Gravitational Waves from Neutron Stars Lecture 4 by Nils Andersson

[00:00:11]so today the theme will be instabilities and I hope to get to the point of talking about why we really don't know if some instabilities that we think should work and operate in neutron stars if they are actually active in real neutron stars or not so that's what I want to get to I've got to cover lots of ground some of it is going to be a little bit on the mathematical side for a bit you can fall asleep then if you want to but I think it's quite important to understand both the history and some of the machinery that you need to use if you want to really understand these things at a deeper level okay but I'm going to start by talking about something that really laid the foundation and that's the work by Chandra on ellipsoids so this is pretty much going to explain in a way everything or much of what comes after and much of the work that people did later so Chandler's work was early in the 70s ish 70s and the work from but in particular John Friedman and Bernard shirts there's a sequence of paper starting in the early 1970s culminating in 1978 the famous papers are from 1978 there are two of them many people in this business would cite these papers automatically without I believe having actually read them the most interesting papers in many ways and not the papers from 1978 but the earlier papers leading up into this because you can read the whole sequence you can see how they struggled with the ideas and what they were trying to do and then finally and actually John Friedman and Bernard separately wrote papers leading up to this you can see there's only when they combine their forces and come together and join their thinking that these white papers which are really masterpieces in some sense or also confusing but okay they are fantastic papers but you can see that when they come together that that that comes that works out okay but I wanted to start drawing a diagram and talking about this background from for for sometimes I'm going to draw it books don't know why I'm going books but what oh I'm not going dribbles do it like this and so I'm going to draw this axis here it's not going to have any labels on it because I think will be confusing but what I want to draw is a book it is a diagram for rotating oh yes a uniform density bodies that are not necessarily no sphere so ellipsoids and this is in some sense is the ratio if I rotate with respect to the three axis in my body so that this is the three axis this is ratio of the two other two axis a2 a1 whatever and it grows in this direction and it also grows in this direction okay so it's kind of a funny diagram and then what I want to do is I want to quantify this you know in a variable I'm going to better speak that is a measure that says I take the kick energy which I'm going to call T and then I take ratio between this to the absolute value because it's negative the potential energy okay so this is a measure of how much rotation you've got um you would think perhaps they will be more natural to write down something like a rotation velocity right but for the the diagram on drawing that's not such a natural measure because we will have systems with differential rotation on this plot so then that you can still measure these energies and get a sort of number okay and now it's useful for your preference that you calculated for me in splendid fashion yesterday the break-up velocity for a neutral star and you came to an answer about the Hertz their bells now in this measure just for our reference for later if you worry about such things then that's going to sit at some value here which is about maybe say 0.1 I'm just putting that there in order to relate what we did yesterday in the tutorial to what's happening in this diagram okay now you're going to start immediately ask well why have i drawn this line above this point because clearly you demonstrated yesterday that no star will spin faster than this because if it does it will throw the matter off of the equator so you can't go higher than this okay but the assumption in the ellipsoid is you don't worry about that thing there is no such thing as throwing matter off of this ellipsoid the matter is locked into it okay and you just talk how fast can I spin this thing okay so the first thing we really need to do is we need to give a name to objects that live on this axis so uniform that's where these we have just a feudal type deformation so these axes are are the same actually is probably decreasing this way by doesn't matter right so they're equal to one here and then decreasing this way so you make one of them bigger than the other so these guys are called the Maclaurin spheroids that go through some ethic names in mathematical physics in in a way this is this sequence here where the two actives are the same so it looks like oh taping flap and sort of think that if you keep on going it becomes some sort of pancake I circular pancake but then what you find is if you try working this out you get to a point at this point one say point 14 a very important point where there's a bifurcation of this sequence and you have a branch off to another guy another set the guys over here these are called the Jacobi ellipsoids again Jacobi is a famous name so these are different see because these have got not the same value of these two axes so they got three there all the three different axes in the body are different right but so we need to figure out somewhere happen we do that and the way we can do it if by all I introducing some kind of differential rotation okay so put in some either internal velocity field or let it push it some other way okay so these guys rotate around the three axes as before but if it's only when you look at them in the rotating frame if you rotate with them that they appear stationary which means that they have some kind of differential rotation okay in the inertial frame basically these have lower energy it's going to be important and these guys but no circulation so there's no internal flow and then in another way I can do this I can break this this way and go off to something called a Dedekind sequence these guys achieve different taxes here by how the internal circular flow right so you can get the set up by basically saying that I let the fluid inside move differentially these have internal motion and because they have internal motion they have circulation and they have essentially the same energy you can arrange it to have the same energy as these guys here well I've drawn this diagram for the simple reason that when I get to this point if I try to march on along this uniforms Maclaurin sequence alright once I get above this point it would be cheaper for me to move over here because I can find a lower energy state with the same angular momentum okay so provided I can find some way of losing energy while conserving angular momentum I can move over here okay and that means effectively that this is this sequence the Maclaurin is not stable if you introduce some extra piece of physics so if