## 3 Dynamic Link Establishment of Mega-Constellations
### Constraints on Establishing Dynamic Links  
In the previous section, three fixed links were established. Now, it is necessary to commence the construction of dynamic temporary links. As we can see in Figure 6, the red dashed lines in the following diagram represent the dynamic temporary links under discussion. Figure 5: Diagram of the fixed permanent link connection. Based on the current number of lasers equipped on Starlink satellites, it is stipulated that each satellite can only connect to four other satellites with its available lasers. Presently, each satellite can utilize one of these lasers for establishing a dynamic link at any given time; if a laser is connected to another satellite at a particular moment, both satellites cannot connect to any other satellite at that same instant. Thus, the problem can be abstracted into a two-by-two pairing issue between satellites at every moment. Given the numerous potential connections around each satellite at any given time, the following constraints are necessary to identify the most suitable connections:  
1. The link duration should be maximized to minimize the additional overhead caused by the frequent re-establishment of links. 2. The link distance must not be excessively large, since smaller distances reduce the signal transmission latency. 3. Angular velocities between satellites should not exceed maximum thresholds, as frequent angular velocity changes would incur extra energy consumption for altitude adjustments. 4. Topological changes (re-linking events) should be minimized; ideally, multiple satellites should switch links synchronously. Note that the link duration evaluations should occur after all other constraints are met, i.e., selecting satellites that satisfy all previous conditions before assessing the link duration. Specifically, to enhance the flexibility of dynamic links and enable interconnection among satellites crossing orbits, it is imperative first to avoid connecting to satellites on adjacent or identical orbits during the selection process for each satellite's potential connections. Moreover, to achieve a lower latency, reliance solely on \"mutual visibility\" between satellites for determining connectivity is insufficient, especially when satellites are in close spatial proximity. Therefore, when defining dynamic link connections, it is mandated that the connection distance be less than the maximum distance found in the fixed links. Through an analysis of the fixed links, the longest distance between neighboring orbit satellites over a complete orbital period in the network was found to be 1620.06 km. Consequently, when constraining the relative distance of the dynamic links, satellites within a distance of less than 1620.06 km must be selected to meet the inter-satellite distance constraint. Additionally, the establishment of dynamic links also necessitates limiting the relative angular velocity to an acceptable maximum. Using the existing Formulas (4) and (5), the elevation angle and azimuth angle between any two satellites can be calculated. From the simulation in the previous section, it is observed that elevation angle variations are generally small and unlikely to exceed limits, especially when satellites are close. Hence, the primary focus is on the azimuth angle. Given its significant variation, based on the previous section's findings, it is crucial to ensure that the azimuthal angular velocity does not exceed 0.5\({}^{\circ}\)/s. To improve computational efficiency, it is possible to perform azimuth angular velocity filtering on the satellites after completing the orbit and distance filtering. In order to achieve  
Figure 6: Simplified abstract diagram of the fixed permanent link connection. optimal satellite pairing, a satellite state connectivity matrix \(\alpha(t)\) can be established based on the constraints outlined in this section. \[\alpha(t)=