1920AJ. equal thatofJupiter'sorbit(assumedcircular)the enter aspheredescribedabouttheSun,withradius number whichhavetheirorbitschangedintoellipses the encounterisveryclose,sothattheywillberare, the cometmaybestoppeddead,sothatitfallsinto smaller, andtheperiodafterencounterislonger. the Sun. a muchlargernumberinwhichtheperturbationsare and foreverycaseinwhichtheyhappen,therewillbe period shorterthanthatoftheplanet,andeven been entensivelystudied,notablybyH.A.Newton results tocometsofstilllongerperiod. VOli. XXXIIIALBANY,N.Y.,1920,SEPTEMBER15NO.7 one ofthemajorplanetsintoanellipticorbitwith quoted: the validityofcommonsuppositionasregards five andnineyearsareoftendescribedasbelongingto may betransformedbyasingleencounterwithany Neptune. Strongevidenceagainsttherealexistenceof the cometsofsomewhatlongerperiodbelong,in captures bytheouterplanets,andtoextend Neptune's familyhashoweverbeenpresentedbyH.C. same sense,tothe“families”ofSaturn,Uranusand Jupiter's “family.”Itisalsogenerallysupposedthat this reasonthenumerouscometswithperiodsbetween encounters withJupiter,previoustowhichtheir Wilson (1). periods weremuchlongerthanatpresent;andfor have beendivertedintotheirpresentorbitsbyclose (la) fromwhosediscussionthefollowingresultsmaybe 9 Newton calculatesthat,outof10cometswhich The theoryofthecapturecometsbyplanetshas (b) Suchgreatperturbationswilloccuronlywhen It iswellknownthatcertainshortperiodcomets It isthepurposeofpresentdiscussiontoexamine (a) Itispossiblethatanoriginallyparabolicorbit ASTEOSOMICAL JOÜENAI © American Astronomical Society I —Theory ON THEORIGINOFPERIODICCOMETS, By HENRYNORRISRUSSELL. FOUNDED BYB.A.GOULD 3 g r?Q5p No. 775 Provided bytheNASA Astrophysics DataSystem fraction Fofthewholenumberwhichwillhavetheir orbits, approachtheSunwithindistancer, number ofcomets,movingatrandominparabolic moves inacircularorbitofradiusr,andverylarge tioned. ton, butfollowingfromhisequations,maybemen- and smallinclinations. number ofcometswithshortperiod,directmotion, operate tocauseaveryconsiderableincreaseinthe possible iftheinclinationoforbitissmallafter random, andencounterswithasecondplanetare capture. Thesefactors,asNewtonshows,may greater foracapturedcometthanonemovingat such additionalencounters,however,isenormously planet whichitlastencounters.Theprobabilityof should haveinclinationslessthan30°,andonly51 with aperiodlessthanhalfthatofJupitershouldbe ment, ifweagreetoassignacometthefamilyof inclinations exceeding150°.Moreover,thosewith moving indirectorbits.Newtoncalculatesthatout number 839:withperiodslessthantwiceJupiter's some otherplanet,maybeexcludedfromthisstate- perturbations oftheordinarytype.Furtherclose tured cometswillcontinuetopassnearthoseofthe of the839cometswithperiodlessthanJupiter's257 encounters withtheoriginalcapturingplanet,or greater distancebythecumulativeeffectofplanetary reverse isthecaseforcometswithretrograde 2670: andsoon. capturing planet,untiltheyaregraduallyshiftedtoa motion. counter thantohavethemlengthened,whilethe periods stillfurthershortenedatasubsequenten- direct motionsaremuchmorelikelytohavetheir 126. ThosewithperiodslessthanJupiter'sshould (e) Newtonprovedthat,ifaplanetofmassm Certain otherresults,notexpresslystatedbyNew- (c) Ofthecapturedcometsmajoritywillbe (d) Finally,andobviously,theorbitsofcap- HAF0RD c HAVE Zl*m H ^FôRD, PA (49) 1 00 00 O'! ^r 1920AJ. tí and inthelatter that X<2—\/2,butare\/21and+if the limitsofintegrationare1—Xand+provided into ellipsesofmeandistanceslessthanAwillbe Newton, musthavebeenusedbyhiminderivingthe positive. Ifinthisexpressionweset where therangeofintegrationextendsoverallvalues purposes wemaywrite which issorapidlyconvergentthatforallpractical in parenthesis(3)maybeexpandedaseries results givenunder(6)above. exceeds thislimit.Intheformercase of sgreaterthan\/2-1,whichmaketheintegrand orbits changed,byasingleencounterwiththeplanet, whenever a>3r. 50 the fractionofthiswholenumberwhichhaveperi- ever, greaterfortheremoterplanets.Newtonshows val oftime,aredivertedbyagivenplanetintoelliptic helion distanceslessthanqwillbe^.Hencethe will beproportionaltor—fromwhichitfollowsthat that, ifthecometsaremovingatrandominparabolic orbits, thenumberwhichenterasphereofradiusr, the numberwhichapproachSunwithinunit orbits ofmeandistancelessthanais^times whole numberofcometswhichduringagiveninter- described abouttheSunascentre,perunitoftime of distanceduringthisinterval.Butmanythese These equations,thoughnotgivenexpresslyby When aisconsiderablygreaterthanRthefactor F = The numberofcometsavailableforcaptureis,how- F = 23 23 © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem 4 2+\/2r 3 8a96 2 1+V2 a —r=(Xl)a, F 2 ^ 4ma 4s2 2 X +g- 4 ma 3 ~F‘ 2 3 r THE ASTRONOMICALJOURNAL 3 X 2 4 ma \/ 2-1 (3) (2) (1) a 2 2 2? a" a'aAmAcos6-\-hsin-^ given bytheequations JL =14-:Lf_s_A+dhsin6 large, beingcompensatedbytheconcentrationof vectors: hthedistancewhichplanethasstillto to theSun;6anglebetweenthesetwovelocity relative totheplanet,thatof planet’s orbitandtheundisturbedofcomet: predominance ofJupiterwillbestillgreater. perihelion distancestowardsmallvalueswhichnec- decrease inthecoefficientF,whichis good approximationforsmallervaluesofa,—the with r;but,ifqissmallcomparedritremainsa s theratioofundisturbedvelocitycomet, parabolic, themeandistanceafterencounterisa. beyond whichcometsarelikelytoescapeobservation, 3 r 4 maq the realsemi-axisofhyperbolicorbitwhich go toreachthepointonitsorbitwhichisnearest In (4)a'anda"arethecomeLsmeandistancesbefore by Uranus14;andNeptune8. equal totheEarth’smeandistance,numberwhich which enteraspheredescribedabouttheSunofradius and a=100,wefindthat,outof100,000,000comets comet’s undisturbedorbitattheinstantwhen In equation(5)clistheleast'distancebetween and aftertheencounter.Iforiginalorbitis Earth, andofperiodlessthan1000years,bythe are changedbycaptureintocometsvisiblefromthe essarily occurswhenaislessthanr. own orbitwhichisnearesttheplanet’sorbit;andA comet, ifundisturbed,wouldreachthepointonits action ofJupiterwillbe90,000;bySaturn2400; This expressionholdsgoodwhenaislargecompared than aandperiheliondistanceslessqthefraction the perturbationsaresmall)wefind,fornumberof with theusualapproximations,(neglectingper- counter. comet describesabouttheplanetduringen- captured cometswhichwillhavemeandistancesless fraction ofthemhaveperiheliondistanceslessthanq turbations ofthecometbyplanetwhenitis should benearlytrueinthemajorityofcases,where as inthecaseoforiginalparaboliccomets,(which and escapeobservation.Ifweassumethatthesame captured cometswillhavelargeperiheliondistances, = If thelimitingperiodismadeshorter,relative If wesetq=2,correspondingroughlytothelimit (/) Theeffectsofanindividualencounterare These equationshavebeenderivedbyNewton of allthosewbichentertheunit-sphere. 