Two part physics question

[IXP]

I made a jesting post in another thread about using tidal energy and how it would cause the moon to eventually spiral into the earth. Someone pointed out, correctly, that the tides cause the moon to recede in its orbit.

I found this line in Wikipedia:

http://en.wikipedia.org/wiki/Moon
The energy dissipated in generating tides is directly responsible for the reduction in potential energy in the Moon-Earth orbit around the barycentre, resulting in a 3.8 cm yearly increase in the distance between the two bodies.

How can increasing the distance between the two bodies reduce the potential energy? The gravitational potential energy, mgh, increases with increasing distance. Is there another potential energy component in orbitting bodies, or is the article incorrect and should read kinetic energy instead? If you reduce the orbital speed I would expect the distance to increase which would support that it is the kinetic energy that is being reduced.

Part two is: What would be the effect of extracting tidal energy? Would it increase or reduce the rate at which the moon is receding?

IXP

[John Hewitt]

I made a jesting post in another thread about using tidal energy and how it would cause the moon to eventually spiral into the earth. Someone pointed out, correctly, that the tides cause the moon to recede in its orbit.

I found this line in Wikipedia:



How can increasing the distance between the two bodies reduce the potential energy? The gravitational potential energy, mgh, increases with increasing distance. Is there another potential energy component in orbitting bodies, or is the article incorrect and should read kinetic energy instead? If you reduce the orbital speed I would expect the distance to increase which would support that it is the kinetic energy that is being reduced.

Part two is: What would be the effect of extracting tidal energy? Would it increase or reduce the rate at which the moon is receding?

IXP

As I understand it (amd I may be wrong) the energy driving the moon away from the earth comes from a decline in the earth's rate of spin. Tidal friction slows the earth's spin and therefore reduces the rotational energy due to that spin; at the same time, the distance from the earth to the moon increases as does the moon's potential energy.

[JoeTheJuggler]

Part two is: What would be the effect of extracting tidal energy? Would it increase or reduce the rate at which the moon is receding?


I wouldn't think so, unless extracting tidal energy changed the shape of the ocean enough to create a greater tidal effect somehow.

[Thabiguy]

The angular velocity at which the Earth rotates differs from the angular velocity at which the Moon orbits. Tides serve as a form of "remote friction" that works to equalize these angular velocities. As the worlds "remotely rub" against each other, the Earth's surface slows down, braked by the slower orbiting Moon, and the Moon recedes, accelerated by the faster rotating Earth, which makes it enter a higher orbit (and paradoxically, in final effect, slow down even more).

Two laws are at work here: conservation of energy and conservation of angular momentum. The latter dictates that the total angular momentum of the Earth-Moon system must be conserved; whenever Earth's rotation slows down a bit, the Moon must recede by a specific corresponding amount to keep the total momentum constant. The Earth's rotational energy decreases as it slows down, and the Moon's orbital energy increases as it recedes. Earth's loss is greater than Moon's gain; the surplus energy is dissipated in the friction that makes the two worlds gradually match angular velocities. Therefore, dissipated energy is directly correlated with angular momentum transferred from the Earth to the Moon; the amount of one determines the amount of the other.

Part two is: What would be the effect of extracting tidal energy? Would it increase or reduce the rate at which the moon is receding?

Friction within tidal bulges liberates energy. Whether you capture and utilize it or let it dissipate unharnessed is irrelevant. If you just capture energy that is being freed anyway, there won't be a difference.

If you build big brakes that make it harder for the tidal bulges to pass through, increasing friction and thus the amount of liberated energy, then you will increase the rate at which the Earth's rotation is slowing down and the Moon receding. Considering the orders of magnitude that we are talking here (the rotational energy of the Earth is some 2.6 x 1029 J), unless you were extracting mindbogglingly vast amounts of energy, the effect would probably be negligible.

[Schneibster]

The Laws of orbital mechanics:
East takes you Out; Out takes you West; West takes you In; In takes you East. North and South bring you back. (Stolen from Larry Niven.)

