BASIC THEORY OF ASTRONOMICAL NAVIGATION

Astro navigation is a simple process which has been in use for hundreds of years. The following is a description of what happens when you take a sight.

A Normal Sight

• You measure the vertical angle from the horizon to the celestial body you choose: sun, moon, planets or stars. (You have to be able to see the horizon when using a normal sextant - i.e. it needs to be daylight, dusk or very bright moonlight. Also, do not try to measure a body that is almost overhead - it is too difficult to get an accurate result.)

• note as accurately as possible the time when that measurement is made

• make some small corrections to the measured angle - the index error of the sextant (which you can measure), and adjustments for height of the observation above sea level and for refraction of the atmosphere (you look these up in tables)

• for a position close to where you are (called the assumed position), you work out the predicted angle of the body above the horizon at that time, and the body’s predicted bearing from the assumed position. This is the complicated bit, requiring quite a lot of looking up in tables etc.

• you then compare the measured angle to the predicted angle for the assumed position, which tells you how far you are from the assumed position, and therefore enables you to draw a position line

• one position line does not give your position: you can get another one, and hence your position where they cross, by simultaneously taking a sight on another body. Alternatively take the sun (say) a few hours later when it has moved. In this case (known as the Sun-Run-Sun procedure) you have to transfer the first position line by the distance run between the sights, as you do with a running fix.

A Noon Sight

Wherever you are in the world, there will be a time when the sun is exactly north or south of you, and at its highest point in the sky that day. This is your local noon: also known as the sun’s meridian passage. At this time you can:

• measure the angle of the sun above the horizon and correct the measurement as above

• look up the declination of the sun at that time

• perform a simple calculation to determine your latitude.

The same method can be used at the meridian passage of any other body - moon, planets, stars - but the sun is generally most convenient because its meridian passage occurs during daylight and is fairly straightforward to predict when it will occur.

The Basic Theory

You can skip this section.

You don’t need to know any of this in order to follow the “recipe” for sight reduction in section 3. However some people will want to know more about what is going on, and this bit is for them.

Movement of the Earth, Moon and Planets

The earth follows a nearly-circular path around the sun, taking a year to complete a full rotation. Simultaneously the earth spins on an axis going through the north and south poles.

You can think of the earth’s path around the sun as drawn on a flat plate with the sun in the middle. The earth’s axis of spinning is almost at right-angles to that plate, but not quite. It is, in fact, angled off at about 23°, and this is why the sun is higher in the sky in the summer (when the pole of the hemisphere you are in is angled towards the sun) and lower in the winter (when the pole is angled away from the sun).

The moon goes round the earth taking 28 days to complete the trip. The moon stays close to the same flat plate that the earth moves on, and occasionally the sun, moon and earth line up (which is when you get an eclipse as the moon throws a shadow on the earth or vice versa).

The planets are circulating round the sun like the earth is, so it is more complex (but still possible) to predict where they are going to be in our sky.

Meanwhile the stars are so far away that we can hardly detect any change in their position, even though they are rushing away from us at phenomenal speeds. They are in the same place all the time, but we can only see them when we are facing away from the sun on our spin. That’s why we see different stars in the winter to the ones we see in the summer - we’re at opposite sides of the sun and it’s blanking out a different part of the sky for us.

The View from the Earth

If you are sitting on the earth, it’s a lot easier to imagine that the earth is stationary and everything else is moving around you.

This is exactly how the Nautical Almanac is arranged. It gives us the positions of the sun, moon, planets and stars for every hour of every day of the year - and from that we can work out their positions for every second of every minute of every hour of every day of the year.

It gives these positions in terms of two angles. These are declination and Greenwich hour angle. It’s easiest to explain each of these using the sun as the example. The moon, planets and stars work in exactly the same way.

Declination

As I have explained, the spin axis of the earth is at an angle to the plate on which the earth moves round the sun.

Twice in the year the sun is directly overhead at the equator; this is around 21st March and 22nd September - the equinoxes. At this time the declination is zero.

Moving into October/November, the southerly declination of the sun increases, until it reaches its furthest southerly point at the winter solstice, 21st / 22nd December. The southerly declination then reduces, the sun crosses the equator in March and then the northerly declination increases until 20th / 21st June, which is the summer solstice.

The figure shows how declination is measured - it is in fact the latitude of the place on the earth where the sun is directly overhead.

Declination changes continuously, and it is tabulated for every hour of every day in the Nautical Almanac. You then apply a small correction to get it for the nearest minute of time.

Greenwich Hour Angle - GHA

The other angle, GHA, changes much more quickly. The Greenwich hour angle is the angle that changes due to the earth’s spin.

When the GHA is zero, it is noon at the Greenwich meridian. Every hour after that it increases by 15°, so that it goes full circle in 24 hours.

Like the declination the exact angle is tabulated in the Nautical Almanac for every hour of every day, and you then have to apply quite a large correction to get the GHA for a particular second in that hour.

The figure above shows how GHA is measured. It is measured in a similar way to longitude, except that it goes from 0° to 360° in a westerly direction (whereas longitude is measured from 0° to 180° East or West).

