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Astrometric Reference Frames in the Solar System and Beyond Mem. S.A.It. Vol. 83, 1001 c SAIt 2012 Memorie della Astrometric reference frames in the solar system and beyond S. Kopeikin Department of Physics & Astronomy, University of Missouri, 322 Physics Bldg., Columbia, MO 65211, USA Abstract. This presentation discusses the mathematical principles of constructing coordi- nates on curved spacetime manifold in order to build a hierarchy of astrometric frames in the solar system and beyond which can be used in future practical applications as well as for testing the fundamentals of the gravitational physics - general theory of relativity. 1. Introduction equations in order to find out the metric tensor corresponding to the coordinate charts cover- Fundamental astrometry is an essential ingre- ing either a local domain or the entire space- dient of modern gravitational physics. It mea- time manifold. sures positions and proper motions of vari- ous celestial objects and establishes a corre- The standard theory adopted by the spondence between theory and observations. International Astronomical Union (IAU) pos- Theoretical foundation of astrometry is general tulates that spacetime is asymptotically-flat theory of relativity which operates on curved and the only source of gravitational field is the spacetime manifold covered by a set of local matter of the solar system (Soffel et al. 2003; coordinates which are used to identify posi- Kopeikin 2007). This approach completely ig- tions and to parametrize the motion of celestial nores the presence of the huge distribution of objects and observer. The set of the local co- mass in our own galaxy – the Milky Way, the ordinates is structured in accordance with the local cluster of galaxies, and the other visi- hierarchical clustering of the gravitating bod- ble matter of the entire universe. Moreover, the ies. The astrometric frame is derived after mak- current mathematical approach to the construc- ing observational connection between the the- tion of astrometric local and global coordinates oretical and observed coordinates of the celes- does not take into account the presence of the tial objects. The frame can be either dynam- dark matter and the dark energy which, alto- ical or kinematic depending on the class of gether, make up to 96% of the total energy celestial objects having been used for build- of the observed universe. It requires a certain ing the frame. The local astrometric frames are extension of the current IAU paradigm on as- not fully independent – they must be matched trometric frames to account for recent astro- to a global astrometric frame which is usu- physical discoveries. The updated astrometric- ally build by making use of quasars as bench- frame theory has to answer another important marks. Mathematical description of the local question about how the Hubble expansion of and global frames require to solve Einstein’s the universe affects our measurement of the spacetime properties. This problem has been Send offprint requests to: S. Kopeikin extensively discussed recently by Krasinsky & 1002 Sergei Kopeikin: Astrometric reference frames in the solar system and beyond Brumberg (2004), and especially by Carrera to and receives back from retroreflectors and & Giulini (2010) on the basis of exact solu- or transponders placed on other celestial bod- tions of Einstein’s equations. However, their ies. Spatial axes of the observer’s local frame arguments can not be fully accepted so far are not rotating in dynamic sense and their ori- as the post-Newtonian approximation methods entation is maintained by the gyroscopes. The have to confirm (or disprove) their mathemati- proper time, τ, and the spatial axes, ξi, of the cal derivations. local frame are considered as four-dimensional coordinates ξα = fτ; xig. Relative motion of observer is usually mea- 2. Astrometric frames in the solar sured, and is supposed to be well-known, with system respect to another local frame associated with the massive body at the surface of which (or The current IAU paradigm suggests that the by which) the observer is located. The most solar system is the only source of gravita- theoretically-elaborated example of such a lo- tional field and that at infinity the spacetime cal frame is the geocentric frame. is asymptotically flat. This paradigm has been worked out over time to the degree of practical implementation by a number of researchers. 