OPTICAL ASTROMETRY FROM SPACE

Astrometry plays a very particular part in the realm of astronomy. On the one hand, it is essentially an ensemble of techniques that provides some essential data to astronomers and astrophysicists about celestial objects. On the other hand, until the second half of the nineteenth century, what is now called astrometric observations were the only astronomical activity that existed. Actually, astronomy has a tradition that goes back to Egyptian, Assyrian, and Greek astronomy. Astrometry is the oldest of all sciences and still is a scientific domain of its own, encompassed by theoretical developments such as stellar dynamics and celestial mechanics now supported by the theory of general relativity.
One can define astrometry as the part of astronomy that measures the apparent positions of celestial bodies on the sky. And, because these positions vary with time, the objective is to describe and study these motions that, for stars, provide two essential parameters: the proper motion and the parallax from which the distance is derived. As an extension, one ascribes also to astrometry the measurement of apparent dimensions and shapes of celestial bodies. However, in this article, we shall consider the determination of star positions, the primary goal of space astrometry.
The physical quantities that are measured by astrometry are angles that are often very small. Radians are not used in astrometry; the basic units are degrees and seconds of arc (denoted”). Smaller units are necessary, and as-trometrists are currently using milliseconds of arc (denoted mas, that is, a milli-arcsecond) and now are starting to use one millionth part of a second of arc (mas). Their respective values are close to 5 x 10 ~ 9 and 5 x 10 ~12 radians.


Essentials of Astrometry

Before discussing how astrometric measurements are performed, it is appropriate to present some basics that have to be known for further understanding. Reference Systems and Frames. The position of a point in the sky is defined by its two spherical coordinates. The most frequently used are the equatorial system. The principal plane is the celestial equator coplanar with Earth’s equator. Starting from the vernal equinox at the intersection of the equator and the ecliptic, the right ascensions (denoted a) are reckoned counterclockwise. The second angular coordinate is the declination (d), counted from the equator, positive to the North, negative to the South (Fig. 1). We shall also use the ecliptic system. The principal plane is the ecliptic, and the celestial longitudes (1) are reckoned counterclockwise from the vernal equinox. The second coordinate is the celestial latitude (b), also shown in Fig. 1. Because both the physical equator and ecliptic are moving, the principal planes have a conventional fixed position (sometimes called mean equator or ecliptic).
These coordinate or reference systems are virtual and are obviously not actually located in the sky. A reference system is actually determined by assigning a consistent set of coordinates to a number of objects (fiducial points). Such a catalog of positions is said to be a reference frame. The position of any object is deduced from relative measurements with respect to fiducial points. Another important condition is that the coordinate systems must be fixed in time, so that the apparent motions of celestial bodies are not falsified by spurious rotation. This is realized now by choosing as fiducial points very distant objects (quasars or galaxies) whose motions are slower than the speed of light and so appear negligible as seen from Earth. The system so defined is the International Celestial Reference System (ICRS) (1), and the catalog of fiducial extragalactic objects is the International Celestial Reference Frame (ICRF), a radio-source catalog extended (2) to optical wavelengths by the Hipparcos catalog (see later). Apparent and True Positions. The direction from which the light arrives at the instrument has undergone a series of deviations. For this reason, it arrives from an apparent position of the star, not the true one. There are three causes for this deviation.
Equatorial (a, d, pole P) and ecliptic (b, l, pole Q) coordinate systems.
Figure 1. Equatorial (a, d, pole P) and ecliptic (b, l, pole Q) coordinate systems.
Atmospheric Refraction. Light from outer space is bent progressively as it enters the atmosphere which is composed of layers of different refractive indexes. The integrated effect depends upon the pressure, temperature, and humidity of the atmosphere and the wavelength of the light. The lower the object in the sky, the larger and the more uncertain the correction to be applied. In practice, ground-based astrometric observations are not performed below 60° zenith distance.
Aberration. The apparent direction of a source is a combination of the direction from which the light arrives and the the velocity of the observer. In ground-based observations, one distinguishes the diurnal aberration due to the motion of the observer as a consequence of Earth’s rotation and stellar aberration due to the motion of Earth around the Sun. In astrometry from space, the diurnal aberration is replaced by the orbital aberration due to the motion of the satellite in its orbit.
In the Newtonian approach, the apparent direction r’ of a star is linked to the undeviated direction r by
tmp115_thumb
where V is the velocity of the observer and c is the speed of light. For very precise astrometry, one must use a more complex formulation based on the theory of general relativity (3).
Relativistic Light Deflection. Following the theory of general relativity, a massive body produces a curvature of space, and the geodesic followed by the light ray is not a straight line: it deviates by a small amount toward the massive body. For the Sun, the deviation is
tmp116_thumb
where 6 is the angle between the directions of the star and of the Sun. Parallactic Displacements. The true position obtained after applying the corrections described above refers to a moving observing site. For positional comparisons, this is not convenient, and it is necessary to refer to a more stable origin of the coordinate system. The correction to be applied to get the direction viewed from another point is the parallactic displacement or correction. Two cases are useful.
Geocentric Coordinates. The parallactic correction necessary to shift from ground-based or satellite-based observations to the center of Earth is totally negligible for a star. This is not the case for observations of objects in the solar system.
Barycentric Coordinates. This coordinate system is centered at the bary-center of the solar system. It is the only point whose motion in space is linear with very high accuracy, because it corresponds to an orbit around the center of the Galaxy described in 280 million years. For all practical applications, it can indeed be considered linear without any dynamic effect on the members of the solar system. The construction of the parallactic correction is sketched in Fig. 2. Let B be the barycenter of the solar system, E the center of Earth in its orbit C around B. Let S be the actual position of a star and r its distance; r = BS. From Earth, S is seen along the vector r’ = ES. The apparent direction ES differs from the barycentric direction BS by the angle
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If we call y the angle (EB, BS),
tmp118_thumb
where R is the length BE. So the variation of the paralllactic displacement p with time is a function of the motion of Earth on C, usually taken as an ellipse but may be made more precise using ephemerides.
Stellar Parallax. The angle p is of the order of or smaller than R/r. The convention is to express R/r not in radians, but in seconds of arc and define a quantity called stellar parallax, or simply parallax, which is equal to the angle p when R is equal to the mean Earth-Sun distance, that is one astronomical unit (149,597,870 km). Because 1″ is equal to 2p/(360 x 3600) radians, the distance for which the parallax is equal to 1″ is 206, 265 AU or 3.2616 light-years. This distance is called the parsec (parallax-second, abbreviated as ps) and is the commonly used distance unit outside the solar system. With this unit, the distance r is simply
tmp1110_thumb
where $ is the parallax p expressed in seconds of arc. Note that the nearest star, Proxima Centauri, has a parallax of 0.762″. Very few stars are at distances smaller than 10 ps and most of the stars of astrophysical interest have parallaxes of the order of a few hundredths or even thousandths of an arc second. This implies that to be significant, their parallaxes should be determined at least at an accuracy of 1 mas, and even much more. This is the major challenge to astrometry nowadays, and this is the main driver for very accurate astrometrical measurements, possible only from space.
Stellar parallactic correction.
Figure 2. Stellar parallactic correction.
Motion of Stars. Stars move in space, and observing their apparent motion in the sky allows us to access dynamic properties of groups of stars (double and multiple stars, star clusters and the Galaxy itself). Two types of motion can be distinguished.
Proper Motions. The position of a star with respect to a fixed celestial reference frame varies with time. Very often, the motion is linear and is expressed as yearly variations of the coordinates, called proper motion:
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In local coordinates centered at the star, the components of proper motion are ma cos d and ms. It is often useful to express the tangential velocity in kilometers per second. This is possible only if the distance is known and, after some transformations of units, for any component m of the proper motion,
tmp1113_thumb
This is the projection of the actual velocity of the star on the plane perpendicular to the direction of the star. The third component of space velocity is radial velocity, which is measured by spectroscopic techniques. It is obtained by measuring the Doppler shift Al of spectral lines at a wavelength A0:
tmp1114_thumb
where c is the speed of light.
Sometimes, the path of a star is not linear. This means that it is attracted by some invisible body, generally a companion of the star such as another faint star, a brown dwarf, or a planet. The star is then called an astrometric double star.
Relative Motions. One star moves with respect to another one, close to it. The observation of such motions is of particular importance in the case of double stars, when one of them revolves around the other following Newton’s law of universal gravitation. If M1 and M2 are the masses of the components, the force that attracts M2 by M1 is
tmp1115_thumb
where k is the gravitational constant and q is the radius vector between the components. Observing double stars is one of the main activities in astrometry. If the distance to the star is known, the sum of their masses can be determined by modeling the apparent path as the projection of a Keplerian orbit. In addition, simultaneous knowledge of the radial velocities or the actual absolute path of both components, as shown in Fig. 3, taken from Reference 4, also allows us to determine M1/M2 and hence obtain the values of both masses (5).
The determination of relative motions in a star cluster is the material from which one can study the kinematic and dynamic properties of the cluster and compare the results to models. An example of what can be achieved with the best presently available astrometric data and radial velocities is found in Reference 6.
Geocentric path of the components of a double star (99 Herculis).
Figure 3. Geocentric path of the components of a double star (99 Herculis).

