A&A 378, 279-294 (2001)

DOI: 10.1051/0004-6361:20011080

**I. Lerche ^{1} - R. Schlickeiser ^{2}**

1 - Department of Geological Sciences, University of South
Carolina, Columbia, SC 29208, USA

2 -
Institut für Theoretische Physik, Lehrstuhl IV:
Weltraum- und Astrophysik, Ruhr-Universität Bochum,
44780 Bochum, Germany

Received 21 March 2001 / Accepted 23 July 2001

**Abstract**

Observations of interstellar scintillations, general
theoretical considerations and comparison of interstellar radiative cooling
in HII-regions, and in the diffuse interstellar medium, with
linear Landau damping estimates for
fast-mode decay, all strongly imply that
the power spectrum of fast-mode wave turbulence in the interstellar medium must
be highly anisotropic. It is not clear from the observations whether the
turbulence spectrum is oriented mainly parallel or mainly perpendicular to the
ambient magnetic field, either will satisfy the needs of balancing wave damping
energy input against radiative cooling. This anisotropy must be included
when transport of high energy cosmic rays in
the Galaxy is discussed. Here we evaluate the relevant cosmic ray
transport parameters in the presence of anisotropic wave turbulence.
Using the estimates of the anisotropy parameter in the strongly parallel and
perpendicular regimes, based on linear Landau damping balancing radiative
loss in the diffuse interstellar medium,
we show that in nearly all situations the pitch-angle scattering of relativistic
cosmic rays by fast magnetosonic waves at pitch-angle cosines
is dominated by the transit-time damping
interaction.
The momentum diffusion coefficient of cosmic ray particles is calculated
by averaging the
respective Fokker-Planck coefficient over the particle pitch-angle for
the relevant anisotropy parameters within
values of
.
For strongly perpendicular
turbulence (
)
the cosmic ray momentum
diffusion coefficient is enhanced with respect to the case of
isotropic (
)
turbulence by the large factor
.
For strongly parallel turbulence (
)
the momentum diffusion
coefficient is reduced with respect to isotropic turbulence by the
large factor
.

**Key words: **magnetohydrodynamics (MHD) - plasmas - turbulence -
cosmic rays - ISM: magnetic fields

This anisotropy must be included when transport of high energy cosmic rays in the Galaxy is discussed. So far, with the noteworthy exception of Jaekel & Schlickeiser (1992), in all the literature concerning the determination of cosmic ray transport parameters, there appears to be consideration given only to turbulence which has a power spectrum either slab-like along the ordered magnetic field or isotropically distributed in wavenumber (e.g., Schlickeiser & Miller 1998, hereafter referred to as SM). The purpose of the present paper is to remedy this defect to some extent by evaluating the relevant cosmic ray transport parameters in the presence of anisotropic wave turbulence.

satisfies the needs of the interstellar scintillation observations, the balance of wave energy dissipation and radiative cooling in HII-regions and in the diffuse interstellar medium, and is in accord with the general theoretical arguments advanced by Goldreich & Sridhar (1995). According to Rickett (1990) and Spangler (1991) Eq. (1) is valid for larger than a minimum wavenumber, , and less than a maximum . Spangler (1991) identifies these wavenumbers as due to an inner scale length, , and an outer scale length . Observations indicate that the power spectral index,

where , , with being the cosine of the propagation angle of a plasma wave with respect to the ambient magnetic field. Moreover, is the fluctuation strenth in the magnetic field, and the constant accounts for the turbulence anisotropy.

Note that if the turbulence is isotropic(
)
then

while for non-isotropic turbulence

with the integral

which can be expressed in terms of the hypergeometric function.

Quasilinear transport equations for magnetohydrodynamic plasma waves were formulated originally by Kennel & Engelmann (1966), Hall & Sturrock (1967) and Lerche (1968). The quasilinear approach to the interaction of energetic charged particles with partially random electromagnetic fields ( ) is a first-order perturbation calculation in the ratio and requires smallness of this ratio with respect to unity. In most cosmic plasmas this requirement is well satisfied as has been established by direct in-situ measurements in interplanetary plasmas, or due to saturation effects in the growth of fluctuating fields. Comparison with Monte Carlo simulations of the transport of charged particles with different plasma wave fields (e.g., Michalek & Ostrowski 1996) demonstrates that the quasilinear theory provides an accurate description of cosmic ray transport for ratios .

