AN ESTIMATION OF THE ACOUSTIC CUTOFF FREQUENCY OF THE SUN BASED ON THE PROPERTIES OF THE LOW-DEGREE PSEUDOMODES

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1 The Astrophysical Journal, 646: , 2006 August 1 # The American Astronomical Society. All rights reserved. Printed in U.S.A. AN ESTIMATION OF THE ACOUSTIC CUTOFF FREQUENCY OF THE SUN BASED ON THE PROPERTIES OF THE LOW-DEGREE PSEUDOMODES A. Jiménez 1 Received 2005 November 17; accepted 2006 April 13 ABSTRACT The acoustic cutoff frequency ( ac ) is an important atmospheric parameter whose estimation has in the past been based on the study of power spectra that have yielded a wide range of values between 5300 and 5700 Hz. The discovery of a solar signal well beyond the acoustic cutoff frequency (pseudomodes) might lead one to think that the determination of ac would be even more complicated because, for example, looking for a sudden drop in the power density signal could no be longer used. Contrary to what might be thought at first sight, the existence of pseudomodes helps to provide a good estimation of ac, because the frequency pattern of pseudomodes is shifted with respect to that of p-modes. In this study a bivariate analysis (coherence and phase shift) between the intensity signals of VIRGO and the velocity signal of GOLF (both instruments on board the SOHO probe) is carried out over the frequency range of p-modes and pseudomodes. The results shows clear evidence that the acoustic cutoff frequency of the Sun is close to the theoretical value of 5300 Hz; specifically, a value around 5100 Hz is found in this research. Subject headinggs: Sun: helioseismology Sun: oscillations 1. INTRODUCTION Solar p-modes are essentially resonant acoustic waves that can be regarded as a superposition of outwardly and inwardly propagating waves that interfere constructively. At certain discrete frequencies the interference is maximally constructive, yielding the eigenfrequencies of the acoustic cavity within which the p-modes propagate. The lower boundary of these resonant cavities lies at a depth inside the Sun at which the horizontal phase speed of the wave equals the local sound speed. The upper boundary of the p-mode cavities lies near the solar surface and is a complicated function of frequency and wavenumber. An important parameter for locating the upper boundary of p-mode cavities is the Lamb acoustic cutoff frequency ( ac ); ac can also lead to a determination of the mean molecular weight provided the temperature is known (Isaak 1983). Only acoustic oscillations with frequencies < ac are trapped in the resonant cavities beneath the photosphere. For an isothermal atmosphere, this frequency is approximately the ratio of the sound speed to twice the density scale height. Solar atmospheric theoretical approaches by Balmforth & Gough (1990) show that the acoustic cutoff frequency for the solar atmosphere is ac ¼ 5300 Hz. For > ac acoustic disturbances are no longer trapped and propagate as traveling waves through the chromosphere to the base of the solar corona. In oscillation power spectra, these highfrequency peaks ( hereafter pseudomodes) show a clear structure well beyond ac (Jefferies et al. 1988; Libbrecht 1988; Duvall et al. 1991; García et al. 1998; Chaplin et al. 2002; Jiménez et al. 2005). Several models have been proposed to explain the nature of the pseudomodes (Balmforth & Gough 1990; Kumar et al. 1990; Kumar 1994; Jain & Roberts 1996; García et al. 1998). Nowadays, it is generally believed that the full-disk integrated pseudomodes ( low-degree) arise from geometric interference between direct waves emitted from a subphotospheric source and indirect waves produced by partial reflection on the far side of the Sun (García et al. 1998). For higher degree modes the indirect waves are 1 Instituto de Astrofísica de Canarias, E La Laguna, Tenerife, Spain those emitted downwards toward the solar interior and refracted back to the solar surface (Kumar et al. 1990; Kumar 1994). The acoustic cutoff frequency represents the frequency boundary between p-modes and pseudomodes. Observationally, some estimates have already been obtained (Claverie et al. 1981; Pallé et al. 1986, 1992; Duval et al. 1991; Fossat et al. 1992) yielding different results, which lie between 5300 and 5700 Hz. Before the discovery of pseudomodes, it might have been expected that a way to find a measurement of ac would be to look for a sudden drop in the power density signal on the highfrequency side of the spectrum. The discovery of pseudomodes means that the cutoff frequency cannot be estimated from an examination of the power spectrum of acoustic oscillations. Although, in principle, the existence of solar pseudomodes does complicate the determination of the acoustic cutoff frequency, they nevertheless really help in finding it. In this study certain properties of pseudomodes are used to determine the acoustic cutoff frequency: 1. The frequency shift of pseudomodes with respect to p-modes. 