DOE/ER/02A-40561-INT99-31 UF-IFT-HEP-99-7 July 1999 Simple Seismic Tests of the Solar Core 1 Dallas C. Kennedy Institute for Nuclear Theory, University of Washington and University of Florida ABSTRACT A model-independent reconstruction of hyostatic profiles (density, pressure) of the solar interior is outlined using the adiabatic sound speed and bouyancy frequency profiles. These can be inferred from helioseismology if both p- and g-mode frequencies are measured. With homology applied to the solar core only and some additional assumptions, but without a full solar model, the chemical profile of the evolved core can be deduced from the hyostatic profiles as well. Subject headings: Sun: abundances Sun: interior Sun: oscillations Submitted to Astrophysical Journal Letters Standard solar models (SSMs) are now very accurate, exemplified by the work of Bahcall et al. (Bahcall et al. 1998; Pinsonneault 1998; Bahcall 1999) and Turck-Chièze et al. (Brun et al. 1998). But confidence in predicted solar neutrino fluxes is enhanced by checks of the solar core independent of model details, depending on only a few general assumptions. This can be done with helioseismology, although the g-modes are needed. Serious claims of g-modes observations have been put forward (Hill & Gu 1990; Thomson et al. 1995) but remain controversial (Guenther & Sills 1995; Lou 1996). The Bahcall-Pinsonneault 1998 SSM is used here for specific numerical results. 1 Contribution to the INT Low-Energy Neutrino Physics Workshop, Univ. Washington, Seattle (June- September 1999).
2 We assume negligible radiation pressure P γ P gas and no significant departures from an ideal gas (P = P gas = ρrt/µ). Γ=dln P/dln ρ, Γ ad =( ln P/ ρ) ad, (1) = d ln T/dln P, ad =( ln T/ ln P ) ad. Γ ad = 5/3, ad = 2/5 for a monatomic gas. The Brunt-Väisälä (bouyancy) frequency N(r) : N 2 = g [1 λ P Γ 1 ] g = [ ad + µ ], λ 1 P = dln P/, (2) Γ ad λ P with µ = d ln µ/d ln P, is real for propagating g-waves, outside any convective zones. For the Sun, the base of the convective zone R CZ (0.71)R. The adiabatic square sound speed c 2 ad =Γ ad P/ρ. Adiabatic seismology here follows Unno et al. (1989), with some differences in notation. The seismic modes are labeled by radial index n (number of radial nodes) and angular indices l and m; frequencies degenerate in the last if rotation is ignored. The spectrum exhibits middle f (fundamental or n =0,l>1) modes and the p- (g-) modes rising above (falling below) the f mode in ν for n>0, l>0. For large n, thep-andg-modes are concentrated near the surface and the center, respectively. Their frequencies ν p and ν g are ν 1 p (n, l) 2 n + 1 l(l +1) 2 R 0 c ad (r), ν g (n, l) l(l +1) 2nπ 2 RCZ 0 r N(r), (3) for large n. (lcannot be large in ν p.) A large set of measured frequencies allow reconstruction of the c ad (r) andn(r) profiles. Let Ω g the second integral in (3). 1. Some Tests of Hyostatic Equilibrium Correct prediction of the ν g (n, l), n 1, would rule out central convection, although the Sun is not far from this state ( 1.1M ), as N 2 < 0 otherwise (Kippenhahn & Weigert 1990). Present helioseismic data rule out the Γ = 5/3 convective polytrope almost to the center (Basu et al. 1997; Christensen-Dalsgaard 1997). A second test: dc 2 ad = gγ ad (1 1 ) P + Γ ρ dγ ad, 1 1 Γ = µ. (4) In addition, assume dγ ad / = 0 in the core and a non-singular mass distribution, so that g = Gm/r 2 =(4πG/3)ρ c r 0asr 0. (5)