2.2. The geocentric reference frame The most comprehensive theoretical descrip- tion of the IAU reference frames paradigm is Geocentric reference frame is a local frame presented in (Soffel et al. 2003) and in a re- parametrised with coordinates wα = fu; wig cent monograph (Kopeikin et al. 2011). The where u is the coordinate time and wi are spa- IAU paradigm operates with a number of local tial coordinates with origin at the center of and one global reference frames which are con- mass of the earth. The spatial axes have no ro- nected to each other by post-Newtonian coor- tation in kinematic sense with respect to the dinate transformations. The most important for BRF, and stretch over space out to the distance description of astrometric measurements is the being determined by the acceleration of the local frame of observer, the geocentric frame world line of geocenter (Kopeikin et al. 2011). (GRF), and the barycentric frame (BRF) of the Besides the geocentric frame, there exists a entire solar system (Kopeikin 2011). number of other local frames in the solar sys- tem associated with the massive planets and their satellites (moons). These frames are re- 2.1. The proper frame of observer quired to describe the orbital motion of plan- An ideal observer in general relativity must etary satellites, for spacecraft’s navigation, or be understood as a point-like test parti- for measuring gravitational field of the plan- cle equipped with measuring devices like ets. The geocentric and all planetocentric ref- clocks, rulers, gyroscopes, lasers, receivers, erences frames are intimately related to a sin- and other possible measuring devises (Klioner gle coordinate chart known as the barycentric 2004). Observer moves along timelike world- reference frame of the solar system. line which may be or may be not a geodesic. For example, observer located on the surface 2.3. The barycentric reference frame of the Earth is subject to Earth’s gravity force that incurs non-geodesic acceleration of grav- The barycentric reference frame of the solar ity g = 9:81 m/s2. On the other hand, ob- system is parametrized with coordinates xα = server located on board of a drag-free satellite ft; xig having the origin at the center of mass is not subject to any external force, and moves of the solar system. Because the spacetime is along a geodesic. The observer is always lo- assumed to be asymptotically-flat the spatial cated at the origin of the local (topocentric) coordinate axes of the barycentric frame go to frame. He measures the proper time τ with the (spatial) infinity. Moreover, they have no kine- help of the ideal clock, and the proper length matic rotation or deformation. Practical real- ` with the help of the ideal rules calibrated by ization of this frame is given in the form of means of light (laser) pulses which he sends ICRF that represents a catalogue of several Sergei Kopeikin: Astrometric reference frames in the solar system and beyond 1003 i hundred radio quasars uniformly distributed where VB is the velocity of the barycenter with over the whole sky (Lanyi et al. 2010). respect to the center of mass of the Milky Way (VB ' 250 km/s), and the matrix 2.4. Transformation between the frames ! ! i Ugal i j −2 1 i j [i j] L j = 1 + δ + c V VB + F ; (3) Coordinates corresponding to the astrometric c2 2 B frames of the solar system are connected to describes the Lorentz and gravitational length each other by a smooth post-Newtonian trans- i formations defined in the domain of overlap- contractions due to the velocity V and the ping the coordinate charts. These transforma- gravitational potential Ugal of the Milky Way as tions generalize the Lorentz transformation of well as a rotation of spatial axes of the barycen- tric frame with respect to the axes of the galac- special theory of relativity by taking into ac- [i j] count the effects of the gravitational field of the tocentric frame (the term with F ). The or- solar system objects. bital motion of the solar system with respect to The IAU paradigm of the astrometric refer- the galactocentric coordinates is slow. Hence, ence frames works very well in the solar sys- all quantities depending on the position of the barycenter of the solar system, Xi can be ex- tem. However, it remains unclear to what ex- B panded in the polynomial series of time. For tent the paradigm can be hold beyond the so- 1 lar system. The problem is that the solar sys- example , tem is not fully isolated from the external en- i i i 1 i 2 vironment – it moves around the center of XB(T) = XB(T0) + VBT + ABT + ::: ; (4) our own galaxy with some finite acceleration. 2 1 2 Moreover, a more important argument is that B(T) = B(T0) + B˙(T0)T + B¨(T0)T + :::(5) ; the spacetime is not asymptotically flat but is 2 described by the Robertosn-Walker metric of and so on. Here, the expanding universe. It motivates us to ex- tend the framework of the IAU paradigm and 1 B˙ = V2 + U ; (6) to build the coordinates associated with the 2 B gal galaxy and with the expanding universe. ¨ ˙ B = VB · AB + Ugal : (7) The velocity-dependent term in (1) causes 3. The galactocentric reference frame the change in the rate of the barycentric time i i Definition of the galactocentric coordinates (TCB) as measured at distance R = jX − XBj Xα = fT; Xig is a fairly straightforward gen- from the barycenter.
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