Source: https://guthlabs.ai/docs/instruments/apparent-stellar-motion-proper-motion-projector/mechanism/
instrument mechanism / verified-clean
# Apparent Stellar Motion / Proper Motion Projector mechanism
The inspectable method record for Apparent Stellar Motion / Proper Motion Projector, kept separate from implementation and qualification.
Status: **VERIFIED_CLEAN**Lifecycle: **verified-clean**
## 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
[01**Overview**]()[03**Implementation**]()[04**Qualification**]()
## 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`