Ground-Based Astrometry

Before describing what space has brought to astrometry and what it is expected to bring in the future, it is necessary, for comparison, to present the achievements of ground-based astrometric observations. One must distinguish two classes of instruments that differ by the area of the sky measured. Detailed descriptions of the instruments mentioned in this section can be found in Reference 7.
Small-Field Astrometry. The positions of the celestial bodies are in this case measured with respect to neighboring stars instrument’s in the field of view. The typical small-field instrument is the telescope that has either a photographic plate or—more generally now—charge-coupled device (CCD) arrays at its focal plane. The field of view of an array is a fraction of a square degree, but there is way to increase it by the scan mode. In this observing method, the telescope is fixed, and the charge transfer in the CCD is continuous at the rate of the diurnal motion. By this technique, it is possible to scan a narrow but long band along a declination circle.
The field of view of photographic plates, depends on the focal distance of the telescope. In Schmidt telescopes, it is as large as 5° x 5°, and the precision of position measurements is limited to 0.2″. In long-focus telescopes (10 to 20 meters), the field is reduced to a fraction of a square degree, but the relative position of a star with respect to another is as close as a few hundredths of a second of arc. By combining several tens of long-focus observations, one obtains the best ground-based parallaxes to a few mas precision (8). Michelson interferometry that has a coherent field of a few seconds of arc is used to measure star dimensions and very close double stars at accuracies of the order of one mas (9). Speckle interferometry which allows a larger field, is perfect for double star observations, and reaches precision of a few mas (10).
Semiglobal Astrometry. Instruments in this category are designed to determine relative positions of widely separated celestial bodies, much farther apart than their fields of view. However, because one cannot see the whole sky from any place on Earth, one is constrained to some regions; the corresponding astrometry is called semiglobal, rather than global.
The oldest, and still the most used, instrument of this kind is the transit. It consists of a refractor telescope that can rotate around an east-west axis (6). The optical axis can be set to any direction on the local meridian. A micrometer registers the path of the star image on the focal plane and, by interpolation, one gets the time t of the transit of the star through the meridian. Then t is transformed into T, the Greenwich sidereal time and one obtains the right ascension of the star by
tmp1117_thumb
where L is the longitude of the observatory. Simultaneously, the inclination 6 of the axis of the tube is measured using a divided circle, and one gets the declination d by a formula such as
tmp1118_thumb
for a southern transit in the northern hemisphere, or similar formulas in other configurations, where f is the latitude of the observatory. After determining all of the instrumental parameters, one obtains precisions of the order of 0.1″ or slightly better for stars that can be as much as 120° apart in observations that last the whole night.
Astrolabes have been also used for semiglobal astrometry. They observe star transits through a horizontal celestial small circle a little more accurately, but they are much less efficient (6). Recently, Michelson interferometry has been tested to determine relative star positions in various directions (11). Although there are hopes that this technique can give much more accurate results, the only existing instrument (the Navy Prototype Optical Interferometer in Arizona) is too new to permit definite statements on its performance. In any case, the number of observations per night remains quite limited in comparison with a transit instrument which may observe several stars during a single night. Limitations of Ground-Based Astrometry. Except for limited instances in long-focus or interferometric small-field astrometry, the actual uncertainties in observations are of the order of 0.1″. Even if the same stars are reobserved many times, the resulting precision is hardly improved by more than a factor of 3 due to the presence of systematic errors. Compared with the milliseconds of arc required for astrophysically significant results, at least one order ofmagnitude is to be gained. Several reasons exist for this fundamental limitation. Let us examine them.
Atmospheric Refraction. As already mentioned, refraction is not fully predictable. It varies with time and position in the sky, and the correction applied is not perfectly modeled. The result is that, in semiglobal astrometry, the remaining refractive error is generally of the order of a few hundredths of a second of arc and has some systematic component. The use of multicolor observations, which is practical only in interferometric techniques, improves the situation, but not to the level of milliseconds of arc.
Atmospheric Turbulence. The atmosphere is not a smooth medium. Atmospheric stratifications move, and unstable vortices develop and evolve with time. They produce variations in refractive indexes and in the inclination of equally dense layers. The largest affect the angle of refraction. The dimension of the smallest turbulent cells are in the range of 5-30 cm and are produced by the temperature difference between the ground and the air and by the irregularities of the surface. They move with the wind so that the light is randomly deviated and seems to originate from different points in the sky. In addition, rays interfere, and the resulting instantaneous image of a star, called speckle, is deformed and moves rapidly around some central position on timescales of a few hun-dredths of a second. The resulting accumulated image is a disk whose size is at best 0.5″ and is generally of the order of one second of arc on good nights, 2-3″ on others. These numbers characterize the visibility and indicate that, however small the theoretical resolving power of a telescope may be, the images are always larger than 0.5″. So, whatever is the care with which the photocenter of such an image is determined, the pointing precision is necessarily limited to a few hundredths of a second of arc.
Mechanical Properties of the Instrument. The structure of a telescope, in particular of the transit, is subject to torques that depend on its inclination. In practice, it is impossible to model it so that its effects introduce biases in determining refraction. In addition, again in the case of a transit, declination is determined by using a divided circle. The accuracy and the precision of the readout of the  are limited, not to mention the deformations of the circle due to temperature. These effects bias the observations as a function of the time of observation during the night. These causes of errors, together with other perturbations specific to individual instruments, constitute an ensemble of limitations to the accuracy of astrometric observations that are of the same order of magnitude as those due to the atmosphere.
Sky Coverage. It is important for all global studies (kinematics and dynamics of the Galaxy, for instance) that positions and proper motions be referred to a single frame, independently oftheir situation on the sky. To achieve a global astrometric catalog, it is necessary to compile it from regional catalogs produced by many semiglobal instruments. Despite all of the efforts that are made to reduce the systematic differences among them, inevitably not all are corrected, especially if there are undiscovered correlations or similar systematic effects. The last—and best—global catalog is the FK5, produced in 1988 (12), that contains 1535 stars. The accuracies are about 1 mas in proper motions per year and 0.08″ in position at the date of the catalog. The latter figure is an enhanced marker of the uncertainties of the proper motions used to update the positions from the mean epoch of observations (1950) to the present. The systematic differences with accurate space observations by Hipparcos, shown in Fig. 4, illustrate the complex structure of the biases due to the various causes described before. The figure also shows the actual intrinsic limitations of ground-based astrometry. Only astrometry from space can improve the situation significantly.
Space Astrometry. Going to space to perform astrometric observations has long been a dream of astrometrists. A principle that could be used was first published in 1966 by P. Lacroute (13). The method proposed was actually retained for Hipparcos and is now adopted for several future projects. However, at that time, space technology could not meet the accuracy challenges. Reproposed in 1973 to the European Space Agency (ESA), a feasibility study of such a mission was approved in 1976, and the project was included in the ESA mandatory science program in 1980. The project was delayed because of the priority given to the Halley comet space mission Giotto, so that the detailed design study was completed only in December 1983, and the hardware development started immediately after. Another delay was caused by a failure of the Ariane launcher, so that the actual launch of the satellite occurred in only August 1989.
What was required from space astrometry was first to eliminate the limitations described earlier. Clearly, the absence of atmosphere was the first objective, but at the same time, the possibility of homogeneously scanning the whole sky and giving rise for the first time to true global astrometry was achieved. In addition, the very small gravitational and radiative pressure torques that exist in space do not affect the shape of the instrument. Finally, in the absence of atmospheric turbulence, the shape of the star image is entirely defined by optics and can be modeled with extreme accuracy.
However, a new very serious difficulty appeared: how to monitor the orientation of the satellite. In ground-based astrometry, the orientation of the instrument in space is a function of the parameters of Earth’s rotation which are determined independently and very accurately by a specialized service, the International Earth Rotation Service (IERS). They include Universal Time from which sidereal time is computed, polar motion and, in space, precession and nutation. All are known with superabundant accuracy. In space, there is no such external reference and, at least for global astrometry, the orientation (or attitude) of the satellite must be determined simultaneously as accurately as the expected accuracy of star observations.
Differences in proper motions in right ascension and declination between the FK5 and the Hipparcos Catalogue as functions of declination. The solid line is a robust smoothing of the data.
Figure 4. Differences in proper motions in right ascension and declination between the FK5 and the Hipparcos Catalogue as functions of declination. The solid line is a robust smoothing of the data.
Until now (year 2000), two space astrometry missions were launched successfully. The Hipparcos mission was a global astrometric mission. The other comprises the astrometric facilities on board the Hubble Space Telescope (HST) all directed toward small-field astrometry. Both are described in the following sections. In addition, there are several space astrometric projects expected to be launched, if approved, during the first decade of the twenty-first century. Their principles are mentioned in the preceding section.

