# Autonomous Navigation Based on the Earth-Shadow Observation near the Sun-Earth L2 Point
## 2 Design of Navigation System
### The Earth-Shadow Measurement Model
In the situation that the Sun is totally blocked by the Earth, the overlapping ratio is defined as follows:  
\[ratio_{theory}=\frac{\pi b_{eb}^{2}}{\pi b_{sb}^{2}};ratio_{error}= \frac{\pi a_{ea}b_{eb}}{\pi a_{sa}b_{sb}} \tag{6}\] \[\Delta_{overlap}=\frac{ratio_{theory}}{ratio_{error}}=\frac{\pi b _{eb}^{2}\pi a_{sa}b_{sb}}{\pi b_{sb}^{2}\pi a_{ea}b_{eb}}, \tag{7}\]  
where \(ratio_{theory}\) is the ratio of the overlapping area in theory condition, and \(ratio_{error}\) is the ratio of the overlapping area while considering the atmosphere and oblateness of the Earth and the Sun in space. \(\Delta_{overlap}\) is the observation model error between the theoretical and real space environment. From the above error analysis, we define the observation model errors \(\mathbf{e}_{obs}\) as follows:  
\[\mathbf{e}_{obs}=f(h)+\Delta_{overlap}. \tag{8}\]  
As shown in Figure 4, the Earth-shadow area can be divided into the umbra shadow area and the penumbra shadow area. The umbra shadow is the area where the Sun is completely shielded by the Earth, and the penumbra shadow is the area where the Sun is partially shielded by the Earth. When the spacecraft passes through the penumbra shadow area, the proportion of the Sun being shielded by the Earth will change continuously, which provides the related information of the spacecraft orbit. The information is directly related to the light intensity of the current position and can be measured by the electric energy production of the solar panel or by optical sensors. As shown in Figure 5, it is assumed that the spacecraft is located at the point O. The Earth and the Sun are located at point E and S. Points \(E_{1}\) and \(S_{1}\) are the tangent points of the two circles, respectively. OE and OS represent the distance from the spacecraft to the Earth and to the Sun, respectively. \(EE^{\prime}\) and \(SS^{\prime}\) represent the radius of the Earth and the Sun, respectively. \(\angle EOS\) is the angle between the Earth, the spacecraft, and the Sun. It is apparent that the values of all the above parameters can be calculated through the ephemeris. The line AG is located in the plane composed of the Earth, the Sun, and the spacecraft and is one astronomical unit away from the spacecraft. In order to characterize the proportion of the Sun being shielded by the Earth, the Sun and the Earth are projected onto a plane, which is defined as the reference plane. The reference plane passes through the straight line AG and is perpendicular to the plane composed of the Earth, the Sun, and the spacecraft. The distance between the spacecraft and the Earth and the distance between the spacecraft and the Sun are far longer than the scale of the spacecraft orbit around the Earth, and also longer than the radius of the Earth and the Sun. Therefore, it can be considered  
Figure 4: Schematic diagram of Earth-shadow area. that \(\angle EOE^{\prime}\) and \(\angle SOS^{\prime}\) are very small, approximately equal to 0, such that the following relations are obtained:  
\[AO\approx BO\approx FO\approx GO\approx 1,\angle EOS=\angle AOF\approx \angle BOF. \tag{9}\]  
Perform auxiliary lines \(AC\perp OE^{\prime}\). It can be deduced, from the geometric relationships, that  
\[\angle GOF=\angle SOS^{\prime},\angle OFG=\angle OS^{\prime}S,\angle OE^{ \prime}E=\angle OCA \tag{10}\] \[f(h)=(\mu_{2}-\mu_{1})\frac{h}{H_{0}}+\mu_{1}, \tag{5}\]0  
According to the principle of triangle similarity, \(\angle OSS^{\prime}\simeq\angle OGF,\angle OEE^{\prime}\simeq\angle OAC\), \(\angle BAC\simeq\angle BOF\). Therefore,  
\[f(h)=(\mu_{2}-\mu_{1})\frac{h}{H_{0}}+\mu_{1}, \tag{5}\]1 \[f(h)=(\mu_{2}-\mu_{1})\frac{h}{H_{0}}+\mu_{1}, \tag{5}\]2 \[f(h)=(\mu_{2}-\mu_{1})\frac{h}{H_{0}}+\mu_{1}, \tag{5}\]3  
In the right triangle \(\triangle ABC\) and \(\angle OGF\), AB and FG are given as  
\[f(h)=(\mu_{2}-\mu_{1})\frac{h}{H_{0}}+\mu_{1}, \tag{5}\]4 \[f(h)=(\mu_{2}-\mu_{1})\frac{h}{H_{0}}+\mu_{1}, \tag{5}\]5  
and the lines AB and FG represent the radius of the Earth and the Sun, respectively.