2 0 0 5MNRAS.358.1273V -3o 2 2 1,23 1 pressed asapowerofthedistancefromcentre,toapsidal * E-mail:[email protected] to lowerapsemovesthrougheithermoreorlessthan180°,andthe n /1.Inthelattercaseorbitingbodyingoingfromupper tude /xr",0=n/*/ñ180/y7z.Theformula,byitsmannerof Newton derivesanimportantformularelatingcentripetalforce,ex- In Proposition45ofBookIthe‘Principia’(Newton1687), Accepted 2005January5.Received2004September29 Mon. Not.R.Astron.Soc.358,1273-1284(2005) we shallsaythattheorbitprecesses. GivenNewton’sformula,the derivation, isrestrictedtoorbits‘approachingverynearcircles’ DepartmentofPhilosophy,TuftsUniversity,Medford,MA,02155-7059,USA DepartmentofAppliedMathematics,TheUniversityWesternOntario,London,CanadaN6A3K7 An extensionofNewton’sapsidalprecessiontheorem © 2005RAS amount ofprecessionmeasuresthe exponentintheforcelaw.Zero same angleisrepeatedinthereturn totheupperapse.Inthiscase (e «1). apsis (apocentreandpericentre).Foracentripetalforceofmagni- angle, 0,orangleattheforcecentrebetweenhighestandlowest 1 INTRODUCTION S. R.Valluri,*RYu,G.E.SmithandP.A.Wiegert DepartmentofPhysicsandAstronomy,TheUniversityWesternOntario,London,CanadaN6A3K7 The apsesaretermed‘quiescent’ifn=1,and‘moving’ © Royal Astronomical Society•Provided bythe NASA Astrophysics DataSystem pressed intheneighbourhoodofTV=2(caseinversesquarelaw).Thevalue30/9Afis by theapsidalangle0(N,e),whereNisindexforcentripetalforcelaw,varying interesting newlaboratoriesforstudiesofgravity. istic galaxymodels.Thepossibilityofapsidalprecessionwasalsoexaminedforafewcases in termsofalgebraicandlogarithmicfunctions.Noevidencewasfoundforisolatedcases integrals withsingularitiesatbothlimits.Someoftheintegrals,especiallyforTV=2,canbe the solutionofthisequation,inneighbourhoodN—2whichhappenstobecase theorem. Wehaveexaminedtheprecessionoforbitsundervaryingforcelawsasmeasured mathematical, physical,astronomicalandhistoricalinterest.Thelunartheorythepreces- Newton’s apsidalprecessiontheoreminProposition45ofBookIthe‘Principia’hasgreat ABSTRACT Key words:gravitation-celestialmechanicsSolarsystem:general. discussed inviewofitsinterestforthedynamicseccentricorbitsandrelevancetoreal- zero precessionasewasincreased.TheTV=1caseofthelogarithmicpotentialisalsobriefly given inclosedformtermsofgeneralizedhypergeometricfunctionswhicharereducible computed numericallyaswellfori^TV<3.Theresultingintegralsareinterestingimproper derivatives d0/dNandd0/dh[whereh(N,e)istheangularmomentum]areanalyticallyex- eccentricity e.Thepaperderivesageneralfunctionfortheapsidalangle,dependentonlyon of high-eccentricityasteroidsandextrasolarplanets.Wefindthatthesesystemsmayprovide of greatesthistoricalinterest.Exactsolutionsarederivedwheretheypossible.Thefirst e andNasthepotentialissphericallysymmetric.Further,weexploreapproximatewaysof sion oftheperihelionplanetMercuryarebuttwoexamplesapplicationsthis rapidly withdistancethandoesaninverse-squareforce;theexpo- rograde, wheren>1)isequivalenttohavingtheforcefalloffless precession (where«=1)isequivalenttohavingtheforcevaryas next century,asthetimeperiodover whichreliableplanetaryobser- the planetswerenotdetectablyprecessing; duringthecourseof ing thattheforceofSunonplanetsisinverselyassquare yields ameasureoftheexponentinforcelaw. nent intheforcelawisgreaterthan—2.WhereNewton’sformula the inversesquareofdistance.Positiveprecession(progradeor vations hadbeenmadegrew,itbecame clearthattheseorbitswere whence theforceisinversesquare). AtthetimeNewtonhadreason exponent intheforcelawislessthan—2.Negativeprecession(ret- of thedistance(Proposition2:planetaryapheliaare‘quiescent’, applicable, anempiricaldeterminationoftheamountprecession off morerapidlywithdistancethandoesaninverse-squareforce;the advancing orbitwheren<1)isequivalenttohavingtheforcefall slowly precessing.