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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 :

• 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]:
$\frac{f_0'-f_0}{f_0}=\frac{1}{2} \left( \frac{v}{c} \right) ^2+\frac{\triangle U}{c^2} \simeq -4.464 \cdot 10^{-10} \qquad \mbox{(1)}$

• A periodical component due to the orbit eccentricity (that must be applied by the user receiver software):
$\Delta_{rel}=- 2\, \frac{\mathbf{r}^{sat} \cdot \mathbf{v}^{sat}}{c^2} \qquad \mbox{(2)}$

where ${\mathbf r}^{sat}$ and ${\mathbf v}^{sat}$ are the satellite position ($\displaystyle m$) and velocity ($\displaystyle m/s$) vectors, in an inertial system[footnotes 2]. The scalar ${\mathbf r}^{sat}\cdot{\mathbf v}^{sat}$ can be evaluated either in a CRS or TRS (i.e., ECEF) system. Notice that in a ECEF system, the earth rotation ${\boldsymbol \omega}_E \times {\mathbf r}^{sat}$ should be discounted from ${\mathbf v}^{sat}$, but it cancels in the scalar product with ${\mathbf r}^{sat}$.

Unlike in GPS, relativistic corrections to GLONASS orbits eccentricity are transmitted within the navigation message into the satellite clock corrections ($\displaystyle \tau_n$, $\displaystyle \gamma_n$). Thence, (2) is not needed with such broadcast message .

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 $13$ meters and vertical errors over $20$ 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.