Parametric stellar convection models

Parametric stellar

convection models

Tony Barbosa Da SilvaMestrado em AstronomiaDepartamento de Física e Astronomia

2015

Orientador

Mário João Pires Fernandes Garcia Monteiro,

Professor Associado, Faculdade de Ciências da Universidade do Porto

Coorientador

Michaël Bazot,

Investigador, Centro de Astrofísica da Universidade do Porto

Todas as correções determinadas

pelo júri, e só essas, foram

efetuadas.

O Presidente do Júri,

Porto, ______/______/_________

Acknowledgement

Firstly, I would like to express my sincere gratitude both my supervisors, Dr. Michaël Bazot and Prof.Dr. Mário João Monteiro, without whom the realization of this thesis would not have been possible.

I gratefully acknowledge my host institution, Centro de Astrofísica da Universidade do Porto, for thefacilities provided me during my thesis.

Lastly , I would like to thank my mother for her patience and my family and relatives for their faith.During last year, I was a junior researcher at Centro de Astrofísica da Universidade do Porto (CAUP).

The support for that period has been provided by CAUP grant CAUP-12/2014-BI, with funds from theSpaceInn project, funded by the European Union under the Seventh Framework Programme (FP7).

This research has made use of NASA’s Astrophysics Data System.I like to acknowledge the work done by people who have contributed towards the development of the

GNU project and for helping the development of free software.

3

FCUP 4Parametric stellar convection models

Abstract

This dissertation has been submitted to the Faculdade de Ciências da Universidade do Porto in partialfulfillment of the requirements for the Master degree in Astronomy.

The dissertation is mainly composed of four chapters and a list of appendices.Chapter 1 serves as a general introduction to the field of asteroseismology of solar-like stars. It starts

with an historical account of the field of asteroseismology followed by a general review of the basic physicsand properties of stellar pulsations. The chapter closes with a discussion about the motivations and issuesaddressed in this work.

In Chapter 2, a general introduction to solar seismology and convection are provided. An implementationand validation of a new parametrization for convection is tested for the Sun, by comparing modelled Sun’sfrequencies with observed ones.

In Chapter 3, I address the same problem for solar-like stars, in this case the binary α Centauri.Finally, in Chapter 4, I summarize the results obtained in the two previous chapters. We have shown

that we do not found a region of values of the new parametrization that can cancel the near-surface effectsin the p-modes frequencies.

Keywords

Keywords. asteroseismology – stars: solar-type – stars: evolution – stars: interiors – stars: oscillations– methods: data analysis

5


Resumo

Esta dissertação foi submetida à Faculdade de Ciências da Universidade do Porto no cumprimentoparcial dos requisitos necessários à obtenção do grau de Mestre.

A dissertação é composta principalmente de quatro capıtulos.O primeiro capitulo serve de introdução ao campo da asterossismologia de estrelas do tipo solar. Começa

com um relato histórico do campo da asterossismologia seguido de uma revisão da fısica básica e pro-priedades das pulsações estelares.

No capítulo 2, uma introdução à sismologia solar é fornecida. É testada uma nova implementação deuma parametrização da convecção para o Sol, comparando as frequências téoricas com as observadas.

No capítulo 3, eu abordo o mesmo problema para estrelas similares ao sol, neste caso o binário α

Centauri.Finalmente, no capítulo 4, é sumariado os resultados obtidos nos dois capítulos anteriores. Foi mostrado

que não existe uma região de valores onde a nova parametrização cancela os efeitos provocados pelacamada superficial das estrela para as frequências do tipo p.

Palavras chave

Palavras-chave. asterosismologia – estrelas: tipo solar – estrelas: evolução – estrelas: interiores –estrelas: oscilações – métodos: análise de dados

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Contents

1 Solar Structure and Oscillations 151.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151.2 Stellar modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161.3 Stellar evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171.4 Convection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181.5 Stellar oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211.6 Properties of oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221.7 The asymptotic relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231.8 Motivations and Issues Addressed in this Work . . . . . . . . . . . . . . . . . . . . . . . 24

2 Stellar models 272.1 Input physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272.2 Solar models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282.3 Surface corrections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292.4 Parametrization of convection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3 Solar-like pulsators 393.1 Modelling Solar-like pulstators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393.2 Fundamental properties of α Cen A and B . . . . . . . . . . . . . . . . . . . . . . . . . 403.3 Calibration method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423.4 New parametrization applied to α Cen AB . . . . . . . . . . . . . . . . . . . . . . . . . 43

4 Conclusions and future prospects 454.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

Appendices 59

9


List of Figures

1.1 Computed eigenfrequencies for a model of the present Sun. Figure courtesy of Christensen-Dalsgaard [2002] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.1 Frequency differences between a solar model calibrated with ASTEC using MLT (blue) andCM (red) formulation and observed GOLF frequencies. . . . . . . . . . . . . . . . . . . . 28

2.2 Propagation diagram for the Sun. The profiles of the buoyancy N 2BV acoustic frequency S2

l

, this latter for different values of the angular degree l, are plotted. Horizontal dotted linesdelimit the frequency range of observed solar oscillations. From top to bottom, dashedlines span the cavities where p-modes with l = 20 and l = 2 are trapped, and the regionof propagation of a gravity mode. [From Lebreton and Montalbán [2009].] . . . . . . . . 30

2.3 The surface term between the Sun and different published solar models (INVBB [Antia,1996], BSB(GS98) [Bahcall et al., 2006], Model S [Christensen-Dalsgaard et al., 1996],BP04 [Bahcall et al., 2005], and STD [Basu et al., 2000]) fit with different models of thesurface term (KBCD08 [Kjeldsen et al., 2008], BG14-1 and BG14-2 from [Ball and Gizon,2014]) in the range ±5Δν. In each panel, the points are the data, and the lines are thefits. The different colors and symbols indicate different solar models. Figure 8 of [Schmittand Basu, 2015] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.4 Values of a1 and a2 around the MLT case, (αw,αθ,αh) = (1.0, 1√3, 1.0). Each colour

represents a diferent value for αh: red for αh = 1, blue for αh = 0.5 and green for αh = 1.5 342.5 Values of a1 and a2 around the CM case, (a1, a2) = (24.868, 0.097666). Each colour

represents a diferent value for αh: red for αh = 1, blue for αh = 0.5 and green for αh = 2.0 352.6 a Temperature at the top of convection zone for MLT (continuos line), (a1, a2) =

(1.125, 1.0), models around MLT (dotted lines): blue (2.1747, 0.8083); red (4.6731, 0.5956);green (6.1507, 0.7621) and around CM (dot dashed lines): cian (19.0289, 0.0924); magenta(118.0944, 0.0706); green (9.7428, 0.1155) .b Plot of temperature gradient for the samemodels as in a. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

2.7 Relative differences in Δc/c and Δv/v of models around the MLT relative to the MLT atconstant mass fraction changing one α at time. . . . . . . . . . . . . . . . . . . . . . . . 37

2.8 Frequency differences around the MLT case for different values of m and z. . . . . . . . . 382.9 Frequency differences around the CM case for different values of m and z = 0.017 . . . . 38

3.1 Hertzsprung-Russell diagram showing stars with solar-like oscillations for which Δν hasbeen measured. Filled symbols indicate observations from ground-based spectroscopy andopen symbols indicate space-based photometry (some stars were observed using bothmethods). Figure 1.3 of [Bedding, 2014] made by Dennis Stello . . . . . . . . . . . . . . 40

3.2 Stellar evolution tracks from 0.0 Gyr until 8.8 Gyr for different values of X and Z . . . . 43

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3.3 Large and small separations for the A component of α Cen system. Lines correspond todifferent models from 3.4. For clarity only l = 1 large separations are shown. . . . . . . . 44

3.4 Same as figure 3.3 for the B component of α Cen system . . . . . . . . . . . . . . . . . 44

List of Tables

1.1 Solar characteristics with their values in CGS units . . . . . . . . . . . . . . . . . . . . . 16

3.1 Observational constraints for α Cen A and B. References: (1) [Söderhjelm, 1999], (2)[Pourbaix et al., 2002], (3) [Eggenberger et al., 2004], (4) [Kervella et al., 2003], (5)[Bouchy and Carrier, 2002] and (6) [Carrier and Bourban, 2003]. . . . . . . . . . . . . . 41

3.2 Oscillation frequencies for α Cen A (µHz). Table 1 from Bedding et al. [2004]. . . . . . . 413.3 Oscillation frequencies for α Cen B (µHz). Table 1 from Kjeldsen et al. [2005] . . . . . . 423.4 α Cen A B Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

.1 Values around the MLT case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

.2 Values around the CM case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

.3 Values around the PTGb6 case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

13


The following table describes the significance of various abbreviations and acronyms used throughoutthe thesis.

