Jottings on Celestial and Great Circle Navigation

In response to a network news item I posted, several people asked for a simple explanation of navigation. What follows is my reply.

Shape of the Earth

The shape of the earth is irregular, being more like a pear (thicker at the top) than a sphere. For the purposes of measurement, an imaginary surface called the 'geoid' is used. This surface is the level to which the oceans would settle when the forces of gravity and centrifugal forces of rotation are taken into account. It is rarely used for navigation, but surveyors and people who write code for satellite navigation and mapping systems need to use this model. A reasonable approximation for navigation purposes is to model the earth as an oblate spheroid. The spheroid is usually specified as a semimajor axis and a flattening factor. From these values the semiminor axis and eccentricity can be calculated. These values are formalised in the "World Geodesic System" and the current values are :- 

 			WGS 1984

           a  = 6378137 metres (semimajor axis)

                      1
           f  = _____________         (flattening factor)
                298.257223563
                      
                a - b
              = _____                 (where b is semiminor axis)
                  a

          e^2  = 2f - f^2             (eccentricity)

Unless you require absolute accuracy (surveys, satellite mapping), most calculations of great circle distances use straight spherical geometry. For short distances (less than 600 nautical miles (NM)), distances are calculated using meridional parts either from tables or with a computer program (and brief documentation). A rhumb line is a straight line on a Mercator map projection. It intersects all the parallels of latitude at a constant angle. Except for courses that are East/West along the equator or North/South (which would be great circles), these lines would be a spiral on the earth's surface (a loxodrome). A great circle track would be a curve on a Mercator projection and a straight line on a Gnomonic projection (usually only used for high latitudes).

Mercator Projection

Gnomonic Projection

For navigation, great circles are only worth while for long.... passages. If you left Durban (South Africa) and sailed/flew to Perth (Australia), the great circle distance is 4244 nautical miles, while the rumb distance is 4376, a saving of 132 nautical miles. The compass heading that must be steered to achieve a great circle changes continuously throughout the voyage. It is usual practice to break the voyage into segments and calculate the rhumb approximation for each segment. If you are sailing you could make greater gains in speed by taking advantage of known currents and winds rather than using a great circle track.

Celestial Navigation

Many people make the mistake of thinking that is is possible to calculate your position at any time using sun/star fixes. Except for a few special cases, it is only possible to use stars to make a direct fix when both the stars and the horizon are visible ie just before sunrise and just after sunset. During the day the sun is available but a single sight can only give a line of position on a map, not your exact position. A second sun sight taken some hours latter is required to get two intersecting lines (the earlier line is 'run on' to account for the change in position; commonly called a sun-run-sun). If the moon or a planet is available then is possible to take them in conjunction with the sun and establish an observed position.

The data required to make a position fix are as follows :-

  1. Accurate knowledge of the time, either GMT (UTC) or local time and time zone. An error of four seconds is equal to one nautical mile at the equator.
  2. An accurate measurement of the altitude (angle to the horizon) of the body (sun. moon, planet or star). One minute of arc error is equal to one nautical mile. This measurement has to be corrected for height of eye, dip of the horizon, refraction of the atmosphere and possibly parallax (moon and some planets) and unusual temperatures or pressures.
  3. The declination of the body from tables of ephemerals like a nautical almanac, air navigation tables or a computer program.
  4. An estimate of your position.

Sight Reduction

The taking of a sight and the calculation using that data is called 'sight reduction'. For a single sight, the result is a bearing and distance from your estimated position to a line that you are on (called a line of position). You could be at any position along the line. A second sight taken of another object at least 40 degrees away from the first will give a second line of position. Where the two intersect is your observed position. Taking a third sight will give another line, though they will probably not all intersect at the same point. You will probably get a small triangle (or cocked hat).

At sunrise/sunset you can probably take several sights of stars in the available time (about 20 minutes depending of latitude). During the day you will usually only have the sun. In this case you have to take measurements some hours apart and plot them. If you have moved between the observations, then you have to compensate for the movement when you plot the positions.

