A massive photon perturbs the Shapiro gravitational time delay of spacecraft radio links through mass-dependent corrections of order α2\alpha^2, where α=mγc2/E\alpha=m_\gamma c^2/E is the ratio of the photon rest energy to the carrier photon energy. Using 1279 Doppler residuals from the 2002 Cassini superior conjunction, Malta and Zarro (Physical Review D 110, 095017 (2024)) set the bound mγ<4.9×10−7m_\gamma<4.9\times10^{-7} eV/c² at 95% confidence, but explicitly left the decisive follow-up undone: the full iterative spacecraft orbit-determination fit was not re-run with mγm_\gamma promoted to a free estimated parameter (an improvement of at most about two orders of magnitude, toward 10−910^{-9} eV/c², was estimated), and the ultimate sensitivity of the Shapiro observable — intrinsically limited because its dominant logarithmic term carries no leading-order mass dependence — was not quantified against carrier frequency, measurement noise, and link geometry. Complete the mγm_\gamma-aware orbit-determination re-analysis of the Cassini data, and determine within the massive-photon time-delay framework which configurations, if any, could make time-delay bounds competitive with the 10−1810^{-18} eV/c² best existing limits. No follow-up has performed this analysis.

Background

In Maxwell electrodynamics the photon is exactly massless, and a nonzero rest mass mγm_\gamma can be accommodated only in the Proca extension of the theory, which breaks electromagnetic gauge invariance. Detectable effects of a finite mγm_\gamma scale as O(mγ2d2)\mathcal{O}(m_\gamma^2 d^2), where dd is the size of the system, so meaningful tests require astronomical baselines or extreme precision. The tightest accepted limit, mγ<10−18m_\gamma<10^{-18} eV/c², comes from Ryutov's magnetohydrodynamic analysis of the solar wind, and is the value quoted by the Particle Data Group (Navas et al. 2024, 2025 update); older limits from the gravitational deflection of GHz radio waves by the Sun and from laboratory Coulomb-law tests are many orders of magnitude weaker (Goldhaber and Nieto 2010; Accioly, Helayël-Neto and Scatena 2010). Reviews of the field emphasize that genuinely independent observables are valuable because each has its own systematics.

The Shapiro effect is the excess round-trip travel time of an electromagnetic signal passing near the Sun, proposed in 1964 as the fourth classical test of general relativity: for a ray grazing the solar limb the extra delay reaches about 200 microseconds. In the parametrized post-Newtonian (PPN) framework — a standard parametrization of metric theories of gravity in the weak-field limit — the decisive coefficient is Eddington's parameter γ\gamma, which multiplies the spatial curvature term of the metric and equals 1 in general relativity. The 2002 Cassini superior conjunction experiment (when the spacecraft passed almost directly behind the Sun as seen from Earth) measured γ−1=(2.1±2.3)×10−5\gamma-1=(2.1\pm2.3)\times10^{-5} using multi-frequency X/Ka-band radio links (Bertotti, Iess and Tortora 2003). Its observable was the fractional frequency shift (FFS), the time derivative of the gravitationally induced delay of the two-way (uplink–downlink) signal, referenced to hydrogen-maser frequency standards; the measurement noise is characterized by the Allan deviation (the fractional frequency instability of the link over the averaging time), about 2×10−142\times10^{-14} for Cassini. Behind the measurement stands orbit determination: an iterative procedure in which the spacecraft trajectory and a set of model parameters — for Cassini, twelve in total, comprising the spacecraft state vector at a reference epoch, non-gravitational accelerations from the radioisotope thermoelectric generators, the thermo-optical properties of the high-gain antenna, and the PPN parameter γ\gamma — are fitted simultaneously to tracking data (ranges and range rates) until the residuals converge.

