## 2 Short-Term and Long-Term Collision Probability Calculation Models
### Long-Term Collision Probability Calculation Model
#### 2.2.2 Space Object Density Calculation Model  
Spatial density refers to the statistical value of the number of space objects contained in each cubic kilometer of volume at a certain time and a certain spatial location; the unit is \(1/\text{km}^{3}\), and the calculation method is as follows [31]:  
\[s(x)=\lim_{\Delta V\to V^{*}}\frac{\Delta N}{\Delta V} \tag{32}\]  
where \(V^{*}\) is the size of the volume element related to the distribution of space objects, and \(\Delta N\) is the number of space objects in the space volume element. \(V^{*}\) needs to choose a reasonable value to reflect the distribution characteristics of space objects. The above formula shows that that the spatial density is related to the size of the volume element and the number of objects in the volume element. However, the space objects move relative to the inertial space. At different times, the numbers of objects in the space volume element are different, the spatial density changes, and the low orbital target moves at a fast speed, so a certain instantaneous spatial density change has no practical reference importance. To better describe the density of space objects, \(\tilde{O}pik\) [PERSON] nd Alessandroet al. [32] introduced the concept of stay probability, averaged the instantaneous probability of space objects into the average probability over a period of time, and obtained the average density of space objects in the volume element as  
\[s(x)=\lim_{\Delta V\to V^{*}}\frac{\Delta t}{T\cdot\Delta V} \tag{33}\]  
where \(T\) is the orbital period of the space target, and \(\Delta t\) is the elapsed time of the space target entering and leaving the space volume element. On this basis, [PERSON] et al. made assumptions about the parameters of the target orbit and gave the calculation method for the spatial density of the object [33]:  
\[s(r)=\left\{ \begin{array}{l}\frac{1}{4\pi^{2}n\sqrt{(r-r_{p})(r_{a}-r)}};r_ {p}<r<r_{a}\\ 0;r>r_{a},r<r_{p}\end{array} \right. \tag{34}\]  
\[f(\beta)=\frac{2}{\pi\sqrt{\sin^{2}i-\sin^{2}\beta}},|\beta|<i \tag{35}\]  
where \(r\) is the geocentric distance; \(a\) is the semi-major axis of the orbit; \(r_{a}\) and \(r_{p}\) are the geocentric distances of the orbit apogee and perigee, respectively. \(i\) is the inclination of the target orbit, and \(\beta\) is the declination of the target location. From the above formula, the spatial density of the space target at any height and latitude in space is obtained as  
\[s(r,\beta)=s(r)f(\beta)=\frac{1}{2\pi^{3}ra\sqrt{\left(\sin^{2}i-\sin^{2}\beta \right)\left(r-r_{p}\right)\left(r_{a}-r\right)}} \tag{36}\]  
This formula requires that \(r_{p}<r<r_{a}\) and \(\beta<i\). \(r>r_{a},r<r_{p}\) when or \(\beta>i\). For the spatial density of the orbital height within the range of \([r_{i}^{m},r_{i}^{M}]\), \(r_{i}^{m}\) and \(r_{i}^{M}\) represent the geocentric distance of the upper and lower boundaries of the \(t\)th height interval, respectively. [PERSON] derived the following four formulas for solving the spatial density by averaging over the height range [25]. (1) When \(r_{p}\
eq r_{a}\) and \([r_{i}^{m},r_{i}^{M}]\subseteq[r_{p},r_{a}]\), the formula for calculating the spatial density is  
\[s\left(r_  
The decay rate of the same space debris in the shell is assumed to be the same as when the debris is in the center of the shell, and the atmosphere is stationary relative to the inertial frame. The orbital decay of space debris in a period can be integrated through the perturbation equation to obtain the average decay speed of the satellite in the shell. In accordance with the calculation formula of atmospheric drag acceleration, we have  
\[a_{d}=-\frac{1}{2}C_{d}\frac{A}{m}\rho\Big{|}\overrightarrow{v}\Big{|} \overrightarrow{v} \tag{41}\]  
where \(C_{d}\) is the atmospheric drag coefficient, \(A\) is the target area, \(m\) is the target mass, \(\rho\) is the atmospheric density, and \(\overrightarrow{v}\) is the velocity vector of space debris. Figure 6: The density of LEO space targets varies with orbital altitude.