Fundamental physics with the ESA mission GAIA

C. R. Acad. Sci. Paris, t. 2, Série IV, p. 1299–1311, 2001 Relativité, gravitation/Relativity, gravitation (Métrologie, instrumentation/Metrology, instrumentation)

MISSIONS SPATIALES EN PHYSIQUE FONDAMENTALE SPACE MISSIONS FOR FUNDAMENTAL PHYSICS

Fundamental physics with the ESA mission GAIA François MIGNARD

DOSSIER

CERGA, av. Copernic 06130 Grasse, France E-mail: [email protected] (Reçu le 6 juin 2001, accepté le 20 juillet 2001)

Abstract.

ESA advisory committees have selected the astrometry mission GAIA as one of the future cornerstones of the science program. GAIA is primarily a mission to study the history, formation and evolution of the Galaxy from astrometric, photometric and spectroscopic measurements, but it also displays remarkable capabilities in some areas of fundamental physics. After introducing briefly the measurement principles and the expected accuracy, I focus on the three topics most relevant for the physicists: the definition and realization of a quasi-inertial machian frame, the testing of the PPN formulation of the gravitational theory and finally a determination of the possible time change of the gravitational constant.  2001 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS

astrometry / general relativity / GAIA / reference frames

Physique fondamentale avec la mission GAIA Résumé.

Les comités de sélection de l’Agence Spatiale Européenne ont retenu la mission astrométrique GAIA pour une des futures pierres angulaires du programme scientifique. A la base GAIA a pour objectif l’étude de la formation, l’histoire et l’évolution de notre Galaxie à partir de mesures astrométriques, photométriques et spectroscopiques. Mais cette mission présente également un grand intérêt pour la physique fondamentale. Dans cet article j’introduirai brièvement le principe des mesures et leur précision attendue avant de me concentrer sur les domaines plus spécifiques de la physique : la définition et la réalisation d’un système de référence machien quasi-inertiel, les tests de la formulation PPN de la théorie de la gravitation et enfin la mesure d’une variation éventuelle de la constante de la gravitation.  2001 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS astrométrie / relativité générale / GAIA / systèmes de référence

Version française abrégée La mission spatiale de l’Agence Spatiale Européenne (ESA) HIPPARCOS a ouvert une nouvelle voie dans une discipline vénérable de l’astronomie presque tombée en désuétude, à savoir la mesure des positions et déplacements des étoiles du voisinage solaire. Aujourd’hui quatre nouvelles missions d’astrométrie spatiale ont été approuvées en Europe et aux Etats-Unis, dont la mission GAIA de l’ESA. Cette mission a pour objectif de réaliser une cartographie de la Galaxie pour toutes les étoiles jusqu’à la magnitude 20 avec une précision nominale de 10 µas (∼ 0.05 nrad) pour V = 15. Elle s’appuie sur le Note présentée par Pierre E NCRENAZ. S1296-2147(01)01268-9/FLA  2001 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS. Tous droits réservés

