Midgley Je 1963ocr

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CALCULATION BY MOM NT TBCHNIQUB OF THB PERTURBATION

OJ THB GBOMAGNBTIC JIBLD BY THB SOLAR WIND

Thes i s by

James B. Midgley

In P ar t i a l Ful f i l lment of the Requirementsor the Degree of

Doctor of Phi losophy

al i fornia I n s t i t u t e of TechnologyPasadena. a l i f o r n i a

196


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ACKNOWLBDGMBNTS

Both the i n i t i a l problem and the idea oso lv ing t us ing moments o the sur face cu r ren t swere sugges ted to me by Dr. Levere t t Davis J r .

am deeply g ra t e fu l to him not only for t h i s buta l s o Cor h i s cons tan t i n t e r e s t in the problem andnumerous valuable suggest ions in overcoming theobs tac les which a rose dur ing ts so lu t ion .

am a l so g ra t e fu l to the Na t iona l ScienceFoundat ion for fe l lowships dur ing pa r t of thet ime worked on t h i s r e sea rch and to the Na t iona lAeronaut ics and Space Adminis t ra t ion who have suppor ted the comple t ion of the r e sea rch under GrantNsG-151-61.


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BSTR CTs

n i t e r a t i ve method i s developed by which one canca lcu la te approximately the boundary of a magnetic f i e ldconf ined by a plasma. This method cons i s t s e s s e n t i a l l y ofvarying an assumed surface u n t i l the magnetic mult ipolemoments of the cu r ren t s , which would f low on t ha t surfaceto balance the plasma p r e s s u r ~ cancel the correspondingmoments of the magnetic sources within the sur face . Themethod i s appl ied to two problems.

For a dipole source of moment M emu in a plasma ofuniform pressure dynes/cm2 t ha t does not pene t ra te themagnetic f i e l d , the approximate equat ion of the surfacei s r -0 .82615 Ml/J p - l / 6 l O . l 2 0 0 ) 9 ~ 2 - . o 0 4 t 8 0 ~ 4 - . o o l 0 8 5 ~ 6

+ . o o o 2 o o ~ o 0 0 5 9 7 ~ 1 0 + . 0 0 0 ) 2 6 ~ 1 2 - . o o o 0 9 4 ~ 1 4 em, where ~i s the l a t i tude in radians from the plane normal to

The sur face formed by a cold plasma of dens i ty N0 andpa i r massve loc i ty

Mt moving pas t a dipole of moment-u e extends to i n f i n i t y downwind.~ z

with aa coor-

dina te system x, y, z) centered a t the dipole , neu t ra lpo in ts , where the sur face i s pa ra l l e l to the wind d i rec t ipn ,occur a t the poin ts O,Rn,.27Rn), and other poin ts on thesurface a re O , O , l . 0 2 R n ) , 0 , ~ 2 R n , ~ ) and ~ 1 . 9 7 R n , O , - o o ) .Rn l .OOJ5 M/ MtN 0 u ) f ) l / J i s about 9 ear th r a d i i for theso la r wind case.


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TABLB OF CONTENTS

TITLE1 In t roduct ion2 Model for the Calcula t ions3 The ~ f o m n t Technique4 Solut ion for the Uniform Pressu re Case5 Rela t ionship of the Current and Surface

for the Wind Case6 Calcula t ion of the Moments for the Wind Case7 Speci f i c Solut ion for a Dipole Source8 Resul t s and Conclusions

\PPBNDIX TITLBI Determinat ion of the Surface ThicknessI I Fie ld Ins ide the CaTityI I I Rela t ion of Vector Moments to Scalar MomentsIV Choice of the T r i a l Flux Funct ionV Program for Numerical Calcula t ions

ill

86

21

JOJ6428

773758499


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LISI OF ILLUSI RATIONS

FIGURE TITLE P GB

1 . Exter ior view oC the bounding surCaceof the e a r t h s dipole f i e l d .

2 . Plo t of magnetic f i e ld s t r eng th andpa r t i c l e t r a j e c to r i e s within the sur face reg ion . 9

3. Cross sec t ion of the sur face boundinga dipole f i e ld in a uniform pressure plasma. 24

4. Front view of the geomagnetic f i e ld boundaryshowing curren t l i ne s . SJ

S Side view of the geomagnetic f i e l d boundaryshowing curren t l i ne s . s4

6. Top view of the geomagnetic f i e ld boundaryshowing current l i ne s . SS

7 Front view of the geomagnetic f i e ld boundaryshowing contours of constant z . S6

B Actual ca lcu la ted cross sec t ionsshowing nature of smoothing modi f i ca t ions . 7

9. Fie ld l i nes and f i e ld magnitude contoursin the noon-midnight meridian plane. 60

10. Fie ld magnitude contours in the equa tor ia l plane . 6111. Comparison of Beard s sur face with the

moment sur face in the planes of symmetry. 64


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1 . In t roduc t i on .t has long been be l ieved t ha t there ex i s t s a Clow oCplasma Crom the sun which, because o its high conduct iv i ty ,

compresses the e a r t h ' s magnetic f i e ld , conf in ing it to at e a r -d rop shaped cav i ty , such as i l l u s t r a t e d in Figure 1 .

Sola r plasma burs t s were f i r s t sugges ted by Chapmanand Fer ra ro (1) as an explanat ion for magnetic s torms-- thesudden a r r i v a l o the plasma s t ream g iv ing r i s e to thesudden commencement of the s torm. Late r Biermann 's (2)obse rva t ions of coaet t a i l s suppor ted the exi s t ence of aso la r plasma f lux and i nd i ca t ed tha t it was probably acont inua l phenomenon. Fol lowing Unsold and Chapman ( ) ) hees t imated its ve loc i ty a t 1000 Km/sec and its p a r t i c l edens i ty a t anywhere from 100 pa r t i c l e s / co in qu ie t t imesto 105 pa r t i c l e s / co i n ac t ive t imes . He assumed the s t reamto have a t empera ture of 10 K.

Parker (4) developed a hydrodynamic theory of the so la rcorona which inc luded heat ing out to about e i gh t sun r a d i iby hydromagnet ic waves. His theory i nd i ca t ed t h a t thecorona should be in a s t a t e of cons tan t expansion giv ingr i s e to a so la r wind with a ve loc i ty of )00 Km/sec anddens i ty of )0 protons /co a t the r ad i us of the e a r t h ' s or b i t .Chamberlain (5) objec ted tha t a hydrodynamic approach wasnot appropr ia te and t ha t the l o s s of mat te r from the coronawas l imi ted by evapora t ion of pa r t i c l e s from the tail othe Maxwellian d i s t r i bu t i on . His theory a l s o ind ica te s a


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igure Exte r io r view of the bounding su r faceof the ear th s d ipo le f i e ld or ien ted in the yd i r e c t i o n ) for a pl sm wind in the -z d i r e c t i o n .


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dens i ty ~ about JO protons/co but pr e d i c t s the Yeloci ty a tthe rad ius ~ the e a r t h to be only about 20 Km/sec. Therecent r e su l t s ~ r o m the Mariner I I plasma de tec to r (6)i nd ica te t h a t the stream probably has a mean ve loc i ty ~about 5 Km/sec. a dens i ty between 2 .5 and ions /co and atemperature in excess of lOS K. This of course favorsParke r s theory over Chamberla in s .

The qua l i t a t ive aspects of the t r ans ien t phenomenainvolved when a plasma burs t impinges on the e a r t h s mag-ne t i c f i e ld have been s tudied by considera t ion of severa li dea l i zed problems. Chapman and Ferra ro (7) f i r s t con-s idered the two dimensional a x i a l l y symmetric problem ofplasma i n j ec ted r a d i a l l y i n to a magnetic f i e l d which f e l loff r a d i a l l y as r - 3 They deduced t h a t a t h in sheath ,which would screen the plasma from the f i e ld , would formand move inward u n t i l the pressure of the f i e ld ~ s t ins idet was su f f i c i en t to balance the plasma pressure . La te r

Ferraro (8) solved the i dea l i zed one dimensional problem,where the f i e ld f a l l s off as -JX in cons iderab le d e t a i land came to the same genera l conclusion concerning theformation of a cur ren t sheath and ts decelera t ion to r e s t .

The ques t ion of the t r ans i en t dis turbances involvedwhen the so l a r wind changes ts i n t e n s i t y i s not , however,with in the scope of t h i s paper . Cer ta in ly before anyquan t i t a t i ve work could be done on t ha t fo r the r ea l threedimensional problem,one must be ab le to so lve quan t i t a t ive lythe s impler problem of the s teady s t a t e i n t e rac t ion of the


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-4 -

e a r t h s Cield wi th constant i n t e ns i t y plasma stream.Dungey 9) seems to have been the Cirs t one to r ea l i ze

tha t the cav i ty must c e r t a in ly close on the night s ide dueto the Cini t e plasma pressure , and t h a t thereCore thee a r t h s Cield must be en t i r e l y conCined by the s o l a r wind.

The topologica l desc r ip t ion o the Cield with in thecav i ty i s due to Johnson 10) who in t roduced the idea t ha twith in the cavi ty those Cield l i ne s t ha t l i e near the po lesdo not ro t a t e r i g i d l y with the ea r th as do the Cield l i ne sa t lower l a t i t ude s but ins tead remain in the t a i l of thecav i ty and coun te r - ro t a t e as descr ibed in seot. ion 8 .

Zhigulev and Romishevskii 11) seem to have been thef i r s t to have sugges ted t ha t the wind i s supersonic and t ha tthere fore detached bow shock should be formed upstreamCrom the cav i ty . The plasma i t s e lC i s es s en t i a l l y c o l l i s ion less , so , in order to have such s h o k ~ i t i s necessaryto have magnetic Cields in the plasma which can serve torandomize the pa r t i c l e motions, and the necessary condi t ionfor shock i s t ha t the Clow ve loc i ty exceed the AlCvenve loc i ty . Lees 12) has shown tha t iC there i s r ad i a l

Crom the sun) magnetic f i e ld giv ing r i s e to such bowIshock t ha t the plasma which has become subalvenic on pass ing

through the shock w i l l acce le ra t e again to supera lvenicve l oc i t i e s on f lowing around the cav i ty , and t ha t t h i s con-verging plasma w i l l there fore form con i ca l wake shocktta t the t a i l of the cavi ty .


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-.5-Once the genera l pr inc ip le s governing the conf inement

o t h e , e a r t h s f i e ld were well unders tood, numerous i n v e s t i -gators se t to work to t r y to obtain a more quant i t a t ivepic ture or the r e s u l t i ng cavi ty . I.t tu rns out t ha t ther e l a t ed two dimensional problem of plasma f low pas t a l i nedipole can be done ana l y t i c a l l y by the technique of a conformal t ransformat ion . This was done t o r the stream normalto the dipole ax i s by Dungey, whose e a r l i e r so lu t ion wasnot publ ished un t i l 1961 ( l J ) , and or a rb i t r a ry or ien ta t ionby Zhigulev and Romishevskii (11) . La ter Hurley 14) solTedthe same problem but by a s l i gh t ly d i f f e r e n t method.

