194 9ApJ. . .109. .308K 34 3_12 3 retrograde orbitsorhighinclinationsareomitted.Asomewhatbettersecondapproximationisgivenin distance (intermsofthemeantosun)beA;thenlawplanetarydistancesafirst of theprotoplanetsorprotosatellites,whichyieldedandretained,onaverage,onlyabout1percent equation (5).Theresultsareinterpretedasduetothecombinedeffectofgravitationalbreakupgase- approximation is/i/A^10~.Thesamerelationholdsforeachofthesatellitesystemsifobjectswith “minor” amongtheplanets,andtoabsenceoftrue“gaps”inplanetaryorsatellitesystems. has arousedrenewedinterestinconnectionwithseveralnewtheoriesontheoriginof in spiteofthefactthatthisexpressioncontainsthreeadjustableconstants.Itrequires ous ringsatthecriticaldensity,whichwouldleadto/¿/A=10or10“,andofincompletecondensation of eachmass. simplified termswemayregardtheattractionbetweentwocompanionsascompeting Let themassesexpressedintermsofprimarybe/¿iandju,withfxastheiraverage.It this problem.Itisfoundthatasinglerelationwilldescribeallthe“normal”planetary geometric series(noconstantterm). also beensuggestedfortheplanetsaswellsatellites,replacingequation(1)bya of severalirregularitieshascastdoubtonthevaluetheseresults.Distancelawshave Jupiter, andrepresentsNeptune(nottomentionPluto)poorly.RecentlyBode’slaw and satellitedistances,includingtheso-called“gaps,”withaprobableerrorofabout the arbitraryrulethat^=—ooforMercury,needsanothergapbetweenMarsand this simpleexpressionisasgoodas,orbetterthan, Bode’slaw.Butacloserfitmaystill is clearthatthesmallertwocompanions,i.e.,valueof/x, latter requiresm/A<1.Weshallfindthat with thetidalforceofprimary;conditionthatformerbesmallerthan ellites). Letaiand0%bethesemi-majoraxesoforbitsA=(#2—ai)/i(#i+(h)- the solarsystem. be obtained. gives afairlycloserepresentationoftheobservations, bothforplanetsandsatellites; the relativeseparation,A,canbewithoutdangertostabilityoftwoorbits.In 10 percent. 2 © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem Attention iscalledtotheexistenceoftwoclassessatellites,analogous“major”and Consider twoconsecutiveplanets;lettheiraveragemassratio(intermsofthesun)bemand Relations similartoBode’shavebeensuggestedforthesatellites,butpresence The discoveryofthefifthsatelliteUranushasinducedwritertore-examine 2. Considerthesunwithtwoconsecutiveplanets(oraplanetsat- 1. PlanetarydistancesareonlyimperfectlyrepresentedbytheBode-Titiuslaw, * ContributionsfromtheMcDonaldObservatory,Universityof Texas,No.166. THE LAWOFPLANETARYANDSATELLITEDISTANCES* Yerkes andMcDonaldObservatories Received January26,1949 Gerard P.Kuiper ¿=0.4+ 0.3.2”(w=0,1,2,...),(i) ABSTRACT 308 194 9ApJ. . .109. .308K 1 3 therefore obtainaroughapproximationforthemasses oftheprotoplanetsifwemodify higher) meandensity.Thismeansthattheprocessofsegregationwouldtakeplacees- in whichtoexpressplanetarydistances.Wemayalsovisualizetheprocessofplanet ure 1;onemayvisualizemanysmallspheresinsidethedisk,eitherseparating,ifmean have hadessentiallythesamenumericalvalueasthatdeterminedbymodelofFig- fore consideredfirst,consistingoftwosphericalmasses(“protoplanets”)innearcon- bodies andcouldnotreadilybeappliedtocompressiblebodies.Asimplemodelisthere- formation. Onlythoseobjectswillbeconsideredwhichcouldhaveoriginatedinaflat- equation (3)soastoallowforthesmallthickness ofthedisk: density ofthefieldwerebelowcriticalvalue,orcombining,ifitabove.Weshall low orbitalinclinationsshowthatthediskfromwhichplanetsoriginatedwasvery Roche limitleadstotheequation large-scale