TrojAI Program: Securing AI from Trojan Attacks

• TrojAI Program is a focused initiative that identifies and challenges vulnerabilities in AI systems caused by Trojan attacks.
• The program employs advanced adversarial testing and robust evaluation methods to uncover and mitigate security threats in machine learning models.
• It fosters interdisciplinary collaboration between academia and industry to accelerate the development of secure and resilient AI technologies.

Gaia–Hipparcos absolute astrometry is the practice of determining stellar positions, parallaxes, and proper motions in an inertial coordinate system, the International Celestial Reference System (ICRS), by means of space-based, wide-angle, all-sky scanning and a global solution rather than local differential referencing. Hipparcos established the first space-based global astrometric framework, and Gaia extends the same logic to the billion-source regime with microarcsecond performance, making absolute astrometry a foundational data layer for stellar astrophysics, Galactic dynamics, Solar-system studies, reference-frame realization, and long-baseline orbital inference (Høg, 2014).

1. Definition and conceptual basis

In this context, absolute astrometry means that the astrometric parameters are determined in an inertial system rather than relative to nearby field stars. The standard source model comprises the position at a reference epoch $(\alpha,\delta)$, the trigonometric parallax $\varpi$, and the two proper-motion components $(\mu_{\alpha *},\mu_\delta)$, with $\mu_{\alpha *}=\mu_\alpha \cos\delta$. Relative astrometry, by contrast, measures motion with respect to stars in the same small field and is therefore vulnerable to local distortions and to the residual parallaxes and internal motions of the reference stars (Høg, 2014).

The methodological distinction is inseparable from the observing geometry. Hipparcos and Gaia do not infer parallaxes from a local plate model; they measure wide angles between targets in many orientations while scanning the whole sky over years. The whole sphere is then reconstructed globally, so the solution does not depend on any locally “fixed” field. This global reconstruction is what makes the parallaxes absolute and the proper motions globally consistent. Repeated observations are essential because they disentangle the constant sky position, the annual parallax signature, and the linear proper motion (Eyer et al., 2012).

A common misconception is that absolute astrometry is merely very accurate relative astrometry. The mission architecture described for Hipparcos and Gaia indicates otherwise: the decisive element is not only precision but the global, self-calibrated linkage of distant fields into a single inertial frame. A plausible implication is that absolute astrometry is as much a problem of global geometry and attitude reconstruction as of centroiding precision.

2. Measurement architecture and global sphere reconstruction

Hipparcos and Gaia share the same high-level principle—continuous scanning and global sphere reconstruction—but implement it with different detectors, focal-plane architectures, and calibration systems. Hipparcos executed continuous great-circle scanning, modulating starlight through a grid and detecting signals via photon counting astrometry. Gaia uses two viewing directions separated by a fixed basic angle, direct imaging on CCDs operated in time-delayed integration mode, and on-board metrology via the Basic Angle Monitor. Gaia also adds photometry and radial velocities to the astrometric payload (Høg, 2014, Bruijne, 2012).

Aspect	Hipparcos	Gaia
Detection principle	Modulating grid and photon counting astrometry	Direct CCD imaging in TDI mode
Sky coverage strategy	Continuous great-circle scanning	Two telescopes with fixed basic angle and scanning law
Mission-scale solution	Global sphere reconstruction	AGIS block-iterative global solution
Typical scale	118,000 stars over 3.3 years	More than one billion sources to $G\approx20$ over 5–6 years
Auxiliary data	Tycho/Tycho-2 photometry	Photometry and radial velocities

Each Hipparcos scan yielded one-dimensional along-scan measurements along great circles. A global sphere reconstruction was then obtained by solving simultaneously for source parameters and the time-dependent satellite attitude, using overlapping scans to tie the sphere rigidly. Gaia operationalizes the same principle through the Astrometric Global Iterative Solution (AGIS), which solves in blocks for source parameters, attitude, instrument calibration, and global frame parameters in a weighted least-squares scheme (Høg, 2014).

Gaia’s observing law is quantitatively specified in pre-launch performance studies: the two fields of view are separated by $106.5^\circ$, the spacecraft rotates at $1^\circ$ per minute, the spin axis maintains a solar-aspect angle of $45^\circ$, and the scan axis precesses with a period of 63 days. The focal plane contains 106 CCDs, and the sky-average number of focal-plane transits is about 70 over five years. These design choices produce the highly interconnected observation network required for a globally rigid solution (Bruijne, 2012).