MLT . . . . . Mixed Length Theory

CE . . . . . . Convective Envelope

EOS . . . . . . Equation Of State

SSM . . . . . Standard Solar Model

FST . . . . . . Full Spectrum of Turbulence

SAL . . . . . . Super Adiabatic Layer

Chapter 1.

Solar Structure and Oscillations

1.1 INTRODUCTION

Stars are the fundamental units of the observed Universe. For that reason, the understanding of stellarstructure and evolution is a important aspect in the field of astrophysics. Until some decades ago, theonly measures that were obtained from the stars was the radiation emitted by the surface layers and gaveno direct knowledge about their interior. This problem is expressed by Sir Arthur Stanley Eddington inthe opening paragraph of The Internal Constitution of the Stars, Eddington [1926],

At first sight it would seem that the deep interior of the Sun and stars is less accessibleto scientific investigation than any other region of the universe. Our telescopes may probefarther and farther into the depths of space; but how can we ever obtain certain knowledge ofthat which is hidden behind substantial barriers? What appliance can pierce through the outerlayers of a star and test the conditions within?

Asteroseismology focuses on the propagation of waves through stars. An Asteroseismologist observeeffects of these waves in luminosity variations and shifts in radial velocity. The observational effects implynormal modes of oscillation and each mode contains information about the interior stellar structure.

In the case of the Sun, a detailed picture of its internal structure was obtained thanks to helioseismology,i.e. the understanding of the Sun’s internal structure and dynamics through the study of solar globaloscillations.

Leighton et al. [1962] observed small modulations in the surface velocity of the sun that occured on arougly five minute timescale, and Ulrich [1970]; Leibacher and Stein [1971] interpreted as global acousticoscillation modes that later Deubner [1975] confirmed as manifestations of global modes.

Since the first detection of the global five minute oscillation modes [Claverie et al., 1980], helioseismologyhas been providing a wealth of high-quality information about the structure and proved to be very successfulin probing the internal structure of our Sun [Chaplin and Miglio, 2013]. The massive amount of data onsolar oscillations collected over the past two decades made possible a considerably accurate determinationof the Sun’s internal sound speed, density profiles and helium content, the determination of the locationof the solar helium second ionization zone and the base of the convective envelope [Christensen-Dalsgaardet al., 1991] [Basu and Antia, 1997], the detailed testing of the equation of state and the inference ofthe solar internal rotation (e.g., Christensen-Dalsgaard [2002]; Basu and Antia [2008]; Chaplin and Basu[2008]; Howe [2009]).

Adding precision to the determination of global parameters is certainly an interesting application. Thedetermination of parameters relies on the stellar models we use to compute oscillation frequencies.

15


Instead of assuming accepted models and deriving parameters, with asteroseismolgy and observationaldata from ground and space based telescopes like Corot [Baglin et al., 2002] and Kepler [Koch et al.,2010], we can prove if our models are wrong or inaccurate. The objective of asteroseismology is to usestars as laboratories to test physical inputs of the models. This would add accuracy and precision to thedetermination of global parameters and, on the other hand, it would allow a deeper knowledge of physicsin the conditions reached in the stellar interiors.

1.2 STELLAR MODELLING

In order to test stellar evolutionary models, we need high-precision and accurate measurements offundamental parameters for the star, such as: mass M , luminosity L and the radius R. With thoseparameters and using basic physics we can calculate the effective temperature Teff , the surface gravitylog(g) and the mean density ρ.

The Sun is an unique star that its properties are known with high precision. The Sun mass, luminosityand radius are know to 0,1%, the age to 1 % and the composition to 10% [Guenther and Demarque,1997]. The commonly used values are available in table 1.1 , taken from Lang [1980]

Name Symbol Value UnitsMass M� 1.989× 1033 gRadius R� 6.9599× 1010 cmLuminosity L� 3.846× 1033 erg s−1

Effective temperature Teff 5778 k

Table 1.1: Solar characteristics with their values in CGS units

In most astronomical objects, hydrogen and helium are the dominant elements. Thereby the chemicalcomposition of the stars are represented by X, Y and Z, respectively the abundances by mass of hydrogen,helium and the sum of all the remaining heavy elements. By definition, we have

1 = X(r) + Y (r) + Z(r) (1.1)

Solar modelling depends on two unknown parameters: the initial helium abundance, Y0, and a parametercharacterizing the efficacy of convective energy transport near the solar surface. These parameters canbe adjusted to provide a model of solar mass, matching the solar radius and luminosity at the age of theSun.

One of the best and most widely adopted solar models, that combine all the generally accepted physics,is Model S discussed by Christensen-Dalsgaard et al. [1996] 1. It was computed with a local mixing-length theory (Böhm-Vitense [1958] MLT formulation) ignoring turbulent pressure and initial hydrogenand heavy-element abundances, X0 = 0.7091, Z0 = 0.0196, chosen to match the current solar valuestable 1.1.

1The model is available at http://astro.phys.au.dk/~jcd/solar_models/


1.3 STELLAR EVOLUTION

Here I very briefly describe the basic equations of stellar structure and evolution. The equations forstellar structure and evolution, assuming spherically symmetric, neglecting rotation and magnetic fields,comprise four first order non linear differential equations for hydrostatic equilibrium, conservation of mass,conservation of energy and energy transport (for more details see for example Schwarzschild [1958], Cox[1968], Clayton [1968], Kippenhahn et al. [2012]):

dp

dr= −Gmρ

r2, (1.2)

dm

dr= 4πr2ρ , (1.3)

dT

dr= ∇T

p

dp

dr, (1.4)

dL

dr= 4πr2

�ρ�− ρ

d

dt

�u

ρ

�+

p

ρ

dρ

dt

�. (1.5)

Here r is distance to the center of the star, p is pressure, m(r) is the mass of the sphere interior tor, ρ is density, T is temperature, L is the flow of energy per unit time, � is the rate of nuclear energygeneration per unit mass and time, u is the internal energy per unit volume and G is the gravitationalconstant. The temperature gradient is characterized by ∇ = d lnT/d ln p and is determined by the typeof energy transport.

It is necessary to consider that during the evolution of the star, the composition changes via nuclearreactions and convective process. The rate of change of the abundance X by mass of hydrogen is thereforegiven by

dX

dt= RH +

1

r2ρ

∂

∂r

�r2ρ

�DH

∂X

∂r+ VHX

��(1.6)

where RH is the rate of change in the hydrogen abundance from nuclear reactions, DH is the diffusioncoefficient and VH is the settling speed.

The equations must be supplemented by what is called the microphysics of the stellar interior. In otherwords, an equation of state (EOS) that relates ρ, T and other thermodynamic functions,

ρ = ρ(r, P, T,X, Y ) , (1.7)

∇ad = ∇ad(r, P, T,X, Y ) , (1.8)

Q = Q(r, P, T,X, Y ) , (1.9)

Cp = Cp(r, P, T,X, Y ) , (1.10)

Γ1 = Γ1(r, P, T,X, Y ) . (1.11)


where ∇ad, Q, Cp and Γ1 are respectively the adiabatic gradient, expansion coefficient, specific heat atconstant pressure and the adiabatic exponent.

Also the nuclear reaction rates and the opacities of the assumed elements,

� = �(r, P, T,X, Y ) , (1.12)

k = k(r, P, T,X, Y ) . (1.13)

and a theory for energy transport inside the star given by the temperature gradient

∇(r) ≡ d log T

d logP= ∇(r, P, T,X, Y ) , (1.14)

1.4 CONVECTION

The energy can be transported by convection, conduction and radiation.When the energy transport is dominated by radiation, ∇ = ∇rad, where the radiative gradient is given

by

∇rad =3κp

16πac̃GT 4

L(r)

m(r). (1.15)

Here c̃ is the speed of light, a is the radiation density constant and κ is the opacity defined such that1/(κρ) is the mean free path of a photon.

