D. F. SMART ,M.A.SHEA and E. O....

MAGNETOSPHERIC MODELS AND TRAJECTORY COMPUTATIONS

D. F. SMART1, M. A. SHEA1 and E. O. FLÜCKIGER21Air Force Research Laboratory, Space Vehicles Directorate, Bedford, MA 01731, U.S.A.

2Physikalisches Institut, CH-3012, Bern, Switzerland

(Received 22 February 2000; accepted 16 June 2000)

Abstract. The calculation of particle trajectories in the Earth’s magnetic field has been a subjectof interest since the time of Störmer. The fundamental problem is that the trajectory-tracing processinvolves using mathematical equations that have ‘no solution in closed form’. This difficulty hasforced researchers to use the ‘brute force’ technique of numerical integration of many individualtrajectories to ascertain the behavior of trajectory families or groups. As the power of computers hasimproved over the decades, the numerical integration procedure has grown more tractable and whilethe problem is still formidable, thousands of trajectories can be computed without the expenditureof excessive resources. As particle trajectories are computed and the characteristics analyzed wecan determine the cutoff rigidity of a specific location and viewing direction and direction anddeduce the direction in space of various cosmic ray anisotropies. Unfortunately, cutoff rigiditiesare not simple parameters due to the chaotic behavior of the cosmic-ray trajectories in the cosmic raypenumbral region. As the computational problem becomes more manageable, there is still the issueof the accuracy of the magnetic field models. Over the decades, magnetic field models of increasingcomplexity have been developed and utilized. The accuracy of trajectory calculations employingcontemporary magnetic field models is sufficient that cosmic ray experiments can be designed on thebasis of trajectory calculations. However, the Earth’s magnetosphere is dynamic and the most widelyused magnetospheric models currently available are static. This means that the greatest uncertainlyin the application of charged particle trajectories occurs at low energies.

1. Historical Background

The integration of the equation of motion of a charged particle in a magnetic fieldis a problem that has no solution in a closed form. The first numerical efforts atintegration of the equations of particle motion began with Störmer (1930) who uti-lized a dipole representation of the Earth’s magnetic field. (The legend is that therewere rooms of students manually doing the computations.) The work of Störmer issummarized in his book ‘The Polar Aurora’ (Störmer, 1950). The first applicationof computers to obtain solutions for particle trajectories was done by Lemaitre andVallarta (1936a, b) who used a ‘Bush differential analyzer’ (what would now becalled an analog computer) to obtain solutions for entire families of trajectories.Their definitions and classic work on the ‘allowed cone of cosmic radiation’ arestill in use (Vallarta, 1938, 1961, 1978). The problem of defining particle trajec-tories in a magnetic field was so difficult that ‘terella’ experiments (large vacuumchambers with scale size simulations of the Earth’s magnetic field and evaluation

Space Science Reviews93: 305–333, 2000.© 2000Kluwer Academic Publishers. Printed in the Netherlands.

306 D. F. SMART ET AL.

of electron trajectories in the magnetic field) were a preferred method of approachfor a number of years (Brunberg, 1953, 1956; Brunberg and Dattner, 1953).

Researchers at the University of Chicago employing the AVIDAC computer atArgonne National Laboratories did the first application of the digital computer tothe cosmic ray trajectory problem in the United States. Jory (1956) calculated aset of 663 particle orbits in a dipole magnetic field. Lust (1957) calculated some1,500 orbits to better define the concept of impact zones. Kasper (1959) did amore extensive set of trajectory calculations on a digital computer, some 2000 tra-jectories in a dipole magnetic field. More advanced magnetic field models wereutilized by McCracken and his co-workers who became very successful in thedigital computer calculation of cosmic-ray trajectories in high order simulationsof the geomagnetic field to describe observed cosmic ray phenomena. They cal-culated particle access to specific cosmic ray stations on the Earth to describe thecosmic ray anisotropy (McCrackenet al., 1962, 1965, 1968). They also showedthat the observed cosmic ray intensity could be well ordered by geomagnetic cutoffrigidities derived from cosmic ray trajectories calculated in high order simulationof the Earth’s magnetic field (Sheaet al., 1965). They also demonstrated that theEarth’s internal magnetic field is evolving (quite rapidly on geologic time scales),and that updated cutoff rigidity calculations are necessary to explain the changesobserved in some areas of the world (Shea and Smart, 1970, 1990; Mischkeet al.,1979). However, the Earth’s geomagnetic field evolution is not uniform. Geomag-netic ‘jerks’ have been found in the Earth’s magnetic field (Langelet al., 1986;Macmillan, 1996).

Gall and co-workers were the first to utilize magnetospheric models to improvethe calculations of asymptotic directions and high-latitude cutoff rigidities. Theyobtained a better resolution of the asymptotic cones of acceptance and calculatedthe range of the daily variation in both asymptotic directions and cutoff rigiditiesat high latitudes (Gallet al., 1968; 1969; 1971a, b; Smartet al., 1969). It alsobecame possible to delineate solar particle access to regions of the magnetosphereby tracing allowed particles (see Morfill and Scholer, 1973, for a review of thisperiod.) However, it became evident that the early models of the magnetosphericfields were deficient in that they were unable to adequately explain the low-altitudeearth-orbiting spacecraft observations of energetic particle access into the Earth’shigh polar regions during very anisotropic solar particle events. (Gall and Bravo,1973; Morfill and Quenby, 1971; Morfill and Scholer, 1972a, b; Thomaset al.,1974). The general result of these interchanges was a realization of the inade-quacies of the early magnetospheric field models. Paulikas (1974) noted that theearly magnetospheric model trajectory calculations could delineate the general re-gions of solar particle access to the magnetosphere, but were not capable (at thattime) of resolving the fine spatial structure noted by polar orbiting spacecraft. Theresults obtained by the trajectory tracing calculations, particularly in the magne-tospheric tail, during the 1970s and early 1980s were reflecting the topology of the

MAGNETOSPHERIC MODELS AND TRAJECTORY COMPUTATIONS 307

cartoon-like magnetic fields in the early magnetospheric models and not the resultof physical processes.

