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Sergei A. Klioner

Publications and source records attributed to Sergei A. Klioner.

33 records · Page 2Linked to original sources
arXiv2009

Formal proof of some inequalities used in the analysis of the post-post-Newtonian light propagation theory

A rigorous analytical solution of light propagation in Schwarzschild metric in post-post Newtonian approximation has been presented in \cite{report1}. In \cite{report2} it has been asserted that the sum of all those terms which are of order ${{\cal O} (\frac{m^2}{d^2})}$ and ${{\cal O}(\frac{m^2}{d_σ^2})}$ is not greater than $15/4 π\frac{m^2}{d^2}}$ and $15/4 π\frac{m^2}{d_σ^2}}$, respectively. All these terms can be neglected on microarcsecond level of accuracy, leading to considerably simplified analytical transformations of light propagation. In this report, we give formal mathematical proofs for the inequalities used in the appendices of \cite{report2}.

astro-ph.IM↗
arXiv2009

Parametrized post-post-Newtonian analytical solution for light propagation

An analytical solution for light propagation in the post-post-Newtonian approximation is given for the Schwarzschild metric in harmonic gauge augmented by PPN and post-linear parameters $β$, $γ$ and $ε$. The solutions of both Cauchy and boundary problem are given. The Cauchy problem is posed using the initial position of the photon $\ve{x}_0 = \ve{x}(t_0)$ and its propagation direction \veσ at minus infinity: $\veσ = {1\over c} \lim\limits_{t \to -\infty}\dot{\ve{x}}(t)$. An analytical expression for the total light deflection is given. The solutions for $t - t_0$ and $\veσ$ are given in terms of boundary conditions $\ve{x}_0 = \ve{x} (t_0)$ and $\ve{x} = \ve{x}(t)$.

astro-ph.IM↗
arXiv2008

Explicit exact expression for the Thomas precession

This work gives an explicit exact expression for the Thomas precession arising in the framework of Special Theory of Relativity as the spatial rotation resulting from two subsequence Lorentz boosts. The final result for the orthogonal matrix of Thomas precession is given by Eqs. (21)--(25). A trivial calculation leads to the compact formula (26) for the angle of rotation due to Thomas precession. In the framework of Gaia the special-relativistic Thomas precession is an important step in the derivation of an aberrational formula with the Mansouri-Sexl parameters. The latter formula will be used to test the Local Lorentz Invariance with Gaia data as will be explained elsewhere.

astro-ph↗
arXiv2008

Determination of the barycentric velocity of an astrometric satellite using its own observational data

The problem of determination of the orbital velocity of an astrometric satellite from its own observational data is studied. It is well known that data processing of microarcsecond-level astrometric observations imposes very stringent requirements on the accuracy of the orbital velocity of the satellite (a velocity correction of 1.45 mm/s implies an aberrational correction of 1 $μ$as). Because of a number of degeneracies the orbital velocity cannot be fully restored from observations provided by the satellite. Seven constraints that must be applied on a velocity parameterization are discussed and formulated mathematically. It is shown what part of velocity can be recovered from astrometric data by a combined fit of both velocity parameters and astrometric parameters of the sources. Numerical simulations show that, with the seven constraints applied, the velocity and astrometric parameters can be reliably estimated from observational data. It is also argued that the idea to improve the velocity of an astrometric satellite from its own observational data is only useful if the a priori information on the orbital velocity justifies the applicability of the velocity constraints. The proposed model takes into account only translational motion of the satellite and ignores any satellite-specific parameters. Therefore, the results of this study are equally applicable to both scanning missions similar to Gaia, and pointing ones like SIM, provided that enough sources were observed sufficiently uniformly.

astro-ph↗
arXiv2007

Relativistic scaling of astronomical quantities and the system of astronomical units

For relativistic modelling of high-accuracy astronomical data several time scales are used: barycentric and geocentric coordinate times, TCB and TCG, as well as two additional time scales, TDB and TT, that are defined as linear functions of TCB and TCG, respectively. The paper is devoted to a concise but still detailed explanation of the reasons and the implications of the relativistic scalings of astronomical quantities induced by the time scales TDB and TT. We consequently distinguish between quantities and their numerical values expressed in some units. It is argued that the scaled time scales, the scaled spatial coordinates and the scaled masses should be considered as distinct quantities which themselves can be expressed in any units, and not as numerical values of the same quantities expressed in some different, non-SI units (``TDB units'' and ``TT units''). Along the same lines of argumentation the system of astronomical units is discussed in the relativistic framework. The whole freedom in the definitions of the systems of astronomical units for TCB and TDB is demonstrated. A number of possible ways to freeze the freedom are shown and discussed. It is argued that in the future one should think about converting AU into a defined quantity by fixing its value in SI meters.