If you speed up in your orbit (by applying thrust to move faster in the direction you're orbiting), you wind up in a bigger orbit. If you move your orbit out, you fall behind. If you slow down your orbit, you move to a smaller, slower orbit. And if you apply thrust to move downward, you'll speed ahead.

So when the friction Thabiguy spoke of makes the Moon move faster, the laws of orbital mechanics make its orbit larger. East takes you Out.

The statement that this is "tidal energy" is misleading; the tides provide the friction, but the laws of orbital mechanics provide the increasing distance.

[Beausoleil]

The angular velocity at which the Earth rotates differs from the angular velocity at which the Moon orbits. Tides serve as a form of "remote friction" that works to equalize these angular velocities. As the worlds "remotely rub" against each other, the Earth's surface slows down, braked by the slower orbiting Moon, and the Moon recedes, accelerated by the faster rotating Earth, which makes it enter a higher orbit (and paradoxically, in final effect, slow down even more).

Two laws are at work here: conservation of energy and conservation of angular momentum. The latter dictates that the total angular momentum of the Earth-Moon system must be conserved; whenever Earth's rotation slows down a bit, the Moon must recede by a specific corresponding amount to keep the total momentum constant. The Earth's rotational energy decreases as it slows down, and the Moon's orbital energy increases as it recedes. Earth's loss is greater than Moon's gain; the surplus energy is dissipated in the friction that makes the two worlds gradually match angular velocities. Therefore, dissipated energy is directly correlated with angular momentum transferred from the Earth to the Moon; the amount of one determines the amount of the other.

Friction within tidal bulges liberates energy. Whether you capture and utilize it or let it dissipate unharnessed is irrelevant. If you just capture energy that is being freed anyway, there won't be a difference.

If you build big brakes that make it harder for the tidal bulges to pass through, increasing friction and thus the amount of liberated energy, then you will increase the rate at which the Earth's rotation is slowing down and the Moon receding. Considering the orders of magnitude that we are talking here (the rotational energy of the Earth is some 2.6 x 1029 J), unless you were extracting mindbogglingly vast amounts of energy, the effect would probably be negligible.




Excellent explanation!

[Smike]

How can increasing the distance between the two bodies reduce the potential energy? The gravitational potential energy, mgh, increases with increasing distance.

Simply, gravitational potential energy is not mgh. This is an approximation that assumes that the value of g is constant. On Earth's surface, this is a good approximation.

The actual formula is slightly more complicated:

http://en.wikipedia.org/wiki/Potential_energy

[Brown]

Read the book Bad Astronomy. Some guy wrote it. On a lark, I think.

Anyway, there's an "explanation" of sorts in there. I put "explanation" in quotes because the author neglected to mention the influence of aliens, elves, werewolves, the Almighty and Captain Kirk.

[Thabiguy]

If you speed up in your orbit (by applying thrust to move faster in the direction you're orbiting), you wind up in a bigger orbit. If you move your orbit out, you fall behind. If you slow down your orbit, you move to a smaller, slower faster orbit. And if you apply thrust to move downward, you'll speed ahead.
Very true and well said, with just one small correction, no doubt an innocent oversight, so that readers new to orbits aren't confused.

[IXP]

Friction within tidal bulges liberates energy. Whether you capture and utilize it or let it dissipate unharnessed is irrelevant. If you just capture energy that is being freed anyway, there won't be a difference.

If you build big brakes that make it harder for the tidal bulges to pass through, increasing friction and thus the amount of liberated energy, then you will increase the rate at which the Earth's rotation is slowing down and the Moon receding. Considering the orders of magnitude that we are talking here (the rotational energy of the Earth is some 2.6 x 1029 J), unless you were extracting mindbogglingly vast amounts of energy, the effect would probably be negligible.
Thank you for this great explanation.

Since tidal energy is captured by impeding the flow of tides and turning generators as the waters flow through, I think the second paragraph above sums it up. The rate of recession of the moon would be increased.

