected in the linewidth. It is the checking of the coupling between the amplitudes and the linewidths that allows one to find and address this problem. This effect is easier to track on the linewidths, which are very well defined, and without this new noise component, they can be changed by as much as 15% in the 3 mHz range. Changes in this quantity between the P3 and P4 periods are being investigated, but we currently have no answer to this point. Our model of the velocity background noise in the spectra relies on the hypothesis of independence of the sources of noise we have listed, e.g. granulation, supergranulation, and photon noise level, with the oscillation. This extra noise component may come from the weakness of this hypothesis: if there were some correlation between the modes and the solar noise, this would possibly explain the non-random spectral shape of this new component.
|Figure 4: a) -- is the average relative frequency difference for an between the known value and the output of the fit. b) is the average relative linewidth difference for the same modes. - - - - is the standard deviation of the value (frequency difference or linewidth difference) computed over 30 realizations, and - - - - is the average of the error-bars we estimate in the MSIM process for these values.|
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For the p-mode parameter extraction, we have been testing two options, either assuming a pure Lorentz profile for each p-mode component, or assuming asymmetrical profiles, (Nigam & Kosovichev 1999a) better representing the physics of the excitation of the modes. Neither of these formulae did actually suit our needs:
Also, note that given the difficulty in determining properly the value of B for frequencies higher than 3.7 mHz, all modes profiles were taken symmetric above this value. This problem is similar to the determination of the splitting: the increase in the linewidths and the decrease in the distance between modes of even or odd greatly bias the result and decrease its confidence, so that at that point we prefer to return to a simpler scheme.
We fit the pseudo-modes in the 5800 to 7000 UpsymnmHz frequency range
using the following 6-parameter model:
We have checked for the reliability of our parameter extraction technique by synthesizing artificial oscillation spectra and feeding them into the extraction routines.
Artificial spectra are created using a known p-mode parameters table, and known solar and photon noise characteristics which allow us to generate a complete power spectrum from 20 UpsymnmHz up to the sampling cutoff frequency. The randomness of the excitation is introduced using a normal noise distribution pattern in the complex Fourier spectrum as in Fierry Fraillon et al. (1998). The resulting square modulus (spectral density) is then used as the input of the peak-finding routines. Given the duration of the extraction process, we had to limit the number of realizations providing the statistics of the results to 30 spectra. There are two underlying physical assumptions to this simulation process:
We have reasons to believe the second assumption poorly justified, because of the fact that modes are asymmetric, and that the explanation of this asymmetry suggests that some noise has to be correlated with the mode to explain the reversal of the sign of the asymmetry parameter between velocity and intensity measurements (Nigam et al. 1998; Nigam & Kosovichev 1999b; Severino et al. 2001).
However, our goal here was to test the MSIM algorithm for its accuracy, and our artificial spectra, representing only some of the best known properties of the signal, provide enough features to perform this test.
|Figure 5: Experimental splitting restitution. -- is the average splitting difference between the known value and the output of the fit, for an mode. - - - - is the standard deviation of this value computed over 30 realizations, and - - - - is the average of the error-bars we estimate in the program.|
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Figure 4a shows the average frequency difference between 30 realizations of an artificial spectrum computed as in Sect. 4.1 and differing only by their random noise. Results are divided by the input value, so that we actually plot the average relative difference. The retrieval of the input frequency is quite good: the largest errors are of about 10-4 at 4300 UpsymnmHz which means about 0.5 UpsymnmHz at worst, and the magnitude of the error is always less than the size of the corresponding error-bars. Above 4400 Hz the combination of the increase in the linewidths and the decrease in the separation of the modes degrades seriously the quality of the frequency restitution, although it is important to note that there is no obvious bias, even above that point. Figure 4b is the same for the linewidths: there is a hint of a bias at the lowest and highest frequencies, which means that the fit would slightly underestimate the linewidths at those points, but for most of the range the linewidths are correct and the error is inside the error-bars.
