A conjugate gradient algorithm for the astrometric core solution of Gaia

The ESA space astrometry mission Gaia, planned to be launched in 2013, has been designed to make angular measurements on a global scale with micro-arcsecond accuracy. A key component of the data processing for Gaia is the astrometric core solution, which must implement an efficient and accurate numerical algorithm to solve the resulting, extremely large least-squares problem. The Astrometric Global Iterative Solution (AGIS) is a framework that allows to implement a range of different iterative solution schemes suitable for a scanning astrometric satellite. In order to find a computationally efficient and numerically accurate iteration scheme for the astrometric solution, compatible with the AGIS framework, we study an adaptation of the classical conjugate gradient (CG) algorithm, and compare it to the so-called simple iteration (SI) scheme that was previously known to converge for this problem, although very slowly. The different schemes are implemented within a software test bed for AGIS known as AGISLab, which allows to define, simulate and study scaled astrometric core solutions. After successful testing in AGISLab, the CG scheme has been implemented also in AGIS. The two algorithms CG and SI eventually converge to identical solutions, to within the numerical noise (of the order of 0.00001 micro-arcsec). These solutions are independent of the starting values (initial star catalogue), and we conclude that they are equivalent to a rigorous least-squares estimation of the astrometric parameters. The CG scheme converges up to a factor four faster than SI in the tested cases, and in particular spatially correlated truncation errors are much more efficiently damped out with the CG scheme.Comment: 24 pages, 16 figures. Accepted for publication in Astronomy & Astrophysic

Gaia astrometry for stars with too few observations - a Bayesian approach

Gaia's astrometric solution aims to determine at least five parameters for each star, together with appropriate estimates of their uncertainties and correlations. This requires at least five distinct observations per star. In the early data reductions the number of observations may be insufficient for a five-parameter solution, and even after the full mission many stars will remain under-observed, including faint stars at the detection limit and transient objects. In such cases it is reasonable to determine only the two position parameters. Their formal uncertainties would however grossly underestimate the actual errors, due to the neglected parallax and proper motion. We aim to develop a recipe to calculate sensible formal uncertainties that can be used in all cases of under-observed stars. Prior information about the typical ranges of stellar parallaxes and proper motions is incorporated in the astrometric solution by means of Bayes' rule. Numerical simulations based on the Gaia Universe Model Snapshot (GUMS) are used to investigate how the prior influences the actual errors and formal uncertainties when different amounts of Gaia observations are available. We develop a criterion for the optimum choice of priors, apply it to a wide range of cases, and derive a global approximation of the optimum prior as a function of magnitude and galactic coordinates. The feasibility of the Bayesian approach is demonstrated through global astrometric solutions of simulated Gaia observations. With an appropriate prior it is possible to derive sensible positions with realistic error estimates for any number of available observations. Even though this recipe works also for well-observed stars it should not be used where a good five-parameter astrometric solution can be obtained without a prior. Parallaxes and proper motions from a solution using priors are always biased and should not be used.Comment: Revised version, accepted 21st of August 2015 for publication in A&

The Hipparcos Transit Data: What, why and how?

The Hipparcos Transit Data are a collection of partially reduced, fully calibrated observations of (mostly) double and multiple stars obtained with the ESA Hipparcos astrometry satellite. The data are publicly available, as part of the CD-ROM set distributed with the Hipparcos and Tycho Catalogues (ESA SP--1200, 1997), for about a third of the Hipparcos Catalogue entries including all confirmed or suspected non-single stars. The Transit Data consist of signal modulation parameters derived from the individual transits of the targets across the Hipparcos focal grid. The Transit Data permit re-reduction of the satellite data for individual objects, using arbitrarily complex object models in which time-variable photometric as well as geometric characteristics may be taken into account. We describe the structure and contents of the Transit Data files and give examples of how the data can be used. Some of the applications use standard astronomical software: Difmap or AIPS for aperture synthesis imaging, and GaussFit for detailed model fitting. Fortran code converting the data into formats suitable for these application programs has been made public in order to encourage and facilitate the use of Hipparcos Transit Data.Comment: A&AS, accepted for publication, 17 pages, 9 figures, 1 Table, Software available via http://www.astro.lu.se/~lennart/TD/index.html, Figures 4, 5, 6 and 7 need to copied separately, A complete postscript file can be found at http://www.astro.lu.se/~lennart/TD/ds1699.ps.g

