The study of cosmic ray access to locations inside the geomagnetic field has evolved in a manner that has led to some misunderstanding and misapplication of the terminology originally developed to describe particle access. This paper presents what is believed to be a useful set of definitions for cosmic ray cutoff terminology for use in theoretical and experimental cosmic ray studies

Byrnak, B.

Cooke, D. J.

Goret, P.

Humble, J. E.

Lund, N.

Petrou, N.

Rasmussen, I. L.

Shea, M. A.

Smart, D. F.

English

NASA Technical Reports Server

328
SH 6.1-11
RE-EVALUATION OF COSMIC RAY CUTOFF TERMINOLOGY
D. J. Cooke I, J. E. Humble 2, M. A. Shea 3, D. F. Smart 3,
N. Lund 4, I. L. Rasmussen 4, B. Byrnak 4, P. Goret 5, N. Petrou 5
ABSTRACT
The study of cosmic ray access to locations inside the geomag-
netic field has evolved in a manner that has led to some mis-
understanding and misapplication of the terminology originally
developed to describe particle access. This paper presents what
is believed to be a useful set of definitions for cosmic ray cut-
off terminology for use in theoretical and experimental cosmic
ray studies.
I. Introduction. Early work on cosmic ray motion in axially symmetric
representations of the field (Stormer, 1930; Lemaitre and Vallarta (1936a,
b) was limited to analytic considerations. More recent studies, which
rely on digital computer calculations of charged particle trajectories in
high order mathematical representations of the field (Shea et al., 1965)
emphasize the study of access to locations within the geomagnetic field as
a function of particle rigidity, in contrast to the earlier studies which
examined the broader directional access picture. The terms used in the
early work to refer to the access regions were carried over into the nu-
merical computations, although often in an imprecise way. Whereas the
early theoretical workers viewed access conditions in what may be called
a "direction picture", describing the directions from which particles of
a specified rigidity could or could not arrive, the later computer calcu-
lations have usually used a "rigidity picture" in which accessibility is
considered as a function of particle rigidity in a single arrival direc-
tion. The use of the old geometrical terms, appropriate to the direction
picture, has caused considerable confusion in rigidity picture studies.
A re-examination of the characteristics of the access regions dis-
tinguished by Stormer and by Lemaitre and Vallarta, together with a com-
parison between the properties of these analytically distinguished regions
and those detected by the digital computer method has resulted in defini-
tions which, it is hoped, will meet present needs.
2. Definitions. The definitions required may conveniently be subdivided
by viewpoint, as indicated in Table I. The list is not exhaustive, but
seeks to portray the most useful quantities in each situation.
1. Physics Dept., University of Utah, Salt Lake City, Utah 84112, USA
2. Physics Dept., University of Tasmania, Hobart, Tasmania, Australia 7001
3. Air Force Geophysics Laboratory, Hanscom AFB, Bedford, MA 01731, USA
4. Danish Space Research Inst., Lundtoftevej 7, DK-2800, Lyngby, Denmark
5. Sect. d'Astrophys., Centre d'Etudes Nucl. de Saclay, Saclay, France
https://ntrs.nasa.gov/search.jsp?R=19850026771 2020-03-20T16:50:58+00:00Z
a29
SH 6.1-11
TABLE I.
Summary of terms used in cutoff calculations. Quantities describing phe-
nomena which are equivalent in both pictures are listed on the same line.
Direction Picture Ri_idit[ Picture
Cutoff Rigidity
Allowed Cone
Main Cone Main Cutoff Rigidity
First Discontinuity Rigidity
Shadow Cone Shadow Cutoff Rigidity
Penumbra Penumbra
Penumbral Band Penumbral Band
Primary Band
Stormer Cone Stormer Cutoff Rigidity
Forbidden Cone
Upper Cutoff Rigidity
Lower Cutoff Rigidity
Horizon Limited Rigidity
Effective Cutoff Rigidity
Estilnated Cutoff Rigidity
2.1. Directional Definitions. The following definitions appear to be ap-
propriate for use with the directional picture. Each definition is for
charged particles of a single specified rigidity value arriving at a par-
ticular point in the geomagnetic field.
Allowed Cone: The solid angle containing the directions of arrival
