Spacetime astrometry and gravitational experiments in the solar system

Transcription

1 Spacetime astrometry and gravitational experiments in the solar system Sergei Kopeikin University of Missouri 1

2 Abstract Astrometry is the branch of astronomy that involves precise measurements of the positions and movements of stars and other celestial bodies. The main goal of spacetime astrometry is to build the inertial coordinate system in the sky and to test general theory of relativity as well as other fundamental theories. Modern astrometry uses the sophisticated technologies and techniques including the satellites in deep space, ultraprecise atomic clocks, very long baseline interferometry (VLBI) and Doppler tracking. We overview the current astrometric space missions and discuss the theoretical principles of the gravitational experiments utilizing the light propagation through the gravitational field of the massive bodies in the solar system. We pay a special attention to the goals and results of the light-propagation experiments in time-dependent gravitational field of planets and Sun which were conducted in the last decade. We will also touch upon a possibility of the local measurement of the Hubble constant with spacecraft s Doppler tracking without making a direct observation of cosmological objects (quasars, supernova).

3 Contents 1. Astrometric Experiments. Gravitational Field Model 3. Light-ray Propagation 4. Light-ray Deflection Angle 5. Gravitomagnetism and the speed of gravity 6. Gravitational Time Delay 7. The idea of the speed-of-gravity experiment 8. Jovian 00 and Cronian 009 experiments 9. Cassini gravitomagnetic experiment 10. Pioneer anomaly - Local measurement of the Hubble constant? 3

4 Astrometry in Space 4

5 SIM SIM PlanetQuest has been designed as a space-based 9-m baseline optical Michelson interferometer operating in the visible waveband. This mission might open up many areas of astrophysics, via astrometry with unprecedented accuracy. Over a narrow field of view (1 ), SIM aimed to achieve an accuracy of 1 µas in a single measurement! Colloquium at the University of Mississippi, 5

6 GAIA Gaia: was launched in 013. It scans the sky continuously according to a pre-defined pattern. The satellite rotates around its spin axis at a rate of 60 arcsec/s, equivalent to a spin period of 6 hours. The spin axis itself precesses at a fixed angle of 45 degrees to the Sun. The line of sight of the two astrometric instruments are separated by the 'basic angle', which is degrees. Astrometric precision 10 μas. Colloquium at the University of Mississippi, 6

7 JASMINE = Japan Astrometry Satellite Mission for INfrared Exploration. It will survey the Milky Way and its bulge in the infrared band around 1 milli-micron, measure positions, distances, and proper motion of several hundred million stars at high accuracy approaching 10 μas. Launch date: Colloquium at the University of Mississippi, 7

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9 Square Kilometer Array (SKA) The SKA will be an interferometric array of individual antenna stations, synthesizing an aperture with a diameter of up to several thousand kilometers. The SKA is a new generation radio telescope that will be 100 times as sensitive as the best present-day instruments. It will unlock information from the very early Universe and, using novel capabilities, be able to undertake entirely new classes of observation including VLBI with a micro-arcsecond resolution. Colloquium at the University of Mississippi, 9

10 Mauna Kea Hawaii Owens Valley California Brewster Washington North Liberty Iowa Hancock New Hampshire at the University of Mississippi, Kitt Peak Arizona Pie Town New Mexico Fort Davis Texas Los Alamos New Mexico St. Croix Virgin Islands 10

11 VERA VLBI Exploration of Radio Astrometry is the first VLBI array dedicated to phase-referencing micro-arcsecond astrometry. S69 (Sharpless 69) is a massive star forming region toward constellation Orion. VERA has successfully measured its trigonometric parallax of 189 +/- 8 micro-arcsecond. This is the smallest parallax ever measured, corresponding to a source distance to 17,50 light year (~ 5.3 Kpc). Colloquium at the University of Mississippi, 11

12 Gravitational Field Model 1

13 Existing and incoming astrometric facilities demand new approach in theoretical understanding of light propagation through the variable gravitational fields generated by moving, oscillating, and rotating massive bodies as well as the field of gravitational waves. 13