Beard 15) was the f i r s t one to a t tempt a so lu t ion ofthe three dimensional problem. e s impl i f ied the problemby assuming t ha t a t any point j us t i n s ide the sur face thef i e ld i s j us t twice the t angen t ia l component of the undis -tu rbed d i po l e f i e l d . e ju s t iC ie s t h i s by poin t ing out t ha tit would be exact i the sur face were an i n f i n i t e plane,which of course it i s f a r from being. However, t h i s s impl i -f i ca t ion enabled him to wri te down a p a r t i a l d i f f e r e n t i a lequat ion for the sur face , and the so lu t ion of t h i s equat ionseemed to give a reasonable shape for the sur face . Beardonly appl i ed h i s method to the non-polar regions on the sunlit side of the ea r th fo r normal inc idence of the s t ream;however, soon papers began to appear applying t h i s approximate boundary condi t ion to the so lu t ion of more and morecomplex problems. For ins tance Spre i t e r and Briggs 16)


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-6-extended the so lu t ion to the night s ide and consideredvar ious or ien ta t ions of the d ipole r e l a t i ve to the stream,but solved only ~ o r the t r a ce of the s u r ~ a o e in the meridianplane conta in ing the ear th-sun l i ne . Beard (17) at temptedto improve h i s approximat ion y i nc1ud ing as pa r t o h i s

source f i e ld the f i e ld of a cur ren t system on the s un l i tpor t ion of h i s s u r ~ a c e hen he ca r r i ed t h i s out , itchanged h i s r e s u l t s Tery little Spre i t e r and Alksne (18)r eca l cu la t ed the meridian and equa to r ia l cross sec t ions forthe case when there i s a westward f lowing r ing curren t ofabout f ive mi l l ion amperes a t a dis tance of about ten ea r thr a d i i .

In the meantime o t he r s who were unsa t i s f i ed withBeard ' s approximation have at tempted to obta in so lu t ions bymore r igorous methods, two of which have been proposed. Bothof these methods es s en t i a l l y involve s e t t i ng up a t r i a l surface, t e s t i ng to see if the surface s a t i s f i e s the completeboundary condi t ions , modifying the surface in such a way asto improve the agreement , and i t e r a t i n g the process of t e s t ing and modif ica t ion un t i l the r e s u l t converges to thecor rec t answer . Slutz (19) proposed to solve for the sca la rpo t en t i a l of the f i e l d i n s ide the sur face , t r e a t i ng it as acavi ty in a diamagnet ic medium, and then t e s t the surface bysee ing whether the f i e ld had the co r rec t value a t each po i n tju s t ins ide the su r face . Levere t t Davis , J r . a n ~ theauthor (20) proposed to solve for the cur ren ts ~ r o p o r t i o n l


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-7-to the f i e ld ju s t i n s ide which must f low on the sur face inorder to balance the pressure and then t e s t the accuracy ofthe sur face by computing the moments of the f i e ld outs ide .The two papers j u s t c i t e d apply these methods to the simpleth ree dimensiona l problem of d i po l e f i e ld in uniformpressure plasma which served pr i mar l l y to t e s t the convergence of the methods. In what fo l lows, the moment t echniquew i l l be extended to the wind problem.


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-8 -

2. Model or the Calcu la t ions .

Despi te the long h i s t o r y o the problem and the l a rgeamount of e f f o r t t ha t has been given it t he re i s still agrea t dea l about the so la r wind i n t e rac t ion with the mag-netosphere t ha t i s e i t h e r unknown or contes ted . One of thefew th ings t ha t i s genera l ly agreed upon i s t ha t the surfacebounding the magnetosphere i s r e l a t iv e ly t h in .

Ferra ro 8) was the f i r s t to quan t i t a t i ve ly ca lcu la tethe th ickness of t h i s surace by cons ider ing an i dea l i zed ,one-dimensional problem. Dungey 21) s t reamlined h i s c a l -cu la t ion and el iminated some ambigu i t ies which it contained.The same r e s u l t s .can be obta ined by a d i f f e r e n t metho4 usedby Davis , Lust and Schlu te r 22) in ca lcu la t ing the s t ruc -t u re of hydromagnetic shock waves. This l a t t e r method,which s t r e s ses more the ind iv idua l p a r t i c l e approach andenables one to obta in the t r a j ec to r i e s of the p a r t i c l e s asa func t ion o t ime, i s given in Appendix I . There it i sshown tha t the t r a j ec to r i e s of the p a r t i c l e s of a coldplasma, whose pa i r ion e lec t ron mass i s Mt and p a i rdens i ty i s N0 pro jec ted normally with ve loc i ty U0 i n toa region of cons tan t f i e l d B0 . 16rrMtN 0u;)t are as shownin Figure 2 . In add i t ion it i s shown t ha t the magneticf i e l d f a l l s o in the plasma in d i r e c t propor t ion to thedisplacement of the p a r t i c l e t r a j e c t o ry from its asymptote .

Thus it i s c l ea r from Figure 2 t h a t for a dens i ty of2 .5 pro tons /co the f i e l d has f a l l en to s of its i n i t i a l


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-10-value in a dis tance of only about f ive ki lometers . andequat ion I -22 shows t ha t t he r ea f t e r t decreases by afac to r of two every 1.65 Km These d i s t ances a re of courseneg l ig ib l e compared to the sca l e of the su r face .

Knowing tha t the surface i s neg l ig ib ly th in , t i snext necessary to decide what pressure i s exer ted on thesur face by the streaming plasma outs ide .

For the model assumed in Appendix l specular r e f l e c t ion of normal ly d i rec ted pa r t i c l e s ) the pressure i s eas i lyin fe r red by a momentum ba lance .

2.1)

In genera l the pa r t i c l e s a re inc iden t upon the sur faceobl ique ly r a th e r than normal ly but t h i s does not changes i gn i f i c an t l y the r e s u l t s a r r ived a t in Appendix v ALorentz t r ans format ion based on a r e l a t i ve veloc i ty pa ra l l e lto the in te r face w i l l reduce the problem to one of normalincidence. Thus any constant ve loc i ty which i s p a r a l l e l tothe sur face and small compared to the ve loc i ty of l i gh t maybe superimposed on the given so lu t ion without a l t e r i ng thesca le and s t ruc tu re normal to the sur face . The only modif i c a t i on necessary in equat ion Z l i s to rep lace the t o t a lve loc i ty u by i t s normal component U c o s r , where ~ ri s the angle which the wind makes with the normal to thesur face . Using the abbrevia t ion Mt Mi+M8 for the t o t a lpa i r mass. the pressure law fo r a r b i t r a r y angle of inc idence


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-11 -then becomesa

I f the surface i s a c t ua l l y curved r a the r than f l a t ,then the tens ion in the magnetic f i e l d l i n e s ly ing in thesur face w i l l he lp to balance the p ressure of the f i e ld j us ti ns ide the surface and equat ion 2.2 i s not pr e c i s e l y cor -r ec t . However, t h i s cor rec t ion i s c l ea r ly very smal l b e-cause the normal force exer ted on the sur face by the f i e ldl i n e s in the surface i s propor t iona l to the r a t i o of thee f fec t ive sur face th ickness to its rad ius of curva tu re .For the magnetopause t h i s r a t io i s about 10-4.

A more se r ious objec t ion to equat ion 2.2 a r i s e s fromthe assumption made throughout the ca lcu la t ions t ha t theoutgoing s t ream passes unimpeded through the ingoing s t ream.From an ind iv idua l p a r t i c l e viewpoint t h i s assumption wouldc e r t a i n l y be qui t e va l id if the re were no magnetic f i e l d s inthe plasma, fo r the dis tance which a s ing le r e f l ec t ed protonwould t r a v e l back through the s t ream before it s cumulat ivede f l e c t i on approached 90 i s of the order of 10 A .U . v i r t u a l l y i n f in i t e ) fo r a so la r wind of 5 Km/sec and 2 .5pro tons /co . Also, us ing the formula given by Sp i t ze r 2J,p.78) fo r the r e l axa t ion t ime in a plasma def ined as theaverage t ime for a t yp i c a l p a r t i c l e to be de f l ec t ed 90) ,one f inds t ha t if the wind has a t empera ture of 105 K, itsown in te rna l r e l axa t ion t ime i s of the order of 105 seconds.


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Since the length of the agnetosphere cav i ty i s of the orderof 4xto5 Km the wind passes it in about lOJ seconds or onlyone hundredth of its own i n t e r na l r e l axa t ion t ime.

However, objec t ions do a r i s e from the randomizinge f f ec t of any magnet io f i e1ds conta ined in the wind andfrom the pos s ib i l i t y of a co l lec t ive in te rac t ion such as atwo s tream i n s t a b i l i t y . Parker 24) worked out the problemof two in te rpene t ra t ing cold plasma s t reams and came to theconclus ion t ha t the so la r wind f lowing through a s ta t iona ryin te rp lane ta ry gas would be uns tab le and would l ead to ashock f ron t only about 100 meters th ick between the , two.Presumably, then the counterf lowing st ream of r e f l e c t edpa r t i c l e s might s imi la r ly reac t with the incoming s t reamthus providing the d i s s i pa t i ve mechanism needed to have ath in s tandoff shock. Noerdl inger 25) a l so t rea ted t h i sproblem in a very genera l manner. n the o the r hand ad e t a i l e d t rea tment by Kellogg and Liemohn 26) has shownt ha t two cont ra s t reaming plasmas are not necessa r i ly un-s tab le i t h e i r i n t e rna l temperatures a re high enough compared to t he i r r e l a t i ve k in e t i c energy. For i n s t ance theyshow tha t two equal dens i ty plasmas each with i n t e r na lt empera ture T and streaming through each othe r wi thr e l a t i ve ve loc i ty u a re s tab le i

2.J)

For a 5 Km/sec wind t h i s i nd ica t e s t ha t there i s no


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- 1 ) -in te rac t ion with the r e f l e c t e d plasma as long as its tempe ra tu re i s grea te r than about ) ,000K, which i s more thanan order r magnitude below presen t e s t i ma t es of its temp-e ra tu re .

The d e f l e c t ~ o n and r n d o m ~ z t i o n of pa r t i c l e s y f i e l d sconta ined in the wind i s the most ser ious objec t ion to thehypothes i s of in te rpene t ra t ion . t has been shown by spacec ra f t da t a 27) t ha t there are f i e ld s with in the so la r wind,However, it i s not wi th in the scope of t h i s paper to t r y todecide if there i s or i s not a s teady s t a t e shock envelop-ing the magnetosphere . e wi l l use the assumption t ha t thepa r t i c l e s are specular ly r e f l e c t e d i . e . do not i n t e r a c twith the incoming s t ream) because the pressure law it givesi s as good as any other and it bas the fu r the r advantage ofsimpl i fying the ca lcu la t ions .

As a f i na l defense of the pressure law der ived from theassumption of specula r r e f l e c t i on it i s worthwhile to notet ha t ordina ry hypersonic f low pas t a blunt body r e s u l t s inju s t such a pressure d i s t r i bu t i on 28) . The only changenecessary i s the subs t i tu t ion of the pressure a t the s t a g -

2nat ion poin t for the fac to r 2MtN00 ) . This change a l t e r sonly the sca le of the so lu t ion and not its shape.

