How accurate is the cancelation of the first even zonal harmonic of the geopotential in the present and future LAGEOS-based Lense-Thirring tests?

The strategy followed so far in the performed or proposed tests of the general relativistic Lense-Thirring effect in the gravitational field of the Earth with laser-ranged satellites of LAGEOS type relies upon the cancelation of the disturbing huge precessions induced by the first even zonal harmonic coefficient J_2 of the multipolar expansion of the Newtonian part of the terrestrial gravitational potential by means of suitably designed linear combinations of the nodes \Omega of more than one spacecraft. Actually, such a removal does depend on the accuracy with which the coefficients of the combinations adopted can be realistically known. Uncertainties of the order of 2 cm in the semimajor axes a and 0.5 milliarcseconds in the inclinations I of LAGEOS and LAGEOS II, entering the expression of the coefficient c_1 of the combination of their nodes used so far, yield an uncertainty \delta c_1 = 1.30 10^-8. It gives an imperfectly canceled J_2 signal of 10.8 milliarcseconds per year corresponding to 23% of the Lense-Thirring signature. Uncertainties of the order of 10-30 microarcseconds in the inclinations yield \delta c_1=7.9 10^-9 which corresponds to an uncanceled J_2 signature of 6.5 milliarcseconds per year, i.e. 14% of the Lense-Thirring signal. Concerning a future LAGEOS-LAGEOS II-LARES combination with coefficients k_1 and k_2, the same uncertainties in a and the less accurate uncertainties in I as before yield \delta k_1=1.1 10^-8, \delta k_2=2 10^-9; they imply a residual J_2 combined precession of 14.7 milliarcseconds per year corresponding to 29% of the Lense-Thirring trend. Uncertainties in the inclinations at about 10 microarcseconds level give \delta k_1=5 10^-9, \delta k_2 = 2 10^-9; the uncanceled J_2 effect is 7.9 milliarcseconds per year, i.e. 16% of the relativistic effect.

💡 Research Summary

The paper revisits the widely‑used strategy of canceling the dominant classical precession caused by the Earth’s first even zonal harmonic (J₂) in satellite‑based tests of the Lense‑Thirring (LT) effect. The method relies on forming linear combinations of the nodal longitudes (Ω) of two or more laser‑ranged LAGEOS‑type satellites so that the J₂‑induced rates vanish analytically. For the traditional two‑satellite combination (LAGEOS and LAGEOS II) the coefficient c₁ is defined as the ratio of the partial derivatives of the nodal rates with respect to J₂, which in turn depend on the satellites’ semimajor axes (a) and inclinations (I). Consequently, any uncertainty in a or I propagates into an uncertainty δc₁, leaving a residual J₂ signal.

Using realistic present‑day uncertainties—≈2 cm in a and ≈0.5 mas in I for both LAGEOS satellites—the author computes δc₁ ≈ 1.3 × 10⁻⁸. This translates into an uncanceled J₂ nodal precession of about 10.8 mas yr⁻¹, i.e. roughly 23 % of the expected LT signal (≈48 mas yr⁻¹). If the inclination determination could be improved to the 10–30 µas level, δc₁ would shrink to 7.9 × 10⁻⁹, reducing the residual J₂ effect to 6.5 mas yr⁻¹ (≈14 % of LT). These numbers illustrate that even modest errors in the orbital elements generate a non‑negligible systematic bias.

The analysis is then extended to a prospective three‑satellite configuration involving LAGEOS, LAGEOS II and LARES. The combination Ω = Ω₁ + k₁Ω₂ + k₂Ω₃ introduces two new coefficients, k₁ and k₂, each with a more intricate dependence on a and I. Applying the same a‑uncertainty and the current I‑uncertainty yields δk₁ ≈ 1.1 × 10⁻⁸ and δk₂ ≈ 2 × 10⁻⁹, which correspond to a residual J₂ precession of 14.7 mas yr⁻¹ (≈29 % of LT). With inclination errors reduced to ≈10 µas, δk₁ falls to 5 × 10⁻⁹ and δk₂ to 2 × 10⁻⁹, giving a remaining J₂ contribution of 7.9 mas yr⁻¹ (≈16 % of LT).

The key insight is that the effectiveness of J₂ cancellation is fundamentally limited by the achievable precision in the satellites’ orbital parameters. Present laser‑ranging technology cannot guarantee the sub‑10 % systematic error budget required for a high‑precision LT test. Moreover, the residual J₂ bias is comparable to, or larger than, the contributions from higher‑order even zonals (J₄, J₆, …) and non‑gravitational perturbations (atmospheric drag, solar radiation pressure, thermal effects). Therefore, future experiments must adopt a more holistic approach: improving the measurement of a and I (potentially via inter‑satellite ranging or next‑generation tracking), refining Earth‑gravity models, and explicitly modeling or mitigating additional perturbations.

In conclusion, the paper quantifies the imperfect cancellation of the J₂ term in current and planned LAGEOS‑based LT experiments, showing that realistic orbital‑parameter uncertainties leave a residual J₂ signal amounting to 14–29 % of the relativistic effect. This systematic error sets a lower bound on the achievable accuracy of LT measurements with the existing methodology and underscores the need for both technological advances in satellite tracking and more sophisticated combination schemes that can simultaneously address multiple sources of error.