Interaction of Moons, Asteroids, & Comets with their Surroundings #1 | Margaret Kivelson

Okay, first place, can you hear me in the back? Okay, good. And Fran, I didn't really expect to be introduced. Thank you for that very lovely introduction. So I need not tell you what I'm talking about, but the important thing from my perspective is that all of these bodies that I'm going to be talking about are embedded in plasma seas. Maybe it would be better to say in plasma rivers or brooks, because these plasmas are also flowing in with respect to the bodies. The plasmas, as you have discovered in the last ten days, are extremely complex, highly nonlinear, and one of the important features that I—I feel a little embarrassed because I haven't been present for all of the talks, so some of this may be quite repetitive of what you've heard before—but to me, the importance of long-range forces is that they are what keep the plasmas charge neutral to a very large degree. On any scale that contains many particles, Coulomb forces fall off like 1 over R squared. The number of particles that are at any distance R tends to go up like a R squared, so no matter how far you go away, there will be that many particles interacting, and that means these forces are really very long range, and that's what keeps the plasma on average charge neutral. The charged particles, of course, can carry electrical current. The plasmas are threaded by magnetic fields, and they not only respond to the magnetic fields but they develop currents that create magnetic fields themselves. So we have perturbation magnetic fields, and a lot of what I'm talking about has to do with the perturbation magnetic fields.

So the form of the interaction depends heavily on the plasma properties: the density, the flow velocity, the magnetic fields intensity, but it also depends on the properties of the body. And the properties of the body include its size—and I will soon talk about what I mean by it depends on its size—its conductivity, the presence of an atmosphere or other source of neutrals. So there's an interplay between the two elements of this interaction. So let's start by a refresher course on the physical principles that govern the behavior of the plasma. And I know that we started with Tomasombo C's elegant presentation of the basic laws, but I'm going to strip them down to the parts that really are critical for the systems I'm looking at.

So in a neutral gas, of course, the dynamics are largely controlled by collisions. In space plasmas, collisions are infrequent. How many of you know what's the collision scale for a collision in the solar wind near Earth, or the frequent—that how—how what is the time between collisions? Well, it's of the order of days—is the time scale—and the length scales of the order of astronomical units. So collisions are really infrequent in the plasmas we talk about, and so it's the electromagnetic interactions that control the system: the interaction between the charges in the magnetic field and the fact that the electric fields developed to keep them charged neutral. So as I've said several times already, charge neutrality, but that doesn't rule out currents. We can maintain charge neutrality while having the electrons, which are very mobile, move through the ions. Typically, in most space plasmas, it is the electrons that carry the current. And Benjamin Franklin got the sign wrong, but it—there are actually places in the solar system where negative ions play an important role, and so that—that the statement that it—the electrons are the current carriers—not always true. The spatial and temporal scales that dominate the interaction are large. And now comes an important statement: what this large mean? A good physicist is not allowed to use a term like large or small without saying compared with what. Large means absolutely nothing. We can have scales of astronomical units that are not large because it depends on what you're talking about. So I've asked the question, I answered, and so the scales are long in both space and time. Particle properties on scales that are large compared with reference time scales can be described by MHD fluid equations. Can anybody suggest a time scale that is not large in a space plasma? What—what would you be comparing the—the time scales or the spatial scales of MHD? What would the spatial or temporal scale you're comparing with be? Anybody have a suggestion? Gyro radius—that's a very good one. Of course, with that goes gyro period. So it's large compared with the largest gyro radius and gyro period. There are other numbers you can compare with, but though that's probably the most important. We have to be looking at dimensions that contain many ion gyro radii.

So then we start with the most simple of the equations, which is the conservation of mass. This is, of course, mass density, but that's the same thing. And the conservation of momentum, and here we have a form in which I've—what—excuse me, I would push the wrong button. I assume is a pointer—no, I'm gonna get my—let's try my pointer and see if it works. Okay, I've written this in the form that—in which it is the momentum that comes into the time derivative, the momentum density, and it involves the usual pressure gradient force, grad—minus grad P—that's thermal pressure—but now we have added in an electromagnetic force term, J cross B. I think I didn't—I think I didn't go back far enough. Let me just go back because I really wanted to show this first without sources and losses and just add the J cross B and then show you that in the—in the near vicinity of the bodies we're considering, there may be sources and losses of both number density, mass density, and momentum. So S and L are source and loss. I noticed we used P for production in one of the talks, but that's what I mean. And S_P and L_P are just sources and losses of momentum. And I think I later on will tell you a little bit about how that can happen, but to me the most interesting thing about this equation is that it's been around for a century or more before anybody thought of putting that J cross B in, because really people weren't terribly interested in ionized systems. And it was Alfvén who came along and said, "You know, if you have an ionized gas, we ought to put in the—the electromagnetic forces." And then, of course, we have a relationship between flows and density and the J and B that come in here, so we'd better put in also Maxwell's equations, and he put these two together and made MHD. And nobody really believed him.

