Visibility study in a chief-deputy formation for CMB polarization missions

Vol.:(0123456789) https://doi.org/10.1007/s40295-022-00325-z 1 3 ORIGINAL ARTICLE Visibility Study inaChief‑Deputy Formation forCMB Polarization Missions JuanBermejo‑Ballesteros1 · JavierCubas1 · FranciscoCasas2 · EnriqueMartínez‑González2 Accepted: 5 April 2022 © The Author(s) 2022 Abstract Scientific instruments on board satellites are becoming increasingly sensitive, making it imperative to submit these instruments to a thorough calibration. In-flight calibration could be largely improved by using an ancillary microsatellite flying in formation with the main satellite and emitting a well-defined and known reference signal. Due to the main satellite attitude motion, the calibration satellite and therefore, its calibration signal, will only enter the instrument FoV (Field of View) at certain instants. It is not intuitive how frequently and during how much time this will happen, or how this depends on the scan strategy. In the present work, the available time for calibration and its characteristics in terms of total, mean, and maximum duration are studied, deriving analytical expressions for these quantities. These expressions are validated numerically and allow us to assess the impact of different scan strategies and to evaluate the most suitable region to locate the calibration satellite. The focal plane of the instrument is also modelled to evaluate the calibration process at detector level, calculating the number of detectors viewed and the direction of the polarized signal that they received. For this last analysis, only numerical methods have been employed. The tools are finally used in a case study in order to show how they can be employed to test, evaluate, and optimize scanning strategies and relative positions. The tools presented in this work can be easily adapted to evaluate more generally the characteristics of the observation of each point in the sky for a given scan strategy and instrument FoV. Keywords formation flight· Visibility· Access time· Calibration· Microwaves calibration· Scan strategy * Juan Bermejo-Ballesteros [email protected] 1 Instituto Universitario “Ignacio Da Riva” (IDR/UPM), Universidad Politécnica de Madrid, Plaza Cardenal Cisneros 3, MadridE-28040, Spain 2 Instituto de Física de Cantabria (CSIC-UC), Avda. de los Castros s/n, SantanderE-39005, Spain Published online: 29 April 2022 The Journal of the Astronautical Sciences (2022) 69:651–691 / 1 3 Nomenclature Ttotal Total access time. Tmean Mean access time. Tmax Maximum access time. Tsim Simulated time. Tcomb Period of the combined motion of precession and spin motions. Tprec Precession motion period. Tspin Spin motion period. Taccess Access time. 𝜙 Spin angle. 𝜓 Precession angle. Ω Precession speed. 𝜔 Spin speed. 𝛼 Precession axis angle. 𝛽 Instrument axis angle. 𝛿 Half-angle of instrument FoV. 𝜁 Spatial separation between sequential trace rings. FoV Field of View. LoS Line-of-sight. 𝜌 Percentage of viewed detectors. G(𝜉) Angular coverage parameter. C0,b Satellite attitude matrix. Cb,inst Instrument mounting matrix. C0,inst Instrument attitude matrix. q(t) Instrument pointing direction. 𝐫cd Deputy position relative to the chief. 𝐩cd Unit vector of the deputy position relative to the chief. 𝐫deputy Deputy position vector. 𝐫chief Chief position vector. 𝜑,𝜃 Coordinates in the pseudo-inertial reference frame. 𝜑∗ ,𝜃∗ Coordinates in the instrument frame. 𝜑v Angle between instrument direction and X0 -axis. 𝜃𝜑 Half-arc angle of directions with the same 𝜑 coordinate inside the FoV 𝜃∗ 𝜑 Half-arc angle of a direction trajectory inside the FoV. 𝐩𝜑 Arbitrary direction. 𝐪∗ Equivalent vertical motion of the instrument. 𝐪0 Starting position of the instrument direction. 𝐪 Instrument pointing direction. 𝐮spin Spin axis direction. R Rotation matrix around the spin axis. ft Average fraction of directions inside the FoV in one spin period. fm Fraction of directions that have been inside the FoV after one spin period. 𝜑e,𝜃e Coordinates of exterior curve. 𝜑i,𝜃i Coordinates of interior curve. N Number of accesses. 652 The Journal of the Astronautical Sciences (2022) 69:651–691 1 3 𝐪e Exterior curve direction. 𝐪i Interior curve direction. 𝐧 Unitary normal to the trace. xe,ye,ze Cartesian coordinates of exterior curve. xi,yi,zi Cartesian coordinates of interior curve. 𝐪′ Derivate of instrument pointing direction. tmax Optimum access time. 𝜑∗ tmax Value of 𝜑∗ coordinate to achieve tmax . t1,t2,t3 Maximum access time according to 𝜑 coordinate. 𝜏 Auxiliary angle to describe 𝐯prec . 𝛾 Projection of 𝐯prec over 𝐯spin . 𝐯prec Precession velocity unitary vector. 𝐯spin Spin velocity unitary vector. D Distance between satellites. d Size of emitting source. 𝜀 Angular size. 𝜆 Wavelength of signal. CMB Cosmic Microwave Background. Re[x] Real part of x. dT Time step. 1 Introduction The scientific instruments on board satellites are becoming increasingly sensitive. To obtain measurements with an accuracy in line with such sensitivity, it is necessary to apply exhaustive calibration processes and error reduction algorithms to mitigate the systematic errors. The calibration process is normally carried out prior to the flight and also during the mission. Each case presents its own limitations. Pre-flight calibration is not always capable of reproducing the mission environment or the signal to be measured due to the laboratory constraints (size, technology, etc.). Furthermore, after calibration, the satellite experiences different environmental conditions during launch, orbital transfer and eclipses, with subsequent thermal loads and vibrations, which can result in slight calibration drifts. During the mission, the main obstacle is that the signals from natural sources (moons, planets, compact objects, or even galaxies) are not known with the required accuracy for ultra-sensitive polarization missions observing in the microwave range [6, 12, 16] and, due to their scanning strategies to cover the whole sky, they do not always allow for the calibration of the instrumentation as required. For example, precision of the order of arc-minutes in the polarization angle are needed in such experiments, while the main polarization calibrator, Messier-1 (Tau A), is known with an error of about 0.5 degrees. If more precise scientific goals are to be achieved with sufficient success, a more robust in-flight calibration must be addressed for future missions. 