I put in viscosity right what does miss Casa T do it lowers energy viscosity will drive me this way so and because we need to put in some extra physics and the rate at which this instability proceeds depends on how fast these physics mechanism is is called the secular instability secular here means simply associated with some other time scale the viscose is an example but it's a way that you can go from the maclaurins to the Jacobi sequence okay and this is the classic totally a very classic result which point what you need to add okay while you get it all okay well you can just imagine you're trying to sit here but this is the top of the ridge okay so it will be very nice for you to stay down here because this is lower energy need to work less to go down here okay we're cheaper live right but you need to figure out a way of spending the excess cash you've got all right if you find someone that they ask you to pay tax at a certain rate and you call that guy viscosity because the positive taps energy from the system while conserving no then you can move down here okay and in fact if you put any such mechanism in how to move down all right so that means that this branch here isn't stable you would have to set the system up in a very specific way and avoid all viscosity to continue up okay so a real system will not be able to do that but then you have to ask okay how what rates of tax is it am i living in my home country Sweden which is the highest tax cut in the world whoo-hoo or am i living in in a country where tax is illegal and no if I love living in the taxi legal nation which would be a bad place it would need a hospital to school and stuff like that and worry about it then I can carry on if I live in Sweden I go down here immediately okay yeah so now you can ask okay well that does one way now how can I go down here well that will all be cheaper but now I have to confine some way of going that conserves circulation okay now this corset II doesn't do that but applicational way the patient does so if you have have because these are body that is deform it will rotate gravitation waves right well actually not here it is not but if you just perturb it a tiny little bit right then it has different axes here so it has one of these at salons we talked about and therefore it will have to go so as long as you fine-tune it so it's perfectly symmetric will carry on up but if you just kick it a tiny bit like gravitational wave deformation it's going to kick it off and it will fall down to this branch so okay I see a third circulation along the top so I usually talk about circulation in in gravitational wave standards right when you can show that gravitational waves conserve the circulation in the flow and therefore that that's the past you're going to follow so what we learn if we're learning this I love this diagram learning a huge amount from it about the ideas I mean and but if it is messy so we learned that this guy the McLaurin guy is insensitive ok you don't have to push him very hard for him to fault art but let's at all just for a experiment that I very own bravely anyway it tried to I don't have any viscosity and I avoid deviating from this perfect symmetry I carry on marching upwards how long how far can I go will there be an end and yes there will be an end when you get to about twice this about a point two seven basically the end we have a dynamic instability and here dynamical means that it will and even if you don't have gravitational waves or viscosity it just relies on if you like the equations without any extra forces okay and at that point this this body that was symmetric will deform into a bar on its own without any strategy or the bomb with instability okay and in principle if that instability acted in real systems this would be very interesting because it grows very quickly deforms the system in rotating bar essentially you've done calculation for rotating balls you know they radiate gravitation waves right so if it grows like that spontaneously that will be very interesting right of course you can see from my diagram that that might be tricky might be tricky to get there so how do you get a real star up to this kind of thing where I've said it's actually difficult to stay there okay so one idea could be that you get this in Mathare is born in the collapse so for some reason the collapse is such that you get a whooping large amount of angular momentum so this U is large from the beginning and people have shot lead in other different Nova collapse proceeds you get up to this threshold in some situations so you trigger this instability okay but unfortunately what happens next is that you hit the high density region to get to what they call the core bounds so then the system expands again so you trigger this instability and then immediately quench it by having the growth you can get this in collapse supernovae but it might not have time to do anything another situation but this can happen is in neutral mergers because now you're taking the two stars wacom in together there's a lot of angular momentum here and actually the system naturally forms the bar anyway I'll show you some that's going to come tomorrow some art snapshots from simulations but essentially you can imagine taking the two stars they go around one other is it when they come together that looks like a bar in some sense and that could also be above this threshold so people in numerical relativity have looked at this quite a lot and there is a very simple reason why they do that this is a simple simulation to do because you don't have to add any fancy physics the fluid will do this on its own so you have to just set up your code whatever it is so that you have a body rotating fast enough and this should happen and for some time people thought this could be quite a good gravitational wave mechanism now I think people are not so convinced because it turns out that similar to the mechanism we're going to talk about later there will be other molds that interfere with this and basically stop this thing from growing it means that it doesn't last very long and you remember when we talked about rotating deform stars the signal itself is weak so you need to have many cycles to make this detectable and so if you disrupt this thing before it builds up these many cycles you don't win okay so it's not clear certainly not in my mind whether this is going we're actually going to see this kind of thing energy but it could happen in principle okay okay so now talked a lot about this you're thinking okay but real star not yes yes that's right so over here you still have uniform rotation in the sense but over here you have in turn that you have still have differential rotation if you look at it from the inertial frame and over here you have internal circulation so there's an internal flow I could have shown you a picture of a contra member right I've got it anyway but it's got sort of little arrows or the velocities going around at different different rates it's like a football American football rugby ball whatever you like