Light propagation in the gravitational field of $N$ arbitrarily moving bodies in the 1.5PN approximation for high-precision astrometry

High-precision astrometry on sub-micro-arcsecond level in angular resolution requires accurate determination of the trajectory of a light-signal from the celestial light source through the gravitational field of the Solar System toward the observer. In this investigation the light trajectory in the gravitational field of $N$ moving bodies is determined in the 1.5 post-Newtonian approximation. In the approach presented two specific issues of particular importance are accounted for: (1) According to the recommendations of International Astronomical Union, the metric of the Solar System is expressed in terms of intrinsic mass-multipoles and intrinsic spin-multipoles of the massive bodies, allowing for arbitrary shape, inner structure and rotational motion of the massive bodies of the Solar System. (2) The Solar System bodies move along arbitrary world lines which can later be specified by Solar System ephemeris. The presented analytical solution for light trajectory is a primary requirement for extremely high-precision astrometry on sub-micro-arcsecond level of accuracy and associated massive computations in astrometric data reduction. An estimation of the numerical magnitude for time delay and light deflection of the leading multipoles is given.

Figure 1

A geometrical representation of the light trajectory through the Solar System (only one massive body $A$ of the $N$-body Solar System is depicted) in terms of the new variables $\mathit{\xi }$ and $\tau$. The impact vector $\mathit{\xi }$ is defined by Eq. (78) and points from the origin of global system to the point of closest approach of the unperturbed light ray to that origin, and is time-independent. The impact vector ${\mathit{d}}_A(\tau +t^{*})$ is defined by Eq. (85) and points from the origin of local system of body $A$ toward the point of closest approach of unperturbed light ray to that origin, and is time-dependent due to the motion of the body. Furthermore, $\mathit{x}(\tau +t^{*})$ is the global spatial coordinate of the photon of the light trajectory, while ${\mathit{x}}_{\mathrm{N}}(\tau +t^{*})$ is the unperturbed light ray. The world line of massive body $A$ in the global system is given by ${\mathit{x}}_A(\tau +t^{*})$, and ${\mathit{r}}_A(\tau +t^{*})$ points from the origin of local system towards the exact photon’s position, while ${\mathit{r}}_A^{\mathrm{N}}(\tau +t^{*})$ points from the origin of local system toward the unperturbed light ray.