## 3 Ranging Error Source Analysis and Modeling
### Analysis and Modeling of LEO Satellite Orbit Disturbance  
LEO satellites are affected by various forces during their movement around the Earth. These forces can be divided into two categories: conservative forces and nonconservative forces (divergent forces). The former can be described by a \"potential function\", while the latter force system does not have a \"potential function\" and can only use the expressions of these forces directly. Applying [PERSON]'s second law to obtain the motion equation of an artificial satellite is as follows [34]:  
\[\tilde{r}=F_{TB}+F_{NB}+F_{NS}+F_{TD}+F_{RL}+F_{SRP}+F_{AL}+F_{DG}+F_{TH} \tag{16}\]  
where \(F_{TB}\) is the two-body gravitation, the attraction of the center of the Earth to the LEO satellite; \(F_{NB}\) is the gravitational attraction of the sun, moon and other planets, except the Earth on the LEO satellite; \(F_{NS}\) is the gravitational attraction of the non-spherical part of the Earth to the LEO satellite; \(F_{TD}\) is the change in the Earth's gravitational force on the LEO satellite caused by the Earth's tide; \(F_{RL}\) is the influence of the relativistic effect; \(F_{SRP}\) is the force of sunlight pressure on the LEO satellite; \(F_{AL}\) is the pressure of the Earth's infrared radiation and the Earth's reflected light on the LEO satellite; \(F_{DG}\) is the resistance of the Earth's atmosphere to LEO satellites; and \(F_{TH}\) is other forces acting on the LEO satellite, such as the satellite attitude control force. For a specific problem, we do not need to consider all the mechanical factors. When the accuracy requirements (denoted as \(\kappa^{*}\)) are given, the corresponding perturbation factors should be estimated:  
\[\kappa=\frac{F_{\kappa}}{F_{0}} \tag{17}\]  
where \(F_{\kappa}\) is a certain kind of perturbation force mentioned above, \(F_{0}\) is the central gravity and \(\kappa\) is the magnitude of the perturbation force relative to the gravity of the central body. Table 2[34] shows the main perturbation magnitude data for four typical GEO satellites, IGSO satellites, MEO satellites and LEO satellites. We follow the principle of selection of perturbation factors given in references [35]. For the SpaceX constellation, we select the perturbation accuracy \(\kappa^{*}=10^{-7}\). For conservative and nonconservative forces, there are the following selection principles:  
1. If the perturbation force is conservative and if it satisfies  
\[2\kappa(n\Delta t)\geq\kappa^{*} \tag{18}\] \begin{table}
\begin{tabular}{l l l l l} \hline  & **GEO** & **IGSO** & **MEO** & **LEO** \\ \hline Earth aspheric perturbation & \(3.68\times 10^{-5}\) & \(4.00\times 10^{-5}\) & \(9.27\times 10^{-5}\) & \(1.59\times 10^{-3}\) \\ Solar gravitational perturbation & \(1.05\times 10^{-6}\) & \(7.42\times 10^{-6}\) & \(2.96\times 10^{-6}\) & \(4.34\times 10^{-8}\) \\ Lunar gravitational perturbation & \(2.02\times 10^{-5}\) & \(2.10\times 10^{-5}\) & \(5.73\times 10^{-6}\) & \(7.48\times 10^{-8}\) \\ Solar pressure perturbation & \(4.49\times 10^{-8}\) & \(4.49\times 10^{-8}\) & \(1.93\times 10^{-8}\) & \(1.01\times 10^{-9}\) \\ Solid tidal perturbation & \(3.19\times 10^{-10}\) & \(1.69\times 10^{-10}\) & \(4.01\times 10^{-10}\) & \(7.34\times 10^{-9}\) \\ Perturbation of relativistic effect & \(3.17\times 10^{-10}\) & \(3.17\times 10^{-10}\) & \(4.82\times 10^{-10}\) & \(2.01\times 10^{-9}\) \\ Atmospheric drag perturbation & \(5.04\times 10^{-21}\) & \(5.42\times 10^{-15}\) & \(7.20\times 10^{-15}\) & \(3.02\times 10^{-7}\) \\ \hline \end{tabular}
\end{table} 
Table 2: The magnitude of the perturbation force of several typical orbiting satellites. then the perturbation factor must be considered. For the situation that only causes a short period of change, the condition becomes:  
\[2\kappa\geq\kappa* \tag{19}\]  
2 If the perturbation force is a nonconservative force (dissipation force) and if it satisfies  
\[\frac{3}{2}\kappa(n\Delta t)^{2}\geq\kappa* \tag{20}\]  
where \(n\Delta t\) is the arc length experienced by the satellite movement.