4 3 when ais N°- 775 1 ^ N°-775TWWA‘BTRO-.N-QMICAL;J0UNL,51 O'! ^r 1920AJ. tí from aparabolicorbitintooneofmeandistancea. at pericentre,then,asNewtonshows, planet acometmustapproach,inordertobediverted and thelineofapsides,pperpendiculardistance of thehyperbolicorbitcometaboutplanet If aistheacuteanglebetweenoneofasymptotes of theplanetfromasymptote,andqdistance we mayneglectAincomparisonwithd,andwrite tude ofmai,andthisquantityisverysmall.Hence the hyperbola. unless thedistancebetweenorbitsisverysmall, and negativesigntothemaximumindirectionof from theSunattimeofencounter,andaiits latter iscircular,andofradiuscq,thenA=; remote fromthelatter,andby¿Shmwhenitis maximum perturbationinthedirectionofellipse, which gives in themoregeneralcase,ifrisdistanceofplanet The positivesignoftheradicalcorrespondsto termined astomaketheperturbationamaximum. whatever theeccentricityofplanet’sorbit.If small quantities,andapplicabletoallencounters, near theplanet)butarecorrecttofirstorderof; planet isgiven,allthequantitieswhichappearfin mean' distance circumstances oftheencounter,andmaybesode- If theorbitalongwhichcometapproaches (5) arefixedwithexceptionofK,whichdefinesthe (g) Itisofinteresttodeterminehownearthe It followsfrom(6)thatAisoftheordermagni- The conditionforthisisreadilyfoundtobe 2 © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem _s_ A+d 2 2 A cos6±a/A+dsinB A-\-d 2 q —A(seca1). 2 2 =d+hsinB, p A cosB±a/A+dsin a = p =Atana, Ö^A(8) 2 v s (2ai—r) 2m sm& m air s d (7) 6 1 2 to asingleencounter,thenumberwithmeandis- for onlyoneinfourthousand. comet inforty,andUranusNeptunetogether of thecaptures.Saturnaccountingforonlyone should showthefollowingcharacteristics: originally movingatrandominparabolicorbits,they planetary perturbationfromagreatnumberofcomets the periodiccometsowetheirorigintocaptureby an encounter,however,isnegligiblysmall. and aperiodof2.6years.'Theprobabilitysuch collision mighthaveameandistanceassmall1.9 meteorite capturedbytheEarthwhichjustescaped radius is43xl0~astronomicalunits..Hencea them. planets maycapturecometswithoutcollidingwith the.>SW) theactualvalueofqisma^-y/^—1). equatorial radiiare4.3forJupiter,and2.8Saturn, ton showsthatinthiscase(whenthecometfallsinto At thelimitwhena=(hwehaveqAmai.New- value ofcosai(l—a)isJAwhena=60°.Hence equal tounitywhen=0and6>a.Themaximum If nowwesetd=■psin\p,hcos.(5)be- we have comes <:v a m The lastfactoristhisexpressionamaximum,and We thenfind orbit ascircular,andset For thepresentpurposewemay,treatthe,planet’s (or 1.25timestheouterradiusofrings,)2.0for Uranus and3.6îoyNeptune.Henceallthemajor q _16m 6 2. Ofthecometswhichowetheirpresentorbits Summarizing theresultsoftheory,wefindthatif 1. Jupitershouldberesponsibleforalmostall In thecaseofEarth,m=3.0x10~while The correspondingdistancesintermsoftheplanets’ 2 cos a(1—a)(cos(9+sin6^) 2 - ._sA,seea:., Am cos6+tanasinB\f/., r 2 4m a ai (9) 1920AJ. 2 responding columns,givingtheminimumdistances with thecometsoflongerperiod,therearefourcor- tance betweentheorbitofeachshort-period elements arenotdefinitive,butbasedonlong tances, andthefifthinclinationoforbitto from theorbitsoffourmajorplanets.The the thirdandfourthperihelionapheliondis- comets andJupiter’sorbit.InTableII,whichdeals enough observedarcstogivereliableresults. the correspondingapheliondistancesisusuallycon- perihelion. Theuncertaintyofthelongerperiodsand elements, and,representtheosculatingorbitnear usual byacolon.Forfewofthelatestcomets siderable. Themostdoubtfulcasesareindicatedas ecliptic. Theseareusuallytakenfromthedefinitive return beingdistinguishedbythenamesoftheir apparition. Thesecondcolumngivestheperiod, — cometswhichhavebeenobservedatmorethanone des Tempsfor1915,exceptwhenotherwisenoticed. periodic cometsintheAppendixConnaissance discoverers, andthoseseenonlyoncebytheyearof been computedaresummarizedinTablesIandII. The dataaretakenfromthetableofelements few whichneverapproachitclosely,thaninthecase passing neartheplanet’sorbit,andcorrespondingly in suchacase,thereshouldbemanymorecomets comets forwhichperiodsoflessthan2000yearshave the accumulationofordinaryperturbations,but,even of arandomdistribution. minimum distanceislikelytohavebeenincreasedby random distributionoftheirorbitplanes. have beenrelativelysmall,shouldexhibitnearlya effects ofsubsequentencountersareconsidered.These tions, whichislikelytobeverypronouncedwhenthe of cometswithperiodslessthanPwhichcometo The remoterthedateofcapture,moreoriginal close tothatoftheplanetwhichlastcapturedit. of longperiod,formostwhichtheperturbations preponderance ofdirectmotionsandsmallinclina- fore beproportionaltothecuberootofperiod. proportions ofshortperiods. Subsequent encounterstendtoincreasetherelative perihelion withinagivenshortintervalshouldthere- the numberofperiodslessthanPtoP%.The tances lessthanashouldbeproportionalto—or 52 The lastcolumnofTableIgivestheminimumdis- The firstcolumngivesthedesignationofcomet, The observedcharacteristicsoftheorbitsthose 4. Theorbitofeverycapturedcometshouldpass 3. Thecometsofshortperiodshouldshowa © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem II —ObservationalData THE ASTRONOMICALJOURNAL