These are the things that keep me up at night.

The Moon is gradually moving away from the Earth at a rate of a few cm a year. This has the additional effect of causing the Earth to rotate on its axis more slowly. Eventually the Earth will find itself tidally locked with the Moon. Both bodies will show only one face to each other.

So here's the question, in several parts. 1) What will be the distance from the Earth to the Moon when they become tidally locked together? 2) How long will this take? 3) Since there is also a tide from the Sun (although smaller than the tide from the Moon) I presume that the Earth will become tidally locked to the Sun. How long will this take? and 4)will the Earth's orbital radius increase and if so by how much?

I'm working on the math on my own, but I thought I'd see what you guys can come up with.

Thanks

__________________
Jeff Schwarz
__________________________________________________
Argh!! They booby-trapped their sun!!****--Invader ZIM

I don't have the answers...but I have another question...

Think about the day, ever so long ago, when the moon *first* became tidally locked to the earth. It was rotating, but more and more slowly...

Does it just slow down to a complete halt, or is there a period of nutation, rocking forward and backward, perhaps as the earth tugs at some largish gravitational unevenness somewhere in the moon's crust?

Is it an asymptotical decline, or is there a "catastrophe" of some sort?

I've been thinking on this for some time, but don't know how to put together the model...

Silas

Are you familiar with lunar libration? The moon still does not maintain an absolutely constant orientation with respect to the Earth. Its rotation is nearly constant, but its orbit is not circular--so it moves faster in some parts of orbit, and the rotation cannot compensate.

for those who doubt libration of lunation...

Cool pic, but it's giving me a headache.

Back to the original question, I quote from the ultimate authority -- namely, our own BA in his new book.

He discusses tidal evolution, that is, the way the tides affect the Earth-Moon system. He explains that Earth's rotation is slowing, and that eventually it will rotate once a month, keeping the same face toward the moon at all times.

But...
This suggests we're talking about several billion years.

I'll buy that, but the geek in me wants more concrete figures.

I've lost the links I had to some pages that more completely dealt with the Earth-Moon tidal system and the forces invoved. Anyone still have them?

__________________
Jeff Schwarz
__________________________________________________
Argh!! They booby-trapped their sun!!****--Invader ZIM

Glad to meet all of you. This being my first post (perhaps last) please forgive any faux pas.

This is to address the first post by Geo3gh.

"These are the things that keep me up at night."
Perhaps you are asking the wrong questions. Sorry!
BTW I never sleep well.

"The Moon is gradually moving away from the Earth at a rate of a few cm a year."
The Earth-Moon system moving apart by 3-4 cm per year is experimental fact.

However the following is (astronomy book) conjecture:
"This has the additional effect of causing the Earth to rotate on its axis more slowly. Eventually the Earth will find itself tidally locked with the Moon. Both bodies will show only one face to each other."

The facts are:
The Moon does run slow (recession of nodes), however there is no evidence that it is "slowing".
The Earth does run slow (leap second), again there is no evidence that it is "slowing".
I'm not completely sure on this, but I don't believe that the gravitational mass can effect the rotation of the inertial mass.
If you locate the center of gravity of a mass (point around which all its weight is equally distributed) and then suspend that mass from that point it will be in perfect balance and will not rotate, or will continue to rotate indefinitely. The Earth's center of gravity suspends the Moon's center of gravity and vice versa. Thus the Earth cannot rotate the Moon, and the Moon cannot rotate the Earth. Furthermore, tides are internal to a system so therefore they cannot change the angular momentum of the system.

"3) Since there is also a tide from the Sun (although smaller than the tide from the Moon) I presume that the Earth will become tidally locked to the Sun. How long will this take?"

Forever!
If the Earth were locked to the Sun, the Moon were locked to the Earth, and the Earth were locked to the Moon, the Earth Moon system would no-longer rotate the Moon would crash straight down and the Earth and Moon would become one mass:-)


Gary

Actually, the Earth's rotation IS slowing. See http://www.badastronomy.com/bad/misc/tides.html for a discussion on this.

__________________
Jeff Schwarz
__________________________________________________
Argh!! They booby-trapped their sun!!****--Invader ZIM

It's an AWSOME pic, and the headache is worth it! [img]/phpBB/images/smiles/icon_biggrin.gif[/img]

<a name="redlink"><hb>
The total angular momentum stays the same, but the Earth's momenutum is less and the moon's is more. The slowing of the Earth in ancient times has been established by referencing daily and yearly coral growth patterns, and also by study of ancient eclipse observation.

<font size=-1>[Added name]</font>

<font size=-1>[ This Message was edited by: GrapesOfWrath on 2002-03-20 11:09 ]</font>

I did this whole calculation as part of my PhD oral exam back in 1995. Let's see how much I remember. [img]/phpBB/images/smiles/icon_smile.gif[/img]

Um, well, it's not so much conjecture as extrapolation.

Both not true. Think about it, if the moon is moving further away, it *has* to slow down. Kepler's third law.

This is imprecisely stated. First of all, neither the earth nor the moon are spherically symmetric. It is the *torque* produced by the tidal bulges that drags
on the two bodies and changes the angular momentum of the joint system (torque = dL/dt). See below.

