It is impossible to draw the solar system to scale on (any) paper!

 By MUNGAI KIHANYA

The Sunday Nation

Nairobi,

15 April 2012

Last week we saw that it is impossible to make a scale drawing of the solar system on a newspaper page. It turned out that, when everything is scaled down to fit in the paper, the heavenly bodies (sun and planets) will be too small to be drawn using the thinnest pencil available

The largest object in the solar system (the sun) would be only 0.05mm wide in the scale drawing – only visible with the help of a microscope. This is much smaller than the 0.2mm tip of the thinnest pencil. So the obvious problem is that there would be nothing to draw it with!

Still, it would be interesting to find out the size of the smallest paper that can fit a scale drawing of the main objects in the solar system – that is, the sun and planets; ignoring the moons, asteroids and other minor bodies.

To do this, we start with the smallest planet – Mercury – and draw it as a dot using the thinnest pencil available. Now, even though last week’s conclusions were made assuming that we can get the thinnest pencil ever made (0.2mm), the fact is that general purpose pencils only go down to 0.5mm. So it is only sensible to do the calculations using this limit.

Mercury measures about 4,800km across, and we shall represent it with a 0.5mm dot. Thus the scale of the new drawing is 1mm represents 9,600km (4,800 divided by 0.5). We can round that result and use 1mm-to-10,000km in order to simplify the arithmetic.

In this scale, the 1.4million-kilometre sun will be a circle measuring 140mm, or 14cm (1,4000,000 divided by 10,000). At 12,800km home planet, the Earth will be another dot 1.28mm (1.3mm) in diameter. The sizes of the other planets will be as follows: Venus – 1.2mm (12,000km), Mars – 0.7mm (6,800km), Jupiter – 14mm (140,000km), Saturn – 12mm (120,000km), Uranus – 5mm (50,000km), and Neptune – 5mm (48,000km).

The next step is to draw the orbits of the planets using the same scale of 1mm representing 10,000km. This time, however, we start from the farthest planet (Neptune) and work backwards. This way, we shall immediately know the size of the paper required. Now, Neptune revolves around the sun at a distance of about 4.5 billion kilometres. Therefore its orbit is a circle measuring 9bn km across. In our scale, this will be 900,000mm (9,000,000,000 divided by 10,000).

That looks like a large number, but how big is it in “normal” units? Well, one metre has 1,000mm, therefore, 900,000mm is the same as 900m. Now pause for a moment and think about it: it is almost one kilometre!

In other words, to make a scale drawing of the solar system using the thinnest commercially available general purpose pencil, you would need a piece of paper measuring almost one kilometre by one kilometre. It’s stupendous!

For that reason, I think we should revise last week’s conclusion and say categorically that: it is completely impossible to draw the solar system to scale. The problem being that one cannot find a paper large enough to fit it, nor a pencil thin enough to draw it on a smaller paper!