Final Report

Grant NAG5-8009

Prepared by D. F. Smart and M. A. Shea

Center for Space Plasmas and Aeronomic Research

The University of Alabama in Huntsville

Huntsville, Alabama 35889

**Synopsis**. The access of charged particles to the earth from space
through the geomagnetic field has been of interest since the discovery
of the cosmic radiation. The early cosmic ray measurements found that
cosmic ray intensity was ordered by the magnetic latitude and the
concept of cutoff rigidity was developed. The pioneering work of Störmer
(1930, 1955) resulted in the theory of particle motion in the
geomagnetic field, but the fundamental mathematical equations developed
have \"no solution in closed form\". This difficulty has forced
researchers to use the \"brute force\" technique of numerical
integration of individual trajectories to ascertain the behavior of
trajectory families or groups. This requires that many of trajectories
must be traced in order to determine what energy (or rigidity) a charged
particle must have to penetrate the magnetic field and arrive at a
specified position. It turned out the cutoff rigidity was not a simple
quantity but had many unanticipated complexities that required many
hundreds if not thousands of individual trajectory calculations to
resolve. The accurate calculation of particle trajectories in the
earth\'s magnetic field is a fundamental problem that limited the
efficient utilization of cosmic ray measurements during the early years
of cosmic ray research.

As the power of computers has improved over the decades, the numerical
integration procedure has grown more tractable, and magnetic field
models of increasing accuracy and complexity have been utilized. These
improvements have made the general application of the numerical
integration procedure more practicable and while the cutoff rigidity
problem is still formidable, thousands of trajectories can be computed
without the expenditure of excessive resources.

This report is documentation of a general FORTRAN computer program to
trace the trajectory of a charged particle of a specified rigidity
(momentum per unit charge) from a specified position and direction
through a model of the geomagnetic field. This software has been
incorporated into a general control program that makes the computation
of a number of trajectories to scan through a rigidity interval to
determine the cutoff rigidity of a specified location. The input control
file may contain as many locations as deemed necessary for a specific
study.

This report is organized in sections. Section I gives a scientific
background. Section II gives program documentation for three versions of
the program. Section III provides examples of the data input and
examples of the program output for the station selected. Section IV is
an appendix describing rigidity to energy conversion with tables and
demonstration programs. The final section is a listing of the trajectory
program FORTRAN source codes (with added line numbers).

This document is also included on a IOMEGA 100 MB ZIP disk, and the
FORTRAN source code is also provided on a 1.44 MB \'floppy\' disk.

**1.1** **Historical Background**

The integration of the equation of motion of a charged particle in a
magnetic field is a problem that has no solution in a closed form. The
first numerical efforts at integration of the equations of particle
motion began with Störmer (1930) who utilized a dipole representation of
the earth\'s magnetic field. The work of Störmer is summarized in his
book \'The Polar Aurora\' (Störmer, 1950). Lemaitre and Vallarta (1936
a,b) used a \"Bush differential analyzer\" (what would now be called an
analog computer) to obtain solutions for entire families of
trajectories. Jory (1956), Lust (1957), and Kasper (1959) were among the
first researchers to utilize the digital computer as a tool for
trajectory calculations in a dipole magnetic field. More advanced
magnetic field models were utilized by McCracken and his co-workers
(McCracken *et al.*, 1962, 1965, 1968). These workers were very
successful in the use of high speed digital computers for the
calculation of cosmic ray trajectories in high order simulations of the
geomagnetic field. They calculated particle access to specific cosmic
ray stations on the earth to describe the cosmic ray anisotropy and also
showed that the observed cosmic ray intensity could be well ordered by
geomagnetic cutoff rigidities derived from cosmic ray trajectories
calculated in high order simulations of the earth\'s magnetic field
(Shea *et al.*, 1965). They also demonstrated that the earth\'s internal
magnetic field is evolving (quite rapidly on geologic time scales), and
that the use of updated magnetic field models is necessary to explain
the changes observed in cosmic ray intensity in some areas of the world
(Shea and Smart, 1970, 1990; Mischke *et al.*, 1979). This is necessary
because the earth's geomagnetic field evolution is not uniform, and
sudden changes (called geomagnetic "jerks") have been found in the
Earth's magnetic field (Langel *et al.*, 1986; Macmillan, 1996).

