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 coordinates 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 = {τ, xi}. 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 description 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 = {u, wi} 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 system 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 {t, xi} 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 barycentric 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 = |X − XB| Xα = {T, Xi} is a fairly straightforward gen- from the barycenter. For terrestrial observer it eralization of the IAU paradigm. We consider reveals as an annual periodic variation amount- now the whole galaxy as an isolated system ing to 0.37 s. Nevertheless, this large relativis- with asymptotically flat spacetime. Then, the tic effect is extremely difficult to observe as it barycentric reference frame of the solar sys- is hidden in the astrometric positions of stars in tem becomes one of the local frames associated the form of the secular aberration (Kopeikin & with stars or clusters of stars in the galaxy. We Makarov 2006). Function B(T) in (1) causes put the origin of the galactocentric coordinates divergence of TCB from the more uniform, at the center of mass of the Milky Way and pos- galactocentric time T. The derivative B˙ ' 8.4× tulate that the coordinate spatial axes do not ro- 10−7 redefines the unit of time (SI second) as tate in kinematic sense with respect to quasars. measured in TCB. It causes re-scaling the as- Transformation between the barycentric coor- tronomical unit of length and mass. This point dinates of the solar system and the galactocen- has been never discussed by the IAU, and its tric coordinates is given by practical consequences for astrometric observations are not quite clear. The second deriva- h i −2 i i i tive B¨ ' 2 × 10−15 yr−1 causes systematic t = T − c B(T) − VB X − XB , (1) i i j j 1 i ˙ i x = L j X − XB , (2) An overdot denotes a time derivative. AB = VB 1004 Sergei Kopeikin: Astrometric reference frames in the solar system and beyond quadratic drift of TCB with respect to the uni- cosmology has developed a powerful approx- form galactic time. It may be important in pre- imation scheme for doing this. Unfortunately, cise astrometric observations of quasars to en- it can not be applied for calculation cosmo- sure the stability of the ICRF and the space logical perturbations caused by self-gravitating astrometry catalogues. The rate of relativistic isolated systems. This is because physical cos- precession of the spatial axes of the barycen- mology assumes the density contrast of the tric frame of the solar system with respect to perturbation, δρ/ρ¯ 1, whereρ ¯ is the mean the galactocentic frame, is small and amounts density of the universe 2. However, the den- to 0.004 µas/yr−1. It can be ignored in process- sity contrast in isolated astronomical system, ing of astrometric observations. δρ/ρ¯ 1, that makes unacceptable any approximation method previously developed in cosmology to find out the metric perturbation 4. Astrometric frame in cosmology hαβ for the purposes of celestial mechanics and Cosmological observations have established astrometry. that the universe is expanding with accel- To overcome the problem some researchers eration. Current cosmological paradigm ex- resorted to exact methods of embedding a plains this acceleration by the presence of spherically-symmetric Schwarzschild solution the dark energy. The nature of the dark en- to the expanding FLRW universe. The most ergy is yet unknown but the common wis- famous model was worked out by Einstein & dom is that it is made of a scalar field with Straus (1945, 1946) and is known as the Swiss self-interaction described by some form of cheese or vacuole solution (Schucking¨ 1954) potential. Cosmological observations also re- in a dust-dominated universe. The vacuole so- veal the presence of the dark matter in halos lution is unsatisfactory from theoretical point of galaxies and the clusters of galaxies. The of view since it is unstable and can be de- overall spacetime in cosmology is curved and stroyed by imposing an infinitesimal pertur- time-dependent with space cross-sections be- bation. Furthermore, the vacuole model does ing flat. The background Friedmann-Lemetre- not provide a realistic model of spacetime at Robertson-Walker (FLRW) metricg ¯αβ of the the scales below the scale of galaxy clusters. universe is conformal to the flat spacetime. This is because the radius of the vacuole for In the isotropic coordinates the FLRW metric a central object like a star or a galaxy signif- reads icantly exceeds the average distance between 2 stars/galaxies (Bonnor 2000). g¯αβ(η) = a (η)ηαβ , (8) For this reason other exact solutions were where a(η) is the scale factor depending on the sought. However, the complexity and non- linearity of Einstein’s