Shapiro Delay and Radio Propagation

Shapiro delay is an excess light time produced by a gravitational potential along a radio path. A conjunction experiment does not read that delay from one clock: it models transmit and receive epochs, spacecraft and planetary orbits, station and transponder delays, propagation media, oscillator behavior, and the relativistic light-time equation, often using Doppler as the most sensitive observable.

Required background. Weak-Field Gravity supplies the PPN potentials; Geodesics and Jacobi Fields supplies null propagation and endpoint geometry. Helpful background. Probability and Statistical Inference supplies correlated likelihoods and nuisance parameters; Tests of Gravity supplies the measurement-to-conclusion structure.

Logarithmic gravitational light time

For emitter and receiver distances $r_{\rm e}$ and $r_{\rm r}$ from a spherical mass $M$, with Euclidean endpoint separation $R$, the leading one-way PPN delay is

$$\Delta t_{\rm grav} = (1+\gamma)GM \ln\left( \frac{r_{\rm e}+r_{\rm r}+R} {r_{\rm e}+r_{\rm r}-R} \right).$$

The logarithm is dimensionless and $GM$ has time dimensions when $c=1$. The delay grows near conjunction, where the denominator encodes a small impact parameter. General relativity has $\gamma=1$. A two-way link contains an uplink and downlink evaluated at different epochs; it is not simply twice one static expression.

Radio tracking commonly measures fractional frequency shift,

$$y(t)=\frac{\nu_{\rm rec}-\nu_{\rm ref}}{\nu_{\rm ref}} \simeq-\frac{d}{dt}\Delta t_{\rm total},$$

so the conjunction signature is the time derivative of the logarithm. This emphasizes changes and suppresses some constant delays, but not drifting clocks, orbit error, or plasma. The PPN derivation and measurement geometry are given in Will 2014, §4.1.2 and Eq. (62).

Delay and Doppler near conjunction

In the weak-field metric, the null condition gives $dt\simeq[1+(1+\gamma)GM/(c^2r)]\,dl/c$ along the leading straight path. Integrating its extra term, $\int dl/r$, produces the logarithm. When both endpoints lie far from closest approach on opposite sides of the mass, $r_{\rm e},r_{\rm r}\gg b$,

$$\Delta t_{\rm grav}\simeq\frac{(1+\gamma)GM}{c^3} \ln\!\left(\frac{4r_{\rm e}r_{\rm r}}{b^2}\right), \qquad y_{\rm grav}\simeq\frac{2(1+\gamma)GM}{c^3}\frac{\dot b}{b}.$$

The Doppler expression neglects the slower endpoint-distance changes. Before closest approach $\dot b\lt0$, so the growing delay produces a negative fractional-frequency perturbation in the convention above; afterward the sign reverses. A two-way prediction must sum the two leg derivatives at their actual epochs. With $r_{\rm e}=r_{\rm r}=1$ au and a grazing solar $b$, the static one-way GR formula gives roughly $119\ \mu{\rm s}$, illustrating the light-time scale without modeling a particular mission.

Separate gravity from the radio path

Solar-corona delay is dispersive. To leading order its group delay scales with the electron column divided by radio frequency squared, whereas the metric delay is achromatic. Simultaneous links at two or more frequencies can estimate plasma, provided their ray paths, calibrations, and time sampling are sufficiently common. Tropospheric and ionospheric delays, antenna phase centers, station coordinates, oscillator stability, transponder group delay, and solar multipoles also enter.

The 2002 Cassini conjunction used a multi-frequency radio system and a global Doppler fit, not a single peak-delay measurement; its observable, plasma calibration, selection, and fit are described in Bertotti, Iess, and Tortora 2003, Eqs. (1)–(3) and Figs. 1–2. The durable lesson is experimental design. A quoted number from one mission cannot be transferred to another link without its orbit, frequencies, clock system, media model, and data editing.

The evidence basis is current through 2026-08-19. A result must identify the tracking dataset and arc version, radiometric calibration, transmit and receive time scales, radio bands, spacecraft and planetary ephemerides, station network, transponder model, selection mask, force model, likelihood, covariance, software, and retrieval date. Deep-space navigation observables and their corrections are defined in Moyer 2000, Chapters 8–13, PDF.

Selection includes solar elongation, link lock, maneuvers, station quality, frequency availability, solar-weather intervals, data gaps, and residual thresholds fixed before the gravity fit. Completeness means completeness within those usable arcs. Removing the closest conjunction data may reduce plasma bias but also removes much of the relativistic leverage; sensitivity must be recomputed rather than inferred from the retained point count.

Joint orbit and propagation likelihood

For Doppler or range observations $\mathbf d$,

$$\mathbf m = \mathbf m_{\rm orbit}(\boldsymbol\xi) +\mathbf m_{\rm grav}(\gamma,\boldsymbol\xi) +\mathbf m_{\rm plasma}(\boldsymbol\eta) +\mathbf m_{\rm link}(\boldsymbol\kappa).$$

The orbital state $\boldsymbol\xi$ enters both geometry and gravitational delay, so it must be fit jointly. Plasma parameters $\boldsymbol\eta$ may vary by epoch and impact parameter; link variables $\boldsymbol\kappa$ include clocks, stations, and transponder delays. A Gaussian likelihood uses the full time- and station-correlated covariance, with auxiliary priors from calibration data. Residuals should be checked against frequency, impact parameter, station, solar activity, and pre- versus post-conjunction time.

Systematic uncertainty is assessed by alternate corona profiles, tracking-arc boundaries, ephemerides, force and maneuver models, clock bases, and data-editing rules. A fit that fixes the spacecraft orbit under GR and then attributes the remaining conjunction residual to $\gamma$ can understate covariance. The light-time law and dynamics should be changed consistently and refit, following the ephemeris construction and residual-reinterpretation warning in Fienga and Minazzoli 2024, §§3.4.1–3.5, §4.1.1, and §5.

Reproduction should generate synthetic multifrequency range and Doppler through a conjunction, inject a known $\gamma$ shift and plasma history, and recover both while refitting the orbit. Alternate plasma bases and withheld arcs test robustness. The retained software, data product, calibration, selections, covariance, and machine-readable residuals define what was reproduced.

Exercises

A one-way conjunction delay is approximated by $\Delta t=A\ln(b_0/b)$, while the impact parameter varies as $b(t)=\sqrt{b_{\min}^2+v^2t^2}$. Derive the gravitational contribution to $y=-d\Delta t/dt$ and identify its symmetry about conjunction.

Solution

Differentiate the logarithm: $d\Delta t/dt=-A\,\dot b/b$. Since $\dot b=v^2t/b$, we obtain $y(t)=A v^2t/(b_{\min}^2+v^2t^2)$. The function is odd in $t$, so it changes sign across conjunction and vanishes at closest approach even though the delay itself is largest there. This antisymmetry helps distinguish the relativistic Doppler signature from constant delays. It does not remove an asymmetric plasma history or orbit error, which must be constrained by multiple frequencies and a joint orbital fit.

References

• Bertotti, B., Iess, L., and Tortora, P., “A Test of General Relativity Using Radio Links with the Cassini Spacecraft,” Nature 425 (2003) 374–376, DOI.
• Fienga, A., and Minazzoli, O., “Testing Theories of Gravity with Planetary Ephemerides,” Living Reviews in Relativity 27 (2024) article 1, DOI.
• Moyer, T. D., Formulation for Observed and Computed Values of Deep Space Network Data Types for Navigation, JPL Publication 00-7 (2000), official NASA/JPL PDF.