1 +4
3 V
c
≈+4 3
V
c. (1.22)
The probability to escape the acceleration zone Pesc using classical kinetic theory with Fig. 1.7 is Uc = 4V3c, so
P = 1−Pesc ⇒ln P = ln(1−Pesc) = ln
1−4V 3c
≈ −4V
3c (1.23)
and Eq. 1.17 becomes:
N(E)dE∝E[−1+(ln P/lnβ)]dE∝E−2dE. (1.24) This provides an excellent physical justification why a power-law with a unique spectral index should be expected from cosmic rays. The fact that we observe a spectral index of
∼2.7 in cosmic rays instead of a 2 comes form the influence of the magnetic field inside the source and many types of possibles shocks (parallel or perpendicular to the magnetic field). A steeper observed spectrum is correlated with the fact that the ratio v1/v2 = 4 in the basic diffusive model becomes lower than 4 because of the effects mentioned above.
1.3 Experimental Observations
The observed differential flux Φ of cosmic rays in an energy interval [E, E+dE] is defined as follows:
Φ(E,E + dE) = N(E,E + dE)
∆T×A×dE (1.25)
where N(E,E + dE) is the number of cosmic rays between E and E + dE, ∆T is the observation time, A is the geometrical acceptance and dE is the energy bin. The observed energy spectrum integrated over all types of cosmic rays is shown in Fig. 1.8. The differential spectrum spans over many orders of magnitude in energy, from less than 1 GeV to 1020 eV. The flux φ is approximated between 10 GeV and 1015 eV by φ ∝ E−2.7, where the exponent 2.7 is the spectral index and for energies above 1016 eV until
∼1019 eV it is ∼3.1. The steepening of the flux is called the“knee”. After 5·1019 eV the evolution in energy of the spectrum is not so clear from the experiments. The highest energy cosmic rays have such large Lorentz factors that photons of the cosmic microwave background radiation (CMB) have very high energies in the rest frame of the cosmic ray and the photo-pion and photo-pair production can take place, which degrade the energy of the cosmic ray. This energy threshold is known as GZK cut-off (from the names Greisen, Zatsepin and Kuzmin). The existence of the cut-off depends upon the value ofγ of the cosmic rays. If the highest energy cosmic rays are dominated by iron nuclei rather than protons, the photo-pion production process would not be responsible for the cut-off.
One interpretation of the break in the cosmic-ray spectrum at 1015 eV is that it
CHAPTER 1. COSMIC RAYS IN THE UNIVERSE (1 particle/m2
Knee -year) (1 particle/m2
Ankle -year) (1 particle/km2
-century) (1 particle/km2
FNAL Tevatron (2 TeV)CERN LHC (14 TeV)
LEAP - satellite Proton - satellite Yakustk - ground array Haverah Park - ground array Akeno - ground array AGASA - ground array Fly’s Eye - air fluorescence HiRes1 mono - air fluorescence HiRes2 mono - air fluorescence HiRes Stereo - air fluorescence Auger - hybrid
Cosmic Ray Spectra of Various Experiments
28. Cosmic rays 15
[eV]
Figure 28.8:The all-particle spectrum as a function ofE(energy-per-nucleus) from air shower measurements [90–105].
and confinement in the galaxy [109] also need to be considered. The Kascade-Grande experiment [100] has reported observation of a second steepening of the spectrum near 8×1016eV, with evidence that this structure is accompanied a transition to heavy primaries.
Concerning the ankle, one possibility is that it is the result of a higher energy population of particles overtaking a lower energy population, for example an extragalactic flux beginning to dominate over the galactic flux (e.g. Ref. 106). Another possibility is that the dip structure in the region of the ankle is due topγ→e++e−energy losses of extragalactic protons on the 2.7 K cosmic microwave radiation (CMB) [111]. This dip structure has been cited as a robust signature of both the protonic and extragalactic nature of the highest energy cosmic rays [110]. If this interpretation is correct, then the galactic cosmic rays do not contribute significantly to the flux above 1018eV, consistent with the maximum expected range of acceleration by supernova remnants.
The energy-dependence of the composition from the knee through the ankle is useful
February 8, 2016 19:55
Figure 1.8: Left: Flux of cosmic rays measured by various experiments shown together [10].
The energy scale of CERN and Tevatron are also marked. The green dashed line represents an energy spectrum E−3. Right: zoom of the cosmic-ray flux at higher energies observed on ground and multiplied by E2.6[11]. φ(E)·E2.6should be more independent of the energy of the particle and it can show additional features in the spectrum.
represents the energy at which cosmic rays can escape from the galaxy. It is expected that the cosmic rays with greater energies would display significant anisotropy if they originated from within the galaxy. The diffusive acceleration model of cosmic rays can explain the observation of cosmic rays until 1016−1017eV. In fact from the first Maxwell equation, the electric and magnetic fied in order of magnitude are:
E∼B·U (1.26)
where U is the velocity of the shock. The maximum acceleration energy of a particle of charge Ze, subject to the force F of this electric field will be then:
Emax= where L is the accelerator size. For a young supernova the acceleration time is t≈103 years, B∼ 10−10T and U ∼104km/s. For a nucleon at the speed of light, Emax ≈ 1016 eV.
