## 2 Short-Term and Long-Term Collision Probability Calculation Models
### Long-Term Collision Probability Calculation Model
#### 2.2.4 Space Target Collision Disintegration Model  
During the long-term evolution of the constellation shell, space objects in the shell may collide, and collisions between different space objects will produce different amounts of space debris, causing different effects on the environment. Space debris is classified in accordance with the different sizes to study the specific influence of the collision of different space objects on the space environment, and the amount of space debris generated by  
Figure 9: Distribution of eccentricity along the semi-major axis of the track. collisions between different types of space objects and the distribution of debris in space are analyzed. (1) Collision type investigation  
In this study, the space debris was divided into two types: cataloged debris and complete objects. In accordance with the classification, six types of collisions between space objects, which are collisions between cataloged debris and collisions between cataloged debris and space intact objects, can be obtained to catalog collisions between fragments and loads, collisions between space intact objects, collisions between space intact objects and loads, and collisions between loads and loads. In accordance with the close encounter events of space objects published on the Spacetrack website, 1086 events were selected for statistical analysis, and the distribution of different types of collision events was studied [20]. Figure 10 shows the statistics for the rendezvous target types and the numbers of rendezvous. In accordance with the statistics in Figure 10, the close encounters between cataloged fragments accounted for 79.19% of the total number of encounters, the collisions between cataloged fragments and complete objects were 7.64%, and the collisions between cataloged fragments and loads were 12.34%. The collisions between loads were 0.74%, and the collisions between loads were 0.09%. Therefore, the proportions of the collision probability between various types of space objects to the total collision probability were assumed to be as follows: the collision probability between cataloged fragments accounts for 79.19% of the total, the collision probability between cataloged fragments and complete objects accounts for 19.98% of the total proportion, and the collision probability between complete objects and complete objects accounts for 0.74% of the total proportion. (2) Spacecraft breakup model  
The amounts of debris produced by the collisions of different types of space targets are different. This condition is mainly because the collision energy is different. The calculation of the collision energy can be expressed as  
\[E_{p}=\frac{m_{p}v_{r}^{2}}{2m_{t}} \tag{48}\]  
where \(E_{p}\) represents the collision energy, \(m_{p}\) is the target with smaller mass, \(m_{t}\) is the target with larger mass, and \(v_{r}\) is the relative motion speed. Under the condition of the same  
Figure 10: Rendezvous target types and rendezvous statistics. relative collision speed, the collision energy is mainly related to the mass between the two collision targets. When the collision energy is small, the collision will not cause the target to disintegrate. At present, the two targets can be completely disintegrated when the energy generated by the collision of the center of mass is greater than 40 kJ/kg [34]. The collision and disintegration model of space targets adopts the American NASA collision disintegration model.