198 8ApJ. . .331L. .213 3 1 37 solar neutrinoproblemascosmions. massive particlesasWIMPs,andto thesubsetofparticlesthatresolve must alsoremainintheSun,if theyaretobeeffectiveinenergy the p-modesplittingdiscrepancy. “ solutions”tothesolarneutrinoproblem,whichexacerbate transport.Thiscontrastswithallotherextantintrinsic crepancy betweenobservedlow-degreep-modefrequency splittings andtheoryinsolarmodelsthatincludeWIMP Christensen-Dalsgaard (1986)haveshownthatthereisno dis- Gough, andVahia(1986)Däppen,Gilliland, and energy lossmechanism.Therefore,whilecoolingthecentral outer, moremassiveportionsofthecoremodestly.Faulkner, significantly,theyalsoraisetheof the mions serveasanenergytransportmechanism,not an high-energy neutrinos.Itisimportanttonotethatthecos- the reducedcoretemperaturelowersrateofproduction detected byDavis,isstronglytemperature-dependent.Thus, which isexpectedtobethesourceofmostneutrinos central temperature.ThereactionrateofthePPIIIchain, and Spergel1985;Gillilandetal.1986). transport energyeffectivelyinthecoreandreduceSun’s matter. Throughoutitslifetime,theSunaccretescosmions, could besolvedbythesameparticle(Steigmanetal.1978; accumulating ~5x10~Minitscore.Theseparticles Spergel andPress1985;FaulknerGilliland and ourcollaboratorshavesuggestedthatbothproblems neutrinos intheC1detector(BahcallandUlrich1988).We standard modelsofthesolarinteriorandobservedflux do weunderstandthesourceofdiscrepancybetween not knowwhatcomprises90%ofthemassourGalaxy,nor missing massproblemsandthesolarneutrinoproblem.Wedo 2 3 e 1 The AstrophysicalJournal,331:L21-L24,1988August1 © 1988.TheAmericanAstronomicalSociety.Allrightsreserved.PrintedinU.S.A. OnleavefromPrincetonUniversity Observatory. InthisLetter,wewillrefertothe entireclassofweaklyinteracting, LickObservatoryBulletin,No.1108. The cosmionsmustnotonlybecapturedbytheSun,but In thisscenario,thecosmionscompriseGalacticdark Two oftheoutstandingproblemsinastrophysicsare 18Thls Letter © American Astronomical Society •Provided bythe NASAAstrophysics Data System Subject headings:darkmatter—nuclearreactionsstars:horizontal-branchinteriors no effectonthelaterstagesofstellarevolution. WIMPs eitherrapidlyevaporatefromhorizontal-branchstarsorareinefficientatenergytransport.Theyhave Ii/tI'™' ,reconsiderstheroleofcosmionsinhorizontal-branchstars.Wedemonstratethat isothermal^e thecoresofhorizontal-branchstars;thiswouldsuppressconvectionandalter core oftheSunandsolvesolarneutrinoproblem.Renzinihassuggestedthattheseparticleswouldalso msm mstars.Inpreviouspapers,weandourcollaboratorshaveshownthatWIMPscanisothermalizethe Weakly interacting,massiveparticles(WIMPsorcosmions)canbeanimportantenergytransportmecha- 1 WEAKLY INTERACTING,MASSIVEPARTICLESINHORIZONTAL-BRANCHSTARS I. INTRODUCTION Lick Observatory,BoardofStudiesinAstronomyandAstrophysics,UniversityCalifornia,SantaCruz Received 1988March1;acceptedApril28 Institute forAdvancedStudy 2 David N.Spergel John Faulkner ABSTRACT L21 AND 2 rated inavery short timeonreachingtheHB stage(in<10yr HB convectivecores—isprecisely whytheparticlesareevapo- scale heightisoftheorder or exceedsthephysicalsizeof the strongestpointinhisargument—the factthatthecosmion comparable totheirorbitallengths. Includingthesecorrections vitiates Renzini’sconclusions. WIMP energytransportwhentheirmeanfreepathsare not ration ofWIMPsfromHBstars,and(2)thesuppression of has, however,ignoredtwoimportanteffects:(1)theevapo- of thecosmionexplanationforsolarneutrinoproblem. He ages inferredfromstarcounts. branch lifetimewouldproduceadiscrepancywiththerelative the horizontal-branchlifetimerelativetoasymptoticgiant horizontal branchtotheasymptoticgiantbranch.Shortening would rapidlyexhaustitssupplyofheliumandevolvefrom the core convectionandconsequentsemiconvection,the core cores