Due to the high conductivity of most cosmic plasmas,
large-scale steady electric fields are absent, so that
the interest concentrates on
magnetized plasma. By linear stability calculations it has been established
that these systems contain low-frequency magnetohydrodynamic turbulence such
as shear Alfvén waves and fast and slow magnetosonic waves.
For these plasma waves the magnetic part of the Lorentz force
is much larger than the electric part of the Lorentz force, so that the
time scale
for rapid pitch angle scattering of energetic charged particles
is much shorter than the time scale
for energy changes. In this case the particle's gyrotropic distribution
function adjusts rapidly to quasi-equilibrium, which is close to the
isotropic distribution function, in excellent agreement with the observational
fact of the isotropy of cosmic ray particles. For nonrelativistic ()
bulk speed of the turbulence-carrying background plasma the diffusion-convection
transport equation for the isotropic part of the phase space density
*F*(*z*,*p*,*t*) can be derived by a well-known approximation scheme
(Jokipii 1966; Hasselmann & Wibberenz 1968;
Earl 1973; Schlickeiser 1989)
from the quasilinear Fokker-Planck equation

where the spatial diffusion coefficient , the cosmic ray bulk speed

In Eq. (6)

The three Fokker-Planck coefficients, describing particle-wave interaction
processes and entering Eqs. (7), (8), (9) are calculated (Hall & Sturrock 1967;
Krommes 1984; Achatz et al. 1991) from ensemble-averaged first-order corrections
to the particle orbit. Therefore they
depend on the tensor components of
the plasma wave power spectrum
.
For a magnetic turbulence
tensor with no preferred direction, Batchelor (1953) notes that
can be written in the general
form

Application of Cramer's theorem requires for all , and for all . Then

where .

For fast-mode waves propagating either forward (phase velocity
,
*j*=+1) or backward (phase velocity
,
*j*=-1) to the ambient
magnetic field an index *j* is used to track the wave direction (SM) and, in
principle, the magnetic helicity
can also be included in the
evaluation of the Fokker-Planck coefficients. However, little is known about any
magnetic helicity term in the interstellar turbulence so, in this first
investigation of the effects of wave turbulence anisotropy on the cosmic ray
transport parameters, we restrict our attention to the anisotropy factor
.

With the identification

it follows that the anisotropic variants of Eqs. (27)-(29) of SM take the form

where

The general Fokker-Planck coefficients represented through Eqs.
(13)-(15) can be split into two parts: components with *n*=0(customarily referred to as transit-time contributions), and components with
(customarily referred to as gyroresonant contributions). We consider
each in turn.

(18) |

where and where

Using Eqs. (3) and (4) for
Eq. (17) can be written

Relative to the isotropic ( ) situation one can write

In this case the contributions are more complex, as also noted in the isotropic
case by SM, due to the fact that the argument of the -function now
involves the wavenumber *k* explicitly. Following the same sense of argument as
given by SM, after some algebra, one can write the gyroresonance contributions
as

together with

where we introduced the cosmic ray particle gyroradius .

Likewise we obtain

(24) |

and

again together with the restriction (23).

Without further information on the relative strengths of *I*_{0}^{-} to *I*_{0}^{+} it
is not possible to take the gyroresonance contributions much further. In Sect. 5 we treat with the symmetric case where
*I*_{0}^{-}=*I*_{0}^{+}, to illuminate the
changes in the Fokker-Planck coefficients brought about by the anisotropic
nature of the plasma wave turbulence. However, for the transit-time
contributions (Eqs. (19)-(21)) it is possible to evaluate the
effects of anisotropy directly without needing to make any further assumptions
on *I*_{0}^{-} and *I*_{0}^{+}. This aspect is discussed next.

we obtain from Eq. (20)

in .

Three cases provide insight into the anisotropic effects:

**(i)**- weak anisotropy
;
**(ii)**- strongly ribbon-like anisotropy
;
**(iii)**- strongly perpendicular anisotropy .

Consider each in turn.

in so that the anisotropy component changes sign as crosses .

