A&A 467, 15-20 (2007)

DOI: 10.1051/0004-6361:20066932

**M. Vukcevic ^{1,2} - R. Schlickeiser^{1}**

1 - Institut für Theoretische Physik, Lehrstuhl IV:
Weltraum- und Astrophysik, Ruhr-Universität Bochum, 44780 Bochum, Germany

2 - Department of Physics, University of Montenegro, Podgorica, Montenegro

Received 14 December 2006 / Accepted 22 February 2007

**Abstract**
*Context.* The mean free path and anisotropy of galactic cosmic rays is calculated in weak plasma wave turbulence that is isotropically distributed with respect to the ordered uniform magnetic field.

*Aims.* The modifications on the value of the Hillas energy, above which cosmic rays are not confined to the Galaxy, are calculated. The original determination of the Hillas limit has been based on the case of slab turbulence where only parallel propagating plasma waves are allowed.

*Methods.* We use quasilinear cosmic ray Fokker-Planck coefficients to calculate the mean free path and the anisotropy in isotropic plasma wave turbulence.

*Results.* In isotropic plasma wave turbulence the Hillas limit is enhanced by about four orders of magnitude to
PeV resulting from the dominating influence of transit-time damping interactions of cosmic rays with obliquely propagating magnetosonic waves.

*Conclusions.* Below the energy
the cosmic ray mean free path and the anisotropy exhibit the well known *E*^{1/3} energy dependence. At energies higher than
both transport parameters steepen to a *E*^{3}-dependence. This implies that cosmic rays even with ultrahigh energies of several hundreds of EeV can be rapidly pitch-angle scattered by interstellar plasma turbulence, and are thus confined to the Galaxy.

**Key words: **ISM: cosmic rays - ISM: magnetic fields - plasmas - scattering

To unravel the nature of cosmic sources that accelerate cosmic rays to ultrahigh
energies has been identified as one of the eleven
fundamental science questions for the new century (Turner et al. 2002). Cosmic rays with energies up to at least 10^{14} eV are likely accelerated at the shock fronts associated with supernova remnants
(for review see Blandford & Eichler 1987). Radio emissions and X-rays give conclusive
evidence that electrons are accelerated there to near-light speed (Koyama et al. 1995, 1997; Tanimori et al. 2001; Allen et al. 1997; Slane et al. 1999;
Borkowski et al. 2001). The HESS observations of supernova remnants up to 100 TeV
provide direct evidence of very high energy particle acceleration in the shocks
(Aharonian et al. 2004,2005), while the
leptonic or hadronic nature of these gamma-rays is currently being disputed (e.g. Enomoto et al. 2002; Reimer & Pohl 2002). The supernova remnant origin would be consistent
with the observed GeV excess of diffuse galactic gamma radiation from the inner Galaxy
(Büsching et al. 2001), although the GeV excess has been found to be present in all directions including galactic latitudes where no supernova remnants are present and the outer Galaxy (Strong et al. 2004).
This indicates that the origin of the GeV excess is more complex and is not straightforwardly
connected with supernova remnants in the inner Galaxy.

More puzzling are the much higher energy cosmic rays
with energies as large as 10^{20.5} eV. It has been argued (Lucek & Bell 2000; Bell & Lucek 2001; Hillas 2006) that, due to the amplification of the magnetic field in the shock, the acceleration of cosmic rays in
young supernova remnants is possible up to 10^{18} eV. This implies that such particles may
have a Galactic origin. For ultrahigh-energy (
10^{18}-10^{20.5} eV) cosmic rays an extragalatic origin is favored by many researchers.
Extragalactic ultrahigh-energy cosmic rays (UHECRs) coming
from cosmological distances 50 Mpc should interact with the universal
cosmic microwave background radiation (CMBR) and produce pions. For an extragalactic origin
of UHECRs the detection or non-direction of the Greisen-Kuzmin-Zatsepin cutoff resulting
from the photopion attenuation in the CMBR will have far-reaching consequences not only for
astrophysics but also for fundamental particle physics as e.g. the breakup of Lorentz symmetry (Coleman & Glashow 1997) or the non-commutative quantum picture of
spacetime (Amelio-Camelia et al. 1998).

