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Relativity in space astrometry projects
Adv. Space Res. Vol.9, No.9. pp. (9)61—(9)70, 1989 Printed in Great Britain. All rights reserved.
0273—1177/89 80.00 + .50 Copyright© 1989 COSPAR.
RELATIVITY IN SPACE ASTROMETRY PROJECTS F. Mignard and J. Kovalevsky CERGA, Observatoire de Ia Cote d’Azur, Grasse, France
ABSTRACT We outline the main features and capabilities of several space astrometric projects approved or currently under review. The most ambitious aim at microarcsecond accuracy in absolute astrometry. Relativistic deflection of light will be detectable in the irregular displacement of a distant star nearly aligned with a massive deflector. Likewise in some instances measurements of star diameters will have to be interpreted by Including relativistic effects.
1. INTRODUCTION General relativity theory ( GRT) has become an Indispensable tool In the hands of astrometrists quite recently, as witnessed by the appearence of several reference texts in the last ten years (Moritz 1979, Murray 1983, Kovaievsky and Brumberg 1986). The main goal of these books is to bridge the gap between practitioner astrometrists, with no specific expertise In GRr, and theoretical physicists proficient In this theory but with no marked interest In the construction of reference frame or In the definition of convenient procedures to go from coordinate-defined quantities to Invariant observables. Yet the first successfully tested prediction of GRT was the measurement of light de flection during the solar eclipse of 1919; as noted by A. Murray (1983). this was a purely astrometric result which had a deep resonance In the scientific community and in the general public as well, but with no immediate fallout as to the status of the GRT In astrometry. Major changes have occured lately to such an extent that GR’F must be introduced at the outset of the construction of reference frames and the Interpretation of the most accurate stellar observations. The foundations of classical astrometry rest heavily on the existence of an absolute eudlidlan three dimensional space, orthogonal to the absolute time as they were defined by Newton. Now all these once Undisputable postulates are reexamined from a relativistic standpoint. This includes some basics as the notion of space and time, but also the meaning of coordinates and their use over extended regions of the universe, the clockreadlng and its bearing with the parameter t of the equations, the link between photon trajectories and angular measurements, definition and practical realization of physical units to be used by astronomers, etc... This paper will be restricted to the current projects In observational astrometry which are accurate enough to involve relativistic effects in the data processing.
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F. Mignard and J. Kovalevsky
2-PRESENT STATUS OF HIGH PRECISION ASTROME’rRY In the years to come. astrometry in optical wavelengths will undergo a dramatic Increase of the precision with which the positions of celestial objects are observed. Presently, one can set the current accuracy limit of a single ground based observation of a star to something between 0”. 1 and 0”. 15 for most ofthe Instruments used: astrographs, Schmidt cameras, transit circles or astrolabes. Somewhat better resuits are obtained by long focus telescopes and some prototypes based upon other principles such as optical Interferometry (Shao et al., 1987) or the multichannel astrometric photometer (Gatewood et al.. 1986. Gatewood, 1987). The limiting precision could be of the order of 3 mas (milliseconds of arc) for the connections between stars in the same field of view and 10 mas for stars in different parts of the sky. In the latter case, the limitation is essentially due to refraction models. Let us also mention that very long baseline radio-interferometry currently achieves accuracies of the order ofone or several mas in global astrometry. In 1989, the first two satellites equipped to do astrometry from space will be launched: (i) The Hubble Space Telescope which has an astrometric focus In which the relative positions of two stars can be measured in a small field with a precision of 2 mas (Duncombe et al., 1982). However astrornetry is not the leading objective of the Instrument, and only a minImal time of observation will be allocated to astrometric programs. (ii) HIPPARCOS which is a dedicated global astrometry satellite (Kovalevsky, 1984). Its objective is to determine positions and parallaxes of about 115.000 stars with an accuracy of 2 mas and proper motions with an accuracy of 2 rnas per year for magnitude 9. degraded by a factor 2 for magnitude 12. The actual precision of an individual measurement ranges from 5 to 10 mas. Hipparcos data reduction Includes the light deflection by the Sun, which reaches 4 inas at solar angular distance of 90 degrees. No test of general relativity with significant accuracy will be possible with HIPPARCOS. Instead the relativistic deflection will be removed from the raw data and computed with the PPN parameters y and f3 set to 1, (FalIn et al. 1984). But several - stifi not approved - space projects in USA and in Europe call for another gain of two, and maybe three, orders of magnitude in relative positional accuracy. This will permit us to measure very small displacements.
3- SUB-MILLISECOND OF ARC SPACE PRWECTS Before discussing some relativistic applications of space astrometry, let us describe schematically the existing projects and their expected precision. They fall natu rally Into four groups, because of the different technologies Involved in their deve lopment. 3.1. Astrometric telescopes Several long focus telescopes In space have been proposed with different focal Instrumentation. The one that has presently the most chances to be flown someday is the ‘Astrometric Telescope Facility’ (Gatewood, 1988) which is proposed to be attached to the NASA space station. The telescope is designed as a 1.25 meter aperture reflector opened at f/d= 13. The star images are modulated by a parallel grid while the telescope moves slowly. The images of up to 32 stars, modulated by the grid are Independently detected and their relative distance perpendicularly to the grid can be obtained from the relative phases of the modulations. Finally, a relative precision of0.01 mas is expected for each dimensional measurement. The gain in precision with respect to HIPPARCOS whose detection system is based on the same principles is due to a larger telescope aperture, a longer focal distance and a slower rotation rate.
