Non-linear Cosmic Ray acceleration in Supernova Remnants

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Non-l inear Cosmic Ray

accel er at ion in

Super nova Remnant s

Damiano Caprioli

Supervisors

Prof. P. Blasi,

INAF–Osservatorio di Arcetri, Firenze

Prof. M. Vietri,

Scuola Normale Superiore, Pisa

Tesi di Perfezionamento – Ph.D. Thesis 2008

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Ever tried. Ever failed. No matter. Try again. Fail again. Fail better. Samuel Beckett

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Nomenclature

Roman Symbols

A = Atomic mass number

A(t, p) = Function defined in Eq. 8.24 B = Magnetic field

c = Speed of light cs = Speed of sound

D (x, p) = Parallel diffusion coefficient ¯

D = Mean diffusion coefficient (Eq. 7.14)

D0(p) = Bohm diffusion coefficient in units of p/ mpc (Eq. 7.3)

DB(E ) = Bohm diffusion coefficient (Eq. 4.22)

Dgal(p) = Galactic diffusion coefficient

Dsg(x, p) = Diffusion coefficient self-generated by CRs

E = Energy / Electric field

E51 = Kinetic energy of SN ejecta in units of 1051 erg

Eadv = Energy in advected CRs

Eesc = Energy in escaping CRs

Eesc(p) = Energy in escaping CRs with momentum p

Emax = Maximum CR energy

ESN = Kinetic energy of SN ejecta

f (t, ~x, ~p) = CR distribution function

fth(p) = Thermal plasma distribution function

fS(v/ vej = Ejecta structure function (Eq. 8.37)

F = Energy flux

F (t, x) = Energy flux normalized to ρ0u30/ 2

F (x, k) = Magnetic energy density per logarithmic bandwidth g(s) = Laplace transform of f with respect to time

h gi = Average column density (grammage) G = Gravitational constant

J = Flux of CRs (in §2.4.3) h(s, p) = Function defined in Eq. 5.22

H (t, tf in) = Decompression factor from t to tf in

Hc = Alfv´en wave helicity

k = Wavenumber

kB = Boltzmann constant

Kep = Electron to proton ratio

K (p) = Function defined in Eq. 5.21 I (s, p) = Function defined in Eq. 5.6

M = Mach number (u/ cs)

MA = Alfv´enic Mach number (u/ cs)

Mc = Star core mass

Mej = Ejecta mass

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NOMENCLATURE

n = Slope of the structure function fS(v/ vej)

n(x) = Number density

N (p) = Number of particles with momentum p

N (t, ~x, ~p) = Number density per momentum unit (in §4.1.1) ncsm = CSM number density in units of one particle/cm3

p = Particle momentum p(x) = Pressure

p∗ = Momentum of particles which diffuse up to x0

pinj = Injection momentum

pmax = Maximum CR momentum

pth = Characteristic thermal momentum (§4.2)

P = Energy pressure per unit logarithmic bandwidth P (x) = Pressure normalized to ρ0u20

PT H(x) = Normalized plasma pressure including TH

q(p) = CR spectral slope at the shock

qD = Diffusion coefficient energy slope (Eq. 4.21)

QΣ = Rate of CR production for unit area

Q(x, p) = Injection energy flux

R = Compression ratio in a test-particle shock Rsh = SNR forward shock radius

Rsub = Subshock compression ratio (ρ2/ ρ1)

Rtot = Total compression ratio (ρ2/ ρ0)

R = Reflection coefficient Rc = Star core radius

Rej = Ejecta radius (Eq. 8.37)

rL = Larmor radius

rth

L = Larmor radius of particles with momentum pth

s = Laplace transform conjugate of time S = Magnetic rigidity

t = Time coordinate T = Temperature

T = Transmission coefficient

tacc = Acceleration time (h ti in Eq. 5.26)

T (p) = Kinetic energy of a particle with momentum p TST = Beginning of the Sedov-Taylor phase

u(x) = Fluid velocity ˜

u(x) = Effective velocity felt by CRs (u − vA)

up(p) = Effective u for a particle with momentum p

U(x) = Fluid velocity normalized to u0

Up(p) = up normalized to shock velocity u0

v = Particle velocity

V (x) = Magnetized cloud velocity (in §2.4.2) vA = Alfv´en velocity

VA = Alfv´en velocity normalized to u0

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NOMENCLATURE Vsh = SNR forward shock velocity

vej = Ejecta velocity

vw = Wave velocity (in §7.2)

V(t) = Volume of the shocked CSM shell (Eq. 8.50) x = Spatial coordinate (the shock lies at x = 0) x∗ = Characteristic length-scale (Eq. 4.35)

x0 = Upstream boundary

X0 = Upstream boundary normalized to Rsh (Eq. 8.16)

Xmax = Depth of EAS maximum (see §1.1.2)

xp(p) = Characteristic length defined in Eq. 6.30

W = Magnetic to gas pressure ratio at x = 0−

W (k) = Energy density in Alfv´en waves (in §4.1.1)

Greek Symbols

α(x) = Magnetic energy density normalized to ρ0u20

β(v) = Velocity normalized to c β = Parameter defined in Eq. 6.53 γ = Plasma adiabatic index γ(p) = Lorentz factor

γp = Momentum spectral slope

γE = Energy spectral slope

Γ = Magnetic energy density damping rate δX = Perturbation of the quantity X

∆ = Quantity defined in Eq. 6.12 ∆′ = Quantity defined in Eq. 6.18 ∆X = Variation in the quantity X

∆R = Thickness of the shocked CSM shell (Eq. 8.33) ε(x) = Energy density

ǫ = Thermal emissivity

ǫtp = Thermal emissivity in the test-particle regime

ζ = Non-linear Landau damping efficiency (Eq. 6.46) η = Fraction of particles which are injected as CR ϑ = Angle between particle and cloud velocity (in §2.4.2) κ(s, p) = Function defined in Eq. 5.12

λ = Shock thickness ˜

λ = Shock thickness normalized to rth L

Λ(p) = Function defined in Eq. 5.19

ΛB = Magnetic feedback parameter (Eq. 6.16)

ΛT H = Turbulent heating feedback parameter (Eq. 6.50)

µ = Parameter defining a time scaling

µesc = Parameter defining the escape flux time scaling

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NOMENCLATURE

ν = Parameter defining the Rsh time scaling (Eq. 8.6)

ξ = CR pressure normalized to ρ0u20

Ξ(x) = Function defined in Eq. 6.33 ̟ (x, p) = Function defined in Eq. 4.25

ρ(x) = Mass density

σ(x) = Magnetic energy density growth rate Σ = Surface mass density

τ = SNR age

Υ(p) = Function defined in Eq. 8.30 φ(x, p) = CR flux

Φ2→1 = Flux of particles crossing the shock (§4.2)

χ = CR to magnetic pressure ratio (Eq. 6.52) ψ = Injection parameter (Eq. 4.17)

Other Symbols

≡ = Is by definition equal to k = Parallel ⊥ = Perpendicular ~∇ x = Spatial gradient ~∇ p = Momentum gradient [X ]2₁ = X2− X1

log = Base 10 logarithm ln = Natural logarithm M⊙ = Solar mass

Subscripts

0 = Far upstream (x = − ∞ ) quantity

1 = Immediately upstream of the subshock (x = 0−_{) quantity}

2 = Immediately downstream of the subshock (x = 0+) quantity adv = Quantity referred to particles advected downstream

