Why the Moon always shows the same face

[PeterDonis]

Music to my ears!

Don't rejoice too soon; the Moon is not like a pendulum in some key respects. See my post #61.

 

[sophiecentaur]

The equilibrium point is not a potential energy minimum, as it is with a pendulum.

Of course, because there is motion in any orbit too (unlike with a stationary pendulum) - but the total Energy reduces due to losses until it reaches a situation where the Energy will not be dissipated any more. That is 'a' minimum and I have already made this point. The pendulum analogy is simpler but you can't just kick the Energy approach because it's hard to include. There are certain principles which run through all of Physics; it's only a matter of finding where they are lurking.
Let's face it, no one here has demonstrated in detail how the 'torque' argument produces stable locking. Wiki has a very nice animation but does that constitute a proof?

 

[PeterDonis]

the total Energy reduces due to losses until it reaches a situation where the Energy will not be dissipated any more.

No, because, strictly speaking, there is no such situation. At least, not as long as we want the Earth and the Moon to be intact. If you wait long enough and dissipation reduces the total energy enough, the Moon will end up inside the Roche limit and will break apart and the Earth will have a system of rings. (This is considering the Earth-Moon system in isolation only; in our actual case, I believe the Sun will become a red giant long, long before the Earth-Moon system gives up enough energy from dissipation to bring the Moon inside the Roche limit.)

no one here has demonstrated in detail how the 'torque' argument produces stable locking.

I have already said, several times now, that the equilibrium of zero torque is not stable. The zero torque equilibrium is not a potential energy minimum. So the reason no one has demonstrated how torques produce stable locking is that they don't.

 

[PeterDonis]

If you wait long enough and dissipation reduces the total energy enough, the Moon will end up inside the Roche limit

I should probably clarify this. This is a very, very long-term effect of dissipation. Perhaps it will help to describe how we would expect the Earth-Moon system to evolve in the future, assuming it to be isolated (i.e., ignoring things like the Sun becoming a red giant).

Right now, the Moon's rotation and revolution periods match, but the Moon's orbit is elliptical and is inclined to the Earth's equatorial plane. So there are unbalanced torques in the system that tend to make the Moon's orbit circular and to bring its orbital plane into Earth's equatorial plane. Eventually those things will happen.

However, there are also unbalanced torques in the system because the Earth is rotating much faster than the Moon's rate of revolution about the Earth. These torques tend to slow the Earth's rotation and move the Moon's orbit further out. Eventually, the Earth's rotation will have slowed to match the Moon's rate of revolution, at which point there will indeed be a zero torque equilibrium (assuming that the Moon's orbit has also become circular and in the Earth's equatorial plane by that time). If we ignore dissipation, the system will oscillate with some small amplitude about this equilibrium forever.

Dissipation, while the above equilibrium has not yet been reached, will mainly be friction inside the Earth and Moon due to them not being perfectly elastic. This will act to reduce the total energy of the system, which I think will make the zero torque equilibrium described above happen at a slightly smaller Earth-Moon separation than it would have in the absence of dissipation (but still a significantly larger separation than exists today).

Once the system has reached the point where it is oscillating about the zero torque equilibrium, dissipation will be, I think, much smaller than it is today, because the difference at any point in time between the actual rotation/revolution rates and their equilibrium ones will be much smaller than any differences today, so there will be much less frictional damping. Dissipation will act to reduce the amplitude of the oscillations, but it will also still be reducing the total energy of the system, which will act to reduce the Earth-Moon separation, making the system more tightly bound.

If a point is reached at which the oscillations have been damped away, there will still be dissipation within the system due to the emission of gravitational waves, but this will be much, much smaller than any frictional damping was. Even so, it will still act to (very, very slowly) reduce the Earth-Moon separation by reducing the total energy of the system. And at some point, the separation will be small enough that the Moon will be inside the Roche limit. That is what I was referring to in the statement of mine quoted above.

 

[sophiecentaur]

I should probably clarify this. This is a very, very long-term effect of dissipation. Perhaps it will help to describe how we would expect the Earth-Moon system to evolve in the future, assuming it to be isolated (i.e., ignoring things like the Sun becoming a red giant).

