The heavens declare the glory of God; and the firmament sheweth His handiwork. --Psalm 19:1--

Are we coming down to the end then?

Yes, indeed!

In a sense, I've put the cart before the horse by obtaining computed altitude first. In practice, of course, we make observations on deck, and find computed altitudes for the times we have taken the observations. I chose to save this for last because it is really the least interesting. There are no intriguing ideas here, unfortunately. It just reads like a long laundry-list of sextant corrections, which it is.

What needs correcting? Did you make some mistake?

The word "correction" here just falls under that now-familiar category of peculiar and archaic vocabulary.

The corrections are needed to put our purely-mathematical computed altitude and our real-world observed altitude on a like basis of comparison. You see, the computed altitude is based on a number of conditions, or assumptions, some of which are:

• we are making observations from an Earth which has no physical extent, or said another way, our eye is located at the center of the Earth;
• all light rays come to the Earth from infinitely far away, and hence are all exactly parallel to one another;
• air has no index of refraction, or said another way, the Earth has no atmosphere;
• orbits of celestial bodies are perfect circles;
• all sextants are mechanically and optically perfect.

Clearly, none of these assumptions are literally true. They are "close enough to true" that the errors these assumptions introduce are small; however, the errors are measurable and taken all together, can add up to a very important discrepancy.

We could make a lot of corrections to our calculations to bring them back to reality, but for historical reasons entirely unknown to me, we leave our calculations "as is" and instead apply all the "corrections" to our sextant observations, as if Nature were somehow wrong and our pretty mathematical abstractions were the Truth of the Universe. Ah, the arrogance of science...

Anyway, this state of affairs will give rise to a number of names of altitudes, all of which are needed by the Navigator. He will find the corrections to be applied on still-more tables in our very useful Nautical Almanac.

When we actually take a reading from the sextant, the number obtained is called sextant altitude and denoted hs. (Note lower case "h")

Most sextants are imperfect things, and there will be some misalignment between the optical system and the mechanical index arm where the scale is found. The resulting error is called index error and there will be some Index Correction, denoted I.C., placarded in the sextant case. This correction must be added to or subtracted from every sextant altitude.

Next, an "error" results from the height of the observer's eye. We are all aware that we can see farther from atop a tall building than on the ground--the distance to the horizon therefore changes with the height of the eye of the observer. Obviously, since our altitudes are nothing but the angle between the horizon and a celestial body, anything that alters the horizon is going to cause us trouble...In navigation, we call this trouble the dip of the horizon and for it we must make a Dip Correction. The amount of the correction is tabulated in the Nautical Almanac.

When we take sextant altitude and correct it for index error and dip, the sum is called apparent altitude and denoted Ha, except some folks use just H. (Note upper case "H")

Apparent altitude is a very useful number, which we need as an entering argument in other tables in the Nautical Almanac. Exactly which table depends on the body we are observing--there are different tables for the Sun in winter and in summer, for the Moon, and for the stars and planets.

These tables take into account many things. For example, the Moon appears to change size as its orbit approaches the Earth or recedes from it. It also goes through a familiar cycle of phases--the horizontal parallax number, mentioned in passing earlier, is used as an argument for the Moon tables to take these things into account. Likewise, the Sun appears to change size as the Earth's orbit approaches it in fall and recedes from it in spring. This accounted for by splitting the tables for the Sun into appropriate months of the year. The "infinite distance" assumption hold best for the stars and outer planets, which are so extremely far from the Earth; yet our neighbor planets Venus and Mars are still close enough that they get specific corrections during certain months of the year. Lastly, sometimes the navigator must take into account the fact that the index of refraction of air, and hence the bending of light rays in it, varies with barometric pressure and temperature. A table exists to correct for this problem, too, although it is usually unimportant unless the body was observed less than 10 degrees above the horizon. These necessary corrections are all found in the Nautical Almanac.

When one takes an apparent altitude and corrects it using the appropriate altitude correction tables in the Nautical Almanac, the result is at last observed altitude, Ho.

Now after that very dry explanation, it must be time to return to our example navigation problem. We will keep all as before, and this time we will add the sextant altitudes. From this information, we will have a typical problem facing the Navigators of the old days...

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 lower limb of the Sun at sextant altitude 55^o 54'.4, and 9h 43m 32s when we observe the upper limb of the Moon at sextant altitude 9^o 45'.2 Our sextant takes I.C. (+) 0'.6; and sitting on the cabin trunk of our vessel for the observation makes our height of eye 9 feet above waterline. It is a beautiful morning in my hypothetical world: temperature 62^o F, barometric pressure 30.08 in. Hg (inches of mercury)

Required: The celestial fix for 0943 on the morning of May 9th.

This sounds pretty routine, except for the low-altitude observation of the Moon. Most Navigators are a bit uncomfortable with altitudes under 15 degrees because the atmosphere can have some really unusual quirks to change refraction and ruin our observations. This is why we could regard this fix as only "pretty reasonable", not "really terrific". I would probably begin by consulting Table A4 of the Nautical Almanac to see if the refraction correction for the Moon is expected to be big just based on temperature and air pressure. We see that prevailing conditions put us in Column H of the Table, and so we would not have a refraction correction bigger than 0'.2, so this is not worth bothering about. Now we can set to work...

