 All right, I'm gonna go ahead and get us started here today. So welcome to everybody online and of course, physically in the room. It's my pleasure today to introduce our colloquium seminar speaker. This is Dr. Nevin Weinberg. So he comes to us from UT Arlington and he earned his bachelor's degree from the University of Chicago and his PhD in astrophysics from Caltech. He was a postdoctoral fellow at the Cavalry Institute for Theoretical Physics at Santa Barbara. If you've never gotten a chance to visit there, find an excuse, participate in one of their programs and go. That place is amazing. And also at the University of California at Berkeley. He joined the MIT physics department as an assistant professor in 2011 and became an associate professor in 2016. He then joined the faculty at UT Arlington in 2020 where he is currently an associate professor of physics. His research interests include stellar fluid dynamics, tidal physics, gravitational wave sources and thermonuclear explosion but they just get more dramatic as you go up the list. Thermonuclear explosions on a creating neutron stars or X-ray bursts as they're politely called. So we're very pleased to have Dr. Weinberg here today to lecture on when stars go non-linear, large amplitude tides and stellar oscillations. Please welcome our speaker today. All right. Definitely audience online. Okay, great. Thank you. Thank you very much for the invitation and for the introduction. So let's see the picture you're looking at here which maybe some of you have been looking at now for a few minutes is a picture of photograph really of Jupiter's moon Io taken by the Voyager 1 mission in 1979 as it passed near Jupiter. And so Io is on a slightly eccentric orbit which means that it experiences a time-dependent compression and expansion as it passes closer and further from Jupiter and that compression and expansion heats up Io's interior. And so Io is actually the most geologically active body in the solar system decided small size and age. And a nice illustration of that geological activity is this volcanic plume. You can see spewing out the top of Io here. And so in this talk, I'm gonna be describing tides in completely fluid bodies, not rocky bodies like Io. But I like this picture and this idea because I think it nicely illustrates an important theme of the work I'll be talking about which is that sometimes tides can have a really dramatic influence on a binary and its components as I'll describe. And so before I get going, I just wanted to acknowledge my collaborators who have been working with me on this problem for a number of years now especially the students that have worked with me. So Reed Essek who's now a postdoc at the Perimeter Institute, Mohamed Morbit who's now a graduate student at Youth Yarlington working with me. Deborah DJ Promenade who's a graduate student at Princeton and Hong Yu who is a student who worked with me at MIT and is now a postdoc at Caltech. And also of course the funding resources which have supported this work. Let's see. Oh, that's right. Oh, let's see. Well, I can always just click with my hand, that's fine. Okay, so this talk sort of can be broken up into two parts. In the first part, I'll be focusing on tides in stellar binaries and in exoplanet systems. So systems where you have two sun-like stars orbiting each other with short period orbits and orbital periods of days. So those are called solar binaries. I'll talk just very briefly actually about neutron star binaries which are important gravitational wave sources and probably spend the most time on short period exoplanets systems called hot Jupiters. And then in the second part of the talk I'll switch gears a little bit and talk about seismology. Sometimes referred to as asterose seismology. And these are stars that are isolated but have waves excited within them. And by studying those waves, observing those waves with telescopes and then analyzing those observations you can learn about the interior of the star. So basically these waves are providing a window into the stellar interior that you wouldn't get from just the photons which of course are just coming from the outermost layers of the star. It's not working. Oops. I see there's a delay. So there we go. So yeah, tides. So I think a nice place to start when you're talking about tides is the classic tide system, the one that Newton studied, the Earth and the Moon. So obviously this is very exaggerated but the idea is that the Moon of course raises a tidal bulge on the Earth in the Earth's ocean. And that tidal bulge leads the Moon's orbit, right? The Moon is orbiting in this counterclockwise direction and the Earth is spinning in this counterclockwise direction and the tidal bulge or this angle delta leads the Moon's orbit. And the reason it leads is because the Earth is spinning faster than the Moon orbits, right? The Earth is spinning on a time scale of a day and the Moon is orbiting on a time scale of 28 days. And so you can sort of think of this as the Earth is dragging the ocean at the base of the ocean in the crust. It's dragging the water with it and the Moon is pulling it back. And the net effect is that this bulge slightly leads the Moon. And if you now decompose the forces acting on the Moon, there's a component of the force that's pointing towards the tidal bulge. And if you now think about it, there's an R cross F, there's a torque acting on the system. And that torque leads to an exchange of angular momentum. So the Earth's spin is slowing down, is losing angular momentum and the Moon is gaining that angular momentum. And so as a result, the Earth's spin is slowing down by something like two milliseconds per century. And the Moon is moving away at something like four meters per century. And we can actually measure that motion. Small as it is, we can actually measure that motion because the Apollo astronauts left mirrors on the Moon and we can shine laser lights from Earth at those mirrors and time how long it takes for that light to reflect. And that time allows us to measure the distance of the Moon to a centimeter accuracy. In fact, we need to account for where on the mirror the laser light hits, right? Because we're measuring it so accurately. And so this four meters per century is roughly a nanometer per second, which is roughly the rate at which your fingernails grow. And so this guy showing how far the Moon has moved away over the last 10 years. I think that used to lag. Okay, good. So another interesting system where tides are important is hot Jupiters. So these are Jupiter mass planets with orbital periods of just a few days. In 2019, the Nobel Prize in physics, half the Nobel Prize in physics was awarded to two astronomers, Mayor and Kalos for their discovery of the first exoplanet orbiting a sun-like star. And that planet was a hot Jupiter. So these are, Nobel Prize physics worthy systems. The other half of the Nobel Prize went to a cosmologist, Jim