B = Magnetic field / Bohm diffusion c = CR quantity

esc = Quantity referred to particles escaping from x0

f i n = Final age g = Gas quantity

w = Magnetic turbulence quantity

ST = Quantity taken at the beginning of the Sedov–Taylor phase

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NOMENCLATURE

Acronyms

AGN = Active Galactic Nucleus

CSM = Circumstellar Medium (also csm) CD = Contact Discontinuity

CMB = Cosmic Microwave Background CR = Cosmic Ray

DSA = Diffusive Shock Acceleration EAS = Extensive Air Shower

FS = Forward Shock

GZK = Greisen–Zatsepin–Kuz’min MFA = Magnetic Field Amplification MHD = Magneto-Hydro-Dynamics

NLDSA = Non-Linear Diffusive Shock Acceleration PIC = Particle In Cells

RS = Reverse Shock SI = Streaming Instability SN = Supernova

SNR = Supernova Remnant SSM = Solar System Medium

ST = Sedov–Taylor TH = Turbulent Heating UHE = Ultra High Energy

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Abstract

Cosmic ray phenomenology

Long time has passed since 1912, when some balloon experiments of the Austrian physicist Victor Hess found that natural radioactivity increases again above 600 m from the ground, despite the larger distance from the radioactive elements in the Earth crust. For the first time an unknown, extraterrestrial, source of radiation usually shielded by the atmosphere had been observed.

During the past century this cosmic radiation has played a very relevant role in Physics, leading to the discovery of new particles (like the positron, the muon and the pion) and providing a clue to the behaviour of particles with energies several orders of magnitude larger than those achievable even in modern laboratories. A big effort has gone into measuring the flux, the en-ergy spectrum, the chemical composition and the anisotropy of these cosmic rays (CRs), with the final goal of understanding the mechanisms responsible for their acceleration in several astrophysical sources and the propagation through the Galaxy. Chapter 1 is dedicated to a brief summary of the main experimental technologies and of the present observational evidences.

CRs show an energy spectrum which is very close to a power-law (∼ E−3) over several decades, extending from about 1 GeV up to more than 1011

GeV. This striking evidence has driven physicists to research very general mechanisms able to accelerate CRs, also trying to identify astrophysical objects which could work as accelerators.

Electromagnetic processes are very likely responsible for CR accelera-tion, hence a limit to the maximum energy achievable by particles in a given source can be obtained by imposing the Larmor radius of accelerated particles to be smaller than the typical object size (Hillas criterion). Simple estimates suggest that galactic sources like pulsars and supernova remnants (SNRs) may be, in principle, consistent with the production of CRs up to ∼ 108 GeV, while the most energetic CRs (above 109 GeV) have to be of extragalactic origin, as discussed in §1.3.

The all-particle spectrum (Fig. 1.1) shows a clear steepening from about E−2.7 _{to E}−3.1_{, the so-called knee, around 3 × 10}6 _{GeV. Recent}

observa-tions able to resolve the CR chemical composition up to 108 GeV (see e.g. Antoni et al., 2005) provided evidences that the knee is actually a result of the superposition of different element spectra. In fact the proton spec-trum shows a very marked steepening above 3 × 106 _{GeV, while spectra of}

heavier nuclei (with charge Z ) show cut-offs at energies a factor Z higher (Fig. 1.8). These findings are consistent with a rigidity-dependent acceler-ation mechanism and may suggest ∼ Z · 3 × 106 _{GeV to be the maximum}

energy achievable by a galactic CR with charge Z , or, at least, that there is a very important class of sources with this property.

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ABSTRACT

Acceleration of galactic cosmic rays

The SNR paradigm for galactic CRs

In this work the attention is focused on CRs produced in galactic sources and more specifically in SNRs, which are widely believed to be the most relevant CR factories in the Milky Way.

The first hint about the validity of the so-called SNR paradigm for the acceleration of galactic CRs was provided by Baade and Zwicky in 1934 (Baade & Zwicky, 1934): in their paper they suggested either supernova (SN) explosions to be due to the release of a huge amount of gravitational binding energy during the transition from an ordinary star to a neutron star or even SNe to be responsible for the galactic population of CRs. The latter statement was based on an energetic argument which estimated a fraction of 20–30 per cent of the SN ejecta kinetic energy (∼ 1051_{erg) to be channelled}

into relativistic particles in order to account for the energetics of galactic CRs (§2.2).

Acceleration at shocks

Some years later Enrico Fermi took a step forward proposing a very general mechanism for accelerating particles in magnetized plasmas (Fermi, 1949, 1954). He noticed that in an elastic head-on collision with a magnetic mir-ror a particle actually gains some energy, while in a bumping it loses some. However, head-on collisions are statistically more frequent than bumpings, hence a particle repetitively scattered by magnetic irregularities (which typ-ically fill common astrophysical plasmas), turns out to be accelerated as a net result (see §2.4.2).

During the late ’70s various authors independently realized that, when Fermi mechanism is applied to SN forward shocks, it leads to a very ef-ficient acceleration of CRs (see e.g. Krymskii, 1977; Axford et al., 1978; Blandford & Ostriker, 1978; Bell, 1978a,b). This process, often referred to as Diffusive Shock Acceleration (DSA), is based on the fact that diffusion in the SNR magnetic field makes particles scatter back and forth across the shock. Since each encounter with the shock can be viewed as a head-on collision, particles are expected to gain energy very efficiently (see §2.4.3).

On top of that, the spectrum of the particles accelerated via DSA turns out to be a power-law E−γE_{. The slope γ}

E = (R + 2)/ (R − 1) depends only

on the shock compression ratio R, i.e. the ratio between the density of the hotter, shocked plasma and of the colder, unperturbed one. For a strong shock, namely a shock with Mach number M0≫ 1, R is equal to 4 and the

energy spectrum turns out to be E−2_{. Such a value is in good agreement}

with the observations outlined above when corrected by the effects of CR propagation in the Galaxy, as discussed in §1.3.

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ABSTRACT

DSA is therefore a very powerful method for generating power-law spec-tra with roughly the expected slope. However, a spectrum ∝ E−2 being mildly-divergent, a maximum energy Emax is required in order to keep the

total energy in CRs finite. This energy limit may be due to the time needed to accelerate a CR up to Emax or to the size of the accelerator, or even to

energy losses particles may suffer.

The most relevant factor limiting the energy of cosmic hadrons is very likely the acceleration time, which is strictly related to the time a particle takes to diffuse back to the shock. It is easy to show (see e.g. Blasi, 2005) that, if particle diffusion occurred in SNRs at the same rate as in the in-terstellar medium, the maximum energy achievable by CRs would not be larger than 1 GeV. Also considering more efficient models of diffusion (like the Bohm one), if in SNRs magnetic fields were comparable to typical in-terstellar ones (1–10 µG), it would be hard to account for energies as high as the knee (Lagage & Cesarsky, 1983a) and thus the SNR paradigm would fail.

Magnetic field amplification in young SNRs

A fundamental tile of the mosaic has been provided by the present generation of X-ray telescopes, such as Chandra and XMM–Newton. Their high spatial resolution has revealed that young SNRs exhibit very bright rims, whose emission is currently interpreted as due to synchrotron radiation by electrons with energies as high as 1–10 TeV.

These results are very important for at least two reasons: first, they are a strong evidence that in SNRs electron acceleration is efficient, and second, the measurement of the rim thickness permits to estimate a lower limit to the magnetization of the post-shock medium. The inferred magnetic fields turn out to be a factor 10–100 larger than typical interstellar ones, implying that some kind of magnetic field amplification has to occur in young SNRs (V¨olk et al., 2005; Parizot et al., 2006).

