Download Centered and Eccentric Geomagnetic Dipoles and their

REVIEWS
OF GEOPHYSICS,
VOL. 25, NO.
1, PAGES
1-16, FEBRUARY
1987
Centered and Eccentric Geomagnetic Dipoles and Their Poles, 1600-1985
A. C. FRASER-SMITH
Space,Telecommunications,
andRadioscience
Laboratory,StanfordUniversity,California
Using a unified approach, expressionsare derived for the various pole positions and other dipole
parametersfor the centeredand eccentricdipole models of the earth's magnetic field. The pole positions
andotherParameters
arethencalculated
usingthe 1945-i985International
Geomagnetic
Reference
Field Gauss coefficientsand coefficientsfrom models of the earth's field for earlier epochs.Comparison is
made between(1) the recentpole positionsand thosepertainingsince1600 and (2) the various theoretical
pole positionsand the observeddip pole positions.
1.
that
INTRODUCTION
In response to questions concerning the most recent positions of the various magnetic poles in the Arctic and Antarctic, where the Space, Telecommunications, and Radioscience
Laboratory is currently operating ELF/VLF noise measurement equipment [Fraser-Smith and Helliwell, 1985], I recently
computed the positions of the poles for the centered dipole
and eccentric dipole models of the earth's magnetic field, using
the Gauss coefficients for the nine available
main field models
of the International Geomagnetic Reference Field (IGRF) for
the Gauss
coefficients
are in Schmidt-normalized
form
(unlike the Gaussian normalization used by Jensen and Cain
[1962], for example).
Because the 1945-1985 IGRF coefficientsare predomi-
nantlygivento foursignificant
figures,
andthelarges•is given
to five figures, I will usually provide four significant figures for
the dipole parameters computed from the IGRF field models,
and I will assumethat the fourth figure is meaningful.
The International Association for Geomagnetism and
Aeronomy(IAGA) is primarily responsiblefor the IGRF, and
the yearsi945-1985,astabulatedby Barkeret al. [19863and it has been particularly active in recent years with revisionsof
Barraclough [1985-]. In the processof deriving the pole positions I also computed, for each IGRF field model, the scalar
moment and orientation of the centered and eccentricdipoles
and the;positionof the eccentricdipole. Further, in order to
provide some perspectiveon the likely changesin pole positions and other geomagneticdipole parametersover the next
few decades I extended the computations to representative
earlier years for which the necessaryGauss coefficientswere
available' the results of these computations, when combined
with those for the 1945-1985 IGRF data, give a comprehensive picture of the changesin pole positions and other dipole
parametersthat are likely in the near future. Since this updated information on pole positions and other properties of the
centered and eccentric dipoles does not appear to be readily
available and is of general interest, I present it here, along
with somedetails of the computations.
The eccentric dipole computations are based on formulas
originally derived by Schmidt!-1934-]and describedin English
past field models, the issuanceof new IGRF models for past
and current years,and the provision of Definitive Geomagnetic Reference Field (DGRF) models, the l,tter consistingessentially of IGRF models that have been revised and probably
will not be altered substantially in the future. The first IGRF
model was adopted by IAGA in 1968 for the main field at
epoch 1965.0 [Peddie, 1982] and the current, or "fourth generation," IGRF now includes IGRF models for 1945, 1950, 1955,
and 1960; DGRF
models for 1965, 1970, 1975, and 1980; and
an IGRF model for 1985 [Working Group 1, 1981; Barraclough, 1985; Barker et al., 1986]. The data in Table 1 are
taken
from
the DGRF
1980 and IGRF
1985 models.
IAGA's
activity is undoubtedly having a strong influence on studies of
the earth's magneticfield, and the fact that updated reference
fields are likely to be issuedmore regularly in the future than
hasbeenthe casein the pasthasinfluencedthis work.
The traditional approach in papers treating the centered
and eccentricdipole modelsof the geomagneticfield is to list
of thedipoleswithoutspecification
of the
by Bartels1-1936]
andChapman
andBartels[-•940].Theywere computedproperties
used by Parkinson and Cleary [1958], whose derivation of the
details of the eccentric dipole for epoch 1955 provided a
model for this work, and they require only the first eight
Gauss coefficientsin each spherical harmonic field model. To
illustrate, Table 1 lists the first eight Gauss coefficientsfor the
mathematical procedures that are involved in their derivation.
This approach saves space but makes it difficult for researchers interested in computing up-to-date values of magnetic fields on and above the earth's surface according to
1980 and 1985 IGRF
ence to the literature. It is, in fact, very simple to obtain the
centereddipole parametersfrom the sphericalharmonic representations, but the procedures are no longer well documented
and can be time consuming to retrieve. The eccentric dipole
parameters are more difficult to compute, and the procedures
appear never to have been completely documented. Further,
one of the best descriptionsof the eccentricdipole approach to
modeling the earth's magnetic field contains an error (seesection 3.1). I have therefore described the steps required to
models; a full list of the coefficients
through 1'0 orders (rn = n = 10) is given by Barl•er et al.
[1986] and Barraclough [1985]. In accordance with modern
practice the coefficientslisted in Table 1 are given in nanoteslas,and I similarly use SI units throughout the derivation
of dipole parameters,which necessitatessome small changesin
the original formulas [Schmidt, 1934; Bartels, 1936' Chapman
and Bartels, 1940]. It will be assumed throughout this work
either of the dipole modelsto do so without extensiverefer-
obtainthe dipoleparameters
fromthe Gausscoefficients
Copyright 1987 by the AmericanGeophysicalUnion.
Paper number 6R0586.
8755-1209/87/006R-0586515.00
that they may be quickly computed from future IGRF or
DGRF
field models.
Another problem faced by a nonspecialistdesiring to utilize
2
FRASER-SMITH:
CENTERED AND ECCENTRIC GEOMAGNETIC DIPOLES
TABLE 1. The First Eight Gauss Coefficientsin the 1980 and 1985
Field
Models
of the IGRF
will also be used here.
1980
n
m
gmn
1
1
2
2
2
0
1
0
1
2
--29,992
-- 1,956
-- 1,997
3,027
1,663
1985
hmn
gmn
hmn
--29,877
-- 1,903
--2,073
3,045
1,691
5,604
--2,129
--200
longitudesto be given a negative sign, and that convention
5,497
--2,191
--309
The geographicrectangularcoordinate systemx, y, z, also
shown in Figure 1, will not be used widely in this work, since
sphericalpolar coordinatesprovide a simpler representation
when spherical geometries are involved. However, the rectangular systemis the conventionalreferencefor the position
of the eccentric dipole. From the discussionabove it can be
seenthat the positive x axis points toward 0ø of longitude,the
y axispointsto 90ø eastlongitude,and the z axis pointsto the
north.
The units are nanoteslas.
the dipole models to obtain up-to-date values of the earth's
magnetic field involves the coordinate systemsin which they
are defined. It is not easy to obtain the magnetic field at a
given geographical position (which is likely to be the most
common requirement) from a simple listing of the dipole parameters. Changes of coordinate systemsare required (one
change, a rotation, for the centered dipole; two changes, a
rotation and a translation, for the eccentricdipole) that can be
time consumingand difficult for someonenot freshly acquainted with the procedures involved. In this work, in addition to
listing the dipole parameters and showing the current pole
positions, I document most of the stepsrequired to obtain the
magnetic field components at any geographical location from
either the centered or eccentric dipole models of the earth's
field.
Finally, it is well known that the earth's magnetic field is
undergoing a secular variation [e.g., Parkinson, 1982; Merrill
and McElhinny, 1983], and it is of course becauseof this variation that updated dipole field parameters are required from
time to time. The change can impact significantly upon the
choice
of locations
for certain
measurements
within
a decade
or two [e.g., Stassinopouloset al., 1984], which is well within
the professional lifetime of a scientist. Thus in addition to
providing up-to-date dipole parameters I have also endeavored to put the parameters into an historical perspectiveby
briefly indicating some of the changesthat have taken place in
the parameters over the last few centuries. Much has been
written on these changes[e.g., Adam et al., 1970; Barraclough,
1974; Dawson and Newitt, 1982] and on the changesthat have
taken place over larger time scales [e.g., McElhinny and Sen-
The other basic coordinate systemis a spherical polar coordinate systembased on the centered magnetic dipole. In this
system the field is symmetric about the axis of the dipole,
which, as indicated by the description,is located at the center
of the earth, and the position of a point P is given by (r, O, •),
where r is the same radial coordinate as in the geographic
system,O is the colatitude measuredfrom the centereddipole
axis in its extension through the northern hemisphereof the
earth (the centereddipole latitude, denoted by A, is given by
90ø -O), and ß is the longitudemeasuredeastwardfrom the
meridian half plane bounded by the dipole axis and containing the south geographicpole. A variety of coordinatesystems
are used in the literature to describethe geographicand centered dipole systems,so it is important to note the conventions involved here: The basic coordinate systems are both
sphericalpolar, and with the exceptionof the commonradial
coordinate r the geographicalcoordinatesare denoted by lowercasesymbolsand the centereddipole coordinatesby the
same symbolsin uppercase.
It is common for the coordinate pair (O, •) or equivalently
(A, •) to be referredto as the "geomagneticcoordinates"of a
point on the earth's surface [Schmidt, 1918, 1934; Chapman
and Bartels, 1940; Matsushita and Campbell, 1967; Parkinson,
1982] and for the two points where the axis of the centered
dipole crossesthe surfaceof the earth to be called the "geomagneticpoles."This restriction of the generalterm "geomagnetic" (that is, denoting "relative to the magnetism of the
N
B•'
anayake,1982]; my purposeis to indicatethe directionof the
changesthat are likely over the next few decades.
2.
