JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 86, NO. A13, PAGES 11,401-11,418, DECEMBER 1, 1981
Solar Wind Flow About the Terrestrial Planets 1. Modeling Bow ShockPosition and Shape
JAMES A. SLAVIN AND ROBERT E. HOLZER
Instituteof Geophysicsand PlanetaryPhysics, University of Californiaat LosAngeles, Los Angeles, California 90024
Generaltechnique for modeling the position and shape of planetarybow waves are reviewed. A three-parameter methodwas selected to model the near portion (i.e., x' > - 1Roo)of theVenus, earth, and Mars bow shocks and the resultscompared with existing models using 1 to6 freevariables. By limitingconsideration tothe forward part of thebow wave, only the region of theshock surface that is mostsensitive to obstacleshape and size was examined. In contrast,most other studies include portions of themore distant downstream shock, thus tending to reducethe planetarymagnetosphere in question to a pointsource and constrain the resultant model surfaces to be paraboloid or hyperboloidin shapeto avoiddownstream closure. It wasfound by thisinvestigation that the relative effective shapesof the near Martian, Cytherean, and terrestrial bow shocks are ellipsoidal, paraboloidal, and hyperboloidal, respectively,in responseto theincreasing bluntness of theobstacles that Mars, Venus, and earth present to the solarwind. The position of theterrestrial shock over the years 1965 to 1972showed only a weakdependence onthe phaseof thesolar cycle after the effects of solarwind dynamic pressure on magnetopause location were taken into account.However, the bow waveof Venuswas considerably more distant around solar maximum in 1979than at minimumin 1975- 6 suggestinga solar cycle variation in itsinteraction with the solar wind. Finally, no significant deviationsfrom axial symmetry were found when the near bow waves of theearth and Venus were mapped into the aberratedterminator plane. This finding is in agreementwith the predictions of gasdynamic theory which neglects theeffects of theIMF onthe grounds of theirsmallness. Farther downstream where the bow wave position is being limitedby theMHD fastmode Mach cone, an elliptical cross section is expectedand noted in theresults of other investigations.
INTRODUCTION vieet al., )977;Russelland Greenstadt, 1979; Slavin et al., 1980]. Themagnetic field observations in Figure 1 showa seriesof quasi- Amongthe majorearly discoveries of the spaceprogram was the perpendicularshock crossings by, from top to bottom, Mariner 10, presenceof a bow shockupstream of the earth(e.g., Spreiterand PioneerVenus, Isee 1, and Mariner 4. In each casethe upstream Alksne[1970], Dryer [1970], and referencestherein). Given the precursorsare quite small in amplitude,the transition layer thick- largecollisional mean free path of solarwind particles (i.e., h -,• ! nesson the order of theion inertiallength, and the jump in thefield AU) it mighthave been expected at thetime that the magnetosphere magnitude'strong(i.e.,approaching (y + 1)/(y- 1)for the trans- wouldrepresent a small (i.e. ,'--, 10 -3 •.) scatteringcenter as opposed versecomponent). Under these favorable conditions, bow shock to an obstacledeflecting fluid throughthe formationof a standing crossingsmay be unambiguously identified in the experimental data bow wave. Thus planetarybow shockscomprise some of our recordswith precision limited only by the instrument sampling rate earliest,and perhaps most striking, observational evidence for the andthe relative velocity of theshock. However, it mustbe noted existenceof the microscaleplasma processes that allow the solar that the determinationof bow wave location is complicatedon wind to exhibitbulk fluid propertieson spatialscales much smaller occasionby "quasi-parallel"conditions when the angle between thanthe physicaldimensions of the planets.Further, the thickness theshock surface normal and the upstream interplanetary magnetic of the transitionlayer within whicha portionof the plasmaflow field becomessmall. In theseinstances there is a broadeningof the energyis convertedto internalenergy, turbulence, waves, and regionover which the plasmais "shocked"and an increasein suprathermalparticles has beenfound to be small in comparison turbulenceboth upstream and downstream [e.g., Greenstadt, 1970] with the shockstand-off distance. Hence, the bow wave may, for makinga judgement as to theshock location less certain and some- somepurposes, b• considered a mathematical discontinuity ashas timesdifficult in the absenceof plasmaas well as magneticfield beendone in obtainingboth gasdynamic [Spreiter and Jones, 1963; data. However,the relativenumber of passesduring which a dis- Dryer and Faye-Petersen,1966; Spreiter et al., 1966;Dryer and tinctcrossing fails to be recorded is notlarge [e.g.,Fairfield, 1971; Heckman,! 967; Spreiterand Stahara, 1980a,b] andMHD [Sprei- Olsonand Holzer, 1975]and the study of minorchanges in shock ter andRizzi, 1974] solutionsto the continuumdescription. positionas a functionof itsstructure [Formisano etal., 1973;Auer, Theseremarks find relevancein Figure 1 which bringstogether 1974]is outside the intended domain of thiswork. Accordingly, the samplesof theshock observations that have been made at eachof the resultsof this studymay not be strictlyapplicable to the highly terrestrialplanets: Mercury [Ness et al., 1975],Venus [Slavin et al., quasi-parallelportions of planetarybow waves. Observations have 1980], earth [Russelland Greenstadt,1979], and Mars [Smith, alsoshown that shock motion may occur at speeds of 10-102km/s 1969].Magnetometer observations are preferentially displayed here [Holzeret al., 1966;Greenstadt et al., 1972]in responseto chang- becauseof their ability to makehigh time resolution(e.g., 1-10 ing upstreamparameters and interactionswith solar wind Hz) measurementsrelative to those of the plasma instruments. discontinuitiesto produce multiple encounters between spacecraft However,with someexceptions that will be discussedlater, there is andthe shock even when the probe velocity is as great as ,-• 11 km/s generallygood agreementamong the particlesand fields experi- suchas was the casefor Mariner 10 at the time of the observationsin mentsas to the locationof theshock over length scales greater than Figure1. The treatmentof multipleencounters in boundarymap- thetransition layer thickness, •c/topi [e.g.,Vaisberg, 1976; Ogil- pingproblems varies from study to studyand is consideredin the next section.Mean valuesof the variousinterplanetary parameters changewith distancefrom the sunas can be seen,for example,in Copyright¸ 1981by the AmericanGeophysical Union. . Papernumber 1A 1147. 11,401 0148-0227/81/001A- 1147 $01.00 11,402 SLAV•NAND HOLZER: FLOW ABOUT THE PLANETS
6O be highlydesirable to explorethe regionbetween the bow wave and the denselower atmosphereof eachplanet with particlesand fields 4O experiments.However, suchprograms have been restricted largely to the earth. Further,the examinationof flow throughany volume, suchas the magnetosheath,is limited to statisticalstudies if the 20 boundarylocation and boundaryconditions are not known at the time of the individualobservations as is the casewith singlespace- craft measurements[e.g., Hundhausenet al., 1969; Howe and 40 Binsack, 1972]. Owing to the naturalvariability of the solarwind and the planetaryresponse both the bow shocksand the magneto/ 20 ionopausesof planetsare in nearlycontinuous motion [e.g., Holzer et al., 1966;Slavin et al., 1980]. This pointis illustratednot only by the multiple boundarycrossings in Figure 1, but also Figure 2, La 40 whichplots average daily locationof the Venusbow shockmapped o.,..,,. into the solarwind aberratedterminator plane as observedby Pio- 20 neer Venus during its first 225 days in orbit. As is the caseat the earth, large fluctuationsare present.They have been studiedby Slavin et al. [1979a,b; 1980] and attributedto changesin not only ionopauseheight and solarwind Mach number,but alsothe solar wind interactionthrough the effects of varyingsolar corpuscular and electromagneticradiation on the ionosphereand neutral atmos- 5 Mars phere.At Mercury andMars still lessdata are availablewith noneof it havingbeen taken at low altitudeson the dayside(see reviews by Ness [ 1979],Russell [ 1979], andSiscoe and Slavin [ 1979]). Hence, Time at this time the study of solar wind flow about theseplanets as a Fig. 1. Quasi-perpendicularbow shocksat Mercury,Venus, Earth, and group is still reducedthe temporal variability of these physical Marsare displayed as adapted fromNess et al. [ 1975],Slavin et al. [ 1980], systems,the overall paucity of observations,and the limitationsof Russelland Greenstadt [1970], and Smith [ 1969].In timethe top 3 panels spanseveral minutes each, while the Mars encounter covers about 12 hours. single spacecraftmeasurements to an examinationof bow wave Multiple encountersdue to shockmotion are evidentin the Mercury and position, shape,and variability. Venusprofiles. The time spentin themagnetosheath at Mars appears small In this paperwe reporton the findingsof the first of the two part owing to the compressedtime scaleand the grazingnature of the high- studyof this subject.As will be describedbelow, Part 1 modelsthe altitudeMariner 4 fly-by aswell asany variations associated with bow wave motionat the time of the meaurements(Note: Greenstadt [ 1970] hasdeter- shapeand location of the bow wavesof Venus,earth, and Mars by a minedthat the final (i.e., exit) shockcrossing at Mars mayhave fulfilled the singlestandard technique for thepurpose of comparingtheir relative requirementsfor being quasi-parallel.) shapesand positions.This is in contrastto the many other studies [e.g., Bogdanovand Vaisberg, 1975; Vaisberg, 1976; Russell, the decreasein the total upstreammagnetic field magnitudein 1977;Verigin et al., 1978] thathave considered different aspects of Figure 1 from almost20 nT at Mercury(i.e., 0.46 AU) to below5 thisproblem but with fewer observationsand in lesscomprehensive nT at Mars (i.e., 1.52 AU). However, the alterationsin the relevant and methodicalmanners than employedhere. Figure 3 showsthe quantitiessuch as the IMF spiralangle, sonicand Alfv6nic Mach timing of the variousspace missions that will be usedrelative to the numbers,or othersappearing in Tables 1 and 2 are not large in sunspotcycle. The reason for such a display is that the in- comparisonwith the statisticaldeviations about their mean levels terplanetarymedium appears to exhibit variationswith solarcycle and have not been found to result in any substantialdifferences in phase[e.g.,Diodato et al., 1974;King, 1979] whichare known to shockstructure among these planets [Greenstadt, 1970; FairfieM affect the natureof solar wind interactionwith the earth (e.g., andBehannon, 1976; Slavin et al., 1980]as is evidentfrom the very Holzer and Slavin [ 1981] andreferences therein). Similar modula- similar quasi-perpendicularmagnetic field profiles assembledin tions are also expectedat the other planetsbut so far have been Figure 1. reportedon thebasis of in situobservations only at Venus[Wolff et What hasbeen found to vary from planetto planetis the natureof al., 1979; Slavin et al., 1979b]. Since the variousmissions to their respectiveinteractions with the solar wind and hence the Mercury, Venus, and Mars have occurredat different times with variousaspects of flow aboutthese bodies. For thisreason it would respectto the solarcycle, observationsof the terrestrialbow wave
TABLE 1. InterplanetaryConditions Planet R,AU Vsw,Km/S n/,,cm -3 B,nT T•o,104øK Te,104øK
Mercury 0.310.47 430 3273 4621 1713 2219
Venus 0.72 430 14 10 10 17 Earth 1.00 430 7 6 8 15 Mars 1.52 430 3.0 3.3 6.1 13 (Scaling) ooo Rø R-2 R- 1(2R-2+2) • R-2/3 R- 1/3
A set of 'typical' long term meanparameters at 1 AU have been scaledto the orbitsof the otherterrestrial planets by usingthe approximateradial dependenciesdisplayed in thelast row. The least certain scalings are those for temperature where the values of Gazis et al. [ 1981] for T•, and Sittier and Scudder[1980] for Te havebeen adopted. SLAVINAND HOLZER: FLOW ABotrr TaE PLANETS 11,403
TABLE 2. Bow Wave Parameters
Psw,10-8 Planet R,AU dynes/cm2 Ms M.4 fl Q c/tOpi, km SpiralAngle, deg 0.31 26 5.5 3.9 0.5 15 27 17 Mercury 0.47 11 6.1 5.7 0.9 32 40 25 Venus 0.72 5.0 6.6 7.9 1.4 62 61 36 Earth 1.00 2.5 7.2 9.4 1.7 88 86 45 Mars 1.52 1.1 7.9 11.1 2.0 120 130 57 Usingthe basic parameters fromthe first table aseries ofplasma quantities relevant toplanetary bowshocks have been computed; Psw= 1.16n•n•,Vs2w, Ms = Vsw/(2kB(1• .....14Tp + 108Te)/1 16mp) • M.4 = Vsw(116n t,rnp 4,n')•/B , fl-8'n'npkB(1 - ....14Tp + 1 08re)/B2 Q = M.4= •,6Psw8'n'/BZc/%,i_ = 228/n, andIMF spiral angle = tan-](R).In allcases, the solar wind plasma has been assumed tocontain 4% He ++ with THe = 3.5Tp.