Another object ion to t h i s s i mpl i f i ed model a r i s e s fromthe fac t t h a t for a cold plasma the sur face must extend toi n f i n i t y on the night s ide , whereas the r e a l wind has at empera ture of the order of 105 K and there fore would closeof f the cavi ty a t a f i n i t e dis tance due to its thermal


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-14-pressure . The maximum rad ius o ~ the cavi ty i s determined

almost en t i r e ly by the momentum ~ l u x o ~ the wind, but ~ o ra given momentum ~ l u x the loca t ion o ~ t h i s maximum rad iusi s determined by the thermal pressure which must there ba l -ance the pressure o ~ the ie ld jus t i ns ide . Past t ha t pointthe i e l ins ide ~ a l l s o ~ ~ so rap id ly t ha t the shape i sdetermined pr imar i ly by the ra te a t which the gas can expandi n to vacuum. According to Lees 12) the r e su l t i ng cavi ty i sabout 60 ear th rad i i in length. This pressure r e su l t i ng

~ r o m the plasma temperature wi l l be ignored , however, simplybecause i t s inc lus ion would se r ious ly complicate the problem.

T h e r e ~ o r e the computed s u r ~ a c e wi l l have littl r e l a t i on tothe ac tua l magnetosphere on the ~ a r night side o ~ the ear thbut it should still give a good approximation to it on thedayl igh t s ide .

The quest ion of i n s t ab i l i t i e s in the surface i s animportant one, but one about which there i s no generalagreement. Parker 29) considered the two dimensionalproblem of a tenuous ionized gas inc ident upon the sur face

an incompressible conduct ing ~ l u i d in which i s embeddeda uniform magnetic ~ i e l d e found it to be uns tab le anddeduced t h e r e ~ r o m t ha t the surface of the magnetosphere i suns tab le . Dessler 30) concluded ~ r o magnetic da ta a t thesurface of the ea r th tha t the surface must be s t ab le butColeman and Sonet t 31) took except ion with the bas i s o ~ h i sargument. Later Dessler 32) advanced an independent andvery convincing argument for the s t ab i l i t y of the sur face .


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The present au thor f ee l s t h a t the i n s t a b i l i t y of Parke r ' smodel proves noth ing concerning the r ea l sur face , f i r s t b e-cause the outer f r inges of ~ h magnetosphere are not loadedwith mat te r l i ke the f i e l d in h i s problem and second becauseh i s problem ignores the s t ab i l i z i ng curva ture o the Cieldl i ne s . Having t h i s demonstra ted the moot na ture of thes t a b i l i t y problem we w i l l now ignore it and assume the su r -face i s s t ab le i n order to ca lcu la t e its s teady s t a t e shape .If l a t e r i nves t iga t ions should demonst ra te tha t it i s indeeduns tab le , the s teady s t a t e so l u t i on w i l l a t l e a s t provide

va luable zero order approximation to itIn the numerical ca lcu la t ions of t h i s paper , the r ing

curren t desc r ibed by Sone t t , i JJ) w i l l be ignored .It could be eas i l y inc luded, but it was not f e l t t h a t itwas advisable a t t h i s t ime to expend the computer t imewhich would be requi red to so lve the problem fo r var iousr ing cur ren t s t r eng t hs and diamete rs .

In summary then , it w i l l be assumed t h a t the so la rwind problem has s teady s t a t e so l u t i on in which an i n f i -

n i t e l y th in curren t shea th t e rmina tes the e a r t h ' s magneticf i e ld , assumed to be simple d ipo le ; and t ha t the pressureexer ted on t h i s sur face by the wind i s given by equat ion


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-16-J. The oment Technique.

The moment technique i s a genera l method which can, inpr inc ip le , be used to dete rmine the shape of the sur face ofsepara t ion in any problem involv ing an i n f i n i t e l y conduct ingplasma sepa ra ted from a magnetic f i e ld by an i n f in i t e s imal lyth in curren t shea th . Of course any such problem involvestwo sub-problems. Fi r s t , one must be able to compute thepressure P exer ted by the plasma on the surface fo r anyassumed sur face shape. This i s a problem in kine t i c theoryand in the discuss ion which fo l lows its so lu t ion w i l l betaken as given. Second, one must be able to solve fo r themagnetic f i e ld ins ide any assumed surface shape and asce r -t a i n whether its pressure balances the plasma pressure . Theboundary condi t ions on the magnetic f i e ld j us t i n s ide thesur face r ~ as follows&

J . l )

which amounts to saying tha t the f i e l d i s excluded from theplasma, and

J .2 )which i s necessary for dynamic equi l ibr ium.

The bas ic idea o the moment technique i s to rep laceequat ions J l and J .2 by two d i f f e r en t but equiva len t con-d i t i ons . Fi r s t , if the f i e l d i s everywhere zero in theplasma reg ion as equat ion J l impl ies , then the sur face


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-17-curren t a t each poin t of the sur face must be Bt/47T , whereBt i s the magnetic f i e ld j us t i n s ide tha t po in t . Usingt h i s f ac t , equat ion J .2 may be wr i t t en in terms of the su r -face cu r ren t .

J . J )This f ixes the magnitude of J.. . a t every poin t on the su r -face , and then in pr inc ip le the d i rec t ion of i s de te r -mined i f we know its d i rec t ion on one l i ne of the sur face)by the requi rement t h a t be divergence f ree . Howeverthe d e t a i l s of the process for determining the d i r e c t i o n of

w i l l depend e n t i r e ly upon the pa r t i cu l a r problem; fo rins tance , see sec t ion 4 for the uniform pressure problemand sec t ion 5 for the plasma wind problem.

Fina l ly equat ion J l i s rep laced by the condi t ion t h a tthe magnetic f i e ld vanish everywhere in the plasma reg ion .This w i l l be t rue if each of the magnetic mul t ipole momentsof the sources in the f i e ld region i s cance l led by the cor-responding moment of the sur face cu r ren t . Actua l ly thef i e ld wi l l vanish to a very high order of accuracy if onlythe lower moments cancel , and it i s t h i s fac t tha t makesthe moment technique usefu l .

\ V \ 1 f f , ( i) ;\> . > ( ftT h e - ~ - S m a ~ r e - g i - o n i s assumed cur ren t . f ree so t ha t

VX a.o and the magnetic f i e ld may be decomposed i n to mul--t i p o l e moments e i t he r in terms of its sca la r or vectorpoten t i a l .

The sca la r po ten t i a l , def ined to be the func t ion


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-18 -whose grad ien t i s ~ i s cer t a in ly a so lu t ion o ~ Laplace s

2equat ion, since VB V p .o. Likewise, i ~ we d e ~ i n e the...vec tor poten t i a l A..._ to be the ~ u n c t i o n wht cu r l i s Band choose a gauge in which VAO each o ~ ts componentswi l l s a t i s ~ y L a p l a c e equat ion s ince \ 7 X B V X V X A \7 A ) -V 2 .A.o. The ~ o l l o w i n g ~ u n c t i o n s ~ o r m a se t ofso lu t ions o ~ Laplace s equat ion in terms o ~ which any so lu-t ion which vanishes a t i n ~ i n i t y such as cf or .Ax maybe expanded.

n nm - nO ,l , ry:JmO,l , npO,l J .4)I ~ the f i e ld region surrounds the plasma region, then so lu t ions vanishing a t the or ig in a re needed ins tead , but t h i scase wi l l not be considered fu r the r . Thus we may wri te thefol lowing express ions fo r the poten t ia l s .

n 1L : s nnm nm

J .5)

(J .6)

Here Rn has the uni t s of a l ength and J 0 the un i t s o ~curren t -per-un i t -wid th . These ~ a c t o r s have been wri t t ene xp l i c i t l y so tha t the remainder of the r igh t hand sidemight be dimensionless . In general , lower case l e t t e r s wi l ldenote dimensionless var iab les and cap i t a l l e t t e r s wi l ldenote dimensioned var iab les except ~ o r tha moments andthe funct ions such as Dp and Pm h i h b i 1c are o v ousnm n


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-19-dimensionless) . Thus

J .7)

Since there a re th ree t imes as many vec tor moments assca la r moments and yet e i t he r s e t of moments i s adequate todescr ibe the f i e ld , it fol lows t ha t the vec tor moments cannot a l l be independent quan t i t i e s . In Appendix I I I , 2n+J)re la t ionsh ips are der ived which must hold between the vec tormoments for each value of n , and it i s pointed out t ha tthere a re (2n- l ) more re la t ionsh ips which wi l l depend onthe gauge o (s ince spec i fy ing the cu r l and divergenceof still l eaves one f ree to add to the gradien t ofany sca la r func t ion which s a t i s f i e s Lap l ace s equa t i on ) .Thus t here a re r e a l l y only 2n+l) independent vec t o r momentsfo r each value of n , ju s t as there are 2n+l) sca la rmoments. The equa t ions r e l a t i ng the sca la r moments to thevec tor moments are a l so der ived in Appendix I I I . The 4n+4)re la t ionsh ips given by equat ion I I I -2 J can be summarized asfo l lows:

l s m : 6 n pO, l

yP (2p- l )x1 -Pnn nnJ .8)

t i s c lea r from t hese equat ions why the sca la r moments must


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be considered. In order t ha t the magnetic f i e ld vanishouts ide the sur face it i s only necessary t ha t its s c a l a rmoments vanish . and t h i s c l ea r l y does not imply tha t itsvec t o r moments vanish . The only reason t ha t the vec t o rmoments a re considered a t a l l i s t ha t they a re cons i de r -a b ly eas i e r to ca lcu la te d i r e c t l y than the sca l a r momentsa re , and by equat ion J .8 the sca la r moments can be eas i l yobta ined from them.

Insummary, then, the bas ic out l ine of the momenttechnique i s as fol lows: F i r s t ca lcu la t e the moments ofthe sca la r po ten t i a l of the f ixed sources with in the s u r -face . Then assume a tri l shape for the sur face and de te r -mine the r esu l t ing f lu id forces i t i s assumed t ha t t h i s i sposs ib le ) . Next ca lcu la t e the sur face cur ren t which woulds a t i s fy equat ion J . J on tha t sur face , and f i na l l y ca lcu la tethe sca la r moments of t h i s sur face cu r ren t . I f these j us tcancel the moments of the f ixed sources , the problem i ssolved; if not , vary the sur face appropr ia te ly and r epea tthe process un t i l an adequate ly accura te so lu t ion i sobta ined.

In Sect ion t h i s method w i l l be appl ied to the t e s tcase of a dipole in a uniform pressure plasma, and in suc -ceeding sec t i ons it w i l l be appl ied to the more impor tantcase of a dipole in a plasma wind.


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4. Soluti .on for t he Uniform Pressure Case.Consider a magnetic dipole of moment M z emu su r

rounded by a s ta t iona ry plasma of uniform pressure Pdynes/cm2 The so lu t ion of t h i s problem i s di scussed ina paper wri t t en by Levere t t Davis . J r . and the author 20)but t wi l l be repea ted here in terms of the more genera lno ta t ion of Sect ion ) .

The un i t of l ength Rn w i l l be chosen to be therad ius in the equa to r i a l plane to the poin t where the mag-ne t i c pressure of the undis turbed dipole f i e l d equa ls thegas pressure J _. .R M 8rrP)n 4.1)From equat ion ) . J t i s c lea r t ha t ~ has t he constant mag

. .ni tude P/2rr) 2 so t h i s w i l l be chosen as the un i t curren tJ 0 Obviously the bounding surface and ~ u s t have ax ia lsymmetry so must be in the d i r ec t i on .