While advances in computer technology over the past decades have allowedresearchers to more fully utilize the trajectory-tracing technique for various cosmicray analyses, this approach continues to present a formidable problem. As comput-ers become more powerful, magnetic field models of increasing complexity, whichbetter represent the Earth’s magnetic topology, are being developed. Consequently,these more complex geomagnetic field representations must be utilized for analysesof the higher precision measurements of cosmic radiation phenomena. As long asthe measurement techniques increase in accuracy and as long as the geomagneticfield models continue to improve, the trajectory-tracing process will be used forcosmic radiation research. This paper presents the mathematical equations used inthe trajectory tracing procedure, identifies the various geomagnetic field represen-tations, explains the determination of the cutoff rigidity values, and summarizeshow these calculations have been and continue to be used for cosmic radiationstudies.

2. The Equations Involved

2.1. THE CHARGED PARTICLE EQUATION OF MOTION

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

r = (e/mc) rB .

In this equation,r is the particle acceleration,r the particle velocity, andB themagnetic field vector. The electronic charge is denoted bye,m is the particle’srelativistic mass, andc is the speed of light. This equation, when expressed inr, θ, φ coordinates, results in three simultaneous differential equations with sixunknowns.

dvrdt= e

mc(vθBφ − vφBθ)+ v

2θ

r+ v

2φ

r,

dvθdt= e

mc(vφBr − vrBφ)− vrvθ

r+ v2

φ

r tanθ,

dvφdt= e

mc(vrBθ − vθBr)− vrvφ

r− vθvφ

r tanθ.

In these equations the particle velocity terms are

dr

dt= vr, dθ

dt= vθ

r,

dφ

dt= vφ

r sinθ.


This system of simultaneous linear differential equations can be integrated nu-merically if the components of magnetic inductionBr, Bθ, Bφ , are known as ex-plicit functions ofr, θ, φ. The method chosen by McCrackenet al. (1962) to solvethe above system of equations was fourth order Runge-Kutta integration (Ralstonand Wilf, 1960). In this numerical integration process, when the magnetic field isknown (see next section), a knowledge of the position and velocity coordinateson one point of the trajectory is used with the differential equations of motion togive the coordinates of subsequent points along the trajectory. Repeated applicationgives sufficient points to locate the trajectory in space.

The equation of charged particle motion in a magnetic field may also be writtenin x, y, z coordinates. This was done by Fermi (1950), and the resulting equa-tions will not be duplicated here. However, we note that thex, y, z coordinatesare applicable when using the magnetospheric coordinates which are right handed,orthogonal, Earth centered coordinate systems. The most commonly used mag-netospheric coordinates are GSE (geocentric solar-ecliptic) and GSM (geocentricsolar-magnetospheric). In GSE coordinates thex axis is the radial direction towardthe sun center and thez axis is pointed ‘north’ perpendicular to the ecliptic plane.In GSM coordinates thex axis is oriented in the direction of the sun and thezaxis is the projection of the Earth’s dipole axis on the GSEy − z plane. Furtherdefinitions of magnetospheric coordinate systems are given by Hapgood (1992).

2.2. COMPUTING THE EARTH’ S MAGNETIC FIELD

Computation of a high order simulation of the Earth’s magnetic field is a computerintensive process and to the surprise of many, even more demanding of computerresources than integration of particle trajectories.

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

V (r, θ, φ) = a∞∑n=1

(a/r)n+1n∑

m=0

[gmn cosmφ + hmn sinmφPmn (cosθ).

In this equationgmn andhmn are the Gauss coefficients describing the magnetic field,Pmn (cosθ) are the Schmidt-normalized associated Legendre polynomials, anda isthe average radius of the Earth. In the dipole case, the expansion results in simplealgebraic equations inr, θ, φ that can be repeatedly evaluated to quickly find asolution for a specific trajectory initiated from a specified direction at a specificenergy. However, as the complexity of the magnetic field expansion increases,the number of terms to be evaluated increases asn. For a 10th order descriptionof the Earth’s main magnetic field as provided by the International GeomagneticReference Field (IGRF, 1992; Sabkaet al., 1997), about 90% of the computerprocessing time is consumed in evaluating the magnetic field and only about 10%of the CPU time utilized in integrating the particle equation of motion. The most


efficient computer techniques available for evaluating the Legendre polynomial ex-pansion 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 orderof magnitude slower. (All attempts to develop a very efficient parallel-processingalgorithm to evaluate magnetic fields have so far met with failure.)

The Earth’s internal magnetic field is normally expressed in magneticx, y, z

coordinates∗ , or in r, θ, φ coordinates. The magnetospheric field is often expressedin GSM coordinates. It is necessary to convert to a common coordinate system todo the particle trajectory calculations.

2.3. METHODS FOR EFFICIENT COMPUTATION OF COSMIC-RAY

TRAJECTORIES

It is difficult to calculate the trajectory of an incoming cosmic ray particle throughthe magnetic field and expect to intersect the exact location for which the calcu-lation was desired. Since the path of a negatively charged particle of a specificmagnetic rigidity is identical (except for the sign of the velocity vector) to thatof a positively charged particle reaching the same location in space, the commonmethod of calculating cosmic ray trajectories in the Earth’s magnetic field is tocalculate the trajectory in the reverse direction. Thus the ‘starting point’ of thereverse trajectory calculation is given by the geographic coordinates, direction andaltitude of the location in question.

