The Gaia astrometric measurement model: error budget and calibration issues.

Transcription

1 Università degli Studi di Siena Facoltà di Scienze M.F.N. Tesi di Dottorato in Fisica sperimentale PhD. Thesis in Experimental Physics The Gaia astrometric measurement model: error budget and calibration issues. Candidate Deborah BUSONERO Supervisor Dott. Mario GAI Tutor Prof. Pier Simone MARROCCHESI XVII ciclo

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3 Contents Contents i Introduction 1 1 Goals of modern Astrometry Overview on Astrometry The five types of Astrometry Limitations of ground-based Astrometry Astrometric requirements from Astrophysics: the Galaxy Astrometry from space Hipparcos: the precursor The Gaia mission The Payload Scientific case Cosmic Distance Scale Galaxies, Quasars and Reference Systems Measurement concept General design considerations Nominal scanning law and orbit L2 orbit Reference System and Reference Frame Center of mass reference system The Scanning Reference System (SRS) The Field of View Reference System (FoVRS) Mechanical Spacecraft Reference System (SCRS) Focal-Plane Reference System (FPRS) Observation strategy

4 ii Contents Abscissa determination on the current great circle Reconstruction of the sphere Improvement Image formation Monochromatic PSF Real optics Detected signal FOV discrimination: concept The Gaia system: spacecraft and payload The Astro instrument Optical design Base Angle definition and stability requirements Astrometric Focal Plane CCD detectors for Astrophysics Back-illuminated CCD TDI: Time Delay Integration mode Gaia CCD detectors Properties of the Gaia CCDs Detector Modulation Transfer Function The Spectro instrument Spectro focal plane RVS objective RVS measurement principle Gaia measurement process The Gaia measurement chain Raw data Elementary measurement Elementary exposure precision Measurement composition End-of-mission accuracy Data processing Windowing and Sampling in Astro Location algorithms

5 Contents iii 5 Astrometric error budget Astrometric performance requirements Main operational assumptions Elementary measurement error: random and systematic errors Astrometric performance contributors Critical parameters identification Requirements in the design, integration, alignment phase and data reduction Blurring of the effective Line Spread Function Datation and synchronisation requirements Selected calibration aspects Gaia transit diagnostics Focal plane to sky mapping Measurement equations CCD position and orientation Auxiliary parameters Detector geometry and operation Procedure Numerical model implementation and validation The ASTRO optical response model Building the signal model Assumptions and approximations The optical model Numerical implementation Cross-validation between numerical model and optical code Domain of the investigation Description of the results Astrometric performance of Baseline configuration Focal Plane to Sky Mapping The Baseline FPSM Differential FPSM: FoV discrimination Chromaticity Astrometric Chromaticity Potential Sources of Chromaticity Aberrations of Real Optical Configurations

6 iv Contents Wavefront Errors related to Optical Quality Baseline chromaticity Chromatic astrometric error WFE expansion: identification of the critical terms Compensation by aberration selection Transit level compensation Partial compensation on misaligned systems Optical engineering aspects Conclusions Gaia alternative configuration and engineering aspects Payload concept Performance analysis on Gaia alternative Configuration FOV discrimination: realization Requirements for field discrimination Model set-up Computational requirements Conclusions Acronyms and Abbreviations 161 Bibliografy 167

7 Introduction Gaia is an ESA mission aiming at unprecedented improvements to our current understanding of the structure, content and dynamics of our Galaxy, by means of a set of combined astrometric, photometric and spectrometric measurements over its five years lifetime. Gaia will measure every object between magnitudes V = 10 and V = 20. The sample of about one billion objects includes stellar populations, asteroids, extra-solar planets, and extra-galactic objects, providing also the context for verification of general relativity to The precision objective of Gaia is 10 µas on parallax, position and annual proper motion for V = 15 objects, with natural photon driven precision degradation for fainter stars and a goal of 3 4 µas at the bright end. This achievement is an improvement of two orders of magnitude with respect to the current best result available, from the mission Hipparcos, which produced a catalogue of 100,000 stars down to magnitude V = 12 with milliarcsec level errors. In order to reach this challenging goal, it is necessary to implement and validate design and analysis tools with an adequate level of reliability. The reference precision of 10 µas corresponds to about 2.3 nm on the focal plane, and the analysis must be validated to at least one order of magnitude better for intermediate brightness objects. For comparison, the image diameter is about 262 mas, so that the relative measurement precision is 10 5 and the model precision is The photon budget is compatible with this goal, but the instrument model must be defined with corresponding detail, to provide a comparably accurate link among measurements. Even the software packages currently used for optical, mechanical and thermal modelling, which have adequate formal precision, must be validated with respect to consistency at this level, defining suitable procedures for verification of the model implementation. The current work is devoted to implementation of a representative mea-

8 2 Introduction surement model, starting from the high-level description of the mission concept and operations. The model is used for definition of the astrometric error budget, describing the expected random and systematic contributions, and of a set of calibration procedures. Several of the basic assumptions are derived from the current mission definition framework (e.g. payload and satellite structure, orbit, and operation), and are included in the measurement model as such. The overall mission description is provided for the sake of completeness, starting from the scientific drivers and discussing the measurement concept and its main consequences. The measurement model is implemented as a set of procedures in the IDL programming environment. The optical description of the telescopes is derived from the configuration implementation on a ray tracing code (Code V), verifying the basic properties and implementing the diffractive optics characteristics. The astrophysical sources are represented by blackbody distributions at suitable effective temperatures; this is an adequate simplification for the evaluation of the instrument performance, and may be replaced by detailed spectral distributions for specific astronomical investigations. For the current purposes, only point-like sources are used, but resolved objects can be derived in a straightforward way. The simulator is able to represent the optical response over any position of the field of view and for objects with any effective temperature, by superposition of the monochromatic images obtained by the local wavefront error map through diffraction theory and Fourier techniques. The local image profile is used to build the signal corresponding to the elementary exposure, introducing the detector response and operation and their modulation effects. The detected signals are evaluated with respect to local astrometric performance and mutual connection of the measurements. This detailed model allows identification of the impact of several parameters on the measurement; several of them can be monitored through their signature on the science data. Examples of methods for calibration of significant contributions are provided. A peculiar systematic error is the chromatic displacement suffered by different spectral type objects, due to the realistic optical response of the instrument. We analyse the distribution associated to the nominal configuration, its origin, and methods for minimisation of its effect, either by construction or by correction in data reduction. The measurement model is implemented for both instrument configurations

9 Introduction 3 currently considered for the Gaia payload. This allows cross-validation of the methods and procedures, and provides valuable information on the capability of the two options to meet the mission objectives. The current level of development of the measurement model, the error budget and the calibration procedures is described, introducing the possible further improvements.

10 4 Introduction

11 Chapter 1 Goals of modern Astrometry Our understanding of many physical and astrophysical subjects, ranging from general relativity and the nature of mass to the mechanism of formation of the galaxies, to the evolution of our Galaxy and the distribution of planetary systems in our neighbourhood, will greatly benefit from each of several space based experiments. Since the measurement goals are extremely ambitious with respect to current technologies, however, most of such experiments are highly challenging and require correspondingly ambitious improvements in our engineering capabilities. The astrophysical requirements are often associated to high angular resolution, in the microarcsecond (hereafter, µas) range; it is thus important to remind that 5prad=1µas, and on a 1 m length bar it corresponds to an edge displacement by only 5 pm. This level of precision makes accessible a set of important features, from the structure of remote galaxies in their primordial states, to the induced motion from extra-solar planets. Astronomical objects and systems are large, but their image is small due to the cosmic distances involved. Earth s orbital semi-axis is defined as one astronomical unit (AU), and it corresponds to 150 million km; it is useful to describe with manageable numbers the geometry of our Solar system. Besides, nearby stars are still separated from us by few parsecs (1 parsec = 3.086e16 m), the distance to which one AU subtends the angle of 1 arcsecond; this was selected as operational unit because distance was measured by apparent displacement of star position during the year with respect to remote galaxies, which in turn are removed by millions of parsecs. A requirement of stability in the µas range, for an instrument of few m size, corresponds to a fraction of atomic size (order of 100 pm). The actual meaning of such challenging number is usually referred to collective pa-

12 6 Goals of modern Astrometry rameters, like the equivalent curvature of a mirror or length of a laser cavity, rather than to each component atom. This kind of requirement is applied to the mission GAIA, described below in Section 1.6. In the following sections we give some explanation about astrometry in order to provide a feeling on the subject. 1.1 Overview on Astrometry The Astrometry, or position astronomy, is devoted to the measurement of positions, distances, masses and motions of celestial bodies. It is the most ancient part of Astronomy, still highly promising, given the recent improvements in instrument performance. Astrometry has two main scientific objectives: 1. to provide a non-rotating stellar reference frame to which the motions of objects in the Solar System and stars in the Galaxy may be referred, and which can be used as a reference framework for relating optical observations to those in other regions of the electromagnetic spectrum; 2. to provide, through the use of such a framework, basic observational data for the studies of stellar properties (luminosity, mass, etc.), the spatial distribution of stars within the Galaxy, and their motions. The astrometric techniques can be divided into fives types according to the size of the field in which a single instrument can produce measurements. 1.2 The five types of Astrometry In addition to the creation of the fundamental catalog, functional for the setting of a reference system as inertial as possible, the astrometric measurements are useful for solving a large range of problems. The different techniques are characterised by the different problems addressed. The principal factor that distinguishes the different techniques is the size of the field of view needed to obtain the desired result. This size can vary between few arcseconds to the whole celestial sphere. Let us show the different techniques as function of the extension of the field of view. 1. Very narrow field astrometry: θ Narrow field astrometry: θ 0.5

13 1.2 The five types of Astrometry 7 3. Wide field astrometry: θ 5 4. Semi-global astrometry: a wide part of the sky. 5. Global astrometry: the whole sky. Orbits Exo planets Parallaxes ASTROMETRY Galactic dinamics Clusters Global semi Global Wide Field Reference System Narrow Field Very Narrow Field Figure 1.1: Let us list the different methods for each of the five cases: Very narrow field astrometry This kind of astrometry studies single or multiple stars without paying any particular attention to the other objects in the neighbour. A typical request for such kind of astrometry is the study of the relative motion of system constituted of a couple of stars with an angular separation smaller than 0.1 Another application of this technique is the measurement of the diameter of a star within few mas of accuracy. Narrow field astrometry It is used when it is needed to link the position of a particular star to a given set of neighbour stars. Or also in the determination of the parallaxis of a star with respect to some other stars supposed to be fixed. It is also used in the quest of a possible invisible partner which introduces non-linear component in the motion of the visible partner.

14 8 Goals of modern Astrometry Wide field astrometry It is essentially relative astronomy: the field of study is the determination of the quasi-inertial coordinate of sky-objects by using reference stars available in the field of view, or for the study of the internal motions of stellar cluster. Semi-global astrometry It represents a first step toward the realization of a fundamental catalo. Global astrometry The position and the relative motion of the stars obtained by the observations need to be comparable whatever the sky-region. It is mandatory that all objects are referred to a unique coherent system of reference. Establishing such a reference frame, and establishing which are the more appropriate objects needed to achieve such a result, is the objective of the Global astrometry. 1.3 Limitations of ground-based Astrometry Ground-based Astrometry is affected by different and drastic limitations that forbid the possibility to reach accuracies some millisecond of arc. These limitations are due to: Atmospheric turbulence The wavefront coming from star is disrupt by atmosphere. In the best cases the resolution is within few hundred milliarcsecond. In order to reach better accuracies, it is necessary to use interferometric methods, but they are only usable for the very narrow field astrometry, with a reachable accuracy of some milliarcsecond. Atmospherical refraction Errors due to this phenomenon go beyond 0.02 and reach 0.1 on measurements obtained with methods typical of the semi-global astrometry. Earth s movements The apparent position of a star depends from the Earth s movement around the Sun (annual parallaxis), from the rotational movement around its own axis (diurnal parallaxis), from its velocity in the space that cause the phenomenon of aberration, from regular and irregular movements of the terrestrial axis, respectively precession and nutation,and polar motion. The observation conditions are not isotropic directed towards any point in the sky. The direction observed is referred to the Earth and is subject to errors due to an insufficient knowledge of Earth rotation parameters. Homogeneity limitations

15 1.4 Astrometric requirements from Astrophysics: the Galaxy 9 The usage of various instruments in different ambiental conditions cause the errors called regional, unavoidable in building a basilar catalog and of entity longer than the best precision reachable by current instruments. In this situation the final accuracy is imposed by regional errors, so it is clear the impossibility of performing ground-based measurements of global astronomy in optical band, more accurately than some millisecond of arc. 1.4 Astrometric requirements from Astrophysics: the Galaxy For example, determining the key-parameters of the Galaxy model, representing the structure and evolution of the Galaxy in the CDM (Cold dark Matter) cosmological context, requires three complementary observational approaches: i. a complete and unbiased census of the contents of a large part of the Galaxy, including tracers representative of all the relevant components listed in Table 1.4; ii. quantification of the present spatial structure, from positions and distances; iii. knowledge of the three-dimensional space motions, to determine the gravitational field and the stellar orbits. That is, one requires complementary astrometry, photometry, and spectroscopy. Astrometric measurements (l, b, µ l, µ b, π i.e. celestial coordinates, proper motions and trigonometric parallaxes) uniquely provide 3D spatial distributions (R, Φ, z) based on model independent distances, plus the transverse kinematics of the targets. Photometry, with appropriate astrometric and astrophysical calibration, gives a knowledge of extinction (Av), and hence, combined with astrometry, provides intrinsic luminosities (Mv), stellar temperatures (Teff) and chemical abundance ([M/H]), as well as age information. Spectroscopic radial velocities complete the kinematic triad and the 6D phase space (R, Φ, z; Vr, VΦ, Vz) distribution, allowing determination of gravitational forces, and the distribution of invisible mass. An all-sky (4π sterad) survey providing astrometric parameters (l, b, µ l, µ b, π) and spectro-photometric observations (V rad, [Fe/M], multi-band photometry) for large and unbiased

16 10 Goals of modern Astrometry samples of stellar tracers, non-kinematically or chemically selected, and belonging to all different galactic components, up to several tens of kiloparsec from the Galactic center, appears to be the ideal tool to investigate properly the problems of local cosmology. Position and kinematic requirements For each tracer, Table 1.4 reports the expected signal, in terms of measurable parallax, proper motion and line of sight velocity, as a function of the typical distance and velocity dispersion of its parent population. In order to attain significant Signal-to-Noise measurements for most of the cases, we require ac-

17 1.4 Astrometric requirements from Astrophysics: the Galaxy 11 curacy at the following levels: trigonometric parallax accuracy: σ π 10 µas, proper motion accuracy: σ µ µas/yr radial velocity accuracy: σ Vrad 5-10 km/s An all-sky stellar survey complete down to V=18-20, with the precision on the astrometric parameters and radial velocities reported above, will permit to determine the 6D space-velocity distribution in a sphere of 20 kpc about the Sun and provide accurate determinations of the structure and kinematic parameters the thin disk, thick disk, bulge and halo. Inner halo streams can also be easily detected and their members identified as clumps in energy and momenta, although their internal kinematics will not be resolved in the case of the tidally dissipated systems (σ V 1 km/s). In addition, kinematic studies may also be extended to the outer halo (d>20 kpc), beyond the horizon of the trigonometric parallaxes, with accuracy better than 20% by means of bright tracers (BHB, gm) for which reliable distances can be derived from the spectrophotometric observations. Finally, we notice that the availability of position accuracy of 10 µas will also allow the astrometric detection of extra-solar systems formed by Jupiter like planets around G-K dwarfs up to a distance of pc. We expect to discover and determine accurate orbital parameters of a few thousands new planetary systems with periods P<10-15 years. E 1 φ 1 ρ abs ρ 0 = 1 ( φ 2 φ 1 ) 2 E 1 φ1 2ρ abs ρ abs = 1 2 ( φ 2 φ 1 ) 2ρ abs S S φ 2 E 2 φ 2 E 2 ρ 2 0 S 0 S 0 (a) The relative parallax, (φ 1 φ 2 )/2, of star S with respect to the background reference star S0, requires to be statistically corrected for the true parallax, ρ 0, of the reference star(s). E 1 and E 2 indicates the Earth positions after a period of months. (b) The absolute parallax ρ abs = φ 1 φ 2 /2 can be directly derived if large angular separations can be accurately measured. Figure 1.2: Parallaxis concept.

18 12 Goals of modern Astrometry We point out that astrometric accuracy at such a level ( 10 µas) cannot be attained by ground based observations because of the degradation from atmosphere turbulence and refraction. In particular, the relative parallaxes (Fig. 1.2 a) from ground are affected by systematic errors of about 1 mas, due to the uncertainty on the distance of the fixed stars, which need to be adopted as reference stars in medium field astrometry (0.5-1 ). The only way to attain distances accurate to 10 6 arcsec is by means of absolute parallaxes (Fig. 1.2 b) that can only be carried out from space because it requires simultaneous observations of stars located in different parts of the sky with very different parallax factors (Kovalewsky 2002). 1.5 Astrometry from space Let us stop for a while to think what can be the effects of astrometric measurement from space on astronomy and astrophysics fields. A space astrometry mission has a unique capability to perform global measurements, such that positions, and changes in positions caused by proper motion and parallax, are determined in a reference system consistently defined over the whole sky, for very large numbers of objects. In fact, in space an instrument will not be subject to the errors induced by the four effect mentioned in the previous section and will be able to scan the whole sky. We have only one instrument in uniform geometrical and ambiental condition. The only limitations to reach the desired accuracy are now those imposed by the instrument itself and by the procedure of measurement and data analysis. Keeping in mind what above described, the great potentialities and scientific possibilities of an astrometric mission from the space become clear. In the 1989 the idea became real: the European Space Agency (ESA) launched Hipparcos, the first space astrometry mission performing global astrometric measurements Hipparcos: the precursor Hipparcos was a pioneering space experiment dedicated to the precise measurement of the positions, parallaxes and proper motions of the stars. It demonstrated that it was possible to perform global measurements with milliarcsecond accuracy by means of a continuously scanning satellite which observes two directions simultaneously. Turning slowly on its axis and repeat-

19 1.5 Astrometry from space 13 edly scanning the sky on different slants, it measured angles between widely separated stars, and recorded their brightness, which were often variable from one visit to the next. Each star selected for study was typically visited about 100 times over four years. The intended goal was to measure the five astrometric parameters of some 120,000 primary programme stars to a precision of some 2 to 4 milliarcsec, over a planned mission lifetime of 2.5 years, and the astrometric and two-colour photometric properties of some 400,000 additional stars (the Tycho experiment) to a somewhat lower astrometric precision. The project was accepted within the ESA scientific programme in The satellite was launched by Ariane from Kourou, French Guyana, into a geostationary transfer orbit, in August 1989, and after collecting more than three years of extremely high-quality scientific data, communications were terminated with the satellite in August All of the mission goals have been significantly exceeded. The final Hipparcos Catalogue (120,000 stars with 1 milliarcsec level astrometry) and the final Tycho Catalogue (more than one million stars with milliarcsec astrometry and two-colour photometry) were completed in August Several features combine to make the Hipparcos mission unique and the corresponding results so dramatic. The satellite was able to observe the entire celestial sphere from its location in space. This advantage was combined with the absence of a perturbing atmosphere, and the instrumental stability brought about by the absence of gravitational instrumental flexure as well as an actively controlled thermal environment. Differential angular measurements were made over large angles, at many different orientations, and at many different epochs. The parallaxes are consequently absolute, and regional or systematic errors in positions and annual proper motions are expected to be well below the milliarcsec level. A rotation-free, or quasi-inertial, system will be represented by relating the final catalogue to observations of extragalactic objects, and the link to the radio reference frame by direct or indirect observations of radio stars or quasars observed by VLBI. The payload was centred around an optical all-reflective Schmidt telescope. A novel feature of the telescope was the beam combining mirror, which brought the light from the two fields of view of dimension , separated by about 58, to a common focal surface, and thus achieved both largeand small-field measurements simultaneously. The satellite swept out great circles over the celestial sphere (see section 2.2), and the star images from two

20 14 Goals of modern Astrometry fields of view were modulated by a highly regular grid of 2688 transparent parallel slits located at the focal surface and covering an area of cm 2. Hipparcos detectors were photomultiplier tubes. For more details about the satellite, the Input Catalogue and the data reductions see ESA SP The Gaia mission Hipparcos pioneered the techniques of space astrometry and placed Europe at the forefront of this scientific discipline. With current technology the same Hipparcos principle can be applied with a gain of a factor of more than 100 improvement in accuracy, a factor 1000 improvement in limiting magnitude, and a factor of 10,000 in the numbers of stars observed. Gaia builds on this expertise to create a satellite capable of addressing one of the most difficult challenges in modern astronomy: to create an extraordinarily precise three-dimensional map of more than one billion stars throughout our Galaxy and beyond. The first ideas for Gaia began circulating in the early 1990 s, culminating in a proposal for a cornerstone mission within ESA s science programme submitted in 1993, a workshop in Cambridge in June 1995 to discuss possibilities and the selection of the Gaia mission by the ESA Science Programme Committee in 2000 as one of the next two cornerstones of the ESA science programme Horizon 2000 Plus. The results of the preliminary technology Study indicated that a launch in 2009/10 was feasible, and that a period of technological development would be required in advance of the detailed design and construction phase, in order to gain full confidence in the required technology. This technology phase was scheduled to run between Gaia was originally foreseen to be launched by Ariane 5, but a further industrial study was carried out in early 2002 to assess accommodation within the smaller (and cheaper) Soyuz launcher. The studies were completed, a workable design with only small accuracy degradation was identified, and the Gaia mission was reconfirmed within the ESA Cosmic Vision programme in May 2002 and again in November The launch date is scheduled to be no later than The technological studies and scientific definition are still in progress (fig. 1.3). Gaia is a spinning satellite (6 hour period, scan rate=60 arcsec/s), with continuous full-sky observation. The measurements cover repeatedly the whole sky, by composition of rotation, precession and orbital motion of the satellite.

21 1.6 The Gaia mission 15 Figure 1.3: present time. The figure shows the overall Gaia mission timeline at the The orbit is positioned at the external libration point (L2) of the Earth-Sun system, at about 1.7 million km from Earth. As Hipparcos, Gaia will perform global astrometry, generating a three dimensional map of the sky, using simultaneous observations on two different lines of sight (LOS). The measurement concept, inherited from the previous mission Hipparcos but with much more ambitious scientific goals, requires observation along two lines of sight, separated by a large base angle (BA), along the equatorial great circle and optical and electronic instrumentation at the state of art (see sec.3). The mission is optimised for Milky Way formation, structure and evolution studies; chemical composition and evolution of stellar populations. Gaia will provide measurements of global astrometry on all the celestial objects with magnitude brighter than V=20. It will perform astrometric, photometric and spectrometric measurements with an accuracy of 10 microarcsecond (µas) (50 prad) in star parallaxes and absolute positions and some µas/yr in proper motion for star of visual magnitude V=15. Photometric measurements will have an accuracy of 0.02mag at V=20 in all bands (four bands for the astrometric instrument and eleven for the spectroscopic one). We remark that a 10 µas accuracy means to know the position object on the moon within few centimeter. A similar accuracy on parallax means to know

22 16 Goals of modern Astrometry the stars distance better than 10% within 10,000 parsec (pc),i.e. to define the structure of the Galaxy. Today we obtain a similar accuracy only within 100 pc The Payload The spacecraft is composed of several subsystems and all of them are crucial to the mission performance. The fundamental subsystem is the Payload that performs the scientific measurements. Gaia s payload shall consist of three instruments: an astrometric instrument, ASTRO, that perform astrometric measurements and broad band photometry; a spectrometer, SPECTRO, that performs spectrometric measurements and a multi-band photometer with five broad- and 11 medium-band filters shared between ASTRO and SPECTRO instruments. Currently, the detailed design is in progress, in a competitive framework, and two configurations are evaluated: the one, named Baseline, developed by the industrial team Astrium; the other, named Alternative, developed by the industrial team Alcatel/Alenia. The selection will be made in November In chapter 3 we describe in detail the payload and the Baseline configuration, while we devote the chapter 9 to the Alternative configuration. The main astrometric payload requires two telescopes with a Base Angle BA = 99.4, with a large focal plane (FP) detector: a CCD mosaic of 170 CCD operated in Time Delay Integration (TDI) mode. The principal feature of the mission is a highly stable payload. This payload requires extensive technological development prior to full implementation into the project (for more detail see chapter 3) Scientific case The Gaia project offer a rich crop of results providing benefits in various field of Astrophysics and Astronomy. This project aims to achieve a complete astrophysical study of the Milky Way structure and evolution, trying to cover a wide group of interesting scientific fields. The main objective of the mission is, indeed, the acquisition of data useful for understanding the structure and evolution of our Galaxy. Three different and complementary kind of observation approaches are requested: a census of a big and representative part of the celestial bodies in the Galaxy;

23 1.6 The Gaia mission 17 a quantification of the actual spatial structure through direct measurements of distance; knowledge of spatial tridimensional motions to determine stellar orbits and map the gravitational field. To this purpose, it requests three different, but complementary, kind of data: astrometric measurements, multi-band photometry and spectroscopic measurements, in order to determine the radial velocity with an independent method. Astrometric measurements provides distances to the objects and their tranverse velocity, i.e. the starting point for the cosmic distance scale. The multiband photometry provides spectral classification, information about the extinction/reddening rate. In conjuction with the astrometric data, it allow us to extract informations about intrinsical luminosities, spatial distribution functions, chemical abundances and ages of stars. Radial velocity measurements are needed in order to find the third component of velocity, the one along the visual line. From the 3D kinematics, we can determine the gravitational forces and dark matter distribution, i.e. the dynamics of our Galaxy and the Local Group. Gaia performance will allow the realisation of a highly accurate tridimensional map (both in position and in velocity) of about a billion of stars of our Galaxy and of the Local Group. It will describe their motions, giving an important instrument in order to codify the origin and the evolution of the Galaxy and the distribution of the dark matter. Through the on-board photometric measurements, Gaia will provide physical properties of each observed star: luminosity, temperature, gravity, chemical composition. This will give us information on star formation and on chemical enrichment of the Galaxy. The range of scientific subjects affected by Gaia data is wide, and will contribute to improve knowledge not only in modern astronomy but also in fundamental physics. Gaia will find with great precision a large number of exotic objects: thousands of extra-solar planets, with the possibility, in several cases, of estimating precisely the orbits; it will identify several thousands on brown and white dwarves. It has been estimated that Gaia will be able to find one hundred thousand extragalactic supernovae in a time useful to observe them with large ground-based telescopes. With its high performance, Gaia will allow us to make a travel in the Milky Way starting from Earth s proximity, reaching distances as for as 20,000 pc,

24 18 Goals of modern Astrometry i.e. to 65,000 lightyears (1 pc = 3.26 lightyears). 1. Gaia will observe the minor bodies of the solar system, giving orbits very precise for hundred of thousands of asteroids, mainly new observed one, included the potentially dangerous NEA (Near Earth Asteroids) with elliptical orbits that cross the Earth s one. Gaia will be also able to determine diameters of a thousand of major asteroids, giving a measurement of the density and other important informations on the physical conditions of the solar proto-planetary cloud. 2. Coming out from the solar system, and within a sphere of 200 pc of diameter, Gaia will observe hundred thousands of stars, for most of which it will be able to report the presence of companions of planetary mass, from little perturbations on the measured trajectory of the star. According to the actual models of Galaxy, there are at least 500,000 interesting candidates within 200 pc, stars relatively bright (V < 13) with a mass of more than 0.7 solar mass. Recent studies have shown that within these the satellite could discover and measure the orbit of several thousands of giant planets, those with a mass similar to Jupiter and orbital period less than 10 years. 3. At a distance of 500 pc, Gaia will find million of stars, and in particular will reach some tens of stellar systems, with about 30 open clusters, and various stellar associations and regions of stellar formation. For the first time Gaia will allow to observe and give an absolute calibration for all the Main Sequence stars, starting from heavier and rare O-B star, and down to weaker star of later spectral type (dwarves M), arriving also at the cold brown dwarves and including many stars at other evolutive phases. Moreover the luminosity and mass distribution of cluster stars and associations will allow to have the so-called Initial Mass Function (IMF), base element in the evolution theories of the Galaxy. 4. In the region between 500 pc and 2,000-3,000 pc the structure of the disc become visible. The disc extends over a radius of 15,000 pc but with a thickness of some hundred parsec. The matching between astrometric, photometric and spectroscopic measurements will allow to understand the complex evolutive processes of the disc (dynamic and chemical), overcoming the difficulties about the strong and irregular interstellar extinction given by gas and dusts on the galactic plane. At

