# Non-Cooperative Target Ranging Based on High-Orbit Single-Star Temporal-Spatial Characteristics
## 3 Experimental Results  
We used simulation software (STK 11.0) to establish a scenario for the primary and target satellites to validate the algorithm proposed in this paper. First, we set the orbital parameters for the primary satellite and the different target satellites. At different times, the orbital parameters of the primary satellite and the pointing vector toward the target satellite are known. The actual distance between the two satellites can be obtained through the simulation software, and this distance serves as the standard answer for validating the algorithm presented in this paper. Our algorithm does not know the orbital parameters of the target satellite. We compared the distance calculated by the algorithm with the standard answer to analyze the algorithm's error.  
The orbital parameters for the host satellite in the ranging experiment were as follows: semimajor axis, 42,166 km; eccentricity, 0.0005; and inclination, 5\({}^{\circ}\). For the target satellite, the corresponding values were as follows: semimajor axis, 42,176 km; eccentricity, 0.0002; and inclination, 0.0\({}^{\circ}\)/0.1\({}^{\circ}\)/0.2\({}^{\circ}\)/0.3\({}^{\circ}\)/0.4\({}^{\circ}\)/1.0\({}^{\circ}\).  
Figures 8-12 display the error curves for inclinations of 0.0\({}^{\circ}\), 0.2\({}^{\circ}\), 0.4\({}^{\circ}\), and 1.0\({}^{\circ}\). Ranging accuracy within 600 km was superior to 5% of the detection distance for inclinations between 0.0\({}^{\circ}\) and 0.4\({}^{\circ}\). Furthermore, for inclinations between 0\({}^{\circ}\) and 1\({}^{\circ}\), a ranging accuracy within 100 km range was guaranteed to be superior to 5% of the detection distance.  
Figure 8: Relationship between the calculated distance of the model and the real distance when the inclination is 0\({}^{\circ}\).  
Figure 9: Relationship between distance and error at an inclination of 0\({}^{\circ}\).  
The two \"V\" shapes occur because the ground track of the host satellite forms a figure-eight shape, resulting in two approaches and two retreat events between the two satellites. In Figures 9-14, we represent these two events with the symbols Move-1 and Move-2. Furthermore, we assume that the satellite (S) is an inclined geosynchronous satellite and that the target satellite (T) is a geosynchronous satellite with a small eccentricity (ranging from 0 to 0.001) and a small inclination (ranging from 0.0\({}^{\circ}\) to 0.4\({}^{\circ}\)). The ground track of the satellite (S) forms a figure-eight pattern, which is the trace created by the line connecting the satellite to the center of the Earth and its intersection with the Earth's surface. The satellite (S) experiences two approaches and retreats from the target satellite (T), resulting in the red and black lines shown. In Figure 8, the black dots represent the calculated distance values from the model, the red dots represent the actual distances, and the green dots indicate the error.  
Figure 11: Relationship between distance and error at an inclination of 0.4\({}^{\circ}\).  
Figure 10: Relationship between distance and error at an inclination of 0.2\({}^{\circ}\).  
As shown in Figure 13, under the first set of movements (Move-1), the error curves for different inclinations (0.0\({}^{\circ}\) to 0.4\({}^{\circ}\)) indicate a gradual reduction in error from 0.0\({}^{\circ}\) to 0.2\({}^{\circ}\), followed by a gradual increase from 0.2\({}^{\circ}\) to 0.4\({}^{\circ}\), with a maximum error of approximately 10 km at around 500 km. The ranging error is less than 2% of the measured distance. Figure 14 demonstrates that under the second set of movements (Move-2), the maximum error at a distance of 600 km is less than 30 km. The ranging error is less than 5% of the measured distance.  
Figure 12: Relationship between distance and error at an inclination of 1\({}^{\circ}\).  
Figure 13: Error curve in the first group (inclination 0.0\({}^{\circ}\)–0.4\({}^{\circ}\)).