equations make it a dif- conformal time η, and ηαβ is the Minkowski metric. The presence of the solar system (and ficult task to construct a suitable exact solution other massive bodies - stars, galaxies, etc.) which may serve as realistic model for actual causes perturbation of the background metric physical situations. Indeed, as a rule, exact solutions of Einstein’s equations are known for 2 gαβ(η, x) = a (η) ηαβ + hαβ , (9) idealized situations typically admitting high degree of symmetry. At the same time, re- which should be compared with the current alistic environment of isolated gravitational post-Newtonian paradigm of the IAU that pos- systems has no symmetry. The problem then tulates that the scale factor a(η) ≡ 1. This pos- is how to immerse the non-symmetric (and tulate is not valid, at least, in principle, thus, time-dependent) localized distribution of mass challenging theorists for deeper exploration of smoothly to the expanding FLRW universe. the impact of cosmological expansion on as- The predominant idea was to glue several exact trometric measurements and on dynamics of solutions across suitably chosen hypersurface self-gravitating systems like the solar system, with the appropriate matching conditions im- binary pulsars, galaxies and their clusters. posed on the metric tensor and its derivatives. The task is to calculate the perturbation 2 −30 3 hαβ by solving Einstein’s equations. Physical ρ¯ ' 10 g/cm in the present epoch Sergei Kopeikin: Astrometric reference frames in the solar system and beyond 1005

Such an approach was pioneered isolated astronomical system along with the by McVittie (1933) who has taken the perturbation of the background universe. The Schwarzschild solution and embedded it to a perturbation of the background geometry cou- spatially-flat FLRW universe by multiplying ples with the source of gravitational field of the spatial part of Schwarzschild’s metric the isolated system and makes the effective by a time-dependent scale factor a(t). Such mass of the system depending on time even “marriage” of two metrics is possible, if if the bare mass of the system was taken as and only if, the mass of the central body constant. This resembles and may explain the is a function of time, m = m0/a(t), where physical origin of the time-dependence of the m0 is a constant mass. Scrutiny analysis of central mass in the McVittie metric. The scalar Einstein’s equations reveals that in the most fields describe the structure of matter. In par- general case, McVittie’s metric is physically ticular, the dark matter is modelled as a per- implausible because imposing the condition fect fluid described by the Clebsch potential, of a non-vanishing central mass m0 and the Φ, that is the specific enthalpy, µ, of the fluid time-dependence scale factor a(t) of the FLRW (Schutz 1970). The dark energy is modelled as spacetime leads to an unrealistic equation of a quintessence scalar field, Ψ, with an arbitrary state of matter in universe. Similar to the potential W(Ψ). Other scalar fields being per- case of the vacuole solution, the McVittie tinent to the problem may be accounted for, metric is spherically-symmetric and, hence, if necessary. The localized isolated system is inappropriate for analysing more realistic also described by a Lagrangian of a continu- situations having no spherical symmetry like ous distribution of matter. It serves as a source binary pulsars or planetary systems. of gravitational field of the isolated system that The only way to surpass mathematical dif- is a bare perturbation of the background cos- ficulties of embedding an isolated astronomical mological spacetime. system to the cosmological environment is to The overall Lagrangian of the system is resort to a suitable method of approximations presented as a Taylor (asymptotic) series with which generalizes the post-Newtonian approx- respect to the perturbations of all variables, imations in asymptotically-flat spacetime to ¯ the case of curved and expanding cosmologi- L = L + L1 + L2 + ..., (10) cal manifold. where L¯ is the Lagrangian of the background FLRW universe, and Li (i = 1, 2,...) are linear, quadratic, etc. perturbations. The field 5. Post-Newtonian approximations in equations for the metric perturbation and the cosmology dynamic matter variables are obtained by tak- We have worked out a new method of finding ing variational derivatives from L with respect cosmological perturbations induced by an iso- to the variables hαβ, Φ, Ψ, etc. We do not im- lated astronomical system (Kopeikin & Petrov pose any limitation on the equation of state of 2012) which is valid for sufficiently high con- the background FLRW universe so that the the- trasts of matter density but singularities, and ory is applicable for wide range of cosmologi- for arbitrary equation of state of the back- cal models. ground matter. It is based on the Lagrangian As an example, we write