The extreme requirement of intensive computation to obtain a sufficient numberof particle trajectories to evaluate cosmic ray access to a specific location on theEarth or its magnetosphere may involve obtaining solutions to millions of individ-ual cosmic ray trajectories. Therefore efficient computation is essential (and a veryfast computer desirable). One approach developed by Smart and Shea (1981a) wasto compute a dynamic variable step length that was of the order of one percent ofa particle gyro-distance in the magnetic field. This process allows computation ofa simple cosmic ray trajectory from the ‘top’ of the atmosphere to interplanetaryspace in about 100 Runge–Kutta iterations. Complex trajectories, or trajectoriesof low rigidity (rigidity is momentum per unit charge) take correspondingly moreiterations. The gyro-radius of a charged particle in a magnetic field is given by

ρ = 33.33R/B.

In this equationρ is the particle gyro-radius in km,R is the particle rigidity in unitsof GV, andB 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 thespeed of light(v/c) and designated by the symbolβ which can be derived fromthe relativistic factor,γ , as follows:

∗Magnetic coordinates a local coordinate system in which thez axis is down (toward the nadir),thex axis is north, and they axis is toward the east


β = 1.0− (1.0/γ 2)1/2 and γ = (RZ)/(m0c2A)2+ 1.01/2,

noindent whereR is the particle rigidity†, Z the atomic charge,A the atomicnumber andm0c

2 is the rest mass energy.In a major computational effort where the objective may be to calculate a world

grid of cutoff rigidities or to analyze spacecraft data, any method to reduce thecomputational effort required for a trajectory calculation becomes important. Therehave been a number of attempts to do this. The ‘guiding center’ approximation canbe used at low energies when the magnetic field gradient is small over a gyro-radius. At higher energies numerical integration of the actual particle path is therecommended method of approach.

Fourth order Runge–Kutta iteration Ralston and Wilf, 1960) involves four mag-netic field computations per step. Other techniques of the class called ‘predictor-corrector’ only involve two magnetic field evaluations per step. The disadvan-tage of the predictor-corrector method is that it is a linear process involving uni-form step lengths. It has some difficulty with trajectories moving away from theEarth because of the 1/r3 magnetic field gradient allows the particle gyro-radiusto grow very rapidly. The technique developed by Byrnak (1979) employed a helixpredictor-corrector with the Runge–Kutta technique being utilized to re-initiate thecalculation whenever the step length required adjustment. A more recent techniqueutilized by Kobel (1990) and Flückiger and Kobel (1990) employs the Bulirsch-Stoer numerical integration technique (Stoer and Bulirsch, 1980; Presset al., 1989)to minimize the number of steps required in a charged particle trajectory computa-tion.

3. Characteristics of Cosmic-Ray Trajectories in the Earth’s Magnetic Field

To examine the characteristic behavior of cosmic-ray trajectories in the Earth’smagnetic field we consider trajectories of cosmic ray particles with different ener-gies as these trajectories are calculated from a location at the top of the atmosphereoutward into the magnetic environment surrounding the Earth. The trajectory for avery high-energy particle propagating outward through the Earth’s magnetic fieldwill reach interplanetary space with a minimum of geomagnetic bending. As thecharged particle energy decreases, then it will undergo more geomagnetic bendingbefore it can escape. At some lower energy, it will no longer have sufficient momen-tum to escape the magnetic field and in these cases the particle trajectory initiatedin an outward direction near the top of the atmosphere, will re-enter (i.e., intersectthe solid Earth). The presence of a solid object in the magnetic field complicates

†Rigidity is momentum per unit charge and is a canonical unit that is especially useful in char-acterizing 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, independentof elemental or isotopic composition, particle mass or atomic charge.


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

the problem, and an analytical description of the phenomena becomes even morecomplicated if the solid object is not centered in the magnetic field.

Some actual trajectory calculations are illustrated in Figure 1. All of the trajec-tories in this figure were initiated in the vertical direction from the same location.The trajectories labeled 1, 2, and 3 show increasing geomagnetic bending beforeescaping into space. The trajectory labeled 4 develops intermediate loops beforeescaping. The lower energy trajectory labeled 5 develops complex loops near theEarth before it escapes. As the charged particle energy is further reduced, thereare a series of trajectories that intersect the Earth (i.e., re-entrant trajectories). In apure dipole field that does not have a physical barrier embedded in the field, thesetrajectories may be allowed, illustrating one of the differences between Störmertheory and trajectory calculations in the Earth’s magnetic field. Finally the stilllower energy trajectory labeled 15 escapes after a series of complex loops near theEarth. These series of allowed and forbidden bands of particle access are calledthe cosmic ray penumbra. They also illustrate an often-ignored fact that cosmicray geomagnetic cutoffs are not sharp (except for special cases in the equatorialregions).

3.1. CUTOFF RIGIDITIES

Our procedure for determining geomagnetic cutoff rigidities is to make trajectorycalculations at discrete intervals through the rigidity spectrum with the assumptionthat the results of a specific trajectory at a specific rigidity are characteristic of adja-


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

cent trajectories at very slightly different rigidities or direction. These calculationsbegin at high rigidities (at a value above the highest possible cutoff) and progressdown through the rigidity spectrum until the lowest possible allowed trajectory hasbeen found. An examination of the characteristics of particle trajectories from highrigidities to low rigidities will show definitive fiducial marks. These are the firstdiscontinuity in asymptotic direction, the first forbidden trajectory, and perhaps arange of allowed and forbidden trajectories called the cosmic ray penumbra, andthe lowest allowed trajectory. In the cosmic ray penumbra, the highest rigidity for-bidden band is called the ‘first forbidden band’ which is discussed later in Sect. 6.We currently use three parameters to describe a geomagnetic cutoff rigidity. Theseare:– Ru: the upper cutoff which is the rigidity of the last allowed before the first

forbidden trajectory;– Rl : the lower cutoff which is the rigidity of the last allowed trajectory in a

decreasing rigidity scan, and– Rc: the effective cutoff which is an average betweenRu andRl that accounts

for the transparency of the penumbra.A more detailed explanation of the characteristics of geomagnetic cutoffs derivedfrom trajectory calculations is given by Cookeet al. (1991). Figure 2 illustratescosmic ray penumbra structure and geomagnetic cutoffs determined by trajectorycalculations for three North American neutron monitor stations.

