Solar Models with Revised Abundance
aa r X i v : . [ a s t r o - ph . S R ] A p r SOLAR MODELS WITH REVISED ABUNDANCE
S. L. Bi , , T. D. Li , L. H. Li , & W. M. Yang , ABSTRACT
We present new solar models in which we use the latest low abundances and wefurther include the effects of rotation, magnetic fields and extra-mixing processes.We assume that the extra-element mixing can be treated as a diffusion process,with the diffusion coefficient depending mainly on the solar internal configurationof rotation and magnetic fields. We find that such models can well reproduce theobserved solar rotation profile in the radiative region. Furthermore the proposedmodels can match the seismic constraints better than the standard solar models,also when these include the latest abundances, but neglect the effects of rotationand magnetic fields.
Subject headings:
Sun: abundances — Sun: oscillations — Sun: interior
1. Introduction
The standard solar models with the latest input physics are well known to yield the solarstructure to an amazing degree of precision, and agree with the helioseismic inversions (seee.g., Christensen-Dalsgaard et al. 1996; Bahcall et al. 2001). Those models use the old abun-dance values (Grevesse & Savual 1998, hereafter GS98). However, the standard solar modelswith the new solar mixture AGS05 (Asplund et al. 2005, hereafter AGS05) disagree withhelioseismic constraints, e.g., the position of the base of convection zone (CZ) is too shallowand the surface helium abundance is lower than in the Sun (Christensen-Dalsgaard et al.1991; Basu 1998; Basu & Antia 2004). Larger discrepancies are in the sound-speed anddensity profiles between the Sun and the models with low Z (Bahcall & Pinsonneault 2004;Guzik et al. 2005). See Basu & Antia (2008) for a detailed review paper. Department of Astronomy, Beijing Normal University, Beijing 100875, China; [email protected] Key Laboratory of Solar Activity, National Astronomical Observatories, Chinese Academy of Sciences Department of Astronomy, Yale University, P.O. Box 208101, New Haven, CT 06520-8101, USA School of Physics and Chemistry, Henan Polytechnic University, Jiaozuo 454000, Henan, China
2. Beyond the standard solar model
Helioseismology has revealed that the Sun is rotating differentially at the surface,slowly in the core and almost uniformly in the radiative region (Chaplin et al. 1999). Thepurely rotation-induced mixing has been considered in modeling rotating stars, as given inZahn (1992), Maeder & Zahn (1998), Maeder & Meynet (2000), Palacios et al. (2003) andMathis & Zahn (2004). However, these models appear insufficient to reproduce the helio-seismically inferred internal solar rotation profile. This suggests that other effects should beconsidered in extracting angular momentum from the central core of the Sun.Recently, two main mechanisms have been proposed to explain the solar flat rotation pro-file, namely internal gravity waves (e.g. Charbonnel & Talon 2005) and magnetic fields (e.g.Eggenberger et al. 2005). Here, we mainly describe the efficiency of the extra-mixing causedby rotation and magnetic fields, as prescribed by the Tayler-Spruit dynamo (Pitts & Tayler1986; Spruit 2002). The theoretical formulation of this dynamo is still a matter of debate(Denissenkov & Pinsonneault 2007; Zahn et al. 2007), however, Eggenberger et al. (2005)found that the model with the Tayler-Spruit dynamo-type field successfully reproduces theobserved solar rotation profile. Therefore, it is particularly interesting to investigate theeffects of rotation and magnetic fields on the solar interior and global parameters.We present a simple scheme for dealing with angular momentum transport and elementmixing in the solar interior. It is based on the stellar structure equations which include rota-tion and magnetic fields (Pinsonneault et al. 1989; Li et al. 2003). The detailed derivationis given in Yang & Bi (2006, 2008). This formulation allows us to estimate the effects ofrotation and magnetic fields on the Sun properties. The angular momentum transport andelements mixing can be