A Short Guide to Celestial Navigation.pdf

Revised January 28, 2001

Carpe diem.

Horace

Preface

Why should anybody still use celestial navigation in the era of electronics and GPS? You might as well ask why some

people still develop black and white photos in their darkroom instead of using a high-color digital camera and image

processing software. The answer would be the same: because it is a noble art, and because it is fun. Reading a GPS

display is easy and not very exciting as soon as you have got used to it. Celestial navigation, however, will always be a

challenge because each scenario is different. Finding your geographic position by means of astronomical observations

requires knowledge, judgement, and the ability to handle delicate instruments. In other words, you play an active part

during the whole process, and you have to use your brains. Everyone who ever reduced a sight knows the thrill I am

talking about. The way is the goal.

It took centuries and generations of navigators, astronomers, geographers, mathematicians, and instrument makers to

develop the art and science of celestial navigation to its present state, and the knowledge thus accumulated is too

precious to be forgotten. After all, celestial navigation will always be a valuable alternative if a GPS receiver happens to

fail.

Years ago, when I read my first book on navigation, the chapter on celestial navigation with its fascinating diagrams and

formulas immediately caught my particular interest although I was a little deterred by its complexity at first. As I became

more advanced, I realized that celestial navigation is not as difficult as it seems to be at first glance. Further, I found that

many publications on this subject, although packed with information, are more confusing than enlightening, probably

because most of them have been written by experts and for experts.

I decided to write something like a compact guide-book for my personal use which had to include operating instructions

as well as all important formulas and diagrams. The idea to publish it came in 1997 when I became interested in the

internet and found that it is the ideal medium to share one's knowledge with others. I took my manuscript, rewrote it in

the form of a structured manual, and redesigned the layout to make it more attractive to the public. After converting

everything to the HTML format, I published it on my web site. Since then, I have revised text and graphic images

several times and added a couple of new chapters. People seem to like it, at least I get approving e-mails now and then.

Following the recent trend, I decided to convert the manual to the PDF format, which has become an established

standard for internet publishing. In contrast to HTML documents, the page-oriented PDF documents retain their layout

when printed. The HTML version is no longer available since keeping two versions in different formats synchronized

was too much work. In my opinion, a printed manual is more useful anyway.

Since people keep asking me how I wrote the documents and how I created the graphic images, a short description of

the procedure and software used is given below:

Drawings and diagrams were made with good old CorelDraw! 3.0 and exported as GIF files.

The manual was designed and written with Star Office. The Star Office (.sdw) documents were then converted to

Postscript (.ps) files with the AdobePS printer driver (available at www.adobe.com). Finally, the Postscript files were

converted to PDF files with GsView and Ghostscript.

I apologize for misspellings, grammar errors, and wrong punctuation. I did my best, but after all, English is not my

native language.

I hope the new version will find as many readers as the old one. Hints and suggestions are always welcome. Since I am

very busy, I may not always be able to answer incoming e-mails immediately. Be patient.

Last but not least, I owe my wife an apology for spending countless hours in front of the PC, staying up late, neglecting

household chores, etc. I'll try to mend my ways. Some day ...

January 28, 2001

Henning Umland

Correspondence address:

Dr. Henning Umland

Rabenhorst 6

21244 Buchholz i. d. N.

Germany

Fax +49 89 2443 66455

E-mail h.umland@online.de

Chapter 1

Introduction

Celestial navigation is the art of finding one's current geographic position by means of astronomical observations,

particularly by measuring altitudes of celestial bodies – sun, moon, planets, or stars.

The apparent position of a body in the sky is defined by the horizon system of coordinates ( Fig. 1-1 ). The altitude , H ,

is the vertical angle between the line of sight to the body and the horizontal plane . The zenith distance , z , is the

corresponding angular distance between the body and the zenith – an imaginary point vertically overhead. H and z are

complementary angles (H + z = 90°). The azimuth , Az N , is the horizontal direction of the body with respect to the

geographic (true) north point on the horizon, measured clockwise from 0° through 360°.

Three imaginary (invisible) horizontal planes which are parallel to each other are relevant to celestial navigation ( Fig. 1-

2 ):

The sensible horizon is the horizontal plane passing through the observer's eye.

The geoidal horizon is the horizontal plane tangent to the earth at the observer's position.

