# Reduction of Satellite Signature Effect in High-Accuracy Satellite Laser Ranging to Etalon
## 2 Satellite Signature Effect  
Satellite laser ranging measures the distance between the telescope reference point of the terrestrial observatory and the center of mass of a satellite equipped with multiple retroreflectors. It is ideal for a satellite to have a spherical shape and to behave as an optical point-of-light target. However, Etalon is a highly dense spherical satellite in reality. It consists of 12 segments that are connected via welding on both the internal and external surfaces. A titanium and aluminum body composes Etalon satellite (diameter 1.294 m, weight 1415 kg). It has a hollow spherical shape and a wall thickness of 100 mm. A total of six germanium and 2140 fused-quartz corner cube reflectors of the same dimensions make up the array of retroreflectors. The positions of the corner cube reflectors on Etalon area units are shown in Figure 1[10]. It can be seen that the distribution of arrays on Etalon satellite is not homogeneous. Retro-reflectors are positioned in every bracket with a large distance between them. Based upon the geometrical characteristics of Etalon satellite, we can deduce the contribution of each retroreflector to the range measurement by using the retroreflector uneven distribution model [11]:  
\[\mathrm{P_{i}(t)=n_{i}\times S_{i}\times I(t)}, \tag{1}\]  
where \(n_{i}\) is the reflectance of the retroreflector, \(S_{i}\) is the reflection area, and \(I(t)\) is the incident beam. The relative time delay \(\Delta R_{delay}\) of each retroreflector can be written as [12]:  
\[\Delta R_{delay}=R\times\cos\theta_{i}-L\times\sqrt{n_{i}^{2}-\sin^{2}\theta}, \tag{2}\]  
Figure 1: Arrangement of corner cube reflectors installed on the surface of Etalon. where R represents the radius from the satellite center to the surface of cube corner and \(\theta_{i}\) is the angle between the normal of the retroreflector and the incident beam. Equations (1) and (2) indicate that the contribution of each retroreflector participated to the reflection is different. The response \(P_{r}(t)\) of Etalon can be obtained by summing up these individual returns with neglecting the interference between the return signals. \[P_{r}(t)=\sum_{i=1}^{k}P_{i}(t), \tag{3}\]  
The final signal \(P(t)\) received by the SLR system is obtained with the convolution of the system errors and the model of SPAD [12]:  
\[P(t)=P_{r}(t)\otimes P_{e}(t)\otimes P_{j}(t), \tag{4}\]  
where \(P_{e}(t)\) is the SLR system error function and \(P_{j}(t)\) is the time jitter or the SPAD in SLR system. Figure 2 illustrates how the brightness of each retroreflector on Etalon satellite varies with different laser incident angles. The reflected signals from the nearer reflectors are stronger and arrive earlier than those from the farther ones. Reflector illumination patterns vary according to the angle of the incident laser. A recent report has found that the retroreflector area on Etalon is affected by the incident angles [13]. Consequently, the geometrical parameters of the retroreflectors on Etalon are supposed to be responsible for the apparent difference in reflected energy density. Further, the inhomogeneous retroreflector distribution (some space must be allocated to the holders and retroreflector separation) is a possible factor causing the serious satellite signature effect. We simulate the response of Etalon satellite by using different incident angles, as shown in Figure 3. Variations in retro segments participating in the reflection result in different distributions of echo signals, as can be seen from the graph. It is apparent that Etalon's return signal distribution is significantly skewed towards shorter ranges with a long tail. There are still several reflections occurring at distances of up to 250 mm from the peak of the distribution. The waveform produced by the observations has been deformed and widened, and it exhibits multiple peaks at certain angles of incidence. Figure 2: Distribution of laser reflection intensity of Etalon satellite angular reflector in different incident directions.