# Space Object and Background Polarization Models and Detectability Analysis
## 3 Characteristics of the Sky Background Polarization Distribution
### Sky Background Polarization Scattering Model Complex Spherical Projection Representation  
According to the polarization degree and polarization angle formula, the polarization information at any position in the sky background in the three-dimensional horizon coordinate system can be treated by the pseudo-color algorithm to distinguish the polarization information differences in different locations. In order to make the results of this study generalizable, the three-dimensional polarization distribution is projected into a two-dimensional polarization distribution. Figures 4 and 5 are the simulation results of the model under the conditions of \(0^{\circ}\) solar altitude and azimuth. Figure 4: 3–D polarization distribution characteristics of Rayleigh scattering sky background, (**a**) Polarization degree (**b**) Polarization angle. The direct vertical projection method is commonly used to convert the three-dimensional polarization distribution of the simulated sky background into a two-dimensional distribution. However, the projection method lacks a rigorous correspondence, resulting in information loss, which cannot be reverse projected, and also makes the proportion of the zenith area greater than the actual distribution. In this paper, the polarization distribution characteristics of the sky background theory are expressed using the complex spherical mapping theory, which effectively solves the problem of losing polarization information during direct vertical projection [17]. Each location on the celestial sphere can be projected to the point on the plane where the optical system is placed, according to the theory of complex spherical mapping. At this point, the sophisticated spherical projection-based two-dimensional polarization distribution model of the sky background can be constructed. Figure 6 depicts the complex sphere's particular projection method. The complex sphere \(Z=R\) establishes the \(x_{m}o_{m}y_{m}\) coordinate system by using \((0,0,r)\) as the coordinate origin. L is the line formed by joining any two points \(R(x_{0},y_{0},z_{0})\) and \(z^{\prime}(0,0,-r)\) on the celestial sphere E. Because the normal of complex sphere \(z=r\) is not orthogonal to the line L, the complex sphere \(z=r\) must meet the line L at \(R_{m}(x_{m},y_{m})\). According to the complex sphere mapping theory, each point on the celestial sphere E can be projected onto the complex sphere \(z=r\) using the above technique. Figure 5: Polarization distribution characteristics of 2–D vertical projection of Rayleigh scattering, (**a**) Polarization degree (**b**) Polarization angle. Figure 6: Complex sphere mapping relation diagram. In a coordinate system \(\alpha xyz\), a complex spherical coordinate system \(x_{m}\)\(oy_{m}\) is represented as:  
\[z=r \tag{31}\]  
The equation for a straight line L is expressed as:  
\[\frac{x}{x_{0}}=\frac{y}{y_{0}}=\frac{z+r}{z_{0}+r} \tag{32}\]  
When actually imaging the foundation of a space target, it is only necessary to consider the impact of the sky background in hemispherical space. Therefore, it can be deduced from the spatial geometry that any point \(R(r,\theta,\varphi)\) in the hemisphere space on the celestial sphere E corresponds to the coordinates of the points \(R_{m}(x_{m},y_{m})\) projected onto the complex sphere. Since only the influence of the sky background in the hemispheric space needs to be considered when actually performing ground-based detection imaging of space targets. Therefore, the coordinates of the point \(R(r,\theta,\varphi)\) corresponding to the projection of any point \(R_{m}(x_{m},y_{m})\) in hemispheric space on the celestial sphere E onto the complex sphere \(z=r\) can be deduced from the spatial geometric relations as Equation (33):  
\[x_{m}+iy_{m}=2r\tan(\theta/2)\exp(i\varphi) \tag{33}\]  
In summary, a two-dimensional polarization distribution model of the sky background based on the complex spherical projection theory can be established.