from a seriesof five satellitesspanning most of cycle 20 were magnetosheath,or solar wind. In principal, the locationsof the selectedas a controlgroup against which the more limited Mars and magnetopauseand bow wave could then be recoveredboth in the Venusmeasurements will be compared.Part 2, whichis in prepara- meanand under specific conditions through appropriate selection, tion, examinesthe abilityof existinggasdynamic and MHD models averaging,and weightingof the data. The main advantageof such to describethe meanflow conditionsas impliedby shocklocation an approachis thatall of theinformation gathered, as opposed to just andmakes use of thesemodels in a comparativestudy of the solar the boundarycrossings, may be used in formulatingprobability wind interactionwith the terrestrialplanets. Mercury has been for distributionsfor boundarylocation. However, its usefulnessis themost part excluded from this modeling study due not only to the limitedby the needfor a statisticallysignificant number of observa- limitedamount of datacollected during the Mariner 10 fly-bysbut tionsper unit volume.The resultantgrid at thistime is still too large alsobecause those encounters may havetaken place during abnor- to be of usein studyingthe shocklocation. At somefuture date this mal interplanetaryand magnetosphericconditions (Slavin and will, hopefully, no longerbe the case. , Holzer[ 1979]and references therein). While Mariner 10 provided a For the reasonsgiven above,the approachused since the begin- wealth of information on the basic nature of the solar wind interac- ningof suchstudies is to identifyshock crossings in theparticles and tionwith the Hermaeanmagnetosphere, its observationsare insuf- field observationsand utilize curve fitting techniquesto model ficientto giveus a viewof shockshape and stand-off distance under locationand shape.Its principledisadvantage is that care mustbe the typical0.3-0.5 AU interplanetaryparameters shown in Table exercisedin data selectionto avoid producinga spatiallybiased 2. Examinationof Jupiterand Saturnhas beendeferred until the representation.In general,shock observations taken along satellite largebody of Voyagerobservations at Jupiterand Saturn are more trajectoriesthat make a small angle with the boundarysurface fully reducedand disseminated. and/ordo notbegin well below(above) and end well above(below) the altitudesat whichthe shockresides must be omitted.There is MODELING THE BOW SHOCK
The idealexperiment to determinethe shapeand location of the MERCURY: MARINER 10 bow shockfor a givenset of interplanetaryand obstacle conditions wouldinvolve a largenumber of probessimultaneously crossing the Z VENUS: VENERA 9,10 boundaryat nearnormal incidence angles with highrelative speeds to obtaina" snapshot"of theshock surface. In theabsence of such PIONEER EARTH: IMP 3 4 VENUS anexperiment, the next choice would be to followthe trajectories of a largenumber of satellitesthrough a three-dimensionalgrid classi- HEOS 1 fying each unit volumeas being locatedin the magnetosphere, T PROGNOZ1,2 MARS: MARS 2,3
4 VENUSBOW SHOCK LOCATION DURING PVO's MARS 5 FIRST SIDEREAL YEAR IN ORBIT 200
N 150 • - . - r ' • I.' - •J• . ß'r 100 0 0 ' •' •0' •o •o ,00 ,• •0',•0',;0'•0'2•0' ;• 50 PIONEERVENUS ORBIT NUMBER Fig. 2. With its 24-hour orbit the PioneerVenus orbiter crossesthe bow { I I i I I I I I I [ I I i i I i I I I [ I i i i ! i 1955 1960 1965 1970 1975 t980 shocktwice daily except occasionally when apoapsis is nearlocal midnight andthe spacecraft doesn't get far enough away from the x' axisto penetrate YEAR into the solarwind. Displayedabove are the bow shockcrossings from Fig. 3. Smoothedsunspot number, Rz (Solar.-GeophysicalReports, 1980), Slavinet al. [1980]. Theyhave been mapped into the aberratedterminator andthe activeperiods of the variousplanetary missions used in thiswork to planeby usingthe modelsurface from that studywith the daily average modelbow wavesare plotted against time. As shown,Pioneer Venus orbiter distancefrom the centerof the planetto the shockplotted against orbit may offer our first opportunityto studythe solarwind interactionwith a number(Note: Orbit 1 = December5, 1978). planetother than earth over a completesolar cycle. 11,404 SLAVINAND HOLZER: FLOW ABOUT THE PLANETS
TABLE 3. Summaryof PrincipalEarth Bow ShockModels (Cartesian Form)
Study S/C #Passes Period Domain k1 k2 k3 k4 k5 k6 Fairfield [ 1971] 'Meridian 4 ø' Imp 1 Imp 3 Imp 4 Exp 33 Exp35 389 1963-8 x -45Re 1 0.0296 -0.0381 - 1.280 45.644 -652.10 'Meridian NO 4 ø' Same Same Same Same 1 0.2012 -0.1023 -4.76 44.466 -629.03 'X ROT. NO 4 ø' Same Same Same Same 1 0.2164 -0.0986 -4.26 44.916 -623.77
Formisano[ 1979] (With-•= O) 'PswUnnormalized' Imp 1 Imp 3 Imp 4 Exp 33 Heos 1 Heos 2 700 1963-73 Same 1 0.12 0.06 -4.92 43.9 -634 'PswNormalized' Same '• 450 Same Same 1 0.18 0.45 -4.16 46.6 -618 Informationon the respective bow shock data sets of Fairfieldand Formisano ispresented and the resultant coefficients for their second order surfaces listed.While the data set of Formisanois thatof Fairfieldwith Heos 1 and2 crossingsadded, no exact number of passesis availablefor theFormisano study becauseeach shock encounter was considered separately. In addition,the Formisano models were three dimensional sothat for the sake of thecomparisons conductedin thisinvestigation only the traces of hismodels in theecliptic plane are considered (i.e., z = 0).
alsothe question of multiplecrossings with the shock on a single on the long term meanshock location of suchphenomena are orbitthat can take place at theearth, for example,over distances of expected,consistent with previous analyses [Gosling et al., 1967], manyplanetary radii. It hasbeen the policy of thisstudy to fit the to be small.Of far greaterimportance is "aberration"due to the averageshock location per inboundor outboundorbit leg as is orbitalmotion of eachplanet which makes the apparent direction of generallydone [e.g., Holzer et al., 1966;Fairfield, 1971 ]. The theaverage solar wind velocity of theplanetary rest frame deviate reasonfor this decision is that the individual multiple encounters are from the anti-sunwarddirection. Past studies, such as Fairfield not usuallystatistically independent of eachother with respectto [1971]and Formisano [1979], have usually taken the approach of obstacleand solarwind conditions.Multiple encounters with the fittingthe shock crossings in unaberratedsolar ecliptic coordinates. shockmay take placeover intervalsof manyhours, but are often The orientationof the bestfit is then interpretedin termsof the separatedby only minutes, or less. By comparison,the in- averageeffects of aberrationand otherphenomena which might terplanetaryparameters are most commonly available as 1- to producea lackof symmetryabout the x axis.However, as the solar 3-houraverages [King, 1977] andare statisticallyindependent only wind speedis variablewe can reduce"noise" associatedwith overperiods of tensof hours,or more, dependingupon the physical aberrationby aberratingeach individual shock crossing at a position variable considered[e.g., see Gosling and Bame, 1972; King, (x,y, z) relativeto a planetwith a meanorbital speed of Vpduring a 1979]. Further, the rate of occurrenceof multiple crossingswill periodof solarwind speedVsw (Vsw = 430 km/s assumedin the vary with the individual spacecrafttrajectory and tend to bias the absenceof upstreamsolar wind observations)into a new space data set by more heavily weightingthe lessdesirable observations where made along orbits with smaller incidenceangles to the boundary surfacedue to the large numberof encountersgenerated by small r = X/x2 + y2 amplitudeshock motions. An excellentexample of sucha problem c•= tan-l(Vr/Vsw ) + cos-i(x/r) is containedin thestudy of shockposition by Formisano [ 1979] in y
12- nonuniform,the objectiveis to use as few free parametersas is GosdynomicEorth Bow Shock X'>-I Roe necessaryto producea fit with therequisite accuracy. As a straight 10 (Ms=8, line is a verybad representation of shockshape over any substantial length,the standardpractice is to usea generalsecond order curve:
kty2 + k2xy+ k3x2 + k4y+ ksx+ k6 = 0 (2)
0.6 • Xo=0.25BestFitRo• 0.4 • •=0.89 Somemethods of fitting to thiscurve are discussed in bothFairfield ,,,+,' L: 1.92Roe [ 1971] andForrnisano [ 1979], who alsoincludes a z2 term, with 0.2 • , , , their results listed in Table 3. In consideringthe nature of the -0.5 0 solutionsto thisequation it mustbe notedthat the k 2 termrepresents cos a rotation of the symmetry axes of the solutions(i.e., circles, Fig. 4. A three parameterbest fit to six pointstaken off the gasdynamic ellipses, parabolas,and hyperbolas,if we ignore the degenerate modelof theearth bow shock by Spreiter et al. [1966]is shownin the 1/r - cases)about their center by an amount cos0plane with distancein unitsof obstacleradii.
