# IAP Seminar: Celestial Navigation at Sea -- Notes & Outline

(c) Douglas S. J. De Couto, January 2000

## Definitions and Terms

Degrees, Minutes, Seconds
360 degrees in a circle, 60 minutes in a degree, 60 seconds in a minute. Note this is different than time! Write as deg o min' sec'', or with decimal minutes.
Latitude
North-South angle between our location, the center of the earth, and the equator. Equator is latitude 0, latitude increases in North and South directions to 90 degrees at the poles. We can also think of North as +, and South as -. Lines of latitude are circles that run parallel to the equator.
Longitude
East-West angle between our location, the center of the earth, and the Greenwich meridian. Greenwich is chosen as longitude 0 for historical reasons. Longitude is measured in the East and West directions, increasing to 180 degrees. We can think of East as + and west as -. Circles of longitude are also known as meridians; they form vertical wedges of the earth, like orange wedges.
Small Circle
All points at the same latitude. Small circles are parallel, and decrease in circumference as you move further North or South from the equator.
Great Circle
All points on a circle on the earth's surface, around the center of the earth. The equator and all meridians are great circles. There are other great circles at different angles. Lines of latitude are not great circles; they are not centered around the center of the earth (except for the equator).
Geographical Position (GP)
This is the location on the earth immediately underneath a heavenly body (e.g. the sun or a star). That is, the GP is the point where a line from the center of the body to the center of the earth intersects the surface of the earth
Declination
The latitude of a body's GP at some time. We look this up in a table (in the almanac).
Hour Angle
The longitude of a body's GP at some point in time. Greenwich Hour Angle (GHA) is the East-West angle between the body's GP and the Greenwich meridian. Local Hour Angle (LHA) is the East-West angle between our position and the GP of the body. Hour angle is different from longitude in the following ways:
1. Hour angle is always measured positive to the west.
2. Hour angle can be expressed as time: 24 hours = 360 degrees, 4 minutes of time is 1 degree of angle. But don't confuse this with clock time. Although the sun moves through about 360 degrees in 24 hours, its position doesn't always correspond to the time of day as the sun sometimes moves faster or slower than the clock time.
3. We can talk about the LHA, but there is no notion of `local' longitude.
GHA is tabulated in the almanac. Given our longitude, we can work out the sun's LHA by adding or subtracting out longitude to or from the GHA as appropriate.
Zenith
The point in space immediately above our position; that is, our position is the GP of our zenith.
Horizon
The plane tangent to the surface of the earth at our position. We pretend the earth is a perfect sphere. Although it isn't, this stuff all works out anyway... The horizon and the line from the center of the earth to our zenith are perpendicular.
Altitude
What the sextant measures. This is the angle between the center of the sun (or other body), our position, and the horizon. We pretend all light rays from the sun or other body are parallel.
Zenith Distance
The complement of the altitude; 90 degrees - altitude. Angle between the sun, our location, and our zenith.
Azimuth
Bearing from our position to the GP of the heavenly body. 0 to 360 degrees, increasing from 0 at North through East. Note that this is `true' degrees, not magnetic as a compass would measure. Magnetic bearing needs to corrected to get true bearing as the earth's magnetic field varies from place to place. We can look up the azimuth of a body in tables or calculate it if we know our position.
Universal Time (UT), Universal Coordinated Time (UTC), and UT1
.
To over-simplify, this is what GMT (the time at the Greenwich meridian, or Greenwich Mean Time) is now called. UT, UTC, and UT1 are not quite the same, but we won't worry about that for our purposes...

## Facts

1. All places with the same altitude to the sun lie on a circle centered around the sun's GP. If we could accurately measure the azimuth to the sun, then we could locate ourselves exactly on that circle, and hence in the world. But, we can't measure the azimuth, we have to calculate it or look it up, and even if we could, we wouldn't be able to easily draw that circle on a map. Instead we can approximate the circle near our position as a straight line (as the circle has such a huge circumference). We call this the position line. Since we are not able to exactly determine the azimuth of the sun, we only know that we are somewhere on the position line.