When ∇rad exceeds the adiabatic gradient ∇ad = (∂ lnT/∂ ln p)s, the layer becomes unstable to con-vection. In that case energy transport is predominantly by convective motion. This is the Schwarzschildcriterion [Schwarzschild, 1958] that states that convection occurs in regions where the adiabatic temper-ature gradient is smaller than the radiative gradient.

The favourable conditions for convection in main sequence stars are: large opacities, which leads tolarge ∇rad; low γ 2 , which leads to small ∇ad; partial ionisation zones, which brings γ close to 1 andhence small ∇ad ( ∇ad = 1− 1

γ);

Convection is one of the most important physics processes for mixing in stars. Stellar convection ischaracterized by large upflowing regions (granules) separated by cool fast downdrafts (intergranular lanes).At the present, convection remains a source of uncertainty in stellar model computations. In the caseof the Sun, convection occurs in the outer about 29% of the solar radius. In the most part of the Sun,convection is nearly adiabatic with ∇ slightly greater than ∇ad, due to the high efficacy of convection.However convection becomes inefficient near the upper boundary of the solar convection zone, and the

2in the case of idea gas γ = Γ1, then ∇ad = γCp


temperature gradient ∇ deviates significantly from its adiabatic value ∇ad. This transition region betweenthe adiabatic region and the radiative atmosphere is called the superadiabatic layer (SAL) because ∇−∇ad

is of the order unity Demarque et al. [1997, 1999]. The difference between those two gradients provides ameasure of the inefficiency of convection, and reveals the depth at which there is a significant departurefrom efficient convection Tanner et al. [2013].

There are three main analytical models of convection using to modelling a star: mixing-length theory(MLT), non-local MLT and turbulent convection models [Trampedach, 2010].

The standard and simplest description of convection used in stellar modelling is the Mixing LengthTheory introduced by Vitense [1953] and Böhm-Vitense [1958], which are based on earlier works on theconcept of convective motion by Taylor [1915] and later by Biermann [1951]. The basic idea is to modelthe complicated pattern of structure and motion of convective elements by a suitable mean element withconvenient velocity. There are two characteristic lengths: the mixing length and the dimension of theelement. In the MLT both are assumed to be equally proportional to the local pressure scale height,Hp = P

ρg, with the proportionality coefficient, αMLT taken as a free parameter to be fixed by comparing

model results with observations. Normally αMLT , the "mixing length ratio", is between 1 and 2. Whichmeans that each of convective elements is supposed to travel on average over the distance lm , the mixinglength, and dissolve into the surrounding medium losing its identity

lm = αMLT Hp (1.16)

The mixing length lm, is a free parameter that is not determined within the theory and varies across theHertzsprung–Russell diagram according to 3D convection simulations (Abbett et al. [1997] Ludwig et al.[1999]; Freytag et al. [1999]).

The problem is that convective elements may have noticeable different scales. In the MLT, all theelements possess the same physical properties if generated at a certain radial distance r from the center.Also, the dimensions of the average elements are assumed to be unique (which is lm) and the shape ofthe elements is not specified. As mention by Trampedach and Stein [2011], the standard MLT is a localtheory, meaning that energy flux is derived purely from local thermodynamical properties, ignoring thusany nonlocal properties such as overshooting , also, it does not allow for turbulence, it does not allow forasymmetry between upflows and downflows, especially at large temperature inhomogeneities present atthe surface layer and its effects on opacities and radiative transfer.

The MLT formalism only provides a very simplified description of convection. To solve some of theproblems of the classic MLT, like a consistent description of overshooting, some theories inclued non-localeffects, have been developed [Spiegel, 1963]. There have been several attempts to introduce non-localityin the MLT [Gough, 1977]. A series of non-local MLT have been developed by Balmforth [1992a,b,c]Grossman et al. [1993], Grossman and Narayan [1993], Grossman [1996], Deng et al. [1996], Grigahcèneet al. [2005], Dupret et al. [2006]. Forestini et al. [1991] have produced an MLT-type model incorporatinga measure of asymmetry between upflows and downflows. Demarque et al. [1997] introduced a variablemixing length parameter derived from the simulations of Kim et al. [1996].

A lot of effort has been devoted to the development of a non-local theory of convection, Gough [1977].More reasonable theories of stellar convection are the turbulent convection models such as the proposed


by Canuto and Mazzitelli [1991], hereinnafter CMT. In Canuto and Mazzitelli [1991] have attempted togeneralize the MLT by taking into account the whole spectrum of convective wavelengths. The parametersof CMT of convection are based on the results of laboratory experiments of incompressible convectionextrapolated to stellar conditions. The CMT also contains a free length-scale Λ = αz , where z is thedistance to upper boundary of the convective envelope. A scaling factor α is also required to producesolar models with the correct radius.

It is worth to notice that this kind of parameters exist also in some sophisticated models of stellarconvection. More turbulent Convection models are described in Xiong et al. [1997]; Canuto [1997];Canuto and Dubovikov [1998]; Deng et al. [2006]; Li and Yang [2007].

An ideal self-consistent theory of convection should derive those typical scales from first principlesinstead of assuming them as parameters. Recently Pasetto et al. [2014] have presented an analytical,non-local, time-dependent model for the convective energy transport that does not depend on any freeparameter. Arnett et al. [2015] argue that the Pasetto et al. [2014] theory may not yet be applicable tothe SAL. However this new model has not been implemented in any stellar evolution code.

Comprehensive 3D hydrodynamic simulations have become an important tool to study solar and stellarconvection. The mixing length parameter can be deduced from hydrodynamic simulations, where convec-tion emerges from first principles. The first attempt to incorporate the results of numerical simulations ofconvection into solar models is attributed to Lydon et al. [1992, 1993a], the convective flux approximation(CFA), based on the simulations of Chan and Sofia [1989]. Other works that that attempt a connectionbetween MLT and numerical simulations are Kim et al. [1996], Robinson et al. [2003] and Freytag et al.[1999].

There is an extensive literature that compare local mixing-lenth versions of convection with 3D simu-lations of the hydrodynamics of convection Chan and Sofia [1987]; Stein and Nordlund [1989]; Freytaget al. [1996]; Nordlund and Stein [1997]; Abbett et al. [1997]; Nordlund and Stein [2000]; Trampedach[2010]; Trampedach and Stein [2011]; Trampedach et al. [2014]. Nordlund and Dravins [1990] and Freytaget al. [1996] have reported on detailed comparisons of the results of such simulations with the predictionsof MLT. Numerical simulations of near-surface turbulent convection have also been able to explain theobserved frequency dependence of the energy input and damping rates [Stein and Nordlund, 2001].

Li et al. [2002] present a method to include turbulence in solar modeling within the framework of theMLT, using the turbulent velocity obtained from 3D numerical simulations of the outer layers of the Sunof Robinson et al. [2001]. This method produces p-mode frequencies better than the MLT and CM.

These three-dimensional calculations are extremely time-consuming because convection zone motionsare characterized by turbulence that is chaotic and it involves non-linear interactions over many disparatelength scales. To circumvent this problem, simplifications of the governing equations are often used.

Results show that numerical simulations and MLT produce very different pictures of convection onthe SAL part of the convection zone [Demarque et al., 2007], that convection is driven on the scaleof granulation by radiative cooling at the surface layers. There are no convective bubbles, but highlyasymmetric convective motions.

Second, superadiabatic peak is displaced outward because no turbulent pressure term is present in theMLT. The turbulent pressure is caused by turbulent fluctuations in the velocity field and has a significantcontribution to hydrostatic balance through the SAL.


