AC01 · Stellar astrometry
Measuring the depth of the sky
Earth’s orbit turns a tiny annual shift into a distance. Begin with that geometric idea, then learn how noisy positions become a fitted motion and how real Gaia parallaxes become distance distributions.
How far apart can neighbouring stars really be?
Learn how Earth’s orbit gives the sky depth, why nearby stars wobble more, and how proper motion differs from parallax.
02 · L2Derive a parallax ellipse from Earth’s orbit
Define the tangent plane, derive the annual parallax factors, and combine the ellipse with proper motion through worked examples.
03 · L3Separate parallax and proper motion in noisy positions
Build a five-parameter linear fit, understand correlated position errors, and test uncertainty with repeated synthetic observations.
04 · L3Compare two distance estimates for twenty-four Gaia stars
Reproduce a frozen Gaia DR3 sample, retain negative parallaxes, reconstruct covariances and compare inverse parallax with prior-based distances.
05 · L4Read Gaia’s parallax zero point and scanning errors in the original papers
Follow fixed Gaia papers through source models, parameter correlations, scanning patterns and the range of parallax zero-point corrections.
Sources and calculation scope
- Lindegren et al., Gaia EDR3 — The astrometric solution, arXiv:2012.03380v1
Sections 3.1, 5.3–5.7; equation 23 and figures 10–12. The fixed single-source model and covariance conventions apply to the basic DR3 astrometry inherited from EDR3.
- Bailer-Jones, Estimating distances from parallaxes, arXiv:1507.02105v1
Sections 2–3 and 7; equations 1, 4 and 17–18. Used for the Gaussian parallax likelihood and an explicitly pedagogical exponentially decreasing space-density prior.
- Bailer-Jones et al., Estimating distances from parallaxes V, arXiv:2012.05220v1
Sections II.1–II.3, V.2 and V.5. The published direction-dependent Galactic prior differs from this tutorial’s simple EDSD prior.
- Lindegren et al., Parallax bias versus magnitude, colour, and position, arXiv:2012.01742v2
Sections 2.3, 4.1 and 8; figures 2, 5, 21–22 and tables 9–10. The tutorial applies no zero-point correction and does not claim calibrated catalogue distances.
- ARI Gaia Services — TAP documentation
Synchronous CSV queries against gaiadr3.gaia_source. Query, source IDs, raw CSV and checksum frozen on 28 October 2026.
- Gaia DR3 documentation release 1.3 — 14.2.2 Astrometry
Validation discussion of five- and six-parameter solutions and negative parallaxes. A sign cut is not used in the frozen tutorial selection.
- JPL Horizons system manual
Reference-frame and observer-vector conventions for a future ephemeris-based extension. The browser uses a declared circular orbit, not a Horizons ephemeris.