## 3 Numerical Simulations
### Orbit Determination Refinement  
The analysis presented in the previous section was focused on the observation capabilities of BIRALET. The goal of this second section is to assess the performance of the sensor while performing Orbit Determination (OD) refinement. Specifically, the accuracy of the state estimates at both the OD epoch and later epochs is investigated. Starting from the list of detectable passages, a reduced list of passages is obtained by sorting them in chronological order and imposing a minimum time distance between two consecutive passages of 20 minutes. The list is thus reduced to 355 passages. For all the objects of the list, we assume that an estimate for their state vector (i.e., position and velocity) is available at time epoch \(t_{0}\) (26 July 2019 00:00:00 UTC). This estimate, which may be obtained by other sensors capable of performing initial orbit determination (e.g., BIRALES radar sensor), can be modelled as a normal random variable (RV)  
\[\hat{\mathbf{X}}_{t_{0}}^{ECI}\sim\mathcal{N}\left(\hat{\mathbf{\mu}}_{t_{0}}^{ECI},\ \hat{\mathbf{\mathcal{C}}}_{t_{0}}^{ECI}\right) \tag{2}\]  
where \(\hat{\mathbf{\mu}}_{t_{0}}^{ECI}\) is the mean value, whereas \(\hat{\mathbf{\mathcal{C}}}_{t_{0}}^{ECI}\) is the associated covariance matrix. The superscript \"ECI\" indicates that the variable is expressed in the Earth Centered Inertial reference frame, whereas the subscript \(t_{0}\) indicates the epoch the estimate refers to. The mean value of the estimate \(\hat{\mathbf{\mu}}_{t_{0}}^{ECI}\) is computed by propagating, for each object, its TLE to \(t=t_{0}\), computing its state vector in ECI reference frame \(\mathbf{x}_{t0}^{ECI}\), computing the rotation matrix \(A^{ECI2 RSW}\) from ECI to radial, along-track, cross track (RSW) reference frame [22], and perturbing the state vector with a random vector \(\mathbf{\epsilon}_{t_{0}}^{ECI}\)  
\[\hat{\mathbf{\mu}}_{t_{0}}^{ECI}=\mathbf{x}_{t_{0}}^{ECI}+\mathbf{\epsilon}_{t_{0}}^{ECI} \tag{3}\]  
where  
\[\mathbf{\epsilon}_{t_{0}}^{ECI}=\left[ \begin{array}{cc}\mathbf{A}^{ECI2 RSW}&0\\ 0&\mathbf{A}^{ECI2 RSW^{T}}\end{array} \right]\left\{ \begin{array}{c}\mathbf{\epsilon}_ {t_{0},p}^{RSW}\\ \mathbf{\epsilon}_{t_{0},p}^{RSW}\end{array} \right\} \tag{4}\]  
and  
\[\mathbf{\epsilon}_{t_{0},p}^{RSW}=\left\{ \begin{array}{c}\mathcal{U}(-00.1,0.01) \\ \mathcal{U}(-0.05,0.05)\\ \mathcal{U}(-0.01,0.01)\end{array} \right\}\text{km}\mathbf{\epsilon}_{t_{0},p}^{RSW }=\left\{ \begin{array}{c}\mathcal{U}(-0.5,0.5)\\ \mathcal{U}(-0.5,0.5)\\ \mathcal{U}(-0.5,0.5)\end{array} \right\}\text{m/s} \tag{5}\]  
The variables \(\mathbf{\epsilon}_{t_{0},p}^{RSW}\) and \(\mathbf{\epsilon}_{t_{0},p}^{RSW}\) represent random errors in position and velocity expressed in the RSW reference frame. The covariance matrix is estimated by assuming reasonable values for the  
Figure 5: SRT observation capabilities: (**a**) orbital inclination as a function of the perigee height, (**b**) number of passages per object. standard deviations in position (\(\sigma_{p}\)) and velocity (\(\sigma_{v}\)) in the RSW reference frame, building a diagonal covariance in RSW reference frame, and expressing this matrix in the ECI reference frame.