The Earth, spinning down, and the Moon, spiraling away, all by the Moon's tides on the Earth

[lpetrich]

On the Tidal History and Future of the Earth–Moon Orbital System - IOPscience -- PDF version: On the Tidal History and Future of the Earth–Moon Orbital System - Tyler_2021_Planet._Sci._J._2_70.pdf

The Moon makes very noticeable tides on the Earth, and the Earth's rotation drags them forward. This pulls the Moon forward, giving it more overall energy and making it spiral outward, and from Newton's third law, the Earth's rotation is slowed down.

At the present, the Moon is moving away at 3.8 centimeters per year, with its period increasing by 35 milliseconds per century. The Earth's rotation period is also increasing, at 2 msec/cy.

The Moon cannot get too close to the Earth, or else it will be pulled apart. Its minimum distance is the  Roche limit and it can be estimated with the constant-density approximation:

\( \displaystyle{ a \simeq 2.44 R \left( \frac{ \rho_{planet}}{\rho _{moon}} \right)^{1/3} } \)

for a moon at distance a from a planet with radius R. For the Earth's Moon, that is 2.88 Earth radii or 18,000 kilometers. The Moon would have a period of 6.8 hours and an angular size of 11 degrees, about that of a fist at arm's length. Its present angular size is 0.5 d.

 

[lpetrich]

If one assumes a linear rate of increase, one finds that the Moon was at the Earth's surface about 10 billion years ago. But tidal effects get stronger with decreasing distance, so the Moon was receding faster in the past.

Explicitly modelled deep-time tidal dissipation and its implication for Lunar history - ScienceDirect contains an estimate of how much faster:

da/dt = f * a^-11/2

where f has a lot of parameters in it, including a tidal-dissipation factor.

That makes the actual starting time about 2/13 this naive estimate, or 1.5 billion years ago. That is very implausible, so the Earth must have had less tidal drag in the past.  Earth's rotation and  Orbit of the Moon and  Tidal acceleration and  Tidal locking

The Earth's spindown rate has been roughly 2 ms/yr over the last 640 million years, then was roughly 1/5 of that for the 2 billion years before that. From day-length estimates from seashell growth rings and tidal deposits like at Elatina, one finds a peak at roughly 400 million years ago. What was going on back then? Being at the peak of some resonance effect?

 

[lpetrich]

I measured off the OP's paper, and I did some other calculations, to make this table:

Time	Ear Day	Dys in Yr	Mn Dst	Mn Siz	Mn SidP	Mn SynP	Mn SP Dy	Mnth in Yr
Present	24	365.256363	3.844	0.52	27.321662	29.530589	29.530589	12.368746
P-Tr 251.9 Mya	23.0	381	3.79	0.53	26.76	28.88	30.1	12.65
Ed-Cb 538.8 Mya	20.9	419	3.58	0.56	24.63	26.41	30.3	13.83
Tn-Cy 720 Mya	20.1	436	3.54	0.56	24.12	25.83	30.8	14.14
Hd-Ar 4 Gya	15.6	562	3.09	0.64	19.66	20.78	32.0	17.58
Mid Hd 4.25 Gya	14.3	613	2.91	0.68	18.01	18.94	31.8	19.28

What's what?

Ear Day = Earth Solar Day, in hours
Dys in Yr = Earth Days in a Year
Mn Dst = Moon Distance in 10⁵ km = 10⁸ m
Mn Siz = Moon observed size in degrees of arc
Mn SidP = Moon sidereal period: relative to the stars, in present-day days
Mn SynP = Moon synodic period: relative to the Sun, in present-day days
Mn SP Dy = Moon synodic period: days back then
Mnth in Yr = synodic months in a year

P = Permian, Tr = Triassic
Ed = Ediacaran, Cb = Cambrian
Tn = Tonian, Cy = Cryogenian (both Proterozoic)
Hd = Hadean, Ar = Archean

The OP's paper then showed slowing down after about 4.25 Gya (billion years ago), when it ought to have shown much faster change. That paper's numbers for 4.5 Gya: day length 13.5 hours, Moon distance 280,000 km, sidereal month 17 days.

The most plausible hypothesis to date is the  Giant-impact hypothesis where the Moon formed from fragments from the collision of a Mars-sized planet with the early Earth. In attempts to reconstruct this event with simulations, the Earth ends up with a rotation period of around 5 hours. That is not much less than Moon's Roche-limit period of around 7 hours, meaning that tidal drag will pull the Moon outward and not inward, what would happen with too-slow rotation. I also calculated that its Roche-limit distance is 18,000 km.

