Intercept method: Difference between revisions

Revision as of 19:45, 24 January 2011

Revision as of 06:10, 4 May 2011

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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 (See 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]].

Next, the altitude and azimuth of the celestial body are computed for a selected position (assumed position or AP). This involves resolving a spherical triangle. Given the three magnitudes local hour angle, LHA, observed body's declination, dec., and assumed latitude, lat, the altitude Hc and azimuth Zn must be computed.

Next, the altitude and azimuth of the celestial body are computed for a selected position (assumed position or AP). This involves resolving a spherical triangle. Given the three magnitudes: local hour angle, (LHA), observed body's declination, (dec.), and assumed latitude, (lat), the altitude Hc and azimuth Zn must be computed. The local hour angle, LHA, is the difference between the AP [[longitude]] and the hour angle of the observed object. It is always measured in a westerly direction from the assumed position.

The relevant formulas are

The relevant formulas (derived using the [[Spherical_trigonometry#Identities|Spherical trigonometric identities]]) are:

Sin(Hc) = Sin(lat) * Sin(dec) + Cos(lat) * Cos(dec) Cos(LHA)

: {{math|size=large|sin(Hc) {{=}} sin(lat) · sin(dec) + cos(lat) · cos(dec) · cos(LHA)}}

Sin(LHA)

Tan(Z) = -------------------------------------------

: <math>\mathrm{tan(Z) = Sin\frac{sin(LHA)}{sin(lat) *\cdot cos(LHA) - Coscos(lat) *\cdot Tantan(dec) }}</math>

or, alternatively,

sin(dec) - sin(lat) * sin(Hc)

Cos(Z) = -----------------------------

cos(lat) * cos(Hc)

: <math>\mathrm{ cos(Z) = \frac{sin(dec) - sin(lat) \cdot sin(Hc)}{cos(lat) \cdot cos(Hc)}}</math>

Where

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

: <math>\mathrm{hav}\,CompHc = haversin(\mathrmoverline{havHc}\,) = haversin(LHA) \cdot cos{(lat}) \cdot cos{(dec}) + \mathrm{hav}haversin(lat \pm dec) }</math>

Where CompHc is the zenith distance or complement of Hc. CompHc = 90º - Hc

Where CompHc{{overline|Hc}} is the zenith distance, or complement of Hc.: CompHc{{overline|Hc}} = 90º - Hc.

The relevant formula for Z is

cos (lat - Hc) - sin(dec)

Hav Z = -------------------------

2 * cos(lat) * cos(Hc)

The relevant formulasformula arefor Z is

: <math>\mathrm{ hav(Z) = \frac{ cos(lat - Hc) - sin(dec)}{2 \cdot cos(lat) \cdot cos(Hc)} }</math>

The local hour angle, LHA, is the difference between the observer's DR [[longitude]] and the hour angle of the observed object. This is always measured in a westerly direction from the observers DR position.

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.