Advances in computer technology over the past decades have allowed
researchers to more fully utilize the trajectory-tracing technique. As
computers become more powerful, magnetic field models of increasing
complexity, which better represent the earth\'s magnetic topology, have
been developed and must be utilized for analyses of the higher precision
measurements of cosmic radiation phenomena. As long as the measurement
techniques increase in accuracy and as long as the geomagnetic field
models continue to improve, the trajectory-tracing process will be used
for cosmic radiation research.

**1.2. The Equations Involved**

1.2.1 The Charged Particle Equation of Motion

The equation of charged particle motion in a magnetic field may be
written in vector form as

> **r** = (e / m c) **r B.**

In this equation, **r** is the particle acceleration, **r** the particle
velocity, and **B** the magnetic field vector. The electronic charge is
denoted by e, m is the particle's relativistic mass, and c is the speed
of light.v This equation, when expressed in r, , coordinates, results in
three simultaneous differential equations with six unknowns.

dv~r~ = e (v B - v B) + v^2^ + v^2^

dt m c r r

dv = e (v B~r~ - v~r~ B) - v~r~ v + v^2^

dt m c r r tan

dv = e (v~r~ B - v B~r~) - v~r~ v - v v

dt m c r r tan

In these equations the particle velocity terms are

dr = v~r~

dt

d = v

dt r

d = v

dt r sin

This system of simultaneous linear differential equations can be
integrated numerically if the components of magnetic induction B~r,~
B~,~ B~,~ are known as explicit functions of r, , . The method chosen by
McCracken *et al.* (1962) to solve the above system of equations was
fourth order Runge-Kutta integration (Ralston and Wilf, 1960). In this
numerical integration process, when the magnetic field is known (see
next section), a knowledge of the position and velocity coordinates on
one point of the trajectory is used with the differential equations of
motion to give the coordinates of subsequent points along the
trajectory. Repeated application gives sufficient points to locate the
trajectory in space. Adaptive step size control (see section 1.2.3.1)
can make the process more efficient. This is sometimes called fifth
order Runge-Kutta (see Press *et al.*, 1989).

1.2.2 Computing the Earth\'s Magnetic Field

Computation of a high order simulation of the earth\'s magnetic field is
a computer intensive process and to the surprise of many, even more
demanding of computer resources than integration of particle
trajectories.

If the field being modeled is composed of only internal sources, then it
is possible to define a magnetic potential, V, that can be expanded in
spherical harmonics.

~n~

V(r, , ) = a (a/r)^n+1^ \[ g^m^~n~ cos m + h^m^~n~ sin m \] P^m^~n~ (
cos )

^n=1\ m=0^

In this equation g^m^~n~ and h^m^~n~ are the Gauss coefficients
describing the magnetic field, P^m^~n~ ( cos ) are the
Schmidt-normalized associated Legendre polynomials, and a is the average
radius of the earth. In the dipole case, the expansion results in simple
algebraic equations in r, , that can be repeatedly evaluated to quickly
find a solution for a specific trajectory initiated from a specified
direction at a specific energy. However, as the complexity of the
magnetic field expansion increases, the number of terms to be evaluated
increases as n!. For a 10^th^ order description of the earth\'s main
magnetic field as provided by the International Geomagnetic Reference
Field (IGRF, 1992; Sabaka, 1997), about 90 percent of the computer
processing time is consumed in evaluating the magnetic field and only
about 10 percent of the CPU time utilized in integrating the particle
equation of motion. The most efficient computer techniques available for
evaluating the Legendre polynomial expansion involve using the
derivative of the previous term to obtain the current term, a process
that is inherently serial. The use of the recursion process is about an
order of magnitude slower. (All attempts to develop a very efficient
parallel-processing algorithm to evaluate magnetic fields have so far
met with failure.)

1.2.3 Methods for Efficient Computation of Cosmic-Ray Trajectories.

It is difficult to calculate the trajectory of an incoming cosmic ray
particle through the magnetic field and expect to intersect the exact
location for which the calculation was desired. Since the path of a
negatively charged particle of a specific magnetic rigidity is identical
(except for the sign of the velocity vector) to that of a positively
charged particle reaching the same location in space, the common method
of calculating cosmic ray trajectories in the earth's magnetic field is
to calculate the trajectory in the reverse direction. Thus for cosmic
ray trajectory calculations the "starting point" of the reverse
trajectory calculation is given by the geographic coordinates, direction
and altitude of the location in question.

