Mercury Orbiter Radio-Science Experiment (MORE)

• MORE is a high-precision radio-science experiment that employs multi-frequency tracking and rigorous relativistic modeling to test metric theories of gravity.
• It utilizes a coherent 5-leg X-/Ka-band link with two-way range and Doppler observations to achieve cm-level precision for key PPN parameters.
• The methodology integrates full 2PN light-time formulations, batch-wise SRP estimation, and multi-arc orbit determination to constrain gravity parameters at the 10⁻⁶ level.

The Mercury Orbiter Radio-Science Experiment (MORE) is an advanced radio science investigation embarked on the ESA/JAXA BepiColombo mission to Mercury. As one of the mission’s fundamental-physics experiments, MORE combines high-precision, multi-frequency Ka-band tracking with rigorous post-Newtonian dynamical modeling to enable unprecedented tests of general relativity, refined measurement of post-Newtonian (PN) parameters, and detailed investigation of Mercury’s geodesy and rotation. Utilizing both two-way range and Doppler (range-rate) observations, MORE aims to attain formal parameter uncertainties at or below the 10⁻⁶ level for the Eddington parameter γ, the Nordtvedt parameter η, and associated parameters, thereby pushing the precision frontier for solar system gravity tests.

1. Scientific Objectives and Instrumentation

MORE’s primary scientific objective is to test metric theories of gravity in the solar system, with emphasis on measuring the PPN parameter γ via superior solar conjunctions, exploiting the Shapiro gravitational time delay and Doppler tracking of radio signals passing near the Sun (Cappuccio et al., 2022, Stefano et al., 2022, Schettino et al., 2020, Schettino et al., 2018, Marchi et al., 2016). Secondary objectives encompass Mercury’s high-precision gravimetry, geodesy, and characterization of its gravity field and rotation state.

Key elements of the radio tracking system:

• Coherent 5-leg link in X- and Ka-band: Architecture enables dual-frequency plasma calibration, significantly mitigating ionospheric and solar corona noise.
• Ka-band transponder (KaT): Operates near 32 GHz uplink/37 GHz downlink; together with X-band, supports PN ranging at 24 Mcps for ∼1 cm 2-way range precision.
• Ultra-stable oscillator: Ensures phase and timing coherence.
• Ground antennas: ESA DSA-3 (Malargüe) and NASA DSS-25 (Goldstone) provide deep-space coverage.
• Measurement precision: 2-way range-rate (Doppler) accuracy of 0.01 mm/s at 60 s integration, and 2-way PN range of cm-level in a few seconds.

These capabilities allow near-continuous tracking at solar elongation angles down to ∼7–8 R_⊙, placing MORE at the forefront of interplanetary radiometric experiments.

2. Relativistic Light-Time Formulations and the Shapiro Effect

The theoretical core of MORE’s relativity experiment lies in computing the gravitational time delay—primarily the Shapiro effect—as radio signals propagate through the Sun’s field. The spacetime metric, up to second post-Newtonian (2PN) order, is expressed in isotropic coordinates as:

$$ds^2 = [1-2U/c^2 + 2β U^2/c^4 + …]\,c^2 dt^2 - [1+2γ U/c^2 + 3ε U^2/(2c^4) + …]\,d\ell^2$$

with $U = GM/r$, where $γ, β$ are PPN parameters.

Key Light-Time Formulae

• First-Order (Shapiro) Delay:

$$Δt_{1{\rm PN}} = (1 + γ)\frac{GM}{c^3}\ln \left( \frac{r_1 + r_2 + r_{12}}{r_1 + r_2 - r_{12}} \right)$$

For the inaugural BepiColombo superior conjunction (Mar 2021), this term peaks at 50.6 km (2-way).

• Moyer’s Approximation (MONTE/ODP):

$$t_{\rm ODP} = \frac{r_{12}}{c} + (1 + γ)\frac{GM}{c^3} \ln \left( \frac{r_1 + r_2 + r_{12} + (1 + γ)\frac{GM}{c^2}}{r_1 + r_2 - r_{12} + (1 + γ)\frac{GM}{c^2}} \right)$$

This approach includes the major “enhanced” second-order contribution (St_ApN) but omits higher-order effects.

• Full 2PN Expansion:

$$t_{2{\rm PN}} = \frac{r_{12}}{c} + (1 + γ)\frac{GM}{c^3}\ln \left( \frac{r_1 + r_2 + r_{12}}{r_1 + r_2 - r_{12}} \right) + \frac{(1+γ)^2 G^2 M^2 r_{12}}{c^5 r_1 r_2} + \frac{8(1+γ) - 4β + 3ε}{4} \frac{G^2 M^2}{c^5} \frac{\arccos(\hat{r}_1 \cdot \hat{r}_2)}{|\hat{r}_1 × \hat{r}_2|}$$

The last two terms represent the enhanced and non-enhanced 2PN corrections (St_ApN, St_ppN), which become critical at cm-level measurement noise.

