Non-Thermal Messengers from the Universe: Lecture I | Cosmic Rays

[00:00:01][Music] [Applause] yes thanks thanks for the introduction welcome everybody I hope you are not too tired um so yes so I I'll cover non-thermal Messengers for the from from from the universe in general and um just as a preliminary remark on methodology um I might not be the best clear uh Blackboard Rider so especially to help those who are following remotely I prepared some notes uh uh hopefully with all the passages and so on so that should be available after the lecture in the afternoon uh so the the the my suggestion is that you focus on the concepts rather than frantically taking notes you will find hopefully all the content in a more readable form uh and um uh that's for the technical part uh of course there will be two exercise sessions tomorrow afternoon and on Thursday by syia mancone um and uh uh I looked at the program you have a very dense uh school program so you won't have time for very deep uh uh conceptual problems so the goal of the problems I I've suggested is really to consolidate the Notions that I introduce they should be rather straightforward uh just applying the the formula and and the concepts used and I also um uh add some more advanced problems uh uh or if you wish reading suggestions for you after the school maybe to go back to some of these Notions and and uh explore for further um um some of these uh Notions um so the topic of the of the course is non-thermal Messenger so the key word here is nonthermal uh which means that we will uh deal with the evolution in space and in uh in uh in momentum so in whole configuration space um of particles which do not interact fast enough if you wish to attain uh balsman or or um Maxwell balsman equilibrium or you know a fider or BOS Einstein distribution for for Quantum uh particles um and so the Notions of thermodynamic equilibrium are not going to be applicable and uh we will uh rather um um investigate um the evolution in nmal in a normal way uh subject to collisionless and collisional type of process processes so these are the two big uh type of processes we will focus on mostly collisionless one in the first part of the first maybe three uh lectures and and collisional one in particular for the last uh two um in general when I give this type of classes I find a rather heterogeneous audience so just for a quick poll at least for those following uh here how many of you have actually followed an astrophysics course in your studies okay roughly what I have and how many of you have follow the plasma Physics course okay so um yeah that that's a very typical situation so uh some of you might find the first part boring uh some of you might find too advanced I don't know uh um so again I will have to go a bit fast on some fundamental concepts that I introduce in the notes you find references also to hopefully introductory textbooks and reviews on some of the Notions and more advanced ones if you uh want to go uh deeper um and that's a typical problem in the in the curriculum on many modern physics uh courses there is less and less attention to Advanced classical physics Notions and uh this is unfortunately what you would need to to to go deeper in this in this uh in this real so uh um a quick overview of the plan so uh today um I will certainly cover what I would call a preliminary ah this is not going to be very nice preliminary Notions um which are not directly related actually to uh uh to um non-thermal Messengers but are some Notions that I need you to to grasp otherwise you will be uh uh sort of lost uh then some uh proper introduction which means some some history some Basics uh things like that uh and I will start covering probably this certainly this part two uh which uh is about uh Co cosmic ray propagation perhaps again about nomenclature um when I talk about non-thermal Messengers I will refer both to charged particles and neutral ones huh but the charged particles are the one the object of most of the uh the early lectures at least for the simple reason that they are the ones uh being being accelerated and propagating non-trivially um the neutral ones photons and neutrinos are typically byproducts of their interactions and they are very very useful for Diagnostics but we will cover um some Concepts later um so Cosmic rate propagation I was saying um there are some Elementary considerations and I would say something like istics uh these are meant to convince you of some basic properties and then uh we will um deal with this problem in a more let's say first principal way uh here is more or less where I expect the transition from today lecture to to end um and then a a a third big chapter is cosmic ray acceleration the basics what you need to to actually uh be able at all to accelerate uh in astrophysical environment and then a little bit how do you describe it in terms of uh equations um the fourth is is module is is basically collisions both leptonic and adonic collisions and these are important as I said for because they are involved in the in the production of these Diagnostics tools like gamma rays and neutrinos but also because they alter the chemical composition and the Spectra of charged particles so we we need to cover them um and then eventually um well let's say multi messenger aspects gamma neutrinos uh diffuse fluxes things like that okay that's the menu um okay in terms of units I will be using natural units mostly I am any way a theoretical physicist by by training which means balsman constant equal to c equal to H bar equal to 1 uh sometimes I will rate C for uh for for clarity because we need to to to compare some velocities so uh I hope is not confusing I will keep I'm not uh I'm not doing gravitational physics so I will keep in general G Newton uh um Dimension full uh this is also equal to one over blank mass square um since we are doing some astrophysics so um there are also astrophysical units which will pop up um in the most frequently used astrophysical units is certainly the ones concerning distances these are um related to the Parc uh kilop Parc MEAP Parc gigap Parc um and roughly this is 3.2 uh three uh light years and these are essentially geometric units in in their in their origin it has to do with the Parallax method to determine distance of to the Stars huh so this is the earth in its orbit that's a star uh far away uh six months apart on the orbit around the sun we will see this star position with respect to further stars move and basically a parsec is the distance at which this angle is is basically one Arc Okay so that's rough slightly more than three Li ear and these are multiples uh and I will give you shortly the link with the typical scales that we deal with in in astrophysics uh another system of units that will uh unavoidably appear um are electromagnetic units and in most of these literature uh Goan convention is used which means that the four pies are not in Kum and BOS low uh they are rather in Max equations uh uh hopefully this doesn't I I I will stick to the convention that you will find in most of the textbooks and the review that's why I use them uh which basically means that things like the the magnetic energy density is is b² over 8 Pi uh it's not b² over 2 mu0 we will not use this one but hopefully you will be able to convert them back and forth if you need and I also provide you some basic conversions for natural units so uh magnetic fields um so magnetic fields intensity uh has the units of an energy Square because this is an energy density which has a unit of energy to the fourth um and uh the typical units in astrophysics for magnetic fields is the gaus and uh actually microa for typical Galactic values um which is 10us 4 Tesla um if you are not familiar with natural units there are few exercises that won't be described tomorrow for you to get used to it uh uh hopefully but by now you have encountered them quite often I hope um in terms of scales we will deal with uh Galactic and extragalactic environments okay so for those of you who have astrophysical Notions that's trivial but for those of you who have never uh been exposed to them um be aware that both environment are extremely rarified as far as a comparison with with terrestrial or even laboratory media are concerned when I say uh extremely rarified I mean that uh you know typical density in of matter in a galactic environment so in a galactic environment a density typical densities are typically below one particle per CM Cube maybe higher in the towards galactic center