The extreme requirement of intensive computation to obtain a sufficient
number of particle trajectories to evaluate cosmic ray access to a
specific location on the earth or in the earth\'s magnetosphere may
involve obtaining solutions to millions of individual cosmic ray
trajectories. Therefore efficient computation is essential (and a fast
computer desirable).

1.2.3.1 Variable Step Size Methods for Computation of Cosmic-Ray
Trajectories.

One approach developed by Smart and Shea (1981a) was to compute a
dynamic variable step length that was of the order of one percent of a
particle gyro-distance in the magnetic field. This process allows
computation of a simple cosmic ray trajectory from the \"top\" of the
atmosphere to interplanetary space in about 100 Runge-Kutta iterations.
Complex trajectories, or trajectories of low rigidity (rigidity is
momentum per unit charge) take correspondingly more iterations. The
gyro-radius of a charged particle in a magnetic field is given by

> = 33.33 R / B.

In this equation is the particle gyro-radius in km, R is the particle
rigidity in units of GV, and B is the magnitude of the magnetic field in
units of Gauss.

The particle velocity can be specified as the ratio of the particle
speed to the speed of light (v/c) and designated by the symbol which can
be derived from the relativistic factor, , as follows:

> = \[ 1.0 - ( 1.0 / ^2^ ) \]^1/2^,

and

> = { \[ ( R Z ) / ( m~o~c^2^ A ) \]^2^ + 1.0 }^1/2^,

where R is the particle rigidity[^1], Z the atomic charge, A the atomic
number and m~o~c^2^ is the rest mass energy.

**1.3. Characteristics of Cosmic-Ray Trajectories in the Earth's
Magnetic Field**

To examine the characteristic behavior of cosmic ray trajectories in the
earth's magnetic field we consider trajectories of cosmic ray particles
with different energies as these trajectories are calculated from a
location at the top of the atmosphere outward into the magnetic
environment surrounding the earth. The trajectory for a very high-energy
particle propagating outward through the earth's magnetic field will
reach interplanetary space with a minimum of geomagnetic bending. As the
charged particle energy decreases, then it will undergo more geomagnetic
bending before it can escape. At some lower energy, it will no longer
have sufficient momentum to escape the magnetic field and in these cases
the particle trajectory initiated in an outward direction near the top
of the atmosphere, will re-enter (i.e. intersect the solid earth). The
presence of a solid object in the magnetic field complicates the
problem, and an analytical description of the phenomena becomes even
more complicated if the solid object is not centered in the magnetic
field.

![](media/image1.wmf){width="3.9993055555555554in"
height="2.5479166666666666in"}

> *Figure* 1. Illustration of charged particle trajectories of different
> energies (rigidities) traced out in the vertical direction from the
> same location. The trajectories undergo increased geomagnetic bending
> as the particle energy (rigidity) is decreased. Charged particle
> trajectories near the cutoff rigidity develop intermediate loops and
> become complex. In the cosmic ray penumbra, some trajectories are
> re-entrant, and some are allowed. See text for more details.

Some actual trajectory calculations are illustrated in Figure 1. All of
the trajectories in this Figure were initiated in the vertical direction
from the same location. The trajectories labeled 1, 2, and 3 show
increasing geomagnetic bending before escaping into space. The
trajectory labeled 4 develops intermediate loops before escaping. The
lower energy trajectory labeled 5 develops complex loops near the earth
before it escapes. As the charged particle energy is further reduced,
there are a series of trajectories that intersect the earth (i.e.
re-entrant trajectories). In a pure dipole field that does not have a
physical barrier embedded in the field, these trajectories may be
allowed, illustrating one of the differences between Störmer theory and
trajectory calculations in the earth's magnetic field. Finally the still
lower energy trajectory labeled 15 escapes after a series of complex
loops near the earth. These series of allowed and forbidden bands of
particle access are called the cosmic ray penumbra. They also illustrate
an often-ignored fact that cosmic ray geomagnetic cutoffs are not sharp
(except for special cases in the equatorial regions).