653The Journal of the Astronautical Sciences (2022) 69:651–691 1 3 One well-known example of this issue is the case of Cosmic Microwave Background (CMB) space missions. These telescopes measure the microwave signals that arrive from all directions of space, and that contain very valuable information about the origins of the universe. The primordial CMB signal was produced when the universe was about 380,000 years old and has a black-boby spectrum that, at present, peaks in the microwave range at about 160 GHz. Its tiny anisotropies in temperature and even dimmer ones in polarisation contain a wealth of information about the origin, matter and energy content, and the evolution of the universe. However, their measurement represents a very difficult challenge and requires the use of ultra sensitive telescopes with very accurate calibration procedures and an exquisite control of systematic effects. Some telescopes use the signal arriving from Jupiter as a reference for CMB intensity calibration, but it cannot be used for polarization calibration as it is not polarized. In-flight calibration could be largely improved by using an ancillary microsatellite which would fly in formation with the main satellite and would emit a precisely characterised reference signal. This concept was initially proposed to calibrate Ground-based CMB telescopes [13] and has been extended for CMB satellites located in Sun-Earth L 2 [5]. This location is commonly used for astronomical missions due to its thermally stable environment (ideal for cryogenic missions), low acceleration, and the possibility of observing the sky continuously without interference from the Earth, or the Moon [7, 12, 16]. These CMB telescopes have a given scan strategy, which defines its attitude motion in order to cover all the sky and collect data according to its scientific objectives [4, 9, 20]. The formation would be a chief/deputy formation, in which the chief (main satellite) moves independently while the deputy (ancillary satellite) locates itself around the chief, keeping the formation. During operations, the calibration satellite would stay out of sight, so it does not interfere with the scientific observations. Then, when calibration is required, the deputy would move to a predefined location so the calibration signal could reach the chief’s instrument. The extent of the sky region that is seen by the instrument at a given moment is called Field of View (FoV). Therefore, assuming that the deputy emits its calibration signal pointing always to the main satellite, the calibration will only occur when the deputy is inside the instrument FoV. Due to the chief’s scan strategy, this only occurs in certain instants, and it is not intuitive when and how frequently it will happen. Such an event will be called here an ‘access’ and it can be characterized by its frequency, duration, and how effective it can be for calibration. These features depend mainly on the relative position of the deputy and the scan strategy followed by the chief. The main goal of this article is to study the available time for calibration and its characteristics in terms of maximum and mean duration. This study will enable us to know which regions of space are more suitable to place the calibration satellite relative to the main satellite during the calibration phase. These aspects have been partially studied previously [3] and here a more complete study is presented. To carry out the analysis, the direction at which the instrument is pointing over time has been modelled (Sect.2) as well as the detectors geometrical disposition in the focal plane and the trajectory of the calibration signal through it (Sect.4). In order to quantify the performance of a certain relative position, a set of parameters have been defined 654 The Journal of the Astronautical Sciences (2022) 69:651–691 1 3 based on the access features. These parameters describe two main aspects of the accesses: their temporal characteristics and how effective they can be for calibration. The former aspect is assessed in Sect.3 through an analytical and numerical analysis that reports the number, frequency, and duration of the accesses. For the later aspect, analysed in Sect.4, the signal received by the main satellite instrument is evaluated regarding the percentage of detectors that receive the signal and whether they are calibrated at different polarization angles. The complexity added by the instrument model implies the use of numerical methods for this aspect. The methodology described has been applied in Sect.5 to an example mission, assessing the evolution of the parameters according to the variation of the scan strategy parameters and identifying suitable regions where the deputy can be placed. Finally, the conclusions are summarized in Sect. 6. All the methods presented here have been implemented in Python. Both the code and the results are accessible in an open GitLab repository [2], where other related works can also be found. 2 Scanning Strategy andRelative Position The first step has been the modelling of the main satellite attitude motion to determine the instrument attitude and its pointing over time. The model used to describe the main satellite attitude is based on a simplified version of the one presented in [20]. In this case, only four parameters are used: the spin motion period ( Tspin ), the precession motion period ( Tprec ), the precession axis angle ( 𝛼 ), and the instrument axis angle ( 𝛽 ). The satellite spins around its main axis (spin axis) with a time period of Tspin . Simultaneously, the spin axis, which is separated by angle 𝛼 from the precession axis, rotates around it with a time period of Tprec . The instrument line-of-sight (LoS) is separated by angle 𝛽 from the spin axis and has a circular FoV with a halfangle of 𝛿 . The attitude of the satellite is defined with respect to a pseudo-inertial frame of reference (0) centered in the main satellite. Although this frame would not be inertial due to the satellite orbital motion, it can be considered as such for shorts period of time. The frame is shown along an scheme of the attitude motion in Fig.1. The X0 -axis of the pseudo-inertial frame, which coincides with the precession axis, is parallel to the Sun direction and it is positive towards deep space. The Y0 and Z0 axes complete the right-handed frame and their direction can be chosen arbitrarily. The attitude of the satellite can be described with an Euler angle sequence (1-3-1) in which the three angles ( 𝜓 , 𝛼 , 𝜙 ) are directly related to the scan strategy parameters, with 𝜙 being the angle rotated due to spin (2) and 𝜓 the angle rotated due to precession (3). The equations (2) and (3) also define the spin speed ( 𝜔 ) and the precession speed ( Ω ). Then, the attitude matrix, as a function of three of the scan strategy parameters ( Tspin , Tprec and 𝛼 ), is [8] where (1) C 0,b= ⎡ ⎢ ⎢ ⎣ c𝛼c𝜙s𝛼s𝜙s𝛼 −c𝜓s𝛼c𝜓c𝛼c𝜙−s𝜓s𝜙c𝜓c𝛼s𝜙+s𝜓c𝜙 s𝜓s𝛼−s𝜓c𝛼c𝜙−c𝜓s𝜙−s𝜓c𝛼s𝜙+c𝜓c𝜙 ⎤ ⎥ ⎥ ⎦ , 655The Journal of the Astronautical