Celestial Navigation Theory - The Celestial Sphere & PZX Triangle

Prerequisites

This lesson is part of the RYA/MCA Yachtmaster Ocean syllabus. Before starting you should:

• Hold, or be working towards, the RYA/MCA Yachtmaster Offshore Certificate of Competence (or MCA OOW Yachts less than 3000 gt). The Ocean exam cannot be sat without one of these.
• Be fully competent in Day Skipper and Coastal Skipper chartwork: latitude and longitude, true/magnetic conversion, variation and deviation, dead reckoning (DR) and estimated position (EP), running fixes and transferred position lines.
• Understand the basic astro awareness covered at Offshore level: what a sextant does, the idea of a noon sight for latitude and a compass check on the sun.
• Have studied the companion lessons on Sextant Corrections (Hs to Ho) and the Nautical Almanac and Sight Reduction Tables, or study them alongside this one.

The RYA/MCA Yachtmaster Ocean shorebased course (the usual route to the theory) covers astro navigation, sextant use, measurement of time, compass checking and sun position fixing, and its written paper is passed at 60 per cent. Candidates must be 18 or over, and the qualifying ocean passage is 600 nm or more, with at least 200 nm more than 50 nm from land, lasting 96 hours or more. The astro evidence is gathered on that passage, or on another passage out of sight of land.

Remember what the examiner will ask for. To qualify for Yachtmaster Ocean you must have navigated a yacht by astro, out of sight of land, and you must bring the records: as a minimum, the planning, reduction and plotting of a sun-run-meridian altitude (or sun-run-sun) and a compass check using the bearing of the sun, moon, a star or a planet. Everything in this lesson builds towards producing that evidence confidently.

Learning Objectives

By the end of this lesson you will be able to do the following. The RYA Yachtmaster Ocean syllabus items covered are named in brackets: astro navigation, measurement of time, sextant use and corrections, compass checking, and sun position fixing.

• Describe the celestial sphere and the coordinates used on it: declination, Greenwich Hour Angle (GHA), Local Hour Angle (LHA), Sidereal Hour Angle (SHA) and the First Point of Aries.
• Explain the geographical position (GP) of a body and why a single altitude places you on a circle of equal altitude.
• Relate zenith distance to distance from the GP (1 minute of arc = 1 nautical mile).
• Handle time correctly: UT, zone time, chronometer error and rate.
• Select an assumed position (AP) and reduce a sun sight by the Marcq St Hilaire intercept method.
• Work a meridian altitude for latitude, and a Polaris latitude.
• Combine a morning sun sight with a noon latitude to produce a sun-run-meridian fix.
• Carry out a compass check by the bearing of a celestial body.
• Plan twilight star sights with a star finder, identify the main navigational stars and constellations, and plot a multi-body fix.
• Take a good sextant sight, record it correctly and judge its quality (syllabus: sextant use).
• Reduce moon and planet sights and find the true bearing of the sun at rising and setting by amplitude (syllabus: compass checking, astro navigation).
• Choose sights that give a good angle of cut for a sun-run-sun or sun-run-meridian fix, and assess the accuracy of the result (syllabus: sun position fixing).
• Prepare the written evidence of astro navigation that the Ocean examiner requires (syllabus: qualifying passage documentation).

Why Astro Still Matters

GPS is superb, but it is a single point of failure: lightning strike, flooded electrics, a failed antenna, jamming or spoofing, or a satellite system outage can all leave an ocean yacht with no position 1,500 miles from land. A sextant, an accurate watch, a current Nautical Almanac, sight reduction tables and plotting sheets need no power and cannot be jammed. A navigator who can get two good position lines a day and a daily compass check can cross any ocean safely, and the discipline of astro (keeping a DR, logging the barometer, checking the compass) makes you a better navigator even when the GPS is working.

The RYA-recommended text for the Ocean course, Tom Cunliffe's Celestial Navigation, makes the point that a competent offshore skipper can be navigating by the sun long before finishing the book, and that the stars, planets and even the moon are easier than their reputation; the only chapter that cannot be skipped is the one on concepts and definitions, because everything else is built on it. The same is true of this lesson: get the celestial sphere, GP, hour angles and zenith distance straight first, and the rest is routine.

Astro accuracy is modest by modern standards. From a small yacht in a seaway, a competent observer achieves position lines good to about 1 to 3 miles, and a fix good to a few miles. That is more than enough in mid-ocean; it is not enough for reef pilotage, which is why landfalls are planned for daylight with a margin.

The Celestial Sphere

To navigate by the sky we use a simple model: imagine all the heavenly bodies painted on the inside of an enormous sphere centred on the Earth. The real distances do not matter for navigation; only directions do. On this sphere we draw lines that mirror those on Earth.

• The Earth's axis, extended, meets the sphere at the north and south celestial poles (NCP and SCP). Polaris lies less than 1 degree from the NCP.
• The Earth's equator, projected outwards, becomes the celestial equator (also called the equinoctial).
• The sun appears to travel round the sphere once a year along the ecliptic, which is tilted about 23 degrees 26 minutes to the celestial equator. The point where the sun crosses the celestial equator heading north (around 20 March) is the First Point of Aries, the zero point for star positions.

The diagram below shows the celestial sphere with the poles, the celestial equator and a star's declination measured north of it.

Click to zoom

Celestial coordinates

Two points to note. First, hour angles are always measured westward through the full 360 degrees, unlike longitude which stops at 180 E or W. Second, the GHA of every body increases by roughly 15 degrees per hour as the Earth turns (the sun almost exactly 15 degrees; Aries and the stars slightly more, 15 degrees 02.5 minutes; the moon about 14 degrees 19 to 14 degrees 36 minutes).

The key relationships are:

• LHA = GHA minus West longitude, or LHA = GHA plus East longitude. If the answer exceeds 360 degrees, subtract 360; if it is negative, add 360. Memory aid: "West subtract, East add". A sense-check: with the body on your meridian, LHA is 0 and GHA equals your West longitude.

