198 7ApJS. . .64. .2 6 9G The AstrophysicalJournalSupplementSeries,64:269-304,1987May <■' 1987.TheAmericanAstronomicalSociety.Allrightsreserved.PrintedinU.S.A. relative amountsofnuclearfragmentationandradioactive which reachthesolarsystemprovideinformationon decay. Therefore,propagationmodelswhichsatisfythe ob- major physicalcharacteristicmodifyingboththecomposition stellar mediumarecentraltoreachinganunderstanding of server. Thispath-lengthdistribution(PLD)determines the and thespectraisdistributionofpathlengthsmatter gation throughinterstellarmatterandmagneticfields. A sources andaccelerationofcosmicrays,theirpropa- the originofcosmicradiation. servational constraintsofboththecosmicraysandinter- traversed bythenucleibetweentheirsourcesand ob- topic compositionandenergyspectraofthecosmic-raynuclei fornia. 2 The analysisandinterpretationoftheelementaliso- Presentaddress:TheAerospace Corporation,LosAngeles,Cali- Also DepartmentofPhysics;John SimonGuggenheimFellow,1985. -1 is notthecontrollingprocess.Thesub-Fe/Feratiosensitivetorelativeabundanceofshortpathlengthsin heliospheric propagation.Thedetaileddescriptionofthepropagationmodelandinputparametersisgiven weighted-slab techniqueinGalacticpropagationandasphericallysymmetricmodelofsolarmodulation The dataarecomparedwiththeresultsofafullyenergy-dependentpropagationcalculationusing including UniversityofChicagodatafromtheIMP8satelliteatlowenergyandacollectionhigher propagation. Itisseenthatatlowenergiesthepredominantfactorlimitingaccuracyindeterminationof may beindicativeofmattertraversedinthevicinitycosmic-raysourceregionsbeforeGalactic energy dependencesetslimitsonthenatureofcosmic-raydiffusionand,atlowenergies,indicatesthat models fortheGalaxy,butdetailedfitrequiresalargevelocityGalacticwind(>20kms).This is concludedthatthemeanfreepathinGalacticpropagationmustbefullyenergy-dependent,withof in theAppendix.FromanalysisofB/Cratio,whichisbest-measuredsecondary-to-primaryit energies hasbeenusedtostudytheenergydependenceofsecondary-to-primaryratiosinGalacticcosmicrays. distribution isnotpurelyexponentialbutdepletedinshortpathlengthsatlowenergies.Thisdepletion decreasing withincreasingenergyabove1GeVpernucleon.Suchbehaviorisconsistentdynamichalo shown tobeenergy-dependent,beinglargestatlowenergiesanddecreasingwithincreasingenergy.Thisresult an exponentialpath-lengthdistributionincreasingwithenergybelow1GeVpernucleonand the cosmic-raypathlengthisscarcityofpreciselymeasuredcrosssectionsfornuclearfragmentation. the path-lengthdistribution,andsimultaneousanalysisofbothB/Csub-Fe/Ferevealsthat Subject headings:cosmicrays:abundances—particleacceleration A compilationofdatafrommeasurementscarriedoutmostlyattimessolarminimummodulationand © American Astronomical Society • Provided by theNASA Astrophysics Data System COSMIC-RAY PROPAGATIONINTHEGALAXYANDHELIOSPHERE: I. INTRODUCTION THE PATH-LENGTHDISTRIBUTIONATLOWENERGY Department ofPhysicsandAstronomy,LouisianaStateUniversity,BatonRouge Department ofPhysics,WashingtonUniversity,St.Louis 1 Received 1986August25;acceptedNovember4 M. Garcia-MunozandJ.A.Simpson Enrico FermiInstitute,UniversityofChicago T. G.GuzikandJ.P.Wefel 2 S. H.Margolis ABSTRACT AND cosmic-ray nucleiwhichoriginate inthesources(primary of the“light”(L)elementsLi,Be,andB,werefarinexcess of propagation intermsofanenergy-dependentpath-length planetary mediumaswellrecent,newmeasurements of wind. Thisstudyusessatellitemeasurementsintheinter- Peters 1948).Theseobserved excessesareproducedinspalla- rays beganwiththediscoverythatcosmicradiation con- distribution. nuclear crosssections,andleadstoadescriptionofcosmic-ray cosmic-ray confinementandpropagationintheGalaxy relativistic velocities,anditsimplicationsformodelsof path-length distributionofcosmicraysatlowenergies,below their solarsystemabundances(Freieretal1948;Bradt and abundances ofsomecosmic-rayelements,inparticularthose Galactic halo,includingtheeffectsofapossible tion reactionsresultingfrom the interactionofpropagated tains nucleiheavierthanhydrogen,andthattherelative The informationontheinterstellarpropagationofcosmic The presentpaperstudiestheenergydependenceof 198 7ApJS. . .64. .2 6 9G 3 10 2 field ofelementalandisotopicmeasurements wasIMP1,launchedin Space FlightCenterandtheUniversity ofChicago. 1963, withexperimentscapableof isotopicseparation,bytheGoddard 270 isotopic composition.Thesatelliteexperimentsimposed new possible toextendcosmic-raymeasurementslowenergies rigidity. matter traversedbytheobservedcosmicrays,e.g.,decreasing constraints onpropagationmodels.Forexample,theisotopic nucleon thesecondary-to-primaryratiosdecreasewithin- and obtainhigh-resolutionmeasurementsofelemental and as aninversepowerlawintotalenergyormagnetic et al.1973).Thisfactisinterpretedtobetheconsequenceof creasing energy(Juliusson,Meyer,andMuller1972;Smith exponential inform.AtenergiesgreaterthanafewGeVper niques formodelingcosmic-raypropagationandtherequired leaking outoftheboundariescontainmentregion(e.g., be changedfromadeltafunctiontocontinuousfunction. of about20millionyears.Thisresultsinanaverageinter- component ofthecosmicrayshadaconfinementlifetime abundance ofradioactiveBeshowedthatthelow-energy and Tsao1984.)Inaddition,ithasbecomeapparentrecently atomic andnucleardatahasbeengivenbyLetaw,Silberberg, agation isaverycomplexprocess.(Adescriptionoftech- are assumedtobeinequilibrium(Davis1960;Cowsiketal Various formswereinvestigated,includingGaussian(e.g., particular theratioofScthroughMn(sub-Fe)secondaries satellites formeasurementsininterplanetaryspacemade it the decreasewithincreasingenergyofmeanamount that thePLDisneitherindependentofenergynorapure 1968; GloecklerandJokipii1969). a “leakybox”),andinwhichcosmic-rayproductionloss function (Fig.la)resultsfrompropagationmodelsinwhich nent inthecosmicradiationatlowenergies(e.g.,Comstock, of sources(e.g.,Hayakawa,Ito,andTerashima1958;Davis necessitates thattheypropagatethroughaslabofmaterial cosmic rays.Cosmic-rayLi,Be,andBresultfromthespalla- Balasubrahmanyan, Boldt,andPalmeira1965)exponen- Fan, andSimpson,1966;Comstock1969).ThePLDhadto material couldnotsimultaneouslyreproducethemeasured confinement andpropagationassumedthis“slab”distribu- and O.Ifallcosmic-raynucleiareassumedtotraversethe the cosmicrayshaveasmallandconstantprobabilityof tial (e.g.,Cowsiketal1967).Anexponentialdistribution for theobservationsunlesstherewasmorethanonecompo- 1983). Simpleslabmodelsofpropagationcouldnotaccount L/M andsub-Fe/Feratios(Garcia-MunozSimpson and statisticalaccuracy,othersecondary-to-primaryratios,in approximately 6gcm'(“slab”PLD).Earlymodelsof same amountofinterstellarmatter,theobservedL/Mratio tion ofmainlythecosmic-ray“medium”(M)elementsC,N, cosmic rays)withinterstellarmatter,andarecalledsecondary 1970; seereviewsbyShapiroandSilberbergSimpson to Fe,weremeasured,anditbecameclearthatasingleslabof 1960; GinzburgandSyrovatskii1964). tion ofpathlengthstogetherwithahomogeneousdistribution 3 Thefirstspacecraftinstrumentation whichopenedtheexperimental Beginning intheearly1960s,introductionofEarth In recentyearsithasbecomeclearthatcosmic-rayprop- As cosmic-raymeasurementsimprovedinbothresolution © American Astronomical Society • Provided by theNASA Astrophysics Data System GARCIA-MUNOZ ETAL. -3 Munoz etal.(1979,1981a,1984)inaseriesofpapersshowed Hsieh 1970). vection, diffusion,andadiabaticdeceleration(e.g.,Parker high-energy datawhenitwas proposedthatadepletionof low energiesthemeanpathlengthdecreaseswithdecreasing new propagationalgorithm,itisshowninthispaperthat at confirmed byotherinvestigators(OrmesandProtheroe1983). distribution isalsoenergy-dependent.Thisconclusion was at lowenergies(e.g.,Fan,Gloeckler,andSimpson1965,1966; of heliosphericpropagation—includingmechanismscon- (PLDs) studiedinthispaper:(a)pureexponentialPLD;(b)zero-short Using anup-to-datecompilationofexperimentaldataand a origin (e.g.,Fan,Meyer,andSimpson1960)thatamodel of Earth.Itwasshownthatthemodulationisheliosphericin short pathlengthsintheexponential distributionisneededto energy. that atenergiesbelow1GeVpernucleon,thepath-length 1965; Hsieh1970)—couldexplaintheobservedspectralslopes and spectraofthelow-energycosmicraysarrivingatorbit truncated exponentialPLD;and(c)doublePLD. propagation paths(e.g.,Garcia-Munoz,Mason,andSimpson gations showedthatsolarmodulationmodifiestheintensity stellar matterdensityofonly-0.2atomcmalongthe and Greiner1980,referencestherein).Theearlyinvesti- 1911a\ Garcia-Munoz,Simpson,andWefel1981;Wiedenbeck Taking intoaccounttheeffectofsolarmodulation,Garcia- Claims thatthePLDisnota pureexponentialstartedwith Fig. 1.—Schematicrepresentationofthepath-lengthdistributions Vol. 64 198 7ApJS. . .64. .2 6 9G ratio ispredominantlysensitive tothelongpathlengthsand cosmic-ray confinementand propagationconditions.Inpar- Appendix. input parametersusedintheanalysisisrelegatedto the sub-Fe/Feratioispredominantly sensitivetotheshort on theanalysisofcosmic-raypath-lengthdistribution. The detaileddescriptionofthepropagationmodeland ticular, weusetheB/Cand sub-Fe/Fe ratios,sincetheB/C Garcia-Munoz etal.1983,1984).Thepresentpaperfocuses it isconcludedthatthemeanpathlengthdecreaseswith No. 1,1987 tion utilizessecondary-to-primaryratiosasprobesof the and Wefel1980;Guziketal.Dwyeretai1981; Munoz etal.1975,1979,1981a,¿>;Garcia-Munoz,Simpson, aspects ofthisworkhavebeenpresentedpreviously(Garcia- els ofcosmic-rayconfinementandtransportintheGalaxy. discusses theimplicationsoftruncationfordifferentmod- of confinementandpropagationcosmicraysintheGalaxy. The energydependenceofthePLDunderassumption parameters usedinthesecalculations,withspecialemphasis sources andtheroleofcosmicraysinourGalaxy.Some forms ofthisdepletionarepresentedin§VIII,andIX deficiency decreaseswithincreasingenergy.Thefunctional sources oferrorinthecalculations.SectionVIisdevotedto decreasing energybelowabout1GeVpernucleon,andthe modulation modelandparametersappliedtothisanalysis. current understandingofsolarmodulationandpresentsthe on thenuclearcrosssections.SectionIVsummarizesour presents thecompilationofexperimentaldataonsec- ing inmagnitudewithincreasingenergy. paper confirmsthisdeficiencyandfurthermoresuggestsa length decreasestozero,asinFigure1c. tinuing programtounderstandthenatureofcosmic-ray Sections XandXIcontainadiscussiontheconclusions. the PLDisdeficientinshortpathlengthsandthatthis short pathlengthsinthePLD.Itisfoundthatatlowenergies strength ofthisconclusionisinvestigatedbyconsideringthe ondary-to-primary abundanceratiosusedinthisstudy.Sec- resolution ofthetruncationcontroversy:itisshownthat Section VIIstudiestheproblemofrelativeabundance the studyofimplicationsabovefindingsformodels that thePLDisapureexponentialpresentedin§V,where deficiency ofshortpathlengthsisenergy-dependent,decreas- Using thebestsetofparameterscurrentlyavailable,this depletion canberepresentedbyacutoffoflowpathlengths explain simultaneouslybothsecondary-to-primaryratiosL/M tion IIIdealswiththepropagationcalculationsandinput the PLDisdeficientinshortpathlengthsatlowenergy. Garcia-Munoz etal.