or um in molecular clouds a couple of or higher but these are extremely rarified much better than the best vacuum you can get in a lab uh and in the extra Galactic environment you may have densities which are even below 10 to Theus 6 particles per CM Cube today um in terms of U distances typical Galactic distances are let's say in the kilop Parc to 100 kilop Parc range roughly this is the size of the structure in the local spiral arms and this is the size of a dark matter hello around a Galaxy like like ours just to to give you numbers um a radi the radius of the Milky the visible disc of the Milky Way is roughly 15 kilop Parc and the Sun is 8.3 kilop Parc from the galactic center um in terms of extra Galactic scale we are rather dealing with larger than say 10 MEAP Parc the closest galaxy like Andromeda is almost a MEAP Parc away close big Galaxy spiral galaxy outside uh hours and and and you can get up to tens of Kil G Parc for cosmologically distance but still closely connected um uh objects so these are these are the typical scales you deal with in the CMV um in terms of time scales as well uh a revolution of the Sun in the Galaxy uh takes about 240 million years this gives you an idea and uh a cosmological time scale is the age of the universe is almost for 14 gig years so you see that these are scales very extreme compared to what we have uh in laboratory experiments um I won't use think very much um in the exercise for instance um coordinate systems I try to avoid uh them but for you to be aware if you for instance uh browse some cataloges of objects uh you should be at least aware that there are a couple of of reference systems that you will face one is the equatorial uh which is the general purpose type of uh uh uh reference system that you find in astronomy and this is nothing but the projection on the celestial Sky uh uh frame of the terrestrial uh latitude and longitude system okay so uh uh this is the Earth this is the equator you can look at the same structure in the sky you have a Celestial equator which is the projection of the terrestrial one into the sky the only difference is that Latitude uh the latitude is replaced by the the the declination but that's the same criteria you have a positive latitude a positive declination above the equator negative one below and then the longitude in this Frame is replaced by by right Ascension um here is a little bit trickier for the zero because on Earth the convention has varied over time but it was usually due to geopolitical reasons right it was elero the zero latitude since uh tonomy uh later was green witch because of the uh British power uh over the seas in in in space uh we use more uh natural type of CH uh criteria for the zero basically the zero is chosen um as the intersection of the uh equator with the ecliptic the ecliptic is the apparent orbit of the sun around the earth if you wish is the orbit of the Earth around the Sun and it's the Vernal uh Equinox about the two points where this intersect is the Vernal one the spring one which is chosen as a zero latitude okay this and usually you find these in degree and you find these in terms of equivalent uh um hours minutes and seconds okay uh you divide uh uh uh the the the the 360 uh degree in in in the in time equivalent which is very practical because a distance of 1 hour in right Ascension really means that basically your star takes one hour to cover that type of during the night that's that's the the and by the way this is the reason why we still have the the division of the the clock in multiples of 12 rather than than than 10 uh it's related to the to the angles um besides this this this is a general purpose uh uh frame the ones that are usually used in in Galactic astrophysics U uh tend to um uh use the galactic plane as a reference because you might want to see by eye if an object correlates with the galactic plane the galactic center or it's very far uh off so uh uh one one one coordinate system that you use it's the galactic coordinate system and this is Center on the Sun so if we are here in the Sun and this is the galactic center um basically you you measure angles which is the galactic latitude as the angle between it's indicated with b and it's the angle U above the the the the direction connecting the Sun to the galactic center for positive latitudes and negative latitudes are below and um and you have then a a a a uh uh a longitude and the longitude okay there is a a little um let me see from the from the bottom if this is the sun this is the galactic center this is the C solar position um basically you have 90° is here this is instead 270° in longitude yes you the a little bit more people not able okay let me erase this so this is the convention used okay 90 170 and this is the galactic center sorry C okay for the longitude this is for the latitude and then there is a galactocentric uh system as well that is used sometimes s um this is a cylindrical coordinate system centered on galactic center huh what I mean is that now you have in the spiral Arms This is the galactic center this is the Sun so uh the coordinate R is the the radial distance in the plane H then you have a coordinate Z which is the height up above or below the plane and then you have another um uh coordinate the angular one which is uh usually in this is not Universal but usually is measured clockwise Theta is measured clockwise from the with respect to the the the direction connecting galactic center to the sun okay all these are are in the notes just for for for you to to get familiar with those if you there are some exercise where I suggest you to go and look in some cataloges uh and you might find some of these coordinates I also give you a link with with a set of routines in Python uh astropy is very used so don't need to learn this by Art but at least to have an idea what they are uh uh so that you are not lost um in terms of um content here I I mentioned briefly the density of matter uh but these these environment are also filled with with uh magnetic fields and Photon Fields okay so as far as we know all environment where we have been able to measure sensitively enough we have found magnetic fields in astrophysics basically um and the typical um how how do we diagnose them uh by techniques like far rotation or you have synchrotron radiation and you have uh uh polarized light emission uh Zan splitting for for more intense Fields so if you are not familiar with those uh I suggest you to to go to general purpose type of reference like uh if you are really curious you want won't need that to understand what I'm going to say but if you are curious uh books like u i find particularly useful for instance books like keep thorns and Roger blanford uh modern classical physics there you find a qu quite accessible General introduction um just to give you an idea in terms of magnetic fields uh in the Galaxy you get magnetic fields which are few microa in intensity this can be uh lower for the regular component of the field the regular component field more or less follows the spiral arms in the dis um it's much more uncertain what happens at high latitude uh there might be a fountain toward the inner Galaxy and so on and it's larger and you have a comparable if not larger uh turbulent field so it's a small scale more incoherent type of fields and in terms of extragalactic fields um uh well we have detected similar intensity maybe a slightly lower micro gous intensity field in clusters of galaxies um for the really extragalactic space or should I say uh Intergalactic space there are indirect observations that suggest that there is a there is a magnetization uh the lower limit is very weak we are more or less sure that it should be above 10 to minus 19 18 gaus which is a very loose lower limit we have more or less robust upper limited nanog gous uh scale for Mega par cerence length what exactly is the field we don't know we suspect based on simulation that it should be correlated with the filaments and large scale structure connecting uh you know over densities but we don't know probably even the vacu the the the the the the the almost vacuum uh volumes the voids uh uh should be magnetized but we don't know exactly how how strongly um and uh and concerning Photon fields sorry if I do some zoology but uh I want things to be more or less clear so the photon fields on the extragalactic uh level they are dominated by the CNB so in the extra Galactic the CMB which is a black body Spectrum with a temperature of 2.7 Kelvin an energy density