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Algorithm

Given:

• L a t {\displaystyle Lat} , the latitude (North - positive, South - negative), L o n {\displaystyle Lon} the longitude (East - positive, West - negative), both approximate (assumed);
• D e c {\displaystyle Dec} , the declination of the body observed;
• G H A {\displaystyle GHA} , the Greenwich hour angle of the body observed;
• L H A = G H A + L o n {\displaystyle LHA=GHA+Lon} , the local hour angle of the body observed.

First calculate the altitude of the celestial body H c {\displaystyle Hc} using the equation of circle of equal altitude:

sin ⁡ ( H c ) = sin ⁡ ( L a t ) ⋅ sin ⁡ ( D e c ) + cos ⁡ ( L a t ) ⋅ cos ⁡ ( D e c ) ⋅ cos ⁡ ( L H A ) . {\displaystyle \sin(Hc)=\sin(Lat)\cdot \sin(Dec)+\cos(Lat)\cdot \cos(Dec)\cdot \cos(LHA).}

The azimuth Z {\displaystyle Z} or Z n {\displaystyle Zn} (Zn=0 at North, measured eastward) is then calculated by:

cos ⁡ ( Z ) = sin ⁡ ( D e c ) − sin ⁡ ( H c ) ⋅ sin ⁡ ( L a t ) cos ⁡ ( H c ) ⋅ cos ⁡ ( L a t ) = sin ⁡ ( D e c ) cos ⁡ ( H c ) ⋅ cos ⁡ ( L a t ) − tan ⁡ ( H c ) ⋅ tan ⁡ ( L a t ) . {\displaystyle \cos(Z)={\frac {\sin(Dec)-\sin(Hc)\cdot \sin(Lat)}{\cos(Hc)\cdot \cos(Lat)}}={\frac {\sin(Dec)}{\cos(Hc)\cdot \cos(Lat)}}-\tan(Hc)\cdot \tan(Lat).}

These values are contrasted with the observed altitude H o {\displaystyle Ho} . H o {\displaystyle Ho} , Z {\displaystyle Z} , and H c {\displaystyle Hc} 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:

• The Nautical Almanac Sight Reduction (NASR, originally known as Concise Tables for Sight Reduction or Davies, 1984, 22pg)
• Pub. 249 (formerly H.O. 249, Sight Reduction Tables for Air Navigation, A.P. 3270 in the UK, 1947–53, 1+2 volumes)²
• Pub. 229 (formerly H.O. 229, Sight Reduction Tables for Marine Navigation, H.D. 605/NP 401 in the UK, 1970, 6 volumes.³
• The variant of HO-229: Sight Reduction Tables for Small Boat Navigation, known as Schlereth, 1983, 1 volume)
• H.O. 214 (Tables of Computed Altitude and Azimuth, H.D. 486 in the UK, 1936–46, 9 vol.)
• H.O. 211 (Dead Reckoning Altitude and Azimuth Table, known as Ageton, 1931, 36pg. And 2 variants of H.O. 211: Compact Sight Reduction Table, also known as Ageton–Bayless, 1980, 9+ pg. S-Table, also known as Pepperday, 1992, 9+ pg.)
• H.O. 208 (Navigation Tables for Mariners and Aviators, known as Dreisonstok, 1928, 113pg.)

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 1955⁴ and involved haversines. The altitude is derived from sin ⁡ ( H c ) = n − a ⋅ ( m + n ) {\displaystyle \sin(Hc)=n-a\cdot (m+n)} , in which n = cos ⁡ ( L a t − D e c ) {\displaystyle n=\cos(Lat-Dec)} , m = cos ⁡ ( L a t + D e c ) {\displaystyle m=\cos(Lat+Dec)} , a = hav ⁡ ( L H A ) {\displaystyle a=\operatorname {hav} (LHA)} .

