 Dobro, povedaj. So, we had the third lecture about cosmic rays. Ok. I left you yesterday with the suspense of what is happening to a charged particle in a magnetic field, which has a large regular component oriented and then there is some stochastic fluctuation on top of it, which it does sense to analyze in the direction which is orthogonal to the direction of orientation of the large scale field. And we also made this, well, I haven't given you the full justification, but we suspect that this stochastic fluctuation is associated to some velocity, which is smaller compared to the speed of light and the velocity of the particle itself. And we said, ok, it's a particle in a magnetic field, so its energy is fixed. So, we don't have to look at how the modulus of the momentum varies, but we have to look how the direction of this particle is changing and the equation, we've write it in form of an equation for the pitch angle, which just gives the projection of the momentum on this direction parallel to the magnetic field. So, the question revolved here and now I want to make a little bit of a gymnastic to see when this variation of the pitch angle gets significant. So, I want to make the computation on the blackboard, so I have to go in some simple limit and let me restrain first of all a stochastic perturbation which is associated to one single mode. Ok? One mode, which has some given wavelength, so it's associated to some given frequency and then let me write this component, so I took b in the direction z, let me write the component of delta bx, delta by and I've write it in form, which is the following as there is some amplitude and then I just glue here a cosine of kz plus some phase psi and here I take this to be sine of kz plus psi. So, this psi is some arbitrary phase I'm going to average on in a second. So, if I just plug it in here, ok, I chose the phase in such a way that I get a nice cosine of omega t, cosine of this argument plus sine, the two sine of the same arguments, so then I can go back to my trigonometry course, I have the cosine of angle b, a cosine of the angle b plus sine of the angle a, sine the angle b and then this is what, this is just the cosine of angle a minus b. Ok, so if I take that and I substitute in this, I get that the evolution of the pitch angle is just q divided by m gamma c square root of 1 minus mu square and then I have this cosine of omega t minus kz minus psi. Ok, and then I remember that the motion in the z direction is not touched by the large scale magnetic field so this z I will just write as the velocity of the particle in the direction of the magnetic field so I projected out again with the cosine of the angle and then times t. Ok, so this is just q divided by m gamma c 1 minus mu square and then the cosine of omega minus v mu k t minus psi. Ok, fine. So I have something that now I need to average over this psi which is, as I said before, a random phase. I want to perform the average of what is this delta mu that I get connected to that so you see if I just average it like that then I have an average over psi of this cosine of psi so clearly the average of the variation of the pitch angle is zero. Ok, that's not really what I'm after because this is a variation which has zero average but what about the variance. Ok, so to compute the variance I need to compute this product here. This product here is squared, m squared, gamma squared, c squared 1 minus mu squared and then, ok, I'm getting some finite delta mu and I have to integrate over some time t prime this cosine of omega minus v mu k t prime minus psi and then integrate over d cosine of omega minus v mu k t second minus psi ok and just notice that I did a little trick so I applied this integration over time to the terms that are fastly varying in time so there is some time variation also here but I'm assuming that this difference here is small so over that fast variation I can take that off that out of the integration. Ok, and then I wrote it like this because I go back again to my trigonometry book and the cosine of A cosine of B is equal to one half of the cosine of A plus B plus the cosine of A minus B ok it's very possible that I get some of these wrong so you should spot mistakes just wave the end ok so ok, so this is something that goes in like m squared, gamma squared, c squared 1 minus mu squared the integration over dT prime d second prime and then I have the two terms so 2 here I have the term in which I have the sum cosine of just like that cosine of sorry omega minus mu v k I have T prime plus T second prime minus 2 psi plus the cosine of omega minus v mu k T prime minus T second prime and the psi is gone ok, so this is this is parenthesis ok then clearly I did like that because when I take the variance of this this random phase the term that still contains the psi goes while the other stays ok, so this first term here just drops this second term let me do some extra little basic gymnastics which is the following let me use the representation of the cosine of some product small a times t this is just the exponential of i at plus e to the i to the minus i at divided by 2 and then you see that t is integrated over so I can use the integral representation of the delta function the delta function of small a is 1 divided by 2 pi the integral in dt between minus infinity and plus infinity of this e to the i at ok so when I take this