ABC Tables

ABC 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) ]

Example

Latitude           20° N

Declination       45° S

LHA                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; Difference

The 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 Line

The difference between the True (TZD) and Calculated (CZD) Zenith Distances is the Intercept.

TRUE, TINY, TOWARDS

If 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 Lines

A 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 Section

Corrections to a Sextant Altitude

Corrections to a Sextant Altitude

Index Error

This error can be found using the horizon. The sextant’s altitude is set to zero and then the two images of the horizon are aligned. The Index Error can then be read off.

If the sextant altitude reads high, the correction is subtractive and termed “On the Arc.” “Off the Arc” is the opposite.

After Index Error has been applied, the Sextant Altitude it is referred to as the Observed Altitude.

Dip/ Height of Eye

The True Horizon is at 90° to the Earth’s gravitational field. It coincides with the apparent horizon at sea level. However the Apparent Horizon starts to dip below the horizontal plane as the height of (the observer’s) eye increases.

Dip includes an allowance for Refraction below the horizontal plane.

The formulae are;

 Dip = 0.97 x Square Root (Ht of Eye in feet)

 Dip = 1.76 x Square Root (Ht of Eye in meters)

Dip is subtracted from the Observed Altitude to give Apparent Altitude.

Refraction

The deflection of light as it enters/ passes through the atmosphere is known as Refraction.

Refraction is stable and therefore predictable above about 15°, below that one needs to consider the characteristics of the atmospheric layers through which the light passes at that time. (Taking the altitude of bodies at less than 15° is usually avoided for this reason.)

For altitudes above 15°, a simplified formula is adequate (± 0’.02)

  Refraction = 0.96/ Tan (Altitude)

Refraction tables make assumptions on the layers for low altitudes and should be treated with caution. +/- 2 is not uncommon at an altitude of 2.

Refraction is subtracted from the Apparent Altitude to obtain the True Altitude.

Temperature and Pressure Correction for Refraction

The correction for Refraction assumes a temperature of 10° C and pressure of 1010mb. This may be modified for actual temperature and pressure. A temperature difference of 10° C will alter Refraction by 3% and a 10mb pressure difference will change Refraction by 1%. (0’05 and 0’.02 for an altitude of 30°)

The multiplier to correct for Temperature (°C) and Pressure (mb)

               = Pressure/ 1010 * 283/ (Temperature + 273)

Semi-Diameter

When measuring the altitudes of the Sun, Moon, Venus and Mars, it is usual to align either the top (Upper Limb) or bottom (Lower Limb) of the body with horizon. This offset must then be removed before comparison with the calculated value.

The angular diameter of a body depends on its distance from the Earth. Thus for the Sun the Semi-Diameter varies between 16’.3 in January, when the Sun is closest and 15’.7 in June when it is furthest away.

For a lower limb observation, the Semi-Diameter should be added to the True altitude.

Augmentation of the Moon’s Semi-Diameter

The Earth’s radius is about 1/ 60^th of the distance to the Moon. The reduction in distance compared to when on the horizon, has a measurable effect on its size. In contrast the Sun’s distance is 23,000 times the Earth’s radius and the effect is negligible.

 Augmentation = Sin (Altitude) x Horizontal Parallax

Horizontal Parallax is used in the formula as the lunar distance is not provided in a Nautical Almanac.

This correction is typically 0’.15 and should be added to the Moon’s Semi-Diameter before applying the Semi-Diameter to the True Altitude.

Parallax in Altitude

The Parallax correction allows for the difference in the altitude measured on the Earth’s surface versus the altitude that would be measured at the centre of the Earth.

The effect of parallax reduces with altitude. It is greatest when the body is at the horizon (Horizontal Parallax) and declines to zero when the body is overhead.

The effect is also proportional to the distance of the body. Thus the Horizontal Parallax for the Moon is about 1° but only 0’.15 for the Sun. This correction must be included for the Moon but is usually ignored for the Sun. It can be significant for Venus and Mars, depending on their distance, but is always insignificant for Jupiter and Saturn. (< 0’.05)

Sin (Horizontal Parallax) = Earth’s Radius/ Distance of Body

After correcting for altitude, the correction is known as Parallax in Altitude.

Parallax in Altitude = Horizontal Parallax x Cos (Altitude)

Parallax in Altitude should be added to the True Altitude.

Reduction of the Moon’s Horizontal Parallax

Horizontal Parallax is proportional to the Earth’s radius. Therefore as the Earth’s radius declines with latitude, so does Horizontal Parallax.

Correction = Horizontal Parallax * [Sin (Lat) ^ 2] / 298.3

This should be subtracted from Horizontal Parallax before calculating Parallax in Altitude.

Examples of Corrections to a Sextant Observation

                                                           Add or Subtract

For a Star

Sextant Altitude            31° 22’.0

Index Error                           2’.0                       Depends on the error

Observed Altitude         31° 24’.0

Dip                                   -3’.0                       Subtract

Apparent Altitude        31° 21’.0

Refraction                        - 1’.6                        Subtract

True Altitude              31° 19’.4

                                90° 00’.0

True Zenith Distance   58° 40’.6

For the Moon

Sextant Altitude           31° 22’.0

Index Error                          2’.0                    Depends on the error

Observed Altitude         31° 24’.0

Dip                                   -3’.0                     Subtract

Apparent Altitude         31° 21’.0

Refraction                        - 1’.6                      Subtract

True Altitude                31° 19’.4

Semi-Diameter                    16’.5                     Add for Lower Limb

Parallax (in Altitude)            51’.1                     Add

True Altitude                 32° 26’.8

                                   90° 00’.0

True Zenith Distance       57° 33’.2

Moon’s Additional Corrections

Tabulated Horizontal Parallax       59’.9

Latitude Correction                     - 0’.1             e.g. 52° N

Horizontal Parallax                    59’.8

Tabulated Semi-Diameter            16’.3

Augmentation of Semi-Diameter + 0’.15

Moon’s Semi-Diameter                 16’.5

The Horizontal Parallax must be included but the Latitude correction is often ignored. Similarly the Semi-Dia

Next Section NAUTICAL ALMANAC DATA

meter must be included but Augmentation is often ignored.