CENTERED
z
CD Axis
DIPOLE
,
2.1.
y
Coordinate Systems
Two basic coordinate systems are used in this work. The
primary, or reference,systemis based on the earth's geographic coordinates. Some variation of choice is possible; I will
assumethat it is a geographicallybased sphericalpolar coordinate system with its origin at the center of the earth (assumed spherical),in which the position of a point P is given
by (r, 0, •p), where r is the radial coordinate, 0 is the polar
angle measuredfrom the north polar axis, and •b is the azirduthal angle, equivalent to the longitude, measured to the
east from the Greenwich meridian (Figure 1). Thus
180ø > 0 > 0ø, and 360ø _>•b > 0ø. The angle 0 is the colatitude and is related to the geographic latitude 2 through
with the earth's surface,representedby the sphere r • Re in this
figure, is the north CD pole (RE, On, •Pn)'N is the north geographic
0 = 90 ø-
pole, and P is a generalpoint.
2. It
is usual
for
southern
latitudes
and
western
x
I
I
Fig. 1. The geographically based spherical polar coordinate
system r, 0, •b that is used as a reference in this work for the CD
coordinate system. In the associated Cartesian system x, y, z the
positive x axis points to 0ø of longitude, the positive y axis points to
90 ø east longitude, and the z axis points to the north. The coincident
origins for the two systemsare located at the center of the earth, O.
Only the northern part of the CD axis is shown; its intersectionB
FRASER-SMITH'
CENTERED AND ECCENTRIC
GEOMAGNETIC
DIPOLES
3
earth") to the specialcaseof the centereddipole model of the
earth's field has disadvantages,as pointed out by Chapman
[1963], and in this work the problems pointed out by
Chapman are even more acute because of the use of two
different dipole models for the earth's field. I therefore build
on Chapman's suggestion(also see Matsushita and Campbell
[1967]) and, instead of "geomagnetic,"use "centereddipole"
(or CD) and "eccentricdipole" (or ED) to describequantities
relating to their respectivefield models.Thus the CD polesare
the intersectionsof the CD axis with the earth's surface, with
the north CD pole being the intersection in the northern
hemisphere.
The geographicand CD coordinatescan be related through
the use of the cosine and sine rules for spherical trigonometry,
as is shown by Chapmanand Bartels [1940] and Mead [1970],
in particular. To effect a transfer between the coordinate systems, it is necessaryfor the orientation of the magnetic axis of
the centered dipole to be specifiedin the geographic coordinate system. I will denote the orientation of that part of the
magnetic axis intersecting the earth's surface in the northern
hemisphereby 0,, 4•, (Figure 1) and of the part intersectingthe
surfacein the southern hemisphereby 0s, (ks,where 0s = 180ø
-0•, and 4•s= 180ø+ 4•. The distinction may seem trivial,
but it is a primary source of confusion in computations of the
earth's magnetic field from the dipole models because of the
confoundingcircumstancethat the southward directed pole of
the dipole is actually a north magnetic pole and the part of the
magnetic axis extending out from the north pole of the dipole
actually intersects the earth's surface in the southern hemisphere. It follows from the above choice of notation that the
coordinatesof the north CD pole are (Re, 0,,, •b,,),and for the
south CD pole they are (Re, 0s,Cks).
A usefulquantity in CD field computationsis the CD declination ½. It is an idealization of the conventional declination
used in geomagnetism,which is defined to be the angle between true north and magnetic north, taken to be positive
when magnetic north is to the east of true north. In CD caseit
is the (spherical) angle between geographic north and the
north CD pole, taken to be positive when the CD pole is to
the east of geographicnorth.
Applying the sine rule to the spherical triangle on the
earth's surface defined by the point P, the north geographic
pole, and the north CD pole (Figure 2), we obtain
sin 0
sin 0,
sin (9
- •
sin (180ø -- •)
sin (--½)
sin (& -- &.)
(1)
In addition, the cosinerule gives
Fig. 2. The sphericaltriangle usedto convert betweengeographic
and CD spherical polar coordinates. N representsthe north geographic pole, B is the north CD pole, and P is a general point with
geographiccoordinatesr (= Re), 0, •b.
above equationsprovide its CD coordinates(r, (9, tI)) and the
CD declination ½. Two different equations are given for each
of tI) and ½ in order to avoid the ambiguity in angle that
occurswhen an inversesine or cosineis evaluated for a possible angular range of -180 ø to + 180ø: For each value of the
argumentthere are two possibleangles(for example,cos0.9397 can be either 20ø or -20ø). The ambiguity is unimportant if a guide to the expectedvaluesis available (a world map
of CD coordinates, for example). However, if the computations are being conducted without such a guide, both the
inverse sine and inverse cosine should be computed, giving
two pairs of possibleangles;the correct value is the one angle
that is common to the two pairs. The ambiguity does not
occur for (9 in (3) becauseits range is restricted to 0ø-180ø.
The inverse transformation,from CD coordinatesto geographic, also follows from (1) and (2); the relevant equations
are
0 -- COSI[COSOncos(• .4-sinOnsin0 cos(180ø-
4•= 4•n4-cos-•[(cos(9 -- cosOncos0)/sinOnsin0]
(4)
4•= 4• + sin-• [sin (9 sintI)/sin0]
2.2.
Derivation of Centered Dipole Parameters
The parameters of the centered dipole model of the earth's
magnetic fields are specified completely by the first three
Gausscoefficients
gxo,gxx,hxX.The formulasrequiredfor the
derivation of the moment M and orientation 0•, 4•nof the
cos 0,,= cos0 cos0 + sin 0 sin 0 cos(-½)
cos0 = cosO. cos 0 + sin O. sin 0 cos(& - &.)
B,
(2)
cos0 = cos 0,,cos 0 + sin O. sin 0 cos(180ø - •)
north magnetic axis of the centered dipole in the geographically based sphericalpolar coordinate systemare
Bo2 = (g•o)2+ (g••)2 + (h•)2
From theseequations we obtain
cos0. = --g•ø/Bo
0 = cos-•[cosO.cos0 + sin O.sin 0 cos(& - &.)-[
(5)
tan ok.= h• •/g• •
ß = cos-•[-(cos 0 - cosO.cosO)/sinO. sinO]
where Bo, a referencemagnetic field (termed the "reduced
tI) - sin-•[sin 0 sin(4•- •bn)/sin
(9]
moment" by Bartels [1936], in a different system of units),
gives the dipole moment M through the equation
(3)
½ = cos-X[(cos0, - cos0 cosO)/sin0 sin (9]
½ = sin-xI-sin 0, sin(4•- 4•,)/sin(9]
Given any point P with geographiccoordinates(r, 0, 40, the
4•
M =•
BoRe3
(6)
/.to
where Re is the radius of the earth, which will be assumedto
4
FRASER-SMITH: CENTERED AND ECCENTRIC GEOMAGNETIC DIPOLEG
have a mean value of 6371.0 km, as specified by Geodetic
Reference System 1980 [International Union of Geodesyand
Geophysics,1980].
fieldcomponents
(Be),.,
(Be)
0,and(Be)
,, (hatis,thevertical
component (positive when directed outward), the geographic
north-south component (positive in the direction of increasing
Substitutingthe DGRF 1980 valuesof g2ø, g22, and h22 0, that is, when directed to the south), and the geographic
from Table 1, (5) gives
east-west component (positive when directed to the east). The
following procedure, based on a transform of geographic to
Bo - 3.057x 104nT
CD coordinates, makes this possible:
M--7.906
x 1022 A m 2
from
1. (5)
First,
andderive
(6), using
the dipole
the chosen
parameters
spherical
M (or
harmonic
Bo),0n and
repre•Pn
(7)
0n= 11.19ø
sentation of the earth's field.
•Pn-- --70.76ø
Similarly,substituting
the IGRF 1985valuesof g2ø, g2•, and
h22fromTable 1, (5) gives
Bo-- 3.044x 104nT
M-
2. Next, supposing the geographical location of the point
is (r, •, •p), where • = 90ø-- 0 is the latitude and •p the east
longitude, the CD colatitude (9 of the point is computed from
the expressionfor cos (9 in (2). For example, if the DGRF
1980 field model is used, the expressionfor cos (9 is
cos (9 = [0.9810 cos (90ø - •)
7.871 x 1022A m 2
(8)
+ 0.1941 sin (90ø - •) cos (•p + 70.76)]
on- 11.02ø
(11)
where appropriate substitutionshave been made from (7).
•Pn- -70.90ø
3.
The values of (9 and radial distance r are now substitu-
Given the above 1980 values of 0nand •Pn the following ted into either (9) or (10) to obtain the magnetic field quangeographic coordinates are obtained for the 1980 centered tities IBel,(Be),.,and (Be)o, which apply in the CD coordinate
dipole (geomagnetic) poles: north CD pole is 78.81øN, system.
70.76øW, and south CD pole is 78.81øS,109.2øE.
4. The field quantitiesIBeland (Be),.also apply in the geoSimilarly, from the 1985 valuesof On and •Pnthe geographic graphic coordinate system;(Be)o does not, but it can be re-
coordinates obtained for the 1985 centered dipole pole are
solvedinto thetwo geographic
components
(Be)oand(Be), by
north CD pole of 78.98øN, 70.90øW, south CD pole of 78.98øS, using
109.1øE.
(Be)o = (Be)o cos ½t
2.3.
Centered Dipole Magnetic Field
(•2)
The magnetic field components produced by the earth's
equivalent magnetic dipole take their simplestform in the CD
coordinate system, since the field is symmetric about the axis
and there is no dependenceon azimuthal angle. In terms of
the CD (or "geomagnetic") coordinates the centered dipole
approximation to the earth's magnetic field takes the form
IBel
-
•oM(3 cos20 + 1)2/2
4•rr
3
(Be),.
= --
2•oM cos O
(Be)
o= --
4rrr
3
(Be), = --(Be)o sin½
where ½tis the CD declination given by (3).
