A perpetual motion machine in space

[brianvds]

Here's one for our resident physics nerds. Imagine this scenario:

You are living on a planet that is part of a double planet system, somewhere in intergalactic space. You would have frozen but for the fact that the planets' orbits around one another are elliptical, and hence they get "tidally squeezed" in the same way as happens to Jupiter's Galilean moons. And as with Jupiter's moons, this makes them volcanic, so you power everything by geothermal power. A neat little perpetual motion machine that will keep on turning out power forever.

Now if I understand the laws of thermodynamics correctly, the above scenario is impossible. No system can keep on producing energy forever; the system must be "running down" in some way. Something must be happening to the two planets.

The question is what? And is the same thing happening to Jupiter's Galilean moons?

[Mefiante]

The tidal deformations of orbiting bodies occur by virtue of their motion where the differential in gravitational potential across a body varies with position along the orbit.  In a perfect circular orbit, tidal squeeze is at a minimum because there is no variation in the differential in gravitational potential.  The more eccentric an elliptical orbit becomes, the greater the variation along the orbit.  At perigee, the differential gravitational potential is at a maximum, and at apogee at a minimum.  The tidal energy generation effect is enhanced if the orbiting bodies also rotate about their own axes when these are more or less perpendicular to the plane containing their orbit.

The cyclical squeezing and relaxing robs the system of angular momentum (rotational energy) which is converted to heat.  In doing so, the rotations are gradually slowed down and the orbit shrinks.  (As you may know, Earth’s days are slowly getting longer, the result of the planet’s spin slowing down.)  The gist of it is that a combination of gravitational potential energy and angular momentum decays in a systematic way to feed the “geothermal” energy.  Taken to its limit, the orbiting bodies will not continue indefinitely, but will eventually crash into each other.  (Note that on Earth, geothermal energy is primarily the result of nuclear decay processes deep in the planet’s core and mantle.)

A full treatment of the dynamics in play in a binary system requires General Relativity.

'Luthon64

[cr1t]

Mefiante is very correct.

I once had the idea that if you just spin up a magnet in zero G with no
friction and with a coil around it you should have free energy for life.
But turns out the magnetic waves will stop the magnet eventually. No free lunch.

[Mefiante]

The cyclical squeezing and relaxing robs the system of angular momentum (rotational energy) which is converted to heat.

I really should have added here that the underlying mechanism is that deforming such an orbiting body changes the mass distribution of the system, and this in turn manifests as changes in the bodies’ inertia (resistance to acceleration).  If the system comprised only perfectly rigid bodies, no deformation could occur and no inertial-energy-to-heat processes could occur inside the bodies.  Such a system would also eventually decay after enormous spans of time, owing to the tiny drag of spacetime itself on accelerating bodies.

'Luthon64

[brianvds]

Now can we then conclude from the above that Jupiter's Galilean moons will eventually crash into the planet? How long will it take? Can we observe that their orbits are shrinking, or is it happening so slowly that it cannot at present be measured with existing equipment?

What I also wonder is how exactly the planets are slowed in their orbits. I notice Mefiante says one needs to understand some pretty advanced physics to get it (and I never could wrap my puny mind around even very simplified popular accounts of relativity). But is there no somewhat more simple explanation? E.g. I often see the fact that the earth's moon is receding explained as the result of the earth's tidal bulge, in combination with earth's spin on its axis, exerting a slight pull on the moon's tidal bulge, which slightly accelerates the moon and hence makes it recede. Is something similar perhaps at work in my hypothetical double planet system (or, for that matter, the Galilean moons), except that here the planets are slowed down in their orbits?

It occurs to me that in a perfectly circular orbit, the tidal bulges will point to one another all the time. But in an elliptical orbit, the effect of libration will mean that that tidal bulges will not point exactly at one another through the whole orbit, and so may exert drag on one another, but I now find it a bit tricky to visualize, or to work out whether this drag will accelerate or decelerate the planets.

Another thing that occurs to me is that in a perfectly circular orbit, there will no longer be any tidal squeezing effect. So is that not perhaps what happens? I.e. instead of the two planets spiraling into one another, their orbits are circularized until there is no more tidal squeezing and thus no more energy can be obtained out of the system?

What a fascinating place the universe is, when even such a seemingly simple setup has so many subtleties to take note of. :-)

[brianvds]

Incidentally, it is partly through the questions of the pseudopaths and others on LitNet that I began to get interested in such things as orbital mechanics, but my interest will have to remain mostly non-mathematical because my knowledge of physics is almost non-existent. What's more, I cannot make head or tails of a mathematical description if I do not first understand it in an intuitive sort of way. (Despite the fact that I think I more or less get the tidal effect now, I still cannot make head or tails of the explanation of it on Wikipedia). Nevertheless, I thank our crazy correspondent Kobus de Klerk on LitNet for getting me to improve my own education, even as he himself sinks further into denial of reality. :-)

[Mefiante]

Now can we then conclude from the above that Jupiter's Galilean moons will eventually crash into the planet?