The Hipparcos Mission

Hipparcos is the acronym for high precision parallax collecting satellite, which points out the main astrophysical objective, but recalls also Hipparchus, the Greek astronomer who discovered precession and is the author of the first star catalog. The satellite was launched by ESA on 8 August 1989. A geostationary orbit was aimed at, but due to the failure of the apogee boost motor, the final orbit was very elongated; the perigee was at an altitude of 500 km, and the apogee was at 36,500 km.The period was 10 hours and 40 minutes. Communications with Earth were secured by three stations in Odenwald (Germany), Perth (Australia), and Goldstone (USA). They ensured direct visibility that covered 97% of the orbit and 93% of the useful observing time. Satellite control and pretreatment of the data were provided by the ESA Operation Center (ESOC in Darmstadt, Germany).
Because of the difficulties that arose from the change from the nominal to the actual orbit, the operations started only at the end of November 1989. They stopped in March 1993, after the failure of several onboard gyroscopes. Taking into account several interruptions, the total useful data collected represents an accumulated 37 months of observations. However, instead of quasi-continuous 24 hour a day observations anticipated for the nominal mission, only 7 to 9 hours and sometimes less observation time per orbital revolution could be achieved because occultation times by Earth, passage through radiation belts which induces strong Cerenkov radiations in the optics, and illumination by the Moon when it was near a field of view that produced a noise that masked the signal had to be excluded.
Principle of Hipparcos. The principle of Hipparcos is sketched in Fig. 5. The main characteristic is that two fields of view are focused on a single focal surface. The optical axes from the center of each field are combined by two glued half-mirrors called a beam combiner whose angle sets the angular reference g, known as the basic angle. On the focal surface is a grid composed of slits parallel to the intersection of the mirrors, a fundamental direction that we shall call vertical. The satellite revolves slowly around an axis parallel to this direction, and the light of the star is modulated by the grid. A photoelectric receiver registers the resulting signal. On both sides of the main grid are two systems of vertical and chevron slits, called star-mappers. Another photoelectric system registers the light crossing a star-mapper that provides data for determining the satellite’s attitude. These data are also used for astrometry in the framework of the Tycho program. Note that, except in part for the star-mapper, the observations are one-dimensional, they amount to determining the transit time of star images through the grid. To cover the whole sky, it is necessary to modify the satellite’s attitude in a predetermined way.
 Principle of Hipparcos showing the motion of the images Ii and I2 of the stars Si and S2 in different fields of view.
Figure 5. Principle of Hipparcos showing the motion of the images Ii and I2 of the stars Si and S2 in different fields of view.
Description of the Hipparcos Payload. Construction of the payload followed the principles described before. The basic angle is g = 58°31.25″. The beam combiner was produced by cutting a 29-cm Schmidt corrected mirror into two halves. The refractive Schmidt configuration was chosen to have a large astro-metrically good field of view across more than 1.3° in diameter. The space structure of the dual telescope and its baffles that protect it from stray solar light is shown in Fig. 6. The light paths from the two fields to the optical block on which the grids are engraved are shown in the figure. The equivalent focal distance is 140 cm. A star produces a diffraction pattern on the grid that is elongated along the vertical direction. The along-track dimensions range between 0.5 and 0.7 seconds of arc in the sky, depending on the color of the star. The grids are engraved on the front side of the optical block which is curved so that it matches the focal surface of the telescope. The main grid consists of 2688 regularly spaced slits whose period is 8.20 mm (1.208″ in the sky); the transparent width is 3.13 mm (0.46″ in the sky). It covers two fields of view in the sky of 0.9° x 0.9°. The vertical extension of the star-mapper grids is limited to 0.7°. Each grid is composed of four 0.9″ wide slits that have pseudorandom separations respectively of a,3a, and 2a (a = 5.675″ in the sky). Only the preceding star-mapper was operated. The second, which was redundant, was actually never used.
Space configuration of the Hipparcos optics.
Figure 6. Space configuration of the Hipparcos optics.
The receiving systems were quite different for the main grid and for the star-mapper. The rear side of the main grid was curved to serve as a field lens of the optical system used to image the full grid on the entrance of an image dissector. This special kind of photomultiplier produces an electronic image on the back wall of the tube. A small hole leaves the way open only to electrons coming from a tiny portion of the electronic image. A set of magnetic deflectors can be controlled to shift any point of the electronic image on the hole. As a consequence, only the light coming from a 30″ radius circle in the whole focal image is recorded. The way this is used will be described later. In contrast, all of the light that enters the star-mapper grids is transmitted to a dichroic mirror that splits the light into two wavelength ranges that correspond roughly to the B and V filters of the Johnson UBV photometric system, called here BT and VT Then each channel is directed toward a different photomultiplier. The photoelectrons are recorded at a rate of 1200 Hz for the main grid and 600 Hz for each of the star-mapper channels. More details on the Hipparcos instrumentation and operations are given in Volume 2 of the Hipparcos and Tycho Catalogue (14).
Scanning the Sky. Because observations are made along a scan in a narrow band 0.9° wide, it is necessary to modify the attitude of the satellite so that all of the sky is scanned as homogeneously as possible and all of the stars are observed for roughly the same amount of time. Various scanning laws can do this, but there are additional constraints. First, and this is the most important condition, the angle between the observed fields of view and the Sun must be at least 45° to minimize stray light. However, the inclination of the scan with respect to the ecliptic should be as small as possible. The parallactic deviation is parallel to the plane of the ecliptic. One wishes to maximize its projection along track because this quantity is used to determine parallaxes. So the inclination chosen is at the limit of acceptance by the first condition. Finally, the attitude should change slowly so that there are overlaps between successive scans.
Nominal Scanning Law. As a compromise between these conditions, the following nominal scanning law was adopted. The satellite rotates in 2 hours 8 minutes allowing 19 seconds for each star to cross the main grid. The rotational axis circles the direction of the Sun in 57 days, keeping an angular distance of43° from the Sun. Figure 7 shows the motion of the axis of rotation in one year and the part of the sky scanned in 70 days.
Left to itself, the rotational axis would drift rapidly due to the various torques that are applied to the satellite (gravitation, radiative pressure, reaction of the gyroscopes, etc.). To follow the scanning law demands active attitude control that is realized by six gas-jet thrusters using compressed nitrogen. The satellite attitude is monitored onboard (see earlier section). When it deviates by 10 minutes of arc from the nominal scanning, gasjets were actuated to reverse the natural attitude drift. In practice, this happened four to six times an hour during the observation conditions. When the satellite was in Earth’s shadow or in radiation belts, attitude control was sometimes very bad, special scanning law recovery procedures had to be applied, and observations were not possible during these maneuvers.
Hipparcos sky scanning law in ecliptic coordinates. Above: Motion of the satellite axis in one year. Below: Part of the sky scanned in 70 consecutive days.
Figure 7. Hipparcos sky scanning law in ecliptic coordinates. Above: Motion of the satellite axis in one year. Below: Part of the sky scanned in 70 consecutive days.
Onboard Attitude Determination. The attitude had to be known continuously during the observations at an accuracy better than one second of arc. The onboard attitude determination used five rate-integrating gyroscopes calibrated in real time using star-mapper data from star crossings. When, a star image crosses a star-mapper slit at a time t, one may write an equation stating that the star lies on the projection of the slit in the sky. Knowing the celestial coordinates of the star corrected for stellar and orbital aberrations, the equation sets one condition on the parameters describing the attitude at time t. If the evolution of the attitude between two gas-jet actuations is smooth, a few equations of condition are sufficient to calibrate the gyroscope drifts and therefore to know the attitude with the desired precision.
The Input Catalogue. As just stated, the necessary real-time orbit was determined on the basis of rough knowledge of the position of some stars. Actually, there was also the same need in the operations of the main grid: to shift the image of a star on the hole of the image dissector, it is necessary to know the position of the star as well as the attitude of the satellite. In addition, because the main grid is periodic, there is no other method to know on what slit the image of a star lies than to know the position significantly better than the grid period. Finally, if all of the observable (up to magnitude 12.5) stars were to be observed, the time allocated to each star would have been insufficient. This led to a limit of the number of stars in the observing program and to creating a list of program stars. Finally, 118,322 stars were selected after a considerable amount of optimization between the astrophysical return and the operational requirements of an even distribution throughout the sky.
From this list of stars, an Input Catalogue was constructed by an international consortium led by C. Turon (15). This catalogue provided the positions of the program stars to a mean uncertainty of 7 0.25″. The preparation involved a large number of astrometrists, most of whose work had to be reobserved on transits or measured on photographic plates. To optimize the observation time, the magnitudes of the stars also had to be known to 7 0.5 magnitude. This objective involved many photometrists, particularly because the time variations of irregular, long-period large-amplitude, variable stars had to be monitored, work that was pursued during the entire mission. Finally, the Input Catalogue provided other known parameters of the stars such as the description of double and multiple stars, many of which were reobserved on this occasion, variability types and amplitudes, parallaxes, proper motions and various identifications in major star catalogues. If Hipparcos had not been successful, the Input Catalogue would still have been a very useful database for many astronomical investigations.