Independent verification of the global sphere reconstruction became part of Gaia’s processing philosophy because large-scale systematics in an absolute solution are difficult to diagnose from the final catalog alone. The AVU Global Sphere Reconstruction pipeline was therefore developed as an independent implementation parallel to AGIS and was shown on simulated data to reproduce AGIS results at the sub-$\mu$as level in the relevant demonstration run (Vecchiato et al., 2018).

3. Astrometric model, observation equations, and frame realization

The single-star model is commonly expressed through five parameters. The basic geometric relation is

$$d\,[\mathrm{pc}] = \frac{1}{\varpi\,[\mathrm{arcsec}]},$$

with the observed direction perturbed by the observer’s barycentric motion according to

$\varpi$0

In scanning astrometry the fundamental observable is the along-scan field angle $\varpi$1. After linearization around the current parameter estimates, the residual can be written schematically as

$\varpi$2

where the terms encode sensitivities to attitude, parallax, proper motion, calibration, and measurement noise (Høg, 2014).

The global solution is posed as a weighted least-squares problem,

$\varpi$3

with unknowns spanning source parameters, attitude, calibration, and global frame parameters. AGIS updates these blocks cyclically until convergence. This formulation is the mathematical expression of “global sphere reconstruction”: the catalog is not a post-processing of local astrometric fits but the direct solution of a coupled world-wide inverse problem (Høg, 2014).

The “absolute” character of the catalog depends on the reference-frame realization. Hipparcos delivered an optical frame linked to the ICRS through linking procedures, with the Hipparcos Celestial Reference Frame aligned with ICRS to 0.6 mas on the three axes and deviating from inertiality by about $\varpi$4 mas yr$\varpi$5 on each axis. Gaia aligns its optical frame to the International Celestial Reference Frame using quasars and targets an inertiality of about $\varpi$6 $\varpi$7as yr$\varpi$8, while also accounting for the secular aberration drift produced by the Solar-system barycentric acceleration, of order $\varpi$9 (Høg, 2014).

Early Gaia catalog results already demonstrated the operational realization of this frame concept. In Gaia DR1, positions and proper motions were aligned with the ICRF to better than 0.1 mas at epoch J2015.0 and non-rotating with respect to ICRF to within 0.03 mas yr$(\mu_{\alpha *},\mu_\delta)$0. The Hipparcos frame was found to rotate with respect to the Gaia DR1 frame at a rate of 0.24 mas yr$(\mu_{\alpha *},\mu_\delta)$1, quantifying the gain from Gaia’s quasar-tied realization of the optical frame (Lindegren et al., 2016).

4. Precision, catalogs, and astrophysical consequences

Hipparcos transformed stellar distance work by delivering parallaxes and proper motions with space-based precision for 118,000 stars and by extending the astrometric basis for bright stars through Tycho and Tycho-2. Distances with 1.0% accuracy were obtained for 719 stars, and the Tycho-2 catalogue provided positions, proper motions, and $(\mu_{\alpha *},\mu_\delta)$2 photometry for 2.5 million bright stars. In the broader distance-scale literature, Hipparcos raised the number of stars with trigonometric parallaxes known better than 10% from about a thousand to 22,396 in the original catalogue and 30,579 in the re-reduction (Høg, 2014, Turon et al., 2012).

Gaia scales the same program by orders of magnitude. Pre-launch descriptions specify a survey of more than one billion stars to $(\mu_{\alpha *},\mu_\delta)$3, with photometry and radial velocities, and astrometric accuracies of about 10 micro-arcseconds at 14 mag, or, in sky-averaged end-of-mission parallax form, less than 10 $(\mu_{\alpha *},\mu_\delta)$4as at $(\mu_{\alpha *},\mu_\delta)$5, about 25 $(\mu_{\alpha *},\mu_\delta)$6as at $(\mu_{\alpha *},\mu_\delta)$7, and about 300 $(\mu_{\alpha *},\mu_\delta)$8as at $(\mu_{\alpha *},\mu_\delta)$9. The expected science yield included distances with 1.0% accuracy for 10 million stars and about 150 million stars with expected distances better than 10% (Høg, 2014, Bruijne, 2012, Turon et al., 2012).

