3 Astrophysics of Cosmic rays

3.1 Relevance of cosmic rays

The energy density of cosmic rays in the Galaxy is dominated by low-energy cosmic-rays which are not detectable within the solar system. Thesolar wind expands into the interstellar medium and shields low energycosmic-rays. The lower energy cut-off is modulated by the varying so-lar activity. Extrapolating the energy spectrum backwards to smaller en-ergies provides a similar energy density as obtained from modelling thediffuse gamma-ray emission (see Section 2.2) from the Galactic plane tobe ucr ≈ 1 eV/cm3. The energy density in the Galactic magnetic fields isof similar magnitude, hinting at a deeper connection between cosmic rayacceleration/transport and magnetic field structure and strength in the in-terstellar medium. Before addressing these issues, we highlight the impor-tance of cosmic rays in the context of astrochemistry, star formation, andfinally possible connections between Earth’s climate changes in the pastand modulation of the cosmic-ray bombardment of Earth’s atmosphere.

Astrochemistry: The formation of complex molecules requires an en-vironment shielded from intense sources of photo-ionization with suffi-ciently dense gas densities. The ideal place for astromechanical reactorsare the dense and cold cores of molecular clouds. Deeply embedded in gasand dust, only cosmic-rays provide the source of ionization (at densities> 103 cm−3, radio-activity provide additional ionization power) whichleads to the formation of free radicals (mainly H+) which initiate the for-mation of molecules with as many as 13 atoms14. In total, more than 150(230 including isotopomere) molecules are known to exist in the interstel-lar medium. The commonly assumed ionisation rate of cosmic-rays is10−17 s−1. The formation of molecules proceeds along a molecular net-work in which the formation/destruction of molecules leads to an evolu-tion of the abundances with time.

Star formation: The generally accepted picture of the formation of mas-sive stars includes a period of accretion, in which an accretion disk forms.

14For a recent list of molecules found see http://www.cv.nrao.edu/ awoot-ten/astrophysics.html

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The accretion of matter from the accretion disk is directly linked to thestate of ionisation of the gas in the disk. Gammie (1996) concluded thata layered accretion takes place in which the outer layers of the accretiondisk are ionized by cosmic-rays, leading to a sufficiently large viscosity inorder to form an accretion flow. The source of ionization is however notthe star itself but rather cosmic-rays which can penetrate sufficiently deepinto the accretion disk (a few 100 g/cm2 column density). The inner coreof the accretion disk may remain neutral and therefore inactive without anaccretion flow.

Cosmic-rays and climate changes in Earth’s past (for a review see Kirkby2007): (Non-anthrogenic) climate changes in the past appear to be relatedto solar cycles (e.g. the small ice age in the 16th and 17th century whichtook place during a phase of pronounced minimum in the solar activity).Recent data suggest a possible anticorrelation between cloud coverage inthe lower troposphere and the solar activity (Svensmark et al. 1997). Thesolar irradiance is unlikely to cause a change in the low cloud coveragegiven that the main changes are in the UV spectral band which is mostlyabsorbed in the stratosphere. The alternative possibility of cosmic-rayinduced cloud formation would provide an explanation of the observedanti-correlation as in the case of a solar minimum the rigidity dependentcut-off at the lower energy end of the spectrum shifts towards lower ener-gies leading to increased cosmic-ray fluxes at Earth while during the solarmaximum, the cut-off shifts to higher energies, reducing the cosmic-rayrate. The actual mechanism of cosmic-ray induced cloud formation is notentirely settled and is currently investigated in the frame-work of an ac-celerator experiment (CLOUD: Cosmics Leaving OUtdoor Droplets). Eventhough this paleo-climatology theory is disputed, it is an elegant way toexplain the fact that even at time-scales of the order of 100 Myrs, possiblecycles of ice-age/warm time appear to be correlated with the passage ofthe solar system through spiral arms.

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3.2 Transport of Cosmic rays in the interstellar medium

Note, the following discussion follows quite closely the Wefel overview article inCosmic rays, supernovae and the interstellar medium, eds. M.M. Shapiro,R. Silberberg, and J.P. Wefel, p44, Dordrecht: Kluver Academic Publishers. Asummary of data is given in Simpson’s article from 1983. Many of the calcu-lations are also discussed in Sections 9 and 20 of the Longair book. More re-cent data in the review by Strong A.W., Moskalenko, I.V., and Ptuskin, V.S.Ann.Rev.Nucl.Part.Sci. 57, 285-327, 2007, as preprint arxiv:astro-ph/0701517In order to set the stage, let us review what we know from measurementsof the local cosmic-ray population:

1. The spectral energy distribution of the all-particle spectrum in dN =N(0) · (E/E0)−Γ with Γ ≈ 2.7. The broad-band power-law stretchesfrom the energy of the heliospheric cut-off up to the so-called kneeat an energy of a few PeV (1015 eV). At higher energies, the energyspectrum softens to Γ ≈ 3.1 (see Fig. 13), recovering the value of 2.7at the so-called ankle at a few EeV (1018 eV), finally dropping off at40-60 EeV.

2. The arrival direction is (almost) isotropic. Up to knee energies, theisotropy is good within 0.1 %, while at higher energies, the anisotropyincreases, reaching a value of about 1 % at energies of 1018 eV (seeFig. 14).

3. The composition (chemical abundance) of the cosmic rays arrivingat Earth is at low energies similar (but not equal) to the solar metal-licity. The cosmic-ray abundance of rare elements like Lithium (Li),Berrylium (Be), and Boron (B) is significantly higher than the solarabundance (see Fig. 15). Furthermore, isotopic composition deviatessignificantly in the neutron rich isotopes.

