Solar Core Homology, Solar Neutrinos, and Helioseismology1

Precise numerical standard solar models (SSMs) now agree with one another and with helioseismological observations in the convective and outer radiative zones. Nevertheless, these models obscure how luminosity, neutrino production, and g-mode core helioseismology depend on such inputs as opacity and nuclear cross sections. Although the Sun is not homologous, its inner core by itself is chemically evolved and almost homologous, because of its compactness, radiative energy transport, and hydrogen-burningÈdominated luminosity production. We apply luminosity-Ðxed homology transformations to the core to estimate theoretical uncertainties in the SSM and to obtain a broad class of nonSSMs, parameterized by central temperature and density and purely radiative energy transport in the core. Subject headings : elementary particles È Sun: interior È Sun: oscillations 1. HOMOLOGY AND THE SOLAR CORE After more than three decades of nuclear cross section measurements, opacity calculations, and detailed computer evolutionary calculations, standard solar models (SSMs) with the same inputs now agree in their neutrino Ñux predictions to within about 1%. The theoretical models are now also consistent with precise p-mode helioseismological observations of the SunÏs outer radiative zone x 4 r/R _ \ 0.26È0.71 and convective zone x [ 0.71. If the g-mode helioseismological oscillations have sufficient amplitude, their observation can be expected soon to calibrate the solar inner core, where the thermonuclear luminosity and neutrino production take place. While necessary in the complex convective zone and justiÐed by the precise helioseismological observations, the complexity and numerical form of precise SSMs obscure the simplicity of the solar core and the determinants of solar neutrino Ñuxes. In order to understand standard and nonstandard solar models, we return to the homology methods of Schwarzschild (1958) and Iben used before the advent of fast com(1969, 1991) puters, but with three new features. Castellani et al. have found that changing (1993a, 1993b) input parameters by factors as large as 2 leads to only homologous changes over 60% by mass of the Sun. In this paper, we explain this remarkable homology and demonstrate that, while the entire Sun is certainly not homologous, the core is homologous enough to be parameterized by its central temperature, and density This T c o c . (T c , o c ) parameterization subsumes all astrophysical e†ects of opacity, composition, and the ppI nuclear cross section factors and (the pp and 3He reactions) into the two S11 S33 parameters one representation of the central bound(T c , o c ), ary conditions of solar structure. Indeed, any standard or 1 An earlier version appeared as University Pennsylvania preprint UPRÈ0615-T (1994) and in Proc. INT Solar Modeling Workshop (1994 March). nonstandard solar model that depends principally on radiative energy transport can be parameterized by and (T c , o c ) the remaining nuclear cross section factors and (the S34 S17 3HeÈ4He and pÈ7Be reactions) Even (DegliÏInnocenti 1994). solar models with a nonstandard low opacity or low metallicity Z are all essentially parameterized by or by (T c , o c ) the principal cross section factor determining (See S11, Tc. Fig. 2 of et al. or Fig. 2 of & Langacker Hata 1994 Hata see also As the dependence of the 1995 ; Hata 1994.) o c neutrino Ñuxes is weak (see below), & Langacker ° 4 Hata were able to show that the 0.7% theoretical uncer(1994) tainty in together with the remaining nuclear cross T c , section uncertainties, provide the same theoretical neutrino Ñux and rate uncertainties and correlations that & Bahcall Ulrich obtained from 1000 Monte Carlo SSM simu(1988) lations. (See Figs. 2È4 and 6È8 of & Langacker Hata 1995.) The parameterization allows analytic estimation (T c , o c ) of the logarithmic derivatives of the b j (i) 4 L ln /(i)/L ln S j principal neutrino Ñuxes /(i) with respect to input parameters which & Ulrich obtained from 1000 S j , Bahcall (1988) Monte Carlo SSMs calculated with small changes in input parameters. Because homology makes these logarithmic derivatives constants, from any precise SSM, we cannot only estimate theoretical uncertainties, as did & Bahcall but we can now also extrapolate to non-SSMs, so Ulrich, long as the energy transport is primarily radiative. Our application of homology di†ers from earlier ones in three ways : (1) We apply homology only to the solar inner core, not to the whole Sun ; (2) we do not assume o D T 3 or any polytropic relation ; and (3) we use homology to derive the dependence of core temperature and density on opacity, nuclear energy generation, and mean molecular weight, at Ðxed luminosity, instead of the dependence of e†ective temperature on luminosity, with Ðxed opacity and nuclear energy generation & Giuli (Cox 1968). After accounting for the di†erent energies released when pp, 7Be, 8B, and CNO neutrinos are produced, the known solar luminosity Ðxes the total photon energy proL _