# Non-Cooperative Target Ranging Based on High-Orbit Single-Star Temporal-Spatial Characteristics
## 2 Establishment of the Distance Measurement Model
\[k_{1}\mathbf{v_{1}} =S1-T1 \tag{5}\] \[k_{2}\mathbf{v_{2}} =S2-T2 \tag{6}\]  
Assuming a short time interval between the two samples (e.g., 1 min) and a near-zero orbital inclination, the motion of target satellite T in the Z-axis direction can be approximated as nearly 0 (the high-orbit target is on a relatively stable orbit with minimal movement in the Z-axis direction) [20; 21]. Variation along the Z-axis is illustrated in Figure 4. Figure 5 shows the 24 h orbital Z-axis variation relative to the X and Y-axes. The analysis during ranging, with a 1 min sampling interval, revealed minimal changes in the Z-axis (0.4\({}^{\circ}\) inclination within 600 km at 0.2 km/min). Consequently, we assumed Equation (7):  
\[z_{t1}=z_{t2}\]  
Figure 2: Direction of the camera’s optical axis. \[\mathbf{v_{T}}=\mathbf{M}_{\mathrm{J2000}}\mathbf{U} \tag{2}\]0 High\(-\)orbit satellite at 0.161\({}^{\circ}\) inclination and change in \(Z\)-axis in the 24 h orbit (J2000 coordinate system). At short sampling intervals, such as 1 min, the variation in the \(Z\)-axis is approximated according to Equation (7). \[\mathbf{v_{T}}=\mathbf{M}_{\mathrm{J2000}}\mathbf{U} \tag{2}\]1 High\(-\)orbit satellite at 0.161\({}^{\circ}\) inclination and the change in the 24 h orbit (J2000 coordinate system). \[\mathbf{v_{T}}=\mathbf{M}_{\mathrm{J2000}}\mathbf{U} \tag{2}\]2 Motion process and direction change of the two satellites. The blue line represents the orbit of the target satellite (T), the green line represents the orbit of the satellite (S), and OXYZ represents the celestial coordinate system. According to [PERSON]'s first law, the satellite's orbit is elliptical [22; 23], as shown in Equation (8), where \(a\) represents the semimajor axis, \(e\) indicates the eccentricity, \(r\) denotes the polar radius, and \(\theta\) represents the true anomaly, as shown in \[\mathbf{v_{T}}=\mathbf{M}_{\mathrm{J2000}}\mathbf{U} \tag{2}\]3  
\[\mathbf{v_{T}}=\mathbf{M}_{\mathrm{J2000}}\mathbf{U} \tag{2}\]4  
Based on the characteristics of high\(-\)orbit satellites, when the eccentricity \(e\) is small, the elliptical orbit can be approximated as a circular orbit [24]. Additionally, when the inclination is small, its projection onto the XY plane forms a circular orbit, as shown in Figures 5 and 7. Therefore, an approximation can be made, as shown in Equation (9). \[\mathbf{v_{T}}=\mathbf{M}_{\mathrm{J2000}}\mathbf{U} \tag{2}\]5  
Furthermore, the variation in XYZ coordinates in the J2000 coordinate system can be calculated by converting the orbital elements to the J2000 coordinate system. First, the orbital elements are defined as follows: semi-major axis (a), eccentricity (e), inclination (i), right ascension of the ascending node (\(\Omega\)), argument of periapsis (\(\omega\)), and mean anomaly (M). Additionally, the eccentric anomaly (E) and true anomaly (f) are defined for the computational process. From [PERSON]'s equation, we can obtain the following:  
\[\mathbf{v_{T}}=\mathbf{M}_{\mathrm{J2000}}\mathbf{U} \tag{2}\]6  
\[\mathbf{v_{T}}=\mathbf{M}_{\mathrm{J2000}}\mathbf{U} \tag{2}\]7 Schematic of [PERSON]’s elliptical orbit. \[\mathbf{v_{T}}=\mathbf{M}_{\mathrm{J2000}}\mathbf{U} \tag{2}\]8 Orbital projection on the XY plane at 0.161\({}^{\circ}\) inclination, as described by Equation (9). where the eccentric anomaly (E) can be calculated iteratively from the mean anomaly (M). The satellite's position in the J2000 coordinate system can be calculated [25]:  
\[ \begin{pmatrix}x_{J2000}\\ y_{J2000}\\ z_{J2000}\end{pmatrix} =\frac{a(1-e^{2})}{1+e\cos(f)} \begin{pmatrix}\cos(\Omega) \cos(\omega+f)-\sin(\Omega)\sin(\omega+f)\cos(i)\\ \sin(\Omega)\cos(\omega+f)+\cos(\Omega)\sin(\omega+f)\cos(i)\\ \sin(\omega+f)\sin(i)\end{pmatrix}  \tag{11}\]  
Equation (11) includes the exact expression for \(z_{J2000}\). When the eccentricity (e) approaches 0, we have the following relationship:  
\[z_{J2000}=a\sin(i)\sin(\omega+f) \tag{12}\]  
In this case, a, i, and \(\omega\) are constants, and the value of f ranges from 0 to \(2\pi\).