Peebles. And so if you now analyze the tides in these systems, you can show that the torque is in the opposite direction from the Earth and the Moon. So the star is spinning more slowly than the planet orbits. The planet orbits with a time of maybe days, whereas the star spins if it's a star like our sun with periods of tens of days a month. And so the bulge that the planet raises on the star lags the planet's orbit, it points behind it. And so if you now analyze the torques acting on the system, the planet is spinning up the star. It's transferring the orbital angular momentum to the spin of the star. The planet is losing angular momentum. And that means the planet is falling into the star at some rate, it's inspiring into the star. And the question is, and that, this was something that people noticed or realized right away, back in the early 90s when they first detected these systems. And there were big surprise when they first found them and they quickly realized that the tides would be dramatic in the system. But the question for all these years has been, what's the rate at which that in spiral occurs, that infall occurs? And that depends on the efficiency of tidal dissipation in the star. Essentially, what's the effective viscosity of the star? And so I'll be describing work with my students that have addressed exactly that question. What's the rate of infall? Okay, so how do you go about studying these problems? So you can do something, what you usually do is something that I like to call tidal perturbation theory. So the idea is you take this spherical object, just gonna make sure I have a timer so I can see what time it is. So you have this spherical object, a star, and you put a companion near it and the spherical object becomes ellipsoidal. It takes on this kind of football-like shape. And the amplitude of that distortion, delta R over R, depends on the ratio of the masses of the companion, say the planet to the star, little m over big M, times the ratio of the radius of the star to the separation of the system cubed, right? And so if you now think about the fluid forces in the star, the fluid response of the star, the fluid equations are inherently non-linear. So there's restoring forces due to pressure, due to buoyancy within the star, and you're perturbing those forces with this delta R over R. And so just like in a Taylor series expansion of a non-linear equation, you get terms that are linear, non-linear, and so on. Same is true of the fluid forces. There's a linear term. There's a term that's quadratic in delta R over R, cubic in delta R over R. So there's this full expansion in terms of delta R over R that you can express the fluid response in terms of. And so what most studies of tides, dating back decades have assumed is that you can ignore these non-linear terms. They've basically said for the systems that we're interested in, we're going to assume that we can ignore those non-linear terms and just do what you might call linear title theory. And really the motivation was one of simplicity. It's a lot simpler to just keep the linear terms. The non-linear terms, as I'll give you a feel for, are complicated and in an ideal world, we can ignore them. But the question, of course, is that true for the systems that we're studying? Is that a good description of nature? And so there's both, I think, observational motivations and theory motivations for not dropping those non-linear terms in many of these systems. So from the observational side, some of the best measurements we have of title dissipation in binary systems comes from these observations of solar binaries, sun-like stars. And so we see these systems in sometimes eccentric orbits. And you can show that over time, tides will circularize in orbit. And so you can use this collection of solar binaries that astronomers have observed in clusters of stars and place constraints on how efficient tides will circularize a system. And when you look at the numbers, you find that linear theory predicts what are called quality factors. So if you strike a bell and you ask how many times does a ring before the amplitude falls off by one e-folding, you say the Q of the bell is a hundred or a thousand. Linear theory predicts that the Q of the star is 10 to the nine. So it's a measure of the viscosity of the star, the title dissipation efficiency. And so the larger the number, the less efficient. And so linear theory says 10 to the nine, but to fit the observations, to fit the circularization of these binaries, you need Qs more like a million, which tells you that linear theory is underpredicting the efficiency of title dissipation by factors of a thousand or more. Question. They're out of contact, they're separated. Yeah, so they're orbital periods of a few days to tens of days. And so they're not Rho-Slob overflow, or anything like that. They're pretty close. Yeah, and the title effects are strong, but there's no mass transfer between them. So another system, another type of system where there's indications that linear theory is getting things wrong are these hot Jupiters. So when you look at the orbital distribution of hot Jupiters, how many planets there are as a function of orbital period, you find that there's evidence of orbital decay in these systems. There's measurements you can place on, or constraints anyways that you can place on the Q of the star that is orders of magnitude smaller than the linear theory estimates. Again, just like with the solar binaries. And then finally, there's stars that some people call heartbeat stars. These are binaries that the Kepler Space Telescope saw that you see individual modes of oscillation, waves in the star are excited. And there's pairs of waves that they see in any given in some of these stars that whose frequencies add up to the frequency of a third wave, which is exactly what you expect from non-linear interactions of waves, triplets of waves interacting with each other. And so you can only explain those observations if you account for the non-linear effects I'll be talking about. So those are all the sort of observational motivations. On the theory side, you can do calculations of tides and stars and show that the waves are gonna reach such large amplitudes in portions of the star and regions of the star that they're gonna become unstable to various non-linear interactions and excite secondary waves. So the primary wave reaches a large amplitude and excites secondary waves. And I'll be talking more about this in a moment, but you can do these calculations that basically can convince you you need to account for these non-linear effects that the linear title theory is not enough in many of these systems. And then when you account for those non-linear effects you can show that they significantly enhance the rate of tidal dissipation exactly in the direction that the observations are pointing that the efficiency is greater than the linear theory predictions. So these