A scale for the solar system
Problem
The Earth is further from the Sun than Venus, but how much further? Twice as far? Ten times?
When we measure something we use a scale : we consider the size of one thing in terms of another.
Make a fist with your hand (it's about the size of a large orange isn't it?), and then locate a point about 10m away from you.
If your fist was the Sun, the Earth would be more than 10 metres away and less than 1 mm in diameter (the tip of a ball-point pen).
Venus orbits the Sun three times in roughly two Earth years. On rare occasions Venus can be seen (from Earth) passing across the face of the Sun, this is called the Transit of Venus.
Usually Venus appears to pass either above or below the Sun because the plane of Venus' orbit is slightly tilted to the Earth's own orbit around the Sun. The Transit of Venus has only been observed and recorded six times since telescopes became available early in the 17th century.
The Transit of Venus last happened in 2004 and 2012, and will not happen again until 2117.
The Transit of Venus was a valuable observation because it provided the data with which the Earth to Sun distance could be calculated. This distance is called the Astronomical Unit and is used like a scale for the Solar System.
But even without a Transit of Venus to provide data astronomers could know the ratio between the Venus to Sun distance and the Earth to Sun distance. What could they observe and what calculation would they need to do?
Google will give you all sorts of interesting things connected with the Astronomical Unit and the Transit of Venus, but before you do that imagine you are living in the 17th century and don't have the internet, how might this Venus to Sun : Earth to Sun ratio be known?
Getting Started
Draw the plan view. Imagine you are looking down on the plane containing the Sun, Earth and Venus. Put in circles to represent the orbits, none of the sizes will be right of course but the diagram can still be useful.
Choose a position for the Earth and then imagine Venus in all possible positions relative to Earth.
What's that view going to look like to an observer on Earth?
Student Solutions
Lukaz Smith, from Parkside School, sent in the following correct approach to finding the ratio of the Earth-Sun and Venus-Sun distances (diagram added to help clarify the solution)
If you said the Earth to sun distance was 1 astronomical unit ($a$ in diagram), then you could simply measure the angle ($\theta$ in diagram) between Sun, Earth and Venus, when you knew that there was a 90 degree angle between Earth, Sun and Venus. Then you just find the tangent of the angle, so it's 1 : $\tan \theta$ as the ratio of Earth to Sun distance and Venus to Sun distance.
Teachers' Resources
This accessible problem will hopefully lead students into the whole area of solar and astronomical scales. The internet is full of sites offering information. Google anything and you'll start a trail of enquiry.
The methods often take some thinking about and nearly all of them require the imagination of different viewpoints. Grasping the methods and especially the required adjustment of perspective will not be immediate for most students and serves as a useful reminder that mathematical understanding isn't an "instant" or "not-at-all" experience.