Pressure Balance Between Solar Wind and Magnetosphere

•OURNALOF GEOPHYSICALRESEARCH, SPACE PHYSICS VOL. ?4, NO. 5, MARCH 1, 1959

Pressure Balance between Solar Wind and Magnetosphere

M. A. SCHIELD 2

Department of Space Science Rice University, Houston, Texas 77001

An evaluationof the subsolarpressure balance between the solarwind and the geomagnetic field showsthat the average proton density of the quiet-day solar wind (300-400 km/sec) shouldbe between6 and 10 p/cma. Even duringstorm times the protondensity should always be between2 p/cm3 and 70 p/cm3. The relation betweenthe interplanetarysolar-wind parameters and the stagnationpressure is reviewed.The subsolargeomagnetic field, including the quiet-dayring currentfield, is evaluatedas a functionof subsolardistance. The quiet-dayring current is based on the dipole gradient-drift motion of the low-energyprotons observedby Davis and Williamson and by Frank. This quiet-day ring current has a magneticmoment of 0,?6 MB and producesa 41-? decreaseat the earth'ssurface. A geomagneticfield normalization of observedboundary distancesis also proposedto remove the effectsof the dipole tilt to the solar wind.

INTRODUCTION man-Ferraro theory, which assumesthat the The size of the magnetosphereand the pres- solar plasmais noninteractingand field-freeand sureof the solarwind are relatedby the pressure assumesthat the geomagneticfield is free of in- balanceequation p + B2/2/• -- constant.For a ternal plasma. Observationsof the solar wind field-freesolar wind and a plasma-freegeomag- [Wilcoxet al.,1967], of plasmaenergy densities neticfield this equationreduces to po--- B,2/2/•o. within the magnetosphere[Davis and William. The subsolarsolar-wind pressure po was first son, 1963; Frank, 1967], and of the geomag- calculatedby assumingspecular reflection of the netic field [Mead and Cahill, 1967] indicate noninteractingions and electronsand was given theseassumptions are not satisfied.The purpose by po = 2 pv• wherep is the massdensity and v of this paper is to evaluatethe pressurebalance is the solarwind velocity.The subsolargeomag- between the shockedsolar wind and the magnetic field Bg was first assumedto be twice the netospherewith its internal plasma. The solar tangentialcomponent of the subsolardipole field wind pressure necessaryto confine the mag- in analogywith the confinementof a dipolefield netosphereto a given subsolardistance is cal- by a plane boundary. A self-consistentsolution culated.The proton number density of the solar of the Chapman-Ferraro problem [Mead and wind is deduced from observations of the size Beard, 1964] indicated that the geomagnetic of the magnetosphereand the velocity and com- field was actually 2.44 times the dipole field at positionof the solarwind and is comparedwith the subsolar point [Mead, 1964]. See Beard observations. [1964] for a review. On these bases a 400- SOLAR WIND PRESSURE km/sec solar wind containingabout 2.4 p/cm• would confinethe geomagneticfield to a subsolar The body pressure po of the shockedsolar distance of 11 RB. wind on the subsolarpoint of the magnetosphere However, this conclusionis based on Chap- may be related to the interplanetary solar wind momentumflux pv• at 1 AU by a solar wind pressurecoefficient K definedby K = po/pV•. x Based on thesis submitted in partial fulfillment The value of K dependsupon the modelof the of the requirements for the Ph.D. at Rice Univer- solar wind. See Spreiter et al. [1966] for a re- sity. view. In the steady-state condition the zero- 2 Present address:Physics Research Center, Uni- temperature, field-free solar wind has a K of 2 versity of Iowa, Iowa City, Iowa 52240. appropriate to an elastic collision.In the gas- Copyright ¸ 1969 by the American Geophysical Union. dynamic approximation(B = 0) for a blunt- 1275 1276 M. A. SCHIELD

nosed obstaclein a hypersonicstream, K is +18 • I i I I I I 0.844 for y = 2 and 0.881 for y = 5/3. When the solarwind passesa magneticfield in alignedflow the stagnationpressure is givenby pm,,•/(pv•) = 1 -- (%)e [1 + 2/(( 7 -- 1)M•)] •-,o • • .-,o whereß = (7 -- 1)/(7 + 1) andp• = p + B•/2•o [Lees,1964; Hayes and Probstein,1959, p. 14]. For hypersonicflow (M' >> 1) this rela- • -•o- {190e• tion reducesto 5/6 (0.832)for 7 = 2 andto 7/8 _401• •EQUATORIALFIELDOFQUIET- DAY •--•0 (0.875) for 7 = 5/3. If the interplanetary field has a component • 50• I I I I I I I I I [ I I/• - 50 perpendicularto the free stream velocity vector, I 2 5 4 5 6 7 8 9 I0 II 12 R/R E the gasdynamicapproximation breaks down at the stagnationpoint [Alksne, 1967]. When the •i•. •. •qu•[oriM field of eurren• obtained as the •um of [he fie]d• pro- interplanetaryfield is antiparallelto the subsolar ducedb• •he hi•h- •nd ]o•-ener• component• geomagneticfield, the total stagnationpressure Lhe ma•neLo•pheric plasma. Inclusion of proto• is (2/3)pv • for 7 = 2 [Lees, 1964]. No coefti- be[•een 50 •nd 100 key •hould smoo[h ou• eient is presently available for the parallel or bump • • = 5. perpendicularcases. sphericmodels and will be designatedby 2] GEOMAGNETIC FmLD PaESSUaE [Ferraro, 1960]. In an image dipole magneto- The geomagneticfield is the sum of the earth's sphere] equals 1, while for a two-dimensional internal field, the magnetospheric surface- line dipoleconfined by a streamingplasma ] is currentfield, and the field producedby plasmas less than I [Chapman, 1963]. In the Mead- within the magnetosphereof which the quiet-day Beardmagnetosphere, which contains no internal and storm-timering currentsare examples.The plasmaand no neutral sheet,Bg/B• equals2.44 ratio of the subsolargeomagnetic field B, to so that ] equals1.22 [Mead, 1964]. the subsolardipole field B• is nearly independent The field of the quiet-day ring current is of the subsolardistance in simple magneto- derived and discussedin the Appendix. This ring current is based on low-energy proton energy densitiesmeasured by Davis and Wil- ,o-6 . ..• liamson[1963]andby Frank [1967].These I '..'--:I,•!:!??•i•i'!17•i? I I12øøenergy densities, illustrated inFigure 1,are I '' e.'-",,".'' "•"('"'• •. v'__•F'•'• •ve)x=•ol-I'øø I0 10-7•-- "," ,J...... fittn dntn I--I50.) I /oE=i38kev/cm...... I ,ooo- - r, . ..";i---- _t< ....

