Completing the time-transfer-function relativistic model for Gaia's global astrometric sphere reconstruction: higher-order light propagation, attitude and calibration co-estimation, and the PPN gamma sensitivity

Gaia's global astrometric sphere reconstruction requires a relativistic model of light propagation and observation accurate at the microarcsecond level. The 2017 validation of GSR-TTF, the implementation of the time transfer function (TTF) formalism inside the Global Sphere Reconstruction (GSR) pipeline of Gaia's Astrometric Verification Unit, demonstrated recovery of a five-parameter stellar sphere from simulated observations, but left a well-defined completion agenda open: extending the light-propagation model beyond monopole and $J_2$ terms at order $(v/c)^3$ (moving axisymmetric bodies, higher post-Minkowskian/post-Newtonian orders), incorporating simultaneous estimation of satellite attitude and instrument calibration parameters into large-scale TTF-based sphere solutions, establishing the iteration-for-non-linearity procedure and thresholds for solver convergence, confirming the extrapolated Monte-Carlo sensitivity to the post-Newtonian parameter $\gamma$ at full scale with realistic attitude and calibration estimation, and tracing residual sub-microarcsecond differences with respect to the baseline Gaia model GREM to their implementation-level sources. The published follow-up literature (companion GSR paper, AVU-GSR solver papers, PPN-$\gamma$ coupling studies, real-data qualification reports) provides substantial infrastructure and partial progress but does not close this agenda: the problem remains open.

Background

Gaia is the European Space Agency's scanning space astrometry mission, measuring positions, parallaxes and proper motions of about a billion stars from the time-delayed integration of source transits across CCDs (charge-coupled-device imaging sensors). Because the satellite measures only angles on the sky, absolute positions and parallaxes can be recovered only through a 'global sphere reconstruction': a simultaneous least-squares solution for the astrometric parameters of a set of primary stars together with the satellite attitude (the time-dependent orientation of the instrument reference frame, represented by spline coefficients), instrument calibration parameters (large- and small-scale geometric and chromatic corrections), and possibly a few global parameters. At Gaia's target accuracy of a few tens of microarcseconds ($\mu$as), light deflection by the Sun, planets and the Moon is a first-order effect, so the observation equations must be based on a fully relativistic model of light propagation in the Solar System, formulated in the barycentric reference system of general relativity.

The model baselined for Gaia's operational reduction is GREM (Gaia RElativistic Model), accurate through order $(v/c)^3$. The Astrometric Verification Unit (AVU) of the data processing consortium independently reproduces the sphere reconstruction with the GSR (Global Sphere Reconstruction) pipeline, which uses a different relativistic model and different solution algorithms so that it can validate the baseline catalog. Within GSR two relativistic approaches coexist: RAMOD, a family of models based on the null-geodesic formulation, and the time transfer function (TTF) approach, in which the light path between a source and the observer is described by auxiliary functions solving the eikonal equation (the high-frequency limit of the wave equation) in the post-Minkowskian (weak-field, iteration in powers of the gravitational constant) framework. The TTF formalism provides 'ready-to-use' analytical formulas for the deflection caused by multiple axisymmetric bodies, moving with arbitrary velocities (Hees et al. 2014), and to second and third post-Newtonian (expansion in powers of $v/c$) order for static bodies; its analytical equivalence with RAMOD and GREM in restricted cases was verified by Bertone et al. (2014).

Bertone et al. (2017) reported the first application of the TTF approach, 'GSR-TTF', to a global sphere reconstruction: on five years of simulated Gaia observations of about 132,000 primary stars, the linearized least-squares solution recovered the astrometric parameters with the expected accuracy, and a Monte-Carlo experiment combining 12,000 primaries estimated the parameter $\gamma$ of the parametrized post-Newtonian (PPN) formalism (which multiplies the curvature per unit mass and fixes the strength of gravitational light deflection; general relativity predicts $\gamma=1$) with a statistical uncertainty $\sigma_\gamma^{\rm MC}=1.05\times10^{-4}$, extrapolable to $\sim1.3\times10^{-6}$ for a realistic primary sample of about one million stars. The implementation, however, was deliberately restricted: light propagation retained only the monopole and $J_2$ (quadrupole) terms at order $(v/c)^3$, evaluated at retarded body positions to leading approximation $t_C=t_B-c^{-1}|\mathbf{x}_B-\mathbf{x}_P(t_B)|+O(c^{-2})$ with gravitomagnetic (velocity-dependent) contributions neglected; the satellite attitude and instrument calibration were treated as perfectly known rather than co-estimated; the RAMOD solver inside GSR showed convergence problems requiring an 'iteration for non-linearity' (a re-linearization of the observation equations when the starting catalog is not accurate enough, since the equations are solved in linearized form) whose thresholds and implementation were left to a promised dedicated study; and residual sub-$\mu$as differences between GSR-TTF and GREM abscissae (the along-scan position measured for each stellar transit), attributable to retarded-time handling or attitude representation, were observed but not traced. These restrictions define the completion problem at hand.

Problem Statement

The problem is to complete and validate the time-transfer-function relativistic model inside the Gaia global sphere reconstruction (GSR-TTF) so that it can serve as a full-accuracy, independent relativistic solution for Gaia's astrometric catalog, as framed by the open items of Bertone et al. (2017). The common objective uniting the requested results is an operational TTF-based global sphere reconstruction whose modelling accuracy, solution completeness and error budget match the Gaia mission requirements. Concretely, the requested results are: (i) extend the light-propagation modelling of GSR-TTF beyond the present monopole and $J_2$ terms at order $(v/c)^3$ with the leading-order retarded-time approximation — by incorporating moving axisymmetric bodies (whose formulation already exists) and any post-Minkowskian/post-Newtonian terms (e.g. gravitomagnetic contributions, higher multipoles) needed at the target accuracy; (ii) incorporate the simultaneous estimation of satellite attitude parameters and instrument calibration parameters, together with the astrometric and global unknowns, into large-scale TTF-based sphere solutions, and demonstrate their convergence and accuracy on Gaia-like (or real) data — including establishing the iteration-for-non-linearity procedure and its operational thresholds for cases where the linearization point is not sufficiently accurate; (iii) confirm the extrapolated Monte-Carlo sensitivity to the PPN parameter $\gamma$ ($\sigma_\gamma\sim10^{-6}$ for a realistic primary population) with full-scale simulations that include realistic attitude and calibration estimation, and quantify all sources of systematic error in the $\gamma$ estimate; and (iv) reduce, track and document the residual sub-$\mu$as differences between GSR-TTF and the baseline model GREM — attributed to retarded-time handling or attitude representation — and their impact on catalog-level systematics. The four facets together are what would bring the TTF-based global sphere reconstruction to the mission's microarcsecond modelling accuracy, solution completeness and error budget; each facet is a step in bringing the same relativistic model from validated prototype to qualified scientific instrument.

References

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