turbulence)formingplanetsandbreaks,respectively.Applicationofthe sentiafly attheRochelimit,withaccidentalmaximaandminima(probablyproducedby the meandensityincombinedvolume.Eachoftwomassesdoesnotbreakup,by tact, locatedinsidethegaseousdisksurroundingsun(Fig.1).Byassumption, therefore beconsidered.Suchcomputationsashavebeenmaderefertoincompressible two massesremainseparatedandarethereforeinsidetheRochelimitappropriatefor tional andhydrodynamicalforces.Thestabilityoftoroidsflattenedringsshould tened disk,innearlycircularmotion.Suchadiskmusthavebeenbrokenupbygravita- be explainedbyassumingthat,ontheaverage,somewhat lessthan1percentofthemass The discrepancybetweentheobservedrelation(2) andthepredictedrelation(4)may where C=(2.44/2)forspheres;2.44istheconstantwhichoccursinRochelimit. assumption, andmustthereforebeoutsidetheRochelimitappropriateforits(somewhat of theprotoplanetscondensedintoplanetsthemselves. Thisfractionisquitereason- However, thedensitywithindiskatwhichgravitationalbreakupoccurredmuststill flat andwouldnothavebeenabletocontainthehugespheresvisualizedinFigure1. able inviewoftheknownplanetarycomposition. disks willbereflectedonlybythoseobjectswhich now moveinorbitsoflowinclination tvith respecttothefundamental planeofeachsystemandinnearlycircular orbits. 1 F.Tisserand,Mécaniquecéleste, Vol.2(Paris:Gauthier-Villars,1891),chaps,ix-xi. © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem The significantfeaturesofplanetsandsatellites originating withinhighlyflattened The stabilityoftheorbitsisnotonlyapproachtofindingproperparameters The modelasitstandsisnotadequatetodescribethesystemofplanetsbecause Fig. 1.—Thesun{blackcircle)andtwoprotoplanets LAW OFDISTANCES 3 A 1 10-2-10-. (4) 309 194 9ApJ. . .109. .308K 2 3 3 5 4 -14 310 yield twopairsofcompanionswhoseinclusionheremightbelegitimate.Twoother: objects wereomittedbecausethepresentdistancesbearnoknownrelationto“origi-I In thediscussionofactualdistancesweshallthereforehavetoexcludeanumber to separatethecasesofveryunequalcompanionsfromthoseinwhich different masses;clearly,theproblemthenbecomesasymmetrical.Itisthereforeprudent clined andeccentricorbits(Pluto,JupiterVI,VII,X). limits 1<510oo. are essentiallyequal.Threeclassesdistinguished,A,B,andC,withthemass-ratio to bodilytides;andHyperion,whichisstronglyperturbedbythemassivesatellite, except wheretheyoccurredingroups(JupiterVI,VQ,X;VIII,IX,XI),whicheach showing planetandsateUitedistancesareixA.Figures23showtheresults; and theearth;itmeasures truediameterifdifferencesinalbedoareneglected. has beenimplicitinthederivationofeq.(3);aplotwasalso madewiththeratio02/aiofconsecutive form pairsofsatellitesbelongingtodifferentgroupsorbital inclination. interval 2.5-3.2a.u.isthemostdenselypopulatedandwillbeusedhere.First,largest tio, ju-Furthermore,itisnotedthatthe(/x,A)relationslightlyshiftedforgroup: with thedatadescribedbelow.Bode’slawcorrespondsroughlyto02/^1^2orA§ Titan. nal” ones:Phobos(thesatelliteofMars),whichiscontinuallyclosinginonMars,owing subsequent perturbations.Itwouldobviouslybecontrary tothediscussioninsection2ifwewere therefore refertoadifferentprocess.Thesamedoubt attachestotheuseofaverage inclinations andeccentricitiesarehigherthanfor theplanets;(A,fx)relationmay fainter). Wefind0.0007a.u.astheiraverageseparation,orlogÄ=—3.6;while/x> case isthatinwhichtheprimarystarhasbothacloseanddistantcompanion.In ratio, ¡X,willbeconsidered.Thecaseof/x^1occursinmultiplestars.simplest side