2. A significant change between the I-V phase differences of p-modes and pseudomodes. The physical motivation of these methods are as follows Frequency Shift Acoustic waves in the Sun are trapped in resonant cavities below the chromosphere when the frequency is less than the photospheric acoustic cutoff frequency ( ac ), thus giving rise to the well-known p-mode spectrum. Waves with frequency beyond ac propagate in the solar chromosphere as traveling waves. In the acoustic spectrum of the Sun, the frequency separation between consecutive modes of the same degree n;l ¼ n;l n 1;l is approximately equal to the inverse of the sound travel time from the upper reflection point to the lower turning point and back. This n;l decreases if the lower turning point moves inward (increasing or decreasing l ), or if the outer reflection point moves outward. At a given spherical harmonic, the observed frequency spacing between peaks decrease with increasing frequency. However several authors ( Kumar et al. 1994;

2 ESTIMATION OF ACOUSTIC CUTOFF FREQUENCY OF SUN 1399 Nigam & Kosovichev1996) have pointed out that between 5000 and 5500 Hz the frequency spacing increases slightly, this feature probably being associated with the acoustic cutoff frequency, indicating the transition from trapped to traveling waves. If this frequency separation ( n;l ) increases around the acoustic cutofffrequency ( ac ), all the peaks with frequencies > ac will be shifted with respect to the peaks with frequencies < ac.ifit is possible to find the frequency at which this frequency shifts takes place, it would be a good measurement of the acoustic cutoff frequency. In this research the coherence function between intensity and velocity signals are used instead of the power spectra to avoid an intensity contamination, as is explained in x I-V Phase Differences Phase relationships can be a very useful tool in investigating the oscillatory behavior of the solar atmosphere. These phase shifts can correspond to velocity-velocity or intensity-intensity measurements at two different spectral lines (two different formation heights), or to intensity-velocity measurements at spectral lines, narrowband photometry, etc. For velocity-velocity (or intensity-intensity) observations a phase difference of around 0 should be expected in the frequency range corresponding to standing (trapped) waves (the p-mode range), as has been measured by various authors (Fossat et al. 1992; Pallé et al. 1992; Jiménez et al. 1999). For waves with frequencies beyond ac (the traveling wave range), a nonzero phase difference should be expected. Staiger (1987) also measured these near-zero V-V phase differences between 2500 and 5000 Hz, but for frequencies between 5000 and 7000 Hz, he also found that the phase difference changes nearly linearly and that these values were in good agreement with theoretical values calculated by Schmieder (1977, 1978) under the assumption that vertically traveling waves exist beyond the acoustic cutoff frequency. For intensity-velocity observations the phase differences give information about the adiabatic or nonadiabatic behavior of the solar atmosphere. In the adiabatic case a value of 90 (downward velocity positive) or 90 (upward velocity positive) is expected for the p-mode range and a value of 0 for a model close to isothermal ( Marmolino & Severino 1991). For nonadiabatic conditions the phase differences change with frequency depending on the model used (Gough 1985; Houdek et al. 1995). The results of Jiménez (2002) show that in the p-mode range the I-V phase differences do not show an exactly adiabatic behavior but one close to it. Our concern in the present research is that these I-V phase differences in the p-mode range are close to 90 with no important changes with frequency. The theoretical results of Schmieder (1978) shows that the I-V phase differences for frequencies beyond ac also change nearly linearly in the frequency range between 5000 and 6500 Hz because of the traveling waves. Another good determination of the acoustic cutoff frequency would be to find the frequency at which the I-V phase difference changes in a significant (probably linear) way. 2. INSTRUMENTATION 2.1. VIRGO SPM ( Intensities) The Variability of Irradiance and Gravity Oscillations ( VIRGO) Sun Photometer (SPM) (Fröhlich et al. 1995, 1997) comprises three Sun photometers, used independently, at 402 ( blue), 500 (green), and 862 nm (red) that look at the Sun as a star with a 60 s cadence. The bandwidth of the filters is 5 nm. In 1998 June the Solar and Heliospheric Observatory (SOHO) was lost for several months but after a search campaign was finally found and resumed operations around 1998 October. The VIRGO data after SOHO s vacations show the same high quality as before the temporary loss of the probe GOLF ( Velocity) GOLF (Global Oscillations at Low Frequency) is a resonant scattering spectrophotometer (Gabriel