2. As we move closer to the sun, the altitude of the sun increases; as we move away, the altitude decreases. Think of sunrise and sunset and noon: at sunrise and sunset, the altitude is 0 (and the zenith distance is 90 degrees), and at noon the sun is highest.

3. Noon sight: So we can't get a position fix with one altitude measurement; at most we get a line somewhere on which we lie. However, at noon we can determine our latitude with one measurement. At noon, the sun crosses our meridian (our line of longitude), and will be at its highest altitude when it crosses. So we just watch the sun; when it is at its highest point (it will stay there for a relatively long time), we note the time. Then we can look up the sun's declination at that time, and by adding or subtracting the observed altitude to or from the declination as appropriate, we obtain our latitude. There are a few different cases, you can look them up in the Blewitt book

4. The general navigation triangle: In the general case, we can locate three points that form a sort of triangle (a spherical triangle) on the surface of the earth. These are: our assumed position, the pole closest to us, and the GP of the sun. We can solve this triangle using formulas or tables to calculate what the altitude of the sun should be and what the azimuth is, using our assumed position. Here are the formulas:

• sin CalcAlt = (sin Lat)(sin Dec) + (cos Lat)(cos Dec)(cos LHA)
• cos Az = (sin Dec - (sin Alt)(sin Lat)) / (cos Alt)(cos Lat)
Lat is latitude, Dec is the sun's declination, Az is the azimuth. If the latitude and declination are in different hemispheres, treat the declination as negative.

## Determining our position

But there is a problem here: we have 3 unknowns in those equations. We don't actually know what the real azimuth is, nor what our real latitude and longitude are. We only have two equations with the measured altitude as a known. So instead we get a position line (not an actual position) using our measured altitude, by performing a kind of approximation:

1. First, we need to make a guess at where we are. Then we can refine our guess. This guess is known as the assumed position (AP). Normally we can use our current dead reckoning (DR) position at the time of our sun sight as the AP. We assume that since we have been navigating a boat we have at least some idea of where we are; that we have been keeping a log, etc.

2. Then, using our AP and the time of the sight we solve the spherical triangle to find out what the altitude and azimuth of the sun should be for our AP. We find the declination and GHA of the sun from the almanac (see below), then we plug them into the formulas above, along with the latitude and longitude of our AP to get the calculated altitude. Since the formulas require LHA, we use our assumed position and the tabulated GHA to calculate our LHA. It turns out that we only need the relative angle between the AP and the sun; since the formulas use cos LHA it doesn't matter if this angle is negative, or if it is the interior or exterior angle between the sun's GHA and our assumed longitude.

3. The Intercept: Solving the spherical triangle gives us a calculated altitude. Unless we have guessed our actual position perfectly when working out our AP, the calculated altitude is going to be different than the sun's true altitude at our position (which we have measured with the sextant). If the calculated altitude is greater than the measured altitude, then we are further away from the sun's GP than we thought; if the calculated altitude is smaller, than we are closer to the sun's GP (calculated greater away). Think of the position circle defined by an altitude around the sun's GP which we discussed above.

How far away or toward the sun from our position are we? The distance is exactly the difference in the calculated and measured altitudes. The geometry works out so that 1 minute of difference in the altitude means one nautical mile in distance on the surface of the earth. This distance is the intercept.

4. The Azimuth: We use the calculated azimuth to figure out what direction the sun (i.e. the sun's GP) is from our AP. We draw a line on our chart along this direction through the AP. This is our azimuth line. We then measure the intercept distance along the azimuth line from the AP, toward or away from the sun as appropriate (calculated greater away), and then draw a line perpendicular to the azimuth at this point. The perpendicular line is our position line, somewhere on which we lie. It may seem that we are being very inaccurate by using the AP instead of our actual position and measured altitude to calculate out the azimuth; this may be so, but we could not measure or construct an angle on our chart to within better than a degree due to our tools, so the calculated azimuth is ``good enough''.