variance I take this variance in a limit in which I'm integrating over a time interval which is so to speak large and then what do I get from here I get this first term q squared divided by twice m squared gamma squared c squared 1 minus mu squared and then I get from this integration over this term I just get 2 pi delta of omega minus mu v k so I pick a prefer the frequency and exactly I can define some resonance wave number which is just defined like the combination that I get from this delta function omega divided by each angle times velocity and remind you that this omega is what is reflecting the large scale magnetic field I'm sitting e to is the larger frequency so it's b is q b not divided by gamma mc so I get sorry, you see I did a mistake here I integrated over one one of the two time integration to get this delta function then I have the integration over the other time which gives me a delta t so now if I just compute this variance per unit time that's as an expression which is just what I get it from from here so this delta function of omega minus v mu k is just one divided by the mu delta function of k minus k resonance put everything together and find that I have this variance pi q squared v 1 minus mu squared divided by c squared p squared mu and then delta b, the modulus of delta b squared delta of k minus k resonance so mysteriously the delta b is reappeering back and I forgot it since the very beginning sorry, there was a modulus of delta b here and then I had it delta b squared here sorry and since I copied it this should be ok ok, so if I want to write explicitly that something in terms of delta b squared normalized to the background field squared then you see there is some combination which comes in and ok, this b squared is hidden in omega, I can introduce omega and at the end of the day I get k resonance times omega and then a factor of pi 1 minus mu squared in front delta of k minus k resonance ok, ok so I'm almost there because ok, now I did this exercise for one mode and I get a variance which is non-negligible when this mode resonate on this k resonance in general I will have a power spectrum of modes right, so I will have a p of k which will be some distribution of I'm not sure you see it here maybe I should go back to the other side so I have a superposition of alpha in waves with a given power spectrum of perturbations with respect to the mean field then the correct way of normalizing it is to put 1 over 4 pi there and 1 over 8 pi here and then you see when I go and compute this variance I'm going I have this resonance criteria which puts me on having the variance to be something like pi 1 minus mu squared and I have omega k resonance the integral over k of this power spectrum so that's why I wrote this combination here that's this except for factor 2 and then I have the resonance condition so this variance is picking for the variance modes this resonant mode and I just get that the final formula is this with power spectrum computed at the resonance k this result we just got is a result for the variance of the pitch angle which is usually defined to be twice this is called diffusion coefficient in pitch angle this is telling you on average p mu mu is telling you which is on average the variance over delta theta the angle itself in a time interval per unit time so if I am looking at delta theta I am looking at this delta mu over the sign of the angle which is here so that is just equal to pi divided by 4 omega k resonance p of k resonance so ok so we get that this particle is responding to the background in this way what's the physical meaning of the formula that I wrote it there what is this k resonance again this k resonance is just what I wrote there is just omega divided by mu v so it's 1 divided by mu times the larma radius and then why am I looking at modes that match this larma radius why am I looking at physical scales which are of this kind well you can visually look at that in this way suppose first that you are sitting on a case in which k to the minus 1 is much larger than the larma radius so you have your large scale and then some perturbation which is orthogonal to that that is taking your magnetic field and bending it a little bit and then you have a particle whose larma radius is much smaller than this bending and what is this is in this plot is like saying that this particle is having a zero radius and like this so effectively what this is doing this particle is doing is just surfing this perturbation you have a surfing of the wave and then the net effect is that the direction of propagation is not changing suppose you are in stead in the opposite limit you are in the limit in which the scales of the perturbation are much smaller than the larma radius then again you have your large scale magnetic field and then you have perturbations that are like this and your larma radius now is in this scale here so again what the particle does is that it doesn't see the perturbation so to have a significant variation of the direction you nearly need to have this matching of k to the minus one with the larma radius so in that case you have something that is of this kind and your particle is trying to do a generation that is of that kind the two matches and then you can get out of this perturbation with a direction that is significantly changed compared to the incoming direction once you have that you have