Well…The answer is 700 million years from now, but the details are interesting!
Since the dawn of recorded history, humans have had a love-hate relationship with total solar eclipses. For most of human history, these events were feared and taken as omens of the downfall of empires or the end of the world. Only in the last thousand years or so have people settled down and viewed them as the beautiful and bizarre events that they are. By the 19th Century, scientists and artists traveled the world over to capture them with sketches at the telescope eyepiece. Among the first images taken by primitive cameras were those of total solar eclipses.
Predicting total solar eclipses
Today, the physics and mathematics of these events are known with such detail that they can be predicted to within minutes from 2000 BCE to 3000 CE [1]. They can even be used to track the slowing down of earth’s rotation by comparing the predicted time and place with historical observations [2]. But total solar eclipses require a precise geometric circumstance to exist. Our moon has a diameter of 3,475 km at a perigee distance of 363,300 km, while the sun has a diameter of 1.4 million km at a distance of 150 million km. This means that, although the sun has a diameter that is 403 times the moon, it is 412 times farther away so that the apparent size of the dark lunar disk completely covers the blinding disk of the sun in the sky. Depending on the exact timing of the moon in its orbit, this ratio of 403/412 can be made to be exactly equal to 1.00 so that the disk of the moon exactly covers the sun to give the classic total solar eclipse shown in the picture above. But this precise geometric circumstance is not written in stone. In fact, to get a proper prediction far into the fiuture you need a supercomputer!
Earth orbit evolution
Currently the distance from earth to the sun has an average value of 150 million km, but because Earth’s orbit is an ellipse, it varies from 152 million km in July to 147 million km in January. This leads to the ironic circumstance that in the Northern Hemisphere, the sun is actually farthest away from the sun in the summer and closest in the winter! Only the Southern Hemisphere with its reversed seasons gets it right!
For many decades, researchers have modeled the evolution of the orbit of the Moon and Earth with supercomputers and pretty much nailed down what we can expect to happen for the next few billion years. As it turns out, this is a fiendishly difficult calculation because it depends on an exact knowledge of the interiors of the moon and earth, the location of the continents, and the influences of the other planets. The inner solar system is dynamically unstable and displays a chaotic behavior over times of 100 million years or longer. A consequence of this is that even changing the location of Mercury in its orbit by 1 meter today causes a variety of different outcomes in a billion years including its collision with Venus and ejection from the solar system. Earth, however, seems to exist in a remarkably stable gravitational balance such that its orbit changes only insignificantly from what we see today. [3] It will drift outwards from the sun by a few thousand kilometers due to the sun itself losing mass. The sun converts 4 million tons of mass into radiant energy every second and added up over millions of years, this causes the sun’s gravitational hold on Earth to weaken and its orbit to drift outwards by 1.5 cm/year [4].
The outward drift of Earth in its orbit is entirely negligable so we won’t bother including it. We will assume that the average perihelion and aphelion distances will still remain close to 147 and 152 million km. This means that from Earth the angular diameter of the sun from the surface will vary between 1,964 seconds of arc at perihelion to 1,900 seconds of arc at aphelion, where 3600 seconds of arc equals 1 angular degree.
Lunar orbit evolution.
We know from geologic data that our moon was formed some 4.4 billion years ago and orbited Earth at a distance of only about 30 Earth Radii ( 190,000 km) causing Earth to have a rotation period of about 12 hours in a ‘day’. [5]. Since its formation, it has drifted out to its present distance at a current rate of about 3.8 cm/year based on lunar laser metrology [6]. But this outward drift continues today so that in the future the moon will be even farther from Earth. This means that at some time in the future, the ratio of lunar:solar size and lunar:solar distance will fall below the magic 1.000 needed for a total solar eclipse. The moon will simply be too small in apparent size to perfectly cover the disk of the sun. We can’t predict the exact date when we will see the very, very, very last total solar eclipse from Earths surface, but we can get a pretty good idea what timescales are involved.
