Download 6. ACCELERATION MECHANISMS FOR NON

6. ACCELERATION MECHANISMS FOR NON-THERMAL
PARTICLES
We have discussed at length in the previous Section about the widespread evidence
for the presence of huge flows of non-thermal, highly energetic particles in essentially
all cosmic environments, and particularly present in extra-galactic radio-sources.
The energy spectra of the emitting particles that we have inferred from the study of
extra-galactic radio-sources turn out to be completely consistent with those of the fast
particles that come from the outside of the atmosphere, the terrestrial cosmic rays.
The present Section is fully dedicated to try understanding how such energetic
particles can be separated from the thermal particles and accelerated to enormous
energies.
6.1 Cosmic Rays
The discovery and study of cosmic rays have played a fundamental role in the
evolution of physics during XX century. Extraterrestrial cosmic rays have been the
first high-energy particles to be discovered. Although laboratory accelerators
currently produce huge fluxes of high-energy particles, the highest energy particles,
up to energies of about 3 1020 eV , still come from the cosmos (and the same will
keep true in the future).
The energy spectrum of extra-terrestrial cosmic rays is reported in Figure 1.
As we see from the figure, cosmic rays in the Galaxy show indeed energy spectral
slopes between 2 and 3 over an enormous interval of particle energies. This is
completely consistent with the observed synchrotron spectra of radio-galaxies
(similar acceleration mechanisms appear to operate on such different scales, from the
parsec to the Mpc ones).
Some more information on the cosmic rays that we directly collected near the Earth
are reported in the following figure, including their observed composition. Note these
are fraction of numbers, not mass fractions. The Helium fraction is higher than the
estimated primordial numerical abundance (6%), the difference is to be attributed to
the stellar activity.
Note in particular that the flux of electrons is about 100 times less than for protons, a
situation that has been used by us in Sect. 5.14. This much lower number is explained
by the fact that electrons are much more emissive than protons or Helium nuclei and a
large fraction of them has decayed in energy from the cosmic-ray source to the Earth.
6.1
Figure 1. Global energy spectrum of extra-terrestrial cosmic rays measured by a number of
experiments.
6.2
Figure 2. Another representation of the energy spectrum of extra-terrestrial cosmic rays. The
relative fractions of the various kinds of particles are also reported. These are fraction of
numbers, not mass fractions. The Helium fraction is higher than the estimated primordial
abundance (6%), the difference is to be attributed to the stellar activity.
These energy distributions are then strongly departing from the classical MaxwellBoltzmann Gaussian (thermal) distributions, and are so called non-thermal
distributions, with a power-law form:
N (ε )d ε Cε − p d ε
=
ε1 < ε < ε 2
1
One of the very specific and apparently surprising properties of these particle energy
spectra is their apparent universality, with a spectral index p that is quite confined to
p ≈ 2−3
2
Other observational features that the process of acceleration of high energy particles
must account are the acceleration of cosmic rays to energies E ∼ 1020 eV and the
chemical abundances of the primary cosmic rays.
6.3
6.2 Particle acceleration. General principles
The acceleration mechanisms may be classified as dynamical, hydrodynamical and
electromagnetic (Longair, HEA). Of course, the distinction may often be fictitious.
In some models, the acceleration is purely dynamical, for example, in those cases
where acceleration takes place through the collision of particles with clouds.
Hydrodynamic models can involve the acceleration of whole layers of plasma to high
velocities. The electromagnetic processes include those in which particles are
accelerated by electric fields, for example, in neutral sheets, in electromagnetic or
plasma waves and in the magnetospheres of neutron stars.
The general expression for the acceleration of a charged particle in electric and
magnetic fields is given by eq. (5.0) Sect.5:
d
d
d F d
d qd d
a=
(γ mv)= v × B + qE;
=
m dt
c
γ≡
1
1 − v /c
2
2
.
In most astrophysical environments, static electric fields cannot be maintained
because of the very high electrical conductivity of ionised gases – any electric field is
very rapidly short-circuited by the motion of free charge. Therefore, electromagnetic
mechanisms of acceleration can only be associated either with non-stationary electric
fields, for example, strong electromagnetic waves, or with time-varying magnetic
fields. In a static magnetic field, no work is done on the particle but, if the magnetic
field is time-varying, work can be done by the induced electric field, that is, the
electric field E given by Maxwell’s equation curl E = −∂ B/∂t . It has been suggested
that phenomena such as the betatron effect might be applicable in some astrophysical
environments. For example, the collapse of a cloud of ionised gas with a frozen-in
magnetic field could lead to the acceleration of charged particles since they conserve
their adiabatic invariants in a variable magnetic field.
6.3 Particle acceleration. Fermi second-order mechanism
The problem of a physical understanding of high-energy particle acceleration in
astrophysics has fundamentally two aspects.
The first one is to explain how a small subset of particles are selected and separated
from the large number of the low-energy thermal ones to become the high energy
cosmic ray population, with energies enormously higher. This is called the problem
of the particle injection.
The second aspect is to understand how this small subset of particles is brought to the
large energies that we know.
6.4
We will start addressing the second question, which is currently better understood,
while the first one is quite less understood.
It is first of all interesting to note that, once a particle has detached from the thermal
bath and has become even mildly relativistic,
, then from eq. 3.55 (Sect.3) it can
be immediately seen that the timescale for significant exchange of kinetic energy
with the other particles become very large and tending to infinity:
3
for an electron, and a particle density of 1
. This is a very long time for the
timescales at play. So this particle stops exchanging its energy with the lower energy
particles, and effectively decouples from the ambient.
The only significant interaction of such relativistic particles is with the magnetic
field, that we know is pervasive in all astrophysical media. But because the magnetic
fields preserves the energy, these particles would maintain their energy mostly
unaltered in principle.
Figure 3. Cosmic ray "scattering" in a magnetic cloud moving with velocity V. [From Longair HEA].
Fermi in 1954 proposed a simple idea of how a magnetic field could accelerate
particles in spite of the fact that it conserves the energy (see for this our discussion in
Sect. 5.1 about charged particles in e.m. fields). Let us look at this apparent paradox.
Let us consider a relativistic particle of energy and momentum p in the lab system.
Let us assume that the particle moves inside a medium characterized by the presence
of a number of clouds moving chaotically at non-relativistic speed v and containing
a frozen magnetic field. The particle entering the cloud will feel the magnetic field
6.5
and will be randomly deflected by it. Because the cloud mass is infinitely superior to
that of the particle, the collision will be completely elastic, and the particle will
receive a small amount of kinetic energy any time it is reflected by a cloud, and this
will happen many times.
Let us work out a quantitative calculation about it. This is a relatively simple exercise
of special relativity. Assuming without loss of generality the cloud is moving with
velocity –v along the x direction while the particle velocity makes a total deflection
angle θ= θ1 + θ 2 with x (see Fig. 3), it is immediate to calculate the energy of the
particle in the cloud reference system with a first Lorentz transformation:
ε=′ γ V ( ε + pv cos θ ) ,
γ V ≡ (1 − v 2 / c 2 )
−1/2
4
(as usual, we label with the apex the quantities in the cloud system and without it in
the laboratory system), while its momentum will be given by the transformation
vε 

=
px′ γ V  p cos θ + 2 
c 

5
After the shock, because of the elasticity of the collision, the energy will be
conserved and the momentum reversed
ε 2′ = ε ′ , p2′ = − px′ .
6
Now, in the laboratory system, after the collision we have for the particle energy
(
) (
ε 2 = γ V ε 2′ − p2′ v = γ V ε ′ + px′ v
)
7
and, upon substitution of (4) and (5) and setting close to c the particle velocity, 1
 2v cosθ v 2 
ε 2 =γ ε 1 +
+ 2
c
c 

We can approximate this for v << c to obtain the fractional energy increase per
2
V
8
single collision
1
With eq.6, we substiture 7 into 4:
ε2 =
γ {γ (ε + pv cosθ + γ ( pv cosθ + v 2ε / c 2 )} =
=
γ 2ε {1 + pv cosθ / ε + pv cosθ / ε + v 2 c 2 )} =
γ 2ε {1 + 2v cosθ / c + v 2 c 2 )} , because
ε
p
≈ c for the
almost relativistic particle.
6.6
ε2
2v cos θ v 2 
2
= − 1 =−1 + γ V 1 +
+ 2 
ε
ε
c
c 

2v cos θ
2
2
2
2
∆ε

−1 + v c +1 + v c +
1 − v2 c2
c

9
2v cos θ 2v
+ 2
c
c
2
If all collisions might be head-on, then cos θ = 1 and the fractional energy increase
for each collision of the particle with the clouds would be of the first order in the
small ratio v c . However, the cloud velocities are randomly distributed, and the first
addendum in (9) vanishes after angular average (front-tail symmetry). It remains only
the quadratic term in v 2 c 2 , a rather small number and a very marginal energy gain,
unfortunately. Note that there is in any case such small gain because the probability
of a head-on encounter with clouds is slightly larger than that for tail bumps. Indeed
consider that on average a particle feels many more clouds coming towards itself than
the clouds it encounters traveling in its own direction (an effect we can name "cars in
the highway coming in opposite sense to that of a fast traveling car"). Considering in
detail the probability of an encounter with a cloud with velocity direction θ with
respect to x and averaging over angles 2, we get the average energy gain to be:
∆ε
ε
8 v2
=
3 c2
10
(the famous Fermi result). Particles gain energy only at the second order in the
velocity ratio (second-order Fermi mechanism).
This process of energy gain is very slow. Furthermore, this theory does not explain
the characteristic p value in eq. 2, because it can be shown that any p value would be
possible from this theory.
In spite of these problems and limitations, the basic ideas of the Fermi theory are
essentially correct and make the foundation of other improved models that are
discussed in the next subsection.
6.4 Particle acceleration. First-order theory (diffusive shock acceleration).
For many years, the second-order mechanism was the only one being considered.
More recently (e.g. Blandford & Ostriker 1978), the Fermi concept has evolved in the
2
See M. Vietri, Astrofisica delle Alte Energie, Bollati. As remarked there, there is no paradox in considering the effects
of the interaction of the charged particles with purely the magnetic field. As already remarked, the latter interacts with
particles but cannot do work on them (just deflecting them), so in principle there should be no particle acceleration. As a
matter of fact, because of the relativistic transformation, there is a small electric field in the observer frame, which is
responsible for the acceleration.
6.7
direction of obtaining a model able to accelerate with first-order dependence on the
. This new acceleration model was obtained for high-speed particles moving
ratio
in the proximity of shock waves.
The first order Fermi mechanism, also mentioned as diffusive shock acceleration,
consists in the acceleration experienced by a particle when crossing and re-crossing a
shock front, as illustrated in Figure 4. In a supernova event a shock wave is formed
by the fast outflowing gas layers of the star, with high (although non-relativistic)
velocities, see Sect. 3.10. The shock makes a discontinuity in the pressure, density
and temperature of the surrounding medium, which assume higher values behind the
shock wave (up-stream) and lower values beyond the shock wave (down-stream).
Scattering ensures that the particle distribution is isotropic in the frame of reference in
which the gas is at rest. Let's set ourselves in the reference system of the shock
(shock at rest at
), with matter advancing through the medium with velocity
.
from left to right, and will exit the shock with velocity
Figure 4. Scheme illustrating the diffusive shock acceleration of a particle when crossing and recrossing a shock front.
relative to the
So the gas behind the shock travels at a velocity
upstream gas (Fig. 4). A particle crossing the shock, because of the Lorenz
6.8
transformation due to the change of velocity, acquires a small energy gain ∝ v c , as
we see below. After shock crossing, the particle velocity is again randomized by
collisions with other particles or magnetic fields. A particle (that may be the same
particle as before) making the same shock crossing from right (downstream) to left
(upstream) will encounter a gas moving towards its own downstream reference
system will see gas approaching from the other side of the shock with the same
velocity v1 -v 2 = 3v1 4 and will undergo exactly the same process of receiving a
small increase in energy ∝ v c on crossing the shock from the downstream to the
upstream flow as it did in travelling from upstream to downstream.
Figure 5. Scheme further illustrating the diffusive shock acceleration of a particle when crossing and recrossing a shock front. 1) Shock is moving towards the unperturbed fluid with velocity
-v1 . 2)
Viceversa, in
the shock reference frame, matter enters the shock with velocity v1 and exits it with velocity v 2 = v1 4 .
3) In the upstream region 1, particles are randomized and cross again the shock. They see the material on the
other side of the shock approaching with velocity v=v1 − v 2 =
3v 4 . 4) Once the particle has first crossed
the shock front, it is randomized in region 2 and eventually will cross the shock a second time. Any time the
front is crossed, the particle improves its energy by an average amount given by eq. 13.
Every time the particle crosses the shock front it receives an increase of energy –
there are never crossings in which the particles lose energy – and the increment in
energy is the same going in both directions. This is because in both cases the frame
wherein the particle is moving sees the region behind the shock to approach with the
same velocity: in the reference system in position 2 in the figure below this
differential velocity is v = 3v1 4 ; similarly, the frame in position 3 moves with
6.9
exactly the same velocity v = 3v1 4 with respect to the other side of the shock.
Thus, unlike the original Fermi mechanism in which there are both head-on and tail
collisions, in the case of the shock front the collisions are always head-on and energy
is transferred to the particles. The situation is illustrated in the scheme above.
Let us now consider in more detail a fast-moving particle with energy ε in the
reference system of the pre-shock fluid, moving towards the shock with a velocity
making an angle θ with the plane surface of the shock. This particle will see the
shock and post-shock material approaching with velocity v1 -v 2 = 3v 4 , which is
the equivalent of a galactic cloud in approach. In the reference frame of the postshock (downstream) material, it will have an energy ε ′ given by eq. (4).
Note that the high-energy particles scarcely notice the shock at all since its thickness
is normally very much smaller than the gyro-radius of the high-energy particle.
Now, because of the presence of clouds with frozen-in magnetic fields in the
downstream region, the particle will be reflected back across the shock and again in
the pre-shock region, where it will have the same energy ε ′ than in the post-shock
frame, while the energy in the pre-shock frame will be given by a second Lorenz
 2v cosθ v 2 
transformation, i.e. by eq. (8), ε 2 =γ v ε  1 +
+ 2  . However, similarly to
c
c 

2
the other crossing, now again the condition on the impact angle such that the particle
crosses the shock front will be cos θ ≥ 0 . It follows that, once we average over
angles, the first order term with v c does not vanish now, and the energy gain will be
∆ε ε = ( ε 2 − ε ) ε  2v cosθ c
12
because v 2 c  v c and γ v  1 . So, when moving both from upstream to
downstream and viceversa, the particle will see the material approaching with the
same velocity v = 3v1 4 . The process is then now of first order, which means that it
proceeds much faster than based on the classical Fermi mechanism.
2
Assume that the particle moves from downstream towards the non relativistic shock
in the x direction with relativistic velocity, forming an angle θ with the plane of the
shock. For isotropically distributed directions of the particles, the probability that a
particle will cross the shock wave with an angle of incidence between θ and θ + dθ
is proportional to sinθ dθ and the rate at which the particles approach the front is
proportional to v ⋅ cos θ . So the energy gain after one shock crossing results from
integration of eq. (12) weighted by the factor v ⋅ sin θ cos θ :
∆ε
ε
=2β
π /2
2v
2β
∫ cos θ sinθ dθ = 3c = 3
0
2
.
13
6.10
For a complete round trip across the shock with two crossing this fractional energy is
∆ε ε = 4 β 3 .
14
We will refer in the following to the first (shock acceleration) and second order as the
Fermi mechanisms in general.
6.5 The particle energy spectrum from diffusive shock acceleration.
Let us first define as α the parameter setting the energy of a particle with initial
energy ε 0 and given by: ε = α ε 0 . From what we have previously seen the
parameter α can be immediately calculated from eq. (14) as
α=
ε ∆ε
4
=
+1 =1+ β
ε0 ε0
3
15
We also define as P the probability that the particle remains within the accelerating
region after one collision: after n collisions, in this region there will be
N (> ε ) =
N 0 P n particles with energy equal to ε = α nε 0 or greater. So, eliminating
n in the two defining relations, we have simply
ln ( N [> ε ] N 0 ) ln P
=
ln (ε ε 0 )
ln α
so the energy spectrum can be expressed as
N (> ε )  ε 
= 
N0
 ε0 
ln P ln α
16
while in differential units (number of particles per unit interval of energy, which is
the usual representation of energy spectra) we have
ε 
N (ε )d ε = cost  
 ε0 
ln P ln α −1
dε .
17
Next we need to calculate the escape probability P (see e.g. Bell 1978). We first
consider that, if N is the number density of particles moving with velocity almost c,
the number of particles in any direction is Nc (particle velocities are quickly
isotropized). Those crossing the shock per unit time are Nc cos θ , with the condition
that cos θ > 0 . So the total rate of particle crossing is given by
6.11
J
=
dΩ
Nc cosθ
∫=
θ
4π
cos > 0
Nc
4
18
In the post-shock region the particles are removed because there is a bulk flow of the
fluid outside the region with velocity v 4 . On one side, the particles are deviated and
isotropized by the presence of magnetic fields, on the other these fields are frozen in
the fluid that exits the shock, so there is a classical diffusion process removing
particles systematically from the downstream region: considering what we discussed
e.g. in Fig. 5, the outflowing velocity here is v 4 . So the particles are lost by the
region at a rate of Nv 4 . Altogether, the fraction of particles lost by the system,
with respect to those that enter it, is
Nv 4 v
=
Nc 4 c
such that the probability they remain inside the region is P = 1 − ( v c ) , which is a
number quite close to unity. So
v
 v
 4v  v
ln P =ln 1 −   − ; ln α =ln 1 +  
c
 c
 3c  c
19
ln P
 −1
ln α
−2
and from (17)
ε 
N (ε )d ε ∝   d ε ∝ ε − p d ε , p  2 .
 ε0 
20
This is the classical result for the predicted spectrum of the particles accelerated at the
shock. It is a remarkably stable and robust result in which all physical details of the
process have disappeared. The important fact is that the spectrum is very “stable”
and, contrary to what happens for the Fermi second order process, it has a very
precise spectral shape that is very close, though slightly flatter, to what we observed
for radio galaxies and what is measured in Galactic cosmic rays (Sect. 6.1).
Note that any small energy loss, particularly for electrons, (e.g. radiative losses or
more efficient escaping of particles from the acceleration region) will steepen the
spectrum, towards the slightly steeper observed values. So the spectrum in (20) can
be considered as an asymptotic value in the immediate vicinity of the acceleration
site.
The results of numerical simulations based on the assumption of relativistic shock is
reported in Fig. 6.
6.12
Figure 6. Simulated spectrum of non-thermal particles accelerated in a shock. Thinner lines show
spectra that have made 1, 2, 3, … n cycles around the shock [Lemoine & Peletier 2003].
6.6 The problem of particle injection.
We know that sufficiently energetic particles do not exchange energy with the
thermal background particles. The only effective interaction of such particles is with
the magnetic fields, as we have discusses. So these energetic particles are decoupled
from the bath of thermal particles, and once they become sufficiently energetic, they
don’t loose energy by interacting with the huge number of low energy ones and can
start to be effectively accelerated from Fermi. So the basic problem is to understand
how these particles get separated from the sea of thermal ones and are hence brought
to sufficiently high energies to experience the final enormous acceleration towards
the cosmic ray energies. As we have already remarked, these particles, more energetic
they become, the less they interact with the surrounding medium. This was remarked
from consideration of eq. (3) and can also be immediately seen by considering the
rate of energy loss as a function of the particle energy given by eq. (2.4) for the nonrelativistic Bremsstrahlung case, for which this rate is approximately proportional to
6.13
∝ ε −1 . For the relativistic case the situation is a bit more complex, still a rough
dependence ∝ ε −1 is found.
The starting point for this differentiation is in any case assumed to originate in the
non-relativistic Maxwell-Boltzmann distribution of energies (collisional processes
like in thermal plasmas naturally produce such distribution). This distribution extends
to the highest energies, and it is the particles in the extreme tail of the distribution that
start progressively differentiating from the others.
Then the various Fermi acceleration mechanisms start to operate. One additional
factor (the so-called drift mechanism) that has been considered is the case in which
there is a component in the magnetic field not exactly perpendicular to the surface of
the shock. In this case, assuming the ideal magneto-hydrodynamical condition, in the
shock frame we have the occurrence of an electric field component with intensity 3

v
21
E ≈ − ×B
c

where v is the shock propagation speed and B the field in the pre-shock frame. Eq.
21 is immediately understood as the occurrence of an electric field component from
Lorentz transformation as discussed in eq. (2.18). Hence producing an acceleration to
the charged particle increasing its energy to
e = ZevB c ,
22
where  is the distance travelled by the particle before exiting the shock region. This
latter quantity is not likely to be very large, but the whole process may significantly
contribute to the early stages of particle acceleration (and it has been sometimes
proposed in alternative to the Fermi mechanisms themselves).
6.7 Limitations to the maximum energy for the Fermi process.
The Fermi mechanisms are formally able to produce infinitely energetic particles (eq.
20 may extend to ε → ∞ ). Of course this is an highly idealistic situation that does
not consider energy losses by the particles for radiation, the finite sizes of the shock
region and the fact that the shock piston cannot operate for an infinite amount of time.
Indeed the supernova shock wave decelerates and enters the Sedov evolutionary
phase once it has swept a mass of interstellar matter of the order of that of the
supernova ejected gas remnant.
3
This result can also be inferred, with an order of magnitude estimate, from the third Maxwell (eq. 1.4),


1 ∂B : assuming v to be the shock velocity and  the size of the region, we have for the induced electric
∇× E = −
c ∂t
field E  ≈ B c (  / v ) ⇒ E ≈ Bv c .
6.14
Altogether, the Fermi acceleration processes, both of them, are not very rapid: the
particles have to scatter back and forth many times across the shock to achieve
sufficient energy (see illustration in the simulation of Fig. 6), and particles gain
energy just by a few percent at most in each shock crossing.
Even more fundamental, these acceleration mechanisms have been demonstrated to
achieve some maximum energy for the particle, that may be estimated from (22).
Adopting typical values for the process, such as a magnetic field intensity
B ≈ 10−6 G , v=5000 Km/s, t=1000 years as the duration of the acceleration
phenomenon (shock duration) such that   vt  5 pc , we obtain from (22) 4
e ZevB c ≈ 3 1014 eV
=
23
as a maximum energy obtainable per nucleon by these acceleration processes. As we
can see e.g. in Figs. 1 and 2, the observed energy spectra of cosmic rays extends well
beyond this limit to 1020 eV at least (the maximum energy of cosmic rays is not
accessible by observation because of their energy loss by pair production, as
discussed in Sect. 10).
We defer to Vietri, AAE, for a more complete discussion of this problem.
6.8 The unipolar accelerator.
We are left from the previous Sect. with the problem that the acceleration processes
so far discussed are able to explain the spectra of cosmic rays over a substantial range
of energies, but not quite to explain the most energetic of such particles. This is the
case by several orders of magnitude in energy. Other acceleration mechanisms need
to be invoked to explain this.
The natural alternative to the role of magnetic fields that can be considered is the
electric field. We have seen that the general rule in astrophysics is that, given the
typically enormous numbers of particles involved, any significant E field on an
astrophysical scale (i.e. outside the Debye radius) is immediately neutralized by even
small displacements of a fraction of the particles. An exception to this is the case in
which static electric fields are induced by the systematic motion of magnetic fields.
By Lorentz transformation (eq. 2.18) an electric field is generated in the reference
frame in which the B is moving, whose intensity is E ≈ vB c (if v << c ).
The most classical expected occurrence is the case of a massive rotating object
hosting a frozen magnetic field, like we discussed in Sect. 5 for the Crab pulsar,
where the electric component is generated on the surface of the star and the
surrounding environment. Massive flows of charged particles (typically electrons)
4
The calculation in CGS units is:
e ≈ 4.8e-10[e] ×1e-6[B] × 5e8[v] ×1.5e19[]/3e10[c]/1.6e-12[erg to eV] ≈ 7.5e13 eV .
6.15
then move on the surface to attempt neutralizing the E field. The electric force on the
charge e will be given by the Lorentz transform

  B
eE= e Ω × r × .
c
(
)
24
The moved charges will attempt to neutralize this field with its opposite, and an
equilibrium condition will be verified when

  B
E = − Ω×r × .
c
(
)
This implies that an electric potential is set between different positions over the
surface of the sphere. In particular, if the B field is a north-south dipole with respect
to the rotation radius, an electric potential will be generated with north-south
differential potential of
d
BΩR 2
,
=
V ∫=
ERdθ
0
c
π
25
where R is the radius of the sphere. If there is a conductive plasma around the object,
then the circuit will be closed and an electric current will be established.
If we apply this theory to a fast rotating pulsar, we get the following situation.
R  10 Km  106 cm , enormous magnetic field B  1012 − 1014 G frozen in the
collapsed object, rotational period of 0.03 sec, hence obtaining energies of about
e =eV ≈ 1016 − 1017 eV , not enough to explain the highest energy cosmic rays.
However, the younger pulsars tend to be those rotating faster, and rotational periods
of 1 milli-sec are plausible and energies up to almost 1019 eV for a proton are not
implausible.
Looking this in detail, these values are not entirely realistic, because there are
processes that tend to limit, once more, the electric fields and the acceleration.
6.9 Explaining the highest energy cosmic rays.
Figure 7 summarizes some fundamental information about potential sites for the
acceleration of the highest energy cosmic rays. The figure is based on eqs. 22 and 23
by setting the maximal energy to that observed for the highest energy particles
=
e max ZevB c ≈ 1020 eV , a condition that can be written as
1020 eV ≈ ZevB c ⇒
B

0.1
≈
Gauss pc Z β
28
6.16
This is the condition on the magnetic field intensity and the size of the acceleration
5
region to obtain particles of
. For example, if you want to accelerate a
proton (Z=1) to such an energy you need that the product is at least 0.1. This means
that if potential sources lie to the left of the locus Fig. 7, protons cannot be
accelerated to 1020 eV in these objects. To achieve that energy for a proton, in any
case, we need a shock piston velocity that should be close to c (β≈1), a condition that
is not met, for example, in supernova ejecta’s shock waves.
Figure 7. The combinations of length-scale and magnetic flux density necessary to accelerate particles to
energy
where β = v/c is the shock piston velocity. Crosses indicate the typical magnetic flux
densities and sizes found in different classes of astronomical objects on different scales. (Hillas, 1984).
5
Let us do the full calculation:
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However, in the Galaxy, the condition β≈1 can be obtained in pulsar
magnetospheres. Similar conditions can be obtained in γ -ray bursts and active
galactic nuclei and radio jets.
However, for more massive nuclei, like those of heavy elements, iron and others, the
achievable energy benefit by an increase of Z in eq. 28, where it is possible to get
even in Galactic sources, depending on the particle mass.
The crucial issues concern the masses and charges of the highest energy cosmic rays.
If they are protons, an origin in active galaxies seems the most natural explanation.
As of now, unfortunately, the composition of the highest energy cosmic rays is still
unknown (their number flux is too low to obtain a mass measurement). There are
indications about an increase of the atomic number with energy, as illustrated in
Figure 8, but what happens in detail at an energy higher than
is still very
uncertain.
Figure 8. Illustrating how the overall energy spectrum of cosmic rays could be accounted for as the
superposition of the energy spectra of different species accelerated in the strong magnetic fields in
supernova shock waves. The data are taken from the survey by Nagano and Watson (2000). (From Longair,
HEA).
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It is interesting to consider here the gyration radius of a cosmic ray. As discussed in
Sect. 5, from eq. [5.7] we have for ultra-relativistic particles
30
where
, and finally, for a proton of 1 GeV:
31
. For a proton of
this is 1 Kpc,
For a proton of 1000 GeV,
while for the highest energy particles it can get up to a Mpc. Considering that the
height of the Galaxy is about 300 pc, it is clear that there is a good probability that a
cosmic ray with
is generated outside the Galaxy, and very likely by
AGNs in the closeby Universe.
The possibility that the very high-energy cosmic rays are produced by low-redshift
galaxies has been extensively investigated by the Pierre Auger observatory, an array
of one hundred square kilometers on ground (in Argentina). This is one of the
observatories able to identify the extended air showers triggered by the arrival of
6.19
high-energy cosmic rays that collide with molecules in the atmosphere. The arriving
high energy particle generates a cascade whose depth inside the atmosphere depends
on the energy of the original. The cascade includes both particles and photons (see
more details in Sects. 10 and 11).
Auger, as well as other similar ground based observatories, are able to detect with
good precision the direction of arrival of cosmic rays by making careful use of the
times of arrival of the signals in the array. Auger has a precision of slightly better
than 1 degree in the determination of the cosmic ray arrival direction. Examples of
showers from gamma-ray photons and hadrons are shown here below.
The Auger collaboration has reported the detection of 58 events with energies E ≥ 6 ×
1019 eV observed between 2004 and 2009, with positional accuracy of better than
0.9◦. These arrival directions have been correlated with the objects in various
catalogues of nearby objects (an original report of a correlation with nearby AGNs
contained in the compilation of Veron-Cetty & Veron was not later confirmed).
Instead a significant correlation was found with a bright catalogue of galaxies with
magnitudes brighter than K = 11.25 (Huchra et al., 2005), or a distance of roughly 80
Mpc, has been shown to be just as strong as the correlation with active galaxies. The
correlation is indicated by an excess of pairs of galaxies-cosmic ray at separation
angles of few degrees, as shown in the figure below.
This result is taken as a confirmation that cosmic rays of the highest energies likely
originate outside our Galaxy.
6.20