Tom Cunliffe's advice in the RYA-recommended text on the subject is to draw a diagram whenever in doubt, because without a grasp of LHA "the rest of the book may as well be written in Hottentot". The four cases that arise:

• GHA of a star = GHA Aries + SHA of the star (subtract 360 if over 360).

The Geographical Position (GP)

At any instant each body is directly overhead somewhere on Earth. That point is its geographical position. Its latitude equals the body's declination, and its longitude is given by its GHA (GHA up to 180 degrees = that longitude West; GHA over 180 = 360 minus GHA, East). The almanac therefore tells us, for any second of UT, exactly where on Earth each body's GP lies.

The diagram below shows the body, its GP on the Earth's surface, and the GHA and declination that define it.

Click to zoom

The sun's GP races westward at about 900 knots (15 degrees per hour x 60 miles at the equator). That is why time is so critical: an error of 4 seconds of time moves the GP by 1 minute of longitude, which is up to 1 mile of position error.

Altitude, Zenith Distance and the Circle of Equal Altitude

Your zenith is the point on the celestial sphere directly above your head. The angle from your zenith down to a body is the zenith distance (ZD), and the angle from the horizon up to the body is its altitude. They always add up to 90 degrees:

ZD = 90 degrees minus Ho (Ho = corrected, true observed altitude)

Click to zoom

Because one minute of arc of a great circle on the Earth's surface is one nautical mile, the zenith distance is your distance from the GP. If the sun's true altitude is 40 degrees 00 minutes, ZD is 50 degrees = 3,000 nautical miles. You are 3,000 miles from the sun's GP.

But you could be anywhere on a circle of radius 3,000 miles around the GP. Every observer on that circle sees the sun at exactly the same altitude at the same instant. This is the circle of equal altitude, the astro equivalent of a range ring from a radar.

Click to zoom

Because the radius is so large, the small part of the circle that passes near you is, for chartwork purposes, a straight line at right angles to the direction of the body. That straight segment is your position line (LOP). Two LOPs from different bodies (or one body at different times, with the first line transferred) give a fix.

The PZX triangle

The mathematics behind sight reduction is the spherical triangle formed by the elevated Pole, your Zenith and the body's position X:

• PZ = co-latitude (90 minus your latitude)
• PX = polar distance (90 minus declination if the same name as your latitude, 90 plus declination if contrary)
• ZX = zenith distance (90 minus altitude)
• the angle at P is the LHA (or 360 minus LHA)
• the angle at Z is the azimuth angle, which gives the body's true bearing

Given your (assumed) latitude, the declination and the LHA, the triangle can be solved for the altitude and azimuth you would see. That is exactly what the sight reduction tables do for you.

Time

All almanac data is tabulated against UT (Universal Time, for navigation the same as GMT/UTC). Ship's clocks normally keep zone time. The zone description is the number of hours to add to zone time to get UT: in West longitudes it is positive, in East longitudes negative. Each zone is 15 degrees wide, centred on a multiple of 15 degrees; for example 042 W lies in zone +3 (37.5 W to 52.5 W), so 0840 zone time is 1140 UT.

Your sight watch (the "chronometer") must be checked against a radio time signal or GPS UT daily. Record:

• Chronometer error: how much the watch is fast or slow, e.g. "2 seconds slow".
• Rate: how much the error changes per day, e.g. "gaining 0.5 seconds per day".

Without a time check for 20 days at 0.5 seconds per day you would accumulate 10 seconds of error, which is 2.5 minutes of longitude. That is why the rate is logged: you can still predict the error accurately if your time source fails.

Practical tips: use a digital watch set to UT, start a stopwatch at the moment of the sight ("Mark!") if a helper is not available, and always note seconds. Watch out for the date line in UT: an evening sight in West longitude may be the next UT date. Keep a navigation clock set permanently to UT as the final arbiter when your mind blanks on a zone sum, and when a conversion crosses midnight work it out logically: in zone +8 at 1830 zone time on 25 March, UT is 1830 plus 8 hours = 2630, which is 0230 UT on 26 March.

The mean sun and the apparent sun. The almanac tabulates everything against UT, which is kept by an imaginary "mean sun" moving at a perfectly regular rate. The real (apparent) sun runs ahead of or behind it by up to about 16 minutes through the year, a difference called the equation of time, printed on the daily pages beside the time of the sun's meridian passage at Greenwich. That is why local noon is not at 1200 plus or minus your longitude in time, and why you check the time of meridian passage each day before planning the noon sight.

Finding the time of local noon. The sun crosses 15 degrees of longitude an hour, one degree every four minutes. Take the meridian passage time from the almanac and add four minutes for every degree of West longitude (subtract for East). Example: meridian passage 1157 and DR longitude 4 W gives local noon at 1157 plus 16 minutes = 1213 UT; for 73 E it is 1144 minus 4 hours 52 minutes = 0652 UT. Use a whole degree of DR longitude and err on the early side: you do not want to miss it. Go on deck ten minutes beforehand and follow the sun up with the micrometer until it stops rising; the maximum is the meridian altitude. Note that a vessel moving fast north or south can see a maximum altitude slightly before or after true meridian passage; at 20 knots due north or south the error can reach several minutes of arc, but at a yacht's 5 knots it is rarely worth considering, unless your course has a large north-south component, in which case take the sight at the calculated time of meridian passage rather than waiting for the sun to stop.

Taking a Good Sight with the Sextant

The arithmetic is only as good as the observation. The sextant itself, its index error and the corrections from Hs to Ho are covered in the companion lesson; this section covers technique. The photograph below shows a sextant being used at sea.

