Will the Moon be invisible in 500 million years?
Do you need to correct the word going around from a professor in India that the Moon will be totally invisible in half a billion years when it is only 10,000 miles further distant than it is now from us? Or maybe I am missing something.
You aren't missing anything -- the Moon will most certainly not be invisible in 500 million years! As stated in a previously answered question, the fact that the Moon is moving away from the Earth at a rate of around 3.8 centimeters per year might mean that we will have no more total solar eclipses (when the Moon passes in front of and entirely blocks out the disk of the Sun) in 500 million years or a bit longer, but that is a totally different matter.
As you rightly suspect, changing the Moon's distance by a few percent won't have any significant effect on our ability to see it. In fact, the Moon's distance from us changes by a few percent every month as it orbits us (see a previously answered question for more information on this), and that certainly doesn't affect our ability to see it! Changing the Moon's average distance by a few percent (which is what will happen over the next 500 million years or so) will similarly not prevent us from being able to see the Moon, and to see it quite easily with the unaided eye.
To get an appropriate perspective on this, consider the fact that the amount of light we now receive from a full Moon is tens of thousands of times more than what we get from even the brightest star in the night sky, Sirius. The Moon would have to move to many times its current distance for it to even drop below that level, let alone be so faint that we couldn't see it at all.
The only real difference we can expect to notice 500 million years from now is that the average size of the Moon will appear a few percent smaller. The average "surface brightness" of the Moon (basically how bright it appears when we look at it) will be the same as it is now -- this is because although we will be receiving less total light from the Moon since it is farther from Earth, that light will get concentrated into a smaller region of our field of view, and the two effects cancel out. (This is true for any object that's big enough for your eyes to resolve -- for example, the computer screen doesn't look fainter when you move your head farther away from it.)
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- Why do the size and brightness of the full moon change?
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- Will we ever stop having solar eclipses because of the moon's motion away from the Earth?
- How can we see galaxies if their stars are so faint?
- How close was the Moon to the Earth when it formed?
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