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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 calcu-

lation 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 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 its magnetosphere may involve obtaining solutions to millions of individ-

ual cosmic ray trajectories. Therefore efﬁcient computation is essential (and a very

fast computer desirable). 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

[equation]

In this equation is the particle gyro-radius in km, is the particle rigidity in units

of GV, and 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 and designated by the symbol which can be derived from

the relativistic factor, , as follows:

[equation]

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 the

computational effort required for a trajectory calculation becomes important. There

have been a number of attempts to do this. The ‘guiding center’ approximation can

be 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 the

recommended 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 the

Earth because of the 13 magnetic ﬁeld gradient allows the particle gyro-radius

to grow very rapidly. The technique developed by Byrnak (1979) employed a helix

predictor-corrector with the Runge–Kutta technique being utilized to re-initiate the

calculation whenever the step length required adjustment. A more recent technique

utilized by Kobel (1990) and Flückiger and Kobel (1990) employs the Bulirsch-

the 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 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 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 its magnetosphere may involve obtaining solutions to millions of individ-

ual cosmic ray trajectories. Therefore efﬁcient computation is essential (and a very

fast computer desirable). 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

[equation]

In this equation is the particle gyro-radius in km, is the particle rigidity in units

of GV, and 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 and designated by the symbol which can be derived from

the relativistic factor, , as follows:

[equation]

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 the

computational effort required for a trajectory calculation becomes important. There

have been a number of attempts to do this. The ‘guiding center’ approximation can

be 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 the

recommended 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 the

Earth because of the 13 magnetic ﬁeld gradient allows the particle gyro-radius

to grow very rapidly. The technique developed by Byrnak (1979) employed a helix

predictor-corrector with the Runge–Kutta technique being utilized to re-initiate the

calculation whenever the step length required adjustment. A more recent technique

utilized by Kobel (1990) and Flückiger and Kobel (1990) employs the Bulirsch-

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