## 2 Methods of Atmospheric Density Inversion for Low Earth
### Comparison of Three Inversion Methods  
According to the classification results of atmospheric density acquisition methods in Table 1, a comparative analysis is conducted from the aspects of acquisition principle, inversion process, required data, and method advantages. The instrument detection method derives atmospheric density from parameters such as temperature, humidity, and barometric pressure through sensors integrated into aviation equipment. Gradiometers and atmospheric density probes, such as those carried on low-orbiting satellites, are capable of covering a large area of the Earth's surface and detecting it at high altitudes, directly acquiring atmospheric density, but the launch and maintenance costs are high, and there may be time delays in data processing and transmission, which affects the ability to monitor it in real-time. The inversion of atmospheric density from precision orbital data provided by satellites includes two methods, the half-length axis decay method and the energy conservation method. In Low Earth Orbit satellite operation, atmospheric drag force results in orbit height decline, usually with the satellite half-length axis of the change in the description of the orbit height change [28], and the half-length axis with the rate of change of time directly reflects the size of the atmospheric drag force, as the atmospheric drag force is related to the atmospheric density. The atmospheric drag force in Equation (1) is described by the change in the half-length axis per unit of time in Equation (5) [29]. \[\frac{da}{dt}=-\rho*\frac{C_{D}*A}{M_{sat}}v_{r}^{2}\frac{\left(1+2 ecosf+e^{2} \right)^{\frac{1}{2}}}{n\sqrt{1-e^{2}}} \tag{5}\]  
\[\frac{da}{dt}=\frac{a^{\prime}-a}{\Delta t} \tag{6}\] \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Method** & **Sensor Detections** & \multicolumn{2}{c}{**Precision Orbits**} & **Accelerometer Data** \\ \hline Theory & Instrument detection & \begin{tabular}{c} Semi-long axis \\ attenuation \\ \end{tabular} & \begin{tabular}{c} Conservation of \\ mechanical energy \\ \end{tabular} & \begin{tabular}{c} Acceleration \\ calibration \\ \end{tabular} \\ \hline Data & Atmospheric density & Positional velocity & \begin{tabular}{c} Gravity field model \\ Gravitational \\ potential \\ \end{tabular} & \begin{tabular}{c} Scale, bias, and \\ temperature factors \\ \end{tabular} \\ \hline Factors affecting & \begin{tabular}{c} Atmospheric \\ environment \\ signal source \\ \end{tabular} & \begin{tabular}{c} Quality and period \\ of orbital data \\ \end{tabular} & \begin{tabular}{c} Model assumptions \\ and parameter \\ \end{tabular} & \begin{tabular}{c} Instrument noise \\ and error \\ \end{tabular} \\ \hline Conditions of & Widespread & \begin{tabular}{c} Long-term \\ monitoring and \\ large-scale \\ \end{tabular} & \
\begin{tabular}{c} Scenes of \\ high-precision \\ inversion \\ \end{tabular} &
\begin{tabular}{c} Real-time monitoring \\ and dynamic \\ inversion \\ \end{tabular} \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Comparison of three methods for obtaining atmospheric density, based on these methods’ principles, accuracy-influencing factors, and applicable situations in different environments for obtaining atmospheric density. To calculate the decay rate of the half-length axis by the atmospheric drag, we need to consider the position and velocity state quantity of the LEO satellite at the beginning and end moments; the initial moment is designated as \(t_{0}\), the satellite state quantity is represented by (\(r_{0}\), \(v_{0}\)), and the corresponding instantaneous half-length axis is denoted as \(a_{0}\). Subsequently, the satellite is subjected to the central gravitational field in addition to all regenerative forces, resulting in a change in the satellite's position and velocity vector at the final moment. The satellite state quantity at the final moment, designated as \(t_{1}\), is recorded as (\(r_{1}\), \(v_{1}\)), with the corresponding instantaneous half-length axis noted as \(a_{1}\).