These precision gains underpin several domains already identified as core science drivers: calibration of the cosmic distance scale and stellar luminosities via precise parallaxes; cluster membership and internal dynamics; Galactic rotation, local kinematics, disk warp and breathing modes; dark-matter constraints through tangential velocities; Solar-system dynamics and asteroid masses; an optical ICRS anchored by quasars; and exoplanet astrometry for long-period giant planets (Høg, 2014).

Systematics remain structurally important. Hipparcos reductions had to contend with attitude discontinuities, scan-phase jumps, grid distortions, and localized glitches. Gaia’s systematics include basic-angle variations, chromaticity and color-dependent PSF/LSF effects, CCD charge-transfer inefficiency, gating for bright stars, stray light, and geometric-distortion calibration. A mission-level parallax zero-point can arise from couplings between basic-angle behavior and calibration, so external validations using quasars and standard candles are part of the mitigation strategy (Høg, 2014).

Early Gaia data made these issues explicit. For the roughly two million TGAS stars in DR1, systematic errors depending on position and color were at a level of 0.3 mas, and users were advised to treat TGAS parallaxes as having a systematic floor of about 0.3 mas. This does not contradict the absolute-astrometry concept; it shows that an inertial, globally solved frame can still carry mission-phase calibration systematics, especially when based on less than a quarter of the nominal mission duration (Lindegren et al., 2016).

Hipparcos also left scientifically important controversies. The re-reduction improved bright-star accuracy and reduced error correlations, but the Pleiades mean distance of $\mu_{\alpha *}=\mu_\alpha \cos\delta$0 pc remained controversial with respect to main-sequence fitting. Gaia was explicitly expected to close that debate, illustrating how absolute astrometry can expose discrepancies in external astrophysical calibrations rather than merely confirm them (Turon et al., 2012).

5. Long-baseline combination, accelerations, and orbital inference

The combination of Hipparcos and Gaia extends absolute astrometry from static catalog production to dynamical inference. The central idea is that two precise absolute positions separated by about 24–25 years yield greatly improved proper motions and can reveal curvature in the sky motion caused by binaries or planets. In the general joint-solution framework, if $\mu_{\alpha *}=\mu_\alpha \cos\delta$1 and $\mu_{\alpha *}=\mu_\alpha \cos\delta$2 are the normal equations from the two missions, then

$\mu_{\alpha *}=\mu_\alpha \cos\delta$3

with the combined covariance $\mu_{\alpha *}=\mu_\alpha \cos\delta$4. This is the statistically proper combination because it preserves the full covariance structure rather than using only catalog differences (Michalik et al., 2012).

This logic underpinned the Hundred Thousand Proper Motions project. Simulations for an early Gaia-plus-Hipparcos joint solution predicted proper-motion accuracies between 14 and 134 $\mu_{\alpha *}=\mu_\alpha \cos\delta$5as yr$\mu_{\alpha *}=\mu_\alpha \cos\delta$6, depending on magnitude and Gaia data volume, and introduced a goodness-of-fit statistic sensitive to deviations from uniform space motion caused by binaries with periods of 10–50 years (Michalik et al., 2014). Gaia DR1 then operationalized the same baseline idea through TGAS, producing five-parameter solutions for 2,057,050 bright stars; for the 93,635 Hipparcos entries used in TGAS, the median proper-motion uncertainty was 0.064 mas yr$\mu_{\alpha *}=\mu_\alpha \cos\delta$7 thanks to the 1991.25 to 2015.0 baseline (Lindegren et al., 2016).

The modern acceleration formalism is encapsulated by the Hipparcos–Gaia Catalog of Accelerations. In the Gaia EDR3 edition, each Hipparcos star receives three proper motions on the Gaia EDR3 frame: the Hipparcos proper motion $\mu_{\alpha *}=\mu_\alpha \cos\delta$8, the Gaia EDR3 proper motion $\mu_{\alpha *}=\mu_\alpha \cos\delta$9, and the long-term proper motion $G\approx20$0 from the Hipparcos–Gaia position difference over the baseline. The catalog offers a factor of about 3 improvement in precision relative to its DR2 edition, uses a 60/40 mixture of the two Hipparcos reductions, corrects color- and magnitude-dependent frame rotations at up to about 50 $G\approx20$1as yr$G\approx20$2, and calibrates Gaia EDR3 uncertainties with an error inflation factor of 1.37 (Brandt, 2021).