et al. 1995, 1997) that measures the line-of-sight velocity using the sodium doublet, in a manner similar to the International Research on the Interior of the Sun ( IRIS) and Birmingham Solar Oscillation Network (BiSON) ground-based networks. The GOLF window was opened in 1996 January and became fully operative by the end of that month. Over the following months occasional malfunctions in its rotating polarizing elements were noticed that led to the decision to stop them in a predetermined position; truly nonstop observations began by 1996 mid-april. Since then, GOLF has been continuously and satisfactorily operating in a mode unforeseen before launch, showing fewer limitations than anticipated. The signal, therefore, consists of two close monochromatic photometric measurements in a very narrow band (25 m8) on the blue wing of the sodium doublet (Gabriel et al. 1995). This signal has been calibrated as velocity and is indeed similar in nature to other known velocity measurements, such as those of IRIS and BiSON (Pallé et al. 1999). The sampling of the GOLF data used in this paper is 60 s. Before SOHO s vacations (1998 June), GOLF data were obtained in the blue wing of the sodium line; thus, after the SOHO vacations the GOLF team decided change to the red wing of the sodium line (see García et al. [2005] for the latest report on the GOLF instrument). 3. DATA SETS To get a high visibility in the pseudomodes frequency range, 200 consecutive time series of 4 days are used, from 1996 April 11 through 1998 June 20, up to the SOHO vacations. This period covers the solar activity minimum. After the recovery of SOHO, GOLF changed to the red wing of the sodium line. This time span of 800 days divided into 4 day time series shows a very clear pseudomode spectrum, as is seen below. No data after the SOHO vacations have been used in this study for the following reasons. (1) The red-wing data of GOLF after the SOHO vacations have demonstrated less sensitivity to pseudomodes, because their power spectra have between 1.5 and 2 times less intrinsic power than in the blue wing. (2) The photon noise also increases by a factor of 2 because of the aging of the instrument (see García et al. [2004] for a complete description of GOLF behavior in both wings of the sodium doublet). (3) Finally, the fact that the pseudomode properties seem not to depend on the solar activity cycle (Jiménez et al. 2005) makes it more sensible to use only the best available data of GOLF for pseudomode-related works. To study the transition frequency range between p-modes and pseudomodes, i.e., to find a determination of the acoustic cutoff frequency ( ac ), the coherence and phase shift between the intensity and velocity signal is used. In phase analysis the data timing is of crucial importance. While VIRGO data have no time shift during the mission, GOLF suffered two delays during the time span used in this study. The GOLF detection chain was designed to make every measure integration 4 s in 5 s cycles. This 5 s signal is triggered by the GOLF local on-board time (LOBT), which is being driven by the SOHO OBT. This OBT is ensured always to be within 20 ms of TAI (Temps Atomique International). However, the GOLF temporal sampling suffered the two delays mentioned above. The data between 1997 February 16 and 1998 March 2 are shifted by s with respect to the previous ones. From 1998 March 4 up to SOHO s 1998 vacations,

3 1400 JIMÉNEZ Vol. 646 the data have an additional shift of s. The scaled data sets provided by the GOLF team are partially corrected for these delays. In order to avoid any sort of interpolation, which may affect the quality of the final series, the segments corresponding to the two delays have been shifted by multiples of 10 s (the original sampling of the data). More precisely, by +10 and +30 s, respectively. The residual shifts of and s are not critical for normal analysis like the estimation of power spectra, but other types of calculations, like any phase studies, may require a higher degree of accuracy in the timing of the GOLF signal. If we compute the I-V phase differences with the velocity data shifted by the previous residuals shifts, the computed I-V phase difference would change, as function of the frequency (2T, T being the time shift) by (using the larger time shift of s): 3N6, 4N8, and 8N3 at 3, 5, and 7.0 mhz, respectively. Although these quantities are not very large, they have been corrected for in the present analysis. After computing the I-V phase differences for each series (see x 4), the artificial phase shift as function of the frequency has been added to the original values, removing this small effect completely. 