25 1.6 The Gaia mission 19 these distances the warp of the galactic disc can be seen, i.e. the curvative of the disc seen on about half of the spiral galaxies, which cause is not so clear by now. 5. Within 10,000 pc, Gaia will observe the bulge in the center of the Galaxy and the galactic halo. 6. The theories about the hierarchical formation forecast in the external halo, i.e. for distances of 20,000 pc from the center of the Galaxy, the presence of some fossil remnants of dwarf galaxies cannibalized by the Milky Way, whose stars are merged with the halo ones but recognizable kinematically, because of the conservation of the orbital motion of the original galaxy; Gaia will measure precisely their velocity and derive the signature in phase from mergers. 7. Standard Candles. Within a volume of 10,000 pc of radius, thousands of variable stars Cepheides and RR Lyrae will be observable. These stars are rare in our Galaxy so most of them are at over pc from Sun. That is the reason why Hipparcos didn t observed them. Gaia instead will be able to evaluate absolute parallaxes useful to calibrate the period-luminosity relationship with a high accuracy. Due to their high luminosity Cepheides are used as standard candles for measuring the distance of galaxies up to ten millions of lightyears. In some of these galaxies there is the possibility of an explosion of some supernovae (Gaia itself should discover about 100 thousand supernovae in the 5 years of mission duration), that will be used like derived indicators to measure distances up to billions of lightyears Cosmic Distance Scale Gaia will provide accurate distances (and proper motions) for such huge numbers of each category of stellar distance indicators that the analysis methods can be drastically changed. The sampling of open and globular clusters in age, metal, oxygen or helium content will be complete all over the Galaxy. Parallel improvement in the transformation between the observational and the theoretical H-R diagram will be required to take full benefit of these accuracies in terms of stellar evolution and age determination: photometric and/or spectroscopic data should

26 20 Goals of modern Astrometry allow the determination of the bolometric magnitude and of the effective temperature from the observed magnitudes and colours. For pulsating variables, the sampling versus period, populations, colours, and metal content will be as good as possible as excellent distance determinations will be obtained for all observable galactic stars, and a first reliable estimation of the intrinsic dispersion of the period-luminosity relations will be possible. Moreover, a first check of the universality of these relations (not only the slopes, but also the zero-points) will be possible, directly for LMC Cepheids, or using Gaia mean distances for the closest galaxies of the Local Group, at least LMC, SMC and Sagittarius. We show in figure 1.4 the different methods used for determination of distances and that are needed for the realization of the Distance Scale, at progressively increasing distance from us. Gravitational lens time delay Sunyaev- Zel dovich Supernovae Ia standard candles Elliptical galaxy kinematics Spiral galaxy kinematics Supernova Baade-Wesselink Globular cluster luminosity function Surface brightness fluctuations Novae as standard candles Planetary nebula luminosity function Cepheids PL and PLC relations VLBI proper motions RR Lyrae starts as standard candles Stellar Baade- Wesselink Main sequence fitting Statistical parallax Moving cluster Trigonometric parallax Hipparcos ground based Redshift Distanza Figure 1.4: Brief summary of the cosmic distance scale. In graphic are shown the relative and absolute measurements of distance: full lines show the gap of distances on which the method shown on the side is applied; the dotted lines show that the indicated method is supported by an absolute calibration. (pc)

27 1.6 The Gaia mission Galaxies, Quasars and Reference Systems Not only Gaia will provide an important census of the Milky Way stars, but also will contribute in a significant way to the extragalactic astronomy, with information about structure, dynamics and star populations of the Magellan Clouds and other satellite galaxies of M31 and M33. Moreover the magnitude limit at V=20 mag, and the whole sky coverage, will give the possibility to make unique and fundamental cosmological studies analysing spatial movements of galaxies of the Local Group and studying a large number of supernovae, galactical nuclei and quasars. Supernovae: Gaia will be able to detect all compact objects brighter than V=20 mag, so that, in principle, supernovae could be detected up to redshift of z 0.1 (500 Mpc).The more useful kind of supernovae as scale distance indicators are the Ia, which light curve are indicators of distance very accurate within a ± 5%. Quasar: The estimate is to obtain a census of 500,000 quasar. Between intermediate and high galactic latitudes, these will provide a direct link between the astrometric reference system of Gaia and the quasi-inertial reference system in radio realized by the VLBI Catalog. The observation of quasar has a direct astrophysics interest, for example the study of the (macro) gravitational lenses between the quasar populations. International Celestial Reference System: Currently the International Celestial Reference System (ICRS) is realized by International Celestial Reference Frame (ICRF). The definition and the creation of the ICRF was a joint cooperative effort of a sub-group of the International Astronomical Union (IAU) Working Group on Reference Frames. It was formed expressly for the purpose of creating the definitive catalogue of extragalactic radio source positions using the best data and methods available at the time the work was done. The ICRF was adopted by IAU as the fundamental celestial reference frame, replacing the FK5 optical frame as of 1998 January 1. The first expansion of ICRF at the optical band is given with Hipparcos Catalog (1997), which includes all the FK5 stars, with precisions evaluated about 0.25 mas/year for each component of ω vector, which describes the rotation of the system and 0.6 mas on the components of the ɛ vector, which describes how the system is oriented. Gaia will create a much denser frame directly in the visible with an average of 20 sources per square degree outside the galactic plane, an increase by a factor

28 22 Goals of modern Astrometry Figure 1.5: Sky distribution in galactic coordinates of the approximately 660 extragalactic sources that makes up the International Celestial Reference Frame as adopted by IAU in These are radio sources. Gaia will create a much denser frame by a factor of almost a thousand from the current ICRF. of almost a thousand from the current ICRF and more accurate by two order of magnitude with respect to Hipparcos: it will be possible refine ICRS at best of 60 µas in the orientation of the frame and some µas/year in rotation [1]. The figure 1.5 shows the sky distribution in galactic coordinates of the approximately 660 extragalactic sources, with a precision in position between 100 and 500 µas, that make up the International Celestial Reference Frame (ICRF). These are radio sources observed by long baseline interferometry and are primarily very faint in the visible.

29 Chapter 2 Measurement concept The Gaia measurement concept is, as for Hipparcos, the complete and repeated coverage of the sky by reduction of the endless strip obtained by its CCD detectors operating in TDI mode. The objective of the Gaia mission is to perform global or wide angle astrometry as opposed to local or narrow field astrometry. In local astrometry a star position is measured with respect to neighbouring stars in the same field. Even with an accurate instrument, the errors become prohibitive when making a survey, due to the need of combining measurements obtained in different fields, and thus affected by systematic errors and noise (section 1.3). The principle of the global astrometry is instead to link stars with large angular distances in a network where each star is connected to a large number of other stars in every direction. In order to implement this approach, the measurement of large angular distances through the simultaneous observation of two fields of view separated by a large angle is required. This leads to the determination of absolute trigonometric parallaxes, and thereby circumvents the problem which has plagued ground-based parallax determinations, namely the transformation of relative parallaxes to absolute distances. The Gaia payload must therefore provides two lines of sight (two telescope, one focal plane concept is adopted (section 3.1). 2.1 General design considerations The ultimate accuracy with which the direction to a point source of light can be determined is set by the dual nature of electromagnetic radiation, namely as waves (causing diffraction) and particles (causing a finite signal-to-noise ratio

30 24 Measurement concept in the detection process). Consider the observation of a distant monochromatic point source by means of an optical telescope with an idealised detector. The instrument generates a diffraction image in the focal plane and the detector records the precise location of each detected photon in the diffraction pattern. If λ is the wavelength and D the overall size of the instrument aperture (diameter), then the characteristic angular size of features in the diffraction pattern that can be used to localise the image is of order λ/d radians. If a total of N detected photons are available for localising the image, then the theoretically achievable angular accuracy will be of order (λ/d) N 1/2 radians. A realistic size figure for non-deployable space instruments is of order a few metres, say D 2m. Operating in visible light (λ 0.5µm) then gives diffraction features of order λ/d 0.05 arcsec. To achieve a final astrometric accuracy of 10 µas it is therefore necessary that the diffraction features are localised to within 1/5000 of their characteristic size. Two obvious requirements follow: firstly, that at least some 25 million detected photons are needed to beat down the statistical noise by this factor; secondly, that extreme care is needed to achieve such a huge improvement in practice. Elementary calculations show that the first requirement (number of photons) can be satisfied for objects around 15 mag with reasonable assumptions on collecting area and bandwidth. The second requirement is clearly a technical challenge, but the conclusion from the GAIA study is that this condition, too, can be met with two general constraints: two telescopes, and a scanning satellite. The measurements conducted by a continuously scanning satellite can be shown to be almost optimally efficient, with each photon acquired during a scan contributing to the precision of the resulting astrometric parameters. Pointed observations cannot provide the overriding benefit of global astrometry using a scanning satellite, which is that a global instrument calibration can be performed in parallel, and the interconnection of observations over the celestial sphere provides the rigidity and reference system, immediately connected to an extragalactic reference system. Quantifying and generalising from these basic design considerations, the general principles of the proposed mission can be summarised as follows: (i) it is a continuously scanning instrument, capable of measuring simultaneously the angular separations of thousands of star images as they pass across a field of view of about 1 diameter. Simultaneous multi-colour photometry of all astrometric targets is a necessary and integral part of the concept; (ii) high angular resolution in the scanning direction is provided by a mono-

31 2.2 Nominal scanning law and orbit 25 lithic mirror of dimension 1.4m; (iii) the wide-angle measuring capability is provided by two viewing directions at large angles to each other and scanning the same great circle on the sky. The precise basic angle between two viewing directions is determined from the 360 closure condition on each great-circle scan, while short-term ( 6 hours) variations are passively controlled, and monitored by internal metrology; (iv) the whole sky is systematically scanned in such a manner that observations extending over several years permit a complete separation of the astrometric parameters describing the motions and distances of the stars. The chosen observative strategy is due also to technological and data storage/processing requirements. 2.2 Nominal scanning law and orbit GAIA, as Hipparcos, scans continuously the sky according to pre-defined scanning law, resulting from the composition of two motion of the satellite: the slow motion of precession of the rotation axis around the Sun at a fixed angle (ζ = 50 ), covering /hr, equivalent at a precession period of 70 days, and a rapid spin motion around the own rotation axis with a spin rate of 60 arcsec/s, three hours for a complete rotation equivalent at 4 full scanning in a day (figure 2.1). In a day the satellite covers on its orbit around the sun an angle of 4.08 ; the rotational speed can be assumed as constant on the whole period of the mission. This because the mareal effects due to the gravitational perturbations of the sun and the planets have a longer scale time. Beyond the two motion described, the satellite show a libration motion around the Lagrangian point L 2 of the Sun-Earth system. Denoting the ecliptic as the xy plane with the x-axis that link the Sun with the Earth and ẑ the axis perpendicular at the ecliptic plane, the libration motion describe an orbit characterized by an harmonic motion in the xy plane, with an uncoupled oscillation in ẑ with a different period. The frame system rotates with the Earth around the Sun. The period of these two motions, that made up the so called Lissajous orbit, is around 6 months. The selection of the specific value of the ζ angle between the rotational axis of the satellite and the sun-satellite direction is also given by particular requirements for the scanning law, but is imposed more strictly by the

32 26 Measurement concept Figure 2.1: Schematical raffiguration of the satellite motion. needing of avoid the direct radiation by the sun (straylight) and the termical perturbations (see below). The ζ value has an impact on the design of the shield required for protect the satellite from the direct sun radiation, with implications on the dimension and the whole mass of the satellite and so on the design of the module that will carry it. It has its weight in the calculation of the dimension of the field of view and of the instrumental parameters present in the final expression of the error budget (see section 2.5 and section 4.1). Independently by the adopted angle, the scanning law can be defined in order to maximise the covering of the sky, reducing at maximum the possible precession motion of rotational axis. The selected value fore the rotational period around its axis, longer than Hipparcos one, increase the performances of the astrometric instrument, making more feasible the reaching of limit magnitude, because the photon number collected for each pixel in integration time is greater. The evaluation of the exact scanning period is done during the mission time. In fact there isn t the certainty that the satellite will have the exact period, a verification is necessary during the mission; in principle deviations within 1% are acceptable, larger deviation must be corrected. Gaia will perform its observations from a controlled Lissajous-type orbit around the L2 Lagrange point of the Sun and Earth-Moon system. During its 5-year operational lifetime, the satellite will continuously spin around its axis, with a constant speed of 60 arcsec/s. As a result, over a period of 6 hours, the two astrometric fields of view will scan across all objects located along

33 2.2 Nominal scanning law and orbit 27 Figure 2.2: L2 position with respect to Earth-Sun system. the great circle (section 2.4.1) perpendicular to the spin axis. As a result of the basic angle (section 3.1.2) separating the astrometric fields of view on the sky, objects transit the second field of view with a delay of about 100 minutes compared to the first field. Gaia s spin axis does not point to a fixed direction in space (or on the sky), but is carefully controlled so as to move slowly on the sky. As a result, the great circle that is mapped out by the two fields of view every 6 hours, also changes slowly with time, allowing repeated full sky coverage over the mission lifetime. The scanning law prescribes how the satellite s spin axis evolves with time during the mission. The optimum scanning law (i) maximises the angle ζ between the Sun and the spin axis at all times; (ii) maximises the uniformity of the sky coverage after 5 years of operation. The first requirement results from the fact that the parallactic displacement of transiting stars is proportional to sin ζ; a higher value of ζ thus leads to larger measurable parallaxes and higher end-of-mission astrometric accuracies. Thermal stability and power requirements, however, limit ζ to about 50. The best strategy is thus to let the spin axis precess around the solar direction with a fixed angle ζ = 50. This combination of a spinning satellite, scanning the sky along great circles, and a precession of the spin axis is referred to as revolving scanning, and was actually used for the Hipparcos mission. The actual speed of precession of the spin axis on the sky should be small enough that consecutive great-circle scans overlap sufficiently, and large

34 28 Measurement concept enough that all stars on the sky transit the astrometric fields sufficiently often. The above requirements have been worked out in detail for Gaia, leading to an optimum nominal scanning law. For a spin rate of 60 arcsec/s and a solar aspect angle of 50, the precession speed is of order of arcsec s 1, such that 5 years of operation corresponds to 26 revolutions of the spin axis around the solar direction; the precessional period thus equals 70 days. With this scanning law, precession motion, nominal spin period and nominal basic angle value, the dimension of the field of view in the across-scan direction for each Astro instrument shall be larger than On average, each object on the sky is observed 83 times (astrometric fields combined) L2 orbit Figure 2.3: Gaia Lissajous orbit over 6.3 years. Gaia will operate in the vicinity of the second Lagrange point L2, approximately 1.5 million km from the Earth, along the Sun-Earth line in the direction opposite to the Sun. The region around L2 is a gravitational saddle point, where spacecraft can be maintained at roughly constant distance from the Earth for several years by small and cheap manoeuvres. Around L2 there is a circular zone of radius 13,000 km where the Sun is always eclipsed by the Earth. Here the solar panels of a spacecraft would be unable to generate sufficient power since they would not receive enough sunlight. In addition, even

35 2.3 Reference System and Reference Frame 29 entering this region for a few minutes would generate a detrimental thermal shock in the spacecraft. Therefore, Gaia will be placed in a large Lissajous orbit ( 300,000 km) around L2 to ensure that it stays away from the eclipse zone for at least 6 years (fig. 2.2). The constant pull exerted by the Sun and the Earth will cause Gaia to swing around L2 on a nearly periodic orbit and 6 months will be needed to complete a full cycle (fig. 2.3.) The selection of the orbit arises from a trade-off between communication, operations, cost, thermal and radiation environment, and accessibility with current rockets. Around L2 one benefits from a virtually unchanging environment with very stable thermal conditions, an essential asset for the success of the mission. The optics are so sensitive to minute changes of temperature that a variation of less than 1/1000 of degree over a few hours would disturb the alignment of the mirrors and degrade the quality of the images. So it is evident the benefit of a quite environment as L Reference System and Reference Frame A number of different coordinate systems are needed for the description of different aspects of the spacecraft, the instrument and the measurements, and equally for the description of the universe to be observed. Present-day terminology distinguishes between reference system and reference frames. In short, a reference system gives a more or less ideal, abstract definition of a coordinate system, while a reference frame is a practical realisation of such a system. Astronomical reference system and reference frames play two very different roles: 1) They are used to represent actual measurement made by real instruments in a standardised, uniform way that is independent of the specific instrument and circumstances: the direction of light rays measured by Gaia somewhere on its orbit will ultimately be represented by coordinates (α and δ) in the barycentric ICRS - or rather in a practical realisation of it, the ICRF (see section 1.6.4). 2) They are used to represent the paths of celestial bodies (artificial or natural) through the solar system: The orbits of Gaia end Earth around the sun will be represented in a barycentric reference frame whose axes are (nominally) the same as those of the ICRF. At the precision level reached by the Gaia measurements it is mandatory that the applied astronomical reference systems are fully relativistic, i.e. 4-

36 30 Measurement concept dimensional and in accordance with General Relativity. The choice of such systems is also made necessary by IAU conventions, to which the Gaia project will adhere. In addition to astronomical reference system there are mechanical (spacecraft or payload) reference system. These are used to describe the locations and orientations of hardware parts of the actual Gaia spacecraft and payload. As before, they may come in two flavours, namely as abstract definitions and as practical realisations. However, this distinction is less fundamental here than in the case of astronomical systems. The mechanical reference systems are spatially Euclidean, as there is, of course, no noticeable space curvature over the few meters that the spacecraft extends within the relativistic universe. The natural time coordinate would be proper time of the spacecraft, but in practical usage the mechanical reference systems will be just three-dimensional Center of mass reference system The CoMRS is a kinematically non-rotating system that moves with the Gaia spacecraft. In the terminology of [2] it is the proper system of Gaia. In the terminology of [1] it is the local rest frame of the Gaia instrument. The significance of this system is expressed by the fact that the measurable angle ψ i,j between the light rays coming from two stars scanning the two different field of view (FoV1 and FoV2), is simply given by the Euclidean product of their cartesian direction vectors (unit vectors) r i, r j - if these are expressed in the CoMRS: cosψ i,j = r i r j (2.1) In other words: this is the natural system in which to describe the measurements made by Gaia. The transformation of the ICRS coordinate direction towards a source into the CoMRS coordinate direction of a light ray coming from that source (and vice versa) is one of the central topics of the Gaia astrometric data reduction. In astrometric terms it is the computation of proper directions (in technical Gaia documents it frequently appears under the heading of astrometric modelling ). The details are subject of intense study by the Gaia Working Group on Relativistic Model and Quasi-Inertial Reference Frame.

37 2.3 Reference System and Reference Frame 31 z γ S j η j LOS2 LOS1 x ψij IGC S i η i l ij RGC The Scanning Reference System (SRS) In the previous section we introduced a system co-moving with Gaia. Now we are going to introduce a system co-moving and co-rotating with the body of the Gaia spacecraft, the Scanning Reference System (SRS). It is mainly used to define the satellite attitude. It is an intermediate system between the CoMRS and the Field of View Reference System. Definition of SRS The SRS is rigidly connected to the body of the Gaia spacecraft (which in turn is assumed to be a rigid body). The origin of the system is at the center of masses of Gaia. The natural time coordinate is the proper time of the spacecraft. SRS, celestial coordinates Celestial coordinates in the SRS differ from those in the CoMRS only by Euclidean rotation, given by the attitude of the satellite. They are expressed by cartesian unit vectors. The unit vectors along the principal axes are called x, y, z. The z axis is the nominal rotation axis of the satellite; with direction towards the sun being at an angle of 50 ( ) from the z axis during Gaia operations. The x axis is in the plane of the two Astro viewing directions (i.e. the two projections of the optical axis of Gaia s Astro telescope onto the sky), half a basic angle (γ/2 i.e. 50 ) away from each of them. The y axis is also in the plane of the two viewing directions such that the system x, y, z is right-handed.

38 32 Measurement concept The Field of View Reference System (FoVRS) There are three Field of View Reference System, one for each sky field seen by a Gaia instrument (i.e. the Spectro field, the Astro1 field and the Astro2 field). All three systems are simply rotated version of the SRS. They are defined for convenience of the modelling of the observations and instruments. FoVRS, definition The FoVRS are, like the SRS, rigidly connected to the body of the Gaia spacecraft (which in turn is assumed to be a rigid body). The origins of the systems are at the center of mass of Gaia. Their natural time coordinate is the proper time of the spacecraft. The difference to the SRS is a rotation around the z axis such that in each of the three system the first of the principal coordinate axes point along the viewing direction (i.e. projection of the optical axis onto the sky) for the respective field of view. More precisely, the viewing directions, and thus the origins of the field coordinates (and field angles) are defined to (nominally) coincide with the optical projections of the origins of the focal-plane coordinates onto the celestial sphere. The actual operational definition of these field coordinate origin in the course of the scientific data reduction is a part of the calibration processing. The celestial coordinates are expressed by field angles η, ζ (spherical longitude and latitude coordinates) or by field coordinates f, w, z (direction cosines of cartesian unit vectors) Mechanical Spacecraft Reference System (SCRS) The SCRS is the mechanical system equivalent to the astronomical system SRS. It is derived from the engineers tradition to simplify the communication between astronomers and engineers. Both the SRS and the SCRS are rigidly connected to the body of Gaia spacecraft. The SCRS differs from the SRS only for the different spatial origin and for the orientation of the axes with respect to the satellite body. The spatial origin is defined by easily accessible and invariant fiducial point in the spacecraft body. It will be not the center of masses. The coordinate axes X S, Y S, Z S are so defined: a) the Z S lies in the plane of the two viewing directions of the Astro telescope,

39 2.4 Observation strategy 33 half a basic angle (γ/2 i.e. 50 ) away from each of them (corresponding to the x axis in the SRS); b) the - X S points along the nominal rotation axis of the spacecraft (corresponding to the z axis in the SRS); c) the Y S axis also is in the plane of the two viewing directions such that the system X S, Y S, Z S is right-handed Focal-Plane Reference System (FPRS) There are three Focal-Plane Reference Systems, one for each focal-plane assembly (FPA). They are the RVS FPA and MBP FPA of the Spectro instrument, and the Astro FPA (see chapter 3). The FPRS are defined to describe the locations of the individual CCDs on the various FPAs and other properties of the FPAs. The issue is discussed in detail in the chapter 3. For details about the others see [3]. 2.4 Observation strategy By measuring the instantaneous image centroids (section 4.2.2) from the data sent to ground, Gaia measurements the relative separations of the thousands of stars simultaneously present in the combined fields. The spacecraft operates in a continuously scanning motion, such that a constant stream of relative angular measurements is built up as the fields of view sweep across the sky. High angular resolution (and hence high positional precision) in the scanning direction is provided by the primary mirror of each telescope, of dimension m 2 (along-scan across-scan). The wide-angle measurements provide high rigidity of the resulting reference system. The whole sky is then systematically scanned such that observations extending over several years yield about 100 sets of relative measurements for each star. These permit a complete determination of each star s astrometric parameters: two specifying the angular position, two specifying the proper motion, and one (the parallax) specifying the star s distance. A 5-year mission permits the determination of additional parameters, for example those relevant to orbital binaries, extra-solar planets, and Solar System objects. During the scan the Gaia CCDs integrate continuously the star images crossing the field of view. The continuous integration is achieved by the CCD TDI mode (section 3.2.2). Without any on-board read-out strategy the raw

40 34 Measurement concept data should be a continuous and long stream of image-data with signal level over the detection threshold, acquired overall the five year mission. Due to technology, data storage and calculus needs it is mandatory to subdivided this long stream according to the so called Data Processing (section 4.2) Abscissa determination on the current great circle The following step is to bring back the unidimensional linear coordinate evaluated on the focal plane of the system, into the angular coordinate on the celestial sphere along the Instantaneous Great Circle (IGC). Due to the rotation of the instrument there is the needing of bring back the measure to a fixed reference system, that consider the orientation of the satellite in the space, or the satellite reference system. So the definition of the attitude and of the motion of the satellite is a crucial aspect of the measure. This is the main goal of the reduction at the Great Circles system, that transform the position measurements, taken on the focal plane during a rotational period, in an ensemble of unidimensional coordinates, called abscissae, defined on a given Reference Great Circle (RGC). The choice is to process together the star position measurements obtained during six following rotation of the telescope, corresponding to a period of around 24 hours [4]. This fixes one of the parameter that determine the final accuracy of the mission, N transiti =6. The measurements acquired during this short time period can be referred to the same RGC and it is possible recover for each star a single angular coordinate, called abscissa of the star. The RGC is chosen in the middle of a strip of the celestial sphere scanned by the telescope in 36 hours. The RGC defines the fundamental plane of an intermediate celestial reference system. It is chosen an arbitrary origin on the intersection of that plane with the sphere of unitary radius; the abscissa of the star is calculated starting from the origin. The ecliptic longitude of the RGC ascending node and its inclination with respect to the Ecliptic define the orientation of the RGC in the space. The figure 2.4 shows the passage from the projection of the star position from a generic IGC to the corresponding RGC. In this passage are calculated the following input data: the mean time t S n at which the star S is observed and the measure of the

41 2.4 Observation strategy 35 z Ζ γ y γ 2 γ 2 η (F) η X Ψ (F) ξ FOV2 x (P) IGC RGC (P) ξ Y FOV1 Figure 2.4: Geometry point of view for the abscissa determination on the current great circle. field coordinates (x S, y S ) of the stellar image photocenter. About 100 million of observation are estimated for each RGC, distributed on about 860,000 stars; the reconstruction on ground of the attitude, given analysing the measurements of preselected stars; the Input Catalogue and the orbital data of the satellite useful in order to calculate the apparent position of all the star with the necessary accuracy: the transformation G that link the coordinates on the CCD to the field ones, G op (ξ, η) for the first field of view, FoV1 or preceding, and G of (ξ, η) for the second field of view, FoV2 or following, and the parametric model G op (ξ, η, t; d) and G of (ξ, η, t; d) that takes in account the dependence from the time, where d is the vector of the unknown parameters to evaluate. Once rebuilt the orientation of the satellite, given by the three Eulero s angles Φ(t), Θ(t) and Ψ(t), at the central time of the observation and given the orbital data, it is possible define the right transformation in order to pass from the coordinates of the star in the instrument to the satellite ones and vice versa.

42 36 Measurement concept So, introducing the set of the instrumental parameters d that enter in the transformation linking the field coordinates on the CCD, it is possible to obtain the following measure equation: x S (t) = x(ξ S, η S, Φ(t), Θ(t), Ψ(t), d,...) + ε S (t), (2.2) where x S (t) is the coordinate along the scanning direction taken on the CCD. The equation above is linearised around the values of the stellar coordinates given by the Input Catalogue, i.e. to the values of Eulero angles obtained processing a set of preselected star, which position is assumed known, and around the nominal values of instrumental parameters: x S (t) = x ξ ξ S + x x Ψ(t) + Ψ d d + ɛ S(t), (2.3) where it is assumed that the variation of the η coordinate of the star and the Φ(t) and Θ(t) angles, that define the direction of the rotational axis, is negligible in the considered time interval. The output is one-hundred million of equations, whose variables to determine are: about 860,000 abscissae ξ, one for each observed star; the parameters to define the satellite orientation, with high precision; the variation of the instrumental parameters d with respect to the nominal value; for example, in Hipparcos, were 25 parameters. It is important keep in mind that the star coordinate that is determined at the end of the procedure just described, is only the abscissa of the star on a RGC. It is important underline that at this step of data reduction, the RGC origins respect to the defined abscissae, are arbitrary [4] Reconstruction of the sphere This step helps in establishing a consistent system of RGC origins. After determining the origin of the abscissa, the astrometric parameters of each star can be determined. During the mission time, a star appear mainly on 41 RGC. A the end of the mission we have a 2433 RGC and each of them is an autoconsistent reference system.