This piece of information perfectly fits in the problem we are studying, especially because the super-Alfv´enic streaming of CRs is expected to excite magnetic turbulence via plasma instabilities. Such streaming instabilities may lead to a rapid growth of different modes, either resonant (Skilling, 1975a; Bell, 1978a) or non-resonant (Bell, 2004; Amato & Blasi, 2008) with the Larmor radius of the relativistic particles. A brief summary of very popular magnetic field amplification models is provided in §3.2.

It is worth noticing that an efficient streaming instability (for which the magnetic field is expected to grow well beyond the background value) leads to a very fashionable scenario in which CRs provide by themselves a magnetic turbulence enhancing their diffusion. As it will be widely discussed in the following, accounting for magnetic field amplification allows SNRs to accelerate protons up to the knee and heavier ions up to even higher energies.

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ABSTRACT

The need for a non-linear theory

According to the SNR paradigm for the acceleration of galactic CRs, a substantial fraction of the SN kinetic energy is converted into CRs. As a consequence, at some point the validity of a test-particle approach regard-ing accelerated particles as passive spectators of the shock dynamics has intrinsically to break off.

In fact, as soon as first quantitative calculations of the DSA efficiency were worked out, it became clear that pressure and energy in the shape of accelerated particles could no longer be neglected with respect to fluid ones, revealing the need for a non-linear theory of DSA (NLDSA) in which particle acceleration and shock dynamics are self-consistently calculated.

The first attempts to carry out a study of NLDSA, often referred to as two-fluid models, treat CRs as a fluid of relativistic particles (Axford et al., 1977; O’C. Drury & V¨olk, 1981a,b; Achterberg et al., 1984). This approach is of great interest for pointing out the main effects of the non-linear CR backreaction on the shock.

As a consequence of the CR pressure, the upstream develops a precursor in which the fluid approaching the shock gradually slows down: the net result is a weaker effective shock (now called subshock ) and in turn a heating of the downstream plasma less efficient than in the test-particle case. The contribution of relativistic particles to the upstream pressure makes the fluid more compressible, in such a way that the unperturbed to downstream density ratio Rtot easily becomes much larger than 4, going with the Mach

number of the shock as M₀3/4. The downstream to upstream of the subshock density ratio (Rsub), instead, is typically in the range 3–4.

Within a two-fluid approach it is possible to work out the amount of energy and pressure converted into CRs and to study the dynamical prop-erties of CR modified shocks, but no information about the spectrum of the accelerated particles can be provided.

A deeper comprehension of NLDSA has to rely on the development of a kinetic approach to the problem, in which CRs are described by means of a distribution function in both space and momentum. Different ways of deal-ing with a kinetic analysis have been proposed in the literature, spanndeal-ing from the semi-analytic models by Malkov (1997); Malkov et al. (2000); Blasi (2002, 2004); Amato & Blasi (2005, 2006), to the Monte Carlo approach by Ellison & Eichler (1985); Ellison et al. (1990); Jones & Ellison (1991); Ellison et al. (1995, 1996); Vladimirov et al. (2006), and to the numeri-cal procedures by Bell (1987); Falle & Giddings (1987); Berezhko & V¨olk (1997); Kang & Jones (1997, 2005, 2006, 2007); Berezhko & Ellison (1999); Kang et al. (2002).

Different approaches to the problem confirmed some very general prop-erties of CR modified shocks. First of all, the spectrum of the accelerated particles is expected to be concave rather of being a straight power-law. This

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ABSTRACT

is a consequence of the fact that, the more energetic a CR is, the farther it diffuses away from the shock, in turn feeling higher effective compres-sion ratios than lower-energy particles which remain confined close to the subshock. Since the spectral slope is determined by the compression ratio, Rtot > 4 (Rsub < 4) implies a spectrum flatter (steeper) than E−2 at the

highest (lowest) energies.

NLDSA is extremely efficient in accelerating particles: typical Mach numbers of young SN shocks may be as high as 50–200, easily leading to total compression ratios of order 50–100. These strong shock modifications correspond to a conversion of more than 90 per cent of the SN kinetic energy into CRs and to very concave spectra, as flat as E−1.2 near Emax.

Limits of the standard non-linear theory

Recent X-ray observations of some shell SNRs has provided a clue to the level of modification of their forward shocks. In fact, the inferred thickness of the shocked ejecta shell in Tycho’s SNR (Warren et al., 2005) and in SN1006 (Cassam-Chena¨ı et al., 2008) turns out to be not consistent with ordinary gaseous shocks described by Rtot= 4. The inferred hydrodynamics indicate

instead mildly modified shocks with Rtot in the range 6–10.

The multi-wavelength (from radio to TeV) emission of SNRs contains information about the broadband spectra of both thermal and accelerated particles and thus it can, in principle, provide a benchmark for investigat-ing the curvature of the CR spectrum. The emission of many SNRs has been fit by either numerical (see e.g. Berezhko & V¨olk, 2004b, and papers based on it) or semi-analytical (Morlino et al., 2008) models for NLDSA, but evidences of very concave spectra have not been found. Nevertheless, also multi-wavelength studies are able to account for the observations only by invoking mildly modified shocks and moderately concave spectra, hence suggesting the relevance of an efficient CR acceleration.

The consistency between standard NLDSA models and recent observa-tions is usually recovered by invoking some kind of non-adiabatic heating in the precursor which makes the upstream plasma hotter and thus less compressible. A widely adopted non-adiabatic heating mechanism is the so-called Alfv´en heating, which consists in a very efficient damping of the magnetic turbulence into thermal energy. In fact, it has been originally in-troduced in order to avoid the growth of magnetic perturbations beyond the linear limit (McKenzie & V¨olk, 1982). Such a mechanism, if on one hand may heal the issue of obtaining strongly modified shocks, on the other hand can hardly account for the evidences of strongly amplified magnetic field in young SNRs.

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ABSTRACT

The original contribution

Aim of the present work

The scenario broadly outlined above is the point of departure for this Ph.D. thesis, which aims to investigate the non-linear acceleration of CRs in SNRs. In particular, NLDSA at SN forward shocks is tackled by means of a kinetic approach based on the semi-analytic method proposed by Malkov (1997); Blasi (2002, 2004) and Amato & Blasi (2005, 2006). In this model CR acceleration, magnetic field amplification and their backreactions on the shock dynamics are taken into account, while the maximum momentum particles can achieve at a given SNR age is self-consistently computed.

This approach also permit to calculate how CRs are effectively accel-erated and then released into the Galaxy during the SNR evolution, thus providing an interesting tool to probe the SNR paradigm for the acceleration of galactic CRs.

The original part of the work is articulated as follows.

The maximum momentum of accelerated particles

In Chapter 5 the problem of calculating the maximum momentum achievable by particles in a given lapse of time is addressed, generalizing the quasi-stationary treatment of Lagage & Cesarsky (1983a,b) to the case of a CR modified shock, also including the effects of the magnetic field amplification induced by streaming instability.

Two main opposite effects are found to be relevant for CR modified shocks: on one hand the presence of a precursor makes the upstream res-idence time longer, thus slowing down the acceleration process, while, on the other hand, a strong magnetic turbulence favours the CR diffusion and in turn a faster return to the shock. Eq. 5.28 provides the time needed to accelerate a particle from the thermal bath up to a momentum p, thus rep-resenting a very general recipe for estimating the instantaneous maximum momentum of the CR distribution in any quasi-stationary shock.

