Please forgive me for boring you all to tears with more of my celestial navigation stuff, so if you’ve had enough, please feel free to skip this. What follows is an introduction to a course in celestial navigation I am developing, The outline is not designed to teach you how to navigate by the stars, but to outline what’s involved before digging into the meat and bones of it all. The idea is to let you know how its done, not how to do it. If you’ve ever been curious how a ship is navigated by the stars, but are not necessarily keen on actually learning how, this one’s for you.

Introduction

The bare minimum equipment required for celestial naviation is
a) A sextant
b) A timepiece correctable to UT (Greenwich Mean Time).
c) A current Nautical Almanac
d) A chart or plotting sheet of your area
e) A calculator capable of trigonometry or sight reduction tables.
f) Simple drafting tools (dividers, parallel ruler), pencil, paper

The only mathematical skill you will need is the ability to add and subtract angles expressed in degrees and minutes.
Every celestial object, whether it be Sun, Moon, star or planet, is directly above some location on Earth. We call this spot the Geographical Position, or GP, of that heavenly body. The objects are in constant motion, and the Earth itself is spinning on its axis and flying about the sun, so the GP of a body is constantly moving, but astronomers study these objects and their motions and can predict the GP of the navigational stars with high precision. This information is coded in the Nautical Almanac in such a way that you can correct for all these motions if you know the exact date and time.
When we measure with a sextant the altitude in degrees of a star above the horizon, we immediately know how far away we are from its GP. For example, if the star is directly overhead (altitude 90 degrees) we are at the GP. If its altitude is zero degrees (directly on the horizon) we are 90 degrees away from it A degree is 60 nautical miles. That’s all there is to it.

Of course, knowing how far you are from some point on Earth still doesn’t tell you where you are, but it does tell you where a Line Of Position (LOP) happens to be. If a sextant reading tells you are X degrees away from a known spot on Earth, all you can say for sure is you are somewhere on a huge circle drawn on the Earth where your sextant would show 90-X degrees.

To determine where you are you have to measure the sextant altitude of some other body and generate another LOP. Where the two LOPs intersect is your location. Of course, two giant LOP circles will usually intersect in two places, but one of those is so far away from the other you can usually pick the right one. To be safe, you need to measure three stars, and the point where all three circles intersect is your exact location.

Fortunately, it turns out that you don’t have to worry about these huge LOP circles. They are so big that all you need be concerned with is the tiny bit near your ship, and the tiny piece of that circle is, for all practical purposes, a straight line. So when you derive an LOP from a sextant reading on a star, you draw a straight line on your chart and you are somewhere on that line. The LOP from the second star will intersect the first and where they cross is your position, or “fix”.

So that’s the theory. What’s involved in practice? We will list the steps or each sextant observation and its corresponding sight reduction and plotting. I will discuss all these steps later in more detail.

1) Sextant Observation. Observe the star with the sextant, note the sextant reading Hs and the exact time. Correct the sextant reading value with the necessary corrections to find the Observed Height (Ho).

2) Almanac Lookup. Find the Almanac GHA and Dec values for that star at that time and date, add the corrections in the Almanac if necessary. These values are the star’s position in the sky, and the location on earth of its GP.

3) AP Selection. Find a convenient Assumed Position (and mark it on your chart) at Assumed Latitude (ALAT) and Assumed Longitude (ALON). It need not be too accurate. Anywhere within a hundred miles or so of your true position (which you don’t know yet) is good enough.

4) Sight Reduction Do the calculation using tables, calculator or computer with Ho, GHA, Dec, ALAT, ALON as input. The output of the algorithm will be the Azimuth and the Intercept for that sight. (Five numbers go in, only two come out). The sight reduction tells you where the star SHOULD appear to you IF you were at the AP.

5) Plotting. Draw the Azimuth through the ALAT, ALON of the AP on the chart or plotting sheet. The Azimuth is the direction of that star’s GP from your AP, At the AP, mark off the Intercept either towards or away from the GP of the body as required. At that spot, draw a perpendicular to the Azimuth. That perpendicular is your Line of Position for that heavenly body. Its a little tiny piece of the giant circle on earth where your sextant will read that altitude

6) Repeat steps 1) through 5) for additional LOPs based on other sextant observations. Two LOPs is the bare minimum, but any fool can get two lines to cross. If one or more is in error you will never know it. Three LOPs will tell you if at least one is in error, but not which one. Four will allow you to eliminate the bad LOP. Take as many sextant readings of as many stars as possible, just in case., if three LOPs cross at the same spot, you need not take the trouble to reduce the others.

To sum up, creating an LOP involves calculating the elevation of a star as if you were at the AP and then comparing that value with what you actually measure with your sextant. The difference, or intercept, is how far you are from the AP. Where all your LOPs intersect is your actual position. How closely they intersect gives you a measure of your possible error.

