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Astron. Astrophys. 364, 799-815 (2000)

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6. Velocities and detrending

The procedures described in the preceeding sections provide two functions [FORMULA] and [FORMULA] which are dependent on the sun-spacecraft line-of-sight velocity (note that the subscript indicating the application to the blue wing is dropped here since these applications will only be to the period between 11-Apr-1996 and 24-Jun-1998 for which the data was exclusively from the blue wing). The signal functions [FORMULA] have been corrected for the line intensity variation and some instrumental effects but not for instrument aging and other possible instrument drifts. There are four separate signals to correct [FORMULA], [FORMULA], [FORMULA] and [FORMULA] and these are treated independently. For the purpose of describing the process, the present discussion will be limited to only one of them: [FORMULA]. Results will be discussed for all channels in the following section. Because most of the effects treated by the detrending occur on long time scales, the figures presented here are for low pass filtered data sampled once per hour. The full reduced data set retains the 10 second sampling and is detrended for long time effects by the methods described here.

Fig. 10 shows S as a function of time. The nature of the variation is not understood, however, it appears to be the result of a decay modulated by some nearly sinusoidal process. Since we do not have any constraints we can place on the nearly sinusoidal process, we model this decay as the product of an exponential-like decay and a sinusoid with a phase function to be adjusted. To find the exponential-like decay, we need to examine periods when the nearly sinusoidal function has the same value. Lacking knowledge of times of equal phase for this function, we choose times when the uncertain phase will have the least impact, namely, those times when this function is at maxima or minima. Examination of Fig. 10 shows that there are two maxima and two minima. Imposing a requirement that the decay-corrected signal should be the same at the two maxima and the same at the two minima yields two conditions that allow us to adjust two constants in the decay formulation. Based on this plan we take the gain factor to be a quadratic exponential:


Fig. 10 shows the product between S and this gain factor [FORMULA].

[FIGURE] Fig. 10. The derived signal function [FORMULA] (solid) and the gain corrected function [FORMULA] as a function of time (dashed). For comparison the orbital velocity is shown as well offset by 1 km s-1 plus the Einstein gravitational shift of 636 m s-1 (dotted).

Although the residual variation in gS is nearly sinusoidal and also nearly synchronous with the orbital velocity variations the initial and final turning points of gS do not coincide with the orbital velocity turning points and the orbital velocity is much further from a sinusoid than is gS. The cause for this variation is not understood but one can speculate that some interference effect perhaps a gradual change in the index of refraction of one cavity of the multi-layer band-pass filter is gradually changing to produce this sinusoid-like variation in the overall throughput. Regardless of the cause, the trends can be removed by taking advantage of the near sinusoidal behavior. To accomplish this a phase function [FORMULA] is derived so that a fitting function [FORMULA] of the form [FORMULA] matches the time dependence of gS. The fit is carried out iteratively in the following sequence: 1) coefficients [FORMULA] and [FORMULA] in Eq. (39) are guessed and a trial value for gS is computed; 2) the turning points in gS are found by fitting a 4th order polynomial to gS in the vicinity of each turning point and then finding either maximum or minimum of this polynomial; 3) the two coefficients [FORMULA] and [FORMULA] are adjusted to cause the two maxima and two minima to have equal values of gS. Following step 3) the process can be repeated until the values of [FORMULA] and [FORMULA] are adequately stable. The interdependence between the fitting function z and gS depends on the proximity of the turning points: near the turning points z mostly depends on [FORMULA] and [FORMULA] but is independent of [FORMULA] while midway between the turning points z is most sensitive to [FORMULA].

The quantity [FORMULA] is the average of gS at the maximum and minimum turning points. While the fitting parameters [FORMULA], [FORMULA] and [FORMULA] all must be determined numerically to high precision in order to leave a small remainder, the exact values are of little interest. To modest precision, all four time series are fitted by the values [FORMULA] (days)-1, [FORMULA] (days)-2, and [FORMULA]. The phase function is found from [FORMULA]. This quantity is smoothed by replacing the point by point values with a 10th order polynomial [FORMULA]. The phase function for PM2 in the minus magnetic modulation state is shown in Fig. 11. The other phase functions are indistinguishable on the scale of this plot.

[FIGURE] Fig. 11. The phase function [FORMULA] used to detrend the nearly sinusoidal drift of the intensity from the GOLF instrument. Shown is the phase function for photomultiplier 2 in the minus state of magnetic modulation.

