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# Theory of GPS

Our research is focused on the “time transfer” between atomic clocks on satellites, moving on two adjacent orbits. For example, in the satellite experiment GRACE (Gravity Recover and Climate Experiment) the two satellites are 220 km apart, but the range rate between the two satellites must be known to better than $10^{-6} m s^{-1}$. A new mathematical model has been created in which the time transfer between the satellites takes into account the ellipticity of the orbits. A new physical conception has been implemented, according to which for the synchronization of clocks it is of key importance to have an inverse time transformation, which in the case is ensured by elliptic integrals and Jacobi elliptic functions, possessing this property For values of a typical GPS-orbit, the range of the proper and coordinate time is calculated to be $28136.7619\times 10^{-6} s$ and $-18.7924\times 10^{-6}s$.

### References

1. Bogdan G. Dimitrov, Manoff’s Generalized Deviation Equation and its Possible Applications in Celestial Mechanics and Relativistic Astrometry, Bulgarian Chemical Communications, Volume 42 (pp. 1–7) 2015.

GPS (Global Positioning System), based on satellite technology, has a wide range of applications – sea and land navigation, low Earth orbit (LEO) satellite determination, static and kinematic positioning etc. The breakthrough in geophysical applications of atomic clocks have stimulated the advanced research in relativistic geodesy. The obtained overall fractional frequency uncertainty of for atomic clocks enables to perform height difference on the Earth surface with unprecedented absolute level of 1 cm, providing the opportunity thus to compare different see-level changes – probably caused by melting continental ice sheets (satellite GOCE- “the Gravity field and steady-state ocean circulation explorer”).