4. The cosmic-ray composition changes with energy and the positionof the knee seems to be correlated with mass or charge (see Fig. 16).

These are the most important observational findings. We will re-visitthese issues in the following and derive an ansatz for an interpretation.We are mainly interested in the following questions:

• What is the confinement volume of cosmic-rays (as a function of en-ergy)?

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10-10

10-8

10-6

10-4

10-2

100

100 102 104 106 108 1010 1012

E2 dN

/dE

(

GeV

cm

-2sr

-1s-1

)

Ekin (GeV / particle)

Energies and rates of the cosmic-ray particles

protons only

all-particle

electrons

positrons

antiprotons

CAPRICEAMS

BESS98Ryan et al.

GrigorovJACEEAkeno

Tien ShanMSU

KASCADECASA-BLANCA

DICEHEGRA

CasaMiaTibet

Fly EyeHaverahYakutskAGASA

HiRes

Figure 13: Energy spectrum of cosmic rays from a recent compilation.Note, the energy dependent scaling on the flux.

• What is the confinement time of cosmic-rays (as a function of en-ergy)?

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Figure 14: Observed relative anisotropy of cosmic-rays (Ambrosio et al.PRD 2003), compared with predictions.

• What is the total power required to sustain the cosmic-ray popula-tion?

Note, these questions are mainly related to the nucleonic component ofcosmic-rays. The origin of primary cosmic-ray electrons (and possiblypositrons) requires a different treatment15 We can identify different ap-proaches to address the questions given above:

• Analysis of cosmic-ray spallation leading to the observed secondarynuclei Be, Li, B

15The main difference is the importance of radiative losses for electrons which can beneglected for cosmic-ray nuclei.

72

Figure 15: Comparison of solar abundances (open symbols) and cosmic-ray abundances (filled symbols). Note, the logarithmic scale.

• Analysis of radio-isotopes (and other rare isotopes) and the relativeabundance of cosmic-ray clocks

• Measurement of Photons and neutrinos from cosmic-ray interaction.

73

Figure 16: Compilation of broad band energy spectra for H, He, Fe (topto bottom). The compilation combines direct measurements with indirect(air shower based) measurements. Taken from Horandel (2005).

Again, cosmic-ray electrons require a separate treatment as a consequenceof radiative losses. The overall scheme on how to proceed is shown schemat-ically in the cartoon (Fig. 17).

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Figure 17: Cartoon of the cosmic ray transport in our Galaxy- taken fromMoskalenko et al. 2001.

3.2.1 Origin of light elements (Li, Be, B) in cosmic-rays

Obviously, the cosmic-ray interaction with the interstellar medium requiressome consideration of nuclear physics. The interaction of a cosmic-ray nu-cleus with a hydrogren atom (and to some extent and much rarer an inter-action with a He or a metall in the interstellar medium) leads to a spalla-tion reaction in which the original nucleus breaks up in potentially excitednuclear fragments. The cross section for these processes are in principleaccessible to lab experiments, however, the data-base of spallation crosssections is not complete. As it turns out, some of the analysis of cosmic-raydata is limited by the lack of available cross section data. For now, we as-sume, that cross sections for processes like σ(C + p→ He + X) are knownor phenomenologically calculated. The general equation for a particle dis-tribution of type i with uniform spatial diffusion (neglecting convection

75

and any form of momentum diffusion) is

∂Ni

∂t= D∇2Ni +

∂

∂E(biNi) + Qi −

Ni

τi+ ∑

j>i

Pji

τjNj. (106)

In the following, we will argue to use a simplified version of this equation.The energy losses are negligible for typical gas densities of n = 1 cm−3

and for the following discussion, we neglect the source term. This corre-sponds to an instantaneous injection at t = 0. Furthermore, we neglectthe diffusion which would correspond to a homogeneous distribution inthe considered medium. Generally, this is not valid, but sufficient for thepurpose of demonstrating the basic properties of the effect of spallation(and the limitation of this approximation). Furthermore, we change thevariable from time to the traversed column density ξ = ρ · x = ρ · v · t. Inthis case, after substituting, we arrive at the following equation:

∂Ni

∂ξ= −Ni

ξi+ ∑

j>i

Pji

ξ j· Nj. (107)

In this case, we assume, that all nuclei traverse the same column density(slab), this approximation is therefore known as the single slab model (wewill see shortly, that this model is too simple). For now, let us considerthe group of light (L) nuclei (Li, Be, B) and the most abundant group ofmedium (M) nuclei (C, N, O). The boundary condition can be simply setto NL(ξ = 0) = 0, because the light nuclei are not enriched by stellarfusion processes (Li is only produced in the primordial nucleosynthesis).For the M-group, we ignore the additional contribution from spallation ofheavier elements (we will see shortly, that this assumed decoupling is areasonable approximation):

dNM

dξ= −NM

ξM(108)

dNL

dξ= −NL

ξL+

PML

ξMNM. (109)

The solution to Eqn.108 with the boundary condition that NM(ξ = 0) =NM(0) is

NM(ξ) = NM(0) · exp(−ξ/ξM). (110)

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We can find a solution to Eqn.109 after substituting Eqn.110, expandingwith exp(ξ/ξL), and integrating (exercise). The ratio of the groups is thensimply:

NL(ξ)NM(ξ)

=PML · ξL

ξL − ξM·[

exp(

ξ

ξM− ξ

ξL

)− 1]

. (111)

Before comparing with the measurements, let us consider briefly whichvalues to use for ξL, ξM, PML. The column density ξ is related to the inte-gral cross section σ and the molar mass mA:

ξ =mA

NA · σ, (112)

with NA the Loschmidt/Avogadro number (6.02 × 1023 mol−1). As anexample, consider a typical cross section of 30 mbarn for inelastic pp-scattering. In this case, ξpp = 55 g/cm2.The total cross section for the spallation of an element of the M-group issimply the sum of all partial cross sections (e.g. σ(C + p → B + X) etc.).The values of the partial cross sections are listed in Table 4 in units ofmbarn. In order to calculate the values for ξM and PML, we calculate theabundance weighted cross section. The abundances are the actual valuesmeasured for the individual species C, N, and O (using the values fromTable 5). The averaged cross section is simply 〈σ〉 = ∑i wiσi/ ∑i wi withthe weights taken from the relative abundance (600-1000 MeV/n) and σithe total cross section for species i ∈ C, N, O. The resulting values are〈σM〉 = 280 mbarn and for 〈σL〉 = 200 mbarn16. The probability for pro-duction of an element from the L-group is estimated by taking the ratio ofPML = ∑i∈Li,Be,B σMi/〈σM〉 = 0.28.The measurement (see also Table 5) indicates for the ratio

NL

NM= 0.25 → ξ = 4.8 g cm−2. (113)

This result is obviously consistent with the naive expectation that the col-umn density is of similar magnitude as ξM.When looking into the relative abundance of the individual elements inthe L-group ([Li] = 136, [Be] = 67, [B] = 23317), the ratio of elements is

16note, that the values for L are not included in Table 417The abundances are always considered relative to the abundance of Si

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consistent with the respective weighted production cross sections (σ(M→Li) = 24 mbarn, σ(M→ Be) = 16.4 mbarn, σ(M→ B) = 35 mbarn). Thisis reassuring and shows that the simple, one slab model is sufficient toexplain the observed abundance of light elements. The same exercise canbe done to estimate the slab thickness for the production of 3He which isagain roughly 5 g/cm2.The simple one slab model works surprisingly well for the spallation ofthe M-group elements into the light nuclei. However, when looking intothe details of the spallation of iron nuclei, a simple one slab model doesnot work.The spallation cross section for iron nuclei is very high (see Table 4): σFe =763.4 mbarn, this corresponds to ξFe = 2.2 g/cm2. If the slab thicknesswere constant at 5 g/cm2, the iron nuclei abundance observed in cosmicrays should be strongly depleted with respect to the average abundance.Independent of the absolute value of the abundance, the ratio of spalla-tion products to the observed abundance of iron in cosmic rays should bestrongly dominated by the spallation products:

[products][primaries]

=1− exp(−ξ/ξFe)

exp(−ξ/ξFe)= 8.7 (114)

for ξ = 5 g/cm2. This is clearly not consistent with the observed abun-dances: The spallation of iron results mainly in the production of nuclei inthe range from Cl. . .V (see also the partial cross sections listed in Table 4).Using the values given in the appendix, the observations are

[Cl, . . . , V][Fe]

= 1.5 (115)

clearly in contradiction with the higher value expected. The discrepancyof the simple one slab model can be resolved, when treating the transportof cosmic rays more realistically.There are in principle two ways of treating the cosmic ray transport. Themost simple approach is to consider a distribution of path lengths and tryto find the distribution that matches the observed ratio of secondaries toprimaries (phenomenological approach). A different approach is basedupon a more elaborate, physics oriented model of cosmic ray transport inthe Galaxy which then in return results in a prediction of the path lengthdistribution (theoretical approach). In principle, both approaches have

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strengths and short comings. Let us for the sake of clarity consider twoextreme cases for the distribution of path lengths:

1. Leaky box: In this approach, the confinement volume of free stream-ing cosmic rays is defined by a boundary region with a finite escapeprobability. In this case the entire population of cosmic rays can bedescribed by

∂N∂t

= − Nτe(E)

. (116)

In the most simple case, the escape time would be energy indepen-dent (we will see later, that the data suggest an energy dependence)which would simplify the solution to be N ∝ exp(−t/τe) ∝ exp(−ξ/ξe).Exponential path length distribution.

2. Diffusion in infinite volume: The other extreme would be an infiniteescape time τe → ∞ with a diffusive transport:

∂N∂t

= D∇2N (117)

with a solution which corresponds to a Gaussian distribution ofpath lengths.

Before comparing the expectations from various models (variants of thetheme suggested above), let us summarize the observational results forsecondary to primary ratios.The most important measurements of the secondary to primary ratio arethe B/C (Boron to Carbon) and (Sc+Ti+V)/Fe-measurements. The mainresult of both measurements is an energy dependence of the secondary/primaryratio which indicates that the path length distribution changes with energy(see Fig. 18 for a recent compilation of measurements). The energy depen-dence is such that the secondary/primary ratio increases with increasingenergy until it reaches a peak and subsequently for increasing energy itdrops. The behaviour for the two measurements is very similar, indicat-ing that the path length distribution is similar for heavy and for mediumgroup.In order to appreciate the relevance of the observations, we take a step

79

back and consider the path length distribution required to match the ob-servational data. Again, we simplify the underlying equation by assum-ing a steady-state case (i.e. ∂N

∂t = 0). In this case, we can simply solve thefollowing equation:

− NL

ξe(E)+

PML

ξMNM(ξ)− NL

ξL= 0 (118)

NL =PML · NM(ξ)/ξM

ξe(E)−1 + ξ−1L

, (119)

which we simplify by assuming (realistically) ξe ξL:

NL(ξ)NM(ξ)

= PML ·ξe(E)

ξM. (120)

This effectively means that the secondary-to-primary ratio represents theeffective path-length as a function of energy (ξM is fairly constant overenergy).Returning to the measured secondary-to-primary ratio we clearly see thatfor large energies an exponential drop-off in the path-length with energyis a fairly good description of the model. However, at the low energy endas well as at the high energy end, there are markable differences visible.Let us summarize the main results:

• The increase of the secondary production with increasing energy atthe low energy end is not naively expected. It requires an ad-hocassumption in the leaky box model. The exponential path lengthdistribution is however a good approximation at high energies.