Click to zoom

Image: U.S. Navy photo by Photographer's Mate 3rd Class Kevin S. O'Brien, public domain, via Wikimedia Commons.jpg) (downscaled)

Technique

1. Check the index error before every session, by bringing the horizon and its reflection into line (or the star and its image) and reading the arc. Record whether it is on or off the arc.
2. Choose a steady position low on the boat, near the centre where the motion is least, braced against the backstay or a shroud and clipped on. Hold the sextant in the right hand with the frame vertical.
3. Set the shades for the sun before raising the instrument. Use the dark filters on both mirrors to avoid eye damage.
4. Find the body. With the arm at zero, look at the body through the telescope or sight tube, then pull the index arm down while tilting the frame to follow the body to the horizon. Then lower the body to the horizon with the micrometer drum.
5. Swing the arc. Rock the sextant gently from side to side about the line of sight: the body appears to swing along an arc, and the correct reading is when the lower limb just touches the horizon at the lowest point of that arc (the bottom of the swing). Never take a sight in a tilted state.
6. Note the time to the second at the instant the lower limb touches the horizon: call "mark" and stop the stopwatch, or ask a helper to read the watch and record the reading.
7. Take several sights (three to five) in quick succession, record each time and altitude, and use the average, or plot altitude against time and draw a best-fit line. A sight that is out of line is an error: discard it.

Choosing the best conditions

• Take sights when the horizon is crisp. In haze, the horizon is indistinct and the sight is unreliable.
• Keep the body between about 15 and 70 degrees altitude (see below).
• Use the height of eye you will be at when you take the sight, not the height of the deck. A low position gives a smaller dip and a sharper horizon.
• In a heavy sea, wait until you are on a wave crest to take the sight: in a trough the horizon is the top of the next wave, so the altitude is too small.

The Intercept (Marcq St Hilaire) Method

We cannot draw a 3,000-mile circle on a plotting sheet, so we work from a nearby point we choose ourselves.

1. Choose an assumed position (AP) close to your DR.
2. Use tables to calculate the altitude (Hc) and true bearing (Zn) the body would have from the AP at the time of the sight.
3. Compare Hc with your corrected observed altitude Ho. The difference in minutes is the intercept, in nautical miles.
4. If Ho is greater than Hc you are closer to the GP than the AP is, so the intercept is plotted towards the body. If Hc is greater than Ho, plot it away. Memory aids: "Calculated Greater Away" and "Ho Mo To" (Ho more, toward).
5. At the end of the intercept, draw the position line at right angles to the azimuth.

Click to zoom

The two cases are compared side by side below. Notice that the position line is always perpendicular to the azimuth line, whichever way the intercept goes.

Click to zoom

Choosing the assumed position

Tables such as AP3270/NP303 (HO 249) are entered with whole degrees of latitude and LHA. So we choose the AP to make the arithmetic clean:

• Assumed latitude: the whole degree nearest the DR latitude.
• Assumed longitude: within 30 minutes of the DR longitude, chosen so that LHA comes out as a whole number of degrees. In West longitude this means the assumed longitude has the same minutes as the GHA; in East longitude its minutes are 60 minus the GHA minutes.

Click to zoom

The AP may be up to 30 miles in latitude and 30 minutes of longitude from the DR. That is fine: the intercept method corrects for it. Do not be alarmed by an intercept of 20 to 30 miles; it is the fix that matters, not the length of the intercept. Very long intercepts (say over 30 miles) do introduce small curvature errors, so if one appears, check your DR and arithmetic first.

The complete reduction sequence

Every sun sight follows the same nine steps. Learn them as a drill.

Click to zoom

1. Take the sight; record Hs, UT (to the second), log reading and height of eye.
2. Correct Hs to Ho (index error, dip, then the altitude correction table: refraction, semi-diameter, parallax).
3. Extract GHA and declination for the whole hour of UT from the daily pages.
4. Add the increment for minutes and seconds (and the d correction to declination).
5. Choose the AP and find LHA.
6. Enter the tables with assumed latitude, LHA and declination (whole degrees, same or contrary name): extract Hc, d and Z.
7. Apply the d correction for the declination minutes to Hc; convert Z to Zn.
8. Intercept = Ho minus Hc, towards or away.
9. Plot from the AP.

Click to zoom

Plotting Sheets and Plotting the Position Line

On a passage you plot astro lines on a blank Mercator plotting sheet (for example Admiralty or Reeds 'Plotting Sheet') rather than a chart, since the sheet can be used at any latitude and gives plenty of room. The sheet has latitude lines marked but the longitude lines are left blank for you to scale.

1. Choose the scale, for example 1 minute of latitude = 1 cm, and mark the latitudes along the left.
2. Scale the longitude using the cosine of the middle latitude of the sheet. One minute of longitude is cos(latitude) nautical miles. At 34 degrees North, cos 34 = 0.83, so 1 minute of longitude is 0.83 nm, and 10 minutes of longitude spans 8.3 nm on the latitude scale.
3. Mark the assumed position (AP) for each sight at its own latitude and assumed longitude. Each sight has a different AP because the assumed longitude is chosen to give a whole LHA.
4. From the AP draw a line towards the sun's azimuth (Zn) for a Towards intercept, or the reciprocal for an Away intercept, and measure the intercept distance on the latitude scale.
5. At the end of the intercept draw the position line at right angles.
6. Label each line with the body, the UT and the date.

Do the plotting with a sharp pencil and divider. A line drawn 2 degrees off the right angle can move your fix by a mile on a long intercept, so be careful.

Worked Example: Sun-Run-Meridian

This is the exact type of record the examiner will want to see. The almanac values below are illustrative of a day in mid-May; always use the almanac for the actual year and date.

Situation. Mid-Atlantic, bound west. Height of eye 3 m. Index error 2.0 minutes off the arc (so add 2.0). Course 250 T, speed 6.0 knots through the water, no known current. Ship's clocks keep zone +3.