The sca la r poten t i a l of the d ipole a t the f i e ld poin tR2 R r 2 i s :.. n....,

4. 2)' :...... tObviously, then a l l sca la r moments of the-sur face must

vanish except 4 r r see equat ions J.4 and J .6) .I f the coord ina tes of sur face poin ts a re spec i f ied by

~ a n r the vec tor po ten t i a l of the f i e ld due to the sur facecur ren ts i s :

4.J)


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-22-Use equat ion 6 .2 to express -1- l as an i n ~ i n i t e se r i e s ,each term ~ which i s separable i n to i t s and r

2depend

ence. The symmetry about the pola r ax i s enables the -in tegra t ion to be done e a s i l y with the r e su l t :

wheren

( ~ 2 ) R n J o ~ ~ ~n l27T rr n+l [ 2 dr>2] 1 ) n(n+l ) r r + cnr Pn cose s in9d8

0

4.4)

4 .5)

Set e ~ = ( c o s ~ e - s in -e ) in equation 4.4 and it becomes c l ea rP ~ ,...Xby comparison with equat ion J 5 t ha t Y0 1--x1 1 - I and a l ln n nother vector moments are zero . This means ( r e fe r to equa-t ion J .8) tha t s ~ n i n and a l l other sca lar moments are

0iden t i ca l ly zero . Actual ly even sn 0 a for n even, because1Pn cose) i s an odd funct ion fo r n even. Thus the problemreduces to choosing a funct ion r (9 ) such tha t :

~ . ~ 4rr nJ, .5,7,9 4.6)

Since the surface has cusps a t the poles and i s sym-metric about the equa to r i a l plane it i s ba t t e r to expressr as a func t ion of the magnetic l a t i t u d e ~ r t h e r than thepolar angle e.

4.7)

To solve for the parameters , se t C l a t f i r s t and ignore I 1 Consider the next N n o n - t r i v i a l I ( i . e . those for nJ ,5 ,7 ,n 2N+l) . t i s easy to d i f f e r en t i a t e the In under the


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- 2 J -in t egra l s ign and obtain ana ly t i c express ions Cor the r a t e so change o the In with respec t to the various c 5 Hencethe General ized Newton s Method was used to determine thec which reduced the In to zero . f i na l l y I i s madeequal to ~ y adjus t ing C, which i s seen to be the equa-t o r i a l rad ius . The computation was carr ied out on a Bur-roughs 220 computer Cor various values o N up to seven.For the case N7 the numerical r e s u l t s are given in Table

and the r esu l t ing cross sec t ion i s p lo t t ed in Figure J .

Table 1. Coef f ic ien ts in the Equation for the Surface c 1.41J95 CJ a 0.001085 c6 -O.OOOJ26c l 0.120039 04 --0 .000200 07 = 0.000094c2 0.004180 c 0.000597

t i s t rue t ha t a t the pole the l a s t few terms of equat ion 4.7 are of the order o 7 o the f i r s t term, but t h i sdoes not ind ica te an e r ro r of t ha t order the re . The coe f f i -c ien t s in Table are not the f i r s t seven terms in the powerse r i es expansion o the t rue sur face . They are . the coeCCi-c ien t s of the polynomial of degree fourteen which mostc lose ly approximates the t rue sur face . There are two r ea -sons Cor bel iev ing t ha t the so lu t ion i s very accura te evennear the pole . Fi r s t when the computation was carr ied outwith only four parameters , the rad ius of the computed su r -face near T~ where agreement was wors t , was only aboutone percent grea te r than the corresponding rad ius of the


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1 . 0

5

z

+--_._--- ---L--- ----_L_-__L_______l__L_. __ __ _ __. L_ _.__-- JU----J.5 1 . 0Uni t s of Rn

Figure J Cross sec t ion {one quadran t of tho sur faceo u n d i n ~ a d ipo le f i e l d in a uniform pressure plasma.The dashed l i n e was ca lcu la t ed by us ing Beard s condi -

t i on ; the so l id l i ne , by using t he moment t echnique .

1.5


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seven parameter su r face . Second, when c was changed soas to decrease the rad ius to the surface by only 0.1 a t thepole , the r e s idua l f i e ld s a t d i s t ances grea te r than O.JRnouts ide the sur face ca lcu la ted as descr ibed in the t e s t ofthe next sec t ion) were inc reased by a fac to r of ten or more.A major f ea tu re of i n t e r e s t in t h i s computat ion, in addi t ionto providing a t e s t , o f the moment technique , i s t ha t itind ica te s tha t the surface very de f in i t e ly has cusps a t thepoles and tha t these cusps do not go c lea r to the or ig in ashas been suggested, but r a the r i n t e r s ec t the ax i s a t af i n i t e d i s t ance . The cusps undoubtedly i n t e r s ec t the ax i st angen t ia l ly in r e a l i t y , but such a sur face could not .berepresented by a polynomial with a f i n i t e number of termssuch as was used. However, the grea t e r the number of pa r -ameters t ha t were used the s teeper the angle of i n t e r s e c t i o nwas. t i s easy to see t h a t these a re the r e s u l t s t ha tshould be expected . Consider a cavi ty in a medium of zeropermeabi l i ty . I f there were a f i n i t e angle between the s u r -face and the ax i s , the f i e l d there would be zero, and if thecusp were a t the d i po l e the f i e ld would be i n f i n i t e ; ine i t he r case the f i e ld would not be in equi l ibr ium with theplasma pressure .

I f we def ine the f i e l d ju s t i n s ide the sur face to beBs(87TP)f , then it i s a simple matter to see t ha t thechange in the f i e ld , a t the or ig in due to the sur facecur ren ts i s


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TTr cos cJ -26- dCX (4.8)For a sphere the i n t eg r a l i s j u s t rr/4, and fo r any othersur face t would be s l i gh t l y grea te r . For the computed sur -face t i s 0.769JJ . Thus a 10 Y di s turbance in the geomag-ne t i c f i e ld a t the ea r th could a r i s e from a sudden change ofpressure o 2.52 x lo - 10 dynes/cm2 on the surface ( i . e . apa r t i c l e dens i ty t imes tempera ture of 1 .8J x 106 K0 /cmJ ora kine t i c energy dens i ty of 1 .58 x 102 ev/cmJ).

For comparison purposes the uniform pressure problemwas a l so solved by Beard s d i f f e r e n t i a l equat ion technique.To get the equat ion for R( ;, )2](4.10)

Cal l r (O) re and note t ha t d r / d ~ cO a t ~ o by symmetry.Inse r t ing these values , and t he value o Rn from equat ion4.1. i n t o equat ion 4.10 one obtains the r e l a t ion

(4.11)and the d i f f e r e n t i a l equat ion


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r d r ]coscx. -2s in x rdcx -27-

( :-)6 [1 ( d r > jr +Fci'C:(:""eWhen equat ion 4.12 i s solved t gives r(O()/r e Then

4 .12)

r ei s determined by the condi t ion tha t I 1 -4. Since equat ion4.12 i s of second degree there a re two such so lu t ions . Theappropr ia te so lu t ion i s p lo t t ed in Figure and t i s seent h a t t d i f f e r s s i gn i f i c an t l y from the moment t echniquer e s u l t near the po le .

There i s a l so an i n t e r e s t i ng s ide l igh t tha t can begleaned from these ca lcu la t ions . There has been some d i s -cuss ion r ecen t ly as to whether the fac to r f which Beardassumes to be should not be c loser to 1/J. From equat ion4 .11 we see tha t in t h i s th ree dimensional case

f r-J (1.39577)-J a O.J6775e (4.1.3)To determine the r e l a t i ve accuracy of the methods, the

f i e l d due to the sur face was ca l cu l a t ed a t var ious r a d i ialong the po l a r ax i s and in the equa to r i a l p lane) , subt r a c ted from the f i e ld of the dipole loca ted a t the or ig in ,and then div ided by the dipole f i e ld . This g ives a number\17hich would be zero everywhere outs ide the sur face fo r thet r ue sur face and would be one everywhere for the d i po l ef i e ld alone. The computations for t h i s t e s t were ca r r i edout on a Burroughs 220 computer, rep lac ing the sur face byn ine ty -e igh t curren t loops . The r e s u l t s of t h i s t e s t forthe two sur faces are given in Table 2 . The values on the


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-28-polar axis may be inco r rec t by as much as .5 due to t runca-t ion e r ro r . The t runca t ion e r ro r was removed from the equa-t o r i a l va les by subt rac t ing the so lu t ion for a sphere witha o s ~ curren t var i a t ion , which should t heo r e t i c a l l y bezero everywhere and which there fore equa1s the t runca t ione r ro r in prac t i ce . Since the sur face approximates spherenear the equator and the o s ~ cur ren t approximates a u n i -form cur ren t near the equator , the t runca t ion er ror must bevery near ly the same for both cases near the sur face a t theequator . The inheren t roundoff e r ro r in the ca lcu la t ion wasabout .2 x 10-5.

Table 2. Rat io of Net Fie ld to Dipole Fie ld x 105Distance from Moment Surface Beard Surfacethe su r face - -Frac t ion of On the In the On the In theEquator i a l Polar Equator i a l Polar Equator i a lRadius Axis Plane Axis Plane

o. o4 905 -0.4 -61078 7126o.o8 -222 0.2 -42966 67210.16 - 2J o.6 -27676 5997o.J2 - 2.7 0 5 -1.5913 4844o.64 - 0.9 0.5 - 7947 JJ241.28 - 0.2 0.2 - JJ78 18172 56 - 0.1 O.J 0 1222 77J5.12 o.o 0.5 - J86 26710.24 0.2 o.o - 110 81

Clear ly , the moment technique gives a net f i e l d ou t s ide which i s about 0.001 of t ha t given by the surfacederived using Beard s boundary condi t ion .


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-29-Slutz 19) has a l so solved t h i s i den t i c a l problem by

an i t e r a t i ve procedure which beg ins wi th a t r i a l surace .However, h i s procedure involved so lv ing for the sca la rpoten t i a l of the ie ld in s ide the sur face , t r e a t i ng t asa cavi ty in a diamagnet ic medium, and then comparing ther esu l t an t f i e ld s j us t ins ide the sur face wi th the f i e ld sgiven by the pressure law to ind ica te how to change thesur face fo r the nex t i t e r a t i o n . The r e s u l t he obta ined i svery c lose to tha t given by the moment technique exceptnear the equator where h i s cross sec t ion i s near ly f l a tand l i e s about J i n s ide the moment r e su l t . hen thef i e ld s fo r S l u t z s sur face were ca l cu l a t ed they were muchl a rge r than those fo r the moment sur face , espec ia l ly inthe equa to r i a l plane .


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30

5. Rela t ionsh ip of the Current and Surface fo r the Wind Caset i s c l e a r from the proceeding t ha t before the moment

t echnique can be appl ied to t e s t and improve a surface , thecur ren t s f lowing on tha t sur face must be known. Consider ana x i a l l y symmetric source of magnetic f i e ld loca ted a t theor ig in and or ien ted along the y di rec t ion and a plasma mov-ing in ~ -z di rec t ion . surface z x ~ y ) such as the oneshown in Figure 1 w i l l be formed. Choosine x and y as theindependent var iab les enables the surface to be descr ibed ya s ing le valued func t ion , r e s t r i c t s the independent v a r i a -bles to a f i n i t e range and s impl i f i e s ce r t a in formulas in thederived l a t e r . Adopting the nota t ion z = z z ez theX oX y cy,outward normal to the surface has the fol lowing form.

