Measuring the Distance to the Sun with Aristarchus

Earth Science Extras

by Russ Colson

 

Aristarchus' measurement of the distance to the Sun began with observing and timing a lunar eclipse. Here is my family watching a lunar eclipse in May 2003 with me.

 

We humans have always wanted to think ourselves to be important (what's wrong with that, right?). Early on, that desire for importance meant that most humans believed that Earth must be a really big and important place and everything else in existence relatively smaller. All that spatial importance disappeared forever when Aristarchus measured the distance to the Sun over 2200 years ago and realized that Earth is really a small speck in a very large universe.

 

2200 hundred years ago? Really? How could he do that? He didn't even have a space ship,or a telescope, or a computer, or, or, well anything, really. He only had what is available to most high school students and teachers today. And yet, he figured out, to a pretty close approximation, the distance to the Sun, without even looking it up on the internet.

When I was in 4rth grade, my teacher said that she could remember when one person could know all there was to know. She seemed really old to me when I was in 4rth grade, so I beleived her. But since then, I've come to understand that she couldn't have possibly been that old. She just didn't know how much people knew even long ago.

She should have read about Aristarchus and his amazing, intuitive, creative, mathematical, and really pretty cool measurement of the distance to the Sun.

 

Let's take a look at how he did it. His method involves lots of geometry, so you might want to get yourself ready to do some math.

Ooohhmmmm..Ooohhmmmm. Ooohhmmmm.

Ok, I'm ready.

One of the strengths of science is found in taking a very complex problem and breaking it down into smaller, bite-sized chunks. We are going to do that in following Aristarchus' calculation, making this math problem an exercise in scientific thinking. This exercise is also an example of how math and science can work together. Remember, mathematical thinking is not about the abiility to do complex calculations, but rather it is about the ability to set up a mathematical problem in order to solve it!

Aristarchus' Hypotheses

Aristarchus started his calculation with 6 'hypotheses', which were actually scientific deductions based on observational evidence. The first three of these hypotheses we have already studied in a previous lesson on phases of the Moon. You can go back and review that lesson to figure out what observational evidence he might have had on which to base his ideas.

Hypothesis 1) The Moon receives its light rom the Sun (how could he know that?)

Hypothesis 2) The Moon moves as though following a sphere with the Earth at its center

Hypothesis 3) At the time of half Moon (1rst and 3rd quarter), we are viewing the shadowed/lit circle edge on

 

For the remaining hypotheses, we are going to figure out how Aristarchus came up with each one.

Hypothesis 4) We will come back to hypothesis 4 later

Hypothesis 5) The breadth of the Earth's shadow at the distance where the Moon passes through it during an eclipse of the Moon is the breadth of _________ Moons. Can you figure out how to fill in the blank using the data that Aristarchus used?

 

Of course, the Moon doesn't always pass through the middle of Earth's shadow. Sometimes it cuts across one edge or the other, making it take less time to cross the shadow. Aritarchus' eclipse was one of those, so his estimate of the width of the shadow was a little bit on the low side. We now think Earth's shadow at the distance of the Moon is about 2.5 Moon widths (although it varies somewhat due to the Moon's not-perfectly-circular orbit around the Earth). But we are still going to follow Aristarchus' calculation by using the value of 2 Moon-widths.

 

Hypothesis 6: The Moon covers _______ of the sky (angular size of the Moon in the sky)

This is a lab exercise that you can do yourself to determine the angular size of the Moon in the sky. There are two ways to do it: 1) measure the size of the Moon or 2) measure the size of the Sun which we know (and Aristarchus knew) is the same angular size as the Moon because the Moon just exactly covers the Sun during a solar eclipse

Before attempting the problem farther below, first measure the size of the Moon in the sky using one of the following two methods.

