LFI design, qualification, and performance

Instrument description

The Low-Frequency Instrument (see Fig. 1 in the LFI overview) consists of a 20-K focal plane unit hosting the corrugated feed horns, the orthomode transducers (OMTs) and the receiver front-end modules (FEMs). Forty-four composite waveguides are interfaced with three conical thermal shields and connect the front-end modules to the warm (roughly 300 K) back-end unit (BEU) containing a further radio frequency amplification stage, detector diodes, and all the electronics for data acquisition and bias supply.

Best LFI noise performance is obtained with receivers based on InP High Electron Mobility Transistor (HEMT) low noise amplifiers (LNAs) for minimal power dissipation and best performance. To further minimize power consumption in the focal plane, the radiometers are split into two sub-assemblies connected by waveguides, one located at the telescope focal area, the other on the 300 K portion of the Planck satellite. These design features allow the entire front-end LNAs dissipation to be <0.55 W, which enables the active cooling of the focal assembly. This is achieved with a vibration-less hydrogen sorption cooler, which also provides 18 K pre-cooling to the HFI helium J-T cooler. Two sorption cooler units are included in the flight hardware.

As shown schematically in Fig. 1 below, the LFI consists of the following subsystems:

• Radiometer Array Assembly (RAA);
• Sorption Cooler Subsystem (SCS);
• Radiometer Electronics Box Assembly (REBA).

The RAA includes the Front End Unit (FEU) and the Back End Unit (BEU), connected via waveguides. The FEU is located at the focus of the telescope, as one component of the joint LFI/HFI focal assembly (see sections below). The BEU is mounted on the top of the Planck service module (SVM). The REBA (Radiometer Electronics Box Assembly) and the warm parts of the Sorption Cooler System (SCS) are located on one of the lateral panels of the SVM. The FEU and the Sorption Cooler Compressor (SCC) are connected by concentric stainless steel tubes. The smaller tube carries hydrogen at approximately 60 atmospheres from the cooler compressors to the FEU, while the larger tube returns the hydrogen at about 0.3 atmospheres. These units are described in following sections and in the Annexes, while the SCS is described in details in the Sorption Cooler section. All LFI units are linked together by the LFI harness, which also connects to the spacecraft interface.

Figure 1. Block Diagram of LFI.

Radiometer Array Assembly (RAA)

The Radiometer Array Assembly (RAA) consists of two main units (the front end unit, FPU and the Back End Unit, BEU), connected by a set of waveguides. The Focal Plane Unit (FPU) is the heart of the LFI instrument and it contains the feed array and associated orthomode transducers (OMTs) and FEMs, all cooled to 20 K by the sorption cooler. The FPU comprises a set of 11 modules, which are mounted on a mechanical support which meets the thermo-mechanical requirements of the instrument and adds thermal inertia. The BEU comprises the radiometer Back End Modules (BEM) and the Data Acquisition Electronics (DAE), which are connected by an internal harness. The HFI Unit is located inside the LFI FPU and supported by the LFI structure. The LFI structure gives the mechanical and thermal interface to the HFI unit with the proper stiffness and thermal de-coupling. The LFI structure also guarantees the proper alignment of the HFI detector with the telescope focal plane.

The timescale of the stability of the receiver is driven by the 1 rpm rotation speed of the spacecraft, which requires a very low 1/f-noise or gain variation of the low noise amplifiers and other components. The LFI uses a pseudo-correlation receiver concept (Fig. 2 below). This radiometer concept is chosen to maximize the stability of the instrument by reducing the effect of non-white noise generated in the radiometer itself. In this scheme, the difference between the inputs to each of the chains (the signal from the telescope and that from a reference blackbody, respectively) is continuously being observed. To remove the effect of instability in the back-end amplifiers and detector diodes, it is necessary to switch the signal detected at the diodes at a high rate. The signals from the sky and from a reference load are combined by a hybrid coupler, amplified in two independent amplifier chains, and separated out by another hybrid. The sky and the reference load power can then be measured and differenced. Since the reference signal has been subject to the same gain variations in the two amplifier chains as the sky signal, the true sky power can be recovered. The differencing receiver greatly improves the stability if the two input signals are almost equal, at a cost of a factor of √2 in sensitivity compared to a perfectly stable total-power radiometer with the same noise temperature and bandwidth. This radiometer concept is capable of greatly reducing the knee frequency. We define as the Radiometer Chain Assembly (RCA, see Fig. 2) each functional unit from the feed horn to the BEM. The RAA therefore includes a set of 11 RCAs and the Data Acquisition Electronics (see also Fig. 1 above), all mounted on a suitable mechanical structure. Although there are differences in the details of the radiometer chains at different frequencies, their overall configuration is similar, and a general description of its design is provided in this section. Planck LFI has 11 Radiometer Chain Assemblies (RCAs). Each RCA is constituted by feed horn and FEM in the FEU (at 20 K), BEM (at 300 K) in the BEU, and four waveguides that connect each FEM-BEM couple. The frequency distribution of the RCA is the following:

• 2 RCAs at 30 GHz;
• 3 RCAs at 44 GHz;
• 6 RCAs at 70 GHz.