(1979,1981¿z,¿>,1983,1984)indicatethat troversy. Thelow-energysatellitestudiesintroducedby tion,” inanexponentialPLDhasbeenthesubjectofcon- the relativeabundanceofshortpathlengthsas at X,asillustratedinFigure1Z?,orbyagradualdecrease and sub-Fe/Fe(e.g.,ShapiroSilberberg1970).This c This investigationofthecosmic-raypathlengthdistribu- The resultsandanalysisreportedherearepartofacon- The presenceorabsenceofshortpathlengths,“trunca- The organizationofthispaperisasfollows.SectionII II. SECONDARY-TO-PRIMARYABUNDANCERATIOS © American Astronomical Society • Provided by theNASA Astrophysics Data System COSMIC-RAY PROPAGATION 710 cross sections,weusetheaccumulated dataforBe/C,along . Inaddition,twoofthe berylliumisotopes(Be,e) analysis. Nevertheless,since haswell-studiedpartial are radioactive,andtheirdecay mustbeincludedinthe of berylliumaresignificantlypoorerqualitythanthose of production crosssectionshavebeenstudiedextensively(e.g., Raisbeck andYiou19756),butthecosmic-raymeasurements knowledge ofthecrosssectionsforproduction the secondary species.Forexample,theberylliumisotopepartial relatively lowaccuracyofthemeasurementsorbypoor general, theusefulnessofmostratiosislimitedby the e.g., Li/C,Be/C,F/Ne,butnoneofthesehavebeenmea- employed inplaceoftheB/Cratiothistypeanalysis, sured inthecosmicradiationaswellB/Cratio. In ment. ratios obtainedfromtheUniversityofChicagoIMP8experi- reduced uncertaintylimits.TheBe/C,sub-Fe/Fe,Sc/Fe, N/O ratiosareidenticalwithpreviousresults,but previously fortheIMP8experiment.TheB/C,Li/C,and systematic uncertaintiesareincluded,withintheuncer- and V/Feratiosreportedhere,forwhichbothstatistical primary ratios,thefullenergyintervaldataonupperpart tainties oftheearlierresultsandrepresentmostaccurate of thetableareusedindataplots. in Figure2andforN/O8.Forothersecondary-to- lower portionofthetablecontainsdataplottedforB/C interval usinganenergycalibrationderivedfromthe response ofthesolidstatedetectorsininstrument.The ratios, theresultsobtainedbysubdividingfullenergy The lowerpartofthetableshows,forB/CandN/O portion ofthetablegivesrelevantratiosderivedfor ergy intervalnormalizations,andbackgrounduncertainties. has thegreatestaccuracy,anduncertaintiesquotedin- been describedbyGarcia-MunozandSimpson(1979) clude statisticalerrors,energycalibrationuncertainties,en- full energyintervalcoveredbytheCsltotal-energydetectorin ment ontheIMP8satellite{filledcircles).Thisdatasethas most precisedataarefromtheUniversityofChicagoexperi- 19776). Thisinterval,definedbyrange-energyrelationships, the instrument(seeGarcia-Munoz,Mason,andSimpson over aboutthreedecadesinenergy.Atthelowestenergies Simpson (1983)andisgivenforreferenceinTable1.Thetop energy pointsweremeasuredinthe1980s.The1974-1978 points arederivedfromalargenumberofdifferentexperi- production oftheirsecondariesarerelativelywellknown. energies belowabout1GeVpernucleon.Someofthehigher ments performedmainlyinthetimeperiod1974-1978at path lengthsinthePLD.Theseratioshavebeenwellmea- have beensuppressedforlegibility. by anapproximatelyconstantlevelofsolarmodulation(see includesthelastsolarminimumandwascharacterized sured inthecosmicradiation,andcrosssectionsfor and sub-Fe/Fe{bottom)ratiosasafunctionofenergy.The 2. Theenergyintervalscorrespondingtoeachmeasurement § IV).OnlydifferentialmeasurementsareincludedinFigure There areothersecondary-to-primaryratioswhichcouldbe The resultsinTable1areanupdateofreported The B/CdatainFigure2showreasonableconsistency Figure 2showsacompilationofdatafortheB/C{top) 271 198 7ApJS. . .64. .2 6 9G 2 parameter of490MV. The meansoftheexponentialPLDs rangefrom4to12gcmasindicatedonthefigure.Thecalculations includesolarmodulationwitha {b) Measurementsarefromthefollowing: dent exponentialpath-lengthdistributions.(¿0Measuredvaluesarefromthefollowing: Fig. 2.—Compiledmeasurementsof(¿/)theB/Cand(b)Sc-fTi+V+Cr+Mn/Feratios,comparedwithcalculatedresultsforenergy-indepen- © American Astronomical Society • Provided by theNASA Astrophysics Data System 4 Thiswork(seealsoGarcia-MunozandSimpson1979) O Scarlett,Freier,andWaddington (1978) Q Younga/.(1981) Q Engelmannetal.(1981) 0 DwyerandMeyer(1981) H Maehletal.(1977) □ Lundetal.(1975) 0 Buffington,Orth,andMast(1978) 4 Orthetal.(1978) ^ Mewaldtetal.(1981) © CaldwellandMeyer(1977) O Dwyer(1978) H Fishercta/.(1976) O Juliusson(1974) 3 DwyerandMeyer(1981) • Garcia-MunozandSimpson(1979) \2 Hagen,Fisher,andOrmes(1977) v Israeletal.(1979) ^ Webber,Damle,andKish(1972) a. LezniakandWebber(1978/?) ▼ Benegasetal.(1975) ^ Julliot,Koch,andPetrou(1975) ^ Webber(1982) ^ Webber,Damle,andKish(1972) a. ChappellandWebber(1981) ▼ Engelmanna/.(1981) v Lundera/.(1975) E Simon(1980) A Webberc/i//.(1977) x Fisherö/.(1976) □ Maehletal.(1977) Webber, Kish,andSimpson(1979) Lezniak andWebber(1978¿/) 198 7ApJS. . .64. .2 6 9G 54 54 There maywellbesourcecomponentsofMn,Cr,andTi, Appendix andKochetal.1981a).Thepartialproduction lated results.TheisotopeMnhasanalloweddecaychannel which, iftheyhavetherelativeabundanceofsolarsystem measure, andthatitisnotapuresecondary-to-primaryratio. primary ratio),asapartiallyindependentcheckfortheresults with Li/CandN/O(amixedsecondary-plus-primary/ primary ratioswouldbeSc/FeandV/Fe,sinceasolar (about 3%;Cameron1981),makeacontributiontothecalcu- negligible contributiontothe finalcalculatedratio.Themain results (e.g.,Perron1976;Dwyeretal.1981; and used toanalyzecosmic-raydata,butoftenwithconflicting cross sectionsfromironspallationhavebeenstudied and path lengthsinthePLD,sincetotalinelasticcrosssection derived fromthedetailedanalysisofB/Cratio. compiled cosmic-raydatafor whichthesystematicuncertain- difficulty withtheSc/Feand V/Feratiosisthequalityof Koch 1981;Westfalletal.1979a;Webber1983; ties arestilllarge.Nonetheless, wewillusetheSc/Feand system sourceabundance for theseelementsproducesa Raisbeck etal.1979).Themostrehableheavysecondary-to- to Feforwhichthehalf-lifeisverypoorlyknown (see for ironisabout3timesthecorrespondingcrosssection per nucleon.Atlowenergies,onlytheIMP8point{filled do notcoveraslargearangeinenergytheB/Cdata,and . Thecollecteddataforthesub-Fe/Feratio(Fig.2b) tive thantheB/Cratiotorelativeabundanceofshort diamond'. Table1)hasbeenreported,andthispointisplotted energy dependenceofthisratiofrom~0.5toabout10GeV measurement. However, theavailabledatadodefine,reasonablywell, there islessagreementamongthedifferentmeasurements. showing theenergyinterval,87-398MeVpernucleonof Energy interval(MeV/pernucleon). Mean energy(MeV/pernucleon)... Ratio 130.7. 113.9. 46.7. 63.4. 29 .. 96.9. 80.1. The sub-Fe/Feratiohasthedrawbacksthatitisdifficultto The sub-Fe(Sc+Ti+VCr+Mn)/Feratioismoresensi- (MeV pernucleon) © American Astronomical Society • Provided by theNASA Astrophysics Data System Parameter (MeV pernucleon) Element RatiosfromtheUniversityofChicagoExperimentonIMP8Satellite 122.4- 139.0 105.4- 122.4 71.8-88.5 38.4- 5.1 88.5- 105.4 55.1-71.8 B/C 21-37 kE 0.12 +0.01 29-114 Li/C 72 B. ResultsObtainedbySubdividingInterval COSMIC-RAY PROPAGATION 1974 January-1978October 0.25 +0.02 0.25 +0.02 0.22 +0.02 0.20 +0.02 0.21 +0.03 0.24 +0.02 0.20 +0.02 0.053 +0.01 A. ResultsforFullInterval 39-151 Ratio Be/C 95 TABLE 1 0.23 +0.2 38-139 12 B/C bution, weusetheweighted-slabtechnique(Fichteland propagation calculation.Startingwithasetofrelativeabun- inelastic crosssections.Figure3showsanexampleof the integrated overanassumedPLDtoobtaintheparticledensi- common sourceenergyspectrum,thedifferentisotopicspecies dances ofelementsandisotopesatthesourceassuminga Reames 1968)inwhichthePLDisaninputparameter with twoexcitationfunctions. Thedashedcurveshavebeen Experimental datareported in theliteratureareshownalong cross sectionsandtheirenergydependence,also,at low excitation functionfortheproduction oftheboronisotopes, calculations requireknowledgeofboththepartialcross sec- energies, theionizationenergyloss.Accuratepropagation in theAppendixtothispaper. The procedureisrepeatedwithdifferentPLDsuntilagood fit calculations arethencomparedwiththeexperimentaldata. densities arethenmodifiedbysolarmodulationinthehelio- and theirradioactiveprogenitors, fromCfragmentation. calculate theratiosatorbitofEarth.Theresultsthese are propagatedthroughslabsofinterstellarmatterand traced throughtheexperimental pointsandaretheexcitation tions fortheproductionofsecondarynucleiandtotal to-primary ratiosinGalacticpropagationarethenuclear sphere (usingamodelofsolarmodulation;see§IV)to ties inlocalinterstellarspaceoutsidetheheliosphere.These to thedataisobtained.Thedetailsofcalculational technique andtheinputparameterstocodearedescribed V/Fe ratiosbothasaguideandcheckontheresults derived fromouranalysisofthesub-Fe/Fedata. 89 (MeV pernucleon) For theinvestigationofcosmic-raypathlengthdistri- The mostimportantparametersforthestudyofsecondary- 127.9 173.0 150.6 105.2 60.2 34.1 82.7 0.25 +0.02 49-184 N/O 117 HI. PROPAGATIONCALCULATIONS 0.048 +0.009 87-366 Sc/Fe N/O 227 (MeV pernucleon) 1973 October-1978October 161.8-184.2 116.4-139.3 139.3-161.8 49.0- 7 1.4 94.0- 1 16.4 71.4-94.0 25-43 0.094 +0.01 90-380 V/Fe 235 Sub-Fe/Fe 0.62 +0.04 0.27 ±0.02 0.27 +0.02 0.24 +0.02 0.26 +0.02 0.28 +0.04 0.24 +0.02 0.24 +0.02 87-398 Ratio 236 273 198 7ApJS. . .64. .2 6 9G 11 10 -2 274 pirical formulaeofSilberbergandTsao(1973a,b).ForB production thecurvesareverysimilar,differingmainlyin with thedifferenceincreasingdecreasingenergy. energy interval0.6-2GeVpernucleon.ForB,however,the by number).Twoconclusions areimmediatelyevidentfrom poorly measuredthantheonesinexampleofFigure 3. cross sectionsarequitedifferentbelow1GeVpernucleon, solid curvesaretheexcitationfunctionsgivenbysemiem- functions adoptedforuseinthisanalysis.Forcomparisonthe values ofXingcminterstellar matter(93%H,7%He literature. Whenexperimentaldataareunavailable,semiem- energy-independent. Thenumbers onthecurvesgive gation calculationsforanexponential PLDwhosemeanXis proach hasbeenadoptedforthetotalinelasticcrosssections pirical excitationfunctionshavebeenused.Asimilar ap- generated forallreactionswhichreliabledataexistin the Curves similartothedashedlineinFigure3havebeen dressed intheAppendix. (Guzik etal.1980),andthisaspectoftheproblemis ad- 0 0 circles, Davids,Laumer,andAustin(1970);diamonds,Rocheetai(1976);triangles,Fontes(1977);invertedLindstromal.(1975a); squares. the excitationfunctionadoptedforpresentanalysis. Silberberg andTsao(1973a).Thesolidcurvegivestheresultsofsemiempiricalcalculations(Silberberg19736),dashed curveshows 12 The curvesshowninFigure 2givetheresultsofpropa- Most oftheneededexcitationfunctionsaremuchmore Fig. 3.