Hipparcos Gaia Catalogue of Accelerations

• The Hipparcos–Gaia Catalogue of Accelerations is a comprehensive dataset that cross-calibrates stellar astrometry to detect small changes in proper motions over multi-decade baselines.
• It integrates three independent measurements—Hipparcos proper motion, Gaia proper motion, and scaled positional differences—while correcting systematics via techniques like spherical harmonics.
• The catalog enables robust orbit fitting and dynamical mass estimates, revealing binary and substellar companions through statistically significant acceleration detections.

The Hipparcos–Gaia Catalogue of Accelerations (HGCA) is a comprehensive, cross-calibrated compilation of stellar astrometric measurements from the Hipparcos and Gaia missions, designed to identify and quantify changes in proper motion—i.e., accelerations—in the plane of the sky. By leveraging the multi-decade baseline separating Hipparcos (epoch ~1991.25) and Gaia (epochs ~2015.5 for DR2 and EDR3), the HGCA provides rigorously frame-tied, statistically independent proper-motion vectors and their covariances for ~115,000 stars, facilitating dynamical studies of binary, planetary, and otherwise perturbed stellar systems (Brandt, 2018, Brandt, 2021).

1. Reference Frames, Cross-Calibration, and Proper-Motion Construction

The core objective of the HGCA is to place all proper-motion measurements onto the Gaia ICRS frame, with consistent calibration and error characterization. Three nearly independent astrometric quantities are constructed for each star:

1. Hipparcos Proper Motion ($\mu_H$): Defined as a statistically optimal $f/(1-f)$ linear combination of the two Hipparcos reductions (van Leeuwen 2007, $f \approx 0.6$; ESA 1997, $1-f \approx 0.4$), supplemented by locally-fit residual rotation corrections and perspective-acceleration terms for nearby stars:

$$\mu_H = f\,\mu_{\mathrm{vL2007}} + (1-f)\,\mu_{\mathrm{ESA1997}} + \boldsymbol{\xi}_H(\alpha,\delta) + 2\boldsymbol{\gamma}(\alpha,\delta)$$

1. Gaia Proper Motion ($\mu_G$): The five-parameter instantaneous proper motion from Gaia DR2 or EDR3.
2. Scaled Positional-Difference Proper Motion ($\mu_{HG}$): The long-term average motion between Hipparcos and Gaia positions, scaled by the epoch difference $\Delta t$ and corrected for local systematics:

$$\mu_{HG} = \frac{f\,\boldsymbol{\theta}_{\mathrm{vL2007}} + (1-f)\,\boldsymbol{\theta}_{\mathrm{ESA1997}} - \boldsymbol{\theta}_{\mathrm{Gaia}}}{\Delta t} + \boldsymbol{\xi}_{HG}(\alpha, \delta) + \boldsymbol{\gamma}(\alpha, \delta)$$

Locally-varying rotational misalignments, magnitude and color dependencies, and perspective corrections are modeled and mitigated for all three measurements using spherical-harmonic or Matern-kernel-based smoothing techniques (Brandt, 2021, Feng et al., 7 Feb 2024).

2. Calibration of Uncertainties and Systematics

Accurate companion and acceleration inference requires rigorously calibrated uncertainties:

• Hipparcos (Composite) Covariances: Additive inflation to the formal covariance matrix, e.g. $b^2 = (0.199~\mathrm{mas~yr}^{-1})^2$ (Brandt, 2018), or a jitter term of $\sim$2.16 mas for Hipparcos 2007 IAD (Feng et al., 7 Feb 2024).
• Gaia Covariances: Multiplicative inflations; for DR2, $a(\alpha,\delta) \approx 1.74$; for EDR3, $a = 1.37$ globally (Brandt, 2021).
• Systematics Correction: Large-scale, correlated systematics (e.g., due to scanning law or instrument model errors) are corrected using vector spherical harmonics (VSH) decompositions up to degree $L=7$ (126 terms), with median systematics of $\sim$9 mas yr$^{-1}$ subtracted from all stars (Mignard et al., 2012, Makarov et al., 20 May 2024).

The resulting proper-motion differences achieve Gaussian residual distributions to within a few percent, enabling high-confidence identification of physically real accelerations.

3. Acceleration Measurement and Statistical Significance

Plane-of-sky accelerations are computed as angular derivatives of the difference between instantaneous and long-term proper motions, typically:

$$\boldsymbol{a} = \frac{\boldsymbol{\mu}_G - \boldsymbol{\mu}_H}{\Delta t}$$

with full covariance propagation:

$$\mathbf{C}_a = \frac{\mathbf{C}'_G + \mathbf{C}'_H}{\Delta t^2}$$

Detection confidence is quantified by the Mahalanobis norm of the acceleration vector,

$$\chi_\mu = \sqrt{\Delta \boldsymbol{\mu}^T\,\mathbf{C}_{\Delta\mu}^{-1}\,\Delta\boldsymbol{\mu}}$$

which is compared to $\chi^2$ thresholds (e.g., $\chi^2 > 13.8$ for 99.9% confidence, or $\chi^2 = 28.75$ for a formal $5\sigma$ detection in two dimensions) (Whiting et al., 2023, Makarov et al., 20 May 2024).

