They that go down to the sea in ships, that do business in great waters; these see the works of the LORD, and His wonders in the deep. --Psalm 107: 23-4--

OK, Where will the discussion go from here?

First, we need to talk about coordinate systems. Then we need to understand a little about the motions of bodies in the sky, known as celestial mechanics. Armed with all this information, we will be ready to tackle the problem of actually computing a "computed altitude." Lastly, we need to iron-out all the fine wrinkles of turning a sextant observation into an actual observed altitude. (No, you cannot just read it right off the instrument--I warned you that the work of celestial navigation is very meticulous and demanding!!) You already know by now that once we have a computed altitude and an observed altitude in hand, finding the intercept is a snap. And plotting 2 or more intercepts yields the fix.

It is plenty of ground to cover, but you will be surprised at how fast it will go. With only a little practice, you will see that it all makes perfect sense and fits together in a most remarkable way...

Is this going to get complicated?

Not right away. We are starting with coordinate systems because they are the most familiar and we'll branch out from there.

As I said a few pages ago, the navigator must be familiar with 3 coordinate systems. Each one stands alone, but they are all related. Each one invokes bizarre, alien names, but don't be frightened by that; it is only more of that peculiar and archaic vocabulary to learn!

Why must we have 3 systems?

Because we have one system for each of 3 things that interest us:

• The Earth
• The Sky
• Ourself

The Earth's coordinate system is that of Latitude, denoted L. and Longitude, denoted Lo.. I start here since most of us were introduced to this long ago, making it a little familiar. We all know that the Earth rotates about an axis, and this axis can be located at 2 points on the surface of the Earth, namely the North Pole and the South Pole. These two poles are 180 degrees apart from each other, so we can pick an Equator right in between the 2 poles. We call it 0^o Latitude and measure to 90 degrees North and 90 degrees South. If you look at this on a globe, and I encourage you to do so, you see that every parallel of Latitude makes a big ring around the world, with the exception of the 2 poles, which are only points. You may also notice that the Equator is the biggest of the rings, and all others get ever smaller as you get closer to the Poles.

If we want to define an exact point on the Earth, clearly Latitude alone is not enough (unless you happen to be speaking of one of the 2 poles, which would be rare indeed for a navigator at sea!) To pick a specific point from a Latitude circle, we can designate a Longitude. "Lines" of Longitude, called meridians, are drawn to cross all parallels of Latitude in right angles, and so all meridians run exactly North-South and pass through both Poles. We use them to mark points East and West of the Prime Meridian, which is the meridian running through the Royal Greenwich Observatory, England. A very important point here: Since Royal Greenwich Observatory is rotating with the Earth, and our coordinate system is referenced to the Observatory, clearly the coordinate system is rotating with the Earth. That may not seem important now, but it will be when we talk about celestial mechanics. So if you want a true model, just spin your handy globe. Not surprisingly, all the lines printed on it will spin with the Earth! Although this comment may sound silly, as usual I have something interesting to point out by it. It is very natural to think of those reference lines as exactly that, lines, but a careful observation will reveal that they are actually circles. This notion of drawing "lines" that really come out as circles when extended far enough is a very general notion that will come up in all 3 coordinate systems.

Every meridian cuts the Earth into the biggest possible ring, called a Great Circle. I didn't mention it before, but the Equator is also a Great Circle, and the only Latitude circle to win that distinction. All other Latitude rings are smaller than a Great Circle, so they get the imaginative name of Small Circle.

With this coordinate system, we can find any point on the surface of the Earth by calling out 2 numbers: a Latitude which will be between 0 degrees to 90 degrees North or South of the Equator, and a Longitude which will be between 0 degrees and 180 degrees East or West of the Prime Meridian.

And since the whole objective of Celestial Navigation is to decide the position of your vessel on the surface of the Earth, this coordinate system is very useful for us.

Yes, I remember all that stuff...Can't we use it for the sky, too?

Unfortunately not. It would be very appealing to do so, and we have devised the system as close as we can, but you must understand clearly that the Latitude-Longitude system pertains only to the Earth. The reason for this will lead us into celestial mechanics, which we will only hint at right now.

As you recall from the previous page, to an observer on the Earth the sky appears to be in a rather slow but elaborate motion. Of course, science class tells us that it is the Earth which is in elaborate motion, but just go outside one evening and look up for a few hours. You will have to agree that the Earth certainly seems solid and secure and absolutely immobile, while you walk about on the Earth and watch the stars make their slow, silent pinwheel above. This Ptolemaic point of view will be the basis of the mariner's version of celestial mechanics, but it also tells us why the sky gets its own coordinate system--the stars appear to be stationary or "fixed" into their constellations, and all the constellations are fixed in relation to each other, as if painted on some immense cathedral dome over our heads.

