3
3.1
Astrophysics of Cosmic rays
Relevance of cosmic rays
The energy density of cosmic rays in the Galaxy is dominated by lowenergy cosmic-rays which are not detectable within the solar system. The
solar wind expands into the interstellar medium and shields low energy
cosmic-rays. The lower energy cut-off is modulated by the varying solar activity. Extrapolating the energy spectrum backwards to smaller energies provides a similar energy density as obtained from modelling the
diffuse gamma-ray emission (see Section 2.2) from the Galactic plane to
be ucr ≈ 1 eV/cm3 . The energy density in the Galactic magnetic fields is
of similar magnitude, hinting at a deeper connection between cosmic ray
acceleration/transport and magnetic field structure and strength in the interstellar medium. Before addressing these issues, we highlight the importance of cosmic rays in the context of astrochemistry, star formation, and
finally possible connections between Earth’s climate changes in the past
and modulation of the cosmic-ray bombardment of Earth’s atmosphere.
Astrochemistry: The formation of complex molecules requires an environment shielded from intense sources of photo-ionization with sufficiently dense gas densities. The ideal place for astromechanical reactors
are the dense and cold cores of molecular clouds. Deeply embedded in gas
and dust, only cosmic-rays provide the source of ionization (at densities
> 103 cm−3 , radio-activity provide additional ionization power) which
leads to the formation of free radicals (mainly H + ) which initiate the formation of molecules with as many as 13 atoms14 . In total, more than 150
(230 including isotopomere) molecules are known to exist in the interstellar medium. The commonly assumed ionisation rate of cosmic-rays is
10−17 s−1 . The formation of molecules proceeds along a molecular network in which the formation/destruction of molecules leads to an evolution of the abundances with time.
Star formation: The generally accepted picture of the formation of massive stars includes a period of accretion, in which an accretion disk forms.
14 For a recent list of molecules found see http://www.cv.nrao.edu/ awootten/astrophysics.html
68
The accretion of matter from the accretion disk is directly linked to the
state of ionisation of the gas in the disk. Gammie (1996) concluded that
a layered accretion takes place in which the outer layers of the accretion
disk are ionized by cosmic-rays, leading to a sufficiently large viscosity in
order to form an accretion flow. The source of ionization is however not
the star itself but rather cosmic-rays which can penetrate sufficiently deep
into the accretion disk (a few 100 g/cm2 column density). The inner core
of the accretion disk may remain neutral and therefore inactive without an
accretion flow.
Cosmic-rays and climate changes in Earth’s past (for a review see Kirkby
2007): (Non-anthrogenic) climate changes in the past appear to be related
to solar cycles (e.g. the small ice age in the 16th and 17th century which
took place during a phase of pronounced minimum in the solar activity).
Recent data suggest a possible anticorrelation between cloud coverage in
the lower troposphere and the solar activity (Svensmark et al. 1997). The
solar irradiance is unlikely to cause a change in the low cloud coverage
given that the main changes are in the UV spectral band which is mostly
absorbed in the stratosphere. The alternative possibility of cosmic-ray
induced cloud formation would provide an explanation of the observed
anti-correlation as in the case of a solar minimum the rigidity dependent
cut-off at the lower energy end of the spectrum shifts towards lower energies leading to increased cosmic-ray fluxes at Earth while during the solar
maximum, the cut-off shifts to higher energies, reducing the cosmic-ray
rate. The actual mechanism of cosmic-ray induced cloud formation is not
entirely settled and is currently investigated in the frame-work of an accelerator experiment (CLOUD: Cosmics Leaving OUtdoor Droplets). Even
though this paleo-climatology theory is disputed, it is an elegant way to
explain the fact that even at time-scales of the order of 100 Myrs, possible
cycles of ice-age/warm time appear to be correlated with the passage of
the solar system through spiral arms.
69
3.2
Transport of Cosmic rays in the interstellar medium
Note, the following discussion follows quite closely the Wefel overview article in
Cosmic rays, supernovae and the interstellar medium, eds. M.M. Shapiro,
R. Silberberg, and J.P. Wefel, p44, Dordrecht: Kluver Academic Publishers. A
summary of data is given in Simpson’s article from 1983. Many of the calculations are also discussed in Sections 9 and 20 of the Longair book. More recent 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/0701517
In order to set the stage, let us review what we know from measurements
of 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 stretches
from the energy of the heliospheric cut-off up to the so-called knee
at an energy of a few PeV (1015 eV). At higher energies, the energy
spectrum softens to Γ ≈ 3.1 (see Fig. 13), recovering the value of 2.7
at the so-called ankle at a few EeV (1018 eV), finally dropping off at
40-60 EeV.
2. The arrival direction is (almost) isotropic. Up to knee energies, the
isotropy is good within 0.1 %, while at higher energies, the anisotropy
increases, reaching a value of about 1 % at energies of 1018 eV (see
Fig. 14).
