Click
Here
JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 112, A06216, doi:10.1029/2006JA012224, 2007
for
Full
Article
Energetic particles in the paleomagnetosphere:
Reduced dipole configurations and quadrupolar
contributions
J. Vogt,1 B. Zieger,1,5 K.-H. Glassmeier,2 A. Stadelmann,2 M.-B. Kallenrode,3
M. Sinnhuber,4 and H. Winkler4
Received 14 December 2006; revised 5 February 2007; accepted 20 February 2007; published 13 June 2007.
[1] The Earth’s magnetosphere acts as a shield against highly energetic particles of
cosmic and solar origin. On geological timescales, variations of the internal geomagnetic
field can drastically alter the magnetospheric structure and dynamics. This paper deals
with energetic particles in the paleomagnetosphere. Scaling relations for cutoff energies
and differential particle fluxes during periods of reduced dipole moment are derived.
Particular attention is paid to high rigidity galactic cosmic ray particles in the GV range
and to lower rigidity solar energetic particles. We find that in the paleomagnetosphere of a
strongly reduced dipole moment, solar protons of several tens of MeV may access the
Earth’s atmosphere even at midlatitudes. Higher-order core-field components can widen
the polar caps and open new particle-entry regions around the equator. Potential field
modeling of the paleomagnetosphere and magnetohydrodynamic (MHD) simulations are
utilized to study the impact of solar particles on the Earth’s atmosphere for more complex
field configurations that are likely to occur during geomagnetic polarity transitions.
Citation: Vogt, J., B. Zieger, K.-H. Glassmeier, A. Stadelmann, M.-B. Kallenrode, M. Sinnhuber, and H. Winkler (2007), Energetic
particles in the paleomagnetosphere: Reduced dipole configurations and quadrupolar contributions, J. Geophys. Res., 112, A06216,
doi:10.1029/2006JA012224.
1. Introduction
[2] Energetic charged particles of solar and cosmic
origin may affect geospace in various ways. Svensmark
and Friis-Christensen [1997] suggested that galactic
cosmic ray (GCR) modulations are associated with the
formation of tropospheric clouds which in turn influence
the Earth’s climate [see also Svensmark, 2000 and
Usoskin et al., 2004]. In order to reach the Earth’s
troposphere from space, energetic particles must have
energies in the GeV range where GCRs dominate the
spectrum. Solar energetic particle (SEP) events yield
considerable fluxes at lower energies in the MeV range
(up to about 100 MeV for large events). Through
dissociation and ionization of N2 and O2, SEPs contribute
to the formation of NO x and HO x in the middle
atmosphere [Crutzen et al., 1975; Porter et al., 1976;
Solomon et al., 1981; Randall et al., 2001], which are
key agents in the depletion of stratospheric ozone. NOx
formation and subsequent ozone losses have been mea1
School of Engineering and Science, Jacobs University Bremen
(formerly International University Bremen), Bremen, Germany.
2
Institut für Geophysik und extraterrestrische Physik, Technische
Universität Braunschweig, Braunschweig, Germany.
3
Fachbereich Physik, Universität Osnabrück, Osnabrück, Germany.
4
Institut für Umweltphysik, Universität Bremen, Bremen, Germany.
5
Geodetic and Geophysical Research Institute, Hungarian Academy of
Sciences, Sopron, Hungary.
Copyright 2007 by the American Geophysical Union.
0148-0227/07/2006JA012224$09.00
sured during several large SEP events and reproduced by
atmospheric chemistry models [Solomon et al., 1983;
Jackman et al., 2001; Rohen et al., 2005; Jackman et
al., 2005].
[3] The Earth’s atmosphere is not our only shield against
energetic particles from space. Before arriving at the upper
atmosphere, GCRs and SEPs have to pass through the
geomagnetic field that in fact acts as a huge particle
spectrometer. The main part of the magnetic field that we
measure on the Earth’s surface is generated in the liquid
outer core of our planet where thermally driven convection
sustains the field through dynamo action. Core field and
crustal magnetization constitute the so-called internal geomagnetic field that interacts with the supermagnetosonic
and magnetized solar wind plasma to create the Earth’s
magnetosphere. Large-scale current systems in the magnetosphere and also in the Earth’s ionized upper atmosphere, the
ionosphere, add a smaller but much more variable external
contribution to the surface field. At distances of several
Earth radii, external and internal contributions to the geomagnetic field become equally important.
[4] Størmer [1955] carried out a very comprehensive
analysis of particle orbits in a dipole field that shows why
the present-day geomagnetic field protects large parts of the
Earth’s surface from energetic particles in the range of up to
several GeVs. Numerical-particle orbit integrations in more
detailed model magnetic fields began in the 1960s [e.g.,
Shea et al., 1965] and first concentrated on typical GCR
energies and particles in the internal geomagnetic (core)
A06216
1 of 13
A06216
VOGT ET AL.: PARTICLES IN THE PALEOMAGNETOSPHERE
field only. Magnetospheric models were employed to describe daily variations of particle fluxes [e.g., Fanselow and
Stone, 1972; Smart and Shea, 1972] and to quantify cutoff
parameters in the lower part of the energy spectrum where
SEPs dominate (e.g., Flueckiger and Kobel [1990], Smart
and Shea [2001], see also the review by Smart et al.
[2000]).
[5] The magnetosphere is by no means a static entity.
Solar wind variations drive magnetospheric dynamics on
timescales like hours or days and up to years. The basic
mechanisms and key solar wind control parameters are
explained in textbooks on space plasma physics [e.g.,
Baumjohann and Treumann, 1997; Kallenrode, 1998].
Solar wind-induced dynamics gives rise to a wide range
of so-called ‘‘space weather’’ phenomena including auroral
emissions, geomagnetic storms and substorms, and atmospheric effects caused by solar energetic particles.
[6] On timescales much larger than decades, the internally generated magnetic field itself is subject to variations which influence the size and geometry of the
magnetosphere. Siscoe and Chen [1975] coined the term
paleomagnetosphere and worked out how a changing
dipole moment alters structural magnetospheric parameters. The most dramatic events in the magnetic history of
our planet were geomagnetic polarity transitions which
are well documented in paleomagnetic records [Merrill
and Mcfadden, 1999]. They must have drastically affected
all aspects of the Earth’s magnetosphere. The phenomena
associated with geomagnetic and solar variability on large
timescales can be summarized in the term ‘‘space climate’’. Numerical particle tracing schemes were used to
determine cutoff parameters for the past 2000 years by
Flueckiger et al. [2003] and for the past 400 years by
Shea and Smart [2004]. Possible biological implications
of magnetic reversals were discussed by Biernat et al.
[2002]. In the context of the Priority Programme SPP
1097 ‘‘Geomagnetic variations’’ funded by the Deutsche
Forschungsgemeinschaft (DFG), various aspects of the
paleomagnetosphere have been studied, ranging from
scaling relations for plasma boundary layers and current
systems in dipolar configurations [Vogt and Glassmeier,
2001; Glassmeier et al., 2004] and an analytical treatment
of trapped particle populations in quadrupolar magnetospheres [Vogt and Glassmeier, 2000] to energetic particle
parameters determined through trajectory tracing in a
magnetospheric potential field model [Stadelmann, 2004]
and large-scale magnetohydrodynamic (MHD) simulations
of reduced dipole paleomagnetospheres [Zieger et al.,
2006a, 2006b], equatorial dipole magnetospheres [Zieger
et al., 2004], as well as quadrupolar configurations [Vogt
et al., 2004]. Effects of energetic particle flux changes on
the middle paleoatmosphere were studied in the same
context [Quack et al., 2001; Sinnhuber et al., 2003;
Schröter et al., 2006].
[7] This paper deals with scaling relations for particle
intensities and cutoff parameters in the Earth’s paleomagnetosphere. We are most interested in particles affecting the
middle atmosphere and concentrate on protons in the range
of several tens of MeV. Section 2 summarizes useful
theoretical concepts and explains the methodology of our
approach. To characterize particle fluxes in a dipolar paleomagnetosphere, scaling relations for particle intensities and
A06216
cutoff parameters with changing dipole moment are compiled in section 3. Selected aspects of particle populations in
nondipolar configurations during geomagnetic polarity
reversals and issues related to numerical modeling and
particle tracing are discussed in section 4. We conclude in
section 5.