Quantitative Assessment

Comparison of Moyer’s ODP and full 2PN expansion demonstrates that the neglected 2PN term (St_ppN) can reach +17 mm in 2-way light-time—on the order of the PN ranging noise floor (∼1 cm). Without including the full 2PN terms, systematic errors in estimating γ at the desired 10⁻⁶ level or below will ensue. Both the Lorentz transformation term (St_ssB, due to solar barycentric velocity) and 2PN contributions from planetary Shapiro delays are negligible (≲10⁻³ mm and <0.02 mm, respectively), as are corrections for solar quadrupole (J₂, ≲0.07 mm) and angular momentum (spin, ≲0.003 mm) (Cappuccio et al., 2022).

3. Parameter Estimation, Orbit Determination, and Systematic Effects

Orbit Determination Approach

A multi-arc strategy segments the spacecraft’s Mercury-centric trajectory into daily arcs, absorbing orbit perturbations on hour-timescales and enhancing resilience to instrument drifts (Schettino et al., 2018, Schettino et al., 2020). Orbit determination is formulated as a nonlinear least-squares minimization problem:

$\min_{u}\, Q(u) = \frac{1}{m}\,\xi^T W \xi$

subject to $\xi(u) = O - C(u)$, where O are observables (range, Doppler), C computed values, W the weighting matrix, and m the number of observations.

The design matrix (B), normal matrix (C = B^TWB), and covariance (Γ = C⁻¹) are constructed for simultaneous estimation of:

• Spacecraft state vectors for each arc
• Mercury and Earth-Moon barycenter (EMB) heliocentric states
• Global parameters: PPN parameters (γ, β, η, α₁, α₂), solar mass (μ☉), quadrupole (J{2☉}), time derivative (ζ), and solar angular momentum (GS_☉)

Rank Deficiencies and Mitigation

The combined estimation problem exhibits approximate four-fold rank deficiency due to near-SO(3) rotational symmetry and scaling degeneracies (Kepler's third law). Remedies include:

• Descoping: Remove (fix) four state parameters (e.g., EMB’s position and a velocity component).
• A priori constraints: Impose weak, near-null constraints on rotational and scaling degrees of freedom, restoring the full rank of the normal matrix and ensuring parameter convergence (Schettino et al., 2018).

Error Budget

Assuming dominant statistical errors from Ka-band noise and optimally calibrated systematics, forecasted formal uncertainties for a one-year data span are:

Parameter	1-year σ (Baseline)	Method
γ	7.6×10⁻⁷ – 1.3×10⁻⁵	Range/Doppler, ODP/2PN, batch SRP
β	2.5×10⁻⁶ – few×10⁻⁵	Orbit fit, covariance
η (Nordtvedt)	6×10⁻⁵ – 4.5×10⁻⁵	PN, covariance
α₁, α₂	∼6×10⁻⁷, ∼1×10⁻⁷	Orbit fit
J_{2☉}	~2×10⁻⁹ – 5×10⁻⁹	Orbit fit
μ_☉	∼10¹⁴ cm³/s² (~10⁻¹⁴ rel.)	Orbit fit
Mercury/EMB position	0.6–4 cm	Covariance
Mercury/EMB velocity	1–8×10⁻⁷ cm/s	Covariance

Systematic degradations due to ephemeris and asteroid mass uncertainties increase σ(η) to about ~4.5×10⁻⁵, yet this remains an order of magnitude better than available pre-BepiColombo measurements (Marchi et al., 2016).

4. Modeling and Mitigating Non-Gravitational Accelerations

Solar Radiation Pressure (SRP)

The fine proximity of BepiColombo to the Sun exposes the MPO spacecraft to solar radiation pressure fluctuations with amplitudes δI(t) in the 10⁻⁴–10⁻³ range (from sunspots/faculae), and nominal SRP acceleration ~10⁻⁷ m/s² at 0.5 AU (Stefano et al., 2022). The SRP on surface element k is given by:

$$a_{SRP,k}(t) = \frac{F_{⊙}(R_0/R(t))^2 A_k}{m}\left[ (1-C_{s,k})\hat s + 2(C_{s,k} (\hat s\cdot\hat n_k)+C_{d,k})\hat n_k \right]$$

To address stochastic SRP, MORE employs:

• Exponentially correlated random-variable (ECRV) model: First-order Gauss–Markov process for δI(t) with time constant τ ≈ 27 days.
• Batch-wise SRP scale factors (s_j): Fitted for each time batch (Δt ~24 hr), modeled as ECRV with prior covariance Q_s to capture SRP variability.
• Integration in orbit fit: H^T W H + Q⁻¹ regularization in normal equations ensures GR signatures (γ) are not absorbed by SRP noise.

Solar Plasma and Wind

Dual-frequency (X-/Ka-band) “five-link” radio tracking enables linear calibration and subtraction of plasma-induced dispersive delays. This calibration is reliable for impact parameters ≳7 R_⊙, degrading at smaller elongation due to coronal scintillation.