—Partialproductioncrosssectionfortheisotopesofboron(plusradioactiveprogenitors)inp+Cinteractions.Datapointsare the following: © American Astronomical Society • Provided by theNASA Astrophysics Data System GARCIA-MUNOZ ETAL. Fe/Fe ratio. B/C observationdoesnot,simultaneously,explainthesub- 5 GeVpernucleon,thevalueofXneededtoreproduce the measurements.Second,inanyenergyintervalbelowabout the experimentaldataoverfullenergyrangecoveredby varied, withintheuncertainty limitsfortheindividualparam- reproduce theB/Cobservations,(c)compareresults with propagation calculation(thisisthesubjectofAppendix plish theseobjectives,aniterativeprocedureisadopted: (a) Figure 2.First,nosingleconstantvalueofXcanreproduce eters, toassesstheoveralluncertainty inthederivedPLD. (d) modifytheshapeof PLDtosatisfyalltheratios. establish thebestvaluesforallofinputparametersin the reproduce bothoftheratiosshownonFigure2.Toaccom- exponential PLDovertheentireenergyrangeshown,and (2) remainder ofthispaper:(1)theenergydependence an Finally, thesetofbestvalues fortheinputparametersis and §IV),(b)adoptanenergydependenceofXinorder to the sub-Fe/Fedataandwith theratiosSc/FeandV/Fe, the shapeofPLD,asafunctionenergy,needed to 0 0 0 Figure 2illustratesthedualprobleminvestigatedin Vol. 64 198 7ApJS. . .64. .2 6 9G No. 1,1987 where V(r')isthesolarwindvelocity,K(r')radial part intensity andthespectralshapeofcosmicraysarrivingat Axford 1968)thismodulationparametercorresponds to a noting thatinthe“force-field”approximation(Gleeson and of thediffusioncoefficient,andRisradiushelio- have analyzedthesimultaneousmodulationofelectrons,pro- been developed(Parker1965;Jokipii1971;UrchandGleeson toevenlargerdisagreement. length isconstantbelow1GeVpernucleonfailstoagreewith on theinterpretationofpropagationcalculations.Itis shown thatapropagationcalculationinwhichthemeanpath diffusion coefficientisproportionaltoparticlerigiditytakes sphere. Aninsightintothephysicalmeaningofisobtained, radius risgivenbythemodulationparameter more thanonesolarcycle.Theyfindthatingeneralthemodel condition. eters arethesolarwindvelocity,diffusioncoefficientand represented byaFokker-Planckequationinwhichtheparam- librium intheheliosphere.Quantitativelytheseeffectsare drifts duetothegradientandcurvatureofmagneticfield) “potential energy”Othatintheparticularcasewhich the Garcia-Munoz etal1986),solvingthisequationnumerically, energy spectruminlocalinterstellarspaceasaboundary and assumesthatthesethreephysicalprocessesareinequi- diffusion, convection,andadiabaticdeceleration(butnot energy totheexpandingfield(adiabaticdeceleration),and out oftheheliosphere.Inthisprocess,cosmicrayslose outward-flowing solarwind,carryingthefrozen-ininterplane- the orbitofEartharesignificantlymodifiedbysolarmodula- 1979) thatatenergiesbelowafewGeVpernucleonthe around thesolarminimum.Astrongermodulationlevelwill tion atenergiesbelowseveralGeVpernucleon,anditseffects the interstellarsecondary-to-primaryratiosbysolarmodula- the simpleform fits thedatawell. the radiusofheliosphere,withcosmic-raydifferential tary magneticfield,whichtendstoconvectthecosmicrays tion. Thecosmicraysdiffuseintotheheliosphereagainst the modulationlevelcharacteristicof1974-1978period the experimentaldataifpropagationcalculationassumes tons, andheliumnucleioverthe1965-1979periodinvolving the modulationprocess.Thismodelincludeseffectsof 1972; Fisk1979)whichexplainsmostofthegrossfeatures their energyspectrumismodified. been identifiedasthemean energylossthattheparticles where Zeistheparticlecharge. Thispotentialenergyhas We discussinthissectionthemodificationintroduced A sphericallysymmetricmodelofsolarmodulationhas It iswellknown(seereviewsbyJokipii1971andFisk In thismodelthedepthofmodulationatheliospheric Recently Evensonetal.(1983)(see,formoredetails, © American Astronomical Society • Provided by theNASA Astrophysics Data System IV. SOLARMODULATION $ =|Ze|for1974-1978period in thefigurebyvaluesofparameter<£equation(2), has beencalculatedatdifferentlevelsofmodulationindicated lation forFigure5usedanexponentialPLDinwhichthe with thecorrespondingvalueofmodulationparameter<#>, demonstrated onFigure5isthatthecalculationagreeswith expressed inunitsofMeVpernucleon.Theessentialpoint mean pathlengthXwasconstantbelow1GeVpernucleon gation followedbysolarmodulation.Thepropagationcalcu- curves aretheresultsofcalculationsinterstellarpropa- or thecorresponding“potentialenergy”0.Figure5demon- B/C attheorbitofEarth.Wehavesolvednumerically (1983) tocalculatetheeffectofsolarmodulationonratio 490 MV,or0=245MeVpernucleonforA/Z2particles. pared withthetimedependenceofClimaxNeutron parameter usedbyEvensonetal(1983)for1973-1980,com- 75 MeVpernucleon.However,sincewehaveshownabove strates theeffectofmodulationfordifferentvalues0.The developed byFisk(1971),andwehavelabeledthesolutions complete Fokker-Planckequationusingthecomputercode nearly constantlevelofsolarmodulationforwhichtheaver- used valuesofthemodulationparameter<¡>tospecifydiffer- modulated nucleonicspectrawhichwillbeverycloselyequal the dataonlyformodulationlevelslessthanabout= and variedathigherenergiesasXccE~.TheB/Cratio Monitor countrate(givenasthepercentdecreasebelow ent levelsofmodulation. r, V(r),andK(r)givingthesamevalueofwillleadto modulation parameter.CombinationsoftheparametersR, age valueofthemodulationparameterisapproximately= sponding tothedatausedinthispaperischaracterizedbya of rigiditytimestheratioA/Zparticle. 0 inunitsofkineticenergypernucleonisequalto 1954 solarminimumlevel).Theperiod1974-1978corre- to eachother(UrchandGleeson1972).Therefore,wehave tion arealmostcompletelydeterminedbythevalueof tained bythenumericalsolutionofFokker-Planckequa- experience inpenetratingtheheliospheretoaradiusr.Thus ratios below~1GeVpernucleon, anditwillbeshownthat length decreaseswithincreasingenergyasapowerlaw in ponential path-lengthdistributioninwhichthemeanpath interval. Thedecreaseintheratioswithenergyabovea few an explanationofthedatarequires thatinthisenergyinterval gation reportedinthispaper istheenergydependenceof the meanpathlengthdecreases withdecreasingenergy. can bemodeledinpropagationcalculationsusingan ex- GeV pernucleon(e.g.,Juliusson,Meyer,andMuller1972) total energyorrigiditywithanexponentintherange—0.4 to to-primary ratiosB/Candsub-Fe/Feoverawideenergy Q Q — 0.7.HereadependenceE~ - isadopted.Thenewinvesti- We haveusedthemodelandtechniquesofEvensonetai Figure 4showsthetimedependenceofmodulation The modulatednucleonicdifferentialenergyspectraob- Figure 2showstheenergydependenceofsecondary- V. ENERGYDEPENDENCEOFTHEPLD 275 o UDO'! CM 1 KO^r h) a oor-

Fig. 4.—Time dependence of solar modulation in the heliosphere, (a) Modulation parameter ^>(MV) as derived by Evenson et al. (1983) from the measured flux of 70-95 MeV per nucleon cosmic-ray for the period 1973-1980. (b) The Climax Neutron Monitor 27 day average count rate for the period 1967-1985 (J. A. Simpson 1985, private communication).

Fig. 5.—Calculated B/C ratio using different levels of solar modulation. The PLD used is an exponential with X cc £~° 6 for 7 > 1 GeV per nucleon -2 0 and X0 constant for 7< 1 GeV per nucleon. The constant value of X0 = 9.25 gem adopted for 7< 1 GeV per nucleon is determined by requiring that the calculated ratio reproduce the weighted mean of the B/C observations in the interval ~ 0.5-2 GeV per nucleon. Different levels of solar modulation are indicated by the value of the modulation parameter O expressed as mean adiabatic deceleration in MeV per nucleon for A /Z = 2.0 particles. (Data references as in Fig. 2a).

© American Astronomical Society • Provided by the NASA Astrophysics Data System 198 7ApJS. . .64. .2 6 9G 2-06 -0 6 with themeasuredlow-energyB/Cratios. given bytheempiricalformula source spectrumintotalenergygivesthelargestdifference representation oftheexperimentaldata. ulation. Ofthethreedependences,onlycurve1isanadequate combined effectsofionizationenergylossandsolarmod- The flatteningofthecurvesatlowenergyisdueto the source spectrumusedinourcalculationsisverysimilar propagation andsolarmodulationforthreedifferentenergy to asourcespectruminrigidity.AsshowntheAppendix, a the formofenergyspectrumatcosmic-raysource. The ponential PLDhavebeensuggestedatenergiesbelowafew dependences ofatlowenergy. experimental B/CdatawiththecalculatedratioafterGalactic a constantinthisenergyinterval.Figure6comparesthe GeV pernucleon.ThemostfrequentassumptionisthatX Q per nucleon.Solarmodulationfor$=490MVhasbeenincludedallcurves. 1 GeVpernucleon:curveisfora£;2X=constant9.25gcmand3-,whereEthetotal particleenergy the energydependenceatlowofmeanXanexponentialPLD.Inallcases,forenergiesabove~1GeVpernucleon, aF.Below 0 Q Q 1 06 The variationofXwithenergy employedforcurve1is The resultsofthecalculationsareonlymildlysensitiveto by severalauthors(Westergaard1979;Kochetal.19816; Perron etal.1981). per nucleon,X850MeVpernucleon 0 0 The energydependenceforthemean,X,ofexponen- The partialcrosssectionuncertainties wereestimatedfrom 0 b) UncertaintiesintheDerivedEnergyDependence 277 198 7ApJS. . .64. .2 6 9G 242856 20 12131416 -1 Mg,Si,Fe)wasnegligible. line inthecentergivesenergy dependenceevaluatedfrom importance, theuncertaintyinpartialcrosssections for 278 equation (3).Thecrosssection-induced uncertainties,just cross sectionsforelements heavier thanoxygen(e.g.,Ne, nuclear reactionsandinwhichallthecrosssectionswere without ionizationenergyloss)whichincludedallof the carbon spallation,oxygentotalinelasticcross sec- contributions tothiserrorinXwerefoundbe,order of determined withasimplepropagationcode(leakyboxmodel tions, andnitrogenspallation.Theuncertaintycontributed by assigned anuncertaintyasafunctionofenergy.Themain little totheoveralluncertaintyinX,whichisdominated by spallation oftheprimaryCNOnuclei. from about15%to30%forthelesswellmeasuredreactions. boron productioncrosssectionsfrominteractionsofcosmic- these unmeasuredcrosssectionswerefoundtocontribute 1981; butseealsoLetaw,Silberberg,andTsao1985), sections (StoneandWiedenbeck1979;PerronKoch 30% uncertaintyusuallyquotedforthesemiempiricalcross For othercrosssections,auniform15%errorwasassignedat ray C,N,andO.Theseuncertaintiesrangedfrom of 9kms(curve1)and172). 0 the errorsinmeasuredpartialcrosssectionsfor the all energies.Thisvalueissomewhatlowcomparedwiththe 15%-20% atafewGeVpernucleonforthebestcase,and 3 orFigure17.Thisprovidedexperimentaluncertaintiesfor outer uncertaintyenvelopeisreproducedasthehatchedregionin(/?)andcomparedwithresultsofdynamichalomodelforconvection velocities results withtheoreticalcalculationsforthedynamicalhalomodel.In(a)actualvalueofXusedtofitB/Cratioisshownby heavy line,while the uncertaintyduetocrosssectionerrorisshownbydottedlines.TheouterdashedlinesincludeinB/Cmeasurements, andthis 0 — 10%atenergiesofafewhundredMeVpernucleonto Q The uncertaintyinA"duetothecrosssectionerrorswas Figure lashowstheenergy dependence ofX.Theheavy 0 0 Fig. 7.—MeanoftheexponentialPLDusedtofitB/Cdataasafunctionenergy,{a)showinguncertaintiesinXand(b)comparing the Q © American Astronomical Society • Provided by theNASA Astrophysics Data System Kinetic Energy(MeV/Nucleon) GARCIA-MUNOZ ETAL. of uncertaintyaretheionization energylossrelation(mea- gest contributorstotheuncertainty inX.Additionalsources at lowerenergies. with thecrosssectionuncertainty,asafunctionofenergy, to were thencomparedwiththebestfitgivenbyequation (3), boundaries ofthedata,andanenergy-dependentXwas section measurementsarerequired toreducetheerrorinV ( -50%).Improvedcosmic-raydatacanreducetheuncer- ments providesthelargestcontributiontototalerror but athigherenergiesthespreadinexperimentalmeasure- curves inFigurelaandalsoastheshadedregion give thetotaluncertaintyinX,whichisshownasdashed errors intheexperimentaldata.Thisuncertaintywasfolded tainty inXathighenergies,butmoreaccuratenuclearcross lb. BelowseveralGeVpernucleontheoveralluncertainty in points, followingapproximatelythe2alevelfromweighted and thedifferenceusedtodetermineanuncertaintydue the mean oftheobservations.Forexample,at1GeVpernucleon measurements oftheB/Cratio,curvesweredrawnonFigure Xq 30%)isdominatedbythecrosssectionuncertainties, fitted toeachlimit.TheseupperandlowerboundsforX(E) the upperandlowerlimitswereB/C=0.340.30,respec- 2 correspondingtoanupperandalowerenvelopeofthedata curves. Theouterdashedbandgivestheoveralluncertainty discussed, producetheerrorbandshownasinnerdotted tively. Theseenvelopeswerethenassumedtorepresentthe after foldingintheerrorscosmic-raymeasurements. 