1/2 tan-i[k2/(k3- kl) ] (3) treatments,that by Formisano[ 1979] whichfits the shockin three dimensionssets the least constrainingrequirement in that only from a configurationin whichthe symmetryaxes parallel the coor- symmetrywith respect to theecliptic plane (i.e., z = Izl)is assumed. dinateaxes. Since we are aberratingthe crossinglocations before Fairfield [ 1971] incorporatedquasi-axial symmetry into the model fitting andhave found no largeasymmetries, we canassume h = a, by rotatingthe crossinglocations about the x axis into the dawn the tilt due to the aberration.Rotating the axesby the amountgiven (dusk)half of thex-y planewhen they coordinateof thedatum was in equation(2) eliminatesthe crossterm so that we can then have negative(positive). In anothermodel he assumedspherical symme- try for the shocksurface on the dayside.These procedureswere adoptedby Fairfield sothat any east-westasymmetry in flow about kt'y'2 + k3'x'2 + k4'y'+ ks'x'+ k6' = 0 (4) the earth might be detected. The existence of significant asymmetriesassociated with the spiral configurationof the IMF where within the magnetosheathregion have been both predicted [Walters,1964] and disputed [Shen, 1972] on theoretical grounds as kt' = k3sin2a- k2sina cosa + klCOS2a will be discussedfurther in a later section.Observationally, only k2' = k2(1- 2sin2a)+ 2(kI - k3) sina cosa = 0 smallasymmetries beyond aberration were found in the studiesby k3' = k3cos2a-- k2cosa sina + k•sin2a (5) Goslinget al. [1967] andFairfield [1971], whereasthe work by k4' = - kssina + k4cosa Formisanoproduced mixed results that will be discussedlater. ks' = kscosa + k4sina Hence, on the basisof the Goslinget al. and Fairfield resultswe k6' = k6 haveassumed axial symmetryabout the aberratedxaxis (i.e., thex' axis)and modeled the shock in thex'- (y,2+ z,2)1/2plane. A search Equation(4) can then be arrangedinto a more familiar form for biasesdue to violationsof thissymmetry was conducted without finding any significant asymmetriesas will be reported in the sectiondevoted to the subject.The advantageof our symmetry assumptionis thatit enhancesthe density of datapoints and thereby minimizesthe problemof nonuniformcoverage that can bias fitting , ---- GASDYNAMIC .-o I• --,- EARTH in boththe eclipticplane and three dimensions when the numberof •WS-U N, [ BOW SHOCK crossingsper unit directionis variable.In particular,the orientation of the model surfacecan be very sensitiveto the nature of the coverage.For thesereasons the assumptionof axial symmetryis ß especiallyappropriate for comparativeinvestigations, such as this Xo:+0.25 ROB one, given the paucityof observationsavailable at Mars. Further, we will limited ourselvesin this studyto modelingshock crossings '- .... OB :*"-VENUS H/R o: o.o,, forwardof approximatelyone obstacle radius behind the terminator planeat eachplanet: Venusx' > - 1Rv,earthx' > - 1OR e and Marsx' > - 1R•ts.Besides avoiding the morepoorly sampleddownstream regions,this requirementalso has the effect of constrainingus to that portion of the bow shock which is influencedmost by the boundaryconditions at the daysideobstacle [e.g., seeSpreiter et // Xo=+O.45ROBF0.5 al., 1966] as will be addressedin part 2, whichis concernedmore II e=o.9• / with the solar wind interactions. , i,•,/,L:"45, Ro• l , , The optimumfit to the dataon shockposition in termsof produc- 1.5 0.5 -0.5 ing theminimum rms (i.e., "root-mean-square")deviation normal to the model surfacewould most certainlybe a representationin X' (Ro) termsof somecomplete polynomial setwith an infinite number of Fig.5. Bestfits to the forward portions ofthe gasdynamic bowshock modelsof $preiter et al. [1966] and $preiter and Stahara [1980] are coefficientsto be determined(e.g., see the ionopausemodel of displayed.If downstreamdata were includedthe resultingcurves would be Theiset al. [1980a]). However, interms of usefulness, particularly hyperboloid, butat the expense ofthe goodness offit alongthe forward if any extrapolationsare to be made or if the coverage is section. 11,406 SLAVINAND HOLZER: FLOW ABOUT THE PLANETS
TABLE 4. Earth Bow Shock Models (Conic Form)
Study SymmetryAssumption Domain 3, Xo,Re Y0,Re •5 L, Re ConicType Fairfield [ 1971] Spherical(x' > O) 'Meridian 4 ø' AxialWRT x'(x' < O) x > -45Re-0.8 ø +3.4 +0.3 1.02 22.3 Hyperbola=-4.0 ø 'Meridian, NO 4 ø' Same Same -5.2 ø + 1.5 +0.4 1.07 27.2 Same • 0 ø 'X ROT., NO 4 ø' Axial WRT x Same -5.6 ø +1.2 -0.1 1.07 27.2 Same = 0 ø
Formisano[ 1979] (With z = 0) 'PswUnnormalized' WRT Ecliptic(z = •l) Same -3.6 ø +2.6 +1.1 0.97 22.8 Ellipse • 0ø 'PswNormalized' Same Same -9.1 ø -4.1 - 1.6 0.76 28.1 Same • 0 ø
This Study Imp 3 AxialWRTx' x > -10Re =0 ø +3 =0 1.19 23.1 Hyperbolatan-l(Ve/Vsw) Imp 4 Same Same = 0 ø + 3 = 0 1.15 23.5 Same Same Heos 1 Same Same = 0 ø + 3 = 0 1.10 23.5 Same Same Pognoz 1,2 Same Same = 0 ø + 3 = 0 1.20 22.9 Same Same (+__3%) (+__1%) Mean #6-9 Same Same = 0 ø +3 = 0 1.16 23.3 Same Same
Equations(2)-(6) havebeen used to transformthe coefficientsin Table 3 into geometricquantities for comparisonwith the resultsof this study. Alternatively,the resultsof this work couldhave been expressed in the sameform asused by Fairfieldand Formisano, but the termsabove are bothmore insightfuland allow for greaterease in use by avoidingthe needto solvesecond-order equations as must be donein the Cartesianform.
(X' q-ks'/2k3') 2 pointx0 = (-ks'/2k3') _ c, Y0= (-k4'/2kl') wherethe plus sign is for (k4'2/4kl'k3') + (ks'2/4k3'2) - (k6'/k3') an ellipseand the minussign is takenfor a hyperbola.Thus so long as thelocation of the foci are arbitrary,equation (7) is equivalentto (y' + k4'/2kl')2 (1) with 5 free parameterseach. By assumingaxial symmetryabout = 1 (6) (k4'2/4kl'2) + (ks'2/4kl'k3') - (k6'/kl') thex' axis,two freeparameters are removed to leavethe eccentricity ,5,the semi-latusrectumL, and the positionof the focusalong thex' Finally, this expressioncan be put into a convenientpolar form axisx 0. Thusfor this model, three coefficients are determined by fitting the data. As demonstratedby Slavinet al. [ 1980], allowing r=L/(1 +Ecos0) (7) the locationof the focuson thex' axis to be a free parameterresults in betterfits over the original2 parameter,5-L modelsofHolzer et wherethe semi-latusrectum, L = b2/a, andthe eccentricity, E = al. [1966, 1972]. This slightgeneralization is of particularadvan- c/a, aredetermined from a = I(k4'e/4k•'k3') + (ks'2/4k3'2) - (k6'/ tagein thiswork wherethe shapesof threedifferent planetary bow k3')l•/2, b = I(k4'2/4k•'2) + (ks'2/4•'k3') - (ka'/kl')l•/2, wavesare being compared. Equation (7) may be rearrangedso that andc 2 = a2 ___b2 (plussign for a hyperbolaand a minussign for an 1 ,5 1 elipse)with thefocus about which (r, O)are measured located at the -7-=Z-cos 0 +Z- (8)
... 40- ,-,. 40-
EARTH BOW SHOCK EARTHBOWSHOCK [• IMP 3 (6/65 - f/67) IMP4 (6/67- f2/65) + . / 30 + (Psw)= 2.3xfO' dyne$/cm + 30' (Psw): f.Sx fO'e dynes/cm a (Ms> = Z2 + ( Ms>=Z4 /
+ + +
+ .*:;,'.... [ Xo +3Rz œ 1.19 + + ++-i•' + I œ= 1.15
+ R• 2,5.t1.9R zRE
I I i 20 •5 •0 5 0 -5 orE -;o Fig. 6. Threeparameter second-order fit to the Imp 3 shockcrossings. Fig. 7. Threeparameter second-order fit to the Imp 4 shockcrossings. SLAVINAND HOLZER: FLOW ABOUT THE PLANETS 1 1,407
.-. 40 .-. 40-
EARTH BOW SHOCK PROGNOZ 1-2 (4/72-11/72) EARTH BOW SHOCK HEOS f (f2/68- f/70) (Psw)--2.9x 10'e dynes/cm • 30 (Psw):f. 5 x fO-edynes/cm• 3O + + + + + + 20, +/ + 20 + + +7' ++ + + N: 123 + + X0: +3 RE E: t20 N= 78 L = ,02.9 RE + •-+++ fO' XO= +3 RE 2-D Normal RMS = 2.0 RE E = 1.10 L = 23.5 RE 2-D NormalRMS = 1.8R E +1;f5 fOI+ 5I 0 -5 -10 I Fig. 9. Three parametersecond-order fit to the Prognoz1 and 2 shock 15 •0 5 0 -5 -f0 crossings. Fig. 8. Three parametersecond-order fit to the Heos 1 shockcrossings. MODELING RESULTS Earth and linearregressions in 1/r-cos 0 spaceperformed for different focuslocations to identifythe fit for whichthe rms deviation normal As wasmentioned in the introductiona series of earthorbiting to the modelsurface in thex' - (y,2+ z,2)1/2plane is least. satellitesactive during solar cycle 20 (seeFigure 3) wereselected This methodis illustratedin Figures4 and 5 whereit has been for usein examiningthe terrestrial bow wave in orderto noteany testedby fitting six pointstaken off theSpreiter et al. [ 1966] and solarcycle effects which might influence the position of theshock. Spreiterand Stahara [ 1980b]gasdynamic models of shocklocation Imp 3 and4 (D.H. Fairfieldand NSSDC, privatecommunication, at the earthand Venus, respectively. In Figure4 the bestfit for the 1978), Heos 1 (V.V. Formisano,private communication, 1980), earth model in "conic coordinates"(i.e., conics are all straight andPrognoz 1 and2 (O.L. Vaisbergand V.N. Smirnov,private lines in 1/r-cos 0 plane) is displayedwhile Figure 5 showsthe communication,1980) bow shockcrossings were obtained, aber- resultsof theexercise in x' - (y,2q_ z,2)1/2 coordinates. Based upon rated,and modeled by thethree parameter method described previ- the observedgoodness of fit in thesetwo 'test' cases,and past ouslywith the results displayed in Table4 andFigures 6- 9. Table5 experience[Slavin et al., 1980], we haveconcluded that the pro- providesa listingof the numberof passes,mean upstream condi- posedthree parameter fitting method will fulfill itsintended purpose tions[King, 1977], andother information pertaining to eachdata of modelingthe bow waves in the solar wind associatedwith set. The individualsatellite observations were fitted separately differing obstacleshapes. Finally, the coefficientsof the second exceptin thecase of Prognoz1 and2, whichoverlapped in timeand ordermodels of Fairfield and Formisanohave beenexpressed via weremodeled together. With theinterval of timeduring which each equations(2)-(6) in termsof moregeometrically meaningful quan- satelliteprovided information being only 6-18 months,no large tities suchas focuslocation, eccentricity, and semi-latusrectum, rapidchanges in meansolar wind parameters were found during the which are shown in Table 4. In the sections to follow these models individualmissions that couldbias the results.Such problems andothers will be comparedwith the resultsobtained by this study wouldarise if, forexample, the subsolar crossings took place during usingthe 3 parametersecond order methoddescribed above. a periodof significantlydifferent Mach number, or dynamicpres- TABLE 5. Earth Bow Shock Data Set Satellite#Passes Averageupstream parameters have been tabulated, including the average dynamic pressure and the Alfv6nic, sonic, and magnetosonic Mach numbers. Thesonic Mach numbers were computed asin Table 2 butwith an assumed Te of 1.5x 105øK. In eachcase, the number, N of instances inwhich hourly averagedupstream parameters [King, 1977]were available is alsolisted. Note the relatively small number of cases(i.e., < 50%) in whichMach numbers maybe calculatedeven when hourly averages are utilized. The distance to thenose of thebow wave Rss varies monotonically with Psw, but there are significantcontributions from solarwind Mach numbersas may be seenin Figure 10. 