today of roughly 3 electron volt per CM Cube to give you an idea um this is in the microwaves huh uh but there is also another background which is of interest and it's often denoted as ebl let me put it here ebl extragalactic background light uh now this is way less um dense in terms of energy a couple of ERS of magnitude than the uh in terms of partic sorry particle density is way less uh important than the than the CMB it's less than one order of man actually lower even in energy density uh but it's important because this falls in the ultraviolet Optical and infrared range as opposed to to the to the microwaves here um the scaling with red shift CNB scales as 1 plus Z Cub for the number density of particles as 1 plus Z to the 4 for the energy density uh this is less um uh trivial because this ebl where it comes from it comes from the The Starlight first of all Stellar objects everywhere in the universe they they they shine and they meit ultraviolet Optical and near infrared photons and eventually there is a background due to all that but there is also some reprocessing coming from absorption from dust grains the universe is Dusty if you have never heard about that but you should take into account account especially if you do CNB uh physics um and they they Reit uh typically in the longer wavelength so infrared and far infrared um in the galactic case in the galactic case of course you always have the CNB but it's usually less important for dynamical considerations unless you go to High energies like PV energies uh in the galactic case you have also a a a a UV uh Optical and and infrared background uh which is slightly less more dense in terms of energy density it's it's at the level of you know electron volt per CM Cube and this become this is contributed of course by all Stellar objects and the dust in the Galaxy um and the difference is that while these are basically isotropic this is not so the density of these guys is pecking towards the inner Galaxy okay so if you calculation of absorption uh you should take that into account and I I in the notes you will find some plots uh to give you an idea of this uh how they look like in terms of wavelength uh Spectra um a fact that you know might have a dynamical uh explanation but there is no consensual uh explanation yet uh at least in the details is the fact that in the galactic environment the energy density in Starlight in um in uh magnetic fields and in cosmic rays are roughly equal they are this is sometimes called the manifestation of the equip partition it's of the order of fractions of electron volt per CM Cube that might be related to to coupling between these things but uh there are also people that believe this is just a coincidence and uh it has not no no deep reason okay I I just mention you because you might find this expression in some some textbooks okay so uh so far so good for things which are nothing to do especially with with Cosmic race but are some Notions that I really need you to to to have heard at least once uh uh in order to understand what I'm going to say um uh let me start now the cosmic cray part with with um with some history so um I won't do an historical review but you must be familiar about some uh dates and even some Concepts including the name the origin of the name uh I don't know but when I I I started learning about Cosmic race I was wondering why the hell are they called Rays right these are mostly charged particles and still they are called cosmic rays why why this so uh and this has a historical origin in fact so um the the the first phenomena that are um attributed to what we call today Cosmic race uh go back at least to the to the 18th century for instance Charles colum was was experiencing some unexplained discharge of uh um electroscopes that were as long as it could say isolated uh we know uh we now know that uh Cosmic race contribute a lot to this discharge phenomena um the first theories related to this phenomena appeared at the Turning of the uh 19 to 20th century when people discover radioactivity uh so there were speculations that this disar and could be related to a spontaneous emission from from the ground of radioactive material from the ground that would discharge them and the first pioneering experiences in Cosmic race uh uh were were done around 1911 1912 by people like uh Pacini s h so uh uh these people looked in in particular Pini looked at how this ionization rate changed when he performed experience is um underground uh sorry underwater uh in the in the Gulf of Genoa and he found that they were declining when he went down rather than going up if it's coming from the ground you should go up when when you go deep uh uh s did the opposite he took a balloon and was measuring how it changes when when you go up and he found that instead of decreasing it went up okay he was lucky because eventually you would find a decrease if you go up enough but uh these experiments experiments convince people that this phenomenon is astrophysical it's not geophysical so you have to find an explanation in in astrophysics uh has also um tested if he could find a correlation with the with the presence of the Sun in the sky or not at night or day and he didn't find any significant variation so uh he concluded that this is a phenomenon that comes from outside the solar system it's not a local astrophysical phenomenon now we know that it's more complicated even the sun can produce solar flares and accelerate low energy cosmic rays but yeah the the basic picture is correct um the the name according to the name so there were two basic theories in the you know the 20s or so uh uh about the cosmic race one was the main proponent was milikan the one that you have probably heard in part physics uh studies he thought that these were essentially gamma rays photons The Cry of the birth of the elements that's how they he thought of of this gamma rays so gamma rays Associated to the production of the elements where elements were produced and if this was a real serious problem scientific problem at the time was disputed According to some this was an initial condition meaningless to think about where elements originate uh he was of a different advice so we called them Cosmic race um the opponent the main opponent was Compton that you might have also heard about uh hopefully uh and he thought they were charge particles this big debate was settled in favor of Compton but the name Cosmic R remained so Milan got a saying about the name and and uh the same has happened in cosmology you might know that big bang actually was a a a a a a a name used by oil to ridiculizar how do they prove thatuh how could they infer that they are charged particles essentially because of uh the early in the early times in the late 20s and 30s in particular it was the latitude effect so people started u campaigns uh all over the world um um es especially by European uh physicists uh taking profit of the fact that this was the end of the colonial period so there were colonies scattered a little bit everywhere uh so they they run campaigns of measurements of cosmic rides at different latitudes and they found systematically that there is an increase of the cosmic rate intensity when you go to High geomagnetic latitude so there is an exercise very simple you can do it more elabor ly or in a very oneline type of calculation uh that I propose that you will discuss tomorrow that that explains why it is so and uh uh there were also other studies so people involved in these experiments for inance you might have heard about especially for the Italian students you might heard about Rossi Rossi was one uh astrophysic that pioneered these studies in particular there was some campaign in arria other physicist went to to the spard uh uh and measured there so and you and that was a compilation there is still a nice review by the way uh by um by by Compton uh you can find on physics review if you are interested on these historical aspects um yeah yeah it's from here that's why I mentioned uh uh you might heard um about these campaigns um and uh uh and by the way also brought a review uh on showers we will talk shortly about about that uh that you might find uh very pedagogical actually it's a very nice introduction to this topic um roughly in the same period uh the fact that they were predominantly positively charged was discovered because there is an asymmetry between East and West if you look at the rate of arrival from the East and from the West you find an asymmetry and this is consistent