0 0 0 0 0 0 The crosssectionsandtheexperimental dataarethebig- In ordertoassesstheuncertaintydueerrorsin Vol. 64 198 7ApJS. . .64. .2 6 9G 14 1415 14 cosmic rays. employed asdetailedtracersofthepropagationprocess the better agreementwiththatdataset. and Li/Cwillbenecessarybeforethesetworatioscan be propagation processisthesameforallthesespeciesand that effect ofincreasing,slightly,thecalculatedBe/Cratioat experimental dataorthecrosssectionsused.Figures8h and little self-consistency.Noconclusionscanbedrawnfromthis HEAO C-2energies,therebybringingthecalculationsinto were collectedin1979-1980underheliosphericconditions high statisticsHEAOC-2data(Engelmannetal.1981)at flicting conclusionsoriginatemainlyfromthequalityof the compilations ofthefightelementsandN/Oarguethat the ments ofcosmic-raylithium. comparison, excepttopointtheneedforbettermeasure- higher energy.ItshouldbenotedthattheHEAOC-2data increasing thesourceratioN/Owillmoveentirecurve is inreasonableagreementwiththecompiledexperimental 8 csuggestthatsignificantlyimprovedmeasurementsofBe/C some previousinterpretationsintheliteraturedrawingcon- sured crosssectionswhereverpossible)andtheup-to-date time period.Thelargerlevelofmodulationwouldhavethe the calculationofcurveappliesstrictlyto1974-1978 approaching solarmaximum.Themodulationusedin calculations agreewiththedataatlowenergyand cross sectiondataexist,asdiscussedintheAppendix), upward. Thus,theN/Oratioprovidesatestforshapeof shape oftheexperimentaldata. mean ofthemeasurementsatallenergies.However,slightly data inFigure8,butappearstofallsomewhatbelowthe components ofbothNandN.OurcalculatedN/Oratio with calculationsassumingthree possibledependencesofthe mixed ratio,secondary-plus-primarytoprimary,sincethere tion, andtheresultsshowninFigure8areproducewell the calculatedcurveratherthanforabsolutenormaliza- are bothasourcecomponentofNandlargesecondary calculations whichproducedcurve1inFigure6,i.e.,using the energydependencegivenbyequation(3).TheN/Oisa Li/C ratioscomparedwiththeresultsofpropagation nucleon for^4/Z=2particles).Inclusionoftheseadditional energy. drawn fromthisanalysisisthatevenwithalloftheuncertain- error inXshowFigure7.Theimportantconclusiontobe uncorrelated errorsdoesnotsignificantlyaltertheoverall ulation employedinthecalculations(AO<40MeVper GeV pernucleon,and(2)itmustdecreasewithdecreasing ties included,(1)Xisnotenergy-independentbelow-1 spectrum (<5%;seeAppendix),andthelevelofsolarmod- sured toanaccuracyof<3%),thecosmic-raysourceenergy No. 1,1987 0 0 The abovecomparisonsbetweencalculations(usingmea- For theLi/Cratiocosmic-raydataarescarceandshow In thecaseofBe/Cratio(forwhichgoodexperimental We showinFigure6acomparison ofexperimentaldata Figure 8showscompileddatafortheN/O,Be/C,and VI. IMPLICATIONSOFTHEENERGY DEPENDENCEFOR COSMIC-RAY CONFINEMENTAND PROPAGATION © American Astronomical Society • Provided by theNASA Astrophysics Data System c) OtherSecondary-to-PrimaryRatios IN THEGALAXY COSMIC-RAY PROPAGATION -1 -1 -1 -0 6 wind velocityof9kmsand curve2forawindvelocityof process, leadingtoasmallereffectivevalueofXand a particles willberemovedmosteffectivelybytheconvection wind. Ifconvectiondominatesdiffusion,thelowestenergy derived fromtheB/Cdata {shaded region).Awindvelocity wind speedanddimensionsoftheGalactichalo.Figure 76 decrease ofXwithdecreasingenergy. where theydiffusewhilebeingconvectedbytheGalactic hydromagnetic waves,drivegasandmagneticfieldoutofthe halo modelsinvolvingaGalacticwindenergizedbythe interstellar hydromagneticwavesmightshowabreakintheir disk inaGalacticwindformingdynamicalhalo, important atlowparticleenergies. power-law spectrumatlowrigidityorbecomeinefficientin 17 kms—comparedwith theenergydependenceofX shows theresultsoftwothesecalculations—curve1for a for theenergydependenceofmeanamountmatter energy-dependent diffusioncoefficients,obtainingpredictions (1976) andJones(19796),haveperformeddetailedintegra- cosmic raysthemselves(JohnsonandAxford1971;Ipavich of Xmustdecreasewithdecreasingenergybelowabout density haloregion)orthatanothermechanismismore either thatsuchself-limitingconfinementdoesnotoperatein produce hydromagneticwavesandare,inturn,scatteredby below afewGeVpernucleon.Ifthelow-energyparticles controlling particleescapefromtheconfinementregion. index thatisrelatedtothepowerspectrumofwaves pected tobeapower-lawfunctionofrigiditywithspectral traversed bythecosmicraysfordifferentvaluesofGalactic region oflowparticleenergy.However,suchaconstant escape lengthwhichisconstantwithenergy(rigidity)inthe velocity (Holmes1974).Thisargumentleadsdirectlytoan involves modelsinwhichtheescapeisindependentofrigidity overwhelmed byanotherphysicaleffect.Forexample,the must changeatenergiesbelowafewGeVpernucleonorbe (energy) dependenceofthetrapping/scatteringmechanism (Jokipii andHigdon1979;Cesarsky1980).However,wehave these modelsthecosmicrays,whicharecoupledtogasby 1975; Jokipii1976;Jones19796;Freedmanetai1980).In 1 GeVpernucleon.Suchbehaviorispredictedbydynamical the Galacticdisk(althoughitmaybeeffectiveinlow- a goodfittothelow-energyB/Cobservations,suggesting escape lengthbelowafewGeVpernucleonalsodoesnotgive cosmic raysareconstrainedtostreamatabouttheAlfvén able fittotheexperimentaldata.Thisimpliesthatrigidity shown thatatlowenergiescurve3doesnotgiveanaccept- waves, thentheenergy(rigidity)dependenceofXisex- has beenexploredbyanumberofworkers,sinceifcosmic-ray of 17kmsappearstobe marginallyconsistentwiththe tions oftheequationsforadynamicalhalomodelincluding these waves,theprocessmaybecomeself-limiting,and interaction oftheparticleswithinterstellarhydromagnetic shaded region,butthelower windvelocitypredictstoolittle escape fromtheconfinementregioniscontrolledby mean YofanexponentialPLDonenergybelowabout dependence proportionaltoFatallenergies.Thiscase 1 GeVpernucleon.Incurve3,Xfollowsthesameenergy 0 0 Q Q 0 0 0 The experimentaldatadiscussedaboveshowthatthevalue An alternateapproach,representedbycurve2onFigure6, Kota andOwens(1980),followingtheworkofJokipii 279 198 7ApJS. . .64. .2 6 9G experimental datafortheratios:(a) N/O,(b)Be/C,and(c)Li/C.(DatareferencesasinFig.2a.) See textfordetails. Fig. 8.—Comparisonofthecalculated ratios(usingthe“best-fit”energy-dependentexponential PLD,curve1ofFig.6)withthecompiled © American Astronomical Society • Provided by theNASA Astrophysics Data System 198 7ApJS. . .64. .2 6 9G -1 -1 10 10 -1 10 -2 109 -2 _1 n ing crudelyfromthesecurves,thevariationinXdeduced Wiedenbeck andGreiner1980;Garcia-Munoz,Simpson, be somewhatflatterthantheobservations. below 1GeVpernucleonappearstodecreasefasterwith variation withenergy.Also,theexperimentalformofX{E) excess of25kms.Notealsothattherigiditydependence decreasing energythanthetheoreticalpredictions.Extrapolat- velocity of8kmsandusingthemeasurements Wefel 1981)canbeused,alongwiththemeanamountof assumed athighenergybyKotaandOwens(1980)appearsto from theB/Cmeasurementswouldimplywindvelocitiesin velocity assumingasurvivingfractionofBelessthany,and matter traversedattheequivalentenergy,tocomputelimits such asBe(Garcia-Munoz,Mason,andSimpson1977Z?; derived anupperlimitof16kmsfortheGalacticwind coefficient. Freedmanetal.(1980)adoptedthisapproachand on thewindvelocity,sizeofhalo,anddiffusion age derivedfromthesurvivingfractionofaradioactivespecies large topermitdefiniteconclusionsanddidnotinvestigate increased. ter inthedynamicalhalomodel.Therefore,cosmic-ray of thehaloanddiffusioncoefficient.Heconcludedthat curve 1inFig.6),theupperlimitforwindvelocityis surviving fractionofBetoplacelimitsontheeffectivesize However, ifXislessthan6gcm(asthecaseforour the spreadinmeasuredvaluesforBe/Beratiowastoo other valuesforthevelocityofGalacticwind. Y =6gcmofinterstellarmatterat300MeVpernucleon. mechanism forparticlelossfromtheconfinementregion,we from thecosmic-rayB/Cobservationswheninterpretedin inflate themagneticfield,leadingtoformationofmag- can bemadeconsistentwiththedata,butothermechanisms the frameofcurrentdynamicalhalomodelsindicatesconvec- upper andloweredgesoftheGalacticdiskcosmicrays may beinvolved.Parker(1969,1976)assumedthatatthe see thatparticleescapedrivenbyaconvectiveGalacticwind tive velocitiesfortheGalacticwindof>20kms. 0 netic “bubbles”whichextendfarabovetheGalacticdiskinto 0 which impliesthatthemeanamountofmattertraversed by loss oflow-energyparticles,thentheprobabilityescape per unit timeshouldbecomeconstantatlowenergies(rigidities), by fittingthevaluesofYderivedatvariousenergies to a that, ifsuchamechanismisprincipallyresponsiblefor the trapped cosmicrays.BlandfordandOstriker(1980)argue the Galactichalo,wheretheymayrupture,releasing which thelow-energycosmicraysarelostviamagnetic“bub- power lawof«=0.98±0.15isobtained,showingthat the power lawinparticlevelocity(X

)andhavetheeffectofdecreasing These reducedvaluesfallwithinouruncertaintylimitsforthe nucleon, WebberandKish(1985)reportvaluesforB+C would decreaselessstronglywithdecreasingenergybelow into betteragreementwithGalacticwindcalculations(Fig. results beverified,then,inthepresentanalysis,PLD and B+Cproductionwhichare32%13%,respec- and radioactiveC,whichdecaytoboron.At415MeVper particularly atenergiesbelowseveralGeVpernucleon.Use in ananalysisofHEAOC-2,balloon,andsatellitedata(such tively, belowthevaluesemployedinthisanalysis(cf.Fig.3). length proportionaltoparticlevelocityreproducestheexperi- the calculatedB/Cratio.Shouldthesesmallercrosssection suggested byWebber(1985;butseealsoLau,Mewaldt,and mental data,confirmingtheconclusionreachedin§VI. lb) forwindvelocities<20kms.Webberetal(1985)and 1 GeVpernucleon,bringingtheempiricallydeterminedPLD found thatatenergiesbelow1GeVpernucleonanescape as ourcompileddatainFig.2)aleakyboxmodelandhave Soutoul etal(1985)haveemployedthesenewcrosssections intestellar heliuminthecalculations(whichhasalreadybeen of theescapelengthneeded,inaleakyboxmodel,torepro- of thesecrosssectionsgivesalargediscrepancyinthevalues Stone 1985)leadtoareducedproductionofsub-Feelements, (Soutoul etal1985).Investigatingtheeffectofincluding duce separatelytheexperimentalB/Candsub-Fe/Feratios low energy(Fig.13)andtoextendhighertherange veloped indetailthispaper,thatanenergy-dependent done inallthecalculationsreportedthispaper),Ferrando, path lengthsinthePLDinvolvesultraheavy(UH;Z>30) ratios. Thedetailedeffectofthesenewironfragmentation reproduce simultaneouslythemeasuredB/Candsub-Fe/Fe depletion ofshortpathlengthsinthePLDisneededto conclusion presentedbyGarcia-Munozetal(1984),andde- escape lengthsbecameevenlarger.Thesepapersverifiedthe cross sectionsistoincreaseslightlythevalueofXorat Goret, andSoutoul(1985)showedthatthediscrepancyin cosmic rayswhosetotalinelasticcrosssectionsarelargerthan over whichtheXorcomponentisimportant. (Fowler etal1985ú,¿>)doprovidedataatdifferentenergies ments HEAOC-3(Binnsetal1981,1984)andAriel 6 rays, butathigherenergiesthetwoexistingsatelliteexperi- sensitive totheshortpathlengthsinPLD.Unfortunately, owing tothedifferencesintheirorbits.Themeanenergy for there havebeennorehablestudiesoflow-energyUHcosmic that ofiron,therebymakingtheUHcosmicraysmore and mightthereforebeexpectedtoshowmoreeffectsdue to the ArieldatasetislowerthanthatforHEAO set the depletionofshortpathlengths.Klarmannetal(1985) have analyzedtwosecondary-to-primaryratios,(70< Z < high energies.Quahtatively, these observationsareexpected