HOW FAR AWAY IS THAT?

Hiya, Sean Lally Physics Guy here. Today on “How do we know that?” we are looking at distances in the universe. So, how DO we know how far things are away from us? How far away is the Moon? How about the Sun? Stars? How can we know the distance to anything we can’t easily visit?

Think about this for a minute. If you want to know how far away your pencil is from you right now, how would you determine that?

How about if you want to know how far your science teacher is from you?

These aren’t super tricky. You’d probably use some type of measuring stick.

How about if you want to know how far you are from home? That might require a different type of “measuring stick” or different method altogether. Think about that one and come up with an idea or two.

I suppose you could drive there and just watch the mileage on your car’s odometer (trip counter). Or maybe you could just map it online and see what the program claims the mileage to be. Or perhaps you could use a GPS.

Here’s a tougher one: Imagine that you want to know how far across a river is, but (and this is the catch) you can’t actually cross it to measure directly. Hmmmmmm – how would you do that? We’ll go one step trickier – let’s say that you’re limited by the technology of 2500 years ago. That means, no GPS, no computers, no modern technology whatsoever.

And while you’re thinking about that one, imagine how exactly one would measure the distance to the Moon without going there. That means that, no, we can’t drive a spaceship there with a long measuring tape behind us. We also can’t drive our spaceship there and check the odometer when we’re done!

OK, both of these problems require the same type of thinking. Take a minute to discuss with your partner how one might be able to attack the river problem.

Have an idea? Here’s one low-tech way to “solve” the problem. (Hey, why did I put solve in quotation marks?)

(See image 1 above.)


So, you’re standing where the diamond is and you notice a tree across the river, indicated by the star. (It doesn’t have to be a tree, but you get the idea.)

Now let’s say that the only equipment you have is a protractor and meter stick. How might you find the distance across the river to the tree? Think about it. Discuss with your partner.

Here’s one approach that will work as long as you’re careful. It involves the use of a simple instrument that you construct, similar to a surveyor’s tool called a transit.

(See image 2 above.)


In this example, you mark a spot close to the shore directly across from the tree. Then you walk a measured distance away from this spot, ending at the diamond spot on the diagram. Here is where the protractor comes to use; also, it may be easier to put it on a notebook so that you can mark off the angle as you measure it. Get on the ground as low as you can. Aim the protractor such that the center (90 degrees) is aimed directly across the river. Try your best to determine the angle (θ) that the tree makes with respect to the shoreline – see the picture above. You may want to use a straw as a “sighting tube”. If so, make sure that the straw crosses the “origin” of your protractor and line it up with the tree. Make sure that you record the angle that the straw makes with respect to the horizontal base of the protractor (which is parallel to the shoreline). You may find it best to ask your teacher for help.

For most students, using a river won’t be very realistic – instead, use this method to calculate the distance across a road (but do very careful – watch for traffic!).

Yes, it’s a little tricky. That’s why I used the word solve in quotes above – this method, like any measuring method, is only as accurate as the tools and the person taking the measurements. In principle, though, it can be quite accurate.

Now, construct a scale diagram on paper. You measured a distance along the shoreline above. Decide upon a realistic scale for the size of your paper. For example, 1 m (of outside distance)= 1 cm (on paper).

Use your protractor to construct the angle that you measured above, in the same position on the diagram that it was outside. Draw additional lines to make a triangle, as shown above.

Now, measure the side of the triangle that represents the distance across the river. Convert it back to meters, using your scale. For example, if your scale was 1 m = 1 cm, and the line is 20 cm – the distance would be 20 m. Get it?

Alternate method – trigonometry!

An alternate method that does not require a scale diagram is to use the mathematics of trigonometry. Everything that you did above is repeated, but a scale diagram is not needed. Since you know one side of a right triangle and an angle adjacent to it, you may use the trigonometric ratios – specifically, the tangent function.

In case you do not know:

In a right triangle, the three sides can be defined as a, b and c. However, it’s sometimes more useful to call them opposite, adjacent and hypotenuse. The hypotenuse is the longest side of the right triangle – the side directly across from the right angle. But what side is opposite and what side is adjacent? This depends on the angle that you’re thinking about – a so-called reference angle.