astro-ph↗
arXiv2005

Refining the relativistic model for Gaia: cosmological effects in the BCRS

This paper represents a first attempt of embedding the Barycentric Celestial Reference System, the fundamental relativistic reference system to be used for the modeling of Gaia observations, into some cosmological background. The general Robertson-Walker metric is transformed into local coordinates where the cosmological effects are represented as tidal potentials. A version of a cosmological BCRS is then suggested to lowest order. The effects of cosmological background on the motion of the solar system are estimated and found to be completely negligible. The relation to the de Sitter and Schwarzschild-de Sitter solutions is discussed.

astro-ph↗
arXiv2004

Relativistic formulation and reference frame

After a short review of experimental foundations of metric theories of gravity, the choice of general relativity as a theory to be used for the routine modeling of Gaia observations is justified. General principles of relativistic modeling of astronomical observations are then sketched and compared to the corresponding Newtonian principles. The fundamental reference system -- Barycentric Celestial Reference System, which has been chosen to be the relativistic reference system underlying the future Gaia reference frame is presented. Principal relativistic effects in each constituent of a relativistic model of astronomical observations are briefly elucidated. The structure of a relativistic model of positional observations which can be used as a standard relativistic model for Gaia is sketched. The physical meaning of the Gaia reference frame is discussed. It is discussed also how Gaia observations can be used to verify general relativity.

astro-ph↗
arXiv2004

Independent Components of an Indexed Object with Linear Symmetries

The problem of finding independent components of an indexed object (e.g., a tensor) with arbitrary number of indices and arbitrary linear symmetries is discussed. It is proved that the number of independent components $f(k)$ is a polynomial of degree not greater than the number of indices $n$, $k$ being the dimension of the space. Several algorithms to compute $f(k)$ for arbitrary $k$ are described and discussed. It is shown that in the worst case finding $f(k)$ for arbitrary $k$ requires solving at most P(n) systems of linear equations with at most $(n!)^2$ equations for at most of $n!$ unknowns, P(n) being the number of partitions of $n$. As a by-product, an efficient algorithm to parametrize all components of the object through its independent components is found and implemented in \Mathematica.

gr-qc↗
arXiv2004

Physically adequate proper reference system of a test observer and relativistic description of the GAIA attitude

A relativistic definition of the physically adequate proper reference system of a test observer is suggested within the framework of the PPN formalism. According to the nomenclature accepted within the GAIA project this reference system is called Center-of-Mass Reference System (CoMRS). The interrelation between the suggested definition of the CoMRS and the Resolutions 2000 on relativity of the International Astronomical Union (IAU) are elucidated. The tetrad representation of the CoMRS at its origin is also explicated. It is demonstrated how to use that tetrad representation to calculate the relation between the observed direction of a light ray and the corresponding coordinate direction in the Barycentric Celestial Reference System of the IAU. It is argued that the kinematically non-rotating CoMRS is the natural choice of the reference system where the attitude of the observer (e.g. of the GAIA satellite) should be modeled. The relativistic equations of rotational motion of a satellite relative to its CoMRS are briefly discussed. A simple algorithm for the attitude description of the satellite is proposed.

astro-ph↗
arXiv2003

Numerical Simulations of the Light Propagation in the Gravitational Field of Moving Bodies