Step 1: Convert hs to Ha

This is speedy--we just apply the index correction which we get from the sextant case, and the Dip Correction for 9 feet which we get from the bookmark of the Nautical Almanac.

Notice that the bookmark contains several useful tables. We will be returning to the Altitude Corrections for the Sun in a few moments. To keep the image size reasonable, I "cut out" most of the center of the table, since all information required for this problem can be found near the upper and lower edges. This is the meaning of the jagged line across the center of the image.

Let's put the data in a table for convenience in arithmetic:

Step 2: Convert Ha to Ho

Now things get a little more involved, but not horribly so.

The altitude correction table for the Sun is conveniently located on the bookmark. The altitude correction tables for the Moon are inside the back cover of the Nautical Almanac. They have entirely different look-up systems, so it is important to read carefully the instructions located with the Moon Tables.

To use these Tables, we must also understand another word in our peculiar and archaic vocabulary: the limb of the Sun or Moon. The limb is nothing more than the "rounded edge" of the Sun or Moon. When we are taking sextant observations, the Sun and Moon do appear to have a physical extent, unlike the stars and planets. To make an observation, we adjust the sextant until the limb of the body appears to be exactly on the horizon. Since all our calculations are taken to the center of the body, and we have measured to the edge, there is a correction to make up the difference. With the Sun we get to choose whether to observe the upper or lower limb; but the phases of the Moon usually preclude us from choosing in its case--we must use whichever limb exists for us at observation time. In the present example, that is the upper limb of the Moon. Our calculations will follow accordingly.

We look on the bookmark for the Altitude Correction Table for the Sun in the months Apr.-Sept. We glance down the column of Apparent Altitudes until we find 2 entries bracketing our present Ha for the Sun. The center column provides for us the altitude correction. Since we may neglect the Additional Correction for Non-Standard Refraction, the sum of apparent altitude and the altitude correction is our goal: observed altitude, Ho

Turning to the back inside cover of the almanac, we see that the Moon Tables are much more extensive. So much so, that I had to "Crop" most of them from the digitized image, and leave only the part of immediate interest. We will need the left-hand page, which covers altitudes 0-35 degrees. We see entries for apparent altitudes broken down into ranges by degrees. Within each column, the apparent altitudes are listed for each bold-faced degree (in this case 9). Each degree is broken-down by increments of 10'. We will pick the correction for the table entry closest to our apparent altitude (in this case 40'), and note the correction. Then we look down that same column to the bottom half of the page, where we will find an additional correction for horizontal parallax, which depends on whether the observation was an upper-limb or lower-limb sight. This table is where we use the H.P. number tabulated for the Moon in the main pages, which we mentioned earlier. In the present case, we will enter the correction table with H.P. of 59.3 for an upper-limb observation (these are the columns labeled with the U). We see that our H.P. argument is between 2 tabulated values, so we must estimate whether to mentally change the respondent to "fit in-between" as well, or just pick the closest one. In this case, it is more appropriate to pick the respondent for H.P. 59.4 since we are very close. Last but not least, we observe the strict warning in the Table instructions which states that all corrections are to be added, but for upper-limb observations 30' must be subtracted. This is an important point!

Now let's make a table to apply all corrections to apparent altitude and find the necessary observed altitudes:

Notice that this process yields no azimuth information at all. Azimuth data comes strictly from computation, as we have already seen.

Step 3: Compute Hc and compare with Ho

In solving a real problem, this would be the time to begin our calculations for Hc, but since we just completed that on the previous page, we can simply copy the results over here and find the intercepts.

Step 4: Plot intercepts to obtain the fix

With intercepts and assumed positions in hand, we are ready to plot the fix! I should note here that to be a strictly correct fix, the assumed positions should be "advanced" to the exact same time to account for the movement of the vessel during the time interval between observations. However, these observations were taken only about one minute apart, and at the speed of sailboats, one minute will not make a whole lot of practical difference. This is why I provided no course or speed information about the vessel in this example.

Since my computer graphics programs are rather primitive, I actually plotted the answer by hand on a plotting sheet; I attempt to recreate the result here. The final answer to the problem, and indeed this Web Site, is: The celestial fix for 0943 on 1996 May 9 is L. N 38o 24'.9, Lo. W 42o 05'.6 We have found our position at sea by means of observation of astronomical objects far away in space, an accurate clock, and a bit of mathematical elegance.

Wow! So that's it?

Yes, but no.

In these Web Pages, I have tried to show you just a hint of the work that goes into Celestial Navigation. I hope I have answered questions in generality enough that you see important concepts, and in detail enough that you see it really does work to solve a practical problem--this isn't just some academic exercise.

I sincerely hope you have enjoyed your visit with me, and that you will consider buying any of the many excellent books and video tapes now available to deal with many of the 10 000 details I simply skipped over in order to get through this one example. There is still so much left to talk about!

For further reading here, I have prepared 3 Appendices to deal with topics which are interesting and important, but not essential to the fundamental work of the altitude-intercept method.

"I wish you a pleasant voyage."

Go to the Celestial Navigation Appendices

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Go to the previous Celestial Navigation page

Go to the Entry Point to this tutorial