Angular Size ↔ Physical Size mechanism

Model scope

The stated formula is standard for the small-angle approximation, and the honest label correctly identifies that limitation. Review is required because angular units, diameter-versus-radius semantics, validity bounds, and zero/invalid-input behavior are not fully specified.

• The small-angle relation θ = size/distance is correct only when θ is small and size is the full physical diameter corresponding to the full angular diameter.
• For larger angular sizes, the exact full-diameter relation is θ = 2·atan((size/2)/distance); the specification does not state the allowable small-angle range.
• The input units for angular size are unspecified even though the formula requires radians.

Units and conventions

• Angular size θ must be in radians before applying the formula; degrees must be converted using θ_rad = θ_deg·π/180, and arcseconds using θ_rad = θ_arcsec·π/(180·3600).
• Physical size and distance must use the same linear units.
• size is assumed to mean full linear diameter, and angular size is assumed to mean the corresponding full angular diameter.
• The geometry is Euclidean and the object is sufficiently distant for the small-angle approximation.

Inspectable vector

One of 4 source-bound review vectors:

{
  "inputs": {
    "physical_size_km": 3474,
    "distance_km": 384400
  },
  "expected_outputs": {
    "angular_size_rad": 0.00903746098
  },
  "working": "θ = size/distance = 3474 km / 384400 km = 0.00903746098 rad."
}

Failure boundary

BREAK_CLAIMED · SELF_FLAGGED

A zero-distance input produces an undefined angular size and must be rejected; additionally, an angular-size unit ambiguity can change the computed physical size from approximately 1.74533 m to 100 m.

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-tranche4/tranche4-results.json
• BOUNDARY_HAZARD_CATALOG.json