Completing the second post-Newtonian light-propagation model for sub-microarcsecond astrometry: coordinate-independent enhanced terms, operational implementation, and observability of quadrupole signals

The rigorous second post-Newtonian (2PN) solution of the boundary value problem of light propagation in the monopole and quadrupole field of a body at rest — the three basic transformations between the source–observer direction, the ray tangent at past infinity, and the ray tangent at the observer — was completed in 2025 by Zschocke, providing the closed-form ingredients needed for sub-microarcsecond relativistic astrometric modeling. This problem collects the issues that this completion leaves open: (1) settling the conjectured coordinate origin of the anomalously large ('enhanced') 2PN quadrupole terms and providing a coordinate-independent, or systematically coordinate-controlled, formulation of the 2PN quadrupole light deflection; (2) incorporation, testing, and validation of the 2PN monopole–quadrupole and quadrupole–quadrupole terms in an operational GREM-type data-reduction pipeline; and (3) determining whether quadrupole-induced 1PN and 2PN deflection signals can be disentangled from the dominant monopole deflection in realistic astrometric data through their characteristic angular-distance scaling.

Background

Astrometry measures the directions of celestial objects by tracing light rays backwards through the curved space-time of the solar system. Angular precision has advanced from the milli-arcsecond level (Hipparcos) to the micro-arcsecond ( $\mu$ as; $1\,\mu\mathrm{as}\simeq4.85\times10^{-12}$ rad) level with ESA's Gaia mission, and proposed missions such as Gaia-NIR and Theia aim at the sub- $\mu$ as scale. Since the light deflection caused by the Sun is about $1.75$ arcseconds at the solar limb and falls off with the impact distance of the ray, attaining sub- $\mu$ as accuracy requires modeling light propagation far beyond the leading deflection term. The standard tool is the post-Newtonian (PN) scheme: an expansion of the metric tensor and of the light trajectory in powers of $c^{-1}$ , where $c$ is the speed of light, usually carried out in harmonic coordinates of the Barycentric Celestial Reference System (BCRS), the global coordinate standard adopted by the International Astronomical Union for solar-system ephemerides and astrometry. The first post-Newtonian (1PN) light deflection of a body of mass $M$ scales as $GM/(c^{2}d)$ , where $d$ is the impact distance of the ray; the second post-Newtonian (2PN) contribution scales as $(GM/(c^{2}d))^{2}$ . For grazing rays at Jupiter the 2PN monopole deflection reaches about $16\,\mu$ as and the 2PN contribution caused by the planet's mass quadrupole about $0.95\,\mu$ as — both above the sub- $\mu$ as target, which is why the 2PN scheme with at least the monopole $M$ and the symmetric-trace-free (STF) quadrupole moment $M_{ab}$ (the coefficient of the $P_{2}$ distortion of the body's gravitational field, proportional to $-MJ_{2}P^{2}$ with $J_{2}$ the dimensionless quadrupole coefficient and $P$ the body radius) must be modeled. The body-centered 2PN metric in harmonic gauge contains monopole–monopole, monopole–quadrupole, and quadrupole–quadrupole terms of order $c^{-4}$ . The 2PN monopole problem was already solved decades ago for bodies at rest and revived around 2010 by several independent groups (Klioner and Zschocke; Ashby and Bertotti; Teyssandier), which also discovered so-called enhanced terms: 2PN contributions that carry an additional large factor $x_{1}/d\gg1$ , where $x_{1}$ is the distance between the gravitating body and the observer. These factors do not diverge, because realistic observers reside inside the solar system so $x_{1}$ is bounded, but they make the 2PN deflection much larger than a naive order-of-magnitude estimate. For the monopole, Klioner and Zschocke (2010) traced the enhanced terms to the use of a coordinate-dependent impact parameter (the length $d$ constructed from coordinate positions of the unperturbed ray), rather than a coordinate-independent one. Solving the trajectory as an initial value problem (given a tangent direction at past infinity) is not enough for astrometry: real sources and observers sit at finite distances, so the light path must be determined as a boundary value problem, fixed by the spatial position $\boldsymbol{x}_{0}$ of the source and $\boldsymbol{x}_{1}$ of the observer. A relativistic data-reduction model then needs the unit tangent vector $\boldsymbol{n}$ of the ray at the observer, obtained through three basic transformations: $\boldsymbol{k}\to\boldsymbol{\sigma}$ , $\boldsymbol{\sigma}\to\boldsymbol{n}$ , and $\boldsymbol{k}\to\boldsymbol{n}$ , where $\boldsymbol{k}$ is the unit vector from the source toward the observer, $\boldsymbol{\sigma}$ the unit tangent at past infinity, and $\boldsymbol{n}$ the unit tangent at the observer. These transformations are the backbone of GREM (Gaia RElativistic Model), the relativistic model of light propagation used for the data reduction of Gaia, built by Klioner (2003) and subsequently refined by implementing simplified 1PN quadrupole formulas (Zschocke and Klioner 2011) and the 2PN monopole. In a series of papers, Zschocke solved the 2PN initial value problem for the monopole-plus-quadrupole field of a body at rest (2022, finding enhanced quadrupole terms of up to $0.95\,\mu$ as at Jupiter and $0.29\,\mu$ as at Saturn for grazing rays), the boundary value problem (2025), and finally, in Physical Review D 112 (2025) 064087, the three basic transformations, together with upper limits for the 2PN light deflection and a prescription (translation of spatial coordinates from the body's barycenter to the solar-system barycenter) for implementing them into GREM. The 1PN and 2PN monopole limits of these transformations reproduce the earlier results of Klioner (2003) and Klioner and Zschocke (2010), whereas the 2PN monopole–quadrupole and quadrupole–quadrupole coefficients have not been independently verified by an alternative re-derivation or by numerical integration of the geodesic equation. The result holds for a single body at rest: no spin multipoles $S_{L}$ (which encode rotation and inner currents) are included, and only $M$ and $M_{ab}$ are retained. Several gaps remain between this solution and an operational relativistic model for sub- $\mu$ as astrometry. First, solar-system bodies move and their multipole orientations are time-dependent; the 2PN moving-body problem has been solved only for monopoles (Zschocke 2016–2019), while the moving-quadrupole problem is solved only at 1PN, where it underwrites the retarded-time prescription used in Gaia data reduction. Second, the enhanced 2PN quadrupole terms are conjectured, by analogy with the monopole case, to originate from coordinate-dependent impact vectors, but this has been neither proven nor circumvented by a coordinate-independent formulation. Third, the 2PN monopole–quadrupole and quadrupole–quadrupole terms have not actually been implemented, tested, and validated in an operational data-reduction pipeline of the GREM type. Fourth, the observed deflection is an integral effect of monopole and quadrupole contributions; separating them observationally requires exploiting their different characteristic scalings with the angular distance between the source and the body's barycenter, which existing detectability studies (Crosta and Mignard 2006; Kopeikin and Makarov 2007; Abbas et al. 2019, 2022) have addressed only for the 1PN quadrupole light bending by Jupiter with Gaia.