No. I think about it this way: the tides cause a deformation of the earth along the direction of the earth-moon axis. However, the earth does not have perfect elasticity, so as it rotates, the bulges are pulled ahead of the earth-moon axis. Therefore, if you look at the force the bulges exert on the moon (Newton's third law), that force is mis-aligned with said axis. Therefore, the torque (|t| = |r x F| = |r||F|sin(theta)) is non-zero, therefore the angular momentum of the system changes. This will manifest in two ways: the earth's rotation period will slow and the moon will shift into a higher orbit and revolve more slowly.

No. It's still rotating, it's just that from the point of view of an observer on the earth, the moon would *appear* to hover over the same spot on the earth. By your argument, geostationary satellites should come crashing down. When the earth/moon system tidally locks, the moon will be at the geosynchronous orbit, which will be about 12 times further out than current geosynchronous orbits, or half a million km. I remember that the asymptotic value for the final rotation period is about 42 (current) days. Douglas Adams would have been so proud. The moon is currently about 3.8e5 km, so if its drift away is linear (which it isn't), it would take about 3 billion years to drift that far. Actually, it will take much longer, since the drift rate will slow down as the moon drifts away and the torque decreases.

I haven't ever tried to calculate what will happen when the earth-moon system tidally locks with the sun. Safe to say it's not going to happen for a long, long time, but let me think about it...

My instinct is that over the long haul, the tiny tidal bulge will work towards locking the earth with the sun, which will cause the earth to move away (Neptune is demonstrably moving away from the sun, which is why Pluto and other KBOs are trapped in resonance orbits with Neptune) and slow down, which will in turn cause the moon to slow down and move away from the earth to keep up. Unless the moon is orbiting in the opposite sense, which I don't remember off the top of my head. Anyway, I suspect that eventually the moon will get stripped off the earth into its own solar orbit, but I haven't tried to caluclate that for sure.

I have to go to a meeting,

Don

I found this from Phil's page on tides. Warning, lots o' math, but it's good.

http://www.jal.cc.il.us/~mikolajsawicki/tides_new2.pdf

Here's a bit on what it has on my question:

Wheee! We get a ring!! OK, we'll all be dead by then, but it's the principle of the thing.

[img]/phpBB/images/smiles/icon_smile.gif[/img]



_________________
Jeff Schwarz

<font size=-1>[ This Message was edited by: Geo3gh on 2002-03-18 15:44 ]</font>

I think what he was talking about is a situation where both are locked together with the sun--but there is a couple of lagrange points with that criteria.

Hmmm... maybe we should anticipate the future Earth-Moon configuration, and go to a 50-day month right now. Each year is 7 months long, with 15 holidays - say, two between each pair of months, with an extra one thrown in at solstice time.

The months could be ten weeks of five days each. Three working days and two weekend days?

I'm starting to like this a lot!

I don't know where I ran across this, but...
I remember that when I was a kid I learned your out-then-back-in projection for tidal evolution, but I've recently read (wish I remembered where) that it's no longer expected to work that way. Possibly because it takes so long for the outbound leg that the sun does its red giant bit in the meantime, which seems likely to throw a monkey wrench into the whole scenario... but I don't know if that's the right explanation.

Well, it's not *my* out and back again model.
I don't see how slowing the system down further would cause the moon to come back in and speed up. Wait a minute.... the earth's tidal bulges lead the moon, increasing its angular momentum and causing it to move out and slow down. If the earth's bulges were to lag the moon, it would want to *decrease* the moon's angular momentum, causing it to move in and speed up. This would cause the earth to speed up again, because the time scale for earth-moon locking is so much shorter for earth-sun locking.

Okay. I'm convinced. :-)

Well that's *certainly* true. I don't know if that's the explanation you were thinking of, but it's definitely true.

Don Smith

Darn. And here I was preparing for the spectacle of having our own ring system.

__________________
Jeff Schwarz
__________________________________________________
Argh!! They booby-trapped their sun!!****--Invader ZIM

Well if the Moon is in geosynch orbit at 432,000km (12 x 36,000km) for a 42 hour day and it takes more than 3 billion years to reach that point, it would likely be very near there when the Sun becomes a red giant. At that distance, the escape velocty from the Earth isn't terribly great, so I have little doubt that the Moon would be stripped from the Earth, even if the Earth manages to survive the Sun's red giant phase.

Of course, people living on the tropical moon Europa would not care much about such a lifeless world as the Earth by then, especially since the lifeless Mercury and Venus would have been consumed by the Sun long ago.

__________________
If E = MC², why do I have less energy the more mass my body acquires?
That is all.

--Azpod... Formerly known as James Justin

That's general situation when a moon orbits the planet faster than the planet rotates. Phobos does that, except it is inside Mars' Roche limit already, so the fact Phobos is holding together implies it is a solid rock rather than a rubble pile. In any case, I read that Phobos will crash into Mars in 40 million years.

BTW, if a moon orbits the planet against the planet's rotation, the tides also bring the moon closer. In 100 million years Triton will reach Neptune's Roche limit and will become a spectacular ring...