# Space Object and Background Polarization Models and Detectability Analysis
## 4 Imaging Contrast Simulation Analysis
Finally, the detection signal-to-noise ratio expression can be derived as:  
\[SNR=\frac{N_{s}}{N} \tag{38}\]  
The system's limit detection capability will be expressed in the concept of star magnitude, and its transformation relationship is:  
\[M=2.5\log\left(\frac{\int_{\lambda_{1}}^{\lambda_{2}}\frac{1}{T_{0}^{4}\cdot \lambda^{4}}\cdot\frac{8.48\times 10^{43}\times 10^{-44c}}{\epsilon^{(144 \times 10^{8}/\lambda T_{0})}-1}d\lambda}{\Phi_{s}}\right) \tag{39}\]  
where \(T_{0}\) is the stellar equivalent temperature, and bc is the bolometric correction, bc takes 0.08, and \(T_{0}\) is 5900 K.  
Let the target be at the orbit radius of 500 km with the same distribution of sky background in the field of view. By substituting the zenith angle and azimuth angle of the space target (A = 10\({}^{\circ}\), E = 101\({}^{\circ}\)28\({}^{\prime}\) and A = 60\({}^{\circ}\), E = 10\({}^{\circ}\)8\({}^{\prime}\), respectively) into the complex spherical coordinate system, the polarization value \(P_{B}\) of the space target surface material can be found in Figure 9, and the corresponding sky background polarization value \(P_{s}\) can be obtained from Figure 10. Referring to Equation (38), we know the range of the ultimate imaging capability of the system, the range of the target-background contrast of the original intensity imaging under the ultimate detection conditions, and the range of the target-background contrast in the polarization degree image obtained by the polarization imaging technique. The calculation results are shown in Table 3.  
When the system reaches the limit of detection capability, the space target brightness and sky background brightness difference of 3-4 magnitude, the brightness of the sky background is generally 15.85-39.82 times the brightness of the space target, when the signal-to-noise ratio is greater than 6, the detector can be considered corresponding, but the effective signal under this condition is susceptible to background noise interference, imaging contrast is extremely poor. The polarization technology, by acquiring the polarization image of the space target and the sky background, converts the contrast between the sky background and the space target from a value with a large difference in magnitude to a value with the same magnitude at certain angles, so that the system can greatly improve the target-background contrast index at a certain detection angle when detecting the space target with a weak signal-to-noise ratio. \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & **Polarization Value** & **Extreme Imaging** & **Raw Intensity** & **Polarization Imaging** \\  & **Capability Range** & **Imaging Contrast** & **Contrast** \\ \hline A = 10\({}^{\circ}\), E = 101\({}^{\circ}\)28\({}^{\prime}\) & 0.32 & [8.86, 9.28] & [0.028, 0.042] & [0.006, 1.5] \\ A = 60\({}^{\circ}\), E = 101\({}^{\circ}\)28\({}^{\prime}\) & 0.89 & [8.83, 9.23] & [0.029, 0.043] & [0.002, 0.539] \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Contrast calculation result. For example, the polarization of the polyimide plate can reach about 0.48 when the macroscopic surface normal of the polyimide plate is at an angle of 62\({}^{\circ}\) from the center of the field of view of the ground-based optical system at a position of 10\({}^{\circ}\) zenith angle and 101\({}^{\circ}\)28\({}^{\prime}\) azimuth angle, while the polarization of the sky background is only 0.32. Sometimes, the polarization contrast will indeed be lower than the contrast of intensity imaging due to the low brightness of the polarized image caused by the addition of a micropolarizer array, but a higher contrast of polarized imaging can be obtained by choosing the most appropriate satellite observation angle. The highest value of spatial target-background contrast can reach 1.5, which is improved by 35.71 times compared to the maximum value of target-background contrast of 0.042 for the original intensity imaging technique at this condition.