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Total solar eclipses occur because the Moon and the Sun have the same apparent size in Earth's sky — the Sun is about 400 times wider than the Moon, but the Moon is about 400 times closer.

It's my understanding that the Moon is currently receding from the Earth at about the rate our fingernails grow (about 1.6 inches, or 4 centimeters, per year). Lunar retreat rates in the distant past may have been slower than this.

At some point, the Moon will no longer be able to fully block the Sun. (This question on StackExchange addresses the period from now onwards; one solution offered is that the Earth will have its last total solar eclipse in about 600 million years.)

However, what I'm interested in is determing the total duration of that period when it will ever be possible to perceive the Sun's corona (and other effects such as Baily's beads) from Earth by means of a total solar eclipse. This requires knowing both how long it is until the last eclipse will occur (~600 million years) and the time that has elapsed since the first such event happened. Note that the critical period I'm trying to discover is that window of time during which the apparent sizes of the Sun and the Moon, as seen from Earth, are approximately the same size, as they are now.

(In case anyone's interested, my reason for asking this is to try to determine the probability that we humans should have evolved at that point in time when it's possible to perceive a total solar eclipse, to further a pet theory of mine that suggests that it may not be a coincidence.)

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  • $\begingroup$ The unthinking answer to your question is given in your question itself, "600 million years". So for those of us who don't think, could you please clarify? Is it that you are asking for duration backwards as well as forwards, "from X years ago to 600 million years' time"? Is it that the time of the final solar eclipse is not exactly the same as the time of the final sighting of the solar corona? $\endgroup$ Dec 9 '20 at 15:35
  • $\begingroup$ @MartinKochanski Many thanks! I hadn't spotted that this was unclear. I've edited the question, hopefully making it clearer that what I'm trying to find is the "duration backwards as well as forwards, from X years ago to 600 million years' time," as you say :) $\endgroup$
    – Pendantry
    Dec 9 '20 at 16:22
  • $\begingroup$ The first such event happened near when the moon formed. $\endgroup$
    – Joe
    Dec 10 '20 at 0:50
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    $\begingroup$ I think I like your pet theory. :) In any case, I'm sure total eclipses were rather awe-inspiring to proto-human proto-scientists. They can have a big emotional impact even on scientifically educated modern humans. $\endgroup$
    – PM 2Ring
    Dec 10 '20 at 0:53
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    $\begingroup$ "Exactly the same" sounds suspicious. The apparent size of the Sun varies by ~3% and that of the Moon by ~12%. $\endgroup$ Dec 10 '20 at 13:18
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There are a lot of competing models for how fast the Moon is retreating from the Earth, but they all agree the rate of distance increase is non-linear. That means any answer to this question is going to involve a lot of error. I think there are two different questions here and I will address them separately:

1. How long ago could the whole circumference of the Corona be seen from a hypothetical observer on Earth?

No matter how close the orbiting moon was to the Earth, Bailey's beads and some of the corona would be visible just as the Sun became fully obscured by the Moon.

If you are wondering when the entire circumference of the corona might be visible during a solar eclipse, NASA says the corona extends to about 12 solar radii.

enter image description here

So the whole circumference of the corona presumably wouldn't be visible if the moon was 12 times closer to the earth. There a few different guesses for the Earth-Moon distance in the past, and Franco and Hinnev show Webb's TD (Tidal Dissipation) model here contrasted against constant extrapolated recession models:

enter image description here

Here, some of the points on the plot denote geological tidal evidence for Earth/Moon proximity. If we believe their TD model, the answer is about 3.9 billion years.

1. How long ago could the angular diameter of the Sun and Moon be the same in the sky as viewed from Earth?

Remember that angular diameter of a sphere is $\delta = \arcsin(d/2D)$, where $d$ is the diameter of the sphere and $D$ is the distance from the observer to the sphere. The angular diameter of the Sun as seen from the Earth is then $\delta_s = \arcsin(d_s/2D_s)$ and the angular diameter of the Moon is $\delta_m = \arcsin(d_m/2D_m)$, where the subscripts $s$ and $m$ denote the Sun and Moon. Setting them equal and simplifying yields the common rule of thumb for determining if an exact full eclipse is possible: $D_m/d_m=D_s/d_s$.

Wiki gives Earth's perihelion at $D_s = 1.47095e8 km$, the Sun diameter $d_s = 695,700 km$, the Moon diameter $d_m = 1737.4 km$ so solving for $D_M$ gives us 367,346 km. This means that the Moon would block the Sun entirely if the Earth was at perihelion and the Moon was 367,346 km from the Earth and all three bodies were colinear.

The current moon's apogee is 405,400 km, so it still would barely eclipse at 38,054 km closer. This is about 6 Earth radii, so if the above chart is to be believed using their 'TD' model, the earliest time the sun and the moon could have had the same angular size is 1.2 billion years ago. Of course, this guesstimate doesn't account for the fact that the sun's size isn't constant. It also doesn't account for the fact that the Moon and Sun aren't perfectly spherical.

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