Schaumasse Faye Borrelly 1889 VI Brooks Biela 1881 V Holmes Wolf Finlay 1906 VI Kopff (4) D’Arrest TeMPELi 1894 I 1895 II 1918 d(5) Perrine 1858 III 1896 V 1892 V Giacobini De Vico WlNNECKE Tempel-Swift 1909 IV Brorsen 1890 VII 1783 1915 e(3) 1770 I 1886 IV 1916 a(3) Tempel 1743 I(2) Encke 1884 II 1819 IV(2) 1766 II minimum ofthedistancebetweenanypointon tabular quantities,strictlyspeaking,representthe 2 Comet Table I—Short-PeriodComets Period 8.69 7.59 7.44 7.20 6.68 6.58 6.55 8.92 8.07 7.42 7.10 6.93 6.86 6.80 6.69 6.54 6.54 6.66 6.61 6.52 6.51 6.45 6.40 6.40 6.36 6.37 5.89 5.89: 5.68 5.60 5.59 5.5/0 5.46 5.43: 5.40 5.17 5.10: 5.03: 3.30 0.88 0.73 2.12 2.09 0.98 1.23 1.67 1.15 1.30 1.40 1.59 1.89 1.15 1.70 1.27 0.97 1.36 1.63 1.96 1.01 1.45 1.43 1.17 0.68 1.38 1.67 0.59 0.86 1.55 1.46 1.82 1.15 1.33 1.34 1.28 0.89 0.40 1.32 0.34 4.90 6.15 5.32 2a —q 7.25 6.82 6.09 5.10 5.59 6.22 5.21 5.55 7.72 5.97 6.46 5.43 5.87 6.08 5.89 5.72 5.55 6.00 5.51 5.22 5.76 5.05 5.06 4.97 5.29 5.55 5.21 5.63 4.89 5.61 5.31 4.87 4.66 5.03 5.46 4.09 T5.7 25.3 30.4 20.8 31.3 45.1 29.9 12.4 10.8 19.5 11.4 15.8 14.5 10.6 10.3 17.7 19.4 12.8 18.3 15.5 29.4 12.7 10.7 12.7 12.6 5.6 3.4 3.0 6.1 8.7 6.9 5.5 3.6 5.4 1.6 1.9 5.1 9.0 8.0 Least Distance from Jupiter + 0.05 + 0.42 + 0.47 + 0.01 + 0.11 + 0.12 +0.13 + 0.07 +0.22 +0.33: + 0.15 +0.20 + 0.13 -0.08 -0.02 -0.46 -0.36 -0.07 -0.04 -0.56 -0.16 +0.65 -0.10 -0.16 +0.26 -0.09 -0.34 -0.08 -0.58 -0.43 -0.00 -0.02 +0.55 -0.14 -0.02: +0.63 +0.31: +0.92 -0.08: N°* 775 d 1 00 00 O'! ^r 1920AJ. tí N°- 775 two pointsontheorbits,ifplanetaryorbits which hasthesameheliocentriclongitude.This would berigorouslytheleastdistancebetweenany comet’s orbitandthatpointontheplanet’s were circular.Theactualeccentricitiesofthese orbits aresosmallthatnosensibleerroriscommitted logarithms wereused. few casesofverycloseapproach,whenfive-place by thisapproximation,whichsavesmuchlabor.The calculations weremadewithaslide-rule,exceptfor than thetabularquantity. borne inmind,however,thatthelinejoining that thedistancefromthisplanetopointunder perpendicular totheplaneofplanet’sorbit,so nearest pointsofthetwoorbitsisoftenveryfarfrom proach, isnorthorsouthoftheplanet.Itmustbe according asthecometatpointofnearestap- consideration onthecomet’sorbitmaybemuchless The plusorminussignisprefixedtothesedistances Halley 76.00.5935.3162.2-0.77-1.73-4.7 Brorsen-Metcalf (8)72.10.4834.219.2—0.48—1.67—3.86 Pons-Brooks 71.60.7733.774.0—2.05—1.69+1.16 Westphal 61.51.2529.940.9-0.36-2.48-7.0 1857 IV235.0.7575.4 32.8 +1.87+2.38+3.5 1862 III120.0.9647.6113.6+1.57+0.80-1.98 Olbers 72.71.2033.644.6—0.70—3.39—9.3 Tuttle 12.11.039.555.0—0.66+2.15 1793 II422:1.50110.9 51.5 -0.59-3.20-8.7 1861 14150.92110.4 79.8 -1.22+0.11+3.7 1861 II4090.82109.3 85.4 -1.40-0.49+2.4 1840 IV3671.48101.1 58.0 +1.12-0.60-4.4 1874 IV3061.6989.1 34.1 +1.71+1.10+0.030 1905 III2971.1287.8 40.2 -2.58-3.23-4.7 1885 III275.0.7583.7 59.1 +0.21-1.16-6.2 1855 II(10)252.0.5679.3156.9 +0.54+0.48-0.65 1917 a(9)189.0.1965.732.6-1.53-2.96-7.8 1889 III128.1.1049.831.2-0.33-1.76-5.8 1846 IV75.70.6635.185.1+2.25+1.84-0.54 1827 11(7)63.80.9531.1136.5+1.32+0.85-1.00 1867 I40.11.5821.818.2+1.37+1.49-0.83 1866 I33.20.9819.7162.7+0.79+0.45-0.38 1913 c(6)17.61.5312.014.8+0.94-0.71 1846 VI13.31.539.730.7-1.17+0.52 1898 I4171.10110.6 72.5 +0.57-0.97-6.4 Comet © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem THE ASTRONOMICALJOURNAL Period Table II—CometsofLongerPeriod 2a —d period comets,givingtheminimumdistancesfrom orbit ofonetheouterplanetscorresponding distance betweentheorbitsisomittedintable values forthecometsofperiodlessthantenyears, Jupiter’s orbitforallthoseknownupto1911.The served atmorethanonereturnhadalreadybeen paper, (exceptforthesewhichhaveappearedsince and observedatonlyonereturn,aretakenfromhis is considerablylessthanthevariationsinthisdistance (or, inafewcases,placedparentheses.) when thesameorbitalelementsareused,difference attention. Comparisonoftheresultsshowsthat, with theslide-ruleandbyFayet’smuchmoreexact between thevaluesofminimumdistanceobtained calculated whenFayet’spapercametothewriter’s computations was0.008astronomicalunits,—which 1911). Thecorrespondingdataforthecometsob- 2 When theaphelionofcometfallsfarinside Fayet haspublishedadiscussionoftheshort- Jupiter SaturnUranusNeptune Least Distancedfrom + 7.2 -17.4 + 8.6 -11.5 - 4.9 -11.0 + 3.3 - 3.2 -10.8 - 1.2 - 7.7 + 5.4 - 7.8 -11.8 -11.0 - 6.5 - 3.8 -16.4 - 4.9 -15.0 ( 8.5) (10.8) 53 1 oo oo O'! ^r 1920AJ. tí this, thedifferencesinnumbersareprobablyno small groups. greater thanmightarisefromchanceamongsuch period, andhence,onthesimpletheoryofcapture, regarded bycomputersas“parabolic.”.Apartfrom of therecord,—manythesecomets,havingbeen of verylongperiodisdoubtlessduetotheimperfection same. Thegradualfallingoffinthenumberofcomets respond toequalincrementsofthecuberoot comets inthesuccessiveintervalsareroughly from theoutset. should containequalnumbersofcomets. of 10,000years;andthesuccessiveintervalscor- of theConnaissancedesTemps,toalimitingperiod in whichthedatahavebeenextended,withaid period” cometsandthoseoflongperiodisconspicuous cally thefamiliardistinctionbetween“short which mayoftenbecausedbyperturbationsduringa single revolutionofthecomet. 54 For periodsgreaterthantenyearsthenumbersof The distributionoftheperiodsisshowninTableIII When theobservationaldataarediscussedstatisti- Ill —ComparisonofObservationwiththe 1858 YI 1807 1785 II 1853 I 1894 II 1854 IY 1855 I 1887 II. 1854 Y 1881 YIII 1853 III 1882 II 1886 Y 1811 II 1906 YII(11) 1846 VII 1843 I *Closest approachnearcomet’sperihelion.Asecondaryat+3.89. Comet © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem Capture Theory 1880 1215: 1143 1089 1059 1714 1326: Period 755 999 994 771 793 772 782 583 537 512: THE ASTRONOMICALJOURNAL 0.80 0.98 2.20 0.58 0.65 0.43 1.09 0.27 0.91 1,63 1.36 0.008 1.58 0.63 0.006 1.92 1.21 q 226.7. 