And that inverted Bowl they call the Sky -- Rubiayat of Omar Khayyam

The dome as a whole appears to turn through the course of a night, and somehow "shift" westward over the course of a year. When extended far enough, such a dome would seem like a magnificent sphere surrounding the whole world, namely the celestial sphere. This is not the true nature of the Universe, of course; but the celestial sphere serves a remarkably useful purpose for us. Mapping out the positions of all objects (including the Sun, Moon, and planets) on the surface of the celestial sphere is the job of a coordinate system.

To introduce this coordinate system, let us once more begin with something familiar and then press on to less familiar ground...

Navigators living in the Northern Hemisphere will all have heard of the "North Star", Polaris. This is the star that doesn't seem to move in the sky (to the unaided eye, anyway) and hangs high over the North Pole of the Earth.

So if I were standing on the North Pole, Polaris would be in my zenith?

Brilliant! Absolutely correct! That level of understanding will be important after all 3 of our coordinate systems are in place. For now, let us say that it suggests we can make an analog of Latitude in the sky. Since Polaris is in the zenith of the North Pole of the Earth, why not just go ahead and call that spot the North Pole of the sky, or North Celestial Pole? And so we have. Now having decided that our celestial sphere has a North Pole, then 180 degrees away it must have a South Pole, which will be exactly in the zenith of the Earth's South Pole. To carry this idea even further, we could say that all along the Earth's Equator we could pick points in the Equator's zenith, and collectively call them a Celestial Equator, which many still refer to by its older name, the Equinoctial.

Our Celestial Equator divides the sky into Northern and Southern Hemispheres, entirely analogous to the Earth's Northern and Southern Hemispheres. When we plot the position of celestial objects North and South of the Celestial Equator, we refer to this as the Declination, denoted Dec. of the body. Since we are building a nice analog of Latitude, you may expect that declination is measured in degrees North and South of the Celestial Equator. You would be exactly correct in thinking that!

Declination data for many celestial objects is part of the data compiled in our handy reference book, the Nautical Almanac.

Guess that covers North and South in the sky, but what about East and West?

Yes, we must consider that, too. This part is confusing to many beginners, because we have some really arcane language to describe something which will be unfamiliar to begin with--a double whammy!!! So take heart and we will hit it straight on.

As in all coordinate systems, we have to pick some special point to be "zero". In the sky we call this the First Point of Aries, which gets a symbol I cannot reproduce in html, so we will just call it Aries. It refers to that spot on the Celestial Sphere where the Sun appears to be when it crosses the Celestial Equator from South declination to North. The day when this occurs is the vernal equinox (or spring equinox or the now-politically-correct March equinox) and is popularly called "The First Day of Spring" on many calendars and in the media. But we are interested in location in the sky, not time, right now.

So the First Point of Aries marks one exact spot on the Celestial Equator?

Yes! You are doing fine so far. Now we must use a bit of imagination. Suppose I could draw a "line" in the sky at right angles to the Celestial Equator at this First Point of Aries and extend it all the way around through both Celestial Poles. Clearly, we would have something analogous to a meridian on the Earth. But remember that a meridian refers only to the Earth! When we have its analog in the sky, it is referred to as an Hour Circle. The particular hour circle that includes the First Point of Aries is properly called the Hour Circle of Aries. Where the Nautical Almanac marks columns of data "Aries", the column presents information about this Hour Circle.

The Hour Circle of Aries is the sky's analog of the Prime Meridian, and we designate it as 0 degrees sidereal hour angle, denoted SHA. This sounds appalling, I know, to use 3 words to be the analog of earthly Longitude, but that is just how it goes--the peculiar and archaic language again.

Now the next interesting tidbit about it: We measure sidereal hour angle (SHA) only westward from Aries, so it runs between 0 degrees and 360 degrees. Since we are measuring in one direction only, it is not necessary to specify E or W as with Longitude.

The Declination-SHA coordinate system for the sky is used by mariners. Astronomers use a system of Declination-right ascension to map the heavens. Right ascension is measured in units of time, not degrees, and this is historically how we pick up all this dreadful "hour circle" vocabulary. For the astronomer, this system makes sense because his observatory is fixed in one location and he is interested in knowing when objects will move into optimal observing postions. For the mariner however, right ascension is too big a nuisance since the "observatory" (the vessel) is moving around and the mariner is looking for answers as to where--and the answers will be written in degrees. So we skip an inconvenient step of converting between time units and arc units by simply starting in arc units and staying with them.