4. DATA ANALYSIS For each pair of simultaneous intensity and velocity time series of 4 days, the corresponding power spectra ( ¼ 2:89 Hz), the coherence, and phase difference between them are obtained using a bivariate analysis. This method is briefly described in Koopmans (1974). Let A and B be two time series of length Tand sin A, cos A,sinB, and cos B be the sine and cosine amplitudes of the spectra for series A and B. The power spectral densities, P A () and P B (), the cospectral density, C AB (), the quadrature spectral density, q AB (), and the complex cross-spectral density, P AB (), are defined as P A () ¼ T 2 sin2 A() þ cos 2 A() ; ð1þ P B () ¼ T 2 sin2 B() þ cos 2 B() ; ð2þ C AB () ¼ T ½sin A()sin B() þ cos A()cos B()Š; 2 ð3þ q AB () ¼ T ½sin A()cos B() sin B()cos A()Š; 2 ð4þ P AB () ¼ C AB () iq AB (): After applying 5 bin smoothing (indicated by h...i in the following equations), the coherence (the analog of the linear correlation coefficient between the two time series A and B in linear regression analysis) and the phase difference, AB (), between series A and B are given by Coh 2 AB () ¼ hc AB()i 2 þhq AB ()i 2 ¼ hp A ()ihp B ()i AB () ¼ tan 1 j hp AB()ij 2 hp A ()ihp B ()i ; hq AB ()i hc AB ()i and the errors for the phase difference are given by ð5þ ð6þ ; ð7þ " sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi # AB () ¼ sin 1 1 Coh 2 AB () (2n 2)Coh 2 AB () t 2n 2 ; ð8þ 2 Fig. 1. Power spectra showing the clear structure of pseudomodes for the signals used in this work. Bottom to top: Three channels of VIRGO SPM, red, green, and blue; the GOLF spectra; and the average of the three VIRGO SPM channels multiplied by 8 for reasons of clarity. where n represents the equivalent degrees of freedom (EDF), t 2n 2 (/2) is the Student t-distribution, and is the confidence level at which the errors are computed ( ¼ 0:8 for phase differences). After the computation of the power spectra and bivariate parameters for each pair of the 200 I-V time series, they are averaged, obtaining four final power spectra (one for each of the three VIRGO SPM channels and one for GOLF) and three sets of bivariate parameters (coherence and phase shift) between each of the VIRGO SPM channels and GOLF. Finally, the three VIRGO SPM power spectra and the three sets of bivariate parameters are averaged. This result is referred in this article as VIRGO SPM. 5. RESULTS Figure 1 shows the power spectra of the three channels of VIRGO SPM (red, green, and blue), their average (VIRGO SPM) multiplied by 8 for clarity, and the GOLF spectrum from 3000 Hz (including the right side of the 5 minute oscillations) to the Nyquist frequency. The pseudomode structure is clearly seen in all the signals. The VIRGO signals show a peak (highest in the green channel) at just 5555 Hz, corresponding to 3 minutes. This is just the period of the calibration reference used by the Data Acquisition Signal of VIRGO. This is an electronic artifact that in principle could contaminate the determination of the acoustic cutoff frequency but, as is seen in xx 5.1 and 5.2, is not a problem because the coherence and phase shift are not affected by this ghost signal. From now on, only the results of the SPM average (VIRGO SPM) and GOLF are presented, because they are equivalent to those obtained using each one of the three VIRGO SPM channels separately. Figure 2 illustrates how the coherence function is not affected by the electronic signal at 5555 Hz. Figure 2a shows the power spectra between 5000 and 8000 Hz, after removal of the exponential decay by fitting a Legrendre polynomial (of order 4) to the logarithm of the power versus frequency. Both spectra have been separated by adding 0.1 to VIRGO SPM and subtracting 0.1 to GOLF for reasons of clarity. Figure 2b shows the coherence function in this frequency range. The shape of the coherence function is in phase with the power spectra; the maxima in

4 No. 2, 2006 ESTIMATION OF ACOUSTIC CUTOFF FREQUENCY OF SUN 1401 Fig. 2. (a) Power spectra in the pseudomodes frequency range after removing exponential decay. Top line: VIRGO SPM; bottom line: GOLF. Both spectra have been separated by adding and subtracting 0.1, respectively. (b) Coherence function in the pseudomodes frequency range. The maxima of the coherence coincides with the maxima of the power spectra and the minima with the noise signal between consecutive pseudomodes. Note that in the coherence function the contamination of the spurious peak (electronic artifact) at 5555 Hz does not exist, because this contamination is not present in GOLF data. The coherence value at this contaminated frequency is just at the same level as the other minima of the coherence function. the coherence function coincide with the maxima in the power spectra. The minima in the coherence functions correspond to the noise signal between two consecutive pseudomodes. The high spurious peak at 5555 Hz is present in VIRGO SPM but not in the coherence function, because this signal does not exist in GOLF data, yielding a low value of the coherence at this frequency, at the same level of the other minima of the coherence function. Use the power spectra to determine the acoustic cutoff frequency would require special care in the treatment of