Impact of basic angle variations on the parallax zero point for a scanning astrometric satellite

Determination of absolute parallaxes by means of a scanning astrometric satellite such as Hipparcos or Gaia relies on the short-term stability of the so-called basic angle between the two viewing directions. Uncalibrated variations of the basic angle may produce systematic errors in the computed parallaxes. We examine the coupling between a global parallax shift and specific variations of the basic angle, namely those related to the satellite attitude with respect to the Sun. The changes in observables produced by small perturbations of the basic angle, attitude, and parallaxes are calculated analytically. We then look for a combination of perturbations that has no net effect on the observables. In the approximation of infinitely small fields of view, it is shown that certain perturbations of the basic angle are observationally indistinguishable from a global shift of the parallaxes. If such perturbations exist, they cannot be calibrated from the astrometric observations but will produce a global parallax bias. Numerical simulations of the astrometric solution, using both direct and iterative methods, confirm this theoretical result. For a given amplitude of the basic angle perturbation, the parallax bias is smaller for a larger basic angle and a larger solar aspect angle. In both these respects Gaia has a more favourable geometry than Hipparcos. In the case of Gaia, internal metrology is used to monitor basic angle variations. Additionally, Gaia has the advantage of detecting numerous quasars, which can be used to verify the parallax zero point.Comment: 8 pages, 2 figures; Accepted for publication in Astronomy & Astrophysic

Joint astrometric solution of Hipparcos and Gaia: A recipe for the Hundred Thousand Proper Motions project

The first release of astrometric data from Gaia is expected in 2016. It will contain the mean stellar positions and magnitudes from the first year of observations. For more than 100 000 stars in common with the Hipparcos Catalogue it will be possible to compute very accurate proper motions due to the time difference of about 24 years between the two missions. This Hundred Thousand Proper Motions (HTPM) project will be part of the first release. Our aim is to investigate how early Gaia data can be optimally combined with information from the Hipparcos Catalogue in order to provide the most accurate and reliable results for HTPM. The Astrometric Global Iterative Solution (AGIS) was developed to compute the astrometric core solution based on the Gaia observations and will be used for all releases of astrometric data from Gaia. We adapt AGIS to process Hipparcos data in addition to Gaia observations, and use simulations to verify and study the joint solution method. For the HTPM stars we predict proper motion accuracies between 14 and 134 muas/yr, depending on stellar magnitude and amount of Gaia data available. Perspective effects will be important for a significant number of HTPM stars, and in order to treat these effects accurately we introduce a scaled model of kinematics. We define a goodness-of-fit statistic which is sensitive to deviations from uniform space motion, caused for example by binaries with periods of 10-50 years. HTPM will significantly improve the proper motions of the Hipparcos Catalogue well before highly accurate Gaia- only results become available. Also, HTPM will allow us to detect long period binary and exoplanetary candidates which would be impossible to detect from Gaia data alone. The full sensitivity will not be reached with the first Gaia release but with subsequent data releases. Therefore HTPM should be repeated when more Gaia data become available.Comment: Revised manuscript following referee report. Accepted for publication in A&

Reducing "Structure From Motion": a General Framework for Dynamic Vision - Part 2: Experimental Evaluation

A number of methods have been proposed in the literature for estimating scene-structure and ego-motion from a sequence of images using dynamical models. Although all methods may be derived from a "natural" dynamical model within a unified framework, from an engineering perspective there are a number of trade-offs that lead to different strategies depending upon the specific applications and the goals one is targeting. Which one is the winning strategy? In this paper we analyze the properties of the dynamical models that originate from each strategy under a variety of experimental conditions. For each model we assess the accuracy of the estimates, their robustness to measurement noise, sensitivity to initial conditions and visual angle, effects of the bas-relief ambiguity and occlusions, dependence upon the number of image measurements and their sampling rate

The Gaia Project - technique, performance and status

Gaia is a satellite mission of the ESA, aiming at absolute astrometric measurements of about one billion stars (all stars down to 20th magnitude, with unprecedented accuracy. Additionally, magnitudes and colors will be obtained for all these stars, while radial-velocities and spectral properties will be determined only for bright objects (V<17.5). At 15th magnitude Gaia aims at an angular accuracy of 20 microarcseconds (muas). This goal can only be reached if the geometry of the telescopes, the detectors, and the pointing of Gaia at each moment ("attitude") can be inferred from the Gaia measurements itself with muas accuracy.Comment: six pages, invited talk at the conference "Galactic & Stellar Dynamics in the era of high resolution surveys", Strasbourg 16-20 March 200