(Laplacechooses nottoinfertheinversesquare (from Streete’s‘AstronomiaCarolina’) tothinkthattheorbitsof In BookIIIofthePrincipia,NewtoncitesProposition45inargu- doi: 10.111l/j.1365-2966.2005.08819.x 2 0 0 5MNRAS.358.1273V results. Itisdesirablethentoexaminethederivatived0/dN=d0/d8 become increasinglyrelevant,owingtorecentphotometricobser- power indexN.Wealsostudytheforcelawforinverse promises toshow-theradialcomponentofperturbingaction per revolution,whencen—3=—2(4/243),thesmalldeviation near thecentresofmanynearlyelliptical galaxieshaverekindled decay ofblackhole(BH)binaries (Peters &Mathews1963;Peters the orbitofMercury,gravitational wavesemittedduringtheorbital the absenceofisolatedcasesapsidal quiescence.Thelunartheory, illustrate foreacheccentricityethe dependenceof6{N)andconfirm integrals offeranimportantmathematicalstudy.Moreover,they can bedifferentiatedundertheintegralsign.Theresultingimproper mathematical interestoftheirown.Theexpressionfortheangle6 from anapproachsimilartothatofClairautinhisstudythelunar insight. Thedifferentialequationoftheorbitinthispaperisstudied careful study.Inaddition,theanalyticalcalculationofapsidal numerically evaluatedforN=3¡2,1¡A,2,9/4and5/2.Adetailed was calculatednumericallyonlyforthecase5^0and0(N,e) integration forverysmall8toobtain0withoutprovidingnumerical vations ofnearlyellipticalgalaxiesandspiralgalaxies,aspointed from theinversesquare,whathappenstoapsidalangleas laws otherthantheinversesquareaswelldifferingbutslightly it wouldseem,thecaseforProposition45. the orbitsofcomets,henodoubthasinmindHalley’scomet,which, effect appliesittotheorbitofMercury,whoseeccentricityis0.2056, is notclosetozero.InProposition2ofBookIII,forinstance,hein entrant, haveagreatestandleastdistancefromthecentreofforce, the apsides’beingonlyapproximate).Theseorbits,thoughnon-re- dal angleinallcasesdiffersfrom180°,Newton’s‘quiescenceof the planetaryorbits,noneofwhichisstrictlyanellipse(theapsi- for anellipse,andhastheadditionalmeritofbeingapplicableto least. Thisdefinitionreducestotheusualmeasureofeccentricity r), whereisthegreatestdistancefromcentre,and infinity, wecandefineorbitaleccentricitybye=(r—r)/(+ from theinverse-squareproportionbeingattributable-soNewton tion (Proposition3:thelunarapsismovesonaverage3:3arcmin force oftheEarthonMoonvariesininverse-squarepropor- Proposition 2.)Againciting45,Newtonarguesthatthe could nolongerrunthesimpleargumentNewtonhasinBook3, for theplanetsfromprecessiontheorem,probablybecausehe orbit (Brown1895).Theresultingseriessolutionshaveintrinsic angle anditsderivative,when8isverysmall,offersadditional analysis ofthederivativeininterval1Mercury? Andmoregenerally,doesprecessionmeasuredeviation instead apaperbyJosephBertrandfromwhichtheformulacanbe unaware ofNewton’sderivationtheformula180°/^/^;hecited law inwhichtheexponentN=2.00000016.Hallwasapparently general solutionisthereforethesum ofthecomplementaryfunction depends ontheeccentricitye,aresonant termthatbuildsupthe can beputintheform when 8differsfromzero. we wanttoexamine,5,whichcanbepositiveornegative,will where N—2=8(Yallurietal.1997).Hereh(angularmomentum) 0 asr—>oo.Thestandarddifferentialequationfortheshapeof differ infinitesimally.Istheformulavalidlyapplicabletoorbitof derived (Bertrand1873).ButBertrand’sargument,likeNewton’s, was ninNewton’snotation,todeducefrom43arcsecpercenturya given by—iir~Hallusedtheformula7t/V3—N,where3N © 2005RAS,MNRAS 358,1273-1284 (— l)Vcos{0—O^/i.Thisisequivalent toadrivingforcewhich (McLachlan 1947).Inclusionofthissecondtermresultsinthedif- solution ofequation(3)withthesecondtermonrightdeleted 0{8), oncewehaveshownthatAyisof0{8).Weshalldiscussthe side. Alltermswiththecoefficient8{Ay)/hwillprovetobeof shall thustakeintoaccounthigherpowersoftheeccentricitythan and Ayisaperturbationintroducedtotakeaccountofthedifference of equation(1)inthiscase,wesupposeu=uq+Ay,whereis small enoughinabsolutevaluesothatwecanignoreitssquare.