Radio synchrotron radiation intensity and polarisation surveys
of our own and external galaxies (for review see Sofue et al. 1986) have revealed that
the interstellar medium is transversed by large-scale ordered
magnetic fields with superposed plasma wave turbulence. The Galactic magnetic field
has a regular and a random component of about equal strength. The turbulent field has a broad spectrum of scales with the largest one being 10-100 pc (e.g. Beck 2007, and references therein). This could be compared with the gyroradius of 1 pc for 10^{15} eV particles, or 1 kpc for 10^{18} eV particles. The conventional size of the Galactic halo derived from abundances of radioactive isotopes in
cosmic rays is about 4-6 kpc (Ptuskin & Soutoul 1998; Strong & Moskalenko 1998;
Webber & Soutoul 1998). The turbulent magnetic field may thus present a mechanism for isotropization of Galactic cosmic rays up to
10^{17}-10^{18} eV (see, e.g., Candia et al.
2003).

According to the current understanding (reviewed in Schlickeiser 2002) the relativistic
charged particles (hereafter referred to
as cosmic ray particles) in these space plasmas are confined and accelerated by
resonant interactions in these weakly random electromagnetic fields.
In the presence of low-frequency magnetohydrodynamic plasma waves, whose
magnetic field component is much larger than their electric field
component, the particle's phase space distribution function
adjusts rapidly to a quasi-equilibrium through pitch-angle diffusion, which
is close to the isotropic distribution. The isotropic part
of the phase space distribution function *F*(*z*,*p*,*t*) obeys the *diffusion-convection-equation*

where the parallel spatial diffusion coefficient , the cosmic ray bulk speed

In Eq. (1) the space coordinate

For many years the theoretical development of the resonant wave-particle
interactions has mainly concentrated on the special case that the plasma waves
propagate only parallel or antiparallel to the ordered magnetic field - the socalled slab turbulence. In this case only cosmic ray particles with gyroradii
smaller than the
longest parallel wavelength
of the plasma waves
can resonantly interact. Obviously this condition is
equivalent to a limit on the maximum particle rigidity *R*:

An alternative way to express the condition (5) is

where

It is the purpose of this work to investigate how the Hillas limit (6) is affected if we discard the assumption of purely slab plasma waves, i.e. if we allow for oblique propagation angles of the plasma waves with respect to the ordered magnetic field component. There is ample observational evidence that obliquely propagating magnetohydrodynamic plasma waves exist in the interstellar medium (Armstrong et al. 1995; Lithwick & Goldreich 2001; Cho et al. 2002). In particular, we will consider the alternative extreme limit that the plasma waves propagation angles are isotropically distributed around the magnetic field direction. It has been emphasised before by Schlickeiser & Miller (1998) referred to as SM) that oblique propagation angles of fast magnetosonic waves leads to an order of magnitude quicker stochastic acceleration rate as compared to the slab case, since the compressional component of the obliquely propagating fast mode waves allows the effect of transit-time damping acceleration of cosmic ray particles. Here we will demonstrate that the obliqueness of fast mode and shear Alfven wave propagation also modifies the resulting parallel spatial diffusion coefficient and the Hillas limit.

Most cosmic plasmas have a small value of the plasma beta , which is defined by the ratio of the ion sound to Alfven speed , and thus indicates the ratio of thermal to magnetic pressure. For low-beta plasmas the two relevant magnetohydrodynamic wave modes are the

**(1)**- incompressional
*shear Alfven waves*with dispersion relation

(7)

at parallel wavenumbers , which have no magnetic field component along the ordered background magnetic field , **(2)**- the
*fast magnetosonic waves*with dispersion relation

(8)

for wavenumbers , which have a compressive magnetic field component for oblique propagation angles .

Schlickeiser & Miller (1998) investigated the quasilinear interactions
of charged particles with these two plasma waves. In case of negligible
wave damping the interactions are of resonant nature: a cosmic ray
particle of given velocity *v*, pitch angle cosine
and
gyrofrequency
interacts with waves
whose wavenumber and real frequencies obey the condition

(9) |

for entire

For shear Alfven waves only interactions with
are possible.
These are referred to as *gyroresonances* because inserting the
dispersion relation (7) in the resonance condition (9) yields for the
resonance parallel wavenumber

(10) |

which apart from very small values of typically equals the inverse of the cosmic ray particle's gyroradius, and higher harmonics.

In contrast, for fast magnetosonic waves the *n*=0 resonance is possible
for oblique propagation due its compressive magnetic field component.
The *n*=0 interactions are referred to as *transit-time damping*,
hereafter TTD. Inserting the dispersion relation (8) into the resonance
condition (9) in the case *n*=0 yields

(11) |

as necessary condition which is independent from the wavenumber value

(12) |

Additionally, fast mode waves also allow gyroresonances () at wavenumbers

(13) |

which is very similar to Eq. (10).

The simple considerations of the last two subsections allow us the following immediate conclusions:

truecm

(1) With TTD-interactions alone, it would not be possible to scatter particles with , i.e., particles with pitch angles near 90. Obviously, these particles have basically no parallel velocity and cannot catch up with fast mode waves that propagate with the small but finite speeds . In particular this implies that with TTD alone it is not possible to establish an isotropic cosmic ray distribution function. Gyroresonances are needed to provide the crucial finite scattering at small values of .

