Phase Differences and Gains between Intensity and Velocity in Low

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THE ASTROPHYSICAL JOURNAL, 525 : 1042È1055, 1999 November 10 ( 1999. The American Astronomical Society. All rights reserved. Printed in U.S.A.

PHASE DIFFERENCES AND GAINS BETWEEN INTENSITY AND VELOCITY IN LOW-DEGREE ACOUSTIC MODES MEASURED BY SOHO1 ANTONIO JIME NEZ,2 TEODORO ROCA CORTE S,2,3 GIUSEPPE SEVERINO,4 AND CIRO MARMOLINO5 Received 1999 February 12 ; accepted 1999 July 2

ABSTRACT Helioseismic instruments aboard SOHO are making possible a more accurate way of investigating the internal structure of the Sun. Making use of the di†erent techniques and characteristics of these instruments, it is possible to measure solar oscillations as variations of the photospheric velocity (GOLF, MDI) or as irradiance and radiance Ñuctuations (VIRGO, MDI). Among the other advantages of observing solar oscillations simultaneously with di†erent instruments and techniques, the study of velocity and irradiance measurements provides information on nonadiabatic e†ects in the radiatively cooled solar atmosphere. The thermodynamical properties of the atmosphere determine a phase shift between intensity and velocity (downward positive) oscillations of [90¡ in the case of an adiabatic atmosphere. Here we compute the phase di†erences and gains between intensity and velocity acoustic modes measured by SOHO to quantify the nonadiabatic degree of the solar atmosphere. After correcting the observed phase di†erences of the solar background inÑuence, we Ðnd not exactly an adiabactic behavior, but close to it. Finally, we compare our results with three di†erent theoretical models of the solar atmosphere, Ðnding the best agreement with a model that includes turbulent pressure associated with convection and Ñuctuations of the superadiabatic temperature gradient. Subject headings : Sun : atmosphere È Sun : atmospheric motions È Sun : oscillations

1.

INTRODUCTION

time series results agreed well with the former study and revealed an IÈV phase di†erence between [120¡ and [130¡. In spite of this, two problems arose. While in the Ðrst observation the Ñight time of the balloon was only about 7 hr, giving a poor frequency resolution, in the second observation the study of low-degree p-modes is very difficult due to the high atmospheric noise level in groundbased intensity observations ; therefore, it became evident that intensity observations from space were needed. In 1988, the cruise phase of the USSR mission to the Martian satellite Phobos o†ered the opportunity to measure such phase shift using space intensity data of the IPHIR instrument (FroŽhlich et al. 1988) aboard Phobos. In 1991, Schrijver, Jimenez, & Dappen (1991, hereafter IPHIR) used these intensity data and the simultaneous velocity data from the Mark I instrument (three periods of 1 week and one period of 3 weeks) to obtain the most accurate measurements of the phase shift, [119¡ ^ 3¡, which agreed with previous results to within the errors. Now, with the SOHO mission (Domingo, Fleck, & Poland 1995), it is possible to get uninterrupted simultaneous intensity and velocity measurements from space and drastically improve the calculations of these phase shifts as has never been possible previously. In this article we compute the phase di†erences and gains between IÈV low-degree p-modes using data from VIRGO (FroŽhlich et al. 1995), GOLF (Gabriel et al. 1995), and MDI (Scherrer et al. 1995).

The helioseismic instruments on board SOHO (VIRGO, variability of solar irradiance and gravity oscillations ; GOLF, global oscillations at low frequencies ; and MDI, Michelson Doppler Imager) have obtained the highest quality data to date, thereby facilitating a more accurate way of investigating the internal structure of the Sun. Due to the di†erent techniques used by these instruments, it is possible to measure the solar oscillations simultaneously, either as variations in the photospheric velocity (GOLF and MDI) or as intensity Ñuctuations (VIRGO and MDI). The main objective of this work is to obtain the gain and phase di†erences between the p modes measured as intensity and velocity Ñuctuations and compare them with predictions of several theoretical models to quantify the adiabatic behavior of the solar atmosphere. FroŽhlich & van der Raay (1984) measured these phase shifts using velocity data obtained from ground observations with the well-known Mark I resonant spectrophotometer at the Observatorio del Teide (Tenerife) since 1976, and intensity data from a stratospheric balloon Ñight with a very short span of data. Jimenez et al. (1990) used the velocity data of the same instrument (Mark I) and simultaneous intensity observations obtained with two ground stations (Tenerife and San Pedro Martir, Baja California) with the SLOT Sun photometer (Andersen et al. 1988). These longer 1 Based on observations with the VIRGO, GOLF, and MDI instruments on board SOHO. 2 Instituto de Astrof• sica de Canarias, E-38200 La Laguna, Tenerife, Spain. 3 Departamento de Astrof• sica, Universidad de La Laguna, La Laguna, Tenerife, Spain. 4 Osservatorio Astronomico di Capodimonte, via Moiariello 16, I-80131 Napoli, Italy. 5 Dipartimento di Scienze Fisiche della Universita` di Napoli, Mostra d“Oltremare Pad. 19, I-80125 Napoli, Italy.

2.

