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Standard Solar Model: Status and Prospects
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Nuclear Physics B (Proc. Suppl.) 235–236 (2013) 41–48 www.elsevier.com/locate/npbps
Standard Solar Model: Status and Prospects A. Serenellia a Institute
of Space Sciences (CSIC-IEEC), Campus UAB, Bellaterra, Spain
Abstract I present a personal perspective of the current status of standard solar models, the comparison with current constraints from helioseismology and neutrino fluxes derived from available data. Results are discussed in the context of the solar abundance problem, and it is shown that current solar neutrino experiments can not discriminate between different alternatives for the solar composition available in the literature. The importance of a measurement of the neutrino fluxes from the CN-cycle is emphasized, not only because of the possibility of discerning between solar compositions, but also in connection to the formation of the planetary system. Keywords: Sun, neutrinos, helioseismology, solar abundances, accretion
1. Introduction For about four decades, the solar neutrino problem was a powerful driving force in neutrino and solar physics research. Albeit the unexpected and wonderful discoveries of neutrino properties, the original motivation for establishing solar neutrino experiments, namely probing how the Sun shines and the internal solar structure, has been only partially fulfilled. It is only in the last few years, with the precise measurements of the 8 B and 7 Be neutrino fluxes and the prospects of probing pep (indirectly determining pp) and CN neutrinos, that solar neutrinos are becoming a very competitive tool to constraint solar interior properties. At present, standard solar models (SSMs) come in two flavors, depending on the choice of the solar surface metallicity (Z) used to construct the model. HighZ models use solar abundance determinations that reproduce helioseismic probes of the solar interior but are based on previous generation of spectroscopic abundance determinations. Low-Z solar models, on the contrary, are based on the most refined solar abundance determinations but do not reproduce helioseismic constraints. This is the solar abundance problem. Here I will present the most recent results for the two flavors of standard solar models (SSM) and dis0920-5632/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.nuclphysbps.2013.03.009
cuss the problems and possibilities to discriminate between solar compositions using current and future data, both from helioseismology and solar neutrinos. I also present some speculative arguments that link the apparently unconnected process of planetary system formation and solar neutrinos. 2. Standard Solar Models Results presented here are based on the SSM that include the most up-to-date microscopic input physics, particularly regarding nuclear reaction cross sections, for which we adopt both the central values and uncertainties recommended in the recent review Solar Fusion II (SFII; [1]). A complete description of input physics of the models is given in [2]. The only difference between the two solar models upon which I base the discussion is the choice of the solar surface abundance determination. The two choices used are as follows: • SFII-GS98: model constructed using the solar composition given in [3], derived from 1D atmosphere models. In a nutshell, this model is characterized by the present-day surface metal-tohydrogen ratio (Z/X) = 0.0229 and it is representative of high-Z SSMs;
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• SFII-AGSS09: model that incorporates solar abundances from [4], obtained from 3D model atmospheres. The reduction by 30 − 40% in the abundances CNO and Ne presented in [4] leads to (Z/X) = 0.0178, the value used for this low-Z SSM. The present-day solar composition is of crucial importance for SSM because, aside from the 10 to 12% change in the surface metal abundances induced by gravitational settling during the evolution of the Sun, the adopted (Z/X) basically determines the internal composition of the model. 3. Results 3.1. Helioseismology The impact of the different solar abundances on solar models is most readily seen in properties of the solar structure that can be probed with helioseismic data. There is a large amount of literature devoted to the solar abundance problem [5, 6, 7]. Here we just present a summary. The most striking change in solar models brought up by the low-Z abundances is the mismatch of the predicted sound speed profile between low-Z models and the Sun. This is illustrated in the left panel of Figure 1 where the relative difference of sound speed between the Sun (derived from inversions [8]) and the models is shown as a function of solar radius. It is evident from this plot that the low-Z SFII-AGSS09 SSMs is much worse than the high-Z SFII-GS98 model in reproducing the solar sound speed. Error bars in the solar sound speed from the helioseismic data are shown for reference. The most prominent feature in this plot is the peak right below the convective envelope (CZ; grey area in the plot). The origin of this peak mostly lies in the wrong location of the boundary of the convective envelope in the SFII-AGSS09 model. In fact, from helioseismology we know RCZ = 0.713 ± 0.001R [9] while RCZ = 0.723R in the SFII-AGSS09 model. On the contrary, the SFII-GS98 predicts the right location of the convective envelope boundary and it does a much better job on the sound speed right below the CZ, the agreement being about 4 times better than with SFIIAGSS09. The right panel of Figure 1 shows analogous results but for the relative difference in density profiles. Here the disagreement between SFII-AGSS09 and the Sun is very large in the convective envelope, but the argument against low-Z models is less compelling. The reason is
that the total solar mass is a constraint imposed on density inversions and a small difference between the Sun and models in the high density central regions translates into a large difference (of opposite sign) in the outer low density regions. The are other helioseismic probes that can be used to test the internal structure of solar models. For reference, below is a summary of the current state of affairs for constraints I consider robust. • Boundary of convective envelope RCZ . Helioseismology: 0.713 ± 0.001R [9], SFII-GS98: 0.712R , SFII-AGSS09: 0.723R , model uncertainty: ±0.004R [10]. • Sound speed. Maximum relative difference - SFIIGS98: 0.3%, SFII- AGSS09: 1.0%. Average rms deviation - SFII-GS98: 0.09%, SFII-AGSS09: 0.37%. • Density. Maximum relative difference - SFIIGS98: 2%, SFII-AGSS09: 7.3%. Average rms deviation - SFII-GS98: 1.1%, SFII-AGSS09: 4%. • Surface helium abundance YS . Helioseismology: 0.2485 ± 0.0034 [5], SFII-GS98: 0.243, SFIIAGSS09: 0.232, model uncertainty: ±0.004 • Core averaged mean molecular weight μC 1 . Helioseismology: 0.723±0.002 [12], SFII-GS98: 0.720, SFII-AGSS09: 0.713, model uncertainty: ±0.003 [12]. The solar abundance problem can be understood from the list given above. There is a systematic failure of low-Z SSMs to reproduce solar properties as determined from helioseismology and, on the other hand, high-Z models, of which SFII-GS98 is representative, agreement is quite good. 3.2. Neutrinos Solar neutrinos are also affected by the choice of solar composition. A low-Z solar model has a lower core temperature than a model with higher-Z and, because solar models have to reproduce L , the effect is that low-Z models have a slight increase in the neutrino fluxes with smallest temperature sensitivity and a decrease in the most sensitive ones. Results for SFII-GS98 and SFIIAGSS09 are given in Table 1. Changes in the fluxes from the pp-chains simply show the response to the 1 Based on using small separation ratios of low-degree oscillation modes[11]
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Figure 1: Sound speed and density relative differences between the Sun and solar models obtained from inversions of helioseismic data [8]. The convective envelope is shown with the grey area.
