# Parallax Arc Seconds

Can you really make the Kessel Run in less than 12 parsecs?

British astronomer Herbert Hall Turner combined the words “parallax” and “second” in 1913 and created the term “parsec.” Defined as the distance from the Sun to an object that has an Earth-Sun parallax of one arc second. It sounds so simple when you say it, and it’s really a simple idea but that definition isn’t precise enough for the International Astronomical Union. They fixed the distance in 2015 to be exactly 648,000/pi astronomical units (the mean distance from the Earth to the Sun). In short, the math works out to approximately 3.26 light years, about 1.92×10^13 miles, or about 31 petameters.

That’s pretty far. It’s still not far enough to get to the next star system though, which is about 4.3 light years away, or more than 1.3 parsecs. The nearest galaxy to our Milky Way, the Andromeda Galaxy, is about 765,000 parsecs away. The edge of the observable universe is 28.5 gigaparsecs (that’s 28,500,000,000 parsecs). So even though one parsec is pretty big, in universal scales of distances, it’s still a fraction of the space between objects in the cosmos.

Let’s take a deeper look at parallax for a “second” (see what I did there?). Parallax is what you experience if you close one eye, hold out your hand, and line up your thumb with a distant object. Now switch eyes but don’t move your thumb. It appears that your thumb moves (or the object does). There is an angle that the two “lines of sight” make from the object, to your thumb, to each eye, that could theoretically be measured. This angle would be the “angular separation between your eyes (from the perspective of the distance object). If one of your eyes were the Sun, the other the Earth, and that angle was 1 arc second, then the object would be one parsec away!

Using parallax, Eratosthenes of Cyrene (c. 276 BC – c. 195/194 BC) used a form of parallax to determine that the Earth was about 44,100 km in circumference. A number that was surprisingly accurate (it’s actually about 40,075 at the equator and 40,007 at the poles). Once you have a general idea of how big the Earth is, you can use observations made of distant objects from different locations on the Earth (or different locations in its orbit around the Sun) to figure out the parallax of other objects. Think back to your trigonometry days for a moment (Sohcahtoa, anyone?) and if you do this with enough objects – some of which you actually know the distances – then you can figure out the actual distance to many things.

This is, in fact, how the first accurate measurements of the size of the Earth, Moon, Sun, Venus, and other planets were made. Same for how far away they are. It was the Transit of Venus across the Sun that solidified these distances and sizes of other objects, and truly made us scientifically certain of how big everything was (except us).

So the next time you feel like doing some backyard science, try this experiment:

• Find a fence post, lamp post, tree, or anything reasonably straight and narrow in your yard that’s more than 20 feet away
• Mark a spot on the ground with a rock or other object
• Stand there, and look at the distant object
• Find a spot about 10 feet away along the line to the distant object and jam a stick or other sharp, pointy thing into the ground such that it appears to be in line with this first spot and the distant object
• Take two big steps away to the right (or left) from the first spot and mark a second spot
• Stand at spot 2 and do another stick in the ground along the line of sight to the distant object
• Try to get the two sticks to be the same distance from the observation spots so that a line between the two sticks is parallel to a line between the two observation rocks
• Now measure the distance between the two sticks and the distance between your two observation rocks
• Lastly, measure the distance from each observation rock to their respective sticks

What you have created is a trapezoid. You know the length of all the sides of the trapezoid but you don’t (yet) know the distance to the distant object. If you imagine extending the sides of the trapezoid until they intersect, they should intersect at that distant object. So using trigonometry, or better yet, this online calculator (https://www.analyzemath.com/Geometry_calculators/trapezoid_calculator.html) you can see the angles A and B. Hopefully, they’re pretty close to each other. For the sake of this backyard experiment, take the average of the two angles for the next step.

You now have an isosceles triangle with one known side and two known angles. But if you drew a straight line from the distant object to the bottom of this triangle, it would be a right triangle and that line represents how far away the object is. That line would also cut the bottom of your triangle exactly in half, so 1/2 of the distance between the two observation rocks becomes the base of this new right triangle.

Remember, in all triangles, the angles add up to 180 degrees. In a right triangle, one angle is 90 degrees (that’s the “right” one) and you know the other, which we’ll call A. The remaining one is 180 – 90 – A. It’s probably pretty small. Using this calculator (https://www.analyzemath.com/Geometry_calculators/right_triangle_calculator.html) head to option 3 (you know one side and the angle opposite it) and enter the 1/2 distance between the two rocks and the angle you just calculated, and it will tell you the length of “b” which is the distance to your distant object!

Now, if possible, go walk to the distant object and count your steps. Then measure how long each of your steps are (or guess) and multiply number of steps by your step size and these two numbers should be relatively close! Congratulations, you just used parallax to measure the distance to a distant object!