5. Finally, if required, the CD coordinates of the geographical point and the CD declination at the point can be
obtained by using the expressionsin (3).
If it is desired to extend the above procedure to obtain CD
estimatesof the conventional elementsof the earth's magnetic
field [e.g., Parkinson, 1982; Merrill and McElhinny, 1983], the
following further relationsare required:
(9)
•o M sin {9
Bo(3cos20 + 1)2/2
I = tan-2(2 cot (9) = tan-2 (2 tan A)
D=½
(13)
H = I(Be)01-I[(Be)o
2 + (Be),212
/21
V(or Z)= --(Be),.
(r/Re)
3
X = --(Be) o
2Bo cos (9
(Be)r---(r/Re)
3
IB•I
I = tan-2 [(Be),./(Be)o]
4•rr
3
where the negative signsresult from the inversion of the dipole
moment relative to the polar axis.
Substituting for M from (6),
IBel
-
F-
(10)
Y = (Be),
where F is the total magnetic intensity (always positive), I is
the inclinationor magneticdip (positivewhen (Be), is directed
toward the earth's center), D is the magnetic variation or decliFrom these latter equationswe see that the referencefield Bo nation (positive when the magnetic north is to the east of true
is simply the horizontal surface field at the CD equator (r =
north), H is the intensity of the horizontal component of the
earth's field (always positive), V (or Z) is the intensity of the
R e, (9 = 90ø).
Most users of the centered dipole approximation to the
vertical component of the earth's field (same sign as I), X is
earth's field will wish to enter the geographic coordinates of the north directed component of H (positive when directed to
the location in question in the appropriate formulas and the geographicnorth), and Y is the east directed componentof
obtain valuesof the total magneticfield IBel and the magnetic H (positive when directed to the geographiceast).
Bo sin (9
(Be)o=-(r/Re)3
FRASER-SMITH: CENTERED AND ECCENTRIC GEOMAGNETIC DIPOLES
8.6
i
i
i
i
i
I
i
5
of the dipole moment M with time, startingwith the value
givenby Gauss'coefficients
for epoch1835 and endingwith
I
the value given by the IGRF 1985 coefficients.In between
thosetwo extremesthe M valuesare also plotted for all the
IGRFs and DGRFs in the interval 1945-1980[Barraclough,
8.4 ß
1985], as well as the M valuesfor 1885 and 1922 accordingto
the sphericalharmoniccoefficientsderivedfor thoseepochsby
Schmidt [Schmidt, 1934; Chapmanand Bartels, 1940]. Table 2
lists the correspondingnumerical valuesfor M. Figure 3 clearly showsthe known decline of M with time [e.g., Merrill and
McElhinny, 1983]. A straight line has been fitted to the points
using the least squaresmethod, and its closefit showsthat the
decline was essentiallylinear with time over the interval covered by the display.
The data in Figure 3 suggestthat the dipole moment will
continue to decline in the near future at the rate given by the
(Am2)
8.2-
8.0-
7.8
1820
1860
1900
1940
1980
slopeof the leastsquaresfitted line, whichis -0.45 A m2 per
YEAR
Fig. 3. Variation of the dipole magnetic moment M with time
during the interval 1835-1985 A.D. The first data point is derived
from the original Gauss coefficientsfor the year 1835, and the final
point comesfrom the IGRF 1985 field model.
2.4. Secular Change in Dipole Parameters
It is interestingto make a brief historical comparison of the
above 1980 and 1985 IGRF centered dipole parameters with
those that follow from the Gauss coefficientsderived by Gauss
himselffor epoch 1835 [Chapman and Bartels, 1940] and those
derived by A. Schmidt, the geomagnetician who introduced
geomagnetic coordinates, for epoch 1922 [Schmidt, 1934;
Bartels, 1936]. First, from the coefficientsderived by Gauss
(epoch 1835) we obtain
Bo=3.31 x 10'•nT
M=8.56
x 1022Am
2
(14)
0,, = 12.2ø
century, or roughly a 5% drop each century. However, the
data in Figure 3 also show what appears to be an acceleration
of the rate of decline starting around 1975. To place this accelerated decline in a more historical perspective,Figure 4 extends the time scale of Figure 3 back to the year 1600 by
adding the dipole moments for eight epochsbetween 1600 and
1910 that result from the Gauss coefficientsderived by Barraclough 1-1974] (see Table 2 for the numerical values of M; the
g•O coefficientsin Barraclough'smodels for epochsbefore
1850 were derived by linear extrapolation from later coefficients, and thus the resulting values of M are not independent and would be expected to have a linear trend). The expanded set of points is still closely fitted by a straight line, but
it now has a slope of -0.385 A m2 per century, and the
apparent recent acceleration of the decline is seenmore clearly
to start around 1970. There is an interesting possibility that
the start of the accelerationof the decline in M may relate to
the magnetic "jerk" [Courtillot et al., 1978; Malin and Hodder,
1982] observedin 1970, but the relation must remain a specu-
-63.5 ø
TABLE 2.
and from the coefficientsderived by Schmidt (epoch 1922) we
CD Parameters for the Indicated Spherical Harmonic
Models of the Earth's Magnetic Field
obtain
Date
Bo=3.15 x 104nT
M = 8.15 x 1022 A m 2
(15)
0. = 11.5ø
-68.8 ø
Care must be taken in interpreting the changes in the
properties of the centered dipole that are implied by a comparison of the data in (7), (8), (14), and (15), since the magnetic
surveyson which the Gauss coefficientsare based have improved greatly over the years (Gauss's data were merely adequate for a first trial of his spherical harmonic analysis,
Schmidt's data depended heavily on surveys made by the
wooden vessel Carnegie, and the 1980 and 1985 IGRF data
benefit from satellite observations), and the changes may
relate more to our improved knowledge of the magnetic field
than to the secularchange.However, there are consistenciesto
the changes,which suggestthat they have some geophysical
significance.
To illustrate the consistencyin the changesand to place the
most recent changesin context, Figure 3 shows the variation
Model
x 1022 A m 2
øN
øE
1985
1980
IGRF
DGRF
7.871
7.906
78.98
78.81
289.1
289.2
1975
1970
1965
1960
1955
DGRF
DGRF
DGRF
IGRF
IGRF
7.938
7.972
8.004
8.025
8.049
78.69
78.59
78.53
78.53
78.55
289.5
289.8
290.1
290.5
290.2
1955
1950
1945
1922
1910
FL
IGRF
IGRF
Schmidt
Barr.
8.068
8.068
8.084
8.15
8.25
78.31
78.49
78.52
78.5
78.4
291.0
291.1
291.1
291.2
291.8
1890
1885
Barr.
Schmidt
8.36
8.36
78.7
78.7
294.8
290.5
1850
Barr.
8.50
78.7
296.0
1835
1800
1750
1700
1650
1600
Gauss
Barr.
Barr.
Barr.
Barr.
Barr.
8.56
8.65
8.84
9.00
9.17
9.38
77.8
79.2
79.9
81.5
82.7
82.7
296.5
302.3
307.3
312.9
319.2
318.2
FL denotes Finch and Leaton [1957], and Barr. is short for Barraclough [1974].
6
FRASER-SMITH' CENTERED AND ECCENTRIC GEOMAGNETIC DIPOLES
9,4
I
I
I
i
9.0
m
75 ø
M
(Am2)
8.6
o
./'....:•}';:. .'.::'?;'
:,.--.-:!':;'
-..... i!•
......-.-!::i.:},:•::.
X
z
-,,:,.
8.2-
1700
1800
'-'-v"
70 ø
x
©\
260 ø
I
1600
X
1900
270 ø
Fig. 6.
YEAR
280 ø
290 ø
300 ø
EAST LONGITUDE
2000
Comparisonof the locationsof the north CD pole and the
observeddip pole, 1830-1985 A.D.
Fig. 4. Variation of the dipole magnetic moment M with time
during the interval 1600-1985 A.D. Additional points from the field
models derived by Barraclough [1974] for the years 1600-1910 A.D.
have been incorporated in the data set illustrated in Figure 3. G
now apparentlymovingawayfrom Thule in the generaldirec-
indicates
tion of the north geographicpole. As already noted, the exact
the Gauss moment
for 1835.
positionof the CD pole given by the original Gausscoeflation until further data are available; another proposed
ficients for epoch 1835 must be treated with caution, even
thoughthe value of M givenby the coefficients
is completely
"jerk" around 1912 doesnot appear to have influencedthe
in accord with values of M derived for earlier and later epochs
(Figure 3); it is shownbecauseit is quite remarkablycloseto
Figure 5 showsthe positionsof the north CD pole for the positionsgivenby muchlater field models.
Figure 6 placesthe north CD pole data shownin Figure 5
variousepochs,usingthe sphericalharmonicdata that were
utilizedin the preparationof Figure 3, that is, data applicable into the appropriategeomagneticcontext.It will be recalled
to epochsfrom 1835 to 1985 (seeTable 2). It is commonly that the objectof usingthe CD modelfor the earth'smagnetic
decline of the dipole moment.
observedthat the positionsof the CD polesdo not vary much
with time [Merrill and Mcœ1hinny,1983], and the data in
Figure 5 clearlysupportthat observation.On the otherhand,
there is some progressivevariation in position, with the pole
80 ø
field is to have a simple and perhaps reasonably accurate
representation
for the earth'sfield.One testof the accuracyof
the CD model is its ability to reproducethe observed"magnetic" poles,that is, the actual measuredlocationswherethe
earth'smagneticfield is vertical,which I will hereafterrefer to
as the dip poles (or observeddip poles).Figure 6 showsthe
north CD polestogetherwith the actuallocationsof the north
dip pole as recordedsinceits discoverynear BoothiaPeninsula by J. C. Rossin June 1831 [Ross, 1834]. The coordinates
are taken from Dawson and Newitt [1982] and include the
1904.5 observationby Amundsen,the 1948.0 observationby
Serson and Clark, the 1962.5 observation by Dawson and
Loomer, and the 1973.5 observation by Niblett and
Charbonneau. To these I have added the 1984.4 position re-
portedby Canadianscientists
[seeNewitt, 1985;Newitt and
78 ø
Niblett, 1986]. It is clear from the data in Figure 6 that the
CD pole is only a very crudeapproximationto the observed
dip pole.
