Light propagation in 2PN approximation in the monopole and quadrupole field of a body at rest: The basic transformations

TL;DR

The paper advances sub-μas astrometry by delivering a complete 2PN boundary-value framework for light propagation in the solar system’s monopole and quadrupole fields, including both finite source and observer positions. It derives and consolidates three fundamental transformations—k→σ, σ→n, and k→n—within the Gaia-relativistic model (GREM), furnishing explicit 2PN expressions with monopole and quadrupole contributions and identifying enhanced terms. Numerical results for axisymmetric giant planets quantify the 1PN and 2PN deflections, including cross-terms such as M×M, M×Q, and Q×Q, and establish upper limits for grazing rays that guide mission requirements. The work enables precise data reduction for current and future space astrometry missions (e.g., Gaia, GaiaNIR, NEAT, Theia) by embedding the 2PN quadrupole structure into standard transformations and providing practical guidance on implementing these corrections into GREM.

Abstract

Todays astrometry has reached the micro-arcsecond level in angular measurements of celestial objects. The next generations of astrometric facilities are aiming at the sub-micro-arcsecond scale. Sub-micro-arcsecond astrometry requires a considerable improvement in the theory of light propagation in the curved space-time of the solar system. In particular, it is indispensable to determine light trajectories to the second order of the post-Newtonian scheme, where the monopole and quadrupole structure of some solar system bodies need to be taken into account. In reality, both the light source as well as the observer are located at finite spatial distances from the gravitating body. This fact implies for the need to solve the boundary value problem of light propagation, where the light trajectory is fully determined by the spatial positions of source and observer and its unit direction at past infinity. This problem has been solved in a recent investigation. A practical relativistic model of observational data reduction necessitates the determination of the unit tangent vector along the light trajectory at the spatial position of the observer, which is determined by a sequence of several basic transformations. The determination of this unit tangent vector allows one to calculate the impact of the monopole and quadrupole structure of solar system bodies on light deflection on the sub-micro-arcsecond level, both for stellar light sources as well as for light sources located in the solar system. Numerical values for the magnitude of light deflection caused by the monopole and quadrupole structure of the body are given for grazing light rays at the giant planets. The model GREM is presently used for data reduction of the ESA astrometry mission Gaia. It is shown how the implementation of these basic transformations into GREM would proceed for possible future space astrometry missions.

Figures (2)

• Figure 1: A geometrical representation of the propagation of a light signal through the gravitational field of a solar system body at rest. The origin of the spatial coordinates is located at the barycenter of the body. The spatial position of the light signal $\hbox{\boldmath$x$}(t)$ is here with respect to the barycenter of the solar system body. The light source is located at $\hbox{\boldmath$x$}_0$ and the observer is located at spatial position $\hbox{\boldmath$x$}_1$, both these vectors are with respect to the barycenter of the body. The light signal is emitted by the light source at $\hbox{\boldmath$x$}_0$ and propagates along the exact light trajectory $\hbox{\boldmath$x$}\left(t\right)$ (solid line). The unperturbed light ray $\hbox{\boldmath$x$}_{\rm N}\left(t\right)$ is given by Eq. (\ref{['unperturbed_lightray']}) and propagates in the direction of $\hbox{\boldmath$\sigma$}$ along a straight line through the position of the light source at $\hbox{\boldmath$x$}_0$ (dotted line). The spatial positions of source and observer are connected by a straight line (fine dashed line). The unit tangent vector $\hbox{\boldmath$\sigma$}$ along the light trajectory at past null infinity is defined by Eq. (\ref{['Boundary_Condition']}). The unit tangent vector $\hbox{\boldmath$n$}$ along the light trajectory at the observers position is defined by Eq. (\ref{['vector_n']}). The unit tangent vector $\hbox{\boldmath$k$}$ points from the source toward the observer and is defined by Eq. (\ref{['vector_k']}). The impact vector $\hbox{\boldmath$d$}_{\sigma}$ of the unperturbed light ray is given by Eq. (\ref{['impact_vector_unperturbed']}). The impact vector $\hbox{\boldmath$d$}_k$ from the center of mass of the body toward the connecting line between source and observer is given by Eq. (\ref{['impact_vector_k']}).
• Figure 2: A geometrical representation of the General Relativistic Model (GREM) according to Klioner2003aKlioner2004, which is used for data reduction in the ESA astrometry mission Gaia Gaia1Gaia2. The diagram illustrates how the implementation of the basic transformations into GREM proceeds. The five unit vectors, $\hbox{\boldmath$s$}$, $\hbox{\boldmath$n$}$, $\hbox{\boldmath$k$}$, $\hbox{\boldmath$\sigma$}$, $\hbox{\boldmath$l$}$, of the model GREM are explained in the main text. The origin of the spatial coordinates is located at the barycenter of the solar system. The spatial position of the light signal $\hbox{\boldmath$x$}(t)$ and of the source and observer, $\hbox{\boldmath$x$}_0$ and $\hbox{\boldmath$x$}_1$, are with respect to the barycenter of the solar system. The light signal is emitted at coordinate time $t_0$ and received at coordinate time $t_1$. The spatial position of the body in the BCRS is denoted by $\hbox{\boldmath$x$}_A$. In order to implement the transformations $\hbox{\boldmath$k$} \rightarrow \hbox{\boldmath$\sigma$}$, $\hbox{\boldmath$\sigma$} \rightarrow \hbox{\boldmath$n$}$, $\hbox{\boldmath$k$} \rightarrow \hbox{\boldmath$n$}$, as given by Eqs. (\ref{['transformation_k_to_sigma']}), (\ref{['transformation_sigma_to_n']}), (\ref{['transformation_k_to_n']}), into GREM, one has to perform a translation of the spatial coordinates from the barycenter of the solar system body into the barycenter of the solar system, which implies a replacement $\hbox{\boldmath$x$}_0$ by $\hbox{\boldmath$r$}_0^A = \hbox{\boldmath$x$}_0 - \hbox{\boldmath$x$}_A$ and $\hbox{\boldmath$x$}_1$ by $\hbox{\boldmath$r$}_1^A = \hbox{\boldmath$x$}_1 - \hbox{\boldmath$x$}_A$ in these transformations. These replacements would have to be performed in the scalar functions as well as in the tensorial coefficients. In particular, the impact vector $\hbox{\boldmath$d$}_k$ in Eq. (\ref{['impact_vector_k']}) has to be replaced by $\hbox{\boldmath$d$}_k^A = \hbox{\boldmath$k$} \times (\hbox{\boldmath$r$}_0^A \times \hbox{\boldmath$k$}) = \hbox{\boldmath$k$} \times (\hbox{\boldmath$r$}_1^A \times \hbox{\boldmath$k$})$.