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MAGNETOSPHERIC MODELS AND TRAJECTORY COMPUTATIONS

1 1 2

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

1Air 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 subject

of interest since the time of Störmer. The fundamental problem is that the trajectory-tracing process

involves using mathematical equations that have ‘no solution in closed form’. This difﬁculty has

forced researchers to use the ‘brute force’ technique of numerical integration of many individual

trajectories to ascertain the behavior of trajectory families or groups. As the power of computers has

improved over the decades, the numerical integration procedure has grown more tractable and while

the problem is still formidable, thousands of trajectories can be computed without the expenditure

of excessive resources. As particle trajectories are computed and the characteristics analyzed we

can determine the cutoff rigidity of a speciﬁc location and viewing direction and direction and

deduce the direction in space of various cosmic ray anisotropies. Unfortunately, cutoff rigidities

are not simple parameters due to the chaotic behavior of the cosmic-ray trajectories in the cosmic ray

penumbral region. As the computational problem becomes more manageable, there is still the issue

of the accuracy of the magnetic ﬁeld models. Over the decades, magnetic ﬁeld models of increasing

complexity have been developed and utilized. The accuracy of trajectory calculations employing

contemporary magnetic ﬁeld models is sufﬁcient that cosmic ray experiments can be designed on the

basis of trajectory calculations. However, the Earth’s magnetosphere is dynamic and the most widely

used magnetospheric models currently available are static. This means that the greatest uncertainly

in 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 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 uti-

lized a dipole representation of the Earth’s magnetic ﬁeld. (The legend is that there

were rooms of students manually doing the computations.) The work of Störmer is

summarized in his book ‘The Polar Aurora’ (Störmer, 1950). The ﬁrst application

of computers to obtain solutions for particle trajectories was done by Lemaitre and

Vallarta (1936a, b) who used a ‘Bush differential analyzer’ (what would now be

called an analog computer) to obtain solutions for entire families of trajectories.

Their deﬁnitions and classic work on the ‘allowed cone of cosmic radiation’ are

still in use (Vallarta, 1938, 1961, 1978). The problem of deﬁning particle trajec-

tories in a magnetic ﬁeld was so difﬁcult that ‘terella’ experiments (large vacuum

chambers with scale size simulations of the Earth’s magnetic ﬁeld and evaluation of electron trajectories in the magnetic ﬁeld) were a preferred method of approach

for a number of years (Brunberg, 1953, 1956; Brunberg and Dattner, 1953).

1 1 2

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

1Air 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 subject

of interest since the time of Störmer. The fundamental problem is that the trajectory-tracing process

involves using mathematical equations that have ‘no solution in closed form’. This difﬁculty has

forced researchers to use the ‘brute force’ technique of numerical integration of many individual

trajectories to ascertain the behavior of trajectory families or groups. As the power of computers has

improved over the decades, the numerical integration procedure has grown more tractable and while

the problem is still formidable, thousands of trajectories can be computed without the expenditure

of excessive resources. As particle trajectories are computed and the characteristics analyzed we

can determine the cutoff rigidity of a speciﬁc location and viewing direction and direction and

deduce the direction in space of various cosmic ray anisotropies. Unfortunately, cutoff rigidities

are not simple parameters due to the chaotic behavior of the cosmic-ray trajectories in the cosmic ray

penumbral region. As the computational problem becomes more manageable, there is still the issue

of the accuracy of the magnetic ﬁeld models. Over the decades, magnetic ﬁeld models of increasing

complexity have been developed and utilized. The accuracy of trajectory calculations employing

contemporary magnetic ﬁeld models is sufﬁcient that cosmic ray experiments can be designed on the

basis of trajectory calculations. However, the Earth’s magnetosphere is dynamic and the most widely

used magnetospheric models currently available are static. This means that the greatest uncertainly

in 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 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 uti-

lized a dipole representation of the Earth’s magnetic ﬁeld. (The legend is that there

were rooms of students manually doing the computations.) The work of Störmer is

summarized in his book ‘The Polar Aurora’ (Störmer, 1950). The ﬁrst application

of computers to obtain solutions for particle trajectories was done by Lemaitre and

Vallarta (1936a, b) who used a ‘Bush differential analyzer’ (what would now be

called an analog computer) to obtain solutions for entire families of trajectories.

Their deﬁnitions and classic work on the ‘allowed cone of cosmic radiation’ are

still in use (Vallarta, 1938, 1961, 1978). The problem of deﬁning particle trajec-

tories in a magnetic ﬁeld was so difﬁcult that ‘terella’ experiments (large vacuum

chambers with scale size simulations of the Earth’s magnetic ﬁeld and evaluation of electron trajectories in the magnetic ﬁeld) were a preferred method of approach

for a number of years (Brunberg, 1953, 1956; Brunberg and Dattner, 1953).

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