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Astron. Astrophys. 336, 769-775 (1998)

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3. Data analysis

After applying the usual procedures of dark current subtraction, flat fielding and cosmic event corrections, the inaccuracy of the telescope guiding was compensated by shifting the frames: For the time series of coma images, the maxima of the [FORMULA] crosscorrelation functions were used; for the tail positions, we recorded coma pictures with a fixed distance from the tail position before and after the observing run, correcting their drift with a linear interpolation. Thus we obtained movies at a fixed position relative to the comet's nucleus. Fig. 3 displays typical single frames of these movies both taken from the coma with the Strömgren v and the Gunn g filter.

[FIGURE] Fig. 3. Intensity maps of the comet's coma seen through the Gunn g filter (a ) and the Strömgren v filter (b ). Each contourline marks an increase of the count rate by 25 %, beginning at 1000 counts (a ) and 316 counts (b ). The brightness of the night sky contributes with about 10 counts. At the comet's geocentric distance of about 1.4 AU, 1 arcsec corresponds to about 1000 km. Compare with Figs. 6 and 8.

In a first step we binned [FORMULA] pixels - i.e. [FORMULA] - to derive fields of view comparable to those presented two decades ago by Isserstedt & Schlosser (1975). These authors obtained light curves with a photometer aperture of 45 arcsec when observing comet Kohoutek at about the same heliocentric distance as Hale-Bopp had during our observations in April 1997, namely between 0.92 and 0.94 AU (1.38 - 1.48 AU geocentric distance).

Sample light curves from 40" [FORMULA] 40" subfields of the comet's coma are shown in Fig. 4. These unprocessed light curves are dominated by long term trends due to changes in the brightness of the sky, the airmass, and the comet's activity during the observations. In order to concentrate on the periods adressed in this study and again, to allow for comparison with the measurements of comet Kohoutek's oscillations, we applied in our data sets the same reduction procedure which was used by Isserstedt & Schlosser, i.e. we subtracted a 5.5 minutes running mean from a 1.5 minutes running mean, reducing the high frequency noise and suppressing all changes on long time scales. Obviously, the processed data has to be treated carefully to avoid misinterpretations due to reduction effects. Some of the processed lightcurves are shown in Fig. 5. The reader should compare these curves with those in Fig. 3 of Isserstedt & Schlosser (1975) which are indeed remarkably similar both in period and amplitude. A first guess to explain this similarity might be to assume that the signal is an artefact either due to the similarity of the applied frequency filter or due to the earth's atmosphere. We will argue below, that neither explanation is satisfactory.

[FIGURE] Fig. 4. Light curves of the comet's coma. In these unprocessed light curves (see also Fig. 5 below) changes on long time scales dominate. Nevertheless, some fluctuations on the time scale of minutes can also be seen. For further processing we have used only the data within the time interval, marked by dotted lines, when the observations were neither contaminated too much by daylight nor by strong irregular variations due to large zenith distance.

[FIGURE] Fig. 5. Selected light curves derived from the original data sets (like the one shown in Fig. 4) by smoothing and subtracting the long-time trend (see the text for further explanation). Each curve represents the intensity averaged over 40 x 40 arcsec corresponding to the aperture of the photometer used by Isserstedt & Schlosser (1975). Both in coma and ion tail one can see an oscillation in C3 (b , d ) with an amplitude distinctly higher than fluctuations in the reflected sunlight (a , c ).

In the next step we calculated the power spectrum for each recorded pixel. In Fig. 6 we display the power corresponding to the lightcurves oscillating with a notable amplitude in Fig. 5, i.e. those observed through the Strömgren v filter and thus representing C3 emission. Both the power of the unprocessed lightcurves and the power of the filtered lightcurves - as shown in Fig. 5 - is displayed. The power around 5 mHz - corresponding to a period of about 3 minutes - is well enhanced by the filter method in particular with respect to lower frequency contributions. The similarity of the peaks in the processed and unprocessed data shows that they are not due to the filter transmission. This excludes the first potential artefact mentioned above.

[FIGURE] Fig. 6. Power spectra of the C3 emission both in the coma and ion tail. The power was averaged over an area equal to that used for the light curves in Figs. 4 and 5. Displayed are power spectra of the original data (a , c ) and of the data after applying the procedure adapted from Isserstedt & Schlosser (b , d ). The dotted lines indicate the smoothing frequencies (see the text). Apart from the power at lower frequencies which is not our subject, there are two prominent peaks in the coma's power spectrum: a double-peak at about 3.7 mHz and one at 5.5 mHz, corresponding to periods of 4.5 and 3.0 min, respectively. The signal from the tail has a distinct power peak at about 4.8 mHz (3.5 min). We note that both positions were observed in different nights.

At any given frequency, one can rearrange the power values taken from all power spectra to form a power map in the spatial domain (Fig. 7). These maps are a tool quite helpful in discussing the power spectra. For calculating the power, the intensity fluctuations were normalized to the mean intensity of the corresponding pixel. Consequently, the noise signal will be reduced with the square root of the intensity. On the left side of Fig. 7(a and c) we show a power map derived from data taken with the Gunn g filter together with a map of the average gradient of the intensity in the same data set. Here, the power is generally reduced with enhanced intensity (see Fig. 3a) and only at the loci of high intensity gradients the power rises again in the inner coma. One can argue straightforward from such a powermap that noise and image motion are the dominant contributors to the oscillations seen through the Gunn g filter 2.

[FIGURE] Fig. 7. Power maps of the comet's coma seen through the Gunn g filter (a ) and the Strömgren v filter (b ) together with corresponding maps (c , d ) of the gradient of the local intensity (see Fig. 3). For the power calculation the intensity fluctuations were normalized to a 5.5 minutes running mean per pixel. Both maps display power on a logarithmic scale around 3.5 mHz in an interval corresponding to the frequency resolution. In a the power is concentrated mostly at loci of high intensity gradients (c ), which means that this power signal is probably produced by image motion. In contrast, we see in b a (higher) power signal that spreads smoothly over the coma of the comet with no distinct connection to local intensity gradients (d ). Compare panel b also with Fig. 8. Note that 1 arcsec corresponds to about 1000 km at the comet's geocentric distance of about 1.4 AU.

In contrast, the right side of Fig. 7(b and d) shows a power signal that spreads smoothly over the inner coma, gaining amplitude with intensity (see Fig. 3b and Fig. 8), and which apparently is hardly at all influenced by local intensity gradients resulting from the well-known onion peel structure of the coma. Comparing panels b and d with a and c we find that the artefacts made responsible for the power in a will not be the main contributers in b. This excludes the second potential artefact with respect to the oscillations shown in Fig. 5.

[FIGURE] Fig. 8. Power spectra of the coma intensity fluctuations arranged in correspondence to Figs. 3b and 7b. Each spectrum represents an independent 40" x 40" bin. The scaling of the power axis is the same for all positions. These diagrams demonstrate the similarity of the peak frequencies over a wide field of view. Since the intensity signals were normalized, the fact that the power signal increases near the comet's nucleus demonstrates that its dependence from the intensity is stronger than linear. See the greyscale panel where the power at the peak frequency is displayed at full resolution on a linear scale for direct comparison with the logarithmical scaled panel b in Fig. 7.

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© European Southern Observatory (ESO) 1998

Online publication: July 20, 1998