1.5 STELLAR OSCILLATIONS

In order to understand the diagnostic potential of solar oscillations, some basic insight into the propertiesof stellar oscillations is required. Here I will briefly point out the main characteristics of stellar oscillations.A more complete and detailed discussion can be found in the monographs of Cox [1968], Unno et al.[1989] , Aerts et al. [2010] and J. Christensen-Dalsgaard’s Lecture Notes on Stellar Oscillations 3. Recentreviews on solar and stellar oscillations are given for example by Christensen-Dalsgaard [2002], Di Mauro[2003] Cunha et al. [2007], Basu and Antia [2008] , Christensen-Dalsgaard and Houdek [2010], Kosovichev[2011], Chaplin and Miglio [2013], Bedding [2014] and Christensen-Dalsgaard [2014].

A simple theoretical model of solar oscillations can be derived from the equation of conservation of massand momentum.

The conservation of mass is expressed by the equation of continuity:

d�

dt= −� div�v (1.17)

The momentum equation (conservation of momentum of a fluid element) is:

ρd�v

dt= −∇P + ρ�g, (1.18)

The gravitational acceleration g, is calculated from the gradient of the gravitational potential Φ

g = −∇�Φ (1.19)

where �Φ satisfies the Poisson’s Equation

∇2�Φ = −4πG� , (1.20)

To complete the description, it is necessary to relate p and ρ. Since in the star time scale, the energyexchange is much longer than the relevant pulsation periods and so the motion of the fluid element isessentially adiabatic, satisfying the adiabatic approximation

dS

dt=

d

dt

�P

ργ

�= 0, (1.21)

3http://users-phys.au.dk/jcd/oscilnotes/


or

dP

dt= c2

dρ

dt, (1.22)

where c2 = γP/ρ is the squared adiabatic sound speed and d/dt is the material derivative, followingthe motion of the fluid element; given by

d

dt=

∂

∂t+ v ·∇, (1.23)

This approximation shouldn’t be applied in the surface of the sun, where density is low and the pertur-bations are no longer adiabatic.

1.6 PROPERTIES OF OSCILLATIONS

The frequencies of the modes are determined by the characteristic frequencies: acoustic (Lamb) fre-quency, Sl , and the buoyancy (Brunt-Vaisala) frequency, N , which are given below.

S2l =

l(l + 1)c2

r2, (1.24)

N2 = g

�1

Γ1p

dp

dr− 1

�

d�

dr

�(1.25)

Based on the nature of the restoring force, we can distinguish between pressure (p) modes, surfacegravity (f) modes and internal gravity (g) modes.

Mathematically, each p mode can be described by three integers: the radial order n, that specifies thenumber of nodal shells of the standing wave; the angular degree l, that specifies the number of nodal linesat the surface, and the angular order m, where −l ≤ m ≤ l, so for each degree l are 2l+1 modes. Modeswith l = 0 are called radial modes, and those with l ≥ 1 are non-radial modes. The eigenfrequenciescomputed for the standard model of the present Sun using are presented in fig 1.1.

Different oscillation modes are sensitive to different regions of the solar interior; for example, modeswith small n and l penetrate more deeply into the Sun whereas high modes sample only the near-surfacelayers.

Since the Sun is not spherically symmetric and rotate, the frequencies of the 2l + 1 modes with thesame values of l and n are not the same, the so called degeneracy of the eigenfrequencies is removed.


Figure 1.1: Computed eigenfrequencies for a model of the present Sun. Figure courtesy ofChristensen-Dalsgaard [2002]

1.7 THE ASYMPTOTIC RELATION

The asymptotic relations are very important in many pulsating stars. According to the asymptotic theoryTassoul [1980] the oscillation spectrum can be characterized by two frequency separations, the large andthe small spacing.

νn,l ≈ Δν0(n+l

2+ �) (1.26)

where Δν0 is the the mean frequency separation between two modes of consecutive radial orders, and


is also defined by

Δν0 ≈�2

� R

0

dr

c

�(1.27)

where c is the local speed of sound at radius r and R is the photospheric stellar radius.The large frequency spacing corresponds to differences between frequencies of modes with the same

angular degree l and consecutive radial order. It is

Δνn,l = νn+1,l − νn,l (1.28)

The small spacing is the difference between the frequencies of modes with an angular degree l of sameparity and with consecutive radial order. This small spacing is very sensitive to the structure of the coreand mainly to its hydrogen content, i.e. the age of the star.

δνn,l = νn,l − νn−1,l+2, (1.29)

1.8 MOTIVATIONS AND ISSUES ADDRESSED IN THIS WORK

The usual way to determine the properties of a stars is to construct models that reproduce not just theclassical parameters of the star i.e., Teff , L , but also the pulsation frequencies. Since these frequenciesare usually extremely precise, 10−5 [Libbrecht et al., 1990], the stellar properties can also be determinedprecisely.

However a significant source of uncertainty in stellar modelling is the treatment of convection. We arestill unable to model the near-surface layers of a star properly because of the super-adiabatic layer, whereconvection is inefficient.

We now investigate the effect of modifying the convective treatment using a new parametric model.In this work, we focus on new parametric model of convection that can reproduce stellar models with

smaller frequency differences from observed data. This is important to calibrate the stellar models toother stars and determine the fundamental global parameters, such as mass and radius.

The dissertation is organized as follows: in the following chapter (Chapter 2), I describe the parametricmodel of convection of Monteiro et al. [1996] and applied to Sun. Next I extend the parametric model toinclude more degrees of freedom.

In chapter 3, I address the same problem for solar-like stars, in this case the binary α Centauri.The thesis ends with some general conclusions, a summary of the results from the presented investiga-

tions and a discussion about their importance to the field, along with an overview of our plans for the


future.


Chapter 2.

Stellar models

2.1 INPUT PHYSICS

The goal of this thesis is to adopt the state-of-the-art physics using a robust and already tested versionof the code. The stellar models used in this work were obtained using the Aarhus STellar EvolutionCode ( ASTEC, Christensen-Dalsgaard [2008]). In the interior model, we used OPAL opacities tablesfor high temperatures computed by [Iglesias and Rogers, 1996] and atmospheric opacities from Kurucz[1991], Alexander and Ferguson [1994] and low temperatures Ferguson et al. [2005] using the interpolationscheme developed by G. Houdek [Houdek and Rogl, 1996], the Livermore OPAL equation of state [Rogerset al., 1996], the NACRE nuclear reaction rates [Angulo et al., 1999] and the solar mixture from Grevesseand Noels [1993], that the ratio of the heavy elements to hydrogen Z/X = 0.0245(1± 0.1) at the solarsurface and is in good agreement with the helioseismological properties. This abundances are at themoment, subject to debate. Asplund et al. [2004] have derived new solar abundances (AGS abundances)where solar metallicity is reduced by 30 %.However, including these new abundances in solar models hasled to sound-speed profiles in disagreement with helioseismology.

We consider sound-speed, density, pressure, and gravity are functions of height only. Unlike standardsolar models, we did not include magnetic fields or core overshoot in this reference model. Also mass loss,difusion and rotation are not considered.

Eigenfrequencies and eigenfunctions are computed using the Aarhus adiabatic oscillation package (ADIPLS1, Christensen-Dalsgaard [2011]) for a star in the adiabatic approximation.

By comparing the theoretical and observed frequencies and looking for the best fitting model, we canplace constraints on the physical parameters of the observed star, assuming that the physics in the stellarmodel is accurate enough to model the processes inside the star.

To validate our models, we compared the results of the simulations from ASTEC and ADIPLS with theobservations of the Sun with the GOLF 2 instrument (Global Oscillations at Low Frequencies, García et al.[2005]) onboard of the Solar and Heliospheric Observatory (SoHO Domingo et al. [1995]) spacecraft. Thisinstrument measures disk-integrated radial velocities as a function of time.

1Available at http://users-phys.au.dk/jcd/adipack.n/2http://www.ias.u-psud.fr/golf/

27


2.2 SOLAR MODELS

There are several seismic representations of the Sun in the literature (Shibahashi and Takata [1996];Turck-chièze et al. [2001]; Couvidat et al. [2003]).

Although, there is no accepted theory of stellar convection, generally, stellar structure calculations treatthe superficial layers with simplified atmosphere models and the stratification of the superadiabatic regionby means of a local, time-independent mixing-length approach, the mixing length theory (MLT).

Figure 2.1: Frequency differences between a solar model calibrated with ASTEC using MLT(blue) and CM (red) formulation and observed GOLF frequencies.