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(2) Conditions (11) and (12) reveal that TTD is
no gyroradius effect. It involves fast mode waves at all wavenumbers
provided the cosmic ray particles are super-Alfvenic and have large
enough values of
as required by Eq. (12). Because gyroresonances
occur at single resonant wavenumbers only, see Eqs. (10) and (13), their
contribution to the value of the Fokker-Planck coefficients in the
interval
is much smaller than the contribution from
TTD. Therefore for comparable intensities of fast mode and shear Alfven waves, TTD will provide the overwhelming contribution to all
Fokker-Planck coefficients
,
and *D*_{pp} in
the interval
.
At small values of
only
gyroresonances contribute to the values of the Fokker-Planck
coefficients involving according to Eqs. (10) and (13) wavenumbers at
.

truecm

(3)
The momentum diffusion coefficient (4)

(14) |

has contributions both from transit-time damping of fast mode waves,

(15) |

and from second-order Fermi gyroresonant acceleration by shear Alfven waves (Schlickeiser 1989)

(16) |

truecm

(4)
On the other hand, the spatial diffusion coefficient (2)

(17) |

is given by the integral over the

The gyroresonances can be due to shear Alfven waves or fast magnetosonic waves. For relativistic cosmic rays the relevant range of pitch angle cosines is very small allowing us the approximation so that

(5)
According to Eq. (90) of Schlickeiser (1989) the streaming cosmic ray anisotropy due to spatial gradients in the cosmic ray density is given by

which also is determined by the smallest value of around . Approximating again for we derive with Eq. (19) the direct proportionality of the cosmic ray anisotropy with the parallel mean free path, i.e.

Introducing the characteristic spatial gradient of the cosmic ray density Eq. (21) reads

Cosmic ray gradients derived from diffuse galactic GeV gamma-ray emissivities (Strong & Mattox 1996) suggest a value of 2 kpc.

Throughout this work we consider isotropic linearly polarised
magnetohydrodynamic turbulence so that the components of
the magnetic turbulence tensor for plasma mode *j* is

The magetic energy density in wave component

We adopt a Kolmogorov-like power law dependence (index

The normalisation (24) then implies

Moreover we adopt a vanishing cross helicity of each plasma mode, i.e. equal intensity of forward and backward moving waves, so that

According to Eq. (30) of SM the Fokker-Planck coefficients
and
with
for fast mode waves are
the sum of contributions from transit-time damping (T) and
gyroresonant interactions (G):

with

where the lower integration boundary is

and . denotes the gyrofrequency of the cosmic ray particle,

The gyroresonant contribution from fast mode waves is

On the other hand shear Alfven waves provide only gyroresonant () interactions yielding

According to SM at particle pitch-angles outside the interval transit-time damping provides the dominant and overwhelming contribution to these Fokker-Planck coefficients. This justifies the approximations to derive Eqs. (19) and (21) for the cosmic ray mean free path and anisotropy, respectively, Both transport parameters are primarily fixed by the small but finite scattering due to gyroresonant interactions in the interval . We then derive

and

In the following, we consider both transport coefficients for positively charged cosmic ray particles with especially in the limit .

At
the contribution from shear Alfven waves to the
pitch-angle Fokker-Planck coefficient is according to Eq. (23)

where we readily performed the -integration. Substituting , and using , Eq. (34) reduces to

where

Likewise the contribution from gyroresonant interactions with fast mode waves is according to Eqs. (27) and (30)

where we performed the

The Bessel function integral in Eq. (38)

has been calculated asymptotically by SM to lowest order in the small quantity as

yielding

In Appendix B we evaluate the Bessel function integral in Eq. (35)

for small and large values of .

For values
we obtain approximately

yielding

in terms of the zeta and the generalised zeta functions of Riemann (Whittaker & Watson 1978).

For values of
we obtain Eq. (43)
for values of ,
where
is the
largest integer smaller than
,
while
for smaller *n*

and

According to Eq. (35) this yields

Comparing the Fokker-Planck coefficients from fast mode waves (41) and Alfven waves (Eqs. (44) and (47)) we note that the latter one is always smaller by the small ratio than the first one:

so that the gyroresonant contribution from Alfven waves can be neglected in comparison to the gyroresonant contribution from fast mode waves.

Neglecting
we obtain for the cosmic ray mean free path (32)

which exhibits the familiar Lorentzfactor dependence at Lorentzfactors below a critical Lorentz factor defined by

with being the inverse ion skin length. The Lorentzfactor dependence especially holds at rigidities , in a rigidity range where the slab turbulence model would predict an infinitely large mean free path.