11,408 SLAWNAND HOLZER: FLOW ABotrr THE PLANETS leastsensitive of themodel parameters was the focus locationx 0. In T eachcase the best fits wereobtained for x0 = + 3 Re, but with only slightlypoorer representations at+3.5 and+2.5 Re. This finding is similarto the'meridian 4 ø' modelofFairfield [ 1981] listedin Table I .//,' .....' 4, whichshows an x0 valueof +3.4 Re whena 4ø aberrationis assumedbefore fitting the data. However, there is a largevariation $.oc ; :5...... inx0 from +2.6 to-4.1 Reinthe Forrnisano [1979] model when the 2.I x 10'edynes/cm •) . / ..' dataare scaledby Psw1/6 to takeinto accountthe dependence of (ScoledtoPsw-" / •/,,•/ ..- ...... /.• .." magnetopauseradius on upstreamdynamic pressure. For a given focuslocation in ourstudy significant degradation in thegoodness of fit is observedwith changes from the optimum in theeccentricity e, andsemi-latus rectum L, at the--, 3% and"' 1% levels,respec- IMP 3 ' ,•C." -- I (6/65 - f/6 7) tively,as indicated in Table4. In eachcase the type of secondorder IMP 4 curveobtained was a hyperbolawith the eccentricities ranging from (6/6?42/68) 1.10 for Heos 1 to 1.20 for Prognoz.These results compared HE:OS I ...... (12/68 -1/70) favorablywith the 1.02-1.07 values from the Fairfield study. The PROGNOZ f-2 Formisanomodel also produced a similar result, ß = 0.97, whenno (4/72-ff/72) I t.": scalingfor varyingPsw was included. However, the normalized modelwithx 0 = -4.1R efound a farmore elliptical shock surface of eccentricity0.76. The possible reasons for the similarity among the /// ß andx 0 valuesobtained here and the P sw unnormalized models of Fairfieldand Formisano, as contrasted with the Formisanonormal- izedcase, will be consideredin thediscussion section. Finally, the I I two dimensional rms deviation normal to the model surfaces in 15 10 o -5 40 Figures6-9 arerelatively constant with a maximumof 2.0 Re for X'(RE) Prognoz1-2, anda minimumof 1.7 Re for Imp 4. Fig. 10. Earthbow shock models from Imp 3, Imp4, Heos1, andPrognoz The bow shockstand-off distance along the x' axis 1-2 all scaledto the meansolar wind dynamicpressure are displayed. L • (8) sure,conditions from thosepresent when the flank observations Rss=Xø+1+ ß weremade. This difficulty may be avoidedaltogether by selecting dataon the basisof interplanetaryconditions, but sufficientmea- is tabulatedin Table 5 and shownto vary monotonicallywith the surementsto conductsuch a studydo not exist for Mars and are only highestdynamic pressure, which occurs before and after solar justbecoming available atVenus [seeSlavin etal., 1980].Since for maximum,producing the minimumRss and the lowestdynamic thesake of comparisonit is desirable to analyze the earth bow wave pressure,near solarmaximum, associated with the largestRss. in the samemanner as at the other planets,we have modeled Figure10 shows the 4 modelshocks scaled (i.e., scalingbothx 0 and separatelythe average terrestrial bow wave observed by eachmis- L) tothe mean pressure by assumingaP 1/6sw magnetopause pressure 'sion asopposed to usinga singlemerged data set. dependence[e.g., Holzer and Slavin, 1978]. As will be examined in The samplingof theboundary surface is quitegood between part2 of thisstudy, both the stand-off distance and shock shape 20ø and• 95ø in sun-planet-satelliteangle. Few subsolar observa- (i.e., forx 0 constant,the eccentricity) for thebow wave models in tionsare available owing to thegenerally nonequatorial latitudes of Figure10 appearordered by theaverage sonic Mach numbers in satelliteapogees and the assumption of axial symmetry as opposed Table6 withthe highest TABLE 6. Summaryof PrincipalSecond Order Venus Bow ShockModels (Conic Form) Symmetry Study S/C Period #Passes Assumption Domain h xo(Rv) yo(Rv) e L(Rv) ConicType a 1. SlaVin et al. PVO 1978-9 86 Axial WRTx' x' > -1R v -=0 =0 •0 0.80 2.44 Ellipsetan-1(½) [1979a] •r SW• 2. Slavin et al. PVO 1978-9 172 Axial WRTx' x' > - 1Rv =0 +0.2 =0 0.88 2.21 Ellipse Same [19801 (_+4%) (_+0.5%) 3. Srnirnov et al. V9-10 1975-6 45-62 Axial WRT x' x' > -16R v =0 +0.41 -0 1.04 1.70 Hyperbola Same [19801 PVO 1978-9 86 Same x' > - 1Rv =0 •-0 0.80 2.44 Ellipse Same PVO Same 86 Same Same =0 +0.29 =0 1.02 2.17 Hyperbola Same 4. This study V9-10 1975-6 48 AxialWRTx' x' > - 1Rv =0 +0.2 =-0 0.89 1.95 Ellipse Same (+_3%) (_+1%) 5. Mean of 2 and ...... x'>-lR v •0 +0.2 --=0 0.89 2.08 Ellipse 4 (Solar Cycle average?) As a resultof thelarge number of bowshock observations at Venus which have been made over the last five yearsmuch progress has taken place in the modelingof thisboundary. For this reason the earlier model surfaces based upon a smallnumberof shock encounters [e.g., Russell, 1977] have not been listed above. SLAVINAND HOLZER: FLOW ABOUT THE PLANETS 11,409 Prognoz1-2 with Mars Althoughthe Red Planet has been the subjectof an intensive researcheffort by both the U.S. and U.S.S.R. over the past two Venus Bow Shock I ,•././•' ...? decades,including six orbiter missions,less is known about the particlesand fields environmentof Mars than any of the other planetsprobed thus far. The reasonis thatonly Mariner 4 fly-by and the Mars 2, 3, and 5 orbiters were instrumentedto cary out such measurementsnear Mars, and even then no data were obtained at ": ,/•'"' r (Verigine/c/,f978) low altitudes,beneath --• 103 km, or deepin the wake region ,' ,I ' Vener•(Smirnov e/c/,f980) ...... [Vaisberget al., 1976; Dolginov et al., 1976; Gringauzet al., 1976]. More specifically,it haslong been known that Mars deflects thesolar wind, asopposed to absorbingit like ourmoon, because of the strongbow shockdetected by Mariner 4 (seeFigure 1). How- ever, it has not been possibleto identify unambiguouslythe pro- 2 • 0 -• -2 X'(Rv) cessesby whichthe flow is divertedin the absenceof low altitude Fi[. ]3. Venusbow shock m•cls dc•vcdf•om Vcnc•a 9 and]0 and observations[e.g., Russell,1978a,b; Dolginov, 1978a,b]. For the Pioncc•Venus shockobservations •c displayedalan[ with the bow wave purposeof modelingthe Martian bow wave a set of boundary c•ossin•sof Y•r•m •t •1. []978]. crossingsidentified by the Mars 2, 3, and 5 RIEP plasmainstru- ments(O.L. Vaisbergand V.N. Smirnov,private communication, wind took place (note: The Verigin et al. crossingshave not been 1979) in generalagreement with the otherexperiments [Vaisberg, aberrated. However, if similar numbers of dawn and dusk hemi- 1976]were obtained. In Figure14 theaverage crossing location per spherecrossings were included, then the effect is just to increasethe passis modeledwith thethree parameter fit usedpreviously on the width of the distributionin Figure 13. SeeGringauz [ 1980]). The Earth and Venus data. Figure 15 displaysboth the model and the dashedcurve in thatfigure is a gasdynamicshock that Verigin et al. intervals over which the transition between shocked and unshocked passedthrough their crossings.It is at onceapparent that although solarwind took placea long the individualsatellite trajectories. In the Smirnovet al. Venera orbitermodel and ours show agreement, one case,several multiple encounters by Mars 3 are connectedby the meancurve and individualshock positions of Verigin et al. lie dashedlines to avoid attributingthem to three separatepasses and somewhatcloser to the planet. This is especiallytroublesome as weighingthem more heavily. The remotecrossing labeling 'A' was bothSmirnov et al. andVerigin et al. claimgood agreement with the not includedin the fitting althoughits presence,when allowed, did magnetometerexperiment [Dolginov et al., 1976] asto the location notcause any large changes in thebest fit parameters.In Part2, this of the bow wave. The reasonmay then be one of data selectionin crossingand one otherjust downstreamof thex' = -1 Rm•plane thesestudies, but only a carefulcomparison of their measurements (also marked with 'A' in Figure 17) are modeled as low Mach could reveal it. However, regardlessof this unexplaineddiscrep- numberevents. This possibilitywas first suggestedby Vaisberg ancy it is clear in both data setsthat the positionof the bow shock [1976].Accordingly, their usually larger distances from the planet wasindeed closer to Venusduring the epoch ofVenera 9 and 10 than maynot be soindicative of thesolar wind interaction being unusual that of PVO. on thoseoccasions as the upstreamconditions being atypical (e.g., Also shownin Figure 13 are all threemodels of the Venus shock low Ms). Despite the limited nature of the coverage,the shock basedupon PVO data that have beenpublished thus far. Those of locationdoes appear to be adequatelysampled, particularly in terms Slavinet al. [ 1980]and $rnirnov et al. [ 1980]were discussed earlier of SMS angle,by thesethree orbiter missions, and there is reasonto and showvery similarresults. The third, by Theiset al. [ 1980b], have confidencein the results.Sources of spatialbiasing by the employsa novel approachin that it usesa polar curvewhich they spacecrafttrajectories will be discussedbelow. terman 'ArchimedianHyperboloid' formed from the sumof hyper- Listed in Table 7 and plottedin Figure 16 are a numberof bolic and Archimedianspirals. This two free parametermodel is different models of the Mars bow shock. Both Bogdanov and seento predictshock locations similar to thoseof the othertwo but Vaisberg[ 1975]and Russell [ 1977]used the E-L shockmodeling with the subsolarpoint somewhatmore distant.This latter differ- methodof Holzer et al. [1966, 1972] to fit differentsets of orbiter encemay be duein partto not only the spiralshape assumed in their shockcrossings, but produceddifferent mean surfaces due to their two-parameterfit, but also the lack of any data selectionin that study. Slavin et al. [1979a, 1980] omitted crossingsin certain regionsto limit biasingof the representationby the orbit of Pioneer Mars2,3 and 5 Bow ShockCrossings Venus parallelingthe shocksurface at smallersun-planet-satellite WithX'>-I R•s in Aberroted Xo: +O. 50 R•s Centered angles.In ge9eral,the use of anyreasonable prescribed shape such as spiralsis little different than assuming,for example, that the shocksurface is paraboloid[e.g., Egidi et al., 1970]. Many differ- . •" ConicCoordinote•+ A + ent typesof curvesexist which may be usedunless it is desiredto • O5 actuallydetermine the shapeof the bow wave as is the casein this • +• +• Best Fil work anda few of the otherstudies [e.g., Fairfield, 1971'Russell, f Xo: +0.5 RMS 1977'Forrnisano, 1979]. To do this, the modelsurface shape must • :O94 MAES 23 •+ L =I 94 RMS be variablewith at least,in ourexperience, three free parameters to MAES5 • 2- D NormelRMS =015 RMs fit theobserved variation among the planet' s bow waves. The model -0.5 0 0 5 +1 ' of Theiset al. demonstratesgraphically the lack of anyuniqueness cos in the frequentlyused model formulations that assume the shapeof Fi•. ]4. Three p•amctcr fit to the •s 2, 3, and 5 shockcrossings the shocksurface prior to fitting the measurements. fo•d of the planex' = - 1•. SLAVINAND HOLZER: FLOW ABOUT T•IE PLANETS 11,411 Mars 2 andMars 3 crossingswas only about7 hours.In addition,the two Mars 3 bow wave crossingsclosest to the planetin the vicinity of