ratios. Further,theHEAO ratiosareinagreementwith calculations usingarigidity-dependent escapelengthatthese 73)/(74 1 crosssection);aisthepartial average atomofinterstellargas; ßistheparticlevelocityinunitsoflight; yistheparticleLorentzfactor;o 1968; Garcia-Munoz,Mason,andSimpson1977a;Lezniak 1979)usingaLagrangiancomputeralgorithmwrittenbyoneofus spectrum. Ineachsubsection,theresultspresentedarethoserelevanttoanalysisofshapeandenergy dependence of the ionizationenergylossrate forspeciesi.Thus,equation(Al)isactuallyacoupledset offirst-orderdifferentialequationswhich section fortheproductionof speciesifrominteractionsofy;Tisthemeanlifetime ofspeciesiforradioactivedecay;Tj the cosmic-raypath-lengthdistributiondiscussedinbodyofthispaper. ondary-to-primary ratioscanbemeasuredoveralargeenergy energy-dependent depletioneffectinthePLDonceUHsec- data. Thus,theUHcosmicrayscanprovideatestfor quently, thesecondary-to-primaryratios,asobservedin rigidity (energy)dependentmean.AtthelowerArielenergies, altitude non-polar-orbitingsatellites). range includinglowenergies(sofarnotaccessibletolow- the HEAOenergies,PLDiseffectivelyexponentialwitha No. 1,1987 there isstillanimportantcontributionfromthetruncation, with particlelossfromtheGalaxyvia“magneticbubbles” model, wehavestudiedthecosmic-raypath-lengthdistribu- and acompleteGalacticpropagationplussolarmodulation The energydependenceoftheGalacticconfinement/propa- gation isconsistentwithdynamicalhalomodelsinvolving tion atlowenergyandfindthefollowing: Galactic windswithvelocitiesinexcessof20kmsandalso i max iJ i t We employtheweighted-slabtechniqueofcosmic-raypropagation (e.g.,GinzburgandSyrovatskii1964;FichtelReames Cosmic-ray propagationcalculationsbydifferentgroupshaveoftenproducedconflictingresults,evenwhen thesame Using compiledcosmic-raydataasafunctionofenergy bution inwhichthereisadepletionofshortpathlengths which decreaseswithincreasingenergy. full rangeofenergiesinvestigated(0.1-50GeVpernucleon). 2. ThePLDcanberepresentedbyanexponentialdistri- 1. ThemeanofthePLDvarieswithenergyover © American Astronomical Society • Provided by theNASA Astrophysics Data System XI. CONCLUSIONS THE GALACTICPROPAGATIONMODELANDINPUTPARAMETERS dx ¿A'Kg,*) d IdE COSMIC-RAY PROPAGATION I. CALCULATIONALTECHNIQUE 1 ^N 0 APPENDIX NAG-8848 andTheMcDonnellCenteratWashingtonUni- NAGW-550 andtheDepartmentofPhysicsAstronomy 001-006, contractNAS5-25731,andNSFgrantATM84- This workwassupportedinpartbyNASAgrantNGL-14- versity. J.A.SimpsonwouldliketothanktheJohnSimon at LouisianaStateUniversity;andinpartbyNASAgrant and M.Wiedenbeckforusefuldiscussions,V.Zueil,G. largest amountofmaterial.Thenewfactorintroducedbythe 12382 attheUniversityofChicago;inpartbyNASAgrant Sutton, andM.Proutyforhelpinmanuscriptpreparation. source/acceleration process. confinement canexplaintheexistingbodyofcosmic-raydata ergy-dependent trappingofthelow-energycosmicrays,with Guggenheim Foundationforafellowshipin1984-1985. and providesnewconstraintsformodelsofthecosmic-ray two confinementregionsandtheenergy-dependentparticle gion andthenininterstellarspace. energies. Theshortpathlengthdepletionmaybetheresultof inflated bythecosmicrays.Therigidity-dependentlossof the lowestenergyparticlesconstrainedtopassthrough energies doesnotcontinuetobedominantatlowparticle cosmic raysfromtheconfinementregionobservedathigh a two-stagepropagation,firstinthesource/accelerationre- We wouldliketothankS.Hayakawa,H.Volk,R.Mewaldt, The source/accelerationregionischaracterizedbyanen- yßcnAT, /Ç,yßmAT/ N J (Al) 287 198 7ApJS. . .64. .2 6 9G -2 4 -2 which havebeendiscussedintheliterature,allowingourresultstobecompareddirectlywithpreviouswork.Afuller description where ionizationenergylossislessimportant,thesecorrectiontermsarenegligible.Atlowerenergies,itcanbeshownthatthe Accuracy ismaintainedbyusinganalyticsolutionsforthecalculationofsuchproductswithineachstep.Crosssections foreach in eachenergybinisvariedaccordingtolossesbynuclearinteractionsandradioactivedecaygainsdue interactions or because oftheincreasingionizationenergylosswithdecreasingparticleenergy.Foreachstep,numberparticles ofspeciesi is filledwithparticlesasdeterminedbythesourcerelativeabundancesandenergyspectrum.Thecalculation then width ofeachenergybinandthenumberbinsrequiredtocoverregionspectruminterest.Each energybin magnitude ofthecorrections,evaluatedintermssecondary-to-primaryratiosdiscussedhere,aresmallerthanother below thelowerenergylimitascalculationprogresses.Thelatterproblemishandledbycollectinglowest particles boundaries oftheenergybinsarepermittedtomoveinspace;lowerboundarymovesfasterthanupper boundary The overallspeedofthecalculationisdeterminedbyspecieshighestchargeatlowestenergies.For each step,the proceeds insmallstepsAx,withthesizeofAxdeterminedbyratiogridspacingAEtomaximumenergy lossrate. established initially,consistentwitharesolutionparameter,LE/E,whichissetexternally.ThesizeofkE/Edetermines the of themathematicalformanenergy-dependentPLDderivedfromdiffusionequationswillbepresentedelsewhere. uncertainties inthecalculations—forexample,crosssectionuncertainties.ThePLDsusedthispaperaresimilarto (Ginzburg andSyrovatskii1964).However,inthepresenceofenergyloss,Lezniak(1979)hasshownthatweighted-slab which subdividesenergybinshavebecometoowide(asdeterminedbythehE/Ecriterion).Thecoderesembles others in asinglebinbelowthelowerenergylimitofcalculation.Inaddition,codeincludesanautomatic“rezoning” procedure medium. SpeciesdecayingbyelectroncapturepresentaspecialproblemandarediscussedinAppendix§V. radioactive decaycorrespondingtoastepAxiscalculatedassuminguniformdensityandcompositionofthe propagation cross sections(seeAppendix§III)andbyinterpolationinalargematrixforthefragmentationpartialsections. Thetimefor decay ofallspeciesjaffectingi,assumingAß=0forsecondaryparticleproductioninnuclearinteractions (Heckman technique doesnotyieldanexactsolutiontothediffusionequationunlessseveralcorrectiontermsareincluded.Athighenergies, both primary(i.e.,source-produced)andsecondary produced bynuclearinteractionsduringpropagation)particlespecies. species arecalculatedattheenergycorrespondingtobincenter(inenergy)fromananalyticexpressionfor totalinelastic et al1972).Fragmentationanddecayproductsgeneratedwithineachsteparepermittedtothroughthenetwork ofisotopes. the calculationsreportedhere,numericalsolutionsofequations(Al)aretailoredtothiscondition.Aset energy binsis characteristics oftheenergy-lossrelation.Thistechnique maintainsanaccurateenergydependenceforthenumberdensitiesof employing aLagrangianformulationofhydrodynamics(RichtmyerandMorton1967)inthatparticlesare evolved along 288 GARCIA-MUNOZETALVol.64 back toafixedenergygrid.Theaccuracyofthistechnique islimited,again,bythestepsizeAxandnumberofpointsin residual range)andusedinmuchofourpreviouswork, involvesinvertingthetransformationmatrixformedforpropagation fixed energygrid.Usingidenticalsetsofinputparameters, wehavecomparedtheresultsofbothcodesandfoundthemtobein energy loss,theleakyboxequations canbesolvedanalyticallyandcomparedwiththe computercalculations,givinginourcase of computerstorageandCPUtime.Asapracticalmatter, itisnecessarytoadjustAE/Eand/orthestepAxfitagiven (Garcia-Munoz, Mason,andSimpson1977a;Blakeetal. 1978; seealsoJones1979aforasimilarsolutiontechniqueemploying calculation andafinitecomputer budget. agreement canbeimprovedbyreducingAE/Eandthestep sizeAxinbothcodes,butatacostofprohibitivelylargeamount agreement towithinafewpercent,withthelargestdeviations occurring,asexpected,atthelowestenergypoints.Thelevelof agreement tobetterthan2%. Apseudo-energyloss(constantdE/dx)canbeincluded andstillpermitananalyticsolutiontothe this techniqueareequivalent to aleakyboxmodelinwhichtheescapelength\equals x(Cowsiketal.1967).Inthecaseofno through astepAx,accountingforfragmentation,,andenergyloss,theninterpolatingtocorrectthefluxes contributes lessthan1%tothetotalintegral.AllofPLDsinvestigatedinthispaperarebaseduponanexponentialfunction upper limitofintegrationinequation(A2)isreplacedwithx,whichselectedsothattheintegralbetweenandinfinity energy Etohavepassedthroughanamountofmaterialx,andN^E^x)aretheintermediateslabdistributions.Inpractice, by numericalintegrationas must besolvedsimultaneouslyforallspecies/.Currently,weconsider96isotopic“species”(seeAppendix§§IIandV)ranging for which,at1GeVpernucleon,themeanofexponentialisinrange8-9.5gcm,andwehaveadoptedx= from Hethroughtheisotopesofelementnickel. 30 gcmastheoperationalupperlimit. for eachenergypointofinterest.P{x,E)isthepath-lengthdistribution(PLD),aprobabilityparticlesobservedat e 0 max max The weighted-slabmethod,asusedhere,isanexactsolutionofthediffusionequationwhenionizationenergylossneglected Below severalGeVpernucleon,theionizationenergylosscauseslargestchangeinparticlenumberduringpropagation. In Two consequencesofthistechniquearethattheenergybinscontinuetogrowinwidthandlowest binmoves Another techniquetosolveequations(Al),developeda number ofyearsagoattheUniversityChicagobyJ.D.Anglin For anenergy-independentexponential PLD,P(x)=(l/x)exp(—x/x,wherex is the“mean”ofPLD,results Once thesetofequations(Al)havebeensolved,numberdensityspeciesiatboundaryheliosphereisobtained 0 © American Astronomical Society • Provided by theNASA Astrophysics Data System h =/P(x,E)N(E,x)d(A2) zi rOO 198 7ApJS. . .64. .2 6 9G 5 u No. 1,1987 leaky boxequations,andinthistestcasetheanalyticalresultcomparesequallywellwithfullcomputercalculation.Finally, Eskola etal.1980;EwanKekelis1978;Norman,Davids,andMoss1978)indicatedotherwise.Known branches Decay by^-particleemissionwasassumedunlessenergyleveldataordecaystudies(Freedmanetal.1978;Gronemeyeretai interaction ofspecies/,then,iscomputedasthesumcrosssectionfordirectproductionj(z->j)plus propagation calculation.Fortheresultsreportedhere,anddisplayedinFigures1415,7^^is10years. with shadingintheupperrightandlowerleftcomers.Radioactiveisotopesmusthavehalf-livesT>rtobeincluded long-lived radioactiveisotopes(^-unstable)andelectroncaptureisotopes.Figures1415presentacosmic-rayprogenitor” Wiedenbeck 1981;Ormes1981). 