An interesting finding is that, for typical SNR parameters, proton ener-gies as high as 106GeV are achievable within the first 1000 yr of the remnant evolution. This fact is of primary relevance for explaining the evolution of the acceleration of CRs during the evolution of a SNR, as it will be discussed in the following.

Magnetic dynamical feedback

Chapter 6 is dedicated to the original implementation of an ingredient previ-ously neglected in NLDSA theory, namely the feedback of the self-amplified magnetic field on the shock dynamics.

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ABSTRACT

Standard treatments of NLDSA in fact do not consider magnetic pressure and energy density in the conservation equations, since even the amplified magnetic fields inferred by X-ray observations of young SNRs typically cor-respond to magnetic pressures as large as 2–4 per cent of the fluid ram pressure (V¨olk et al., 2005; Parizot et al., 2006).

Nevertheless, it is possible to show by means of purely hydrodynamical considerations (the three fluid model described in §6.1) that inferred mag-netic fields do provide a pressure and energy contributions which upstream are comparable to, or even larger than, thermal plasma ones. This very solid estimate calls for the inclusion of magnetic terms into the conservation equa-tions, which in turn leads to a modification of standard Rankine–Hugoniot conditions (see e.g. Scholer & Belcher, 1971; Vainio & Schlickeiser, 1999).

The effect of the magnetic pressure is to make the upstream fluid less compressible than in the unmagnetized case. It is possible to show that inferred magnetic fields imply a strong reduction of the shock modifica-tion, naturally returning total compression ratios in the range 5–15, in good agreement with the evidences outlined above. It is worth stressing that magnetic feedback relies only on first principles, namely Maxwell equations and conservation equations for mass, momentum and energy, being thus an unavoidable, solid effect and not a phenomenological fine-tunable recipe, unlike some implementations of the Alfv´en heating.

A kinetic approach to NLDSA including particle acceleration, magnetic field amplification and their dynamical feedback on the shock dynamics is also carried out in order to assess the achievable levels of magnetization and the effects of the magnetic feedback on the CR spectrum (§6.3)

For typical SNR parameters, the found levels of shock modification and magnetic turbulence (§6.5) are in close agreement with the observational evidences discusses above. In particular, CR acceleration is found to be still efficient (a fraction of 40–50 per cent of the kinetic energy is channelled into non-thermal particles) and CR spectra are found to be only slightly concave, going like p−3.5–p−3.7 at the highest momenta. On top of it, the maximum momentum particles can achieve at a given SNR age is found to be slightly larger than in the case without magnetic feedback.

It is worth noticing that the magnetic feedback also affects pressure and temperature of the downstream plasma, these quantities being relevant to an accurate description of the remnant evolution and to explain the mea-surements of thermal X-ray emission from some young SNRs.

The effect of adding the Alfv´en heating to the treatment is also ad-dressed, finding that, when the magnetic feedback is taken into account, for strong shocks (young SNRs) the Alfv´en heating does not contribute further to the smoothening of the precursor (§6.7). Instead, it suppresses magnetic field amplification rather dramatically, in turn providing maximum momenta significantly lower than the knee.

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ABSTRACT

of SNR RX J1713.7–3946, whose multi-wavelength emission has been suc-cessfully fit without invoking any non-adiabatic heating but only including the magnetic feedback (Morlino et al., 2008).

The escape flux

In Chapter 7 the assumption of stationarity widely used in NLDSA theories is investigated. It particular, its relation with the presence of a flux of particles escaping the system towards upstream infinity is assessed.

In order to better understanding this point, it is useful to sketch out the evolution of the instantaneous maximum momentum of the CR spectrum (pmax) during a typical SNR evolution. After the SN explosion, the ejecta

expand almost-freely, with very high velocities (∼ 10000–20000 km/s): in the course of this stage the production of CRs and magnetic turbulence is very efficient; pmax is expected to increase with time up to ∼ 106 GeV/c,

corresponding to the knee of the all-particle spectrum detected at Earth. When the mass of the swept-up material becomes comparable with the ejecta one, the so-called Sedov–Taylor phase begins: the shock slows down and magnetic field damping is expected to become more effective, as pointed out in §3.3. As a consequence, the maximum momentum of particles that can return to the shock is expected to decrease: at each time t, particles with momentum exceeding the current pmax(t) do not make it back to the shock

and leave the system from upstream generating the so-called escape flux. The escape flux plays an essential role in the formation of the CR spec-trum detected at Earth. In fact, if there were no escape from upstream during the Sedov–Taylor phase, all the particles accelerated in a SNR would be advected downstream, hence suffering some adiabatic energy losses as a consequence of the SNR expansion before being injected into the Galaxy. If this were the case, the SNR paradigm would fail to account for CRs with energies around the knee.

Standard approaches to NLDSA are based on the assumption that the acceleration process and the shock modification may be assumed to reach some sort of stationarity. All these calculations, independently of the tech-niques used to solve the equations, predict an escape flux of energy towards upstream infinity: the shock becomes radiative, which is one of the very reasons why the shock becomes modified and Rtot becomes larger than 4.

It is interesting to notice that also in test-particle approaches the escape of the highest-energy CRs is expected. However, these particles carry away a substantial fraction of the energy only if their spectrum is flatter than p−4, thus energy escape may be relevant only for modified shocks.

In §7.1 the ingredients for understanding when the escape flux has a physical meaning and when it is rather an artifact of the assumption of stationarity are provided. During the Sedov–Taylor phase, in particular, escape is physically expected as a consequence of the reduction of the

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ABSTRACT

fining power of the system, due to the slowing down of the shock and to the suppression of magnetic field amplification.

The contribution of the escape flux to the galactic CR spectrum has been first estimated by Ptuskin & Zirakashvili (2005). They included the magnetic field damping and the consequent decrease of pmax during the ST

expansion, but in their study the shock structure and the particle spectrum are fixed rather than self-consistently calculated.

In §7.4, instead, a calculation of the escape flux within the framework of a kinetic approach to NLDSA is provided, self-consistently including parti-cle acceleration, magnetic field amplification and the dynamical reaction of both on the shock. The attention is focused on shell SNR shocks: the phys-ical meaning of the escape flux during the different phases of their evolution is discussed and the flux of energetic particles leaving the remnant during the Sedov–Taylor phase is explicitly computed, highlighting its phenomeno-logical implications for the origin of CRs.

The diffuse spectrum of galactic CRs

In Chapter 8 the findings above are applied to an attempt to calculate the contribution provided by a typical SNR to the diffuse spectrum of the galactic CRs.

CRs accelerated at SNR forward shocks may be injected into the sur-rounding medium in two ways: on one hand, by escaping from the upstream boundary when the confining power of the system drops or, on the other hand, at some later time after having been advected downstream and hav-ing lost part of their energy because of adiabatic losses due to the remnant expansion.

It is interesting to start with some basic estimates about the energetics and the spectrum of the escaping particles. First, we show by means of simple back-of-the-envelope calculations that the escape flux is expected to carry away a large fraction of the SN kinetic energy (§8.1). Second, assuming a standard self-similar adiabatic evolution for the remnant in the Sedov–Taylor stage, the spectrum of escaping particles is found to be close to p−4 (§8.2).

This result is indeed striking since the escape flux involves only particles with momentum very close to pmax(t), thus being in principle unrelated to

the spectrum of the confined CRs. In this scenario, the expected generality of the CR spectrum is due to the SNR evolution rather than to the acceleration via Fermi mechanism.

