Sergei A. Klioner

Gaia Data Release 1 - Astrometry: one billion positions, two million proper motions and parallaxes

Gaia Data Release 1 (Gaia DR1) contains astrometric results for more than 1 billion stars brighter than magnitude 20.7 based on observations collected by the Gaia satellite during the first 14 months of its operational phase. We give a brief overview of the astrometric content of the data release and of the model assumptions, data processing, and validation of the results. For stars in common with the Hipparcos and Tycho-2 catalogues, complete astrometric single-star solutions are obtained by incorporating positional information from the earlier catalogues. For other stars only their positions are obtained by neglecting their proper motions and parallaxes. The results are validated by an analysis of the residuals, through special validation runs, and by comparison with external data. Results. For about two million of the brighter stars (down to magnitude ~11.5) we obtain positions, parallaxes, and proper motions to Hipparcos-type precision or better. For these stars, systematic errors depending e.g. on position and colour are at a level of 0.3 milliarcsecond (mas). For the remaining stars we obtain positions at epoch J2015.0 accurate to ~10 mas. Positions and proper motions are given in a reference frame that is aligned with the International Celestial Reference Frame (ICRF) to better than 0.1 mas at epoch J2015.0, and non-rotating with respect to ICRF to within 0.03 mas/yr. The Hipparcos reference frame is found to rotate with respect to the Gaia DR1 frame at a rate of 0.24 mas/yr. Based on less than a quarter of the nominal mission length and on very provisional and incomplete calibrations, the quality and completeness of the astrometric data in Gaia DR1 are far from what is expected for the final mission products. The results nevertheless represent a huge improvement in the available fundamental stellar data and practical definition of the optical reference frame.

The IAU 2000 Resolutions for Astrometry, Celestial Mechanics, and Metrology in the Relativistic Framework: Explanatory Supplement

We discuss the IAU resolutions B1.3, B1.4, B1.5, and B1.9 that were adopted during the 24th General Assembly in Manchester, 2000, and provides details on and explanations for these resolutions. It is explained why they present significant progress over the corresponding IAU 1991 resolutions and why they are necessary in the light of present accuracies in astrometry, celestial mechanics, and metrology. In fact, most of these resolutions are consistent with astronomical models and software already in use. The metric tensors and gravitational potentials of both the Barycentric Celestial Reference System and the Geocentric Celestial Reference System are defined and discussed. The necessity and relevance of the two celestial reference systems are explained. The transformations of coordinates and gravitational potentials are discussed. Potential coefficients parameterizing the post-Newtonian gravitational potentials are expounded. Simplified versions of the time transformations suitable for modern clock accuracies are elucidated. Various approximations used in the resolutions are explicated and justified. Some models (e.g., for higher spin moments) that serve the purpose of estimating orders of magnitude have actually never been published before.

Gaia Data Release 2 - The astrometric solution

Context. Gaia Data Release 2 (Gaia DR2) contains results for 1693 million sources in the magnitude range 3 to 21 based on observations collected by the European Space Agency Gaia satellite during the first 22 months of its operational phase. Aims. We describe the input data, models, and processing used for the astrometric content of Gaia DR2, and the validation of these resultsperformed within the astrometry task. Methods. Some 320 billion centroid positions from the pre-processed astrometric CCD observations were used to estimate the five astrometric parameters (positions, parallaxes, and proper motions) for 1332 million sources, and approximate positions at the reference epoch J2015.5 for an additional 361 million mostly faint sources. These data were calculated in two steps. First, the satellite attitude and the astrometric calibration parameters of the CCDs were obtained in an astrometric global iterative solution for 16 million selected sources, using about 1% of the input data. This primary solution was tied to the extragalactic International Celestial Reference System (ICRS) by means of quasars. The resulting attitude and calibration were then used to calculate the astrometric parameters of all the sources. Special validation solutions were used to characterise the random and systematic errors in parallax and proper motion. Results. For the sources with five-parameter astrometric solutions, the median uncertainty in parallax and position at the reference epoch J2015.5 is about 0.04 mas for bright (G < 14 mag) sources, 0.1 mas at G = 17 mag, and 0.7 masat G = 20 mag. In the proper motion components the corresponding uncertainties are 0.05, 0.2, and 1.2 mas yr−1, respectively.The optical reference frame defined by Gaia DR2 is aligned with ICRS and is non-rotating with respect to the quasars to within 0.15 mas yr−1. From the quasars and validation solutions we estimate that systematics in the parallaxes depending on position, magnitude, and colour are generally below 0.1 mas, but the parallaxes are on the whole too small by about 0.03 mas. Significant spatial correlations of up to 0.04 mas in parallax and 0.07 mas yr−1 in proper motion are seen on small (< 1 deg) and intermediate (20 deg) angular scales. Important statistics and information for the users of the Gaia DR2 astrometry are given in the appendices.