sort of considerations motivated us about 10 years ago to develop some of the formalism and computational techniques to study this problem of non-linear tides. And so over the last few years we've kind of been systematically applying these tools to a range of problems, including hot Jupiters, compact object binaries, binary neutron stars and white dwarfs, which are important gravitational wave sources and also isolated stars in which waves are excited through internal motions within the star, astro-sizing measurements of red giants and other types of variable stars. And so there's this range of examples. In all of these cases, the question is that we're interested in is when are non-linear effects important in which systems? And when they are important, how does it impact the system? How does it impact the evolution of the binary, for example? And so an important component of the story that I'll be telling you is that when the tide perturbs the star it excites oscillations within the star, waves in the star, modes of oscillation. This is sometimes referred to as the dynamical tide. And so I just wanted to give a very brief primer on modes of stars and the types of oscillations they can support. So there's something known as P modes. The P stands for pressure. So pressure is the restoring force. So basically these are sound waves in the star. They tend to have short periods in the sun periods of order minutes. There's what are called G modes where G stands for gravity. So there the restoring force is buoyancy or it's sort of like gravity waves that you see when you throw a rock in a pond. These are internal gravity waves within the star and they attempt to have long periods, orbital periods of hours to days. An important property of these G modes is that they only propagate in radiative regions of the star. So this is supposed to be like a sunlight star and this sort of granular region is supposed to represent the convection zone. So the outer third of the sun is convective. So these G modes can only propagate in the interior of a sunlight star. And then finally another class of modes that's important if the star is rotating rapidly enough are inertial modes where the restoring force is the Coriolis force. And those tend to have periods of order the spin period of the star. And so a particularly important class of modes for tides are the G modes. And that's because they can have these long periods, periods of days, which means they can be resonantly excited by the planet or the companion star, which also has a period of days. So you get into a situation where you have basically a driven damped harmonic oscillator that you're driving resonantly, you're driving these modes resonantly. So there's a really neat terrestrial example of this resonant dynamical tide here on earth, the Bay of Fundy in Nova Scotia, Canada. So this is a Bay that's about 300 kilometers in length. And the natural oscillation frequency of this Bay is about 12.4 hours. If you think of the oscillation frequency of a bathtub, it's maybe a second, right? This is because it's so big, it's 12.4 hours. And that just so happens to be the tidal forcing period due to the moon. So the moon is resonantly exciting the water in this Bay. And as a result, you get this huge tidal range, 15 meters between high tide and low tide, vertical difference between high tide and low tide, twice a day. So twice a day, you get this and twice a day, you get that, right? All this water comes in and out of the Bay due to this resonant dynamical tide here on earth. And so in getting back to stars, the tide can excite many waves in a star, but in linear theory, those waves all pass through each other and don't interact. They sort of obey the principle of superposition, but pass right through each other like ships in the night. Whereas when you start accounting for non-linear effects, these second order and third order terms in your fluid forces, you start getting wave-wave interactions. And so at second order, an important type of wave-wave interaction is something known as a parametric instability in which you have a, we can call a parent wave that's directly excited by the tide, by the forcing of the companion. And if it reaches large enough amplitude, it can excite daughter waves of half its frequency. And whether that happens or not depends on the amplitude of the parent wave, the strength of the coupling and the vertex of this Feynman-like diagram. I know there's some high energy physicists here. So a shout out to them, right? And then there's the frequency of the daughter waves whether there's really pairs of daughter waves that can add up to the parent wave frequency. But the important thing is that if this happens, if this instability occurs in the star, you have a new source of dissipation in the system, right? Because suddenly this parent wave can lose energy to these daughter waves. And those daughter waves themselves can have very short wavelength, much shorter than the parent. And so they can be much more efficient at thermalizing the wave energy and just depositing it as heat within the star. And so this is a new source of dissipation that you have in nonlinear theory that's not there in linear theory and could potentially lead to a faster rate of tight induced orbital evolution. So in a moment, I'll describe where you have many, tens of thousands of interacting waves but as a sort of toy example, a toy model of the problem, I thought I'd start by just showing you what happens if you have just a three wave system. And so this plot here shows the energy of a parent wave as a function of time. And I initialize the parent to have basically zero energy, turned on the tide, and the parent starts growing in energy due to the tidal driving. And if I didn't have three wave interactions, the parent would follow this blue line here and reach some linear equilibrium. Basically it's the solution to the driven damped harmonic oscillator problem that probably you've seen in your undergraduate physics classes. But when you now allow for three wave interactions, when you now allow the parent to couple to other modes or a pair of other modes in the star, what happens is the parent can reach a critical energy sort of shown by this dashed line here, horizontal line. And rather than reach the linear energy, it starts to undergo this kind of sort of complicated limit cycle interaction with a daughter pair, which initially has zero energy, but then grows exponentially, right? And so the non-linear solution to this system is very different from the linear one. You have a parent that doesn't quite reach the same energy and a daughter pair that in linear theory would have effectively zero energy, excited to large energies. And so this can obviously then affect the efficiency or rate of dissipation in your system. Okay, and so in terms of the parameters of a binary, you can ask the question, which systems are unstable to these three wave