So the next thing is we've—I've seen a lot of appearance of the statement—yes—I'm using the—the symbol U for the flow velocity to distinguish it from V, which is the velocity of an individual particle. So we've seen a lot of E as equal to minus U cross B, but I don't know if anybody has asked you to think about why we have that equation. It's because plasma electrons are extremely mobile, so in the rest frame of the plasma they short out any magnetic—any electric field, and so we can say that in the rest frame of the plasma the electric field is zero. But there are transformation laws if you go from one frame to another, and in a frame that's flowing at velocity U, the—the electric field is E in the rest frame minus U cross B, but since E in the rest frame is zero, E is minus U cross B, and that's basically where this idea comes from—that we can represent the—the effect of the flow of the plasma is an electric field that's equal to minus U cross B. Why don't—you have to change B when I go from the rest frame to the frame of the body that the plasma is flowing on? It turns out that the correction in a flowing frame is proportional to the velocity over the speed of light, and the flows we're talking about are very very much—very small compared with the speed of light, so B is the same in both frames, but E is not. Okay, a few more important equations. I took Faraday's law and I put in the form of E as minus U cross B, and then we get a Faraday's law in this form. The energy equation that Tomas Cambo C showed you was extremely complex, and it is customary in simplifying the equations—one approach is to simply assume that every—all the processes are adiabatic and to take the simple form P times Rho or n to the minus 5/3 is constant, and we usually address our problems in that form, although there are systems in which that's not a good approximation. And then another thing that's of interest is that one can expand J cross B by using—I think I'm missing a μ₀ but in this—in this—but I can use Ampere's law to expand J cross B, and the reason that that's of interest is that J cross B is a force that can be thought of as the sum of two forces, and I think I want to go to the next form here where now I have grouped the first term here with the pressure, and you see it acts like a pressure. So the thermal pressure plus the magnetic pressure plus the flow dynamic pressure can be put into the same gradient term, and then what's left is a term that depends on how much B changes along B, and that's the curvature. So that's called the curvature force, and I think I heard people talking earlier today about the magnetic field acting like a string that's pulling with tension. So this expresses the tension now. So we have the thermal pressure gradient, the magnetic pressure gradient, the dynamic pressure gradient, the curvature force, and then I have the sources or the losses of momentum.

Okay, now the behavior of flowing plasma in the MHD limit is often described in terms of moving magnetic flux tubes, and that's an important concept. And I think one way to think about it is by linking what happens to the motion of particles in a magnetic field. We have a Lorentz force, so that—some—I have so many buttons, I don't know which one to press—the Lorentz force is—tells me the V of the particle—I'm distinguishing V, the particle, from U of the flow—and that the force is proportional—is the product of the charge times V cross B—that imparts a circular motion. It doesn't affect the parallel direction. Magnetic force does not change the parallel velocity. So we have a V_parallel that is not changed in a uniform field, in any case, by the magnetic forces, but the—the transverse component forces the particle to move in a gyro orbit. You notice that the—the charges in the denominator, and if you do this correctly with vectors, you'll find that it means that the electrons circle the magnetic field in the right-handed sense, the ions in the left-handed sense, and the net effect is—in the MHD limit—the particles that start on a field line just can't get away from it; they keep moving in circles. And I've tried to show it here with the ions in a larger gyro radius because they've got a bigger mass and the electrons in a smaller gyro radius, and I think I've got the arrows in the right direction, I think, and that's called the frozen-in field. It means that if you take all the particles that are in a field line at one time, then they lie on a common field line at all times, and that's discussed in volume of the wonderful collection of books from this summer school. So there—there are exceptions to the frozen field: one is that energetic particles gradient and curvature drift. We think of them as a relatively small fraction of the total particle population, but in—in some—in some context they're very important, but it means that they don't remain frozen to the flux tube. When we're talking about a frozen flux tube, we mean that the particles of very low thermal energy that are on the field lines stay on it. MHD does not do a good job of treating the energetic tail of the plasma. Tricks have been developed for dealing with them. There are ways in which people use simulations and then add energetic particles to see how they behave. That's called large-scale kinetics, but I just wanted an aside to point this out. Okay, so the basic interaction involves moving flux tubes that's frozen to the plasma, and you should always think of the flow of the plasma and then paint the magnetic field on it. Don't—try to do it the other way around. So the—if—if you have the plasma moving toward an obstacle and getting slowed by the presence of the obstacle, that will put a bend in the flux tube, and all the particles that were on this flux tube stay on the flux tube, but we get a kink in the flux tube, and that can also happen if the flux tube that we're talking about runs into a magnetic barrier as well as a solid barrier. I'll go into this a little bit more, but let's for a minute forget that the plasmas charged and suppose that the flowing obstacle is a neutral gas. So the question is, what sort of interaction would you expect? It's just a neutral gas flowing onto a body, and I've drawn it as a sphere—it can be anything else you want it to be. Anybody want to make a comment about what kind of disturbance—in particular, what kind of disturbance would you see to the left of the dashed red line? You'd have a buildup of density in front of the object, and would—what if you build up the density and in some place—what happens? Well, that's a possible answer. So he said—one of them said you build up density, the other one said there might be a shock. What would—what decides whether they would or wouldn't be a shock? Yeah, whether the relative speed is supersonic or not. Okay, so if it's more than the sound speed, you get a shock. What happens if it's less than the sound speed? Would you say a little—an over denser—would it just hover? It would propagate at the sound speed, right? It would propagate away at the sound speed. That's why the sound speed is important. If it's less than the sound speed, it means that it can move away from the body faster than the flow is going toward the body, so it just propagates away. I built up the density, it propagated—it propagated away; it's still there. So there was no question of whether it would make a shock or not, but it just propagates away. So you would get perturbations upstream if your sub-magnetosonic and you would not if you're super-magnetosonic, but not far upstream, but you'd get a shock, and then a way beyond the shock there would be no evidence to see that the obstacle was there. So I think—so the—the question of whether it—would it matter how fast the gas is flowing? It's a question of whether it—the—the signal that's moving at the sound speed can move upstream faster than the flow is bringing it back.

