A new eccentric geomagnetic dipole to give the correct dip pole locations

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.