Figure 2. A complete RCA from feed-horn to analogue voltage output. The insets show the OMT, the details of the 20 K pseudo-correlator and of the back-end radio-frequency amplification, low-pass filtering, detection, and DC amplification.

Radiometer Chain Assembly (RCA)

Every RCA consists of two radiometers, each feeding two diode detectors (see Fig. 2 above), for a total of 44 detectors. The 11 RCAs are labelled by a numbers from 18 to 28, as outlined in Fig. 1 in LFI overview, right panel.

Fig. 2 provides a more detailed description of each radiometric receiver. In each RCA, the two perpendicular linear polarization components split by the OMT propagate through two independent pseudo-correlation differential radiometers, labelled as M or S, depending on the arm of the OMT they are connected to (Main or Side, see lower-left inset of Fig. 2).

In each radiometer the sky signal coming from the OMT output is continuously compared with a stable 4 K blackbody reference load mounted on the external shield of the HFI 4-K box . After being summed by a first hybrid coupler, the two signals are amplified by approximately 30 dB (see upper-left inset of Fig. 2). The amplifiers were selected for best operation at low drain voltages and for gain and phase match between paired radiometer legs, which is crucial for good balance. Each amplifier is labelled with codes 1 or 2 so that the four outputs of the LNAs can be named with the sequence: M1, M2 (radiometer M) and S1, S2 (radiometer S). Tight mass and power constraints called for a simple design of the Data Acquisition Electronics (DAE) box so that power bias lines were divided into five common-grounded power groups with no bias voltage readouts; only the total drain current flowing through the front-end amplifiers is measured and is available to the house-keeping telemetry (this design has important implications for front-end bias tuning, which depends critically on the satellite electrical and thermal configuration and was repeated at all integration stages, during on-ground and in-flight satellite tests). A phase shift (or phase switch) alternating between 0° and 180° at the frequency of 4096 Hz is applied in one of the two amplification chains and then a second hybrid coupler separates back the sky and reference load components that are further amplified and detected in the warm BEU, with a voltage output ranging from -2.5 V to +2.5 V.

Each radiometer has two output diodes which are labelled with binary codes 00, 01 (radiometer M) and 10, 11 (radiometer S), so that the four outputs of each radiometric chain can be named with the sequence M-00, M-01, S-10, S-11.

After detection, an analogue circuit in the DAE box removes a programmable offset in order to obtain a nearly null DC output voltage and a programmable gain is applied to increase the signal dynamics and optimally exploit the ADC input range. After the ADC, data are digitally down-sampled, re-quantized and compressed in the REBA according to a scheme described in Refs. , before preparing telemetry packets. On the ground, telemetry packets are converted to sky and reference load time ordered data after calibrating the ADU (Analog-to-Digital Unit) samples into volts considering the applied offset and gain factors.

To first order, the mean differential power output for each of the four receiver diodes can be written as follows (Planck-PreLaunch-II):

[math] \label{eq:power} P_{\rm out}^{\rm diode} = a\, G_{\rm tot}\,k\,\beta \left[ T_{\rm sky} + T_{\rm noise} - r\left( T_{\rm ref} + T_{\rm noise}\right) \right] [/math]

where G_tot is the total gain, k is the Boltzmann constant, β the receiver bandwidth, and a is the diode constant. T_sky and T_ref are the average sky and reference load antenna temperatures at the inputs of the first hybrid and T_noise is the receiver noise temperature.

The gain modulation factor Planck-Early-V, r, is defined by

[math] \label{eq:erre1} r = \frac{T_{\rm sky} + T_{\rm noise}}{T_{\rm ref} + T_{\rm noise}} , [/math]

and is used to balance (in software) the temperature offset between the sky and reference load signals and minimize the residual 1/f noise in the differential data stream. This parameter is calculated from the average uncalibrated total power data using the relationship

[math] \label{eq:erre2} r = \langle V_{\rm sky} \rangle/ \langle V_{\rm ref}\rangle, [/math]

where <V_sky> and <V_ref> are the average sky and reference voltages calculated in a defined time range. The white noise spectral density at the output of each diode is essentially independent from the reference-load absolute temperature and is given by

[math] \label{eq:dt1} \Delta T_0^{\rm diode} = \frac{2\,(T_{\rm sky}+T_{\rm noise})}{\sqrt{\beta}}. [/math]

If the front-end components are not perfectly balanced, then the separation of the sky and reference load signals after the second hybrid is not perfect and the outputs are mixed. First-order deviations in white noise sensitivity from the ideal behaviour are caused mainly by noise temperature and phase-switch amplitude mismatches. Following the notation used in Ref. , we define ε_Tn, the imbalance in front end noise temperature, and ε_A1 and ε_A2, the imbalance in signal attenuation in the two states of the phase switch. Equation above for the two diodes of a slightly imbalanced radiometer then becomes

