How many Earths can fit inside the Sun?
The Sunday Nation
Nairobi,
28 October 2012
Joseph Kimani was watching a documentary on TV recently and he heard the presenter say that one million Earths can fit inside the Sun. He writes: “This sounded unbelievable especially because I remembered that you had written a while back that the sun is 100 times bigger than the Earth. What is the correct position – is one million or one hundred?”
My response will probably sound even stranger and more unbelievable: Both positions are correct! What I wrote is that the diameter of the Sun is about 109 times that of the Earth. The Sun measures about 1.4 million kilometres across while the Earth is 12,800km.
What this means is that if you lined up 109 bodies each the size of the Earth, they would form a continuous line whose total length would be equal to the diameter of the Sun. But that doe NOT mean that they would completely cover the Sun from view – obviously not!
The quick way to find out how many Earths can cover the Sun is to start by working out the cross-sectional areas of the two bodies. The formula is the well-known “pi-r-squared’. But bear in mind that the figures given above are diameters, that is, double the radius.
The results are 129 million square kilometres for the Earth and 1.54 trillion sq-km for the Sun. Note that these are the cross-sectional areas of the bodies; they are NOT the surface areas!
The next step is to simply divide the cross-section of the Sun by that of the Earth; that is, 1.54 trillion divided by 129 million. The answer is approximately 12,000. There is a quick way of getting to this result without having to work out the areas: it is the square of 109.
Now this is an “idealised” result because it is based on the assumption that the smaller circles will not leave any spaces between them. But that is not true: Circles do not tessellate.
The tightest arrangement of circles leaves about 10 per cent uncovered. Therefore, we should reasonably expect that about 10,800 Earths (90 per cent of 12,000) can shield the Sun from view. But that number is quite far from the one million quoted in the TV documentary.
The reason is that we have treated the two bodies as circles while they are actually spheres. So, we should be concerned with their volumes; not their cross-sectional areas.
The volume of a solid is given by the well-known formula: “four-thirds-pi-r-cubed”. The result for the Earth is about 1.1 trillion cubic km and that for the Sun is 1.44 million-trillion cubic km. Dividing the two as before we find that the volume of the Sun is about 1.3 million times that of the earth. There is a short-cut method to get this result as well – it is the cube of 109.
Again, we have to bear in mind that spheres do not tessellate. It turns out that the tightest arrangement of spheres leaves about 26 per cent of the space unoccupied. Thus we can realistically hope to fit about 960,000 Earths inside the Sun. Therefore, the TV documentary was not very wrong in calling it one million.
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