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Speed-Error Cross-Correlation Dating of Ancient Star Catalogues, with Application to the Almagest

Carlos Baiget Orts

Abstract

We present SESCC (Speed-Error Signals Cross-Correlation), a method for dating ancient star catalogues from the cross-correlation between stellar proper-motion speeds and positional residuals. At the true epoch, residuals are independent of proper-motion speed; the epoch estimate is the trial date that minimises this cross-correlation. For ecliptic latitudes, SESCC applies the dot product between speeds and residuals across all catalogue stars without subset selection or linear modelling. For ecliptic longitudes, SESCC-pairs uses pairwise longitude differences between neighbouring stars, making the method immune to any global longitude offset by algebraic construction. Validated against Tycho Brahe (1547 CE, true ~1580 CE) and Ulugh Beg (1452 CE, true 1437 CE), and confirmed invariant under offsets of +-6 deg, the method is applied to the Almagest. Both coordinates yield bootstrap distributions with 74% pre-Christian minima, consistent with a Hipparchan origin and inconsistent with a Ptolemaic one. The near-absence of quarter-degree fractions in the Almagest longitudes, explained as the deterministic consequence of Ptolemy's precession correction, provides independent corroboration.

Speed-Error Cross-Correlation Dating of Ancient Star Catalogues, with Application to the Almagest

Abstract

We present SESCC (Speed-Error Signals Cross-Correlation), a method for dating ancient star catalogues from the cross-correlation between stellar proper-motion speeds and positional residuals. At the true epoch, residuals are independent of proper-motion speed; the epoch estimate is the trial date that minimises this cross-correlation. For ecliptic latitudes, SESCC applies the dot product between speeds and residuals across all catalogue stars without subset selection or linear modelling. For ecliptic longitudes, SESCC-pairs uses pairwise longitude differences between neighbouring stars, making the method immune to any global longitude offset by algebraic construction. Validated against Tycho Brahe (1547 CE, true ~1580 CE) and Ulugh Beg (1452 CE, true 1437 CE), and confirmed invariant under offsets of +-6 deg, the method is applied to the Almagest. Both coordinates yield bootstrap distributions with 74% pre-Christian minima, consistent with a Hipparchan origin and inconsistent with a Ptolemaic one. The near-absence of quarter-degree fractions in the Almagest longitudes, explained as the deterministic consequence of Ptolemy's precession correction, provides independent corroboration.

Paper Structure

This paper contains 23 sections, 2 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. The SESCC Method for Ecliptic Latitudes
  3. Formulation
  4. Practical implementation
  5. Relation to previous approaches
  6. Bootstrap resampling
  7. SESCC-pairs: Extension to Ecliptic Longitudes
  8. The problem of longitude dating
  9. Pairwise longitude differences
  10. Proximity criteria and pair selection
  11. Longitude positions in the J2000 inertial frame
  12. Stars with large parallax
  13. Validation
  14. Validation strategy
  15. Tycho Brahe
  16. ...and 8 more sections

Figures (4)

  • Figure 1: SESCC latitude dating curves $C(T)$, normalised to [0,1000], for Tycho Brahe (left), Ulugh Beg (centre), and the Almagest (right). Vertical dashed lines indicate the known observational epoch (Brahe and Ulugh Beg) or the Hipparchan ($-127$ BCE) and Ptolemaic ($+137$ CE) reference epochs (Almagest).
  • Figure 2: SESCC-pairs longitude dating curves $C_{\rm p}(T)$, normalised to [0,1000], for Tycho Brahe (left), Ulugh Beg (centre), and the Almagest (right). Vertical dashed lines as in Figure \ref{['fig:curves_lat']}.
  • Figure 3: Bootstrap distributions of the epoch estimate for the Almagest, from SESCC latitudes (left) and SESCC-pairs longitudes (right). Vertical dashed lines mark the Hipparchan reference epoch ($-127$ BCE) and the CE boundary.
  • Figure 4: Distribution of sexagesimal fractional minutes in the Almagest for ecliptic latitudes (top) and ecliptic longitudes (bottom). The bars at $M = 15'$ and $M = 45'$ (quarter-degree fractions, shaded) are prominent in latitudes but nearly absent in longitudes.