Juno spacecraft gravity measurements provide evidence for normal modes of Jupiter

Juno has been orbiting Jupiter in a highly eccentric 53.5-day orbit since July 2016. During the prime mission, the hemispherically symmetric part of the gravity field was used to infer a dilute core, and the north–south asymmetric component implied that zonal winds extend a few thousand kilometers deep, under the assumption of a purely zonal (axisymmetric) field. However, analysis of Doppler data up to mid-prime mission indicated additional accelerations of 2–5 × 10⁻⁸ m/s² and Doppler signatures up to ~0.1 mm/s over 10–15 minutes that are not explained by zonal harmonics alone. Similar but larger unexplained accelerations were observed during Cassini’s Grand Finale at Saturn. Potential causes include time-dependent density anomalies in the differentially rotating outer shell, deep non-zonal density anomalies, or internal oscillations (normal modes). Because normal modes can displace large masses and generate time-variable gravity, the study investigates whether Jupiter’s normal modes can account for the unexplained Juno Doppler signatures and seeks to characterize the corresponding amplitude spectrum.

Prior work established Jupiter’s zonal gravity field and suggested a dilute core and deep-reaching zonal winds from Juno gravity (e.g., hemispherically symmetric and north–south asymmetric components). Unexplained accelerations were noted in Juno data and in Cassini data at Saturn. A localized gravity anomaly associated with the Great Red Spot was detected during a targeted overflight, illustrating that atmospheric features can produce measurable gravity signals. Ground-based observations by Gaulme et al. reported excess power consistent with Jovian p-modes at 800–1500 μHz, while lower-frequency f-modes likely eluded detection due to instrumental limitations. At Saturn, ring seismology revealed numerous internal f-modes from density waves in the rings and constrained interior structure, and Cassini gravity residuals provided evidence for p-modes, though f-modes were not seen in gravity (consistent with the rings’ sensitivity to f-modes and insensitivity to p-modes). Numerical simulations had predicted that Jupiter’s normal modes should be observable in Juno Doppler data, motivating a targeted search in the expanded Juno gravity dataset.

Data and preprocessing: The analysis uses 22 gravity-dedicated Juno passes up to PJ33, comprising 12,299 two-way Doppler points at 60 s integration. Tracking employed dual X/Ka-band links when available to reduce plasma noise (up to 75% mitigation), with some passes lacking Ka uplink. Wet tropospheric noise was calibrated using AWVR data when available, reducing Doppler noise by ~34% on average (up to 65%). Both perijove and outbound (pre-OTM) data segments were used per pass. Orbit determination and dynamical model: Doppler data were fit with JPL’s MONTE software using a multi-arc least-squares filter. Each perijove pass is an arc with its own local parameters: spacecraft initial state, velocity changes associated with Earth repointing turns, and normal-mode amplitudes (estimated as cosine and sine components per mode, per arc). Global parameters common to all arcs include Jupiter GM, zonal harmonics up to degree 30 (even and odd), full degree-2 tesseral field, tidal Love numbers up to degree/order 4 (even l−m), Jupiter’s spin axis initial position and polar moment of inertia factor, a dipole model for the Great Red Spot mass anomaly, and a solar radiation pressure scale factor. The orbit model accounts for third-body gravitation (planets, Jupiter’s satellites) in 1-PN relativity, solar and satellite tides on Jupiter, Jupiter’s spin-axis motion (numerically integrated with torques), non-gravitational forces (solar radiation pressure, albedo/IR emission, anisotropic thermal acceleration of solar arrays), relativistic light-time and oblateness effects, solar-array bending Doppler effects, and ground station delays. Except for normal-mode amplitudes (with a priori covariance), no a priori constraints were imposed on parameters. Normal-modes gravity model: Normal-mode eigenfrequencies and radial eigenfunctions were computed for a non-rotating n=1 polytropic Jupiter using GYRE. The density perturbation is proportional to the radial eigenfunction times the density gradient and a spherical harmonic Y_lm, oscillating at each mode’s angular frequency. Perturbations to gravitational harmonic coefficients were obtained by volume integration over the planet. The model was limited to zonal coefficients (m=0) to avoid overparameterization and because Juno’s polar, highly eccentric orbit confounds separation of high-order zonal from tesseral terms; robustness to including tesseral terms was checked. To compute mode amplitudes, the surface radial velocity v(f) was modeled as a Gaussian function of frequency with parameters: v_min (noise floor), v_max (peak velocity) at peak frequency f_peak, and Gaussian width σ, optionally including a background. This choice reflects stochastically excited, intrinsically damped oscillators. Parameter search: A grid over {v_min, v_max, f_peak, σ} was explored. For each grid model, only modes with l ≤ 8 and periods ≥ 10 min (roughly up to p-mode radial order n ≲ 7) were included due to sampling and noise limits. A mode was estimated on an arc only if its expected spherical harmonic coefficient amplitude exceeded 1e−10 in magnitude. Because of long gaps between perijoves and frequency uncertainties, phase coherence across arcs was not assumed; amplitudes and phases were treated as local per-arc parameters. Model selection and degrees of freedom: Solutions were evaluated via Akaike Information Criterion (AIC), with ΔAIC used to compute Akaike weights. The regression-effective degrees of freedom were computed from the trace of the influence matrix H = (H^T W H + P)^−1 H^T W, accounting for the a priori covariance P on normal-mode amplitudes. Alternative hypothesis tests: In addition to Gaussian p-mode dominated spectra, a flat velocity profile v(f)=v_uniform (energy-equipartition scenario) was tested. Static tesseral fields and localized deep density anomalies were also examined (details in Supplementary Information).

