Celestial with the Nautical Almanac and Pub. 249

Prerequisites

• The ASA 105 Coastal Navigation skills: plotting, dead reckoning, variation and deviation.
• The ideas behind a sight (geographical position, the intercept method, plotting sheets), covered in Celestial Navigation.
• Correcting Hs to Ho, covered in Sextant Corrections.
• Taking GHA and declination from the almanac, covered in Nautical Almanac and Sight Reduction Tables.

Learning Objectives

This lesson fills the gaps the other celestial lessons leave in the ASA 107 Celestial Navigation standard, using the American books: the Nautical Almanac and NGA's sight reduction tables. By the end you will be able to:

• Convert between longitude and time, and between zone time and UT.
• Find chronometer error on any day from a previous error and the daily rate.
• Choose between Pub. 249 and Pub. 229.
• Find latitude from Polaris, and give sunrise, sunset and twilight in ship's time and UT.
• Check the compass by amplitude or azimuth, and work a sun-run-sun running fix.

Time and Longitude

The earth turns 360° in 24 hours, so time and longitude are two ways of saying the same thing:

Longitude to time. Example: 071° 37' W.

• 71 x 4 minutes = 284 minutes = 4 h 44 m.
• 37 x 4 seconds = 148 seconds = 2 m 28 s.
• Total: 4 h 46 m 28 s. Your local noon comes about that long after noon at Greenwich (before it, in East longitude).

Time to longitude runs the other way: 1 h 10 m is 15° + 2° 30' = 17° 30'.

One second of time is a quarter of a minute of longitude. That is the link between your watch and your position.

Zone Time, Standard Time and UT

The almanac is tabulated in UT (Universal Time, the same as GMT for our purposes). The ship's clock keeps zone time (ZT).

• Each nautical zone is 15° wide, centred on a meridian that is a multiple of 15°.
• The zone description (ZD) is what you add to ZT to get UT: plus in West longitude, minus in East.
• To find the ZD, divide the longitude by 15 and round. At 071° 37' W, 71.6 / 15 = 4.8, so ZD = +5. Zone +5 runs from 067° 30' W to 082° 30' W.
• Zone 12 is split at the 180th meridian: +12 on the West side, minus 12 on the East side.

Standard time ashore follows legal boundaries, often with daylight saving. US Eastern Standard Time is UT minus 5 (ZD +5); Eastern Daylight Time is UT minus 4 (ZD +4). At sea, set the ship's clocks to the nautical zone and log the ZD.

The date is the trap. An evening sight in West longitude is often the next day in UT. Always write down the UT date before opening the almanac.

Chronometer Error and Daily Rate

Your "chronometer" is probably a good quartz watch. It need not be perfect; you need to know how wrong it is.

• Chronometer error (CE): how far the watch is fast or slow of UT.
• Daily rate: how much the error changes each day.

Check the watch against a reliable source (GPS showing UTC, or the NIST time stations WWV and WWVH) at the same time each day and log it. If the source fails, the rate lets you predict the error.

Finding the rate:

• 1 October, 0000 UT: watch 6 seconds slow.
• 11 October, 0000 UT: watch 9 seconds fast.
• From 6 slow to 9 fast is 15 seconds gained in 10 days: gaining 1.5 seconds per day.

Predicting the error. By 13 October it gains another 2 x 1.5 = 3 seconds: CE = 12 seconds fast.

Applying it. The watch reads 14 h 22 m 40 s at a sight. Fast, so subtract: UT = 14 h 22 m 28 s.

Left uncorrected, 12 seconds is 12 x 15 = 180 seconds of arc: 3 minutes of longitude. With a 12-hour watch, also be sure whether it is 0222 or 1422 UT.

Pub. 249 or Pub. 229?

US navigators solve the navigational triangle with one of two sets of tables published by the National Geospatial-Intelligence Agency (NGA). Both are US government works, also reprinted commercially. How to enter and interpolate them is in Nautical Almanac and Sight Reduction Tables; here is how to choose.

• Sun, planets and most moon sights on a yacht: Pub. 249 Vols 2 and 3. A sight in a seaway is rarely better than a minute of arc, so extra precision is wasted.
• Twilight star fixes: Pub. 249 Vol 1, which lists seven well-placed stars with Hc and Zn ready to use: a ready-made star plan.
• Declination over 29° (a star not in Vol 1, or the moon at times): Pub. 229 or a calculator.
• Full working to a tenth of a minute: Pub. 229.

