⚛️ general relativity

Light deflection in the gravitational field of a solar system body with finite distance of source and observer for sub-micro-arcsecond astrometry

This paper derives a simplified, high-precision formula for light deflection by solar system bodies with finite source and observer distances, incorporating full time-dependent mass and spin multipoles to achieve sub-micro-arcsecond astrometric accuracy.

Original authors: Sven Zschocke

Published 2026-08-18

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Sven Zschocke

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

To understand the position of a star or a distant galaxy, astronomers must measure the direction from which its light arrives. For centuries, this was treated as a straight line, but Albert Einstein's theory of general relativity revealed that massive objects like the Sun and Jupiter warp the space around them. When a beam of light passes near such a body, it does not travel in a straight line; it curves, much like a car turning on a curved road. This bending of light, known as gravitational deflection, means that the star appears to be in a slightly different spot than it actually is. For most of history, this shift was too small to matter for everyday navigation, but modern astronomy has reached a level of precision where even the tiniest shift is critical.

Today, space missions are mapping billions of stars with an accuracy measured in micro-arcseconds, a unit so small that a human hair seen from ten kilometers away would appear wider. The next generation of these missions aims for even greater precision, targeting the sub-micro-arcsecond level. At this scale, the rules of measurement become incredibly strict. To map the universe correctly, scientists must account for every possible factor that could nudge a light beam, including the complex shapes of planets, their internal density, and their rotation. If the mathematical models used to correct these measurements are not precise enough, the resulting map of the cosmos will be flawed.

A researcher at the Technical University of Dresden has tackled the challenge of calculating these light deflections for the most demanding future missions. The work focuses on a specific scenario: a light signal traveling from a distant source to an observer, such as a telescope located near Earth, passing by a solar system body like the Sun or Jupiter. In the past, calculations often assumed the light source and the observer were infinitely far away, a simplification that breaks down when dealing with the actual distances in our solar system. The new study provides a rigorous method to calculate the bending of light when both the source and the observer are at finite, real-world distances from the massive body.

The core of the research involves deriving a simplified mathematical description of the light's path. The full equations governing this path are incredibly complex, filled with thousands of terms that describe the subtle gravitational influence of a planet's shape and spin. However, the researcher found that for the specific goal of sub-micro-arcsecond accuracy, most of these terms are irrelevant. They contribute so little to the final angle of deflection—often less than a few billionths of an arcsecond—that they can be safely ignored without compromising the accuracy of the mission. By stripping away these negligible terms, the study produces a streamlined formula that is much faster to compute. This efficiency is vital, as future missions will need to process data for billions of stars, a task that would be impossible if every calculation required solving the full, cumbersome equations.

The study also reveals a beautiful mathematical structure hidden within the physics of light bending. When the massive body is symmetric, like a spinning planet, the remaining terms in the simplified formula can be expressed using a specific family of curves known as Chebyshev polynomials. This discovery allows scientists to calculate the effect of light deflection for any level of detail regarding the planet's shape and rotation, simply by adding more terms to this polynomial series. The research confirms that the maximum possible bending of light occurs when the source and observer are infinitely far away, meaning that the finite-distance calculations for real missions will always be smaller than this theoretical maximum.

Furthermore, the paper establishes clear boundaries for what must be included in these calculations. It demonstrates that to reach the sub-micro-arcsecond goal, astronomers must account for the mass distribution of planets up to a certain level of detail, specifically including terms up to the eighth order of mass variation and the third order of spin variation. The study also briefly examines even higher-order effects from the theory of relativity, showing that while some of these are significant for the Sun, others are negligible for the planets. Ultimately, this work provides the essential toolkit for the next era of astronomy, ensuring that when future telescopes like GaiaNIR or Theia begin their surveys, they can translate the curved paths of light into a perfectly accurate map of our universe.

Technical Summary: Light Deflection in the Gravitational Field of a Solar System Body with Finite Distance of Source and Observer for Sub-Micro-Arcsecond Astrometry

Problem Statement
The primary objective of high-precision astrometry is the determination of the spatial positions of celestial objects based on the observed direction of light signals. Current missions like Gaia have achieved micro-arcsecond ( $\mu$ as) accuracy, but proposed future missions (e.g., GaiaNIR, Theia, TOLIMAN, SHERA) aim for sub-micro-arcsecond (sub- $\mu$ as) precision, targeting approximately 0.1 $\mu$ as. To achieve a theoretical model accuracy roughly 10 times better than the measurement goal (a threshold of 0.01 $\mu$ as or 10 nano-arcseconds), the relativistic modeling of light propagation in the solar system must be refined.

A critical challenge arises because existing analytical solutions for light trajectories often assume infinite distances for the source and observer, or contain cumbersome terms that are negligible at the sub- $\mu$ as level. Furthermore, numerical integration of geodesic equations is computationally unfeasible for the billions of observations expected from future missions. The specific problem addressed is the derivation of a simplified, analytical expression for the unit tangent vector of a light ray at the observer's position, accounting for:

Finite distances of both the light source and the observer from the gravitating body.
The full set of time-dependent mass-multipoles and spin-multipoles, allowing for arbitrary body shapes, internal structures, and rotational motions.
A reduction of terms to only those contributing $\ge$ 10 nano-arcseconds to the light deflection angle.

