"Black Hole Magnetospheres"-AlexanderTchekhovskoy

hello everyone I was told that it's time to start to have you all here it's been a fantastic school so far I've enjoyed all the lecturers have attended and I'm excited to share my own experience with you guys but six years ago I was an attendee of this school back in 2009 and it's an honor for me to be back here as a lecturer so let me kick it off by showing you a rendering of a 3d simulation of an accreting black hole that's a disc that's a whirlpool of gas from which the black hole feeds itself and naturally as we will see throughout the talk black holes consuming gas are often accompanied by these collimated outflows or jets and I will focus on the magnetic mechanism of launching of these Jets hence the name of the talk the title of the talk black hole mania spheres because that's where the Jets originate in the magnetosphere very close to the black hole event horizon to set the record straight let us briefly overview where we see black holes around us in the universe they come in two broad classes supermassive black holes with masses ranging between millions and billions of solar masses these are found at the Centers of galaxies and stellar-mass black holes with masses ranging between a few and maybe 10 or 20 solar masses and these are found in binary systems such as a normal star orbiting a stellar mass black hole and donating as gas either through a Roche lobe overflow or through a wind these are called black hole binaries or simply x-ray binaries you can have a neutron star instead of a black hole or another object by the way gamma-ray bursts which are short duration gamma bursts in this case are as form as a result of a binary merger of two neutron stars or a neutron star in a black hole and long-duration gamma-ray bursts so core collapse gamma-ray bursts they are the end product of the evolution of massive stars where the star exhaust is fuel its core collapses to form a black hole or a neutron star that is rapidly spinning and the rotation of the central object can launch a pair of relativistic Jets and if you happen to be along the line of sight of this jet if this jet points towards us then we will see a bright flash on the sky which is where the name for the gamma ray bursts came from bright bursts and gamma rays recently there is evidence for an intermediate mass black hole population not surprised and it's called intermediate because it fills in the map gap between the supermassive and stellar-mass black holes and it's really a topic of a really hot topic really hotly debated what is the origin of this intermediate as black holes they can be either a low mass and distribution of supermassive black holes or a high mass and distribution of stellar-mass black holes probably it's a combination of both this is a really exciting topic I will tell you about a few developments that happen just in the past few years I in a few moments but let me take a pause here and and draw your attention to the fact that all of these black hole systems produce relativistic Jets all of these are artists depictions except this one which is an actual simulation and this is the only one that doesn't have a jaw drawn here I decided not to draw it by hand but Jets is how we know about these binary mergers as they appear as a short gamma-ray burst if the jet is pointed toward us I just to be fair you do not have to have a black hole event horizon to produce a jet the neutron stars and white dwarfs produce Jets just fine and we observe them and this is an actual observation of a jet produced by a star so doors do that as well so this suggests that there is something very simple about the physics of jet production that doesn't really care about the physics of the central the nature of the central object and this is something that it will focus on in a moment but before I go there let me convey the excitement and the dynamical nature of this field because in just a few years we discovered completely new manifestations of black holes for instance if a star is so unlikely that it comes too close to the black hole it would get tidally disrupted and that was predicted a long time ago but only recently did we get evidence that these tidal disruption events can actually produce jets which allows us to detect these events very far because Jets beam radiation and make objects appear brighter recently there was a surprise the new star satellite detected pulsations in one of the ultraluminous x-ray sources which proved that it was actually a neutron star instead of a an intermediate mass black hole so neutron stars can be powering these really bright sources that produced too many too much x-ray emission for their mass even though they are so small recently there was an observation of what we think would be a signature of our process nuclear synthesis in the ejecta of the binary merger disk and that opens up exciting new ways of hunting for the binary mergers in the local universe which is particularly a hot topic because of the recent lack of detection of merging black holes so when a black hole neutron star our interest our neutron star mergers detected the hunt will be for these smoking guns signatures in the in the optical and UV spectra and finally very recently there was a surprising observation of a long-duration extremely long duration long a gamma-ray burst that was accompanied by a magnetic explosion usually we think of jets usually if you think of magnetic fields as either power in a Christiaan sorry either powering a jet or an explosion in this case they were both happening at the same time and how this happens is a mystery so a lot of puzzles to be worked out not many things are actually known in this field and I will try and tell you a little bit about what we think we know so what do we know how do we know that Jets move how do we know that they are collimated well because we see those Jets and there is a lot of wonderful work observe that observers do for us that allows us to compare a modest tube so for instance in this case showing you data for beautiful active galactic nucleus to recede to 79 and on the Left panel I'll show you snapshots in the radio of the source ranging from 1992 to year 1998 and what you're seeing here is that there is a central component which doesn't really move in time you see it's at the same location that is associated with the central black hole and then you can see the jet and one component that is moving away from the center you can draw a straight line to measure it slow to see how fast it moves on the sky and you find that it moves faster than the speed of light that sounds ridiculous at first but actually there is nothing about the theory of Einstein realize the theory of relativity that is broken here nothing physically is moving faster than the speed of light this is just a pattern speed as it appears on the sky and the fact that these apparent velocity on the sky exceeds the speed of light is actually telling us about the velocity of intrinsic velocity of this jet if the jet points almost toward us as it does in the system and it's moving very close to the speed of light the jet is catching up with the radiation it produces with the photons it produces and this is what leads to this apparent superluminal motion in fact the underlying physical velocity of this jet is inferred to be ninety-nine point seven percent of the speed of light and this jet is I think just a couple of degrees away from pointed away from us so we are basically right in the laser beam of this jet because the Jets are so tightly collimated and focusing all their power in this really tight opening involved it means that they will appear very bright to us moreover because they move toward us the Doppler boosting makes this Jets bright across a wide range of wavelengths including the high-energy emission so for instance here is a broadband spectrum of this source and it extends from a low frequency radio to the very high-energy gamma-ray emission and there are a lot of instruments that we can use to study the emission of this jet and similar jets for instance in the radio there is a beautiful Alma instrument and many other instruments just showing you the latest and greatest in the infrared optical and UV we have a lot of ground-based facilities including space telescopes of course of in addition to ground-based the Hubble Space Telescope is an example in x-rays we have set a lineup of x-ray satellites Chandra xmm-newton and on higher energies new star when we move to gamma rays we of course have swift and Fermi and the TVV gamma rays can be observed using Hawk and Magic telescopes as an example Jets do not have to talk to us in black holes don't have to talk to us via radiation they can like like we can we can see the radiation by using the Cherenkov telescopes but we can also detect cosmic rays which we think are produced in at least some of the Jets we can also detect high-energy neutrinos for which the Jets effectively technically are a potential source and of course we can detect gravitational waves which are signatures of emerging black holes and probably the Jets that those black holes produce but we should think and study Jets not just because we can see them or detect them but also because they play an important physical role in the evolution of the universe around us for instance here I show you an exam all very well-known image of the Perseus cluster where you see the Jets influencing the surroundings in a dramatic fashion they blowing out bubbles filled