Abstract
Helioseismology has been widely acclaimed as having been a great success: it appears to have answered nearly all the questions that we originally asked, some with unexpectedly high precision. We have learned how the sound speed and matter density vary throughout almost all of the solar interior – something which not so very long ago was generally considered to be impossible – we have learned how the Sun rotates, and we have a beautiful picture, on a coffee cup, of the thermal stratification of a sunspot, and also an indication of the material flow around it. We have tried, with some success at times, to apply our findings to issues of broader relevance: the test of the General Theory of Relativity via planetary orbit precession (now almost forgotten because the issue has convincingly been closed, albeit no doubt temporarily) the solar neutrino problem, the manner of the transport of energy from the centre to the surface of the Sun, the mechanisms of angular-momentum redistribution, and the workings of the solar dynamo. The first two were of general interest to the broad scientific community beyond astronomy, and were, quite rightly, principally responsible for our acclaimed success; the others are still in a state of flux.
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Notes
I consider the seismic surface r=R of the Sun (assumed here to be spherically symmetrical) to be the radius at which c 2, regarded as a function of r, or c, regarded as a function of acoustic radius τ(r)=∫c −1 dr – both of which are close to being linear functions in the outer adiabatically stratified layers of the convection zone (Balmforth and Gough 1990; Lopes and Gough 2001) – extrapolate to zero. In the Sun, according to Model S of Christensen-Dalsgaard et al. (1996), it lies about 1000 km above the photosphere, the precise value depending on exactly how the extrapolation is carried out. There is nothing special about the structure of the actual atmosphere in its vicinity, which is well inside the outer evanescent zone of most of the seismic modes and therefore has little significant influence on the dynamics. Instead, it acts simply as a (virtual) singularity in the acoustic wave equation, providing a convenient parametrisation of conditions (well below the photosphere) in the vicinity of the upper turning points of the modes. Put another way, it provides a convenient fiducial location with respect to which the acoustic phase in the propagating zone beneath is related. Unlike the photosphere, which has no acoustic significance, it shares a relation with the deeper solar interior that is robust, and is insensitive to the non-seismic, thermal and radiative, properties of the outer convective boundary layer, whose structure changes with the solar cycle (Antia and Basu, 2004; Dziembowski and Goode, 2004, 2005; Lefebvre and Kosovichev, 2005; Lefebvre, Kosovichev, and Rozelot, 2007). In contrast to other, non-seismic, radii (cf. Bahcall and Ulrich, 1988), it provides a stable outer limit to the effective total acoustic-radius integral τ(R), which determines the large frequency separation; in Model S it is some 200 seconds or so greater than the actual acoustic radius of the photosphere.
Included for comparison is one of the models with low Y that was used in the original calibration with low-degree modes by Christensen-Dalsgaard and Gough (1981). The qualitative differences between the models can be appreciated by realising first that the radiative envelopes are roughly polytropic (with index 3.5), and that it is adequate to approximate the equation of state by the perfect-gas law. Then it is clear that the magnitude of T(r) (and ρ) must be greater in the higher-Y model, because the total energy generation rate, which is an increasing function of X, ρ, and T, is the same for the two models. Polytropic scaling (e.g. Gough, 1990) indicates that Z is a steeply decreasing function of X, so Z is much greater in the higher-Y model. In addition, the polytropic radius scale is greater for the higher-Y model, as is evident from Figure 1 by imagining an extrapolation of the functional form of T(r) outwards from the radiative zone. Consequently the convection zone, which steepens the gradient, has more truncating from the radiative structure to perform in order to maintain the observed photospheric radius, and is therefore deeper. An additional scaling in magnitude is required to convert T to c 2∝T/μ, raising the dotted curve (corresponding to the model with the lower Y) relative to the continuous curve. It also depresses both curves near the centre of the star, in the energy-generating core where μ has been augmented by nuclear transmutation, providing a diagnostic of main-sequence age.
