Is there a possible explanation for the apparent equal size of sun and moon or is this a coincidence?
(An explanation can involve something like tide-lock effects or the anthropic principle.)
It just happens to be a coincidence.
The current popular theory for how the Moon formed was a glancing impact on the Earth, late in the planet buiding process, by a Mars sized object. This caused the break up of the impactor and debris from both the impactor and the proto-Earth was flung into orbit to later coallesce into the Moon. So the Moon's size just happens to be random.
Plus the Moon was formed closer to the Earth and due to tidal interactions is slowly drifting away. Over time (astronomical time, millions and millions of years) it will appear smaller and smaller in the sky. It will still always be roughly the size of the Sun but total solar eclipses will become rarer and rarer (they will be more and more annular or partial). Likewise in the past, it was larger and total eclipses were both longer and more common.
Yes, similar apparent size equates to similar tidal forces from astronomical bodies. The fact that the moon exerts similar tidal forces as the sun is correlated with the existence of life on Earth.
I will write a short list of parameters. These are attributes of a planet that may reasonably be necessary for life to evolve. Let me be clear that these are not understood well, but that's the job of exo-biology among many other sciences.
Now, we must also consult the numerical evidence. The day length today is 24 hours by definition. Around 3 billion years ago, the day length was closer to 17 hours, give or take an hour (I've referenced a paper that says this in a prior answer, so I might be able to find it later). The reason the day became longer was due to the Earth throwing the moon further out, as well as the interaction with the sun.
Let me introduce another constraint. No planet is formed from the proto-planetary disk with a rotation anywhere close to 2 hours. This value of 2 hours is special in terms of universal constants because it represents (roughly) the shortest day length at which a body may hold itself together through self-gravitation. The argument can be made looking at the divergence of the gravitational field compared to that of the false gravitational field from acceleration. The divergence of the gravitational field reduces to a simple term containing only density. The false field from acceleration can be found from the rotational potential $d/dr 1/2 \omega^2 r^2 = \omega^2 r$, and the divergence of this goes to a constant $\omega^2$. Equate the two divergences, assume Earth density, rearrange for period, and you find a value like 48 minutes. This is the period at which the average surface gravity is zero. Of course this is absurd because that would imply negative gravity regions, which is impossible for large bodies. Practically, you set a better limit by asking for the rotation at which the equator on a sphere would have zero gravity, and this would be close to two hours. This is wrong because the planet deforms, but that is a very mathematically challenging complication. In reality though, you would never approach the 2 hour spin in the hectic planetary formation disk.
What I'm getting to is that you extrapolate from two different directions. One, our predicted spin of proto-Earth, and the speeds we would reasonably expect from an early planet, working back from the limits imposed by gravitational divergence. You probably get something close to 12 hours on the Earth historical limit, and something close to 6 hours with the physics limit.
My lengthy wording here is to answer "why not faster?" The answer is because of physical limits. It is difficult to envision a planet starting with a rotation much faster than the Earth, and certainly not by an order of magnitude.
The question of "why not slower?" is easier to answer from a biology standpoint. Complex life on Earth has always been challenged to cope with the varying day/night temperature differences, as shown in the different (cold-blooded vs warm-blooded) approaches to the problem. Any longer day would make this challenge all the more difficult for life, and could prevent its emergence altogether.
Now let's use math. An astronomical body has a certain diameter $\theta R$, the angle we see it occupy times its distance from us. The density of astronomical bodies varies, but not by an order of magnitude. In the case of the sun and the moon it's about a factor of two. To keep the answer short, we'll call it even, if we take the cubed root it won't matter much. So now we can approximate the relative mass.
$$ M \propto \theta^3 R^3$$
The force of gravity is a $1/r^2$ force, this tells us that the force of the sun on Earth is greater than the force of the moon on Earth. This is correct. The tidal force, however, varies by $1/r^3$. This tells us that the tidal force from the moon on the Earth is about the same as the force of the sun on the Earth. This is also correct.
Whether the sun is closer or further is irrelevant with this simplification. The only parameter that matters to tidal forces is the distance to diameter ratio. Now we can start asking:
Why not a larger (apparent) moon - The empirical arguments here clearly point to the fact that a larger moon in terms of appearance would result in a longer day length, and this would not be as good for the evolution of life. You can't solve this problem by starting out with a faster rotation at formation because gravity isn't strong enough. It turns out the moon slows down the Earth at about the same rate as the sun, with other anthropic factors wanting to see a larger moon, this is the point of diminishing marginal returns for getting the shorter day. It would be unhelpful for the moon to appear much smaller because Earth would still only rotate a few hours faster since the sun slows the Earth down anyway.
Why not a smaller (apparent) moon - Not my area, but probably asteroid protection. With asteroids as major a threat as they are, life probably needs a moon as large as possible, and it looks like that's what it got.
Both the moon and the earth's orbits are eccentric, and so the ratio between the sun's and moon's apparent diameter varies with the time of year. When the moon is at perigee, and the earth at aphelion, the moon will seem larger than the sun than when the moon is at apogee and earth at perihelion.
However, the eccentricities of these orbits are low, and the moon always seems "about the same" as the sun. This is a coincidence, both that its size is what it is, and that we're here to observe it. The moon is and will continue to recede from the earth. Eventually, the moon will appear smaller than the sun's disk and won't be able to completely eclipse it anymore.
It seems like this question has its roots in intelligent design (I've heard this argument made in favor of ID before). Were the moon designed to be the same size as the sun's disk, you would think that the earth and moon's orbits wouldn't be eccentric, and that the moon wouldn't gradually receded from the earth, in order to preserve that symmetry.
The real value of the moon vis-a-vis life is that its large mass stabilizes the earth's obliquity (the 23.5deg tilt to the sun that stabilizes climate). Mars on the other hand, has effectively no moon, and has a chaotic obliquity, which can wreak havoc on long-term climate.
Of course the easy answer is "Purely a coincidence" but for those that think there is no such thing, the situation does lend itself to some fascinating speculation. In this case, let us rephrase the question, "With the Moon slowly and steadily orbiting farther and farther away from the Earth [It's picking up angular momentum from Earth's rotation], why is it that sentient, arguably civilized creatures happened to evolve at just the precise geologic moment that the two most prominent objects in the sky are approximately the same angular diameter?"
Without venturing too far into "2001: A Space Odyssey" territory, it's not out of the realm of possibility to wonder if the spectacular phenomenon of total eclipses might have nudged a fledgling creature in the direction of more sophisticated thought processes or a primitive tribe into considering more carefully the passage of time and patterns of nature, a critical component in first developing agriculture.
I can't imagine any kind of anthropological proof more convincing than noting the coincidental timing as above, so I think the theory will always be nothing more than a speculative flight of fancy. You never know, though. ;)
Don't take this too seriously, or anything but a poetical divagation.
maybe the presence of a corona in solar eclipse and the existence of an almost perfect sun-light-day and moon-darkness-night symmetry was a trigger of abstract thinking in the primitive mammals that were ready for benefit from such stimulus, and the first suggestion of the existence of profound symmetries in our universe.
According to the above, even if the symmetry is just an illusion, it was the trigger for primate brains to conceive religious abstract thought. Religious thinking maybe is so deeply ingrained in our genomes precisely because it provided the right mixture of social cohesiveness and the roots of abstract thinking required for survival enough to stablish agricultural societies.
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