Using the Nautical Almanac - Sight Reduction & Marcq St Hilaire Method

Prerequisites

This lesson covers the "Nautical Almanac and sight reduction tables" element of the RYA/MCA Yachtmaster Ocean syllabus. Before starting you should:

• Hold, or be working towards, Yachtmaster Offshore (or MCA OOW Yachts), the prerequisite certificate for the Ocean exam.
• Understand the celestial sphere, GP, GHA, LHA, SHA, declination and the intercept method (Celestial Navigation lesson).
• Be able to correct Hs to Ho (Sextant Corrections lesson).
• Be confident with time: UT, zone time, chronometer error.

You will need the current year's Nautical Almanac (UKHO NP314, identical in content to the US edition), AP3270/NP303 (the UK edition of US Pub. 249, "Sight Reduction Tables for Air Navigation") and ideally sight reduction forms. Reeds Nautical Almanac also contains abridged astro tables for the sun, moon, planets and selected stars, adequate for yacht navigation.

Learning Objectives

By the end of this lesson you will be able to:

• Find your way around the Nautical Almanac: daily pages, increments and corrections, altitude correction tables, star lists and Pole Star tables.
• Extract GHA and declination for the sun, moon, planets and stars for any UT, applying increments, v and d corrections.
• Use SHA with GHA Aries to find a star's GHA.
• Enter AP3270 (Pub. 249) with assumed latitude, LHA and declination; extract Hc, d and Z; interpolate for declination; and convert Z to Zn.
• Describe NP401 (Pub. 229) and when you would use it instead.
• Reduce a sight by calculator using the cosine or haversine formula when tables are unavailable.
• Recognise and avoid the common extraction errors.
• Find sunrise, sunset, twilight and moonrise times, and use Polaris tables for a latitude check.

RYA Yachtmaster Ocean syllabus items covered: use of the Nautical Almanac (daily pages, increments and corrections, altitude correction tables, star data, Pole Star tables, twilight and rising and setting tables); use of sight reduction tables (AP3270 / Pub. 249 and awareness of NP401 / Pub. 229); the calculator and electronic alternatives; and accuracy checks. It supports the Celestial Navigation lesson (intercept method) and the Sextant Corrections lesson (Hs to Ho).

What the examiner expects: that you can complete a sight reduction form from raw sextant altitude to a plotted position line in about 15 minutes, with no errors, using only the almanac, tables and a plain scientific calculator. Examiners often hand a candidate an almanac page and a UT, and ask for GHA and declination of a named body, or set a quick check of the entering arguments for the tables. Neatness and a consistent form count: a tidy sheet shows the habit of mind they want.

The Nautical Almanac

The Nautical Almanac is published every year. It gives the positions of the sun, moon, four navigational planets and the First Point of Aries for every hour of UT, the positions of the navigational stars, and all the tables needed to correct altitudes. It is only valid for its year: always check the year on the cover and at the top of each page. (In an emergency an old almanac can be used for the sun with small corrections, but never for the moon or planets.)

The daily pages

Each opening of the daily pages covers three days.

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Left-hand page:

• Aries: GHA for every hour.
• Venus, Mars, Jupiter, Saturn: GHA and declination for every hour, with the v and d values at the foot of each column, and each planet's magnitude at the top.
• Stars: SHA and declination of the 57 selected navigational stars (valid for the three days).

Right-hand page:

• Sun: GHA and declination for every hour; at the foot, the d value (hourly change of declination) and the semi-diameter (SD).
• Moon: GHA, v, declination, d and horizontal parallax (HP) for every hour.
• Twilight, sunrise and sunset: local mean times for latitudes from 72 N to 60 S, for the middle day of the three.
• Moonrise and moonset: for each of the three days.
• Equation of Time, meridian passage of the sun and moon, moon's age and phase.

The increments and corrections pages

The tinted pages near the back have one block for each minute of time (0 to 59 minutes), each with 60 rows of seconds. For each second there are three columns:

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Beside them, in the same block, are the v or d correction tables: enter with the v or d value from the daily page to get the correction for that many minutes past the hour.