The Hipparcos Data Reduction

Reduction of the Hipparcos data was undertaken by two different international consortia (FAST led by J. Kovalevsky, and NDAC led by E. H0g and later by L. Lindegren). It is a very complex process that cannot be presented in the scope of this topic. Details of the methods used by each consortium can be found in several publications: Reference 16 for FAST, Reference 17 for NDAC as well as a review on Hipparcos by Van Leeuwen (18), and a very detailed account of both is given in Volume 3 of the final catalog (14). So, we present here only a sketch without giving the rationale and the mathematics of the methods adopted.
Roughly, the reduction procedure followed by both consortia is divided into three steps. In the first, only the data acquired during one orbit were considered. During that time, the scanning law restricts the observations to a band in the sky that had a maximum width of 3°. All of the observations were treated together and projected on a single celestial great circle, called the Reference Great Circle (RGC), chosen in the observation band. The abcissas of all observed stars were reckoned from a conventional origin on the RGC. In practice, this step included analyzing photon counts on the main grid and determining the positions on the grid at some defined times, analyzing photon counts on the star-mapper and calculating the attitude, and finally computing the abcissas on the RGC. Some details of these three procedures are given in the following subsections. The second step involves a large number of RGCs to shift the origins of individual RGCs, so that they form a single consistent reference system. In the third step, the RGC abcissas of a given star are combined to determine the five astrometric parameters, namely, the position at a common reference epoch (1991.25), the parallax, and the two components of the proper motion. Later, the three steps are iterated to improve the solution. In addition, several off-line tasks were performed to determine the elements of double and multiple stars and the magnitudes and positions of observed minor planets. Finally, the results obtained by the two consortia were merged and rotated to match the International Celestial Reference System (ICRS).
Reduction on a Reference Great Circle. As stated before, Hipparcos data reduction consists of three distinct phases that are briefly presented here.
Grid Coordinates. They are computed for the central time of a 2.133-second frame during which main grid photon counts are registered. Each count is the integral of the modulated light curve during 1/1200 second. The counts are used to compute the intensity modulation in the following form:
tmp1123_thumb
where I0 is the mean intensity of the star, B is the background noise, M1 and M2 are the modulation coefficients, and f 1 and f 2 are the phases reduced to the mean time of the frame. For a single star, f 1 = f 2, but this is no longer the case for double stars for which the observed intensity is the sum of the intensities of the components. Figure 8 shows some shapes of modulation curves. A conventional phase,
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with
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was used to represent the position of the star within a grid step. Then, the horizontal coordinate on the grid is
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where s = 1.208″ is the grid step. The integer N is deduced from what is known about the coordinates of the star and the attitude of the satellite. It may be in error by 71, rarely more. This is called a grid-step error. The vertical Y coordinate is computed from the same data and a field-to-grid transformation that is calibrated separately.
Attitude Determination. The principle of the attitude determination has already been given earlier. During the reduction phase, it is almost uniquely based on star-mapper observations. See References 19 and 20. The photon counts recorded during the transit of the star image through one of the grid systems are correlated with a calibrated response curve obtained by analyzing many transits of single stars. This gives the transit time through a conventional mean line for which the condition equation mentioned earlier is written. For the interval of time between two gas-jet actuations, the attitude varies smoothly and can be represented by some analytical formula (trigonometric series, polynomials, or cubic splines). The parameters of these formulas are determined from all of the equations of condition in the interval, and the result is used to interpolate the attitude for any time.
Theoretical modulation curves of a single star and two different configurations of double stars that have equal luminosity components.
Figure 8. Theoretical modulation curves of a single star and two different configurations of double stars that have equal luminosity components.
Abscissas on the RGC. This is a rather long and complex task (21). First, the celestial coordinates of the star are deduced from the grid coordinates and the attitude, using grid-to-field transformations, which are third-order polynomials linking the X and Y coordinates to the tangential coordinates in the sky at the central point of each field of view. The parameters At of the transformation also depend on the color and the magnitude of the star. Then, the position so determined is projected on the RGC giving an abscissa which therefore depends on the value of the attitude at the time of the grid coordinate, the parameters of the grid-to-field transformation, the coordinates of the star, and the basic angle g. In the equation that represents this dependence, the attitude along the RGC is considered unknown and is represented by the coefficients of an ensemble of cubic splines that cover the entire set of observations. Some 1500 parameters are needed to cover it. Errors in other components of the attitude as well as the vertical coordinate of the star have only a limited influence on the error budget. So they are considered known and will be improved only during the iterations.
One abscissa observation provides one equation: 1000 to 1500 stars are present on a RGC. Each star is observed several times during an orbit, and generally nine times during one grid transit. Overall, this represents of the order of 40,000 equations that have about 3000 unknowns, including the field-to-grid transformation coefficients and a correction of the basic angle. The system of equations is largely over-determined, and all of the unknowns are straightforwardly determined by the least squares method. The values of the coefficients At are kept as calibrated values of the grid-to-field transformation and its inverse, the field-to-grid transformation. The fact that the basic angle is determined (actually with an uncertainty of 0.2 to 0.3 mas) shows that, in reality, the yardstick for angles is not this material artifact, but 2p in the attitude determination. However, its existence is fundamental because it is the basis of the rigidity of the solution. After iterations, the uncertainty of the abscissas ranges between 3 mas for bright stars to 5 mas for fainter stars. The along-track attitude is obtained with a similar uncertainty of a few mas.
Sphere Solution. Once a sufficient number of RGCs is processed and covers the whole sky, the second reduction step can be undertaken. Each star, observed at different times, moves in the sky, as shown in Fig. 9. The motion is the combination of parallactic displacement and proper motion. Therefore, the abscissa of the star on an RGC is a function of five astrometric parameters and of the inconsistency of the position of the origin of the RGC. If one considers not the position, but a correction to the position in the reference catalog, one can write a linear equation involving the five star parameters and a correction Aaj to the position of the origin of the RGC(/’). At this stage, it is not necessary to take all of the stars, so only the 40,000 well-observed single stars are considered. At the end of the mission, each star was observed in the mean between 20 and 40 times, so that there is a total of more than a million equations with 200,000 star unknowns and 2500 Aaj. If the star unknowns are eliminated, a large linear system that has 2500 unknowns remains which is solved by the method of least squares. With the new origins, the RGCs constitute a rigid mesh that represents the provisional Hipparcos reference system.
Path of a star and the projections on the RGCs on which it was observed.
Figure 9. Path of a star and the projections on the RGCs on which it was observed.
Determination of the Astrometric Parameters. The same equations are now written with the new RGC origins for all stars and sorted star by star. There are generally 20 to 40 equations with five unknowns that are readily solved, and the astrometric parameters are obtained with the complete variance-covariance matrix. In some cases, the residuals of the solution show that the proper motion linear model is not adequate. A polynomial representation is then adopted, meaning that there is an unseen companion that perturbs the motion of the star. It is thus classified as an astrometric double star. Generally, the companion is a faint star, but in a few cases, it was shown that it was a brown dwarf (22).
If the star is recognized as double or multiple by examining its modulation curve, the parameters of the system (magnitudes of each component and their relative positions) must be determined before, or together with, the computation of the astrometric parameters. In the great majority of cases, the assumption that it is a double star is found true, and the magnitudes and the relative position are determined from the time variation of the modulation coefficients. Then, the equations for astrometric parameters are rewritten and solved with an especially adapted algorithm.
Final Steps and Results. The ground-based data used to initialize the reduction process, to determine the attitude, for the first time and to compute the grid coordinates are too inaccurate to be kept throughout the reduction. They must be improved to suppress the effect of their errors on the final result. This is done by iterating the reduction using the latest calibrations and star astrometric parameters obtained in the preceding reduction. Actually, the first complete solution could be obtained with 18 months of data, and this already greatly improved the Input Catalogue information. By performing an iteration, the treatment of photon counts did not need to be redone, so in the first step only the grid coordinates, the attitude, and the reduction on the RGC had to be repeated. The last two steps were of course totally reprocessed.
Merging the Catalogs. When both consortia had finalized their last catalog, the question arose how to present a single Hipparcos set of results to the scientific community. The idea of just making a weighted average of the astro-metric parameters obtained by the two consortia was rejected because there was no statistically justified means to compute the resulting correlations between the merged parameters. Although the consortia were justified in considering the observations as independent, the two solutions obtained were highly correlated. So, it was decided to return to step 2, take the abscissas obtained by the consortia together, and treat them as correlated observations (20). The correlations were computed for a number of stars, and an analytical representation of these correlations was determined as a function of time, magnitude, and the standard errors of abscissas obtained by each consortium. Applying this formula, steps 2 and 3 were reprocessed taking these correlations into account. The standard errors and correlation coefficients obtained this way are those published in the Hipparcos Catalogue.
Link to the ICRS. The system to which the merged positions and proper motions were referred was close to the FK5 system. The objective was that the reference system should be the ICRS. To do this, it was necessary to determine a rotation of the Hipparcos reference that would fit it to the ICRS or, better, its realization, the ICRF. Actually, two rotations are necessary: one for the reference epoch and another proportional to time.