Right now, the Moon's rotation and revolution periods match, but the Moon's orbit is elliptical and is inclined to the Earth's equatorial plane. So there are unbalanced torques in the system that tend to make the Moon's orbit circular and to bring its orbital plane into Earth's equatorial plane. Eventually those things will happen.

However, there are also unbalanced torques in the system because the Earth is rotating much faster than the Moon's rate of revolution about the Earth. These torques tend to slow the Earth's rotation and move the Moon's orbit further out. Eventually, the Earth's rotation will have slowed to match the Moon's rate of revolution, at which point there will indeed be a zero torque equilibrium (assuming that the Moon's orbit has also become circular and in the Earth's equatorial plane by that time). If we ignore dissipation, the system will oscillate with some small amplitude about this equilibrium forever.

Dissipation, while the above equilibrium has not yet been reached, will mainly be friction inside the Earth and Moon due to them not being perfectly elastic. This will act to reduce the total energy of the system, which I think will make the zero torque equilibrium described above happen at a slightly smaller Earth-Moon separation than it would have in the absence of dissipation (but still a significantly larger separation than exists today).

Once the system has reached the point where it is oscillating about the zero torque equilibrium, dissipation will be, I think, much smaller than it is today, because the difference at any point in time between the actual rotation/revolution rates and their equilibrium ones will be much smaller than any differences today, so there will be much less frictional damping. Dissipation will act to reduce the amplitude of the oscillations, but it will also still be reducing the total energy of the system, which will act to reduce the Earth-Moon separation, making the system more tightly bound.

If a point is reached at which the oscillations have been damped away, there will still be dissipation within the system due to the emission of gravitational waves, but this will be much, much smaller than any frictional damping was. Even so, it will still act to (very, very slowly) reduce the Earth-Moon separation by reducing the total energy of the system. And at some point, the separation will be small enough that the Moon will be inside the Roche limit. That is what I was referring to in the statement of mine quoted above.

I am quite happy to accept that the situation as it is and how it will be is more or less as you describe it. Many of these things have already all been mentioned in the thread. However, I am more interested in what happens in the very simplest system than in a load of instances because the whole thing becomes divergent instead of homing in on some actual understanding. Plenty of time to expand when the basics have been established.
What you are saying does not disprove what I suggest, at all. I realize that you don't want to look at the problem this way but it surprises me that you don't find the Energy analysis approach is of use when you find it elsewhere, all over Physics. You talk about dissipation getting less but you don't want to acknowledge a parallel between this sort of system and other damped oscillating systems. Never mind, I have found out a lot from you and others.

Once the system has reached the point where it is oscillating about the zero torque equilibrium, dissipation will be, I think, much smaller than it is today,

Is it surprising that the rate of Energy Dissipation will follow a basically exponential law? It applies to other systems that we come across.

 

[PeterDonis]

I realize that you don't want to look at the problem this way but it surprises me that you don't find the Energy analysis approach is of use

Didn't I say that dissipation will reduce the total energy of the system? How is that ignoring energy?

What I did say is that the zero torque equilibrium is not a minimum of potential energy, and you agreed with that. That means you can't analyze that equilibrium the way you would analyze an equilibrium that is a minimum of potential energy, like a pendulum.

You talk about dissipation getting less but you don't want to acknowledge a parallel between this sort of system and other damped oscillating systems.

It seems to me that you are the one failing to acknowledge something: the key difference between this system and other damped oscillating systems, which I have just repeated above.

 

[PeterDonis]

it surprises me that you don't find the Energy analysis approach is of use when you find it elsewhere, all over Physics

I've given a pretty complete description of what the system will do. How do you think an energy analysis approach would improve it?

 

[PeterDonis]

Is it surprising that the rate of Energy Dissipation will follow a basically exponential law?

I'm not sure it will in the regime I was referring to in what you quoted (after the Earth and Moon are both tidally locked and the system is oscillating with small amplitude about the zero torque equilibrium). In that regime, as I said, dissipation doesn't just reduce the amplitude of oscillations; it moves the equilibrium, because the equilibrium depends on the total energy of the system (since that determines the Earth-Moon distance, which affects the rotation/revolution rate). So, again, it's not the same as a system like a pendulum, where the equilibrium does not depend on the total energy.

