# Rethinking LEO Mega-Constellation Routing to Provide Fast Internet Access Services
## 2 Background
### Satellite Network Model  
This paper focuses on routing algorithms applied to a constellation called _Walker-Delta_. According to published regulatory filings, this configuration will be used in the mega-constellation designs proposed by many companies. In this constellation, the satellites will move in different circular orbital planes. Before mathematical modeling of the entire constellation, we first modeled the spatial position information of the satellite.  
The position of a satellite in space can be represented by classical orbital elements. In this paper, we are concerned about the _true anomaly_, \(\
u\), and the _longitude of the ascending node_, \(\Omega\). Since the satellite orbit is circular and the periapsis cannot be calculated, the _phase angle_, \(u\), is used as a substitute parameter for the _true anomaly_, which represents the angle between the satellite and ascending node. The \(\Omega\) at time \(t\), is given by \(\Omega=L_{0}-\omega_{e}\cdot t\), where \(L_{0}\) is the initial longitude of the ascending node and \(\omega_{e}\) is the rotational angular velocity of the Earth. The remaining elements, _inclination_, \(\alpha\), and _semi-major axis_, \(a\), are constants of the constellation. The _semi-major axis_, \(a\), is equal to the radius of the orbit, which is the sum of the orbit's _altitude_, \(h\), and the radius of the Earth. A schematic diagram of satellite orbit parameters is shown in Figure 1.  
A Walker-Delta constellation described by \(\alpha:PQ/P/F\), contains \(P\) orbital planes. Each of these planes consist of \(Q\) equally spaced satellites, and the relative distance between adjacent plane satellites of this constellation is indicated by \(F\). \((i,j)\) is used to denoted the \(j\)-th satellite in orbital plane \(i\), in the subsequent sections of this paper.  
The spacing between adjacent planes is specified by the right ascension difference of ascending nodes \(\Delta\Omega=\frac{2\pi}{P}\), and the phase difference between adjacent satellites in an orbital plane \(\Delta\Phi=\frac{2\pi}{Q}\), can be calculated by the number of satellites in each orbit. Finally, due to the existence of _the phase factor_, \(F\), in the constellation parameters, the latitude of two horizontal neighbor satellites located in adjacent orbits are different, and the difference value is expressed as _phase offset_, \(\Delta f=\frac{2\pi F}{PQ}\). A visualization of the above constellation parameters can be seen in Figure 2. Meanwhile, the satellite numbered \((i,j)\) can be represented by these parameters as \((u=\mathcal{N}(j\cdot\Delta\Phi+i\cdot\Delta f),L_{0}=\mathcal{N}(i\cdot \Delta\Omega))\), where \(\mathcal{N}(x)\) is a normalized function ensuring that its result is in the interval \([-\pi,\pi]\).  
In the hypothesis of this paper, each satellite is equipped with four wireless terminals, to establish two intra-plane links and two inter-plane links with neighboring satellites. The notations for the preceding and subsequent satellites relative to the satellite \((i,j)\) are denoted as \((i,(j-1)modQ)\) and \((i,(j+1)modQ)\), respectively. The neighbor to the left is denoted as \((i-1,j)\) if \(i\
eq 0\), otherwise \((P-1,(j-F)modQ)\). Homoplastically, the neighbor to the right is \((i+1,j)\) if \(i\
eq P-1\), otherwise \((0,(j+F)modQ)\).