with hot magnetized plasma and sending shockwaves enter the surrounding medium thereby heating the medium and perhaps preventing the black hole from accreting too fast also perhaps keeping the gas hot which otherwise would under go around the way cooling in the m87 galaxy that we'll talk more about later in the talk in the next talk as well we see more of the same shocks and cavities blown out by the Jets and in this dramatic example of the cluster MS 0-7 35.6 we see huge cavities blown out presumably by the Jets powered by the central black hole and what's really puzzling is that the entire power the entire energy associated with those cavities is comparable to the rest mass energy of the black hole it's almost as if if this black hole was nearly maximally spinning the Jets somehow managed to extract all of that rotational energy and dump it in that the ambient medium over the past maybe ten million years or so how that can happen is really not an issue and this motivates the understanding of what determines the power of these magnificent outflows we can try and look at how the Jets are produced and what's that's their power on a more quantitative level by plotting for instance the mass of the central black holes versus the velocity dispersion of the stars and the Bulge of the galaxies here is an example of a plot by Scott and collaborators back from 2002 and what we see here is a tight correlation between the mass of the central black hole which is a tiny little dot of the very center of the galaxy and the central part of the galaxy which is huge much bigger than the sphere of influence of the black hole so the black hole cannot talk to the stars whose this version is plotted on the x axis yet it somehow knows very well about how fast the rotation in the galaxy is and how do black holes talk to their surroundings is not an issue it could be a product of coevolution of black holes in the galaxy or it could be perhaps due to some mechanism which allows the Jets to talk to the surroundings perhaps due to jet feedback or radiative feedback yet another way of quantitatively looking at Jets and their interaction with the surroundings is to plot the radial luminosity of active galaxies versus their nuclear B band luminosity essentially the optical light and both of the axes are on the log scale and so what we see here is we see two tracks of galaxies very well separated tracks the top track is called the radio loud track and the bottom called radio quiet track and that's because for the same optical luminosity emanating from the sources and therefore we think the same rates of accretion at which these black holes consume the mass we see very different radio luminosities different by as much as three orders of magnitude on average and if radio luminosity is a tracer of jet power as many people suggest then what this tells us is that in addition to the black hole mass which is roughly the same between these two tracks and the mass accretion rate which is the same as well at the given luminosity at the given value of the x-axis there must be some third hidden parameter which tells a black hole whether it lands on the top or in the bottom track what this parameter is is not agreed upon there are several possibilities that have been suggested over the years and listen here just a few it could be that the magnetic flux is different between these two tracks stronger magnetic fields will produce stronger Jets so the top track may have more field of a higher magnitude and the bottom the bottom population has a weaker field it could be that there is a difference in the ambient medium maybe the radio emission trace is not just a but the interaction with the ambient medium so if there is more material to the jet interacts with that will lead to more radio emission produced and so this is just an illusion not a real dichotomy in the jet power and finally the third possibility is maybe the black hole spin is different between these two populations if a black hole is spinning faster it produces more jet power and therefore more radio emission it's probably combination of all these three possibilities and it's not clear how we will be able to resolve this perhaps we could try and take a look at this jet with our eyes and that will give us some suggestions as to what can be going on so these are two of my most favorite jets this is a Cygnus a galaxy jet and this is the m87 galaxy jet both of them are host similar comparatively large black holes about 10 billion solar masses each and you can see that even though the black holes are rather similar the morphologies of these jets are really rather different you can see that here these jets are really thin and straight and end up in a bright hot spot whereas in the other case in the case of the m87 galaxy the Jets are kind of Wiggly and unstable and disappear into nothing ending up in a swoosh this dichotomy in jet morphology has been known for a long time since 1974 for more than 40 years now and the reason no well agreed explanation for this which I will get back to try and answer what is causing this dichotomy in the second talk meanwhile what we've seen here is that for very similar black holes we find in very different morphologies and it's unclear what causes this could it be differences in the central engine or could it be differences in the surroundings so maybe if we could only zoom in onto the very center the very central black hole and make an image of it we would be able to answer all the questions right away maybe we will see signature of black hole spins so we will be able to measure it we will know how massive the black hole is and how much gas is falling there and all of the questions will be resolved in fact we can do this already from the scale of 3,000 light years we can zoom in to the scale of only one light here that is by factor about 3000 and this is this observation is performed by very long baseline interferometry that is a bunch of dishes a spread out over a large area that correlate the detections and act together as a bigger dish that allows to be down on the diffraction limit and what you see here is that the jet is produced somewhere here which this is called the radio core which is actually not where the black hole is the black hole is a little bit to the left you can see the jet going to the right and up and you can see pattern moving away from the center so we think is what we think is going on here is the jets of produced very close to the black hole with a rather small velocity and as they move away they accelerate faster and faster and in fact by the time this jet would have gotten here it would have been already moving highly relativistically you can get a hint that this jet is moving relativistically not because of the proper motions which suggests that but because you do not see the contra jet - well in fact there is almost no emission here right this jet is booming and this jet is almost not there but you see its effect on the ambient medium so what is happening here is that this jet is pointing almost toward us it's I think misaligned with our line of sight by about 15 degrees so we are seeing significant Doppler boosting of this one jet but in in tribute and intrigue here is that we see signs of contra jet very close to the black hole so by looking at the ratios of brightnesses on both sides for the two Jets we could perhaps reconstruct how fast the jet is moving so let's try and zoom in even further let's try and make an image of an actual black hole and see if we can try and get out something useful out of it so this is an image on the scale of thousand black hole gravitational radiate these are the images on the scale of just 10 black hole gravitational radii these are not the usual images these are actually Fourier transforms of images in the direction where we were able to spread the dishes on two different continents so this is also very long baseline interferometry but it is on the baselines between Hawaii and California and Hawaii and Arizona so this is the triangle of base stations that allowed to make this Fourier transform yes for which one oh it's almost in plane for the Sigma say for the Sigma say it's almost in plane so like here what we're looking at is a Fourier transform of an of an image and if we were looking at the Gaussian blob and we made a Fourier transform for navigation block we would see a Gaussian blob in a furious pace for a transform of a Gaussian is a Gaussian what we're seeing here is this several sets of points for each of the source that correspond to different combinations of the base stations in there for different baselines and the the Gaussian fit is shown with the solid line and the dotted line shows a Gaussian with a circle which you expected there is a source behind the block form you would see an Einstein ring lancing the the result of the lensing by the black hole of a point source from behind it and you can see that in this in these both cases we are not able to tell apart between these two models which suggest that we better have a good idea of what we're looking at before we try to interpret these observations you can put it this way data is interpretation limited we need to have a prior a good guess for what we expect this topic of imaging the black hole on small scales is really hot right now because in April next year we expect to have observations the Sagittarius a star hope I forgot to tell you that these two black holes are the two largest black holes as they appear on the sky this