# The Optimal Deployment Strategy of Mega-Constellation Based on Markov Decision Process
## 2 Space Orbital Debris Flux and Collision Probability  
### Space Orbital Debris Flux  
As the variety of astronautic activities and the ability of the orbit-insert is enhanced, the numbers of on-orbit spacecrafts are significantly increased. Considering the limit of the accommodating capacity of orbital planes, the collision risk could also be increased and then accompany when a catastrophe occurs, such as satellites failure. Space orbital debris could be produced from retired spacecrafts, collision debris, and launch vehicles bodies; space particles are also included. According to the size of the orbital debris, it could be divided into \"big\" debris and \"small\" debris; the detriments are included in Table 1.  
Before calculating the collision probability of orbital debris, the debris space flux model should be built based on the space cells. The space orbital operation environment could be divided into multiple space unit cells by taking advantage of the orbital height, longitude, and latitude, as shown in Figure 2.  
Let \(h\) be the orbital height, \(\alpha\) be the longitude, and \(\beta\) be the latitude. Assuming \(i\) is the object in the space environment, \(j\) is the space unit cell and \(V_{j}\) is space volume of the cell. When the object \(i\) flies through the cell \(j\), \(t_{ij}\) is the flying time during the orbital period \(T_{i}\); then, the space density \(\rho_{ji}\) could be denoted by the following equation:  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Scale Range** & **Orbital Debris** & **Detriments and Influence** \\ \hline More than 1 cm & Big debris & Orbit changed; Structure damaged; Mission failed \\ Less than 1 cm & Small debris & Payload hurt; Performance degraded; Life shortened \\ \hline \hline \end{tabular}
\end{table}
Table 1: The scale and detriments of orbital debris.  
Figure 2: Space unit cell of orbital debris flux model.  
\[\rho_{ji}=\frac{t_{ji}}{T_{i}V_{j}} \tag{1}\]  
Taking into account the existence of \(N\) objects, the space density can be calculated by the following equation:  
\[\rho_{j}=\sum_{i=1}^{N}\rho_{ji} \tag{2}\]  
Let \(\vec{v}_{ji}\) be the flying-through velocity and \(\vec{n}\) be the pointing unit vector of the cross profile, then the space cell flux can be computed by:  
\[F_{ji}(\vec{n})=\rho_{ji}\vec{v}_{ji} \tag{3}\]  
Given \(N\) objects, the above equation can be expressed by:  
\[F_{j}(\vec{n})=\sum_{i=1}^{N}F_{ji}(\vec{n}) \tag{4}\]  
Assuming \(P\) is the point in the space unit cell and \(S\) is the infinitesimal element of a spatial spherical centered at \(P\), then the solid angle \(d\Omega_{SP}\) from \(P\) to \(S\) is denoted by:  
\[\oint\limits_{S}d\Omega_{SP}=4\pi \tag{5}\]  
Furthermore, the total orbital debris cross profile flux can be calculated by the following equation:  
\[F_{j}=\oint\limits_{S}F_{j}(\vec{n})d\Omega_{SP} \tag{6}\]  
The mega-constellation projects deploy satellites on the low earth orbit in most cases, so the influence of atmospheric drag cabnot be neglected; therefore, it is necessary to consider the solar radiant index when calculating space debris flux. It is difficult to forecast the solar radiant activity in the future but thiw is not the emphasis of this article; the solar radiant factor can be approximately computed by:  
\[RF=75\cos\{a(y-b)+0.35\sin[a(y-b)]\}+145 \tag{7}\]  
In the equation, let \(a=0.001696\), \(b=44,\!605\), and \(y\) be the modified Julian date. The relationship between space debris flux and different size of debris varying by years can be seen in Figure 3; the relationship between the space debris flux and different orbital heights varying by years can be seen in Figure 4.