Many astronomers contributed to this task by providing positions and/or proper motions of Hipparcos stars with respect to extragalactic objects. Several ground-based techniques were used:

• observations of radio-stars by long-base or connected radio interferometry with respect to quasars of the ICRF,
• photographic plates taken at different epochs providing proper motions referred to galaxies, and
• data taken from several catalogs of proper motions referred to galaxies.
In addition, direct observations were made at the astrometric focus of the Hubble Space Telescope. A general weighted solution was made to determine the rotations (23) which were then applied to the merged catalog. The resulting astrometric parameters are those published in the Hipparcos Catalogue. The uncertainty of the rotations that provide this link to the ICRS is 0.6 mas for the position at epoch and 0.25 mas/yr for the time-dependent rotation.
Accuracy of the Hipparcos Astrometry. The published catalog (14) contains astrometric parameters of 117,955 stars. The median uncertainties for stars brighter than magnitude 9 (two thirds of the stars) are the following:
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Several attempts were made to evaluate possible biases in the results. Several tests were made, including the parallaxes of stars in the Magellanic Clouds, O- and B-stars in clusters, and a model of distribution of negative parallaxes (24). All concluded that a general bias in parallaxes should not exceed 0.1 mas. However, very locally, and particularly in clusters where the observations are partially correlated, larger systematic errors may exist, but in any case no more than a few tenths of a mas.
Another set of results concerns double and multiple stars. Using the modulation curves, one can detect them and solve for the respective positions and luminosities of the components of double (25) and sometimes even multiple stars (26). As a result, 12,430 double or multiple systems were solved, out of which almost 3000 are newly discovered systems. One must also mention the discovery or confirmation of 2910 astrometric double stars, although 8542 stars were suspected to be nonsingle, but no solution was found.
The results of the Hipparcos mission were immediately used in a large number of investigations in all fields of astronomy and astrophysics. The first results are published in Reference 27. A synthetic account is given in Reference 28.
Hipparcos Photometry. Although nominally Hipparcos was an astromet-ric mission, the payload was a remarkable photometer. The analysis of modulation coefficients by different methods (29,30) also provides the magnitudes of the stars. The system of Hipparcos magnitudes is a wide-band photometric system defined by the transmission of the optics and of the image dissector. A great effort in calibration had to be made to correct for the inhomogeneities of the sensitive surface and aging of the optics and the detector. The sensitivity of the latter decreased with a marked chromatic dependence. One also had to estimate and remove the effect ofthe background. The calibrations were based upon about 22,000 standard single stars all of which had multicolor photometric observations in various systems. The resulting Hipparcos magnitudes were derived by M. Grenon in Geneva.
The photometric data were reduced by the same two consortia. After numerous comparisons, the merger of the results simply consisted of computing the mean. The results are published in the same catalog as the astrometric results. There are 118,204 entries. The median uncertainty for stars of magnitude smaller than 9 is +0.0015 magnitude. In addition, the Hipparcos Epoch Photometry Annexes, available in machine-readable form, give 13 million individual measurements, one per main grid transit. A total of 11,597 stars has been recognized as variable, out of which 8237 have been discovered as such from Hip-parcos data. Among them, 2712 were recognized as periodic and for most of them, the catalog provides folded light curves in addition to light curves for 1101 other objects.
The Tycho Project. Only a very small part of the observations made with the star-mapper are used in the Hipparcos data reduction for determining attitude. Actually, all of the stars that appeared in both fields of view gave signals that were recorded. The objective of the Tycho project was to recover all of these data and use them to obtain the astrometric and photometric information they contain. The data were treated by the international consortium TDAC led by E. H0g. An overview of the reduction method is given in Reference 31, and a detailed description is given in the Hipparcos and Tycho Catalogue, Volume 4 (14).
Principle of Tycho. We have seen earlier that when a star image crosses the mean line of one of the grid systems at some time t, one may write an equation that links the position of the star in the sky and the attitude. If the position of the star is known, the equation constrains the attitude at time t. Conversely, if the attitude is known, the same equation becomes a constraint to the position of the star. Assuming that the positition of the star is known approximately, the condition is represented on the plane tangent to the sky by a straight line parallel to the slit. Each grid crossing produces one such line. A shift perpendicular to the line corresponds to a difference in transit time. Figure 10 shows how these lines roughly converge near a point that represents the actual position of the star. A few lines correspond to misidentified transits. The problem is to find this point from these lines. Before, one had to identify the lines that pertain to the same star. This identification is actually the most difficult part of the reduction.
Prediction and Identification of Transits. As in the case of Hipparcos, but for different reasons, it was necessary to have a Tycho Input Catalogue (TIC). It first contained the 3.26. million brightest stars in the sky to a limit of Johnson B = 12.8 or V = 12.1 magnitudes and was constructed by merging the Hubble Space Telescope Guide Star Catalog (see later) and the Hipparcos Input Catalogue.
The transit times obtained during the first months of the mission were compared to the predictions for all stars of the TIC computed using the first versions of the attitude obtained by the Hipparcos data reduction. This showed that more than 60% of the TIC stars were not found probably because most of them did not satisfy the thresholds for acceptance of a transit. The remaining stars constituted a new revised TIC (TICR) that contains some 1.26 millon stars. Then the transit times for all of the objects of the TICR were predicted using the best available Hipparcos attitude description provided by the Hipparcos consortia and applying, of course, a grid-to-field calibrated transformation and the corrections for stellar and orbital aberrations.
The identification of transits consisted of pairing the observed transit time with that predicted (32). The accuracy of the predicted positions was around 0.2 second of arc. When a transit differed by more than 1″ from the prediction, the transit was rejected and later used for a search of companion stars. After processing these data, 6600 new companion stars were discovered.
Loci of positions of a star derived from Tycho observations. Three lines correspond to misidentifications.
Figure 10. Loci of positions of a star derived from Tycho observations. Three lines correspond to misidentifications.
Astrometric Reduction. The relation between the transit time and the position of the star is a transformation that involves, for each type of grid (vertical, upper, and lower chevrons) and each field of view, a set of coefficients that describes the grid-to-field transformation and corrections to the astrometric parameters given in TICR (33). The grid-to-field transformation was periodically calibrated using the basic equations applied to observations of Hipparcos stars whose positions were known with a superabundant precision for this particular objective. It included a global transformation analogous to that used for the main grid plus 68 medium-scale values as functions of the positions on the grids.
The determination of the astrometric parameters again used the same equations, but with the calibrated values of the field-to-grid transformation at the time of the observation. All of the identified transits of the star were collected and the position of the star described by its five astrometric parameters, as for Hipparcos, was determined by minimizing the sum of the squares of the distances to the lines shown in Fig. 10.
The final Tycho catalogue (14) contains 1,058,332 entries and is practically complete up to VT magnitude 10.5. The median astrometric precision is 25 mas, if one considers all stars, and 7 mas if only bright stars (VT>9) are considered. Evaluation of systematic errors showed that they are smaller than 1 mas. This is, of course, far from the precision of the Hipparcos Catalogue, but the gain is that it concerns nine times more stars. Actually, the difference in precision is easily explained if one considers that the cumulated observation time for a star is 25 to 30 times smaller than that on the main grid. It is expected that a new reduction of the data now in progress with lower thresholds for the acceptance of transits may add another million stars of magnitudes VT in the range of 10.5 to 11.5.
Photometry of Tycho. The data collected by each of the two channels are treated separately in a manner similar to the Hipparcos photometric observations (34). The calibrations were performed from the observations of about 10,000 standard stars with magnitudes between 4.5. and 9 and included dependences on position along the slits, field of view, star color, and channel. The median photometric precisions are as follows in magnitudes:
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Astrometry with the HST