Sagittarius a star black hole is the black hole at our galactic center and it's about twice as large in its apparent size as the black hole at the center of the m87 galaxy the amazing seven galaxy is about thousand times further away than the center of galaxies but it also hosts a black hole that is thousand times more massive or so so these two black holes are just within a factor of two of each other in size that which makes them primary targets for these sorts of observations why this is so important right now in time Li is because in April there will be more base stations added to the to the array which is called the event horizon telescope and we expect to have images not like this but approaching that going in that direction produced of the accretion flow around Sagittarius a star and this is one of the simulated images carried about out by student Shan wrestler who is here in the audience and we're trying to try and understand what sorts of morphologies we're going to expect what sorts of spectra and how can we invert the observe a shinto the physics that underlines the model let me now take a step back and yeah so this is showing a 230 gigahertz image that you would predict to see with the event horizon telescope and what you've seen here is a combination of emission from the jet and an accretion disk you cannot really tell the two apart - well actually in fact most of this emission is coming from the jet at least in the model that I'm showing here so this was not filtered through the actual baseline stations response so this is an image as we would see if we were to come there and look but in reality this image will be blurred both by the limited TV coverage therefore for airplane coverage because there will be only so many stations this image also will be blurred by the ferraday screen by the interstellar medium and also a little bit by the scattering even though at these high frequencies probably scattering is not a very for you the black hole is right there in the middle so it's in the middle of the image any other questions ok let me move on I will take one step back right now and I will show you in very simple analogy how jets are produced by magnetic fields so let us consider a perfectly conducting sphere which is meant to represent the central object be it a neutron star a black hole or a normal star it could be why to work as well and let us consider a seeding which is meant to represent the mbn medium it's also perfectly conducting this is important because if we take a field line and we attach by one end the field line to the ceiling and by the other end to the sphere then if we move the ceiling and we move the sphere the field line foot point will have to move together with the surface so if we now switch on the rotation of the central object at an angular frequency Omega and wait for n rotations then the initially straight field line will develop and toroidal field loops so instead of a straight line we have gotten a magnetic spring what does it mean it means that we now have toroidal magnetic field in this direction addition to the vertical field toroidal magnetic field has pressure given by the well-known formula B Phi squared over 8 pi so this magnetic spring is pushing on the ceiling if we wait for longer and longer then more and more toroidal field loops will be put on them on the spring it will become stronger and will eventually push the ceiling aside break through and expand vertically under its own pressure which is a jet it is expanding under its own pressure and accelerating any plasma that is attached to the magnetic field lines this is very similar to acceleration in a hydrodynamic nozzle where the flow expands sideways it's pressure drops and the pressure gradient pushes it out here the same thing is happening you have sideways expansion of the jet the pressure above is lower than the pressure below and this pressure gradient pushes it out except for one complication which makes the systems really complicated as you will see in a moment that there is also tension of the field it's because the magnetic pressure is anisotropic so it's described by tensor as opposed to a scalar and this is what makes a study in Jets really complicated for instance if it were a hydro outflow then all of the thermal energy for instance would have converted into kinetic energy at large distances and we would have ended up with a flow that is completely cold and moving at the maximum possible velocity as we will see in the case of Jets because the hoop stress counteracts the acceleration of the Jets these Jets tend to lock in some fraction of the magnetic energy in the form of magnetic energy as opposed to converting it into kinetic energy a convenient way of looking at these Jets is the following you can think of the jet as continuously converting the initially vertical magnetic field reprocessing it via rotation in the toroidal loops at a certain rate so these loops pop up from the black hole and fly out sliding along the original magnetic field line and as they do though so they expand the pressure drops and this pressure gradient pushes them out modulo the caveat that I mentioned that there is tension this loop so they are doing so reluctantly so let us try and figure out what is the power of the outflow that is produced here and here I will focus on an example on the case of a neutron star I will not start with a black hole right away I will ease it in because we have a surface we're comfortable with this hard surface let's try and see what kind of power will come out yes yeah I I think this was also what surprised me when I ran a lot of simulations that we'll be talking about in the next talk when you actually include the accretion disk I will not be including a Christian disk in this first lecture that the system doesn't break the symmetry above below the midplane by a lot and that's probably that's probably because the disk doesn't move around too much above and below the midplane so the things that are above and below that is don't know about each other too much and they kind of evolve independently and jet production is rather a bust in both cases the Jets are collimated by the same accretion flow just on different sides so the pressures are similar the force balance is similar and therefore the jet power is similar that's really my understanding of this but if anybody has any suggestions I'm happy to to talk to you any other questions okay so we have a neutron star and we have magnetic field lines sticking out of the neutron star and I'm expecting to get a question there are no mana poles in the universe why are you put in a monopole neutron star because half of these field lines should be going in and half should be going out because the total magnetic charge on the inner star should be zero and that's true that is completely true what I'm doing here is I am considering a really simple case when I don't have to worry about sign changes in fact we can make this monopole into what we call split monopole by changing the orientations of half of these field lines such that they point in and I will get to that in just a moment for now we have a neutron star spin at angular frequency Omega and it has magnetic field strength on the surface that's equal to B how do we go about computing the power carried out by the wind or the Jets that the this rotation causes well the first concept we will look at is the concept of light cylinder it is a distance at which if you were to rotate together with the star you would be moving at the speed of light it's really important beyond this distance the outflow essentially separates from the neutron star and you can consider it as a as a wave and it's really convenient to compute the power of the outflow at the light cylinder so the light cylinder is such that Omega times R of the light cylinder is the speed of light and the magnetic field strength on the light cylinder can be estimated by taking the magnetic flux thread in let's say the Northern Hemisphere and dividing it by there should have been a square here missing so divided by the area of the Northern Hemisphere how can we estimate the spin-down power well we can integrate the pointing flux over the area of a sphere over radius of the light cylinder and then we get an expression which is very simple it's a characteristic magnetic energy density at the light cylinder B square that lights from number times the characteristic length scale times the speed at which everything is flying out which is the speed of light and then if we plug in the magnetic field in terms of the magnetic flux we are going to get expression for the power and what we see is that not surprisingly the power is some numerical pre-factor which is important if you want to quantitative estimate what the power output of a black hole is times the square of the magnetic flux times the square of the rotation it's not a surprise that we have gotten wet we ended up with a bunch of squares because if we were to flip the sign of the magnetic flux nothing would change in terms of power so there's got to be at least a square there and the same thing about the rotation if we change the rotation to make it rotate in his clockwise instead of counterclockwise nothing would change that's why you have to have a square here at least question okay now this was a very simple derivation the actual exact answer is six instead of four so when I was making these integrals and made assumptions and approximations and was rather loose but it's pretty amazing this