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Wow, I wonder what the max duration of totality was back then? Currently, it's around seven and a half minutes.
As it so happens, I had just been working on a program to compute the evolution of the Moon's orbit, and from which get the duration of totalities.
Here's what I find:
Totalities lasted much longer in the past, and in the future there will only be annular eclipses (which can be thought of as a totality with negative duration, as the smaller disk of the Moon crosses in front of the Sun.)
How did I calculate this? For the sake of sanity, I made a large number of simplifying assumptions (like treating the Earth and the Moon's orbits as circular and coplanar, and ignoring contribution of solar tides, obliquity changes, and so forth.) Also, computing how the Moon's orbit changed over time
is somewhat complicated, so I instead find it as a function of a conserved quantity: the angular momentum. I transfer some angular momentum between the Earth's spin and the Moon's orbit to see how both change, and from there can find relevant quantities as a function of the Moon's orbital distance. With some geometry it is then fairly simple to get the duration of totality for solar eclipses. To maximize the duration of any eclipse, I treat the viewer as being on the equator and with the Sun at zenith, so that they are closest to the Moon and also rotating fastest along with the shadow.A subtle flaw in the above plot for length of total eclipses:
If you look closely, it might seem like the maximum duration of totalities now (when the Moon is at perigee) is too small. In fact, this predicts they should last just over 5 minutes, instead of 7.5. Why? Because I pretended the Earth's orbit is circular with a 1AU orbital distance. The longer totalities of up to 7.5 minutes require the Earth to be near aphelion, so that the Sun's apparent size is a little smaller (as FastFourierTransform
showed just above). At aphelion, the Earth is at about 1.02AU, and if I plug that in for the Earth-Sun distance then it confirms the 7.5 minute max length of totality.
Accounting for the Earth's orbital eccentricity here would be a horrendous task, since it changes over time in a very complicated way (a part of the Milankovitch cycles
.) Instead, I think it makes a lot more sense to show it in these "average" terms for the Earth at 1AU.
Going back to the tidal evolution, here's what I find for the evolution of the Earth's spin and Moon's orbit (in terms of their angular velocities), as a function of the angular momentum transferred from the Earth to the Moon (compared to the present day):
0 on the x-axis, marked with a vertical line, represents today. To the right, positive angular momentum transfer raises the Moon's orbit and slow's the Earth's spin, so the angular velocities of both are decreased, until eventually they would end up being equal to each other in a mutual tidal lock of about 1.39 microradians per second, or a spin and orbit period of about 52 days. (Note the plot is on a semi-log scale, which distorts the shape of the curves. The change in Earth's spin rate vs. angular momentum transfer is actually linear, since rotational angular momentum of a sphere is proportional to the angular velocity. The Moon's orbital angular velocity vs. angular momentum is inverse cubic, and these two curves intersect twice.)
To the left, negative momentum transfer (i.e. from the Moon to the Earth, or backwards in time) brings the Moon closer, and both it and the Earth revolve faster. Extended all the way, we might predict they would have been in a tidal lock in the past, with the Moon much closer to Earth and the Earth spinning much faster. But such extrapolation can't work that far out. If they had started out mutually locked, then they would have stayed that way. The Moon didn't really form quite that close, and the Earth was never spinning quite that fast.
Here's the same figure as above again, except recast more intuitively in terms of the Moon's orbital distance (instead of angular momentum), and in terms of spin/orbit periods in hours (instead of angular velocities.) Again I have chosen a log vertical scale.
Here we can see more easily the changing length of the Earth's day and the lunar month as the Moon's distance changes. We can also clearly see why the Moon never escapes the Earth. The tidal interaction leads to them being tidally locked with the Moon a little less than 600,000km away. But that won't even happen in the next 5 billion years before the Sun dies. By then the Moon will only recede to roughly 475,000km or so.Explanation of how the duration of totality changes with Moon distance:
Intuitively, the Moon being closer to Earth would make it larger on the sky, so it should cover the Sun longer. But closer orbits are also faster. The increased angular size matters more, so the duration of totality increases with decreasing distance to the Moon.
But there are more effects to consider. One of the most important is that the Earth's rotation rate is changing! During an eclipse, the shadow of the Moon sweeps over the Earth (from west to east -- the same direction the Moon is orbiting), but the Earth's rotation also sweeps the viewer from west to east. The Moon's orbital velocity is faster than the Earth's rotational velocity (even at the equator), so the shadow still sweeps over from west to east, but not as quickly. In the past, the Earth rotated faster, which more than compensated the Moon's faster orbital velocity when it was closer. This helped further increase the duration of totalities in the past.If we extrapolate the Moon's distance as close as it could have been (at the Roche limit), at left side of the figure, then the duration of totalities explodes to over an hour! This is again due to the combination of the Moon's angular size being much larger as it gets closer (1/r dependence), along with the Moon orbital angular velocity starting to closely match the Earth's rotational angular velocity. That would mean the Moon would not only have looked huge, but also move very slowly across the sky, and thus cover the Sun for a much longer time.Another insight to the question of the "coincidence", that the Moon and Sun appear similar in size on the sky.
This cosmic coincidence is perhaps most striking when we look at the first plot for the totality duration. Again, the Moon's orbit is elliptical, so the range of distances it covers helps make the match more likely. This range is shown with the perigee/apogee lines in yellow and blue. Still, this range seems to be fairly narrow compared to the full range of distances that the Moon's orbit could cover, and the point where the Moon and Sun's apparent sizes are equal runs right through the middle of it. So maybe it still looks like a really strong coincidence.
The expansion of the Moon's orbit over time means this match was almost bound to happen sooner or later, but that still doesn't explain why we exist during the time that it does. The key (or at least another part of it) is that evolution does not occur at a constant rate! When the Moon was closer, the tidal interaction was stronger, so angular momentum was transferred more quickly and the Moon's orbit was expanding faster. So the Moon did not spend as much time very close to Earth as it will spend at larger distances. There is an inherent bias for an observer at a random moment in time to find the Moon farther away where the evolution is slower. Hence this "coincidence" is actually a lot more likely than it seems.
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Do you think there was a time in the extremely distant past when the moon appeared so large it actually blocked out the corona?
Shortly after it formed (a few hundred million years, maybe even a billion), it might have been close enough to do this during the mid-point of totality. A competing effect however is that as Salvo says, when the Moon was closer and larger on the sky, its shadow on the Earth was larger, so the sky during totality would be darker. This would make the Sun's corona appear even larger, since the fainter extended parts would be more visible. For an idea, you can compare images of the Sun's corona taken during eclipses to those taken from space using coronagraphs: