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Schematic of an LFI radiometer, taken from \protect\OldPlanckLFICalPaper. The two linearized polarization components are separated by an orthomode transducer (OMT), and each of them enters a twin radiometer, only one of which is shown in the figure. A first amplification stage is provided in the cold (20\,K) focal plane, where the signal is combined with a reference signal originating in a thermally stable 4.5-K thermal load. The radio frequency signal is then propagated through a set of composite waveguides to the warm (300\,K) back-end, where it is further amplified and filtered, and finally converted into a sequence of digitized numbers by an analogue-to-digital converter. The numbers are then compressed into packets and sent to Earth.

Quantities used in the determination of the value of $\phisky$ (Eq.~\protect\ref{eq:phisky}) for radiometer LFI-27M (30\,GHz) during a short time span (2\,min). \textit{Panel A}: the quantities $B_{\mathrm{mb}} * T_{\mathrm{CMB}}$ and $B_{\mathrm{sl}} * T_{\mathrm{CMB}}$ are compared. Fluctuations in the latter term are much smaller than those in the former. \textit{Panel B}: the quantity $B_{\mathrm{sl}} * T_{\mathrm{CMB}}$ shown in the previous panel is replotted here to highlight the features in its tiny fluctuations. The fact that the pattern of fluctuations repeats twice depends on the scanning strategy of \Planck, which observes the sky along the same circle many times. \textit{Panel C}: Value of $\phisky$ calculated using Eq.~\protect\eqref{eq:phisky}. There are several values that diverge to infinity, which is due to the denominator in the equation going to zero. \textit{Panel D}: distribution of the values of $\phisky$ plotted in panel C. The majority of the values fall around the number $+0.02\,\%$.

Variation in time of a few quantities relevant for calibration, for radiometer LFI-21M (70\,GHz, left) and LFI-27M (30\,GHz, right). Grey/white bands mark complete sky surveys. All temperatures are thermodynamic. \textit{Panel A}: calibration constant $K$ estimated using the expected amplitude of the CMB dipole. Note that the uncertainty associated with the estimate changes with time, according to the amplitude of the dipole as seen in each ring. \textit{Panel B}: expected peak-to-peak difference of the dipole signal (solar + orbital). The shape of the curve depends on the scanning strategy of \Planck, and it is strongly correlated with the uncertainty in the gain constant (see panel A). Note that the deepest minima happen during Surveys~2 and 4; because of the higher uncertainties in the calibration (and the consequent bias in the maps), these surveys have been neglected in some of the analyses in this \Planck\ data release \protect\citep[see e.g.,][]{planck2014-a15}. \textit{Panel C:} the calibration constants $K$ used to actually calibrate the data for this \Planck\ data release are derived by applying a smoothing filter to the raw gains in panel A. Details regarding the smoothing filter are presented in Appendix~\ref{sec:smoothing}.

Dipole amplitudes and directions for different radiometers. \emph{Top}: errors in the estimation of the Solar dipole direction are represented as ellipses. \emph{Bottom}: estimates of the amplitude of the Solar dipole signal; the errors here are dominated by gain uncertainties. \emph{Inset}: the linear trend (recalling that the numbering of horns is approximately from left to right in the focal plane with respect to the scan direction), most likely caused by a slight symmetric sidelobe residual, is removed when we pair the 70\,GHz horns.

\emph{Top}: estimate of the value of $\phid$ (Eq.~\protect\ref{eq:phid}) for each LFI radiometer during the whole mission. The plot shows the median value of $\phid$ over all the samples and the 25th and 75th percentiles (upper and lower bar). Such bars provide an idea of the range of variability of the quantity during the mission; they are not an error estimate of the quantity itself. \emph{Bottom}: estimate of the value of $\phisky$ (Eq.~\protect\ref{eq:phisky}). The points and bars have the same meaning as in the plot above. Because of the smallness of the value for the 44 and 70\,GHz channels, the inset shows a zoom of their median values. The large bars for the 30\,GHz channels are motivated by the coupling between the stronger foregrounds and the relatively large power falling in the sidelobes.

\emph{Top}: comparison between the measured and estimated ratios of the $a_{\ell m}$ harmonic coefficients for the nominal maps (produced using the full knowledge of the beam $B$ over the $4\pi$ sphere) and the maps produced under the assumption of a pencil beam. The estimate has been computed using Eq.~\protect\eqref{eq:GalacticPickupRemoval}. \emph{Bottom}: difference between the measured ratio and the estimate. The agreement is better than 0.03\,\% for all 22 LFI radiometers.

\emph{Top}: impact on the average value of the $a_{\ell m}$ spherical harmonic coefficients (computed using Eq.~\ref{eq:measuredChangeInT}, with $100 \leq \ell \leq 250$) due to a number of improvements in the LFI calibration pipeline, from the first to the second data release. \emph{Bottom}: measured change in the $a_{\ell m}$ harmonic coefficients between the first and the second data release. No beam window function has been applied. These values are compared with the estimates produced using Eq.~\eqref{eq:estimatedChangeInT}, which assumes perfect independence among the effects.

