## 3 Simulation Results and Analysis
### Simulation and Analysis of the Long-Term Collision Probability
#### 3.2.2 Results and Analysis  
The number of space debris objects in a shell is the number of debris objects in this shell plus the number of debris objects entering the higher shell and the number of debris objects falling out of this shell. In accordance with Tables 4 and 5, the amount of space debris in the constellation life cycle of the first shell due to orbital decay changed as follows:  
\[\left\{ \begin{array}{c}N=653-\frac{N_{1,l}}{40000}v_{1,l}t-\frac{N_{1,D}}{40000 }v_{1,D}t+\frac{N_{2,l}}{40000}v_{2,l}t+\frac{N_{2,D}}{40000}v_{2,D}t,t\leq 1444 \\ N=N_{1444}+\left(\frac{N_{2,l}}{40000}v_{2,l}+\frac{N_{2,D}}{40000}v_{2,D}- \frac{N_{1,l}}{40000}v_{1,l}+\frac{N_{2,D}}{40000}v_{2,D}t\right)(t-1444),t\leq 2 083\\ N=N_{2083}+\left(\frac{N_{2,l}}{40000}v_{2,l}+\frac{N_{3,D}}{40000}v_{3,D}- \frac{N_{1,l}}{40000}v_{1,l}-\frac{N_{2,D}}{40000}v_{2,D}t\right)(t-2083),t\leq 2 919\end{array} \right. \tag{51}\]  
where \(N\) is the number of debris objects, subscripts \(i,j\) represent the \(j\) type of the \(i\)-th shell fragment, \(v\) is the decay speed, and \(t\) is the number of days. In accordance with the definition of density \(\rho=N/V\), the evolution relationship of the space environment density with time during the life cycle of the Starlink constellation could be obtained. Figure 16 shows the spatial density evolution of background debris in the shell. In accordance with the assumption, the space debris only collided with the failed Starlink satellites. Up to 32 satellites failed each year at the failure ratio of 2%. Given that the satellites decayed at a rate of 165.4 m per day in the shell, the satellites were assumed to be evenly distributed in the shell. In 4 months, half of the satellites fell out of the shell due to attenuation. Although 32 satellite failures occurred in the first and second year, the spatial density of the failed satellites reached a maximum of \(2.6526\times 10^{-9}\), which was still lower than the background. If no collision event occurred in space during the lifetime of the constellation, the spatial density was reduced by 4-16.7% compared with that before the deployment of the constellation. Starlink satellites and intact objects generated new space debris due to collision. Therefore, the collision disintegration events were divided into six types, and the amount of space debris generated by different collision events was calculated. Figure 17 shows the amount of space debris generated by different collision types. As shown in Table 6, a large amount of debris was generated in one collision, the least one collision generated was approximately 200 space debris objects, and the density increased. In accordance with the average value of the debris generated, the generated space debris objects reached 658, the density increased, and the space debris increased. The density was double that when the constellation was not deployed. In accordance with Table 4, the generated space debris has a small surface mass ratio, a low orbital decay rate, and the debris has a long residence time in the orbit, resulting in long-term collision risks  
Figure 16: Spatial density evolution of background debris in the shell. for the constellation in orbit. The risks are multiplied, and in-orbit collisions with satellites should be avoided as much as possible. Taking the Starlink constellation as an example, the short-term collision probability and long-term collision probability of the constellation were calculated, and the results are as follows:  
(1) The probability of collision between Starlink Phase I constellation satellites and space debris is greater than seven times a day. Compared with that before the deployment of the constellation, the collision frequency in the shell increases by about 30-40%, indicating that the collision risk faced by a satellite is higher than that before the deployment of the constellation. The satellite needs to have better orbital maneuvering and collision warning abilities.