change in the temperature, which is on average about 1.2% lower in the SFII-AGSS09 model in the neutrino producing region (R < 0.30R ). The largest changes in these fluxes of course correspond to 7 Be and 8 B, which differ by about 10% and 20% respectively in the two models. Figure 2 summarizes the current status of experiments and a direct comparison with solar models. Different determinations of solar fluxes from available experimental data have been presented [2, 13] and are in good agreement. Results given in the last column of Table 1 have been obtained using methods used in previous studies [14]. The remarkable results from Borexino for 7 Be, together with the measurements of 8 B by SNO and SK allow a very precise determination of these fluxes to 4.5% and 3% respectively. Borexino also allows for a precise determination of the pp and pep fluxes [13] when the luminosity constraint is used, as the uncertainties of 0.6% and 1.2% for these fluxes reflect. The precision of the experimental solar neutrino determinations that has been achieved were unthought of just some years ago. The situation is ripe for the original goal of Davies and Bahcall to be fulfilled. A natural first step in this direction is to contrast solar fluxes with lowZ and high-Z models. The last row in Table 1 gives the results of this comparison for the SFII-GS98 and SFIIAGSS09 models. In both cases models agree very well with the solar values and, unfortunately, it is at present not possible to discriminate between models using solar neutrinos. In Figure 3 the comparison between solar and model fluxes is shown for 7 Be and 8 B. The experimental result falls right in the region where error ellipses (1σ)
Flux pp pep hep 7 Be 8B 13 N 15 O 17 F χ2 /Pagr
SFII-GS98 5.98(1 ± 0.006) 1.44(1 ± 0.011) 8.04(1 ± 0.30) 5.00(1 ± 0.07) 5.58(1 ± 0.14) 2.96(1 ± 0.14) 2.23(1 ± 0.15) 5.52(1 ± 0.17) 3.5 / 90%
SFII-AGSS09 6.03(1 ± 0.006) 1.47(1 ± 0.012) 8.31(1 ± 0.30) 4.56(1 ± 0.07) 4.59(1 ± 0.14) 2.17(1 ± 0.14) 1.56(1 ± 0.15) 3.40(1 ± 0.16) 3.4 / 90%
Solar 6.05(1 ± 0.006) 1.46(1 ± 0.012) 18(1 ± 0.45) 4.82(1 ± 0.045) 5.00(1 ± 0.03) ≤ 6.7 ≤ 3.2 ≤ 59 —
Table 1: Results for solar neutrino fluxes. First two columns for solar models, as indicated; last column solar values determined from all available neutrino data and the luminosity constraint. Units are, as usual, 1010 (pp), 109 (7 Be), 108 (pep, 13 N, 15 O), 106 (8 B, 17 F), 103 (hep) cm−2 s−1 . Last row shows results of a χ2 test comparing the models solar the solar data.
of models overlap. The solar fluxes used here were not derived using solar neutrino data collected over the last year, e.g. SuperK IV results (Smy, this volume) or the upper limit on CNO fluxes from Borexino [15]. However, it is not expected that solar values for fluxes from pp-chains will change significantly because of this.
4. Discussion As presented in the previous section, helioseismology strongly favors SSMs with high-Z, as the comparison of results of the SFII-GS98 and SFII-AGSS09 models against helioseismology show. On the other hand, solar neutrino fluxes are well reproduced by both models. As it has been pointed out previously [16], helioseismic probes listed above are not directly sensitive to the
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Figure 2: Direct comparison (no oscillations accounted for) of the available data from solar neutrino experiments and detailed comparison with the SFII-GS98 solar model as indicated. Also, total expected fluxes for the low-Z SFII-AGSS09 model are shown. Results for the radiochemical experiments are given in SNU; in all other cases fluxes are normalized to predictions from the SFII-GS98 model.
metal abundance of solar models but rather to the opacity of stellar matter to radiation, which determines the efficiency of energy transport in the solar interior below the CZ. Therefore, as has already been discussed in detail [17] the most relevant effect of modifying the abundance of metals in the Sun is the change this produces in the radiative opacity. This introduces a degeneracy in solar models with respect to variations in the metal composition and changes, e.g. due to uncertainties, in radiative opacities.
Figure 3: 7 Be vs 8 B neutrino fluxes. For the model 1σ regions are shown, including correlations. Experimental data (cross) is as given in Table 1.
The reader can understand the complications introduced by this degeneracy by inspecting Figure 4, where the relative contributions to the total Rosseland mean opacity κ are shown for some relevant elements. Clearly, there is not a one-to-one correspondence between κ and the total metallicity, and this makes it even a more complex task to extract information about the solar composition from either helioseismology or solar neutrinos. In fact, the best that can be done with the available information is to constrain the variations in κ that have to be introduced in a reference solar model to match observational constraints as it has been done in [17]. Transforming a derived opacity change into a change
A. Serenelli / Nuclear Physics B (Proc. Suppl.) 235–236 (2013) 41–48
Figure 4: Fractional contribution to the Rosseland mean opacity of a few selected elements. The bottom boundary of the convective envelope is shown with a vertical dotted line.
