page_title: Lunar theory
section_title: Digital computers and lunar laser ranging
hierarchical_section_title: Lunar theory / Modern developments / Digital computers and lunar laser ranging
caption_attribution_description: English: This photograph shows the Laser Ranging Facility at the Geophysical and Astronomical Observatory at NASA's Goddard Spaceflight Center in Greenbelt, Md. The observatory helps NASA keep track of orbiting satellites. In this image, the lowerforeground? of the two green beams is from the Lunar Reconnaissance Orbiter's dedicated tracker. The other laser-beam originates from another ground system at the facility. Both beams are pointed at the LRO in orbit.
context_page_description: Lunar theory attempts to account for the motions of the Moon. There are many small variations in the Moon's motion, and many attempts have been made to account for them. After centuries of being problematic, lunar motion is now modeled to a very high degree of accuracy.
  Lunar theory includes:
  the background of general theory; including mathematical techniques used to analyze the Moon's motion and to generate formulae and algorithms for predicting its movements; and also
  quantitative formulae, algorithms, and geometrical diagrams that may be used to compute the Moon's position for a given time; often by the help of tables based on the algorithms.
  Lunar theory has a history of over 2000 years of investigation. Its more modern developments have been used over the last three centuries for fundamental scientific and technological purposes, and are still being used in that way.
context_section_description: Since the Second World War and especially since the 1960s, lunar theory has been further developed in a somewhat different way. This has been stimulated in two ways: on the one hand, by the use of automatic digital computation, and on the other hand, by modern observational data-types, with greatly increased accuracy and precision.
  Wallace John Eckert, a student of Ernest William Brown who worked at IBM, used the experimental digital computers developed there after the Second World War for computation of astronomical ephemerides. One of the projects was to put Brown's lunar theory into the machine and evaluate the expressions directly. Another project was something entirely new: a numerical integration of the equations of motion for the Sun and the four major planets. This became feasible only after electronic digital computers became available. Eventually this led to the Jet Propulsion Laboratory Development Ephemeris series.
  In the meantime, Brown's theory was improved with better constants and the introduction of Ephemeris Time and the removal of some empirical corrections associated with this. This led to the Improved Lunar Ephemeris (ILE), which, with some minor successive improvements, was used in the astronomical almanacs from 1960 through 1983 and was used in lunar landing missions.
  The most significant improvement of position observations of the Moon have been the Lunar Laser Ranging measurements, obtained using Earth-bound lasers and special retroreflectors placed on the surface of the Moon. The time-of-flight of a pulse of laser light to one of the retroreflectors and back gives a measure of the Moon's distance at that time. The first of five retroreflectors that are operational today was taken to the Moon in the Apollo 11 spacecraft in July 1969 and placed in a suitable position on the Moon's surface by Neil Armstrong.
  Its precision is still being extended further by the Apache Point Observatory Lunar Laser-ranging Operation, established in 2005.
caption: The Laser Ranging Facility at Goddard Spaceflight Center
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