It is found that envelope models which use the Canuto-Mazzitelli (CM) formulation for calculating theconvective flux give significantly better agreement with observations than models constructed using theMLT formalism Basu and Antia [1994], Monteiro et al. [1996], Montalbán et al. [2001], Samadi et al.[2006], see figure 2.1.

Much effort has been made to modify the Mixing Length Theory of convection to extend beyond theformal boundary in order to calculate frequencies that match with observed data.

Frequency differences are very small at low frequency. This is related to the fact that low-frequencymodes have very small amplitudes in the surface region Christensen-Dalsgaard and Thompson [1997].This is not the case for high-frequency p-modes. Many properties of the p-modes can be interpreted interms of standard physical models of the solar interior. However it is increasingly evident that the outerlayers must be better understood for the theoretical models better predict the observed frequencies moreprecisely. The solar p-mode frequencies, particularly the higher frequency modes, are sensitive to thedetailed structure of the superadiabatic layer.

MLT solar models suffer from several basic inconsistencies. The problem lies mosltly in the superadia-batic layer, where the standard treatment of stellar convection based on the MLT leads to an underestimateof the efficiency of radiative transfer in the superficially part of the superadiabatic layer. Two areas of


refinement that could improve the calculated values of the p-mode frequencies are: the inclusion of nonadiabatic effects from radiation and turbulence in the pulsation calculations; and the adoption of a moresophisticated calculation of the structure and thermodynamics of the superadiabatic layer.

2.3 SURFACE CORRECTIONS

Our poor understanding of the physics of the stellar surface layers causes a major problem in the analysisof global solar-like oscillation, because the properties of the near-surface layers have a major impact onglobal oscillation frequencies. This processes are collectively known as the surface effects or surface term[Aerts et al., 2010, Chapter 7.1].

As discussed in Bhattacharya et al. [2015] computation of oscillation frequencies of stellar models usu-ally assumes adiabaticity, a valid approximation in mostly of the stellar interior, i.e. in regions where thethermal timescale is longer than the period of the oscillations. This is not the case in the near-surfaceregion, where non adiabatic effects and the fluctuations of the turbulent fluxes (heat and momentum)do modify the modelled pulsation eigenfunctions and the oscillation frequencies [Gough, 1980, Balmforth,1992a, Houdek, 1996, 2010]. Besides, the dynamical effects of convection and turbulent pressure fluctua-tions are usually neglected (see the discussion of Rosenthal et al. [1999] for the various reasons why theseshifts are observed).

This improper modelling of the near-surface layers gives rise to an offset between observed and computedoscillation frequencies which increases with frequency (Christensen-Dalsgaard et al. [1996], Christensen-Dalsgaard and Thompson [1997]).

In a recent paper, [Piau et al., 2014] have explored the impact of various parametrization of stellarconvection.

The p-modes waves are bounded from below by a lower turning point, rt set by the Lamb frequency,where Sl(rt) = ω, and from above by an upper turning point set by the acoustic-cutoff frequency, νac. Inthe case of the Sun νac � 5.5mHz, figure 2.2.

For modes with frequencies ν much less than νac reflection takes place so deep in the star that themodes are essentially unaffected by the near-surface structure. However for frequencies between 2 mHzand 5 mHz, the inertia of the near-surface layers is a considerable fraction of the total mass above thereflecting layer, leading to a greater modification to the phase shift in the spatial oscillation eigenfunctionsand also to a change in frequency Houdek [2010].

These are the modes for which the measured frequencies display a systematic deviation from model,hence it seems likely that the missing physics lies close to the surface, and the deviation is often termed"surface effect" [Christensen-Dalsgaard and Berthomieu, 1991]. The deviation is predominantly a functionof the frequency of a mode, hence corrective terms should also be functions of the mode frequencyBhattacharya et al. [2015]. Modes with higher degree penetrate less deeply and therefore have a smallerinertia at given surface displacement. As a consequence of this their frequencies are more susceptible tochanges in the model [Christensen-Dalsgaard and Thompson, 1997].

Kjeldsen et al. [2008] devised an empirical correction for the near-surface offset in the form of a powerlaw in frequency, calibrated to radial oscillations in the Sun around the frequency of maximum oscillationpower νmax , and fit to the differences between observed and modelled frequencies.


0 0.2 0.4 0.6 0.8 1r/R

50

100

500

1000

5000

ν (

µH

z)

g-mode

p-modeS5

S20

S100

NBV

S1

S2

p-mode

PRESENT SUN

Figure 2.2: Propagation diagram for the Sun. The profiles of the buoyancy N 2BV acoustic

frequency S2l , this latter for different values of the angular degree l, are plotted. Horizontal

dotted lines delimit the frequency range of observed solar oscillations. From top to bottom,dashed lines span the cavities where p-modes with l = 20 and l = 2 are trapped, and theregion of propagation of a gravity mode. [From Lebreton and Montalbán [2009].]

νcorr − νmod = a

�νmod

νref

�b

, (2.1)

where νmod is the frequency of any given mode of a model, νcorr is the corrected frequency, a is thecorrection at νmod = νref , and b is the slope from fitting the solar frequency differences.

However, the fit over predicts the magnitude of the surface effect at lower frequencies.

Motivated by a lack of a leading method for modelling surface effects Ball and Gizon [2014], inspiredby Gough [1990], propose a method in which the surface effects are modelled by one or both of termsproportional to ν−1

Iand ν3

I, where I is the mode inertia 3, normalized by the total displacement at the

photosphere. This method is able to fit the deviations better than Kjeldsen’s power law, figure 2.3.

δν(ν) =a3I

�ν

νac

�3

, (2.2)

3mode inertia is a measure of the total interior mass that is affected by the oscillation


δν(ν) =a3I

�ν

νac

�3

+a−1

I

�ν

νac

�−1

, (2.3)

These corrections are satisfactory from a model-fitting viewpoint, however the surface term is notuniversal and depends on the physics used, and by eliminating the frequency differences altogether we loseinformation about their origin of this oscilations.

Other approach is to use frequency ratios [Roxburgh and Vorontsov, 2003]. Using the frequency ratiosinstead of the individual modes of oscillation, the surface term is suppressed [Roxburgh, 2005, Otí Floraneset al., 2005]. This way we can directly compare theoretical frequencies with the observed quantities.

1500 2000 2500 3000 3500 4000

ν �µ��

0

−5

−10

−15

−20

−25

−30

δν �µ

��

(a)

KBCD08 fit

INVBB

BSB

Model S

BP04

STD

1500 2000 2500 3000 3500 4000

ν �µ��

(b)

BG14��

��

ν��µ��

��

��

Figure 2.3: The surface term between the Sun and different published solar models (INVBB[Antia, 1996], BSB(GS98) [Bahcall et al., 2006], Model S [Christensen-Dalsgaard et al., 1996],BP04 [Bahcall et al., 2005], and STD [Basu et al., 2000]) fit with different models of thesurface term (KBCD08 [Kjeldsen et al., 2008], BG14-1 and BG14-2 from [Ball and Gizon,2014]) in the range ±5Δν. In each panel, the points are the data, and the lines are the fits.The different colors and symbols indicate different solar models. Figure 8 of [Schmitt andBasu, 2015]

2.4 PARAMETRIZATION OF CONVECTION

Here we present solar models constructed first with the simple parametrization described by Monteiroet al. [1996], hereafter MJM parametrization, and then with a new extended parametrization that followsthe work of Grossman and Narayan [1993].

As discussed in the chapter one, modelling one star consists of adjusting input parameters, in our caseα and X, that give us a model which the output is close to observational parameters: Lobs, Teffobs atpresent age t� = (4.566± 0.005)× 109 yr. The calibration is achieved to better than 106 in radius andluminosity.