Stochastic Model Impact

Monte Carlo simulations with realistic solar activity reveal that, for conjunctions at b_cut=7 R_⊙, 2 cm range noise, and 14-day arcs, a combined analysis across multiple SSCs affords σ(γ−1) ≈ 1.3×10⁻⁵ (best cases in [6–13]×10⁻⁶). Reducing b_cut or leveraging low-activity intervals further improves precision (Stefano et al., 2022).

5. Solar System Perturbations, Alternative Gravity, and Torsion

Planetary and Solar System Effects

• Planetary Shapiro: Jupiter, Earth, and Saturn induce two-way delays up to 162 cm, 15 cm, and 25.2 cm, respectively. First-order (1PN) corrections suffice; 2PN corrections and barycentric-frame effects are negligible (<0.02 mm) (Cappuccio et al., 2022).
• Solar Oblateness/Spin: Quadrupole J₂~2.246×10⁻⁷ yields ≲0.07 mm effect; gravitomagnetic (spin) of S_☉~1.92×10⁴¹ kg·m²/s imparts ≲0.003 mm—both below the experiment’s noise floor.

Solar Lense–Thirring Effect

The solar Lense–Thirring (LT) or gravitomagnetic precession contributes a perturbing acceleration:

$a_{LT} = \frac{2G}{c^2 r^3}[J_☉ \times r]$

This yields tens-of-centimeter secular effects over one year, and its neglect introduces biases in related parameters (notably J_{2☉}). Inclusion in the orbit fit is required for unbiased estimation; the amplitude coefficient GS_☉ is solved for concurrently, despite moderate correlations with μ☉ and J{2☉} (0.5–0.6) (Schettino et al., 2020, Schettino et al., 2018).

Extended Theories: Torsion Constraints

Within the Riemann–Cartan (torsion) spacetime framework, test particle motion follows autoparallels, characterized by up to three independent torsion parameters (t₁, t₂, t₃):

$$a^{tor}_i = 2[t₁(1+γ)-t₃]\mu_☉^2\frac{r_i}{r^4} + (t₁+t₂)\mu_☉\frac{\dot{r}_i}{r^3}v_i - t₂\mu_☉\frac{v_i^2}{r^3}r_i$$

Estimation scenarios allow for solving either (t₁, t₂, β-t₃) or (t₁, t₂, t₃, β) with a-priori constraints, resulting in anticipated uncertainties down to 1–2×10⁻⁵ for t₁, t₂ (1-year mission). Extending to 2 years further improves constraints, providing bounds on spacetime torsion parameters competitive with, or superior to, existing planetary data (Schettino et al., 2020).

6. Recommendations, Limitations, and Future Prospects

• Full 2PN expansion of light-time should be adopted for experiments with cm-level and better ranging precision, superseding the MONTE/ODP (Moyer) approximation which remains suitable for m-level or Doppler-only experiments (e.g., Cassini) (Cappuccio et al., 2022).
• Include planetary monopole (1PN) Shapiro delays; higher-order and solar J₂/spin can be neglected unless sub-millimeter precision is required.
• Explicit modeling and batch-wise estimation of stochastic SRP are critical to avoid systematic biases in γ and related parameters. The combination of daily-resolved SRP scale factors with Gauss–Markov priors is essential for achieving unbiased γ at the (few)×10⁻⁶ level (Stefano et al., 2022).
• Solar Lense–Thirring effect must be included in the model to avoid σ-level parameter biases and to permit simultaneous determination of the Sun’s angular momentum (Schettino et al., 2018).
• Parameter degeneracies require a-priori constraints or parameter descoping using physically motivated relations (e.g., the Nordtvedt relation for η).
• Systematic effects from asteroid/Ephemeris mass uncertainties remain the dominant limitation for η, but are an order of magnitude smaller than current best constraints.

Simulations and design choices reflect that with careful handling of non-gravitational systematics and relativistic corrections, MORE is expected to deliver landmark tests of relativity in the weak-field regime, with γ expected to be constrained at the 10⁻⁶ level, η at ≲4.5×10⁻⁵, and strong bounds on alternative gravity models including torsion (Stefano et al., 2022, Cappuccio et al., 2022, Marchi et al., 2016, Schettino et al., 2020, Schettino et al., 2018).

7. Context and Comparison with Prior Radio-Science Experiments

In comparison to the Cassini experiment, which attained γ−1 = (2.1 ± 2.3) × 10⁻⁵ via Doppler-only methods with ∼0.3 m 1-way light-time accuracy, MORE’s utilization of multi-frequency plasma-calibrated range and Doppler links, full 2PN light-time modeling, and a multi-arc, covariance-based orbital determination framework represents a >10× improvement in attainable parameter precisions. The formal error budgets for leading PPN parameters (γ, β, η) and related solar parameters indicate an advance by one to two orders of magnitude over previous solar system radio-science gravity tests (Stefano et al., 2022, Schettino et al., 2018).