Planet Migration in Protoplanetary Disks (Lecture 1) by Alessandro Morbidelli

Añadido: 2026-05-12

[00:00:16]Uh so with this uh we will uh uh formally start the program. So just one quick announcement like so as many of you would have seen in the schedule this first week we have uh series of pedagogical lectures. Uh we also have posters uh which has to be put up. You all would have received an email uh on that. Uh so please do put up the posters and this evening we have these flash talks of posters where you can just in one minute um introduce your posters right so please do send your poster slides to work by 2 p.m. Okay. Uh with that uh the first uh integration lecture uh professor Alexandra Moideli uh all of you might have heard of the NI model.

[00:01:03]You know he's the world expert on know how our solar system formed and ended up in what we have in the system. I will not take much time. So you know let's start with the lecture series. Uh professor Moelli.

[00:01:18]>> Thank you.

[00:01:25]of the lectures we have allotted one and a half hours but feel free to ask question we have enough time in between also during the session to ask questions so please keep it as interactive as you want >> okay thank you can you hear me yes okay thanks for the invitation it's a honor to be here to this school I'm impressed by the infrastructure the quality of this school really workass so okay So I've been asked to give two lectures.

[00:01:51]The first one will be about the migration of planets in protolanetary discs. The second one tomorrow will be about the effect of this migration on the orbital architecture of planetary system capturing resonance formation resonance chains global instability.

[00:02:09]Let's start from the the very fundamentals why planets should migrate in this.

[00:02:19]Okay, the the key point is that if we embed a planet in a disk, the planet perturbs the distribution of gas in the disk. So if initially you have a disc with a uniform density which is represented here by this uniform blue color, the moment you inject you insert the planet in the disc, the disc create the planet create a spiral wave where the gas density is enhanced. You see here that the color is bright white which means higher gustings. So this is what we call a spiral density wave and the generation of this wave is what drives the migration of the planet in general towards the star. So the first question is why should the planet generate a wave? So uh h first this wave has as a clear pattern. So the discs around the star turn in a capillarian fashion. So are are characterized [clears throat] by what is called the caparian shear. So that means that we respect the motion of the planet. The disc inside the planet rotates faster and the disc outside the planet rotates more slowly. So in a reference frame corrotating with the planet, the fluid elements inside the orbit of the planet move this way. The fluid elements outside the orbit of the planet move that way and so the wave is bended in the direction of the motion of the gas relative to the planet. But why there should be a wave in first place?

[00:03:58]So it's very simple. You have a planet again we are in a reference frame rotating with the planet. This is the star. This is the radial direction. This is the direction. So the planet is fixed. You have a fluid element say in the little bit in the outer disc. So relative to the planet this fluid element comes in vertically in a circular orbit. So it moves this way with a relative velocity with respect to the planet.

[00:04:25]And because of the encounter of with the planet and the gravity of the planet the orbit of the fluid element is bended. So the way it works during an encounter is that the bending does not change the relative velocity vector just rotates the relative velocity. This velocity is the same as this velocity in absolute magnitude but it is rotated which means that now it has a radial component that it didn't have before and the asimutal component is necessarily reduced with respect to the initial one because the full vector is the same but now we have a radial component. So this means that this encounter has created a change in the zimutal velocity of the fluid element directed in this direction which means that the fluid element is accelerated along its orbit receives a positive force and if you accelerate something in orbit what happens happens that it goes out and so as a result of this encounter the the fluid element trajectory is bended it's acceler accelerated and then it has to go out and it goes out further at a larger distance from the star than its initial distance when it gains a distance delta R and this is true you can check it in hydrodnamical simulations. So this is again a radial direction in a frame corating with the planet and this is what is called the streamline. So the trajectory of a fluid element in the disc. So it comes in, it's accelerated by the planet, it dives towards the planet, goes out here. This is the delta R that I described before and then the internal forces of the disc bring it back to the original location, the stream lines are closed. But the the [clears throat] the what is important to note is is is this deflection of the fluid element due to the encounter with the as a result of this deflection you have a compression of the gas which create an over density here where the stream lines are from from this edge and this over density then propagates radially uh at the as a pressure wave at the speed of sound because the way gas communicate with itself is through pressure and so all density perturbations propagate as pressure waves at the sound speed. So this is the radial propagation of this compression that the gas suffers but at the same time the the gas is also in rotation relative to the planet because it's in the outer disc. And this is what generates the wave. So it's a combination of the radial propagation and the radial speed of the over density created by the encounter with the planet with the caparian ship. And this is what leads to.

[00:07:35]And now that we have understood why the wave is generated, we can try to understand why planet migrate. So the effect of the planet is to break the axis symmetric symmetry. the the the axial symmetry of the disc because now we have this wave which acts as an over density. So let's consider first okay no first okay before we go so let's see first the wave so we this is a disk uh with a uniform density inject the planet in in the simulation and you see that the planet generate the waves the planet rotates around the star and the wave which is launched by the planet moves at the same time as the planet around the stars Why we say that the wave is in cor rotation with the planet around the star and this means that if we go in a rotating reference frame so that the planet is fixed and we don't see it turning around the star anymore then the wave is stationed in a rotating reference frame.

[00:08:47]So now we are armed to try to understand why the planet has to m as I said the planet breaks the axial symmetry in the disc because there is this over density which is doesn't have an axial. So let's consider first the outer part of the disc. So the disc beyond the location of the planet. What does the planet see?

[00:09:06]Rather uniform disc plus an over density which is no over density by gravity exerts a force on the planet. Right?

[00:09:17]Because the density the the wave trains the planet. The force that this wave will ex this over density will exert on the planet is a force directed like this. So it's a force that opposes to the motion of the planet around the star. So exerts a negative torque on the planet which means it slows the planet down and tend to if it slows the planet down the planet will lose energy and angular momentum and will spiral towards the star. But then we have the inner disc. In the inner disc it's the opposite, right? the density wave needs the planet and so the gravitational force that it exerts on the planet will be directed in this direction. So it will push the planet along its orbit will exert a positive torque on the planet. So we try to accelerate the planet and if you accelerate the planet will push it away from the star. So you have two opposite effects. The outer part of the wave is slowing down the planet and forcing inward migration. The inner part of the wave is accelerating the planet and forcing outward migration. So the question is w and in general it is the outer wave that wins and the reason is that the disc is slightly sub in subcapillarian rotation because the disc has an internal pressure. It feels a pressure force directed from the sun out which subtracts from the solar gravity. So to be in equilibrium on circular orbits around the sun, the gas rotates around the sun a little bit more slowly than the capillarian.

[00:11:02]So if the this were not the case, if the disc were in caparian rotation, the dynamics of the gas relative to the planet, the planet is here. Here you see the steam lines and the arrow showing the direction of the motion of the gas will be perfectly symmetric.

[00:11:17]But even if uh if the disc has a uniform density it is subilarium and so the the dynamics that that occur is is like this. So first let me show why the disc is subcon because it may not be obvious right the the pressure is uh the product of the sound speed times the density of the disc and the sound speed is the scale height of the squared sorry times the density of the disc and the sound speed is the product of the scale height of the disk times the capillarium frequency and row is the disc.