Morning sight

At 0840 zone time (1140 UT) the DR is 34 20 N, 042 25 W. You take a lower limb sun sight: Hs 46 02.5, at 11h 40m 00s UT, sun bearing roughly east.

Step 1: Hs to Ho.

(In practice the almanac's sun altitude correction table combines refraction, SD and parallax into one figure, here +15.0.)

Step 2: Almanac.

Step 3: AP and LHA. Assumed latitude 34 N. DR longitude is 042 25 W; we need a West longitude with minutes .2 to cancel the GHA minutes, within 30 minutes of the DR: 042 55.2 W. LHA = 355 55.2 minus 042 55.2 = 313 degrees. (LHA greater than 180 means the body is east of the meridian, which agrees with a morning sun.)

Step 4: Tables (AP3270 Vol 2 / HO 249). Enter latitude 34, declination 19 same name (both North), LHA 313. Extract: Hc 45 47, d +29, Z 097.

Step 5: Interpolate and convert. Declination minutes are 04.3. Correction = 29 x 4.3 / 60 = +2.1, round to +2. Hc = 45 49. North latitude, LHA greater than 180, so Zn = Z = 097 T.

Step 6: Intercept. Ho 46 16.5 minus Hc 45 49.0 = 27.5 miles towards 097 T.

Step 7: Plot. From the AP (34 00 N, 042 55.2 W) lay off 27.5 miles in direction 097 T. Through that point draw the position line at right angles, running 007/187 T. You are somewhere on that line at 1140 UT.

Running the line on to noon

Local apparent noon (LAN) is when the sun's GHA equals your longitude (for West longitude). The DR longitude for noon is about 042 40 W. GHA at 12h is 000 55.2, so the sun must move another 41 44.8: that takes 2h 47m. LAN is about 14h 47m UT (1147 ship's time). Start watching the sun ten to fifteen minutes before.

Between 1140 and 1447 UT is 3h 07m at 6.0 knots: 18.7 miles on 250 T. Advance the morning position line by moving any point on it 18.7 miles along 250 T and redrawing the line parallel to the original. Mark it with double arrowheads to show it is a transferred line.

Noon sight (meridian altitude)

At noon the sun stops rising, "hangs", then starts to dip. Keep bringing it down to the horizon until it no longer rises; that maximum is the meridian altitude. The exact time is not critical for a meridian altitude, which is one of its great virtues.

Declination at 14h 47m UT: N 19 05.7 at 14h plus d correction +0.5 = N 19 06.2.

The sun bears south at noon (your latitude is north of its declination), so:

ZD = 90 00.0 minus 74 55.9 = 15 04.1, and Latitude = ZD + Dec = 15 04.1 + 19 06.2 = 34 10.3 N.

Click to zoom

The fix

Draw the parallel of 34 10.3 N. Where it crosses the transferred morning line is the sun-run-meridian fix: 34 10.3 N, 042 40.5 W at 1447 UT. Compare it with the DR, note the difference as a "DR error" in the log, and use it to estimate the current you have experienced.

Meridian altitude rules

For a sun at its meridian:

A reliable approach is to draw a quick sketch: the equator, the sun's declination, the zenith, and the direction the sun bears. You cannot go wrong if you draw it. Name ZD the opposite of the body's bearing (sun bears south, ZD is named North), then add if same names and subtract if different.

If cloud covers the sun at noon, an ex-meridian sight (taken within a few minutes of LAN, the allowed time depending on latitude and declination) can be reduced to the meridian altitude using ex-meridian tables, or simply treated as an ordinary sight and reduced by the intercept method.

Sun-Run-Sun and the Angle of Cut

Sun-run-meridian is the classic fix, but it needs a noon sight. A sun-run-sun fix uses two sun sights at different times, with the first line advanced by the run. Because each line lies at right angles to the sun's azimuth, the angle of cut between the two lines equals the difference in azimuth between the sights.

• A cut of 90 degrees is best; at least 30 degrees is needed for a useful fix, and the fix is poor if the azimuths differ by less than 30 degrees or by more than 150 degrees (the lines are then almost parallel).
• The forenoon sight should be taken not less than about an hour and a half before noon. Earlier gives a better cut but a longer run-up for the transferred line, and the longer the run the greater the error, especially if the current is in doubt. Cunliffe's practical routine is a forenoon sight at about 0930 local time run up to the noon latitude, checked by an afternoon sight between 1400 and 1500 run back to the noon line; in good conditions expect an area of probable position two to four miles across, and remember that at best this is only a running fix.
• In mid-latitudes the sun's azimuth changes by about 15 degrees an hour on average, but fastest near noon. A morning sight three hours before noon and an afternoon sight three hours after noon have azimuths differing by about 160 degrees at 34 degrees North in summer: the lines are almost parallel and the fix is poor.
• Better: a morning sight and a noon latitude (sun-run-meridian, with a cut of about 90 degrees), or two sights taken when the azimuth has changed by 60 to 120 degrees.

Example: morning sun Zn 097 and afternoon sun Zn 228. The difference is 131 degrees, so the lines cross at 49 degrees. The run between sights, say 5 hours 20 minutes at 6 knots on 250 degrees True, is 32 nm. Advance the morning line by 32 nm along 250 degrees True, exactly as in the sun-run-meridian example, and take the fix where it crosses the afternoon line. Any error in the run (log, leeway, current) goes straight into the fix, so shorter intervals are better.

Latitude by Polaris

Polaris lies within a degree of the north celestial pole, so its altitude is very nearly your latitude. The geometry is shown below: the pole's altitude above your horizon equals your latitude.

Click to zoom

The almanac's Pole Star tables give three small corrections (a0 from LHA Aries, a1 from latitude, a2 from the month):

Latitude = Ho minus 1 degree + a0 + a1 + a2

Polaris is only magnitude 2 and is useful between about 5 and 60 degrees North in twilight. It gives a latitude line that can be combined with other star sights. It is of no use in the Southern Hemisphere, where there is no bright pole star; there, the Southern Cross is used to find the south pole by eye but not for a direct latitude.

