# Broadcast Ephemeris with Centimetric Accuracy: Test Results for GPS, Galileo, Beidou and Glonass
## 3 Mathematical Model and Results for Glonass  
In the Glonass navigation message, the clock corrections of the satellite time to the Glonass time is, as is for GPS, expressed in terms of a second order polynomial. The instantaneous ECEF position and velocity of a Glonass satellite is instead computed by numerically integrating the equations of motion. These are conveniently formulated in terms of six ordinary differential equations of first order. The vector of state of the satellite (position and velocity) is therefore broadcast at a reference epoch Toe. The numerical integrator, normally a 4 th order Runge-Kutta, maps this vector of state from time Toe to any other epoch within the validity interval of the message. The force field consists of the gradient of the Earth's gravitational potential truncated to the second zonal harmonic \(j_{2}\) plus the centrifugal and Coriolis terms, since the equations of motion are integrated in a rotating frame. Glonass broadcasts three additional terms, the Lunisolar accelerations, which \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & **Tx (m)** & **Ty (m)** & **Tz (m)** & **Rx (mas)** & **Ry (mas)** & **Rz (mas)** & **Scale** \\ \hline Estimated & 0.26 & 0.02 & \(-0.70\) & \(-0.6\) & 1.9 & \(-2.8\) & \(1.25\times 10^{-8}\) \\
1 sigma formal error & 0.06 & 0.05 & 0.02 & 0.53 & 0.69 & 0.12 & \(1.17\times 10^{-9}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Helmert parameters relating the origin, orientation and scale of the Broadcast reference frame of C12 relative to the IGS14 frame of the SP3 precise ephemeris for SV C12, 2 January 2020. The scale factor accounts for the Center of Mass–Antenna Phase Center correction in the radial direction. Figure 9: Best fitting corrections to (**a**) the clock polynomial parameters, (**b**) Mean anomaly at Toe and amplitude of the cosine and sine perturbations along track due to the second zonal harmonic (Cuc, Cus), and amplitude of the cosine and sine perturbations in the (**c**) radial (Crc, Crs) and (**d**) cross track (Cic, Cis) directions. To convert to radians the scale factor \(26\times 10^{6}\) m should be used. Error bars are 1 sigma formal uncertainties. Satellite C12 for day 2 January 2020.  
are constant accelerations during the validity time of the broadcast message, normally 30 min. Tuning the broadcast parameters on a precise orbit requires therefore the adjustment of the clock and vector of state in arcs of at least 30 min. In the sample broadcast ephemeris data set analyzed here ([[https://igs.bkg.bund.de/root_ftp/MGEX/BRDC_v3/2020/002/brdm0020.20p_Z](https://igs.bkg.bund.de/root_ftp/MGEX/BRDC_v3/2020/002/brdm0020.20p_Z)]([https://igs.bkg.bund.de/root_ftp/MGEX/BRDC_v3/2020/002/brdm0020.20p_Z](https://igs.bkg.bund.de/root_ftp/MGEX/BRDC_v3/2020/002/brdm0020.20p_Z)), accessed on 10 October 2021), the clock drift and clock drift rate (\(a1\), \(a2\)) were zero, and the lunisolar accelerations were allowed to change in fixed increments. It is also important to note that the Glonass message constrains the terrestrial reference frame by providing ECEF initial coordinates and velocities. Therefore, for Glonass it is not necessary to estimate Reference Frame parameters. For the CoM to APC correction the z-bias of 2.450 m appropriate for Glonass M was used ([[https://files.igs.org/pub/station/general/pcv_archive/igs14_2056.atx](https://files.igs.org/pub/station/general/pcv_archive/igs14_2056.atx)]([https://files.igs.org/pub/station/general/pcv_archive/igs14_2056.atx](https://files.igs.org/pub/station/general/pcv_archive/igs14_2056.atx)), accessed on 10 October 2021).