INSTRUMENTATION AND OBSERVATIONS 2.1. V IRGO (Intensities) 2.1.1. Sun Photometers (SPM)

The performance of the VIRGO instrument has been reported by FroŽhlich et al. (1997a). The VIRGO/SPM comprises three Sun photometers at 402 nm (blue), 500 nm (green), and 862 nm (red) that look at the Sun as a star (the 1042

PHASE DIFFERENCES IN LOW-DEGREE ACOUSTIC MODES formation height for all three is a few tens of kilometers). The sampling rate of VIRGO/SPM is 60 s. The VIRGO sampling has an o†set of 30 s with respect to the other two SOHO helioseismic instruments, because VIRGO is the only instrument synchronized to a daily pulse sent every day at midnight. This daily pulse was synchronized to universal time (UT) before 1996 March 27, and then to temps atomique international (TAI), which is 30 s apart in the time span used in this work. This timing conditions of VIRGO have obviously been taken into account in the data analysis. The three VIRGO channels of VIRGO/SPM will be referred to in this paper as VIRGO(red), VIRGO(green), and VIRGO(blue). 2.1.2. T otal Solar Irradiance (T SI)

There are two di†erent absolute radiometers on VIRGO that measure the total solar irradiance (TSI), also called the ““ solar constant ÏÏ : DIARAD and PMO6 V (FroŽhlich et al. 1997b). The sensitivity of the radiometers goes from 100 nm to 100 km. The DIARAD instrument has a sampling rate of 180 s and is therefore not suitable for p-mode analysis. On the other hand, the PMO6 V has been sampled at 60 s since a couple of weeks after VIRGO went into operation. This modiÐcation of the original mode of operation (FroŽhlich et al. 1995) was implemented because the shutter stopped working and remains open. Since then, VIRGO uses the PMO6 V to provide coverage three times per day to get the dark-cavity measurements. As a result, only the PMO6 V is used for the p-mode comparison. The TSI measurements of the VIRGO PMO6 V radiometer will be referred to in this paper as VIRGO(TSI). 2.1.3. L uminosity Oscillation Imager (L OI)

The performance of the VIRGO/LOI instrument has been reported by Appourchaux et al. (1997). The VIRGO/ LOI images the Sun at 500 nm (formation height same as SPM[green]) on a 12 pixel photodiode allow the detection of l \ 8. The sampling rate is 60 s, being in phase with all the other instruments of VIRGO. The LOI pixel data have been also combined to obtain a signal of the Sun seen as a star. This instrument will be referred to in this paper as VIRGO(LOI). 2.2. MDI (V elocity and Intensity) The MDI instrument has been described by Scherrer et al. (1995) and its performance reported by Kosovichev et al. (1997). This instrument measures velocity in the Ni I 676.8 nm line using a double Michelson interferometer with a narrow Ðxed passband Lyot Ðlter. The velocity signal is from Ðltergrams taken at four wavelengths across the spectral line (the equivalent formation height is 250È300 km). The continuum intensity and line depth are computed from these and an additional Ðltergram at another wavelength. The data used in this work come from a proxy that matches the pixel size of the LOI in VIRGO (Appourchaux et al. 1997). Each of the 12 LOI scientiÐc pixels is divided into a di†erent number of bins : the four north-south pixels are each divided into 17 bins, the four east-west and the four central pixels into nine bins, and the four guiding pixels into ten bins (see Appourchaux et al. 1997 for the nomenclature of the pixels). For comparison with the full-disk integrated signals of VIRGO/SPM and GOLF, the 140 nonlimb LOIproxy bins used to determine a ““ full-disk ÏÏ signal have been averaged. The Ðnal sampling rate is 60 s. The intensity and

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velocity measurements of the MDI instrument will be referred to in this paper as MDI(I) and MDI(V ), respectively. 2.3. GOL F (V elocity) GOLF is a resonant-scattering spectrophotometer (Gabriel et al. 1995) that measures the line-of-sight velocity using the sodium doublet, similar to the IRIS or BISON ground-based networks. The GOLF door was deÐnitively opened in 1996 January and became fully operative by the end of that month. Over the following months, occasional malfunctionings of its rotating polarizing elements were noticed, which led to the decision to stop them in a predetermined position, and truly nonstop observations began by 1996 mid-April. Since then, GOLF has been continuously and satisfactorily operating in a mode unforeseen before launch, which has been found to have fewer limitations than anticipated. The signal, then, consists of two close monochromatic photometric measurements in a very narrow band (25 mAŽ ) on the left wing of the sodium doublet (Gabriel et al. 1997). This signal has been calibrated as velocity and has been shown to have a nature similar to other known velocity measurements, such as those of IRIS, BISON, etc. (Palle et al. 1999). The sampling of the GOLF data used in this work is also 60 s. 2.4. Data Sets For the reasons mentioned above regarding the GOLF signals, we will Ðrst use the time series from 1996 February 19 to March 19 (before the rotating polarizing element malfunction started), in which GOLF measured the usual Doppler velocity, which we will call the ““ ratio ÏÏ velocity, GOLF(ratio). The spectral resolution of these times series is 0.38 kHz. We will compare this GOLF time series with the corresponding VIRGO one, mainly with the green channel, to compute phase di†erences and gains and compare the results with those of the IPHIR paper. In a second step we will use the simultaneous 2 month time series of VIRGO, GOLF, and MDI between 1996 May 24 and July 24 for a full comparison between all signals with a spectral resolution of 0.19 kHz. During this period, GOLF measurements correspond to the blue wing of the sodium doublet ; we will call this velocity the ““ blue-wing velocity,ÏÏ GOLF(BW) (for a full account see Palle et al. 1999). In phase-di†erence studies, the absolute timing is a very important parameter ; in fact, an error in the timing produces a frequency-dependent phase shift, a straight line in the IÈV phase diagram whose slope is proportional to the timing error. In all the time series used in this work, the absolute timing has been checked and veriÐed very carefully. 3.