Since there are chaotic structures in the penumbral region with very small fea-tures there is no certainty that all features are identified in a rigidity scan. It ispossible that we might not identify very small penumbral bands near the cutoff.When scanning the asymptotic directions that represent the interplanetary terminus


Figure 3.Illustration of the characteristic fiducial marks in trajectory derived cutoff rigidities. Theseare the first discontinuity in asymptotic direction, the first forbidden band, and the lowest allowedtrajectory, and the effective cutoff (Sheaet al., 1965). These results of calculations in a modelmagnetosphere also demonstrate the local time variations.

of these trajectory calculations as a function of rigidity, there is a systematic in-crease in asymptotic longitude as the rigidity is decreased, until very near the cutoffthere is a discontinuity in asymptotic direction. We have found that whenever thereis a discontinuity in asymptotic direction and we investigate the rigidity regionin minute detail, there is a forbidden (re-entrant) trajectory associated with thediscontinuity. Therefore, the first discontinuity in asymptotic direction is alwaysthe start of the penumbra. Continuing downward through the penumbra and calcu-lating trajectories for particles having successively lower rigidities results in a lastallowed trajectory that identifies the lower rigidity end of the cosmic-ray penumbra.These features are graphically illustrated in Figure 3. The discontinuities in thefeatures illustrated reflect the difficulty of exactly identifying all of the features ina rigidity scan at 0.01 GV intervals. This figure also illustrates the daily variationof the changes in cutoff of a neutron monitor location at an effective cutoff ofabout 1.5 GV. The relative magnitude of the daily variations becomes smaller asthe cutoff rigidity increases. The daily variation dominates the cutoff change athigh and polar latitudes and becomes insignificant at mid-and equatorial latitudes.


Figure 4. World map projection of the asymptotic directions of approach computed for cos-mic-ray muon detectors for high-energy solar cosmic-ray events.Left: 23 February 1956.Right:29 September 1989.

3.2. ASYMPTOTIC DIRECTIONS OF APPROACH

If we follow a charged particle trajectory away from the Earth, the amount ofgeomagnetic bending per unit path length decreases. In a magnetic field extendingto infinity, it can be said that the particle direction asymptotically approaches itsfinal direction. If we introduce a boundary such as the magnetopause, we oftenuse the same terms to describe the direction of the particle velocity vector at thepenetration location. (Ruth Gall in her work was most specific that these weredirections of approach.) McCrackenet al.(1968) and Sheaet al.(1965) performedcalculations in internal magnetic fields and utilized the particle velocity vector(expressed in geocentric coordinates at radial distance of 25 Earth radii) to specifythe asymptotic direction of approach. The set of asymptotic directions accessibleto a specific location on the Earth defines the asymptotic cone of acceptance. Forpolar or even mid-latitude muon detectors that only respond to high-energy parti-cles, these asymptotic cones of acceptance are restricted to specific regions of thecelestial sphere. Thus if multiple stations simultaneously observe an anisotropicsolar cosmic ray-flux, it is possible to deconvolve the flux direction in space andthe anisotropy (see Crampet al., 1995). If these stations are located at different geo-magnetic cutoffs, it is possible to deduce the solar particle spectra. (This techniqueis also called the global spectrograph method by Russian scientists.) Similarly,if a number of cosmic-ray stations, each having asymptotic cones of acceptanceviewing a different portion of the celestial sphere, rotate through a slowly evolvingcosmic ray anisotropy, then it is possible to deconvolve the spatial anisotropy. (seeNagashima and Fujimoto, 1994) for an example of this application.) The asymp-totic directions of approach in the rigidity range from 20 GV to 5 GV computedfor cosmic ray muon detectors for the maximum of the 23 February 1956 and 29September 1989 high-energy solar cosmic ray events are illustrated in Figure 4.

Cosmic ray neutron monitors located at polar latitudes have asymptotic conesof acceptance (over the range of rigidities that cover the response of the neutronmonitor to solar or galactic cosmic ray nuclei) that scan a restricted portion of thecelestial sphere. These characteristics are currently being aggressively exploited


Figure 5. Asymptotic directions of approach computed for selected cosmic ray neutron monitorsmapped on a spherical projection of the Earth. These projections are oriented on the probable inter-planetary magnetic field direction for two specific solar cosmic ray events.Left: 29 September 1989.Right:19 October 1989. See text for more details.

by the Bartol Research Institute in its cosmic ray program (Bieber and Evenson,1995). In a rigidity scan of the trajectories allowed at a specific location (cosmic-ray detector) the geomagnetic bending of the particle trajectory increases as theparticle rigidity decreases. The amount of geomagnetic bending becomes verylarge as the particle rigidity approaches the geomagnetic cutoff rigidity, perhapsinvolving several circum-navigations of the Earth. The result is an extremely broadasymptotic cone of acceptance for mid- or low-latitude stations with a large rangeof asymptotic longitudes involved. Figure 5 illustrates asymptotic cones of accep-tance for selected neutron monitor stations projected on a spherical mapping of theEarth. Note the longitudinal extent of the asymptotic cones for the Calgary, DeepRiver, and Goose Bay, Canada and the Hobart, Australia cosmic-ray stations.

