## Will we ever stop having solar eclipses because of the moon's motion away from the Earth? (Intermediate)

I read somewhere that since the moon is receding from the earth, someday there will be no more total eclipses of the sun. Do you know when that will occur?

It won't happen for a long while, 500 million years or more as best I can tell!

The earth-moon and earth-sun orbits are both elliptical, so the distances between these objects change over the course of an orbit. The "best" possible time for a solar eclipse is when the earth and moon are as close as they possibly can be (so the moon appears large in the sky) while the earth and sun as as far away as they possibly can be (so the sun appears small in the sky). It turns out that at this point, the size of the moon in the sky is about 4.6 percent larger than the size of the sun.

This means that the average earth-moon distance will need to increase by 4.6 percent for there to be no more total solar eclipses - when this happens, the moon will always be smaller than the sun, even at the most favorable time for an eclipse.

4.6 percent of the earth-moon orbital distance is around 17,000 kilometers, and given the current rate at which the moon moves away from the earth (3.8 centimeters per year), it will be around 500 million years before the moon has reached the required distance.

That is a lower limit, I think, since in reality, the rate at which the moon moves away from the earth will slow down with time. (Basically, it's the effect of tidal bulges on the earth's surface pulling on the moon that cause its orbit to increase, so as the moon moves farther away, these forces get smaller, and its orbit won't increase as fast.) It's a little too involved for me to look up all the equations and try to calculate a more accurate number, but I would guess something like 1 or 2 billion years, maybe.

Incidentally, the moon won't move away from the earth forever. The whole process will stop around 15 billion years from now, when the earth's rate of rotation, the moon's rate of rotation and the moon's orbital period are all around 55 days long - at this point, the moon will be about 1.6 times farther from the earth than it is now (see this website for more information -- this also assumes that the earth and moon actually exist for 15 billion more years, which they probably won't due to the Sun's evolution into a red giant star).