Astro Navigation: the orthodox method

What follows is an introduction to Astro Navigation. I have written it as a way of revising and remembering what it is and how to do it. It will not replace the textbooks. To make it simpler I have related it only to the use of the sun (not stars or planets) and only for a trip such as ours at this time of year. Modification for other seasons and parts of the world is not difficult once you have the basics and a textbook.

Contents

1. Introduction
2. Taking a noonsight (with example)
3. Basic concepts in taking a general sunsight
4. Procedure for taking a general sunsight
5. Calculations for general sunsight (with example)
6. Appendix – Sunsight proforma

Part One: Introduction

Using a sextant I can measure the angle that the sun makes with the horizon. This is called a sunsight. There are two ways in which I can use this measurement to give me an indication of where I am on the surface of the earth.

• Firstly I can take a sight at noon to give my latitude (in effect a position line parallel to the equator); this is a relatively simple calculation but can only be taken once a day.

• Secondly I can take a general sunsight at anytime of the day and using a different and more complex calculation I can deduce a position line. A position line is a line on the chart along which I lie, and the point where two position lines cross marks my position. Practically speaking, this means I must take two sun sights in a day (ideally at least 2hours apart). This will give me 2 position lines and so enable me to fix my position (using the technique of a “running fix”).

We will now look at the simpler noonsight first, then approach the general sunsight in stages. Both include my calculations from the beach here in Lagos . Finally there is a proforma which may be used to aid the general sunsight calculations.

The calculations are done using the versine method and the page numbers relate to Reeds Astro Navigation Tables 2005 – this book gives all required tables in one slim volume which is replaced yearly. Different types of table and methods are available.

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Part Two: Taking a Noonsight .

Basic Concepts

Tables tell me how high in the sky the sun is at noon on each day from the equator, this is called the sun’s declination(DEC). Using a sextant I can observe the height of the sun in the sky (altitude) where I am, compare it with the figure in the tables and then use it to calculate my latitude.

Taking the sight ( measurements as taken on 4 th October 2005 )

Noon is when the sun is highest in the sky and using the sextant I can see when it stops going up and starts to fall again. I must record the altitude at its highest point.

It helps (but is not essential) if I know the time of noon so looking in the tables for October 4th (p35), I find that noon at Greenwich (near London ) is 11:49UT. I’m not in Greenwich , in fact Lagos is 8degs West of it. I can use a table to tell me how much later it will be here (arc to time on p53) but I can do this one in my head. I know that for the earth to spin 1deg takes 4 mins so the calculation is easy. Noon in Lagos is 32 (8x4) minutes later than 11:49 , which is 12.21UT.

Just before this time I sat on the beach and began taking sights with the sextant to record the sun at its highest point. At that point the sextant said 48degs 7.8mins.

Calculations

I adjust this (using the tables on p45) for height of eye, diffraction and other factors. It says I must add 12.3mins so the true altitude is 48degs 20.1.

I can now use this to get something called Zenith Distance (ZD) which is the angle from vertical to the sun rather than from horizontal to the sun. This one is easy. I just subtract the true altitude from 90degs.

So ZD = 90 - 48.20.1 = 41degs 39.9mins.

I then need to adjust for declination (Dec) of the sun (angle north or south of the equator).

The tables give declination of the sun on p35. For 12.21UT today it is about 4degs 30.0mins south.

I must subtract this southerly declination from my ZD to give my latitude.

So the latitude is 41degs 39.9mins (ZD) – 4degs 30.0 (Dec) = 37.09.9mins North.

Lagos is actually about 37degs 05 north so I’m 4 miles out. But hey it’s my first sun sight.

Discussion

A daily noon site to give latitude is very useful and fairly simple (?). Navigating to the Caribbean would be possible using this alone and this is how it was done for many years before accurate clocks were available on boats. Simply sail southwest until you get to a latitude of St Lucia (14degs North) and then steer along this line of latitude in a westerly direction, remembering to keep a sharp lookout after about 25 days!

It’s probably a good idea to confirm the name of island on arrival .

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Part Three: Basic concepts for General Sunsights.

Ok, this starts to get a bit more tricky.

To start with I need to estimate my new position (by adding the boat’s travelled distance and direction to my last fix) - that’s easy.

I can then use the tables to calculate the direction and height (altitude) of the sun if I were at that estimated position (EP) at the current time (that’s when you need an accurate clock).

Now, for the clever bit. The direction (bearing) of the sun is called the “Azimuth” and the difference between what the tables say the altitude should be (from my EP) and what I can observe with my sextant can be used to calculate how much closer or further from the sun along that azimuth my boat is in relation to that EP– this is called the “intercept”. I can then draw a position line across the Azimuth (at right angles) at this intercept distance. I am somewhere along this line.

To get a fix I need at least two of these position lines, just like for coastal navigation, so I need to do this more than once.

Practically speaking if I was using the sun, I could take a morning sight, a noon latitude sight (see part 1) and an afternoon sight. This will give me 3 lines and with knowledge of my boats direction and speed I can do things called a running fixes between these. This is called a “sun-run-sun”.

