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Thursday, September 15, 2011

Celestial Navigation 101: Lesson 1, Circles of Equal Altitude

We're going to start with the easiest and most fundamental concepts first, and then build upon these. Throughout these lessons I will occasionally flag something in bold as a Very Important Concept. In some cases these may at first seem trivial, but they represent conceptual pitfalls I have encountered often in the 20+ years I've taught celestial navigation. Bear with; if I'm flagging a concept, however seemingly obvious or mundane, there's probably a point to it.

Okay, let's begin.

Very Important Concept: The stars are very far away.

Imagine that there is a bright object in space which is pretty close by. Perhaps at the distance of the orbits of the GPS satellites. Let's imagine that there is a giant desk lamp floating at the distance of the orbits of the GPS satellites. Seen from space, it looks something like this:

The light from the desk lamp arrives at the earth from different directions in space, depending on where on the earth you are observing the desk lamp from. This difference is called parallax.

However, the stars, and incidentally the sun and most of the planets, are so far away from the earth that the light rays coming from them are parallel to each other, so that wherever you are on earth they are seen to be coming from the same direction in space. Like this:

Note that because the earth is roughly spherical, the angle which the light from the star is seen from earth changes, depending where the observer is on the sphere.
We'll come back to this point in a moment.

Not only is the earth roughly spherical, it is spinning. Because it is spinning, we can define directions and locations on the sphere relative to that spin. Without that spin, we would have no reference for direction. We call the two places where the hub of the spin intersects the surface of the earth the poles, which we then arbitrarily name "north" and "south". The part of the sphere which is farthest away from both the north pole and the south pole, the part which is spinning the fastest, we call the equator.

We call the direction the sphere is spinning toward "east" and the direction the sphere is spinning away from "west". We won't be addressing east and west for a while yet, that's pretty advanced stuff for a later date. But I wanted to get those terms out there.

Because the earth is kind of like a sphere, and because a sphere is basically a three-dimensional circle, we divide the perimeter of the earth into 360 equal units. This is also arbitrary; it could just as easily be a hundred units, or a thousand, or seventy-three. But 360 degrees is the convention we have used since Euclid, so we'll roll with it. So to speak.

The convention we use is to call the equator 0° and the two poles 90°, north and south, respectively. Between these in each direction are 90° of latitude. Each degree of latitude is divided into 60 equal minutes.

Very Important Concept: A minute of latitude, and any other minute of arc on the earth, is equal to one nautical mile.

This, in fact, is what a "nautical mile" is, and why we have such a thing. Just remember "a mile a minute" as a mnemonic. Incidentally, from this point forward in this series, at any point you see the word "mile" by itself, assume that it means a nautical mile. We will have no reason to ever use the other kind in this series.

Okay, so. The earth is spinning, we're on it. Imagine now that you are standing at the north pole, exactly, 90° north, the very hub of the spinning planet. Now, let's further imagine that there is a star EXACTLY over the north pole. We'll call it Polaris. Now, in reality, the real Polaris is NOT exactly over the north pole. It happens to be pretty close, but "pretty close" isn't good enough for this demonstration. That's okay. We're going to pretend that Polaris is at exactly 90° north, just for a little bit.

So, standing at the north pole, which is 90° north, we look at Polaris, which is also 90° north. By definition, Polaris must be directly overhead. Another way to say this is that Polaris is at our zenith; "zenith" is an astronomical term which means "up". Yet another way of stating this is to say that if we were to measure the angular distance in the sky from the horizon to Polaris, it would be 90°, as 90° defines the zenith, or up, or "directly overhead", or however you want to say that.

Very Important Concept: Every star, and also every other celestial body, at any given moment in time has some point on the earth which is directly "below" it.
This point is called the Geographic Position, or GP.

If you now walk one mile south from the north pole (every direction is "south" from the north pole) and again measure the angular distance from the horizon to Polaris, you find that Polaris is now 89° 59' from the horizon. And, your latitude is 89° 59' north of the equator. And, you are now one nautical mile away from the GP of Polaris. You could be anywhere on a circle which is exactly one nautical mile around the GP of Polaris, but you are definitely somewhere on that circle. This is called a Circle of Equal Altitude.

That was also a Very Important Concept.