interactions, to this parametric instability? And the sort of relevant parameter space of the binary is the mass ratio of the binary shown on the vertical scale. So the mass of the planet or companion star to the main star as a function of the orbital period. And this jagged black line here is the threshold for the parametric instability. So anything below that line is stable to the parametric instability, and anything above it is unstable. And so as a point of reference, I put a hot Jupiter on this diagram. So Jupiter has a mass of about 10 to the minus three, the mass of the sun. And if it's a hot Jupiter, it has an orbital period of a few days. And so you can see that a hot Jupiter is well inside the unstable region, well above that jagged line. If you think about a solar binary, those have mass ratios of order unity, right? There's two stars, maybe two sun-like stars orbiting each other. So they're sort of off the scale of this diagram. They'd be up here, but they'd be unstable out to tens of days to this parametric instability. So nonlinear effects are extremely important in both these hot Jupiters and solar binaries. And you can do the same kind of analysis in different types of systems, white dwarf binaries, neutral star binaries, and show that in many cases, for the short period systems, the nonlinear effects are important. The linear approximation fails, that you can't just ignore those nonlinear terms because it's simpler, right? You have to account for them if you really want to understand what's going on in your system, right? And so the question then is, if you're unstable to these wave interactions, what are the implications? How does it impact the evolution of these systems? And so now I'm gonna sort of starting getting into the specific systems that we've studied and applied some of these ideas to. And so this is a problem that my former student, Reed Essek, investigated the orbital decay of hot Jupiters. So as I mentioned, these planets are falling into, inspiring into their host star, but the question is at what rate, right? Are they sitting out there for essentially eternity, right? Much longer to say than the main sequence lifetime of the star, or are they falling in relatively rapidly? And we're only catching them in a moment before they basically merge with their star. And so if you go back and look at this diagram again, right? These hot Jupiters are well inside this unstable region. So it's not as if there's just a single daughter pair that gets excited by the parent wave. You have hundreds of daughter pairs that are unstable. Those daughter pairs reach such large energies that they excite granddaughters. And those granddaughters reach such large energies that they excite great granddaughters. And you get this kind of cascade of modes of waves excited in the star. And so what we did is simulate the resonant excitation of tens of thousands of these interacting modes by solving a large set of non-linear coupled amplitude equations for these systems. And so here's just one example of one of his simulations. So this is showing mode energy versus time for some of the modes in one of his mode networks. We're not, I think it's one out of every hundred modes. If we showed all of them, it would just be a blur and you wouldn't see anything. So we just picked out one out of every hundred randomly. And what you see is that the energies fluctuate in kind of this chaotic way. At any given moment, there's the different colors of different modes. So some of the modes are excited some of the time and then they die off and another mode is excited. So there's kind of this randomness to the energy of the modes. But if you then ask what's the dissipation in the system and the way you can calculate that is by calculating the damping rate through to each mode times this energy and summing all those up, right? And so each mode contributes a small amount. You sum up each of the modes contributions and you get this plot on the right. This is showing the total dissipation as a function of time. And after a kind of burn-in period for our run, you sort of, you get a saturation of the dissipation. You basically reach a non-linear equilibrium that you can turn into a measurement of the total dissipation in your system, which in turn tells you how much energy is being pulled out of the orbit and thermalized inside the star, which then tells you the rate of the orbital decay of the planet, how quickly its period is decreasing with time. And so this is what we found. So in terms of, again, this quality factor, which is again a measurement of how efficient the dissipation is in the star, we found that for a Jupiter mass planet in a one-day orbit, the Q of the star is around 10 to the five, 10 to the six, right? Which is orders of magnitude smaller than linear theory prediction. Linear theory, if you did the same kind of analysis, but ignored the non-linear terms, would give you Qs of 10 to the nine, 10 to the 10, which would say that the planet should basically sit there for all time, the decay so slow that you shouldn't have to worry about it or think that it's gonna happen on a time scale of the star's evolution at all. Whereas when you account for these non-linear effects and account for the fact that Q is this small, you find that the orbital decay time is less than a billion years for planets on orbital periods less than about two days. Meaning that the orbit is gonna decay in a significant way on less than the time scale of the age of the star. And then the question of course is there observational evidence for such orbital decay? And so there is some indirect evidence. So you can do a statistical analysis of the number distribution of hot Jupiters. How many hot Jupiters there are as a function of orbital period. And there's indications that there's fewer very short period planets than you would expect if only linear theory was at play. And so there's the evidence of a clearing out of the shortest period systems. But there's not that many hot Jupiters with orbital periods less than two days. There's maybe a dozen or so. And so you're sort of in the small number statistics regime. So you can't do a really precise measurement of Q in this way. There's some hope that the test space mission, I know some of you take observational data with tests. So not in the context of exoplanets, but obviously one of the goals of tests is to find exoplanets and to find hot Jupiters. And so it's finding many more hot Jupiters. And so there's some hope that we'll have better statistics in the coming years and be able to do this kind of analysis more definitively. That said, there is a really interesting hot Jupiters system called WASP 12B, which shows direct evidence of orbit decay. So this is a planet on an especially short orbital period or just a little over a day. And what astronomers have been able to do, so they detected this planet in something like 15 years ago and measured its orbital period