## 2 Short-Term and Long-Term Collision Probability Calculation Models
### Short-Term Collision Probability Calculation Model
#### 2.1.5 Proximity Analysis Based on Lagrangian Interpolation  
Two space targets filtered out by relative positions may collide. Given that the two targets are most likely to collide when the orbits are closest, the collision probability at the closest orbit is regarded as the probability of collision between the two space targets. In reality, orbit prediction usually requires a certain step size. The step size cannot be infinitely small, and the trajectory cannot be continuous. Therefore, the predicted trajectory point is not necessarily the closest position of the orbit. Interpolation in accordance with the trajectory of the space target in the space cube must be performed to fit the curve equation of the orbit and find the position closest to the moment. Orbits are usually fitted by using Lagrangian interpolation. Suppose the moment when the two targets in the space volume element are closest to the trajectory point is \(t_{i}\), and let the position vector of one of the space targets be \(\overset{\rightarrow}{r}(t_{i})\). Two trajectory points are selected before and after the closest trajectory point, and five points with the closest trajectory point are used as interpolation points to perform polynomial interpolation. The constructor is obtained in accordance with the Lagrangian interpolation method. \[f_{k}(t)=\overset{1\leq j\leq 5}{\underset{j\
eq k}{\overset{\rightarrow}{ \Pi}}}\frac{\left(t-t_{j}\right)}{\left(t_{k}-t_{j}\right)} \tag{13}\]  
The Lagrangian interpolation fitting curve equation of the target orbit can be obtained as:  
\[\overset{\rightarrow}{f}(t)=\overset{5}{\underset{k=1}{\overset{\rightarrow }{r}}}(t_{k})f_{k}(t) \tag{14}\]  
Given that the fitting curve is a quartic polynomial and the interpolation points are known, the Lagrangian interpolation fitting curve equation can be expressed as:  
\[\overset{\rightarrow}{f}(t)=\overset{\rightarrow}{a}_{0}+\overset{ \rightarrow}{a}_{1}t+\overset{\rightarrow}{a}_{2}t^{2}+\overset{ \rightarrow}{a}_{3}t^{3}+\overset{\rightarrow}{a}_{4}t^{4} \tag{15}\]  
calculation requirements. However, the traditional Chan method approximates the integral domain, and the obtained new integral model has a deviation that cannot be accurately measured from the original initial integral model [25]. Reference [26] improves the defects in the Chan method and proposes a Laplace transform-based method. The space target collision probability calculation method improves the calculation accuracy and speed. Therefore, this method is used to solve the collision probability in this study. \[P_{c}=e^{(-pR^{2})}\cdot\sum_{k=0}^{+\infty}\frac{a_{k}R^{2(k+1)}}{(k+1)!} \tag{22}\]  
\[a_{0}=\frac{1}{2\sigma_{x}\sigma_{y}}\exp\left(-\frac{1}{2}\Bigg{(}\frac{\mu_ {x}^{2}}{\sigma_{x}^{4}}+\frac{\mu_{y}^{2}}{\sigma_{y}^{4}}\Bigg{)}\right) \tag{23}\]  
\[a_{1}=a_{0}p\bigg{(}1+\frac{\eta}{2}+\frac{Q_{x}+Q_{y}}{p}\bigg{)} \tag{24}\]  
\[a_{2}=\frac{a_{0}}{2}p^{2}\Bigg{[}\bigg{(}1+\frac{\eta}{2}+\frac{Q_{x}+Q_{y}}{ p}\bigg{)}^{2}+\bigg{(}1+\frac{\eta^{2}}{2}+\frac{\eta Q_{x}}{p}\bigg{)}\Bigg{]} \tag{25}\]  
\[ \begin{array}{l}a_{3}=\frac{a_{0}}{6}p^{3}\bigg{[}\Big{(}1+\frac{\eta}{2}+ \frac{Q_{x}+Q_{y}}{p}\Big{)}^{3}+3\Big{(}1+\frac{\eta}{2}+\frac{Q_{x}+Q_{y}}{ p}\Big{)}+\Big{(}1+\frac{\eta^{2}}{2}+\frac{2\eta Q_{x}}{p}\Big{)}\bigg{]}+\\ \frac{a_{0}}{3}p^{3}\Big{(}1+\frac{\eta^{3}}{2}+\frac{3\eta Q_{x}}{p}\Big{)} \end{array}  \tag{26}\]  
\[ \begin{array}{l}a_{k+4}=\frac{1}{k+4}\big{(}-p^{3}\eta^{2}Q_{y}a_{k}+p^{2} \eta\big{(}p\eta\big{(}k+\frac{5}{2}\big{)}+2Q_{y}\big{(}\frac{\eta}{2}+1\big{)} a_{k+1}\big{)}-\\ \frac{p}{k+4}\big{(}p\eta\big{(}\frac{\eta}{2}+1\big{)}(2k+5)+\eta\big{(}2Q_{y }+\frac{3p}{2}\big{)}+Q_{x}+Q_{y}\big{)}a_{k+2}+\\ \frac{1}{k+4}\big{(}p(2\eta+1)(k+3)+p\big{(}\frac{\eta}{2}+1\big{)}+Q_{x}+Q_{y }\big{)}a_{k+3}\end{array}