3 Thus dc 2 ad(0)/ must vanish. A third test: assume a hyostatic unperturbed state with no convection (N 2 > 0or Γ<Γ ad ). Hyostatic equilibrium, dp = ρg = Gmρ r 2, (6) with 4πr 2 ρ = dm/, implies a static unperturbed state, the simplest case. But a more complex unperturbed state can include a non-zero steady velocity field v 0, taken here as radial for simplicity. Ignoring rotation and magnetic effects, also calculable, the correction linear in v 0 to frequency ν(n, l) is the expectation value in the (n, l) modeofk r v 0 /2π, with k r = radial wavenumber. The related length scale R /few for low-n modes, but for high-n g-modes, is the central scale R 0 : R 2 0 =3P c /2πGρ 2 c, with R 0 /R = 0.12 for the present SSM (Kennedy & Bludman 1999). For large n, k r n R 0 Ω g 8π3 ρ c G 3 ( 1 1 ) Γ c Γ ad. (7) A conservative relative measurement error for ν g is 0.003, while the theoretical uncertainty 0.001. Thus a direct bound of the core v 0 < 2πν g(n, l)/k r (7.5 10 4 msec 1 ) l(l+1)/n 2, or 120 m sec 1 for l =1,n= 30, is feasible, independent of azimuthal splitting by m. 2 2. Mechanical Profiles The N(r) andc ad (r) profiles are sufficient to reconstruct the mechanical (hyostatic) profiles. N(r) anddc ad (r)/ are proportional, either one controlling the ν g : dc ad (r) N(r)/2 = Γ ad /Γ(r) 1 [ 1 1 ] Γad. (8) Γ(r) Assuming dγ ad / = 0, Γ satisfies a quaatic equation, (1 + D)Γ 2 (2 D Γ ad )Γ + 1 = 0, D(r) = [ ] 2 2 dcad (r)/, (9) Γ ad N(r) 2 This is too large by about 10 7 to detect directly the heavy element mixing invoked to solve the solar neutrino deficit (Cumming & Haxton 1996), although such mixing is disfavored by the measured c ad profile (Bahcall et al. 1997).
4 with one physical root, Γ(r) = 2 D(r) Γ ad + [2 D(r) Γ ad ] 2 4[1 + D(r)] 2[1 + D(r)]. (10) Thus the Γ(r) profile is reconstructible with N(r) and a numerical derivative of c ad (r). The mass and pressure profiles then follow: Gm(r) = r2 dc 2 ad(r)/ Γ ad [1 1/Γ(r)] 4πGρ c = 6 dc2 ad (0)/2 Γ ad (1 1/Γ c ) P (r) = ρ(r) c2 ad (r), Γ ad,, (11) where ρ(r) is obtained from m(r). Note that m>0forr>0asγ>1ifdc 2 ad/ < 0 and Γ < 1ifdc 2 ad / > 0, the present SSM inner core being an example of the latter case. Dimensionless structure in terms of homology variables can also be derived (Kennedy & Bludman 1999). 3. Chemical Profile of the Evolved Solar Core As the SSM evolves, falls while µ increases from zero, the core becoming more concentrated. Helioseismology is adiabatic and non-evolutionary and thus cannot give the T or chemical profiles without a detailed solar model. However, many SSM core features can be understood via mechanical-thermal core homology without a detailed model (Bludman & Kennedy 1996). Homology applied to the core only is valid for power-law opacity and luminosity generation and does not require a polytrope, not even approximately valid for the present SSM core. Unlike ordinary homology, which relates stars of different mass and luminosity, core homology varies the core properties of one star, leaving the total luminosity and mass fixed. The homology in reality in violated somewhat, as the exponents are not constant and the luminosity is not exclusively produced by one reaction chain. Assume luminosity in equilibrium, with power-law dependences: in erg gm 1 sec 1, ε = ε o ρ λ T η ; and luminosity transported by radiative diffusion, with opacity in cm 2 gm 1 κ = κ o ρ n T s. Reasoning similarly to Section 3 of Bludman & Kennedy (1996) yields the scaling laws r η s+3λ+3n ε o κ o µ η s 4 m η s+λ+n 2, (12)