Relativity in Space Astrometry Projects
3.2. Optical space Interferometry The development of ground based optical interferometry inspired several propo sals in USA and in Europe to launch an interferometer in space. A ‘classical’ ap proach is to place the two end mirrors on a solid rail that limits their distance to about 20 meters (Shao et al., 1984). Similar distances are also proposed for multimirror aperture synthesis projects like OASIS (Noordarn et al.. 1987). Depending upon the accuracy with which the phases can be measured, the precision ranges between 5 mas and a few tenths of a mas. The possibility of measuring accurately the angular distance between two stars decreases with the separation and is limited by the stability of the structure, the pointing accuracy and the availability of accu rately calibrated delay lines. For these reasons, this class of Interferometers is restricted to small field astrometry and imaging. To achieve higher resolutions, one has to imagine that the light collecting mirrors are situated on independent satellites. This Is the case of project TRIO (Labeyrle et al.. 1985) in which several free flying satellites driven by solar sails move around a central light collecting station on a spiral slowly expanding from 5 meters to 1 Idlometer distance. An alternate design is TRIANGLE (Vakili, 1985) in which each of the three satellites is the central station for the two others. A precision of 0.01 mas or better is claimed, but the stability requirements are so difficult to meet that, as for the connected receivers, the measurements can be made only in very small fields. 3.3. SpaceVLBI The same type of conclusion can also be drawn from the description of space radiointerferometry projects and among them, one may mention QUASAT In which a space radio telescope is separated from the ground based telescope by a maximum of 47,000 kilometers. However, because of the VLBI concept in which there is no direct connection between the dishes, an astrometric link can be expected between sources separated by several degrees using the differential VLBI technique. 3.4. Double field optical interferometry The project POINTS (Reasenberg, 1986 and Reasenberg et al., 1988) Is the only genuine global astrometry proposal after HIPPARCOS. It consists of two coupled op tical interferometers that observe simultaneously in two directions 90°±3° apart. A very sophisticated system of thermal control and laser driven optical Interferome ters to determine changes in critical dimensions are designed in order to control and monitor the stabffity of the instrument so that the authors claim a single ob servation precision of 0.005 mas. With a sufficient number of observations relative positions and parallaxes could be obtained to better than one m.tcroarcsecond. Because of funding problem only one, if any, of the Astrometric Telescope and POINTS will be ultimately built. (Reasenberg 1988, personal communication). 4- APPLICATION TO GENERAL RELATIVI’IY
As soon as measurements In the submilliarcsecond level are achievable, observa tions must be reduced In the framework of general relativity. The most promising applications are In small field astrometry, although the most exciting might lie in global astrometry. A careful comparison of the kinematical and dynamical refe rence frames should yield valuable information on the Mach’s principle ( Bertotti et al. 1984). We know, thanks to the accurate measurements of the cosmic background radiation (Weiss. 1980). that the anisotropy is less than 0.001. Already in 1969, S. Hawking pointed out that this indicates that the universe is rotating very little. If at all...This could possibly be regarded as an experimental verification of Mach’s principle
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Mignard and J. Kovalevsky
The rotation ofthe universe has been probably less than few degrees since the bigbang, or io-3 to iO~mas /year. This would be the order ofmagnitude of the rotation of the dynamical frame with respect to a frame geometrically linked to extra galactic sources. Such a level of precision is out of reach to modern techniques by at least three orders ofmagnitude (Dickey. 1989), prImarily because the dynamical frame cannot be realized to a better accuracy. Thus the small field astrometry will remain for some years the main domain where relativistic effects are observable with significant accuracy. In the following we consider effects connected to the light deflection that are accessible to mllliarc second Interferometry. 4.1 Apparent direction of a star. In the Newtonian framework there Is an Immediate association between the space vector k of the incoming photon and that of an euclidlan straight line, that is to say a geometrical object. In this statement we assume that all corrections that modlfy k (aberration, refraction) are properly computed, so that we can refer to a geo metrical direction. In GRT we must consider first the concept of direction as a geometrical entity and then link the spacetime coordinates of a luminous source to the local measurement of k in a Lorentz frame. In fact the first point Is of little Interest because we have no access to the photon path between its emission by the source and reception by the observer. It is therefore rather academic to speak of the tangent to the photon geodesic thousands of light years away. Instead we will rely on observables and define the direction In space by the space components of the 4-vector kit, which lie In the 3-space orthogonal to the observer’s 4-velocity. We consider the motion of a photon from a source to a detector In the vicln.tty of the earth. Though this question is standard In GRT. we need to apply the results In situations that differ for two reasons from those usually considered in the computations found in textbooks: The source Is not necessarily at a distance very large compared with that of the deflector. For example a binary star where the mutual distance between the components is very small with respect to the distance from the solar system. Take an other situation: when the light of a distant star is deflected by an object located somewhere between the sun and the star, say halfway between, the dis tances are ofcomparable magnitude and the source cannot be considered at infinity. The result is that the photon path must be determined with boundary conditions rather than with Initial conditions( one point and an Initial directon). -
The deflections of interest In space astrometry will be very small in absolute 2 r = 1 to 10 mas) but not small in comparison with the angu values (- GM/c lar separation between the source and the deflector. Hence one cannot work with the linear problem when the deflection is determined from the path followedby the unperturbed ray and added linearly to the euclidlan direction. -
Deflector
~I~I=roIX 1~=r1
Earth
FIg 1. Geometry and notations of the light deflection.
Relativity in Space Astrometry Projects
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2 The basic equation of motion of a photon in the field of a mass m, i.e. m = GM/c in dimensional terminology, is derived In Will (TEGP 1981) and we follow closely his notations. The essential definitions are shown in Fig. 1. The path ofa photon starting at the PPN coordinates X 0, ~
with the Initial direc -
ton n Is Zt)=X0+n(t-t)+X~ (1) where accounts for the departure from the Euclidian straight line. All 3-D vec tors are to be only interpreted 2y2+x3y3. as compact In addition writingthe forstandard the tripletmeaning (xl.x2,xs)ofof vecto PPNrial notationsand Is recovered in the newtonian limit. coordinates, X.Y=x’y’÷x
From Will (Eq. 7.9) we have (with y dX~
1):
md X.n
1 0.fl\
ar
~2Un.2_~_~~)
where U is the Newtonian potential of the deflector at X, d the Impact vector along the unperturbed rays; r = Xl and similarly for the initial position. By integrating (2) between ~ and t. it is possible to obtain explicitely X(t) as a function of X0 and n. Then by solving for a one obtains the Initial direction for the ray which goes through the boundaries X,~and X1. In this way the equation of motion Is solved with boundary conditions. When Inserted into (2) one gets the tangent vector at X1: I
=
~(l-2U)
-
2md(ro
-
x1.
