Source: https://guthlabs.ai/docs/instruments/orbital-period-kepler-s-3rd-law/mechanism/

instrument mechanism / verified-clean

# Orbital Period (Kepler's 3rd Law) mechanism

The inspectable method record for Orbital Period (Kepler's 3rd Law), kept separate from implementation and qualification.

Status: **VERIFIED_CLEAN**Lifecycle: **verified-clean**

## Model scope

The displayed formula is standard and correct under the stated two-body approximation. The catalog claim is incomplete because reverse computation is advertised but its inverse formula and input constraints are absent. The specification therefore needs review before release.

- The forward formula T² = 4π²a³/(GM) is correct for an ideal two-body Keplerian orbit when a is the semi-major axis and GM is the standard gravitational parameter of the central body.

- The specification claims computation in reverse (semi-major axis from period), but does not provide the inverse formula a = (GMT²/(4π²))^(1/3).

- The input is described as central body mass, while the formula requires GM. The implementation must either multiply mass by G or accept GM directly; this distinction is not specified.

- The formula assumes a bound Keplerian orbit with a > 0. It is not valid as an unrestricted calculator for a = 0 or non-orbital trajectories.

## Units and conventions

- Use one consistent unit system: a in metres, GM in m³/s², and T in seconds; or a in kilometres, GM in km³/s², and T in seconds.

- If central body mass M is supplied instead of GM, use G = 6.67430 × 10^-11 m³ kg^-1 s^-2 and convert a to metres.

- Reported days are T/86400, using exactly 86400 s per day.

- Reported years require a declared convention; the test vectors use the Julian year of 365.25 days = 31557600 s.

- For preset bodies, values should be identified as GM values, not merely masses, and should use a documented JPL/IAU epoch or reference source.

## Inspectable vector

One of 4 source-bound review vectors:

````
{
  "inputs": {
    "semi_major_axis": 7000,
    "semi_major_axis_unit": "km",
    "central_body": "Earth",
    "GM": 398600.4418,
    "GM_unit": "km^3/s^2"
  },
  "expected_outputs": {
    "period_s": 5828.52,
    "period_days": 0.06746
  },
  "working": "T = 2π√(a³/GM)\na³ = 7000³ = 343,000,000,000 km³ = 3.43 × 10^11 km³\na³/GM = 343,000,000,000 / 398,600.4418 = 860,510.85 s²\n√(860,510.85) = 927.637 s\nT = 2π × 927.637 = 5828.52 s\nT_days = 5828.52 / 86400 = 0.06746 days"
}
````

## Failure boundary

- No explicit constraint that semi-major axis must be finite and strictly positive; reject a <= 0 for an orbital-period instrument rather than presenting zero as a valid orbit.

- Reject GM <= 0 and central-body mass <= 0; otherwise the square root is undefined or nonphysical.

- Guard against division by zero if a custom central-body mass or GM is zero.

- Guard against unit mismatch, especially supplying a in kilometres with GM in m³/s² or vice versa; this causes a factor of 1000^(3/2) error in period.

- A mass-only input requires G and consistent SI conversion; a GM preset is preferable because it avoids introducing an additional rounded constant.

- Elliptic-orbit terminology should be explicit: for a two-body bound orbit, a > 0 and the period is independent of eccentricity, but perturbations, oblateness, third bodies, and relativistic effects are ignored.

- The reverse calculation requires T > 0 and GM > 0; it also needs an explicit period-unit conversion before applying the inverse formula.

## Other records

[01**Overview**](<https://guthlabs.ai/docs/instruments/orbital-period-kepler-s-3rd-law/>)[03**Implementation**](<https://guthlabs.ai/docs/instruments/orbital-period-kepler-s-3rd-law/implementation/>)[04**Qualification**](<https://guthlabs.ai/docs/instruments/orbital-period-kepler-s-3rd-law/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-tranche4/tranche4-results.json`