,o_.[.,,,,,,; , .,,,,,fi, r•9ioo /d - •o-*I , I, I :",,,,,,,,,,,,t: • 50

20 Fig. 1. Radial distribution of plasma energy densitiesobserved by Davisand Williamson [1963]and by Frank[1967]. The parameters and the resultant functional forms used to describe theseenergy densities are also illustrated. Future observationof protons between 50 and 100 key Fig. 3. Sum of the quiet-day ring currentfield shouldfill in the gap in energy density around and the earth'sdipole field versusequatorial dia- L = 5. rance. MAGNETOSPHERE PRESSURE BALANCE 1277 assumed to be symmetric and moving in a sistent with the external field statistically de- dipolar magneticfield. The equatorialquiet-day rived from ground-basedobservations [Cain, ring current field, illustrated in Figure 2, pro- 1966]. The quiet-day ring current has a total ducesa 41-7 decreaseat the earth'ssurface and magnetic moment of 0.26 M•. Between 10 and a maximum decreaseof 58 7 at 3.5 R,. The 12 R•, the ring current field enhances the equatorial ring-current field changessign at 9 equatorial dipole field by 25% to 35%, whereas R, and enhancesthe dipole field by 8 to 10 7 alongthe polar axis, the dipolefield is enhanced between 10 and 11.5 R,. The sum of the dipole by 15% to 17% over the sameradial region. field and quiet-dayring current field is shownin To calculate the subsolargeomagnetic field Figure 3. This quiet-day ring current field is while including internal plasmas, the entire consistentwith satellite observations[Mead and problem should be reworked self-consistently. Cahill, 1967], illustrated in Figure 4. The net However, the zeroth-ordereffect of this ring external field causedby the ring current and current field is to enhancethe earth's dipole magnetopausecurrents is shown to be con- moment by 26% beyond 10 R•. By neglecting

