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4. Discussion and comments

4.1. Finding an optimal scanning strategy?

It is clear that for the methods discussed above to work there must be some optimal way of scanning the sky. The scanning strategy sets the position of the points of intersection of the different circles described by the main beam of PLANCK SURVEYOR.

For instance, scanning the sky with great circles while the spin axis is kept rigorously anti-solar insures that every circle crosses every other circle in exactly two points, namely the North and South ecliptic poles. For this scanning strategy, pixels close to the ecliptic are seen by only one circle (obtained from adding together a great number of scans, 120 scans for PLANCK) every 6 months.

If instead of great circles the main beam scans 140 degree diameter circles while keeping the spin axis anti-solar, as for the nominal PLANCK mission, there is no common "reference" pixel seen by all circles. Circles cross each other on points that are spread out all over their length. Circles observed within a small time difference cross at high ecliptic latitudes. Circles observed at a time interval of about 140 days cross near the ecliptic plane, as shown in Fig. 1 (click here). For readjusting offsets, this scanning strategy should be superior, as all the pixels along one circle contribute to the evaluation of the offset of that circle.

This is not the case for great circles, and as emphasised by Wright (1996), the estimated offset for a great circle that crosses reference circles in only two points will depend on the realization of the noise at these two points. Of course, one could rely on more than two points, by using also points along the scan close to the pole which overlap substantially (and thus compare measurements in more extended "polar caps"). Some significant improvement can probably be obtained in this way, as after all consecutive great circles have a significant area in common, but one may have to worry about high spatial frequency signals on the sky (especially point sources) for fields of view that do not exactly coincide. Iterative correction of oversampled maps could be a solution, but this may lead to complications. Finally, this scanning strategy using great circles with anti-solar spin axis does not allow the natural improvement of the method where the low-frequency noise realization along each circle is fitted by sampling it at its own Nyquist frequency, as there are points of the circle (near the ecliptic plane) which are not seen by any other circle (here we talk about circles obtained by adding consecutive scans, not about individual scans).

In both cases, the destriping method can be improved by allowing the spin axis to move away from the anti-solar. This is especially true if great circles are used to scan the sky, as then circles would not cross other circles in two points only. A small displacement of the spin axis of a circle along the ecliptic makes it cross other circles near the ecliptic poles, and a small displacement of the spin axis perpendicular to the ecliptic makes the circle cross nearby circles near the ecliptic plane. Thus, in order to insure that all points along a circle have other circles to be compared to, a reasonable solution as far as destriping is concerned could be to move the spin axis on circles around the anti-solar. This has the additional characteristic that the sun aspect angle is kept constant, which may help monitoring the thermal stability of the payload if the satellite is reasonably symmetric around its spin axis. Sinusoidal or tooth-saw motion out of the ecliptic while the spin axis turns around the ecliptic are also possible solutions.

It has been argued by some authors (Wright et al. 1996) that based on the inherent difficulty of stabilising the sensitivity of an instrument to enough precision, it is desirable for the raw data to be collected in differential form. However, we would like to stress that even differential measurements can potentially suffer from low-frequency noise, because any asymmetric source of fluctuations (as for instance thermal fluctuations of optical elements which are not commonly shared) generates low-frequency instabilities in the measurements. Differential measurements are not just desirable: in a way or another they are, on short time scales, a necessity for radiometers, less stable than bolometers by orders of magnitude for CMB applications. These authors proved that it is possible to invert megapixel differential data. This task may be much harder if in addition some unforeseen low-frequency drifts contribute to the data stream.

Finally, whereas the method works better if the spin-axis is not kept rigorously anti-solar, it should be kept in mind than moving the spin axis could generate increased thermal instabilities of the payload or increased sidelobe contamination. For PLANCK, it has been shown that thermal specifications could be fulfilled if this angle does not exceed 15 degrees, which is sufficient for destriping and for sky integration time optimisation if desired. The freedom to move the spin axis also allows one to readjust the scanning strategy after the preliminary analysis of the first few days or weeks of data if necessary. This flexibility is a powerful tool for monitoring systematic effects.

4.2. Comments on the method

Our method has the very nice property that the correction of low frequency drifts does not require the inversion of a huge matrix, contrarily to methods which try to make a least square fit of both the signal on all pixels of the sky and all additional parameters as those which allow for the correction of low-frequency drifts. A (relatively!) small matrix of a few thousands by a few thousands needs to be inverted, which is an easy task for modern computers.

However, the method relies heavily on the assumption that different measurements on the same pixel of the sky should generate the same "useful" signal, apart from different contributions of low-frequency drifts. If this is not the case, it is still possible to fit the noise in much the same way, but this will require the inversion of huge matrices of typically a few million by a few million entries. Iterative methods make this task possible if there are not too many non-vanishing elements. Potential sources of trouble include far sidelobe straylight, which depends on the orientation of the satellite and on the position of bright sources in the sky, polarisation (for detectors sensitive to polarisation, because again what is measured on a given pixel with a polarisation-sensitive detector depends on the orientation of the satellite, as shown in Fig. 10 (click here)), and all other potential systematic effects. The remark about the complications induced by the measurement of polarised light is also true when one wants to invert differential data (e.g. Wright et al. 1996). These authors' method for inverting the differential signal expected from the MAP satellite also relies on the assumption that the useful signal from a given pixel is independent of satellite orientation. This assumption is not correct for polarisation sensitive measurements, obviously, and again this can make the inversion significantly harder than expected.

Bolometers are not sensitive to polarisation unless one places a polariser in front of them.

Figure 10: Representation of the effect of polarisation-sensitive measurements on the comparison of relative measured signal on different scans. Small circles with arrows inside represent different fields of view of the instrument (main beam). On the common pixel in the middle of the figure, measurements on different scans do not see the same polarised light, as indicated by the arrows. This may complicate attempts to measure polarisation and/or the inversion of polarised differential data

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