In §8.5 the total contribution (escaping plus advected flux) a SNR is ex-pected to provide to the galactic CRs is worked out within the semi-analytic kinetic approach outlined above. Energies as high as the knee are achieved at the end of the ejecta-dominated phase, and the bulk of CR production occurs at the beginning of the Sedov–Taylor stage (about 1000 yr after the

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ABSTRACT

explosion), i.e. when the shock begins to slow down as a consequence of the inertia of swept-up circumstellar medium.

Even more interestingly, the superposition of advected and escaping fluxes (respectively dominant at the highest and at the lowest energies) provides a total CR spectrum which is very similar to p−4. This not trivial result shows how the convolution of many concave spectra (those instanta-neously advected downstream) and of very peaked distributions (the escape occurring close to pmax) may lead to a global power-law distribution with

the expected spectral slope.

Moreover, the total CR energetics is found to be fully consistent with the standard expectations, the fraction of the SN kinetic energy converted in the shape of CRs being around 20–30 per cent.

The outlined results have been carried out for the first time within a semi-analytic, kinetic, approach to NLDSA hence representing a substantial contribution to the understanding of the origin of galactic CRs as predicted by the very popular SNR paradigm.

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1

Cosmic ray phenomenology

1.1 The detection

Everywhere, at any moment, the Earth is relentlessly hit by a radiation of extraterrestrial origin consisting of very energetic particles, accelerated in some place in the Universe; physicists refer to them as cosmic rays (CRs).

During the last century people have widely investigated their properties, providing very accurate measurements of their flux, their energy spectrum and, in many cases, even of their direction of arrival and of their chemical composition.

Probably the most striking property of CRs is that their spectrum (num-ber of particles for unit energy) is well fit by a broken power-law, not very far from E−3_{, extending over an extremely broad range of energies: from about}

1 GeV up to above 1011 GeV (Fig. 1.1). Such a general result claims for equally general mechanisms able to accelerate particles from typical thermal energies up to ultra-relativistic ones. These mechanisms, and in particular the ones responsible for the acceleration of galactic CRs, will be widely investigated in the following chapters. In these sections a brief review of the main experimental techniques and of the observational evidences will be presented in order to outline the scenario theoretical models have to explain. The detection of CRs is usually carried out by means of direct observa-tions for what concerns relatively low energy particles (below ∼ 105 _GeV),

or, for higher energies, by the study of the particle interaction with the ter-restrial atmosphere. This is a natural consequence of the fact that for low fluxes the detectors have to be vary large to collect an appropriate statistics. A huge amount of experiments has been built in order to study the fluxes, the spectrum, the composition and the anisotropy of cosmic radiation1_{. In}

this chapter the attention is focused on the features of accelerated hadrons, avoiding a deep discussion about lepton and photon astrophysics, since it would deserve an equally wide treatment.

1_{A comprehensive list of CR, gamma ray and neutrino detection experiments can be}

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CHAPTER 1. COSMIC RAY PHENOMENOLOGY

1.1.1 The direct observations

Direct detections of accelerated particles are usually performed with bal-loons and spacecrafts carrying Geiger counter, scintillators and/or calorime-ters able to measure the energy and the electric charge of incident CRs. In order to directly probe the raw galactic composition, detections are carried out outside the atmosphere by spacecrafts or immediately within it by bal-loons, which typically operate with residual atmosphere 5 g/cm2. The main limitation to this kind of investigations is, however, the relatively small vol-ume of a flying detector: in fact, as the energy increases, the CR flux reduces from one particle m−2sr−1s−1 at 100 GeV to only one particle m−2sr−1yr−1 at 106 GeV (see Fig. 1.1). For a review of current and recent balloon instru-ments measuring CR spectrum and chemical composition see Cherry (2006) and Blasi (2008).

1.1.2 Extensive Air Showers

When a primary CR, i.e. a proton or a nucleus of a heavier element acceler-ated in an astrophysical source, enters the terrestrial atmosphere, it collides with an atom of nitrogen or Oxygen, thereby inducing a huge cascade of sec-ondary particles, usually referred to as Extensive Air Shower (EAS). The study of EASs can provide a lot of information about the characteristics of primary CRs, and it is particularly precious because it is the only way to investigate the CR spectrum from 106 GeV up to the highest energies.

The pioneer contribution in this field is due to the work of Bruno Rossi, at the Osservatorio di Arcetri, Florence, in the ’30s. He in fact assembled the first practical electronic coincident circuit (a precursor of the modern AND logic circuits) able to assess the simultaneity of several Geiger counter triggers, posing the grounds for the study of CR-induced showers (Rossi, 1930).

This kind of observation was then extensively carried out by Pierre Auger who, collecting a large number of quasi-contemporary triggers in several distant detectors, realized that the number of particles involved in these showers may be huge, hence inferring that some EASs must be very energetic (Auger et al., 1939).

The development of an EAS begins with the nuclear disintegrations of the primary and the target nuclei, leading to a hadronic cascade of smaller nuclei, nucleons and pions: these processes represent the main energy loss mechanisms at high energies, while below 10 GeV nucleons lose energy mainly for ionization. Short-living pions are instead unaffected by ion-ization and tend to produce more pions in a process called pionion-ization (π± → π±+ π±+ π0 + π∓) or to decay into muons (π∓ → µ∓+ ¯νµ) or

photons (π0 _{→ γ + γ). Muons produced at early stages of the cascade}

generation usually have energy enough to reach the ground, forming the

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1.1. THE DETECTION

Figure 1.1: All particle CR spectrum. The hashed regions indicates the energy regions sensitive to satellites, balloon borne detectors and ground based air-shower detectors (from Boyle, 2008)

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Figure 1.2: Schematic EAS development; the collimation of nucleonic compo-nent with respect to incident ray direction has been schematically made evident (Ferrari & Szuszkiewicz, 2006).

radiation called hard component, while later-stage ones contribute to the electromagnetic shower of electrons, positrons, photons, and neutrinos (soft component). Finally, below about 80 MeV, light leptons release their en-ergy into the atmosphere via bremsstrahlung. In Fig. 1.2 a typical EAS is shown, and the nucleonic and electromagnetic contributions to the cascade are emphasized. In order to give an idea of the length-scales involved in the process, at ground level a typical EAS is about 2 m thick and 0.1–10 km wide, depending on the energy of the primary.

An accurate observation of an EAS can provide very interesting informa-tion about the energy, the direcinforma-tion of arrival and the chemical composiinforma-tion of the primary radiation. Roughly speaking, the primary energy can be in-ferred from the number of secondary particles generated in the shower and detected at Earth using wide arrays of scintillators or by the fluorescence light the cascade emits.

The direction of arrival of the primary is usually well defined, since the primary momentum is so much larger than any transverse momentum gener-ated in the shower that EAS axis and primary direction are always collinear. Finally, the chemical composition of the incident radiation is inferable from the way an EAS develops, namely its geometrical displacement and the ra-tios of abundances of secondary particles.

It is worth spending some words about the two principal techniques used to study EASs, i.e. the ground-based arrays and the atmospheric fluo-rescence detectors, which can be used separately or in hybrid experiments.

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1.1. THE DETECTION

Figure 1.3: Various detection techniques adopted for the study of an EAS, includ-ing ground-based detectors, Cherenkov and fluorescence telescopes.

The direction of arrival can be deduced or by the relative timing of the various detectors swept by the shower (ground-based arrays) or by a 3D reconstruction of the shower when two or more fluorescence detectors are available.