ofHBstars,thussuppressingcoreconvection.Without cores. Renziniclaimsthatcosmionswillisothermalizethe branch starstransformheliumintocarbonintheirconvective port inthecoresofhorizontal-branch(HB).Horizontal- more advancedevolutionarystagesoflow-massstars.Hehas suggested thatcosmionswouldsignificantlyalterenergytrans- posed thatsatisfytheannihilationconstraint. et al.1985).Severalextensionsofthestandardmodel(Gelmini, Hall, andLin1987;RabyWest1987a,b)havebeenpro- below thatneededtosolvethesolarneutrinoproblem(Krauss and theirresultingequilibriumabundancewouldbereduced massive neutrinos)wouldannihilatesignificantlyintheSun cle physicsdarkmattercandidates(photinos,higgsinos,and metry betweencosmionandanticosmion.Manypopularparti- to theirscatteringcrosssectionortheremustbeanetasym- lation crosssectionatlowenergiesmustbesuppressedrelative cosmions aretoavoidself-annihilationintheSun,theirannihi- Gilliland 1985;GriestandSeckel1987;Gould1987a).Ifthe will beunimportant(SpergelandPress1985;Faulkner transport. Ifthecosmionmassexceeds~4GeV,evaporation Indeed, in§II,weshallshow thatwhatRenziniconsiders Renzini’s arguments,ifcorrect,wouldthreatentheviability Renzini (1987)hasstudiedtheeffectsofcosmionson 198 8ApJ. . .331L. .213 41216 2 L22 for themostdesirablecasesfromsolarneutrinopointof view). are optimalfortransportingenergyeffectivelyintheSun,need coupled onlythroughaspin-dependentaxialcoupling,thenits cant inthecoreofanHBstar,whichiscomposedprimarily effective (abundance-weighted)crosssectionwouldbeinsignifi- not beefficientatenergytransportinHBstars.Ifthecosmion cosmion coupledthroughvectorcouplings,itseffectivecross spin-zero nuclei,He,C,and0.Ontheotherhand,if horizontal-branch stars,notonlyarecosmionsineffectual,but reasons, energytransportissuppressed. than inthesolarcore.Inbothcases,thoughforopposite section wouldbemuchlargerintheheavynuclei-richHBcore the entireclassofWIMPsdonotalterHBevolution. estimated therateofcaptureWIMPsmass,m,withan WIMPs withbaryonsinthestars.PressandSpergel(1985) particles willbecapturedintostarsthroughcollisionsofthe mass, M*,andradius,R*: elastic scatteringcrosssectionoffnucleons,istheinversevelocityaver- aged overtheWIMPhalovelocitydistributionviewedin where mostofthemassisincore,R*coreradius. has beenincludedineq.[1].)Inahorizontal-branchstar, ’s frame.(Thefactor2correctionnotedbyGould[1987h] x This equationisvalidonlywhenmostWIMP-nucleoncolli- xN cussion ofthecapturemassiveWIMPs. sions leadtoWIMPcapture.SeeGould(19876,c)fordis- cussed theeffectsofannihilationonrelativestellarabun- dance ofWIMPs.Severalauthorshaveexploredtherate stars: annihilationandevaporation.Kraussetal(1985)dis- WIMP evaporationfromtheSun(SpergelandPress1985; evaporation ratedependsonlyonWIMPpropertiesand the Gaisser, Steigman,andTilar1986;Nauenberg1987;Gould N WIMPs canescapedirectlyfromthecoreofSun,then the transfers enoughenergysothattheWIMPscanescapefrom number densityofnucleiinthecore,n,temperature,T and the .IfWIMPcrosssectionissmallenoughthat the the depthofpotential,: If theWIMP-nucleoncross section islarger,WIMPsevapo- rate fromtheradiusoflastscatter, R,where 1987a). EnergeticnucleiscatteringoffWIMPsoccasionally Chamberlain (1978) estimatestheevaporation rateintheopti- c C last In §III,weshallshowthatcosmions,whosecrosssections We willgeneralizeourdiscussionandfindthatin If thehaloofourgalaxyiscomposedWIMPs,thenthese There aretwoprocessesthatdepleteWIMPabundancein II. ABUNDANCEOFWIMPSINHORIZONTAL-BRANCHSTARS 29 V %9x10 © American Astronomical Society •Provided bythe NASAAstrophysics Data System ¿log AT dt 1 GeV/Ro m, 3 VO.Ol Mpc J^last q i n(r)adr=I. (3) NxN 1 M*R* Px mm 300 kms ™®c\ X kT J c -, 1 SPERGEL ANDFAULKNER (1) (2) 1 HB 8 8 2 cally thickregime, These ratesdefineacharacteristicevaporationtimescale, the horizontal-branchlifetime,thenN%ÑT,where