3. The composition (chemical abundance) of the cosmic rays arriving
at Earth is at low energies similar (but not equal) to the solar metallicity. The cosmic-ray abundance of rare elements like Lithium (Li),
Berrylium (Be), and Boron (B) is significantly higher than the solar
abundance (see Fig. 15). Furthermore, isotopic composition deviates
significantly in the neutron rich isotopes.
4. The cosmic-ray composition changes with energy and the position
of the knee seems to be correlated with mass or charge (see Fig. 16).
These are the most important observational findings. We will re-visit
these 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 energy)?
70
Energies and rates of the cosmic-ray particles
100
protons only
E2dN/dE
(GeV cm-2sr-1s-1)
all-particle
-2
10
electrons
positrons
10-4
10-6
CAPRICE
AMS
BESS98
Ryan et al.
Grigorov
JACEE
Akeno
Tien Shan
MSU
KASCADE
CASA-BLANCA
DICE
HEGRA
CasaMia
Tibet
Fly Eye
Haverah
Yakutsk
AGASA
HiRes
antiprotons
10-8
10-10 0
10
102
104
106
108
1010
1012
Ekin (GeV / particle)
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 energy)?
71
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 population?
Note, these questions are mainly related to the nucleonic component of
cosmic-rays. The origin of primary cosmic-ray electrons (and possibly
positrons) requires a different treatment15 We can identify different approaches to address the questions given above:
• Analysis of cosmic-ray spallation leading to the observed secondary
nuclei Be, Li, B
15 The
main difference is the importance of radiative losses for electrons which can be
neglected for cosmic-ray nuclei.
72
Figure 15: Comparison of solar abundances (open symbols) and cosmicray abundances (filled symbols). Note, the logarithmic scale.
• Analysis of radio-isotopes (and other rare isotopes) and the relative
abundance 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 (top
to bottom). The compilation combines direct measurements with indirect
(air shower based) measurements. Taken from Hörandel (2005).
Again, cosmic-ray electrons require a separate treatment as a consequence
of radiative losses. The overall scheme on how to proceed is shown schematically in the cartoon (Fig. 17).
74
Figure 17: Cartoon of the cosmic ray transport in our Galaxy- taken from
Moskalenko 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 requires
some consideration of nuclear physics. The interaction of a cosmic-ray nucleus with a hydrogren atom (and to some extent and much rarer an interaction with a He or a metall in the interstellar medium) leads to a spallation reaction in which the original nucleus breaks up in potentially excited
nuclear fragments. The cross section for these processes are in principle
accessible to lab experiments, however, the data-base of spallation cross
sections is not complete. As it turns out, some of the analysis of cosmic-ray
data is limited by the lack of available cross section data. For now, we assume, that cross sections for processes like σ (C + p → He + X ) are known
or phenomenologically calculated. The general equation for a particle distribution of type i with uniform spatial diffusion (neglecting convection
75
and any form of momentum diffusion) is
∂Ni
∂t
= D ∇2 Ni +
Pji
∂
N
(bi Ni ) + Qi − i + ∑ Nj .
∂E
τi
τ
j >i j
(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 corresponds to an instantaneous injection at t = 0. Furthermore, we neglect
the diffusion which would correspond to a homogeneous distribution in
the considered medium. Generally, this is not valid, but sufficient for the
purpose of demonstrating the basic properties of the effect of spallation
(and the limitation of this approximation). Furthermore, we change the
variable from time to the traversed column density ξ = ρ · x = ρ · v · t. In
this case, after substituting, we arrive at the following equation:
∂Ni
∂ξ
= −
Pji
Ni
+∑
· Nj .
ξi
ξ
j >i j
(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 (we
will see shortly, that this model is too simple). For now, let us consider
the group of light (L) nuclei (Li, Be, B) and the most abundant group of
medium (M) nuclei (C, N, O). The boundary condition can be simply set
to NL (ξ = 0) = 0, because the light nuclei are not enriched by stellar
fusion processes (Li is only produced in the primordial nucleosynthesis).
For the M-group, we ignore the additional contribution from spallation of
heavier elements (we will see shortly, that this assumed decoupling is a
reasonable approximation):
N
dNM
= − M
dξ
ξM
dNL
N
P
= − L + ML NM .
dξ
ξL
ξM
(108)
(109)
The solution to Eqn.108 with the boundary condition that NM (ξ = 0) =
NM (0) is
NM (ξ ) = NM (0) · exp(−ξ/ξ M ).
76
(110)
We can find a solution to Eqn.109 after substituting Eqn.110, expanding
with exp(ξ/ξ L ), and integrating (exercise). The ratio of the groups is then
simply:
ξ
PML · ξ L
ξ
NL (ξ )
=
· exp
−
−1 .
(111)
NM ( ξ )
ξL − ξM
ξM ξL
Before comparing with the measurements, let us consider briefly which
values to use for ξ L , ξ M , PML . The column density ξ is related to the integral cross section σ and the molar mass m A :
ξ =
mA
,
NA · σ
(112)
with NA the Loschmidt/Avogadro number (6.02 × 1023 mol−1 ). As an
example, consider a typical cross section of 30 mbarn for inelastic ppscattering. In this case, ξ pp = 55 g/cm2 .
