Andrea Miglio - uliege.beastrotheor3.astro.ulg.ac.be/miglio/tesi.pdf · Seismic analysis of solar-type stars Andrea Miglio Universit´a degli Studi di Milano Facolt´a di Scienze

Seismic analysis of

solar-type stars

Andrea Miglio

Universita degli Studi di Milano

Facolta di Scienze Matematiche Fisiche e Naturali

Corso di Laurea in Fisica

November 2002

2

Universita degli Studi di Milano

Facolta di Scienze Matematiche Fisiche e Naturali

Corso di Laurea in Fisica

Analisi sismica di stelle di tipo solare

PACS 97.30

Relatore Interno: Prof. Laura Pasinetti

Relatore Esterno: Prof. Jørgen Christensen-Dalsgaard

Correlatore: Dott. Elio Antonello

Candidato: Andrea Miglio

Matricola: 552397

anno accademico 2001/2002

2

Acknowledgments

Thanks to:

• my supervisor in Aarhus Jørgen Christensen-Dalsgaard, for giving me very useful

advice and for encouraging me during all the time I spent in Denmark,

• my supervisors in Milan: prof. Laura Pasinetti and dott. Elio Antonello for their

support and the opportunity they gave me to work on my thesis abroad,

• Frank, Teresa, Joris, Torben, Brandon, Niels, Anette and all the people I know at

IFA for making me feel at ease and introducing me to different cultures,

• Maria Pia Di Mauro for helping me every time she was in Aarhus and Mario

Monteiro for his kindness and help while in Porto,

• all my family for its limitless support and encouragement, especially my sister

Anna,

• Sara for tolerating my “grizzly” habits and for her affection.

2

Contents

1 Introduction 7

2 Introduction to observational asteroseismology 9

2.1 The detection of oscillation modes . . . . . . . . . . . . . . . . . . . . . . 9

2.1.1 Observable quantities . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.1.2 Time series analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.1.3 Mode identification . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.2 Classes of oscillators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.2.1 Solar type stars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.2.2 Intrinsically excited pulsators . . . . . . . . . . . . . . . . . . . . . 14

3 Stellar modelling foundation 15

3.1 Basic equations of stellar structure . . . . . . . . . . . . . . . . . . . . . . 15

3.1.1 Hydrostatic equilibrium . . . . . . . . . . . . . . . . . . . . . . . . 15

3.1.2 Energy conservation equation . . . . . . . . . . . . . . . . . . . . . 16

3.1.3 Energy transport equation . . . . . . . . . . . . . . . . . . . . . . . 16

3.1.4 The chemical composition . . . . . . . . . . . . . . . . . . . . . . . 18

3.1.5 The overall problem . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.2 Physics involved . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.2.1 Equation of state . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.2.2 Treatment of convection . . . . . . . . . . . . . . . . . . . . . . . . 22

3.2.3 Opacity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.2.4 Nuclear energy generation . . . . . . . . . . . . . . . . . . . . . . . 24

3.3 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.3.1 Set of models considered . . . . . . . . . . . . . . . . . . . . . . . . 25

3.3.2 Some thermodynamic variables . . . . . . . . . . . . . . . . . . . . 30

4 Modelling linear adiabatic oscillations 35

4.1 Hydrodynamics equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

4.1.1 Basic equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3

CONTENTS

4.1.2 Perturbation analysis . . . . . . . . . . . . . . . . . . . . . . . . . 36

4.1.3 Pressure waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

4.1.4 Surface gravity waves . . . . . . . . . . . . . . . . . . . . . . . . . 38

4.1.5 Internal gravity waves . . . . . . . . . . . . . . . . . . . . . . . . . 38

4.2 Equations of stellar linear oscillations . . . . . . . . . . . . . . . . . . . . 39

4.2.1 Equations of stellar linear adiabatic oscillations . . . . . . . . . . . 40

4.2.2 Separation of variables . . . . . . . . . . . . . . . . . . . . . . . . . 40

4.2.3 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . 42

4.3 Functional analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

4.3.1 Effects on frequencies of a small change in the oscillations equations. 44

4.4 Properties of oscillation modes . . . . . . . . . . . . . . . . . . . . . . . . 44

4.4.1 Physical nature of the modes . . . . . . . . . . . . . . . . . . . . . 45

4.4.2 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

4.5 Rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

4.5.1 Effects of slow rotation . . . . . . . . . . . . . . . . . . . . . . . . 50

4.5.2 Effects of moderate rotation on the oscillations equations . . . . . 51

4.6 Excitation and damping of the oscillations . . . . . . . . . . . . . . . . . . 51

4.6.1 The quasi adiabatic approximation . . . . . . . . . . . . . . . . . . 52

4.6.2 Self excitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4.6.3 Stochastic excitation . . . . . . . . . . . . . . . . . . . . . . . . . . 53

