## 4 GPS L1/L5 Receiver: L5 Pattern Assumption
### Navigation Solution  
#### 4.4.1 Least Squares Estimation  
Figure 29 shows the zoomed least squares navigation solution for the three assumed L5 patterns. As we can see from this figure, for the entire MTO, when we assume L5 pattern 1 the navigation solution has lower error, while the largest error occurs when we assume L5 pattern 3. Indeed, this was expected as the number of satellites is bigger for L5 pattern 1, leading to better GDOP values. In the case of L5 pattern 3, the relative geometry between the receiver and the transmitters is quite bad, quite often reaching values above 10,000 at the end of trajectory (after 330,000 km), as explained in Section 4.2. Moreover, also the GDOP of L5 pattern 2 has more peaks compared to the one of L5 pattern 1, leading to worse positioning accuracy. In Table 7 are shown the quantitative statistics of the navigation solution, for the three defined portions and the full MTO. The statistics for the full MTO confirm what can be seen in Figure 29. However, as shown in Table 7, the statistics for the entire MTO are strongly affected by the third portion, where the navigation performance is much worse for the third defined L5 pattern. Indeed, in the third portion the mean and standard deviation (\(1\sigma\)) of the position error estimation are less than 800 m, less than 1.6 km and several km for L5 pattern 1, L5 pattern 2, and L5 pattern 3, respectively. Also, the maximal error reflects a very big difference; indeed, it is only 6.6 km for the first pattern, more than 22 km for the second one, while for the last pattern it is more than 2000 km. For the first and the second portion, respectively, the difference of the position error estimation is about 0.2 m and less than 20 m among the three assumptions, in terms of mean and standard deviation (\(1\sigma\)). From Table 7 we notice that the velocity error statistics also reflect what was already shown in the position error statistics. Figure 28: Max code tracking thermal noise range error for the assumed MTO. The thermal noise range error is computed for each available PRN and the max is plotted. Even though the differences of the error statistics are smaller in absolute value, we can still notice that the smaller availability in the case of L5 pattern 3 is reflected in higher error values for the filtered solution in the third assumed portion. Once more, the differences of the position estimation error in the first two portions are very small among the three different assumptions (always less than 1 m in terms of mean and standard deviation), while in the third portion the differences are significant. Indeed, in terms of standard deviation, for the third portion, the errors are approximately 15 times more and 5 times more for the third assumed L5 pattern compared to the first and second assumed L5 patterns, respectively. On the other hand, the velocity statistics are smaller than 0.01 m/s in the first two portions. In the third portion, the difference is smaller than 0.04 m/s in terms of mean and standard deviation and smaller than 0.13 m/s in terms of maximal error. Obviously, filtering the GPS measurements with an orbital filter reduces the impact of the transmitter pattern assumption on the velocity error solution. Number of Available Global Navigation Satellite System (GNSS) Observations for Each Receiver Antenna Elevation  
In our previous studies, as well as here, we assumed there was always 10 dB antenna gain at the receiver, which could be provided by one steerable antenna or more than one antenna placed on different faces of the spacecraft (as assumed in [19]). In this section, assuming the L5 pattern 3 (which is the most conservative among the considered antenna patterns defined in Section 2.3), we analyzed the elevations at which the available signals are received by the receiver's antenna. It is important to highlight that in our simulations, as illustrated in Figure 31, the spacecraft that hosts the receiver is assumed to be pointing towards Earth, with the same body axis always directed towards the earth's center.