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19 69ApJ. . .155. .587R -2 -2 2 andMars,intheprotonalpha- fluxes(O’GallagherandSimpson dicated bythemeasurementsofaninterplanetary gradient,betweentheorbitsof path lengthsdeducedfromthesestudiesrange afewgcmtomorethan10 by cosmicraysatlowenergiesisseriouslycomplicated byanunknownresidualsolar cm, andtheymayornotbeenergydependent. Eventhoughthesevaluesareof ma (1958),FosterandMulvey(1963),BadhwarDanielDahanayake, gested, therefore,thattheseisotopesareofsecondaryorigin,producedbynuclearinter- nificantly largerthanthatexpectedfromthestudyofuniversalabundance B (ShapiroandSilverberg1968),thedetermination oftheamountmaterialtraversed the sameorderaspathlengthdeducedfrom observedabundancesofLi,Be,and Sreenivasan (1968),andMeyer,Hagge,McDonald (1968).Themeancosmic-ray Kaplon, andLavakare(1964),Kuzhevskii(1966),Fanco-workers(1966a,b), and calculatedintensitiesofdeuteriumhelium-3coulddeterminethemeanamount modulation. Fichtel andReames(1966),BadhwarKaplon (1966),Biswas,Ramadurai,and of materialtraversedbycosmicraysininterstellarspace. actions ofcosmicrayswiththeinterstellargasandthatcomparisonobserved Biswas elal.1967;DennisFreierandWaddington1968).Ithasbeensug- elements (Fanelal.1966a;Fan,Gloeckler,andSimpson1966b;O’Dellet1966; planetary fieldmodulationatsolarminimumwhichabove600MVisoftheformexp(—ri/Rß)with unique determinationofboththecosmic-raypathlengthandresidualinterplanetaryfieldmodulation The AstrophysicalJournal,Vol.155,February1969 nuclei. Thisdeterminationdoesnotdependonanyassumptionsregardingeitherthesourcespectraor at solarminimummaybemadefromacomparisonofthecalculatedandmeasuredintensitiesthesetwo and -3bycosmic-rayinteractionsintheGalaxyiscalculatedconsideringenergy-dependent 7] —0.35±0.15BV,whereRandßarerigidityvelocity,respectively. stood intermsofanenergy-independent,cosmic-raypathlength4±1gcm-andaresidualinter- tensities attheEarth,weconcludethatwithinexperimentaluncertaintiesalldatacanbeunder- cross-sections, interactionkinematics,anddemodulatedcosmic-rayspectra.Theresultingfluxatthe the unmodulatedproton-to-alphaparticleratioofprimarycosmicrays.Theproductiondeuterium modulation. Fromacomparisonofthesecalculationswiththemeasureddeuteriumandhelium-3in- Earth isobtainedbytakingintoaccountleakagefromtheGalaxy,ionizationlosses,nuclearbreakup,and 3 COSMIC-RAY DEUTERIUMANDHELIUM-3OFSECONDARYORIGIN © American Astronomical Society •Provided bytheNASA DataSystem The existenceofsucharesidualmodulation,active evenatsolarminimum,isin- Assuming thatcosmic-raydeuteronsandhelium-3nucleiareofsecondaryorigin,weshowa Previous studiesofdandHeproductionweremadebyHayakawa,Ito,Terashi- The abundanceofdeuteriumandhelium-3inthecosmicrayswasfoundtobesig- * NAS-NASAPostdoctoralResident ResearchAssociate. f Researchsupportedbythe National ScienceFoundationundergrantGP-849, Institute ofGeophysicsandPlanetaryPhysics,UniversityCalifornia,LosAngeles AND THERESIDUALMODULATIONOFCOSMICRAYS NASA, GoddardSpaceFlightCenter,Greenbelt,Maryland Received April19,1968;revisedJuly22,1968 Richard E.LiNGENFELTERf Reuven Ramaty* I. INTRODUCTION ABSTRACT AND 587 19 69ApJ. . .155. .587R 3 3 3 3 3 -1-3 production (RamatyandLingenfelter1968),wesuggestedthatbecauseofthesecondary total residualmodulation,which,shortofdirectmeasurementsoverlargedistancesin kinematics andthesecondaryenergydistributionsofvariousinteractionsproducing calculated bytakingintoaccounttheenergydependenceofcross-sections, cosmic-ray ,alphaparticles,andmediumnucleiwiththeinterstellargas,are charge ratiosorspectralshapesoftheprimarycosmicrayseitherininterstellarspace the relativemodulationoveradistanceofroughlyanastronomicalunitandnot lating theresultantsecondaryequilibriumfluxeswithsamemodulatingfunctionas served primarycosmic-rayfluxesattheEarth,thenbyusingthesedemodulatedintensi- deuterons andhelium-3nuclei.Usingthesespectra,weevaluatetheequilibriumd modulation, thistechniquereliessolelyontheassumptionthatobserveddandHe study ofdandHeinthecosmicraysallowsanessentiallyindependentdetermination origin ofthesenucleiandbecausetheirdifferentcharge-to-massratio,acomparative the solarsystem,remainsessentiallyundetermined. 