• More realistic models have been considered in the literature (see e.g.Jones, Lukasiak, and Ptuskin 2001). The increase in path length is anatural consequence in a model where the cosmic rays are injectedin an infinitesimally thin disk with a halo in which cosmic rays movewith a convective or turbulent wind. The wind speed can be tunedsuch that the cosmic rays are removed from the disk before they caninteract with the gas in the disk. This naturally leads to the peak inthe secondary/primary ratio.

• The observed drop off in secondary-to-primary ratio indicates thatcosmic rays with increasing energy leave the galactic disk and tra-

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verse on average a smaller column density on the way to the ob-server on Earth.

Figure 18: Compilation of secondary to primary ratios (from Strong,Moskalenko, and Ptuskin 2008). The left panel shows the B/C ratio repre-sentative for the spallation induced light nuclei abundance, the right panelshows the production of heavy nuclei in spallation from iron.

The general result of the stable secondaries indicates that the meanpath length is of the order of 5 g/cm2, decreasing with increasing energyroughly as ξe(E) ∝ E−0.3...0.6. When considering a value of 5g/cm2, whatgeometrical path length does this amount to?

As a rough estimate, consider a medium with average density ρ (inunits of g/cm3), the traversed column density relates to the spatial dis-tance travelled x: ξ = x · ρ. With a typical gas density of 1 proton/cm3,the corresponding mass density ρ = 1.6× 10−24 g/cm3. For ξ = 5 g/cm2,the corresponding path length is x = 3.12× 1024 cm≈ 1 Mpc. When com-paring this with the typical radius of the gas disk in our Galaxy of 10 kpc,it is obvious that the propagation of cosmic rays is not rectilinear/free-streaming but is clearly related to a diffusive transport of cosmic-rays inthe entangled magnetic field of the interstellar medium. Similar num-bers can be estimated by considering the propagation time. The parti-cles move roughly with the speed of light. Therefore, the time is givenby τe = 3× 106 n/(1 cm−3) yrs. We will compare this number with theestimate derived from the cosmic clocks.

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3.2.2 Cosmic-ray clocks

Now, what is the confinement volume? In order to investigate this point,the most elegant approach is the use of cosmic-ray clocks.The best-measured cosmic-clock is the radio active isotope 10Be. The life-time of 10Be is τ(10Be) = 3.9× 106 yrs for the decay of 10Be →10 B + e− +νe. 10Be is mainly produced in the spallation of C and O.By measuring the ratio of stable 7Be to 10Be, it is possible to measure theaverage time for the propagation of cosmic-rays from the source to theobserver.Quantitatively, we consider again a leaky box model (steady-state). Inorder to shorten the notation, we define

Ci := ∑j>i

Pij

τjNj (121)

− Ni

τe(i)+ Ci −

Ni

τspal.(i)− Ni

τr(i)= 0. (122)

The most important change is the introduction of the last term which takesinto account the decay of the radio isotopes. The equation for the stableisotopes:

− Nkτe(k)

+ Ck −Nk

τspal.(k)= 0. (123)

Combining both equations:

N(10Be)N(7Be)

=C(10Be)C(7Be)

·τe(7Be)−1 + τspal(7Be)−1

τe(10Be)−1 + τspal(10Be)−1 + τr(10Be)−1 ,(124)

assuming that τspal τe, this simplifies to:

N(10Be)N(7Be)

=C(10Be)C(7Be)

· τe(7Be)−1

τe(10Be)−1 + τr(10Be)−1 . (125)

Inserting the measurements, which show a slight variation with energy(see Fig. 19), the typical value for τe derived is τe ≈ 107 years for [10Be]/[7Be +9

Be +10 Be] ≈ 0.028. When comparing this with the column density tra-versed, cosmic-rays obviously are on average propagating in a medium

82

which has a density of n ≈ 0.3 cm−3, considerably smaller than the av-erage density in the Galactic disk. This indicates that cosmic rays verylikely spend a considerable time propagating outside the disk in an ex-tended halo. It is (at this point) not possible to conclude the actual extentof the halo but it certainly extends beyond the scale height of cold molec-ular gas.

Figure 19: Measured values for the ratio of radioactive Be to stable Be-isotopes.

3.2.3 Comments on particle diffusion

Diffusive transport of particles is a well-known phenomena that has beenstudied intensively in the general context of transportation processes (e.g.heat conductivity). The diffusive transport of cosmic-rays is however amore complicated process as it requires the treatment of charged particlesmoving through a (partially) ionised medium with magnetic field. Thisis a highly non-linear problem which we do not introduce here in all itsdepth. Let us highlight a number of important issus.The general principle expressed in Fick’s second law applies to the generalproblem of diffusive transport:

∂N∂t

= ∇Dxx∇N (126)

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which describes the temporal change of a particle (concentration, density)that is subject to a diffusive (random) transport. The spatial diffusion co-efficent Dxx, given in units of length2 time−1, is related to the mean freepath length λ:

Dxx = 〈v〉 · λ

3. (127)

The mean free path length is commonly estimated to be the distance trav-elled at which the particle has changed the direction considerably and thegeneral distribution of pitch angles is isotropic. The most common valueused is the gyro-radius of a particle moving in a homogeneous magneticfield. Note however, that the motion of a particle is usually not as sim-ple as a gyrating motion. The gyro-radius rg = p/(ZeB) can be readilycalculated with the following equation:

rG = 0.4 pc Z−1 E1015 eV

(B

3 µG

)−1

. (128)

The net motion of diffusing particles is towards regions with smaller val-ues of density (see also Eqn. 126). The distance R travelled in a time τ isgiven by

R2 = D · τ, (129)

which is characteristic for diffusive motion. Before we estimate the dif-fusion coefficient of cosmic-ray propagation, we discuss the microscopicproperties of diffusive transport.