217.6 210.7 304.0 205.6 285.8 240.8 198.3 197.9 164.2 169.3 167.9 138.5 128.0 2a —q 168.9 168.6 131.6 159.8 117.0 128.7 104.3 144.8 142.0 144.3 122.2 150.7 40.9 87.0 63.2 87.7 31.3 92.6 84.8 14.2 tween 10and2,000yearshasaninclinationgreater is radicallydifferentinthetwogroups,asshown with 0.5yearsoftheirmeanvalue,6.39years. interval, butclusteredclosely,—halfofthemlying theory ofcaptureatasingleencounter,andtheir periods arenotuniformlydistributedoverthetabular more numerousthanwastobeanticipatedonthe Table IY..Everyoneofthecometswithperiodsbe- Limiting Periods The short-periodcomets,onthecontraryarefar It iswellknownthedistributionofinclinations + 1.19 +3.46* -1.24 -1.10 + 2.92 -0.13 + 0.31 + 0.39 -1.47 -2.09 + 3.36 Jupiter -0.90 -2.13 -0.49 -2.91 -3.00 -1.52 10000 , 640 7290 5120 3430 2160 1250 270 . Table III.DistributionofPeriods 80 10 0 Least distancedfrom + 3.03 + 0.18 + 3.95 + 3.91 + 0.14 -3.18 -1.80 + 1.47 +4.00 -0.70 + 0.020 -3.73 -1.19 -1.83 -5.01 Saturn Uranus -5.02 -3.69 + 6.7 + 3.9 + 3.8 + + 2.7 - 5.7 + 2.4 + 3.6 + 5.1 -10.8 - 1.05 + 2.4 - 4.6 -10.7 - 8.3 -10,9 8.2 5.0 + 10.4 + 6.3 + 8.9 + 1.5 + 7.5 Neptune Number ofComets -10.5 -15.3 -17.6 + 1.2 + 6.5 + 4.3 -10.6 + 5.0 -16.6 + 4.9 -13.1 -16.8 39 11 11 12 + 12 + 20 + 17 + 14 + 3 6 3 -30 -36 -20 + 12 + 2 5 8 5 + 10 + 8 5 -20 -34 + 9 -26 -34 N°-' 775 A/? 00 00 O'! 1920AJ. cC times theperiodofJupiter,andtheirmeanaphelion nomical units.Theseperiods arerespectively0.49, rather conspicuousdivisionintothreegroups,com- periods betweentenandonehundredyearsshowa of longerperiod?Superficially,theevidencefortheir distance (5.55)is1.07timesJupiter'smeandistance. with periodslessthantenyearsbelongtoJupiter's very little—confirmthegeneralbeliefthatcomets vestigations, —towhichthepresentdiscussionadds may thushavehadacommonorigin.Thesein- been workedoutbyCallandreau(12),andFayet broken uptheoriginalcometsintomuchmorenumer- mean apheliondistancesof10.4,20.8and33.3astro- with meanperiodsof14.3,36.6and70.5years, prising respectivelythree,twoandsevenmembers, ous fragments.-Thetheoryofsuchdisintegrationhas existence seemsveryfavorable.Thecometswith and indicatesthattheyowetheirpresentorbitstothe into twoequalparts,itappearsthatofthefirsthalf, inclinations exceedingthis.Dividingthelattergroup 40 to501 “family.” Themeanperiodofthesecometsis0.54 disruptive tidalforcesatthetimeofencounterhave be expectedasaresultofcapturebysingleencounters. 45°, andtwo-thirdsofthecometslongerperiodhave than themeaninclinationofshort-periodcomets. 30 to402 Newton) whiletheirlargenumbersuggeststhat counters withJupiter(ashasbeenmaintainedby to periodandinclination,therefore,thedistribution 20 to304 (13) haspointedoutseveralgroupsofcometswhich cumulative effectofperturbationsatsuccessiveen- That ofthecometsshortperiodisglaringlydifferent, of thecometslongerperiodisverymuchwhatmight and 5retrograde,themeaninclinationis66°.8, Total 39 The greatestinclinationforanyshort-periodcometis N°- 775 and themeaninclinationis97°.5.Bothwithregard while outofthesecometshavingperiodsfrom400to with periodslessthan400years,16aremovingdirect Mean Inclination13°.9 2000 years,11aremovingdirectand10retrograde, 10 to2018 0° to10°14 Short Periods Are similarfamiliestobefoundamongthecomets Table IV.DistributionofInclinations © American Astronomical Society THE ASTRONOMICALJOURNAL Mean Inclination88°.1 Total .42 120 to1506 150 to1805 90 to1204 60 to909 30 to60’14 Longer Periods 0° to30°4 Provided bytheNASA Astrophysics DataSystem u u of approachesislessforthe outerplanets,sincethe the assumptionsexplainedbelow.Thewholenumber the numberofapproacheswithingiven.limits lies betweenspecifiedlimitsistabulated,separating planet andcomettothemeandistanceaof the leastdistancedbetweenpresentorbitsof proach tothemajorplanetsareassummarizedin than 2000years,thecircumstancesofpossibleap- to otherplanetsthanNeptune. the planet.Thelastcolumnforeach,planetgives the caseswherecometpassesnorthandsouthof the caseofHalley’sComet,whileW.H.Pickering to havebeenfirstpointedoutbyCrommelin,(14)in families —isthusrenderedverydoubtful.Thisseems distance whichmightbeexpectedtheoreticallyunder Table V.Thenumberofcasesinwhichtheratio case oftheothermembersgroup,andWilson to hisorbit;butnotoneofthesevenmembers 3.8 astronomicalunits,whileallofthemmaycome two whichareassignedtoUranuscomeprettyclose family” passfairlyneartheorbitofplanet; found thattwooutofthethreecometsSaturn's (1) showedthatmostofthemmightbetterbeassigned closer thanthistoSaturnandmuchJupiter. (15) observedthatthesamedifficultyoccurredin but thelatterhappensineverycaseof“capture” Neptune's family”cancomenearertohimthan indeed, eveniftheorbitisrenderedhyperbolic. when theperturbationsareexceptionallygreat,—or whether theresultingperiodisshortorlong,—and, that itsorbitshallpassclosetotheplanet’sOrbit. cumulative inthesamedirectionatseveralencounters; The formerresultisattainedonlyintherarecases little greaterthanthatofthecapturingplanet,but fact thatthetruecriterionfordetectingacaptured comet isnotthatitsapheliondistanceshallbebut undiscovered remoterplanets? is thereforegenerallybelievedthatthesethreegroups 270 years.Maytheseperhapsbethefamiliesof 775 years,andanother,lesssharplydefined,near Neptune. present orbitsbyencounterwithSaturn,Uranusand conspicuous groupswithmeanperiodsof406and and 1.11timesthemeandistancesofthreeplanets Neptune, whiletheapheliondistancesare1.09,1.08 0.44 and0.43timestheperiodsofSaturn,Uranus of cometshaveactuallybeendivertedintotheir (as against1.07forJupiter'sundoubtedfamily).It Extending thestudytoallcometswithperiodsless The realityofthis—thelargestsupposed LTpon applyingthistesttothedataofTableIIitis Little attentionappearstohavebeengiventhe Among thecometsofstilllongerperiodtherearetwo 55 1 O'! ^r 1920AJ. tí 56 All 0.15 to0.20 0.10 to0.15 0.05 to0.10 0.0 to0.05 0.5 0.4 0.0 All Over 0.5 0.3 0.2 0.1 0.4 0.1 All Over 0.3 0.2 0.0 All 0.5 0.4 0.3 0.2 0.0 Over 0.1 All 0.6 0.5 0.4 0.3 0.2 0.0 0.1 Ratio to 0.6 to 0.5 to 0.4 to 0.3 to 0.2 to 0.1 to 0.5 to 0.3 to 0.2 