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concept validé par HIPPARCOS d’un satellite à balayage observant simultanément dans deux directions à 106◦ l’une de l’autre, mais en ajoutant une photométrie multi-époque sur une dizaine de bandes spectrales et des mesures de vitesses radiales avec un instrument effectuant de la spectroscopie à basse résolution. L’objectif fédérateur est l’étude de la composition, formation et évolution de la Galaxie grâce au recensement très complet de pratiquement toutes les populations stellaires, définies soit par leur physique, leur âge ou leurs propriétés cinématiques. Comme tout relevé astrométrique précis et systématique, le relevé GAIA a des retombées scientifiques plus ou moins éloignées de son objectif premier, en particulier en physique fondamentale. En premier lieu l’observation dans le visible des radiosources extragalactiques va permettre de construire un système de référence inertiel à partir du concept machien d’absence de rotation de la matière lointaine. La précision devrait atteindre 0.5 µas/an pour la rotation résiduelle. Ce système devrait remplacer en 2015 la réalisation primaire actuelle dans le domaine radio et permettre aux astronomes d’y avoir un accès direct dans le visible, soit au travers des radiosources les plus brillantes (V < 16–17) ou bien à partir du catalogue stellaire de GAIA . La réduction des données astrométriques de haute précision nécessite un traitement relativiste de la propagation de la lumière, tant pour les effets cinématiques du mouvement du satellite par rapport au référentiel barycentrique du système solaire, que pour les effets gravitationnels dans le champ solaire. Dans ce dernier cas, il est aisé de montrer que la déviation gravitationnelle de la lumière par le Soleil est aisément mise en évidence et ceci de façon répétée sur toutes les observations de toutes les étoiles. Rappelons qu’à 90◦ du Soleil, l’effet atteint 0 .004. On peut espérer tester des écarts à la version Einsteinienne de la gravitation aussi faibles que quelques 10−7 en exploitant l’ensemble des mesures sur les étoiles brillantes. De même la précession du périhélie des petites planètes devrait pour la première fois permettre de généraliser dans le système solaire le test si bien vérifié sur Mercure. Le problème du moment quadrupolaire du Soleil ne devrait pas être un facteur limitant en 2015. Plusieurs petites planètes déjà bien connues ont une excentricité suffisamment élevée pour que la précession atteigne environ 25% de la valeur mercurienne, en dépit de l’éloignement au soleil. A la condition que les mesures de GAIA permettent une bonne détermination orbitale, on peut espérer tester la non linéarité de la gravitation au niveau de 10−4 voir 10−5 . En dernier lieu, un recensement profond des naines blanches devrait conduire à une étude plus précise de leur fonction de luminosité (nombre d’objets par classe de magnitude absolue) dont la structure est fortement liée à la vitesse de refroidissement de ces astres inertes. Comme la vitesse de refroidissement dépend de la constante de Newton, et que les étoiles ne peuvent être plus âgées que la Galaxie, il y a là potentiellement un moyen de tester la variation éventuelle de G à l’échelle cosmologique. D’autres mesures de GAIA, comme les effets de lentilles gravitationnelles, la distribution spatiale des quasars ou bien l’échelle des distances et l’âge des étoiles, entretiennent des relations étroites avec la physique, mais ne sont pas abordées dans cet article faute de place.

1. Introduction The success of HIPPARCOS has demonstrated the scientific interest of accurate global astrometry in space and its major impact on many branches of astrophysics and even in physics. It was clear that the potential of this technique had only been glimpsed with the 1 mas accuracy for a sample of rather bright stars (V < 9) and that there was room for improvement both in number of sources, brightness and accuracy, thanks to the advent of electronic digital detectors. Following this remarkable success several space astrometry missions have been studied aiming to explore the µas (1 µas = 10−6 arcsec) world for much fainter sources. Regarding this endless race toward extreme precision, the unique situation of the present epoch can only be fully appreciated when viewed in a historical perspective encompassing the last two millennia. Between the time of Hipparchus and 1980, the art of fixing stellar position on the sky has progressed by about four

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to five orders of magnitude, from a precision of half a degree to something close to 0 .05 for the last fundamental catalogue before HIPPARCOS. At the horizon of 2015, when the GAIA and SIM missions are completed, astronomers will have definitely set foot in the realm of microarcsecond astrometry, something unthinkable just 30 years earlier, after a progress of another four orders of magnitude, this time in less than three decades. This achievement will place the best ground based astrometry of the 1980s, still so common to most astronomers, just mid-way between Hipparchus and the state of the art of space astrometry. Accordingly, it is not exaggerated to refer to this unique period we are living in as the ‘New Golden Age of Astrometry’, maybe even more exciting and promising than that experienced in the XVIIth century with the invention of the telescope and its adaptation for precise astronomy. A space astrometry mission has a unique capability to perform global measurements, such that positions, and changes in positions caused by proper motion and parallax, are determined in a reference system consistently defined over the whole sky, for a very large numbers of objects. Four missions are now planned to be launched within the coming decade, two from the USA and two from Europe, GAIA being the most ambitious. These new space astrometry missions share common features both in objectives and, to a lesser extent, in design. Astrometric missions produce virtually no usable data until the mission is over and all the data sets are analyzed, which happens to be a very complex undertaking. Typically the direct results of a global space astrometry mission resulting from the processing of the raw measurements are: • accurate positions, absolute parallaxes, proper motions for millions of stars; • multi-epochs photometry in several bands at the milli-magnitude level; • radial velocity; • spectrophotometry; • high-resolution imaging. Not every mission is able to perform all this variety of observations. It depends on the instruments and detectors included in the payload, on the observation strategy and the on-board storage capacities. While astrometric measurements are the baseline, in particular with the direct measure of the absolute parallaxes, it remains that the major support and main drive follow from the indirect results and applications to stellar and galactic physics. The list below bears witness of the extraordinary versatility of global astrometry which touches just about every aspect of astronomy. In brief, the major areas that benefit from space astrometry are primarily the: • recalibration of the extragalactic distance scale (cepheids, RR Lyrae); • determination of the age of the globular clusters; • determination of the absolute luminosities of a wide range of spectral types; • detections of companion stars, brown dwarfs, and giant planets; • detailed study of the structure, content and kinematics of our galaxy; • materialization an optical inertial reference frame; • mapping of the interstellar matter; • contribution to fundamental physics (GR testing). It happens that each of the above items would be nearly sufficient to build a well balanced proposal for a space program, and in fact there are such proposals or even approved mission, like the French (and ESA supported) COROT. Probably the most awaited and easily publicized question that Space astrometry may answer is the detection of companion stars either in the substellar range (< 0.1 M ) or in the planetary range (< 10 MJ ). As appealing this topic may seem, I consider the impact on galactic and extragalactic physics as deeper and more far-reaching with a recalibration of the extragalactic distance scale and a high resolution study of the Cepheid period-luminosity relation. Fundamental physics, the central theme of this paper, will also benefit from the astrometry through the accurate observation of the light bending in the solar gravitational field. Again HIPPARCOS has opened the way [1] by demonstrating convincingly the possibility of using observations at wide angles from the Sun to estimate the curvature of the spacetime.