Scattering of cosmic-rays on self-generated Alfven waves: The inter-stellar medium can be simplified by assuming that a static, backgroundmagnetic field B0 is present throughout the medium perturbed by an addi-tional fluctuating field component B1. The fluctuation B1 can be describedby a spectrum of turbulences, where the power (B2

1) is a function of thelength scale (λ). For cosmic-rays, the background field B0 leads to a reg-ular motion of charged particles which singles out a characteristic lengthscale similar to rG. For λ rG or λ rG, the effect of B1 is averaged eitherover many cycles (λ rG) or there are not sufficient cycles (λ rG) to

84

affect the motion of the particle. If however, particles and irregularities areclose in phase and wave-length, a resonant scattering takes place, whereenergy is transferred from plasma waves to cosmic-rays or vice versa.The scattering of the particle will lead after one cycle to a change of theangle

δϕ ≈ B1

B0. (130)

Since the change of the angle is random, it will take N = (δφ)−2 scatteringevents to change the angle by one radian. During the N cycles, the particlewill have moved

λsc ≈ Nλ ≈ rG(δφ)−2 ≈ rG

(B0

B1

)2

. (131)

In the following, we want to show that these irregularities are produced bythe cosmic-rays themselves (this is why frequently, cosmic-rays are calledself-confining particles). The irregularities turn out to be Alfven (plasma)waves. Let us turn for a moment to plasma waves. A plasma is definedas a medium which is (partially) ionised. On average (or large scales),the plasma is electrically neutral. In astrophysics, the plasma is very di-lute such that actual particle-particle collisions are negligible. The long-distance Coulomb interaction however still leads to scattering of particlesin electrical and magnetic fields. This type of plasma is often referred toas a collisionless plasma. There is a large number of plasma waves as wellas instabilities known. For our purposes, we consider only one particular(and relevant) type of plasma waves which are called Alfven-waves18

This particular wave propagates in the plasma with density ρ = Np · mpwith a velocity:

vA =B0√4πρ0

. (132)

For the typical interstellar medium with Np = 1 cm−3 and a magnetic fieldB0 = 3 µG, we can calculate vA = 8 km/s. The growth rate of Alfvenic

18Biographical note: Hannes Alfven (*1908,†1995) was a Swedish Physicist who wasawarded the Nobel price in 1970, recognizing his fundamental work in Magnetohydro-dynamics and plasma physics.

85

waves depend on the streaming velocity of cosmic-rays. Without derivingthe result (see Longair, Section 20.4), the energy density in Alfven wavesis U = U0 exp(Γ t) with a growth factor

Γ(λ) = Ω0N(> E(λ))

Np

(−1 +

|v|vA

). (133)

The particles N(> E) which resonantly scatter with the wave length λ leadto a growth in the perturbance spectrum for streaming velocities largerthan vA. For cosmic-rays N(> E) ∝ E−1.7 implies that the time-scaleτ = Γ−1 for the growth of Alfven waves increases with E1.7. This im-plies that particles with increasing energy are less efficiently confined.We have so far neglected processed leading to wave-damping, which ef-fectively requires energy to be removed from the waves. Among theseprocesses, interactions with the neutral phase of the interstellar mediumlead to a damping on time-scales short in comparison with the growthtime scale (Kulsrud and Pierce, 1969).The interstellar medium in the Galactic plane consists of a mixture of cool,mostly molecular but also neutral atomic (hydrogen) gas with embeddedregions with a high level of ionisation. Outside the disk, the gas is mostlyionised. This leads to the following, simple picture for cosmic-ray trans-port in our Galaxy:

• Inside the (mostly neutral) gas in the Galactic disk, cosmic-rays aremostly free-streaming, possibly drifting along field lines.

• When encountering a region of highly ionized gas (as e.g. outsidethe Galactic plane), the cosmic-rays start moving diffusively, with anet streaming velocity approaching the Alfven velocity.

At the interface between the two regions, particle conservation shouldhold, and therefore:

Nint · c = Next · vA. (134)

This readily translates into the time it takes for cosmic rays to leave thedisk with thickness L (this can only take place at the velocity vA):

τ ≈ LvA

= 5× 107 yrsL

kpc

(Np

0.1 cm−3

)−1/2

· B3 µG

. (135)

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Again, this value is fairly close to the value we have derived using thecosmic-ray clocks. The remaining difference could in principle be an indi-cation that the magnetic field may be on average smaller along the propa-gation path.

Estimate of the Diffusion coefficient for Galactic cosmic-rays: The heightof the halo that is occupied by cosmic-rays can be estimated from the al-ready discussed average density of the traversed interstellar medium. Thetotal column density of (neutral) gas in the direction of the Galactic polesis NH ≈ 1.5 × 1021 cm−2. Strictly speaking this is a lower limit as theionised gas is not accounted for. The height of the halo is then

H =NH

〈n〉 =1.5× 1021

0.3cm = 1.6 kpc. (136)

Using this value in combination with the estimated time travelled of 107 yrs:

D =H2

τ' 3× 1028 cm2

s. (137)

The energy dependence of the B/C-secondary-to-primary-ratio implies,that this value obviously changes with energy. A common parameteriza-tion (independent of any underlying model) is:

D(E) = D0 ×(

E10 GeV

)α

(138)

with α = 0.5 . . . 0.7 (derived from the energy dependence of the B/C-ratio).