5 Asymptotic solutions of stellar oscillations equations 55

5.1 Asymptotic theory of stellar oscillations . . . . . . . . . . . . . . . . . . . 55

5.1.1 The JWKB approximation . . . . . . . . . . . . . . . . . . . . . . 57

5.2 Asymptotic theory for p modes . . . . . . . . . . . . . . . . . . . . . . . . 58

5.2.1 The Duvall law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

5.2.2 Tassoul’s expression for δn,l. . . . . . . . . . . . . . . . . . . . . . . 59

5.2.3 Gough’s expression for δn,l . . . . . . . . . . . . . . . . . . . . . . 60

5.2.4 Deviation from asymptotic expressions . . . . . . . . . . . . . . . . 61

6 Results on computed models 65

6.1 Asteroseismic HR diagrams. . . . . . . . . . . . . . . . . . . . . . . . . . . 65

6.1.1 Large frequency separation. . . . . . . . . . . . . . . . . . . . . . . 66

6.1.2 Small frequency separation . . . . . . . . . . . . . . . . . . . . . . 68

6.1.3 Diagrams for main sequence models without convective cores (M <

1.2 M⊙) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

6.1.4 Diagrams including models with convective cores (M ≥ 1.2 M⊙) . . 74

6.1.5 Solar data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

6.2 Seismic analysis of helium ionization zones. . . . . . . . . . . . . . . . . . 80

4

CONTENTS

6.2.1 Expected signal on first order differences. . . . . . . . . . . . . . . 86

6.2.2 Using numerical kernels. . . . . . . . . . . . . . . . . . . . . . . . . 87

6.2.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

6.3 Onset of degeneracy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

6.3.1 Sound speed in a partly degenerate stellar core. . . . . . . . . . . . 96

6.3.2 Expected consequences on “small separations” and possible aster-

oseismic inference on the presence of degeneracy in the stellar core. 100

7 Conclusions and prospects 107

8 Riassunto in italiano 109

8.1 Introduzione . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

8.2 Equazioni di struttura ed evoluzione stellare . . . . . . . . . . . . . . . . . 110

8.3 Teoria delle oscillazioni stellari non radiali . . . . . . . . . . . . . . . . . . 111

8.3.1 Le equazioni di oscillazione . . . . . . . . . . . . . . . . . . . . . . 111

8.3.2 Approssimazione asintotica . . . . . . . . . . . . . . . . . . . . . . 112

8.4 Risultati . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

8.4.1 Diagrammi C-D . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

8.4.2 Analisi sismica delle regioni di ionizzazione dell’elio . . . . . . . . . 117

8.5 Conclusioni e prospettive . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

A Fitting procedure. 127

B Seismic analysis of the helium ionization zones in low- and moderate-

mass stars. 131

5

CONTENTS

6

Chapter 1

Introduction

Stars represent the fundamental units of the observed universe, therefore understanding

stellar structure and evolution has a privileged role in astrophysics. Classical observa-

tions of stars provide information about the thermodynamic state of the stellar atmo-

sphere, nonetheless, excluding the flux of neutrinos generated in the solar core, we do

not have information about the interior of the star.

Several classes of stars show photometric and spectroscopic variations in time ex-

plainable as non radial oscillations of the equilibrium structure (acoustic and gravity

waves).

The goal of studying stellar oscillation modes (asteroseismology) is not only to observe

stellar variability, but also to disclose the information carried by the frequencies of os-

cillation. Each mode of oscillation sounds differently the inner regions of the star and

represents then a unique tool to probe stellar interiors.

The seismic properties of solar-type stars, excited in several non-radial modes, will be

accurately determined by the forthcoming space observations. The aim of the thesis is

to study and extend inversion techniques that could be used to relate the characteristics

of oscillation spectra with either global and localized properties of solar-type stars.

This thesis is structured as follows:

• Chapter 2 is meant to provide a brief general introduction to observational aster-

oseismology; it also includes a reference list of the latest observations of non radial

oscillations in solar-type stars.

• Chapter 3 explains the basic equations of stellar structure and evolution, as well

as presents the set of models considered and some examples of the output variables

of numeric calculations.

• Chapter 4 is a description of the theory of stellar oscillations.

7

CHAPTER 1. INTRODUCTION

• Chapter 5 is a presentation of the asymptotic theory of stellar oscillations. The

expressions of the large and small frequency separation derived under this approx-

imation have been widely applied in this thesis.

• Chapter 6 concerns the results achieved in this thesis. The C-D diagrams are

calculated and their properties described, as well as a seismic analysis of the helium

ionization regions is presented.

• Chapter 7 provides a brief summary and outlines future prospects and develop-

ments of this work.

• Chapter 8 is a summary written in Italian.

• Appendix A describes the fitting procedure applied in Chapter 6.

• Appendix B is paper following a poster presented in Asteroseismology across the

HR diagram, Porto, 1-5 July, 2002.

8

Chapter 2

Introduction to observational

asteroseismology

2.1 The detection of oscillation modes

From stars others than our sun, we observe light integrated over the whole stellar disk:

this poses an intrinsic limit to the detection of high degree modes, since regions of

positive and negative fluctuations approximatively cancel. This argument could be made

quantitative evaluating the root mean squared intensity/velocity perturbation over the

surface. Every scalar perturbation, as presented in Chapter 4, can be written in the

form:

I(θ, φ ; t) =√

4πℜ [I0 Y ml (θ, φ) exp[−i(ω0t − δ0)]]

where ℜ(z) is the real part of the complex quantity z.

Intensity observations, for instance, are described, neglecting limb darkening1, by:

I(t) = I0(t)S(I)l cos (ω0t − δ0)

where S(I)l is the spatial response function. The rapid decrease of S

(I)l with increasing

degree l is shown in Figure 2.1, this behaviour explains why it is unlikely to detect high

degree modes from observations of light integrated over a stellar disk. A spatial response

function for velocity measurements could be defined as well, this time the perturbations

of the vectorial quantity v have to be projected onto the line of sight when averaging

them on the surface.