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2. The GAIA project 2.1. Overview The proposed concept of GAIA, approved in October 2000 within the ESA science programme aims to create an extraordinarily precise three dimensional map of about one billion stars throughout our Galaxy with a target astrometric accuracy of 10 µas (∼ 50 prad) for a star of 15th magnitude, yielding modelindependent distances accurate to 10 percent at several kiloparcecs, typically the distance of the globular clusters, the best tracers of the oldest stellar populations. Through continuous scanning over five years GAIA will provide detailed information on every star observed thanks to the on-board photometry in more than ten carefully selected wavebands. This includes the determination of the luminosity (combination of distances and measured radiation fluxes), temperature (with the spectroscopy), gravity on their surface (with line widths) and the possibility to infer their size and masses. The chemical composition will be also an outstanding by-product of the photometric and spectroscopic measurements. In addition to accurate position in a non rotating frame, GAIA will trace with unprecedented details the kinematics of the Galaxy through the determination of the stellar proper motion and that of their radial velocity (for a significant sample) providing essential clues on the formation and history of the various populations making up the Galaxy. The science objectives touch virtually any aspect of astrophysics from stars to quasars, through young and old clusters, binary and multiple systems and galaxies of the local group, without omitting the census of the minor bodies of our solar system and the detection of Jupiter-like extrasolar planets. Detailed information is available in the GAIA study report [2] and in a summary paper covering most of the aspects of the mission [3]. 2.2. Design The current instrument design has arisen from the constraints on astrometric precision, the completeness of the survey down to 20th magnitude and the need to carry out both photometric and spectroscopic measurements. Hipparcos has brilliantly demonstrated that global astrometry could be performed in space provided one uses a dedicated scanning instrument with at least two widely separated viewing directions and a well defined and stable angular reference. An elementary analysis of the signal-to-noise ratio yields already interesting design elements. For diffraction limited optics with aperture size D, the diffraction spot on the focal plane has a typical angular size of λ/D. Then if a total of N photons is available for localizing the center of the image, the achievable accuracy is of the order of: λ 1 √ D N Hence for an integration time τ over a source of photon flux Φ we have:
100 100 D=1m =⇒ σ = √ mas = √ λ = 0.6 µm N Φτ σ≈

(1)

which yields for a sun-like star of 15th magnitude:  V = 15 Φ = 4 · 104 ph/(m2 · s) σ = 10 µas

=⇒

τ = 2000 s

Therefore a typical star should by allocated about 2000 s of observation over the mission length to reach the targeted astrometric accuracy of 10 µas.

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Figure 1. Layout of the GAIA focal plane assembly.

