Shapiro Delay Calculator

Weak-field one-body gravitational light-time delay near a massive body.

📊 Sample calculation — Earth–Mars superior conjunction at solar grazing: lab agrees with Will (2018) within stated tolerance — textbook 250 μs to 1 %

Inputs

Mode Multi-body = additive weak-field sum over selected bodies. Not a full multi-body ephemeris.

Central body

Distance units

$r_1$ — emitter to body

$r_2$ — receiver to body

$R$ — emitter to receiver (Euclidean) Same units as $r_1, r_2$.

PPN $\gamma$ GR = 1. Solar-system bounds: $|\gamma - 1| \lesssim 2\times10^{-5}$.

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

Result

Delay

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Range equivalent

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$d/dγ$ sensitivity

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Tolerance

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Formula drawer — show derivation, assumptions, sources

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Formula

Weak-field one-body Shapiro delay $$\Delta t_{\rm Shapiro} \;=\; (1+\gamma)\,\frac{GM}{c^3}\,\ln\!\left(\frac{r_1 + r_2 + R}{r_1 + r_2 - R}\right)$$ Range-equivalent $$\Delta \rho \;=\; c\,\Delta t_{\rm Shapiro}$$

Assumptions

One-body weak-field approximation: $GM / (c^2 r_{\min}) \ll 1$ along the photon path.
Static central body, spherically symmetric (no spin or higher multipoles).
Standard PPN form; $\gamma = 1$ is general relativity.
$r_1, r_2$ are coordinate distances from the body; $R$ is the Euclidean emitter–receiver distance.
No atmospheric, plasma, or instrument-frame contributions; this is the GR term only.

What this would need for mission-grade use

Show requirements

SPICE ephemerides for both bodies + Earth
Full relativistic time-scale handling (TDB ↔ TT)
Station coordinates + Earth orientation parameters
Plasma / atmospheric path delay model
Pointing-control model, instrument & detector models
Covariance / uncertainty propagation
Mission-specific calibration

Warnings

Near solar conjunction the photon path may pass close to the body; the weak-field expansion can break down for $r_{\min} \lesssim R_\odot$ and is replaced by higher-order PPN or a full ray-trace.
The denominator $r_1 + r_2 - R$ approaches zero when the receiver lies behind the emitter relative to the body. The calculator returns a warning in that regime.
This calculator is a public engineering surrogate. It does not reproduce mission-specific internal pipelines (full multi-body relativistic ephemeris, frame transformations, scheme).

Source notes

Formula: standard weak-field PPN one-body Shapiro delay; see Will, Theory and Experiment in Gravitational Physics, §6 (2018 ed.), and any standard GR text.
Constants from CODATA / IAU; full table on the Validation page.

Certificate

Each computed result can be exported as a JSON or Markdown certificate. The result_hash is deterministic (same inputs + version + constants + formulas → same hash); the receipt_hash additionally includes the export timestamp.