Ad ances in Elec ical and Elec onic Enginee ing
316
A NEW ECCENTRIC GEOMAGNETIC DIPOLE TO GIVE THE CORRECT DIP
POLE LOCATIONS
A. de Pao , E. Bu ke
School o Elec ical, Elec onic and Mechanical Enginee ing, Na ional Uni e si y o I eland,
Dublin (UCD), Bel ield, Dublin 4, I eland
e-mail:ann aoi.depao @ucd.ie
Summa y
In his pape , we desc ibe a new eccen ic dipole model o he Ea h’s magne ic ield. The cons ain s unde
which he con en ional eccen ic dipole model is de ined esul in p edic ed dip pole loca ions ha di e signi ican ly om
he measu ed loca ions. He e, we gi e a p elimina y exposi ion o a new dipole model which, because i is cons ained by he
obse ed dip pole loca ions, o e comes his p oblem.
1. INTRODUCTION
I is cus oma y o desc ibe he geomagne ic
ield as he g adien o scala po en ial
V
, which is
exp essed as a se ies expansion o o hogonal
sphe ical ha monics [1]. Ideally, his is an in ini e
se ies in e ms o coe icien s
m
n
g
and
m
n
h
, whe e
m
uns om
0
o
n
and
n
uns om
1
o
∞
. In
p ac ice, he se ies is ypically unca ed a
10
=
n
o
12
=
n
.
T unca ed a
1
=
n
, he se ies includes e ms
con aining only
0
1
g
,
1
1
g
and
1
1
h
. This gi es he ield
o a dipole cen ed a he geog aphic cen e o he
ea h, ha ing an axis inclined wi h espec o he
geog aphic axis. This is he con en ional cen ed
dipole model. T unca ing he se ies a
2
=
n
p oduces a highe o de app oxima ion, e aining he
8 coe icien s
0
1
g
,
1
1
g
,
0
2
g
,
1
2
g
,
2
2
g
and
1
1
h
,
1
2
h
,
2
2
h
.
This is he con en ional eccen ic dipole model,
which has been he subjec o pape s by Schmid [2],
Ba els [3] and F ase -Smi h [4]. This eccen ic
dipole is es ic ed inso a as i s axis is pa allel wi h
ha o he cen ed dipole. One consequence o his is
ha he associa ed dip poles a e qui e a om he
measu ed dip pole posi ions. Since he dip poles a e
such a p ominen and measu able ea u e o he
geomagne ic ield, we el ha i migh be o alue
o in oduce ano he eccen ic dipole model which
gi es he co ec dip pole loca ions. To ep oduce
dip poles a any wo speci ied loca ions, he e a e an
in ini e numbe o poin s a which he dipole can be
si ua ed. Howe e , by in oducing an addi ional
cons ain - ha he dipole posi ion be equidis an
om he wo dip poles – a unique solu ion o he
dipole’s posi ion and o ien a ion is ob ained. We
gi e he e a p elimina y exposi ion o his
de elopmen . I emains o he u u e o es ablish
how accu a ely o he ea u es o he geomagne ic
ield a e ep oduced by his new dipole model.
2. DEVELOPMENT OF THEORY
A poin
S
in Fig. 1, he magne ic lux densi y
ec o is eadily calcula ed as
(
)
( )
}cos422cos31
,cossin3{
22
θθ
θ
θ
d d
d k
+−−
−
=
B
(1)
whe e
[ ]
2
5
22
0
cos24
θπ
µ
d d
m
k
−+
=
(2)
We now conside ha
S
is in ac he Sou he n dip
pole and
N
he no he n dip pole. A
S
he e o e,
he lux is along he adius ec o
(
)
θθ
sin,cos =
. The e o e,
S
sa is ies he
ela ions
(
)
( )
[ ]
θγ
θθ
θ
γ
θ
θ
sin
2
cos422cos31
coscossin3.
22
d d
k
d k
=
+−−
=
−
(3)
Fig. 1. A c oss-sec ion o he Ea h con aining he dip
poles S and N and he geog aphic cen e. The
p oposed eccen ic dipole, D2, is aligned pa allel o
SN ( he line joining he dip poles) and loca ed a a
dis ance d om he geog aphic cen e.
A new eccen ic geomagne ic dipole o gi e he co ec dip pole loca ions
317
Elimina ing
k
and
γ
be ween hese equa ions leads
a e some manipula ion o
03coscos
2
=−