with internal flow where are we getting it from those numbers or how do you calculate it the T is the kinetic energy suggest a rotational kinetic energy and of course if you got in differential rotation internal rotation that's a bit more complicated calculate the W is just the gravitational potential energy so that's it okay so you might wonder if this has in connection with what we've done what you've done in fact it does so let me show you this this diagram so here is a rather complicated plot that I won't talk about this one for a little while you could start asking okay what what is the relation between these deformed ellipsoids and the stability here and the moulds of the star right so we looked at the estimate the fundamental F mode yesterday and it turns out this is a very close connection with this poutine with this picture and the calculation that you did so now we need to just ask a different question what happens to the F mode if we rotate the star well first of all we saw already that we had some frequency that went like L or M and we know that these are going to split up into two let's do em when we take the square root to go to plus 1 and the minus 1 and then we looked at that pattern speed which is minus Omega over m in a non rotating star this is a general statement but if we add rotation then we need to add or subtract depending on what frame were using an extra piece that just says that if we rotate there is an angular bit that comes in when you go from the rotating frame to the inertial frame and then there's also going to be some other bit but we don't know which depends on what kind of fluid it is so this is just translating the frames okay and this piece is saying well depending on what the fluid is it's going to react to the rotation in different ways say through deforming and some depending on how compressibility says oh I don't know what that number is because it says what the function is we know that centrifugal deformation comes in at Omega squared and so on I don't I'm not going to worry about it okay so in this diagram I'm showing a number of things it up here is one of those modes the plus one say and down here is the minus one both of them have two branches you can essentially think of this is a complex thing or two of these but then so this is for one value of M but I have the opposite sign of M that gives me another pair okay so I can think of either the two learners really it should be want them and the other one down here correspond to the same value of M and then the other ones are the opposite okay there's a symmetry in here and then you see what happens this is I spin the star up in this beta okay one of them let's take the top arc goes up a bit and then it bends down again the other one is this one goes up and it goes through zero okay it goes through zero at this value here which you can probably eyeball is point 14 okay now let's trace what happens next it carries on up and it meets that first guy they come together and suddenly there's only one line here and that is because this frequency of the airflow becomes complex and so here is the imaginary part and you can see it picks up an imaginary part positive and negative but because we have assumed eetu the I Omega T if you've got a guy with a positive imaginary part that's fine because that's going to decay but a negative imaginary part here is going to be unstable okay so in this diagram you see the ball mode instability here at 0.27 and you see this point where it changes sign okay so this is where you could have that first bifurcation and here is the ball mode instability in the F mode of the start okay it's the same plot for just one of the pairs but not Ian and in beta but in frequency but in pattern speed and that's just to show that this pattern speed go drew changes sign here okay this takes this is the hard regulation the one we did but it extends what the stuff that you did to rotation and it connects cells the mode picture with this ellipsoid picture okay it's that big step so now what I want to do is I'm going to write down some stuff I have no idea how long it's going to take it could be all day but I want to write down a piece of mathematics that proves rigorously what we have these two instabilities okay and that's going to give us something that I kind of already know now so this next exercise will be a complete waste of time okay apart from this is the machinery that you have to use if you're going to do these kinds of problems okay and even though it's not clearly spelled out in all the paper this is where everything builds on this is the masterpiece this first thing was a masterpiece of Chandra and this next pitch is Friedman insurance what they added to the picture so we already see what happened at point 1 for the pattern speed changes sign what does that mean it meets that a mode that started out going backwards or forwards we started out going backwards is now dragged forwards by the rotation basically imagine sitting on a non rotating star you have two s modes one goes this way one goes that way it doesn't matter to you this way or that way because the star isn't rotating you have no reference direction okay now this starts rotating suddenly the mode that goes backwards will appear to you to go a little bit slower backwards right because I'm dragging it forwards this way and then we get to this point 14 that F mode will now appear to you to be forwards or to me to be backwards okay so now let's ask the following question let's suppose the radiates angular momentum okay because use it got edged clearly for you it will radiate positive angular momentum because it's going forwards right okay I'm sitting on the star is going backwards you're taking away positive angular momentum it's negative already so you're taking something positive from something negative I'm now in more debt than what I was before right so the amplitude of this mode will grow okay that's the picture so that's why this pattern speed change is important so what I want to do is I want to sketch talked a little bit about the machinery and then sketch how you prove this and this is a I think certainly for me because I'm a little bit first of all I'm very sloppy and secondly I'm marginally incompetent so when it comes to calculations I always have to do things 100 times and so for me going through this calculation what we do now because this is now I know it but the first couple of times this is hard okay so don't worry if you think I never want to see that again okay that's not but you can't leave because we've locked the door right so we're going to do now is um it'll be closer at the court tip that I wrote down before but now we're going to try to make it precise and we need to do that for a very specific reason I will explain that so we're going to take a quantity Q it can be anything and we're going to define the Lagrangian pair perturbation the one sort of also long with the flow all right as this oil Irian perturbation which is at the fixed point which we use before and a piece that is that