really well through the transit timing. Basically the planet eclipses its host star. And what these transit timing observations do is they come back 10 years later, 15 years later and look at where the planet is. And they can say if the period was perfectly constant, the planet should be exactly at this spot 15 years later. It'll have orbited the star thousands of times in that time scale, but if the period is really constant, you can be able to say exactly where it should be. And instead what they find for this system is that it's a little bit ahead of where it ought to have been if the period were constant. It's consistent with the period decreasing at a rate of something like 30 milliseconds per year. Not much, but look at the precision at which they can measure them. Again, because they're sort of integrating over this long time scale. And so if you turn that into a time scale of P over P dot, you get a decay time scale of three million years. So much less than a billion years, much less than the lifetime of a star. And so then the natural question asked is, can you understand this as being the result of tidal dissipation of orbital decay? And so we studied this problem because it seemed like this is the ideal system for the kind of problems, kind of questions we're interested in. And so we analyzed the system specifically and found that if the star is a sub-giant, then what can happen is you excite a wave, internal gravity wave in the outer parts of the star at the radius of convective interface for subtle reasons. If you ask me, I can tell you them. And what happens is that internal gravity starts propagating towards the center of the star. And as it propagates, its amplitude deep increases due to geometric focusing. So basically you're trying to put the same amount of wave luminosity in a smaller and smaller volume of the star. And so the amplitude increases. And what we found is that the amplitude gets so large as it approaches the center of the star that the amplitude becomes larger than the wavelength of the wave. Delta R becomes larger than lambda. And that's the condition for what's called wave breaking. So we've all seen this when we go to the beach and the waves come to the shore and they turn over and crash. So these waves propagated thousands of kilometers in the open ocean and then deposit all their energy right at the shoreline. Just like this wave propagates throughout the whole star and then deposits all of its energy in angular momentum right in the center or very close to the center of the star. So this is sort of maximally efficient tidal dissipation. This is an example of strongly nonlinear, not even weakly nonlinear, like the parametric instability that I was talking about, strongly nonlinear wave breaking. And so you can do a calculation of this and what we found is that the implied dissipation is very close to the one that you observe, that there's a good agreement between the theory estimate and the observation of period change. And so I think this is a really nice example of nonlinear tidal dissipation. The one sort of outstanding question that remains is is this star truly a sub giant? So from the parameters of the system, we can't quite rule out the possibility that it's not a main sequence star, right? There's sort of an open question whether the star is on the main sequence or has just evolved off the main sequence in a sub giant and that matters because it changes the structure of the star and therefore the efficiency of tidal dissipation. And so really to answer that question, we need more precise measurements of its stellar parameters. People have tried to do that with Gaia and seismic measurements. There isn't, I don't think great prospects of doing much better than they've done, but it sort of again is an open question that remains to be answered. Okay, so another aspect of Hot Jupiters that I think is interesting and of course important is how they form, right? How do you form a planet so massive, so close to its host star? So you can show that in the protoplanetary disk, that close to the proto star, the temperatures are so high that you don't expect gravitational instabilities in the disk to form a Hot Jupiter. You don't expect the direct formation of a Jupiter mass planet that close to a star. So the preferred explanation for how you form these systems is that they form somewhere further out, maybe where our Jupiter forms. And then somehow some mechanism brings them into this very short period orbit close to the star. And so one mechanism that people consider is something known as high eccentricity migration. The idea is you have this planet on initially a circular orbit far out. It gets kicked into a very eccentric orbit by another planet in the system. There's a third planet or second planet system, a third body in a system that perturbs the Jupiter and puts it on in a very eccentric orbit, eccentricities of 0.99. And as a result, the planet passes really close to the host star. The paracenter passage is only maybe 10 stellar radii or even less. And as a result, the tides are really strong. The tides not just in the star, I've been focusing on tides in the star, but the tides also in the planet become really strong. And the idea is that those tides will circularize the system, tides remove, tend to circularize systems, and you end up with a Hot Jupiters we see, these nice circular short period systems that we see. The problem with this high eccentricity migration scenario is that if you assume the tidal dissipation in the planet is like our own Jupiters corresponding to queues of around 100,000, the prediction before Kepler went up was that there should be several high eccentricity systems in the Kepler sample, whereas Kepler didn't see any. And so, well, what does that tell you? Well, one possibility is that maybe the high eccentricity migration scenario isn't responsible for the formation of most Hot Jupiters. Maybe it's just, there's another mechanism. And there are others, people talk about disk migration where the planet in spirals in a disk as opposed to high eccentricity migration. But the other possible solution is that maybe this assumption that our planet, that these Hot Jupiters have the same quality factor as our own Jupiters is incorrect. Maybe their queues are much less than 10 to the five and they circularize so fast that it's not surprising that Kepler didn't see them because they spend so short a time in that phase of their evolution that you just wouldn't have expected Kepler to see any question. So how many Hot Jupiters? Oh, and I forgot to repeat the question. So how many, the question was how many Hot Jupiters there are in the Kepler sample? I think there's, on the order of dozens, tens of, yeah. So what do you say separately? Yeah, three or four or five, yeah. So the error bars are actually big enough that you can say, well, maybe we just got unlucky because you're sort of small numbers in six. That