Okay, well, here I've written some dimensionless parameters, because now we have the idea that it matters how fast the flow is relative to the propagation speed of waves. Now I'm going to go to talk more about MHD waves, but in the meantime I'm giving you some dimensionless quantities: the Alfvén Mach number, the magnetosonic Mach number, and here I'm just using the sound speed to represent it, so that's really the sonic Mach number, and then the ratio of the thermal pressure to the magnetic pressure. And the important question is not particularly how big they are, but whether they're bigger or smaller than one. And I've given it to you for several different situations: for magnetospheric plasma relative to a moon—this is the speed of the rotation of the internally trapped plasma relative to the moons that are the obstacles I'm talking about—and in all cases they—the Mach numbers are less than one, with some marginally possible exceptions, but generally less than one is a good suggestion, and typically the thermal pressure is larger than the magnetic pressure at those distances, but they're not very far from one. This is for solar wind plasma is relative to all the obstacles; it's always far bigger than one for all of these things. At comets—I put—I'll explain this a roughly equal number later on—and at the moon—just think about the moon's orbit; it takes it from the magnetotail of Earth, which is basically no flow, into the solar wind where there's a super-magnetosonic flow. Okay, so I've—I've introduced the idea of the Alfvén Mach number, but I—magnetosonic Mach number, but I haven't defined them. I now—β is a quantity that's useful in telling us whether it is the thermal pressure or the magnetic pressure that is dominating the interaction, and that also plays a role in determining what the symmetry of the interaction region will be. When the magnetic pressure dominates, it's the magnetic geometry that imposes itself to zeroth order on the interaction region. When it is—when β is large compared with one, the symmetry tends to be about the flow direction. So those are very—that's a very useful quantity to know, but the other parameters relate to waves, the transmitted information in a plasma, and that will require me to introduce MHD waves. And MHD waves are important because they communicate information through a fluid, just the way the ripples on the surface of a lake tell you—communicate the fact that a rock has just been dropped into the water. So the characteristic waves differ for different situations, but I'm going to look at a uniform plasma with a constant background magnetic field, constant pressure, constant density in equilibrium and at rest. So I'm assuming that the flow velocity is zero, and I'm going to make small changes in these properties, and the question is what are the MHD equations that govern the small perturbations in magnetic field, which I use small B, and here I'm using V for the perturbations about zero in the velocity, and Δρ and ΔP, the perturbations about the density and the pressure. I'm going to assume that the perturbations vary with space and time like e to the minus i K dot X minus ωt, and K is the wave vector and it's inversely proportional to the wavelength of the disturbance, and ω is 2π over the period of the waves, and I've given you a homework assignment, but I'm not going to make you derive the whole thing, but what you'll find is you've got a product of two functions of ω and K that have to equal zero in order for the equations that come out of doing this to the MHD equations and keeping only the lowest order in the perturbations—this quantity has to be zero. Well, evidently you can make it zero either by making this thing zero and having this nonzero or by having this thing zero and this nonzero. So I'm first going to talk about the solution that has this factor of the product equal to zero. That's called a dispersion relation; it relates the frequency to the wave number or the wavelength, and—and by the way, in—in this, C_S² is the sound speed and V_A² is the Alfvén speed, and you were talking about β earlier today. I give it in the square form, so it's B² over 2μ₀ times the mass density. Okay, now it turns out that for the waves that satisfy ω² minus V_A² K² cos²θ is zero, there is a requirement on the—on the polarization of the—of the perturbations. The magnetic perturbation is transverse to the plane containing the propagation vector, the wave vector and B, the background B, and a perturbation and magnetic field that's perpendicular to B bends B but doesn't necessarily change its magnitude. The other thing I want you to look at in this diagram is where J is found. J is the current density. This wave carries current. What is particularly important about it is that if there's a non-vanishing component of J along the background field, the Alfvén wave carries field line current. That's why Alfvén waves are so important in a magnetospheric interaction because they carry a current from wherever in the magnetosphere a perturbation arises; they can send a current along the magnetic field down into the ionosphere. Now why do I emphasize that so strongly? Because the other factor in the—that it could be zero instead has a different polarization of the wave. The background—the perturbation magnetic field has a component that has a finite projection on the background field. It changes the field magnitude. This guy can't change the field magnitude to lowest order. Remember, if I want to look at the total field magnitude, it would be Big B² plus 2 Big B dot little B, but that's zero because the little B is perpendicular here, so Big B dot little B is zero plus little B², but that's higher order, so we drop it. So the two—to first order, there's no change in the field magnitude here. Here, little B dot Big B is a finite component, so these waves are compressional. You notice this is fourth order, this is second order. The…

