# The Symmetry of GPS Orbit Ascending Nodes
## 1 Introduction  
Studies on the space segments of global navigation satellite systems (GNSS) in most cases are focused on the accuracy of satellite orbit determination. Researchers have considered satellites in low Earth orbit (LEO), medium Earth orbit (MEO), and geostationary orbit (GEO) systems. [PERSON] et al. [1] showed the accuracy of determining orbits in the Multi-GNSS Experiment (MGEX). A study regarding the Galileo system was conducted by [PERSON] and [PERSON] [2]. [PERSON] et al. [3] published the results of analogous research on the GLONASS system. [PERSON] et al. [4] analyzed the Beidou system broadcast navigation messages. [PERSON] and [PERSON] [5] conducted a long-term analysis of satellite orbits based on Standard Product 3 (SP3). A similar study was carried out by [PERSON] et al. [6]. [PERSON] et al. [7] described broadcast and precise GPS ephemerides from a historical perspective. Some papers presented the results of research on the constellations of satellites. [PERSON] and [PERSON] [8] presented a comparison of the GLONASS system constellation satellite parameters. [PERSON] et al. [9] analyzed the Multi-GNSS constellation for the detection of anomalies using SP3 orbits. [PERSON] et al. [10] developed a method for determining the yaw attitude of Beidou satellites. Several authors conducted research based on almanacs. The accuracy of GPS almanacs relative to precise SP3 orbits was studied by [PERSON] and [PERSON] [11]. Almanac analysis was also performed by [PERSON] et al. [12]. This work presents the spatial relationships of the GPS orbits, with emphasis on the symmetry of the ascending nodes. The study complements the current knowledge in the field of satellite geodesy with a broader perspective on the GPS constellation. In a previous study, [PERSON] and [PERSON] [13] discussed the separation from the orbital plane D of the orbit of the Satellite Vehicle Number 46 (SVN46) transmitting the Pseudo-Random Noise \(11\) (PRN11) code. The authors indicated a systematic separation of the satellite orbit fromthe orbital plane. This long-term process resulted in an anomalous position relative to the GPS satellite constellation. Due to the limitations of the previous analysis, the present work was extended to all orbital planes and all almanacs available online. Excluding almanacs published before the 847 th GPS week, the source data consisted of 7035 System Effectiveness Model (SEM) almanac files. The SEM almanacs contain Keplerian elements and other GPS satellites' orbital parameters determined for a given epoch. The initial GPS constellation consisted of concept validation block I satellites launched into three 63\({}^{\circ}\) inclined orbital planes [14]. Hence, the comparative analysis based on the current constellation of six orbital planes with a 55\({}^{\circ}\) inclination used only satellites of block II and later. In the nominal form of the GPS space segment, the orbital planes are distributed symmetrically and separated by about 60\({}^{\circ}\) on the equator [15]. The satellite orbit parameters achieved for each launch are never perfect and deviate from nominal values within certain limits. For instance, SPS [16] informs that the achieved orbit inclination for any specific GPS satellite can be anywhere within 3\({}^{\circ}\) of the nominal inclination of 55\({}^{\circ}\). The time derivative of the longitude of the ascending node (\(\Omega\)) is a function of the achieved orbit inclination and orbit semi-major axis. As the GPS control segment does not remain passive due to changes in the constellation of satellites, some constellation tolerances and management practices need to be indicated. From time to time, orbit adjustments (thrusts) are performed to manage the constellation. Orbit maneuvers require a pause in the satellite mission as they cause discontinuities in the satellite track. The scheduled satellite outage times are provided by the control segment in the form of the Forecast Delta-V (FCSTDV) messages.