BIVARIATE SPECTRAL ANALYSIS OF TIME SERIES

This powerful method, described in Koopmans (1974), permits the computation of the coherence, phase shift, and gain between two time series of the same length. It has already been used by several authors (Kendall, Stuart, & Ord 1983 ; FroŽhlich & van der Raay 1984 ; Schrijver et al. 1991). Let A and B be two time series of length T , and let sin A, cos A, sin B, and cos B be the sine and cosine amplitudes of the spectra for series A and B, respectively. The power spec-

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FIG. 1.ÈTwo examples of the bivariate spectral analysis between intensity and velocity time series. From bottom to top : intensity amplitude spectrum, velocity amplitude spectrum, coherence, phase di†erence, and gain. The left and right panels correspond to two di†erent frequency intervals in the p-mode range.

tral densities, P (l) and P (l), the cospectral density, C (l), AB the quadratureA spectral Bdensity, q (l), and the complex AB cross-spectral density, P (l), are deÐned as AB T P (l) \ [sin2 A(l) ] cos2 A(l)] , (1) A 2 T P (l) \ [sin2 B(l) ] cos2 B(l)] , B 2

coherence can be computed as SC (l)T2 ] Sq (l)T2 AB AB SP (l)TSP (l)T A B o SP (l)T o2 AB \ . SP (l)TSP (l)T A B

Coh2 (l) \ AB

(2)

T C (l) \ [sin A(l) sin B(l) ] cos A(l) cos B(l)] , (3) AB 2 T q (l) \ [sin A(l) cos B(l) [ sin B(l) cos A(l)] , (4) AB 2 P (l) \ C (l) [ iq (l) . (5) AB AB AB Before computing coherence, phase shift, and gains, these bivariate parameters need to be smoothed ; the length of the smoothing is not too critical, but needs to be chosen adequately. If it is too short, the parameters are noisy from bin to bin, and if it is too large the frequency resolution is lost. After some tests varying the length of the smoothing, we concluded that a smoothing length of 3.8 kHz is the best tradeo† for the p-mode range. After apply the smoothing (indicated by angle brackets in the following equations) to all the above deÐned parameters of the spectra of all the time series used in this work, the

(6)

The coherence is the analog of the linear correlation coefÐcient between the two time series A and B in linear regression analysis : Coh2 measures the degree to which the AB series B can be represented as the output of a linear Ðlter whose input is series A, or, more precisely, the proportion of the power at a given frequency that can be explained by its linear regression on the other. The phase di†erence, */ (l), and gain, g (l), between series A and B are given AB by AB

C

D

Sq (l)T AB , */ (l) \ tan~1 AB SC (l)T AB o SP (l)T o AB g (l) \ , AB SP (l)T A and the corresponding errors by (l) \ sin~1 v *ÕAB

CS

(7) (8)

A BD

1 [ Coh2 (l) a AB t 2n~2 (2n [ 2)Coh2 (l) 2 AB

,

(9)

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PHASE DIFFERENCES IN LOW-DEGREE ACOUSTIC MODES

S

[1 [ Coh (l)]2g (l) AB AB F (a) , (10) 2,2n~2 gAB (n [ 1)Coh (l) AB where n is the equivalent degrees of freedom (EDF), t (a/2) is the Student t-distribution, F (a/2) is the 2n~2 F-distribution, and a is the conÐdence 2,2n~2 Fisher level at which the errors are computed. We chose a \ 0.8 for phase di†erences errors and a \ 0.9 for gain errors. A best-deÐned gain can be computed by dividing the gain by the coherence, as in Schrivjer et al. (1991). As in linear regression analysis, the results depend on whether an x variable is Ðtted against a y variable or vice versa. In fact, the ratio of the slopes (g and 1/g ) is equal to the square of AB AB the correlation coefficient in linear regression, as is also the case here : v

(l) \

C2 (l) \ g (l)g (l) . (11) AB AB BA If the relative errors in both time series are the same, a useful slope in linear regression terms is deÐned as the ratio of the standard deviations in the x and y directions. By analogy, the best-Ðt amplitude ratio is

S

S

g (l) SP (l)T g (l) (12) AB \ B \ AB . g (l) SP (l)T Coh (l) BA A AB Figure 1 shows an example of the bivariate analysis procedure between intensity and velocity time series for a frequency range in the central part of the p-mode spectrum (left) and for a higher frequency one (right). ( (l) \ AB

4.

INTENSITY-VELOCITY COMPARISONS : RESULTS

In this section we compute phase di†erences and gains between intensity and velocity acoustic modes measured

1045

simultaneously by VIRGO, GOLF, and MDI. Only IÈV p-modes with coherence higher than 50% have been used in the analysis. 4.1. Phase Di†erences Using GOL F(Ratio) V elocities As mentioned in ° 2, we will Ðrst use the time series between 1996 February 19 and March 19, in which GOLF was fully operative and measuring ratio velocity, for a comparison with the results of IPHIR, which were also obtained with the ratio velocity of the Mark I instrument. The intensity data correspond to the green channel of VIRGO at 500 nm, the same as IPHIR. The Mark I instrument uses the K I 769.9 nm potassium line, and GOLF uses the sodium doublet (D1 at 589.6 nm and D2 at 589.0 nm). The di†erence in the working spectral line is not a problem, since it has already been proved using calibrated ratios obtained simultaneously for both lines ; the mean phase di†erence between solar p-modes of low degree is zero within the errors (Palle et al. 1992), as expected for standing waves. Therefore, the phase di†erences obtained in this article with GOLF(ratio) velocities must agree with those obtained by IPHIR. The phase di†erences obtained with bivariate analysis are really constant although the p-modes peaks, and we average three bins around the maximum of the peak to get an averaged value. Figure 2 shows the computed IÈV phase di†erences obtained for the VIRGO(green) and GOLF(ratio) data. The di†erent symbols correspond to di†erent l values (0, 1, 2, and 3). For the Ðrst time, a degree dependence of the phase di†erence going toward smaller (more negative) values is clearly seen when the mode degree increases. Looking at the mean values on the Ðgure insets, this degree dependence increases with l (the di†erence between l \ 0, 1,