1980; Hardyetal.Alburger,Millener,andWilkinson1981;AystoBenenson1978;Bogatin etal.1980; Fintz etal.1976;Poskanzer1966,1968;Roeckl1974;Symons1979;ThomasWestfall1919a). nuclei offthevalleyofßstability(Arthukhetal1969,1970^,b,1971;Augereial.1979;Butleret1977;Detrax cross sectionsforalloftheradioactiveprogenitorsspeciesj.Allknownnuclideswereconsideredas(Lederer assuming decayimmediatelyuponformationasaninteractionproduct.Thecrosssectionfortheproductionofspeciesjfrom chart ofthenuchdes.Thestableisotopesareshownasshadedsquares,andradioactivespeciessquareswhichopenor the computerresultscanbecomparedwithcalculationsofothergroups,usingapredeterminedsetinputparameters,and and Shirley1978;Ajzenberg-Selove1977,1978,1979,1980,1981;EndtvanderLeun1978)includingmanynewlydiscovered again ourcodeisingoodagreementwiththeresultsfromothergroupsforcasesbothandwithoutenergyloss(Freier1981; Hmit 464 The speciesactuallyfollowedinthepropagationcalculationsconsistofallstableisotopesbetweenHeandNiplus withT<(inFigs.14and15,unshadedsquares)areincludedinthecalculationofcrosssectiondatafileby ]hrái © American Astronomical Society • Provided by theNASA Astrophysics Data System II. REACTIONLISTANDRADIOACTIVEDECAY COSMIC-RAY PROPAGATION 289 198 7ApJS. . .64. .2 6 9G LO CVI O CVJ O lo ro LO ro American Astronomical Society •Provided bythe NASA Astrophysics Data System jaqwriN uo+ojd CVJ LO CVI LO CVI o LO ro O lo lo ro CVJ O cvj 198 7ApJS. . .64. .2 6 9G 5456 59 problem withtheseapproachesistheneglectofactual energydependenceofthecrosssection,whichisimportantforany proposed byKirkbyandLink(1966)orRenbergetai (1972),whichcontainaweakenergydependence.Themostserious renewed inthepastfewyears bytheincreasingamountofnucleus-nucleuscrosssection dataemergingfromheavy-ionaccelerator Wiedenbeck 1979)withdifferentformsfortheoverlap correction parameter.Alternatively,somegroupshaveusedformulae overlap model(BradtandPeters1950;DanielDurgaprasad 1962;SilberbergandTsao1911b]Hagen1976;Stone experiments. Anumberofrecent analysesusingGlaubertheoryand/ortheopticalmodel havebeensuccessfulinreproducingthe discussion oftheenergyvariationcosmic-raydata (Garcia-Munoz, Simpson,andWefel1980;Guziketal.1980). cross sectionatalloftheenergiesinterest.Traditionally, thesecrosssectionshavebeencalculatedfromtheenergy-independent shape ofproton-nucleusand nucleus-nucleuscrosssectionresultsoverlimitedenergy intervals. Karol(1975)usedasemiclassical the individualnucleon-nucleon crosssections.Interestinthestudyoftotalandinelastic orreactioncrosssectionshasbeen very muchneeded. lifetime andcanbeneglected.ForMnNi,weadopttheestimatesofCasse(1973a,b)withunderstanding (Wilson half-life forfullystrippednucleiinthecosmic-raybeam.Thedifferencesarise(1)fromsmallelectroncapturebranches whichare in thedecayschemesofprogenitors(forexample,Vieira,Gough,andCemy1979)havebeenincludedare indicatedin decays whosehalf-livesarelongenoughtobeeffectivelyinoperativeforthecosmic-raypropagationproblem. energeticaly allowed(butnotobservedinterrestrialexperiments)ßdecay.Thiswasdoneusingtheatomicmasstable ofWapstra S shallelectronstoparticipateinelectroncapture,and(3)fromscreeningeffects(Wilson1978).Note4Table2indicates those calculated (Casse1973a,Wilson1978)fordecayinthelaboratory.Thecolumnheaded“CosmicRay”gives theexpected cosmic-ray daughtersareindicatedalongwiththehalf-lives.Thecolumnlabeled“Laboratory”giveshalf-life measuredor 1978) thatthesehalf-livesmaybeinerrorbyasmuch a factorof5!Clearly,experimentalmeasurementsthesehalf-livesare and Gove(1971)withbranchingratiosfromMartin(1971).ForNianypositrondecaywouldhavean extremely long assumed tobeinactive(comparedwiththefasterßdecay)incosmicrays,(2)fromfactthatlaboratory therearetwo Figures 14and15. The energydependenceofproton-nucleus totalinelasticcrosssectionsarisesnaturally asaresultoftheenergydependencein For eachisotopeconsideredinthecalculations,itisnecessary tohavevaluesofthetotalinelasticormass-changing(>1) Table 2listsalloftheradioactiveisotopesincludedasspeciestobepropagated.Thepossibledecaymodes andstable For isotopesthatareobservedtodecayinthelaboratoryonlybyelectroncapture,itisnecessaryestimatehalf-life ofany © American Astronomical Society • Provided by theNASA Astrophysics Data System 41 49 44 + 36+ 36 26+ 7 60 53 51 50 50 10 57 55 54 54 54+ 59 56+ 56 Ca£ V£ Ti£ “»K ß Clß Clß~ A1 ß Be£ Feß~ Mne Cr£ V ß- V £ Beß~ Co £ Fe£ Mnß~ Mn£ Mnß cosmic rays.(4)Canbeomittedfromconsiderationasadecay,ifdesired. Ni E Ni ß Ni £ state unnecessary.(3)Assumingelectroncapturebranchisinactivein ‘‘Ar £ Nucleus DecayDaughterLaboratoryCosmicRayNotes Notes-(I) Electronshellsandscreeningeffects.(2)TreatmentofX* ß~ III. TOTALINELASTICCROSSSECTIONS COSMIC-RAY PROPAGATION 6060 5656 5656 (Co) Ni (Co)Fe (Co)Fe 40 40 49 41 50 50 54 54 37 36 26 10 54 53 51 55 59 57 7 Ca ^Ca Ar Ti K Cr Ti 36o Cr Cr C1 Ar Mg B Fe Cr V Mn Co Fe Radioactive Species Li TABLE 2 5 16 9 4 5 6 5 6 6 5 4 0.904 0.099 0.146 48 0.83 0.0762 3.07 X10 0.017 4X10 1.28 XlO 2.6 8 XlO 7.4 XlO 2X10 2xl0 1.5 X10 3.7 XlO IXlO 8 XlO Half-Life (years) 5 5 6 16 9 5 12 12 17 5 6 5 9 6 5 0.209 0.307 0.160 2.2 XlO 3.07 XlO 0.0358 7.7 XlO 5.25 XlO 1.90 1.43 XlO 1.7 XlO 100 1.24X10 1.8 XlO 5.47 1.75 1.33 X10 8.8 XlO 1.6 XlO 1.55 1X10 ~10 ~ 2XlO ~ 2X10 4 4 3,4 4 1 3,4 3,4 3 3 1,2 1,2 1,2 1,2 1 1,2 1,2 1,2 1,2 1,2 291 198 7ApJS. . .64. .2 6 9G 40 011 292 with thecollectedexperimentaldatadowntoabout40MeVpernucleon.Notethatneutron-nucleusin Figure 16are has beenfound.Additionaldataforoxygen,Ca,,andcoppershowaformsimilartothattheelements inFigure16. below about300MeVpernucleon.Thisempiricalterm,representingacompromisebetweenanexactformulation and was incorporatedforthelightelements,sinceuseofequation(4)ledtoanunderestimatecrosssections forLiandBe lower energies.WilsonandTownsend(1981)investigatednuclearscatteringinacompleteopticalmodelusingrepresentationsfor electron scatteringexperiments.Hewasabletoreproducetheshapeofnucleon-nucleusornucleus-nucleustotalreactioncross optical modelapproachinvolvingaveragednucleon-nucleoncrosssectionsandnucleardensitydistributionsobtainedfrom points. Clearly,furtherhigh-precisionmeasurementsareneededtoestablishthesecurvesfirmly. The solidcurvesshowtheresultofourtotalinelasticcrosssectioncalculations,asdescribedabove,andareingood agreement Jacob (1979)wereemployed. compiled fromnumeroussources(AmaldiandSchubert1980;GiacomelliJacob1979;Carrolletal.1976;Galbraith etal. Karol (1975)scaledfromequation(4)ofKirkbyandLink(1966)evaluatedat145MeVpernucleon.Thelattergivesa energy dependenceinthetotalinelasticcrosssections.Forhigherenergieswehaveadoptedscalingformulaof cross sectionbetween~30and-300MeVpernucleon. nucleus, (2)theFermimomentumofnucleonsinand(3)Pauliexclusionprinciplenucleon-nucleon energies. DeVriesandPeng(1980;seealsoPeng,DeVries,DiGiacomo1982)analyzedlow-energynucleus-nucleusdata neglected muchofthephysicsinteractionprocess.Suchanapproachisjustifiedathighenergybutrapidlybreaksdown cross sectionatanyenergytobedeterminedfromaknownmeasuredparticularenergy. Peng (1979),andSchwalleretal(1979).However,Karol(1975)providedausefulscalingformulawhichallowsthetotalreaction be obtainedbycomparingthemass-changingcrosssections measuredwithaheavy-ionbeam(Bräutigametal.1981;Jarosetal. problem, therelevantparameteristotalmass-changing crosssection.Thedifference,ifany,isinthesectionfor represented assquareswithdifferentshadings,andallothersymbolsrefertoproton-nucleusresults.Onepoint mentionwith reaction, totalinelastic,absorption,andmass-changingcrosssectionswerecompiledfromtheliterature.These areshownin data describedinthispaper. calculational simplicity,givesagoodrepresentationofthedatadowntoabout50MeVpernucleon,whichissufficiently lowfor MeV to2TeVandwerefittedsmoothcurves.Forhigherenergiestheenergy-dependentextrapolationsgivenbyGiacomelli and (see discussionbyKirkbyandLink1966).The{p,p)(/>,n)totalcrosssectionsneededintheKarol(1975) formulawere good representationtothedetailedsetoftotalreactioncrosssectiondataat-100MeVpernucleon.Asmallcorrection factor scattering. Thiscomplexcomputercalculationwasabletoreproducetheenergydependenceofnucleon-nucleustotalreaction concluded thattherewas“noevidenceforanenergyindependent,geometrictotalreactioncrosssection.”Fornucleus-nucleus sections forkineticenergiesabove300MeVpernucleon,buttheabsolutevalueofcalculatedcrosswaslargerthan respect toFigure16isthescatterinsomeofexperimentalmeasurementsandlargeuncertaintylimitson manyofthe Figure 16forthetargetelementsberyllium,carbon,aluminum,andiron,whichmostcomprehensivesetof measurements an empiricalcorrectiontermwasappliedwhichreducedthecalculatedcrosssectionasafunctionofenergy(ccE ) forenergies 1965; Bellettinietal.Foleyetal.1967;Schwaller1979;Giacomelli1969,1974;Ashmore1960; Amaldietal. total reactioncrosssections,themostcompletecalculationsarethoseofDiGiacomo,DeVries,andPeng(1980),whoincorporated all two-bodyscatterings.Theresultsgiveareasonablefittothedataathighenergiesbutcouldnotbeextrapolatedlower the experimentalvalue(Heckmanetal1978).SimilarcalculationshavebeenreportedbyRay(1979),ErnstDeVriesand energies. Suchacomparisonispossibleonlyat1-2GeV per nucleonandshowsthatthetwotypesofcrosssectionsareingood “inelastic butintact”interactionsinwhichthetargetdoes notloseanucleon.Onemeasureoftheimportancesucheffectscan 1977; Kloetetal.Denisovetal.1971;Hansen1970;ParticleDataGroup1980)coveringthelaboratory energy range1 the effectsof(1)Coulombandrealnuclearpotentialsindeterminingtrajectoryincidentnucleonthroughtarget mass-changing crosssectionforcosmic-raypropagation studies. energies of50-100MeVpernucleon.Thus,totalreaction crosssectiondatarepresentagoodapproximationtothetotal “quasi-elastic” crosssection(Acquardoetal.1981;Frahn 1978)whichimplythatitis<10%ofthetotalreactioncrosssectionat agreement, withthemaximumeffectbeing-5%atiron. Atlowerenergiestherehavebeenseveralcalculationsofthis 1978; Heckmanetal.Westfall1919b;Lindstrom etal.1975h)withtotalreactioncrosssectionsmeasuredatsimilar comparison betweencalculations andtheorywillrequiremoreexperimentalmeasurements. et al.(1980).Therearefew experimental dataavailableforhelium-nucleustotalreaction crosssections,andamorecomplete comparable totheresultsreportedhere,butdonotinvolve theuseofnucleon-nucleoncrosssectiondata. agreement withthemeasurements ofJarosetal.(1978)andbetweenthemeasurements of Millbumetal.(1954)andAbdrahmanov are scaledfromthefitgiven byWestfalletal.(1919b)evaluatedat1GeVpernucleon. Thisgivescalculatedcrosssectionsin Silberberg, andTsao(1983).Theirequationsreproduce the energyvariationobservedforthesecrosssectionstoaprecision The failureofKarol’s(1975)methodbelow~300MeVpernucleoncanbetracedtohissemiclassicalapproachwhich For cosmic-raypropagationstudiesoftheenergydependenceprocess,itisnecessarytoincludereal The datapointsinFigure16arederivedprincipallyfrommeasurementsoftotalreactioncrosssections.Forthe cosmic-ray In ordertocomparethecalculatedcrosssectionswithexperimentaldata,valuesofproton-nucleusandneutron-nucleus total For lowenergiesitwasnotpossibletoreproducethedetailedcalculationsofDiGiacomo,DeVries,andPeng(1980). Instead, A semiempiricalequationdesignedtofitproton-nucleustotal reactioncrosssectiondatahasbeenpresentedrecentlybyLetaw, For cosmic-rayinteractionswith interstellarheliumweusethesameformalismdescribed aboveexceptthatthecrosssections © American Astronomical Society • Provided by theNASA Astrophysics Data System GARCIA-MUNOZ ETAL. o UDO'! CM 1 KO^r h) a oor-

en en c en

o LU CO

CO CO O en o

KINETIC ENERGY (MeV/NUCLEON) Fig. 16.—Total inelastic (reaction) cross sections for proton- and neutron-induced reactions compared with the predictions of the formula described in the text for Be, C, Al, and Fe. Data points for protons are as follows: ® Wilkins and Igo (1963) v Pollock and Schrank (1965) C Kirschbaum (1953) ♦ Millbum et ai (1954) 3 Chen, Leavitt, and Shapiro (1955) Booth 67 a/. (1957) + Longo and Moyer (1962) V Heckman et ai (1978) ^ Renberg et ai (1972) 4> Cassels and Lawson (1954) A Bellettini et ai (1966) Ô Menet c/ü/. (1971) • Kirkby and Link (1966) v Jaros c/a/. (1978) Johansson, Svanberg, and Sundberg (1961) O Abdrahmanov c/^/. (1980) 0 McGill et ai (1974) x Bearpark, Graham, and Jones (1965) ^ Ashmore et ai (1960) ▼ Westfall et ai (19796) ^ Makino, Waddell, and Eisberg (1964) Bräutigam