[math] \label{eq:dt2} (\Delta T^{\rm diode} )^2 ≈ (\Delta T_0^{\rm diode})^2 ( 1± \frac{\epsilon_{A1}- \epsilon_{A2}}{2} + \alpha \epsilon_{T{\rm n}}) [/math]

which is identical for the two diodes apart from the sign of the term ε_A1 − ε_A2, representing the phase switch amplitude imbalance. This indicates that the isolation loss caused by this imbalance generates an anti-correlation between the white noise levels of the single-diode data streams. For this reason, the LFI scientific data streams are obtained by averaging the voltage outputs from the two diodes in each radiometer:

[math] \label{eq:v1} V^{\rm rad}_{\rm out} = w_1 V^{\rm diode\,1}_{\rm out} +w_2 V^{\rm diode\,2}_{\rm out} [/math],

where ω₁ and ω₂ are inverse-variance weights calculated from the data as discussed in Ref. Planck-Early-V. This way, the diode-diode anti-correlation is cancelled, and the radiometer white noise becomes

[math] \label{eq:dt3} \Delta T^{\rm rad} ≈ \frac{ T_0^{\rm diode}}{\sqrt{2}} (1+\alpha \epsilon_{T{\rm n}})^{1/2}. [/math]

In Equations above, ε«1, while α is a term ≈1 given by

[math] \label{eq:alpha} \alpha = \frac{ T_{\rm noise} (2 \; T_{\rm noise} + T_{\rm sky} + T_{\rm ref} } {2 (T_{\rm sky} + T_{\rm noise} )(T_{\rm ref} + T_{\rm noise})} [/math]

See Diode Combination section for the details of the diode combination procedure.

In Fig. 3 below we show a close-up of the two front end modules of an RCA with the four phase switches, which are labelled with the four letters A and B (main arm), and C and D (side arm). Each phase switch is characterized by two states: state 0 (no phase shift applied to the incoming wave); and state 1 (180° phase shift applied) and can either stay fixed in a state or switch at 4 kHz between the two states.

Phase switches are clocked and biased by the DAE and their configuration can be programmed via telecommand. In order to simplify the instrument electronics, phase switches are configured and operated in pairs, A/C and B/D. This means that if phase switches A and C are switching at 4 kHz then B and D are fixed both in the same state (either 0 or 1). This simplification, required during the design phase to comply with mass and power budgets, comes at the price of losing some setup redundancy.

Figure 3. Close-up of the two front-end modules of an RCA. There are four phase switches, labelled A, B, C, and D. Each switch can be fixed in one of its two positions (labelled as 0, 1) or switch at 4 kHz between 0 and 1. Phase switches are clocked and biased by the DAE in pairs: A/C and B/D.

Feed Horns (FHs)

Dual profiled corrugated horns have been selected at all LFI frequencies as the best design in terms of shape of the main lobe, level of the side lobes, control of the phase centre, and compactness. Details of the design, flight model and tests of Planck-LFI feed horns can be found in Ref. and in the corresponding Annex section.

Ortho-Mode Transducers (OMTs)

The Ortho–Mode Transducer (OMTs) separates the radiation collected by the feedhorn into two orthogonal polarization components. It consists of a circular to square waveguide transition (directly connected to the FH), a square waveguide section and two separate rectangular waveguides (the main and side arms, which separate and pick up the orthogonal polarization, connected with the FEU). A 90° bend is always present on the side arm, while a twist is also necessary on the main (30 and 44 GHz) and side (70 GHz) arms, in order to math the FEU polarization.

The details of the flight models and measurements of the Planck LFI ortho-mode transducers can be found in Ref. and in the corresponding Annex section.

Front End Modules (FEMs)

Front End Modules are located in the FPU, just behind the Feed Horn and the Ortho Mode Transducers. 70 GHz FEMs are mounted onto the inner wall of the main frame (the wall facing HFI instrument) from the HFI side. 44 and 30 GHz FEMs are inserted into the main frame from the waveguide (WG) side and fixed to the bottom plate. Screws to the bottom plate are inserted from the WG side. The LFI FEMs are the first active stage of amplification of the radiometer chain. Each FEM contains four amplification paths, each of which is composed by several cascaded LNAs followed by a phase switch. Two passive hybrids, at the input and output of the FEM, are used to mix couples of signals of the same radiometer (see Fig. 3). This makes the instabilities of each chain to be applied to both the sky and load signals.

The passive hybrid coupler ("magic-tee") combines the signals from the sky and cold load with a fixed phase offset of either 90° or 180° between them. It has a 20% bandwidth, low loss, and amplitude balance needed at the output to ensure adequate signal isolation.

The details of the design, development and verification of the 70 GHz front-end modules for the Planck Low Frequency Instrument can be found in Ref. and in the corresponding Annex section.