to 0.1 to 0.6 to 0.4 to 0.6 to 0.5, to 0.1 to 0.4 to 0.3 to 0.2 to 0.7 to 0.3 to 0.6 to 0.5 to 0.4 to 0.2 to 0.1 Table V—ApproachestoPlanets Short-Period CometsandJupiter 0.6 0.6 0.6 © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem North 19 10 14 14 19 19 0 2 0 6 5 0 0 3 8 2 0 0 4 9 1 0 0 3 2 0 3 6 4 3 1 2 1 1 Neptune Uranus Jupiter Saturn THE ASTRONOMICALJOURNAL 20 South 25 14 23 23 23 0 7 3 4 4 2 4 4 2 0 0 6 4 6 6 1 4 1 0 2 0 9 7 1 6 0 3 3 5 5 1 39 39 24 42 37 42 13 13 12 All 2 9 3 4 5 9 0 4 9 9 8 9 1 0 6 8 3 5 0 3 2 8 4 4 2 3 9 Theory 39 42 37 42 10 3 4 4 4 3 6 7 8 3 7 7 5 6 7 5 3 6 7 8 4 5 3 6 6 7 T 57 is thereforetheprobabilitythatatleastoneof for thecomet’sorbit,andsometimesgreater,depend- least distancebetweenthislineandtheplanet’sorbit, less thanthedistanceofAfromplanet’sorbit. the orbitofcometandplanetwillingeneralbe the planet’sorbit:thenleastdistancebetween points AandBshallbewithin thedistancexfrom latter; butthetwowillusuallybeverynearlyequal, ing onthecurvatureandorientationinspaceof vdll sometimesbelessthanthecorrespondingdistance If astraightlineisdrawn,joiningAtotheSun, be obtained.Treattheplanet’sorbitascircular, tion, goodenoughforthepresentpurpose,caneasily of theturnpointsAandB,desiredprobability radius ofthesphereis-wellknovmtobesimplyx. Let Âbethatoneofthesetvopointswhichisnearest orbit cutsthespheredescribedaboutSunas and considerthepointsABatwhichcomet’s be difficulttoevaluaterigorously:butanapproxima- the orbitsofcometandplanetshouldbelessthan random, acertainfractionofthemshould,merelyby responding percentageis31forSaturn,20Uranus, But Ahasbeendefinedastheneareronetothisplane given diametralplaneshallbelessthan.vtimesthe planet’s orbit.IfthepointAlayatrandomon accuracy enoughforthepresentpurpose. and thelinemaybesubstitutedfororbit,with a givenfractionofthepianeCsmeandistancewould twenty percent,caneverapproachJupiterwith sphere, theprobabilitythatitsdistancefrom dicular distanceofthepointAfromplane orbit andthislineisobviouslyequaltotheperpen- centre, withradiusequaltothedistanceofplanet. chance, passfairlyclosetotheorbitsofplanets. even ifthecometsorbitsweredistributedquiteat their presentorbitsremainunaltered).Thecor- retarding force,wdiichmusthavediminishedthesize The probabilitythattheminimumdistancebetween and only8forNeptune.Nowitisobviousthat, of itsorbit. one-tenth ofitsdistancefromtheßun(providedthat which isexceptionalinmanyways,andsubjecttoa Jupiter’s meandistanceisthatofEncke’sComet,— the distanceofclosestapproachexceedsone-eighth of itsdistancefromtheSun.Theonlycaseinwdiich encounters withJupiter—85percent,ofthembeing They showthewell-knowntendencytowardclose capable ofapproachingtheplanetwithinone-tenth orbits. aphelia ofafewthecometsdonotreachtotheir But theminimumdistancebetweenplanet’s Of thecometsoflongerperiod,however,only The short-periodcometsaretabulatedseparately. N°- 775 1 00 ^ N°*775 O'! ^r 1920AJ. tí 2 o the distanceofatleastonetwopoints,AandB, in whichthetabularquantityisprobabilitythat two pointswereplacedatrandom,andindependently to theareaofwholesphere.Thisratiomaybe four majorplanetsis20percent.Buttheassumption of useinotherproblemstheyaregivenTableVI, area ofthesphereoccupiedbytwozonestogether that AandBaredistributedindependentlyoverthe while themeanofobservedfrequencyfor This givesanexpectationof19percentforx—0.10, on thesphere,thisprobabilitywouldbe2x—x. diametral plane,(takingtheradinsasunit).If calculated withoutdifficulty.Astheresultsmaybe ability ofthisoccurenceisequaltotheratio less thanxfromthediametralplane.Theprob- for theconditionsofobservationlimitourstudyto sphere isincorrectinthecaseofactualcomets, one orotherofthepointsAandBwillbeatadistance angle 2C.IfPliesanywherewithineitherofthem, pole. Thecentrallinesofthesetwozonescutatan from thediametralplane,Pmustbewithinasimilar zone, whichfollowsthegreatcirclehavingBasits for Saturnis32°.3;for'Uranus2I.3;Neptune from thevaluesoftrueanomalyattime B willbesmallincomparisontotheradiusof means thatthedistanceseparatingpointsAand orbits ofrelativelysmallperiheliondistance,andthis (counting thepartcommontotwoonlyonce) girdling thesphere.IfBiswithindistancex therefore, twopoints,AandB,onthesphere,ata not great.Forexample,inthecaseof is smallest,:butthedispersionaboutthe,meanvalue sphere onwhichtheylie,providedthatthelatteris pole, —thatiswithinacertainzoneofwidth2d, values arewithin8°of;this.,-Themeanvalue0 mean value^of,0isA7°,andhalfofthe.individual denoted by20.Thevaluesofthisquantityforeach responding greatcircleonthesphere,(whenx=sind). fixed distance2Capart,butotherwiselyingatrandom. each planet,thearcABissameforallcomets, may beobtainedbyassumingthat,intheCaseof considerably, ■beingleastwhentheperiheliondistance For thesameplanet,anddifferentcomets,theyvary closest approach,whichhadalreadybeencomputed. comet andalltheplanetsmaybeimmediatelyderived considerable. LetthearcABuponspherebe distance dfromthegreatcirclewhichhasAasits plane, itwillbewithinadistance6fromthecor- and equaltotwicethemeanvalueofC.Consider The polePofthisgreatcirclemustbewithinthe If Aiswithinthedistancexfromfixeddiametral 14°.6. ■, ; ; A verygoodapproximationtotheactualprobability © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem THE ASTRONOMICALJOURNAL influence thefartherAisfromplanet’sorbit, on aunitsphere,separatedbydistance2C,shall planets’ orbitsgotothesouthofthem,especiallyin this distanceisgreat.Secondly,itnoteworthy which wasneglectedabove,willclearlyhaveagreater all onlytwocasesinwhich-exceeds0.6,asagainst that mostofthecometswhichpassfarfrom will thereforeexaggeratethenumberofcasesinwhich of closestapproach,sothattheapproximatetheory and willtendonthewholetodiminishdistance comet’s orbitbetweenthepointAandperihelion, explained. Inthefirstplacecurvatureof 26 casespredicted. which passveryfarfromtheplanets.Therearein theoretical andactualdistributionisalackofcomets random. Theonlyseriousdifferencebetweenthe be foundbygraphicalinterpolation. be withintheperpendiculardistancexfromafixed Neptune. TheprobabilityforothervaluesofCmay approximately tothecasesofJupiter,Saturnand gether thenumberofpossibleapproacheswithin near toNeptune.Forthefourmajorplanetsto- inevitably beratherragged.Thereisapparently diametral plane. 