The calculation is:

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

Ultra compact sight reduction

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 derived⁶ using haversines, hav ⁡ ( ) {\displaystyle \operatorname {hav} ()} , for all the terms of the equation: hav ⁡ ( Z D ) = hav ⁡ ( L a t − D e c ) + ( 1 − hav ⁡ ( L a t − D e c ) − hav ⁡ ( L a t + D e c ) ) ⋅ hav ⁡ ( L H A ) {\displaystyle \operatorname {hav} (ZD)=\operatorname {hav} (Lat-Dec)+\left(1-\operatorname {hav} (Lat-Dec)-\operatorname {hav} (Lat+Dec)\right)\cdot \operatorname {hav} (LHA)}

where Z D {\displaystyle ZD} is the zenith distance,

H c = ( 90 ∘ − Z D ) {\displaystyle Hc=(90^{\circ }-ZD)} 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(|Lat| − |Dec|) m = hav(|Lat| + |Dec|) if contrary name (one is North the other is South) n = hav(|Lat| + |Dec|) m = hav(|Lat| − |Dec|) 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 diagram⁷ was developed for a faster solution without calculation, and with an accuracy of 1°.

This diagram could be used also for star identification.⁸

An ambiguity in the value of azimuth may arise since in the diagram 0 ∘ ⩽ Z ⩽ 90 ∘ {\displaystyle 0^{\circ }\leqslant Z\leqslant 90^{\circ }} . Z {\displaystyle Z} 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 formula⁹ should be used:

hav ⁡ ( Z ) = hav ⁡ ( 90 ∘ ± | D e c | ) − hav ⁡ ( | L a t | − H c ) 1 − hav ⁡ ( | L a t | − H c ) − hav ⁡ ( | L a t | + H c ) {\displaystyle \operatorname {hav} (Z)={\frac {\operatorname {hav} (90^{\circ }\pm \vert Dec\vert )-\operatorname {hav} (\vert Lat\vert -Hc)}{1-\operatorname {hav} (\vert Lat\vert -Hc)-\operatorname {hav} (\vert Lat\vert +Hc)}}}

The algorithm if absolute values are used is:

if same name for latitude and declination (both are North or South) a = hav(90° − |Dec|) if contrary name (one is North the other is South) a = hav(90° + |Dec|) m = hav(|Lat| + Hc) n = hav(|Lat| − 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.¹⁰¹¹

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

• Navigation
• Celestial navigation
• Circle of equal altitude
• Intercept method

External links

• Navigational Algorithms: AstroNavigation - Free App for Windows
• Navigational Algorithms: resources for Longhand Haversine Sight Reduction
• NavList A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Position-Finding
• Celestial Tools for the USPS/CPS JN/N Student
• Graphical all-haversine Hc reduction Archived 2016-05-16 at the Wayback Machine
• Sight Reduction - free App for android
• Vector Solution for the intersection of two Circles of Equal Altitude - free App for android

References

1. The American Practical Navigator (2002) /wiki/The_American_Practical_Navigator ↩

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 https://www.thenauticalalmanac.com/Pub.%20249%20Vol.%201-2020-Dec.pdf ↩

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°; Pub. 229 Volume 3. Latitudes 30° to 45°; 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. https://msi.nga.mil/MSISiteContent/StaticFiles/NAV_PUBS/SRTM/Pub229/Vol_1/Pub229Vol1.pdf ↩

4. Table de point miniature (Hauteur et azimut), by R. Doniol, Navigation IFN Vol. III Nº 10, Avril 1955 Paper http://fer3.com/arc/m2.aspx/Table-De-Point-Miniature-R-Doniol-FrankReed-jul-2015-g32063 ↩

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. http://issuu.com/navigatorpublishing/docs/on227_download_edition ↩

6. Altitude haversine formula by Hanno Ix http://fer3.com/arc/m2.aspx/Longhand-Sight-Reduction-HannoIx-nov-2014-g29121 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 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 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 http://fer3.com/arc/m2.aspx/Longhand-Sight-Reduction-Bergman-nov-2014-g29441 ↩

10. "NavList: Re: Longhand Sight Reduction (129172)". http://fer3.com/arc/m2.aspx/Longhand-Sight-Reduction-HannoIx-nov-2014-g29172 ↩

11. Natural-Haversine 4-place Table; PDF; 51kB https://yadi.sk/i/4MmOYyXhUshbxA ↩