So, let's continue with the exercise. Continue walking (and swimming and whatever it takes) southward until you get to the equator. Now turn around, and look at Polaris again. Now, Polaris is just touching the horizon. The angular distance between Polaris and the horizon, which is also called the altitude of Polaris, is 0°. And your latitude is also 0°. Another way to say this is that the distance from your zenith down to Polaris is 90°, and the distance from the GP of Polaris to where you are standing is also 90° on the globe. And so you are standing somewhere on a circle of equal altitude 90° away from the GP of Polaris.

Now, imagine walking back north to the place where you live (if you don't happen to live in the northern hemisphere, just play along for a minute). I happen to live in Seattle, so I'm going to use Seattle for this part of the exercise, but by all means try this with your own actual latitude. Seattle is at about 48° north. As you walk northward, Polaris will move higher and higher in the sky, until it is at 48° above the horizon. The altitude of Polaris is 48°, and your latitude is 48° north. The other way to say this is that the distance from your zenith to Polaris is 42° (90°-48°=42°) and the distance between yourself and the GP of Polaris is also 42°. And so you are standing somewhere on a circle of equal altitude 42° away from the GP of Polaris.

From here, it is an easy step to see that the altitude of any celestial object defines a circle of equal altitude around the Geographic Position of that object. The distance from the GP is simply 90° minus the altitude.

In this illustration, you are somewhere on the circle of equal altitude around the GP of the object. But where? That circle could be thousands of miles in circumference. The answer is simple; measure the altitude of two more celestial bodies. Where the three circles of equal altitude intersect is your location on the globe.

In a very real sense, that's all there is to it. In principle, with a globe in-hand, you could go out on deck with your sextant, measure the altitude of three stars, plot the GPs of those stars on the globe, subtract the altitudes you shot from 90° to determine the size of the circle, in degrees, to draw around each of the respective GPs. Where the three circles intersect is your position.

This does work, and in fact was one of the ways early European navigators did such things. However, it can only yield a position accurate to within several hundred miles. Which isn't horrible out in the middle of the ocean; it would at least give you some idea of which course to steer to get back to your own country.

But we can do much, much better.

The rest of this course will focus on how to take these concepts and turn them into a position which is functionally as accurate as one derived from GPS.


  1. Welcome back, sailor! You were missed over the summer.

    Interesting new series, looking forward to seeing more of it. Learned enough celnav in the Air Force to pass a test and then promptly forgot most of it. Your approach is very different, I think if you'd been my instructor I might have learned more.

    Surprisingly good graphics for Paint. I especially like the Death Star. JK!

  2. That the stars are very far away is actually why Tycho Brahe rejected the Copernican notion that the Earth moved around the Sun. He was capable of measuring points in the sky to an accuracy of about 1-2 arcminutes, before telescopes had been invented, and yet did not observe any seasonal stellar parallax. This meant that if the Sun were the centre of the universe, the crystal sphere holding the stars would have to be absurdly far away, many many times the distance of the furthest planet, Saturn...

  3. @ Space Cowgirl-- Thank you, it's good to be back online. I missed quite an interesting summer in your neck of the industry; sad for that, but happy for the gainful employment thing.

    Yes, it does look like the Death Star, realized that about a minute after posting. Oh well, it gets the point across, I hope.

    @Ashley-- Tycho was an interesting case of someone who wasn't exceptionally successful as a theorist, but whose meticulous gathering of data ultimately allowed other scientists to advance our understanding of the universe a thousand-fold. Kepler, for example, based nearly all of his theories on Tycho's data rather than his own direct observation.

    The other problem with the Copernican model waa that it relied on epicycles even more convoluted than Ptolemy's to account for what Kepler later realized was the elliptical nature of the planetary orbits. Also, Copernicus had no clue what the planets were, and assumed that since Venus and Jupiter were much brighter than the "fixed" stars, they must necessarily be larger.

    Only vaguely related but interesting, the Anseireh nomads in what is now Turkey thought the stars were just holes in the fabric of the celestial sphere, which allowed tiny portions of the infinite light of heaven in. Presumably living in tents in the desert would afford one such an insight.

  4. Oh, you know all this stuff already. I've been reading up on it because I'm preparing a piece for my team about how Kepler might have used our product. (Which is a bit odd, given that I'm in the Life Sciences group.)