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lpetrich
2017-Aug-25, 11:56 AM
I've attempted to estimate that, but I've had trouble with uncertainties in the numbers that I can find.

First, the orbit eccentricities. The Moon's is 0.0549, and that might be a relic of a resonance effect that happened long ago -- a speculation that I recall reading somewhere. So I will treat it as constant.

The Earth's is a more difficult problem. Its present-day value is 0.0167, but it varies quite a bit due to perturbations from the other planets. Over the last million years (Milankovitch Orbital Data Viewer (http://biocycle.atmos.colostate.edu/shiny/Milankovitch/)), I find:

Q0: 0.0024
Q1: 0.0158
Q2: 0.0272
Q3: 0.0370
Q4: 0.0578

Q = quartile, Q0 = minimum, Q2 = median, Q4 = maximum. The mean value is 0.0272, close to the median value. I'll use the full range of eccentricities.

The numbers for the Moon's orbit in the past are not very good, it must be conceded. Precambrian length of day and the validity of tidal rhythmite paleotidal values - Williams - 1997 - Geophysical Research Letters - Wiley Online Library (http://onlinelibrary.wiley.com/doi/10.1029/97GL00234/pdf) quotes one set of values for the Elatina formation in Australia (620 million years ago), and two each for the Big Cottonwood formation (900 Mya), and for the Weeli Wolli formation in Australia (2500 Mya). For Elatina, the Moon was 3.5% closer to the Earth than today, for Big Cottonwood, 5% or 10%, and for Weeli Wolli, 9% or 14%.

The Moon is slowly spiraling away, pulled outward by tidal drag. The recession rate is given by da/dt = K * a-11/2, where K depends on the amount of tidal drag, among other things. So the Moon likely got close to its present distance rather early. The present-day tidal-recession rate is 3.82 cm/yr, while averaged over Elatina - present, it was 2.16 cm/yr. For Big Cottonwood, the average is 2.19 or 4.38 cm/yr, and for Weeli Wolli, the average is 1.45 or 2.11 cm/yr. This suggests that tidal drag has varied by sizable amounts, likely due to the presence or absence of big shallow ocean areas like the Bering Sea.

There is an additional effect: proximity due to the size of the Earth. It is not very big, but I've included it for the last total eclipse, something that would happen when the Sun is at the local zenith. The first annular eclipse would happen at local sunrise or sunset, and the Earth's size would not contribute.

Here are the numbers, for the various Earth eccentricities:

Last total eclipse
Present-day recession: 480, 620, 740, 840, 1060 million years in the future
Elatina average recession: 860, 1100, 1310, 1490, 1870 million years in the future
Avg distance = 63.2, 64.1, 64.8, 65.4, 66.7 Earth radii

First annular eclipse
Present-day recession: 820, 940, 1050, 1140, 1330 million years in the past
Elatina average recession: 1440, 1660, 1850, 2010, 2350 million years in the past
Avg distance = 55.4, 54.7, 54.1, 53.5, 52.3 Earth radii

Present avg distance = 60.3 Earth radii.
Avg distance for Elatina: 58.2, Big Cottonwood: 54.1, 57.2, Weeli Wolli: 52, 54.6

lpetrich
2017-Aug-25, 12:46 PM
I had neglected to include the Sun's increasing radius: Future of Earth - Wikipedia (https://en.wikipedia.org/wiki/Future_of_Earth). Reading off of the graph, I find that the Sun is expanding at a rate of 5.2% every 1.5 billion years. I measured over the 1.5 billion years past to 1.5 billion years future.

Last Total Eclipse
Present-day recession: 360, 460, 540, 620, 770 million years in the future
Elatina average recession: 530, 680, 800, 910, 1140 million years in the future

First Annular Eclipse
Present-day recession: 610, 710, 790, 860, 1010 million years in the past
Elatina average recession: 910, 1050, 1170, 1280, 1500 million years in the past

So the Earth's first annular solar eclipse occurred at sunrise some 1500 million years ago or thereabouts. It was borderline annular, and it turned into a borderline total eclipse for most of its duration, becoming borderline annular again at sunset. The Moon was at apogee, the Earth at perihelion, and its eccentricity at maximum. Several thousand years later, the Earth's eccentricity had declined, and solar eclipses were always total again. By about a billion years ago, annular eclipses happened no matter what the Earth's eccentricity was.