2. Useful Theoretical Concepts
[8] In this section, we compile selected theoretical
concepts and tools which will be used in the subsequent
analyses. In brief, we review the definition of the differential
particle flux and discuss how its values in the Earth’s upper
atmosphere can be reconstructed from the source distribution
in interplanetary space and the topology of particle orbits in
the geomagnetic field. Scaling relations for particle parameters and fluxes can be derived for a particular class of
magnetic-field variations. We finally stress the importance of
the rigidity concept and note its relation to the kinetic energy
of a particle.
2.1. Differential Particle Flux and Distribution
Function
[9] Measurements of particle spectrometers on spacecraft
are usually expressed in terms of the differential particle
flux J ¼ J ðW ; ^
v; r; tÞ [e.g., Baumjohann and Treumann,
1997; Kallenrode, 1998]. Here W denotes the (kinetic)
energy of particles coming from direction ^
v, r is position,
and t is time. The differential particle flux gives the number
of particles per energy band, time interval, detector area
element, and solid angle, so a convenient physical unit of
J is particles per (m2 sr s MeV).
[10] A key concept in plasma kinetic theory is the
(particle) distribution function f that depends on r, t, and
velocity v (or, alternatively, on r, t, and momentum p).
The distribution function is defined as the number
density in phase space. In textbooks on space plasma
physics such as the works of Baumjohann and Treumann
[1997] and Kallenrode [1998], it is shown that the
function f and the differential particle flux J differ only
by a factor that depends on energy W and particle speed
v. Both quantities are constant along particle orbits in
time-independent magnetic fields, and since in the collisionless plasma of the Earth’s magnetosphere this kind of
invariance also applies to the distribution function f
(Vlasov equation, Liouville theorem), we can conclude
that particle orbits in phase space are lines of constant
differential particle flux J.
[11] For a given source distribution in interplanetary
space and a geomagnetic paleofield configuration, we are
interested in determining the differential flux of energetic
particles into the Earth’s upper atmosphere. Only those
particle orbits which connect to interplanetary space actually contribute to the atmospheric flux; hence the phase flow
topology in combination with the source distribution fully
controls the particle flux into the upper atmosphere. If the
interplanetary source distribution is isotropic and homogeneous, it is sufficient to identify whether or not an atmospheric location can be reached by particles with a certain
energy and direction of approach, and one can directly
assign the value of the differential particle flux for this
particular set of parameters. It is only the connectivity of a
2 of 13
VOGT ET AL.: PARTICLES IN THE PALEOMAGNETOSPHERE
A06216
particle orbit that matters, i.e., the topology of the flow in
phase space.
2.2. The Normalized Equation of Motion
[12] The equation of motion for a particle of mass
m = m(v), momentum p = mv, and charge q in a
magnetic field B = B (r) reads (SI units)
dp dðmvÞ
¼
¼ qv B:
dt
dt
d^v qvB
^ ¼ 1 ^v B
^
^v B
¼
ds mv2
r
ð2Þ
where the parameter
r¼
p
mv
¼
qB qB
ð3Þ
is equal to the gyroradius of particles with a pitch angle
of 90°. The equation for d^v/ds must be solved in parallel
with
dr
¼ ^v
ds
ð4Þ
to yield the particle orbit in configuration space in the
normalized form.
[13] For convenience, the equation of motion can be
normalized further using a system length scale R0 and a
reference magnetic field value B0. Let B0 = |B(r0)|, R0 = |r0|,
and r0 be the gyroradius in the reference field B0 measured
in units of R0, i.e.,
r0 ¼
p
mv
¼
;
qB0 R0 qB0 R0
and
ð5Þ
B*0
Bðr=R0 Þ
¼
B0
ð10Þ
immediately from these definitions of W and p is
2
2
m0 c2 þð pcÞ2 ¼ m0 c2 þ W
ð11Þ
which leads to
ð6Þ
ð7Þ
ð8Þ
P
¼
MV
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
m0 c2
W 2
m0 c2 2
þ
MeV MeV
MeV
W
¼
MeV
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ffi
m0 c2 2
P 2 m0 c2
þ
:
MeV
MeV
MV
ð12Þ
and
Hence two magnetic fields B and B* support topologically
the same particle orbits in phase space if
B* r=R0*
p p* B0 R0
:
¼
q q * B* R*
0 0
2.3. Rigidity and Energy
[15] Particle parameters enter the equation of motion in
magnetic fields only through the ratio P = p/q = mv/q
(momentum per charge) commonly referred to as the
(magnetic) rigidity of the particle. P is usually expressed
in units of volts which can be explained by the
corresponding form in the cgs system of units where the
Lorentz force contains the speed of light c as an additional
constant, and c is absorbed in the definition of rigidity as
P = pc/q = mvc/q (cgs).
[ 16 ] The rigidity of a particle with rest mass m 0
can be converted into kinetic energy W using the relativistic
expressions for energy mc2 = m0 c2 + W and momentum
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
p = m0v/ 1 ðv=cÞ2 . One convenient form that follows
and
dðr=R0 Þ
¼ ^v:
dðs=R0 Þ
ð9Þ
This relationship is the basis for the scaling studies in
section 3.
[14] Condition (8) means that a magnetic field configuration B is geometrically similar to B* up to a scaling
transformation in the parameters R0 and B0. In this sense,
the magnetic field can be understood to transform in a ‘‘selfsimilar’’ way: B ! B* as R0 ! R0* and B0 ! B0* . In the
following, condition (8) will thus be referred to as a selfsimilarity assumption, and the relations that follow from
such an assumption will be termed scaling relations or
mapping relations. In the paleomagnetospheric context,
Siscoe and Chen [1975] were the first ones to apply a
scaling approach to dipolar magnetospheres using B0 / M
and R 0 / M 1/3 where M denotes the magnitude of
the geomagnetic dipole moment. With regard to particle
orbit scaling, condition (8) has to be supplemented by
equation (10).
then
d^v
1 B
^
^v B
¼
dðs=R0 Þ r0 B0
r0 ¼ r*0
which implies
ð1Þ
This equation can be normalized to yield a more
convenient form for studying the particle orbit connec^ = B/B, ^
tivity. We use the unit vectors B
v = v/v, and the
arc length parameter s = v t. Particle motion in a
magnetic field conserves the speed v and thus also the
mass m = m(v). This implies that dt = v1 ds and dp =
d(mv^v) = mvd^v, so dp/dt = mv2d^v/ds. Furthermore, qv ^ and we finally obtain
B = qvB^v B,
A06216
ð13Þ
Inserting the rest energy m0c2 for the particle of interest
(protons, 938 MeV; electrons, 0.511 MeV) yields handy
conversion formulas. The resulting rigidity-to-energy con-
3 of 13
A06216
VOGT ET AL.: PARTICLES IN THE PALEOMAGNETOSPHERE
Figure 1. Rigidity to kinetic energy conversion curves for
protons (solid line) and electrons (dashed line).
version curves for protons and electrons are shown in
Figure 1.
[17] Particle spectra in the form of differential fluxes or
distribution functions are generally quite complex, and their
information content can be rather difficult to comprehend.
Therefore the shielding efficiency of the geomagnetic field
is often condensed in a few meaningful cutoff parameters
such as the cutoff rigidity introduced by Størmer [1955]. In
Størmer theory, particles at a given location with rigidities
lower than the local cutoff value cannot escape the Earth’s
magnetic field (forbidden orbits). The vertical cutoff rigidity
PCV refers to trajectories that arrive at a given location from
the radial direction. However, as neutron monitor measurements and numerical trajectory calculations show, the two
classes of particle orbits are generally separated by a rigidity
band (the so-called cosmic ray penumbra) rather than a
sharp threshold value. The effective cutoff rigidity is then
understood as an appropriate average of the edge values of
the transition band. For details, the reader is referred to the
review of Smart et al. [2000].
Earth, whereas the outer magnetosphere (bow shock, magnetosheath, magnetopause, magnetotail) can be considered
stationary in the solar wind frame of reference. Viewed from
the Earth’s surface, the outer magnetosphere is changing
in the course of day even during geomagnetically quiet
periods. Accordingly, energetic particle populations reaching
the Earth’s atmosphere consist of a high-rigidity component
that does not change with daily rotation and, at high geomagnetic latitudes where cutoff rigidities are much smaller
than in the equatorial region, a diurnally modulated component with low rigidities. The dividing line between the
diurnally varying low rigidity population and the invariant
high rigidity component is not well defined. Experimentally
observed daily variations by Fanselow and Stone [1972]
indicated that the separation occurs at proton energies of a
few tens of MeV (equivalent to rigidities of a few hundred
MV). Numerical trajectory calculations [e.g., Flueckiger and
Kobel, 1990; Smart et al., 1999; Smart and Shea, 2001] put
the threshold a bit higher (at rigidities of about 1 GV),
especially during geomagnetically disturbed periods. See
the review by Smart et al. [2000] for more information.