AUGUST 28, 1961 INBOUND EXPLORER 12

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Jensen and Cain -30 ø •-J. and Co plusboundary currents I I I I I I I I ,•,t 1'* R/RE; 4 5 6 7 8 9 10 11 12 km 4 ø -4 ø - 9 ø - l l ø - 9 ø zSEP 37 ø 24 ø 17 ø 15 ø 18 ø / SED 90 ø 91 o 91 o 89 ø 82 o L.T. 14:30 13:35 13:05 12:40 12:20 U.T. 6:30 5:25 4:05 2:l0 23:l0 Bj. &C. 510y 150Y 65Y 337 197 Fig. 4. Difference between the observedgeomagnetic field within 30ø of the subsolaraxis for DST _• 10 ,• and the theoretically predicted geomagneticfield [Mead, 1964] assuming no internal plasma[Mead and Cahill, 1'967].The field of the quiet-dayring current (Figure 2) is capableof explaining this difference. 1278 M.A. SCHIELD higher-ordereffects, the shape of the magneto- solar distanceby about 20%. However, the dis- sphere remainsunchanged from that obtained turbance field at the earth can be enhancedby by Mead and Beard. In this case,the subsolar up to a 100 ¾ if the magnetopauseexpands from geomagneticfield continuesto be 2.44 timesthe 6 to 10 R• [Mead, 1964]. internal field, which now includes both the Large subsolardistances often occur during dipole and ring current fields.The ratio Bg/B• the recovery phases of geomagneticstorms is now 26% greater than 2.44 so that in the [Freeman, 1964]. Unless the storm is quite self-consistentmagnetosphere with this quiet- large (DST • --180 ¾), it appearsthat a de- day ring current field f equals1.54. Considera- creasein the solarwind pressureafter the high- tion of the nondipolaraspect of the ring current pressure geomagneticstorm condition must be field would increaseboth the ring current field the primary causeof these large subsolardis- and the surfacecurrents in the equatorial plane, tancesrather than the internal pressureof the thusincreasing thee subsolar geomagnetic field at main-phasering current. the boundary. Tail currentfield. The field producedby the Normalization of subsolarboundary distances. tail currentsforming the neutral sheet opposes Since the subsolargeomagnetic field is also a the subsolargeomagnetic field. A split solenoid function of subsolarlatitude Xss,the subsolar model of these tail currents is illustrated in boundarydistances must be normalizedto re- Figure 5. The field a tail radius ahead of this move this variation. When the dipole axis is configurationalong the axis is 5% of the tail perpendicularto the solarwind the geomagnetic field. For tail fields of lessthan 30 ¾ with an field pressurein the noon meridianplane is inner edge at 10 R•, and a tail radius of 20 Rr proportionalto cos•Xover the regionwhere the the subsolartail currentfield is lessthan 1.5 ¾. magnetosphericboundary is circular[Figure 7; This southward-directed tail field is less than Mead,1964]. This relationis basedon the equal- the northward-directednondipolar component ity of the geomagneticpressure and the solar of the quiet-dayring currentfield and is, there- wind pressure.If the subsolargeomagnetic field fore, neglected. is assumedto be equal to the geomagneticfield Under the same conditions the truncated cur- at that samelatitude for Xs, = 0 ø, the normali- rent sheet model [Williams and Mead, 1965] zation factor is cos-•X**. This total-field nor- would producea 22-7 tail field at the subsolar malization increases observed boundary dis- point when the sheethas an outer edge of 200 tancesby up to 7%. Rr. However, this model is unsatisfactorybe- The dipolenormalization used by Nesset al. causethe current loop is completedentirely at [1964] and by Patel and Dessler [1966] de- infinity and not on the surfaceof the magneto- creasedboundary distances by up to 12%. Their sphere,giving the length of the geomagnetic normalizationassumed the subsolargeomagnetic field wasproportional to the total magnitudeof the subsolardipole field. The total-field nor- malizationpresented here is equivalentto assuming that the subsolargeomagnetic field is proportionalto the angularcomponent of the subsolardipole field. Storm-time ring current field. Beyond 7 Rr the ratio between the storm-time ting-current field and the earth's dipole field may be describedby (--DST/700 ¾) in the equatorial plane. This ratio assumesthe inductionfield Zo producesa third of the total DST but no sig- nificant field beyond 7 Rr [Kavanagh, 19677 Fig. 5. Split solenoid model of the tail cur- 1968]. For a constant solar wind pressurea rents. The field measured on the axis ahead of storm-timering-current field with DST _• --180 this semi-infinite configurationis given by B(zo) -- (•oJ/2•r)[ln[(1 -•- a)/(1 -- a)]--2a] where 7 wouldincrease the subsolardistance by about a -- [1 • (zo/a)•](-•/2• -- sin •0 (J. Midgley, per- 8%, a DST of --500 ¾ wouldincrease the sub- sonal communication). MAGNETOSPHERE PRESSURE BALANCE 1279 TABLE 1. Solutionof n•* = [2BoRb-S]2/(2t•om,v•) Where B0 = 0.31 Gauss

Solar Wind Subsolar Distance in Earth Radii Velocity, km/sec 12 11 10 9 8 7 6

300 3.40 5.74 10.2 19.1 38.8 86.4 218. 400 1.91 3.23 5.72 10.8 21.8 48.6 123. 500 1.23 2.06 3.66 6.88 14.0 31.1 78.4 600 .85 1.43 2.54 4.78 9.69 21.6 54.4 700 .63 1.05 1.87 3.51 7.12 15.9 40.0 800 .48 .81 1.43 2.69 5.45 12.1 30.6 tail an unrealistic influence on the subsolar tail quired number density n? for the inelastic field. collision(K -- 1) and image dipole (f -- 1)

PRESSURE BALANCE modelsat selectedsubsolar, distances and solar wind velocities. The resultant pressure balance equation is The effective number densities for other values given by of K and f are relatedto •_• by n• ----(f/K)n• •'. Table 2 illustrates various values of the factor (f/K) basedon various modelspreviously dis- where K and f are based on the modelsof the cussed.These resultsmay be summarizedby solar wind and geomagneticfield, respectively. Bo is the equatorial dipole field and Rb is the n*(no./cmS)v2(100km/sec)(R•,/10) 6 subsolarboundary distance. The assumption -- 91.45(f/K) that the magnetopauseis a tangential discon- OBSERVATIONSAND •)ISCUSSIONS tinuity appears to be valid most of the time [Sonnerupand Cahill, 1968]. The pressureof The average solar wind velocity ranges from the plasmasjust inside the magnetopausehas 300 to 800 km/sec but during quiet times is been neglectedalthough their presencewill in- between 300 and 400 km/sec. See Liist [1967] creasethe solar wind pressurenecessary to for a review of the properties of interplanetary confinethe geomagneticfield. space. The helium-hydrogenratio varies be- The mass density may be expressedin terms tween 0% and 15% and averages about 5%. of an effectivenumber density n• where n• ---- The solar wind is normally hypersonic(M s •>P p/m• [Mead, 1964]. For a neutral proton-elec- 1) since the thermal energy density nkT is tron gas, n• is the number density of each, 1%-2% of the streaming energy density pv2 neglecting mdmp. Table i illustrates the re- and there is generally equipartition of energy

TABLE 2. Values of the Factor (f/K) for Various Models

f = Bo/2B.