in solar composition with respect to the reference solar model used can be attempted in simplified forms [18, 19], but it strongly relies on the assumption made for the intrinsic uncertainty in radiative opacities calculations. It is very unsettling that such information is completely missing in the literature and we have to comfort ourselves by assuming that comparison between different sets of opacity calculations give us a measure of the actual uncertainty in opacity calculations. In fact, recent work [20] has shown that small differences, of a few percent, in the mean Rosseland opacity are the result of cancellation of much larger differences in the opacity contributions of single elements. It may well be that agreement between different opacity calculations for conditions appropriate to solar radiative interior (OPAL [21], OP [22], OPAS[20]) is just mere chance. The almost perfect degeneracy between composition and opacity regarding helioseismic probes of solar structure has been shown previously [16]. For the 7 Be and 8 B results have also been presented [23] which we update here in Figure 3, where the dashed line, indicated in the figure as SFII-AGSS09+κ shows results for a solar model with AGSS09 composition with the only modification that the opacity has been increased to match, as a function of radius, that of the SFII-GS98 model. As shown, 7 Be and 8 B fluxes from this model almost coincide with results from model SFII-GS98. The question therefore is: how can the degeneracy between radiative opacity and metallicity be broken? Probes of the solar composition that do not depend on opacity, or at least not only on opacity, are needed. Helioseismology offers a possibility that has been investigated [24] and yields Z = 0.0172 ± 0.002, in good
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agreement with the GS98 result. This result depends, however, on a subtle signal imprinted on the sound speed by the partial ionization of metals and therefore, to extract the metal abundance from the helioseismic signal an accurate equation of state (EoS) is needed. Also, even the high-Z solar models do not perform particularly well in reproducing the solar adiabatic index [25, see Fig. 4] in the region of partial ionization of metals, which may be an indicator of deficiencies in the EoS. Potentially, solar neutrinos from the CNO-bicycle offer another probe of solar metallicity that is only mildly degenerate with stellar opacity. In Figure 5 we show the added 13 N and 15 O neutrino fluxes (CN for short), the two CNO fluxes that can be experimentally detected, for the two reference models and the SFII-AGSS09 model with increased opacity. The shaded area represents the experimental determination of the solar 8 B flux. The difference between the SFII-AGSS09 and SFII-AGSS09+κ models reflects the κ-dependence of the fluxes. Contrary to fluxes from pp-chains, fluxes from the CN-cycle carry an extra (dominating) linear dependence on the C+N solar core abundance because the added abundance of these elements catalyze the nuclear cycle. Additionally, we have included the upper experimental limit recently announced by the Borexino collaboration [15], which sets 7.7 × 108 cm−2 s−1 as the most stringent upper limit on the added 13 N+15 O flux to date . A crucial problem in determining the CN neutrino flux in liquid scintillator detectors is the 210 Bi background. In this regard, a very interesting possibility for removing this background has recently been put forward [26], but has not yet been included in the analysis of Borexino, that could allow, with the data already available, an actual measurement of the CN flux. A rather direct way of determining the C+N abundance in the solar core relies on the similar temperature sensitiviy these fluxes share with 8 B [27]. The correlation between these fluxes can be used to isolate the linear dependence of the CN fluxes with the C+N abundance and use the experimental determination of the 8 B flux as a thermometer to fix the solar core temperature. The role of solar models is merely that of a normalization of the relation between fluxes and composition. Updating the relation from the original publication, it can shown that exp 8 0.828 Φexp (CN) Y(C + N) Φ ( B) × (1) = SSM 8 ΦSSM (CN) Φ ( B) YSSM (C + N) 1 ± 0.03(8 B) ± 0.028(env) ± 0.03(θ12 ) ± 0.10(nuc) . In this expression Y(C+N) is proportional to the number
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5. Prospects
solar properties. 8 B can now be used as a very good thermometer of the solar core and 7 Be, together with the solar luminosity constraint allow for the determination of the pp and pep neutrino fluxes to precisions of the order of 1%. The next goal is the determination of the CN fluxes, and Borexino has given the first solid step in this direction by setting the most stringent upper limit to date, only 1.5 times larger than predictions from solar models with high metallicity. Further improvements in the, already astonishing, radiopurity level of Borexino and, very importantly, a new technique to extract the 210 Bi background [26], raise the expectations that a measurement of the combined CN flux will happen in the near future. On the negative side, current plans for SNO+ seem to indicate such a measurement has a very low priority for the collaboration and may not be attempted at all. One can only regret that such a great opportunity could be missed by the Sudbury Observatory. It may well be that Borexino will be the only solar neutrino experiment running for the next few years with the capability of measuring CN fluxes. The importance of detecting CN neutrinos can not be overstated. The current answer to the original question ”How does the Sun shine?” is that, within 15% precision, it does by nuclear fusion of hydrogen into helium by the different pp-chains [13]. The answer can probably extended to other stars as well, but only to those less massive than 1.2 M and only during their main sequence evolution. There is no direct evidence yet that the CNO-bicycle is the channel by which more massive stars burn hydrogen. Only the detection of CN neutrinos will give us a definite answer to the question How do the stars shine?”. As stated above, CN fluxes can also provide a determination of the metal abundances, particularly C and N, in the solar core. Such a measurement has multiple implications. A solution to the solar abundance problem is the first one that comes to our minds, as it has been discussed to some extent here and in quite some detail in the literature. If CN neutrinos point towards a large C+N content of the solar core, abundance determinations from the solar spectrum will require profound revisions. Alternatively, if the a low C+N abundance results, then the input physics of solar (and stellar) models has to be revised, particularly radiative opacities, as well as the basic hypothesis entering solar model calculations2 .
As stated in the introduction, only now solar neutrino experiments have been able to provide the necessary precision to make solar neutrinos a practical probe of
2 We remark here that, although standard solar models are a simplified picture of the solar interior, non-standard solar models (which
Figure 5: Added 13 N + 15 O vs 8 B neutrino fluxes. For models, 1σ regions are shown, including correlations. Experimental determination of 8 B is shown as a grey area (1σ). The upper limit from Borexino for the combined 13 N + 15 O flux is shown as a horizontal solid line.
of C+N nuclei per unit mass in the solar core and is simply computed as Y(C+N)= X(C)/12 + X(N)/14, where X denotes the mass fraction of a given element (in the primordial solar composition only the isotopes 12 C and 14 N are abundant at the relevant level of accuracy). Uncertainties in Eq. 1 above are, in order: 1) the experimental uncertainty in the determination of 8 B flux, 2) the residual uncertainty from parameters of solar model calculations - dominated almost completely by microscopic diffusion, 3) neutrino parameters - dominated by the current uncertainty in the θ12 and, 4) the experimental determination of nuclear reaction rates - in particular the astrophysical factors of the 7 Be+p (S17 ) and the 14 N+p (S1,14 ) reactions. The nuclear contribution to the total error is clearly the dominant uncertainty, but it is experimental and can therefore be improved. As a simple exercise, if we use the upper limit of the CN flux determined from Borexino, the solar 8 B flux (Table 1), and results for either SFII-GS98 or SFIIAGSS09 solar models, the limit Y(C+N) < 0.00056 is obtained. SFII-GS98 and SFII-AGSS09 values are, respectively, 0.00035 and 0.00029. Therefore, the upper limit for the solar core C+N abundance set from Borexino data is only 1.6 times the SFII-GS98 SSM value. It is still not possible to discriminate different solar abundance compositions, but it is a goal within reach in the near future.