The most time consuming (and error prone) of all these steps is 4), the sight reduction. The sight reduction is the trigonometric solution of the nautical triangle, a huge triangle on the earth’s surface with the pole, the AP and GP at its corners. Sight reductions can be done on a navigational computer, a programmable calculator, a student/scientific calculator or even using paper trig tables (in descending order of convenience). The first three choices are the easiest, but all three involve you relying on an expensive, delicate or hard-to-replace piece of electronic equipment. The scientific, calculator is cheap (you can afford to carry several of them as backups) and can be purchased anywhere in the world so that is my preferred method. The tables are a backup to the calculator, its nice to say you can reduce a sight with no mechanical aid whatsoever, but looking up and writing down long lists of numbers is an error-prone and cumbersome task, particularly for an exhausted, scared sailor on a cramped and uncomfortable boat at night off a lee shore.

You don’t really need to know anything to navigate except steps 1) through 6). You will be guided through the whole process by pre-printed forms which remind you what comes next and keep track of all the numbers. But knowing a bit of the theory gives you a better understanding of what’s happening and helps you remember your place if you get lost.or confused. You can determine a position by just memorizing the steps and following them like a recipe, but you will never really trust your results or recover easily from an error. Trust me on this. And if you make a really big blunder (like shooting the wrong star) a knowledge of the theory will help you locate the problem and correct it without having to start all over from scratch. Also, a knowledge of WHY you are doing what you’re doing will allow you to take short-cuts and work-arounds with confidence. These will become obvious as you become familiar with the formal procedure.
Celestial navigation doesn’t mean you can’t get lost. But it does mean you will never be afraid of being lost again. You will always have the confidence that you’ll be able to figure something out eventually.

1)Sextant
There’s a lot to sextant technique and you need to know how to take care of your sextant and how to adjust it; but we’ll get to that later. I’ll also postpone to a later chapter how we decide which celestial bodies we choose to observe, and how to locate them. For the time being, let’s concentrate on the observation itself.
After noting the exact Universal Time of your sight (you will either have a stopwatch, or an assistant to record it), write down the measured angle indicated on your sextant. It is important to remember that these are the two raw data items provided by the astronomical part of a star shot: the reading and the time. There are two personal corrections to the reading that must be made. The first is index error. The second is dip correction. I call them “personal” corrections because they involve only you and your situation. There is no way they can be predicted or calculated ahead of time. Index error is the result of your sextant having some small residual error that cannot be removed by adjustment, that is, your sextant will not read zero when it is set at zero. This is easily determined (I will show you how later) but the index error must be added or subtracted from the sextant reading to account for it. The second correction to the reading is the dip correction. This is the error introduced into your reading by your height above sea level. In other words, the horizon is further away from you if you are on the bridge of a supertanker than on the deck of a yacht. Iit must be subtracted from your sextant reading after you correct for index.
The dip correction for different elevations above sea level is listed in the right hand column of page A2 in the Nautical Almanac.

For example, if your sextant is between 5.0 and 5.2 meters (16.5 and 17.4 feet) above sea level the dip correction is –4′.0. Dip is always negative.

Sextant altitude (Hs) +/- Index – Dip = Apparent altitude (Ha).

But you’re not done yet.The Ha must be corrected for refraction, the earth’s atmosphere distorts the position of a celestial body, making it appear higher in the sky than it actually is. Furthermore, this effect is more pronounced the lower in the sky the object is, so the error increases as the Hs decreases. Pages A2 and A3 of the Almanac give you this Main correction for different sextant elevations when observing the sun, stars and planets
(The moon is handled separately on pages xxxiv and xxxv).

These tables also take into account several other minor astronomical corrections so that you need not concern yourself with them. For example, if you are observing the upper limb of the sun from October to March of 2023 and your Ha is 9d 50′ your main correction is –21′.4. In other words, your fully corrected Ho is 9d 28′.6.
Observed altitude (Ho) = Apparent altitude (Ha) +/- Main correction
There are additional altitude corrections for extreme temperature and barometric pressure on page A4, but these are rarely necessary.

The Nautical Almanac will provide you with a complete explanation and examples of these procedures on page 258 (Altitude Correction Tables).

2) Almanac

There are two types of information in the Nautical Almanac : permanent and ephemeral. Permanent is information of use to mariners that doesn’t change, like instructions on how to do a calculation. In fact, the Almanac also serves as a navigation textbook. Everything you need to know is included there, although it is not necessarily in a way that is easily accessible to the student. The publishers make an effort to keep each yearly edition as similar as possible to previous years so navigators will remember where to look things up. Ephemeral data is information referring to astronomical bodies, which changes every year due to the motions of the bodies. The worked examples of calculations in the Almanac are as similar as possible to previous editions, but they do use ephemeral data relevant to the year of that issue. Keep this in mind.

