# Near-Real-Time Global Thermospheric Density Variations Unveiled by Starlink Ephemeris
## 2 Data and Methods
Conversely, the Maneuver phase (red data points) shows adjacent SMA differences exceeding 10 m, with the mean SMA gradually increasing. The irregular patterns observed in gray data points in the third day align with [PERSON]'s findings, which attribute this phenomenon to SpaceX's simplified orbital design strategy that considers only \(J_{2}\) perturbations [38]. Additionally, the mean SMA (pink data points) derived from the subsequent batch is also shown in Figure 2b. The decay rates of the Natural Decay phase differ from those of the reference batch, suggesting that each batch is generated from a new orbital extrapolation. Considering the accumulation of errors over time in extrapolation [38] and the updated features of the Starlink ephemeris, we employ a data integration approach in which subsequent batches replace overlapping arcs from previous data. This iterative process continues until a continuous ephemeris sequence is established, as illustrated in Figure 2c,d, where color-coded data represent different batches. Notably, the updated features of the Starlink ephemeris induce discontinuities at batch boundaries that may compromise maneuver detection accuracy. Therefore, we exclude data within transition zones from subsequent analyses. A satellite's orbital motion is governed by both conservative and non-conservative forces. From an energy conservation perspective, when a satellite is subjected exclusively to conservative forces--such as Earth gravitational field, tidal perturbations, gravitational influences from the Sun and Moon, and relativistic effects--its total mechanical energy remains constant. In contrast, when non-conservative forces come into play, including atmospheric drag, solar radiation pressure, and Earth's albedo pressure, the satellite experiences a gradual decrease in mechanical energy. This energy loss corresponds precisely to the work done by these non-conservative forces. Generally, for low-orbit satellites, atmospheric drag accounts for a major proportion of the total non-conservative force in most cases [1; 2]. Consider two consecutive epochs, the change in mechanical energy can be expressed as follows [28]:  
\[\Delta E=\left(V_{K_{2}}-V_{Sum_{2}}-\ V_{Moon_{2}}\right)-\left(V_{K_{1}}-\ V_{Sun_{1}}-V_{Moon_{1}}\right)-\int_{r_{1}}^{r_{2}} \overrightarrow{g}_{ECI}\cdot d\overrightarrow{r} \tag{1}\]  
In this equation, the subscripts 1 and 2 denote the parameters corresponding to the preceding and subsequent epochs, respectively. Here, \(\overrightarrow{r}\) represents the position of the satellite, and \(\overrightarrow{g}_{ECI}\) is the Earth gravitational force in the Earth-centered inertial frame. The term \(V_{K}\) is the satellite's kinetic energy, which can be calculated directly by the velocity \(v\) provided in the ephemeris through the equation \(V_{K}=\frac{1}{2}v^{2}\). \(V_{Sun}\) and \(V_{Moon}\) corresponded to the central gravitational potentials of the Sun and Moon [39]:  
\[V_{i}=GM_{i}\times\left(\frac{1}{s}-\frac{1}{\left|\overrightarrow{r}_{i} \right|}-\frac{\overrightarrow{r}\cdot\overrightarrow{r}_{i}}{\left| \overrightarrow{r}_{i}\right|^{3}}\right) \tag{2}\]where \(GM_{i}\) denotes the gravitational constant of the celestial body (Sun or Moon), \(s\) represents the distance between the satellite and the celestial body, and \(\overset{\rightarrow}{r}_{i}\) represents the geocentric position vectors of the celestial body. The positions of the Sun and Moon can be interpolated from the planetary and lunar ephemerides (DE440) [40] provided by the Jet Propulsion Laboratory. The last integral term on the right-hand side of Equation (1) represents the cumulative work performed by Earth's gravitational force on the satellite throughout its orbital motion. The Earth's gravitational force is computed through the geopotential spherical harmonic model [2] after gravitational potential coefficient corrections for Earth tide effects, including both solid Earth tides and ocean tides.