## Essential concepts

One minute of arc over the surface of the Earth is exactly one nautical mile. This means that one minute of latitude is one nautical mile but one minute of longitude is ONLY one nautical mile along the equator.

You must distinguish instinctively between “degrees and decimals”, “degrees, minutes and decimals” and “degrees, minutes and seconds”.  We use degrees minutes and decimals.  You must also distinguish instinctively between minutes of longitude and minutes of time.

The Earth rotates 360° every 24 hours, 15° every hour, 15' of longitude every minute of time. This means that, if your watch is one second slow and you are near the equator, your astro sight will put you a quarter of a mile east of your true position.

One sight does not give you a fix; it gives you one position line which is, in reality, a range ring (albeit virtually a straight line).

Before you can produce a position line you need to know where you are! – at least approximately.  And you may have to pretend you are somewhere else (the “chosen position” - CP) to suit the tables you are using.

You cannot work in local time (J or "Mickey Mouse") - you have to work in Zulu (Z, GMT, UT or UTC).  TimeZone codes

You can only shoot stars and planets during "navigational twilight".

We sometimes think in terms of the celestial sphere. Positions on Earth are projected at infinite distance on to the celestial sphere. This is not necessary but some texts rely on it and it may be confusing.

## Clocks and time

### accuracy

With the exception of Meridian Passage and Polaris sights, you must have an accurate time reference. Once you are out of sight of land, you are reliant on SSB time signals or the time output from your GPS. You must take this seriously. Every minute of time error will put you 15 minutes of longitude out. Near the equator that's 15 miles.

### deck watch error

It is normal practice to track the error of the watch you use for navigation (the deck watch) by plotting it graphically over a period of time and noting the date and time each time you correct it to a reliable time reference. When you shoot a sight, note the time on your deck watch and then add or subtract the deck watch error extracted from the resulting graph.

### world time signals

#### Single Sideband

SSB time signals can be received from the following transmitting stations.

WWV in Fort Collins, USA (2.5, 5, 10, 15, 20 MHz),

WWVH in Kauai, Hawaii, USA (2.5, 5, 10, 15 MHz)

CHU in Ottawa, Canada (3330, 7850 and 14670 kHz)

#### AM

AM radio, especially long wave, may be heard when you are not too far out of sight of land so you may be able to hear the pips.

#### FM

FM radio is very short range and is unlikely to provide a time reference useable for ocean navigation. It's good for checking your watch ashore and contstructing a deck watch error graph.

#### DAB & DVB-T

Likewise DAB radio is very short range. But, in any case, don't use DAB radio or DVB-T (digital television) to check your watch ashore. There is a finite delay between the pips and your hearing of them at the loudspeaker due to the decoding time required for the signal. The delay is known as "latency" and is variable but can be up to 5 seconds.

#### GPS

Of course, if you have GPS you don't need astro navigation! However, while you wait for it to fail or for Mr Obama to switch the satellites off, your GPS receiver gives you an extremely accurate time reference and so is a good source to reset your watch at sea and to construct your deck watch error graph.

### timezone letters TimeZone codes

Zulu is the timezone code for Universal Time Co-ordinated or Univeral Time, which used to be called Greenwich Mean Time.

All astro navigation must be done in Zulu. You can't work in local time. You must keep your deck watch on Zulu. Some of your crew will not like sunrise occuring at "mid-day" but they have to get over it.

## The PZX triangle

This spherical triangle is the fundamental trigonometrical problem that we have to solve to determine a position line. It is the spherical triangle joining the North or South Celestial Pole, the Zenith of observer and the GHA/declination of the body. Their corresponding positions on the surface of the Earth are the North or South Pole (P), the observer's latitude and longitude (Z) and the ground position (GP) of the body (X). PZX diagram

We know where the pole (P) is; we can look up the GP of the body (X) in the Almanac and we can guess our position (Z).  So we can calculate our zenith distance.  Then we can measure our true zenith distance using the sextant.  Our position line is towards or away from X by the difference between our estimated zenith distance and our true zenith distance.   However, for simplicity of practical application, we work in altitude, which we can read off the sextant, rather than zenith distance, which we can’t.

### Work in altitude rather than zenith distance

We can measure altitude easily relative to the visible horizon. We can't measure zenith distance easily because it would mean referencing the local gravitational vertical, which is impossible on a boat that is heaving up and down. It can be done in an aircraft where we use a bubble sextant, which establishes the local vertical using a spirit level. The altitude we observe with the sextant is called Ho.

### Sight reduction – a two-stage process

First we calculate the bearing (azi or Zn) and altitude of the body (Hc) from our chosen or DR position (Z) to to the ground position of the body (X). Then we measure that distance using the sextant. Then we move our position line towards or away from X along the azimuth or its reciprocal. We do so by the amount of the difference between the calculated altitude (Hc) and the observed altitude (Ho).

