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
Abstract
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.
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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.
Period | P2 | P3 | P4 |
begin | Feb. 18, 1996 | Apr. 10, 1996 | Oct. 15, 1998 |
end | Mar. 31, 1996 | Jun. 24, 1998 | Dec. 31, 2001 |
days | 43 | 805 | 1174 |
coverage | 79% | 99.35% | 94.09% |
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
and
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:
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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:
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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.
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Figure 4:
a)
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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.
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Figure 5:
Experimental splitting restitution.
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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
.
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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.
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Each frequency value
in Tables A.1, A.2,
or A.3 is the weighted average of n measurements
:
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(6) |
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(7) |
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(8) |
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.
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Figure 8:
Amplitudes and noise background in 1979 days of GOLF data for
low-degree modes. From top to bottom: ![]() ![]() ![]() ![]() |
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Figure 9:
Ratio
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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
,
and is fainter, although real, on
.
To check this
feature, we have computed the ratio
:
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(9) |
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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.
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Figure 11:
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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:
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Figure 12:
Per-mode splitting determination plotted against
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n | ![]() |
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7 | 432 | ![]() | 2 | ||||||
8 | 432 | ![]() | 2 | 433 | ![]() | 5 | |||
9 | 432 | ![]() | 6 | 437 | ![]() | 10 | 435 | ![]() | 8 |
10 | 444 | ![]() | 12 | 430 | ![]() | 16 | 429 | ![]() | 17 |
11 | 432 | ![]() | 16 | 438 | ![]() | 9 | 430 | ![]() | 15 |
12 | 442 | ![]() | 28 | 422 | ![]() | 26 | 428 | ![]() | 25 |
13 | 453 | ![]() | 42 | 429 | ![]() | 22 | 438 | ![]() | 13 |
14 | 431 | ![]() | 42 | 425 | ![]() | 14 | 436 | ![]() | 28 |
15 | 447 | ![]() | 66 | 425 | ![]() | 21 | 437 | ![]() | 22 |
16 | 406 | ![]() | 38 | 436 | ![]() | 24 | 448 | ![]() | 29 |
17 | 449 | ![]() | 57 | 462 | ![]() | 22 | 437 | ![]() | 28 |
18 | 450 | ![]() | 66 | 430 | ![]() | 20 | 442 | ![]() | 14 |
19 | 467 | ![]() | 31 | 433 | ![]() | 20 | 458 | ![]() | 13 |
20 | 455 | ![]() | 28 | 425 | ![]() | 28 | 440 | ![]() | 14 |
21 | 449 | ![]() | 47 | 417 | ![]() | 47 | 444 | ![]() | 18 |
22 | 457 | ![]() | 50 | 429 | ![]() | 35 | 456 | ![]() | 25 |
23 | 460 | ![]() | 60 | 410 | ![]() | 50 | 465 | ![]() | 40 |
24 | 437 | ![]() | 60 | 440 | ![]() | 50 | |||
25 | 450 | ![]() | 52 |
BiSON (![]() |
Bertello et al. (![]() |
This work (![]() |
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435.4![]() over
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436![]() over
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433![]() over
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436.0![]() over
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426![]() over
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434![]() over
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435.4![]() over
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459![]() over
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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.
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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.
Acknowledgements
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.
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