(You can print the following pages and read them at your leisure)
Reprinted from "The Navigator’s Newsletter", Issue 64, Summer 1999.
Foundation for the Art of Navigation
Box 1126, Rockville, MD, USA.
The Complete On-Board Celestial Navigator by George G. Bennett.
An International Marine / McGraw-Hill Book
Camden, Maine, USA.
ISBN 0-07-007110-1. 176 pages. $US27.95 www.internationalmarine.com
A navigator who puts to sea these days without some means of fixing position electronically, in my view, would be most imprudent - some might even say foolhardy. The safety and lives of the crew are at stake and must be of prime consideration. There can be no reasonable argument for not providing some such facility. The cost of purchasing GPS (Global Positioning System) equipment, expressed as a fraction of the value of even a small ocean-going sailing boat plus its maintenance and running costs, is minute. However, the navigator who relies solely on GPS for position fixing is just as imprudent as the navigator who disdains any "new-fangled paraphernalia". The advantages of an all-weather positioning system should be self-evident, but as with all navigation systems a healthy skepticism of the reliability of such devices and the data they provide is essential. It brings to mind a report which stated that the skipper of a high-speed catamaran ferry was relying entirely on GPS to guide him into port in bad weather and ended up running the vessel into a sea wall. To rely totally on GPS positions surely is the height of folly. Plotting one’s course and taking heed of other navigational information as well is paramount in the practice of a responsible navigator. An understanding of accuracy standards, particularly the deliberate downgrading of satellite position and/or clock data, and the uncertainties in charted information, particularly when dealing with old and new datums, is of the utmost importance.
When the navigator is unable to fix position out of sight of land from some external source - e.g. a hyperbolic or satellite system - then, apart from bearings obtained from shore-based radio stations, which are often sparsely located, signs from the weather and condition of the sea - e.g. its color, temperature, movement - all that remains are observations to those celestial objects which can be seen when the horizon is also visible during the day, and at morning and evening twilight. There are, of course, exceptions such as sextant observations of lunar distances, but these are rarely, if ever, used. The norm will be, weather permitting, observations of the Sun, Moon, bright stars and the four brightest planets. Then equipped with a sextant, watch, radio, almanac, and the means for reducing and plotting those observations, the position of the vessel can be determined.
Celestial navigation will normally require the purchase of a text book, an almanac for the current year, a set of tables for reducing sights and, useful though not essential, some means of predicting and/or identifying celestial bodies. The almanac will normally need to be replaced annually. To this writer’s knowledge, no-one has produced, under one cover, the above requirements. Now we have a publication that covers a period of five years containing all the essential elements required for celestial navigation in about 160 pages. The scope of the work encompasses an explanation of the care and adjustment of equipment and details of practical methods for predicting, observing and reducing observations for fixing position and determining azimuth (true bearing). In particular:
Text books vary in quality, length and price, and to overcome the limitations of time, usually include extracts from old almanacs as illustrative examples. This latter problem is avoided here because all the examples use the current data from the almanac section. The explanations in the text book section of this work embrace the standard and accepted methods of position fixing and compass checking, using celestial observations. Some novel treatments of, for example, multiple sextant observations, noon sun-sights etc are included in this work.
The book provides the astronomical coordinates of the Sun, Moon, four navigational planets and 58 stars. This has been achieved by giving all data to the nearest minute of arc (1' ) and at less frequent intervals than conventional almanacs. Some old salts might throw up their hands in horror at this loss of accuracy, but no data item will be in error by more than about 0.5' and, on average, about 0.25'. This standard of accuracy is in keeping with the requirements of what is essentially a back-up system.
Tables are given for interpolating the above coordinates and for correcting sextant observations. For the latter, the corrections are given in the form of critical tables. These tables require no interpolation and limit the errors to a maximum of half a unit in the quantity extracted.
If a calculator/computer is not available, tables and graphs for sight reduction by the intercept method are provided. Those who have made a study of this topic may be interested in what follows, otherwise go straight to (5).
The history of the last two centuries of the development of aids for reducing celestial observations made to determine position has been chronicled in Bowditch. Although there is no claim to completeness in that account, the highlights in the development of a variety of methods has been set out to give the reader a good understanding of how so many of the best minds have been directed to this important navigational problem. The interest in devising a simple practical solution when only basic calculating devices (mainly mechanical and tabular) were available has receded with the establishment of hyperbolic and, later, satellite navigational systems and the calculator/computer. We look on many of these methods in Bowditch as historical curios, as indeed they are. However, we shall still need to provide some means of sight reduction which is independent of electronic or outside agencies.
