1961ApJ. . .133. .6575 -1 1 1 are notdistinguishedinthiscommunication. distances, a,andeccentricitieshavebeencalculated.Cometsurfacetemperaturehascalculatedas of theparaboliccometshasbeeninvestigated,andresultingcorrectionstoobservedaphelion by occasionalcollisionswithpassingstars. ing theapparentorbitaleccentricityovertrueinallcases.Foracometwithperihelion dius oftheorder50000-150000a.u.whichbecome visibleastheirperiheliaarelessened perature at1a.u.isroughly180°K,andrepresentativevaluesforthemassesradiiofcometary as much+0.0005auandthattheapparentshiftin1/aispositiveforeverycomet,therebyincreas- given evidenceofacloudhundredsmillions cometssurroundingthesunatara- the cometsareobservedononlytheirfirstor fewpassagesaboutthesun.Hehas and 1a.u.Thistheoryisnotinagreementwith observationswhichshownearlyall of 1au,thecorrectiontoobservedeccentricitymaybeasmuch—0.0005.Thecometsurfacetem- It wasfoundthatparaboliccometswithperiheliaof1au.mayexperienceanapparentshiftin1/a a functionofsolardistance,andestimateshavebeenmadethemassesradiicometarynuclei. the surfacematerialinsolardirectionTheeffectofthisunidirectionalrapidmasslossonorbits aphelia willrangefrom25to100a.u.forcometswithperiheliondistancesbetween4.5 aphelia distributionofthecometsshouldbeconstantforequalintervals1/a,and comets tohavenegligible1/a.Oort(1950a,b)has concludedthatthegreatmajorityof that thecumulativeeffectofthisperturbationissuchafteramillionyears a.u.“ forcometswithperiheliaof1and4.5a.u.,respectively.VanWoerkemhasshown possibility existsthatseeminglyparaboliccometsactuallyapproachthesunalongellipti- nuclei are6X10"gmand300meters. solar system.Cometsarevisibleonlywhilenearthesun(<8a.u.),andtheirobserved or reciprocaloftheaphelion,1/a,onanaveragepassageis±0.00053and±0.00071 tion oncometaryorbitsrevealedthattheresultantchangeinreciprocalmajoraxis pass closetothesun. cal trajectoriesandhavetheirapparenteccentricityincreasedbyperturbationsasthey orbital eccentricitieshavebeencomputedfromonlyasmallfractionoftheirorbits.The not clearlyresolved.BecauseofthemotionsolarsysteminourGalaxy,ab- ing theiroriginandwhetherornottheyarepermanentmembersofthesolarsystem orbits mustbeexplainedifcometsaretoconsideredaslong-timemembersofthe origin, buttheexistenceofapproximatelyparabolicandsometimesslightlyhyperbolic sence ofanysignificantlyhyperbolicorbitcastsconsiderabledoubtonaninterstellar 1 5 © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem Themajoraxisofanearlyparaboliccometandtheaphelion differbylessthanonepartin10and As acometapproachesperihelionandthesurfacefacingsuniswarmed,thereevaporationof * Partofthisworkwascompletedinpartialfulfilmentthe M.A.degreeinphysicsforR.E.Squires Investigations (Sinding1948;VanWoerkem1948)oftheeffectJupiter’sperturba- Since mostcometsareobservedtomoveinnearlyparabolicorbits,questionsconcern- PHYSICAL ANDORBITALBEHAVIOROFCOMETS* University ofCalifornia,Davis,andtheAerojet-General Received August11,I960;revisedOctober14,1960 University ofCalifornia,Davis,California Corporation, Sacramento,California DaVid B.Beard Roy E.Squires I. INTRODUCTION ABSTRACT AND 657 1961ApJ. . .133. .6575 2 2 2 4 sun isEi=eiE*/47rr,where€itheemissivityofsurfaceatsolartemperature. know thebehaviorofreactionforceduetovaporpressure,Fasafunctionsolar the effectofrapidmasslossontrajectoriesparaboliccomets,ashasbeenin- may wellbecomeunobservableafterafewpassages,dependingontheiroriginalsizeand is subjecttoslowrotationsonly,resultingintheevolutionofmaterialpredominantly indeed survivebutafewpassagesaboutthesunbecauseoftheirrapidevolutionmat- monia, andpossiblymethane,carbondioxide,cyanogen,mixedwithmeteoricma- low-density, rathercobwebbyconglomerationoficesmoleculessuchaswater,am- apparitions, probablyloseupto1percentormoreoftheirmassineachapparition. dim cometsinJupiter’sfamily,whichhavelostmuchoftheirvolatilematterrepeated ter inthesolarheat.Ithasbeenshown(Whipple1950,1951)thatevenold,extremely the forceinsolardirection duetotheevaporationofmaterialfromsurface ofthe per gram.Thustheexpressionforconservationofenergyatanyunitareanormal unit area,misthemassofthesemolecules,andRtheiraverageheatsublimation is givenby=nmU,wherenthenumberofmoleculesevaporatedpersecond comet temperatureand Since \nwmwvw = pw and vx/vw = the vapor pressure of the mixture finally co oc (5) Thus the force becomes (6) where the subscript w has been dropped and p i s understood to be the vapor pressure of water ice {mx/mw remains 10/9). For convenience in comparing this force with the 2 sun’s gravitational attraction for the comet, it can be written as Fev = K{r)/r and the quantity K(r) determined. Thus K(r) (7) irb2 Figure 2 is a plot of K(r)/irb2 in dynes versus solar distance in a.u., and, as in Figure 1, the broken line is for pure water (j = 0) and the solid line for/ = 2, using the exponent A = 2675° K. As seen from Figure 2, K (r) is very well approximated by a straight line such that K{r) = ^o( i-jQ (r Fig 1 —Graph of comet surface temperature as a function of solar distance The dashed line is for a comet composed of pure water; the solid line is for a comet composed of two molecules of more volatile material for every molecule of water. © American Astronomical Society • Provided by the NASA Astrophysics Data System 1961ApJ. . .133. .6575 252 225 2 of solardistanceThelowercurveisforacometcomposedpurewater;theupper composed oftwomorevolatilemoleculesforeverymoleculewater.Thedashedstraightlineisthe velocity ofabodyinparabolic orbitaboutthesunis approximation-curve, K=Kq—r/R),whereR7.00auand4.85AX10dynes/cm where qandaaretheorbitperihelionaphelion, respectively.Since due tosublimatedvaporalonefromacometofmassmduringoneapparitionisgivenby mass lossisassumedtooccursymmetricallyabouttheperihelion,total where visthethermalroot-mean-squaresurface molecule escapevelocity.Theradial force, AV&,mustbeestimatedintermsofthemassloss,asWhipple(1951)didforcon- comets resultedinaneffectivereductionthesolargravitationalattraction.If mating the^=2curveareKq—4.857rZ>X10dynes/cmandR7.00a.u. stant K.Incalculationsbasedonthetrajectoriesofsome64comets,HamidandWhipple where Kisindynes/cmandRa.u.Thevaluesfortheinterceptsoflineapproxi- c (1953) obtainedpositiveevidencethattherapidevolutionofmassundergonebyvisible c tional attractionforacomet,theratioofvapor-pressureforcetogravitational © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem Fig 2.—Graphofevaporativereactionforceparameterdividedbyprojectedcometareaasafunction In ordertodeterminethemagnitudeofresultingreductioninsolargravita- IV. EVALUATIONOETHEREACTIONFORCEFROMEVAPORATION SOLAR DISTANCEINASTRONOMICALUNITS = (11) -dT T7>' dm 2K QC - K,70-1dm_ Qc 12 a 2==V il0) Anigc —2I—-Jr--t~dr, (9) BEHAVIOR OFCOMETS661 d r[2GMq(r—q)l/ _rdmdt dt r = —=\nimrbv2~dT’ AQC = 1^1—, (12) J dtdr q 1961ApJ. . .133. .6575 -3 -5 4 for qabout1.5a.u. in gaseousevolutiononeapparition.Thispointisdiscussedfurtherthenextsec- fraction oftheirtotalmassinoneapparition. lation oftheirresultstofainterandpresumablysmaller cometswhichmaylosealarger brightest, best-observedcometsfortheirstudy;they havecautionedagainsttheextrapo- more rapidlywithdistancethanthesolargravitationalforceandthat,asshownlater, vestigated couldhavehadanobservedchangeinAk/koftheorder—10,aver- bolic cometswhichlastforveryfewapparitionsloseasmuch40percentoftheirmass to ourexamplewerenotobservedbeyondabout2.5 a.u.HamidandWhippleselectedthe for allcometsobservedoutto6or7a.u.,while theircometswhichgivevaluesclose must beborneinmind,however,thatweknowofnosureevidenceonthispointasyet the calculationpresentedinthispaper.Ineffect,theyassumedthatKequation(8) gaseous evolutioninoneapparition,itisnotunreasonabletosupposethatthenewpara- since k=GMonic-Substitutingequation(8)forK(r)into(13),oneobtains obtain valuesfortheevaporativeforceoflessthan 1percentofourillustrativevalue culations thanasconclusionsbasedonthem.One maynotethatHamidandWhipple at aphelion,thusobtainingexactlyanellipticorbitonwhichtobasetheircalculations of thispaperwasindependentrandweakenedthesolargravitationalconstanteven age valuewasmorelike10.Itisunfortunatelydifficulttocomparetheirresultwith tion, butiftheydidlose40percentoftheirmassingaseousevolutiononeapparition, sumed tobe4.6X10cm/secforanaveragemolecularweightof19. The thermalvelocityvwasassumedtobeconstant,sincethesurfacetemperatureisa and thatourvaluesfororbitalparametersareproposedmoreasillustrationsofcal- of theobservations.Inthispaperitisconcludedthatevaporativeforcediminishes