# A Tutorial on Retroreflectors and Arrays Used in Satellite and Lunar Laser Ranging
## 7 Lunar Laser Ranging  
Lunar Laser Ranging (LLR: h-384,000 km) tracks the distance and angular position of the surface retroreflector array relative to the Earth-based ground station and contributes to a wide range of applications, including astronomy, lunar science, gravitational physics, geodesy, and geodynamics [10]. From Equation (1), the return signal strength varies as 1/R\({}^{4}\), and typically meter-class telescopes, combined with relatively high-energy lasers, were required to achieve detectable signals. The earliest lunar-range measurements were conducted in 1962 by MIT Lincoln Laboratories by reflecting high-energy laser pulses (50 Joules) off the lunar surface and recording the received photons [11]. Subsequently, in 1969, in order to reduce the ground station telescope and laser energy requirements and simultaneously improve range accuracy to a specific lunar location, an array of one hundred 38 mm-diameter retroreflectors was placed on the lunar surface by the manned NASA Apollo 11 mission. In subsequent years, two other manned Apollo missions (14 and 15) also placed retroreflector panels on the Moon. Of these, the Apollo 15 array was the largest (300 vs. 100 cubes), thereby increasing the effective array cross-section (and return signal strength) by a factor of three relative to the earlier Apollo 11 and 14 arrays. Two unmanned Soviet Lunakhod (17 and 21) missions also landed arrays provided by France. Images of the current arrays and their locations on the lunar surface are provided in Figure 11.  
Figure 10: Example of “clocking” where two adjacent “spoiled” retroreflectors are rotated by 30 degrees with respect to one another in order to fill in potential gaps in the array FFDP due to “spoiling”. The angular radius to the inner and outer dark circles in the figure correspond to the minimum and maximum values of a plotted in Figure 8.  
Although the Moon presents the same face to the Earth at all times, the angular velocity of the arrays as viewed by a particular Earth station varies slightly on a monthly basis due to a varying lunar velocity resulting from the Moon's slightly elliptical orbit about the Earth and, on a daily basis, due to the Earth's rotation about its polar axis. The Earth station velocity decreases with station latitude and also with time as the Moon approaches the station horizon. Fortunately, the beams reflected from the small-diameter (\(D_{cc}=38\) mm) cubes used in the current lunar arrays are highly divergent (on the order of \(4\lambda/D_{cc}=4\) (532 nm/38 mm) = 56 \(\upmu\)rad for the widely used 532 nm wavelength), and therefore velocity aberration has minimal effect on the cube performance. On the negative side, the high divergence of the retros also meant that many reflectors would be required to achieve measurable return signals, and, as discussed previously in Section 5 of this paper, the larger panels produce more pulse spreading at increasingly large incidence angles, resulting in diminished range precision. Using the Earth-moon distance, \(R_{EM}=h+R_{E}=384.4\times 10^{6}\) m, in Equation (12), we obtain \(\alpha_{max}=6.74\upmu\)rad or 1.40 arcsec and \(\alpha_{min}=6.68\upmu\)rad or 1.39 arcsec at an elevation angle of 20 degrees, where the relative velocity between the lunar target and the Earth ground station due to lunar orbital motion \(v=1\) km/s. However, the latter equations ignore the smaller contribution of the station's motion due to the Earth's rotation (\(\sim\)0.46 km/s) to the relative velocity, which typically reduces \(\alpha\) to 4 or 5 \(\upmu\)rad for LLR but is negligible for LEO to GEO satellites.