Until 1998 the reference frame was defined in terms of the Solar System and the dynamics induced by the motions of Solar System bodies and their shapes. The two reference planes were the equator of the Earth and the ecliptic, the mean plane of the Earth's orbit. The intersection of these two planes, the equinox, was the fiducial point. Since both the equator and the ecliptic move owing to solar and lunar gravitational forces on the shape of the Earth and perturbations by the planets on the Earth's motion, the equinox moves with time. Therefore, each dynamical reference frame had to be defined for a specific epoch and observations and predictions were transformed to and from standard epochs to the chosen times of observations.

We have presented, in Chapter 7, the new adopted International Celestial Reference System (ICRS) based on the extragalactic radio sources. Now, the fundamental reference frame is fixed for all times. The IAU introduced in 2000 two space fixed systems. The Barycentric Celestial Reference System (BCRS) and the Geocentric Celestial Reference System (GCRS) have been defined in terms of metric tensors and the generalized Lorentz transformation between them, which contains the acceleration of the geocenter and the gravitational potential. The ICRS is to be understood as defining the orientation of the axes of both these systems for each of the origins. The International Celestial Reference Frame, (ICRF), determined from VLBI observations is the realization of the ICRS and can similarly be geocentric and barycentric.

As a rule, an astrometric observation does not directly provide the quantity sought, but rather some function of it and of a number of other parameters. The procedure that is used to obtain it is called reduction of observations. Generally, one needs to use several observations taken at different times and conditions, and treat them together. Astrometric data reduction includes taking into account a variety of effects that are described in the following chapters. But it also consists of a mathematical treatment of the data, for which one generally assumes some statistical properties. Among other things, one must be aware of the fact that a numerical result has no physical significance, and is not useful, unless there is an indication of how good the numbers are. The results must be supplemented by another number, called uncertainty, which provides an interval (sometimes called error bars) within which there is a stated probability that it includes the true value. Sometimes, when several quantities are computed simultaneously, and are mutually dependent or correlated, other numbers called correlation coefficients must be determined.

This chapter presents succinctly the most commonly used statistical tools to perform the data reduction. There are many textbooks in statistics or data treatment to which one should refer in order to go deeper in the understanding of the procedures or get more information.

Astrometry is undergoing fundamental changes. The celestial reference frame based on bright optical stars is being replaced by the extragalactic reference frame observed at radio wavelengths by Very Long Baseline Interferometry (VLBI); Hipparcos has proven the capabilities of astrometry from space; photographic plates are being replaced by charge coupled devices (CCDs) as the detectors of choice; optical interferometers are replacing transit circles and astrolabes; accuracies of tenths of arcseconds are being replaced by milli- and microarcseconds; the dynamical reference frame is being replaced by a kinematic reference frame; Global Positioning System observations are changing the determinations of Earth Orientation Parameters; and the theory of general relativity is required as the basis of astrometry.

This book is designed to provide the fundamentals for the new astrometry at the milli- and microarcsecond accuracies. The intent is to start from basic principles, without approximations, and develop the fundamentals of astrometry at microarcsecond accuracies. It is based on the general theory of relativity and the concepts introduced by the International Astronomical Union (IAU) in the past decade. The book provides the definitions and applications of the International Celestial Reference System (ICRS) to astronomy and astrometry. It is also designed to provide the philosophy and concepts of astrometry for the future, the principles behind the algorithms, the reasons for astrometry and its relationships with astronomy, geophysics, planetary sciences, astrophysics, cosmology, and celestial mechanics.

The previous chapters have provided the bases for the applications to observations. This chapter will provide examples of the reduction and analysis of observational data. Software for coordinate conversions and individual effects, such as precession, nutation, proper motions, aberration, refraction, etc., is available as NOVAS at the US Naval Observatory website. A software program that performs on-line the full reduction of a CCD plate, called PRIAM, is available at the IMCCE web site of the Paris Observatory.

Observing differences, classical and new systems

In Chapters 7, 8, and 10, we introduced fixed and moving reference systems. At present, we are in a transition period, when old and new systems may be used, although it is to be expected that the new system will progressively remain the only one used, because they have important advantages. However, it is still necessary to be acquainted with both.

Referring positions to the fixed reference frame at epoch J2000.0, namely the International Celestial Reference Frame (ICRF), instead of the reference frame and positions referred to the equinox and the ecliptic and based on the FK5, improves the accuracies of the positions. The ephemerides are now on the ICRF. Astrometric observations made with respect to ICRF positions only require the corrections for proper motions of the reference stars and appropriate differential corrections to reduce the magnitudes of the plate solution parameters (Section 14.5).

What characterizes the new observational techniques is, of course, not only that they appeared recently – say in the past couple of decades – but also that they strive to reach one or two orders of magnitude better accuracy than the classical techniques. The unit for describing the new astrometric capabilities is no longer one tenth of a second of arc, but one thousandth of a second of arc, a subunit that we shall designate throughout this book by the abbreviation mas (milliarcsecond). Plans are underway to reach microarcsecond (μas) accuracies. Such a gain in precision impacts directly on all aspects of astronomy and, particularly, on the reduction procedures. One may divide these new techniques into three major groups: interferometry, timemeasuring techniques, and space astrometry.

New detectors

Many astrometric techniques take advantage of the development of much more sensitive new detectors. Among them, one must mention the following.