1The choice of the polar axis is free in a spherically symmetric star, it is computationally convenient

to have the axis pointing towards the observer.

9

CHAPTER 2. INTRODUCTION TO OBSERVATIONAL ASTEROSEISMOLOGY

0 1 2 3 4 5 6 7 8 9 10l

0.2

0.4

0.6

0.8

1

S Intensity

0 1 2 3 4 5 6 7 8 9 10l

0.2

0.4

0.6

0.8

S Velocity

Figure 2.1: Spatial response functions for intensity (left panel) and velocity (right panel) ob-

servations of whole-disk light.

2.1.1 Observable quantities

There are essentially two methods of detecting stellar oscillations: photometry and

Doppler spectroscopy.

Luminosity fluctuations are mainly due to temperature variations on the surface of

the star, combined observations of luminosity (sensitive to both temperature and surface

changes) and color (only sensitive to temperature variations) could help identifying the

observed mode ([2]).

Oscillations generate periodic velocity fields on the surface of the star that will influ-

ence the profile of spectral lines. Since photometric observations from earth are severely

limited by atmospheric scintillation noise, Doppler ground-based measurement repre-

sent the primary tool for detecting small amplitude pulsations (e.g. solar-type stars

oscillation, as described in Section 2.2.1).

2.1.2 Time series analysis

Extracting a frequency spectrum from time strings, generated from photometric or spec-

troscopic measurements, is usually done with refined methods based on the Fourier anal-

ysis; the major uncertainties come from the finite total time of observation, the presence

of data gaps and by instrumental and intrinsic noise. I will here briefly and qualitatively

recall them.

• The finite extent of observations is responsible for the profile of the peak in the

power spectrum of the time series and also limits the frequency separation that

can be resolved.

• The presence of gaps in the data sets introduces aliases and side-bands in the power

10

2.2. CLASSES OF OSCILLATORS

spectra of time strings.

• The amplitudes of oscillation are not time-independent: damping rates, obtained

by including non-adiabatic effects in the equations describing stellar oscillations,

cause a broadening in the power spectrum peaks (see Section 4.6).

• Several sources of noise affect observations: instrumental and photon noise, stellar

granulation, atmospheric scintillation and sources of stellar variability others than

oscillations.

The forthcoming observations from space will significantly lower most of the noise sources

listed above, and will hopefully reveal many more excited modes then the ones detected

from Earth.

2.1.3 Mode identification

A pulsation mode is described by three integers: the radial order n, the angular degree

l and the azimuthal order m. How can we associate each frequency extracted from the

time series analysis with the correct (n, l,m) set? There are basically two classes of mode

identification techniques: photometric and spectroscopic method.

The photometric method is based on the comparison of light curves observed in

different frequency bands, while the spectroscopic method associates the characteristics

of a line profile with the degrees l and m of the mode. This could be accomplished

by fitting only a few parameters of a line profile (moment method) or by generating

synthetic spectra and directly fitting the observed line profile (line profile modelling)

(see [33]). Both photometric and spectroscopic approaches help identifying only l and

m: the radial order n has to be inferred comparing observed and synthetic frequency

spectra.

2.2 Classes of oscillators

In Figure 2.2 are shown the known classes of pulsating stars on a HR diagram.

The different classes of variable stars differ in mass range, evolutionary stage and

mainly in the characteristics and driving mechanisms of the observed oscillations. A

review of opacity driven oscillators is given in [33]. Solar-type pulsators are described in

the following paragraph.

2.2.1 Solar type stars

A clear definition of solar-type stars is given in [41]: this class comprises “. . . F, G and

possibly K stars on the main sequence, or subgiants, which are expected to display tur-

11


Figure 2.2: A HR diagram showing the positions of many classes of pulsating stars.

12

2.2. CLASSES OF OSCILLATORS

Figure 2.3: Power spectrum of radial velocity measurements in α Cen A. (from [19])

bulently excited modes: a defining feature is the expected low amplitudes of pulsation”.

This class of pulsators is the one the techniques studied in this thesis will apply to, their

modes of oscillation are mainly acoustic waves (p modes), excited by turbulent convec-

tive motions. The amplitude of a single mode is of the order of 10 cm s−1 in velocity

and 10 − 100 × 10−6 in relative fluctuations in intensity.

Multiple acoustic oscillations have been detected in several solar-type stars:

• The star with the richest solar-type spectrum, identified so far, is certainly α

Centauri A. The recent observation of p modes, reported in [19], led to the

identification of 28 modes of oscillation, in a frequency range between 1.8 and 2.9

mHz and with amplitudes in the range 12 to 44 cm s−1. The power spectrum of

the radial velocity measurements is shown in Figure 2.3.

• η Bootis is a G0 subgiant, confirmation of the detection of p modes is given in

[24] where an estimate of the large (40.06± 0.02 µHz) and small (3.85± 0.28 µHz)

frequency splitting is given.

• Solar-like oscillations have also been detected in Procyon, a F5IV star, both

[32] and [18] observational campaigns suggest a large splitting of approximatively

55 µHz

13


• The evidence of a solar-like oscillation spectrum in the G2 subgiant β Hydri has

been recently confirmed in [38]; in [20] the large frequency separation is estimated

in 58 µHz.

Solar-like oscillations have also been recently observed in giants:

• The giant G7 star ξ Hydrae clearly shows solar type oscillations in the frequency

range 50-130 µHz. ([36])

• The variability in the red giant α Bootis could also be explained by acoustic

waves, as suggested in [1].