Figure 7 showsthe shift of the south CD pole from 1850
until the present.There are no significantgeographical
features in the vicinity of the CD pole except for the Soviet
station ¾ostok (-78.45 ø, 106.87ø).Consideringits motion
76 ø
relative to ¾ostok, the CD pole appears to have moved in
282 o
286 o
290 o
294 ø
298 o
EAST LONGITUDE
Fig. 5. Positionsof the north CD pole for the interval 1850-1985
A.D., using the IGRF/DGRF models for 1945-1985 and the Barraclough [1974] models for 1890 and 1850. The position given by the
original Gausscoefficients
for 1835is denotedby G. The CD pole has
been located in western Greenland for well over a century, but it has
now moved out into Nares Straight,which separatesEllesmereIsland
(on the left) from Greenland.
sucha way as to reduceits distancefrom the station until
about 1975, but it is now beginning to move away in the
generaldirectionof the southgeographic
pole.
3.
ECCENTRIC
DIPOLE
3.1. Derivation of the ED Position
Coordinates
Once it is decided to approximate the earth's field by a
magneticdipole not necessarilylocated at the geographic
FRASER-SMITH: CENTERED AND ECCENTRIC GEOMAGNETIC DIPOLES
/
_----- --- --- '-'-
102o
__--__-----/--/
7
whereBo is the referencefielddefinedin (5), and
Lo = 2gløg2
ø q-(3)l/2[gllg21q-hllh21]
76øS
L 1 = --gllg2ø + (3)l/2[gløg21
+ gllg22+ hllh22]
78øS
I
106o-----Ir
(17)
L 2 = --hllg2 ø q-(3)l/2[gløh21
-- h•lg22 q-gllh22]
E = (Loglø + L•gi • + L2hll)/4Bo2
1985
+ __+
+-t- •
I
I
The total shift of the dipole from the center is g, given by
110OE--
_
• __(•2 q-r/2q- •2)l/2RE
1945-H'
191e J
+1890 /
+
.....
.....
It followsfrom the aboveequationsthat the eccentricdipole is
completelyspecifiedby first eight Gausscoefficients.
Substitutingthe valuesof the 1980 Gausscoefficients
(Table
1) in (17), the following numerical valuesare obtained:
114
ø---- -
/
1850
/ +G
(18)
/
L o=8.887 x 107nT2
/
t
'"---118ø-._
_•/-
L x = --1.687 x 108nT2
(19)
Fig. 7. The southernequivalentof Figure 5, showingthe south
CD pole positionsin Antarctica for the interval 1850-1985 A.D.,
usingthe samefieldmodelsaswereusedfor Figure5. The locationof
L 2 = 1.063x 108nT2
E = -4.652
x 102 nT
the Soviet Antarctic station ¾ostok(¾O) is also shown.
which give r/= -0.06049, • = 0.03884, and • = 0.02671. The
shifts Ax, Ay, and Az in the x, y, and z coordinate directions
are therefore -385.4
center, the question then arises as to what criterion is to be
usedto judge the bestfit to the observedfield. The criterion
adopted by Schmidt[1934] and describedby Barrels [1936] is
to minimize the terms of secondorder in the potential used in
the spherical harmonic representationof the field. The eccentric dipole so obtained has the same moment as the centered
dipole and the same orientation of its axis, but in terms of the
geographicrectangular coordinate systemx, y, z (Figure 1) it
is locatedat a positionAx = r/Re,Ay = •Re, Az = •Re, where
the quantities r/, •, • can be derived from the Gauss coefficients, as described in the following paragraph. It might be
km, 247.5 km, and 170.2 km, respec-
tively, and the total distance shifted by the dipole is 6 =
488.6 km. The direction of the shift is given by
cos-x (170.2/488.6)
= 69.61ø,and •pd= 90ø + tan- I (385.4/
247.5)- 147.3ø, that is, it is toward the point 20.39øN,
147.3øE.This point is in the northwestPacific,at the northern
end of the Mariana
3.2.
Islands.
Secular Change in ED Position
If the IGRF 1985 Gauss coefficientsare substituted in (17),
the position parametersfor the eccentricdipole are found to
be Ax =-391.9
km, Ay = 257.7 km, Az = 178.9 km, and
notedat this point that the rectangularcoordinatedesigna- 6-- 502.0 km. This result suggeststhat the dipole is moving
tions usedby Schmidt[1934] and Barrels[1936] differ from away from the earth's center.Indeed, computationswith the
those now conventionally used,for example, the x axis is used completeset of !GRF and DGRF data for the interval 1945for what is now conventionally the z axis, and in the work by
1985 indicate that the dipole has been gradually drifting away
Chapman and Barrels [1940] this circumstancehas led to an
from the earth's center since 1945. To put this drift into per-
erroneousdesignationof the shifts Ax, Ay, and Az. When
referenceis made to the eccentric dipole model of the earth's
magnetic field, it is now generally understoodthat the Schmidt
spective,I have compute
d the positionof the dipole since
1600,usingthe sameGausscoefficientdata setsthat were used
to investigatethe secularvariation of the dipole moment M in
[1934] criterionand its resultingmathematical'
formulation section 2.4 (note that the same secular variation of dipole
are applicable,
e..ven
thoughothereccentric
dipolemodelsare moment applies in the case of the eccentricdipole, since the
possible[e.g., Bochev,1969a], and it is the Schmidt eccentric ED and CD moments are identical). The ED position data
dipole model that is describedin this work. There is not an
obtained from these computations, together with the correextensive
literaturetreatingtheeccentric
dipoleformalism;the spondingdistance6 from the earth'scenter,are listedin Table
major works are thoseby Schmidt[1934], Barrels[1936], and
Chapmanand Barrels[1940], togetherwith valuablecontributions by Akasofu and Chapman [1972], Ben'kot,a et al.
[1964] and James and Winch [1967]. Other relevant articles
include those by Vestine[1953], Parkinsonand Cleary [1958],
3, and the resultsare illustratedin Figures8, 9, and 10.
Figure 8 showsthe secularvariation of the distance6 of the
eccentricdipole from the earth's center. There appear to be
threedifferent
regimes
overthetimeinterval
covered
bythe
display'(1) a steadydeclineof 6 throughoutthe interval 1600Cole[1963],Kahleet al. [1969],Parkinson
[1982],andWallis 1800, (2) a steady increasefrom 1800 to around 1920, and (3)
et al. [1982].
'
an acceleratedsteadyincreasefrom 1920 until the present.As
The dimensionlesscoordinate quantities •, r/, and • are can be seen, the eccentric dipole is now farther from the
given by
earth's center than it has been at any other time since at least
1600; at roughly 500 km the distancei•sabout 7.8% of the
• = (Lo -- gløE)/3Bo
2
r/= (L1 -- gl 1E)/3Bo
2
• __(L2 _ hi 1E)/3Bo
2
earth's radius. On the basis of its recent trend we can expect
(16)
the distance 6 to continue increasing in the near furture at
what appearsto be an historicallysubstantialrate. Thus the
distance,already nearly twice its average value during the
FRASER-SMITH'
TABLE
3.
CENTERED AND ECCENTRIC
GEOMAGNETIC
ED Position Coordinates, as Measured in the
I
Geographic
Rectangular
Coordinate
System,
andthe
200
Distanceg of the Dipole From the Earth's Center
for the Indicated Spherical Harmonic Models of
the Earth's Magnetic Field
Date
DIPOLES
Model
Ax,
Ay,
Az,
g,
km
km
km
km
lOO
1985
IGRF
-- 391.9
257.7
178.9
502.0
1980
DGRF
-- 385.4
247.5
170.2
488.6
1975
1970
DGRF
DGRF
-- 378.6
-- 373.1
237.0
231.0
159.8
146.4
474.4
462.6
1965
1960
1955
1955
1950
1945
DGRF
IGRF
IGRF
FL
IGRF
IGRF
- 368.8
-- 366.3
-- 361.5
-- 366.8
--356.3
- 351.9
223.8
212.9
204.0
204.8
190.7
174.3
133.6
121.8
110.6
117.9
100.3
89.7
451.6
440.9
429.6
436.3
416.4
402.8
1922
Schmidt
- 324.4
107.0
39.1
343.9
1910
Barr.
- 325.1
88.8
39.3
339.3
1890
Barr.
- 311.8
67.1
1885
Schmidt
- 286.4
59.9
28.7
293.9
the geographicequatorial plane during the interval 1600-1985 A.D.
1850
1835
Barr.
Gauss
- 279.2
--278.4
Barr.
Barr.
Barr.
Barr.
Barr.
- 222.4
-236.6
-256.2
-280.1
-- 214.2
1.4
-65.2
- 21.7
- 51.7
- 107.3
-60.2
15.0
279.3
288.8
1800
1750
1700
1650
1600
4.8
-40.4
- 83.6
- 106.7
-99.1
- 161.8
- 105.8
As can be seen, the dipole was located below the plane for much of
the interval, but it is now at its greatest distance above the plane for
the last four centuries. The IGRF/DGRF models for 1945-1985 and
the field models of Barraclough [1974] for 1600-1910 were used for
this display.
-0.8
319.0
238.6
264.7
294.9
329.0
239.3
FL denotes Finch and Leaton [1957], and Barr. is short for Barraclough [ 1974].
interval 1600-1900, should continue to set new records for
some years to come.
