Apparent Stellar Motion / Proper Motion Projector mechanism

Model scope

The core multiplication is valid only after defining μ as a scalar angular rate in arcsec/yr. Because the spec claims RA/Dec components and new coordinates without defining the RA convention or requiring initial coordinates, it needs review before implementation or catalog publication.

• Δθ = μ·t is correct for a scalar proper-motion magnitude μ in arcsec/yr and a time span t in years, but the specification supplies RA/Dec components and does not define how they are combined. Under the usual tangent-plane convention, Δθ = t·sqrt((μ_α* )² + μ_δ²), where μ_α* = μ_α cos(δ).
• The new-coordinate calculation is underspecified: the instrument must state whether the supplied RA component is μ_α* in arcsec/yr or the raw coordinate rate dα/dt. If it is μ_α*, the RA coordinate increment is Δα = μ_α*t/cos(δ), while Δδ = μ_δ*t.
• Initial RA and Dec are required to compute new coordinates but are absent from the declared inputs.
• The linear tangent-plane approximation is not sufficient near the celestial poles without an explicit spherical-coordinate treatment.

Units and conventions

• Proper-motion components are in arcsec/yr.
• Time span is converted to Julian years before multiplication; for example, 18 months is treated as 1.5 years.
• The RA component in the test vectors is μ_α* = μ_α cos(δ), expressed as an angular displacement rate on the tangent plane.
• Angular displacements are converted from arcseconds to degrees by dividing by 3600 before adding them to degree-valued coordinates.
• RA is normalized to the range [0°, 360°).
• The reported angular displacement is the tangent-plane magnitude sqrt(Δα*² + Δδ²), not the exact great-circle separation.

Inspectable vector

One of 4 source-bound review vectors:

{
  "inputs": {
    "initial_ra_deg": 120,
    "initial_dec_deg": 30,
    "proper_motion_ra_cos_dec_arcsec_per_year": 0.15,
    "proper_motion_dec_arcsec_per_year": -0.2,
    "time_years": 10
  },
  "expected_outputs": {
    "angular_displacement_arcsec": 2.5,
    "new_ra_deg": 120.0004811252,
    "new_dec_deg": 29.9994444444
  },
  "working": "ΔRA* = 0.15 × 10 = 1.5 arcsec. ΔDec = -0.2 × 10 = -2.0 arcsec. Angular displacement = sqrt(1.5² + (-2.0)²) = sqrt(2.25 + 4.00) = sqrt(6.25) = 2.5 arcsec. cos(30°) = 0.8660254038, so the RA coordinate shift = 1.5 / 0.8660254038 = 1.7320508076 arcsec = 1.7320508076 / 3600 = 0.0004811252°. The Dec shift = -2.0 / 3600 = -0.0005555556°. New RA = 120.0000000000 + 0.0004811252 = 120.0004811252°. New Dec = 30.0000000000 - 0.0005555556 = 29.9994444444°."
}

Failure boundary

• Division by cos(δ) is required when converting μ_α* to an RA-coordinate rate and becomes singular or numerically unstable at δ = ±90°.
• The specification does not constrain or validate time units; using months, days, or seconds without conversion to years produces incorrect results.
• The specification does not require initial RA/Dec even though they are necessary for new coordinates.
• RA wraparound at 0°/360° must be handled.
• Declination must remain within [-90°, 90°]; simple addition can produce invalid coordinates near a pole.
• Long-baseline extrapolation omits radial-velocity perspective acceleration, as acknowledged by the honest label, and also omits exact spherical-coordinate propagation.
• A zero or near-zero time span is valid, but non-finite inputs and excessively large values need validation.

Other records

Sources and evidence

These pointers bind the page to evidence. They do not grant authority or collapse verification into deployment.

• VERIFIED_LEDGER.json
• evidence-tranche5/tranche5-results.json