A unified theory of cosmic ray diurnal variations has been proposed in which the first 3 harmonics of the cosmic ray daily variation all results from a single anisotropy produced by the combined effects of adiabatic focusing and anisotropic pitch angle scattering. The theoretical description of steady state cosmic ray anisotropies are simplified and improved. Preliminary results of a study of correlations between cosmic ray diurnal variations and the fluctuation characteristics of the interplanetary magnetic field are presented and discussed in light of the theory

Bieber, J. W.

Pomerantz, M. A.

English

NASA Technical Reports Server

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MAGNETICFLUCTUATIONSAND COSMICRAY DIURNALVARIATIONS
J. W. Bieber and M. A. Pomerantz
Bartol Research Foundation of The _ranklin Institute
University of Delaware, Newark, Delaware 19716
ABSTRACT
A unified theory of cosmic ray diurnal variations
has been proposed in which the first 3 harmonics
of the cosmic ray daily variation all result from
a single anisotropy produced by the combined ef-
fects of adiabatic focusing and anisotropic pitch
angle scattering. The purpose of the present re-
port is twofold: (i) to simplify and improve the
theoretical description of steady-state cosmic
ray anisotropies, and (2) to present and discuss,
in light of the theory, preliminary results of a
study of correlations between cosmic ray diurnal
variations and the fluctuation characteristics of
the interplanetary magnetic field.
1. Theory. Several theoretical investigations of energetic particle
anisotropies (Roelof, 1969; Kunstmann, 1979; Earl, 1981) suggest that
observed anisotropies may appropriately be cast in the form of a simple
analytic solution of the steady-state Boltzmann equation, which,
according to Earl (1981), is given by
f = c° + cIB exp{ (4-q)_ p3 L l_ll-q} " (1)
Here, f is the phase space density, Co and cI are constants, B is
the magnetic field magnitude, q is a parameter characterizing the
anisotropy of pitch angle scattering, x is the scattering mean free
path, L is the focusing length (- 0.94 AU for the nominal Parker field
at Earth), and p is the cosine of the particle pitch angle. The
property of (i) relevant to the present work is that it has a non-
sinusoidal form, and hence will naturally contribute higher order
harmonics to the cosmic ray daily variation observed by ground-based
° instrumentation.
For reasons connected with geomagnetic correction of neutron
, monitor observations (Yasue et al., 1982), it is convenient to express
the free space anisotropy as a Legendre series. Figure i shows the
relationship between the parameters xlL and q and the coefficients ai
of the Legendre expansion of (I). Note that a measurement of the 3
coefficients al, a2, and a3 serves to determine uniquely the
values of both x/L and q. Figure 1 thus allows values of inter-
planetary scattering parameters to be inferred from measurements of
cosmic ray diurnal variations.
https://ntrs.nasa.gov/search.jsp?R=19850026725 2020-03-20T16:52:23+00:00Z
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3 - Fig. i. Lines of constant x/L
O.B- (dashed) and constant q (solid)
+0.5 are plotted in a coordinate
I A/_I __i£ i system representing ratios of03 • the coefficients ai of a
_1._//q i/1.4 q Legendre series expansion of the
°T : ,",' exponential anisotropy (1). The
0 ' ' - coefficients al, a2, and a3
'L__;/_" _ . are closely related to the
first, second, and third
harmonics, respectively, of the
cosmic ray diurnal variation.
-0.5
0 0.5 1.0
02/01
Bieber and Pomerantz (19831 used essentially this procedure to
infer values for x and q from an analysis of 12 years of neutron
monitor data. However, that earlier analysis expressed the anisotropy
as a first-order focusing eigenfunction,which is closely related to
the exponential (1), but which differs somewhat in the quantitative
prediction of harmonic amplitudes. Because the exponential anisotropy
is derived from the steady-state Boltzmann equation which includes th_
effect of a field-aligned density gradient, as wel] as the effects of
focusing and scattering, it should be the more accurate approximation
to actual cosmic ray anisotropies in most circumstances. If the
observational results of Bieber and Pomerantz are re-interpreted in
terms of the exponential anisotropy, then the long-term average of the
scattering mean free path at 10 GV changes from 0.5 to 0.3 AU, while
the long-term average of q remains unchanged at 1.1.
2. Magnetic Fluctuations and Diurnal Variations. In light of the
theory described above, it might be expected that cosmic ray diurnal
variations would be correlated with the fluctuation characteristics of
the interplanetarymagnetic field. In particular, one might expect the
amplitude of the second harmonic relative to the first to decrease as
the fluctuation level increases, since an increase in fluctuations
should correspond to a decrease in x and, according to Figure 1, a
decrease in a2/a1.
A thorough investigationof this effect would require that the
scattering mean free path be calculated from the observed power