++
θθ
d
d
(4)
We now use he simpli ied no a ion
d
e=
whe e
e
is he ( ac ional) eccen ici y o he dipole
and
θ
cos
=
,
whe e
is he ( ac ional) eccen ici y o he
NS
axis. Equa ion (4) can hen be w i en
e
e−=+ 31
(5)
In p ac ice
is known (as we shall show below)
om he posi ions o
S
and
N
measu ed on he
Ea h’s su ace and so, sol ing o he app op ia e
oo o he esul ing quad a ic o e gi es
[
]
)1)(9(3
2
1
222
e−−−−=
(6)
This de ines he eccen ici y o ou p oposed dipole.
I s axis is pa allel o
NS
and lies in he plane
con aining
N
,
S
and he geog aphic cen e o he
Ea h.
The ac ional eccen ici y,
, o he
NS
axis
is measu ed as ollows. No malising he adius o
he Ea h o uni y, a poin on he su ace o he Ea h
may be speci ied in e ms o i s la i ude,
la
, and
longi ude,
lo
, as shown in Fig 2.
In Fig. 2 he
xz
plane con ains he g ea ci cle
°
−
1800
longi ude. The coo dina es o
P
a e
laz
lolay
lolax
sin
sin.cos
cos.cos
=
=
=
(7)
I we deno e he coo dina es o he no h and sou h
dip poles by
(
)
nnn
zyx ,,
and
(
)
sss
zyx ,,
espec i ely, he coo dina es o he cen e poin o
NS
, which co esponds o poin
Q
in Fig. 1 a e






+++
2
,
2
,
2
snsnsn
zzyyxx
No e ha in ou no malized coo dina e sys em,
is
equal o he dis ance o poin
Q
om he o igin.
222
222






+
+






+
+






+
=
snnsns
zzyyxx
(8)
Scaling he coo dina es o
Q
by
e
gi es he
coo dina es o he dipole loca ion as
( )