Compass Check by Celestial Body

The Ocean exam requires a compass check by the bearing of a celestial body. The principle is simple: the tables give the body's true bearing (Zn) for the moment of observation; compare it with what the compass reads.

1. Take a compass bearing of the body, noting the UT to the nearest minute (and your heading, if using the steering compass).
2. Work out the GHA, declination and LHA for that time, using the DR position (or the AP from a sight just taken).
3. Extract Z from the tables and convert to Zn.
4. Total error = difference between compass bearing and Zn. If the compass reads higher than true, the error is West; if lower, East ("Error West, compass best").
5. Remove the chart variation to isolate the deviation for that heading.

In the morning sight above, Zn was 097 T. Suppose the bearing over the steering compass at 1140 UT was 111 C and the chart variation is 12 W. Total error 14 W; deviation = 14 W minus 12 W = 2 W on heading 250 C.

The diagram below shows another example, near noon: Zn 174 T, compass reads 181 C, total error 7 W.

Click to zoom

Practical points:

• The best time is when the body is low (altitude below about 15 to 20 degrees): its bearing is easy to take accurately and changes slowly. At sunrise and sunset the bearing can be found with amplitude tables (true bearing at rising or setting, from latitude and declination). Take the amplitude bearing when the sun's lower limb is about half a diameter above the horizon, to allow for refraction and dip.
• A body near the meridian at high altitude changes bearing very fast and is hard to sight over a compass: avoid it for compass checks.
• If the sun rises or sets close to the ship's heading, a small alteration to bring it dead ahead or astern lets you read its bearing straight off the steering compass; the deviation on that heading will be almost the same as on your course. If you cannot sight across the steering compass, set a chart plotter or protractor on deck with zero on the ship's head and use its arm as a pelorus to measure the sun's relative bearing, aiming at the sun's average position as the boat yaws, then add the relative bearing to the compass heading. Enter every result in the back of the log book and keep a running check on the deviation card.
• A hand-bearing compass, used away from steel and electronics, should show variation only. The steering compass shows variation plus deviation for the current heading. Check it daily and whenever the heading changes significantly; log every result.

Amplitude: the bearing of a rising or setting sun

The true bearing of a body at rising or setting can be found without the almanac's hour angles, using its declination and your latitude:

sin (Amplitude) = sin (Declination) / cos (Latitude)

The amplitude is the angle from True East (rising) or True West (setting) towards the North or South, named the same as the declination. Many almanacs include an amplitude table that avoids the calculation.

Example: declination N 19 degrees, latitude 34 degrees North. sin (Amp) = sin 19 / cos 34 = 0.3256 / 0.8290 = 0.3928, so the amplitude is 23.1 degrees. A rising sun with a northerly declination bears East 23.1 degrees North, which is 090 minus 23.1 = 066.9, say 067 degrees True. You observe the sun when its lower limb is about half a diameter above the sea horizon and read 080 degrees on the steering compass. The compass reads higher than True, so the total error is 13 degrees West. If the chart variation is 12 degrees West, the deviation is 1 degree West on that heading.

This is a quick, powerful check: it needs no sextant, no chronometer and only a rough idea of your latitude.

Star and Planet Sights at Twilight

Sun sights alone require a run between sights, so the fix depends on the accuracy of your DR. Twilight star sights give several position lines within 20 minutes, with no run needed, so they are usually the most accurate fix of the day.

Stars can only be shot when both the star and the horizon are visible: from about sunset to the end of civil twilight (sun 6 degrees below the horizon) and slightly into nautical twilight in the evening, and the reverse in the morning. The almanac tabulates sunrise, sunset and twilight times by latitude.

Reduction is identical to the sun except:

• GHA star = GHA Aries + SHA star (SHA and Dec from the star list on the daily pages).
• No semi-diameter or parallax: correct Hs for index error and dip, then the star refraction correction only.
• Bright planets (Venus, Jupiter, Mars, Saturn) have their own GHA, v and d values on the daily pages and a small additional altitude correction for Venus and Mars.
• AP3270 Volume 1 (Selected Stars) lets you enter with LHA Aries and read Hc and Zn for seven well-placed stars directly, which is the fastest method for twilight sights.

Geometry of the fix

Choose three (better four) bodies spread around the horizon. With three bodies, about 120 degrees apart in azimuth is ideal. Two bodies are best 90 degrees apart; anything closer than about 30 degrees gives a poorly defined fix.

Click to zoom

When the three lines are plotted, they rarely meet at a point. They form a small triangle, the cocked hat. With stars spread evenly round the horizon, the fix is taken at the centre of the triangle, and a systematic error (for example an unknown index error, or an abnormal dip) shifts all lines equally and does not move the centre. A cocked hat smaller than about 3 miles across indicates good observation and reduction.

Click to zoom

Click to zoom

If sights are taken a few minutes apart while the yacht is moving fast, advance the earlier lines to the time of the last sight (6 knots for 10 minutes is 1 mile, which is worth allowing for). If you are in no hurry, heave-to for the twilight session: it stops the boat, so the sights are truly simultaneous, and it reduces the motion so the work goes faster. Otherwise note the log at the start, middle and end of the set, run the early lines up to the middle one and the late ones back, so that the fix is timed at the middle of the session. Shoot the eastern stars first in the evening, because the eastern horizon fades first, and the western ones first in the morning, when the eastern horizon sharpens up first; and when the plot shows one line well adrift of the others, as happens when Castor is shot in mistake for Pollux, discard it rather than stretching the fix to include it.

Choosing altitudes

Bodies between about 15 and 70 degrees are easiest and most reliable. Below 15 degrees refraction is large and variable; above 70 degrees the body is hard to swing accurately and the position circle has a small radius, so the straight-line approximation fails.