Now, if Ω is the field size in square degrees and T the effective mission length, we have for the available time for a specific star: Ω τ = T 4π which yields a strong constraint on the field (with T = 5 years and τ = 2000 s): Ω=

4π τ ≈ 0.7 deg2 T

This ensures that for a uniform scanning of the sky, a star will be repeatedly observed during the mission so that the observation time will total about 2000 s. For GAIA this time will be distributed over an average of 130 observations of ∼ 15 s scattered over the five years of the mission, allowing the sampling of the parallactic ellipse. These overall design constraints have been investigated in detail and resulted in a proposed payload meeting the set of requirements [4]. The system consists of: • Two astrometric viewing directions separated by an angle of 106 degrees. Each of these astrometric instruments comprises an all-reflective three-mirror telescope with an aperture of 1.7 × 0.7 m2 . Each astrometric field is divided into an astrometric sky mapper, the astrometric field proper, and a broadband photometer. The main focal plane assembly ( figure 1) employs CCD technology, with about 250 CCDs and accompanying video chains per focal plane, a pixel size 9 µm along scan, TDI (time-delayed integration) operation, and an integration time of 0.9 s per CCD; • An integrated radial velocity spectrometer and photometric instrument, comprising an all-reflective threemirror telescope of aperture 0.75 × 0.70 m2 . The field of view is separated into a dedicated sky mapper, the radial velocity spectrometer, and a medium-band photometer. Both instrument focal planes are based on CCD technology operating in TDI mode. The optics and focal planes are mounted on a single structural torus employing SiC for both mirrors and structure. A deployable Sun shield protects from direct Sun illumination and rotating shadows on the payload module. 2.3. Operations and performances The choice of the orbit results also from a tradeoff between easy and cheap communications and a stable thermal environment without eclipses. A Lissajous orbit around the Earth–Sun Lagrange L2 point at 1.5

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Table 1. Sky averaged astrometric accuracy expected with GAIA for a mission of 5 years. The last line gives the number of stars brighter than G. G (∼ V )

mag

10

11

12

13

14

15

16

17

18

19

20

Parallax

µas

4

4

4

5

7

11

17

27

45

80

160

Position

µas

3

3

3

4

6

9

15

23

39

70

140

Proper motions

µas/an

3

3

3

4

5

8

13

20

34

60

120

0.3

1.0

2.5

7

16

35

80

200

400

700

1000

N

6

(10 stars)

million kilometers from the Earth in the anti-sun direction was eventually preferred to a geostationary orbit. A launch with Ariane 5 between 2010 and 2012 is the current baseline. The satellite will follow a strictly prescribed rotation enabling a continuous sky scanning. The spin axis of the satellite (perpendicular to the two viewing directions) maintains a constant orientation with respect to the Sun. Over a revolution period of 3 hours, a small zone 0◦ .4 wide on either side of a great circle of the celestial sphere is scanned by the fields of view. The spin axis lies permanently at 55◦ from the Sun, so that stars are observed at any solar distance between 35◦ to 135◦ . Regarding the overall performances, they have been assessed during the study phase for the astrometry, photometry, spectrometry, sky-coverage, star density. The only relevant parameters for the rest of this paper are the astrometric accuracy as a function of the magnitude of the stars and the number of objects to be observed in each interval of magnitude. The important results are given in table 1. There are basically three regimes in the accuracy: the bright stars up to 13 mag, where the performances are limited by the saturation of the CCDs leading to a precision constant with the magnitude; the intermediate stars, which make the core of the mission, from 14 to ∼ 19 mag, where the photon noise is the major source of uncertainty; finally at the faint end, the readout noise becomes significant compared to the signal. About 3 million stars will be observed with the highest astrometric accuracy of ∼ 3 µas, much better than the target accuracy at 15 mag. This fact has a deep impact in the relativity testing and in the realization of an easily accessed reference frame in the visible. 3. Fundamental physics with astrometry Each of the four astrometric missions under study has a section or a chapter of its proposal related to its impact on fundamental physics, meaning that the proposers expect side results, not so directly related to the astronomical objectives. However it is not so obvious to delineate sharply scientific results as belonging to astronomy or to physics. Historically this was a nonsense, and today the history of the Universe, the physics of the big-bang and particle physics seem deeply intermingled. I will adopt my own list, which, as any selection, is not above criticism. I will consider in succession the definition and realization of a quasi-inertial frame, the testing of the PPN formulation of gravitation theory, and the time variation of the Newton constant G. I could have added the age of the Universe, the cosmological distances, detection of dark matter, etc. 3.1. The inertial frame The role of high quality reference frames in astronomy has been recognized early by both theoreticians and practitioners, and most of the classical astronomy until the emergence of physical astronomy was concerned with the realization of good reference frames. Astrometric data, positional or kinematical, rely on observations referred to a frame, either local or global. The choice of the reference frame may be driven by instrumental consideration or be built from deeper theoretical grounds, as is currently done with the definition of the ICRS which bore a fatal blow to the long astronomical tradition of materializing the inertial frame from observations attached in various ways to the apparent motion of the Sun.