The 1955 Finch and Leaton magnetic field model is included in the ED computationsreported here and in the previous
CD computations (seeTable 2) to provide a check against the
results of Parkinson and Cleary [1958], who used the Finch
and Leaton model. Comparing the results for •5, Parkinson
and Cleary report a value of "about 436 km" as compared
with g - 436.3 km in Table 3. The displacementof the dipole
is toward a point at 15.6øN, 150.9øE according to Parkinson
5OO
400
-lOO
-2oo
16oo
-
i
i
18oo
19oo
2000
YEAR
Fig. 9.
VariatiOn of the distance Az of the eccentric dipole above
and Cleary, while the data in Table 3 imply a displacement
toward 15.7øN, 150.8øE. The agreement between these numbers is close; further, it is as close as might be expected,since
the numerical values for the dipole moment and ED coordinates depend on the numerical value that is chosenfor the
earth's radius and Parkinson and Cleary do not document the
precisevalue usedin their work.
Figure 9 shows the variation of the distance Az, that is, the
distance of the eccentric dipole above the geographic equatorial plane, since 1600. For much of the interval the dipole
has been below the equatorial plane, but it moved above the
plane around the end of last century, and it is now at its
largest distance above the plane. On the basis of the trend
since 1900 it can be expectedto move to new record distances
above the plane over the next few decades.
Finally, Figure 10 shows the variation since 1600 of the
point of projection of the eccentric dipole position on the
geographic equatorial plane. The data prior to 1800 do not
show any steady trend, but the point of projection appearsto
have been moving steadily toward the western Pacific for
roughly the last 200 years.
3.3.
300
i
17oo
ED Axial
Poles
The eccentricdipole model for the earth's magnetic field
producestwo differentvarietiesof poles.The first of theseare
what I will refer to as the axial poles (the two points on the
ß
Ge
earth's surface where the ED axis intersects the surface). Be-
cause of the displacementof the eccentric dipole away from
the earth's center the ED axis and the ED magnetic field, in
ß
200
I
I
I
I
1600
1700
1800
1900
2000
:
YEAR
Fig. 8. Variation of the distance 6 of the eccentricdipole from the
earth's center over the interval 1600-1985 A.D. The distanceimplied
by the original Gauss coefficientsfor 1835 is denoted by G. Three
regimeshave been indicatedby straight lines. The IGRF/DGRF
models for 1945-1985 and the field models of Barraclough[1974] for
1600-1910 were usedprimarily for this display.
particular,are not perpendicularto the surfaceat the ED axial
poles.There are, however,two points where the ED magnetic
field is perpendicular to the surface,and I will refer to these
pointsasthe ED dip poles.In thissection,expressions
will be
derivedfor thepositions
of theaxialpoles.
We know that the axisof the eccentricdipoleis parallelto
the CD axis. This fact and the knowledge that the ED axis
passesthrough the point (Ax, Ay, Az) in geographicrectangular coordinates enable us to derive an expressionfor the ED
FRASER-SMITH: CENTERED AND ECCENTRIC GEOMAGNETIC DIPOLES
-400-
Re are first substituted in (22a) and (23a) to obtain the 4•
values for the two poles. These 4• values, together with the
!985•
1955,
•,•._1975
,ßJß
ß J1945
1910
1650
ß
z.00
1600
1965
given data, can then be usedin the appropriateexpressionfor
-300
'18y1922
0 to obtain the polar colatitudes.
1850
3.4. Recent Positionsof the ED Axial Poles
and Their Secular Change
-200
Table 4 lists the computed ED axial pole positions for all
the IGRF and DGRF field models and for the original Gauss
coefficientsfor epoch 1835. In addition, for comparison with
the results of Parkinsonand Cleary [1958], the pole positions
are also listed for the spherical harmonic field model of Finch
and Leaton [1957] for epoch 1955, the field model used by
-lOO
I
-100
'
'
100
0
• -•'-y (km)
200
300
Parkinsonand Cleary [1958]. Finally, to provide information
about their likely change over the next few decades,the pole
positions are tabulated for the earlier field models of Schmidt
[Schmidt, 1934' Chapman and Bartels, 1940] and Barraclough
100
t
x (kin)
(• :o o)
Fig. 10. Variation of the projection of the eccentricdipole's position on the geographicequatorial plane during the interval 1600-1985
A.D. Once again the IGRF/DGRF models for 1945-1985 and the
field modelsof Barraclough[1974] for 160.0-1910were usedprimarily
for the display.
axis in our basic geographic spherical polar coordinate
system. The ED axial poles are then found comparatively
simply by finding the points of intersection of the axis with the
surfacer - Re, representing
the earth.
The equation for a line passingthrough a point (Ax, Ay, Az)
in the geographicrectangularcoordinatesystemis
x - Ax
I
-
y-
Ay
m
-
z - Az
(20)
n
where 1, m, and n are the direction cosines of the line. Convert-
ing to geographicsphericalpolar coordinatesand substituting
1= sin 0. cos •b., m = sin 0. sin •b., and n = cos 0., which
follow from the known orientation of the dipole axis in geographic coordinates(Figure 1), we obtain
r sin 0 cos •b - Ax
sin 0. cos 4•.
9
=
r sin 0 sin •b- Ay
sin 0. sin •b.
=
r cos 0-
cos 0.
Az
(21)
as the equation of the ED axis.
Substitutingr = Re in (21) and carryingout the appropriate
algebraic manipulations, the following equations are obtained
for the points of intersection of the ED axis with the earth's
surface,that is, for the ED axial poles:
[1974].
Comparing the results of the axial pole computations for
the Finch and Leaton [1957] field model for epoch 1955 with
the results obtained for the same model by Parkinson and
Cleary [1958], Table 4 shows the north ED axial pole at
80.90øN, 275.6øE, whereas Parkinson and Cleary obtained
81.0øN, 275.3øE (84.7øW). There is similar close agreement for
the south poles.
The north ED axial pole is currently located in the seajust
off the northwest coast of Ellesmere Island, in the Canadian
Arctic.Figure11 showsits 1985position,togetherwith previous positions back to the year 1600. It has moved over a
greater distancein the time interval than has the CD pole. The
south ED axial pole is now located in a remote part of the
Antarctic continent, as shown in Figure 12. It is about 400 km
from Vostok along the line joining Vostok with Porpoise Bay.
Interestingly,it was probably located on the Ross Ice Shelf
prior to 1600.
3.5.
ED Dip Poles
As pointed out in section3.3, the ED magneticfield is not
perpendicular
to the earth'ssurfaceat the ED axialpoles,due
to the offset of the dipole from the earth's center. However,
there are two points where the ED magneticfield is perpendicular to the surface.One of these points is near the north
ED axial pole and the other near the south ED axial pole;
they will be called the north and south ED dip poles,respectively. Both dip poles are located on the great circle defined by
the intersectionof the plane containingboth the CD and ED
axeswith the earth'ssurface;they are separatedfrom their
corresponding ED axial poles by small angular distances
along the great circle, with the direction of the separation
being away from the local CD pole (the CD and ED poles all
lie on the great circle).At thesepoints there is enoughcurvature of the dipole field lines away from the ED axis to compensatefor the small angle made by the axis with the earth's
surfaceand thus to bring the field lines perpendicular to the
surface(Figure 13a).
It is not difficult to compute the geographic locations of the
ED dip poles, but the computations are involved, and ultimately, as we will see,the equation for the pole positionsmust
be solved numerically.The procedurethat was used here consistsof the following severalsteps:
In the first step a transform is made into the CD coordinate
system.The only significantfeature of this step is a change of
•b=tan-'[L(Re(Re
Az)
sin
4•.
tan
0.
--Az)
cos
•b.
tan
0.+
+AY
•x1 (22a)
O=sin-'
[.Ax
sin
ck"Ay
cøs
ck" (22b)
R e sin (•b. -- •b)
for the north ED axial pole, and
L(Re
+
•b.tan
tan0.0.- A
•x
•b
=tan-'
[('Re
+Az)cos
Az)
sin
•b.
y] (23a)
O=180ø-sin-Z[
Axsinck"--Aycøsck".]
,23b)
Re sin (•b. -- •b)
for the south ED axial pole. To use theseequationsto derive
the locationsof the poles,the quantitiesAx, Ay, Az, 0., 4•.,and
10
FRASER-SMITH: CENTERED AND ECCENTRIC GEOMAGNETIC DIPOLES
TABLE 4. Geographic Coordinates of the Axial and Dip Pole Positions for the Eccentric Dipoles
ResultingFrom the Indicated SphericalHarmonic Models of the Earth's Magnetic Field
Axial
North
Date
Model
Dip
South
North
South
Latitude, Longitude, Latitude, Longitude, Latitude, Longitude, Latitude, Longitude,
deg
øE
deg
øE
deg
øE
deg
øE
1985 IGRF
82.05
270.2
-- 74.79
118.9
82.64
204.3
--66.68
129.2
1980 DGRF
81.78
27i.2
--74.72
118.9
82.65
208.•
--66.88
129.2
1975
81.56
272.3
-74.72
!19.0
82.67
211.7
--67.16
129.3
!970 DGRF
81.40
•73.2
-74.70
119.2
82.70
214.3
-67.37
129.4
1965 DGRF
1960 IGRF
81.28
81.17
274.0
274.6
-74.73
--74.83
119.4
119.8
82.69
82.60
216.6
219.0
-67.62
--67.94
129.6
130.1
1955
DGRF
81.08
274.4
-74.98
119.6
82.47
220.8
-68.33
130.2
1955 FL
IGRF
80.90
275.6
-74.64
120.2
82.46
222.0
-67.88
130.6
1950 IGRF
80.92
275.9
-75.04
120.3
82.44
224.5
--68.64
130.9
1945
1922
1910
1890
80.77
80.0
79.8
79.9
276.0
277.3
277.3
281.6
--75.24
-76.0
--76.0
--76.5
120.5
121.0
121.8
124.7
82.19
81.0
80.7
80.7
227.3
239.1
241.3
246.5
--69.14
-71.2
-71.5
--72.4
131.5
133.6
1885 Schmidt 79.7
278.0
-76.8
120.2
80.3
247.0
--•3.1
133.8
1850
79.4
283.8
--77.1
126.1
79.6
255.4
--74.0
140.7
1835 Gauss
78.0
284.9
-76.6
126.9
77.5
260.3
-74.3
143.2
1800 Barr.
1750 Barr.
1700 Barr.
79.2
79.9
81.9
291.0
294.1
296.5
-78.4
--79.1
-80.4
132.8
139.3
146.7
78.5
78.9
80.6
268.7
268.7
263.3
--76.6
--76.8
--77.5
150.0
157.9
164.9
1650 Barr.
83.0
296.0
--81.3
157.7
80.5
257.6
--77.3
178.6
1600
83.2
30!.2
-81.6
152.0
82.3
267.2
--78.6
170.0
IGRF
Schmidt
Barr.