43 2.4 Observation strategy 37 The objective of the sphere reconstruction process is to obtain a bi-dimensional solution for the spheric coordinates of each star, starting from the knowledge of the star abscissae (unidimensional). The abscissae are measured on different RGC, having different arbitrary origins, due to the unknowledge of the RGC ascending node of RGC on the ecliptic (figure 2.5). In order to make consistent Figure 2.5: Z 1 e Z 2 are the arbitrary origins of the two RGC, S 1 and S 2 are the projections of the star s position S on single RGC, and N 1 and N 2 determine the position on the ecliptic on the ascending node of each RGC. the system of abscissae relative to the star S, we introduce the translations c k on the k-th RGC. The stars are in movement, consequently the abscissae result a function of this motion, as shown schematically in figure 2.6. In ecliptic coordinates, the star s motion is defined by the longitude λ, latitude β and parallaxis π, by own motions in longitude and latitude (µ λ,µ β ) and by radial speed V, at given age t 0. At last we obtain the expression (2.4), that link the abscissa ξ Sk of the star S taken on the RGC k-th, at the five astrometric parameters to determine, two angular position, parallaxis and own motion: ξ Sk = F (λ S, β S, µ λs, µ βs, π S, V S, t 0 ; r k, t k ) c k + d, (2.4) where c k is the translation on k-th RGC, r k define the direction of the k-th RGC pole and t k is the central time of the k-th RGC. In the figure (2.6) is shown the reconstruction of the trajectory of a star using five abscissae on five different RGC. It is a bent line passing, in the situation shown, through five star geometric positions on the sphere; their projection on five different RGC, at the RGC mean time, gives the abscissa value.

44 38 Measurement concept RGC2 RGC1 Z3 Z4 RGC5 Z5 5 4 Traiettoria dellastella 3 2 RGC3 Z1 1 Z2 RGC4 Figure 2.6: Trajectory described by the star observed on the sphere. Z i is the origin of the i-esim RGC. 2.5 Improvement The possibility of passing from accuracy of the order of the milliarcsecond to the order of the microarcsecond is mainly due to the use of the CCD instead of the fotomultiplier used by Hipparcos. Hipparcos made counting of photons emitted by the object observed and tuned by a grid of modulation unidimensional, while GAIA acquire images of the object observed in continuous mode (section 3.2.2). The procedure followed to determine the position of the celestial body on the focal plane and its relative accuracy, is therefore different. In order to extract the wanted information, the obtained images are processed using theoretical model and algorithms of analysis of images [5] (see also section 4.2.2). Let s start giving a brief but complete description, that must be kept in mind for all the discussion developed from here. The theoric accuracy limit that we may use to determine the direction of a point luminous source is given by the nature of the electromagnetic radiation, that as a wave is ruled by the diffraction, while as a particle in collecting process cause a signal noise ratio (SNR) finite. The lower limit for the angular accuracy on the single measure, given by an exposition, result to be [6]: σ el λ eff 2πB eff SNR, (2.5) where λ eff us the wavelength, B eff is the effective baseline and SNR is the

45 2.5 Improvement 39 signal-noise ratio. Is valid the parity in the ideal situation of an entire poissionian noise SNR = N, or the noise only due to the photon statistic, with N the number of photoelectrons revealed, and in the infinite sampling case. In real situations SNR N and the finite sampling of the signal introduce a degradation on the accuracy reachable [5]. For an optical configuration, as Gaia one, with rectangular primary of dimension L along the scan direction B eff = L 3, with circular primary B eff = d. 2 In order to understand where come from the parameter B eff we should penetrate the optical theory of diffraction. In fact the B eff parameter consider the geometrical characteristics of the telescope. It is binded at the order two moment of the aperture x rms : B eff = 2 x rms, where x rms 2 1 = (x x) 2 T (x, y)dxdy, T (x, y)dxdy A A where T(x,y) is the function that describe the outline of the aperture and assume the value 1 inside and 0 outside. For each observed star the final accuracy on parallaxis σ π depends by the total weight of the single measurements and by numerical factor determined by geometry and temporal distribution of the measurements: λ eff σ π = g π 2πB eff SNR, (2.6) g π is the factor that reminds that the geometry of the scanning law affects the evaluation of the parallaxis. The simulations of the scanning law are used to determine the mean value, and so g π = 2.1(sin ζ) 1. Also the positions and the proper motions show this factor within the expression for the calculation of the accuracy with similar values. We want that the final error on parallaxis is about some µas. Using eq.(2.6) we can make an easy estimation of what can be the signal-noise ratio useful to reach the required accuracy. For example, for B eff = 2.45 m, λ eff = 700 nm we obtain g π λ eff 2πB eff = 18.2 mas. (2.7) We want an error thousand times smaller. This require a SNR for each star, that imposes a minimum value limit at the total number of photons that can be collected, in this situation we need N Fora a star,

46 40 Measurement concept of V=15 mag, assuming a 1000 photons/s flux, and assuming a throughput A T λ Q λ dλ = 100m 2 nm the above condition mean a requirement of total time of observation of τ 500 s. For a continual scanning instrument the total time spent by an object within the field of view result to be: τ = Ω 4π L (2.8) where Ω is the solid angle subtended by the field of view (FOV) and L is the mission time. For τ 500s and L 4 years eq.(2.8) gives Ω = = as minimum field of view necessary for obtaining the final wanted precision. Moreover, the need of having an uniform coverage of the sky so that every celestial body can be detected in the observation, and so the request of partial overlapping of sky strips covered by the FOV on different crossing, mean further imposition to the choice of the dimension of the astrometric field of view. This overlapping is obtained only if the dimension of the FOV perpendicular at the direction of the scanning result to be greater than the angular spacing between two scanning great circle [7]. 2.6 Image formation Monochromatic PSF Accordingly to the diffraction theory for optical system (see e.g. Born & Wolf, 1980), the amplitude of a monochromatic electromagnetic wave after a linear aberration-free optical system, called Amplitude Response Function (ARF), is the Fourier transform of the generalised pupil function P g with area A, i.e., ARF (x, y) = q(λ) A dζdηp g (ξ, η)e i 2π λf (xζ+yη), (2.9) where q(λ) is constant over the pupil, described by the coordinates (ξ, η); the FP coordinates are (x, y). The integral is performed over the pupil, in Gaia case rectangular. The point spread function (hereafter, PSF), i.e. the energy distribution, is the squared modulus of the ARF. For a rectangular pupil the integral can be solved analytical to provide: I(x, y) = I 0 [ sin( πxl X ) λf πxl X λf ] 2 [ sin( πyl Y λf ) πyl Y λf ] 2 (2.10)

47 2.6 Image formation 41 where x, y are the focal plane coordinates; L X, L Y : X and Y aperture dimensions; F the effective focal length and λ the wavelength. The central lobe size along the direction î is 2λ/Lî. The ideal non-aberrated diffraction figure from a rectangular aperture have a bi-axial symmetry Real optics Any real system is an aberrated system, i.e. the reference wave exiting from the pupil is no longer spherical. The deviation from the ideal wavefront across the pupil are described by the wavefront error (WFE) function. So the pupil function P g is function of the WFE W (ξ, η): P g (ξ, η) = exp [i 2πλ ] W (ξ, η), (2.11) and the spectral dependence is evidenced: at longer wavelength, the WFE has smaller effect, as the phase contribution is a smaller fraction of the period. The monochromatic aberrated PSF is given by the expression: P SF M (x, y; λ) = I M (x, y; λ) = ARF (x, y) 2 = = 1 2 λ 2 A dζdη e iφ(ζ,η) e i 2π λf (xζ+yη) where Φ(ζ, η) = 2π W F E(ζ, η) is the pupil function; ζ, η are the pupil coordinates, λ the reference wavelength, F the effective focal length, A the pupil λ area and (x,y) the focal plane coordinates Detected signal The detected polychromatic PSF is generated by the contribution of several monochromatic PSFs, weighted according to the source spectral distribution S(λ), optical transmission T (λ) and quantum efficiency QE(λ) and integrated over the spectral range λ. The detected energy distribution I P becomes I P (x, y) = A λ dλi M(x, y; λ) S(λ)T (λ)qe(λ) λ dλs(λ)t (λ)qe(λ) (2.12) and the PSF can then be normalised to the expected average number of photoelectrons. The aberrations modify the PSF profile over the FP and, consequently, the

48 42 Measurement concept root mean square width value L RMS. Moreover the image quality (and L RMS ) changes smoothly over the field of view, giving a variation of the PSF photocenter value over the FP. On the other hand, the entrance pupil dimension, the optical transmission T(λ) and the quantum efficiency QE(λ) of the detector define the signal intensity and the SNR value. 2.7 FOV discrimination: concept The FP detects images from both fields, which in the simplest case of ideal telescopes and scan law, without precession, are indistinguishable and follow the same trajectory. Due to the TDI operation, matching clock rate and sidereal speed, each object is observed subsequently by 11 CCDs, producing a detected signal quite similar to the corresponding pointed exposure; the 11 measurements can be associated to the time of transit of the object on each device. However, even retaining the ideal optics and motion, but simply introducing the precession, required to displace progressively the current great circle over the celestial sphere, we get a transverse motion component. It has a peak value of about 170 mas/s, corresponding to the precession rate, and it features a nearly sinusoidal behaviour, because it is zero at the nodes (intersection of the current great circle and precession plane), and maximum at the anti-nodes (i.e. at 90 degrees separation on the current great circle). Moreover, since the two fields are close to quadrature, due to the BA value ( 99 ), the transverse motion is different for the objects from each field, and in some cases it has opposite sign. It should be noted that the real along scan velocity, taking precession into account, basically is not changed: the mean value is still sufficiently accurate to provide acceptable TDI exposures over the whole spin period. Besides, the transverse velocity generates a blurring of the integrated image in the across scan direction, with a peak value of about 120 µm, i.e. four pixels. This does not degrade significantly the detected signal, due to across scan binning (marginal degradation is associated to increased background contribution). Windowing and readout of selected CCD regions around the predicted position of each object is foreseen, as described above, in order to optimise the electronics and storage requirements. Due to the variable across scan image size, along the great circle, and to simplify operation, the across scan size of the readout window is set to 12 pixels, to account for the peak image displacement

49 2.7 FOV discrimination: concept 43 with one pixel margins. Besides, setting the window size to a value compatible with the peak differential velocity (about 260 mas/s, inducing a peak position difference of 860 mas or 195 µm, i.e. 6.5 additional pixels, was considered excessive with respect to background increase and the risk of confusion among stars in crowded regions. It is therefore necessary to identify the field associated to each detected target, in order to enable proper readout window selection by the on board logic, taking advantage of the knowledge on satellite attitude (i.e. of the current transverse speed). The key point of the baseline strategy for field discrimination is based on identification of the different residual along scan velocity, due to the different optical response for the two fields. The implementation is different for the baseline configuration, described in section 3.1, and the alternative configuration, described in section 9. In both cases, the displacement in the images detected in the ASM and on AF1 is evaluated to identify the associated viewing direction. In the former case (baseline), a shift between the individual telescope focal planes is applied, by design, so that each CCD observes a different section of the individual field. Due to the optical response variation over the field, discussed in chapter 6, this results in a sufficient signature on the detected signals, which allows discrimination: the amount of along scan displacement of an object must be consistent with the value associated to either field 1 or to field 2. In the latter case (alternative), the focal plane is common, by definition, but there is a natural separation at pupil level of the telescope regions used by either field. Due to the natural variation of aberrations over the pupil, it is possible to impose at optical design level a signature on the two sub-pupils, sufficient to induce a detectable differential displacement on the focal plane. The alternative of direct detection on each object of its transverse velocity is considered, but at the moment it is not considered convenient from the standpoint of precision and of real time computational load. However, this has the advantage of improving the performance on the asteroid population, allowing efficient differential tracking for a subset of objects with high proper motion. However, nominal operations already provide adequate performance on an interesting fraction of the expected observation sample.

50 44 Measurement concept

51 Chapter 3 The Gaia system: spacecraft and payload The satellite is composed by two separate modules with clear interfaces, the Payload module (PLM) and the service module (SVM). The SVM provides all functions and necessary services to the payload in order to meet the overall mission objectives. The satellite is protected from the direct Sun illumination by a large and flat sun-shield that also limits the thermal variations. The satellite mass shall not exceed 2030 kg at launch and the power consumption shall be lower than 2200 W. The opto-mechanical-thermal assembly of the Payload module comprise: 1. a single structural torus supporting all mirrors and focal planes, employing SiC for both mirrors and structure. There is a symmetrical configuration for the two astrometric viewing directions, with the spectrometric telescope accommodated within the same structure, between the two astrometric viewing directions; 2. a deployable Sun shield to avoid direct Sun illumination and rotating shadows on the payload module, combined with the Solar array assembly; 3. control of the heat injection from the service module into the payload module, and control of the focal plane assembly power dissipation in order to provide an ultra-stable internal thermal environment; 4. an alignment mechanism on the secondary mirror for each astrometric instrument, with micron-level positional accuracy and 200 µm range, to

52 46 The Gaia system: spacecraft and payload correct for telescope aberration and mirror misalignment at the beginning of life; 5. a permanent monitoring of the basic angle, but without active control on board. The targeted measurement precision of 10 µas, i.e. 50 prad, in a structure with characteristic size 2 m, can easily be associated to an instrument stability requirement corresponding to internal linear displacements significantly smaller than 2 50 e 12 = 100pm. The instrument configuration must be retained over a time elapse sufficient to ensure that slower perturbations are identified and corrected in the data reduction; the typical reference is the satellite spin period, i.e. 6 hours. To this purpose, the design relies on high stability thermoelastic structural properties, in the comparably quiet operating environment. In-orbit realignment is achieved by inchworm actuators mounted on each telescope secondary mirror, to recover misalignments of telescope optics induced by launch effects and ensure the best image quality in orbit. Telescope realignment is performed at the beginning of life, based on the data from a Shack-Hartmann wave front analyser which shall be implemented within the FPA. The payload module consists of two kinds of instruments in order to provide the six standard astrometric parameters: two angular position (right ascension and declination), the two components of proper motion, parallax and radial velocity. In addition photometric measurements are provided to complete the astrophysical information about the objects observed. The Gaia instruments are: The Astrometric instrument (Astro) The Spectrometer (Spectro) Angular position, proper motion, parallax and broad band photometer measurements are estimated using Astro data. Spectro instrument includes star s radial velocity and medium band photometric measurements.

53 3.1 The Astro instrument 47 Figure 3.1: Schematic figure of the Gaia payload. The whole payload module assembly shows the common focal plane in the plane of symmetry of the two telescopes. 3.1 The Astro instrument Optical design The telescope apertures are rectangular, with size 1.4m 0.5m and a collecting area of 0.7m 2. The corresponding image size is between minima at the reference wavelength of λ = 700nm. The optical system consists in two identical physically separated telescopes, mounted on a common torus structure, called Astro1 and Astro2, looking at different lines of sight (LOS) to perform global or wide angle astrometry, combined to share the same focal plane. The angle between Astro 1 and Astro 2 lines of sight defines what is called the Basic Angle (BA). A dedicated Basic Angle Monitoring (BAM) device (section 3.1.2) shall provide during all the mission the variation of this angle in order to perform the associated on ground calibration. Each optical train of the two telescopes consists of seven main elements: three mirrors with optical power (M1, M2, M3), three flat mirrors (M4, M5, M6) and the Focal Plane (FP).

54 48 The Gaia system: spacecraft and payload Combination of the two optical channels is performed at the level of M4, so the physical realisation of M5 and M6 and of the FP is common to both telescopes. The mirrors M5 and M6 are necessary to fold the beams in order to allow the system fit within the 3.8m envelope of the launcher. Figure 3.2: The overall optical path (on the right) of each astrometric telescope is made of six reflectors, two of which are common for the two telescopes. The superposition of the two LOS at the FP is not perfect, they are shifted across-scan by 64 mm (about a CCD column), obtained by a field stop placed along the beams path in a position that must be still defined; it cut off one FP CCD column above and one below so to have about one column of CCDs in each part of the detector array where only one Field of View (FoV) is imaged. Also the field stop is made so that the ASM first line received light only from FoV1 and the ASM second line from FoV2. This for identifying the LOS of each target mapping the targets before they enter the astrometric part of the detectors array. The effective focal length of size m results in an optical scale (os)of mas/µm (arcsec/mm). The nominal diffraction image is binned in the across scan (low resolution) direction, producing one-dimensional signals sampled with about 6 pixels on the central lobe. The real image is affected by aberrations from the optical configuration, finite spectral bandwidth and other

55 3.1 The Astro instrument 49 instrumental effects. The spectral bandwidth is limited mainly by the stellar emission plus detector sensitivity curve (no filter present); the intrinsic stars emission provides in most cases the bandwidth is λ = 300 nm, at different effective wavelength. CCD readout is tagged against a common time scale, so that the pixel value distribution can be centred to find the transit instant of the star photo-center over reference positions on the FP (e.g. the CCD edge). The scanning law maps time to the sky position through the scan velocity: and we can express the measurement requirements either in angular units on the sky, or timing or linear units on the FP, equivalently. In chapter 6, we discuss some of the requirements on the design, implementation, verification and in-flight calibration of the GAIA astrometric payload Base Angle definition and stability requirements Our definition of the BA is the angle on the sky separating the two directions associated to a given FP point, through the two telescopes, measured along the scan direction. This relative orientation of the two telescopes is a key aspect of the measurement: the separation between stars in different fields is the focal plane coordinate difference plus the BA (see section 6.2); configuration perturbations of few nm, negligible with respect to the optical response, are critical to astrometry as they induce several ten µas systematic errors. This also sets stringent constraints on payload thermal stability [8]. The BA fluctuations over the nominal spin period (6 hours) must be lower than 7 µas RMS. It is required high stability for each telescope and for the supporting structure to preserve the BA value: 1 µas x 1 m = 5 pm. Instrument stability at sub-atomic scale is required! A dedicated Basic Angle Monitoring (BAM) device is foreseen, in order to keep track of residual instrument perturbations and provide correction information to the data reduction. Base Angle monitoring device The absolute parallax determination at the µas level is a primary goal of Gaia, but to reach this goal, it is of crucial importance that the BA is kept constant to a very high level. In the current design the base angle stability is passively ensured by: 1. Avoiding thermal load fluctuations anywhere in the spacecraft and particularly in the PLM

56 50 The Gaia system: spacecraft and payload 2. A passive thermal control of the payload aiming at insulating the PLM from the rest of the spacecraft 3. The use of ultra-homogeneous pressure-less sintered Silicon Carbide material. This material is used both for the reflectors and the structure. It provides low thermal expansion with high thermal conductivity and simultaneously good optical and structural properties. The overall payload is therefore a-thermal and LOS variations can only result from thermal gradient fluctuations within the payload. The payload mechanical/thermal design aims at obtaining the required opto-mechanical stability passively (i.e. without using re-positioning systems). Figure 3.3: Monitoring system of the base angle variations. The only mechanism in the astrometric telescopes is the secondary mirror mechanism, the purpose of which is to compensate at begin of life potential misalignments due to launch and residual on-ground alignment errors. In case the passive base angle control approach turns out not to be adequate to ensure the required stability, an Optics Active Control System will have to be implemented. This system consists of an ultra high precision laser metrology system and ultra high resolution tip-tilt actuator systems. The system should compensate to a very high level the thermo-elastic deformations of the GAIA support structure. The development of the Optics Active Control System is

57 3.1 The Astro instrument 51 the follow on of the APLT & AMTS TRP study that was performed by Alenia in During this study the COSI test bed already demonstrated that under laboratory conditions such a system can fulfil the GAIA requirements for base angle stability. For more detail see [9], [10], [11], and [12] Astrometric Focal Plane The FP is composed of 170 chips: 10 CCD horizontal columns placed side by side across scan and 17 CCD vertical strips along scan direction. Two CCD strips at their entrance on the detector devoted to target identification, the Astrometric Sky Mapper [ASM], 11 CCD strips for main astrometry, the main Astrometric Field [AF], 4 CCD strips to obtain colour information for all objects observed in AF, the Broad Band Photometry [BBP]. The ASM permits the on-board autonomous star detection and discrimination functions, without the support of any pre-loaded star catalogue. The main functions of the sky mappers are: 1) on-board detection and selection of stars to be measured in the astrometric field, including a discrimination for cosmic ray events; 2) scan-rate determination about two axes (along-scan and across-scan) which are provided to the satellite attitude control subsystem. In the ASM all pixels are read, selecting for follow-up the regions of interest (ROI), with signal above detection threshold. Two detections on subsequent chips are required for confirmation, to reject cosmic rays and spurious effects; they also estimate the satellite scan velocity and, for solar system objects, their differential speed. The first strip of the AF is used in cold redundancy to replace ASM1/2 in case of need. The AF provides 11 independent exposures of each selected target, on subsequent chips; the AF operates in windowing mode (section 4.2.1), on the ROI identified by the ASM, thus alleviating the electronics requirements. The acquisition time on the useful pixels is maximized by discarding the empty regions, thus improving on effective bandwidth and read-out noise. The star then crosses the BBP region, with four CCDs endowed with colour filters, providing spectral classification information down to the limiting magnitude. The transit of each star is thus observed by 17 CCDs. A single CCD model (section 3.2) is used for the whole FP, with specialised operating mode; the pixel size is 10µm 30µm respectively along and across scan. A large FoV of is required to achieve sufficient exposure time on every

58 52 The Gaia system: spacecraft and payload target, and to reach the desired astrometric precision; in the Gaia case the FoV is along scan (AL) (ASM + AF + BBP) and across scan (AC) ; also, the FP breadth ensures superposition between subsequent TDI passages. ASM1 ASM2 AF1 11 BBP1 4 row number Astro1 FoV Astro2 FoV + + Z fpa X fpa A row + Y fpa Star motion Figure 3.4: Astro focal plane layout and notations. The coordinate arrows indicate only the orientation of the coordinate system, but not its origin. The red and the blue rectangle represent the Astro1 FoV and the Astro2 FoV respectively. A strip Focal-Plane Reference System (FPRS) Now we define the FPRS of the Astro FP (figure 3.4): FPA coordinates are three-dimensional cartesian coordinates measured in units of millimeters; the origin of the FPA coordinates is in the nominal centre of the gap between the CCD n.5 and n.6 and the gap between the CCD strips AF6 and AF7, on the nominal plane of the CCD s illuminated silicon surface;

59 3.2 CCD detectors for Astrophysics 53 the positive Xfpa axis lies in the nominal plane of the CCD s illuminated silicon surface. It points into the nominal cross-scan direction, towards CCD row n.10. the positive Yfpa axis lies in the nominal plane of the CCD s illuminated silicon surface. It points into the nominal along-scan direction, in the direction of star image motion across the CCDs. the positive Zfpa axis is perpendicular to the nominal plane of the CCD illuminated silicon surface. It points down into the CCD and FPA substrates, i.e. in the direction of the light rays. 3.2 CCD detectors for Astrophysics The CCD detector is well-known and widely used in optical astronomy. One of the most fundamental advantages of the CCD devices is the possibility to acquire simultaneously images of all the stars that cross the focal plane and at the same time to achieve the detection of a large fraction of photons collected for each celestial body. If we want to integrate the signal in a continuous way, following an observation strategy based on a continuous sky scan, the CCDs must operate in TDI mode. The TDI operation mode has been used from ground for both imaging and spectroscopic observations. For examples, Schneider et al. ([13]) carried out a survey to search for faint, high-redshift QSOs at the Mount Palomar 200-inch reflector, by performing low-resolution slit-less spectroscopy of all objects drifting across the field of view at the sidereal rate. The Sloan survey also uses the same principle for the photometric sky scans ([14]) Back-illuminated CCD A class of CCD called the back-illuminated CCD has uniform spectral response throughout the visible, UV, and NIR spectral regions. They assure a high quantum efficiency (QE), i.e. 0.7, for a large spectral range. The CCD s are fabricated in a conventional triple-polysilicon, single-metal, 10-mask process, using n-type, high-resistivity silicon (EEV). For more details see in literature [15].

