# Aristarchus ≈ 250 BC, the size of the sun

In various cultures throughout history, the ratio between sides of a triangle has been used to measure distances. Some of these distances have had a major impact on science and on our conception of the world.

## The sun, the earth and the moon

The sun and the moon appear to have the same size.

That observation was made by Aristarchus when he observed a solar eclipse. He also realized that they do not have to be equal in size if only the sun is further away than the moon.

Let `d` be the distance to the moon and `D` the distance
to the sun. Let `r` be the radius of the moon and `R` the
radius of the sun. If the angular sizes of the moon and the sun are equal, the
triangles are similar.

Aristarchus wanted to compare the sizes of the sun and the earth.

## Exercise 1

Find the ratio of `R` and `r` in terms of `D` and `d`!

## Exercise 2

Aristarchus knew what the moon looked like when the angle `β`
was 90`°`. (What does it look like?). He then estimated the angle
`α`.

Aristarchus estimated the angle `α` to 87`°`. During
a lunar eclipse he estimated that the radius of the moon was about one third
of the radius of the earth. Using these numbers, calculate the ratio of the
radius of the sun and the radius of the earth! Using these numbers, how much
larger is the sun than the earth (Now it is a comparison of volumes, not radii)?

Aristarchus then concluded:

the giant sun does not
move around the tiny earth,

it is probably the
other way around!

## Exercise 3

Aristarchus made a minor mistake when comparing the radius of the earth with the radius of the moon, otherwise his method was correct.

The data used by Aristarchus, on the other hand, was not correct. Aristarchus'
estimation of the angle `α` was 87`°`, the modern
estimation is 89`°`50'. (89 + 50/60 degrees). Find the ratio `D:d`
using the correct angle! What was his relative error when measuring the angles?
What is the relative error in distances?

1 sun ≈ 1 300 000 earths

image from: NASA

# references:

Aristarchus' sizes: http://en.wikipedia.org/wiki/Aristarchus_On_the_Sizes_and_Distances

by Malin Christersson under a Creative Commons Attribution-Noncommercial-Share Alike 2.5 Sweden License