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Astrometric Reference Frames in Cosmology

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Astrometric Reference Frames in Cosmology Astrometric Reference Frames in the solar system and beyond Sergei Kopeikin University of Missouri-Columbia GREAT-ESF workshop 6-9 June 2011 1 University of Porto, Portugal Why to bother about the “beyond” ? Curiosity is as old as humankind. Ever since humans exist, they want to explore the world they are living in. They want to reach the “Edge of the World” to see what’s beyond it, even if it puts their lives in great danger (Maggelan’s circumnavigation of the Earth) GREAT-ESF workshop 6-9 June 2011 2 University of Porto, Portugal Outline: Definitions and Resolutions Role of boundary conditions Galactocentric reference system Examples Local Group Standard of Rest (LGSR) Cosmological reference frame Gravitational potential, Gdot, Pioneer,… Concusions GREAT-ESF workshop 6-9 June 2011 3 University of Porto, Portugal Basic definitions A reference system is the complete specification of how a (celestial) coordinate system is to be formed. The specification consists of a set of rules, resolutions, recommendations, etc. which are based on a solid theory. A reference frame consists of a set of identifiable fiducial points (objects) on the sky along with their coordinates, which serve as the practical realization of a reference system. The current standard is the celestial reference system adopted by the IAU. Its origin is at the barycenter of the solar system, through appropriate modeling of VLBI observations in the framework of GR, with axes that are intended to be fixed with respect to space (time-independent,kinematically-nonrotaing orientation) GREAT-ESF workshop 6-9 June 2011 4 University of Porto, Portugal Reference System Specification in GR: Manifold + Metric + Connection + Curvature x y Local Coordinates x (x1, x2 ,..., xn ) Local Coordinates y (y1, y2 ,..., yn ) Local Coordinates z (z1, z 2 ,..., z n ) Global Coordinates z y x z GREAT-ESF workshop 6-9 June 2011 5 University of Porto, Portugal Field Equations + Boundary Conditions Cosmological and the galactic Solar System relativity effects are rather large Galaxy UNIVERSE but almost static. Difficult to stars measure but important for astrometry and gravitational GREAT-ESF workshop 6-9 June 2011 6 University of Porto, Portugal physics. The IAU 2000 paradigm: Space-time is flat at infinity. The only source Solar System of gravity is a matter stars of the solar system. GREAT-ESF workshop 6-9 June 2011 7 University of Porto, Portugal 8 Galactic Dynamics: - stars - globular clusters - dark matter halo - structure of the disc, bulge,… - propagation of light - fundamental gravitational physics (the central BH, MOND, gravitational waves) GREAT-ESF workshop 6-9 June 2011 9 University of Porto, Portugal Galactocentric Reference System • A rotational transformation of the Barycentric Coordinate Reference System supplemented by a translation of the BCRS origin to the galactic center. • The X-Y plane is the transverse plane of the disc of the Milky Way (the galactic plane ). • A “small” uncertainty in the distance to the galactic centre prevents this system M. J. Reid et al., NRAO eNews:March 2009, 2, Issue 3 from being widely used. • Galactic coordinates (no translation to the center of the galaxy) are much more common. GREAT-ESF workshop 6-9 June 2011 10 University of Porto, Portugal References to the galactocentric coordinates • Classic works by Lindblad, Oort, Ogorodnikoff, Milne • The Hipparcos Space Astrometry Mission Team • R0, 0, 0 measurements by Reid, Ghez et al., Genzel et al. • Relativistic aspects: – Damour & Taylor (1991) “On the orbital period change of the binary pulsar PSR 1913+16”, ApJ, 366, 501-511 – Brumberg & Kopeikin (1990) “Relativistic time scales in the solar system”, CMDA, 48, 23-44 – Kopeikin (1994) “Supplementary parameters in the parameterized post-Keplerian formalism”, 434, 67-70 – Kopeikin & Ozernoy (1999) “Post-Newtonian Theory for Precision Doppler Measurements of Binary Star Orbits”, ApJ, 523, 771-785 – Kopeikin & Makarov (2007) “Astrometric Effects of Secular Aberration”, AJ, 131, 1471-1478 GREAT-ESF workshop 6-9 June 2011 11 University of Porto, Portugal Galactocentric Reference System t - the barycentric coordinate time (T CB) x - the barycentric spatial coordinates (TCB coordinates)) T - the galactocentric dynamical time ( TDG) X - the galactocentric spatial coordinat es (TDG coordinates) Spacetime at infinity is asymptotically flat. The metric tensor: g tx, - the barycentric coordinates of th e solar system G TX, - the galactocentric coordinates of the Milky Way The metric tensors are found by solving the gravity field equations of Einstein's theory of general relativity or by a valid alternative theory. GREAT-ESF workshop 6-9 June 2011 12 University of Porto, Portugal Spacetime transformation xx GTX, g t, x matching the two metrics X X -15 -7 ~ 2×10 yr⁻¹ annual variation ~ 0.37 s ~ 8.4×10 1 1 2 t TBTVXXBTBBTTBTT ... c2 0 0 02 0 0 U 11 xi ijX jX j ij1 G ijVVF i j[] ij cc222 dB 1 VU2 time delay of TCB versus TDG dT 2 G dF []ij 3 VUVUi j j i the de Sitter precession of Laplace's invariable plane dT 2 GG The precession rate is ~ 0.004 µas· yr⁻¹ 1 2 XTXTVTTTATTT ... 