(See image 3 above.)

Note that opposite means “opposite the reference angle”, and adjacent means “adjacent to the reference angle.”

In a right triangle, several ratios can be defined:

Sine (sin), cosine (cos) and tangent (tan) are the most common. For our purpose, tangent will be most useful. Here are the definitions of the ratios:

sin (θ) = opposite / hypotenuse

cos (θ) = adjacent / hypotenuse

tan (θ) = opposite / adjacent

(This is easy to remember with the pneumonic SOH CAH TOA: Sin equals Opposite over Hypotenuse, Cos equals Adjacent over Hypotenuse, Tan equals Opposite over Adjacent.)

Literally, we read this as (for example):

“Sine of theta (θ) is equal to the opposite side divided by the hypotenuse.” What this means is that no matter how big or small the triangle, the ratio of the sides associated with this angle will always have the same value. For example, the sine of 30 degrees is 0.5 – this means that no matter what the actual size of a right triangle, if it has a 30-degree angle in it, the ratio of side opposite this angle to the hypotenuse of this triangle will always be 0.5. Pretty neat, eh? Sin(θ), cos(θ) and tan(θ) are simply ratios of sides associated with particular angles (θ).

For tangent, our trig ratio of choice, “tangent of theta (θ) is equal to the opposite side divided by the adjacent.”

But how do we apply this to our river problem above?

You have the angle, measured with your protractor. Determine the tangent of this angle by using a graphing calculator (make sure it is in “degrees mode”). Your teacher may help you with this. You know the adjacent side of the triangle and now you can calculate the opposite side:

tan (θ) = opposite / adjacent

Using algebra, we can find that:

opposite = (adjacent) x [tan (θ)]

Try it!

How well did you do? Well, if you were measuring the distance across a river, it may be tricky to get the actual distance. However, if you used a road you could carefully measure the actual distance across the road – use a meter stick, trundle wheel, string or gullible friend to find the actual distance across the road.

Now find the percent that your value is different from the actual value:

[ (Your value) – (actual value) ] / (actual value)

Multiply this by 100 to make it a percent.

Now what is a good percent? That’s hard to say. Different experiments and methods have different amounts of acceptable error. There are no real absolutes here – usually, the scientific community decides on what is acceptable. But here is a rough guide for this particular experiment:

If your value is under 25% (which means that you measured at least 75% of the actual value), that is not bad for a quick experiment of this sort. Higher than that? Try it again.

What are sources of your error? Think about this and write some down. It will be very helpful to talk this over with your partner or teacher.

Astronomical Distances

But what does this have to do with astronomical measurements? The same principle applies, believe it or not. We can measure star angles from the Earth, using instruments that are similar to protractors – sextants, quadrants, etc. This was a classic technique used for centuries. It is usually easier, however, to measure how far stars are from other stars in the sky – this can be viewed as an angle if you think about one line from you to a star, and then another line from you to a different star. Imagine the angle that exists between these two lines.

But there is a little problem.

We still need a measurable distance like the shoreline above, and moving a short distance on the Earth doesn’t give us a significant distance (especially when the stars are pretty far away). So, we change the game a little – we wait for the Earth to move to another part of its orbit, having gone 2 AU. (An Astronomical Unit, or AU, represents the size of the semi-major axis of Earth’s orbit. It is also close to the average distance between Earth and Sun). The 2 AU becomes our known baseline. We take angular measurements at the two points in the orbit and come up with a parallax angle with respect to background stars. The technology is a little different, but the approach is very similar. This technique (called astrometry) works very well for many stars, particularly the nearby ones. Further stars have much smaller parallax angles, making this technique less useful; for the stars farther away, we have other methods for measuring the distances.

By the way, this technique was used to measure the distance to the Moon nearly 2500 years ago by Eratosthenes, among others. Measuring cosmic distances is tricky business, but it’s certainly not new.

That’s all for now – see ya soon, see ya on the Moon!

Further Reading

Kitty Ferguson – Measuring the Universe
Dava Sobel – Longitude
Barbara Ryden - Cosmology


ALL TEXT AND IMAGES COPYRIGHT SEAN LALLY 2009