Sciences (2022) 69:651–691 1 3 The attitude matrix (1) relates the orientation of the satellite body frame (b) and the pseudo-inertial reference frame (0). The instrument frame (inst) is defined as follows: its Xinst -axis is the instrument line of sight and the YinstZinst -plane corresponds to the plane where the detectors are located. The orientation of the instrument frame with regard to the satellite body frame is expressed by the mounting matrix which consist of a rotation of angle 𝛽 around satellite Zb -axis. Thus, the attitude of the instrument is obtained by combining the satellite attitude matrix (1) and the mounting matrix (4) like (2) 𝜙 =𝜔t= 2 𝜋 T spin t , (3) 𝜓 =Ωt= 2𝜋 T prec t . (4) C b,inst = ⎡ ⎢ ⎢ ⎣ c𝛽s𝛽0 −s𝛽c𝛽0 0 01 ⎤ ⎥ ⎥ ⎦ , (5) C0,inst =C0,bCb,inst. Fig. 1 Pseudo-inertial coordinate frame (0) and attitude motion scheme 656 The Journal of the Astronautical Sciences (2022) 69:651–691 1 3 The instrument pointing direction can be obtained directly from the orientation of the instrument Xinst –axis as As time passes, the line of sight of the instrument creates a pattern of observation over the celestial sphere. The morphology of this trace depends on the four parameters of the scan strategy. The orthographic projection of the instrument axis on the reference frame Y0Z0 -plane is shown in Fig.2 as an example of scan pattern. Note that the area swept by the FoV is not plotted. The scan strategy parameters choice is subjected to some constraints to ensure a complete and adequate observation of the sky [20]. For example, the sum of 𝛼 and 𝛽 defines the extent of the celestial sphere observed for a fixed direction of the precession axis. For a satellite orbiting L 2 , where the precession axis usually points towards anti-Sun direction, it is necessary that 𝛼+𝛽≥90◦ to observe the entire sky after six months. However, this sum should not be much higher to avoid pointing towards the Sun. Other constraints are the ratio between Tspin and Tprec , which defines the spatial separation between the sequential rings that are mapped in each spin turn and can be related to the FoV size, or the value of the sampling frequency, that must be high enough to sample the sky with the required accuracy and thus establishes a lower limit on Tspin . The scan strategy parameters used in the following examples are similar to to those used in previous and future CMB missions [6, 7, 11, 12, 17]. (6) 𝐪 (t)=C0i ⎧ ⎪ ⎨ ⎪ ⎩ 1 0 0 ⎫ ⎪ ⎬ ⎪ ⎭ . Fig. 2 Orthographic projection of the scan pattern for 𝛼=45◦ , 𝛽=50◦ , Tspin =10 min, and Tprec =90 min 657The Journal of the Astronautical Sciences (2022) 69:651–691 1 3 Once the sum constraint of 𝛼 and 𝛽 is set, the trace geometry will differ according to whether 𝛼<𝛽 or the contrary, as shown in Fig.3. In the former case, the instrument line of sight encircles the precession axis in each spin rotation; while in the latter case it does not. In both cases, the minimum angle that the line of sight forms with the precession axis (the origin) is |𝛼−𝛽| . Due to this, if the difference between both angles is high, the trace does not pass close to the precession axis, leaving a circular region without any crossover. The relation between the periods of the spin and precession motions will also influence the geometry of the trace and is chosen carefully to avoid undesired effects and to ensure adequate coverage of the sky [18, 20]. As previously said, the ratio between the period of both motions defines the spatial separation 𝜁 between the sequential rings of the trace This separation can be chosen to be less than the FoV size ( 𝛿 ) to allow continuous mapping of the sky. Additionally, as both motions are periodic, their combined motion will also be periodic as long as the periods are related by a rational number (i.e., they are commensurable), in which case the pattern will repeat itself. In general, a higher period of the combined movement will produce a finer pattern, providing a more uniform coverage of the sky, since it will be necessary a larger amount of time until the trace starts repeating itself. Thus, the access duration of those points in the sky at the same angular distance from the precession axis will become more uniform. The combined period ( Tcomb ) can be calculated as the least common multiple of both periods. This behaviour can be observed in Fig.4, where a slight variation of the parameter Tprec changes the coverage considerably as the (7) T spin Tprec = 𝜁 2𝜋sin 𝛼 . Fig. 3 Change of the scan pattern morphology according to the relation between angles 𝛼 and 𝛽 . In both cases Tspin =10 min and Tprec =90 min 658 The Journal of the Astronautical Sciences (2022) 69:651–691 1 3 combined motion period rises. In Fig.4(a), the combined period is Tcomb =190 min while in Fig.4(b) the resulting period is Tcomb =970 min. In the first case, for points separated by 45 ◦ from the precession axis and 𝛿=7.5◦ , the variations of the mean duration of the accesses is much higher, reaching a difference of 14 s, than in the second case, where the maximum differences is roughly 3 s. As shown in Fig. 4, the pattern of the trace becomes increasingly uniform as the combined period of the two motions grows. Ideally, if such a period is infinite, this distribution will be fully axial-symmetric around the precession axis. Under this condition, the results of the visibility parameters can also be expected to have axial-symmetry around the precession axis. This simplification will be assumed in the analytical study. Once the scan strategy is fixed, the access features will vary according to the relative positioning between deputy and chief. The deputy position relative to that of the chief is defined by the vector connecting both satellites In this analysis, the distance separating the satellites is not relevant. For clarity the unit vector, defined as is used instead. This unit vector represents the direction of the deputy on the sky as seen from the chief’s perspective (5). This direction can be expressed in terms of two coordinates ( 𝜑 , 𝜃 ) as (8) 𝐫cd =𝐫deputy −𝐫chief . (9) 𝐩 cd = 𝐫 cd |𝐫cd| Fig. 4 Change in the scan pattern due to the value of the combined motion period. For (a), the resulting period of the combined motions is Tcomb =190 min while for (b), the resulting period is Tcomb =970 min. In both cases 𝛼=45◦ and 𝛽=50◦ 659The Journal of the Astronautical Sciences (2022) 69:651–691 1 3 numerical results for total access time are compared. The numerical results have been averaged over 𝜃 . Overall, the error is low except in the deep slope zones of the curve, due to the limited precision of the bilinear interpolation. The criterion to consider that both results coincide has been established in an RMSE (rootmean-square error) that is lower than 1E-3 % of Tsim . This is fulfilled for all the scan strategies tested. Fig. 9 Total access time. Parameters considered: Tsim =1 day, 𝛼=45◦ , 𝛽=50◦ , Tspin =10 min, Tprec =93 min, and 𝛿=7.5◦ Fig. 10 Comparison (a) and error (b) between the analytical and numerical results for the variation of the total access time as a function of coordinate 𝜑 . Parameters considered: Tsim =1 day, 𝛼=45◦ , 𝛽=50◦ , Tspin =10 min, Tprec =93 min, and 𝛿=7.5◦ 666 The Journal of the Astronautical Sciences (2022) 69:651–691 1 3 3.2 Mean Access Time The mean access time, Tmean , can be defined as the total access time over a period of time divided by the number of accesses. As before, for a direction 𝐩 , if spin and precession are chosen properly, it is possible to assure that this parameter does not depend on 𝜃 . 