The total cross section for the spallation of an element of the M-group is
simply 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 of
mbarn. In order to calculate the values for ξ M and PML , we calculate the
abundance weighted cross section. The abundances are the actual values
measured for the individual species C, N, and O (using the values from
Table 5). The averaged cross section is simply hσ i = ∑i wi σi / ∑i wi with
the weights taken from the relative abundance (600-1000 MeV/n) and σi
the total cross section for species i ∈ {C, N, O}. The resulting values are
hσM i = 280 mbarn and for hσL i = 200 mbarn16 . The probability for production of an element from the L-group is estimated by taking the ratio of
PML = ∑i∈{ Li,Be,B} σMi /hσM i = 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 column density is of similar magnitude as ξ M .
When looking into the relative abundance of the individual elements in
the L-group ([ Li ] = 136, [ Be] = 67, [ B] = 23317 ), the ratio of elements is
16 note,
17 The
that the values for L are not included in Table 4
abundances are always considered relative to the abundance of Si
77
consistent with the respective weighted production cross sections (σ ( M →
Li ) = 24 mbarn, σ( M → Be) = 16.4 mbarn, σ ( M → B) = 35 mbarn). This
is reassuring and shows that the simple, one slab model is sufficient to
explain the observed abundance of light elements. The same exercise can
be done to estimate the slab thickness for the production of 3 He which is
again roughly 5 g/cm2 .
The simple one slab model works surprisingly well for the spallation of
the M-group elements into the light nuclei. However, when looking into
the details of the spallation of iron nuclei, a simple one slab model does
not work.
The spallation cross section for iron nuclei is very high (see Table 4): σFe =
763.4 mbarn, this corresponds to ξ Fe = 2.2 g/cm2 . If the slab thickness
were constant at 5 g/cm2 , the iron nuclei abundance observed in cosmic
rays should be strongly depleted with respect to the average abundance.
Independent of the absolute value of the abundance, the ratio of spallation products to the observed abundance of iron in cosmic rays should be
strongly dominated by the spallation products:
1 − exp(−ξ/ξ Fe )
[ products]
=
= 8.7
[ primaries]
exp(−ξ/ξ Fe )
(114)
for ξ = 5 g/cm2 . This is clearly not consistent with the observed abundances: The spallation of iron results mainly in the production of nuclei in
the 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 ]
= 1.5
[ Fe]
(115)
clearly in contradiction with the higher value expected. The discrepancy
of the simple one slab model can be resolved, when treating the transport
of cosmic rays more realistically.
There are in principle two ways of treating the cosmic ray transport. The
most simple approach is to consider a distribution of path lengths and try
to find the distribution that matches the observed ratio of secondaries to
primaries (phenomenological approach). A different approach is based
upon a more elaborate, physics oriented model of cosmic ray transport in
the Galaxy which then in return results in a prediction of the path length
distribution (theoretical approach). In principle, both approaches have
78
strengths and short comings. Let us for the sake of clarity consider two
extreme cases for the distribution of path lengths:
1. Leaky box: In this approach, the confinement volume of free streaming cosmic rays is defined by a boundary region with a finite escape
probability. In this case the entire population of cosmic rays can be
described by
N
∂N
=−
.
∂t
τe ( E)
(116)
In the most simple case, the escape time would be energy independent (we will see later, that the data suggest an energy dependence)
which would simplify the solution to be N ∝ exp(−t/τe ) ∝ exp(−ξ/ξ e ).
Exponential path length distribution.
2. Diffusion in infinite volume: The other extreme would be an infinite
escape time τe → ∞ with a diffusive transport:
∂N
= D ∇2 N
∂t
(117)
with a solution which corresponds to a Gaussian distribution of
path lengths.
Before comparing the expectations from various models (variants of the
theme suggested above), let us summarize the observational results for
secondary to primary ratios.
The most important measurements of the secondary to primary ratio are
the B/C (Boron to Carbon) and (Sc+Ti+V)/Fe-measurements. The main
result of both measurements is an energy dependence of the secondary/primary
ratio which indicates that the path length distribution changes with energy
(see Fig. 18 for a recent compilation of measurements). The energy dependence is such that the secondary/primary ratio increases with increasing
energy until it reaches a peak and subsequently for increasing energy it
drops. The behaviour for the two measurements is very similar, indicating that the path length distribution is similar for heavy and for medium
group.
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 observational data. Again, we simplify the underlying equation by assuming a steady-state case (i.e. ∂N
∂t = 0). In this case, we can simply solve the
following equation:
−
P
N
NL
+ ML NM (ξ ) − L = 0
ξ e ( E)
ξM
ξL
PML · NM (ξ )/ξ M
NL =
,
1
ξ e ( E ) −1 + ξ −
L
(118)
(119)
which we simplify by assuming (realistically) ξ e ξ L :
NL (ξ )
ξ e ( E)
= PML ·
.