Barr.
Barr.
Barr.
135.0
137.6
FL denotesFinch and Leaton [1957], and Barr. is short for Barraclough[1974].
polar coordinates(6, 0d,4•d),where
the coordinatesof the eccentricdipole. To operate in the CD
system,it is necessaryfor the position coordinatesof the ec6 = (Ax2 + Ay2 + Az2)•/2
centricdipole to be convertedto their appropriateCD coordi0,•= 90ø-- 3,,•= cos-• (Az16)
nate systemform. Thus insteadof the geographicrectangular
coordinates(Ax, Ay, Az) or equivalent geographicspherical
t24)
•ba= tan- x (Ay/Ax)
the eccentricdipolenow hasthe CD positioncoordinates(AX,
A Y, AZ) and (6, Oa,•d), whereuppercase
is used,as before,to
EAST LONGITUDE
Fig.
11. Positions of the north ED axial pole for the interval
1600-1985 A.D., using the IGRF/DGRF
models for 1945-1985 and
Fig. 12. The southernequivalentof Figure 11, showingthe positions of the south ED axial pole (solid squares)for the interval 16501985 A.D., using the IGRF/DGRF models for 1945-1985 and the
Barraclough[1974] models for 1600-1910. Also shown are the CD
the Barraclough[1974] modelsfor 1600-1910.EllesmereIsland is in
polepositions
(crosses)
for the sameinterval.In additionto Vostok
the center of the display, and Greenland is to the right. G is the ED
axial pole position given by the original Gauss coefficientsfor 1835.
Resolute Bay is denoted by RE.
(VO) the positionsof the Soviet station Mirny (MI), the Australian
station Casey(CA), the French station Dumont D'Urville (DU), and
the U.S. station McMurdo (MM) are shown.
FRASER-SMITH: CENTERED AND ECCENTRIC GEOMAGNETIC DIPOLES
11
13b. The dip pole condition is then
Bresin(© -- 0e)-- Boecos(© -- 0e)= 0
(27)
which,aftersubstitution
for Br•andBo•,becomes
2 tan (© - 0e)= tan 0e
This equation now has to be solvedfor ©.
Applyingthe sinerule to the triangleOEP, we have
re
6
=
cos(©+Aa)
sin(©-0e)
a
=
Re
cos(0e+Aa)
(28)
(29)
b
Fig. 13. Condition for the occurrenceof an ED pole. Both panels
of this figure show part of the plane defined by the CD and ED axes
and its intersection with the earth's surface, represented here by a
segment of a circle. O is the earth's center, E is the location of the
eccentric dipole, OB is part of the CD axis with B the north CD pole,
DEA is part of the ED axis with A the north ED axial pole. (a) The
magnetic condition for the occurrence of the north ED dip pole is
illustrated (the figure is not drawn to scale).One of the dipole field
lines is perpendicularto the earth's surface.(b) The geometry that is
used in the derivation of the equation for the dip poles. P is a general
point in this panel and not necessarilythe point where the dipole field
is perpendicular to the surface.
Further, if F is the foot of the perpendicularfrom E onto the
line OP, we have Re - OF + FP, whichgives
Re = 6 sin (O + Aa)+ re cos(O -- 0e)
(30)
From (29) and (30) we can write
sin (O -- 0e)=
6 cos (O + Ad)
re
(31)
cos (!9 - 0e)=
RE -- 6 sin (O + Aa)
re
giving
distinguish the specificallyCD quantities. Equations analogousto (24) also apply for the CD position coordinatesof the
tan (0 -- 0e)=
5 cos (19 + Aa)
Re --6 sin (© + Aa)
(32)
dipole'
The point P in Figure 13b has CD coordinates(Re, ©) and
g--(AX 2 3. AY2 3- AZ2)1/2
Od= 90ø- Aa = cos-1 (AZ/6)
ED coordinates(%, 0e) which, from the geometryof Figure
(25)
13b,implies the following two results:
Re cos 19= AZ + re cos 0e
tan- 1 (AY/AX)
(33)
Re sin© = (AX2 + Ay2)l/2+ resin0e
The changeof position coordinatesfrom (6, 0
(1)a)is easilycarriedout by usingthe proceduresdetailedin
giving
section 2.
To illustrate this particular changeof position coordinates,
let us take the IGRF
tan 0e =
1985 field model as an example. In
geographicrectangularcoordinatesthe eccentricdipole is located at (-391.9 km, 257.7 km, 178.9 km), as shownin Table
3. The equivalentgeographicsphericalpolar coordinatesare
(502.0 km, 69.12ø, 146.7ø).In the CD coordinatesystemthe
rectangularcoordinatesare (-399.1 kin, --286.1 kin, 104.6
km), and the polar coordinatesare (502.0 km, 77.98ø,215.6ø).
The secondstepin the derivation is to obtain the CD coordinates of the ED dip poles. Figure 13b shows the geometry
requiredin this step of the derivation,that is, the sameplane
is involved as in Figure 13a, and it follows that the CD azimuthal coordinatesfor the two ED dip polesare the sameand
equal to tI)a' the azimuthalangledoesnot play a further role
at this stage of the derivation. The other CD coordinatesof
the ED dip polesare now obtainedby resolvingthe magnetic
field of the eccentricdipole, located at E in Figure 13b, along
the tangent to the circle (representingthe earth's surface)at
the general point P, equating the resolvedfield to zero, and
then rearranging the resulting equations to obtain an expressionfor ©. Rememberingthat the dipole at E is oriented
along the axis DEA in Figure 13b,the two componentsof the
dipole field at P are
2poM cos 0•
Re sin 19-- 6 cos Aa
Re cos 19-- 6 sin Aa
(34)
Substituting the expressionsfor tan (0- 0e) and tan 0e,
given by (32) and (34), into the dip condition(28), and carrying
out the necessaryalgebraic manipulation, the following expressionfor 19is obtained:
COS
2 (•}- K• cos19sin 19- K 2 cosO-- K 3 sin 19+ 2 = 0
(35)
where
K• = tan Aa
K 2 = 36 sin Ad/Re
K 3 --
(36)
(Re2 + 62) -- 362sin2 Aa
fir e cos Aa
Equation (35) must be solvednumericallyfor ©, and being of
the second order in cos O, it gives two values, O x and 0 2,
correspondingto the north and south ED dip poles.
The third and final step in the derivation is to convert the
CD coordinates(Re, (•}1,2'Od)of the ED dip polesinto geographic coordinatesusingthe proceduresdescribedin section
2.
4tOre
3
(26)
laoM sin O•
4rcr
e3
wherere and 0e are the ED polar coordinatesof P in Figure
3.6. Recent Positionsof the ED Dip Poles
and Their Secular Change
Table 4 lists the computed ED dip pole positions for the
same field models that were used to obtain the ED axial pole
12
FRASER-SMITH: CENTERED AND ECCENTRIC GEOMAGNETIC DIPOLES
o =1985, ß = FL, ß = 1955,
o=
axial and dip poles for the Finch and Leaton [1957] field
model as well as the same pole positions for the 1945, 1955,
and 1985 IGRF field models. One purpose of Figure 14 is to
show the magnitude and direction of the shift of the poles
away from the geographic south pole as the pole models are
progressivelyrefined from centereddipole to eccentricdipole
(axial pole) to eccentricdipole (dip pole). It does not appear to
be generallyrecognizedthat in each polar region the CD, ED
axial, and ED dip polesderived from a particular field model
1945
•75øS
• 70•• •65
o
•J
\\
CD
Poles
•
?;::-..•
• 100
ø
ED.
,xia
/
•
lie approximatelyalong a straightline, dependingon the map
projectionthat is used,a result that followsimmediatelyfrom
their colocation on the great circle segmentshown in Figure
13. It also followsthat the geographicpolesare not in general
aligned, even approximately, with their respectivetriad of
magneticpoles.Another purposeis to showthe comparatively
small movements of all the magnetic poles in the interval
1945-1985. The ED dip poleshave moved the most, with the
1985 pole located in the waters of Porpoise Bay in Wilkes
I
200
/ EDDip
•:-•-
• ' • //Poles
•::'/130•
Fig. ]4. The Antarctic positionsof the CD and ED axial and dip
poles for the Fi•c• a• L•ato• []957] field model used
•n• Cleary []958]. Also shown arc the sam• pole positionsfor the
]945, ]955, and ]985 ]GEF [•ld models. Note how the three
varietiesof •coma•ncticpolesarc approximatelycolinca•in this map
projection.
positions (section 3.4). Once again comparing the results obtained for the Finch and Leaton [1957] model for epoch 1955
with those of Parkinson and Cleary [1958] for the same model,
Table 4 shows the north ED dip pole at 82.46øN, 222.0øE,
whereas Parkinson and Cleary obtained 82.4øN, 222.7øE
(137.3øW). There is therefore close agreement between the
computed coordinates for this pole and similar close agreement between those for the south dip pole.