3.1. Magnetopause Scaling
[20] To a good approximation, the magnetopause is
characterized by a balance of the solar wind ram pressure
psw on the sunward side of the boundary and the magnetic
pressure on the earthward side [Martyn, 1951; Siscoe and
Chen, 1975]. For the dayside magnetopause stand-off
distance, the pressure balance condition yields the scaling
1=6
relation Rmp / M1/3psw which means that the system
length scale Rmp varies moderately with the dipole moment
M. The magnetic field value at the magnetopause, however,
is entirely controlled by solar wind parameters and scales
1=2
as Bmp / psw . For the product of Bmp and Rmp, we thus obtain
1=3
BmpRmp / M1/3psw .
[21] The self-similarity assumption now means that the
magnetospheric field behaves as in equation (8) with the
subscript ‘‘0’’ replaced by ‘‘mp’’, and particle orbits are
similar if
Bmp Rmp
~ 1=3 ~
P ¼ P*
¼ P* M
p1=3
sw ;
*
*
Bmp Rmp
3. Dipolar Paleomagnetospheres
[18] The present-day magnetosphere of the Earth can be
characterized as an ‘‘equator-on’’ configuration in the sense
that the dipole axis is quasi-perpendicular to the solar wind
flow direction. Assuming that the overall configuration
changes in a self-similar way with the magnitude of the
dipole moment M, Siscoe and Chen [1975] and later Vogt
and Glassmeier [2001] as well as Glassmeier et al. [2004]
derived scaling relations for the location of magnetospheric
boundary layers and for the strength of major magnetospheric and ionospheric current systems. Here we pursue the
same approach and look at variations of particle properties
with changing dipole moment M. Reference values of the
present-day magnetosphere carry the superscript *.
[19] Before we start with the mathematics, it is useful to
recall the overall magnetospheric configuration (see also
textbooks on space plasma physics, e.g., Baumjohann and
Treumann [1997], Kallenrode [1998]) and to distinguish
between its outer part and its inner part. The inner part
(plasmasphere and radiation belt) is corotating with the
A06216
ð14Þ
according to equation (10) and the definition of rigidity. Here
~ = M/M* and ~
M
psw ¼ psw =p*sw where the reference parameter
p
sw is not fixed but can assume a range of values. The
corresponding dependence of kinetic energy on the dipole
moment and the solar wind ram pressure is obtained from
the rigidity-to-energy conversion formulas (13) and (12):
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
h
i2 ~ 2=3 ~
w¼ 1þM
1 þ w* 1 1:
p2=3
sw ð15Þ
Here w is kinetic energy expressed in units of the
rest energy: w = W/(m0c2). In the limiting cases W <<
m0c2 (P / W1/2) and W m0c2 (P / W + m0c2 W), we
can approximate W / M2/3 and W / M1/3, respectively.
[22] To illustrate the physical meaning of the similarity
approach and the mapping relations obtained above,
4 of 13
A06216
VOGT ET AL.: PARTICLES IN THE PALEOMAGNETOSPHERE
consider a particle of rigidity P* in today’s magnetospheric field B* coming from the interplanetary medium
and reaching a sphere at geocentric distance r = aR*mp
* ).
(i.e., at the fraction a of the magnetopause distance Rmp
In the paleofield B and using properly scaled length units,
~ 1=3 ~
a particle of rigidity P ¼ P* M
p1=3
sw would follow the
same orbit and thus reach the sphere at r = aRmp, i.e., at
the same fraction a of the magnetopause distance. In other
words, particle transmission in the field B* for particles of
rigidity P* or, equivalently, kinetic energy W* is the same
as for particles of rigidity P in the paleofield B. We can
now follow the physical reasoning explained in section 2.1
and map differential particle fluxes according to
J * W * ; r ¼ aR*mp
J W ; r ¼ aRmp
¼
J1 ðW Þ
* W*
J1
ð16Þ
where W = m0c2w and W* = m0c2w* are related through
equation (15), and J1 (W) = J (W,r!1) is the differential
particle flux in interplanetary space.
[23] The mapping relations can be used in several ways to
describe particle populations in the paleomagnetosphere
using particle characteristics in the present-day magnetospheric configuration and also to save computational resources in numerical particle tracing schemes.
3.1.1. Solar Wind Variations
[24] Since the system length scale is proportional to
(M 2/psw)1/6 [e.g., Martyn, 1951; Siscoe and Chen, 1975],
the size of the magnetosphere does not change if M2/psw =
constant. For a reduced dipolar paleomagnetosphere at
average solar wind conditions psw, we can identify a matching present-day configuration as the result of an increased
solar wind ram pressure p*sw ¼ psw (M*/M) 2 , hence
~ 2 . A particle of rigidity P* during such periods of
~psw ¼ M
enhanced p*sw follows the same trajectory as a particle of
1=3
~ 1=3 ~
~ 1=3 ðM
~ in the
~ 2 Þ1=3 ¼ P* M
rigidity P ¼ P* M
¼ P* M
psw
paleomagnetosphere. The differential particle flux at energy
W into the Earth’s upper paleoatmosphere at altitude h can be
obtained as
J ðW Þ
1
:
J ðW ; r ¼ RE þ hÞ ¼ J * W * ; r ¼ RE þ h
* W*
J1
conditions), then a = k/10 in equation (16), which yields
the particle spectrum at the Earth’s surface in a paleofield of
dipole moment M = M*/k3.
3.1.3. The Low Rigidity Population
[26] The energy mapping relation (15) allows the discussion of characteristics of the low rigidity population modulated by the outer magnetosphere as described above.
Considering solar protons with typical energies in the range
of 1 – 100 MeVs, we can safely assume that W* m0c2 and
~ 2/3. For example, a
approximate equation (15) as W = W* M
dipole-moment decrease to one third of its present value
implies that the characteristic energies are reduced by more
than a factor of two. Since solar particle events can have
rather steep spectra with power law exponents of typically
g 3 but up to g 5 or more [Mewaldt et al., 2005],
the differential particle flux at low rigidities in the outer
paleomagnetosphere can be much larger than today.
3.1.4. Numerical Particle Tracing Schemes
[27] Numerical trajectory integration schemes may also
take advantage of the magnetopause scaling approach used
here when particles are launched from interplanetary space
and monitored as they pass different spherical surfaces on
their way toward the Earth [e.g., Stadelmann, 2004].
3.2. Dipole Field Scaling
[28] The solar wind influence on the geomagnetic field
goes down with decreasing geocentric distance, and close to
the Earth, the magnetosphere is dominated by the dipolar
component of the internal field. Since the field-line topology of a dipole field does not change if the dipole moment
is reduced from M* to M, the two configurations are related
through
B* ðrÞ BðrÞ
¼
:
BE
B*
ð18Þ
E
Here B E and B E* are surface values taken at the
geomagnetic pole. In equation (8), we identify R0 = RE =
R*0 , B0 = BE, and B*0 = BE* to conclude that particle orbits
are similar if
BE
M
~
¼ P* M
P ¼ P*
¼ P*
M*
B*E
ð17Þ
Here RE denotes the radius of the Earth. Because of the
scaling law P / M, cutoff rigidities in the paleomagnetosphere can be significantly lower than today, and not only
GCRs but also particles from the upper end of the SEP
spectrum may precipitate into the middle atmosphere at
midlatitudes. This issue will be addressed in more detail in
section 3.2.
3.1.2. Particle Measurements on Spacecraft
[25] If the solar wind ram pressure is not sufficiently
modulated to reduce the extent of the magnetosphere as a
corresponding reduction of the dipole moment would do,
particle orbit scaling with magnetopause parameters offers a
possibility to deduce paleomagnetospheric particle spectra
from spacecraft data. For example, suppose such observations yield particle spectra in today’s magnetosphere at a
geocentric distance of k Earth radii and the dayside magnetopause resides at R*mp = 10 RE (average solar wind
A06216
ð19Þ
according to equation (10). Combining this rigidity-scaling
relation with the conversion formulas (12) and (13) yields
the following expression for the normalized kinetic energy
w = W/(m0c2) of similar particle orbits:
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
h
i2 ~ 2 1 þ w* 1 1:
w¼ 1þM
ð20Þ
Since rigidity scales directly with dipole moment (as for
~ 2 discussed in the previous
the special case ~
psw ¼ M
subsection), differential particle flux mapping is given by
equation (17).