• Juno Doppler residuals exhibit signatures near perijove that are better explained by time-variable gravity from internal normal modes than by static tesseral fields or localized deep density anomalies.
• Best-fit amplitude spectra concentrate power in p-modes with peak surface radial velocity v_max ≈ 10–50 cm/s at f_peak ≈ 900–1200 μHz, Gaussian width σ ≈ 300 μHz (FWHM ≈ 0.5σ), consistent with ground-based detections of excess power.
• f-modes are constrained to small amplitudes: upper bounds imply radial velocities smaller than ~1 cm/s (also characterized as ~1–10 mm/s in the recovered spectra), i.e., p-modes exceed f-modes by factors of roughly 30–100 in velocity, similar to the Sun’s p-to-f amplitude ratio.
• The p-mode dominated models substantially reduce Doppler residuals compared to zonal-only gravity; across the search grid, the most probable solutions cluster around f_peak ~1000–1200 μHz with v_max 10–50 cm/s and v_min < 1 mm/s.
• A flat velocity profile (energy-equipartition) fails to fit the data: the minimum ΔAIC for v_uniform models is ~1520 (at v_uniform = 1 mm/s), far worse than the best p-mode model (ΔAIC = 0). In such flat models, estimated f-mode amplitudes are driven below the assumed profile, indicating that power in higher-frequency p-modes is required by the data.
• The recovered amplitude spectrum implies surface displacements of order 15–80 m for p-modes and only a few meters for f-modes.

Attributing the unexplained Juno accelerations to Jupiter’s internal normal modes provides a coherent explanation for the observed time-variable Doppler signatures near perijove and yields an amplitude spectrum compatible with independent ground-based indications of p-modes. Alternative explanations were examined and found less satisfactory: a small set of low-degree static tesseral harmonics cannot reproduce the encounter-to-encounter variability; deep, spectrally rich, non-zonal anomalies are not presently constrained by convection or dynamo models and do not align with the changing anomaly pattern across perijoves. The preferred solutions place most power in p-modes at ~900–1200 μHz with v_max ~10–50 cm/s, while f-modes are small (<1 cm/s). The ratio of p- to f-mode amplitudes resembles the solar spectrum, though whether this indicates common excitation mechanisms is unclear given the vastly different luminosities and convective environments. The findings highlight the potential of planetary seismology to probe gas giant interiors beyond what static gravity can resolve, offering sensitivity to core structure and stratification. Rejecting energy-equipartition underscores the need for models in which excitation and damping produce a peaked spectrum in the p-mode band. These results motivate dedicated strategies and instrumentation for future missions to measure time-variable gravity and directly observe normal modes.

This study demonstrates that Juno’s Doppler gravity measurements contain a time-variable component consistent with Jupiter’s internal normal modes. A p-mode dominated spectrum with peak surface radial velocities of 10–50 cm/s at ~900–1200 μHz explains the data, while f-modes are constrained to small amplitudes (<1 cm/s). Flat, energy-equipartition velocity spectra and simple static tesseral-field explanations do not adequately fit the observations. Methodologically, the work introduces a multi-arc estimation framework that retrieves normal-mode amplitudes from spacecraft tracking, coupled with a parameterized frequency–amplitude model and AIC-based model selection. Future research should (i) obtain precise mode frequencies and phases through mission designs optimized for time-variable gravity, (ii) employ onboard instruments dedicated to seismology for direct mode detection at higher angular degrees, and (iii) integrate gravity-based mode constraints with improved theoretical models of excitation and damping to infer interior structure (e.g., core properties and stratification) of gas and icy giants.

• Mode identification: Unambiguous identification of individual modes is not possible if multiple modes contribute; phases cannot be coherently tracked across arcs due to 53.5-day gaps and frequency uncertainties.
• Observational geometry: Juno’s highly eccentric, polar orbit concentrates sensitivity to a few hours around perijove, limiting mapping of time variability; short 6–8 h windows and 60 s sampling reduce sensitivity to higher-frequency modes.
• Model truncation: Analysis restricts to l ≤ 8 and periods ≥ 10 minutes (roughly p-mode n ≤ 7), potentially neglecting higher-degree/frequency contributions that are less accessible with current data.
• Interior model simplification: Eigenfunctions and frequencies are computed for a non-rotating n=1 polytrope; while coefficient differences are estimated to be within 20–50% over relevant frequencies, rotation and more realistic EOS could refine results.
• Parameterization: The amplitude–frequency relation is assumed Gaussian with four parameters; different spectral shapes or excitation/damping models could alter inferences. A priori covariance is used for mode amplitudes.
• Alternative sources: While static tesseral fields and simple localized anomalies are disfavored, complex deep, time-dependent density or magnetic-field-related anomalies are not fully constrained by current models.
• Data availability and noise: Some passes lacked full dual-link calibration; residual plasma/tropospheric noise and thermal/solar-array systematics may affect sensitivity to weak modes, especially low-frequency f-modes.