## 2 Short-Term and Long-Term Collision Probability Calculation Models
### Long-Term Collision Probability Calculation Model
#### 2.2.2 Space Object Density Calculation Model
Figure 7: Schematic of space debris attenuation. The atmospheric drag acceleration is projected along the velocity direction \(f_{u}\), the geocenter radial direction \(f_{n}\), and the orbital plane normal direction \(f_{h}\). The projected components shown in Figure 8 are obtained. Given that the velocity of the spacecraft along the radial vector direction and the normal direction of the orbital plane is zero, the components of atmospheric drag in the three directions can be obtained as  
\[\left\{ \begin{array}{c}f_{u}=-\frac{1}{2}C_{d}\frac{A}{m}\rho v^{2}\\ f_{n}=0\\ f_{h}=0\end{array} \right. \tag{42}\]  
This equation is brought into the Gaussian perturbation equation of motion  
\[\left\{ \begin{array}{l}\dot{a}=\frac{2}{n\sqrt{1-e^{2}}}\big{(}1+e^{2}+2e \cos f\big{)}^{1/2}f_{u}\\ \dot{e}=\frac{\sqrt{1-e^{2}}}{na}\big{(}1+e^{2}+2e\cos f\big{)}^{-1/2}(\cos f+ e)\cdot f_{u}\\ \dot{i}=0\\ \dot{\Omega}=0\\ \dot{\omega}=\frac{\sqrt{1-e^{2}}}{nae}\big{(}1+e^{2}+2e\cos f\big{)}^{-1/2} \sin f\cdot f_{u}\\ \dot{M}=n-\frac{\sqrt{1-e^{2}}\big{(}1+e^{2}+2e\cos f\big{)}^{-1/2}}{nae} \Big{(}2\sin f+\frac{2e^{2}}{\sqrt{1-e^{2}}}\sin E\Big{)}\cdot f_{u}\end{array} \right. \tag{43}\]  
where \(E\) is the anomaly angle, \(n=\sqrt{\mu/a^{3}}\) is the average angular velocity, and \(\mu\) is the Earth's gravitational constant. The perturbation motion equation is averaged over the orbital period to obtain  
\[\left\{ \begin{array}{l}\overline{a}=\frac{1}{T}\int_{0}^{T}\bigg{(}\frac{2} {n(1-e^{2})^{1/2}}\big{(}1+e^{2}+2e\cos f\big{)}^{1/2}f_{u}\bigg{)}dt\\ \overline{e}=\frac{1}{T}\int_{0}^{T}\bigg{(}\frac{\sqrt{1-e^{2}}}{na}\big{(} 1+e^{2}+2e\cos f\big{)}^{-1/2}(\cos f+e)\cdot f_{u}\bigg{)}dt\\ \overline{l}=0,\overline{\Omega}=0\\ \overline{\omega}=\frac{1}{T}\int_{0}^{T}\bigg{(}\frac{\sqrt{1-e^{2}}}{na} \big{(}1+e^{2}+2e\cos f\big{)}^{-1/2}\sin f\cdot f_{u}\bigg{)}dt\\ \overline{M}=\frac{1}{T}\int_{0}^{T}\bigg{(}n-\frac{2e\sin f}{na}f_{u}\bigg{)}dt \end{array} \right. \tag{44}\]  
Figure 8: Components of atmospheric drag along the direction of velocity, radial, and orbital surface normal. Using the Spacetrack website, 19,839 cataloged targets with an orbital height of less than 2000 km were found. The eccentricity of these cataloged targets can be counted, and the distribution of eccentricity along the semi-major axis of the orbit can be obtained, as shown in Figure 9.  
As shown in Figure 9, the eccentricity of most low-orbit targets is less than 0.05, and the space debris moves in approximately circular orbits in the low-orbit region. Therefore, the eccentricity of space debris can be set to zero, and the average perturbation motion of space debris in one cycle under the action of atmospheric resistance is  
\[\left\{ \begin{array}{l}\overline{a}=\frac{2}{n}f_{u}\\ \overline{M}=n\\ \overline{e}=0,\overline{i}=0,\overline{\Omega}=0,\overline{\omega}=0\end{array} \right. \tag{45}\]  
In accordance with Equation (42), the average decay rate of space debris in the corresponding shell is  
\[v_{r}=-\rho C_{d}\frac{A}{m}\sqrt{\mu a} \tag{46}\]  
The static atmospheric resistance in the above formula is determined by the exponential model  
\[\rho=\rho_{0}e^{-\frac{r-r_{0}}{\Omega_{0}+\frac{1}{2}(r-r_{0})}} \tag{47}\]  
where \(\chi\approx 0.1\), \(H_{0}=37.4\)km, \(r_{0}=H_{0}+6378.137\), and \(\rho_{0}=3.6\times 10^{-10}\) kg \(\cdot\) m\({}^{-3}\). The decay rate of space debris is relatively slow, and the heights of the far and near points do not change much during a 7-year simulation time due to the large difference in the heights of the far and near points of the large elliptical orbit, so the influence of the space density in this part of the space debris shell is ignored.