Likewise, the Earth's last total eclipse will occur at noon some 1000 million years in the future or thereabouts. It was borderline total, and the eclipse before and after noon were all borderline annular. The Moon was at perigee, the Earth at aphelion, and its eccentricity at maximum. Total eclipses were becoming rarer and rarer as they became more and more dependent on the Earth's eccentricity being high, and this was the last one before the Sun became a white dwarf and thus much easier to eclipse. That's if the Earth survives the Sun's red gianthood.

Amber Robot
2017-Aug-25, 04:14 PM
Doesn't whether the eclipse is at sunset or noon depend on where it's being viewed from?

lpetrich
2017-Aug-25, 05:24 PM
Doesn't whether the eclipse is at sunset or noon depend on where it's being viewed from?
Yes it does. I had in mind sunrise, noon, and sunset in the local time of the eclipse's observer, and observers at different locations along the totality/annularity track will see it at different local times.

I should add that in that last total eclipse, the Sun and the Moon would be directly above the observer of that eclipse, at that observer's zenith. That's to be as close as possible.

Arneb
2017-Aug-25, 07:58 PM
Doesn't whether the eclipse is at sunset or noon depend on where it's being viewed from?

Oh yes. The eclipses in question would all be hybrid eclipses, with a fleeting moment of annularity at susnset/sunrise in the first case, and a last, fleeting moment of totality at noon in the second.

lpetrich
2017-Aug-26, 03:18 PM
Arneb, that's right.

I tried to do the calculation very carefully, including effects that are apparently omitted in a lot of calculations. For instance, The Last Total Solar Eclipse...Ever! - 4Page28.pdf (https://spacemath.gsfc.nasa.gov/weekly/4Page28.pdf), lacks expansion of the Sun and uses the present-day values of the Moon's recession and the Earth's eccentricity.

StupendousMan
2017-Aug-26, 07:28 PM
I did a quick and simple calculation, assuming a) the moon's orbital radius increases at a constant 3.8 cm per year indefinitely, and b) the Sun's radius increases according to the evolutionary models at the Dartmouth Stellar Evolution Database. My values for the maximum and minimum apparent angular sizes of the Sun and Moon are shown below:

22584

This simple calculation suggests that the last total eclipse will occur about 450 million years from now.

lpetrich
2017-Aug-27, 12:42 AM
I don't know how you got that solar-radius expansion rate -- it's about 10% per billion years, much larger than mine. I used values from a graph in Future of Earth - Wikipedia (https://en.wikipedia.org/wiki/Future_of_Earth), in turn from The Sun and stars as the primary energy input in planetary atmospheres | Proceedings of the International Astronomical Union | Cambridge Core (https://www.cambridge.org/core/journals/proceedings-of-the-international-astronomical-union/article/sun-and-stars-as-the-primary-energy-input-in-planetary-atmospheres/293C0314C44A1A4AAF8175A3C288B50B) ([0911.4872] The Sun and stars as the primary energy input in planetary atmospheres (https://arxiv.org/abs/0911.4872)).

Also, you seem to have used the Earth's present-day eccentricity. But it's a nice sort of graph.

lpetrich
2017-Aug-27, 12:54 AM
[0801.3807] Solar Mass Loss, the Astronomical Unit, and the Scale of the Solar System (https://arxiv.org/abs/0801.3807) gives the Sun's mass loss as 9.13*10^(-14) solar masses per year. That's about 5.75 million metric tons per second. Of that, 4.26 million metric tons per second is its electromagnetic emission, and about 100 thousand metric tons per second is its neutrino emission. The remainder, 1.39 million metric tons per second, is its particle emission, in the solar wind.

Over a billion years, that is about 10-4 of the Sun's original mass. The planets' orbits expand from the Sun losing mass, with (orbit size) ~ 1/(solar mass). So the Earth's orbit was 0.999 times its present size a billion years ago, and will be 1.001 times its present size a billion years from now. So it is an insigificant factor for when the first annular eclipse occurred and when the last total eclipse will occur.