60 54 The Gaia system: spacecraft and payload TDI: Time Delay Integration mode The time delay integration mode of operation of charge couple devices (CCD) was one of the earliest modes used for imaging applications where an object is tracked by synchronising the charge transfer rate with the speed of motion of the object. This method allows for the integration of charge as the image moves across the focal plane producing a final image which has improved statistics. Accurately determining the position of and measuring the intensity (as a function of wavelength) of faint objects is of primary interest to observational astronomers; therefore, instruments composed of arrays of CCDs can be used effectively to track objects and accumulate charge in order to improve these measurements. This is particularly relevant to the Gaia observatory as it will have the task of tracking and classifying over one billion objects in the Galaxy over its six year mission. So let us spend some words to describe better what is the TDI mode. When a CCD is pointed at the sky and an exposure is taken, an object appears as a streak on the detector, the length of which is dependent on both the duration of the exposure and the relative speed of the object. In order to get a still picture, the CCD telescope must be pointed at the object, track the object at it is rate movement (sidereal for a star), and integrate for a specified amount of time. At the end of the integration time the shutter is closed, and the CCD is read out. Obviously this presents a problem for those telescopes incapable of tracking the sky or any other fixed telescope. TDI mode is a technique which allows a fixed telescope to track an object while eliminating most of the problems associated with flat field calibration and device uniformity. In TDI the CCD does not move at all and is read out at the sidereal rate, or at the rate of movement of the object being observed. The motion of the image on the focal plane is matched by the CCD clock rate: each potential well follows the current position of the associated point in object space, observed by the optical system. So only the electrons in the CCD are moved, and hence TDI tracking has also been referred to as electro-optical tracking. The CCD used in commercial applications is a full frame image sensor, which is essentially the frame transfer architecture without the memory zone. The CCDs in the array are oriented so that the vertical columns are aligned with the direction of the object movement (e.g. in the Gaia case opposite to the satellite rotation). The charge is then shifted down the vertical columns, in the east to west direction, on ground, at the sidereal rate appropriate to the

61 3.2 CCD detectors for Astrophysics 55 telescope declination, in space at the spin rate of the satellite. When the image point move to the adjacent line of pixels, a line transfer clock cycle is sent, so the integration continues until the pixels reach the western edge of the array, where they are shifted into the horizontal register, and then quickly into the output electronics. In such a way, the continuous motion is matched to a step-by-step process, since the CCD potential well is displaced by one pixel per clock cycle. The conventional ideal pixel, associated to a specific device location, is replaced by a logical pixel generated by the superposition of the contributions from all subsequent steps of the elementary exposure (sec. 4.1). With this mode all columns are optically sensitive and only the horizontal output register is shielded from the light. The resulting integration time is the time the image takes to pass over the entire array. The exposure time can be increased by rescanning the same area and adding it to previous scans. In TDI the integration time per object is determined by the size of the CCD. The clock cycles can be produced by hardware logic or software logic. Advantages of the TDI mode The main advantage of TDI perhaps comes from the fact that any imaged object are sampled by all the pixels in a column. Each logical pixel scans all physical electrodes along one CCD column; this averages out the spectral and spatial responses across the column, leading to a very good flat fielding performance. All objects observed on the same column are measured with the same response. As each imaged object is sampled by every pixel in the column, it is essentially detected with the mean efficiency of all the pixels in the column. Therefore, non-uniformities between pixels are reduced or eliminated, and the resulting radiometric quality is excellent. The good flat fielding performance with TDI results from the fact that all flat field information comes from the sky radiation that is detected at the same time as the objects that are being observed. The sensitivity of both astrometric and photometric performance to local device parameters is therefore reduced. The geometric calibration of a CCD used for pointed observation, in principle, requires characterisation of every pixel, which is an heavy task for the large logical format of most modern devices. For a CCD in TDI mode, thanks to the uniformity of logical pixels from each CCD column, the number of individual parameters drops from the order N 2

62 56 The Gaia system: spacecraft and payload to N. A relevant case of effects due to local CCD characteristics has been investigated in detail, concerning the astrometric and photometric calibration of the Wide Field/Planetary Camera 2 on board the Hubble Space Telescope Gaia CCD detectors One of the most important hardware components that allows Gaia to reach the desired performance is the CCD. CCD detectors form the core of Gaia Payload and their development and manufacture is one of the key challenges of the mission schedule. The Astro instrument are a very large focal plane, so, giving the restrictions on the CCD size, we have to cover the focal plane with a elevate number of detectors (170) having an active area of 45mm 59mm. All of the Gaia CCDs are large area, back illuminated, full frame devices. They all have a 4-phase electrode structure in the image section and a 2-phase structure in the readout register leading to a single output node. Gaia will observe objects over a very wide range of apparent magnitude and the CCDs must therefore be capable of handling a wide dynamic signal range. In order to observe most objects as efficiently as possible, the CCD QE must be optimised. At the same time, a number of features are incorporated in order to cope with bright stars. These include a large full-well capacity (> electrons), an anti-blooming drain and TDI gates which effectively reduce the integration time for very bright objects. Due to the TDI operation mode the CCD electrodes are clocked at the projected sidereal speed, with pixels following the image along the focal plane. In the astrometric focal plane, the image will cross 4500 pixels in seconds so the TDI period between each line transfer must be T T DI = 736µs. Furthermore, the along scan dimensions of the four phases of a single astrometric CCD pixel are 2, 3, 2 and 3 µm respectively. The T T DI period is therefore sub-divided between the four phases as µs, µs, µs and µs respectively to give the closest possible match between the motion of the incoming optical image and the motion of the integrated charge distribution providing the measured image. The exposure time is the time that a star image spends on the light-sensitive area of a CCD. The transit time is the along-scan coordinate of an optical image in the data pixel space. To first order, it corresponds to the time instant when the centroid

63 3.2 CCD detectors for Astrophysics 57 of the image (section 4.2.2) reaches the trailing edge of a CCD. CCD-related terminology ROA parallel clocks working in TDI mode 1996 pixels TDI direction across scan binning pixels readout direction 10 * 30 µm 2 a column serial clocks 4500 pixels a line Transit direction = along scan direction = TDI direction Figure 3.5: CCD related terminology. SR = summing register, ROR = read-out register also called serial register, ROA = read-out amplifier, TDI = time delay integration. SR ROR Properties of the Gaia CCDs Each ASM, AF or BBP CCD shall be made of 1966 columns and 4500 lines. Required properties of all CCDs in the Astro focal plane are: 1. Pixel pitch 10µm along scan and 30µm across scan; 2. Operation in TDI mode, with charge motion along CCD columns; 3. Inclusion of pre-scan and post-scan pixels; 4. Read-out register and operation consistent with windowing strategy (sec.4.2.1); 5. Four-phase image transfer in the image zone; 6. Availability of a supplementary buried channel;

64 58 The Gaia system: spacecraft and payload 7. Provision for charge injection; 8. Total detection noise compliant with requirements from Section ; 9. Inclusion of TDI gates. ASM, AF, and BBP CCDs shall include TDI gates in order to reduce the effective integration time for bright objects. TDI gates replace CCD lines at intervals given in the tables below and will normally be clocked like any other line. However, when a bright object is detected (G<12 mag), an appropriate integration time will be calculated on-board in order to avoid saturation and the corresponding TDI gate will not be clocked. It shall be possible to activate any number of TDI gates simultaneously. This procedure blocks the charge integrated before the gate(s), so that the integration is effectively reset to zero. Quantum Efficiency Figure 3.6: The detector QE: green- Baseline QE for AF, blue - MBP, red - MBP CCDs [from GAIA-AS-002-issue1 (technical note)] Detector Modulation Transfer Function The Modulation Transfer Function (MTF) is used to model the effects of various phenomena on the TDI process: the effects of the finite pixel size (integration MTF), TDI smoothing, along-scan (attitude) rate errors, charge diffusion (diffusion MTF), and transverse source motion.

65 3.2 CCD detectors for Astrophysics 59 The electronic MTF impacts on the effective image MTF introducing a degradation of the image quality and, as consequence, on the final angular resolution. Integration MTF The integration MTF is the fundamental MTF due to the geometry: the CCD samples a continuously varying image with an array of finite pixel elements. At the Nyquist frequency, for an ideal pixel the MTF due to the signal integration on pixel is 2/π = Lateral diffusion of charges, fringing fields, and capacitance between electrodes causes one electrode to collect charge from another, degrading MTF. Diffusion MTF x Illuminated (back) side x ff Electrons diffuse slowly in field free region Electrons liberated in field free region x d Electrons move quickly through depletion region to buried channel Electrons liberated in depletion region t buried channel depth V Electrode side Charge shared between several pixels (degraded MTF) Figure 3.7: Diffusion MTF is the component of MTF due to charge diffusion or spreading which is dominated by behaviour in the field-free region (figure 3.7). Photon interactions generate electrons with a density profile dictated by the PSF. The resulting clouds of the electrons expand due to diffusion. The degree of spreading depends on the collection time, the field strength and hence the interaction depth of the incident photons. Charge liberated in a given pixel may therefore be shared between two or more adjacent pixels, rather than being confined to one. This represents a blurring of the image, i.e. degradation of the overall MTF.

66 60 The Gaia system: spacecraft and payload The MTF is more degraded at blue wavelengths than at red ones. This happens because the red photons generates the electron in the depletion region while the blue photons generates the electron in the field-free region. So in the absence of a strong electric field the blue electrons are also able to spread laterally before the capture by the electrodes. We can found a simple model of diffusion MTF for a back-illuminated CCD in [16]. TDI smearing In the TDI mode, the charge image is being integrated for a short time τ, named TDI window, before it is shifted forward in order to follow the star image on focal plane. τ is related to the number of CCD phases n phase, to the spin angular rate ω and to the along-scan pixel size l px (in radiant) according to the equation τ = l px /(n phase ω). The results is a smearing of collected image along the scan, resulting from the convolution of PSF with a rectangular window with width equal to ω τ in the scan direction: f MT F T DI = sinc( ω τ ) (3.1) where f is the spatial frequency along scan in radiant on meters. Charge Transfer Inefficiency MTF The Charge Transfer Inefficiency (CTI) is the fraction of charge lost during the transfer from one pixel to the next. During the charge movement process, a little part of the charges are for example captured by traps and re-emitted at a later time, thus ending up in the wrong packet at the output. The main effect observed is image smearing. For Gaia at the beginning of life we may consider the charge transfer MTF to be equal to 1. Multiply integration MTF, charge transfer MTF, diffusion MTF, TDI MTF and so on we have a complete MTF model. In the de Bruijne treatment the CCD MTF is included in the form of a parametrised function. This function was determined empirically so as to reproduce the measured MTF values for Gaia CCD.

67 3.3 The Spectro instrument The Spectro instrument The radial velocity spectrometer (RVS) [17] is an integral field spectrograph with a field of view 2.0 x1.6. It disperses the light of all sources entering the field of view with a resolving power R = λ = over the wavelength range [848,874]nm with an along scan dispersion orientation. The RVS continuously and repeatedly scan the sky during the 5 years of the mission so that each source will be observed 102 times over this period. It will collect the spectra of million stars up to magnitude V and provide radial velocities with precision of 2 kms 1 at V=15 and kms 1 at V=17, for a solar-metalicity G5 dwarf. The Spectro telescope which illuminates both the RVS and the medium-band photometer is made of three silicon carbide (SiC), off-axis rectangular mirrors. It has an entrance pupil of 0.25 square metres and a focal length of 2.1 m. The 11 medium bands of the photometer cover a wavelength range from the near UV (300 nm) to the near infrared (1000 nm). The mirrors are coated with aluminium, which unlike other metals, exhibits a reflectivity higher than 85 per cent over the whole photometer wavelength range. Two optical configurations are under consideration for the RVS. The Spectro telescope illuminates three different instruments located in two distinct physical planes. The RVS sky-mapper (RVSM) and the photometric instrument are located at the telescope focus. The RVSM field of view precedes that of the RVS detectors (with respect to the scan orientation). The RVS CCDs are located at the spectrograph focus. In the current baseline the RVSM is composed by 5 CCDs strips. A dedicated telescope, the spectroscopic telescope, feeds the three instruments located in the spectroscopic focal plane. Its overall field of view is split into a central area of 2 x1.6 devoted to the spectrograph, and three outer areas (covering a total surface of 8 square degrees) devoted to the RVS skymappers (whose role is to detect the sources before they enter the spectrograph field of view) and the medium band photometer. The viewing direction of the spectroscopic telescope is located in the same plane as the two astrometric telescopes and precedes them by respectively 38 and 144 (corresponding to 38 and 144 minutes of time, the spin velocity of the satellite being 1 arc-min per second). The spectroscopic telescope consists of three off-axis mirrors. The optical combination presents no intermediate image. It has an entrance pupil of 0.25 m 2 and a focal length of 2.1 m. The image quality at the telescope focus

68 62 The Gaia system: spacecraft and payload is such that 10x15µm pixels (along across scan) can be used, corresponding to a spatial resolution of 1x1.5 arcsec. In order to guarantee the thermal and mechanical stability of the satellite, the spectroscopic mirrors, like the torus and the astrometric mirrors, will be made of Silicon-Carbide. The mirrors of the spectroscopic telescope will be coated with aluminium. This coating has been chosen because of its very good performance in the near ultra-violet (its reflective index at 300 nm is about 5 times larger than that of Silver). The overall reflective index (three reflections) of the spectroscopic telescope is 78% at 300 nm, 66 % at 861 nm (central wavelength of the spectrograph) and 77% at 1 µm. A high reflective index in the near ultra-violet is of great importance for the astrophysical parameterisation of the observed objects through the photometric system, which will have one or two filters below 400 nm. The colour indices resulting from the combinations of the ultra-violet bands with the filters will mainly provide accurate estimates of the luminosity and interstellar absorption of early type stars and of the metal contents of late type stars Spectro focal plane The spectroscopic focal plane is divided into two distinct physical planes. The Radial Velocity Spectrometer (RVS) sky-mappers and the photometric instrument detectors are located at the telescope focus, i.e. at the entrance of the spectrograph, while the spectroscopic instrument CCDs are located at the spectrograph focus. Yet, the three instruments share the same telescope and scan adjacent areas of the celestial sphere.the sky-mappers are placed before (with respect to the orientation of the scanning law), the spectrograph CCD and the two sets of photometric CCD are placed above and below. The sky-mappers, the photometer and the spectrograph detectors are all operated in Time Delay Integration (TDI) mode: i.e. the charges are continuously transfered from column to column as the sources cross the eld of view. The main role of the RVS sky-mappers is to detect the sources that will enter the spectrograph eld of view. This information will then be used to transmit to Earth only the pixels which are illuminated by a source. The RVS sky-mappers have three other functions: derive the magnitudes of the sources in the spectrograph photometric band, map the background with a resolution of about half an arc-minute and detect and characterise Near Earth Objects (NEOs). The RVS sky-mappers are made of 8 CCDs of 336 x 3930 pixels (along x across

69 3.3 The Spectro instrument 63 scan). The dimensions of the CCD pixels are 10 µm x 15 µm corresponding to a spatial resolution of 1 x 1.5 arcsec. Five of the CCDs operate in white light, one is coated with a neutral filter in order to record unsaturated images of the brightest stars and 2 CCDs are coated with the same filter as the spectrograph. The medium-band photometer (MBP) detectors are split into two blocks of two modules of 8 CCDs. The two blocks display the same distribution of filters: the first (or first two) CCD is a sky-mapper (without coating) and the 11 filters are distributed on the 15 (or 14) remaining CCDs (some filters are repeated on adjacent detectors in order to increase the exposure time which is 5.5 s per chip). In each CCD block, one of the modules (of 8 chips) is red-enhanced and the other is blue-enhanced. The dimensions of the medium-band photometer CCDs and pixels are the same as the ones of the RVS sky-mappers. The spectrograph focal plane is divided into 6 Low Light Level CCD (i.e. CCD displaying a very low readout noise) of 1010 x 3930 pixels. The spectra are dispersed along the scan direction and extend over about 700 pixels RVS objective The primary objective of the Radial Velocity Spectrometer (RVS) is the acquisition of radial velocities. These line-of-sight velocities complement the proper motion measurements provided by the astrometric instrument. Together they provide the means of deciphering the kinematical state and dynamical history of our Galaxy. The RVS will provide the radial velocities of about 100 to 250 million stars up to magnitude V = with precisions ranging from km/s at the faint end to 1 km/s or better at the bright end. Gaia s kinematical information will radically improve our understanding of the Milky Way. It will allow us to probe the galactic gravitational potential and the distribution of dark matter, to map the spiral structure of the disc, to disentangle, characterise and constrain the origin and evolution of the stellar populations of the Galaxy, to recover the history of the halo accretion events and, finally, to test the paradigm of the hierarchical formation of galaxies. The RVS will collect, on average, 100 spectra per star over the 5 years of the mission. The multi-epoch radial velocity information will be ideally suited for identification and characterisation of double and multiple systems. In particular, Gaia will provide mass and radii accurate to a few per cent for several thousand eclipsing binaries. The RVS will also monitor the radial motions of the outer layers

70 64 The Gaia system: spacecraft and payload of pulsating stars. It will provide pulsation curves for RR Lyrae, Cepheids and Miras up to V= Radial velocities will also be used to correct the parallaxes and proper motions of nearby, fast moving stars for the effects of perspective acceleration. The RVS wavelength range, nm, is a very rich domain. It will not only provide radial velocities, but also many stellar and interstellar diagnostics. The RVS data will very effectively complement the astrometric and photometric observations of Gaia s targets improving object classification. RVS data will also contribute to the derivation of stellar atmosphere parameters, in particular effective temperature, surface gravity, overall metal abundances. Individual abundances of key chemical elements, e.g. Ca, Mg, Si, will be derived for 4 to 8 million stars up to V = 12-13, bringing major improvement in our knowledge of the chemical history and the enrichment processes of the Galaxy. Information on many facets of stellar physics will be extracted from the spectroscopic observations, for example, stellar rotation, chromospheric activity, mass loss. Finally, from observations in the 862 nm Diffuse Interstellar Band, RVS will derive the 3 dimensional map of galactic interstellar reddening RVS measurement principle The Radial Velocity Spectrometer (RVS) is an integral field spectrograph dispersing the light of its 2 x1.6 field of view with a resolving power, R = The RVS operates in scan mode (as do the astrometric and photometric instruments) observing each source 100 times, on average, over the 5 years of the mission. The RVS wavelength range, nm, is close to the energy distribution peaks of G and K type stars, the majority of the RVS targets. In late type stars this interval displays 3 strong core saturated ionised Calcium lines (849.80, , nm), which allow radial velocities to be derived at very low signal-to-noise ratios. Numerous weak lines, for example, Fe, Si, Mg, are also present. In hotter stars, RVS spectra are dominated by the Hydrogen Paschen lines. Spectra of early type stars also contain some weak lines: e.g. Ca II, He I, He II, N I. Over the 5 years of the mission, the RVS will observe between 10 and 25 billion spectra of 100 to 250 million stars. The analysis of this data will be a complex and highly challenging task, not only because of the huge volume of spectroscopic data but also because it will rely on the multi-epoch photometric (5 broad band and 11 medium band) and astrometric information. As a consequence, the extraction of the astrophysical parameters

71 3.3 The Spectro instrument 65 should be performed in a fully automated fashion. For bright enough stars (e.g. V 15 for a G5 main sequence star), radial velocities will be derived for each observation. Summation of the spectra collected over the mission will allow the determination of mean radial velocities of the faintest stars (V 17 18). The radial velocities will be obtained by cross-correlating the observed spectra with either a template or a mask. A first estimate of the source atmospheric parameters, derived from the astrometric and photometric data, will be used to select the most appropriate template or mask.

72 66 The Gaia system: spacecraft and payload

73 Chapter 4 Gaia measurement process 4.1 The Gaia measurement chain The planar wavefront from a remote point-like (unresolved) source passes through the optical system, at a given time, and it is focused onto the FP as an instantaneous photon distribution, i.e. the Point Spread Function (PSF). The images from the two telescopes, Astro1 and Astro2, pointing in two directions separated by the basic angle, are superposed onto the same FP. The nominal optical configuration is the same for both telescopes, but the flight configuration is in general different from the nominal case, with residual differences between the telescopes. Also, the in flight value of the BA is in general different from the nominal value. The PSF generates a distribution of photo-electrons onto a CCD, accordingly to the local detector sensitivity, i.e. the QE. The effective charge distribution collected in the CCD pixel array is also affected by the local Modulation Transfer Function of the device. Even ideal pixels, due to their finite size, introduces a little smoothing of the optical PSF. The satellite spins at a nominal rate of 60 /s = 1 /s = 1 /minute, corresponding to a 6 hour period; the base angle is nearly orthogonal to the spin axis, i.e. set onto the equatorial plane of the scan. The precession motion provides an evolution of the instantaneous scan circle over the celestial sphere. The scan law provides repeated observation of each object in different directions along five years; an average value on the sky of the transits of a generic object in each FoV is of order of 42. Due to the scan operation of Gaia, the instantaneous image moves across the detector, in a direction which is mostly co-linear with the CCD columns

74 68 Gaia measurement process (section 3.2.3) and the y axis of the Gaia reference frame. In order to achieve a useful integrated image, the detector is operated in Time-Delay Integration (TDI) mode, providing the matching between the continuous satellite motion (resulting in a continuous apparent motion of the image on FP) and the logical pixel array, transferred in a discrete or stepwise way over the detector between the CCD edges. The instantaneous image is integrated throughout the whole transit over the CCD, as each logical pixel is generated at the leading edge and transferred in step with the corresponding point in object space up to the trailing edge of the device, where the readout process takes place. For each star a window is selected for read-out according to the windowing strategy (sec.4.2.1). The image is compressed by binning in the across scan (low resolution) direction to minimise telemetry, so that the output data is a one-dimensional signal. The time tag of the exposure is recorded for data processing. Time of observation and position on the sky are linked by the scan law plus optical response. For each transit a target is observed in equal conditions on 11 elementary exposures. The independent composition of the 11 elementary exposures provides the transit-level accuracy. To obtain the final mission accuracy we must take into account data reduction factor for parallax, position and proper motion. During the integration, the effective PSF is affected by a small degradation due to the nominal TDI process, plus additional degradation associated to the TDI and attitude errors, corresponding to a fraction of a pixel, i.e. in the mas range. The readout process conveys the last row of logical pixels to the CCD outputs, and it converts the charge signal into a voltage amplitude signal, introducing the read-out noise (RON). The electric signal is amplified by the front-end electronics and then converted in digital form for storage and download to ground. The electronics noise is expressed in terms of total equivalent detector RON. Readout on the Astrometric Field (AF) is restricted to the regions of interest (ROI) (see sec.4.2.1) identified by the ASM (section3.1.3). Each star is observed 11 times during its transit over the FP, by equivalent CCDs; the readout mode depends upon the target brightness, in particular using across scan binning to improve upon signal to noise ratio (SNR) and reduce the data volume (improving storage and telemetry requirements). The subsequent measurements allow to reduce the random error on target position. In the ideal case of equal optical performance over the field, the elementary exposure preci-

75 4.1 The Gaia measurement chain 69 sion results in a transit level uncertainty improved accordingly to the number of exposures, as 11. The different astrometric performance, for a given object and associated photon budget, arises from the variation of optical quality over the field and the disturbances affecting each individual measurement. A weighted average is thus likely to be the most convenient approach for practical implementation, taking into account the optical response and detector geometry. The measurements collected throughout the mission lifetime are combined together in the data reduction process to provide for each object an estimate of parallax and the two components of angular position and proper motion. From simulation of the scan law and reduction scheme, the transit level precision is improved for parallax by an average factor 2.1/ Similar expressions hold for the other astrometric parameters, taking into account the average number of transits and the two fields of view through a different geometric factor Raw data The effective PSF, resulting from optics + detector (QE, MTF and TDI), is integrated by binning in the across scan direction, for most targets, during CCD readout. This provides a one-dimensional intensity signal as a function of time, measured in digital units, so that its centre (defined e.g. by barycentre or centre of gravity COG, see section 4.2.2) can be considered as the time of transit of the object by a reference position of the CCD. According to readout operation, the reference is the edge of the CCD readout register, assuming the time stamp of the whole CCD line is set to the clock value reached at the final moment of transfer of the last CCD row into the horizontal register. Appropriate transformations can be defined to different references, e.g. the CCD central position, if convenient Elementary measurement Now we analyse in more detail what happens during one transit: Identification of the star The payload operates without the support of any pre-loaded star catalogue, because the astrometric field focal plane includes an on-board autonomous star detection mode. The star object enters the FoV at the first strip of the Sky Mapper. On this strip there are the detection and the selection of the stars to be measured in

76 70 Gaia measurement process the astrometric field. This first measure allows to stabilise the operation mode for the astrometric focal plane (windowing, etc., see section 4.2.1). All pixels are read, without binning in the across scan direction, so the output is a bidimensional matrix of intensity values. It is possible to use these raw measurements to calculate a mean across scan position as input to calibration and data reduction. The next step is the star transit on AF1. Two operation mode are selected for AF1: in confirmation mode to confirm the star transit or in combination with the transit data on the SM to compute the angular velocity component values, essential input to the attitude determination; in standard mode as AF2-AF11. Elementary measurement The elementary measurement is the primary Gaia measurement, on each CCD in the AF1-AF11 row. In principle we can retain the elementary measurements not correlated measurements. For each star is selected a window which dimension dependent by the star magnitude (see section 4.2.1) and the binning is made in the scan direction. For each transit star the output data is an array of size equal to the along-scan number pixels of the selected window, i.e. 12. A time t i is associated to each pixel i and a signal value I i. From the output data we can reconstruct the one-dimensional LSF of the star image. Note that there is a time coordinate on the abscissa axis, not a space coordinate. The center algorithm give so a transit mean time of the star at the CCD serial register. The relation between time and pixel position is achieved for the nominal scan rate. In such way we obtain the longitudinal position along the selected strip. The sampled LSF consist of a set of intensity values at subsequent time intervals of T T DI =.736 msec. I x (t k ) = k k 1 dt k 1 ε 2 dt k 2 I x (t k ) = where ε n = n m=0 ε m. N N 1 N 1 dyp (y, t, φ) + ɛ k 1 k 2 dt N 1 N 2 N 1 dyp (y, t, ϕ) + k (y, t, ϕ) + ε N dt dyp (y, t, φ) (4.1) N 2dyP k N N 2 N [ k n N n ] ε N dt dyω n (y)p (y, t, φ) (4.2) n=0 k n 1 N n 1 I x (t k ) = (clock) (px) P (y k ωt k + φ i ) 1 ε 1 ε N N (4.3)

77 4.1 The Gaia measurement chain 71 where T exp = N (clock) Elementary exposure precision The elementary exposure image can be centred, to deduce the one-dimensional along scan abscissa, with a precision given by the Cramér-Rao bound: σ p = ηλ SNR D, (4.4) where η is an efficiency parameter related to the telescope (throughput and image quality over the focal plane) and the detection system (quantum efficiency, sampling resolution), including the location algorithm ([6]; [18]); D is the RMS telescope size and SNR is the photometric signal to noise ratio Measurement composition The several elementary measurements are composed to obtain the final astrometric parameters with the desired accuracy. At this level we must take in account the different degrade effects due to the optics, the focal plane geometry and the attitude. Zero point, focal plane geometrical parameter, optical and attitude reconstruction parameter calibrations enter at this level. Transit level precision is scaled by geometric compression factor as derived by data reduction simulations - e.g. g π = 2.1 for parallax, mean on sky. During each transit, the images of the same target on subsequent CCDs are binned across scan, and centred along scan, providing independent onedimensional location estimates, with comparable precision. The transit level coordinate is thus improved by 11. The composition of subsequent measurements, on the whole sky, in different directions, allows determination of the bi-dimensional angular coordinates in a common reference system ( global sphere reconstruction, section 2.4). The evolution of the apparent source position during the mission lifetime ( 5 years) gives the absolute parallax and the proper motion End-of-mission accuracy The end-of-mission parallax accuracy σ π is: σ π = mg π t el ( N i τp det (G) (σ2 ξ + σ2 cal )), (4.5)

78 72 Gaia measurement process where m denotes an overall, end-of-mission contingency margin. A value m = 1:2 (20% margin) should be used; g π denotes the dimensionless geometrical parallax factor which relates the scanning geometry to the determination of the astrometric parameters. The quantity g π varies as function of position in the sky, mainly as function of ecliptic latitude, but a sky-averaged value should be used. For a given scanning law, the sky-average value of g π depends (mainly) on the solar aspect angle; N i is the number of astrometric instruments (viewing directions/telescopes); τ = LΩ/4π is the (average) total observing time available per object and per instrument, expressed in units of seconds; L denotes the nominal mission length, counted from the start of scientific observations (i.e., at the end of in-orbit calibration), expressed in units of seconds; Ω, expressed in units of steradians, denotes the effective solid angle of all AF CCDs per instrument, where effective refers to the light-sensitive CCD area exclusively ( thus excludes CCD dead zones in both alongand across-scan directions); τ 1 denotes the single-ccd transit time, expressed in units of second; p det = p det (G) denotes the star detection and confirmation probability as function of G magnitude, which should only affect stars fainter than G = 20 mag. This factor may be ignored in the end-of-mission astrometric accuracy budget at G = 15 mag, on assumption thatp det (G = 15 mag) = 1; σ 2 el (G), expressed in units of µas 2, denotes the variance of the LSF centroiding (localisation process, section 4.2.2) as function of G magnitude for a single-ccd transit. It is the so called the elementary exposure precision; σ calc, expressed in units of µas, is a calibration term.

79 4.2 Data processing Data processing Windowing and Sampling in Astro Samples Neither equatorial coordinates nor field coordinates of celestial objects, nor their magnitudes are directly observable by Gaia. The primary signal of Gaia are streams of charge pixels ( samples ) produced by the TDI operation of the detector chips. The primary astrometric observable thus is the location of a stellar image within one such stream of samples. The primary astrometric observable is the total amount of photo-electric charge in the set of samples constituting a particular stellar image. Note that samples are not physical entities on a CCD chip, but data items in the output data stream from a CCD. Usually there is no real danger of confusion between these two concepts, but a clear distinction can be made (whenever necessary) by calling them samples and physical pixels, respectively. A sample may contain the sum of the photo-electric charges from several physical pixels on CCD. There are two basic possibilities for adding ( binning ) the charges from several pixels into one sample: on-chip binning and numerical binning. The former operates on the CCD chip itself, combining the physical charges before read-out, amplification and digitisation. The latter operates in the instrument computer, simply adding the digitised numerical values representing the original charges. Windowing The total stream of samples that the Gaia CCDs could in principle produce is much too big to be treated by the instrument computer, let alone to be telemetered to the ground entirely. But it mostly contains dark, empty sky. Therefore it is both necessary and useful to select specific cut-outs containing actual stellar images of that stream for inclusion into the telemetry data stream sent to the ground. Such cut-outs are called windows. A window by definition is a rectangular sub-array of the stream of samples produced by a specific CCD and selected from that stream to represent a specific optical image of some celestial source. The samples in a window are uniform, i.e. they all combine the same number of pixels (both along-scan and across-scan). The unwanted samples containing dark, empty sky can be avoided in two

80 74 Gaia measurement process ASM AF1-10 AF11 Sample width (centre) - 2 (edges) Sample height Window width (centre) (edges) Window height 1 1 Number of sample/ccd Table 4.1: Windowing and sampling strategy for 12 < G A < 16mag objects. All windows are rectangular shape, apart from ASM case. ASM AF1-10 AF11 Sample width (centre) - 3 (edges) Sample height Window width variable 6 6(centre) (edges) Window height variable 1 1 Number of sample/ccd Table 4.2: Windowing and sampling strategy for 16 < G A < 20mag objects. All windows are rectangular shape, apart from ASM case. different ways: they can either be skipped in the read-out amplifier already (i.e. the charge is flushed electronically without being read and digitised) or else be deleted from the memory of the instrument computer later on. Both ways will be used by Gaia. The latter reduces only the telemetry stream while the former directly reduces the data rate load on the instrument computer - and indirectly even the read-out noise in the remaining samples. In fact, windowing reduces read-out requirements, too. The complete specifications of all sorts of windows for the various Gaia instruments, the so-called sampling strategy which is still under scientific development, can be found in [19]. In the Tables 4.1 and 4.2 we list the different basic windows. The window shape is star magnitude dependent. Samples are read from each CCD inside a window centred on each of the stars detected at the beginning of a field transit. Some samples are numerically binned, in other words they are added to become one sample before being transmitted to ground.