A Practical Relativistic Model for Microarcsecond Astrometry in Space

We develop a practical model for relativistic reduction of positional observations with an accuracy of 1 μas, which is expected to be attained in future space astrometry missions. All relativistic effects that are caused by the gravitational field of the solar system and are of practical importance for this accuracy level are thoroughly calculated and discussed. The model includes relativistic modeling of the motion of the observer and modeling of relativistic aberration and gravitational light deflection, as well as a relativistic treatment of parallax and proper motion suitable for the accuracy of 1 μas. The model is formulated both for remote sources (stars, quasars, etc.) and for solar system objects (asteroids, etc.). The suggested model is formulated within the framework of the parameterized post-Newtonian formalism, with parameters β and γ. However, for general relativity (β = γ = 1) the model is fully compatible with the year 2000 IAU resolutions on relativity in celestial mechanics, astrometry, and metrology. The model is presented in a form suitable for implementation in a software system for data processing or simulation. The changes that should be applied to the model to attain an accuracy of 0.1 μas are reviewed. Potentially important relativistic effects caused by additional gravitational fields that are generated outside of the solar system are also briefly discussed.

Considerations concerning the non-rigid Earth nutation theory.

This paper presents the reflections of the Working Group of which the tasks were to examine the non-rigid Earth nutation theory. To this aim, six different levels have been identified: Level 1 concerns the input model (giving profiles of the Earths density and theological properties) for the calculation of the Earths transfer function of Level 2; Level 2 concerns the integration inside the Earth in order to obtain the Earths transfer function for the nutations at different frequencies; Level 3 concerns the rigid Earth nutations; Level 4 examines the convolution (products in the frequency domain) between the Earths nutation transfer function obtained in Level 2, and the rigid Earth nutation (obtained in Level 3). This is for an Earth without ocean and atmosphere; Level 5 concerns the effects of the atmosphere and the oceans on the precession, obliquity rate, and nutations; Level 6 concerns the comparison with the VLBI observations, of the theoretical results obtained in Level 4, corrected for the effects obtained in Level 5.Each level is discussed at the state of the art of the developments.

Microarcsecond astrometry in space - Relativistic effects and reduction of observations

The framework of General Relativity is applied to the problem of reduction of high-precision astrometric observations of the order of 1 μarcsec. Such precision is expected to be attained in the not so distant future by means of the space optical interferometers orbiting the Earth. Theoretical methods are described enabling one to construct astrometric reference systems involving the barycentric system, geocentric system, and satellite (observer) system. The relativistic transformations between the employed reference systems have been derived. The equations of geometric optics for the nonstationary gravitational field of the Solar system have been deduced

Analysis of astrometric catalogues with vector spherical harmonics

Aims. We compare stellar catalogues with position and proper motion components using a decomposition on a set of orthogonal vector spherical harmonics. We aim to show the theoretical and practical advantages of this technique as a result of invariance properties and the independence of the decomposition from a prior model. Methods. We describe the mathematical principles used to perform the spectral decomposition, evaluate the level of significance of the multipolar components, and examine the transformation properties under space rotation. Results. The principles are illustrated with a characterisation of systematic effects in the FK5 catalogue compared to Hipparcos and with an application to extraction of the rotation and dipole acceleration in the astrometric solution of QSOs expected from Gaia.