3.2.4 Anisotropy of cosmic-rays

The anisotropy of cosmic-rays is an important ingredient on the way totrace the sources of cosmic-rays. The measurements indicate the presenceof a relative anisotropy δ < 10−3, with the following definition of δ:

δ :=Imax − Imin

Imax + Imin. (139)

Generally, the measurement of such a small anisotropy is experimentallyvery challenging. Especially indirect (air-shower) measurements show

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variations of the detection rate orders of magnitude larger than the level ofanisotropy searched for. Furthermore, a large scale (e.g. dipole) anisotropyrequires a reasonably homogeneous exposure of the entire sky, which againis not trivial for an individual detector located on Earth. Generally, the ob-served level of anisotropy should be (cautiously) considered as an upperlimit. Nevertheless, we can at least qualitatively discuss the expectations.

The degree of anisotropy depends on the location of the observer withrespect to the sources. We would start with the assumption that the ac-celerators of cosmic rays are probably following the general mass distri-bution of the Galaxy. Most of the accelerators would therefore be locatedwithin the disk and fairly close to the inner Galaxy. The solar system is lo-cated at the exterior part of the Galaxy and therefore, we would expect ananisotropy with Imax located grossly in the direction of the Galactic centerand Imin probably in the opposite hemisphere or even directed towardsthe Galactic poles.The fact, that the observed relative anisotropy is well below unity, impliesalready that cosmic-rays do not propagate rectilinear but obviously dif-fusively (no surprise after the discussion in the previous sections). Forthe diffusive transport, we can readily estimate the relative anisotropy.Based upon the estimate of the diffusion coefficient in the Galaxy (D ≈3× 1028 cm2 s−1 ), the net streaming velocity of the cosmic-ray gas is there-fore

V ≈ DR

= 107 cms

= 100km

s≈ 3× 10−4 c. (140)

The cosmic-rays are streaming in the direction of the gradient of the cosmic-ray density. In the rest frame of the streaming motion, the cosmic-ray dis-tribution is isotropic.Given that the nuclei travel roughly with the speed of light, the relativeanisotropy for an observer δ ≈ V/c ≈ 3× 10−4. A more detailed treat-ment would have to take into account that the relative motion leads to ashift in the observed energy such that δ = v/c · (p + 2) with p the power-law index of the differential energy spectrum. For the cosmic-rays, p = 2.7which leads to δ = 1.2× 10−3, which in turn is fairly close to the observedanisotropy.The observed degree of anisotropy is in agreement with a diffusive trans-port of cosmic-rays in our Galaxy.However, there are a number of buts which (in the light of the good agree-

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ment of the simple estimate with the measurement) should not be forgot-ten:

• The sun is not at the edge of the cosmic-ray source distribution. Wecan certainly expect, that cosmic ray sources are also active in thedirection of the Galactic anti-center.

• Even more important is the fact that the sun is located close to thecenter of a local active star forming region called Gould belt. Here,we have e.g. a high number of massive stars as well as supernovaremnants which in principle inject cosmic rays in the local medium.

• The sun is located in the so-called local bubble, ie. the interstellarneigbhorhood of the sun19 is characterized by a hot, ionised, andtenuous gas, possibly heated up by multiple supernova explosions.

• The presence of spiral arms. In the spiral arm structure the magneticfield lines are more ordered (as we can derive from the polarizationdirection of radio-synchrotron emission seen predominantly in spi-ral galaxies). This ordered magnetic field component leads to a driftof the cosmic-rays parallel to the field lines (the particles gyrate alongthe field line and move parallel to the field). This intrinsically shouldlead to an anisotropy larger than the rough estimate given above.

• The motion of the sun around the Galactic center as well as the mo-tion of Earth around the sun: The net streaming velocity of the cos-mic rays is smaller than the rotational velocity of the sun (> 200 km/s).This leads to an additional dipole-moment that is annualy modu-lated by the motion of the Earth around the sun (≈ 30 km/s) thatis either parallel or anti-parallel to the orbital motion of the solarsystem. Both effects need to be removed from the data to infer thestreaming velocity. This is commonly done by considering the anisotropyperpendicular to the direction of motion.

3.2.5 Anti-matter from cosmic-rays in the Galaxy

The search for anti-matter is mainly of interest for two reasons: (i)acceleratedanti-matter could be the result of acceleration of anti-nuclei in a anti-matter

19As the name implies, the interstellar medium in the solar environment is not typicalfor the interstellar medium.

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galaxy and (ii) anti-protons (as well as positrons) are produced in cosmic-ray interactions with gas and provide insights into the cosmic-ray trans-port. Deviations from the p-flux expected from secondary production maylead us to conclude that additional, primary anti-proton-production is nec-essary as e.g. suggested in models of evaporating black holes as well asfrom self-annihilating dark matter (e.g. WIMP-dark matter).