In early work on trajectory calculations Kasper (1959) found the ‘focusing ef-fect’ of the magnetic field where trajectories initiated outward from the Earth withdifferent azimuth and zenith angles of incidence (at high and mid-latitudes, within afactor of two above the cutoff rigidity) reached a similar final asymptotic directionat distances far from the Earth. This ‘focusing effect’ which is valid when the scalesize 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 arrivingparticles are a good approximation of the entire asymptotic cone of acceptance.

In the trajectory calculations, we compute a trajectory at a specific rigidity anddirection and then assume that this result is representative of a finite domain ofrigidity or angular space. The question then arises, is this sample truly represen-tative? In an earlier section we discussed the possibility that sampling the rigidityspectrum at uniform intervals of 0.01 GV might not identify the first transitionfrom the continuously allowed rigidities to the cosmic ray penumbral regions of


alternating allowed and forbidden rigidity bands There is also the question of howvalid is the approximation that a sample in one direction is truly representative ofthe entire asymptotic cone of acceptance for a wide variety of directions. We haveno definitive answer as yet to these questions.

The problem of determining which trajectories are allowed and which are for-bidden is not as simple as it might initially seem. In the internal magnetic fieldrepresentations and especially in the more complex magnetospheric fields, there isa set of low rigidity trajectories that have very long path length, consisting of manycomplex loops. Often for the sake of economy of computer resources, trajectorycalculations are terminated after a large number of steps. This results in groupsof indeterminate trajectories whose fate is not resolved. In Störmer theory there isa special set of trajectories which will have an arbitrary number of loops beforereaching a final solution. In a simple dipole field, these low rigidity trajectorieshaving many loops were generally forbidden. Sheaet al. (1965) adopted the con-vention of declaring these indeterminate solutions as forbidden. This conventionis questionable, especially since these trajectory paths are the result of a stablemagnetic 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 raydetector in a balloon flown at high latitudes was consistent with defining these lowrigidity indeterminate trajectories as representing allowed charged particle accessthrough the Earth’s magnetosphere. Boberget al. (l995) considered any trajectorythat originated at low altitudes and reached the altitude of a geosynchronous satel-lite to be allowed. Tylkaet al. (1995) and Smartet al. (1999 a–c) adopted theBoberget al. (1995) definition in their recent work for calculating geomagneticcutoff 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 beambeing simulated may encounter a penumbral structure that is not truly representa-tive of the cosmic-ray access over a wider solid angle of acceptance. This leadsto a ‘lumpy’ structure that may not properly order the counting rate acquired by aneutron monitor during a latitude survey. However, the time requirements of com-puting a complete world grid of cutoff rigidities for a variety of directions has beenso formidable that the vertical cutoff approximation is the most widely used setof cutoff rigidities. Figure 6 shows the result of a trajectory derived vertical cutoffrigidity at one-degree intervals along the 285 deg East meridian from low rigiditynorthern polar values to the maximum cutoff value at the cosmic ray equator. Notethe irregular character of the cutoff values.

To more completely characterize the neutron monitor response, Raoet al.(1963)and McCrackenet al. (1965, 1968) used a 9-direction trajectory calculation se-quence (the vertical, and a sequence in the north, east, south and west directions atzenith angles at 16 degrees and 32 degrees) to approximate the angular responseof a neutron monitor. Shea and Smart (1970) used this approach to derive what


Figure 6.A set of trajectory derived vertical cutoff rigidity values, calculated at one-degree intervalsin the northern hemisphere along the 285-deg east meridian. Note the irregular character of the cutoffvalues. The upper computed cutoff,Ru, is indicated by the upper boundary of the shaded area; thelower computed cutoff,Rl , (the last allowed trajectory) is indicated by the lower boundary of theshaded area. The solid line is the ‘effective cutoff’,Rc, attempting to account for the transparency ofthe penumbra in the method as defined by Sheaet al. (1965).

they considered more truly representative effective cutoff rigidities for neutronmonitors.

The Hobart, Australia group has preferred to use the 9-direction approximationas being a better representation of the asymptotic cone of acceptance than the ver-tical approximation in the analysis of solar cosmic ray ground-level events (Crampet al., (1995). The Bartol group has used an average of 41 directions as beingnecessary for the determination of cutoff rigidities over the neutron monitor accep-tance cone for their recent sea-level surveys (Clemet al., 1997). However, this issuch a computer intensive requirement that Bieberet al. (1997) introduced anothersimpler 9-direction value, averaging the cutoff rigidity in the vertical direction andthe values calculated for 8 azimuthal directions at 30 degrees zenith.

4. Accuracy of the Calculations

The trajectory calculations are presumably as accurate as the geomagnetic fieldmodels utilized assuming that the numerical techniques yield an exact solutionand the computers involved have sufficient numerical accuracy. The high ordersimulations of the Earth’s magnetic field are better representations than the simplemodels.

For precise trajectories involving exact locations on the Earth, then the ini-tial directions must be specified in geodetic coordinates. See, for example, Smart


Figure 7.Illustration of the magnetic field line topology derived from the Tsyganenko model of themagnetosphere in GSM coordinates projected on theXsm Zsm plane. (Adapted from Flüeckiger andKobel, (1990.)

and Shea (1981b) for calculation of the termination of muon trajectories fromthe Batavia, Illinois, USA high- energy particle accelerator. For this mid-latitudelocation, the geodetic horizon is at an elevation angle of about1

2 degree whentransformed into geocentric coordinates. Shea and Smart (1983), Sheaet al.(1987),and Smart and Shea (1997a, b) use geodetic coordinates when calculating cutoffrigidities for locations on the surface of the Earth or in its atmosphere, but usegeocentric coordinates when calculating particle access or geomagnetic cutoff forspacecraft Smart and Shea (1997a, b). We have found a few noxious cases where,in complex particle trajectories near the cutoff rigidity, there were sudden, verysmall loops in the trajectory and the step size adjustment algorithm did not respondwith sufficient agility to faithfully trace the trajectory. However, these cases arerelatively rare. (The classic method to check the accuracy of a numerical integrationprocedure is to half the step length, repeat the calculation, and verify that the samesolution is obtained.)