14 1. Linearized general relativity g. The harmonic gauge h h The gravity field equation (c = 1) h t h 0 Colloquium at the University of Mississippi, 14

15 Retarded gravitational potentials h h h 00 0i ij i ij M I ( s) I ( s)... i i j r x r x x r i ij 4 I ( s) I ( s)... j r x r ij I ( s) ijh00... r the retarded time: s t r i i ij i j ij I ( s) Mx ( s) I ( s) Mx ( s) x ( s) J ( s) P P P Colloquium at the University of Mississippi, 15

16 Light-ray Propagation 16

17 The light-ray perturbation The light-ray geodesic dk d K K 0 The wave vector decomposition K dx k d unpertrurbed null vector perturbation The Christoffel symbols 1 h h h x x x The unperturbed equation of light ray dk d 0 The perturbed equation of light ray d d 1 x h k h k k Colloquium at the University of Mississippi, 17

18 The unperturbed light-ray trajectory x i N ( ) i i k r d Colloquium at the University of Mississippi, 18

19 Light-ray Deflection Angle 19

20 Colloquium at the University of Mississippi, 0 The light-ray deflection angle j p i p j i p j i p j i jp i Q i j j j i j i j i D i i M i Q i D i M i Sun i i j j i i j i jp p j i i i i i n m m n m m m m n n n n d s I n d s I k m m n n d s I n d M d dx k k h k k k h k h k d d h k k d x d 3 00 ) ( 4 ) ( 4 ) ( Time argument is the retarded time: s = t - r Gravitational field of a moving planet is localized on null cone and interacts with light with retardation.

21 Colloquium at the University of Mississippi, 1 The deflection equations and the central inverse mapping R M v k d R m m z n z n m z n z d L m m s n s n m s n s d R J m m z n n z d L n P Q D M 4 1 ) )( ( ) ( ) ( ) )( ( ) ( ) ( ) ( ) ( cos 1 limb limb

22 Snapshot deflection patterns Monopole Dipole Quadrupole Colloquium at the University of Mississippi,

23 Dynamic deflection patterns Circle Cardioid Cayley s sextic r cos p1 cos q cos 3 3cos L r M X March 1, 1988 Treuhaft & Lowe DSN JPL NASA 0 p r L X Colloquium at the University of Mississippi, 0 September 8, 00 Fomalont & Kopeikin VLBA+MPfRA q L r L X 0 Not measured yet (SIM, SKA, Gaia, JASMINE, VERA?) 3

24 Gravitomagnetism and the speed of gravity 4

25 Gravitomagnetism GRAVITOMAGNETIC FIELD arises from moving masses just as a magnetic field arises from moving electric charges. g h The metric tensor c h 00 The gravitoelectric potential The leading term is U=GM/r. c 4 Ai h0i The gravitomagnetic potential The leading term is (v/c)u. 5

26 Two types of gravitomagnetic field Intrinsic (Lense-Thirring): caused by rotating currents of matter induced by angular momentum of the massive body Extrinsic (Lorentz-Einstein): caused by translational currents of matter induced by motion of massive bodies in space with respect to observer 6

27 Speed-of-gravity Parameterization of Gravitomagnetism Gravity Fields Gauge condition Einstein s Field Equations Post-Newtonian parameter labels timedependent gravitational effects and characterizes the speed of the respond of the gravitational field to the positional changes of a massive body. We call it the speed of gravity parameter c c/ ε Hence, g c ε c The speed of gravity is the speed of light entering the gravity sector of the fundamental interactions. g 7

28 Gravitational Time Delay 8

29 Gravitational Time Delay 9

30 Extrinsic gravitomagnetic force on a test particle dv v 4 extrinsic 1 v v Fgm F dt c c noise extrinsic 4 Fgm v A c 4 A v 1 v v v A c t c c t c c c c t these terms vanish in the field of a rotating mass being at rest Massive body must move wrt observer to generate the extrinsic GM. How to measure it? USE PHOTONS! For photons v ck that amplifies the PN terms depending on v/c = O(1) dk c dt 4 k k F F "Newtonian" force extrinsic gm noise 4 A 1 F 4k A k 4k k A c t c t c extrinsic gm post-newtonian force of the order of V/ c t post-newtonian force of the order of V /c 30