the aberratedterminator plane are absentfrom all of the studies MARS BOWSHOCK citedsave the one by Bogdanov and Vaisberg[ 1975]. Thus,we find (X'>-I that blunter bow wave modelsobtained by most of the previous Mars 2 (1971-2) studiesare due largely to the inclusionof the two crossingsjust Mars ,.3 (1971-2) anti-sunwardofx = - 1RMS.Shock crossings may be recordedonly Mars 5 (1974) along the pathsof the available spacecraft.In this casethe very limited natureof the Mars 2 and 3 spatialcoverage downstream of the terminator,as shownin Figure 17, has led to the other models tendingto produceshock surfaces which follow the orbitsof these I satellites.The Mars 5 bow wave encountersnear x' - 5 RMSfrom a ! • 1- 6 =0.94 trajectorywhich is lessparallel to the boundaryclearly supportthe .I ,• I L= 1.94 more slender shock model we have arrived at after data selection. - -- ' S TESTING AXIAL SYMMETRY By comparisonwith the two and three dimensional2nd order modelsof Fairfield [ 1971] and Formisano[1979], we have re- I +1 0 -1 moved2 and3 freeparameters, respectively, with theassumption of X' ½R•s) axial symmetryabout the x' axis. Thesestudies do lend supportto thisassumption that iny0, they coordinateof focusposition after the Fig.15. B•stfit to the Mars bow shock crossings from the preceding figure conichas been rotated by the angleh, tendsto be smalland much displayedin aberratedsolar ecliptic coordinatescentered on Mars. The shockcrossing marked 'A' hasnot beenconsidered in obtainingthe fit, but lessis absolutemagnitude thanx0 as shownin Table 4. Further,the only minor modificationsresult when it is included. modelsof Fairfield find only 0.8 ø and 0.6 ø tilt beyondthe mean aberrationangle (i.e., h = tan-•(30/430) = 4ø), whilethe models of differentdata setS. The Gringauz et al. [ 1975]and Gringauz [ 1975] Formisanogive much more, h = 9.1 o, or slightlyless, h = 3.6 ø, curvesare previouslypublished gasdynamic models scaled to their than expected for aberrationdepending upon whether or not a observations(i.e., one free parameter)and as such are not very scalingfactor for solar wind dynamicpressure is added. As was sensitiveto the actualobservational data in termsof determiningthe discussedearlier, the additionof the solarwind dynamicpressure shapeof the bow waves. In comparingthese models, the most scalinginto the Formisanomodel brings about large changes in the significantdifference is thatthe shocksurface obtained in thisstudy shockmodel surface that conflict with theresults ofFairfield [ 1971] is seen to be much less blunt than those obtained in the other works. andthis study.In the next sectionwe will presentarguments to the The reasonappears to be lessthe natureof the modelingtechnique effect that these anomalous results are an artifact due to the model- thanthe setof crossingsutilized. In Figure 17 we haveplotted all of ing methodemployed in that studyvis-•-vis the ,-, 103Heos 2 the Mars 2, 3, and5 shockcrossings along with somerepresentative multiple shock encountersincluded in his shock data set. orbitalpaths. First, it is apparentthat by limiting the studyto those The theoreticalexpectation of a lack of axial symmetryoriginates crossingsforward of - 1 RMSwe do not includetwo crossingsjust with theobservational fact thatthe interplanetarymagnetic field is anti-sunwardof thisplane at leastone of whichis muchfarther from onlyrarely aligned with the solarwind velocityvector. In thisevent the x' axis than is consistentwith typical Mach numberconditions B and ¾ would remain parallel in the magnetosheathdue to the and the shock observations farther downstream. In Part 2 the two 'frozen flux' condition (barring interactions with the geo- crossingsmarked 'A' in Figure17 areboth modeled as a singlelow magnetic field and non-MHD processes)and yield an axially Mach numberevent given thatthe differencein time betweenthese symmetricflow. For all otherIMF orientations,such as the average TABLE 7. Mars Bow Shock Models ? Symmetry Study S/C Period # Assumption Domain h x0(RMs) Y0(RMs) e L(RMs) Conic Type Bogdanovand Mariner4 7/15/65 2 Axial WRT x x > -13RMs =0 =0 =0 1.08 2.83 Hyp. -0 ø Vaisberg[1975] Mars 2 1971-2 7 (_+32%) (_+4%) Mars 3 Same 9 (total 18) Russell[1977] Mars2 1971-2 3 Axial WRTx x > -3RMs zO ø 0 =0 0.99 3.00 Ell. =0 ø Mars 3 1971-2 4 (_+11%) (_+4%) Mars 5 1974 4 (total 11) Thisstudy Mars 23 1971-2 46 Axial WRTx' x' > - 1RMs=0ø +0.5 =0 (_+4%)0.94 (_+1%)1.94 Ell. tan-•(-•sw•) Mars 5 1974 4 (total 14) The degreeto whichthe solarwind interactionith Marshas been ignored in the planningof missionsis very apparentin the smallnumber of shock observationsrecorded above. The 1965Mariner 4 fly-by representsthe entire American contribution. According, fewer quantitative modeling efforts have been aimed at the Mars shock surface. 11,412 SLAVINAND HOLZER: F•.OW ABOUT THE PLANETS andenergy across the bow wave.Quasi-parallel structure, for ex- ample,can result in somedeceleration of the solarwind upstreamof theshock through wave-particle interactions in theforeshock region [e.g., Bonifaziet al., 1980], while the enhancedturbulence down- 3 streamrelative to quasi-perpendicularconditions may act as an MarsBow Shock :'.½ f"" additionalenergy sink and result in a smalldecrease in local magne- tosheaththickness [e.g., Formisanoet al., 1973;Chao and Wisker- .?;',?//q Oogdonov chen, 1974]. ...:';'/'/ • ½,r'ngouz e/a/(•975o) ...... We examinethe observationalvalidity of the axial symmetry assumptionused in thisstudy further in Table 8. ThoseImp 3, Imp i:...z.•' / //' I Russell, (•977) 4, Heos 1 and Prognoz1-2 crossingsfor which measuredsolar windspeeds were available to transformthem into (x', y', z') coordi- ///,' nateshave been mappedinto the aberratedterminator plane by meansof theirrespective model surfaces (i.e., Table4). Theyhave beenseparated by they' coordinateinto dawnand dusk groupings I i'1111 i with their mean distances and statistical uncertainties listed in the 2 1 0 -• -2 table. As shownin 3 of the 4 setsof crossings,the difference/5 X'(RMs) between the mean dusk radius and the mean dawn value is less than Fig. 16. Marsb(•w shock models derived from the Mars 2, 3, and5 the uncertainty.However, in all casesthe distanceto the shockon observationswith the one from this studybeing most differentdue to its considerationof onlythe better sampled forward region (i.e., x' > - IRMs). the dusksideis indeedslightly greater than or equalto thatat dawn sothat it may be that someslight asymmetry exists on the average whicha muchlarger number of crossingscould resolve. Neverthe- Parker spiral configuration,draping of the field lines produces less,it is clearthat the assumptionof axial symmetry,and hence our magneticstresses possessing symmetry only with respectto a plane three-parametermodeling method, is valid to withinthe statistical containingthe x' axis and parallel to the magneticfield. Unfortu- resolution of the data set. nately, the only completeMHD flow modelof the magnetosheathis In additionto the two-dimensionaldawn-dusk asymmetry model that of Spreiter and Rizzi [1974] for aligned B and ¾, which is proposedby Walters[ 1964],Cloutier [ 1976]suggested that devia- accordinglyaxisymmetric about x'. Walters [1964] examinedthe tionsfrom axial symmetry would also occur due to drapedconfigu- MHD jump conditionsacross the bow shockin 2 dimensionsand rationof magnetosheathmagnetic field linescausing the flow to noted that the spiral orientationof the IMF causedthe apparent behavelike a fluid with two degreesof freedom(i.e., 7 = 2) in stagnationpoint (i.e., the locationon the shocksurface where flow regionswhere flow velocityis perpendicularto themagnetic field crossingthe shock is slowed, but not deflected) to shift from and three degreesof freedom(7 = 5/3) when flow velocityis coincidencewith the x' axis by an amount dependentupon the alignedwithB. Romanov et al. [ 1978]reported evidence for suchan upstream/•, Q, and the IMF direction with respectto solar wind asymmetryin theirVenera 9 and10 shockcrossings after mapping velocity and the shock surface. For a Parker spiral IMF with/• theminto the terminatorplane and aligningthe componentof the = 0.5and Q = 40(i.e., M,• = • = 6.3)he found a7.5 ø tilt in the interplanetarymagnetic field in they'-z' planeat thetime of each same direction as, but in addition to, the effects of aberration. crossingwith a singledirection as shown in Figure18. Theyfound However, as can be seenin Tables 1,2, and 5 more typical values thetrace of the bow wavein theterminator plane, as displayed with wouldbe fl -,• 1- 2 andQ --• 90-100. Hencethe choiceof mean parametersby Waltersbased upon the limited interplanetarymea- , i , i , i , surementsof his day significantlyover estimatedthe relative MARS ORBITER BOW SHOCK CROSSINGS strengthof theIMF. Whenthese mean parameters are coupled with Mars 2 (1971-2) the actuallarge variability of the IMF aboutthe Parkerspiral, it is Mars 3 (1971-2) not clearthat the modelof Walterswould not in fact, predictonly a 6 Mars 5 (1974) smalladditional tilt near-,- 1o. The optimalmethod of testingthis z•A Mars 3 hypothesiswith the observations is to selectonly high interplanetary o o magneticfield intensityconditions (i.e., low r, M,•) with orienta- tionsclose to the averagespiral for modelingin orderto examine onlymaximum asymmetry conditions. While suchan investigation is beyondthe scopeof this study,the fact remainsthat Walters' analysisof the jump conditionsalong the bow wave does not o , necessarilypredict any significantasymmetry in the meanwith ß • Mars5 respectto the natural distributionin upstreamparameters. This considerationhas apparentlybeen overlooked in previousstudies. 