The gravitational attraction associated with the energy density [−GM/r] of the gravitational field surrounding the Sun, absent in Newton’s theory, causes the total gravitational attraction to increase: very roughly speaking, as a result of energy conservation the apparent gravitational mass of a planet at distance r from the Sun, in a flat representation of space, is augmented by approximately GM/c 2 r per unit mass of planet above what it would have appeared to have been at infinity; M is the mass of the Sun, and here c is the speed of light. Similarly, the energy, hence the frequency, of a photon is multiplied by a factor Γ=1+GM/c 2 r – that causes the familiar gravitational redshift. Consequently, the orbit equation is modified simply by multiplying the Newtonian gravitational force on the planet by Γ3. After linearisation and rewriting M in terms of the orbital specific angular momentum \(h=\sqrt{(GMr)}\), valid for nearly circular orbits, the effective attractive force becomes −(1+3h 2/c 2 r 2)GM/r 2; it increases with increasing proximity more rapidly than Newton’s inverse square. It is easy to see also that the gravitational field in the equatorial plane of a rotating (oblate) axisymmetric self-gravitating body (like the Sun) also increases with decreasing distance faster than the field around a corresponding spherically symmetrical body: the act of flattening a spherical body takes equal amounts of material from the poles towards the near and the far sides of the Equator, the increase in gravitational attraction by closer nearside matter exceeding the lesser decrease by farside matter. Therefore the net gravitational attraction is increased, by an amount which increases as r decreases and the shape of the Sun becomes more apparent. The force is given approximately by \(-(1+\frac{3}{2}J_{2}R^{2}/r^{2})GM/r^{2}\); J 2 is the quadrupole moment. In both cases, therefore, a planet is drawn towards the Sun more strongly with increasing proximity than it would have been in an inverse-square field. Conserving its angular momentum, it is thereby caused to rotate through a greater angle near perihelion because its orbital angular velocity is augmented, distorting an otherwise approximately elliptical bound Newtonian orbit in such a manner as to appear to make it simply precess in the same direction as the angular velocity of the planet. (Near aphelion the oppositely directed contribution to the precession is lesser, therefore too small to annul the contribution from near perihelion.) Use of planetary (or spacecraft) orbital precession rates to calibrate the relativistically induced deviation from the inverse-square gravitational field surrounding the Sun therefore requires one to know the contribution from the distortion from spherical symmetry of the mass distribution in the Sun, such as is produced by centrifugal acceleration due to rotation. The precession rate is most easily calculated by perturbation theory (e.g. Ramsey, 1937).
I am assuming that the seismic data processers have judged their errors correctly, and that the error correlations in the frequency data sets, normally ignored, are not unduly severe. I am also assuming that, where it is appropriate, asphericity of equilibrium structure is taken correctly into account.
The original analysis employed simulations in a model atmosphere with the previously accepted chemical composition; from 2009 onwards, newer simulations by Asplund and his collaborators were carried out with the Fe abundance inferred from the 2005 abundance determinations; the abundances of C, N, and O were not changed because their spectral lines are weak and have no significant impact on the radiative energy flux (R. Trampedach, personal communication, 2012).