3.2.1 Analytical Approach forMean Access Time In the previous section, the total access time has been obtained, and now the number of accesses is calculated following a similar approach. Although, for calculating the total access time the precession motion can be omitted, it will influence the number of accesses for a certain 𝜑 and it has to be taken into account if precession speed Ω is not negligible. In order to calculate the number of accesses, it is not relevant how much time one given direction stays inside the FoV, but whether after one spin period it has been seen or not. Thus, the approach to solve the problem is to calculate the proportion of directions, fm , with the same 𝜑 coordinate that have been inside the FoV after one spin period . As before, the motion is symmetrical with regard to the axis of rotation after a period, therefore, this proportion must remain constant for subsequent periods. If accesses are equally distributed along all the points with the same 𝜑 , the total number of accesses during a given period can be obtained by multiplying the number of spin cycles by the aforementioned proportion (22) N (𝜑)=fm(𝜑) T sim T spin . Fig. 11 Geometrical scheme for the mean access time when precession is negligible 667The Journal of the Astronautical Sciences (2022) 69:651–691 1 3 First, the case for Ω negligible is presented and then it will be extended to the general case that takes into account the precession speed. A geometrical scheme of half spin period when Ω is negligible is shown in Fig.11. Every half spin period, an arc of Δ𝜃 is seen from the instrument. Over one spin, this makes a proportion of which is equal to the mean number of accesses for all the directions with the same 𝜑 coordinate. The value of Δ𝜃 can be obtained as where 𝜃e and 𝜃i are the limits of the highlighted arc in Fig.11. If Ω is negligible, this arc is limited by the two circles of radius 𝛽+𝛿 and 𝛽−𝛿 . Therefore, following the same approach that in (19), the limit values are and Once 𝜃e and 𝜃i are obtained, the distance 2Δ𝜃 can be calculated and using (23) in (22), the number of accesses will be (23) fm= 2Δ𝜃 2𝜋, (24) Δ𝜃=𝜃e−𝜃i, (25) 𝜃 i | 𝜑= Re ( arccos ( cos(𝛽−𝛿)−cos(𝛼)cos(𝜑) sin(𝛼)sin(𝜑) )), (26) 𝜃 e | 𝜑= Re ( arccos ( cos(𝛽+𝛿)−cos(𝛼)cos(𝜑) sin( 𝛼 )sin( 𝜑 ))). (27) N (𝜑)=2Δ𝜃 2𝜋 T sim T spin . Fig. 12 Variation of the mean access time as a function of coordinate 𝜑 for negligible Ω . Parameters considered: Tsim =1 day, 𝛼=45◦ , 𝛽 = 50◦ , Tspin =10 min, and 𝛿=7.5◦ 668 The Journal of the Astronautical Sciences (2022) 69:651–691 1 3 Retrieving the total access time (21), the mean access time is The mean access time profile for the case of negligible Ω is shown in Fig.12. However, if Ω<𝜔 but not negligible, the trace cannot be simplified as a circle. An example of how the scheme can look is shown in Fig.13. In this case, in order to calculate the value of 𝜃i and 𝜃e it is necessary to obtain the side curves of the trace. These curves are separated by a 2𝛿 distance in the perpendicular direction to the trace. If the trace expression is 𝐪(t) (6), the equations of the exterior and interior curves are respectively and with 𝐧 being the unitary normal to the trace, whose expression is: (28) T mean(𝜑)= T total (𝜑) N( 𝜑 ) =Tspin f t (𝜑) fm( 𝜑 ) . (29) 𝐪 e(t)=𝐪cos(𝛿)+𝐧sin(𝛿)= ⎧ ⎪ ⎨ ⎪ ⎩ xe(t) ye(t) ze(t) ⎫ ⎪ ⎬ ⎪ ⎭ (30) 𝐪 i(t)=𝐪cos(𝛿)−𝐧sin(𝛿)= ⎧ ⎪ ⎨ ⎪ ⎩ xi(t) yi(t) zi(t) ⎫ ⎪ ⎬ ⎪ ⎭ . Fig. 13 Geometrical scheme for mean access time when precession speed is significant. Only the trace for half Tspin period is shown 669The Journal of the Astronautical Sciences (2022) 69:651–691 1 3 It is more useful to express the external and internal curves in terms of its 𝜑 and 𝜃 components. These can be calculated from the Cartesian coordinates from (29) and (30) as and With the equations of the external and internal curve, the procedure to calculate the 2Δ𝜃 for a given 𝜑 is as follows: first, using equations (32) and (34) the instants ti and te when the traces have respectively 𝜑i=𝜑 and 𝜑e =𝜑 are calculated. Second, these instants are used to calculate 𝜃e and 𝜃i with equations (33) and (35). Once these angles are obtained, the distance 2Δ𝜃 can be determined and the number of accesses is obtained through equation (28). An example of the resulting profile obtained from equation (28) is shown in Fig.14. (31) 𝐧 =𝐪 �( t )∧ 𝐪 ( t ) |𝐪 � (t)|. (32) 𝜑e(t) = Re(arccos(ye(t))) (33) 𝜃 e(t)=Re ( arccos (z e (t) sin( 𝜑 e(t)))) (34) 𝜑i(t)=Re(arccos yi(t)) (35) 𝜃 i(t)=Re ( arccos (z i (t) sin( 𝜑 i(t)))). Fig. 14 Variation of the mean access time as a function of coordinate 𝜑 , with Ω<𝜔 but not negligible. Parameters considered: Tsim =1 day, 𝛼=45◦ , 𝛽=50◦ , Tspin =10 min, Tprec =93 min, and 𝛿=7.5◦ 670 The Journal of the Astronautical Sciences (2022) 69:651–691 1 3 3.2.2 Numerical Results fortheMean Access Time The numerical results for the mean access time for the same example on Fig.14 are presented in Fig. 15. As can be seen, the mean access time also presents axial-symmetry. Fig. 15 Mean access time. Parameters considered: Tsim =1 day, 𝛼=45◦ , 𝛽=50◦ , Tspin =10 min, Tprec =93 min, and 𝛿=7.5◦ Fig. 16 Comparison (a) and error (b) between the analytical and numerical results for the mean access time as a function of coordinate 𝜑 . Parameters considered: Tsim =1 day, 𝛼=45◦ , 𝛽=50◦ , Tspin =10 min, Tprec =93 min, and 𝛿=7.5◦ 671The Journal of the Astronautical Sciences (2022) 69:651–691 1 3 In Fig.16 the analytical and numerical results for mean access time are compared. The numerical results have been averaged over 𝜃 . In this case, the criterion to validate the results is that the RMSE is lower than the time step used for the numerical analysis. This is fulfilled for all the scan strategies tested. 3.3 Maximum Access Time The maximum access time ( Tmax ) is the maximum time that a direction p=f(𝜑 , 𝜃) can be inside the FoV of the instrument. This time also depends on the scan strategy and will not depend on 𝜃 if the rotation and precession are chosen properly. 