estimate is giving you actually almost the right answer so there should be an exact equality here here I'm coming to the back to the point of how to make it more realistic if we flip the direction of the magnetic field below the mid plane then we will end up with a zero magnetic charge on the neutron star which is good we don't want to have any magnetic charges but the price we pay for it is we will develop a current sheet in the mid plane because the magnetic field flips the sign and while this is fine for theoretical consideration and changes nothing in terms of this derivation you can make sure that the energy flux is still pointing outwards even though the field orientation changed what this does when you try and simulate the system numerically is it causes you numerical difficulties because the magnetic fields will be reconnecting it introduces time variability to the system that perhaps you don't want to have is you want to study this really simple case and understand what is the power that's coming out you don't want to be bogged down by these extra complications that's why in the exercises which I advertised are really illustrative and simple I will get back to this towards the end of the top you will be able to run this simulation with a black hole at the center for the monopole field there will be no current ship to make lives easy and you will be able to measure the spin down rate and you will be able to compare it to the simple formulas so what about the black hole if we were to replace the neutron star with a black hole what would change well let us first ask ourselves how is that called different than interest or well there is one big difference this black hole is characterized by just three parameters the mass the spin which is the a way to parameterize its angular momentum and the charged black hole has only three hairs in reality Astrophysical black holes we think have a charge that is essentially zero because if they were positively charged they would attract to charge from the surrounding medium and so in reality we just have two hairs in the black hole and I cannot resist but to use the slide by remeshing Orion which makes the point really clear how simple the black holes are Einstein had a lot of hair black hole has only three hairs more likely actually - so only two hairs mass and spin and what is really important is that black hole's know nothing about the magnetic flux they have no magnetic hair at all so if you leave a black hole alone it will not produce any jet or any radiation classically at least not talking about Hawking radiation so you need to have charges and currents around the black hole to give them magnetic flux and for them to produce the Jets so when we switch to the black hole case and this is the same picture as I showed you for the initial star except I replace the black hole the neutron star with a black hole here you see you have to have currents outside and so you would have to supply this currents by some means so keep that in mind another important difference is that the black hole doesn't have a surface so if we have a field line that's sticking out from the black hole it won't have a surface to be attached to what it means is that black hole must do something else to cause these field lines to wrap around and rotate and that is done through the rotation of space-time so the frame the frame dragon frequency which I call here Omega scales with radius roughly is R to the minus 3 at the event horizon it's given by Omega sub H which is proportional to spin give a take a factor of 2 because the black hole event horizon radius changes as the black hole is spun up but this is what it is so if I am near the black hole and very close then I will feel like I'm my head of spin because I'm going to be going around the black hole had this frequency if I go further and further out the space-time rotation slows down and at infinity there will be no rotation so you can think of a field line that started a black hole and ended up at infinity as it is sheared between the black hole and infinity and because the field line is nice and tries to both what it does it picks out roughly an average between the two so it slips through the black hole and through infinity at roughly the same rate this is a very simplified consideration you can show this more rigorously that the this is the right answer but what this allows us is to fall back to the machinery that we developed for the neutron star because for the neutron star we had this expression where we had Omega and now we know what the expression for this Omega is we can plug it in and here you go we're getting the power output of an actual black hole without having done any gr whatsoever except for this very simple approach so black holes are actually very simple even though intrinsically they might seem quite hard so let me go back to this problem of black holes having no magnetic flux because if I were to leave the black hole with this current sheet like this what will happen is that the current sheet would reconnect and magnetic field would fly away and we will end up with a black hole without any magnetic charge as the no-hair theorem stipulates yeah oh sorry well as soon as I leave it to its own devices the current rate will start reconnecting so it will not be steady state anymore you will develop you can develop plasmids and Terran modes and so on but once the transient is gone you lost the magnetic field and if you lost the magnetic field and there is nothing to study because the Bloch call is just two parameters so what happens in the reality is that the reason accretion disk that is doing a very important job at transporting the large scale magnetic flux from large distances from the scales of a galaxy down to the black hole and at the same time it's preventing the reconnection between the field lines from above to below the black hole so the disk is what hall these magnetic field lines together in the black hole and allows the black hole to continuously produce the outflow and there are this while this is the accepted picture that there is a disc that holds the magnetic flux on the black hole where this magnetic flux came from is an open question there are several possibilities for the origin of the magnetic flux that could be a patch of large-scale magnetic flux that the disc grabbed from very large distances on the scale of the galaxy and delivered it and dumped it on the black hole or it could be that there was really no large-scale magnetic flux and turbulence in the disc through some process I came to dynamos generated large-scale magnetic field loop that then ended up on the black hole I will get back to this important problem in the next lecture but the bottom line here is that black hole must be accreting there must be material around the black hole to keep the magnetic field on the black hole event horizon and that's all I'm going to talk about the accretion for this lecture I'm going to focus just on the properties of the outflow I'm going to postulate that there is some magnetic flux thread in the black hole event horizon and I'm going to ask what consequences are we getting what are the properties of the magnetosphere in the outflow among it is very Cod flow that results from that so let us now switch back to the neutron star now that we know that neutron stars and black holes are very similar you just need to plug in a different value of Omega and then I'm going to look from above down the rotational axis onto the neutron star what is going to happen here is if you have an if you have a magnetic field lines so I'm the neutron star and there are field lines that are sticking out from the equatorial plane and I'm spinning the field lines will be lagging behind me so if you if you look down the barrel of a gun the rotational axis the equatorial plane is rotating and the straight field lines will be lagging behind this way just like I draw here so now let us zoom in onto this very small patch at the surface so this is the blow-up of the surface of the star this field line is sticking out from the surface of the star and this is the initial condition this is neutron star without an irritation yet the field line is sticking out radially no winding of the field by the rotation yet the surface is flying to the left with the velocity V Phi which is given by the rotation rate and let us wait for time period delta T the foot point of the field line will be displaced by the product of velocity and the time interval and this information will propagate along the magnetic field line at the velocity of the waves which is the Elven speed here so this distance over which the field line will be able to communicate the fact that the surface started moving is given by the product of the element velocity times the period of time delta T let me make a key simplification for a little bit I'm going to assume that this plasma is really highly magnetized there is really no inertia to this plasma so I'm going to assume that the plasma is massless or as it is called force free so magnetic fields is what carries all the energy and there is no energy in the particles all the particles do they just provide the charges that instantaneously screen whatever magnetic field electric field is developed in the frame of the charges so this is a force free approximation and in this case the the fast mode and the Elven mode travel at the speed of light along the magnetic field lines so I can replace the Elven velocity here in fact we already