According to Eq. (43) of Lerche & Schlickeiser (2001)
the integral (5) for
is approximately
.
Moreover, in this case
except from the small range of
in
.
Then

Thus, in most of the range of , and only in a very narrow range , is . Consequently, in

Thus the pitch angle transit-time Fokker-Planck component is massively reduced as is

With respect to the first factor in Eq. (27), in this case the factor dominates except when . The outer pair of inequalities require to be obeyed. So two limits exist: either () , in which case the factor dominates everywhere; or () , in which case the factor dominates in while dominates in . Consider each case in turn.

which is much smaller unity.

which is larger unity.

and

Note that is antisymmetric in , so that the rate of adiabatic acceleration

is identically zero in the symmetric wave intensity case . In deriving this result we have used Eq. (16) that there is no transit-time contribution to .

Note that as , the integral over in Eq. (33) also tends to zero, so that is small compared to .

We further reduce Eq. (37) by changing variables in the first term of the bracket to with while in the second term one writes with . We obtain

where and . Eq. (38) is discussed in Appendix A.

with

involving the inner and outer scale of the turbulence spectrum and the interstellar ion skin length, respectively. We introduced the gyroresonance anisotropy ratio for small pitch angle cosines

in terms of the integral (5). Obviously

the ratio

For isotropic turbulence ( ) Eq. (39) (apart from the factor ) agrees with Eqs. (44) and (58a) of SM.

For large pitch angles
we restrict our analysis to
cosmic ray particles with gyroradii less than
.
In this case from Appendix A we obtain from Eq. (108)

and

with

where

(46) |

and

respectively. The two gyroresonance anisotropy ratios are given by

and

It is remarkable that the three gyroresonance anisotropy ratios (Eqs. (41), (48) and (49)) are independent of cosmic ray particle properties and solely determined by the turbulence parameters

With the asymptotic behaviour of
for small and large arguments
we immediately find for the asymptotic behaviour of the two gyroresonance
anisotropy factors

The asymptotics of the anisotropy factor

and

so that the second factor in the bracket of Eq. (48) can be neglected, leaving

In the opposite case the first hypergeometric function in Eq. (48) is approximated as

whereas the second is written as the integral

with where . Substituting

The replacement of the upper integration boundary with is allowed because . Accordingly, we obtain

In summary then

complementing Eqs. (50) and (51).

We are now in the position to discuss the influence of the turbulence anisotropy on the ratio of the contributions from transit-time damping and gyroresonances in the pitch-angle interval .

where the indices 2,3 refer to the intervals and , respectively. The functions

refer to the corresponding ratios for isotropic turbulence, and are obtained from Eqs. (19), (45) and (47) under the assumptions that have been made as

where

at where

is again of order unity. Eq. (60) reproduces the result of SM that for isotropic turbulence transit-time damping provides the dominant contribution to pitch angle scattering in the interval .

which can be reduced further using the approximations (29), (31), (32) and (56).

For strongly perpendicular anisotropy we obtain

which is much larger unity unless is extremely small.

For strongly parallel anisotropy we derive

which in the small interval is always much larger unity. Outside this interval, i.e. the ratio is much larger unity unless becomes extremely large.

For strongly perpendicular anisotropy we obtain

which is much larger unity unless is extremely small.

For strongly parallel anisotropy we derive

which is much smaller unity.

(a) For massively parallel ( ) situations, the ratio of the transit-time contribution to the gyroresonance contribution to pitch-angle scattering in the interval of cosmic ray particles with gyroradii behaves as follows:

- for large the ratio is smaller than unity indicating that the gyroresonance contribution dominates the transit-time damping contribution,

- in the small interval the ratio is larger than unity indicating that the transit-time contribution dominates the gyroresonance contribution,

- in the interval the ratio is larger than unity (i.e. dominance of the transit-time damping contribution) for anisotropy values smaller than whereas for extremely large values of the ratio is smaller than unity (i.e. dominance of the gyroresonance contribution).

(b) For massively perpendicular ( ) situations, the ratio of the transit-time contribution to the gyroresoance contribution to pitch-angle scattering in the interval of cosmic ray particles with gyroradii is much larger than unity for anisotropy values larger than , indicating that the transit-time damping contribution dominates the gyroresonance contribution.

For extremely small anisotropy values the ratio is smaller than unity indicating the dominance of the gyroresonance contribution over the transit-time damping contribution.

Now, estimates of the anisotropy parameter in the strongly parallel situation ( ) based on linear Landau damping balancing radiative loss in the diffuse interstellar medium, provide the value (Lerche & Schlickeiser 2001) which is much smaller than . Hence, it would seem that in the diffuse interstellar medium the transit-time damping contribution to is dominant in the pitch-angle angle interval whereas the gyroresonant contribution dominates in the interval . The same conclusion holds in HII-regions (the fluctiferous domain of Spangler 1991), for which Lerche & Schlickeiser (2001) estimated .

