# ATLAS: Latest Advancements and First Observations
## 4 Discussion
### Revisiting the Radar Equation  
By adapting one of the many forms of the radar equation, the single pulse signal-to-noise ratio (SNR\({}_{p}\)) after pulse compression is given by [39]:  
\[\text{SNR}_{p}=\frac{P_{peak}G^{2}\lambda^{2}\sigma\tau\beta}{4\pi^{3}R^{4}k_{ B}TN_{F}BL_{s}} \tag{14}\]  
where \(P_{peak}\) is the transmitted peak power (W), \(G\) is the antenna gain, \(\lambda\) is the operating wavelength (m), \(\sigma\) is the target radar cross-section (RCS) (m\({}^{2}\)), \(\tau\) is the pulse width (s), \(\beta\) is the bandwidth of the waveform (Hz), \(R\) is the distance to the object (m), \(N_{F}\) is the noise factor of the receiver, \(k\) is the Boltzmann constant (\(\frac{1}{K}\)), \(T\) is the receiver temperature (K), \(B\) is the bandwidth of the receiver (Hz), and \(L_{s}\) accounts for all losses.  
Figure 12: Range-Doppler map with detection at 502.0 km of range and 0.014 m/s of Doppler velocity.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline
**NORAD ID** & **Timestamp** & **Azimuth (\({}^{\circ}\))** & **Elevation (\({}^{\circ}\))** & **Range (km)** & \
\begin{tabular}{c} **Doppler** \\ **Velocity (m/s)** \\ \end{tabular} &
\begin{tabular}{c} **Velocity (From** \\ **Line Detection)** \\ **(m/s)** \\ \end{tabular} \\ \hline
25544 & 2023-07-04T06:51:05 & 91.94 & 57.37 & 502.0 & 0.014 & 623.115 \\ \hline
25544 & 2023-07-06T23:34:16 & 258.87 & 45.27 & 589.5 & \(-\)0.105 & 4407.915 \\ \hline
25544 & 2023-07-13T21:11:20 & 27.68 & 46.97 & 573.5 & \(-\)0.040 & 3931.333 \\ \hline
54216 & 2024-01-31T12:40:09 & 287.22 & 45.39 & 541.5 & 0.052 & 4653.335 \\ \hline
25544 & 2024-02-02T18:44:46 & 303.62 & 45.84 & 583.9 & 0.037 & 4593.325 \\ \hline \end{tabular}
\end{table}
Table 3: Some examples of observations.  
The loss term \(L_{s}\) represents all the factors that are difficult to obtain/identify, e.g., system losses and atmospheric losses. While an educated guess can be a useful starting point to evaluate the future performance of a radar system, ideally, this term should be updated by performing measurements with the system. Figure 13 represents the theoretical curves obtained for ATLAS for different RCS values, represented as solid lines. Since we have now made observations of the ISS, we decided to use these to revisit the equation and improve it. To do this, we fixed the equation terms for an RCS of 400 m\({}^{2}\) and fitted the loss term \(L_{s}\) to the observations using a non-linear least-squares method. The fitting resulted in an \(L_{s}\) of 15.94 dB with a variance of 6.09 dB. The fit curves are represented as dashed lines. The variance of the estimate is high due to the dispersion and number of measurements. We can see that for ranges around 500 km, the detections have variations of around 5 dB in the SNR, which can be due to many factors such as the scintillation of the object or variations in the gains of the system during pulse emission/reception. We are currently investigating the cause of these differences in the SNR for the same range and RCS. Currently, the number of observations is limited, and the objective is to gather more observations at different ranges and RCSs to obtain a better estimate of \(L_{s}\).