A boat is called "she" because she is graceful and attractive, spirited and fun, and she has her own ways of doing things. She won't tolerate your clumsiness or sloppiness, and yelling she'll just ignore. But if you know just how to ask her for what you want, and treat her right, then she'll make you a good home and never let you down. --old traditional lore, Source Unknown (and have times changed since then!!)--

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

Must I remember trig for this part?

NO! This will be only simple addition and subtraction, sketching our Diagram on the Plane of the Equator, followed by more simple addition and subtraction. Sound easy? Good, because it is!!

Here is the situation: We estimate our position at L. N 38^o 25'.5 , Lo. W 42^o 12'.8 on 1996 May 9. The Moon has waned to a half-moon, so we can make a morning observation of both Sun and Moon for a pretty reasonable celestial fix. We are keeping zone time (Z.D. W (+) 3h), and our accurate watch tells us it is 9h 42m 18s when we observe the Sun, and 9h 43m 32s when we observe the Moon.

Required: The LHAs of both the Sun and the Moon at the times observed.

Bonus:     The declinations of both the Sun and the Moon at the times observed.

First of all, pay close attention to the notation in this "story problem." Notice how we use symbols for Latitude and Longitude and that we designate N S E or W before the numerals; we write dates out as shown, and we record time broken down into hours, minutes, and seconds in a very specific way to avoid confusing time minutes and seconds with arc minutes and seconds.

Second, notice that we begin a celestial navigation problem with some estimate of our present position. In a sense, we never use celestial navigation to "find our position", we use it to "refine our estimate of our position."

Let's begin to solve this problem...

Step 1: Convert watch time to UT. In this case, there is no watch error, so the minutes and seconds recorded are fine.

We are keeping zone time, however, so we will need to make a correction to hours. We are West of Greenwich, so our zone description tells us to add 3 hours to watch time to make the observation times this:

That was a simple beginning...let's press on.

Step 2: We'll open the Nautical Almanac and start looking for information. The Nautical Almanac is organized by date, recording 3 days down each set of pages. We are looking for May 9. When we find it, we notice that information about the stars and planets is on the left-hand page, and information about the Sun and Moon is on the right-hand page. Glancing down the Sun and Moon columns, we quickly notice that the entries are made for each hour of UT for every day of the year, exactly on the hour.

Here is a digitized section of the page we need. I enlarged it to be legible on a computer monitor:

As an aside, let me apologize now for the large image sizes and hence slow load times of the next few pages--it was the best I could do with the image processing tools at hand. I felt it important that you see for yourself not only how, but also where a Navigator gets vital information. I highlighted nothing in the tables, so that you may practice finding the information required by the problem.

Does it matter that our observations weren't taken exactly on the hour?

Absolutely yes! We must always know the correct time of our observations down to the second. So now I should mention how we get the data we need down to the second, when they print it in the Almanac only for the hours.

In the back section of the Nautical Almanac is a set of pages with grey edges. These are the Increments and Corrections pages. There are 3600 entries in these pages, one for each second of an hour. So the game is to look up in the main pages the declination and GHA for the hour of UT the observation was made, then add to that main page value for GHA the increment for the minutes and seconds of time past the hour of our observations. Declination is handled a bit differently...

There is also another important point in using the Increments and Corrections tables. Since we just discussed the "Increments", you have probably guessed that now we will speak of the "Corrections". Notice that for each minute table, there is a table associated with it for v or d Correction. The small italicized v or d refers to the speed with which a planet, the Moon, or the Sun is moving against the background of the stars. These values are found in the main tables and used as entries or arguments for the Increments and Corrections tables. The d is the change in declination from one hour to the next, while the v tells the change of the body's GHA.

A little study of the main tables reveals that the Moon changes in declination and GHA quite a bit even from hour to hour, while the Sun and planets change much more gradually.

Some very important notes here:

• There is never a v correction for the Sun. The Sun's v is already "built into" the main table values.
• Main table values for v are usually positive and the correction is to be added to GHA. Exceptions sometimes occur for the planet Venus and expressly note the minus sign (-)
• Main table values for d may be positive or negative and they don't say expressly which! You must glance at all the declinations listed to see if they are increasing or decreasing and decide for yourself whether to add or subtract the correction.