[00:12:00]Now uh the sc say the scale height of the disc is more or less proportional to r omega is proportional to 1 / r to the power three half. So the the product of these two is proportional to 1 / r. And if you have a flat disc, so the surface density of the disc is the same at every radius. The density decays as one / r because the scale height increases. And so this the pressure is typically even for a flat disc. Typically this are not flat. They also decay. But it's at least like sigma. It's like sigma divided by r². And so the pressure gradient is negative from the from the sun out. And this is why the disc is in subarian. So being in subarian rotation means that the corotation region is made by the streamlines that make a Uturn at the location of the planet. This corotation region is not located at the planet location but is shifted inward because the disc is moving a little bit more slowly. So to move at the same speed of the planet, you cannot be at the distance of the planet from the star. You have to be a little bit closer to the star.

[00:13:10]And so at the planet location there is a net flux of gas that goes in this way.

[00:13:16]And the the gas moving upwards is much further from the planet than the gas moving downwards.

[00:13:24]Which means that the interaction with the planet is much stronger for this gas going downward and for this gas going upward. And so the wave that will be generated by the encounter of these downgoing streamlines will be much stronger than the wave generated by the encounter with the planet with these upstream uh streamlines. And so this wave is will win will exert a bigger torque and this exerts a negative torque. O drives the migration of the planet towards uh [clears throat] so because the over density of the wave is proportional to the mass of the planet because the planet is the perturb the density of the wave is proportional to the mass of the planet and to the overall density of the gas. It's a fraction the overall density is a fraction of the density gas. Then the planet migration speed is to be proportional to these two quantities.

[00:14:26]migration of a planet is linearly proportional to its own mass the mass of the disc.

[00:14:32]So I just said that it's typically inwards because this branch of the wave winds. But then somebody may say okay but let's take a steep disc a disc with a steep surface density profile like this right and and then I will have much more much higher density inside than outside and because the torque is proportional to the density of the disc maybe the inner wave will start to win but this is not the case because of a process called the pressure So if you have a disc with a given density as a function of r here this is the location of the planet. Uh the the same disc has a given rotation curve. So this is the rotation frequency of the disc as a function of distance and the location of the planet the the frequency the capillarian frequency is this one because the disc is sub kaparian. Once again at the location of the planet, the disc is rotating less fast and it's in corotation with the planet closer to the star. If I take now a steeper disc so to enhance the density in the inner part so to hope that the inner wave inner branch of the wave can win. What happens is that the disc becomes even more subillary.

[00:15:56]So the corotation region becomes even further from the star which weakens the inner part of the wave even more. And so at the end you can make this disc as steep as you want and you actually never make the inner wave win. On the contrary, the inner wave becomes weaker and weaker as the disc becomes steeper because the disc becomes more and more sub and the corotation region sh and uh indeed one can demonstrate that the net torque on the planet which is negative okay it becomes even more negative with increasing alpha and alpha is the power law here of the surface density. this to be. So the steeper is the disc, the higher is alpha, the more negative that is the to. So it's the opposite of what one would naively expect that the steeper disc because it has more gas inside will eventually drive the planet out. Actually the steeper is the disc. The faster is in more vibration.

[00:17:04]Uh few things about the properties of the wave. So the wave is an over density in the gas. So to create an over density you have to fight the internal pressure of the gas. So the higher is the scale height of the disk the stronger is the pressure. So the smaller is the over density in the w and also the higher is the scale height in the disc or the aspect ratio the faster is the sound speed. So that means that in a cyanotic period which is a a cycle of the wave the the pressure the the over density pulse has traveled faster because it travels at the speed of of the sound. And so the the the wave will be less red.

[00:17:51]And now I will show an animation showing how the wave changes with the aspect ratio of the disc increasing. Of course a disc has a given aspect ratio. This does not change the aspect ratio. But in a simulation, you know, you have a planet within a given disc and then you change in the simulation the aspect ratio of the disc to see how the wave changes. And this is how it goes. So as you can see with increasing aspect ratio of the disc, the wave becomes more and more pale, less pronounced, and it's also much less white than it was before.

[00:18:26]So one can demonstrate that as a consequence the torque felt by the planet decays with the aspect ratio in the disc. The higher is the aspect ratio the weaker is the torque because the weaker is the wave and it's actually proportional to h over r. So the aspect ratio of the disc. The power minus. So thinner disc exert a faster migration on the planet than thicker disc.

[00:18:56]So we can summarize this initial part of what is called type one migration. Type one migration is the theory for planets that perturb the disk but not enough to change the overall radial density distribution.

[00:19:12]Yes.

[00:19:20]Then uh as the the pressure the density propagate out by pressure it tends to be moved away by viscosity and and so after you know it's a competition between viscosity that tends to spread the disk and the the sound speed probably out. So in an invasive disc the wave goes very far out. In a more visitive disc you see just the first two.

[00:19:50]So uh type one migration concerns the planets like those that we have seen that generates a wave but don't open a gap in the distance. Then I will talk about gap opening a bit later. So planet migration as we said is proportional to the mass of the planet. That's the main property of type one migration. So the bigger is the planet the faster it moves. It's proportional to the surface density of the disc. The more mass is the disc more faster the planet moves.

[00:20:17]But it's also the migration the speed of migration it's it's inverse of the square of the dis aspect ratio.

[00:20:26]So this is the general formula right the change in specific angular momentum of the planet. The specific angular momentum is the square root of the location of the planet is the the angular momentum per unit mass of the planet. So it's general negative. So there is this coefficient that becomes stronger and stronger the steeper is the disc. Alpha is the exponent of the power law of this proportional to the mass of the planet. Then you can normalize to the mass of the star or mass of star appear where this here for dimensional reasons. is proportional to h r minus 2. So r power two sigma and it's proportional to the location of the planet the power four and the frequency uh rotation frequency orital frequency at the power two. So it applies only for relatively small planets that don't change the global structure of the disc. And as you can see in this formula there is no mention of the viscosity. Prove the viscosity uh matters for final damping of the wave. But the the what drives the migration is the interaction between the planet and the beginning of the wave.

[00:21:41]Gravity of course proportional to one distance square. So what matters is the wave close to the planet. So how far the wave can propagate is not really important in the second order effect on the migration form.

[00:21:57]So in this video we see uh nicely uh an example of migration. So it's in a rotating frame with the planet. So you don't see the planet turning around the star but you see the planet moving. And in this plot you see the distance of the planet from the star as a function of time as the movie is being played. So as you can see migration is a fairly linear function. So the planet decays towards the star and relatively quickly. So if you apply the formula I gave before for a planet of one earth mass at 1 AU in a fairly typical disc what we call the minimum mass solar nebula disc a reference this that everybody uses in the community doesn't mean that the disc was exactly this one but uh it's it's a good reference so the density at 1 AU would be 1,7 of gas 1,700 g per square cm. mters and the aspect ratio would be 5%.

[00:23:01]So then the migration time scale would be only 200,000 years. That means that a dot over a is 200 one over 200,000. Basically the planet would move towards the star in in in on this time scale. It's very short. H this lifetime is of the order of a few million years. So there is ample time for planets already of an earth mass to lose them into the star if something else happens. If nothing else happens, we see later what could be uh reasons possibilities to slow down migration or track migration. And because of the formula I gave you, you can use this number and scale it up, scale it right easily. So for instance when you wonder what is the migration time scale of a planet that is not one earth mass but five earth masses okay in a disk which is three times more massive and two twice thicker then we know how things scale so it scales linearly with the mass so we take our 200,000 years migration scales linearly with the mass so we divide by five this case linearly with the disk and the disk is three times more massive so we divide eight, but it's inversively proportional to the square of the thickness. The thickness is double. So, we multiply by four. Then you take this product and it to find 53,000 years. So, this is very useful formula, right? So, once you remember these numbers, they're very easy easy to scale to any kind of problem you have.

