# Broadcast Ephemeris with Centimetric Accuracy: Test Results for GPS, Galileo, Beidou and Glonass
## 5 Improving the Broadcast Model for GPS-like and Glonass-like Messages  
We have seen indications that both the GPS-like and Glonass-like navigation messages have a potential for high accuracy, in the sense that the model coefficients can be tuned to a precise ephemeris with centimetric rms spread relative to a precise SP3 ephemeris. At these levels of accuracy, it is meaningful to ask if the model implemented in both types of Broadcast Ephemeris is adequate to such accuracies. For GPS-like messages, we have secular perturbations in inclination and node, and periodic perturbations in mean anomaly, in the radial direction and inclination. The period is one half the Keplerian period, roughly 6 h, implying that the periodic perturbations caused by the \(J_{2}\) part of the Earth's gravity field are modeled. For Glonass-like messages, the Glonass ICD (Interface Control Document) prescribes a force field which includes, for the Earth's gravity, the monopole and quadrupole (\(J_{2}\)) components. Figure 13: Plots of the differences between IGS high rate clocks and the prediction of our clock polynomials based on 15 min sampling. Update rate of the clock polynomials are 2 h for (**a**) G01, (**b**) E01, (**c**) 1 h for R01, (**d**) C07, and (**e**) C12. The mean and rms of the residuals are provided for each satellite. To investigate the effect of higher order terms of the gravitational potential on the orbital elements, we refer to [27]. The differential equations relating a disturbing potential \(U\) to the temporal changes of the orbital elements \(\varepsilon^{k}\) (\(k=1\):\(6\)) have the general form:  
\[\frac{de^{k}}{dt}=\mathcal{L}^{k}(a,e,I)U \tag{3}\]  
where \(\mathcal{L}^{k}(a,e,I)\) is a linear differential operator dependent on the semimajor axis \(a\) eccentricity \(e\) and inclination \(I\). If we assume that interaction of perturbations can be ignored (this is appropriate for all the harmonics except \(I_{2}\)), then we can write:  
\[\varepsilon^{k}=\varepsilon^{k}{}_{0}+\sum_{l=2}^{\infty}\sum_{m=0}^{l} \delta\varepsilon^{k}{}_{lm} \tag{4}\]  
where \(\delta\varepsilon^{k}{}_{lm}\) is the perturbation of the k-th orbital element due to the harmonic \(C_{lm}\) and \(\varepsilon^{k}{}_{0}\) is the unperturbed orbital element. The general form of the perturbed orbital elements is  
\[\delta\varepsilon^{k}{}_{lm}=Re\Bigg{[}\mathcal{L}^{k}(a,e,I)\sum_{p=0}^{l} \sum_{q=-\infty}^{\infty}C_{lm}A_{Impq}(a_{0},e_{0},I_{0})\frac{\varepsilon ^{[q_{0}-\pi/2]}}{\dot{\psi}_{0}}\Bigg{]} \tag{5}\]  
\[\dot{\psi}_{0}=(l-2p)(\omega_{0}+\dot{\omega}t)+(l-2p+q)M+m(\Omega_{0}+\dot{ \Omega}t-\dot{\theta}) \tag{6}\]  
\[\dot{\psi}_{0}=(l-2p)\dot{\omega}+(l-2p+q)n+m(\dot{\Omega}-\dot{\theta}) \tag{7}\]  
The coefficients \(A_{Impq}\) scale as \((a_{e}/a)^{l}\), \(e^{q}\) and \(sin^{p}(I)\), \(cos^{p}(I)\), and abs(\(q\)) < 5 typically. \(\dot{\psi}_{0}\), \(\dot{\psi}_{0}\) are the phase and frequency of the harmonic perturbation, and \(M,\dot{\omega},n,\dot{\Omega},\dot{\theta}\) are respectively the mean anomaly, the rate of the perigee, the mean motion, the rate of the ascending node and the Earth rotation rate. Secular terms arise when \(m=0\) and \((l-2p)=q=0\) that is for even zonal harmonics. Because the rate of perigee and node are of the order of \(10^{-3}n\) and the period is \(n\cong 2\dot{\theta}\), the period P of a perturbation is primarily determined by:  
\[\frac{2\pi}{P}\approx[2(l-2p+q)-m]\dot{\theta} \tag{8}\]  
It is observed that the amplitude of the harmonics \(C_{lm}\) decrease with the degree \(l\) according to the [PERSON]'s rule:  
\[|C_{lm}|\approx\frac{10^{-5}}{l^{2}} \tag{9}\]  
Equation (9) implies that, except for resonant terms, we have to consider small degree terms. For example, considering the \(J_{3}\) zonal (\(l=3\), \(m=0\)), for \(p=q=0\) we have perturbations with period close to 4 h. The \(J_{3}\) term, accounts for the pear-shaped figure of equilibrium of the Earth, that is, the lack of symmetry between north and south hemispheres.