81 4.2 Data processing 75 Patches A patch is a one-dimensional along-scan sub-array of adjacent and identical samples belonging to a window, i.e. the set of all samples having the same across-scan coordinate. For a rectangular window all patches have the same size. The size of a window is conventionally given in units of samples, with the along-scan size named first. Thus a 12 x 6 window consist of 72 samples, composed of 6 patches with 12 samples each. A more detailed illustration can be found in [19] Location algorithms The elementary measurement precision goal of Gaia is a small fraction of the size of the detected image, or of the pixel size. The precision scales with the photometric signal to noise ratio (SNR); the location algorithm provides the definition of the photocenter and the scaling factor, as discussed in [6]. We have chosen the centre of gravity (COG) for the present work, since its performance in the class of configurations considered for Gaia is quite comparable with the optimal estimator in the least square sense ([5]). If f n is the sampled signal from pixel n at x n position, the photocenter x COG and the location estimate variance σexp 2 (COG) are: x COG = xn f n fn (4.6) σ 2 exp (COG) = (xn x COG ) 2 σ 2 (f n ) fn (4.7) In the photon limit, we have σ 2 (f n ) = f n and the Signal to Noise Ratio SNR = fn so that we retrieve the elementary expression for the COG noise: σ exp (COG) = L RMS SNR, (4.8)

82 76 Gaia measurement process

83 Chapter 5 Astrometric error budget An high stability and accurate knowledge of the variation of the base angle between the two lines of sight is required in order to calibrate the astrometric measurement (section 3.1.2). The star parameters determination requires the knowledge of the spacescraft attitude and payload calibration. Attitude and focal plane calibrations are estimated thanks to the great number of available measurement for each stars with respect to number of astrometric parameters of standard model. The calibration model takes into account a detailed description of the instrument and of the measurement process, as derived from the error tree (figure 5.1), and including the telescope optical response, described in terms of aberrations, the geometrical and functional representation of the FPA and other instruments parameter to be determined throughout the following years of study. 5.1 Astrometric performance requirements Astrometric performances are assessed taking into account stars with visual Johnson magnitude V=15 mag and with the following spectral types and interstellar extinction: B1V (A V =0 mag), G2V (A V =0 mag), M6V (A V =0 mag), B1V (A V =5 mag), K3III (A V =5 mag), and M0III (A V =5 mag). Until starting phase B (2004) the selected spectral types were B3V (A V =0 mag), G2V (A V =0 mag), M8V (A V =0 mag). The medium stellar density to take into account is about 25,000 stars/deg 2, while the maximum is about 3,000,000 stars/deg 2. We list in Table 5.1 the value of several astrometric instrument parameters. The Basic Angle between the two Astro telescopes is 99.4, this value is retain

84 78 Astrometric error budget Astrometric Instrument Parameters Value Observing time L L=5 years Mission time L L=6 years Scan rate ω 60 arcsec/sec (6 hours period) Precession period ω p 70 days Rotation axis 50 from sun direction Star Population 25000stars/ 2 Total number of observed star 1 bilion Basic Angle 99.4 Entrance pupil 1.4 m x 0.5 m Focal length m Total FOV (ASM+AF+BBP) Ω = Total FOV Field of view(af) Ω AF = FOV AL (AF) H y = FOV AC (AF) H x = Focal plane scale 1arcsec = 226µm Star speed along scan 13.6mm/s(60arcsec/s) Star speed across scan 60.9µm/s(270mas/s) Pixel size 10µm 30µm(44.2mas 133mas) Airy λ = 0.7µm T A = 103 mas 289 mas CCD active area AL 45 mm= arcsec= circ CCD active area AC mm = arcsec = circ dead zone AL 3.9 mm along scan (transit time = sec) dead zone AC 1.27 mm across scan Star speed along scan 60 arcsec/s 13.6 mm/s Star speed across scan 171 mas/s 38.6µm/s TDI Integration time sec Transit Integration time 38.2 s Avarage FP passage/star/telescope 41 Star number and flow Astro AF CCD number 17 CCD AL 10 CCD AC Table 5.1: Configuration parameter value.

85 5.1 Astrometric performance requirements 79 within a precision of half degree. The Basic angle required stability (or knowledge error) in orbit must be < 7µas. An accuracy of 0.5 arcsec/s is required on attitude in order to allow the data acquisition. Requirements for the attitude and orbit control system: A precise stability of the line of sight of the instrument respect the the nominal scan law is required, in order to avoid the blurring of the image during each integration period of 3.3 s; 4.5 mas along scan 4.5 mas, 20 mas across scan. The rotational speed must be stable during the integration period in order to guarantee a perfect synchronization between the scanning motion and the transfer charge operations within the CCDs (TDI); 1.5 mas/s along scan, 10 mas/s across scan Main operational assumptions A CCD strip refers to a set of 10 CCD aligned along the X axis. There are 17 CCD strips: 2 ASM, 11 AF and 4 BBP strips. A CCD row refers to a set of 17 CCD aligned along the Y axis. There are 10 CCD rows and 17 CCD strips: the 10 rows are identical and each is populated by 2 ASM CCDS, 11 AF CCDS and 4 BBP CCDs. For each CCD, similar definitions are set: the CCD columns are oriented in the FPA row direction (along scan), the CCD rows are oriented in the strip direction (across-scan). Consequently, each ASM, AF, BBP CCD has 1966 columns and 4500 lines. The two ASM CCDs strips are distributed between the two telescopes: the ASM1 CCDs act as sky mappers for the Astro1, the ASM2 have the same role for the Astro2. The data collected by the ASM CCDs are necessary to: detect and select the stars which will be tracked by the AF and BBP CCDs (windows determination); adjust the AF/BBP TDI gain in order to avoid any saturation (TDI gates programming). The AF and BBP CCDs receive the photons collected by both telescopes n1 and n2. The analysis of the data coming from the ASM1 and ASM2 are necessary to discriminate the stars coming from the telescope n1 or n2. All AF and BBP CCDs are operated in windowing mode. Within the AF CCDs, AF1 and AF2 CCDs have specific roles: AF1 is used to provide angular rates measurements

86 80 Astrometric error budget AF1 and AF2 are used to confirm the presence of the stars selected on the basis of the ASM data BBP CCDs are used to extract colour information of the stars by means of optical filters located in front of the detectors. Along-scan velocity and integration The spin of the satellite generates a scan rate at instrument level whose period is about 6 hours; this is equivalent to a scan rate equal to 60 arcsec/s or 13.6 mm/s at focal plane level. The integration is then fixed to the time necessary for one pixel to cover its own size. As the pixel size is 10 µm, the integration time is equal to 736 µs. Across-scan velocity and windowing The spin axis of the satellite is not in a fixed direction. The resulting is a across-scan velocity which can reach 176 mas/sec (equivalent to 38.6 µm/sec at focal plane level), and with a period of 6 hours. It is important to note that a star will not generally have, on the FPA, a travel parallel to the Y axis. That is to say that, during the whole FPA travel, whose duration is around 1 minute (few seconds per CCD), the star motion along the X axis can reach several pixels (up to 4 per CCD leading to several tens of pixels for the whole FPA). Consequently, the windowing strategy of the AF CCDs has to take this variable deviation into account (simple pipeline encoding of the AF CCDs is inefficient). 5.2 Elementary measurement error: random and systematic errors The astrometric accuracy can be separated in two independent terms, the random part induced by photo-electron statistics on the localisation process accuracy, and a bias error which is independent of the number of the collected photons. In figure 5.1 we show the astrometric error tree, in the following we give some comments in order to fully clarify the philosophy subtended at this kind of formulation.

87 Figure 5.1: Error budget Instrument Related Errors Optical System Performance Parameters Aperture size Focal Length Transmissivity Optical Aberrations Field Distortion CCD Performance Parameters CCD and Pixel Size QE CTE Pixel MTF RON Dark Current Elementary Measurement Error Spacecraft Related Errors LOS Jitter (Relative Pointing Error) Disturbance Sources Attitude Control Performance Scan Rate Measurement Error Centroiding Algorithm Errors Star Parallax Standard Deviation Instrument Related Errors Basic Angle Stability Medium Term Management combination error Optical System Tolerance Mechanical Stability Thermal Stabilitty Star Relative Position on the FP Stability Optical System Tolerance Mechanical Stability Thermal Stabilitty Processing Related Errors On ground Attitude Estimation Error Astrometric Model Mission Related Errors Overall Observation Time Spin Rate of the Scanning Law Precession Rate of the Scanning Law Sun Spin Axis Angle Mission Parameters Instrument Related Parameters Field of View Amplitude 5.2 Elementary measurement error: random and systematic errors81

88 82 Astrometric error budget The GAIA fundamental measurement is the single exposure over a CCD, called elementary measurement. The eleven elementary measurements acquired on a single transit are composed to obtain the desired information. The data processing is formulated on opportune time scale i.e. parallax, proper motion. The data processing wraps the various elementary exposures on proper scaletimes; respect Hipparcos, there are different needs of calibration and monitoring of the payload The cell Medium Term Measurement combination error indicates the analysis of the instrument on time period to be defined for the verification of the stability of itself, where with instrument we mean both the attitude and the instrument part. In the specific, with the expression Medium Term are identified the scale-times between tens of second (FoV transit) to some hour or more. It is still in development the model useful to plan the calibrations. It will define how to join together the various elementary exposures in order to keep the necessary informations on various parameters. In order to take in account these needs, we have inserted a cell called Calibration. The cell Attitude Estimation Error will be subsequently detailed. Concerning the left side of the scheme, the called Elementary Measurement Error, we have added, with respect to the old formulation, a cell to take in account the performances of the TDI technique. Under the nomenclature Source Related Errors we consider not only the centroiding algorithm errors, as formulated in a precedent writing out, but also other error sources linked with the kind of source. In the end we have considered the need to add a cell called Astrometric model. This because in the analysis of the error budget will be take in account also the precision of the functional model that link the elementary observation properly combined with the astrometric parameters of positioning, speed and distance of each star. In other words, all the astronomical/astrophysical effects (i.e. aberration, relativistic deflection, perspective acceleration, scale times and so on) will be estimated and properly modelized, and in each case must be taken in account in the evaluation of the error budget. We also want put in evidence the importance of an analysis of correlation between theese noises of astronomical nature and others purely instrumental, in particular periodical ones. An investigation of this kind will help in defining the requirements of accuracy for the instrumental calibrations both in flying

89 5.3 Astrometric performance contributors 83 and on ground. The right part of Mission Parameters become an input of the new cell Astrometric Model. The Mission Parameters have a nominal value of implementation that must be in any case monitored in fly in order to know the effective value. At the level of the elementary measurement, the whole left column contribute as random error, while at transit level and more, in presence of some kind of combination of the elementary measurements, some voices listed give also a contribution at bias level (systematic error). At the level of the elementary measurement the whole right column contribute as random error. 5.3 Astrometric performance contributors Now we list the several contributors to take into consideration to achieve the astrometric performance goal. We list the contributors to the determination of the astrometric performance. There are contributors at PLM, SVM and S/C level. Source Photon noise, sky background, location algorithm, source structure and variability PLM contributors PLM alignment in-orbit stability, Spectral response for BBP and MBP bands Optical System Aperture size, Focal length, Transmissivity, Optical Aberrations (in particular Distortion), Chromaticity, Mirror polishing and mounting, Telescope alignment, WFS accuracy, FPA alignment, thermoelastic stability, Stray-light, ageing Basic Angle Nominal value, Thermal and mechanical stability, optical system tolerance, Basic angle stability in orbit, BAM system accuracy, ageing Detection system CCD QE, CCD linearity/dynamics, Linear pixel size CCD integration time, CTI, MTF, RON, Dark current, Detection chain gain, Focal plane geometry (location accuracy and

90 84 Astrometric error budget stability), TDI timing accuracy, Clocking knowledge accuracy and stability, data timing accuracy. Ageing and radiation impact S/C contributors Scan rate error and LOS jitter, On-board timing knowledge, EMC disturbances AOCS and Attitude error AOCS rate error, Absolute pointing error, Absolute rate error, Scan Law, High frequency attitude disturbances, Disturbance sources: Solar pressure, micrometeoroids, gravity gradients, Star tracker performance, alignment and stability, additional contributions TBD Operations Important contribution to the astrometric performance comes out also from the kind of operation defined at level of astrometric focal plane that define the way of acquiring the data (binning, readout windows, crowding, bright star detection) Critical parameters identification Focal length In construction phase the requirement on the focal is within the specifications guaranteed from the constructor of about ± 1mm. The alignment result to be a more delicate phase, because the focals of the two telescopes must result more possible equal, in order to guarantee an optical quality of the image useful to reach the wanted accuracy. The apparent sidereal velocity and the charge drift velocity must be in agreement for both the field of view. A difference on the values of the focals means a difference on the optical scale between the two field of view. The focal plane is in common with both the telescopes, so there is the possibility to fix an unique value for the charge drift speed so that given the optical scale, it will be in agreement with the sidereal velocity value. The effect is global over all the field, it is common mode effect and contribute to the random error. In facts the precision required for Gaia of about a hundred of µas on the single elementary exposure correspond to a small part of the dimension of the image and so to a small part of the measure of the pixel.

91 5.3 Astrometric performance contributors 85 Field distortion Field distortion is a kind of aberration which effects the image quality. It has impacts on: - Displacement of the image photocenter from the nominal image point. - Non constant motion of the image through the whole focal plane. The distortion is field dependent. This means that the image doesn t move through the field of view at constant speed. In presence of distortion, TDI and image velocity can not match over the whole field of view, thus causing a smearing of image. Definition An ideal optical system would provide a uniform matching between the focal plane segments and the corresponding on sky ones, i.e. a uniform optical scale; real instruments are affected by distortion, which provides progressive variation of the optical scale along the field of view. This is one of elements which must be taken into account for focal plane design, since each detector must be matched to the local value of apparent sidereal speed, i.e. the satellite rotation velocity, seen through the optical system. The distortion provides a progressive variation in the object speed as seen through the optical system respect to the nominal scanning velocity; the effect is a blurred image. Defining y as the object position on the focal plane and η the corresponding sky position with y = F η, the projected velocity is ẏ = F η = F V s = V f = cost, ẏ = ẏ + A 0 t y 3 ẏ = V f [1 + A 0 t y 3 ]. The star runs through the focal plane with a speed greater or less than the drift charge one and provides a displacement from the star image and the electronic one inducing a PSF blurring. Application requirement After the integration period T exp the displacement must be not greater than a fraction of pixel dimension. It is important to establish how much great must be a tolerable distortion effect to keep the required accuracy. We analyse what is the effect of distortion at the level of request accuracy for GAIA. We ll going to show that the optics tolerances are strictly connected with the tolerance requested for the electronic device. So, if we want relax the request on the optics we must made more pressing conditions on the electronic device. The idea is to find the best matching between the two requests and to give possible solutions.

92 86 Astrometric error budget Attitude and Orbit Control System TDI rate is fixed and identical for both Astro telescopes, and is fixed as well for MBP/RVS. Otherwise, the scanning velocity of the star must be equal to the charge transfer motion under well determined precision to keep the goal of 10 µas. To maintain the machting between the two motion we can only act on the satellite through the Attitude and Orbit Control System (AOCS). In this optics a measure of the across-scan and along-scan rate is fundamental. The angular rates are used by the attitude control system as input measurements that are combined with the auxiliary star tracker measurements for defining the FEEP actuator command. This means that the focal lengths of the telescopes are adjusted with sufficient accuracy so as to be compatible with the TDI fixed rates. The AOCS loop will then drive the system in such a manner that the physical motion of the star is made equal to the charge transfer motion, i.e. by driving to zero the difference between the measured rates and the nominal rates (defined by the scan law and the actual focal length). The star speed is measured with ASM1 and AF1 for telescope Astro1. In the same manner, it is measured with ASM2 and AF1 for telescope Astro2. At first order, the system works as following: With (ASM1, AF1) one gets two angular rate measurements, along and across scan. The across scan measurement gives the angular rotation of the satellite about an axis perpendicular to the plane defined by the telescope Astro1 optical axis and the scan axis. The same holds for (ASM2, AF2), by changing telescope Astro1 in telescope Astro2. By combining the two sets of measurements, one gets four speed measurements, from which one can deduce an estimation of the three angular rates of the satellite (with one redundancy for the angular speed about scan axis, i.e. scan rate). 5.4 Requirements in the design, integration, alignment phase and data reduction We can single out four different mission phases with specific requirements for phase: design, integration, allineament phase that make up the operations phase and data reduction. Some of the critical parameters dealing with have more or less relaxed require-

93 5.4 Requirements in the design, integration, alignment ments according to the different phases Blurring of the effective Line Spread Function In this section we make some considerations about errors that contribute to the blurring of the effective Line Spread Function. The Line Spread Function (LSF) in output is composed by a temporal set of intensity measurements, that come out from an integration of the signal with the TDI method done during the crossing time of the CCD by the star (3.3 s). The integration consists in adding the content of neighbour pixels making that their reading is synchronized with the speed of scanning of the satellite (1px = ms; pixel size =10 µm=44.2 mas). The reconstructed LSF (called temporal LSF) will be equal, in the ideal situation, to the one of an image LSF derived from an single exposure image for a time of 3.3 sec. for 3.3 s. In reality, it will result blurred essentially because of: optics: longitudinal residual aberration (< 2.9µm rms spec., see SR); attitude control: jitter in the scanning direction ( < 0.5µm rms = 2 mas rms spec., see SR); timing of the TDI: error on the scanning speed ( < 0.5µm rms = 2 mas/sec rms spec., see SR) Another effect to be considered is the - Time of transfer of charge between neighbour pixels, during which the star shift of a non negligible amount respect to the pixel width (0.025 ms, or 0.3 µm (to be verified). This effect is counted in the TDI temporisation, while its impact on the temporal LSF, of the TDI technique, is not important if compared with the precedents. With an error corresponding at a movement of the image in the scanning direction of 3 µm rms, the characteristic of the theorical border and the width (200 mas at first minimum) of the LSF are preserved and the astrometric accuracy doesn t result degraded. Further residual discrepancy between spatial LSF and temporal LSF due exclusively to the CCD characteristic (pixel, MTF, charge transfer efficiency, cosmetic,...) can be considered, once flat fielded, of order of magnitude lower than the ones that comes from causes described above.

94 88 Astrometric error budget Above all, the effects on the LSF due to the real motion of the star on the chip are uncorrelated from the ones linked to the response function of CCD, and so can be analysed separately. Another error that enter directly in the evaluation of the position of the star at time t is the one of synchronization of the aging of the pixel values with the Spacecraft Elapsed Time. If this error is of 1 µs/px and a pixel cover 44.2 mas, the error on aging imply a random error of positioning of the pixel of 0.06 mas, corresponding at a sigma of 17.3 microarcsec. This error must be added quadratically to the one of the centering algorithm Datation and synchronisation requirements On-Board Time It is assumed that the timing on board is based on a reference clock having a stability of on scale times of a rotation period of the satellite (6 hours), corresponding at about 20 ns on 6 hours. This requirement correspond at a stability on the basis exposition (3.3 seconds) of 0.25 ns, scaling by the square root of the time. This stability requirement mean a performance of accuracy of about 1 µas on a single period, so the time gap can be calibrated on the sky every 6 hours with this accuracy; the stability over larger periods is not required. The stability requirement, through the calibration, guarantees the accuracy on the time scale and make a reference on the size of systematic errors. Science Data Datation The observation in TDI create a value sequence of intensity of the pixels read at known times, spaced by 736 µs. The centering take into account the mean time of passage, on the border of the CCD, of the photocenter of the star, corresponding at an angular position on the sky stripe observed, through the scanning law and the transformation focal plane-sky. The above requirement of temporal accuracy is related to the resolution and dynamic of timescale compatibly with the needs of system sizing (i.e. the memory size). It is not important define the resolution of the aging with the datation values obtained for the stability of the time base. The truncation of the temporal data at the µs (TBC) introduce an additional random error negligible (1/736

95 5.4 Requirements in the design, integration, alignment pixel), without changing the systematic error, and allow the codification of a whole period over 35 bit. The best implementation will be defined during the detailed design. Synchronisation Requirements TDI-scan rate synchronization About the synchronization between TDI and scan rate the requirement is of 0.3 pixel rms and fixed the speed of the drag at electronic level, the main contribute come from optical system (aberration, see section), from geometry of the detector (right alignment, see section) and from the attitude. AF-SP synchronization Talking of the synchronization requirement between the focal planes Astro and Spectro, we start by the assumption that the Master Clock will be in common. Assumed that the internal requirements of stability and precision of the time scale will be guaranteed, the synchronization between the two focal planes is automatically satisfying. Given the hypothesis of using the sky mapper of RVS for completing the Astro1/2 one for the detection of bright stars, rise a synchronization requirement between the two parts of Payload. The synchronization is important by the point of view of the stability of the attitude, because the speed errors are added, in principle, for long times corresponding to the separation of the fields of view. The effective spacing between RVSM and Astro1/2 must be calibrated during the acquisition phase and verified in operation phase, so that it is possible to include the possible contribution of attitude for defining the reading windows within a pixel (TBC).

96 90 Astrometric error budget

97 Chapter 6 Selected calibration aspects The photometric and geometric calibrations of the Gaia instruments, and attitude determination, constitute an integral and a very central part of the data analysis task. It is important to keep in mind that these calibrations must be derived from exactly the same data used to derive the astrometric and photometric characteristics of the objects. 6.1 Gaia transit diagnostics The sources observed by Gaia cross a significant part of the field of view (FoV), during their transit, with 11 subsequent observations of equal nominal duration (T exp = 3.3 s) over the astrometric FoV (section 4.1). We assume the attitude is ideal, or better corrected outside the current part of data processing. Also, we do not address at the moment the elementary location process, with the only proviso that different location algorithms also have different Focal Plane to Sky Mapping (FPSM). The elementary exposure provides a signal in terms of intensity vs. time, so that the photo-centre is the average crossing time of the star over the CCD reference position corresponding to the readout register. In an ideal optical system, the subsequent observations are equally spaced, i.e. the time separation between measured crossing times is uniform. The real optics modifies this situation, inducing a variation in the measured time separation between subsequent CCDs over the FP. This effect is due to the presence of optical aberrations, that change the optical scale (os) and consequently the measured time separation between different FP positions.

98 92 Selected calibration aspects As the nominal detector geometry is regular, i.e. with equal device spacing, the observed transit is affected by acceleration. The elementary location is affected by an uncertainty associated by the local optical response and the star SNR; for the purposes of the current analysis, we assume that the precision is comparable for the 11 elementary exposures obtained from a star during its transit over the AF. The sidereal motion with speed 60 /s can be modelled as a straight line, which is projected as a more complex curve over the focal plane (FP) due to the varying optical response over the FoV. The simplest terms to be considered are the transversal and longitudinal acceleration or displacement terms, resulting in slightly different FP speed and effective CCD crossing time. The transversal motion is in general small and not observable due to across scan binning. The variation in exposure time is negligible with respect to photometric response and SNR. The longitudinal acceleration is averaged over the elementary exposure, so that the curve derived from instantaneous positions is affected by a smoothing which can be modelled by a boxcar average with size corresponding to one CCD: 3.3 s 198 = 3 18 = 45 mm, given the EFL = 46.7 m, corresponding to an os = /mm = mas/mum. The effective transit curve is sampled on 11 positions, y1, y2,..., y11, spaced correspondingly to the CCD separation, i.e. 50 mm = 221 = s. The whole transit over the AF requires 41 s. At a given across scan field position, the nominal current transit for a reference source can be described by the trajectory t(y), where y is the along scan coordinate, and the internal parameters of the trajectory are a function of the across scan coordinate x. The whole FP response can thus be mapped by a 2D function t(y;x). Since this function represents the discrepancy introduced in the measured data with respect to the uniform distribution associated to the sky by ideal optics, we label this function as Focal Plane to Sky Mapping (FPSM). 6.2 Focal plane to sky mapping Two configurations are currently considered in the industrial studies for phase B1; the concepts discussed herein are valid for both cases, with reasonable changes in the implementation details. The location measurements in pixel coordinates need to be transformed to angular field coordinates through a geometric calibration of the focal plane and of whole Gaia instrument, and subsequently to coordinates on the sky through

99 6.2 Focal plane to sky mapping 93 calibrations of the instrument attitude and basic angle. That is it is necessary to detail the transformation from the FPRS to FoVRS and consequently to the SRS (see section 2.3). The FPSM describes the detailed response of the Astro instrument, including the optical transfer function over the field, the detector response and operations. Its definition at the microarcsecond (µas) level requires good knowledge of in-flight instrument parameters. Science data can be used to trace directly the instrument response, taking advantage of the repeated measurements of stars over the field. Each telescope provides its own characteristic focal plane to sky mapping (FPSM, or FPS Transformation - FPST), i.e. a single-valued function relating the two coordinate sets. In practice, the variation of effective focal length (EFL) or optical scale (os=1/efl) associates an elementary interval on the focal plane to different values of angular interval on the sky, depending upon the specific optical configuration and the selected field position. The on-sky position (η, ζ) of a source observed in the focal plane coordinates (x, y) is thus derived in an equivalent (hypothetical) pointed exposure through a given telescope (e.g. Astro1) is described as η(1) = y os(1; x, y) = y/ef L(1; x, y) (6.1) ξ(1) = x os(1; x, y) = x/ef L(1; x, y) We remark that, for ideal (geometric) optics, the FPSM is the identity function, apart multiplication by a scalar proportionality factor. This suggests the possibility of dealing conveniently with this function in a perturbative approach, suitable to a successive approximation technique. Also, accordingly to the assumption (1) of the astrometric model implementation described in detail in chapter 7, the FPSM is a slowly varying function over the field, and an appropriate effective value can be used for the result of an elementary exposure on each CCD. For telescope Astro2, the same FP position (x, y) is mapped in general into a different position: η(2) = y os(2; x, y) + BA (6.2) ξ(2) = x os(2; x, y) Since the BA definition is referred to the angle on the sky between the two directions associated to a given FP point, through each telescope, along

100 94 Selected calibration aspects the scan, we have that the BA is in general a function of the FP position, depending upon the current optical configurations: δba(x, y) = η(2) η(1) = y [os(2; x, y) os(1; x, y)] (6.3) Besides, this effect is strictly related to the specific optical response of each telescope, so that it may be more convenient to retain a single BA value as representative (e.g., at the centre of the FoV), and carry on explicitly the field dependence as a derivation of each telescope FPSM. Then, two directions on the sky, exactly separated by the BA value, would be mapped in slightly different FP positions. Both approaches are potentially applicable, with some care. Hereafter, the discussion is independent from the BA definition, apart straightforward changes. The formal definition of the FPSM must also take into account the nominal BA definition; the most symmetric adjustment to the above eqs. 6.2 and 6.3 is η(1) = y os(1; x, y) BA/2 ξ(1) = x os(1; x, y) (6.4) η(2) = y os(2; x, y) + BA/2 ξ(2) = y os(2; x, y) We remark that the FP position of each target is derived from the measured intensity-vs.-time data by means of a location process, which may be based on a simple centre-of-gravity (COG) algorithm, a least-square fit method, or different techniques, which are all dependent from a set of parameters describing the instrument In the COG case, which appears to be the simplest, the FPSM still contains the optical configuration and the detector geometry parameters. The discussion below, as far as possible, is independent from the specific location process, and therefore does not take into account explicitly the required instrument parameters Measurement equations Individual field Given the scan observation, two stars from the same field (i.e. imaged by telescope Astro1), measured on a given CCD n, are separated in the measurement direction by a time elapse associated to their angular separation on the sky,

101 6.2 Focal plane to sky mapping 95 through the scan rate (plus the appropriate subset of instrument parameters). We assume that the instrument is not changed during the measurement, accordingly to the assumption (2) (section 7.1.2). If t1 and t2 are the crossing time stamps of the two stars 1 and 2, given the clock rate of CR pixels/s (possibly specified in Hz) and the pixel size ps in mm, the source linear separation in the measurement direction (equivalent to the image distance on a plate made by the nominal Astro telescope in a hypothetical pointed observation) is y(1, 2) = (t 2 t 1 ) ps CR (6.5) The on-sky angular separation η in the scan direction is directly provided by the local values of EFL, or os: η(1, 2) = y(1, 2) os = t(1, 2) ps CR os = t(1, 2) ω (6.6) where the nominal angular spin rate ω is equivalent to the composition of pixel size, clock rate and EFL, as the ratio between angle and time: ω = ps CR os (6.7) However, the spin rate is a uniquely defined physical quantity for the satellite, with a nominal value, and time dependent errors which are measured and limited by the AOCS system on board, and reconstructed to much higher precision on ground during the data reduction process. Thus, another representation of the FPSM is the variable apparent scan speed over the field, due to optics: this means that the same time elapse on different detector positions, or the same linear separation on different FP regions, corresponds to a variable angular distance on the sky. Also, the nominal pixel size is materialised in a variable parameter, taking into account the real FP geometry, e.g. associated to small deviations from nominal CCD position or alignment. Similarly, the clock rate CR is nominally the same, but in principle there may be random and systematic errors in the distribution over the detector, with an effect similar to perturbations to the actual FP geometry, since the time base is used as a reference to provide the corresponding linear or angular positions through the spin rate and FPSM.