Figure 14 showsthe Antarctic positions of the CD and ED
+ = CDpole,
-75 ø
80....
o
.
,_
'oX
•
/
Figures 15 and 16 show the positions of the north and
south ED dip poles for various epochsin the interval 16501985 A.D., and they summarizethe various CD and ED pole
positionsand their motion since1650. The CD, ED axial, and
ED dip poles in these figureswere computed solely from the
1945-1985 IGRF and Barraclou•/h [1974] field models, and
the numerical data are listed in Tables 2 and 4. There is just
one exception,the ED dip pole positionsthat follow from the
original Gauss coefficientsfor epoch 1835, which are included
for continuity with the earlier displays.The pole positionsfor
the 1600 A.D. field model have been excluded becausethey are
less reliable than the others (Barraclou•/h [1974] lists significantly larger standard deviations for the Gauss coefficients),
and unlike the other positions they do not always conform
ß= ED axial pole, ß= EDdippole
,e--•
/
'•
/
/ z, I,
'1
•
/
AN
OCE
.
ent models for the same epoch.
•, 85o
•-• -'n
•
•
Z
•
Land, whereas the 1945 pole was well inland. Finally, comparison of the IGRF 1955 and Finch and Leaton pole positions givesan idea of the variability associatedwith two differ-
-"t'"
• .•.••/
•
• .
1650:
•:•.
.:.....
• -0
:::,.
• '
"::•:,.
, •••••:•.:•:•.•:•G.•••:• J9•5•TM'X X GREENLAND
•
70 ø
......
.
>
240
ø':•..,
' 1831
•t:
:? ':•%.:.
• %,. 32()•
,
250 ø
260 ø
270 ø
EAST
280 ø
290 ø
300 ø
310 ø
LONGITUDE
Fig. 15. Summaryof theCD, ED axial,ED dip,andmeasured
dip polepositions
for the northernpolarregion.All the
polesarecurrentlymovingroughlyto thenorth,or northwest,
asindicatedby arrowsfor theED axialanddip poles.The
measured
dip polepositions,
denotedby smallsolidcircles,are thesameas thoseshownin Figure6, and are described
in
the associated text.
FRASER-SMITH: CENTERED AND ECCENTRIC GEOMAGNETIC DIPOLES
+ = CDPole
/
/
/
/
.=
i'•
_ __
1985'•
/+
1650
/••
•
ß=
•
•
•
60øS
:•::•'•••
I '1650 •
/
•
• ....
•
•.:?'
•
•?'
1985
110OE
•
.•'
• MM?••• G //1962
.•:'
'
• .•1986
'::':'
'
'
•
/ •:t::'t;•
1903 ••••
•90 oE•
:.• __ __ •
I
.•:'::•
\\
• CA•:::..•
I
•s
EDdippole
\
x•:•:"•.::_,..
'•;'
":•.:i.-.'..
/
+ +.•
/'
70øS'%•ii..x
/
//
•
ED axial pole
%:.
/
./
/
•L•
•/
•' 130oE
MacKay party of the British Antarctic Expedition of 19071909, which obtained a position of 72.42øS,155.3øEfor epoch
1909.0 [Fart, 1944] that was later correctedto 71.6øS,152.0øE
[Webb, 1925] and (2) the Bage,Webb, and Hurley party of the
Australasian Antarctic Expedition of 1911-1914, which obtained a position of 71.17øS,150.8øEfor epoch 1912.0 [Webb,
1925]. The fourth position is inferred from the measurements
of Kennedy during the British, Australian, and New Zealand
Antarctic Research Expedition of 1929-1931; it is 70.3øS,
149.0øEfor epoch 1931.0 [Fart, 1944]. The fifth position was
measured by Mayaud [1953a, b] during the French South
Polar Expedition of 1951-1952: 68.10øS, 143.0øE for epoch
1952.0. The sixth position, at 67.5øS, 140.0øE for epoch 1962.1,
was obtained by Burrows and Hanley [Burrows, 1963]. This
appears to have been the last measurementof the dip pole on
land. Already close to the sea during 1952-1962, it must have
moved out to sea around 1965. There appear to have been no
documented attempts in the interval 1962-1986 to measure
the location of the dip pole directly. On January 6, 1986,
scientists from
Fig. ]6. Summar• of the CD, ED axial, ED dip, and measured
dip pole positions for the southern polar region. The arrows indicate
the apparent presentdirection of motion of the poles.
well to the trend establishedby the positions for neighboring
epochs.Also shown in both Figures 15 and 16 are a number of
the measured (or inferred) dip pole positions (usually called
"magnetic poles") that have been obtained as the result of
various expeditions. Needless to say, one of the tests of a
model of the earth's magnetic field is how well it predicts the
actual observedlocations of the north and south dip poles. In
both Figures 15 and 16 it is evident that the ED dip pole
positionsare in much better agreementwith the observeddip
pole positionsthan are the CD pole positions.
Considering the individual figures,in Figure 15 we see that
the paths followed by the ED dip pole and the measured dip
pole have been roughly parallel since the beginning of the
century and that they should continue to be parallel for some
time into the future. Around 1750-1800 there was a comparatively very abrupt change in the direction of the path being
followed by the ED dip pole. The change was so abrupt that it
appeared possiblethat the computations of the pole positions
13
the Australian
Bureau
of Mineral
Resources
aboard the M/V Icebird located the pole at 65.3øS, 140.0øE
[Barton, 1986].
One of the interesting features of both Figures 15 and 16 is
the comparatively abrupt recent change in the direction of
motion of the CD pole. In the south it is now moving away
from Vostok, whereas a few decades ago it was moving
toward the station. None of the other southern geomagnetic
poles show any comparable change in their paths. In fact, for
the map projection that is used, their paths are remarkably
straight. As noted above, there is a considerable distance
(about 400 km) between the ED axial pole and Vostok. From
the point of view of phenomenain the upper atmospherethat
relate to the dipole part of the earth'smagneticfield, Vostok is
not ideally located at its present position near the CD pole,
sincethe dipole axis (the axis of symmetry)is more accurately
that of the eccentric dipole, which intersectsAntarctica at the
ED axial pole. The small tilt of the ED axis relative to the
earth's surfaceis not significantin this context, since the CD
and ED axes are parallel, and the tilt at the ED axial pole is
merely the result of the earth's curvature. For this somewhat
negativereason the motion of the CD pole away from Vostok
is unlikely to have implicationsfor upper atmospherephysics.
were in error, but no error could be found. Confirmation of
the correctnessof the results was obtained by applying the
3.7. CD/ED Coordinate Transformsand
ED Magnetic Fields
"straight-line criterion" for the triad of geomagnetic poles:
despite the change in path, the CD, ED axial, and ED dip
poles continued to be closelycolinear. Since the computations
for the three different poles are independent, the abrupt
change must be considereda genuine feature of the ED dip
pole motion. There is no such feature in the motion of the
ED dip pole in the south.
The measured dip pole positions in Figure 16 come from a
variety of sources, including Dawson and Newitt [1982] in
particular. Some emphasishas been given to relatively localized measurements,and thus many of the early pole positions
inferred solely from field measurementsat sea off the Antarctic
coast have not been included. The first position, for epoch
Cole [1963] has provided the necessarydetails for transforming from geographic to eccentricdipole coordinates,and
he has, in addition, given plots of ED latitude and longitude
for epoch 1955 (using the Finch and Leaton [1957] field
model) superimposedon three geographic grids: one for the
world, usinga Mercator projection,and two coveringeach of
the polar regions. A more general treatment of transforms
between geophysical Cartesian coordinate systems,but one
that does not specificallyinclude the eccentricdipole system
has been given by Russell[1971]. More recently, Wallis et al.
[1982] derived ED coordinatesfor the presentationof Magsat
data. In view of this earlier work I will, in this section,present
only the transform equations necessaryfor conversion between the CD and ED systems.
The transform betweenCD and ED coordinatesis straightforward, sinceonly a translationis involved: The origin of the
ED system is located at the point (AX, A Y, AZ) in the CD
system, but the directions of the Cartesian axes are the same.
1903.2, is 72.9øS, 156.4øE, and it comes from measurements
made during Scott's first expedition, the British Discovery expedition of 1901-1904 [Bernacchi, 1908]. The second and
third positionscom• from measurements
made by partiesthat
attempted to reach the dip pole: (1) the David, Mawson, and
14
FRASER-SMITH' CENTERED AND ECCENTRIC GEOMAGNETIC DIPOLEG
Using this information, the basic equations relating the CD
and ED
coordinates
are
X = r sin O cos ß = xe + AX = Fe sin 0e COS•e q- AX
Y -- r sin O sin •b - Ye+ AY -- re sin 0e sin 4•e+ AY
(37)
ED coordinates will be the most useful, since it is the one
required to compute the ED magnetic field at a given geographic location.
Considering the ED magnetic field, it will be given, in the
ED coordinate system,by
#oM(3cos2 0e q- 1)•/2
4Itre3
Z • F cos {• • ze q- AZ • re cos 0e q- AZ
where the subscript e is used to denote ED coordinates X, Y,
Z' r, •, and ß are the Cartesian and spherical polar coordi-
2#oM cos 0e
(BE)re
=
47ire
3
natesin the CD system,and Xe,Ye,Ze;re, Oe,and •Peare the
correspondingcoordinatesin the ED system.The radial coordinate r is common to both the geographicand CD systems.