The v and d corrections

• d is the hourly change in declination. The declination at your time = declination at the whole hour + (d x minutes past the hour / 60). The sign of the correction depends on whether the declination is increasing or decreasing; the almanac does not print a sign for d, so look at the next hour's declination to see which way it is going. For the sun, whose d is less than a minute, most navigators do the correction by inspection and mental arithmetic (14 minutes past the hour with d 0.9 gives 0.2); for the moon, whose declination changes rapidly, use the table every time. The v and d tables are entered with minutes only, not seconds, which is accurate enough.
• v is the amount by which a body's hourly GHA change exceeds the rate built into the increments table. For the moon, v is the excess over 14 degrees 19.0 minutes per hour, and is always positive. For planets, v is the excess over 15 degrees per hour; it is usually positive but can be negative for Venus (the sign is then printed). The sun needs no v: the increments are based on its rate.

The altitude correction tables

Inside the front cover: the sun correction tables (lower and upper limb, October to March and April to September), the stars and planets table (refraction), the dip table, and the additional corrections for Venus and Mars. A separate table gives additional refraction for non-standard temperature and pressure. Inside the back cover: the moon correction tables.

Other tables

• Star list at the back: SHA and declination of 173 stars, monthly.
• Pole Star tables: a0, a1 and a2 corrections for latitude by Polaris.
• Planet notes: which planets are visible in the morning or evening each month, and which stars they are near.
• Conversion of arc to time table.

Extracting GHA and Declination

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The sun

The diagram below shows how GHA and declination are read for the whole hour, with the d value at the foot of the column. The figures are illustrative.

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Procedure:

1. Write down the UT of the sight (date, hours, minutes, seconds), having corrected the watch for chronometer error.
2. From the daily page for that date, take the GHA and declination for the whole hour, and the d value.
3. From the increments page for the minutes, find the row for the seconds and take the sun/planets increment.
4. From the same increments block, take the d correction.
5. GHA = hourly GHA + increment. Declination = hourly declination plus or minus d correction.

Worked example: sun

Sight at 14h 23m 40s UT. Daily page: GHA at 14h 031 48.2, Dec N 19 05.8, d 0.6 (declination increasing).

Worked example: moon

Sight at 14h 23m 40s UT. Daily page: GHA at 14h 205 17.4, v 11.2, Dec S 12 31.6, d 9.8 (increasing), HP 56.8.

Note the moon's increment (5 38.8) is smaller than the sun's (5 55.0) for the same time, because the moon's GHA increases more slowly; v then adds back the excess for that particular hour. Keep a note of HP for the altitude correction.

Planets

Exactly as the sun, but use the planet's own GHA, declination, v and d, and the sun/planets increment column. Add the v correction (or subtract if v is printed with a minus sign for Venus).

Stars: SHA and GHA Aries

Stars keep their positions relative to each other, so the almanac does not tabulate each star's GHA. Instead it tabulates GHA Aries hourly and each star's SHA, measured westward from Aries.

GHA star = GHA Aries + SHA star (subtract 360 if the sum exceeds 360)

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Example (Sirius): GHA Aries at the time of the sight 150 20.0 (hourly value plus Aries increment); SHA Sirius 258 42.9; GHA Sirius = 409 02.9 minus 360 = 049 02.9. Declination is read straight from the star list (S 16 45 approximately). There is no v or d for stars.

Remember to use the Aries column for the increment, not the sun column: Aries moves about 2.5 minutes of arc per hour faster than the sun, and over 59 minutes the difference reaches nearly 10 seconds of time, worth 2.5 minutes of longitude.

Sight Reduction Tables: AP3270 (Pub. 249)

AP3270 (UK designation NP303; US Pub. No. 249) was designed for air navigators and is the favourite of yachtsmen because it is quick and its accuracy (to the nearest minute of altitude and degree of azimuth) matches what can be observed from a small boat.

Bodies with declination over 29 degrees (the moon occasionally, some stars) cannot be reduced with Volumes 2 and 3; use Volume 1 if the star is listed, NP401, or a calculator.

Entering arguments

Volumes 2 and 3 are entered with three whole-degree arguments:

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1. Assumed latitude: the whole degree nearest the DR latitude.
2. Declination: the whole degrees of the body's declination (the minutes are used afterwards for interpolation), and whether it is the same name as the latitude (both N or both S) or contrary name.
3. LHA: a whole number of degrees, made so by choosing the assumed longitude.

Each latitude has its own set of pages. Declinations of the same name and contrary name to the latitude are tabulated on separate pages (or separate parts of a page), clearly headed; make sure you are in the right one. Across the top are declinations; down the side, LHA.