102 96 Selected calibration aspects Dual field Two stars measured on the same CCD from different fields (i.e. object 1 from Astro1 and object 2 from Astro2) have celestial coordinate separation, along the current scan circle, described by an equation similar to the above, with the introduction of the base angle term: η(1, 2) = BA + y(2) os(2) y(1) os(1) = BA + [t(2) os(2) t(1) os(1)] ps CR = BA + t(2) ω(2) t(1) ω(1) (6.8) This defines the astrometric arc measurement over a large angle. Notably, here we cannot easily factor out the factors multiplying the time difference and label them globally as spin rate, unless taking into account that in principle it is different for the two fields. Also, for stars separated by exactly the BA value, the observation time is the same, in an ideal case, and the separation is only across scan (if they are not very close multiple systems). We can extend the definition to measurement of the two stars on different CCDs (labelled 1 and 2 as the associated objects), either in a common field or in different fields, with further generalisation of the above equations, introducing the along scan separation S(1,2) of the two CCDs considered: η(1, 2) = (BA+)y(2) os(2) y(1) os(1) + S(1, 2) = (BA+)[t(2) os(2) t(1) os(1)] ps CR + S(1, 2) = (BA+)t(2) ω(2) t(1) ω(1) + S(1, 2) (6.9) where the BA term appears only if the two stars are in different fields. We remark that the effective geometric separation among individual CCDs must be included, because it takes into account the linear separation in an equivalent pointed observation. Individual field Calibration methods The measurement equations suggest an approach for implementation of a set of calibration methods based on the measurement of the same object repeatedly over the FP, which is in any case the normal operation mode of Gaia. With reference to the 11 subsequent observations of an object, in the CCD row k,

103 6.2 Focal plane to sky mapping 97 we have η(k; m, n) = y(m) os(k; m) y(n) os(k; n) + S(k; m, n) = [t(m) os(k; m) t(n) os(k; n)] ps CR + S(k; m, n) (6.10) = t(m) ω(k; m) t(n) ω(k; n) + S(k; m, n) 0 since the object is assumed to remain at rest ( η = 0) during the measurement (time scale of one minute). This imposes strict conditions between the individual observations (m,n=1,2,...,11) and the projected angular separation of CCDs m and n in row k (S(k;m,n), describing the detector geometry). The target location is affected by random errors, depending upon the photon budget, and it depends on the local values of the instrument parameters, through the location process. Some of the parameters are explicitly included in the apparent local spin rate. It is possible to use eq as a tool for monitoring some of the instrument parameters, over the transit time scale. With respect to the above eq. 6.8, we remark a slight change in the nomenclature: although the object is the same, it is observed several times, so that the corresponding index is no longer associated to physically distinct stars, but to different observations of the same target over subsequent exposures. The estimated values can then be averaged over several stars, to reduce the noise, assuming the system remains stable over the appropriate time frame (assumption (2)), In particular, it is possible to use the subsequent exposures of the same objects on the 11 CCDs in a longitudinal detector strip to evaluate either the spin rate w or the CCD separation, labelled as S(n;1,2), S(n;2,3),..., S(n;10,11) for the CCD strip n = 1, 2,..., 10 across the scan direction. In general, it is desirable to have an independent assessment of both spin rate AND FP geometry, but assuming the parameters change over different time scale and in different ways it is possible to apply filtering techniques. For example, displacement of CCD AF4n, strip n, results in opposite variations on S(n;3,4) and on S(n;4,5), with no effect on the CCD separation in other strips, i.e. on S(n1;3,4) and on S(n1;4,5). Besides, spin rate errors modify in uncorrelated ways all above quantities (S(n;3,4), S(n;4,5), S(n1;3,4), S(n1;4,5)), since they are measured at different time, whereas the correlation appears with respect to other devices in the across scan direction. Also, the spin rate is the same for all CCD strips, so that it is more convenient to derive it by best fit over the whole FP. The physical concept is that a common mode variation is likely to be associated to a common mode parameter like the spin rate, whereas a local variation is likely due to individual CCD

104 98 Selected calibration aspects displacement or local FP deformation. A possible choice is to define the spin rate estimate as the value providing best fit over the FP, and allocate all parameters describing optical performance to the effective PSF (used within the location process) and to the FP geometry. The case will be analysed in more detail, since the FP structure, with noisy power dissipation localised on each CCD and a fixed distribution of thermal contact / mechanical support, can be expected to have specific deformation modes which could be verified on larger scale prior to launch and detected as collective effects in flight. The above concept of individual field characterisation is insensitive to a common mode, rigid displacement of the whole FP, corresponding to an offset to the common zero point along the instantaneous great circle. This method defines and maintains the FPSM, for each telescope, intrinsically including some aspects of the FP geometry. Dual field Calibration methods A certain fraction of objects, after crossing the FoV of the leading telescope, is also re-observed by the trailing telescope field, after a delay of about two hours, in some cases on different CCD strips k, l. In this case, the relevant equation, derived from eq. 6.9 above, becomes η(k; l; m, n) = BA + y(m) os(k; m) y(n) os(l; n) + S(k; l; m, n) = BA + [t(m) os(k; m) t(n) os(l; n)] ps CR + S(k; l; m, n) (6.11) = BA + t(m) ω(k; m) t(n) ω(l; n) + S(k; l; m, n) 0 again under the assumption that the object is at rest during the measurement, i.e. about two hours. It is assumed that the BA, as above the spin rate, is a common mode parameter, whose value could be estimated by best fit over the data from the whole FP, as an operative definition of BA estimate on a time scale shorter than, or comparable with, the great circle (using Hipparcos nomenclature). This consideration provides a possible approach for base angle monitoring from the science data, suitable to be included in the reduction process. Besides, data from the BAM sub-system could be profitably used at this level. However, the case shall be analysed with utmost care, since the effects of all other instrument parameters (spin rate, optical performance, FP geometry) are entangled with the BA, with the above definition, and the possibility of performing a monitoring at mas level within the scan period is of paramount importance.

105 6.2 Focal plane to sky mapping 99 Assuming the individual field parameters are already identified by the appropriate procedures, as mentioned above, the BA parameter can be uniquely identified. One of the limiting factors is expected to be the compound effect of spin rate (attitude), integrated over the two hour time elapse between measurements. In order to derive a suitable mas-level BA evaluation, it is necessary to maintain over a two hour period a spin rate monitoring to the precision of 1 mas over minutes. Besides, Eq. (11) provides a convenient way to define the relative positioning (local zero points ) of different CCD strips, completing the FP geometry description. This method defines and maintains aspects of the FPSM not observable at the single transit level, thus generalising the closure condition on great circles. Common mode and differential disturbances The spin rate is a common mode parameter, since the whole satellite is assumed to move as a rigid body; also, the clock rate of all CCDs, operated synchronously, is the same. For the purposes of the current discussion, they may be considered as set to given values, close to the nominal ones, and affected by perturbations (noise) which apply in the same way to all FP detectors and to all objects measured within a small time elapse, in which the perturbation can be considered as constant (static). All perturbations are considered as independent, i.e. their contributions can be evaluated separately and composed (in quadrature or linearly, depending upon the specific case). Some of the parameters, e.g. individual CCD position, introduce opposite contributions on different parameters, as described above, but they are not differential in the sense of the astrometric measurement between targets in different fields. The typical differential term is explicitly a BA error, which enters linearly in the astrometric arc measurement. An overall FP displacement acts as a common mode error, as well as the spin rate error. Although the effect on the individual wide angle measurement is negligible, vanishing in the difference, the projection on the sphere might have some residual, due to the projection on the current scan coordinate. This is to be verified with respect to the data reduction approach.

106 100 Selected calibration aspects Monitoring methods: sensitivity vs. bandwidth In principle, the proposed calibration method is suitable to identification of slow perturbations, acting on a time scale comparable with, or longer than, the single field transit period. We assume that, at any given time, the instrument can be described by a set of parameters. The data reduction processes all targets, using the current set of parameters, and outputs the astrometric parameters of each target and new estimates of the instrument parameters, which could be used for reduction of subsequent data. The estimate of one instrument parameter from the data set corresponding to the subsequent exposures of a single star (where applicable) is affected by an error depending at least on the photometric SNR of the target used; it is therefore necessary to average on several objects in order to achieve the precision corresponding to the µas level. The estimate period (EP) corresponds to the amount of time required to accumulate sufficient precision on the estimated value of a given parameter. On a rate corresponding to the reciprocal of the EP, it is possible to safely modify the parameter value in the data reduction chain. This should not be confused with the characteristic time scale of variation of the same parameter, which depends on the instrument stability and perturbation conditions. In case the EP is shorter than the variation time scale, the parameter monitoring mechanism is effective, i.e. able to follow the instrument evolution. In the definition of in-flight calibration procedures, we assume that this is always the case. If for any reason the EP becomes larger than the intrinsic time scale of variation, and engineering requirements can no longer be modified to account for the new stability constraints, additional calibration procedures shall be defined, in order to minimise the Gaia performance degradation. Assuming only comparably bright stars are used for calibration, e.g. down to V = 15 mag, the number of available objects on the celestial sphere is of order of 40 millions. The average density is about 1000 objects per square degree. Therefore, in the 0.45 square degree of Gaia, the number of objects simultaneously observed, from both fields, is again of order of Each object (at V = 15) has individual location precision of order of 160 µas, so that the collective precision is of order of 5 µas per exposure. Taking into account the higher precision of brighter objects, the collective precision is of order of 1 µas. This abstract consideration shall be applied to actual parameter estimate procedures, in order to evaluate the realistic noise expected on

107 6.2 Focal plane to sky mapping 101 the measurements. Random and systematic errors: distortion map Some of the instrument parameters require specific consideration: e.g., some optical aberrations only contribute to random errors, whereas others also induce significant systematic errors, like distortion. Distortion modifies progressively the matching between unit segments in object space and image space; the optical design requirements include a prescription for very low distortion, sufficient to avoid significant image blurring in each point of the FoV due to the mismatch between the local apparent sidereal speed and the uniform nominal TDI clock rate. The local image quality is preserved, at the mas level, i.e. the effective PSF is not affected by large degradation, and the RMS image size is increased by only a small amount (typically a fraction of pixel). Thus, the contribution to the final mission error is also small. Besides, the systematic error at the elementary exposure level is significantly large, with a peak level of several mas, because the photo-center displacement corresponds to about half the peak mismatch between object and image space, during the elementary exposure. In lack of correction, this may result in a residual systematic error corresponding to several µas on the final estimate of the astrometric parameters. Thus, distortion represents a case in which the operating requirements of Gaia, corresponding to preservation of local image quality, and the associated random astrometric error, must be complemented by a correction of the systematic errors through (in this case) a distortion map, providing the estimate position correction. This is a fundamental part of the FPSM, independent from the target spectral type; distortion is mapped on the measured CCD separation, accordingly to the procedure described above. In practice, usage of eq above, for a given FP geometry, allows to monitor the distortion distribution for each telescope. Random and systematic errors: chromaticity map Chromaticity was discussed in chapter 8. The effect at elementary exposure level is a difference in the direction associated to a given FP position, depending upon the local aberration distribution and spectral content, with peak value at the mas level. In lack of correction, this may result in a residual systematic error corresponding to several mas on the final estimate of the as-

108 102 Selected calibration aspects trometric parameters. Usage of eq above, measuring the apparent CCD separation vs. spectral type, allows to evaluate the in-flight chromaticity map and monitor its variations, for each telescope. The CCD separation difference for different known sources (in particular well behaved intermediate type stars) can be used as an indication of the chromatic response of the instrument throughout the mission lifetime. CCD aspects In the above description, the detector is assumed to be modelled by parameters like MTF and QE, which may be different for different devices. Also, each device will be affected by its own distribution of defects, traps etc., degrading with time due to the progressive radiation damage. The main effect is in general a deferred charge, with impact on both random error (due to the larger image width) and on systematic errors (the charge is released after the image, inducing a delay on the photo-center position). Besides, the effective signal is integrated over several CCD columns, so that only a fraction of the total target flux is affected by each defect. Also, if the effect is large, it may be possible to identify the discrepancy between the currently measured signal and the expected signal profile for the current field position and target brightness, therefore sorting out the measurements potentially affected by problems. Another effect is due to the timing distribution: the master clock is disseminated over the different units of the front-end electronics, and the signal may be degraded by propagation over the CCD electrodes. In both cases, random and systematic errors may be induced, corresponding in an apparent modification of the actual FP geometry. The possible effects of CCD aspects on the elementary measurement (effective PSF) should be identified, in order to provide an assessment of potential impact on the measurement and of possible diagnostics tools. 6.3 CCD position and orientation It is necessary to provide an initial measurement of the FPSM, at the beginning of the mission, because launch stress and settling in the operating conditions, quite different from ground ambient, are likely to induce variations in the FP geometry large with respect to the µm scale. Also, the optical response after

109 6.3 CCD position and orientation 103 re-alignment may be quite changed with respect to the on-ground case, which is different from the design case. It may be convenient to derive the FPSM by successive approximation, namely starting from the nominal values and adjusting for one CCD strip at a time, from the sky mapper onwards. The optical contribution, at least, can be expected to have smooth variation, and this initial search can be performed by using larger windows than those foreseen for operations, to provide the required margins. Depending on the selected strategy and resources, it may be performed either on-board or on-ground. After initial measurement, the FPSM may be maintained by monitoring procedures included in the on-ground data reduction, and the relevant data could be uploaded to the satellite when necessary to follow the instrument ageing or other slow variations. In Fig. 6.1 we show the six degrees of freedom of each CCD. The main contribution from CCD alignment is the along scan displacement, labelled y-decenter. It is the largest term of FPSM apart classical optical distortion. It must be known to µm level for on-board operation, and at sub-nm level for on-ground data reduction. Figure 6.1: Translation and rotation degrees of freedom of each CCD. CCD translation along x (across scan coordinate) is not measured in science data due to signal binning, but are relevant to the placement of read-out windows. It must be known to few µm for on-board operation, and at sub-nm level for on-ground data reduction; it could be measured by full resolution, bi-dimensional read-out of a few stars, for initial definition, then similarly maintained throughout the measurements. A convenient approach, for a com-

110 104 Selected calibration aspects parably dense spatial and temporal sampling, may consist in full resolution read-out of a few stars, compatibly with the current star density constraints on operation. CCD translation along z is not easily measured, because its effects are negligible due to the significant depth of focus of the telescope (few hundred µm). Besides, this also means that it is not necessary to detect and maintain such parameters, relying on the FP mechanical stability and including the residual contribution to image quality degradation in the overall budget. CCD rotation vs. z, labelled γ-tilt in Fig. 6.1, is the dominant angular degree of freedom, since its effect is an along scan displacement variable over the device. The threshold is at the arcsec level for operation, and at the mas level for data reduction. This behaviour can also be identified by FPSM monitoring, since the measured discrepancy has a linear trend across the CCD. 6.4 Auxiliary parameters Figure 6.2: Variation over the field of the image skewness. In order to ease the astrometric payload diagnostics, it may be convenient to define auxiliary parameters. The FPSM depends from both optical

111 6.5 Detector geometry and operation 105 response and detector geometry; the factors must be separated, and this requires additional information. The photo-centre is the first-order moment of the signal; the higher order moments, by definition, are independent from the photo-centre. In particular, due to the smooth image profile variation over the field, the CCD position does not affect the higher order moments. Besides, the instrument configuration generates a specific structure of the FP images, which are reflected in the moment distribution. In Fig. 6.2, the distribution of skewness over the FP is shown; this is the normalised third order central moment, and it represents the image asymmetry. The moments, derived from the measured data, are all referred to the effective signal, including the combined effects of optics, detector, and TDI operation. The possibility of deducing the effective instrument aberrations from the moment distribution has been investigated [20], in simple cases, and it represents a promising tool for diagnostics. In practice, specific parameters can be addressed individually; current investigations are focused on diagnostics of chromaticity. 6.5 Detector geometry and operation Hereafter, we select as x coordinate the across scan direction, running along the CCD rows; y is along the scan, i.e. along the CCD columns, due to the TDI operation; z is orthogonal to the nominal CCD surface, and coincident with the optical axis at the appropriate position of the field. We label the rotations around x, y and z as α-tilt, β-tilt and γ-tilt, respectively. The independent measurement along y is the time of transit of the target photo-center over each CCD reference position, with precision depending on the image SNR. The whole CCD row observes subsequently the target, and we can fit the whole data set to a common law of motion in order to get a better estimate of the equivalent time of transit over a common reference. This process is effective if the main uncertainty is associated to the data, with sufficiently good knowledge on the detector geometry. The initial CCD alignment must reduce the dead space and preserve the main measurement characteristics, e.g. that an object detected by SM be observed by all trailing AF CCDs. For operations, we need to retain the image quality and sufficient knowledge of the detector geometry for reading out the ROI. In data reduction, the individual exposures must be merged into a single transit measurement with a mas-level accuracy; this requires much better knowledge on the configuration,

112 106 Selected calibration aspects as well as sufficient stability. Below, we discuss each degree of freedom in the most relevant context. CCD position The x position of each CCD must be set on the FP to within one or a few columns, i.e. few ten µm, to ensure that all devices read the same targets. The y position does not affect the measurement, provided it is known, and the dead space is dominated by the interface to the front-end electronics. We retain the above requirement of few ten µm. They must be known to a few µm, in order to select the ROI to within a fraction of pixel; besides, the ROI is defined with margins, because it is also affected by attitude. The z position requirement can be derived from the image blurring due to a CCD displacement, resulting in defocusing. The image size increment y associated to z is y = z tan α zα = z 1 2 D F (6.12) z 2 5 λ F 2 = 155µm (6.13) D2 CCD rotation Rotation vs. x provides image blurring and smaller effective pixels size; this results in a mismatch between nominal clock rate and projected sidereal velocity. For small angles α, the along scan size L decreases to L cos α L. At the end of the on-chip integration, the displacement between the instantaneous image provided by the optical system and the nominal on-chip image is L = L L, and the image blurring is L, assuming the CCD is driven at 2 the nominal clock rate. Therefore, y = L Lα2 (6.14) 4 α 5 λ F DL (6.15) i.e. a linear displacement of the CCD edge by about 3µm. Rotation vs. y provides differential defocusing across the device, as the CCD columns are placed at different height vs. the nominal FP. The across scan pixel size is reduced; however, the effect on image quality is not critical given the large tolerance on z positioning derived above. Besides, the reduction on the across scan CCD size reduces the coverage of the nominal sky strip. The few ten µm

113 6.5 Detector geometry and operation 107 requirement on x position, in angle, becomes Rotation vs. z induces effects similar but opposite to rotation vs. x. The along scan size L increases to L = L L for a small rotation angle γ. As above, cos γ we deduce, i.e. a linear displacement at the CCD corner of about 2.25µm Procedure We can use the data to extract fundamental parameters from the star images. To obtain such goal we can treat the data making use of different procedures according to the information that we want to obtain. We list some procedure which give us some method to calibrate the system. We can use raw data to keep under monitoring the base angle at the nm precision, the statistic come to help us. We can individuate some good stars and follow them during the transit in the two field of view. The useful targets transit on the perimetric parts of the focal plane, because of the combination of the updated rotation period and the precession. Those data are affected by an abberation degree more greater than in the central part of the focal plane. We can utilise the moments (eq. 6.16) to extract the parameters that characterise the real PSF. The most significant one is the three order moment, or skewness, that weight the asymmetric part of the image; the most problematic aberrations are the asymmetric ones as coma and distortion. µ nm = dxdy( x µ x σ x ) n ( y µy σ y ) m I(x, y), (6.16) dxdyi(x, y) where I(x, y) is the image intensity distribution, µ x and µ y are the photocenter coordinates along x and y. We can compensate the distortion effects by CCD rotation on or out the focal plane. The precision with which we obtain those informations concur to improve sky-focal plane transformation accuracy. The targets must have magnitude fainter than 13 because of the saturation problems. The difference of time of transit of each star between CCD pairs gives an estimate of their separation y( t = y/(ω F ), with precision depending on the target SNR, e.g. for a star. Any other star on the same sky strip provides an independent estimate o the same quantity. The measurements can be averaged, in order to achieve sufficient precision through statistics. Notably, in case of variations, we do not know by a pair-wise measurement which chip is displaced; we can identify

114 108 Selected calibration aspects collective effects, e.g. global shrinking or expansion, or displacement of a single device with respect to all others. We average over the stars contained in one square degree, away from the Galactic plane, in the magnitude range V = 13 20mag, and with some conservative reduction factor. The cumulative precision achieved by such data set, taking into account number of stars and individual precision vs. magnitude, is 5.7µas 1.29nm. Since the across scan size of the CCD is 4.3, the required sky strip has along scan size 15, and it is scanned in TDI mode in 900 s, or 15 minutes. We can reach the microarcsecond accuracy on one complete rotation but it is not so easy because of the possible perturbations. Thus, the normal science data stream allows calibration of relevant FP parameters on a time scale below 20 minutes, i.e. much shorter than the six hour spin period. If the instrument is intrinsically stable over a long period, the calibration data could be averaged, achieving the level of µas in about 90 minutes. This result, although obtained in the framework of in-flight operations, is also relevant for on-ground verification of the FP, which has mechanical proper frequencies above few ten Hz for launch compatibility. Thermal variations may have typical scales of a few or several minutes: it is convenient that their time constant is set by design and manufacturing significantly longer than the above, e.g minutes, to retain the performance of the proposed calibration technique. The FP time constant can be tailored by proper choice of geometry, material and structure, and verified in the laboratory over macroscopic range (a temperature difference of few several K).

115 Chapter 7 Numerical model implementation and validation In the previous chapter we spook about the geometric calibrations of the Astro focal plane and we gave a first treatment on the focal plane to sky transformation. In the present chapter we describe the numerical implementation of the detailed signal model used to perform the Gaia Astro instrument performance analysis. The static model includes the detailed optical response of the nominal configuration and can easily be applied to perturbed cases. The effect of a realistic optical system is described by the input from ray tracing code (e.g. Code V), as tables of aberration coefficients (section 7.2), used to built a monochromatic PSF library over the field, composed to generate the realistic local PSF for different spectral type objects. The signal processing has been developed in the IDL environment. The instantaneous PSF is processed according to the Gaia operations: CCD response, TDI, windowing, across scan binning. In this context the detector is assumed to be modelled by parameters like Modulation Transfer Function (MTF) and Quantum Efficiency (QE). This model can be extended to take into account dynamic contributions, e.g. attitude or TDI errors. In section 7.1 below we describe the numerical implementation of the signal model and list the assumptions made to build the model. In section 7.2 we provide some detail of the ASTRO optical model. In section 7.3 we give a crossvalidation of some model assumptions. In section 7.4 we show the astrometric performance of the baseline configuration. In section 7.5 we recall the FPSM

116 110 Numerical model implementation and validation definition and we give the differential FPSM for the baseline configuration. The instrument must provide sufficient image quality to preserve the elementary measurement precision. The analysis is devoted to evaluate the optical performance in terms of astrometric random errors and systematic errors for the nominal configuration at the elementary exposure level and scaled to equivalent final mission performance by means of a simple transformation. TDI and attitude contributions are neglected, in order to address explicitly the limiting optical performance. The requirements on TDI and attitude are such to retain most of the optical performance, in nominal conditions. The figure of merit for the astrometric performance can be split between the systematic error, and random errors, depending upon the stellar type and magnitude. For the nominal configuration the analysis is referred to the three spectral types B3V, G2V and M8V. Since the baseline configuration is not to be further optimised, the analysis is limited to evaluation of the optical performance in terms of astrometric errors and quantification of the Zernike coefficients in the Baseline nominal configuration, up to a degree to be identified. The analysis of alternative configuration is been performed along the same lines (cap. 9). 7.1 The ASTRO optical response model The Astrium optical configuration is not directly available, so that the analysis can be performed on a reconstructed configuration, considered representative, for the purpose of assessment of the relevant aspects. The effect of the real optics is described by the input from ray tracing code (e.g. Code V), as tables of aberration coefficients, used to built a monochromatic PSF library over the field, composed to generate the realistic local PSF for different spectral type objects. The PSF is processed according to the Gaia operations: CCD response, TDI, windowing, across scan binning. The optical response is described by the Focal plane to sky mapping (FPSM) (cap. 6), i.e. the transformation between object and image space, encoded in terms of discrepancy with respect to the geometric optics. We calculate the FPSM for several spectral types, also deriving the chromaticity map.

117 7.1 The ASTRO optical response model 111 field position λ 1 Monochromatic PSF Functional form of the Zernike or Fringe Zernike expansion. Polynomial coefficients 0 Sources Monocromatic PSF library for each selected field 2 Polychromatic PSF Polichromatic PSF library for each selected field Sampling 4 Readout Detected Polichromatic PSF library 3 Detected PSF Measured PSF library 5 Set of procedures dedicated to performance analysis Figure 7.1: Flux chart Building the signal model The signal processing has been developed in the IDL environment. In this context the detector is assumed to be modelled by parameters like MTF and QE. The numerical implementation of the detailed signal model follows the scheme of figure 7.1. The simulator is developed on three different levels: a) procedures building the monochromatic PSFs library in the wavelength interval [300,1100] nm; b) procedures for simulation of the polychromatic PSFs and of the measured signal; the output data is a bidimensional array; c) procedures devoted to performance analysis.

118 112 Numerical model implementation and validation Sources We compute fluxes according to telescope geometry, operations and blackbody emission of the star for the selected magnitude. Therefore we test that for the spectral types at the edge of the Gaia bandwidth the Black Body model (B.B.) is perfectly valid for the solar spectral types (G2V) and acceptable for the early and later spectral types. Monochromatic PSF The monochromatic PSF is built in the IDL environment according to diffraction models and local optical response. For each field position (1-13) we generate the monochromatic PSFs, according to eq.2.12 given in section 2.6, over the spectral range [300, 1100] nm. The Zernike coefficients describing the WFE are provided by the ray tracing code and are listed in Table 7.1. Polychromatic PSF The generation of polychromatic PSFs requires the spectral distribution S(λ) for the selected source associated to its effective temperature and scaled by G magnitude, the QE(λ) and the instrument transmittance T(λ). For each spectral type and selected field position we generate the polychromatic PSFs, accordingly to eq.2.12 given in section 2.6. The detected PSF At this level we introduce the ideal pixel response (geometry, MTF, TDI - nominal) and all additional effects, some of them wavelength dependent, to transform from the optical polychromatic PSF to the detected PSF The finite pixel size increases L RMS by 18% with respect to the purely optical value The measured signal A further step is required to obtain the simulation of the measured signal: the selection of the read-out region and across scan binning. The measured signal is a one-dimensional PSF restricted to the six/twelve central pixels (see section 4.2.1). The selection of along scan area is required for telemetry needs or to reach a good compromise between the read-out noise level and the signal level.