Click to zoom

Moon and Planet Sights

The moon and the planets are useful additions to the twilight sights, and the moon can be shot in daylight, when the horizon is clear.

The moon

The moon is the nearest body, so it has large parallax and its semi-diameter varies. Its GHA changes by between about 14 degrees 19 minutes and 14 degrees 36 minutes an hour, so the increment must come from the moon column of the Increments and Corrections pages, with the v correction (the difference between the moon's actual and nominal hourly rate) read from the daily page. Declination also needs its d correction.

The altitude correction has two parts: a main correction based on apparent altitude, and a second correction based on the horizontal parallax (HP) given on the daily page, with different tables for the lower and upper limb. The corrections are large, around 50 to 60 minutes, because of parallax. You cannot use the sun's small corrections.

Illustrative example: Hs 40 degrees 10.0 minutes, index error 1.0 minute on the arc, height of eye 3 m (dip 3.0 minutes), lower limb, HP 58.2 minutes.

The almanac combines these into the main and HP corrections to give the same result in one step. If you use the upper limb you subtract the semi-diameter (and the almanac tables are marked for upper limb). Always check that the moon's limb you observe matches the correction you apply. When the moon is a crescent or gibbous, the limb on the horizon is the one that is lit, and the bright limb is the easiest to observe.

Planets

Venus, Mars, Jupiter and Saturn have their own GHA and declination columns on the daily pages, with v and d corrections. Their altitude corrections are those for stars (index error, dip, refraction), plus a small additional correction for Venus and Mars from the almanac table. Planets are bright and do not twinkle, so they are easy to find in twilight, and are excellent for sights. Remember to note which planets the almanac lists as visible on the date.

Planning Sights with a Star Finder

In twilight there is no time to wonder which star is which. The navigator plans the sights in advance using a star finder (the US 2102-D "Rude" star finder, or the equivalent). It consists of a white base plate showing the navigational stars on a polar projection, with a north-pole side and a south-pole side, and a set of transparent templates, each printed with altitude and azimuth curves for one latitude band.

Click to zoom

Click to zoom

Setting it up for twilight

1. Find the time of civil twilight (UT) for your DR position from the almanac (sunset plus the twilight duration, corrected for longitude in time).
2. Calculate LHA Aries for that time and DR longitude.
3. Choose the side of the base plate for your hemisphere (north side for North latitudes).
4. Choose the template for the nearest latitude (they come in 10 degree steps) and place it on the correct side.
5. Rotate the template until its arrow points to the LHA Aries value on the base plate scale.

Click to zoom

1. Read off every star inside the horizon circle with its approximate altitude and azimuth. Choose three or four between 15 and 70 degrees, well spread in azimuth. Note backups in case of cloud.
2. Write the list down with altitudes and true bearings, then preset the sextant to each altitude. Point it along the bearing and the star will appear near the horizon in the telescope.

The order of shooting matters. In the evening the eastern sky darkens first, so its horizon fades first; shoot towards the east early. Dimmer stars become visible later but the horizon is going; the brightest bodies (Venus, Jupiter, Sirius) can be found earliest, sometimes before sunset. A common approach is to shoot the brightest bodies as soon as they appear, then work round the list, finishing before the horizon becomes indistinct. In the morning the order reverses.

Click to zoom

Identifying a planet by pre-computation

Planets are bright enough to be seen in twilight long before the stars, so they are easy to find once you know roughly where to look. Cunliffe's method needs no star finder: work out the time of civil twilight, enter that time on a planet pro-forma as if it were a sight, extract a rough GHA, declination and LHA for the planet (to the nearest degree is fine), and run it through the sight reduction tables for a calculated altitude and azimuth without bothering to correct for declination minutes. At twilight look along the azimuth with the hand-bearing compass (allowing for variation, because azimuths are true) and up to about the calculated altitude: the bright star there is your planet. Once identified, a planet changes position so slowly that you can use it for the rest of the voyage. The same pre-computation applied to the seven stars of Volume 1 lets you preset the sextant to each altitude and sweep the horizon along each bearing; going back one degree of LHA Aries for every four minutes of time lets you start on the brightest stars before the official time of civil twilight. To find the UT of civil twilight at your position, take the tabulated time for your latitude and add the longitude in time if West (subtract if East): at 22 N, 55 W, a tabulated 0504 becomes 0504 plus 3 hours 40 minutes, or 0844 UT; at 10 S, 139 E, a tabulated 1811 becomes 1811 minus 9 hours 16 minutes, or 0855 UT.

Identifying an unknown star

If a gap in cloud shows a bright star you did not plan for, take the sight anyway and note its compass bearing. Afterwards:

1. Convert the compass bearing to true.
2. Set up the star finder for the time of the sight.
3. Find where the measured altitude and the true bearing meet on the template; the nearest star is your candidate.
4. Confirm with the almanac (and check it is not a planet; planets are not printed on the base plate and must be plotted in pencil from their declination and 360 minus SHA, or simply identified by their brightness and position on the ecliptic).

Click to zoom

Star finders are printed for a particular epoch. The stars' SHA and declination drift slowly because of precession of the Earth's axis (a cycle of about 26,000 years), so an old star finder becomes slightly inaccurate after several decades. It is only a planning tool: always use the current almanac for the actual reduction.

Click to zoom

Recognising the Navigational Stars

The almanac lists 57 selected stars plus Polaris. You do not need to know them all, but you must recognise the brightest ones and the main pointer patterns, because the star finder is useless if you cannot find the star in the sky. The brightest navigational stars include Sirius, Canopus, Rigil Kentaurus, Arcturus, Vega, Capella, Rigel, Procyon, Achernar, Betelgeuse, Hadar, Altair, Acrux, Aldebaran, Spica, Antares, Pollux, Fomalhaut, Deneb and Regulus.

