2 Applications of lunar laser ranging

Lunar laser ranging has many applications in various domains including astronomy, lunar science, geodynamics, and gravitational physics (Dickey et al. 1994); (Nordvedt 1996); (Williams et al. 1996). In astronomy, LLR data are used to build lunar ephemeris relying only on these data. The millimetric data provide a determination of radial distance variations with 6 mm accuracy and can improve the angular-rate and the mean distance uncertainties. LLR data contribute to planetary ephemeris and are essential for the positioning of these ephemeris in the fundamental astronomical reference frame at the milliarcsecond level. Ephemerides are used for spacecraft navigation and mission planning, and for every precise astronomical computation such as reference frame connection or asteroid mass determination.

The study of LLR data provides a lot of information concerning the dynamics of the Moon: the gravitational harmonics, the moments of inertia and their differences, the lunar Love number k₂, and variations in the lunar physical librations. As these values are related to the composition of the Moon, we can deduce the mass distribution, the internal dynamics, and obtain information on the Moon's structure. The Love number k₂ measures the tidal changes in the moments of inertia and gravity. The apparent k₂ obtained from LLR analysis, 0.0302$\pm$0.0012, is larger than expected from models, perhaps due to the presence of a small core-boundary ellipticity. The millimetric data could solve this problem by exhibiting smaller periodic terms allowing the separation of the k₂ from the core ellipticity effects. The millimetric data could also improve the separation of the competitive dissipative terms in the secular acceleration of the Moon, and the determination of the 2.9 years arbitrary libration of the Moon in longitude, which is probably due to core boundary effects.

In the field of geodynamics, the analysis of both LLR and VLBI data permitstet to determine the Earth's precession and nutation. LLR permits the faster determination of the Universal Time (Earth rotation). It allows us to determine the Earth's station co-ordinates and motion, the GM of the Earth, and to yield information about the exchange of angular momentum between the solid Earth and the atmosphere, as well as on the tides acceleration of the Moon. The millimetric data would improve the determination of the increase of the Moon's distance, presently 3.82$\pm$0.07 cm/year, and consequently the estimation of the tides which cause the tidal acceleration responsible for the moving away of the Moon and the slowing down of the Earth's rotation.

LLR has contributed to solar system tests in gravitational theories and is at present the best way to test the principle of equivalence for massive bodies. Following Nordtvedt, violation of the principle of equivalence should cause the polarization of the Moon's orbit about the Earth-Moon centre of mass in direction of the Sun. Currently, LLR analysis gives the ratio of the gravitational mass $M_{\rm G}$ to the inertial mass $M_{\rm I}$ for the Earth (as compared to the Moon): $M_{\rm G}/M_{\rm I}-1=(2\pm 5)$ 10^-13, corresponding to $C_{0}\eta =-0.7\pm 1.4~\rm cm$ or $\eta =-0.0005\pm 0.0011$,where C₀ is the characteristic size of the polarized orbit elongation and $\eta$ is the Nordtvedt coefficient. The above result can be interpreted as a test of the parameter $\beta _{\rm R}$ from the Parametrized Post-Newtonian, providing $\beta _{\rm R}=0.9999\pm 0.0006$. The millimetric data could lead to a better than 10^-4 precision of the $\beta _{\rm R}$coefficient measuring a superposition of gravitational effects. A second important test is the measurement of the relativistic precession of the lunar orbit in agreement with the predictions of General Relativity of 0.9%. LLR also provides a test of a possible change in the gravitational constant G.