From theseequations we obtain
rsinOcos•--AX
rsin•sin•--AY
sin 0e COS(pc
r cos 0
#oM sin 0e
(BE)Oe
: __ 471;re
3
Supposenow that the total ED magneticfield is required at
a particular geographic location. To obtain the field, the
sin 0e sin •pe
=
(45)
-
COS0e
AZ
(--re)
(38)
dipolemomentM and north CD pole positioncoordinates0,
and •p, are computedfrom the sphericalharmonicfield model
of choice, using the procedure detailed in section 2.2. Next, the
ED Cartesian position coordinates are computed from the
field model using the equations in section 3.1; these coordinates should immediately be converted to their CD form as
describedin section3.5. The final preparatory step is to convert the geographic coordinates of the location of interest to
giving
4•e
=tan
ß-• [•sinsinOOcossinßA•X
]
(39a)
0e
=tan•I(;rCOS
sin
O--c__os
• •_AX_
1 (390)
(•
AZ)
cos
4e]
ED coordinates
via an intermediate
conversion to CD coordi-
nates; the proceduresare describedin section2.1 (geographic
to
CD) and in this section (CD to ED). With the ED coordiTo transform from CD to ED coordinates,•Peis computed
natesof the point and the scalarmoment M it is then possible
from (39a),followedby 0e from (390) and refrom
r cos O -
re =
AZ
(39c)
cos0e
to calculatethe total magneticfield from the equationfor IB•l
in (45). A similar but more involved procedureis required to
obtain the ED magneticfield components.
which follows from (38).
4.
In the eventthat (Pc-- 90øor 270ø the equationto usefor 0e
is
DISCUSSION
The principalresultsof this work are the pole positionsand
other dipole data presentedin the variousfiguresand tablesin
the text. With the exceptionof the observeddip pole positions,
which provide a check on the applicability of the dipole field
models, the data were all computed, with negligible error,
Oe=tan-•[
rsinOsinrD--AY]
(40)
r cos O-
AZ
Similarly,if 0e = 90øor 270ø,the equationto usefor reis
from the original Gauss coefficient field models. Thus the ac-
re = [(r sin O cosß -- AX)2 + (r sin O sinß - AY)2]1/2
(41)
To transform from ED to CD coordinates, the required
equations are
ß=tan• '•resin
sin0
esin•pe+
0ecos•Pe
+
O = tan- •
(420)
re cos 0e + AZ
cos O
be accuratelymeasurablequantities.
A general question, arising out of this work, concerns the
fidelity with which the dipole field models representthe actual
field of the earth. As is shown by the summary figures,Figures
15 and 16, the computed ED dip pole positions are much
closer to the observed dip pole positions than are the computed CD poles. From this point of view the ED model provides a superior representation of the earth's field, as compared with the CD, or "geomagnetic,"model. However, the
ED dip poles do not closely correspond to the observeddip
pole positions.Better agreementmay be obtained by using the
complete spherical harmonic field models for each epoch (that
is, by using all the Gauss coefficients)to compute the dip pole
positions, or even by combining a number of field models to
produce an average spherical harmonic model for each epoch
and then computing the dip pole positions, as was done by
Dawsonand Newitt [1982]. Even with the best modeling, however, there are still discrepancies between the observed and
computed fields. Under these circumstancesit would be helpful to have a quantitative measure of the agreementbetween a
particular geomagneticfield model and what I will refer to as
(42a)
COS
0eq-AZ)
•g •3]
[r(•eSinOeCOS
CPeq-AX
r =
curacy of the pole positionsrelatesdirectly to the accuracyof
the Gausscoefficients.
However,this is not an importantpoint
at the presenttime, sinceonly the dip pole positionsappearto
(42c)
with equations similar to (40) and (41) when ½Dand O are 90ø
or 270ø:
O=tan-•[
resinOesinqbe+AY]
(43)
reCOS0e+AZ
r = [(re sin0e cos•Pe+ AX)2 + (resin0e sin•Pe+ Ay)2]1/2
(44)
Of these two transformsit is likely that the one from CD to
FRASER-SMITH: CENTERED AND ECCENTRIC GEOMAGNETIC DIPOLES
the measured field, that is, a set of geomagneticfield measurements that, perhaps by general agreement, are sufficiently
timely, accurate, and complete to be used as the basis for a
model. For example, the set of measurements on which the
world magnetic charts and their associatedspherical harmonic
models were based [Barker et al., 1981]. In the following,
interpolation between the measuredvalues may be required to
give the necessaryworldwide coverage.This measure,a goodness of fit index (GFI), would enable a better informed judgment to be made concerning the use of a particular field
model' Under some circumstancesthe centered dipole model
may be entirely adequate; under other circumstancesthe use
of the presumably more accurate eccentric dipole model
would be desirable or, if high accuracy was required, the GFI
would enable the most accurate spherical harmonic model to
be selected.
One possible GFI could be constructed as follows. A partic-
ular field quantity, the total field for example, is computed
from
the model
for each 10 ø intersection
of the latitude
and
longitude lines,starting with the equatorial point on the prime
meridian (that is, with the point 0øN, 0øE) and including the
geographicpoles. The magnitude of the percentagedifference
between the computed value and the measured value of the
field quantity at each point is determined and the median
value, or alternatively the averagevalue, of the magnitudesfor
all 614 points computed. This final computed value would
then serveas the index for goodnessof fit. Becauseof the more
rapid decline with distance of the higher order terms in the
sphericalharmonic field modelsand correspondingchangein
the actual measured field of the earth the GFI for any field
earth'g field is already essentiallyin the required Legendre
polynomialform [e.g., Stratton, 1941] for a generalmultipole
representationof the field, and as shown by James [1967,
1968] and Winch [1968], for example,the strengthsand axial
directions of the geographically centered multipoles can be
computed from the Gauss coefficients.The centered dipole
representationfollowsidenticallyfrom the first-orderterms of
the multipole expansion.However, the next higher order of
the multipole expansion, to magnetic quadrupoles, does not
lead to the eccentricdipole representation,although the magnitude and direction of the eccentric dipole's displacement
from the earth's center can be related to the parameters of the
geomagneticquadrupole[Winch and Slaucitajs,1966b; Winch,
1968]. Since the eccentricdipole is located at a finite distance
from the earth's center and the multipoles of all orders are
located at the center, it would appear unlikely that the eccentric dipole field model could be reproducedas a specialcaseof
the multipole representation. However, since both representations can be derived from the one basic spherical harmonic field model, a relationship is implied. Whatever this
relationshipmay be, the two representationsprovide different
views of the earth's magnetic field, and their relative merits
depend on such practical factors as their accuracy and the
physical insight they give. A complete multipole representation must be as accurate as the original spherical harmonic field model, but taken to its first two orders (dipole plus
quadrupoleterms),it may be no more accurateand is likely to
provide lessphysicalinsightthan the eccentricdipole,particularly in those regions of spacenear the earth where charged
particlemotion and wave particleinteractionsare important.
model will be a function of altitude, and the GFIs for the
centered and eccentric dipole models will tend to approach
those for their originating spherical harmonic models as the
altitude
is increased.
But why bother using a centered dipole or eccentric dipole
model for the earth's magnetic field when, with modern computers, it is nearly as easy and fast to use a full spherical
harmonic
field model?
The
answer
is because
tional Science Foundation, Division of Polar Programs, through
grant NSF-DPP 83-16641.
of the combi-
nation of geophysicalinsight and adequate accuracyfor many
purposesprovided by the dipole models.To give one example,
the motion of charged particles in a magnetic dipole field has
been the subject of much study and is reasonablywell understood. Thus general statements can be made about the expected motions of charged particles in the earth's magnetic
field simply by assumingthat it can be representedby a dipole
REFERENCES
Adam, N. V., T. N. Baranova, N. P. Benkova, and T. N. Cherevko,
Sphericalharmonic analysisof declinationand seculargeomagnetic
variation 1550-1960, Earth Planet. Sci. Lett., 9, 61-67, 1970.
Akasofu, S.-I., and S. Chapman, Solar-Terrestrial Physics,pp. 89-92,
OxfordUniversityPress,New York, 1972.
Barker, F. S., D. R. Barraclough,and S. R. C. Malin, World magnetic
charts for 1980--Spherical harmonic models of the geomagnetic
field and its secular variation, Geophys.J. R. Astron. Soc., 65, 525533, 1981.
field.
There seems little doubt that t•he eccentric dipole representation
Acknowledgments.
This work owesmuch to the stimulationof
RonaldN. Bracewelland the graduatestudentsin EE249.Supportfor
the Arctic and Antarcticresearchis providedby the Officeof Naval
ResearchthroughcontractN00014-81-K-0382,
by RomeAir DevelopmentCenterthroughcontractF19628-84-K-0043,
and by the Na-
of the earth's field is closer to the real field than the
centereddipole representation,and one of the purposesof this
paper is to make the superior eccentricdipole representation
more accessible.However, it must also be pointed out that the
eccentricdipole representationitself has the potential for improvement. The off-center dipole of Bochev[1969a], for example, where the orientation of the dipole axis is no longer restrictedto that of the centereddipole, could conceivablygive a
marginally better approximation to the earth's field. More
substantial improvements would be expected from the inclusion of additional dipole sources [e.g.,Bochev, 1969b] or,
moregenerally,
fromthe useof a multipolerepresentation
of
the field [Umow, 1904' Winch and Slaucitajs, 1966a, b' Zolotoy, 1966' James, 1967, 1968, 1969' Winch, 1968' Winch and
Malin, 1969].
The conventional spherical harmonic representation of the
Barker, F. S. et al., International geomagneticreferencefield revision
1985, Eos Trans. AGU, 67, 523-524, 1986.
Barraclough,D. R., Sphericalharmonic analysesof the geomagnetic
field for eightepochsbetween1600and 1910,Geophys.
J. R. Astron.
Soc., 36, 497-513, 1974.
Barraclough,D. R., Internationalgeomagnetic.
reference
held revision
1985,Int. Assoc.Geornagn.
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(Received June 30, 1986'
revised September29, 1986'
acceptedOctober 6, 1986.)