ABC TablesABC tables are very easy to use and more than adequate for the bearing of a celestial body.  These tables avoid the need to use a calculator or Log tables but are based on the previous formulae.The Rules may seem unwieldy at first but they are printed on each page and quickly become automatic.These transpose the Azimuth formula so that A = Tan(Lat) / Tan(LHA) B = Tan(Dec) / Sin(LHA) C = Difference A ~ B = 1/ [Tan(Azimuth)  x Cos(Lat) ]  ExampleLatitude           20° NDeclination       45° SLHA                30° A               0.63 S                    Opposite to Latitude unless LHA > 180°B              2.00 S                      Same as Declination              --------C             2.63 S                      Same name; Sum. Different names; DifferenceThe C Table gives a bearing of 22°.0. The sign of C means that this bearing is south. It is west because the LHA is less than 180°. The C result would normally be written as "S 22°.0 W" or 158°.The effect of rounding ABC Tables’ values is negligible (+/- 0°.1)  This is not true of the older Sight Reduction Tables where the calculated altitude is rounded to the nearest minute. Furthermore the need to use a plotting sheet with a rounded, estimated position provides considerable scope for inaccuracy. (Sight Reduction Tables were known as the Air Navigation Tables until 2003.)The author’s preferred manual method is a calculator for the Zenith Distance and ABC tables for Azimuths. Without a calculator he would still use the Cosine formula but with log tables. Obtaining a Position LineThe difference between the True (TZD) and Calculated (CZD) Zenith Distances is the Intercept.TRUE, TINY, TOWARDSIf the TZD is less than the CZD then the assumed position must be moved in the direction of the body by the amount of the Intercept. This gives a position of the correct distance from the body. It is known as the Intercept Terminal Position or ITP.As the radius of the circle is normally very large, it is considered to be a straight line near this point. A line at 90 to the direction of the body is the Position Line. Combining Position LinesA single Position Line must be combined with other observations for a fix. This can be achieved using a plotting sheet and then transferring the ITP by the distance to the next sight and redrawing the Transferred Position Line in the same direction as the original. For Sun sights, it is more usual to calculate the ITP of a morning sight and then calculate the transferred position for the Sun's Meridian Passage (Noon.) The difference between calculated and observed latitudes provides a longitude using “Plane Sailing.” With a little practice, this will be found to be a faster, not to mention more accurate method. For Star Sights, many people use a single position and then plot the Position Lines without allowing for the vessel's movement. This may appear a sloppy practice but a few miles error mid-ocean is usually irrelevant. Even if the position at sunset was perfect, there is no guarantee that the position an hour later is within a mile. Indeed even if the position agrees perfectly with a GPS position, there is no guarantee that an intervening military operation has not thrown the GPS position out let alone a fault in the equipment/ aerial. “I am about here,” is a far safer assumption than “My wheelhouse is/ was within 10m of this position.” Next SectionCorrections to a Sextant Altitude
 Nautical Almanac InformationGreenwich Hour Angle (GHA)This is the body’s angular distance west of the Greenwich meridian.The GHA for the Sun, Moon and Planets is tabulated in a Nautical Almanac for each hour. An increment is applied to allow for the minutes and seconds. The main values assume that the GHAs change by consistantly. This assumption is corrected using a “v” correction.
 To find the Increment, the Increments and Corrections pages are entered with the minutes and seconds of the sight. For a Sun sight taken at exactly 30 minutes past the hour, the increment value would be 7° 30’.0On the left hand side of the Increments page for the appropriate minutes, there is the correction to apply based on the tabulated “v” value for the body. The “v” corrections are considered to be linear but are actually tabulated for halfway through the minute.  This is why a “v” of 12.0 produces 6.1 for 30 minutes. The “v” adjustment is always positive for the Moon but can be negative for the planets.The “v” correction for the planets is found at the bottom of the daily pages. The rapid changes in the motion of the Moon mean that a “v” value is tabulated for each hour. No “v” correction is supplied for the Sun. Instead the GHA values for each hour in the tables are massaged.The GHA for stars are treated differently. Here the SHA (Sidereal Hour Angle) or angular distance west of Aries is tabulated for each three day period. The GHA of Aries is then tabulated for each hour and the Increment value for minutes and seconds is found in exactly the same way as for the Sun, Moon and planets. No “v” value is given for simplicity. The GHA for a star is the sum of the GHA for Aries and the SHA.Local Hour Angle (LHA)This is the angular distance west of the observer. Thus we find the GHA of a body and then adjust it for longitude. An easterly longitude is added to the GHA and a westerly longitude is subtracted.LHA = GHA +/- LongitudeDeclination (Dec)The Declination for most bodies is tabulated for each hour. Due to the very small movement of stars, it is only provided once for each three day period.The hourly change of Declination is usually small therefore the adjustment is found using the “d” value at the bottom of the page for the Sun and planets. (It is negligible for stars.) The Moon however moves rapidly which means that its’ “d” value is provided for each hour.The “d” correction is found in the same manner as for “v” by going to the appropriate Increments and Corrections page to obtain the appropriate correction from the right hand “v and d” section.The direction in which to apply the “d” correction (North or South) is determined by examining the next hourly value.  Simplifications Vs Accuracy in Nautical AlmanacsIn the explanation section at the back of the UK/ US Governments’ “The Nautical Almanac,” paragraph 24 provides details of the expected errors in the values of GHA and Declination. The maximum error in each is 0’.2 for the planets, 0’.25 for the Sun and 0’.3 for the Moon. These errors are caused entirely by the need to keep the presentation as simple as possible.It goes on to say; “In practice it may be expected that only one third of the values of GHA and Dec taken out will have errors larger than 0’.05 and less than one tenth will have errors larger than 0’.1.”The superficial attraction of using data from a source such as the Astronomical Almanac in the interests of accuracy is illusory. RA and Declination may be quoted in arc-seconds to several decimal places but the bodies, particularly the Moon, do not necessarily move in a linear manner during the day. Similarly the Hour Angle of Aries does not change linearly. Irregular motions are included in Nautical Almanac data. Example of Calculations for the Moon ~ 00:30:05 GMT, 19th January 2011GHA at 00:00                         10° 09’.4Increment for 30m 05s           + 7° 10’.7“v”   (6’.1 at 00:00)               +       3’.1GHA at 00:30:05                    17° 23’.2Longitude                            10° 00’.0 E         Add for East and subtract if WestLHA                                     27° 23’.2 Declination at 00:00             21° 08’.5 S“d”  (7’.6 at 00:00)                      3’.8 S           South by inspection of the change in the daily pageDeclination at 00:30:05       21° 04’.7 S Example of Calculations for Acamar ~ 00:30:05 GMT, 19th January 2011GHA of Aries at 00:00           118° 02’.7                     (Not visible above)Increment for 30m 05s         +  7° 32’.5SHA                                     315° 19’.5GHA of Acamar at 00:30:05      80° 54’.7                    (440° 51’ - 360°)Longitude                              10° 00’.0 WLHA                                      70° 54’.7The Declination value of 40° 15’.8 in the day’s pages for stars is not adjusted.
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