2.2.2 Intrinsically excited pulsators

Cepheids and RR Lyrae are well known for their period-luminosity relation and large

amplitudes of oscillation. Along the white dwarfs cooling sequence other classes of

oscillators are found. Multiperiodic oscillations are also found in:

β Cephei These are main sequence B0-B2.5 stars, their masses are in the range between

7 and 20 M⊙. They show oscillations that are interpreted as low-order p modes,

excited by the κ-mechanism related to the metal opacity bump.

SPB stars Slowly pulsating B stars are main sequence g mode pulsators, their masses

range approximatively from 3 to 8 M⊙ and spectral types from B3 to B8. The

observed modes are self-excited by the same mechanism as β Cephei stars.

δ Scuti The κ-mechanism related to the second helium ionization zone is responsible

for the excitation of low-order p modes in δ Scuti stars. This class includes main

sequence or slightly post-main sequence stars of 1.5-2.5 M⊙.

γ Doradus γ Doradus stars are located near the main sequence near the red edge of the

classical instability strip, their oscillations are identified with high order g modes,

the excitation mechanism is still uncertain.

roAp stars In rapidly oscillating Ap stars magnetic fields and rapid rotation affect the

properties of the observed p-mode oscillation spectra.

14

Chapter 3

Stellar modelling foundation

In this chapter I will briefly review the foundation of stellar modelling: structure

and evolution equations, boundary conditions and the micro-physics involved. I will

also include a reference list of the set of models computed, and some examples of the

output variables of numeric calculations, with particular emphasis on those aspects

that will influence the properties of the modes of oscillation.

3.1 Basic equations of stellar structure

3.1.1 Hydrostatic equilibrium

The first equation of stellar structure (see [26] or [34] ) comes from star’s assumed static1

mechanical equilibrium, i.e. the balance between gravitational and pressure forces in a

self-gravitating gaseous sphere. Combining Euler’s equation for hydrostatics:

∇p = ρg + f (3.1)

where f are volume forces others than the gravitational force, with Poisson’s Equation

∇2φ = −4πGρ (3.2)

and assuming that the gas is subject only to the self-generated gravitational field (strong

magnetic fields and apparent forces due to rotation are neglected), and that its structure

depends only on the distance r to the center we obtain:

dP

dr= −Gm(r)ρ

r2(3.3)

1Variability on dynamical time-scales in most of the stars may be treated as perturbations on a static

equilibrium structure, as explained in the following chapters.

15

CHAPTER 3. STELLAR MODELLING FOUNDATION

where m(r) is the mass contained in a sphere with radius r, i.e.

m(r) =

∫ r

04πr′2ρ dr′. (3.4)

In the case of strong gravitational fields, as in neutron stars for example, the hydrostatic

equation has to be generalized and obtained from general relativity fluid equations.

3.1.2 Energy conservation equation

If we define L(r) as the net energy per second passing outward through a sphere of

radius r, which comprises the energies transported by convection, radiation and thermal

conduction, we can consider its variation dL between two adjacent layers as provided

by nuclear reactions, cooling or contraction/expansion of the mass shell2. If only ǫ, the

nuclear energy generation rate per unit mass is considered, we obtain:

dL

dr= 4πr2ρǫ (3.5)

Considering changes in the internal energy of the gas, and the work exchanged by com-

pression and expansion we can generalize Eq. (3.15) :

dL

dr= 4πr2

[

ρǫ − ρd

dt

(

u

ρ

)

+P

ρ

dρ

dt

]

(3.6)

The additional terms in Eq. (3.6), while fundamental in certain stages of evolution, are

substantially negligible during the normal nuclear burning phases of the evolution of the

star (see [10]).

3.1.3 Energy transport equation

The energy continuously radiated through the surface of the star is slowly transported

from the inner regions, where nuclear reactions take place, to the external layers thanks

to a non-zero temperature gradient whose local properties are strictly related to the way

energy is transported. Radiation, conduction and convection might play different roles

in various parts of the star.

Radiative transport The radiative flux FR is defined by

dE = FR dAdt (3.7)

where dE represents the energy flowing through a surface element dA in a time inter-

val dt. While deriving the equation of radiative transport in stellar interiors, we could

2Energy decrease by neutrino losses might be considered as an additional term, ǫν , in the energy

equation.

16

3.1. BASIC EQUATIONS OF STELLAR STRUCTURE

assume that the mean free path of a photon is much shorter than the characteristic

dimension of the star, typically its radius R; we could then apply the diffusion approxi-

mation and obtain:

FR = −4ac

3

T 3

κρ

dT

dr(3.8)

where a = 7.57 × 10−15erg cm−3K−4 is the radiation-density constant, κ the opacity3,

related to the mean free path by lph = (κρ)−1. If only energy transport by radiation

is considered, by using the relation between energy flux and local luminosity we obtain

one of the fundamental equations of stellar structure:

dT

dr= − 3κρL(r)

16π a c r2 T 3(3.9)

This simple expression, valid in stellar interiors, breaks down as we approach the surface

of the star, i.e. where the mean free path of a photon increases, becomes comparable

and even larger than the distance to the surface.

Energy transport by convection A sphere of fluid fullfilling the hydrostatic equi-

librium condition (3.3) may not be “realistic” if its equilibrium is unstable. The growth

or damping of an adiabatic disturbance in the fluid at equilibrium distinguishes between

stability and instability of the configuration. The physical situation is that of a fluid

stratified in density and subject to a gravitational field, if the buoyancy force acting on

an element of fluid displaced from its equilibrium position tends to enhance the displace-

ment macroscopic mass currents will appear.