588 tion intheinterplanetarymagneticfield.Unlikepreviousattemptstoestimatetotal of boththeamountmaterialtraversedbycosmicraysandtheirtotalresidualmodula- length andaresidualsolarmodulationwhichisbothvelocityrigiditydependent. fluxes attheEarthcanbebestunderstoodintermsofanenergy-independentpath used intheinitialdemodulationprocess.BycomparingresultantdandHespectra up. Theresidualsolarmodulationistakenintoaccountbyfirstdemodulatingtheob- He fluxesbyconsideringleakagefromtheGalaxy,ionizationlosses,andnuclearbreak- with theobservations,itisdemonstratedthatmeasureddeuteronandhelium-3 and conclusions. at theirsources.Inthepresentpaperwegiveadetaileddiscussionofthesecalculations are ofsecondaryoriginanddoesnotdependonanyarbitraryassumptionaboutthe 1967; O’Gallagher1967).Themeasurementofthisgradient,however,determinesonly ties tocalculatetheproductionofdandHeininterstellarspace,finallybymodu- helium-3, andtritium,whichdecaysinto arelistedinTable1.Eachofthese acting cosmic-raynuclei;nisthenumberoftarget nucleipergramofinterstellarma- stellar materialmaybewrittenas per nucleonindEaroundE. , respectively;jistheequilibriumintensity ininterstellarspaceoftheinter- nuclei intheratioIrlO:!)(SuessandUrey1956; Cameron1959). interactions mayalsoincludethegenerationofpions. cleus, s.Theinteractionswhichwehaveconsidered fortheproductionofdeuterium, cosmic-ray nucleusofenergypernucleonEwillproduce asecondarynucleusofenergy stellar material;Eandaretheenergiespernucleonofprimarysecondary where qistheproductionspectrumofsecondarynucleipersecondgraminter- in theevaluationofequation (1). production cross-sections, andthekinematicssecondaryenergyspectra tobeused terial; cistheinteractioncross-section;andF(E,E )dE istheprobabilitythataprimary 8s 8 s s8 © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem The productionspectraofdeuteriumandhelium-3,resultingfromtheinteraction In apreliminaryreportofratherextensivecalculationsdeuteriumandhelium-3 The rateofproductionsecondarynucleibycosmic-rayinteractionsintheinter- The productionissummedoverallinteractions, i, whichproducethesecondarynu- The interstellarmaterialisassumedtobecomposed ofhydrogen,helium,andCNO We shallnowprovidedetailed discussionsoftheprimarycosmic-ray intensities, REUVEN RAMATYANDRICHARDE.LINGENFELTER (£) =^fdEjmm^FiiEß,),(1) ?ss II. DEUTERONANDHELIUM-3PRODUCTION i Vol. 155 19 69ApJ. . .155. .587R ured inconsiderabledetail fromthresholdtoafewBeVbySchulz(1952), Crawford values obtainedfromthetemporalandspatialvariations mentionedabove,wehave parameter R.Sincethereisnogeneralagreement astothevalueofRo,andsince With thisfunctionalformfor/(R),themodulatingfunctionmentionedabovecanbe modulating parameterswhichdefinethetotalmodulation maybedifferentfromthe explained bysuchamodulationbuthavededuced differentvaluesforthemodulating with RoessentiallytimedependentandoftheorderseveralhundredMVin1965. treated bothr¡andRoasfreeparameterstobedetermined fromthepresentstudy. lagher andSimpson(1967)O’Gallagher havealsoshownthatthemeasured Jokipii (1966,1968)andColeman(1968)demonstratedthatsuchaformis irregularities. ItwasfirstsuggestedbyDorman(1963)that/(R)couldbeapproximated temporal andspatialvariationsofcosmicraysin theinterplanetarymediumcanbe rigidity dependingonthedistributionofmagneticirregularities.Subsequently, by f(R)=RoforRRqwhereisacharacteristictransition The modulatingparameterrç,whichisspaceandtimedependent,definesthetotalre- function, M,oftheformexp[—r)/ßf{R)\,basedonParker’s(1958)solar-windmodel. consistent withthemeasuredpowerspectrumofinterplanetaryfieldvariations, magnetic rigidityanddependsonthedistributionofinterplanetarymagnetic-field sidual solarmodulation,ßistheparticlevelocityinunitsofc,andf(R)afunction equation (1)areobtainedbydemodulatingthesemeasuredspectrawithamodulating and Freier(1966).Theinterstellarprimarycosmic-rayintensitiesusedtoevaluate Gloeckler andJokipii(1966),Balasubrahmanyan, Boldt,andPalmeira(1967),O’Gal- written as Balasubrahmanyan etal.(1966a,£),Comstock,Fan,andSimpson(1966),Fan were measuredneartheEarthatsolarminimumbyFreierandWaddington(1965), No. 2,1969 0 (1966¿>), HofmannandWinckler(1966),OrmesWebberWaddington 0} © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem The deuteron-production cross-sectionforproton-protoninteractionshas been meas- The energyspectraofthevariouschargecomponentsprimarycosmicradiation 11... 10... 4.. 9.. 8.. 6.. . 3.. . 2 . 5.. . 7.. 1 . l 4 4s 1 1 4 4 p+H.e^d+2p+n+(ir) ^-j-CNO—»¿-j- ...d“(ir) p -J-He—-\-p-j-(tt) ^>+He—>¿+He+(ir) CNO+H—... +M a+He—>2¿-j-¿+(*-) a-j-He—>2¿-|-a-i-(x) a-f-He—>d -{-p-\-na-J-(tt) a+IF-íá+HeS+Or) a+HJ^+2ÿ+»+(x) COSMIC-RAY DEUTERIUMANDHELIUM-3 Deuteron andHelium-3Production Deuteron a) PrimaryCosmic-RayIntensities b) ProductionCross-Sections M =exp(—rj/Roß),R<; 0 = exp(—rj/Rß),R>. 0 TABLE 1 10. 9. 8. 6. 4. 3. 2. 5. 7. 1. 3 43 4 43 18 43 13 4 1 13 ^+CNO-»He+ ...-Ktt) p -j-He—>He-\-p-\-n+(x) />-j-He—H-\-2p-\-{Tc) ^+He-^He+¿+(x) CNO +H—>He+...-Ktt) a-|-H—»He +»+a+W a+H-^He -\-p-\-n+(tt) a-j-He—H -\-p-ha-{-(ir) a-j-H—^/-j-2/>-J-(7r) a+H—>He+¿+(7r) e Helium-3 589 (2) 19 69ApJ. . .155. .587R 4 4 3 been studiedbyRosenfeld(1954).Thesecross-sectiondatatogetherwithasmoothed Stadler (1954),Baldonietal.(1962),Guzhavin(1964),NeganovandParfenov curve usedinthepresentcalculationareshownFigure1. 