# Space Object and Background Polarization Models and Detectability Analysis
## 3 Characteristics of the Sky Background Polarization Distribution  
### Establishment of Rayleigh Scattering Model for Sky Background Polarization  
The sun position affects the sky background polarization distribution characteristics, and in order to introduce the relationship between the ground-based optical system and the sun position, the horizontal coordinate system is used. A schematic diagram of the horizontalcoordinate system is shown in Figure 3, which can be obtained from the vertical projection of the celestial coordinate system in the equatorial plane. The position of the Sun is \(S_{1}\), the position of the ground-based optical system is O, the plane where the optical system is located is XOY, and OV is the direction of the space target detected by the ground-based optical system, which intersects with the celestial sphere at point V. The altitude angle and azimuth angle of the Sun are \(\theta_{S}\) and \(\varphi_{s}\), respectively, and the altitude angle and azimuth angle of the space target detected by the ground-based optical system can be expressed by \(\theta\) and \(\varphi\), respectively. South, north, east and west directions are replaced by S, N, E, and W respectively.  
Figure 3: Horizontal coordinate system diagram.  
Figure 2: DOP simulation curves of materials under different models, (**a**) Aluminum. (**b**) Black paint materials.  
According to the study of the Rayleigh scattering process in this paper, the scattering plane in the geodetic coordinate system is the plane where SOV is located, the scattering angle \(\gamma\) is the angle between the ground-based optical system and the sun line OS and the ground-based optical system detection direction OV, and the polarization angle can be expressed by \(\chi=90^{\circ}-\angle ZVS\).  
According to the cosine theorem of spherical triangles, the scattering angle can be derived to satisfy the following geometric relations:  
\[\cos\gamma=\cos\Bigl{(}\frac{\pi}{2}-\theta\Bigr{)}\cos\Bigl{(}\frac{\pi}{2}- \theta_{S}\Bigr{)}+\sin\Bigl{(}\frac{\pi}{2}-\theta\Bigr{)}\sin\Bigl{(}\frac{ \pi}{2}-\theta_{S}\Bigr{)}\cos(\varphi-\varphi_{S}) \tag{26}\]  
The result is:  
\[\cos\gamma=\sin\theta\sin\theta_{S}+\cos\theta\cos\theta_{S}\cos(\varphi- \varphi_{S}) \tag{27}\]  
The expression can be derived from the basic definition of the polarization degree as follows:  
\[P=P_{\text{max}}\biggl{(}\frac{1-\cos^{2}\gamma}{1+\cos^{2}\gamma}\biggr{)} \tag{28}\]  
where, the \(P_{\text{max}}\) maximum polarization of the sky background is the maximum, and the presence of fully polarized light in the sky background light is 1.  
Before finding the polarization angle expression, it is necessary to derive the expression \(R(r,\theta,\varphi)\) of the sunlight vector at the space position according to Rayleigh \(e\) scattering law as follows:  
\[e=v\cos\chi+h\sin\chi \tag{29}\]  
where, \(\
u=-\left(\cos\theta\cos\varphi,-\cos\theta\sin\varphi,\sin\theta\right)\), \(h=\left(-\sin\varphi,\cos\varphi,0\right)\), \(r\), \(\theta\), and \(\varphi\) are the coordinate components of the target in the ground level coordinate system. The expression of the space object polarization angle information \(\chi\) at the \(R(r,\theta,\varphi)\) ground level coordinates is derived by setting the ground level coordinates of the Sun S to \((r,\theta_{S},\varphi_{S})\).  
\[\tan\chi=\frac{\sin\theta\cos\theta_{S}-\cos\theta\sin\theta_{S}\cos(\varphi- \varphi_{S})}{\sin(\varphi-\varphi_{S})\sin\theta_{S}} \tag{30}\]