Self- Self-Consistent Image Consistent Dipole and Dipole Dipole Ring Current K = po/(pv•) 1.00 1.22 1.54

1.000 inelastic collision 1.00 1.49 2.36 0.881 gasdynamic 5/3 1.14 1.69 2.68 0.875 alignedflow 7 = 1.14 1.70 2.70 0.844 gasdynamic 1.18 1.76 2.80 0.832 aligned flow 7 - 2 1.20 1.79 2.84 0.667 antiparallel 1.50 2.23 3.54 ]280 M.A. SCHIELD

betweenthe thermal energy density and the proton densityof about 6 to 10 p/cm•. Further- magneticfield energy density. more,the protondensity must always be greater The measurementsof the proton densityin than 2 p/cm• unlessthe solar wind velocity the quiet-time solar wind appear to fall into exceeds800 km/sec and the subsolarradius two groupsdepending on the type of instrument simultaneouslyexceeds 11 R•. The protonden- utilized. The curved-plate analyzer measure- sity shouldalways be lessthan 70 p/cm• unless ments(Mariner2 and 4, Wolfe on IMP 1 and 2, the subsolarradius is less than 6 R• while the and Pioneer6, Vela 2 and 3) indicatethe quiet- solarwind velocity is lessthan 800 kin/sec. time solar wind proton density is between 1 and 5 p/cm• The Faraday cup measurements APPENI•IX. TI-IE QUIET-DaY RINO CURRENT (Bridge on IMP 1, Mariner 4, and Pioneer6) Plasma energy densitiesand currents. The indicate the proton density of the quiet-time radial distributionof plasmashaving sufficient solar wind is between5 and 10 p/cm• [Wilcox energydensity to contributesignificantly to the ei al., 1967; Lazaruset al., 1966]. quiet-dayring current are illustrated in Figure1. The subsolarextent of the magnetopsherehas Ho#manand Bracken [1965] noted that, owing been analyzed by Ness et al. [1964] and by to a recalibrationof the detectors,the Davis Patel and Dessler [1966]. Both used the dipole and Williamsonfluxes and, therefore,their normalization discussedpreviously. A subsolar energy densities,should be increasedby 25%. radius of 10.25 R• provided the best fit of the However,Davis and Williamson[1966] ob- IMP 1 data to the theoreticalshape of the mag- servedthat the flux has a variation of 25% netosphere.For 60 boundary crossingswithin and that particlesbelow 300 key have decreased 75ø of the subsolarpoint, Patel and Dessler by 25% between1961 and 1965.Therefore, the deduced subsolar radii between 6.9 and 11.5 R•. energy densities deduced by Hoffman and Sinceall of theseobservations were made during Brackenhave been utilizedas originallypre- the winter, the use of the total field normaliza- sented.Frank's data were taken at a geomag- tion would increasethese values by up to 20%. netic latitude of 22 ø _ 2 ø and for 5 • L • 11 On this basis, the average subsolardistance is were taken within 15 ø of the dusk meridian. expectedto be between 10.5 and 11 R•. The analysisof the ring currentsproduced by If the number density ratio of helium to plasmasin the dipole field is simplifiedwhen hydrogenis taken as 10%, the proton number the equatorialenergy density is fit to the form densityis 71.5% of the effectivenumber density NE -- (NoE)exp(--[g, (L -- to)]2) where i sincenp -- n•/(1 •- 4 na/np). On the basisof equals1 for L _• ro and i equals2 for L _• ro the pressurebalance relation, this 400-km/sec (seeAkaso/u and Chapman[1967] and refer- solar wind must have between6.2 p/cm• and ences therein). A flux distribution of the form 8.2 p/cm • to contain the geomagneticfield to a /(B, •) -- sin• •B -a/•,where • is the pitchangle, subsolardistance of 11 R•, and have between has proven to be both convenient and realistic 11.0 p/cm• and 14.5 p/cm• for a subsolardis- in fitting particle distribution. Thus in this tance of 10 R•. modelthe five parametersNoE, to,a, g•, and g2 Althoughthese values are in direct disagree- completely specify the ring current and its ment with somecurved-plate analyzer observa- associated field. tions, they are in agreement with the MIT Hoffman and Bracken determined that the plasmacup measurements[Lazarus et al., 1966] valuesNoE -- 138 kev/cm•, ro -- 3.5, a -- 2.5, and with group path measurements[Eshleman g• -- (1 R•) -•, g• -- (1.25 R•) -• provideda and others,1966]. UsingPioneer 6 the problem 'best fit' to plasmasabove 100 key. They also of correcting for ionospheric electrons was foundthat in this energyrange a decreasedwith solvedby utilizing a sourcethat is continuously decreasingenergy and with increasingdistance. moving away from the earth. Near the earth's Frank's data has been fit by the parameters oribt the average interplanetary electron num- ro-- 7, andg• -- g•(2 R•)-'. A knowledgeof the ber density was determined to be between 8 pitch-angledistribution is necessaryto obtain and 9 e/cm• with an rms deviationof 4.4 e/cm3. the equatorial energy density from that meas- It is, therefore,concluded that the quiet-day ured alongthe samefield line at higherlatitudes. solar wind (350-400 km/sec) has an average Detailed pitch-angleinformation is not yet MAGNETOSPHERE PRESSURE BALANCE 1281 available,but it is likely that a lies between0 rent density [Kavanagh, 1968; Appendix A]. and 2. It hasbeen assumed somewhat arbitrarily A computer calculation,prepared by Dr. Kava- that the maximum equatorial energy density nagh, which assumesthat all particles are con- NoE is 25 kev/cm8 and that a equals2. Since fined to a region bounded by the earth's sur- this choice reflects the minimum equatorial face and the dipole shell with L -- 10 has been energy (a -- 0) and limits the particle distribu- usedto determinethe ring current field. tion to low latitudes (a ----2), future data may Properties o• the ring current field. The enhancethe total energy of the quiet-day ring fields due to these ring currents are illustrated current. Figure 1 also illustrates the values of in Tables 3-6. The field of the quiet-day ring these functions versus L. current is the sum of thesefields. The equatorial The derivation of the ring current field is fields of these ring currents are shown in Fig- based on the integration over the resultant cur- ure 2. The sum of the earth's dipole field and the

TABLE 3 Values of the theta componentof the magnetic field in gammas produced by a ring current band of chargedparticles describedby the parameters' NoE - 25 kev/cm 8, R0: 7.0, g• - g2 -- (2 RE)-•, • - 2.0. Negative values denote opposedin senseto geomagneticfield. By symmetry the theta componentis zero at 90 ø latitude.