Outputs

For each combination the table gives:

• Hc: the calculated altitude for the whole-degree declination.
• d: the change in Hc for a 1 degree increase in declination, with its sign.
• Z: the azimuth angle, to be converted to the true bearing Zn.

Interpolating for declination minutes

Hc is tabulated for whole degrees of declination; your body's declination has minutes as well. The correction is:

Correction = d x (declination minutes / 60), with the sign of d.

The interpolation table (Table 4) at the back of the volume gives the same result without arithmetic: enter with d and the declination minutes.

Example from the Celestial Navigation lesson: latitude 34 N, declination N 19 04.3 (same name), LHA 313. Tabulated Hc 45 47, d +29, Z 097. Correction = 29 x 4.3/60 = +2.1, rounded +2. Hc = 45 49.

Converting Z to Zn

Z is an angle measured from the elevated pole (north in North latitudes, south in South latitudes), east or west, from 0 to 180. Zn is the true bearing, 000 to 360 measured clockwise from true north. The rules are printed on every page:

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Always sense-check: LHA between 180 and 360 means the body is east of your meridian (morning sun); LHA between 0 and 180 means it is west (afternoon sun). If Zn says east in the afternoon, you have made an error.

Worked example: contrary name, South latitude

DR 27 40 S, 152 20 W. Sun GHA 095 32.6, Dec N 12 18.0 (contrary name to latitude).

1. Assumed latitude 28 S.
2. Assumed longitude (West, same minutes as GHA) 152 32.6 W. LHA = 095 32.6 minus 152 32.6 = minus 57, plus 360 = 303.
3. Enter latitude 28, declination 12 contrary name, LHA 303. The table gives Hc 21 53, d minus 36, Z 118.
4. Correction for 18.0 minutes: minus 36 x 18/60 = minus 10.8, rounded minus 11. Hc = 21 42.
5. South latitude, LHA greater than 180: Zn = 180 minus 118 = 062 T. (Morning sun, north-east of you, as you would expect in South latitude with the sun's declination north.)
6. Compare with Ho to get the intercept, and plot from the AP 28 00 S, 152 32.6 W.

Volume 1: selected stars

For twilight star sights, Volume 1 is much quicker. Enter with assumed latitude (whole degree) and LHA Aries (whole degree, by choosing the assumed longitude to cancel the minutes of GHA Aries). The page lists seven stars well spread in azimuth, with Hc and Zn for each, the brightest in capital letters and the three best for a fix marked with a diamond. No SHA or declination is needed, and the same AP serves all the stars shot at one time (if you take them within a few minutes, use the GHA Aries at each sight time to choose separate APs, or advance the lines). Apply the small precession and nutation correction from the table at the back if you are more than a year or so from the epoch.

NP401 (Pub. 229)

NP401, "Sight Reduction Tables for Marine Navigation" (identical to US Pub. No. 229), is the professional's set. It comes in six volumes, each covering a 15 degree band of latitude (with a degree of overlap), and covers all declinations 0 to 90 degrees and all bodies.

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The entering arguments are the same (LHA, assumed latitude, declination), but the page layout is reversed: each opening is for one LHA, with latitudes across the top and declinations down the side. Interpolation uses the "d" and the "Dec Inc" (declination increment) via a tens-and-units table inside the covers; where d is printed in italics followed by a dot, a further double-second-difference correction is needed for full precision.

For a yacht, AP3270 is entirely adequate: a yacht sextant sight in a seaway is rarely better than about 1 minute anyway. NP401 is worth having if you expect to work bodies with high declination or want greater precision, and it is the standard on merchant ships.

Calculator Methods and the Haversine Formula

If tables are lost, soaked, or the body's declination is beyond their range, solve the PZX triangle directly. With a scientific calculator:

sin Hc = sin Lat x sin Dec + cos Lat x cos Dec x cos LHA

cos Z = (sin Dec minus sin Lat x sin Hc) / (cos Lat x cos Hc)

Treat declination as negative when it is contrary name to latitude. For North latitude, Zn = Z if LHA is greater than 180, otherwise Zn = 360 minus Z. (For South latitude, use the absolute values and the southern rules, or simply solve with latitude negative and the same formula.)