= .S.l)

Therefore , s ince y i s def ined as the angle between the normaland the ear th -sun di rec t ion , it fo l lows tha t

5.2)Define the uni t surface cur ren t see equat ion 3.7) as

2 tJ 0 = MtN 0 U0 /rr Then the magnitude of the dimensionlesscur ren t i s e a s i l y obtained from equat ions 3.3 and 2.2.

j cos 1 1 5 .3)The problem now i s to determine the di rec t ion of j .

_, ;


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-Jl-0 ~ course s ince j must l i e in the s u r ~ a c e it must be...

perpendicu lar to ~ the normal to the sur face .n j 0... ,.., (.5.4)

The l a s t condi t ion necessa ry to determine j completely i st ha t it must be divergence ~ r e e or in other words the f lux

o ~ j across any closed curTe on the sur face must be zero ....

I ~ t h i s i s t rue , then there must ex i s t a f lux func t ion ,def ined by the l i ne i n t eg r a l

f(x Y)f x ,y ) n . j X ds( o,o ) ~ .. .5 5)t ha t depends only on x and y and not on the path of i n t e -gra t ion chosen on the s u r ~ a c e

The u s e ~ u l n e s s of t h i s ~ l u x func t ion a r i s e s from the~ a c t tha t if f x ,y ) i s spec i f i ed , then the correspondingj (whioh i s there fore guaranteed to be divergence less) can...

be eas i l y der ived from it us ing equat ions .5 5, .5.1, .5.2and .5.4 in t ha t order .

f n . j xGliJ .. - j . n x ( z e )x ox ~ ... [dx] ,_ r X x,...zy - cos1bj [ +Z e +Z (z e -z e,.. ~ y y,..z X X Y y ~ x J - jy (.5.6)

A s imi l a r ca lcu la t ion shows t ha t the same r e su l t , exceptfo r the minus sign, holds with x and y in terchanged.


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-J2-With jx and jy known, equat ion 5.4 can now be used

to obtain in terms of der iva t ives of f and z .z

Hence the sur face current i s

5. 7)

5.8)

Subs t i tu t ion of t h i s value of i n to equat ion 5.J t r a ns forms it in to a par t i a l d i f f e r e n t i a l equat ion r e l a t i ng thefunc t ions z x ,y) and f x , y ) .

cos t 5.9)

It seems most na tu ra l , in using the moment techniqueto solve any problem, to guess a surface and then computethe cur ren ts t ha t should flow on t ha t sur face . In otherwords, assume z x ,y) i s known and use equat ion 5 9 tosolve for f x , y ) .

Unfortunate ly , t h i s s t ra igh t forward way i s not t r a c t a -b l e . Equation 5 9 as an equat ion for determining f x ,y )from z x ,y) i s non- l inear and it appears from manyt r i a l s ) to be imposs ible to devi se a s tab le numerical methodof solving it Of course ana l y t i c a l methods can be ru ledout from the beginning because of the necessa r i ly complicated func t ions tha t must be assumed for z x ,y ) .

However, there i s nothing inherent in the ove ra l l


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-33-method which requ i res one to begin the process by assuminga surface . I f i n s t ead a f lux func t ion with an appropr ia tenumber of parameters i s assumed, then equat ion 5 9 might beused to obtain the sur face which s a t i s f i e s equat ion 5.3.In fac t if z x ,y ) i s considered to be the unknown funct ionin equat ion 5 9 it then becomes a l i nea r equa t ion .

5.10)

The s ign chosen fo r the square roo t i s the one which i sappropr ia te in the f i r s t quadran t .

t turns out t ha t even t h i s l i n e a r f i r s t order equat ionseems to be numerical ly uns tab le for any s t ra igh t fo rwardmethod of so lu t ion invo lv ing a r egu la r ly spaced gr id . How-ever , the p a r t i c u l a r form of the coef f i c i en t s in t h i s equa-t ion make it poss ib le to reduce it to the problem of so lv ingan ordinary d i f f e r e n t i a l equat ion a long c e r t a i n curves . Tosee why t h i s i s so, rewr i te equat ion 5.10 as fol lows.

P x ,y )zx Q x,y)zy R x ,y) a 0 S . l l )

Refer r ing to equat ion 5 .1 for n t h i s i s c l ea r ly equiv-a l e n t to the equat ion

n . Pe Qe Rez) 0~ ~ ~ y ~ 5.12)

which says t ha t a l i ne with di rec t ion numbers P,Q,R) i sperpendicu la r to the normal to the sur face and i s the re foretangent to the sur face . Thus an i n f in i t e s ima l l i n e element


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. 4with d i rec t ion numbers propro t iona l to these w i l l l i e inthe so lu t ion sur face . Clear ly then the d i f f e r e n t i a l equa-t ions

determine a l i ne , ca l l ed an i n t e g ra l curve, which l i e se n t i r e ly in the so lu t ion surface i any one poin t of itl i e s in the so lu t ion sur face . Thus we could cons t ruc t thesur face , i we knew the value of z x ,y ) along one l i newhich i s not an i n t e g ra l curve, by fol lowing the i n t e g ra lcurves which i n t e r s e c t t ha t l i ne .

The th ing which makes t h i s approach fea s ib le in t h i scase i s t ha t the i n t eg r a l curves a re f a i r l y easy to ob ta in .Rewri t ing equat ion 5.1.) expl i c i ty ,

dx=r .y X , .14)it i s c lea r t ha t the f i r s t equat ion t akes an espec ia l lys imple form.

This s imply says t ha t along any i n t eg r a l curve of the sur face f x , y ) . f 0 , a cons tan t . Thus in pr inc ip le for anycurve one could wri te x -x y , f 0 where i s now j us t aconstant parameter . Subs t i tu t ion of t h i s i n to the secondof equa t ions 5.14 gives a s imple ord inary d i f f e r e n t i a l


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Jsequat ion for z y , x y , f 0 .

In prac t i ce it i s much b e t t e r to use the dis tance sin the xy plane) along the curve, r a the r than e i t h e r x

or y, as the independent var iab le in the so lu t ion of equat ion 5.14. In terms of s , then, equat ion 5.14 becomes

dzds . 1 21] tfy 5.16)In ob ta in ing t h i s , has been chosen to inc rease in

the counter -clockwise di rec t ion around the upper neu t r a lpoint .

As pointed out above, de te rmina t ion of the surfaceunique ly requi res spec i f ica t ion not only of the f lux func-t ion f x ,y) but a l so of one l i ne in the sur face . Clear lythe bes t l i ne to use i s tha t par t of the i n t e r sec t ion of thesurface with the XO plane which l i e s between the sub-so l a r poin t and the upper neut ra l poin t . A few of the var -iable parameters w i l l then be used in speci fy ing the cur ren tfunct ion.

In passing it may be noted t ha t when only the parame t e r s speci fy ing t h i s l i ne are changed the f lux funct ionremaining unchanged) it i s unnecessary to r e in t egra t e equat ion 5.16 before ca lcu la t ing the new moments. This fac t canshor ten the computer t ime required for the problem.

Thus we have a d i r ec t method of obta in ing a surface andsurface cur ren t which are o n ~ i s t e n t with equat ion S.J.


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-J6-6. Calcula t ion of the Moments for the Wind Case.

Consider a source of magnetic f i e ld loca ted a t theor ig in and a zero tempera ture plasma wind moving in the -zdi rec t ion . Assume t ha t a sur face z x ,y) and the f lux funct i on f x ,y ) fo r ts sur face cur ren ts a re given. in t h i ssec t ion the formulae w i l l be der ived for the moments ofthose cu r ren t s .

The proper un i t cur ren t dens i ty J =(MtN u2 /rr)t was d e -o 0 0f ined in Sect ion 5. At t h i s t ime the un i t l ength Rn w i l l bedef ined to be the d i s t ance from e i the r neu t ra l poin t to thez ax i s . With the convent ion tha t fO a t the subsolar po in t ,tha t has the double advantage o making both Yl and f a l a t

dfthe upper neu t ra l poin t , because ay l on the l i ne jo in ingthe subsola r point and the upper neu t ra l poin t see the f i r s tparagraph of Appendix IV) .

Let R2 -R r 2 be the coord ina tes of a f i e ld poin t andn. -R=R r be the coordina tes of a poin t on the sur face . The' n'i n t eg r a l form for the vec t o r po ten t ia l i s :

6.1)

To separa te t h i s i n t eg r a l i n t o ts moments, make use of theexpansion of 1 / jr2 -rj , in assoc i a t ed Legendre func t ions .

1- -lr r1-2 ncoUpon making t h i s subs t i tu t ion and t r ans f e r r i ng every th ing


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7poss ib le through the i n t eg r a l sign, A becomes:

where~ m ( 2 - $ m o > f ~ : : J : < . ) P ~ ( o o s e ) o o s ( m ; - p ~ ) r n d s

s

6.3)

6.4)

t i s c lea r by comparison of equat ions J.S and 6.J t ha tthe components of the ~ m are j us t the vec tor po t en t i a lmoments def ined before .

6 5)

Before proceeding fu r the r the source f i e ld w i l l be spec ia l i zed to one which mirrors in the yz plane and mirrorswith a change of sign in the xz plane . Thi s i s necessaryin order fo r the sur face to be symmetric about these twoplanes and topologica l ly s imi la r to Figure 1 . Since thesurface cur ren t must be perpendicula r to the f i e ld j us t in s ide , it i s c l ea r l y f lowing in the x or -x) d i rec t ion asit crosses e i t he r of the p lanes of symmetry, and jx i s aneven funct ion about e i t he r plane . Visua l iza t ion of thecur ren t flow pa t te rn with the help of Figure 4) shows t ha tjy i s odd about both p lanes of symmetry and jz i s even aboutthe xz plane and odd about the yz plane . These symmetriesof the sur face and cur ren t cause th ree - fou r th s o thevec tor moments to vanish i den t i c a l l y .


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JS-For ins tance , consider the symmetries about the xz

0) plane. Cos i s an even ~ u n t i o n of - while jy i sodd, which causes ~ to vanish, s ince the r e s t of thein tegrand i s even. Sin m- i s an odd func t ion of - while

are even, which causes 1and Znm to vanish .Likewise consider the symmetries about the yz )

plane . About t h i s plane jz and jy a re odd, sowhen m i s even ( s ince cos m- i s then even) and

vanishesvanishes

when m i s odd ( s ince then s in m- i s even) . Simi la r ly jxi s even about t h i s plane, which causes ~ m to vanish whenm i s odd.