3.3.1 Analytical Approach forMaximum Access Time For the maximum access time, the procedure followed is similar to the mean access time section. First, the expression for the duration of an access with no precession motion is obtained. But now, the maximum access time is studied instead of the mean access time. Then, the precession motion is added, distinguishing two cases according to whether the precession speed is negligible or not, like in the mean access time section. First, the case without precession will be analyzed. As there is only rotation around the spin axis, the motion will be analyzed centered in that axis, the new coordinates being 𝜃∗ and 𝜑∗ . Furthermore, in order to obtain the access time for this pure spin motion, it is advantageous to consider that the instrument remains fixed and it is the direction vector the one that rotates around the spin axis ( X∗ in the new coordinates). Without loss of generality, it is assumed that the instrument is located on the X∗Z∗ - plane, separated by an angle 𝛽 from the spin axis, as shown in Fig.17, thus, Fig. 17 New reference system defined 672 The Journal of the Astronautical Sciences (2022) 69:651–691 1 3 In this new frame, the direction vector (37) is separated an angle 𝜑∗ from the spin axis, and its projection over the Y∗Z∗ -plane form an angle 𝜃∗ with the Z∗ -axis, so it can be expressed as The starting condition of an access (18) establishes that the relation between 𝛽 , 𝜃∗ , 𝜑∗ and 𝛿 for a given instant is This equation is similar to (19). Due to how the direction vector is expressed, the angle 𝜃∗ 𝜑 is the half-arc of its trajectory inside the FoV. Therefore, if the trajectory in one spin period has a length of 2𝜋 , the proportion of time inside the FoV is 2 𝜃 ∗ 𝜑∕ 2 𝜋 and the access time is (36) 𝐪 = ⎧ ⎪ ⎨ ⎪ ⎩ cos 𝛽 0 sin 𝛽 ⎫ ⎪ ⎬ ⎪ ⎭ . (37) 𝐩 = ⎧ ⎪ ⎨ ⎪ ⎩ cos(𝜑∗) sin(𝜑∗)sin(𝜃∗) sin(𝜑∗)cos(𝜃∗) ⎫ ⎪ ⎬ ⎪ ⎭ . (38) cos (𝜃∗ 𝜑)= cos(𝛿)−cos(𝜑 v )cos(𝜑∗) sin(𝜑 v )sin(𝜑∗) . (39) T access = T spin 𝜋Re ( arccos ( cos(𝛿)−cos(𝛽)cos(𝜑 ∗ ) sin(𝛽)sin(𝜑∗) )). Fig. 18 Trace of the direction vector over the FoV for different 𝜑∗ coordinates 673The Journal of the Astronautical Sciences (2022) 69:651–691 1 3 The maximum access time can be obtained deriving (39) with regard to 𝜑∗ and equating to zero, which gives This maximum access time, called here optimum access time, is achieved for the directions whose 𝜑∗ coordinate is equal to 𝜑∗ tmax , whose value is For any higher or lower value of 𝜑∗ , the access time will be lower than the optimum access time. In Fig.18 it is shown a scheme of this behaviour. As mentioned before, to add the effect of precession, two cases have been considered. For the case where Ω<< 𝜔 , i.e., the precession motion is much slower than the spin motion, it can be assumed that the trace of the instrument over the sky is nearly a circumference and therefore the previous results can be used directly. As the spin axis rotates around the precession axis, those points in the sky which, at any time, are at an angular distance of 𝜑∗ tmax will have an optimum access time. For any point with 𝜑<𝛼+𝜑∗ t max and 𝜑>| 𝛼 − 𝜑 ∗ t max | there will be an instant in which the trace is at the right point. Therefore, its maximum access time will be the optimum access time. For those directions whose 𝜑 coordinate is out of the aforementioned range, the maximum access time will not be the optimum. However, (39) is still valid assuming that 𝜑∗= 𝛼 − 𝜑⋅ sgn( 𝛼 − 𝜑 ∗ t max ) for 𝜑<| 𝛼 − 𝜑 ∗ t max | and 𝜑∗=𝜑−𝛼 for 𝜑>𝛼+ 𝜑 ∗ t max . Taking all of this into account, according to the value of 𝜑 , its maximum access time is for 0<𝜑<|𝛼−𝜑∗ tmax | : for |𝛼−𝜑∗ tmax |<𝜑<𝛼+𝜑∗ tmax : for 𝛼+𝜑∗ tmax <𝜑<𝜋 : An example of the maximum access time profile in the case of negligible Ω is shown in Fig.19. The three regions can be clearly identified. For the central region, the maximum access time is the highest and is constant. (40) t max =Tspin 𝜋Re � arccos �√ cos2(𝛿)−cos2(𝛽) sin(𝛽) ��. (41) 𝜑 ∗ tmax =arctan �√ cos2(𝛿)−cos2(𝛽) cos(𝛽) �. (42) t 1=Tspin 𝜋Re ( arccos ( cos(𝛿)−cos(𝛽)cos(𝛼−𝜑⋅sgn(𝜑 ∗ tmax )) sin(𝛽)sin(𝛼−𝜑⋅sgn(𝜑∗ t max )) )) (43) t 2=tmax =Tspin 𝜋Re � arccos �√ cos2(𝛿)−cos2(𝛽) sin(𝛽) �� (44) t 3= T spin 𝜋Re ( arccos ( cos(𝛿)−cos(𝛽)cos(𝜑−𝛼) sin(𝛽)sin(𝜑−𝛼) )) 674 The Journal of the Astronautical Sciences (2022) 69:651–691 1 3 For the case of a significant precession speed, no analytical solution has been found. However, its effect can be approximated by modifying the maximum access times (42), (43) and (44). First, lets analyze the points that do not reach 𝜑∗ tmax and therefore, tmax . These points have their own optimum access time when they reach the minimum angular distance from the spin axis. If the precession motion is taken as a rotation of Fig. 19 Variation of the maximum access time as a function of coordinate 𝜑 for Ω negligible. Parameters considered: Tsim =1 day, 𝛼=45◦ , 𝛽=50◦ , Tspin =10 min, and 𝛿=7.5◦ Fig. 20 Scheme of how the precession speed Ω (black) would be added or subtracted to spin speed 𝜔 (red) according to whether 𝛼>𝛽 or 𝛼<𝛽 675The Journal of the Astronautical Sciences (2022) 69:651–691 1 3 The particularities of the telescope optics are omitted and it has been considered that the signal goes through a simple focusing lens [1, 12], thus, parallel rays arriving at the lens will converge to a unique point in the instrument focal plane, which is considered to be in the YinstZinst -plane of the instrument frame, where the array of detectors (see Fig.27) is situated. Thus, each possible direction in the FoV corresponds unequivocally to a point in the circle where the array of detectors is circumscribed. This mapping is done projecting the orientation vector (equivalent to the incoming direction of the calibration signal) to the YinstZinst -plane. The linear polarization of the signal can be defined by a vector perpendicular to the direction of propagation. In this case, the direction of propagation coincides with the LoS and the polarization of the calibration signal is chosen as an arbitrary direction perpendicular to the LoS. During the calibration, it is expected that the reference signal reaches the detectors several times. This will benefit the calibration process, particularly if the relative orientation between the polarization signal and the detectors framework of each measurement is different [5]. In order to assess if the signal arrives with different orientation, the parameter G(𝜉) , derived from the one used in [4] to asses the angular coverage of each point in the sky, is defined for each of the N detectors which are reached by the reference signal at least once ( k≥1 ) This parameter is computed for each detector using