1994AJ 107.1885S 2 sar 8 THE ASTRONOMICALJOURNAL namics ofcometsandasteroidsintheinnersolarsystem.In relativity (Einstein1915)areimportantinmodelingthedy- piro etal(1968),Lieske&Null(1969),andShapiro(1971) found thatthenon-Newtonianmotionforasteroid1566 attempts toverifythepredictionsofgeneralrelativity,Sha- ity, butorbitalsolutionsforIcaruswereofmarginaluse Icarus wasconsistentwiththepredictionsofgeneralrelativ- optical observationsfrom1949through1987. conclusion whenheupdatedtheorbitofIcarususing cantly improvedresults.Sitarski(1992)reachedthesame Icarus usingrelativisticequationsofmotionprovidedsignifi- Mercury. Usingobservationsovertheinterval1949through determining thesolarquadrupolemomentormassof respectively thespeedofobjectanditsdistancefrom vlc {G^MqIvc),wherecisthespeedoflight,andrare mass. Thus,thefractionalfirstorderrelativisticeffectsat tional measurementaccuracyforasteroidsandcometsisof Sun, Gisthegravitationalconstantandi°l bits, smallsemimajoraxes,andlongobservationaldatain- period ofasteroidIcarus,theperihelionprecessionisabout the orderofafewtenthssecondarc.Foroneorbital about 1AUareoftheorderIXHT.Thecurrentposi- 1968, theseauthorspointedoutthatorbitalsolutionsfor bital rmsresidualsarenotimprovedwhenrelativisticequa- the observations. tervals, relativisticeffectswillbenecessarytoproperlyfit the same.Forsomeasteroidsandcometswitheccentricor- motions aresignificantlymodifiedbytheeffectsofgeneral tions ofmotionaresubstitutedforthecommonlyusedNew- relativity. Inaddition,wewillshowthatevenwhentheor- improve theorbitsofsixasteroids,includingIcarus,whose tonian equations,significantorbitalerrorsformanyasteroids motion areemployedinconjunction withmodernplanetary ephemerides. Sincerelativistic correctionsareoftenincor- and cometswillbeavoidedwhen relativisticequationsof rado, Boulder,CO80309-0390. Present address:DepartmentofPhysics, P.O.Box390,UniversityofColo- 1885 Astron.J.107 (5),May1994 0004-6256/94/107(5)/1885/5/$0.90 © American Astronomical Society • Provided by the NASA Astrophysics Data System Relativistic effects,arisingfromthetheoryofgeneral The genericfirstorderrelativisticeffectsareof We haveusedalltheavailableopticalandradardatato We studytheeffectsarisingfromrelativisticperturbationsonmotionofasteroidsandcometsshow to significantimprovementsintheorbitalsolutions.Furthermorewearguethatignoringrelativistic that foranumberofsuchobjects,inclusionrelativisticcontributionsintheequationsmotiongivesrise corrections totheequationsofmotion,whileusingmassesderivedfromrelativisticephemeridesyields incorrect solutionscorrespondingtoaninconsistent,non-Newtonian,nonrelativisticmodel. Jet PropulsionLaboratory,MailCode301-150,CaliforniaInstituteofTechnology,Pasadena,91109 RELATIVISTIC EFFECTSONTHEMOTIONOFASTEROIDSANDCOMETS 1. INTRODUCTION 1 Electronic mail:[email protected],[email protected] Barman Shahid-SalessandDonaldK.Yeomans Received 1993August2;revised1994January6 VOLUME 107,NUMBER5 ABSTRACT 2 rectly assumednegligibleinthemotionsofasteroidsand nation process. tivistic effectsthatcomeintoplayduringtheorbitdetermi- comets, wewillbeginwithanoverviewofthevariousrela- radar observations,thesecorrectionsareusuallymuch However, fortheprocessingofcurrentasteroidandcomet tional potentialoftheSun(Shapiro1964;Standish1990). lay duetothepropagationofradarsignalingravita- models mayhavetobemodifiedbytherelativistictimede- currently performedusingeitheropticalorradartelescopes. observations