3 Convolution of the simulated map with the beam response

We have written a code that simulates the basic properties of Planck observations in order to study the effect of beam distortions on the measured sky temperatures. The sky is simulated adding the CMB and galactic components as described in Sect. 2.

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 a roughly elliptical shape, with more complex asymmetries in the sidelobe structure. We have assumed here 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. This assumption has to be considered here as a simple work-hypothesis, not far from the truth, useful for deriving a general description of the magnitude of the main beam distortion effect as a function of few basic parameters (see Sect. 4). More detailed studies are in progress for taking into account realistic beam shapes for the Planck optic (Mandolesi et al. 1997, 1998).

  

Figure 1: Schematic representation of the observational geometry

Figure 1 shows the schematic representation of the observational geometry. Let be i the angle between a unit vector, $\vec s$, along the satellite spin axis outward the Sun direction and the normal to the ecliptic plane and $\vec p$ the unit vector of the direction of the optical axis of the telescope, at an angle $\alpha$ from the spin axis ($i=90^{\circ}\;$ and $\alpha=70^{\circ}\;$ for the Phase A study, Bersanelli et al. 1996). We choose two coordinates x and y on the plane tangent to the celestial sphere in the optical axis direction, with unit vector $\vec u$ and $\vec v$ respectively; we choose the x axis according to the condition that the unit vector $\vec u$ points always toward the satellite spin axis; indeed, for the standard Planck observational strategy, this condition is preserved as the telescope scans different sky regions. With this choice of reference frame, we have that $\vec v = \vec p \wedge \vec s / \vert\vec p \wedge \vec s\vert$ and $\vec u = \vec v \wedge \vec p / \vert\vec v \wedge \vec p\vert$(here $\wedge$ indicates the vector product). In general the coordinates (x₀,y₀) of the beam centre will be identified by two angles (e.g. the colatitude and the longitude) in the $\vec u , \vec v , \vec p$ reference frame. Then, the beam (elliptical) response in a given point (x,y) is given by:

$$K(x,y)= {\rm exp} \left[ { - {1 \over 2} [((x-x_0)/\sigma_x)^2+((y-y_0)/\sigma_y)^2] } \right].$$ (4)

In practice, for the present study we can take x₀=y₀=0.

The ratio $r={\rm max}(\sigma_x,\sigma_y)/{\rm min}(\sigma_x,\sigma_y)$between major and minor axis of the ellipses of constant response quantifies the amount of beam distortion respect to the case of a pure symmetric beam with $\sigma_x'=\sigma_y'=\sigma'=\sqrt{\sigma_x \sigma_y}$ (we have choosen the major axis along the x axis, but we have verified for a suitable number of cases that our conclusions are unchanged if the major axis is choosen along the y axis). We have convolved the simulated map with this beam response up to the level $(x/\sigma_x)^2+(y/\sigma_y)^2 = 9$, i.e. up to the $3\sigma$ level ($\simeq -20$ dB). The integration has been performed by using a 2-dimensional gaussian quadrature with a grid of $48 \times 48$ 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. We will study the effect introduced by the telescope motion in a future work (see Sect. 4.8). The sky map, obtained by using the COBE-cube pixelisation, 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 ($\simeq 30'$) at 30 GHz and 6 (less than 1) at 100 GHz (FWHM $\simeq 12'$); then the true accuracy of the integration depends not only on the adopted integration technique but on 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 rms of the difference between the temperature observed by an elliptical beam and by a symmetric one. We express it here in terms of thermodynamic temperature, which does not depend on the observational frequency; the present results can be translated in terms of antenna temperature with the relation ${\rm rms_A} = {\rm rms_{th}} {x}^2 \exp({x}) / [\exp({x})-1]^2$ where ${x}=h\nu/kT_0$, where T₀=2.726 K is the CMB thermodynamic temperature (${\rm rms_A} \simeq {\rm rms_{th}}$ at 30 GHz, whereas ${\rm rms_A} \simeq 0.777\; {\rm rms_{th}}$ at 100 GHz).