(In and theangleaboutcentreofforce,0,is orbit, expressedintermsofthereciprocalradiusvector,u=1/r, fi isaconstant(chooseß=1).WeassumeA>0,sothatf{r)^ the orbit.Letcentralforcebegivenbyf(r)=—ßr~,where assumes thatthegreatestandleastdistancestocentreofforce Ä d(Ay) dn u d<7 V’ 0 y 2 1 d<9 To addressthisquestion,weturntothedifferentialequationof Substituting thisexpressionforuintoequation(1),weobtain 28 2 h+ d(Ay) 2 <5 ¿>(Ay) 28 dö + (Ay) h E<-»“ + Ay--(uoAy)-ÿ yV-iyV cos'te-é>'). i=l oo © Royal Astronomical Society•Provided bythe NASA Astrophysics DataSystem f cos*^ —0) s Uq ,8(Ay)u 0 (2) (i) (3) 282S 28 28 l /2 2 2 2 2 rotating apse,whichwecanwriteas lunar theory,weadoptarotatingellipseasstartingpoint ment ofthelunartheoryfoundedonintegrationdifferential ity. Whatroledoestheeccentricityplayinoursolution? Newton’s formula,werecall,isvalidonlyforverysmalleccentric- For 5=0(inversesquarecase)theresultistt,asitshouldbe. formula gives:n/^Jn=n/^/3—Nn/^/\8n/(\8/2). the smallnessof<5.ThisturnsouttobesameresultthatNewton’s mation, X=1—8/2hl8/2,asisclosetounitybecauseof factor 8;neglectingthem,wehavetheequationofanellipsewith The termsof0(8)ontherightaresmallrelativetothosefree tary function=const,cosÀ{0—O').Afterfurthersimplification formulas. Hall, thesquarebracketiseffectively 1,andXisgivenwithgood This formulagivesresultsforthe apsidalanglecloselyagreeing third orderineibytheformula it overtheapsidalangle,from0tott,thendividingbyn. We cangetanaverageresultfortheright-handsidebyintegrating to ahigherpowerthanthefirst.Wefindthat the smallconstant1/h—l/h^,andalltermsinvolving8raised to takeaccountoftheeffecteccentricityonapsidalangle.We equations (Brown1895).InthespiritofClairaut’sapproachinhis we obtain with thoseobtainedbynumerical integration(seeSection3).In ,£3(1/96XA2) 1 -X'=— \-X' 2 =1 Further, inthecourseofourinvestigation,itwasbroughtto Clairaut wasprobablythefirsttopublishamethodfortreat- The apsidalangleisgivenby(0—O')=n/X.Toanearapproxi- À2lß 2 2 ' {(3-IV)[l(IV2)(7V+2)í'/12]} h h 28 28 28 h e h 8 arcsine h 2 8 h 1 + V(-ly-^—cos^i? -6>'). /=o 6"35"IÏ2' 246 e 3e5 (N -2)(N+2) / +1 i 12 Apsidal precession1275 (9) (8) (7) 2 0 0 5MNRAS.358.1273V 18 2 2A 1A 3 1 + result by830.39,thenumberofhalf-cyclesperJuliancentury,we becomes amoresensitiveindexofdeparturefromtheinverse-square by afactorthatincreaseswithincreasinge\however,thevaluesit q (an)0.255 a (an)2.97 e 0.914 Name 1984QY11999XS352000LKSG82001DQ8 2002AJ129PD432003MT92004CK395025P-L Table 1.High-eccentricityasteroids,theirsemimajoraxisa,inclination ialongwiththeirperihelionqandaphelionQdistances. Relativity (GR:Einstein1915;Roseveare1982;Earman&Janssen law (Brown1903).In1915Einsteinshowedthattheanomalous theory wassufficientlydetailedtoruleoutHall’salternativepower tenable: ErnestW.Brown’sdevelopmentoftheHill-Brownlunar tion (13)7t/2. For ourMoon(e=0.05493)thederivativeforequation(11)is found theextraprecessiontobe43.11zb0.45arcsecpercentury.] into accountalltheobservationsofMercuryfrom1765to1940, the precession.