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Scale

Scale or Size Matters!

Being humans we are used to dealing with things on a human scale, a centimetre or a metre (or an inch or yard if you are old school) is something most of us can visualise or even guess at with a reasonable degree of accuracy. We can expand this immediate body sized measurement a thousand fold and still cope with kilometres and miles. If you've walked, run, swam a kilometre (or mile), or seen it marked out on the ground then you can keep that image in your mind and relate to it.

But how about astronomical distances? Lets start at the moon which is on average is about 385,000 km (240,000 miles) from the centre of the Earth. I drive an average of 12 000 miles a year so if I add up all the miles I have driven and all the miles I'm going to drive before I'm too doddery to drive any more then in my life time I may just have driven the equivalent distance of going to the moon and back. Thinking about it, one or two of the vehicles I've owned had probably been that far when I bought them! So the closest astronomical object to Earth is an imaginable distance away. The nearest star, our Sun, is about 150 000 000 km how to we relate to that? The distance to the next nearest star, Proxima Centauri, is about 40 000 000 000 000 km. The number of noughts is already getting out of hand and this is only the nearest star, so for this reason astronomers use light-years as a measure of interstellar distances. Proxima Centauri's distance is about 4.3 light-years by the way. But this doesn't really help us in to coming to terms with these measurements and how they relate. If we really want to get a grip of really big, and really small numbers, that are beyond our normal everyday experience we need scientific notation! A system based on the powers of 10


Instead of writing 100 we write 102 because 10 x 10 = 100 = 102
or 1000 we write 103 because 10 x 10 x 10 = 1000 = 103


You may have noticed that all you have to do to convert from conventional numbers to powers of 10 is count the noughts and write 10 and superscript the number of noughts you counted. To cope with all the numbers between the nice big round numbers we combine ordinary numbers and powers of 10 together.
Instead of writing 250 we can write 2.5 x 102 (2.5 hundreds)
or instead 7750 we write 7.75 x 103 (7.75 thousands)
This way of writing numbers is known as scientific notation. It may seem a bit pointless with numbers of a few hundreds or thousands but it really does help in dealing with the very big or the very small. Lets go back to the distance to the Sun and next nearest star.

Instead of writing 150 000 000 km we write 1.5 x 108 km for the distance to the Sun
and instead of 40 000 000 000 000 km we write 4.0 x 1013 km for the distance to Proxima Centauri..

Just a quick look at those numbers and by just looking at the superscript numbers (8 and 13) we can immediately get a feel for the amount of difference, that is 13 - 8 = 5 orders of magnitude or 100 000 times. Using this scientific notation also means we can divide (or multiply) these numbers quite easily. Maybe you would like to know how many Earth – Sun distances (1 Astronomical Unit) is it to the nearest other star?
Distance to Sun is 1.5 x 108 km
Distance to Proxima Centauri is 4.0 x 1013 km

to divide 4.0 x 1013 km by 1.5 x 108 km
first divide 4.0 by 1.5 and get 2.33
then just subtract the powers of ten 13 – 8 = 5
so our answer is it is 2.33 x 105 Earth – Sun distances to Proxima Centauri
Or in conventional number notation it is about 233 000 times further away than the Sun is.
Scientific notation is also very good at dealing with very small numbers. Lets ask another scale question such as “how much bigger is a pinhead than an atom?”

Lets assume we have a pinhead and we measure it and find conveniently it is 1mm across.
Being scientifically minded we want to write this down in metres and in scientific notation. A millimetre is 1/1000 of a metre that is 0.001m or in scientific notation 1 x 10-3 m.
So our pinhead is 1 x 10-3 m.

To know the actual exact size of any atom is a question way too complicated to deal with here so please accept that roughly approximately an atom of iron is 1 x 10-11 m across.

Pinhead = 1 x 10-3 m.
Atom = 1 x 10-11 m
From these figures we can see a pinhead is 8 orders of magnitude bigger than an atom.
That is 108 or 100 000 000 times bigger than an atom.

Try putting that in everyday object sizes! If an atom was the size of a pea, say 1cm (1 x 10-2 m), then a pinhead would be 8 orders of magnitude bigger, that is 1 x 106 m or 1 000 000m or 1000 km across. That's about 625 miles in old money!

So to get to grips with the scale of the universe all we need is the power of 10!

To see this power illustrated beautifully watch this short video

Power of 10


or the Power Point Presentation

Fantastic Trip

The power of 10 is, with just a little practise, a way to come to some kind of feeling for the true scale of the universe we live in, to imagine the unimaginable.