 

[lpetrich]

The OP paper's authors then extrapolated forward in time, and they found a big resonance effect a little over 4 billion years from now.

The Earth's day length goes from 24 h to 35 h, then jumps to 55 h, making the year length go from 365.256363 days to 250 d, then 160 d. The Moon's distance goes from 384,400 km to 425,000 km, then to 450,000 km, making its angular size go from 0.52 degrees to 0.46 d, then 0.44 d. The Moon's sidereal orbit period goes from 27.321663 d to 31.5 d, then 35.5 d. Its synodic period goes from 29.530589 d to 34.5d, then 39.3 d, and in the days of those future times, 25.8 then 17.2. The number of synodic months per year goes from 12.368746 to 10.6, then 9.3.

A problem with this calculation is that the future Earth is not likely to have liquid water for much longer, maybe about 1 billion years more.  Future of Earth This is from the Sun gradually getting brighter and brighter  Faint young Sun paradox This will also mean that carbon dioxide will accumulate in the Earth's atmosphere, making our homeworld eventually much like Venus. So that will complicate calculations of what kinds of tides the Earth will have.

 

[lpetrich]

How far can the Moon get away from our planet before it has a risk of wandering away?

One can find a simple upper limit by doing (the Sun's tide on the Moon's orbit) ~ (the Earth's pull on the Moon). Leaving aside purely numerical factors, this gives us: (Moon's sidereal orbit period) ~ (Earth's sidereal orbit period) or a Moon-Earth period ratio of ~ 1.

Doing a more careful calculation gives us the  Hill sphere and ignoring the Sun's action on the Moon, one finds a period ratio of ~ 1/sqrt(3) ~ 0.577.

Not much of an improvement.

One can go further by finding a series solution for the Moon's motion under the Sun's influence, especially with making some simplifications, like a coplanar orbit that is circular to lowest order, and only the lowest-order tidal effects.

Thus giving the Hill-Brown  Lunar theory though "solution" is a better word here, since it's not a separate physical theory but the solution of equations of motion.

One starts with a lowest-order-circular solution, then finds the solutions for eccentric and inclined orbits, including their precession rates.

Literal solution for Hill's lunar problem | SpringerLink and (PDF) Literal solution for Hill's lunar problem and https://adsabs.harvard.edu/full/1979CeMec..19..279S

The Moon's line of nodes (where its orbit plane intersects the Earth's) precesses backward with a period of 18.60 years, and its line of apsides (closest-distance and farthest-distance points) precesses forward with a period of 8.85 years. A lowest-order calculation gives 17.825 years, and while the nodal precession is close, the apsidal precession is almost twice as fast. That problem caused a lot of headaches for Isaac Newton, but it was resolved by calculating to higher orders, greater powers of the Moon-to-Earth period ratio.

The above paper goes to power 15 for the nodal precession and power 29 for the apsidal precession, because the series coefficients increase much faster for the latter than the former, requiring more terms for good accuracy.

 

[lpetrich]

Once that was done, I estimated the radius of convergence for the apsidal-precession series, and I estimated a stability bound for both direct (prograde; same direction) and retrograde (opposite direction) orbits.

The moon-planet sidereal-period ratio is (direct) 0.164 and (retrograde) 0.243, or (direct) 1/6.11 and (retrograde) 1/4.11.

The OP paper's calculated sidereal lunar period at 4.5 billion years from now is 35.5 days, giving a period ratio of 0.097 or 1/10.3 . That's within that stability limit of 59.8 days.


Let's look at other planets' moons.

• Jupiter -- (r) S/2022 J 2 - 781.56 d - 0.180 - 1/5.54 -- (d) Valetudo - 527.61 d - 0.121 - 1/8.21
• Saturn -- (r) S/2004 S 26 - 1603.95 d - 0.149 - 1/6.71 -- (d) S/2019 S 6 - 1055.68 d - 0.098 - 1/10.19
• Uranus -- (r) Ferdinand - 2790.03 d - 0.091 - 1/11.00 -- (d) Margaret - 1661.00 d - 0.054 - 1/18.48
• Neptune -- (r) Neso - 9796.67 d - 0.162 - 1/6.14 -- (d) Laomedeia - 3175.65 d - 0.052 - 1/18.96

So these moons are well within the Hill-Brown apsidal stability limits.