Why does the sun have to be a lot further from us than the moon? A deduction.

Remember the hands-on demo of the phase of the moon?

Mondphase
In both pictures: Model of the moon between my fingers on the left, and moon in the background on the right. See how the lit and dark sides of both spheres are in the same position?

Holding a sphere up in the sunlight in the direction of the moon, the sphere will show the same phase as does the moon. Of course it has to, because the sun is so far away that its rays hitting the moon and the ones hitting the sphere are pretty much parallel.

If the sun wasn’t so far away, what would we see?

Schematic of how the Earth, your little sphere you are holding up in the direction of the moon (marked X) and the Moon would be lit if the Sun was not very far away (left) and very far away from Earth and Moon. See how the phase of the moon differs from that of your little sphere when the sun is “close”?

So the only way we can explain that the lit and dark sides of the sphere and the moon are the same is that the light lighting both of them comes in parallel, which can only be the case of the sun is very very very far away compared to the distance of earth, sphere and moon.

Isn’t that a nice little thought experiment?

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