For an unaccelerated single star, $G\approx20$3, $G\approx20$4, and $G\approx20$5 agree within uncertainties. A companion produces statistically significant differences among them, encoding the reflex acceleration over the roughly 25-year baseline. These acceleration measurements can then be combined with radial velocities and relative astrometry in orbit-fitting codes such as orvara, which analytically marginalizes parallax, barycenter proper motion, and instrument-specific radial-velocity zero points, or in mission-aware forward-modeling tools such as htof, which ingest Hipparcos intermediate astrometric data and Gaia scan-angle metadata (Brandt et al., 2021, Brandt et al., 2021).

The empirical impact is already substantial. Joint modeling of Hipparcos–Gaia accelerations with long-baseline radial velocities and direct imaging has yielded precise dynamical masses for very low-mass stellar companions, including cases where face-on geometry would have made radial-velocity-only inference misleading (Rickman et al., 2022). The same strategy has lifted the $G\approx20$6 degeneracy for long-period radial-velocity companions in the planetary regime, showing how absolute astrometry converts $G\approx20$7 into a true mass by constraining the inclination and sky-plane orbit orientation (Philipot et al., 2023). In binaries with significant secondary light or resolved Gaia components, dedicated tools such as BINARYS model Hipparcos transit data, Gaia EDR3 absolute astrometry, and relative astrometry in a single dynamical framework to obtain model-independent masses (Leclerc et al., 2022).

6. Future baseline extension and the fifty-year perspective

The long-baseline logic of Gaia–Hipparcos absolute astrometry naturally extends to the case for a Gaia successor mission. The stated rationale is that a second Gaia-like mission about twenty years after Gaia would produce an astrometric foundation that “cannot be surpassed the next 50 years,” with a microarcsecond-level absolute frame, improved quasar anchoring, and substantially better long-term stability (Høg, 2014).

The quantitative gains proposed for a two-mission architecture are explicit. Positions predicted 50 years from now would have errors more than 20 times smaller than from a single Gaia; tangential velocities would have 10 times smaller errors in a 30 times larger volume; and exoplanet periods up to 40 years would become accessible. In the same framework, two missions would make zero-motion quasar selection about 100 times cleaner, improve sensitivity to dark-matter dynamics from proper motions, and maintain a high-accuracy inertial optical frame for future facilities (Høg, 2014, Høg, 2014).

The argument is partly driven by error propagation. For Gaia alone, position uncertainties at future epochs become dominated by proper-motion errors, with

$G\approx20$8

In the 50-year outlook paper, representative Gaia-alone figures at $G\approx20$9 imply position errors of about 1.76 mas in 2026, 3.5 mas in 2036, and 8.8 mas in 2066. Two Gaia-like missions would reduce those long-horizon position errors by about 20 times and proper motions by about 10 times, preserving sub-mas reference stars for future small-field work (Høg, 2014).

The envisioned future is not limited to a single successor. VLBI maintains the radio realization of the ICRS; MICADO on the ELT is described as capable of about 50 $106.5^\circ$0as precision per hour for stars brighter than $106.5^\circ$1 and of absolute parallaxes around 20 $106.5^\circ$2as for individual stars in a six-epoch strategy; LSST is described as reaching proper-motion accuracy of about 0.2 mas yr$106.5^\circ$3 and parallax accuracy of about 1.0 mas over ten years. These techniques densify and exploit the reference frame, but the stated conclusion is that long-term inertial maintenance and sub-mas all-sky stability still favor a second Gaia-like space mission (Høg, 2014).

A plausible implication is that Gaia–Hipparcos absolute astrometry should be understood not as a completed catalog enterprise but as the first realized segment of a multi-mission inertial metrology program. Hipparcos established the concept, Gaia industrialized it at scale, and the successor-mission case follows directly from the way astrometric information compounds with time baseline.