[00:24:48]Now the wave not only forces the planet to migrate towards the star but also damps the eccentricity and the inclination of all we not speak about the inclination about the eccentricity of the planet. Why is that? So if you have a planet on an eccentric orbit in a frame cor rotating with the mean orbital speed of the planet so that your frame makes one revolution as the planet makes one revolution your planet is no longer stationary because it's on an eccentric orbit. So your planet will have a radial excursion but also the asimutal speed changes. planet is faster at perilion and slower at a. So your planet draws an a cycle in this rotating frame like this faster per lower and has a radial distortion. creates this cycle and uh so what happens is that the planet moves faster than the local gas and at aon it moves more slowly than the local gas.

[00:25:59]So we can take the same figures as before. So because at Perinion the planet moves faster than the local gas the local the gas at the planet location will move from top down. Okay.

[00:26:14]And for this reason as before the the lower part of the branch will be much of the wave will be much stronger than the upper part of the wave. So that means that this planet is slowed down at perilion. But at theion we can rotate the figure. The planet is moving faster than is moving slow more slowly than the gas. Sorry. So the the gas at the location of the planet in the rotating reference frame is moving from bottom up upwards. And so the wave will become stronger here at the top than here at the bottom. So in this case there is a positive torque exerted on the planet at aion which speeds the planet up. So we have a body on a on an eccentric orbit which is slow down at perilion and speeded up at aion. This is what we do for space navigation. So what happens in this case the eccentricity is damped slow because when you are on an eccentric orbit you move faster at perilion and slower at aion. If you slow down at perilion and accelerated you polarize the to show that this is what happens I show the simulation but I need to stop it move here.

[00:27:41]Okay. So, the here is the disk and you see my mouse. You see the mouse or not?

[00:27:51]You don't see the mouse. So, okay. Uh, so you see this is the usual density of the disc. This is the planet and this is the P cycle of the planet. And this is the way that the planet launches. Look at what happens as the planet goes around its AP cycle.

[00:28:08]Okay. So here it just passed through per and you see where the wave is pronounced. It's pronounced behind the planet as I said. So here the entire gravitational source is pulling the planet back. So he's slowing down the planet.

[00:28:26]And if uh we wait up in orbit up here we go the planet is aion. And now you see it's like a comet. planet has an A and you have this the the strong part of the spiral density wave is directed in this direction. So exert a force that tries to accelerate the planet or accelerate the planet.

[00:28:51]So uh the eccentricity so this is why the eccentricity is there the eccentricity denting is proportional to the velocity of the planet relative to the disk because of the dynamics that I showed and and so it's proportional to the eccentricity itself. So the decay rate of the eccentricity is proportional to the eccentricity.

[00:29:14]It's still like before. So it's the the intensity of the wave. So the damping rate is proportional to the mass of the planet and the density of the disc and it has a stronger sensitivity on the scale height of the disc is a little bit complicated to show why this so uh so instead of being proportional to h r to the power minus2 like the torque exerting migration it's actually proportional to h / r to the power minus4 so eccentricity damping is much faster than migration that's what it means H r is a small number 2%. So the eccentricity ding is much faster than migration and so we can write it as e over and so what we have seen before the the what the formula I gave before for the the change in the specific angular momentum of the planet is still valid apart from small corrections and eccentricity. So we can say can apply the same formula to to to the change in angular momentum of the disc but because the angular moment the angular momentum of the planet but because the angular momentum of the planet depends on the sage axis and the eccentricity of the planet the actual damping of the angular momentum which is given by the formula I gave before. It's actually a combination of the A damping and the E damping.

[00:30:46]And the E damping is is given here. The the A the angular momentum damping was given by the formula I gave before. And so the change in semi- major axis is actually given by this. So it's the usual rate in case the planet is circular. But if the planet is eccentric, there is this term that uh changes makes the semi- major axis decay even faster. And this is very important for the resonance dynamics that we will see tomorrow. And it's the key of a process that we call overstability of the motion resonance. So when a planet is eccentric it decays towards the planet faster but the eccentricity decays also extremely faster. Then the planet circularizes and migrates to towards the star at the usual speed.

[00:31:42]Now all I've said so far concerns planets that are small enough not to change the global structure of the disk.

[00:31:52]That is they create this weight but the density overall density in the disc remains the same. But this cannot be true forever. If the planets are massive enough they have to exert a a a stronger effect on the disc. Why? Because we've seen before that the the the wave which is an over density in the disc exerts a gravitational force on the planet as you know from Newton principle that is the action reaction principle. So if the outer wave exert a force on the planet the planet exert the force on the wave.

[00:32:28]So the planet exert a force on the wave directed in this way. So exerts a positive torque on the outer disc and exerts a force on the inner wave directed this way. So tries to slow down the inner disc exerts a negative torque on the inner disc. So which means that the planet tries to push the outer disc out and to slow down the inner disc and so to push the inner disc.

[00:32:58]Of course, the disc has internal forces due to viscosity, due to pressure that try to oppose to this force of the planet. But if the planet is big enough, it can overcome the internal forces of the disc and then a gap can open because indeed the planet manages to force the push the outer disc outward and the inner disc inwards creating a depletion in the gas density along its own.

[00:33:28]This is what we are going to see in this movie. So we are going to see a planet that is inserted in the disc and initially it's a lumass planet. So it just generates a sort of a wave but then in the simulation the planet is grown artificially fast or it doesn't last forever in an unphysical way but just to see the effect and then you will see that as the planet grows in mass it the global surface density of the disc changes. So here we go the frame I think is a corotating frame with the planet.

[00:34:02]So here the planet with its wave and you see that over time the gas along the orbit of the planet is being removed and at the end. So we have a figure which is very different from the one I showed before. We have not only the spiral density wave but we have this big gap along the planet's orbit.

[00:34:25]This is only present at a given distance from the planet outside the the central region which is black.

[00:34:33]It's not physical. It's just that in the idle dynamical simulation if you want to go too close to the star of course we have numerical issues because the frequency orbital frequency becomes so fast. So if you want to make any simulation on a meaningful time scale of hundreds of orbits, you have to limit your this to a patch that doesn't go too close to the star. So this cavity is not physical. It's just a numerical one before. But this gap is physical. It's the one that is okay. So uh can we estimate how big is the gap? In principle, it's easy because we can take our uh formula for the torque. So the same torque that the wave was acting on the planet, the planet is acting on the wave. So this is the torque that the planet acts on a disc on a ring of the disc located at the distance delta from the location of the planet RP.

[00:35:34]So it's you divide your disc in concentric rings and for each ring so it's characterized by a given distance delta and you compute the torque exerted by the planet through the wave on that ring at distance delta.

[00:35:50]At the same time the disc has internal forces particularly due to viscosity and so the as the surface density tends to change the disc exerts a torque on itself which is given by this formula which then depends on the viscosity.

[00:36:07]There is no viscosity on the disc.

