# Autonomous Navigation Based on the Earth-Shadow Observation near the Sun-Earth L2 Point
## 2 Design of Navigation System
### The Earth-Shadow Measurement Model  
Firstly, we introduce the Sun-Earth L2 point. The Sun-Earth Lagrange point refers to the space balance point between the Sun and the Earth. There are 5 points in total, as shown in Figure 2. The important points are L1 and L2 for humans to explore and utilize space resources in the future, which are located on both sides and 1.5 million kilometers away from the Earth. Since the point L1 is located between the Sun and the Earth, it is of great significance to detect the environment of the Sun and the Earth. Point L2 is located on the other side of the Earth, which faces away from the Sun. Typically, Halo orbits in the vicinity of the Sun-Earth Lagrange points, such as L1 or L2, are regarded as suitable locations for astronomical observation because the spacecraft around these places are able to maintain the same orientation and consume little fuel for station keeping. Halo orbit missions such as ISE-3, WMAP, Herschel, and Chang'e-2 are notable successful projects [38; 39]. Because of numerous advantages such as facing away from the Sun, stable lighting conditions and the reduced effect of visible light and infrared radiation, it is worthy exploring this more deeply in future deep space explorations. In some special circumstances, the Sun is blocked by the Earth, forming the Earth-shadow area. This phenomenon is called Earth-shadow [40; 41; 42]. It is apparent that the Earth-shadow is closely related to the relative position of the Sun, the Earth, and the spacecraft. In fact, the Sun is a ball of gases. The diameter of the Sun varies depending on the optical wavelength considered. It is assumed that the errors regarding the diameter of the Sun are ten kilometers. As shown in Figure 3, the Earth and the Sun on the projection plane are both ellipses, and \(O_{e}\) and \(O_{s}\) are the centers of them, respectively. It is assumed that the equivalent radius of the Sun is \(a_{sa}\). The equivalent radius of the Earth is \(a_{ea}\). Generally, astronomers define the boundary of the Sun with the wavelength of 500 nm. The theoretical radius of the Sun and the Earth are noted \(b_{sb}\) and \(b_{eb}\). The errors of the diameter of the Sun between different wavelengths are far smaller than the distance between the spacecraft and the Earth, the distance between the spacecraft and the Sun, and the diameters of the Earth and the Sun. Therefore, the error of the diameter of the Sun between different wavelengths can be considered as a small quantity. Moreover, the Sun also has a little flattening. The Earth itself is surrounded by atmosphere, which blurs the light and radiation transition between the surface of the earth and vacuum [43; 44]. The function \(f(h)\) that describes the radiation reduction coefficient is defined. This function computes the ratio between the solar radiation at the depth of \(h\) in the atmosphere and the radiation outside of the atmosphere. The thickness \(H_{0}\) of the atmosphere on the image is different depending on the position of the satellite. For any point that has a distance \(h\) to the solid Earth in the direction from the center of the Earth \(O_{e}\) to the center of the Sun \(O_{s}\) on the image, \(f(h)\) is given by  
\[f(h)=(\mu_{2}-\mu_{1})\frac{h}{H_{0}}+\mu_{1}, \tag{5}\]  
where \(\mu_{1}\) is the radiation reduction coefficient at the boundary of the solid Earth and \(\mu_{2}\) is the radiation reduction coefficient at the boundary of the atmosphere. Thus, \(\mu_{1}=0\) and \(\mu_{2}=1\). Figure 3: The model considering the atmosphere and oblateness of the Earth and the Sun. Figure 2: Schematic diagram of Lagrange point between the Sun and the Earth.