described with two diffusion equations as follows: ρr ∂ Ω ∂t = f Ω r ∂∂r (cid:20) ρr ( D rot + D m ) ∂ Ω ∂r (cid:21) , (1) 3 – ∂X i ∂t = f C ρr ∂∂r (cid:20) ρr ( D rot + D ′ m ) ∂X i ∂r (cid:21) + (cid:18) ∂X i ∂t (cid:19) nuc + (cid:18) ∂X i ∂t (cid:19) micro , (2)where the adjustable parameters f Ω and f C are introduced to represent some inherent un-certainties in the diffusion equations. The second and third terms on the right-hand-side ofEquation (2) are the nuclear and gravitational settling terms, respectively. In our model,the diffusion coefficient D rot is associated with the rotational instability as described byChaboyer et al. (1995). In the case of a Tayler-Spruit dynamo-type field, the diffusion coef-ficient for the angular momentum transport can be written as (Maeder & Meynet 2003): D m = r Ω q (cid:18) Ω N µ (cid:19) (3)and the one for chemical element transport as: D ′ m = r Ω q (cid:18) Ω N µ (cid:19) . (4)Equations (3) and (4) are valid in the regime of negligible thermal diffusion, namely N µ ≫ N T , where N T ( N µ ) represents the thermal ( µ -) gradients associated with buoyancy fre-quency. When this condition is violated, we should replace Equations (3) and (4) with D m = r Ω (cid:18) Ω N T (cid:19) / (cid:18) Kr N T (cid:19) / (5)and D ′ m = r Ω | q | (cid:18) Ω N T (cid:19) / (cid:18) Kr N T (cid:19) / , (6)respectively, where q = − ∂ ln Ω ∂ ln r and K = 4 acT / κρ c p is the thermal diffusivity.Additionally, in order to reproduce the solar surface angular velocity, we adopt theKawaler (1988) braking law: dJdt = f K K Ω (cid:18) RR ⊙ (cid:19) / (cid:18) MM ⊙ (cid:19) − / Ω , (7)where K Ω ≃ . × g cm s and f K is an adjustable parameter related to the magnitudeof the magnetic fields. 4 –
3. Calculations and results
Our solar models are obtained from the one-dimensional Yale Rotating Stellar EvolutionCode (YREC; Guenther et al. 1992; Li et al. 2003; Yang & Bi 2006), by including rota-tion, magnetic fields and relevant extra-mixing processes. In addition, we use the followingupdated physical quantities: OPAL equation of state tables EOS2005 (Rogers & Nayfonov2002), the opacities (GS98, AGS05 and AGSS09) supplemented by the low-temperature opac-ities (Ferguson et al. 2005), diffusive element settling (Thoul et al. 1994) and the Krishna-Swamy Atmosphere T − τ relation.In order to investigate the influence of rotation and magnetic fields, we constructedthese solar models in accord with different physical processes corresponding to differentsolar compositions. In the numerical calculations, all models are calibrated from the initialzero-age main sequence (ZAMS) to the present solar-age models, for which the radius is6 . × cm, the luminosity 3 . × erg/g, the mass 1 . × g and theadopted photospheric Z/X ratio. The free variables are the initial helium abundance Y , theinitial metallicity Z and the mixing-length parameter, all of which are adjusted to matchthese observational constraints. In addition, we assumed that the convective region rotatesrigidly, as proposed by Pinsonneault et al. (1989). The initial angular velocity is anotherfree parameter that can be tuned so that the surface velocities of solar-age models matchthe observed values.Figure 1 shows the angular velocity as a function of radius r at the ages of 2.0 Gyr and4.57 Gyr. For the purely rotating model, the Ω-gradient clearly appears in the radiativeregion. In the solar interior, the angular velocity increases with increasing age during themain sequence, while in the surface it is just the opposite. As a consequence of this at thepresent age the core rotation velocity is about four times as large as the surface one. On theother hand, the angular velocity profile for the model with magnetic fields is significantlydifferent. During the main-sequence stage, the Sun is a quasi-solid body. It is interesting tonote that at the age of 4.57 Gyr, the surface rotation velocity predicated by both models isapproximately 2 . × − rad/s. However, the total angular momentum is quite different.For the calibrated models, the total angular momentum of the rotating