[00:36:09]Nothing happens. But it depends on the the gradient of the surface density of the viscosity tends to smooth the disc out and to prevent large gradients of the disk density to be established. So as the planet starts to carve a gap, this gradients become steeper and steeper, right? Because you you have a density at the beginning.

[00:36:35]of the disc like this. So relatively smooth gradient but the planet start to open a gap and so the gradient becomes very very high here and here and so the viscous torque becomes stronger and stronger. So you have a competition the planet torque right to push the gas away but as you push the gas away makes the gradient steeper and this enhance the viscous torque. Okay, so at some point you may hope to reach an equilibrium and when the viscous torque and the gravitational torque are equal to each other and so well you do some math which here will take too long and you can solve this equation. So this equal to this for each value of deltas.

[00:37:23]So for each concentric ring in the disk to compute an equilibrium gap profile of the density as a function of delta which is the distance from the and this is what has been done for instance in this paper uh in 2004. It works pretty well for discs that have a strong viscosity and the formula fails when the viscosity is weak because of course the viscosity is weak the the coefficient in front. So this torque becomes very weak and so this formula would predict gigantic gaps very deep and gigantic gaps for small viscosity and even infinite gaps for vanishing viscosity which is not really not really easy. What happens is that the viscosity is weak. So this gradient continuously grow grows right. But at some point if it grows too much the disc becomes unstable to be stable the disk has to satisfy the rail condition which is this one. What is the rail condition? Is simply the requirement that the radial gradient of the specific angular momentum of the disk is negative.

[00:38:43]And uh so basically it means that uh uh yeah so the the so so the radial gra sorry the radial gradient or the specific angular momentum of the disc has to be positive for the disc to be stable. So that means the angular moment of the disc grows in distance from the sun.

[00:39:07]As soon be because because this gradient of the specific angular momentum of the disc is linked to the gradient of the density.

[00:39:20]If the gradient of the density becomes too steep because there is the torque of the planet which is not counterbalanced by the viscosity then at some point the gradient of uh of the angular momentum in the disc becomes negative and then the relay condition is not valid anymore. The disc becomes unstable and what does it mean that the disc becomes unstable is that the the disc develops turbulence which in in practice enhances the disc of the disc. Initially you had a known viscous disc or a low viscosity disc. But if the planet turns it unstable the disc become turbulent and becomes viscous and so this limits the opening of the gap. So there is a maximum slope that the gaps edge can have and these are the slope that are at the limit of the rail instability where the slope given by the gradient of the specific angular momentum of the dist being zero. This is the work that has been published by kanagawal in 2015.

[00:40:25]And so this is what happens in reality in numerical simulation. Take a planet with the mass of Jupiter and let's study the surface density of the disc uh around its orbit for various values of the viscosity.

[00:40:43]So if the viscosity so this is minus log of the viscosity. So red would be relatively high viscosity and 7.5 is a very weak viscosity that is okay in in China. So you see that this is the gas surface density profile normalized to the initial one. So the all effects of gradients have been removed by the normalization and you see that if the viscosity is high the gap is relatively shallow. So it's not even a factor of 10 right the depletion of the gas at the planet location even if this planet is Jupiter because the viscosity is so high the density is just reduced by a factor of five but then if you remove if you reduce the viscosity in the simulation you see that the gap becomes deeper as expected in varnier in varnier paper but then as you remove the decrease the viscosity even further into the limit of the invisitive disc.

[00:41:44]You see that the gap becomes deeper, a bit wider, but not indefinitely deep and indefinitely wide. There is indeed a limit that and the limit width of the gap which is set by the rail in the opening of a gap has a big effect on migration and this is illustrated in this plot. in this op in this movie. So it's a movie like the one before. So in the rotating frame we see the planet that migrates. We project the distance as a function of time. The planet at the same time is grow is uh is opening a gap and so you will see the effect of gap opening as as clearly.

[00:42:32]So go so the planet initially is moving by type one migration. The same time it's opening the gap. Now the gap is open and you see that its migration is still inward but slows down considerably.

[00:42:48]So the fact of gap opening slows down migration does not block migration does not reverse migration but slows down migrations.

[00:42:57]Giant planets migrate more slowly than small planets. So we have this uh situation that in type one migration the fast the bigger is a planet the faster it moves but this does not go on forever. If the planet is too massive it starts to open a gap and then migration slows down. So in some sense the migration rate as a function of the mass of the planet is something like this more or less.

[00:43:31]So this is the type one branch you will see goes to some we go to some migration rate independent of the mass of the planet that we the type two branch and this is sort of transition.

[00:43:58]So why is a planet with a gap forced to move more slowly?

[00:44:05]The reason is that now we are in a situation like this. The planet has opened a deep gap. If the gap is purious just a partial gap, we are in the transitional region which is very messy.

[00:44:19]We show at the very end. But let's make the case clear. The gap is very deep. So the planet has effectively divided the disc in two and there is no gas in between. There is an inner disc and there is an outer disc and the planet is there. So what can the planet do? Okay, the planet is you you could say, okay, it wants to move, but if it goes too close to the inner disc of is is it feels a a a strong force from the inner disc because it approaches the inner disc and the inner disc where the wave is going upwards gives a positive torque. So the planet cannot approach the inner disc because it will be repelled out. And similarly the planet cannot go towards the outer disc because the outer disc the wave takes away angular momentum from the planet and move the planet inward. So the planet is condemned to be in the middle of the gap not go into the inner dis.

[00:45:25]So that means that if the planet want to move wants to move towards the star the entire disc has to move towards the star together with the gap. So the planet cannot move relative to the gap repelled by the edges of the gap.

[00:45:41]So at what speed can the gap move towards the star? It can move only at the speed that the viscous radial speed of the gas because the disc has some viscosity. The gas is drifting towards the star is spiraling towards the star.

[00:45:59]That's why the protolanetary discs are also called accretion discs. They bring gas to the star and the radial motion of the gas is equal it's a famous formula to minus 3 half positive mu over r. So this will be the speed at which the gap can move towards the star. The planet is condemned to live in the middle of its gap. Prisoner of the gap created itself in jail. And so the planet migration will be the same as the migration of gas in the disk will be minus half mu / r.

[00:46:37]And this is the famous formula of type two migration produced by bill word back in 1997.

[00:46:45]And what it is important is that in this simple vision of migration, so the migration speed for a massive planet in type two becomes independent on the mass of the planet and even independent on the density of the immediately you understand that this is sort of a syntoic formula that cannot be absolutely correct because makes no sense physically that the migration is independent of the mass of the distance.

[00:47:13]you have a very thin disc which with a negligible mass the planet will still move at the same the same speed. So of course this formula has some limitations but the first order for a massive disc and a massive planet this is what what happens and it's because the planet is blocked in in the in the gap and so it's locked in the viscous evolution.

[00:47:35]But if you look a little bit broader of course things are a little bit more complicated. This is a result of a series of syn numerical simulations. I forgot the citations but a number of people did this from duffel uh with will group on h. So this shows the migration rate of the planet uh normalized to the the capillarian speed okay as a function of this quantity. This quantity is the ratio between the mass of the disc interior to the orbit of the planet sigma p^ squ about the mass of the disc interior to the orbit of the planet and the mass of the planet.