Northern sky

The Plough (Ursa Major). The two stars at the end of the "pan", Dubhe and Merak, are the Pointers. Extend the line from Merak through Dubhe about five times their separation and you reach Polaris. Follow the curve of the handle to Arcturus ("arc to Arcturus") and on to Spica.

Click to zoom

Cassiopeia is a W (or M) shape of five stars on the opposite side of Polaris from the Plough, so when one is low the other is high. Schedar and Caph are its navigational stars.

Click to zoom

Around the celestial equator

Orion straddles the celestial equator and is visible from both hemispheres (it dominates northern winter evenings). Betelgeuse (reddish) and Bellatrix mark the shoulders, Rigel (blue-white) and Saiph the feet. Follow the belt down to the south-east to Sirius, the brightest star in the sky, and up to the north-west to Aldebaran.

Click to zoom

Southern sky

The Southern Cross (Crux) is small and kite-shaped. Acrux is at its foot and Gacrux at its head; the long axis, from Gacrux through Acrux, points roughly towards the south celestial pole. Beside it, Rigil Kentaurus (Alpha Centauri) and Hadar (Beta Centauri) are the Pointers that identify the true Cross (there is a "False Cross" nearby that lacks pointers). Extending the Cross's long axis about four and a half times its length gives the approximate position of the south celestial pole.

Click to zoom

Planets

Venus and Jupiter are brighter than any star and do not twinkle. They are often the first "stars" visible in evening twilight and are excellent for sights, but they can mislead the unwary: always check the almanac's planet notes for which planets are visible and where.

Worked Example: Twilight Star Fix

Evening civil twilight, DR 22 10 S, 145 30 W. GHA Aries at the time of the first sight is 150 20.0. For Sirius, SHA 258 42.9, Dec S 16 45 (approximately):

• GHA Sirius = 150 20.0 + 258 42.9 = 409 02.9, minus 360 = 049 02.9.
• Assumed longitude (West, same minutes): 145 02.9 W. LHA = 049 02.9 minus 145 02.9 = minus 96 degrees, plus 360 = 264.
• Assumed latitude 22 S. Declination 16 S, same name. Enter the tables with 22, 16 same, LHA 264 and reduce as for the sun, but with the star altitude correction (refraction only after IE and dip).

Repeat for, say, Canopus and Altair or Fomalhaut chosen from the star finder to give three azimuths roughly 120 degrees apart. Plot all three intercepts from their own APs (each will have a different assumed longitude), and take the centre of the cocked hat as the fix.

Accuracy and Error Sources

An astro fix has an error budget just like a DR. The main contributions are:

A competent observer in a seaway therefore achieves a fix good to about 2 to 3 nm, and a single position line good to about 1 to 2 nm. These are the working limits you give the examiner when asked how accurate your position is. A systematic error (such as an unapplied index error) shifts all lines in the same direction, so check it first when three lines from different directions all miss the DR by the same sense.

You can check your technique by comparing the astro fix with GNSS, and logging the difference each day: a consistent bias points to a systematic error, while scatter points to observation error.

Daily Routine on an Ocean Passage

A practical astro day, which also provides exactly the evidence the examiner wants:

1. Morning twilight: three to four star or planet sights for a fix (if the sky allows), and a compass check on a low star or planet.
2. Mid-morning (sun about 3 to 4 hours before noon, bearing well east of the meridian): sun sight, LOP, plus a compass check.
3. Noon: meridian altitude for latitude; transfer the morning line for the sun-run-meridian fix. Many yachts record the noon-to-noon day's run here.
4. Afternoon: sun sight; transfer the noon latitude for a sun-run-sun or noon-run-afternoon fix.
5. Evening twilight: star sights.
6. Throughout: hourly log entries (log, course, barometer), DR maintained, chronometer checked against time signal and its error and rate recorded.

What the Examiner Will Look For

The Ocean examiner assesses the astro records you bring, then asks you to explain them. A good sight record shows every step.

• Planning. The date, DR, and why you chose the time (twilight, morning, noon).
• Observation data. Hs, UT with the chronometer error applied, height of eye and index error, each stated.
• Corrections. The full Hs to Ho calculation, with the source of each correction.
• Almanac extraction. GHA, declination, d, increments and corrections, each shown.
• Reduction. The assumed position, LHA, the tables entered, Hc, d, Z and Zn, and the intercept with its direction.
• Plot. The plotting sheet, with each line labelled, the run advanced, and the fix with a time. A clear statement of the DR error and current implied.
• Compass check. Date and time, body, Zn calculated, the compass bearing, total error, variation and deviation.
• Chronometer log. Daily time checks with error and rate.

You will also be asked questions such as: why was this AP chosen, why did you plot the intercept towards, how accurate do you think this fix is, what if the sun had been behind cloud at noon. Answers should show understanding, not recitation.

Common Mistakes

• Wrong time. Using zone time instead of UT, forgetting the UT date change, or ignoring chronometer error. Four seconds of time is one minute of longitude.
• LHA arithmetic. Adding West longitude or subtracting East; forgetting to bring the answer within 0 to 360.
• Assumed longitude. Choosing minutes that do not cancel the GHA minutes, so LHA is not a whole degree; in East longitude forgetting that the minutes must be 60 minus the GHA minutes.
• Same/contrary name. Entering the wrong half of the table page when declination and latitude have different names.
• Using the AP latitude in the wrong place. The AP latitude enters the tables; the intercept is plotted from the AP, not from the DR.
• Plotting intercept the wrong way. Remember "Calculated Greater Away".
• Z to Zn conversion. In North latitude, LHA greater than 180 means Zn = Z; LHA less than 180 means Zn = 360 minus Z. Southern latitude rules differ (they are printed on every table page). A quick sense-check: a morning body is east of you, an afternoon body west.
• Forgetting the d correction to declination or to Hc.
• Noon sight sign errors. Draw a sketch every time.
• Transferring the line wrongly. Advance the whole line along the course and distance made good, not just the AP; and allow for known current.
• Star misidentification. Shooting a planet thinking it is a star, or the wrong star; confirm with the star finder and the almanac.
• Poor fix geometry. Two bodies only 20 degrees apart in azimuth give a fix that is very sensitive to small errors.