0 -2 -4 In addition,the approximatetwo dimensionalMHD flow solutions x' (RMs) of Shen[1972] suggest that the effect noted by Waltersmay not Fig. 17. Mars bow shockcrossings (O. L. Vaisbergand V. N. Smimov, resultin any net east-west asymmetries inthe shock surface due to privatecommunication, 1979)displayed alongwith sample satellite trajec- the restoringstresses of the drapedmagnetosheath fieldlines. toriesin January 1972 for Mars 2 and3 andFebruary 1974 for Mars 5. The Zhuangetal. [1981] in another approximate MHD calculation find downstreamMars5bow wave encounters havenot been included inany of onlyslight (i.e., < 1o• deviations ofthe equatorial magnetopause themodels shown inthe preceding figure. As a result,there is a strong symmetryaxisfrom the x' axis. Further, any observational resolu- tendencystudy, to beforbiased all of outward the empirical from thex' surfaces,axis by save thelimited the one spatial produced coverage bythis of tionof thisfluid problem will be complicated bythe influence of Mars2 and3. The crossings marked with the letter 'A' are discussed inthe collisionlessshock structureupon the conservationof momentum text. SLAVINAND HOLZER: FLOW ABOUT THE PLANETS 11,413 TABLE 8. Dawn-DuskAsymmetry Search For EarthBow Shock y' > 0 (Dusk) y' < 0 (Dawn) S/C N < r(x'= 0)> N dashes,to be ellipticalin approximateagreement with the predic- downstreamobservations, our model would not flair out from the x' tionsof Cloutier.Slavin et al. [ 1979b, 1980] repeatedthis experi- axis to the degreethat it does and would better representthe mentwith muchlarger sets of observationsfrom PioneerVenus and Explorer33 and 35 shockencounters at lunar distances.But, by foundthe Venusbow shockto be axisymmetricabout x' to within omittingthe dow, nstream crossings a superior fit to thenear shock the statisticalerrors shown in Figure 18. By usingthe Imp 4 bow surfaceis obtainedwhich is the declaredgoal of this modeling wave crossingsand hourly averagedmagnetic field observations, exercise. thisprocess was repeated at earthwith thesame results as displayed Themodel which has not been plotted in Figure19 is thePsw in thefigure. Thus, again we concludethat our assumptionof axial normalizedone ofForrnisano [ 1979]. As shownin Tables3 and4, symmetryfor this studyis valid. In the caseof the Romanovet al. that studyfound a very differentmodel surface when they scaled finding,the lack of symmetrymay be dueto the inclusionof shock theirobservations to a commondynamic pressure. Performing such crossingsup to'--,8Rvdownstream of Venus. At suchdistances the a scalingis, in fact,highly desirable when merging shock crossings bow waveis beginningto asymptotewith the expectedresult being fromdifferent periods of thesolar cycle due to Paw (which has a solar an ellipticalcross section whose eccentricity is determinedby the cycledependence; Fairfield [ 1979]) beingthe dominantinfluence ratioof Mr•sll to Mr•sñ (i.e., thefast mode Mach numbers parallel on shockposition by largelycontrolling magnetopause altitude (see and perpendicularto the IMF direction). Egidi et al. [1970], Holzer and Slavin [1978], andTable 5/Figure 10 of this report).The expectationis that this procedureshould DISCUSSION decreasethe 'noise' (i.e., reducethe rms deviation from the average surface)and allow refinementsin the bow wave model. Under these In thepreceding sections three parameter second order bow wave circumstancesit might then be possibleto resolve lower order modelshave beendeveloped for Venus, earth, and Mars and con- effectsinfluencing shock position. However, when Formisano did trastedwith existingmodels. Figure 19 continuesthis process by this, all of the modelcoefficients changed markedly and the tilt h displayingthe terrestrialbow wavemodel from this study along increasedby over 250%. Given the good agreementbetween the with the five-parametermodel of Fairfieldand the eclipticplane unnormalizedmodel and the resultsof the Fairfieldstudy and our traceof thesix parameter three-dimensional unnormalized model of own, the inference would then be that the normalized model had in Formisano. No corrections have been made to allow for diferences betweenthe three models in themean upstream parameters, such as BOW SHOCKS OF EARTH AND VENUS Psw,that each represent. The Fairfield'Meridian 4 ø' modelhas been MAPPED INTO THE TERMINATOR PLANE usedbecause it wasbased upon aberrated data while the ecliptic traceof theFormisano unnormalized model has been aberrated by a 3.6ø rotationto makeit symmetricabout a lineparallel to thex' axis eventhough the implied 480 km/s solar wind speed is ,-• 10% higher thanexpected. The agreementbetween the threemodels over the daysideportion of the bow wave is goodwith the unnormalized modelof Formisanoand the mean surface from this study bracket- ing the Fairfield model further downstream. The dawn-dusk asymmetryin the unnormalizedFormisano surface is due to the + 1.1 Re displacementof the focustoward dusk shown in Table 4, whilethat of theFairfield model is splitbetween a +0.3 Re offset in thefocus and a 0.8 ø tilt beyondthe assumed 4ø dueto aberration.If 4 ø had been taken as the rotation associated with aberration in the Formisanomodel, then a 0.4ø tilt in a senseopposite to that of Fairfield'smodel would have resulted. In comparingthese curves it shouldbe noted that the downstream regions, in whichthe effects of Fig. 18. As describedin thetext shock crossings by PVO andImp 4 have anyasymmetry in thebow wave are greatest, are only slightly better beenmapped into the aberrated terminator plane and rotated about thex' axis sampledin the Formisanoand Fairfield modelswhich includesome untilthe component ofthe IMF inhe Y'-Z' planeat the time of thecrossing is Explorer33 and 35 observations.The presentstudy, as stated alignedwith the Y' axis.The larger sample error bar in eachcase refers to the previously,terminates atx'= - 10Re. However, it isthe presence of averagestandard deviation about the meanin eachangular sector while the theobservations between -45 Re < x' • - 10Rethat results in the smallererror bar is the averageuncertainty of the meanobserved for each angularsector. In contrastto the resultsof Romanovet al. [1978] no slightlylower eccentricitiesof the Fairfield and Formisanomodels significantdeviations from axial symmetry are apparent in thePVO and Imp relativeto the onesproduced by thisstudy. Had we includedthe observations. 11,414 SLAVIN AND HOLZER:FLOW ABOUT THE PLANETS EARTHBOW SHOCK T-4ø observedby Whangand Ness [1970] at the Moon and possibly MODEL Romanovet al. [1978] at Venus.Thus further downstream than consideredhere, where shock position is becomi•nglimited by the fast wave signalspeed which is a functionof B, it may still be :..:..:..:.':":'"expected that the effectof the IMF on bow wave shapewill be apparent. The locationof the bow shocksof theterrestrial planets have been scaledto a singledynamic pressure at 1 AU of 3.5 x 10-8 dynes/ cm2 in Figure20 whichcorresponds to 1.5 x 10-8 at 1.5 AU, the meandynamic pressure during the Mars observations used to create the shockmodel in Figure 16 [Dolginovet al., 1976;Dolginov, X'•R•)n I ,,, , 1978a].In thisway we have avoided scaling the size of theMartian obstaclewith pressurebecause of its unknownresponse to upstream 20• 10 -10 -20 conditions.The mean, or possiblysolar cycle average,bow wave for Venusis plottedwith its weak dependenceupon dynamic pres- sure[Slavin et ai., 1980]assumed negligible relative to thescale of • -10 thefigure. At Mercurythe uneroded solar wind stand-off distance of Slavinand Holzer [1979] has been scaled to a pressure of 14x 10-8 Fairfield (1971) ...... ,•-20 dynes/cm2assuming a sixth root pressure dependence. The width of Forrnisano (1979) ...... the magnetosheathand eccentricityhave beenincreased slightly (Unnorrnalized) relative to the terrestrial case due to the lower Mach number condi- This St.udy tionslisted in Table2 for 0.3-0.5 AU in orderto givea qualitatively Y'(RE) correctpicture of theHermaean bow wave. Finally, the mean earth bow shockfrom this studyis alsoshown after being scaled to the Fig. 19. Themean earth bow shock from this study is compared with the indicatedcommon pressure. Displayed in thisway the immense size 'unnormalized'model ofFormisano [ 1979],which we have rotated by 3.6ø to includethe suggested effects of aberration,and the 'meridian 4 ø aberra- of theterrestrial magnetosphere relative to theobstacles to thesolar tion'model ofFair•½M [ ! 97! ], bothof whichinclude downstream crossings wind at the other terrestrialplanets is quite evident.In addition, at lunar orbit. whencombined with the existenceof compressional[Siscoe et al., 1969]and shocklike features [Schubert and Lichtenstein, 1974] in factbeen highly biased by thecorrection. The probable cause is the largenumber, --, 103, of multipleshock crossings athigh latitudes by Heos 2 in the Formisano data set. While these data allow him to BOW SHOCKS OF THE modelthe shockin threedimensions, they do notprovide informa- TERRESTRIAL PLANETS tion on the lower latitude shock. At the lower latitudesthe data set of SCALEDTO Psw=3.5x10 -edynes/cm 2at 1 AU) Formisanois essentiallythat of Fairfieldwith only the addition of • 30 Heos 1, a-• 35% enhancementin the numberof crossings.Hence, the ecliptictrace of the three-dimensionalmodel shouldagree with thetwo-dimensional models of Fairfield andthis study. As shownin Figure 19 and Table 4, this is indeedthe situationfor the unnor- malized Formisanoshock surface, but not the normalizedmodel. In fact, judgingfrom the figuresin Formisano[1979] it appearsthat 20 the normalizationprocedure worsens the fit to the data near the ecliptic.It is the Heos2 portionof the datathat appears to be smoothedby the P • scaling.Thus, we feel there is justification for our suggestionthat the Heos2 high-latitudedata, even though it has beenweighted, is dominatingthe Formisanonormalized model near the plane of the ecliptic. When more measurementsare available EARTH from different orbits at high latitudes, it shouldbe possibleto 10 performa morecareful data selection and avoid some of the prob- lemsexperienced by thisfirst pioneeringeffort to createa 3 dimen- sional model. 5 Hence, on the basisof the threemodels in Figure 19 andTable 4 we concludethat the upper limits on deviationsfrom axial symmetry of the forwardshock surface under average solar wind conditions I are,,-, 1 o in orientationand --, 1Re in offsetalong the y' axis.This is t•5 1o 5 0 -5 not to saythat whenthe interplanetarymagnetic field is relatively X' (Rp) strong(e.g.,/3 • 1,MA • 5, andQ • 25)the bow wave will remain Fig. 20. The bowshocks of all fourterrestrial planets are plotted in terms axisymmetricin the face of large nonaxiallysymmetric magnetic of planetaryradii to showtheir relative stand-off distances. In thecase of stresses.However, it doesstate that the typicalstrength of the IMF Venusthe mean shock position from Venera and PVO, whichmay represent is insufficientto causelarge deviationsfrom axial symmetry.In a solarcycle average, is displayed.For Mercurythe bow wavesurface shownis basedupon the uneroded obstacle height determined by Slavin and addition,it shouldbe noted again that the presenceof the spiral Holzer[1979] with the magnetosheath thickness and eccentricity increased magneticfield in the solar wind shouldresult in a nonaxially slightlyrelative to theterrestrial case to givea qualitativerepresentation of symmetricMach cone [e.g. ,Dryer andHeckman, 1967] as has been theexpected reduced Mach numberconditions at 0.3-0.5 AU. SLAVINAND HOLZER: FLOW ABOUT THE PLANETS 11,415 the solar wind flow about the limbs of the Moon and the vast bow Body Bluntness Vs. wavesof Jupiterand Saturn, this figurepoints up the largerange in Bow Shock Bluntness scalelengths over which the solar wind exhibits fluid characteris- Bb tics. Mercury, while possessingan effective obstacleof greater T+I IMs:cø ,' diameter than Mars and Venus, is still much closer in size to those planetsthan to earth. It is alsonoted that the averagelocation of the ;• Cythereanbow wave is at a lower relativealtitude than that of Mars but that nearsolar maximum their positions become comparable as Ms~7- , judged by the Pioneer Venus results. This finding weakensthe argumentthat Mars possessesa significantintrinsic magnetic mo- +l ment simplydue to its bow wave stand-offdistance being greater t Bs thanthat observed at Venusnear solar minimum by Venera9 and 10 [seeSlavin and Holzer, 1981]. Finally, it remainsto be determined (VinDyk