[00:48:20]Basically, it's the ratio between the density in the disc and the mass of the planet the the the number a dimensional multiply by R² and uh and this is the measure migration rate and this is the viscous drift rate the famous minus half m.

[00:48:41]So as you can see the planet does not move exactly that speed not too far in the limit of a massive disc relative to the planet. So the planet is massive to open a gap but the disc overall has a lot of mass. Indeed the migration rate does not seem to depend on this quantity as predicted by the word formula only depends on viscosity and not on the mass of the planet nor sigma. And indeed we have this sort of asintotic behavior.

[00:49:13]This asintotic behavior curiously the migration rate is not exactly the viscous drift rate is a factor of two higher and this is because the viscous motion of the outer disc goes a bit faster because the the existence of the grad of the gap has changed the gradients in the disc but it's a little ch little on the other hand if you go towards the low disk mass so sigma is small compared to the planet then of Of course, a light disc cannot move the planet at the same speed. And so you go into this branch where migration is proportional to this ratio and linearly proportional to this ratio.

[00:49:56]And only the coefficient depends on this column. So a viscous disc will have a a larger coefficient. So a steeper dependence than the low viscosity. But of course there is this this aspic range and this is called the inertial limit or a very massive planet in a light m in a light disc has to slow down. For instance, when Jupiter formed, probably Jupiter formed, the disc was still massive, was moving fast. But as the gas is removed from the disc, for instance, in photo evaporation, by photo evaporation, the planet has lost.

[00:50:38]Okay, so far so good. So I explained the classic vision of type one and type two migration.

[00:50:45]But to have a more complete view of how the planets evolve and be able to understand their resonant dynamics tomorrow, we need to go a little bit into some computation.

[00:50:59]So let's go back to small planets, planets that don't open gaps. Okay?

[00:51:03][clears throat] So they have gas everywhere also along their orbit. And because they have gas along their orbit, uh we need to worry about what the gas along the orbit is doing which creates a torque which is called the core orbital torque that we neglected up to now just by considering the spiral bas. So you have a fluid element that is share the same orbit of a planet and what does this do? It does this kind of uh motion relative to the planet again in a rotating plane with the planet. So this is what is called a horseshoe motion because it draws these curves which look like horseshoe and what happens is that these elements so vibrate back and forth and when one element comes here encounters the planet and goes from the outer part of the horseshoe to the inside part of the horseshoe. So doing this uturn it uh loses angular momentum. So it gives it to the planet exerts a positive torque to the planet. But of course when it the opposite occurs and this other U-turn and the fluid element goes from inside of the shoe to the outside of the shoe.

[00:52:21]The fluid element gains angular momentum and takes away angular momentum from the plant.

[00:52:29]So in principle you would say okay this equ equilibrium and one effect we cancel the other one. Yes. And so to understand that let's change of coordinates. So let's go from this system of coordinates which is the polar preference frame with the radius and the aim to this system of coordinates where we introduce simply a radial direction and the nasimutal direction and analyze what happens in this coordinate much more clear and this is what we get. So this is the other direction this is direction the star is in that direction the planet is here are the same [clears throat] and these are the ocean project.

[00:53:19]So let's take a fluid element that makes a U-turn and this fluid element is characterized by a given delta R in the radial direction and a given delta V in the aimutal direction. So there is a theorem in mechanics it's theorem that says that the canonical volume which is the product of DV times DJ where J is the specific angular momentum is conserved. So in doing this U-turn this quantity is conserved.

[00:53:53]Now in doing this U-turn d is conserved because you have 180 degrees on this side and 180 degrees on this side. You cannot have compression or militation in the field in the F direction.

[00:54:06]But because the angular momentum specific angular momentum is the square root of the radius then a change in in radius is equal to the change in angular momentum times the square root of r.

[00:54:23]So con consequently when moving from R to R prime right because the interval in a specific angular momentum DJ is the same because DJ must be conserved because D must be conserved and the product must be conserved. So the the the interval of angular momentum in the fluid element is conserved. then the the radial uh extension of the fluid element has to expand by a factor square root of r prime / r. So this fluid element has the same extension t is stretched in r by this ratio square root of r prime over r.

[00:55:09]The other hand the volume the this the circle is still 2 pi but we are further from the sun. So in physical units this circumference is a factor of RP prime / R longer than this circumference.

[00:55:26]So this fluid element already is stretched by a square root of R prime / R. But then you have to distribute it over a longer circumference. So the the surface density here is smaller than the surface density of the gas here by a factor R prime / R. Again R prime is here, R is here to the power minus three hot and this is the key. So during the the orho uturn the fluid elements stretch and their surface density decreases when they go from inside out or compress and their surface density increases when they go from outside in.

[00:56:08]And so in other words this means that if the surface density of the disc across the horseshoe region changes as one / r to the power three half then you have the same mass per unit time encountering the planet in this direction or in this direction. So the torque is but if the surface density of the disc across the region is different from one of our rup is steeper or shallower then you have some the effects don't don't don't uh uh don't cancel and in particular if uh uh if the density the disc is steeper you have more mass doing this uturn and this U-turn so that means that the planet is accelerating ated towards the star and if the disc is shallower than this then you have more mass doing this U-turn than this U-turn and so the planet is actually slowed down.

[00:57:07]Uh so in particular if the density gradient is positive so it's not like 1 / r ^ 3 half but it's even a positive function of the distance then this coro orbital torque can become positive and very strong and can win over the negative torque exerted by the spiral density wave. So first of all is it realistic to have a positive gradient in the disk? It can because there are regions of the disk in particularly close to the inner part of the disc where gas is depleted. It can be depleted by magnetic effect related to the star and also be depleted by a change in viscosity because close to the star the disc becomes very hot. So becomes ionized. So it can ex become magnet rotationally unstable and become very viscous. So you can expect this positive surface gradient of the disc in the inner part of the disc. You can expect them also at the outer edge of a gap of a major planet which would be here. So you have a Jupiter gap another planet that comes and fills this positive gradient density gradient. So the positive density gradients can exist in the disc and so they exert this strong um corrotation torque that can stop the planet and this is a representation of what happens. So the planet migrate inwards reaches the branch where the the region where the surface density of the disc start to be positive and stops there with some oscillations that I will show in a second. So this is a cartisian representation of migration of planets of different masses. Each simulation it's only one planet. It's not all the planets simulated together. So there are no interactions among the planet in a same disc. And you see that the planets all stop at the same location. The location of what we call the planet trap is independent of the mass of the planet. So the planet go there very fast if they are massive slowly if they have a low mass because this is the type one migration rate. But they all stop at where the surface density of the disk becomes positive enough. And then you have these oscillations that you've seen in the movie as well. And this is because when you have a surface density that is like the one I I drew. So with the positive gradient and then a negative gradient, sometimes in some cases it's Rosby unstable. So it creates a vortex near the density maximum and then you have a gravitational interaction of the planet with the vortex and this is what is giving this giving giving this uh there is a caveat though that it's true that if I have more [clears throat] gas out that in in principle I have more elements doing this uturn so giving a positive to the planet than is in the other utter term.