Following the end of the time period shown here, the observations have been interrupted by the SOHO loss of attitude control and communications. Just prior to the end of the sequence shown, it is evident in Fig. 11 that the pattern of evolution of [FORMULA] has changed. Following the recovery of SOHO, the GOLF instrument is operating in the red wing and the new data can be treated with an appropriately applied version of the method described here. This new data shows that the change in the [FORMULA] evolution is caused by a decrease in [FORMULA]. In the new data the times of the turning points continues to follow the pattern shown in Fig. 10 with g being extended from Eq. (39) with approximately constant [FORMULA] and [FORMULA]. Thus the evolution of the system throughput due to the effects discussed in this section is largely independent of the operation of the instrument which was off during the SOHO loss of control period. Consequently, it is unlikely that the throughput loss is due to aging of the photomultipliers or a degradation in the sodium cell optics. The change in [FORMULA] means that the fitting function derived above cannot be used in the reduction of the new GOLF time series.

The above detrending procedure reduces the amplitude of temporal components having periods longer than about one tenth of the full time period or about 80 days. We can now calculate the velocity from Eq. (25) by taking [FORMULA] to be


and using [FORMULA] calculated from the orbital velocity as shown in Fig. 9. The resulting quantity [FORMULA] has dimensions of a velocity and is a mixture of line shift and intensity effects from large scale convection and magnetic activity as well as possible instrumental drifts. The independent treatment of the two states of magnetic modulation provides an indication of the differing effects of solar magnetic activity on the velocity derived from the intensity and the velocity derived from a local slope of the solar line. Fig. 12 shows the final velocity function, the difference in the velocity functions between the two states of magnetic modulation, and for reference a scaled and inverted plot of the MPSI(see the next paragraph). The difference function has been offset by -20 m s-1 in order to permit it to be seen easily. Each curve shows the result of a low pass filtered time series sampled once per hour. The above detrending procedure causes the overall trend lines to be of nearly zero slope. In the case of the difference velocity, it is of interest to point out that the temporal detrending is not required to bring the differences to near zero. The difference between the fitting functions relative to the fitting function itself is [FORMULA] so that the line intensity determination is accurate to within one part per 500.

[FIGURE] Fig. 12. This figure shows the velocity v and difference in velocity dv derived for PM2 according to the methods described in the text. The difference is the velocity from the plus state of magnetic modulation minus that for the minus state of magnetic modulation. The time series was subject to a low pass filter and then resampled to once per hour for this plot. The difference has been offset by -20 m s-1.

The long-time variation shown in Fig. 12 evidently has a periodicity close to the solar synodic rotation rate and is a consequence of the persistence of solar longitude zones of magnetic activity. In order to quantify this effect, we calculate the lagged cross-correlation coefficient between several velocity measures and an index of solar activity which has a high correlation with plage regions. This index, shown on Fig. 12, is called the Magnetic Plage Strength Index (MPSI) and has been developed by Chapman & Boyden (1986), Ulrich (1991) and Parker et al. (1998). Fig. 13 shows the cross-correlation function between the MPSI and the velocity v, the velocity difference [FORMULA], and the velocity derived from the quantity [FORMULA] where the angle bracket notation implies a smoothing over one day. The velocities derived from X show the strongest correlation with solar activity but at a shifted lag time whereas the velocity derived directly from the intensity has the greatest correlation with no lag time. Evidently the darkening of the surface by sunspots and plages (for the GOLF working point of sodium) causes the effective velocity to be most negative when the spots and plages are nearest the center of the disk. For the quantities which depend on the line slope, dv and [FORMULA] the time of greatest cross-correlation is shifted from the time when these features are nearest disk center. Grec et al. (2000) have discussed a quantity [FORMULA] which is similar to the inverse of X and found that it provides a means of studying the location of solar activity measured in a global fashion. A detailed comparison between the MPSI and specific excursions in the velocity derived from S shows that the features do line up as indicated by the cross-correlation but that the maximal excursions are a combination of uncorrected instrumental drift and the signal induced by the magnetic field features.

[FIGURE] Fig. 13. The cross correlation functional dependency on the lag time [FORMULA]. Each curve is labeled by the two functions which are included in the cross-correlation function. The lag [FORMULA] is the amount by which the first function leads the second - i.e. where both functions change sign from positive to negative, a positive [FORMULA] means the first function crosses zero before the second. The three velocity functions are explained in the text.

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© European Southern Observatory (ESO) 2000

Online publication: January 29, 2001