Methodology
The investigation utilizes the Post-Newtonian (PN) scheme up to the 1.5PN approximation (terms of order $O(c^{-4})$ ). The gravitational field of an isolated body is described by a metric tensor expanded in terms of mass-multipoles ( $M_L$ ) and spin-multipoles ( $S_L$ ).

Geodesic Equation and Integration: The geodesic equation for light propagation is solved using advanced integration methods based on new variables ( $\tau$ and $\xi$ ) introduced in previous works. This transforms the problem into a boundary value problem where the light source ( $x_0$ ) and observer ( $x_1$ ) are at finite distances.
Derivation of the Unit Tangent Vector: The unit tangent vector ( $n$ ) at the observer is derived by integrating the geodesic equation and transforming the initial direction ( $\sigma$ ) to the observed direction ( $k$ ). The solution is expressed as a sum of a background direction ( $k$ ) and perturbations due to mass-multipoles ( $\Delta n^{ML}$ ) and spin-multipoles ( $\Delta n^{SL}$ ).
Magnitude Analysis and Simplification: The paper systematically analyzes the magnitude of individual terms in the perturbations. Terms are categorized by their scaling with the impact parameter ( $d_k$ $d_{k}$ ), the distance to the observer ( $r_1$ $r_{1}$ ), and the body's radius ( $P$ $P$ ).
- Terms contributing less than 10 nano-arcseconds are identified and neglected.
- Two specific differential operations ( $U^L$ and $V^L$ ) are analyzed, and specific components ( $U^2_L$ and $V^2_L$ ) are shown to be negligible for sub- $\mu$ as astrometry.
Axi-Symmetric Case: For bodies approximated as axi-symmetric (e.g., the Sun and giant planets), the mass and spin multipoles are expressed using zonal harmonic coefficients ( $J_l$ ). The resulting expressions for the unit tangent vector are shown to be representable using Chebyshev polynomials of the first and second kinds.

Key Contributions

Simplified Unit Tangent Vector: The paper derives a "simplified unit tangent vector" (Eq. 133) for an arbitrary body at rest. This expression retains all terms relevant for sub- $\mu$ as accuracy while discarding those contributing less than 10 nano-arcseconds. It is valid for arbitrary finite distances of the source and observer.
Chebyshev Polynomial Representation: For axi-symmetric bodies, the paper demonstrates that the light deflection terms can be compactly expressed using Chebyshev polynomials ( $T_l$ and $U_{l-1}$ ). This allows for the determination of light deflection effects up to any order of mass and spin multipoles efficiently.
Upper Limit Proof: It is proven that the total light deflection (calculated for infinite source and observer distances) represents a strict upper limit for the light deflection effect when the source and observer are at finite distances, provided the observer is located near Earth (e.g., at Lagrange point L2).
Multipole Order Requirements: Numerical analysis establishes that for sub- $\mu$ as accuracy, mass-multipoles up to order $l=8$ and spin-multipoles up to order $l=3$ must be included in the model for solar system bodies.

Results

Negligible Terms: The study quantifies that terms involving higher-order derivatives of multipoles and specific spin-multipole terms (e.g., $l=4, 6, \dots$ for spin) contribute less than 1 nano-arcsecond and can be safely neglected.
Jupiter and Saturn Grazing Rays: For grazing rays at Jupiter and Saturn, the sum of all neglected terms is calculated. For Saturn, the sum is $\le$ 14.9 nas. For Jupiter, it is $\le$ 36.2 nas. The paper notes that for Jupiter, if the impact parameter is slightly larger than the radius ( $d_k \ge 1.7 P$ ), the neglected terms drop below the 10 nas threshold. In extreme grazing cases at Jupiter, the exact mass-quadrupole term (1PN) may need to be retained separately to maintain the 10 nas accuracy goal.
2PN and 3PN Terms: The paper briefly evaluates higher-order Post-Newtonian terms. It finds that 2PN mass-monopole and mass-quadrupole terms can reach magnitudes of several micro-arcseconds (e.g., ~~3136 $\mu$ as for the Sun) and must be included. 3PN terms are generally smaller but non-negligible for the Sun (~~11.2 $\mu$ as).
Numerical Tables: The paper provides comprehensive tables (IV, V, VI) listing the upper limits of light deflection for the Sun and giant planets due to mass-multipoles, spin-multipoles, and higher-order PN terms.

Significance
The paper claims that the derived simplified unit tangent vector is essential for the effective reduction of observational data for future space astrometry missions like GaiaNIR and Theia. By providing an analytical solution that is both compact and sufficiently accurate (neglecting only terms below the 10 nano-arcsecond threshold), the work addresses the computational bottleneck of handling billions of astrometric observations. The demonstration that Chebyshev polynomials can describe the deflection for axi-symmetric bodies offers a computationally efficient method for modeling light propagation in the gravitational fields of solar system bodies, facilitating the transition from micro-arcsecond to sub-micro-arcsecond astrometry.

More like this