# Design of Mega-Constellations for Global Uniform Coverage with Inter-Satellite Links
## 2 Mega-Constellation Design Model
### Mega-Constellation Configuration Design  
To achieve uniform global coverage, this paper constructs a mega-constellation composed of multiple sub-Walker constellations with the same semimajor axis, eccentricity, argument of perigee, and different orbital inclination. For the Walker constellation, the satellite's inclination limits the latitude range of the sub-satellite points distribution, and the latitude boundary is \([-i,i]\). Moreover, its sub-satellite points are densely distributed in high latitudes while sparsely in low. Based on these characteristics, the following mega-constellation design strategy is proposed. Firstly, utilize a large inclination Walker constellation to cover the high latitude region. Then, based on the density of sub-satellite points in high latitudes, the parameters of sub-constellation with small inclination are deduced in turn. A mixed Walker constellation with the same sub-satellite points density at each latitude zone will eventually be obtained. In addition, all satellites are designed to be in prograde orbit to ensure orderly operation within the mega-constellation. Figure 2 depicts the process of developing a mega-constellation using the method proposed in this paper. The specific implementation steps are shown as follows:  
Step1: Set initial parameters for mega-constellations design. Dividing the Earth's surface into m latitude zones, the latitude span of each latitude zone is: \(\Delta\delta=\frac{\pi}{m}\). The area of latitude zone from north to south is: \(S_{1},S_{2},\ldots,S_{m}\). Then, with \(\Delta\delta\) as the unit, the sub-constellation is constructed in turn. The inclination of sub-constellation in descending order is: \(\frac{\pi}{2},\ \frac{\pi}{2}-\Delta\delta,\ \frac{\pi}{2}-2\Delta\delta,\ldots, \frac{\pi}{2}-\left(\frac{m}{2}-1\right)\Delta\delta\). In this order, the sub-constellations are denoted as \(C_{1},C_{2},\ldots,C_{m/2}\). (Mega-constellation \(C\) consists of \(\frac{m}{2}\) sub-constellations since the Walker constellation is symmetrical in the southern and northern hemispheres. In other words, the constellation design only needs to meet the uniform distribution of sub-satellite points density in the northern hemisphere to achieve global uniform distribution.) Step2: Set the total number of satellites \(N_{C_{1}}\) in the Walker constellation with maximum orbital inclination \((i=90^{\circ})\). Calculate the average density \(\overline{D}_{k,C_{1}}\) of the sub-constellations \(C_{1}\) in each latitude zone. \(j=2\). Step3: Based on the initial sub-constellations and Equation (14), the average number of sub-satellite points in the northernmost latitude zone of the \(j\)-th sub-constellation can be determined by Equation (15). \[\overline{D}_{1,C}=\overline{D}_{j,C}+\overline{D}_{1,C_{j}} \tag{14}\]  
\[n_{1,C_{j}}=\frac{\overline{n}_{1,C_{1}}\cdot S_{j}}{S_{1}}-\overline{n}_{j,C} \tag{15}\]  
where \(\overline{n}_{j,C}\) is the average number of sub-satellite points of mixed constellation \(C\) falling on the \(j\)-th latitude zone. Step4: According to \(n_{1,C_{j}}\), the average distribution number of the \(C_{j}\) in the \(k\)-th latitude zone \(k=2,\;3,\dots,\;m-j+1\) can be obtained:  
\[n_{k,C_{j}}=\frac{n_{1,C_{j}}\cdot S_{k+j-1}\cdot f_{k,C_{j}}}{S_{j}\cdot f_{1,C_ {j}}} \tag{16}\]  
where \(f_{k,C_{j}}\) represents the probability density of \(C_{j}\) for the density of sub-satellite points in the \(k\)-th latitude zone. Total satellites number of \(C_{j}\) can be calculated by adding the number of sub-satellite points on each latitude zone. \[N_{C_{j}}=\sum_{k=1}^{m}n_{k,\;C_{j}} \tag{17}\]  
Step5: The mixed constellation \(C\) composed of \(j\) sub-constellations is obtained. Calculate the average number of sub-satellite points in each latitude zone of \(C\) and update \(\overline{n}_{1,C},\;\overline{n}_{2,C}\;,\dots,\;\overline{n}_{m,C}\). \[\overline{n}_{m,C}=\sum_{k=1}^{m}n_{k,\;C_{m-k+1}} \tag{18}\]  
Step6: If \(j<m/2\), repeat Step3-Step5, \(j=j+1\). Otherwise, continue in step 7.