[01:00:08]But by keeping doing this libration, the material should homogenize, right? And at least inside the libration region, we should redistribute the gas. So to achieve the famous 1 / r to the power three density gradient that cancels the corbital torque. And if this happens, this is called the saturation of the orbital torque. So the orbital torque is present at the beginning but after a number of librations then the surface density in the corital region has been changed to this one and there is no more rotation. So the only way the corotation torque can continuously exert its effect is that there is something counteracting this saturation. And so you need that the internal forces of the disc that establish this this positive surface density gradient that can come from magnetic effect from turbulence from winds from whatever they manage to they oppose to the fact that the local surface density across the corotation region wants to eat. the planet would like to change the surface density to something like this and you need that internal forces of the disc prevent the planet from doing so only if the disc is strong enough then the orbital torque is desaturated and continuously acting. So in an invisive disc well these forces do not exist and uh and and and so the rotation torque can be desaturated but maybe the density gradient is due not to viscosity but to magnetic fields and then you can have the saturation by the magnetic fields. Uh I mean it's the the bottom line is that whatever acts on the disc to create this positive surface density gradient has to be strong enough to prevent the planet from changing the slope in the coordinates.

[01:02:08]The orbital torque is important not only to stop planets at the edges of the disc but can also change significantly the migration of the planet as they move and this is called the dynamical corbital torque. So trying to illustrate this again in a bunch of movies. So we have again our scheme. This is the radial direction from the sun. The neutral direction from 0 to 2 pi. This is the planet 0 and 2 pi. And the planet is moving inwards. And look at what happens to this fluid element. Okay. So the fluid element initially will move upwards at a fixed location because this is what a fluid element does. So it's moving it's in inside the orbit of the planet. So moving faster than the planet. So from bottom up until the planet runs over this fluid element and then there will be a change in in location radial direction in radial distance of the fluid element and when the fluid element will go beyond the planet we start of course to move in the other direction because of the let's see we're moving up. Ah here it is the planet comes what did the fluid element do? It can be show it again.

[01:03:33]It Yes, there go. Okay. Enters the corotation zone, makes a one U-turn and goes out of the cor rotation zone and it's left behind. So as the planet is moving, these fluid elements encounter the planet only once are kicked out. So they kick the planet in. So they give a positive feedback to migration. Right?

[01:03:57]planet is moving inwards and the fluid element is kicked outwards. So it pushes the planet even more inwards. What we call a positive feedback migration.

[01:04:08]Let's now consider its brother which is a fluid element which is just inside the corotation region. So this is the boundary of the corotation and so as the planet migrates it moves with the planet and look what it does.

[01:04:22]It does a small U-turn here and a large U-turn here. loop again.

[01:04:28]Oops. This one we know. Okay. One kick and you're out. So, positive feedback.

[01:04:35]Look at this. A small kick at the top, big kick at the bottom. So, that means that this fluid element loses angular momentum more here than it gains here. So, the net effect of the planet is to give angular momentum.

[01:04:55]So the the the a positive the the fluid element trapped in the orbital zone gives a positive torque. So a negative feedback on planet's migration that wants to slow down planet migration. So we have two counter effects. The fluid elements outside of the corotation torque as they pass through the corotation torque give a positive feedback on migration. Those trapped in the corotation torque give a negative feedback on migration. So who wins?

[01:05:25]Again, if the density is uniform across the orbital region, the two effects cancel out and the positive or negative feedback is only possible if the corotation region is less dense or more dense than the surrounding disc. But as the planet moves the situation of the density in the corotation region versus the density of the in the disc changes because this corotation region is is moving with the planet going in another region of the disc which can be different. So again during migration the canonical volume DV DJ J being the specific angular moment of the disk is constant.

[01:06:05]Uh and um but but the angular moment the uh sorry as no as the planet moves in the width of the corotation region shrinks with r but there is a relationship between the width in angular momentum and the width r which is one / r delta r. So if delta r shrinks with r, delta j the interval the width in the specific angular momentum of the rotation region shrinks as square root of r.

[01:06:47]And so that means that there is less space for the gas in the corotation region to stay in the corotation region as the planet moves. So there is some mass which goes has to go out of the corotation region because the corotation region shrinks. At the same time the surface area of the corotation ring region shrinks like R squared because the width shrinks like R but the circumference shrinks like R as well. So you have that the mass trapped in the corotation region decays a square root of r decreases by square root of r. But this the area on which this mass is distributed decays like r squared. And so the surface density in the corital region increases like 1 / r the^ 3/2.

[01:07:40]And this is the key. So if your planet is moving in a disk whose surface density profile is shallower than one / r to the^ 3 half then as the planet migrates at some point the density in its corotation region becomes higher than the environment density in the disk and so the negative feedback from the corotation wing region wins and the planet slows down and this is illustrated here. So we have a planet initially at one that migrates and this is the migration rate of the planet normalized some quantities and this is what would happen without this dynamical orbital torque the so the planet is at one and migrates down 2.5 and its migration rate slows down a little bit because of the star but it's for this one. And now you take into account the cor rotation torque. We are in a disc with a flat surface density.

[01:08:47]So shallow the disc is constant. So it's shallower than 1 / r to the power r.

[01:08:54]There is no viscosity in the disc. And as the you increase your total disk mass then the effect of this corbital torque dynamical corbital torque becomes stronger and stronger. And as you can see, this slows down the planet. And in particular, if the disc is is massive, the planet moves a little bit, but then its migration rate drops to a very small fraction of the initial migration rate.

[01:09:22]So this is in a very effective way to slow down the migration of planets and uh but it requires that the disk has a low a small viscosity because if the disc is viscous it tends to oppose to this density contrast because basically you are in a situation where so the this is the surface density of the disc and this is the distance your planet is here and because the planet is moving and carrying along its its orbital region then the density of the disc would do something like this and the viscosity doesn't like any steep gradient. So the viscosity will tend to spread the disc and make it more more uniform. So this dynamical corotation torque works well only in low viscosity discs and in high viscosity discs it can be completely neglected because the surface density throughout the corotation region maintained at the same level of the environment in the disc.

[01:10:32]uh the opposite is true right if the disk profile is steeper than one / r to the^ 3 hat migrations can step up can speeds up now this steep bit can be difficult to envision but if you have a planet that starts to be not really a planet that goes in type two migration but in a transition zone so where you open a partial gap and so the migration re the the mass in the corbital region is reduced with respect to the mass in the ambient disc. Then of course the negative feedback on the corbital region is is reduced and the positive feedback from the non-orbital disk is the same and so you have a positive feedback on migration and uh this is what happens for planets.

[01:11:29]So this is the distance of the planets as a function of time for planets of different masses. So they have deeper and deeper gaps and at some point you can go into a runaway migration where migration even accelerates exponentially and this happens when the mass deficit in the corital region the mass that is missing in the partial gap. So now we are in a situation like this. This is the distance from the planet. This is the disc. Okay. And because there is a planet here, there is a partial gap.

[01:12:04]Okay. When the mass missing here with respect to the disk is so this is called the mass deficit is larger than the mass of the planet then you enter into this runaway mode because the faster goes migrates the planet the faster is the positive torque exerted by the fluid elements that pass from one side to the other of the corrotation region. So you have this extremely fast.

[01:12:32]Okay, the last few minutes I will show you another strange effect that can happen in uh in in disks and this concerns the case where you have two planets opening gaps a little bit like Jupiter and Saturn and this is relevant for our solar system as we will see tomorrow. So uh let's see what happens in the first by looking at a simulation a first simulation done in 2001 uh simulating at the same time the migration of not one planet but two the years 2000 simulations hydrodnamical simulations were still complicated and people to understand what happened was just simulating one planet at a time.