Second order means you get plus or minus, so you can propagate one way or the opposite way; it doesn't matter. Here you have fourth order; you have really two solutions: Omega squared equals this or Omega squared equals that, and each of those can propagate up or down, so you have two different types of waves, but four different waves because each of them can propagate both ways. So, first place, this wave is compressional; it can change the magnitude of B. But now take a look at where J is. J is strictly perpendicular. These waves can communicate with the ionosphere, but they can't send current directly to the ionosphere; that's really important.

I: You have to talk.

Capital yes.

I: Isn't that interesting?

I: Yes, I think it must be the pointing.

I: I think it's the pointing back, but why isn't it in the other one?

Oh yeah, it is. Okay, it's it's it's the it's the rate, the direction in which information is propagating.

Fran: You know, I have a hearing problem, so I have to come closer to the pointing point, to the pointing.

Okay, I shall do that.

Okay, it was an S here and there's an S here. I can't see it.

Yeah, okay. Is that all you wanted?

Yes.

Okay, alright.

So the dimensionless parameters that I was giving you characterize aspects of the interaction, such as the presence of shocks. So it turns out that those compressional waves I talked about, there are two of them, and very unimaginatively we call them the fast and the slow, and guess which one goes faster. So we have the magnetosonic Mach number is the flow velocity divided by the fastest velocity of the fast mode wave, and if that's smaller than one, compressional waves can propagate upstream into the side and that produces a bow wave, but not necessarily a bad shock. So here's a picture of a situation in which waves are propagating upstream and and are and and are diverting the flow, but they're not making a a a shock. But if the magnetosonic Mach number is greater than one, even the fastest MHD waves can't propagate upstream because the flowing plasma keeps carrying the signals back downstream. The result is that a shock forms upstream of the flowing obstacle. So that's really important, and plasma flowing through the shock compresses, decelerates, and deflects the plasma, and that means that downstream of the shock it is sub-magnetosonic. So it's always sub-magnetosonic after it's gone down. Now it still could be super Alfvénic because the Alfvén wave might be traveling; that might have a smaller amount. You notice the magnetosonic Mach number is always greater than or equal to the Alfvénic Mach number because you have to add in the denominator the sound speed, and the smallest that can be is zero. Okay, so so it, and I just comment here that it's very much analogous to a supersonic jet during takeoff. You're going slower than the sound speed; you have no shock. If you go fast enough, you develop a shock. The only question is the relative speed of the two; it doesn't matter whether it's the plane moving into the air or the air moving into the plane; you've got the same physical situation. So that's a question of what frame you're in, how you see it. Okay.