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The main novelty in the ICRS lies in the adoption of a kinematical system which assumes that the visible Universe does not rotate, so that the most distant sources have no individual motion relative to each other. As a whole these sources define and materialize a non-rotating reference system. The extragalactic reference frame is assumed to approximate an inertial frame, defined within the context of general relativity, through Mach’s Principle. The ICRF was formally adopted as the primary materialization of the International Celestial Reference System as of 1st January 1998 as a fundamental catalogue of carefully observed radiosources [5]. The whole catalogue comprises 608 compact radiosources, of which only 212 constitute the defining sources, corresponding to the best observed subset with an internal positional accuracy in the range of 0.2–0.5 mas. Aside from this primary system, an optical counterpart was also adopted by the IAU in 1997, from the HIPPARCOS catalogue and its content was refined in a resolution adopted by the IAU in August 2000. The global nature of the HIPPARCOS observations produced a very well defined freely rotating sphere materialized by the positions and proper motions of the stars. Eventually it was linked to the extragalactic frame through a set of radio stars common to the two frames, with a residual rotation estimated to be 0.25 mas/yr [6]. A mission like GAIA will permit a realization of the ICRS more accurate by two or three orders of magnitude, from direct observations of the defining sources brighter than magnitude 20 and also by adding thousands of QSOs. QSOs will be recognized from ordinary stars or white dwarfs with the broad- and narrow-band photometry of GAIA up to redshifts z ∼ 4–5. The sky coverage will be fairly uniform outside a zone of ±25 degrees centered on the galactic plane. This sample will yield an outstanding non-rotating frame at the level of 0.5 µas/yr, provided the source random instability is less than 20 µas. As GAIA will survey the quasars down to an apparent magnitude of 20, there will be plenty of material (about 500 000 from current estimates based on local surveys) to select a small sample, maybe less than 10 000 to construct the primary reference frame in the visible. The frame orientation will be obtained from a comparison with the ICRF sources, so that there will be no discontinuity within the uncertainty of the current version. With ∼ 200 sources usable for the comparison, one should be able to maintain the continuity with an uncertainty less than 50 µas. The spin vector will be constrained by imposing that the selected sample of ∼ 10 000 extragalactic sources exhibits no overall rotation. The testing of this feature will be crucial to detect any departure to this essential assumption for the reference frame but also for our knowledge of the quasars. An iterative process would yield the final selection of primary sources and pinpoint the sources with detected transverse motion, a result of major cosmological significance. 3.2. The space curvature Space curvature can be determined from astrometric measurements from the careful monitoring of the path of the photons in the gravitational field of the Sun. 3.2.1. The effect and its magnitude The first step into a relativistic modelling of the light path consists of determining the direction of the incoming photon as measured by an observer located in the solar system as a function of the barycentric coordinate position of the light source. This point has been addressed in several books and publications and will not be discussed here. Apart from second order aberration the only other sizeable effect is linked to the bending of light rays in the gravitational field of solar system bodies, planets and satellites. The relevant geometry and notations are shown in figure 2. The star is located at very large distance compared to the Sun and χ is the angular separation between the Sun and the star. With the space observations to be carried out by GAIA, χ is not necessarily a small angle. In fact, it can be very small for the planets where grazing observations are feasible, but remains always larger than χmin = 35◦ for the Sun. The impact parameter of the unperturbed ray is denoted by d and the distance between the observer (on the earth or spaceborne somewhere in the solar system) is r. The

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Figure 2. Deflection of starlight by the sun or the planets.

deflector has a mass M and a radius R. To the first order in GM/c2 and by neglecting any departure from the spherical symmetry the deflection angle is given by: δα =

1 1 + γ 4GM R 2 2 c R r 2 tan χ2

(2)

When the angular separation χ is 1, this expression reduces to: δα =

1 + γ 4GM R 2 c2 R d

(3)

Then it is convenient to introduce the grazing deflection for a ray just passing at the surface of the deflector, or equivalently when d = R, which yields the classical result: δαg =

4GM 1 + γ 4GM ≈ 2 2 c2 R c R

(4)

