## 2 Methods of Atmospheric Density Inversion for Low Earth
### Comparison of Three Inversion Methods
The value of \(t_{1}\) is calculated as \(t_{1}=t_{0}+\Delta t\). During the interval of out, the atmospheric drag force causes a decay in the satellite's half-length axis, with the resulting change recorded as \(\Delta a\). In the following section, the mechanical model is employed to simulate the actual uptake force, with the satellite state quantity (\(r_{1}^{\prime}\), \(v_{1}^{\prime}\)) at the time of \(t_{1}\) being calculated based on the satellite state quantity at the initial moment of the motion. This is achieved through numerical integration within the interval of the mechanical model, which serves to remove the atmospheric drag force. The satellite's position and velocity vector at the final moment, denoted (\(r_{1}\), \(v_{1}\)), is then used to calculate the corresponding instantaneous half-length axis \(a_{1}^{\prime}\). The attenuation of the satellite's half-length axis by the atmospheric drag force during the time interval is then given by \(a_{1}^{\prime}-a_{1}\)[30]. Semi-long-axis decay is a kinetic method that requires orbital extrapolation when calculating the change in the semi-long-axis, which not only adds to the calculation but also introduces an error in the numerical integration. The error in numerical integration is related to the time of extrapolation, and since the extrapolation time is very short, the error in numerical integration introduced is small. According to the law of conservation of energy, LEO satellites are subjected to central gravitational and regenerative forces in their operation, and the mechanical energy changes under the action of non-conservative regenerative forces, which are mainly the atmospheric drag force and the light pressure regenerative force, which can be inverted to the atmospheric density according to the attenuation of the overall energy occurring in the adjacent moments of LEO satellite operation. The LEO satellite ephemeris moment has the total energy calculation shown in Equation (7), where \(V_{SUN}\) and \(V_{MOON}\) are the effects on the LEO satellite by the sun and the moon's gravitational potential energy calculated according to the gravitational potential formula, respectively [29; 31]. \(V_{R}\) is the rotational potential energy due to the rotation of the Earth [32], while \(V\) is the gravitational potential energy of the satellite corrected using the model [33]. \(E_{n}\) is the non-conservative force including the atmospheric drag force and the optical pressure uptake force doing work expressed as an integral of the non-conservative force power in time (8). The integral over time of the atmospheric drag force can be expressed as the work done by the atmospheric mean drag force over time (9). The atmospheric density can be calculated by bringing Equations (1) into (9). \[E_{0}=\frac{1}{2}|v|^{2}-V_{SUN}-V_{MOON}-V_{R}-V-E_{n} \tag{7}\]  
\[E_{n}=\int a_{drag}*vdt+\int a_{solarpress}*vdt \tag{8}\]  
\[\int a_{drag}*vdt=a_{drag}|v|\Delta t \tag{9}\]  
The accuracy of onboard accelerometers is very high, and gravity satellites CHAMP, GRACE, and SWARM are all equipped with this payload. Since the 1960s, satellites from the United States, Italy, and France have successively been equipped with onboard accelerometers for studying the dense modeling of the thermosphere atmosphere. On 15 July, 2000, the German Geological Research Center (GFZ) launched the CHAMP (CHAllenging Mini satellite payload) satellite with an orbital altitude of 456 km, which operated in orbit for ten years. On 17 March 2002, two identical satellites, named GRACE (Gravity Recoveryand Climate Experiment), developed in collaboration between NASA and the German Aerospace Center (DLR) and located approximately 220 km apart, were launched into orbit at an initial altitude of 485 km. As a continuation of the CHAMP satellite, the accuracy of ranging and accelerometers was improved by an order of magnitude. In 2009, GOCE (Gravity Field and Steady-State Ocean Circulation Explorer) was the world's first satellite to use gravity gradient measurement technology.