Intercept method: Difference between revisions

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TheIn "Intercept[[astronomical Method"navigation]], orthe "Marcq St Hilaire'''intercept method"''', also known as it'''Marcq isSt. alsoHilaire knownmethod''', is an astronomical [[navigation]]a method of calculating an observer's position on earthEarth ([[geopositioning]]). It was originally called the ''azimuth intercept'' method because the process involves drawing a line which intercepts the [[azimuth]] line. This name was shortened to ''intercept'' method and the ''intercept distance'' was shortened to "'intercept"'.

The method yields a [[line of position]] (LOP) on which the observer is situated. The intersection of two or more such lines will define the observer's position, called a "fix". Sights may be taken at short intervals, usually during hours of twilight, or they may be taken at an interval of an hour or more (as in observing the Sun during the day). In either case, the lines of position, if taken at different times, must be advanced or retired to correct for the movement of the ship during the interval between observations. If observations are taken at short intervals, a few minutes at most, the corrected lines of position by convention yield a "fix". If the lines of position must be advanced or retired by an hour or more, convention dictates that the result is referred to as a "running fix".

== Summary ==

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==Methodology==

[[Image:Intercept Sight Reduction-00.png|thumb|304px|right|Diagram illustrating the intercept sight reduction process]]

Suitable bodies for celestial sights are selected, often using a Rude Star Finder. Using a [[sextant]], an altitude is obtained of the sunSun, the moonMoon, a star or a planet. The name of the body and the precise time of the sight in [[UTC]] is recorded. Then the sextant is read and the altitude (''Hs'') of the body is recorded. Once all sights are taken and recorded, the navigator is ready to start the process of [[sight reduction]] and plotting.

The first step in sight reduction is to correct the sextant altitude for various errors and corrections. The instrument may have an error, IC or index correction (Seesee article on [[Sextant adjustment|adjusting a sextant]]). Refraction by the atmosphere is corrected for with the aid of a table or calculation and the observer's height of eye above sea level results in a "dip" correction, (as the observer's eye is raised the horizon dips below the horizontal). If the Sun or Moon was observed, a semidiameter correction is also applied to find the centre of the object. The resulting value is "observed altitude" (''Ho'').

Next, using an accurate clock, the observed celestial object's geographic position (''GP'') is looked up in an almanac. That's the point on the Earth's surface directly below it (where the object is in the [[zenith]]). The latitude of the geographic position is called declination, and the longitude is usually called the [[hour angle]].

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: <math>\sin(Hc) = \sin(lat) \cdot \sin(dec) + \cos(lat) \cdot \cos(dec) \cdot \cos(LHA) </math>

: <math>\begin{align}
\tan(Zn) = \tan(Zn \pm 180) &= \frac{\sin(LHA)}{\sin(lat) \cdot \cos(LHA) - \cos(lat) \cdot \tan(dec)}</math> \triangleq tanZ \\

Z &= arctan(tanZ) \text{ } \in [-90,+90] \\

Zn &= \begin{cases}

Z & \text{if }LHA \in [0,90] \\

Z+180 & \text{if }LHA \in [90,270]\\

Z+360 & \text{if }LHA \in [270,360] && \equiv \text{ mod 360}\\

\end{cases} \\

\end{align}</math>

:The adjustment from Z to Zn (which is in <math>[0,360]</math>, and measured from North) has two reasons:

::(1)The angles in [0,360] with the same <math>\tan</math> is not unique (since <math>\tan(X) = tan (X \pm 180)</math>), but <math>\arctan</math> is defined only in <math>[-90,90]</math>.

::(2)The negative angle must be adjusted to positive angle.

or, alternatively,

: <math>\begin{align}
\cos(\pm Zn) &= \frac{\sin(dec) - \sin(lat) \cdot \sin(Hc)}{\cos(lat) \cdot \cos(Hc)}</math> \triangleq cosZ \\

Z & = \arccos(cosZ) \text { } \in [0,180] \\

Zn &= \begin{cases}

+Z & \text{if }LHA \in [180,360] \\

-Z+360 & \text{if }LHA \in [0,180] \\

\end{cases} \\

\end{align}</math>

:The adjustment for disambiguating <math>\cos</math> values has similar reasons.

