Context. We present the second Gaia data release, Gaia DR2, consisting of astrometry, photometry, radial velocities, and information on astrophysical parameters and variability, for sources brighter than magnitude 21. In addition epoch astrometry and photometry are provided for a modest sample of minor planets in the solar system. Aims. A summary of the contents of Gaia DR2 is presented, accompanied by a discussion on the differences with respect to Gaia DR1 and an overview of the main limitations which are still present in the survey. Recommendations are made on the responsible use of Gaia DR2 results. Methods. The raw data collected with the Gaia instruments during the first 22 months of the mission have been processed by the Gaia Data Processing and Analysis Consortium (DPAC) and turned into this second data release, which represents a major advance with respect to Gaia DR1 in terms of completeness, performance, and richness of the data products. Results. Gaia DR2 contains celestial positions and the apparent brightness in G for approximately 1.7 billion sources. For 1.3 billion of those sources, parallaxes and proper motions are in addition available. The sample of sources for which variability information is provided is expanded to 0.5 million stars. This data release contains four new elements: broad-band colour information in the form of the apparent brightness in the GBP (330–680 nm) and GRP (630–1050 nm) bands is available for 1.4 billion sources; median radial velocities for some 7 million sources are presented; for between 77 and 161 million sources estimates are provided of the stellar effective temperature, extinction, reddening, and radius and luminosity; and for a pre-selected list of 14 000 minor planets in the solar system epoch astrometry and photometry are presented. Finally, Gaia DR2 also represents a new materialisation of the celestial reference frame in the optical, the Gaia-CRF2, which is the first optical reference frame based solely on extragalactic sources. There are notable changes in the photometric system and the catalogue source list with respect to Gaia DR1, and we stress the need to consider the two data releases as independent. Conclusions. Gaia DR2 represents a major achievement for the Gaia mission, delivering on the long standing promise to provide parallaxes and proper motions for over 1 billion stars, and representing a first step in the availability of complementary radial velocity and source astrophysical information for a sample of stars in the Gaia survey which covers a very substantial fraction of the volume of our galaxy.

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 results performed 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 mas at 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.

Context. Gaia Early Data Release 3 (Gaia EDR3) gives trigonometric parallaxes for nearly 1.5 billion sources. Inspection of the EDR3 data for sources identified as quasars reveals that their parallaxes are biased, that is, they are systematically offset from the expected distribution around zero, by a few tens of microarcseconds. Aims. We attempt to map the main dependences of the parallax bias in EDR3. In principle, this could provide a recipe for correcting the EDR3 parallaxes. Methods. Quasars provide the most direct way for estimating the parallax bias for faint sources. In order to extend this to brighter sources and a broader range of colours, we used differential methods based on physical pairs (binaries) and sources in the Large Magellanic Cloud. The functional forms of the dependences were explored by mapping the systematic differences between EDR3 and DR2 parallaxes. Results. The parallax bias is found to depend in a non-trivial way on (at least) the magnitude, colour, and ecliptic latitude of the source. Different dependences apply to the five- and six-parameter solutions in EDR3. While it is not possible to derive a definitive recipe for the parallax correction, we give tentative expressions to be used at the researcher’s discretion and point out some possible paths towards future improvements.

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.

Context. We present the early installment of the third Gaia data release, Gaia EDR3, consisting of astrometry and photometry for 1.8 billion sources brighter than magnitude 21, complemented with the list of radial velocities from Gaia DR2. Aims. A summary of the contents of Gaia EDR3 is presented, accompanied by a discussion on the differences with respect to Gaia DR2 and an overview of the main limitations which are present in the survey. Recommendations are made on the responsible use of Gaia EDR3 results. Methods. The raw data collected with the Gaia instruments during the first 34 months of the mission have been processed by the Gaia Data Processing and Analysis Consortium and turned into this early third data release, which represents a major advance with respect to Gaia DR2 in terms of astrometric and photometric precision, accuracy, and homogeneity. Results. Gaia EDR3 contains celestial positions and the apparent brightness in G for approximately 1.8 billion sources. For 1.5 billion of those sources, parallaxes, proper motions, and the (GBP − GRP) colour are also available. The passbands for G, GBP, and GRP are provided as part of the release. For ease of use, the 7 million radial velocities from Gaia DR2 are included in this release, after the removal of a small number of spurious values. New radial velocities will appear as part of Gaia DR3. Finally, Gaia EDR3 represents an updated materialisation of the celestial reference frame (CRF) in the optical, the Gaia-CRF3, which is based solely on extragalactic sources. The creation of the source list for Gaia EDR3 includes enhancements that make it more robust with respect to high proper motion stars, and the disturbing effects of spurious and partially resolved sources. The source list is largely the same as that for Gaia DR2, but it does feature new sources and there are some notable changes. The source list will not change for Gaia DR3. Conclusions. Gaia EDR3 represents a significant advance over Gaia DR2, with parallax precisions increased by 30 per cent, proper motion precisions increased by a factor of 2, and the systematic errors in the astrometry suppressed by 30–40% for the parallaxes and by a factor ~2.5 for the proper motions. The photometry also features increased precision, but above all much better homogeneity across colour, magnitude, and celestial position. A single passband for G, GBP, and GRP is valid over the entire magnitude and colour range, with no systematics above the 1% level

Context. Gaia Data Release 1 (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. Aims. 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. Results. For about two million of the brighter stars (down to magnitude ∼11.5) we obtain positions, parallaxes, and proper motions to Hipparcostype precision or better. For these stars, systematic errors depending for example 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 −1 . The Hipparcos reference frame is found to rotate with respect to the Gaia DR1 frame at a rate of 0.24 mas yr −1 . Conclusions. 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 present results nevertheless represent a huge improvement in the available fundamental stellar data and practical definition of the optical reference frame.

This paper is devoted to a practical model for relativistic reduction of positional observations with an accuracy of 1 µas which is expected to be attained in the future space astrometry missions. All relativistic effects which are caused by the gravitational field of the Solar system and which are of practical importance for this accuracy level are thoroughly calculated and discussed. The model includes relativistic modeling of the motion of observer, 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 Parametrized Post-Newtonian Formalism with parameters β and γ. However, for general relativity (β = γ = 1) the model is fully compatible with the IAU Resolutions (2000) 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 which should be applied to the model to attain the accuracy of 0.1 µas are reviewed. Potentially important relativistic effects caused by additional gravitational fields which are generated outside of the Solar system are also briefly discussed. Subject headings: astrometry, reference systems, relativity• the 3-dimensional coordinate quantities ("3-vectors") referred to the spatial axes of the corresponding reference system are set in boldface: a = a i ;• the absolute value (Euclidean norm) of a "3-vector" a is denoted as |a| and can be computed as |a| = (a 1 a 1 + a 2 a 2 + a 3 a 3 ) 1/2 ;

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