with a mostly positively charged uh um nature um and in this period say from the 30s to the early 50s this is really the golden period for cosmic ray research uh also for fundamental physics okay so you might heard about the discovery of the positron and the discovery of the muon and the discovery of the Pion uh and also strange particles chons and sigma and Lambda and so on all these were discovered with cosmic rays they were basically no accelerators at the time so particle physics was essentially using these natural beam and studying the byproducts of this in the atmosphere uh this per and there were many Nobel prizes awarded you know Anderson hes etc etc um o and uh rosi uh Pioneer the the use of um you know coincident detectors to study extensive air showers these are induced when Cosmic Rayes interact in the up atmosphere and there are multiplicative processes we will uh discuss shortly a a a toy model of that uh and this allowed what is today known as indirect detection technique for Cosmic race um this this sort of uh golden age uh of um twin development of cosmic race and particle physics ended sort of at the um if you want a date is 1953 there was a cosmic ray conference uh in the Baner the biger in in France uh where more or less the two communities got split and uh particle physicist started War you know building accelerators and studying uh uh controlled uh condition uh Productions and Cosmic cray physicists basically devoted more on to the astrophysics uh of it it doesn't mean that the the the same person could not be interested in both topics one great example is firy just after uh World War II uh uh uh and the development of nuclear power and the Manan project and so on he wrote the first paper proposing more or less a a a a a a meaningful explanation for the acceleration of cosmic cray it's the process that is today known as second order firmy acceleration um so some people worked on both but it's fair to say that the methods sort of decoupled and this situation has I mean I don't want to be too sharp but I sort of started to reverse in the80s uh when people started to think again of cosmic Messengers as a way to probe um uh fundamental physics uh this has to do with research like indirect searches and uh even direct searches of Dark Matter uh and um uh you might be familiar with the neutrinos cations atmospheric neutrinos they require knowledge of uh uh cosmic ray physics um and then there have been many discovers in what is now known the multimessenger uh approach for instance the over the last decade you might have heard about the discovery of a diffuse astrophysical neutrino flux uh by Ice Cube and we will mention that at the end of the lectures okay so this is for a short historical overview let me go a little bit ahead before stopping um um what about what about some phenomenological aspects concerning Cosmic race namely uh how do they look like in in energy space huh if you look at the Flux Of cosmic ray as a function of energy this flux looks like I mean varies by roughly 10 orders or or more over the range of energies where we have detected them from from let's say GV energies up to 10 to the 20 electron volt or so and this is roughly above a few GV this is looking like a power low I oversimplified it in the sense that it's not all always having the same spectral index there are tiny variation of the spectral index um but the key thing I want to uh uh mention is the fact that basically the fluxes here are of the order of one particle per cm squar per second which means these are very frequent events uh and these are the type of events you might worry about if you do accelerator physics or uh Dark Matter uh direct detection or so you want to be shielded by the bulk of these events um if you go to tens or 100 uh TV uh uh the typical flux is is at level of a particle per day per meter square and this is sort of the limiting size of experiments that you can Fly Above the atmosphere or at the top of the atmosphere or you can dock to the International Space Station or put on a satellite um and beyond that you have flux is as low as one particle per kilometer squar per century and and clearly this realm is only possible to probe in an indirect way this is really possible to probe in a direct way and by direct I mean that here you can really identify particle by particle uh what this particle is and basically you fly particle physes detectors on balloons on satellites and uh with spectrometers calorimeters chank of detectors you measure the the the charge the mass the velocity etc etc uh example of these are I don't know filot for gamar Ray uh astrophysics or uh which is on a satellite or AMS O2 which is do it's a cosmic charged Cosmic race predominantly detector docked on the International Space Station there are peculiarities I don't want to oversimplify the problem compared to the typical experimental problems that you find in in um in acceler ators uh of course you cannot go there and uh study very precisely your alignment so you have to devise indirect techniques to to align your uh your uh detector for spectrometric uh type of considerations uh the background is very different for fmy think of fmy uh you know this is a gamar ray telescope but most of the background that you have uh is adonic in nature in fact uh um this is may be very different in a collider where you might have many more more electromagnetically interacting particles than adonic interacting particles but uh morally speaking we are talking about partical physics experiment uh in the sky uh for indirect techniques the situation is different what do I mean by indirect technique what I mean by indirect think is that let me draw with my limited capabilities some some little uh cartoon by indirect technique I mean that you do not study the the particle that is coming from outer space directly you study the byproducts of the particle interaction in the atmosphere so there is a very simple model due to itler that roughly um Works uh uh um for electromagnetically induced shower in a reasonable way so the idea for instance imagine you have a photon uh impinging for Simplicity vertically here on the atmosphere uh after a characteristic length uh let's call it Lambda this Lambda depends on on the on particle physics and the environmental conditions of the atmosphere the composition the density Etc um uh uh um Lambda actually the typical variable used here is is is called gramage huh and this is uh the the integral over the path length of your of your part particle of the density encountered okay so this is measured in U this is measured in gram over cm square typically so after Crossing some material your Photon can P Produce so it will produce e plus e minus E plus e minus eventually after a a distance which is comparable it can uh radiate a a a photon and so you have another multiplication of particles and so on and so forth this is um um the type of process multiplicative process that you have huh eventually the number of particles grows and the average energy per particle declines and this goes on until the average energy drops below a critical energy which is an energy depending only on on particle uh uh Nuclear Physics basically this energy is the energy below which energy losses proceed essentially through ionization uh not by multiplicative processes okay um so this does not depend on you can measure it if you you can even comput it for QED uh but uh but that's that's that's the idea of this hler model um and uh before going into little math concerning this hler model um indirect technique mix means that you have access to either observables like the the number of particles as a function of the depth in the atmosphere um this is usually um computed in terms of the gramage crossed let me let me draw it like that you can think of the number of particles um this is the number of particles once you go to through the atmosphere the number of particle grows and eventually declines huh because you you run below this condition and um there are observables which are sensitive to this shower development uh quantity uh examples are the scintillation light that these particles induce when they collide with molecules in the atmosphere or um the chank of radiation so uh for instance the the the for telescopes in the OJ Observatory or telescope array Observatory measure this type of um profile development shower development observable Imaging air cherenov telescopes uh measure the cherenov light but the idea is that they are sensitive to the development