Thus, us ing the symbol without the superscr ip t toind ica te the non-zero moment for t ha t n and m, the onlynon-zero i n t eg r a l s of equat ion 6.4 a re as fol lows:

Xnm ~ m m 0,2 ,4 . . nynm 1 ynm m 2,4 n nl ,2 , .3 znm ' zonm m 1,.3,.5 . . n 6.6)

Accordingly, when pO or m i s even in equat ions .3.8 thevec t o r moments vanish , and so the assoc ia ted sca la r momentsmust vanish . 1Thus the only non-zero sca l a r moments a re Snmm odd) and these w i l l hence for th be denoted by the symbolSnm For a proper ly symmetric source, then, equat ions .3.8reduce to :5nm znm

--znm6.7)

ml,J,.S n


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i de n t i t i e s .cos f /grad f j

o s ~ Igrad f j

-40-

[l - f 2 r2] ty2 2f + fX y

dxds

ds

dzds

(6.9)

Subs t i tu t ing these i de n t i t i e s i n to equat ion 6.8, the non-zero moments, equation 6.6, become

1 S f )Xnm 4 2 - S ) fnm t d r ds(dx U CmO n+m I ds nm m0 01 S f )Y 8 ~ d f [ d s ~ ) U Snm Tn+M1T ds nm m0 0 (6.10)

1 S f )Z 8 n -m ) 1- d t i d s ( d z ) U Cnm (n+m) t , ds nm m0 0where S f ) i s the t o t a l length in the f i r s t quadrant of thef lux curve f cons tan t , and the U,S and C a r e def ined asfo l lows:

(6.11)

In the computer program these func t ions are eas i ly genera tedby the following recurs ion re la t ions :


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U . 2 n - l ) l lnn U 1 . 2 n - l ) l l znn-

unm

c l .x

2 22 m+l)zUn+1 - x Y )Unm+2n- .. ) n+m+l)

c . c 1c 1-s 1 s1m- m-

6.12)

S =S 11 C 11m- m-

From these r e l a t ions , t i s c lea r tha t the Unmcm and Unmsm~ a c t o r s o ~ the in tegrands o ~ equat ion 6 .10 are simply poly-nomials in x, y and z each term o ~ which i s o ~ degree n . Thehighes t degree o ~ z in any o ~ these terms i s n-m. Since xand y are bounded while z--oo. t i s c lea r tha t the l a rge rthe value o ~ m ~ o r a given n) , the more accura te ly thei n t eg r a l may be evaluated. This leads to the conclusiont ha t the ~ i r s t o ~ equat ions6 .7 i s the be t t e r one to use inca lcu la t ing the sca la r moments. Subs t i tu t ion o ~ equat ions6 .10 in to t h i s equation gives the expl ic i t re la t ion .

6.1J)Thus to t h i s point the machinery has been se t up ~ o r obta ining a s u r ~ a c e z x ,y) and ca lcu la t ing a l l i t s mult ipolemoments. B e ~ o r e proceeding ~ u r t h e r t i s necessary to spec-i a l i z e to a pa r t i c u l a r source ~ i e l d


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-42-7. Speci f i c Solu t ion fo r a Dipole Source.

This sec t ion w i l l begin with a summary of a l l thoseformulae der ived in previous sec t i ons which a re necessa ryfo r programming a computer to obta in a numerical so lu t ion .For the case when the source f i e l d mir rors in the yz planeand mir rors wi th a change o s ign in the xz plane , thesca la r po ten t i a l of the sur face cur ren ts i s

n- R J s Dln o n l ml nm nm m odd only) (7.1)where J (MtN u /rr t, R i s the y coordina te in cen t i -o o o nmeters ) of the neu t ra l poin t and the S a re obta ined fromnm

(7.2)The coordina tes f , s ) a re the value of the curren t func t ionand the d i s t ance i n the xy plane a long the l i n e s f con-s t an t . measured from XO. S f 0 i s the l ength in the f i r s tquadrant of the l i ne Bquations 6.9 give the d e r iv -oa t i v e s of the coord ina tes with r e spec t to s .

dx lrs Igrad r,- f X 7.J)ds grad j

dz [jgra: I ticrs f l2 - l iJ


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-4J-And z a t any point on a curve f cons t an t i s found by i n t e gra t ing dz/ds along t ha t curve . r i n a l l y the func t ions U,C and S are given by equat ion 6.12.

c l -x

U 1 - 2n- l ) JJ znn-2 22 m+l)zU 1- x +Y U 2m+ nm+ 7.11-)

Consider now a dipole source. The sca la r po t en t i a l ofa dipole of moment Me i s-y

Since the po ten t i a l of equat ion 7.1 must be equal and

7.5)

opposi te to t h i s , t i s c lea r t ha t for the t rue sur face

s - 0m n2, J , 4 ml,J, .S n 7.6)and equat ing c o e f f i c i e n t s of the terms gives the sca l ingr e l a t i on

?.?)

which w i l l be used to determine Rn a f t e r the sur face hasbeen made to s a t i s fy equat ion 7.6 approximate ly .

The f i r s t s tep in the so lu t ion of the problem i s thechoice of a func t ion of x, y and some parameters Ai, which


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44

i s s u r C i c i e n ~ ~ Y r e ~ t r i c t e d in func t i ona l Corm tha t fo r anyreasonable va lues oC the Ai t he , r e s u l t i ng func t ion f (x ,y )has a.11 tue qua l i t a t i ve 1 ea tures t h a t the curren t func t ionmust have (as seen from Figure ~ . However, a t the samet ime the parameters must permi t enough v a r i a b i l i t y in 1 tobr ing it su1 f i c i en t ly c lose to the t r ue 1 unction fo r somese t oC va lues of the Ai. Actua l ly the Choice of a parame t r i z ed form for t h i s func t ion vas one of the most t imeconsuming aspec t s of the en t i r e problem, and it i s not herepre tended t ha t the bes t poss i b l e func t ion has been deve l -oped, only t ha t a sa t i sCac t o ry one has . I f any inves t iga to rshould des i r e in the fu t u re to improve on the r e s u l t s p r e -sen ted in t h i s paper , be could sure ly do so by working outa d i f f e r e n t ana ly t i c form fo r f which has the a b i l i t y tocome c lose r to the t r ue f , whatever t h a t is.

Without fu r the r apology then , the cur ren t func t ion usedin t h i s work w i l l be of the Collowing form

(?.8)where (p.-) a re the usua l polar coordina tes in the xy plane ;

2vcos ;; a (v) i s h a l f the rad ius o the sur face a t oOIup /a (v ) : and g and h a r e double power s e r i e s in u and v ,given by equa t ions IV-)9 and IV-54 to IV-56. The motivat ions l ead ing to t h i s Corm fo r f , a s wel l a s the condi t ionson g and h and t h e i r de r iva t ions , are discussed in AppendixIV and w i l l not be considered here , except to note t ha t


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4 spermi t t ing h u , v ) to conta in terms up to u8 and y S al lows22 f ree parameters among the coe f f i c ien t s a f t e r a l l thecondi t ions are appl i ed . These parameters a re denoted byA i 1 ~ i L 2 2 . Likewise it i s shown in Appendix IV tha t ig u ,v) conta ins only terms up to u4 and v4 but i s o the rwiseas unres t r i c ted as poss ib le cons is ten t with the condi t ionson f ) it conta ins l S f ree parameters among its coef f i c i en t s .These a re denoted by Ai, J l ~ i ~ 4 . S . The remaining a rb i t r a ryfunc t ion in equat ion 7.8 wi l l be parametr ized as fol lowsa

(7.9)

As pointed out on page J S t h i s f lux func t ion does notuniquely spec i fy an assoc ia ted sur face , but the pr o f i l e ofthe surface must a l so be spec i f i ed . The pr o f i l e wil l bedef ined to be tha t pa r t of the cross sec t ion o the sur facein the merid ian p lane which l i e s between the subsola r poin tand the neu t ra l poin t . This pr o f i l e w i l l be parametr izedas fol lows:

The dis tance from the dipole to the subsola r poin t ofthe surface i s given by A71 The z dis tance from the sub-so la r poin t to the neu t r a l poin t i s given by A72 A75governs the pla teau in the immediate neighborhood of theneu t r a l po in t . The remaining terms a i d i n ad jus t ing theove ra l l shape proper ly .


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-46-As a l ready mentioned most ~ the qua l i t a t i ve r e s t r i c

t i ons which can be placed on as consequence o ~ thephysics of the problem have been incorpora ted automat ica l lyby the r e s t r i c t i o n s placed on h a n d g in Appendix IV. However , there i s one very importan t r e s t r i c t i o n which can notbe so ~ s i l y f u l f i l l e d . This i s the condi t ion, obviousfrom equat ion 5.9 tha t

jvf < t 7.11)Clear ly every parameter w i l l a f fec t the gradien t of f inway which wi l l depend non- l inear ly on every other parameter .Thus t would be imposs ib le to der ive se t of reasonabler e s t r i c t i o n s which would guarantee t ha t equat ion 7.11 i ss a t i s f i e d . The bes t t h a t can be done i s to t e s t each t r i a lse t os parameters aga ins t equat ion 7.11 and r e j e c t thosese t s which v io l a t e i t . s i g n i f i c a n t l y . In prac t i ce t wasfound tha t t was d i f f i c u l t to f ind se t of parameterswhich d i d n t v io la te t h i s condi t ion a t some poin t in thexy plane , even when the shape and moments of the r esu l t ingsur face were ignored. Thus t was decided to t o l e r a t egrad ien ts grea te r than one as long as they occurred overonly smal l percentage of the sur face ; and in these casesequat ion 5.16 was kept from becoming imaginary by the simpleexpedient of se t t ing the gradien t equal to one. Thi s comp l ica ted the convergence process in tha t cons tan t manualadjus tments were needed in the parameters to minimize theseunphysica l grad ien ts , but t cou ldn t be avoided.


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-47-The obvious way to go about reduc ing the moments i s by

the genera l ized Newton method (used in the uniform pressurecase . s everyone knows who has used the method exten-s ive ly , however, t i s very prone to wandering when thenumber ~ va r i ab le s exceeds or 6, unless t h e problem i s awel l condi t ioned one. From what has been sa id a l ready ,though, t should be obvious tha t t h i s problem i s not awel l condi t ioned one and, indeed, t was found t ha t Newton smethod was v i r tua l l y use l e s s fo r as few as f i ve parametersand moments. One th ing which cont r ibu tes heavi ly to t h i sd i f f i c u l t y i s tha t t he re i s no na tu ra l order ing of theparameters as to impor tance . That i s to say , with 46 p a r -ameters occurr ing in four d i f f e r en t power se r ie s (two ofwhich are double s e r i e s ; which f ive parameters should bechosen to reduce the f i r s t f ive moments? In a l l l i ke l i hoodsome 15 or so of these parameters should r e a l l y be var i edin order to reduce the f i r s t f ive moments smoothly to ze ro .

Therefore s ince t was unr ea l i s t i c to work with l e s sthan about 15 parameters a t a t ime, but even more unrea l i s t i c to t r y to reduce 15 Moments a t a t ime by Newton smethod, t was necessary to work out a new method by whichN parameters Ak) could be used to reduce quant i t i e s (Vi) ,where M


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Where HkidVk/dAi i s assumed to be a cons tan t . Since N>Mt h i s system o equat ions does not have a unique so lu t ionunless an add i t iona l condi t ion i s imposed. The na tu ra lcondi t ion i s to requi re t ha t

7.13)

be a minimum, where wk are approximately chosen weight ingfac to rs for the parameters . There are two advantages tothus minimizing the l ength of the a i vectorr 1) the assumpt ion of the constancy e f the aki i s more va l id , and 2) thecondi t ions such as equat ion 7.11 which have been manuallyoptomized wi l l be i n t e r fe red with as little as poss1b1e.

To solve equa t ions 7.12 and 7.13 toge ther , f i r s t solveequat ion 7.12 for the f i r s t M of the a i in terms of theremaining a i .

