# Rethinking LEO Mega-Constellation Routing to Provide Fast Internet Access Services
## 3 Our Proposed Algorithm
### Minimum Hop Count  
In the following, we present a model to determine the theoretical minimum number of hops between two satellites, based on the inter-satellite link (ISL) model put forth by [PERSON] et al. [16]. We divide the hops between the two satellites into two parts, the intra-orbit hops and the inter-orbit hops.  
The minimum number of inter-orbit hops, \(N_{inter}\), of two satellites, can be calculated from the horizontal distance of the orbits of the satellites. Specifically, only the longitude difference of the ascending nodes is needed to calculate \(N_{inter}\):  
\[\Delta L_{0}=(L_{0}^{1}-L_{0}^{2})mod2\pi. \tag{1}\]  
\(\Delta L_{0}\) is the angular difference of longitude from the source orbit to the destination orbit in the east direction. Therefore, in the west direction, the angular difference of longitude can be expressed as \(2\pi-\Delta L_{0}\). Since the longitude difference of adjacent satellites between orbital planes is \(\Delta\Omega\), the minimum number of inter-orbit hops in the east and west directions are respectively given by:  
\[N_{inter}^{\rightarrow}=\frac{\Delta L_{0}}{\Delta\Omega}\hskip 28.452756 ptN_{ inter}^{\leftarrow}=\frac{2\pi-\Delta L_{0}}{\Delta\Omega}. \tag{2}\]  
When calculating the minimum number of intra-orbit hops connecting two satellites, it is necessary to consider the phase angle of the satellites. A hop to the neighbor satellite in an easterly direction between orbits increases the phase angle by \(\Delta f\), while a hop to the subsequent satellite in the orbit increases the phase angle by \(\Delta\Phi\). When only the hops to the adjacent satellites in the right orbit and the hops to subsequent satellites are taken into account, the following formula is true:  
\[u_{2}=u_{1}+\Delta\overleftarrow{u}+(N_{inter}^{\rightarrow}\cdot\Delta f) \hskip 28.452756 pt\Delta\overrightarrow{u}=\Big{(}N_{intra}^{\
earrow} \cdot\Delta\Phi\Big{)}. \tag{3}\]  
where \(N_{intra}\) is the minimum number of intra-orbit hops we need to calculate. The phase angle difference caused by intra-orbit hops in one direction, \(\Delta\overrightarrow{u}\), can be obtained by converting the above formula:  
\[\Delta\overrightarrow{u}=u_{2}-N_{inter}^{\rightarrow}\cdot\Delta f-u_{1}. \tag{4}\]  
Similarly, the phase angle difference in the other direction, \(\Delta\overleftarrow{u}\), can be obtained by the following formula:  
\[\Delta\overleftarrow{u}=u_{2}+N_{inter}^{\leftarrow}\cdot\Delta f-u_{1}. \tag{5}\]  
Finally, the minimum number of intra-orbit hops, \(N_{intra}\), can be obtained by the phase difference between adjacent satellites in orbit and the phase difference between the source and destination satellites in orbit:  
\[\begin{array}{l}N_{intra}^{\leftarrow}=\frac{\Delta\overleftarrow{u}}{ \Delta\Phi}\hskip 28.452756 ptN_{intra}^{\
earrow}=\frac{\Delta \overrightarrow{u}}{\Delta\Phi}\\ N_{intra}^{\leftarrow}=\frac{2\pi-\Delta\overleftarrow{u}}{\Delta\Phi}\hskip 28.452756 pt N_{intra}^{\searrow}=\frac{2\pi-\Delta\overrightarrow{u}}{\Delta\Phi}.\end{array} \tag{6}\]  
Therefore, the theoretical minimum number of hops between the two satellites, \(N\), is given by the minimum in the following combination:  
\[\min\{N_{inter}^{\leftarrow}+N_{intra}^{\
earrow},N_{inter}^{\leftarrow}+N_{intra }^{\
earrow},N_{inter}^{\rightarrow}+N_{intra}^{\
earrow},N_{inter}^{ \rightarrow}+N_{intra}^{\searrow}\}. \tag{7}\]Meanwhile, the minimum value in the forwarding direction combination above represents the reference forwarding direction of the minimum hops path.