we take the lead riveted along a displacement vector so the difference between these two points where I am now and where it was before of Q so how many people are familiar with lead riveted and supply please come on don't be shy it's not a shame just means that you know a little bit more than I do that's okay okay so I'm not going to this the differential geometry of this that'd be kind of pointless but if you want to think about what I'm going to write down it is sort of meant have a mental picture I think the best mental picture of League relative I have is from the mathematician Arnold you say called it the fisherman's derivative okay so you can imagine there are two different ways of taking derivatives derivatives there's a river and you can stand on the side and do your fishing that's just the normal derivative you look at the flow and see how it changes as normal derivative or you can do your fishing from both that flows around and flows along with the flop okay that's the leader so what I do I use this fancy thing here well I use this fancy thing here because if this Q is now a geometric object not a scaler a vector tensor something like that then it will now suddenly be pet on the make a difference if it's coal contravariant and so on I need to be careful about the geometry because the way I drag the coordinate systems depends on how on this flow okay that's why we have to make this distinction okay so what happens I'm just write it down if you have a scalar then this Delta F is just normal it's just as any derivative then the leader in v2 of F is just the VEX that vectors I contracted with the gradient of s this is what we used before because we used it for the pressure alright if you have a velocity velocity field V I say contravariance because it's a vector then this guy is the set of the I sorry the same piece the one that's easy to remember you just put the V I instead of the F while it has a second bit which is not so easy to remember and then you have the minus sign and you know what happens in minus signs I put one int into the exercise yesterday where I shouldn't have so that's what happens two minus signs already has the same form so it's also easy to remember just take exactly the same thing just swap over the exercise and the B's okay and you learn something immediately but if I take this derivative along V of V itself this is going to vanish because this becomes V J graduate VI VJ graduate is zero now we're squeezing the covariance done here that sounds easy but unfortunately this leader rivet Evac ting on the metric that would lower this index even the flat space-time metric is not zero so that messes up what happens to the core and object okay still has the same first piece which is always the same whatever the object like whatever you know number of indices you have and so on but then there's a cloth and the second piece is a bit messy it still has a V but now the index the free index is on the derivative okay and this is simply a essentially you can now figure out for me it could be your exercise figure out what is the leader it've of the metric right because you know this is lowered by a metric so you can work out from the product rule what would that have to be for these two to be consistent and I see why this becomes much much harder to work with okay because you need to keep track of our way upstairs Downstairs is not just simply thing out with you've got covariant derivatives or through the like that's not the goal because the metric is now affected by this so this is important because I'm doing is in with IJ indices because I'm going to work in three dimensions at the moment that is not zero whereas the beauty of covariant derivatives in relativity and flat space whatever is that whatever metric you've got just goes straight through now we need to worry about it and you can see if you're familiar with any relativity calculations but this is not going to be useful it's going to be terrible okay so now we have everything we need in principle we know how to work out the perturbations of a velocity field and so on I'm obviously not going to crank through the whole thing just want to show you what the essence is so I'm not going to crank through the algebra but we can show one very very important thing and for a moment brief moment I thought that we could do this in the tutorial but then I had someone one of my able assistants have a go at it and I think you're still working on it right I think he will report back before 2 o'clock if this is doable by mankind and if it's not then we'll change our mind ok I just thought it would be nice to do something related and we'll find some version of something that's doable Y has to be doable sort of within a limited time some of these calculators are just hard ok so the reason one one one reason why these calculations are not so bad and you are not so bad they're still not so good but they're not so bad if you can show that the combination that you can run it achieved from the Euler equations and so on namely the time derivative plus the Z derivative of V of say the downstairs table this is true for all objects but I just take the soundstage velocity field so this is something I can build alright I just put this guy definition I have it down here put the V instead of X I copy down I know what this means right and now I act with this on this Delta well I know what that means I know I need to take this and then ask what is this object how many indices have I got on this object I've got one free index downstairs that means I should apply this rule and now I have to hit this whole object with this this whole thing goes in there and here okay and then the other two is X eyes just like that so I can just almost cut them paste okay then I have to crank through try to move the derivative derivatives around and that's messy part but if you do that you can show allegedly I have done this but not today that this operation the Lagrangian variation amuse with this operator and that means that any time you see this you just bring that there that's straight through the continuity equation and the first stop is to say okay let's get rid of as much as I can and so I rewrite this as DT roll or maybe DT plus D i grad I roll that's a piece or acting on roll and then I have another piece which is VI grad I roll okay and then I go back and I compare notes with over here and I see that this is a scalar of course so this piece here is just dt plus li of v acting on roll like that all right and now i know if i hit this with this delta that just goes straight through this piece so this piece is sorted I don't know just need to worry about what happens to this piece okay so that's one way that you can make use of this and what happens at the end which we may or may not work out later today it's just that it works out that this is a overall derivative of an object that looks like this and that just says that the Lagrangian variation of the density depends of how much expansion and contraction you've got of this de specter of course the big