Although one talks about light deflection or bending, there is no way to measure this effect directly since the initial direction is not known. In fact one must have access to the proper direction in the observer’s frame, and only the variation of this proper direction with time, due to varying geometry with respect to the Sun, is accessible, which eventually permits a determination of the deflection proper. The magnitude of the deflection is given in table 2 for various observing conditions likely to occur with GAIA. Considering an astrometric accuracy in the range of few µas, the magnitude of the effect is considerable, even for the planets in grazing conditions. However the observations of a star nearly aligned with a giant planet will be a rare event, and while this must be allowed in the modelling, these observations will not be very useful to determine the space curvature. On the other hand, there will be numerous and regular observations at wide angle from the Sun, up to χ = χmax = 145◦ , carrying the signature of the deflection. Additional terms are needed to allow for gravitational effect on the light ray at the µas level [7]. The post-PPN term for the grazing ray is no more than 7 µas for the Sun and could be modeled with the Einstein theory. The effect of the solar quadrupole is shown in table 2 and will not be a problem. It is significant for Jupiter and Saturn, but falls off very rapidly with the angular separation and only a handful of observations are concerned.

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Sun

Grazing

χmin

(mas)

(mas)

χmax

GAIA

χ = 45◦

χ = 90◦

J2

Grazing

χ = 1◦

(µas)

(µas)

(mas) −7


10

1750

13

10 mas

4.1 mas

2.1

0.3

–

Earth

0.5

0.5

2.5 µas

1.1 µas

0

0.001

1

0.01

Jupiter

16

16

2.0 µas

0.7 µas

0

0.015

500

7 · 10−5

Saturn

6

6

0.3 µas

0.1 µas

0

0.016

200

3 · 10−6

Table 3. Determination of the PPN parameter γ with space astrometry. HIPPARCOS

σH /σG

GAIA

10 stars V < 10

8 · 10 stars V < 13

2.5 · 10 abscissas

2.5 · 108 abscissas

σ ∼ 3 to 8 mas χ > 47◦

5

6

⇓

6

=⇒

10

σ ∼ 10 µas

=⇒

400

χ > 35◦

=⇒

3

⇓ −3

σγ ∼ 3 · 10

⇓ −7

σγ ∼ 5 · 10

⇐=

12 000

3.2.2. Space curvature determination Like HIPPARCOS, the astrometric observations of GAIA will be modeled by including the effect of the light deflection in the computation of the apparent direction. Even in the case of HIPPARCOS, the magnitude of the solar deflection at wide angle was large enough (4 mas at 90 degrees) to introduce the PPN parameter γ as an additional unknown to be solved as a general parameter within the astrometric solution. Although the strong correlation between the deflection and parallactic displacement limited the precision, γ was found in agreement with general relativity within ±0.003 [1]. With GAIA the individual observations will be much more precise and the solar avoidance angle smaller, which at the same time increases the signal and decreases the correlation with the parallax. Scaling on the HIPPARCOS result, table 3 shows that a determination of γ better than one part in a million could be achieved with GAIA. The last column of the table gives the ratio of the standard deviation for each scaling factor. This estimate is based on the 3 · 106 stars brighter than the 13th magnitude for which the expected astrometric accuracy is better than 4 µas (table 1) and a correlation of 0.85 has been considered between the zero point of the parallaxes and γ. The extreme accuracy of the astrometric measurements will allow determination of γ to an unprecedented accuracy of few 10−7 . This rests upon the ability to determine global parameters affecting in more or less the same way all the observations on all the stars, yielding an improvement with the square root of the number of observations. The difficulty is not to solve for γ, but to make sure that any other global parameter, not orthogonal to the subspace determined by the condition equations of γ is properly accounted for in the model. Otherwise, the real meaning of the parameter so determined will be questionable. 3.3. Non linearity of gravitation GAIA will observe and discover many hundreds of thousands of minor planets during its five year mission. Most of these will belong to the asteroidal main-belt comprising objects, primarily located between Mars and Jupiter and having small to moderate eccentricities. However there are also other populations exhibiting much smaller semi-major axes and/or very large eccentricities. For example, planets of the Amor group have perihelia between 1 to 1.3 AU and approach or cross the orbit of the Earth.

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Table 2. Light deflection (monopole: left, quadrupole: right with J2 = 10−6 ).