 

Method 1: Measuring the size of the Sun in degrees (which is roughly equal to the angular size of the Moon):

Create a "lens" to project the light of the Sun by poking a small hole in the middle of a sheet of file folder paper (e.g. with a large metal paper clip or steel awl).  Hold the paper perpendicular to the sun's rays, and hold a second piece of paper, also perpendicular to the sun's rays, 1-3 feet away.   An image of the sun will be cast onto the second paper as shown in the illustration below. (This type of projection even works to see an image of a crescent sun during a partial solar eclipse!). The angular diameter of the sun can be determined by measuring the distance between the two sheets (labled with a cursive "L" in the figure below) and the diameter of the sun image (d).   The angular diamter can the be determined from trigonometry (Aristarchus would have used geometry) from the expression

Tan (1/2 Sun's angular size) = (1/2d/L). Solving for the Sun's angular size gives us

Sun's angular size = 2 times Inverse tangent (1/2d/L). (Inverse tangent might be called invtan, arc tangent, Atan, or tan^-1 on your calculator). 

 

Method 2: Measuring the angular size of the Moon with a ruler (best if the Moon phase is between 1rst and 3rd quarters):

Take a ruler and hold it at arm length toward the Moon.   Measure the apparent size of the widest diameter of the Moon.   Then measure the distance from your eye to the ruler.  

 

 

 

 

 

 

This sets up a trigonometry problem similar to the calculation of the Sun's diameter above. The angular size of the Moon can be calculated from the expression

tan (1/2 Moon's angular size) = 1/2 its apparent width on the ruler / distance from the ruler to your eye. (you will need to rearrange the equation to solve for the Moon's angular size and use an inverse tangent or arctangent function)

When you have completed at least ONE of these measurements, proceed to the question below.

 

Aristarchus' actual Hypothesis 6 is the following: The Moon covers 1/15 of a sign of the Zodiac

 

Why did Aristarchus use 2 degrees instead of 0.52 degrees? This was a very easy measurement to make even in Aristarchus' day. We also know that Aristarchus knew the correct size of the Moon in the sky because he reports it correctly elsewhere. Did he just make a mistake? Did other scholars think that 2 degrees was right so he wanted to make them happy? Did he think that 0.5 degrees would make the Sun so far away that no one would believe it, so he 'played it safe' by picking the biggest possible number? We can't know now. We only know that he chose 2 degrees for his Hypothesis 6.

 

So, back to Hypothesis 4.

Aristarchus' hypothesis 4) states that at the time of half Moon (1rst or third quarter) the angle formed by the Sun-Earth-Moon is 87 degrees.

The modern value is 89.853 degrees, which might not sound like a very big difference, but it makes a HUGE difference in calculating the distance to the Sun. This was the largest source of error in Aristarchus' calculation.

In fact, this was a very difficult measurement to make because it is very difficult to know the exact moment when the Moon is at exactly half Moon. Aristarchus's did not have adequate technology to make it any more accurately. It is likely that he chose the value of 87 degrees because it was the smallest possible angle given his ability to measure. The smallest angle meant that his calculated distance to the Sun yielded the lowest possible value, and since he was already showing that the Sun was really far away, he wanted to be sure that everyone knew that this was the closest possible distance to the Sun!

Like Aristarchus, you won't be able to measure this angle very accurately . However, you can make some rough measurements of the angle if you like, using the method illustrated below. Remember, to derive this value, you can only measure the angle at the exact moment the Moon is half full (1rst or 3rd quarter).

 

How Aristarchus used his Hypotheses to Calculate the Distance to the Sun

Step 1: Calculating the size of the Moon

Using Hypotheses 5 (2 Moons fit into Earth's shadow at the distance the Moon orbits from the Earth), and recognizing that the angular sizes of the Sun and Moon in the sky are the same (based on observed solar eclipses), and recognizing that as long as the Sun is far enough away, the angle of Earth's shadow will be the same as the angular size of the Sun in the sky, Aristarchus constructed the following model.

 

Notice that the two right triangles (shown in red and yellow) are similar triangles, meaning that all of their angles are the same. Since they are both right triangles, that angle has to be 90 degrees. The acute angles are half of the angular size of the Sun and Moon in the sky, and thus equal, and since all the angles have to add up to 180 degrees, the third angles have to be equal as well, making the triangles similar. This means that the ratios of the lengths of the sides must be equal. With this information, Aristarchus figured out the proportional distance that the Moon orbits from the Earth to the point where Earth's shadow vanishes. Can you figure it out? (For our calculation, we are going to ignore the diameter of the Earth, because, despite how big it looks in the illustration above, it is pretty small in real life, although Aristarchus took it into account)

The Moon orbits the Earth at what proportional distance toward the 'end' of the Earth's shadow? This is a mathematical thinking exericise--that is, it requires you to figure out how to set up and solve a problem, not simply do a calculation or punch numbers into a calculator. Here's a hint: Remember that the ratios of sides of similar triangles are equal, and use that to figure out the proportional distances of A and B.