a variation of your direction on a time scale that is typical time scale which is of the order of this one divided by k resonance sorry, omega k resonance p of k resonance and what we did is just to look at this in a frame that is the frame in which these alpha and waves are sitting but so in the frame of the alpha and waves you have isotropization of the pitch angle on a time scale which is of this kind to go back to the lab frame and have a scattering of the particle in real space so you will have a scattering length that is of the order the propagation length which is of the order of the velocity of the particle times this tau scale and then you will have a analogously to pitch angle diffusion you will have a real space diffusion coefficient which is of the order of this variance say in one dimension delta z delta z per unit time and that will be what it will be just of the order of this lambda squared divided by tau so this will be of the order of v squared times tau and again this will be the order of v squared divided by omega k resonance p of k resonance so we find isotropization of pitch angles in the alpha and wave frame we find Brownian motion in real space so this connection here between this dxs and the pitch angle diffusion coefficient here it's sort of heuristic level but ok would I have I don't know at least a couple of hours I could derive the more detail on the blackboard let me give you at least a flavor of what really one does so now what you have to do is to really take an ensemble of particles just not one particle so describe the ensemble and the phase space distribution function for these pieces this will be something that will depend on x on position on momenta on time and then what you have to do is to sketch the probability of having this scattering into p plus delta p within this what we just sketch within this pitch angle diffusion as I said that is not terribly hard but would require little time so let me give you a result flavor of the result is the following first do it it's particularly it's essentially the same exercise I did before but just encoding better this transition probability and an ensemble of state if I do it in the alpha and wave so I have something that is of the following form get a partial derivative of with respect to time of f plus some diffusion the convection term v scalar df over dv and this is equal to one half some diffusion in momentum term so something that looks like a diffusion coefficient and this is applied derivative in momentum of the phase phase density so that is what I would define just like d over dp of dpp df over dp so some pp diffusion coefficient and really you understand here the connection with what I did before before I started like that and then I observed that momentum did not change in modulus so if I do the same in one dimension as I did before this is just d over dmu of the same pitch angle diffusion coefficient I introduced before and the derivative with respect to pitch angle so this is in this frame it's slightly more lengthy to go formally from the alpha wave frame to the lab frame so in the lab so to speak in the galaxy frame then effectively what you get for this question I forgot to say this question is really familiar right because this question has a form which pretty much looks like on the left hand side is like to have a UV operator acting on this term on f and this right hand side is like having some kind of collision operator so this is really looking like a Boltzmann equation for f then when I translated into the galaxy frame the feature that it takes will be similar so I have again a UV operator so I have some velocity u df over this dx what did I write here I told you, shout at me you have this term there then this comes in so in this u I am hiding the fact that there is this mismatch in velocity between the alpha wave frame and the galaxy frame so you have this term which corresponds to the alpha in velocity plus some kind of overall convective effect if you have that the full plasma is actually having some preferred direction in the galaxy frame so that's why I wrote it more generally than just plugging in da this bulk emotion is to induce a term which corresponds to adiabatic losses which takes this form and then this right inside really the isotropization of pitch angle in the alpha wave frame is as I was saying in effect of spatial diffusion in the lab frame so you get it in form which is this one where now there is a connection between the diffusion spatial diffusion coefficient and the pitch angle coefficient diffusion coefficient which again one can derive with some care and in one dimension is just taking a form which is the one that I sketched before the right numbers are you have a v square divided by 8 my integral in pitch angle from minus 1 to plus 1 of this 1 minus mu squared squared divided by d mu mu ok and this is essentially the structure that has the equation that describes spatial diffusion plus convection for charged particles in environment which is this setup with larger scale magnetic fields and small perturbation on tops in reality there are some extra term that so let me put this on the left hand side with minus sign and some extra terms that one has to add when one thinks about the physics more carefully for instance there is an effect that comes in this you will term by noticing that you have an acceleration term