Where

:''Hc'' = Computed altitude

:''Zn'' = Computed azimuth (Zn=0 at North)

:''Z'' = preliminary result for Zn (in some nautical almanacs)<ref name="Training_Movie_CelNav">{{cite web

| title = Celestial Navigation

| website = youtube.com

| date = 11 January 2015

| url = https://www.youtube.com/watch?v=fn9xMkNUMmY&t=1592s

| at = about 26m32s (1h33m31s)

| language=en

| access-date=July 1, 2022

}}</ref>

:''lat'' = Latitude

:''dec'' = Declination

:''LHA'' = Local Hour Angle

Since recently theseThese computations can be done easily done using electronic calculators or computers but traditionally there were methods which used logarithm or haversine tables. Some of these methods were H.O. 211 (Ageton), Davies, [[haversine]], etc. The relevant [[haversine formula]] for ''Hc'' is

: <math>\operatorname{hav}(\overline{Hc}) = \operatorname{hav}(LHA) \cdot cos(lat) \cdot cos(dec) + \operatorname{hav}(lat \pm dec) </math>

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When using such tables or a computer or scientific calculator, the navigation triangle is solved directly, so any assumed position can be used. Often the dead reckoning DR position is used. This simplifies plotting and also reduces any slight error caused by plotting a segment of a circle as a straight line.

With the use of astroastral navigation for air navigation, faster methods needed to be developed and tables of precomputed triangles were developed. When using precomputed sight reduction tables, selection of the assumed position is one of the trickier steps for the fledgling navigator to master. Sight reduction tables provide solutions for navigation triangles of integral degree values. When using precomputed sight reduction tables, such as H.O. 229, the assumed position must be selected to yield integer degree values for ''LHA'' (local hour angle) and latitude. West longitudes are subtracted and east longitudes are added to ''GHA'' to derive ''LHA'', so AP's must be selected accordingly. When using precomputed sight reduction tables each observation and each body will require a different assumed position.

Professional navigators are somewhat splitdivided in usage between sight reduction tables on the one hand, and handheld computers or scientific calculators on the other. EitherThe methodmethods isare equally accurate. It is simply a matter of personal preference which method is used. An experienced navigator can reduce a sight from start to finish in about 5five minutes using nautical tables or a scientific calculator.

The precise ___location of the assumed position has no great impact on the result, as long as it is reasonably close to the observer's actual position. An assumed position within 1 degree of arc of the observer's actual position is usually considered acceptable.

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==Sights==

Until the age of satellite navigation ships usually took sights at dawn, during the forenoon, at noon (meridian transit of the Sun) and dusk. The morning and evening sights were taken during twilight while the [[horizon]] was visible and the stars, planets and/or moonMoon were visible, at least through the telescope of a [[sextant]]. Two observations are always required to give a position accurate to within a mile under favourable conditions. Three are always sufficient.

==Running fix==

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* [[Celestial navigation]]

* [[Circle of equal altitude]]

* [[Sight reduction]]

* [[Intersection (air navigation)]]

* [[Intersection (land navigation)]]

* [[Navigation]]

* [[Latitude]]

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==References==

{{Reflist}}

*''Nicholls's Concise Guide, Volume 1'', by Charles H. Brown F.R.S.G.S. Extra Master

*''Norie's Nautical Tables'', edited by Capt. A.G. Blance

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==External links==

* ''Navigational Algorithms'' http://sites.google.com/site/navigationalalgorithms/

*[https://web.archive.org/web/20080430060810/http://www.smartcomsoftware.com/winastro.html WinAstro sight reduction software]

* Correction to the sextant altitude http://es.wikipedia.org/wiki/Archivo:CorrecionHs.jpg

* Marcq St Hilaire intercept for the line of position http://es.wikipedia.org/wiki/Archivo:MarcqSaintHilaire.jpg

*[http://www.smartcomsoftware.com/winastro.html WinAstro sight reduction software]

*[http://www.navigation-spreadsheets.com/navigation_triangles.html Navigation Spreadsheets: Navigation Triangles]