# 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
The conventional values from the JGM-3 gravity model [28] are:  
\[\text{JGM}-3\text{ zonals}:J_{2}=-0.10826*10^{-2};J_{3}=0.25324*10^{-5} \tag{10}\]  
We can estimate a maximum value of the perturbation caused by \(J_{3}\) for an orbit of roughly 26,000 km by scaling the maximum periodic perturbation due to \(J_{2}\), which along track has a maximum amplitude of ca. 200 m:  
\[\frac{J_{3}}{J_{2}}200\approx 0.47\ m \tag{11}\]  
The period of the associated perturbation is \(1/3\) of the orbital frequency, which is close to 4 h. We suggest that both GPS-like and Glonass-like messages should account for such term. The changes in the respective messages which could accommodate the improvedgravity model are described by the following equations for a GPS-like ephemeris and a Glonass-like model. For GPS, denoting by \(\mathrm{uk}\) the true anomaly from the node, we need to introduce six additional coefficients \(c3_{uc}\), \(c3_{us}\), \(c3_{tc}\), \(c3_{is}\), \(c3_{rc}\), \(c3_{rs}\) which define the amplitude of the cosine and sine component of the third harmonic periodic perturbation in the along track, cross track and radial directions, respectively. This notation is similar to that adopted (\(c_{uc}\),..., \(c_{is}\)) to represent the amplitudes of the perturbations caused by the second harmonic. This approach was first proposed by [PERSON] et al. [29]. The total periodic perturbations due to \(J_{2}\) and \(J_{3}\) are then given by:  
\[\frac{de^{k}}{dt}=\mathcal{L}^{k}(a,e,I)U \tag{3}\]0 \[\frac{de^{k}}{dt}=\mathcal{L}^{k}(a,e,I)U \tag{3}\]1 (13) \[\frac{de^{k}}{dt}=\mathcal{L}^{k}(a,e,I)U \tag{3}\]2  
For Glonass, we simply need to modify the force function used in the numerical integration of the equations of motion in a co-rotating frame. The additional terms are marked in red. \[\frac{de^{k}}{dt}=\mathcal{L}^{k}(a,e,I)U \tag{3}\]3 \[\frac{de^{k}}{dt}=\mathcal{L}^{k}(a,e,I)U \tag{3}\]4 (16) \[\frac{de^{k}}{dt}=\mathcal{L}^{k}(a,e,I)U \tag{3}\]5  
For the GPS-like approach, the amplitude of the 3 rd harmonic perturbations need to be estimated by including them in the set of solve for parameters (Equation (1)) and the relevant partials in the partial derivative matrix. These six additional parameters would justify an increase in length of the validity interval to 4 h or perhaps more, so that in a daily fit the total number of parameters to be estimated with 4-h arcs is 7 (Helmert) + 6 (arcs) * [3 (clocks) + 6 (orbit)] = 103 parameters, smaller than the 127 parameters of Section 2. Appropriate words would have to be defined in the Navigation message to allocate space to the six additional terms. For Glonass, there would be no change in the vector of state, nor in the navigation message. Only the function which describes the force needs to be modified. We have seen in Figure 2 that the fit interval of the first arc of G01 was extended to 4 h to test the possibility of extending to 4 h the nominal validity time of 2 h. Figure 14 shows the detail of the post-fit residuals in this 4-h arc. The blue dots represent the best fitting signal we computed by modifying the GPS navigation message according to Equations (12)-(14). The similarity is evident, but for this short arc it is difficult to draw a sufficiently motivated conclusion. \[\frac{de^{k}}{dt}=\mathcal{L}^{k}(a,e,I)U \tag{3}\]6 Zoom of the first 4 h of the post-fit residuals of G01 shown in Figure 2 (red diamonds) showing an oscillatory pattern ((**a**) X, (**b**) Y, and (**c**) Z). The blue dots represent an attempt to model the oscillation with a signal driven by the third zonal harmonics of the Earth’s gravity field \(J_{3}\).