Problem Statement

The complete, rigorous 2PN solution of the boundary value problem of light propagation — the three basic transformations $\boldsymbol{k}\to\boldsymbol{\sigma}$ , $\boldsymbol{\sigma}\to\boldsymbol{n}$ , $\boldsymbol{k}\to\boldsymbol{n}$ between the unit vector $\boldsymbol{k}$ from the light source toward the observer, the unit tangent $\boldsymbol{\sigma}$ of the ray at past infinity, and the unit tangent $\boldsymbol{n}$ of the ray at the observer's position — is known only for a single body at rest with mass monopole $M$ and mass quadrupole $M_{ab}$ in harmonic coordinates. The problem is to develop this static-body theory into the form needed by sub-microarcsecond astrometric data reduction, under the single objective of an operational, coordinate-controlled 2PN relativistic model of light propagation for solar-system astrometry. The required results and their relationships are:

(1) Coordinate content of the enhanced 2PN quadrupole terms. Prove or refute the conjecture that the enhanced 2PN quadrupole terms (those carrying large factors $x_{1}/d\gg1$ ) originate in the choice of coordinate-dependent impact vectors, and provide either a formulation of the 2PN monopole–quadrupole and quadrupole–quadrupole light deflection in terms of coordinate-independent quantities, analogous to the coordinate-independent impact-parameter treatment available for the monopole, or a systematic account of how the coordinate dependence enters and how it behaves in the observable quantities.

(2) Operational implementation. Incorporate the 2PN monopole–quadrupole and quadrupole–quadrupole terms into a GREM-type relativistic data-reduction model and validate them.

(3) Observational disentanglement. Determine whether the quadrupole-induced 1PN and 2PN deflection contributions can be separated from the dominant monopole deflection in realistic astrometric data (simulated or actual), exploiting the characteristic scaling of each multipole term with the angular distance between the source and the barycenter of the gravitating body.

Strand (1) settles the theoretical coordinate content of the enhanced 2PN quadrupole terms, strand (2) operationalizes the resulting expressions in a data-reduction model, and strand (3) establishes their observational content, all within the solar-system regime (weak fields, slow motion, observers within the solar system).