Estimates of the anisotropy parameter in the strongly perpendicular situation ( ) based on linear Landau damping balancing radiative loss in the diffuse interstellar medium, provide the value (Lerche & Schlickeiser 2001) which is much larger than . The transit-time damping contribution then dominates the gyroresonance contribution throughout the whole pitch-angle interval in the diffuse interstellar medium. The same conclusion holds in HII-regions, for which Lerche & Schlickeiser (2001) estimated in this case.

These estimates have direct consequences for the cosmic ray transport parameters in the interstellar medium, as the parallel mean free path and the momentum diffusion coeffient. However, before the parallel mean free path can be calculated, we have to determine the influence of the anisotropy parameter on the Fokker-Planck coefficients in case of shear Alfven waves, because interstellar plasma turbulence is a mixture of fast magnetosonic waves and shear Alfven waves (SM). This analysis will be the subject of the second paper of this series. Here, we restrict our analysis to the momentum diffusion coefficient which, for the relevant range , is solely determined by the transit-time damping contribution.

Using Eqs. (26), (19), (21) and (9)
we obtain for the momentum diffusion coefficient of cosmic rays with gyroradii
much less than

with

and the anisotropy function

We calculate the anisotropy function Eq. (68) using the respective approximations of the ratio from Sect. 4 in the relevant anisotropy range .

This case
has been considered before by SM who derived

Here we use Eq. (29) to obtain

with the two integrals

and

with the substitution .

We note that the -integration interval in Eq. (71) is very small
so that we approximate the integrand by its value at
to obtain approximately

For the integral (72) we substitute to derive

(74) |

which after the substitution can be expressed in terms of hypergeometric functions,

According to Eq. (15.3.10) of Abramowitz & Stegun (1972) we use

to approximate the two hypergeometric functions in Eq. (75) as

and

We then find

Collecting terms in Eq. (70) we find

which is strongly reduced compared to the isotropic value.

Substituting in the first integral yields

where

and

Because the integral (81) is well approximated by

Apart from the minor difference in the lower integration boundary the integral (82) is identical to the integral (72). According to Eq. (75) we obtain

Using again Eq. (76) we find

Collecting terms in Eq. (80) we obtain

which is enhanced compared to the isotropic value.

We summarize
the asymptotic behaviour of the anisotropy function
in Table 1.

With this first paper we have started to evaluate the relevant cosmic ray transport parameters in the presence of anisotropic fast magnetosonic plasma wave turbulence. All technical details of the calculation of Fokker-Planck coefficients in this case are presented, in particular the deviations from the case of isotropic turbulence are identified. Using the estimates of the anisotropy parameter in the strongly parallel and perpendicular regimes, based on linear Landau damping balancing radiative loss in the diffuse interstellar medium, we have calculated the Fokker-Planck coefficients needed to infer the parallel mean free path, the rate of adiabatic deceleration and the momentum diffusion coefficient of cosmic ray particles. We show that in nearly all situations the pitch-angle scattering of relativistic cosmic rays by fast magnetosonic waves at pitch-angle cosines is dominated by the transit-time damping interaction.

These results have direct consequences for the cosmic ray transport parameters in the interstellar medium, as the parallel mean free path and the momentum diffusion coefficient. In order to calculate the parallel mean free path, we have to determine the influence of the anisotropy parameter on the Fokker-Planck coefficients in case of shear Alfven waves, because interstellar plasma turbulence is a mixture of fast magnetosonic waves and shear Alfven waves. This analysis will be the subject of the second paper of this series.

Without considering the influence of the
anisotropy parameter on
the Fokker-Planck coefficients in case of shear Alfven waves, we are able
to calculate
the momentum diffusion coefficient *a*_{2} of cosmic ray particles by
averaging the
respective Fokker-Planck coefficient over the particle pitch-angle for
the relevant anisotropy parameters within
values of
.
For strongly perpendicular
turbulence (
)
the cosmic ray momentum
diffusion coefficient is enhanced with
respect to isotropic (
)
turbulence by the large factor
.
For strongly parallel turbulence (
)
the momentum diffusion
coefficient is reduced with respect to isotropic turbulence by the
large factor
.
This implies that the acceleration time
scale of cosmic ray particles by momentum diffusion for anisotropic
turbulence is shorter (strongly perpendicular turbulence) or longer
(strongly parallel turbulence) by the same factors with respect to the case
of isotropic turbulence. Hence, depending on small or large enough anisotropy
factors ,
reacceleration effects
in the transport of galactic cosmic rays become much stronger (
)
or weaker (
), respectively.