A. Serenelli / Nuclear Physics B (Proc. Suppl.) 235–236 (2013) 41–48
Before closing, I touch upon a speculative, but potentially very interesting and important, topic. If we leave the solar abundance problem aside and assume a robust determination of the solar surface metallicity is available, the determination of the core C+N abundance provides a way of testing, for example, a basic assumption of stellar evolution. This is, that the very young Sun (and other stars) formed as a chemically homogenous body. The underlying idea leading to this assumption is that young stars lie on the Hayashi track and are fully convective, and therefore homogenous. Hydrodynamic simulations of star formation have put this traditional ”wisdom” under doubt [28], showing that an accreting protostar similar to the young Sun has an effective temperature higher than found in hydrostatic contracting pre-main sequence models, and that the entropy deposited by the accretion process itself prevents the star from becoming fully convective. Stellar models with episodic phases of high accretion rates [29] point towards the same conclusion: that convective envelopes in young accreting stars are in reality relatively thin. These results suggest that, if matter in the disk around the young Sun is chemically processed (e.g. some elements are preferentially evaporated from the disk or others retained in the disk, for example, after forming rocky planetary cores) and later on accreted onto the forming Sun, it is possible that the formation process leads to the Sun not being initially homogeneous. Theoretically, helioseismic tests of such solar models have been explored [30, 31], and it has also been shown that even current solar neutrino data offer an alternative, sometimes even more sensitive probe of hypothetical accretion histories than helioseismology [2]. This is a very interesting result in itself but, it is even more so, when considered under the light of the differences in the composition between the Sun and the socalled solar-twins, stars with the same atmospheric parameters - effective temperature, gravity and iron abundance - as the Sun. Given that solar twins have the same surface properties as the Sun, a differential spectroscopic study of metal abundances of the solar twins against the Sun is basically a systematics-free analysis that allows a very precise differential abundance determination. Interestingly, some works [32, 33] find that solar twins with no detected planetary companions have an abundance pattern different from the Sun such that the solar surface seems to be enriched in volatile eleusually involve some extra number of free parameters) offer no solution to the solar abundance problem and usually perform, actually, much worse than standard solar problems when contrasted against helioseismic probes.
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ments (such as CNO) with respect to the solar twins. In fact, the difference of element abundances between the Sun and the solar twins seems to be very well correlated to the condensation temperature of the chemical elements. The most likely interpretation, according to the authors, is that in our Solar System refractory elements were preferentially locked in the cores of the rocky planets during the process of planet formation and, the subsequent accretion of chemically processed material from the protoplanetary disk onto the Sun led to the observed pattern of abundances. Order of magnitude estimations of the inventory of excess volatiles in the solar envelope and refractories in the rocky planets [27, 34] support the plausibility of the argument if the Sun had a thin convective envelope at the time the accretion episode took place. If the reasoning turns out to be correct, it implies the Sun has a convective envelope that is rich in volatiles compared to its own interior. As derived from solar twins, the estimation is that the solar surface could be between 15 to 20% more CN rich than the core. We should therefore expect a proportionally lower CN flux than predicted by solar models. By deriving the core C+N abundance using solar neutrinos, it could be possible, by comparing with the surface abundance, to actually determine if such a segregation of material in the protoplanetary disk took place or, at least, put a constraint to the amount of segregated material the young Sun might have accreted. The possibility of connecting solar neutrinos and planetary formation is tantalizing. Acknowledgments I thank the organizers of the Neutrino 2012 Conference for the invitation to participate in this wonderful event and also for financial support. This work has been partially supported by the European Union International Reintegration Grant PIRG-GA-2009-247732 and the MICINN grant AYA2011-24704. References [1] E. G. Adelberger et al., Rev. Mod. Phys. 83, 195 (2011) [2] A. M. Serenelli, H. C. Haxton, & C. Pe˜na-Garay, Astrophys. J. 743, 24 (2011) [3] N. Grevesse, & J. Sauval, Space Sci. Rev. 85, 161 (1998) [4] M. Asplund, N. Grevesse, J. Sauval, & P. Scott, Annu. Rev. Astron. Astr. 47, 481 (2009) [5] S. Basu, & H. M. Antia, Astrophys. J. 606, L85 (2004) [6] J. N. Bahcall, A. M. Serenelli, & S. Basu, Astrophys. J. 621, L85 (2005) [7] J. A. Guzik, L. S. Watson, & A. N. Cox, Astrophys. J. 627, 1049 (2005) [8] S. Basu et al., Astrophys. J. 699, 1403 (2009)
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[9] S. Basu, & H. M. Antia, Mon. Not. R. Astron. Soc. 287, 189 (1997) [10] J. N. Bahcall, A. M. Serenelli, & S. Basu, Astron Astrophys. Sup. 165, 400 (2006) [11] S. Basu et al., Astrophys. J. 655, 660 (2007) [12] W. J. Chaplin et al., Astrophys. J. 670, 872 (2007) [13] M. C. Gonz´alez-Garcı´ıa, M. Maltoni, & J. Salvado, J. High. Energy Phys. 5, 72 (2010) [14] J. N. Bahcall, & C. Pe˜na-Garay, J. High Energy Phys. 11, 4 (2003) [15] G. Bellini et al. for the Borexino collaboration, Phys. Rev. Lett. 108, 51302 (2012) [16] J. Christensen-Dalsgaard, M. P. di Mauro, G. Houdek, & F. Pijpers, Astron. Astrophys. 494, 205 (2009) [17] F. Villante, Astrophys. J. 724, 98 (2010) [18] F. Delayahe, & M. H. Pinsonneault, Astrophys. J., 649, 529 (2006) [19] F. Delahaye, M. H. Pinsonneault, L. Pinsounneault, & C. J. Zeippen, arXiv1005.0423 (2010) [20] C. Blancard, P. Coss`e, & G. Faussurier, Astrophys. J. 745, 10 (2012) [21] C. A. Iglesias, & F. J. Rogers, Astrophys. J. 464, 943 (1996) [22] N. R. Badnell et al., Mon. Not. R. Astron. Soc. 360, 1458 (2005) [23] A. M. Serenelli, Astrophys. Space Sci. 328, 13 (2010) [24] H. M. Antia & S. Basu, Astrophys. J. 644, 1292 (2006) [25] C.-H. Lin, H. M. Antia & S. Basu, Astrophys. J. 668, 603 (2007) [26] F. Villante, Phys. Lett. B 701, 336 (2011) [27] W. C. Haxton & A. M. Serenelli, Astrophys. J. 687, 678 (2008) [28] G. Wuchterl & R. S. Klessen, Astrophys. J. 560, L185 (2001) [29] I. Baraffe & G. Chabrier, Astron. Astrophys. 521, 44 (2010) [30] M. Castro, S. Vauclair, & O. Richard, Astron. Astrophys. 463, 755 (2007) [31] J. A. Guzik, & K. Mussack, Astrophys. J. 713, 1108 (2010) [32] J. Mel´endez, M. Asplund, B. Gustafsson, & D. Yong, Astrophys. J. 704, L66 (2009) [33] I. Ram´ırez et al., Astron. Astrophys. 521, 33 (2010) [34] A. Nordlund, arXiv:0908.3479 (2009)