Simple Linear Model
Suppose we just used the current perihelion and aphelion distances and then assumed that the moon is moving away from Earth at a constant rate of 3.8 cm/year. If we calculate the angular sizes of the moon and sun from Earth we get the following figure.
Explanation: The orange line is the angular size of the sun viewed from Earth when Earth is closest to the sun (perihelion) and the yellow line is the same calculation from when Earth is farthest from the sun (aphelion). The black line is the angular diameter of the moon at its farthest distance from Earth (apogee) and the green line is for its closest distance to Earth (perigee). What you see is that the lunar curves cross the solar curves and indicate when these two diameters are equal, allowing a total solar eclipse to be viewed. So long as the solar lines are between the two lunar lines, you will have a total solar eclipse.
What this graph says is that 1044 million years ago, the sun at perihelion matched the moons size at apogee when it had the smallest angular size. After this ‘year’ the moons size at apogee was too small to cover the sun at perihelion and so total solar eclipses at lunar apogee ceased to happen when the solar disk was largest at perihelion. Notice that before 1044 million years the lunar lines were above the solar lines. This means that the disk of the moon was always much greater than the disk of the sun at any time in the lunar orbit. In fact, the lunar disk was so big that not only was the disk of the sun covered by the moon but much of the inner corona too. You would still have total solar eclipses before 1044 million years ago, but they would look dramatically different than the ones we see today.
By the time we get to 710 million years ago, the moon at apogee was also too small to cover the sun at aphelion when the solar disk is smallest. Between 1044 and 710 million years ago, the small apogee moon could still cover the sun when the sun was between aphelion and perihelion, but after 710 million years ago, there would never again be a total solar eclipse of the sun when the moon was at apogee. This was before the emergence of multi-cellular life on Earth during the Cambrian Explosion. Only annular eclipses will be viewed from then on during lunar apogee.
Now the second lunar curve in green is more interesting. It shows the angular size of the perigee moon, and it is pretty clear that today (Time-0) the size of the perigee moon is larger that the sun at both perihelion and aphelion. So we get total solar eclipses no matter if Earth is at perihelion or aphelion. However, by 280 million years from now, the moon will start to become smaller than the solar disk at perihelion and so eclipses will stop being total solar eclipses when the sun is closest to earth and the moon is also closest to earth. After 613 million years from now, you will no longer have total solar eclipses for the perigee moon and the smaller aphelion sun. After 613 million years the lunar disk will never again be big enough to completely cover the solar disk. This is the estimate you are likely to find in many popularizations of this Final Event such as a SpaceMath problem at NASA, and NASAs lunar scientst Dr. Richard von Drak.
A More Accurate Calculation.
The previous linear calculation was based on the moon maintaining its outward 3.8 cm/yr motion for the next 600 million years, but detailed supercomputer calculations of the evolution of the Earth-Moon system give a more accurate result. I used the model published in 2021 by Prof. Houraa Daher and her team at the University of Michigan [7], and specifically used their Figure 5a, which gave the past value for the lunar orbit semi-major axis. I also used the 2020 data from the published work by Dr. Bijay Sharma [8] at the National Institute of Technology in India, specifically Figure 7, which gave the recession speed (cm/yr) with lunar semi-major axis. Ideally, both of these data should be derived from the same calculations but unfortunately this was not possible to obtain at the time of this writing. However, if they are both faithful to the same underlying physics, then the results should be consistent.
The application of these detailed models to the lunar size evolution is shown in the next figure.
The straight, linear extrapolations have now been replaced by more realistic curved predctions. Here we see along the black line that at 700 million years ago, the lunar size at apogee matched the solar disk size at perihelion (1952 arc-seconds) , some 300 million years later than the linear model. By 500 million years ago the apogee lunar disk no longer covered the disk of the sun at aphelion, so from this time forward there were no longer any total solar eclipses when the moon was at its farthest apogee distance. This happened around the time of the Cambrian Explosion.