One of the most subtle points in the modern relativistic models for microarcsecond astrometrical observations is the treatment of the influence of translational motion of gravitating bodies on the light propagation. This paper describes numerical simulations of the light propagation in the gravitational field of moving gravitating bodies as well as summarizes the underlying theory. The simulations include high-precision numerical integrations of both post-Newtonian and post-Minkowskian differential equations of light propagation and a detailed comparison of the results of the numerical integrations with various available approximate analytical formulas. The simulations has been performed both for hypothetical bodies with various parameters of trajectories as well as for all the major bodies of the solar system using the JPL ephemeris DE405/LE405 to calculate their motion. It is shown that for the accuracy of $\sim 0.2$ $μ$as it is sufficient to use the well-known solution for the light propagation in the field of a motionless mass monopole and substitute in that solution the position of the body at the moment of closest approach between the actual trajectory of the body and the unperturbed light path (as it was first suggested by Hellings (1986)). For a higher accuracy one should use either the post-Newtonian solution for uniformly moving bodies (Klioner & Kopeikin, 1992) or the post-Minkowskian solution for arbitrarily moving bodies (Kopeikin & Schaefer, 1999). For astrometric observations performed from within the solar system these two solutions guarantee the accuracy of $\sim 0.002$ $μ$as and are virtually indistinguishable from each other.

astro-ph↗
arXiv2003

Light Propagation in the Gravitational Field of Moving Bodies by means of Lorentz Transformation. I. Mass monopoles moving with constant velocities

We show how to derive the equations of light propagation in the gravitational field of uniformly moving mass monopoles without formulating and integrating the differential equations of light propagation in that field. The well-known equations of light propagation in the gravitational field of a motionless mass monopole are combined with a suitable Lorentz transformation. The possibility to generalize this technique for the more complicated case of uniformly moving body of arbitrary multipole structure is discussed.

astro-ph↗
arXiv2002

A Practical Relativistic Model of Microarcsecond Astrometry in Space

This paper is devoted to a practical model for relativistic reduction of positional observations with an accuracy of 1 \muas which is expected to be attained in the future space astrometry missions. All relativistic effects which are caused by the gravitational field of the Solar system and which are of practical importance for this accuracy level are thoroughly calculated and discussed. The model includes relativistic modeling of the motion of observer, modeling of relativistic aberration and gravitational light deflection as well as a relativistic treatment of parallax and proper motion suitable for the accuracy of 1 \muas. The model is formulated both for remote sources (stars, quasars, etc.) and for Solar system objects (asteroids, etc.). The suggested model is formulated within the framework of Parametrized Post-Newtonian Formalism with parameters $β$ and $γ$. However, for general relativity ($β=γ=1$) the model is fully compatible with the IAU Resolutions (2000) on relativity in celestial mechanics, astrometry and metrology. The model is presented in a form suitable for implementation in a software system for data processing or simulation. The changes which should be applied to the model to attain the accuracy of 0.1 \muas are reviewed. Potentially important relativistic effects caused by additional gravitational fields which are generated outside of the Solar system are also briefly discussed.

astro-ph↗
arXiv1999

Relativistic Celestial Mechanics with PPN Parameters

Starting from the global parametrized post-Newtonian (PPN) reference system with two PPN parameters $γ$ and $β$ we consider a space-bounded subsystem of matter and construct a local reference system for that subsystem in which the influence of external masses reduces to tidal effects. Both the metric tensor of the local PPN reference system in the first post-Newtonian approximation as well as the coordinate transformations between the global PPN reference system and the local one are constructed in explicit form. The terms proportional to $η=4β-γ-3$ reflecting a violation of the equivalence principle are discussed in detail. We suggest an empirical definition of multipole moments which are intended to play the same role in PPN celestial mechanics as the Blanchet-Damour moments in General Relativity. Starting with the metric tensor in the local PPN reference system we derive translational equations of motion of a test particle in that system. The translational and rotational equations of motion for center of mass and spin of each of $N$ extended massive bodies possessing arbitrary multipole structure are derived. As an application of the general equations of motion a monopole-spin dipole model is considered and the known PPN equations of motion of mass monopoles with spins are rederived.

gr-qc↗
arXiv1998

Nordtvedt effect in rotational motion

Rotational motion of extended celestial bodies is discussed in the framework of the Parametrized Post-Newtonian (PPN) formalism with the two parameters $γ$ and $β$. A local PPN reference system of a massive extended body being a member of a system of N massive extended bodies is constructed. In the local PPN reference system the external gravitational field manifests itself only in the form of tidal potentials. Rotational equations of motion, which are then derived in the local reference system, reveal a special term in the torque analogous to the Nordtvedt effect in the translational equations of motion: it is proportional to $4β-γ-3$, to the acceleration of the body relative to the global PPN reference system and to some quantity characterizing the distribution of inertial gravitational energy within the body. This term is a direct consequence of the violation of the Strong Equivalence Principle in alternative theories of gravitation.

gr-qc↗