The traditional alternative, devised for logarithm tables because it avoids awkward signs, is the haversine formula:

hav ZD = hav (Lat minus Dec) + cos Lat x cos Dec x hav LHA, where hav x = sin squared (x/2), and Hc = 90 minus ZD.

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Worked example (calculator)

Latitude 34 N, declination N 19 04.3, LHA 313 (same sight as above, but using the exact declination).

1. Lat minus Dec = 14 55.7 = 14.928 degrees; hav = sin squared (7.464) = 0.01688.
2. cos 34 x cos 19.072 = 0.82904 x 0.94511 = 0.78353.
3. hav 313 = sin squared (156.5) = 0.15900.
4. hav ZD = 0.01688 + 0.78353 x 0.15900 = 0.01688 + 0.12458 = 0.14146.
5. ZD = 2 x arcsin (square root of 0.14146) = 44 11.1; Hc = 45 48.9.
6. Z from the cosine formula: 97.4 degrees; LHA greater than 180, North latitude, so Zn = 097.4 T.

The table answer was 45 49 and 097: the calculator confirms it. Many navigators carry a calculator programmed with these formulas as a backup, and use it to check table work. The examiner may ask you to explain how you would reduce a sight if you lost your tables.

Electronic aids

Dedicated astro apps and calculators reduce sights instantly. They are excellent for checking your work, but the Ocean exam requires navigation without electronic aids, and an app on a phone shares all the failure modes of the GPS it is backing up. Learn the tables first; use the app to check.

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Worked Example: Full Star Reduction with Volume 2

Evening twilight. DR 22 10 S, 145 30 W. Star: Sirius. Sight at 05h 12m 30s UT (next UT date; local time is around 1910 the previous evening).

1. GHA Aries 05h from the daily page, plus the Aries increment for 12m 30s, gives (say) 150 20.0.
2. SHA Sirius 258 42.9; GHA Sirius = 049 02.9. Dec S 16 45.
3. Assumed latitude 22 S; assumed longitude 145 02.9 W; LHA = 049 02.9 minus 145 02.9 = minus 96, plus 360 = 264.
4. Enter Vol 2, latitude 22, declination 16 same name (S and S), LHA 264: extract Hc, d and Z.
5. Correct for 45 minutes of declination: d x 45/60.
6. South latitude, LHA greater than 180: Zn = 180 minus Z.
7. Intercept = Ho minus Hc; plot from 22 00 S, 145 02.9 W.

Notice the UT date. A sight at 1910 local in zone +10 is 0510 UT the next day; use that day's daily page.

Worked Example: A Planet and a Star

Both examples use illustrative almanac figures. Work them with your own almanac for practice.

Venus (a planet with a negative v)

Sight of Venus at 05h 18m 40s UT. Daily page at 05h: GHA 211 04.6, v minus 1.1, Dec N 6 41.2, d 0.4 (declination decreasing).

Notice that v for Venus is printed with its sign, and that the correction is subtracted here. For Mars, Jupiter and Saturn v is normally positive and is added.

A star, with the Aries increment

Sight of Altair at 05h 18m 40s UT on the same date. Daily page: GHA Aries at 05h 123 29.8; SHA Altair 062 07.2, Dec N 8 54.7.

1. Increment of Aries for 18m 40s (Aries column) = 4 40.8.
2. GHA Aries = 123 29.8 + 4 40.8 = 128 10.6.
3. GHA Altair = 128 10.6 + 062 07.2 = 190 17.8. Dec N 8 54.7 (no correction).
4. DR 35 20 N, 030 10 W. Assumed latitude 35 N. Assumed longitude 030 17.8 W (minutes equal to the GHA minutes, West).
5. LHA = 190 17.8 minus 030 17.8 = 160.
6. Enter AP3270 Volume 2 (latitudes 0 to 39) with latitude 35, declination 8 same name, LHA 160. Extract Hc, d and Z, apply d x 54.7/60, and, because LHA is less than 180 in North latitude, convert Zn = 360 minus Z.

More Worked Reductions with AP3270 Volumes 2 and 3

The tabulated Hc, d and Z figures below were checked against the cosine formula, so they behave exactly as real table entries do. Use them to rehearse the interpolation and the Zn rules, then repeat the exercise with your own tables.