its measurements of the polarization ( 𝜉i) , which is constant during the simulation, and then the mean value between all viewed detectors is calculated. If a detector is reached by the signal only once or several times but with similar orientation (or parallel), the G(𝜉) parameter will be equal or close to 1. Conversely, several measures with roughly perpendicular orientations will bring the parameter close to 0. (55) G j(𝜉)= ( 1 k k ∑ i=1 cos(2𝜉i) )2 + ( 1 k k ∑ i=1 sin(2𝜉i) )2 for j=1, …, N Fig. 28 Numerical results for the percentage of viewed detectors and the G(𝜉) parameter. Parameters considered: Tsim =1 day, 𝛼=45◦ , 𝛽=50◦ , Tspin =10 min, Tprec =93 min, and 𝛿=7.5◦ 682 The Journal of the Astronautical Sciences (2022) 69:651–691 1 3 4.2 Numerical Results ofDetectors Analysis Once the events have been detected, they are analyzed in detail. The percentage of viewed detectors ( 𝜌 ) is obtained similarly to the access detection. For each access, it is checked which detectors receive the signal. These data can be further processed to obtain how many times the detectors are illuminated and for how long. The followed approach assumes that only one detector is illuminated simultaneously. Additionally, the polarization of the arriving signal is computed and stored. After considering all of the time period, the parameter G(𝜉) is computed. This process is done for all the directions obtained in the discretization. In Fig.28 the results for Tsim =1 day are shown. As in the accesses analysis, both results present axial-symmetry. It can be observed that for relative angles between 10 ◦ and 15 ◦ degrees the majority of the detectors (around 80%) are reached and with different polarization orientation ( G(𝜉) around 0.5). These results will be discussed in more detail in Sect.5. 5 Application Example In this section, the above methodology is applied to an example mission, showing how it can be used to select the best location to place a calibration satellite or to vary the scan strategy in order to obtain a better calibration performance. The baseline scan strategy has been chosen to be similar to those employed in previous and future CMB missions [6, 7, 11, 12, 17]: 𝛼=45◦ , 𝛽=50◦ , Tspin =10 min, Tprec =93 min, and 𝛿=7.5◦ . The detectors geometrical configuration used is the same as in Sect.4.1. Unless otherwise stated, these results correspond to a simulation period of 1 day. This condition has been selected considering that the calibration process should not last longer to limit interference in the chief’s observations. Fig. 29 Variation of the total access time as a function of coordinate 𝜑 for different values of 𝛼 parameter. In all cases 𝛼+𝛽=95◦ . Parameters considered: Tsim =1 day, Tspin =10 min, Tprec =93 min and 𝛿=7.5◦ 683The Journal of the Astronautical Sciences (2022) 69:651–691 1 3 As shown in Equation (21), the total access time is proportional to the period of time considered ( Tsim ). The total access time will be a fraction of the period that is defined by (20). This will be the fraction of time available for calibration and it is presented in Fig.8. The percentage of total access time has significantly higher values in the region near the precession axis, where it reaches roughly 4% and then descends steeply, remaining around 0.5% until it reaches the border, when it increases to 1% before becoming zero. This behaviour is due to the trace geometry. As is shown in Fig.2, in the region around the precession axis and the border, the trace overlaps more than in the intermediate regions. This translates into accesses happening more frequently. This dependency on the trace geometry is shown in equation (20), where only the scan strategy parameters 𝛼 and 𝛽 are present. The instrument FoV half-angle ( 𝛿 ) is also present, as it will affect the length of the accesses. However, it has been assumed as fixed. None of the attitude motion periods are present as their variations will cause no effect on the total access time. To show the effect of varying 𝛼 and 𝛽 , a parametric sweep has been carried out, with the constraint 𝛼+𝛽=95◦ . Such constraint is established so that the whole sky is observed, ensuring that the instrument never points at the Sun. As shown in Fig.29, the maximum value is always near 𝜑=|𝛼−𝛽| , reaching its highest value when 𝛼=𝛽 , which means that the trace crosses exactly the precession axis in each spin cycle. As 𝛼 decreases, so does the maximum value until a region with no access time emerges around the precession axis. The effect of decreasing 𝛽 is symmetrical. That means, from the total access time point of view, a solution with 𝛼=X and 𝛽=95◦ −X is equal to a solution with 𝛽=X and 𝛼=95◦ −X . In Fig.30 the mean access time and the maximum access time are compared. Neither of the two quantities depends on the period of time considered and they are rather uniform along 𝜑 coordinate. This can be seen from the mean access time in Equation (28), where Tsim disappears when dividing Ttotal by N(𝜑) . For the maximum access time, Tsim is not even considered in the approach followed to obtain its analytical expression. It should be noted that although these parameters do not depend Fig. 30 Comparison between the variation of Tmean and the variation of Tmax as functions of coordinate 𝜑 684 The Journal of the Astronautical Sciences (2022) 69:651–691 1 3 on Tsim , it has been assumed that Tsim is large enough to achieve axial-symmetry in the results. Fig. 31 Variation of Tmean and Tmax as functions of coordinate 𝜑 for different values of the Tspin parameter Fig. 32 Variation of Tmean and Tmax as functions of coordinate 𝜑 for different values of the Tprec parameter Fig. 33 Variation of Tmean and Tmax as functions of coordinate 𝜑 for different values of 𝛼 685The Journal of the Astronautical Sciences (2022) 69:651–691 1 3 The period of spin motion will have a direct effect on both parameters since, as shown in equations (28), (45), (47) and (46), they scale proportionally to it, as shown in Fig. 31. In comparison, the precession period has little effect, being responsible for the slight negative slope as 𝜑 increases. As Tprec grows, this slope becomes steeper (Fig.32). As for the effect of varying 𝛼 and 𝛽 , the outcome is similar to that experienced by the total access time. Decreasing 𝛼 reduces the value of the parameters in general and the no-access-zone emerges in the precession axis. This effect is shown in Fig.33. The results for 𝜌 and G(𝜉) are shown in Fig.34. The percentage of viewed detectors presents two maximums, the higher being close to the precession axis and the lower near the border. The G(𝜉) parameter has a different behaviour. Its lower point is in the precession axis, from where it raises steeply, staying close to 