Sunday, June 29, 2008

Solving Navigational Triangles

This article will show how to "solve" the navigational triangle to find computed altitude (Hc) and azimuth (Zn) for a celestial sight. By comparing Hc with observed altitude (Ho), the navigator finds the altitude intercept (a). With the assumed position coordinates, azimuth, and inter­cept, a line of position can be established.

The two basic methods for solving a navigational triangle or other spherical tri­angle are by sight reduction table or by mathematical solution.

A sight reduction table presents solutions of the navigational triangle for com­putation of the altitude and azimuth of a celestial body. The variable components are co-latitude, polar distance, and meridian angle "t". Their values depend on time of the observation and the observer's position. The parts of the triangle we wish to find are computed co-altitude (in order to find Hc) and azimuth angle (in order to determine Zn).

Mathematical solutions use the same formulas used to pre-compute the values found in the sight reduction tables. Final results, found through the correct use of either method, seldom differ by more than one-tenth minute of arc for intercept values, or one tenth degree for azimuth values.

The Sight Reductton Table for Marine Navtgatton (Pub 229) is a form of sight reduction table which is produced by the Defense Mapping Agency. They are designed to facilitate solution of navigational triangles to find computed altitude (Hc) and azimuth (Zn).

There are six volumes of Publication 229, each volume covering 15 degrees of latitude. All of the problems used in USCG license exams, which may be solved with Publication 229, are in latitudes 15 degrees to 30 degrees covered by Volume II of the tables.

The entering arguments for 229 are assumed latitude, the whole degree LHA which results when assumed longitude is applied to GHA, and declination.

The main part of the tables is divided in half. The first half of Volume II covers latitudes 15 degrees through 22 degrees, and the second half covers latitudes 23 through 30 degrees. The latitude which is used to enter the table may be either north or south. Each half covers LHA's from 0 degrees through 360 degrees.

There are two pages for each LHA value from 0 through 90 degrees, and 270 through 360 degrees. Left-hand pages are for situations where declination and latitude are both north or both south, they are said to have the "same name." Each "same name" page covers declinations from 0 through 90 degrees.

The adjacent right-hand pages cover "contrary name" situations (meaning the declination is north and latitude south or vice versa) for the LHA values at the top of the page, and "same name" situations when the LHA values are between 90 degrees and 270 degrees located at the bottom of the page. The dividing line between "contrary" and "same name" situations is the step-like line across the right-hand page.

The step-like line across the right-hand page represents the horizon. Entry values of LHA, latitude, and declination (same or contrary) should never result in crossing this line, as this would mean the body is below the horizon.

The inside front and back covers of each 229 are an interpolation table to be used for correcting the the altitude and azimuth angle obtained in the main part of the table for the declination increment. Declination increment is the minutes and tenths of minutes part of the declination of the body. For example, if the body's declination is N 43°­23.7', the declination increment is 23.7'. In other words, declination is tabulated on each page in whole degrees, and you must interpolate between those whole degrees for your actual declination value.

Declination increments 00.0 through 31.9 minutes are located inside the front cover, and declination increments 28.0 through 59.9 minutes are inside the back cover.

Detailed, step-by-step, procedures for finding computed altitude (Hc) and azimuth (Zn) will be presented later in another article in a series of example problems. I will cover all the different possible situations you might find in working with Publication 229.