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More angles today… sigh – I do wish I had paid more attention in 5th form maths. Sorry Mr Sanders…

Let’s learn about Azimuth angles. As at the stage of typing this first paragraph of the post I personally have no idea what they are. Time to learn!

Remember that the angle we measured with our sextant, along with the Sun Almanac will give us the information required to draw a position circle around the ground position of the sun, and we are somewhere on that circle. However that circle is likely to be really large, so we need to narrow it down a bit.

So we use an Assumed Position – something close to our Dead Reckoning Position, but rounded off in to whole numbers. We then take that Assumed Position and look up the Sight Reduction Tables (see below) to determine the bearing of the sun at that particular time.

We can then draw a preliminary Line of Position at 90º from this bearing.

Finally we can calculate how far off the Assumed Position we actually are – either towards or away from the sun’s ground position. We do this by comparing the altitude we measured with the sextant (ho) with the altitude of the sun at our Assumed Position from the Sight Reduction Tables.

The Azimuth Angle – also known as Z is the bearing of the sun’s ground position from our boat’s position.

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The best picture I can find to explain it is this one below from the book Celestial Navigation by Dominique Prinet.

In the example below, you can see that the bearing between our Assumed Position AP, the North Pole and the Ground Position GP of the Sun is 100º.

This angle is measured from the nearest pole – so in this example – The north pole.

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OK so this is where things start to get even more mathematical and smoke starts to come out of my ears. The Spherical Triangle or PZX Triangle is a measurement of the angles between the boat – in one corner, the pole in the second corner and the GP of the sun or star in the the third corner.

So you remember ages ago a post I did about Great Circles? Well spherical triangles are based on great circle lines – or lines that slice the earth in half – like meridians of Longitude.

I don’t know that I am explaining this particularly well – so it might be worth doing some googling of your own to make sure you understand the concept.

Sight Reduction Tables

Thankfully another very clever person has worked out all these angles in some more tables. These are called the Sight Reduction Tables and they can be downloaded for free here.

Each page represents a different Latitude. So go to the Sight Reduction table book and then turn to the page of your Assumed Latitude.

Remember we altered our Dead Reckoning Position to an Assumed Latitude and LHA – a nice round number, that is because only round numbers are offered in the Sight Reduction Tables. Click here to refresh yourself on how to work out your LHA or Click here to refresh yourself on Dead Reckoning Positions.

Then you need to decide if the sun’s Declination is the same as our boat’s Latitude – i.e. if our Latitude is 8ºS and the sun’s Declination is also South, then we will be looking at the tables for Declination being the SAME as our Latitude. Click here to refresh yourself on how to find the sun’s declination.

There is another table to use if the Declination is CONTRARY to our Latitude. i.e. if we are at 8ºS and the suns Declination is North, then use the other table.

Along the top line is the Declination – along the sides of the table is the LHA 1-90º listed on the left hand side and then 360-291º on the other side.

So if the Declination was 4º we would go to that column, and then scroll down the column to get to the line corresponding with the appropriate LHA.

The numbers that we get from the table are the Hc – or theoretical altitude (which we compare with Ho – the corrected altitude from our sextant)

We are also looking for the Z number which is the Azimuth Angle.

Once we have got the Azimuth Angle of Z from the table – we can then work out what the Zn – or Azimuth Bearing is. Check out the bottom boxes in the sight reduction form below on how to work that out.

Z is the Angle between the sun and the nearest pole, whereas Zn is the bearing of the sun from the boat. The bit where it gets complicated is when then boat is in the opposite hemisphere to the sun, and whether it is morning or afternoon. Hence if you have a look at the calculations in the table below it changes depending on the LHA and Latitude N or S… (who on earth worked this all out!?)

So finally we have reached the bottom of the Sight Reduction Form! That must mean we are nearly ready to actually put some lines on a chart! Tune in next time to find out how we do this.

Does your brain hurt? Mine does. However I do think it is sinking in.

If you aren’t understanding my blog posts. I highly recommend getting this book: Celestial Navigation by Dominique F Prinet. It seems to be the one that is making the most sense for me at this stage, and it has got lots of good examples to work through as well.