The Hubble Space telescope (HST) is not primarily designed to perform astro-metric observations, but to support several scientific instruments whose common requirement is that the telescope must be able to point in any given direction in the sky and stay pointed with very high stability as long as necessary (35). This task is allocated to three fine guidance sensors (FGS). Actually two FGSs are sufficient to locate a target and stay pointed at it. The third one remains free with the possibility to do astrometry within the field of view. So, in general, astro-metric measurements are confined to a certain field in the vicinity of the region studied by other instruments. However, some astrometric programs are scheduled for their own sake, and then the choice of targets is left to the discretion of the astrometrist. Occasionnaly, one of the scientific instruments, the wide-field planetary camera (WFPC) is used to perform, despite its name, very narrow field astrometry. In even more exceptional cases, the faint object camera (FOC) may also be used for this purpose.
Figure 11 shows how the focal surface of the HST is divided among the various instruments. The WFPC occupies the central part, surrounded by four fields from which the light is transferred to four scientific instruments. The three FGSs occupy the outer ring of the focal surface. The fields allocated to the FGSs are three 90° segments of an annulus of inner and outer radii that correspond to 10.2 and 14 minutes of arc in the sky.
It is well known that the main mirror has an important spherical aberration and that the secondary mirror is slightly tilted and decentered. In addition, there was an important jitter due to the excitation of the solar panel assemblies when the satellite passed into or out of direct sunlight. All of this significantly impaired the astrometric quality of the telescope. The Hubble repair mission in December 1993 suppressed the jitter and the WFPC was replaced by a new camera with modified optics to correct the defects of the telescope. The faint object camera, was corrected by the additional optics provided by the multicorrector COSTAR. But the FGSs remained untouched, and the only improvement came from replacing the solar cell arrays that suppressed the jitter. Otherwise, the situation remained characterized by the major spherical aberration added to the expected astigmatism inherent in a Ritchey-Chretien telescope configuration (a configuration chosen to avoid an important coma that would have been a more severe penalty to astrometry from the FGS). The point-spread function has significant features even a few seconds of arc away from the center of an image and in addition depends strongly on the position in the field of view. Only 15% of the light is concentrated in the 0.1″ central circle instead of more than 50%, as anticipated (36). This corresponds to a loss of more than one magnitude in access to fainter stars and some general degradation of astrometric capabilities. However, even it could have been better, the FGS with careful calibrations, remains a remarkable tool for astrometry.
The focal surface of the Hubble Space Telescope.
Figure 11. The focal surface of the Hubble Space Telescope.
Description of the FGS. The light that reaches the focal surface is deflected by a pickoff mirror and an aspheric collimator into a first star selector that provides exact collimation and correction for nominal astigmatism, spherical aberration, and field curvature. When its two mirrors rotate about an encoding axis, it produces a rotational angle 8a at a fixed calibrated deviation angle 8a that leads to point T (Fig. 12). A second selector produces a rotational angle 8b around axis T at another calibrated deviation angle 8b. The composition of these two rotations allows reaching any point of the FGS and selecting around it a 5″ x 5″ instantaneous field of view in which the fine pointing is performed. The coordinates of this point are
tmp1133_thumbCoordinate systems of the FGS and relative measurements.
Figure 12. Coordinate systems of the FGS and relative measurements.
The image of this field is divided by a beam splitter and sent to two interfero-metric Koester prisms which sense, in two perpendicular directions, whether the incoming wave front is parallel to the front side of the prism. The outputs of each prism are sensed by photomultipliers. If 11 and 12 are the intensities sensed, an error signal S is produced:
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It is null if the wave front is exactly parallel to the entry face of the prism. If both signals are null, pointing is accomplished; otherwise, a signal is sent to the selectors to correct the pointing. The value of S as a function of the depointing angle is the transfer function. Figure 13 shows the shape of the transfer function for images of increasing diameters. It pictures the effect of enlarging the image as the point-spread function widens. Because of the dependence of position on the optical properties, the transfer function must be calibrated by scanning across a number of known single and double stars of different colors in different spots of the field.
HST Pointing. Two FGSs are dedicated to point the telescope to guide stars. Pointing to two stars is sufficient to ensure a unique direction of the axis of the telescope. Because of the reduced fields of view, it was necessary to have a very dense ensemble of star positions all over the sky. This was the objective of preparing the Guide Star Catalog (GSC).
The Guide Star Catalog. A list of star positions and magnitudes of some 20 million stars in the 9 to 15 magnitude range (37). It is based upon micro-densitometer scans of Schmidt telescope photographic plates taken by the 48-inch Palomar Schmidt telescope and the UK Schmidt telescope in Australia. These plates were measured with a precision of 0.25″, but the positions of reference stars for the reduction of the plates were taken from an old catalog (SAO Catalog published in 1966 and hence compiled from much older observations).
FGS transfer functions as functions of the diameter of the image.
Figure 13. FGS transfer functions as functions of the diameter of the image.
The proper motion uncertainties were such that the positions in 1985 were strongly degraded, and the GSC had a mean uncertainty of 1.5 to 2 seconds of arc. However, this was sufficient for HST pointing.
HST Pointing Procedure (38). In the first stage, the telescope is roughly pointed toward the target by its spinning reaction wheels, using the references provided by the gyroscopes. Three fixed-head, star-trackers, independent of the telescope, are used to determine the position and the corresponding FGS coordinates of the two candidate guiding stars. The attitude control system keeps the direction of the telescope fixed, and a first FGS starts searching the guide star using a spiral scan from the indicated coordinates. When it finds it, the second FGS searches the second star. When it also finds it and if the relative positions are compatible with the GSC, the acquisition is confirmed. Then, the Koester prism interferometers send error signals to the attitude control system so that the error signals remain null. The stability of the pointing is about 7 mas rms, limited essentially by the reaction wheel jitter. Because of the uncertainty of the GSC positions, and the limited precision of the head star-trackers, the target may not be exactly in the center of the field of view of the active instrument. This is unimportant for the WFPC and the FOC but may be too large for other focal instruments. Therefore, all have a method of automatically centering the object or may send a signal to the ground-based control station which sends correction instructions.