© American Astronomical Society • Provided by the NASA Astrophysics Data System 198 7ApJS. . .64. .2 6 9G 16 10 7 10 11 10n 12,13 n10 272428 14 294 very differentresultsbelowabout500MeVpernucleon.Atlowenergies(<100nucleon)theexperimental data Albouy etal.1962;Valentin etal.1963;Lindstrometal1975u;Moyle1979;Heydegger etal1976;SilberbergandTsao etal. 1974;Radin,Gradsztajn, andSmith1979;RaisbeckYiou1974),O(Raisbeck andYiou1975ucorrections1977; experimental crosssectiondatafortheberylliumisotopes isduetotheOrsaygroup,summarizedbyRaisbeckandYiou(19756). using radioactiveBeasacosmic-raychronometerandbecause Be,decayinginthelaboratorybyelectroncapture(seeFig.14), measurements, whichwouldallowtheexcitationfunction tobedrawnabout20%higherorlowerthanshown.Thesituationin energy. Atlowenergythedashedcurvesareemployed.The difficultywiththedataofFigure17islargeerrorbarson determine theshapeofexcitationfunctionquitewell.Figure17showsequivalentsituationforproton- interactions. oxygen arethemostimportantsources.Figure3showscollectedexperimentaldataforproductionofB andB(plus In addition,dataareavailablefortargetsof’B(Radin, Gradsztajn,andSmith1979),C(Lindstrometal.1975u;Fontes Figure 16issomewhatbetter,owingtothelargernumber of experimentalmeasurements,butthereisstillaresidualuncertainty Here thereisasmalleramountofexperimentaldata,and the semiempiricalrelationsgiveareasonablygoodfittodataathigh of B,thereisonlyasmalldifferencebetweenthecalculatedandadoptedexcitationfunctions,butfortwo curvesgive solid fines,andthedashedcurvesgive“experimental”excitationfunctionsadoptedforthesecalculations.For the production calculated byourformalismappeartobeaccurate<10%overtheenergyregionofinterest.ForHe-nucleuscrosssections to experimentalexcitationfunctions oftendifferingfromthecurvespredictedby semiempirical formulae.Table3givesour 1973 a),and,finally,ironand nickel(seecompilationbySilberbergandTsao1973a). The analysisofthesecollecteddatahasled 1973tf), A1,Mg,andSi (Heydeggeretal.1976;Westfall1978;Raisbeckand Yiou1974,1977;SilberbergandTsao and Fullmer1967;Obergetal. 1975;seealsodatacompilationbySilberbergandTsao 1973u),N(Laumeretal.1973;Jacobs 1975; Rocheétal.1976;Davids,Laumer,andAustin1970; Westfalletal.1978;Radin,Gradsztajn,andSmith1979;Williams forms aspallationproductrelativelyeasytomeasureand useasareactionnormalizationstandard.Alargeportionofthe ratios (B/C,Sc/Fe,V/Fe)astracersofthepropagationprocess.Forboronisotopeproduction,interactions carbonand calculations. their progenitors)inproton-carboninteractions.Theexcitationfunctionsgivenbythesemiempiricalequationsare shownasthe so thattheuncertaintyinhelium-nucleustotalinelasticcrosssectionshasonlyaminoreffectonresultofpropagation the uncertaintyislarger,beingatleast20%-30%.Fortunately,heliumarelativelyminorconstituentofinterstellarmedium, near thevalleyofßstabilityandnotfarremovedinmassfromtargetnucleus,30%(toafactor2-3)forotherproducts. experimental crosssectionfile.Inthismanner,thepartialfileforuseinpropagationcalculations canbekept experimentally measuredandsemiempiricallycalculatedpartialcrosssectionresults.Asnewmeasurementsbecome available, calculated usingthesemiempiricalformulae.Afileofexperimentalcrosssectionvalues,asafunctionenergy,is then read,and become available.Theparticularcodeusedtocalculatethecrosssectionsforthisinvestigationwasextensivelycross-checked with most importantcontribution. nuclear interactionprocess.Theagreementwithexperimentaldata,generally,isfoundtobewithinabout20%-30%forproducts experimental crosssectionresults—andtheformulaemodifiedwherenecessary—andagainstMonteCarlocalculationsof Unfortunately, suchasetofcrosssectionshasnotbeenmeasuredexperimentally.However,theregularitiesinknown Fortunately, fortheproductionofsecondaryspeciesincosmic-rayproblem,itisformercrosssectionswhichmake actinide region.Belownickel,thepredictionsoftheseformulaehavebeencheckedextensivelyagainstemergingbody sections canbecombinedwithnuclearstructureconsiderationstopredictvaluesforunmeasuredcross(Rudstam1955). the speciesinquestion(plusallofitsprogenitors;seeAppendix§II)fromnuclearinteractionsheavierspecies. current withboththechangingsemiempiricalformulaeandevolvingexperimentaldataset. cross sectionsto<2%andfortheminor±1millibamwasobtained,givingconfidencethat present results the calculationofaforproton-nucleusinteractionsenergiesaboveabout100MeVpernucleonandallisotopesupto Silberberg andTsao(19736,1911a,b,c;1979)havedevelopedasetofsemiempiricalformulaewhichpermit approach, acompletematrixofcrosssectionvalues(aseparateforeachpointintheenergy grid)isfirst and usingthis,whereverpossible,inplaceofthesemiempiricalcalculations(see,forexample,Garcia-Munozetal. 1979).Inthis should beontheexperimentallymeasuredvalues.Foranumberofyearswehavebeencompilingexperimentalcross sectiondata are inaccordwiththecurrent,“best”semiempiricalestimates. the currentcalculationsofSilberbergandTsaowithresultsotherworkersinthisarea.Agreementforall ofthemajor — 15%.Morepreciseexperimentalmeasurements,especially below1GeVpernucleon,aremuchneeded. they arecomparedwiththeexistingbodyofdata/calculations,andnewormodifiedentriesmade,ifappropriate, inthe these valuesaresubstitutedforthesemiempiricalinmatrices.Thisgivesacrosssectionfileconsisting ofboth lJ The mostwidelystudiedelement,fromthepointofview of crosssectionmeasurements,isberyllium,becausetheinterestin The resultsdiscussedinthispaperusethefightelements(e.g.,B,C)andsub-FeSc,V)secondary-to-primary Considering theindividualmeasurementsinFigure16andcalculatedcurve,protontotalinelasticcrosssections The semiempiricalformulaeprovideanindispensabletoolforpropagationstudies,buttheprimaryrebaneecross sections The semiempiricalequationsareanevolvingtechniqueforcalculatingcrosssections,changingasnewexperimental data For eachisotopicspeciesincludedinthepropagationcalculation,itisnecessarytodetermineproductioncrosssectionfor © American Astronomical Society • Provided by theNASA Astrophysics Data System IV. PARTIALPRODUCTIONCROSSSECTIONS GARCIA-MUNOZ ETAL. Vol. 64 198 7ApJS. . .64. .2 6 9G 4549 50,51 45 No. 1,1987 cosmic raysencounterboth andhelium.Crosssectionsforhelium-nucleus interactionsarecalculatedinamanner may havenonnegligiblesource components. paper isbasedmainlyontheB/Cratio,withBe/Cdata usedasatestoftheresults. interactions, andFigure19givessimilardatafortheproduction ofSc.ForVproductionthedatashowconsiderablescatter, are usedinthecomputationof thesub-Fe/Feratio.Thislatterratioisnotatruesecondary-to-primary ratio,sinceMn,Cr,andTi Vthesemiempiricalcurveappearshighabove1GeV per nucleonandisreplacedwiththedashedcurve.Below600MeV giving anoveralluncertaintyofabout30%,andthesemiempirical formulaegiveagoodrepresentationofthedatapoints.For radioactive species.Figure18showstheavailableexperimental datafortheproductionofisotopesvanadiuminproton- etal. 19816),itisnecessarytoemployasecondary-to-primary ratiointhevicinityofiron.Idealforthispurposeareratios . Additionalinformationforothertargetsisavailable. only stableisotopeofscandium, andthedashedcurveisemployedathighenergy. nucleon therearenoexperimentaldata,andthesemiempirical curvesmustbeemployed.AsimilarsituationexistsforSc,the Sc/Fe andV/Fe,sincebothScVareessentiallysecondary nucleiandtheirisotopesarelargelyunaffectedbylong-lived the cosmic-rayberylliumdataarefarinferiortoboron measurements(cf.Figs.2aand86).Therefore,theanalysisinthis “experimental” valuesofthepartialproductioncrosssectionsforberylliumisotopesfromimportanttargets oftheCNO (1979). Thesolidcurveistheresultofsemiempiricalcalculations(SilberbergandTsao19736),dashedshowsexcitation function circles, RaisbeckandYiou(1975/?);invertedtriangles,Lindstrometai(1975a);Laumer,Austin,Panggabean(1974);squares, Moyleetal. adopted inthispaper. 16 The foregoingdiscussionhas dealtonlywithproton-nucleusinteractions,butinpassing throughtheinterstellarmedium Although thecrosssectionsforberylliumaremorecompletely measuredthanforboron,boththequantityandqualityof Plots similartoFigures18and 19havebeengeneratedfortheisotopesofmanganese, , andtitanium,theresults For thestudyofshapepath-lengthdistribution, inparticularthecontributionofshortpathlengths(Garcia-Munoz Fig. 17.—Partialproductioncrosssectionfortheisotopesofboron(plusprogenitors)from+0interactions.Datapointsare the following: © American Astronomical Society • Provided by theNASA Astrophysics Data System COSMIC-RAY PROPAGATION Kinetic Energy(MeV/nucleon) 295 198 7ApJS. . .64. .2 6 9G 7 67 6 74 7 67 7 67 7 propagation calculations. were combinedtoobtainthecrosssectionsperatomofinterstellarmedium,whichareactuallyemployed inthe et al.1969;Baixeras-Aiguabella1970;FulmerandGoldberg1975;Rudy1972;Jacobs1974;Glascock etal.1979; et al.1979).Theremainingexperimentalmeasurementsareconcentratedatlowkineticenergies,<50MeVper nucleon (Jung cross sectionvalues(SilberbergandTsaoWild).Experimentaldataforhelium-nucleussectionsarerelatively rareexcept compiled, andexperimentalexcitationcurvesweredeterminedwherepossible.Finally,thehydrogenhelium cross sections Homyak étal.1979),anddeterminethelow-energyendofexcitationcurve.Thehelium-nucleusexperimental datawere similar totheprotoncaseusingsemiempiricalrelationsdevelopedallowscalingfromproton-inducedalphaparticle-induced for theworkofOrsaygroup(RaisbeckandYiou1975a,b,1977;Fontes1975;Yiou,Raisbeck,Quechon Raisbeck 296 half-lives giveninTable2,which listsalloftheelectroncaptureisotopesimportant inthepropagationofcosmicraysbelow isotopes areformedwithnoorbitalelectronsattached.Thus, electroncapturedecayisforbiddenunlessthenucleuscan nickel. Both thenormalandelectron-attached statesareassignedthesametotalinelastic cross sectionandthesameor/?”decay stripped nucleusandagainas anucleuswithoneelectronattached(e.g.,Beand *, thelatterhavinganattachedelectron). Li >(Montmerle1977)andfor<(Yiou,Raisbeck, andQuechon1977).Therefore,weassumethatathighenergiesthe pass throughtheGlagolaetal.(1978)results.Atlowenergies, Glagolaetal.(1978)showthattheLicrosssectionislargerthan value of-1millibamaboveabout400MeVpernucleon.Morerecentwork(Glagolaetal1978)hasshownthat, at leastforthe an electronfromtheinterstellar medium.Ifsuchanelectroniscaptured,thendecay canproceedfollowingthecosmic-ray The crosssectionsfromearlywork(Reeves,Fowler,andHoyle1970;Meneguzzi,Audouze,Reeves1971; Mitler1972; the spallationofotherspecies.Atcosmic-rayenergies parentnucleiarecompletelystrippedofelectrons,andthedaughter interval 20-40MeVpernucleon,LiandBedoindeedhaveidenticalcrosssections.MeasurementsofH+ Be byYiou, excitation functionsforLiandproductioninalpha-alpha interactionsarethesame. nucleon, andupontheoreticalextrapolationsforhigherenergies.TheseanalysesconcludedthatLiBe,being mirror nuclei, Kozlovsky andRamaty1974)werebasedupontheprincipleofdetailedbalance,appliedatenergiesbelowabout 20MeVper function forbothmass7isotopeshavingaconstantvalue of0.01millibamsabove200MeVpernucleonandrisingsmoothlyto Raisbeck, andQuechon(1977)at100-250MeVpernucleongaveaconstantvalueofonly0.01millibams.Weadopt anexcitation should havethesamecrosssection,andLisectionbebelowvalueuntilallsectionsreach aconstant the Livalue,contrarytoearlyassumptions.Nohigh-energy measurementsareavailable,andthereargumentsbothfor 76 In ordertotreattheseelectron capturespecieseffectively,eachlong-livedisotopewasincluded twiceinthecode,firstasafully Isotopes thatdecaybyelectroncapturepresentaunique set ofproblems.Thesearenearlyallsecondaryisotopesproducedby The importanceofthefusiontwoalphaparticlestoproductioncosmic-rayLi,Be,andLihaslongbeen disputed. © American Astronomical Society • Provided by theNASA Astrophysics Data System 10000 9.10 (MeV pernucleon) 4000. 2000. 3000. 1500. 5000. 1000. 600. 400. 900. 800. 700. 200. 500. 300. 100. 150. 60. 90. 70. 