You only need the volumes for the latitudes you will sail.

Latitude by Polaris

Polaris sits less than a degree from the north celestial pole, so its corrected altitude is almost your latitude. ASA 107 asks for it at twilight, when star and horizon are both visible. The rule, explained in Celestial Navigation:

Latitude = Ho minus 1° + a0 + a1 + a2, with a0, a1 and a2 from the Pole Star tables (entered with LHA Aries, latitude and month).

Example (example values, not from a real almanac): Hs 40° 18.6', index error 2.0' on the arc, height of eye 9 feet.

1. Index error on the arc is subtracted: 40° 16.6'.
2. Dip: 0.97 x the square root of 9 = 2.9', subtracted. Ha = 40° 13.7'.
3. Star refraction near 40°: minus 1.1'. Ho = 40° 12.6'.
4. Example table values: a0 = 1° 05.2', a1 = 0.5', a2 = 0.4'.
5. Latitude = 40° 12.6' minus 1° 00.0' + 1° 05.2' + 0.5' + 0.4' = 40° 18.7' N.

A Polaris line crosses an east or west star line almost at right angles, a strong partner in a twilight fix.

Sunrise, Sunset and Twilight in Ship's Time

The almanac's daily pages give sunrise, sunset and civil and nautical twilight in local mean time (LMT) for a list of latitudes. The US Naval Observatory defines sunrise and sunset by the sun's upper limb on the horizon, civil twilight as the sun's centre 6° below the horizon and nautical twilight as 12° below. Star sights belong between the end of civil and the end of nautical twilight, when stars show and the horizon is still sharp.

ASA 107 wants the times in ship's time and UT:

1. Interpolate the LMT for your DR latitude.
2. LMT to UT: add West longitude in time, subtract East.
3. UT to ZT: subtract the ZD (a minus ZD is added).

Shortcut: east of your zone's central meridian, events happen earlier than LMT by the difference in time; west of it, later.

Example (example table values): evening, DR 38° N, 068° 45' W, ZD +5.

• Longitude in time: 68° = 4 h 32 m, 45' = 3 m. Total 4 h 35 m.
• Sunset: 1721 LMT + 4 h 35 m = 2156 UT, minus 5 h = 1656 ZT.
• Shortcut check: 068° 45' W is 6° 15' east of the 075° W zone meridian, worth 25 minutes: 1721 minus 25 = 1656 ZT.
• Civil twilight ends 1722 ZT; nautical twilight ends 1754 ZT.

So be on deck with the sextant from about 1720 to 1755 ship's time. Use the DR longitude for the time of the event, not the noon position.

Compass Check by Amplitude and Azimuth

ASA 107 asks for the true bearing of a low body to find compass error and deviation. A low body is easy to sight and its bearing changes slowly. The theory is in Celestial Navigation.

Amplitude (body on the horizon): sin amplitude = sin declination / cos latitude.

Example: sunset, latitude 38° N, declination S 8° (example value).

• sin 8° = 0.1392; cos 38° = 0.7880; 0.1392 / 0.7880 = 0.1766.
• Amplitude = 10.2°, named West (setting) and South (as the declination): W 10.2° S.
• True bearing = 270 minus 10.2 = 259.8, call it 260° T.
• The steering compass reads 272°. Compass higher than true: total error 12° W ("error West, compass best").
• Variation 15° W, so deviation = 12° W minus 15° W = 3° E on this heading.

The formula assumes the sun's centre is on the celestial horizon, which happens while the sun is still a little above the sea horizon, so take the bearing then.

Azimuth (any low body): reduce it with Pub. 249 or Pub. 229 for the time of the bearing, convert Z to Zn and compare with the compass. It works for a low star or planet too, and needs only time and DR, no sextant altitude.

Worked Example

Sun-run-sun running fix. Course 075° T, 6.0 knots, ZD +5. All almanac and table figures are example values, worked to be consistent with each other.

Morning sight, 0930 ZT (1430 UT). DR 37° 57' N, 069° 00' W.