We thank Dipl.-Phys. A. Teufel for a careful reading of the manuscript. We gratefully acknowledge support by the Deutsche Forschungsgemeinschaft through Sonderforschungsbereich 191.

For fast cosmic ray particles so that we may use the approximation of Bessel functions for large arguments (Abramowitz & Stegun 1972)

implying

for the argument . We then obtain

With Eqs. (3)-(4) for we find

where we introduced the gyroresonance anisotropy ratio for small pitch angle cosines

in terms of the integral (5). Finally,we note that

(93) |

can be expressed in terms of the ion skin length . The two Heaviside step functions in Eq. (91) then imply the restriction on the value of the cosmic ray particle energy

with

involving the inner and outer scale of the turbulence spectrum and the interstellar ion skin length, respectively. Equation (91) then takes the form

where . We first note the symmetry in Eq. (97).

so that we can restrict our discussion on positive values of .

Next, the two step functions tell us that the integration
range of *q* is limited to

Comparing these boundaries with the upper integration limit in Eq. (97) restricts the possible values of

Changing from the integration variable

with the integral

where

Now it is convenient to introduce the tangent of the wave propagation angle and the absolute value of the tangent of the pitch angle , respectively. With

with

The expression is evaluated in Appendix B. The general calculation is very involved, but for energetic particles with super-Alfvenic ( ) velocities and gyroradii smaller than , which is of order 1 pc, we obtain approximations for small ( ) and large ( ) pitch-angle cosines. In the former region we obtain

where we introduced the Riemann zeta-function . For large we find the constant value

For cosmic ray particle with gyrodii less than
we obtain with Eqs. (3)-(4) for

with

Obviously, Eq. (104) is equal to

with

and

To evaluate these two integrals approximately, we will use the approximation (89) for values of , while for small values of

(112) |

implying

We consider both integrals in turn.

Here we have to compare the value of *M*^{-2} with the upper integration
boundary of the integral (110). If

we can use approximation (113) throughout to obtain

In the opposite case we use approximation (113) in the range and approximation (89) in the range with the result

The condition (114) translates into

Since the maximum value of the left hand side of this inequality is less than 1/4 the inequality is always fulfilled for values of implying

If

and

implying again Eq. (118) in this range.

In the intermediate pitch angle range approximation (116) holds.

Here we have to compare the value of *M*^{-2} with the upper integration
boundary of the integral (111). If

we can use approximation (113) throughout to obtain

In the opposite case we use approximation (113) in the range and approximation (89) in the range with the result

The condition (120) translates into

Again the maximum value of the left hand side of this inequality is less than 1/4, so that the inequality is always fulfilled for values of implying

If the inequality (123) is fulfilled in the pitch angle ranges

and

implying again Eq. (124) in this range.

In the intermediate pitch angle range approximation (122) holds.

Because we are concerned with the transport of very energetic particles
we do not lose much generality if we extend the turbulence power
spectrum to infinitely large wavenumbers, i.e.
.
In this case
*U*=0 according to Eq. (99) and

As a consequence, the general expression (109) simplifies enormously to

with

and

Restricting the analysis to cosmic ray particles with gyroradii which is of order 1 pc, the second sum in Eq. (127) vanishes, and we obtain

Obviously, we obtain with the approximations (89) and (113)

with

and

where we neglected the oscillating part in approximation (113). We consider the cases and

In this case
is a small quantity, and the integral (132)
is approximately

in terms of the hypergeometric function. In deriving Eq. (134) we have used the transformation formula

Likewise, the integral (133) can be approximated as

Collecting terms in Eq. (131) we obtain to lowest order in the small quanitity

Here
is large, so that the integral (133) is approximately

where we substituted

Likewise, the integral (132) can be approximated as

In terms of hypergeometric functions we obtain

Collecting terms in Eq. (131) we obtain

Because the leading terms of Eq. (142) are

For the two hypergeometric functions in Eq. (143) can be approximated by

and for

yielding factors of order unity, whereas for

which, because of the dependence, dominates the bracket of Eq. (143). Consequently, we obtain

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