FIG. 2.ÈPhase di†erences between the VIRGO(green) channel and GOLF(ratio) velocity p-modes for degrees l \ 0, 1, 2, and 3 (averages denoted by angle brackets), and compared with the IPHIR results.

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l \ 1, 2, and l \ 2, 3 are around 7¡, 14¡, and 26¡, respectively). The IPHIR results are plotted in Figure 2 with a dotted line (below 2500 kHz) and a solid line (above 2500 kHz), representing the averaged values (Fig. 11 of IPHIR) obtained in that paper for l \ 0, 1, and 2. The reason for plotting below and above 2500 kHz is that in the IPHIR paper the results below that frequency were questioned by the authors because of the low coherence in that frequency range. The average value found by IPHIR was [119¡ ^ 3¡, while in this paper, excluding l \ 3, the value is [121¡.28 ^ 1¡.23, which agrees very well to within the errors.

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4.2. Phase Di†erences Using GOL F Blue-W ing V elocities We will now use the GOLF(BW) velocity data from 1996 May 24 to July 24 and the corresponding simultaneous VIRGO (TSI, LOI, and green) and MDI intensity data. Figure 3 shows these phase di†erences. Although the degree dependence is found to be the same as before, the absolute values are lower. In the GOLF time series between 1996 February 19 and March 19, during which period GOLF was fully operative, it is possible to compute the ratio and blue-wing velocities simultaneously and to compare the phases. Figure 4 shows these phase di†erences, and a small

FIG. 3.ÈPhase di†erences between intensities and GOLF(BW) velocity p-modes for degrees l \ 0, 1, 2, and 3 (averages denoted by angle brackets). The intensity signals of (a) VIRGO(green), (b) VIRGO(LOI), (c) VIRGO(TSI), and (d) MDI(I) have been used.

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FIG. 4.ÈPhase di†erences between the GOLF(BW) velocity and GOLF(ratio) velocity for di†erent degrees l \ 0, 1, 2, and 3 (averages denoted by angle brackets) in the time span during which both were available (1996 February 19ÈMarch 19).

phase shift of around [7¡ is clearly seen for the whole spectrum, which agrees with the work of Palle et al. (1999). Note also that there is a small degree dependence, very probably as a consequence of the intensity-like signal leaking into the GOLF blue-wing signal. If we add the average values of Figure 4 to the average values of Figure 3a, we get what would be the phase di†erences for VIRGO(green)ÈGOLF(ratio) obtained from blue-wing measurements. The approximate results are [121¡, [113¡, [118¡, [134¡, and [171¡ for SallT, l \ 0, 1, 2, and 3, respectively, in perfect agreement with Figure 2. Only l \ 3 seems a little di†erent (around 10¡) ; this is because in Figure 2 there are l \ 3 modes with frequencies higher than 3800 kHz that have smaller phase di†erences. Note that values for l \ 3 in Figures 2 and 3a, for the frequency range between 2600 and 3800 kHz, are all very similar to within the errors, approximately [170¡, because the bluewing[ratio phase di†erence for l \ 3 is less than 2¡ (Fig. 4). 4.3. Phase Di†erences Using MDI V elocity We have also used the MDI intensity and velocity data together with VIRGO (TSI, LOI, and green) to compute phase di†erences, as shown in Figure 5. The results are congruent, although some small di†erences appear with respect to the previous ones. Comparing Figure 5a with Figure 2 and the average values therein, and taking into account that the intensity data are the same (VIRGO[green]) and the phase di†erences between the calibrated ratios obtained simultaneously in the sodium and potassium lines is zero to within the errors (Palle et al. 1992), the most plausible explanation for these small di†erences is the way in which the Doppler shift velocity has been

computed. In the next subsection we will discuss this point further. 4.4. Some Remarks on the IÈV Phase Di†erences Using Di†erent Sources of V elocity Data : GOL F(BW ), GOL F(Ratio), and MDI(V ) Palle et al. (1999) have shown that the ratio velocity is not a pure Doppler velocity. In that paper it is shown that the phase shift between the measurements of the blue (BW) and red (RW) wings of the sodium lines used by GOLF is not 180¡, as would correspond to a pure Doppler velocity shift of the line, but 163¡. Comparing the blue and red wing measurements with the calibrated ratio velocity, the phase di†erences, BWÈratio and RWÈratio, are around [9¡ and 8¡, respectively (for l \ 0), which provide the missing 17¡ to get the required 180¡ for a pure velocity signal. This explains why we need to subtract the average values of Figure 4 (GOLF[BW]ÈGOLF[ratio]) from those of Figure 3a (VIRGO[green]ÈGOLF[BW]) to obtain the values of Figure 2 (VIRGO[green]ÈGOLF[ratio]), as explained in ° 4.2 (we know how to convert IÈGOLF[BW] to IÈratio phase di†erences). When the GOLF BW signal is compared with that obtained from a well-known ground instrument, Mark I, an identical behavior is obtained to the case of the GOLF ratio, giving a phase di†erence, GOLF(BW)ÈMark I(ratio), of around [9¡ for the l \ 0 modes, with the result that the ratio measurements of Mark I and GOLF are of the same nature (Palle et al. 1999). Nevertheless, this e†ect is not yet completely understood ; it could be related to the physics of the oscillations, which would introduce a small intensity-like variation (probably opacity) mixed with the Doppler shift, or a distortion of the line proÐle. When com-