Nuclear Physics B (Proc. Suppl.) 235–236 (2013) 41–48 www.elsevier.com/locate/npbps
Standard Solar Model: Status and Prospects A. Serenellia a Institute
of Space Sciences (CSIC-IEEC), Campus UAB, Bellaterra, Spain
Abstract I present a personal perspective of the current status of standard solar models, the comparison with current constraints from helioseismology and neutrino fluxes derived from available data. Results are discussed in the context of the solar abundance problem, and it is shown that current solar neutrino experiments can not discriminate between different alternatives for the solar composition available in the literature. The importance of a measurement of the neutrino fluxes from the CN-cycle is emphasized, not only because of the possibility of discerning between solar compositions, but also in connection to the formation of the planetary system. Keywords: Sun, neutrinos, helioseismology, solar abundances, accretion
1. Introduction For about four decades, the solar neutrino problem was a powerful driving force in neutrino and solar physics research. Albeit the unexpected and wonderful discoveries of neutrino properties, the original motivation for establishing solar neutrino experiments, namely probing how the Sun shines and the internal solar structure, has been only partially fulfilled. It is only in the last few years, with the precise measurements of the 8 B and 7 Be neutrino fluxes and the prospects of probing pep (indirectly determining pp) and CN neutrinos, that solar neutrinos are becoming a very competitive tool to constraint solar interior properties. At present, standard solar models (SSMs) come in two flavors, depending on the choice of the solar surface metallicity (Z) used to construct the model. HighZ models use solar abundance determinations that reproduce helioseismic probes of the solar interior but are based on previous generation of spectroscopic abundance determinations. Low-Z solar models, on the contrary, are based on the most refined solar abundance determinations but do not reproduce helioseismic constraints. This is the solar abundance problem. Here I will present the most recent results for the two flavors of standard solar models (SSM) and dis0920-5632/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.nuclphysbps.2013.03.009
cuss the problems and possibilities to discriminate between solar compositions using current and future data, both from helioseismology and solar neutrinos. I also present some speculative arguments that link the apparently unconnected process of planetary system formation and solar neutrinos. 2. Standard Solar Models Results presented here are based on the SSM that include the most up-to-date microscopic input physics, particularly regarding nuclear reaction cross sections, for which we adopt both the central values and uncertainties recommended in the recent review Solar Fusion II (SFII; [1]). A complete description of input physics of the models is given in [2]. The only difference between the two solar models upon which I base the discussion is the choice of the solar surface abundance determination. The two choices used are as follows: • SFII-GS98: model constructed using the solar composition given in [3], derived from 1D atmosphere models. In a nutshell, this model is characterized by the present-day surface metal-tohydrogen ratio (Z/X) = 0.0229 and it is representative of high-Z SSMs;
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• SFII-AGSS09: model that incorporates solar abundances from [4], obtained from 3D model atmospheres. The reduction by 30 − 40% in the abundances CNO and Ne presented in [4] leads to (Z/X) = 0.0178, the value used for this low-Z SSM. The present-day solar composition is of crucial importance for SSM because, aside from the 10 to 12% change in the surface metal abundances induced by gravitational settling during the evolution of the Sun, the adopted (Z/X) basically determines the internal composition of the model. 3. Results 3.1. Helioseismology The impact of the different solar abundances on solar models is most readily seen in properties of the solar structure that can be probed with helioseismic data. There is a large amount of literature devoted to the solar abundance problem [5, 6, 7]. Here we just present a summary. The most striking change in solar models brought up by the low-Z abundances is the mismatch of the predicted sound speed profile between low-Z models and the Sun. This is illustrated in the left panel of Figure 1 where the relative difference of sound speed between the Sun (derived from inversions [8]) and the models is shown as a function of solar radius. It is evident from this plot that the low-Z SFII-AGSS09 SSMs is much worse than the high-Z SFII-GS98 model in reproducing the solar sound speed. Error bars in the solar sound speed from the helioseismic data are shown for reference. The most prominent feature in this plot is the peak right below the convective envelope (CZ; grey area in the plot). The origin of this peak mostly lies in the wrong location of the boundary of the convective envelope in the SFII-AGSS09 model. In fact, from helioseismology we know RCZ = 0.713 ± 0.001R [9] while RCZ = 0.723R in the SFII-AGSS09 model. On the contrary, the SFII-GS98 predicts the right location of the convective envelope boundary and it does a much better job on the sound speed right below the CZ, the agreement being about 4 times better than with SFIIAGSS09. The right panel of Figure 1 shows analogous results but for the relative difference in density profiles. Here the disagreement between SFII-AGSS09 and the Sun is very large in the convective envelope, but the argument against low-Z models is less compelling. The reason is