• Ground-based arrays detect the final products of the electromagnetic cascade, i.e. secondary hadrons, muons, electrons and photons which hit the Earth surface. They mainly consist of slabs of scintillators with an accurate shielding able to separate the contribution of hadrons and muons from the one of electrons and photons. They are also usu-ally paired with tracking detectors and water tanks able to measure the Cherenkov radiation emitted by muons passing through them (see e.g. Fig. 1.3). The primary energy is usually estimated by its pro-portionality to the shower density at a fixed distance from the core (typically of 600–1000 m), in order to reduce the fluctuations.

Big advantages of this kind of detectors are that they can run con-tinuously, in any weather conditions, and that they can cover large surfaces, turning out to be crucial for the observation of Ultra-High-Energy CRs (the acronym UHECR refers to CRs with energies above 1018 _{eV), whose statistics is very low.}

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• Fluorescence detectors, instead, collect the fluorescent light generated as shower particles excite air nitrogen molecules, actually using the atmosphere as a calorimeter for inferring the primary energy. In this case it is possible to measure one parameter relevant to the determina-tion of the primary nature, i.e. the depth of the atmosphere where the shower reaches its maximum size, often called Xmax, which typically

corresponds to 2–3 km over sea level. The drawback of this technique is that a fluorescence detector is quite sensitive to atmospheric con-ditions (temperature, moisture) and can properly work only on dark moonless nights.

• Air Cherenkov telescopes, finally, are ground-based instruments able to observe EASs by detecting the Cherenkov radiation emitted by cas-cading particle moving with velocities larger than the speed of light in the terrestrial atmosphere. They represent very precious instruments especially for studying high-energy photons (1–100 TeV), which are effectively shielded by the atmosphere and whose flux is too low to be detected by flying experiments. Actually, both hadrons and photons induce cascades emitting Cherenkov light, but a careful analysis of the shower development is able to account for the primary nature, hence allowing the direct observation of γ-ray sources.

However, it is worth stressing that EAS energy measurement and, above all, composition determination are all but straightforward. In order to recon-struct the properties of a shower, very stiff calculations of strong and weak interactions have to be carried out, in addition in a energy range which is not directly accessible neither by modern laboratories. When dealing with these very challenging observations, one has to bear in mind that systematic uncertainties due to normalization and attenuation corrections may be even of order 20 per cent. For an excellent review on detectors of CRs between 104 and 1011 GeV, see e.g. H¨orandel (2005) and references therein.

1.2 The spectrum and the chemical composition

In Fig. 1.4 the CR spectrum measured by several experiments, multiplied by E3in order to emphasize its discrepancies from a single-power-law spectrum, is shown.

There are basically three features to be noticed:

• the knee at energy Ek ≈ 3 × 106 GeV, i.e. a steepening of the

all-particle spectrum from E−2.7 to E−3.1

• a further steepening (second knee) around energy E2k ≈ 6 × 108 GeV

followed by a flattening called ankle at Ea ≈ 5 × 109 GeV. Some

authors, however, prefer to consider the depression of the spectrum

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1.2. THE SPECTRUM AND THE CHEMICAL COMPOSITION

Figure 1.4: All-particle spectrum of CRs as observed at Earth, showing several data points of direct and indirect experiments. The collection of data is from H¨orandel (2005), where it is also possible to find the references to the listed exper-iments.

(multiplied by E3_{) between 10}9 _{and 4× 10}10_{GeV as an unique feature}

called the dip, in order to emphasize its coherent origin;

• the end of the detected spectrum above 1011_{GeV, in which recent}

mea-surements have recognized a cut-off as predicted by Greisen, Zatsepin and Kuzmin (GZK).

At lower energies (say below 108 GeV), various experiments typically measure the same spectral slope, even if the fluxes may differ by a factor 2 or 3 because of the chosen energy normalization. At higher energies, instead, the experiment-to-experiment discrepancies are slightly more marked, even if it is possible to estimate the systematic errors on the absolute energy to be of only 15–20 per cent for EAS detectors. In fact, renormalizing (when it is possible) the energy scales of each experiment in order to match direct ob-servations at energies around 106 _{GeV typically requires correction factors}

of order 10 per cent (H¨orandel, 2003). In the literature, these discrepan-cies among different data fittings may reflect in differences of order 0.1 in the inferred spectral slopes. Nevertheless, the spectral features identified above remain rather evident and claim for an explanation, which may be found either in acceleration mechanisms or in processes related to the CR propagation in galactic and intergalactic medium.

It is worth considering in some details these spectral features, also notic-ing their correlation with some variations in the chemical composition.

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1.2.1 Below the knee

Below 50–100 GeV the spectrum is affected by very local phenomena (a daily variation of CR flux is also observed) and shows a significant anti-correlation with the 11-yr solar cycle. This fact suggests a strong relevance of the solar activity, also related to the solar wind intensity, and of the terrestrial magnetic field.

An accurate account for CR flux at these energies, however, is well be-yond the goal of this work because it would require a detailed knowledge of the terrestrial magnetosphere and of the Solar wind as well.

The chemical composition of CRs around 1 TeV is shown in Fig. 1.5, as normalized to Silicon one (empty circles). For comprehensive reviews of the chemical composition of the galactic CRs, see e.g. Simpson (1983) and Biermann & Sigl (2001).

In first approximation, TeV-CR chemical composition is quite similar to the typical Solar System Medium (SSM) one (and in turn to the interstellar medium one), even if some differences come to the eye:

• the CR abundance of Hydrogen (as well as the H/He ratio) is signifi-cantly lower than in SSM;

• light elements like Lithium, Beryllium and Boron are orders of mag-nitude overabundant with respect to SSM;

• the abundances of sub-Iron elements with respect to the Iron one are also larger;

• there is an excess of odd-Z elements;

• elements with a low first ionization potential are systematically more abundant;

• there is a large overabundance of the stable isotope22_{Ne (and probably}

of 58Fe).

In addiction, isotopic ratios for a given element are sometimes very different in CRs and in the SSM.

It is worth noticing that radioactive elements, such as 14C, 10Be, 26Al,

36_{Cl and}54_{Mn, are very likely secondary, instable, nuclei which can be very}

useful as probes of CR propagation in the Galaxy (see §1.3.1).

These discrepancies with respect to the interstellar medium, especially the abundance of odd-Z sub-Iron elements, are usually explained as pro-duced by nuclear disintegration (spallation) of heavier nuclei during CR propagation across the Milky Way. However, this may not be the case for the 22Ne: its abundance has been rather explained by invoking an origin from a region locally enriched by Wolf–Rayet star nucleosynthesis products. The CR chemical composition may be influenced also by selective injection

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Figure 1.5: 1 TeV CR and Solar System chemical compositions, relative to Silicon (left axis) and Iron (right axis), as function of Z (Biermann & Sigl, 2001).

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Figure 1.6: Mean logarithmic mass inferred from measurement of electrons, muons and hadrons at ground level (H¨orandel, 2005)

processes, for instance due to the charge to mass ratio, which may naturally explain the abundances of elements with low first ionization potentials. A much more detailed discussion of the chemical evolution of CRs during their propagation in the Galaxy can be found in Strong et al. (2007).

1.2.2 The knee

The knee is the most evident feature in the whole CR spectrum. It shows up in an energy region which, at the present date, is not accessible to direct observation yet, even if some balloon borne and satellite experiment are rapidly approaching the energy of some TeV per nucleon. See also Blasi (2008) for an accurate review of this topic.

While for direct observations resolving different nuclei is quite straight-forward, it is much more difficult for air shower experiments to provide some information about the chemical composition of the CR flux.