30 mightbeexpectedifthecomets’orbitslayat one-tenth oftheplanet’smeandistanceis32,while but thisisbalancedbyadeficiencyofthosecoming it isevidentthattheredittlelefttobeexplained, a slightexcessofcometspassingclosetoSaturn; a numberofcasestheobserveddistributionwill especially whenitisconsideredthatamongsosmall puted, andisgiveninthelastcolumnforeachplanet. might beexpectedfromchancealone,hasbeencom- approach withinthelimitsgiveninTableVwhich Upon comparingthesewiththeobserveddistribution 0.9 0.1 0.0 0.8 0.7 0.6 0.5 0.4 0.3 0.2 This discrepancycanbelargely,ifnotwholly, The tabularvaluesofOarechosenastocorrespond 1.0 With theaidofthistable,number-ofcases C =45° 0.000 0.540 0.374 0.194 0.996 0.824 0.693 0.932 1.000 1.000 1.000 Table VI 0.990 0.678 0.535 0.000 0.369 0.195 0.945 0.879 1.000 1.000 0.800 32°.5 0.969 0.898 0.818 0.554 0.459 0.346 0.000 0.732 0.644 0.186 1.000 15° 0.900 0.800 0.700 0.300 0.600 0.500 0.400 0.200 1.000 0.100 0.000 0° 57 1 00 00 O'! ^r 1920AJ. tí for which-exceeds0.35passestothesouthward. the casesofUranus,andNeptune,whereeverycomet favorable circumstances.Theappropriateequations 58 in thecomet’smeandistancewhichcouldbeproduced by asingleencounterwiththeplanetundermost by Newton’sformulawhatisthegreatestalteration remote fromtheSun—thusincreasingnumber number whichmightbeexpectedtodosobyaccident. pass closetothoseofthemajorplanetsabove however, isthesmallexcessofcometswhoseorbits aphelia arenearthepolesofecliptic. is thereforegoodreasontobelievethat,ifobservation and intheseinstancesitisworthwhiletocompute The mostpromisingonesaretheclosestapproaches: There canclearlybeveryfewcasesofrealcapture. made, arealdeficiencyofperiodiccometswhose of cometswhoseorbitspassfarfromtheplanetsto in theEarth’ssouthernhemispherehadbeenasassid- to beobservedlongenoughpermitthecalculation tion ofmostastronomersinthenorthernhemisphere something morelikethetheoreticalexpectation. southern periheliaandhighnorthernlatitudeswhen would beextendedbytheadditionofmemberswith uous asinthenorthern,ourlistofperiodiccomets of southernlatitudesamongtheaphelia,andtoa more likelytobediscovered,—and,if mical significance,butismerelyaresultofthesitua- There may,however,be,aftertheseallowances,are comet isatthedistanceofremoterplanets.There nearly equaldegree,ofsouthernlatitudeswhenthe comets whicharenorthoftheSunatperihelion of theearth.AsHoletschek(16)haspointedout, This observationalpreferenceresultsinapredominance are southofit,(unlesstheperiheliondistanceissmall). of anellipticorbit,—thanthosewhoseperihelia It isprobablethatthissingularfacthasnocos- Of muchmoreimportanceforthepresentpurpose, 1854 IV 1885 III 1853 II 1861 I 1874 IV 1886 V Comet a © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem Saturn Jupiter Jupiter Neptune Saturn Uranus Planet 0.020 0.21 0.13 0.110 0.030 1.22 Table VII—PerturbationsatClosestApproaches THE ASTRONOMICALJOURNAL 1.71 1.59 1.54 1.46 1.46 1.75 132 119 123 118 127 114 0.0005 0.0008 0.0010 0.0009 0.0019 0.0022 +580 +370 + 79 + 194 + 111 the periodbyasingleencounterisatmostlittlegreater the direction6,andmagnitudeS,ofvelocity than theuncertaintyindeterminationof velocity. Thelastfourcolumnsgivethepresent to themaximumaccelerationandretardationofits the planet’sorbitwastakenintoaccount.Two the comet,relativetoplanet,eccentricityof are (4),(6)and(7)ofPartIthenecessarydata period itselffromtheobservationsofasingleappari- encounters, —onewithJupiterandSaturn. there areonlytwoofthe42cometswhich,iffollowing insignificant theperturbationsarewhendistanceof to Neptunethananyother,isinsertedshowhow which itwouldhaveaftermaximumperturbationin mean distanceandperiodofthecomet,periods the comet’smeandistance)aregiven,corresponding values ofa(thereciprocaltheperturbation and resultsaregiveninTableVII.Incomputing nomena areopen: present formsoftheirorbitsdirectlytocapture. was unusuallywellobserved). tion, (theonlyexceptionbeingComet1861I,which their presentorbits,couldbe“released”(thatis, approach isconsiderable.Itappears,therefore,that either direction.Thelastcomet,whichcomesnearer tions oftheordinarytypesothattheynolongerpass their orbitsfurthermodifiedbyplanetaryperturba- Several alternativewaysofaccountingforthephe- In theothercases,maximumpossiblechangein parabolic orbitsatsometimesinthepast,andhad directed intosensiblyparabolicorbits)byfuture has beenmadebyW.H.PickeringjT7)who,how- near theoriginalpointofencounter.Thissuggestion revolving inanorbithighlyinclinedtotheecliptic. ever, attributedtheprincipalperturbationto action ofahypotheticaldistantplanetgreatmass, The otherfortycometscertainlydonotowethe (1) Theymayhavebeen“captured”fromsensibly ±27000 -570 -362 - 74 -192 -109 105.8 Mean Dist. 45.4 84.9 42.2 55.7 84.1 Present 1089 ±100 415 782 ±200 306 ±14 771 274 ±90 Period Yrs. 25 6 778 336 274 263 204 395 Yrs. Perturbation Period after 216000 N°- 775 Yrs. 786 346 533 398 oo 1 00 00 O'! ^r 1920AJ. tí the node(19)andnearlysameforperihelion. some degrees,andtheoriginal closeapproachofits revolutions, itsperihelionandnodewillhaveshifted irregular, aredecidedlyprogressive,theaverage after acapturedcomethascompletedhundred turbations areprobablylargerthantheaverage. etary systemthaninmostothercases;soitsper- the numbertobeexpectedonabasisofchance;and changes perrevolutionofthecometbeing+0°.15for longitude ofperihelion.But,inelongatedelliptical turbations sincetheencounterhaschangedorbit this isnotthecase.If,then,explanationtrue, whose orbitspassedfairlynearthatofJupiter,above there wouldremainaconsiderableexcessofcomets comet passesnearerJupiter’sorbitthantheaverage in thevelocity(whichareofcourseinitialeffects greatly, withrespect,atleast,toitsplane,orthe we