Spacecraft Tracking and Radio Science

Spacecraft radio science infers gravity from a coupled navigation problem. Two-way range locates a spacecraft along the line of sight, Doppler measures accumulated phase and range rate, very-long-baseline interferometry (VLBI) supplies plane-of-sky direction, and an onboard accelerometer may constrain forces that are not gravitational. No one stream cleanly measures a relativistic parameter: the orbit, light path, hardware, propagation media, and force model must be solved together.

Required background. Weak-Field Gravity supplies post-Newtonian dynamics and propagation; Geodesics and Jacobi Fields supplies endpoint-sensitive light paths. Helpful background. Probability and Statistical Inference supplies correlated likelihoods and priors; Tests of Gravity supplies the inference hierarchy.

Tracking observables and their geometry

For a coherent two-way link, let $\tau_{2\rm w}$ be modeled elapsed light time, including the transponder event. A convenient sign convention gives fractional Doppler

$$y(t)=\frac{\nu_{\rm rec}-\nu_{\rm ref}}{\nu_{\rm ref}} \simeq-\frac{d\tau_{2\rm w}}{dt}, \qquad \dot\rho\simeq-\frac{c}{2}y.$$

Both sides of the second relation have velocity dimensions. The sign reverses if the navigation product defines received-minus-predicted phase oppositely, so a release must preserve its convention. Actual Doppler is a phase count over a finite integration interval, not an instantaneous derivative. Range is sensitive to nearly constant delays that Doppler largely differentiates away. A VLBI baseline $\mathbf b$ measures a differential delay with leading geometric term $\mathbf b\boldsymbol\cdot\hat{\mathbf s}/c$, which constrains the direction $\hat{\mathbf s}$ and breaks line-of-sight orbit degeneracies.

The spacecraft state obeys

$$\ddot{\mathbf x} =\mathbf a_{\rm N} +\mathbf a_{\rm PN}(\boldsymbol\theta) +\mathbf a_{\rm ng}(\mathbf x,t;\boldsymbol\kappa) +\sum_j\Delta\mathbf v_j\,\delta(t-t_j).$$

Relativity parameters $\boldsymbol\theta$ affect dynamics and light time. The nuisance vector $\boldsymbol\kappa$ describes solar-radiation pressure, thermal recoil, outgassing, drag where relevant, antenna recoil, and stochastic acceleration; $\Delta\mathbf v_j$ represents maneuvers. An accelerometer measures a calibrated projection of $\mathbf a_{\rm ng}$ plus bias, scale, misalignment, noise, and thermal drift. It is therefore an auxiliary force measurement, not a direct subtraction with zero uncertainty.

The count interval changes the measured signal. For a constant reference frequency and small perturbations, a count centered at $t$ with duration $T_c$ measures

$$\bar y(t)\simeq-\frac{\tau_{2\rm w}(t+T_c/2)-\tau_{2\rm w}(t-T_c/2)}{T_c}.$$

If the instantaneous perturbation is $y_0\cos\omega t$, averaging multiplies its amplitude by $\sin(\omega T_c/2)/(\omega T_c/2)$. Slow signals survive nearly unchanged; fast oscillations can be strongly attenuated or vanish at zeros of this response. Treating all counts as instantaneous samples would then give the wrong gravity sensitivity even with noiseless data. Frequency ramps and real transponder events require the full count model. Operational definitions are developed in Moyer 2000, Chapters 8–13, PDF.

Release, calibration, and selection

The evidence basis is current through 2026-08-19. Before quoting a result, pin the mission data release or archive identifier, retrieval date, tracking file hashes, station and frequency metadata, accelerometer and attitude products, maneuver file, spacecraft mass and optical-thermal model, clock and transponder calibration, planetary ephemeris, Earth-orientation series, and software version. A kernel filename alone is insufficient: coverage, frame chain, time system, and superseding status are part of the data identity. The inputs and fitted data types used for a modern ephemeris are documented in Park et al. 2021, §§2–5, Tables 1–3, official JPL PDF.

Radiometric calibration includes station coordinates, antenna phase center, cable and receiver delays, transmitted frequency ramps, clock transfer, transponder turnaround and group delay, troposphere, ionosphere, and solar plasma. Accelerometer calibration includes bias and scale matrices, axis alignment, temperature, gaps, saturation, and conversion from body to inertial frame. Shared station, clock, or plasma parameters induce covariance across observations and sometimes across missions.

Selection must state tracking arcs, stations, radio bands, elevation and solar-elongation masks, lock and signal-to-noise thresholds, maneuver exclusion windows, attitude modes, accelerometer-valid intervals, outlier rule, and treatment of gaps. Completeness means all usable records within that declared release and mask. Removing conjunction data suppresses plasma contamination but also removes relativistic propagation leverage; removing maneuvers may leave a nonrepresentative force environment.