)
2r~. where D = X1 - Xj and d = (I) x (X0 x D ))/ (i~ Equatlon(3) can be applied also to a photon emitted by the deflector, so as to define In an operational way what the observer means by the direction of the deflector. The angle between the two photons at their reception will be the observable quantity measured by an imaging device. Remark
- In general for a point-mass deflector there are two rays saUs~r1ng the boundary conditions, passing on each side of the deflector. We have chosen the path closest to the undisturbed ray. The other gives a faint secondaiy Image very close to the deflecting star.
If we neglect the observer velocity in the PPN frame ( cf. Will),
cos e = (-g 00)
(4)
In PPM approximation: = (-1
+ 2 U)
g11=(1+2U)8~ which yields the important result: cose ~
2m
d.x1
I
X0X1 r1
(5)
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F. Mignard and J. Kovaievsky
Remark - The computation of the observable angle by Eq.(4) is essential to ensure that the measurable quantity is coordinate-free. It was noted in the past (see for example Clube. 1982) that the functional expression of the light deflection was dependent upon the space-time metric assumed, for instance Schwarzchlld v.s Isotropic coordinates, and taken as an argument against the predictions of general relativity. While this is true for the coordinate angles, for the choice of which we have much freedom, this is absolutely untrue for the observable angle between two well defined directions In spacetime. 4.2 LimitIng cases. Equation (5)ls valid for all separations and mutual distances between the observer, the deflector, and the source. When r 0 —~ the classical deflection for an InlI nitely remote source is recovered. For an alternative derivation see Brumberg (1980,1989). For small separatIons O,O~<< 1 and (5) takes the simpler form, ~
If r0 >or
—
~
(6)
r1thenwehave X~.X1/r0r1
-l
hence
2
o
=
2 00
8mrO +—~——
When the deflection Is small in comparison with the separation, 0—00 << 00, the following linear expression applies also, 2mrO _____
o
_Oo+_x~_~_l~rorij
(8)
which with the same conditions as in (7) gives, r
0= ~0 +
~
(9)
5- APPLICATIONS
The observational consequences of the relativIstic light deflection, or gravitatio nal lensing. have received considerable attention since Einstein reported the resuits of his calculations in science In1936. Lately the interest was renewed after the first detection of a lensed quasar (Walsh et al. 1979). In addition to generate ghost images of distant objects, extended objects are strongly distorted by gravitational lensing (Nconan 1983), to such an extent that it should be detectable by space Interferomters. One of the most conspicuous phenomena that may occur when observing stars in the sky is the modification of the apparent path of a star that happens to pass almost behind another one. (Choilet 1979). The underlying geometry Is illustrated In Fig. 2.
Relativity in Space Astrometry Projects
Figure 2. Geometric and apparent path on the sky of a distant star nearly aligned with a massive deflector. The very accurate observatIons permitted by hIgh resolution techniques as described in the previous sections, will enable us to observe the phenomena and then to deduce the mass of at least one of the stars. Actually, there are two completely different situations that are described by two distinct algorithms. The first case occurs when the two stars are unrelated and by virtue of relative proper motions. their position on the sky are very close. The other case corresponds to physical double stars: one of the components revolves around the other and passes behind it. Let us describe both situations. 5.1. Relative apparent motion of two stars.
In this case Eq.(7) applies and leads to: 92
where a = la.u and w, ~‘ are respectively the parallax of the deflector and of the source. When angles and parallaxes are expressed in mas we have, with sufficient accuracy, the convenient numerical expression with M in solar mass: 0~+
92
We have plotted in fig.(3) the apparent path of a distant star with respect to a deflector kept fixed at the origin of the axes. The y-axis is chosen in the direction of the relative proper motion before the closest approach. ~ 40
_Para!lax
•
10 mas _____________________________________ 15m
ISmas
Figure 3. Apparent motion of stars with different impact parameters with respect to a deflector placed at the origin of coordinates. The scale is in mas in both axes.
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F. Mignard and 1. Kovalevsky
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The illustration is drawn for a parallax difference of 10 mas and a deflector of 1 solar mass. Closest approaches are defined along the undisturbed ray. It is seen that at the mllliarcsecond level the departure from the euclidian straight line. measured with an accuracy of a few microarcseconds would provide the deflector mass to better than i~-~provided that the measurements of the respective apparent positions of the two stars are made over a sufficiently long timespan to obtain also the parallax difference and the asymptotic proper motion. The components of the proper motion are shown in fig.4 for a star with an impact parameter of 3 mas. One should not forget that the signature is striking here only because the axes are well chosen with the y-axls parallel to the relative asymptotIc motion. On an astronomical axis one would get a linear combination of the two curves. The time scale is not given here, as It is directly linked to the actual proper motion. The two units are scaled to the value of dy/dt at large distance.
Parallax:
10
reas
Figure 4. Components of the proper motion of a star whose llghtray is bent by a front deflector. The abscissae is affinely connected to the time. The deflector being the closer of the two objects is likely to exhIbit the larger proper motion and thus in an absolute frame it is the deflector which is passing In the light of sight of the source. In this frame the absolute motion of the source is easily computed from (7) and turns to be almost a circle. (This interesting property was found by Gorenstein and Shapiro In 1986 but not published; see also Reasenberg et ai 1988). This can be understood as follows: In the linear regime valid for small deflections, the deflection 89 is. 30
1/d
where d Is the angular distance between the source and the deflector. This is also the angular separation between the source and its image. For a rectilinear motion of the deflector with respect to the source we have d=dm/cos~ where tim Is the closest approach and ~ a polar angle. Hence 30 ac cos ~ which is nothing but the polar equation of a circle with the source as a pole. One might fear that this motion could be mistaken for an orbital motion, but fortunately it is quite nonuniform and thus will be identified if observed over enough time. Deflection might be produced also by an unseen massive star or even a black hole (Schutz, 1982).