119 7.2 The optical model 113 Performance analysis We made up several procedures devoted to the computation of random noise and systematic errors, e. g. chromaticity, and of optical performance Assumptions and approximations We assume that: the PSF is locally invariant, i.e. it has slow variation over the FP the PSF is temporally invariant over short time scale at mas level the BA is temporally invariant over short time scale at µas level the QE and MTF are nearly uniform over a single CCD small perturbations are negligible below 1µas level the random and systematic errors can be analysed separately The model is based on the current mission profile, payload design, operation and parameter values. For the detector, we also assume uniformity of response and in particular we retain an average MTF value (wavelength dependent); the introduction of the pixel MTF degrades the performance by about 10%. For most aspects, the signal associated to an elementary exposure on one CCD can be represented by the PSF in the middle of the CCD itself. This assumption was verified in reference cases, and small discrepancies arise only at large angular distance from the optical axis. We retain negligible the perturbation effects at 1µas level. Only a remark: Due to the binning in the across-scan direction, the detected signal is a one-dimensional signal called LSF, the integral along across-scan direction of the PSF. The detected signal is described by the local LSF of the particular selected field position, it is not a mean LSF over the selected CCD. 7.2 The optical model We implement in the simulator a realistic version of the baseline optical configuration, i.e. representative of the Gaia astrometric payload. It was derived with the CODEV optical ray tracing package and optimised over a FOV of

120 114 Numerical model implementation and validation The configuration is re-implemented in IDL environment. The FOV covers the angular ranges [ 0.33; 0.33] in the along scan direction, and [0.20; 0.86] in the across scan direction; the off-axis designs is required to avoid vignetting. The monochromatic real PSF I M is generated according to N.(CODEV) (m,n) Name Functional form 1 (0,0) Piston 1 2 (1,1) Tilt(x) ρcosθ 3 (1,1) Tilt(y) ρsinθ 4 (2,2) Astig(x) ρ 2 cos2θ 5 (2,0) Defocus 6ρ 4 6ρ (2,2) Astig(y) ρ 2 sin2θ 7 (3,3) 3-Coma(x) ρ 3 cos3θ 8 (3,1) Coma(x) (10ρ 4 12ρ 2 + 3) ρcosθ 9 (3,1) Coma(y) (10ρ 4 12ρ 2 +3) ρ sinθ 10 (3,3) 3-Coma(y) ρ 3 sin3θ 11 (4,4) 4Astig(x) ρ 4 cos4θ 12 (4,2) (15ρ 4 20ρ 2 + 6) ρ 2 cos2θ 13 (4,0) Spherical 70ρ ρ ρ 4 20ρ (4,2) (15ρ 4 20ρ 2 + 6) ρ 2 sin2θ 15 (4,4) 4-Astig(y) ρ 4 sin4θ 16 (5,5) 5-Coma(x) ρ 5 cos5θ 17 (5,3) (21ρ 4 30ρ )ρ 3 cos3θ 18 (5,1) (126ρ ρ ρ 4 60ρ 2 +5) ρcosθ 19 (5,1) (126ρ ρ ρ 4 60ρ 2 +5) ρsinθ 20 (5,3) (21ρ 4 30ρ )ρ 3 21 (5,5) 5-Coma(y) ρ 5 sin5θ Table 7.1: In the table we show the functional form of the 21 Standard Zernike terms of the WFE serial expansion made by CODEV. First column: CODEV ordering; second column: Legendre polynomial index; third column: classical optical aberrations; fourth column: functional form the diffraction theory as square modulus of the complex amplitude response function, the Fourier Transform of the generalized pupil function (eq. 2.12). A ray tracing code as Code V computes the WFE distribution associated to each field position for the selected optical configuration; it is thus possible to build the PSF for any desired source, using the eqs. 2.12, 2.12 and for any selected field position. We use a numeric implementation of this model to provide the description of the imaging performance of the Gaia telescope. Additional contributions describing the realistic detector response and the effect

121 7.2 The optical model 115 Figure 7.2: Example of PSF profiles in field position n 4 at λ = 660nm with different set of Zernike terms. of TDI observations can be included, as well as the across scan binning used for Gaia, to build the effective measured signal. The WFE is described by the Zernike expansion provided by the ray tracing code, truncated to 21 terms. The 21 Zernike polynomial of the WFE serial expansion made by CODEV are listed in the Table 7.1. For any field position we define a set of local coordinates centred on the geometric optics position, i.e. the projection on the focal plane (image space) of the selected direction on the sky (object space) trhough an ideal optical system. From a mathematical standpoint, geometric optics is described by the so called gnomonic projection, described by a single parameter: the optical scale (section 6.2). The aberration decribing a real optical system induce both a modification of image profile and a displacement of the photocenter from the zero position. This is a fundamental contribution to the astrometric measurement. The detected PSF is generated according eq Our analysis requires identification of L RMS and SNR over several FOV positions and for several star spectral types to compute the FPSM and the chromaticity map. In figure 7.5 we give a visual rappresentation of the verification of the hypothesis made in the section on the local PSF small change and extensively

122 116 Numerical model implementation and validation explained in the section 7.3. In figure 7.2 we show a case of optical performance analysis used to evaluated the different weight of aberration terms on the image quality and consequently on the exposure precision. For futher details on this analysis see section Numerical implementation The computation is performed in 13 field position as showed in figure 7.2.1; in principle it could be performed in any point, but given the assumptions (sec ) the sampling is sufficient. In Table 7.2 we list the coordinates od the 13 selected field positions. We generate the monochromatic PSFs according to Figure 7.3: Astro focal plane with the dimension of the effective field of view of the telescope shown by points 1-13 the CCD spectral response λ = [300, 1100] nm with 10 nm resolution. We use the FFT function in IDL, with fixed sampling step of 2 µm on the focal plane, and pupil plane sampling ranging from 74x204 points at λ = 300 nm to 20x56 points at λ = 1100 nm. Each PSF is an array of dimension 1024x1024 with a size of 16MB. Due to storage problems the PSFs are saved as a matrix of dimension 257x257 with a size of 1MB, including several Airy lobes. The total number of the monochromatic PSFs result equal to 81 for each of the 13 selected field positions for a total of 1053.

123 7.3 Cross-validation between numerical model and optical code 117 Field X position Y position position [degree] [degree] F F F F F F F F F F F F F Table 7.2: Selected field position over the FP. For each spectral type and selected field position we generate the polychromatic PSFs, accordingly to eq. 2.12, and derived the detected and measured signal. 7.3 Cross-validation between numerical model and optical code In this section we describe part of the validation of the measurement model and of the simulation, with particular reference to the optical description. One of the main concepts to be tested is the smooth variation of the optical response over the field, accordingly to basic physical principles (first assumption in section 7.1.2). Within such assumption, it is possible to describe the optical response over the whole FP by means of suitable parameters defined over a grid of positions with small separation with respect to the tipical variation scale. It is then possible to perform the analysis of a number of parameters over this grid, and interpolate for the intermediate positions. Of course, in future a denser grid may improved on resolution, but at the moment we adopt the current 13 points grid as a reasonable trade-off between model precision and computing load. Also, based on this validation, we may be confident in the results of subsequent analysis in our numerical model without the need of constant input from e.g. ray tracing codes. We remind that a reliable description of the instrument

124 118 Numerical model implementation and validation response over the field is fundamental to provide an accurate link among measurements, i.e. to retain control of systematic error. Since the goal is the validation of the optical model, we truncate the contribution from additional sources (detector, operation). Moreover, for ease of implementation, we perform monochromatic analysis, for a set of wavelengths, rather than the complete polychromatic representation. We selected eight wavelengths, uniformly spaced in the range [300,1000]nm. First of all we validate the input data by verifying that the ray tracing code actually provides smooth variation of the Zernike coefficients over the region investigated. The parameters considered as adequate for image description are the center of gravity, the RMS image width and the low order along scan moments (section 6.5.1) Domain of the investigation We select an intermediate across-scan position (0.53 ), and a eccentric along scan interval ([0.2, 0.3 ]), corresponding to about two CCDs. The size is considered sufficient to verify the assumption of small variation of the image profile over an elementary exposure; large variations would result in a significant contribution to image degradation by blurring, i.e. to the random error. We select six positions in this region, uniformly spaced, with separation 0.02 listed in Table 7.3. The positions (in angular and linear dimensions, respectively across and along scan) are referred to the optical axis, which falls outside the detector due to the off-axis configuration (chapter 3). We build the set of monochromatic PSFs for each position and wavelength. Field name F0a F1a F2a F3a F4a F5a Field Coordinates (0.53 ; 0.20 ) (0.4625;0.1632) cm (0.53 ; 0.22 ) (0.4625;0.1790) cm (0.53 ; 0.24 ) (0.4625;0.1959) cm (0.53 ; 0.26 ) (0.4625;0.2122) cm (0.53 ; 0.28 ) (0.4625;0.2285) cm (0.53 ; 0.30 ) (0.4625;0.2448) cm Table 7.3: Array of field positions for verification of smooth variation hypothesis

125 7.3 Cross-validation between numerical model and optical code Description of the results Intrinsic optical variation We verify that the Zernike coefficients from CODEV indeed feature small and/or smooth variation. In some cases (figure 7.4 b, coefficient 10), the variation is quite linear over a range of 0.35 units; in other cases (figure 7.4 a, coefficient 8) it is more complex but smaller (peak to valley: units). The two cases mentioned above are respectively the 3-coma(y) and the coma(x); the behaviour is therefore in accordance with expectations from optical engineering. Coefficients 2, 16 and 19 have behaviour similar to coefficient 8, while the others are all similar to coefficient 10. We omit reporting the values and plots of each coefficient. In figure 7.5 we show the monochromatic (a) Coefficient 8 (b) Coefficient 10 Figure 7.4: Variation of the Zernike coefficients 8 and 10 over the selected region. PSFs at λ = 600nm for three of the reference positions, superposed on the same reference frame (section 7.2). The small displacement among the PSFs is mostly due to classical distortion and induces a small image degradation in the elementary exposure (section 5.3.1), which in this case is acceptable, i.e. within the requirements. The smooth variation of image profile is confirmed by the distribution of RMS image width and low order moments vs. field and wavelength.

126 120 Numerical model implementation and validation Image intensity [arbitrary units] Short range variation Along scan position [µm] Figure 7.5: Progressive variation of the image quality over the field; position across scan: 0.53; position along scan: 0.20, 0.26, PSF at λ = 600nm. Focal plane resolution The PSF is built with 2 µm resolution, and we see that for the analysis it is convenient to resample them at 0.1 µm resolution to avoid numerical artefacts. In figure we show the plot of center of gravity vs. field position with resolution 2 µm (left) and 0.1 µm (right). The discontinuity in the former case is due to the combination of two aspects: 1. the coarse selection of the read-out region due to the resolution; 2. the higher profile modulation of monochromatic with respect to polychromatic PSF. In terms of MTF, 2 µm correspond to 20% of the pixel size, which induces a significant blurring; the 0.1 µm case (1% of the pixel size) corresponds to a negligible contribution. The 2 µm resolution in the optical PSF construction is a necessary trade-off of the FFT tools used; we verify that interpolation of the optical PSF to 0.1 µm resolution before introduction of additional effects is a viable solution.

127 7.3 Cross-validation between numerical model and optical code 121 (a) 2µ resolution (b) 0.1 µ resolution Figure 7.6: Center of gravity vs. field position at λ = 600nm Pupil resolution For any given field, we evaluate the distribution of c.o.g. vs. wavelength (figure 7.7). We remark a comparably smooth behaviour, with superposed irregularities much smaller than in the previous case: few tenth µm. Figure 7.7: Distribution of c.o.g. vs. wavelength for field F0a. An immediate justification is derived from the PSF construction mechanism: the pupil sampling is wavelength dependent and ranges between and (across along scan). The different sampling may introduce artefacts superposed to the natural weighting of the wavefront with the wavelength. Additional contributions may be induced by the combination with focal plane

128 122 Numerical model implementation and validation sampling and windowing, also due to the variation with wavelength of the diffraction image size. This aspect may be further investigated in the future. Figure 7.8: C.o.g. behaviour vs. wavelength for each field position. Black=F0a, purple=f1a, red=f2a, blue=f3a, orange=f4a, green=f5a In figure 7.8 we show the distribution of c.o.g. vs. wavelength for all fields. We remark that the variation with wavelength is significantly smaller than that with the field position. Conclusions We have verified that intrinsic optical variation is acceptably small and that it features a reasonable distribution. We show that the focal plane resolution of 2µm, while adequate for the optical PSF construction, requires resampling to higher resolution for proper description of detector and operation aspects. The wavelength dependence of the PSF is still affected by minor artefacts, which will be further analysed in future, and are considered negligible in the context of realistic polychromatic images.

129 7.4 Astrometric performance of Baseline configuration Astrometric performance of Baseline configuration We derive the elementary exposure precision σ el from the COG estimator (section 4.2.2) and we are taken in account only the random error contribution. We retain only the 4µas residual systematic floor at the end of mission. The term is added in quadrature. The final astrometric accuracy is obtained scaling σ el with suitable factors (section 4.1.5). Figure 7.9: Astrometric performance of Baseline configuration for the three spectral types B3V, G2V, M8V. 7.5 Focal Plane to Sky Mapping The astro optical response is encoded by the Focal Plane to Sky Mapping (FPSM) function. For an ideal telescope there is a linear relation between the angular coordinate on the sky and the linear coordinate on focal plane, with the proportional factor given by the focal length value F. In the ideal case the sequence of measurements of a star during its transit on the FP generate a linear trajectory due to the fact that the photo-center positions are equally spaced. Instead, a realistic instrument present a non linear relation due to aberrations. The simplest case is the classical distortion

130 124 Numerical model implementation and validation that produces a variable geometric contraction or expansion over the FP. Due to the off-axis nature of the Gaia telescope, the displacement from the ideal position is particularly evident at the field corners as showed in figure 7.10, where the FPSM map for a B3V spectral type is showed. Over most of the Field, the response is close to the ideal telescope, however the effect is large at µas level, becoming relevant at the level of the reduction operation phase. We have to take in account the real star trajectory given by the FPSM map to select the read-out windows following the star during its transit. In addition to the effect already described the blurring of image due to the mismatch between the TDI rate and the scanning speed is also present. An accurate description of the FPSM is in chapter 6, its measurement on science data and its maintenance are described in the next section Figure 7.10: FPSM map for a B3V spectral type The Baseline FPSM The Gaia measurement model is used within this framework for the purpose of identifying critical instrument and operation parameters and evaluating methods for their measurement. We discuss the sensitivity of simple data analysis

131 7.5 Focal Plane to Sky Mapping x = deg x = deg x = deg FPSM [mas] Along scan [degrees] Figure 7.11: FPSM: trajectories of stars at different across scan positions. procedures to several instrument parameters, which can be either disentangled from each other or estimated as collective contributions in case of degeneration. As we know (see section 6.1 and 6.2), the separation of two stars is estimated on the FP by the photo-centre difference deduced from their images. With an ideal optical system, described by the effective focal length (EFL) F only, and an ideal detector, the relationship between linear coordinate y on the FP and angular coordinate η on the sky is simply linear: y = η F (gnomonic projection). The nominal EFL (46.67 m) corresponds to an optical scale s = 1/EF L 4 /mm. Due to the aberrations of a realistic instrument, the true function is no longer linear and the angular position on the sky is affected by a displacement vs. the ideal position, depending on the focal plane position. The simplest case of aberration influencing the image position is the classical distortion, inducing a variable contraction (or expansion) over the field. It is convenient to use the image position discrepancy with respect to geometric optics, since its values are comparably small. The sequence of measurements of a star, during its transit, is not equally spaced, but displaced accordingly to FPSM. It can be described as a trajectory, associated to the stellar transit, describing the variations with respect to the ideal case (F P SM = 0). A representation of the trajectories associated to the baseline design, with nominal values for all parameters, is shown in figure The cases considered are the extreme borders and the central section of the FP, in the across scan direction x; the lower edge (x = 0.20) is closest to zero, the discrepancy increases slowly

132 126 Numerical model implementation and validation in the lower half of the field, then the variation is remarkably steeper up to the upper edge (x = 0.86). Here lower values of the across scan coordinate are closer to the optical axis; using an off-axis configuration for allocation in the satellite, an off-axis field is imposed by vignetting and optical optimisation reasons. The sequence of observations of each star measures the actual FPSM, since the photo-centres are derived according to the current signal profile, for each image, corresponding to the system configuration. Basically, we take advantage from the fact that stars do not move during the transit level observation; therefore, the measured changes are only due to the instrument. Any star provides a set of photo-centre values, with magnitude depending precision; from the above considerations, stars brighter than V = 15 mag have sub-mas precision at the elementary exposure level; even at the faint limiting magnitude (V = 20 mag), the transit level precision is quite comparable with the minimum signature, close to the optical axis. The desired precision on FPSM monitoring, at the µas level, can be achieved by averaging over a sufficient number of bright objects; of course, this requires a calibration framework able to factor out the different contributions, e.g. chromaticity, TDI, spin rate, etc. A significant number of bright sources is in any case required, to cover the whole field of view in each line of sight Differential FPSM: FoV discrimination Our definition of the BA is the angle on the sky separating the two directions associated to a given FP point, through the two telescopes, measured along the scan direction. Besides, the two telescopes have different FPSM, due to the across scan offset corresponding to one CCD strip, so that a given FP point is projected on the sky with different deviations from gnomonic representation by the two instruments, even in the nominal design. Such difference, in the common part of the FP, is shown in figure 7.12; as for the FPSM, the values are quite small in the part of the field closer to the optical axis, and rapidly increasing at larger distance. Additional differences are induced by the unavoidable individual errors associated to manufacturing, on-ground integration, and in-flight re-alignment. This could be represented as a BA variation, carrying the contribution from one term to the other in the equation η = y/ef L + F P SM ± BA/2. The contribution of optics and system can be separated, in principle, since the

133 7.5 Focal Plane to Sky Mapping 127 field dependent effect described above is purely optical, whereas the BA can be considered as a global instrument parameter, which could change e.g. by moving vs. each other two unperturbed telescopes. 40 Differential FPSM [mas] Across [degrees] Along [degrees] 1 Figure 7.12: Differential FPSM distribution between ASTRO1 and AS- TRO2. The differential FPSM does have an impact on operations, since the trajectories of stars from either field are slightly different from each other; three sections of the differential FPSM, corresponding to the extreme values of the superposed FP area and its central part, are shown in figure The peak value, in the FP region at large off-axis distance (dotted line), is close to ±1 pixel. As for the FPSM itself, the effect is measurable, since each object will provide signals centred accordingly to its own FPSM, including field signature: the trajectory difference of stars from either field is thus directly derived. Averaging the measurement over several stars, the necessary precision and field coverage can be achieved. The on-board representation must be maintained to preserve operation, i.e. read the appropriate pixels for each object. A large fraction of stars are observed in both fields within a temporal interval of less than two hours (about 100 minutes), and they are expected to retain

134 128 Numerical model implementation and validation FPSM [mas] x = deg x = deg x = deg Along scan [degrees] Figure 7.13: Differential trajectories of stars from ASTRO1 and ASTRO2, at different across scan positions. the same position on sky over this brief period. The difference of average transit coordinates from the leading to the trailing field provides a combined measurement of the BA plus additional global contributions, i.e. scan rate and residual optical terms from each telescope. Assuming the individual field parameters are monitored by suitable procedures, the BA could be uniquely identified. In order to factor out the spin rate (attitude), integrated over the two hour time elapse between transit level measurements, it is necessary to to retain the 1 µas precision in attitude reconstruction on-ground over this time scale.

135 Chapter 8 Chromaticity Chromatism is usually defined for refractive optics as an aberration due to the light dispersion of the glass with refractive index; this induce a perturbation on the image profile, depending upon the source colour. In astronomy, it provides a variation of the apparent star position, i.e. an astrometric error. This effect can be reduced by using more complex refractive systems (doublets, triplets, etc) taking advantage of the different dispersion of glasses to achieve a certain degree of compensation. For most purpose the only, and reflective telescopes are assumed to be free from it. We show that all-reflective optics still bears significant levels of chromatic effects, potentially critical to modern micro-arcsecond astrometric experiments. We analyse the image formation and measurement process, to derive a precise definition of the chromatic variation of the image position, and we evaluate the critical aspects of optical design with respect to chromaticity. The key requirement related to chromaticity is the symmetry of the optical design and of the wavefront errors. In this chapter, we show that, at the demanding level of modern astrometry, reflective optics is still affected by significant chromatic aberrations, which we will refer to as chromaticity. Hipparcos was already affected by chromaticity at the mas level, but this was not entirely surprising due to the fact that it had a refractive component in its modified Schmidt configuration. It was hoped that Gaia might feature a chromatic-free response thanks to full reflective optics, but during the design phase evidences build up that this was not the case (Lattanzi et al., 1998). We will describe how the aberrations of a realistic optical system may induce chromaticity, due to the diffraction of the light, in spite of the propagation in a dispersive medium as vacuum. Due to the larger size of the optics, the peak

136 130 Chromaticity chromaticity for the Gaia optical design is still at mas level, so that it should be reduced by two or three order of magnitude to reach the µas level accuracy. 8.1 Astrometric Chromaticity We define chromaticity the colour-dependent position variation explained in the next. Respect to what we can think, different spectral type stars, set in the same nominal position, do not have the same estimated position, even with the simplest possible algorithm; i.e. the same location on the sky is not uniquely mapped on the focal plane. It is important to remove this systematic error, dependent on target spectral type and position on the focal plane, to preserve the desired mission accuracy. The PSF is asymmetric because of aberrations, modifying the energy distribution. Hereafter we investigate the different aberration contributions, addressing the most common expansion sets in optical design. The purpose is to derive guidelines for minimisation of chromaticity in optical design, manufacturing and alignment. 8.2 Potential Sources of Chromaticity Aberrations of Real Optical Configurations Dealing with a real system, the effect of aberrations cannot be neglected, and this is usually wavelength dependent. The design of an optical configuration is optimised over a specified passband, in order to achieve a sufficient image quality, defined by means of appropriate thresholds of suitable parameters (e.g. spot diagram diameter, encircled energy). For geometric reasons, most optical systems are closer to the ideal case in a region surrounding the optical axis; further on, aberrations arise in non-linear way, e.g. as a power of the angular distance from the axis. The FOV is defined as the region where image quality is preserved, accordingly to the aforementioned quality figures, in the system band. We note that the shape of the PSF changes along the field, in particular assuming an asymmetric distribution. As shown hereafter, this provides the most relevant chromatic effect. The spectral type of the targets can be represented in terms of a different equivalent wavelength; in case of an asymmetric polychromatic PSF, the ge-

137 8.3 Baseline chromaticity 131 ometrical scaling with λ results in an overall chromatic error, with variations of the apparent star position depending on the target colour, i.e. introducing a systematic term not dependent on photon statistics. Since accurate image location resorts on the exploitation of the photon distribution, different algorithms may be affected to a different degree. The problem can be assessed by means of analysis of the PSFs produced by the nominal optical configuration for different spectral type sources Wavefront Errors related to Optical Quality Real optical components feature surface defects due to manufacturing which provide distortions in the propagated wavefront. This, in turn, provides a degraded image quality because a fraction of the incoming photons is diffracted. In principle, because of the random nature of the surface defects, this increases stray-light but should not modify the photocenter position, since on average most of the flux is distributed accordingly to the nominal PSF. However, surface quality enters the aberration budget, and therefore contributes to the chromatic effects buildup. Surface polishing is defined in terms of a fraction of the operating wavelength, λ/n, therefore the optical configuration can be expected to induce smaller wavefront errors on targets featuring a spectrum peaked at a longer wavelength. This might be investigated by evaluation of PSFs produced by ray tracing evaluation of configurations with specified surface quality. 8.3 Baseline chromaticity Different spectral type stars, set in the same nominal position, do not have the same estimated position; the same location on the sky is not uniquely mapped on the focal plane. We define chromaticity this colour dependent position variation. It is important to remove this systematic error to preserve the desired mission accuracy. The chromaticity has been evaluated using the realistic blackbody spectrum associated to the source, either B3V or M8V, deriving the COG and defining the chromaticity as the difference between The on-board representation must be maintained to p the B3V and M8V COG. Here only the nominal aberrations are considered, manufacturing and alignment errors are not yet included.

138 132 Chromaticity In the figure 8.1 we show the derived chromaticity map. As we can see, the distribution is symmetric with respect to the y axis due to the symmetry of the optical configuration; the mean chromaticity value is µas with RMS value 1.19 mas and a peak values higher than ±2mas. At this level, the compensation of opposite errors along the trajectory provides a reduction of the overall chromaticity. Figure 8.1: Chromaticity distribution over the astrometric field in the nominal configuration of the Gaia instrument. 8.4 Chromatic astrometric error Using the optical representation of the baseline, we derive the images of two stars, respectively of spectral type B3V and M8V, for the field position 0.33 (along scan) and 0.86 (across scan); they are placed in the same position on the sky and it would be desirable to have the same position estimate on the conjugate plane imaged on the detector. The two one-dimensional optical signals are shown in figure 8.2 (left), normalised to the peak; it is apparent that the M8V image (dotted line) is not only a geometrically scaled version of the B3V case (solid line). The different spectral content weighs the aberrations from optical design, and the diffraction image is also affected by profile variations, in addition to the magnification associated to the intrinsic factor λ/d. The signal difference is shown in figure 8.2 (right): it is not symmetric, as would be the case of ideal images, and reaches 15% of the initial peak value. The coordinates system is fixed so that the barycentre of a not aberrated PSF has abscissa equal to 0; the large common mode displacement is due mostly to

139 8.4 Chromatic astrometric error B3V M8V Relative intensity [%] Relative intensity [%] Along scan position [µm] Along scan position [µm] Figure 8.2: Left: PSF for B3V (solid line) and M8V (dotted line) stars in the field position [0.33; 0.86]; right: image difference classical distortion. Herein, the photo-centre estimate is mostly based COG definition (section 4.2.2). It can be shown that different algorithms are affected by similar chromatic effects; however, due to the effective image profile of the Gaia telescope, the performance difference is small (few percent), so that we privilege the simplicity of COG [5]. The barycentre of the two signals is respectively µm (B3V) and µm (M8V), with a difference of 57 nm; given the optical scale of Gaia, about 4 /mm, this results in more than 200 µas, i.e. comparable with the random location error for stars brighter than V = 15 mag. Different spectral type stars, set in the same nominal position, do not have the same estimated position, even with the simplest possible algorithm; i.e., the same location on the sky is not uniquely mapped on the focal plane. We define chromaticity this colour-dependent position variation. It is important to remove this systematic error, dependent on target spectral type and position in the focal plane, to preserve the desired mission accuracy. The PSF is asymmetric because of aberrations, modifying the energy distribution. Hereafter, we investigate the different aberration contributions, addressing the most common expansion sets in optical design. The purpose is to derive guidelines for minimisation of chromaticity in optical design, manufacturing and alignment.