Example 1: same name, North latitude (Volume 3)

Afternoon sun, DR 52 10 N, 004 30 W. Sun GHA 054 20.0, Dec N 15 42.0, observed Ho 36 30.0.

1. Assumed latitude 52 N.
2. Assumed longitude West with the same minutes as the GHA: 004 20.0 W. LHA = 054 20.0 minus 004 20.0 = 050.
3. Enter Volume 3, latitude 52, declination 15 same name, LHA 050. Tabulated: Hc 35 53, d +49, Z 114.
4. Correction = 49 x 42.0 / 60 = +34 (d is positive, so Hc increases as declination increases). Hc = 35 53 + 34 = 36 27.
5. North latitude, LHA less than 180: Zn = 360 minus 114 = 246 T. South-west, which is right for an afternoon sun in northern summer.
6. Intercept = Ho minus Hc = 36 30 minus 36 27 = 3 miles Towards, along 246 T from the AP 52 00 N, 004 20.0 W. Plot the AP, step 3 miles towards 246 T, and draw the position line at right angles.

Example 2: contrary name, North latitude (Volume 3)

Morning sun, DR 50 05 N, 010 40 W. Sun GHA 310 10.0, Dec S 10 17.0, Ho 10 15.0.

1. Assumed latitude 50 N.
2. Choose the assumed longitude so that its minutes equal the minutes of the GHA, which makes the minutes cancel: 010 10.0 W. LHA = 310 10.0 minus 010 10.0 = 300.
3. Enter Volume 3, latitude 50, declination 10 contrary name, LHA 300. Tabulated: Hc 10 34, d minus 50, Z 120.
4. Correction = minus 50 x 17.0 / 60 = minus 14. Hc = 10 34 minus 14 = 10 20. The sign is negative because, with a contrary-name declination, Hc falls as the declination increases.
5. North latitude, LHA greater than 180: Zn = Z = 120 T. South-east, as expected for a morning sun with a southerly declination.
6. Intercept = Ho minus Hc = 10 15 minus 10 20 = 5 miles Away from the sun, along 120 T from the AP 50 00 N, 010 10.0 W. Move 5 miles in the reciprocal direction, 300 T, and draw the line at right angles.

A classic slip is to take the assumed longitude minutes from the DR instead of from the GHA. The LHA then has leftover minutes and is not a whole number of degrees.

Reading d with its sign

The tables print the sign of d. In contrary-name sections d is always negative. In same-name sections it is usually positive, but the sign can change, so always read it from the page and add the correction algebraically to Hc.

Interpolation Table 4 versus mental arithmetic

For an exam, mental arithmetic is quick enough: d x minutes / 60 can be done as (d x minutes) divided by 60, or as d x (minutes / 10) / 6. Round to the nearest minute. Table 4 at the back of Volumes 2 and 3 gives the same figure; it is useful when tired or when the sea is rough.

Using Volume 1 for a three-star fix (outline)

Because Volume 1 gives Hc and Zn directly, the full evening twilight routine takes about ten minutes for three stars:

1. Before twilight, work out the DR for the time of the sights and the GHA Aries at that time.
2. For each star, choose an assumed longitude so that LHA Aries is a whole number of degrees (the minutes of longitude equal the minutes of GHA Aries).
3. Enter Volume 1 with assumed latitude and LHA Aries. Choose the three stars marked best for a fix, or the three with azimuths about 120 degrees apart.
4. Take each sight, correct to Ho, and compare with the tabulated Hc to get an intercept. Because each star has a different sight time, calculate LHA Aries separately for each, or advance or retire the lines to a common time using the run of the vessel.
5. Plot the three lines. The cocked hat shows how good the sights were. Small cocked hats mean good observations; a large one means a timing or arithmetic error.

The volume is calculated for an epoch (for example 2025.0) and the small correction in Table 5 for the effects of precession and nutation (the star positions drift slowly) is applied to the fix, as a distance and direction, when the sights are not made close to the epoch.

Moon Corrections from the Almanac

The moon is the one body whose altitude correction needs extra care, because its horizontal parallax is large (about 57 minutes of arc). After index error and dip give the apparent altitude Ha, use the moon tables on the inside back pages: enter with Ha for a main correction, then add a small correction (L for the lower limb, U for the upper limb) taken with the horizontal parallax (HP) from the daily page. For the upper limb, subtract 30.0 minutes from the total, the moon's mean semi-diameter being already built into the lower limb figures. The moon corrections can approach a full degree, so a missed table or the wrong limb gives a gross error.