1.3.1 Cutoff Rigidities

Our procedure for determining geomagnetic cutoff rigidities is to make
trajectory calculations at discrete intervals through the rigidity
spectrum with the assumption that the results of a specific trajectory
at a specific rigidity are characteristic of adjacent trajectories at
very slightly different rigidities or direction. These calculations
begin at high rigidities (at a value above the highest possible cutoff)
and progress down through the rigidity spectrum until the lowest
possible allowed trajectory has been found. An examination of the
characteristics of particle trajectories from high rigidities to low
rigidities will show definitive fiducial marks. These are the first
discontinuity in asymptotic direction, the first forbidden trajectory,
and perhaps a range of allowed and forbidden trajectories called the
cosmic ray penumbra, and the lowest allowed trajectory. In the cosmic
ray penumbra, the highest rigidity forbidden band is called the "first
forbidden band" (see Smart *et al.,* 2000, for more discussion). We
currently use three parameters to describe a geomagnetic cutoff
rigidity. These are:

R~u~ The upper cutoff which is the rigidity of the last allowed before
the first forbidden trajectory,

R*~l~* The lower cutoff which is the rigidity of the last allowed
trajectory in a decreasing rigidity scan, and

R~c~ The effective cutoff which is an average between R~u~ and R*~l~*
that accounts for the transparency of the penumbra.

A more detailed explanation of the characteristics of geomagnetic
cutoffs derived from trajectory calculations is given by Cooke *et al.*
(1991). Figure 2 illustrates cosmic ray penumbra structure and
geomagnetic cutoffs determined by trajectory calculations for three
North American neutron monitor stations.

![](media/image2.wmf){width="4.0in" height="2.786111111111111in"}

> *Figure* 2. Illustration of trajectory-derived cosmic ray cutoff and
> the cosmic ray penumbra structure in the vertical direction. The
> calculations have been done for three North American neutron monitor
> stations. White indicates allowed rigidities, black indicates
> forbidden rigidities.

Since there are chaotic structures in the penumbral region with very
small features there is no certainty that all features are identified in
a rigidity scan. It is possible that we might not identify very small
penumbral bands near the cutoff. When scanning the asymptotic directions
that represent the interplanetary terminus of these trajectory
calculations as a function of rigidity, there is a systematic increase
in asymptotic longitude as the rigidity is decreased, until very near
the cutoff there is a discontinuity in asymptotic direction. We have
found that whenever there is a discontinuity in asymptotic direction and
we investigate the rigidity region in minute detail, there is a
forbidden (re-entrant) trajectory associated with the discontinuity.
Therefore, the first discontinuity in asymptotic direction is always the
start of the penumbra. Continuing downward through the penumbra and
calculating trajectories for particles having successively lower
rigidities results in a last allowed trajectory that identifies the
lower rigidity end of the cosmic ray penumbra.

1.3.2 Asymptotic Directions of Approach

If we follow a charged particle trajectory away from the earth, the
amount of geomagnetic bending per unit path length decreases. In a
magnetic field extending to infinity, it can be said that the particle
direction asymptotically approaches its final direction. If we introduce
a boundary such as the magnetopause, we often use the same terms to
describe the direction of the particle velocity vector at the
penetration location. (Ruth Gall in her work was most specific that
these were directions of approach.) McCracken and co-workers (McCracken
*et al.*, 1968; Shea *et al.*, 1965), performed calculations in internal
magnetic fields and utilized the particle velocity vector (expressed in
geocentric coordinates at radial distance of 25 earth radii) to specify
the asymptotic direction of approach. The set of asymptotic directions
accessible to a specific location on the earth defines the asymptotic
cone of acceptance. The asymptotic longitude can exceed 360 degrees.
Large asymptotic longitudes are indicative of how many times the
trajectory has circumnavigated the earth during its transit.

In early work on trajectory calculations Kasper (1959) found the
"focusing effect" of the magnetic field where trajectories initiated
outward from the earth with different azimuth and zenith angles of
incidence (at high latitudes, within a factor of two above the cutoff
rigidity) reached a similar final asymptotic direction at distances far
from the earth. This "focusing effect" which is valid when the scale
size of the gradient in the earth's magnetic field is less than the
particle gyro-radii, also leads to the concept that asymptotic
directions computed for vertically arriving particles are a good
approximation of the entire asymptotic cone of acceptance.