[29] We are interested in identifying the geographic locations and the dipole moments for which particles in the
energy range 1– 100 MeV can penetrate the geomagnetic
field to cause ionization in the middle atmosphere. Spatial
distributions of cutoff rigidity values in the present-day field
5 of 13
A06216
VOGT ET AL.: PARTICLES IN THE PALEOMAGNETOSPHERE
Figure 2. Contours of the proton (solid lines) and electron
(dashed lines) paleo-cutoff kinetic energy in the range 1 –
100 MeV as a function of relative dipole moment and cutoff
rigidity values in the geomagnetic field of today.
are available for a broad range of geomagnetic activity
levels from neutron monitor measurements and numerical
trajectory calculations [e.g., Smart et al., 2006], so the
desired information can be deduced if kinetic energy W in
the paleomagnetosphere is expressed in terms of the pres~.
ent-day rigidity P* and the relative dipole moment M
~
Inserting the scaling law P = P* M into the general rigidity
to energy conversion formula (13) yields the function W =
~ ,P*) displayed in Figure 2. If a cutoff rigidity value at
W(M
a location in the present-day geomagnetic field is known,
the figure shows whether or not particles in the energy range
1 – 100 MeV may penetrate the paleofield and arrive at the
given location.
[30] For a pure dipole field, analytical formulas for cutoff
rigidities were derived by Størmer [1955] (see also Smart
and Shea [2005]), who demonstrated that cutoff rigidities
scale linearly with the dipole moment M. The vertical cutoff
rigidity Pcv at the Earth’s surface can be expressed as
Pcv
M
~ cos4 b
cos4 b ¼ 14:5M
¼ 14:5
GV
M*
A06216
field lines. Hence in order to penetrate the Earth’s magnetosphere and to find a path from space down to the
atmosphere, an adiabatically behaving particle has to move
along an open geomagnetic-field line, i.e., a terrestrial field
line connecting to interplanetary space. The impact areas of
the adiabatic population can be found by studying the
topology of geomagnetic-field lines.
[32] Open field lines in dipolar magnetospheres emanate
from regions around the geomagnetic poles usually referred
to as the polar caps. With b 0 denoting the geomagnetic
latitude of the open field line (polar cap) boundary, Siscoe
and Chen [1975] derived the scaling relation cosb 0 / M1/6
for a dipolar paleomagnetosphere that is subject to selfsimilar scaling with Rmp / M1/3, and they gave b*0 = 72° as
an average reference value for the present-day magnetosphere. However, this picture is incomplete because the size
of the polar cap depends on the field line merging efficiency,
which in turn is mainly controlled by the north-south
component Bz of the interplanetary magnetic field (IMF),
and the polar cap extends in latitudinal direction for strong
southward IMF (negative Bz). The dependence of polar cap
size on both the dipole moment and the north-south component of the IMF was quantified by Zieger et al. [2006b] on
the basis of a large number of MHD simulations. For strong
negative Bz, the polar cap size increases considerably faster
with the decreasing dipole moment than in the self-similar
model.
[33] As explained in section 2, collisionless trajectories of
particles moving in time-independent magnetic fields are
phase space contours of the particle distribution function f
and also of the differential particle flux J. Since adiabatically moving particles follow the magnetic field, the differential particle flux J in the adiabatic regime on open field
lines is expected to be the same as in interplanetary space
although the magnetic field close to the Earth is much
stronger than in the solar wind. The particles that are
reflected at parallel magnetic field gradients reduce the
differential particle flux J at the same rate as the cross
ð21Þ
where b denotes geomagnetic latitude. If we combine this
formula with equation (13), the cutoff energy is expressed in
terms of the relative dipole moment and the latitude to yield
~ ,b) displayed in Figure 3. For small
the function W = W(M
dipole moments, large parts of midlatitude middle atmosphere can be accessed by protons at the upper end of the
energy range 1 – 100 MeV.
3.3. Adiabatic Population
[31] The outer magnetosphere has not much effect on the
motion of high-rigidity particles in the GV range and above,
and the dipole scaling carried out in the previous subsection
is appropriate. For particles at the low end of the rigidity
spectrum where the gyroradii r = P/B are small enough to
consider the field variations during a gyrocycle as first-order
perturbations, particle dynamics is constrained by adiabatic
invariants such as the magnetic moment, and particles are
essentially gyrating around and moving along magnetic
Figure 3. Energy cutoff levels in a dipole field: Contours
of the proton (solid lines) and electron (dashed lines) paleocutoff kinetic energy in the range 1 –100 MeV as a function
of relative dipole moment and geomagnetic latitude.
6 of 13
A06216
VOGT ET AL.: PARTICLES IN THE PALEOMAGNETOSPHERE
A06216
Figure 4. Superposition of an axisymmetric dipole-quadrupole magnetic field (representing the Earth’s
core field) and a uniform field (representing the interplanetary magnetic field) to illustrate the
geomagnetic field line topology. Left: Dipolar core field, resulting in two polar caps of equal size. Right:
Mixed dipole-quadrupole case with the two contributions being equal at the Earth’s pole in the Northern
hemisphere.
section of the magnetic flux tube increases, so J does indeed
remain constant.
4. Nondipolar Paleomagnetospheres
[34] Reversals of the geomagnetic field are likely to
produce significant nondipolar core field components. In
paleomagnetospheric research, higher-order multipoles
were considered in several theoretical studies, and the
case of an internal quadrupole field received special
attention. Auroral zones in a quadrupolar magnetosphere
were addressed by Siscoe and Crooker [1976], and the
quadrupolar paleoionosphere was studied by Rishbeth
[1985]. Two-dimensional analytical models of quadrupolar
paleomagnetospheres were constructed by several authors
[Biernat et al., 1985; Leubner and Zollner, 1985; Starchenko
and Shcherbakov, 1991]. In a highly idealized spherically
symmetric geometry, magnetopause shielding of multipoles
of even higher order was studied by Willis et al. [2000].
Trapped particle populations in quadrupolar paleomagnetosphere were investigated by Vogt and Glassmeier [2000].
Stadelmann [2004] developed a potential field model that
allows the study of the effect of magnetopause currents on the
overall paleomagnetospheric configuration for arbitrary
combinations of dipolar and quadrupolar core fields. MHD
simulations of the quadrupolar paleomagnetosphere were
carried out by Vogt et al. [2004].
[35] To quantify energetic particle characteristics in
paleomagnetospheres with significant nondipolar core field
components, we first generalize the polar cap scaling
relation to include a zonal quadrupole contribution. This
approach allows the identification of entry regions for the
adiabatic (low rigidity) particle population in mixed dipolequadrupole configurations. In MHD simulations of more
complex geometries, such regions can be determined
through tracing of open field lines from interplanetary space
to the upper atmosphere. The high rigidity particle population in higher-order axisymmetric multipoles can be studied
using generalized Størmer theory. For complex core field
topologies that may occur during polarity transitions, numerical particle orbit integration in the potential field model of
Stadelmann [2004] allows the quantification of high rigidity
access regions and cutoff parameters.
4.1. Generalized Polar Cap Scaling
[36] In a dipolar magnetosphere, the open field-line
region around a geomagnetic pole changes with solar wind
parameters and with the Earth’s dipole moment. The size
of such a polar cap is determined basically by the amount
of magnetic flux on open solar wind field lines on the one
hand and by the magnetic field strength at the Earth’s
surface on the other hand. Before we introduce a generalized model of polar cap scaling that takes into account
an axisymmetric quadrupole contribution, we look at the
superposition of a dipole-quadrupole (Earth’s core) field
and a homogeneous (solar wind) magnetic field to illustrate what happens to the field topology. If in the dipoleonly case the direction of the homogeneous magnetic field
is chosen to be parallel to the dipole moment, we obtain
an open dipolar configuration characterized by two polar
caps of equal size as shown in the left panel of Figure 4.
Note that reversing the direction of the interplanetary
magnetic field would result in a closed-type magnetosphere where field lines emanating from regions around
the poles trace back to the Earth’s surface rather than
connecting to interplanetary space. In comparison with the
open dipolar configuration, a significantly larger exposure
of a polar region to open solar wind magnetic field lines is
present in the mixed dipole-quadrupole case shown in the
right panel of Figure 4. At one of the geomagnetic poles,
the quadrupole field and the dipole field point into
opposite directions, which leads to a decrease in the total
field strength, so the open solar wind flux has to be
distributed over a larger surface area. At the opposite pole,
the total field strength increases, and the polar cap
becomes smaller in size than without the quadrupolar
contribution.
