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Orbital Resonance
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0:00
(upbeat music)
0:06
- This is a fact that blew my mind when I first heard it, and that's not an expression I use very often. Actually, it's something that Derek from Veritasium touched on very briefly in his most recent video. So maybe you've had your mind blown by this too. Consider how long it takes for planets in our solar system to go around the Sun. In other words, what their orbital period is. The Earth, for example, has an orbital period of one year. It takes one year to go around the Sun.
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Here are the other orbital periods of the planets in our solar system expressed in Earth years. And you'll be hard pressed to find a pattern in those numbers. But now switch from planets going around stars to moons going around planets. Specifically consider the three largest moons of Jupiter. Io takes this many days to go around Jupiter. Let's call that one Io month. This is how many days it takes Europa to go around Jupiter, or exactly two Io months. And this is how many days it takes Ganymede
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to go around Jupiter. In other words, exactly four Io months. So Europa takes exactly twice as long as Io to go around Jupiter. And Ganymede takes exactly twice as long as that. How can that be? You'll know from Derek's video that this is an example of synchronization but what's the mechanism behind it? That's what this video is about. It's an explanation of something called orbital resonance. You know, I'm a big fan of resonance, I made a whole video explaining what resonance is.
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And it seemed to me that the word resonance, as it's used here in this context of orbital resonance, is different. To add even more intrigue think about the rings of Saturn. You might know that there are some gaps in the rings where there aren't many rocks. There's this big one here. You might also know that as you get further away from a planet the longer it takes to orbit at that distance. And it turns out that if you travel away from that gap in Saturn's rings, further away from Saturn,
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until you reach a distance where the orbital period is twice as long as the orbital period of a rock in that gap, you find something. You find the moon, Mimas. So in some situations orbital resonance will cause celestial bodies to be locked in position. But in other situations orbital resonance will cause celestial bodies to be removed from their position. I always thought that the gap in Saturn's rings was the easier example of orbital resonance to explain
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but it turns out it's one of those situations where the simple description is easy to understand and it makes intuitive sense but when you look at it closer it stops making sense again. Think about two objects orbiting around a planet. They're at different distances from the planet so their orbital periods are different which also means that their distance from each other is always changing. They drift apart, they come together, they drift apart. Sometimes they're at their furthest distance
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from each other, sometimes they're at their closest distance to each other. When they're at their closest point, I'm gonna call that a meeting. Two objects meet when they are at their closest approach. Let's apply that to the moon Mimas and a rock in Saturn's rings. We're choosing a rock that is just at the right distance so that the orbital period of that rock is twice the orbital period of Mimas. So let's start when Mimas and the rock are at their closest
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approach and then play the animation forward. When will they next meet? Well they'll meet when Mimas has completed a full orbit. The rock will have completed two orbits. And you'll notice that they meet at the same point in space and that will keep happening again and again. So there's your resonance. You're getting the timing just right so the two objects meet at the same time and the same point in space every single time. And every time they meet the rock
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gets a little gravitational kick from Mimas until the rock is expelled completely. That happens to all the rocks at that distance and you end up with a gap in Saturn's rings. It makes intuitive sense, but it's wrong. Think about it like this. If the explanation rests on the fact that the rock gets a regular gravitational kick from Mimas every time they meet then Mimas doesn't need to have an orbital period that's exactly half the orbital period of the rock.
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They don't need to be in any kind of whole number ratio for that matter. Look, here are two objects that aren't in orbital resonance. They still meet regularly with a fixed interval between meetings. It's just that the meeting place moves around with each orbit. And so what? The rock is still getting a regular gravitational kick from Mimas so who cares where the meeting place happens? The system has circular symmetry anyway. So by the regular gravitational kick explanation
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all the rocks in Saturn's rings should now be expelled by the gravitational kicks of Saturn's moons, but they're not. So something else is going on. Honestly, I got really stuck at this point. Like, I couldn't understand the papers that I was reading about this stuff. I didn't have enough of a baseline knowledge. So I decided to speak to an expert. - If you've got an elliptical orbit you've got a different speed at different points in the ellipse. - Yeah. - Right? - Yeah.
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- So where it's sort of narrower it speeds up and where it sort of, I guess you could call it shallower, it slows down slightly and then it will speed up again. So it's like the swing analogy, right? You're gonna push the kid sort of at its fastest point, right, and push it so that it keeps going. - That was Dr. Becky Smethurst. We'll hear more from her in a minute. But that's the key, isn't it? Elliptical orbits. Orbital resonance is found with elliptical orbits only.
