## 2 Mathematical Modeling
### Model of Orbits  
In this study, a circular orbit is considered. The circle is a particular case of the ellipse, with the eccentricity \(e\) equal to 0: \(e_{1}=e_{2}=\cdots=e_{\rm k}=\cdots=e_{\rm n}=0\). The semi-major axis \(a\) becomes equal to the radius of the orbit \(r\): \(a_{1}=r_{1}\), \(a_{2}=r_{2}\),..., \(a_{\rm k}=r_{\rm k}\),..., \(a_{\rm n}=r_{\rm n}\).  
Considering this assumption, (1) becomes as in (29).  
\[\mathbf{x}_{\rm S_{k}}\big{|}_{\rm PQW}=r_{\rm k}\begin{pmatrix}\cos\left(v_{ \rm k}\right)\\ \sin\left(v_{\rm k}\right)\\ 0\end{pmatrix}\quad k=1,2,\ldots,n \tag{29}\]  
For a \(k\)-th satellite in a circular orbit, it is possible to define the height of a satellite as \(h_{S_{\rm k}}=r_{\rm k}-r_{\oplus}\).  
Another consequence of the assumption of circular orbit is that the argument of the perigee is undefined: \(\omega_{1}+v_{1}=v_{1}\), \(\omega_{2}+v_{2}=v_{2}\),..., \(\omega_{\rm k}+v_{\rm k}=v_{\rm k}\),..., \(\omega_{\rm n}+v_{\rm n}=v_{\rm n}\); for some authors, it is common to denote this sum as \"argument of latitude\", but, apparently, there is not a standard de facto letter to denote it.  
This also means that the first matrix in (2) is an identity matrix: \(\mathbf{R}_{(\omega_{1},3)}=\mathbf{R}_{(\omega_{2},3)}=\cdots=\mathbf{R}_{( \omega_{\rm n},3)}=\cdots=\mathbf{R}_{(\omega_{\rm n},3)}=\mathbf{I}_{3}\).  
Another assumption of this study is to consider an Earth scanning mission. This affects the value of the orbit inclination in considering a polar orbit: \(i_{1}=i_{2}=\cdots=i_{\rm k}=\cdots=i_{\rm n}=\pi/2\). Therefore, the second matrix in (2) is given by (30) for every satellite.  
\[\mathbf{R}_{(i_{\rm k}=\frac{\pi}{2},1)}=\begin{bmatrix}1&0&0\\ 0&\cos\left(i_{\rm k}=\frac{\pi}{2}\right)&-\sin\left(i_{\rm k}=\frac{\pi}{2} \right)\\ 0&\sin\left(i_{\rm k}=\frac{\pi}{2}\right)&\cos\left(i_{\rm k}=\frac{\pi}{2} \right)\end{bmatrix}=\begin{bmatrix}1&0&0\\ 0&0&-1\\ 0&1&0\end{bmatrix},\quad k=1,2,\ldots,n \tag{30}\]  
The inclination of the orbit of a satellite is one of the three angles that defines the orientation of the orbit plane into the space. An inclination of \(\pi/2\) means that the satellite goes through the south-north direction and passes over the North and South Poles. This value of inclination combined with the rotation of the Earth guarantees that the satellite can scan all the surface of the Earth. It is pointed out that this value is chosen for a first-level evaluation, and actual values of the inclination are both mission-dependent (e.g., in some applications, the scans of some regions are not of interest) and epoch-dependent (i.e., the axis of rotation of the Earth is slightly inclined and changes over time).  
Another assumption is to consider the same height of the orbits, so \(r_{1}=r_{2}=\cdots=r_{\rm k}=\cdots=r_{\rm n}=r\) and \(h_{S_{1}}=h_{S_{2}}=\cdots=h_{S_{\rm k}}=\cdots=h_{S_{\rm n}}=h_{S}\).  
It can be recalled that the period of an orbit depends on the value of the radius of the orbit, so this assumptions guarantees that all the satellites have the same period.  
The last hypothesis of this study considers a signal emitter on the Earth's surface, so \(h=0\) and (6) becomes as in (31).  
\[\mathbf{x}_{\rm E}|_{\rm ECEF}=\begin{pmatrix}r_{\oplus}\cos\left(\Lambda \right)\cos\left(\Lambda\right)\\ r_{\oplus}\cos\left(\Lambda\right)\sin\left(\Lambda\right)\\ r_{\oplus}\sin\left(\Lambda\right)\end{pmatrix} \tag{31}\]