Waveguides (WGs)

The LFI Front End Unit (FEU) is connected to the Back End Unit (BEU) by 44 rectangular waveguides approximately 1.5-2.0 m long. Each waveguide exhibits a low voltage standing wave ratio, low thermal conductivity, low insertion loss, and low mass. In addition, the waveguide path permits the LFI/HFI integration and the electrical bonding between the FPU and the BEU. Because of the Focal Plane Unit arrangement, the waveguides are in general twisted and bent in different planes and with different angles, depending on the particular waveguide. From the thermal point of view the waveguides have to connect two systems (the BEM and FEM) that are at very different temperatures. At BEM level the waveguides are at a temperature of 300 K, while at FEM level the temperature is 20 K. The waveguides have to reduce the thermal flow from 300 K to 20 K. In Fig. 1 in the LFI overview (left panel) a conceptual sketch of the LFI configuration is shown.

Details of the Planck-LFI flight model of the composite waveguides can be found in Ref. and in the corresponding Annex section.

Back End Modules (BEMs)

The BEMs are composed of four identical channels each made of Low Noise Amplifiers (LNAs), an RF bandpass Filter, RF to DC diode detector, and DC amplifiers. The FEM output signals are connected by waveguides from the Focal Plane Unit (FPU) assembly to the Back End Modules (BEMs) housed adjacent to the Data Acquisition Electronics (DAE) assembly. To maintain compatibility with the FEMs, each BEM accommodates four receiver channels from the four waveguide outputs of each FEM. The BEM internal signal routes are not cross-coupled and can be regarded as four identical parallel circuits. Each BEM is constructed as two mirror halves. The two amplifier/detector assemblies each contain two amplifier/detector circuits. Each is supplied from one of a pair of printed circuit boards, which also house two DC output amplifiers.

The details of the design, development and verification of the 30 and 44 GHz back-end modules for the Planck Low Frequency Instrument can be found in Ref. . Details of the design, development and verification of the 70 GHz back-end modules for the Planck Low Frequency Instrument can be found in Ref.. Details are also reported in the corresponding Annex section.

4-K Load

The purpose of the 4-K reference load is to provide the radiometer with a stable reference signal. Reducing the input offset (the radiometric temperature difference between the sky and the reference load) reduces the minimum achievable radiometer 1/f noise knee frequency for a given amplifier fluctuation spectrum. A reference load temperature that matches the sky temperature (approximately 2.7 K) would be ideal.

Details of the design, characteristics, and performance of the LFI 4-K reference load units are given in Ref. and in the corresponding section of the Annexes.

Naming Convention

The naming of all the LFI elements has been described in the previous sections, but here is summarized again for clarity.

The 11 RCAs are labelled by numbers from 18 to 28, as outlined in Fig. 1 in the LFI overview (right panel). In each RCA, the two perpendicular linear polarization components are labelled as M or S according to the arm of the OMT they are connected to (Main or Side, see lower-left inset of Fig. 2).

Each front-end amplifier (see upper-left inset of Fig. 2) is labelled with the codes 1, 2, so that the four outputs of the FEM LNAs can be named with the sequence M1, M2 (radiometer M) and S1, S2 (radiometer S).

Each radiometer has two output diodes (see upper-right inset of Fig. 2), which are labelled with binary codes 00, 01 (radiometer M) and 10, 11 (radiometer S), so that the four outputs of each radiometric chain can be named with the sequence M-00, M-01, S-10, S-11.

REBA

The Radiometer Electronics Box Assembly (REBA) is the electronics unit in charge of processing the digitized scientific data and managing the overall instrument. It is also in charge of communication with the spacecraft. There are two REBA boxes, one nominal and one redundant. The redundancy concept is cold, which means that both boxes are never ON at the same time; the operation of each unit is managed by the spacecraft switching on the corresponding unit. The REBA ASW (Application SoftWare) is the same in each REBA box.

A detailed description of the Planck LFI REBA can be found in Ref. and in the corresponding section of the Annexes.

Instrument On-board Software

The REBA software is the on-board software of LFI. It is installed in the two computing subunits of REBA: the DPU (Digital Processing Unit), responsible of the control and monitoring of the instrument and the interface with the spacecraft; and the SPU (Signal Processing Unit), responsible for the data reduction and compression.

Details can be found in the corresponding section of the Annexes.

Reduction and Compression of Science Data

To asses stability against 1/f noise, the Low Frequency Instrument (LFI) on board the Planck mission acquired data at a rate much higher than the data rate allowed by the science telemetry bandwidth of 35.5 kbps. The data were processed by an on-board pipeline, followed on-ground by a decoding and reconstruction step, to reduce the volume of data to a level compatible with the bandwidth, while minimizing the loss of information. The on-board processing of the scientific data used by Planck/LFI to fit the allowed data-rate is an intrinsically lossy process which distorts the signal in a manner that depends on a set of five free parameters (N_aver, r₁, <i,>r₂, q, O) for each of the 44 LFI detectors. A brief description of the characteristics of this algorithm and the level of distortion introduced by the on-board processing as a function of these parameters can be found in the corresponding section of the Annexes, while a full description of the Planck LFI on-board data handling system and the tuning and optimization method of the on-board processing chain can be found in Ref. .