Using Bowditch (NGA Pub. 9)

The American Practical Navigator, universally called Bowditch, is published free by the US National Geospatial-Intelligence Agency and is on the Ocean reading list. It is not an almanac but a reference textbook, and its chapters on celestial navigation reinforce everything in this lesson.

• Chapters on celestial navigation explain the celestial coordinate systems, the navigational triangle (the PZX triangle used by all sight reduction), time, the almanac and the principal methods of sight reduction. Read them alongside your almanac.
• Sight reduction methods: the cosine-haversine formula; Pub. 249 and Pub. 229 tables; the older Ageton and Dreisonstok tabular methods; the calculator solution of the navigational triangle. Bowditch gives a complete, worked sight form for each.
• Special sights: the meridian passage (noon) sight, the ex-meridian sight and the Polaris sight, each with worked examples.
• Accuracy and error theory: Bowditch explains the sources of error in a line of position (instrument, personal, timing, tabular) and how a running fix and a three-body fix reveal them.
• Appendices and tables: the dip, refraction, parallax and altitude-correction tables are reproduced with their derivations; the glossary is a good source for exam vocabulary.

Treat Bowditch as the authority when the almanac instructions or an exam question leave you unsure why a rule works. The PDF runs to several hundred pages, so save it to the passage laptop and print the chapters you want.

Meridian Passage and the Noon Latitude

A sight at local apparent noon needs no tables beyond the almanac and is the simplest sight to teach and check. The almanac gives the time of the sun's meridian passage (on the right-hand daily page, for the middle day), in local mean time. Convert to UT with your longitude, start observing about ten minutes early, and follow the sun up with the sextant until it stops rising: that maximum altitude is the meridian altitude.

1. Correct the sextant altitude to Ho.
2. Zenith distance ZD = 90 minus Ho.
3. Name ZD in the opposite direction to the sun's bearing (the sun bears south if you are north of it, so ZD is named N).
4. Latitude = ZD plus declination if they have the same name; ZD minus declination if opposite names and ZD is the greater; and so on.

Example: Ho 52 20.0 with the sun bearing south; declination N 8 15.0. ZD = 37 40.0 N. Both N, so latitude = 37 40.0 + 8 15.0 = 45 55.0 N.

The longitude from a noon sight comes from the time of meridian passage: if the sun transits at 12 20 UT on a day when the almanac says meridian passage is at 12 04 LMT, then the difference of 16 minutes of time is 4 degrees of longitude, west of Greenwich in this case.

Rising, Setting and Twilight Tables

The tables on the daily pages give the local mean time (LMT) of sunrise, sunset, civil twilight (sun 6 degrees below the horizon) and nautical twilight (12 degrees below) for latitudes 72 N to 60 S, for the middle date of the three days. The star sights are taken in nautical twilight, when the horizon is still visible and the brightest stars are out. Use these tables to plan when to be on deck with the sextant.

Method:

1. Enter with the date, then find the latitude (interpolate linearly between the tabulated latitudes: 70, 68, 66, 64, 62, 60, 58, 56, 54, 52, 50, 45, 40, 35, 30, 20, 10, 0, minus 10 and so on).
2. Take the LMT.
3. Convert LMT to UT: UT = LMT plus longitude west (in time) or minus longitude east, using 15 degrees = 1 hour, 1 degree = 4 minutes.
4. Convert UT to zone time if wanted.

Example (illustrative): the table gives sunrise 07 29 at latitude 50 N and 07 15 at latitude 45 N. For latitude 47 N: 47 is three-fifths of the way from 50 to 45 (3 of the 5 degrees), so sunrise = 07 29 minus 0.6 x 14 = 07 29 minus 8.4 = 07 21 LMT. In longitude 025 W the correction to UT is plus 1h 40m, so sunrise is at 09 01 UT.

The sunrise and sunset times also tell you when to expect the sun on the horizon for a compass check by amplitude. For sights, avoid altitudes below about 10 degrees, where refraction is large and uncertain.

The moonrise and moonset pages for each of the three days are used the same way. Moon sights are very useful when the moon is clear of the sun and visible in daylight; the almanac also gives the moon's phase and age.