arenotpossibleneartheSun,radarobservation only whenthelineofsightgrazesSun.Althoughoptical In suchmeasurements,relativisticeffectsaremostprominent smaller thantheerrorsindelaymeasurementsthemselves to themotionofacometorasteroid.Hereweconsider from theplanetaryperturbations.Thesecanbeincludedina gravitational fieldoftheSun.Weignorealleffectsarising simple modelofanasteroidorcometfallingfreelyinthe the slow-motion,weak-fieldapproximationofSchwarzs- spherically-symmetric gravitationalsourcecanbewrittenas the presenceofSun.Themetrictensorforasingle moment wenotethatthelargestrelativisticeffectsarisefrom straightforward mannerinamoregeneralschemebutatthe (Yeomans etal1992). child (1916)metric: Here ¡xistheSchwarzschildradius oftheSungiveninterms of thepost-Newtonianmass of theSun;ix=G^Slc m = — Measurements ofthepositionanasteroidorcometare There areanumberofrelativisticeffectswhichcontribute 3. RELATIVISTICEFFECTSINTHEEQUATIONSOFMOTION goo1 +2——2-^2,(3.1) =5(l +2^). (3.3) g=0, (3.2) gi)i7 0i 2. RELATIVISTICEFFECTSINMEASUREMENTMODELS 2 LL © 1994Am.Astron. Soc.1885 MAY 1994 1994AJ 107.1885S 2 l 2ß 2 ' dr g provides1/3oftherelativisticprecessionorbital where misthemassofparticleinmotion.Fromclassical particle. where ¡jlandvarethespace-timeindicesrunningfrom0to four dimensionsisgivenby gravitational fielddescribedbytheabovemetriccanbede- of theSun,Sun’svelocityisnegligible.Ifonewereto metric representcontributionsarisingfromthemotionof line ofapsides,orperihelion.Thegcomponentsthe rise totheNewtoniangravitationalforce.Thelasttermin Sun. Thefirsttwotermsingandthetermgive particle: rived fromtheleastactionprinciple.Theinvariantintervalin frame (Lieskeetal1977;Shahid-Saless&Ashby1988). where v=dx/dx°.Theequationsofmotioncannowbe Thus comparingEq.(3.8)with(3.7)andusing(3.5), gral oftheLagrangianLovertime,alongpath Lagrangian formalism,weknowthattheactionisinte- 3 andtheEinsteinsummationconventionisused.Theclas- 3, iandjarethespatialcoordinateindicesrunningfrom1to ds= -gdxdx’'(3.4) lion precessioneffectandothercurvatureeffectssuchasthe the curvatureofspace.Thistermprovides2/3perihe- non-negligible contributions.Thelastterminrepresents include theplanetsasgravitationalsources,inJPL source. Inthecaseofasingleobjectfreelyfallinginfield sical actionAistheintegralofdsalongpath ephemeris developmentmodels,thesetermswouldhave which agreeswiththesingle-source relativisticequationsof We get the Lagrangianisgivenby derived usingtheEuler-Lagrangeequations: 19 milliarcsec/yrgeodeticprecessionoftheearth’sinertial 1886 B.SHAHID-SALESSANDD.K.YEOMANS:RELATIVISTICEFFECTS motion usedinmodelingsolar system dynamics(Anderson = G'ÆqIc^1.5kmandristheobject’sdistancefrom 0 m 0i; /JiU 2 1 2 021 L =mc The equationsofmotionforanobjectfreelyfallinginthe dt dx cdt d IdL\dL = 11-2^+^2-j(dx)(7)djjdx'dx^(3.5) © American Astronomical Society • Provided by theNASA Astrophysics Data System r+ 3 7 Q dx. ix tx fX V‘ r+4 l} SijVv (r-v)v 1/2 (3.10) (3.11) (3.8) (3.9) 2 The Hamiltonianisnowgivenby perihelion canbestbederivedusingtheHamiltonianformu- precession oftheperihelion.Therelativisticadvance where wehaveusedtheenergyconservationequation: Newtonian gravitationalforce.Wecannowtreattheremain- The firsttwotermsinthetotalHamiltoniancorrespondto lation oftheequationsmotion.Thecanonicalmomenta tivistic correctionswhich,inpart,giverisetotherelativistic gravitational acceleration.Theremainingtermsaretherela- the