Geocentric Latitude, deg Distance, RE 80 70 60 50 40 30 20 10 0

1.0 --3.7 --7.2 --10.5 --13.4 --15.9 --18.0 --19.4 --20.3 --20.6 1.4 --3.7 --7.3 --10.6 --13.5 --16.0 --18.0 --19.4 --20.2 --20.5 1.8 --3.7 --7.3 --10.7 --13.7 --16.1 --18.0 --19.3 --20.1 --20.3 2.2 --3.7 --7.3 --10.8 --13.9 --16.4 --18.2 --19.3 --19.9 --20.1 2.6 --3.6 --7.3 --10.9 --14.3 --16.8 --18.3 --19.3 --19.7 --19.9

3.0 --3.5 --7.1 --10.9 --14.8 --17.4 --18.5 --19.3 --19.5 --19.6 3.4 --3.2 --6.7 --10.5 --15.0 --18.0 --18.8 --19.5 --19.4 --19.4 3.8 --2.9 --6.1 --9.8 --14.5 --18.0 --19.4 --20.1 --19.4 --19.3 4.2 --2.6 --5.5 --8.9 --13.2 --17.5 --20.6 --21.3 --19.6 --19.5 4.6 --2.3 --4.8 --7.8 --11.5 --16.8 --22.3 --23.0 --20.3 --20.4

5.0 --2.0 --4.2 --6.9 --10.1 --16.1 --24.3 --25.0 --21.7 --22.1 5.4 --1.7 --3.6 --5.9 --9.1 --15.4 --25.4 --26.5 --23.9 --24.6 5.8 --1.4 --3.1 --5.1 --8.1 --14.0 --24.8 --26.9 --26.6 --27.5 6.2 --1.2 --2.5 --4.2 --6.8 --11.9 --21.8 --25.6 --29.2 --30.0 6.6 --1.0 --2.1 --3.5 --5.5 --9.3 --16.9 --22.8 --30.9 --31.2

7.0 --0.8 --1.7 --2.8 --4.3 --6.7 --11.1 --18.7 --30.7 --30.4 7.4 --0.6 --1.3 --2.1 --3.2 --4.4 --5.6 --14.1 --28.0 --27.2 7.8 --0.5 --1.0 --1.6 --2.2 --2.5 --1.6 --9.6 --22.8 --21.9 8.2 --0.4 --0.8 --1.2 --1.5 --1.2 0.6 --5.6 --15.7 --15.2 8.6 --0.3 --0.6 --0.8 --0.9 --0.3 1.5 --2.3 --8.0 --8.2

9.0 --0.2 --0.4 --0.5 --0.4 0.3 1.8 0.4 --0.7 --1.9 9.4 --0.1 --0.3 --0.3 --0.1 0.7 2.0 2.4 4.8 2.9 9.8 --0.1 --0.1 --0.1 0.2 0.9 2.2 3.7 8.0 6.1 10.2 0.0 --0..1 0.1 0.4 1.1 2.3 4.4 9.0 7.6 10.6 0.0 0.0 0.2 0.5 1.2 2.4 4.6 8.7 8.0

11.0 0.0 0.1 0.2 0.6 1.3 2.4 4.5 7.9 7.7 11.4 0.0 0.1 0.3 0.7 1.3 2.4 4.3 7.0 7.1 11.8 0.1 0.2 0.4 0.7 1.3 2.3 4.0 6.2 6.4 12.2 0.1 0.2 0.4 0.7 1.3 2.2 3.7 5.4 5.8 12.6 0.1 0.2 0.4 0.7 1.3 2.1 3.4 4.8 5.2

13.0 0.1 0.2 0.4 0.7 1.2 2.0 3.1 4.3 4.6 1282 M.A. SCHIELD

TABLE 4 Valuesof the radial componentof the magneticfield in gammasproduced by a ring currentband of chargedparticles described by the parameters'N• -- 25 kev/cm•, R• -- 7.0,g• -- g• -- (2 R•)-•, • -• 2.0. Negativevalues denote opposed in senseto geomagneticfield. By symmetry•he radialcomponent is zero at 0 ø latitude.