Sight Reduction Tables for Celestial Navigation

Sight Reduction is the process of solving the navigational trianglefor an assumed position and measured Altitude to obtain a Line-of-Position.The Sight Reduction Tables available here are very similar to the tables issuedby the (former) 'Defence Mapping Agency of the Hydrographic/Topographic Center' as publication 'H.O. 229'. Alternatively, the navigational triangle can be solved according to the 'Ageton Method'.
The difference between the 'H.O. 229' Tables and the Tables presented here,is that the range for the Declination is limited to 0° to 29°.This is compliant with the Nautical Almanac available from this site, which does not include star data. The Declination of the Sun, the Moon and the planets is always below 30°.
Omitting the Declination range above 30° reduces the number of pages (and the weight!) of the Sight Reduction Tables by 60%!

Purpose and Scope

The main purpose of these tables is to facilitate the practice ofcelestial navigation at sea.The tables have been designed for use with the 'Marcq Saint Hilaire'or 'Altitude-Intercept' method of Sight Reduction, utilizing a positionassumed or chosen such that interpolation for Latitude and Local Hour Angleis not required.
For each of the entry combination of integral degrees of Latitude, Declination and Local Hour Angle, the correspondingvalues for Altitude and Azimuth are tabulated.The Altitudes and their increments are tabulated to the nearest tenth of a minute.The results for the Azimuth Angles are tabulated to the nearest tenth of a degree.The tables are designed for a linear interpolation of the Altitude forDeclination only.This interpolation can be performed by means of the interpolation tables also used for interpolating the Nautical Almanac data.

Arrangement

Each page of the table contains the Altitude (H) and Azimuth Angle (Z)values for the following ranges of arguments:

• Latitude: one (integral) value
• Local-Hour_Angle: 10 successive (integral) values
• Declination: from 0° to 29° in integral values
  (Declination same as OR contrary to Latitude: NN/SS or NS/SN )

Tables

Interactive Tables

Pre-compiled Tables

The pre-compiled Sight Reduction Tables are available in PDF format. The complete Tables for 0° to 79° of Latitude (about 1000 pages) are divided over 16 Volumes, which can be selected from the table below.Each Volume has a Latitude range of 10°.

A worksheet for this table-based method of Sight Reduction is available as printer ready pdf file:worksheet for Sight Reduction using compiled Tables.

Background Information

The Sight Reduction Tables as produced for the above scripts and files containtabulated values for the solutions of a spherical triangle of which two sides(90°-Latitude and 90°-Declination) and the including angle(Local Hour Angle) are known and the values of the third side (Altitude Hc)and adjacent angle (Azimuth Angle Zc) are required.
The solution can be analytically described by the following Sight Reduction equations:Altitude Hc is calculated by:

The arguments for these formulas are Latitude (Lat), Declination (Dec)and Local Hour Angle (LHA).The values of the interactive tables are calculated with a JavaScript script usingthe build-in 'Math'-trigonometric functions.I have compared at random about 100 sets of values from different argument combinationswith the corresponding values from the 'H.O. 229' publications and found no significantdeviations other than 'rounding errors'.
The values from the 'H.O. 229' were calculated (anno 1981) with 9 significantfigures to obtain an accuracy of 0.1' for the Altitude and 0.1° for the Azimuth Angle.To my knowledge, the 'Number' class and the 'Math' class the JavaScriptinterpreter allow a number representation and processing, which isconsiderable better than the ones used for the 'H.O. 229' publications.
The 'rounding errors' mentioned above are ±0.1' for the Altitude and±0.1° for the Azimuth Angle values.
The pre-compiled Sight Reduction Tables from the section above, have been generated with a C-programusing double-precision number representation and processing for all intermediate variables.The accuracy of these calculations is much higher that the accuracy of the calculationsused for the 'H.O. 229' publications.Since for both calculations, the results are rounded to 0.1' for the Altitude and 0.1°for the Azimuth, differences in the results of the two methods of calculation,are due to 'rounding effects'.