In order to tackle this question, we have to understand a bit about the geometry of the solar system, and how both the planets and the moons which orbit those planets behave.
Most of the planets in the solar system circle the Sun in a very thin plane, meaning that if you drew the orbits out on a sheet of paper, you wouldn’t be missing out on any hidden geometry of our solar system. With the exception of Pluto, there’s very little vertical motion in the solar system that would be obscured by drawing it out on flat paper.
Moons are under no particular obligation to follow this pattern, and it’s often thought that if a moon of a particular planet is doing something particularly odd with its orbit, we can use that information to guess that it might have arrived at that planet in an unusual way, rather than forming around that planet. Neptune’s moon Triton is a good example; not only is it angled quite sharply with regards to the plane of the solar system, but it also goes “backwards” - it orbits in the opposite direction of Neptune’s rotation. These have been taken as hints that Triton didn’t form around Neptune, but formed elsewhere, and got trapped around Neptune after being jostled too near to Neptune's gravitational well.
If the Moon happened to orbit in a circle the way a hula hoop rolled on its edge moves, forever tumbling along the direction of the Earth’s travel, then the Moon would never get any closer to the Sun at any point in its orbit. These sorts of orbits aren’t impossible, though in our solar system, they're non-standard.
Our Moon's orbit is, in fact, quite close to perfectly flat with respect to the direction that the Earth travels. It’s tilted by only five degrees relative to Earth’s orbit around the sun. If your arms, like mine, are about six feet from fingertip to fingertip, five degrees is about 3 inches away from horizontal. If you hold both arms out sideways, point one index finger up, and one index finger down, the tips of your fingers are about five degrees offset from the line drawn by your arms.
Five degrees of an offset means that the distance between the Moon and the Sun will vary almost exactly by the distance between the Earth and the Moon. Everything is moving in the same plane, so drawing it out on a sheet of paper won't be ignoring much geometry. The Moon’s orbit is also quite close to circular, which again helps with this - there’s no long, comet-like orbit for our Moon, which is why we see it as very nearly the same size in our skies. So with all that behind us, how close could the Moon get?
The Moon orbits the Earth at a distance of about 238,900 miles from our home planet. The Earth, in its turn, orbits the Sun once every year (by definition), at a distance of about 93 million miles from the Sun. Because we know the distance between the Sun and the Earth, and the distance to the Moon from the Earth, if we line everything up just right, then we can place the Moon directly in between the Earth and the Sun. We know this situation happens - this is how we get solar eclipses, when the Moon lines up exactly between the Earth and the Sun.
This configuration subtracts 238,800 miles off of the 93 million miles which separate the Earth from the Sun. So in the end, even though we have a pretty ideal setup, the Moon can’t ever get that much closer to the Sun. At best, the Moon manages to get a grand total of 0.25% closer to the Sun than the Earth.
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