MNRAS 000, 000–000 (0000) Preprint 15 October 2018 Compiled using MNRAS LATEX style file v3.0 On the hydrostatic stratification of the solar tachocline J. Christensen-Dalsgaard1⋆, D.O. Gough2†, E. Knudstrup1‡ 1Stellar Astrophysics Centre and Department of Physics and Astronomy, Aarhus University, DK 8000 Aarhus C, Denmark 2Institute of Astronomy and Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge CB3 0HA, UK 15 October 2018 ABSTRACT We present an attempt to reconcile the solar tachocline glitch, a thin layer immedi- ately beneath the convection zone in which the seismically inferred sound speed in the Sun exceeds corresponding values in standard solar models, with a degree of partial material mixing which we presume to have resulted from a combination of convective overshoot, wave transport and tachocline circulation. We first summarize the effects of either modifying in the models the opacity in the radiative interior or of incorporating either slow or fast tachocline circulation. Neither alone is successful. We then consider, without physical justification, incomplete material redistribution immediately beneath the convection zone which is slow enough not to disturb radiative equilibrium. It is modelled simply as a diffusion process. We find that, in combination with an appropri- ate opacity modification, it is possible to find a density-dependent diffusion coefficient that removes the glitch almost entirely, with a radiative envelope that is consistent with seismology. Key words: Sun: helioseismology – Sun: interior – Sun: abundances 1 INTRODUCTION of only the relatively small difference between the Sun and a standard theoretical model. The latter is naturally more There is some ambiguity in helioseismological inferences precise, but it depends in a not wholly obvious manner on about the spherically averaged stratification of the Sun in the approximations adopted in constructing the reference so- the vicinity of the base of the convection zone. Consequently lar model. The conclusion from the differential method was there is an associated uncertainty in the location of the rc = 0.713 ± 0.003R, where R is the (photospheric) radius base of the convection zone itself. Problems in interpret- of the Sun. Subsequently, Basu & Antia (1997) carried out ing helioseismological inversions of global frequency data a broadly similar asymptotic differential analysis with more arise because we are unsure of the dynamical consequences precise frequency data, obtaining rc = 0.713 ± 0.001R. It is of convective overshoot, and of the influence of the mag- essential to understand that the quoted errors indicate only netic field and of chemical-element segregation by gravi- the precision of the analysis, and may have little bearing on tational settling moderated by the large-scale meridional the accuracy of the calibration. Unfortunately, the distinc- flow in the tachocline. Issues such as these were recognized tion between the two has not always been appreciated. arXiv:1803.08675v1 [astro-ph.SR] 23 Mar 2018 by Christensen-Dalsgaard et al. (1991) in their original at- tempt to locate the base of the convection zone, causing Gyroscopic pumping associated with rotational shear them to suggest that the real uncertainty in their estimate in an essentially radiatively stratified layer beneath the con- is substantially greater than the formal precision of their vection zone (Spiegel 1972; Spiegel & Zahn 1992) exchanges data analysis, by a factor 3 or even more. material in that layer with the convection zone, thereby op- Christensen-Dalsgaard et al. adopted two different pro- posing the tendency of gravitational settling to establish a cedures for determining the radius rc of the base of the con- gradient of helium and heavier elements. This process, which vection zone: an absolute method, which used an asymp- must inevitably be taking place, although possibly in tan- totic formulation of the entire sound-speed stratification of dem with other mixing processes, tends to reduce abundance the star and which did not rely on a solar model, and a gradients on a timescale almost certainly less than the local differential method, which used an asymptotic description characteristic settling time (Gough & McIntyre 1998). The outcome is to make the spatial variation of the helium abun- dance Y rather smoother than in standard theoretical mod- ⋆ e-mail: [email protected] els, resulting in a positive anomaly (acoustical glitch) in the † e-mail: [email protected] difference between the sound speeds in the Sun and the mod- ‡ e-mail: [email protected] els. Such a glitch is evident in differential inversions of helio- c 0000 The Authors 2 J. Christensen-Dalsgaard, D.O. Gough and E. Knudstrup seismic data against standard solar models, as is illustrated the abundance profiles. No such assumption appears to have in Fig. 1, and has been recognized also by Antia & Chitre succeeded in removing the acoustic anomaly entirely (e.g. (1998) as being a signature of material mixing . Brun et al. 2002; Brun & Zahn 2006). If essentially laminar