[37] The two parts of the core field are conveniently
quantified using their surface values at the geomagnetic
North pole relative to the present-day reference value BE*.
We write d BE* for the dipole contribution to the surface
value at the North pole, and kBE* for the quadrupole
contribution. To evaluate the open magnetic flux through
the Earth’s surface (r = RE), we have to consider the radial
7 of 13
VOGT ET AL.: PARTICLES IN THE PALEOMAGNETOSPHERE
A06216
A06216
than one fifth of the dipole contribution even if the two field
components were equal in size at the Earth’s surface. For
more moderate values, the expected error of this assumption
should be around 10% or less. The open solar wind flux can
thus be taken from the ‘‘dipole-only’’ scaling model where it
is balanced by the magnetic flux through the polar cap
characterized by a reference polar cap boundary colatitude
qdip
0 :
Ysw ¼ pR2E dBE* sin2 qdip
0 :
ð25Þ
The ‘‘dipole-only’’ polar cap reference boundary follows
dip
1/6
[e.g., Siscoe
the scaling relation sin qdip
0 ¼ cos b 0 / d
and Chen, 1975] or
Figure 5. Polar cap boundary latitudes as predicted by the
generalized dipole-quadrupole polar cap scaling model. The
boundary latitude is given as a function of e (strength of
the quadrupole field relative to the dipole field at the
geomagnetic pole) for three different values of the dipole
moment normalized by its present-day value: d = 0.1, 0.3, 1.
The present-day reference boundary latitude chosen in this
calculation was b*0 = 65° (corresponding to q*0 = 25°
colatitude) which stands for rather strong solar windmagnetosphere coupling conditions.
components of the dipole field and the quadrupole field
which vary with colatitude q as follows:
*
Bdip
r ¼ dBE cos q;
3 cos2 q 1
:
2
ð23Þ
In Appendix A, these expressions for Br are derived and
then integrated from the North pole (q = 0) to the polar cap
boundary at q = q0< p/2 to arrive at the open magnetic flux
Ypc in the North polar region for an axisymmetric multipole
field of arbitrary order n. The combined dipole-quadrupole
field considered here yields
Ypc ¼ pR2E BE* sin2 q0 ðd þ k cos q0 Þ:
ð24Þ
The same formula holds for the South polar region. Note
that in the latter case, the flux integral covers the colatitude
range from q = p to q = q0 > p/2.
[38] A generalized polar cap scaling relation can be
constructed if the amount of open solar wind magnetic flux
is known. Since a quadrupole field decreases faster with
distance than a dipole field, solar wind coupling processes
at the magnetopause are still dominated by the dipole part as
long as the quadrupolar contribution to the mixed dipolequadrupole magnetosphere is not too large. Note that even
for an extreme dipole moment decrease to about one tenth
of its present value, the dayside magnetopause stand-off
distance would still be at more than five Earth radii, and the
quadrupole field strength at this point would thus be less
ð26Þ
The open solar wind flux can thus be written as
Ysw ¼ pR2E d2=3 BE* sin2 q*0 :
ð27Þ
[39] As the present-day reference value q*0 depends on
solar wind parameters, it implicitly parametrizes the
strength of solar wind-magnetosphere coupling. As the open
solar wind magnetic flux given by equation (27) equals the
flux through the polar cap given by equation (24), we finally
obtain
ð1 þ e cos q0 Þ sin2 q0 ¼ d1=3 sin2 q*0
ð22Þ
and
Bquad
¼ kBE*
r
1=6
sin qdip
sin q*0 :
0 ¼ d
ð28Þ
where the parameter e = k/d measures the contribution of
the quadrupole field at the pole relative to the dipole field.
This is an implicit equation to be solved for the polar cap
boundary colatitude q0 in terms of the input parameters d
and e. To display such solutions, one may rewrite the
equation to obtain
ð1 þ e cos q0 Þ sin2 q0
sin2 q*0
!3
¼ d;
ð29Þ
and then plot selected contours of the expression on the lefthand side as was done in the preparation of Figure 5. The
quadrupolar contribution has a strong influence on the polar
cap boundary latitude, and the open field-line regions may
extend down to about 30° geomagnetic latitude for extreme
geomagnetic and solar wind parameters.
4.2. Generalized Størmer Theory
[40] The approach of Størmer [1955] to particle trajectory
classification in a dipole field resides on an exact invariant
that exists in general axisymmetric fields, namely, the
canoncical momentum P8 associated with the azimuthal
angle 8. Stern [1964] computed corrections to dipole field
cutoff rigidities due to axisymmetric multipole field of
higher orders by means of perturbation theory. Shebalin
[2004] classified the particle orbits in steady axisymmetric
multipole fields B = r [anPn(cosq)/rn+1] of order n and
used the 8 component of the vector potential to construct a
8 of 13
A06216
VOGT ET AL.: PARTICLES IN THE PALEOMAGNETOSPHERE
A06216
for the vertical cutoff rigidity as a function of radial distance
r and geomagnetic colatitude q in a multipole field of
order n:
nþ1 Pcv
116
BE
RE
sin qPn0 ðqÞ ðnþ1Þ=n
:
¼
ðn þ 1Þg c
GV nðn þ 1Þ B*E
r
ð31Þ
Here Pn0 ðqÞ is a short-hand notation for dPn(cosq)/dq.
[42] For a dipole field (n = 1), Shebalin [2004] computed
g c = 1, and we recover equation (21). In a quadrupole field
with azimuthal symmetry (n = 2), the critical value is g c =
0.5988, and the vertical cutoff rigidity can be expressed as
Pcv
BE
¼ 50:1
GV
B*E
Figure 6. Energy cutoff levels in an axisymmetric
quadrupole field: Contours of the proton (solid lines, large
contour labels) and electron (dashed lines, small contour
labels) paleo-cutoff kinetic energy as a function of relative
surface magnetic field strength at the pole and of
geomagnetic latitude. Contour levels are at 1, 10, and
100 MeV.
quasipotential that effectively constrains particle motion.
Here q denotes geomagnetic colatitude, and Pn is the
Legendre polynomial of order n. The quasipotential allows
to distinguish between forbidden and allowed regions in
configuration space on the basis of a control parameter g
that is essentially the normalized canonical momentum P8.
The separation between such regions occurs when g
assumes a critical value g c. Shebalin derived all critical
values for single axisymmetric multipole fields up to order
five (hecadecupole).
[41] In order to specify cutoff rigidities and energies in
nondipolar paleomagnetospheres, the condition g = g c has
to be rewritten using dimensional variables and physical
units. These computations are straightforward but rather
lengthy and will not be carried out in every detail here. The
main steps are as follows. We choose the surface magnetic
field strength at the geomagnetic pole BE to represent the
paleofield, and let BE* denote the polar surface value
obtained for the present-day dipole field. Shebalin’s coefficients an, the Gauss coefficients g0n, and BE are related
through BE = (n + 1)g0n= (n + 1)an/RnE. We are interested in
the vertical cutoff rigidity Pcv which allows the simplification of the equations by setting f_ = 0. We combine
Shebalin’s equations (7), (22), and (31) to write the condition g = g c in the form
rn sin q dPn ðcos qÞ an n=ðnþ1Þ
¼ gc :
nPcv
nþ1
dq
ð30Þ
The equation can be solved for Pcv. The expansion
coefficients an used by Shebalin are replaced by the more
familiar Gauss coefficients g0n = an/RnE and then by the
resulting magnetic field value BE at the North pole.
Numerical values of the present-day magnetic field allow
to normalize the equation and to write it in convenient
physical units. We finally arrive at the following expression
!
3
3=2
sin2 q cos q
;
ð32Þ
3 2
3=2
sin q cos q
pffiffiffiffiffi
:
2= 27
ð33Þ
RE
r
or, equivalently,
Pcv
BE
¼ 12
GV
B*E
!
RE
r
pffiffiffiffiffi
The factor 2/ 27 in the denominator is the peak value of
sin2qcosq, so the second version of the quadrupole cutoff
rigidity equation can be better compared with the dipole
case (21). The resulting regions on the globe that can be
accessed by 1– 100 MeV protons and electrons are shown in
Figure 6. In comparison with the cutoff latitudes in a dipolar
configuration (Figure 3), the area around the poles in an
axisymmetric quadrupolar paleomagnetosphere is smaller.
An additional entry region appears at equatorial latitudes
that can make a substantial contribution to the total impact
area as the surface element scales with sinq = cosb.