590 MeV byWickersham(1957),at31Bunch,Forster,andKim(1964),32 the breakupofHeweremeasuredinprotoninteractionswithalphaparticlesat28 (1958), Chapmanetal.(1964),Fickinger(1962),Pickup,Robinson,andSalant (1953), StevensonDurbin,Loar,andSteinberger(1951),Fieldsetal.(1954), Hayakawa etal.(1964),andat95MeVbySeloveTeem(1958),inneutroninter- MeV byBenvenisteandCork(1953),at53Cairnsetal.(1964),55 actions withalphaparticlesat90MeVbyTannenwald(1953),and300 (1962), Smithetal.(1961),andHartthetheoryofinteractionhas portant, andtheproton-pickupcross-sectionis much lowerthanthatforneutrons, barrier forprotonsisnotimportantattheseenergies andthatthesecross-sectionsare Moulthrop (1955)andInnés(1957).ThesedataareshowninFigure2.Wehaveassumed as forneutrons.Themeasurementofthepickupcross-section byTannenwald(1953)of equal. Atlowerenergies,particularlynearthethreshold, theCoulombbarrierisim- that theHebreakupcross-sectionsattheseenergiesaresameforincidentprotons differentiate betweenreactionshavingtwo,three, ormorechargeparticlesinthefinal et al.(1960)andat970MeVbyRiddifordWilliams (1960).Thesemeasurements which showsaresonanceatabout22.15MeV. state butdonotcompletelydistinguishbetween thevariousmodes.Therefore,some charged particlesinthefinal state)theratiosbetweenalpha-particlebreakup modes It wasshownbyMoulthrop (1955)andInnés(1957)thatin3-prongreactions (three assumptions mustbemade inordertoobtainthedeuteron,triton,andhelium-3 yields. (1958) of15+3mbfor95MeVinthereaction pa —»JHesuggestthattheCoulomb 12 ±2.5mbfor90MeVneutronsinthereaction na—>dtandbySeloveTeem © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem The cross-sectionsfortheproductionofdeuterons,tritons,andhelium-3nucleifrom Measurements ofalpha-particlebreakupwerealso madeat630MeVbyKozadaev REUVEN RAMATYANDRICHARDE.LINGENFELTER + Fig. 1.—Deuteron-productioncross-sectionforthereactionW(pd)'K ) Vol. 155 19 69ApJ. . .155. .587R 0+ + 3 0 3z both -andnon-pion-producingreactionsthattheseratesareroughlyinde- into singlychargedfragments,pt:pnd:dd:2p2n«0.58:0.26:0.12:0.04,arethesamein pionic ptbreakup,wecanseparatetheir3-prongcross-sectionintopartialcross-sections pendent ofenergy. for thesebreakupmodesbothpion-andnon-pion-producingreactions.Asimilar No. 2,1969 breakup constitutesroughly10percentofthetotalpion-productioncross-sectionat final states,wemakethreeassumptions:(a)pionproductionwithoutalpha-particle assumption thatttandproductionareroughlyequal. treatment isappliedtothedataofRiddifordandWilliams(1960)withadditional probable inpion-producingreactions,basedona similarequalityfoundfornon-pionic in apinteractionsatthesameavailablekineticenergy inthecenterofmassframe.Since parentheses inFigure2,togetherwiththeinferred -dependentpartialcross- our calculationssuggestthatainteractionsmake anon-negligiblecontributiontod section forpainteractions. modes inthelower-energymeasurements.Thecross-sections thusdeducedareshownin greaterthan300MeV,basedonthemeasurements ofInnés(1957);(6)7rand and Heproduction,ameasurement ofthebreakupcross-sectionsin energy range the samebreakupmodesoccurandthattheircross-sections aresimplyfourtimesthose around 100MeVwouldbe particularlyvaluable. tt productionareroughlyequal;and{c)the¿He andpriKzfinalstatesareequally 3z © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem Using thisrelationship,andthefactthatKozadaevetal.(1960)didmeasurenon- In ordertoreducethe2-pronginelasticreactionswhichinclude¿HeandpriH.z The deuteron-production cross-sectioninprotoninteractionswithCNO nuclei was In theabsenceofdataonhelium-4breakupina interactions, wehaveassumedthat 4 Fig. 2.—Deuteron-,triton-,andhelium-3-productioncross-sectionsfor^Hereactions COSMIC-RAY DEUTERIUMANDHELIUM-3 KINETIC ENERGY/ 591 19 69ApJ. . .155. .587R 3 3 -production cross-sectionfortheseinteractionswasmeasuredat43MeVby energies thecross-sectioncanbeestimatedfromd/tyieldscalculatedbyFraenkel by Kavaloski,Bassani,andHintz(1963)at190MeVBailey(1956).Athigher measured at18.5MeVbyNadiandRiad(1964),20Legg(1963),40 592 by FiremanandRowland(1955),at6.2BeVCurrie(1959).Thehelium-3-produc- ments ofBailey(1956)at190MeVandtheMonteCarlocalculationsFraenkel tion cross-sectioninproton-CNOinteractionsisnotwellmeasured,butthemeasure- Cherny andFehl(personalcommunication),at150MeVbyBrun,Leforth,Tarrago (personal communication)andmeasuredbySchwarzschildZupancic(1963).The (I960), at450MeVand2.05BeVbyCurrie,Libby,Wolfgang(1956),2.2 (1962), at190MeVbyBailey(1956),224,300,400,and750HondaLai protons ofenergiesbetweenahundredandseveral hundredMeV,andwehaveassumed that thisratioholdsforallenergies.Thesecross-sections, averagedtoameanvaluefor interactions. Considertheinteractionofanincident particleofmassMiandLorentz in Figure3. factor,7 =1+E/m^withastationarytargetnucleus ofmassm.Suchareaction angular andenergydistributionsofdeuterons helium-3nucleiproducedinnuclear CNO inthecosmic-raynuclearratioof0.45:0.10:0.45 (Comstocketal.1966)areshown (personal communication)indicatethattheHe/H yieldsareoftheorder1.7from emitted intodcos0*around cos0*withLorentzfactorindy*aroundy*. Intermsof p(y,ys*, cos6*)dy*d 0*betheprobabilitythatincenter-of- frameitis may producetwoormore secondaryparticles.Consideroneoftheseparticles andlet t 9 8 © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem The distributionfunctionsF(E,E)canbedetermined frommeasurementsofthe S Fig. 3.