3.2.6 On the total power required to sustain the cosmic-ray population

Pulling all pieces together, we are ready to estimate the total power re-quired to sustain the cosmic-ray population. Starting with a cylindricalvolume of V = πR2 · H with R = 10 kpc, H ' 1.5 kpc: V ≈ 2× 1067 cm3.The energy density of cosmic rays is simply taken to be the locally mea-sured energy density (with a correction of the effect of solar modulation).The canonical value is usually taken to be ucr ' 1 eV/cm3. The totalenergy stored in the form of cosmic rays in the Galaxy is therefore E =ucr · V = 2× 1067 eV' 1055 ergs. The power required to balance the es-cape losses is P = E/τesc = 3× 1040 ergs/s= 107 L (in words 10 millionsolar luminosities) in the form of cosmic-rays.Generally, the most attractive class of objects suggested to explain thecosmic-ray population are supernova remnants. Each supernova remnantreleases roughly 1051 ergs in the form of kinetic energy in the interstellarmedium. Only a fraction η < 1 is released in the form of cosmic-rays.Together with the super-nova remnant rate nsnr = 1/100 yrs, we can esti-mate the efficiency η required to match Lcr:

Lcr = η · ESNR · nsnr → (141)

η = 9 %(

Lcr

3× 1040 ergs/s

)·(

ESNR

1051 ergs

)−1

·(

1/n100 yrs

)(142)

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Table 4: Partial cross section for spallation reactions. The table is takenfrom Longair, Chapter 5.

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Table 5: Measured abundances of cosmic rays, solar system, and localGalactic. Table is taken from Longair, chapter 9.

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3.3 Acceleration of cosmic rays

The presence of a non-thermal distribution of relativistic nuclei and elec-trons in the Galaxy obviously implies an underlying acceleration mecha-nism which needs to explain a number of relevant observations:

• Power-law shape of the energy distribution with a power-law whichshould be similar to the observed power-law shape.

• Maximum energy should reach at least 1020 eV.

• Chemical abundance of the accelerated nuclei should be similar tothe solar abundances.

• Sufficiently efficient and fast acceleration to balance the injection powerrequired to balance the cosmic ray escape from the Galaxy.

A number of schemes have been investigated in order to achieve particleacceleration in astrophysical environments which follow in principle oneof the following approaches or flavours thereof:

1. Stochastic acceleration in randomly moving clouds of magnetizedplasma (2nd order Fermi acceleration).

2. Acceleration at a hydrodynamic/magneto-hydrodynamic shock front1st order Fermi acceleration.

3. One-shot acceleration in electrical field gradients.

4. Acceleration in magnetic reconnection regions.

5. Acceleration by particle/wave resonances (“surfing”).

In the following we will focus on Fermi type acceleration and leave thediscussion of the other acceleration types to the literature.Enrico Fermi described in his seminal paper from 1949 the notion of parti-cle acceleration through scattering on randomly moving magnetized clouds(Fermi 1949). In this approach, the particles gain energy in head on colli-sions while losing energy in trailing collisions. The energy gain per scatter-ing is ∆E/E ∝ β2 with β = vcloud/c 1 which implies that acceleration israther slow for the average velocity of clouds in the inter-stellar medium.

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The energy gain of the particle is at the expense of macroscopic kinetic en-ergy carried by the cloud.In the presence of strong shocks (Mach number 1), faster acceleration(∆E/E ∝ β with β = Ushock/c = O(%)) is possible. The balance betweenparticles convected away from the shock and the probability to undergomultiple cycles of shock crossings, leads to the formation of a broad bandpower-law with dN/dE ∝ E−s with s = (γ + 1)/(γ− 1) with γ = cp/cvthe polytropic index for an adiabatic process. In the well-justified case ofan ideal monoatomic gas γ = 5/3 and therefore s = 2.The achievable maximum energy of the acceleration process depends mainlyupon the size and life-time of the shock (ultimately the limit would bethe Hubble time) which vary widely between the shocks present in astro-physical environments. Additionally, energy loss mechanisms constrainthe maximum energy achievable. This is particularly important for theacceleration of electrons where radiative energy losses are efficient. Fornuclei, energy losses become only relevant at ultra-high (E > 1019 eV) en-ergies. Generally, acceleration can only proceed until the particle is notconfined to the environment of the shock. Taking the gyroradius of theparticle and comparing it with the size of the shock, a general conditionfor confinement of the particle is that the size of the shock L > 2rg withrg = 1 pc E15/(ZBµG) for a particle with energy E = E15 × 1015 eV in aregion with magnetic field B = BµG µG:

Lpc > 2E15

Z BµG. (143)

This fundamental relation is also met by terrestial accelerators (e.g. LHC).A well-studied example for an astrophysical object which drives a strongshock is the super-nova remnant. In both general types of supernova ex-plosion, the ejecta drives a strong shock in the ambient medium whichis expanding with an initial velocity of 10 000 km/s. Following the free(balistic) expansion, the shock slows down in the Sedov stage of the SNRevolution, with Ush ∝ t−3/5 (see also Section 1.5). The maximum energythat can be attained during the dynamic evolution of the shock is smallerthan the limit given in Eqn. 143 because the shock has a finite life-time.Ultimately, at the end of shock evolution, the maximum energy that canbe reached depends on the diffusion coefficient in the downstream andupstream region. The smaller the diffusion coefficient, the smaller dis-tance the particles diffuse in the downstream and upstream region before

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crossing the shock again, therefore increasing the number of cycles untilthe particles leave the shock region. The upper limit (see eg. Lagage &Cesarsky 1983) is

Emax = 1014 eV Z BµG. (144)