For polar or even mid-latitude muon detectors that only respond to
high-energy particles, these asymptotic cones of acceptance are
restricted to specific regions of the celestial sphere. Thus if multiple
stations simultaneously observe an anisotropic solar cosmic ray flux, it
is possible to deconvolve the flux direction in space and the anisotropy
(see Cramp *et al.*, 1995). If these stations are located at different
geomagnetic cutoffs, it is possible to deduce the solar particle
spectra. Similarly, if a number of cosmic ray stations, each having
asymptotic cones of acceptance viewing a different portion of the
celestial sphere, rotate through a slowly evolving cosmic ray
anisotropy, then it is possible to deconvolve the spatial anisotropy.
(See Nagashima and Fujimoto, 1994, for an example of this application.)
The asymptotic directions of approach in the rigidity range from 20 GV
to 5 GV computed for cosmic ray muon detectors for the maximum of the 29
September 1989 high-energy solar cosmic ray events are illustrated in
Figure 3.

In a rigidity scan of the trajectories allowed at a specific location
(cosmic ray detector) the geomagnetic bending of the particle trajectory
increases as the particle rigidity decreases as illustrated in Figure 1.
The amount of geomagnetic bending becomes very large as the particle
rigidity approaches the geomagnetic cutoff rigidity, perhaps involving
several circum-navigations of the earth. The result is an extremely
broad asymptotic cone of acceptance for mid- or low-latitude stations
with a large range of asymptotic longitudes involved. Figure 5
illustrates asymptotic cones of acceptance for selected neutron monitor
stations projected on a spherical mapping of the earth. Note the
longitudinal extent of the asymptotic cones for the Calgary, Deep River,
and Goose Bay, Canada and the Hobart, Australia cosmic ray stations.

![](media/image3.wmf){width="4.0in" height="2.1590277777777778in"}

> *Figure* 3. World map projection of the asymptotic directions of
> approach computed for cosmic ray muon detectors for the 29 September
> 1989 high-energy solar cosmic ray events.

![](media/image4.wmf){width="3.0902777777777777in"
height="1.4479166666666667in"}

> *Figure* 4. Asymptotic directions of approach computed for selected
> cosmic ray neutron monitors mapped on a spherical projection of the
> earth. These projections are oriented on the probable interplanetary
> magnetic field direction for two specific solar cosmic ray events.\
> *Left*: 29 September 1989. *Right*: 19 October 1989.

In the trajectory calculations, we compute a trajectory at a specific
rigidity and direction and then assume that this result is
representative of a finite domain of rigidity or angular space. There is
the possibility that sampling the rigidity spectrum at uniform intervals
such as 0.01 GV might not identify the first transition from the
continuously allowed rigidities to the cosmic ray penumbral regions of
alternating allowed and forbidden rigidity bands There is also the
question of how valid is the approximation that a sample in one
direction is truly representative of the entire asymptotic cone of
acceptance for a wide variety of directions. We have no definitive
answer as yet to these questions.

The problem of determining which trajectories are allowed and which are
forbidden is not as simple as it might initially seem. In the internal
magnetic field representations and especially in the more complex
magnetospheric fields, there is a set of low rigidity trajectories that
has very long path lengths, consisting of many complex loops. Often for
the sake of economy of computer resources, trajectory calculations are
terminated after a large number of steps. This results in groups of
indeterminate trajectories whose fate is not resolved. In Störmer theory
there is a special set of trajectories which will have an arbitrary
number of loops before reaching a final solution. In a simple dipole
field, these low rigidity trajectories having many loops were generally
forbidden. Shea *et al.* (1965) adopted the convention of declaring
these indeterminate solutions as forbidden. This convention is
questionable, especially since these trajectory paths are the result of
a stable magnetic field and the magnetosphere is a domain of dynamic
plasma processes. Lin *et al.* (1995) found that their result of charged
particle access to a cosmic ray detector in a balloon flown at high
latitudes was consistent with defining these low rigidity indeterminate
trajectories as representing allowed charged particle access through the
earth's magnetosphere. Boberg *et al.* (1995) considered any trajectory
that originated at low altitudes and reached the altitude of a
geosynchronous satellite to be allowed. Tylka *et al.* (1995) and Smart
*et al.* (1999a,b,c) adopted the Boberg *et al.* (1995) definition in
their recent work for calculating geomagnetic cutoff rigidities.