5. Sun-Run-Sun: So we now know how to get a single position line from a single sun sight. But we need at least two position lines to get an actual position fix, or position estimate. But there is only one sun, so we have to wait for the sun to move to take another sun sight and get another position line. Ideally, we would take a sun sight in the morning and obtain a position line, and take another sight in the afternoon for a second position line. This will result in two almost perpendicular lines, which will give us the most reliable fix with two lines (the closer the lines are to perpendicular, the less susceptible their crossing point is to error).

But if we are really on a boat, we are moving during the time that we take our morning and afternoon sights and obtain our morning and afternoon position lines. What we need to do is move our morning position line to allow us to cross it with our afternoon position line. Just as we perform DR to obtain our AP, we perform DR to move the morning position line up to the afternoon position line. If we estimate that we have traveled e.g. 5 miles at 320 degrees between taking our morning and afternoon sights, then if we transfer our whole morning position line 5 miles at 320 degrees, then we are still (roughly) on the transferred position line. Of course, this will introduce more error into the whole process depending on the accuracy of our DR. And of course, just as we can transfer our morning position line forward to cross it with the afternoon line, we can transfer the afternoon line backward to the morning line, or even transfer them both to some intermediate position. For instance, we might have a noon sight: we can transfer the morning and afternoon lines to the time of the noon sight and get a 3 position fix. This is more reliable than a two position fix.

6. Reliability: The more fixes and position lines we can apply to form a position, the more we can trust our position. Also, we will be more likely to spot position lines with great errors in them. All the same principles for getting a sun fix can be used with the stars and planets: in this case we can obtain 3 or more position lines almost simultaneously, and can plot them all at once on the chart. Since we are not perfect, these lines will not all intersect at a point; instead they might form a triangle (for 3 sights). We would like to say we lie somewhere inside that triangle: the smaller the triangle, the more confident we are.

But we must be careful: if we have introduced a systematic error into our calculations, then the triangle itself could be completely shifted to one side or the other of our real position. For this reason we keep track of our intercepts and the directions to the celestial body being measured for each sight. We would like our sights to be spread out in enough different directions that the errors ``cancel out''. e.g. take 3 sights at azimuths that are each separated by 120 degrees. If all of our sights are in the same general azimuth from us, then each sight could suffer from the same systematic error, which will not be revealed to us when we plot the intercepts and azimuths.

One of the most important jobs of a navigator is to evaluate the accuracy and usefulness of information presented to her, including GPS data, chart data, as well as celestial fixes etc.

## Using the Nautical Almanac

The almanac tabulates the GHA and declination of the sun and other bodies for every day of the year. You need a new copy of the almanac for each year. Luckily, the almanac includes a section titled "Explanation" that explains how to use it, although often in greater detail than you might wish!

### Looking up GHA

1. The sun's GHA is tabulated for each hour and each day. We use the UT hour of our sight to look up the GHA under the appropriate day and hour. If our sight was in the middle of an hour, just use the hour part.

2. We use the minute and second part of the time of our sight to look up an increment. The back of the almanac includes a section "Increments and Corrections" with one page for each minute: we look up the increment for the minutes and seconds of our sight, and add add it to the tabulated GHA for the hour. You can also perform a linear interpolation on the tabulated GHA to get the value for the minutes and seconds of time.

### Looking up Declination

1. The declination is also tabulated for the sun at each hour. We look up the declination for the hour part of the time.

2. Note that at the bottom of the sun column for that day in the almanac there is a number labelled d. Look in the increments and corrections page for the minute of the time. There is also a table giving a correction for each value of d; there is no sign. If the declination is decreasing during the given hour (determined by looking at the declinations for that day), then subtract the d correction, else add it. Like GHA, the declination can be linarly interpolated between hours.

## Measuring the Sun's Altitude

We measure the sun's altitude using the sextant; but we must apply some corrections to the reading from the sextant to get the actual measured altitude.