that the total solar mass is a constraint imposed on density inversions and a small difference between the Sun and models in the high density central regions translates into a large difference (of opposite sign) in the outer low density regions. The are other helioseismic probes that can be used to test the internal structure of solar models. For reference, below is a summary of the current state of affairs for constraints I consider robust. • Boundary of convective envelope RCZ . Helioseismology: 0.713 ± 0.001R [9], SFII-GS98: 0.712R , SFII-AGSS09: 0.723R , model uncertainty: ±0.004R [10]. • Sound speed. Maximum relative difference - SFIIGS98: 0.3%, SFII- AGSS09: 1.0%. Average rms deviation - SFII-GS98: 0.09%, SFII-AGSS09: 0.37%. • Density. Maximum relative difference - SFIIGS98: 2%, SFII-AGSS09: 7.3%. Average rms deviation - SFII-GS98: 1.1%, SFII-AGSS09: 4%. • Surface helium abundance YS . Helioseismology: 0.2485 ± 0.0034 [5], SFII-GS98: 0.243, SFIIAGSS09: 0.232, model uncertainty: ±0.004 • Core averaged mean molecular weight μC 1 . Helioseismology: 0.723±0.002 [12], SFII-GS98: 0.720, SFII-AGSS09: 0.713, model uncertainty: ±0.003 [12]. The solar abundance problem can be understood from the list given above. There is a systematic failure of low-Z SSMs to reproduce solar properties as determined from helioseismology and, on the other hand, high-Z models, of which SFII-GS98 is representative, agreement is quite good. 3.2. Neutrinos Solar neutrinos are also affected by the choice of solar composition. A low-Z solar model has a lower core temperature than a model with higher-Z and, because solar models have to reproduce L , the effect is that low-Z models have a slight increase in the neutrino fluxes with smallest temperature sensitivity and a decrease in the most sensitive ones. Results for SFII-GS98 and SFIIAGSS09 are given in Table 1. Changes in the fluxes from the pp-chains simply show the response to the 1 Based on using small separation ratios of low-degree oscillation modes[11]
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Figure 1: Sound speed and density relative differences between the Sun and solar models obtained from inversions of helioseismic data [8]. The convective envelope is shown with the grey area.
change in the temperature, which is on average about 1.2% lower in the SFII-AGSS09 model in the neutrino producing region (R < 0.30R ). The largest changes in these fluxes of course correspond to 7 Be and 8 B, which differ by about 10% and 20% respectively in the two models. Figure 2 summarizes the current status of experiments and a direct comparison with solar models. Different determinations of solar fluxes from available experimental data have been presented [2, 13] and are in good agreement. Results given in the last column of Table 1 have been obtained using methods used in previous studies [14]. The remarkable results from Borexino for 7 Be, together with the measurements of 8 B by SNO and SK allow a very precise determination of these fluxes to 4.5% and 3% respectively. Borexino also allows for a precise determination of the pp and pep fluxes [13] when the luminosity constraint is used, as the uncertainties of 0.6% and 1.2% for these fluxes reflect. The precision of the experimental solar neutrino determinations that has been achieved were unthought of just some years ago. The situation is ripe for the original goal of Davies and Bahcall to be fulfilled. A natural first step in this direction is to contrast solar fluxes with lowZ and high-Z models. The last row in Table 1 gives the results of this comparison for the SFII-GS98 and SFIIAGSS09 models. In both cases models agree very well with the solar values and, unfortunately, it is at present not possible to discriminate between models using solar neutrinos. In Figure 3 the comparison between solar and model fluxes is shown for 7 Be and 8 B. The experimental result falls right in the region where error ellipses (1σ)
Flux pp pep hep 7 Be 8B 13 N 15 O 17 F χ2 /Pagr
SFII-GS98 5.98(1 ± 0.006) 1.44(1 ± 0.011) 8.04(1 ± 0.30) 5.00(1 ± 0.07) 5.58(1 ± 0.14) 2.96(1 ± 0.14) 2.23(1 ± 0.15) 5.52(1 ± 0.17) 3.5 / 90%
SFII-AGSS09 6.03(1 ± 0.006) 1.47(1 ± 0.012) 8.31(1 ± 0.30) 4.56(1 ± 0.07) 4.59(1 ± 0.14) 2.17(1 ± 0.14) 1.56(1 ± 0.15) 3.40(1 ± 0.16) 3.4 / 90%
Solar 6.05(1 ± 0.006) 1.46(1 ± 0.012) 18(1 ± 0.45) 4.82(1 ± 0.045) 5.00(1 ± 0.03) ≤ 6.7 ≤ 3.2 ≤ 59 —
Table 1: Results for solar neutrino fluxes. First two columns for solar models, as indicated; last column solar values determined from all available neutrino data and the luminosity constraint. Units are, as usual, 1010 (pp), 109 (7 Be), 108 (pep, 13 N, 15 O), 106 (8 B, 17 F), 103 (hep) cm−2 s−1 . Last row shows results of a χ2 test comparing the models solar the solar data.
of models overlap. The solar fluxes used here were not derived using solar neutrino data collected over the last year, e.g. SuperK IV results (Smy, this volume) or the upper limit on CNO fluxes from Borexino [15]. However, it is not expected that solar values for fluxes from pp-chains will change significantly because of this.
4. Discussion As presented in the previous section, helioseismology strongly favors SSMs with high-Z, as the comparison of results of the SFII-GS98 and SFII-AGSS09 models against helioseismology show. On the other hand, solar neutrino fluxes are well reproduced by both models. As it has been pointed out previously [16], helioseismic probes listed above are not directly sensitive to the
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Figure 2: Direct comparison (no oscillations accounted for) of the available data from solar neutrino experiments and detailed comparison with the SFII-GS98 solar model as indicated. Also, total expected fluxes for the low-Z SFII-AGSS09 model are shown. Results for the radiochemical experiments are given in SNU; in all other cases fluxes are normalized to predictions from the SFII-GS98 model.
metal abundance of solar models but rather to the opacity of stellar matter to radiation, which determines the efficiency of energy transport in the solar interior below the CZ. Therefore, as has already been discussed in detail [17] the most relevant effect of modifying the abundance of metals in the Sun is the change this produces in the radiative opacity. This introduces a degeneracy in solar models with respect to variations in the metal composition and changes, e.g. due to uncertainties, in radiative opacities.
Figure 3: 7 Be vs 8 B neutrino fluxes. For the model 1σ regions are shown, including correlations. Experimental data (cross) is as given in Table 1.
