Sight reduction

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In astronavigation, sight reduction is the process of deriving from a sight (in celestial navigation usually obtained using a sextant) the information needed for establishing a line of position, generally by intercept method.

Contents

Sight is defined as the observation of the altitude, and sometimes also the azimuth, of a celestial body for a line of position; or the data obtained by such observation. [1]

The mathematical basis of sight reduction is the circle of equal altitude. The calculation can be done by computer, or by hand via tabular methods and longhand methods.

Algorithm

Steps for measuring and correcting Ho using a sextant. Corrections for Sextant Altitude.en.jpg
Steps for measuring and correcting Ho using a sextant.
Using Ho, Z, Hc in intercept method. MarcqSaintHilaire.en.jpg
Using Ho, Z, Hc in intercept method.

Given:

First calculate the altitude of the celestial body using the equation of circle of equal altitude:

The azimuth or (Zn=0 at North, measured eastward) is then calculated by:

These values are contrasted with the observed altitude . , , and are the three inputs to the intercept method (Marcq St Hilaire method), which uses the difference in observed and calculated altitudes to ascertain one's relative location to the assumed point.

Tabular sight reduction

The methods included are:

Longhand haversine sight reduction

This method is a practical procedure to reduce celestial sights with the needed accuracy, without using electronic tools such as calculator or a computer. And it could serve as a backup in case of malfunction of the positioning system aboard.

Doniol

The first approach of a compact and concise method was published by R. Doniol in 1955 [4] and involved haversines. The altitude is derived from , in which , , .

The calculation is:

n = cos(LatDec) m = cos(Lat + Dec) a = hav(LHA) Hc = arcsin(na ⋅ (m + n))

Ultra compact sight reduction

Haversine Sight Reduction algorithm Haversine Sight Reduction.jpg
Haversine Sight Reduction algorithm

A practical and friendly method using only haversines was developed between 2014 and 2015, [5] and published in NavList.

A compact expression for the altitude was derived [6] using haversines, , for all the terms of the equation:

where is the zenith distance,

is the calculated altitude.

The algorithm if absolute values are used is:

if same name for latitude and declination (both are North or South)   ''n'' = hav(|<span class="nowrap" style="padding-left:0.1em; padding-right:0.1em;">Lat</span>| − |<span class="nowrap" style="padding-left:0.1em; padding-right:0.1em;">Dec</span>|)   ''m'' = hav(|<span class="nowrap" style="padding-left:0.1em; padding-right:0.1em;">Lat</span>| + |<span class="nowrap" style="padding-left:0.1em; padding-right:0.1em;">Dec</span>|) if contrary name (one is North the other is South)   ''n'' = hav(|<span class="nowrap" style="padding-left:0.1em; padding-right:0.1em;">Lat</span>| + |<span class="nowrap" style="padding-left:0.1em; padding-right:0.1em;">Dec</span>|)   ''m'' = hav(|<span class="nowrap" style="padding-left:0.1em; padding-right:0.1em;">Lat</span>| − |<span class="nowrap" style="padding-left:0.1em; padding-right:0.1em;">Dec</span>|) ''q'' = ''n'' + ''m'' ''a'' = hav(''LHA'') hav(''ZD'') = ''n'' + ''a'' · (1 − ''q'') ''ZD'' = archav() -> inverse look-up at the haversine tables ''Hc'' = 90° − ''ZD''

For the azimuth a diagram [7] was developed for a faster solution without calculation, and with an accuracy of 1°.

Azimuth diagram by Hanno Ix Azimuth diagram by Hanno Ix.jpg
Azimuth diagram by Hanno Ix

This diagram could be used also for star identification. [8]

An ambiguity in the value of azimuth may arise since in the diagram . is E↔W as the name of the meridian angle, but the N↕S name is not determined. In most situations azimuth ambiguities are resolved simply by observation.