sneakily replaced it with the speed of light now we can actually compute what this angle is and figure out what are the components what is the component of the magnetic field which is responsible for energy extraction from the neutron star and that is the V Phi component which is given by the ratio of these two sides of the triangle V Phi over C times the radial component the initial component of the magnetic field so that's how an Alvin wave launched out from the surface of the neutron star is establishing the twist of the magnetic field lines and by plugging in what V Phi is in terms of Omega we get in this expression for what the it'll feel this the electric field turns out to be exactly the same as the toroidal field by the freezing condition we can write that YZ minus Vickers B over C what this means is that there is no electric field in the fluid frame so if you were to jump into the fluid frame then the electric field would be zero there so that's just a Lorentz transformation and we are finding that Ian's be fire the same in magnitude why do I care about this I want to figure out how fast the Jets move what I need to have for that I need to know what the product of ich he cross B is and divided by B squared that tells me what the velocity of the field is in terms of speed of light and that is nothing but the ratio of e over B so I know e I know B I now can compute the ratio and it's convenient to recast if you ever see in this form as you will see in just a moment I'm doing really basic algebra here what I want is to ultimately compute the Lorentz factor because that really is a convenient measure of velocity of relativistic outflows when velocity goes to the speed of light they didn't know in the denominator vanishes and the Lorentz factor blows up so large Lorentz factor means velocity very close to the speed of light and by plugging in this at the bottom here we are getting going to get this really simple expression which actually is really deep several papers are based just on this expression so do not underestimate it so what we have at the top is this B square it goes to the top and I will decompose it into the radial component of the toroidal component and at the bottom we have B squared minus V squared so in in our case we know that E is equal to B Phi ok so this term goes away and what we are left with is the R squared plus B Phi squared divided by B R squared or just like that one plus the ratio of the components which we know is equal to Omega over C in a square so we just derived how this radial magnetosphere accelerating as a function of Freitas add the surface of the sphere where radius is essentially zero we're going to get Lorentz factor equal to one well for a nonrelativistic rotation rate as is for most of the pulsers that is the case we basically have Lorentz factor of one at the surface as we move away from the surface from the neutron star then the Lorentz factor will be increasing and I sympathetically it will be going linearly so you can plot it on the log log scale and you can see that Lorentz factor starts at one and then once we cross the light cylinder radius where this expression goes to unity it becomes essentially linear by the way I drew it not in Python I drew it really in keynote because I've looked at this plots so many times I know exactly how they look like you can you can over plot it that the true solution you will not find any difference I assure you so this is the key takeaway from this plot that the Lorentz factor asymptotically scales linearly with distance if you think about it there is something fishy about it right because real system will not be able to accelerate up to infinite Lorentz factor it wouldn't apply infinite kinetic energy but this is possible here because there is really no mass attached to the field lines we made the force free approximation there is no inertia associated with the gas so in order to include this inertia and figure out how a real system will behave whether it will accelerate all the way up to infinity indeed which one do or it will level off at some level is an important question because that's what we observe we observe terminal Lawrence factors things stopped accelerating so how does it happen in order to address that we need to look at the complications associated with having inertia in the flow and the most convenient way of doing this that I know of is to look at the conserved quantities along the Jets instead of integrating differential equations along the jet we're going to look at a bunch of numbers bunch of values that are supposed to be conserved along the field lines so one of these is the magnetic flux which I will denote as f sub-p and the other one is the mass flux these both of these are conserved because mass doesn't get destroyed or created in the jet it just flows together with the flow and so does the magnetic flux so if we divide the choose we consider their ratio which I will call a de then the ratio will be constant independent of the radius I can similarly do the same for the flux of the total energy which is the electromagnetic plus the kinetic energy electromagnetic energy is the pointing flux and the kinetic energy is Lorentz factor times the mass flux so far so good make sense there will be a little bit of algebra here but I I spent a lot of time clicking in key mode to make sure that all the animations work correctly so I think what you will see if it works so why am i doing all this there's going to be an amazing answer in just a moment so if we look at the ratio of F II over F M the energy flux to the mass flux this is a really important quantity what is telling us is what does our energy budget if we were to convert all energy flux into mass flux how large would Lorentz factor be that really tells us the maximum Lorentz factor that the flow can have and you will see it in a different way in just a second so the first term here is the ratio of electromagnetic energy flux to the mass flux but mass flux is kinetic energy flux over gamma so we get getting a gamma times the ratio of electromagnetic to kinetic energy flux in the first term and the second term is simply gamma because kinetic over mass flux is just gamma by this equality this ratio of electromagnetic to kinetic energy flux is really important it's called magnetization and denoted by Sigma it's it can also be expressed as the ratio of magnetic energy density in the fluid frame which is B squared I call the the fluid frame magnetic field with a small letter B so it's B squared over 4pi that's the magnetic energy density in the fluid frame actually its enthalpy enthalpy in the fluid frame there is a factor of two between the two divided by them the rasmus energy density so it really tells you how strong the magnetic field is energy wise and now we've gotten really simple result that the total energy flux can be decomposed into the electromagnetic part which is characterized by magnetization Sigma and the kinetic part which is just the Lorentz factor why am I talking about this well because magnetization is always positive definite that means we have proven that the maximum value of Lorentz factor cannot be greater than the than the value of MU that's one first result so if we know in a simulation on a real source what the value of MU is we know that Lorentz factor better be below that the next result is that Sigma is also very convenient parameter because it sets the Lorentz factor of the fast waves it's essentially the ratio of gamma times P of the magnetic pressure divided by Rho and that is really similar to the sound speed which is square root of gamma times P gas over Rho so that also means that we already have the fast waves in our consideration for free now in force free the magnetization is infinity because there is no kinetic energy flux Rho is zero and the fast waves travel the speed of light which we can see from here if we set Sigma to infinity the Lorentz factor of the fast waves gamma F is infinity now remember that funny result when the Lorentz factor was growing out of bounds when do you think that approximation will break let me think so even force free the fast waves travel the speed of light means that you can never catch up with the fast waves means that if they travel not at the fest of the speed of light but at a finite velocity given by this Lorentz factor it means once we caught up with this fast waves our force free approximation is broken that's probably the Lorentz factor at which we cannot apply the force free results anymore what Lorentz factor will this occur at well that that happens when Lorentz factor is equal to gamma F or Sigma to the one-half okay how can we back out what the value of Lorentz factor is here well we can use this relationship here and neglect one assuming that the flow is highly magnetized we will get an equation for gamma on both sides and by solving it we obtain the answer that force free breaks down at a really low that rather low Lorentz factor mu the maximum possible Lorentz factor to the 1/3 this is really a draconian constraint on the outflow acceleration efficiency let me explain why let us imagine that we created a really nice highly magnetized jet where this mu parameter is 1,000 so we expect to get an outflow with Lorentz factor of up to 1,000 as a result that would be fantastic if it points to us it will be so bright everything is so being amazing we can see very far and study sources otherwise not seen the only problem is that we are going to get a Lorentz factor of only 10 as