T =(dlogN/dty.Iftheevaporationtimeisshorterthan Ñ isthecapturerateofWIMPsbyahorizontal-branchstar. and WIMPsdonotevaporateduringtheheliumflash,then If theevaporationtimeexceedshorizontal-branchlifetime nearly evaporatesitsheliumcoreduringtheflash,we and Tisthestar’smain-sequencelifetime.Sincestar N äÑi,whereisthemain-sequencecapturerate vatively assumethatWIMPsremainboundduringthisphase. suspect thatmostWIMPsarelost.However,weshallconser- index, n(Chandrasekhar1939): only onthemeanmolecularweight,/i,andpolytropic where (/>(luminosity as We representthecrosssection dependenceofpeakWIMP out ofthecentralcore—this also suppressesenergytransport. III. WIMPENERGYTRANSPORTINHORIZONTAL-BRANCHSTARS In apolytropicmodel,thedepthofpotentialdepends Despite theirlowabundanceinstars,WIMPscanbe d logN dt m T(r)] «(RlascK 04 ^-* r m hL ‘ V^He * P r X (n+l) /2 xN^ 2 \nGmjimn) 3 /fcTy xpc c mRu exp m dd> L /cT(R)J• F _x^(^last)!(4) last Nr 2 1 +K’ K Vol. 331 (5) (6) (7) 198 8ApJ. . .331L. .213 8 4-3 8 4-3 jump intheluminositycarried by7.5mWIMPsintheradi- remain inthestar. Figure2showsthatineither case,particles core asafunctionofcrosssectionfor5,7.5,and10mWIMPs. (solid line)showsthepeakWIMPluminosityinaconvective core luminosityofthestar(15 L).ThedottedlineinFigure2 The WIMPluminosityisexpressed asafractionoftheinitial the HBlifetimeandWIMPs capturedduringtheMSphase ative coremodelsoccurswhen theevaporationtimeexceeds shows thepeakluminosityin aradiativecore(n=3.25).The model ofSweigartandGrossmodel,1.18x10K.Figure 2 ß =4/3polytrope.Thecentraltemperatureofthepolytrope, p of 2x10gmcm.WefittedtheHBcorewithann= 1.5, estimate ofcosmionluminosity. corresponding efficiencylimitationsonenergytransportin his tal branch,itscoreisnearlypureheliumwithacentraldensity “ standard”0.66MHBstar.Whenastarentersthehorizon- alternatively, Nauenberg[1987]).Renzini(1987)ignored the discussion ofeqs.[2.4]-[2.10]inSpergelandPress[1985], or form betweenthelongandshortmeanfreepathregimes(see for peakluminosity(eq.[7])representsaconvenientbridging where riiisthenumberdensityofisotopei.Theexpression No19 1.27 x10K,iscomparabletothecentraltemperatureof the p cosmion musttravelbeforeitsenergyequilibratesthroughcol- 0 lision toitsscaleheight,r: Knudsen number,isdefinedastheratioofdistancea where T(r)isthestellartemperatureatradiusr.K, polytrope. ThetopaxisshowstherelationshipbetweenKnudsennumberandcrosssectionforcentraldensity,p«2x10gem. 0 x x c - 18WIMPsINHORIZONTAL-BRANCHSTARS FGLwaomtlontime asa i!' F;^functionofmassandcrosssection.Theevaporationtimewasestimatedusingeqs.(3)-(5)fora0.66pure n=1.5 Sweigart andGross(1984)describetheevolutionof a rrt © American Astronomical Society •Provided bythe NASAAstrophysics Data System 1 K- rna = xY.iix i 2 (m +m*)’ x 4m nii x Knudsen # (8) and NJHighTechnology grant86-240090-2. J. F.wassupportedinpart bygrantsfromtheNSF(AST Research; D.N. S.wassupportedbyNSFgrant PHY-8620266 83-11406 A01),CalSpace, and theUCSCCommitteeon helpful discussionandJohnN. Bahcallforhisencouragement. Bailyn, BohdanPaczynski,Stuart Raby,andMartinReesfor begun. BothauthorsthanktheInstituteforTheoretical the LargeScaleStructureWorkshopatwhichthiswork was Physics (U.C.SantaBarbara) foritshospitalityduringthe 1987 WorkshoponSolarNeutrinos. D.N.S.thanksCharles WIMPs inhorizontal-branchstarsaretrulyineffectual. port alargefractionofthestar’sluminosity.Weconclude that mean freepathistooshorttotransportenergyefficiently. Renzini’s arguments,liketheWIMPs,evaporate. either evaporationseverelydepletestheirabundanceor More massiveWIMPsaretoocentrallyconcentratedtotrans- stars. Particleslessmassivethan8GeVareineffective,since effective thansuggestedbyFigure2. evaporate morequicklyinolderHBstarsandareevenless for thelowerpartofthisrange. quency separation),requires4m