# Research on the Efficient Space Debris Observation Method Based on Optical Satellite Constellations
## 2 Constellation Design
### Theoretical Analysis  
The constellation design was based on the walker-\(\delta\) constellation [21; 22]. All the satellite orbits in the constellation were circular orbits, with the same orbital altitude and inclination. The right ascension of all the orbital planes were evenly distributed. Through the establishment of the constellation airspace coverage model, using the appropriate objective functions and constraint conditions and optimizing the design using the genetic algorithm, the best constellation parameters that met the requirements were obtained, and the constellation configuration was determined. This constellation design mainly considered the coverage of the airspace in the orbital altitude range of 300-1000 km.  
The assessment of the constellation's coverage performance in the target airspace was measured in terms of time and space. At the same time, the distribution density of space objects in the grid area corresponding to different right ascensions, declinations, and altitudes in the airspace was considered, as shown in Figure 4. The resulting comprehensive coverage of the constellation was the ratio of the statistical sum of the volume, time, and density of all the observable grids of the constellation to the corresponding total.  
The comprehensive coverage performance of the constellation can be expressed by the following formula, which is also used as the objective function of the constellation's optimization design.  
\[M_{s}=-\frac{\sum\limits_{j=1}^{s}\left(\rho_{j}V_{j}\frac{T_{C}}{l=1}^{T_{C}} d_{t}C_{j}\right)}{\sum\limits_{j=1}^{s}\rho_{j}V_{j}T_{C}} \tag{10}\]  
In the above formula, \(\rho_{j}\) is the spatial debris density corresponding to each grid area, \(V_{j}\) is the volume of each grid area, \(d_{t}\) is the simulation time step, \(C_{j}\) is the apparent value of each grid, \(S\) is the total number of grids, \(T\) is the number of satellites in the constellation, and \(T_{C}\) is the constellation period.  
The constraints are as follows.  
\[\begin{array}{l}12<T<24,\;3\leq P\leq 6,\;F=1\\ 0\leq i\leq\pi,\;300\;\mathrm{km}\leq a-R_{e}\leq 1000\;\mathrm{km}\\ 0\leq\Omega_{0},\Omega_{m}\leq 2\pi,\;0\leq M_{0,0},M_{m,n}\leq 2\pi\end{array} \tag{11}\]  
Finally, the constellation configuration and its parameters were determined as the following parameters: \(T/P/F\), \(a\), \(i\), \(\Omega_{1}\), \(M_{1,1}\).  
Here, \(T\) is the number of satellites in the constellation, \(P\) is the orbital plane number of the constellation, \(F\) is the phase factor, \(a\) is the orbital radius of the satellite, \(i\) is the orbital inclination, \(\Omega_{1}\) is the right ascension of the ascending nodes of the fiducial satellite in the constellation, \(M_{1,1}\) is the mean anomaly of the fiducial satellite, and \(R_{e}\) is the earth radius.  
According to the walker-\(\delta\) constellation configuration, the positions of the satellites in the constellation can be represented by the following formula [23].  
\[\left\{\begin{array}{l}\Omega_{m}=\Omega_{1}+(m-1)\frac{3\delta 0}{P}\\ M_{m,n}=M_{1,1}+(m-1)F\frac{3\delta 0}{T}+(n-1)P\frac{3\delta 0}{T}\end{array}\right. \tag{12}\]  
Here, \(\Omega_{m}\) is the right ascension of the ascending nodes of the \(m\)-th orbital plane in the constellation and \(1\leq M\leq P\), \(M_{m,n}\) is the mean anomaly of the \(n\)-th satellite on the \(m\)-th orbital plane, \(1\leq n\leq\frac{T}{P}\).  
Figure 4: Distribution density of the space objects at different orbital altitudes.