tachocline circulation were the Although the difference between the sound speed in the only homogenizing agent, it would be possible in principle Sun and that in standard solar models is, by many astro- to determine the thickness of the tachocline from the prop- physical standards, extremely small, it is very much larger erties of the acoustical glitch. A calibration of the dimen- than the standard errors in the determinations of the sound sions of the glitch is likely to be more precise than analyses speed (in places by as much as 20 formal standard errors of rotational splitting, for example, because the mean mul- in the helioseismological inversions). Therefore the discrep- tiplet oscillation frequencies that are employed have been ancies are very significant, and demand explanation. Those determined from observation more precisely than has ro- discrepancies can quite easily be separated into two compo- tational splitting. However, the outcome is not necessarily nents: one that varies on a lengthscale comparable with the more accurate, because it is conditional on the assumptions size of the Sun, and is no doubt a consequence of gross, albeit of the stellar modelling, both on the physical approxima- small, errors in properties that influence the global struc- tions adopted in constructing the standard reference solar ture, such as the primordial chemical composition, the treat- model and on the structure adopted for the tachocline, now ment adopted for the opacity which controls the radiative and throughout the earlier evolution of the Sun. transfer of energy, or the dominant nuclear cross-sections, A calibration of the glitch was attempted by and one that is local, and is produced by errors in the treat- Elliott & Gough (1999) and Elliott et al. (1998), who ment of processes that influence the sound speed directly. adopted what one might regard as the simplest of as- The tachocline anomaly is the more prominent member of sumptions: namely that the stratification of the Sun is the latter component, and is the subject of the investigation spherically symmetrical, that material mixing is complete reported here. down to the base of the tachocline at a rate that does It has long been appreciated that a spatially confined not disturb radiative equilibrium, essentially in conformity sound-speed anomaly is most likely to be a result of an with the tachocline model of Gough and McIntyre (1998), error in the pressure-density relation adopted in the ref- sometimes referred to as a slow tachocline, and that the erence solar model, whose influence on the sound speed 1/2 thickness ∆ of the tachocline has not changed through- c =(γ1p/ρ) is direct. Here p is pressure, ρ is density and out the main-sequence lifetime of the Sun, together, of γ1 = (∂ ln p/∂ lnρ)ad is the first adiabatic exponent. Such course, with the other standard assumptions of solar mod- an error could arise from an error in the concentrations of elling (Christensen-Dalsgaard et al. 1996). They found that the abundant elements, which influences the mean molecular ∆ = 0.020R, the formal uncertainty in that value, resulting mass µ. In contrast, an error in the opacity or a controlling from only the quoted uncertainties in the seismic frequency nuclear reaction rate influences the stratification only via data, being about 5%. coefficients that multiply derivatives in the governing differ- Elliott and Gough were encouraged by their finding that ential equations, and therefore has more spatially extensive not only the height and width of the acoustic glitch but also consequences (e.g. Elliott & Gough 1999). Although there is its functional form could be reproduced within the formal well appreciated uncertainty in the physics of the equation errors by adjusting but a single parameter ∆. That pro- of state relating p to ρ, which might well be of a magnitude vided some, yet unjustified, modicum of faith in their result. comparable with the sound-speed anomaly considered here However, the glitch in the calibrated model was displaced (e.g. D¨appen et al. 1990; Christensen-Dalsgaard & D¨appen upwards by 0.015R from that in the reference model, the 1992; Baturin et al. 2000), it is quite unlikely that it would consequences of which we discuss in the next section. With be restricted to a narrow region in the p–ρ relation, partic- what now appears to be the false confidence in their partial ularly the region that happens (by chance, from a thermo- success, they presumed that the calibration of ∆ was reli- dynamical point of view) to coincide with the base of the able, and that the fully mixed region was deeper than earlier solar convection zone. Therefore the most likely culprit is analyses had indicated. the spatial variation of the relative abundances of the most It was never intended to be believed that the simplis- abundant chemical elements, helium and hydrogen, for only tic tachocline modelling of Elliot and his colleagues, based they have a sufficiently strong influence on the equation of on the minimalist analysis of Gough and McIntyre (1998), state.