The reader can understand the complications introduced by this degeneracy by inspecting Figure 4, where the relative contributions to the total Rosseland mean opacity κ are shown for some relevant elements. Clearly, there is not a one-to-one correspondence between κ and the total metallicity, and this makes it even a more complex task to extract information about the solar composition from either helioseismology or solar neutrinos. In fact, the best that can be done with the available information is to constrain the variations in κ that have to be introduced in a reference solar model to match observational constraints as it has been done in [17]. Transforming a derived opacity change into a change
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Figure 4: Fractional contribution to the Rosseland mean opacity of a few selected elements. The bottom boundary of the convective envelope is shown with a vertical dotted line.
in solar composition with respect to the reference solar model used can be attempted in simplified forms [18, 19], but it strongly relies on the assumption made for the intrinsic uncertainty in radiative opacities calculations. It is very unsettling that such information is completely missing in the literature and we have to comfort ourselves by assuming that comparison between different sets of opacity calculations give us a measure of the actual uncertainty in opacity calculations. In fact, recent work [20] has shown that small differences, of a few percent, in the mean Rosseland opacity are the result of cancellation of much larger differences in the opacity contributions of single elements. It may well be that agreement between different opacity calculations for conditions appropriate to solar radiative interior (OPAL [21], OP [22], OPAS[20]) is just mere chance. The almost perfect degeneracy between composition and opacity regarding helioseismic probes of solar structure has been shown previously [16]. For the 7 Be and 8 B results have also been presented [23] which we update here in Figure 3, where the dashed line, indicated in the figure as SFII-AGSS09+κ shows results for a solar model with AGSS09 composition with the only modification that the opacity has been increased to match, as a function of radius, that of the SFII-GS98 model. As shown, 7 Be and 8 B fluxes from this model almost coincide with results from model SFII-GS98. The question therefore is: how can the degeneracy between radiative opacity and metallicity be broken? Probes of the solar composition that do not depend on opacity, or at least not only on opacity, are needed. Helioseismology offers a possibility that has been investigated [24] and yields Z = 0.0172 ± 0.002, in good
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agreement with the GS98 result. This result depends, however, on a subtle signal imprinted on the sound speed by the partial ionization of metals and therefore, to extract the metal abundance from the helioseismic signal an accurate equation of state (EoS) is needed. Also, even the high-Z solar models do not perform particularly well in reproducing the solar adiabatic index [25, see Fig. 4] in the region of partial ionization of metals, which may be an indicator of deficiencies in the EoS. Potentially, solar neutrinos from the CNO-bicycle offer another probe of solar metallicity that is only mildly degenerate with stellar opacity. In Figure 5 we show the added 13 N and 15 O neutrino fluxes (CN for short), the two CNO fluxes that can be experimentally detected, for the two reference models and the SFII-AGSS09 model with increased opacity. The shaded area represents the experimental determination of the solar 8 B flux. The difference between the SFII-AGSS09 and SFII-AGSS09+κ models reflects the κ-dependence of the fluxes. Contrary to fluxes from pp-chains, fluxes from the CN-cycle carry an extra (dominating) linear dependence on the C+N solar core abundance because the added abundance of these elements catalyze the nuclear cycle. Additionally, we have included the upper experimental limit recently announced by the Borexino collaboration [15], which sets 7.7 × 108 cm−2 s−1 as the most stringent upper limit on the added 13 N+15 O flux to date . A crucial problem in determining the CN neutrino flux in liquid scintillator detectors is the 210 Bi background. In this regard, a very interesting possibility for removing this background has recently been put forward [26], but has not yet been included in the analysis of Borexino, that could allow, with the data already available, an actual measurement of the CN flux. A rather direct way of determining the C+N abundance in the solar core relies on the similar temperature sensitiviy these fluxes share with 8 B [27]. The correlation between these fluxes can be used to isolate the linear dependence of the CN fluxes with the C+N abundance and use the experimental determination of the 8 B flux as a thermometer to fix the solar core temperature. The role of solar models is merely that of a normalization of the relation between fluxes and composition. Updating the relation from the original publication, it can shown that exp 8 0.828 Φexp (CN) Y(C + N) Φ ( B) × (1) = SSM 8 ΦSSM (CN) Φ ( B) YSSM (C + N) 1 ± 0.03(8 B) ± 0.028(env) ± 0.03(θ12 ) ± 0.10(nuc) . In this expression Y(C+N) is proportional to the number
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5. Prospects
solar properties. 8 B can now be used as a very good thermometer of the solar core and 7 Be, together with the solar luminosity constraint allow for the determination of the pp and pep neutrino fluxes to precisions of the order of 1%. The next goal is the determination of the CN fluxes, and Borexino has given the first solid step in this direction by setting the most stringent upper limit to date, only 1.5 times larger than predictions from solar models with high metallicity. Further improvements in the, already astonishing, radiopurity level of Borexino and, very importantly, a new technique to extract the 210 Bi background [26], raise the expectations that a measurement of the combined CN flux will happen in the near future. On the negative side, current plans for SNO+ seem to indicate such a measurement has a very low priority for the collaboration and may not be attempted at all. One can only regret that such a great opportunity could be missed by the Sudbury Observatory. It may well be that Borexino will be the only solar neutrino experiment running for the next few years with the capability of measuring CN fluxes. The importance of detecting CN neutrinos can not be overstated. The current answer to the original question ”How does the Sun shine?” is that, within 15% precision, it does by nuclear fusion of hydrogen into helium by the different pp-chains [13]. The answer can probably extended to other stars as well, but only to those less massive than 1.2 M and only during their main sequence evolution. There is no direct evidence yet that the CNO-bicycle is the channel by which more massive stars burn hydrogen. Only the detection of CN neutrinos will give us a definite answer to the question How do the stars shine?”. As stated above, CN fluxes can also provide a determination of the metal abundances, particularly C and N, in the solar core. Such a measurement has multiple implications. A solution to the solar abundance problem is the first one that comes to our minds, as it has been discussed to some extent here and in quite some detail in the literature. If CN neutrinos point towards a large C+N content of the solar core, abundance determinations from the solar spectrum will require profound revisions. Alternatively, if the a low C+N abundance results, then the input physics of solar (and stellar) models has to be revised, particularly radiative opacities, as well as the basic hypothesis entering solar model calculations2 .