The next two quantities we will need for our sight reduction are the Greenwich Hour Angle and the Declination of the observed body at the exact moment we took our sextant sight. The GHA is the number of degrees measured west from the Greenwich Meridian to the meridian of the celestial body. The Dec is the number of degrees north or south of the equator the body is. Declination is the same as the latitude. GHA is comparable to the longitude, except it is measured only to the west and it is constantly changing as the earth rotates .Fortunately Dec changes very little over the course of an hour. This information is coded cleverly into the daily pages of the Almanac and it takes a little effort to extract it. The daily pages start on page 10 and as you open each page you will see columns of figures for Aries (I’ll explain what that is shortly), the four planets and the 57 navigational stars on the left, and the sun, moon and twilight data on the right. Each page contains data for three days. Down the left hand side of each page you will see the lines of each of the even 24 hours of each day. So at the beginning of the almanac, opened to pages 210 and 211, you will see information for Jan 1, 2 and 3, on Aries, planets and stars on the left and sun, moon and twilight on the right. Each of the celestial bodies has a column for GHA and one for Dec and their values for each of the even 24 hours of the day.
Suppose you are interested in the GHA and Dec of the moon on Jan 20 of 2023 at 21h 31m and 18s UT. Go to the daily page for Jan 20 (page 23 in my copy), go the column for moon on the right hand side, and scroll down to hour 21. The GHA for hour 21 is 146d 42′.6. The Dec is S 27d 00′.4. You will also note there is a v value of 1′.0 and a d value of 3′.6 listed after the GHA and Dec, respectively. These are for minor corrections in GHA and Dec respectively which some (but not all!) of the bodies need. You will note the moon and planets have v and d corrections, Aries and the stars do not. The sun has a d correction but no v. We will see in a moment how v and d are used. V and d corrections are usually minor and can be ignored, but there are exceptions so until you become more familiar with this drill make sure you apply them.
So far, we have determined the GHA and Dec of the moon for the even 21st hour of Jan 20, but we now need to add the increment for 31m and 18s. For this we must turn to the “gray pages” near the back of the almanac (pp i – xxxi) and look for minute 31 and second 18 (page xxvii) increment under ‘Moon’, which is 7d 28′.1. The v value is 1′.0, and looking that up on that same page shows a v of 1.0′ which corresponds to a correction of 0′.5, But do we add or subtract it? The GHA values are increasing, so we add it. The final result for GHA is:

146d 42′.6 + 7d 28′.1 + 0′.5 = 154d 11′.2

The Dec correction requires no minutes and seconds increment, because Dec changes very slowly over time, but it does have a d of 3′.6 corresponding to a correction of 1′.9. But do we add or subtract it? Dec is decreasing, so Dec is

S 27d 00′.4 – 1′.9 = S 26d 58 ‘.5

For stars and Aries there is no v or d correction but the GHA of a star is its Sidereal Hour Angle (SHA in the daily pages) added to the GHA of Aries (calculated to the exact second).The “fixed” stars are so far away that their SHA and Dec changes imperceptibly, but the earth still rotates so their GHA changes by the second. Aries is just the zero point in x of the celestial coordinate grid.

The Nautical Almanac has worked examples for all planets, stars and moon/sun on page 256.

3) Assumed Position

The AP is your seed position, it is the point your sight reduction uses to determine how far away from it your ship is.. When you construct your LOP you get a line which is includes all the points on earth equidistant from the GP of your star.
The AP need not be accurate. Anywhere within a degree or so of your true (but unknown) position is good enough. Most navigators use their best estimate, or dead reckoning, position, as an AP. Navigators who use certain sight reduction tables which require certain entry values pick APs which honor those requirements. I use the nearest even number to my DR position. For example, if my DR tells me I’m near 26d 15′ N and 79 55′ W, I pick 26N, 80W as my AP. It will be a nice, easily located spot on my chart or plotting sheet, which reduces the chances of plotting errors or other mistakes. The AP will be the spot on the chart or plot sheet where all the Azimuths are drawn from for each sextant observation; like spokes on a wheel. And it will usually fall neatly on a grid intersection. It makes for a cleaner plot and less clutter. Since I use a pocket calculator for sight reduction, it means less keystrokes and less chances to make a mistake. Your AP goes into the sight reduction calculation as an ALAT, ALON.

There is nothing sacred or magic about an AP. Its chosen for convenience. As long its close to where you are, how close doesn’t really matter.