Expresing
in terms of the longest wavelength of
isotropic fast mode waves
pc yields

The corresponding cosmic ray hadron energy is

which is four orders of magnitude larger than the Hillas limit (6) for equal values of the maximum wavelength. This difference demonstrates the dramatic influence of the plasma turbulence geometry (slab versus isotropically distributed waves) on the confinement of cosmic rays in the Galaxy. With isotropically distributed fast mode waves, even ultrahigh energy cosmic rays obey the scaling .

Only, at ultrahigh Lorentzfactors
or energies
the mean free path (49) approaches the much steeper dependence

(53) |

independent from the turbulence spectral index

Because of the direct proportionality between mean free path and anisotropy, the cosmic ray anisotropy (33) shows the same behaviour as a function of energy:

which is proportional at energies below and at energies above . In particular we obtain no drastic change in the energy dependence of the anisotropy at PeV energies. Quantitatively, with Eq. (22),

At EeV energies we calculate an anisotropy of less than 15 percent, whereas at smaller energies the anisotropy values decrease proportional to .

We have investigated the implications of isotropically distributed interstellar magnetohydrodynamic plasma waves on the scattering mean free path and the spatial anisotropy of high-energy cosmic rays. We demonstrate a drastic modification of the energy dependence of both cosmic ray transport parameters compared to previous calculations that have assumed that the plasma waves propagate only parallel or antiparallel to the ordered magnetic field (slab turbulence). In case of slab turbulence cosmic rays with Larmor radius resonantly interact with plasma waves with wave vectors at . If the slab wave turbulence power spectrum vanishes for wavenumbers less than , as a consequence then cosmic rays with Larmor radii larger than cannot be scattered in pitch-angle, causing the socalled Hillas limit for the maximum energy of cosmic rays being confined in the Galaxy. At about these energies this would imply a drastic increase in the spatial anisotropy of cosmic rays that has not been detected by KASKADE and other air shower experiments.

In case of isotropically distributed interstellar magnetohydrodynamic waves we demonstrated that
the Hillas energy
is modified to a limiting total energy that is about 4 orders of magnitude larger
PeV, where *A* denotes the mass number and
the maximum wavenumber of isotropic plasma waves.
Below this energy the cosmic ray mean free path and the anisotropy exhibit the well known *E*^{2-q}energy dependence, where *q*=5/3 denotes the spectral index of the Kolmogorov spectrum. At energies higher than
both transport parameters steepen to a *E*^{3}-dependence. This implies that cosmic rays even with ultrahigh energies of several tens of EeV can be rapidly pitch-angle scattered by interstellar plasma turbulence, and are
thus confined to the Galaxy.

The physical reason for the four orders of magnitude higher value of the limiting energy is the occurrence of dominating transit-time damping interactions of cosmic rays with magnetosonic plasma waves due to their compressive
magnetic field component along the ordered magnetic field. This *n*=0 resonance is not a gyroresonance implying
that cosmic rays interact with plasma waves at all wavenumbers provided that the cosmic ray parallel speed (transit speed)
equals the parallel phase speed of magnetosonic waves. Only at small values of the cosmic ray pitch-angle
cosine
,
where the cosmic ray particles spiral at nearly ninety degrees with very small parallel speeds less than
the minimum magnetosonic phase speed ,
gyroresonant interactions are necessary to scatter csomic rays. However, the gyroresonance condition of cosmic rays at
reads
instead of the slab condition
causing the limiting energy enhancement from
to by the large factor
.

Partial support by the Deutsche Forschungsgemeinschaft through Sonderforschungsbereich 591 is acknowledged.

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Online Material

The task is to calculate the integral (42)

for small and large values of using the approximations of Bessel functions for small and large arguments (Abramowitz & Stegun 1972), yielding

and

According to Eq. (36)

the lower integration boundary in the case which includes in particular the limit because .

With the identity

we obtain

where

With the asymptotics (57) and (58) we obtain

where we use

and where

in terms of the incomplete gamma function. For large arguments we obtain asymptotically

Collecting terms we find to lowest order in

so that

In this case
for ,
and
for ,
where

(67) |

denotes the largest integer smaller than . Hence we obtain again Eq. (66) for

For values of we find that

where

We may express

with , so that the lower integration boundary in (70) is

In cases where , Eq. (72) yields that for all values of

In the remaining case

is smaller unity, so that we approximate Eq. (70) in this case by

where we approximate

by its upper limit to obtain

Collecting terms in Eq. (69) we derive

and

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