model at the age4.57 Gyr is 8 . × g cm s − , which is about five times as large as the seismic result(1 . ± . × g cm s − (Komm & Howe 2003); while for the model with magneticfields at the same age, the total angular momentum is 2 . × g cm s − , which is in goodagreement with the result obtained by helioseismology at 1 σ level. The magnetic field thusconstitutes a more efficient process to transport angular momentum because it enhances thecoupling between the radiative zone and convective one.Rotation and magnetic fields have important consequences on the chemical composition 5 –profile of the outer convective envelope, as shown in Figure 2. By comparing the models withand without rotation and magnetic fields, we find that the extra-mixing process counteractsthe effect of diffusive settling in the outer envelope. Hence, the model with rotation andmagnetic fields has a smoother helium abundance profile than the standard solar model withthe same abundance. This leads to a change in the CZ structure, as well as improvementsin the sound speed and density profiles.For further investigation of the magnetic field’s role, Figure 3 shows in detail the differ-ences between the calculated and inferred sound-speed and density profile (Basu et al. 2009).The different lines refer to calibrated evolved models at the age of 4.57 Gyr, each using adifferent abundance, indicated as GS98, AGS05 and AGSS09. It is clearly visible that thediscrepancy between seismic inferences and solar models with the new lower abundances ismuch larger than one with the old abundance. As illustrated by the different curves, theeffects of rotation and magnetic fields on the stellar structure equations change the hydro-static equilibrium and thermodynamic variables on the solar interior, and therefore also havea significant impact on the solar models. Table 1 summarizes the main characteristics of ourcalibrated models. Interestingly, among all the models listed in the table, model AGSS09cshows the best agreement with the inversions. This model reproduces the sound speed anddensity profiles in within 0 . . R CZ = 0 . R ⊙ whichshows a 8 σ discrepancy, while AGSS09a model shows a 10 σ one. For the surface heliumabundance the situation is analogous: model AGSS09c predicts Y s = 0 .
243 with a 1 . σ dis-crepancy, while model AGSS09a shows the discrepancy at 3 . σ level. Although the modelsincluding rotation and magnetic fields show some improvements with respect to the standardsolar model, they still disagree with the seismic constraints.
4. Conclusions
We have investigated the effects of rotation and magnetic fields on the solar models, andhave found that when these effects are included, alongside the new abundances, the revisedsolar models can better reproduce the helioseismic constraints. However, we see that it isdifficult to match simultaneously the new abundances and helioseismology data for soundspeed, density profiles, convection zone depth and surface helium abundance. Althoughthe Tayler-Spruit dynamo type magnetic field still needs to be studied further, our resultsshow that it does provide a possible explanation for the solar abundance problem. We haveneglected turbulence, which may feed the differential rotation and sustain magnetic fieldsin the convection zone, and other interactions. These physical processes will be considered 6 –in our future work. The results obtained in this paper are encouraging and we intend toapply our model to solar-type stars to get a proper interpretation of the existing helioseismicobservations and the coming asteroseismic ones.S.L.B. acknowledges grant 2007CB815406 of the Ministry of Science and Technology ofthe Peoples Republic of China and grants 10773003 and 10933002 from the National NaturalScience Science Foundation of China. L.H.L. acknowledges the financial support of GrantATM 073770 by NSF and the Vetlesen Foundation of USA.
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This preprint was prepared with the AAS L A TEX macros v5.2.
Model (
Z/X ) s Z s Y s R cz /R ⊙ < δc/c > < δρ/ρ > Y c Z c Y ini Z ini α MLT