The accuracy of the magnetic field models employed may be the limiting factorin charged particle trajectory calculations. The magnetospheric models derivedfrom spacecraft measurements are better representations of the magnetic fieldsin the Earth’s magnetosphere than previous models. A common procedure is tocombine the internal magnetic field of the Earth (usually represented by the IGRFmagnetic field model) with a model of the external field based on the analysisof spacecraft measurements. Figure 7 is an illustration of the magnetic field linetopology derived from the Tsyganenko magnetospheric magnetic field model. The


Figure 8. An extended world map illustrating the difference in the penetration directions at themagnetopause boundary for low rigidity charged particles within the rigidity range of the neutronmonitor response function incident at the Deep River, Canada neutron monitor. The dashed lineindicates trajectories calculated employing only the IGRF 1980 internal magnetic field model. Thedotted line indicates trajectories calculated by combining the same internal magnetic field modelwith the Tsyganenko magnetospheric model, and the solid line indicates the calculations omittingthe magnetospheric tail. (Adapted from Flüeckiger and Kobel, 1990.)

inclusion of these magnetospheric fields radically changes the topology of thedistant magnetic field from that expected from a field of purely internal sources,especially on the night side of the magnetosphere. This results in changes to theparticle trajectories, which are illustrated in figure 8. This figure is an extendedworld map to illustrate the difference in the approach directions at the magne-topause boundary for low rigidity charged particles within the neutron monitorresponse rigidity range. The coupling of the internal and external magnetic fields asthe Earth rotates within a magnetosphere that is oriented in a solar wind flow coor-dinate system results in a daily variation in the high latitude cosmic ray trajectoriesand consequently a daily variation in the high latitude geomagnetic cutoff rigidity(Smartet al., 1969). There are also seasonal effects (Danilova and Tyasto, 1991)attributed to the manner in which the 23.5 deg tilt of the Earth’s internal magneticfield combines with the external magnetospheric fields which are controlled by thesolar wind flow around the Earth.

The early work of McCracken and Ness (1966) found general agreement be-tween the average interplanetary magnetic field direction measured by spacecraftand the average direction of the maximum solar particle flux derived from theanalysis of the global response of the neutron monitor network to an anisotropicsolar cosmic-ray event McCrackenet al.(1967). In more recent work attempting toobtain better resolution in the maximum flux directions of anisotropic high energysolar particle events, we have found systematic slight differences in the derived


Figure 9.Conceptual illustration of the magnetic fields between the magnetopause boundary and theEarth’s bow shock. (Adapted from Bütikoferet al., 1995.)

maximum flux direction from the measured interplanetary magnetic field direc-tion. In considering possible explanations for these differences, we note that thereare additional fields between the magnetopause boundary and the interplanetarymedium beyond the Earth’s bow shock as illustrated in Figure 9. These magneticfields may be on the order of 10 nT and extend a few Earth radii. This constitutesa sufficient contribution of additional geomagnetic bending for low rigidity solarparticles that the direction of approach at the magnetopause boundary may notbe truly representative of the approach directions at the bow shock. Some initialinvestigations of these possible effects by Bütikoferet al. (1995) have attempted toobtain a better understanding of the possible effects of the magnetic field betweenthe magnetopause and the bow shock. Figure 10 illustrates the change in approachdirection found by including a simple model of the magnetic field between themagnetopause and the bow shock in the trajectory calculation of particle accessfrom interplanetary space. The changes in the approach direction illustrated in thisfigure occur primarily at low rigidities, below the response of a sea level neutronmonitor.

There is no general consensus model of the precise extent or character of thefields in the magnetospheric tail. Furthermore, this is an extremely dynamic por-


Figure 10.Illustration of the change in approach direction found by including a simple model of themagnetic field between the magnetopause and the bow shock (as shown in Figure 9) in the trajectorycalculation of particle access from interplanetary space. The solid line indicates approach directionsat the magnetopause boundary. The dashed and dotted lines indicate approach directions of the sametrajectories at the bow shock boundary forBIMF

xsm = +10 nT andBIMFxsm = −10 nT, respectively.

(Adapted from Bütikoferet al., 1995.)

tion of the Earth’s magnetosphere. In this respect some of the results obtained bytrajectory calculations in magnetospheric models are still model dependent andshould be interpreted within the limitations of the models utilized.

5. Checks on the Accuracy of the Cutoff Calculations Derived by TrajectoryTracing

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

Similarly, the trajectory-derived cutoff has been utilized to determine the ioniza-tion state of anomalous cosmic rays measured during space missions (Mitraet al.,1989; Leskeet al., 1996) and the ionic charge start of solar energetic particles(Kleckeret al., 1995; Leskeet al., 1995).

The use of trajectory derived cutoff rigidities at space shuttle altitudes has provento be a effective method of calculating astronaut dose during solar particle events.Golightly and Weyland (1997) reported that a minute by minute dose calculatedby utilizing the GOES synchronous orbit solar particle flux measurements filteredthrough the geomagnetic cutoff (Smart and Shea, 1997b) was generally within 10%


Figure 11.Illustration of the concept of the first forbidden band in the cosmic ray penumbra. Thisis a relatively stable feature that extends over a significant portion of the allowed cone of access toa cosmic ray detector. This figure illustrates the cosmic ray penumbra at various zenith and azimuthangles for balloon experiments from two different North American locations. The dark areas indicateforbidden rigidities; the white areas indicate allowed rigidities.Top: penumbra structure at SiouxFalls.Bottem: penumbra structure at Cape Girardeau.

of the observed dosimeter readings during the time when the STS-28 space shuttlemission with an orbital inclination of 57 deg encountered the 12 August 1989 solarparticle event.