was for linear title. That's for linear title theory where you assume that queue is 10 to the five. That's correct. So there's an important class or type of title and interaction that takes place in very eccentric systems. This is something that goes by the name of Diffusive Tide. It was first discovered in the mid-90s in the context of stellar binaries but more recently has been applied in the context of these highly eccentric Hot Jupiters and the high eccentricity migration scenario. The idea is to follow them. As the planet passes through Paris Center and gets really close to the host star, the title kick of the planet, the title perturbation of the planet excites the F mode of the planet. And F mode is the fundamental, F stands for fundamental mode of the planet. And that title kick is so strong that it excites the F mode to a large amplitude that F mode can back react on the orbit. Basically, the energy of the F mode is so large it takes so much energy out of the orbit that it changes the orbital period enough that the next time the planet comes around, the phase of the F mode is slightly different from the phase it was in the previous Paris Center passage. And so you get this kind of random feedback interaction between the phase of the mode, each Paris Center passage. And you get, you can show it's, you can show that the energy of the F mode undergoes a kind of random walk, diffusive walk so that the energy increases linearly with orbital cycle. And then the question, you know, this is something that people appreciated a few years ago and we got to think about this problem and realize that the mode energy can become so large that again, you start wondering about nonlinear effects and how important those might be and understanding what's going on. And so with my former student who's again now a postdoc at Caltech, he and Phil Harris and I studied this problem and found that indeed nonlinear mode coupling can be important in these systems and lead to a very efficient dissipation of mode energy. So efficient that you would expect the cue of the planet during this high eccentricity phase to be only around 10 or 100. So, so small, the efficiency so great that it very rapidly circularizes orbit on timescales of tens of thousands of years. And that timescale is short enough that it's then not surprising that Kepler didn't see any high eccentricity systems. The phase that they're in lasts for such a brief time given the sufficient dissipation that it's not surprising Kepler didn't detect any of these systems. Okay, so I'm not gonna have time here to go into tides in binary neutrosaur as I still wanna leave time for the seismology of red giants. I'll just briefly mention some of the key things that we've thought about and why tides might be important in these very interesting systems. So these are of course systems that the LIGO gravitational wave detector has observed. There's been two binary neutrosaur systems that the LIGO saw and to a first approximation you can treat these neutrosaur as point masses but since they're actually merging at some point in the in spiral tides are gonna be important. And the question is how does that impact the gravitational wave signal? So tides will speed up the rate of in spiral by some amount. They remove energy and angular momentum from the orbit on top of the loss of energy and angular momentum associated with the gravitational waves. And so that is a signal that in principle you can hope to see with gravitational waves. And if you can detect it, you can use it as a constraint on the properties of the neutron star and say something perhaps interesting about the equation of state in the core of the neutron star, which is significant unknown because the densities are several times the nucleus of terrestrial matter and we can't probe such high densities in the lab. And if we can use astrophysics of neutron stars to do it we learn something not just interesting astrophysically but about fundamental physics about the strong force and QCD. And so people have of course wondered about this problem but just like with the stellar and binaries and hot Jupiters that I talked about they've in large part ignored nonlinear effects. They've said, well, let's just do the linear problem. Let's also treat the neutron star as a normal fluid as opposed to a super fluid which we think they are because they're old and cold. And so in some studies that we've been doing we showed that both effects can be potentially important that super fluid effects might impact the tidal signature and the gravitational wave signal as well as nonlinear effects. But as I said, I'm not gonna have time here to go into those details but I'm happy to talk to anyone afterwards if they have questions. Instead, yeah, what I'd like to do for my remaining 10 minutes here so is to switch gears a little bit and talk about waves in isolated stars stars that are not in a binary so the waves are not excited because of tides but instead are excited due to internal motions within the star. For example, convective motions within the star. And so in the 1960s, astronomers made the unexpected discovery that our own sun is awash in sound waves. The photosphere of the sun is just oscillating in sound waves. This is an actual image of a small patch of their sun taken by the Stanford Solar Telescope. And those variations are basically sound waves that are excited by turbulent motions in the convective zone of the star. So again, the outer third of the sun is convective and these sort of bubbling eddies excite sound waves that we see at the surface. The characteristic time scale of those oscillations is minutes. There's what's the famous five minute oscillation of the sun which is the characteristic period of these oscillations. So this movie that I just showed is sped up significantly time would be kind of boring to watch a movie of something very on a five minute time scale but the idea is the same. But what you can then do is do a Fourier transform of those fluctuations and get a power spectrum. And this is what you find. So this is power as a function of frequency and you see individual spikes at specific frequencies in the power spectrum. And those frequencies correspond to oscillation modes of the sun. So you're seeing the ringing of the sun much like an instrument has a characteristic modes of oscillation. You're seeing the characteristic modes of oscillation of our sun. And what's interesting about that by analyzing where those spikes occur and their frequency spacings, you can learn about the internal properties of the sun. So even though you're seeing the only the surface aspect of the waves, those waves propagate throughout the sun and their internal and the spectrum depends on how the material they're propagating through the internal properties of the sun. Very much like you can use earthquakes and the seismic measurements of earthquakes to learn about the crust of the earth, the mantle, the