—Deuteron-,triton-,andhelium-3-productioncross-sectionsfor^CNOreactions REUVEN RAMATYANDRICHARDE.LINGENFELTER c) KinematicsandSecondaryEnergySpectra KINETIC ENERGY/NUCLEON. Vol. 155 19 69ApJ. . .155. .587R -1 No. 2,1969COSMIC-RAYDEUTERIUMANDHELIUM-3 factor andemissionangleby this probability,thedistributionfunctionF(y,y)maybewrittenas Lorentz factory*whichcancontributetoagivenlaboratory.Since and thaty*,fory7™7c,wherezisthevalueofatwhich mc sr mc mC m 8ctsC2C © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem Fig. 4.—RangesoftheLorentzfactorscentermass, andofthesecondaryparticlein As theincidentenergyincreasesabovethreshold,yand*alsoincreasetheir The dashedcurvesinFigure4areplotsofy*asafunction7,determinedfrom cm mC REUVEN RAMATYANDRICHARDE.LINGENFELTER 2 Q—m?){m?Mi)^ =s 21/ 2{(m —Mt)[mt(mmf)m{?m¿)]}* s ct ^ 2m(m+M) tsr 2 _ Os+Mr)m?—mi Vol. 155 19 69ApJ. . .155. .587R 43 0 21/ 0 satisfy equation(15),y(min)and*min),are obtainedfromequations(5)and(8) The firsttermwillnevervanishsincey*<;the laboratoryLorentzfactorywillgo particle isemittedintothebackwarddirectionincenter-of-massframe,0*=—tt. minimal, laboratoryLorentzfactorisobtainedwheny*=y*,andthesecondary and and aregivenby vanishes when through aminimumwhenthesecondtermvanishes. Thevaluesofyand*which From equation(4)withy/=y*and0*— tt,wefindthatthederivativeofy corresponds toparticlesbeingemittedinthecenter-of-massframeintoforwardor the curvesofy*forreactionsHe(ÿ,d)Heand/>(He,d)HeinFigure4.This once, whereasforMthecurvey*willintersecteveryhyperbolaatleast bola correspondingtoy,regardlessofthenumberparticlesinfinalstate. mined bythesecondintersectionofy*andhyperbolacorrespondingto.Ifthere M andtwoparticlesinthefinalstate,upperlimityisdeter- is determinedbythefirstintersectionofy*withhyperbolacorrespondingto secondary particlewillalwaysbeemittedintotheforwardconewithnon-vanishing equivalent toy>y^C),wherey^isequal\{m/m+m/m¡)andatlargeinci- the twobranchesofhyperbolacorrespondingtoagivenyare[±(y—1)] given 7s,ascanbeseeninFigure4.Dependingonwhethery>ory* dent energiesy()istheasymptoticvalueofLorentzfactorsecondaryparticles, y* willremainalwaysinsidethehyperbolacorrespondingto.Thisinequalityis and, asmentionedabove,forlargey,*~Therefore,if tivistic additionofvelocities,y(<»)isindependenttheincidentenergy,andtherefore For valuesofygreaterthan*,secondaryparticlesmaybeemittedinthelaboratory y(o° ),providedtheyareemittedcloseenoughtothebackwarddirectionincenter- energy. y, emittedinthebackwarddirectioncenter-of-massframe.Becauseofrela- No. 2,1969 cw mc 8 w cm m 8 m t85 tscwm tsmc 8c m 8tm s w8 t8cw w 8t 8cs ct cw cw 8 s ms cm s c s s 0 © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem The lowerlimit,yz,oftherangevalueswhichmaycontributetoagiven Finally, afurtherdistinctionexistsbetweenreactionshavingMgreaterorsmaller When yor The upperlimitony,however,isnotnecessarilyfinite.asymptoticformsof cs t ct8 c 21/i (7m2 17c(7c -1)-7m*[(7t»*)ir| [?_~=0.(15) 7 y* (min)= m 7c (min)= 21/ COSMIC-RAY DEUTERIUMANDHELIUM-3 7s -(T®1)<—+(7s,(14) 2 2 2 M(t +Mr~— Mj) +ÏMi{M—M Mi(M +M—MjMr)Im^M?~ rs 8t 21 2 21 2m\Mí{m? —M){m?Mi)] r 2m[Mí{m —) (m —Mi)} s r (16) (17) 595 19 69ApJ. . .155. .587R 0 3 2 596 ¿-producing reactions represent thekinematiclimitsdeterminedfromequation(4)forcos6*=±1andform £i(min) and£*(<»),correspondingtoyt,7(min),TcC)areshowninTable2. cos 6*.ThesignificanceofthevariousminimaandasymptoticlimitslistedinTable2 the envelopeofpermissiblesecondaryenergiescorrespondingtointermediatevalues energy isshowninFigure5foravarietyofdeuteron-producingreactions.Thecurves may alsobeseeninFigure5. 7c(min), aswellthesecondarylaboratorykineticenergiespernucleon-E(th), well measuredandhasbeenstudiedbyRosenfeld(1954)whoshowedthatthe(§,§) He-producing reactions (Note thatatthresholdy=t-) interaction wasdominantandthatthedistributionthereforegivenby^+cos6*. distorted-wave Bornapproximationwithdiffuse-well nuclearopticalpotentials.For function isthengivendirectlyfromtheangulardistribution andequation(9): and theycanbefittedindetailbythemodelof Smith andIvash(1962)basedonthe and at95MeVbySeloveTeem(1958).These measurementsareshowninFigure6, was measuredat31MeVbyBunchetal.(1964), 55MeVbyHayakawaetal.