Sagnac Correction Calculator

Rotating-frame timing correction $\Delta t_{\rm Sag} = \vec\Omega \cdot (\vec r_1 \times \vec r_2)/c^2$ for Earth-based or rotating-body links.

📊 Sample calculation — Equatorial Sagnac: lab × 2π agrees with Ashby (2003) within stated tolerance — closed-loop 207 ns to 0.1 %

Inputs

Rotating frame

Emitter position $\vec r_1$ (m)

Default: Goldstone-like ground station (lat ~35°N, longitude 0°).

$r_{1x}$

$r_{1y}$

$r_{1z}$

Receiver position $\vec r_2$ (m)

Default: Madrid-like ground station (lat ~35°N, longitude 120°E).

$r_{2x}$

$r_{2y}$

$r_{2z}$

Tolerance (ns) Above tolerance → KEEP. Below 1% → IGNORE.

Result

Sagnac delay

—

Range equivalent

—

Sign

—

Tolerance

—

Formula registry entry — sagnac_chord_v1

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Vector breakdown

$|\vec r_1|$: —

$|\vec r_2|$: —

$\vec r_1 \times \vec r_2$: —

$|\vec r_1 \times \vec r_2|$: —

$\vec\Omega \cdot (\vec r_1 \times \vec r_2)$: — m²/s

Enclosed-area projection ($A_\perp = |\vec r_1\times\vec r_2|/2$, normal to $\vec\Omega$): —

Formula

Sagnac timing correction (rotating frame, first order in $\Omega R/c$) $$\Delta t_{\rm Sag} \;=\; \frac{\vec\Omega \cdot (\vec r_1 \times \vec r_2)}{c^2}$$ Range-equivalent $$\Delta \rho \;=\; c\,\Delta t_{\rm Sag}$$

Frame and sign convention

Sagnac signs depend on frame convention and vector order. The certificate records the convention used. This calculator uses the right-handed ECEF convention ($\vec\Omega$ points along $+z$ toward the celestial north pole) with $\vec r_1$, $\vec r_2$ taken from emitter and receiver positions in that frame. Swapping $\vec r_1 \leftrightarrow \vec r_2$ flips the sign; this corresponds to reversing the propagation direction.

Assumptions

Rigid uniformly rotating frame with angular velocity $\vec\Omega$ constant during the light-travel time.
First-order Sagnac: $|\Omega| \cdot |\vec r|/c \ll 1$ (well satisfied for Earth-radius geometries).
Simplified Earth-surface model: $\vec r_1, \vec r_2$ are ECEF coordinates; geoid undulations not modelled.
No relativistic frame transformation beyond first-order Sagnac (no time-dependent $\vec\Omega$, no centrifugal/Coriolis terms in the photon equation of motion beyond this order).
Single one-way link; for a closed loop the result is doubled (round-trip Sagnac).

What this would need for mission-grade use

Show requirements

Full Earth-orientation parameters (UT1, polar motion)
Station coordinates in the current ITRF realisation
Troposphere / ionosphere delay model
Clock-bias model
GNSS operational frame transformations (ITRF ↔ TRF ↔ inertial)

Warnings

Sign-flipped if the convention is reversed (e.g. receiver-to-emitter vector, or left-handed frame).
For Earth–satellite links the satellite position must be in the same rotating frame at the reception epoch; using the inertial-frame satellite vector gives a different value.
This is a one-way geometric Sagnac; tropospheric, ionospheric, plate-tectonic, and clock-bias terms are not included.

Source notes

Formula: standard rotating-frame Sagnac correction; see Ashby, Relativity in the Global Positioning System, Living Reviews in Relativity (2003), §2 and §4.
Earth rotation rate $\Omega_\oplus$ from WGS-84 / IERS Conventions 2010; constants table on the Validation page.

Certificate

Each computed result can be exported as a JSON or Markdown certificate. Certificate v2 splits the deterministic result_hash (same inputs + version + constants + formulas → same hash) from a timestamped receipt_hash (unique per export).