inner and outer core. You can do the same with the sun. You can basically see inside the star. And so for 50 years or so, the sun was really the only star where you can do such a measurement with significant precision. But about a decade ago, 15 years ago, this field really underwent a revolution. And that was in large part due to the Kepler space telescope. So Kepler was designed to see detected exoplanets by looking at for transits of these planets in front of the host star. And it needed really good photometry to do that. And one of the sort of secondary pieces of science that came out of that was that it could, it was really good at looking for luminosity fluctuations in isolated stars. So we can measure fluctuations of like a part per million in the stars, enough to see the waves, the impact of the waves on the brightness of the star. And so it made such astro-sizing measurements for tens of thousands of stars, including tens of thousands of red giants. So of course, a red giant is a star that has evolved off the main sequence. It's no longer burning hydrogen, it's core. And it's gotten really big and puffy. And it has a convection zone that excites these waves. And so just like with the sun, you can take a Fourier transform of the light curve of the stars and you see something similar. You see these spikes at specific frequencies corresponding to oscillation modes of the red giant. And so these kind of measurements for again, tens of thousands of red giants has allowed astronomers to probe the evolutionary state of these red giants, measure the rotation profile, which has interesting implications for the angular momentum evolution of these stars and also provided really powerful scaling relations that tell you from just any one star, it's mass, radius, and age with significant accuracy. I mean, if you think about it, when you look at a star, you really have no idea what its mass and radius is, right? But suddenly we have this way of measuring those properties for tens of thousands of stars across our galaxy. So this has impacted not just stellar physics, but also galactic physics. There's a field called galactic archeology that sort of underwent a revolution too because of these measurements. Because suddenly now you can age and measure the mass and composition of stars all across our galaxy with great precision. And so one of the things that Kepler saw, noticed, is that the amplitude of the oscillations of the red giants increases significantly the more evolved the red giant is. The more evolved the red giant is, the larger it is, it gets bigger and puffier, the older it gets. And the larger the observed amplitude of the oscillations more. So that's sort of shown schematically in the plot at the top, more quantitatively in the plot at the bottom, this is showing oscillation amplitude delta L over L, luminosity fluctuations in parts per million as a function of the luminosity to mass ratio of the star. And so as a reference point, here's the sun. So it has luminosity fluctuations of about a part per million. Whereas the red giants have luminosity fluctuations of tens to hundreds of parts per million. So the energy is that squared. So the energies of the modes in these red giants is thousands of times larger, tens of thousands of times larger than in the sun. And so people have wondered whether nonlinear effects are important in the sun and they're sort of maybe marginally important. But given that the energies are so much larger in these red giants, the question is that we were interested in is are nonlinear effects important in these systems? And do we need to account for them if we wanna fully understand what's going on in these stars? These are just red, so what I'm gonna be talking about is just red giants. But of course Kepler saw made astrocytes measurements for a wide variety of class of variables including AGM stars. Right, and so one of the sort of observational motivations for considering these nonlinear effects in these red giants is a mystery sometimes referred to as the suppressed dipole modes. So here are the power spectra of two red giants. They're both very similar in mass and radius and age. But what you can see, and so the color coding is a different degree of the oscillation mode. So black is L equals zero. So if you remember your spherical harmonics those are radial modes. L equals one is dipole modes. L equals two is quadruple. And so the one at the top shows power in the dipole modes and the L equals one modes. Whereas the one at the bottom shows very little power in the dipole modes. But otherwise these two stars are very similar. So it's kind of mysterious as to why one would show power in a dipole mode and the other one not. And so one of the explanations put forward is that maybe in the one that shows suppressed dipole modes there's significant, there's a large magnetic field. And that magnetic field through a process known as the magnetic greenhouse effect basically traps the energy of the dipole modes in the interior, scatters the mode, the waves and prevents them from having large amplitudes at their surface. And so if that's the right explanation that would be extremely interesting because it would be a way of measuring the magnetic fields in the cores of these red giants. We don't even know the magnetic field in the interior of the sun, right? And suddenly this is saying you can measure magnetic fields in these thousands of red giants across our galaxy. But one question we were wondering is, well, what if it's not some sort of magnetic effect but maybe it's nonlinear mode dampening, maybe nonlinear interactions are suppressing the modes in some of these stars and that's the true explanation for the suppressed dipole modes in these systems. And so we basically did an analysis very similar to the one we did for the hot Jupiter problem. We showed first that the modes in these red giants have such large energies that they would excite secondary modes. Parent waves have such large energies that they excite daughter waves which excite granddaughters. So this is again a plot of energy versus time. So the black line is a parent wave who's being directly driven by turbulent motions in the convective zone of the star. And it has such large energy that it excites daughter waves in red. Those daughter waves reach large energies, excite granddaughters waves. And again, you have this cascade of modes excited. And so we analyzed these systems, these mode networks for a range of red giant masses and a range of evolutionary states and just sort of to cut to the chase of what we found. So you can ask the question by how much is the energy of the modes suppressed due to the nonlinear interactions compared to the linear theory predictions? So you can talk in terms of the suppression factor of the energy of the modes due to nonlinear damping. And so the different colors here are for red giants all of a two solar mass but at different stages of