# Research on the Efficient Space Debris Observation Method Based on Optical Satellite Constellations
## 2 Constellation Design
### Model Establishment
Since the Sun is far from the Earth, the solar beam is regarded as parallel light and the sunlight that is obscured by the Earth produces a cylindrical shadow [19], as shown in Figure 3.  
The following formula can be used to discover whether the grid point is in the shadow of the Earth. Satellites are not visible to space objects when they are in the Earth's shadow zone, and satellites are visible to space objects when they are not in the Earth shadow zone. The value of the cylindrical shadow factor is \(-1\) when the space object is in the shadow zone, otherwise it is 1. \[\left\{ \begin{array}{l}-1,\psi>\frac{\pi}{2}\&\&r<\frac{R_{e}}{\sin\psi}\\ 1,\text{else}\end{array} \right. \tag{2}\]  
where \(r\) is the connection between the grid points and the center of the earth, \(r^{\prime}\) is the connection between the Sun and the center of the Earth, and \(\psi\) is the angle between \(r\) and \(r^{\prime}\). The angle \(\psi\) can be expressed using the following formula. \[\cos\psi=\sin\delta_{sun}\sin\delta_{obj}+\cos\delta_{sun}\cos\delta_{obj} \cos(\alpha_{obj}-\alpha_{sun}) \tag{3}\]  
where \(\alpha_{sun}\) and \(\delta_{sun}\) are the right ascension and declination of the Sun, \(\alpha_{obj}\) and \(\delta_{obj}\) are the right ascension and declination of the space objects. The position of the Sun changes with the seasons, so it is necessary to obtain multiple coordinate positions of the Sun, calculate the constellation's space coverage performance for each position and identify the average. You can get the average coverage performance of the constellation in a year. In this model, the four positions of the Sun at the spring equinox (\(0^{\circ},0^{\circ}\)), the summer solstice (\(90^{\circ},23^{\circ}26^{\prime}\)), the autumnal equinox (\(180^{\circ},0^{\circ}\)), and the winter solstice (\(270^{\circ},-23^{\circ}26^{\prime}\)) were taken to calculate the observation performance of the space objects by the constellations at different positions of the Sun. Figure 3: Cylindrical shadow model. (4) Visibility function  
Define the visual function of F: When the sensor is visible to the grid points, \(F>0\) is the opposite of \(F<0\). The sub-apparent function is defined by the two conditions of Formula (1), respectively. Define the visibility function of F: When the sensor is visible to the grid points, \(F>0\), and vice versa \(F<0\). According to the two conditions in Formula (1), the sub-visibility functions are defined separately as shown in the following formula [20]. \[f_{1}=\varphi-\beta \tag{4}\]  
\[f_{2}=\frac{L_{\max}-d_{st}}{L_{\max}} \tag{5}\]  
Define another sub-visibility function based on the Earth's shadow condition as shown in the following formula. \[f_{3}=\left\{ \begin{array}{l}-1,\Psi>\frac{\pi}{2}\delta r<\frac{R_{c}}{\sin \Psi}\\ 1,\text{else}\end{array} \right. \tag{6}\]  
It can be seen that when the values of the three Formulas (4)-(6) are greater than zero at the same time, the satellite can observe the grid point. Define the visibility function as the following. \[F=\min\{\text{sign}(f_{1}),\text{sign}(f_{2}),\text{sign}(f_{3})\} \tag{7}\]  
Define the visibility for each grid area as its visual value, denoted by \(C_{j}\), which is 1 if the satellites are visible to the grid, otherwise it is 0. The visual value of each grid can be calculated using Formula (8). \[C_{j}=\left\{ \begin{array}{l}1,(F>0)\\ 0,(F<0)\end{array} \right. \tag{8}\]  
Visual value calculation rules of the grid: When the same grid can be detected by one satellite in the simulation time, the visual value of the satellite to the grid point is 1. When it is detected by the second satellite, the visual value is \(1/2\), and so on. When it is detected by the \(n\)-th satellite, the visual value is \(1/n\). (5) Constellation period calculation  
For a specific grid area, the time interval between the recurrence of the corresponding satellite constellation geometry is the coverage period of the constellation to the grid area, and the smallest repeat interval is called the minimum coverage recurrence period.