Geocentrio Latitude, deg Distance, 90 80 70 60 50 40 30 20 10

1.0 23.3 20.6 19.6 18.0 15.9 13.3 10.3 7.0 3.5 1.4 22.8 20.7 19.7 18.0 15.9 13.2 10.2 6.9 3.5 1.8 22.5 20.8 19.9 18.1 15.9 13.2 10.1 6.8 3.4 2.2 22.3 20.9 20.0 18.3 15.9 13.1 9.9 6.6 3.3 2.6 22.1 20.9 20.1 18.6 16.1 13.1 9.6 6.4 3.1 3.0 21.7 20.8 20.2 19.0 16.6 12.9 9.3 6.1 2.9 3.4 21.2 20.4 20.1 19.4 17.6 12.5 8.9 5.8 2.6 3.8 20.5 19.9 19.9 19.7 18.7 12.3 8.7 5.5 2.2 4.2 19.7 19.2 19.4 19.7 19.5 12.8 8.9 4.9 1.6 4.6 18.7 18.4 18.7 19.3 19.6 14.2 9.8 4.1 1.0 5.0 17.7 17.5 17.9 18.7 19.3 16.4 11.6 2.9 0.4 5.4 16.7 16.5 17.0 17.9 18.9 18.5 14.0 1.8 0.1 5.8 15.7 15.5 16.1 17.1 18.6 20.2 16.7 1.2 0.4 6.2 14.7 14.5 15.2 16.3 18.1 21.2 19.2 1.6 1.3 6.6 13.7 13.6 14.2 15.4 17.4 21.4 20.8 3.3 2.9 7.0 12.7 12.6 13.2 14.4 16.5 20.7 21.3 6.1 4.9 7.4 11.8 11.7 12.3 13.4 15.5 19.4 20.6 9.5 6.9 7.8 11.0 10.9 11.4 12.5 14.3 17.7 19.0 12.7 8.5 8.2 10.2 10.1 10.6 11.5 13.2 15.9 17.1 14.9 9.6 8.6 9.4 9.4 9.8 10.6 12.0 14.2 15.3 15.8 10.0 9.0 8.7 8.7 9.1 9.8 11.0 12.7 13.7 15.5 9.6 9.4 8.1 8.0 8.4 9.0 10.0 11.3 12.3 14.4 8.8 9.8 7.5 7.4 7.7 8.3 9.1 10.1 11.0 12.7 7.7 10.2 7.0 6.9 7.1 7.6 8.3 9.1 9.8 11.0 6.5 10.6 6.5 6.4 6.6 7.0 7.5 8.2 8.7 9.4 5.5 11.0 6.0 5.9 6.1 6.4 6.9 7.4 7.8 8.0 4.7 11.4 5.6 5.5 5.7 5.9 6.3 6.7 6.9 6.8 4.0 11.8 5.2 5.1 5.2 5.5 5.8 6.0 6.2 5.8 3.4 12.2 4.8 4.8 4.9 5.1 5.3 5.5 5.5 5.0 2.9 12.6 4.5 4.4 4.5 4.7 4.8 5.0 4.9 4.4 2.5 13.0 4.2 4.1 4.2 4.3 4.4 4.5 4.4 3.8 2.2

quiet-day ring current field in the equatorial netic moment could have considerable effect on plane is shownin Figure 3. It is interestingto cosmic-rayobservations except for the nearness note that this first-order calculation may be of the outer ring current belt to the magneto- rather self-consistentin the sensethat the gradi- sphericboundary. ent of the total field never changessign. The total energy of this quiet-day ring cur- The magnetic moment of this quiet-day ring rent is 12.7 X 10• ergs, 58% of which is con- current is 0.26 Mr, based on the value of the tained by the lower energy protons (E _• 50 ring currentfield at largedistances (50-130 Rr). key). The differencebetween the 33.5-7 field The low-energyparticles (E _• 50 key) being predictedby Sckopke's[1966] first-orderrela- located at twice the radius of the higher energy tion and the 41-7 field obtainedby integrating particles produce about 85% of this magnetic over the currentscould be reducedby including moment. This 26% increasein the earth's mag- the field energy of the ring current. MAGNETOSPHERE PRESSURE BALANCE 1283 Comparisonwith obsekvations.Magnetic ob- in the surface current field due to the presence servationsby Explorer 12 indicatedthat there of these internal plasmas. weretwo deficienciesin the Mead [1964]'model Basedon the analysisof terrestrial magnetic of the geomagneticfield [Mead and Cahill, data Cain [1966] determinedthe existenceof a 1967]. First, the inclusionof a neutral-sheettail statisticallysignificant external field of 26.41 7. field wouldprobably explain the enhancedtwist- The component of this field parallel to the ing and expansionof field linesalong the flanks dipole axis is 20.8 7 and is directedsouthward. of the magnetosphere.Second, the field of the The 41-7 southward-directedfield of the quiet- quiet-dayring current (Figure 2) wouldexplain day ring current is necessaryto explain these the observed deviations from Mead's model surface observations. The field of the surface illustrated in Figure 4. This conclusionis not currents is directed northward and is about significantlyaltered by the 26% increase(4-7 7) 26 7 , when enhancedby 26%, for a subsolar

TABLE 5 ¾aluesof the theta componentof the magnetic field in gammas producedby a ring current band of chargedparticles described by the parameters'NoE.- 138kev/cm 3, Ro = 3.5, g• - (1 Rr) -•, g2 - (1.2 Rr) -• a - 2.5. Negativevalues denote opposed in senseto geomagneticfield. By symmetrythe theta component is zero at 90 ø latitude.

Geocentric Latitude, deg Distance, Rr 80 70 60 50 40 30 20 10 0

1.0 --3.6 --7.2 --10.5 -•13.6 --16.1 --18.1 --19.5 --20.3 --20.6 1.4 --3.5 --7.0 --10.5 --13.8 --16.5 --18.3 --19.5 --20.1 --20.3 1.8 --3.1 --6.4 --9.9 --13.9 --17.0 --18.9 --19.9 --19.9 --20.0 2.2 --2.6 --5.5 --8.7 --12.7 --16.9 --20.4 --21.4 --20.3 --20.4 2.6 --2.1 --4.4 --7.2 --10.7 --15.9 --22.5 --23.9 --22.3 --22.9