In the regime the actual Earth-Moon system is in now, where the Moon is tidally locked (mostly--its orbit is still not circular in the Earth's equatorial plane) but the Earth isn't, I don't think the rate of energy dissipation is even close to following an exponential law.

 

[PeterDonis]

In actual objects there is also dissipation going on, but its effects are very small compared to the effects of the torques.

Looking at the Wikipedia page on tidal acceleration...

https://en.wikipedia.org/wiki/Tidal_acceleration
...which gives a detailed treatment of the slowing of the Earth's rotation by the Moon (and the corresponding increase in the Moon's orbital distance), it seems that, at least for the case of the Earth's rotation, my intuition here was way off. According to this article (which gives several references to papers describing detailed measurements of tidal dissipation in the Earth), only a small fraction (about 1/30 if I'm reading the article correctly) of the total rotational kinetic energy lost by the Earth as its rotation slows is transferred to the Moon along with angular momentum; the rest is converted to heat.

 

[sophiecentaur]

I've given a pretty complete description of what the system will do. How do you think an energy analysis approach would improve it?

Because it would demonstrate the commonality between such phenomena. Why exclude it from all the other mechanical, electrical, thermal and vibrational studies? The problem here seems to be that you are finding reasons why not to consider what I suggest. I really don't see why parallels between phenomena don't strike you as 'insightful'.

 

[PeterDonis]

Because it would demonstrate the commonality between such phenomena.

Well, it seems like the key point from this discussion is an important difference between this case and other phenomena where energy and angular momentum are involved, namely, that this case does not have an equilibrium that is a minimum of potential energy.

The problem here seems to be that you are finding reasons why not to consider what I suggest.

No, the "problem" is that I see an important difference between this case and other cases that have been brought up (which I have just restated, again, above), and you either don't see it or don't understand why I think it's important. You have not made a single comment that I can find responding to this important difference, even though I have pointed it out multiple times. Do you seriously not see it as important?

I really don't see why parallels between phenomena don't strike you as 'insightful'.

If you can show me such a parallel that adds something to the analysis I've already done, I'll be happy to consider it. But the fact that I haven't come up with such a parallel myself does not mean that I am "finding reasons" not to. It just means that I see a particular feature of this case that makes it different from other cases as a key to the analysis.

 

[sophiecentaur]

that this case does not have an equilibrium that is a minimum of potential energy.

Post #36 and later:

You are right, of course - the total orbital Energy needs to be considered and it will go to a minimum when locking is reached.

I acknowledged the difference between planetary motion and pendulums, 36 posts ago and I asked for some ideas for combining the two concepts. I suggested a `Total Energy' minimum but I think you must have missed that. Certainly, you have brought up and rejected the PE Minimum idea regularly since my retraction of the idea.
There are plenty of thought experiments that could give the same sort of conditions - in which losses in one mode strongly outweigh the losses in another. Planetary situations tend to differ from many others (as Newton recognised) because the actual orbital losses are so much lower than internal losses so, yes, it's a different type of problem - but hardly fundamentally different.

 

[PeterDonis]

I acknowledged the difference between planetary motion and pendulums, 36 posts ago

Not really. You still said then that total orbital energy is a minimum when tidal locking is reached (I shouldn't have specified "potential energy" in post #72). That's not the case. As I responded in post #41:

Actually, the main mechanism of tidal locking is torques on the tidal bulges, for which no dissipation is required, so the total energy of the system actually remains constant.

We then went back and forth about whether there was an energy minimum. The answer is that there isn't. Dissipation will gradually reduce the total energy of the system, but there is never an equilibrium point where the energy is minimized, because, as I've remarked several times now, dissipation changes the equilibrium point (the point of zero torque) since the equilibrium point depends on the total energy. Which, once more, is the key difference between this case and other cases such as a pendulum.

Planetary situations tend to differ from many others (as Newton recognised) because the actual orbital losses are so much lower than internal losses so, yes, it's a different type of problem - but hardly fundamentally different.

You don't think that the zero torque equilibrium point depending on the total energy, and the consequent lack of a minimum energy equilibrium, is a "fundamental" difference?