[In1947G.M.Clemence(Clemence1947),taking 0.3 percent,ratherlessthantheuncertaintyinempiricalvalueof the factor1.0072,resultwouldbe43.05arcsec,adifferenceof tra precessionperhalfcycle,namely0.052arcsec.Multiplyingthis is 1.0072.Multiplyingby8/2and180x3600,wefindtheex- law astheeccentricityincreases. increases. Itfollowsthatdepartureoftheapsidalanglefrom180° for almostvanishinge.Forgiven5,precession,orthedeparture Equation (13)isindependentofeandgivesaccurateresultsonly Clairaut typeofapproach,theperturbativeanalysis(Vallurietal. with respecttothepower-lawindexNorintermsof8,from increasing withe.Thederivativeoftheapsidalangle0 gives fortheapsidalanglesareabitshyoftruevalues,error later, equation(8)isqualitativelycorrectinthat8/hmultiplied Newton’s formulaisaccuratetofirstorderine.Asweshallshow lowest powerofeenteringintotheformulaisc,whichimpliesthat Here arcsine/e>\,andincreaseswithincreasingeccentricity.The X =1, / (deg)17.8 apsidal precessioncouldbederivedfromhisTheoryofGeneral of anomalousprecessionintheapsisMercuryhadbecomeun- obtain 43.20arcsec,theextraprecessionperJuliancentury.Without of theapsidalanglefrom180°,becomesgreateraseccentricity and theapsidalanglen/k'isapproximatelytt(1+<5arcsine/2e). as unity, Q (an)5.69 (7t/2)(l.000504), forequation(12)(7t/2)(1.000756)andequa- 1997) andNewton’sformulaaregivenbelow. 1993). 1276 S.R.Vallurietal. 90 7T1 d0 n(e 90 7Tarcsinen(^3 d8 ~2(1-5)/2• 9á ~2(JJ- 2S , 8arcsine By 1903Hall’salternativepowerlawtoaccountforthe43àrcsec In thecaseofMercury,withe=0.2056,fractionarcsine/e By thiscalculation,forSsmallenoughsothathcanbetaken 1 + ~ 2¡V"67cT’ 2 £ © Royal Astronomical Society•Provided bythe NASA Astrophysics DataSystem 0.954 0.947 34.9 19.4 17.9 4.41 0.119 2.27 0.947 17.4 4.66 0.242 24.0 2.45 0.901 0.181 0.902 3.51 13.0 1.84 (13) (10) (12) (11) -5 rather onitseccentricity.Boththeprecessionrateofanorbitand precessing) nucleusitself(Wilhelm1987).Thuscometsarenotideal Additionally, themagnitudeofthisforcevarieswithdistance to haveexperiencedsuchbackreactionforcesofmagnituderoughly For example,duringits1986appearance,CometHalleyisthought the easewithwhichsuchprecessionisobservedincreaseas Equation (10)isindependentofthesizeanorbit,butdepends force laws,however,isnotconfinedtotheregioncloseSun. largely withinthecontextofprecessionperihelionMer- fects ofperturbationsfromthisplanet,anditwillmakeanumber measurements is2002AJ129,whichhasbeenobservedforover the exceptionof5025P-Land1984QY1,discoveredin1960 Table 1).Allofthesehavebeendiscoveredinthelast5yr,with Though asteroidsaretypicallyonlow-eccentricityorbitsthepop- number ofsuchobjectsislikelyquitesmall(Levisonetal.2002). fect theirmotion,buttheeffectwouldbegreatlyreducedand mination ofthemotionacomet.Someextinctcometsmaylurk the pre-andpost-perihelionlegsofitsorbit(Sekanina1964;Festou its determinationcomplicatedbysucheffectsasvariationsbetween the cometfromSunanditsinstantaneousdirectionisunknown, grown. with differentproperties,suchashigheandlargermass,hasalso fects, thenumberofopportunitiesfortesting8directlyinsystems eccentricity oftheorbitincreases.Asaresult,observationallimits eccentric orbitsuffersthelargestGR-inducedprecessionofany cury. ThissmallplanetbeingclosesttotheSunandonarelatively close approachestowithin0.1auofEarth(in2010,2018and2026) 2004 CK39havearclengthsfromtwotoseveralweeks.Thelongest Center lists151objectswithe>0.75,and100.9(see ulation knowntobeonorbitsofhigherehasincreasedmarkedly volatile materialswhoseoutgassingcomplicatestheprecisedeter- candidates formeasurementsof8. accelerations (i.e.owingtothebackreactionofgases orbits arecomets.However,thesebodiessuffernon-gravitational orbits