of this and then through um through steroscopic view uh uh if you look at the profile development um uh from different direction you can not only reconstruct the the the projected Direction but really the 3D uh direction of arrival so these are ways also to determine where your initial primary particles comes from um and then there are other experiments which are also indirect experiments which are basically uh measuring a slice of this thing at a given uh depth in the atmosphere because they are on ground they are like muon tanks or scintillators and so even measuring as SCE and measuring some timing of arrival you can reconstruct the shock the shock you can reconstruct the front of these particles and so try to to to reconstruct the arrival Direction and also since you know the the the direction and you know the the age of the shower how is it called uh so at which point if you wish in this curve the thing is located you can try to reconstruct other properties of the the shower so this is the logic now I show you in a simple model find minutes uh how this works to give you an idea uh don't take them too seriously in a quantitative uh meaning because now these things are done we via simulation tools but at least we have an idea um the the the first thing that you might notice in this simple model let's put it there is that the the if you have a particle if you have a particle with an initial energy e zero sorry yes two things first of all what what is the label of the Y AIS this is for instance the number of particles sorry the why yeah the No No the number of the x- axis is the number of partic the y- axis is X is the is the depth is the depth in the atmosphere you can think if for a vertical shower it would be just the height above the ground for an inclined shower you have real a a a a conversion factor in the case where the the atmosphere is uh uniform you have a constant density then you can use them interchangeably okay but otherwise you have to use a model of atmosphere and convert height into a variable X so you can think of it this is X or some proxy of it okay think is that what does it mean e what so yes I I I'll try to explain here maybe in a better way it's more obvious so you have an initial energy in not for say your Photon that is impinging okay after a single pair production you have that the energy is in not over two if for each electron and positron after n steps in this Cascade the average energy is e not over 2 to the N for each particle okay and this corresponds to a depth a gramage crossed which is n times your Lambda so in that picture you see that each step takes the same Lambda but doesn't mean that the same lambar corresponds to the same uh uh geometric uh uh distance in the atmosphere because the atmosphere has not a a uniform uh density okay it's uniform in that variable in that grammage variable um so you have a maximum number of particles in this multiplicative process when when uh basically your Photon or electron prefer to lose energy in non- multiplicative type of process for instance ionization okay and this is something I have some value it's roughly 80 me in the atmosphere this this this EC so um is roughly 80 M and just to give you an idea the Lambda for a photon is roughly 35 grams per C over cenm Square I don't know if this talks to you but these are things that you can measure you find in the particle Data Book Etc that not very conceptually deep and U uh uh uh one thing that is important is that eventually once this average energy drops below that the number of particles that you have reaches his maximum h so this is nothing but e0 over e critical so if you measure the maximum number of particles in your shower development since you know that from the lab you can infert what's the energy of the impinging particle in this simple model when does it happen it happens at the depth x max which is uh Lambda time the log base 2 of e over uh e not over EC because of this multiplicative so take they take the uh it's equivalent to this type of relation okay now um equivalently since since you know this constant you know this constant if you determine the x max you can determine the energy of your shower so this is one way you can have access to the energy information from this type of indirect uh detection before taking a short break what changes if you have an adonic shower if you have a cosmic a charge Cosmic rate that that interacts say a proton or a nucleus huh uh this model doesn't work so well quantitatively anymore uh but the basic idea is still correct so you can model um uh roughly the adonic interaction we will come back to these collisional effects but you can model the adonic interaction as producing a some number of p in general adonic interacting particles uh at each interaction you have some multiplicity new plus new minus of of Pion that you produce and uh um uh what happens is that in the high energy collisions you have an isospin symmetry so the number of pi+ pi minus and Pi 0 that you produce is roughly equal huh so 1/3 of the energy on average at each Collision stage is going into Pi 0 and 2/3 into charged particles charged pons so the part of um uh energy in the and this repeats now once you go into Pi zos Pi Z decays into gamma gamma so they initiate a an electromagnetic sub shower the adonic part can keep interacting okay so at each stage in your Cascade development what happens is that uh you only retain an adonic fra of your energy which is 2/3 of in not and an electromagnetic which is 1/3 of in not this is the first stage H but at following stages at following stages you have an adonic interaction which is eventually 2/3 to the power of n of e KN and an electromagnetic interaction energy which is 1 - 2/3 to the n of in not so your shower becomes more and more electromagnetically uh Rich uh uh because you are draining into electromagnetic particles and you get less and less INF fraction of of uh Pion okay so the number sorry the average energy of your charge part parles your charge bons let's denote it by e+ E minus uh is going to be e not divided by basically 3 Al of new + minus to the power of n this is the multiplicity of charged particles three Alp of it is the total multiplicity to the power of n is the total number of particles after n stages so this is the the the the typical average uh uh energies and uh eventually what happens again the Pion there is a critical energy uh below which a Pion rather decays into a muon rather than uh um undergoing multiplicative adonic interaction okay so the maximum is reached to when this average energy uh uh again be Becomes of the order of this uh decaying critical energy and uh uh you deduce that n Max plus minus is equal to to uh uh basically log e0 over e d this is exactly the same relation apart for this scaling factor that we had before okay why this is important because for each charge particle for each charge P you have eventually a muon coming from the decay of the p and you can observe the muon the muon is a very penetrating particle typically doesn't doesn't interact much and this is something you can measure on the ground and uh the number of muons is nothing but basically the number of pi plus pi minuses which is in turn the multiplicity of Pi minus pi plus per uh interaction raised to the power of this n plus and minus Max huh this one which means that the log of n muons is given by um n +- Max time log of new +us which is in turn something like beta log e over e d okay and this beta coefficient is nothing but the log of new plus new minus over the log of three Al new plus new minus this multiplicity I don't remember exactly the value but it's of the order of six or seven something like that so this beta is roughly 085 so this is just to say you that you can also measure the energy in principle by measuring how many muons you get on the ground this is something new you don't have this in the electromagnetic shower um also imagine that instead and then we take a break uh uh uh imagine that you up to you I don't know we can even end a little bit earlier up to you uh depending on how tired you are um basically the if you take a superposition model and you say that a nucleus is just a nucleons the the energy available is the same let's make these hypothesis so the total number of particles available is going to be the same at the max this is nothing but e not over e uh C or E D whatever this critical energy however um since you have now n particles they will interact earlier so in our in our picture the cross-section