When will the LAST total solar eclipse be?

Okay, based on random bits I’ve heard (and maybe totally misunderstood) over the years and especially the current mania about the coming eclipse:

For a total solar eclipse the moon’s apparent size must be at least as large as the sun’s.

The moon’s orbit is slowly increasing because the Earth’s rotation is slowing down. (I don’t get why these are related, apparently due to ‘conservation of momentum’, which phrase means nothing to me, but nevermind, I’ll take their word for it.)

As the moon’s orbit increases, it will appear smaller from our POV here on the Earth’s surface.

Sooo… At some point the moon will be ‘too small’ to ever totally block the sun from sight, yes? When will that be? I don’t expect an exact date, more like “in three thousand years” or “6 billion years” or something like that.

Going the other way, in the past the moon’s orbit was smaller, and it appeared larger, so it was easier for the magic alignments to happen. How much easier? Was there ever a point where every new moon caused a total eclipse somewhere on the earth? About how long ago would that have been? Like, before humans existed to see it? Or any form of life with eyes? Or any form of life, period?

In around 600 million years.

I was looking into this a few days ago, when someone on a local Facebook group said the last eclipse would be in 600 thousand years. That sounded very wrong, I googled to make sure, and I pointed out that he was missing three zeros. He replied that it depended on the figures used (in other words, sticking by the short number) so I started working on a elaborate devastating takedown based on the maximum and minimum current angular diameters of the sun and moon and the current rate of lunar recession. But I got a figure of just a handful of millions of years before there could be no more total eclipses and I don’t know why my math is an order of magnitude off the “official” figure so I abandoned the reply.

On the other hand, Meeus, in his book, says 1210 million years, assuming the Moon’s recession stays at 3.8 cm/a

His figures are: minimum semidiameter of the Sun with the Earth’s eccentricity at 0.06: 905.31 seconds
Moon’s semidiameter reaches this value when distance from the observer to the centre of the Moon = 395968 km
current distance with the Moon at zenith: 349993 km

I found the math I was doing that came up with a figure of 8 million years, given a claimed figure of 1 inch per year recession for the moon. Obviously I was wrong in some fundamental way, but I don’t know what.

Paraphrased from the abandoned text:

The angular diameter of the moon in the sky varies from around 29.3 arcminutes to 34.1 arcminutes during the course of an orbit.

The angular diameter for the sun ranges from 31.6 arcminutes to 32.7 arcminutes.

For the moon to no longer ever be able to cause a total solar eclipse the moon’s largest angular diameter would have to become smaller than the sun’s smallest angular diameter, meaning (at the Earth’s current orbit and the sun’s current diameter) the moon would have to have a largest angular diameter of less than 31.6 arcminutes. The moon’s largest angular diameter is at perigee, or around 225,300 miles. To drop the moon’s angular diameter at perigee from 34.1 to 31.6 arcminutes would require the moon to move around 8% further from the Earth, or around 18,000 miles. 18,000 miles is almost 8 million inches, or 8 million years.

Your calculation is rougher than his, ignoring things like changes in the Earth’s orbit, but it should still give the same order of magnitude. Let’s see:

Your value for the largest angular diameter of the moon at perigee: 34.1 arcminutes. That is the same as his value, based on the smallest distance between centres of the Earth and the Moon during the period A.D. 1000–3000 being 356371 km.

For the Sun, your value for the smallest angular diameter is 31.6 arcminutes. His value, for the Earth’s current orbit: 31.46

Let’s use your values. For the Moon’s angular diameter to drop from 34.1 to 31.6 arcminutes would require the Moon to move around 28000 km, so 18,000 miles, whatever. 18,000 miles is 1.14 billion inches, so at “1 inch per year” that takes a billion years…

Hm. I wonder what I multiplied that by? Feet in 18,000 miles doesn’t fit…

Small but important nitpick re cause and effect – it’s the other way around. The moon’s gravity creates tidal bulges in the earth’s oceans that steal some of the earth’s angular momentum and impart that energy to the moon’s orbit, sending it higher. In approximate round numbers all eclipses will be annular in about a billion years. But no worries – a few billion years after that the sun itself will start expanding as it enters its red giant phase, so eclipses wouldn’t be possible anyway! :wink:

I don’t think we can just use the average recession rate, because these sorts of thing usually affect the perigee and apogee differently. I think that the greatest effect would be on the apogee (i.e., it’s mostly the apogee that’s getting further away, while the perigee moves out more slowly), since the effect would manifest as an increase in the Moon’s speed, it’d happen most strongly when the Moon is closest, and an increase in speed at perigee moves the apogee out (though there are other effects that would work the other way, but I think those effects would be smaller). This means that an estimate based on the current perigee and the overall rate of recession (average of apogee and perigee) would be a significant underestimate.

Oh, and to address this: Roughly speaking, angular momentum is a measure of how much rotation a system has, and it can only be changed by transferring some of that rotation outside of the system. It’s extremely difficult for the Earth-Moon system to transfer rotation to the Sun or any of the other planets, but not quite so difficult for them to transfer it between each other.

Right now, the Moon’s rotational period is much longer than the Earth’s (about one month vs. one day), so there’s something like friction between them. Imagine if you had two wheels mounted loosely on the same shaft, but right next to each other and rubbing up against each other: If they started off spinning at different rates, friction would speed up the slower one and slow down the faster one, and eventually (in that case, it probably wouldn’t take very long) end up rotating at the same speed. Something similar is happening with the Earth and the Moon (though complicated by orbital dynamics).

