# The Symmetry of GPS Orbit Ascending Nodes
## 3 Results  
The research algorithm consists of two basic levels. The first one processes data from a single SEM almanac and determines the epoch-specific descriptive parameters. To calculate the instantaneous angular separation d\(\Omega\) of a particular GPS satellite, it is necessary to indicate a reference quantity. In the present study, a robust estimation method and the least-squares method with geometric conditions for the adjusted observations were used. The obtained results were saved to separate files grouped with respect to OPs and identified by the SVN and PRN identifiers of the satellites. Based on the time-tagged parameters of the satellites, the algorithm implemented second-level procedures using data from the research timeframe. Due to the large amount of data processed, here are presented the results on one OP (D) standing out from the constellation. The interested reader will find detailed information on the other five OPs in the Appendix A. General conclusions about the GPS constellation are presented in the second part of this section. The analysis describes the satellite \(\Omega\) parameter values and the OP ANs' symmetry as a function of time. To improve the d\(\Omega\) separation data interpretation, the angular unit used in the study is degrees converted from the SEM almanacs' semicircles, and years converted from the GPS time notation are the time unit. The week was selected as the time unit for dot(\(\Omega\)). The results of the algorithm application are presented in Figure 5. The upper panel of the figure shows the values of d\(\Omega\) separations from the two-stage estimated position of the OP D AN over the period 1996-2020. The lower panel shows the synchronous values of the dot(\(\Omega\)) parameter of the satellites. An anomaly in orbital motion in the set of GPS satellites was detected. The SVN46 satellite was launched in October 1999 and is still an active part of the space segment. The position of SVN46 AN is delayed relative to the remaining OP D satellites and is systematically closing to OP C (Figure 2). Additional confirmation of the unique separation trend of the SVN46 satellite orbit was found in a significantly smaller value of dot(\(\Omega\)), which resulted in a slower shift of the orbit AN. While the remaining OP D satellites showed d\(\Omega\) separation not exceeding several degrees in most cases, SVN46 obtained a change of \(26.2^{\circ}\) from an initial value of \(-1.2^{\circ}\) in 2000 to \(-27.4^{\circ}\) in 2020. The first and second week number rollovers' vertical markers represent the midnights of 21-22 August 1999, and 6-7 April 2019, respectively. At the beginning of every 1024 th GPS week, the week number in the navigation message is set to zero since only 10 bits are reserved for the week number [29]. Based on the time-tagged values of the d\(\Omega\) parameter, the satellite AN separation trend from the OP was determined (Table 2). Each satellite in the list is described according to the current activity in the GPS space segment (SVA), satellite orbit AN separation rate from the OP (_a_), space mission duration (MD), number of SEM almanacs analyzed (n), and current d\(\Omega\) value. The graphical interpretation of the d\(\Omega\) trend in the research timeframe is shown in Figure 6.  
The results of the analysis indicate that the magnitude of the satellite AN separation from the OP does not exceed, in seven out of ten cases, the absolute value of 0.3\({}^{\circ}\)/year. The remaining three satellites obtained values of \(-\)0.45\({}^{\circ}\) (SVN45), 0.40\({}^{\circ}\) (SVN75), and \(-\)1.34\({}^{\circ}\)/year (SVN46). Among the three largest trend values is the satellite of block IIIA (SVN75, PRN18), launched on 22 August 2019. Two obtained separation values represented only 29.9% (SVN75) and 33.6% (SVN45) of the absolute value of the AN separation rate compared to the third anomalous SVN46.