ofTitan.Wearedealinghere,evidently,withasystematiceffect,notananomaly. place oftheso-called“gaps”customarilyassumed:betweenMercuryandVenus;be- and 3logA^—0.53;Thislawisseentomisstheimportanteffectofmassra-| the dataaregiveninTable1.The“irregular”objectslistedabovewereomitted orbits, buttheuseofAwasfound tobemoreconvenient. of thefourpairsJupitersatellites,alsoentered inFigure3. asteroids mightbeconsideredseparately.Therearesevenofthemwithg<6^0be- schematically. case thegeometricmeanratioofdistancestoprimaryisabout100,whichimplies tween MarsandJupiter;oneachsideoftheGalileansatellitesi “irregular” objects,suchasthosehavingretrogradeorbits(fivesatellites)orhighlyin- common originwithineachgroup.Thescatterinlongitude nowobservedcouldwellbeattributedto log fx=—10.Second,weconsiderallasteroidsdowntog<11^0(onehundredtimes tered inFigure3,withtheprobableareaoccupiedbysmaller¡xvaluesindicated tween 2.5and3.2a.u.ForthesesevenlargeasteroidswefindlogÄ=—1.5. 5 10.Thisshifttakes class B,andFigure2,c,C,while3showsallthreecombined,together A ^2,whilefxmightbeanywherebetween1andabout0.1.Thisvaluehasbeenen- 2 •10andlog/x=—12J.Theuseoftheseasteroid dataisdoubtfulbecausetheorbital 5 2 4 3 © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem Thequantitygisthemagnitude whichtheasteroidwouldhaveatunitdistancefrom both thesun| It isnotobviouswhattheeffectwouldbe,ifany,twocompanionshadvery Theinclinationsandsemi-majoraxeswithineachgroup are similarandsuggestthepossibilityof 3. Itfollowsfromtheprecedingdiscussionthat“natural”co-ordinatesforaplot The otherextremevalueof/xisfoundamongtheasteroids.Inasteroidbelt' Figure 2,a,showspairsofclassAonly(nearlyequalcompanions);b,' In ordertoextendtherelationshipsshowninFigure3,twoextremesofmass InthediscussionofFig.3wedonotassumeA<1,although thisistrueinmostofthediagramand G.P.Kuiper,Ap.95,212,1942. GERARD P.KUIPER 194 9ApJ. . .109. .308K scales arelogarithmic.Subdivisionbymassratioau/a^intothree groups,1<5100. Fig. 3{below).-.—Combinedplot Fig. 2{above).—Fractionaldistances,A,forplanetsandsatellites versusfractionalmasses,At;both © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem -7 -6-5-4-3 194 9ApJ. . .109. .308K 9 9 Similarly, thepairUranus andNeptunecompareswithloEuropa withEuropai counterpart inthepairs,Dione andRheaArielUmbriel,inthesatellite systems.^ and Ganymede. numerous asminorplanets. include allthesixoutersatellitesofJupiter;they arecharacterizedby/x<10~and unequal mass.Thesamerelationalsoholdsforsatellites. The constantAequals1.8forplanetsofcomparablemassand3.6very only: 312 A <0.05.Itisconceivablethattheirnumber quitelarge,thoughprobablynotso The worst-fittingobjectsareMimasandEnceladus. p <10~,respectively.Thereare,similarly,twosystems ofsatellites.“Minor”satellites -37 © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem d) Itisnotedthatthetwo sister-planets,VenusandEarth,havetheir dynamical ; The followingstatementswillsummarizethedistributionshowninFigure3. c) Therearetwosystemsofplanets,majorand minor, with10>p10~and b) TheprobableerrorofpredictionAfrom¡jlis,ineachcase,about10percent. a) Fortwoconsecutiveplanetstherelativedistance,A,isafunctionofmasses Average, largeasteroids.... Average, mediumasteriods. Jupiter, Saturn.. Jupiter VIIIandIX. Jupiter V,lo Mars, Jupiter.., Miranda, Ariel.. Jupiter XIandVIII. Jupiter VIIandX.. Io, Europa Venus, Earth Mercury, Venus.. (Titan, lapetus)*.. Mimas, Enceladus. Titania, Oberon. Tethys, Dione.... Enceladus, Tethys Jupiter VIandVII.. Europa, Ganymede.. Uranus, Neptune. Saturn, Uranus.. Earth, Mars Ariel, Umbriel... Dione, Rhea Rhea, Titan Umbriel, Titania. Ganymede, Callisto. Relative Distances,a,andMeanMasses,m,forPlanetsSatellites * Inclusionuncertain;relativeinclination14°. Average oflastfourpairs, Pairs ofCompanions TçtAy 3 ¡ logn=2.5A—^4.