Safety Notes

• Never look at the sun through the sextant telescope without adequate shades in place; set the shades before you raise the instrument. Permanent eye damage occurs in a fraction of a second.
• Clip on or brace securely when taking sights on deck at night.
• Astro positions are accurate to a few miles: plan landfalls in daylight, with a generous margin off reefs and low islands, and use every other source (depth, radar, visual bearings) as you close the land.
• Keep paper copies of the almanac and tables and spare pencils; do not rely on a phone app as your only reduction method.

Summary

• A body's GP is defined by its GHA and declination; your zenith distance (90 minus Ho) is your distance from the GP, 1 minute = 1 mile.
• One altitude places you on a circle of equal altitude; near you this is a straight position line at right angles to the azimuth.
• LHA = GHA minus West longitude, or plus East longitude. GHA star = GHA Aries + SHA.
• The intercept method: choose an AP, calculate Hc and Zn, intercept = Ho minus Hc, Towards if Ho is greater, Away if Hc is greater.
• The noon meridian altitude gives latitude directly: Lat = ZD plus or minus Dec. Combined with a transferred morning sun line it gives the sun-run-meridian fix required for the Ocean exam.
• Compass checks compare calculated Zn with an observed compass bearing; do them daily on a low body.
• Twilight star sights give the best fix: plan them with a star finder, choose bodies 15 to 70 degrees high and well spread in azimuth, and take the centre of the cocked hat.
• Know the main pointers: Plough to Polaris, Orion to Sirius and Aldebaran, Centauri pointers to the Southern Cross.

Check Your Understanding

1. The sun's corrected altitude is 52 30.0. How far are you from its GP?

Answer: ZD = 90 00.0 minus 52 30.0 = 37 30.0 = 37.5 x 60 = 2,250 nautical miles.

1. GHA of the sun is 038 14.6 and your DR longitude is 027 40 E. Choose an assumed longitude and find LHA.

Answer: In East longitude the assumed longitude's minutes must be 60 minus 14.6 = 45.4, within 30 minutes of the DR: 027 45.4 E. LHA = 038 14.6 + 027 45.4 = 066 degrees.

1. Ho is 33 12.0 and Hc is 33 20.0. What is the intercept and which way is it plotted?

Answer: 8.0 miles Away from the body (Calculated Greater Away), plotted from the AP along the reciprocal of Zn, with the LOP drawn at right angles.

1. At noon you are in North latitude, the sun bears south, Ho is 60 20.0 and declination is S 05 15.0. What is your latitude?

Answer: ZD = 29 40.0 (named N because the sun bears south). Declination is contrary name, so Lat = ZD minus Dec = 29 40.0 minus 05 15.0 = 24 25.0 N.

1. Why does a chronometer error of 20 seconds matter, and what should you record daily?

Answer: The sun's GP moves 15 minutes of arc of longitude per minute of time, so 20 seconds gives 5 minutes of longitude error (up to 5 miles). Record the chronometer error and its daily rate against a radio time signal or GPS UT.

1. Your sun sight gives Zn 245 T. At the same moment the hand-bearing compass reads 258 C and the variation is 11 W. What is the error, and what does the remaining 2 degrees indicate?

Answer: Total error 13 W (compass reads higher than true). Variation accounts for 11 W, leaving 2 W, which for a hand-bearing compass indicates local magnetic interference (deviation) at the position where it was used, or an observation error. Repeat away from metal and electronics.

1. Why are three stars about 120 degrees apart in azimuth preferred to three stars all in the western sky?

Answer: Widely spaced lines cross at good angles, giving a small, well-defined cocked hat; systematic errors (such as an unknown index error) shift all three lines equally and the centre of the triangle remains correct. Stars bunched together give shallow intersections and a large, unreliable fix.

1. How do you set up a 2102-D star finder for evening twilight?

Answer: Find the UT of civil twilight for your DR, calculate LHA Aries for that time, choose the base plate side for your hemisphere and the template for your nearest latitude, then rotate the template so its arrow points to LHA Aries. Read off altitude and azimuth of suitable stars (15 to 70 degrees high, well spread in azimuth).

1. In the Northern Hemisphere, how do you find Polaris, and how is it used?

Answer: Extend the line from Merak through Dubhe (the Plough's Pointers) about five times their separation. Its altitude, corrected and with the three Pole Star table corrections applied (Lat = Ho minus 1 degree + a0 + a1 + a2), gives latitude.

1. What minimum astro evidence must you bring to the Yachtmaster Ocean oral?

Answer: Records, made on board out of sight of land without electronic aids, of the planning, reduction and plotting of a sun-run-meridian altitude (or sun-run-sun) sight, and of a compass check by the bearing of the sun, moon, a star or a planet.

1. What is the true bearing of the sun at rising if its declination is N 10 degrees and your latitude is 40 degrees North?

Answer: sin (Amp) = sin 10 / cos 40 = 0.1736 / 0.7660 = 0.2267, so the amplitude is 13.1 degrees. A rising sun with northerly declination bears East 13.1 degrees North, or 090 minus 13.1 = 076.9, say 077 degrees True.

1. Two sun sights give azimuths of 110 and 130 degrees. Why is this a poor choice for a fix, and what would you do?

Answer: The azimuths differ by only 20 degrees, so the position lines cross at 20 degrees, below the 30 degree minimum, and the fix is very sensitive to small errors. Wait for the sun to move further in azimuth (a difference of 60 to 90 degrees is better), or take a noon meridian altitude for a latitude line and use a sun-run-meridian fix.