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FIG. 5.ÈPhase di†erences between intensities and MDI velocity p-modes for degrees l \ 0, 1, 2, and 3 (averages denoted by angle brackets). The intensity signals of (a) VIRGO(green), (b) VIRGO(LOI), (c) VIRGO(TSI), and (d) MDI(I) have been used.

puting the calibrated ratio velocities, this e†ect remains to some degree, and we are obviously interested in knowing the magnitude of this e†ect precisely, or at least in having some idea of the precision of the observed IÈV phase di†erences. Comparing Figure 2 (VIRGO[green]ÈGOLF[ratio]) with Figure 5a (VIRGO[green]ÈMDI[V ]), we see di†erences of about [4¡, [7¡, [4¡, and [7¡ for SallT and l \ 0, 1, and 2, respectively (for l \ 3 it is higher due to the lack of points in Fig. 5 at high frequencies, as explained in ° 4.2). Assuming that the ratio velocity measurement is not a pure Doppler shift and taking into account that the way in which the MDI velocity is computed di†ers from that of the ratio and between both of the IÈV phase-di†erence diagrams

(Figs. 2 and 5a), there is a mean di†erence of around 5¡ ; this ““ contamination ÏÏ seems to be very small, and we can select a value of around 5¡ as the uncertainty of the computed IÈV phase di†erences due to this unknown e†ect. 4.5. Amplitude Gains The best amplitude gains between p-modes observed in intensity (VIRGO[green]) and velocity (MDI, GOLF[BW], and GOLF[ratio]) have also been computed, as described in ° 3 (() and plotted in Figure 6. The most interesting feature is contained in Figure 6a, corresponding to the gain of VIRGO(green) and GOLF(ratio) velocity, in which we compare with the results

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FIG. 6.ÈGains between VIRGO(green) channel and (a) GOLF(ratio), (b) GOLF(BW), and (c) MDI(V ) velocities for degrees l \ 0, 1, 2, and 3 (averages denoted by angle brackets). In all panels, the gain of the noise between the p-mode groups have been plotted. In (a) the results are compared with IPHIR (see text).

of IPHIR (represented by a dotted line below 2500 kHz and a solid line above that value, as explained above), which have been divided by 1.4, the gain factor between the solar radial velocities measured by the sodium and potassium resonant scattering spectrophotometers (Palle et al. 1992). The small hump around 4100 kHz in the IPHIR data is not observed now, and the increase in the gain below 2500 kHz is found to be somewhat lower with SOHO data. The Ðlled circles in Figure 6 represent the gain of the noise computed between the p-mode groups, which is higher than the p-mode gains (to avoid confusion, the error bars of the noise gain have not been plotted, since they are

obviously large). It is interesting to note that below 2500 kHz the noise gain is also higher than those of the p-modes, indicating that the small increase in the p-mode gain below this frequency seems to be real. In conclusion, it can be said that the general trend of the p-mode gains is a fairly weak variation with frequency, although below 2200 kHz the background noise in intensity measurements is too high and is probably responsible for the small increase. 5.

INTENSITYÈINTENSITY COMPARISONS : RESULTS

In this section we compute phase di†erences and gains between acoustic modes measured as intensity variations at

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di†erent wavelengths obtained by VIRGO(blue, green, red, TSI) and MDI(I). VIRGO(LOI) has not been included because the observations are equivalent to VIRGO(green). Only IÈI p-modes with coherence higher than 50% have been used in the analysis. 5.1. Phase Di†erences The phase di†erences between intensity p-modes measured by SOHO are plotted in Figure 7. We have used the blue, green, and red channels of VIRGO SPM, the TSI of the VIRGO radiometers, and intensity data from MDI. We expect a null mean value for these phase di†erences, since they are standing waves as measured by several authors

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(Isaak et al. 1989 ; Jimenez et al. 1990 ; Palle et al. 1992). In Figure 7 some interesting features appear. Figures 7a and 7d show that the phase di†erences are zero to within the errors ; the wavelengths of these data are 402 nm (VIRGO[blue]), 500 nm (VIRGO[green]), and 678.8 nm (MDI). The phase di†erences between VIRGO(green) and VIRGO(red) (at 862 nm) are clearly nonzero above 2500 kHz, showing a frequency and degree dependence (Fig. 7b). The phase di†erence is zero up to 2500 kHz, decreases to a minimum around 3600 kHz, and then increases again. This frequency dependence seems to be the same for l \ 0 and l \ 1, less noticeable for l \ 2, and practically Ñat for l \ 3. This feature was previously noted by IPHIR (see Fig. 4 of

FIG. 7.ÈPhase di†erences between VIRGO(green) and (a) VIRGO(blue), (b) VIRGO(red), (c) VIRGO(TSI), and (d) MDI(I) intensities for degrees l \ 0, 1, 2, and 3 (averages denoted by angle brackets).