the signal, but the coherence and phase shift (as are seen below) on which this work is based do not need any special considerations I-V Phase Differences of p-modes and Pseudomodes The coherence function and phase shift between the intensities of VIRGO SPM and velocity of GOLF are plotted in Figures 3a and 3b, respectively, covering the frequency range of p-modes and pseudomodes. The coherence function obviously shows high and sharp peaks in the 5 minute range, reaching values of 0.9 at 3200 Hz. This coherence decreases with frequency up to around Hz. From 5300 to 6000 Hz it seems to be more or less stable. From 6000 Hz to the Nyquist frequency the coherence decreases again until reaching the minimum value of The I-V phase-shift function ( Fig. 3b) behaves similarly to the coherence; it has alternate maximum and minimum values. The exact value of the phase shift for a certain peak in the power spectra correspond to the phase-shift value at the frequency where the coherence has its maximum. In the p-mode range, the I-V phase shift depends on the degree of the modes because of the contaminations of the solar background ( Jiménez 2002), as lower p-mode power means higher background contamination because of the lower signal-to-noise ratio. In power spectra with enough frequency resolution, the coherence function shows maxima for each l ¼ 0, 1, 2, and 3 p-mode, being higher for l ¼ 0and1thanforl ¼ 2and3.The phase shift for each p-mode is computed at the frequency of its corresponding maximum in the coherence function. Fig. 3. (a) Coherence function covering the p-mode and pseudomode frequency range. (b) Phase-shift function for the same frequency range. The circles are the phase-shift values corresponding to the maxima of the coherence function. These phase-shift values change from one side of the phase-shift function to the other side at around 5200 Hz, showing a shift in the coherence function at around this frequency. The frequency resolution used in this article ( ¼ 2:89 Hz), together with the 5 bin smoothing, make the different p-mode degrees indistinguishable and produce coherence and phase-shift functions as shown in Figure 3, whose shapes, in the p-mode range, are produced basically by the l ¼ 0 and 1 p-modes. An interesting feature is observed in Figure 3. As mentioned above, the phase-shift values must be taken at the frequency of the maxima of the coherence function. The circles of Figure 3b show these phase shifts at the maxima of the coherence. From 2600 to 4300 Hz the phase-shift values are located mainly just to the left of the maxima of the phase-shift function. From 4300 to 5200 Hz the phase-shift values are located in the maxima of the phase-shift function. From 5200 Hz to higher frequencies the phase-shift values are on the right-hand side of the phase-shift function, even reaching the minima of the phase-shift function. A shift in the coherence function would produce this behavior. To illustrate how I-V phase differences change because of a shift in the coherence function, a moving correlation between coherence function and phase-shift function has been performed. Starting at an initial frequency of 2600 Hz, a 70 Hz interval is chosen, and the lag corresponding to the maximum of the crosscorrelation between coherence and phase shift is computed. This maximum cross-correlation lag is associated with the mean frequency of the interval. The initial frequency is then increased by 50 Hz and the corresponding maximum cross-correlation lag computed. This process is extended up to 7800 Hz. Figure 4a shows those results ( points) corroborating what was observed in Figure 3. In the p-mode range the lag of the maximum correlation is slightly positive or zero. Around 5000 Hz it starts to become negative and increase its negative value with frequency up to around 7000 Hz, where it seems to be stable again, although noisier. Figure 4b shows the values of the I-V phase differences for p-modes and pseudomodes. For p-modes (at 2600 Hz) the phase differences are around 120 and increase up to 60 at 4800 Hz (these values agree perfectly with those obtained with higher resolution spectra by Jiménez et al. [1999] and Jiménez [2002]). At around Hz a significant change in the I-V phase differences takes place. The phase differences decrease (almost

