4 Simulations results and scanning strategy

We first consider an angle $\alpha =90^{\circ }$ between telescope and spin axes and a beam location with ( $\theta_B,\phi_B)=(2.8^\circ,45^\circ$). Here $\theta_B$ is the angle from the optical axis: this is a typical value for the 100 GHz (with a $\simeq 1.5$ m aperture off-axis Gregorian telescope) horns while the 30 GHz beams are placed at larger $\theta_B$ values. This assumption is therefore conservative with respect to the destriping efficiency, since in this case the region of crossings between scan circles is smaller and closer to Ecliptic poles. The angle $\phi_B$ is the beam center displacement from the axis given by the intersection between the sky field of view plane and the plane containing the telescope and spin axes (see Burigana et al.[1998]). Our choice of $\phi_B=45^\circ$ is intermediate between 0$^{\circ }$ (or 180$^{\circ }$) and 90$^{\circ }$ which are equivalent for the destriping respectively to an on-axis beam (with crossings only at Ecliptic poles for $\alpha =90^{\circ }$) and a beam which spreads crossings over the wider possible region. The adopted beam location is therefore a non-degenerate case although non optimal.

Figure 2: Top panel: noise power spectra at 30 GHz before destriping. Simulations parameters are: $\alpha =90^{\circ }$, $(\theta _B,\phi _B)=(2.8^{\circ },45^{\circ })$, $f_{\rm k}=0.05$ Hz and spin-axis always on the Ecliptic plane. The white noise spectrum and its theoretical level are also reported for comparison. The excess of noise is about 40% over the white noise level. Bottom panel: noise power spectra after destriping. Now the added noise is only 1-2% of the white noise level

We evaluated the impact of 1/f noise both in terms of added rms noise and of angular power spectrum.

Figure 3: Baselines recovered from destriping algorithm for the case reported in Fig. 1

The white noise power spectrum can be derived analytically knowing the total number of pixels in the sky and pixel sensitivity:

$$C_{\ell,wn} = \frac{4\pi}{N_{\rm pix}^2}\sum_{i=1}^{N_{\rm pi... ...i^2 = \frac{4\pi}{N_{\rm pix}} \langle \sigma^2 \rangle \, .$$ (5)

4.1 Simulation results

In the top panel of Fig. 2 we show the square root of the power spectrum (which is roughly proportional to rms contribution of temperature per $\ell$-bin) before applying the destriping technique: the solid line is the white noise level as derived from Eq. (5) and the superimposed spectrum is derived from a simulation with only white noise included. The agreement is very good which confirms the accuracy of our simulator and map-making algorithm. The gray line is the global noise spectrum (white and 1/f noise together). The spectrum is clearly non-white: blobs are present. The excess of noise in terms of both rms and $\sqrt{C_\ell}$ is about 40% of white noise level. Figure 3 shows the recovered baselines and Fig. 4 shows the noise map after applying our destriping code: stripes are no more evident. This situation can be quantified by computing the noise power spectrum of the destripped map, as shown in the bottom panel of Fig. 2. Now the spectrum is considerably flatter and no more blobs are present, with an overall noise excess of 1-2% over the white noise level. This confirms the efficiency of the destriping algorithm, under the above mentioned simplifying assumptions. On the other hand it is clear from the bottom panel of Fig. 2 that at large scales, corresponding to $\ell\,\lower2truept\hbox{${< \atop\hbox{\raise4truept\hbox{$\sim$ }}}$ }\,100$, a non-negligible residual contribution is present in the noise spectrum. A lower knee frequency (achievable with the reference loads at 4 K) is necessary to further reduce the effect.

Figure 4: Noise map after destriping for the same case illustrated in Fig. 1: stripes are no more visible

We also investigated if different scanning strategies may help in destriping efficiency. Possible periodic motion of the spin-axis away from the Ecliptic plane have the effect of broadening the region of crossings between different circles. This goes in the direction of removing possible degeneracies in the destriping system. We implemented both sinusoidal oscillations and precessions: the first does not preserve the spacecraft solar illumination and this is likely to induce thermal effects and drifts in the data.

The second is then preferable, since it keeps the solar angle constant as the satellite moves the spin-axis. For both the motions we performed 10 complete oscillations per year of mission with 10$^{\circ }$ amplitude. The results before and after destriping are nearly the same for the two cases and Fig. 5 reports noise power spectra for the precession motion: before destriping blobs are still present and disappear after destriping. The excess of noise is $\sim$ 40% and 1-2% before and after destriping respectively.

