Kaula’s degree-1 gravitational potential, geocenter and application to Genesis mission.

In this study, we revisit Kaula’s orbit perturbation theory to evaluate the capability of satellite orbits to sense degree-1 gravitational signatures. We explicitly derive the Kaula-based expressions for the degree-1 potential terms and compute the corresponding inclination F_1mp(i), and eccentricity functions G_1mp(e), providing—for the first time—complete tables for degree-1.

As a study case, we applied this to the ESA's Genesis mission set to be launched in 2029. Genesis will be the first in orbit geodetic observatory carrying  onboard the 4 co-located geodetic techniques (GNSS, SLR, DORIS and VLBI). 

The dependence on orbital inclination of the expressions for the degree-1 potential coefficients V₁₀  and V₁₁ is examined. Our analysis shows that V₁₀, associated with the geocenter’s z-component, scales with sin i, implying that polar orbits maximize sensitivity to vertical geocenter variations, while equatorial orbits remain largely insensitive to them. Conversely, the two-term structure of V₁₁  enhances the detectability of the geocenter’s x- and y-components for equatorial or near-equatorial orbits, with reduced but non-negligible sensitivity in polar configurations.

This work demonstrates how Kaula’s theory can guide the design of future gravimetry missions by identifying orbital parameters that optimize degree-1 recovery, thereby improving geocenter estimation and strengthening the link between satellite gravimetry and terrestrial reference frame realization.