by both theoreticaland observationalstudies to what degreethe VENUS differentabsolute shock dimensions displayed in Figure20 affect MARS the physicalprocesses of the foreshock,main transitionlayer, and magnetosheath. ,,,,,,,'•--1 EARTH Figure21 complementsthe preceding view, whichexamined the Fig.22. Theeffective bluntnesses (i.e., 1 - E2) of the obstacie surfaces at relativesizes of the bow waves,by scalingthe subsolarpoint on Venus and earth are plotted againstbow shock bluntnesswith a linear eachto unityfor thepurpose of comparingtheir respective shapes. extrapolation to Martian system indicated. Also displayed are the Eachof theplanets is displayedto providea measureof the scaling gasdynamicresults of VanDyke [1958] for 3' = 7/5,Ms = o• as well asthe factor.This would be equivalentto plottingthe shock waves in units limitingcase oft = 1, M s = o•.In thisway, theobservations of flow pastthe of obstacleradii if it were not for the fact that magnetosheath terrestrialplanets appear qualitatively well orderedby existingblunt body hypersonicaerodynamic theory. thicknessvaries with obstacleshape. As shown,the terrestrialbow wave flaresout from the symmetryaxis more than thoseof Venus deviationsdisplayed are similar, with the high compressibilityof and finally Mars. This is in contrastto the E-L modelsused in a theterrestrial obstacle producing only a marginallygreater variabil- preliminarystudy by Slavin et al. [ 1979a] with a more restricted ity in shockposition than found at Venus and Mars. data setwhich couldnot detectthe differencein shapebetween the As will be consideredin detail in part 2 of this study, these Venusand earth bow stocks.Thus, usingthe samethree parameter findingshave implications for the natureof flow pastthese planets. modelin eachcase it is foundthat Mars possessesthe mostslender In the final figure the relationshipbetween the shockand obstacle bow wave followed by Venus and earth in that order. The rms shapesis examinedin termsof a bluntnessparameter introduced by Van Dyke [1958] for the purposeof parameterizinggasdynamic shock/obstacleshape B = 1 - E2 (9) In termsof thisparameter a sphericalsurface corresponds to B = 1, while ellipsoidshave B > 0 andhyperboloids B < 0. In thelimit of BOW SHOCKS OF THE infiniteMach number the density jump acrossa normalgasdynamic TERRESTRIAL PLANETS shockgoes as (•/+ 1)/(•/- 1), where•/is theusual ratio of specific heatsat constantpressure and volume. Spreiter et al. [ 1966]found thatthe subsolar thickness of themagnetosheath was proportional to "-• MARS (•/- 1)/(•/+ 1) consistentwith the need to conservemass in the magnetosheathflow. Thus in the limit of infinite Mach numberand infinite thermodynamicdegrees of freedom(i.e., •/=(n + 2)/n, wheren = numberof degreesof freedom)the jump in density becomesinfinite With the shock degenerating into an infinitesimal layer adjacentto the obstacleso that the shockand obstaclehave identicalshapes (i.e., Bo = Bs)as shownby the dashedline with a slopeof unity in Figure22. Also displayedis the gasdynamicrela- MARS tionshipbetween shock bluntness Bs and the obstacle or bodyblunt- nessB b for M s = o• and •/= 7/5 from Van Dyke's study.Insofar asgasdynamics has applicability to themean flow aboutthe planets, EARTH it is possibleto includeVenus, earth, and Mars in this picture. Becausethe model determined eccentricity of a surfaceis dependent I 05 0 -0.5 uponthe point at whichthe focus is placed,it is necessaryto create X' an effectiveeccentricity for eachof theterrestrial planets referenced Fig. 21. The bow shocksof Venus, earth, and Mars have been scaledso to a standardfocus position. This hasbeen done in Table 9 wherethe their subsolarpoints are all the same distancefrom the center of their effectivevalue of x0 is takensomewhat arbitrarily to be 1/4 the respectiveplanets for easein comparingthe relative shapesof thesethree bow waves. The error bars normal to the shock surfacesgive the rms shockstand-off distance which is nearthe meanfor theseplanets. deviations of the observations from the model boundaries while the smaller The body shapesat Venus and earth have been taken from the bars parallel to the vertical axis representthe uncertaintyin the best fit. inopause/mantle-boundarylayer [Braceet al., 1980;Spenner et 11,416 SLAVINAND HOLZER: FLOW ABOUT THE PLANETS TABE 9. Effective Bow Shock and ObstacleShapes Planet •ARss EffectiveBow Shock EccentricityObstacle Bs Bb Venus 0.33RV 1.0 0.65 0.0 +0.55 Earth 3.5Re 1.2 0.9 -0.4 +0.2 Mars 0.38RMs 0.8 0.3 +0.4 (+0.9) Thebluntness parameter ofVan Dyke [1958], B - 1 - e2, has been used to characterize theshapes ofthe bow waves at Venus, earth, and Mars as well as the Cythereanmantle/ionopause, the terrestrialmagnetopause, and, by linearextrapolation, the obstacleto the solarwind at Mars (seeFigure 21). al., 1980;Perez-de-Tejada, 1980] and magnetopause[e.g., Fair- 3. Examinationof theearth shock crossings showed no signifi- field, 1971;Holzer and Slavin, 1978; Formisanoet al., 1979] canteast-west asymmetry in the bow waveorientation and y' coor- model surfaces. In the case of Mars we lack the in situ observations dinateoffset of thefocus at the,-• 1o and•-' 1Re levelsin approximate to determinethe obstacleshape directly (e.g., Slavin and Holzer agreementwith the previousstudies by Goslinget al. [1967], [ 1981] andreferences therein). However, a qualitativedetermina- Fairfield [1971], and the unnormalizedmodel of Formisano tionof shapeis impliedby thelinear extrapolation from the Venus [ 1979].Further, given the large variability of theIMF orientation andearth results shown with a dashedline. Inspectionof Figure 22 aboutthe Parker spiral angle and the typical values of/3 = 1 - 2 and shows that our modeling results appear consistentwith the M A = 8-10 at 1 AU in contrastthe smallervalues assumed by gasdynamicexpectations fora hypersonic(i.e. ,M s •> 5) flowwith y Walters[ 1964],it is notclear that any theoretical grounds exist for ) 7/5 (e.g., y = 2 if theinterplanetary magnetic field removesone expectingan east-west asymmetry in theaverage earth' s bow shock degreeof freedomto leaveonly two) aboutthree moderately blunt largerthan the limitsset down above. It remainsto be shownthat bodies. The three casesshown (i.e., Venus-earth-Marsin the solar theyexist during low MA and/3conditions when the IMF is strong. wind, an idealgas with y = 7/5 andMs = o•, andan idealgas with y This problemis currentlyunder study with a significantlylarger = 1 andM s - c•) differ mostlyin the adiabaticexponent y appropri- observational data base. ate to eachof the fluids. As the y factor increasesfrom the degener- 4. Mappingof the PioneerVenus and Imp 4 bow shockcross- ate y = 1 situation,the sheathregion betweenthe shockand the ingsinto the aberratedterminator plane relative to the transverse obstacleincreases in width everywhereand increasesthe rate at componentof theIMF producedno significant deviations from axial which it widenswith growingdistance downstream. The net effect symmetrycontrary to the predictionsof Cloutier[1976] andthe in Figure 22 is that for a given eccentricityobstacle surface, the experimentalfindings of Veriginet al. [ 1977]using a smallVenera eccentricityof theresultant bow waveincreases with increasingy. dataset. It is suggestedthat the Veneraresults are associated with Hence the y "• 2 traceappropriate to planetsin the solarwind is asymmetricMach cone effects due to theuse of distantdownstream displacedupward on they - 7/5 casefor a gaswith five degreesof crossingsin that study. freedom.In Part2 of thisstudy the abilityof existingflow modelsto 5. Finally,in shapethe forward portion (i.e. ,x > - 1Rob) of the matchthese observations quantitatively and the implications for the Mars shockis the leastblunt followedby the shocksof Venus and solar wind interactionwith the terrestrialplanets is examinedin theearth, respectively, implying the sameorder for thebluntnesses detail. of theirrespective obstacles. Part 2 of this studywill examinethe abilityof existinggasdynamic and MHD flow calculationsto model CONCLUSIONS theseresults and consider the implications for their respectiveinter- actions with the solar wind. 1. The threeparameter second order shock model method used in this studyproduces good fits to the forwardportions of the bow Acknowledgments.The authorswish to thankD. H. Fairfieldand the NSSDCfor a listingof Imp 3 and4 bowwave crossings, O. L. Vaisbergand wave surfacesat Venus, earth, andMars. Discrepanciesamong the V. N. Smirnovfor thePrognoz 1 and2, Venera9 and10, andMars 2, 3, and varioussecond order fits that have beenproposed appear to be due 5 shockdata, and V. Formisanofor the Heos 1 shockencounter data set. In moreto differencesin theregions included in themodels than in the addition, we have made use of the UCLA Pioneer Venus magnetometer modelingtechniques and assumptions employed (e.g., inclusionof observations,C. T. Russell,principal investigator. This researchhas been distantdownstream shock crossings will forcethe modelsurface to supportedby NASA grantNAGW-74. haveß • 1 independentof the actualshape of the nearplanet bow wave). REFERENCES 2. After scalingfor theeffects of differingsolar wind dynamic pressure,the terrestrial bow wave showed only minor variations in Auer,R.-D., MHD aspectsof theearth's bow shock, 2, Motionsinduced by positionandshape during the period 1965-1972 ß This finding Bavassano,directional B., discontinuities, F. Mariani, U.J. Viiiante,Geophys. andRes.,N. F.79, Ness, 5121, Multiple 1974. crossings supportsthe similarconclusion reached by Egidiet al. [1970] ofthe earth's bow shock atlarge geocentric distances, J. Geophys. Res., reachedby comparingthe position of thebow shock seen near solar 76, 5970,1971. minimumby Imp 1 with the solar maximum observations ofHeos 1. Bogdanov,A.V., andO. L. Vaisberg, Structure andvariations ofthe solar It isfurther found bythis study that the variations inshock stand-off Bonifazi,wind-Mars C., interactionA. Egidi, G.region, Moreno,J. Geophys. and S. Res.,Orsini,80, Backstreaming 487, 1975. ions distanceand shape remaining after correcting forPsw are ordered by outsidetheearth' sbow shock and their interactions withthe solar wind, J. thesonic Mach number and not M•, MMSii,or MMS ñ in agreement Geophys.Res., 85, 3461, 1980. withthe gasdynamic approximations of $preiter et al. [1966]and Brace,L. H., R.F. Theis,W. R.Hoegy, J. H. Wolfe,J. D. Mihalov,C. T. Dryerand Heckman [1967]. In contrast,the Venus bow shock Russell,R.C. Elphic, and A. F. Nagy, The dynamic behavior ofthe Venusionosphere in responseto solarwind interactions,J. Geophys. becamesignificantly more distant between 1975 and 1979. This Res., 85, 7663, 1980. changehas been attributed by Slavin et al. [ 1979b,1980] to a solar Breus,T. K., Venus:Review of present understanding ofsolar wind inter- cyclerelatedvariationinthesolarwindinteractionwiththatplanet.action, Space Sci. Rev., 23, 153,1979. SLAVINAND HOLZER: FLOW ABOUT THE PLANETS 11,417 Chao, J. K., and M. J. Wiskerchen,The ratio of specificheats for post- Howe, H. C., Jr., andJ. H. Binsack,Explorer 33 and35 plasmaobserva- shockplasmas of a detachedbow shock:An MHD model,J. Geophys. tions of magnetosheathflow, J. Geophys.Res., 77, 334, 1972. Res., 79, 4769, 1974. Hundhausen,A. J., S. J. Bame,and J. R. Asbridge,Plasma flow patternin Cloutier,P. A., Solarwind interactionwith planetaryionospheres, in Solar the magnetosheath,J. Geophys.Res., 74, 2799, 1969. WindInteractions with thePlanets Mercury, Venus,and Mars, editedby King, J. H., InterplanetaryMedium Data Book,Rep. NSSDC 7704, NASA N. F. Ness,NASA SP-397, pp. 111- 119, 1976. GoddardSpace Flight Center, Greenbelt, Md., 1977. Diodatio, L., G. Moreno, G. Signorini, and K. W. Ogilvie, Long-term King, J. H., Solarcycle variations in IMF intensity,J. Geophys.Res., 84, variationsof the solar wind plasmaparameters, J. Geophys.Res., 79, 5938, 1979. 