Geopotential harmonics of order 29, 30 and 31 from analysis of resonant orbits

The Earth's gravitational potential is usually expressed as an infinite series of tesseral harmonics, and it is possible to evaluate “lumped harmonics” of a particular order m by analyses of resonant satellite orbits—orbits with tracks over the Earth that repeat after m revolutions. In this paper we review results on 30th-order harmonics from analyses of 15th-order resonance, and results on 29th- and 31st-order harmonics from 29:2 and 31:2 resonance.The values available for 30th-order lumped harmonics of even degree are numerous enough to allow a solution for individual coefficients of degree up to 40. The best-determined coefficients are those of degree 30, namely

$\text{10}{}^9\text{C}{}_{\mathrm{30,30}}\text{= ?1.2±1.1 10}{}^9\text{S}{}_{\mathrm{30,30}}\text{= 9.6±1.3}$

The standard deviations here are equivalent to 1 cm in geoid height.For the 29th- and 31st-order harmonics, and for the 30th-order harmonics of odd degree, there are not enough values to determine individual coefficients, but the lumped values from particular satellites can be used for “resonance testing” of gravity field models, particularly the Goddard Earth Model 10B (up to degree 36) and 10C (for degree greater than 36). The results of applying these tests are mixed. GEM 10B/C emerges well for order 30, with s.d. about 3×10^?9; for order 31, the GEM 10B values are probably good but the GEM 10C values are probably not; for order 29, the test is indecisive.