Gaia Data Release 1. Reference frame and optical properties of ICRF sources

Context. As part of the data processing for Gaia Data Release 1 (Gaia DR1) a special astrometric solution was computed, the so-called auxiliary quasar solution. This gives positions for selected extragalactic objects, including radio sources in the second realisation of the International Celestial Reference Frame (ICRF2) that have optical counterparts bright enough to be observed with Gaia. A subset of these positions was used to align the positional reference frame of Gaia DR1 with the ICRF2. Although the auxiliary quasar solution was important for internal validation and calibration purposes, the resulting positions are in general not published in Gaia DR1. Aims. We describe the properties of the Gaia auxiliary quasar solution for a subset of sources matched to ICRF2, and compare their optical and radio positions at the sub-mas level. Methods. Descriptive statistics are used to characterise the optical data for the ICRF sources and the optical-radio differences. The most discrepant cases are examined using online resources to find possible alternative explanations than a physical optical-radio offset of the quasars. Results. In the auxiliary quasar solution 2191 sources have good optical positions matched to ICRF2 sources with high probability. Their formal standard errors are better than 0.76 milliarcsec (mas) for 50% of the sources and better than 3.35 mas for 90%. Optical magnitudes are obtained in Gaia’s unfiltered photometric G band. The Gaia results for these sources are given as a separate table in Gaia DR1. The comparison with the radio positions of the defining sources shows no systematic differences larger than a few tenths of a mas. The fraction of questionable solutions, not readily accounted for by the statistics, is less than 6%. Normalised differences have extended tails requiring case-by-case investigations for around 100 sources, but we have not seen any difference indisputably linked to an optical-radio offset in the sources. Conclusions. With less than a quarter of the data expected from the nominal mission it has been possible to obtain positions at the sub-mas level for most of the ICRF sources having an optical counterpart brighter than 20.5 mag.

Relativistic scaling of astronomical quantities and the system of astronomical units

Context. For relativistic modeling of high-accuracy astronomical data, several time scales are used: barycentric and geocentric coordinate times (TCB and TCG) and two additional time scales (TDB and TT), which are defined as linear functions of TCB and TCG, respectively. Aims. The paper is devoted to a concise, but still detailed, explanation of the reasons and the implications of the relativistic scalings of astronomical quantities induced by the time scales TDB and TT. Methods. We consequently distinguish between quantities and their numerical values expressed in some units. Results. It is argued that the scaled time scales, the scaled spatial coordinates, and the scaled masses should be considered as distinct quantities that can be expressed themselves in any units and not as numerical values of the same quantities expressed in some different, non-SI units (“TDB units” and “TT units”). Along the same lines, the system of astronomical units is discussed in the relativistic framework. The whole freedom in the definitions of the systems of astronomical units for TCB and TDB is demonstrated. A number of possible ways to freeze the freedom are shown and discussed. It is argued that in the future one should think about converting AU into a defined quantity by fixing its value in SI meters.

Physically adequate proper reference system of a test observer and relativistic description of the GAIA attitude

A relativistic definition of the physically adequate proper reference system of a test observer is suggested within the framework of the parametrized post-Newtonian formalism. According to the nomenclature accepted within the GAIA project this reference system is called the center-of-mass reference system (CoMRS). The interrelation between the suggested definition of the CoMRS and the International Astronomical Union (IAU) Resolutions 2000 on relativity are elucidated. The tetrad representation of the CoMRS at its origin is also explicated. It is demonstrated how to use this tetrad representation to calculate the relation between the observed direction of a light ray and the corresponding coordinate direction in the barycentric celestial reference system of the IAU. It is argued that the kinematically nonrotating CoMRS is the natural choice of the reference system where the attitude of the observer (e.g., of the GAIA satellite) should be modeled. The relativistic equations of rotational motion of a satellite relative to its CoMRS are briefly discussed. A simple algorithm for the attitude description of the satellite is proposed.