## 3 Simulation Results and Analysis
### Simulation and Analysis of the Long-Term Collision Probability  
#### 3.2.1 Numerical Experimental Setup  
The long-term collision probability simulation calculation of the first phase of the Starlink constellation was performed. The known constellation parameters were Walker \(-\)\(\delta\) 1584/72/33, the orbit height was 550 km, the orbit inclination was 53\({}^{\circ}\), the satellite mass was 260 kg, and the satellite size was 3.2m \(\times\) 1.6m. The simulation time was 7 years and the constellation shell height \(H\) = 40 km. Considering that the atmospheric resistance can be approximately ignored when the orbit height is greater than 800 km, the shell was divided into seven intervals in accordance with the orbit height of the Starlink constellation, and the interval height range was 530-810 km. From Assumption 6 and Formula (48), the mass of space debris \(c\) that reaches the collision disintegration energy was 0.208 kg. From Assumption 4, the cross-sectional area of the space debris of type \(c\) could be obtained as 0.0065 m\({}^{2}\), and the radius of the space debris of type \(c\) could be obtained as 0.045 m. In accordance with the definition of the characteristic size, the attenuation speed of the space target in each shell could be obtained by using the static atmospheric drag density model after the surface-to-mass ratio and orbit height of the space target were known. Figure 14 shows the different decay rates of the targets in each shell. Table 4 presents the decay rates of various targets within each shell. Notably, the decay rates of the Starlink satellites surpass that of other targets by a significant margin. Figure 14: The different decay rates of the targets in each shell. \begin{table}
\begin{tabular}{c c c c c c c} \hline
**Collision Target Type** & **Day One** & **Day Two** & **Day Three** & **Day Four** & **Day Five** & **Day Six** & **Day Seven** \\ \hline Between constellation satellites & 0 & 0 & 0 & 0 & 0 & 0 \\ Between constellation satellites and space debris & 1 & 3 & 7 & 6 & 3 & 3 & 1 \\ Between space debris & 2 & 12 & 23 & 17 & 13 & 8 & 4 \\ \hline \end{tabular}
\end{table} 
Table 3: Collision frequency of space rendezvous targets greater than \(10^{-5}\) times. The space debris environments in the shells could be obtained from the existing cataloging data. The cataloging target data could be downloaded from the Spacetrack website, and the download date was 12 March 2022. The downloaded cataloging targets were screened in terms of the shells, and each item at the initial moment could be obtained. The numbers of space debris objects were cataloged for each shell. Figure 15 shows the distribution of the numbers of space debris objects in each shell. In accordance with the statistics in [10], the number of rocket bodies in the low-orbit area was 916 and the number of loads was 6420 as of 4 April 2022. Assuming that the distribution of rocket bodies was proportional to the number of loads, Starlink could be obtained through the Spacetrack website. The number of satellites in the first shell was 1495, and the number of other payloads and space debris objects in the first shell was 653. Therefore, the number distribution of the rocket bodies in each shell and the statistics on the number of satellites in each shell could be obtained, and the number of complete objects in each shell could be calculated. Table 5 shows the distribution of complete debris types \(I\) and \(D\) in the shells. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Target** & **Shell One** & **Shell Two** & **Shell Three** & **Shell Four** & **Shell Five** & **Shell Six** & **Shell Seven** \\
**Type** & **(m/day)** & **(m/day)** & **(m/day)** & **(m/day)** & **(m/day)** & **(m/day)** & **(m/day)** \\ \hline Starlink satellites & 165.4 & 115.2 & 82 & 59.5 & 44 & 33 & 25.2 \\ Whole object \(I\) & 12 & 8.3 & 5.9 & 4.3 & 3.2 & 2.4 & 1.8 \\ Space debris \(D\) & 27.7 & 19.2 & 13.7 & 10 & 7.4 & 5.5 & 4.2 \\ Space debris \(c\) & 32.3 & 22.5 & 16 & 11.6 & 8.58 & 6.5 & 4.9 \\ \hline \end{tabular}
\end{table}