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PHASE DIFFERENCES IN LOW-DEGREE ACOUSTIC MODES

that paper), although no degree dependence was detected (l \ 3 was not observed in the IPHIR data). The frequency dependence was treated as linear by IPHIR, and Ðtted the straight line */ \ [22l (mHz) ] 56¡. If we use this GR model in our linear range (2500È3700 kHz), they agree well. This feature seems to be a characteristic of the infrared range. In Figure 7c, where the phase di†erences between VIRGO(green) and the total solar irradiance (TSI) are plotted, this feature is also observed, although less clearly than before ; since the absolute radiometers are based on the measurements of the heat Ñux (FroŽhlich et al. 1995), they are sensitive to a wide range of wavelengths (from 100 nm to 100 km), which includes the optical and infrared ranges. If,

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as mentioned above, this feature seems to be an infrared characteristic, what we observe in Figure 7c is to be expected. 5.2. Amplitude Gains The amplitude gains between the VIRGO(blue, green, red, TSI) and MDI intensity are shown in Figure 8, compared to the VIRGO(green) channel. The gain in Figure 8a (VIRGO[red]/VIRGO[green]) agrees with the result of the IPHIR paper. A slight increase in the gain with frequency is observed. In the IPHIR paper, this was described by ( \ 0.045l (mHz) ] 0.4, which agrees with our results. GR For the VIRGO(blue)/VIRGO(green) gains this slight

FIG. 8.ÈGains of VIRGO(red), VIRGO(blue), VIRGO(TSI), and MDI(I) related to VIRGO(green) for degrees l \ 0, 1, 2, and 3 (averages denoted by angle brackets).

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FIG. 9.ÈPhase di†erences corresponding to Fig. 2 after the correction of the background phase inÑuence. The IPHIR results, although they have not been corrected, are also plotted for a easy comparison with Fig. 2.

increase is not observed (Fig. 8b), but it is enlarged for the VIRGO(TSI)/VIRGO(green) results (Fig. 8c) and even for MDI(I)/VIRGO(green) gains (Fig. 8d). In this case, however, it is not too signiÐcant because there are only three points at high frequencies with very large errors bars. It seems that, again, an e†ect related to the infrared range is observed in the results. 6.

ON THE INFLUENCE OF THE BACKGROUND AND S/N IN THE PHASE-DIFFERENCE DETERMINATIONS

We have seen that the IÈV phase di†erences obtained in this work are in agreement with previous ones and give values that di†er signiÐcantly from [90¡. Moreover, a clear degree dependence for the IÈV phase di†erences is found that is difficult to explain. One e†ect not considered up to now is the inÑuence of the solar background noise ; if it is not uncorrelated with the modes, it could change the IÈV phase di†erences measured (Severino, Strauss, & Je†eries 1998 ; Oliviero et al. 1999). If the phase of the background is similar to that of the p-modes, only the mode amplitude would be altered, but if it is substantially di†erent, the p-mode phase will be also altered. The magnitude of this change will depend on the relative amplitude of the modes and the background, that is, on the signal-to-noise ratio (S/N), being larger for lower S/N because signals of similar amplitude (p-modes and background) are then added. In our results, the degree dependence of the phase di†erences increase with l, just as the S/N decreases, suggesting the possibility that the solar background could be partially responsible for this e†ect. An evaluation of the extent to which the phase di†erences are a†ected by the background can be made using a simple simulation based on the following formulations. We built

the two time-string series

A

S (t) \ I sin 2nlt [ 1 ]I

noise

A

A

sin 2nlt [

S (t) \ V sin 2nlt ] 2 ]V

noise

B

*/ IV 2

A

B

(13)

B

(14)

*/ IVnoise , 2

B

*/ IV 2

sin 2nlt ]

*/

IVnoise , 2

where I and V are the mode amplitude and I and V noise are the background amplitude in intensity noise and velocity, respectively. */ is a Ðxed value of the IÈV phase di†erence of the signalIVin the absence of noise, and */ is the IÈV phase di†erence of the noise background. IVnoise After the time series are built, the power spectra are computed, and we obtain the resulting phase di†erence, */ . S1 S2 In the absence of noise (I \ 0, V \ 0) we obviously obtained */ , but with noise noise the noise obtained value di†ers IV from */ . We compute */ [ */ and consider this IV produced S1 S2 by a particular quantity IVas the contamination noise (I ,V , or */ ) over a particular signal (I, V , noise IV noise or */ noise , respectively). After several tests of this simple IV simulation, changing the amplitudes of the signals and noise, the phase of the noise, and the percentage of the phase assigned to the signals and to the noise, we can conclude that the phase results mainly depend on the amplitudes (S/N) and on the phase of the noise. To get a realistic evaluation of the contamination and therefore the value of */ [ */ that we will use to IV di†erences, S1 S2 we computed the correct the measured phase

No. 2, 1999

PHASE DIFFERENCES IN LOW-DEGREE ACOUSTIC MODES

phase di†erence and amplitude of the noise in the frequency intervals between the p-modes using the I and V spectra used to get the results of Figure 2. (VIRGO[green] and GOLF[ratio]). The phase di†erence of the noise is always lower than the p-modes and changes almost linearly from [260¡ at 2000 kHz to [180¡ at around 4300 kHz. At higher frequencies, the phase di†erence of the noise increases ; this is most probably due to the fact that the line width of the modes increases drastically and reduces the background phase differences between the p-mode groups. With these realistic parameters, amplitudes, and phase di†erences of the noise and p-mode amplitudes, we carry out the simulation for each mode and compute the phase correction, which is added to the already measured phase di†erence (see ° 4). The results are shown in Figure 9, in which the IPHIR line result has also been plotted for a easier comparison with Figure 2. The averaged phase di†erence has changed by about 16¡ ; the l \ 0 and l \ 2 have changed by a similar quantity (18¡ and 21¡), l \ 1 has a lesser change (12¡), and l \ 3 has changed by around 55¡, the Ðnal result being that the degree dependence has disappeared given the standard errors. All of this is in agreement with what we can expect from the S/N. Therefore, the results obtained previously must also be corrected for such an e†ect. 7.