[01:13:18]Masses Negro to my knowledge for the first time put two planets in the disc and so they had the interactions of the planet with the disc and the planet with themselves and they chose two masses like Jupiter and Saturn and they put them random distances put Jupiter at one in normalized coordinates they put Saturn twice as far they simulated and that's what happens Jupiter migrates inwards in type two migration Saturn migrates initially inwards then it has this runaway migration that we've seen before because Saturn opens a gap but not super deep gap. So it's not really in type two migration. It's more a transition planet or which has a a gap which can be have a mass deficit bigger than the mass of Saturn itself. And so it goes into this runaway migration.

[01:14:12]And uh that at the the end here it crosses a resonance with Jupiter one to two resonance which breaks and makes the planet temporarily eccentric. So breaks this accelerated migration mode. This is not so important and eventually the planet enters in resonance the 322 resonance. We will talk about resonance capture tomorrow. So I will not go into that but notice that once Saturn becomes close enough to Jupiter and eventually it is captured in resonance Jupiter migration stops and even reverse a little bit.

[01:14:50]And so this is a way the first way that was ever pointed out in which planets may stop migrating towards the star and these simulations was done in a disc with a large aspect ratio. So a disc with high temperature if you reduce the temperature in the disc so you reduce the scale height of the disc the aspect ratio of the disc the same happens but the outward migration of the planet is even more pronounced and then this of course it's intriguing for our solar system history and become really important. So it's been studied by a number of people with different codes and everybody found this this is true effect.

[01:15:31]Why do we have the true effect? First let's see what is the surface density of the disc when Jupiter and Saturn join each other in this 3 to2 resonance. So 3 to2 resonance happens when Jupiter makes three or or revolutions around the star in the time that Saturn makes two revolution around the star. And so this is a plot the high density is in red.

[01:15:54]The low density is in deep blue. So you see the inner disc. You see the outer disc. You see that the g the gap of Jupiter. Saturn is sitting at the edge of the gap of Jupiter. There are spiral density waves merge. The gap of Jupiter is not very very clean, right? Compared to the animations I showed before where Jupiter was alone. You see that there is warp inside the gap. So the gap is not very very deep. So the situation is clearly complicated. But we can sketch it a little bit to see what.

[01:16:30]So when uh you have Jupiter and Saturn, the surface density of gas in the disc looks like this. Not a sketch comes from the simulation. So Jupiter is more massive and opens a deep gap. Right?

[01:16:45]Saturn is less massive and opens a partial gap and at some point the planets are close enough that the two gaps overlap. So we have this common asymmetric strange feature.

[01:17:01]So first this strange feature ensures that the gas can pass from the outer disc into the inner disc. Because sat Jupiter would like to make his gap deeper. So would like to take this gas and and and and remove it. But this amount of gas is the amount of gas is the depth of the gap that Saturn can sustain. So as Jupiter removes this gas, well, new gas comes into Saturn's gap because G Saturn is not too strong and so it cannot sustain by itself a deep gap. So there is a continuous flow from the outer disc into the gap of Saturn because Jupiter is stealing the gas from Saturn and pushing it into the inner part of the disc. So this allows a channel of gas going from the outer disc into the inner disc. And this is why the gap of Jupiter is not very deep as we also saw before because there is a continuous passage of gas through the waves into the inner part of the disc. And so the inner part of the disc does not get repeated because of this refill of gas from the outside. And so this means the planets are not locked in the gap anymore because the gas can pass from one side to the common gap to the other. This is the first key point. The second key point is the torque balance. Now for Jupiter, Jupiter is is a massive inner disc that tends to push the Jupiter out and a a partially depleted outer disc that tends to push Jupiter in. But because this is partially depleted, of course, the inner disc wins and Jupiter tends to be pushed out. For Saturn is the opposite. Saturn sees a massive outer disc outside and basically no disc inside because there is the gap of Jupiter. So Saturn wants to go in. So Jupiter and Saturn are pushed by this common gap in opposite direction and they do converge until they go in resonance. Once they are in resonance they are in resonance and they are locked in resonance. They cannot move.

[01:19:12]They can only move together in order to preserve their periodation. So they have to decide who wins and who wins is uh Jupiter because it's more massive than than Saturn uh exerts a stronger wave and so in principle tends to migrate out feels a stronger positive torque than than Saturn. So in principle it's Jupiter that wins but it depends actually on the depth of Saturn gaps. So that's why you have this dependence of HR. If HR r the disk is thin the gap of Saturn is relatively deep and so really Sat Jupiter feels a strong imbalance and want to go out and takes Jupiter Saturn with it. But if you have a hot disc then the gap of Saturn is very shallow and so the imbalance felt by Jupiter becomes weaker and that's why the planets can just stop migrating and not migrate out because Saturn the the the imbalance on Jupiter is weaker and so Saturn can so let's wrap up. It's almost the end.

[01:20:22]So a few takeaway points.

[01:20:25]Those small planets don't change the radial surface density of the disc but nevertheless break its axial symmetry by generating a spiral density wave and the torque exerted by the wave forces the planet to migrate in. This is the key of type one migration. So type one migration is proportional as a speed proportional to the mass of the planet and the density of the disk and it's inversely proportional to the square of the disc aspect ratio and the disc also the planet eccentricity on a much faster time scale inversely proportional to the aspect ratio to the power four not two and the conservation of angular momentum tells you that when the eccentricity is damped the semi- major axis is damped as well at the rate of the eccentricity damping rate. This rate which overcomes the normal migration rate until the eccentricity becomes zero. Then again at the normal migration the coorbital torque can block type one migration in where the disc has a steep surface density distribution detail. So typically the inner edge of the disc or at the outer edge of the galaxy and it also acts to slow down migrating planets but only if the disc has a low viscosity and the shallow surface density grid shallow one half to the power.

[01:21:53]So this is for the small planet.

[01:21:56]Giant planets open gaps in the surface density of the disc and then they are trapped in their own gap and so they have to move with their gap at approximately viscous speed unless the disc has a little mass and this is the essence of type two migration and two giant planets with a mass ratio comparable of Jupiter and Saturn once once in resonance can halt or reverse their migration. And this is very important because of course Jupiter and Saturn exist in the solar system. So this is what they probably did and we see this elsewhere like for instance the two planet only two planets which are imaged in the disc like PDS7B PDS70C are in a sort of Jupiter sat mass ratio and they are in the 2 to1 resonance and indeed people who tried to take the observed surface density distribution and simulate the evolution of the planet show that the planet don't migrate. So this is a very important point for giant planet migration. So we stop here and then happy to take any of your question.

[01:23:17]Not in these ones. There are simulations that put the planets off plane, but I did not mention any because So in principle the the the Z evolution and the rest are decoupled. So if a planet is inclined, it starts to launch bending waves in the disc and that sort of warp the disc and they this waves damp the inclination of the planet as the epicyclic waves damp the eccentricity of the planets. So the planet tend to become to go back to the to the mid plane of the disc and there are formula for Creswell and Nelson and so on about the damping rate of the inclination and then of course as long as the planet remains inclined there is a feedback of migration little bit like the feedback of the eccentricity of migration before the eent is completely damp again there are factors in the formula you can check the paper by Preswell and Nelson the most updated paper there's also a recent paper I don't but it's maybe only for like revised works of I think also it's not very explored because yeah the inclination that being so fast uh typically one consider planets to be complainer with the disk and then it's harder to to excite the inclination of a planet. Eccentricity of the planet can be excited by resonances as we will see tomorrow those encounter with planet inclination is much harder once the system is planer. So there is much less rich literature on Yes.