Finally, Fig. 5 represents our capability to correctly fit the splitting of an mode with realistic input values, taken from previous determinations. We never miss the splitting value by more than , and again our error-bars are coherent with this error. We note that 3000 Hz appears as a quality boundary for the splitting determinations.
Concerning the error-bars, we have plotted in Figs. 4 and 5 two curves with identical meaning: one is the average of the error-bars given by the fitting program, corresponding to the inversion of the numerical Hessian of the system, and the other is the standard deviation for our 30 realizations (i.e. ). The similarity between the two curves makes us confident in the robustness of our fits.
Another key point is that we did not introduce the extra noise component of Sect. 3.3 in our simulation. However, in forcing the software to fit, we never succeed in getting anything other than random amplitudes at a very low level. This is an a contrario evidence that our result on real spectra was not a spurious effect created by some hidden numerical error.
This work on artificial spectra has proven that the process of automatically extracting the parameters with error-bars from the power spectra has reached a high level of reliability, provided that our hypothesis on the nature of all the contributions to the signal are valid.
Tables A.1-A.3 are p-mode frequencies computed as described in Sect. 3.4 on 16-month time-series. Following our variability study (see below in the same section), the frequencies below 2500 UpsymnmHz seem little affected by solar cycle effects, so in Table A.1 we have averaged these frequencies over the P3 and P4 periods in order to increase their precision. For UpsymnmHz, Table A.2 is the weighted average of our determinations over 16 months (with a 12-month overlap) over the P3 period ("blue wing'' period). Table A.3 is the same determination over the P4 period ("red wing'' period). Above 4.4 mHz, we have tagged the frequencies for and because the increase in their linewidths combined with the decrease in the separation of these modes affects very much their determination, as previously mentioned in Sect. 4.2. The situation is quite different for the group of odd peaks, because although they are also indistinguishable, their difference in amplitude suggests that we are measuring mostly energy from the .
|Figure 6: Frequency changes vs. frequency for 4 low-degree modes; -- : , - - : , - - : , - - :|
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|Figure 7: a) Linewidths vs. frequencies, determined over 16-month time-series. represents all the individual determinations over all the available data. - - is the 16-month average of the P3 period determination, and - - is the 16-month average for the P4 period. b) Linewidth changes vs. frequency. - - : and 3 (simultaneously), -- : 0 and 2 (simultaneously).|
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Each frequency value
in Tables A.1, A.2,
or A.3 is the weighted average of n measurements
We have computed the frequency differences between P4 and P3 periods. Figure 6 shows the frequency changes versus frequency. There is an obvious effect that can be separated in several features:
Figure 7a shows our determination of the p-mode linewidths on 16-month sections, using asymmetrical profiles and forcing and . We emphasize that we have checked that using asymmetrical profiles (Nigam & Kosovichev 1998) as described in Sect. 3.4 instead of simple Lorentz profile helps very much in increasing the coherence of the result, as was previously shown by Thiery et al. (2000). The general shape of the linewidths is in good agreement with former determinations like Chaplin et al. (1997) or Fierry Fraillon et al. (1998). An interesting feature of Fig. 7a is the clear change in the linewidth slopes at 3.9 mHz, both for and . It is not a fitting artifact due, for instance, to the separation being of the same magnitude than the linewidths, otherwise the measurement would not display the same trend. In addition, such a bias, if present, would have been detected in Sect. 4.1. This leads us to think that whatever damping effect is producing the "plateau'' level is still efficient up to 3.9 mHz, where possibly there is a return to another damping rate.
Also, one can already see that the linewidths determined on the P4 period are slightly higher than those determined on P3. There is more p-mode damping at high-activity than at low activity, or the coherence-time of the oscillation is shortened. Figure 7b shows the relative amount of change for the linewidths. They do change by 20% at most at about 3500 UpsymnmHz. Linewidths at higher frequencies do not appear to change significantly in time. Although not obvious, this plot also supports the possibility of some increase in the coherence time of the oscillations below 2000 UpsymnmHz.