3.0 --1.6 --3.4 --5.6 --8.6 --13.8 --22.0 --25.0 --26.2 --26.9 3.4 --1.2 --2.5 --4.1 --6.4 --10.5 --17.6 --22.8 --28.8 --29.3 3.8 --0.8 --1.8 --2.9 --4.4 --6.8 --10.9 -17.5 --27.0 --27.1 4.2 --0.6 --1.2 --1.9 --2.7 --3.7 --5.0 -11.0 --20.2 --20.1 4.6 --0.4 --0.8 --1.1 --1.5 --1.6 --1.3 -5.0 --10.6 --10.9

5.0 --0.2 --0.4 --0.6 --0.7 --0.3 0.6 -0.6 --1.9 --2.6 5.4 --0.1 --0.2 --0.2 --0.1 0.4 1.5 2.1 3.8 2.9 5.8 0.0 --0.1 0.0 0.2 0.8 1.9 3.4 6.2 5.6 6.2 0.0 0.1 0.2 0.5 1.0 2.0 3.7 6.5 6.2 6.6 0.0 0.1 0.3 0.6 1.1 2.0 3.6 5.8 5.8

7.0 0.1 0.2 0.3 0.6 1.1 1.9 3.2 4.9 5.1 7.4 0.1 0.2 0.4 0.6 1.1 1.8 2.9 4.0 4.3 7.8 0.1 0.2 0.4 0.6 1.0 1.6 2.5 3.3 3.6 8.2 0.1 0.2 0.4 0.6 1.0 1.5 2.2 2.8 3.0 8.6 0.1 0.2 0.4 0.6 0.9 1.3 1.9 2.4 2.5

9.0 0.1 0.2 0.4 0.6 0.8 1.2 1.6 2.0 2.1 9.4 0.1 0.2 0.3 0.5 0.8 1.1 1.4 1.7 1.8 9.8 0.1 0.2 0.3 0.5 0.7 1.0 1.3 1.5 1.6 10.2 0.1 0.2 0.3 0.5 0.6 0.8 1.1 1.3 1.4 10.6 0.1 0.2 0.3 0.4 0.6 0.8 1.0 1.2 1.2

11.0 0.1 0.2 0.3 0.4 0.5 0.7 0.9 1.0 1.1 11.4 0.1 0.2 0.2 0.4 0.5 0.6 0.8 0.9 0.9 11.8 0.1 0.1 0.2 0.3 0.5 0.6 0.7 0.8 0.8 12.2 0.1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 12.6 0.1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.7

13.0 0.1 0.1 0.2 0.3 0.4 0.4 0.5 0.6 0.6 284 M.A. SCHIELD

TABLE 6 Values of the radial componentof the magnetic field in gammas produced by a ring current band of of chargedparticles describedby the parameters:N0E = 138 key/eras, R0 = 3.5, g• = (1 Rr) -•, g• - 1.2 Rr)-•, • = 2.5. Negative values denoteopposed in senseto geomagneticfield. By symmetry the radial componentis zero at 0 ø latitude.

Geocentric Latitude, deg Distance, 90 80 70 60 50 40 30 20 10

1.0 22.2 20.6 19.7 18.1 16.0 13.3 10.2 6.9 3.5 1.4 21.6 20.4 19.7 18.4 16.3 13.3 10.0 6.7 3.3 1.8 20.6 19.8 19.5 18.8 17.2 13.3 9.7 6.2 2.9 2.2 19.3 18.7 18.8 18.8 18.2 14.1 10.2 5.4 2.1 2.6 17.6 17.3 17.6 18.1 18.5 16.3 12.3 4.2 1.4

3.0 15.9 15.6 16.1 17.0 18.1 18.6 15.6 3.8 1.8 3.4 14.1 13.9 14.5 15.5 17.2 19.7 18.4 5.7 3.7 3.8 12.4 12.3 12.9 13.9 15.7 18.9 19.3 9.4 6.4 4.2 10.9 10.8 11.3 12.3 14.0 16.9 18.1 13.0 8.5 4.6 9.5 9.5 9.9 10.7 12.1 14.5 15.9 14.7 9.4

5.0 8.3 8.3 8.6 9.3 10.4 12.1 13.4 14.3 8.9 5.4 7.3 7.2 7.5 8.0 8.9 10.0 11.1 12.4 7.6 5.8 6.4 6.3 6.5 7.0 7.6 8.4 9.1 10.0 6.1 6.2 5.6 5.5 5.7 6.0 6.5 7.0 7.5 7.8 4.7 6.6 4.9 4.9 5.0 5.2 5.5 5.9 6.1 6.0 3.6

7.0 4.4 4.3 4.4 4.5 4.8 4.9 5.0 4.7 2.8 7.4 3.9 3.8 3.9 4.0 4.1 4.2 4.2 3.7 2.2 7.8 3.4 3.4 3.4 3.5 3.6 3.6 3.5 3.0 1.7 8.2 3.1 3.0 3.0 3.1 3.1 3.1 2.9 2.4 1.4 8.6 2.7 2.7 2.7 2.7 2.7 2.7 2.5 2.0 1.1

9.0 2.4 2.4 2.4 2.4 2.4 2.3 2.1 1.7 0.9 9.4 2.2 2.1 2.1 2.1 2.1 2.0 1.8 1.4 0.8 9.8 2.0 1.9 1.9 1.9 1.9 1.8 1.6 1.2 0.7 10.2 1.8 1.7 1.7 1.7 1.7 1.6 1.4 1.1 0.6 10.6 1.6 1.6 1.6 1.5 1.5 1.4 1.2 0.9 0.5