[01:25:41]Yeah. Yeah. Sure. So these were example simulations. So we the planet was fixed uh except in the few where the mass was changed artificially to see the opening kind of stuff was not done in a physical manner. Uh of course the planet can repeat. So the the hemisphere of the planet is uh is filled of gas and and and so the planet can capture part of that gas and grow. So as a function of this the the hemisphere get get emptied and can be refilled by new gas coming into the hemisphere and the mass of the planet changes. So then it can open up so it can strengthen this wave.

[01:26:34]There are also effects that I did not have the time to talk about. It's about the fact that the planet that gas heats heats up and so it generates what is called the thermal torque. This was studied by Venit Lambbe and Mi and others initially in a nature paper and then there is a number of papers on this effect and uh so basically a planet that is very hot because it's accreting either solid or gas heats the disc around it and and this changes because of the equation of perfect gases the density of the gas the gas is hotter and This changes the form balance. Maybe this is what you alluded to. And uh yeah is the thermal effects in the discs is something I do not have the time to cover more saddle and here all the discs I considered were locally isopromal.

[01:27:36]Uh I think in general there's been a lot of hope in this thermal torqus to block planet migration at the end. This is just temporary effects. So I decided this were not so important to put in the first lecture.

[01:28:12]Uh no not really because uh I mean in my pressure gradient formula I mean the change radial change in temperature is included when we say that the scale height of the disk is a function of R.

[01:28:30]This includes temperature uh decaying like one hour and um um so the the gradient of temperature in the disco was included there. Um the fact that there is grains that grains change of composition not impacting migration. Now what can change with the disk is the opacity and the cooling grades. So how appropriate is the isopirmal approximation?

[01:29:06]Then again we go into this issue of thermally driven torqus that I did not mention I'm sorry and uh and so locally these torqus can be important right close to a passive transition when you have a steep drop in radian temperature the radius of the disc you can have a corbital torque driven by the temperature gradient that excites and stops the temporarily but these processes are very temporary and so I had to make the choice but there is indeed a literature about these formal tors which we later this week remotely so maybe we'll mention those yeah I didn't but they don't really change the overall everything that is important Yeah, I will start.

[01:30:12]Maybe I we should both speak migration by planet.

[01:30:25]No.

[01:30:30]Ah, planet engulfves into the star. Uh we don't know. It depends how the disc ends close to the star, right? If the disc was a continuous entity to the star, then the planet will go to the star.

[01:30:47]But close enough to the star, the disc is super hot. So it becomes viscous.

[01:30:52]This comes ionized. So if it is ionized with interaction, the magnetic field becomes strongly turbulent. So strongly viscous. This tends to create a drop in the surface density and then even closer to the star there are there is a so-called magnetic truncation. It's a second edge of the where this corotation can actually stop the plants and and that's why we think there are so many hot plants why they all really fallen on the star.

[01:31:22]We should find just a few lucky ones that is the last of the first idea but there are two meanings.

[01:31:29]Okay, they are not elastic. So in most cases they stop probably torque or they are big enough they open their own cavity they go outside of the disc and then there is also the tidal effect of the star point the planet out balance and there is something we've not asked but is understood there is also the question about the balance between gasdriven migration and planetimal driven migration. If this has planet and the and the the planet is migrating through the disk is migrating also through the planet. This could attack on the planet and can counteract a little bit as long as there is a significant amount of gas.

[01:32:17]The gas migration wins but then eventually planet migration can take over. And I hope I will have time to talk Well, the outer boundary is more vanent is the distant disc is probably set by external photo evaporation in the disc.

[01:32:44]Uh can planets exist so close to the outer boundary? Can they go there? There is this tendency to move in.

[01:32:52]So don't expect really to see planets at the outer boundary.

[01:33:08]Not really because type two migration follows the the viscous evolution of the disc and the disc is an accretion disc.

[01:33:16]So it tends to go inward as well except in the very outer part. So this is what you may allude to with your question.

[01:33:25]Um the standard viscous evolution theory of the gas.

[01:33:36]This is radius. This is sigma. You start with you know narrow a narrow disc something like this. And then because of viscous evolution the disc tends to to spread like this and uh until it cannot spread anymore because you have external footpring times 01 three at each time there is a distance that separates the disc in two. Inwards, the motion of the gas is inward and outward of this distance, the motion of the gas is out.

[01:34:30]So if you have a giant planet opening a gap here, it will move out. But it has to be very close to the outer disc to move out. But as time goes on, this the location of this point also moves and and most of the disc becomes accretional. So it's likely that even a planet temporarily going out eventually finds itself in the accretion part of the disc and start to move in again. So the outward type two migration is typically a temporary effect not a sustainable if we have time I'm happy to go on this is flashing chairman what should we do okay Okay.

[01:35:30]>> Okay.

[01:35:32]Okay. Okay.

[01:35:39]Yeah. Yeah. Migration is inevitable.

[01:35:43]Yes.

[01:35:49]Oh, yes.

[01:36:03]Well, I see first there is can be an effect of this truncation. So if you truncate the disk, all of a sudden your disc will want to spread again visorously out. So maybe this can help in trapping a giant planet in the expanding phase of the disc. And these things can all happen. I think they are not generic but can happen. The flyby will also probably excite the orbit of the planet making it eccentric inclined these two. So everything becomes messy until certain the flyby goes away. But eventually the the viscous forces in the disc red the disc to a new disc and the eentriced inclinations are damped and things should come back to normal.

[01:37:11]They call this Yeah.

[01:37:16]Yeah. There is more.

[01:37:28]What part of outward? Yeah. So because it's dominated by the wave that Jupiter exerts in the inner disc because of the partial gap opened by Saturn that imbalances the surface density distribution that the gas in the Jupiter sees an inner disc close to Jupiter. The outer disc is partially depleted by sun. So the inner wave of Jupiter in the inner disc pushes the planet out. And because there is the gas freely passing through the gap, the planets are not blocked in the gap. So they can go upstream against the distance.

[01:38:22]This is the really the only case of sustained out migration because all the rest is temporary.

[01:38:46]Well, they are inside. So if they are already in resonance, it's the same the final resultant method. They are closer than the 322 they become unstable.

[01:38:55]They're too close to each other. So by scattering they separate and then they migrate back towards a common rising cannot be necessary. The 222 can also be the 221. It depends on the migration rate of Saturn on the density of the disc and it's a certain density could also be to one.

[01:39:15]uh in normal viscous discs it's 3 to2 in low viscosity disc it's more to one if the planets are more massive like PDS 70B and 17C definitely to one and uh and if the planet are born closer as they grow they become unstable they separate on eccentric orbits the eccentricities are damped they go back to simpler orbits in the disc they migrate together until they find their The we will see that tomorrow but the resonance state is the only stable relative configuration in the dis. So whatever mess happens unless a collision occurs planets one planet disappear and the planet find a way to go in a rest.