The strategy adopted to fit into the bandwidth relies on three on-board processing steps: downsampling; pre-processing the data to ensure loss-less compression; and loss-less compression itself. To demonstrate these steps, a model of the input signal was used. It has to be noted that while the compression is loss-less, the pre-processing is not, due to the need to rescale the data and convert them to integers (a process named data re-quantization). However, the whole strategy is designed to keep strict control over the way in which lossy operations are done, and to quantify the amount of information loss in order to assess the optimal compression rate with minimal information loss.

Instrument Operations

Operational Modes

The operations of the LFI are designed to be automatic and require little if any intervention from the ground. A small number of commands are required for operating the instrument and eventually for diagnostic and reconfiguration purposes. Each sky survey is conducted by the LFI with the instrument in the Normal Operations Mode. No deployable elements, or mechanically moving parts are included in the instrument. The scanning of the sky is achieved by progressive repointing of the satellite spin axis, with the Sun direction always within a cone 10° from the spin axis. Within the Normal Science Mode the instrument can be configured in order to fit with different science or diagnostic needs without changing the power consumption and thus the temperature in the FPU. Changes in power consumption in the FPU are minimized and should occur only in the case that failures in the radiometers that could create interference problems require an RCA to be switched off. Power adjustments on the first stage of the HEMT amplifiers (which were contemplated), require extremely small power level variations.

A brief summary of the LFI Operational Modes and the transitions between them is given in the corresponding section of the Annexes.

Ground Tests

During its development, the LFI flight model was calibrated and tested at various integration levels from sub-systems to individual integrated receivers Planck-PreLaunch-XIII and the whole receiver array Planck-PreLaunch-VII. In every campaign we performed tests according to the following classification:

• functionality tests, performed to verify the instrument functionality;
• tuning tests, to tune radiometer parameters (biases, DC electronics gain and offset, digital quantization and compression) for optimal performance in flight-like thermal conditions;
• basic calibration and noise performance tests, to characterize instrument performance (photometric calibration, isolation, linearity, noise and stability) in tuned conditions;
• susceptibility tests, to characterize instrument susceptibility to thermal and electrical variations.

Where possible, the same tests were repeated in several test campaigns, in order to ensure enough redundancy and confidence in the instrument behaviour repeatability. A matrix showing the instrument parameters measured in the various test campaigns is provided in Table 1 of Refs. Planck-PreLaunch-VII.

The ground test campaign was developed in three main phases: cryogenic tests on the individual RCAs; cryogenic tests on the integrated receiver array (the so-called radiometer array assembly, RAA); and system-level tests after the integration of the LFI and HFI instruments onto the satellite. The first two phases were carried out at the Thales Alenia Space - Italia laboratories located in Vimodrone (Milano, Italy) (note that receiver tests on 70 GHz RCAs were carried out in Finlad, at Yilinen laboratories), while system level tests (SLTs) were conducted in a dedicated cryofacility at the Centre Spatiale de Liège (CSL) located in Liege (Belgium).

In Table 1 below we list the temperature of the main cold thermal stages during ground tests compared to in-flight nominal values. These values show that system-level tests were conducted in conditions that were as much as possible flight-representative, while results obtained during RCA and RAA tests needed to be extrapolated to flight conditions to allow comparison. Details about the RCA test campaign are discussed in Planck-PreLaunch-XIII, while the RAA tests and the extrapolation methods are presented in Planck-PreLaunch-VII.

Table 1. Temperatures of the main cold stages during the various ground test campaigns compared to in-flight nominal values.

Temperature	Nominal	RCA tests	RAA tests	System-level
Sky	~ 3 K	≳ 8 K	≳ 18.5 K	~ 4 K
Ref. load	~ 4.5 K	≳ 8 K	≳ 18.5 K	~ 4.5 K
Front-end unit	~ 20 K	~ 20 K	~ 26 K	~ 20 K

During the various test campaigns the instrument was switched off and moved several times in a time period of about three years. A series of functional tests were always repeated at each location and also in-flight, in order to verify the instrument functionality and the response repeatability. No failures or major problems have been identified due to transport and integration procedures.

The expected Planck LFI scientific performance, resulting mainly from cryogenic system level tests, are described in Planck-PreLaunch-VII .

In-flight Calibration

The LFI Commissioning and Calibration and Performance Verification (CPV) phases started on 4 June 2009 and lasted until 12 August 2009 when Planck started scanning the sky in nominal mode. At the onset of CPV, the active cooling started when the radiating surfaces on the payload module reached their working temperatures (approximately 50 K on the third V-groove, and 40 K on the reflectors) by passive cooling. This was achieved during the transfer phase. Nominal temperatures were achieved on 3 July 2009, when the dilution cooler temperature reached 0.1 K Planck-Early-II. The cooldown of the HFI 4-K stage (see Fig. 4 below), was key during CPV for the LFI, because it provided a variable input signal that was exploited during bias tuning.