Sight Reduction

For any position on the earth at a particular time, it is possible to calculate the angle of the sun (or other celestial body) above the horizon, which is called the altitude, and its bearing from that position, which is called the azimuth.

All you need to know is:

• the latitude and longitude of the position

• the body’s declination and GHA at that time.

I have no intention of explaining how the mathematics works except to say it can either be done by calculation using trigonometry (sin, cosine etc) or by looking up in special tables. The “recipe” in section 3 uses tables.

The Position Line

It is helpful to understand how taking a sight - i.e. measuring the sun’s angle above the horizon - can give a position line.

Imagine the earth as a simple large sphere - not spinning or moving or anything, i.e. a snapshot at a particular time with the sun in one place.

At this time there will be one point on the earth where the sun is vertically overhead. It is at 90° to the horizon whichever way you measure it.

If you walk away from that point in any direction, you will get to a point where you have gone 1° round the earth. Now measure the angle of the sun to the horizon, and it will be 89° in the direction you have walked from.

Carry on for a further 1° round the earth and the sun will be at 88° to the horizon.

If you walked off in a different direction, the same thing would happen - you would get to the same distance away (2° round the earth) and the sun would be at 88° to the horizon in the direction you had come from.

So you can imagine a series of circles centred on the point on the earth where the sun is vertically overhead.

At any point on each of these circles the sun will be at the same angle to the horizon (altitude) - say 88° for a particular circle. So a measurement of 88° means you are somewhere on that position circle.

Now when the altitude is smaller, say 45°, the position circle you are standing on is very large - several thousand miles in diameter - so the part of it you are interested in will be pretty much like a straight line.

So, how do you draw the position line for where you are?

As mentioned in the section above, you can work out the altitude of the sun at an assumed position close to where you are.

Suppose that the angle you work out is 44°, and suppose that you have measured the sun, and its angle was 44° 30'. Well that means that you are exactly 30' closer to the point on the earth where the sun is directly overhead.

30 minutes on this great circle is 30 nautical miles. So you can draw a position line exactly 30 nautical miles from the assumed position, towards the sun. The position line is part of a huge circle with the sun directly above the centre, so it is drawn at right-angles to the direction of the sun.

Your sight reduction procedure will also give you the direction of the sun at that point (its azimuth) and this is all the information you need to draw the position line.

The Noon Sight

The noon (or meridian passage) sight is the simplest form of sight, by which you can work out your latitude with a very simple formula. This section explains how that works - although you don’t need to know how it works in order to use the formula.

Quite simply, at midday (not necessarily 1200 on your watch, though) the sun is at its highest point in the sky, and is due south if you are in northern latitudes.

Suppose the sun’s declination at that moment is 10° North. If you were on the equator, the sun would not be exactly overhead - it would in fact be north of you, and would be 10° away from vertically overhead, so in other words it would be (90-10)=80° above the horizon.

If you stood at 10° North it would be exactly overhead, or 90° to the horizon, and if you were at 20° north it would now be south of you, at 80° to the horizon again.

At 30° North it would be 70° to the horizon and so on. So it is fairly simple to see the formula. 90° minus the altitude of the sun is your angular distance from the point where the sun is directly overhead.

The figure shows how you derive the formula, in the case where both latitude and declination are North.

So, if latitude and declination are the same direction, i.e. both North (as in the example) or both South, then:

latitude = 90 - altitude + declination

If latitude and declination are contrary, i.e. one is North and the other South, then:

latitude = 90 - altitude - declination

This only works for local meridian passage, requiring two adjustments from 12.00 on your watch:

a) the sun’s meridian passage to Greenwich isn’t precisely at 12.00 GMT. The Nautical Almanac gives you the difference (called the Equation of Time) for each day of the year.

b) allow for your longitude: if you are at 4° west, local noon happens about 16 minutes later than at Greenwich.

Alternatively, take a series of sights and select the greatest altitude.

The altitude needs to be corrected in the same way as a normal sight (see 5.2), and the declination is looked up in the Nautical Almanac for the day and hour of the observation (see 5.3).

Example

Your yacht strikes a submerged container and sinks, on your first Atlantic crossing, 18 days out from Gran Canaria bound for St Lucia. You evacuate into the liferaft and the trade wind is blowing you steadily west. In the grab bag is a sextant and The Nautical Almanac, and you want to know whether to paddle north or south.

Just before your local noon on 3rd December (you reckon you are about 57° west, so that’s at just under 4 hours after Greenwich noon) you take a series of altitude sights, and the greatest is 53° 18´. The sextant’s index error is zero.

• Applying the corrections for Dip and Altitude Correction (see 5.2) gives 53° 18´ - 1.8´ + 15.3´ = 53° 31.5´

• Looking up the sun’s declination for 1600 on 3rd December 1999 gives South 22° 05.7´

• So your latitude is 90° - altitude - declination = 90° - 53° 31.5´ - 22° 05.7´ = 14° 22.8´ North. So you paddle south to make it to the ARC celebrations - your latitude is currently about 17 miles north of the northern most tip of St Lucia.