Sight Reduction Tables for Celestial Navigation
Sight Reduction is the process of solving the navigational triangle
for an assumed position and measured Altitude to obtain a Line-of-Position.
The Sight Reduction Tables available here are very similar to the tables issued
by the (former) "Defence Mapping Agency of the Hydrographic/Topographic Center"
as publication "H.O. 229".
Alternatively, the navigational triangle can be solved according to the
"Ageton Method".
The difference between the "H.O. 229" Tables and the Tables presented here,
is that the range for the Declination is limited to 0° to 29°.
This is compliant with the Nautical Almanac available from this site,
which does not include star data.
The Declination of the Sun, the Moon and the planets is always
below 30°.
Omitting the Declination range above 30° reduces the number of
pages (and the weight!) of the Sight Reduction Tables by 60%!
Purpose and Scope
The main purpose of these tables is to facilitate the practice of
celestial navigation at sea.
The tables have been designed for use with the "Marcq Saint Hilaire"
or "Altitude-Intercept" method of Sight Reduction, utilizing a position
assumed or chosen such that interpolation for Latitude and Local Hour Angle
is not required.
For each of the entry combination of integral degrees of Latitude,
Declination and Local Hour Angle, the corresponding
values for Altitude and Azimuth are tabulated.
The Altitudes and their increments are tabulated to the nearest
tenth of a minute.
The results for the Azimuth Angles are tabulated to the nearest
tenth of a degree.
The tables are designed for a linear interpolation of the Altitude for
Declination only.
This interpolation can be performed by means of the interpolation
tables also used for interpolating the Nautical Almanac data.
Arrangement
Each page of the table contains the Altitude (H) and Azimuth Angle (Z)
values for the following ranges of arguments:
- Latitude: one (integral) value
- Local-Hour_Angle: 10 successive (integral) values
- Declination: from 0° to 29° in integral values
(Declination same as OR contrary to Latitude: NN/SS or NS/SN )
Also the increment or decrement of the Altitude for the Declination
interval of 1° is tabulated (dH). This value can be used for
linear interpolation to obtain the correct Altitude value (Hc) for
the fractional (Degree/Minute) Declination value.
Tables
Interactive Tables
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With the links on the left, a specific page of the Sight-Reduction Tables can
be generated. The required entries are the integral values of Latitude and
Local Hour Angle (LHA) as well as whether Declination and Latitude have the
same or contrary signs. The range of Declination is from 0° to 29°,
which is valid for the Sun, Moon and Planets as well as for most Stars
that can be used for celestial navigation.
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Pre-compiled Tables
The pre-compiled Sight Reduction Tables are available in PDF format.
The complete Tables for 0° to 79° of Latitude (about 1000 pages)
are divided over 16 Volumes, which can be selected from the table below.
Each Volume has a Latitude range of 10°.
A worksheet for this table-based method of Sight Reduction is available as printer ready pdf file:
worksheet for Sight Reduction using compiled Tables.
Background Information
The Sight Reduction Tables as produced for the above scripts and files contain
tabulated values for the solutions of a spherical triangle of which two sides
(90°-Latitude and 90°-Declination) and the including angle
(Local Hour Angle) are known and the values of the third side (Altitude Hc)
and adjacent angle (Azimuth Angle Zc) are required.
The solution can be analytically described by the following Sight Reduction equations:
Altitude Hc is calculated by:
Hc = asin[ sin(Lat) * sin(Dec) + cos(Lat) * cos(Dec) * cos(LHA) ] |
And the Azimuth Angle Z is:
Z = atan[ sin(LHA) / ( sin(Lat) * cos(LHA) - tan(Dec) * cos(Lat) ) ] |
The arguments for these formulas are Latitude (Lat), Declination (Dec)
and Local Hour Angle (LHA).
The values of the interactive tables are calculated with a JavaScript script using
the build-in "Math"-trigonometric functions.
I have compared at random about 100 sets of values from different argument combinations
with the corresponding values from the "H.O. 229" publications and found no significant
deviations other than "rounding errors".
The values from the "H.O. 229" were calculated (anno 1981) with 9 significant
figures to obtain an accuracy of 0.1' for the Altitude and 0.1° for the
Azimuth Angle.
To my knowledge, the "Number" class and the "Math" class the JavaScript
interpreter allow a number representation and processing, which is
considerable better than the ones used for the "H.O. 229" publications.
The "rounding errors" mentioned above are ±0.1' for the Altitude and
±0.1° for the Azimuth Angle values.
The pre-compiled Sight Reduction Tables from the section above, have been generated with a C-program
using double-precision number representation and processing for all intermediate variables.
The accuracy of these calculations is much higher that the accuracy of the calculations
used for the "H.O. 229" publications.
Since for both calculations, the results are rounded to 0.1' for the Altitude and 0.1°
for the Azimuth, differences in the results of the two methods of calculation,
are due to "rounding effects".
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