4) Sight Reduction

This is the calculation that solves the astronomical triangle and allows you to plot a line of position for every celestial body you observe.
The sight reduction requires as input Ho, GHA, Dec, ALAT and ALON, tT returns an Azimuth (a line from your AP to the GP of the star).and an Intercept (a distance towards or away, depending on whether it is positive or negative, from your AP to the star where you will draw (at right angles to the Azimuth) your LOP. There are commercially available programs that will do this for you, or you can program your own pocket calculator to do it. There are numerous tabular methods which can be employed, or you can use a student calculator to do the job. I prefer the latter, it is the best combination of simplicity and robustness. I have also programmed my HP-48G calculator to do this, it prompts me for the inputs and quickly returns the answers, neatly labeled. But if it fails on me or I lose it, it is impossible to replace or repair underway. I prefer relying on a cheap, solar powered Casio student calculator (I carry two), It’s a bit more cumbersome, but it is more dependable. I also carry paper tables as a backup. I will discuss sight reduction by calculator, at length, later.

The sight reduction algorithm I use on my pocket calculator is the one on page 279 of the Nautical Almanac.
The Almanac also includes a “concise” sight reduction table, (p 284) but I find it cumbersome and difficult to use. I prefer “Pepperday’s S-Table” (Paradise Cay Publications) instead.

5) Plotting

Now you have all the information you need to plot your position. You have a list of all the sextant shots, the Azimuths and Intercepts your sight reductions generated for each one, and you have carefully plotted the ALAT and ALON of your AP on the chart. Draw a line in the direction of the heavenly body along the Azimuth through the AP. Label it with the name of the body and the time of observation (eg., MoonLL, 21:31:18 UT). You can do this on your chart, or if you prefer to spare it from scribbles and erasures, you can use a plotting sheet. A plotting sheet is a paper form with a compass rose on it where you can construct a miniature Mercator projection chart of the ocean for your latitude around your AP. If the Intercept is positive, mark off from the AP TOWARD the celestial body the number of miles or minutes of arc. If your Intercept is negative, mark off the number of miles from the AP AWAY from the body .Use the mnemonic, “Coast Guard Academy” or “CGA” for “Computed Greater Away” . If the Computed elevation is greater than the Observed the intercept will be drawn on the Azimuth side on the side of the AP AWAY from the star. If the Observed is greater, the intercept will be positive, or TOWARD the GP. Use triangles or parallel ruler to construct a line perpendicular to the Azimuth at the marked Intercept. That line is your LOP. It is a tiny segment of the huge circle on the earth’s surface where your sextant would read Ho.

If you are traveling at only a few knots speed and all your sextant shots were taken within a few minutes, all your Azimuths can be plotted from the same AP, like spokes in a wheel. If your shots were separated by any substantial length of time, you may need to use several different APs and you’ll have to advance or DR your LOPs by the appropriate course and speed, the technique known as a “running fix”. I like to use different color ink pens for the AZs and LOPs to keep my plotting sheets from becoming too cluttered.

Where all the LOPs cross is your position. As a rule, the LOPs won’t cross at precisely the same spot. This is because of unavoidable errors in observing, plotting or calculation. But if they all cross within a few miles of each other you have a good fix, and you’ll notice immediately if one of them is in wild disagreement with the others. Discard that one. Navigators like to call the little triangle formed by three crossing LOPs as the “cocked hat”. If the hat is small, the center of the triangle is your fix. A good sextant man in ideal conditions can get himself to within a mile or so of his true position. Anywhere under a few miles in a small boat is considered excellent.

Bringing it all back home
These remarks have left out a lot of important details, not to mention a lot of shortcuts, work-arounds, and tips and tricks, but they should serve as a reminder of the important points involved in celestial navigation. Other important details (like how to use and adjust your sextant, or how to find a star useful for an LOP) will become clear through further study and practice. It is important to keep in mind how important practice is to learning this skill. A lot of classroom remarks make perfect sense when you hear them, but at sea, under pressure and perhaps exhausted or scared and in difficult conditions, you will just draw a blank. Only practice and drill will let it all sink in so it will be there when you need it. The upside of celestial navigation is that even if you never actually use it, it will give you an in-depth, fundamental appreciation of the process of wayfinding, how it relates to the entire problem of determining where you are, even if you are using electronics, dead reckoning, or some other method. It’s the confidence builder. You will understand the astronomy, geography, timekeeping, calendar, plotting, cartography, piloting, dead reckoning and mathematics and how they all interrelate to one another. Once you have mastered this skill, it will be all there at your fingertips, whenever you need it. It will give you the situational awareness and intuition that will guide you through the process and provide the subconscious alarm that something is off or you might be making a mistake. You can still get lost, but it will never terrify you again. You will always know what you need to try next, and how to go about it.