• The CCD detector. The CCD (for charge coupled device) has become a major tool in astronomy since the 1970s (Monet, 1988). It consists of a semiconductor device, where a photoelectric effect takes place when light reaches it, producing an electronic image. This image is preserved by arrays of small positive electrodes, which attract photoelectrons and keep them in a similar array of potential wells, providing the possibility of long exposure times by adding photoelectrons in the same pack (Kovalevsky, 2002). […]

In contrast with celestial reference frames, which have existed since the time when catalogs of stars over the whole sky became available, a global reference system for the positions on the Earth did not exist until direct geodesic links could be performed between regions separated by oceans and, more generally, by geographically or politically impassable barriers. So there were a number of local, geodetic coordinate systems, called datums, to which the positions of terrestrial sites were referred. They were given under the form of parameters defining the shape and the size of a reference ellipsoid, as well as its orientation with respect to some conventional features such as a mean pole and a zero meridian. The ellipsoid parameters were determined to best fit the local geoid (equipotential surface corresponding to the mean ocean level), and attached to the Earth by conventional coordinates of an initial point.

Introduction

At the beginning of the space age, positional and, later, laser or Doppler observations of satellites were used to link the individual datums and to place them in a unique terrestrial coordinate system. However, locally, countries continued (and many still do) to use their own datums for surveying and legal objectives. But for scientific purposes, this was a much too complex system, and a global terrestrial reference system had to be developed.

Motions involved

The combinations of different motions of Solar System bodies, involving orbital motions with their respective eccentricities and inclinations, as well as rotations with different planes and periods, lead to a variety of phenomena.

These range from the obvious, such as the rising and setting of the Moon and planets, seasons on the Earth, and the phases of the Moon, to the less-obvious phases of the planets and apparent stationary and retrograde motions of the planets. The ellipticities and inclinations of the orbital motions of the bodies also lead to some variations of these phenomena.

The motions of the Earth, Moon, planets, and satellites lead to objects obscuring other objects and the sunlight onto those objects. The lack of visibility due to the proximity to the Sun, the visibility due to elongation from the Sun, or the positioning opposite to the Sun, are also resulting phenomena.

The motion of the Earth in its orbit determines the seasons and, with the Moon's motion, is the basis for the different calendars. The seasons are defined from the times of equinoxes and solstices. While the equinox may not be used for the reference frame anymore, it will continue to be the basis of defining the seasons, and can be determined from the Solar System ephemerides being used. Likewise, the times of perihelion and aphelion for the Earth and all the planets are determined from the ephemerides.

The main objective of stellar astrometry is to determine the positions of stars in space at some epoch and to describe their displacements in time. The instruments used for observations, and how the positions are determined, are given in Kovalevsky (2002). The reference frames have been discussed in Chapters 7–10 of the present book. The observational reduction procedures and corrections for apparent displacements were described in Chapter 6.

Star positions are not an objective per se. What is of interest are the motions and distances of celestial bodies. The first are the proper motions that describe the apparent displacements due to the actual motion of stars with respect to the barycenter of the Solar System. As we have already seen in Chapter 6, in order to transfer Earth-based observations to a barycentric position, one has to correct for annual parallax. However, the value of the parallax coefficient has a major importance in astronomy, because it is the basic source of distances in the Universe. This is why the determination of distances is discussed in detail in this chapter. By adding radial velocity to a combination of proper motion and distance, one obtains the space motion of a star. In addition, a section on magnitudes and spectra is given, not only because they provide important information to be used together with astrometric parameters, but also because they enter in the reduction of astrometric observations.

Preliminary remark. The numbers that are given in this appendix should not be considered as reference values. Although the present authors have tried to provide numbers that are as close as possible to what is estimated to be the best values, they do not guarantee either that they are the best possible, or that all are mutually consistent. In several cases, authorities that publish values of astronomical constants do not agree, and the choice of one or the other is necessarily subjective. Furthermore, as time goes on, better values will become available. For these reasons, we do not associate uncertainties to values, and one should consider these lists as providing orders of magnitude of the parameters and not as a basis for accurate and dependable calculations.

IAU system of astronomical constants, best estimates

SI units

The units meter (m), kilogram (kg), and second (s) are the units of length, mass and time in the International System of Units (SI).

Astronomical units

The astronomical unit of time is a time interval of one day (D) of 86 400 seconds. An interval of 36 525 days is one Julian century.

The astronomical unit of mass is the mass of the Sun (S).

Does the world really need a new textbook on general relativity? I feel that my first duty in presenting this book should be to provide a convincing affirmative answer to this question.

There already exists a vast array of available books. I will not attempt here to make an exhaustive list, but I will mention three of my favourites. For its unsurpassed pedagogical presentation of the elementary aspects of general relativity, I like Schutz's A first course in general relativity. For its unsurpassed completeness, I like Gravitation by Misner, Thorne, and Wheeler. And for its unsurpassed elegance and rigour, I like Wald's General Relativity. In my view, a serious student could do no better than start with Schutz for an outstanding introductory course, then move on to Misner, Thorne, and Wheeler to get a broad coverage of many different topics and techniques, and then finish off with Wald to gain access to the more modern topics and the mathematical standard that Wald has since imposed on this field. This is a long route, but with this book I hope to help the student along. I see my place as being somewhere between Schutz and Wald – more advanced than Schutz but less sophisticated than Wald – and I cover some of the few topics that are not handled by Misner, Thorne, and Wheeler.

In the winter of 1998 I was given the responsibility of creating an advanced course in general relativity. The course was intended for graduate students working in the Gravitation Group of the Guelph-Waterloo Physics Institute, a joint graduate programme in Physics shared by the Universities of Guelph and Waterloo.