Neutron Star Perturbation Theory & Tidal Dynamics | Lecture 3

[00:00:18]slow go slow go slow please please go slow don't want to go fast maybe there nothing in it for me you know I go at the place I go and when the Friday comes just stop doesn't matter is that right you're the boss you tell me now she's gonna say I'm just a student that means you have to sit the exam have you told them about the exam did you tell them about the exam oh yeah oh yeah instead of preparing for the exam it's not time we're not starting until it says recording started okay right then um good morning you look very cheerful today I don't know why there something wrong with you it's Wednesday you should be exhausted right okay so um we have talked a bit about fluid dynamics we have talked about modes so what we're going to do today is we're going to spend a little bit of time talking about perturbation Theory a little bit of time because I'm going to set it up so that we can do an application where modes enter okay and that application it's all going to be in Newtonian gravity today uh there are reasons for that I can come clean straight from the beginning the first part I could do in general activity in fact I came in and had a sneak peek at the like Bo from the previous lecture there's a strong connection with what you've seen because it's going to in in Connect things like conserved quantities and simplec products and stuff like that which you may or may not care about but that that that will come in okay I could do that in general activity I could just copy down those lectures and do some things they're related that's fine the problem I'm going to do as an illustration is the problem of tithes how a binary companion raises a tidee in a neutral star I don't know how to do that problem in generality when we get to that point I'll tell you why well I know how to do some parts of it but not all so that's to be done something that people are working we're going to get up to as close to what people are thinking think about as possible today which I think is good so all right so the first thing we're going to do oh the volume came up great can whisper first thing we're going to do is we're going to talk a little bit about L Gran perturbation theory in Newtonian okay extension to generativity is not super difficult but the difference is in newtonium we always pull out the time as being special right other than that ingredients are the same okay but maybe um the natural thing to ask is something like this why should we do this so so depending on your inclination the answer could be well you know the mathematics is beautiful but if you're a physicist or especially an astrophysicist beautiful mathematics might not be the thing that gets you excited okay and so that might not work but there's a nice physical intuition for why we need lran perations I've drawn this before imagine a little fluid element piece of fluid that flows okay now this is Newtonian so I'm going to draw that as a something with a velocity V say okay so this is just in space it moves a velocity V now we've seen that when we do the thermodynamics we do that in this box right that's where the equation of State lives okay and as we move the Box we try to keep track of where the box is what's in the box stays in the Box okay that's their logic and so I need to keep track of the fluid elements for the thermodynamics we've seen that yesterday I introduced sneakily the Lum perturbation I used it in the calculation so this is important if you make statements like I did yesterday that something is Frozen say the number of electrons is Frozen in this box I need to make sure I'm tracking the same box because have changeed the volume I Chang the number of particles and clearly the number of electrons is not going to stay fixed right so I need to keep track of this box so that's the first part now uh you could ask yourself in the oiler equations so the momentum equation has effectively a directional derivative right we have something like VJ rad J V up or down C we saw that yesterday due the plutonium physics this comes up okay now when you did general relativity differential geometry whatever you did you probably talked about the meaning of this directional derivative right maybe people say yes people in the front say yes I'm going to assume everyone else is saying what so now you want might want to recall that this combination has to do with parallel transport right that that's where this kind of thing comes into the geodesic equation in 4D and things like that and that's indeed the kind of thing that is used here right so are now being incredibly Advanced let's say we have a red vector what we're saying is we track this Vector it still says parallel to itself right that's the parallel transport but now imagine you're trying to keep track of a box of fluid that shape changes shape as it moves right so for example let's say that that box is a bigger box just otherwise I can't so it connects so this this this is one edge of the Box let's call it one this is the other edge of the Box call it two okay and the box is one side of the box is here okay now a bit later the fluid flows so up here point one has moved over here point two is slowed down so it moves over there so the edge of the box is here right I want to keep track of this books now clearly if this is this is a vector right but this idea of parallel transport doesn't capture how this has changed because it parallel transports so this is not the derivative I want right the derivative I want is the lead derivative because the lead derivative is a derivative that keeps track of this difference oh careful if I write that with the vi upstairs then it's zero okay so this is not hav't done any calculations so just has tried to motivate why when you start doing lran perturbation Theory neonian or relativistic the lead derivative enters there's no no real way around it people that come from classical physics that never did generativity don't like it but what they miss is for example when you derive in fluid dynamics you derive the equation for verticity conservation really natural to write it in terms of a lead derivative instead of the convective material derivative you used to so when people do this in fluid dynamics textbooks they add in that extra piece you get from the lead derivative in the equation and say oh look we need we get this extra piece don't worry about it that extra piece is just the lead derivative so if you know any derivatives from the top which you know maybe you do maybe you don't then you're winning and it's much more natural you understand what's going on so that's the Preamble okay so now we're going to do some U little calculations is now we're going to define a lum perturbation of some quantity Q it could be a scalar it could be a vector it could be a CO Vector it could be anything it could be a tensor it doesn't matter simply as the oion perturbation at the point so this is lran this oion plus the lead derivative of the flow oh sorry lead derivative along some Vector but this is the displacement vector right that's the definition now I want to do fluid dynamics I just talked about how it's really important to me to keep track of where these boxes would shape is got and things like that because I want to do thermodynamics but I also want to do gravity right and gravity doesn't have a sense of flow like this so the problem is for fluids I need to be lran gravity I need to be oion basically fix my observers in space and I see what gravity does I kind of have a conflict here and you're going to see that I'm going to write the equation as if they were oan I'm going derive them here write them as if they were oan and keep track of this vect okay I'm G to try to describe everything in terms of this okay so it gets a bit you know it's like a mixture of different things so that this is the first definition and the second definition is we say that lran variation in the velocity the upstairs velocity components is just the time derivative of this guy okay so obviously if this is a change in position this is clearly a velocity right and so this is the perturbed velocity in the lran sense but if that's true the metric doesn't vanish under the this definition right because in flat space so this is Newtonian the metric is fixed there no real variations there but the lead derivative of this is going to give you the usual grad side J Plus or symmet you can symmetrize this if you want to be fancy that doesn't matter right and so that means that I need to be careful now like everything when you have a derivative or something that involves the metric not Vanishing you need to keep track of if you're Co or contravariant if it's a momentum or a velocity things like that so here sure you get the thing you expect but you also get the difference in the um between the covariant and contravariant the derivatives something like this okay so that's the machinery now if you want this could be your exercise to work on at two in the morning right when we have the inspection you don't know about the inspection they're looking shock sorry I'm just talking nonsense don't worry about just try to wake you up we're not going to come and check that you're working at 2 o'clock in the morning as am using as that would be actually we should send the campus security that's the thing to do knock a really hard knock on your door at 2:00 in the morning a big man saying are you calculating but the problem is he'll probably come knock on my door and then I don't know what would happen okay now you could prove this we're not going to do that because it's going to take us too long but I want to use this because this is a shortcut okay you can show that if you you take the Le derivative on or the the grandium perturbation of DT plus Le of v as an operator then these two commute it's probably easiest to show in examples but we not we're just I'm just going to use it okay this is true so maybe I should box that and then we want to look at things like Marion number conservation Aron density number density so we had DN plus rad i n that's what we had yesterday right so now I want to make use of this as much as I can obviously so now I can look at this and ask okay so what is this then so I know for a scalar I know that LE of V of n is just V K gr of maybe I N I have that sitting here but it's not this whole thing so I rewrite this as DT plus Le of v on N plus another bit the Divergence okay and then I go to my trick and I say now I want to perturb this so I hit this with a Delta the Delta goes straight through this is easy and then I need to spend some time doing this I think that's one of the problems with the tutorial so I'm going to leave that to my able hel he's thinking what me and now he's GNA you're going to see him start to calculating anyway so this gives me straight away um this piece and then the thing to work out which is not so nice I have to tell you this is not super nice is that okay and then you put do a little bit of calculation now I have to tell you a story about this you know it's very very common in papers and books that people people write think a little bit of calculation a bit of algebra does it leads to this okay what do you feel when you read that you think it's true that it's a little bit or do you think this is long I don't know how many of you have read or seen Chandra seer's book on black hole perturbations and stuff you can put your hand up high you should be proud okay it's not uh it's not a sort of geekery this is real science in chandra's book you'll find there is a statement like this and a footnote okay and the footnote says something like this I paraphrase I can't remember the exact word it says 200 Pages showing this has been deposited in the Chicago University Library a little bit of algebra so sometimes this true this kind of just a little calculation sometimes is 200 pages that you have to leave in the library so you know maybe that was chandra's humor I'm not sure but okay anyway we do this calculation you can see that there's going to be some relationship here this is going to certainly give us gradient or Divergence of Delta VI which was this time derivative here there only one little missing bit which is what we're going to do with this Delta head that's that's the tricky bit so what comes out of this is that Delta n is just minus there's a plus here n gr on XI or Divergence XI or if you want that Delta n Orion is minus Divergence on N okay so that's an awful lot of work for something I could just have written down without calculating if I didn't think too hard because I could just say okay look let's just be oion that goes through all the derivatives without any problems let's go through here we suppose it's a static model it vanishes some background the only thing that that that just comes out straight away the difference is this calculation allows me to use this result when the background velocity doesn't vanish so I can use this for rotating stars or collapsing stars or whatever I like okay so this is General but it's easy to show it in the static or easy to see that it has to be the result in start again okay now um I'm not going to write down am I maybe yeah let's do it the gravitational potential we have the poron equation or M mass per particle and number density right so now if I do a lum variation on this clearly I have the right hand side because these are just numers done that and then you can show that the LR variation commutes with the lapasan again it's a little bit of work okay but so that's quite easy the end of the day that's not true that's not the way to do it sorry I take that back absolutely take back everything I said rewind the rewind the tape sorry tape that was like 19 80s I'm old okay you push that button on streaming service Services where it says go back 15 seconds that's what we should do so we're going to use oium because as I said gravity doesn't care about flows so then the Orion commutes immediately with this I get an oion thing over over here but I just gave you a result for theum thing there that's this is how we collec okay now if you want this is a derivative or the derivative there's a derivative there you can actually integrate that right once if you want and get something that says how the gradient of Delta F behaves right but you have to pick up some constant Vector that you might need to figure out what it does a constant in the sense of the covariant dity we don't need that now so now we have the momentum equation woohoo uh so we need um um something that's we saw used yesterday and now I'm going back here I am absolutely committed to this expression so I want to make use of it right so the first thing I do is I rewrite this so you can if know lead derivatives you know this is a piece of the lead derivative what's missing for the downstairs object is V VJ r i VJ upstairs that's gradient of v^ s okay and so you can rewrite this as DT LV on VI plus the gradient of some object I'm going to call a plus a 5 plus a half b 2 minus and what I've done is I've cheated as I wanted a total gradient I've cheated by saying there will exist an object such that this is true okay this AG is called the enthalpy it's one of these thermodynamical words that it's defined like that and you can see it's defined simply to be able to pull things through here so that I get an overall gradient and then if I have a Time independent flow this whole thing is gone right and then I get Boli's law straight away from that yeah that's right this is exactly right this is now basically an energy statement so that's that's what trick it now might be helpful because this isn't new word and it's a little bit mysterious as to why this should be true so you can um connect I think with what we did before where a single component we had um rad P was in grad new right but this was the chemical potential right and now you can see the only difference here is that I have an N instead of a row it's just a particle Mass so this enthalpy is really in the single fluid just chemical potential scaled with the mass okay so these are connected now in a two components of something with heat that's clearly not going to be true anymore it's more complicated and then the question is is there such a thing there is such a thing but it's much harder to connect with with the ingredients okay but for a single component this is not crazy okay that's a just an indication illustration why this kind of exist now this is all completely primed because the lran variation commutes with this and it also commutes with the covariant derivative of scalers that's easy to show and so and that it commutes with the co D scalers for the usual reason that the covariant derivative on a scaler is really just a partial no Christoper play around with and then that saves a lot of effort okay and so now we get straight away oh sorry h h plus 5 so that's the perturbation equation and this trick made life really really easy for me okay so if you take this on faith or if you go away and calculate 2 o' to prove that this is true it's up to you it does work work out but for now this is just real quick shortcut to this point okay now I want to write this in terms of the displacement Vector side okay but before we start writing that's this thing okay what's going to happen here sometimes thinking before you do something is a good idea sometimes it's better not to think and just do it because if you thought about it you'll never do it right if you think a calculation is going to be too hard or very hard you probably wouldn't get started it could be better to get started and then find out how hard it is somewhere in the line when you're committed right so the fact that it's hard just makes you angry I don't know okay so let's see what we've got this is clearly involving background velocity and no perturbations this guy here I know which I've written it down over here and I know it's going to be a little bit messy but I can write it in terms of the velocity and size so this I'm happy with that I have that clearly I'll get a lot of stuff that I need to expand but it's going to be a second derivative of thei it's an acceleration right as it should be this guy here is a bit messy because I said we didn't like this but actually this is just a scaler so I need the bit I have which I know I can already Express in size because it's kind of here right you need to use the greens function probably to integrate this things but it can be done okay and then although actually we need the gradient of this and if we integrate this once we have this gradient here so that's probably F this Delta V squ is clearly going to be now remember I have to be careful the upstairs and downstairs right so this is really has a metric in it is a contraction so I need both the upstairs and downstairs it's going to be some combination of the two that are written down it's fine okay but I have them and then this guy well this guy is going to be something from here or something that depends on let's say again in a single component case it depends only on the number density and I have the number density over here so that that's also okay so I you see I can I can do this I'm just going to be a little bit messy I'm not going to go through this the steps because I really want to write it in a schematic form okay so what I'm going to do I'm going to write down an expression and I'm really sorry but it looks quite hairy and I'm just going to write it down for the single component case it looks the same in general with a few bits where it says H it's going to be a little bit different okay don't worry about it I want to write it down because I want to point out where the different pieces are in the equation okay it's a little bit long I'm sorry about it so I said we're going to get an acceleration so we know that we get some bits that are the material derivative on this displacement it's not really surprising because we had this thing in the or or equations it should be something like that but then we start getting messy bits where these things combine things like the square of this acting on x i that's a bit mess here not clear what it comes from we get the gradient of the oan gravitation potential that's expected with a really messy piece side J this is why I have to copy it down because there's no way I remember this double derivatives double derivatives because effectively I need the derivative here and I have a derivative Le derivative floating around in here already so it's a double derivative there acting on five plus h that's this combination here and then one more time this is where I've converted this H into M okay so it's just going to be something like this DH n j s j now we step back and admire the work oh no no so this is the general expression for a barotropic uh fluid with arbitrary background velocity made no assumptions okay and so for example if the star is rotating this is the background rotation and you can see how that's going to come in okay and so I think the first time this was written down was in a paper by John fredman and Bernard schutz in 1978 it's a very very famous paper it's particularly famous for what we're going to talk talk about tomorrow it proves that rotating stars are generically unstable improving on work that was done by Shandra in the late 60s beginning of the 70s what turned out was that a lot of people have done that kind of work but because they didn't use the proper lran framework they ran into trouble with some inconsistency this paper sorted that out by using this framework and I think in Southampton the students joke about I have done for a long time the fact that they're all forced to read that paper and they basically they don't it's not like we tell them to read them at the same time right but at some point during their PHD they have to read this paper and then the older students come in and say ah you've reached the point of the paper so that's our particular flavor and that's the way it is so this is a little bit too messy for us to keep on writing down right so what we're going to do is we're going to write this down in a schematic form okay we're just going to say what are the actually I've missed the thing here sorry there's a ton derivative there otherwise this is disaster what is on the board is is clearly something let's call it a which happens to be one times the second derivative right that's this term here can put an a because I can rescale this equation multiply by some number right if I want so it's going to look like that sometimes people want to have a density here to make this you know a proper momentum variable but I'm going to take that to be one for now then we have something that's an operator acting on time derivative ofai that's this here okay and then we've got some stuff all this stuff that acts onai this I can rewrite as something that act on I even it's a bit implicit because I need to go through the root sitting here this has XI and this has XI so I can rewrite this as some Gunk let's call it C aai okay so the equation will look like this and of course if you want to be a purist you'd say well I could have guessed that why do we spend 35 minutes on writing that down I could you guess that that had to happen right because clearly it's a perturbed velocity displacement velocity is time derivative of displacement I have to have an acceleration so the first term is obvious is V time this is clearly going to give me a v time some time derivative second term is obvious and the fact that I can rewrite the rest this is Nob brainer it has to look like that okay but there is Method even maybe not good method but there is method to these Partners we're going to see this soon okay L ends part one I need to find part two I don't find part two dispers okay so keep this in mind because what we're going to do now is today we're going to do an application of this and tomorrow we're going to use this to prove to carry out the Freedman shirs proof from 1978 the stars are rotating stars are unstaple okay calculations are the same ingredients but the calculation I'm going to do now is on what's called dynamical tides and we're going to simplify this by assuming that the star is not rotating so we're going to get rid of this B tomorrow the B will come back okay and so the setup is a little bit like this we have a mass m Prime is in Orbit with another Mass which I'm going to call M these are stars okay we want to know what is the tide that this guy races on this guy okay it's a Newtonian orbit problem when you do this in uh you know astronomy no capillarian orbits 101 you learn that as a f Central Mass kind of thing work out the reduced mass of some single or object orbiting around this you get Cap's law that the frequency orbital frequency squared goes like GM over distance VI you're done that's for part Point particles now what we're asking is this guy M Prime can still be a point particle not going to worry about that now it's not in reality but the other guy has finite extent it reacts it deforms because of the tide we want to work out how and what I want to try to convince to you convince you is that with this framework you can work out how the modes of the problem the modes we saw yesterday the fpg modes and other modes enter into this problem that's the purpose of the exercise okay so it's really is a gory application of everything we done so first so we assume that this is non rotating so that b guy over there is gone so now we're going to cross him out sorry but now you're gone and then need some bits and now we're going to say look I've got this equation suppose I take two solutions to this equation let label this maybe let's call them cyan for now okay they both s solutions to this okay now let's define an inner product for these two solutions basically let's write it like uh this and I'm going to be really sloppy and leave out indices because there's not going to be any real scope for confusion later so this is defined as the integral of the complex conjugate OFA integrated over the volume that's the definition okay no sorry I also going to put in a row here because I'm going to take if I take this a here to be one so the equation is this okay and I need to compensate by putting the density in in product doesn't make any difference it's just the same now with this in the product this operator is sub Mission so this operator here that acts side and what that means is now that we set a to one it means we can focus on this it just means that this guy ITA seek I which is taking this equation and hitting it with it that's what I get right is the same name asai c conjugate so what I get if I take the equation for e and hit it with side and conjugate that's the usual kind of mission argument okay so now why is this useful to me well let's see now I can build this so that's call it a structure this is beginning to sound a little bit familiar to the other things you've heard right in the first thing in the morning I'm sorry I'm getting I'm seeming to converge in a disturbing fashion but I'm going to diverge soon no worry but the language is the same because these structures are the same this is where it comes from right so for fluid dynamics it's very very difficult to write down a hamiltonian picture okay especially for relativistic fluids I don't I'm not sure but I don't think a consistent complete framework for hamiltonian fluid dynamics exists the Gran variational fluid Dynamics exists and is very healthy i' claimed that because I've written a 260 page Living reviews article in that it's a big framework that people are using hamiltonian is really difficult so for fluids most things if at all tend to be L now even that is difficult because if you want to do a variational principle you canot write down a very an unconstrained variational principle for fluids you have to build in something like this so you have to have a constrain variation and that causes difficulties there are different versions of doing that and then the other thing is for fluids we are mostly interested in non ideal fluids the fluids with viscosity and then you can ask yourself is it possible to write down a variational principle for dissipative systems and the general statement in the literature is no that's absolutely Fable and don't go that but myself and others have done that nevertheless because we don't listen to advice and actually sorry I'm going off on the tangent doesn't matter actually intuitively this makes sense because you know that general relativity is dissipative theory in the sense that gravitational waves are emitted from systems right and yet general relativity can be written down in the variational approach so doesn't that tell me that as long as I count all the energy even the stuff that I lose in my system I should be able to do that so some people say no I would say yes and then we can keep on arguing but you know it's it's a piece of mathematics it's kind of interesting but not completely developed I would say okay so we can invent this simplec structure which Builds on this in a product so I've got these two solutions now I'm just going to say well uh because my problem is non- rotating all these operators actually the C is time independent right and that means that ifai solves this then the time derivative ofi solves this also okay so I said that these two guys were two solutions so I'm going to say well I'm going to take one of them to be this guy so here is one of these products and then I'm going to subtract the other guy so DT okay and then let's ask what happens to the time D you so or DT maybe straight D it doesn't matter suppose for today we want to bring that inside this product but the volume is essentially fixed the background is not time dependent so that's really just becomes the time of these guys and so this becomes EA plusa second derivative ofi minus second derivative of XI minus EA e the first term and the last term are the same so get rid of them right this guy and this guy are both dictated by this equation over here so the second derivative is just the c operat times that okay so let's use that so this is minus this this guy is minus c times this guy minus minus EA 6 I and this guy is minus minus it's plus C side but then we go back and look at the definition of the inner product the second thing is this guy first thing is complex conjugated I conjugate the whole thing and you just swap these over that becomes conjugate that one is not conjugate right and so let's do that on this guy I can swap this over that conjugating then just means moving these two guys over okay this is actually minus EA 6 I plus s c conjugate but I just said this is her Mission so this guy here is's that guy there and therefore that's good and this is really big because that means we now have a handle on conserved quantities right I'm not going to go in that direction now but tomorrow we're going to write down a conserved energy conserved angular momentum for the perturbation okay and if you know anything about anything in physics you know that conserved quantities they're like gold dust right because every conserved quantity allows you to simplify the problem by one variable right and so that's why we always look at symmetries and conserve quantities associated with the symmetries and now I should say notter currents or something like that you know every Nutter great contribution from AG gr and variational calculus how we get from symmetries to conserve quantities and this is at the heart of this right so I don't want to do that now I want to do something slightly different what we're going to do is we're going to say now I'm just saying that these are two solutions right but yesterday we learned about most right so now suppose that this ITA is a one of the modes it could be a an F mode I don't really care too much and the other one is some other mode well they could be the same so we just say that it is say and now I'm going to mess it up a little bit but don't worry about it e is this is just notation I'm changing this and then put a label on it let's call him U no let's e CER vectorai isai label it Alpha and then with some time dependence like this so this is the mode frequency that we tried to calculate yesterday with something that doesn't depend on time okay and then ether and this is where I'm changing the notation I'm call all the moon functions Cai just with different labels okay maybe I should not have used there but that's a small thing and this has frequency Omega beta okay squeezes in app and now we plug this in here right but now I know this CI Alpha and beta these are not time dependent so when I take this time derivative it only affects this exponential it brings down the frequency right and it's kind of easy to see that if I take the time derivative of this which are WR out and ignore the fact first I know that it vanishes I just hit these guys time derivative this mixed bit is still going to disappear right but these guys this is going to give me a frequency Square this is going to give me a frequency square with the minus sign because there's an i in each I squar in each of them and then I just get the product of these C Alpha and beta so with this s this guy becomes just the difference between the frequencies squared and then the inner product of these guys so that didn't depend on on time and then e to the I oh I'm going to run over the line oh no people online I'm not going to see the final Mysteries oh okay tough luck guys online in the internet world so this comes out so this should vanish because I shall prove that in general so what does that mean I'll give you a minute to think about it because I need to do some work over here well what does it mean it means that if I assume that the modes never have the same frequency right so that the mode spectrum is not degenerate and this can't vanish unless I have the same mode alpha is beta it has to vanish this can't ever do it but this has to vanish right if on the other hand Alpha and beta are the same this doesn't have to vanish because this finishes okay so what does this tell me Well in the US usual sense of oscillating system this tells me that the two modes are orthogonal under this inner product okay so it's a shame I have to separate that statement from that statement by as much distance as possible but anyway so we have an inner product and we just proved that two modes are orthogonal okay so that means that if we take s Alpha beta just the spatial part then this it could be something like an a alpha say we Square it to get the dimensions right times the Delta and this is just some normalization that I don't really care about too much set it to one what does this mean well this means I have a basis I have a basis in which I can express all the solutions if this is a complete set which I'm going to assume doesn't have to be for rotating stars there some extra subtleties that probably is much harder so that may seem like a fair bit to work but actually that's really brilliant because now I have a handle on a way I can represent any solution to the problem right which is good so this brings us to ties so in the title problem the equation is exactly the same apart from we have the title interaction some title potential C from the partner drives the motion on the in the fluid of this star over here right so just to tell you what this is the kai is the usual have the gravitational potential from the other guy some distance D say which depends on time because it could be orbiting and spiraling whatever and then I expand is it's just the usual GM over R potential but you got M Prime here of M over here and this XI is at distance right but I want to reexpress this in in this coordinate system here right because that's where I'm working and then what happens is you just expand that as yesterday sperical harmonics okay you think this is actually a solution to the pluses equation which we met yesterday so we know that the solution is going to go like radial coordinate to the elf power and some coefficients okay so this is going to look something like this we don't need to work out the coefficients it's going to be some sum over L's some sum over M's some coefficients let's call it v r to the L spherical harmonic and then e to the I IM some orbital phase like that I so this is our problem I want to solve this but they have a basis so I can just say well let XI here simply be a sum over all the modes with some amplitude a t and then these Solutions m i of X just to make make it explicit this does not depend on T there are the solutions there which don't care about time but now I'm putting in a time dependen here okay so also we know that the solves minus Omega n² s n n i let's do it like that be a bit liberal with where the N sits it's not an index it's just a label Okay so just to keep it out of the way you know this right and so in this equation for the modes I can just put minus Omega and TI instead of that guy okay and now it's easy to see let's go that we plug this in yeah um we get time derivatives is second time derivative of this guy because this doesn't care then this guy for the modes this doesn't care about the time let's just go that front this is acting on the cyen the cyen are given here okay and then I get EX out front so it's easy to see this this leads to a actually sum all second time derivative of a and two dots plus Omega n s i is minus sorry what modes remember so this is like um the same problem which you probably seen in mechanics right where you have a driven oscillator the mod the modes provide a complete basis so I can expand everything in them and now I'm doing a driven problem it's exactly like a driven res okay I can expand it in this basis which is assumed to be complete so it can describe expands all the directions right it's totally fair so what happens now is I need to get rid of this side but I have this orthogonality so I just multiply it by some other side right and pull it through so this then leads to um minus I pick up that that guy in the product of s and gra I I oh like that okay so now then I have a an equation that says I do does the amplitude of these modes change because of the tide in this expansion and if the orbit evolves that will change okay if there's no tide this is just the mod equation right because then this is Omega n^ s minus Omega n squ just trivial it's G okay now I don't want to keep on writing all of this because fting enough anyway um probably run out of time also so let's see now what we're going to do is we're going to define a few things first we're going to introduce something called the overlap integral and let's just say saying it's like a projection right this guy here is like a projection saying how much of this points in this direction in the solution space so we're going to call that an overlap integral you just say this is q for the mode n and that is just that thing R and then if you look at what happens here I can do an integration by part to make this K and that's the derivative of s and then the thing that I have now eras sounds good but basically the density perturbation is going to show up so this is just equal to minus Delta rowar KV and this is beginning to feel a little bit like multiple moments because remember that went like R to the L and then we expand everything like this right clearly here and know this should be Delta r n Delta row N is just an expansion of this object that you expand in spherical harmonics this guy is expanded in spherical harmonics over there so this just brings out basically the multiple moments of the body so actually this becomes something like qn is this V that I defined VM I am there's a star there so it's a star there e to the minus i m Omega T if F dot is Omega so F to Omega T I'm just putting in orbital frequency because we need frequencies later okay and this I is just integral over the body Delta row N N for some given l r to the L + 2 DV these are the mass multiple moments now we're getting very close to solving this we should make it without speeding up too much okay so I want to solve these are just redefinitions but they're helpful so this is the thing I have in the right hand side right I just short hand not notation for that but this guy contains a lot of stuff that doesn't really end to the problem like this this thing right and then I want to pull out the time dependence so I'm going to assume that the distance here changes very slowly okay so we're going to take that to be fixed so we have SL slowly evolving circular orbits effectively okay just take that otherwise it get messy if that's true then uh on the right hand side here the only time dependence we have is there okay this inner product is in space it doesn't care about time I can take that outside okay now you have a driven oscillator right with a driving term in non-homogeneous differential ordinary differential equation some time dependent term over here what you typically do is you guess the solution and say well solution is going to have the same time dependency look similar to a driving force it's going to have to go like e to the minus I Omega T that means that the A's go like that and that means that for this guy you put get an m s Omega squ right so as long as we're not in resonance so let's just write that down as a warning to ourselves it has to be that the solution picks up that coefficient B from here picks up this thing picks up that coefficient a n s pick up the difference between the frequencies and then time dependence and that's in principle the answer it says that there will be resonances in this problem but the orbital frequency approaches in this sense the modes okay and this is going to blow up the solution is not Val but we can try to fix that some other time okay but the way from resonance is this is and actually you put this in with all the modes that we know in this expansion and there's your solution but that's not quite what people do so let's see if we can take the final step we should be able to so instead in principle we're done but we have 13 minutes or something to waste right so let's do that at this point people tend to introduce something called The Love number from a geophysicist called Arthur love that did this for the earth a long time ago and then Tanya hindra and AA Flanagan did this in gravity not so long ago not as long ago as arov um and so the idea here is this you introduce a number let's call it K LM because we're carrying both the L's and the M's for now where there should be a factor of two I think for convention but doesn't really matter that mediat this guy here is the love number um that mediates between the tidal potential which we know and the response of the star okay how the gravitational potential of a star changes because it's lives in the gravitational potential of the companion and then there's some proportionality constant okay that's the love number but we have everything we need for this because we have the full solution we know how the star responds right and so all we need to do is translate that into this and we can compare because we have this we can write down an expression for this that's what we're going to do okay I hope that's what we're going to do so how does that go well we need that guy so what do we know so we know that the full gravitation of potential perturbation is a sum over all the gravitational potential perturbations associated with each of the modes right that has to be true was our expansion is in terms of the modes with I'm amplitude take all the modes out them up that has to be true and then I look and go well I know Delta fi because that's also related to this Mass multiple I have the Plus on equation which I can integrate with green function okay that just says at the surface we get Delta f n of R is some number 4 Pi G from the pon equation there's some two L + one number there's an r l to the + one and there's the ion the same mass multiple is before okay and then I know for each mode I know this right the a are here I have that expression okay and then have more because I know that the whole thing is expanded in spherical harmonics right so there's also ym attached to this okay and then I just compare to this I'm done and I'm not sure how much of that we should write out actually that's just no it's not forth directing that it's just a long expression I write out the whole thing let's not do that because it's too much so if we compare we get two expressions first that KLM the coefficient in between the two things that we know comes out as a number probably isn't that important what the number is it's 2 pi g 2 L + one R 2 l+ one the r is the radius of the star VM VM is good sorry the sum over all the modes let's say this n Prime modes for given l and m before we summed over absolutely all the modes right now we want just the modes for a given l and m so it's called them M Prime okay just to make a distinction and then we have that funny sum we had this this funny combination we had here there's an ion there there's an ion star so that's just going to be an I squar it's going to be an ion squ over normalization which we could could have dropped but it's there Omega nus M Omega so this is not the usual Expression For The Love number that people have in gravitational wave physics because because this depends on the orbital frequency this is more this is a part of the dynamical tidee that evolves as the orbit evolves okay and it's actually the equilibrium part because we haven't taken into account the resonances yet we're not going to because that would take much much longer the static love number the number that was people using gravitational wave searches um follows from this in the limit but the orbital frequency is really slow so far away okay so we just drop this piece and you see that there's a k km that goes like 2 pi g 2 L + 1 R to the 2 L + 1 some numbers some of the modes that overlap multiple thing squared over normalization squared mode frequencies squared and you see that this is not dependent on say the m there's only one value for all the different DS so this is a long long exercise in a way starting from the need to do lran perturbations writing down the mechan mechanism the calculations how it goes up to proving the orthogonality for the modes turning to a real problem and applying this fality right just to see how it goes okay and it's taking us to something that's being used actually not this form but is relevant for current gravitational wave observations and something that will be relevant for future gravitational wave observations because we will need to worry about the fact that this there's an enhancement here as you approach the resonances we need to worry about what happens as the mod go through resum that's something that a few people are working on right now with the eyes on Next Generation gravitational wave instruments okay two comments if you just want to calculate this static love number this C what I've done is just idiotic okay because all you need to do is solve the static perturbation equation a and then this is just a couple lines so you wouldn't go through all of this the problem is once you've done that you're done you're not going to get any further this argument gives me this and it also gives me the framework for working out what happens at PR so I'm going much further on reaching higher it does one more thing I'll just write it down and then maybe to this is a hint right you can take this sum over all the modes try to calculate these numbers right the frequencies and thing plug it into this and ask if I do that much simpler calculation for the static perturbation I should get the same answer this sum should converge so you can prove that we've done that okay and it's quite an interesting thing because the modes could be different depending on things like um the what we talked about those G modes and so on right but the static perturbation doesn't know that it doesn't involve any of that stuff we talked about yesterday that's effectively in the infinitely where there are no reactions or anything okay the star is kind of always in equilibrium does know about p g modes all they nothing but I I can take different stars with different G modes here and then still always convert that number now that is an indication that there is a universal relation between the mode sum and different properties and that I think someone that's SM should be able to prove that the mode sum is a step towards what people call Universal relations and the most fashionable in that respect is I love Q where the moment of inertia the love number and the quadruple moment are related with different equations of state but it seems to be a very strong relation between them I think you can prove those things with this machine hasn't been done so there's work to do finally 10 seconds this calculation I don't know how to do in general relativity I'd love to because we should for realistic equations of State Etc I have no idea I know where to start is difficult because I don't have a proof of mode orthogonality and I have serious doubts that the from black hole perturbation theory that the modes will be complete so that is unfinished business it's a difficult problem that me and others are working on but I don't think we're going to solve it soon there's an open problem you're welcome to it I think for that fin on time we stop any questions before you're allowed to go yes Sho um if you have rotation no what happens is so there's been work on the rotating problem this exactly the same thing every thing gets more complicated we will see tomorrow how rotation splits the modes into there more modes if you like rotation also enters into these all these orthogonal properties again we're going to see that how this changes tomorrow because I'm going to use this for slowly rotating systems um and you can but you can do absolutely everything up to here it just gets messy because you have to be very very very careful with those orthogonality properties in which modes are independent and things like that but there's no dumping to go with it it's just exactly the same but longer ah okay okay so this is uh obviously set up for one star right you take the other star it has the same kind of expression as long as they don't talk to each other so if this star races a tide on this guy then this guy races it you just add them up but then the deformation of this guy is going to deform this guy so the deformations are going to start talking about talk talking to each other but if this is a small effect and that's going to be quadratically small so probably you can ignore it as the stars come closer in ins spiral at some point this problem is nonlinear and you can't throw away those pieces that's numerical relativity which should be numeric relativity so for now in perturbation Theory you just add them but at some point you can't get away with that okay okay anyone else great well go for lunch