Meanwhile, the green line for the perigee moon shows that it has a disk size greater then the size of the large perihelion sun (1952 arcseconds) disk until 300 million years from today. At this time, the lunar diameter varies from 1718 arcseconds (black line) to 1952 arcseconds (green line) so we can still have total solar eclipses so long as the moon is close to its perigee when the sun passes through one of the lunar ‘nodes’ during the equinoxes. At about 700 million years from now the large perigee moon with a diameter of 1952 arcseconds covers the sun at perihelion, but after this time, its diameter continues to decrease until from this time forward all we ever see are annular eclipses. So this critical ‘date’ is about 80 million years later than the linear model.
By 700 million years from now, the moon will continue to drift away from Earth, but at a slower rate of 3.0 cm/year. Its distance from Earth will have grown from 60.2 Re (384,400 km) to 63.8 Re (407,155 km). The moon will then take 28.4 days to orbit Earth having gained about 26.4 hours since today. This means that the time between one full moon and the next will be 30.7 days instead of the current 29.5 days. Meanwhile, the Earth’s rotation has changed from its current 23h 56m to about 26h 25m as the lunar tides continue to do their work. What this means is that an Earth Year at 700 million years from today will only about 330 days long!
Will there be anyone there to care? Probably not.
Our sun continues to evolve and grow in luminosity so by then it will be about 10% more luminous than it is today. This means the average global temperature will be 117o F and not the 57o F we enjoy today. By this time, the level of carbon dioxide will have fallen below the level needed to sustain C3 carbon fixation photosynthesis used by trees. Some plants use the C4 carbon fixation method to persist at carbon dioxide concentrations as low as ten parts per million. However, the long-term trend is for surface plant life to die off altogether. The extinction of plants will be the demise of almost all animal life since plants are the base of much of the animal food chain on Earth. Climate models suggest that by about this time Earth will be hot enough to cause the slow evaporation of the oceans into the atmosphere. This will be the start of what is called the “moist greenhouse” phase, resulting in a runaway evaporation of the oceans and Earth becoming Venus. Meanwhile, the current continents will have merged and separated and merged again into yet another supercontinent with its own lethal contribution to global heating and weather [9].
So basically by about 700 million years from now, Earth will be a humid, desert world with no complex living organisms to appreciate total solar eclipses except perhaps extremophile bacteria…and cockroaches?
Have a nice day!
[1] Five Millennium Catalog of Solar eclipses https://eclipse.gsfc.nasa.gov/SEcat5/catkey.html
[2] Ancient eclipses Reveal How Earths Rotation has Changed https://www.space.com/ancient-eclipse-records-earth-rotation-history
[3] Highly Stable Evolution of Earths Future Orbit Despite Chaotic Behaavior of Solar System https://iopscience.iop.org/article/10.1088/0004-637X/811/1/9
[4] https://www.forbes.com/sites/startswithabang/2020/04/09/earth-is-spiraling-away-from-the-sun-for-now-but-will-eventually-crash-into-it/?sh=863220238580
[5] Long-Term Earth-Moon Evolution With High-Level Orbit and Ocean Tide Models https://agupubs.onlinelibrary.wiley.com/doi/10.1029/2021JE006875 figure 6
[6] The moon has been drifting away from Earth for 4.5 billion years. A stunning animation shows how far it has gone. https://www.businessinsider.com/video-moon-drifts-away-earth-4-billion-years-2019-9
[7] Long‐Term Earth‐Moon Evolution With High‐Level Orbit and Ocean Tide Models, https://www.ncbi.nlm.nih.gov/pmc/articles/PMC9285098/
[8] The Past, Present and the Futuristic Earth-Moon Orbital-Global Dynamics – and its habitability – https://www.proquest.com/openview/c945a68d9b4a2354aaea7cf859b776ba/1?pq-origsite=gscholar&cbl=4882998
[9] What if You Traveled One Billion Years into the Future? https://whatifshow.com/what-if-you-traveled-one-billion-years-into-the-future/