References

Sven Zschocke, Light propagation in 2PN approximation in the monopole and quadrupole field of a body at rest: The basic transformations, Physical Review D 112 (2025) 064087. DOI: 10.1103/zmsk-wb2k, https://doi.org/10.1103/zmsk-wb2k; arXiv:2509.26164, https://arxiv.org/abs/2509.26164
Sven Zschocke, Light deflection in the gravitational field of a solar system body with finite distance of source and observer for sub-micro-arcsecond astrometry, Physical Review D 114 (2026) 044045. DOI: 10.1103/hyck-h4qr, https://doi.org/10.1103/hyck-h4qr; arXiv:2608.15658, https://arxiv.org/abs/2608.15658
Sven Zschocke, Light propagation in the 2PN approximation in the monopole and quadrupole field of a body at rest: Boundary value problem, Physical Review D 111 (2025) 104069. DOI: 10.1103/PhysRevD.111.104069, https://doi.org/10.1103/PhysRevD.111.104069; arXiv:2505.19963, https://arxiv.org/abs/2505.19963
Sven Zschocke, Light propagation in 2PN approximation in the monopole and quadrupole field of a body at rest: Initial value problem, Physical Review D 105 (2022) 024040. DOI: 10.1103/PhysRevD.105.024040, https://doi.org/10.1103/PhysRevD.105.024040; arXiv:2201.06296, https://arxiv.org/abs/2201.06296
Sven Zschocke, Light propagation in the 2PN approximation in the field of one moving monopole II. Boundary value problem, Classical and Quantum Gravity 36 (2019) 015007. DOI: 10.1088/1361-6382/aaeb4c, https://doi.org/10.1088/1361-6382/aaeb4c; arXiv:1812.00728, https://arxiv.org/abs/1812.00728
Sven Zschocke, Light propagation in the 2PN approximation in the field of one moving monopole I. Initial value problem, Classical and Quantum Gravity 35 (2018) 055013. DOI: 10.1088/1361-6382/aa9976, https://doi.org/10.1088/1361-6382/aa9976; arXiv:1802.02764, https://arxiv.org/abs/1802.02764
Sven Zschocke, Light propagation in the gravitational field of one arbitrarily moving pointlike body in the 2PN approximation, Physical Review D 94 (2016) 124007. DOI: 10.1103/PhysRevD.94.124007, https://doi.org/10.1103/PhysRevD.94.124007
Sven Zschocke, Light-propagation in the gravitational field of moving quadrupoles, arXiv:1105.3117 (2011), https://arxiv.org/abs/1105.3117
Sergei A. Klioner, Sven Zschocke, Numerical versus analytical accuracy of the formulae for light propagation, Classical and Quantum Gravity 27 (2010) 075015. DOI: 10.1088/0264-9381/27/7/075015, https://doi.org/10.1088/0264-9381/27/7/075015
Sven Zschocke, Sergei A. Klioner, On the efficient computation of the quadrupole light deflection, Classical and Quantum Gravity 28 (2011) 015009. DOI: 10.1088/0264-9381/28/1/015009, https://doi.org/10.1088/0264-9381/28/1/015009
Sergei A. Klioner, A Practical Relativistic Model for Microarcsecond Astrometry in Space, The Astronomical Journal 125 (2003) 1580. DOI: 10.1086/367593, https://doi.org/10.1086/367593
M. T. Crosta, F. Mignard, Microarcsecond light bending by Jupiter, Classical and Quantum Gravity 23 (2006) 4853. DOI: 10.1088/0264-9381/23/15/006, https://doi.org/10.1088/0264-9381/23/15/006
Sergei M. Kopeikin, Valeri V. Makarov, Gravitational bending of light by planetary multipoles and its measurement with microarcsecond astronomical interferometers, Physical Review D 75 (2007) 062002. DOI: 10.1103/PhysRevD.75.062002, https://doi.org/10.1103/PhysRevD.75.062002
Ummi Abbas, Beatrice Bucciarelli, Mario G. Lattanzi, M. T. Crosta, R. Morbidelli, D. Busonero, L. Bramante, R. Messineo, Differential astrometry with Gaia: Investigating relativistic light deflection close to Jupiter, Astronomy & Astrophysics 664 (2022) A143. DOI: 10.1051/0004-6361/202243972, https://doi.org/10.1051/0004-6361/202243972; arXiv:2206.09886, https://arxiv.org/abs/2206.09886
Ummi Abbas, Beatrice Bucciarelli, Mario G. Lattanzi, Differential astrometric framework for the Jupiter relativistic experiment with Gaia, Monthly Notices of the Royal Astronomical Society 485 (2019) 1147. DOI: 10.1093/mnras/stz452, https://doi.org/10.1093/mnras/stz452
Neil Ashby, Bruno Bertotti, Accurate light-time correction due to a gravitating mass, Classical and Quantum Gravity 27 (2010) 145013. DOI: 10.1088/0264-9381/27/14/145013, https://doi.org/10.1088/0264-9381/27/14/145013
Pierre Teyssandier, Direction of light propagation to order $G^{2}$ in static, spherically symmetric spacetimes: a new derivation, Classical and Quantum Gravity 29 (2012) 245010. DOI: 10.1088/0264-9381/29/24/245010, https://doi.org/10.1088/0264-9381/29/24/245010