of a nucleus with the with the with atmospheric material is larger which means that if you have a proton doing that a nucleus typically will do that H so although the energy available is the same the depth will will be uh shorter at which you reach your maximum and and uh in fact it's it's trivial to show that the maximum for a nucleus a is going to be X proton uh Max sorry x max proton minus Lambda of a proton times the log of a so this is just to say that if you measure now the both the total energy through through the number of particles produced in the sh and the depth at which this maximum happens you can try to infer the chemical composition of your of your shower okay so you have the same number of particles for a nucleus but the the distribution of development is is is narrowed and you can also prove that the variance of this part this is a stochastic uh variable the variance also is smaller for a nucleus than for a proton okay and um and that's it so uh let's stop here there is an exercise just to consolidate these Notions so tomorrow you will see more uh some quantitative details of that uh let me just conclude about the fact that this model this generalized um uh itler model is not actually very good at quantitative prediction is one for ad dronic interactions and in fact um now we rely on on extensive air shower simulators unfortunately there are uh uh adonic uncertainties is associated with that uh because these type of processes are non-perturbative qcd processes these are what are called forward physics uh uh processes in collider jargon we will come back to that um and U and basically uh there are two difficulties one is that we don't have a a a first principle theory for those and second difficulty is that for the highest energy showers uh we are even above the energies proved by LHC so we don't even have data uh to C too uh so um that's why measurements of say chemical composition in Ultra energy Cosmic R experiments are very hard okay we have a very loose idea if there is a trend of dominating protons rather than dominating irons and so on and so forth so let's stop maybe 10 minutes and then we have half half an hour to conclude okay maybe before we start I just want to answer collectively to a question that was raised why did I insist on the fact that these are non- termal messengers okay one way to realize why why these are non-thermal messengers is the plot I I made before right the flux versus energy this flux here this is a power load type of flux it's very far from a Maxwell bolman type of uh distribution okay so clearly these objects are not energized in a thermal Environ environment where you I don't know hit up some astrophysical objects up to PV or even TV or GV that's not how it works so it's a it's a physics question how do you accelerate these particles up to these high energies why do do they have this shape in energy space and that's exactly what happens also on earth right if you want to study high energy physics you do not heat up a medium up to TV energies that's not a very efficient way not only the fact that we don't know how to do it but that's not really how you do and actually there are experiments trying to do high energy collisions of nuclei try to produce uh things like a quar Glu and plasma Etc so that's not the the way we accelerate protons of to TV in a collider on Earth and that's clearly not the way nature accelerate these particles so we have to study the the tools through which this can happen okay that was just a comment let me um if it was not clear to to to some of you so uh another point I want to mention uh is the chemical the chemical uh composition okay what are these fluxes made of so they are made mostly of protons actually uh we um tra well traces there is a sizable fraction of helium nuclei of course these are fully ionized given the energies uh and actually all nuclei uh all at least all stable nuclei of the periodic table as far as I know have been detected inra in in cosmic rays although nuclei Beyond iron are way less abundant okay and uh um there are also electrons uh uh at level of percentage of the protons and you will understand why it is so it should be so um and because naively you would expect since the universe is neutral right do you have as many electrons roughly as protons huh assuming that the hydrogen dominates how is it that the flux of electrons and the flux of protons are not comparable this is a puzzle a physics uh question whatever the mechanism for acceleration is um and then there are nuclei up to at least iron roughly the abundances of these species so how many I don't know carbons per oxygen they roughly match what we measure say in the solar system you might have seen this nice picture of the astronauts on the moon uh uh deploying these sort of uh Nets uh uh on the surface of the Moon and they were trying to collect dust uh um now this is done with micrometeorites etc etc one of the purposes of these experiments is to know the chemical composition of the material in the solar system but uh for Cosmic race roughly they match uh there are a few notable differences one notable difference is that elements like lithium berium Boron are over represented in Cosmic race with respect to what we know in the solar system sorry these are very rare elements in the solar system and in most thermal astrophysical environments and we think we understand why because these type of elements have a very low binding energy per nucleon so they are fragile fragile in a thermonuclear sense it means that once they are presence in hot environments like stars they tend to be burned out very efficiently okay so they are their abundance is almost zero for for what matters for for instance for us for the injection of cosmic rays um but they are present in cosmic rays and they are you know maybe at the level of 10 percentage to 30 percentage of the more abundant nuclei like carbon oxygen Etc h and the reason this is interpreted as the result of frag you know spalation so this nuclei in cosmic ray differently from solar system material these have energies of GV TV and and above uh so once they hit uh Interstellar material they can uh uh Spate they can lose some nucleons in the collision and produce these lighter elements that's the the way this overabundance is interpreted is interpreted as the result of the gramage that this primary Cosmic race experience in the interstellar medium so there is another exercise uh for tomorrow where just using this fact uh you can infer that Cosmic race cannot propagate in straight line in the Galaxy otherwise the these observable would be badly uh uh predicted okay and then we will come back to this problem once we have the full propagation machinery to compute more from first principle what you should expect for that in a simple geometry but already that without knowing much tells you that they cannot propagate in straight lines okay uh and the same is true for what is called subon with respect to iron and also for some uh isotopic species like dyum and the helium tree H they are overabundant with respect to what we find in the solar system by by like some four orders of magnitude or so so um I won't talk about isotopic studies in Cosmic race but if you find them don't be surprised about that um okay um final notational aspect before mentioning something about ristics on cosmic ray propagation is about units uh fluxes of cosmic rays are typically um either plotted or measured versus energy usually this is done when you want to focus on particle physics processes that can be induced by this cosmic ray energy is a meaningful variable alternatively you can use momentum plot fluxes with momentum this is usually done in theoretical for theoretical studies because this is the natural variable in face space you want to use uh they can be done also in terms of energy per nucleon if they are nuclear species the reason why this is used and sometimes also kinetic energy nucleon or so this is done because in spallation processes the one producing these guys we will come back to that but the ones produc these guys roughly the energy per nucleon is conserved so it makes sense if you want to compare you know um spal objects and also rigidity this is another variable which is extremely used so rigidity is nothing but momentum over over the the the charge okay Z is the uh Atomic uh number e is the Elementary positron charge so it's momentum over charge and we will see in a second that this is the key variable if you want to focus on propagation