(1964), Since thisreactionalsoresultsintwobodiesthe finalstate,thedeuterondistribution cs the purposesofthiscalculation,however,wehave fittedthedistributionbyfunctions of theform: 8 8c + 3 16 3 3 13 3 16 8 016 + 43 PP—>¿7T ^a—*He d p 0->¿ pa—^dd p... par-^d 2pn. pa-^d He Pa—H 2p />a—»He pn. O p^d ap—^dd p ap—>d 2pn.. ap—*d He. p °—»{jjgV;' ap—2p ... a/>—»He pn © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem The incidentkineticenergiespernucleon£h,-^mincorrespondingtoyt,Jcz,and For illustration,therangeofpossiblesecondaryenergiesasafunctionincident The center-of-massangulardistributionofdouteronsfromthereaction~H.(p,d)wwas The center-of-massangulardistributionofdouterons fromthereactionHe(ÿ,d)He tc 3 »He d REUVEN RAMATYANDRICHARDE.LINGENFELTER 2 1/2 ^(7,7.) =(1+COS6*){(7c-l)[(7m*)1]}- . (18) P(7, cos6*) 286.5 23 0 29.9 31 7 23 0 24.8 24.8 23.0 29 9 31 7 23 0 24.8 14 3 14 3 24.8 £th 16.2 16 2 16.2 16 2 -1 Kinetic Energy(MeVnucleon) exp [—(1—€cosfl*)/cos flp] cos 0o[l—exp(—2/cos 0)] ’ O 27.7 38.2 40'.2 37.3 36 1 40 2 UÀ Í6 4 16.4 E z TABLE 2 312 4 48*6' 51 6 2Ï.Ï 37*3 29.9 29 9 27.7 23 0 23 0 E(th) 20 69.1 s 14 12 19 0 15 8 14 15 8 14 14. 0 9 0 0 0 0 1 1 1 0 L 1 1 E(min) s Ü.8 63.9 Í2*5 Í3Ü 1.2.2 13 1 10 1 10 1 9.i 6*6 2847.9 1639 0 1639 0 234 230 230 234 234.1 234 622 8 622.8 234.1 230 622 8 622.8 622 8 37 8 37.8 37.8 (19) r-PÉ¡ coLO LO

O^i OLLO

INCIDENT KINETIC ENERGY (MEV/NUCLEON) Fig. 5.—Ranges of the incident kinetic energies per nucleon which can contribute to a given secondary energy for the indicated deuteron-producing reactions. These ranges are determined from the require- ments that cos 0* < 1 and that the deuteron Lorentz factor in the center of mass be less than or equal to its maximum value determined from the conservation of energy and momentum.

Fig. 6.—Angular distribution in the center of mass of deuterons from the reaction He4(/>,d)He3 for various incident energies.

© American Astronomical Society • Provided by the NASA Astrophysics Data System 19 69ApJ. . .155. .587R 4 34 A3 1 1 4 + -1 3+ 43 Lorentz-factor distributionsinthecenterofmassarepeakedbackwarddirection exp (—Fi/Fo),whereEisthetritonkineticenergypernucleonand1/Foequalto uncertainties, thetritonspectrumfrommultibodyptbreakupofHeisindependent equation (18),whereyand*aredeterminedbyequations(5)(8)withMi= exponential distributioninthecenterofmassangle,0*.Weassumed,therefore,thatfor of theincidentneutronenergyandcanbefittedbyasimpleexponentialform: shown inFigure7,normalizedtoaunitintegral.Ascanbeseen,withintheexperimental in thereactionsH^a^HeandHe(/>,He)d. F(7,7) forbothdeuteronsandhelium-3nucleiisdeterminedfromequations(9) where cosdodependsontheincidentLorentzfactor7.Inënergyregionof the multibodyreactions,He(p,p')npd,dd,ptandwHe,probabilitydistribution mass. Assumingthatthecharacteristicenergy,Fo,isindependentofincident Innés (1957)forincidentneutronsof90and300MeV,respectively.Thesedataare in proton-heliumreactionsareejectedasfromthecollisionoffreenucleons.Because In termsofthecharacteristic energy,E,definedabove,cos0isgivenby laboratory distributionfunctionF(y,y)becomes the exponentialdistributioninkineticenergypernucleonmaybetransformedintoan and aroundLorentzfactorswhichareaboutequaltothefactorofcenter tial with1/Fo=0.063(MeVnucleon“)“. 0.27 (MeVnucleon“)“.Similarly,thedeuteronspectrummaybefittedbyanexponen- and oftritonsfromptbreakupHeweremeasuredbyTannenwald(1953) to thereactionH(ÿ,d)7r,andthereforedistributionfunctionsF{y,y^)aregivenby and weassumedthisenergydependenceforallvaluesof7considered. 30-90 MeVnucleon^,cos0ocanbewellfittedby to beproducedpredominantlywithlowenergies.Thiswillhappeniftheangularand 2. IthasbeenshownbyRiddifordandWilliams(1960),however,thatpionsproduced and dtTT*alsobecomesignificant,ascanbeseenfromthecross-sectionsshowninFigure where inpainteractions,theparametereequals —1 and(7/)=y*for< function F(7,7s*,cos0*)isgivenby of mass. M =MeMd,andMr. this, wehaveassumedthat,kinematically,thedHe7r°anddt7rreactionsaresimilar 598 REUVENRAMATYANDRICHARDE.LINGENFELTERVol.155 (19). Theparametereis+1inthereactionsHe(ÿ,d)HeandH^ajHe)^—1 t cm S 0O s mccz (7s*) =7cfor>7c«,andinapinteractions,e =+1,andthemeanLorentzfactor tpyv (7s*) isthesameasinpainteractionsfor totalavailableenergyinthecenter 4 3 © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem These spectraindicatethatinthelaboratoryframemultinucleonfragmentstend The energyspectrainthelaboratoryframeofdeuteronsfrompndbreakupHe Above thepion-productionthresholdmorecomplexreactionsleadingtoJHe7r° Since thisreactionresultsintwobodiesthefinalstate,distributionfunction Using equations(4)and(9)withbothy*y* replacedby(73*),wefindthatthe m F(y,7s) =exp[- 2 P(y,7s*, cos0*) X {cos0o[l—exp(—2/cos0o)][(7cD((7/) ~DP}• cos 0o 2 cos 0o[(7c-1)((7s*YDP cos 0o=exp[9.38(7—1)]>(20) l7s}) exp [—(1—ecos0*)/cos0],*/*\\ ^ Eo cos 0o[l-exp(-2/cos0)]^’ O O 2 ^[(7c -D«7.