Discrepancy among the radiometers of the same frequency in the height of the power spectrum $C_\ell$ near the first peak. For a discussion of how these values were computed, see the text. \emph{Inset}: to better understand the linear trend in the 70\,GHz radiometers, we have computed the weighted average between pairs of radiometers whose position in the focal plane is symmetric. The six points refer to the combinations 18M/23M, 18S/23S, 19M/22M, 19S/22S, 20M/21M, and 20S/21S, respectively. Note that all six points are consistent with zero within $1\sigma$; see also Fig.~\protect\ref{fig:dipoleParametersPlot}.

Estimate of $\Delta_{\ell}^{\text{70 GHz,other}}$ (Eq.~\protect\ref{eq:interfreqRatio}), which quantifies the discrepancy between the level of the 70\,GHz power spectrum and the level of another map. \emph{Top}: comparison between the 70\,GHz map and the 30\,GHz map in the range of multipoles $100 \leq \ell \leq 250$. The error bars show the rms of the ratio within each bin of width 15. \emph{Bottom}: the same comparison done between the 70\,GHz map and the 44\,GHz map. A 60\,\% mask was applied before computing the spectra.

Difference in the application of the full $4\pi$ beam model versus the pencil beam approximation. \textit{Panel A}: difference between survey 1 and survey 2 for a 30\,GHz radiometer (LFI-27S) with the $4\pi$ model, smoothed to $15^\circ$. We do not show the same difference with pencil beam approximation, as it would appear indistinguishable from the $4\pi$ map. \textit{Panel B}: double difference between the $4\pi$ $1-2$ survey difference map in panel A and the pencil difference map (not shown here). This map shows what changes when one drops the pencil approximation and uses the full shape of the beam in the calibration. \textit{Panel C}: zoom on the blue spot visible at the top of the map in panel A. \textit{Panel D}: same zoom for the pencil approximation map. The comparison between panel C and D shows that the $4\pi$ calibration produces better results.

Visual timeline of Jupiters's crossings with LFI beams. Here ``SS'' lables sky surveys.

Time dependence of the angle between Jupiter's direction and the spin axis of the \Planck\ spacecraft. The darker horizontal bar indicates the angular region of the 11 LFI beam axes, and the lighter bar is enlarged by $\pm 5^\circ$.

Distribution of the values of $\Tb$ for Jupiter measured by the 22 LFI radiometers during each of the seven transits. The histogram has been produced using five bins per frequency. The lack of Gaussianity in the three distributions is evident.

Distribution of the values of $T_b$ for Jupiter as a function of the central frequency $\Fcent$ of each radiometer.

\emph{Top}: brightness temperature of Jupiter ($T_\mathrm{br}$) compared with the data from \protect\citet{weiland2010}, linearly rescaled in frequency to match LFI's central frequencies $\Fcent$ and corrected for difference between LFI's and WMAP's dipoles. \emph{Bottom}: deviation from unity of the ratio between LFI's estimate for $T_\mathrm{br}$ and WMAP's. The agreement is excellent among the three frequencies.

Example of application of the algorithm for the detection of jumps in the stream of $K=G^{-1}$ values (Eq.~\protect\ref{eq:radiometerEquation}) for LFI-27M during \Planck's first year. \textit{Panel A:} the set of calibration constants $K$ computed by \texttt{Da Capo} for LFI27M. The $x$-axis here is cropped to the first two sky surveys (one year of data). \textit{Panel B:} the amplitude of the dipole signal, $\delta T_{\rm dip}$, as seen within each ring by the \Planck\ spacecraft. The two horizontal lines mark the thresholds for regions where the signal is considered either ``strong'' or ``weak.'' \textit{Panel C:} the size $N$ of the window used to compute the moving average of $K$ (there are roughly $N=30$ values of $K$ per day). In regions where the dipole signal is weak or strong, the window width is 1200 or 400 samples, respectively; outside such regions, we use a linear interpolation between these two values. \textit{Panel D:} the result of applying a moving average with the variable window size (Panel C) to the series of data shown in Panel A. This is \emph{not} the smoothed series used for calibration; only the rms of the moving average is used (see next panel). \textit{Panel E:} the figure of merit used to detect jumps is the product of the dipole amplitude and the rms of the moving average. The threshold used to detect jumps in this particular example (LFI-27M) is equal to the 99th percentile of such values (grey dashed line). In this case, two jumps have been found (days 257 and 450).

Comparison between the behaviour of the smoothing code in the nominal case and in the case where the jump near day 257 is considered to be a statistical fluctuation. \textit{Panel A}: value of the calibration constant in the two cases. Note that fast variations have been removed from the data in order to make the plot clearer. If the jump is not considered to be real by the algorithm, the smoothing stage introduces an increasing slope in the values (thin grey line). \textit{Panel B}: map of the difference between the maps from Surveys~1 and 2 (the jump happened during the first survey), in ecliptic coordinates. Residual systematic effects produce stripes that are either aligned with the direction of the scan (i.e., perpendicular to the ecliptic plane) or with the Galactic plane. The black square highlights a region in the map that has been observed during the jump near day 257. \textit{Panel C}: zoom into the region highlighted in panel B (the difference between maps from Surveys~1 and 2), when the jump near day 257 has been considered real (see Panel A, thick black line). \textit{Panel D}: the same as the previous panel, but data have been calibrated assuming no real jump near day 257 (Panel A, thin grey line). Features are sharper in the latter case, and therefore we can conclude that the former calibration produces better results.