aspects of cosmic rays in magnetic fields because cosmic rays having the same rigidity uh um have the same trajectory uh in a given magnetic field okay so these are the typical quantities used if you want you can train yourself to convert these simple uh kinematical variables one into another so you can compute yeah C of transformation of these if you want to convert the flux in one in the flux of the other of course there are codes that can do that for you but you know if you do there are papers even famous papers where there are mistakes on these things so uh the trial things are the ones where people get confused uh the most I think um anyway um what I want to focus on now is actually some Notions of propagation of cosmic race h so let me introduce this this this notion of propagation of cosmic race um hopefully you know how a charge particle propagates in a magnetic field constant magnetic field just as a reminder uh the the equational motion is rather trivial you have just the Lawrence Force so let's start by this uh with this simple problem B 0 is this constant magnetic field that I will assume oriented along the Z Direction in this type of uh equation of motion you have that uh V is conserved uh P which is M gamma V is also conserved huh uh and then you have that VZ the projection of the Velocity along the direction of the magnetic field is also conserved uh and as a consequence even V orthogonal which is the the orthogonal component is also conserved uh mu which is is nothing but the the ratio of pz over P or VZ over V is also conserved because these ones are each conserved and as a consequence also P orthogonal which isqu < TK 1 - mu s or p is also conserved okay uh in term of motion you know that this type of equational motion will describe an El coidal motion around your constant magnetic field which means that in the direction orthogonal to the magnetic field you have a circular movement huh circular uniform motion uh DV orthogonal over DT uh is q v orthogonal b 0 over uh M gamma this must be equal DV gamma over DT must be uh equal also to v² orthogonal over R huh this is a centripetal uh acceleration so you have a radius of your circular motion in the XY Direction which is uh M gamma V orthogonal over qb0 okay so in the non- relativistic case that's the only non-trivial part that maybe you have not in mind in the non- relativistic case you can define a a a a a phase frequency uh of geration which which is QB 0 over uh M and Associate to that radius of duration or Gyro radius which is V orthogonal over Omega G in the relativistic case you have what is called larma radius which is nothing but gamma RG and you have Associated uh uh angular frequency Capital Omega which is Omega generation over gamma and these frequencies are very uh these time scales or one over Omega is a very fast time scale for astrophysical purposes for GV energies and charge one is of the level of maybe 100 seconds or so or 100 inverse seconds for Omega so these are very short time scale compared to the millions of years of that I mentioned before um the the okay the equational motion of course here you can integrate them okay X of T will be given by some initial condition XG + r with some choice of phas you can always write them like that Omega t y of T is equal to y g plus or let's say minus RL cinus Omega T and Z of T is some zg uh plus VZ * uh t uh which you can also write zg uh plus v mu T so these are just describing this type of motion okay and a a a movement with there is one fixes particle of coordinates XG y g z of T this is called guiding Center and this is the particle that moves uniformly along the magnetic field okay so you can describe the movement of cosmic ray like a a a circular movement around this um guiding Center um now if I modify this picture and I add a perturbation to my magnetic field for Simplicity sorry for Simplicity let me choose my perturbation Delta B uh uh orthogonal to B KN and also Delta B much smaller than b not in magnitude okay so you can think of it with respect to that picture which now I put in the vert iCal axis because I chose Z to be the magnetic field I am now sort of perturbing my field like that okay so I have a small component orthogonal to uh to Z I can choose for instance Delta b equal to in components cinus minus KZ plus some phase five or S sorry sinus minus k z plus SI and let me put it to zero along the direction z uh I want to choose this type of fluctuations in the sense that the motion along X and Y is any way controlled by the leading term b0 but I want to know what is the first correction to the motion around Zed which is um uh um happening when I add now a fluctuation to this constant magnetic field okay so we will focus on the change of trajectory in the Z Direction and we will consider that what X and Y is unchanged okay this is a manifestation of some that can be formalized and made more more again more formal but for now take it in an intuitive uh way so we can compute the the the the the evolution of the the pitch angle or rather the cosine of the pitch angle this is the ratio of pz with respect to to P so it's the cosine of the uh angle given by the momentum with respect to the um constant field I just write down the the equational motion with each choice 1 minus mu s over e modulus Delta B and then you have uh something which is cinus Omega t uh you will find all these formul uh in the notes no no no need to really write down everything in detail minus sinus Omega t uh sinus minus KZ + so I can also rewrite this as by definition as a constant times something like cosinus sorry not omega let's say w t plus SI where I introduce this constant C which is nothing but q s < TK of 1us mu s remember this is a constant because this is a constant uh because mu is a constant Delta p over energy and W is by definition Omega minus mu uh k K which is the wave number on my perturbation time V okay so far so good now I can compute okay so I have a random wave with an arbitrary phas P which I have had it well not a wave sorry a fluctuation this is not propagated this is all magnetostatic which I have added so what happens to the evolution along the Z Direction which I express in terms of the evolution of this component pz or rather pz over P mu of course this will be just a statistical description because I have no idea how the field departs from the the from the constant um from the constant um approximation so on average if I average over for instance the phase of these fluctuation I will get that the average of D mu over DT is equal to zero average with respect to the P this is Trivial it's the average of a cosinus is is is basically zero however I can um I can compute Delta Mu Square in a Time T the variance of this guy huh and this is not going to vanish so this is given by c² integral up to T from whatever say zero I have DT Prime DT second of basically cinus WT prime plus cosinus WTC plus now this product of cosinus is basic Trion omry I can rewrite in terms of cosinus of the sum and cosinus of the difference when I average the cosinus of the sum I still have the random phase so I get zero the difference however this phase disappears so in that case when I average over the phase I won't get zero there is a finite contribution to the variance of this uh quantity if you further remember the the fact that um basically um okay when you integrate okay I can take let me State first the the the result and then I will show you where it comes from so the derivative with respect to time of this average value of Delta Mu Square which means I get rid of one integral will tend to this result pi c² Delta of w and the reason why you can prove more formally but this has to do when when you integrate over one variable you get that you get a result like sinus sinus W you get a sinus w t over uh W remember that when sinus x/ Epsilon over x ts to uh Pi Delta X when Epsilon goes to zero okay so in the limit where I have that uh delta T is much larger than Omega to Theus one I can use this approximation to reexpress the the the the Primitive of this function and it gives me my result here okay so this is an approximation valid when physically when you are looking at time scales which are much longer than a generation period okay if you look at time scales which are much longer than a generation period basically this variance behaves in a resonant way otherwise this is a final value which is not a Delta it has a width okay so now there are two qualitative considerations that you have to take into account