(5) GERARD P.KUIPER TABLE 1 B-C Class B-C (C) (B) A A A A A A A B B B C C A A A B A B C C C C C C 10.6 12J 10. 10.2 10.5 10.5 4.32 3.21 3.32 4.18 4.27 4.50 3.78 9.4 4.72 5.56 5.89 3.90 3.90 5.82 5.78 5.53 6.19 6.96 5.2 5.6 5.7 5.1 log 0.605 0.671 0.321 0.00 0.025 0.550 0.458 0.0003 0.441 0.588 0.414 0.246 0.019 0.04 0.456 0.797 0.03 0.978 0.795 0.331 0.212 0.247 0.01 0.288 0.485 0.328 0.379 1.093 3 — logA 10.8 0.52 0.69 0.12 0.65 0.03 0.78 0.30 0.30 4.5 0.94 2.02 5.2 1.48 1.07 1.14 1.02 1.44 1.83 1.02 1.26 1.82 1.62 1.45 194 9ApJ. . .109. .308K 3 6 3 -1-23-4 than 10.Forthehypotheticalsecondsatellitetoearthju/Awouldbeonly0.2ifbasedonpre- XI, thevalueisonly18.Itmaybesurmisedthatnosatelliteswillfoundforwhichless has thereforebeencarriedtotheextreme. sent distanceofthemoon. had togetexceedinglyflatbeforethedensityreachedcriticalvalue.Theorbitalin- quantity isabout100;forSaturn-Phoebe,190;theleaststableorbits,thoseofJupiterVIII,IX,and never attained,notevenuponcondensationofthering;inthiscaseflatteningprocess clinations arethusameasureoftheverticalconcentrationthatwasrequiredbefore unknown factorasaresultofthetides.Ifthiswerelessthan4,outersolution Roche densitywasreached.InthecaseofringsSaturn, developments insection2,thegaseousdisksurroundingsunflatteneduntilitreached diameter oftheobjectislessthan10meters. for thesecondsatellitewouldberuledoutbecauseorbitunstableowingto known satellitewouldaccordinglyeitherbeabouteighttimesfartheror predict thedistancetoearthofahypotheticalsecondsatellite.Suchsatellitehas pared toequation(4);ifthisinterpretationiscorrect,itmeansthatthelargestplanets turbulence. Ifonestartedwithagascloudofverylowdensity,suchwouldhave the critical(Roche)density,afterwhichitbrokeupintoapatternprobablysetby solar perturbations.Theinnersolutionisruledoutbydirectobservationunlessthe which, extrapolated,giveslogA=0.56(since/x—2.2),or1.54.Theun- not beenfoundandmustbeassumedtosmall.Therefore,A=3.6inequation(5), smallest ones(lO-!!)and10~-10',respectively). collected aboutonehundredtimesaslargeafractionoftheirprotoplanetsdidthe ones collectedalargerfractionoftheprotoplanetthandidsmallerones.Thismay closer thanthemoon.Thelatterhasincreaseditsoriginaldistancefromearthbyan If itwereabonafidesatelliteratherthancapturedplanet,onecoulduseFigure3to well betheinterpretationofdifferenceinslopepresentedbyequation(5)ascom- centage wasundoubtedlydifferentforplanets,inthesensethatmassive mass ofeachprotoplanetappearstohavecondensedintothepresentplanets.Thisper- retical prediction(4).Ithasbeenstatedthat,ontheaverage,roughly1percentof for outsideCallistoandoneachsideofTitan. to discoverwhetherthesatellite“gaps”containminorsatellites;theyshouldbelooked 63 © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem Itfollowsfromsec.2thatstablesatelliteorbitsrequireju/A>1.Fortheearth-moonsystemthis One finalremarkmaybemadeontheoriginoforbitalinclinations.Accordingto Inspection ofFigure3showshowpeculiarasatelliteourmoonis,withlog=—1.9. e) The“gap”betweenMarsandJupitercontainsminorplanets.Itwillbeofinterest In conclusionwereturntotheinterpretationofequation(5)ascomparedtheo- LAW OFDISTANCES 313