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5.2 Apparent shape of a star. Another effect will be conspicuous if the farther star, S. has a significant apparent diameter. Then when the angular separation becomes small, the circular shape of S Is deformed and becomes oval or crescent-like, the shortest diameter b being In the direction of the deflector and the largest perpendicular to this direction. Figure 5 gives an example of this deformation for a representative case. We have used the same parameter as before: 1 solar mass for the deflector, 10 mas for the parallax difference and a giant star with an apparent diameter of 2 mas. These figures are consistent with the present domain of application ofground-based interferometry.
D = 100 parsecs R=50R
3
=
sun
6
2mas
12
15
.4 10 mas
FIgure 5. Deformation of the apparent shape of a star with sizeable apparent diameter by a deflector placed at the origin of coordinates. The shaded circle is the true shape of the star and the white oval is its apparent contour. The luminosity is no longer uniform in the apparent oval.The photocenter is displaced as explained in sect. 5.1. The distortion is significant in many cases of interest even with present technolo gy Interferometiy. The data reduction should take this phenomena into consideration when assuming an a-priori luminosity distribution to carry out the deconvolution of the signal, otherwise the meaning of a star diameter would become less obvious. When small deformations are involved the actual apparent shape Is well approximated by a spheroid with a flattening very weakly dependent upon the angular separation of the star with the deflector. If this is not the case the apparent contour is very distorted.
5.3 Physical double stars. In the case of a double star one expects intuitively to observe an enlarged apparent orbit, with a magnification comparable to the deflection itself. It turns out that this apparently sound view is false. Eqs. 6 or 8 apply to a physical double star, with r 0 << r1 and r1= ~. The dot product in the brackets can take any value between -1 and 1. Eq. 6 shows that the deflection is reduced by a factor r0/~as compared to the same ray passing at a closest distance d from the deflector but with the source at very large distance, when we would have r0/Ea = 1. Actually it Is not really the de flection which is reduced since the deflection along the ray has still the value predicted by the GRr, but the difference between the geometric and apparent directions. This accounts also for the fact that the apparent diameter of the sun is not increased by 1.75 arcsec by light deflection, despite the fact that grazing rays are deflected half this quantity. Hence, in case of physical double stars, although the is noseparation hope at present to the detect any signifideflection reaches several 7arcsec. mas. theThere apparent between components is modified by less effect than 10cant relativistic due to the lIght deflection in the apparent orbit of physical binaries. Neither do we expect to see relativistic dynamical effect in the near futu ret the case of the binary pulsar PSR 1913 +16 is In this respect very peculiar.
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F. Mignard and J. Kovalevsky
REFERENCES Bertotti, B., Farinella, P., MilanI, A., Nobili, A.M., Sacerdote, F., 1984, .Astron. Astrophys., 133. 231-238. Brumberg. V.A., 1981, “Relativistic Reduction of astronomical measurements” in Reference Coordinate Systems for Earth dynamics, E.M. Gaposhkin, B. Kolaczek (eds.). Reidel, Chollet, F., 1979, Comptes rendus Academic des ScIences, ParIs, 288, ser.B, p. 163. Clube, 1982, in Scientific aspects of the Hipparcos Mission, In ESA SP-177, p. 187. Dickey, J.O., 1989, “Intercomparison between kinematic and dynamical systems”. To appear in Reference Systems, Kolaczek B. and Kovalevsky J. (eds.). Duncombe, RL., Benedict, G.F., Hemenway, P.D., Jefferys, W.H. and Shelus, P.D.. 1982, in N.B. Hall (ed.). “The Space Telescope Observatory”, NASA Publication CP2244. p. 114. Falin, J.L., Froeschl& M., Mignard, F., 1984, “Calcul des Positions Apparentes des ~toiles pour la mission Hipparcos”, CERGA Note Technique n°10. Gatewood, G.D., Stein, J., Kiewiet de Jorige, J., Di Fatta, C. and BreakLron, L, in “Astrometric Techniques”. H.K. Elchhorn and RJ. Leacock (eds.), Reidel PubI. Co., Dordrecht, p. 341. Gatewood, G.D., 1987, Astron. J., 94, p.213. Gatewood, G,D.. 1988, ‘The astrometric telescope falicity”, in Fundamentals of Astrometxy, LAU Collquium 100, Belgrade (In press). HawkIng, 5., 1969, In ‘The rotation ofthe Universe”, The Observatory, 89, 38-39. Kovalevsky, J., 1984, Space Science Reviews, 39, p. 1. KovalevskyJ., Brumberg V.A., 1986. Relativity in Celestial Mechanics and Astrometry, Reidel. Labeyrle, A., Authier, B., Dc Graauw, Th., Kibblewhite, E, and Weigelt, G.. 1985. in “Kilometric Optical Arrays in Space”, ESA SP-226, p. 27. Moritz, H., 1979, “Concept in Geodetic reference Frames”, Reports of the department of Geodetic Science, n°294. Murray, C.A.. 1983. Vectorial Astrometry, Adam Hilger, Bristol. Noonan, T.W., 1983, Ap. J. 270, 245-249. Noordam, J.E., Greenway, A.H., Bregman. J.A. and Le Poole, RS., 1987, In “ESA workshop on Optical Interferometry In Space”, ESA SP-273, p. 51. Reasenberg. RD., 1986, in “Astrometric Techniques”, H.K. Eichhorn and RJ. Leacock (eds.), Heidel Publ. Co.. Dordrecht, p. 321. Reasenberg. RD. and 14 other authors. 1988. Astron. J. 96, 1731-1745. Schutz B.F., 1982 . Relativistic effects in Hipparcos data. In Scientific Aspects of the Hipparcos Mission. ESA SP-177, p.181. Shao, M., Colavita, M., Staelln, D. and Johnson, K., 1984, Bulletin of the American Astron. Soc., 16, p. 750. Shao, M., Colavita, M., Staelin, D.H., Johnston, K.J., Simon. RS., Hughes, J.A. and Hershey, J.L. 1987, Aslron. J., 93, p. 1287. Vakifi, F., 1985, In ‘~KilometricOptical Arrays in Space”. ESA SP-226, p. 107. Walsh D., Carswell, RF., Weymann RJ. 1979. Nature 279,381. Weiss. R, 1980, “Measurements of the Cosmic Background Radiation”, Ann. Rev. of Astron. and Astrophys., 18, 489-535. Will, C., 1981, press).