Astrometry with the FGS. The third FGS does not differ from the two used for pointing, except for a different choice of filters. Once the telescope is rigidly pointed, as described in the preceding section, astrometric measurements may start but are restricted to the field of view of the third FGS, which is 3.8′ wide and the maximum distance between objects cannot exceed 18′. The preparation of an astrometric observation is very complicated. First, it is necessary to determine how to place the field of view to include all objects of interest. From this, the direction of the optical axis of the telescope is to be determined, the guide stars chosen for each FGS, and their positions in the fields of view are computed in terms of the rotational angles 6a and 9b. Similarly, the positions of the targets must be defined in the same manner, so that the instantaneous fields of view of the Koester prisms are automatically set in the correct place. Finally, the order in which the objects are to be observed must be given together with the duration of each observation. There are three modes of observation (39).
The Lock-On Mode. In this mode, objects are successively viewed by interferometers, and the x and y coordinates are determined for each of them when the corresponding error signals are zero. Each star is measured several times in different sequences to avoid possible systematic effects. The positions on the FGS have then to be transferred onto the sky by a transformation that must be calibrated. This includes several successive calibrations.
• The distortion of the optical field angle is obtained by measuring 25 to 30 stars in two crowded fields (star clusters). This transformation is described by a polynomial.
• The plate scale calibration is obtained by measuring the positions of an asteroid as it moves through the field of view with respect to the background stars which are also measured. The motion of an asteroid in a short time is very well known and serves as a standard of angle in the sky.
• The filter wedge error occurs when objects of different magnitudes are to be measured one with respect to another. The light of the brightest one is dimmed by a filter which may be slightly inclined. This is calibrated like the distortion of the optical field angle on a bright star cluster observed through different filters.
• The lateral color effect is produced by misalignments of the optics and a few mas chromatic effect has been observed. This calibration is accomplished by observing a couple consisting of one blue and one red star in different directions and positions in the field.
After applying the results of these calibrations, one must, in addition, correct for differential orbital aberration because the observations are not simultaneous and the orbital motion and hence the space velocity of the space vehicle change. Only then can one deduce the differences in right ascension and declination between the objects. The final uncertainties obtained are of the order of 3 mas for bright stars (magnitudes 0 to 15) and then degrade rapidly till the observing limit of magnitude 17.
The Transfer Function Mode. This mode is aimed at obtaining information about the structure of an object (40). Figure 13 has shown that the transfer function depends on the apparent diameter of the object. Similarly, it also depends on the shape of the object and particularly whether it is a double star. Calibration of the transfer function in various conditions was used to check the models describing it for different geometries of double stars.
In this mode, the object is slowly moved diagonally across the instantaneous field of view, and the error function is recorded as a function of the target’s position. The analysis of the transfer function thus obtained with the help of calibrated models provides the relative positions and luminosities of the components. One may similarly obtain the diameter of the star, but note that the results obtained by the HST in the transfer mode are not better than those obtained from the ground by speckle interferometry and are less precise than those using Michelson interferometry. However, the great advantage of the HST is that they concern much fainter stars.
Moving Target Mode. This mode is used to track moving objects like minor planets. The FGS keeps locked on the object, and its position is periodically sampled. The preparation of observations in this mode must describe the expected path through the field of view.
Wide-Field Planetary Camera. The field of view used by the WFPC is a 3′ x 3′ field centered at the optical axis of the telescope and then deflected to the relay optics and the receiving units. The original WFPC observation capacities were particularly hampered by telescope aberrations, so that it was changed during the repair mission. The new one is corrected for them and now has its nominal performance (Fig. 14).
The incoming f/24 light beam passes through a filter and a shutter and is then focused on a shallow four-faced, mirrored pyramid that can be locked into two positions (41). In one of them, it splits the field of view into four quadrants,and each is directed toward Cassegrain telescopes that convert them into /712.8 beams focused on four 800 x 800 pixel CCD arrays that provide a total field of 160″ x 160″ in the sky. This configuration is the wide-field camera. To tie the four images collected into a single image, the pyramid has some uncoated small spots along the ridges. They are backlit, and the light coming through these spots produces markings on adjoining CCDs, providing fiducial points for the connection. Another 1.23″ diameter nonreflecting Baum spot on the pyramid attenuates the luminosity of the object falling into it by about six magnitudes, to reduce the stray light if it is a bright star that allows surrounding faint objects to appear.
Optical path in the wide-field planetary camera.
Figure 14. Optical path in the wide-field planetary camera.
If the pyramid is rotated by 45°, the original beam is directed into four other Cassegrain telescopes that convert them into / 30 beams focused on four other 800 x 800 pixel CCDs. The field of view is reduced to 68″ x 68″ in the sky. This is the planetary camera configuration.
The observing sequence is quasi-indentical to that followed in ground-based telescopes equipped with CCDs. A preflash is executed to wipe off ghost images that may have remained from the preceding exposure. To reduce the dark current, the CCDs are cooled below -95°C by thermoelectric coolers. Several preliminary calibrations must be made. One is the field-to-array transformation described by a third-order polynomial of the coordinates for each filter and the respective position. This is done by observing cluster stars previously measured with the FGS. Other calibrations include drawing a sensitivity map of the arrays and of the thermal noise. This is done on Earth but is checked inflight.
The reduction of the data allows obtaining positions to about one-fortieth of the pixel size, that is, about 5 mas for the wide-field camera and 3 mas for the planetary camera. This is comparable to the accuracy of the FGS, but the use of the WFPC is different; although the field of view is much smaller, the limiting magnitude is 21 for a few second exposure and fainter for longer. The astrometric objectives concern precise proper motions for recognizing astrometric binaries. Simultaneous observations of the same field by the WF7CA and the FGS are not possible because they see different parts of the telescope field of view, but observations of the part of the sky made successively combine the advantages of both.
Faint-Object Camera. Let us only mention the faint-object camera (FOC) that is a versatile imaging instrument, essentially devoted to investigations involving very faint and very remote objects (42). Two receivers that are imaging detectors work in a photon-counting mode. They are placed at the foci of two optical relays that correct for the residual astigmatism and field curvature of the ensemble telescope, COSTAR. They convert the f/24 telescope beam into f/48 and f/96 beams, giving pixel sizes, respectively, of 44 and 22 mas, compared with the theoretical angular resolution of 66 mas at 633 nm. So, to exploit the full resolution capability, a facility for imaging at f/288 can be inserted in the f/96 path, giving a 7.5″ x 7.5″ field of view. Several additional instrumentats can also be inserted in the beams, such as various filters, an objective prism, a polarizer, a coronograph, and a long-slit spectrograph. So, in the program of work of the FOC, astrometry is just one of the many techniques to investigate a very small field and is not a primary objective for its own sake.