Testing general relativity by micro-arcsecond global astrometry

The global astrometric observations of a GAIA-like satellite were modeled within the PPN formulation of Post-Newtonian gravitation. An extensive experimental campaign based on realistic end-to-end simulations was conducted to establish the sensitivity of global astrometry to the PPN parameter \gamma, which measures the amount of space curvature produced by unit rest mass. The results show that, with just a few thousands of relatively bright, photometrically stable, and astrometrically well behaved single stars, among the ~10^9 objects that will be observed by GAIA, \gamma can be estimated after 1 year of continuous observations with an accuracy of ~10^{-5} at the 3\sigma level. Extrapolation to the full 5-year mission of these results based on the scaling properties of the adjustment procedure utilized suggests that the accuracy of \simeq 2x10^{-7}, at the same 3\sigma level, can be reached with \~10^6 single stars, again chosen as the most astrometrically stable among the millions available in the magnitude range V=12-13. These accuracies compare quite favorably with recent findings of scalar-tensor cosmological models, which predict for \gamma a present-time deviation, |1-\gamma|, from the General Relativity value between 10^{-5} and 10^{-7}.Comment: 7 pages, 2 figures, to be published in A&

GAIA: Composition, Formation and Evolution of the Galaxy

The GAIA astrometric mission has recently been approved as one of the next two `cornerstones' of ESA's science programme, with a launch date target of not later than mid-2012. GAIA will provide positional and radial velocity measurements with the accuracies needed to produce a stereoscopic and kinematic census of about one billion stars throughout our Galaxy (and into the Local Group), amounting to about 1 per cent of the Galactic stellar population. GAIA's main scientific goal is to clarify the origin and history of our Galaxy, from a quantitative census of the stellar populations. It will advance questions such as when the stars in our Galaxy formed, when and how it was assembled, and its distribution of dark matter. The survey aims for completeness to V=20 mag, with accuracies of about 10 microarcsec at 15 mag. Combined with astrophysical information for each star, provided by on-board multi-colour photometry and (limited) spectroscopy, these data will have the precision necessary to quantify the early formation, and subsequent dynamical, chemical and star formation evolution of our Galaxy. Additional products include detection and orbital classification of tens of thousands of extra-Solar planetary systems, and a comprehensive survey of some 10^5-10^6 minor bodies in our Solar System, through galaxies in the nearby Universe, to some 500,000 distant quasars. It will provide a number of stringent new tests of general relativity and cosmology. The complete satellite system was evaluated as part of a detailed technology study, including a detailed payload design, corresponding accuracy assesments, and results from a prototype data reduction development.Comment: Accepted by A&A: 25 pages, 8 figure

The Global sphere reconstruction (GSR) - Demonstrating an independent implementation of the astrometric core solution for Gaia

Context. The Gaia ESA mission will estimate the astrometric and physical data of more than one billion objects, providing the largest and most precise catalog of absolute astrometry in the history of Astronomy. The core of this process, the so-called global sphere reconstruction, is represented by the reduction of a subset of these objects which will be used to define the celestial reference frame. As the Hipparcos mission showed, and as is inherent to all kinds of absolute measurements, possible errors in the data reduction can hardly be identified from the catalog, thus potentially introducing systematic errors in all derived work. Aims. Following up on the lessons learned from Hipparcos, our aim is thus to develop an independent sphere reconstruction method that contributes to guarantee the quality of the astrometric results without fully reproducing the main processing chain. Methods. Indeed, given the unfeasibility of a complete replica of the data reduction pipeline, an astrometric verification unit (AVU) was instituted by the Gaia Data Processing and Analysis Consortium (DPAC). One of its jobs is to implement and operate an independent global sphere reconstruction (GSR), parallel to the baseline one (AGIS, namely Astrometric Global Iterative Solution) but limited to the primary stars and for validation purposes, to compare the two results, and to report on any significant differences. Results. Tests performed on simulated data show that GSR is able to reproduce at the sub-$\mu$as level the results of the AGIS demonstration run presented in Lindegren et al. (2012). Conclusions. Further development is ongoing to improve on the treatment of real data and on the software modules that compare the AGIS and GSR solutions to identify possible discrepancies above the tolerance level set by the accuracy of the Gaia catalog.Comment: Accepted for publication on Astronomy & Astrophysic