31 Parameterized Time Delay Equation 1 t1 t0 x1 x0 ( t1, t0 ) xn ( t ) x0 ck ( t t0 ) c 1 t 1 t ( t1, t0) dt k k h ( t, N ( t)) 1 d k k t x 0 t0 h (, x) xx N ( ) Kopeikin S. (004) Class. Quant. Grav., 1, 351 Kopeikin S. (006) Int. J. Mod. Phys. D, 15, 305 Kopeikin S. & Fomalont E. (006) Found. Phys., No. 1, pp. 1-4 Kopeikin & Makarov (007) Phys. Rev. D, 75,

32 Gravitational Time Delay by a moving body GM GMij 4GM v h00 hij h0 i x z( t) x z( t) x z( t) c g photon: x x ( t) x ck( t t ) massive body: z( t) z v( t t ) N GM 1 x1 z( s1) k x1 z( s1) ( t1, t0) 1 ln 3 k v c c g x0 z( s0) k x0 z( s0 ) v v v v z( s1 ) z( t1) x1 z( t1) O ( z s0) z( t0) x0 z( t0) O c g c g c g c g Look like a retarded time 1 s t x z ( t 1 ) s ( ) 0 t0 x0 z t0 c cg g 3

33 The idea of the speed-of-gravity experiment 33

34 The Minkowski diagram of the light-gravity field interaction Kip s world line Future gravity null cone Leonid observes. Future gravity null cone Future gravity null cone Future gravity null cone Future gravity null cone Kip emits light Planet s world line Leonid s world line Colloquium at the University of Mississippi, 34

35 The null cones for gravitational field and light Observer and planet are at rest Planet moves uniformly relative to observer Colloquium at the University of Mississippi, 35

36 Jovian 00 and Cronian 009 experiments 36

37 The Jovian 00 experiment Position of Jupiter taken from the JPL ephemerides Position of Jupiter determined from the gravitational deflection of light by Jupiter 10 microarcseconds = the width of a typical strand of a human hair from a distance of 650 miles!!! The retardation effect was measured with 0% of accuracy, thus, proving that the null cone for gravity and light coincides (Fomalont & Kopeikin 003) Colloquium at the University of Mississippi, 37

38 The speed-of-gravity experiment (00) Edward B. Fomalont (observation, data processing) Sergei M. Kopeikin (theory, interpretation) VLBA support: NRAO and MPIfR (Bonn) Albuquerque 00 38

39 Basic Interferometry (in one minute) 39

40 Limitations to Positional Accuracy Location of Radio Telescope Position on earth (1 cm) Earth Rotation and orientation (5 cm) Time synchronization (50 psec) Array stability (5 cm) Propagation in troposphere and ionosphere Very variable in time and space (5 cm in 10 min) CONVERSION FACTORS for astrometry: 1 cm = 30 psec = 300 microarcsec 0.03cm = 1 psec = 10 microarcsec Phase-referencing VLBI technique can achieve 10 microarcsec! 40

41 Interpreting the speed-of-gravity experiment Kopeikin & Fomalont - gravity sector of GR is compatible with SR speed of gravity = speed of light [ = 1 ] gravitomagnetic (velocity-induced) field of moving Jupiter 1. Will aberration of light (radiowaves) from the quasar. Asada, Carlip speed of light (radiowaves) from the quasar 3. Nordtvedt retardation of radio waves from the quasar in Jovian s magnetosphere 4. Pascual-Sanchez the Römer delay of light (already known since 1676) 5. Samuel retardation of radio waves emitted by Jupiter itself 6. Van Flandern the quantity measured was already known to propagate at the speed of light 41

42 Light Deflection Experiment with Saturn and Cassini spacecraft as a calibrator (Proc. IAU Symp. 61, 009) 4

43 Cassini Gravitomagnetic Experiment 43

44 Gravitomagnetic Field in the Cassini Experiment (Kopeikin et al., Phys. Lett. A, 007) Gravitomagnetic Doppler shift due to the orbital motion of the Sun Bertotti-Iess-Tortora, Nature, (.1.3) 10 However, the gravitomagnetic contribution was not analyzed 44

