# How does jupiter's moon get so hot?

by ehab_s
Tags: jupiter, moon
P: 41
 Quote by D H That's just wrong. Look at it this way: The Galilean moons have been in existence for over 4.5 billion years. With regard to your calculations, where's the Q and k2 Love numbers of Jupiter and the Galilean moons?
that is what my math suggests will be the end of the system from the current date so they could have been much further from jupiter longer ago
for such a length of time they would have to be about 7 times further than they currently are assuming a continuing exponential increase in the rate at which they affect each other
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P: 12,870
 Quote by pixelpuffin I would expect the process to go exponentially faster as they get closer to Jupiter... ... [the Jovian sattelites] could have been much further from Jupiter longer ago...
You are imagining that the Jovian moons are getting closer to Jupiter over time then?

 Quote by ehab_s Correct me if I'm wrong but I thought that the total angular momentum of Jupiter's system would be lost if the total [mechanical] energy is lost.
I think you need to be careful about what energy, where it comes from, and where it goes.

How does energy leave a gravitationally bound system again?
How big is this effect?

 So why is it moving away from Jupiter, surely the planet couldn't be losing angular momentum faster than Io (After all, Jupiter is massive compared to it)?
The relative masses have nothing to do with the change in angular momentum.

The Jovian system is very complicated - but basically Io moves away for the same reason the Moon moves away from the Earth. The elliptical orbit is because Io gets a periodic kick from the other moons to keep it like that. Without that, the orbit would settle towards a circle with slowly increasing radius.

The fun part would if the closest approach was inside a certain limit...
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P: 3,085
 Quote by ehab_s Correct me if I'm wrong but I thought that the total angular momentum of Jupiter's system would be lost if the total energy is lost.
That is indeed wrong. Someone posted words to that effect earlier, but D H cleared that up-- energy and angular momentum are two quite different things, and are changed in a non-closed system in different ways. Energy is changed by heat loss or work done, angular momentum is changed by external torques. It turns out the latter is actually more difficult to accomplish, so in the Jupiter-plus-moons system, angular momentum is a better conserved quantity than energy. (To deflect the objections to that language raised above, one could say more precisely, "the Jupiter-plus-moons systems acts much more like a closed system in regard to its angular momentum than in regard to its energy." The "better conserved" language D H used is a standard way to say that in fewer words, though keep the full statement in the back of your mind.)
 What is really confusing to me is that if Io's orbit is eccentric, it must have to be losing its kinetic energy at a quicker rate than the other moons around jupiter. So why is it moving away from Jupiter, surely the planet couldn't be losing angular momentum faster than Io (After all, Jupiter is massive compared to it)?
This question shows you have some misconceptions that are causing you to pose it (which is what is so great about posing questions). An eccentric orbit is not an example of an orbit losing energy, it is an example of an orbit that has some energy sloshing periodically back and forth between kinetic energy (the energy of motion, seen in the speed of the planet) and gravitational potential energy (as calculated from the changing distance to the gravity source). Energy loss from the system (say, by radiated heat from the warmed planet) manifests differently, in a slowing down of the spin and a change in the orbit. As our Moon spirals away from Earth, the Moon itself is actually rising in energy, but the Earth's spin is losing even more kinetic energy than that, because the whole system has to be losing energy in the form of radiated heat. The total angular momentum of Earth spin, plus Moon spin, plus orbital motion of both, is holding fairly constant. What is mostly happening there is the Earth spin angular momentum is being converted, one for one, into Moon orbital angular momentum. Jupiter and Io is a more complicated subsystem, because of those two other Galilean moons that maintain resonances, and Callisto to boot. Hence I imagine that Jupiter is losing spin angular momentum, as all the Galilean moons increase in orbital angular momentum. That probably means most of Jupiter's lost spin angular momentum ends up in Ganymede's orbital angular momentum as the resonances move out, though it may have some effect on Callisto too, I don't know.
P: 5
 An eccentric orbit is not an example of an orbit losing energy, it is an example of an orbit that has some energy sloshing periodically back and forth between kinetic energy (the energy of motion, seen in the speed of the planet) and gravitational potential energy (as calculated from the changing distance to the gravity source)
I get that energy in this case is sloshing back and forth from KE to GPE, but then where does the heat in Io's core come from? I assumed that there is force acting on Io which caused a tidal effect on the moon's core, creating friction and heat. So it looks at first glance as if the energy comes from the constant pulling from the angular momentum of the moon. Where else would the energy come from?? Although I was wrong about the fact that an eccentric orbit loses energy faster, (the orbit converts the same amount of energy through the tidal effect whether it's eccentric or circular, only less spread out when eccentric) I still can't get my head around this angular momentum being conserved while energy being lost even when it looks as if this energy is coming from the angular momentum in the first place thing. It can't simply be shifted around in a 'closed' system if it is heating the planet and the moon by causing tides. There have to be countering forces in order to create friction in the moon's core, the only thing I can think of is angular momentum and gravity. Either there is something else I'm missing or I fail to understand what 'angular momentum' actually is..
P: 23,568
 Quote by ehab_s I still can't get my head around this angular momentum being conserved while energy being lost
Momentum and energy are separate things, conserved separately - period.