NM ( ξ )
ξM
(120)
This effectively means that the secondary-to-primary ratio represents the
effective path-length as a function of energy (ξ M is fairly constant over
energy).
Returning to the measured secondary-to-primary ratio we clearly see that
for large energies an exponential drop-off in the path-length with energy
is a fairly good description of the model. However, at the low energy end
as 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 at
the low energy end is not naively expected. It requires an ad-hoc
assumption in the leaky box model. The exponential path length
distribution 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 a
natural consequence in a model where the cosmic rays are injected
in an infinitesimally thin disk with a halo in which cosmic rays move
with a convective or turbulent wind. The wind speed can be tuned
such that the cosmic rays are removed from the disk before they can
interact with the gas in the disk. This naturally leads to the peak in
the secondary/primary ratio.
• The observed drop off in secondary-to-primary ratio indicates that
cosmic rays with increasing energy leave the galactic disk and tra80
verse on average a smaller column density on the way to the observer on Earth.
Figure 18: Compilation of secondary to primary ratios (from Strong,
Moskalenko, and Ptuskin 2008). The left panel shows the B/C ratio representative for the spallation induced light nuclei abundance, the right panel
shows the production of heavy nuclei in spallation from iron.
The general result of the stable secondaries indicates that the mean
path length is of the order of 5 g/cm2 , decreasing with increasing energy
roughly as ξ e ( E) ∝ E−0.3...0.6 . When considering a value of 5g/cm2 , what
geometrical path length does this amount to?
As a rough estimate, consider a medium with average density ρ (in
units of g/cm3 ), the traversed column density relates to the spatial distance 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 comparing 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/freestreaming but is clearly related to a diffusive transport of cosmic-rays in
the entangled magnetic field of the interstellar medium. Similar numbers can be estimated by considering the propagation time. The particles move roughly with the speed of light. Therefore, the time is given
by τe = 3 × 106 n/(1 cm−3 ) yrs. We will compare this number with the
estimate derived from the cosmic clocks.
81
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 10 Be. The lifetime of 10 Be is τ (10 Be) = 3.9 × 106 yrs for the decay of 10 Be →10 B + e− +
ν¯e . 10 Be is mainly produced in the spallation of C and O.
By measuring the ratio of stable 7 Be to 10 Be, it is possible to measure the
average time for the propagation of cosmic-rays from the source to the
observer.
Quantitatively, we consider again a leaky box model (steady-state). In
order to shorten the notation, we define
Ci :=
∑
j >i
−
Pij
N
τj j
Ni
Ni
N
+ Ci −
− i = 0.
τe (i )
τspal. (i ) τr (i )
(121)
(122)
The most important change is the introduction of the last term which takes
into account the decay of the radio isotopes. The equation for the stable
isotopes:
−
Nk
Nk
+ Ck −
= 0.
τe (k)
τspal. (k )
(123)
Combining both equations:
τe (7 Be)−1 + τspal (7 Be)−1
N (10 Be)
C (10 Be)
=
·
,(124)
N (7 Be)
C (7 Be) τe (10 Be)−1 + τspal (10 Be)−1 + τr (10 Be)−1
assuming that τspal τe , this simplifies to:
C (10 Be)
τe (7 Be)−1
N (10 Be)
=
·
.
N (7 Be)
C (7 Be) τe (10 Be)−1 + τr (10 Be)−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 [10 Be]/[7 Be +9
Be +10 Be] ≈ 0.028. When comparing this with the column density traversed, 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 average density in the Galactic disk. This indicates that cosmic rays very
likely spend a considerable time propagating outside the disk in an extended halo. It is (at this point) not possible to conclude the actual extent
of the halo but it certainly extends beyond the scale height of cold molecular gas.
Figure 19: Measured values for the ratio of radioactive Be to stable Beisotopes.
3.2.3
Comments on particle diffusion
Diffusive transport of particles is a well-known phenomena that has been
studied intensively in the general context of transportation processes (e.g.
heat conductivity). The diffusive transport of cosmic-rays is however a
more complicated process as it requires the treatment of charged particles
moving through a (partially) ionised medium with magnetic field. This
is a highly non-linear problem which we do not introduce here in all its
depth. Let us highlight a number of important issus.
The general principle expressed in Fick’s second law applies to the general
problem of diffusive transport:
∂N
= ∇ Dxx ∇ N
∂t
83
(126)
which describes the temporal change of a particle (concentration, density)
that is subject to a diffusive (random) transport. The spatial diffusion coefficent Dxx , given in units of length2 time−1 , is related to the mean free
path length λ:
Dxx = hvi ·
λ
.
