A method for dating ancient star catalogues by cross-correlating stellar proper-motion speeds with positional residuals.
Author: Carlos Baiget Orts (asinfreedom@gmail.com)
A star catalogue records positions at the epoch of observation. When compared to modern positions computed from the Hipparcos catalogue with proper motion propagated to a trial epoch T, the residual |pos_cat − pos_mod(T)| is minimised when T equals the true observational epoch.
At the true epoch, residuals reflect random measurement errors independent of proper-motion speed. At any other epoch, fast-moving stars accumulate systematically larger residuals. The SESCC statistic exploits this:
For ecliptic latitudes (--lat):
C(T) = Σᵢ |µβᵢ| · |βᵢᶜᵃᵗ − βᵢᵐᵒᵈ(T)|
Precession does not affect ecliptic latitudes, so no correction is needed.
For ecliptic longitudes (--lon, SESCC-pairs):
Cₚ(T) = Σ₍ᵢ,ⱼ₎ |µλᵢ·cosβᵢ − µλⱼ·cosβⱼ| · |(λᵢᶜᵃᵗ − λⱼᶜᵃᵗ) − (λᵢᵐᵒᵈ(T) − λⱼᵐᵒᵈ(T))|
Pairwise differences cancel any global longitude offset — including precession corrections — exactly. SESCC-pairs is immune to systematic longitude shifts by algebraic construction.
The epoch estimate is the T that minimises C(T) or Cₚ(T).
Ready-to-run scripts are provided for the three historical catalogues. Each script shows the C(T) curve and reports the epoch estimate.
Prerequisites: catalogues must be prepared first (see Catalogue preparation).
bash date_almagest_lat.sh # by ecliptic latitudes → expected ~-49 BCE
bash date_almagest_lon.sh # by ecliptic longitudes → expected ~-165 BCE
bash date_ulugh_lat.sh # by ecliptic latitudes → expected ~1177 CE
bash date_ulugh_lon.sh # by ecliptic longitudes → expected ~1452 CE
bash date_brahe_lat.sh # by ecliptic latitudes → expected ~1570 CE
bash date_brahe_lon.sh # by ecliptic longitudes → expected ~1547 CE
Download the source data files and convert them to SESCC format before running the dating scripts.
Almagest and Ulugh Beg — from Verbunt & van Gent (2012), A&A 544, A31:
# Download from CDS: https://cdsarc.cds.unistra.fr/viz-bin/cat/J/A+A/544/A31
python3 verbunt2sescc.py ptolema.dat > catalogs/almagest_verbunt.csv
python3 verbunt2sescc.py ulughbeg.dat > catalogs/ulughbeg_verbunt.csvTycho Brahe — from Verbunt & van Gent (2010), A&A 516, A28:
# Download KeplerE.dat from CDS: https://cdsarc.cds.unistra.fr/viz-bin/cat/J/A+A/516/A28
python3 verbunt2sescc_brahe.py keplere.dat > catalogs/brahe_verbunt.csvAll dating is done through a single script with a --lat or --lon flag.
cat catalog.csv | python3 sescc.py --lat [OPTIONS]
cat catalog.csv | python3 sescc.py --lon [OPTIONS]
Common options:
--bootstrap N run N bootstrap resamples and report distribution (default 0)
--noplot suppress the plot
Latitude options (--lat):
--subset N use a random subset of N stars
--maxmag M exclude stars fainter than magnitude M
Longitude options (--lon):
--dlat D max latitude difference for pairing (default 5.0 deg)
--dlon D max longitude difference for pairing (default 30.0 deg)
--minvel V min relative proper motion for a pair (default 0)
--top N show top N contributing pairs at minimum (default 10)
Output files:
sescc_lat_abs.csv C(T) curve (latitudes)
sescc_lon_pairs.csv Cₚ(T) curve (longitudes)
sescc_lat_abs_bootstrap.csv bootstrap distribution (if --bootstrap N)
sescc_lon_pairs_bootstrap.csv bootstrap distribution (if --bootstrap N)
Important for longitude dating: exclude stars with very large parallax (Keid HIP 19849, α Cen HIP 71681) and filter by positional error:
# Almagest (dist column = col 10)
awk -F';' 'substr($1,1,1)=="#" || $10+0 < 60' catalogs/almagest_verbunt.csv | \
grep -v "^19849;" | grep -v "^71681;" | \
python3 sescc.py --lon --dlon 30
# Tycho Brahe (dist column = col 8)
awk -F';' 'substr($1,1,1)=="#" || $8+0 < 60' catalogs/brahe_verbunt.csv | \
grep -v "^19849;" | grep -v "^71681;" | \
python3 sescc.py --lon --dlon 30Converts Verbunt & van Gent (2012) fixed-width files to SESCC CSV format.
python3 verbunt2sescc.py ptolema.dat > catalogs/almagest_verbunt.csv
python3 verbunt2sescc.py ulughbeg.dat > catalogs/ulughbeg_verbunt.csv
Column 10 (dist_arcmin) gives the positional error for filtering.