11.0 1.5 1.4 1.4 1.4 1.3 1.2 1.1 0.8 0.4 11.4 1.4 1.3 1.3 1.3 1.2 1.1 1.0 0.7 0.4 11.8 1.2 1.2 1.2 1.1 1.1 1.0 0.8 0.6 0.3 12.2 1.1 1.1 1.1 1.0 1.0 0.9 0.8 0.6 0.3 12.6 1.0 1.0 1.0 1.0 0.9 0.8 0.7 0.5 0.3

13.0 1.0 0.9 0.9 0.9 0.8 0.7 0.6 0.5 0.2

distanceof 11 R•. The field of the split solenoid field is nearly dipolar in both magnitude and tail field model, measured half a tail ,radius configurationfor L _< 7 on the daysideand for ahead of the configuration,is about a sixth of L _< 6 on the nightsideSchield [1968]. Beyond the field within the geomagnetictail and is these distancesthe geomagneticfield will en- directed southward.For an 18-7 tail field the hancethe nightsidering current while decreasing predictedexternal field at the earth is 18 7 and the dayside ring current relative to the current is directed southward in excellent agreement based on a dipole field configuration.No com- with Cain's observations. pensationfor these effects has been made in view Analysiso[ premises. A self-consistentsolu- of the conservative evaluation of the maximum tion of the geomagneticfield includingthis quiet- equatorialenergy density of the low-energypar- day ring current field, to which a realistictail ticles forming the outermostportion of the ring field has been joined, indicatesthe geomagnetic current. MAGNETOSPHERE PRESSURE BALANCE 1285 Although the exact configurationof the mag- Ferraro, V. C. A., An approximate method of esti- netosphericelectric field is unknown,estimates mating the size and shape of the stationary hollow carved out in a neutral ionized stream of its magnitudeare all below a mv/m in the of corpusclesimpinging on the geomagneticfield, equatorialplane. A 1-mv/m field wouldbalance J. Geophys. Res., 65, 3951, 1960. the gradient drift of particlesbelow 15 key Frank, L. A., Several observations of low-energy beyond7 R• of the earth.Since at leasthalf of protons and electrons in the earth's magnetosphere with OGO 3, J. Geophys. Res., 72, 1905, the energy density at L = 6.0 is carried by 1967. protonsabove 30 key [Frank, 1967;Figure 11], Freeman, J. W., Jr., The morphologyof the elec- the electricfield may be considereda perturba- tron distribution in the outer radiation zone tion, albeit a very large perturbation.It does and near the magnetosphericboundary as ob- not seem reasonable,however, to evaluate the served by Explorer 12, J. Geophys. Res., 69, 1691, 1964. quiet-dayring current and field in any greater Hayes, W. D., and R. F. Probstein, Hypersonic detail [Hol•man and Bracken, 1967] until the Flow Theory, Academic Press, New York, 1959. energydensity distributions are knownfor vari- Hoffman, R. A., and P. A. Bracken, Magnetic ous local times and the configurationand mag- effects of the quiet time proton belt, J. Geo- nitude of the electric field are more adequately phys. Res., 70, 3541, 1965. Hoffman, R. A., and P. A. Bracken, Higher-order defined. ring currents and particle energy storagein the Acknowledgments. The author is indebted to magnetosphere,J. Geophys.Res, 72, 6039, 1967. Dr. A. J. Dessler for suggestingthis problem and Kavanagh, L. D., Jr., Predicted and observeddis- for commenting on this manuscript. The author placementsand energy changesof chargedpar- would like to thank Dr. L. Kavanagh for the use ticles after distention of the magnetosphereby a of his ring current program and Dr. J. Midgley symmetric ring current, Ph.D. thesis, Rice Uni- for providing the exact solution of the split- versity, Houston, Texas, 1967. solenoid tail field along the axis of symmetry. Kavanagh, L. D., Jr., A revised (B, L) coordinate This work was supportedin part by the U.S. system to allow for the distention of the mag- Air Force Cambridge ResearchLaboratories under netosphereby a symmetricring current, J. Geo- contract AF19 (628)-3858. phys. Res., 73, 185, 1968. Lazarus, A. J., H. S. Bridge, and J. Davis, Pre- l•EFERENCES liminary resultsfrom the Pioneer 6 MIT plasma experiment,J. Geophys.Res., 71, 3787, 1966. Akasofu, S.-I., and S. Chapman, Corrections to Lees, L., Interaction between the solar plasma papersconcerning magnetic effects of modelring wind and the geomagneticcavity, Am. Inst. ol currents,J. Geophys. Res., 72, 445, 1967. Aeronautics and Astronautics, 2, 1576, 1964. Alksne,A. Y., The steady-statemagnetic field in Liist, R., The properties of interplanetary space, the transition region between the magnetosphere in Solar-Terrestrial Physics, edited by J. W. and the bow shock, Planetary Space Sci., 15, King and W. S. Newman, p. 1, AcademicPress, 239, 1967. New York, 1967. Beard, D. B., The solar wind geomagneticfield Mead, G. D., Deformation of the geomagnetic boundary, Rev. Geophys.,2, 335, 1964. field by the solar wind, J. Geophys. Res., 69, Cain,J. C., Modelsof the earth'smagnetic field, in 1181, 1964, Radiation Trapped in the Earth's Magnetic Mead, G. D., and D. B. 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Alksne, netosphere configuration as obtained from Hydromagnetic flow around the magnetosphere, trapped electronsat 1100 kilometers,J. Geo- Planetary Space$ci., 14, 223, 1966. phys. Res., 70, 3017, 1965. Wilcox, J. M., K. It. Schatten, and N. F. Ness, (Received June 27, 1968; Influence of interplanetary magnetic field and revised November 5, 1968.)