## 2 Determination of the Satellite Observation Mode  
According to whether the sensor pointing is controlled with the operation of the satellite, the observation mode of the satellite can be divided into two categories:  
1. The passive observation mode (the natural rendezvous mode). In the passive observation mode, the observation satellite does not need to control the pointing of the sensor, which is usually fixed on the satellite and only relies on the characteristics of its own orbital motion to passively observe the GEO target. The Sapphire Project in Canada, the surveillance satellite constellation built by Lockheed Martin Space Systems in the United States, and the SBO telescope photographed by ESA scientists all adopt this observation mode. 2. The active observation mode. In the active observation mode, the pointing of the satellite sensor is adjusted through different strategies in order to cover certain space areas. The PP area gaze observation mode proposed by the Lincoln Laboratory adopts the active observation mode. In order to choose an observation mode suitable for the satellite studied in this paper, we first divided GEO targets into three categories: the valid satellites, the failed satellites, and other space debris. Among them, the valid satellites can maintain a \(0^{\circ}\) inclination angle through continuous orbital maneuvers, while the failed satellites and other space debris undergo periodic evolution under the influence of perturbation forces. Next, we analyzed the evolution of the failed satellites and other space debris. In inertial space, the orbital plane normal vector expressed as \(I\) determines the position and orientation of the orbit plane (see Figure 1a). \[I= \begin{bmatrix}i_{x}\\ i_{y}\\ i_{z}\end{bmatrix} = \begin{bmatrix}\sin i\sin\Omega\\ -\sin i\cos\Omega\\ \cos i\end{bmatrix} , \tag{1}\]  
where \(i\) is expressed as the orbital inclination, and \(\Omega\) is expressed as the right ascension of the ascending node (RAAN). Under the combined influence of the Earth's oblateness, the zonal sector, and the lunisolar gravitation, the orbital inclination vector of the GEO celestial bodies will drift clockwise around the pole at coordinate \((0,-7.4^{\circ})\) within a period of 54 years [21]. The circular curve in Figure 1b represents the approximate relationship between the RAAN and the orbital inclination of the invalid GEO target at any time. In the Earth-centered inertial (ECI), the target's right ascension \(\alpha\) and declination \(\delta\) are generally used to represent the target's trajectory. Since \(\sin i>0\), and \(\Omega\) of GEO objects beyond control are in the region of \(\Omega\in(-90^{\circ},90^{\circ})\), the result leads to [21]. \[\sin i=2B\cos\Omega \tag{2}\]  
In addition, through the relationship between \(i\), \(\Omega\), \(\alpha\), and \(\delta\), the extreme condition of \(\delta\) is deduced. The result shows that the \(\delta\) of GEO targets are always between envelopes \(\delta_{\text{max}}\) and \(\delta_{\text{min}}\)[21]. \[ \begin{cases}\delta_{\text{max}}=\sin^{-1}[\text{B}(\sin\alpha+1)]=\text{B}/( \sin\alpha+1)\\ \delta_{\text{min}}=\sin^{-1}[\text{B}(\sin\alpha-1)]=\text{B}/(\sin\alpha-1) \end{cases}  \tag{3}\]  
Let B be equal to \(7.4^{\circ}\)[14; 21], then the value range of B\({}^{\prime}\) can be obtained from Equation (3) as \((7.4^{\circ},7.5^{\circ})\). Considering the limitation of the value range of \(\alpha\), and in order to make the envelope interval cover most of the GEO targets, the B is set to be approximately \(7.5^{\circ}\)[21]. Figure 2a shows the relationship between \(i\) and \(\Omega\). Figure 2b shows the theoretical distribution of GEO objects beyond control. Therefore, the trajectories of GEO targets can be surrounded by envelopes \(\delta_{\text{max}}\) and \(\delta_{\text{min}}\). The trajectories of the GEO target present a band-like distribution of sine equal width, and the \(\delta\) span of the GEO band at any \(\alpha\) is approximately \(15^{\circ}\).