1. Although the most suitable regions of both parameters are close to the precession axis, they have very different behaviours since the maximum of 𝜌 occurs during the steep change of G(𝜉) where a slight variation of 𝜑 has a significant effect on this parameter. Fig. 34 Comparison between the variation of 𝜌 and the variation of G(𝜉) as functions of coordinate 𝜑 Fig. 35 Variation of 𝜌 and G(𝜉) as functions of coordinate 𝜑 for different values of Tsim 686 The Journal of the Astronautical Sciences (2022) 69:651–691 1 3 Both parameters are mainly and similarly affected by Tsim and Tcomb . As shown in Fig. 35(a), when Tsim increases, the percentage of viewed detectors grows for all values except for 𝜑=0◦ , where it remains constant. In this particular position, the trace of the deputy over the instrument is repeated in each access, so the same detectors are viewed each time, causing 𝜌 to stay constant independently of Tsim . The same happens with G(𝜉) (Fig.35(b)), increasing Tsim improves its value for all of the interval. The spin and precession periods influence 𝜌 and G(𝜉) through how they change Tcomb . Slight variations in Tprec and Tspin change radically Tcomb , which have a significant impact on 𝜌 and G(𝜉) . This behaviour is shown in Fig.36. A variation of roughly 3 minutes in the precession period while keeping Tspin fixed causes very different results on both parameters. The higher the value of Tcomb , the better coverage of all detectors. Lastly, the variation of 𝛼 and 𝛽 also impact 𝜌 and G(𝜉) (see Fig. 37). In the no-access-zone near the precession axis for a low 𝛼 , the value of 𝜌 drops to zero and G(𝜉) is not defined. As for Ttotal , the point of maximum percentage of viewed Fig. 36 Variation of 𝜌 and G(𝜉) as functions of coordinate 𝜑 for different values of Tcomb , which are achieved keeping Tspin constant and using a Tprec of 90 min, 93 min and 93.1 min for Tcomb cases of 430 min, 930 min and 93100 min, respectively Fig. 37 Variation of 𝜌 and G(𝜉) as functions of coordinate 𝜑 for different values of 𝛼 687The Journal of the Astronautical Sciences (2022) 69:651–691 1 3 detectors is displaced and its value reduced although the value is increased in general. An additional zone with 𝜌=0 appears at the border. This occurs because for those conditions, the deputy crosses the instrument FoV in the lateral gaps that lie between the detectors array and the FoV border (Fig.26). This behaviour is also experienced by G(𝜉) although in this case it is reduced in general when 𝛼 decreases while its minimum value increases. 5.1 Discussion ofResults The results calculated above can be used to propose suitable locations for a calibration satellite and to evaluate its performance. Results show that a suitable location of the deputy should provide enough time to calibrate the detectors of the instrument, and that this available time is enough to perform a successful calibration, i.e., reaching a high enough number of detectors and doing so with different relative polarization orientations. This translates into having large values in all the parameters except G(𝜉) , for which the ideal value is 0. For the baseline scan strategy studied, the results indicate that the region near the precession axis presents the most advantageous characteristics. This is more noticeable in the case of Ttotal , 𝜌 , and G(𝜉) as they experience significant variations along 𝜑 . Although Tmean and Tmax remain in a narrower range of values, they also present higher values in this region. While all the parameters present adequate values near the precession axis, there is not an optimal point where the best value of each parameter is achieved simultaneously. Moreover, a trade-off analysis should be performed according to the characteristics of the calibration process, which will indicate which parameter is more valuable. The effect of the variation of 𝛼 and 𝛽 in the region is mainly the displacement of the parameters curves, with slight variations along 𝜑 . This is caused by the trace geometry. The high concentration of the trace lines when it reaches its minimum distance with the precision axis causes the peak values in the parameters. Such lines get as close as |𝛼−𝛽| and as this value grows, the region of maximum values is displaced to higher 𝜑 . In comparison, other parameters have a much significant impact. The most important factor for Tmean and Tmax is the value of Tspin as they grow proportionally to it. Although not calculated here, the time that a detector receives the signal will also be proportional to Tspin . Conversely, it is the Tcomb of the two motions which mainly drives G(𝜉) and 𝜌 . A high value will ensure an elevated percentage of viewed detectors and a region of low G(𝜉) . It is clear that the higher that Tsim is, i.e., the longer the calibration process lasts, better values of Ttotal , G(𝜉) and 𝜌 will be achieved, while Tmean and Tmax will not be affected. Although G(𝜉) and 𝜌 results depend on the detectors layout, they will be similar for other layouts as long as the number of detectors is roughly the same and they cover the FoV with axial-symmetry. This condition will keep the effective area of the FoV similar and centered, which will produce analogous results as the trace of the deputy over the FoV is independent of the detectors layout. 688 The Journal of the Astronautical Sciences (2022) 69:651–691 1 3 6 Conclusions In the present work, the visibility between a chief-deputy formation for a telescope of a scientific mission and its calibration satellite, has been studied. A typical spin and precession scan strategy has been assumed for the chief telescope. For the deputy, its visibility and access availability depends on its relative position, the chief’s scanning strategy parameters and the FoV of the instrument to be calibrated. In order to study the visibility and obtain a tool to evaluate and locate the best position for the deputy, three different time parameters have been analyzed: total access time, mean access time, and maximum access time. For each of these parameters and depending on the scanning strategy, an analytical calculation method has been presented. These analytical solutions have been validated against the results from numerical simulations of the problem, obtaining satisfactory results. The advantage of the analytical formulation is that it allows for the evaluation of the performance of different scanning strategies quickly and easily compared to the numerical approach. It also allows for sensitivity analysis to be carried out by providing a clear view of the effect that the scan strategy and relative position have in the parameter variation. In Sect.4, a second level of analysis has been carried out to test other characteristics of