3
(127)
The mean free path length is commonly estimated to be the distance travelled at which the particle has changed the direction considerably and the
general distribution of pitch angles is isotropic. The most common value
used is the gyro-radius of a particle moving in a homogeneous magnetic
field. Note however, that the motion of a particle is usually not as simple as a gyrating motion. The gyro-radius r g = p/( ZeB) can be readily
calculated with the following equation:
rG = 0.4 pc Z
E
−1
1015 eV
B
3 µG
−1
.
(128)
The net motion of diffusing particles is towards regions with smaller values of density (see also Eqn. 126). The distance R travelled in a time τ is
given by
R2 = D · τ,
(129)
which is characteristic for diffusive motion. Before we estimate the diffusion coefficient of cosmic-ray propagation, we discuss the microscopic
properties of diffusive transport.
Scattering of cosmic-rays on self-generated Alfvén waves: The interstellar medium can be simplified by assuming that a static, background
magnetic field B0 is present throughout the medium perturbed by an additional fluctuating field component B1 . The fluctuation B1 can be described
by a spectrum of turbulences, where the power (B12 ) is a function of the
length scale (λ). For cosmic-rays, the background field B0 leads to a regular motion of charged particles which singles out a characteristic length
scale similar to rG . For λ rG or λ rG , the effect of B1 is averaged either
over 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 are
close in phase and wave-length, a resonant scattering takes place, where
energy 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 the
angle
δϕ ≈
B1
.
B0
(130)
Since the change of the angle is random, it will take N = (δφ)−2 scattering
events to change the angle by one radian. During the N cycles, the particle
will have moved
2
B0
−2
λsc ≈ Nλ ≈ rG (δφ) ≈ rG
.
(131)
B1
In the following, we want to show that these irregularities are produced by
the cosmic-rays themselves (this is why frequently, cosmic-rays are called
self-confining particles). The irregularities turn out to be Alfvén (plasma)
waves. Let us turn for a moment to plasma waves. A plasma is defined
as a medium which is (partially) ionised. On average (or large scales),
the plasma is electrically neutral. In astrophysics, the plasma is very dilute such that actual particle-particle collisions are negligible. The longdistance Coulomb interaction however still leads to scattering of particles
in electrical and magnetic fields. This type of plasma is often referred to
as a collisionless plasma. There is a large number of plasma waves as well
as instabilities known. For our purposes, we consider only one particular
(and relevant) type of plasma waves which are called Alfvén-waves18
This particular wave propagates in the plasma with density ρ = Np · m p
with a velocity:
vA = p
B0
.
4πρ0
(132)
For the typical interstellar medium with Np = 1 cm−3 and a magnetic field
B0 = 3 µG, we can calculate v A = 8 km/s. The growth rate of Alfvénic
18 Biographical
note: Hannes Alfvén (*1908,†1995) was a Swedish Physicist who was
awarded the Nobel price in 1970, recognizing his fundamental work in Magnetohydrodynamics and plasma physics.
85
waves depend on the streaming velocity of cosmic-rays. Without deriving
the result (see Longair, Section 20.4), the energy density in Alfvén waves
is U = U0 exp(Γ t) with a growth factor
|v|
N (> E(λ))
−1 +
.
(133)
Γ ( λ ) = Ω0
Np
vA
The particles N (> E) which resonantly scatter with the wave length λ lead
to a growth in the perturbance spectrum for streaming velocities larger
than v A . For cosmic-rays N (> E) ∝ E−1.7 implies that the time-scale
τ = Γ−1 for the growth of Alfvén waves increases with E1.7 . This implies that particles with increasing energy are less efficiently confined.
We have so far neglected processed leading to wave-damping, which effectively requires energy to be removed from the waves. Among these
processes, interactions with the neutral phase of the interstellar medium
lead to a damping on time-scales short in comparison with the growth
time 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 embedded
regions with a high level of ionisation. Outside the disk, the gas is mostly
ionised. This leads to the following, simple picture for cosmic-ray transport in our Galaxy:
• Inside the (mostly neutral) gas in the Galactic disk, cosmic-rays are
mostly free-streaming, possibly drifting along field lines.
• When encountering a region of highly ionized gas (as e.g. outside
the Galactic plane), the cosmic-rays start moving diffusively, with a
net streaming velocity approaching the Alfvén velocity.
At the interface between the two regions, particle conservation should
hold, and therefore:
Nint · c = Next · v A .
(134)
This readily translates into the time it takes for cosmic rays to leave the
disk with thickness L (this can only take place at the velocity v A ):
L
L
= 5 × 107 yrs
τ ≈
vA
kpc
86
Np
0.1 cm−3
−1/2
·
B
.
3 µG
(135)
Again, this value is fairly close to the value we have derived using the
cosmic-ray clocks. The remaining difference could in principle be an indication that the magnetic field may be on average smaller along the propagation path.
Estimate of the Diffusion coefficient for Galactic cosmic-rays: The height
of the halo that is occupied by cosmic-rays can be estimated from the already discussed average density of the traversed interstellar medium. The
total column density of (neutral) gas in the direction of the Galactic poles
is NH ≈ 1.5 × 1021 cm− 2. Strictly speaking this is a lower limit as the
ionised gas is not accounted for. The height of the halo is then
H =
1.5 × 1021
NH
=
cm = 1.6 kpc.
hni
0.3
(136)
Using this value in combination with the estimated time travelled of 107 yrs:
D =
cm2
H2
' 3 × 1028
.