*>DP ' «(t. —7c(7/» COS 0oJ/ 1/ (21) (23) (22) 19 69ApJ. . .155. .587R -1 _1 3 3 4 4 3 be fittedbyanexponential characterizedby1/E=0.14(MeVnucleon) . Assuming by 190MeVprotonsoncarbonareshowninFigure 8,normalizedforunitintegral.As by equations(22)and(23). energy, thed,t,andHe distributions for^CNOreactionsarethesameas thosegiven can beseen,thesespectrahavethesameformas those showninFigure7,andcanalso that thetritondistribution alsohasthisformandthatEisindependent ofincident d, t,andHeproducedinaareactionswithwithout pionproduction: breakup ofHe. equations (4)-(9),withMi=m,we derivethedistributionfunctionfor 0 0 t8r sumed arbitrarilythatthebreakupproductsofeachHenucleuseffectivelycluster ters” havinganisotropicangulardistributioninthecenter-of-masssystem.Thus,from together allowingustotreattheinteractionasatwo-bodyprocesswithtwo“clus- the multibodyreactionsnpd,dd,ptand/¿Hewhenpionsareproduced. Since Eoisindependentoftheincidentenergy,energydependencecos0ode- No. 2,1969COSMIC-RAYDEUTERIUMANDHELIUM-3599 termined solelyby7andy. C8 3 © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem The measurementsofBailey(1956)thespectra of deuteronsandhelium-3produced Fig. 7.—Energydistributioninthelaboratoryframeofdeuterons andtritonsfromthemultibody There arenomeasurementsofd,t,orHeproductionbyaareactions.Wehaveas- We havealsoassumedthesamedistributionfunction,givenbyequation(22),for 0 10203040506070 DEUTERON KINETICENERGY,E.(Mev/Nucleon) -1 F(y,y) =(t-l)• (24) s d 19 69ApJ. . .155. .587R 43 -1 -1 -1 + reactions, anuncertainty ofthisorderisintroducedinthecalculationfrom thissource because ofthelackexperimental dataonthecross-sectionsandkinematics ofthese probability bythemultibodybreakupreactions of cosmic-rayhelium-4onhydrogen. alone. It shouldalsobenotedthattheaareactions,under theassumptionsmadeabove,con- tribute approximately20percentofthetotaldeuteron productionatallenergies,and, H(He, d)Heandbycosmic-rayprotonsinthemultibody breakupofhelium;deuterons with equalprobabilitybycosmic-rayhelium-4 on hydrogeninthepickupreaction helium-3-production spectra,respectively,forrj= 350MVandRo=0.Ascanbeseen in theenergyrangefromabout70to200MeVnucleon areproducedmainlyinthe from Figure9,deuteronsofenergieslessthanabout 70MeVnucleonareproduced of energygreaterthanabout200MeVnucleon areproducedwithroughlyequal interactions. (§>§) resonancereactionofcosmic-rayprotonson ,H(ÿ,d)7r;anddeuterons using thedemodulatedcosmic-rayspectra,productioncross-sections,andsecondary with increasingt?butisquiteinsensitivetovariationsinR.Therelativecontribution energy distributionsdiscussedabove.Wehaveperformedthesecalculationsforavariety nuclei asafunctionofsecondarykineticenergypernucleoncannowbecalculatedby modulating parameters.Forillustration,weshowinFigures9and10thedeuteron- of modulatingparametersrjandi?o,ingeneraltheabsoluteproductionincreases of thevariousproductionreactions,however,isnotstronglydependentonassumed 600 0 © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem Fig. 8.—Energydistributionsinthelaboratoryframeofdeuteronsandhelium-3nucleifromproton- The rateofproductionpergraminterstellarmaterialdeuteronsandhelium-3 REUVEN RAMATYANDRICHARDE.LINGENFELTER Deuteron andHelium-3KineticEnergy,E(Mev/Nucleon) $ d) ProductionSpectra Vol. 155 19 69ApJ. . .155. .587R 43 3 4 -1 where uisthesecondary cosmic-rayequilibriumdensity;qisthesecondary production rate calculatedabove,per gramofinterstellarmaterial;pisthedensity of interstellar reaction H^He,He)d,asarethelow-energydeuterons,whilehelium-3nucleiof account leakagefromtheGalaxy,ionizationlosses, andnuclearbreakupofthesecond- nuclei attheEarth.Theequilibriumdensitiesof thesenucleiintheGalaxycanbeob- Rq ^0. tained bysolvingasteady-statecontinuityequation inenergyspacewhichtakesinto in theGalaxyandthen,takingintoaccountmodulation, thefluxofdandHe calculated above,wemaynowdeterminetheequilibrium densityofthesesecondaries Ro —0. in thereactionH^He,t)2p,tritiumdecayingtohelium-3.Theareactionsalso higher energiesareproducedpredominantlybythebreakupofcosmic-rayhelium-4 nucleon areproducedprincipallybycosmic-rayhelium-4onhydrogeninthepickup produce about20percentofthehelium-3andintroduceacorrespondinguncertainty aries. in Table1.Thesespectrawerecomputedforademodulatedcosmic-rayintensitywithrj=350MVand No. 2,1969 in Table1.Thesespectrawerecomputedforademodulatedcosmic-rayintensitywith77=350MVand in thetotalproduction. 8 s © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem Fig. 10.—Helium-3-productionspectrapergramofinterstellarmaterialforthevariousmodesshown From Figure10wealsoseethathelium-3nucleiofenergieslessthanabout40MeV From theratesofdeuteriumandhelium-3production intheinterstellarmedium, Fig. 9.