So when the earth’s rotational speed aligns with the moon’s orbit (as the moon has done with earth) will there no longer be tidal forces disrupting the earth that cause the moon to recede? Everything should be static.

If it were just the Earth and the Moon, then yes (minus some EXTREMELY slow processes that would eventually cause the orbit to decay again). But before it got to that point, other factors would interfere, like the Sun becoming a red giant.

You can’t extrapolate easily from the Moon’s current recession rate in the long term.

As the Moon gets further away, it’s tidal drag becomes less so the rate decreases. It’s not clear if the Space.com figure of 600M years takes this into account but I wouldn’t be surprised if it was close.

Note that once the Moon and Earth a tidally locked (the Moon is ~stationary in the sky), then the effect sort of reverses and the Moon starts approaching the Earth.

A classic example of conservation of momentum that actually somewhat relates here is a skater doing a spin on one point. They move their arms out and they slow down, move them in they speed up.

Since the Moon is slowing down the Earth’s rotation, that is like the skater moving their arms (the Moon) further away. (Yeah, cause and effect are reversed here but the Physics is the same.)

Very unlikely. The Moon’s orbit is tilted about 5 degrees from the path the Sun takes along the sky. That’s enough so that, even when much closer, it wouldn’t always cause an eclipse.

Not only that, but the path the sun takes across the sky moves north-to-south-to-north every year, due to the Earth’s axial tilt of about 21º. This is one reason solar eclipses only happen at certain times of the year, in different places.

As to the effects of the lunar orbit, one should consider the barycenter. “Barycenter” is the axis of an orbit. In most cases, it is pretty close to the center of the larger body (because most systems we know of tend to be a big thing with a very, very small thing going around it. The two best known examples of a barycenter that lies outside the larger body are the orbit of Jupiter around the Sun and the orbit of Charon around Pluto.
       The barycenter of the Earth/Moon system is pretty far from the center of the Earth. In fact, the Moon is the second largest satellite relative to its primary, after Charon. So the Earth turns around the lunar barycenter, which is about a thousand miles below the Earth’s surface. And since the axis of rotation is not aligned with the lunar barycenter, Earth’s rotation is more complex than just a regular spinning.
       Thus, the moon pulls the Earth around, which places its orbit slightly ahead of the Earth’s center of mass: the latter would be pulling down on the moon (speeding it up), but the former would be slowing the moon. Apparently the balance of the equation is toward the former: a slower orbit is a higher orbit.
       As we approach the Sun’s red giant phase, in about three to five billion years, the outer atmosphere of the Sun will envelop the Earth/Moon system. It will be extremely tenuous, but it will add drag to the moon’s orbit, pushing it farther out. It may even reach a point where the moon is far enough out that it loses its tidal lock.

The moon now occupies about one-half of a degree in the sky. So, in order to block the sun every revolution, it would have to ten times closer than it is now. It may be that in the immediate aftermath of the formation of the moon via collision with a rogue planet, it solidified that close to the Earth. But maybe not. I don’t think we know for sure.

There was a discussion of this question in Tuesday’s Science Times. The 1.5 inches per year, if it kept up forever, would lead to the answer of 600 megayears. But that rate has not been constant, sometimes less than 1/2" per year, sometimes as much as 4" per year. It depends, among other things on how rough is the ocean bottom since it is tidal friction that drives the recession. But in a few billion years, the sun will start to grow and then there can be no eclipses (or we would call them transits).

A bit OT, but I find this (from Darren_Garrison’s space.com link, dating to 2017) rather strange:

During the total solar eclipse on Aug. 21, skywatchers will direct most of their attention to the sun, but don’t forget about the moon …

I would have said that most TSE skywatchers are paying close attention to both the sun and the moon.

The last statement is correct but your description of how to get there is backwards. Suppose you had a rocket in a circular orbit around the earth. Consider what would happen if you briefly fired thrusters to accelerate it. It would start to rise to a higher orbit, eventually reaching apogee on the opposite side of what now would be an elliptical orbit. At that point it would be at the highest and also the slowest point of its orbit. As it fell to the lowest point of its now-elliptical orbit, its speed would increase as it traded the potential energy of a higher orbit for kinetic energy, until it returned to its original point where the thrusters were fired, but going faster than before.

Conversely, firing thrusters in the opposite direction to slow it down would send it into a lower orbit, and ultimately would de-orbit it entirely.

The point being that although a satellite in a higher orbit moves more slowly than one in a lower orbit, it’s in a higher energy state and it takes extra energy to get there. The moon is rising to a higher orbit not because the earth is slowing it down, but because it’s speeding it up, sending it into a higher albeit ultimately slower orbit.

Separately, I’m not sure that the location of the earth-moon barycenter has much to do with this. My understanding is that tidal bulges running ahead of the moon and pulling at it via extra gravitational attraction is what is speeding it up while at the same time slowing the earth’s rotation.

Yeah, that’s what I was referring to in post #9, where I ended with “…though complicated by orbital dynamics”. Orbits are weird, in that speeding you up ultimately slows you down, but I thought that would be more complication than was appropriate for that post.