τ
s
(137)
The energy dependence of the B/C-secondary-to-primary-ratio implies,
that this value obviously changes with energy. A common parameterization (independent of any underlying model) is:
α
E
D ( E) = D0 ×
(138)
10 GeV
with α = 0.5 . . . 0.7 (derived from the energy dependence of the B/Cratio).
3.2.4
Anisotropy of cosmic-rays
The anisotropy of cosmic-rays is an important ingredient on the way to
trace the sources of cosmic-rays. The measurements indicate the presence
of 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 experimentally
very challenging. Especially indirect (air-shower) measurements show
87
variations of the detection rate orders of magnitude larger than the level of
anisotropy searched for. Furthermore, a large scale (e.g. dipole) anisotropy
requires a reasonably homogeneous exposure of the entire sky, which again
is not trivial for an individual detector located on Earth. Generally, the observed level of anisotropy should be (cautiously) considered as an upper
limit. Nevertheless, we can at least qualitatively discuss the expectations.
The degree of anisotropy depends on the location of the observer with
respect to the sources. We would start with the assumption that the accelerators of cosmic rays are probably following the general mass distribution of the Galaxy. Most of the accelerators would therefore be located
within the disk and fairly close to the inner Galaxy. The solar system is located at the exterior part of the Galaxy and therefore, we would expect an
anisotropy with Imax located grossly in the direction of the Galactic center
and Imin probably in the opposite hemisphere or even directed towards
the Galactic poles.
The fact, that the observed relative anisotropy is well below unity, implies
already that cosmic-rays do not propagate rectilinear but obviously diffusively (no surprise after the discussion in the previous sections). For
the 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 therefore
V ≈
cm
km
D
= 107
= 100
≈ 3 × 10−4 c.
R
s
s
(140)
The cosmic-rays are streaming in the direction of the gradient of the cosmicray density. In the rest frame of the streaming motion, the cosmic-ray distribution is isotropic.
Given that the nuclei travel roughly with the speed of light, the relative
anisotropy for an observer δ ≈ V/c ≈ 3 × 10−4 . A more detailed treatment would have to take into account that the relative motion leads to a
shift in the observed energy such that δ = v/c · ( p + 2) with p the powerlaw index of the differential energy spectrum. For the cosmic-rays, p = 2.7
which leads to δ = 1.2 × 10−3 , which in turn is fairly close to the observed
anisotropy.
The observed degree of anisotropy is in agreement with a diffusive transport of cosmic-rays in our Galaxy.
However, there are a number of buts which (in the light of the good agree88
ment of the simple estimate with the measurement) should not be forgotten:
• The sun is not at the edge of the cosmic-ray source distribution. We
can certainly expect, that cosmic ray sources are also active in the
direction of the Galactic anti-center.
• Even more important is the fact that the sun is located close to the
center of a local active star forming region called Gould belt. Here,
we have e.g. a high number of massive stars as well as supernova
remnants which in principle inject cosmic rays in the local medium.
• The sun is located in the so-called local bubble, ie. the interstellar
neigbhorhood of the sun19 is characterized by a hot, ionised, and
tenuous gas, possibly heated up by multiple supernova explosions.
• The presence of spiral arms. In the spiral arm structure the magnetic
field lines are more ordered (as we can derive from the polarization
direction of radio-synchrotron emission seen predominantly in spiral galaxies). This ordered magnetic field component leads to a drift
of the cosmic-rays parallel to the field lines (the particles gyrate along
the field line and move parallel to the field). This intrinsically should
lead to an anisotropy larger than the rough estimate given above.
• The motion of the sun around the Galactic center as well as the motion of Earth around the sun: The net streaming velocity of the cosmic rays is smaller than the rotational velocity of the sun (> 200 km/s).
This leads to an additional dipole-moment that is annualy modulated by the motion of the Earth around the sun (≈ 30 km/s) that
is either parallel or anti-parallel to the orbital motion of the solar
system. Both effects need to be removed from the data to infer the
streaming velocity. This is commonly done by considering the anisotropy
perpendicular 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)accelerated
anti-matter could be the result of acceleration of anti-nuclei in a anti-matter
19 As
the name implies, the interstellar medium in the solar environment is not typical
for the interstellar medium.
89
galaxy and (ii) anti-protons (as well as positrons) are produced in cosmicray interactions with gas and provide insights into the cosmic-ray transport. Deviations from the p̄-flux expected from secondary production may
lead us to conclude that additional, primary anti-proton-production is necessary as e.g. suggested in models of evaporating black holes as well as
from 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 required to sustain the cosmic-ray population. Starting with a cylindrical
volume 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 measured energy density (with a correction of the effect of solar modulation).