and why is it the case that you have these magnetic in homogeneities that have some velocity with respect to this frame so you have a magnetic field that is moving compared to the frame in which you do physics so a time value magnetic field as an electric field in this galactic frame so you can have an acceleration effect which is connected to this time value in electric field and in fact you get a term that you can write in this form one over p squared d over now this p is the modulus which is an acceleration so you have a change in the modulus and you have it in a form which is really like having diffusion in momentum with again diffusion coefficient and this diffusion coefficient will be related as I said to this velocity in respect to the galactic frame so it's something that goes with this velocity a squared and divided by the spatial diffusion coefficient and then to get correct dimension you have a p squared there so you have a direct link between the two since we are connected to the same physical effect you can have a term which is related to kind of net energy losses we were talking yesterday about electrons losing energy by synchronous radiation or by branch tralong or by inverse counten so you will have to put in a term which takes that into account and that will go with a coefficient that is like this you have a p squared and then this is dp over dt which is this continuous loss rate and then this is applied to f and finally I put on the right inside the effect that you have sources of your cosmic rays so there will be some source eventually depending on x and momentum and you may have some losses which is related for instance to the fact that you have a primary particle that is generated into a secondary so from the point of view the primary is gone so the secondary is in this source the fact that you lose the primary will be put in here term which is proportional to f and then to normalize to some loss time scale so I took the whole backboard to write for describing the diffusion of cosmic rays into the galaxy so the jargon is that okay you have this df over dt sometimes we look at the case in which there is vibration between sources, the number of particles that you get from the sources and the number that escape from your diffusion volume so in the static limit we just put this to be zero so that's the condition we use in most solution of the diffusion equation there is this term here which as I said is some term which describes convection so you have a particle which gets glued to some wind of plasma and is carried away from the diffusion region while it does that you have this term that corresponds to adiabatic losses you have this term which corresponds to special diffusion you have this term which corresponds to some kind of energy diffusion and the jargon here is to call this term reacceleration term because this term is a term that effectively can give you acceleration of cosmic rays along propagation except that I showed tomorrow that this cannot be the main acceleration source for cosmic rays so cosmic rays are accelerated in the sources most likely and then we call it reacceleration because it's some slight bump on top of the initial spectrum which scales with this alpha and velocity squared so it depends on whether in your model there is a larger alpha and velocity or that's negligible and again these terms I already wrote what they are the energy loss term the source and the species loss so with some effort and taking some shortcut we got to this master equation then I would like to give you a flavor for what this equation implies for what you observe for cosmic rays first of all I should revert in most cases we are not really interested in phase space distribution function but we are more interested in number density for cosmic rays that is number per momentum interval so one introduces a variable n which is nothing but the integral over all angles in the momentum space phase space distribution function which was dependent on ideas and then you multiply by p squared that gives you the number of particles in p p plus delta p and in unit volume so I I should just take this equation and do the kind of integration ok, it is not that interesting that I write it on the blackboard but let me jump as to some specific solution of this equation so let me assume first I have an example in which what I look for is some source which is dependent on time I will take an impulse for the source and let me assume that special diffusion is the only relevant effect so I have a variation of n with time and then this term here which just gives me special diffusion and I also suppose that is constant in x so I can take the diffusion coefficient derivative and I get d laplacian of n equal to this source function of t so this is a particularly simple equation for which I can just get the green function the green function of this equation is what is just x t is 1 divided by 4 pi d t to the three halves and then I have an exponential of minus x square divided by 4 d t so you have this combination of I didn't stress it but the dimension of this diffusion coefficient is of course a length squared divided by a time so this combination d t gives you a length squared and it's the typical propagation scale squared so if I just take as I was saying an impulse source so I just take q being some delta function some q naught that is happening at