—Deuteron-productionspectrapergramofinterstellarmaterialforthevariousmodesshown III. DEUTERONANDHELIUM-3FLUXESATTHEEARTH COSMIC-RAY DEUTERIUMANDHELIUM-3 Fig 910 + ^x7 slb°-J-w-®' u .vd[dE“I s 601 (25 19 69ApJ. . .155. .587R -3 t,T 2 -1 \dE/dt\ loss duetoionization;and\d/visthemeanlifetimeagainstnuclearbreakup.Thesolu- material ingcm;tlistheleakagelifetimefromGalaxy;dE/dtrateofenergy was producedatatimetinthepast.Intermsofthisprobability,numberparticles where ristheeffectivelifetimeagainstleakageandnuclearbreakupgivenby satisfy thefollowingrelationship: where theenergiesEand£',timeintervaltenergylossratedE/dthaveto of energyindEaroundEwhichwereproducedthetimeintervaldt/,with per unittime,/>(/),suchthatP{t)dtistheprobabilityanobservedparticle(at/=0) Equation (28)canbeintegratedbyassumingthattherateofparticleproduction energies indE'around£',isgivenby to equation(26)givenabove,canbeobtainedbyintroducingaprobabilitydistribution 602 tion ofthisequationcanbewrittenas diffusion causedbyscatteringoffmagneticirregularities. Theleakagelifetimetx, formly distributedinatrappingvolumeofradius a,fromwhichcosmicraysescapeby e~. Suchaformisvalidforthesuddenlossesresulting fromnuclearbreakup,andit expression, however,dependsonavarietyoffactors,suchasthespatialdistribution pq{E) istimeindependent.BychangingthevariableofintegrationfromttoE',wefind dE/dt duetoionizationcanbereplacedbypvdE/dR, whereR(E)arechargedparticle mean freepath.Forsuchanexponentialdistribution, equation(30)becomesequivalent mentioned above,maythenbewrittenasrz,= a/(2Xz,fl) whereXlisthescattering, might beagoodapproximationtophysicalsituation inwhichthesourcesareuni- properties ofthetrappingvolumeparticles. tribution isreplacedbythemoregeneralexpression,P{t).Thefunctionalformofthis Equation (30)issimilartoequation(26)givenabove,exceptthattheexponentialdis- that theequilibriumdensityu(E)isgivenby ranges givenbyBarkasandBerger(1964).The deuteron-breakup cross-sectionvaries the sources,natureofparticlepropagationfromsourcestoEarth,and with energybutsincewe findthatthelifetimerisdeterminedprincipally byleakage, we havetakenittobea constantof170mbwhichcorrespondstotheaverage value to equation(26). over theenergyrange10-100 MeVnucleon(VanOers1963;Davisonetal. 1963).The 8 © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem A moregeneralexpressionforwhichunderanappropriatesimplificationreduces In thepresentstudywehaveassumedasimple exponentialdistribution,P(t)= We nowconsiderthenumericalevaluationofequation (26).Theenergylossrate REUVEN RAMATYANDRICHARDE.LINGENFELTER “•® ■-Rwi“>>(-/- u{E) = s du{E) =pqE')^P(t)dt,(28) s \dE/dt\ P t = fdE'q(E')p(f a E \ E 4 \dE/dt\‘ dE" r I dE/dt/’ dE" \ Vol. 155 (26) (29) (27) (30) r-p¿¡ coLO LO No. 2, 1969 COSMIC-RAY AND HELIUM-3 603

helium-3-breakup cross-section is much smaller yet and hence was neglected. The break- l LOO^i up mean path length, Xd, is given by X¿ = {(TdNap)~ , where Na is Avogadro’s number. OL Introducing a mean path length against leakage, x = pvri, we can rewrite equation (26) as

m = ^ / iE,.(E) exp [- /(i + „N) § dE'] , (31)

where ys(E) is the secondary cosmic-ray equilibrium intensity in interstellar space. We have evaluated equation (31) for a variety of production spectra, qs(-E), corresponding to a range of values of the modulating parameters rj and Ro, defined above. We have

DEUTERON KINETIC ENERGY (MEV/NUCLEON) HELIUM-3 KINETIC ENERGY (MEV/NUCLEON ) Fig 11 Fig 12 Fig. 11.—Observed intensity of deuterons, together with the calculated deuteron intensities at the Earth for various values of interstellar path lengths and modulating parameters. Fig. 12.—Observed intensity of helium-3 nuclei, together with the calculated helium-3 intensities at the Earth for various values of interstellar path lengths and modulating parameters.

then obtained the d and He3 intensities at the Earth by modulating the resultant equilib- rium spectra in interstellar space with the same modulating function as that used in the demodulation process. The results of these calculations, for 7?0 = 0 and for several values of the mean path length x and the modulating parameter rj, are shown in Figures 11 and 12 for deuterons and helium-3 nuclei, respectively. The net effect of the process of demodulation and the subsequent modulation, de- scribed above, is a decrease of the secondary intensity, because secondaries of a given energy are produced by primaries of higher energy which are therefore less modulated than the secondaries which they produce. This can be seen in Figures 11 and 12 from a comparison of curves A and C, for zero modulation and modulation characterized by 7] = 0.35 BV. Also shown in Figures 11 and 12 are measurements at the Earth of deuter- ons and He3 nuclei.