The LFI Commissioning and CPV was carried out in four phases:

• LFI switch-on and basic functionality verification (Commissioning);
• tuning of front-end biases and back-end electronics (CPV);
• preliminary calibration tests (CPV);
• thermal tests (CPV).

Figure 4. Functional tests performed during CPV against timeline. Curves from thermal sensors monitoring the 4-K stage and the FPU temperature are superimposed.

Details of the LFI Commissioning and CPV test campaign are given in (Gregorio et al. 2013 ).

Performance Summary

A summary of the LFI performance parameters is given in the table 1 of Summary of LFI data characteristics.

Instrument scientific performance

Optical parameters

The most accurate measurements of the LFI main beams were made with Jupiter, the most powerful unresolved (to Planck) celestial source in the LFI frequency range. Since the LFI feedhorns point to different positions on the sky, they detect the signal at different times. To map the beam, each sample in the selected timelines was projected in the (u, v) plane perpendicular to the nominal line-of-sight (LOS) of the telescope (and at 85° to the satellite spin axis). The u and v coordinates are defined in terms of the usual spherical coordinates (θ, φ) :

[math] u = \sin θ \, \cos φ, [/math]

[math] v = \sin θ \, \sin φ. [/math]

To increase the signal-to-noise ratio, data were binned in an angular region of 2′ for the 70 GHz channels and 4′ for the 30 and 44 GHz channels. We recovered all beams down to −20 dB from the peak. An elliptical Gaussian was fit to each beam for both M and S radiometers. Differences between the M and S beams caused by optics and receiver non-idealities are inevitable at some level, but they appear to be well within the statistical uncertainties, and for the purposes of point-source extraction, the beams may be considered identical. For the details of the typical FWHM and ellipticity averaged over each frequency channel, refer to Planck-Early-III and Planck-2015-A05. Exhaustive details on all LFI beam parameters are presented in the LFI data processing section Beams.

Photometric Calibration

Photometric calibration, i.e., conversion from voltage to antenna temperature, is performed for each radiometer after total power data have been cleaned of 1 Hz frequency spikes (see the LFI data processing section - Spikes Removal section and Planck-Early-V, Planck-2015-A03), and differenced. Here we report a brief overview of the photometric calibration; for the details see the LFI data processing section Photometric Calibration.

Our calibrator is the well-known dipole signal induced by Earth and spacecraft motions with respect to the CMB rest frame. The largest calibration uncertainty comes from the presence of the Galaxy and of the CMB anisotropies in the measured signal. We therefore use an iterative calibration procedure in which the dipole is fitted and subtracted, producing a sky map that is then removed from the original data to enhance the dipole signal for the next iteration. Typically, convergence is obtained after a few tens of iterations.

In our current calibration model we use as calibration signal the sum of the solar dipole ΔT_Sun and the orbital dipole ΔT_orb, which is the contribution from Planck’s orbital velocity around the Sun,

[math] \label{eq:cal1} \Delta T= ( \Delta T_{\rm Sun} + \Delta T_{\rm orb} ) \sin \theta_{\rm axis} [/math]

where θ_axis is the angle between the spacecraft axis and the overall dipole axis (solar + orbital). In this equation, the absolute calibration uncertainty is dominated by the uncertainty in T_Sun = (3.355 mK ± 0.008) mK , which is known to about 0.2%. The modulation of the orbital dipole by Earth's motion around the Sun is known with an uncertainty almost three orders of magnitude smaller; however, at least one complete Planck orbit is needed for its measurement. The accuracy of our current calibration can be estimated by taking into account two components: 1) the statistical uncertainty in the regions of weak dipole; and 2) the systematic uncertainty caused by neglecting gain fluctuations that occur on periods shorter than the smoothing window. In our calibration procedure the gain is estimated for every pointing period; if we call G_i the gain estimate from the ith pointing period, we have that the associated uncertainty is

[math] \label{eq:cal2} \delta G = \sqrt{\frac{ \Sigma_{i=1}^N (G_i-\lt G\gt )^2 }{N-1} } [/math],

where N is the overall number of pointing and <G> is the average of the N gains. We then approximate the effect of the smoothing filter as an average over M consecutive pointings, so that the overall uncertainty can be estimated as

[math] \label{eq:cal3} \delta G |_{\rm stat} = \frac{\delta G}{\sqrt{M}} = \frac{1}{\sqrt{M}} \sqrt{\frac{ \Sigma_{i=1}^N (G_i-\lt G\gt )^2 }{N-1} } [/math].

Noise Properties

The noise characteristics of the LFI data streams are closely reproduced by a simple (white + 1/f ) noise model,

[math] \label{eq:p1} P(f)= \sigma^2 \left[1+ \left(\frac{f}{f_{\rm knee}}\right)^\alpha \right] [/math],

where P(f) is the power spectrum and α≈−1. In this model, noise properties are characterized by three parameters, the white noise limit σ, the knee frequency f_knee, and the exponent of the 1/f component α, also referred to as the slope. Here we give noise performance estimates based on one year of operations, details of the analysis are given in LFI data processing section (Noise estimation) and in Planck-Early-V, Planck-Early-III, Planck-2015-A03.