hasincreaseddramaticallysinceearlydiscussionsoftheef- objects. Asthenumberofbodiesknowntobeonhigh-eccentricity of 8canmosteasilybeconstructedbyobservinghigh-eccentricity of theplanetsinourSolarsystem.Precessionowingtodiffering a year.ItsapheliondistanceisfarfromJupiter,reducingtheef- observational arc,andhenceoneofthebettercandidatesforsuch but 1999XS35,2000SG8,2001DQ8,2002PD43,2003MT9and and arenowconsideredlost.2000LKwasonlyobservedfor7d, over thepastdecades.Forexample,atthiswritingMinorPlanet among theasteroidpopulationandlow-leveloutgassingmayaf- sublimating fromthecometnucleus)thataredifficulttoquantify. (Festou, Rickman&West1993)androtationofthe(potentially 10 ofthegravitationalforceowingtoSun(Rickman1986). 1984, respectively.Bothwereonlyobservedforafew(3-4)days 1986), variationsintheoutgassingontime-scalesoforderaday 2.63 0.117 0.915 15.5 1.37 The longestknownandhencebestobservedobjectsonhigh-0.5currentlyknown.Theseplanetsandtheirpropertiesarede- dia (http://www.obspm.fr/encycl/encycl.html)liststenplanetswith Cutler 2004).Planetsorbitingotherstarsofferanotheropportunity. energy andangularmomentumasgravitationalwaves(cf.Barack& cession astheyundergoorbitalchangesowingtothedissipationof eccentricity oftheorbit6489Golevkaissubstantial(0.6)andit which shouldallowitsorbittobeveryaccuratelydeterminedby © 2005RAS,MNRAS 358,1273-1284 tems likethisone,particularlyifthereisonlyoneplanetpresentor 62.23 d(Fischeretal.2003).If<5=ilO“,then<9~tt±2x10“ sider theextrasolarplanetHD3651bwithe=0.63andaperiod anomalous precessiononareasonabletime-scale.Forexample,con- scribed brieflyinTable2.TheyhavemassestypicallyofaJupiter such aplanetsuffersowingtootherbodiesinthesystemwillbe being observedislikelythemostmassiveinsystemprovides of 8becomessignificantlyeasierasperturbationsowingtoother these starsasnumbersnaivelyimply,thenthedetermination orbiting 105stars.Ifinfactthereisonlyoneplanetmostof ample, areexpectedtodisplaydramaticrelativisticperihelionpre- system. TheinspirallingorbitsofstarsintomassiveBHs,forex- amounting toonly15kmintherangebody,indicatesthat attributed totheYarkovskyeffecthaverecentlybeendetectedby 16 CygniB Anomalous precessionmayalsobeobservedoutsideourSolar We notethatvariationsintheorbitofasteroid6489Golevka M sin/(Mp) ju 4.85 ±1.7 © Royal Astronomical Society•Provided bythe NASA Astrophysics DataSystem 10.35 5.11 3.21 3.41 8.64 1.69 7.2 0.2 1.0 111.78 ±0.21 133.82 ±0.2 540.4 ±4.4 1192 ±150 256 ±0.7 period (d) 2063.8 550.65 798.9 62.23 572.0 0.505 ±0.018 0.927 ±0.012 0.52 ±0.08 0.68 ±0.15 0.7 ±0.02 0.62 0.63 0.67 0.71 0.71 -A regard. period orbitsaroundotherstars,the‘laboratory’formeasure- brief analysisofthedetermination0(e)forthiscase,byTourna to be1andthecaseA/=isexcluded.Theintroductionofa integration. Forourforcelawthepotentialenergyisgivenby We turnnowtocaseswhere8canbeconsiderablylarger,say0.25or 3 THECASEOFFORCELAWSDIFFERING ically overthelastdecade.Theseadvancesprovidepossibility ment ofdeviationsfromNewtoniangravityhasexpandeddramat- teroids andimprovementsinradardetectiontechniques,aswell The resultsintheinverse-square casehavebeenshownearlier lower apse).Thentheanglebetweenhigherandapses,or least rootis«i(forthehigherapse)andgreatestu the angularmomentum.Theresultissimplifiedifwereplacerby The kineticenergy,givenbythebracketedtermonright,is where wehavetakenthegravitationalconstantandcentralmass 0.5. Insuchcaseswemaydeterminetheapsidalanglebynumerical where theradicandonrighthasrealpositiveroots.Suppose cosmological constantAwouldresultin SQUARE and otherparameters,theauthorsencourageobserversinthis of