Figure 8 compares the amplitude determinations of modes , computed over 4-month time-series, to the noise levels at the same frequencies. All modes show a quite broad dispersion in their SNR, up to a factor 3, even over 4-month spans. The thickness of the noise baseline reflects as much changes in the experiment as changes in the solar noise of various origins, but mostly modulation due to the excursion in the line wings, and solar activity. As to the modes and they appear to have roughly the same amplitudes.
|Figure 8: Amplitudes and noise background in 1979 days of GOLF data for low-degree modes. From top to bottom: and , almost superimposed, then , and then Finally the bottom line is the background noise (the extra noise component of Sect. 3.3 is not plotted here).|
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|Figure 9: Ratio of the low resolution power spectra for the "red'' (P4) and the "blue'' (P3) periods. The "blue wing'' signal clearly dominates the "red wing'' one (in log scale).|
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The visible subdivision of the amplitudes in two domains for the
high-frequency end of the curves comes mostly from the switching
of the experiment from the blue to the red slope of the Na lines (from
P3 to P4 period). This effect is mainly present for
and is fainter, although real, on .
To check this
feature, we have computed the ratio
|Figure 10: Amplitudes change between P3 and P4 versus frequency for 3 low-degree modes; -- :, - - : , - - : .|
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Finally, we computed the same kind of ratio as on the P2 period when the experiment was still potentially a differential one. Instead of calibrating a differential velocity, we calibrated separately both the blue and red wings of P2 and we computed . The result is very similar to Fig. 9, indicating that a large majority if not all of the effect seen in this figure is not due to solar activity changes, and probably not due either to experimental changes having occurred between P3 and P4. Its explanation may just be the selective response of the solar atmosphere to the oscillations at different wavelengths inside the same absorption line, giving different levels in the photosphere. For the same reason, the weighting functions of the integration across the solar disc differs and that may be responsible for some of this effect.
Figure 10 shows the amplitude differences computed on 16-month spectra averaged over P3 and over P4. Of course it looks very much like Fig. 9, particularly in its frequency signature, except that the information is coming only from p-mode peaks in high-resolution spectra. No solar cycle information in p-mode amplitudes can usefully come from comparing P3 and P4 in the GOLF data, so far.
|Figure 11: : individual determinations of the B parameter on 16-month time series. - - : Average of the 16 month asymmetries for the P3 period. - - : Average of the 16 month asymmetries for the P4 period.|
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Figure 11 is our determination of the asymmetry parameter B defined in Eq. (3). This figure has a few characteristic features like:
|Figure 12: Per-mode splitting determination plotted against which is comparable to the lower turning point of the modes (only the sound speed value is missing in the numerator): a) GOLF splittings from Bertello et al. (2000a), b) BiSON splittings from Chaplin et al. (2001b), c) our determination.|
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|BiSON (weighted)||Bertello et al. (weighted)||This work (weighted)|
Splittings of the low-order p modes are important because they are the only piece of information we have on the current rotation rate of the solar core, addressing a fundamental problem of angular momentum conservation in this star. Figure 12 shows our determination of the individual rotational splittings for . In our fitting strategy, splittings are computed separately after all other parameters have passed satisfactorily each round of estimation. This ensures a minimum of cross-talk between the splitting computation and the frequency computation. Each point in Fig. 12 is the average of all the individual determinations over 16 months. The error-bars that are shown are compiled from the individual error-bars as in Sect. 5.1. We limit the frequencies where we give the splitting to UpsymnmHz for and 2, and to UpsymnmHz for . Above those limits the error-bars increase dramatically because of the increase in the linewidths which become of the same magnitude as twice the splitting separation, hence making these data irrelevant for core rotation studies. It is important to stress that the higher the frequency of a given the deeper its internal turning point. The deepest "samples'' of the integrated rotation are then provided by the highest frequency splittings of the mode which, unfortunately, are the least accurate that we can provide.