The Pole Star (Polaris) Tables

Polaris lies within about 0.7 degrees of the celestial pole, so its altitude, corrected, is nearly equal to the observer's latitude. The tables at the back of the almanac give three small corrections, entered with LHA Aries, with the latitude and the month. The procedure:

1. Find LHA Aries: GHA Aries (hour plus Aries increment) plus the DR longitude (East plus, West minus).
2. Correct the sextant altitude to Ho in the normal way (index error, dip, refraction; no parallax or SD).
3. Enter the Polaris table with LHA Aries (to the nearest 10 degrees) for a0, then with the LHA Aries and the DR latitude for a1, and with the LHA Aries and the month for a2.
4. Latitude = Ho minus 1 degree plus a0 plus a1 plus a2.

Example (illustrative): Ho 35 12.0, a0 0 50.3, a1 0.6, a2 0.5. Latitude = 35 12.0 minus 1 00.0 plus 0 50.3 plus 0.6 plus 0.5 = 35 03.4 N. This gives a good latitude in twilight in North latitudes, with a pure latitude line that crosses a star position line well. It is of little use in the Southern Hemisphere, where Polaris is below the horizon.

Accuracy and Practical Considerations

• Accuracy of the tables: AP3270 gives Hc to the nearest minute; a good observation from a yacht gives about 1 to 2 minutes of arc, which is 1 to 2 miles. The tables are therefore as accurate as the sight.
• Use of whole-minute rounding: carry tenths of a minute through the almanac work, but round Hc to the nearest minute.
• Time: a 4 second chronometer error is 1 mile of longitude at the equator (4 seconds of time = 1 minute of arc), so keep the time within 2 seconds. Record the error each day.
• A sight reduction form (with printed lines for each item: date, UT, GHA, increments, v, SHA, LHA, assumed latitude and longitude, Dec, d, Hc, Z, Zn, Ho, intercept) forces you to complete each step in order and leaves a record the examiner can read.
• Keep old almanacs only for practice: they are out of date for the moon and planets.

Exam Tips and Practical Habits

• Write UT, date and the body at the top of each sight; underline the date when UT differs from local date.
• Write the three entering arguments in a box before you open the tables.
• For East longitude, subtract the GHA minutes from 60 to give assumed longitude minutes, then add: LHA = GHA plus East longitude.
• For the sun in the Southern Hemisphere, check whether the declination is the same name or contrary name to the latitude.
• Sense-check Zn and the intercept every time.
• State clearly in the exam which table or page you used; if you make a slip, correct it by crossing out neatly and write the correction beside it.

Common Extraction Errors

The diagram below summarises the classic errors examiners see in candidates' sight forms.

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• Wrong year of almanac, or wrong date: in particular forgetting that an evening sight in West longitude may be the next UT date.
• Wrong column: sun instead of a planet, or reading the moon's GHA from the sun's column; Aries GHA from the wrong day.
• Wrong increments column: using sun/planets for a star (should be Aries) or for the moon (should be moon).
• Forgetting v for the moon or planets, or the d correction sign (look at whether declination is increasing or decreasing).
• LHA outside 0 to 360, or assumed longitude not chosen to give a whole-degree LHA. In East longitude the AP minutes are 60 minus the GHA minutes.
• Assumed latitude not the nearest whole degree, or using the DR latitude in the tables.
• Same/contrary name confusion.
• Declination above 29 degrees in AP3270 Vols 2 and 3.
• Z not converted to Zn, or the wrong rule for the hemisphere.
• Hc interpolation sign: the sign of d must be respected.

A neat, consistent sight reduction form, with every line filled in, catches most of these. Always sense-check: does the GP lie in the right direction (morning sun east)? Is Hc close to Ho (within 30 or so minutes)? A huge intercept nearly always means an arithmetic error, not a bad position.

Summary

• The Nautical Almanac is valid for one year. Daily pages (three days per opening): left page Aries, planets and stars; right page sun, moon, twilight and rising/setting times.
• GHA = hourly GHA + increment (+ v for moon and planets). Dec = hourly Dec plus or minus d correction.
• Use the right increments column: sun/planets, Aries, or moon.
• GHA star = GHA Aries + SHA star.
• AP3270 (Pub. 249): Vol 1 selected stars; Vols 2 and 3 for declinations 0 to 29. Enter with assumed latitude, LHA and declination (same/contrary); extract Hc, d, Z; correct Hc by d x dec minutes / 60; convert Z to Zn.
• NP401 (Pub. 229): six volumes, all declinations, precision 0.1 minute.
• Calculator: sin Hc = sin L sin D + cos L cos D cos LHA, or the haversine formula.
• Prevent errors with a standard form and constant sense-checking.