‘Theory and experiment in gravitational physics”, Cambridge Univ.
0273—1177/89 80.00 + .50 Copyright© 1989 COSPAR.
RELATIVITY IN SPACE ASTROMETRY PROJECTS F. Mignard and J. Kovalevsky CERGA, Observatoire de Ia Cote d’Azur, Grasse, France
ABSTRACT We outline the main features and capabilities of several space astrometric projects approved or currently under review. The most ambitious aim at microarcsecond accuracy in absolute astrometry. Relativistic deflection of light will be detectable in the irregular displacement of a distant star nearly aligned with a massive deflector. Likewise in some instances measurements of star diameters will have to be interpreted by Including relativistic effects.
1. INTRODUCTION General relativity theory ( GRT) has become an Indispensable tool In the hands of astrometrists quite recently, as witnessed by the appearence of several reference texts in the last ten years (Moritz 1979, Murray 1983, Kovaievsky and Brumberg 1986). The main goal of these books is to bridge the gap between practitioner astrometrists, with no specific expertise In GRr, and theoretical physicists proficient In this theory but with no marked interest In the construction of reference frame or In the definition of convenient procedures to go from coordinate-defined quantities to Invariant observables. Yet the first successfully tested prediction of GRT was the measurement of light de flection during the solar eclipse of 1919; as noted by A. Murray (1983). this was a purely astrometric result which had a deep resonance In the scientific community and in the general public as well, but with no immediate fallout as to the status of the GRT In astrometry. Major changes have occured lately to such an extent that GR’F must be introduced at the outset of the construction of reference frames and the Interpretation of the most accurate stellar observations. The foundations of classical astrometry rest heavily on the existence of an absolute eudlidlan three dimensional space, orthogonal to the absolute time as they were defined by Newton. Now all these once Undisputable postulates are reexamined from a relativistic standpoint. This includes some basics as the notion of space and time, but also the meaning of coordinates and their use over extended regions of the universe, the clockreadlng and its bearing with the parameter t of the equations, the link between photon trajectories and angular measurements, definition and practical realization of physical units to be used by astronomers, etc... This paper will be restricted to the current projects In observational astrometry which are accurate enough to involve relativistic effects in the data processing.
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F. Mignard and J. Kovalevsky
2-PRESENT STATUS OF HIGH PRECISION ASTROME’rRY In the years to come. astrometry in optical wavelengths will undergo a dramatic Increase of the precision with which the positions of celestial objects are observed. Presently, one can set the current accuracy limit of a single ground based observation of a star to something between 0”. 1 and 0”. 15 for most ofthe Instruments used: astrographs, Schmidt cameras, transit circles or astrolabes. Somewhat better resuits are obtained by long focus telescopes and some prototypes based upon other principles such as optical Interferometry (Shao et al., 1987) or the multichannel astrometric photometer (Gatewood et al.. 1986. Gatewood, 1987). The limiting precision could be of the order of 3 mas (milliseconds of arc) for the connections between stars in the same field of view and 10 mas for stars in different parts of the sky. In the latter case, the limitation is essentially due to refraction models. Let us also mention that very long baseline radio-interferometry currently achieves accuracies of the order ofone or several mas in global astrometry. In 1989, the first two satellites equipped to do astrometry from space will be launched: (i) The Hubble Space Telescope which has an astrometric focus In which the relative positions of two stars can be measured in a small field with a precision of 2 mas (Duncombe et al., 1982). However astrornetry is not the leading objective of the Instrument, and only a minImal time of observation will be allocated to astrometric programs. (ii) HIPPARCOS which is a dedicated global astrometry satellite (Kovalevsky, 1984). Its objective is to determine positions and parallaxes of about 115.000 stars with an accuracy of 2 mas and proper motions with an accuracy of 2 rnas per year for magnitude 9. degraded by a factor 2 for magnitude 12. The actual precision of an individual measurement ranges from 5 to 10 mas. Hipparcos data reduction Includes the light deflection by the Sun, which reaches 4 inas at solar angular distance of 90 degrees. No test of general relativity with significant accuracy will be possible with HIPPARCOS. Instead the relativistic deflection will be removed from the raw data and computed with the PPN parameters y and f3 set to 1, (FalIn et al. 1984). But several - stifi not approved - space projects in USA and in Europe call for another gain of two, and maybe three, orders of magnitude in relative positional accuracy. This will permit us to measure very small displacements.
3- SUB-MILLISECOND OF ARC SPACE PRWECTS Before discussing some relativistic applications of space astrometry, let us describe schematically the existing projects and their expected precision. They fall natu rally Into four groups, because of the different technologies Involved in their deve lopment. 3.1. Astrometric telescopes Several long focus telescopes In space have been proposed with different focal Instrumentation. The one that has presently the most chances to be flown someday is the ‘Astrometric Telescope Facility’ (Gatewood, 1988) which is proposed to be attached to the NASA space station. The telescope is designed as a 1.25 meter aperture reflector opened at f/d= 13. The star images are modulated by a parallel grid while the telescope moves slowly. The images of up to 32 stars, modulated by the grid are Independently detected and their relative distance perpendicularly to the grid can be obtained from the relative phases of the modulations. Finally, a relative precision of0.01 mas is expected for each dimensional measurement. The gain in precision with respect to HIPPARCOS whose detection system is based on the same principles is due to a larger telescope aperture, a longer focal distance and a slower rotation rate.