The condition for convective stability, known as the Ledoux condition is:

(

dρ

dr

)

ad<

dρ

dr(3.10)

where Γ1 = (d ln Pd ln ρ )ad. If we can assume a homogeneous chemical composition the condi-

tion (3.10) can be expressed in terms of change of temperature gradients (Schwarzshild

criterion), as

d ln T

d ln ρ= ∇ < ∇ad =

Γ2 − 1

Γ2(3.11)

where Γ2 is the second thermodynamic coefficient, defined so that Γ2

Γ2−1 = (∂ ln P∂ ln T )ad. In

stellar cores, since convection is very efficient, we can assume ∇ = ∇ad, nonetheless in

general we need an expression for ∇ = ∇conv which is provided by the theory we use to

model convection (see Section 3.2.2).

3The total cross section photon-matter per unit mass averaged on the frequency of the photon.

17


Thermal conduction In thermal conduction the energy transfer, from points where

the temperature is higher to those where it is lower, is due to random thermal motion

of particles (nuclei and electrons in a completely ionized matter).

The contribution to total energy transfer due to conduction is generally neglectable

in stellar interiors, nonetheless it becomes dominant in highly degenerate matter (the

core of an evolved star) where the mean free path and velocity of the degenerate electrons

increase as a result of applying Fermi-Dirac statistics to the stellar gas.

3.1.4 The chemical composition

The variation in time of the chemical composition in stars is obviously important as it

influences directly absorption of radiation and the rate of energy generation by nuclear

reactions. The abundance of elements in a star can be changed by nuclear reactions,

convective motions and diffusion.

If we consider a radiative region, and diffusion is negligible, there is no other way of

changing Xi but through nuclear reactions, therefore we can write:

dXi

dt= rXi , i = 1..I (3.12)

where rXi is the rate of change in time of the Xi abundance, related to the cross section

of the nuclear reactions generating the element Xi.

In the regions where convection takes place, turbulent convective motions, which

occur in a time-scale much shorter than nuclear reaction time-scale, are responsible for

the mixing of the stellar matter; we can therefore assume that the composition in a

convective region is always homogeneous.

The presence of convection in the core, where a gradient in chemical composition is

already created by nuclear reactions, makes this change steeper as result of the mixing,

see Figure 3.4. The characteristics of this change depend on the way convection is

treated and on the stage of nuclear reactions. This gradient in chemical composition,

mean molecular weight µ and so in density, may provide a way to infer the existence and

properties of convective regions in the star, by looking at the consequences it has on the

frequencies of oscillation.

3.1.5 The overall problem

We might rewrite the equations of stellar structure and evolution choosing as independent

variable the mass instead of the position. This is somehow similar to passing from an

Eulerian to a Lagrangian point of view in describing fluid’s equations. Summarizing

the expressions introduced in the previous sections, we obtain the following set of five

18

3.1. BASIC EQUATIONS OF STELLAR STRUCTURE

Figure 3.1: Hertzsprung-Russel diagram showing the evolution tracks of the models considered,

each track is labeled with the mass of the star in units of solar mass.

19


equations:

dP

dr=

1

4πρr2(3.13)

dP

dm= − Gm

4πr4(3.14)

dL

dm= ǫ − [

d

dt

u

ρ− P

ρ2

dρ

dt] (3.15)

dT

dm=

− 3κ4 a cT 3

L16π2r4 radiative energy transport

orΓ2−1Γ2

TP

dPdm convective energy transport

(3.16)

dXi

dt= rXi , i = 1..I (3.17)

where Γ2 is the second adiabatic exponent defined as:

Γ2

Γ2 − 1≡(

∂P

∂T

)

ad

Besides the boundary conditions discussed later, to find a solution of these five coupled

differential equations (3.13)-(3.17) in terms of the five dependent variables r ,P, L, T,

X we must have explicit expressions for ρ, T, u, Γ2, ǫ, κ in terms of the first set

of variables. These are provided by thermodynamics, nuclear and atomic physics: this

fundamental contribution represents the connection between stellar physics and what we

might call “micro-physics”. I will describe these relations in section 3.2.

This set of equations, in order to be solved, has to be supported by a set of boundary

conditions, I will very shortly recall them (see [34] for an exhaustive treatment).

Central conditions In the center of the star, defined by m = 0, we require:

r = 0 and L = 0 (3.18)

since approaching the center ρ has to be finite and non-zero, and the energy sources

finite.

Surface conditions Surface conditions are rather more complicated; taking the pho-

tosphere as the surface to put constraints on, and considering an approximated treatment

of stellar atmosphere (see [10] for details), we demand for m = M :

L = 4πσR2T 4eff and P =

(a + 1)GM

κR2(3.19)

where Teff is the effective temperature, σ the Stefan-Boltzmann constant of radiation,

κ the opacity evaluated at the photosphere and a a parameter describing the density

dependence of the opacity in the atmosphere.

The correct surface conditions can be generally obtained by requiring the solution in

the atmosphere to fit smoothly to the one in the interior of the star.

20

3.2. PHYSICS INVOLVED

3.2 Physics involved

3.2.1 Equation of state

The equation of state gives us an explicit relation P = P (ρ, T, c.c.), where c.c. is the

chemical composition.