righthandsideofaboveequationisNewtonian et al1975;Moyer1971).Wecanseethatthefirsttermon The advanceoftheargumentperihelionisnowgivenby turbation dependsonlyonthreeorbitalelements;themean ing termsintheHamiltonianasaperturbationH.Thisper- are given,tothedesiredorderofaccuracy,by can rewriteFfjas noting thatr=a{l—ecosE),Ebeingtheeccentricanomaly anomaly M,thesemimajoraxisa,andeccentricitye.One perihelion precessionisperhaps themostmeasurableeffect. We get calculated byaveragingoveroneperiodandnotingthat ment ofperihelion.Thecontributionthelasttermcanbe Eq. (3.16)donotgiverisetoasecularadvanceintheargu- (Bertotti &Farinella1990) helion advanceforallcurrently knownasteroidsandcomets This effectismostnoticeablefor eccentricorbitswithsmall orAa> =0.0384/a(l—e)arcsecondsperrevolutionwhena and nbeingthemeanmotionItt/T.Thefirsttwotermsin is inastronomicalunits.Sincemost observationsofasteroids semimajor axes.Onecancompute themagnitudeofperi- and cometsatthistimeareoptical angularpositions,the x =2(319) tfi=-Az+2 —-3^2-,(3.16) 27r ~dt do) Jo 1-ecosEVl-e‘' 12y f dE2ir 2arar’ 'Ih? 2 1 dL 21/ (l-e) dH^ 2 1+- u+3- ÓTTJX 2 r 1 ¡JL 0 de 2 _1 8 2 7T+2 — 6 o A r (320) (3.17) (3.15) (3.14) (3.13) (3.12) (3.18) - 1886 1994AJ 107.1885S Table 1.15asteroidswiththelargestperihelionprecessionrates. period, andthetotalrelativisticprecessiontomid-1993. the numberofyearsforwhichobservationsexist,orbital precession (perihelion)peryear.Thenextthreecolumnsgive eling thedynamicsofsuchobjects.For15objectsmost to seewhetherrelativisticeffectsshouldbeincludedinmod- lion shouldbeclearlydetectablefortheasteroidsnear angular positionofanasteroidisafewtenthssecond and numberoftheasteroidfollowedbyitsamountorbital affected byrelativisticperturbations,Table1givesthename 1887 B.SHAHID-SALESSANDD.K.YEOMANS:RELATIVISTICEFFECTS top ofthelist. arc, therelativisticcontributiontoprecessionofperihe- weak-field, slow-motionsolutiontoEinstein’sfieldequations motion ofsolarsystembodies,theappropriatemetrictouse only whenthereexistsasinglegravitationalsource.Forthe spherically-symmetric Einstein’sfieldequations,relevant for anarbitrarynumberofsourcesandthusabetterrepre- post-Newtonian metrictoincluderelevantrelativisticeffects devised byWill&Nordtvedt(1972).Thismetricisthe is themany-bodypost-Newtonianmetricsuchasone development programutilizesthepoint-massversionof sentation ofthesolarsystem.TheJPLplanetaryephemeris Newtonian order.Incomparingtheobservedpositionsof this developmenteffortarethereforeaccuratetopost- in thesolarsystem.Thesystemparametersderivedby models, onemustalsorealizethat themassofSundeter- comets andasteroidswiththosepredictedfromtheoretical rections devisedbytheIAUtokeep therelativisticequations relativistic correctionsarisingfrom thepost-Newtonianfor- mined bythesolarsystemephemeris effortincludesnotonly of motioninthebarycentricframe simple.Correctionsaris- mulation oftheequationsmotion, butalsoincludescor- Object 2062 Aten 1566 Icarus 433 Eros 2101 Adonis 2100 Ra-Shalom 3753 (1986TO) 5143 Heracles 3200 Phaethon 1862 Apollo 2340 Hathor 1685 Toro 1951 Lick 1865 Cerberus 1620 Geographos 1627 Ivar Assuming thatthecurrentaccuracyindetermining © American Astronomical Society • Provided by the NASA Astrophysics Data System The Schwarzschildmetricisthevacuumsolutionto 4. RELATIVISTICVERSUSNEWTONIANDYNAMICS rs Free. Rate.Obs.IntervalperiodTot.Free,to5/93 0.101 (arc. sec./yr)(yi's.)