GDOP Triangulation Across the ATNF Catalogue

Astrolabe lets you pick any star in the galaxy and renders a Pioneer-plaque-style triangulation map from that star’s perspective. The plaque needs 14 pulsars. There are 3,924 in the catalogue. Which 14?

I tried five selection strategies: fastest (shortest period), closest, longest (longest period), stable (oldest characteristic age), and gdop — geometric dilution of precision.

GDOP is the right answer, and the math is small. For each candidate set, I compute the unit direction vector from observer to each pulsar, build the design matrix H (Nx3 of those vectors), then Q = (HᵀH)⁻¹. PDOP is sqrt(trace(Q)). Lower is better — a low PDOP means the pulsars span the sky enough that an error in any one direction doesn’t blow out the whole solution.

// Diagonal of (HᵀH)⁻¹ via cofactors — only need three values for PDOP
const det = a*(e*k - f*h) - b*(d*k - f*g) + c*(d*h - e*g);
const trace = (e*k-f*h)*invDet + (a*k-c*g)*invDet + (a*e-b*d)*invDet;
return Math.sqrt(trace);

Selection is greedy: start with the highest-scoring single pulsar, then add the candidate that minimizes the new set’s PDOP, repeat to 14. Score factors in characteristic age (τ = P / 2|Ṗ|), period uniqueness (penalize near-duplicates in log-space), and the geometric improvement.

fastest gives you a clump of millisecond pulsars in the galactic plane — terrible for triangulation. gdop gives you a sphere of well-spaced beacons. The plaque output looks like Pioneer’s because the geometry is good, not because the pulsars are familiar.

Source: astrolabe/src/lib/pulsar-selection.ts. Verified against ATNF v2.7.0.