If you calculate the orbital period of an earth sized object around the sun at 1 au, it is 31554651 in seconds (1 year). (Why does Google say a year is 31536000 seconds?!)
If you calculate the orbital period of a moon sized object around the sun at 1 au, it in 31554698 is seconds, that is 47 seconds longer than that of the earth.
That means that the moons orbit is 1.48947932e-6 times longer than earth's. Which means in 1.5 million years the moon would fall behind the earth by an entire orbit, hence it would take a faction of that time for the moon to fall behind the earth enough to no longer orbit the earth.
Is my math wrong? (I did double check it.) Is my understanding of physics wrong? Or is there some strange effect that is causing the moon's orbit to match that of the earth's?
EDIT
So the moon does not fall behind, that is good, I am rather fond of our moon XD
You answered part of my question.
How is the moon captured by the earth? The fact that the moon is close to earth does not mean the sun no longer pulls on it. If that were true the moon would spiral straight and not follow the earth in its path around the sun. Gravity pulls the moon toward the earth (technically both, together), the centripetal force, constantly pulls the same exact amount on the moon (assuming no imperfections in the shape of the earth and moon); when the earth moves toward the sun, from the sun's pull on it, why does the moon get pulled along as well, and more so than its own pull from the sun?
What additional force is causing the moon to stay with the earth? Or is it just a remarkable quirk in how the math works out where the pull of the sun varies as the moon orbits earth, causing it to end up moving along faster?
Best Answer
Expanding my comment into an answer: At any given separation, the common period of revolution of two objects is indirectly proportional to the square root of the sum of their masses. See the Wikipedia plot summary on the gravitational two-body problem for more details.
The Sun has a mass about $300\,000$ times that of the Earth; the Earth has a mass about $81$ times that of the Moon. Therefore, the Sun-Earth system "outmasses" the Sun-Moon system by very nearly one part in $300\,000$ (close enough for our purposes). Thus, we would expect the Sun-Moon system (if isolated) to have a period longer than the Sun-Earth system (if isolated) by about one part in $600\,000$. That is approximately what you found.
The reason the Moon doesn't in fact fall further and further behind is that the systems are not isolated. The Moon is gravitationally bound to the Earth, and is in some hand-wavy sense dragged along with it in their common orbit around the Sun.
To answer your other question: Google's answer of $31\,536\,000$ seconds is obtained by multiplying a simple year of $365$ days by $86\,400$ seconds per year. The discrepancy is mostly covered by the leap year system.