# Atmospheric Density Inversion Based on Swarm-C Satellite Accelerometer
## 2 Materials and Methods
### Method and Model
#### 2.2.2 Swarm-C Satellite Cone Shadow Modeling  
A cylindrical shadow model is used in the existing satellite shadow area modeling [27]. Both the Sun and the Earth are regarded as circular spheres. The Sun's rays are considered to be parallel to the line between the center of the Sun and the center of the Earth's sphere, with the Earth blocking the Sun's rays to create a cylindrical shadow area. However, the radius of the Sun is 109 times that of the Earth. Therefore, the Sun's rays are non-parallel, so the shadow area of the cylindrical model can be further divided into the main shadow area, penumbra area, and artifact area. However, because the artifact area is very far from the Earth's center, the LEO satellites studied in this paper will not involve this area. Therefore, the shadow can be divided into penumbra and umbra to simplify the original cone shadow model. The difficulty in building the simplified conical shadow model lies in the treatment of the transition zone of the satellite from principal shadow to penumbra to full-shadowed or from full-shadowed to penumbra to principal shadow. The simplified conical shadow model shown in Figure 6 does not consider the refraction of the atmosphere and assumes that the solar illumination is smoothly varying. After applying the simplified conical shadow model with the Earth occluding the Swarm-C satellite for the experiment, the acceleration caused by the light radiation pressure along the orbital direction, the cross-orbit direction, and the radial direction, when the Swarm-C satellite leaves the penumbra region and enters the penumbra region can be calculated [27; 28]. In the simulated orbit, the light radiation pressure acceleration in three directions is shown in Figure 7. \begin{table}
\begin{tabular}{l c c c c} \hline
**Panel** & **Area** & **X** & **Y** & **Z** \\ \hline Nadir 1 & 1.540 & 0.0 & 0.0 & 1.0 \\ \hline Nadir 2 & 1.400 & \(-0.19766\) & 0.0 & 0.98027 \\ \hline Nadir 3 & 1.600 & \(-0.13808\) & 0.0 & 0.99042 \\ \hline Solar Array +Y & 3.450 & 0.0 & 0.58779 & \(-0.80902\) \\ \hline Solar Array \(-\)Y & 3.450 & 0.0 & \(-0.58779\) & \(-0.80902\) \\ \hline Zenith & 0.500 & 0.0 & 0.0 & \(-1.0\) \\ \hline Front & 0.560 & 1.0 & 0.0 & 0.0 \\ \hline Side Wall +Y & 0.753 & 0.0 & 1.0 & 0.0 \\ \hline Side Wall \(-\)Y & 0.753 & 0.0 & \(-1.0\) & 0.0 \\ \hline Shear Panel & 0.800 & 1.0 & 0.0 & 0.0 \\ Nadir Front & 0.800 & \(-1.0\) & 0.0 & 0.0 \\ \hline Shear Panel & 0.800 & \(-1.0\) & 0.0 & 0.0 \\ Nadir Back & 0.800 & 0.0 & 1.0 & 0.0 \\ \hline Boom +Y & 0.600 & 0.0 & \(-1.0\) & 0.0 \\ \hline Boom Zenith & 0.600 & \(-0.23924\) & 0.0 & \(-0.97096\) \\ \hline Boom Nadir & 0.600 & 0.22765 & 0.0 & 0.97374 \\ \hline \end{tabular}
\end{table} 
Table 1: Normal vectors of each panel of Swarm-C satellite. It can be seen from Figure 7 that the satellite is completely covered from about 0-20 s. The satellite's along-track direction, cross-track direction, and radial radiation pressure acceleration are all 0, and the satellite is in the umbra area at this time. From 20 s to 40 s, the satellite gradually leaves the state of complete shading and enters the state of partial illumination. At this time, the acceleration curve changes, and there is a positive or negative light radiation acceleration, and the satellite is in the penumbra. With the gradual decrease of the occulted area of the solar disk and the continuous increase of the shadow coefficient of the penumbra, the absolute value of the pressure acceleration of the light radiation received by satellite is also increasing. This process lasts for about 40 s. After 80 s in the figure, the satellite ultimately enters the illuminated area, that is, the shadowless area. At this time, the pressure velocity of the illuminated radiation received by the satellite reaches an almost constant value. The satellite's cross-track direction has the largest change in light radiation pressure for the three directions. In contrast, the satellite's along-track and radial directions are relatively close to the change.