4.3. Open Field Lines in MHD Simulations
[43] MHD simulations have been successfully used to
simulate the interaction of the solar wind plasma with the
Earth’s magnetosphere on large scales in a fully selfconsistent way. No assumptions are needed for the size
and shape of the magnetosphere, like in case of many
analytical models. Although the microphysics of magnetic
merging is not included in the MHD models, the large-scale
magnetic field topology can be reproduced astonishingly
well in numerical simulations. To simulate a hypothetical
polarity reversal with mixed dipolar and quadrupolar internal magnetic field components, we use the MHD code
BATS-R-US [e.g., Powell et al., 1999; Gombosi et al.,
1998]. This fully three-dimensional simulation scheme is
extensively used in the space physics community, and it has
been duly validated with actual spacecraft measurements in
the Earth’s outer magnetosphere. In the paleomagnetospheric context, the BATS-R-US code was used by Zieger
et al. [2004] and Vogt et al. [2004] to study the equatorial
dipole case and quadrupolar configurations, respectively.
[44] Numerical simulations allow the study of open fieldline regions also in more general nondipolar paleomagnetospheres. In order to demonstrate the methodology, we
constructed a hypothetical mixed dipole-quadrupole core
field consisting basically of an equatorial dipole and a nonaxisymmetric quadrupole. The surface field strength of the
combined field was chosen to be about one tenth of the
9 of 13
A06216
VOGT ET AL.: PARTICLES IN THE PALEOMAGNETOSPHERE
Figure 7. Last closed field lines in a simulated paleomagnetosphere. Both panels refer to the same MHD
simulation in GSE coordinates (explained in the text). In
the upper panel, the x axis (Sun-Earth line) is pointing into
the figure plane and to the right, the y axis is into the
figure and to the left, and the z axis is pointing upward.
Rotating the configuration by 180 around the z axis yields
the lower panel where the x axis is out of the figure plane
and to the left, the y axis is out of the figure and to the
right, and the z axis is again pointing upward. The open
field-line regions on the Earth’s surface are exposed to the
adiabatic population of energetic particles.
present value, and the quadrupolar contribution was
slightly larger than the dipolar part. The combined field
has two poles on the Earth’s surface in more or less
antipodal positions close to the equatorial plane, and there
is a so-called magnetic separatrix between the two poles
approximately running along two opposite geographic
longitudes. Magnetic-field lines starting from one side
of the separatrix will end up in one pole, and those
A06216
starting from the other side will converge toward the
opposite pole. Using this particular internal magnetic field
and the average Parker spiral IMF of a relatively low
magnitude of 5 nT directed toward the Sun, we ran a
steady-state MHD simulation to find the open field-line
regions and the corresponding impact areas mapped to the
Earth’s surface. The inner boundary of the MHD simulation was selected to be at a radial distance of 1.5 Earth
radii, as the external field contribution can be safely
neglected within this radial distance, and consequently,
the total field can be replaced with the pure internal field.
We identified and plotted the so-called last closed field
lines at the boundary between open and closed field-line
regions using a three-dimensional field-line integration
scheme.
[45] The two panels in Figure 7 show a number of last
closed field lines viewed from two different sides of the
Earth in the geocentric solar ecliptic (GSE) coordinate
system, where the x axis points toward the Sun, and the
Earth’s rotation axis is in the (x,z) plane. In this particular
simulation, the rotation axis coincided with the z axis.
The last closed field lines define the size and shape of
the paleomagnetosphere in the given hypothetical case of
polarity reversal. The starting points of the last closed
field lines mark the open field-line regions on the
spherical surface at 1.5 RE, where the intensity of the
total magnetic field is plotted as a gray scale contour
map. These open field-line regions have been mapped
along the internal magnetic field to the Earth surface to
yield the impact area for adiabatic particles in the upper
atmosphere (not shown here).
[46] Interestingly enough, there is only one polar cap in
this magnetic field configuration, namely, the smaller quasicircular open field-line region close to the equatorial plane
(lower panel in Figure 7). The other polar region on the
opposite side of the Earth, where the total magnetic field is
much weaker, is actually a closed field-line region. The
banana-shaped open field-line region in the upper panel of
Figure 7 is, in fact, not a polar cap. It is a region opening up
along a part of the magnetic separatrix that roughly follows
a geographic longitude.
[47] On the basis of this case study, we can conclude that
in a general multipolar paleomagnetosphere, not only the
polar regions but also the vicinity of the magnetic separatrix
are potential regions of open field lines. Depending on the
geometry, i.e., the relative directions of terrestrial and
interplanetary magnetic field lines, magnetic merging can
take place at different parts of the magnetopause, resulting
in different open field-line regions at the Earth’s surface.
Open field-line regions can easily open and close up in the
paleomagnetosphere because of the regular diurnal rotation
of the Earth even if the average IMF direction remains
unchanged. Discussing the details of the diurnal variation in
our hypothetical paleomagnetosphere is outside the scope of
this paper.
4.4. High-Rigidity Population in a Simulated Reversal
[48] At the Technische Universität Braunschweig, a particle tracing scheme was combined with a potential field
model of the magnetospheric magnetic field by Stadelmann
[2004] to form the ‘‘Energetic Particle Orbits in Magnetospheres (EPOM)’’ package. The magnetic-field model can
10 of 13
A06216
VOGT ET AL.: PARTICLES IN THE PALEOMAGNETOSPHERE
A06216
identify such a field configuration at least for case studies.
Stadelmann [2004] was provided with the results of the
polarity transition simulation of Glatzmaier and Roberts
[1995]. At the time of the simulated reversal, multipoles of
higher order than the dipole dominated the magnetic energy
spectrum. The dipolar and quadrupolar Gauss coefficients
from this geodynamo simulation were taken as input
parameters for the EPOM potential field model, and energetic particle trajectories and fluxes were computed. Selected results of this case study are displayed in Figure 8. The
parts of the upper atmospheres that become accessible to
256 MeV protons increase significantly in size before the
transition. Energetic particles enter the equatorial region
already 1000 years before the dipole low. During the
reversal, basically the whole upper atmosphere is accessible
to particles in this energy range.
5. Conclusions and Outlook
Figure 8. Regions on the globe that become accessible to
256 MeV protons before and during a simulated polarity
reversal (see text). Inaccessible regions are indicated by
hatched areas. Top: 15,000 years before the reversal. Center:
1000 years before the reversal. Bottom: At the time of the
reversal.
handle arbitrary combinations of dipolar and quadrupolar
core fields. The dayside and nightside magnetopause are
assumed to be spherical and cylindrical, respectively. The
normal component of the magnetic field at the magnetopause can be chosen to model different levels of fieldline merging and solar wind-magnetosphere coupling.
Although the simplified magnetopause geometry and the
limitations of the potential field approach do not allow to
model the magnetic field topology in the outer magnetosphere in sufficient detail, the EPOM package provides
a reasonable description of the inner magnetosphere
where the high-rigidity particle component is affected
most. The particle-tracing scheme as such solves the
equations of motion of a large number of particles using
the Runge-Kutta method and finally computes the flux of
energetic particles into the upper atmosphere. For more
information on the EPOM package, the reader is referred
to Stadelmann [2004].
[49] Paleomagnetic records do not allow to unambiguously reconstruct the geomagnetic-field coefficients during
a polarity transition because of poor spatial coverage and
dating inaccuracies [e.g., Merril and McFadden, 1999].
Although computational resources are still insufficient to
operate in a realistic parameter regime, numerical geodynamo simulations of the transition process can be used to
[50] Energetic particle fluxes into the Earth’s upper and
middle atmosphere are efficiently reduced by the geomagnetic field that reaches out far into space. Variations
of the Earth’s core field on geological timescales can
drastically affect the geomagnetic shielding efficiency.
Dipolar paleomagnetospheres allow to find scaling relations for the incident particle fluxes in terms of geomagnetic and solar wind parameters. Cutoff parameter scaling
indicates that SEPs in the energy range of several tens of
MeV can reach midlatitudes if the Earth’s dipole moment
is considerably reduced. The adiabatic particle population
at the low-rigidity end of the spectrum that enters the
magnetosphere on open field lines can reach low latitudes if higher-order multipoles become important during
polarity transitions. Complex core field configurations
can be studied only through numerical modeling of the
magnetosphere. The methodology was demonstrated in
this paper, and more detailed studies are planned for the
future.
Appendix A: Magnetic Flux Through the Polar
Cap of a General Axisymmetric Multipolar Field
[51] Multipole expansions, spherical harmonics, and
Legendre polynomials are discussed in textbooks of mathematical physics and potential theory. The relationships
used in this appendix can be found, e.g., in the classical
book of Jackson [1963].
