Understanding Astro Navigation

Introduction

I have attempted to understand astro-navigation on a few occasions; I gave up on all but the last of my endeavours. This last time I stuck at it. And found the light. What didn’t help was the lack of a broad overview of what it entails. Too many of the texts started from first principles and definitions and such like. What follows is an overview that I would have wanted to read before attempting to learn the science and the art. Truth be told, I’ve written this to prove to myself that I understand it all!


I make minimal use of all the various navigational terms involved but these can and will need to be understood in the fullness of time. This is not a replacement to proper astro-navigation texts, it is a foreword written by an amateur sailor for an amateur sailor who wants to understand. I’ve also not got any diagrams which typically can be very helpful. Partly because I’m not going to spend that kind of time on this but mainly because I think the thought experiments I work through are better for understanding. Here goes.


Let’s set the scene

An ocean going navigator can, within a few miles, judge his position on the earth’s surface, referred to as “fixes”, if he has the following on board:

  1. A nautical almanac, to tell him facts about the positions of heavenly bodies

  2. An accurate chronometer

  3. A sextant as a means to measure the angle (“altitude”) of a heavenly body

  4. Charts with the usual plotting equipment and a pencil and paper

Historically, over the years, all of these tools gained in accuracy and sophistication to allow ever more accurate fixes. Importantly however, from the perspective of the mariner, these are all simple, reliable tools although in earlier times an accurate chronometer was by far the most complex and precious piece of machinery on board.


Of course astro-navigation has been made all but redundant by the magic of GPS. But even today, it is not deemed prudent to rely on GPS as your only means of navigation for ocean voyages and so the old ways are still actively maintained, taught and used. First, let’s say a little bit more about these tools of the trade.


The Nautical almanac

Nautical Almanac
Thanks, mostly, to Newton we know how to predict where any heavenly body (sun, moon, planets, bright star) is in the sky at any time past, present or future. We can do this with uncanny accuracy (probably with a little help from Einstein too) - using computers of course because the calculations involved are quite hairy and numerous.  The nautical almanac expresses these calculations in an oh-so-clever way as a set of succinct numerical tables that allow a navigator to easily answer one essential question:

Where is a particular heavenly body at a particular instant of time?

The almanac is an annual because these tables change each year. There’s another set of tables that don’t change (much) and they allow the navigator to answer what are essentially trigonometrical problems in a spherical coordinate system. An example of one of these problems is: From a particular position on earth, in what direction (its bearing) and how high (its altitude) would a particular heavenly body appear to be assuming I know where it is?


Now, I need to explain what it means to know “where” a heavenly body is. This is expressed as a Geographic Position or GP for short. It’s reasonably simple to imagine: The GP is the place on the earth’s surface through which a line drawn from the earth’s center to the heavenly body passes. It is expressed as a latitude and longitude position; A position on the surface of the Earth. Familiar territory. I refer to a heavenly body’s GP frequently so make sure you get this picture. Note that this “where” is relative to the Earth and has no connection to “where” we might be at the time.


The key objective of the almanac tables are that they are presented in such a way as to allow the navigator to relatively easily and quickly find answers without the use of a calculator. All he needs to do is look stuff up in rows and columns and add numbers together using a pencil and paper. Easy once you know how.


The chronometer

Chronometer
The chronometer is necessary because it is a vital ingredient to our question we ask the almanac. It has to be set to GMT (now known as UT) as that’s where our longitudes are based. You can imagine the complex relative motions of our Earth, the moon, planets and stars all playing out minute by minute, but the most dramatic of movements is the spin of the Earth, which causes heavenly bodies to appear to move from East to West at a fair pace. Mainly for this reason, our chronometer needs to be pretty accurate - every 4 seconds out can lead to an inaccuracy of up to a mile (we slice the Earth in degrees and minutes and, as we all know, a minute of a degree represents one nautical mile and it takes the Earth just 4 seconds to rotate one minute of a degree).


The sextant

Sextant
The sextant provides a new interesting piece of data for us. It measures the altitude of a heavenly body from the perspective of the sextant user! It tells us how high it is in the sky and is measured in degrees between 0 and 90. You can image that at 0 degrees the body would be on the horizon and at 90 degrees it would be directly over our head. The sextant can measure anything in between. This is useful. The sextant itself is a pretty simple measuring device, nothing to get excited about so far as I can see though it seems to hold a strange brassy, dusty mystique.




Where am I?