Nevertheless, ground-based experiments such as KASCADE, CASA– MIA, EAS–TOP and HEGRA have been able to measure the mean log-arithmic mass at a given energy, defined as h ln Ai =P

iriln Ai, where ri is

the relative fraction of nuclei of mass Ai. The mean logarithmic mass, as

a function of the primary energy, is derivable studying some composition-dependent parameters of the shower development. Good examples of these parameters are the total number of produced electrons, muons and hadrons, as in Fig. 1.6, or also the depth of the shower maximum Xmax, as in Fig. 1.7

(for a discussion see e.g. H¨orandel, 2005, 2007).

An indirect measurements of the chemical composition in the knee region has thus become available, as shown in Fig. 1.8. The resolved spectra of protons, Helium and Iron nuclei up to 108 _{GeV have been measured by}

KASCADE (KArlsruhe Shower Core and Array DEtector), a ground-based

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Figure 1.7: Top panel: average depth of shower maximum extension Xmax as a function of energy, for several experiments. The lines show the simulations of showers induced by protons (upper lines) or by Iron nuclei (lower lines) with dif-ferent hadronic interactions model (solid or dashed lines), as described in Hörandel (2005). Bottom panels: mean logarithmic mass inferred from Xmax for the same two different hadronic interaction models of the top panel. The two panels refer to different experiments (Hörandel, 2005). The lines refer to the phenomenological the poly-gonato model (see e.g. Hörandel, 2003), but for our purposes the solid and dashed lines here and in the following can be used just to guide the eye.

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experiment which exploits careful advanced analysis of electromagnetic and muonic shower components in order to measure separately the fluxes of groups of nuclei with roughly the same atomic number (Antoni et al., 2005; H¨orandel et al., 2006).

It is interesting to notice that each spectrum exhibits a steepening (ac-tually a cut-off) which is much more marked than the steepening of the all-particle spectrum. Where this steepening begins depends on the chem-ical composition. In particular, the proton cut-off matches the all-particle spectrum knee at about 3× 106GeV, while other cut-offs are shifted to higher energies, roughly proportional to the nuclear charge Ecut(Z ) ≃ Z · 3 × 106

GeV. In this scenario, the knee of the all-particle spectrum should be re-garded as a result of a superposition of many spectra with different cut-offs and normalizations, rather than as an intrinsic feature. What is really interesting, hence, is providing an explanation for each chemical-resolved spectrum in terms of the acceleration mechanism (which is expected to be rigidity-dependent) or of the propagation in the Galaxy.

1.2.3 The second knee

The second knee is a feature with a much weaker observational evidence and it has been shown only by four experiments: Akeno (and its improvement AGASA), Haverah Park, Fly’s Eye and HiRes Prototype/MIA. References to these data set, as well as a detailed statistical analysis of them, can be found in Bergman & Belz (2007). Very generally, the second knee is defined as a break at a certain energy, by definition between those corresponding to the first knee and to the ankle, where the spectral slope changes. The various experiments are in good agreement in assessing the existence of a break at 7.5σ confidence level, even if the position of the break and the spectrum normalization may differ by a factor 2–3. It is however interesting to notice that the disagreement seems to be related to the adopted detec-tion technique: Yakutsk is in fact a Cherenkov telescope, while AGASA and Haverah park are ground arrays of detectors and Fly’s Eye and HiRes fluorescence telescopes. The best-fitting to a normalized broken power-law (Bergman & Belz, 2007) returns a break energy of 3× 1017 eV and a slope that varies from 3.02± 0.01 below the break to 3.235± 0.008 above (see Fig. 1.9).

1.2.4 The ankle

The flattening of the UHECR spectrum above 5× 1018_{eV has been detected}

by Yakutsk, HiRes and Fly’s Eye, while Haverah Park re-analysis from one side, and AGASA and Auger data from the other can be considered reliable respectively only below and above the ankle. Among these experiments, only Fly’s Eye has observed both the second knee and the ankle, allowing a

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Figure 1.8: Energy spectra for primary protons, Helium, and Iron nuclei form direct and indirect measurements. QGSJET and SIBYLL are two different KAS-CADE shower models (H¨orandel, 2005). For the solid line, see Fig. 1.7 caption.

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Figure 1.9: Flux measurement in the second knee energy region for different ex-periments (left panel ), scaled to make the flux agree with Fly’s Eye result at 1018 eV (right panel ). The fits are carried out in Bergman & Belz (2007), where it is also possible to find a detailed statistical analysis of the data sets.

Figure 1.10: Flux measurement in the ankle energy region for different experiments (left panel ) and scaled so that the flux agrees with Fly’s Eye results at 1018_{eV or at} 1019_{eV for AGASA and Auger data (right panel ). From Bergman & Belz (2007).}

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measurement of the ratio between their energies.

Fly’s Eye fit marks the two break points at log E2k(ev) = 17.60± 0.06

and log Ea(ev) = 18.52± 0.09, corresponding to a ratio Ea/ E2k = 8.3+2.5−1.8.

The global fitting of these experimental data, as in the case of the second knee, is obtained normalizing the fluxes at 1018 eV where possible, while for AGASA and Auger, which operate above that threshold, the matching is fixed to HiRes flux at 1019 eV. The results are shown in Fig. 1.10 and correspond to an ankle placed at 5× 1018 _{eV and to a changing of the slope}

from 3.242± 0.008 to 2.78± 0.02 (Bergman & Belz, 2007).

1.2.5 The dip and the GZK cut-off

Immediately after the serendipitous discovery of the Cosmic Microwave Background (CMB) radiation in 1965, K. Greisen (Greisen, 1966), G.T. Zat-sepin and V.A. Kuz’min (ZatZat-sepin & Kuz’min, 1966) realized that the CR spectrum detected at Earth has to sharply steepen above 1020 _{eV because}

of the interaction between UHECRs and CMB photons.

More precisely, the GZK cut-off is a consequence of the energy depen-dence of the interaction length and of the large inelasticity of the so-called photopion production (Eq. 1.1). The kinematic threshold for this process is of order 1020 _{eV for scattering of protons on CMB photons with thermal}

energy, and is slightly lower (around 3× 1019eV) for protons interacting with the tail of Planck distribution.

The key point is that the loss length for p − γCMB scattering strongly

depends on the proton energy, as shown in the left panel of Fig. 1.11. For energies above 1020 _{eV the loss length, in fact, drops below 100 Mpc, the}

typical size of the local galaxy cluster, while at energies only two times smaller it is almost as large as the size of the Universe. Since UHECRs can not be of galactic origin (see §1.3), if extragalactic sources have a spatially homogeneous distribution, or simply if there are no peculiar source density enhancements in our galaxy cluster, the UHECR flux has to drop with the energy roughly as the loss length does.

Nonetheless, the GZK effect is not the only result of UHECR interaction with CMB radiation. Basically, there are two processes that can occur:

p + γCMB→ p + π0 , (1.1)

p + γCMB→ n + π+→ ... → p + e+e−+ ... . (1.2)

The first process represents the photopion production discussed above, while the second one is referred to as Bethe–Heitler pair production, or simply pair production. The photopion production is much more inelastic and has a higher threshold than the pair production. The latter is in fact already relevant at energies around 2× 1018 _{eV, since at this energy its loss length}

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Figure 1.11: Left panel: loss length for proton–CMB interactions, both photopion and pair–production (Blasi, 2005). Right panel : modification factor ηee (ηtot) for pair production (pair production plus photopion) process, normalized to the case with adiabatic losses only. Lines labelled with 1 and 2 represent different power-law generation spectra E−γg as in the legend (Aloisio et al., 2007a).