Accelerations are subsequently converted into tangential velocity anomalies and, with parallax, inform constraints on unseen companion masses and separations (Painter et al., 26 Jun 2025).

4. Catalog Content, Structure, and Statistical Properties

The standard HGCA (EDR3 edition) comprises $\sim$115,346 stars, nearly $98\%$ complete relative to the original Hipparcos sample. Key data for each star:

• Proper motions $\mu_H$, $\mu_G$, $\mu_{HG}$ with $2\times2$ covariance matrices
• Central epochs for all quantities
• All cross-calibration parameters (rotations, color/magnitude maps)
• $\chi^2$ statistics and significance flags

Summary statistics under Gaussian error assumptions yield a "Gamma" component for single stars and a log-normal high-acceleration tail containing $\sim$22–28% of systems with clear evidence for binarity or strong perturbations (Makarov et al., 20 May 2024). Outlier handling and residual bias correction (e.g., via RUWE, VSH residuals) are performed, but further individual vetting is recommended for spurious or problematic entries.

5. Orbit Fitting, Dynamical Masses, and Physical Companions

The HGCA is structured to enable case-by-case orbit fitting, particularly for degenerate parameter spaces in radial-velocity (RV) or direct-imaging analyses. By entering the proper-motion differences as acceleration constraints in a joint likelihood function,

$$-2\,\ln \mathcal{L}(\theta) = \chi^2_{\rm sep} + \chi^2_{\rm PA} + \chi^2_{\rm RV} + \chi^2_{\varpi} + \chi^2_{\rm G} + \chi^2_{\rm H} - 2\ln\,\pi(\theta)$$

researchers can robustly break degeneracies between host and companion mass, orbital inclination, and projected separation. The inclusion of orthogonal on-sky acceleration components reduces the uncertainty in dynamical mass estimates for companions to long-period binaries and directly imaged substellar objects (Brandt et al., 2018). Example: Gl 758B’s dynamical mass uncertainty shrank from $\pm6$ to $\pm1.6~M_{\rm Jup}$ with the addition of HGCA accelerations.

6. Extensions, Machine-Learning Approaches, and Future Data Releases

The HGCA framework has been extended to larger and deeper stellar samples via supervised machine-learning classifiers (Random Forest regressors) trained on HGCA, Gaia DR2/EDR3, and other catalog features. The Gaia Nearby Accelerating Star Catalog (GNASC) includes 29,684 G$\le$17.5 mag, $d\le100$ pc stars with high-confidence acceleration signal, capturing $>96\%$ of Gaia’s own acceleration solutions and providing thousands of new candidate binaries and substellar hosts (Whiting et al., 2023).

Ongoing Gaia data releases (e.g., DR4, DR5) are expected to lower proper-motion uncertainties by factors of 2–3, pushing the sensitivity boundary to sub-Jupiter masses at separations of several AU for bright, nearby stars (Painter et al., 26 Jun 2025). Analytical sensitivity estimates support $\sim85\%$ completeness for $5$–$13~M_{\rm Jup}$ companions at $4$–$10$ AU in the latest HGCA (Painter et al., 26 Jun 2025).

7. Systematics, Limitations, and Best Practices

Despite extensive frame, color, and magnitude calibrations, residual systematics persist—especially in stars with large parallax errors, high RUWE, or in the lowest-precision Gaia regime. Vector spherical harmonic correction absorbs large-scale correlated errors, but further improvements may be required for small-scale instrument-induced patterns (Makarov et al., 20 May 2024). The catalog is not intended for population-level statistical analyses of companion frequency or acceleration distribution due to non-uniform selection and heteroscedastic error structure; instead, it is optimized for follow-up and orbit fitting of individual accelerating systems (Brandt, 2018, Brandt, 2021).

Table: Principal Astrometric Quantities in the HGCA

Name	Definition	Typical Precision (bright)
$\mu_H$	Weighted/rotated Hipparcos proper motion	$\sim$0.9 mas yr$^{-1}$
$\mu_G$	Gaia DR2 or EDR3 proper motion	$0.02$–$0.10$ mas yr$^{-1}$
$\mu_{HG}$	(Gaia-Hipparcos position)/$\Delta t$	$0.02$–$0.07$ mas yr$^{-1}$
$\Delta \mu$	$\mu_G - \mu_{HG}$ (or similar)	$0.02$–$0.15$ mas yr$^{-1}$
Acceleration $\boldsymbol{a}$	$(\mu_G-\mu_H)/\Delta t$	$0.001$–$0.01$ mas yr$^{-2}$

A properly cross-calibrated set of these allows precise plane-of-sky accelerations for $\sim$115,000 stars, underlies modern dynamical mass measurements, and supports the efficient identification and characterization of massive and substellar components through joint astrometric and spectroscopic/orbital fits (Brandt, 2018, Brandt, 2021, Brandt et al., 2018, Painter et al., 26 Jun 2025, Feng et al., 7 Feb 2024, Makarov et al., 20 May 2024, Whiting et al., 2023).