Before you jump to the conclusion that what is about to follow is another "new" set of sight reduction tables, you may be surprised that the basis of the calculation of altitude for the intercept method given here is by the tried and proven cosine-haversine formula. In 1979, P F Pfab of the Honorable Cross-Staff Society of Sweden published his so-called PET Tables (reviewed in the US and British Journals of Navigation) which were based on the cosine-haversine formula but only after a detailed investigation of tabular methods had been made. His analysis took into account such things as accuracy, rules, book openings, DR versus assumed position etc. The PET Tables have been modified and adopted here. They are extremely simple to use. There is only one rule to understand and that is for forming the algebraic difference between latitude and declination, L~ D. There are no decimal places and no interpolation is required if the reduced accuracy, referred to before, is acceptable. Headings in the tables correspond directly to the variables latitude, LHA, declination etc. Because the DR position is used in the solution, intercepts are usually short and convenient to plot.
The tables have been investigated using a computer program that simulates a human operator selecting, combining and extracting values from the tables. Some 11,387 tests were made and analyzed over the following ranges of the principle parameters:
Latitude 0° to N80° LHA 0° to 180°
Declination N90° to S90° Altitude 0° to 80°
Input parameters were given to 0.1' . The differences, called errors, between the altitudes calculated accurately and those derived from the computer solution were compiled using two techniques, one with the operator interpolating the tables to 0.1' and the other without interpolation. The results of that investigation are as follows:
Altitude Difference (min.)
Percentage of Errors
Altitude Difference (min.)
Percentage of Errors
* Maximum error
It can be deduced from these tests that there are no geometrical quirks in the solution when, for some methods, certain combinations of data require special treatment. If the tables are not interpolated, then about 94% of reductions have errors not exceeding 1' and if interpolated about 95% have errors not exceeding 0.4' . These accuracies are quite acceptable.
Azimuth calculation must be considered of secondary importance to that of the altitude calculation because it will only be necessary to determine azimuth to a low accuracy for plotting purposes. In practice, errors of a degree or so should be quite adequate, particularly as intercepts are short when the DR, rather than a chosen position, is used. A graphical and a tabular method of solution are given. The former solution is based on the Weir Azimuth Diagram. The original diagram as used by the British Navy (Admiralty Chart No. 5000) has been re-drawn, divided and duplicated to provide a solution that does not require colors and double scales: requirements which were necessary in the original diagram to distinguish between celestial and terrestrial hemispheres – a fruitful source of mistake. The second method of solution is the tabular equivalent of the Rust diagram, which was a graphical solution based on the sine formula. No interpolation is required and the four pages of tables can also be used to find amplitudes.
Although not essential, some form of prediction and identification device e.g. A Rude Star Identifier No. 2102-D or Star Finder and Identifier No. NP 323, will be found to be of considerable value in the planning of observations and for identifying unknown or mistaken bodies. Such devices are rendered unnecessary by the inclusion in the book of tables which list (names and magnitudes) the altitudes and azimuths of 58 stars for every 10° of the LHA of Aries at intervals of 10° of latitude, between N60° and S60° . For the Sun, Moon and planets, a separate tabulation for the same range of latitude at every 10° of the LHA of the body for declinations at 5° intervals is given. The complete tables occupy only 26 pages.
The times of sunrise and sunset, and morning and evening civil twilight, are given in graphical form on two pages. These will be invaluable for predicting times to observe azimuth by the method of amplitudes and to take conventional morning or evening star sights etc at twilight.
The Complete On-Board Celestial Navigator is a practical, convenient and economical solution for the mariner who wants a back-up system to GPS that contains all the essential instructions and data for making and reducing astronomical observations. For navigators who have been used to the Nautical Almanac, the transition to this form of data presentation will require a small adjustment in their technique of interpolating the data. Although the accuracy of the almanac data has been reduced, these errors should be quite acceptable for the circumstances envisaged. Under one cover, in the compass of about 180 pages, one can find, in addition to the five years of almanac data, a description of standard celestial methods of prediction, observation and reduction. Star charts of the northern and southern skies showing constellation outlines and names of the bright stars are included to assist in finding and identifying celestial bodies. Worked examples are provided which illustrate the use of the almanac and associated tables. The book lies open flat (spiral binding), the covers have a weather-resistant coating and the inside pages are on a hardy uncoated paper.