However, there are definite limits to the use of the vertically incident
cosmic ray trajectories to provide an "exact" cutoff rigidity. The
'pencil-thin' particle beam being simulated may encounter a penumbral
structure that is not truly representative of the cosmic ray access over
a wider solid angle of acceptance. This leads to a 'lumpy' structure
that may not properly order the counting rate acquired by a neutron
monitor during a latitude survey. However, the time requirements of
computing a complete world grid of cutoff rigidities for a variety of
directions has been so formidable that the vertical cutoff approximation
is the most widely used set of cutoff rigidities. Figure 5 shows the
result of a trajectory derived vertical cutoff rigidity at one-degree
intervals along the 285 degree East meridian from the cosmic ray knee to
the cosmic ray equator. Note the irregular character of the calculated
cutoff values.

![](media/image5.wmf){width="2.5in" height="2.5972222222222223in"}

> *Figure* 5. A set of trajectory derived vertical cutoff rigidity
> values, calculated at one degree intervals, along the 285-degree east
> meridian from the cosmic ray knee to the cosmic ray equator. Note the
> irregular character of the cutoff values. The upper computed cutoff,
> R~u~, is indicated by the upper boundary of the shaded area; the lower
> computed cutoff, R*~l~*, (the last allowed trajectory) is indicated by
> the lower boundary of the shaded area. The solid line is the
> 'effective cutoff', R~c~, attempting to account for the transparency
> of the penumbra in the method as defined by Shea *et al.* (1965).

**1.4. Accuracy of the Calculations**

The accuracy of the magnetic field models employed is the limiting
factor in charged particle trajectory calculations assuming that the
numerical techniques yield an exact solution and the computers involved
have sufficient numerical accuracy. The high order simulations of the
earth's magnetic field are better representations than the simple
models.

For precise trajectories involving exact locations on the earth, then
the initial directions must be specified in geodetic coordinates. See
for example, Smart and Shea (1981b) for calculation of the termination
of muon trajectories from the Batavia, Illinois, USA high-energy
particle accelerator. For this mid-latitude location, the geodetic
horizon is at an elevation angle of about ½ degree when transformed into
geocentric coordinates. Shea and Smart (1983), Shea *et al.* (1987), and
Smart and Shea (1997a) use geodetic coordinates when calculating cutoff
rigidities for locations on the surface of the earth or in the earth\'s
atmosphere, but use geocentric coordinates when calculating particle
access or geomagnetic cutoff for spacecraft (Smart and Shea, 1997b). We
have found a few noxious cases where, in complex particle trajectories
near the cutoff rigidity, there were sudden, very small loops in the
trajectory and the step size adjustment algorithm did not respond with
sufficient agility to faithfully trace the trajectory. However, these
cases are relatively rare. (The classic method to check the accuracy of
a numerical integration procedure is to half the step length, repeat the
calculation, and verify that the same solution is obtained.)

Some experimenters such as Dryer and Meyer (1975) have used the
prediction of the geomagnetic cutoff derived from trajectory
calculations in the design of experiments that respond to cosmic ray
heavy nuclei in a specific rigidity range. These attempts have been very
successful indicating that there is a general reliability in high-energy
charged particle trajectory calculations in high degree simulations of
the earth's magnetic field.

**1.5. Summary**

The calculation of particle trajectories in the earth\'s magnetic field
was a fundamental problem that limited the efficient utilization of
cosmic ray measurements during the early years of cosmic ray research.
As the power of computers has improved over the decades, the numerical
integration procedure has grown more tractable, and magnetic field
models of increasing accuracy and complexity can be utilized. The
trajectory calculation process is sufficiently mature that it is
possible to do sufficient trajectory calculation to determine and cutoff
rigidities. It is now possible for experiments to be designed on the
basis of trajectory calculations.

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of the IGRF', *J. Geomag. Geoelectr.* **49**, 157-206.