Noise properties have been calculated following two different and complementary approaches: 1) fitting the equation above to time-ordered data for each radiometer; and 2) building normalized noise maps by differencing data from the first half of the pointing period with data from the second half of the pointing period to remove the sky signal (“jackknife” data sets).

Typical uncertainties are 0.5% for the white noise, between 5 and 10% for the slope, and between 10 and 20% for the knee frequency.

White Noise Sensitivity

Details of the white noise sensitivity can be found in Planck-Early-III.

Table 2 summarizes the sensitivity numbers calculated during the first year of operations using methods and procedures described in detail in Planck-Early-V, Planck-2015-A03, compared with scientific requirements. The measured sensitivity is in very good agreement with pre-launch expectations. While the white noise moderately exceeds the design specification, this performance is fully in line with the LFI science objectives.

Table 2. White noise sensitivity of the LFI frequency channels compared with requirements.

Channel	Measurement [μKCMB s1/2]	Requirement [μKCMB s1/2]
70 GHz	152.0	119
44 GHz	174.2	119
30 GHz	148.1	119

Instrument technical performance

Spectral response

The in-band receiver response was thoroughly modelled and measured for all the LFI detectors during ground tests. The complete set of bandpass curves was published in where all the details of the LFI radiometer's spectral response are given. From each curve we have derived the effective centre frequency according to:

[math] \label{eq:spectr} \nu_0 = \frac{ \int_{\nu_{\rm min}}^{\nu_{\rm max}} \nu g(\nu) \; d \nu} {\int_{\nu_{\rm min}}^{\nu_{\rm max}} g(\nu)\; d\nu } [/math]

where Δν = ν_max − ν_min is the receiver bandwidth and g(ν) is the bandpass response. Details about colour corrections, C(α), needed to derive the brightness temperature of a source with a power-law spectral index α, are provided in Ref. Planck-Early-V.

Some details are also given in the corresponding section of the Annexes.

Bandpass estimation

As detailed in Ref. , our most accurate method to measure the LFI bandpasses is based on measurements of individual components integrated into the LFI Advanced RF Model (LARFM) to yield a synthesized radiometer bandpass. The LARFM is a software tool based on the open-source Quasi Universal Circuit Simulator (QUCS). The measured frequency responses of the various subsystems (feed-OMT, FEM, BEM) are considered as lumped S-parameter components. Measurements of single components are obtained with standard methods and provide highly reliable results, with precision of order 0.1-0.2 dB over the entire band. Waveguides are simulated with an analytical model, in order to reproduce the effect of their temperature gradient and the effect of standing waves caused by impedance mismatch at the interfaces between the FEM and BEM. This is because the 1.8-m long waveguides were not measured at unit level in cryogenic conditions. The model provides accurate agreement with the measured waveguide response in the conditions of the test measurements (300 K). The composite bandpasses are estimated to have a precision of about 1.5 to 2 dB.

Some details are also given in the corresponding section of the Annexes.

Stability

Thanks to its differential scheme, the LFI is insensitive to many effects caused by 1/f noise, thermal fluctuations, or electrical instabilities. As detailed in Planck-Early-III, one effect detected during the first survey was the daily temperature fluctuation in the back-end unit induced by the downlink transponder, which was powered on each day for downlinks during the first 258 days of the mission. As expected, the effect is highly correlated between the sky and reference load signals. In the difference, the variation is reduced by a factor around (1 − r), where r is the gain modulation factor defined above (see r definition in RCA section).

A particular class of signal fluctuations occasionally observed during operations was due to electrical instabilities that appeared as abrupt increases in the measured drain current of the front-end amplifiers, with a relaxation time variable from few seconds to some hundreds of seconds. Typically, these events caused a simultaneous change in the sky and reference load signals. Because they are essentially common-mode, their residual on the differenced data is negligible, and the data are suitable for science production. In a few cases the residual fluctuation in the differential output was large enough (a few millikelvin in calibrated antenna temperature units) to be flagged, and the data were not used. The total amount of discarded data for all LFI channels until Operational Day 389 was about 2000 s per detector, or 0.008%.

A further peculiar effect appeared in the 44 GHz detectors, where single isolated samples, either on the sky or the reference voltage output, were clear outliers compared with the rest. Over a reference period of four months, 15 occurrences of single-sample spikes (out of 24 total anomaly events) were discarded, an insignificant loss of data.

Thermal susceptibility

As mentioned in section LFI In-flight Calibration above and detailed in Ref., during the CPV campaign, susceptibility tests were performed in order to characterize the LFI instrument susceptibility to thermal and electrical variations.