5095, 1974. Michel, F. C., Detectabilityof disturbancesin the solarwind, J. Geophys. Dolginov, Sh. Sh., On the magneticfield of Mars: Mars 2 and 3 evidence, Res., 70, 1, 1965. Geophys.Res. Lett., 5, 89, 1978a. Ness, N. F., The magneticfields of Mercury, Mars, and the Moon, Ann. Dolginov, Sh. Sh., On the magnetic field of Mars: Mars 5 evidence, Rev. Earth Planet, Sci., 7, 249, 1979. Geophys.Res. Lett., 5, 93, 1978b. Ness, N. F., K. W. Behannon,R. P. Lepping, and Y. C. Whang, The Dolginov,Sh. Sh., E.G. Eroshenko,D. N. Zhuzgov,V. A. Sharova,K. I. magneticfield of Mercury, J. Geophys.Res., 80, 2708, 1975. Gringauz, V. V. Bezrukikh, T. K. Breus, M. I. Verigin, and A. P. Olilvie, K. W., J. D. Scudder,V. M. Vasyliunas,R. E. Hartle, and G. L. Remizov, Magnetic field and plasmainside and outsideof the Martian Siscoe,Observations at theplanet Mercury by theplasma electron exper- magnetosphee,in Solar Wind Interaction with the Planets Mercury, iment, J. Geophys.Res., 82, 1807, 1977. Venus,andMars, editedby N. F. Ness,NASASP-397, pp. 1-20, 1976. Olson, J. V., and R. E. Holzer, On the local time dependenceof the bow Dryer, M., Solar wind interactions-hypersonicanalogue, Cosmic Electro- shockwave structure,J. Geophys.Res., 79, 939, 1974. dyn., 1, 115, 1970. Perez-de-Tejada,H., Evidencefor a viscousboundary layer at the Venus Dryer, M., andR. Faye-Petersen,Magnetogasdynamic boundary condition inopausefrom the preliminaryPioneer Venus results, J. Geophys.Res., for a self-consistentsolution to theclosed magnetopause, AIAA J., 4, 246, 85, 7709, 1980. 1966. Romanov, S. A., V. N. Smirnov, and O. L. Vaisberg, Interactionof the Dryer, M., andG. R. Heckman,Application of the hypersonicanalogue to solar wind with Venus, Cosmic Res., 16, 603, 1978. the standingshock of Mars, Solar Phys., 2, 112, 1967. Russell, C. T., On the relative locations of the bow shocksof the terrestrial Egidi, A., V. Formisano,F. Palmiotto, P. Saraceno,and G. Moreno, Solar planets, Geophys.Res. Lett., 4, 387, 1977. wind and location of shockfront and magnetopauseat the 1969 solar Russell, C. T., The magneticfield of Mars: Mars 3 evidencereexamined, maximum,J. Geophys.Res., 75, 6999, 1970. Geophys.Res. Lett., 5, 81, 1978a. Fairfield,D. H., Averageand unusual locations of theearth's magnetopause Russell, C. T., The magneticfield of Mars: Mars 5 evidencereexamined, and bow shock,J. Geophys.Res., 76, 6700, 1971. Geophys.Res. Lett., 5, 85, 1978b. Fairfield,D. H., Globalaspects of theearth's magnetopause, Proceedings of Russell, C. t., The interaction of the solar wind with Mars, Venus, and the MagnetosphericBoundary Layers Conference,ESA SP-148, Evr. Mercury, inSolar SystemPlasma Physics, vol. 2, editedby C. F. Kennel, spaceagency, 1979. L. J. Lanzerotti, and E. N. Parker, pp. 208-252, North-Holland, Fairfield, D. H., and K. W. Behannon,Bow shockand magnetosheath Amsterdam, 1979. wavesat Mercury, J. Geophys.Res., 81, 3897, 1976. Russell, C. T., and E. W. Greenstadt,Initial Isee magnetometerresults: Formisano,V., Orientationand shapeof the earth's bow shockin three Shock observations,Space Sci. Rev., 23, 3, 1979. dimensions,Planet. SpaceSci., 27, 1151, 1979. Schubert,G., and B. R. Lichtenstein,Observations of moon-plasmainter- Formisano,V., P. C. Hedgecock,G. Moreno,F. Palmiotto,and J. K. Chao, actionsby orbital and surfaceexperiments, Rev. Geophys.Space Phys., Solar wind interactionwith the earth's magneticfield, 2, MHD bow 12, 592, 1974. shock,J. Geophys.Res., 78, 3731, 1973. Shen, W.-W., The earth's bow shockin an oblique interplanetaryfield, Gazis, P. R., A. J. Lazarus,and J. D. Sullivan, Voyagerobservations of CosmicElectrodyn. , 2, 381, 1972. solarwind temperature1-10 AU (abstract),Eos Trans. 62, 378, 1981. Siscoe,G. L., E. F. Lyon, J. H. Binsack,and H. S. Bridge,Experimental Gosling, J. T., and S. J. Bame, Solar wind speed variations: An evidencefor a detachedlunar compression wave, J. Geophys.Res., 74, autocorrelationanalysis, J. Geophys.Res., 77, 12, 1972. 59, 1969. Gosling,J., J. R. Asbridge,S. J. Bame,and I. Strong,Vela 2 measurements Siscoe,G. L., andJ. A. Slavin, Planetarymagnetospheres, Rev. Geophys. of themagnetopause and bow shockpositions, J. Geophys.Res., 72, 101, SpacePhys., 17, 1677, 1979. 1967. Sittler, E. C., Jr., and J. D. Scudder,An empiricalpolytrope law for solar Greenstadt,E. W., Dependenceof shockstructure at Venus and Mars on wind thermal electronsbetween 0.45 and 4.76 AU: Voyager 2 and orientationof the IMF, CosmicElectrodyn., 1, 380, 1970. Mariner 10, J. Geophys.Res., 85, 5131, 1980. Greenstadt,E. W., P. C. Hedgecock,and C. T. Russell, Large-scale Slavin, J. A., and R. E. Holzer, The effect of erosion on the solar wind coherenceand high velocitiesof the earth'sbow shockon February12, stand-offdistance at Mercury, J. Geophys.Res., 84, 2076, 1979. 1969, J. Geophys.Res., 77, 1116, 1972. Slavin, J. A., and R. E. Holzer, The solar wind interaction with Mars Gringauz,K. I., Reporton 18thCOSPAR meeting, Rep. D-194, SpaceRes. revisted,J. Geophys.Res., in press, 1981. Inst., Moscow, 1975. Slavin, J. A., R. C. Elphic, C. T. Russell, J. H. Wolfe, and D. S. Gringauz, K. I., V. V. Bezrukikh, M. I. Verigin, and A. P. Remizov, Intriligator,Position and shapeof the Venus bow shock:Pioneer Venus Studiesof the solarplasma near Mars andon the earth-Marsroute using orbiter observations,Geophys. Res. Lett., 6, 901, 1979a. chargedparticle traps on Sovietspacecraft in 1971- 3, CosmicRes., 13, Slavin, J. A., R. C. Elphic, and C. T. Russell,A comparisonof Pioneer 107, 1975. Venus and Venera bow shockobservations: Evidence for a solarcycle Gringauz,K. I., V. V. Bezrukikh,M. I. Verigin, and A. P. Remizov, On variation, Geophys.Res. Lett., 6, 905, 1979b. the electronand ion componentsof plasmain the antisolarpart of near- Slavin, J. A., R. C. Elphic, C. T. Russell, F. L. Scarf, J. H. Wolfe, J. D. Martian space,J. Geophys.Res., 81, 3349, 1976. Mihalov, D. S. Intriligator, L. H. Brace, H. A. Taylor, Jr., and R. E. Gringauz,K. I., A comparisonof the magnetospheresof Mars, Venus, and Daniell, Jr., The solar wind interaction with Venus: Pioneer Venus the Earth,Physics of PlanetaryMagnetospheres, vol. 1, Proceedingsof observationof bow shocklocation and structure,J. Geophys.Res., 85, Symposium3 of the 23rd PlenaryMeeting of COSPAR, editedby K. 7625, 1980. Knott, pp. 5-24, Budapest,1980. Smith, E. J., Planetarymagnetic field experiments,in AdvancedSpace Holzer, R. E., and J. A. Slavin, Magnetic flux transferassociated with En¾ironments,edited by O. L. Tiffany and E. M. Zaitzeff, American expansionsand contractionsof the daysidemagnetosphere, J. Geophys. AstronauticalSociety, Tarzana, Calif., 1969. Res., 83, 3831, 1978. Smimov,V. N., O. L. Vaisberg,and D. S. Intriligator,An empiricalmodel Holzer, R. E., and J. A. Slavin, The effect of solar wind structure on of the Venusianouter environment, 2, The shapeand location of the bow magnetosphericenergy supply during solar cycle 20, J. Geophys.Res., shock,J. Geophys.Res., 85, 7651, 1980. 86, 675, 1981. Spenner,K., W. C. Knudsen,K. L. Miller, V. Novak, C. T. Russell,and R. Holzer, R. E., M. G. McLeod, andE. J. Smith,Preliminary results from the C. Elphic, Observationof the Venus mantle, The boundaryregion be- Ogo 1 searchcoil magnetometer:Boundarypositions and magnetic noise tweenthe solarwind andionosphere, J. Geophys.Res., 85, 7655m 1980. spectra,J. Geophys.Res., 71, 1481, 1966. Spreiter,J. R., and A. Y. Alksne,Solar wind flow pastobjects in the solar Holzer, R. E., T. G. Northrop,J. V. Olson, and C. T. Russell,Study of system,Ann. Rev. Fluid Mech., 8, 313, 1970. wavesin the earth'sbow shock,J. Geophys.Res., 77, 2264, 1972. Spreiter,J. R., and W. P. Jones,On the effect of a weak interplanetary 11,418 SLAVINAND HOLZER: FLOW ABOUT THE PLANETS magneticfield on theinteraction between the solarwind andgeomagnetic With the PlanetsMercury, Venus,and Mars, editedby N. F. Ness, pp. field, J. Geophys.Res., 68, 3555, 1963. 21 - 40, NASA SP-397, 1976. Spreiter,J. R., andA. W. Rizzi, AlignedMHD solutionfor solarwind flow Van Dyke, M. D., The supersonicblunt body problem--Review and exten- past the earth's magnetosphere,Acta Astronaut., 1, 15, 1974. sion,J. AeroSp. Sci., 29, 485, 1958. Spreiter,J. R., andS. S. Stahara,A new predictivemodel for determining Verigin,M. I., K. I. Gringauz,T. Gombosi,T. K. Breus,V. V. Bezrukikh, solar wind-terrestrialplanet interactions,J. Geophys.Res., 85, 6769, A. P. Remizov, and G. I. Volkov, Plasmanear Venusfrom the Venera 9 1980a. and 10 wide-angleplasma analyzer data, J. Geophys.Res., 83, 3721, Spreiter,J. R., and S. S. Stahara,Solar wind flow pastVenus: Theory and 1978. comparison,J. Geophys.Res., 85, 7715, 1980b. Walters,G. K., Effectof obliqueinterplanetary magnetic field on the shape Spreiter,J. R., A. L. Summers,and A. Y. Alksne,hydromagnetic flow andbehavior of the magnetosphere,J. Geophys.Res., 69, 1769, 1964. aroundthe magnetosphere,Planet. SpaceSci., 14, 223, 1966. Whang,Y. C.. andN. F. Ness,Observations and interpretations of thelunar Theis, R. F., L. H. Brace, and H. G. Mayr, Empiricalmodels of electron Mach cone,J. Geophys.Res., 75, 6002, 1970. temperatureand density in the Venusionosphere, J. Geophys.Res., 85, Wolfe, J. H., The large scalestructure of the solarwind, in Solar Wind, 7787, 1980a. editedby C. P. Sonett,P. J. Coleman,Jr., andJ. M. Wilcox, pp. 170, Theis, R. F., L. H. Brace, K. H. Schatten,C. T. Russell,J. A. Slavin, andJ. NASA SP-308, 1972. H. Wolfe, The Venusionosphere as an obstacle to the solarwind, Physics Wolff, R. S., B. E. Goldstein,and S. Kumar, A modelof the variabilityof of PlanetaryMagnetospheres, vol. 1, Proceedingsof Symposium3 of the the Venusionopause altitude, Geophys. Res. Lett., 6, 7641, 1979. 23rd Plenary Meeting of COSPAR, edited by K. Knott, pp. 47-60, Zhuang,H. C., C. T. Russell,and R. J. Walker,The influenceof theIMF Budapest,1980. andthermal pressure on the positionand shapeof the magnetopause,In Vaisberg,O. L., Mars-plasmaenvironment, in Physicsof Solar Planetary press,J. Geophys.Res., 1981. Environments,edited by D. J. Williams, pp. 854-871, Wasington,D. C., 1976. (Received March 3, 1981; Vaisberg,O. L., A. V. Bogdanov,V. N. Smirnov,and S. A. Romanov,On revisedJuly 8, 1981; the nature of the solar wind Mars interaction, in Solar Wind Interaction acceptedJuly 9, 1981.)