Do you know what the ballistic pendulum is? It is a perfect example of how to use the fact that these things are conserved separately to determine the velocity of the bullet, regardless of the fact kinetic energy is converted entirely to heat.
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 Quote by ehab_s I get that energy in this case is sloshing back and forth from KE to GPE, but then where does the heat in Io's core come from?
It comes from inelastic changes in the shape of Io, where parts of Io essentially "rub together" as they reconform to the new shape.
 I assumed that there is force acting on Io which caused a tidal effect on the moon's core, creating friction and heat.
Tidal stresses by themselves don't heat anything, they just give the moon a characteristic (tiny) football shape. It is changes in that shape that cause heating-- constant pulling doesn't heat.
 So it looks at first glance as if the energy comes from the constant pulling from the angular momentum of the moon. Where else would the energy come from?? Although I was wrong about the fact that an eccentric orbit loses energy faster, (the orbit converts the same amount of energy through the tidal effect whether it's eccentric or circular, only less spread out when eccentric) I still can't get my head around this angular momentum being conserved while energy being lost even when it looks as if this energy is coming from the angular momentum in the first place thing.
Actually, you were right that an eccentric orbit creates heat and a circular one does not, because the eccentric orbit changes the tidal stresses so changes the shape of the moon. My point is only that it is not the steady component of the force that does this, all that does is cause energy to slosh back and forth between KE and GPE as the orbital distance varies. It is a higher order effect that does the heating, the change in the tidal stresses that cause a change in the shape. An analogy often used is "kneading dough".
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P: 15,164
 Quote by ehab_s I get that energy in this case is sloshing back and forth from KE to GPE, but then where does the heat in Io's core come from? I assumed that there is force acting on Io which caused a tidal effect on the moon's core, creating friction and heat. So it looks at first glance as if the energy comes from the constant pulling from the angular momentum of the moon. Where else would the energy come from??
From gravity. Jupiter's gravity is (approximately) a 1/r2 force. This variation, or gradient, in the gravitational force is what causes the ocean tides on Earth and the heating on Io. Jupiter's gravity pulls on a rock on the near side of Io more than it pulls on a similar rock at Io's center of mass, and it pulls on that rock at Io's center of mass more than it pulls on a rock on the far side of Io. Jupiter "wants" to pull Io apart. Io fights this pulling apart with its own gravity, but not completely. Io is stretched from the ends / squeezed at the middle into a slightly out-of-round shape.

While gravitation is a 1/r2 force, the tidal forces that result from the gradient in gravitational force is a 1/r3 force. The amount by which Io is stretched and squeezed out of round varies because Io's orbit is not circular. Io flexes. Think of what happens when you repeatedly bend and unbend a wire coat hanger. It heats up and eventually gets hot enough to cause the hanger to break. That's what's happening to Io. The flexing between a non-spherical and not quite so non-spherical shape as Io moves from perijove to apojove and back again causes Io to heat up a bit.

So far this sounds like a way to create an over-unity machine. It isn't. Suppose Jupiter didn't have any other moons. What would happen, at least immediately, is that this stretching and squeezing would act to circularize Io's orbit at the perijove distance. Angular momentum is (nearly) conserved, but energy can be dissipated throughout the universe.

One way to restate the second law of thermodynamics is that dissipative systems tend toward a minimum energy configuration. The minimum energy configuration for an orbiting body with constant angular momentum is a circular orbit. Jupiter would eventually circularize Io's orbit, and then the stretching and squeezing would stop. Io would be frozen into a fixed slightly out-of-round shape as it orbits Jupiter and rotates at the same rate as the orbital rate.

There's another, longer term effect. Jupiter rotates rather quickly, much faster than Io's orbital rate. Just as Jupiter causes distortions in Io's shape, Io causes distortions in Jupiter's shape. This slight distortion in Jupiter's shape would be frozen in if Jupiter rotated at the same rate as Io's orbital rate. Since Jupiter rotates much faster than Io orbits Jupiter, Jupiter's rotation pulls the slight bulge caused by Io ahead of Io due to frictional forces. This leading bulge gives a little gravitational kick to Io. Io moves out. At the same time, Jupiter's rotation rate slows down just a bit. The total angular momentum of the Jupiter + Io system is conserved, but energy is not. It decreases. Those frictional forces are dissipative. Friction converts kinetic energy into thermal energy, and that thermal energy eventually radiates away into the universe.