uh first of all let me rewrite this in more physical units I can rewrite this as Pi 1us mu s * Omega * Delta b² over B dot not Square Times what I call K resonant Delta K minus K resonant I'm just using my explicit definition of w and I'm bringing out in terms of a Delta of the wave number and I have defined this K resonant which is important is Omega over mu V okay so what is telling me what is telling me that this evolution along the Z Direction this is the only one we are focusing on once you have fluctuations deviates from this simple you know constant uh uh guiding Center uh movement uniform motion and it becomes a sort of diffusive evolution why diffusive well the first moment vanish the the the the the the average it vanishes but it has a finite variance huh not only behaves like in a diffusive way but it looks resonant in in the limit of long time scale what does it mean physically let me draw a picture again this is the moment where the notes will be handy because I cannot draw so I apologize for my limited skills as a drawer especially in Florence everybody seems to be so so skilled uh anyway so the the picture is the following imagine that you have let me use some color um imagine that you have some um uh perturbation some some fluctuation of the field ah no this is really bad uh which one this one is then then you won't have color effects so it's black and white so imagine that you have this type of fluctuations okay imagine that you have the larmor radius which is very small so that the micay what does it mean it means that it won't diffuse along the Z Direction so your Cosmic cray will basically surf the wave you have an opposite uh regime where the fluctuation is much shorter wavelength than your cosmic ray uh uh uh um Lor radius so imagine a situation like I don't know like that and your cosmic ray is at a larger gyro radius then along Z it won't diffuse it will basically it doesn't really care about the fluctuations it averages them okay and there is a third limit which is the one that we care about when basically the the wavelength of these fluctuations matches the wavelength Associated to the to the to the gyro radius to the larma radius which is given by this expression here basically this is the link huh you can imagine that and this is the challenging one so you you have a fluctuations like that and you have a larm of radius which has the same type of uh uh uh scale in that case you know your Cosmic race will be deflected along Z and this will be corresponding to a diffusion motion along Z that's the idea that's what m physically means this type of result okay so um we have a this is theistic view we will come back to this problem more formally uh but the idea is that um The Cosmic cray will in general diffuse onto uh fluctuations in the magnetic field which are um matching its own specific uh uh wave number if you wish and uh in general it's not like you have a given fluctuation with respect to magnetic field you have an ensemble of fluctuation you can de compose actually in fua uh uh um modes or in general through fua transform whatever configurational magnetic field that you have and so depending on how much energy is basically stored in the modes of that wave number matching the larmor radius of your cosmic ray the cosmic ray will diffuse more or less okay notice that this is a diffusion which is qualitatively different from the diffusion you might have studied of a perfume in a room in that case sure the molecules are diffusing but the diffusion happens through collisions of your molecule with other perfume molecules with the molecules of your the atmosphere the room the air in the room here there is no Collision it's just an interaction in a in a classical sense there is no Collision is just a interaction between the fluctuations of the field and the Char LGE particles okay in a Quantum thinking you can think of these as quasi particles onto which you are colliding but this mode is called of diffusion is called collisionless diffusion by the way in plasma physics in the lab people try to reproduce conditions in which you see this type of phenomena and these are hard to achieve because you need to to decrease the density of whatever gas left to such a level that these are the dominant thing so in in a laboratory plasma typically you are more in an hybrid situation where both collisional effect and collisionless uh one uh matter so let me just conclude with what happens if you you know just write down one formula what happens once you um you have an ensemble of waves you have an ensemble of fluctuations if you have an ensemble of fluctuations basically um your your diffusion in uh pitch angle or mu variable uh will be which is also can be expressed in terms of diffusion with respect to the angle itself H remember this angle is the one specified by my momentum with respect to the B field the regular B field which is our our reference uh this is given essentially that's the result that we have found here forget about prefactors orders of magnitude this is just given by 1 minus mu^ s * Omega if you have an ensemble it will be given essentially by Delta b² over B KN Square evaluated in this resonant uh uh uh uh condition wave number okay the simplest the simplest way to look at this result is that the frequency with which your angle is changing huh this diffusive time scale is basically linked to the frequency the basic frequency huh that we are introducing this is the one with which you rotate along your your magnetic field through a fractor which is the ratio of power or energy of your FL uations with respect to the magnetic field evaluated in this resonant condition and in general you will have an ensemble of that you will have a power spectrum of magnetic fluctuations huh and depending on how this power spectrum is populated as a function of wave number you will get more or less deflection okay um two names that you must be aware of are called mov and Kagan um so that there is no first principle theory for how this turbulence in the magnetic field should look like okay this is still an unsolved problem even in the hydro case this is not really a solved problem according to Fineman turbulence is the most challenging unsolved problem in classical physics um typically how does it look the power spectrum of this mag IC field fluctuation um there are essentially large scales this is K and this is essentially Delta B over B til in Furious space there are large scale K me which are associated to whatever turbulence you inject in the astrophysical setting super NOA explodes motions of gas in the galaxy and things like that kilop par or you know tens of parsec scales and so on and there is a k Max at which this turbulence eventually dissipates heat huh in between typically this is a power L spectrum and this is known as this is an idealization huh in reality things are more like maybe that but this is known as inertial regime for the fluctuations of for the turbulence these are just names you might find whenever this behaves like a power Spectrum or a self invariant scal invariant type of uh Behavior this is known as inertial regime um so you parameterize this guy or the spectral energy density of this guy in terms of a power low index be because of this link between Delta the diffusion coefficient and the power if this is a power low this is also going to become a Power low in terms of Omega and in the notes you find the analytical link between these two okay in particular there is a power low index for the spectral energy density where these or better the spectral energy density Associated to the field scales as K to Theus 5/3 which is known as colog and leads to a diffusion which scales as rigidity to the minus 1/3 we will come back to that but just to introduce the names and then there is another regime that you might find where the density the spectral energy density of these fluctuations is scaling as K to the minus 3 ales and then it leads to a diffusion in pitch angle which is rigidity to minus one Al in terms of R these are just two names so that if you find in the literature kryan or kogoro you know that what people mean uh but I don't think I will use them uh but again uh be aware about this these two behaviors which are related to this inertial regime here so I will stop here sorry for few minutes delay and we will come back to this formalism now will be made more uh mathematically sound after you are convinced of the basic features what you should expect okay that's the next lecture thank you