(y-)sec.) 0.021 0.043 0.022 0.075 0.053 0.019 0.021 0.101 0.025 0.016 0.017 0.040 0.010 0.074 43 37 48 44 57 8 35 56 15 36 30 61 19 7 18 0.950 0.759 1.119 2.567 0.997 1.785 0.775 1.598 2.544 2.484 1.761 1.640 1.423 1.389 1.123 4.34 0.97 0.91 1.20 1.59 0.90 0.61 0.90 1.13 0.52 0.61 0.63 0.72 0.81 0.88 2 where wouldbethemassofSunderivedfromNew- relativistic correctionscanbeunderstoodbycomparingEq. of Eq.(3.11).Discrepenciesarisingfromexclusionthe tion arisingfrompost-Newtonianformulationareoftheform the definitionofspace-timecoordinateswhichwewillcon- ing fromtheIAUchoiceofcoordinatesneedbeutilizedin that ifoneweretousethepost-Newtonianmass,usedinEq. tonian ephemerides.ComparisonwithEq.(3.11)showsus sider shortly.CorrectionstotheNewtonianequationsofmo- made becausetheleadingtermshavesamefunctional yield asemimajoraxisthatisincorrectbyanamount: long periodoftimewiththebrackets(),thenfitwould amount. Ifwedenoteatimeaveragetakenoversuitably (3.11) withNewton’ssecondlaw: (3.11) wouldstillcontributeonaveragebyasignificant (3.11) inNewton’ssecondlaw,afittodatacouldstillbe way, theNewtonianequationsofmotionwould,onaver- by nothavingmodeledtherelativisticprecessionofperihe- 1/r dependenceonr.However,theneglectedtermsinEq. tically formulated,isinerror. relativistic effects,theirdynamicalmodeling,ifnotrelativis- of manyasteroidsandcometsmaynotbesensitivetotrue in general,haveinconsistencies.Thus,althoughthemotions derived fromJPLplanetaryephemerides(i.e.,^)will, Newtonian equationsofmotionbututilizingthesolarmass modeled. Hence,theorbitaldeterminationofobjectsusing the relativisticprecessionofperihelionwouldstillnotbe age, agreewiththerelativisticversionexceptforfactthat to post-Newtonianorderisrescalethesolarmass.Inthis lion. Ofcourseoneremedyforcorrectingthesemimajoraxis barycentric time(t)istoberelatedanidealEarth- physical measurementsareindependentofthechoiceco- formalism developedbyWill&Nordtvedt(1972).This Newtonian coordinates,definedbythepost-Newtonian namic modelingofsolarsystemobjects.Relativisticequa- of usingTDBcoordinatesarealsoimportantincorrectdy- bome clocktime(i)byaperiodiccoefficient,namely ordinates. NeverthelessaccordingtotheIAUconvention, Einstein’s fieldequationsaregenerallycovariantandthus choice ofcoordinatesisonlyamatterconventionsince tions ofmotionareusuallywrittenintermsthepost- TDB by surface. Thepost-Newtoniantime coordinate,whichisused in themodelingofmotion oftheplanets,isrelatedto (Hellings 1986) TDB TDB E This wouldbeinadditiontowhatevererrorcameabout As mentionedearlier,correctionsarisingfromthechoice Sa =a—^ Here Uisthetotalgravitational potentialontheEarth’s ítdb=(i +^+--^U )í. (4.3) rn ? e 2 1 8 a^10 . r (4.1) (4.2) 1887 1994AJ 107.1885S value whenpurelyNewtonianequationsofmotionwereemployedwhilethe Table 2.Orbitalsolutionsforsixasteroidsthatareaffectedbyrelativistic by: matter byrescalingthecomponentsofmetrictensor: However thiswillentertheequationsofmotionordinary changed (i.e.,keepingthespeedoflightconstant)onehasto vided byitsobservationweightinthesameunitsbeforermsvaluewas computed. Forthermsorbitresidual,firstlineforeachobjectgives observations aregivenalongwiththedataintervaloverwhichorbitwas effects. Ineachcase,JPLdevelopmentephemerisDE200wasused.For with therelativisticallydetermined masseswouldingeneral planets derivedfromtheJPLplanetary ephemerides(DE200, TDB masseshavetoberescaledwithrespectPN the formofpost-Newtonianequationsunchanged, also