## 4 GPS L1/L5 Receiver: L5 Pattern Assumption
### Navigation Solution
#### 4.4.1 Least Squares Estimation
Note that this is not the case in a typical lunar mission; however, for the purpose of this analysis (which only aims at investigating the GNSS signals' characteristics and the achievable GNSS-based navigation accuracy) it is equivalent to assume that the spacecraft (despite its attitude) will be equipped by one antenna always pointing towards the earth to provide maximum gain. Also note that the elevation is expressed as the semi aperture of a cone, and thus is always positive. Figure 32 displays the number of available GNSS observations for each receiver's antenna elevation. Figure 33 displays the elevation values at which the four strongest signals are available at the receiver's antenna. While Figure 34 shows the PRNs available for each elevation of the receiver's antenna and for each altitude of the considered trajectory. Figure 35 finally reports a histogram of the available observations for each receiver's antenna elevation. This information can be useful in order to design an antenna system capable of providing a certain required gain (e.g., 10 dB in our assumptions), at least at the receiver elevations where a minimum of four signals can be processed. From Figure 32, we see that above 300,000 km all the signals are received between 0\({}^{\circ}\) and 5\({}^{\circ}\), while between 150,000 and approximately 300,000 km there is no available signal at elevations higher than 10\({}^{\circ}\), which means that above 150,000 km a high directivity antenna pointing towards the earth's center may be sufficient to process most of the signals. From Figure 33, we see that above 100,000 km, the four strongest signals are always received within 10\({}^{\circ}\) elevation and at the same time below the same \begin{table}
\begin{tabular}{l c c c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Portion**} & \multicolumn{4}{c}{**L5 Antenna Pattern 1**} & \multicolumn{4}{c}{**L5 Antenna Pattern 2**} & \multicolumn{4}{c}{**L5 Antenna Pattern 3**} \\ \cline{2-13}  & **1** & **2** & **3** & **All MTO** & **1** & **2** & **3** & **All MTO** & **1** & **2** & **3** & **All MTO** \\ \hline \multirow{3}{*}{\
\begin{tabular}{c} Position \\ error (m) \\ \end{tabular} } & **Mean** & 1.03 & 5.16 & 14.52 & 7.85 & 1.08 & 5.86 & 27.74 & 11.70 & 1.12 & 5.31 & 98.34 & 19.49 \\  & **Std** & 0.88 & 3.39 & 9.83 & 7.23 & 0.89 & 3.88 & 27.84 & 15.41 & 0.96 & 3.44 & 142.58 & 48.25 \\  & **Max** & 7.69 & 20.01 & 63.18 & 63.18 & 5.58 & 22.71 & 239.89 & 239.89 & 7.39 & 20.40 & 856.54 & 856.54 \\ \hline \multirow{3}{*}{
\begin{tabular}{c} Velocity \\ error \\ (m/s) \\ \end{tabular} } & **Mean** & 0.006 & 0.016 & 0.109 & 0.032 & 0.007 & 0.020 & 0.132 & 0.039 & 0.008 & 0.025 & 0.141 & 0.036 \\  & **Std** & 0.004 & 0.011 & 0.040 & 0.031 & 0.005 & 0.010 & 0.049 & 0.036 & 0.006 & 0.012 & 0.079 & 0.040 \\  & **Max** & 0.020 & 0.067 & 0.201 & 0.201 & 0.022 & 0.058 & 0.225 & 0.225 & 0.025 & 0.064 & 0.323 & 0.323 \\ \hline \end{tabular}
\end{table} 
Table 8: Statistics of the range-based orbital filter estimation error. altitude their powers never go below \(-159\) dBm, which is the SANAG receiver sensitivity. Therefore, in principle 10 dB gain from 0\({}^{\circ}\) to 10\({}^{\circ}\) elevation would already allow the processing of at least four signals. Between the GPS constellation altitude and 150,000 km, most of the signals are received within an elevation interval of 10\({}^{\circ}\)-30\({}^{\circ}\); however, as their power is already on average significantly higher than the signals received at higher altitudes (see, for instance, Figure 9), a lower gain for the receiver's antenna may still allow the tracking of a minimum number of signals. Finally, we see that, as expected, only for altitudes below the GPS constellation are there signals reaching the receiver between 90\({}^{\circ}\) and 180\({}^{\circ}\).