### Using the Sextant

The truth of it is that it takes a lot of practice to use a sextant well. Also, it is one thing to use the sextant on land, where you can stand still, but it is anything but easy on a boat, where the boat is moving, there are waves on the horizon, and you are getting wet! But practice makes perfect!

The sextant is an optical instrument that lets you read off the angle between two objects; in celestial navigation we measure the angle between the horizon and a celestial body. Basically, you rotate an arm (the index arm) on the sextant until its mirrors are set so that you can line up two images (of the things you are measuring the angle between, like the edge of the sun and the horizon) in the sextant's viewfinder. There is also a small micrometer drum that lets you make fine adjustments to the angle. There are adjustable filters over the mirrors so that you don't injure your eyes while looking at the sun.

You hold the sextant in the right hand, and adjust the index arm and micrometer, as well as filters, with your left hand. Some sextants can be quite heavy, so I recommend holding the sextant in the following way to keep it steady: Use the index finger of the left hand to support the arc of the sextant; use the thumb and middle finger to operate the micrometer and move the rotating arm. I learned this from a friend of mine who is a merchant marine officer, and it works well.

When taking measurements, take 3 to 5 measurements. Have another person hold the watch (which should read UT, or you should know how to correct the watch time to UT). When you have the sun lined up with horizon, call out "mark" (or something useful like that), at which point the person with the watch should write down the time. When you have a few sights, take the average of the sights and the average time, possible discarding any bad sights. For instance, all the sights will be increasing or decreasing (except around noon); if one sight is decreasing when it should be increasing (or vice versa), it is probably a bad sight. You can also plot the sights and times on graph paper to pick a good average sight using a best fit line. Use this average sight and time for the rest of the calculations.

### Correcting the Sextant Measurements

1. Instrument error: Like any instrument, a sextant can have some inherent error. The primary error in a sextant is index error. This is simply an offset to the measurement we make with the sextant. Index error can change frequently, and should be determined every time we use the sextant as the sextant can change with temperature, humidity, etc. We can work out the error by measuring the angle the horizon makes with itself. i.e. read the value off the sextant when the horizon is lined up with itself in the sextant optics. This is the error, which we write down and add or subtract from out reading as neccessary. There are also other errors which can be checked and adjusted out, but they are smaller and don't change as often.

2. Dip: Since we are not actually standing on the surface of the sea, we have to apply a correction based on our height above sea level. This can be looked up on the front inside cover of the almanac.

3. Atmospheric and semi-diameter corrections: Finally, there are errors from refraction in the atmosphere and from the size of the sun. The theory of celestial navigation talks about the center of the sun, but we can't really find this point. Instead we measure angles from the top or bottom edge of the sun, known as upper or lower limb measurements. The atmospheric and limb corrections are tabulated as a single number, also on the inside front cover of the almanac. We look up the correction according to time of year using the sextant altitude (read off the sextant) corrected for dip and index error.

## Using the Stars and Planets

I don't have too much to say here, except look in a book! The key ideas are the same, although there are a few extra corrections and calculations for the stars and planets. The moon moves very fast, so it can be difficult to get good fixes from; also, the moon so close to the earth that there is a lot of parallax. You have to measure the altitude of stars and planets at dawn or dusk, when there is enough light to see the horizon but not so much light that you can't see the stars or planets.

## Plotting Sheets

In general, if we are at sea, we won't have a chart for our location of the appropriate scale: ocean charts will normally be of such a large scale that we will have trouble plotting our azimuths and intercepts, as they will be too small. Instead we can create our own plotting sheet for our area of the ocean. It is simply a grid, with the ratio between latitude and longitude spacing being: (Lon/Lat) = cos Lat. That is, if one minute of latitude is one inch, then one minute of longitude should be cos Lat, for whatever the latitude is. So we can draw up a grid for the area around our assumed position at whatever scale we want.

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decouto@alum.mit.edu