The basic assumption we make while describing the properties of stellar matter, is

local thermodynamic equilibrium. This hypothesis is justified by the shortness of the

mean free path of particles, compared to the typical distance of changes in the structure

of the star, and by the short time interval between collisions or interactions compared

to the time-scale of variations in the star.

Thanks to the high temperatures in stellar interior, most of the matter could be

considered as fully ionized, and a first choice would be to consider the equation of state

for an ideal gas of a mixture of elements, i.e.

P =ρkbT

µmu(3.20)

where kb is Boltzmann’s constant, mu the atomic mass unit and µ is the mean molecular

weight defined as:

µ−1 =∑

j

XjZj + 1

Aj(3.21)

where for each element j composing the gas, Zj, Aj and Xj are respectively the atomic

number, atomic mass and mass fraction. In reality this equation of state is applicable in

a limited region of the ρ−T plane, as shown in Figure 3.2, generally other contributions

have to be considered:

• Radiation pressure: an additional contribution to gas pressure and internal energy

is provided by radiation pressure, Pr = 13aT 4, which becomes significant at “high”

temperatures and “low” densities (see Figure 3.2)

• Partial ionization: in the uppermost regions of a star helium and hydrogen ion-

ization may not be complete. The partially ionized gas has to be described by

statistics that takes into account the different ionization states of the particles.

Where interaction between particles is negligible, Saha equation is often used.

• Degeneracy : At low temperature and, in the case of stellar interiors, at high den-

sities, quantum mechanical effects have to be considered in the equation of state.

• Coulomb corrections: In the core at high densities or in the envelope at low temper-

atures (see Figure 3.2), electrostatic interaction between the particles constituting

21


Figure 3.2: The ρ−T plane is divided into regions dominated by pressure ionization, degeneracy,

radiation, ideal gas etc. The gas is assumed to be pure hydrogen. (from [25])

the gas has to be considered. Under conditions of high density and low tempera-

ture these interactions may also overwhelm those of thermal agitation and the gas

crystallizes (see [34] for an estimate of this term).

3.2.2 Treatment of convection

Convective energy transport remains one of the major uncertainties in the computation

of stellar models. Energy transport by convection should be described by numerical

solutions of the full set of hydrodynamical equations, however, sufficiently efficient simu-

lation methods are not yet available. What we need from a theory of convection is ∇ad,

i.e. the temperature gradient that will substitute ∇rad when the layers are convectively

unstable (see Eq. (3.11)). I will present only briefly and qualitatively the basic approach

used to model convection.

22

3.2. PHYSICS INVOLVED

The Mixing Length Theory Energy transport by convection is modelled on the

analogy of molecular heat transfer: the transporting particles are macroscopic mass

elements, and their mean free path is the mixing length l, that is, the typical distance

covered by a “bubble” before it dissolves in the surroundings releasing its heat content.

In computations, rather than dealing directly with l, we introduce α, defined as:

l = αλP (3.22)

where λP = −(d ln Pdr )−1 is the pressure scale height. The parameter α is not provided

by this theory, however, it is determined, in the solar case, by requiring that the radius

of the model fits the solar radius; while in other stars it is assumed that the parameter

remains close to the solar value.

Convective Overshoot The extent of a convective region is basically determined

by the Schwarzshild condition, Eq.(3.11), however, since this boundary is defined by

zero condition in the acceleration, not in velocity, convective motions tend to penetrate

beyond the convective unstable region into the region of radiative equilibrium. The

presence and extent of a convective overshoot in the core influences the refuelling of

nuclear reactions, and therefore the evolution of a star.

3.2.3 Opacity

Opacity, defined as the photon absorption coefficient per unit mass, takes into account

all the processes of interaction between radiation and matter. Here I will shortly list the

main sources of opacity:

• The contribution of electron scattering to stellar opacity could be expressed, by

using Thomson scattering cross section, as

κe ≃ 0.20 (1 + X) cm2g−1 (3.23)

• The inverse bremsstrahlung process, called free-free absorption, describes the in-

elastic photon-electron scattering in the proximity of an ion.

• The process of bound-free absorption is related to the absorption of a photon by a

bound electron, where the photon energy is sufficient to remove the electron from

the atom or ion.

• Bound-bound absorption: this source of opacity is associated with the transition

between two bound energy levels in an atom caused by the absorption of a photon.

23


The general expression for opacity has to include all the sources mentioned, and will

therefore involve extensive quantum-mechanical calculations. Approximated expressions

for the opacity in the case of free-free and bound-free absorption, known as Kramer’s

laws, are reported below:

κff ≃ 4 × 1022(X + Y )(1 + X)ρT−3.5 cm2g−1 (3.24)

κbf ≃ 4 × 1025Z(1 + X)ρT−3.5 cm2g−1 (3.25)

The different contributions to opacity in a log ρ − log T plot are shown in Figure 3.3.

Figure 3.3: Summary of the contributions to opacity in a log ρ-log T diagram. (from [25])

3.2.4 Nuclear energy generation

Nuclear and stellar physics are linked by a close bond, which is mainly represented by

Equation 3.15. In order to calculate ǫ, the energy generation rate per unit mass, we need

to:

• compute the cross sections of the nuclear reactions that take place under the typical

conditions and chemical composition of stellar interiors,

• know the amount of energy released by each reaction,

24

3.3. NUMERICAL RESULTS

• calculate the total reaction rate which will also depend on the velocity distribution

of the interacting nuclei.