testingNewtoniangravityinnewregimeseccentricity,mass as thediscoveryofmassivehigh-eccentricityplanetsonshort- other perturbationscomplicatestheanalysis. accuracies canbeobtainedareevenbetter,thoughtheeffectsof and ateachapsewemusthavedu/d0=0,theapsidesbeingdefined & Tremaine(1997),separatelyattheendofthissection. away fromr=0andoo,wemusthaveE<0. always positive,whenceVo)48.80 £ «1 0.8 0.6 0.4 0.2 0.8 139.53057.7-7.439(5.062percent) 0.6 143.60852.7-3.361(2.287percent) 0.4 145.64150.4-1.328(0.904percent) 0.2 146.65749.2-0.312(0.212percent) 1.0 1.0 120.080.0-27.00(18.4percent) S. R.Vallurietal. 144.0 156.438 158.957 160.194 160.808 160.997 in degrees in degrees N -1 © Royal Astronomical Society•Provided bythe NASA Astrophysics DataSystem 2 ~Y~ hu d0/dN 9-0 dO/dN 9-Oo 115.2 -17.00(10.6percent) 0 2 78.5 -4.559(2.832percent) 70.6 -2.040(1.267percent) 66.7 —0.803(0.499percent) 64.7 -0.189(0.117percent) 64.1 0 6u A (per centdeviation) (per centdeviation) (17) N12/ 2 Eccentricity Apsidalangled0/dN0—Oq Eccentricity Apsidalangle tegration: whenI<51isverysmall, mathematica andmaplelabour for verysmall8,althoughnot,it seems, byadirectnumericalin- than 0.2,butrisenearlyto100°for £=1whenN5/2.Thusthe The differences9—0oarelessthanadegreeforeccentricities for N>2,anddecreases<2,withincreasingeccentricity. from 0oisalsoindicatedinthetables. The valuescorrespondingto£=1ofd0/dNfoxN3/2,1/4,9/4 The derivativeof9withrespecttoNthenbecomes for boundorbits)canbeexactlydoneintermsoftheF(gamma)or law force.Numericalintegrationofthisformulaforourfourvalues for motionaboutapoint;itisrathermoretypicaloflinesource that alogarithmicfunctionforthepotentialresultsN=\.This, Table 6.Apsidalangleanditsderivative(N=5/2). Table 5.Apsidalangleanditsderivative(N=9/4). curves 9(e)turnawayfromthehorizontal line9=reuseincreases ß (beta)functionsgivenbelow which leadstotheellipticandhyperellipticintegrals.Weobserve e= f(-urdu.(is) apsidal angle.9isthemoreexactvalueandpercentdeviation and 5/2areshowninTables3-6.0istheNewtonianvaluefor analytical oneswhichwerealsoshownforcomparison(Vallurietal. of N,namely3/2,7/4,9/4,5/2,giveslessaccurateresultsthanthe of randtheapsidalangleisn/2.ThissameforHooke’s of course,isnottheusualpowerlaw.Suchapotentialunusual (16) canbereducedforV>1to (Valluri etal.1997). (Goldstein 1990).ForAf<1,thepotentialisanincreasingfunction 1997). Theintegralfor _ (5)A7t In theparticularcasewheree=l(Vallurietal.1997),integral We canuseequation(16)tocheck theaccuracyofthisformula It canbeobservedfromtheseresultsthatprecessionincreases e <$C1254.558 e <1 0.8 273.275 0.6 262.829 0.4 257.795 0.2 255.316 0.8 0.6 0.4 0.2 E0 354.7 1.0 Jo (3-A0r[l4 +ii^§±i] 239.5 214.929 210.981 209.073 208.132 207.846 in degrees(percentdeviation) in degrees © 2005RAS,MNRAS 358,1273-1284 720.0 279.8 315.1 do/dN o—0q 320.0 +31.65(15.23percent) 173.6 +7.083(3.408percent) 153.0 +3.135(1.508percent) 144.4 +1.227(0.590percent) + 100.1(39.33percent) + 18.717(7.353percent) +8.271 (3.249percent) +2.237(1.272 percent) +0.758 (0.298percent) (per centdeviation) +0.286 (0.138percent) 0 0 (20) 2 0 0 5MNRAS.358.1273V 2 2 x plification beingthatonehastoobtaintherootsofacubicequa- by Hagihara(1931)andChandrasekhar(1983).Itisalsoworth- results in terms oftheWeierstrassellipticfunctions(Boccaletti&Pucacco tion (Schutz1996).Thesolutioninsuchacasewouldbegiven two solutions(whicharecomplexconjugatesofeachother)given It isofinteresttonotethatthecaseimaginaryeccentricity two algebraicequations to equation(15)andsettingáuláO0at«i=1uyieldsthe in termsofthelimitsintegration,and(asresultdependsonly integral sign.Wefirstsimplifyequation(16)byexpressingEandh