5 1402 JIMÉNEZ Vol. 646 computations that expect this behavior when the assumption that traveling waves exist beyond the acoustic cutoff frequency is included in the computations. In x 5.2 an attempt is made to find a good measurement of the acoustic cutoff frequency. Fig. 4. (a) Lags corresponding to the maxima of the cross-correlation between coherence and phase-shift functions computed in 70 Hz intervals and shifted by 50 Hz. Each lag value is associated with the mean frequency of the corresponding interval (see text). (b) I-V phase differences values for p-modes and pseudomodes. Different behavior is observed before and after a frequency of around Hz (see text). linearly) up to around 6200 Hz. From this frequency, the phase differences oscillate upward to reach a more or less stable value at 7800 Hz. In the p-mode range the I-V phase differences show a close to adiabatic behavior without important changes with frequency as expected, but at a certain frequency around Hz the phase differences become linear, in agreement with the theoretical 5.2. Frequency Shift of Pseudomodes with Respect to p-modes As explained in x 1, a frequency shift between pseudomodes and p-modes is expected because of the increase of the frequency separation n;l ¼ n;l n 1;l around the acoustic cutoff frequency. To find where this frequency shift takes place, a physical definition of the acoustic cutoff frequency is derived in this subsection. An exponentially modulated sine wave is fitted to the coherence function between 3500 and 5500 Hz to take into account the decreasing amplitude of p-modes in this frequency range. In Figure 5a the coherence function (thick solid line) is plotted between 3500 and 6500 Hz, together with the modulated sine wave (thin solid line) that is extended to 6500 Hz. The shape of the solar signal in the pseudomode frequency range (between 5000 and 6500 Hz) is like a sine wave, with no clear modulation, so a single sine wave is fitted to the coherence function between 5000 and 6500 Hz. This second fit is also plotted in Figure 5a (dotted line) and extended to lower frequencies, down to 3500 Hz. The interval from 5000 to 5500 Hz is used in both fits because it is the interval where the acoustic cutoff frequency is expected to be found and also because, obviously, it is not possible to separate p-modes from pseudomodes before finding the acoustic cutoff frequency (in several tests this interval has been slightly changed and the same results obtained). Fig. 5. (a) Thick solid line: Coherence function. Thin solid line: Modulated sine wave fitted to the coherence function between 3500 and 5500 Hz (end of the p-mode range) and extended to 6500 Hz. Filled and open circles are the maxima of the coherence function and the fitted modulated sine wave, respectively. Note how the coherence shifts with respect to the modulated sine wave to higher frequencies from Hz upwards ( filled circles vs. open circles). Dotted line: Single sine wave fitted to the coherence function between 5000 and 6500 Hz (first side of the pseudomode range) and extended down to 3500 Hz. Filled squares are the maxima of this fitted function. The opposite effect is observed in this case; the coherence is delayed with respect to the single sine wave toward lower frequencies ( filled circles vs. open squares, respectively). (b) Frequency differences between the maxima of Fig. 5a, that is, between the maxima of the coherence and the maxima of the modulated sine wave (circles) and between the maxima of the coherence and the maxima of the single sine wave (squares). The lines are the best fit to both data sets. The crossing point of both curves is defined as the acoustic cutoff frequency.

6 No. 2, 2006 ESTIMATION OF ACOUSTIC CUTOFF FREQUENCY OF SUN 1403 The maxima of the three curves, coherence, modulated sine wave, and single sine wave have been computed and plotted in Figure 5a. Filled circles, open circles, and open squares correspond to the frequencies of the maxima of the coherence, modulated sine wave, and single sine wave functions, respectively. Up to 5000 Hz the coherence function and the fitted modulated sine wave are in phase, and their maxima have the same frequencies; in fact, no open circles are visible below 5000 Hz, because they are overplotted by the filled ones. Around 5200 Hz the coherence function starts to shift to higher frequencies, and the maxima of the fitted modulated sine wave are delayed with respect to those of the coherence function. Open circles then become visible; in fact, at Hz they are already very clear, and the coherence shift increases with frequency, being out of phase (maxima of the coherence coincides with minima of the fitted modulated sine wave) between 6000 and 6500 Hz. On the other hand, just the opposite effect takes place for the pseudomodes. The coherence function and the fitted single sine wave lose the phase from higher to lower frequencies. From 6500 to around 5000 Hz the coherence and the fitted single sine wave are in phase, and their maxima have the same frequencies. From around 5000 Hz to lower frequencies the coherence function begins to delay with respect to the fitted single sine wave, and in fact, they are out of phase (maxima of the coherence coincides with minima of the fitted function) between 4000 and 3500 Hz. To look in detail at these two opposite effects, the frequency differences between the maxima of Figure 5a are computed; that is, the frequency differences between the maxima of the coherence function and the maxima of the fitted modulated sine wave and between the maxima of the coherence function and