Nevertheless, if we compare Fig. 12 to Fig. 5 in Lazrek et al. (1997), which was also made from GOLF data, we can tell that a lot of improvements come from the much longer available time-series, and that the error-bars have decreased quite significantly.
Comparing this study to the equivalent work of Chaplin et al. (2001b) on BiSON data, we can tell that there is an impressively good agreement between those two independent splitting determinations: only three values (of 69) formally disagree. We can also compare our determination with the work of Bertello et al. (2000a) on GOLF data. The agreement is good, although not as good as for BiSON, some mid-range and modes having a splitting higher than 460 nHz, and that impacts also the averaged values found in Table 3. The agreement is the best at low frequencies for .
Finally, we have looked for splitting changes between the P3 and P4 periods and found none significant, to our current level of accuracy.
Traveling waves above the solar photospheric cutoff frequency have been known to exist as early as the late seventies, but the existence of a regularly spaced peak structure above 5.4 mHz in the p-mode spectrum was first observed by Libbrecht (1988). Kumar & Lu (1991) interpreted it as an interference pattern and proposed the name HIPs (High-frequency Interference Peaks). They showed that the value of the recurrence of the peaks is an indicator of the depth of the acoustic source within the photosphere. HIPs are a remarkable feature of the GOLF signal (García et al. 1998). We find them above the acoustic cutoff frequency, between 5.8 and 7.0 mHz. Figure 13 shows the results of the fit on overlapping 16-month sequences for the P3 and P4 periods, and following Sect. 3.5 specifications. The spectral recurrence of the pseudo-modes peaks (we prefer the term recurrence to the term period in this case) we have measured is (cf. Eq. (6)) UpsymnmHz. It matches quite well the value of Upsymnm Hz given by García et al. (1998) over the same data computed on a shorter time-series. Nevertheless, their amplitude (the parameter in Eq. (6)) was cut off by more than a factor 2 between P3 and P4, until they are barely distinguishable from noise. The coherence in makes us confident that, although weaker, HIPs are still present during P4.
|Figure 13: Pseudo-modes recurrence in frequency and amplitude for the P3 and P4 periods.|
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The frequency dependence of the frequency shifts below 4 mHz associated with solar activity has been known for a long time (Woodard & Noyes 1985; Gelly et al. 1988; Palle et al. 1989; Libbrecht & Woodard 1990; Howe et al. 1999). Its sign, and its regular increase in magnitude up to 4 mHz, can be very well fitted by an inverse mode mass model (Chaplin et al. 2001a; Jiménez-Reyes et al. 2001), and there is a general agreement on the interpretation of this effect as changes taking place in the superadiabatic layer at the top of the convection zone (Balmforth et al. 1996). Still, this explanation does not account properly for the changes that affect the Sun luminosity along its activity cycle. The slope and sign reversal of the p-mode frequency shifts have been previously noticed on intermediate by Ronan et al. (1994) and Jefferies (1998), and this effect was already visible (although unnoticed) in the work of Libbrecht (1988). The analysis is more convincing at higher because the power in the modes is higher and the effect is then obvious. Then, we are left with a challenge: whatever physical effect mentioned to explain the change in the frequencies at 3.5 mHz will possibly have some trouble in explaining the opposite changes at 5. mHz. Several authors have been aware of this problem: Goldreich et al. (1991) explained a frequency decline above 4 mHz because of an entropy increase in the optically thin layers of the solar atmosphere. They suggest a chromospheric resonance to account for the "precipitous nature of the decline [...] at 4.4 mHz'', a statement which is very much in agreement with our results. Jain & Roberts (1993, 1996) propose a model of the convection zone and solar atmosphere in which a uniform horizontal magnetic field is able to reproduce both the downturn and the upturn of the frequency shifts. The key point here is the horizontal nature of the field, which otherwise would require too important changes to be realistic. If we assume that this trend reversal is rather -independent, as suggested by various observations, then the effect will probably be still coming from or even above the surface layers of the Sun: one must start to question the possible influence of the chromosphere on the frequency shifts as suggested by the above-mentioned works.
