## 2 Constellation Model
### Connectivity Requirements  
We now discuss the requirements that need to be imposed to guarantee network connectivity throughout the constellation. This typically involves conditions imposed on both intra- and inter-orbit links. Figure 4 illustrates a typical scenario where a reference satellite C communicates with its adjacent satellites. The satellites \(\mathrm{S_{1}}\) and \(\mathrm{S_{2}}\) are the two nearest satellites in the same orbit \(k\) as C. The satellites \(\mathrm{U_{1}}\), \(\mathrm{U_{2}}\), and \(\mathrm{U_{3}}\) are the first, second, and third closest satellites to C, respectively, located at the next orbit, \(k+1\). Similarly, \(\mathrm{L_{1}}\), \(\mathrm{L_{2}}\), and \(\mathrm{L_{3}}\) are the first, second, and third closest satellites to C, respectively, located in the previous orbit, \(k-1\). This group of satellites forms a swarm \(\mathcal{S}(\mathrm{C})\) of nine satellites around C with a constant line of sight. We assume that the connectivity of C with the rest of the satellites of \(\mathcal{S}(\mathrm{C})\) is the minimum requirement to achieve reliable communication with the rest of the network, providing various alternative paths for data routing. This must be met for the swarm \(\mathcal{S}\) of every satellite in the constellation, as it implies that any two satellites in the communication can communicate with each other using a single hop or multiple hops. Similar to what we discuss in Section 2.2, due to the uniformity of the constellation, it is not necessary to address the swarms formed around every satellite in the constellation. Instead, we can focus on just a single reference satellite, since the distance variation between satellites in the swarm will be the same if we change the reference satellite C, albeit with a time offset.  
Since satellites in the same orbit move at a constant angular velocity, the distance between C and \(\mathrm{S_{1}}\) and \(\mathrm{S_{2}}\) will be constant with \(t\). This is not true for the distances between C and \(\mathrm{U_{i}}\) or \(\mathrm{L_{i}}\) due to the difference in the angles \(\Omega_{k}\). As mentioned in Section 2.2, we introduce a relative initial anomaly difference between orbits. The first orbit begins from the Earth's equatorial frame with a latitude of \(\Omega_{k}=0\). The next orbit shifts in longitude by an amount equal to \(2\pi F/N\), and so on. However, the closest neighboring satellites in the adjacent upper or lower orbits remain fixed relative to any satellite within the same orbit: the nearest upper-orbit neighbor, \(\mathrm{U_{1}}\), maintains its position relative to C throughout C's orbital period, and this pattern holds for other inter-orbit neighbors as well. At time \(t=0\), we can calculate the distances from C to all satellites in adjacent orbits and identify the three nearest satellites in each orbit to determine the swarm \(\mathcal{S}(\mathrm{C})\).  
Figure 3: Illustration of the relationship between the true anomalies \(\theta_{km}\) of the satellites at the same and adjacent orbits, the phasing factor \(F\), and the number of orbits and satellites per orbit, \(N_{\mathrm{orb}}\) and \(N_{\mathrm{sat}}\), respectively, on the Equatorial Plane.