The calibration routine was inspired by the method developed by Christensen-Dalsgaard [1982]. Itconsists of computing partial derivatives of L and Teff with respect to one of the free parameteres: X


and α, keeping Z fixed, by solving the following system:

Lobs = Lmodel +

�∂L

∂X

�(X −Xmodel) +

�∂L

∂α

�(α− αmodel) , (2.4)

Teffobs = Teffmodel+

�∂Teff

∂X

�(X −Xmodel) +

�∂Teff

∂α

�(α− αmodel) (2.5)

The subcript model refers to the values computed from ASTEC and the values of appearing in the lefthand side of the equations are the observed values, table 1.1. The partially derivatives are estimated bylinear regression, changing one parameter and leaving the other constant. Therefore, it is assumed thelinearity of the variation of L and Teff with metalicity and mixing-length parameter. ASTEC has the optionfor iterating on the surface hydrogen abundance (xxh) and the mixing length parameter (alfa) internally.This is flagged by iterlr and the desired values are input in rsfin and alsfin. This option automaticallyrepeats the evolution calculation until convergence flagged in the parameter epslr. By default, and for thecase of the sun’s MLT, iterlr = 1, ASTEC uses the following derivatives:

d logα

d logL= 1.169

d logX

d logL= −0.1536

d logα

d logR= −4.748

d logX

d logR= −0.0446 (2.6)

In the MLT framework, we obtain αMLT = 2.133. The initial composition of the calibrated solar modelis X0 = 0.715167 and Z0 = 0.018194. In the CM model, we obtain αCM = 1.133, the initial compositionof the calibrated CM solar model is nearly the same as that of the calibrated MLT solar model.

In Monteiro et al. [1996], the parametrization of convection has as parameters: the mixing length α, mthat affects the transition between efficient and inefficient regimes of convection and β that it is associatedwith the strength of superadiabatic gradient in inefficient and efficient regions of convection. In ASTEC,the values of m and β are used to calculate the values a1 and a2 to then calculate the convective flux,Fc given by

Fc = KT (∇−∇a)

Hp

a1Σm[(1 + a2Σ)

n − 1]p, (2.7)

n =1

2

1− 2m

2−m, p = 2−m, (2.8)

a1 =9

8

�β5−4m

22m2+m−1

�1− 2m

2−m

�2m2−5m+2�1/3, (2.9)

a2 =

�1

2m+1β2

�2−m

1− 2m

�2−m�2/3(2.10)

The MLT corresponds exactly to m = -1.0 and β = 1.0 and for the CM formulation correspondsapproximately to m = 0.14972 and β = 10.

In the new parametrization, the parameters are: m, αc, and 3 ratios:


αh =lhlv

, (2.11)

αw =lwlv

, (2.12)

αθ =lθlh

(2.13)

in which lw is the length scale for the turbulent loss of momentum, lθ is the length scale for turbulentloss of thermal energy diffusion and lh and lv are respectively the the horizontal and vertical length scalesof the turbulent eddie. Thus αθ ≡ lh

lvdescribes the geometrical asymmetry of the eddies.

a1 =9

8

�αw

3√3α3

θ

��5

2 + α2h

��2 + αh

3

�2

, (2.14)

a2 =

√3αθ

αw

�3 + 2α2

h

5

��3

2 + α2h

�(2.15)

The advantage of this parametrization is that the parameters have more physical meaning that a1 anda2. We proceed in two steps: first, we compute a grid of cases using the three parameters to give therespective values of a1 and a2. For example, for the case of MLT, one of the cases can be archive byimposing the following values (αh,αw,αθ) = (1.0, 1.0, 1.0/

√3). Next we explore the space of parameters

around the MLT case to see the effect of changing one parameter at time (see figure 2.4).

We can apply the same methodology for the CM case (the PTGd4 model of Monteiro et al. [1996]), inwhich (m, a1, a2) = (0.14972, 24.868, 0.097666), figure 2.5.

One of the problems of this parametrization is the huge range of values of a1 and a2 can have. Fromthe table 1 of Monteiro et al. [1996], we can see that a1 varies from 10−9 to 109. It is computationallyinfeasible to explore all the cases. We keep our focus in around three major cases: the MLT, the CM andPTGb6 model of Monteiro et al. [1996], see tables in appendix. For each case we fix Z = 0.017 and testthe parametrization for three different m: -1.0, -0.5 and 0.15. Additionally, for cases around the MLT wetested for Z = 0.015 and Z = 0.019 to check the effect of changing the metallicity, see figure 2.8 and2.9.

In figure 2.6 shows the temperature and the temperature gradient and at the top of convection zonefor some models around the MLT and CM using the new parametrization. By increasing the value of a1,or for low values of αθ and higher values of αω and αh, the convective efficiency is decreased in the SAL.

To investigate the impact of changing the α parameters in the structure of the top layers of the Sun, weshow in figure 2.7 the relative differences in Δc/c and Δv/v. Changing αw has no effect in the internalstructure, but αh and αθ give rise to some differences.

The difference δω between observed frequencies of oscillation of the Sun and frequencies ωnl computedfor solar model can be expressed as [Christensen-Dalsgaard and Thompson, 1995]

δω

ωnl

=

� R

0

[K(c,ρ)n,l

δc

c(r) +K

(ρ,c)n,l

δρ

ρ(r)]dr + I−1

nl G(ωnl) + �nl (2.16)


Figure 2.4: Values of a1 and a2 around the MLT case, (αw,αθ,αh) = (1.0, 1√3, 1.0). Each

colour represents a diferent value for αh: red for αh = 1, blue for αh = 0.5 and green forαh = 1.5

Here δc and δρ are adiabtic and sound speed differences between the Sun and the model at fixes distancer from the center. The kernels K(c,ρ) and K(ρ,c) are determined from the adiabatic eigenfuctions. Thefunction G(ω) accounts for the effect of the near-surface errors in the computation frequency models, I−1

nl

is the mode inertia and �nl is the errors associated from the measurement of the observed frequencies.As we can see, for higher Z values we get better frequencies differences for lower p-mode frequencies.

For some models around the CM, αθ = 0.30, αh = 2.00 and varying αw = between 7 to 8 we get higherp-modes frequencues close to observed frequencies (see models represented by continuous lines in figure2.9). Those are the same models that show an higher superadiabtic peak. However, this is not verifiedfor middle p-modes frequencies. This is an improvement face to current FST theories.


Figure 2.5: Values of a1 and a2 around the CM case, (a1, a2) = (24.868, 0.097666). Eachcolour represents a diferent value for αh: red for αh = 1, blue for αh = 0.5 and green forαh = 2.0


Figure 2.6: a Temperature at the top of convection zone for MLT (continuos line), (a1, a2) =(1.125, 1.0), models around MLT (dotted lines): blue (2.1747, 0.8083); red (4.6731, 0.5956);green (6.1507, 0.7621) and around CM (dot dashed lines): cian (19.0289, 0.0924); magenta(118.0944, 0.0706); green (9.7428, 0.1155) .b Plot of temperature gradient for the same mod-els as in a.


(a) αh : 0.50 for blue, 1.25 for red and 2.00 for green.(b) αθ: 0.40 for blue, 0.55 for red and 0.70 for green.

(c) αω: 0.80 for blue, 1.00 for red and 1.20 for green.

Figure 2.7: Relative differences in Δc/c and Δv/v of models around the MLT relative to theMLT at constant mass fraction changing one α at time.


Figure 2.8: Frequency differences around the MLT case for different values of m and z.

Figure 2.9: Frequency differences around the CM case for different values of m and z = 0.017

Chapter 3.

Solar-like pulsators

3.1 MODELLING SOLAR-LIKE PULSTATORS

Solar-like oscillations are oscillations in other stars that are excited in a similar way as those in the Sun.Stars have solar-like oscillations are also called solar-like stars. Solar-like stars including a wider range ofF and G dwarfs and subgiants.

The search for solar-like oscillations in stars other than the Sun has started some thirty years ago.Solar-like oscillations have within recent years been observed in a number of solar-like stars (see Beddingand Kjeldsen [2006], for a recent review).

The first hint of a hump of excess power with a frequency dependence similar to the one observed in thesolar case was obtained by Brown et al. [1991] from radial-velocity observations of Procyon (α CMi). Thefirst plausible detection of individual oscillation frequencies and a large frequency separation is attributedto Kjeldsen et al. [1995], who observed the G0IV star η Boo an later confirm as a solar-like star Kjeldsenet al. [2003].