Since a star spends most of its lifetime on the hydrogen main sequence and the subject

of the thesis concerns main sequence stars, I will briefly recall the reactions of hydrogen

burning.

Helium can be generated by nuclear reactions involving hydrogen nuclei through

proton-proton chains (ppI, ppII and ppIII) and the CNO cycle (see [25] for example). I

will here recall only ppI chain and the CNO cycle that in the compact notation A+ b →C + d ≡ A(b, d)C respectively become:

1H(1H, e+νe)2D(1H, γ)3He(3He, 21H)4He (3.26)

and:

12C(1H, γ)13N(e+, νe)13C(1H, γ)14N(1H, γ)15O(e+, νe)

15N(1H, 4He)12C (3.27)

The expressions for the energy reaction rates of proton-proton chains and CNO cycle,

approximated over a limited range of temperature around T0, are:

ǫ(pp) = ǫ(pp)0 X2ρ

(

T

T0

)npp

ǫ(CNO) = ǫ(CNO)0 X Z ρ

(

T

T0

)nCNO

where ǫ(pp)0 differs for different pp chains. The energy generation rate from the CNO cycle

is much more temperature-dependent than the one from pp-chains (npp < nCNO). In

high mass stars, where the contribution of CNO reaction becomes significant, convective

instability is generated (see [10]) since the ratio L(r)/m in ∇ increases (see Equation

3.9 and 3.11).

3.3 Numerical results

3.3.1 Set of models considered

In this section I will present the set of models considered later in the thesis. In the

subsequent tables, for each model are listed the

input parameters:

• M/M⊙ the mass of the star in solar mass units,

• X the initial hydrogen abundance,

• Z the initial heavy-element abundance,

25


Input Parameters

M/M⊙ X Z α

0.9 0.70 0.02 1.83

Output Parameters

n age (Gyr) R (cm) Teff (K) L/L⊙ Xc qc

1 0.00000 5.53918e+10 5250.6 0.432 6.92827e-01 0.0000e+00

5 1.90196 5.65408e+10 5303.6 0.469 6.00226e-01 0.0000e+00

10 4.99409 5.89480e+10 5393.8 0.545 4.45652e-01 0.0000e+00

15 7.41434 6.15316e+10 5464.9 0.626 3.20206e-01 0.0000e+00

20 9.20698 6.40584e+10 5514.4 0.703 2.23878e-01 0.0000e+00

25 10.56110 6.64773e+10 5547.5 0.775 1.50109e-01 0.0000e+00

30 11.59640 6.87325e+10 5567.9 0.841 9.27609e-02 0.0000e+00

35 12.34170 7.06132e+10 5577.3 0.894 4.84225e-02 0.0000e+00

40 12.93420 7.23312e+10 5581.5 0.941 1.74465e-02 0.0000e+00

45 13.50900 7.43355e+10 5584.7 0.996 3.65104e-03 0.0000e+00

• α the mixing length parameter.

and some of the basic

output parameters:

• L/L⊙ the ratio between the luminosity of the star and the solar luminosity,

• Teff the effective temperature in Kelvin,

• Age the age of the star in Giga years,

• R the radius of the star in centimeters,

• Xc the central hydrogen abundance,

• qc the fractional mass of the convective core.

Each table describes, for a star of a given mass, the characteristics of the evolved models.

The evolution tracks computed for the models considered are shown in Fig. 3.1 in an

Hertzsprung-Russel diagram (log Teff − log (L/L⊙)).

The whole set of models considered has been computed using stellar structure and

evolution code developed by Christensen-Dalsgaard, see Appendix ?? for details.