their evolution. So the least evolved is in blue. And the suppression factor for the least evolved is pretty minimal, maybe 10% or so due to nonlinear damping. But as you consider more and more evolved red giants the suppression factor can be significant. And it could be as much as 80 or 90% for the most evolved red giants that Kepler sees. So the implication is that these nonlinear effects are significantly damping the modes in the star. The question then of course is does that explain the suppressed dipole modes? And so we found that it maybe can explain the suppressed dipole modes in the more evolved red giants but in the less evolved red giants where you do still see suppressed dipole modes it doesn't seem to work. It doesn't seem to explain the detection of suppressed dipole modes in less evolved red giants. That said, I think there's still outstanding questions that remain about the role of nonlinear effects in these systems. So we focused on this parametric instability that I talked about earlier. There's other types of nonlinear interactions. There's interactions in which you could have two parent waves exciting a daughter wave of twice the parent wave frequency what you might call direct nonlinear forcing. And so with my current graduate student at UTA, Mohamed Morabit, we're planning on studying exactly that problem and addressing whether maybe that effect can explain the suppressed dipole modes. Okay, so to conclude, I think one of the takeaway messages is that the linear approximation is often not a good one. It helps simplify the problem but is not a good enough motivation for assuming it. And that's true in a wide variety of binaries. And when you started counting for these nonlinear effects we're finding that they can significantly enhance the rate of tidal dissipation. So this has implications for the formal formation and evolution of hot Jupiters and also for compact object binaries like binary neutron stars. And then finally, towards the end, I mentioned the role of nonlinear effects on the amplitudes, line widths and frequencies in astro-sizing measurements of evolved stars like red giants. And so this is something that we've been studying now for almost 10 years and continue to study because I think there's a lot of open questions that remain about high X and 50 migration and different types of systems, not just planetary systems, the role of tides in binary white dwarfs which are important sources for future gravitational wave missions like the Lisa space mission and other types of pulsations. So not just red giants, but Cepheids are a Lyrae Delta study, all of which have been detected with Kepler are continuing to be detected with tests. And I think one of the really important takeaways too is that this is a field with an amazing set of data. There's just a wealth of really great data that exists from Kepler and Tess and in LIGO and has a great future. There's gonna be future missions like PLATO which is a future planetary detection but also astro seismic measurement mission and Lisa, this gravitational wave detector in space and future versions of ground-based gravitational wave detectors like the Cosmic Explorer. So I think this is gonna be a really exciting area for many years to come. So I will stop there and thank you for your attention and happy to take questions. Oh, yes, he put it here. I don't know if you know how to think. Oh, I think I may have to turn this off, right? Okay, oh, there we go, yeah. So this might be a bit of a basic question but you have all these nonlinear effects. I presume you calculate all these just numerically or is there analytical solutions and how many orders, I mean, what's sort of the largest order that you need to get to before you can start neglecting them? Question, so you can get analytic solutions as long as you restrict yourself to like three mode systems. You can't assume for these systems because there are so many modes excited. So you really have to just solve them numerically. So we solve a large set of coupled amplitude equations. Let's see, I may have a, so here's a backup slide showing an example of an amplitude equation. So A is the amplitude of the modes. And so the first three terms are, what you might identify as a damped oscillator. And then there's the driving term, which if you include that one, then you have a driven damped harmonic oscillator. And then the last term is this way of coupling. So you have many of these equations that we're solving those all simultaneously. Basically, it's just a large set of coupled ODEs that you're solving on a cluster. And then your next question was, I think, how many terms in the expansion of your perturbation theory do you need to go to? And that's also a great question. So you can construct arguments to show that the next term in the perturbation theory, the cubic term, is unlikely to be important in many cases, but that's not always the case. So for example, in the neutral star binary problem, it turns out that the next term in the perturbation expansion can also be significant. And so really, the only definitive way of showing that is to include the next nonlinear term and see how much is that contributing compared to the previous term, right? But for the most part, as long as you're not strongly nonlinear, you can truncate at a second or third one. Okay, can you go back to your last slide? So I was gonna ask this question anyways, but you sort of led into it. Do you, we have the students and are actually here that we just talked about some of these objects, the RLI res and the Delta Scooties. Do you have any sense for what the effects will be for those? They're very different, obviously, than neutron stars or the red giants. The densities are a lot higher. Delta Scooties have a lot of non-radial modes. Like what do you have a sense for what the impact might be? So we're actually, we're starting with the Delta Scooties because there's actually, there is observations, some groups from UT Austin that show evidence of nonlinear mode coupling systems. That show evidence of nonlinear mode coupling. So you get basically a wave and you have pairs of waves whose frequencies add up to that other wave. And so, and so there's sort of that motivation, observational motivation for Delta Scooties. For Cepheids and RLI, I think it's entirely plausible that nonlinear effects are born. In fact, we know that they're important. The, you know, the radial expansion of these stars is a significant fraction of their actual radius. And people who do, you know, the theory, the simulations of these systems have to account for the fully nonlinear fluid effects. But those are kind of large scale waves, sort of wavelengths of order the size of the star. I think there's interesting questions about whether those waves excite shorter wavelength waves and what role those play in the dissipation of wave energy. But yeah, we haven't gotten to those yet, but I hope to one day. Thank you.