The Moon and Sun are the same apparent size in the sky. The Moon is moving away from us at 3.8 centimeters/year. Isn't it improbable that we live in an epoch in which they appear the same size in the sky?
Given that the Earth-Moon system is a "double planet" with improbable sizes, isn't this whole arrangement HIGHLY IMPROBABLE?
Highly improbable and improbable are awfully hard to quantize. First, I've never heard the Earth-Moon be described as a "double planet," but I suppose its a fair enough definition. After all, we believe the moon became captured by the Earth early in the solar system formation. So, if the moon hadn't been captured, perhaps it would have become its own planet. Calling it a satellite, of course, is a naming convention. And besides, perhaps the moon only stayed its size because it had been captured by the Earth. Perhaps it would have grown larger on its own. In fact, this is why moons tend to be so much smaller than their planets. So, probable, improbable, it's a matter of how its use is intended.
The apparent sizes of the Sun and Moon (i.e., that they are nearly equal, and align so well during an eclipse) is a coincidence. In other words, there's no known mechanism that would tend for this to happen more than chance.
You mention that the Moon is moving away, but it's moving at 3.8 centimeters a year. At that rate, it would take more than 10 billion years to double its orbital radius (it's so slow!) and cut its apparent area down by a factor of four. So given the moon's size, we had a pretty good chance of seeing it as it is now. But you strike on a very important point in the search for new planets: the longevity. Suppose a very massive planet (20x bigger than Jupiter), when it forms, undergoes a few million year period in which it becomes so bright that it could easily be seen from Earth. Why would we have not observed this? Perhaps because -- lasting only a few million years -- as compared to the distribution of stars we see -- ranging in ages from a few million years to 10 billion years -- means we have a low chance of catching any one of those stars producing a planet at that special moment. If, however, that formation lasted ten or a hundred times as long, it'd be the difference between maybe seeing it, and seeing it dozens of times.
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