rescalethespatialcoordinatessuchthat computed. unitless values.Thatis,eachorbitresidual(inarcsec,Hz,or/xs)wasdi- residuals areinarcseconds.Otherwisethermsnormalized, tion wereused.Forthoseasteroidswithoutradardata,theunitsofrms second linegivesthesameinformationwhenrelativisticequationsofmo- each objectlisted,thenumberofopticalandradar(timedelayDoppler) yield asolutionwhichisneither relativisticallycorrect(no et al1985;Ries1988).The massesoftheSunand consistent andhasbeenshowntobesopreviously(Martin motion unlessonerescalesallthemasses.Iftriestokeep etc.) includerelativisticcorrections. equations ofmotionintherelativisticformalismisself- 1888 B.SHAHID-SALESSANDD.K.YEOMANS:RELATIVISTICEFFECTS 3200 Phaethon92 3753 (1986TO)67 1566 Icarus466-906/27/49-10/12/921.42 1862 Apollo99 2100 Ra-Shalom75 =2 x— =— 8 - © American Astronomical Society • Provided by the NASA Astrophysics Data System <^TDB (^)¿>PN*(4-6) ^tdb (1 ^TDB(1 ^)PN*(4.7) In ordertokeeptheequationsoflightpropagationun- The magnitudeofLis1.55052X10~.Integrationthe ¿TDB i1“2?^ Thus, usingtheNewtonianequations ofmotionalong In turn,thiswillaffectthepost-Newtonianequationsof 1 U2 / optical delayDopplerDataIntervalResid. z '2? Observations RMS 1 v 4 804/27/32-12/29/891.33 PN- dx^—(1 L'jdx^'N’ 2 10/03/75-10/09/910.99 10/12/83 -11/29/921.01 10/17/73 -08/25/921.08 12/17/55 -10/06/921.21 0.97 0.99 0.98 1.17 1.06 1.21 (4.5) (4.4) relativistic precession)nornonrelativisticallyconsistent—a riod ofobservation.Table1liststhetop15suchobjects hybrid solution. Table 1.Forsixobjects, 2includesthenumberand orbital fitstothedataavailableforseveralasteroidslistedin oid 1566Icarusisatthetopoflist.Wehaveperformed ordered indecreasingtotalperihelionprecession.Theaster- cession, theobservationalperiod,numberofactualob- studied thefullsetbylookingatrateofperihelionpre- sion oftheperihelionislarge.Thesubsetwaschosenfrom hybrid modelsdescribedabovetotheobservationaldataofa gitude oftheascendingnode,inclination,semimajoraxisinAUand Table 3.Orbitalelementsforsixasteroidswhosemotionissignificantly servations andthetotalamountofprecessionduringpe- some 156cometsandasteroidscommonlyunderstudy.We subset ofobjectsforwhichthecumulativerelativisticpreces- mean anomalyin.degrees. the timeofperihelionpassage(TDB),argumentperihelion,lon- elements are,respectively,theeccentricity,periheliondistanceinAU, corrected intheorbitdeterminationprocess,formalstandarddeviations refer totheeclipticplaneandJ2000equinox.Forsixorbitalelements treated separately.Thenormalizedrmsorbitalresidualsandtheemployed els. Theorderinghereisbythe amountofimprovementin and thermsresidualsforrelativistic andthehybridmod- type oftheavailableobservations, theobservationinterval, astrometric dataaregiveninTable2.Theanglesdegreesand development ephemerisDE200withtheEarthandmoonperturbations affected bygeneralrelativisticeffects.EachorbitwascomputedusingJPL’s evident thattherelativeimprovement totheorbitalsolutions, the rmsresidualwhichforIcarus isasmuch30%.It are giveninparenthesesunitsofthelastdecimalplace.Theorbital i 19.8110943(1086)15.7555290(251)18.9319601(633) Node 126.3950772(4619)170.9613315(273)108.6855238(236) w 43.6374549(3497)355.9442840(504)147.9154590(1198) T 1994Jan15.4233736(1890)Mar31.4564282(705)Jun8.6636837(1129) e 0.826694124(109) T 1994Jan27.7429529(217)19952.7165230(102) Epoch 1994Feb17.0(TDB) i 22.8790200(249)6.3562961(239) w 31.2248612(127)285.6391790(584) q 