Malta and Zarro (2024) computed, in heliocentric isotropic coordinates at first post-Newtonian order, the two-way time delay of a photon of mass mγm_\gamma following the actual geodesic between ground station and spacecraft. Writing α=mγc2/E\alpha=m_\gamma c^2/E with E=hfTE=h f_T the carrier photon energy (about 0.035 meV at the 8.4 GHz X-band carrier), the mass-dependent corrections of order α2\alpha^2 enhance the delay: the group velocity of a massive photon is slightly below cc, and additional subleading terms appear. A crucial structural result is that the dominant, observationally precise logarithmic term of the Shapiro delay is not modified at leading order by α2\alpha^2; the mass sensitivity enters only through suppressed subleading pieces, controlled by a geometric function W(r0,r)≲O(10−2)W(r_0,r)\lesssim\mathcal{O}(10^{-2}) of the impact parameter r0r_0 and the terminal distances. Applying the resulting prediction for the FFS to 1279 Cassini Doppler residuals through a χ2(α2)\chi^2(\alpha^2) profile — which is monotonic toward α2=0\alpha^2=0, so only an upper limit can be set — they obtained mγ<4.9×10−7m_\gamma<4.9\times10^{-7} eV/c² at 95% confidence, comparable to older deflection-based limits (Accioly et al. 2010) but roughly eleven orders of magnitude away from the best 10−1810^{-18} eV/c² limit. The Particle Data Group 2025 tables list this as MALTA 24, among results not used for averages and fits.

What the analysis did not do is the substance of the open problem. The photon mass was not an estimated parameter of the orbit-determination fit: the authors used the Cassini data as already reduced under the standard fit that estimates γ\gamma, arguing that a massive photon remains a test particle whose own dynamics do not perturb the spacecraft orbit, but that mγm_\gamma could still enter the measurement model through the two-way light times underlying ranges and range rates. Their back-of-envelope estimate (Appendix D of the paper) is that with the roughly 10−810^{-8} relative uncertainty of the Cassini position determination at about 8 AU, a full re-fit with γ\gamma fixed to 1 and mγm_\gamma free could reach α2≲10−8\alpha^2\lesssim10^{-8}, i.e. mγ∼10−9m_\gamma\sim10^{-9} eV/c² — an improvement of at most a couple of orders of magnitude, and probably less. Looking forward, they identified three levers and their obstacles: the bound improves at lower carrier frequencies (the naive scaling is mγ∝fTσym_\gamma\propto f_T\sqrt{\sigma_y}), but approaching 10−1410^{-14} eV/c² would require time-delay measurements at sub-MHz frequencies in space, which are technologically remote; the BepiColombo Mercury orbiter radio-science experiment, with a projected ∣γ−1∣≲2×10−6|\gamma-1|\lesssim2\times10^{-6}, would improve the bound by roughly one order of magnitude; and future laser-based time-delay missions, operating at optical frequencies of order 100 THz, lose photon-mass sensitivity unless their Allan deviations or geometric sensitivity improve far beyond current projections. Onboard transponder re-transmission, frequency comparison against the hydrogen-maser standards, and the auxiliary non-gravitational-acceleration measurements were argued in the same appendix to have no meaningful photon-mass sensitivity; the more thorough treatment that was declared technically out of scope, and remains unimplemented, is the full orbit-determination re-fit itself, with mγm_\gamma as the extra free parameter in place of γ\gamma.