job now is to try to work out the Oilers and I'm definitely not going to do that for you so take us a very very long time but I want to do it matically so here we the first step is exactly what I said you leave right what we write your equations as a piece that has exactly these combination again but now because it's a vector or a collector I use this whole expression and then what's left and you see that this picks up the piece of the convective derivative and then you're left with the usual one over Rho grad IP and then off grad I 5 minus 1/2 V squared vanishing so you've got this sort of kinetic energy piece all right let's make a big noise get to bitumen if you should be a rule Liva okay what happens to this equation well lots of work but schematically is quite easy it's going to have three pieces it's going to have some second derivatives of this vector and now I'm not going to put the indices on it to come during ribbon AC okay it's going to have a piece that essentially is the rotation that has a first derivative of sine and then it's going to have a piece that is a function of saw in some sense this piece is really easy it's actually just this Pete it's not so simple there's actually what this piece is is something like to roll VJ grad J of et sigh I so this is what this kind of means schematically means it's some function that depends on time derivatives okay and this guy I'm not going to write down in a hundred million years because it involves tricks with the gravitational potentials and swapping things over and that calculation is a real mess okay but this is a free more interested part one this is a start if you start thinking to doing this calculation just for fun than you think are wealth are clearly my life is clearly you know not busy enough right but they had the plan and the plot was to take these ice and say take two solutions it needs a quick equation so the perturbed order equations and try to build an energy and so the way they did that was by defining essentially a dot product so you need one upstairs one downstairs and then this is simply defined as the volume integral of the complex conjugate of the first one times the second one okay so if you know these two functions you can work this out and then they show that with this definition I've got some conjugation rules for these things so I can show that a solution Etha can take side is the same as a is the times side so that's very very easy this is just Rob alright so in this product it's just a number so clearly I can move that number from sitting with the rock side to sitting with the rock done okay similarly you can show that there is a commutation for B but it's not the same is - sigh or might be Itza and they're all conjugated has to be educated because when you do this thing here with the velocities and just one time derivative that comes out and finally one that's really hard because this expression for C is really messy is just that this one is the same as the first one I should be a conduit here needs a conjugates because you need to move this complex conjugates from one to the other so now what I've got I've got a machinery first for working out the perturbations and second for building out of this equation some drastic bits okay and if you want to build an energy you need squares of velocities and so nothing I can build an energy for the perturbations okay so that's what we're going to do so what you're looking for is some quantity that is constant in time so something that is conserved and one way you can do that given these latest is define a guy called double W say sorry normal racket which is simply that in a product of ETA a time derivative oxide plus B 1/2 of Beeks I - essentially shunting that operation to the first one so why should this be true but what you rely on is the fact that these Triton eaters are solutions to this equation right so when you work out the time derivative here right push the time these are just integrals in volume in example brackets right so push the time derivative through first I hit discover when I hit this guy I get a is not a doesn't depend on T so it's just second derivative plus powerful B times this pun derivative okay but then I know that it's a solution to these equations I can use this equation and express that in 1/2 of B and and C okay and then the same trick over here for these guys but then these and then you end up to do something is quite easy to see that that had they have the counsel so these proof at this level is quite easy okay but this is a very important result cut now you see I didn't tell you what these two guys were yet but you see I can put now in whatever I want let's take this excited to be a solution to this but because the A's B's and C's don't depend on time I could also put in the time derivative oxide that's also solution to this right so what they did was they said let's build the canonical energy let's call it EC and let's just take that to be half of this w acting on time derivative oxide and sigh okay I know from this this rules that this would be a conserved quantity okay so now I have for the perturbation a way of measuring an object that's conserved in time and you'll breathe a sigh of relief in just two minutes or something we're almost done okay these guys also because this is a rotating star if I assume it has actually symmetry then it doesn't depend on the fire angle okay so then if the background is actually symmetric then the a speeds and C's don't depend on fire either that means I can push a Phi derivative through all of this and so the Phi derivative oxide also satisfies this equation and so I can build an economical angular momentum which is simply 1/2 w of partial Phi ox I think sighs ok and this is also conserved and now you can ask what happens if I put in this F mode thing I know it's a solution to the perturbation equations right I can calculate that sigh so I can work out this can economical energy right but I also know that I've assumed that it has a frequencies it goes like the I Omega T well for moles you could show that this canonical energy is just minus Omega over m times J see you find this already this is that pattern speed times J C so apparently so this is result from the one that we'll need and we also showed but this is a messy so I'm just going to give you the result an arc while I the sort of intuition comes from it you can use the angular momentum thing and the mode solution to write down an expression says pattern speed minus Omega has 1 plus 1 over m is always smaller than JC over m squared divided by this kind of thing which is some number some positive number which is also always smaller than pattern speed plot miners another thing 1 minus minus 1 over m this is always bigger and this because you're taking away less you're some more here and you keep doing here ok it's a very weird looking expression that is strange ok well now I need to go through a look that I never get straight in my head by self but now we're going to prove the statement from over there ok the proof is here ok so what you do is you first say what happens if we let the Omega be really small okay well if the Omega is really small then amoled that coal rotates so that this is positive right then basically this is positive we're taking away something really small small enough that this is still positive right so JC is positive so if Omega does not if Sigma P is positive then JC is positive as well right we have positive angular momentum so if we got positive angular momentum here this is positive it positive right this is number two so now let's ask what happens if we get up to the point where if have Sigma P goes to zero in night Omega so what happens then well that's the point we've seen over here that the patterns been changing the mode changes from code two counter-rotating okay so now what happens now this is small zero so this is negative this is more negative so JC is negative this is negative this is small zero and this guy will have to change sign as we cross through right because this guy changes sign and therefore it has to change sign and if we assumed that the mode was stable on the beginning which makes sense we start off with non rotating star then this change says that the star the mode goes unstable and that is the Freedman shows proof of the instability of the F mode and this logic holds for any mode that has fight frequency so they didn't do this patterns argument and you get a change in that pattern speed and since then people don't do this whatever it took me 40 minute calculation and of course I didn't do the meat of it the hard part right nobody does this because the masters have told us that the only thing I need to look at is this pattern speed does it ever pain sign I only have to have this yes it changes sign here and then as well some guy is point in the tour show that this means there will be an instability this is a symptom but I think it's very important to understand where this thing comes from okay there's a complicated logic with some fairly hardcore calculation on the problem and this machinery allows you to do much more than just this final statement more littles are derived fairly trig sounding statement that you then explain remember yesterday I took from simple pictures this Co rotating backwards rotating mode becoming forward rotation rotating egg by the rotation of the star positive angular momentum negative angular momentum the mode has to grow argument is it off the construction okay that comes once you know this you come up with that mental picture and then once you have that mental picture you don't go back okay unless you make it rigorous proof that a given mode is unstable oh you release this to little fear in your clothing their information this stability is generate rollers regardless of how slow the nutrition there is that waiting for my dad which information they does the information that this instability is generally even for okay okay okay now that's a really good question okay so that's really that's that's a really good question so now let me erase this piece we calculated yesterday but the non-rotating mode frequency was went like something like root M for an L equals M mode all right plus or minus and then you have the minus M Omega that you need to do is to go into a rotating frame and then you've got some other stuff which I don't know but let's forget about this stuff for now okay because I just want to sketch to you okay but then you know you want to look at this pattern speed is minus Omega over m all right so let's put that in that just says that we get a plus or minus root them to 1 over root m minus or plus or minus plus Omega okay you need this to change sock at some Rotech 8 so you just say well it can't happen for the plus root but for the minus root so that's going to happen when Omega is 1 over root M but you can make em as large as you want because it's the Sarika harmonic index so that can become arbitrarily small so that's a the argument I am okay now that's what it exists you okay now you have to ask and this is that kind of I want to go to and I want to sketch a little bit for another problem so now this is you know this system is unstable but and you know it can be unset any rotation rate but you know that starts rotating so this cannot up this instability cannot operate because it would suck angular momentum from the star and therefore it would stop the star from rotating so it cannot be they act it all stars all the time because we see them rotating so what happened what physics happen because a shorter wavelength high order M will be harder hit by things like viscosity because viscosity is very efficient for short wavelengths all right and so there will be a balance between when its casa t stops this and when the the growth here is due to gravitational wave permission when gravity wins okay and that's really the answer so the solutions yes yes that's exactly I'm going to give you some numbers but for different case just to give a flavor that but that's what I wanted to get to today and so there's another fact that useful which is this is the quadrupole M equals to 2 s mole this change happens at these beta is point 14 in the real star but you calculated yesterday the break-up velocity for uniform rotation was as I said point 1 which means for the quadruple earth mode this change does not happen until you've surpassed the break-up velocity so that also doesn't happen so then we know that viscosity can kill you M for the quadrupole it shouldn't happen okay in Etonian star the FEV mode instability does not act through the quadrupole it acts through octupole three and predominantly before it's a strongest instability now this changes with gr in gr that brings these the radiation reaction is more efficient and so then you can bring these F Modine stability into play but it acts very close to that breakup that we calculated there was a limb plus one somewhere so notice their legs that's right if you take the quadruple inertial or mode which is a current and we calculated that yesterday it has some frequency and you can see that always satisfies this criteria okay it doesn't have to reach some critical value it always does done that's with it very weird in fact it was completely unexpected and the result was discovered by complete mistake and so I should apologize I did the calculation I did it wrong here's an advice that doing things wrong is not really that bad I found something that was unstable I didn't and that it could be this thing but I didn't understand it I talked to John Freedman who said that can't be true encouraging it that can't be true why don't we walk people and talk about it so I thought about it a bit more we went to Milwaukee he and his student or postdoc Sharon morsing had had looked at look at it they realized that yes it was correct so we agreed that it was correct and we had to write the paper on this um unfortunately the album's possibly agreed I should write the paper and they would ruther paper showing that was really and I had to write the paper saying here's my calculation it's all wrong but the result is true nevertheless because intuitively you can see this argument I just gave you it was obvious all alone okay and then another point so that was kind of difficult paper to write because you're the paper you knew was completely wrong but he shows you something that was potentially important there correct that happens in life but Lane later on the summer that followed I was visiting burnisher in his house is outside Cardiff and we were sitting they were talking about this and he said I think because I've done some more reading later and so do you realize I said that in 1978 you're acknowledged by two people that worked on all modes in a paper they did about gravitation radiation in accreting stars so he had a conversation that that's the point where you see a famous scientist kicking himself because he had all the pieces laid out in front of him and he didn't connect the dots and I think that's also connect an example of how when we focus our minds on this is how it goes everyone talked about F modes because F modes are really good at the meeting gravitation waves nothing else matters this thing was there all along right for twenty years until a mistake led us back to the beginning and some I think this tells you kind of a useful story that you shouldn't you know don't just look at what people tell you all the time don't worry about thinking outside the box don't worry about asking well it's the mother field of fluid dynamics of cosmology or whatever they did this can I use it could this be true over here right is those connections the Fluke connections they come all over and quite regularly it's just sometimes we make discoveries we don't want to talk about the impact part of this I did it by mistake okay that doesn't sound so or glorious somehow you know you stumble on something you would it's better to say yes you know I thought long and hard about this and then one night it came to me said I made a sign error calculation and there realized it wasn't wrong after all that doesn't sound so good right so okay so what you can do now is what ask how does this balance okay and we should have just enough time to get to a result of that so what you do is now you want to ask how fast does an instability grow it is always the first question okay you something that's stable but if it grows on a time scale of a hundred thousand billion years is probably not going to happen in reality right if it happens on a time scale of 1 millisecond you can be sure it's going to happen so you have to calculate this right it's not enough saying in principle this unstable a lot of is in the world are unstable yet they don't seem to fall apart so is important so here this is the only case in gravitational wave physics you will be able to say this I believe for almost we need current multiples the current multiples are leading order so this is not radiation by deforming the deputy is radiation by having currents swirling around okay I think I think it's true that this is the only mechanism that is dominated by the current multiples so when we started doing that again this is something we have never done convict can't you know conventional wisdom tells us a gravitation way to come from density variations end of story this current multiple is something one writes down and says people like Barlow calculate because they might need them for some really high order multiple moments that no one ever heard okay here we have this is leading world we need to calculate it it took a little bit of time to get the head around it okay I'm not going to show you what it is and then if we also calculate what the energy of the mode is we have essentially a way of doing that because we have the canonical in it right or you can take kinetic energy just of the velocities doesn't make much of a difference and then you simply estimate the time scale as saying that whatever energy we've got we're going to start with gravitational waves through the multiples and then you have to put the lobsters value I guess because there's got to be negative right so you take the canonical energy for example the currents multipole formulas as you crank out put in the velocity field that we can you calculate for this guy and then you get a number and this number is about 50 seconds ah I think one killer idea well let's do the scaling with that that's probably worthwhile I'll raise its head spin so I'm ignoring all the numbers masses and radio they're all canonical so typical nutristore values is about 50 seconds at 1 kilohertz spin to a very high power like this that's the spin frequency okay that's far let limit at that break up velocity less than a minute in astrophysics is no time at all neutron stars evolve on timescales of thousands of years millions of years this we could be in plain but we knew from this argument we hear that viscosity could be really bad this is now convective currents this could be bad news okay so in neutral stars what people had worried about so here now what happened was a lot of the work that had been done over here for the earth mode was transported over you know a adapt this thing to over here and figure out what happened that's why there was a flurry of activity around 1998 because there were people already knew how to do all of this alright so that was just a sport over here there two kinds of viscosity shear viscosity which is essentially just particle into particle scattering and the typical number is something like in a neutron star not surprising if nothing makes it different like super fluidity or something new there many neutrons than anything else the neutrons crashing to one another much more often than they crash into something else and when you work this out you get given you got the energy you now have an energy loss from the shear and so you get a shear viscosity timescale which is something like seven times 10 to the 7 times the temperature scaled to 10 to the 9 Kelvin to the squared okay so now you compare these two this one drives the mode this one kills the mold right 50 smaller than 7 times 10 to the 7 therefore the driving winds shear viscosity will not stop it at this temperature if you make the temperature lower you can ask at which temperature do I get to be 50 and the answer is about 10 to the 5 Kelvin that means real neutron stars most neutron stars we see a good content of 610 7 upwards so in the real neutral this is not going to win after higher spin rates then there is another kind of viscosity people worry about we call Bock viscosity which is essentially how do how does the matter react you push the fluid like this okay it's out of equilibrium chemical equilibrium reactions try to push back to equilibrium on some timescale so this depends on how much you compress and expand the fluid okay so the first estimates that were done for this for the for the AR mode were just raw not terribly wrong factor of a sort of hundred wrong and they were wrong because the R mode doesn't do this very much it sees a current so this is a higher order effect and at that time we didn't know