Meridian Passage

Meridian passage is handy since neither the time or an estimate of your position is required (except for hemisphere). The technique is to observe the sun (or moon) and measure its altitude when it is at its highest point (it will always be due North or South). You will have several minutes where the sun will appear to hover, before it's altitude starts decreasing. Your latitude can be calculated directly given this altitude and the declination of the body (from the almanac). With an accurate chronometer, it is possible to calculate longitude. You will have to take a dozen or more sights starting about 20 minutes before meridian passage and 20 minutes after, then plot the altitudes again time and estimate the exact time for the meridian passage. This time can be translated into longitude.

Prime Vertex

This is when the body is rising or setting, usually the Sun. Due to atmospheric distortion the Sun will be one semidiameter above the horizon when it has just risen. The true bearing of the object can be calculated given an estimate of your latitude. It is mainly used as a compass check though the time of this event can be used to calculate longitude. Also known as a true amplitude.

Instruments

The basic instruments required for celestial navigation are an accurate chronometer (read quartz watch) and a sextant. A good instrument should be accurate to 0.1 minutes of arc. It is debatable that it is possible to take sights to this accuracy. The problem with using a sextant is that a clear horizon must be available as a reference point. The only accurate horizon is the sea surface, therefore the marine sextant is useless inland. It is possible to make an artificial horizon using a bath of oil or mercury to reflect the object being measured and halving the angle obtained. There is a variant of the sextant called a 'Bubble Sextant' that has an inbuilt level to determine the horizon. This is only useful if the sextant is on a stable platform and was developed mainly for use in aircraft.

Tables

The calculations required for most celestial navigation can be done on a simple scientific calculator. Replacing the nautical almanac is another matter. The latter provides for each day of the year the following :-
  1. The Greenwich Hour Angle (GHA) and declination for the Sun, Moon, Planets and Aries for each hour of the day.
  2. The declination and Sidereal Hour Angle (SHA) for 57 selected stars.
  3. The times of sunrise, sunset, moonrise, moonset and twilight for various latitudes.
  4. Meridian passage times for the sun, moon and planets.

Supplementary tables are provided for the following:-

  1. Dip and height of eye corrections.
  2. Altitude corrections for the sun, stars and planets.
  3. Additional altitude corrections for non-standard conditions of temperature and air pressure.
  4. Star Charts.
  5. Addition data for 173 stars.
  6. Tables for Polaris.
  7. Sight reduction tables.
  8. Conversion of arc to time.
  9. Increments and corrections to GHA and declinations for minutes and seconds after a whole hour.

Programs

As stated in the previous section, replacing the almanac would be difficult. There have been a few programs on the network, namely 'planets' and xephem that will provide declinations and right-ascensions (RA) for the sun, moon and planets. It is fairly easy to convert RA to GHA. The rest of the calculation for sight reduction involves making all the corrections for the sight and solving a half haversine equation.

Reference Books

American Practical Navigator, Bowditch

The lastest edition is a single volume of 1350 pages, last updated in 1995. It removes a lot of the older navigation methods and tables, and updates various sections, especially electronic navigation and the introduction of Navstar (GPS) which the replaces Navsat (Transit) satellite system.

The older version is a two volume set published by the US Defence Mapping Agency Hydrographic/Topographic Center. It was first published in 1802.

Volume I (1414 pages) covers fundamentals, plotting and dead reckoning, celestial navigation, the practice of navigation, navigational safety, oceanography, weather and electronics (satellite, Omega, Loran and other hyperbolic systems). All for the princely sum of $45 from Boat Books in Sydney.

Volume II (961 pages) has useful tables, mathematics for navigation, and navigational calculations. It is a good source of information for the formulae that are used to generate the various tables.

Nories Nautical Tables

This is a book of tables for use in navigation, not much theory. 570 pages.

Nautical Almanac

The ephemerals are published each year. The yachtsman's edition is $20 or so.

All maps and projections generated by the GMT (Generic Mapping Tools) package, see www.soest.hawaii.edu/gmt.

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