[52] In order to compute the magnetic flux Ypc through a
polar cap for a general axisymmetric magnetic field B = B
(r, q) where r is radial distance and q is colatitude, one has
to integrate the radial component on the Earth’s surface Br =
Br (r = RE, q) over the area from the pole q = 0 to the polar
cap opening angle q0:
Z
Ypc ¼
B dS ¼
2pR2E
Z
q0
Br sin qdq:
ðA1Þ
0
Note that the area element is given by dS = 2pR2E sinq dq ^r.
[53] An elementary axisymmetric multipolar field of
degree n can be written as B = rFn where the multipole
potential Fn = an Pn(cosq)/rn+1, and Pn is the Legendre
11 of 13
VOGT ET AL.: PARTICLES IN THE PALEOMAGNETOSPHERE
A06216
polynomial of degree n. The radial component of the
magnetic field is given by
Br ðr; qÞ ¼ @Fn ðn þ 1Þan
Pn ðcos qÞ
¼
@r
rnþ2
Br ðr ¼ RE ; qÞ ¼ Br;pole Pn ðcos qÞ
[55] In the case of a dipole field (n = 1, dP1(m)/dm = 1),
we obtain
2 dip
2
Ydip
pc ¼ pRE Br;pole sin q0 :
ðA2Þ
which means that Br varies with q only through Pn(cosq).
Note that at the pole, Pn(cos0) = Pn(1) = 1, hence we may
write
ðA3Þ
ðA4Þ
For a quadrupole field (n = 2, P2 = [3cos2q 1]/2),
Bquad
ðr ¼ RE ; qÞ ¼ Bquad
r
r;pole
3 cos2 q 1
:
2
ðA5Þ
[54] Inserting the expression (A3) into equation (A1)
yields
Ypc ¼ 2pR2E Br;pole
Z
q0
Pn ðcos qÞ sin qdq
ðA6Þ
0
for the magnetic flux through a polar cap extending from
q = 0 to q = q0. Substituting m = cosq and dm = sinqdq gives
Ypc ¼ 2pR2E Br;pole
Z
1
Pn ðmÞdm
ðA7Þ
m0
where m0 = cosq0. Since Legendre polynomials satisfy the
differential equation
dPn ðmÞ
d þ nðn þ 1ÞPn ðmÞ ¼ 0;
1 m2
dm
dm
ðA8Þ
or, equivalently,
1
2 dPn ðmÞ
d 1m
¼ Pn ðmÞdm;
nðn þ 1Þ
dm
ðA9Þ
the magnetic flux integral can be written as
Z
dPn ðmÞ
2pR2E Br;pole m¼1 d 1 m2
nðn þ 1Þ m¼m0
dm
m¼1
2pR2E Br;pole dP
ð
m
Þ
n
¼
1 m2
nðn þ 1Þ
dm m¼m0
2pR2E Br;pole dP
ð
m
¼ m0 Þ
n
¼
1 m20
nðn þ 1Þ
dm
dPn ðm ¼ cos q0 Þ
2pR2E Br;pole ¼
1 cos2 q0
nðn þ 1Þ
dm
2pR2E Br;pole
dPn ðm ¼ cos q0 Þ
2
¼
sin q0
:
nðn þ 1Þ
dm
Ypc ¼ ðA10Þ
ðA11Þ
For an axisymmetric quadrupolar field (n = 2, dP2(m)/dm =
3m), the magnetic flux through the polar cap can be written
as
2pR2E Bquad
r;pole
sin2 q0 3 cos q0
6
quad
¼ pR2E Br;pole sin2 q0 cos q0 :
Yquad
pc ¼
where Br,pole = Br(r = RE,q = 0) is the magnetic field at the
pole in radial direction. In the case of a dipole field (n = 1,
P1 = cosq),
dip
Bdip
r ðr ¼ RE ; qÞ ¼ Br;pole cos q:
A06216
ðA12Þ
[56] Acknowledgments. The German Research Council DFG provided financial support for JV, BZ, AS, and KHG, through the grants VO855/2 and GL-142/12, and for MBK, through KA-1297/2. BZ also
acknowledges support by the Hungarian Scientific Research Fund under
contract NI 61013. Simulation results were obtained using BATS-R-US,
developed by the Center for Space Environment Modeling, at the University of Michigan, with funding support from NASA ESS, NASA ESTO-CT,
NSF KDI, and DoD MURI. Simulation runs were carried out on highperformance computing facilities of the CLAMV at Jacobs University
Bremen.
[57] Wolfgang Baumjohann thanks Helfried Biernat & George Siscoe
for their assistance in evaluating this paper.
References
Baumjohann, W., and R. A. Treumann (1997), Basic Space Plasma
Physics, Imperial Coll. Press, London.
Biernat, H. K., N. I. Kömle, and H. I. M. Lichtenegger (1985), Analytical
two-dimensional model of a quadrupole magnetosphere, Planet. Space
Sci., 33, 45 – 52.
Biernat, H. K., N. V. Erkaev, T. Penz, H. Lammer, S. C. Manrubia, F. Selsis,
D. F. Vogl, and S. Mühlbacher (2002), Magnetic field reversals on earth:
Possible implications for the biosphere, in Proceedings of the Second
European Workshop on Exo/Astrobiology, ESA SP-518.
Crutzen, P. J., I. S. A. Isaksen, and G. C. Reid (1975), Solar proton events:
Stratospheric sources of nitric oxide, Science, 189, 457 – 459.
Fanselow, J. L., and E. C. Stone (1972), Geomagnetic cutoffs for cosmic
ray protons for seven energy intervals between 1.2 and 39 meV, J. Geophys. Res., 7, 3999 – 4009.
Flueckiger, E. O., and E. Kobel (1990), Aspects of combining models of the
Earth’s internal and external magnetic field, J. Geomagn. Geoelectr., 42,
1123 – 1136.
Flueckiger, E., R. Buetikofer, L. Desorgher, and M. R. Moser (2003),
Global cosmic ray cutoff rigidities over the past 2000 years, in Proc.
28th Int. Cosmic Ray Conf., pp. 4229 – 4232.
Glassmeier, K., J. Vogt, A. Stadelmann, and S. Buchert (2004), Concerning
long-term geomagnetic variations and space climatology, Ann. Geophys.,
22, 3669 – 3677.
Glatzmaier, G. A., and P. H. Roberts (1995), A three-dimensional selfconsistent computer simulation of a geomagnetic field reversal, Nature,
377, 203 – 209.
Gombosi, T. I., D. L. DeZeeuw, C. P. T. Groth, K. G. Powell, and P. Song
(1998), The length of the magnetotail for northward IMF: Results of
3D MHD simulations, in Physics of Space Plasmas, vol. 15, edited by
T. Chang and J. R. Jasperse, pp. 121 – 128, Mass. Inst. of Technol. Center
for Theoretical Geo/Cosmo Plasma Physics, Cambridge, Mass.
Jackman, C. H., R. D. McPeters, G. J. Labow, E. L. Fleming, C. J. Praderas,
and J. M. Russell (2001), Northern hemisphere atmospheric effects due to
the July 2000 solar proton event, Geophys. Res. Lett., 28, 2883 – 2886,
doi:10.1029/2001GL013221.
Jackman, C. H., M. T. DeLand, G. J. Labow, E. L. Fleming, D. K.
Weisenstein, M. K. W. Ko, M. Sinnhuber, and J. M. Russell (2005), Neutral
atmospheric influences of the solar proton events in October – November
2003, J. Geophys. Res., 110, A09S27, doi:10.1029/2004JA010888.
Jackson, J. D. (1963), Classical Electrodynamics, 98 – 101 pp., John Wiley,
Hoboken, N.J.
Kallenrode, M.-B. (1998), Space Physics: An Introduction to Plasmas and
Particles in the Heliosphere and Magnetospheres, Springer, New York.
Leubner, M. P., and K. Zollner (1985), The quadrupole magnetopause,
J. Geophys. Res., 90, 8265 – 8268.
12 of 13
A06216
VOGT ET AL.: PARTICLES IN THE PALEOMAGNETOSPHERE
Martyn, D. F. (1951), The theory of magnetic storms and auroras, Nature,
167, 92 – 94.
Merrill, R. T., and P. L. McFadden (1999), Geomagnetic polarity transitions, Rev. Geophys., 37, 201 – 226.
Mewaldt, R. A., C. M. S. Cohen, A. W. Labrador, R. A. Leske, G. M.