So what would happen to Io if it was an isolated moon in a slightly eccentric orbit? Initially its orbital energy would decrease so that the eccentricity goes to zero. Then it would very slowly climb to an ever higher orbit as Jupiter's rotation rate slowly slows down.

Io isn't an isolated moon. The next two innermost moons are in a 1:2:4 orbital resonance with Io. This makes the picture a lot more complicated, but the basic principles still apply. Those moons are gradually moving away from Jupiter as they steal a tiny bit of oJupiter's rotational angular momentum.
P: 5
 Quote by Ken G It comes from inelastic changes in the shape of Io, where parts of Io essentially "rub together" as they reconform to the new shape.Tidal stresses by themselves don't heat anything, they just give the moon a characteristic (tiny) football shape. It is changes in that shape that cause heating-- constant pulling doesn't heat. Actually, you were right that an eccentric orbit creates heat and a circular one does not, because the eccentric orbit changes the tidal stresses so changes the shape of the moon. My point is only that it is not the steady component of the force that does this, all that does is cause energy to slosh back and forth between KE and GPE as the orbital distance varies. It is a higher order effect that does the heating, the change in the tidal stresses that cause a change in the shape. An analogy often used is "kneading dough".
I know really, I just didn't mention that clearly
P: 5
 Quote by D H From gravity. Jupiter's gravity is (approximately) a 1/r2 force. This variation, or gradient, in the gravitational force is what causes the ocean tides on Earth and the heating on Io. Jupiter's gravity pulls on a rock on the near side of Io more than it pulls on a similar rock at Io's center of mass, and it pulls on that rock at Io's center of mass more than it pulls on a rock on the far side of Io. Jupiter "wants" to pull Io apart. Io fights this pulling apart with its own gravity, but not completely. Io is stretched from the ends / squeezed at the middle into a slightly out-of-round shape. While gravitation is a 1/r2 force, the tidal forces that result from the gradient in gravitational force is a 1/r3 force. The amount by which Io is stretched and squeezed out of round varies because Io's orbit is not circular. Io flexes. Think of what happens when you repeatedly bend and unbend a wire coat hanger. It heats up and eventually gets hot enough to cause the hanger to break. That's what's happening to Io. The flexing between a non-spherical and not quite so non-spherical shape as Io moves from perijove to apojove and back again causes Io to heat up a bit. So far this sounds like a way to create an over-unity machine. It isn't. Suppose Jupiter didn't have any other moons. What would happen, at least immediately, is that this stretching and squeezing would act to circularize Io's orbit at the perijove distance. Angular momentum is (nearly) conserved, but energy can be dissipated throughout the universe. One way to restate the second law of thermodynamics is that dissipative systems tend toward a minimum energy configuration. The minimum energy configuration for an orbiting body with constant angular momentum is a circular orbit. Jupiter would eventually circularize Io's orbit, and then the stretching and squeezing would stop. Io would be frozen into a fixed slightly out-of-round shape as it orbits Jupiter and rotates at the same rate as the orbital rate. There's another, longer term effect. Jupiter rotates rather quickly, much faster than Io's orbital rate. Just as Jupiter causes distortions in Io's shape, Io causes distortions in Jupiter's shape. This slight distortion in Jupiter's shape would be frozen in if Jupiter rotated at the same rate as Io's orbital rate. Since Jupiter rotates much faster than Io orbits Jupiter, Jupiter's rotation pulls the slight bulge caused by Io ahead of Io due to frictional forces. This leading bulge gives a little gravitational kick to Io. Io moves out. At the same time, Jupiter's rotation rate slows down just a bit. The total angular momentum of the Jupiter + Io system is conserved, but energy is not. It decreases. Those frictional forces are dissipative. Friction converts kinetic energy into thermal energy, and that thermal energy eventually radiates away into the universe. So what would happen to Io if it was an isolated moon in a slightly eccentric orbit? Initially its orbital energy would decrease so that the eccentricity goes to zero. Then it would very slowly climb to an ever higher orbit as Jupiter's rotation rate slowly slows down. Io isn't an isolated moon. The next two innermost moons are in a 1:2:4 orbital resonance with Io. This makes the picture a lot more complicated, but the basic principles still apply. Those moons are gradually moving away from Jupiter as they steal a tiny bit of oJupiter's rotational angular momentum.

Phew, great explanation thanks!

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