5.1. HIGH RIGIDITY CUTOFF VALUES

The late B. Peters and co-workers (Peters, 1974; Lundet al., 1970, 1971) did pio-neering work on the use of geomagnetic cutoff features to measure actual cosmicray phenomena. Lundet al.(1971) noted a feature he called the first forbidden bandthat was generally stable and could be used as a sharp edge for isotope separation(Byrnaket al., 1981; Soutoulet al., 1981). The concept of this first forbidden bandis illustrated in Figure 11. The rigidities illustrated are the relatively simple trajec-tories that intersect the solid Earth as the rigidity scan passes through the uppercutoff rigidity. These relatively simple trajectories forming the first forbidden bandalso form a relatively stable and persistent feature of the cosmic ray penumbra.They generated the sharp edge that the HEAO 3 experimenters used for isotopeseparation (Copenhagen-Saclay, 1981).

The specific feature of the first forbidden band can also be used as a check ofthe absolute accuracy of the trajectory calculations. The concept is that 100 percentof the cosmic ray flux is transmitted at rigidities above the rigidity of the first


Figure 12. Calculated and experimentally observed cosmic ray cutoff at 5 GV by the HEAO-3experiments at 400 km. The heavy line indicates the predicted transmission through the cosmicray penumbra obtained by trajectory calculations in the internal geomagnetic field. The light lineindicates the observed average transmission derived from several thousand oxygen nuclei. (Adaptedfrom Copenhagen-Saclay, 1981.)

forbidden band. The first forbidden band is the fiducial mark that normalizes boththe theoretical and observed transmission. The transmission decreases as a func-tion of rigidity as the forbidden bands in the cosmic ray penumbra block particleaccess. The trajectory calculations offer a prediction of the rigidity of the firstforbidden band and the relative transmission through the cosmic ray penumbra.The difference between the predicted transmission and the observed transmissionis an indication of the accuracy of the trajectory calculations. The HEAO-3 ex-perimenters (Copenhagen-Saclay, 1981) found that at 5 GV the experimentallyobserved first forbidden band in their16O data set was about 5% lower than pre-dicted by the trajectory calculations using the IGRF internal field. These results areshown in Figure 12.

However at about 2 GV they found larger differences between the experimentalobservations of the first forbidden band and the trajectory calculations utilizingthe IGRF field model. There was a larger shift between the predicted and ob-served rigidity of the first forbidden band, as shown in Figure 13, and the observedpenumbra was more transparent than predicted by the trajectory calculations. Theseresults indicate the inadequacy of trajectory calculations using only the internal ge-omagnetic field to describe the trajectory of charged particles in the magnetosphere.These results also strongly suggest that at rigidities below a few GV, the use ofmagnetospheric models is essential for reliable cosmic ray trajectory calculations.


Figure 13.Calculated and experimentally observed cosmic ray cutoff at 2 GV by the HEAO-3 ex-periments at 400 km. The solid line indicates the predicted transmission through the cosmic-raypenumbra obtained by trajectory calculations in the internal geomagnetic field. The dashed lineindicates the observed average transmission derived from several thousand oxygen nuclei. (Adaptedfrom Copenhagen-Saclay, 1981.)

5.2. LOW RIGIDITY CUTOFF VALUES

There is an observable daily variation in high latitude cutoff that is the result of thevariation in the coupling of the internal and external magnetic fields as the Earthrotates within a magnetosphere that is oriented in a solar wind flow coordinatesystem (see Figure 7). Experimentally observed daily variations in the low energygeomagnetic cutoffs are illustrated in Figure 14.

The low energy cutoffs computed for very high latitudes from early magne-tospheric models were not particularly accurate. They reproduced the phase of thedaily variation (Gallet al., 1971a, b; Smart and Shea, 1972), but the magnitudeof the cutoff was not very reliable. In addition there were consistent reports ofparticle access measured by spacecraft (e.g., Gussenhovenet al., 1988) at lati-tudes equatorward of the value of the cutoff derived from the early magnetopshericmodels, especially during magnetic disturbances. It is only in the late 1990s thatmore accurate magnetospheric models such as those of Tsyganenko (1989) startedto be used for the calculation of geomagnetic cutoff values. Other models existsuch as those of Ostapenko and Maltsev (1997) that utilize parameters such as themagneticDst index or the solar wind pressure to parameterize the generators ofa magnetospheric model. However, these models are still statistical averages anddo not include a dynamic response to magnetospheric conditions. It is not clearthat particle trajectory tracing through static magnetospheric models will correctlyreproduce the low energy charged particle access to the polar regions through themagnetospheric tail. This is a particularly dynamic portion of the magnetosphere


Figure 14.The experimentally observed daily variation in high latitude (low energy) geomagneticcutoffs. (Adapted from Fanselow and Stone, 1972.)

and is currently not well modeled. Models of the Earth’s magnetosphere basedon magnetohydrodynamic descriptions of a dynamic magnetosphere are currentlybeing developed. This class of model offers the possibility of a good representationof the topology and magnitude of the magnetic fields in the Earth’s magnetosphere.Only preliminary work of using this class of models for charged particle trajectories(Orloff and Freeman, 1999) and geomagnetic cutoffs (Orloff, 1998) has been doneat this time.

The work of Smartet al. (1999a–c) gives some indication of the change incutoff with magnetic activity documenting the equatorward movement of the cutoffrigidity contours as theKp magnetic activity index becomes larger. These authorscomputed cutoff rigidities utilizing the Tsyganenko (1989) magnetospheric fieldmodel and the Boberget al. (1995) extension to include the probable effect ofadditional ring currents during severe magnetic storm conditions. The change incutoff rigidity at the longitude of the minimum cutoff for the International SpaceStation (ISS) orbit in the northern hemisphere is illustrated in Figure 15.