a given time in a given point so let me just assume this is like this then clearly the solution of my equation I just have to integrate over space and time this green function so I get that the solution is q naught divided by 4 pi d t minus t naught to the free half and then an exponential of minus x squared 4 pi 4 d t minus t naught so it's something that so this is in a case we can apply to a solar flare effect solar flare event so what's a solar flare you get a magnetic reconnection on the sun and the burst of particle that are coming out from the sun so you have a monitor of the sun that receives this burst and then you see there is some typical scale over which the number of particles has this sharp increase and then you have this exponential blowoff so a model like this fits very nicely in event like that for instance you have only one parameter that you have to adjust and put in this x to be the distance from the sun one astronomical unit you get that this diffusion coefficient in environment at this energy which is pretty low energy is something of the order of 10 to the 22 centimeters squared second minus one ok let's try to have some other example which still is a simplified limit but is something we are more after two that is galactic cosmic rays secondary primaries what's going on there so the second example I want to take is the limit of so-called leaky box in the leaky box approximation you just take the same equation before and what you do is to bargain this term here instead of looking at in detail what is happening as a space diffusion you bargain this term in terms of a time of confinement so you take the galaxy as some kind of black box and what you are going to estimate here is how much time the cosmic rays are staying in the box before leaking out from its boundary so you go from this equation to an equation in which dn over dt plus n divided by this diffusion time is a function of momentum and that's q and clearly here you cannot discuss any special variation so you have a source which just depends on p suppose you are also in the steady state limit so the idea is the one I was telling you before this is a game in which you have sources that are filling in the box and you are reaching some equilibrium at which the rate of injection is compensated by the rate of leaking out of the blocks so you just have that this you set it to zero and then ok, the solution of this is trivial, you have that the number density is just the source function times this diffusion time depends on p ok, so why is this interesting at all well because this is the simplest scheme which describes gramage setup for primary over secondaries that I was touching on yesterday because now I can plug in the fact that for a secondary component the number density of the secondary component is going to be the source function of the secondary times this diffusion time and so this is something that is going to be proportional to the conversion probability of a primary into a secondary times the number density of the primary component times this diffusion time and then obviously you have instead that for the primary component ok ok when you look at the secondary over primary then that's just proportional to this diffusion time which in turn is going to be proportional to the inverse of the special diffusion coefficient ok, so you have this match here so if I take the diffusion coefficient to be some generic function of rigidity of the particle so I'm the rigidity of the particle is defined just as the ratio between the momentum of the particle and the charge in unit of electron charge so this is just the Larmor radius times the larger scale magnetic field so as we were discussing before the micro physics is connected to this resonance on Larmor scales so I expect that if that is the mechanism which is set in the diffusion propagation I will have a diffusion coefficient which will mainly depend on this rigidity and let's say that this is just a power low so it's d0 which multiplies r divided by some rigidity scale that we take to some power delta I can go back to this plot and as we were seeing yesterday this plot is connecting secondary to a primary component and you see yesterday I was normalizing somewhere here you have the dependence of this ratio that at the energy which is fairly large is indeed some kind of power low and then if I take the justice plot and normalize this result here to this observed ratio I find out that for the diffusion coefficient I have a scaling which is of the order of delta 0.5 and normalization that here is of the order of 10 to the 28 centimeters cube second minus one so I'm predicting a scenario in which I have a source of a primary component that is nothing that will scale with some power low in energy this source of primary corresponds to a net equilibrium distribution of primary which will scale like this energy minus alpha and then you have a steepening with T diffusion so a factor an extra factor of delta and a secondary component which scales like e to the minus alpha minus 2 delta so that's the general pattern and what people do is to try to solve the diffusion equation in a scenario which is as much realistic as possible still being computationally okay so for instance the typical approximation one makes is the following you model the galaxy like a diffusion region and this diffusion region is defined in a cylindrical