Problem Statement

Determine how tightly solar-system Shapiro time-delay measurements can bound the photon mass, in the two parts left open by Malta and Zarro (2024). (1) Re-analysis of existing data: perform the complete iterative spacecraft orbit-determination fit of the 2002 Cassini superior-conjunction Doppler-tracking data with the photon mass promoted to a free estimated parameter (fixing the PPN parameter γ=1\gamma=1), implementing the mγm_\gamma-dependent modeling of the two-way ranges and range rates, and extract the resulting 95% confidence-level upper bound on mγm_\gamma, or demonstrate with quantified reasons that the fit cannot improve on the published 4.9×10−74.9\times10^{-7} eV/c² limit. If the Cassini residuals or the orbit-determination pipeline cannot be obtained, Part (1) is completed instead by a quantified uncertainty-propagation analysis of the mγm_\gamma-aware fit — using the published orbit-reconstruction uncertainty, the measurement noise, and the link geometry, and accounting for the correlations among the fitted parameters — that yields an explicit numerical bound on mγm_\gamma or an explicit numerical demonstration that the achievable bound cannot improve on the published limit; a qualitative or order-of-magnitude argument does not meet this standard. The authors' own estimate is an improvement of at most about two orders of magnitude (toward mγ∼10−9m_\gamma\sim10^{-9} eV/c², and probably less), driven by the roughly 10−810^{-8} relative uncertainty of the Cassini orbit reconstruction; the actual outcome, including degradation from correlations between α2\alpha^2 and the spacecraft state vector and non-gravitational-acceleration parameters, is unknown. (2) Sensitivity limits of the observable: within the massive-photon time-delay framework, quantify what combinations of carrier frequency fTf_T, measurement noise (Allan deviation σy\sigma_y), and link geometry (amplifying the mass-dependent coefficient yαy_\alpha) are required for time-delay photon-mass bounds to reach the 10−1410^{-14} eV/c² regime of Earth-scale limits or the 10−1810^{-18} eV/c² best limit — including the tension that lower carrier frequencies worsen coronal plasma systematics — and assess whether any realistic or proposed configuration, such as BepiColombo-class radio science or laser time-delay missions at optical frequencies, can achieve this. The two parts jointly answer one question — the true photon-mass reach of the solar-system Shapiro observable.

References

  1. P. C. Malta and C. A. D. Zarro, "Bounds on the photon mass via the Shapiro effect in the solar system", Physical Review D 110, 095017 (2024). DOI: 10.1103/PhysRevD.110.095017; arXiv:2403.12286. https://doi.org/10.1103/PhysRevD.110.095017
  2. B. Bertotti, L. Iess, and P. Tortora, "A test of general relativity using radio links with the Cassini spacecraft", Nature 425, 374–376 (2003). DOI: 10.1038/nature01997. https://doi.org/10.1038/nature01997
  3. D. D. Ryutov, "Using plasma physics to weigh the photon", Plasma Physics and Controlled Fusion 49, B429–B438 (2007). DOI: 10.1088/0741-3335/49/12b/s40. https://doi.org/10.1088/0741-3335/49/12b/s40
  4. S. Navas et al. (Particle Data Group), "Review of Particle Physics", Physical Review D 110, 030001 (2024); photon mass review and listing in the PDG 2025 update. DOI: 10.1103/PhysRevD.110.030001. Photon-mass listing (data-table PDF): https://pdg.lbl.gov/2025/listings/rpp2025-list-photon.pdf
  5. O. del Barco, "Picosecond-level corrections to the Shapiro time delay in a Schwarzschild spacetime", Physica Scripta 100, 095020 (2025). DOI: 10.1088/1402-4896/ae08f9; arXiv:2509.11366. https://arxiv.org/abs/2509.11366
  6. C.-K. Qiao and P. Su, "Time delay of light in the gravitational lensing of supermassive black holes in dark matter halos", The European Physical Journal C 84, 1032 (2024). DOI: 10.1140/epjc/s10052-024-13403-3. https://arxiv.org/abs/2403.05682
  7. J.-F. Glicenstein, "Gravitational lensing time delays with massive photons", The Astrophysical Journal 850, 102 (2017). DOI: 10.3847/1538-4357/aa9439. https://arxiv.org/abs/1710.11587
  8. A. S. Goldhaber and M. M. Nieto, "Photon and graviton mass limits", Reviews of Modern Physics 82, 939–979 (2010). DOI: 10.1103/RevModPhys.82.939. https://doi.org/10.1103/RevModPhys.82.939
  9. A. Accioly, J. Helayël-Neto, and E. Scatena, "Upper bounds on the photon mass", Physical Review D 82, 065026 (2010). DOI: 10.1103/PhysRevD.82.065026. https://doi.org/10.1103/PhysRevD.82.065026