Converts KeplerE.dat (Verbunt & van Gent 2010, A&A 516, A28) to SESCC CSV format. Coordinates are ecliptic, equinox AD 1601.0.
python3 verbunt2sescc_brahe.py keplere.dat > catalogs/brahe_verbunt.csv
Column 8 (dist_arcmin) gives the positional error for filtering.
Generates synthetic catalogues at a known epoch with realistic Gaussian noise,
runs SESCC (--lon) on each, and reports bias, standard deviation, and the
fraction of resamples giving a pre-Christian minimum.
python3 validate_pairs_seeds.py --epoch -127 --seeds 20
python3 validate_pairs_seeds.py --epoch 137 --seeds 20
Options:
--epoch YEAR true epoch of synthetic catalogue (default -127)
--seeds N number of random seeds (default 20)
--noise-lat L latitude noise in arcminutes (default 23)
--noise-lon L longitude noise in arcminutes (default 27)
--dlat D max latitude difference for pairing (default 5.0)
--dlon D max longitude difference for pairing (default 30.0)
--exclude HIPS comma-separated HIPs to exclude (default 19849,71681)
--input CSV star list (default catalogs/almagest_verbunt.csv)
--noplot suppress histogram plot
All computed positions are cached in sescc_positions.pkl.xz (LZMA-compressed
pickle, included in the repository). If the file is not found, positions are
computed on the fly — this may take several hours for a full 2500-year scan.
Key format:
"{hip}_{epoch_index}_lat" ecliptic latitude, milli-degrees (frame of date)
"{hip}_{epoch_index}_lon" ecliptic longitude, milli-degrees (J2000 inertial)
where epoch_index maps to year 1900 − epoch_index (step = 1 year, range
1900 to −600 BCE). Longitude entries use the J2000 inertial frame, essential
for SESCC-pairs to cancel precession exactly. Writes are atomic.
| Catalogue | True epoch | Minimum | Bootstrap 68% | % pre-CE |
|---|---|---|---|---|
| Tycho Brahe | ~1580 CE | 1570 CE | — | 0% |
| Ulugh Beg | 1437 CE | 1177 CE | [1126, 1182] | 0% |
| Almagest | ? | −49 BCE | [−177, +110] | 74% |
| Catalogue | True epoch | Minimum | Bootstrap 68% | Std | % pre-CE |
|---|---|---|---|---|---|
| Tycho Brahe | ~1580 CE | 1547 CE | [1535, 1582] | 27 yr | 0% |
| Ulugh Beg | 1437 CE | 1452 CE | [1186, 1500] | 187 yr | 0% |
| Synth. −127 BCE (mean, 20 seeds) | −127 BCE | −125 BCE | — | 179 yr | 85% |
| Synth. +137 CE (mean, 20 seeds) | +137 CE | +148 CE | — | 170 yr | 20% |
| Almagest | ? | −165 BCE | [−384, +176] | 235 yr | 74% |
The 74% pre-Christian bootstrap fraction for the Almagest (both coordinates independently) is consistent with a Hipparchan origin and inconsistent with a Ptolemaic one. Invariance to global longitude offsets verified empirically: 13 offset versions of both the Almagest and Ulugh Beg (±6°) all give identical epoch estimates.
pip install skyfield numpy matplotlib scipy
Skyfield downloads the DE441 ephemeris and Hipparcos catalogue automatically on first run (~600 MB total).
- Verbunt, F. and van Gent, R. H. (2010). A&A 516, A28. (Tycho Brahe)
- Verbunt, F. and van Gent, R. H. (2012). A&A 544, A31. (Almagest, Ulugh Beg)
- ESA (1997). The Hipparcos and Tycho Catalogues. ESA SP-1200.
- Park et al. (2021). AJ 161, 105. (DE441 ephemeris)
- Rhodes, B. (2019). Skyfield. ASCL 1907.024.