Solar-like are stars with fundamental physical properties very similar or identical to the Sun Cayrel deStrobel [1996]. Some examples of solar-like are: α Cen A and B Kjeldsen et al. [2005], 18 Sco Bazotet al. [2011] and β Hyi Brandão et al. [2011]. A Hertzsprung-Russell diagram showing stars with solar-likeoscillations are present below fig 3.1.

Some recent reviews and articles on asteroseismology of solar-like oscillators include Chaplin and Miglio[2013] and Bedding [2014].

Binary stars are known to provide additional constraints (orbital parameters) to the modelling such thathave been widely used in the literature for testing new theoretical models.

Since the work of Flannery and Ayres [1978], the binary system α Cen has been one of the preferredasteroseismic target because its proximity, only 4.3 light-years away, due to the numerous and preciseseismic and non-seismic data that are available for both stars (see Noels et al. [1991]) and to the similarityof its components to the Sun.

A few tens of modes of global oscillations have been detected for α Cen A, B [Bouchy and Carrier,2002, Bedding et al., 2004, Kjeldsen et al., 2005].

Thévenin et al. [2002] and Thoul et al. [2003] have been compared the frequencies with theoreticalmodels.

The internal structures of α Cen A and B have been considered theoretically in many papers. Recentstudies by Miglio and Montalbán [2005], Eggenberger et al. [2004], Thoul et al. [2003], Morel et al. [2000]and Guenther and Demarque [2000] have analysed the structure and evolution of these stars in detail.

The large and small frequency separations of α Cen A and B have been derived observationally from their

39


Figure 3.1: Hertzsprung-Russell diagram showing stars with solar-like oscillations for whichΔν has been measured. Filled symbols indicate observations from ground-based spectroscopyand open symbols indicate space-based photometry (some stars were observed using bothmethods). Figure 1.3 of [Bedding, 2014] made by Dennis Stello

p-mode oscillations by Bouchy and Carrier [2002] and Carrier and Bourban [2003], using spectrographicmethods Yıldız [2007]. The masses [Pourbaix et al., 2002] and radii [Kervella et al., 2003] of α Cen arewell known from the cited observations, for the radii, the erros are smaller than 0.3 %.

In this chapter, we concentrate our interest on binaries with solar-like pulsating components, and weanalyse in detail the α Centauri A B. We applied the work done in the previous chapter to the solar-likestars α Centauri A B.

3.2 FUNDAMENTAL PROPERTIES OF α CEN A AND B

According on the internal structure studies, α Cen A and B (HD 128620/1), the age of the systemis between 4.85–7.6 Gyr. The lower limit is given by Thévenin et al. [2002] and the upper limit is fromGuenther and Demarque [2000]. However, Miglio and Montalbán [2005] find the age of the system to be8.88 Gyr from the models for non-seismic constraints, close to the end of its main-sequence (MS) lifetime.

According to [Pourbaix et al., 2002] the masses of α Cen A and B are MA = 1.105 ± 0.007M� andMB = 0.934 ± 0.007M�, respectively. Which for the case of α Cen A its mass is very close to the limitabove which main-sequence stars keep the convective core developed during the pre-main-sequence phase.


Table 3.1: Observational constraints for α Cen A and B. References: (1) [Söderhjelm, 1999],(2) [Pourbaix et al., 2002], (3) [Eggenberger et al., 2004], (4) [Kervella et al., 2003], (5)[Bouchy and Carrier, 2002] and (6) [Carrier and Bourban, 2003].

α Cen A α Cen B Referencesπ [mas] 747.1± 1.2 (1)M/M� 1.105± 0.0070 0.934± 0.0061 (2)V [mag] −0.003± 0.006 1.333± 0.014 (3)L/L� 1.522± 0.030 0.503± 0.020 (3)Teff [K] 5810± 50 5260± 50 (3)[Fe/H]s 0.22± 0.05 0.24± 0.05 (3)θ [mas] 8.511± 0.020 6.001± 0.034 (4)R/R� 1.224± 0.003 0.863± 0.005 (4)

Δν0 [µHz] 105.5± 0.1 161.1± 0.1 (5),(6)δν02 [µHz] 5.6± 0.7 8.7± 0.8 (5),(6)

Table 3.2: Oscillation frequencies for α Cen A (µHz). Table 1 from Bedding et al. [2004].

n l = 0 l = 1 l = 2 l = 3

14 . . . 1675.9 . . . . . .15 . . . 1779.7 1828.6 . . .16 1839.2 1885.9 1935.7 . . .17 1943.3 1993.8 2038.9 2082.918 2045.5 2094.6 2146.3 2193.119 2152.9 2203.2 2253.4 2296.320 2258.1 2309.1 2357.3 2404.821 2364.0 2412.4 2463.4 2507.522 2471.5 2522.1 2572.7 2616.823 2572.7 2627.1 2676.8 2723.524 2682.7 2733.2 2783.4 . . .25 . . . 2840.2 . . . . . .26 2895.9 2945.7 2998.3 . . .27 . . . 3055.1 . . . . . .

The radii of the components have been measured by Kervella et al. [2003] as RA = 1.224± 0.003R� andRB = 0.863± 0.005 . Morel et al. [2000] have determined the effective temperatures from the spectra ofα Cen A and B. According to the results of studies based on spectra, the effective temperature of α CenA is between 5830 and 5720 K,and for α Cen B the effective temperatures is between 5250 and 5325 K.For the metallicity of both components there is also no complete agreement in the literature Miglio andMontalbán [2005].

The most recent values of the observational data are present in table 3.1, the frequencies observed forα Cen A and B are present in table 3.2 and 3.3, respectively.

Fernandes and Neuforge [1995] studied the α Cen system to test stellar models based upon the MLTand CM formulation [Canuto and Mazzitelli, 1991, 1992]. Miglio and Montalbán [2005] constrained α

MLT of the α Centauri A B systems using a method from asteroseismology and found an α MLT forcomponent B higher by 10% than of A. Yıldız et al. [2006] by studying binaries in the Hyades found thatα MLT to be strongly dependent on stellar mass (mass-range of 0.77–1.36 M�).


Table 3.3: Oscillation frequencies for α Cen B (µHz). Table 1 from Kjeldsen et al. [2005]

n l = 0 l = 1 l = 2 l = 3

17 . . . 3059.7 . . . . . .18 . . . 3224.2 . . . . . .19 3306.6 3381.9 3456.6 3526.320 3466.9 3544.9 . . . 3685.621 3628.2 . . . 3778.8 3849.322 3789.2 3865.9 . . . 4008.523 3951.1 4025.9 4102.0 . . .24 4109.5 4188.0 4262.0 4333.325 4275.7 4351.0 4425.4 . . .26 . . . . . . 4585.6 . . .27 4598.4 4670.3 4750.8 . . .28 . . . 4835.4 4912.4 . . .29 . . . 4998.8 . . . . . .30 5085.7 5155.6 . . . . . .31 5248.8 . . . . . . . . .32 5411.4 5489.0 . . . . . .

3.3 CALIBRATION METHOD

The calibration of fundamental parameters is only possible for the Sun, because for other stars we havemore parameters to determine than available observations. In the case of α Centauri A B, we face fourunknowns for each star: age, Y , Z, α. To overcome these difficulties, we reduce the number of unknownvariables assuming that both stars share the same age and chemical composition because α Cen A B aremembers of a binary system and presumably formed simultaneously from the same material.

Some authors also suggest an universal α, that αA = αB = αsun, Thévenin et al. [2002]. The questionof about an unique value for the mixing-length parameter has been discussed by Noels et al. [1991];Andersen [1991]; Edmonds et al. [1992]; Lydon et al. [1993b]; Neuforge [1993], Fernandes and Neuforge[1995], Kim et al. [1996]; Morel et al. [2000] and Guenther and Demarque [2000]. It usually assumethat the solar α MLT applies for other stars. Some studies like Ferraro et al. [2006] suggest this isvalid hypothesis for low-mass red giant stars. A recent investigation of solar like Kepler stars by Bonacaet al. [2012], indicated that αMLT correlates more significantly with metallicity than Teff or log g. Howevercalibrations using 2-D simulations [Ludwig et al., 1999] suggest that α should be represented as a functionof log Teff , log g and chemical composition and this was confirmed by 3D simulations of Trampedach[2007], Trampedach et al. [2013].