+++
=2
,
2
,
2
,,
snsnsn
zzyyxx
e
cba
. (9)
The dipole axis is pa allel o
NS
. Thus, i we
de ine
(
)
(
)
snsnsn
zzyyxxihg −−−= ,,,,
, (10)
he dipole axis may be aced ou pa ame ically as
(
)
ichbga
λλλ
+++ ,,
, (11)
whe e
λ
is a eal pa ame e .
The poin s o in e sec ion o he eccen ic dipole
axis wi h he su ace o he no malized Ea h a e
de ined by he condi ion
(
)
(
)
(
)
222
1ichbga
λλλ
+++++=
. (12)
This gi es he equa ion
(
)
222
222
1
ihg
cba
++
++−
±=
λ
, (13)
(whe e i is no ed in he de i a ion ha
0
≡
+
+
cibhag
). The posi i e solu ion o
λ
co esponds o a new no h “geomagne ic pole” and
he nega i e solu ion o
λ
o he sou h. All ha
emains now is o use he equa ion
z
y
x
,
,
in
e e se o ind he ind he la i ude and longi ude o
he new geomagne ic ields.
3. EXAMPLE
His o ic loca ions o
S
and
N
a e summa ized by
Mandea and Do my [5]. The la es measu emen s
gi en by hem a e
S
, 2000 and
N
, 2001. We could
use hese in an example bu we ind in ac , ha hey
a e almos iden ical wi h alues p edic ed by he ull
Fig. 2. Rela ionship be ween no malised Ea h’s
sphe ical coo dina es la (la i ude) and lo (longi ude)
and he ca esian coo dina es x, y and z.
Ad ances in Elec ical and Elec onic Enginee ing
318
geomagne ic se ies, and so we use he igu es o
2006 p edic ed by he ull se ies [6]:
°=°−=
°
−
=
°
=
7.137,5.64:
0.122,8.83:
lolaS
lolaN
(14)
These gi e he ac ional cen e o ou eccen ic
dipole as
(
)
(
)
01559.0,033737.0,063957.0,, −=cba
(15)
and ou new no h and sou h geomagne ic poles a e
°=°−=
°
−
=
°
=
82.130,424.72:
669.66,898.79:
lolaS
lolaN
g
g
(16)
4. DISCUSSION
By de ini ion, ou dipole model ep oduces he
expe imen ally obse ed posi ions o he dip poles.
O he ea u es ha could be expe imen ally
in es iga ed include he p edic ed collinea i y o he
ield a
g
N
and
g
S
, he new geomagne ic poles
(whe e he dipole axis in e sec s he su ace o he
ea h). Fu he mo e, he e exis s a unique ci cula
locus o poin s a which he su ace ield due o he
dipole is pa allel o he dipole axis. This ci cle
coincides wi h he in e sec ion o a plane passing
h ough he dipole posi ion and no mal o i s axis
wi h he ea h’s su ace. In es iga ion o he
co espondence be ween hese p edic ed ea u es and
he obse ed ield would be o in e es . Howe e ,
he ine i able e ec s o local pe u ba ions should be
expec ed.
Ou dipole di e s om he Schmid -Ba els-
F aze -Smi h eccen ic dipole (1985 wi h 1984N and
1986S), bu we no e he d awback ensh ined in
F ase -Smi h’s commen ha “ he ED dip poles do
no closely con o m o he obse ed dip pole poin s”.
We ha e a leas shown how o de ine an eccen ic
dipole which o e comes his limi a ion. How
accu a e i is in o he espec s emains o be
in es iga ed.
Ano he no ewo hy dipole model is ha o
Boche [7]. His is an eccen ic dipole, gene a ed by
minimizing he mean squa e e o be ween he
dipole ield and he ield as measu ed by 61
geomagne ical obse a o ies. He abula ed he
esul s o 1932, 1937, 1942, 1945, 1950, 1955 and
1960. To illus a e his esul s, we conside 1945, o
which he measu ed dip pole loca ions we e:
°=°−=
°
−
=
°
=
5.144,2.68:
2.100,9.73:
lolaS
lolaN
(17)
The co esponding no malised dipole posi ions
gi en by Boche a e:
0027.0,0402.0,0489.0
−
=
=
−
=
cba
.
(18)
This ansla es o an eccen ici y
0634.0
=
e
. The
co esponding igu es o ou dipole a e:
0055.0,0097.0,0594.0
=
−
=
−
=
cba
(19)
gi ing
0605.0
=
e
. The no malised ec o
di ec ion o Boche 's dipole is
(
)
9790.0,1929.0,0653.0 −−
and o ou dipole is
(
)
9601.0,2483.0,1287.0 −−
The angle be ween hese ec o s is
°
9449.4
.
Al hough he posi ion o Boche 's dipole is qui e
di e en o ha o ou s, he eccen ici ies a e e y
close, and he angle be ween he dipole di ec ions is
small. Boche 's 61 geomagne ical obse a o ies gi e
a e y spa se and geog aphically biased co e age o
he Ea h's magne ic ield, pe haps accoun ing o
F ase -Smi h's passing o e his wo k in a single
sen ence. Howe e , i would be e y in e es ing o
see his me hod epea ed wi h he much mo e dense
and widely dispe sed measu emen s now a ailable
om sa elli es, which migh mo e his diploe
coo dina es a ound close o ou s.
Acknowledgemen
The au ho s would like o hank P o esso Denis
O’Sulli an o he Dublin Ins i u e o Ad anced
S udies o supplying he pape s by F ase -Smi h
and Boche , which we e o he wise una ailable.
REFERENCES
[1] Dipole app oxima ions o he geomagne ic ield,
The Space En i onmen In o ma ion Sys em,
h p://www.spen is.oma.be/spen is/help/backg
ound/mag ield/cd.h ml, accessed on 18/2/2006.
[2] Schmid , A., De magen ische Mi elpunk de
E de und seine Bedeu ung, Ge lands Bei .
Geophys., 41, pp. 346-358, 1934.
[3] Ba els, J., The eccen ic dipole app oxima ing
he ea h’s magne ic ield, J. Geophys. Res., 41,
pp. 225-250, 1936.
[4] F ase -Smi h, A. C., Cen ed and Eccen ic
Geomagne ic Dipoles and Thei Poles, 1600-
1985, Re . Geophys., 25, 1, pp. 1-16, 1987.
[5] Mandea, M. and Do my, E., Asymme ic
beha iou o magne ic dip poles, Ea h Plane s
Space, 55, pp. 153-157, 2003.
[6] Magne ic No h, Geomagne ic and Magne ic
Poles, Wo ld Da a Cen e o Geomagne ism,
Kyo o, h p://swdcwww.kugi.kyo o-
u.ac.jp/poles/polesexp.h ml, accessed on
13/1/2006.
[7] Boche , A., A dipole app oxima ing o he
highes possible deg ee he ea h’s magne ic
ield, Pu e Appl. Geophys., 74, pp. 25-28, 1969.