Theories explain pseudo-modes using the acoustic path difference between downward and upward propagating waves to build interferences above the photospheric cutoff frequency (Kumar & Lu 1991). Changes in the reflectivity of the surface layers because of changing magnetic effects would affect the amplitude of those modes, while changes in the source position of the waves would affect the value of the spectral recurrence of the HIPs (Kumar 1993). We observe a very important change in the amplitudes when switching from P3 to P4, but we have a very good stability in the recurrence of the HIPs spectral pattern during the whole run. We conclude that HIPs visibility in the spectra should be sensitive to the wavelength, or rather to the level in the photosphere to which they are observed. Also, from the steadiness of the recurrence, we can infer that the position of the acoustic source remained the same to our precision level on , e.g. within a few kilometers.
We have measured linewidth differences between P3 and P4 and found them changing: between 2 and 3.3 mHz, we see a positive dependence (linewidths increase with activity), and between 3.3 and 4.5 mHz the trend is reversed, although the shift is still positive. We believe that this is a true solar activity signature. Komm et al. (2000a) and Komm et al. (2000b) have reported linewidth shifts in intermediate- and high- in the GONG network data, showing that p-modes are broader with increasing magnetic activity. We agree with them on the frequency dependence and on the sign of the effect. Recently, Houdek et al. (2001) have given an interpretation of the linewidth shifts measured in the BiSON network data in terms of changes in the shape of the convective eddies. They model the shifts in linewidths by changing the eddy shape parameter by 6% between 1991 and 1997. While this is a very interesting way of connecting the p-mode linewidths to some physical phenomenon, we think that the fit of the model giving this result does not account very convincingly for their data, and would not match our own results of Fig. 7b. Anyway, the few available observational results of linewidth shifts all indicate a decrease in the coherence time of the p-modes at high solar activity.
Starting with Roxburgh & Vorontsov (1997), several authors have discussed the possibility of correlation between the modes and the background noise. Such a correlation would explain the reversal of asymmetry of the p-modes between intensity and velocity observations (Toutain et al. 1998; Nigam et al. 1998). Our approach in Sect. 3.3 (i.e. the tentative model of an extra noise component), has been so far purely empirical, and only guided by numerical considerations on the quality of the results on the modes amplitudes and linewidths. Whether asymmetry reversal between intensity and velocity is the only effect of this possible correlation, or some other effects like the one we present here can happen is a problem that we shall be investigating in depth. If confirmed, the amount of correlation would probably be quite significant.
Finally, the p-mode amplitudes and the SNR of the GOLF experiment do change significantly between P3 and P4, and we can state that p-mode radial velocities were better seen on the blue wing of the Na line for a large part of the high frequency domain up to the cutoff frequency. We are careful in saying that our velocities here are rather "scaled intensities'' than pure Doppler shifts converted to radial velocities. However, the blue side of the Na lines looks very different to the red side, and this prevents us from concluding whether there are solar-cycle effects in our amplitudes determinations. This difference not only concerns the modes, but the background seems also implicated up to and even above the cutoff frequency. What progress can be made on the response of the solar photosphere to the p-mode oscillations, or to the dependence of the modes SNR on the experimental wavelength inside a spectral line are questions that would require more observations of an "enhanced GOLF'' instrument.
The auhors wish to thank P. Boumier from IAS, Université Paris Sud, for many constructive remarks and helpful suggestions in the final writing of this paper.
The GOLF instrument is due to a team from a consortium of institutes. SOHO is a space solar observatory from a common project between ESA and NASA.