Relativity in Space Astrometry Projects
3.2. Optical space Interferometry The development of ground based optical interferometry inspired several propo sals in USA and in Europe to launch an interferometer in space. A ‘classical’ ap proach is to place the two end mirrors on a solid rail that limits their distance to about 20 meters (Shao et al., 1984). Similar distances are also proposed for multimirror aperture synthesis projects like OASIS (Noordarn et al.. 1987). Depending upon the accuracy with which the phases can be measured, the precision ranges between 5 mas and a few tenths of a mas. The possibility of measuring accurately the angular distance between two stars decreases with the separation and is limited by the stability of the structure, the pointing accuracy and the availability of accu rately calibrated delay lines. For these reasons, this class of Interferometers is restricted to small field astrometry and imaging. To achieve higher resolutions, one has to imagine that the light collecting mirrors are situated on independent satellites. This Is the case of project TRIO (Labeyrle et al.. 1985) in which several free flying satellites driven by solar sails move around a central light collecting station on a spiral slowly expanding from 5 meters to 1 Idlometer distance. An alternate design is TRIANGLE (Vakili, 1985) in which each of the three satellites is the central station for the two others. A precision of 0.01 mas or better is claimed, but the stability requirements are so difficult to meet that, as for the connected receivers, the measurements can be made only in very small fields. 3.3. SpaceVLBI The same type of conclusion can also be drawn from the description of space radiointerferometry projects and among them, one may mention QUASAT In which a space radio telescope is separated from the ground based telescope by a maximum of 47,000 kilometers. However, because of the VLBI concept in which there is no direct connection between the dishes, an astrometric link can be expected between sources separated by several degrees using the differential VLBI technique. 3.4. Double field optical interferometry The project POINTS (Reasenberg, 1986 and Reasenberg et al., 1988) Is the only genuine global astrometry proposal after HIPPARCOS. It consists of two coupled op tical interferometers that observe simultaneously in two directions 90°±3° apart. A very sophisticated system of thermal control and laser driven optical Interferome ters to determine changes in critical dimensions are designed in order to control and monitor the stabffity of the instrument so that the authors claim a single ob servation precision of 0.005 mas. With a sufficient number of observations relative positions and parallaxes could be obtained to better than one m.tcroarcsecond. Because of funding problem only one, if any, of the Astrometric Telescope and POINTS will be ultimately built. (Reasenberg 1988, personal communication). 4- APPLICATION TO GENERAL RELATIVI’IY
As soon as measurements In the submilliarcsecond level are achievable, observa tions must be reduced In the framework of general relativity. The most promising applications are In small field astrometry, although the most exciting might lie in global astrometry. A careful comparison of the kinematical and dynamical refe rence frames should yield valuable information on the Mach’s principle ( Bertotti et al. 1984). We know, thanks to the accurate measurements of the cosmic background radiation (Weiss. 1980). that the anisotropy is less than 0.001. Already in 1969, S. Hawking pointed out that this indicates that the universe is rotating very little. If at all...This could possibly be regarded as an experimental verification of Mach’s principle
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Mignard and J. Kovalevsky
The rotation ofthe universe has been probably less than few degrees since the bigbang, or io-3 to iO~mas /year. This would be the order ofmagnitude of the rotation of the dynamical frame with respect to a frame geometrically linked to extra galactic sources. Such a level of precision is out of reach to modern techniques by at least three orders ofmagnitude (Dickey. 1989), prImarily because the dynamical frame cannot be realized to a better accuracy. Thus the small field astrometry will remain for some years the main domain where relativistic effects are observable with significant accuracy. In the following we consider effects connected to the light deflection that are accessible to mllliarc second Interferometry. 4.1 Apparent direction of a star. In the Newtonian framework there Is an Immediate association between the space vector k of the incoming photon and that of an euclidlan straight line, that is to say a geometrical object. In this statement we assume that all corrections that modlfy k (aberration, refraction) are properly computed, so that we can refer to a geo metrical direction. In GRT we must consider first the concept of direction as a geometrical entity and then link the spacetime coordinates of a luminous source to the local measurement of k in a Lorentz frame. In fact the first point Is of little Interest because we have no access to the photon path between its emission by the source and reception by the observer. It is therefore rather academic to speak of the tangent to the photon geodesic thousands of light years away. Instead we will rely on observables and define the direction In space by the space components of the 4-vector kit, which lie In the 3-space orthogonal to the observer’s 4-velocity. We consider the motion of a photon from a source to a detector In the vicln.tty of the earth. Though this question is standard In GRT. we need to apply the results In situations that differ for two reasons from those usually considered in the computations found in textbooks: The source Is not necessarily at a distance very large compared with that of the deflector. For example a binary star where the mutual distance between the components is very small with respect to the distance from the solar system. Take an other situation: when the light of a distant star is deflected by an object located somewhere between the sun and the star, say halfway between, the dis tances are ofcomparable magnitude and the source cannot be considered at infinity. The result is that the photon path must be determined with boundary conditions rather than with Initial conditions( one point and an Initial directon). -
The deflections of interest In space astrometry will be very small in absolute 2 r = 1 to 10 mas) but not small in comparison with the angu values (- GM/c lar separation between the source and the deflector. Hence one cannot work with the linear problem when the deflection is determined from the path followedby the unperturbed ray and added linearly to the euclidlan direction. -
Deflector
~I~I=roIX 1~=r1
Earth
FIg 1. Geometry and notations of the light deflection.
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2 The basic equation of motion of a photon in the field of a mass m, i.e. m = GM/c in dimensional terminology, is derived In Will (TEGP 1981) and we follow closely his notations. The essential definitions are shown in Fig. 1. The path ofa photon starting at the PPN coordinates X 0, ~
with the Initial direc -
ton n Is Zt)=X0+n(t-t)+X~ (1) where accounts for the departure from the Euclidian straight line. All 3-D vec tors are to be only interpreted 2y2+x3y3. as compact In addition writingthe forstandard the tripletmeaning (xl.x2,xs)ofof vecto PPNrial notationsand Is recovered in the newtonian limit. coordinates, X.Y=x’y’÷x
From Will (Eq. 7.9) we have (with y dX~
1):
md X.n
1 0.fl\
ar
~2Un.2_~_~~)
where U is the Newtonian potential of the deflector at X, d the Impact vector along the unperturbed rays; r = Xl and similarly for the initial position. By integrating (2) between ~ and t. it is possible to obtain explicitely X(t) as a function of X0 and n. Then by solving for a one obtains the Initial direction for the ray which goes through the boundaries X,~and X1. In this way the equation of motion Is solved with boundary conditions. When Inserted into (2) one gets the tangent vector at X1: I
=
~(l-2U)
-
2md(ro
-
x1.