In order to determine the age, we run several stellar evolution tracks from 0.0 Gyr until 8.8 Gyr, whichis the age of the galaxy thin disk del Peloso et al. [2005]. Then, we verify which model number (timestep) give us the parameters closest to the observed ones ( within the error margin). The best fits for αCen A, is obtained with the parameters: X = 0.73 and Z = 0.023 which give us the age 6.5 Gyr. For αCen B, almost all our combinations of X and z can fit the observations, however none of the models giveus an age close to 6.5 Gyr, but beween 3.7 and 5.7 Gyr. figure 3.2.


Figure 3.2: Stellar evolution tracks from 0.0 Gyr until 8.8 Gyr for different values of X and Z

3.4 NEW PARAMETRIZATION APPLIED TO α CEN AB

In order to full understand the behaviour of the new parametrization, we measure the large frequencyseparation for some cases for the binary α Cen ( see figure 3.3 and 3.4 for the α Cen A and B, respectively).In table 3.4 are present the cases we tested. Like Miglio and Montalbán [2005], we also found thatαB is bigger than αA by 10%. for all cases. The large separation is strongly dependant on radiusΔν ∝ (M/R3)1/2. It is very sensible by the change of the description of convection in the external layers,such as the superadiabatic layer. The higher value of Δν is obtained in MLT case for both stars. For αCen B the lower values of Δν are obtained for parametrization close to the CM formulation, however thisis not verified for the A component.

Table 3.4: α Cen A B Models

Star Model α αw αθ αh

α Cen A A01 2.13 1.0 0.6 1.0α Cen A A02 1.22 1.0 0.4 2.0α Cen A A03 1.63 0.8 0.7 2.0α Cen A A04 0.73 8.1 0.3 2.0α Cen A A05 1.50 7.0 0.5 0.5

α Cen B B01 2.22 1.0 0.6 1.0α Cen B B02 1.29 1.0 0.4 2.0α Cen B B03 1.70 0.8 0.7 2.0α Cen B B04 0.84 8.1 0.3 2.0α Cen B B05 1.62 7.0 0.5 0.5


Figure 3.3: Large and small separations for the A component of α Cen system. Lines corre-spond to different models from 3.4. For clarity only l = 1 large separations are shown.

Figure 3.4: Same as figure 3.3 for the B component of α Cen system

Chapter 4.

Conclusions and future prospects

4.1 SUMMARY

Observed solar oscillation frequencies can differ from modeled frequencies because of an inaccuratebackground model, as well as incomplete description of mode physics. The presence of near-surfaceconvective flows on the Sun results in mode frequencies which are different from ones computed startingfrom a quiet background. The near-surface contribution is usually removed using ad hoc models of whatthe difference should look-like.

We proceeded in three steps. First, we calculated calibrated solar models for the two prescriptions ofconvection: the MLT and the CM version of the full spectrum of turbulence. We then calculated calibratedsolar models for other values of the parametrization for different metallicities. We compared the frequenciesmodels with the observational frequencies from GOLF data. Finally, we calculated calibrated solar-likemodels for the same set of values of parametrization as the Sun.

After doing this work, we have shown that we do not found a region of values of the MJM parametrizationthat can cancel the near-surface effects in the p-modes frequencies. In some cases we get calculatedfrequencies better than present in the literature.

More work have to be done, in order to find a way to get proper p-mode frequencies. From this work,we can see that we have to find a more general parametrization that can cancel the middle bump in ourresults. After that, we can apply bayesian analysis, such as Markov chain Monte Carlo, that are alreadyimplemented in ASTEC, in order to further explore the nature of this parameters.

45


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Appendices

59


Table .1: Values around the MLT case

αw αθ αh a1 a2

0.80 0.40 0.50 2.1747 0.80830.80 0.40 1.25 3.1154 0.89340.80 0.40 2.00 4.9206 0.95260.80 0.55 0.50 0.8366 1.11140.80 0.55 1.25 1.1984 1.22840.80 0.55 2.00 1.8928 1.30990.80 0.70 0.50 0.4058 1.41450.80 0.70 1.25 0.5813 1.56340.80 0.70 2.00 0.9181 1.66711.00 0.40 0.50 2.7184 0.64661.00 0.40 1.25 3.8942 0.71471.00 0.40 2.00 6.1507 0.76211.00 0.55 0.50 1.0457 0.88911.00 0.55 1.25 1.4980 0.98271.00 0.55 2.00 2.3660 1.04791.00 0.70 0.50 0.5072 1.13161.00 0.70 1.25 0.7266 1.25071.00 0.70 2.00 1.1477 1.33371.20 0.40 0.50 3.2621 0.53891.20 0.40 1.25 4.6731 0.59561.20 0.40 2.00 7.3809 0.63511.20 0.55 0.50 1.2548 0.74091.20 0.55 1.25 1.7976 0.81891.20 0.55 2.00 2.8392 0.87321.20 0.70 0.50 0.6087 0.94301.20 0.70 1.25 0.8719 1.04231.20 0.70 2.00 1.3772 1.1114


Table .2: Values around the CM case

αw αθ αh a1 a2

7.00 0.30 0.50 45.1055 0.06937.00 0.30 1.00 56.1313 0.07427.00 0.30 2.00 102.0569 0.08177.00 0.40 0.50 19.0289 0.09247.00 0.40 1.00 23.6804 0.09907.00 0.40 2.00 43.0552 0.10897.00 0.50 0.50 9.7428 0.11557.00 0.50 1.00 12.1244 0.12377.00 0.50 2.00 22.0443 0.13617.55 0.30 0.50 48.6495 0.06427.55 0.30 1.00 60.5416 0.06887.55 0.30 2.00 110.0756 0.07577.55 0.40 0.50 20.5240 0.08567.55 0.40 1.00 25.5410 0.09187.55 0.40 2.00 46.4382 0.10097.55 0.50 0.50 10.5083 0.10717.55 0.50 1.00 13.0770 0.11477.55 0.50 2.00 23.7763 0.12628.10 0.30 0.50 52.1935 0.05998.10 0.30 1.00 64.9519 0.06428.10 0.30 2.00 118.0944 0.07068.10 0.40 0.50 22.0191 0.07988.10 0.40 1.00 27.4016 0.08558.10 0.40 2.00 49.8211 0.09418.10 0.50 0.50 11.2738 0.09988.10 0.50 1.00 14.0296 0.10698.10 0.50 2.00 25.5084 0.1176


Table .3: Values around the PTGb6 case

αw αθ αh a1 a2

20.00 0.0010 0.50 3479566354.5 0.00008120.00 0.0010 1.00 4330127018.9 0.00008720.00 0.0010 2.00 7872958216.2 0.00009520.00 0.0020 0.50 434945794.3 0.00016220.00 0.0020 1.00 541265877.4 0.00017320.00 0.0020 2.00 984119777.0 0.00019120.00 0.0030 0.50 128872827.9 0.00024220.00 0.0030 1.00 160375074.8 0.00026020.00 0.0030 2.00 291591045.0 0.00028630.00 0.0010 0.50 5219349531.7 0.00005430.00 0.0010 1.00 6495190528.4 0.00005830.00 0.0010 2.00 11809437324.3 0.00006430.00 0.0020 0.50 652418691.5 0.00010830.00 0.0020 1.00 811898816.0 0.00011530.00 0.0020 2.00 1476179665.5 0.00012730.00 0.0030 0.50 193309241.9 0.00016230.00 0.0030 1.00 240562612.2 0.00017330.00 0.0030 2.00 437386567.6 0.00019140.00 0.0010 0.50 6959132709.0 0.00004040.00 0.0010 1.00 8660254037.8 0.00004340.00 0.0010 2.00 15745916432.4 0.00004840.00 0.0020 0.50 869891588.6 0.00008140.00 0.0020 1.00 1082531754.7 0.00008740.00 0.0020 2.00 1968239554.1 0.00009540.00 0.0030 0.50 257745655.9 0.00012140.00 0.0030 1.00 320750149.5 0.00013040.00 0.0030 2.00 583182090.1 0.000143

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