0.186836585(118) e 0.514811131(2438) Node 88.1537825(55)35.9330697(579) q 0.484091097(2440) Epoch 1994Feb17.0(TDB) We performedexperimentstofittherelativisticand 32.2170508 0.997737433 3753 (1986TO) 1566 Icarus 1.078074153 17.8364024 5. ORBITALSOLUTIONEXPERIMENTS 304.8653242 0.647353126(49) 0.559941362(33) 0.832048169 1994 Feb17.0(TDB) 1862 Apollo 0.468895485(198) 0.436456323 (241) 2100 Ra-Shalom 1.471061059 1994 Feb17.0(TDB) 183.3868217 244.1941062 0.966622518 0.139653993 (173) 0.890151589 (136) 3200 Phaethon 265.5970894 (370) 321.8104926(355) 22.0974651(363) 1994 Feb17.0(TDB) 2062 Aten 1993 Sep12.7348078(2513) 0.790133591 (363) 0.182583091 (379) 1.271333759 1994 Feb17.0(TDB) 108.1302769 1888 1994AJ 107.1885S 2 bital semilatusrectum[p=a{\—e)]andproportionalto when aconsistentrelativisticformulationisemployedin- Improved orbitalelementsforthesixasteroidsinTable2are the observationintervalandamountofdataavailable. ticular setofobjectswithsmallsemimajoraxesandlarge given inTable3. stead ofthehybridmodel,isinverselycorrelatedtoor- of thisworkhasbeentoshowthatforsuchobjects,exclu- eccentricities, relativisticeffectscanplayamajorrolein inclusion ofgeneralrelativityinmodelingthedynamics sion ofrelativityeffects,whileusingsolar-systemmasses dynamical modelingofsuchobjects.Animportantpurpose asteroids andcomets.Wehavedemonstratedthatforapar- Anderson, J.D.,Esposito,P.B.,MartinW.,&Muhleman,D.O.1975,ApJ, derived fromrelativisticallyephemerides(suchas 1889 B.SHAHID-SALESSANDD.K.YEOMANS:RELATIVISTICEFFECTS Bertotti, B.,&Farinelia,P.1990,PhysicsoftheEarthandSolarSystem Lieske, J.L,Lederle,T,Fricke,W„&Morando,B.1977,A&A,581 Hellings, R.W,1986,AJ,91,650 Einstein, A.1915,Sitz.Preuss.Akad.Wiss.Berlin,47,778;799;844 Moyer, T.D.1971,MathematicalFormulationoftheDouble-PrecisionOrbit Martin, C.F.,Torrence,M.H.,&Misner,1985,J.Geophys.Res.,90, Lieske, J.L,&Null,G.W.1969,AJ,74,297 200, 221 Determination Program,NASATechnicalReport32-1527 9403 (Kluwer, Dordrecht),p.385 © American Astronomical Society • Provided by the NASA Astrophysics Data System We havestudiedtheprominenteffectsarisingfrom 6. CONCLUSIONS REFERENCES DE200), willingeneralresultinconsistentsolutionsfor with acompletelynon-relativisticmodel;i.e.,theycorre- the dynamicalparameters.Wehaveshownthatsuchorbital solutions areneitherrelativisticallycorrectnorconsistent tract withtheNationalAeronauticsandSpaceAdministra- Laboratory, CaliforniaInstituteofTechnology,undercon- The conclusionisthatinthemoreextremerelativisticcases, with boththefullyrelativisticmodelandhybridversion. to derivethermsresidualresultingfromsolutions spond toahybridmodel.Finallyweperformedexperipients tion. discussions. ThisworkwascarriedoutattheJetPropulsion is aslarge30%. the asteroid1566Icarus,improvementinrmsresidual the relativisticmodelyieldssmallerresiduals.Incaseof available datausingthecurrentJPLprograms.Thiswasdone Ries, J.C,Huang,&Watkins,M.1988,Phys.Rev.Lett.,61,903 Will, C.M.,&Nordtvedt,Jr.,K.1972,ApJ,177,757 Shapiro, I.1971,AJ,76,588 Shapiro, I.1964,Phys.Rev.Lett.,13,789 Shahid-Saless, B.,&Ashby,N.1988,Phys.Rev.D,38,1645 Schwarzschild, K.1916,Sitzber.Deut.Akad.Wiss.Berlin,KL.Math-Phys. Yeomans, D.K.,Chodas,P.W.,Keesey,M.S.Ostro,J.,Chandler,J.F., Standish, E.M.1990,A&A,233,252 Sitarski, G.1992,AJ,33,1226 Shapiro, I.I.,Ash,M.E.,&Smith,W.B.1968,Phys.Rev.Lett.,20,1517 Tech., 424 & Shapiro,I.1992,AJ,103,303 The authorswouldliketothankR.W.Hellingsforhelpful 1889