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

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