# Space Object and Background Polarization Models and Detectability Analysis
## 4 Imaging Contrast Simulation Analysis
After the target is irradiated by sunlight, the spherical scattering model simulation is used, and assuming that the incident light is completely passed by polarized light radiation and half of the unpolarized radiation is transmitted, the luminous flux arriving at the pupil is deduced as:  
\[\Phi_{s}=\frac{I_{s}\cdot\lambda}{L^{2}\cdot h\cdot c}=\frac{1.65R^{2}\cdot\rho_ {0}\cdot(1+P_{s})\cdot[(\pi-\phi)\cos\phi+\sin\phi]}{\pi L^{2}}(\text{photons}/ \left(\text{m}^{2}\cdot\text{s}\right)) \tag{34}\]  
where, \(P_{s}\) is the polarization of the space target, \(\rho_{0}\) is the average reflectivity of the target in the detection band, \(L\) is the space target orbit height, Planck's constant is \(h\), \(c\) is the speed of light, \(\lambda\) is the detection of the central band of the working band, and \(\phi\) is the scattered light phase angle, set the band for 1.5\(\sim\)1.8 \(\upmu\)m. When the signal detection effectiveness is guaranteed that the SNR is 6, the minimum imaging luminous flux of space target for ground-based optical system polarization imaging is estimated to be \(\Phi_{s}=8.472\times 10^{5}\text{photons}/\left(\text{m}^{2}\cdot\text{s} \right)\). The expression of the number of electrons of the space target on the target surface of the system detector is as Equation (35) shows, the parameters are show in Table 2.  
\[N_{s}=\frac{1}{k}\cdot\Phi_{s}\cdot A_{0}\cdot\tau_{0}\cdot\tau_{s}\cdot\tau_{ p}\cdot Q_{B}\cdot t_{s}\cdot\left(1-\varepsilon^{2}\right)\cdot f_{s} \tag{35}\]  
Similarly, the expression of the number of photoelectrons that can obtain the sky background signal that reaches the detector target in the real-time imaging method of the micropolar array is:  
\[N_{B}=\Phi_{\text{B}}\cdot A_{0}\cdot\tau_{0}\cdot\tau_{p}\cdot Q_{B}\cdot t_{ B}\cdot f_{B}\cdot A_{p}^{2}\cdot\left(1-\varepsilon^{2}\right)\cdot\frac{1+P_{B}}{2} \tag{36}\] \begin{table}
\begin{tabular}{c c c} \hline \hline  & **Definition** & \\ \hline \(k\) & number of image elements occupied by the image spot on the & 4 \\  & detector target surface & \\ \(A_{0}\) & system pupil area & 2.544 m\({}^{2}\) \\ \(\tau_{0}\) & system transmittance & 0.9 \\ \(\tau_{s}\) & average atmospheric transmittance in the operating band range & 0.7 \\  & average transmittance of the micropolarizer array of the & \\ \(\tau_{p}\) & polarization device in front of the image elements occupied by & 0.7 \\ \(Q_{B}\) & average quantum efficiency of the detector for the effective signal & 0.65 \\ \(t_{s}\) & integration time of the detector to collect the space target & 0.025 s \\  & information & 0.025 s \\ \(\varepsilon\) & blocking ratio existing in the structure of the card-based system & 0.2 \\  & transmittance coefficient of the reflected light from the space & 0.13 \\ \(f_{s}\) & target by the spectral filter & \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The parameter, definition and value of Equation (35). Figure 10: Simulation of the polarization value of the sky background. where \(\Phi_{\rm B}=5.6\cdot 10^{10}/2.512^{M_{B}}\) is the number of background-generated photons, \(M_{\rm B}\) is the deep-sky background equivalent magnitude, which is 22, and \(t_{B}=0.025\)s is the integration time for the detector to collect the spatial target information, \(f_{B}=0.021\) is the transmittance coefficient of the filter for the sky background noise, \(A_{P}=d_{pix}/f^{\prime}=0.047^{\prime\prime}\) is the instantaneous field of view of a single image element of the detector, \(d_{pix}=15\) um is the image element size of the detector and \(f^{\prime}\) is the system focal length. In the case, \(P_{B}\) is the polarization of the sky background. Considering the photon noise caused by the target and the sky background, the total noise signal expression of the system is:  
\[N=\sqrt{N_{s}+N_{B}+N_{D}\cdot t_{\rm s}\cdot k+N_{R}^{2}} \tag{37}\]  
where \(N_{D}=0.183e^{-}\) is the dark current noise of a single pixel of the detector, \(t_{s}\) is the integration time of the detector, \(N_{R}=60e^{-}\) is the readout noise of the detector.