## 2 Short-Term and Long-Term Collision Probability Calculation Models
### Short-Term Collision Probability Calculation Model
#### 2.1.3 Spatial Discrete Volume Element Proximity Analysis  
The probability of collision between space targets and constellation satellites after apogee-perigee filtering and time window filtering is high. If the pairwise calculation method is used to determine whether two targets collide, then the computational complexity will vary in second-order growth with the number of constellation satellites and the number of space targets. Thus, the collision probability calculation of LEO megaconstellations is unsuitable. In this section, the constellation shell is divided into several space volume elements, and the objects that may collide are filtered by judging the distance among the space objects in each volume element. At this time, the computational complexity is in linear growth with the number of constellation satellites and the number of space objects. This condition can greatly reduce the computational complexity of collisions between LEO megaconstellations and space debris, and improve computational efficiency. The space target orbit used in this study is defined under the J2000 inertial coordinate system, so the discrete volume element coordinate system is also defined as the J2000 inertial coordinate system. In the inertial coordinate system, the constellation shell is centered on the center of the Earth, and the spherical coordinates are established along the three directions of geocentric distance, longitude, and latitude. The discrete volume element is the starting point \(r_{\min},\lambda_{\min},\varphi_{\min}\), and the space of the constellation shell is divided into several space volume elements in accordance with the interval \(\Delta h,\Delta\lambda,\Delta\varphi\). Figure 4 shows a schematic of a space volume element. In accordance with Figure 4, the coordinates of the space volume element in three directions are expressed as  
\[\left\{ \begin{array}{l}\widetilde{r}_{j}=r_{\min}+j\Delta h,j=0,1,2\ldots,J \\ \widetilde{\lambda}_{k}=\lambda_{\min}+k\Delta\lambda,k=0,1,2\ldots,K\\ \widetilde{\varphi}_{l}=\varphi_{\min}+l\Delta\varphi,l=0,1,2\ldots,L\end{array} \right. \tag{8}\]  
Figure 3: Projection of the space target orbit on the ground. The center of the space volume element is expressed as  
\[\left\{ \begin{array}{l}r_{j}=\widetilde{r}_{j}-\Delta h/2,j=1,2\ldots,J\\ \lambda_{k}=\widetilde{\lambda}_{k}+\Delta\lambda/2,k=1,2\ldots,K\\ \varphi_{l}=\widetilde{\varphi}_{l}+\Delta\varphi/2,l=1,2\ldots,L\end{array} \right. \tag{9}\]  
The volume element division needs to consider factors such as calculation amount, storage amount, and accuracy. The larger the volume element, the less the number of traversed volume elements, the clearer the trajectory of the space target in the volume element, and the greater the number of targets appearing in the same volume element at the same time, resulting in an increase in the computational complexity of collision discrimination between targets. The smaller the discrete volume element, the fewer objects appear in the same volume element, the more time it takes to traverse the volume element, and the trajectory points of spatial objects in the volume element may be skipped, resulting in a decrease in computational accuracy. Therefore, setting the volume element size reasonably becomes the key to the approach analysis. The orbital planes of LEO megaconstellations are uniformly distributed along the right ascension of ascending node direction, and the phase and latitude differences between satellites in the orbital plane are equal. A parameter slightly smaller than the reference standard is selected to divide the volume element while ensuring a small number of LEO megaconstellation satellites in the volume element. The volume of the volume element is increased so that the trajectory continuity of the space target in the volume element is stronger.