Coupled likelihood and systematic controls

Stack range, Doppler, VLBI, and acceleration residuals into $\mathbf r$. A normalized Gaussian likelihood is

$$-2\ln\mathcal L =\mathbf r^{\mathsf T}C^{-1}\mathbf r +\ln\det C +N\ln(2\pi),$$

with blocks in their native units and the corresponding covariance $C$. Estimated parameters include arc states, gravity parameters, station and link biases, media coefficients, maneuvers, accelerometer calibration, and a bounded stochastic-force model. Priors must be tied to engineering or calibration evidence; an excessively flexible acceleration spline can absorb a relativistic signature, while an overly rigid model can create one. Posterior-predictive checks separate residuals by observable, station, band, solar elongation, attitude, and arc.

Systematic uncertainty is probed by alternative plasma and troposphere models, thermal and radiation-pressure models, antenna phase centers, transponder delays, maneuver solutions, ephemerides, station coordinates, stochastic-force bases, masks, and covariance kernels. Range–Doppler agreement and VLBI cross-track predictions are useful internal checks; accelerometer-off intervals are a stress test rather than missing-at-random data. A claimed gravity deviation must survive withheld arcs and force-model changes without migrating into nuisance parameters.

The dynamical and propagation modification must be integrated and all correlated parameters refitted. Treating post-fit residuals from a general-relativistic ephemeris as observations of another theory is inconsistent because the original fit has already absorbed signal into initial states and nuisance parameters; Fienga and Minazzoli 2024, §§2–3 and §5 explains this failure.

Reproducible orbit-determination benchmark

A useful numerical benchmark combines two known injected effects: a small change in $\gamma$ affecting dynamics and propagation, and a body-axis thermal acceleration. For an illustrative 180-day synthetic heliocentric arc, one could sample daily eight-hour passes with 60-second Doppler counts, range every 300 seconds, weekly VLBI, attitude, and accelerometry. Example injections are $\delta\gamma=3\times10^{-5}$ and an initial thermal acceleration of $2\times10^{-10}\ {\rm m\,s^{-2}}$ with a specified exponential decay. These are freely chosen benchmark signals, not empirical bounds or a mission simulation already performed here. Record the actual software versions, implementation, ephemeris bytes, station states, random seeds, and units.

Integrate states and variational equations with DOP853 at relative tolerances $10^{-9}$, $3\times10^{-11}$, and $10^{-12}$; solve each light time to $10^{-9}$, $10^{-11}$, and $10^{-13}$ seconds. Compare against an independently coded Gauss–Jackson solution. Fit both injections with arc states, plasma, clocks, maneuvers, radiation pressure, accelerometer bias, and stochastic acceleration. Withhold every fifth tracking day and one station. Across 200 fixed noise seeds, require declared interval coverage, less than $0.1$ posterior-standard-deviation change between the finest two levels, and agreement between implementations within propagated covariance. A failure of injection recovery, normalization, coverage, convergence, input counts, or withheld prediction must stop the rerun.

The actual rerun command should produce parameters, covariance, residuals, selections, convergence, coverage, implementation comparisons, and run-environment metadata. This specifies what an independently reproducible orbit fit needs; no executable mission pipeline or fitted gravity limit is supplied here.

Validity boundary

Spacecraft tracking constrains the declared force and propagation model over the mission’s geometry, frequencies, and time span. A null result does not exclude theories whose effects are screened, lie outside that support, or are degenerate with allowed forces. Conversely, an unexplained acceleration is not evidence for modified gravity until hardware, media, thermal history, maneuvers, and orbit-model discrepancy are independently controlled.

Exercises

A spacecraft has an unmodeled constant line-of-sight acceleration $a$ during an arc of duration $T$. To leading order, find its range and two-way fractional-Doppler signatures, taking both perturbations to vanish at the arc start. Explain why combining the observables does not by itself identify the force as gravitational.

Solution

Integrating the acceleration once gives $\delta\dot\rho=a t$, and integrating again gives $\delta\rho=a t^2/2$. Because $y\simeq-2\dot\rho/c$, the Doppler perturbation is $\delta y=-2at/c$. The dimensions are respectively length and dimensionless frequency ratio. The two records confirm a common kinematic history, but solar pressure, thermal recoil, outgassing, or gravity can produce the same line-of-sight acceleration over a short arc. Therefore force direction, attitude and thermal correlations, accelerometry, and cross-track information are needed to identify its origin.

References

• 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.
• Park, R. S., Folkner, W. M., Williams, J. G., and Boggs, D. H., “The JPL Planetary and Lunar Ephemerides DE440 and DE441,” Astronomical Journal 161 (2021) 105, DOI.