the calibration that do not have to do with access time: the percentage of calibrated instrument detectors and the variety of relative polarization orientations with which they observe the calibration source. This analysis was not well suited for analytical methods, but a numerical approach based on an instrument sensor distribution model has been proposed instead. Such a model allows us to evaluate the performance of a given calibration process at detector level and could be easily adapted to obtain other quantities and their distribution over the detectors array. Finally, a case study based on different scanning strategy options for future scientific missions has been studied in order to test the proposed tools. Results show that the region around the precession axis presents the most favorable properties for all the parameters. Nevertheless, the final decision will be subject to a trade-off analysis according to the requirements of the particular calibration process. The results of this study provide tools to do it both rapidly and systematically. It should be noted that, although the tools presented in this work have been used for the analysis of a satellite formation, they can be easily adapted to evaluate more generally the characteristics of the observation of each point in the sky for a given scan strategy and instrument FoV. Acknowledgements This research was funded by the Spanish Agencia Estatal de Investigación (AEI, MICIU) by means of the project with reference PID2019-110610RB-C21. Funding Open Access funding provided thanks to the CRUE-CSIC agreement with Springer Nature. Declarations Conflict of interest On behalf of all authors, the corresponding author states that there is no conflict of interest. 689The Journal of the Astronautical Sciences (2022) 69:651–691 1 3 Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http:// creat iveco mmons. org/ licen ses/ by/4. 0/. References 1. Ade, P.A., Aikin, R., Barkats, D., Benton, S., Bischoff, C.A., Bock, J., Bradford, K., Brevik, J., Buder, I., Bullock, E., etal.:BICEP2/Keck Array. IV.optical characterization and performance of the bicep2 and keck array experiments. Astrophys. J.806(2), 206 (2015) 2. Bermejo-Ballesteros, J.: Nutpy. https:// gitlab. com/ nutpy (2022) 3. Bermejo-Ballesteros, J., García-González, S., Cubas, J., Reinares, F.J.C., Martínez-González, E.: Development of a calibration satellite for a CMB telescope flying in formation about L2 libration point.In: EUCASS 2019, EUCASS2019-697 (2019) 4. Bock, J., Cooray, A., Hanany, S., Keating, B., Lee, A., Matsumura, T., Milligan, M., Ponthieu, N., Renbarger, T., Tran, H.: The Experimental Probe of Inflationary Cosmology (EPIC): a mission concept study for NASA’s Einstein inflation probe. arXiv preprint arXiv: 0805. 4207 (2008) 5. Casas, F.J., Martínez-González, E., Bermejo-Ballesteros, J., García, S., Cubas, J., Vielva, P., Barreiro, R.B., Sanz, A.:L2-CalSat: A calibration satellite for ultra-sensitiveCMBpolarization space missions. Sensors 21(10), 3361 (2021) 6. Delabrouille, J., Abitbol, M.H., Aghanim, N., Ali-Haïmoud, Y., Alonso, D., Alvarez, M., Banday, A.J., Bartlett, J.G., Baselmans, J., Basu, K., etal.: Microwave spectro-polarimetry of matter and radiation across space and time. arXiv preprint arXiv: 1909. 01591 (2019) 7. Delabrouille, J., De Bernardis, P., Bouchet, F., Achúcarro, A., Ade, P., Allison, R., Arroja, F., Artal, E., Ashdown, M., Baccigalupi, C., etal.: Exploring cosmic origins with CORE: Survey requirements and mission design.J. Cosmol. Astropart. Phys.2018(04), 014 (2018) 8. Diebel, J.: Representing attitude: Euler angles, unit quaternions, and rotation vectors. Matrix 58(15– 16), 1–35 (2006) 9. Dupac, X., Tauber, J.: Scanning strategy for mapping the Cosmic Microwave Background anisotropies with Planck.Astron.Astrophys.430(1), 363–371 (2005) 10. Gorski, K.M., Hivon, E., Banday, A.J., Wandelt, B.D., Hansen, F.K., Reinecke, M., Bartelmann, M.: Healpix: A framework for high-resolution discretization and fast analysis of data distributed on the sphere. Astrophys. J.622(2), 759 (2005) 11. Hanany, S., Alvarez, M., Artis, E., Ashton, P., Aumont, J., Aurlien, R., Banerji, R., Barreiro, R.B., Bartlett, J.G., Basak, S., etal.: PICO: probe of inflation and cosmic origins. arXiv preprint arXiv: 1902. 10541 (2019) 12. Hazumi, M., Ade, P.A., Adler, A., Allys, E., Alonso, D., Arnold, K., Auguste, D., Aumont, J., Aurlien, R., Austermann, J., etal.: LiteBIRD satellite: JAXA’s new strategic L-class mission for allsky surveys of cosmic microwave background polarization. In: Space Telescopes and Instrumentation 2020: Optical, Infrared, and Millimeter Wave, vol. 11443, p. 114432F.Int. Soc. Opt.Photonics (2020) 13. Johnson, B.R., Vourch, C.J., Drysdale, T.D., Kalman, A., Fujikawa, S., Keating, B., Kaufman, J.: A CubeSat for calibrating ground-based and sub-orbital millimeter-wave polarimeters (CalSat). J.Astron.Instrum.4(03n04), 1550007 (2015) 14. Price-Whelan, A.M., Sipőcz, B., Günther, H., Lim, P., Crawford, S., Conseil, S., Shupe, D., Craig, M., Dencheva, N., Ginsburg, A., etal.: The Astropy project: Building an open-science project and status of the v2.0 core package. Astron. J.156(3), 123 (2018) 15. Snyder, J.P., Voxland, P.M.: An album of map projections. U.S. Geological Survey professional paper, 1453.US Government Printing Office (1989) 690 The Journal of the Astronautical Sciences (2022) 69:651–691 1 3 16. Sutin, B.M., Alvarez, M., Battaglia, N., Bock, J., Bonato, M., Borrill, J., Chuss, D.T., Cooperrider, J., Crill, B., Delabrouille, J., et al.: PICO - the probeof inflation and cosmic origins. In: Space Telescopes and Instrumentation 2018: Optical, Infrared, and Millimeter Wave, vol. 10698, p. 106984F.Int. Soc. Opt.Photonics (2018) 17. Tauber, J., Nielsen, P., Martín-Polegre, A., Crill, B., Cuttaia, F., Ganga, K., Gudmundsson, J., Jones, W., Lawrence, C., Meinhold, P., etal.: Characterization of the in-flight properties of the Planck telescope.Astron.Astrophys.622, A55 (2019) 18. Thuong Hoang, D., Patanchon, G., Bucher, M., Matsumura, T., Banerji, R., Ishino, H., Hazumi, M., Delabrouille, J.: Bandpass mismatch error for satellite CMB experiments I: estimating the spurious signal.J. Cosmol. Astropart. Phys.2017(12), 015 (2017) 19. Virtanen, P., Gommers, R., Oliphant, T.E., Haberland, M., Reddy, T., Cournapeau, D., Burovski, E., Peterson, P., Weckesser, W., Bright, J., etal.: SciPy 1.0: fundamental algorithms for scientific computing in Python.Nat. Methods17(3), 261–272 (2020) 20. Wallis, C.G., Brown, M.L., Battye, R.A., Delabrouille, J.: Optimal scan strategies for future CMB satellite experiments.Mon. Not. R. Astron. Soc.466(1), 425–442 (2017) 691The Journal of the Astronautical Sciences (2022) 69:651–691