0 0 02 0 0 VTVTATTT0 0 0 ... GREAT-ESF workshop 6-9 June 2011 13 University of Porto, Portugal Light propagation Incoming light rays X V X GREAT-ESF workshop 6-9 June 2011 14 University of Porto, Portugal The astrometric equation 2 Ksα α α αAE αO α non- kinematic catalogue aberration linear parallax terms Intrinsic the solar gravitational observed proper system deflection motion corrections (macrolensing) α μ(TT0 ) ... α α((TTT00)) α ... μμobs α ααAE ααAAA((T00)α TT ) ... ααEEEα ((TTT00) ) ... GREAT-ESF workshop 6-9 June 2011 15 University of Porto, Portugal Numerical Estimates KVK 1 kpc α 51 mas · yr 1 Distance to Sgr A* R R 1 Mpc 51 as · yr 1 Distance to the Local Group R The galactocentric acceleration KAK 1 α A 4.2 as · yr of the solar system c 2 4GM ( d ) KVK M (d ) 1 pc Macrolensing – α 10 as · yr 1 E c26 d d10 M d F. Mignard; 100 μas – α 6.379 0.026 mas · yr 1 (Reid and Brunthaler 2004 ) Sgr A* Ian Browne V αE Probability of a macrolensing event – Francois Finet P=0.006 GREAT-ESF workshop 6-9 June 2011 16 University of Porto, Portugal Pulsar Timing of PSR B1913+16 Observed value of the orbital period the solar system V 1 r c P P V kV V b, obs b 2 r B1913+16 V d = 9.9 ± 3.1 Kpc 1 2 (Weisberg et al. 2008) c PPA 2 R b, obs b, GR rad ... Pb Pcb c J.H. Taylor and collaborators The distribution of the dispersion measures at the locations of pulsars. The PPb- b,gal spiral arm locations are adoptedfrom the = 1.0081 0.0098 Damour & Taylor, 1991 Cordes-Lazio (2002) model (Figure from Pb,GR Ramesh Bhat et al. , 2004) PP- b b,gal = 0.997 0.002 Weisberg, Nice, & Taylor, 2010 Pb,GR GREAT-ESF workshop 6-9 June 2011 17 University of Porto, Portugal Fundamental galactocentric parameters and the central BH (in arcseconds) (in Orbits of stars orbiting the super-massive BH at the The seven masers (red giants), whose radio positions center of the Milky Way. are well measured by Reid et al. (2007) and which are used to establish an absolute reference frame, are circled. Picture credit to: Ghez et al. ApJ (2008) 689, 1044 GREAT-ESF workshop 6-9 June 2011 18 University of Porto, Portugal Mass-distance and mass-velocity correlation relationships Correlation of the measured mass of the central BH with 1.8 the distance to the center of the Milky Way: M R0 and velocity. 6 A. Ghez et al. : M = 4.53 0.55 10 M for R 0 =8.36 0.44 kpc GREAT-ESF workshop 6-9 June 2011 19 University of Porto, Portugal The Local Group (N. Bartel – M81, K. Jonhston – 4 QSOs) Vgal VMW 0.55 ±0.23 mas/yr (Piatek et al. 2005) V 15 - 70 µas/yr (Darling et al. 2010) -29.3±7.6 µas/yr (Brunthaler et al. 2007) 0.61 ±0.23 mas/yr (Dinescuet al. 2004) http://abyss.uoregon.edu/~js/ast223/lectures/lec02.html Measuring the proper motions and geometric distances of galaxies within the Local Group is very important for our understanding of the history formation, evolution, and gravitational potential of the Local Group. Measurements that yield astrometric accuracies of 10 micro-arcseconds make determination of proper motions and angular rotation rates of galaxies out to a distance of 1 Mpc feasible. Supplemented with radial velocity measurements it will allow us to reconstruct a three- dimensional gravitational potential and to set a stringent limits on the parameters of dark matter halo. GREAT-ESF workshop 6-9 June 2011 20 University of Porto, Portugal Panoramic view of the entire near-infrared sky reveals the distribution of galaxies beyond the Milky Way. Blue are the nearest sources (z < 0.01); green are at moderate distances (0.01 < z < 0.04) and red are the most distant sources that 2MASS resolves (0.04 < z < 0.1). The map is projected with an equal area Aitoff in the Galactic system (Milky Way at center). Image credit to: Thomas Jarrett GREAT-ESF workshop 6-9 June 2011 21 University of Porto, Portugal Theoretical aspects of astrometric cosmology (other aspects, talk by M. Lattanzi) The standard IAU resolutions state that the space- time is asymptotically flat at infinity. However, we live in expanding universe, which is described by the cosmological metric of Friedmann- Robertson-Walker fully-symmetric and globally- curved four-dimensional space-time manifold. How to map cosmological space-time to the tangent space of a local astrometric observer? GREAT-ESF workshop 6-9 June 2011 22 University of Porto, Portugal Mapping a curved surface on a plane is not unique GREAT-ESF workshop 6-9 June 2011 23 University of Porto, Portugal Conformal Infinity and Astrometric Catalogue Astrometric catalogue (celestial reference LIGHTLIKE frame) is defined on the FUTURE INFINITY LIGHTLIKE celestial sphere by FUTURE INFINITY directions of light rays from a set of “non- moving” reference celestial objects located at conformal null-like LIGHTLIKE past infinity of space- LIGHTLIKE PAST INFINITY time (Penrose 1963) PAST INFINITY ds2 c 2 dt 2 dr 2 r 2( d 2 sin 2 d 2 ) metric tensor of a unit sphere GREAT-ESF workshop 6-9 June 2011 24 University of Porto, Portugal Space in the Universe Riemann space Euclid space Lobachevsky space Example: sphere Example: plane Example: pseudo-sphere Curvature of space is positive.
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