• Example GHA sun 39° 04', declination S 7° 58'.
• AP 38° 00' N, 069° 04' W, so LHA = 39° 04' + 360° minus 69° 04' = 330°.
• Pub. 249: Hc 36° 12', Zn 142°. Ho 36° 18'.
• Ho is greater: intercept 6 miles toward 142°.

Afternoon sight, 1400 ZT (1900 UT). DR 38° 04' N, 068° 27' W.

• Example GHA sun 106° 34' (67° 30' more: 4.5 hours at 15° an hour), declination S 8° 02'.
• AP 38° 00' N, 068° 34' W, so LHA = 106° 34' minus 68° 34' = 38°.
• Pub. 249: Hc 31° 56', Zn 226°. Ho 31° 52'.
• Ho is less: intercept 4 miles away from 226° (toward 046°).

The run. 0930 to 1400 is 4 h 30 m at 6.0 knots: 27.0 miles on 075° T, which moves the DR 7.0' north and about 33' east (the 1400 DR above).

Plotting:

1. From AP 1, lay off 6 miles along 142° and draw LOP 1 at right angles.
2. Advance LOP 1 by moving a point on it 27.0 miles along 075° and drawing a parallel line through it. Label it with both times.
3. From AP 2, lay off 4 miles along 046° and draw LOP 2.
4. Angle of cut: 226 minus 142 = 84°, close to the ideal 90°.

Result. The lines cross at about 38° 01' N, 068° 28' W, about 3 miles south-southwest of the 1400 DR. Log it as a running fix: any error in the run (leeway, current, log error) moves it. Had the watch been 12 seconds fast and uncorrected, both lines would shift 3' of longitude, about 2.4 miles at this latitude.

Common Mistakes

• Wrong UT date. Evening sights in West longitude often fall on the next UT day.
• ZD sign reversed. West is plus: ZT + ZD = UT.
• Chronometer error the wrong way. Fast watch: subtract. Slow: add. And remember the days of rate since the last check.
• LMT treated as zone time. Almanac sunrise and twilight are LMT; correct for longitude.
• Pub. 249 Vols 2 and 3 for declination over 29°. Use Vol 1, Pub. 229 or a calculator.
• Amplitude named wrongly. Measure from East (rising) or West (setting) toward the declination's name.

Summary

• 15° of longitude is 1 hour; 1' of arc is 4 seconds; 1 second of time is 0.25' of longitude.
• ZD = longitude / 15, rounded; plus in West; ZT + ZD = UT. Watch the date.
• Find the daily rate from two time checks, then predict the error for any day.
• Pub. 249 is quick and suits yachts; Pub. 229 covers all declinations to 0.1'.
• Polaris latitude = Ho minus 1° + a0 + a1 + a2.
• Almanac sunrise and twilight are LMT: longitude gives UT, the ZD gives ship's time.
• Check the compass on a low body by amplitude or azimuth.
• Sun-run-sun needs an accurate run and a cut of at least 30°, ideally near 90°.

Check Your Understanding

1. Convert 052° 15' W into time. 52 x 4 = 208 minutes = 3 h 28 m; 15' = 1 m. Total 3 h 29 m.
2. Your DR is 128° E. What is the ZD, and the UT when the ship's clock shows 0500 on 9 June? 128 / 15 = 8.53, rounded to 9: ZD minus 9. UT = 2000 on 8 June.
3. On 2 May the watch was 4 seconds fast; on 6 May, 2 seconds fast. What is the rate, and the error on 10 May? Losing 0.5 seconds a day. On 10 May it will be 2 minus 2 = 0, correct.
4. A star with declination N 38° is not in Pub. 249 Vol 1. What do you reduce it with? Pub. 229 or a calculator. Pub. 249 Vols 2 and 3 stop at 29°.
5. The almanac gives sunrise 0612 LMT for your latitude. You are at 080° W, ZD +5. What is sunrise in ship's time? 080° W is 5° west of the 075° W zone meridian, worth 20 minutes later: 0632 ZT (1132 UT).

Sources

• https://aa.usno.navy.mil/faq/RST_defs
• https://msi.nga.mil/Publications/SRTM
• https://msi.nga.mil/Publications/APN
• https://www.nist.gov/pml/time-and-frequency-division/time-distribution/radio-station-wwv