In Fig. 1.11 (left panel) the energy dependence of the loss length for both the interactions is shown.

The dip is interpreted as the spectral feature due e+e− pair production for scattering on CMB radiation. The energy losses due to pair production in fact overcome the adiabatic ones from 1018 eV to the onset of photopion production, above 4–5× 1019 eV, as outlined by Berezinskii & Grigor’eva (1988) and by Berezinsky et al. (2006).

Losses due to pair production may in principle be sensitive to the in-jection spectrum, to the maximum energy achievable in the sources, to the cosmological red-shift evolution and to the spatial distribution of the sources. In the right panel of Fig. 1.11 the modification factor, defined as the ratio between the spectrum predicted when all the losses are taken into account and the spectrum obtained when only adiabatic (red-shift) losses are in-cluded, is plotted for different power-law injection spectra E−2 and E−2.7. A noticeable result is that the dip shape is not sensitive to the slope of the spectrum injected by the astrophysical sources. For a deep analysis of the robustness of the dip scenario at the variation of the distribution geome-try and the evolution of the sources, as well as for a wide discussion about UHECR chemical composition and energetics, see Aloisio et al. (2007a).

Detecting UHECRs in order to investigate the dip and the predicted GZK cut-off is indeed a challenging enterprise, because of the difficulties in dealing with very low fluxes, of order of 1 particle km−2 _yr−1_{. In this}

range of energies the calibration of the various experiment becomes a cru-cial issue: historically, significant discrepancies among different experiments probing this spectral region were reported. One possible way to reconcile the findings of the various experiments might be to take into account the

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Figure 1.12: UHECR spectra measured by Akeno–AGASA, HiRes I–II and Yakutsk as viewed separately (left panel ) and after dip calibration (right panel ). From Aloisio et al. (2007a).

ical meaning of the dip, thereby scaling different spectra in order to match their shape in the dip region (Aloisio et al., 2007a). The procedure is very similar to a fit where a floating break point for the GZK cut-off is allowed Very interestingly, the correction factor resulting from the minimum χ2 is less than unity for ground arrays (0.9 for AGASA and 0.75 for Yakutsk) and exceeds unity for fluorescence experiments (1.2 for HiRes), thus suggesting that different experimental techniques provide rather different systematic errors.

Only four present experiments can make detections above 6× 1019 eV: AGASA, Yakutsk, HiRes and Auger (a fifth one, the Telescope Array, is under construction in the Utah desert2_{). Among them, Yakutsk do not}

have enough data to claim an observation of a cut-off, AGASA claimed the absence of the GZK cut-off (Takeda, 1998), while the first HiRes fit was consistent with it (HiRes/Fly’s Eye collaboration, 2004).

Very recently (HiRes/Fly’s Eye collaboration, 2008), the HiRes collabo-ration claimed the first observation (at 5σ confidence level) of the GZK sup-pression, inferring a cut-off energy of 6× 1019 eV. The reader can also refer to Sokolsky & Thomson (2007), where a comprehensive review of UHECR observations can be found.

The dilemma about the GZK suppression has been definitively unraveled by the very large statistics of the newest, biggest and most advanced ground-based experiment: the Pierre Auger Observatory, located near Malargue (Argentina) at 1400 m a.s.l..

Auger is an hybrid experiment which can rely on 1600 water-Cherenkov tanks (laid over 3000 km2_{) in order to detect the flux of secondary particles}

2

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and also on 24 telescopes able to detect the fluorescent light released into the atmosphere by EASs (P.Auger collaboration, 2008).

Combining the implementations of both techniques with its huge ex-posure 2 (3) times larger than HiRes (AGASA) one, Auger confirmed the presence of a flux suppression above 4× 1019eV. More precisely, from 4× 1018 to 4× 1019 _{eV the spectrum is well fit by a power-law whose slope is 2.69 ±}

0.02(stat)± 0.06(syst), the systematic uncertainty coming from the energy calibration. The numbers of event expected if this power-law were to ex-tend above 4× 1019 ₍₁₀20 _{eV), would be 167 ± 3 (35 ± 1), while only 69 (1)}

events have been observed. The spectral index above 4× 1019 eV is found to be 4.2 ± 0.4(stat)± 0.06(syst). A method which is independent of the slope of the energy spectrum is used to reject a single power-law hypothesis above 4× 1018 eV with a significance of more than 6 standard deviations, so the conclusion is very likely independent of the systematic uncertainties currently associated with the energy scale (see P.Auger collaboration, 2008, and references therein).

1.3 Anisotropy, origin and propagation

The features of the CR spectrum outlined in the previous section are not the only information we can extract by observations in order to understand the mechanisms of CR acceleration and propagation through the galactic and intergalactic space.

Another important piece of knowledge is embedded in the anisotropies of the CR flux that hits the Earth. In principle, this flux is affected by the source distribution and by a lot of mechanisms at work in the interstellar medium, such as diffusion in magnetic fields, convection in galactic winds, re-acceleration, nuclear interactions (mainly spallation), radiative losses and photopion production. The observed spectral slope is thus the result of the contributions of both injection and propagation. Any model of acceleration at astrophysical sources aiming to reproduce the spectrum detected at Earth can not be tested without a reliable accounting for propagation effects.

The theory of CR propagation is incredibly wide and it is usually carried out with both numerical and phenomenological models (like for instance the very popular leaky-box model). It is beyond the purposes of this work to provide a detailed discussion of the CR transport in the Galaxy, but, nevertheless, there are some crucial aspects of the problem which have to be mentioned.

1.3.1 The flux anisotropy

As pointed out in §1.2.1, significant excesses of some isotopes, with respect to typical interstellar medium abundances, are usually explained as debris

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1.3. ANISOTROPY, ORIGIN AND PROPAGATION

Figure 1.13: B/C ratio at energies below 100 GeV for nucleon. The data points of listed experiments are fit with four different models for CR propagation in our galaxy, all in agreement with the E−0.6_{dependence at these energies but providing} different extrapolations at higher ones (for more details see Jones et al., 2001).

of heavier accelerated particles undergone nuclear interactions with other nuclei during their propagation across the Milky Way (spallation).

CR unstable nuclei are particularly important because are believed to be mainly secondary particles. For instance, Boron nuclei are typically pro-duced by spallation of Carbon ones. The ratios between the abundances of stable an unstable nuclei thus provide estimates of the galactic residence time, which it turns out to be as long as 3× 107 yr for energies of order of 1 GeV. This residence time is much longer than the free-streaming time, hence implying that CRs diffuse in the Milky Way. Moreover, very likely not for coincidence, this characteristic time is comparable with both sonic and Alfv´enic time scales across the hot thick galactic disc, suggesting a relevant role of some magneto-hydrodynamical processes in the transport of CRs in the Galaxy.

By means of accurate measurements of the abundances of Carbon and Boron in CRs, it is also possible to estimate the energy dependence of the galactic residence time, and in turn of the galactic diffusion coefficient. In Fig. 1.13 the ratio B / C up to 100 GeV is shown: it scales asymptotically as E−0.6_{, thus also the residence time has to scale in a similar way.}

If such an energy dependence were extrapolated up to 1015 eV, reliable source distributions and models for CR transport would predict very strong anisotropies in the observed flux (see e.g. Hillas, 2005). However, such a scenario is not supported by observations: for particles in the energy range

Non-linear Cosmic Ray acceleration in Supernova Remnants