)
2r~. where D = X1 - Xj and d = (I) x (X0 x D ))/ (i~ Equatlon(3) can be applied also to a photon emitted by the deflector, so as to define In an operational way what the observer means by the direction of the deflector. The angle between the two photons at their reception will be the observable quantity measured by an imaging device. Remark
- In general for a point-mass deflector there are two rays saUs~r1ng the boundary conditions, passing on each side of the deflector. We have chosen the path closest to the undisturbed ray. The other gives a faint secondaiy Image very close to the deflecting star.
If we neglect the observer velocity in the PPN frame ( cf. Will),
cos e = (-g 00)
(4)
In PPM approximation: = (-1
+ 2 U)
g11=(1+2U)8~ which yields the important result: cose ~
2m
d.x1
I
X0X1 r1
(5)
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F. Mignard and J. Kovaievsky
Remark - The computation of the observable angle by Eq.(4) is essential to ensure that the measurable quantity is coordinate-free. It was noted in the past (see for example Clube. 1982) that the functional expression of the light deflection was dependent upon the space-time metric assumed, for instance Schwarzchlld v.s Isotropic coordinates, and taken as an argument against the predictions of general relativity. While this is true for the coordinate angles, for the choice of which we have much freedom, this is absolutely untrue for the observable angle between two well defined directions In spacetime. 4.2 LimitIng cases. Equation (5)ls valid for all separations and mutual distances between the observer, the deflector, and the source. When r 0 —~ the classical deflection for an InlI nitely remote source is recovered. For an alternative derivation see Brumberg (1980,1989). For small separatIons O,O~<< 1 and (5) takes the simpler form, ~
If r0 >or
—
~
(6)
r1thenwehave X~.X1/r0r1
-l
hence
2
o
=
2 00
8mrO +—~——
When the deflection Is small in comparison with the separation, 0—00 << 00, the following linear expression applies also, 2mrO _____
o
_Oo+_x~_~_l~rorij
(8)
which with the same conditions as in (7) gives, r
0= ~0 +
~
(9)
5- APPLICATIONS
The observational consequences of the relativIstic light deflection, or gravitatio nal lensing. have received considerable attention since Einstein reported the resuits of his calculations in science In1936. Lately the interest was renewed after the first detection of a lensed quasar (Walsh et al. 1979). In addition to generate ghost images of distant objects, extended objects are strongly distorted by gravitational lensing (Nconan 1983), to such an extent that it should be detectable by space Interferomters. One of the most conspicuous phenomena that may occur when observing stars in the sky is the modification of the apparent path of a star that happens to pass almost behind another one. (Choilet 1979). The underlying geometry Is illustrated In Fig. 2.
Relativity in Space Astrometry Projects
Figure 2. Geometric and apparent path on the sky of a distant star nearly aligned with a massive deflector. The very accurate observatIons permitted by hIgh resolution techniques as described in the previous sections, will enable us to observe the phenomena and then to deduce the mass of at least one of the stars. Actually, there are two completely different situations that are described by two distinct algorithms. The first case occurs when the two stars are unrelated and by virtue of relative proper motions. their position on the sky are very close. The other case corresponds to physical double stars: one of the components revolves around the other and passes behind it. Let us describe both situations. 5.1. Relative apparent motion of two stars.
In this case Eq.(7) applies and leads to: 92
where a = la.u and w, ~‘ are respectively the parallax of the deflector and of the source. When angles and parallaxes are expressed in mas we have, with sufficient accuracy, the convenient numerical expression with M in solar mass: 0~+
92
We have plotted in fig.(3) the apparent path of a distant star with respect to a deflector kept fixed at the origin of the axes. The y-axis is chosen in the direction of the relative proper motion before the closest approach. ~ 40
_Para!lax
•
10 mas _____________________________________ 15m
ISmas
Figure 3. Apparent motion of stars with different impact parameters with respect to a deflector placed at the origin of coordinates. The scale is in mas in both axes.
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F. Mignard and 1. Kovalevsky
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The illustration is drawn for a parallax difference of 10 mas and a deflector of 1 solar mass. Closest approaches are defined along the undisturbed ray. It is seen that at the mllliarcsecond level the departure from the euclidian straight line. measured with an accuracy of a few microarcseconds would provide the deflector mass to better than i~-~provided that the measurements of the respective apparent positions of the two stars are made over a sufficiently long timespan to obtain also the parallax difference and the asymptotic proper motion. The components of the proper motion are shown in fig.4 for a star with an impact parameter of 3 mas. One should not forget that the signature is striking here only because the axes are well chosen with the y-axls parallel to the relative asymptotIc motion. On an astronomical axis one would get a linear combination of the two curves. The time scale is not given here, as It is directly linked to the actual proper motion. The two units are scaled to the value of dy/dt at large distance.
Parallax:
10
reas
Figure 4. Components of the proper motion of a star whose llghtray is bent by a front deflector. The abscissae is affinely connected to the time. The deflector being the closer of the two objects is likely to exhIbit the larger proper motion and thus in an absolute frame it is the deflector which is passing In the light of sight of the source. In this frame the absolute motion of the source is easily computed from (7) and turns to be almost a circle. (This interesting property was found by Gorenstein and Shapiro In 1986 but not published; see also Reasenberg et ai 1988). This can be understood as follows: In the linear regime valid for small deflections, the deflection 89 is. 30
1/d
where d Is the angular distance between the source and the deflector. This is also the angular separation between the source and its image. For a rectilinear motion of the deflector with respect to the source we have d=dm/cos~ where tim Is the closest approach and ~ a polar angle. Hence 30 ac cos ~ which is nothing but the polar equation of a circle with the source as a pole. One might fear that this motion could be mistaken for an orbital motion, but fortunately it is quite nonuniform and thus will be identified if observed over enough time. Deflection might be produced also by an unseen massive star or even a black hole (Schutz, 1982).
Relativity in Space Astrometry Projects
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5.2 Apparent shape of a star. Another effect will be conspicuous if the farther star, S. has a significant apparent diameter. Then when the angular separation becomes small, the circular shape of S Is deformed and becomes oval or crescent-like, the shortest diameter b being In the direction of the deflector and the largest perpendicular to this direction. Figure 5 gives an example of this deformation for a representative case. We have used the same parameter as before: 1 solar mass for the deflector, 10 mas for the parallax difference and a giant star with an apparent diameter of 2 mas. These figures are consistent with the present domain of application ofground-based interferometry.