Shea, M.A., and Smart, D.F.: 1970, 'Secular Variation in Cosmic Ray
Cutoff Rigidities', *J. Geophys. Res.* **75**, 3921-3922.

Shea, M.A., and Smart, D.F.: 1983, 'A World Grid of Calculated Vertical
Cutoff Rigidities for 1980.', *Proc. Int. Cosmic Ray Conf. 18^th^*
**3**, 415-418.

Shea, M.A., and Smart, D.F.: 1990, 'The Influence of the Changing
Geomagnetic Field on Cosmic Ray Cutoff Rigidities', *J. Geomag.
Geoelectr.* **42**, 1107-1121.

Shea, M.A., Smart, D.F., and Gentile, L.C.: 1987, 'Vertical Cutoff
Rigidities Calculated From the Estimated 1985 Geomagnetic Field
Coefficients', *Proc. Int. Cosmic Ray Conf. 20^th^* **4**, 205-207.

Shea, M.A., Smart, D.F., and McCracken, K.G.: 1965, 'A Study of Vertical
Cutoff Rigidities Using Sixth Degree Simulations of the Geomagnetic
Field', *J. Geophys. Res.* **70**, 4117-4130.

Smart, D.F., and Shea, M.A.: 1981a, 'Optimum Step Length Control for
Cosmic Ray Trajectory Calculations', *Proc. Int. Cosmic Ray Conf.
17^th^* **4**, 255-258.

Smart, D.F., and Shea, M.A.: 1981b, 'Muon Trajectories from the Batavia
Accelerator N-E Beam Dump', *Proc. Int. Cosmic Ray Conf. 17^th^* **5**,
6-9.

Smart, D.F., and Shea, M.A.: 1997a, 'World Grid of Cosmic Ray Vertical
Cutoff Rigidities for Epoch 1990.0', *Proc. Int. Cosmic Ray Conf.
25^th^* **2**, 401-404.

Smart, D.F., and Shea, M.A.: 1997b, 'Calculated Cosmic Ray Cutoff
Rigidities at 450 km for Epoch 1990.0', *Proc. Int. Cosmic Ray Conf.
25^th^* **2**, 397-400.

Smart, D.F., Shea, M.A., and Flückiger, E.O.: 1999a, 'Calculated
Vertical Cutoff Rigidities for the International Space Station During
Magnetically Quiet Times', *Proc. Int. Cosmic Ray Conf. 26^th^* **7**,
394-397.

Smart, D.F., Shea, M.A., and Flückiger, E.O.: 2000, 'Magnetospheric
Models and Trajectories Calculations', *Space Science Rev.,* **93***,
281-308.*

Smart, D.F., Shea, M.A., Flückiger, E.O., Tylka, A.J., and Boberg, P.R.:
1999b, 'Calculated Vertical Cutoff Rigidities for the International
Space Station During Magnetically Active Times', *Proc. Int. Cosmic Ray
Conf. 26^th^* **7**, 398-401.

Smart, D.F., Shea, M.A., Flückiger, E.O., Tylka, A.J., and Boberg, P.R.:
1999c, 'Changes in Calculated Vertical Cutoff Rigidities at the Altitude
of the International Space Station as a Function of Magnetic Activity',
*Proc. Int. Cosmic Ray Conf. 26^th^* **7**, 337-340.

Störmer, C.: 1930, 'Periodische Elektronenbabahnen im Feld lines
Elementramagneton und ihre Anwendung auf Bruches Modellverauche und auf
Eschenhagens Elementarwellen des Erdmagnetismus'. *Zeits f. Astrophys*.
**1** 237-274.

Störmer, C.: 1950, '*The Polar Aurora*' Oxford University Press, London.

Tylka., A.J., Boberg, P.R., Adams, J.H., and Beahm, L.P*.*: 1995, 'The
Mean Ionic Charge State of Solar Energetic Fe Ions Above 200 MeV per
Nucleon', *Astophys. J. Lett.*, **444**, L109-L113.

[^1]: Rigidity is momentum per unit charge and is a canonical unit that
    is especially useful in characterizing charged particle access in
    magnetic fields. All particles having the same magnetic rigidity,
    charge sign and initial conditions will have identical trajectories
    in the magnetic field, independent of elemental or isotopic
    composition, particle mass or atomic charge.