We consider also a case of an on-axis beam $(\theta _B,\phi _B)=(0^{\circ },0^{\circ })$with $\alpha =90^{\circ }$. This situation is representative of horns which, in the focal plane arrangement, are placed close to the scanning direction: from the destriping point of view these horns are indeed equivalent to on-axis horns and intersections between different scan circles are nearly only at Ecliptic poles. This represents a degenerate situation with respect to destriping efficiency. Figure 6 shows noise power spectra for this on-axis case before and after destriping: the geometry of the simulation is changed and the same for blob shape which is not well defined. Here the level of added noise is $\sim 42\%$. After destriping we are left with an excess of noise of the order of 5% of the white noise and the noise spectrum is considerable less flat than before. Such excess noise and a non-flat spectrum around $\ell \sim 100-200$ (where the first CMB Doppler peak is expected) after destriping is clearly not acceptable for the feeds aligned with the scan direction. Moving the spin axis is a way to remove this degeneracy, as well as to complete the sky coverage for all channels.

Figure 5: Top panel: noise power spectra at 30 GHz before destriping. Simulation parameters are the same of Fig. 2 except for the scanning strategy which includes here precession (see the text). The added noise is $\sim 40$% of the white noise level. White noise spectra are also reported for comparison. Bottom panel: noise power spectra after destriping. Again, the added noise is now only 1-2% of the white noise level

Figure 6: Top panel: noise power spectra at 30 GHz before destriping. Simulation parameters are the same of Fig. 2 but the beam position is now $(\theta _B,\phi _B)=(0^{\circ },0^{\circ })$. The added noise is $\sim 42$% of the white noise level. White noise spectra are also reported for comparison. Bottom panel: noise power spectra after destriping. The added noise is now larger than previous cases, being 5% of the white noise level

Figure 7: Top panel: noise power spectra at 30 GHz before destriping. Simulation parameters are the same of Fig. 6 but the angle $\alpha$ is now set to 85$^{\circ }$. The added noise is $\sim 42$% of the white noise level. White noise spectra are also reported for comparison. Bottom panel: noise power spectra after destriping. The added noise is now only 2% of the white noise level

Keeping fixed the beam position we move the angle $\alpha$ from 90$^{\circ }$ to 85$^{\circ }$: this of course leaves small regions around Ecliptic poles which are not observed but has the advantage to enlarge the scan circle crossing region. Figure 7 shows noise power spectra before and after destriping for this 85$^{\circ }$, on-axis case: now blobs are clearly visible with dimensions in $\ell$-space different from Fig. 2 due to different geometrical configuration. These blobs are completely removed after destriping leaving a level of added noise after destriping of only $\,\lower2truept\hbox{${< \atop\hbox{\raise4truept\hbox{$\sim$ }}}$ }\,$ 2% over the white noise. This is essentially what we obtained for the off-axis configuration.

Figure 8: Top panel: noise power spectra at 30 GHz before destriping. Simulation parameters are the same of Fig. 2 but now $f_{\rm k}=0.01$ Hz. The added noise is now only 9% of the white noise level. White noise spectra are also reported for comparison. Bottom panel: noise power spectra after destriping. The added noise is now only 0.5% larger the white noise level

All the simulations considered so far were performed with a $f_{\rm k}=0.05$Hz, a "reference'' value close to that predicted by Seiffert et al. ([1997]) for a reference load at 20 K.

To decrease the residual effect, a 4 K reference load is being designed. In this configuration the theoretical knee frequency would be less than 10 mHz. We chose to run a simulation with $f_{\rm k}=0.01$ Hz with the usual off-axis configuration with $\alpha =90^{\circ }$. As for the other cases we report noise power spectra before and after destriping in Fig. 8. It is interesting to note that now the level of added noise is 9% before destriping and reduces only to 0.5% after applying destriping algorithm.

4.2 Morphology of the $C_\ell$ artefacts due to 1/f noise

We now give a short theoretical argument for the expected morphology of the $C_\ell$ power spectra of the simulated striping pattern. For scanning without wobbling or precession, two angular scales fully determine the structure of the striping pattern on the sky. One is the half opening angle $\lambda_1=90^\circ$ of the ecliptic equator which the spin axis traces on the sky as PLANCK completes a full orbit about the Sun. This makes the noise pattern symmetric under parity, on average. The other scale is the half opening angle of the scanning rings $\lambda_2\simeq \alpha - \theta_B\sin(\phi_B) , \; \theta_B\ll \pi/2$.

The combination of these two nearby scales leads to beats in the $C_\ell$, which are visually apparent as the "blob'' in the top panels of Figs. 2, 5, 6, 7, 8. Treating this effect in the same way as the appearance of fringes in an interference pattern we can calculate the width in $\ell$ of each blob as $\Delta \ell = 90/\left(\lambda_2-\lambda_1\right)$. This is in quantitative agreement with our numerically computed spectra. It also explains the absence of beats in the top panel of Fig. 6, where $\lambda_2=\lambda_1=90^\circ$ and the power spectrum can be understood as a single blob with infinite $\Delta \ell$.

For scanning strategies with some kind of "wobble'' the correlation function will still be approximately symmetric if the amplitude of the wobble is small compared to $90^\circ$ and the above statements remain approximately true although a weakly symmetry breaking can be seen in the top panel of Fig. 5, where the beats do not reach all the way down to the white noise level.

The offset of the blobs compared to the white noise level is given by the rms excess power due to the 1/f noise component.