Nuclei like carbon and oxygen are accelerated even to higher energies. In fact the so-calledknee in Fig. 1.8 appears almost at 1015−1016 eV and it is higher for nuclei than
22
protons (the integrated spectrum is dominated by protons). Observed energies of 1020 eV can be reached by protons only if in Eq. 1.27 the accelerator shock has a velocity U∼c. Such velocities can be found in pulsars, active galactic nuclei, γ-ray bursts but not supernova remnants in our galaxy. But if the accelerated particles are iron nuclei, they can reach this energies with sources in our galaxy.
The cosmic-ray spectra for different nuclear species are summarized in Fig. 1.9. The 2 28. Cosmic rays
whereE is the energy-per-nucleon (including rest mass energy) and α (≡γ+ 1) = 2.7 is the differential spectral index of the cosmic-ray flux and γ is the integral spectral index. About 79% of the primary nucleons are free protons and about 70% of the rest are nucleons bound in helium nuclei. The fractions of the primary nuclei are nearly constant over this energy range (possibly with small but interesting variations). Fractions of both primary and secondary incident nuclei are listed in Table 28.1. Figure 28.1 shows the major components for energies greater than 2 GeV/nucleon. A useful compendium of experimental data for cosmic-ray nuclei and electrons is described in [1].
Figure 28.1: Fluxes of nuclei of the primary cosmic radiation in particles per energy-per-nucleus are plotted vs energy-per-nucleus using data from Refs. [2–13].
The figure was created by P. Boyle and D. Muller.
The composition and energy spectra of nuclei are typically interpreted in the context of propagation models, in which the sources of the primary cosmic radiation are located within the Galaxy [14]. The ratio of secondary to primary nuclei is observed to decrease with increasing energy, a fact interpreted to mean that the lifetime of cosmic rays in the galaxy decreases with energy. Measurements of radioactive “clock” isotopes in the low energy cosmic radiation are consistent with a lifetime in the galaxy of about 15 Myr [15].
February 8, 2016 19:55
Figure 1.9: Cosmic-ray fluxes from hydrogen until iron measured by various experiments. Pic-ture taken from [11].
chemical abundance integrated over the energy is shown in Fig. 1.10 in the solar sys-tem and in the galaxy. The most abundant cosmic rays are hydrogen (∼ 90%) and heavier nuclei (mainly helium). Immediately one can see in Fig. 1.10 the difference in the lithium, beryllium and boron abundances in the two environment, as well as in the sub-iron elements. This is due to the spallation of the most abundant cosmic nuclei, such as carbon, oxygen and iron, with the interstellar medium in the galaxy. One detail of the data is particularly important, the ratio of the unstable isotope 10Be to stable
9Be, which is discussed in the next section following [9].
CHAPTER 1. COSMIC RAYS IN THE UNIVERSE
100 T.K. Gaisser, T. Stanev / Nuclear Physics A 777 (2006) 98–110
spectrum above 1018eV that may be related to a transition from particles of galactic origin to those accelerated in extra-galactic sources. The question of whether there is a transition to cosmic rays of extra-galactic origin, and if so where it occurs, is of considerable interest.
Finally, the question of whether the spectrum extends beyond 1020eV is currently the fore-most problem in high-energy particle astrophysics because of the difficulty of accounting for the absence of energy loss by protons in the microwave background radiation if the spectrum is not suppressed above this energy.
In what follows we discuss the three energy regions in order. A common theme is use of the primary composition as a clue to the nature of the sources of the particles.
2. Galactic cosmic raysE <1014eV
A classic problem in nuclear astrophysics is the determination of the cosmic-ray source abundances [5,6]. This requires a model of the galaxy, including the spatial distribution of cosmic-ray sources; density, composition and ionization state of the interstellar medium;
and the strength and topology of the magnetic field. Important recent investigations of cosmic-ray propagation from two different viewpoints may be found in Refs. [7,8]. Here we give only the main points in a simplified form. The starting point is a determination of the abundances of different nuclei near the solar system (but outside the heliosphere).
Fig. 2 shows the elemental abundances in the cosmic radiation. The overabundance by several orders of magnitude of secondary elements such as lithium, beryllium and boron
Fig. 2. Comparison of Solar system [9] and cosmic-ray elemental abundances. Nuclear abundances are from [10];
protons and helium are from [11,12].
Figure 1.10: Comparison between the chemical abundances measured near the solar system (but outside of the heliosphere) in black, and in the solar system in green. Picture taken from [12].
1.3.1 Confinement of cosmic-rays
We define the production rate through spallation of nucleus i from a nucleus j with higher atomic mass, with probability Pji in a timeτj:
Ci=X
j>i
Pij
τjNj. (1.28)
The transfer equation for nucleus i will be (nothing else than a balance equation):
− Ni
τesc(i) + Ci− Ni
τspall(i) = 0⇒Ni= Ci
1
τesc(i)+ τ 1
spall(i)
(1.29) where τspal is the spallation time for nucleus i. For a radioactive nucleus j there is an additional term that takes into account its decay:
− Nj
τesc(j)+ Cj− Nj
τspall(j)− Nj
τr(j) = 0⇒Nj = Cj
1
τesc(j) +τ 1
spall(j)+τ1
r(j)
(1.30) If we assume for the beryllium isotopesτspall τesc:
N(10Be) N(7Be) =
1 τesc(7Be) 1
τesc(10Be)+τ 1
r(10Be)
C(10Be)
C(7Be) . (1.31)
24
In 1983, the ratio 10Be+107BeBe+9Be = 0.28 was measured and reported by J. Simpson [14].
Inserting this value in Eq. 1.31 givesτesc≈2·107 years. This is an example of how the measurements of cosmic rays help in understanding the Universe.