It’s certainly possible; it’s also possible that some other subtle effect(s) is/are playing out that will give a different outcome.  A large part of the problem is that as soon as you have a gravitating system that consists of three or more real bodies, the system is inherently chaotic in the mathematical sense where the tiniest perturbation can produce vastly different results.  The short of it is that I don’t know what the situation is with the Jovian moons.

What I also wonder is how exactly the planets are slowed in their orbits.

As I tried to explain in my second post, the deformation of the bodies changes the mass distribution of the system over time.  One way to think about it is that changes in mass distribution affect the inertial relationships between the bodies by introducing small variations in the forces of gravitational attraction between them.  Thus, those forces of attraction become briefly and slightly unbalanced and so allow the bodies to move slightly apart or closer together, depending on the nature of the imbalance.  Remember that real matter does not react instantaneously to applied forces or stresses; almost invariably, one finds hysteresis, plasticity, elastic and viscosity effects (without which the internal heat generation wouldn’t work, BTW).  It is these material properties and effects which cause those small gravitational and inertial imbalances to occur in the first place, as illustrated with the hypothetical case involving perfectly rigid bodies.  A reverse situation can be envisaged with a child on a swing who imparts internal energy to the swing with carefully synchronised rhythmic accelerations: the child’s internal energy becomes simple harmonic pendulum motion of increasing amplitude.

Another thing that occurs to me is that in a perfectly circular orbit, there will no longer be any tidal squeezing effect.

This is not quite correct.  It will be true in a few special cases, namely (1) the orbiting bodies are point masses; and/or (2) they are perfectly rigid; and/or (3) the bodies rotate around axes perfectly perpendicular to the plane of the orbit and at the same rate as their orbit (that is, the bodies perpetually show one another exactly the same face).  Any changes in aspect from one body to another over the period of the orbit means that there will a differential in gravitational potential which changes its relative location in the body.  More simply, the back of the body (where the gravitational attraction is at a minimum) eventually moves to the front (where it is at a maximum) and then recedes again to the back.  But it is precisely this relative movement of the differential in gravitational potential that produces heat.

What a fascinating place the universe is, when even such a seemingly simple setup has so many subtleties to take note of.

Agreed.  Unreservedly so.

'Luthon64

[BoogieMonster]

I once had the idea that if you just spin up a magnet in zero G with no
friction and with a coil around it you should have free energy for life.
But turns out the magnetic waves will stop the magnet eventually. No free lunch.

Heh, me dad illustrated this to me when I was a wee tyke by doing the following:

1. Take an electric motor (in our case liberated from a tape player).
2. Make sure it's electrical connections are open-circuit and turn it by hand. There is a little resistence, but not a lot.
3. Short the ends, turn it by hand again.

Result: Motor is suddenly much more difficult to turn. There's no such thing as a free lunch.

This was fascinating to me as a kid: to feel electricity and magnetism interact like that to produce a physical force.

[cr1t]

Now can we then conclude from the above that Jupiter's Galilean moons will eventually crash into the planet? How long will it take? Can we observe that their orbits are shrinking, or is it happening so slowly that it cannot at present be measured with existing equipment?

I googled a bit and found this on Wiki

Io orbits Jupiter at a distance of 421,700 km (262,000 mi) from the planet's center and 350,000 km (217,000 mi) from its cloudtops. It is the innermost of the Galilean satellites of Jupiter, its orbit lying between those of Thebe and Europa. Including Jupiter's inner satellites, Io is the fifth moon out from Jupiter. It takes 42.5 hours to complete one orbit (fast enough for its motion to be observed over a single night of observation). Io is in a 2:1 mean-motion orbital resonance with Europa and a 4:1 mean-motion orbital resonance with Ganymede, completing two orbits of Jupiter for every one orbit completed by Europa, and four orbits for every one completed by Ganymede. This resonance helps maintain Io's orbital eccentricity (0.0041), which in turn provides the primary heating source for its geologic activity (see the "Tidal heating" section for a more detailed explanation of the process).[35] Without this forced eccentricity, Io's orbit would circularize through tidal dissipation, leading to a geologically less active world.
Like the other Galilean satellites of Jupiter and the Earth's Moon, Io rotates synchronously with its orbital period, keeping one face nearly pointed toward Jupiter. This synchronicity provides the definition for Io's longitude system. Io's prime meridian intersects the north and south poles, and the equator at the sub-Jovian point. The side of Io that always faces Jupiter is known as the subjovian hemisphere, while the side that always faces away is known as the antijovian hemisphere. The side of Io that always faces in the direction that the moon travels in its orbit is known as the leading hemisphere, while the side that always faces in the opposite direction is known as the trailing hemispher

[brianvds]

Without this forced eccentricity, Io's orbit would circularize through tidal dissipation, leading to a geologically less active world.

Heh, thanks for that. Looks like just for once (and trust me, this is a very rare event), my physical intuition was right. :-)

Of course, now I'll have to go think about how exactly tidal forces circularize an orbit...