The insignificance of an arcsecond

In math, and especially angular degrees, there are several ways to measure angles. Radians (arc length equal to the radius) seem to be the preferred way for the purists, due to the elegance, but in engineering, degrees are the way to go. Everyone knows degrees, a 45° angle would be rising at the same rate it is running horizontally. A 90° angle is almost the universal measure of most corners. It's simple.

In breaking up the fractions of a degree, there are several ways to do this. The most simple mathematical way to do this is to simply stick with decimal degrees. It's easy and requires no special conversion. Any calculator would do it. But the cooler way to do this is with arcminutes and arcseconds.

You might not recognize the names, but you've seen them in action quite frequently. The most apparent use that most see them is in GPS coordinates: 121 degrees, 23 minutes west. Those minutes are also known as arc minutes, and refer to a 60th measure of a degree. Broken further down are arcseconds, which would be the remainder of that arcminute, broken down to a further 60 second measurement of that minute. 121 degrees, 23 minutes, 29 seconds would simply be written 121°23'29". It's very common in navigation, but engineers and surveyors also enjoy using this convention. Recently, it occured to me that a single arcsecond must be a significant amount of angular measurement. But how significant is it?

Let's consider a golf ball. According to wikipedia, a golf ball is 1.68 inches high (approximately). In terms of the pass of an arcsecond, how far away would a point of reference need to be in order to have that golf ball represent the distance passed in the sway of an arcsecond?

We return to our roots of trigonometry. In the triangle

represented here, scale is ignored for the

sake of the ease of explanation. Our frame of reference will be angle A in this case, the unknown distance to the golf ball will be distance b, and the hight of the golf ball will be hight a. Trigonometry will tell you that the tangent function will be of use here, in that the tangent of an angle will be equal to the opposite divided by the adjacent sides. In math form:

tan A = a / b

Let's plug some numbers in and see where this gets us.

tangent (1 arcsecond) = 1.68" / b, so distance b = 1.68 " / tangent (1 arcsecond)

Google now makes math insanely easy, and gives us an answer of 346,525 inches. Divide by 12 inches per foot and 5280 feet per mile, the golfball would have to be 5 and a half miles away to represent the swing of one arcsecond. An arcsecond might as well not even exist, and to think that in surveying class, we'd take things further down, to the hundredth's of a second. That's like putting the golf ball 550 miles away! Ridiculous, don't you think?

But let's not throw the arcsecond away completely. Let's look at the earth, and take the point of reference to be the core. The diameter of the earth is about 8000 miles, so for our study, we'll take the radius to be 4000 miles. We'll still stick with the same arcsecond, and see how much surface area of the earth is represented by one arcsecond:

tan (1 arcsecond) = length / 4000 miles ~ length = 4000 miles * tan (1 arcsecond)

You end up with .0193 miles, which is 102 feet, quite a significant distance on land! You could miss a location completely. In this case, carrying to the hundred's place of a second would get you within a foot of what you're looking for, which is accurate for most non-engineering purposes.

Of course, when you get astronomical, and you're talking in terms of light years, the hundredth of a second could be far from what you need. It could mean missing the location of a star, or even a galaxy completely.

Math is fun.