 

[sophiecentaur]

We then went back and forth about whether there was an energy minimum. The answer is that there isn't. Dissipation will gradually reduce the total energy of the system, but there is never an equilibrium point where the energy is minimized, because, as I've remarked several times now, dissipation changes the equilibrium point (the point of zero torque) since the equilibrium point depends on the total energy. Which, once more, is the key difference between this case and other cases such as a pendulum. etc.. . . .

That's just an assertion of yours, I think - certainly your Wiki source doesn't mention it.
Not an equilibrium POINT but an equilibrium condition.
Dissipation reduces the Energy situation to a minimum. How can it increase the Energy in a system? Of course, once the oscillations have died out there will be zero torque; there will be equilibrium of a sort because (stable) Equilibrium is the state where a disturbance will increase the Energy in a system and that will cause a restoring Force.

You don't think that the zero torque equilibrium point depending on the total energy, and the consequent lack of a minimum energy equilibrium. Please don't insist that a pendulum is the only alternative to tidal locking; I just chose it as a very simple situation - with very obvious differences from a planetary system.

What does that actually mean? It is not necessary to include Forces in an Energy argument. It's been far too long since I 'did' this in Uni but isn't this just the difference between Newtonian and Hamiltonian Mechanics? Your above argument is trying to mix the two together. Even if it's ok to do that, it must be doing things the hard way and would need some good authoritative reference.

 

[PeterDonis]

That's just an assertion of yours, I think

It's an obvious consequence of the fact that the semi-major axis and orbital period of the Earth-Moon system depend on the total energy.

Dissipation reduces the Energy situation

Obviously.

to a minimum

Only if there is one. What is the minimum total energy of the Earth-Moon system? There isn't one, except in the sense that if the Moon gets close enough to the Earth to be inside the Roche limit, it breaks up, and you no longer have a two-body system.

How can it increase the Energy in a system?

I never said it did. You appear to be reading things into my post that aren't there. Please read what I've actually said.

once the oscillations have died out there will be zero torque

Yes, until dissipation has reduced the total energy of the system and hence changed the zero torque equilibrium point.

there will be equilibrium of a sort because (stable) Equilibrium is the state where a disturbance will increase the Energy in a system and that will cause a restoring Force

This is not completely correct, because the "increase the energy in a system" part is not required. And it's not true for the Earth-Moon system, as we've already said multiple times in this thread.

What does that actually mean?

The quote you gave from me appears to be partially incorrect: I did not say "Please don't insist that a pendulum is the only alternative to tidal locking; I just chose it as a very simple situation - with very obvious differences from a planetary system." Perhaps you intended that to be part of your reply? (Also, the sentence before that is cut off.)

As for what it means, again, the fact that the equilibrium of zero torque for the Earth-Moon system depends on the total energy of the system is an obvious consequence of the fact that the semi-major axis and orbital period depend on the total energy, since the orbital period is what determines the rate of rotation required for the zero torque equilibrium.

It is not necessary to include Forces in an Energy argument.

If you're making an energy argument, yes. I'm not making an energy argument. The argument I'm making includes energy as a contributing factor, but it's not an "energy argument" in the sense of analyzing all of the system's dynamics using energy alone.

isn't this just the difference between Newtonian and Hamiltonian Mechanics? Your above argument is trying to mix the two together.

I don't know what you mean by this or how it relates to what I've said.

 

[PeterDonis]

the semi-major axis and orbital period of the Earth-Moon system depend on the total energy

I should probably expand on this some. There are three main stores of energy (more precisely, mechanical energy) in the Earth-Moon system (if we view things from the standpoint of an inertial frame centered on the Earth): the Moon's orbital energy (kinetic + potential), the Moon's spin energy, and the Earth's spin energy. Heuristically, the processes involved in tidal locking and dissipation work like this:

If the Moon's rotation and revolution periods don't match, there will be torques that act to make them the same. This process conserves total mechanical energy, but transfers energy from the Moon's spin energy to the Moon's orbital energy (assuming that the system started with the Moon rotating much faster than it revolves around the Earth). There will also be dissipation within the Moon during this process, which transfers energy from the Moon's spin into heat, reducing the total mechanical energy of the system.