Of course I realize that the effect is miniscule. I just wanted to get the direction right, which I did not in my jesting post about the moon crashing down.

IXP

[IXP]

Very true and well said, with just one small correction, no doubt an innocent oversight, so that readers new to orbits aren't confused.
Yep, it seems counterintuitive that slowing down an orbit actually speeds it up! The additional energy comes from falling toward the other body.

IXP

p.s. No one has answered my first question. Is the Wikipedia wrong to say that the tides are decreasing the potential energy in the moon / earth orbital system? I now see where the the energy to send the moon outwards comes from, but this surely is still increasing the potential energy, is it not? The kinetic energy is lowered to compensate since the orbital velocity is descreased, so isn't it the kinetic energy in the system rather the potential energy that is being dissapated by the tidal friction?

[IXP]

I wouldn't think so, unless extracting tidal energy changed the shape of the ocean enough to create a greater tidal effect somehow.
How much is enough? Any change would have an effect, albeit very very small. And extracting tidal energy does change the shape, it retards the tidal flows.

IXP

[Thabiguy]

p.s. No one has answered my first question. Is the Wikipedia wrong to say that the tides are decreasing the potential energy in the moon / earth orbital system?
Yes, you are right and Wikipedia is wrong. (At least in the article that you quoted; they've got it right in the very detailed article on tidal acceleration (http://en.wikipedia.org/wiki/Tidal_acceleration).)
I now see where the the energy to send the moon outwards comes from, but this surely is still increasing the potential energy, is it not?
Yes. What actually increases is the orbital energy, which is the sum of the kinetic and potential energy (and has the neat property of remaining constant throughout the orbit). As the kinetic energy of the Moon decreases as it recedes, its potential energy must increase even more so to make up for net energy increase.
The kinetic energy is lowered to compensate since the orbital velocity is descreased, so isn't it the kinetic energy in the system rather the potential energy that is being dissapated by the tidal friction?
I'm not sure if I understand that question. Kinetic energy of the Moon is not dissipated in tides, it's the rotational energy of the Earth that is. When the Moon slows down as it recedes, its kinetic energy just changes into its potential energy, it doesn't go anywhere else. The Moon receives energy here, it doesn't give it up.

[DavidS]

...the Almighty and Captain Kirk.

They're different guys?

[IXP]

Yes, you are right and Wikipedia is wrong. (At least in the article that you quoted; they've got it right in the very detailed article on tidal acceleration (http://en.wikipedia.org/wiki/Tidal_acceleration).)

Yes. What actually increases is the orbital energy, which is the sum of the kinetic and potential energy (and has the neat property of remaining constant throughout the orbit). As the kinetic energy of the Moon decreases as it recedes, its potential energy must increase even more so to make up for net energy increase.

I'm not sure if I understand that question. Kinetic energy of the Moon is not dissipated in tides, it's the rotational energy of the Earth that is. When the Moon slows down as it recedes, its kinetic energy just changes into its potential energy, it doesn't go anywhere else. The Moon receives energy here, it doesn't give it up.
Thanks again. Now it all makes sense.


IXP

[JoeTheJuggler]

How much is enough? Any change would have an effect, albeit very very small. And extracting tidal energy does change the shape, it retards the tidal flows.

I'm really bad at thinking of these things. My first thought was sort of like what Thabiguy said:
Friction within tidal bulges liberates energy. Whether you capture and utilize it or let it dissipate unharnessed is irrelevant. If you just capture energy that is being freed anyway, there won't be a difference.
I wanted to liken it to solar energy and photovoltaic panels. Whether or not we collect the solar energy into something usable has no effect on the sun.

But that's not right, because tidal energy isn't "from the moon" the same way solar energy is from the sun. So I think you're right, that it would always have an infinitesimal effect. I imagine the effect would be completely washed out by other things like ocean currents, axial precession, other non-extractive human activity on the coast, changes in river discharges, warming of some waters due to local weather patterns, sunspot activity, and of course that butterfly flapping its wings.