COMPARISON BETWEEN THE OBSERVED IÈV PHASE DIFFERENCES AND THEORY

The results found in the last section, in which the phase di†erences have been corrected by taking into account the S/N of each mode and the phase di†erence of the noise close to them (Fig. 9), show an atmosphere with a behavior close to the adiabatic case. Several points must be taken into account when computing nonadiabatic e†ects on the waves : (1) Ñuctuations of the superadiabatic temperature gradient in the low photosphere induced by oscillations ; (2) radiative damping of the wave temperature Ñuctuations in low photospheric layers ; (3) turbulent pressure associated with convection ; (4) radiative transfer to pass from the wave temperature and pressure perturbation to the observed intensity Ñuctuations ; and (5) integration of intensity over the disk to account for the center-to-limb spatial distribution of the wave perturbations. In this section we compare the results found with three di†erent models : model 1, which takes into account point 2, and models 2 and 3, which take into account points 1 and 3. The numerical code of model 1 is constructed following the approach of Marmolino & Severino (1991). This model assumes that the unperturbed atmosphere is plane-parallel, isothermal, and in hydrostatic equilibrium (which implies an exponential stratiÐcation in density and pressure), and the e†ect of wave radiative damping is taken into account with the Newtonian cooling approximation by means of a constant radiative damping time, t . This code produces R T ÈV phase di†erences for l \ 1, considering upward velocities as positive (Marmolino & Severino 1991). Before any comparisons, two points need to be clariÐed. First, the code uses temperature perturbations, whereas the perturbations we observe are in intensity. Usually I is taken as a proxy of T . To verify this, some computations of the IÈV phase di†erences from theoretical wave and radiative transfer models have been done. These preliminary calcu-

1053

lations indicate that I is a good proxy of T except close to the f-mode (Severino et al. 1998). Second, in order to make the conventions and notations used in the phase di†erences studies uniform, Masiello, Marmolino, & Strauss (1998) suggest using a notation that attributes positive values to the phase di†erences when the intensity perturbation leads the velocity perturbation, taking positive as away from the Sun (upward positive velocity), precisely the convention used in our code to compute phase di†erences. The observational results presented here and in previous papers, as well as other theoretical computations, use the opposite notation ; that is, downward velocity as positive, which gives negative phase di†erences. This change in notation involves merely adding (or subtracting) 180¡. In Figure 10 we use both notations, showing downward positive velocity on the left y-axis and upward positive velocity on the right y-axis. In Figure 10 the results of model 1 are plotted (dashed lines) together with the observational results. The model has been computed for a small value of the horizontal wavenumber, corresponding to l \ 1, and for T \ 5000 and 6000 K. These two results are very similar, so only the results corresponding to T \ 5000 K have been plotted in Figure 10, to avoid confusion. The calculations have been performed for three damping-time values (Fig. 10, bottom to top) : t \ 105 (essentially adiabatic), 50, and 1 s (close to R isothermal). The observational results are those of Figure 9 (the uncertainty of 5¡ explained in ° 4.3 does not change the general trend of the comparisons presented in this section) and correspond to the VIRGO(green) intensity and GOLF(ratio) velocity plotted as di†erent symbols for mode degrees l \ 0, 1, 2, and 3 after the background-phase correction. There is a general discrepancy between model 1 and the observations. The computed T ÈV values (proxy for IÈV ) are [90¡ (downward positive velocity) for the adiabatic case and become much greater than [90¡ if a short damping time is present ; on the other hand, the observed values are sometimes lower than [90¡ (between 2600 and 3300 kHz, precisely where the modes are better deÐned and the background correction less important). One important physical process to be taken into account in the models is convection, not included in model 1. In model 2, Gough (1985) has computed the e†ects of turbulent pressure, as well as the modiÐcations in the structure of the superadiabatic region due to physical changes in the convection process (Fig. 10, dash-dotted line). Model 2 gives results that show strong deviations from the [90¡ adiabatic value and that are signiÐcantly lower than the observations. Finally, we compare the observations with a third model (model 3) by Houdek, Balmforth, & Christensen-Dalsgaard (1995), which includes convective dynamics in the calculations according to a time-dependent mixing-length formulation. These theoretical results are shown in Figure 10 for three di†erent sets of mixing-length parameters (solid lines). The agreement between model 3 and the observations is acceptable. The model produces phase di†erences close to [90¡ for low (below 2500 kHz) and high (above 3500 kHz) frequencies, and a bit lower in the central p-mode frequency range, just as the observational results. Precisely in this central frequency range is where the p-modes are better deÐned, with higher S/N, and the background phase correction is less critical for modes l \ 0, 1, and 2. We can conclude then that the IÈV phase di†erences observed, if it does

1054

JIMEŠNEZ ET AL.