box what's the extent of this cylindrical box well this is the region that we think is filled by this stochastic component of magnetic field which does the job of diffusing cosmic rays the regular component is on a much larger scale but the stochastic component we think is connected to the sources of cosmic rays so it's dying out when you go away from the sources the sources we think are sources distributed in the disk of the Milky Way there is some distribution of most probably supernova remnant as we will discuss tomorrow that is sitting within as the stellar disk of the galaxy and within this thin layer here there is also the gas density which is the one that is doing the job of converting primaries into secondaries and then one puts in some assumption for what you expect for the special diffusion coefficient in this volume what you expect acceleration in this volume whether you expect there is some collective effects which take your cosmic rays and drives it out from the source region so some bulk velocity which convects cosmic rays out of the disk you put in numbers for what you expect for cosmic ray energy losses and so on and so forth you take the question you put it into a computer and try to solve it as carefully as as one can again one can also take the approach of going to simplified models and for instance this is a this leaky box in which we suppressed the special dimension but you can actually solve this diffusion equation in on more general grounds taking constant diffusion and then you have a term like this you can keep convection so you have this u the f over say that convection is in the vertical direction outside out from the source gas disk then you would have a term like this ok you have the adiabatic loss case if you have this ok but let's don't put it and then you have this source function was for instance that the source function again depends just on z and p then what you do is to solve this equation with the boundary condition that at the maximum radius corresponding to diffusion this rd over there radius r equal to rd or when you meet the condition for z being equal to this large h that is the vertical size of the diffusion box you set the density to zero so this means that once you reach the boundary or the diffusion box the particle just escapes then this is a situation in which you can solve again for an analytically I could write the solution on the blackboard since time is short let me write at least one result which is interesting which is that in this model I can compute explicitly what I was using yesterday the grammar of cosmic rays at a given energy or actually rigidity r so you compute it is equal to nh the density of gas in this thin disk this thin disk is supposed to have a height small h which is much smaller than big h and then it's a grammar so I have to plug in a mass which mass of the proton has a velocity and here I'm normalizing to twice the modulus of the convective parameter there and then you have 1 minus the exponential of minus u big h divided by b of rigidity ok so this is something you can work out in some half an hour of calculation so there are I'm finishing soon there are here two effects that I left on this diffusion equation that I would like to compare you see when you have that so there is this combination u versus d over h when you have that the velocity u is much larger than this d over h then this exponential goes to a small number and you get this path length you get the path length doesn't depend on rigidity ok so in this limit here you have x which is not x of r in the opposite limit instead you have that which is the more interesting one you have that u much smaller d over h you see I do the Taylor's function in first order and then I get the dependence of on u which is a drop in so I get x of r which is n h h m p v divided by 2 u times u h divided by d ok so I can group the path length in terms which are what I was using yesterday for my naive estimator of the path length itself so I have a gas this product here times h I normalize this small h over big h and why do I do that I do that because you see the cosmic rays are diffusing part of the time in the region where you get conversion of primaries into secondaries this fin disk but most of the time in the larger region where this conversion doesn't happen so this is the density rescaled on the time so this is the mean density that cosmic rays sees along propagation so the density in the disk is riscated by the time the time fraction that stays in the disk yes ok then there is velocity and then there is this combination here h squared divided by 2d of r ok so again this is combination which corresponds to my diffusion time and you see we can go back to this plot here and have again what I was sketching before with this diffusion time that at high energy so if you have a finite v u we will have since this d is increasing with rigidity you are going from high energy regime to a low energy regime actually low energy to high energy at high energy I am fitting up this slope and finding what is the diffusion coefficient low energy will be sensitive to other kind of effects rather than special diffusion for instance convection or if you have reacceleration you can fit reacceleration and so on and so forth so I apologize for being over time almost on average with respect to other speaker but let's stop here continue tomorrow