the maxima of the single sine wave. In Figure 5b open circles correspond to the former ( p-modes) and open squares to the latter (pseudomodes). Open circles have an approximately constant value close to zero in the frequency range where the p-mode coherence signal is in phase with the fitted function, from 3500 up to 5000 Hz. At this frequency the coherence shifts to higher frequencies and the values of the maxima differences increase up to 38 Hz at the frequencies where coherence and fitted function are out of phase. The frequency differences between the maxima of the coherence and the maxima of the single sine wave fitted to the pseudomode range (squares) are out of phase in the p-mode frequency range. From a value of around 50 Hz at 3500 Hz, these differences decrease with frequency up to, again, around 5000 Hz. From this frequency the differences have an approximately constant value in the frequency range where the coherence and the fitted function are in phase. This constant value is close to zero but not so close as the one in the p-mode range probably because of the irregular shape of the pseudomodes. A physical definition of the acoustic cutoff frequency can be derived from the crossing point of the two data sets shown in Figure 5b. This crossing point is the frequency at which, hypothetically, the maxima of both fitted functions and the maxima of the coherence coincide. To look for this crossing point several functions have been fitted to both data sets in Figure 5b, resulting in a six-degree polynomial providing the best fit. The fitted functions are overplotted as full lines in Figure 5b. The crossing point is at 5106:41 61:53 Hz. In Figure 5b, the symbols are quite coincident at this frequency (see close to 5100 Hz). The closest values to the left of this frequency are inverted with respect to the closest values to the right of this frequency; this means that on the left, the open circle is lower than the square, and on the right the open circle is higher than the square. This is the transition frequency that, following the definition derived in this section, can be called the acoustic cutoff frequency. The acoustic cutoff frequency for low-degree modes found in this research, ac 5100 Hz, is lower than indicated in previous observations, in which a wide range of values were found between 5300 to 5700 Hz. All these previous determinations were performed using ground-based observations, using different techniques and without good visibility of the pseudomodes. A suggested explanation for these discrepancies could be the higher quality of the space-based data of SOHO and that the lack of good visibility of the pseudomodes could yield different techniques to consider the trace of the first pseudomodes as a p-mode signal. 6. CONCLUSIONS The best available data for pseudomode determination of VIRGO and GOLF instruments on board the SOHO probe have been used in this research for a determination of the acoustic cutoff frequency ( ac ). Instead of power spectra, the coherence and phase-shift functions between intensity and velocity signals have been computed using a bivariate analysis and covering the p-mode and pseudomode frequency range. The increase in frequency separation n; l ¼ n; l n 1; l around the acoustic cutoff frequency causes the peaks with > ac (pseudomodes) to be shifted with respect to the peaks with < ac ( p-modes). The I-V phase differences are also expected to be different for p-modes and pseudomodes, with an important change in their behavior around the acoustic cutoff frequency. For p-modes the I-V phase differences are expected to have values close to the adiabatic model ( 90 ) with no important changes with frequency, but a significant change is expected around the acoustic cutoff frequency. For > ac the I-V phase differences are expected to be nearly linear up to around 6500 Hz. The VIRGO SPM signal has a spurious frequency produced by its electronics at a frequency inside the frequency interval in which ac is looked for; this contaminates the intensity power spectra with a high peak at 5555 Hz. For this reason, the coherence between intensity and velocity signals have been used in this research; this contamination does not exist in the coherence nor in the phase-shift functions, because the spurious signal is not present in the GOLF velocity signal. At this spurious frequency, the bivariate parameters (coherence and phase shift) have similar values to those at the noise level. The I-V coherence shifts in frequency for pseudomodes, yielding different values of the I-V phase differences in the I-V phase function. This produces a different behavior in the I-V phase differences for p-modes and for pseudomodes. The cross-correlation between coherence and phase function from 2500 to 7800 Hz shows how the position of the maxima of this cross-correlation is more or less constant for the p-mode range and starts to change at around 5200 Hz. The I-V phase differences