If the Earth's rotation period does not match the Moon's period of revolution around the Earth, there will be torques that act to make them the same. This process conserves total energy, but transfers energy from the Earth's spin energy to the Moon's orbital energy. There will also be dissipation within the Earth during this process, which transfers energy from the Earth's spin into heat, reducing the total mechanical energy of the system.

Once the Earth's rotation, the Moon's rotation, and the Moon's revolution periods all match, we have a zero torque equilibrium. The system might oscillate about this equilibrium, but that process conserves total mechanical energy. However, there will also still be dissipation in the system, both due to the oscillations, if any, and (a much smaller effect) due to the system's emission of gravitational waves, which happens even at the zero torque equilibrium. Dissipation acts to convert all three types of mechanical energy into either heat (if it's due to oscillation about the zero torque equilibrium) or gravitational wave energy. In either case, it reduces the total mechanical energy of the system.

During the first two phases described above, while dissipation is reducing the total mechanical energy of the system, there is also exchange of mechanical energy going on that is increasing the orbital energy of the Moon (at the expense of the Moon's and Earth's spin energy), which increases the semi-major axis and orbital period. So during these phases, the semi-major axis and orbital period are increasing while the total mechanical energy is decreasing.

During the third phase described above, there is no longer any net exchange of mechanical energy (I say "net" because there will still be some exchange during oscillations about the zero torque equilibrium, but they net out to zero over a complete cycle), so the reduction of total mechanical energy decreases the semi-major axis and orbital period. So during this phase, the semi-major axis, orbital period, and total mechanical energy are all decreasing. This is the main phase I was thinking of in the remark of mine that I quoted at the start of this post. And it should be evident that this implies that (1) the zero torque equilibrium point depends on the total mechanical energy, and (2) there is no minimum of the total mechanical energy (other than the Moon getting inside the Roche limit and breaking up).

 

[alantheastronomer]

...if the Moon did not have an asymmetry which side of the Moon...faced the Earth as a result of tidal locking would have been a random result, since no side would have been preferred.

We know this is true because Venus is also tidally locked yet has no mass asymmetry...

 

[PeterDonis]

Venus is also tidally locked

Not the way the Moon is. Venus rotates retrograde, not prograde, and its rotation period is longer than its period of revolution around the Sun. I believe the state it is in is a tidal resonance, but it's not the one usually referred to by the term "tidal locking".

 

[alantheastronomer]

Not the way the Moon is. Venus rotates retrograde, not prograde, and its rotation period is longer than its period of revolution around the Sun. I believe the state it is in is a tidal resonance, but it's not the one usually referred to by the term "tidal locking".

Yes that's certainly true, but assuming that it's rotation rate was much greater and prograde in the distant past, the same mechanism of tidal gravity is what's brought it into it's present resonant state.

 

[PeterDonis]

assuming that it's rotation rate was much greater and prograde in the distant past, the same mechanism of tidal gravity is what's brought it into it's present resonant state.

That's not possible. If it were rotating prograde faster than it revolved around the Sun, tidal torques would drive its prograde rotation rate to be the same as its rate of revolution around the Sun. (Or it could possibly get stuck in another resonance before that, as Mercury is.) Tidal torques at that point would stop changing the rotation rate; they certainly woudn't reverse it.

 

[alantheastronomer]

That's not possible. If it were rotating prograde faster than it revolved around the Sun, tidal torques would drive its prograde rotation rate to be the same as its rate of revolution around the Sun. (Or it could possibly get stuck in another resonance before that, as Mercury is.) Tidal torques at that point would stop changing the rotation rate; they certainly woudn't reverse it.

Yes one would think so... but if -dw/dt (where w is angular rotational velocity) were nonzero, then when it reached corotation it is possible for it to overcompensate and become retrograde. A similar effect occurs in accretion - if the angular momentum of the accreting matter is such that v^2/r is less than GM/r^2, then the deceleration -dv(r)/dt is nonzero and when it reaches the equilibrium radius, it overshoots and continues on to smaller radii until the centripetal acceleration overpowers gravity and it begins to move outward again. Of course then it will collide with other infalling matter and it's outward motion will be damped. I'm not saying this is definitely the case for Venus - it's also possible that it started out with a retrograde rotation with it ending in it's current resonance, or that it started out in that resonance when it first formed, or it's also possible that it formed at some other radii and migrated to it's present location. All I'm saying is that Venus' situation (and, as you point out, Mercury's) illustrates that, as you say, tidal gravity doesn't need to depend on an asymmetric mass distribution.

