# What's the time window for total and narrow eclipses?

Because of the tidal effect angular momentum is transmitted from Earth rotation to the Moons orbit around Earth. This means that perfect eclipses only has occured during a certain time window. Can this window be calculated?

It really doesn't matter how narrow the eclipse should be. It's enough that Moon sometimes appeare as greater and sometimes appear as smaller than the Sun, due to the irregular orbit of Moon around Earth.

In a paper https://arxiv.org/abs/1502.01421 I found this diagram: which suggests that one day on Earth $$700-2000$$ million years ago was about 21 hours. Supposing that the angular momentum in the last $$700$$ million years in the system is approximated as the sum of the angular momentums from the Earth rotation and the Moon orbit around the Earth, one will come upp with that the angular diameters of Moon and Sun could have been comparable $$700$$ million years ago, taking in consideration the variations of the eccentricities in the orbit of Earth and specially in the orbit of Moon.

• What is a perfect eclipse? Mar 15, 2022 at 3:07
• What do you mean by "precisely," then?
– Chris
Mar 15, 2022 at 3:12
• So a total solar eclipse?
– Chris
Mar 15, 2022 at 3:18

Was writing the first two paragraphs before he clarified:

Oh, I think I know what he means.

So nothing in nature is perfect, so I highly doubt that a perfect eclipse will ever happen. At the very least the Sun is spherical and the Moon has craters, so because the two silhouettes have different shapes there's no way they could be identical from the vantage point of the earth.

These paragraphs still apply to the clarified question:

However, if you're asking for when the apparent diameters of the Moon and sun could hypothetically be arbitrarily close to equal, I'm glad to say that that's currently the case.

The apparent diameter of the Moon varies based on where it is in its orbit. Because the Moon's orbit is elliptical, when the Moon is farthest away (it's at its apogee) its apparent diameter is less than the Sun's apparent diameter, and when the Moon is as close as it gets its apparent diameter is greater than the Sun's apparent diameter.

I think that the variation in the Sun's apparent diameter due to the Earth being closest (at perihelion) or farthest (at aphelion) is less than the variation in the Moon's apparent diameter. Regardless, if the conditions are right, the minimum and maximum apparent diameters of the Moon and Sun can overlap each other either way.

In fact, sometimes when the apparent diameters are very close you can see the Sun during an eclipse through the Moon's craters. These points of light are called Baily's beads.

IDK how long it'll take for the Moon's apparent diameter to no longer overlap the Sun's, so let's do the math to estimate. I looked and couldn't find any predictions on how eccentricity will change, so I'd make my first guess assuming it's constant. Solid angles will be your friend. From Wikipedia:

average solid angle of the Sun is 6.794×10−5 steradians

The radius of the Moon is 1,737 km, so its cross-sectional area is 9.5 X 10e6 km2

So setting the perigee of the Moon to be that in steradians we get:

$$\frac{9.5e6\,\mathrm{km^2}}{R^2} = 6.794e-5\,\mathrm{steradian}$$

So $$R = \sqrt{\frac{9.5e6}{6.794e-5}} = 373937\,\mathrm{km}$$

From Wikipedia:

Tidal rhythmites from 620 million years ago show that, over hundreds of millions of years, the Moon receded at an average rate of 22 mm (0.87 in) per year

From NASA:

Its closest point is the perigee, which is an average distance of about 226,000 miles (363,300 kilometers) from Earth

So $$\frac{373937-363300}{22e-6} = 483500000\,\mathrm{years}$$, that is about half a billion years, but by then then the sun will have expanded as well meaning that it'll actually be sooner than that. I recommend looking at the graph of solar radius, using that to make a graph of solar apparent diameter, then look at your graph of the Moon's apparent diameter, and see where they intersect.

Of course, this is assuming that the Moon's tidal drift remains constant over they years. It will be affected by the vaporization of the Earth's oceans limiting tidal acceleration, and if I had to guess the fact that its gravitational potential wrt the Earth is not linear and the attenuation of tidal forces due to distance will have a negligible effect.

• I think this is good upper aproximation. So perhaps the first "perfect eclipses" occured after more than 4 billion years.
– Lehs
Mar 15, 2022 at 9:16
• According to NASA, the last total solar eclipse will be "a bit sooner than 600 million years". sunearthday.nasa.gov/2006/faq.php Mar 15, 2022 at 11:24