Sensitivity of p modes for constraining velocities of microscopic diffusion of the elements
aa r X i v : . [ a s t r o - ph ] O c t **FULL TITLE**ASP Conference Series, Vol. **VOLUME**, **YEAR OF PUBLICATION****NAMES OF EDITORS** Sensitivity of p modes for constraining velocities ofmicroscopic diffusion of the elements
Orlagh L. Creevey
Instituto de Astrof´ısica de Canarias, C/ V´ıa Lactea s/n, La Laguna38205, Tenerife, Spain: Email: [email protected]
Savita Mathur
Indian Institute of Astrophysics, Koramangala, Bangalore 560034, India
Rafael A. Garc´ıa
Laboratoire AIM, CEA/DSM-CNRS-Universit´e Paris Diderot; CEA,IRFU, SAp, F-91191, Gif-sur-Yvette, France
Abstract.
Conventional astrophysical observations have failed to providestringent constraints on physical processes operating in the interior of the stars.However, satellite missions now promise a solution to these problems by provid-ing long-term high-quality continuous data which will allow the application ofseismic techniques. With this in mind, and using the Sun as our astrophysicallaboratory, our aim is to determine if Corot- and Kepler-like asteroseismic datacan constrain physical processes like microscopic diffusion. We test to what ex-tent can the observed atmospheric abundances coupled with p-mode frequenciessafely distinguish between stellar initial chemical composition and diffusion ofthese elements. We present some preliminary results of our analysis.
1. Introduction & Method
If we have the following observations for a solar-type star: a set of p-modefrequencies, measured abundances, and an identified g mode, we can investigatehow these observations may constrain some of the physical processes occuringin the interior of a star. Suppose that the stellar model is a main sequence 1M ⊙ star, and that we can calibrate the age of the star using the radius andluminosity. Then the free parameters that remain to be fitted in the star arethe initial mass fractions of hydrogen and metals X and Z , and the velocitiesof diffusion of these elements V X , V Y and V Z , (assuming that other microscopicprocesses may not be “observable”), where Y denotes helium. Let’s denote theseset of parameters as P . Using P we can calculate a stellar model which willgive us the expected observables , B (expected p-mode frequencies, abundancesetc.). If we have a set of observations O , then in order to find the P that mostadequately reproduce O , we can use the following equation iteratively, until wemake O as close as possible to B : δ P = V ˜W − U O − B ǫ . (1)1 Figure 1. The relative uncertainties as a fraction of the initial values of 3%,24%, 189%, 217%, 422% (details in text) as extra observables are included. δ P are the parameter corrections to make to the initial guess P , ǫ are themeasurement errors, and UWV T is the singular value decomposition of thesensitivity matrix, which can be written as ∂B∂P /ǫ and ˜ W − has some zero valuesto stabilize the inversion (see Creevey (2008, 2009)).Once P have been found, then the theoretical uncertainties σ ( P ) can becalculated for each j = 1 , , ...N parameter σ j ( P ) = N X k V jk ˜W k . (2)The partial derivatives were calculated using the CESAM code Code d’EvolutionStellaire Adaptatif et Modulaire (Morel 1997) and ADIPLS (Christensen-Dalsgaard2007). We took the standard solar model as a reference using the abundances ofGrevesse & Noels (1993), and calibrated these models in radius and luminosityto an accuracy of ∼ − as described in Mathur et al. (2007). The referenceparameters for the models are ( X , Z ) = (0 . , . V X , V Y , V Z ) =(2 . , − . , − . × − cm s − .
2. Results and Discussion2.1. Impact of each observable on the uncertainties
Using only p-mode frequencies to determine the parameter uncertainties (usingEq. 2) the diffusion coefficients V X , V Y and V Z can not be disentangled. Theiruncertainties remain too large (189%, 217% and 422% respectively) when weassume that we have only these observations with errors of the order of 1.3 µ Hz,meanwhile the values for σ ( X ) and σ ( Z ) are 3% and 24%. However, to showthe effect that each of the observations has on the determination of the pa-rameters, Figure 1 shows how the uncertainties in each of the parameters reducerelative to these values, as extra observables such as abundances and g modes areincluded. Each of the bars represents from left to right the following additionsto the set of observations: • Blue reduce the errors on the p-mode frequencies by almost a factor of 2(from 1.3 µ Hz – 0.7 µ Hz) • Green include abundance measurements of He, C, N and O • Yellow include abundance measurements of Li and Be ( ǫ = 0.1 dex for allabundances) • Red include 1 identified g modeAfter including all of these observations, the final parameter uncertainties are:1%, 6%, 28%, 57% and 72% respectively.It is interesting to note the various effects of adding in new observables.Firstly, reducing all of the uncertainties by almost a factor of 2 should resultin a corresponding reduction in the uncertainties by this same amount. Theblue bars clearly indicate that this is the case. When we include the abundancemeasurements of He, C, N, O with an observational error of 0.1 dex, the uncer-tainties in both X and Z reduce by almost a factor of two (green). These sameabundances have little effect for constraining the diffusion velocities. However,including Be and Li measurements has a significant impact on the determina-tion of the metal diffusion velocity V Z (yellow), reducing its uncertainty to 72%( ∼
20% of its original value). Finally, the inclusion of one identified g mode hasmost impact on constraining V X and then V Y (red). The final uncertainties may be sufficiently low that the observations do containenough information to be able to distinguish safely between initial metal massfraction Z and diffusion of this V Z (our original scientific question). Assumethat we have two new sets of observations, and that these O really come from asolar model with 1) lower Z and 2) lower V Z . If we use Eq. 1 to fit the new O (separately) while using the reference set of parameters as P (the initial guess),we will obtain non-zero negative values of δZ and δV Z respectively, only if theobservations contain sufficient information. If there is not enough informationcontained in the observations, then δZ ∼ δV Z ∼ δ P (relative to the original value of P , forscaling purposes) for some of the parameters of the models, when attempting tofit a set of observations generated from a model with a reduced Z (red) and areduced V Z (blue). As the figure shows, some non-zero δ P are needed to accountfor the new observations. • When inverting using the lower Z model observations (red), δZ is nega-tive. This means that we need to reduce the original Z value to a lower Figure 2. The parameter changes δ P needed to make to P to correctly fitthe new observations. These new observations were generated using a modelwith a lower Z (red) and a lower V Z (blue). one — consistent with the input model. However, we also see a nega-tive contribution to V Z ; ideally V Z would not change, because it was notchanged in this model, just as X does not change. • When inverting using the lower V Z model observations (blue), we obtaina much more negative δV Z (but no change in Z ), indicating a necessarydownward revision of only V Z — consistent with the input model.These results are preliminary, but encouraging: there is some indication inthe set of observables that we may be able to distinguish between initial metalabundance and diffusion of this using abundance measurements and seismic data. Acknowledgments.
OLC wishes to acknowledge David Salabert for helpwith MATLAB. RAG is very grateful to the IAC for its visitor support. SMwishes to thank S´ebastien Couvidat for his help with Fortran. This researchwas in part supported by the European Helio- and Asteroseismology Network(HELAS), a major international collaboration funded by the European Com-mission’s Sixth Framework Programme and by the CNES/GOLF grant at SAp,CEA, Saclay.