tive dO/dNalsofromequation(16)bydifferentiatingunderthe formula (11)is90°(arcsine/e)’,this,multipliedby5,willgivethe have proceededasfollows.ThederivativedO/dNobtainedfrom improper integrals,withoutproducingareliableresult.Insteadwe for hoursattheintegrationofequation(16),whichturnouttobe 2. Theintegrandhasdiscontinuities atbothlimits.Theintegralhas where x=N—1.Thisresultistrue forallxintheinterval0< derivative of0withrespecttoxisshownbelow Clearly theapsidalangledependsonlyonNandu(ore).The while pointingoutthattheperihelionprecessionforMercuryin corresponds toorbitsthatplungeintotheforcecentreandhas where uisnowtheratioofgreatesttoleastdistancefrom © 2005RAS,MNRAS 358,1273-1284 a meaningbythelimitprocessand hasafinitevalue.Evaluatingthe Schwarzschild space-timehasananalogousexpression,thesim- u = the centre,thatis,r/,so and Solving forEandh/2simultaneouslyfromthesetwoequations and on theratiou'.ui)stipulatingthatU\=1.Toachievethis,reverting amount ofprecessioninahalf-period.Butwecanobtainthederiva- E + E + 1996). h —1 2 2 2 2 Mm 2 dx -2y/u1 d0 ^2—1 2 = («2-1) With theabovesimplifications,integraltakesform m2m—wJ:2 N -1 N -1 (JV-l) («!-!) u 1 +e : /)+(2l(n^]du. 1 —e 2 N-l 2 1 r -1/2 1/2 2x— (l —nW2)u+«2 ^2] x «2 (1—u)\nuU'2In«2 © Royal Astronomical Society•Provided bythe NASA Astrophysics DataSystem 3/2 du, (22) (21) (27) (26) (25) (23) (24) 71/1 1 result atx=1(sothatN2),wefind tion (29).Thus,forthespecialcaseoíN=2,exactsolutions The expressionfromtheGHGFreducestothatgiveninequa- The integralsontherightcanbegiveninclosedformterms the inducedprecessioncanbemodelledbyaconsiderationofd0/dh. mentum oftheorbit.ThoughhisconstantinKeplercase, ties of0.2,0.4,0.6,0.8,namely2/2,1ß,4and9.Theresultsour than equation(11). not anapproximationintheusualsenseandismuchmoreaccurate for 0intheregionaroundN=2.Equation(12)isunderstandably the paperof(Vallurietal.1997)reallyusesfirstderivativeterm the approximatemethodforobtainingapsidalangle0givenin tion (12).ThisexactsolutionfordO/dN{N=2)clearlyshowsthat given below,thoughmuchofthetediousalgebrahasbeenomitted. expressed intermsofalgebraicandlogarithmicfunctions,whichare 0.8 112.50 0.6 100.00 0.4 93.92 0.2 90.92 Table 7.Derivativesof(dO/dN)(N=2)fordifferenteccentricitye. where wehaveused0=0(N,e)orh).Therefore comparison, indegreesperunitN,areshownTable7. which isidenticaltotheapproximation(forsmalle)givenbyequa- are 0=7tand of generalizedhypergeometricfunctions(GHGFs)andcanalsobe ated equation(28)forthevaluesofuthatcorrespondtoeccentrici- of theTaylorexpansion0{e,N)inobtainingapproximatevalues systems wheredissipationisimportantthisconstraintvanishes,and X <7U/2ln«2— 2 dÑ -V^2-1)Vv^+/ 0x 2(u—1) dÑ dhdÑ d0 _1 dh~ dÑ\dN) 36» _3Ö/dh\ 1; 2 d0 _/w+l\f-\/u21\ dO _dh 7t2 By thechainrule We canalsocomputed0/dh,whereh(e,N)istheangularmo- We havecheckedbynumericalintegration,andthusevalu- = nu\—2n(u+1) 2 2 2 (1 —u)In(1¿4)w (29) 2 1 +a/T^T£ 2 TT 7T(1—a/1e) [(u -u)(u-1)] 2 109.01 99.56 93.89 90.92 (12) 2 (1 —u)Inm(l^2)«w 2 32 - Ill/ 3//2 Quartic Apsidal precession1279 111.50 [ (u—u)(u1)] 2 98.97 92.96 89.98 Í V^2-A \ +1y 111.40 Cubic 98.98 92.94 89.97 du Finite difference 111.39 98.99 92.96 89.98 (31) (29) (28) 2 0 0 5MNRAS.358.1273V — 2 1 related totheLambertWfunctionf(W)=exp(TF)(Corlessetal. variables for0,d0/dN,Eandh,which dependonlyonAande. from theempiricalvalueheassumed (43arcmin)bylessthanthe factor arcsine/egivesanapsidalprecessionthatis0.34percent gration ofequation(28)showsthat,foraneccentricity0.2,our 90° forx=1(A2)irrespectiveoftheeccentricitye.Theinte- finite value(Vallurietal.1999). differential