# The Physics, Astronomy and Mathematics of the Solar System

## The Mathematical Dynamics of Celestial Navigation and Astronavigation

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## THEORY OF NAVIGATION

Position determination in astronavigation is primarily a matter of converting one set of coordinates to the other. This is done by the solution of a spherical triangle called the navigational triangle.

The concept of the spherical navigational triangle is graphically shown in the illustration, a diagram on the plane of the celestial meridian. The celestial meridian passes through the
*
zenith
*
of the observer, and is therefore a vertical circle of the horizon system. Elements of both systems are shown in the illustration below, indicating that an approximate solution can be generated graphically.

(image available in print form)

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Figure
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4, Source: [8].
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The navigational triangle across the spherical surface of Earth.
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The vertices of the navigational triangle are the elevated pole (
*
P
_{
n
}
*
), the zenith (

*Z*), and the celestial body (

*M*). The angles at the vertices are, respectively, the meridian angle (

*t*), the azimuth angle (

*Z*), and the parallactic angle (

*X*). The sides of the triangle are the codeclination of the zenith or the colatitude (colat) of the observer, the coaltitude or zenith distance (

*z*) of the body, and the codeclination or polar distance (

*p*) of the body.

A navigational triangle is solved, usually by computation, and compared with an observed attitude to obtain a line of position by a procedure known as sight reduction. With the emergence of electronic computers and hand-held calculators, sight reduction has been performed increasingly with limited use or elimination of tables.

To establish a celestial line of position, the navigator observes the altitude of a celestial body, noting the time of observation. An observation is made with a sextant, the name of which derives from early instruments that had an arc of one-sixth of a circle. By means of the double reflecting principle, the altitude of the body is double the amount of arc used. The marine sextant uses the visible horizon as the horizontal reference. An air sextant has an artificial, built-in horizontal reference based upon a bubble or occasionally a pendulum or gyroscope. The sextant altitude, however measured, is subject to certain errors, for which corrections are applied. Time is repeatedly mentioned as an important element of a celestial observation because the Earth rotates at the approximate rate of 1 minute of arc each 4 seconds of time. An error of 1second in the timing of an observation might introduce an error in the line of position of as much as one-quarter of a mile. Time directly affects longitude determination, but not latitude. The long search for a method of ascertaining longitude at sea was finally solved two centuries ago by the invention of the marine chronometer, a timepiece with a nearly steady rate.

Coastal Navigation is similar to Celestial Navigation. Instead of celestial reference points, Coastal Navigation determines the location of a moving vessel with reference to a different set of fixed geographical objects such as a lighthouse as illustrated in Figure 5.

(image available in print form)

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**

Figure 5, Source:
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Spinka, 2007. Comparative navigational references: a lighthouse and a celestial object.
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- 1. When navigation references of a lighthouse and a celestial object are compared, the lighthouse always remains at the same geographical location, independent of time.
- 2. The position of the celestial object
CNhowever, is dependent on time, so that the exact time at the moment of the observation and measurement is required for calculations with this variable.- 3. The distance from the celestial object at
CNto the surface of the Earth, at pointCN'is of such a large magnitude that the results are far more difficult to map when compared to those of the lighthouse reference.