how to calculate this higher order effect so people just make it up a little bit okay and they made it up using logic that is over here but actually doesn't apply over here and so that's why early on you will see a debate but what the ball velocity could be at the end of the day is kind of irrelevant because the number was still never going to be big enough to do anything interest the calculate this and it's about for completeness two times 10 to the 11 at again frequency dependence is the same glared and temperature dependence because the reactions depend on temperature through a high order and again you see 10 times 10 to the 11 is not smaller than 50 and so again in this balance these guy wins so now we can do this is actually exactly where I wanted to go to you want to say something like this in general you can ask if I put all things together I get some equation I don't really know what the ingredients are but I write it like the energy the change of the mole he does and it's going to be something like I put a minus sign here just to show that it's going to be is going to be different or maybe we should do it like this the odd gravitation weights that mod makes you want to grow and then all the dissipation mechanism is you can think of the ads like parallel resistors so shear viscosity bulk viscosity anything you can think of can go in here okay so basically what's happened in the last 80 years or so is people have tried to hit us with different things here but before we do that just look at these results okay where I've written them take a given star which I've done I see that what happens here you got something that depends on spin temperature spin and temperature okay so I can plot this balance here as a curve in spin and temperature okay and I know all that if I'm this guy shorter if I'm above the curve where these balance and I have an instability okay so typically this curve looks a bit like this and this is the not in ninety-eight type curve it's perfect so what happened since is a own long story of various suggestions why one mechanism or another breaks changes this dramatically probably wipes it out completely okay because this is a problematic curve and we'll talk about that a little bit tomorrow because I'll show you tomorrow how we get gravitational wave estimates for this and then finish by concluding by looking at some numerical relativity results that goes the connects with the earth mode that's what I want to get to but we can because this is kind of now nicely set up we can calculate some gravitational wave estimates and talk about detectability which i think is nice place together okay but so what happens here is you can start asking what what not putting other physics mechanisms yeah then these things so the first thing that happened was people said one of the first thing let's put in the fact that we've got an elastic crust okay what does the crust do they change the Coriolis force at the rotations interesting in vastly overwhelmed the elasticity of the cross the elasticity is irrelevant but what could be relevant is the fact that you have a mismatch so the fluid below the cross and a bubbling in the crust okay and essentially like the teacup problem when you stir a cup of tea you know why does he stop going round it deforms a little viscous boundary layer very close to the edge of the cup it's called an egg man layer okay so people said okay we can have these near the cross and people show that this is really important really quite strong so it lifts this curve up here somewhere and then our people this actually will take two more minute that it wasn't so strong because as I said the crust actually more this is not the teacup problem this is the sort of wobbling teacup problem where the teacup jiggles about with the fluid that's not the same problem so it really is a mismatch mismatch problem and you lose they have a penalty factor of something like a hundred or something that brings you down because there's this mismatch so that I maybe would be the best estimate today for this problem but I think that cancer is completely wrong and let's answer it's completely wrong because there is a magnetic field penetrating this and the magnetic field does not allow that mismatch at all and so that calculation I really don't know okay it's passive in these lectures at the end of the day I don't know okay people talked about hyper arms hi prawns are important here because they can have very very fast reactions okay but if hi prawns are present in the art your large extent in neutron star cores because they got fast reactions they cool the star really quickly so that that goes against cooling data cooling observations so accelerations of cooling intro stars so we think the hype runs have to be superfluid if the Habra's our superfluid DP actions are slow because the superfluity stops reactions so this has an effect but is very big it's a little bomb somewhere so it is a bit more advanced the mugging for the eff mode there is a classic paper by Greg Mendell and Lille in Rome that shows that thing called superfluid neutral friction which is essentially how the Porta C's by which a super food rotates which I think you talked about last week how they interact with the fluid flow essentially electron scattering or from these vortices that mechanism wipes out the F mode instability completely okay because that is very strong so people say surely it's going to kill this guys well that's not you have to make the coefficients of this viscosity 100 times for a thousand times bigger than we think they are to make a big indent in this and you have to make it that much bigger because of that strong scaling with the spin okay so we don't think this wins either and then we have the final culprit is the magnetic field and in principle this could be very important because the magnetic field is very efficient at propagating information but as I said to you the other day the problem with the magnetic field in neutron star interiors is we don't know what the configuration is so this is hard to calculate so my suspicion is we will fix this problem remove this instability once and rules once we understand this problem but this problem is hard the what I wanted to do today I wanted to get to this point be able to say look here is a problem that has a nice mathematical classical underpinning starting with the ellipsoid some really nice but quite difficult mathematics on the ground young perturbations with the rigorous proof of instability going all the way to physics some estimates and then ending up with quite a large number of outstanding questions and what we learn from these ism authors if you put in the Messier things get but in this problem we actually have to do this because we know if you just take this diagram here we know quite a number of systems these are creating systems again for which you calculated things that sit in here okay and according to our understanding they should not be allowed to be here but they are so this story is unfinished but I'm done thank you [Applause]