This implies that at least in this linear theory (the particles are treated astest-particles without feedback on the shock structure). However, the re-quired efficiency of shock acceleration in supernovae ofO(∞′%) indicatesthat possibly non-linear feedback should be investigated. Here, we onlysketch the main ideas of this theory and evidences from observation.When looking at the hydrodynamic structure of a shock, the presence ofnon-thermal particles exerts additional pressure which changes the shockstructure. So far, the hydrodynamic shock was given by a compressionratio of 1 : 4. With the addition of the cosmic ray pressure, the compres-sion ratio is increased to values up to 7 . . . 10. From a magnetohydrody-namic analysis carried out be Lucek&Bell (2000), the cosmic rays stream-ing ahead of the shock drive instabilities that rapidly increase the magneticfield. This leads to efficient scattering/confinement of the cosmic-rays tohigh energies. The maximum energy achievable via this non-linear effectis probably close to the knee energy of 1015 eV. A telltale sign of nonlinearamplification of magnetic field and therefore presence of cosmic rays areso-called filaments which have been detected with high spatial resolutionobservations with the Chandra X-ray telescope. These filaments are X-raybright regions which in projection are narrow features, mostly aligned par-allel to the shock. Long-term monitoring reveal time-variability in thesefilaments. The common interpretation of the filaments is that magneticfield has to be amplified to values up to 100 µG in order to confine X-ray emitting electrons. The time variability is then expected as the coolingtime of these electrons is accordingly shortened. Indirectly, the presence offilaments in most young X-ray emitting supernova remnants has been in-terpreted as evidence for efficient cosmic-ray acceleration with non-lineareffects. Another peace of observational evidence in favor of non-linearshock acceleration is the observation that e.g. the historical supernovaremnant RCW 86 tends to show a deceleration at the north-eastern rimwhich could be related to the high acceleration efficiency which leads toan observable effect in the evolution of the SNR. For RCW 86 the north-eastern rim is the site of particle acceleration as evident from X-ray syn-chrotron emission. Furthermore, the shock heating in RCW 86 proceeds

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Figure 20: The energy spectrum beyond the knee. The thick line is the av-erage of measurements up to 1017 eV. The lines indicate models for extra-galactic origin of cosmic-rays and different species. Taken from Hillas(2006)

less efficient which again would indicate that cosmic-ray acceleration re-duces the energy available for expansion/shock heating.In summary, the evidence for shock acceleration in supernova remnants

with high efficiency is convincing and indicates indirectly, that a high en-ergy density in cosmic rays must be present in the shock vicinity. The effi-ciency achieved is sufficiently large to explain the overall Galactic cosmicray budget. However, there are a number of interesting problems withrespect to this conclusion from maybe the most sensitive observationalchannel as discussed in the next section.Finally, we turn to extra-galactic cosmic-rays from the knee to the ankleand beyond in the energy spectrum (see Fig. 20). The origin of particles atultra-high energies (> 1019 eV) is very likely extra-galactic given that theseparticles are not confined to the Galaxy (see Fig. 21). On the other hand,

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Figure 21: The curvature of particles (protons and iron nuclei) traversingthe Galaxy for a mean (homogeneous) magnetic field of 2 µG (Hillas 1984).

the origin of particles at energies above ≈ 8× 1019 eV is constraint to bereasonably local since once the threshold for exciting the ∆-resonance andphoto-pion production is exceeded, the mean free path length for inelasticpγCMB converges to λ = (nCMBσpγ)−1 ≈ 1/410 cm3/100 µbarn ≈ 10 Mpc.More accurate calculations take the energy dependence of the cross sec-tion into account, but the basic conclusion remains: catastrophic losses ofcosmic-rays above the so called Greisen-Zatsepin-Kuzmin (GZK) cutoff ef-fectively shield UHECR from cosmological distances. Furthermore, thesmall deflection by a few degrees of the cosmic-rays in the extra-galacticmagnetic field and galactic magnetic field open the opportunity to identifythe accelerators by tracing back the reconstructed direction and correlatingit with (known) nearby source candidates. The criterium on the confine-ment of cosmic-rays to the accelerator imposes quite severe constraints onthe size and magnetic field of the accelerator. Fig. 22 summarizes the re-quirement to achieve a maximum energy of 1020 eV for protons and ironnuclei. The existence of particles beyond 1019 eV requires extreme condi-tions for acceleration which are only met by a few objects including radiogalaxies (specifically their hot spots), galaxy clusters, and possibly AGN.Note however, that compact objects like AGN are disfavored by the highphoton density environment which ultimately limits the acceleration by

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Figure 22: Constraint on the size and magnetic field for accelerators ofultra-high energy cosmic-rays: Objects which confine protons should beabove the solid line. For iron nuclei, the constraint is relaxed by the highercharge number (dashed line). The grey band indicates the requirement fornon-relativstic motion of the scattering centers (upper boundary is for β =1/300). The various potential accelerators/acceleration sites are indicatedin the diagram.

energy losses through photo-pion production inside the compact source.Most interestingly, the first data released by the Pierre-Auger consortiumtaken with the large (3000 km2) surface array (PAO 2007) indicates a closecorrelation of the arrival direction (within a space angular separation ofof data taken during the first 3.5 years. A total of 27 events above ener-gies of 57 EeV qualify for the correlation study. The statistical analysis ofa possible correlation is not straight-forward and a word of caution on thesignificance of the findings is necessary. One of the most serious problemsis very likely the limited number of events and the choice of selection cutsas well as the choice of source catalogues used to conduct the study. How-ever, the consortium claims that a correlation with nearby AGN exist at a

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Figure 23: The Hammer-Aitoff projected sky map in Galactic coordinates.The center of the image is the direction of the Galactic center. North isup, East is left. The blue shaded region indicates equal exposure in equalshades of blue (lighter blue is less exposure). The red stars are positionsof nearby AGN while the circles (3.2 radius) indicate the directions ofthe 27 events used in the correlation study. The dashed line marks thesupergalactic plane (Pierre Auger collaboration 2007).

confidence level of more than 99 %. The corresponding sky plot is shownin Fig. 23 where the finding of a correlation within an angular window of3.7 is apparent.

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