So let's build up our tables for the complete Almanac information required in the problem. First, we simply read information from the main pages shown above:

(You no doubt notice that the Moon listing also tabulates a 5th value, H.P., for Horizontal Parallax. We will need this number to solve an actual navigation problem in its entirety, but let us skip it for now and return to its meaning later.)

We note that 1996 May 9 was one of those tricky days when the Moon was in South declination and moving back toward the Celestial Equator, so we had to insert that (-) for the d value.

Now we turn back to the grey-edged pages and pull out Increments on minutes and seconds after the hour, and Corrections for v and d. We must take care to find the correct increment--the Sun and Planets, Aries, and the Moon each get their own columns in these tables.

Here is the correct page from the almanac for both times in this problem. To reduce image sizes I choose to display information only for the Sun, except for cases when the Moon requires a different set of tables. See if you agree with my answers for the Sun!

Notice that only the minute of the observation time is used for the v or d Correction. The exact second of time does not matter here, as it does for the increment.

The values, or respondents look like this:

So much for the Almanac...that wasn't too terrible, just tedious--exactly like I said in the Introduction!

Step 3: We'll arrange our information in columns to make addition easier. Then we will be ready to make a Diagram on the Plane of the Equator.

So we solved the first question?

Not quite yet. Remember, the problem asked for the LHA of each body, and the Almanac lets us solve only for GHA. Now we need a relationship between LHA and GHA. Of course, you already know how to do that...

Step 4: Just sketch a Diagram on the Plane of the Equator! Remember, we start by drawing our own meridian straight up the page to the top of the circle. Then we put in the Greenwich meridian, in its approximate relationship to our own. This is naturally our Longitude. Next, since we have the GHA for both bodies, so we simply sketch the hour circles of the Sun and Moon in their relationship to the Greenwich meridian. After we draw in arcs for the LHAs, the result will look like this:

The relationships are clear: The Sun has passed the Greenwich meridian (it is almost an hour past noon there.) The Sun has not yet reached our own meridian (it is only just before 10 a.m. on our clock.) The Moon has passed our meridian (it will soon set.) Since Hour Angles are always measured westward, clearly the LHA of the Moon will be some number less than 90 degrees. We obtain it by taking LHA(Moon) = GHA(Moon) - Lo. The LHA of the Sun will be something approaching 360 degrees. We obtain it by taking LHA(Sun) = 360 + GHA(Sun) - Lo.

Study this diagram until you feel confident that it all makes sense to you. For example, you know that it is only just before 10 a.m. on your clock, so the Sun could not possibly be in the western sky--that won't occur until after noon. Another common sense check: the Moon is setting, so it had best be far to the West of a Sun that is still rising. These are the sorts of cross-checks against dumb errors that the Diagram on the Plane of the Equator helps you make.

Step 5: We'll put in these last pieces of information and finish this problem...

There's the result! We can go for the Bonus, too. We really could have done this as soon as we had the Increments and Corrections in hand, but here we were focusing on getting an LHA first.

Is there a "common sense" way of checking this calculation, too?

Yes, indeed! You can turn back to the main pages of the Nautical Almanac and compare your answer with the tabulated value for the next hour of UT. Since your observations were made between 12h and 13h of UT, in this example, the results had best fit in between the values tabulated for 12h and 13h. Also, since the time was actually closer to 13h than 12h, we expect our calculated answers to lie closer to the 13h values than the 12h values. In this case, they do--so we can feel a bit more confident that we haven't made some simple error. It is really easy to do, especially when "crossing over" between the minutes columns and the degrees columns because of the sexagesimal notation. Always remember, a number listed in the minutes columns must always lie between 0'.0 and 59'.9 Any other result is a mistake.

So how did all this help us?

Although it may be difficult to appreciate now, this really helped immensely. The reason is this:

Remember the essence of the altitude-intercept method...we need to compare a computed altitude with an observed altitude for each body observed. Well, to make our calculations, we must have 3 arguments:

• our Latitude--estimated from keeping up with the vessel's progress;
• the Declination of the body--taken from the Almanac, as we just did;
• the LHA of the body--computed from GHA out of the Almanac and our Longitude, as we just demonstrated as well.

Thus, we have everything in hand to actually compute an altitude of a celestial body. (In fact, 2 bodies!!) We'll try this out on the next page...

Go to the next Celestial Navigation page

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