Projects for the Future

The successful entry of astrometry among space astronomical techniques is a powerful incentive to devise and propose to space agencies new more powerful, more effective space astrometric missions with a better science return to cost ratio. Many projects have been presented during the last decade. Some have already been thoroughly studied, and engineered descriptions of a possible realization exist. Other are in a more dormant state. In this section, the two most ambitious projects are described. However, they have not yet been built, so the information given here is provisional. Other projects will only be mentioned, even if some might be launched before those more sophisticated. The Space Interferometry Mission. The primary objective of the Space In-terferometry Mission (SIM), scheduled for launch by NASA in 2005, is to measure stellar distances via parallaxes and apparent proper motions to discover small perturbations of their motions that could be interpreted as caused by planets, in particular Earth-sized planets. A second objective is to measure large angles and to construct a rigid grid ofstar positions that covers the whole sky, as the basis of a global celestial reference frame.
Description of the Instrument. The principle is that of a Michelson phase interferometer already in use on the ground and in a configuration now tested in actual size at Mount Palomar Observatory. The Palomar Testbed Interferometer is now operational and regularly observes some 100 stars per night by remote control from the JPL. Two options of the spacecraft have been studied. The first involved seven siderostats arranged linearly on a 10-meter boom (43). We describe a second one, the so-called RainBird configuration which will probably be chosen for flight. It consists of two collector pads placed symmetrically on a 7-meter, high-precision rail with respect to the combiner pad (Fig. 15). An ensemble of solar arrays and sun shades is placed on the end of a boom to protect the instrument continually from direct sunshine.
Principle Of Measurements. A sketch of the instrument is given in Fig. 16. The two collectors receive light from a star and send it to a beam combiner. One of the beams is directed into a controlled delay line so that the external path delay is equal to the internal one. Thus, interferometric fringes are obtained at the detector. The delay line is activated so that the central fringe remains on the central line of the detector. The reading of the delay line added to the calibrated other internal paths gives the path delay x which is recorded. If D is the baseline, also calibrated internally,
tmp1139_thumb Oblique deployed view of the Space Interferometry Mission.
Figure 15. Oblique deployed view of the Space Interferometry Mission.
where 6 is the angle between the baseline and the direction of the star. Two more interferometers between articulated light collectors on the same baseline measure similarly, the direction of two bright stars with known positions, part of the global rigid grid. From the results, one deduces the space orientation of the baseline at an accuracy of 30 mas. Associated with the main observation, 8 defines a portion of a celestial small circle around the baseline on which the star is located. Observations at different orientations of the baseline provide the position of the star at the intersection of the loci.
Principle of the SIM interferometer. The peak of the interference pattern occurs when the internal path equals the external path delay.
Figure 16. Principle of the SIM interferometer. The peak of the interference pattern occurs when the internal path equals the external path delay.
Calibrations and Expected Results. The objective is to achieve microsecond of arc (mas) astrometry at the end of the 5-year mission. During this time, the spacecraft, which is not on a geocentric orbit, progressively moves away from Earth and reaches finally a distance of 95 million kilometers. This choice minimizes the speed of variation of the aberration which is quite important for an Earth satellite. To perform this correction with a superabundant accuracy of 0.7 mas, it will be necessary to know the velocity of the spacecraft to an accuracy of 4mms ~1. The velocity will be determined from tracking by the NASA Deep Space Network observing two hours a day.
But the major technical challenges are the onboard measurements of distances between various fiducial corner cubes needed to determine internal delays. There are two types of measurements. First, absolute determinations of distance up to 12 meters are to be made with an accuracy of 10 mm. This condition is not so stringent, but it requires a very rigorous stability of the lasers. On the other hand, relative metrology concerned with the variations of the baselines must be accurate to 1 or 2 picometers to achieve the astrometric objectives, that is, 10 million times better than the absolute length measurements. To achieve this, several different methods have been proposed and tested in the laboratory (44,45). The conclusion is that such measurements are possible on the ground and also in space. In all cases, the corresponding calibration cycles will be performed every hour for the relative internal measurement needs and every few days for the external delay. In the error budget, one has to take into account, in addition, thermal effects, even if they are reduced by severe thermal control, fringe measurement errors, and beam walk error produced by mispointing the compressed beams, warping of the pads, shear of the metrology beams, etc.
When these calibrations are performed, it is expected that a 7.5-mas precision measurement of one locus of the star position may be obtained in 0.2 s for stars of magnitude 8, 10 s at magnitude 12, 7 minutes at magnitude 16, and four and a half hours at magnitude 20. The uncertainty also decreases as the inverse of the square root of the exposure time, so that brighter stars may be observed longer without scheduling consequences, because it is essentially the slow (0.25° per second) pointing motion and acquisition time that will limit the scheduling. Finally, an accuracy of 1 mas will be achievable for a majority of the 10,000 stars expected to be on the program, and 4 mas for the global grid of 4000 stars. GAIA. The Global Astrometry Instrument for Astrophysics (GAIA) is a spacecraft proposed to the European Space Agency (ESA) as a follow-up to Hipparcos. It is not yet an approved project. If it was to be programmed by ESA before the year 2001, it could be launched in 2008-2009. As in the case of SIM, several successive versions of the project were studied. Originally, the letter I of GAIA stood for Interferometry because it was proposed that the receiver would be a Fizeau interferometer that produces fringes that would give a more precise measurement (46,47). But engineering studies, financed by ESA, proved that if one replaces the two interferometric apertures by a single mirror encompassing them, the same accuracy of measurements results with a tremendous gain in limiting luminosity and a smaller overall cost. As it is designed now (48), GAIA is a versatile all-sky survey instrument which, in addition to performing very accurate astrometry, will do multicolor photometry, radial velocity measurements, and some narrow-band photometry for all stars up to magnitude 17. The highest priority is, however, astrometry, and the objective is to get astrometric measurements up to magnitude 20, that is, on more than a billion stars.
Description of the Payload. The principle of GAIA is identical to Hip-parcos in the sense that two fields of view separated by a basic angle (here, g = 106°) are simultaneously observed. However, rather than directing the two fields of view on the same focal surface, there are two separate identical telescopes; each has its own receiving subsystem. The invariance of the basic angle is monitored by laser interferometers. The layout of the two full-reflective, three-mirror telescopes, thermally controlled at a temperature below 200 K, is shown in Fig. 17. The space left between the two telescopes is filled by a third telescope adapted for radial velocity measurements and spectrophotometry. The primary mirror of the astrometric telescopes is a rectangule of 1.4 x 0.5 meters. The optics give an equivalent 50-meter focal length so that the useful field of view of 0.66° x 0.66° is projected on a detector whose dimensions are 575 x 700 mm. The satellite rotates around an axis perpendicular to the telescope layout and scans the sky following a law analogous to that of Hipparcos. As in the case of SIM, it was recognized that the observations should be made far from Earth. For GAIA, the choice is a Lissajous type of orbit around the Laplace L2 point of the Sun-Earth system.
The Detector System. The detector (Fig. 18), placed on the focal surface of the telescope, includes 250 CCD arrays of 2100 x 2660 pixels organized in 10 along-scan (horizontal) strips. Each array is 24 mm wide along scan and 57 mm long in the vertical direction. The pixel size along scan is 9 mm (36 mas in the sky) x 27 mm (108 mas), compatible with the shape of the point-spread function. The observing strategy is the scan mode already mentioned earlier. The transfer of charges is done continuously, and the speed corresponds to the rate of rotation of the instrument. The collection of charges is done in the 4-mm dead zones between the arrays.
Layout of the three telescopes of the GAIA project. A1 and A2 are the astrometric telescopes; B is the radial velocity/photometry telescope.
Figure 17. Layout of the three telescopes of the GAIA project. A1 and A2 are the astrometric telescopes; B is the radial velocity/photometry telescope.
Arrangement of the CCDs on the GAIA focal surface.
Figure 18. Arrangement of the CCDs on the GAIA focal surface.
The vertical columns of CCDs do not have all the same functions. The first four in the scan direction form the star-mapper. All signals are processed. Those corresponding to an input catalog are used to control the attitude, but all are analyzed, and the position of those that are sufficiently bright to be measured are used to sort the useful signals in the astrometric CCDs. The latter occupy the next 17 columns in which the astrometric data are acquired. The last four columns are used for broadband photometry in different colors.
Because of the curvature of the focal surface, the arrays will be slightly tilted and individually sequenced to compensate for optical distortion. Each individual CCD features a special operating mode, which may or may not be activated, that allows reducing the integration time and acquiring bright stars with no saturation which occurs for magnitudes brighter than 12.
Expected Results. The astrometric performance depends on magnitude and also on color. Many more photoelectrons will be received from red stars than from blue ones. For intermediate stars (solar type), the accuracy floor up to magnitude 12 is 3 mas. It degrades for fainter stars and is 10 mas for magnitude
15. In other terms, for some 35 million stars, the astrometric accuracy in position, parallaxes, and yearly proper motions will be better than 10 mas. For magnitude
16, the numbers will be, respectively, 60 millon stars and 18 mas; for magnitude 18; 300 million stars and 55 mas; and for magnitude 20, more than a billion stars and 0.2 mas. So even at the limiting magnitude of 20, it is expected that GAIA will be five times more precise than Hipparcos for its bright stars.
For the first time, radial velocities will be systematically measured all over the sky. The wavelength interval provisionally set is 847-879 nm. The expected precision is a few km s ~1 for stars up to magnitude 17 that have spectral lines in this interval. This means that more than one hundred million radial velocities would complete the proper motion and parallax measurements for space velocities.
Other Projects. Among various proposals presented for space astrometry, three have some chance of being realized and launched.
FAME. The Fizeau Astrometric Mapping Explorer (FAME) was conceived to meet the NASA philosophy of”small, fast, and cheap.” It consists of two dilute aperture telescopes operating in the Fizeau mode (49), a concept that was also originally proposed for GAIA (46), but on a smaller scale. The apertures would be approximately 10 x 20 cm and the baseline 50 cm. The angle between the telescopes would be 80°. The operations would be quite similar to Hipparcos with a predefined scanning law. The detectors will be CCD arrays working in scan mode as in GAIA. The expected limiting magnitude is 15, and the accuracies for astrometric parameters would range from 40 mas at magnitude 9 to 180 mas at magnitude 12 and 0.8 mas for the faintest stars.
DIVA. The Deutsches Interferometer fur Vielkanalphotometrie und As-trometrie (DIVA) was submitted to the German Space Agency and is designed to perform astrometric and photometric observations of a million stars (50). The configuration consists of two Fizeau interferometers with a baseline of 10 cm; the receiver is a mosaic of CCDs. The operations would be analogous to Hipparcos. It is expected that the resulting catalog will be complete to magnitude 10.5 with an accuracy of astrometric parameters of the order of 0.3 mas.
LIGHT. The Light Interferometer for the study of Galactic Halo Tracers (LIGHT) is much more ambitious than the two preceding projects. It is a Japanese project for global astrometry (51). It is composed of four sets of Fizeau interferometers with a beam combiner unit of 1 meter baseline. The operations are again similar to Hipparcos. The prime scientific objective, as witnessed by its name, is to monitor parallaxes and proper motions of distant galactic and halo giant red stars up to visual magnitude 18, but all stars in this magnitude range will also be observed. The expected accuracy of the astrometric parameters is 50 mas and should involve some hundred million stars. The satellite would also include photometry in visual and near-infrared bands.

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