The canonical value is usually taken to be ucr ' 1 eV/cm3 . The total
energy 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 escape losses is P = E/τesc = 3 × 1040 ergs/s= 107 L (in words 10 million
solar luminosities) in the form of cosmic-rays.
Generally, the most attractive class of objects suggested to explain the
cosmic-ray population are supernova remnants. Each supernova remnant
releases roughly 1051 ergs in the form of kinetic energy in the interstellar
medium. Only a fraction η < 1 is released in the form of cosmic-rays.
Together with the super-nova remnant rate ṅsnr = 1/100 yrs, we can estimate the efficiency η required to match Lcr :
Lcr = η · ESNR · ṅsnr →
(141)
−1 ESNR
1/ṅ
Lcr
·
·
(142)
η = 9%
40
51
100 yrs
3 × 10 ergs/s
10 ergs
90
Table 4: Partial cross section for spallation reactions. The table is taken
from Longair, Chapter 5.
91
Table 5: Measured abundances of cosmic rays, solar system, and local
Galactic. Table is taken from Longair, chapter 9.
92
3.3
Acceleration of cosmic rays
The presence of a non-thermal distribution of relativistic nuclei and electrons in the Galaxy obviously implies an underlying acceleration mechanism which needs to explain a number of relevant observations:
• Power-law shape of the energy distribution with a power-law which
should 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 to
the solar abundances.
• Sufficiently efficient and fast acceleration to balance the injection power
required to balance the cosmic ray escape from the Galaxy.
A number of schemes have been investigated in order to achieve particle
acceleration in astrophysical environments which follow in principle one
of the following approaches or flavours thereof:
1. Stochastic acceleration in randomly moving clouds of magnetized
plasma (2nd order Fermi acceleration).
2. Acceleration at a hydrodynamic/magneto-hydrodynamic shock front
1st 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 the
discussion of the other acceleration types to the literature.
Enrico Fermi described in his seminal paper from 1949 the notion of particle acceleration through scattering on randomly moving magnetized clouds
(Fermi 1949). In this approach, the particles gain energy in head on collisions while losing energy in trailing collisions. The energy gain per scattering is ∆E/E ∝ β2 with β = vcloud /c 1 which implies that acceleration is
rather slow for the average velocity of clouds in the inter-stellar medium.
93
The energy gain of the particle is at the expense of macroscopic kinetic energy 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 between
particles convected away from the shock and the probability to undergo
multiple cycles of shock crossings, leads to the formation of a broad band
power-law with dN/dE ∝ E−s with s = (γ + 1)/(γ − 1) with γ = c p /cv
the polytropic index for an adiabatic process. In the well-justified case of
an ideal monoatomic gas γ = 5/3 and therefore s = 2.
The achievable maximum energy of the acceleration process depends mainly
upon the size and life-time of the shock (ultimately the limit would be
the Hubble time) which vary widely between the shocks present in astrophysical environments. Additionally, energy loss mechanisms constrain
the maximum energy achievable. This is particularly important for the
acceleration of electrons where radiative energy losses are efficient. For
nuclei, energy losses become only relevant at ultra-high (E > 1019 eV) energies. Generally, acceleration can only proceed until the particle is not
confined to the environment of the shock. Taking the gyroradius of the
particle and comparing it with the size of the shock, a general condition
for confinement of the particle is that the size of the shock L > 2r g with
r g = 1 pc E15 /( ZBµG ) for a particle with energy E = E15 × 1015 eV in a
region with magnetic field B = BµG µG:
Lpc > 2
E15
.
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 strong
shock is the super-nova remnant. In both general types of supernova explosion, the ejecta drives a strong shock in the ambient medium which
is 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 SNR
evolution, with Ush ∝ t−3/5 (see also Section 1.5). The maximum energy
that can be attained during the dynamic evolution of the shock is smaller
than 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 can
be reached depends on the diffusion coefficient in the downstream and
upstream region. The smaller the diffusion coefficient, the smaller distance the particles diffuse in the downstream and upstream region before
94
crossing the shock again, therefore increasing the number of cycles until
the 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 as
test-particles without feedback on the shock structure). However, the required efficiency of shock acceleration in supernovae of O(∞0 %) indicates
that possibly non-linear feedback should be investigated. Here, we only
sketch the main ideas of this theory and evidences from observation.