spectrum of the magnetic field (e.g., Jokipii, 1971). However, for
this preliminary study a simpler procedure was chosen in which the
fluctuation level is characterized by the relative variance of the
magnetic field. Relative variance was calculated on a daily basis by "
taking the vector variance of observed hourly field vectors and
dividing this by the mean-square field. For days in which both toward
and away field polarities occurred, the direction of toward vectors was
reversed before the variance was taken. Days with fewer than 12 hours
of data were excluded from the analysis. The remaining days were then
sh 4.5-21
161
divided into 7 groups according to relative variance, and the first 3
harmonics of the cosmic ray daily variation observed at Swarthmore over
the period 1965-1978 were averaged separately for each group.
Results of the analysis appear in Figure 2, where the amplitudes
and phases of the first 3 harmonics of the diurnal variation are
displayed as a function of relative variance. The only quantity to
exhibit a strong dependence upon magnetic fluctuation level is the
amplitude of the second harmonic, which decreases significantly at the
higher fluctuation levels.
After correction for geomagnetic effects and for the Compton-Getting
anisotropy (Bieber and Pomerantz, 1983), the data of Figure 2 may be
used to infer values of the interplanetary scattering parameters x/L
and q, as described above. For relative variances in the range 0-0.5,
x/L remains nearly constant at a value ~ 0.35, while for ranges 0.5-0.7
and 0.7-1.0 x/L decreases to 0.27 and 0.19, respectively. For all
values of relative variance the q parameter remains near a value of 1.1.
°Io.... ... I!;11n
_-. 0 .... I .... _- 0 ....
L.
"® 0.08 _t . _ _
"0 0
: +_e -r 60.04 ,.., n=2
n=2 _' ®
gJ • • • OlO •E 0 , , , , I .... o 0 ........
•< _c:
0.02 4 . '++++ .
n= n=3
0 i ' ' 0 .... i ....
0 0.5 1.0 0 0.5 1.0
Relative Variance of IMF" Relafive Variance of IMF
Fig. 2. The amplitudes and phases of the first (top panels),
- second (middle panels), and third (bottom panels) harmonics
of the cosmic ray daily variation observed at Swarthmore are
plotted as a function of the relative variance of the
interplanetarymagnetic field. Data are not corrected for
geomagnetic effects or for the Compton-Getting anisotropy.
Error bars are • i_.
SH 4.5-21
162
3. Discussion. The daily variance calculated from hourly-averaged
magnetic field vectors will naturally be most influenced by fluctuations
with time scales in the range of i to 24 hours. This, it turns out, is
a very suitable range insofar as the scattering of cosmic rays of
neutron monitor energies is concerned, as a cosmic ray of 10 GV
rigidity is resonant with waves that have periods ~ 24 hours in the
spacecraft frame. Thus, it is reasonable to suppose that the relative
variance provides a rough measure of the scattering rate that would be
calculated from the actual power spectrum of fluctuations.
Under this hypothesis, the results described above show general
agreement with theoretical expectation, in that the mean free path
inferred from particle anisotropies decreases at the higher fluctuation
levels. There is also a hint of a possible discrepancy, however, in
that the inferred mean free path seems to be independent of fluctuation
level for relative variances < 0.5.
Further analysis is required to see whether any significance can
be attached to this puzzling result. In particular, the
characterization of fluctuation level by daily relative variance should
be replaced by a careful evaluation of observed magnetic power
spectra. This will allow the relationship between interplanetary
scattering parameters and the fluctuation characteristics of the
magnetic field to be studied quantitatively at a level of detail that
has not heretofore been possible.
4. Acknowledgments. C. H. Tsao contributed significantly to the
extensive computer analysis involved in this work. This research was
supported by the National Science Foundation under grant ATM-8303758.
References
Bieber, J. W., and M. A. Pomerantz, (1983), A unified theory of cosmic
ray diurnal variations, Geophys. Res. Lett., i0, 920-923.
Earl, J. A., (1981), Analytical description of charged particle trans-
port along arbitrary guiding field configurations, Astrophys. J., 251,
739-755.
Jokipii, J. R., (1971), Propagation of cosmic rays in the solar wind,
Rev. Geophys. Space Phys., 9, 27-87.
Kunstmann, J. E., (1979), A new transport mode for energetic charged
particles in magnetic field fluctuations superposed on a diverging mean
field, Astrophys. J., 22___99,812-820.
Roelof, E. C., (1969), Propagation of solar cosmic rays in the inter-
planetary magnetic field, Lectures in High-Energy Astrophysics, NASA
Spec. Publ. 199, edited by H. Ogelm_ and J. R. Wayland, Nationa'F---
Aeronautics and Space Administration, Washington, D.C.
Yasue, S., S. Mori, S. Sakakibara, and K. Nagashima, (1982), Coupling
coefficients of cosmic ray daily variations for neutron monitor
stations, R__eportof Cosmic-Ray Research Lab, No. 7, Nagoya, Japan.


Magnetic fluctuation and cosmic ray diurnal variations

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