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That answers the question that I had. Like in my mind I was thinking all of the points around an orbit are the same so who cares if two orbiting bodies meet at the same point in an orbit or at different points around an orbit because they're all the same. Well they're only all the same if the orbits are circular. If the orbits are elliptical then it's a different story. Like a moon orbiting round a planet will experience very different gravitational conditions at this point
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in an elliptical orbit compared to this point in an elliptical orbit. You know, it's one of those situations where you just need the right search terms and armed with elliptical orbits I was able to find a paper that explained it really well. Consider two moons going around a planet. The outer one is elliptical. Let's position the two moons so that the outer moon has an orbital period that is double that of the inner moon, it takes twice as long to go around the host planet
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than the inner one does. Let's also set it up so that the two moons meet when the outer moon is at its furthest distance from the host planet. In other words here. We're also assuming that the inner moon is much larger than the outer moon so the effect of the outer moon on the inner moon is negligible. Right. The inner moon is always pulling on the outer moon a little bit but the angle of that pull changes as the orbits progress. Consider the moment just before the two moons meet.
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Well there's a component of that pulling force that's actually in the opposite direction to the direction of travel of the outer moon. In other words, before meeting the inner moon is actually holding the outer moon back a little bit. It's decreasing the angular momentum of the outer moon. Contrast that to what happens after the moons have met. The inner moon is now slightly ahead of the outer moon and there's now a component of that pulling force
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in the same direction as the motion of the outer moon. In other words, it's pulling the moon forward, it's giving it a little extra angular momentum. So the inner moon reduces the angular momentum of the outer moon prior to meeting and it increases the angular momentum of the outer moon after meeting. And because these orbits are symmetrical around this line we should expect those two effects to cancel out. The increase in angular momentum post-meeting should exactly cancel out the decrease
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in angular momentum pre-meeting. So we shouldn't expect this situation to change over time. Now consider the same two moons but the meeting point is further around on the elliptical orbit. In other words, here. You'll notice that the period leading up to the meeting of the two moons isn't symmetrical with the moments after the meeting of the moons. Consider for example one day before the moons meet and one day after the moon's meet. Just by inspecting the geometry we can see
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that the component of the force in the direction of travel of the outer moon is larger after the two moons have met than before the two moons have met. The consequence of that is that the increase of angular momentum after meeting isn't canceled out by the decrease in angular momentum from before the meeting. In other words, when the meeting point is slightly further around in the orbit the outer moon will gain some angular momentum each time. Angular momentum is just regular momentum
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multiplied by the radius of the orbit. In simple terms r times m times v. So you might think that when you increase the angular momentum of a moon you're increasing the velocity but it's actually the orbital radius that increases. In fact, the velocity goes down a little bit just not as much as the radius goes up. So the overall effect is an increase in angular momentum. All of that can be derived from the equations of gravity but it's probably something you intuitively
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knew anyway like from watching animations of the solar system you know that the outer planets travel more slowly than the inner planets. Right. So in our scenario the outer moon gains a bit of angular momentum from each meeting, which means the orbital radius goes up and the speed goes down. Because the speed goes down that moon won't have traveled as far by the time the next meeting comes around. So the next meeting will happen earlier in the orbit.
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In other words, a little closer to this symmetry line. And that will keep happening until the meeting point is exactly on that symmetry line. And of course the opposite is true when the meeting point occurs before the symmetry line. In that scenario the inner moon takes angular momentum from the outer moon. That causes the orbital radius to decrease and the speed to go up. With increased speed the outer moon will have gone further by the time the next meeting happens.
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In other words, the next meeting will be closer to that symmetry line once again. So we seem to have this restoring mechanism. Any deviation of the meeting point away from the symmetry line will be brought back to the symmetry line through that mechanism. It's a stable equilibrium. This restoring mechanism works for our two moons because the outer moon has an orbital period that's exactly double the orbital period of the inner moon. We chose it to be that way.
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If it wasn't like that then the meeting point would be all over the place and the mechanism would be lost. So the question is how do two moons end up with this two to one ratio in the first place and how does this restoring mechanism stop them from losing it? Well it turns out that moons tend to slowly drift away from their parent planets and it's because of tidal forces. You might know that the Moon causes the Earth to have two tidal bulges. One towards the Moon and one away from the Moon
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which I have shown here massively exaggerated. That second tidal bulge away from the Moon is a little counter-intuitive but I'm not gonna get into that in this video. What you need to know for our purposes is that because the earth is rotating underneath the tidal bulge it actually drags the tidal bulges around a little bit so that the tidal bulge that's near the Moon is actually slightly in front of the Moon and the tidal bulge that's further from the Moon is slightly behind the Moon.
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The tidal bulge that's close to the Moon and in front of it is pulling the Moon around, it's giving the Moon extra angular momentum. The tidal bulge on the far side, that's behind the Moon, is doing the opposite but because it's further away the effect is less strong. The net effect is that this tilted tidal bulge gives the Moon an extra bit of angular momentum. And as we saw earlier, when you give an orbiting body extra angular momentum its orbital radius increases.