So, preamble over, how do I ascertain my position given the above? Imagine we are in the middle of the ocean. Let us take any heavenly body at random currently in view, say the planet Venus. Suppose we then measured its altitude with our sextant and it turned out to be 90 degrees exactly, i.e. directly over our head. How fortuitous! Draw a line from Venus and have it to go straight through our body from head to toe and it will end up at the center of the earth (as any line extended vertically below us would). Recall what the GP is? We are at the GP of Venus. Recall what we can ask of the almanac? Yes, indeed, we can get the lat/long of Venus’s GP for the current time which will be our lat/long since we are sitting right there right then. Job done. We know where we are. That was easy.


Except that Venus, in all probability, will not be directly above us. So freeze frame this imagined directly-over-our-head scenario and lets start sailing. Doesn’t matter which direction you go so long as you keep going in the same direction in a straight line. Keep sailing for miles and miles, the whole set-up frozen in time except for you. What’s going to happen to Venus? As the miles go by you will notice that Venus will slowly descend in the sky behind you because you are going round the Earth. Eventually, when you are a quarter of the way around the Earth, Venus will appear like it’s on the horizon. That’s 90 degrees round the earth and Venus’s altitude has dropped by 90 degrees to 0 degrees altitude. So every minute of a degree you sailed around the world would have resulted in Venus dropping towards the horizon by a minute of a degree. And how far would you have sailed for 1 minute of a degree? 1 mile. So let’s say you sailed 600 miles. That’s 10 degrees. At which point you would measure Venus to have an altitude of 80 degrees. So now unfreeze frame and pretend nothing happened and here we are measuring Venus having an altitude of 80 degrees. Where are we? We are somewhere on a circle that surrounds the GP with a radius (albeit a bent-round-the-earth radius) of 600 miles. We don’t know where on the circle because, as we discovered in the imagined scenario above, we could have sailed in any direction from the GP and still seen Venus drop in altitude behind us.


So, to sum up, when we measure the altitude of a heavenly body (it’s called its True Altitude - TA) we are somewhere on a circle that is (90-TA)*60 miles in radius around the GP of the heavenly body. We can use that. It’s not giving us a fix but at least we’ve narrowed down the possibilities to the perimeter of a big circle on the surface of the Earth. So what would a sensible navigator do now? They’d pick another heavenly body and measure its altitude and draw a circle round it’s GP. Where the circles intersect must be where you are. Of course, you’ll probably have two intersections so go pick another heavenly body and get 3 circles. Job done. You’ve got a fix.


And that’s it folks. That’s how astro navigation works. Now go read the manuals!


Practicalities

The rest is, as they say, just implementation. But these are not to be sniffed at and I’ll shed some light on them. Here’s some issues:

  1. You have to know how to use a sextant and the almanac to get answers to questions

  2. The altitude you measure with your sextant has to be accurate which is not quite so simple as it sounds.

  3. Every second counts when you measure altitudes. So if you want your 3 circles to provide an accurate point of common intersection you’d better do it pretty damn quick and note the time accurately.

  4. During the day, you’ve not got a lot of choice when it comes to heavenly bodies (the Sun if its not cloudy, sometimes the moon).

  5. Drawing circles with diameters in the 1000’s of miles is just not practical even on the smallest of scale charts - that’s a serious problem


So while we have the answer in theory, in practice there’s still work to be done. Points 1 & 2 can be learned & understood using manuals and additional tables in the almanac as well as the almanac’s own instructions. Points 3, 4 and 5 are also covered in the manuals, but I think it’s worth a dummies guide here. It’s all in the intercepts.


Intercepts

It took me a while to understand what an intercept is and the reason is that you can’t simply say what it means - you have to explain it. This is trickier without diagrams, but with a little bit of concentration you should get this too (it all depends how well I can explain it).


In practice the circles you need to draw are just too big. Besides, you’ll always have a pretty reasonable idea of where you are. So even if your circle covers half the Indian ocean and you’re in the pacific, you’ll not be considering the fact that you got so lost that you’ve ended up in the Indian ocean. That’s because you are a sensible navigator and you knew your position pretty well yesterday and you know roughly how fast you’ve been going and in what direction to have some idea of where you are today - a Dead Reckoning position. So really, when it comes to that big circle of possible places to be you can estimate a pretty small segment of it to show where you expect to be - small enough not only to fit on your chart but also to be drawn as a straight line. We just need to know exactly how and where to draw that small segment of circle on our chart. Here’s how:


Let’s be cautious and say you’re lousy at dead reckoning and you can only estimate your position correct to the nearest degree (that’s within an area of sea 60 miles across) - we’ll call it your Assumed Position - AP. The first task is to draw a line that passes through the AP but at a bearing that would take it through the GP of the heavenly body. We then know that our small segment of circle would need to be plotted at right angles to this line (i.e. it is a part of the circle that has this line as its radius). We’ll call this line AP-GP and think of it as the path our imaginary freeze frame journey took when we sailed away from the GP - essentially a radius line. If your chart is showing, say, 100 square miles then the GP is likely many thousands of miles away and thus we can imagine that point might be at the other end of the boat at this chart scale. So even if our AP was off by 60 miles, this line’s bearing would not be so very different. Since it is the bearing that is important for our purposes (so that our small segment of circle is plotted correctly at a right angle), I’m making the point that the AP can afford to be inaccurate. This bearing can be calculated using the good almanac as it’s a simple question: “What is the bearing from my AP to the GP?” We know both these points (lat/long coordinates) so easily answered.


Now we can ask the almanac another simple question. At what altitude would I measure a heavenly body if I am at the AP and I know the GP of the heavenly body? Easily answered and we get a “Calculated Altitude”. Now, we have also measured the Altitude of the body at this GP from our current (but unknown) position and it turns out that it’s exactly the same as the Calculated Altitude. What joy! Because this means my AP was exactly right in terms of how far away I am from the GP (remember that this distance is equivalent to the altitude). So I can draw my small segment of circle at right angles to the AP-GP line at the point of the AP. I am somewhere on that line. Too good to be true? Well, yes, the altitude I measured will most likely be different from the Calculated Altitude (but not by too much if the AP was a decent guess as to where we were). If it was, say, one minute larger that means I must be one mile closer to the GP than my AP plot point and so I’d draw my small segment of circle at right angles to the AP-GP line one mile down the AP-GP line towards the GP. Conversely, If it was, say, 15 minutes smaller that means I must be 15 miles further away from the GP than my AP plot point and so I’d draw my small segment of circle at right angles to the AP-GP line 15 miles up the AP-GP line away the GP.


So the difference between my measured altitude and the Calculated Altitude tells me how far up or down the AP-GP line I need to plot my small segment of circle away from the AP point. And guess what this difference is called? The intercept!


We must remember, however, that we still do not have a fix on our position, we still only know that we are somewhere on that small segment of the circle. I’ve been calling it the “small segment of circle” to keep reminding us what it truly is. The official name for it is Line of Position (LoP).


Fixes

So now we have a practical way of drawing a Line of Position where we might be. And we can do the same for just one other heavenly body to get a fix, so that saves on having to get 3 as it’s highly unlikely that we’d get two intersections on our little Line of Position segments. But still, in day time, we really only have the sun. Which is the story for our final practicality. If we did the whole exercise in getting a Line of Position from the sun twice with, say, just 5 minutes apart, our lines would be pretty parallel and so we’d get no intersection and no fix (i.e. not enough has changed in 5 minutes). But if we left it a couple of hours they’d be at significantly different angles to get a fix (the GP of the sun could have moved up to 1,800 miles, depending on the season). Trouble is we would have moved our position by then - but not by much. You can image in that relatively small period of time we’d have a pretty accurate idea of our passage speed and direction and so we could plot that course on our chart together with the two position lines and then shift them together along that course line to get a reasonable fix of where we were an hour ago. That’s good enough and it’s referred to as a running fix. Keep on doing that and any errors can get factored out and you end up with a pretty accurate set of position plots.


Job done. Now, go read the manuals!


Books


I've read "Celestial Navigation" by his excellency Tom Cunliffe which is very good. The Reeds Astro Navigation Tables annual lists these as good references:

  • Reeds Sextant Simplified by Dag Pike
  • Celestial Navigation For Yachtsmen by Mary Blewitt
Tristan Jones, in his book "One hand for yourself, One hand for the Ship - the essentials of single-handed sailing" has a great chapter on astro navigation as well. Altogether this book is a fun read, though it's meant for single handed ocean going yachtsmen of a past generation!

I'm going to try it during my circumnavigation

I'm guessing that every astro navigator makes his own pro forma for building up a fix and I've done the same for myself (see link below) as I intend to put to practice what I have now learnt in theory. I'll report on progress!
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Dominic Thwaites,
27 Mar 2015, 04:03