5 µ f [ µ +1 1/Γ] ε o κ o ρ g (c 2 ad) f, (13) f =3 η+s, g = n+λ+1, (14) where (12) is dimensional only (correcting an error in Bludman & Kennedy 1996), and (13) keeps separate track of derivatives. Γ ad is assumed constant. Equation (13) is best solved as a differential equation using the mechanical profiles gleaned in Section 2. The boundary conditions at the core edge are µ µ ZAMS 1.26 and µ = 0. A complication arises from the chemical dependencies of ε o and κ o. With H, He, and metal mass fractions X, Y,and X i, ε o X 2, κ o (1 + X)[X + Y + i X i Z 2 i /A i ], 1 µ = X + Y 4 + i X i A i, (15) A i = nuclear mass number, Z i = nuclear charge number (Kippenhahn & Weigert 1990). X + Y + i X i = 1, and the SSM implies i X i 0.02, although the metal contribution to κ o is non-negligible (Bahcall 1989; Böhm-Vitense 1992) and has to be treated separately by hand. With ρ, P,andµknown, T follows. CNO-dominated fusion can be ruled out, as it would raise, not lower, neutrino fluxes (Bahcall 1989). For ppi-dominated luminosity and realistic OPAL opacity, λ = 1, η = 4, n = 1/2, and s = 5/2 are reasonable approximations (see Böhm-Vitense 1992, Rogers 1994, Iglesias & Rogers 1996, and Bludman & Kennedy 1996 for details), yielding f = 3/2 and g = 5/2. 4. Summary Some solar core tests are proposed, independent of model details. Measured p- and g-modes are needed to reconstruct c 2 ad(r) andn 2 (r). Absence of core convection implies N 2 > 0 in (2) and the existence of ν g (n, l) for high n (3). A non-singular mass distribution implies vanishing of the sound speed slope (4) at the center (5). The high-n ν g can set a direct limit O(10 2 )msec 1 on a central velocity field. Assuming a constant Γ ad allows reconstruction of the solar interior mechanical structure (10-11). Chemical and thermal structure reconstruction requires more detailed assumptions or a full solar model. But many model results are derivable from a simple homology construction applied to the core alone (12-15), with fixed solar mass and luminosity. Truly independent tests of thermal structure and chemical evolution require comparison with other Sun-like stars, in particular via asteroseismology (Deubner et al. 1998). With the appropriate changes, the method outlined here can be used to analyze such stars as well.
6 The author thanks John Bahcall (Institute for Advanced Study) and Marc Pinsonneault of (Ohio State Univ.) for the BP98 model tables and Jørgen Christensen-Dalsgaard (Aarhus Univ.) for helioseismic data. This work is partly based on research done in collaboration with Sidney Bludman (Univ. Pennsylvania and DESY) and was supported by the Department of Energy under Grant Nos. DE-FG06-90ER40561 at the Institute for Nuclear Theory (Univ. Washington) and DE-FG05-86ER40272 at the Institute for Fundamental Theory (Univ. Florida). REFERENCES Bahcall, J. N. 1989, Neutrino Astrophysics (Cambridge: Cambridge University Press) Bahcall, J. N. 1999, URL http://www.sns.ias.edu/ jnb/sndata/solarmodels.html Bahcall, J. N., Basu, S., & Pinsonneault, M. H., 1998, Phys. Lett. B433, 1 Bahcall, J. N., Pinsonneault, M. H., Basu, S., & Christensen-Dalsgaard, J. 1997, Phys. Rev. Lett. 78, 171 Basu, S. et al. (BiSON/LOWL) 1997, Mon. Not. R. Astron. Soc. 292, 243 Bludman, S. A. & Kennedy, D. C. 1996, ApJ 472, 412 Böhm-Vitense, E. 1992, Introduction to Stellar Astrophysics, vol. 3 (Cambridge: Cambridge University Press) Brun, A. S., Turck-Chièze, S., & Morel, P. 1998, ApJ 506, 913 Christensen-Dalsgaard, J. 1997, in T. R. Bedding, A. J. Booth, & J. Davis, eds., Proc. IAU Symp. No. 189 (Dorecht: Kluwer Academic) Cumming, A. & Haxton, W. C. 1996, Phys. Rev. Lett. 77, 4286 Deubner, F.-L., Christensen-Dalsgaard, J., & Kurtz, D., eds., 1998, Proc. IAU Symp. No. 185 (Dorecht: Kluwer Academic) Guenther, D. B. & Sills, K. 1995, in A. B. Balantekin & J. N. Bahcall, eds., Proc. Solar Modeling Workshop, INT, Univ. Washington (Singapore: World Scientific) Hill, H. A. & Gu, Y.-M. 1990, Science in China A37, 854 Iglesias, C. A. & Rogers, F. J. 1996, ApJ 464, 943 Kennedy, D. C. & Bludman, S. A. 1999, ApJ 525(2) Kippenhahn, R. & Weigert, A. 1990, Stellar Structure and Evolution (Berlin: Springer- Verlag)
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