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In other words, moons tend to drift away from their planets. So suppose you have a planet, it's got two moons let's say and they've just got arbitrary orbital periods, they're not in a two, one ratio with each other. Maybe they're in a 1.8 to one ratio with each other, let's say. But they're experiencing tidal forces. Let's assume the inner moon isn't effected much by tidal forces, maybe because it's really large, but the outer moon is affected by tidal forces
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and it's slowly drifting away from its host planet. Of course if its orbital radius is increasing then its orbital period is increasing as well. And as its orbital period increases at some point it will have an orbital period that is exactly twice the orbital period of the inner moon. And so long as that outer moon has an elliptical orbit we know that the restoring mechanism we described earlier will bring the meeting point of those two moons onto the symmetry line of that elliptical orbit.
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But those tidal forces are still there so surely the outer moon will continue to drift away. Its orbital period will continue to increase until it's no longer in a two to one ratio. But remember this moon is drifting out because it's gaining angular momentum. And as it gains angular momentum, remember, the orbital radius increases but the speed goes down. And as we saw before a reduced speed means a shift in the meeting point. Except that we know that the restoring mechanism
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we described earlier prevents that from happening. So, so long as the restoring mechanism is stronger than the effect of tidal forces the outer moon, once it's in that position, will stay there.
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There you go, orbital resonance. It's the same mechanism that knocks rocks out of Saturn's rings. Because Mimas is shifting those rocks around in their orbit to line them up with that symmetry line they're moving through this crowded neighborhood, they're colliding with other rocks and it's those collisions that are knocking them out and leaving that area empty. Those aren't the only examples of resonance in our solar system. Neptune and Pluto are in a three, two orbital resonance
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with each other. Because their orbital paths cross if they weren't in this fixed ratio with each other they would eventually collide. Just like the gap in Saturn's rings there are gaps in the asteroid belt and they are orbital resonances with Jupiter. All we've described so far is mean motion resonance. There are other types of resonance, though the one we've described is the main one. - The solar system was long thought to be in resonance with, all the planets were in a resonance.
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And people like hunted for it for years. It almost became like a, "If you can prove the resonance you can prove Heliocentrism." But we obviously- - It's one a classic story in science, isn't it? Where you want the beautiful answer to be the truth and it turns out annoyingly it's not. Becky has her own YouTube channel, it's really good, I'll leave a link in the description. So no interplanetary orbital resonance in our solar system but we have discovered them in other star systems.
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Quite a few of them. Some of them really long strings of resonances. For example TOI-178 star system has five planets all in orbital resonance with each other with a ratio of two to four to six to nine to 12. What? In this video I described the restoring mechanism for a two to one ratio. Mechanisms for other ratios are more complex and I won't go into them for this video. But look, here's the TOI-178 system as an animation. I've actually animated it so all the planets meet up
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in a single line periodically. That might not be the case, actually it probably isn't the case. Like you can have all the planets in pairs being in resonance with each other without them all having to meet at the same time. That's true for the three largest moons of Jupiter, for example. They don't all meet in a line but they do pair up individually. But let's assume they all meet in a line just 'cause it's easier for me to animate and also because I wanna point out something quite interesting.
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Like it's hard visually to see the structure of this resonance just watching the animation. Like occasionally when they all meet up in one go you can see that there are these patterns but it's not so easy to see the individual pairs of resonances. But it turns out you can tap into a different part of your brain. Your auditory system is really good at seeing patterns in time. So like it's very easy to recognize a beat, it's easy to recognize relationships between beats.
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So we can do something really interesting. We can take this orbital resonance data and turn it into sound. It's called the sonification of data. Scientists are doing it more and more because our auditory system has these skills that our visual system doesn't. In this case let's give each planet its own sound and then play that sound every time the planet sweeps through this line. (drum beats)
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And then if we speed that up you can start to hear the structure. (drum beats)
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You might recognize two times UK Beatbox Champion Beardy Man there. Just to close the loop on that one you may remember I put a video up a while back of me doing standup comedy about a friend of mine whose dad was wrong about maths. Well the dad who was wrong about maths is Beardy Man's dad, also Jay Foreman's dad, because they're brothers. Our auditory system doesn't just find patterns in beats it finds patterns in tones as well. In fact, if you take a beat and speed it up
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beyond around 20 Hertz it becomes an audible tone. So what if we took these moon and planetary systems and sped them up until they were tones we could hear? Like if we speed time up until Ganymede is going around Jupiter 261.63 times per second? Well that frequency corresponds to C below middle C. And because Europa has an orbital period that is half that of Ganymede's it has double the frequency. And you might know that when you double the frequency of a tone you're moving up one whole octave
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which means that if Ganymede is C below middle C then Europa is middle C and then Io is C above middle C. And that would sound like this. (piano playing) Neptune and Pluto have frequencies that are in the ratio of two to three and in music theory the distance between two notes that are in that ratio is called the perfect fifth and an example of that is D to A. (piano playing)
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One of my favorite star systems, TRAPPIST-1, would sound like this. (piano playing)
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TOI-178, that we showed earlier with beats, would sound like this. (piano playing)
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