When there are reasons for doubt or for the purpose of checking the following formula [9] should be used:

The algorithm if absolute values are used is:

if same name for latitude and declination (both are North or South)   ''a'' = hav(90° − |<span class="nowrap" style="padding-left:0.1em; padding-right:0.1em;">Dec</span>|) if contrary name (one is North the other is South)   ''a'' = hav(90° + |<span class="nowrap" style="padding-left:0.1em; padding-right:0.1em;">Dec</span>|) ''m'' = hav(|<span class="nowrap" style="padding-left:0.1em; padding-right:0.1em;">Lat</span>| + ''Hc'') ''n'' = hav(|<span class="nowrap" style="padding-left:0.1em; padding-right:0.1em;">Lat</span>| − ''Hc'') ''q'' = ''n'' + ''m'' hav(''Z'') = (''a'' − ''n'') / (1 − ''q'') ''Z'' = archav() -> inverse look-up at the haversine tables if Latitude ''N'':   if ''LHA'' > 180°, ''Zn'' = ''Z''   if ''LHA'' < 180°, ''Zn'' = 360° − ''Z'' if Latitude ''S'':   if ''LHA'' > 180°, ''Zn'' = 180° − ''Z''   if ''LHA'' < 180°, ''Zn'' = 180° + ''Z''

This computation of the altitude and the azimuth needs a haversine table. For a precision of 1 minute of arc, a four figure table is enough. [10] [11]

An example

Data:   ''Lat'' = 34° 10.0′ N (+)   ''Dec'' = 21° 11.0′ S (−)   ''LHA'' = 57° 17.0′ Altitude ''Hc'':   ''a'' = 0.2298   ''m'' = 0.0128   ''n'' = 0.2157   hav(''ZD'') = 0.3930   ''ZD'' = archav(0.3930) = 77° 39′   ''Hc'' = 90° - 77° 39′ = 12° 21′ Azimuth ''Zn'':   ''a'' = 0.6807   ''m'' = 0.1560   ''n'' = 0.0358   hav(''Z'') = 0.7979   ''Z'' = archav(0.7979) = 126.6°   Because ''LHA'' < 180° and Latitude is ''North'': ''Zn'' = 360° - ''Z'' = 233.4°

See also

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References

  1. The American Practical Navigator (2002)
  2. Pub. 249 Volume 1. Stars Archived 2020-11-12 at the Wayback Machine ; Pub. 249 Volume 2. Latitudes 0° to 39° Archived 2022-01-22 at the Wayback Machine ; Pub. 249 Volume 3. Latitudes 40° to 89° Archived 2019-07-13 at the Wayback Machine
  3. Pub. 229 Volume 1. Latitudes 0° to 15° Archived 2017-01-26 at the Wayback Machine ; Pub. 229 Volume 2. Latitudes 15° to 30° [ permanent dead link ]; Pub. 229 Volume 3. Latitudes 30° to 45° [ permanent dead link ]; Pub. 229 Volume 4. Latitudes 45° to 60° Archived 2017-01-30 at the Wayback Machine ; Pub. 229 Volume 5. Latitudes 60° to 75° Archived 2017-01-26 at the Wayback Machine ; Pub. 229 Volume 6. Latitudes 75° to 90° Archived 2017-02-11 at the Wayback Machine .
  4. Table de point miniature (Hauteur et azimut), by R. Doniol, Navigation IFN Vol. III Nº 10, Avril 1955 Paper
  5. Rudzinski, Greg (July 2015). "Ultra compact sight reduction". Ocean Navigator (227). Ix, Hanno. Portland, ME, USA: Navigator Publishing LLC: 42–43. ISSN   0886-0149 . Retrieved 2015-11-07.
  6. Altitude haversine formula by Hanno Ix http://fer3.com/arc/m2.aspx/Longhand-Sight-Reduction-HannoIx-nov-2014-g29121
  7. Azimuth diagram by Hanno Ix. http://fer3.com/arc/m2.aspx/Gregs-article-havDoniol-Ocean-Navigator-HannoIx-jun-2015-g31689
  8. Hc by Azimuth Diagram http://fer3.com/arc/m2.aspx/Hc-Azimuth-Diagram-finally-HannoIx-aug-2013-g24772
  9. Azimuth haversine formula by Lars Bergman http://fer3.com/arc/m2.aspx/Longhand-Sight-Reduction-Bergman-nov-2014-g29441
  10. "NavList: Re: Longhand Sight Reduction (129172)".
  11. Natural-Haversine 4-place Table; PDF; 51kB