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The relativistic effect and the solar quadrupole cause the orbital perihelion to precess at the rate: ∆ =

2 3πJ2 R 6πλGM + a(1 − e2 )c2 a2 (1 − e2 )2

(5)

where λ = (2γ − β + 2)/3 is the PPN precession coefficient and the rate is given in radians per revolution. For the relativistic precession one has in mas/yr: ∆ =

38 a5/2 (1 − e2 )

(6)

where a is in AU. For the main belt this is about seven times smaller than for mercury in rate per revolution and more than a hundred times in absolute rate. The current list of numbered asteroids with good or approximate orbits totals more than 105 objects. Nearly 90 percent have a very small perihelion precession, less than 3 mas/yr, that will not be usable for GR testing. However there are ∼ 300 with ∆  20 mas/yr and a handful for which the precession rate is even larger than 100 mas/yr ( figure 3). A few cases of earth-crossing objects are illustrated in table 4 giving a very significant yearly precession, due to a favorable combination of distance and eccentricity. These objects are predominantly small (0.5 to 3 km in radius), so the accuracy should not be too much

Figure 3. Distribution of the known minor planets in the plane semi-major axis — eccentricity. The level curves give the relativistic perihelion precession rate in mas/yr.

Table 4. Perihelion precession due to general relativity and the solar quadrupole moment for Mercury, a typical main-belt object and relevant objects with large precession rate. ∆ (mas/yr)

J2 (= 10−6 ) (mas/yr)

a (AU)

e

Mercury

0.39

0.21

Asteroid (main belt)

2.70

0.1

1566 Icarus

1.08

0.83

101

0.30

2100 Ra-Shalom

0.83

0.44

76

0.12

Body

423 3.4

1.24 0.001

2340 Hator

0.84

0.45

74

0.12

3200 Phaeton

1.27

0.89

102

0.41

5786 Talos

1.08

0.82

101

0.30

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affected by their apparent diameter. The main difficulty will be the evaluation of a correction to be applied to the astrometric positions for the phase effect to refer the position to the center of mass instead of the center of light and then to determine good orbital parameters from the GAIA observations. The observational conditions for this sample of minor planets must be investigated in detail to determine the fraction which are actually too faint when crossing the astrometric field of view of GAIA, making the orbital fitting difficult to achieve with the required accuracy because of limited coverage or biased coverage. A determination of λ with an accuracy of 10−4 is a reasonable goal with a value closer to 10−5 probably achievable with favorable statistics. Simulations are not in a sufficiently advanced state at the moment to be more precise and one must be cautious as long as the orbit determination problem from GAIA short arc observations is not solved. Compared to the huge experience accumulated with ground-based observations, GAIA scientists must reformulate the problem, nearly from scratch because the positional data will be one dimensional and very accurate but will cover at the beginning a very short timespan, of the order of few hours, from which one will have to compute a preliminary orbit. Then, when an object is identified at the next passes, the orbit will be refined with the best precision allowed by the observations. The situations will be simpler for already known asteroids for which one can benefit for an already good orbit from groundbased observations. This will put the identification on much safer grounds. In any case there is no contender at this level of accuracy in the coming decade, with the exception of the ESA mission to Mercury with the reservations expressed in the study report about the possibility of high precision tracking [8]. 3.4. Variation of the Newton constant The possible time variability of G has been revived with the appearance of the super-string theories. Astronomical observations are best placed to detect a tiny change in G because they can benefit from a cumulative effect, either in planetary motion or in stellar evolution. Recent determinations [9,10] have ˙ shown that G/G < 0.5–1 · 10−11 yr−1 . The most recent solution of lunar ranging data has yielded ˙ G/G = (0 ± 1.1) · 10−12 yr−1 [11]. White dwarfs are good candidates to test the variability of G over a timescale comparable to the age of the Galaxy. A white dwarf is one of the end-states of stellar evolution for low to intermediate mass stars after they have exhausted their nuclear fuel. All stars with initial masses up to about eight solar masses finally end up as white dwarfs when pressure of degenerate electrons balances the gravitational attraction. With typically 0.6 solar masses and radii of about 109 cm the mean densities of white dwarfs are of the order of 105 –106 g/cm3 so that these stars can be considered as laboratories for matter at extreme densities and pressures. Regarding the energy production, white dwarfs are dead stars which loose progressively their internal energy by radiation in the outer atmosphere as they cool down. The slow energy release depends on the chemical composition of the interior (usually dominated by carbon and oxygen) and the properties of the external envelope. The rate of cooling is largely determined by radiative diffusion through this almost insulating layer of helium and/or hydrogen surrounding a largely isothermal interior. Therefore the luminosity as a function of time depends on the rate at which internal and gravitational energy can be evacuated [12]. Because of the long time scales involved, any departure from standard physics (quantum or classical) will result in sizeable effects in the observable properties of the population of white dwarfs. Garcia-Berro [13] ˙ derived from the luminosity function of the white dwarfs an upper bound for G/G
−(1 ± 1) · 10−11 yr−1 . The radius of a classical white-dwarf, made of classical ions and non relativistic degenerate electrons is given by:  R ∝ R

