Can one million Earths fit inside the sun? Yes & no!


The Sunday Nation


19 February 2023


A reader asked me to confirm whether a post he saw on Twitter claiming that 1.3 million Earths can fit inside the sun was correct. Well, it is (somewhat) true; I wrote about it in October 2012.

At 1.5 million kilometres, the diameter of the sun is about 109 times that of Earth (12,800km). Since the volume of a sphere is proportional to the cube of its diameter, it turns out that the volume of the sun is 109 x 109 x 109 = 1,295,029 (about 1.3 million) times that of Earth.

However, this calculation assumes that the 1.3 million Earths are melted and poured into the sun. Earth is spherical and when spheres are packed in a closed place, they leave a lot of empty space. The volume of that unoccupied regions depends the arrangement of the spheres.

Suppose that 1-cm radius (2-cm diameter) spheres are arranged in a cubic pattern; that is, with their centres are at each of the 8 corners of a cube. In that arrangement, the cube will have sides measuring 2cm each and only an eighth (NB: not a quarter) of each sphere will be inside the cube.

The volume of the cube will be 2 x 2 x 2 = 8 cubic cm (8cc). The volume of each sphere is “four-thirds-pi-r-cubed” which comes to about 4.19cc. Now each sphere contributes an eighth inside the cube so eight eighths make a whole. Thus, out of the 8cc volume of the cube, only 4.19cc is occupied by the spheres. This comes to about 52 per cent. The rest (48 per cent) is empty space!

When this cubic arrangement is repeated over a wide space, the fill-factor of 52 per cent is maintained throughout since every cube is identical to all others. Obviously, this is not a very efficient way to arrange spheres. If the Earths are arranged this way inside the sun, only 678,000 would fit.

The most efficient arrangement would start with three spheres arranged in a triangle and then a fourth one on top and the at the centre to form a triangular pyramid. This is also known as a tetrahedron, or a tetra pack. The calculation of the fill-factor is more involved than that of a cubic pattern, but it turns that to be about 74 per cent. Thus, the largest number of Earths that can fit inside the sun is about 958,000…perhaps we call that one million.

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