a result because of the breakdown of our approximation acceleration ceases to be linear beyond this distance in fact instead of this linear nice linear dependence that we saw a few slides ago we're going to get this leveling off of the acceleration and this is one of the problems that I set up for you in the code that are shared with you guys and if you have you gotten the information about the problem sets or assignments or the fun that you can have with the what the code has anybody communicated this to you there is a git repository I heard where everything is contained and if you do git clone that repository we'll be information about the problem sets but if you haven't gotten it I will send you a link to all of you I will ask Allison to send it so there is a code that I'm sharing with you just like Anatoly spit coughs ki I'm going to make my code public for everyone so you guys can use it and do science with it it's really powerful and it will give you more information about it but this is just one of the simplest problems you can run on your laptop and I will show it to you if I have time at the end of the lecture that you actually can't get the result that the Lorentz factor sets your rates very simply so that is actually 1d problem so you say that the magnetic geometry is exactly Radio one dimensional field lines cannot move around and then in this case you get a really flat distribution of Lorentz factor with this terminal value for it if you let the field lines move around in two dimensions then there is a little bit of extra power that comes out and that's one of the questions to try and understand why Lorentz factor doesn't say sure it exactly at me to the one-third it has a weak logarithmic dependence with radius that goes as log to the 1/3 you can actually show it analytically in a really simple way that's one of the problems that I'm asking you to take a look at if you interested but this is really depressing and I didn't mean to I wanted it to be an upbeat presentation so let me tell you right away that this doesn't have to be the case and I will show you how you can get really nice acceleration for the Jets what did it 1 yes badly accelerated yes very bad so why are things so bad but remember let's be optimistic it will all be good at the end like in a true good Hollywood movie if there are any such movies so let's let's try and investigate what kind of assumptions did we make that made the Jets behave so badly so let's revisit them we started out with Lorentz factor expressed in this way there is no problem with that then we expressed it through the field components there is also no problem with that that's that's one-to-one but then we drop this term because we found for them for the split monopole geometry beef I exactly compensated e and that is actually where the problem is so let us try and relax this assumption that actually is breaking in the case even not in the case of the monopole problem when we made this assumption we said that the toroidal pressure pressure associated with the toroidal field in the fluid frame and this is preciousness it with the toroidal field in the fluid frame because you remember that maybe remember that the reality mystic invariant B squared minus Z squared it's the same in all frames so if I jump into the fluid frame where E is zero there is no electric field in the fluid frame I will be left with only be B squared in the fluid frame that will be my pressure that is the magnetic pressure in the fluid frame once that boost into the lab frame in order to compute the fluid frame pressure which is what I care about this is what sets the force balance I need to subtract a squared from it because that's that's that's what's the inverse that that's what is the environment in invariant sorry so this is that pressure so seeded with the toroidal field in the fluid frame so what I said is it's negligible now if it's not negligible if I cannot drop this component then my magnetic pressure is not just be R squared but it's V R squared plus the fluid frame be Phi and this V Phi will be varying laterally and this will mess up my perfectly radial magnetic field lines will make them deviate and change shape and that is an important difference as we will see in just a moment so now let us go in the opposite limit let us neglect not this guy but let's neglect the V R so in the same formula we will cross out VR and what we will end up with is this ratio which really doesn't tell me anything when I look at it in fact I have no idea how to compute this at all but we will do this in just a moment but because the steroidal field is important I want to reiterate that the field lines will change shape and they will do something like this and this will be crucial they will move away from the mid plane and that's what will prevent the flow from accelerating and why is that on an intuitive level imagine that you're driving a car looks at one mile an hour you can make as sharp of a turn as you want imagine now you drive in a car at 100 miles an hour you kind of make a 45-degree turn right away right so the same thing happens with the toroidal field that's flying out along this radial field lines even the field lines start to turn then it means that the toroidal field cannot fly out too fast because otherwise it won't be able to make the turn so this really small change in curvature of the field lines is what limits the Lorentz factor to what we see to the very low value that we see so fast jets can't make sharp turns ok jets also Bay laws of physics just like driving does so except that they can move at the speed of light so the speed limit there is the speed of light okay so force balance across bend magnetic field lines so here it will be I guess the as hard as it gets math wise so see there is a gradient I think Matt had many more gradients than I did and Elliott also did a lot of gradients this will be very relaxed in only one gradient and I'm going to approximate it with a finite difference okay so the two forces that I play now are both associated with the toroidal field one force is the toroidal pressure gradient which is the magnetic energy density most literal field here because that's what we dominated by by the divided by the cylindrical radius because that's the distance that that's that's the that's the distance over which you difference right this is a gradient in that direction and the other force is the central the centripetal force this is the force that wants to straighten this field line and that is the effective inertia associated with the field which is just the energy density times the square of the velocity divided by the radius this is really straight forward generalization of the usual V squared over R center Google force so if we balance these two forces out then we can actually back out what the Lorentz factor is and it turns out to be given by the square root of the ratio of the curvature of this field line which is this guy this is the curvature radius I somehow it disappeared from the slide divided by the cylindrical radius so it turns out that this ratio of radii of curvature in one plane to the to the other plane to the to the toroidal plane is what tells you what the limit of Lorentz factor is now it turns out we can actually combine the limit of this guy of this complicated bent jet limit on Lorentz factor to that really simple linear acceleration regime that's because and I won't go into the details but it's rather simple if you take an expression for for this this expression for gamma gamma squared and then you you say that this part is that and that part sorry here's this then you can write down the total Lorentz factor as the sum of the inverse squares of the two expressions so we'll have a linear regime and this non linear regime summed up in such a way that we are getting the answer so this is really remarkable result we have done almost no hard computations and we have arrived at a first principles expression for what Lorentz factor of an outflow is if you've given the shape of the outflow and you can actually also compute the shape that the curvature of the field lines and that's one of the problem sets any other questions no okay so now what I'm going to try and do is I'm going to understand how can we get the Jets to accelerate better than bad better than bad so how do Mass loaded Jets accelerate and how do they do this efficiently so we've tried to look at the radial outflow I didn't work too well so in order to understand what do we need to do in order to accelerate for the just accelerate better we need to understand what was actually the limitation and it wasn't too clear I mean it was the curvature but what would be the solution so let's try and get a debt so here I show what we already worked through we have the two ratios the the mass flux per magnetic flux the energy flux per per mass flux and we have decomposed it into the magnetic part and the kinetic part magnetic part and the kinetic part let us now forget about what we've done here let us massage the numbers a little bit differently so the electromagnetic part stays the same kinetic part stays the same but we will make use of the fact that asymptotically very far away from the center however Jets will be moving fast beat Lorentz factor of ten or a hundred doesn't matter we are looking at a relativistic jet if the jet is relativistic then electric field and magnetic field will be very comparable to each other so I will make this approximation this is actually supposed to be