The Nautical Almanac tabulates the positions of stars in SHA. If you have a further interest in learning the right ascension system, you will have to find another Web Page!!!

Wow! What a lot of words. It must be time for a diagram. The diagrams on this page look a bit more polished than mine, because they are digitized from Pub. No. 9, American Practical Navigator which is a US Government publication and hence not copyrighted.

The position of objects in space is given by declination and hour angle. Hour angles may be measured from our reference in space, the Hour Circle of Aries (not shown in this diagram); or they may be referenced to some meridian on the Earth, which has been projected onto the sky to make a "celestial meridian". We'll say more about this in a few moments... This diagram shows the second case. If the meridian in question is our own, the hour angle will be a Local Hour Angle, denoted LHA. If the meridian is the Prime Meridian, the hour angle will be a Greenwich Hour Angle, denoted GHA. The Nautical Almanac tabulates the positions of planets, the Sun, and the Moon in GHA.

Now let us move on to the most curious and interesting coordinate system of all, our personal coordinate system.

This one is no surprise now that you understand the concept of the altitude-intercept method. The third important coordinate system is the now-familiar altitude-azimuth system. As you recall, its primary reference point is the spot exactly over your head, the zenith. Since we know that the sky extends all the way around the Earth, we can imagine a point 180 degrees from our zenith. This would be the point directly below our feet all the way out the other side of the Earth. We refer to it as the nadir, although the word itself is today more useful for describing American politics than Celestial Navigation. :)

Of course, we can observe things only if they are above our horizon, so altitudes are normally measured from the horizon, which is 0 degrees, all the way up to the zenith, at 90 degrees. Altitudes below our horizon are simply designated as negative altitudes.

We know from the first lighthouse example that an altitude of a body is insufficient to place it in our local sky. An azimuth is required, too. Azimuths, denoted Zn, we measure by reference to direction around the horizon, so they will run from 0 degrees at True North, the azimuth of the North Pole, and around the horizon clockwise up to 360 degrees.

In the two other coordinate systems, we develop a series of "lines" which are really circles when drawn on the surface of a sphere. We can do this in our personal altitude-azimuth coordinate system as well. Imagine that we could draw a line that crosses the horizon at right angles. If we then extend the line all the way around, it will include both our zenith and our nadir. Such a "line" is called a vertical circle. Every vertical circle has an upper branch which seems to be above us and above the Earth as it soars upward through the zenith, and a lower branch which is blocked from our view where it descends below the horizon.

We designate 2 special vertical circles. If we take the vertical circle that passes through the point due East of us, that is, it has azimuth 90 degrees, it will necessarily trace through the zenith (because that is how we defined a vertical circle) and also connect the due-West point (azimuth 270 degrees) and the nadir all on one circle. This is called the Prime Vertical.

The other special vertical circle is the result of crossing the prime vertical at right angles in the zenith. This vertical circle will necessarily fall on the horizon at exactly the North and South points, azimuths 0 degrees and 180 degrees, respectively. This is formally known as the principal vertical circle, but no navigator ever calls it that...

That confusing tangle of words deserves a diagram!

The position of objects in relation to ourselves is given by altitude and azimuth. Notice the similarity between this diagram and the previous diagram. Note especially the presence of the "celestial meridian".

How interesting! Don't meridians run perfectly North and South along the Earth?

Yes! And that is how navigators name the principal vertical circle--it is the celestial meridian. As you may have also noticed, this is another case of taking some coordinate "line" on the Earth and projecting it outward on the sky, much as we did to make a relationship between the Equator and the Celestial Equator. This is a recurring theme in Celestial Navigation.

The celestial meridian is really at the heart of our whole elaborate system. It can be thought of as 3 things at once:

• a meridian in the Latitude-Longitude system of the Earth
• an hour circle in the Declination-Hour Angle system of the Sky
• a vertical circle in the altitude-azimuth system of the Navigator

This "interchangability" is not just some theoretical abstraction, either! You will use this idea of projecting earthly things on the sky, or conversely, projecting celestial things on the Earth, as routine everyday work in drawing the "Diagram on the Plane of the Equator" to check your figures. That name sounds really pompous, but it is a very useful construction--we will have more to say about it in a moment. I am mentioning it as a "teaser" to keep you interested now.

The Cookbook Summary of Coordinate Systems

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