Mason, M. I. Desai, M. D. Looper, J. E. Mazur, R. S. Selesnick, and D. K.
Haggerty (2005), Proton, helium, and electron spectra during the large
solar particle events of October – November 2003, J. Geophys. Res., 110,
A09S18, doi:10.1029/2005JA011038.
Porter, H. S., C. H. Jackman, and A. E. S. Green (1976), Efficiencies for
production of atomic nitrogen and oxygen by relativistic proton impact in
air, J. Chem. Phys., 65, 154 – 167.
Powell, K. G., P. L. Roe, T. J. Linde, T. I. Gombosi, and D. L. DeZeeuw
(1999), A solution-adaptive upwind scheme for ideal magnetohydrodynamics, J. Comput. Phys., 154, 284 – 309.
Quack, M., M.-B. Kallenrode, M. von König, K. Künzi, J. Burrows,
B. Heber, and E. Wolff (2001), Ground level events and consequences
for stratospheric chemistry, in Proceedings of ICRC 2001, Copernicus
Gesellschaft.
Randall, C. E., D. E. Siskind, and R. M. Bevilacqua (2001), Stratospheric
NOx enhancements in the southern hemisphere vortex in winter/spring of
2000, Geophys. Res. Lett., 28, 2385 – 2388, doi:10.1029/2000GL012746.
Rishbeth, H. (1985), The quadrupole ionosphere, Ann. Geophys., 3, 293 –
298.
Rohen, G., et al. (2005), Ozone depletion during the solar proton events of
October/November 2003 as seen by SCIAMACHY, J. Geophys. Res.,
110, A09S39, doi:10.1029/2004JA010984.
Schröter, J., B. Heber, F. Steinhilber, and M. B. Kallenrode (2006), Energetic particles in the atmosphere: A Monte-Carlo simulation, Adv. Space
Res., 37, 1597 – 1601, doi:10.1016/j.asr.2005.05.085.
Shea, M. A., and D. F. Smart (2004), Preliminary study of cosmic rays,
geomagnetic field changes and possible climate changes, Adv. Space
Res., 34, 420 – 425.
Shea, M. A., D. F. Smart, and K. G. McCracken (1965), A study of vertical
cutoff rigidities using sixth degree simulations of the geomagnetic field,
J. Geophys. Res., 70, 4117.
Shebalin, J. V. (2004), Størmer regions for axisymmetric magnetic multipole fields, Phys. Plasmas, 11, 3472 – 3482.
Sinnhuber, M., J. P. Burrows, M. P. Chipperfield, C. H. Jackman, M.-B.
Kallenrode, K. F. Künzi, and M. Quack (2003), A model study of the
impact of magnetic field structure on atmospheric composition during
solar proton events, Geophys. Res. Lett., 30(11), 1580, doi:10.1029/
2002GL016798.
Siscoe, G. L., and C.-K. Chen (1975), The paleomagnetosphere, J. Geophys. Res., 80, 4675 – 4680.
Siscoe, G. L., and N. J. Crooker (1976), Auroral zones in a quadrupole
magnetosphere, J. Geomagn. Geoelectr., 28, 1 – 9.
Smart, D. F., and M. A. Shea (1972), Daily variation of electron and proton
geomagnetic cutoffs calculated for Fort Churchill, Canada, J. Geophys.
Res., 77, 4595 – 4601.
Smart, D. F., and M. A. Shea (2001), A comparison of the Tsyganenko
model predicted and measured geomagnetic cutoff latitudes, Adv. Space
Res., 28, 1733 – 1738.
Smart, D. F., and M. A. Shea (2005), A review of geomagnetic cutoff
rigidities for earth-orbiting spacecraft, Adv. Space Res., 36, 2012 – 2020.
Smart, D. F., M. A. Shea, E .O. Flückiger, A. J. Tylka, and P. R. Boberg
(1999), Changes in calculated vertical cutoff rigidities at the altitude of
the International Space Station as a function of magnetic activity, Proc.
26th Int. Cosmic Ray Conf., 7, 337 – 340.
Smart, D. F., M. A. Shea, and E. O. Flückiger (2000), Magnetospheric
models and trajectory computations, Space Sci. Rev., 93, 305 – 333.
A06216
Smart, D. F., M. A. Shea, A. J. Tylka, and P. R. Boberg (2006), A geomagnetic cutoff rigidity interpolation tool: Accuracy verification and
application to space weather, Adv. Space Res., 37, 1206 – 1217.
Solomon, S., D. W. Rusch, J. C. Gérard, G. C. Reid, and P. J. Crutzen
(1981), The effect of particle precipitation events on the neutral and ion
chemistry of the middle atmosphere: II. Odd hydrogen, Planet. Space
Sci., 29, 885 – 893, doi:10.1016/0032-0633(81)90078-7.
Solomon, S., G. C. Reid, D. W. Rusch, and R. J. Thomas (1983), Mesospheric ozone depletion during the solar proton event of July 13, 1982.
II—Comparison between theory and measurements, Geophys. Res. Lett.,
10, 257 – 260.
Stadelmann, A. (2004), Globale Effekte einer Erdmagnetfeldumkehr: Magnetosphärenstruktur und kosmische Teilchen, Dissertation, Technische
Universität Braunschweig.
Starchenko, S., and V. Shcherbakov (1991), Inverse magnetosphere, in
Doktorlady Akademii Nauk SSSR, 321, in Russian.
Stern, D. (1964), The vector potential and motion of charged particles
axisymmetric magnetic fields, J. Geophys. Res., 69, 2715 – 2719.
Størmer, C. (1955), The Polar Aurora, Clarendon, Oxford, U.K.
Svensmark, H. (2000), Cosmic rays and Earth’s climate, Space Sci. Rev.,
93, 175 – 185.
Svensmark, H., and E. Friis-Christensen (1997), Variation of cosmic ray
flux and global cloud coverage—A missing link in solar-climate relationship, J. Atmos. Sol.-Terr. Phys., 59, 1225 – 1232.
Usoskin, I. G., N. Marsh, G. A. Kovaltsov, K. Mursula, and O. G. Gladysheva
(2004), Latitudinal dependence of low cloud amount on cosmic ray
induced ionization, Geophys. Res. Lett., 31, L16109, doi:10.1029/
2004GL019507.
Vogt, J., and K.-H. Glassmeier (2000), On the location of trapped particle
populations in quadrupole magnetospheres, J. Geophys. Res., 105,
13,063 – 13,071.
Vogt, J., and K.-H. Glassmeier (2001), Modelling the paleomagnetosphere:
Strategy and first results, Adv. Space Res., 28, 863 – 868.
Vogt, J., B. Zieger, A. Stadelmann, K.-H. Glassmeier, T. I. Gombosi, K. C.
Hansen, and A. J. Ridley (2004), MHD simulations of quadrupolar
paleomagnetospheres, J. Geophys. Res., 109, A12221, doi:10.1029/
2003JA010273.
Willis, D. M., A. C. Holder, and C. J. Dais (2000), Possible configurations
of the magnetic field in the outer magnetosphere during geomagnetic
polarity reversals, Ann. Geophys., 18, 11 – 27.
Zieger, B., J. Vogt, K.-H. Glassmeier, and T. I. Gombosi (2004), Magnetohydrodynamic simulation of an equatorial dipolar paleomagnetosphere,
J. Geophys. Res., 109, A07205, doi:10.1029/2004JA010434.
Zieger, B., J. Vogt, A. J. Ridley, and K.-H. Glassmeier (2006a), A parametric
study of magnetosphere – ionosphere coupling in the paleomagnetosphere,
Adv. Space Res., 38, 1707 – 1712, doi:10.1016/j.asr.2005.04.077.
Zieger, B., J. Vogt, and K.-H. Glassmeier (2006b), Scaling relations in the
paleomagnetosphere derived from MHD simulations, J. Geophys. Res.,
111, A06203, doi:10.1029/2005JA011531.
K.-H. Glassmeier and A. Stadelmann, Institut für Geophysik und extraterrestrische Physik, Technische Universität Braunschweig, Braunschweig,
Germany.
M.-B. Kallenrode, Fachbereich Physik, Universität Osnabrück, Osnabrück,
Germany.
M. Sinnhuber and H. Winkler, Institut für Umweltphysik, Universität
Bremen, Bremen, Germany.
J. Vogt and B. Zieger, School of Engineering and Science, Jacobs
University Bremen (formerly International University Bremen), P.O. Box
750561, 28725 Bremen, Germany. ( [email protected])
13 of 13