Determination of Earth’s gravity field parameters based on orbit perturbations – theoretical approach

Earth’s gravity field parameters can be derived from the perturbations of Keplerian orbit parameters. For example, even-degree zonal harmonics cause secular rates of the right ascension of the ascending node, whereas odd-degree zonal harmonics cause long-term periodic perturbations of the argument of perigee. The observations of changes in these Keplerian parameters can be used to derive Earth’s potential parameters. We investigate the relationship between the orbit perturbations as a function of the satellite height, inclination angle, and eccentricity to find optimum orbit parameters for the gravity field recovery which maximize the orbit perturbations. We employ the Kaula theorem of orbit perturbations based on the expansion of the gravity potential into trigonometric series and derive inclination F(i) and eccentricity functions G(e) for zonal even and odd-degree spherical harmonics. We also employ the satellite visibility function to find the optimum satellite height for the recovery of the global gravitational constant product GM, degree-1 spherical harmonics corresponding to the geocenter motion, Earth’s oblateness term C₂₀, and other low-degree harmonics.

We found that the optimum heights of satellites for GM, geocenter, and degree-2 are in the areas where no geodetic satellites currently are; i.e., between 1700 and 3500 km. The best inclination angles for the even-degree harmonic recovery are in the range of 20-40 degrees for prograde or 140-160 degrees for retrograde orbits. The best separability of odd-degree harmonics is for critical inclinations (63.4, 116.6 deg) or high eccentricities. C₃₀ can be well determined from a satellite at the inclination of 40 or 140 degrees at the height of 1400 km. To support future GRACE/MAGIC missions with C₂₀ and C₃₀, the best inclination would be about 40 or 140 degrees with a height of about 1500-1700 km.  Finally, the best height for superior geocenter recovery and determination of the gravitational constant is about 2300 – 3500 km, which is in between the LAGEOS-1/2&LARES-2 height (5800 km) and LARES-1&Ajisai height (1500 km).