 

 

Aristarchus next figured out the size of the Moon relative to the size of the Earth. He again used Hypothesis 5 and also geometry. He realized that the width of the Earth's shadow = 1 Earth Diameter (ED) at the Earth, it equals zero ED at the pointy end of the shadow, and everywhere in between it varies in a linear fashion, such that, for example, half way toward the end of the shadow, the shadow width is 1/2 ED.

Based on the calculations in the previous question, how large is the Moon?

 

Step 2: Calculating the Distance to the Moon

Using Hypothesis 6 (the angular size of the Moon in the sky is 2 degrees--remember, in reality it is closer to 0.52 degrees, but we are going to follow Aristarchus' calculation), Aristarchus next calculated the distance to the Moon in terms of Earth's size. He envisioned the model as shown below (except that he used geometry rather than the trigonometric function shown to solve the problem, but trigonometry is so much easier in a world with calculators!).

 

Step 3 Calculating the Distance to the Sun

Using Hypothesis 4, At the time of half Moon (1rst or third quarter) the angle formed by the Sun-Earth-Moon is 87 degrees, and his calculated distance to the Moon (determined in the preceding calculations), Aristarchus calculated a distance to the Sun based on the geometric model illustrated below.

 

 

Remember, Aristarchus chose his Hypotheses so that he would be able to say that the Sun is at least this far away! He proved, for the first time so far as we know, that the Earth is a relatively small place in a much, much bigger universe. It changed forever our perception of the scale of humanity in existence.

Of course, we already know that Aristarchus' Hypothesis 4 was only a minimum possible value for the Sun-Earth-Moon angle within the uncertainty that his ability to measure made possible. We can use modern measurements of this value to get a better distance to the Sun, while still using Aristarchus' methods.

 

Step 4: Calculating the Size of the Sun

Aristarchus did not stop at getting the distance to the Sun, but calculated its size as well (although also signiicantly in error due to the error in his Hypothesis 4). Although his size of the Sun is quite far off from the modern value, remember that he chose his Hypotheses so that he would be able to say that the Sun is at least this far away, which meant he could also say that the Sun is at least this big! He proved, for the first time so far as we know, that the Earth was not the largest body in the Universe, but that the Sun was bigger than the Earth. Again, this changed human perspectives on our place in existence.

Aristarchus' geometric model for figuring out the size of the Sun, using similar triangles, is shown below.

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Step 5: The Diameter of the Earth

All the calculations above being done in terms of Earth Diameter (ED) may have led you to think that Aristarchus only calculated the distance to the Sun in terms of another unknown parameter, the size of the Earth. But that would not be the case. Despite all the mythological nonsense that was once taught in schools that people didn't know the Earth was round until Columbus sailed to America and proved it, people at the time of Aristarchus 2200 years ago not only knew that the Earth was roughly spherical but had measured its size. This was measurement was done by another Greek mathematician, Erastosthenes.

Erastosthenes had the following information. He knew that in the town of Syene, Egypt, there was a deep well in which light from the Sun went all the way to the bottom at noon on the Summer Solstice, showing that the Sun was directly overhead at noon on that day (Syene was on the Tropic of Cancer). He knew that Syene was 800 kilometers due south of his own town of Alexandria (he actually was working in units called stadia rather than kilometers, but we are going to use more familiar units of measure). On the very same day, at the very same time, he took measurements of the shadows of pillars in Alexandria, from which he could calculate the angle of the Sun at noon. He recognized that the angle of the Sun at Alexandria had to be the same as the angular distance between Alexandria and Syene. His geometrical model is shown in the two images below.

 

Aristarchus' hypotheses and calculations are reported for us in his small book, On the Sizes and Distances of the Sun and Moon. Aristarchus lived circa 310–230 BCE. During my construction of this activity, I also used the following book as a source: Ferguson, K. 1999. Measuring the universe: Our historic quest to chart the horizons of space and time. New York: Walker Books, along with other souces.

 last updated 8/9/2020.  Text and pictures are the property of Russ Colson.