## Defining X

The ground position of the body is determined from the tables in the Almanac or it can be generated by a computer programme. It is defined by its Hour Angle and Declination.

### Declination (Dec)

The latitude at which the body is directly overhead at the time of observation

### Hour Angle

Longitude measured in a westward direction from a reference point – either Greenwich (Greenwich hour angle or GHA), the longitude of the observer (local hour angle or LHA) or the First Point of Aries (sidereal hour angle or SHA). Eg The GHA of Paris, (2°20’ E) is 357°40’.  Diagram of hour angles

### Greenwich Hour Angle (GHA*)

The hour angle difference from Greenwich and the point where the body is directly overhead at the time of observation

### Local Hour Angle (LHA*)

The hour angle westward from the observer’s longitude to the longitude of the point where the body is directly overhead at the time of observation

### Aries – sometimes the “First Point of Aries”

A mythical point in the heavens at infinite distance from Earth and for which the GHA is given in the Almanac.  It is used as a reference point for all stars.  It co-incides with the point at which the plane of the ecliptic crosses the celestial equator at the moment of the vernal equinox - but don't fret about that!

### Sidereal Hour Angle (SHA)

The hour angle westward from the First Point of Aries to the longitude of the point where the star is directly overhead at the time of observation.  The SHA of a star and its declination are virtually constant. Hour angle test

Our objective in all of this is to determine the LHA of the body that we are shooting.

## Defining Z

Z may be your DR or EP position if you are using the Marc St Hilaire method and Norie's Tables or a computer for sight reduction. However, if you are using the Air Tables for sight reduction, it may have to be a chosen position (CP), which is selected to give a whole number of degrees of LHA and latitude.

Z is defined using latitude and longitude in the normal way.

## Calculating the calculated altitude (Hc) and azimuth (Zn or azi) - The Sight Reduction

AutoAstroSun

### Sight Reduction by Norie's Tables AstroProforma AstroProformaAuto

Norie's Nautical Tables can be used for all latitudes and declinations and can be used with your DR or EP position. You don't need a chosen position with whole numbers of degress of LHA and latitude. The tables provide log haversines, log cosines and natural haversines to calculate zenith distance by the Marc St Hilaire method. The tables also provide values of A, B and C for the determination of azimuth by the ABC method. We use a specially-designed proforma to simplify the calculations and to reduce the risk of error. For interest, the versine of angle θ is 1-cosθ and the haversine is half that, i.e. (1-cosθ)/2. Don't fret about that.

## Determining the true (observed) altitude (Ho) by sextant.

### How it works

It's important to remember that one sight gives you one position line, which is part of a very large radius circle. If you want a fix, you must work two or three sights of two or three different bodies. It is sometimes possible to shoot the sun and moon simultaneously during the day (a "simultaneous sun-moon fix") but normally, during the day, you can only shoot the sun several times two or three hours apart and TPL the results. During navigational twilight it is normal to shoot three stars (with well-displaced azimuths) in quick succession and plot a "cocked hat". This is called a "three-star sandwich fix".

### Practical issues

It's important to select a position as high as possible on the boat but where you can hold a safe and steady posture. This ensures that you are not looking at nearby wavetops rather than the distant visible horizon and that you don't fall overboard. You need two hands to operate the sextant so it's useful to use a harness and position yourself at the mast. Obviously you need a clear view of the body and of the horizon.

### Using an artificial horizon for practice

For the purposes of practising on land you can use what is called an "artificial horizon". This is simply a bowl of water which is allowed to settle so that it acts as a horizontal mirror surface. Using this, you can see two bodies, one in the sky and one in the surface of the water. The angle between the two is exactly twice the sextant altitude (Hs). After dividing by two, apply the corrections (see below) for refraction, semi-diameter and horizontal parallax (moon) or additional correction (planets). Since you are using a true horizon, dip is zero.

### sextant error

Every sextant has minute errors in the construction of the arc. Manufacturers like Zeiss provide a table of the error at every part of the arc. These are sometimes in seconds of arc, which will need to be converted to decimal minutes when you include them in your calculation. The sextant reading corrected for sextant error is known as sextant altitude or Hs

### index error (IE)

This is a small error in the adjustment of the index mirror such that the instrument may not read zero when the horizon is aligned in both mirrors. It can be adjusted out using the key or it can be added to the corrections in the calculation. The sextant altitude, Hs, when corrected for index error and dip gives the apparent altitude, Ha.