Implications from earlier estimates of Y to determine T(r) have been discussed by Elliott (1995) and Tripathy and Christensen-Dalsgaard (1998), the first under the assumption that Y(r) differs from that in a reference solar model by just a constant, the second that it can be obtained simply by scaling the reference-model value by a constant factor, both of which are inaccurate in the core, although Elliott notes that the opacity perturbations produce a predominantly local response, so that these analyses should provide a fair estimate in the radiative envelope. The Tripathy–Christensen-Dalsgaard scaling was used by Tripathy, Basu, and Christensen-Dalsgaard (1998) to obtain the opacity difference from Model S by RLS frequency fitting (J. Christensen-Dalsgaard, personal communication, 2012) assuming that difference can be expressed as a function of T alone, yielding a superficially similar functional form to the continuous curve in Figure 6, but with a magnitude about 50 per cent greater. Bahcall et al. (2005) have estimated the opacity difference by adjusting κ by hand in solar models. Their preferred model had a constant 11 % augmentation over the OPAL values using the most recent abundance determinations (Asplund et al., 2000, 2004; Asplund, 2005; Allende Prieto, Lambert, and Asplund, 2001, 2002; Asplund personal communication with Bahcall et al., 2004) in the radiative envelope down to T=5×106 K, beneath which the augmentation was smoothly reduced to zero (probably by a half Lorentzian function with half-width at half maximum of 2×105 K), and has a seismic structure as close to that of the Sun as does Model S. Regarding that model as a proxy Sun, one would expect the relative opacity difference between it and a model with the unmodified abundances to be comparable with the inference by Christensen-Dalsgaard et al. (2009). The two estimates are depicted in Figure 6. Korzennik and Ulrich (1989) had earlier estimated opacity errors by RLS (L2 norm) data fitting, and Saio (1992) by L1 data fitting, each by scaling κ by a function of T and ignoring the dependence of the relation between T and the seismic variables on chemical composition; they expressed their results as deviations from different reference models, so they cannot easily be compared with those presented in Figure 6. An estimate by OLA (Takata and Gough 2001) of the absolute structure of the Sun, including opacity, using the procedure for determining Y described in the text (together with tachocline homogenisation as calibrated by Elliott, Gough, and Sekii (1998)) is presented by Gough and Scherrer (2002) and Gough (2004, 2006).
Di Mauro et al. quote Y s=0.2539±0.0005 when OPAL is used, Y s=0.2457±0.0005 when MHD is used, the errors being merely formal, representing a precision error that takes no account of the error in the relation between Y s and γ 1 in the reference model. With those values the magnitude of the inferred \(\overline{\delta_{\mathrm{int}} \mathrm{ln}\,\gamma_{1}}\) was found to be the greater for the MHD equation of state beneath the helium ionisation zones, and the lesser above r/R≈0.97 in the He ii ionisation zone. That is perhaps not surprising because MHD is possibly better at taking into account the complicated chemistry that dominates higher up in the solar envelope, whereas the virial expansion used for OPAL is perhaps more reliable where such complications make only a minor non-ideal contribution. The difference between OPAL and MHD must surely offer some estimate of the total uncertainty.
Moreover, the initial helium abundance [Y 0] of the calibrated model is about 0.250, which is dangerously close to the amount [Y p] believed to have been created by Big-Bang nucleosynthesis, whose estimated value has been climbing over the last two decades (Steigman 2007): the latest estimate is Y p=0.2478±0.0006 (G. Steigman and M. Pettini, personal communication, 2011). Subsequent contamination of the interstellar medium by supernovae exacerbates the situation.
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Acknowledgements
I am grateful to Jørgen Christensen-Dalsgaard, Werner Däppen, Günter Houdek, and Sasha Kosovichev for interesting discussion. I thank Jeannette Gilbert and Paula Younger for typing the first draft of this paper, Günter Houdek for providing Figure 4, and Amanda Smith for her help in producing Figures 1 and 6. I thank the Leverhulme Trust for an Emeritus Fellowship, and P.H. Scherrer for support from HMI NASA contract NAS5-02139. This article has benefited from comments on the first draft by the referee who raised the matter of the reliability of the equation of state.
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Invited Article.
Solar Dynamics and Magnetism from the Interior to the Atmosphere
Guest Editors: R. Komm, A. Kosovichev, D. Longcope, and N. Mansour
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Gough, D. What Have We Learned from Helioseismology, What Have We Really Learned, and What Do We Aspire to Learn?. Sol Phys 287, 9–41 (2013). https://doi.org/10.1007/s11207-012-0099-1
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DOI: https://doi.org/10.1007/s11207-012-0099-1