The 2019 three-satellite satellite-laser-ranging (SLR) test of the weak equivalence principle (WEP) using LAGEOS, LAGEOS 2 and LARES confirmed the equivalence of gravitational and inertial mass for these bodies at the 10−910^{-9} level, but the comparison could not separate a composition-dependent WEP violation from a distance-dependent change of the effective terrestrial gravitational parameter, because LARES differs from the two LAGEOS satellites in both composition and orbital radius. The open problem is to extend the orbital method — by incorporating additional well-tracked dense laser-ranged satellites, notably LARES 2, which provides a third composition at a LAGEOS-like orbital radius, and by reducing radial-ranging, center-of-mass, atmospheric and retroreflector systematics with correspondingly tighter control of higher Earth gravity harmonics and non-gravitational perturbations — so that the two hypotheses are estimated separately, and to determine how strongly the resulting orbital constraints can bound specific modified-gravity and Yukawa-type fifth-force models.

Background

The weak equivalence principle (WEP) states that the gravitational mass mgm_g and inertial mass mim_i of a test body are proportional, so that all bodies fall identically in a given gravitational field; a violation is parametrized by δ(mg/mi)\delta(m_g/m_i), the fractional difference of the mass ratio between two materials. Satellite laser ranging (SLR) measures the distance between ground stations and retroreflector-equipped satellites to millimeter–centimeter precision, yielding precise orbits from which small non-Newtonian orbital perturbations can be sought; the inferred range must be corrected for the offset between the satellite's center of mass and the effective optical reflection point of its corner-cube retroreflector array (the center-of-mass correction), for atmospheric propagation delays, and for the optical response of the retroreflector array itself, each correction carrying its own systematic uncertainty. The passive geodetic satellites LAGEOS (launched 1976) and LAGEOS 2 (1992) are aluminum-shelled bodies with brass cores in near-circular orbits at radii of about 12,270 km and 12,160 km, while LARES (2012) is a sphere of sintered tungsten at a radius of 7,820 km. Ciufolini and colleagues analyzed the laser-ranging data of all three satellites (February 2012 to December 2014) with independent orbit determiners and the GGM05S Earth gravity model and confirmed the WEP for these bodies with δ(mg/mi)=2.0×10−10±1.1×10−9\delta(m_g/m_i)=2.0\times10^{-10}\pm1.1\times10^{-9}, the systematic error being dominated by the uncertainty in the measured radial distance (Ciufolini et al. 2019). A conceptual limitation remains: since LARES differs from the LAGEOS satellites in both composition and orbital radius, an apparent composition-dependent anomaly in the three-satellite comparison is degenerate with a distance-dependent change of the effective gravitational parameter of the Earth GM⊕GM_\oplus over the 7,820–12,270 km band — the authors note, citing the shell theorem, that a Yukawa-type correction to Newtonian gravity, V(r)=−Gm1m2/r [1+αe−r/λ]V(r)=-Gm_1m_2/r\,[1+\alpha e^{-r/\lambda}] with range λ\lambda comparable to the orbital radii, would change the effective GM⊕GM_\oplus inferred at different radii and could mimic a WEP signal. The problem predates the 2019 result: Iorio (2007) estimated that a LARES-type satellite could test the WEP at about the 10−910^{-9} level, which is what was achieved, without addressing the composition–radius degeneracy. The degeneracy is not resolved by other existing WEP tests: the MICROSCOPE mission (Touboul et al. 2022), a drag-free satellite whose thrusters continuously cancelled non-gravitational forces so that the test masses followed a purely gravitational trajectory, bounded the Eötvös parameter — the fractional differential acceleration between two freely falling materials, the standard dimensionless WEP-violation measure — for titanium and platinum at 10−1510^{-15}, but at a 710 km altitude and for accelerometer test masses rather than extended bodies in free orbital flight, and atom-interferometric tests (Asenbaum et al. 2020) reach 10−1210^{-12} over laboratory baselines; neither probes the several-thousand-kilometer range band where a Yukawa interaction of range comparable to the Earth's radius would act. The configuration needed to break the degeneracy now exists in orbit: LARES 2, launched 13 July 2022, is a dense body of nickel alloy (Haynes 242; see Pilone et al. 2021) at an altitude of about 5,900 km — a LAGEOS-like radius but a third composition — with inclination supplementary to that of LAGEOS, i.e. the two orbital inclinations sum to 180∘180^\circ (Ciufolini et al. 2023). Its data have been incorporated into combined LAGEOS–LARES geodetic solutions (Sośnica et al. 2024), Earth tidal perturbations on it have been analyzed (Gurzadyan et al. 2025), and its frame-dragging measurement — frame-dragging being the Lense–Thirring precession of an orbit induced by the rotation of the Earth — was published in 2026 (Ciufolini et al. 2026), but no published analysis separates the WEP-violation and distance-dependent-gravity hypotheses. Yukawa-type and modified-gravity constraints from laser-ranged satellites have a separate lineage: Lucchesi (2011) constrained Yukawa-like interactions from LAGEOS orbital residuals (the differences between observed and modeled orbital quantities), and the LARASE/SaToR-G program continues to bound parametrized post-Newtonian (PPN) parameters — the standard framework describing small metric deviations from general relativity — torsion and non-symmetric gravity theories, and Yukawa-like interactions from orbital residuals, bounded by current modeling accuracy (Lucchesi 2026); the a-priori systematic error budget of LAGEOS–LARES 2 experiments remains actively debated (Iorio 2025).

Problem Statement

Extend the multi-satellite laser-ranging test of the weak equivalence principle so that its two currently degenerate interpretations are determined separately. Using the LAGEOS, LAGEOS 2 and LARES satellites together with additional well-tracked dense laser-ranged satellites — notably LARES 2, whose nickel-alloy composition and LAGEOS-like orbital radius make composition and orbital radius vary independently across the satellite set — and with reduced radial-ranging error and correspondingly controlled center-of-mass, atmospheric and retroreflector systematics, and tighter control of higher Earth gravity harmonics and non-gravitational perturbations: (i) separately estimate any composition-dependent violation of the weak equivalence principle, δ(mg/mi)\delta(m_g/m_i), with sensitivity at or below the 10−910^{-9} level, and any distance-dependent change of the effective terrestrial gravitational parameter GM⊕GM_\oplus over the sampled orbital radii (approximately 7,820–12,270 km for the existing fleet), with a quantified error budget for each; and (ii) determine, from these separately estimated effects, how strongly this orbital method can constrain specific modified-gravity and Yukawa-type fifth-force models (for example a Yukawa correction with range λ\lambda comparable to the orbital radii or the Earth's radius). The common objective uniting these requests is a defensible attribution of the orbital residuals: a claim that the equivalence principle holds for the satellite bodies requires excluding, not assuming away, the distance-dependent-gravity alternative.

References

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