approximate is approximately be Phi and it's given by the product of velocity times B this is really nothing than the freezing condition and if I plug both of these in here then I will get the square of this divided by four pi C so I can plug this into here and what I'm going to get is actually quite interesting we have this combination here which is independent of the distance along the field line Omega is the same along the field line if it were different between different radio than the field would start to shear and so it wouldn't be steady state so in steady state on mega is conserved along the jet a de is also conserved along the jet it's a conserved quantity so this is something that doesn't depend on radius so if we go and along the magnetic field and asking how does it accelerate then the only parameter that controls the acceleration efficiency is this expression B pi times the poloidal field strength times the radius squared or the magnetic field strength times the area that's almost the magnetic flux right in the jet so if we had a jet that was that had straight lines then you can immediately see from here that this is the magnetic flux and we can rewrite this expression in sure in this way you can see this by going to the very base of the jet there the magnetic field is uniform so this is the magnetic flux and you know that Lorentz factor is essentially zero compared to to everything else so this entire term should be equal to MU times 1 because this is the magnetic flux that's the magnetic flux this to go away so we know that mu is equal to this and this works at every radius even though we derived it at the base now it works everywhere so what this tells us is if magnetic field poloidal magnetic field stays uniform within the jet these two cancel each other out mu cancels on both ends and gamma stays at zero it doesn't accelerate so there is something very important here if the jet stays really uniform and nice there will be no acceleration this is precisely why this monopole field didn't accelerate at all because monopole is really hard to change its direction because because if everything is moving relativistically you can only change by 1 over a gamma and that's a really change compared to the two the theta angle the poloidal magnetic flux couldn't rearrange it stayed almost uniform and therefore there was no acceleration you can rewrite this and see it even better in this way if we divide both sides by by mu and collect the terms we can define acceleration efficiency which is the ratio of Lorentz factor to the maximum possible Lorentz factor it's 1 minus the ratio of magnetic local magnetic field times the area divided by the magnetic flux again this will be 1 if the jet magnetic field is uniform and therefore acceleration efficiency is 0 but if magnetic field somehow redistributes itself and becomes the uniform maybe we can get something out of it so you need to reduce magnetic field strength of the local density of the field lines in order to accelerate in that region but how do you do this so how does hydro flow do how do hydro flows do that has I already discussed if we have a hydro dynamic allah azza l-- in order to accelerate you need to expand so that's why nozzle cross-section increases as you go forward the pressure gradient pushes you out here just expanding is not enough you need to expand in a non uniform way and how the Jets the actual just do this if you run a numerical simulation is they do this by bunching up the magnetic field lines in one area so they sacrificed the pole at the and then they able to accelerate near the jet edge so the magnetic field lines go away from the edge their density drops poloidal field drops that's how this ratio drops and the value of Lorentz factor increases does that make sense that's called magnetic nozzle it was envisioned by bagel mail in 1994 and then a couple of russians surya komissarov and i independently arrived at that in a numerical simulation so this is really cool we now actually see how Jets can accelerate and they do so quite well if they are collimated so let's let's try and understand when actually can accelerate and what is required for that so here is our spherically symmetric our model of a jet that is not collimated here is the fast point beyond which there is no way for fast waves to to communicate to this Center so if we try to talk from just inside the fast point then the waves can propagate but if we're just outside at this point then the fast waves are confined to the mac cone with an opening angle of Pi which is given by the ratio of the Sound speed to the local speed or the in the relativistic case the Lorentz factor of the fast waves divided by the Lorentz factor local Lorentz factor and as you remember the Lorentz factor of the fast waves is square root of Sigma so this gives us the opening angle of the mat cone so for the for the jet to accelerate we need to get rid of the flux and dump it onto the pole remember in the previous slide so can this point effectively accelerate no because this point cannot talk to the polar regions where all the flux wants to be bundled up by the solution and so these field lines the ones whose matte cones do not cross the pole they will not be able to accelerate however this field line closer to the pole will be able to accelerate because it's opening angle of the matte cone exceeds the polar angle theta and therefore this field line can curve towards the pole and achieve the local verification of the field lines so the condition for efficient acceleration is that the local polar angle the the polar angle is less than the opening goal in that cone and we can rewrite it in this way the product of Lorentz factor in open jungle is less than Sigma to the one half or mu over gamma to the one half by the energy conservation we wrote out a few slides ago so the limit on Lorentz factor now becomes relaxed it's not just new to the one third to remember that we to the one third that was 10 for MU of a thousand now it's me to the 1/3 divided by the local polar angle so if here the polar angle is a for unity therefore a gamma is limited by mu to the one third near the pole where theta is small you can get much closer to the actual true limit of MU so you can actually in fact construct solutions which reach a good fraction of MU let's say for a mu of a thousand you can reach Lawrence factors of 500 in the polar regions where a theta is of order of 0.1 or less great but the jets that we consider you can see that they are nowhere close to being mono polar these Jets are actually collimated in fact this jet looks like a parabola to me in fact people have gone in and measured and they found that this jet accelerates like a parabola over column it's like a parabola over many orders of magnitude about five orders of magnitude so does this all at all apply to the parabolic Jets can parabolic jets accelerate efficiently or not that is the question as we discussed communication is essential we need to make sure that the field lines that want to accelerate well can talk to the polar axis so let me consider two field lines here a and B and what we want is that B is able to communicate with a and then a can communicate but the jeddak's is this way we can bunch up the field lines in here the pole but you can see that if there was no communication across the jet this Filan would have not known the fact about the fact that the boundary is curving and would have run into the boundary there would be collisions and it would be bad the solution avoids these sorts of collisions and so collimating jets naturally required to be in causal contact the jet boundary i cannot have a mac cone like that or else there would be self intersections and collisions side the jet in fact it's guaranteed that this collimated jets will have much larger macros so in order for that to happen we can find a condition just as the same as in the previous slide that the product of Lorentz factor and opening angle is less than Sigma to the one-half okay that is really really good it means that in such collimated jets we expect to have full causal contact so that there are no self intersections and we also have placed an upper limit on something that we can actually observe we can measure the Lorentz factor through the proper motions we can measure the opening annuals through the proper motions as well and we can maybe learn something about the magnetization of digits from just these very simple observations so how does our model fare with the observations in column eight Jetts we expected the product Lorentz factor and opening angle is Sigma to the 1/2 and maybe less than order unity if the Jets are efficient acceleration in active galactic nuclei we observed that the product is 0.1 0.2 so big check means that will win but in gamma-ray bursts this product is huge it can be 10 or a hundred so something is really wrong in the case of gamma-ray bursts and there could be a few possibilities it could be that the gamma-ray bursts are magnetized so everything that told you can forget or at least in the context of gamma-ray burst Jets but maybe there is some ingredient in the model that we're missing and let's look at what that could be so the jet that we looked at right now in the context of active galactic nuclei is looking as such we have a jet with field lines starting out at the black hole and then they continuously column it as a parabola this is the jet that is limited to the product learns vector and opening goal to be less than 1 or 2 however