A first application of our simulation code is the study of the effect of beam distortions on the measured sky temperatures. The sky is simulated adding the CMB and galactic components as described in section 3.
Different feedhorns must be located on different parts of the focal plane. The magnitude and the kind of beam distortion depend on several parameters: the beam FWHM, the observational frequency, the telescope optical scheme and the beam location with respect to the optical axis. Optical simulations (Nielsen & Pontoppidan (1996)) show that the main expected distortion in the off-axis beams has an elliptical shape, with more complex asymmetries in the sidelobe structure. Here we have assumed that the beam is located along the optical axis, but that it can have an elliptical shape, i.e. the curves of equal response are ellipses.
Let be the spin axis vector, which is on the ecliptic plane for the standard scanning strategy, and the vector of the direction of the optical axis of the telescope, at an angle from the spin axis. We choose two coordinates and on the plane tangent to the celestial sphere in the optical axis direction, with vector and respectively; we choose the axis according to the condition that the vector points always toward the satellite spin axis; indeed, for standard PLANCK observational strategy, this condition is preserved as the telescope scans different sky regions. With this frame of reference choice, we have that and (here indicates the vector product). With the simple work assumption that the beam centre is along the optical axis, we have that the beam (elliptical) response in a given point is given by:
The ratio between major and minor axis of the ellipses of constant response quantifies the amount of beam distortion (we have chosen the major axis along the axis, but we have verified for a suitable number of cases that our conclusions are unchanged if the major axis is chosen along the axis). We have convolved the simulated map with this beam kernel up to the level , i.e. up to the level. The integration has been performed by using a 2-dimensional Gaussian quadrature with a grid of points. We have performed the convolution under the assumption that the telescope points always at the same direction during a given integration time; this artificially simplifies the analysis, but it is useful to make the study independent of the scanning strategy and related only to the optical properties of the instrument. The sky map, obtained by using the COBE-CUBE pixelization, has been interpolated in a standard way to have the temperature values at the grid points. For maps at resolution 11 (9) we have about 50 (3) pixels within the FWHM () at 30 GHz and 6 (less than 1) at 100 GHz (FWHM ); then the true accuracy of the integration derives not only from the adopted integration technique but from the map resolution too. For this reason the use of high resolution maps and a careful comparison between beam test results obtained from maps at different resolutions are recommended.
In order to quantify how the beam distortion affects the anisotropy measurements, we use a simple estimator: the of the difference between the temperature observed by an elliptical beam and a symmetric one. We use here thermodynamical temperature which does not depend on the observational frequency; the present results can be translated in terms of antenna temperature with the relation where ( at 30 GHz, whereas at 100 GHz).