As stated in the introduction, only now solar neutrino experiments have been able to provide the necessary precision to make solar neutrinos a practical probe of
2 We remark here that, although standard solar models are a simplified picture of the solar interior, non-standard solar models (which
Figure 5: Added 13 N + 15 O vs 8 B neutrino fluxes. For models, 1σ regions are shown, including correlations. Experimental determination of 8 B is shown as a grey area (1σ). The upper limit from Borexino for the combined 13 N + 15 O flux is shown as a horizontal solid line.
of C+N nuclei per unit mass in the solar core and is simply computed as Y(C+N)= X(C)/12 + X(N)/14, where X denotes the mass fraction of a given element (in the primordial solar composition only the isotopes 12 C and 14 N are abundant at the relevant level of accuracy). Uncertainties in Eq. 1 above are, in order: 1) the experimental uncertainty in the determination of 8 B flux, 2) the residual uncertainty from parameters of solar model calculations - dominated almost completely by microscopic diffusion, 3) neutrino parameters - dominated by the current uncertainty in the θ12 and, 4) the experimental determination of nuclear reaction rates - in particular the astrophysical factors of the 7 Be+p (S17 ) and the 14 N+p (S1,14 ) reactions. The nuclear contribution to the total error is clearly the dominant uncertainty, but it is experimental and can therefore be improved. As a simple exercise, if we use the upper limit of the CN flux determined from Borexino, the solar 8 B flux (Table 1), and results for either SFII-GS98 or SFIIAGSS09 solar models, the limit Y(C+N) < 0.00056 is obtained. SFII-GS98 and SFII-AGSS09 values are, respectively, 0.00035 and 0.00029. Therefore, the upper limit for the solar core C+N abundance set from Borexino data is only 1.6 times the SFII-GS98 SSM value. It is still not possible to discriminate different solar abundance compositions, but it is a goal within reach in the near future.
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Before closing, I touch upon a speculative, but potentially very interesting and important, topic. If we leave the solar abundance problem aside and assume a robust determination of the solar surface metallicity is available, the determination of the core C+N abundance provides a way of testing, for example, a basic assumption of stellar evolution. This is, that the very young Sun (and other stars) formed as a chemically homogenous body. The underlying idea leading to this assumption is that young stars lie on the Hayashi track and are fully convective, and therefore homogenous. Hydrodynamic simulations of star formation have put this traditional ”wisdom” under doubt [28], showing that an accreting protostar similar to the young Sun has an effective temperature higher than found in hydrostatic contracting pre-main sequence models, and that the entropy deposited by the accretion process itself prevents the star from becoming fully convective. Stellar models with episodic phases of high accretion rates [29] point towards the same conclusion: that convective envelopes in young accreting stars are in reality relatively thin. These results suggest that, if matter in the disk around the young Sun is chemically processed (e.g. some elements are preferentially evaporated from the disk or others retained in the disk, for example, after forming rocky planetary cores) and later on accreted onto the forming Sun, it is possible that the formation process leads to the Sun not being initially homogeneous. Theoretically, helioseismic tests of such solar models have been explored [30, 31], and it has also been shown that even current solar neutrino data offer an alternative, sometimes even more sensitive probe of hypothetical accretion histories than helioseismology [2]. This is a very interesting result in itself but, it is even more so, when considered under the light of the differences in the composition between the Sun and the socalled solar-twins, stars with the same atmospheric parameters - effective temperature, gravity and iron abundance - as the Sun. Given that solar twins have the same surface properties as the Sun, a differential spectroscopic study of metal abundances of the solar twins against the Sun is basically a systematics-free analysis that allows a very precise differential abundance determination. Interestingly, some works [32, 33] find that solar twins with no detected planetary companions have an abundance pattern different from the Sun such that the solar surface seems to be enriched in volatile eleusually involve some extra number of free parameters) offer no solution to the solar abundance problem and usually perform, actually, much worse than standard solar problems when contrasted against helioseismic probes.
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ments (such as CNO) with respect to the solar twins. In fact, the difference of element abundances between the Sun and the solar twins seems to be very well correlated to the condensation temperature of the chemical elements. The most likely interpretation, according to the authors, is that in our Solar System refractory elements were preferentially locked in the cores of the rocky planets during the process of planet formation and, the subsequent accretion of chemically processed material from the protoplanetary disk onto the Sun led to the observed pattern of abundances. Order of magnitude estimations of the inventory of excess volatiles in the solar envelope and refractories in the rocky planets [27, 34] support the plausibility of the argument if the Sun had a thin convective envelope at the time the accretion episode took place. If the reasoning turns out to be correct, it implies the Sun has a convective envelope that is rich in volatiles compared to its own interior. As derived from solar twins, the estimation is that the solar surface could be between 15 to 20% more CN rich than the core. We should therefore expect a proportionally lower CN flux than predicted by solar models. By deriving the core C+N abundance using solar neutrinos, it could be possible, by comparing with the surface abundance, to actually determine if such a segregation of material in the protoplanetary disk took place or, at least, put a constraint to the amount of segregated material the young Sun might have accreted. The possibility of connecting solar neutrinos and planetary formation is tantalizing. Acknowledgments I thank the organizers of the Neutrino 2012 Conference for the invitation to participate in this wonderful event and also for financial support. This work has been partially supported by the European Union International Reintegration Grant PIRG-GA-2009-247732 and the MICINN grant AYA2011-24704. References [1] E. G. Adelberger et al., Rev. Mod. Phys. 83, 195 (2011) [2] A. M. Serenelli, H. C. Haxton, & C. Pe˜na-Garay, Astrophys. J. 743, 24 (2011) [3] N. Grevesse, & J. Sauval, Space Sci. Rev. 85, 161 (1998) [4] M. Asplund, N. Grevesse, J. Sauval, & P. Scott, Annu. Rev. Astron. Astr. 47, 481 (2009) [5] S. Basu, & H. M. Antia, Astrophys. J. 606, L85 (2004) [6] J. N. Bahcall, A. M. Serenelli, & S. Basu, Astrophys. J. 621, L85 (2005) [7] J. A. Guzik, L. S. Watson, & A. N. Cox, Astrophys. J. 627, 1049 (2005) [8] S. Basu et al., Astrophys. J. 699, 1403 (2009)
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