equationwithrespectto8(=N—2)givesasmoothand in thevicinityofA=2isasmoothfunctionforalle.Equation tion (30),whichindicatesthatthefirst-orderapproximationof0(e) Another waytoshowthisisusethefirstderivativegiveninequa- cent ofthecorrectvalues.Thisdeviationispossiblyowingto discrepancies fromthecorrectvaluesarelessthanorabout1per The valuesinTable7aregoodagreementwitheachother;the these methodsagreeverynearly(Wilson,privatecommunication). the datapointsofTables3-6.Theuseallfivegives cation ofLagrangianinterpolationandfinitedifferencemethodsto for N=2.Thetrivialsolution¿¿21gives£0whichthe function withbranchindexk=0.Recallingthat«2(1+e)l where fToistheprincipalbranchofmultivaluedLambertW From amathematicalviewpoint,itisinterestingthatthisequation equation (34)iszerowhere itational vV-bodyproblem,weexaminethissituationfurther. tering resonances.Giventheimportanceofresonancestograv- criteria inparticle-scatteringtheorydeterminesthelocationofscat- This equationwillbesingularwherethedenominatoriszero. Combining equation(33)with(30)weobtain From equation(24)wecancompute observational uncertainty. of 8gives43.35arcsecprecession perJuliancentury,differing all e.ThederivativeofthecompletesolutionMathieu-Hill of thefirstfourandlastdatapoints.Theslopecurveat a quarticequationwhereasweobtaintwocubicequationsbyuse apsidal lineisnotdefined. and indicatesthisconditionisnotsatisfiedontheprincipalbranch «2 InW2+1=0.(35) shy ofthecorrectvalue.Withthiscorrection,AsaphHall’svalue (30) canbeusedtodetermine0intheneighbourhoodofA=2for scarcity ofdatapoints.Weconcludethatthecurves0(e)aresmooth. slopes forthecubicatA=2areaveraged.Theresultsfrombothof A =2isobtainedfromasingledifferentiationofthequartic; (1 —e),thesolutiontothisequationisw^—3.5911214or 1.77. Thisisinconflictourearlierassumptionofanellipticorbit, 1996). Rewritingequation(35)wefind l/¿¿2 (=r/rM)forselectvaluesof A.eandAactasindependent 1 =—Wo[—exp(—1)],(36) 1280 S.R.Vallurietal. U2 k 2 dh -(m—— 90 _nh{u+\){.^_-X)({N 3AÍ “7(N-l)(u\ m dh _1Íu*-\N-\)\num"“+ 2 2 2 It isinstructivetopresentplotsof 0,80/dN,E andhversuseor We observethatNewton’sformulaimplies80/3Aisalways We alsopresentthevaluesofderivativesobtainedbyappli- Assuming e<1andtakingthecaseN=2,denominatorof © Royal Astronomical Society•Provided bythe NASA Astrophysics DataSystem the apsidalprecessionmightdiminish continuouslyandfinally,at three-dimensional graphsfor0(e, A) and80/dN(e.A). Figure 1.Thefunction1(e)fordifferentvaluesofN:(a)N=3/2,(b) evidence forsuchisolatedcasesof apsidalquiescence. dimensional graphsforE(e,A)andh(e,A).Figs910showthe A =7/4,(c)9/4,(d)5/2. some valueoftheeccentricity,become zero?Wecouldnotfindany sible that,withtheeccentricityincreasing inacontinuousfashion, We havealsoaskedthefollowing interesting question:isitpos- The valuesofIincreaseforhigherA.Figs7and8showthethree- Parts (a)-(d)ofFigs1-6studythevariationfor (6) /¿asafunctionofAfordifferente. (5) /¿asafunctionofefordifferentA, (4) £asafunctionofAfordifferente, (3) £asafunctionofefordifferentA, (2) 0asafunctionofefordifferentA, (1) d0/dN(=/)asafunctionofefordifferentA, © 2005RAS,MNRAS 358,1273-1284 2 0 0 5MNRAS.358.1273V Figure 2.Thefunction6(e)fordifferentvaluesofN\(a)N=3/2,(b) © 2005RAS,MNRAS 358,1273-1284 N =7/4,(c)9/4,(d)3/2. 6 2.60 2.65 2.70 2.75 2.80 4.4 4.6 4.8 4.0 2.1 2.2 2.3 2.4 2.5 2.6 5.0 5.2 3.6 3.7 3.8 3.9 0 0.20.40.60.81.0 © Royal Astronomical Society•Provided bythe NASA Astrophysics DataSystem Figure 3.Eversuse. e lE r 1/2 2 21/ lar momentum(foracircularorbit withradiusr).Here,L(E= for —1Earth-Moon systemortheprecessionofperihelionMercury the interval1