M M

1/3 (7)

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(the relativistic equivalent leads to the so-called Chandrasekhar limit for M ). Combining equation (7) with the Stefan law, gives for a surface temperature TS , L = 4πR2 σTS4 : 

M L∝ M

2/3 TS4 L

(8)

While a white dwarf of given mass cools down, there is no change of structure and its luminosity declines exactly as the fourth power of its surface temperature, in contrast with a star, whose size changes by two orders of magnitude during its evolution. Then the luminosity can be related to the cooling and the variation of G as [13]: L=−

d(E + U ) G˙ + U dt G

(9)

where E is the internal energy (primarily the thermal energy of the ions) and U the gravitational energy. A careful modeling of the internal structure of the stars and that of its thin atmosphere allows the studying ˙ of L(t) and the determination of the cooling time scale as a function of G/G. Clearly since no white dwarfs can be older than our Galaxy (white dwarfs are much too faint to be observed in external galaxies, even in the local group), the luminosity function (number of objects per bin of brightness and per unit of volume) must exhibit a cut-off corresponding to the luminosity of the oldest ˙ stars. The precise location of the cut-off being sensitive to G/G, this provides the basis for the testing with GAIA . Today only about 2000 white dwarfs are catalogued and GAIA could multiply this number by 100, giving plenty of material to refine the luminosity function and the location of its upper limit. However, the interpretation of the luminosity cut-off involves significant modelling and is also dependent on the age of the stellar populations in the solar neighborhood. Assuming that GAIA can provide this number from ˙ calibration of stellar evolution models, the upper limit of G/G could be decreased to 10−12 –10−13 yr−1 . 4. Conclusion GAIA faces a very bright and exciting future but somewhat remote. In between there will be two smaller missions relying on the same concept pioneered by HIPPARCOS. The first one is DIVA, a small German mission due for launch in early 2004 with an astrometric goal of 200 µas at V = 9 mag for a survey down to V = 15 including about 40 million objects. Then comes FAME, the NASA–USNO mission, aiming to 50 µas at V = 9 mag for the same number of stars and scheduled for the end of 2004. Both missions will be able to determine the space curvature better than HIPPARCOS and probably comparable to the result achievable by GPB in the coming years.

Acknowledgements. This paper relies on the work of the many scientists who have contributed to the report. They are gratefully acknowledged. Daniel Hestroffer suggested the original design of figure 3.

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study

References [1] Froeschlé M., Mignard F., Arenou F., Proc. ESA Symp. Hipparcos – Venice 97, ESA SP-402, 1978, p. 49. [2] ESA 2000, GAIA, Composition, formation and evolution of the Galaxy, ESA-SCI (2000) 4. [3] Perryman M.A.C. et al., GAIA: Composition, formation and evolution of the Galaxy, Astron. Astrophys. 369 (2001) 339–363. [4] Merat P. et al., Baltic Astron. 8 (1999) 1. [5] Ma C. et al., Astron. J. 116 (1998) 516. [6] Kovalevsky J. et al., Astron. Astrophys. 323 (1997) 620–633. [7] Klioner S., in: Johnston K.J., Mc Carthy D.D., Luzum B.J., Kaplan G.H. (Eds.), Proc. IAU Coll. 180, 2000. [8] ESA 2000, BepiColombo, A mission to the planet Mercury, ESA-SCI, 2000, p. 1.

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[9] Damour T. et al., Phys. Rev. Lett. 70 (15) (1988) 2217–2219. [10] Williams J., Phys. Rev. D. 53 (1996) 6730. [11] Williams J.G., Boggs D.H., Dickey J.O., Lunar laser tests of gravitational physics, in: Proc. Ninth Marcel Grossmann Meeting, World Scientific, 2001. [12] Salaris M. et al., Astrophys. J. 486 (1997) 413–419. [13] Garcia-Berro E. et al., Mon. Not. R. Astron. Soc. 277 (1995) 801–810.

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