Circles Of Equal Altitude
Celestial bodies are so far away that rays of light from them reach earth as parallel lines. In solving celestial navigation problems you can assume that all bodies are at the same infinite distance from earth on the celestial sphere. Corrections for bodies close to earth are taken into account when determining the actual altitude of a body in a celestial sight.
The geographical position of a celestial body is the point on the earth directly below the body at any given instant. The geographical position is one of the three corners of the "navigational" triangle, and the side of the triangle between the GP and an observer's position is equal to 90° minus altitude, or co-altitude. Because the light rays from celestial bodies arrive on earth as parallel lines, you measure the same angle between the body's light rays and the horizon are located the same distance from the GP of the body. All observers who measure a celestial body at the same altitude at an instant of time are located on a common circle. This circle is called a circle of equal altitude, and the distance from the center to the circle (radius) is equal to co-altitude.
This circle of equal altitude is a celestial "line of position" like a radar range line of position from piloting. In this case the co-altitude, converted from degrees and minutes of arc to miles, is similar to the radar range. The geographical position (GP) of the body is the center from which you swing an arc with a pencil compass.
A celestial line of position can be established by:
1. Observing the altitude of a celestial body.
2. Recording the instant of time (GMT) of the observation.
3. Determining the GHA and declination of the body from the Nautical Almanac.
4. Locating the GP of the body on a chart using GHA and declination.
5. Subtracting altitude from 90° and converting the resulting co-altitude to minutes of arc or miles.
6. Swinging an arc with a radius equal to co-altitude in miles using the GP as the center.
A celestial fix may be obtained by crossing two or more circles of equal altitude based on observations of celestial bodies.
Plotting Fixes Based on Circles of Equal Altitude
In actual practice fixes based on circles of equal altitude are impractical and are seldom used in marine navigation. The main reason for not using them is that the radii of these circles is too large to plot the circular line of position on the chart unless very high altitude sights are used, and they are difficult to obtain. The altitude of the body must be very close to 90° (87° is the practical minimum) in order for the co-altitude to be small enough to be plotted on a normal celestial plotting sheet or navigation chart. In taking a sight on a moving ship, the navigator would have difficulty determining the actual point on the horizon, below the body vertically, from which to measure altitude when the body is nearly at the zenith. The only real advantage of high altitude sights is that the same body may be used for a fix because the location of the GP will move a considerable distance in only a few minutes, giving a good angle of intersection between the two arcs when plotted.
The USCG requires all unlimited license candidates to be proficient in solving high altitude "running" fixes using the sun. The following is an example solution of a high altitude running fix.
Example Problem
On 15 November 1981, your 1030 ZT DR position is Lat. 19° 41.0' S, Long. 41° 37.0' W. You are on course 239°T, speed 22 knots. You take the following observations of the Sun:
What was the 1200 ZT position?
Zone Time 1128
GHA 40° 50.4
Declination S 18° 33.6
Ho (observed altitude) 88° 18.4
Zone Time 1133
GHA 42° 05.4
DeclinationS 18° 33.6
Ho (observed altitude 88° 37.7
Step 1: Using a plotting sheet constructed for the proper latitudes, label three meridians from right to left 40°W, 41°W, and 42°W. Plot the ship's 1030 DR position and establish a DR based on the course and speed. Determine the 1200 DR position by measuring the distance which would be traveled between 1030 and 1200 along course 239° T at a speed of 22 knots. (Distance = speed x time / 60 or 22kts x 90m / 60m = 33 miles).
Step 2: Plot the geographical positions of the sun and label them with their observation times. Treat declination as latitude and since GHA is less than 180° it is equal to longitude (west). Next, advance each GP in the direction of 239° T at 22 knots to 1200. The first would be advanced from 1128 to 1200 (32m at 22 knots = 11.7 miles). The second would be advanced from 1133 to 1200 (27m at 22 knots = 9.9 miles).
Step 3: The next step is to determine the radius of the circles of equal altitude of the respective sights. To do this subtract each observed altitude from 90 degrees.
1128
89° 60.0 - 88°18.4 = 1° 41.6 co-altitude
1133
89° 60.0 - 88° 37.7 = 1° 22.3 co-altitude
Convert the co-altitudes just found to minutes (miles). Doing the conversion you should find that 1128 is 101.6 miles and 1133 is 82.3 miles. Using a compass, measure each distance on the latitude scale and then swing an arc of that radius from the advanced GP. The circles formed will intersect at two points, but the DR position indicates which is the correct position for 1200. The fix should be labeled and the coordinates you obtain should be close to Lat. 20° 01.0' S, Long. 42° 05.0' W.
