Modelling the Solar System in School. An activity for using maths in science lessons

This is another blog post featuring an activity which uses maths in science teaching.  This one is about making a scale model of the Solar System.  

The Solar System, at least in terms of the main objects of the Sun and the planets, might not be regarded as a problematic thing to teach about.  There are mnemonics for helping children learn the names of the planets in order of their distance from the Sun, and data can help provide some sense of relative sizes and distances.  The thing that is difficult, however, is to get a sense of scale.  

Textbooks will often show images of the planets, with them all on the same diagram to show the order of average distance from the Sun, and usually their relative diameters.  Do children realise these diagrams do not also represent the scale of average distance from the Sun?  I would hope so, but it can't be guaranteed without explicitly finding out.  Just how big would the page have to be if we wanted to accurately represent the scale of both the planets' sizes and their average distance from the Sun?  This is in essence what this exercise is about.

The way I'm going to consider this is to suggest we have a school field that is 450m in length.  At one end of the field is the centre of the Sun.  At the other end is the centre of the furthest known planet, Neptune (we'll not count Pluto, unfortunately relegated to the status of dwarf planet).  This will give us our scaling factor, because we know the actual average distance from the Sun to Neptune, and we are basically scaling this distance down to 450m - quite a significant shrinkage.

Here are the data we'll use:

It's perhaps clear why I've suggested the length of the school field is 450m (look at Neptune's mean distance from the sun) - it will make our scaling factor somewhat easier to handle as it'll be a nice round number.  Obviously you would need to measure or estimate the size of the field you're going to do this on, but many big fields or parks can accommodate 450m pretty easily.

Calculating the scaling factor:

The scale is 450m : 4,501,000,000,000m  (I'm multiplying the distance from the Sun to Neptune by 1,000 to convert it into metres - we have to be working in the same units).  To work out what 1m is on our scale I just need to divide each number in the scale ratio by 450m.  

This gives an approximate scale of 1 : 10,000,000,000  (1 : 10 billion).

In other words, we need to divide every number given in the data in the table by ten billion to work out what it would be on our school field sized model.  There's a relatively simple way to do this (because a calculator's display isn't big enough to handle the numbers): simply move the decimal point ten places to the left.  

Before I do this, I'm also going to multiply every number in the table by 1,000, just so they are all in metres.

The scaled down data 

These are the numbers we get from scaling down by a factor of 10 billion:

It's clear to see that to actually make a model on this scale, the smaller planets would be little more than tiny dots drawn on a piece of card.  Mercury's dot would be half a millimetre across, which would indeed be difficult to measure accurately.  

How big would the Sun be?
The Sun's actual diameter is 1,392,000 km.  This is 1,392,000,000 m.

Hence on our school field scale, dividing this by 10 billion gives a size of 13.9 cm.  Hence you could represent this with a large grapefruit or a small melon.  

Thinking points for children

Doing an exercise like this does not have to involve children doing all the calculations (although they could do).  The activity could be scaffolded so that some of the scaling is already done.  The more powerful learning and thinking would come from actually taking children out on to a large school field and posting them at the positions of the Sun and the planets to give them the sense of how much closer to the Sun the 'inner' planets are, and how far away the outer planets are.  Following my suggestion, above, about drawing circles on pieces of white card, the children positioned at the Earth could look out towards the distant planets and see whether it is possible to actually even see them with their naked eyes.  They could be reminded that when looking at the night sky there will thousands of other, brighter, objects in the sky too,making it very difficult to see any of the planets apart from those closest to us.  This could lead on to a discussion about the prospects of humans ever being able to travel beyond our solar system, given the huge distances involved.

Extending the scaling further

Our Solar System is huge, but even that pales into insignificance next to the distances within our galaxy, and obviously these are then tiny compared to inter-galactic distances.  The next nearest star to Earth after the Sun is called Proxima Centauri.  Its distance from Earth is 4.243 light years.  Where would it be on our scaled down model?

This calculation generates some very large numbers, and it's best to use the number convention of standard form in order to keep them manageable.  

First, we need to know how far 4.243 light years is in metres.

1 light year = the distance light travels in one Earth year.  

Light travels at a speed of 299,792,458 m/s.  

One year is equal to 365 days; each of 24 hours; each hour is 60 minutes; each minute is 60 seconds. 

Hence in one year light will travel 299,792,458 x 365 x 24 x 60 x 60 m 

Therefore 1 light year = 9.454255 x 10¹⁵ m

Therefore distance to Proxima Centauri (4.243 light years) = 4.243 x 9.454255 x 10¹⁵ m

= 4.0114403 x 10¹⁶ m.                     

Scaling this down to our school field size model (dividing by ten billion) would put this star at a distance of 4.0114403 x 10⁶ m from our Earth model.  That is a whopping 4,011 km away.  In other words, if you made the scale model in London, Proxima Centauri would be in the USA!  And that is just the next closest star.  It makes you think... 

Extending children's thinking - and avoiding misconceptions

The distances

It's easy to overlook where misconceptions might arise in the exercise above, and as teachers we should always be thinking about this sort of thing.  The most obvious one is that although we are creating a scale model (with a reasonable degree of accuracy), the model has a high degree of artificiality about it.  Principally this is because we have lined the sun and the planets up in a straight line.  In reality, at any given moment the planets are at particular positions in their orbits around the sun, and this means they are in fact scattered all over, including 'behind' the sun, by which I mean on opposite sides of the sun to each other.  Hence although the distance from the individual planets to the sun remains within relatively small limits defined by their elliptical orbits, the distances between the planets themselves varies much more depending whereabouts on their respective orbits they are.  How would you teach children about this?  I suggest this is where a model orrery would come in handy - a model which dispenses with scale but which demonstrates movement and relative positions well.  

Orbital plane

Another aspect to get children thinking about is the orbital plane of the planets.  It is rather interesting that the known planets of our solar system occupy roughly the same orbital plane.  It is as though they are sitting on the same turntable with the sun at the centre.  This gives some clues to scientists about the origins of the solar system.  It also means however that if we imagine the solar system as a 3D sphere with the sun at the centre, there is a vast amount of space within it with no planetary orbits - the whole remainder of the sphere apart from the known orbital disc of the planets.     

Direction of orbit

The planets all orbit in the same direction.  This is something children could be asked to think about and maybe speculate reasons for.  The reason scientists suggest is because of the formation of the planets from the original dust cloud orbiting the younger sun. 

Other non-planetary objects

Just teaching children about the planets can create a view of the solar system as a somewhat simple thing, but there are many more things present and happening which add up to a rather more bustling and complex picture.  Where would we start?  Perhaps one of the first places might be the moon, and the question of other planets having moons - something children might not be aware of.  Galileo's observations of the moons of Jupiter is one of the key milestones in the growth of knowledge about the solar system, but was also one of the things that got him into trouble with the Catholic church, which was far more powerful then than now.  The story of Pluto might be a place to move on to.  At one time considered a planet, it is now thought to be too small and is referred to as a dwarf planet.  It has several moons.  Then there are the trillions of smaller objects such as asteroids and comets.  This sort of stuff would make a good stimulus for children to go off and do their own independent research.  There are plenty of good internet sources.  I've listed some here:

NASA:  solarsystem.nasa.gov/

National Geographic:  www.nationalgeographic.org/topics/resource-library-solar-system/