down the system theory of perturbations of an arbitrary-curved of the field equations defining the metric tensor and dynamically-evolving spacetime manifold perturbations in case of the universe filled up (Grishchuk et al. 1984; Popova & Petrov with the dark matter in the form of dust and 1988). In our approach both the universe and the dark energy in the form of the cosmological the localized source of perturbation are mathe- constant Λ. They are (Kopeikin & Petrov 2012) matically modelled by a Lagrangian depend- α 2 2 ing on a set of dynamic variables which in- q − 2Hv¯ q,α + H − a Λ q = (11) h i clude the metric perturbation, hαβ, and sev- 2 −1 8πa σ + ρ¯m δm + a Hχm , eral scalar fields Φ, Ψ, .... The metric pertur- β 2 2 bation describes the gravitational field of the pα − 2Hv¯ pα,β + H − a Λ pα = 16πaσα , 1006 Sergei Kopeikin: Astrometric reference frames in the solar system and beyond

γ pαβ − 2Hv¯ pαβ,γ = 16πσαβ , This is because of the presence of the scale factor a in the right side of equation (11) which where H = a˙/a is the Hubble parameter, µν makes the solution dependent on cosmological = η ∂µν is the wave operator in flat spacetime. The magnitude of the “Newtonian” po- time, comma denotes a partial derivative with tential q evolves in the conformal coordinates respect to a corresponding coordinate, q = α β β γ µ ν as the universe expands – the effect which can v¯ v¯ lαβ, pα = −π¯ α v¯ lβγ, pαβ = π¯ α π¯ β lµν, be interpreted as time-dependence of the uni- µ β β β lαβ = −hαβ +(h µ/2)ηαβ, andπ ¯ α = δα +v¯ αv¯ is versal gravitational constant G in the spirit of the operator of projection on the hypersurface the Dirac large numbers hypothesis. However, being orthogonal to 4-velocity vα of observer the conformal coordinates are not observable with respect to the FLRW isotropic frame. The quantities nor any other coordinates. The re- source of the gravitational perturbations is the lationship between the coordinates and obser- tensor of energy-momentum, ταβ, of the local- vations is established by mean of propagating ized system, and photons between worldline of observer and a spacetime event. The round trip of photon is ¯ α ¯ β αβ σ = v v ταβ + π¯ ταβ , (12) described by solution of light geodesics which β γ σα = −π¯ α v¯ τβγ , is given in the first approximation and in the µ ν σαβ = π¯ α π¯ β τµν . conformal coordinates by the same equation as in flat spacetime, We notice that the source of the scalar i i i gravitational perturbation q includes two more x = k (η − η0) + x0 . (15) functions, δm and χm, which describe the den- i sity perturbation of the background FLRW Here, k is the unit vector in the direction of i j spacetime. Equations for these functions are as propagation of photon (δi jk k = 1), η0 is the follows (Kopeikin & Petrov 2012) (conformal) time of emission of photon from the point with coordinates xi . The conformal α β α 2 0 v¯ v¯ δm,α,β + Hv¯ δm,α − 4πa ρ¯mδm = (13) time, η, relates to the proper time of observer, 4πa2σ , τ, by an ordinary differential equation, 1 2 2 α dτ 1/2 χm + 3H − a Λ χm = av¯ δm,α , (14) = a(η) 1 − β2 − h βµβν , (16) 2 dη µν whereρ ¯m is the mean density of the back- i i ground universe. The perturbation δm is in- where β = dx /dη is the velocity of ob- duced by the presence of the localized system. server expressed in the conformal coordinates. However, it can be generated by the primor- Solution of this equation depends on the scale dial cosmological perutbations as well. The factor a(η). Hence, the radar distance ` = impact of the Hubble-induced densityρ ¯m on c(τ3 − τ1) between observer and the point the measured mass of the isolated system is in space is a rather complicated function of small. However, it is accumulated over time the conformal coordinates and the scale factor starting from the origination of the progenitor a(η). The overall technical details of the calcu- of the isolated system making significant con- lation of observables along with the discussion tribution to the current value of the observed of the post-Newtonian dynamics in the expand- mass. ing universe are given in (Kopeikin & Petrov 2012).

6. Connection to astrometric Acknowledgements. I am grateful to the organizers observables of the Great-ESF workshop in Porto – M.-T. Crosta and S. Anton, for providing partial travel support The connection of the theory to astrometric ob- and to the Faculty of Sciences of the University of servations is not straightforward. First of all, Porto for hospitality. This work was partially sup- we notice that q is to be an analogue of the ported by the Faculty Incentive Grant 2011 of the Newtonian potential but solution of (11) does College of Arts and Science of the University of not yield the Newtonian potential right away. Missouri in Columbia. Sergei Kopeikin: Astrometric reference frames in the solar system and beyond 1007

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