30. Geometrical Representation of Gravity Field Determination

Geometrical or kinematic orbit determination, demonstrated for the first time using GPS on board the CHAMP satellite (Švehla and Rothacher 2003), was the basis for the retrieval of the very first determination of the gravitational field of the Earth making use of the energy balance approach, see Gerlach et al. (2003). By means of numerical differentiation, the geometric positions of the CHAMP satellite were used to determine geometrical velocities along the orbit, and making use of the energy integral, the very first geometrical gravity model of the Earth was developed. One advantage of gravity field determination based on the energy balance approach is that we can work directly with the gravitational potential as a scalar field instead of having to integrate the equation of motion. In the case of the GOCE mission, a gravity gradiometer maps gravity gradients along the orbit (Rummel et al. 2011). Geometrical positions determined using GPS are used to position the gravity gradient measurements within the terrestrial reference frame and to estimate low-order gravity field coefficients. Here we present gravity field determination using kinematic orbits, and in addition, introduce a concept of gravity field determination based on gravitational redshift and atom interferometry. The possibility of determining kinematic orbits of LEO satellites has triggered the development of new approaches in gravity field determination, opened up new fields and significantly changed the way we think about the gravity field of the Earth, not only from the point of view of satellite dynamics and numerical integration. One of the most important applications of the metric theories of gravity, such as the General Theory of Relativity, is that a clock moved further away from the source of the gravitational potential will run faster, thus one can measure perturbations in the gravitational potential along an orbit by measuring variations in the optical clock frequency. Very soon mechanical test masses used to observe gravity from space will be replaced by atoms and test particles at quantum level. One advantage of quantum mechanics compared to the classical post-Newtonian framework we use in geodesy is that atoms can be used to directly measure not only the acceleration of motion, but, in addition, also relative frequency offsets, i.e., gravitational redshift. A gravity gradiometer could be constructed based on atom interferometry and this is most likely the next step in the determination of the Earth’s gravity field. On the other hand, the redshift effect for matter waves is by orders of magnitude higher in frequency than the frequencies of standard microwave and optical clocks. The Compton frequency ω_C of matter waves is very high since it includes the rest mass energy multiplied by c², e.g., for cesium one obtains ω_C/2π = 3.2 x 10²⁵ Hz. This is significantly higher than the frequency used to measure time and to define the SI second using cesium atomic clocks. Considering that an orbit error is consistent with an error in the orbit velocity, the net redshift effect for the clock determined from the satellite position is compensated by the second order Doppler effect calculated from the satellite velocity. In size, the net effect on the total redshift effect is smaller and satellite orbit in terms of radial position is required with less accuracy compared to the accuracy of the static position for a ground clock placed on the Earth. A smaller variation in frequency can be measured at higher matter wave frequencies or by an atom gradiometer concept. This symmetry principle could be used to map gravity fields from space and in the construction of an atom gradiometer. Here we discuss the question of how the new relativistic technique based on optical clocks and atom interferometers, in general, can contribute to global, regional and local gravity field determination and the realization of a global height system. We show that there are applications for this new technique in reference frame realization for positioning, time and temporal gravity determination and how this new geometric technique could unify all three fundamental reference frames in geodesy. The principle of error compensation in the calculation of the redshift effect, considering an orbit error in satellite position and the error in the second order Doppler effect calculated from the satellite velocity, has been discussed in the timing community. This is one of the main arguments, why an orbit in space (GEO) offers the best environment to define and establish the standard of frequency and define the SI second using an atomic clock, far better than using the geoid and the surface of the Earth. The main argument is, however, that cold atoms can be observed for a long time in space and are not limited by the free-fall on Earth, gaining an additional 3–4 orders of magnitude in sensitivity for atomic clocks. Thus a GEO or a GNSS orbit could offer the best place to define the datum for time on Earth and be used in supporting definition of the fundamental reference frames in geodesy.