## 2 Constellation Model  
### Constellation Dynamics  
The equations describing the motion of the satellite with a circular orbit can be derived using the general approach of [29]. In three-dimensional space, the satellite's position relative to the Earth's center is described by a three-component vector \(\mathbf{r}=(x,y,z)^{\intercal}\), with \((\cdot)^{\intercal}\) denoting the vector's transpose. Figure 1 shows an Earth-centered coordinate system where the \(xy\) plane contains the equatorial plane (orange dashed line), while the \(z\)-axis aligns with the Earth's rotation axis, pointing north. The figure illustrates the basic parameters involved in estimating the satellite position. In the figure, \(r=\sqrt{x^{2}+y^{2}+z^{2}}\) denotes the distance of the satellite from the Earth's center. In this paper, we study circular orbits, and hence, \(r\) remains constant.  
Assuming a constellation consisting of \(N_{\text{orb}}\) orbits containing \(N_{\text{sat}}\) satellites each. Every satellite \((k,m)\) can be identified by the index \(k\) of its orbit and its index \(m\) within the orbit. Its position \(\mathbf{r}_{km}(t)\) is determined by [29]  
\[\mathbf{r}_{km}(t)=\mathbf{T}(\Omega_{k},i)\mathbf{u}_{km}(t), \tag{1}\]  
where  
\[\mathbf{u}_{km}(t)=\frac{h^{2}}{\mu}(\cos\theta_{km}(t),\sin\theta_{km}(t),0)^ {\intercal}, \tag{2}\]is the position of the satellite with respect to the perifocal frame of reference (i.e., the frame attached to the satellite itself), \(i\) is the inclination angle, and \(\Omega_{k}\) is the angle between the \(x\)-axis and the node line for orbit \(k\) determined by  
\[\Omega_{k}=(k-1)\frac{2\pi}{N_{\mathrm{orb}}}, \tag{3}\]  
while \(h\) is the magnitude of the angular momentum related to the satellite's period \(T\) through \(h=2\pi r^{2}/T\). The period of the circular orbit is given by  
\[T=\frac{2\pi r^{3/2}}{\sqrt{\mu}}, \tag{4}\]  
with \(\mu=GM_{\mathrm{e}}\) being the product of the gravitational constant, \(G=6.6743\times 10^{-11}\) N \(\times\) m\({}^{2}\)/kg\({}^{2}\), and the mass of the earth, \(M_{\mathrm{e}}=5.974\times 10^{24}\) kg. The transformation matrix \(\mathbf{T}\) in (1) is given by  
\[\mathbf{T}(\Omega_{k},i)=\begin{bmatrix}\cos\Omega_{k}&-\sin\Omega_{k}\cos i &\sin\Omega_{k}\sin i\\ \sin\Omega_{k}&\cos\Omega_{k}\cos i&-\cos\Omega_{k}\sin i\\ 0&\sin i&\cos i\end{bmatrix} \tag{5}\]  
The true anomalies \(\theta_{km}(t)\) in circular orbits are determined by  
\[\theta_{km}(t)=\theta_{km}(0)+\frac{2\pi t}{T}. \tag{6}\]  
We can assume that on each orbit \(k\), the satellites are evenly spaced, in which case  
\[\theta_{km}(0)=\theta_{k}(0)+\frac{2\pi(m-1)}{N_{\mathrm{sat}}}, \tag{7}\]  
where \(\theta_{k}(0)\) is the initial anomaly of the first satellite at the \(k^{\mathrm{th}}\) orbit. Given these initial anomalies, we can combine (6) and (7) to obtain the values of \(\theta_{km}(t)\) for every time instance \(t\) and calculate the position of every satellite \(\mathbf{r}_{km}(t)\) in the constellation using (1)-(5).  
Table 1 presents the constellation parameters representative of Starlink's Phase 1 V2, which, unless otherwise specified, serves as a reference in our simulations. Figure 2 illustrates the satellite trajectories for this constellation. Due to the inclination of the constellation, the satellite trajectories are more widely spaced near the equator and progressively converge as they approach higher latitudes. Beyond a certain latitude near the poles, satellite coverage diminishes entirely.  
Figure 1: Satellite frame of reference and various orbital parameters used to determine the orbital dynamics.