 

[PeterDonis]

one would think so

Not just "one would think so". It is so based on the torques vanishing when the rotation and revolution rates are equal. That has already been discussed at length in this thread.

if -dw/dt (where w is angular rotational velocity) were nonzero, then when it reached corotation it is possible for it to overcompensate and become retrograde

First, rotating prograde a bit more slowly than the revolution rate is not at all the same as retrograde rotation.

Second, if prograde rotation overshoots and gets slower than the revolution rate, then the torques are now opposite and will act to speed up the rotation rate until it is equal to the revolution rate. If it overshoots again, you simply get oscillations about the zero torque equilibrium. That has already been discussed at length in this thread as well.

All I'm saying is that Venus' situation (and, as you point out, Mercury's) illustrates that, as you say, tidal gravity doesn't need to depend on an asymmetric mass distribution.

That's certainly not all you are saying. This part of what you are saying is correct. But you are also saying other things that are not correct. See above.

 

[alantheastronomer]

That's certainly not all you are saying.

It is in post #78. Also, "all I'm saying is.." is just an expression; it means "my main point is..." or "the gist of my statement is...", not "the sum total of my account is..."

First, rotating prograde a bit more slowly than the revolution rate is not at all the same as retrograde rotation.

That's very true but - let me give you one more example in physics, along with the accretion situation, where counterintuitively, motion seemingly miraculously appears to reverse; are you familiar with the concept of the magnetic mirror? A charge particle moves in a circular motion in a magnetic field...as it moves towards a region where the magnetic field gets stronger, it's perpendicular velocity increases and it's kinetic energy gets larger. Since energy must be conserved, where does this increase in kinetic energy come from? It "steals" it from the particle's forward motion and it decelerates until the field strength becomes so great that it actually reverses direction!

if prograde rotation overshoots and gets slower than the revolution rate, then the torques are now opposite and will act to speed up the rotation rate until it is equal to the revolution rate. If it overshoots again, you simply get oscillations about the zero torque equilibrium.

Yes that's true, unless the planet has time to "relax" and lose it's tidal bulge so that there are no more restoring torques to bring it back to equilibrium. Venus now is perfectly spherical, so it is pretty safe to say that at some time in it's past, it's shape was restored to it's present condition... If I can go a little off topic for a moment, something else you said caught my interest...From post #65

Dissipation...will mainly be friction inside the Earth...

and post #70

...only a small fraction of the total rotational kinetic energy lost by the Earth as it's rotation slows is transferred to the Moon along with angular momentum; the rest is converted to heat.

I remember reading somewhere that given the best estimates of the amount of uranium in the Earth's core, it's still not enough to keep the outer core fluid and it should have solidified long ago and so there needs to be another source of heat otherwise there would be no magnetic field. Is it possible that tidal friction could be the source of this heat, and that tidal torques on the liquid outer core drives the convection that is necessary for the dynamo responsible for the Earth's magnetic field?

 

[PeterDonis]

It is in post #78.

No, it isn't. Either own everything you said, or don't say it in the first place.

are you familiar with the concept of the magnetic mirror?

This concept is irrelevant to the discussion here, since there is no gravitational analogue to it.

that's true, unless the planet has time to "relax" and lose it's tidal bulge so that there are no more restoring torques to bring it back to equilibrium.

The planet won't "relax" and lose its tidal bulge unless it is no longer subject to tidal forces. See further comments below.

Venus now is perfectly spherical

To the accuracy we can measure. But since it is subject to the Sun's tidal gravity, it must have a tidal bulge of some extent. It's just much smaller than Earth's, because Venus has no ocean and no Moon. But this will have been true throughout its history if it's true now; it's not the result of any "relaxation" process.

Since the thread has run its course anyway, the thread is now closed. If you want to discuss further your question about the heat source of the Earth's core, please PM me and I will spin that part of your post off into a new thread.