Rigidity is not the most convenient unit for use in comparing with energeticparticle data since most energetic particle measurements are in units of energy.


Figure 15.Illustration of the cutoff reduction at various magnetic activity levels. The coordinates arethe locations of the lowest cutoff value experienced by the ISS orbit at the specified latitude in theNorthern Hemisphere. (From Smartet al., 1999c.)

For comparison purposes, we have selected the invariant latitude calculated fromthe internal geomagnetic field as a common parameter. We interpolated throughthe published world grids of vertical geomagnetic cutoff rigidities for eachKp

magnetic activity level (Smartet al., 1999a–c), to determine proton cutoff en-ergy contours as a function of invariant latitude and obtained an average invariantlatitude for each energy. These results are presented in Figure 16. These curvesindicate an almost linear relation between the proton cutoff energy and invariantlatitude in the range from about 10 MeV to a few hundred MeV. We note thatthe change of proton cutoff energy withKp is relatively uniform over the rangeof the original Tsyganenko (1989) model (Kp = 0 to Kp = 5), but the cutoffchanges introduced by the Boberget al. (1995) extension is nonlinear with theDstincrement.

6. Comparison of Contemporary Computed and Measured GeomagneticCutoffs

At the time of this publication the work of Smartet al. (1999a–c) provides themost comprehensive set of calculations of cutoff rigidity values derived from con-temporary magnetic field models. Other work (Smart and Shea, 1967, 1994) hasshown that the cutoff rigidity change with radial distance is proportional toL−2


Figure 16.Calculated changes in the effective vertical cutoff energy for protons at 450 km altitudeas a function of magnetic activity.

whereL is the McIlwain (1961) parameter. We have extrapolated the cutoff rigidityvalues derived at 450 km (Smartet al., 1999a–c) to the altitude of the SAMPEXsatellite (∼600 km) for each observedKp value from day 304 to day 312 of 1992(30 October to 7 November) in order to compare trajectory-drived proton cutofflatitudes with the SAMPEX cutoff latitudes published by Leskeet al.(1995, 1997).A comparison of these proton cutoff latitudes for the 29–64 MeV protons is shownin Figure 17. These results shows a general correspondence between the calculatedproton cutoff latitude and the proton cutoff latitude derived from an analysis of theSAMPEX data. We note a systematic trend that the calculated proton cutoff latitudeis about 1.5◦ higher (poleward in latitude) than the values published by Leskeet al.(1997).

Some of the discrepancy may be explained by the fact that there is a systematicdifference between ‘measured’ and ‘computed’ cutoffs. Measured cutoffs are oftendetermined by a procedure such as finding when the instrument counting rate dur-ing a solar particle event has dropped to a value ofe−1 of the polar cap countingrate. If we use an exponential in rigidity [such asJ = J0 exp(R/R0)] to describethe particle flux spectrum, then this ‘measured’ latitude of the cutoff will always beR0 MV equatorward of the latitude of the ‘computed’ cutoff rigidity. However, thisis not enough to account for all of the systematic differences shown in this figure.

There is one time period when theDst values are exceptionally quiet, prior tothe arrival of the interplanetary shock on 1 November at 21:47 UT (day 306) andthe resultant magnetic storm, when there is relatively good agreement betweenthe magnetospheric model calculated and the SAMPEX derived proton cutoff lati-tudes. There is also a time on 3 November (day 308) when magnetic storm activityindicated by theDst index is not reproduced in theKp magnetic index. Note that


Figure 17.Simulation of the UT averaged proton cutoff energy variation as a function of theKp

index for the time period of 31 October to 7 November 1992.

the 3-hour averaging interval of theKp index was designed to ‘damp out’ thehigher frequency magnetic storm variations (Berthelier, 1993). There is also nocorresponding depression in the computed cutoff following the sudden commence-ment (SC) at 13:12 UT on 4 November (day 309), apparently because the variationin theDst index is not reproduced in theKp magnetic index. Leskeet al. (1997)found the hourlyDst index was a good indicator of the temporal behavior of theobserved cutoff variations.

7. Summary

The accurate calculation of particle trajectories in the Earth’s magnetic field is afundamental problem that limited the efficient utilization of cosmic-ray measure-ments during the early years of cosmic ray research. As the power of computers hasimproved over the decades, the numerical integration procedure has grown moretractable, and magnetic field models of increasing accuracy and complexity havebeen utilized. However, the Earth’s magnetosphere is dynamic and the most widelyused magnetospheric models currently available are static. Recent trajectory andcutoff calculations are being used to identify and deduce cosmic ray anisotropyand solar particle spectra and flux distributions. It is now possible for experimentsto be designed on the basis of trajectory calculations.

The accuracy of the trajectory computations is a function of energy, the greatestprecision being achieved for the highest energies. At low energies where the topol-ogy of the magnetospheric fields dominates the results, reasonable cutoff values


can be obtained which compare favorably with contemporary spacecraft measure-ments. In any case, the magnetic field description is the limiting factor in obtainingaccurate calculations of charged particle trajectories in the Earth’s magnetosphere.

Acknowledgements

The authors greatly appreciate the assistance of Dr Rolf Bütikofer with severalfigures. EOF acknowledges support by the Swiss National Science Foundation,Grants NF 20-050697.97 and 20-057175.99.

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Smart, D. F. and Shea, M. A.: 1997b, ‘Calculated Cosmic Ray Cutoff Rigidities at 450 km for Epoch1990.0’,Proc. 25th Int. Cosmic Ray Conf.2, 397–400.

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

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


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Address for Offprint:D.F. Smart, Air Force Research Laboratory (VSBS), Space Vehicles Direc-

torate, 29 Randolph Road, Bedford, MA 01731 USA; [email protected]

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