for the p-modes range are, as expected, close to the adiabatic model ( 90 ) with no important frequency dependence, but at a certain frequency they come to have a nearly linear behavior, also as expected; this frequency being between 5000 and 5200 Hz. To find the frequency at which the frequency pattern of pseudomodes starts to be shifted with respect to the one of the p-modes, the coherence function between intensity and velocity signals has been used. The coherence function has a modulated sinusoidal shape in the p-mode frequency range and a sinusoidal shape in the pseudomode frequency range. If the coherence function shifts for pseudomodes with respect to the p-modes, a fitted modulated sine wave to the coherence in the p-mode frequency

7 1404 JIMÉNEZ range must be, respectively, in phase and out of phase before and after the acoustic cutoff frequency. The same effect, but in the opposite sense, must take place for a single sine wave fitted in the pseudomodes frequency range. Looking for this, a modulated sine wave is fitted to the final part of the p-mode range ( Hz) and extended to the first side of the pseudomodes (up to 6500 Hz) computing the frequency differences between the maxima of the coherence function and the maxima of the modulated sine wave. On the other hand, a single sine wave is fitted to the pseudomode frequency range between 5000 and 6500 Hz, and extended down to 3500 Hz, also computing the frequency differences between the maxima of the coherence function and the maxima of the single sine wave. These two sets of frequency differences between maxima have a crossing point that can be defined as the acoustic cutoff frequency; that is, the frequency at which the maxima of the coherence function and both fitted functions lie is the same. After fitting a six-degree polynomial to both data sets, this crossing point is found to be at 5106:41 61:53 Hz. All these results show that the acoustic cutoff frequency of the Sun seems to be close to the theoretical value of 5300 Hz. Both instruments benefit from the quiet and well-run SOHO platform built by Matra Marconi Space. SOHO is an international collaboration program of the European Space Agency and the National Aeronautics Space Administration. VIRGO and GOLF are cooperative efforts of many individual scientists and engineers at several institutes in Europe and USA to whom I am deeply indebted. I offer thanks to Rafael García for the GOLF time series and to the referee of this work for the useful revision. Balmforth, N. J., & Gough, D. O. 1990, ApJ, 362, 256 Chaplin, W. J., et al. 2002, in Local and Global Helioseismology: The Present and Future, ed. H. Sawaya-Lacoste ( Noordwijk: ESA), 247 Claverie, A., Isaak, G. R., McLeod, C. P., van der Raay, H. B., & Roca Cortés, T. 1981, Sol. Phys., 74, 51 Duvall, T. L., Harvey, J. W., Jefferies, S., M., & Pomerantz, M., A. 1991, ApJ, 373, 308 Fossat, E., et al. 1992, A&A, 266, 532 Fröhlich, C., et al. 1995, Sol. Phys., 162, , Sol. Phys., 170, 1 Gabriel, A. H., et al. 1995, Sol. Phys., 162, , Sol. Phys., 175, 207 García, R. A., Jiménez-Reyes, S. J., Turck-Chièze, S., & Mathur, S. 2004, in Helio- and Asteroseismology: Towards a Golden Future, ed. D. Danesy ( Noordwijk: ESA), 432 García, R. A., et al. 1998, ApJ, 504, L , A&A, 442, 385 Gough, D. 1985, in Proc. ESA Workshop on Future Missions in Solar, Heliospheric and Space Plasma Physics, ed. E. Rolfe & B. Battrick ( Noordwijk: ESA), 183 Houdek, G., Balmforth, N. J., & Christensen-Dalsgaard, J. 1995, in Proc. Fourth SOHO Workshop: Helioseismology, ed. J. T. Hoeksema et al. ( Noordwijk: ESA), 447 Isaak, G. R., 1983, Sol. Phys., 82, 205 Jain, R., & Roberts, B. 1996, ApJ, 456, 399 REFERENCES Jefferies, S. M., Pomerantz, M. A., Duvall, T. L., Jr., Harvey, J. W., & Jaksha, D. B. 1988, in Seismology of the Sun and Sun-like Stars, ed. E. Rolfe ( Noordwijk: ESA), 279 Jiménez, A. 2002, ApJ, 581, 736 Jiménez, A., Jiménez-Reyes, S. J., & García, R. A. 2005, ApJ, 623, 1215 Jiménez, A., Roca Cortés, T., Severino, G., & Marmolino, C. 1999, ApJ, 525, 1042 Koopmans, L. H. 1974, The Spectral Analysis of Time Series ( New York: Academic) Kumar, P. 1994, ApJ, 428, 827 Kumar, P., Duvall, T. L., Jr., Harvey, J. W., Jefferies, S. M., Pomerantz, M. A., & Thompson, M. J. 1990, in Progress of Seismology of the Sun and Stars, ed. Y. Osaki & H. Shibahashi ( Heidelberg: Springer), 87 Kumar, P., Fardal, M. A., Jefferies, S. M., Duvall, Jr., Harvey, J. W., & Pomerantz, M. A. 1994, ApJ, 422, L29 Libbrecht, K. G. 1988, ApJ, 334, 510 Marmolino, C., & Severino, G. 1991, A&A, 242, 271 Nigam, R., & Kosovichev, A. G. 1996, Bull. Astron. Soc. India, 24, 195 Pallé, P.L.,Pérez, J. C., Régulo, C., Roca Cortés, T., & Isaak, G. R. 1986, A&A, 169, 313 Pallé, P.L.,Régulo, C., Roca Cortés, T., Sanchez Duarte, L., & Schmider, F. X. 1992, A&A, 254, 348 Pallé, P. L., et al. 1999, A&A, 341, 625 Schmieder, B. 1977, Sol. Phys., 54, , Sol. Phys., 57, 245 Staiger, J. 1987, A&A, 175, 263