The effect of temperature fluctuations on the LFI radiometers originates in the Planck cold end interface of the hydrogen sorption cooler to the instrument focal plane. The temperature is actively controlled through a dedicated stage, the Thermal Stabilization Assembly (TSA), providing a first reduction of the effect. The thermal mass of the focal plane strongly contributes to reduce residual fluctuations. The physical temperature fluctuations propagated at the front end modules cause a correlated fluctuation in the radiometer signal, degrading the quality of scientific data. The accurate characterization of this effect is crucial for attempts to remove it from raw data by exploiting the housekeeping information of thermal sensors.

The propagation of the temperature oscillations through the focal plane and the instrument response to thermal changes were characterized through two main tests:

• the thermal dynamic response, aimed at measuring the dynamic thermal behaviour of the LFI Focal Plane;
• the radiometers thermal susceptibility.

Further details are also given in the corresponding section of the Annexes.

Instrument budgets

LFI power, mass and telemetry budget are given in the corresponding section of the Annexes.

Systematic Effects

The LFI design was driven by the need to suppress systematic effects well below instrument white noise. The differential receiver scheme, with reference loads cooled to 4 K, greatly minimizes the effect of 1/f noise and common-mode fluctuations, such as thermal perturbations in the 20-K LFI focal plane. The use of a gain modulation factor (see r definition in the RCA section above) largely compensates for spurious contributions from input offsets. Furthermore, diode averaging (see V^rad_out definition in the RCA section above) allows us to cancel second-order correlations, such as those originating from phase switch imbalances.

We have developed an error budget for systematic effects Planck-PreLaunch-IIPlanck-Early-IIIPlanck-2013-III and Planck-2015-A04 as a reference for both instrument design and data analysis. Our goal is to ensure that each systematic effect is rejected to the specified level, either by design or by robust removal in software. At this stage, the following effects are relevant:

For each of these effects we used flight data and information from ground tests to build timelines, maps, and angular power spectra that represent our best knowledge of their impact on the scientific analysis. The details of the systematic effect analysis are given in section LFI data processing section (Systematic Effects uncertainties).

The Sorption Cooler

The Planck H2 Sorption Cooler is the first stage of the active cryogenic chain. Its task is to maintain the LFI down to the operating temperature while providing a pre-cooling stage for the HFI refrigerators. The system performs a simple thermodynamic cycle based on hydrogen compression, gas pre-cooling by three passive radiators, further cooling due to the heat recovery by the cold low pressure gas stream, expansion through a J-T valve, and evaporation at the cold stage (Planck-Early-II). A schematic of the Planck Sorption Cooler System (SCS) is shown in Fig. 5. The engine of the cryocooler is the compressor. It serves two main functions: to produce the high-pressure hydrogen gas flow; and to maintain a stable gas recovery rate, which keeps the return pressure, hence the liquid temperature, constant. The high pressure gas flowing to the cold end is pre-cooled by exchanging heat with the three passive stages at the V-Grooves and with the evaporated cold gas returning back to the compressor. The gas then expands at the cold end through a J-T valve, producing approximately 1 Watt of cooling power at a temperature of <20K: most of this heat lift is used at the LVHX2 (Liquid Vapour Heat eXchanger 2, see Fig. 5) interface to absorb the LFI heat load at a temperature around 20K. The remaining heat lift is used at the LVHX1 as a pre-cooling stage, at a temperature lower than 19 K, for the two HFI refrigerators. The cooler and its performance are described in detail in Ref..

Figure 5. Planck Sorption Cooler CAD view with labels of the main sub-systems.

Both LVHXs provide a temperature of around 18 K, with fluctuations driven by the cooler instabilities (compressor element variations, cycling, two-phase flow dynamics, etc.). Stabilization of the HFI interface (LVHX1) temperature is not necessary, since thermal control of the subsequent colder stages is more efficient and very effective. To reduce cold end fluctuations directly transmitted to the LFI radiometers, a copper block, designated as the Temperature Stabilization Assembly (TSA), is inserted, as an intermediate stage, between LVHX2 and the LFI Focal Plane Unit (FPU). The TSA comprises a temperature sensor and a heater, controlled by a PID-type feedback loop, working in combination with the passive thermal inertia. The set-point temperature of the TSA is an adjustable parameter of the sorption cooler system, chosen to provide dynamic range for control during mission: as the compressor elements age, the return gas pressure and thus the temperature of LVHX2 rise slowly. To keep the LFI temperature reference stable during operations the set point must be periodically adjusted to maintain the level of oscillations within the required range (Fig. 6).

Figure 6. LVHX2 (black, HK parameter name SD029540) and TSA (red, HK parameter name SD030540) temperature profile from 125 to 563 days after launch.

Two sorption cooler units were integrated on-board the Planck spacecraft. Both units were used to cover the mission lifetime, with one cooler operated for the first two sky surveys. At the end of its lifetime a switchover operation, performed on August 11th 2010 (455 days after launch), activated the second unit that ran since then. The LVHX2 temperature change due to the new cooler start, and the subsequent re-adjustment of the LFI temperature stabilization stage, is clear in Fig. 6.