7.14)

where H-l i s the inverse of the square matr ix formed fromonly the f i r s t M columns of H. These express ions can nowbe inse r ted i n t o equat ion 7.1) to give B in terms of onlythe l a s t N-M of the ak . t i s then a s t ra igh t forward mat te rto d i f f e re n t i a t e the r esu l t ing B with respec t to each ofthe ak. Set t ing these de r iva t ive s equal to zero the cond i t i on for a minimum) gives N-M) l i nea r equat ions for theN-M) des i red ak


54/119

where

Pkiv i

P ~ a 1 1 /k i H H wj m k j jj a l mal

(7.1.5)

(7 .16)

~ t e r these equa t ions a re solTed ~ o r the l a s t N-N o ~ theak , these va lues may be subs t i t u t ed i n to equat ion 7 .14 toobta in the Ci r s t M o the a i While t may appear t h a t Mo the a 1 are t r e a t e d es s en t i a l l y diCCerent ly than the r e maining, t i s c l ea r t h a t the r e s u l t does no t depend onthow the a i a re apport ioned i n to the two groups , because thebas i c equa t ions 7.12 and 7.1) complete ly determine t hena ture o f the so l u t i on and they a r e complete ly symmetr icin the a i .

As expected t was Cound in p rac t i ce t ha t t h i s methodwas very much more s t ab l e than Newton 's method, which s implyamounts to a spec i a l case o equat ion 7.14 wi th MN (whiche l imina tes the second te rm) .

Bven wi th t h i s ~ p r o v e method o f convergence, however,t was found t h a t t was unadvisable to t r y to zero more

than t he -C i r s t to 8 moments (n-4 or S) by t h i s method.Bxperienee with t he uniform pressure problem on the o the rhand i nd i ca t ed t h a t t would be necessary to a t l e a s t reducecons ide rab ly the moments up to about n in o rd e r ' t o achievemuch accuracy in the sur face . Thus as t f i n a l l y workedout the convergence process i t s e l f became semi-manual . Thati s between each cyc le , in which the moments up to n4 o rwere zeroed by t he above t echnique , t was necessa ry to


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5s tudy the Hij ~ o r the n6 and n7 moments, as wel l a s thegrad ien ts of the cur ren t func t ion , in an a t tempt to varythe parameters in such a way as to reduce these aoments andthe excess ive grad ien t s . The process i s a l abor ious and

d i f ~ i c u l t one, but wi th so .e s k i l l i s a convergent one.All of the numerica l ca lcu la t ions were ca r r i ed out on

an IBM 7090 and the f ina l . vers ion of the program fo r theseca lcu la t ions i s given in Appendix V t oge t he r with an exp lana t ion of the program and flow diagrams of the majorsubrout ines . There fore , it i s unnecessary to go i n t o t ha tin any d e t a i l here except to mention one t a c t wbich i ss ign i f i can t in the i n t e rpre t a t ion of the r e su l t s . Sincethe purpose i s to zero all the momenta except the d i po l emoment) it would not in p r inc ip le ~ f e c t th ings if a l l themoments were mul t ip l ied by a r b i t r a r y f i n i t e ~ a c t o r s . However , having accepted our i na b i l i t y to ac tua l ly zero a l lthe m ~ m e n t s and des i r ing r a the r to reduce them a l l to somecommon low l eve l , it becomes s ign i f i can t what fac to rs themoments a re mul t ip l ied by as t h i s wi l l a f f ec t t h e i r r e l a t i ve reduct ion . The th ing which f i na l l y governed thechoice of the proper fac to r was the accuracy wi th which thevar ious moments could be ca lcu la t ed . t was found t ha t ifthe fac to r 2n- l ) l l i s dropped from the de f i n i t i on of Unmand the fac to r n-m)l/ n+m)l in equat ion 7.2 i s rep laced by1 /n l l , then a l l the ca lcu la ted moments w i l l have about thesame number of decimal places of accuracy before t r unca t ioner ror se t s i n . Also t h i s change of fac to r c l ea r l y deemph-


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as i ze s the h ighe r moments as r i gh t fu l l y they should be .The f i na l so lu t ion tha t i s , the so lu t ion beyond which

fu r the r improvement was judged too d i f f i c u l t to be worthwhile) i s i l l u s t r a t e d p i c t o r i a l l y in Figure 1 and topographica l ly in Figure 7 and pro jec t ions of its curren tl i ne s a re given in Figures 4, and 6, which l ikewise gives i lhoue t te s of the sur face . Table J gives the values ofthe va r ious parameters for t h i s su r face , and Table 4 givesthe ca lcu la ted values of the moments up to m7 The i n t e gra t ions were done us ing JO curves and a bas ic i n t e r va ls ize of 0.07 see Appendix V).

t should be noted t ha t the sur face p lo t t ed in thesef igures i s not exac t ly the one ca lcu la t ed , though it d i f f e r sfrom it only s l i gh t l y . F i r s t of a l l , over about J 6 ~ ofthe pro jec ted a rea of the sur face in the xy plane most lynear the subsolar poin t ) the gradien t of f exceeded one.These regions then were considered by the computer to beperpendicula r to the wind, but in p l o t t i ng them I smoothedthem out to conform to the s lope of neighbor ing reg ions .The second change cons i s t ed of smoothing out the sur face inthe region near the dipole-sun meridian p lane above theneu t r a l po in t . There were loca l os c i l l a t i ons of the su r face there r e s u l t i n g probably from a de fec t ive curren tfunc t ion . The exten t of these cor rec t ions on the crosssec t ions in the two planes of symmetry i s shown in Figure 8,and an i nd ica t ion of t he i r ef fec t on the surface as a wholei s given in Figure 7.


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-.52-TABLB 3 . Parameters for the so lu t ion sur face .

Parameters e ~ i n i n g the asymptot ic cross sec t ion .

Parameters def in ing the moridian plane pro f i l e .A71 1.0166 0.7480 A73 0.3370 A74 0.1970

A 0.0300Non-zero parameters in g u ,v ) .A34 0.3000 A36 o. 7soo A40 . -o . l400 A45 o 12ooA3s 0.1000 A3a 0.2000 A43 0.0900Non-zero parameters in h u ,v) Al 1 5388 A6 -0 .0737 A11 1.0400 A16 . -0 .01 )6A2 0.0277 A -0 .7844 A12 .o . 0720 A17 o. 02 0A -1 .6113 As 0. 817 Al) 0.5470 A1a 1.7700A4 -0.24 .5 A9 -0 .0076 A14 1.1)20 A19 1 . )2 )0s -0 .0184 A10 o.1o6o A15 . - o . o243 A2o o.os4o

.A21. 0.0 00

TABLB 4 . Residual moments for the so lu t ion sur face .n m Moment n m Moment n m Moment2 1 -0 .0000 5 1 0.00174 6 5 0.00389

1 -o.oooo4 5 -0.00198 7 1 0.00026J J 0.00002 5 5 -0.00097 7 J 0.000094 1 -0.00008 6 1 -0.00064 7 5 o.ool6J4 J 0.0000 6 J -0.00026 7 7 -0.00077


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2

-sJ-

y

.5 ~ 1.5 .oFigure 4 Front view of su r face showing cur ren t l i n e s .Uni t s of R :0.680 :p.itN 0U20 l/6R ea r t h r a d i i ) . For an e plaHma of 2 5 pro tons /cc ve loc i t y 5 Km/sec Rn=9.16R 8


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z1 .0

Figure

y

1.5

------r------. 5 0 5 1 .0 1 . 5 2 .0

Side view of surface showing current l i nes . Units of R see Figure 4).n

t


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z1.0

Figure 6

-x

5 5 1 .0 1 5 2 .0

Top view of surface showing current l ines . Uni ts of R see Figure 4).n

I'AI


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2

1 .5

1 .

-.56-

yz = 00

-1

5I J

1

7

Figure

1 .0 1..5

Contours o f cons tan t z .Uni t s of Rn see F igure 4) .

2 . 0

Dots show ca l cu l a t ed po in t s , i nd ica t ing exten tto which surraoe was modif ied by smoothing.


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z

X

7

Units of Rn1

Figure Cross sec t ions in yz and xz planes) of ac tua lsurface generated by parameters of Table so l id l i nes )showing the a l t e ra t i ons dot ted l i nes ) made in smoothing.

2


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-58-B Resul t s Conclusions.

The so lu t ion of the problem of a magnetic dipole in acold f i e ld - f r e e plasma wind as obtained in Sec t ion 7 i si l l u s t r a t e d in Figurds 1 , 4 5 6 and 7.

To r e l a t e t h i s to t he geomagnetic case , tak e the d i pole moment M to be Jll gauss - ea r th r a d i i ) J and theplasma to be i on ized hydrogen. Then using the computedmoment s11 - -7 .00JO equat ion 7.7 becomes

where N0 i s in proton/cc and U0 i s in Km/sec. Mariner I Ida ta {6) suggests N .2.5 and u -soo, which gives 9 .16 ,o o nor in other words 9 . J ea r th r a d i i out to the subsolarpoin t . This i s e n t i r e ly cons is ten t with the exper imenta lva lues {J4) .

Since the moment t echnique i s the f i r s t approximatemethod of so lu t ion for t h i s problem which a l so spec i f i e sthe surface cur ren ts , it i s the f i r s t which can be used toca lcu la te the magnetic f i e ld everywhere. Appendix I Idevelops the i n t eg r a l s necessary to ca lcu la te the f i e ldin the two planes of symmetry where Bx vanishes .

These i n t eg r a l s have been evalua ted numerica l ly a tvar ious poin ts , fo r the sur face ca lcu la ted in Sect ion 7and plo t s have been made of the r e s u l t i n g magnetic f i e l d s .The heavy l i ne s in Figure 9 show some rep re sen ta t ive mag-ne t i c f i e ld l i nes in the merid ian p lane and the l i gh t e r


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-.59-l i ne s in t ha t z igure and in Figure 10, which shows theequa to r i a l plane , show contours of cons tan t f i e l d s t rength .The do t t ed l i nes in each f igure give the contours of con-s tan t f i e ld s t r eng th for the unper turbed dipole f i e l d . Ofcourse zor the exact so lu t ion the z ie ld s t r eng th outs ide.the sur face should be zero everywhere, so the f i e lds t r eng t hs which were ca l cu l a t ed outs ide the sur face inthese two f igures give some i dea of the accuracy of t heind ica ted sur face . To t r ans l a t e the r e l a t i ve z i e l d

2 . .s t r eng t hs mul t ip ly them by the fac to r J M t N U / ~1 rhich equa l s 5 77 t 1110-.S gauss) for a 500 Km/sec windwith 2 5 pro tons /co .

For f i e l d s t r eng t hs grea te r than about 64 the contoursdo not depar t from the or ig ina l dipole contours s u f f i c i e n t -ly to show the d i f f e rence . The f i e l d near the or ig inBaJ B(x,y ,z )e i s approximately:,... 0 y

2B x,O,O) 4.)0-0.BOx2B O,y,O) 4.)0+2.17y

B O,O,z) 4 . )0+) . )2z(8 .2)

Thus the compression of the magnetosphere (again us ingMariner I I data ) i nc reases t he e a r t h s f i e l d a t the equat o r by 2 6 .9 I a t noon and 22.8 t a t midnight and decreasest a t the pole by 2 5 0 ~

Before concluding t h i s d i scuss i on of the f i e l d a fewremarks concerning the topology of the z ie ld are in order .


