A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding
From: Frank Reed
Date: 2015 Aug 7, 10:11 -0700
You can calculate the angular size of anything (when the angular size is expected to be smaller than a few degrees) from a simple ratio:
angular size = k·(object diameter)/(distance to object).
For this equation to work, the object diameter and the the distance to object have to be in the same units. The quantity object diameter should really be the distance across the line of sight. So, for example, if there is a boat sailing past you five miles away and it is roughly beam on (side of the boat towards you), then you should use the boat's length for object diameter. The number k that multiplies the ratio of the diameter to the distance depends on the angular units you're using. The best choice for k in mathematical terms is 1. That yields a pure angle or an angle "in radians". But in navigation and astronomy and in most common applications, we prefer angles in degrees and minutes and so on. If you want the result in degrees, k should be 57.3 (memorize that number!). If you want the result in minutes of arc, k should be 3438 (memorize that number, too, or just remember that it's 57.3 multiplied by 60). And if you want the result in seconds of arc, k should be 206265 (you don't need to memorize this one usually). For navigation calculations, I recommend remembering this equation in the "minutes of arc" form:
(angular size in minutes) = 3438·(object diameter)/(distance to object).
If you want to see where these 'k' numbers come from, remember that the circumference of a circle is 2·pi·R where R is the radius. So for a complete circle, the ratio of distance "across" the line of sight to the distance "to object" is just 2·pi. Meanwhile the number of degrees in a complete circle is 360, so the conversion factor from a pure ratio to degrees is 360/(2·pi). If you plug that into a calculator, you'll find that this is 57.30 to four significant digits. And of course to get minutes of arc, you would multiply degrees by 60 and, sure enough, 57.3 multiplied by 60 is 3438.
Notice that you don't need any trig for this sort of calculation (unless we're dealing with angular sizes larger than a few degrees!). Angles are ratios. An angle is just "distance across" divided by "distance out". Two objects can have the same angular size if one object is ten times bigger than the other so long as the distance to the larger object is ten times greater. We don't need degrees and minutes to measure angles, and it would be mathematically simpler in many cases if we worked with angles always as ratios (or "in radians"). Some of our road signs adopt this practice, and we don't even notice. When you see a road sign that says "10% grade ahead" that described an angle: the ratio of the decline in altitude for a given forward distance travelled is 10%. You can immediately convert that to degrees by multiplying by 57.3 (remembering first that 10% is "really" 0.10): a 10% slope is inclined 5.73°.
An example: The Sun is 864,000 miles across. Its approximate distance from Earth is 92,900,000 miles. What's its angular size in minutes of arc?
angular size = 3438·(diameter / distance) = 3438·864/92900 = 32.0'.
Another example: A ship is 400 feet long. We see it nearly abeam in the distance. We measure its angular length with a sextant and find that it is 45' across. How far away is it?
angular size = 45' = 3438·(400 feet)/distance.
If you remember a little introductory algebra, you can invert this equation to solve for distance. Or, even without algebra, you can find a number that works for "distance" by trial and error (easy with a calculator). The result either way is distance = 30560. And remember that the equation works as given when the size and distance are in the same units of length, in this case feet. So that's 30560 feet. You can convert in your head to nautical miles. The final result: the ship is 5.0 nautical miles away.
An angle-measuring example: You hold a common index card at arm's length. Its lines make an angular scale that you can use to measure small angles. What is the angular distance between the lines? On a common (US printed) index card, the lines are 0.25 inches apart. You have to measure how far out qualifies as "arm's length" for you personally, but let's suppose you get 26 inches. Then for minutes of arc:
angular size = 3438·(0.25 in.)/(26 in.) = 33.1'.
You can now use an index card as a makeshift angle-measuring instrument. Mark off the lines in multiples of 33': 0°33', 1°06', 1°39', 2°12', etc.
Conanicut Island USA