The celestial horizon is the horizontal plane passing through the center of the earth.

Sensible and geoidal horizon coincide if the observer's eye is at ground level. Since both horizons are usually very close

to each other, they can be considered as identical under practical conditions.

None of the above horizontal planes coincides with the visible horizon , the line where the earth's surface and the sky

appear to meet. Calculations of celestial navigation are always based upon the altitude with respect to the celestial

horizon. Since this altitude is not accessible through direct measurement, it has to be derived from the altitude with

respect to the visible or sensible horizon (altitude corrections, chapter 2).

Which of both altitudes is obtained, depends on the instrument used. For example, a marine sextant measures the

altitude with reference to the visible horizon, whereas instruments with a built-in artificial horizon measure the altitude

referring to the sensible horizon (chapter 2).

Altitude and zenith distance of a celestial body depend on the distance between a terrestrial observer and the

geographic position of the body , GP . GP is the point where a straight line from the body to the center of the earth, C,

intersects the earth's surface ( Fig. 1-3 ).

A body appears in the zenith (z = 0°, H = 90°) when GP is identical with the observer's position. A terrestrial observer

moving away from GP will observe that the altitude of the body decreases as his distance from GP increases. The body

is on the celestial horizon (H = 0°, z = 90°) when the observer is one quarter of the circumference of the earth away

from GP.

For a given altitude of a body, there is an infinite number of positions having the same distance from GP and forming a

circle on the earth's surface whose center is on the line C–GP, below the earth's surface. Such a circle is called a circle

of equal altitude . An observer traveling along a circle of equal altitude will measure a constant altitude and zenith

distance of the respective body, no matter where on the circle he is. The radius of the circle, r, measured along the

surface of the earth, is directly proportional to the observed zenith distance, z ( Fig 1-4 ).

[ ]

Perimeter

Earth

[ ]

[ °

360

One nautical mile (1 nm = 1.852 km) is the great circle distance of one minute of arc (the definition of a great circle is

given in chapter 3). The mean perimeter of the earth is 40031.6 km.

Note that light rays coming from distant objects (stars) are virtually parallel to each other when reaching the earth.

Therefore, the altitude with respect to the geoidal (sensible) horizon equals the altitude with respect to the celestial

horizon. In contrast, light rays coming from the relatively close bodies of the solar system are diverging. This results in a

measurable difference between both altitudes. The effect is greatest when observing the moon, the body closest to the

earth (parallax, see chapter 2, Fig. 2-4 ).

The azimuth of a body depends on the observer's position on the circle of equal altitude and can assume any value

between 0° and 360°.

Whenever you measure the altitude or zenith distance of a celestial body, you have already gained partial information

about your own geographic position because you know you are standing somewhere on a circle of equal altitude with the

radius r and the center GP, the geographic position of the body.

Of course, the information available so far is still incomplete because you could be anywhere on the circle of equal

altitude which is a typical example of a line of position (see chapter 4).

Let us go one step further now. You are watching two bodies instead of one. Then you are standing on the two

corresponding circles of equal altitude – or lines of position – intersecting each other at two points on the earth's surface,

as is the case when two circles overlap. Logically, one of those two points of intersection must be your own position

( Fig. 1-5a ).

In principle, it is not possible to know which of the two points of intersection – Pos.1 or Pos.2 – is identical with your

actual position unless you have additional information, e.g., a fair estimate of where you are, or the compass bearing of

at least one of the bodies. Solving the problem of ambiguity can also be achieved by observation of a third body

because there is only one point where all three circles of equal altitude intersect ( Fig. 1-5b ).

Circles of equal altitude can be plotted on a map if their radii are small enough. This usually requires observed altitudes

of almost 90°. The method is rarely used since such altitudes are not easy to measure and the risk of ambiguity is higher

than normal. In most cases, circles of equal altitude have diameters of several thousand nautical miles and can not be

plotted directly on maps with appropriate scale, apart from geometric distortion due to map projection. There are,

however, elegant ways of plotting only the relevant parts of the circles (those in the vicinity of the observer's assumed

position ), as will be shown in chapter 4 and 7.

In summary, determination of your position includes three basic steps:

1. Choose two or more celestial bodies and measure their altitudes or zenith distances.

2. Find the geographic position of each body at the time of its observation.

3. Derive your position from the above data.

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