When looking at the hydrodynamic structure of a shock, the presence of
non-thermal particles exerts additional pressure which changes the shock
structure. So far, the hydrodynamic shock was given by a compression
ratio of 1 : 4. With the addition of the cosmic ray pressure, the compression ratio is increased to values up to 7 . . . 10. From a magnetohydrodynamic analysis carried out be Lucek&Bell (2000), the cosmic rays streaming ahead of the shock drive instabilities that rapidly increase the magnetic
field. This leads to efficient scattering/confinement of the cosmic-rays to
high energies. The maximum energy achievable via this non-linear effect
is probably close to the knee energy of 1015 eV. A telltale sign of nonlinear
amplification of magnetic field and therefore presence of cosmic rays are
so-called filaments which have been detected with high spatial resolution
observations with the Chandra X-ray telescope. These filaments are X-ray
bright regions which in projection are narrow features, mostly aligned parallel to the shock. Long-term monitoring reveal time-variability in these
filaments. The common interpretation of the filaments is that magnetic
field has to be amplified to values up to 100 µG in order to confine Xray emitting electrons. The time variability is then expected as the cooling
time of these electrons is accordingly shortened. Indirectly, the presence of
filaments in most young X-ray emitting supernova remnants has been interpreted as evidence for efficient cosmic-ray acceleration with non-linear
effects. Another peace of observational evidence in favor of non-linear
shock acceleration is the observation that e.g. the historical supernova
remnant RCW 86 tends to show a deceleration at the north-eastern rim
which could be related to the high acceleration efficiency which leads to
an observable effect in the evolution of the SNR. For RCW 86 the northeastern rim is the site of particle acceleration as evident from X-ray synchrotron emission. Furthermore, the shock heating in RCW 86 proceeds
95
Figure 20: The energy spectrum beyond the knee. The thick line is the average of measurements up to 1017 eV. The lines indicate models for extragalactic origin of cosmic-rays and different species. Taken from Hillas
(2006)
less efficient which again would indicate that cosmic-ray acceleration reduces 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 energy density in cosmic rays must be present in the shock vicinity. The efficiency achieved is sufficiently large to explain the overall Galactic cosmic
ray budget. However, there are a number of interesting problems with
respect to this conclusion from maybe the most sensitive observational
channel as discussed in the next section.
Finally, we turn to extra-galactic cosmic-rays from the knee to the ankle
and beyond in the energy spectrum (see Fig. 20). The origin of particles at
ultra-high energies (> 1019 eV) is very likely extra-galactic given that these
particles are not confined to the Galaxy (see Fig. 21). On the other hand,
96
Figure 21: The curvature of particles (protons and iron nuclei) traversing
the 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 be
reasonably local since once the threshold for exciting the ∆-resonance and
photo-pion production is exceeded, the mean free path length for inelastic
pγCMB converges to λ = (nCMB σpγ )−1 ≈ 1/410 cm3 /100 µbarn ≈ 10 Mpc.
More accurate calculations take the energy dependence of the cross section into account, but the basic conclusion remains: catastrophic losses of
cosmic-rays above the so called Greisen-Zatsepin-Kuzmin (GZK) cutoff effectively shield UHECR from cosmological distances. Furthermore, the
small deflection by a few degrees of the cosmic-rays in the extra-galactic
magnetic field and galactic magnetic field open the opportunity to identify
the accelerators by tracing back the reconstructed direction and correlating
it with (known) nearby source candidates. The criterium on the confinement of cosmic-rays to the accelerator imposes quite severe constraints on
the size and magnetic field of the accelerator. Fig. 22 summarizes the requirement to achieve a maximum energy of 1020 eV for protons and iron
nuclei. The existence of particles beyond 1019 eV requires extreme conditions for acceleration which are only met by a few objects including radio
galaxies (specifically their hot spots), galaxy clusters, and possibly AGN.
Note however, that compact objects like AGN are disfavored by the high
photon density environment which ultimately limits the acceleration by
97
Figure 22: Constraint on the size and magnetic field for accelerators of
ultra-high energy cosmic-rays: Objects which confine protons should be
above the solid line. For iron nuclei, the constraint is relaxed by the higher
charge number (dashed line). The grey band indicates the requirement for
non-relativstic motion of the scattering centers (upper boundary is for β =
1/300). The various potential accelerators/acceleration sites are indicated
in the diagram.
energy losses through photo-pion production inside the compact source.
Most interestingly, the first data released by the Pierre-Auger consortium
taken with the large (3000 km2 ) surface array (PAO 2007) indicates a close
correlation of the arrival direction (within a space angular separation of
of data taken during the first 3.5 years. A total of 27 events above energies of 57 EeV qualify for the correlation study. The statistical analysis of
a possible correlation is not straight-forward and a word of caution on the
significance of the findings is necessary. One of the most serious problems
is very likely the limited number of events and the choice of selection cuts
as well as the choice of source catalogues used to conduct the study. However, the consortium claims that a correlation with nearby AGN exist at a
98
Figure 23: The Hammer-Aitoff projected sky map in Galactic coordinates.
The center of the image is the direction of the Galactic center. North is
up, East is left. The blue shaded region indicates equal exposure in equal
shades of blue (lighter blue is less exposure). The red stars are positions
of nearby AGN while the circles (3.2◦ radius) indicate the directions of
the 27 events used in the correlation study. The dashed line marks the
supergalactic plane (Pierre Auger collaboration 2007).
confidence level of more than 99 %. The corresponding sky plot is shown
in Fig. 23 where the finding of a correlation within an angular window of
3.7◦ is apparent.
99