D = 100 parsecs R=50R
3
=
sun
6
2mas
12
15
.4 10 mas
FIgure 5. Deformation of the apparent shape of a star with sizeable apparent diameter by a deflector placed at the origin of coordinates. The shaded circle is the true shape of the star and the white oval is its apparent contour. The luminosity is no longer uniform in the apparent oval.The photocenter is displaced as explained in sect. 5.1. The distortion is significant in many cases of interest even with present technolo gy Interferometiy. The data reduction should take this phenomena into consideration when assuming an a-priori luminosity distribution to carry out the deconvolution of the signal, otherwise the meaning of a star diameter would become less obvious. When small deformations are involved the actual apparent shape Is well approximated by a spheroid with a flattening very weakly dependent upon the angular separation of the star with the deflector. If this is not the case the apparent contour is very distorted.
5.3 Physical double stars. In the case of a double star one expects intuitively to observe an enlarged apparent orbit, with a magnification comparable to the deflection itself. It turns out that this apparently sound view is false. Eqs. 6 or 8 apply to a physical double star, with r 0 << r1 and r1= ~. The dot product in the brackets can take any value between -1 and 1. Eq. 6 shows that the deflection is reduced by a factor r0/~as compared to the same ray passing at a closest distance d from the deflector but with the source at very large distance, when we would have r0/Ea = 1. Actually it Is not really the de flection which is reduced since the deflection along the ray has still the value predicted by the GRr, but the difference between the geometric and apparent directions. This accounts also for the fact that the apparent diameter of the sun is not increased by 1.75 arcsec by light deflection, despite the fact that grazing rays are deflected half this quantity. Hence, in case of physical double stars, although the is noseparation hope at present to the detect any signifideflection reaches several 7arcsec. mas. theThere apparent between components is modified by less effect than 10cant relativistic due to the lIght deflection in the apparent orbit of physical binaries. Neither do we expect to see relativistic dynamical effect in the near futu ret the case of the binary pulsar PSR 1913 +16 is In this respect very peculiar.
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F. Mignard and J. Kovalevsky
REFERENCES Bertotti, B., Farinella, P., MilanI, A., Nobili, A.M., Sacerdote, F., 1984, .Astron. Astrophys., 133. 231-238. Brumberg. V.A., 1981, “Relativistic Reduction of astronomical measurements” in Reference Coordinate Systems for Earth dynamics, E.M. Gaposhkin, B. Kolaczek (eds.). Reidel, Chollet, F., 1979, Comptes rendus Academic des ScIences, ParIs, 288, ser.B, p. 163. Clube, 1982, in Scientific aspects of the Hipparcos Mission, In ESA SP-177, p. 187. Dickey, J.O., 1989, “Intercomparison between kinematic and dynamical systems”. To appear in Reference Systems, Kolaczek B. and Kovalevsky J. (eds.). Duncombe, RL., Benedict, G.F., Hemenway, P.D., Jefferys, W.H. and Shelus, P.D.. 1982, in N.B. Hall (ed.). “The Space Telescope Observatory”, NASA Publication CP2244. p. 114. Falin, J.L., Froeschl& M., Mignard, F., 1984, “Calcul des Positions Apparentes des ~toiles pour la mission Hipparcos”, CERGA Note Technique n°10. Gatewood, G.D., Stein, J., Kiewiet de Jorige, J., Di Fatta, C. and BreakLron, L, in “Astrometric Techniques”. H.K. Elchhorn and RJ. Leacock (eds.), Reidel PubI. Co., Dordrecht, p. 341. Gatewood, G.D., 1987, Astron. J., 94, p.213. Gatewood, G,D.. 1988, ‘The astrometric telescope falicity”, in Fundamentals of Astrometxy, LAU Collquium 100, Belgrade (In press). HawkIng, 5., 1969, In ‘The rotation ofthe Universe”, The Observatory, 89, 38-39. Kovalevsky, J., 1984, Space Science Reviews, 39, p. 1. KovalevskyJ., Brumberg V.A., 1986. Relativity in Celestial Mechanics and Astrometry, Reidel. Labeyrle, A., Authier, B., Dc Graauw, Th., Kibblewhite, E, and Weigelt, G.. 1985. in “Kilometric Optical Arrays in Space”, ESA SP-226, p. 27. Moritz, H., 1979, “Concept in Geodetic reference Frames”, Reports of the department of Geodetic Science, n°294. Murray, C.A.. 1983. Vectorial Astrometry, Adam Hilger, Bristol. Noonan, T.W., 1983, Ap. J. 270, 245-249. Noordam, J.E., Greenway, A.H., Bregman. J.A. and Le Poole, RS., 1987, In “ESA workshop on Optical Interferometry In Space”, ESA SP-273, p. 51. Reasenberg. RD., 1986, in “Astrometric Techniques”, H.K. Eichhorn and RJ. Leacock (eds.), Heidel Publ. Co.. Dordrecht, p. 321. Reasenberg. RD. and 14 other authors. 1988. Astron. J. 96, 1731-1745. Schutz B.F., 1982 . Relativistic effects in Hipparcos data. In Scientific Aspects of the Hipparcos Mission. ESA SP-177, p.181. Shao, M., Colavita, M., Staelln, D. and Johnson, K., 1984, Bulletin of the American Astron. Soc., 16, p. 750. Shao, M., Colavita, M., Staelin, D.H., Johnston, K.J., Simon. RS., Hughes, J.A. and Hershey, J.L. 1987, Aslron. J., 93, p. 1287. Vakifi, F., 1985, In ‘~KilometricOptical Arrays in Space”. ESA SP-226, p. 107. Walsh D., Carswell, RF., Weymann RJ. 1979. Nature 279,381. Weiss. R, 1980, “Measurements of the Cosmic Background Radiation”, Ann. Rev. of Astron. and Astrophys., 18, 489-535. Will, C., 1981, press).
‘Theory and experiment in gravitational physics”, Cambridge Univ.