in a core collapse gamma-ray burst we have a star and a jet being collimated inside the star once it leaves the star has nothing to keep it together so it will expand sideways so can this effect be important let's look at what happens in a simulation so you can run idealize models of jets these are beautiful because they allow you to study the properties of acceleration without one worrying about accretion time variability these are steady-state models so we put a sphere at the bottom we take a wall that shapes the jet collimation so we control all the aspects of the jet and we start with the poloidal magnetic field lines color shows the Lorentz factor log of it Blue is a Lorentz factor of 1 and red is Lorentz factor 1000 and we consider two cases in one case the wall continues to column it all the way through in another case wall column it's until the surface of a star and then D column 8 at larger distances and what we will see is that this jet will reach Lorentz factors of 1 hundred and the opening of about point two radians or about a couple degrees and this one will reach much higher Lorentz factor about five hundred and but twice as large opening angle giving the product of Lorentz factor an opening of about twenty much closer to the observed range so let's see what happens once we switch on the rotation and this is what you're going to see in the code if you run the test problems you see that there is an outgoing wave at which the flow realize is that the reason rotation of the base and so the outflow starts to get accelerated and once the wave propagates out you end up with a steady-state solution and you can ask what are the properties of the solution you can see that here Lorentz factor is yellowish so it's about a hundred as I said and you can also compute what the opening angle is and notice that this scale is about five times less than that scale so it's a really strongly stretched out for you to see the structure on the right something interesting happens once the jet exits what I call the star and the wall starts to curve outwards there is a huge increase in the Lorentz factor that is because this field lines suddenly have room to expand right remember acceleration is associated with expansion so if the few lines before reaching the surface they had to redistribute and you can see already signs of this redistribution there are more field lines here per unit length per unit distance than here here the feelings can just expand sideways and accelerate really rapidly another way of thinking about it is if there is more room to expand sideways there is more room for the pressure gradient to develop and so the pressure gradient is much stronger here that pushes the flow the knowledge that I really like it is if you take it toothpaste tube don't do this please though and step on it then the toothpaste will squirt out really fast this is really what's happening here pressure here is huge pressure here is very low and so the jet is going out like a toothpaste from a tube so what is it going on here analytically actually all the three acceleration regimes and the new one are included here at the very bottom we have Lorentz factor increasing linearly with distance as we go further out we go into the regime where which is limited by the curvature of the magnetic field lines and as we go further out then we have this D confined jet the mono polar jet in fact the mono polar solution describes this regime really well and you get the products of Lorentz factor and open Engel much greater than one so here is the summary what have I talked to you about all right if you want to get a jet you want large scale magnetic fields and rotation combine the two magnetic fields get 100 up into magnetic Springs that expand and take away the energy from the center magnetic fields have only two hairs the spin and the mass they do not have magnetic hairs so you need to provide magnetic flux to them this is the topic of my next lecture how does the jet power scale with the parameters the faster the spin the more power comes out the more magnetic flux the more power comes out the dependence is quadratic as we saw the collimated jets do much better at acceleration than unconfined Jets that's because they can bunch up the flux near the pole and if the jet doesn't collimate then there is no access to the pole that it has so it cannot call them a - well now I just wanted to make it clear that in this talk I focused in the magnetic aspects of jets there are other ways of launching jets like radiation pressure for instance and I think that Jim stone will probably be talking about those types of jets now let me spend a couple of minutes discussing what kind of fun you can have with Jets and this is really the only thing that I will tell you if you want to compare how if you want to see how the monopole has trouble accelerating and get your hands dirty you can easily do this you can go to the web page of the code if you don't have it I will arrange for the to be sent to you go into the code and choose the problem name - this one for one dimensional problem or this one for two dimensional problem and then run it and then you can look at the results and I can show to you right now how easy and it is to do so so how am i doing on time okay great so what what I'm going to do now is I'm going to go ahead and show you the full lifecycle of how you would do this just to make sure that there are no questions so this is a github repository for the code which I called harm pie so harm is the code written by Charles gommi and collaborators back in 2003 they made an open-source in 2006 I picked it up and I paralyze that made it three dimensional added a few bells and whistles and then I was struggling to come up with the code name because it's really harm but it's not really that harm so I just added PI to it which means it's which stands for harm MPI and then I I made it one of the aims not really creative but it was it will do so if you go if you go to this repository then there is a tutorial that explains how to get the code and so you get this you are you this command and let's say go to and then you do paste let me make it bigger so you see it can you see it okay so that's it we got the code then what we can do is we can open it up and edit the file oops I need to get into the code code directory okay so at the very top there is a choice for what sort of problem you want to to study and there is a there is an option monopole problem 1d 2d and so on the default is Taurus problem accretion of a Taurus which is a lot of fun I recommend that you run it at least once for yourself too to get the experience but what I'm going to do now is I'm going to run the monopole problem because it's really fast and can run on your laptop in seconds and then all I do is I type make or if you want if you're impatient you can do make minus J 8 which will paralyze it on 8 cores and then the code has just compiled and what I'm going to do is I'm going to run it in parallel on 4 cores 4 1 1 so the syntax here is really simple you say how many cores you want to run it on and say how many cores in each direction and this is a one-dimensional problem so it only makes sense to stuck cores up in radius is there is one cell and theta one cell and Phi so it doesn't really make sense to give any any cells in the direction beyond 1 so you can see that this is the time in the units of flight crossing time this is the time step value so you each every time step takes us about point 0 5 time units and the the grid goes between the black hole event horizon radius and hundreds or thousands already a so it's dynamic range of three orders of magnitude you can you can expand it and run it for longer if you want to get a better better dynamical range and the code will stop at two thousand which is twice as long as the light crossing time of the simulation and now I can run the Python script it's the python script is located inside the code so you can just run it like that once you've got it you can read in the grid and this is all given in the tutorial and then we can read in the lost dump and then we can plot and here I'm cheating I have it in in the history you can plot as a function of radius so it's the greatest the grid is actually in this Kia in in this case it will work for both 2d in 1d grids radius I sliced in the mid plane so the number of cells in the second direction over to and then here I will plot in the Lorentz factor which is alpha times you are party which I also explained and now that reminds me that I haven't computed what alpha is okay so I'm going to plug that in and log and bingo I don't see where the plot is of plot is here okay and now you see immediately the perils of the high-performance computing you can see that the Lorentz factor this is log versus radius it starts to level off but then picks up that's because this is a trail of the switch on wave so in fact what you need to do is you need to run the simulation for longer for this trail to move out but you can see that for a actually magnetization here is about 60 and by a thousand you are only getting to about six or so so you can you can see immediately that the acceleration is pretty inefficient but these are these are just this is just a simple example of how you can make use of this code and solve a real-life problem thank you very much you

Channel: Institute for Advanced Study

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Length: 88min 20sec (5300 seconds)

Published: Thu Jul 21 2016