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# Relativistic Clock Correction

Fundamentals
Title Relativistic Clock Correction
Author(s) J. Sanz Subirana, J.M. Juan Zornoza and M. Hernández-Pajares, Technical University of Catalonia, Spain.
Level Intermediate
Year of Publication 2011

The rate of advance of two identical clocks, placed one in the satellite and the other on the terrestrial surface, will differ due to the difference of the gravitational potential (general relativity) and to the relative speed between them (special relativity). This difference can be split into [1]:

• A constant component that only depends on the nominal value of the semi-major axis of the satellite orbit, which is adjusted modifying (in factory) the clock oscillating frequency of the satellite[footnotes 1]:


• A periodical component due to the orbit eccentricity (that must be applied by the user receiver software):


where  and  are the satellite position () and velocity () vectors, in an inertial system[footnotes 2]. The scalar  can be evaluated either in a CRS or TRS (i.e., ECEF) system. Notice that in a ECEF system, the earth rotation  should be discounted from , but it cancels in the scalar product with .

Unlike in GPS, relativistic corrections to GLONASS orbits eccentricity are transmitted within the navigation message into the satellite clock corrections (, ). Thence, (2) is not needed with such broadcast message [1].

Figure 1 illustrates the effect of neglecting the relativistic correction given by equation (2) on the user position. As it is shown, range errors up to  meters and vertical errors over  meters can be experienced when neglecting this correction.

 First row shows the horizontal (left) and vertical (right) positioning error using (blue) or not using (red) the relativistic correction 2. The variation in range, in meters units, is shown in the second row at left.

## Notes

1. ^ Being , one has  thus the satellite must use . Notice that  is the frequency "emitted" by the satellite and  is the one "received" on the terrestrial surface, i.e., an apparent increase of the frequency is of . That is, the clock on satellite appear to run faster () than on ground [note: ]. This effect is corrected (in factory) decreasing the oscillating frequency of the satellite by this amount .
2. ^ Note: Over the osculating orbit, . Thence, , where  the (geocentric) gravitational constant,  the speed of light in a vacuum,  and  the semi-major axis and eccentricity of the osculating orbit and  the eccentric anomaly.

## References

1. ^ a b Ashby, N., 2003 Relativity in the Global Positioning System