45 Gravitational time delay in the ODP code The linearized w.r.t. v/c time delay equation can be re-formulated as follows ( Kopeikin arxiv: ) GM 1 R R R 1 1 Cassini-Earth 3 1 k vln c c R1 R R1 R1 x1 z( t1 ) R x z( t ) R1 R1 R = z( t ) z v( t t ) z( t ) z v( t t ) Notice that velocity v of the light-ray deflecting body enters the argument of the logarithm in the time delay. 45

46 Numerical Estimates for Cassini Doppler Shift The peak value of the Doppler shift is caused by 10 orbital motion of Earth and reaches R.M.S. error of the measurements is 110 Doppler shift due to the orbital motion of Sun is The value of (-1) would be affected by the solar 4 motion by the amount 1.10 if the gravitomagnetic deflection of light were not in accordance with GR Conclusions Cassini solar conjunction experiment has a potential to detect the gravitomagnetic field of the moving Sun directly!. It requires re-processing of the data 5 3. The announced value for 1 (.1.3) 10 is based on the implicit assumption that the gravitomagnetic deflection of light agrees with GR, but this assumption was not tested

47 PROGRESS IN MEASUREMENTS OF THE GRAVITATIONAL BENDING OF RADIO WAVES USING THE VLBA E. Fomalont, S. Kopeikin, G. Lanyi, and J. Benson The Astrophysical Journal, 699, 1395 (009) γ = ± October

48 Pioneer Anomaly: Local measurement of the Hubble constant? 48

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50 Heat recoil explanation of the Pioneer anomaly 50

51 Background metric Standard assumption of gravitational experimental physics is that spacetime is asymptotically flat where t is the proper time measured by static observers. In fact, we live in the expanding universe described on all scales by the Robertson-Walker metric where t is the proper time measured by the Hubble observers. 51

52 Local Diffeomorphism We introduce the conformal time: where a η R t η. It reduces the RW metric to the conformally-flat form: Now, we look for a local diffeomorphism reducing the RW metric to the Minkowski metric: which means 5

53 Special Conformal Transformation 53

54 Local Minkowski Coordinates Expand the scale factor, and substitute it to the local diffeomorphism. Compare with the Taylor expansion of the special conformal transformation w.r.t. vector b α. It yields Local Minkowski coordinates are defined by the special conformal transformation where t is the proper time measured by the Hubble observer. The Minkowski time coordinate x 0 is not the proper time except for the timelike world line y i = 0 or x i = 0. 54

55 Einstein s principle of equivalence The Christoffel symbols are nil in the local Minkowski coordinates. According to EEP any test particle moves along a geodesic which are straight lines One can prove that σ = x 0 on photon s worldline (but remember that x 0 is not a proper time of observer). We want to parameterize the geodesic with the proper time t measured by the observer along her/his worldline: 55

56 Motion of light in local coordinates EEP, applied to a conformal manifold, tells us that a freely-moving particle experiences a geometric (Finsler-type) force because for a particle moving with the velocity v x 0 = t + 1 Hv t In particular, equation of motion of photons in the local coordinates in cosmology Light (in local coordinates) moves non-uniformly! 56

57 Doppler shift Emitter s world line ω P k ω 1 P 1 Receiver s world line P 0 57

58 Doppler shift Frequency of radio waves: Doppler shift: Light-ray trajectory: Observer s proper time: 58

59 Relation of the proper time of moving clocks to the cosmic time: Time derivatives Light-ray path: Relation of the cosmic time at the point of emission to that at the point of observation 59

60 Doppler tracking experiment Doppler shift equation: predicts gravitational blue shift of frequency for static observers in cosmology: + _ Doppler shift for local (static) observers Integrated Doppler shift: Pioneer anomaly may have a cosmological explanation! Δω ω 1 = magnitude as the Pioneer anomaly. N i=1 Doppler shift for distant quasars δω i ω 1 = H t N t 1 has the same sign and 60

61 Thank you! 61