.213 The Astrophysical Journal, 331 : L21-L24,1988 August 1 © 1988. The American Astronomical Society. All rights reserved. Printed in U.S.A. .331L. 8ApJ. 198 WEAKLY INTERACTING, MASSIVE PARTICLES IN HORIZONTAL-BRANCH STARS1 David N. Spergel2 Institute for Advanced Study AND John Faulkner Lick Observatory, Board of Studies in Astronomy and Astrophysics, University of California, Santa Cruz Received 1988 March 1 ; accepted 1988 April 28 ABSTRACT Weakly interacting, massive particles (WIMPs or cosmions) can be an important energy transport mecha- msm m stars. In previous papers, we and our collaborators have shown that WIMPs can isothermalize the core of the Sun and solve the solar neutrino problem. Renzini has suggested that these particles would also isothermal^e the cores of horizontal-branch stars; this would suppress convection and alter horizontal-branch Ii/tI'1™'8' ,Thls Letter reconsiders the role of cosmions in horizontal-branch stars. We demonstrate that WIMPs either rapidly evaporate from horizontal-branch stars or are inefficient at energy transport. They have no effect on the later stages of stellar evolution. Subject headings: dark matter — nuclear reactions — stars: horizontal-branch — stars: interiors I. INTRODUCTION transport. If the cosmion mass exceeds ~4 GeV, evaporation Two of the outstanding problems in astrophysics are the will be unimportant (Spergel and Press 1985; Faulkner and missing mass problems and the solar neutrino problem. We do Gilliland 1985; Griest and Seckel 1987; Gould 1987a). If the not know what comprises 90% of the mass of our Galaxy, nor cosmions are to avoid self-annihilation in the Sun, their annihi- do we understand the source of the discrepancy between the lation cross section at low energies must be suppressed relative standard models of the solar interior and the observed flux of to their scattering cross section or there must be a net asym- neutrinos in the 37C1 detector (Bahcall and Ulrich 1988). We metry between cosmion and anticosmion. Many popular parti- and our collaborators have suggested that both problems cle physics dark matter candidates (photinos, higgsinos, and could be solved by the same particle3 (Steigman et al. 1978; massive neutrinos) would annihilate significantly in the Sun Spergel and Press 1985; Faulkner and Gilliland 1985; Press and their resulting equilibrium abundance would be reduced and Spergel 1985; Gilliland et al. 1986). below that needed to solve the solar neutrino problem (Krauss In this scenario, the cosmions comprise the Galactic dark et al. 1985). Several extensions of the standard model (Gelmini, matter. Throughout its lifetime, the Sun accretes cosmions, Hall, and Lin 1987; Raby and West 1987a, b) have been pro- 11 accumulating ~ 5 x 10~ Me in its core. These particles posed that satisfy the annihilation constraint. transport energy effectively in the core and reduce the Sun’s Renzini (1987) has studied the effects of cosmions on the central temperature. The reaction rate of the PP III chain, more advanced evolutionary stages of low-mass stars. He has which is expected to be the source of most of the neutrinos suggested that cosmions would significantly alter energy trans- detected by Davis, is strongly temperature-dependent. Thus, port in the cores of horizontal-branch (HB) stars. Horizontal- the reduced core temperature lowers the rate of production of branch stars transform helium into carbon in their convective high-energy neutrinos. It is important to note that the cos- cores. Renzini claims that cosmions will isothermalize the mions serve as an energy transport mechanism, not as an cores of HB stars, thus suppressing core convection. Without energy loss mechanism. Therefore, while cooling the central core convection and consequent semiconvection, the core temperature significantly, they also raise the temperature of the would rapidly exhaust its supply of helium and evolve from the outer, more massive portions of the core modestly. Faulkner, horizontal branch to the asymptotic giant branch. Shortening Gough, and Vahia (1986) and Däppen, Gilliland, and the horizontal-branch lifetime relative to the asymptotic giant Christensen-Dalsgaard (1986) have shown that there is no dis- branch lifetime would produce a discrepancy with the relative crepancy between observed low-degree p-mode frequency ages inferred from star counts. splittings and theory in solar models that include WIMP Renzini’s arguments, if correct, would threaten the viability energy transport. This contrasts with all other extant intrinsic of the cosmion explanation for the solar neutrino problem. He “ solutions ” to the solar neutrino problem, which exacerbate has, however, ignored two important effects: (1) the evapo- the p-mode splitting discrepancy. ration of WIMPs from HB stars, and (2) the suppression of The cosmions must not only be captured by the Sun, but WIMP energy transport when their mean free paths are not must also remain in the Sun, if they are to be effective in energy comparable to their orbital lengths. Including these corrections vitiates Renzini’s conclusions. 