### dip

When you observe the horizon from any point above the surface, it appears lower that the true horizontal. The effect is greater with increasing height above the surface. This must be subtracted from the sextant reading to produce the true altitude of the body being observed. The sextant altitude, Hs, when corrected for index error and dip gives the apparent altitude, Ha.

There is no dip when using a bubble sextant or when practising with an artificial horizon since both use as a reference the true horizontal rather than the visible horizon.

### semi-diameter (SD)

The sun and the moon subtend about 32' of arc at the Earth's surface. It is very difficult to judge the exact centre of either so it is normal to align either the top (the Upper Limb) or the bottom (the Lower Limb) with the horizon and then add or subtract the semi-diameter, which is given in the Almanac. The Almanac combines the correction for refraction and semi-diameter in the Altitude Correction Tables - Sun, Stars, Planets.

### refraction

Light arriving at the Earth's atmosphere is refracted by the denser layers of air towards the surface. This means that an object appears higher in the sky than it really is. This effect is maximal when the body being observed is on the horizon and is zero when the body is overhead. The Almanac combines the correction for refraction and semi-diameter in the Altitude Correction Tables - Sun, Stars, Planets.

### horizontal parallax (Moon’s parallax in altitude)

The moon is close enough to Earth that rays of light from it are not parallel when they reach our surface. Again, this effect is maximal when the moon is on the horizon (when the value is called the "horizontal parallax") and is zero when the moon is overhead. For any given apparent altitude, the Almanac combines the correction for refraction, semi-diameter and parallax in altitude in two stages in the Altitude Correction Tables - Moon. This error is shown in the Moon's parallax diagram.

### additional correction (Venus & Mars)

There is a minute correction for parallax in the case of Venus & Mars. This is given in the Altitude Correction Tables - Sun, Stars, Planets. The geometry is similar to the moon's parallax in altitude. See the

## Plotting the result

Now that you have your calculated altitude (Hc), your calculated azimuth (Zn) and your observed altitude Ho, you can draw a line through your DR or CP position in the direction of the azimuth and its reciprocal. You can then plot your position line according to the difference between Hc and Ho. Each minute of difference represents one nautical mile of error between your DR or CP position and your true position. Remember "GOAT". Greater Observed Altitude: move Towards! Plotting diagram

## Special cases

### Meridian Passage (MerPass or Noon Sight)

You can determine your latitude at local noon without the aid of a watch. You simply observe the sun as it climbs to its maximum altitude. Once you see the sun going down again you stop adjusting the sextant. You then make the corrections for semi-diameter, refraction and dip. Your latitude is co-altitude plus or minus declination. Add declination if it's in the opposite hemisphere to your DR latitude; subtract it if it's the same hemisphere.

### Polaris

Polaris is almost vertically above the North Pole so you can (almost) read your latitude directly off the sextant by observing it. The Polaris Tables in the Almanac give three simple corrections to Ho, which is gives your latitude.

## Other uses for the sextant

### Horizontal sextant angle

For coastal navigation, the sextant can provide very accurate position rings given two or, preferably three clearly identified landmarks. See a text on coastal navigation for a description of this simple technique.

### Vertical sextant angle

For coastal navigation, the sextant can provide approximate distance off a landmark of known height above sea level. See a text on coastal navigation for a description of this simple technique.

## Other considerations for ocean navigation

### The oblate spheroid

The Earth is an oblate spheroid. All map projections are a compromise between conformality (or accuracy of shape) and constancy of scale. The classic map of the world is a Mercator projection, which is produced by projecting the image from the centre of a globe on to a cylinder of paper wrapped around the earth in contact with the equator. This gives a good result at the equator but is seriously distorted near the poles. A transverse Mercator projection is similar but the paper cylinder is in contact with a chosen meridian. This gives good results all along the chosen meridian but not at any distance from it. Oblique Mercator and conical projections can be used to cover specific areas and shapes.

An appropriate chart is required to plot an ocean course. The choice of chart depends on the sailing technique proposed. The shortest distance between two points is part of a great circle, the fragment being called a small circle. For this great circle route to be a straight line on a chart, the chart must be conformal throughout the area of the route. The required compass course will change constantly throughout the journey. A conical chart or oblique Mercator may serve the purpose. However, a rhumb line route, which is one where the required course remains constant throughout the journey, can only be drawn as a straight line on a Mercator projection. The rhumb line route may be considerably longer than the great circle route.

## Other uses for astronavigation techniques

### Compass swing

Any celestial body that appears on the horizon on the nose of the boat can be used to check the compass deviation. Point the boat directly at the body and note the compass heading and time. Using the techniques described above, calculate the azimuth of the body. Correct the noted compass heading for magnetic variation and compare it with the azimuth. Add the difference to your deviation card.