mustassumethatinmostcasestheeffectofper- that, evenafterveryconsiderableperturbations, should mostlyberecentcaptures.But,sinceJupiter it, sincethereisnoreasonwhytheobservedcomets force, andsohadtheirperiodsshortened. be obtainedfromtheperturbationsofHalley’s tions whichmightbeanticipatedwhenthecomet all tracesoftheinfluenceoriginalencounter orbits, themajoraxisismoresensitivetosmallchanges all theotherplanetsputtogether,onewouldexpect must befarmoreeffectiveincapturingcometsthan does notencounteranyoftheplanetscloselymay changed alongwiththeotherelements,andalmost and nevercomeanywherenearanyoftheplanets,) of ordinaryperturbations. wiped out. Hence thecomets’periodsshouldalsohavebeen of thedisturbingforces)thanotherelements. Encke’s Comet.Thefirsthasmuchtorecommend and iscertainlyacontributingfactorinthecaseof ably appliesinthecaseofComets1843Iand1882II the sameorderofmagnitudeasatpresent. nearly parabolicorbits,andhavehadtheiraphelion 1.34) anditsorbitplaneliesnearerthatoftheplan- Crommelin (18)forthepast28revolutions.This distances graduallyreducedbythecumulativeeffect (20) Iftheseratesaretypical,wemayexpectthat, (0,77 astronomicalunits,asagainstanaverageof Comet, whichhavebeencalculatedbyCowelland (which passthroughthesolarcoronaatperihelion N°* 775 The changesintheperihelionandnode,though An ideaoftheordermagnitudeperturba- Of thesefourpossibleexplanations,thethirdprob- (2) Theymayhavebeenmovingoriginallyin (4) Theymayhave“originally”hadperiodsof (3) Theymayhavebeenretardedbysomeother © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem THE ASTRONOMICALJOURNAL alternation oflongerandshorterintervals,differing passages showstwoconspicuousfeatures,—aregular plot ofthesuccessiveintervalsbetweenperihelion by about1)+years,andaslowerchange,apparently period of77.10years,aregiveninTableVIII.A also periodic,completingacycleintenperiodsofthe the intervalsbetweenthem,anddeviationsof years. Thefirstoftheseisobviouslyexplicableby observed timesfromthosecomputedwithauniform the factthattworevolutionsofcometare,on comet, andwithadoubleamplitudeofnearly2^ revolutions since. if originallycaptured,havemademanyhundred average, almostexactlyequaltothirteenofJupiter. fully fiveyears.Thefluctuationisthereforenot from B.C.240toA.D.1759;butintwomorerev- years periodrepresentstheresidualsfortimeof It alternatelyacceleratesandretardstheperihelion random onethatitseemsprobablemostofthem, perihelion passage(aftercorrectionfortheeffectjust fluctuations, asmoothsinusoidalcurveofabout770 passage byaboutfourmonths.Asfortheslower of theorbitsperiodiccometsissonearlya relationship willremain.Theexistingdistribution simply periodic,andisprobably ofaverycomplicated described) withsurprisingaccuracyforthe26returns olutions thedeviationfromthiscurvehasrunupto after athousandrevolutionslittletraceoftheoriginal like one-tenththatoftheplanetfromSun:while widened outintoaminimumdistanceofsomething orbit tothatofthecapturingplanetwillhavebeen 1910.30 1835.88 1759.20 1378.86 1456.44 1531.65 1607.82 1682.71 1066.73 1145.30 1222.69 1301.81 The timesofperihelionpassageHalley’sComet, Date IntervalResidual 837.15 912.55 989.71 Perihelion PassagesofHalley’sComet 74.42 76.68 75.40 77.16 76.52 77.05 77.58 75.21 76.17 74.89 76.49 79.07 77.39 79.12 +0.37 + 2.47 +0.93 + 0.33 + 0.08 + 2.34 + 2.82 -3.24 -1.37 -2.20 -2.82 -5.92 -1.31 -1.89 Table VIII 0.00 -162.62 -219.63 - 86.38 - 11.23 451.50 760.44 837.15 218.27 373.85 684.85 295.27 530.87 607.23 141.23 Date IntervalResidual 66.07 75.59 76.71 76.24 75.15 77.00 79.37 76.36 77.01 77.30 75.16 77.04 78.58 77.65 77.62 +0.72 + 2.86 + 1.71 + 2.23 + 2.95 + 2.00 + 0.05 +0.25 +0.18 + 2.45 +0.33 -1.69 -1.75 -1.85 -0.37 59 1 0 00 -■•^....— oo O'! ^r tí

1920AJ. : 60T-Hß!ASÍROÑOMICALJOURNAL-N-775 turbations 'oftheperiodappear;tobe'fluctuating, that themeanperiodisconstant,asincaseofa and notprogressive,suggestingthatitisprobable íiature. THefactremain^/'however;thatthe’per- length ofonerevolution'andthenextbeing1.55-years. relatively great,—theaveragedifferencebetween planet. The-alterations'intheperiodare,however, preceding (therebyeliminatingaconsiderablepartof If wecompareeachrevolutionwiththenextbutone the perturbationsbyJupiter)averagedifferenceis respond respectivelytoalterationsofthereciprocal still 1.06years.Thesechangesintheperiodcor- tribution of-isclearlythatallvaluesthisquantity at random.Itfollowsthat,foragivencomet,- of themeandistance-by0.00074and0.00050. value. Consideralargenumberofcometswhich will graduallydivergemoreandfromitsinitial the perturbationsatsuccessivereturnstakeplace likely. Supposethatthisisexactlytrue,and ative perturbationsof^areprobablynearlyequally for aperiodof10,000yearsitamountsto3,500years. the lengthofonerevolutionandnextwillbe The conditionofasteadystateasregardsthedis- one perihelionpassageofacomet,wefindthat,for average perturbationsof-whichmaybeexpectedat come toperihelionwithinagivenintervaloftime. great). Atagivenreturn,equalpositiveandneg- very littleontheapheliondistance(providedthisis this partofthecomet'sorbit,andonplanetary the cometcanbesaidtohavea“meanperiod"atall. planets atthecomet'snextperihelionpassagewill perturbations whichitexperiencedatthelastone. period of500years,theaveragedifferencebetween planets. Theiramountwilldependonthepositionof of perturbationsatsuccessivereturnsmayradically It seemsprobablethatinsuchacasetheaccumulation 24 years;foraperiodof2000years,240yearswhile configuration atthetimeofperihelionpassage,but accumulation ofsmallperturbations. alter theoriginalperiod,anditisdoubtfulwhether depend almostentirelyuponthemagnitudeof place, essentially,whenitisfairlyneartheSunand of periodswhichmightbeexpectediftheperiodic above andsuggestsaninquiryintothedistribution comets owedtheirpresentorbitstothegradual Taking thelatterfigureasaroughmeasureof For theselongerperiods,theconfigurationof This leadsbacktothesecondalternativesuggested The perturbationsofacometlongperiodtake © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem Cl a large veluesof“>—sincetheCorrespondingcomets ‘rejected, pendingsomeone'sdetailedexamination the groupsaboveandbelowit.inthe,seriesasloses to them.Thislawofdistributionwould,howmver, would beofshortperiods,andforthesetheperturba- hold goodonlyoveralimitedinterval,—failingfor tions atsuccessivereturnswouldnottakeplace definite limitsofgainingasmanymembersfrom shall beequallyprobable,any-‘givengroup,*between turbations whichmake-negativewillneverreturn, random, —andalsoforverysmallvalues,since intervals definedbyequalincrementsof-. subsequent perturbations.Theperiodshaveprobably comets whichareshiftedoutofthisgroupbyper- In additiontothecometslistedinTableIII,for hypothesis tofindnearlyequalmembersofcometsin thousand years,however,weshouldexpect,onthis so thattheirorbitshavesincebeenmuchalteredby which thenumbersarenearlyequalforin- For periodsrangingfromafewcenturiestoseveral and standnochanceofbeingshiftedbackintoit. be thattheperiodiccometsowetheirperiodicityto lions, —ofperhelionpassagesdemandedonthis periods asatpresent,thereistheobviousdifficulty captures bythemajorplanets,butatnorecentdate, assumption. they formatail,anditseemsimprobablethat that cometslosesomeoftheirmaterialeverytime the periodiccometshave“always"hadaboutsame should lastforthemanythousands,—perhapsmil- may be,thesecondexplanationbetentatively in whichthemotionisstable,(22)andthatcomets is based.Itmaybethat,evenforcometsofvery less accuratelyobserved“parabolic"comets. has shown,haveallapproachedthesolarsystemin of thisinterestingbutdifficultsubject. system, whileothershavebeenlost.Howeverthis which satisfytheseconditionshaveremainedinour long period,theperturbationsofmeandistance wrong abouttheassumptionsonwdiichdiscussion cuments in-v/a,therearemanymorewithstilllonger elliptic orbitsofverylongperiod),andmostthe computed periods,andtotheseshouldbeaddedthe or thattherearecertainpositionsofacomet'sorbit are fluctuatingandnotprogressiveinthelongrun: s “hyperbolic" comets,(which,asStrömgren(21) The actualdistributionofperiodsisverydifferent. The mostprobableconclusionappearsthereforeto With respectfinallytothefourthsuggestion—that This discordanceindicatesthatthereissomething ’a 1 ^ N°-775 O'! ^r 1920AJ. tí time. suffered relativelyunimportantchangesduringthis vast majorityandSaturnforpracticallyalltherest. founded. Jupiteroughttoberesponsibleforthe attribution ofthecapturetothatplanetwhoseaphelion justification. families ofSaturn,UranusandNeptuneonabasis than fiftytooneagainstitinsosmallasample;— distance happenstobenearthatofthecometiswell associated withtheLeonidmeteors)mayhavebeen It mayindeedbethatsomeonecomet,(suchas1866I, tion thattheyhavebeen captured inthepast,but years) bySaturn. planets socloselythattheymayhavebeendiverted position ofmostterrestrialobservers. Jupiter isstronglyconfirmedbythepresentdis- increased bydisruptionduetothetidalactionof north oftheecliptic,canbeexplainedbynorthern the apparentlyshortlivesofsomethemasvisible event (asitshouldbetheoretically).Theproduction of theirapheliondistancesaloneappearstobewithout but theconventionalassignmentofcometsto captured byUranus,—thoughthechancesaremore comets canbesatisfactorilyrepresentedontheassump- IV, (period1089years)byJupiter,and1886V(263 directly intotheirpresentorbitsbycapture—1854 cussion. objects, thetheorythattheirnumberhasbeenlargely of thelargenumbershort-periodcomets,and of ashort-periodcometshouldbestillrarer.Inview been approximatelycomputed,andtheobserved the comets’orbitsweredistributedatrandomhas the majorplanetshavebeencomputedforallcomets deficiency oflargedistancesapproach,especially distribution isincloseagreementwithit.Anapparent to theplanetaryorbitswhichmightbeexpectedif tional tothedifferenceofcuberootslimiting which areknowntohaveperiodslessthan2,000years. planets. periods; andfewofthempassneartheorbits periods betweentwogivenlimitsisnearlypropor- period: theirinclinationsarehigh;thenumberwith an entirelydifferentbehaviorfromthoseofshorter It doesnotfollow,however,thattheconventional The observedcharacteristicsoftheorbitsthese The captureofacometappearstoberatherrare Only twoofthesecometsapproachanythe The circumstancesofclosestpossibleapproachto The 42cometsofperiodexceedingtenyearsshow The distributionofthedistancesclosestapproach © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem Summary THE ASTRONOMICALJOURNAL ventional descriptionofthecometswithaphelion and noneatallofcapturebyNeptune.Thecon- has capturedthegreatmajorityandSaturnrest. Princeton UniversityObservatory, Neptune asbelongingtothe“families”oftheseplanets distances aboutequaltothoseofSaturn,Uranusand except inafewcasesofprobablyrecentcapture. has beeninstrumentalincapturingagivencomet, actually found. many moreshortperiodsandfewerlongonesthanare perturbations. Thelatterleadstoanexpectationof present periodstothegradualaccumulationofsmall appears tobewithoutsecurefoundation. There issomeevidenceofpossiblecapturebyUranus, is probablethattheirperiodshaveundergonerel- so longagotháttheirorbitshavesincebeengreatly than tenyearstoJupiterisfullyconfirmed,andthe satisfactory, asalsodoesthatwhichattributesthe that theperiodshavealwaysbeenshortappearsless shifted inspacebyperturbations.Thedistribution by theactionofplanetisstrengthened. belief thattheirlargenumberisduetodisruption On theoreticalprinciples,itisprobablethatJupiter atively muchlesschangesincecapture.Thetheory of theirperiodsaccordscloselywiththatpredictedfor captured cometsbyH.A.Newton’stheory,andit It isimpossibletosaywithcertaintywhatplanet The well-knownrelationofthecometsperiodless (12) Callandreae',AnnalesdeVObservatoireParis,22, (16) F.J.S.,43,299,1908. (14) J.B.A.A.,17,215,1907. (13) Bull.Ast.,28,170,1911. (11) Waage,A.N.,200,65,1914. (10) VanBiesbroeck,A.J.,29,115. (19) Loc.cil,p.112.. (18) M.N.,68,111,173,375,510and665,1907-8. (17) H.A.,61,217-220,1908. (15) H.4.,61,213,1908. 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