1 Lick Observatory Bulletin, No. 1108. Indeed, in § II, we shall show that what Renzini considers 2 On leave from Princeton University Observatory. the strongest point in his argument—the fact that the cosmion 3 In this Letter, we will refer to the entire class of weakly interacting, scale height is of the order of or exceeds the physical size of the massive particles as WIMPs, and to the subset of particles that resolve the HB convective cores—is precisely why the particles are evapo- solar neutrino problem as cosmions. rated in a very short time on reaching the HB stage (in < 102 yr L21 © American Astronomical Society • Provided by the NASA Astrophysics Data System .213 L22 SPERGEL AND FAULKNER Vol. 331 .331L. for the most desirable cases from the solar neutrino point of cally thick regime, view). d log N mRu m 8ApJ. F _ x ^(^last)! (4) In § III, we shall show that cosmions, whose cross sections «(RlascKxN^ exp are optimal for transporting energy effectively in the Sun, need dt L /cT(R ) J • 198 last not be efficient at energy transport in HB stars. If the cosmion These rates define a characteristic evaporation time scale, coupled only through a spin-dependent axial coupling, then its 1 T = (d log N/dty . If the evaporation time is shorter than effective (abundance-weighted) cross section would be insignifi- evap cant in the core of an HB star, which is composed primarily of the horizontal-branch lifetime, then Nx % ÑHB Tevap, where spin-zero nuclei, 4He, 12C, and 160. On the other hand, if the ÑHB is the capture rate of WIMPs by a horizontal-branch star. cosmion coupled through vector couplings, its effective cross If the evaporation time exceeds the horizontal-branch lifetime section would be much larger in the heavy nuclei-rich HB core and WIMPs do not evaporate during the helium flash, then than in the solar core. In both cases, though for opposite Nx ä Ñms ims, where Ñms is the main-sequence capture rate reasons, energy transport is suppressed. and Tms is the star’s main-sequence lifetime. Since the star nearly evaporates its helium core during the helium flash, we We will generalize our discussion and find that in horizontal-branch stars, not only are cosmions ineffectual, but suspect that most WIMPs are lost. However, we shall conser- vatively assume that WIMPs remain bound during this phase. the entire class of WIMPs do not alter HB evolution. In a polytropic model, the depth of the potential depends only on the mean molecular weight, /i, and the polytropic II. ABUNDANCE OF WIMPS IN HORIZONTAL-BRANCH STARS index, n (Chandrasekhar 1939): If the halo of our galaxy is composed of WIMPs, then these hL dd> particles will be captured into stars through collisions of the m (n+l) (5) WIMPs with baryons in the stars. Press and Spergel (1985) mP estimated the rate of capture of WIMPs of mass, mx, with an where (/>(<!;) is the Lane-Emden function and ^ is the first zero elastic scattering cross section off nucleons, <jxN , into a star of of this function. In the radiative core of the Sun (n = 3.25, mass, M*, and radius, R* : H = 0.7) Oc° = l.SkTJmp. Cosmions with masses greater than ~4 GeV do not evaporate in a solar lifetime. Horizontal- 29 Px 300 km s branch stars have convective cores that are primarily helium V % 9 x 10 3 HB VO.Ol Mq pc {n = 1.5, fi = 4/3); thus Oc æ 33kTc/mp. Figure 1 shows that 1 the evaporation time for WIMPs that interact once per m, M* R* dynamical time is much shorter than the horizontal-branch mm -, 1 (1) 8 1 GeV/ Ro lifetime of ~ 10 yr for mx < 9mp(8 GeV). If cosmions could 2 successfully suppress convection, then the core may become where a* = mNR /M*, and <l/y> is the inverse velocity aver- radiative. For a representative polytropic index of n = 3.25, aged over the WIMP halo velocity distribution viewed in the cosmions with mass less than ~8mp evaporate in less than star’s frame. (The factor 2 correction noted by Gould [1987h] 108 yr. For the cosmions, most efficacious at solving the solar has been included in eq. [1].) In a horizontal-branch star, neutrino problem, the evaporation time scale can be as short as where most of the mass is in the core, R* is the core radius. 102 yr. This equation is valid only when most WIMP-nucleon colli- sions lead to WIMP capture. See Gould (19876, c) for dis- cussion of the capture of massive WIMPs. III. WIMP ENERGY TRANSPORT IN HORIZONTAL-BRANCH STARS There are two processes that deplete WIMP abundance in Despite their low abundance in stars, WIMPs can be stars: annihilation and evaporation.