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USNO Celestial Navigation Algorithms

Three papers, listed below, published in the (U.S.) Institute of Navigation journal Navigation , provide a set of algorithms for celestial navigation that incorporate a moving observer as part of the basic construction. It assumes a set of observations of the altitudes of stars above the horizon, either from a sextant or some sort of automated star tracker. This approach, based on a least-squares analysis of the observations, is closely analogous to "orbit correction" problems familiar to astronomers who deal with the dynamics of bodies (natural or artificial) in the solar system. Although more complex mathematically than previous sight-reduction schemes (a computer is definitely required), the new procedure provides the course and speed of the vessel along with the fix — if enough observations are available, of course. When only a few observations are available, the procedure still provides a good fix, but cannot provide course and speed information.

The second paper listed below describes the method in detail. The first paper provides a piece of the necessary mathematical foundation — relatively simple but precise formulas that describe a vessel's motion in longitude and latitude as a function of time, as it sails along a rhumb-line track. (It was surprising that such a gap existed in the literature.) The third paper extends the method to multi-leg tracks. These three papers assume that the reader has a knowledge of basic calculus and statistical analysis and they contain a fair number of equations.

The fourth paper in the list, published in the Navigator's Newsletter, is a less technical review of how the motion of the observer has been previously dealt with in celestial navigation. It assumes knowledge of standard celestial navigation practice.

  1. Kaplan, G. H. 1995, "Practical Sailing Formulas for Rhumb-Line Tracks on an Oblate Earth," Navigation, Vol. 42, No. 2, pp. 313–326.   ( Abstract and download from ION or PDF preprint)
  2. Kaplan, G. H. 1995, "Determining the Position and Motion of a Vessel from Celestial Observations," Navigation, Vol. 42, No. 4, pp. 631–648.   ( Abstract and download from ION or PDF preprint)
  3. Kaplan, G. H. 1996, "A Navigation Solution Involving Changes to Course and Speed," Navigation, Vol. 43, No. 4, pp. 469–482.   ( Abstract and download from ION or PDF preprint)
  4. Kaplan, G. H. 1996, "The Motion of the Observer in Celestial Navigation," Navigator's Newsletter, Issue 51 (Spring 1996), pp. 10–14.   (PDF preprint)
    Note: the Navigator's Newsletter is published by the Navigation Foundation .