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..___ 0.5 1 8 __...1.5 \ .17 / 11 08A-

- .08 \.16 12 / .19 ~ O - - \'-. I\\ .JO .61 / ~ \ \\

\\ \ \\ - i I \

' - .24 . 67 / y ~ / \ ~ \\\

.J5 , / / ' .. \ I I - I '\ \\ I I\I \

/ JH II I __;___ \. \ I II\IIJ2sJ I : I I . { ) r I 1 I I II I I I . I1.0 .5 0 .5 -1 .0 -1 5 -2 .0

2 .1.Figure 9 Fie ld l i n es magnitude contours in the meridian plane, uni t s of (NtN 0 U0 /rr) 2 Dashed contours a re for the d ipo le a lone . The computed f i e l d magni tudes and d i r e c t i ons a tsevera l po in t s ou ts ide the su r f ace a re inc luded t o ind ica te the accuracy of the so lu t ion .

I0 \I


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+ 006+. 008

+.009

+,010

+. 015

+Ol J

+ OOJ

+.008

+ 020

+ 007

+.012

-------

+.010 +.001 : 02 0 OOJ.

\\\\ \ \+.018 //

/ - \+ .090

.200 IIIIII

1.0

J

II

I

///

/

IIII

I

//I

__.----. .../

5 0

\ \\\I\\I2\ I 1I

1III-1:.5

2 lFigure 10 Fie ld s t reng th contours in the equa tor i a l plane uni t s of (MtN U /rrP.0 0Dashed contours a re fo r the dipole a lone . The computed f i e l d s t reng ths a t severa lpoin t s outs ide the sur f ace a r e inc luded to ind ica te the accuracy of the so lu t ion .

\I

1

I0 \....


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-62-As poin ted out by Johnson (10) the Cield l i ne s div idein to two es s en t i a l l y diCCerent groups: those tha t co-r o t a t e with the ea r th and those t ha t always extend in tothe t a i l of the cav i ty . To see why t h i s must be so con-s ide r the l i ne which passes through the neu t ra l poin t N( see Figure 9); it fans out a t t ha t poin t over the en t i r esur face and in pa r t i cu l a r passes through the subsola rpoin t S and the an t i s o l a r poin t A a t z--oo. Thisl i ne i n t e r s ec t s the ea r th a t some point E on the noonmeridian . Since the ea r th i s ro ta t ing , however, the l i newhich i n t e r s ec t s a t the pa r t i cu l a r poin t E can be theneu t ra l l i ne Cor only an i n s t an t , and twelve hours l a t e rmust i n t e r s ec t the ea r th a t the poin t l abe led E 1 andmake a simple loop in the t il of the cavi ty , i n t e r s ec t i ngthe equa to r i a l plane a t s . The fami ly of a l l such l i neswhich pass through N a t some i n s t a n t each 24 hours Corman envelope which d i v i des the l i ne s i n to two groupsa 1)those which i n t e r s ec t the ea r th a t a l a t i t ude lower than Eand t he re fo re pass through the region out l ined by SN onceeach 24 hours , and 2) those which i n t e r s e c t the ea r thnearer the pole than E and there fore can pass through themeridian plane only in the region out l ined by S 1B1ENATopologica l ly the two regions occupied by these two groupsof l i ne s form i n t e r lock ing t o r i donuts ) . The f i e ld l i ne sof the f i r s t group r o t a t e r i g i d l y with the ea r th . butthe second group i s conCined to the t a i l of the cav i ty and


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-6:J-thereore r o t a t e s i n s t ead about ts own cen te r l ine . Thistype motion i s reer red to by Dungey 21) as tw iddl ing .In r e a l i t y , course , the e a r t h s ax i s of ro t a t i o n doesnot coinc ide with the d i po l e ax i s , and ne i the r are perpen-d ic u l a r to t he wind d i r e c t i o n , as in t he presen t i dea l i zedcase , but t h i s does not qua l i t a t i ve l y change the p ic tu r e .

Since Beard s approximate boundary condi t ion i s theonly othe r method desc r ibed in the l i t e r a t u r e for ob t a i n -ing a so lu t ion to t h i s problem, t i s n a tu ra l l y of i n t e r e s tto compare the two so lu t ions .

Figure 11 gives ha l the equa to r i a l cross sec t ionbelow the z axis ) as given in the or ig ina l a r t i c l e by

Beard 15), and ha l the merid ian cross sec t i on above thez ax i s ) as given in a l a t e r t rea tment by Spre i t e r andBriggs (16) . The Spre i t e r and Briggs sec t ion was used b e-cause Beard gives only a hand drawn guess the nights ide shape in the merid ian p lane in h i s or ig ina l a r t i c l e .The dashed l i ne s in Figure 11 r ep resen t the correspondingcross sec t i ons of the surface ca lcu la ted by t he momenttechnique .

Figure 11 i s p lo t t ed in un i t s n so the height ofthe neu t ra l poin t coinc ides by de1ni t ion) or the twocases , but in order for Be a rd s so l u t i on to correspond tothe same plasma momentum lux dens i ty t i s necessa ry tochoose an approximate value for (def ined to be thef rac t ion of the f i e ld j u s t i n s ide the su r fa9e which i s


69/119

z

' '' ' ' ' '

y

,

-64-

Units of Rn-1 -2

fx , _----------

Figure 11 Cross sec t ions of Be a rd s ~ u r f c e andthe moment su r face dashed} in the equa t o r i a l planebelow l i ne ) and the meridian p lane above l i ne ) .


70/119

-6.5-cont r ibuted by the d ipo le ) . The subsola r poin t fo r Bea rd ssur face i s a t 1.0.58 Rn and the e a r t h s f i e l d a t tha tpoin t i s

f 16 rr MtN u2 {0 0 (8 . ) )

where equat ion 7 7 with s11 - -7 .00JO i s used to obta in thecen t e r express ion. Solving t h i s fo r f gives f 0 .4714.For comparison, the corresponding f for the moment so lu -t ion (obta ined by us ing 1.0166 ra the r than 1.0.58) i sf 0 .5JOJ 1

In a l a t e r a r t i c l e J6) Beard re f ined h i s ca lcu la -t i ons by taking i n t o account some of the surface cu r ren t .He i nd ica t e s t ha t t h i s makes the c ross sec t ion e ~ w e e n thesubsola r poin t and the neu t r a l poin t s l i gh t ly e l l i p t i c a l .decreas ing the rad ius to the subsola r poin t by .B% andi nc reas i ng the he ight of the neu t r a l poin t about J . Thismakes the shape with proper choice of f ) more nea r l y thesame as the dashed curve in t h i s reg ion , but Beard has notye t extended h i s second approximation to any othe r pa r t sof the sur face .

t was not poss i b l e to compare : f ie lds outs ide a s at e s t of the r e l a t i ve accuracy (as was done fo r the uniformpressure case ) because the f u l l t h ree dimensional so l u t i onby Beard s t echnique has not been publ i shed . Ne i the r doesBe a rd s method y ie ld the surface cur ren ts , and these arenecessary to ca lcu la t e the f i e ld s .


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-66- t i s u n ~ o r t u n a t e t ha t the complications encountered

in t h i s problem made it impossible to achieve the so r t ofaccuracy obta ined in the u n i ~ o r m pressure problem butmore accuracy in the ca lcu la t ions i s probably not j u s t i ~ i -ab1e anyway consider ing the inaccuracy in the mode1. Inaddi t ion to a l l the poss ib le objec t ions mentioned inSect ion 2 there i s one e ~ ~ e c t which makes the pressurelaw o ~ equat ion J . J inaccurate even i ~ the plasma weret r u ly co l l i s ion les s ~ i e l d ~ r e e s tab le and t h e r e ~ o r e ~ r e e

o ~ any shock t rans i t ions . This i s the ~ a c t t ha t pa r t i c l ewhich glances o ~ ~ the s u r ~ a c e j us t below the neu t ra l poin twil l be t r ave l ing a t such an angle t ha t it may glance o ~ ~the s u r ~ a c e again jus t above the neu t ra l po in t . Thus thepressure in t h i s region above the neu t ra l poin t wouldexceed t ha t given by equat ion 2.2 .

In conclusion the moment technique i s in pr inc ip lecompletely genera l approach ~ o r determinat ion o ~ the su r -~ a c e o ~ separat ion between p e r ~ e c t l y conduct ing plasmaand magnetic ~ i e l d . However in p rac t i ce it can e n t a i lalmost prohib i t ive d i ~ ~ i c u l t y except in cases o ~ cons ider -able symmetry such as the dipole in u n i ~ o r m pressureplasma. n example o ~ another problem o ~ l ike symmetry

~ o r which the moment technique should be u s e ~ u l i s t ha t o ~grav i t a t ing plasma cloud surrounded by magnetic ~ i e l d

which i s u n i ~ o r m a t i n ~ i n i t y .


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7APPRNDIX I Determinat ion o ~ the S u r ~ a c e Thickness

Consider a cold plasma ~ l o w i n g in the X d i rec t ionwith ve loc i ty U0 a t x - - ~ ) from a f i e ld f ree region in to

region o ~ magnetic ~ i e l d ~ B X ) ~ y Since a steady s t a t eso lu t ion i s des i red , the e l e c t r i c ~ i e l d must be able to beexpressed as the gradien t o ~ a sca la r - ~ . Further , sincenothing var ies in the Y or Z di rec t ions , a l l quan t i t i e sare funct ions only of X. Clear ly the t r a j e c to r i e s des-cribed by the pa r t i c l e s wi l l be symmetrical with respectto t he i r ingoing and outgoing sec t ions , so we need considere xp l i c i t l y only the ingoing pa r t i c l e s . Let the ve loc i ty ofthese pa r t i c l e s be

I-1)

where pe ~ o r the e lec t rons , or p i for the ions . TheY component of ve loc i ty does not ente r the problem and somay be assumed zero without l oss of gene ra l i ty . Further wewi l l assume normal incidence, i . e .

~ u e to the absence o ~ thermal motions, a l l pa r t i c l e sof the same s ign must pene t ra te to the same value of X andso the f lux of pa r t i c l e s must be independent of X for a l lX l e s s than t h i s maximum X. Our l a s t assumption concern-ing the boundary condi t ions on the problem w i l l be t ha t theve loc i ty of the protons and e lec t rons are equal a t - ~ asare the dens i t i e s . T h e r e ~ o r e we may wri te

I -2 )


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-68-

where N Ni(-oo) Ne(-oo)The equat ions of motion for the pa r t i c l e s are

M dV /dT q ( - Vp+V X Bp 'P p ~ ~ I -J)

and the Maxwell s equat ion r e l a t i ng the f i e l d and curren tbecomes:

dB 8 q N idx pI -4)

The ext ra fac to r of two has been i n se r t ed here because boththe ingoing and outgoing pa r t i c l e s cont r ibu ted equal ly tothe curren t in the z d i rec t ion , but N wi l l be used topr e fe r to the pa r t i c l e dens i t y of the ingoing s tream only.I f we were now to impose the remaining Maxwell s equat ion,

2 2 2d dX -4rreo (Ni-Ne), we would have an exact se t of equa-t i ons fo r the system. However t h i s system of equat ionswould be too d i f f i c u l t to so lve . The system of equat ionstha t r e s u l t s if t h i s condi t ion i s replaced by the approx-imate r e l a t ion

(I-5)

i s very much s impler to so lve . This approximation i s c e r -t a i n l y a good one, for the r a t i o of t

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