Sight Calculations

Sight Calculations and obtaining a Position

The stages to resolving a sight are;

- Correct the Sextant Altitude to find the true distance of the body

- Calculate the bearing and distance from an assumed position

- Use the difference in distances to obtain a Position Line

Finally Position Lines are combined to provide a fix.
 


Correcting a Sextant Altitude

An explanation of the corrections is found in the next section under “Corrections to a Sextant Altitude.” All of these, except Index Error, are found in Nautical Tables.

Example for the Sun

Sextant Altitude             31 22’.0
Index Error                             2’.0                   Assuming "Off the Arc"
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
True Altitude                 32 26’.8
                                  90 00’.0        
True Zenith Distance      57 33’.2

Altitudes of Stars do not need a Semi-Diameter correction while the Moon needs more corrections. See examples at the end of the next section.



Calculating the Bearing and Distance

Positions for the observer and position lines can be plotted on a chart or calculated. The section on  “Sailings” deals with the mathematical calculations.

The other terms in the following formulae are derived from a Nautical Almanac. (See Nautical Almanac Information.)

The formulae for calculating the distance of the body and its altitude are

Cos(Zenith Distance) = Sin(Lat) x Sin(Dec) + Cos(Lat) x Cos(Dec) x Cos(LHA)
and
Tan(Azimuth) = Sin(LHA)/ (Cos(Lat) x Tan(Dec) – Sin(Lat) x Cos(LHA))
 

North is positive, South negative and the LHA is negative between 090 and 270.


These formulae can be used without further knowledge however the section on “Celestial Navigation Calculations” provides an introduction to spherical trigonometry.
 

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