Vol. 525

FIG. 10.ÈComparisons between IÈV phase di†erences obtained in this work after background-phase correction and theoretical computations using two notations employed by di†erent authors : downward positive velocity on the left y-axis and upward positive velocity on the right y-axis. The dashed lines correspond to model 1 for three di†erent damping times : from bottom to top, t \ 106 s (adiabatic), t \ 50 s, and t \ 1 s (close to isothermal). The R mixing-lengthR parameters. The observed phases dash-dotted line correspond to model 2, and the solid lines corresponds to model 3Rfor three di†erent sets of di†erences are indicated by di†erent symbols for di†erent degrees l \ 0, 1, 2, and 3, and correspond to the results of Fig. 2 (after the background-phase correction has been applied).

not exactly show an adiabatic behavior, is close to it and Ðts model 3 reasonable well. 8.

CONCLUSIONS

The phase di†erences between intensity and velocity acoustic modes obtained by the helioseismic instruments on board SOHO have been measured with a precision never reached until now. The general trend of the IÈV phasedi†erence diagrams at Ðrst revealed phase di†erences much lower than [90¡ (V downward positive), with a slight frequency dependence, increasing toward higher frequencies. The mean value of these phase di†erences is [120¡ ^ 1¡.39 (see Fig. 2), in perfect agreement with previous observations. For Ðrst time, a mode degree dependence is clearly observed decreasing the phase di†erence with increasing l ([113.01 ^ 1.64 for l \ 0, [119.90 ^ 0.98 for l \ 1, [134.91 ^ 1.81 for l \ 2, and [161.41 ^ 4.49 for l \ 3). Up to now, the phase di†erences have been computed in a direct way, without taking into account the inÑuence of the solar background noise, cleanly measured now with the SOHO space observations. The background noise a†ects the p-modes signals, and if its phase is, for example, much lower than that of the p-modes, it diminishes the resulting phase di†erence, the magnitude of this e†ect being a function of the S/N. This is what is happening in fact. When this e†ect is taken into account, the mean value of the measured phase di†erence is a†ected. Moreover, since the inÑuence of the solar background on the phase di†erences is stronger

for modes with low S/N, this e†ect produces an l dependence on the phase di†erence, as we observe. We have used a simple model to estimate this e†ect and correct our observed phase di†erences for the inÑuence of the background. We computed how much the phase di†erence is modiÐed by the background using the S/N of each observed mode of the spectra and the phase di†erence of the noise close to them. The phase di†erences after this correction (Fig. 9) show that no clear degree dependence exists, thereby establishing that it was produced by the solar background. The mean values after the correction are [94.74 ^ 2.31 for l \ 0, [108.15 ^ 2.25 for l \ 1, [113.24 ^ 2.78 for l \ 2, and [102.4 ^ 5.33 for l \ 3. The phase di†erences are around [90¡ below 2500 kHz and above 4000 kHz, and a bit lower than [90¡ in the central part of the p-mode range ; we conclude that the behavior of the phase di†erences is not exactly adiabatic but close to it. Comparing the results with three di†erent theoretical models (Fig. 10.), we conclude that the model of Houdek et al. (1995), which includes turbulent pressure associated with convection and Ñuctuations of the superadiabatic temperature gradient, seems to Ðt the observations reasonable well. Apart from the background phase e†ect, the blue-wing velocity measurements by GOLF have a small contamination, which introduces a small phase shift that must be taken into account when computing IÈV phase diagrams with these data (see Fig. 4).

No. 2, 1999

PHASE DIFFERENCES IN LOW-DEGREE ACOUSTIC MODES

The ratio velocities, conceived up to now as pure Doppler shifts, are not exactly pure, as concluded also by Palle et al. (1999). This introduces an uncertainty of about 5¡ in the IÈV phase di†erences computed in this paper. The I/V amplitude gains obtained here show a small increase for frequencies lower than 2500 kHz that seems to be real, although the errors are high at these frequencies (the noise gain is always higher than the p-mode gains). Nevertheless, the general trend is a weak variation with frequency in the p-mode range, with no apparent degree dependence (see Fig. 6). This agrees with previous results. When computing the IÈI phase di†erences, a null value for wavelengths lower than 677 nm (MDI intensity) is obtained, as expected for standing waves. Nevertheless, when the infrared parts of the spectrum, VIRGO(red) and VIRGO(TSI), are used, a clear frequency dependence (see Fig. 7b) is observed. The mean value for VIRGO(green)È VIRGO(red) is [10.53 ^ 0¡.72, and takes the values [12.53 ^ 1.24 for l \ 0, [12.57 ^ 0.96 for l \ 1, [6.07 ^ 0.85 for l \ 2, and 5.9 ^ 1.0 for l \ 3. This e†ect is not yet well understood, but could be related to a downward-traveling wave, di†erences in the damping time in di†erent layers of the atmosphere, or a di†erent inÑuence

1055

of the solar background noise in the green and red. Further studies will be made of this point. The I/I amplitude gains obtained here show a weak variation with frequency, increasing slightly to higher frequencies when infrared measurements form part of the observations (see Fig. 8). This e†ect is increased (although large errors are present) when total solar irradiance (TSI) measurements are used. Again, an e†ect related to the infrared is observed, as mentioned above. All three instruments beneÐt 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. We thank G. Houdek for providing the model data, R. Garc• a and P. L. Palle for the GOLF(ratio) time series, and Th. Appourchaux, P. Boumier, and our referee for the useful comments. VIRGO, GOLF, and MDI are cooperative e†orts of many individual scientists and engineers at several institutes in Europe and USA to whom we are deeply indebted. This work has been funded by grant 95-0028-C from DGICYT (Spain).

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