50 20.90 80. Energy Experimental PartialCrossSections(millibams)forthe 9.45 9.30 9.60 9.80 10.00 10.60 10.50 10.60 10.65 10.70 10.60 10.00 10.50 13.30 16.90 18.80 10.45 10.23 14.20 15.20 7 V. ELECTRONCAPTUREANDLOSS Be Production oftheBeIsotopes GARCIA-MUNOZ ETAL. 2 C +p- 4.75 4.48 4.05 5.90 5.85 5.40 2.90 2.70 2.40 2.15 5.90 5.90 5.80 5.70 5.50 5.45 5.30 5.15 3.70 3.20 5.00 3.05 Be TABLE 3 u 0.90 0.82 0.72 0.53 0.43 3.35 3.35 3.35 3.35 3.35 3.32 3.20 3.05 2.94 2.85 2.70 2.45 2.12 0.64 3.15 1.62 1.30 Be 4 N+ p- 7 9.70 9.70 9.70 9.70 6.30 4.35 4.15 4.05 4.20 4.60 9.70 9.70 9.60 9.55 9.50 9.43 9.30 9.15 7.30 5.30 8.83 8.37 Be 7 10.70 13.40 12.30 13.50 14.00 Be 9.70 9.70 9.70 9.70 9.70 9.70 9.65 9.53 9.20 6.75 6.00 9.40 8.85 7.60 5.90 9.10 8.30 6 OH y 4.45 4.45 4.45 4.45 4.45 4.45 4.45 4.45 4.45 4.45 4.45 4.45 4.37 4.19 6.10 7.00 3.40 3.81 3.75 5.10 8.80 9.70 Be J 1.92 1.92 1.92 1.90 1.85 2.08 2.27 1.92 1.92 1.92 1.90 1.89 1.87 1.82 1.78 1.68 1.66 1.67 1.76 1.80 1.85 1.94 Be Vol. 64 198 7ApJS. . .64. .2 6 9G No. 1,1987 probability ofelectronstripping isnegligibleforthestep,thenessentiallyallofthese atoms intheelectron-attachedstatedecay radioactive decayforthenormal . immediately uponformation. Fortheseparticularisotopesitisnotnecessarytopropagate theelectron-attachedstateseparately lifetime (ifapplicable).However,productionbyspallation ofheavierspeciesgoesonlytothenormalstate.Theelectron-attached (see note2toTable2),but ratherthecrosssectionforcapturinganelectronby normalstateistreatedasanadditional state specieshastheadditional possibilityofdecayingviaelectroncapturedecay.If the half-lifeofelectron-attachedstate state isproducedwhenanucleusofthenormadisotopecaptures anelectronfromtheinterstellarmedium.Theelectron-attached against electroncapturedecay ismuchlessthanthepropagationtimethrougha typical stepAxinpathlength,andthe of thesemiempiricalcalculations(SilberbergandTsao19736), the dashedcurvegivesexcitationfunctionadoptedinthispaper.Datapointsare et al.(19796);opencircles,Bräutigamai(1981);diamonds, Lavrukhina etal.(1963);crosses,compilationbySilberbergandTsao(1973¿¿). from thefollowing:filledtriangles.Perron(1976);opentriangles,Raisbeck etal.(1979);opensquares,Estrup(1963);invertedfilledtriangles,Westfall 56 p lg—Partialproductioncrosssectionfortheisotopesofvanadium (plusprogenitors)from/?-fFeinteractions.Thesolidcurveistheprediction IG © American Astronomical Society • Provided by theNASA Astrophysics Data System COSMIC-RAY PROPAGATION Kinetic Energy(MeV/nucleon) 297 198 7ApJS. . .64. .2 6 9G 298 propagation medium.Theresultingformalismisinreasonablygoodagreementwiththeexperimentaldata. higher energiesinthecosmic-rayproblemrequiressomemodificationstoformalism(seereviewsbyRaisbeck 1974;Pratt, The deficiencycanberemediedbyincludingtheeffectsofdistantcolhsions(Fowleretal1970;Wilson1978),which bringsthe is treatedintheOppenheimer-Brinkman-Kramers(OBK)approximation,suitablyextrapolatedtohighenergies (Wilson 1977, modifications. (Crawford 1979;Raisbecketal1977).Thetheoreticalformalismfordescribingthecaptureandlossofelectrons was developed Recently, however,somedatahavebecomeavailableathigherenergies(RaisbeckandYiou1971)forheavy-ion beams comparison withtheanalysisofPratt,Ron,andTseng(1973).OurformulaereproducecalculationsCrawford (1979)and calculated crosssectionsintoagreementwiththedata(seediscussioninCrawford1979). (1948). Thiscrosssectionisimportant,since,afterthecaptureofanelectron,itstrippingwhichcontrols return Ron, andTseng1973).Inaddition,comparisonwiththerecentexperimentalmeasurementshasconfirmed many ofthe for low-velocityions(Bohr1948;Oppenheimer1928;BrinkmanandKramers1930),theapplicationofthese formulaeat Raisbeck andYiou(1971)tobetterthan0.3%. to thenormalstate.TheBohrformalismwasfoundconsistentlyunderestimatethiscrosssectioncomparedwith experiment. 1978; Crawford1979)andincludingcorrectionsforcaptureintoexcitedstateselectronscreeninginthe atomsofthe modifications tothesimplefirst-orderBomapproximationcalculation.Thesehavebeendescribedby Wilson(1978) radiative capturecanbetreatedaccuratelybydetailedbalancefromtheinversephotoionizationprocess(Raisbeck andYiou in Fig.18. and havebeenshownbyCrawford(1979)togiveagoodfittherecentexperimentaldata. 1971). Thehigh-energyphotoionizationcrosssectionhasbeendiscussedbyPratt,Ron,andTseng(1973), whosuggest cosmic-ray energyspectrum,fewerparticlesarrivefromhigher energiesthandepartforlowerenergies.Theeffectofenergylosson especially atlowerenergies.Thisenergylossusuallyacts asanadditionalparticlelossterm,since,foranegative-power-law length. Equation(A3)isfor the caseofnoenergyloss,andinclusionionizationresults intheadditionofanotherterm,a',to where NisAvogadro’snumber, Aisthemeanatomicweightofanatominterstellar medium,aisthepartialproduction secondary andoneprimaryparticle.Inthiscasethesecondary-to-primary ratio,S/P,isgivenby secondary-to-primary ratiosismosteasilyseeninthe“leaky box”formalism(GloecklerandJokipii1969),consideringone cross sectionforthesecondary fromtheprimary,Ojistotalinelasticcrosssectionof thesecondary,andismeanescape 0 /y 4556 The crosssectionsforcaptureandstrippingofelectronsbynucleihavebeenmeasuredindetailonlyatverylow energies. The largestchangeintheformalismwasneededforstripping(electronloss)crosssectionoriginallycalculated byBohr The electroncapturecrosssectioniscomposedoftwoparts,radiativeandnonradiativecapture. case For thecalculationsperformedhere,wehaveadoptedformulationspresentedbyWilson(1978)andCrawford (1979) after In manyapplications,radiativecaptureismoreimportantthanthenonradiativeprocess(low-Zionsatlowenergy), and The energylossofchargedparticlesbyionizationduring traversal oftheinterstellarmediumisanextremelyimportanteffect, Fig. 19.—PartialproductioncrosssectionfortheisotopeSc(plusprogenitors)from/?+Feinteractions.Curvesanddatapointsareasdescribed © American Astronomical Society • Provided by theNASA Astrophysics Data System VI. IONIZATIONENERGYLOSS GARCIA-MUNOZ ETAL P A/\N+a/ e0 Vol. 64 198 7ApJS. . .64. .26 -2 -2 2/3 calculations withenergylosscomparedcurvesbforwhichnowasincluded.AnexponentialPLDaconstant valueofX= 9.25 gcmwasassumed,andtheresultsincludesolarmodulationatalevelof<>=490MV. No. 1,1987 were foundtogiveagoodfit tothedatareportedbyAllisonetal.(1976).Itisexpected that,atveryhighenergies,thepresent in Figure20areforanenergy-independentexponential path lengthdistributionwithaconstantmeanX=9.25gcm,and loss stillhasaneffect,~3%forB/Cand10%inthesub-Fe/Fe ratio. include solarmodulation,whichreducesthesizeofeffect atlowenergies.Athighenergy,about1GeVpernucleon,theenergy differences, especiallyatlowenergy,arestriking,withthe sub-Fe/FeratioshowingalargereffectthantheB/Cratio.Thecurves energy lossincreasesforheavier,higherchargeisotopes. formulation shouldgiveionization energylossvaluesaccurateto<15%. Peierls (1971).Thecalculations werecheckedagainstthemeasurementsandcalculations ofCobb,Allison,andBunch(1976) employed, alongwiththeeffective-charge formulationofPierceandBlann(1968).For energiesgreaterthanthosegiveninthe as ZwhileOjisapproximatelyproportionalto^,and Xiseffectivelyconstantwithparticlespecies,therelativeeffectof 0 the denominatorofequation.Theneteffectistoreduce thesizeofcalculatedsecondary-to-primaryratio.Sinceo'scales table (-5GeV),theanalytic formulaofAhlen(1978)wasused,incorporatingthedensity-effect correctionofStemheimerand 0 e Fig. 20.—B/C{top)andsub-Fe/Fe{bottom)ratiosasafunctionofenergy.Datapointsaredescribedin2.Curvesshow theresultof To calculatetheenergyloss,protonstoppingpower tablesofBarkasandBerger(1967)forhydrogenheliumwere Figure 20showstheresultsoftwoseparaterunscode, (a)withionizationenergylossand(b)withoutloss.The © American Astronomical Society • Provided by theNASA Astrophysics Data System COSMIC-RAY PROPAGATION 299 198 7ApJS. . .64. .2 6 9G 10-3 14 -06 _Y _y 300 ionization potentialofiron,sothatthereshouldbelittledifferential enhancementfortheseelements(CasseandGoret1978).The relative toiron(Cameron1981).Inthelattercase,first ionizationpotentialsofMn,Cr,andTiarealmostidenticalwiththe models. Theseabundancesareinagreement(±20%)withthedeterminedbyothergroupsfrom data sets.For involving aGalactichalo(Parker1976)orcloud-intercloudmedium(McKeeandOstriker1977),butsuchinterpretations have isotope Be,weassumeauniformdensityfortheinterstellarmediumof0.2atomscm.Thismayimplyconfinement region Wiedenbeck andGreiner1980;Garcia-Munoz,Simpson,Wefel1981)theiranalysisofpropagationforthe radioactive nearly, puresecondary-to-primaryratios. described inthetext,buthaveonlyasmalleffecton calculatedSc/FeandV/Feratio,sincetheselatterratiosare,more exact valuesoftheMn/Fe,Cr/Fe,andTi/Fesourceratios makeanimportantdifferenceinthecalculatedsub-Fe/Feratio,as for theseratios,eitherzero,whichassumesthesub-Fe elements tobepurelysecondaryspecies,orsolarsystemabundances and V/Feelementalratios,theimportantsourceabundances aretheMn/Fe,Cr/Fe,andTi/Feratios.Wehaveusedtwovalues data. OflessimportanceistheN/0ratioand of oxygentothemajorprimaryelements,Ne,Mg,Si,S,Ar,Ca,andFe. no effectontheresultsofthesecalculations.Wedoassume,however,thatpropagationmediumiscomposed of coldgas giving anatomicweightfortheaverageatomofA=1.191.FollowingGarcia-Munoz,Mason,andSimpson(1977a; seealso below 1GeVpernucleonandfallsasFforenergiesabovenucleon.Solarmodulation<$>=490MVisincluded.For asourceenergy Mewaldt 1981;Simpson1983)fromaconsiderationofbothhigh-andlow-energydataselectiondifferent propagation uniform averagedensity.Theconsequencesofrelaxingthisassumptionwillbediscussedelsewhere. respectively. Curve3givestheresultsforapower-lawsourcespectruminrigiditywithindex2.6. spectrum oftheform(T+7J)),curves1and2giveresultsforr=931MeVpernucleon,y2.3,400 y=2.6, between arigidityandtotal energyspectrum,andhasbeenadoptedfortheanalysis reportedhere. kinetic energypernucleonand Tandyconstantswhichcanbechangedfordifferent calculations.Garcia-Munoz,Mason,and Small changesinthesesourceratiosgiveanegligiblechange inthecalculatedB/Cratio.Forsub-Fe/FeratioandSc/Fe the B/Cratio,mostimportantsourceabundanceratio isO/C,forwhichtherelittlevariationallowedbytheexperimental source incompositionorenergyspectrumareconsidered. spectrum. Bothoftheseareassumedtoberepresentativea“typical”cosmic-raysource,andnodifferencesfrom sourceto modulation, tothemeasured protonandheliumspectrabelow1GeVpernucleon. This formofthespectrumisintermediate Engelmann etal.1981)oramore generalform.Wehaveadoptedtheform(T±T) for thesourceenergyspectrum,withT Ormes, andComstock1981) assume apowerlawintotalenergypernucleon,whileother groupsuseapowerlawinrigidity(e.g., Simpson (19776)haveshown thatvaluesoiT=400MeVpernucleonandy2.6give agoodfit,afterpropagationandsolar 0 0 0 0 For thesecalculationsweassumeaninterstellarmediumcomposedofhydrogenandheliumwith6.9% by number, For thesourcecompositionweuse“average”abundancesdeterminedbyDwyeretal.(1981;seealsosummary by The calculationsrequireinitialconditionsatthecosmic-raysourceconsistingofrelativeabundancesanda energy Fig. 21.—B/Cratioasafunctionofenergy.Datapointsarein2.ThecurvesshowresultsforanexponentialPLDwithmeanthat isconstant The cosmic-rayenergyspectrumatthesourceisamorecontroversial parameter.Manyanalyses(see,forexample,Protheroe, © American Astronomical Society • Provided by theNASA Astrophysics Data System Kinetic Energy(MeV/Nucleon) VII. 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