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Xi-Eta Coordinates

(glossary entry)


A system for expressing the position of a nearside lunar feature in terms of its X-Y coordinates in a standard zero-libration lunar atlas view. The radius of the Moon is 1 in this system. Xi (a Greek letter) is the horizontal (X) coordinate measured from -1 on the equatorial west limb to +1 on the equatorial east limb. Eta (another Greek letter) is the vertical (Y) coordinate, ranging from +1 at the north pole to -1 at the south pole.

Additional Information

  • Although the IAU convention for east and west on the Moon was reversed in the early 1960's, the convention for Xi is unaffected.
  • Many of the older lunar maps, such as the System of Lunar Craters maps and many others, are plotted on rectangular grids labeled with the Xi-Eta positions.
  • According to the rules of trigonometry, the position of a feature of known selenographic longitude and latitude can be reduced to the Xi-Eta system using the following formulas:
    • Xi = Cos(Lat)*Sin(Lon)
    • Eta = Sin(Lat)
  • A position given in terms of Xi and Eta can be similarly converted to the now more conventional longitude-latitude system:
    • Latitude = ArcSin(Eta) <-- compute this first
    • Longitude = ArcSin[Xi/Cos(Lat)] <-- use computed latitude here
  • The projected positions of a lunar feature as seen on an actual plate (displaced from that feature's standard position by librations) are conventionally referred to as "x" and "y".
  • In their classic book The Moon, Wilkins and Moore use some kind of extended Xi-Eta system to express the positions of features beyond the mean limb, but the rules they use for this are not explained. - Jim Mosher
  • Since the Moon is spherical, rather than flat, a better representation of the three-dimensional position of a lunar surface feature can be obtained by introducing a third parameter Zeta representing the distance "in and out of the paper". Again, this will run from +1 (point closest to Earth) to -1 (point farthest from the Earth). All features that are on the nearside and mean libration will have Zeta>0, while all those on the farside will have Zeta<0. For a spherical Moon of radius 1, the Pythagorean theorem dictates that the sum of the squares of the three parameters will equal one. Hence given Xi and Eta, the magnitude of Zeta must be:

    • Zeta = Sqrt(1 - Sqr(Xi) - Sqr(Eta))

  • with the sign being decided by the features location on the nearside (+) or farside (-). Since the Moon is not perfectly spherical, one might think that the constraint on the sum of squares would be relaxed to permit representing actual points in space with values of the radial distance:

    • R = Sqrt(Sqr(Xi) + Sqr(Eta) + Sqr(Zeta))

  • differing slightly from 1 (indicating small deviations relative to the mean radius). But his does not seem to be the practice among selenographers. See, for example, pages 223-228 of Ralph Baldwin's The Measure of the Moon (1963), where the sum of squares of Xi, Eta and Zeta is always forced to 1, and the deviations from sphericity are represented by a separate parameter, h. - Jim Mosher
  • It might be noted that "east-west" (the Xi coordinate) and "north-south" (the Eta coordinate) in the Xi-Eta system are not always the same as "east-west" (a difference in longitude) and "north-south" (a difference in latitude) in the longitude-latitude system. Indeed, just to the "left" and "right" of the Moon's poles, the relative senses of the two systems can be rotated by as much as 90°. This, and the tendency to think almost always in terms of a projected nearside view, leads to a little-noticed (and perhaps trivial) ambiguity in lunar terminology. The statement that "Nansen is to the east of Peary" is implicitly referring to the projected positions of those two craters in the Xi-Eta system. The equally valid statement that "Nansen is south of Peary" is implicitly referring to the longitude-latitude system. But when the features are unfamiliar, as when one reads that "feature X is located 10 km SE of crater Y" it can be very hard to tell if the writer is thinking in terms of projected Xi-Eta directions, or (what is more properly called a "bearing") a direction relative to the N-S meridian (the line of constant longitude) through crater Y. - Jim Mosher

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