140 134 Chromaticity Term Functional Form Term Functional Form (4ρ 4 3ρ 2 )cos(2θ) 2 ρcos(θ) 13 (4ρ 4 4ρ 2 )sin(2θ) 3 ρsin(θ) 14 (10ρ 5 12ρ 3 + 3ρ)cos(θ) 4 2ρ (10ρ 5 12ρ 3 + 3ρ)sin(θ) 5 ρ 2 os(2θ) 16 20ρ 6 30ρ ρ ρ 2 sin(2θ) 17 ρ 4 cos(4θ) 7 (3ρ 3 2ρ)cos(θ) 18 ρ 4 sin(4θ) 8 (3ρ 3 2ρ)sin(θ) 19 (5ρ 5 4ρ 3 )cos(3θ) 9 6ρ 4 6ρ (5ρ 5 4ρ 3 )sin(3θ) 10 ρ 3 cos(3θ) 21 15ρ 6 20ρ 4 + 6ρ 2 )cos(2θ) 11 ρ 3 sin(3θ) Table 8.1: Functional form of the 21 Fringe Zernike terms WFE expansion: identification of the critical terms The source of chromatic errors is the WFE, which is present even in a nominal optical design with a finite field of view; manufacturing and alignment may only aggravate the problem. Besides, not any WFE contribution is associated to the same chromaticity: below, we analyse the effect of individual aberrations, and the related symmetry properties. We then investigate the minimisation by design of the chromatic errors in any given field position, analysing the effect of partial or total suppression of selected aberration terms from a realistic PSF. Finally, we proceed to evaluate the overall field properties of chromaticity, verifying the possible compensation at transit level. Typical expansions of the WFE, in optical engineering, are the Zernike or Fringe Zernike functions (see Table 8.1), i.e. a set of circular functions, orthogonal and normalised on a circular pupil, expressed in terms of trigonometric functions of the angular coordinate and polynomials of the radial coordinate [21]. The rappresentation in Zernike polynomials is not optimal due to the rectangular pupil of Gaia; more convenient expansions are being studied [22]. The first analysis is performed on the Standard Zernike polynomial wavefront expansion (see Table 7.1 in section 7.2), with the goal of identify the main individual contributors and possible correlations among RMS WFE, image RMS width and chromaticity. The chromaticity is evaluated by the barycentre separation, either using monochromatic PSFs at the reference wavelengths associated to spectral types B3V (blue) and M8V (red), or using the polychromatic spectrum of the same

141 8.4 Chromatic astrometric error 135 Term RMS WFE [nm] Image RMS COG Chromaticity [µas] width increase [%] [mas] non-aberrated Table 8.2: Effects of individual Zernike terms sources, modelled as blackbody emission. The latter description is more precise from the astrophysical standpoint, whereas the former has lower computational requirements, but the results are qualitatively in agreement for both cases. The image RMS width is derived on the monochromatic PSF at λ = 700 nm (roughly representative of solar type objects); the image COG displacement is in any case referred to the ideal, non-aberrated case, which has an image RMS width of µm. Each Zernike term is individually evaluated, with coefficient = 0.1 (i.e. in small aberrations regime): the WFE is built from the selected term, then the PSF is computed accordingly to the above model. The results for the first 21 terms are listed in Table 8.2. In figure 8.3, we show the impact of each individual aberration respectively on the average COG (i.e. an effect corresponding to classical distortion) and on chromaticity. Some terms have a strong contribution to both, but in general, there is little correlation. Similarly, we derive that there is no simple

142 136 Chromaticity relationship between RMS WFE, or image RMS width, both including the overall contribution of all aberrations, and chromaticity. The complex relation between WFE, image RMS width and chromaticity can be understood in terms of symmetry, since the aberrations in general are bidimensional, whereas the parameters relevant to the GAIA measurement are mostly one-dimensional, i.e. referred to the scan direction. Thus, specific aberration terms may contribute significantly to the WFE, across-scan image width and across-scan centroid displacement, with small impact on the location noise and chromaticity, due to the across scan image binning. There is a trend of increasing image RMS width with RMS WFE, but this is not a strict relation, and several cases of large WFE and small image width degradation are evidenced. This could be associated to aberrations inducing significant image degradation in the across scan direction only. The centre of gravity, in some cases, is affected by a large displacement, associated to low chromaticity values. Thus, the images are translated with respect to the non-aberrated position by an amount not depending upon the spectral distribution of the source. The case of classical distortion may fit in this description. From figure 8.3, we see that significant contributions to chromatic- Centre of gravity [mas] Aberration # Chromaticity [µas] Aberration # (a) Image COG variation vs. aberration terms (b) Chromaticity variation vs. aberration terms Figure 8.3: ity, i.e. of order of 100 µas or larger, are all and only provided by aberrations (n. 3, 6, etc.) with a specific functional form: they are all odd (sinusoidal) functions of the angular coordinate, so that they have odd symmetry on the pupil plane. All even (cosinusoidal) terms do not provide net chromaticity;

143 8.5 Compensation by aberration selection 137 Case RMS WFE Image RMS Chromaticity [nm] width [mas] [mas] Table 8.3: Aberration selection residual few ten µas values are probably associated to the limited precision of the numeric implementation. A similar analysis has been carried on with the Fringe Zernike functions. Again, the critical terms for chromaticity are identified as those associated to odd parity of the WFE, i.e. sinusoidal functions, as for the standard Zernike expansion. The set of terms which must be minimised, by design, manufacturing, and alignment, have been identified as those corresponding to odd parity with respect to the along scan axis. Both in Standard as in Fringe Zernike functions, the anti-symmetric terms correspond to sinusoidal functions. 8.5 Compensation by aberration selection The test objective is the verification of chromaticity in a realistic case, with many aberration terms present. The analysis is referred to the field position F4, with coordinates 0.20 (along scan) 0.33 (across scan), which is affected by a significant value of chromaticity. For ease of computation, the chromaticity is evaluated as difference between the monochromatic PSF at 628 nm and 756 nm respectively. The chromaticity derived in the fully polychromatic model is about 2 mas, i.e. larger by a factor 2. Six cases are considered: 1. Non-aberrated (ideal) PSF; 2. Nominal case for field F4; 3. Removal of all symmetric terms;

144 138 Chromaticity Figure 8.4: PSF from aberration selection 4. Removal of all anti-symmetric terms; 5. Removal of anti-symmetric terms AND scaling of symmetric terms to equivalent RMS WFE; 6. Subset of anti-symmetric terms 9 and 10. and the corresponding values of RMS WFE, image RMS width, and chromaticity are listed in Table 8.2. Some of the cases are shown in figure 8.4, represented as central sections of the monochromatic PSF at the reference wavelength of 700 nm. The nominal image (solid line) is quite close to the diffraction limit (dotted line), and the case restricted to only symmetric aberration (dashed line) is even better. Case 6 is shown by the dash-dot line. The ideal PSF provides zero RMS WFE and zero chromaticity. The nominal F4 case provides an image RMS width reasonably close to the diffraction limit, in spite of non negligible RMS WFE (40 nm, i.e. λ/15 at λ= 600 nm); the chromaticity is about 1 mas. Suppression of the symmetric aberrations provides some improvement to the RMS WFE and marginal to image width, but the initial chromaticity is mostly retained. Conversely, removing

145 8.5 Compensation by aberration selection 139 completely the anti-symmetric aberrations (case 4), we achieve an improvement significant on the RMS WFE and limited on the image RMS width, as in the previous case; the chromaticity, however, is reduced to zero, accordingly to expectations. The RMS sum of WFE in cases 3 and 4 restores the nominal value: the mutual orthogonality of symmetric and anti-symmetric function sets is preserved, even if this is no longer true for the individual functions within each set. Even when the symmetric aberrations are scaled to restore the initial WFE level of 40 nm (case 5), the chromaticity is still zero. Besides, when a random subset of anti-symmetric aberrations (9 and 10) is retained, together with the symmetric terms, with the nominal coefficients (case 6), the result is a small improvement in chromaticity, together with a drammatic degradation of WFE and image quality. This shows that the optimisation procedure of the ray tracing code actually achieves some partial compensation (balancing) among different aberrations, with benefit to chromaticity. This is due to the standard optical design optimisation procedures used by ray tracing packages, based on improvement of general image quality parameters as WFE and spot diagram. Since aberration balancing also contributes to WFE minimisation, an optical configuration with good image quality on a selected field is likely to have comparably low chromaticity. However, local values up to several mas are possible, with severe consequences on microarcsecond astrometry. It is possible to introduce custom merit functions in the optimisation procedure of a ray tracing code, which can include computation of the chromaticity or of the critical anti-symmetric contribution to WFE, with some averaging rule over the field. This does not modify necessarily the actual optical configuration in a significant way, since it is convenient to start in any case after standard optimisation, but chromatic aberrations can be further reduced at the expense of the others, and possibly of a small increase in overall WFE, which in many cases is acceptable. The map of chromaticity over the FOV is derived from the optical representation of the baseline configuration of the astrometric payload. Only the nominal design aberrations are considered; manufacturing and alignment errors are not yet included. The result can therefore be considered as a best case. The analysis is performed using monochromatic PSFs in the set of positions listed in Table 8.4, listing also the associated RMS WFE in nanometres (nm), the RMS image size L RMS for the two cases of blue (B3V) and red (M8V) source, and the chromaticity. The values have been interpolated to cover the field with 0.02 resolution, and shown in figure 8.5. The distribution is

146 140 Chromaticity Field position RMS WFE L RMS B3V L RMS M8V Chromaticity [nm] [mas] [mas] [mas] F F F F F F F F F F F F F Table 8.4: Chromatic distribution over the field symmetric with respect to the y axis, due to the intrinsic symmetry of the optical configuration: the mean chromaticity value is 0.05µas, with RMS value 1.19mas and peak values exceeding ±3mas, i.e. three order of magnitude larger than the Gaia measurement goal Transit level compensation The symmetry of chromaticity distribution can be exploited to reduce the overall contribution to a set of measurements, at transit-level, in spite of comparably large local values. We remind that all targets detected by Gaia are observed in TDI mode throughout the whole FP, so that in particular the measurements are performed in opposite positions with respect to the symmetry axis; by composition of photo-centre values from exposures in symmetric along scan positions, the residual chromaticity cumulated over a transit drops to values quite close to zero, The average value of transit chromaticity, over the astrometric field, is 0.10µas, with RMS value 0.21µas. It appears that chromaticity compensation over the transit, in the nominal configuration, is quite effective.

147 8.5 Compensation by aberration selection Chromaticity [µas] Across [degrees] Along [degrees] Figure 8.5: Chromaticity distribution over the astrometric field in the nominal configuration of the Gaia astrometric payload Partial compensation on misaligned systems The real configuration used in flight will not retain the design symmetry, due to manufacturing, integration and re-alignment errors. In order to model the effect, the chromaticity map derived for the nominal configuration is simply shifted by an amount corresponding to the selected offset, in the range from one arcsecond to 0.5 arcminute (a fairly large alignment error). The local chromaticity value is mostly preserved, but the transit level combination, which is very close to zero in the nominal (symmetric) case, is affected by a degradation increasing with the error, as shown in figure 8.6. The statistics of transit-level chromaticity across the FP is shown in Table 8.5. Column 1 lists the offset applied (zero corresponds to the nominal case); in column 2 and 3 we show respectively the average and standard deviation, computed across the field, of transit-level chromaticity. The transit-level chromaticity remains very close to zero in an across scan position of about 0.55, which appears to be a chromatic-free section of the field; the residual has opposite signs on either side of this position. This is due to the structure of the local chromaticity distribution (figure 8.5), with alternate signs in each quadrant. The result is that stars of a given spectral type in different regions of the field have nearly opposite residual chromaticity. This aspect may be exploited for further reduction of the residual chromaticity in the data reduction phase. Upon definition of a threshold of acceptable residual chromaticity, at transit

148 142 Chromaticity Offset Chromaticity Mean Chromaticity RMS arcsec µas µas Table 8.5: Transit-averaged chromaticity as a function of re-alignment error. Field compensated chromaticity [ µas] Across [degrees] Figure 8.6: Transit level chromaticity vs. across scan field in the nominal case and for increasing alignment error. level, it is possible to provide a specification for alignment, both referred to initial telescope integration, and above all to in-orbit re-alignment Optical engineering aspects Symmetry considerations similar to those referred to the individual aberration terms hold for the individual optical components, in particular for their figuring error in the manufacturing phase. Also, the above discussion on alignment, which in the above form is applicable directly to a misalignment of the primary mirror, can be used with reasonable modifications to other cases of perturbed optical configurations. Manufacturing errors inducing an anti-symmetric residual WFE on a telescope mirror result in increased chromaticity; also, displacement from the nominal position, following the projected along scan direction,

149 8.6 Conclusions 143 provide similar impact on the chromatic effect. The real on-flight configuration is thus affected by manufacturing and alignment errors, which can also be described in terms of symmetric and antisymmetric aberrations. If the design is optimised in terms of low chromaticity, small perturbations due to manufacturing and alignment are likely to induce comparably low increase of the chromatic errors, due to larger tolerances. With the configuration considered as representative for Gaia, the relevant values of residual transit chromaticity are of order of few arcseconds at the level of the primary mirror, which is challenging for standard optical techniques. However, it is not necessary to perform re-alignment of each optical component: as for many astronomical instruments, when the errors are not too large, a global re-optimisation can be achieved by adjustment of a small number of degrees of freedom, often localised on a single component (e.g. the secondary mirror). The correction requirements in terms of stroke and resolution are within the range of the actuators considered for Gaia; the main limiting factor is the diagnostics capability, which in any case will benefit from an accurate analysis of the image properties over the field. Quantitative analysis may be performed on specific optical configurations, including manufacturing and alignment aspects based on realistic WFE data from manufacturers. It is thus possible to achieve also some compensation of the in-flight configuration, balancing part the manufacturing and alignment errors with an adjustment of the final alignment. The resulting minimisation of the instrumental chromaticity is an improvement to the overall systematic error budget, which is desirable because the correction procedures based on the science data necessarily have limited performance, and in some way subtract information from astronomy. 8.6 Conclusions We discuss the issue of chromaticity in all-reflective optical systems: in spite of common misconception, avoiding refractive components is not sufficient to achieve an achromatic instrument response, due to basic diffraction considerations. In the scenario of future space mission, aiming at microarcsecond astrometry, the impact of systematic errors two or three orders of magnitude larger than the measurement goal is of fundamental importance. Specific analyses have been performed with reference to the baseline configu-

150 144 Chromaticity ration of the Gaia astrometric payload, but the assumptions, principles and conclusions of our discussion can be applied to any high accuracy astrometric instrument. The chromatic error is defined as the difference in image photo-centre location at different wavelengths, and exact value depends on the selected centring algorithm, but the effect is unavoidable. Independently from the selected WFE expansion (standard Zernike or Fringe Zernike representation), the terms associated to relevant chromaticity are those associated with anti-symmetry of the PSF in the FP, and with anti-symmetric WFE contributions on the pupil plane. Symmetric terms only contribute to the astrometric noise, by increasing the effective image width in the measurement direction. The relation between RMS WFE and chromaticity is complex, since aberrations inducing mainly across scan effects provide negligible impact on the measurement. The first prescription to optical manufacturers is to suppress or at least minimise the anti-symmetric terms. However, it is not necessary to set to zero all chromatic terms: an appropriate combination is still able to provide some local balancing. Standard optical design optimisation techniques are able to provide reasonable results by application of conventional image quality merit functions; optimal results from the standpoint of chromaticity, of course, require definition of ad hoc criteria. The distribution of chromaticity over the field derives some symmetry properties from those intrinsic in the optical system. Deviations from symmetry are induced e.g. by manufacturing and alignment errors on each optical component, which can be kept to reasonable values by smart usage of standard tolerancing techniques. In case of repeated measurements in different parts of the field, as for Gaia, some chromaticity compensation is achieved in the data combination, depending on the symmetry of both instrument and measurement scheme. Each object observed by a symmetric configuration of the Gaia astrometric payload (i.e. correctly re-aligned) along a full transit provides a set of astrometric measurements affected by opposite chromatic errors in symmetric positions; transit-level composition is therefore likely to remove a large fraction of the local chromaticity. The residual chromaticity must be removed in the science data processing; after the best implementation of the astrometric payload, to minimise the initial systematic error. This requires spectral information on each source, provided for Gaia by the photometric and spec-

151 8.6 Conclusions 145 trometric payloads, and a good knowledge of the detailed instrument response. Part of our future work will be devoted to implementation of techniques for diagnostics of the detailed optical performance of the astrometric instrument and monitoring of the relevant parameters, including the chromaticity map.

152 146 Chromaticity

153 Chapter 9 Gaia alternative configuration and engineering aspects 9.1 Payload concept The Gaia alternative configuration has been proposed by the industrial team of Alenia Spazio and Alcatel Space, including the INAF-Osservatorio di Torino as scientific consultant, in the framework of the ESTEC contract Gaia System Level Technical Assistance and Design Study. The rationale for development of a new payload concept arises from the initial studies on the detailed implementation by this team of the baseline configuration concept, which was considered exceedingly sensitive to thermo-elastic perturbations. The issue is also discussed in the literature [8]. The key point is that the power fluctuations on the focal plane electronics, plus any additional contributions from the satellite sub-systems and the residual illumination variation associated to the spin period and due to imperfections on the sun-shield, induce local deformations on the toroidal optical bench, on a time scale of several minutes, i.e. much shorter than the spin period. Such perturbations are propagated to the optical system in an anti-correlated way, i.e. the corresponding individual mirrors (primary, secondary and tertiary) are affected by anti-symmetric perturbations due to their specular position with respect to the detector. Any change in the telescope geometry induces an effect on the optical response, which for small perturbations corresponds to a shift of the focal plane position of all observed objects. Therefore, the resulting displacement of focal plane image from each field is also anti-correlated, i.e. acting in opposite directions, so that the optical pertur-

154 148 Gaia alternative configuration and engineering aspects bation directly affects the astrometric measurement of the angular separation between PSFs, i.e. a noise on the BA, amplified by a factor two. The estimated amplitude of the power fluctuations, in conservative realistic conditions, propagated to the elementary measurement through the overall payload model, corresponds to a systematic error of order of several hundred µas, i.e. beyond the mission objectives. The estimate may be pessimistic, since the performance can be improved by implementation of a number of technical solutions in the detailed design, and the error may be measured during the mission by the BAM for subsequent removal in the data reduction. Besides, this means that the BAM becomes an unavoidable element of the payload, i.e. a single point failure : any fault of the BAM results in severe degradation of the Gaia measurement performance. Increasing the stiffness of the baseline is a potential solution, but it has dramatic impact on the mass budget, since it is necessary to strengthen the whole optical bench. Therefore, the baseline configuration was not considered as sufficiently reliable from an engineering standpoint. The alternative concept proposed retains several of the characteristics developed during the study of the baseline configuration: the mission profile (orbit, scan law, lifetime, etc.) is retained; the collecting aperture and effective focal length are the same; the launcher, satellite and sun-shield are not modified. Therefore, most of the simulations on sky coverage, data reduction, and signal level from astrophysical sources, retain their validity. Moreover, the focal plane structure is exactly the same, and as well as its operation (although some implementation changes may be convenient). Then, assuming a reasonable imaging performance from the telescope, the detected signals and their performance is quite similar to the baseline case. In particular, the simulator discussed as key part of the present work can easily be adapted to the alternative configuration, with minor changes, and the results on error budget and calibration strategy derived in a straightforward way. The alternative payload concept is based on a different telescope configuration, building upon the idea successfully adopted by the Gaia predecessor, Hipparcos: a single common telescope, observing simultaneously with the same detector the objects from two fields of view, superposed on the same focal plane by a beam combiner in front of the optical system.

155 9.1 Payload concept 149 Figure 9.1: The overall optical path of the astrometric telescope. Figure 9.2: Schematic figure of Gaia alternative Payload.

156 150 Gaia alternative configuration and engineering aspects The BC is a set of two flat mirrors, mounted at a suitable angle of 127 with respect to each other and to the telescope axis; the line of sight is folded over two directions on the sky (LOS1, LOS2), corresponding to the desired basic angle of 106. This case can still be represented as two different telescopes, T1 and T2, since two different off-axis regions of the overall entrance pupil are used, so that the optical response is different, at the sensitivity corresponding to the Gaia astrometric measurement. The combination is based on pupil division, i.e. the collecting area of the telescope is split in two areas, each fed by one blade of the BC, and observing on its own line of sight. The two m apertures of the baseline configuration are replaced by a single m aperture of the common telescope of the alternative concept. Since also the size of the BC flat mirrors is of order of m (actually larger due to projection reasons in the selected geometry), there is a significant impact on instrument mass, since the alternative configuration is more massive than the baseline. The performance with respect to random errors, by construction, is quite comparable to the baseline. The main improvement consists in the control of systematic errors, which in this case, accordingly to engineering evaluations, is reliably compatible with the desired µas mission goal. The thermal input associated to internal power fluctuations and variable solar irradiation are not changed, as well as the basic properties of the optical bench (conduction and elastic deformation). However, given the proposed payload structure, the structural perturbations act on a telescope which is used by both LOSs; discrepancies are introduced only at higher order, and for a subset of the mechanical degrees of freedom, due to the different regions of each mirror used by the two beams. The BC is the only part of the payload which appears in an anti-symmetric geometry with respect to the detector, due to the need of merging the two observing directions. However, its impact can be much smaller than in the baseline case: (a) the BC size is significantly smaller than the whole payload; (b) it is therefore affected by a much lower thermal gradient; (c) only few degrees of freedom of the BC are relevant; (d) the BC can be supported by an high rigidity structure;

157 9.1 Payload concept 151 (e) the geometry of the BC is very simple and can easily be monitored. The results have been obtained by a detailed sensitivity analysis from the proposing team, validated by ESTEC. The sensitivity of individual telescopes and of the BC case was verified by direct computation, in a simple geometric case, and from basic principles, in previous literature contributions [23]. Accordingly to current estimates, the astrometric systematic error associated to residual BA fluctuations is 1 2µas, i.e. fully compatible with the most ambitious scientific goals of Gaia. In this alternative payload configuration, the need for BA monitoring is significantly relaxed, since the expected perturbations are within the mission requirements. However, due to the challenging linear and angular values involved, it is commonly felt in the Gaia Science Team that the mission reliability is significantly improved by the presence of an independent verification of the BA stability throughout operation. The BAM should in any case provide a backup to the consolidation of the astrometric measurement in case of unexpected disturbances or failures. The baseline BAM concept is potentially applicable to the alternative configuration, but a significant modification of its geometry is required. Besides, due to the simple BC geometry, a promising alternative solution is considered, based on the results of another ESTEC contract for technological development, Laser Metrology and Active Optics Control, carried on by a team lead by Alenia Spazio and including the INAF-Osservatorio di Torino and the Istituto Metrologico G. Colonnetti of the CNR. The activity is aimed at production of a space-compatible Fabry-Pérot cavity for laser gauging between fiducial points, with precision of few pm, over distances of few meters. Since all components are solid state, the reliability is considered satisfactory; the only element requiring redundancy is the laser source, with an expected lifetime below the mission operation duration. The application to the BC is straightforward: the angle between two flat surfaces is unambiguously defined, from a mathematical standpoint, by the distance of three reference points. The proposed metrology system, to be used for instrument qualification on ground, and proposed also for on board implementation, is therefore based on installation of three/four Fabry-Pérot cavities between reference positions on the beam combiner.

158 152 Gaia alternative configuration and engineering aspects 9.2 Performance analysis on Gaia alternative Configuration The configuration is described by the data set of the first 21 Zernike fringe coefficients provided by Alcatel for each field position (Table 8.1). The performance in each field position is described by the wavefront error (WFE), decomposed in terms of coefficients of the first 21 Zernike fringe polynomials, accordingly to the table below, provided by Alcatel. The coefficients are provided in waves at 630 nm. The angular field positions associated to the 13 field are listed in the table below, in degrees. X is across scan (low resolution direction), and Y is along scan (high resolution direction). 9.3 FOV discrimination: realization The problem considered is the possibility of correct identification of the appropriate field of view (FOV), either 1 or 2, associated to a target observed by the alternative configuration proposed for Gaia by the Alenia/Alcatel team. The concept investigated below takes advantage of the different optical characteristics of the two telescopes in order to derive the information related to discrimination of the LOS associated to each observed target. The two telescopes T1 and T2 have quite different optical response, so that it appears possible to detect a signature of the individual optical channel on the measured data Requirements for field discrimination Field discrimination from observed data removes the need for field stops to provide a region of the focal plane (FP) fed exclusively by a single line of sight; this feature is foreseen in the baseline configuration, and its implementation in the alternative configuration would require a complication of the current design. It is assumed that the on-board operations are ensured by the standard detection / confirmation functions of the sky mapper (SM): each object is detected and the regions of interest (ROI) corresponding to the read-out windows on the subsequent CCDs are derived by projecting the initial SM position accordingly to the current attitude / scan law and the target intrinsic motion. The longitudinal as well as transversal velocity of the target can be derived

159 9.3 FOV discrimination: realization 153 with magnitude dependent precision from SM images, sampled with reduced resolution with respect to AF but with bidimensional sensitivity, sufficient for correct ROI definition. Field discrimination by across scan velocity estimate in the SM is not an appealing option since there are regions with reduced sensitivity, when the values are similar for both fields. In the circumstances, the appropriate read-out regions are acquired, and the need for field discrimination is associated to on-board attitude determination (scan speed monitoring) and on-ground data reduction. The former can be restricted to a subset of the observed targets from simple SNR considerations. The concept is that subsequent observations of each target may be sufficient to identify if a given target is following the trajectory associated to Astro1 or to Astro2. The 11 exposures of each target provide, in the appropriate ROI, a onedimensional distribution of intensity vs. time; the photo-centre position corresponds to the time of transit of the target at the reference position associated to each CCD, and in geometric optics approximation (ideal optics) the measurements are equally spaced. The deviation from ideal optics are described by the focal plane to sky mapping (FPSM). In the simplest description, the contribution is mostly due to distortion, but in practice the overall effect of several aberrations is introduced, taking into account the detailed image profile (PSF or LSF) in the location process. Distortion remains one of the major contributors to the FPSM. The chromatic differential response of the system, treated elsewhere, is a minor contribution due to aberration sensitivity to source spectral distribution. Different location algorithms are associated to different FPSM. The FPSM is measured by each observed target, since the sidereal motion associated to the satellite scan law is common to all of the objects within one field, but each CCD integrates for a slightly different exposure time. The deviation from the simple geometric model is mapped either in terms of discrepancy from uniform spacing in time or in space, projected by the scan velocity. The read-out ROI should, in principle, be placed accordingly to FPSM and individual trajectory, i.e. corrected vs. the geometric optics position. In practice, this is a marginal issue because the design requirements on distortion are set on purpose to about 0.1 pixel for an exposure at the border of the field. Besides, the image displacement with respect to geometric optics may be up to several mas over the field, i.e. a comparably large amount with respect to the elementary exposure precision.

160 154 Gaia alternative configuration and engineering aspects Model set-up We build the FPSM by deriving the apparent FP position of a set of reference directions in object space, propagated through the optical system. The intermediate positions are derived by interpolation. The local zero corresponds to geometric optics, in which the relation between FP and sky is provided simply by the optical scale (or EFL). The location process used is the centre of gravity (COG) because of its simplicity and model independence. Each object follows a trajectory roughly corresponding to a section of the map, for each telescope, at a given across scan position. The FPSM of T1 and T2 are different, as shown in the figure 9.3 and 9.4 b, inducing different trajectories of the observed objects according to their FOV. Simple data analysis can evidence the difference and discriminate between LOS1 and LOS2. The surface plot is in mas units vs. along/cross scan position in degrees. The FOV position closest to the optical axis (x = 0.34) is expected to have the lowest distortion, representing a worst case with respect to discrimination capability. The trajectory of a star, detected in this position of the SM, respectively from field 1 and field 2, is shown in the figures below. The FPSM is set to zero at the SM, in order to show the position discrepancy with respect to the trajectory predicted by geometric optics. The 11 subsequent exposures provide 11 position estimates, which are positioned accordingly to the trajectory associated to the field appropriate to the target. By fit of the 11 position data to each trajectory, available as tabulated data C(1;x,n=1,..,11),C(2;x,n=1,..,11) respectively for telescope 1 and 2 (as represented in figures 9.5), based on the current configuration parameters, it is possible to select the appropriate field by the minimum discrepancy criterion. For the 11 values associated to a given target,we compute the discrepancy with respect to the T1 and T2 trajectory, respectively, and adopt the simple decision criterion: 1.discrepancy target - T1 > discrepancy target - T2 target from LOS1 2.discrepancy target - T1 < discrepancy target - T2 target from LOS2

161 9.3 FOV discrimination: realization 155 Figure 9.3: FPSM of telescope T1 Figure 9.4: FPSM of telescope T2