Check Your Understanding

1. Which bodies are tabulated on the left-hand and right-hand daily pages of the Nautical Almanac?

Answer: Left: Aries, Venus, Mars, Jupiter, Saturn and the 57 navigational stars (SHA and Dec). Right: sun and moon (with v, d and HP for the moon), plus twilight, sunrise and sunset, moonrise and moonset, Equation of Time and meridian passage.

1. Find the sun's GHA and declination at 09h 41m 20s UT, given GHA at 09h 315 58.1, Dec S 22 14.5, d 0.3 (decreasing). The sun increment for 41m 20s is 10 20.0.

Answer: GHA = 315 58.1 + 10 20.0 = 326 18.1. d correction = 0.3 x 41/60 = 0.2, subtracted because declination is decreasing: Dec = S 22 14.3.

1. Why is the moon's increment for a given time smaller than the sun's, and how is the difference made up?

Answer: The increments table for the moon is based on a rate of 14 degrees 19.0 minutes per hour, slower than the sun's 15 degrees. The moon's actual rate varies from hour to hour, so the excess over 14 degrees 19.0 minutes is given as v on the daily page and the v correction is added.

1. GHA Aries is 287 41.3 and SHA of Vega is 080 34.0. What is the GHA of Vega?

Answer: 287 41.3 + 080 34.0 = 368 15.3, minus 360 = 008 15.3.

1. You are in 41 15 N, DR longitude 017 50 E, and the sun's GHA is 338 22.4. Choose the AP and find LHA.

Answer: Assumed latitude 41 N. East longitude: minutes = 60 minus 22.4 = 37.6, so assumed longitude 017 37.6 E. LHA = 338 22.4 + 017 37.6 = 356 00.0 = 356.

1. The table gives Hc 38 12, d minus 44, and the declination minutes are 36. What is Hc?

Answer: Correction = minus 44 x 36/60 = minus 26.4, rounded minus 26. Hc = 38 12 minus 26 = 37 46.

1. In South latitude with LHA 040 the table gives Z 128. What is Zn, and does it make sense?

Answer: South latitude, LHA less than 180: Zn = 180 + 128 = 308 T. LHA less than 180 means the body is west of the meridian, and 308 is north-west, which is consistent.

1. When would you use NP401 rather than AP3270?

Answer: When the declination exceeds 29 degrees (outside AP3270 Vols 2 and 3) for a body not in Vol 1, or when greater precision (0.1 minute) is wanted. NP401 covers all declinations but is bulkier.

1. Your tables have been soaked and are unreadable. How do you reduce a sun sight?

Answer: With a calculator: sin Hc = sin Lat sin Dec + cos Lat cos Dec cos LHA, and cos Z = (sin Dec minus sin Lat sin Hc) / (cos Lat cos Hc), treating contrary-name declination as negative; or use the haversine formula hav ZD = hav (Lat minus Dec) + cos Lat cos Dec hav LHA, then Hc = 90 minus ZD.

1. A star sight is taken at 1915 zone time in zone +9 on 3 March. Which daily page do you use and which increments column?

Answer: UT is 1915 + 9 h = 0415 on 4 March, so the 4 March daily page; use the Aries column of the increments for GHA Aries, then add the star's SHA.

1. A planet has GHA 211 04.6 at the whole hour, v minus 1.1, and the sun and planets increment for 18m 40s is 4 40.0. What is the GHA, and what is the sign of the v correction?

Answer: v correction = minus 1.1 x 19/60 = minus 0.3. GHA = 211 04.6 + 4 40.0 minus 0.3 = 215 44.3.

1. In longitude 040 W, sunset is tabulated as 18 20 LMT. What is the UT of sunset, and which time must you use for a sight at that moment?

Answer: 040 W is 2h 40m behind Greenwich, so UT = 18 20 + 2 40 = 21 00 UT. The sight must be timed and the almanac entered in UT (corrected for chronometer error), not in zone or local time.