of all trajectories which do not intersect the earth and which cannot
possess sections asymptotic to bound periodic orbits (because the rigidity
is too high to permit such sections to exist in the directions of arrival
concerned).
Main Cone: The boundary of the allowed cone. The main cone is
constituted in part by trajectories which are asymptotic to the simplest
bound periodic orbits and in part by trajectories which graze the surface
of the earth. (For this purpose the surface of the earth is generally
taken to be at the top of the effective atmosphere.)
Forbidden Cone: The solid angle region within which all directions
of arrival correspond to trajectories which, in the absence of the solid
earth, would be permanently bound in the geomagnetic field. Access in
- these directions from outside the field is, therefore, impossible.
Stormer Cone: The boundary of the forbidden cone. In an axially
symmetric field the surface forms a right circular cone.
Shadow Cone: The solid angle containing all directions of particle
arrival which are excluded due to short range earth intersections of the
approaching trajectories while loops within the local field line bundle
are being traversed.
Penumbra: The solid angle region contained between the main cone
and the Stormer cone. In general the penumbra contains a complex struc-
ture of allowed and forbidden bands of arrival directions.
Penumbral Band: A contiguous set of directions of arrival within
the penumbra, the members of which are either all allowed or all forbidden.
330 SH 6.1-11
2o2. Ri_idit_ Picture Definitions. The following definitions are proposed
as being applicable to the rigidity picture. Each definition refers to
particles arriving at a particular site within the geomagnetic field from
a specified direction.
Cutoff Rigidity: The location of a transition, in rigidity space,
from allowed to forbidden trajectories, as rigidity is decreased. Unless
otherwise defined, the value normally quoted in representing the results
of computer calculations is, for practical reasons, the rigidity of the
allowed member of the appropriate juxtaposed allowed/forbidden pair of
trajectories computed as part of a spaced series of traces. Sometimes the
term is employed to refer to the location of a notional transition from
one region to another, for example at the Stormer cone, where an allowed
trajectory may not perhaps exist at all.
Main Cutoff Rigidity, RM: The rigidity value at which the direction
concerned is a generator of the main cone as defined in the "direction
picture". The associated trajectory is either one which is asymptotic to
the simplest bound periodic orbit, or (owing to the presence of the solid
Earth) is one which is tangential to the earth's surface.
First discontinuity ri@idit_, RI: The rigidity associated with the
first discontinuity in asymptotic longitude as the trajectory calculations
are performed for successively lower rigidities, starting with some value
within the allowed cone. The value of R I is approximately equal to the
main cutoff as defined above.
Shadow Cutoff Rigidity, RSH: The rigidity value at which the edge
of the shadow cone lies in the direction concerned.
Penumbra: The rigidity range lying between the main and the Stormer
cutoff rigidities.
Penumbral band: A continuous set of rigidity values within the
penumbra, all members of which have the same general access characteris-
tics, either all allowed or all forbidden.
Primary Band: The stable forbidden penumbral band which is associ-
ated with the earth intersection of a low point in the loop which lies at
the last equatorial crossing before the trajectory (or its virtual exten-
sion in the assumed absence of the earth) takes on the characteristic
guiding centre motion down the local field line bundle.
Stormer Cutoff Rigidity, RS: The rigidity value for which the
Stormer cone lies in the given direction. In a dipole field (and perhaps
also in the real geomagnetic field) access for particles of all rigidity
yalues lower than the Stormer cutoff rigidity is forbidden from outside
the field. In a dipole approximation to the geomagnetic field, one form
of the Stormer Equation gives the Stormer cutoff rigidity, in GV, as:
as : M / { r2[, + (, - cos3cosAsinz) 1212}
where M is the dipole moment and has a normalized value of 59.6 when r is
expressed in units of earth radii, I is the magnetic latitude, r is t_e
distance from the dipole in earth radii, A is the azimuthal angle measured
clockwise from the geomagnetic east direction (for positive particles),
and Z is the angle from the local magnetic zenith direction.
Upper Cutoff Rigidity, RU: The rigidity value of the highest detect-
ed allowed/forbidden transition among a set of computed trajectories.
The upper cutoff rigidity can correspond to the main cutoff if and only
if no trajectories asymptotic to bound periodic orbits lie at rigidities
higher than this value. This can be identified from the nature of the
trajectory associated with the main cutoff.
337 SH 6.1-11
Lower Cutoff Rigidity, RL: The lowest detected cutoff value (i.e.,
the rigidity value of the lowest allowed/forbidden transition observed in
a set of computer calculations). If no penumbra exists, R L will equal R U.
Horizon Limited Rigidity, RH: The rigidity value of the most rigid
allowed trajectory found in a set of computer calculations performed for
a below horizon direction at a location above the surface of the earth.
Effective Cutoff Rigidity, RC: The total effect of the penumbral
structure in a given direction may be represented usefully, for many pur-
poses, by the "effective cutoff rigidity" - a single numeric value which
specifies the equivalent total accessible cosmic radiation within the pen-
umbra in a specific direction. "Effective cutoffs" may be either linear
averages of the allowed rigidity intervals in the penumbra or functions
weighted for the cosmic ray spectrum and/or detector response.
Estimated Cutoff Rigidity, Rest: A value obtained by using an empir-
ically normalized equation to approximate the cosmic ray cutoff variation
in the location of a particular point within a magnetic field in order to
estimate a cutoff value pertaining to the point. This value can be found
by use of a variety of interpolation techniques, one of which is applica-
tion of the Stormer equation given above. Because the Stormer equation
characterises the spatial variation of the cutoff rigidity, with appropri-
ate normalization it may be used to obtain useful estimates of the various
cosmic ray cutoff rigidities over intervals of latitude, longitude, zenith
and azimuth, for example. In practice, estimates of the value of any cut-
off can be obtained from adjacent calculated values to a reasonable accu-
racy by employing this method.
3. Discussion and Conclusion. Not all the cutoffs defined in the previous
section exist for every location and direction. It should be borne in
mind that, because the definitions have deliberately been kept usefully
general, the application of the terms may require more detailed qualifica-
tion in sorae individual circumstances. In addition, there is no doubt
that other physically meaningful quantities exist. It is believed, how-
ever, that the cutoff concepts described in this paper presently have the
greatest significance, and that the use of these definitions should alle-
viate most of the existing confusion and satisfy the current requirements
of investigators involved in cosmic ray access studies.
4. Acknowledgements. DJC acknowledges support from the U.S. Air Force
Geophysics Laboratory, under contract # F 19628-81-K0020.
References
- Lemaitre, G., and M. S. Vallarta, "On the geomagnetic analysis of cosmic
radiation", Phys. Rev., 49, 719-726, 1936a.
Lemaitre, G., and M. S. Vallarta, "On the allowed cone of cosmic radia-
tion", Phys. Rev., 50, 493-504, 1936b.
Shea, M. A., D. F. Smart, and K. G. McCracken, "A study of vertical
cutoff rigidities using sixth degree simulations of the geomagnetic
field", J. Geophys. Res., 70, 4117-4130, 1965.
Stormer, C., "Periodische Elektronenbahnen im Felde Lines Elementarmagne-
ton und ihre Anwendung auf Bruches Modellverauche und auf Eschenhagnes
Elementarwellen des Erdmagnetismus", Astrophys., 1, 237-274, 1930.


Re-evaluation of cosmic ray cutoff terminology

Re-evaluation of cosmic ray cutoff terminology

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