## 2 Short-Term and Long-Term Collision Probability Calculation Models
### Long-Term Collision Probability Calculation Model
#### 2.2.1 Model-Simplifying Assumptions  
Educated guesses and rationalized assumptions about relevant variables and their ranges are required to simplify computational models [20]. The assumptions defined in this section are as follows:  
(1) All spatial objects are treated as spheres.  
(2) The objects in the initial environment of LEO are divided into two groups for estimation, where the first group is the \"complete objects\" with a size greater than 10 cm, such as orbiting satellites and abandoned complete payloads. The radius of these objects is uniformly assumed to be 1.9 m, and the mass is 1000 kg [21]. The second group is space debris objects with a size of less than 10 cm. The radius of these debris objects is uniformly assumed to be 10 cm, and the mass is 1.2 kg [29].  
(3) The calculation model for the average collision probability of an LEO megaconstellation in the shell adopts the \"PIB\" model [20]. The collision probability of the constellation in accordance with the density of space objects in the shell can be expressed as  
\[CR_{O-D}=\pi(r_{O}+r_{D})^{2}V_{Rrl}\rho_{O}\rho_{D}V \tag{29}\]  
where \(CR_{O-D}\) is the average collision probability of different targets, \(r_{O}\) is the outer envelope radius of the satellite, \(\rho_{O}\) is the satellite space density, \(V_{Rel}\) is the satellite relative target velocity, \(V\) is the shell volume, \(r_{D}\) is the space debris size, and \(\rho_{D}\) is the space debris density. The formula for calculating the density \(CR_{0-0}\) of the same space object in the shell is:  
\[CR_{O-D}=2\pi r_{0}^{2}\rho_{0}V_{Rel}(\rho_{0}V-1) \tag{30}\]  
(4) Assuming that the mass and cross-sectional area of the LEO megaconstellation satellites are \(M_{0}\) and \(A_{0}\), the mass and cross-sectional area of the new debris generated by the collision of constellation satellites in orbit satisfy the relational expression [30]:  
\[M_{0}[kg]=\left\{\begin{array}{l}62.013A_{0}^{1.13},A_{0}\geq 8.04\times 10^{- 5}m^{2}\\ 2030.33A_{0}^{1.5},A_{0}<8.04\times 10^{-5}m^{2}\end{array}\right. \tag{31}\]  
Figure 5: Long-term collision probability calculation model.  
(5) The collision disintegration model meets the NASA collision disintegration model.  
(6) The collision only produces two types of threatening space debris with a radius of 10 cm and a characteristic length sufficient to meet the collision disintegration energy, and small debris that does not pose a threat to the disintegration of the satellite.  
(7) Space debris collisions with a radius of less than 10 cm do not produce new satellite-threatening debris.  
(8) The debris from the collision is evenly distributed within the shell.  
(9) The average relative velocity of objects colliding in LEO is 10 km/s [17; 23].  
(10) Targets other than the constellation satellites in the shell are considered uncontrolled, and the controlled constellation satellites will not have a collision and disintegration event.  
(11) The annual failure rate of constellation satellites due to events, such as micro debris collisions or equipment failures, is 2%.