6 Discussion and conclusions

In the previous section we have shown that, except for low frequency measurements in regions quite close to the galactic centre, the typical difference between the temperature measured by symmetric and distorted beams is a few $\mu {\rm K}$, depending on the eccentricity of the beam response and on the FWHM. Although the $\rm rms_{th}$ value of this effect may be small compared to the averaged instrumental sensitivity, for some "pixels'' the effect may be significantly higher than the average. In addition it cannot be reduced by repeating the observation with different spin axis directions, because (for scanning strategies similar to that proposed by the Planck mission) a given sky region is typically observed with similar orientations of the plane x,y; therefore this kind of distortion produces systematic and not statistical errors in the temperature measurements of any given resolution element.

Deconvolution techniques are generally well established for symmetric beams but it may be difficult to deconvolve observed maps in the case of asymmetric beams, although possible in principle. On the other hand, by averaging maps deconvolved with standard methods and obtained from different channels at the same frequency, this effect, at least in principle, may be reduced, provided that detectors at the same frequency observe the same sky region with different beam orientations. For example, a set of integrations derived from the average of two sets of integrations along the same scan circle and both with the same distortion parameter r but with the major and minor axes of the elliptical response profiles exchanged in the plane x,y for the two cases, presents very small temperature differences with respect to the case of a symmetric beam. We have compared the differences between the temperature observed by a symmetric beam and the average of the temperatures observed by two beams, one like that in the test 2 in Table 1 and the other like that but with the major axis along the y axis: the corresponding value of $\rm rms_{th}$ is $0.349 \,\mu {\rm K}$,much smaller (by a factor $\sim$6.5) than that obtained in the test 2 and close to the numerical accuracy (see Table 1). Indeed, the average of two elliptical responses with (a) $\sigma_x=\sigma \sqrt{r} \gt \sigma_y=\sigma / \sqrt{r}$ and (b) $\sigma_x=\sigma / \sqrt{r} < \sigma_y=\sigma \sqrt{r}$ respectively is equal to a symmetric response ($\sigma_x=\sigma_y=\sigma$) multiplied by $1-0.25 (x^2+y^2)/\sigma^2 (r-1)^2 + ...$, i.e. the first order effect in (r-1) drops out. On the contrary, as already pointed out in the analysis of IRAS data, a simple average of data sets deriving from detectors with significant differences in sensitivity does not allow an analogous improvement. For example, from the temperature differences respect to the observations from a symmetric beam for the simple average of the temperatures obtained in the case (b) with those obtained in the case of a distorted beam like that in the test 1 in Table 1 we have ${\rm rms_{th}} = 0.750 \,\mu {\rm K}$, a value very close to that of test 1. On the other hand, by averaging this two data sets with weights inversely proportional to the values of $\rm rms_{th}$ shown in Table 1 we obtain ${\rm rms_{th}} = 0.16 \,\mu{\rm K}$, much smaller (by a factor $\sim$5.2) than that obtained in the test 1 and equal to the numerical accuracy. Analogous improvements cannot be reached by averaging the measurements of beams with distortions oriented in similar direction in the plane x,y. For example, a simple average of the data from test 1 and 2 gives ${\rm rms_{th}} = 1.55 \,\mu {\rm K}$; we find only a little improvement from the use of appropriate weights, obtaining in this case ${\rm rms_{th}} = 1.22 \,\mu {\rm K}$. These results are only intermediate between those of the tests 1 and 2.

These considerations indicate as more advantageous in presence of beam distortions a global feedhorns arrangement in which, for the same frequency, the different beams show distortion figures differently orientated in the plane x,y. Further, an arrangement in which the beams at the same frequency have similar distortion parameters is in general more advantageous respect to an arrangement with a wide spread in the beams distortion parameters; if this is not possible, a careful quantification of the error introduced by the beam distortions is demanded, to correctly average data from beams arrays with wide spreads in optical performances (on the other hand it may be not simply to do it in presence of a large set of systematic effects).

The present analysis shows that, for the same beam distortion parameter r, the effect is more important for the low frequency beams than for high frequency ones, due to the different beam widths. This fact indicates that a good solution may be to arrange low frequency feedhorns near to the optical axis and high frequency ones in the outer regions of the focal plane, because beam distortions typically increase with its distance from the optical axis. On the contrary, when this distance is fixed, the distortion parameter r is typically larger for high frequency detectors than for low frequency ones (Nielsen & Pontoppidan 1996; Villa et al. 1998); this fact suggests a focal plane arrangement that goes in a direction opposite to that delined above. Given the present knowledge of the amount of the beam distortion as a function of the distance from the optical axis and as a function of the frequency and given the present estimate of the distortion effect as a function of the FWHM and of the distortion parameter r, the second choice seems to be more advantageous. On the other hand, the goodness of the focal plane arrangement must be checked by evaluating the average global temperature effect, $\rm rms_{th}$, and by minimizing the resulting potential errors introduced by all the distortion effects. An analysis based on preliminary realistic shapes of off-axis beams indicates that elliptical distortions coupled to localized distortions (spots) in the beams responses and to a decreasing in the effective beam angular resolution can produce larger effects on anisotropy measurements, at levels of some $\mu {\rm K}$ (Mandolesi et al. 1997, 1998); in these conditions the global effect may be worst than previous estimates and careful simulations for any given optical design are demanded. Another crucial point is the comparison between the $\rm rms_{th}$ figures and the sensitivities of the receivers. For Planck LFI, low frequency individual channels are more sensitive (Bersanelli et al. 1996); on the other hand, the 70 and 100 GHz channels are more efficient for the primary cosmological goal.

The full success of missions like Planck and MAP require a good control of all the relevant sources of systematic effects. Discrete sources above the detection limit must be carefully removed and accurate models for foregrounds radiation and anisotropies (Franceschini et al. 1994; De Zotti et al. 1996; Bouchet et al. 1996; Toffolatti et al. 1998) are required to keep the sensitivity degradation in the knowledge of CMB anisotropies below few tens percent (Dodelson 1996). Thermal drifts and stripes generated by the 1/f - type noise due to amplifiers gain fluctuations (Janssen et al. 1996; Seiffert et al. 1997) must be minimized by efficient cooling and by optimizing the observational strategy and possibly further reduced in the data analysis (Delabrouille 1998; Burigana et al. 1997).

All in all maximum effort should be addressed to optimize the telescope and the focal plane assembly design in order to minimize the beam distortions effect, one of sources of systematic effects.

Acknowledgements

It is a plesure to thank K. Gorski, E. Hivon, C. Lawrence, M. Malaspina, P. Platania, G. Smoot, L. Toffolatti, F. Villa and M. White for useful discussions and J. Aymon for his exhaustive suggestions on the use of the COBE software. We wish to thank also an anonymous referee for constructive comments.