© American Astronomical Society • Provided by the NASA Astrophysics Data System 19 69ApJ. . .155. .587R -2 3 3 3 3 3-1 3 3 -2 -2 3-2 23 z 3 3 -1 -13 -1 fitting curveCandthelimiting curvesBandDinFigures1112. form exp(—r)/Rß)characterized byrj=0.35±0.15BV.Thiscanbeseen bythebest- which bestfittheobservedintensities.Within experimentaluncertainties,wefind an energy-independentpath length,x=4±1gcm,andaresidualmodulation ofthe that theavailabledataon boththedeuteronsandhelium-3nucleicanbe best fittedby and arangeofvaluesxrj,therebydetermining thevaluesoftheseparameters of about600MV,wehavecalculatedthedand He spectraattheEarthforR=0, spectrum oftheresidualsolarmodulation. parison offuturemeasurementsthe¿/Heratio withthecalculationsdescribedin is about1.3,whereasthemeasuredratioatsameenergypernucleonperhapsas velocity aswellrigiditydependent.Becauseof this, wewishtosuggestthatthecom- residual modulationbutalsothatabovearigidity ofabout600MVthismodulationis dependent modulationwouldincreasethecalculated ¿/Heratio.Thus,thecomparison have ahigherrigidityandhencearelessmodulatedthanHenuclei,suchrigidity- calculated ¿/Heratioat,forexample,50MeVneucleonzeroresidualmodulation resented byapowerlawinRwithvaryingspectralindex,rangingfromabout—0.5 planetary fieldvariationsindicatesthatthefunction/(J?),definedabove,canberep- the presentpapercoulddeterminemoreaccurately thechargedependenceandenergy great as6.Suchadifferencecouldresultfrommodulationoftheformexp(—rj/Rß), deuteron andhelium-3data.AscanbeseenfromcurvesAinFigures1112,the at lowrigiditiesto—2highrigidities.Withintheexperimentalerrors,thiswouldbe According toJokipiiandColeman(1968)theobservedpowerspectrumofinter- as discussedabove,therearesignificantdisagreementsamongthevariousexperimenters by theobservedgradientincosmic-rayintensitybetweenorbitsofEarth as givenbyequation(2)forR>Ro.Sinceatthesameenergypernucleon,douterons 600 MV.Sucharigidity-dependentmodulationisindeeddemandedbytheobserved consistent withthemodulatingfunctiongivenbyequation(2)jRooforder of thedandHemeasurementswithcurveAsuggests notonlythenecessityofsome sidual interplanetaryfieldmodulationatsolarminimum.Thisconclusionissupported leakage pathlength.Ontheotherhand,comparisonofcurvesB,C,andDwith as totheenergydependenceofobservedcosmic-rayspatialandtemporalvariations. and (O’GallagherSimpson1967;O’Gallagher1967). d andHedatasuggeststhatthesespectracanbeunderstoodeasilyintermsofare- clude thatthemeasuredspectraareprobablynotresultofanenergy-dependent lated equilibriumisdeterminedprincipallybyionizationlosses.Ascanbeseenfrom nuclei becomemuchsmallerthanthe4gcmleakagepathlength,andthuscalcu- intensities (O’Delle/a/.1966;Biswase/aZ.1967;DennisFreierandWadding- energy, xwouldhavetobeabout0.1gcm.Thisrequiresaverystrongvariationin where therangeofanHenucleusis1gcm,thisdiscrepancyamountstomorethan the calculatedequilibriumfluxisdeterminedprincipallybyleakagefromGalaxy, length ofabout4gcm“andzeromodulation,thecalculateddHeintensitiesat the pathlength(oforderxE)whichseemsratherunlikely.Wethereforecon- and theintensityisdirectlyproportionaltox.AtlowerenergiesrangesofdHe an orderofmagnitude,andthusintoaccountfortheobservedintensityatthis curve Aattheselowerenergies,thecalculateddandHeintensitiesaremuchgreater against ionizationlossesisgreaterthanthevaluesofxinvestigated.Attheseenergies ton 1968).Asensitivedeterminationofthemeanleakagepathlength,x,canbemadeonly sidual modulation.AscanbeseenfromcurveA,wefindthatforacosmic-raypath 604 than themeasuredvalues(Fanetal.1966a,b).Forexample,at50MeVnucleon, at thesehigherenergies(>150MeVnucleon),wheretherangeofdandHenuclei energies greaterthanabout200MeVnucleonareingoodagreementwiththemeasured 0 © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem Since thepresentcalculationsdonotdependon processes occurringbelowrigidities The rigiditydependenceofthisresidualmodulation,however,isnotwellknown,and In comparingthesedatawithourcalculations,wefirstconsiderthecaseofzerore- REUVEN RAMATYANDRICHARDE.LINGENFELTER Vol. 155 19 69ApJ. . .155. .587R -1 -2 3 3 particle spectrumliesabove thiscurve.Ifsignificant,changeofspectral shapemay modulated protonandalpha-particle spectraareconsistentwithacosmic-ray source calculations cannotbemeaningfully applied.Above200MeVnucleon boththede- per nucleoncorrespondingtoarigidityof600MV, belowwhichtheresultsofpresent which isapowerlawin total energy(curveIII)whereasbelowthis thealpha- solar-minimum protonandalpha-particlespectra withthemodulatingfunctiongiven respectively, for#=4gcm.Forcomparison wehavedemodulatedtheobserved from ahomogeneouslydistributedgalacticsource withagivenenergyspectrum.Such are alsoshowninFigure15.Thesedemodulatedcurves extenddowntokineticenergies rigidity, andtotalenergypernucleonareshown in Figure15bycurves/,//,andIII, spectra resultingfromsourceswhicharepower lawsinkineticenergypernucleon, the interstellarequilibriumintensityofprimary protons andalphaparticlesresulting above (77=0.35BV),andtheresultantinterstellar protonandalpha-particlespectra at theEarthforvariousvaluesofinterstellarpathlengths andmodulatingparameters. the Earthforvariousvaluesofinterstellarpathlengthsandmodulatingparameters. rigidity dependentandisoftheformexp(—rj/Rß). ¿/He ratioscanbeexplainedbyanenergy-independent,path-lengthdistributioncoupled However, ascanbeseenfromthesefigures,ourcalculationsshowthatbothHe/Heand and residualinterplanetarymodulationontheprimarycosmic-raysourcespectra.Using with aresidualsolarmodulation,whichaboveabout600MVisbothvelocityand Figures 13and14.Biswasetal.(1968)haverecentlyconcludedthattheenergydepend- No. 2,1969 the cosmic-raypropagationmodelassumedinprecedingcalculations,wecompute ence oftheHe/Heratiocanbeaccountedforonlybyanenergy-dependentpathlength. 3 3 © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem Fig. 14.—Observedratioofhelium-3toheliumnuclei,togetherwiththecalculatedHe/Heratios Fig. 13.—Observedratioofdouteronstoheliumnuclei,togetherwiththecalculated¿/Heratiosat The ¿/HeandHe/Heratiosderivedfromthepresentcalculationsareshownin Finally weshallexaminetheimplicationsofinterstellarcosmic-raypathlength COSMIC-RAY DEUTERIUMANDHELIUM-3 Fig 13Fig.14 605 19 69ApJ. . .155. .587R 2 interactions ininterstellarspace. reflect eitherasimilarchangeinthespectraofcosmic-raysourcesormaybecaused by avarietyofeffectssuchassolarorstellaraccelerationandproductionnuclear 606 energy pernucleon,respectively,forx—cm-. represent theequilibriumspectrawhichwouldresultfromhomogeneouslydistributedcosmic-ray Biswas, S.,Lavakare,P.J.,Ramadurai, S.,andSreenivasan,N.1967,Proc.IndianAcad. 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