Geodetic Measurements with Lunar Laser-Ranging
Abstract
An analytic work has been undertaken in order to evaluate the possibility of precise determination of the various parameters involved in ranging a fiducial point of the Moon. Such an experiment has been in project for the last two years at the Pic-du-Midi Observatory and the Ecole Polytechnique. Ranging is to be effected by measuring the round trip time of a light pulse refiected from a lunar cube corners-panel. Using an accurate lunar Ephemeris and estimated values for pertinent parameters will yield a theoretical value of observatory to retrorefiectors distance. The analytical expressions giving the distance have been computed and derived, but only the main terms, neglecting the inclination of the Moon equator on the ecliptic, are taken into account for the following discussion. Comparison with effective results will give residuals originating in the theory of the motion of the centre of gravity of the Moon and in errors on the assumed parameters values. Provided the duration of measurements is short enough, one can assume a linear relationship between residuals and the latter errors. These relations are given (formula 4) for the five parameters for which the method is the most sensitive. They include w cos and w sin , where w is the distance of the observatory to the instantaneous rotation axis of the Earth, and is the geocentric declination of the centre of gravity of the Moon. The other parameters are the geocentric hour angle of the centre of the Moon (H), the difference (A - r) between the geocentric distance of the Moon and the radius of the Moon, and the distance of the observatory to the Earth equatorial plane. A first step will call for quasi-continuous ranging, from one observatory, over a period extending from 6 to 8 hours. In this case, we assume differences between estimated and actual values of the various parameters to be constants, or, in the worse case, linear functions of time, during the whole period. The assumption is permissible if a very precise lunar Ephemeris is used such as the LE 5 of the Jet Propulsion Laboratory, where the only errors are due to erroneous constants of integration. It then appears possible to evaluate with great accuracy w cos 6, H and A - r. If the number of distances is large enough to average out fluctuations (typically, 200 are considered), normal equations predict a final 0.6 m error about w, and 0.02" error about H. In a second step, the above determinations will be conducted from two or more observatories, yielding their respective locations since both their distance to Earth axis and longitudes difference will be known. As a consequence, the position of Earth axis with respect to the observatories can be determined, and so will be its displacements, provided of course the observing stations polyedra is rigid. It must be noted, however, that axis displacements having periods shorter than a day are outside the scope of the above method. The errors in the determination of axis position have been studied in function of the geographic positions of the stations, in the case of two or three observatories. It appears that optimum is very broad, yielding accuracies of 0.4 m and 0.02", respectively, for axis position and orientation.
- Publication:
-
Astronomy and Astrophysics
- Pub Date:
- February 1970
- Bibcode:
- 1970A&A.....4...18C