fairly constant165°Koverthevisibleportionofcometaryorbits.Thusvwasas- section, wefavoramuchmorerapidevolutionofgasthanHamidandWhippledo.It the orbitisnotanellipsewhileunderobservation.Asmentionedaboveandinnext where alldistancesaremeasuredina.u.,Ristakentobeabout7andç1.5a.u. equation (17),onecanfind that where pistheaveragedensity ofacomet.Thus,bysubstitutingequation (16)into where GistheuniversalgravitationalconstantandMqsolarmass.Therefore, 662 ROYE.SQUIRESANDDAVIDB.BEARD © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem Although HamidandWhipple(1953)concludedthatafewcometsofthe64theyin- If theoldencrustedcometsofJupiter’sfamilyloseupto1percenttheirmassin From Figure2, 25 25(17) Ko —4.85X107tô= 3.6X lO^ 1t Aw _2(2GAfe)^Kof1-r_/R gc 12 r V. PHYSICALSIZEIMPLICATIONSOFLARGEMASSLOSS m vkJr(r—q)/ cr 2a Aw _2(2GMq)VrKjr)_d^ ge 12(U m vkJr(r—q)/’ cg * \v5/ Ko k -o.aY as) 3 2 X10“ ifl h bp GM®bp (16) (14) 1961ApJ. . .133. .6575 u16 3 3 3 mass islostinanaverageapparition. cent perperihelionpassagemayevenbeconservative, andperhapsalargerfractionof lose roughly75percentofitsmass.Thustheestimated newcometmasslossof40per in cometbrightness.Inorderthatthebrightnessmay changeby1mag.,thecometmust mass wouldchangeonly0.5inmagnitude,whichis wellwithintheobservedfluctuations evaporate lessmaterialthandothebrightestparabolic comets.Itisinterestingtonote from —1to7,dependingmainlyontherelativeconcentrationsofdustandgasesin would beunobservable.Indeed,acometwhichhas lostasmuch50percentofits these values(Vsessviatsky1933,1934),butcometshavelargeperiheliaandhence evaporated material.Themajorityoftheparaboliccometshavemagnitudesgreaterthan immediate tailofthecometatanytime.Thesecalculationsyieldedmagnitudes that thechangeinbrightnessduringoneapparition duetoamasslossof20percent estimates ofthealbedosmaterialsandamountmatterincoma size bothforcometswhichevaporatedustonlyandalsogasesbymakingrough observed magnitudes.Brightnesscalculationswereperformedforthesmallersuggested by evaporationalargegaseousanddustcloudwhichrendersthecometvisiblewithin if cometbrightnessisassumedproportionaltonucleararea. nuclear radiusofonly15km.SinceHalley’scometisolderandlessvolatilesixor seven magsbrighterthantypicalcomets,300metersappearsasaferepresentativeradius apart bythetidalactionofsunastheypassedperihelion.Furthermore,comets represent anupperlimitforthedensityofcometarynuclearmaterial,indicatingthat tion ofsevendifferentkindsobservationalphenomenathatHalley’scomethasa gm forthemass.Vorontsov-Velyaminov(1946)hasconcludedfromacarefulconsidera- values of30000cmor300metersto12kmfortheradiusand6X10gm4 volumes ofmatterhighdensitywouldnotbelikelytodo.Perhapsarepresentative have beenobservedtofadecompletelyonasingleapproachthesun,whichlarge p ^0.05gm/cm.Therehavebeeninstances(Watson1956)ofcometsbeingtorn striking theupperregionsofatmosphere(Jacchia1955;öpik1956),indicatingdensi- low. Meteoritesassociatedwithcometsareobservedtodisintegrateimmediatelyupon is mountingevidencethatthestructuralstrengthanddensityofcometnucleiarevery value fortheaveragedensityofcometarynucleiwouldbep=0.005gm/cm,yielding ties forthiscometdebrisoflessthan0.05gm/cm.Theseintactpiecesmostprobably loss wouldbemuchgreaterthanthatduetogaseousevolutionaloneandmightexplain If largesolidpiecesofcometbreakoffbecauseunevensublimation,thetotalmass why cometswithlittleevolutionofgasstillapparentlymakeveryfewapparitions.There while acometthatlostonly1percentofitsmassingaseousevolutionasproposedby Hamid andWhipplewouldhave and © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem The equationofmotionforacometorbitingabout thesunmaybewritten This ismuchtoominuteanobjecttobevisible,butitofsufficientsizegenerate VI. EFFECTOFMASSEVOLUTIONONOBSERVEDNEAR-PARABOLIC ORBITS (£.),5 + BEHAVIOR OFCOMETS 2 2 bp/^ 6000gm/cm. k bp 140gm/cm, 6 X 10- (18a) (16a) (19) 663 (18) 1961ApJ. . .133. .6575 2 the radiusvectorfromsunmakeswithperihelion,andf(u)issumof gravitational andvapor-pressurereactionforcesactingonthecomet: 664 parabolic observedorbit,etheeccentricityofsolution,Uo,isapproximately1; where u=1/r,Listhecomet’sangularmomentumaboutsun,0isanglethat measured fromtheobservedoractualperihelion.Sincecalculationassumesanearly tricity observedair