Shapiro Time Delay Calculator
Why light arrives late near massive bodies
In the mid-1960s, astrophysicist Irwin Shapiro proposed a radar test of general relativity: send radio waves past the Sun and measure the round-trip time when the receiver sits on the far side of our star. If Einstein was right, the signal should return a few hundred microseconds later than classical physics predicts. The Sun’s gravity curves spacetime, so photons travel along a subtly longer path and experience gravitational time dilation. This Shapiro delay joined the perihelion advance of Mercury, light bending, and gravitational redshift as a cornerstone confirmation of Einstein’s theory.
The calculator models the weak-field limit appropriate for planetary radar. Define the source-to-mass distance as , the mass-to-observer distance as , and the impact parameter as . The additional travel time beyond flat space is
, where is the gravitational constant, is the lensing mass, and is the speed of light. The logarithm shows that even distant geometries contribute when the impact parameter is small. AgentCalc converts your astronomical unit and solar-radius inputs to meters before applying the formula so the output arrives in seconds and microseconds.
Interpreting the calculator output
Enter masses in solar units, distances in astronomical units, and the closest-approach distance in solar radii. The result card reports the extra delay, the flat-space travel time, and how large the delay is as a percentage of the unperturbed journey. Positive values mean light arrived later than a straight-line path would predict; a negative number signals that your geometry falls outside the weak-field assumption. Because the equation scales only logarithmically with distance, shaving the impact parameter by a solar radius can change the delay more than doubling the range of the transmitter or receiver.
Sample Shapiro delays
The table below illustrates three classic scenarios that highlight how the effect depends on mass and geometry. Case A mirrors a Venus radar experiment, Case B considers a signal passing Jupiter, and Case C represents a compact binary pulsar.
| Scenario | Mass (M☉) | r₁ (AU) | r₂ (AU) | b (R☉) | Δt (µs) |
|---|---|---|---|---|---|
| Venus superior conjunction | 1.0 | 1.0 | 0.7 | 5.0 | |
| Jupiter radar pass | 1.0 | 1.0 | 5.2 | 6.5 | |
| Binary pulsar eclipse | 1.4 | 0.01 | 0.01 | 1.0 |
Microsecond-scale delays might seem trivial, yet they drive precise navigation. Spacecraft navigation teams bake the expected logarithmic delay into trajectory solutions, while pulsar astronomers extract companion masses by fitting timing models that include Shapiro terms. Emerging missions that target picosecond timing will probe even subtler departures, testing for tiny deviations from general relativity.
Keep exploring relativity
After experimenting with this tool, continue your journey with the Schwarzschild Radius Calculator, Gravitational Time Dilation Calculator, and the Gravitational Lens Time Delay Calculator to compare how curvature shapes time, distance, and brightness across the cosmos.