Ken G
2017-Aug-27, 02:39 PM
And bear in mind that these are not really the last ever total solar eclipses, they are the last for a long time. The Sun will eventually be a tiny white dwarf, and then almost all its eclipses will be total. Not that it's likely there will be any life here to see them.

StupendousMan
2017-Aug-27, 05:04 PM
I don't know how you got that solar-radius expansion rate -- it's about 10% per billion years, much larger than mine.

I went to the Dartmouth Stellar Evolution Database website:

http://stellar.dartmouth.edu/models/index.html

and generated a model of stellar evolution for a star with initial mass equal to the Sun's, initial metallicity [Fe/H] = 0.07 and [alpha/Fe] = 0.00 -- that was the closest I could find for a match to the Sun. The model provided age, effective temperature, and luminosity, from which one can compute the radius.



Also, you seem to have used the Earth's present-day eccentricity.

Yes, that's right: I used e = 0.0168. But as you imply, the eccentricity oscillates from around 0 to 0.058 over long periods of time. That would clearly make the apparent angular size of the Sun larger (and smaller), expanding the shaded pink area on the graph considerably, and thus extending into the farther future the date of the last possible total eclipse. Good point.

Below is a new version of the graph, computed with this largest value of eccentricity. It pushes the last eclipse out to about 750 million years in the future.

22588


But it's a nice sort of graph.

Thank you. And thanks for bringing to our attention the importance of the variable eccentricity of the Earth's orbit.

lpetrich
2017-Aug-30, 09:56 AM
Seems almost that you are using the farthest distance for both the Sun and the Moon. Which is the mean distance and which is the extreme one? For the best case, it ought to be the farthest distance for the Earth and the nearest distance for the Moon.

lpetrich
2017-Aug-30, 11:35 AM
I found another paper on orbit elements:

Secular evolution of the solar system over 10 million years
J. Laskar
Astron. Astrophys. 198 341-362 (1988)

I had painstakingly OCRed and proofread the numbers in that paper, so that's why I could do this so quickly.

I collected statistics on the elements of the 8 planets for the last 10 million years. The inclination I referred to the Solar System's invariable plane, the plane perpendicular to the planet orbits' combined angular-momentum vector. The statistics are quartiles, 0%, 25%, 50%, 75%, and 100%. Before the quartiles is the present-day value, calculated from the same source.

Mercury:
ecc: 0.207 -- 0.125, 0.180, 0.211, 0.233, 0.301
inc: 6.5 -- 0.1, 4.9, 6.8, 8.5, 10.2

Venus:
ecc: 0.006 -- 0.001, 0.020, 0.031, 0.042, 0.071
inc: 2.2 -- 0.0, 1.0, 1.5, 2.0, 3.4

Earth:
ecc: 0.017 -- 0.000, 0.017, 0.027, 0.037, 0.059
inc: 1.6 -- 0.0, 0.9, 1.3, 1.7, 3.0

Mars:
ecc: 0.092 -- 0.002, 0.055, 0.072, 0.089, 0.125
inc: 1.7 -- 0.0, 3.1, 4.6, 5.5, 6.9

Jupiter:
ecc: 0.048 -- 0.026, 0.035, 0.047, 0.056, 0.061
inc: 0.3 -- 0.2, 0.3, 0.4, 0.4, 0.5

Saturn:
ecc: 0.056 -- 0.012, 0.032, 0.058, 0.076, 0.085
inc: 0.9 -- 0.8, 0.9, 0.9, 0.9, 1.0

Uranus:
ecc: 0.046 -- 0.007, 0.025, 0.049, 0.062, 0.068
inc: 1.0 -- 0.9, 1.0, 1.0, 1.1, 1.2

Neptune:
ecc: 0.009 -- 0.004, 0.007, 0.010, 0.011, 0.015
inc: 0.7 -- 0.5, 0.6, 0.7, 0.8, 0.8

StupendousMan
2017-Aug-30, 01:04 PM
Seems almost that you are using the farthest distance for both the Sun and the Moon. Which is the mean distance and which is the extreme one? For the best case, it ought to be the farthest distance for the Earth and the nearest distance for the Moon.

The graph shows two lines for each body: one represents the maximum angular size as seen from Earth, the other the minimum angular size. My estimate of about 750 million years is the point at which the minimum angular size of the Sun is equal to the maximum angular size of the Moon.