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Part Four: Procedure for taking a General Sunsight

1. Plot EP – normally my estimated position would be on the chart.
2. Take sunsight and record accurate time (UT), correct the sight using table –this is the measurement of the angle between the sun and the horizon
3. Extract Greenwich Hour Angle (GHA) from table and calculate Local Hour Angle (LHA) – GHA is how far west or east the sun is at that time from Greenwich (somebody has helpfully calculated this for us for any day and time) , tables adjust this figure to where I am (LHA).
4. Sun’s declination (DEC) from tables – giving how far north or south of the equator the sun is at that time
5. Calculate Altitude using VerCZD=VerLHAxCosLATxCosDEC+Ver(LAT+DEC) – this complex formula calculates how high the sun should be in the sky from my estimated position using LHA, DEC and estimated latitude(LAT)
6. Calculate Intercept by comparing calc altitude with observed altitude – difference between expected angle and observed angle to the sun gives distance from EP
7. Calculate Azimuth using ABC tables – these tables give the bearing of the sun at the time of the sight.
8. Plot azimuth and intercept on chart and create the latest position line.
9. Use with other position lines to give fix

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Part Five: Example Calculations for a General Sunsight

1. As mentioned previously an estimated position (EP) would be on the chart, I have invented one a few miles away in Alvor. 37deg 07.0N 008deg 37.0W
   
2. I went to the beach to get a sight about 90mins before noon . Before leaving I set my quartz watch using the time readout on the GPS (yes I suppose that is cheating).
   At 1207 and 30 secs I observed the sun at 44deg20.0min to the horizon.
   Convert to UT (GMT) by taking an hour off
   Correct sight using tables (P45), I must add 12.1 mins.
   So after correction 44deg 32.1mins at 1107 and 30secs.
   
3. Using tables on P35 and 47 I extract sun’s GHA for 1107 and 30s UT.
   GHA = 349deg 46.6mins.
   In this instance LHA = GHA minus EP longitude
   LHA = 349deg 46.6 minus 008deg 37.0
   LHA = 341deg 09.6 mins
   
4. Using table on P35 I extract sun’s declination(DEC) at 1107 and 30 s UT
   DEC = 4deg 51.8mins South.
   
5. Now for the calculation (take a deep breath!)
   VerCZD=VerLHA x CosLAT x CosDEC + Ver(LAT+DEC)
   The tables on P55-64 give versines and cosines and their logs. If we use logs then we can add them rather than do that devlish multiplication.
   Log ver LHA = 8.7290
   Log cos LAT = 9.9017
   Log cos DEC = 9.9984
   Add these = 28.6291
   Ignore tens figure = 8.6291 (don’t ask just do it)
   So log ver = 8.6291
   ver = 0.0426 (use table to find antilog) (FIGURE 1)
   
   Now add latitude to sun’s declination
   LAT + DEC = 41deg 58.8 (additive because on north and one south)
   Extract its versine = 0.2567 (FIGURE 2)
   
   Add figures 1 and 2 and use tables to undo the versine function (i.e. use them backwards)
   Calculated Zenith Difference (CZD) = 45deg 31.0 mins
   As before Calculated Altitude(CA) = 90 minus CZD
   
   CA = 44deg 29.0
   
6. So the difference between the sun’s altitude at my EP (the calculated altitude) and the sun’s altitude where I am (seen through my sextant and adjusted to become True Altitude) gives me my distance from the EP. This is called the intercept (see Basic Concepts in Fixing Longitude).
   Intercept = True Altitude minus Calculated Altitude
   Intercept = 44deg 32.1 minus 44deg29.0
   Intercept = 3.1 mins
   
   Because the True Altitude was larger than Calculated Altitude I must be 3.1 mins or nautical miles towards the sun.
   
7. Now I must calculate the Azimuth (the direction of the sun at the time of the sight), using ABC tables.
   LHA and LAT into table A = 2.19
   LHA and DEC into B = 0.270
   A+B = 2.460 (if this was minus I would ignore for now)
   Then, (A+B) and LAT into table C
   
   Azimuth = 27.4deg
   This is in quadrantal notation. The angle is expressed east or west from north or south. Sum A+B was positive so azimuth is south. LHA is greater than 180deg so the angle is east from due south. Basically 180-27.4.
   
   Azimuth bearing in normal notation is 180 minus 27.4 equals 152.6deg
   
   
8. I plotted the azimuth (152.6deg true) from my EP, marked off the intercept (3.1miles toward the sun) and plotted a line at 90 deg across the azimuth at this point. This is my position line, I am somewhere along this line.
   
9. Using my latitude calculation from the previous day (I hadn’t moved much since then) I now had 2 position lines. The point at which they crossed was a fix. It put me 13miles to the north east! But remember this was my first go (err… again).

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Part Six: Appendix : SUNSIGHT PROFORMA

• Plot EP
• Take sunsight and record accurate time (UT), correct the sight using table
• Extract Greenwich Hour Angle (GHA) from table and calculate Local Hour Angle (LHA)
• Sun’s declination from tables
• Calculate Altitude using VerCZD=VerLHAxCosLATxCosDEC+Ver(LAT+DEC)
• Calculate Intercept by comparing calc altitude with observed altitude
• Calculate Azimuth using ABC tables
• Plot azimuth and intercept on chart and create position line
• Use with other position lines to give fix

'The End '

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