... «Se tu segui tua stella,

non puoi fallire a glorïoso porto,

--Dante Alighieri--

Translation: "If thou follow thy star, thou canst not miss the glorious port..."

The Compass Rose takes you to the top page...just in case a Search Engine dropped you in the middle of all this!

Is that all there is to coordinate systems?

Actually no; but these 3 are the only ones of importance to the navigator.

Much of the rest of the theory and indeed, the practical work, will consist of relating the 3 systems of coordinates.

The easiest way to see the relationships is to draw them, so let us begin by introducing that "Diagram on the Plane of the Equator." First, in all our discussions of celestial mechanics, we will suppose that the Earth is a perfect sphere. We will also suppose that all the appearances to our eye are literally true.

What do you mean by that?

That our picture of the Universe described on the last page is really so. Namely, 3 things are true:

I know this flies in the face of centuries of scientific thought, but for the purpose of Celestial Navigation it simplifies matters quite a bit, and rest assured, your calculations will be entirely accurate. So for now, just go along with it. Having a globe close at hand may also help until you are accustomed to visualizing the systems without a physical model.

All that said, let us begin...

First imagine that you may have a "god's-eye view" of the Earth; that is, you may look at the Earth from an extreme distance above the South Pole. What do you see? A ball, of course. And, assuming you are no great artist, how would you draw what you see? As a circle, I'm sure. If you are looking from directly above the South Pole (denoted Ps), the "edge" of the globe you see is not just any circle. It is the biggest possible circle that may be made by cutting the Earth in exact halves. And the circle itself would just so happen to be the Equator. This is how we get the name "Diagram on the Plane of the Equator." It is also very important to note that we can also assign directions to this diagram. Since we are looking down at the South Pole, working clockwise along the Equator will be the same as moving East along the Equator. Clearly then, working anti-clockwise around will be moving West.

This is a very useful picture once we start adding in some details.

Like what?

Meridians!

If you want to draw the meridians of the globe on this flat, 2-dimensional diagram, they will simply look like radii of the circle. Navigators are especially interested in 2 meridians:

So if we want to make a sketch of our own longitude, we customarily place ourselves at the top of the picture, and fill in the Greenwich Meridian (Prime Meridian) in its approximate relationship. It is definitely time to show a couple of examples of this idea:

Notice that this diagram is an approximate sketch. It doesn't need to be draftsman-grade perfect since its only purpose is to help the navigator decide the East-West relationships between objects of interest. In these sketches, the only one shown is the relationship between the observer on his meridian, M, and the Prime Meridian, G. This diagram is hardly worth the trouble if this were the only relationship it could show, but obviously there is much more to come!

Let us continue developing this tool by considering what happens between us and the Sun as one day passes. For now, let us assume we are staying in one spot on the Earth, which is to say our own meridian does not change, for an entire day. At dawn we will see the Sun low in the East. At noon, the Sun will be as high over us as it can get that day. In evening, of course, it will set in the West. Then all night it will seem to travel all the way around the "back-side" of the Earth to rise again in the East the next day. We can sketch these events in this way:

The important part to notice is the line I have drawn from the "center of the Earth" to the Sun out in the Sky. This "line" is actually just the hour circle of the Sun. Notice that shows there is always some angular relationship between my meridian and the hour circle of the Sun. In fact, since both "lines" fall on the Earth at least for part of their length, it is perfectly legitimate to say that the Sun has a meridian! This is another example of projecting a celestial object onto the Earth, and it tells us that we are perfectly free to omit the circle of the Sky, and just treat objects moving around the Earth as if they are moving on the Earth, at least for purposes of finding angular relationships.

So now let us redraw the example of the setting Sun as we actually will in the everyday practice of navigation:

  • The Sun is symbolized by a circle with a dot at its center.
  • GHA Greenwich Hour Angle is that angle measured westward from the Prime Meridian to the hour circle of the Sun, which we represent as a meridian on the Earth.
  • LHA Local Hour Angle is that angle measured westward from the Navigator's Meridian to the hour circle of the Sun.
  • This diagram makes it very clear that LHA and GHA are related to each other by the Longitude of the Observer. In this instance, we must subtract Longitude from GHA to obtain LHA.
  • It is simply understood that Longitude Lo. refers to objects on the Earth, and that GHA and LHA refer to an object in the Sky.

This simple sketch speaks volumes about celestial mechanics! Please notice that I have introduced a couple of very important angles in this diagram.

Yes, and why on Earth did you have to add still more complications to this explanation?

On the contrary, this is a key to making things simpler.

You see, part of the information in the Nautical Almanac is declination data about the celestial bodies. Another part is data on the GHA of those same bodies. We already know that things are in constant motion, so this is the utility of the Almanac--it compiles the declination and GHA of the planets, Sun, and Moon, as well as the GHA of the Hour Circle of Aries, for every hour of the year, so we always know where the objects are in the sky in relationship to a fixed reference on the Earth, namely the Prime Meridian. By drawing Diagrams on the Plane of the Equator, we can use an approximate knowledge of our Longitude to decide on the relationship of the celestial body to us, and we call this relationship LHA.

Here is the sweet part: LHA is a piece of data which allows us to compute that computed altitude needed to work the altitude-intercept method.

So with the Diagram on the Plane of the Equator we have hit a relationship between celestial mechanics, coordinate systems, and known compiled data, plus we are gaining pieces that let us do our computation for needed data. That is really powerful for one little picture!

As if that weren't enough, the sketch also hints at how celestial mechanics defines time on the Earth, and why time zones exist. Let us first make this point perfectly clear, and then we will be ready to work our first mathematical example.

Consider that I am at exactly Longitude 90 W. Also suppose that it is exactly local apparent noon, that time at which the Sun is as high in the sky as it can possibly be on a given day. At this instant, the hour circle of the Sun is exactly over my meridian, or said another way, the Sun and I are "both on the same meridian". This is a common event which will happen once a day for every fixed location on the Earth. Things look like this:

  • It is customary to omit the designation for the South Pole since it is obvious from the rotational sense of East and West that we are "looking down on the South Pole" in these diagrams.

Given this situation, notice what is going on with the Prime Meridian. The Sun is now 90 degrees West of it. In fact, the Sun is setting! To make this more clear, let me draw the diagram over, this time with Greenwich on top.

Of course, in everyday life we all want the Sun "on our meridian" each day at noon. After all, this is what we mean by the word "noon." Therefore, the Sun must be traveling 360 degrees/24 hours = 15 degrees per hour. Or, said another way; to keep all clocks reading approximately right everywhere on Earth, we ought to divide the world into 24 "zones", each 15 degrees of Longitude wide, and make each zone one hour "earlier" than the one East of it. This is exactly what was done, and we call it Zone Time, denoted Z.T.

Daylight Savings Time, or Summer Time, is another complication which I won't say more about. I'll also gloss over the International Date Line by leaving you to think about the implications of zone time and what happens if you make multiple laps around the Earth!

For this Web Page, it will be sufficient to mention that our useful reference, the Nautical Almanac compiles all its data for time at the Prime Meridian, called Universal Time and denoted UT. (This time standard was formerly known as GMT for Greenwich Mean Time.) It is necessary for the Navigator to know which Time Zone he is keeping and apply the appropriate zone description in order to look up correct information. You can figure out your zone description, which is the number of hours difference between your clock and Universal Time if you just get correct time from the US Naval Observatory Master Clock and subtract.

And with that, it is time for our first practical problem...

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