So now let us talk about how wave-related dimensionless parameters determine the geometrical properties of the interaction region. Incidentally, I meant to talk about when I got Alfvén what waves out where we had B over the square root of MU not Row. Two things I wanted to point out: the sound speed is the square root of the thermal pressure divided by the density; the Alfvén wave is the square root of the magnetic pressure divided by the density, so it's they really are very parallel in many ways. The other thing I wanted to tell you is that Alfvén discovered these waves in the early 1940s, and one of the comments that I find actually documented is there could not be such waves because if there had been, Maxwell would have found them. So you're looking at waves that Maxwell didn't find, and Alfvén went around trying to explain this idea to people, and nobody was buying it until he visited the University of Chicago and he told his story to Fermi, and Fermi said, "Of course," and the next day everybody said, "Of course." So those were two things I meant to tell you about. Okay.

So we now have the Alfvén Mach number that compares the velocity of the flow with the Alfvén speed, and here is a schematic of a magnetized plasma flowing from the left and encountering an obstacle. You notice there's a little kink in the field here. That kink tries to move up the flux tube at the Alfvén speed, but as it's moving up the flux tube, remember the Alfvén speed is the velocity in the plasma rest frame, but at the same time the plasma is flowing this way, and so the front that sees the Alfvén perturbation appears at an angle to the place where the where the perturbation was introduced into the flow, and therefore you get this angle, this bend back, and you know that the tangent of that angle is the flow speed divided by the Alfvén speed. So you can, by looking at how far the field is bent, where the bend back of the field starts, you can you can tell what the Alfvén speed of the plasma is or what the flow speed is if you happen to know the Alfvén speed. So that's kind of cute. Okay.

Now remember I said that the Alfvén wave carries a field-aligned current, and in fact we have field-aligned currents that are represented in this diagram with arrows. The dark arrow is on the side closer to you, and the dashed arrow is on the side of the body farthest away from you. So one thing that you can see is that the in this picture, which is sort of a generic moon at a giant planet—think of the planet as being inside, away from you—and the currents always flow into the moon from the side closer to the planet and flow back out on the side farther away from the planet. And this morning Rod Hillis was talking about currents moving and having to close somewhere. We haven't talked about where this current system closes; it's coming in from both sides on one side of the planet, going out on both sides from of the moon, excuse me, and going out on both sides on the other side of the Moon. Somewhere in here it's going to have to close, and that's something we're going to be talking about. All right.

So something that I'm very interested in is—Fran mentioned that Ganymede has a magnetic field, and I'll be talking about it—it makes this strangely shaped magnetosphere inside the magnetosphere of Jupiter. That's because the plasma beta is very small here. In the magnetosphere of Jupiter, the plasma beta is very small; that means the magnetic field is dominating; it dominates the symmetry. So although the flow is causing these fields to bend back, it's still the magnetic fields are so rigid that the magnetosphere takes this almost cylindrical shape. That's as contrasted with the magnetosphere of a planet and the solar wind where the it is the the plasma dynamic pressure that dominates, and the magnetic field is just sort of along for the ride, and the symmetry of the field of the magnetosphere is dominated by the direction of the flow, and you get this interesting bullet-shaped cavity, which you would not get if the solar wind had a much higher plasma magnetic field, if the plasma beta were very very small in the solar wind. So it's playing a very major role. So I went backwards. Oh, isn't that nice? Is this a good time? Is is my timing good, or should I—anybody want to ask any questions on this part before we—yeah. Well, I I think the—hip—yeah, I think it's like Ganymede because I think—yeah.

For those of you who would like to know what's behind this exchange between Fred and me, the studies from the IBEX spacecraft have shown that there's an anomalous region of production of energetic neutral atoms—at least that's what they tell us—that's on the heliopause. It's not centered about the symmetry axis of the flow; it's offset both to the south and—well, what we would call dusk if it were the Earth's magnetosphere—and it makes a ribbon along the boundary of the heliopause. And it turns out that at Ganymede, which is embedded in a flow dominated by the magnetic field, if your magnetic field is tilted a little bit to the right direction, you you find that the field becomes tangent to the magnetopause of Ganymede along a ribbon, and reconnection occurs there preferentially. And if you just assume that the same situation, dimensionless situation, the magnetic pressure is higher than the thermal pressure and the magnetic field is tilted in just the right way, you could account for the ribbon by by having reconnection occur preferentially there and create the neutral atoms that are observed. I don't know that there's anybody except my co-author who believes this picture, but we think that the local interstellar bubble is probably one in which the magnetic pressure dominates the thermal pressure and that the heliosphere is very stretched out; it's not a bullet-shaped thing, Jeff. Is it okay?