26


Input Parameters

M/M⊙ X Z α

1.0 0.70 0.02 1.83

Output Parameters


1 0.00000 6.21361e+10 5627.8 0.718 6.92827e-01 0.0000e+00

5 1.58215 6.41452e+10 5680.1 0.794 5.76877e-01 0.0000e+00

10 3.60810 6.74863e+10 5744.5 0.919 4.19823e-01 0.0000e+00

15 5.03932 7.05897e+10 5783.6 1.033 2.99063e-01 0.0000e+00

20 6.04154 7.32741e+10 5803.9 1.129 2.06369e-01 0.0000e+00

25 6.73825 7.54511e+10 5811.8 1.203 1.34947e-01 0.0000e+00

30 7.21487 7.71017e+10 5812.2 1.257 8.05402e-02 0.0000e+00

35 7.57668 7.84775e+10 5810.0 1.300 4.06785e-02 0.0000e+00

40 7.92978 8.00087e+10 5808.1 1.350 1.48770e-02 0.0000e+00

45 8.29154 8.18171e+10 5806.4 1.410 3.98473e-03 0.0000e+00

Input Parameters

M/M⊙ X Z α

1.1 0.70 0.02 1.83

Output Parameters


1 0.00000 7.02389e+10 5944.9 1.142 6.92827e-01 0.0000e+00

5 1.07674 7.27803e+10 5983.0 1.258 5.75959e-01 0.0000e+00

10 2.39992 7.66817e+10 6022.4 1.433 4.16982e-01 0.0000e+00

15 3.26230 7.98182e+10 6037.6 1.569 2.94539e-01 0.0000e+00

20 3.80668 8.20839e+10 6038.5 1.660 2.00967e-01 0.0000e+00

25 4.15516 8.36580e+10 6033.3 1.718 1.30477e-01 0.0000e+00

30 4.40303 8.48582e+10 6027.0 1.760 7.83043e-02 0.0000e+00

35 4.62219 8.60241e+10 6021.6 1.803 4.02463e-02 0.0000e+00

40 4.85330 8.74034e+10 6017.4 1.856 1.62889e-02 0.0000e+00

45 5.09541 8.90186e+10 6013.6 1.920 5.41272e-03 0.0000e+00

27


Input Parameters

M/M⊙ X Z α

1.2 0.70 0.02 1.83

Output Parameters


1 0.00000 7.92201e+10 6227.6 1.749 6.92827e-01 8.1776e-03

20 0.89260 8.27152e+10 6258.8 1.945 5.60338e-01 4.0230e-03

40 1.66482 8.63628e+10 6271.8 2.138 4.40822e-01 9.7188e-03

60 2.40322 9.05407e+10 6256.7 2.328 3.42848e-01 2.4725e-02

80 3.01574 9.46946e+10 6209.7 2.470 2.61705e-01 3.9761e-02

100 3.43669 9.79960e+10 6153.8 2.552 1.94361e-01 4.7667e-02

120 3.69912 1.00260e+11 6108.7 2.594 1.38912e-01 4.9695e-02

140 3.85988 1.01766e+11 6081.1 2.624 9.51749e-02 4.5808e-02

160 3.98178 1.03041e+11 6069.2 2.669 5.95192e-02 4.3070e-02

180 4.07421 1.04281e+11 6083.0 2.759 3.03641e-02 3.7422e-02

Input Parameters

M/M⊙ X Z α

1.5 0.70 0.02 1.83

Output Parameters


1 0.00000 1.00027e+11 7192.3 4.960 6.92827e-01 6.7895e-02

20 0.55014 1.07393e+11 7108.1 5.455 5.61206e-01 8.4808e-02

40 1.00914 1.16753e+11 6944.2 5.873 4.39364e-01 9.3463e-02

60 1.31721 1.25051e+11 6783.0 6.133 3.41744e-01 8.8380e-02

80 1.52784 1.31749e+11 6646.4 6.276 2.60829e-01 8.4440e-02

100 1.68168 1.37122e+11 6535.1 6.354 1.94821e-01 7.9903e-02

120 1.79582 1.41310e+11 6448.8 6.399 1.40914e-01 7.5072e-02

140 1.88148 1.44519e+11 6387.3 6.441 9.68848e-02 7.0597e-02

160 1.94611 1.46925e+11 6353.2 6.516 6.09023e-02 6.6001e-02

180 1.99466 1.48642e+11 6359.1 6.694 3.14808e-02 6.0658e-02

28


Input Parameters

M/M⊙ X Z α

1.7 0.70 0.02 1.83

Output Parameters


1 0.00000 1.04048e+11 8071.2 8.512 6.92827e-01 1.0494e-01

20 0.39358 1.14846e+11 7842.9 9.246 5.60897e-01 1.1716e-01

40 0.69212 1.26762e+11 7585.8 9.858 4.38984e-01 1.0731e-01

60 0.89563 1.38096e+11 7341.1 10.262 3.40211e-01 1.0175e-01

80 1.04055 1.48672e+11 7117.1 10.508 2.59579e-01 9.3941e-02

100 1.14544 1.57947e+11 6926.9 10.642 1.93771e-01 8.6763e-02

120 1.22260 1.65454e+11 6779.9 10.717 1.40043e-01 8.0247e-02

140 1.28021 1.71073e+11 6678.2 10.786 9.61657e-02 7.4779e-02

160 1.32362 1.74828e+11 6624.8 10.908 6.03159e-02 6.9783e-02

180 1.35632 1.76607e+11 6634.8 11.198 3.10057e-02 6.4572e-02

Input Parameters

M/M⊙ X Z α

2.0 0.70 0.02 1.83

Output Parameters


1 0.00000 1.11169e+11 9237.5 16.673 6.92827e-01 1.4176e-01

20 0.25053 1.25692e+11 8857.7 18.019 5.60073e-01 1.3447e-01

40 0.43721 1.40127e+11 8533.0 19.288 4.38860e-01 1.2457e-01

60 0.56453 1.53766e+11 8236.4 20.161 3.40041e-01 1.1357e-01

80 0.65390 1.66469e+11 7968.7 20.703 2.59374e-01 1.0209e-01

100 0.71778 1.77911e+11 7736.9 21.014 1.93579e-01 9.2787e-02

120 0.76441 1.87795e+11 7546.9 21.197 1.39867e-01 8.4902e-02

140 0.79901 1.95754e+11 7405.8 21.357 9.60136e-02 7.8587e-02

160 0.82496 2.01226e+11 7326.1 21.612 6.01869e-02 7.3010e-02

180 0.84449 2.02896e+11 7343.5 22.181 3.09016e-02 6.7753e-02

29


Input Parameters

M/M⊙ X Z α

3.0 0.70 0.02 1.83

Output Parameters


1 0.00000 1.36818e+11 12446.2 83.226 6.92827e-01 1.9979e-01

20 0.08753 1.59460e+11 11817.9 91.895 5.59419e-01 1.7756e-01

40 0.15044 1.80041e+11 11378.4 100.671 4.38066e-01 1.5472e-01

60 0.19179 1.99368e+11 10982.5 107.138 3.39212e-01 1.3579e-01

80 0.21993 2.17354e+11 10626.4 111.614 2.58637e-01 1.2026e-01

100 0.23967 2.33627e+11 10318.0 114.622 1.92920e-01 1.0775e-01

120 0.25385 2.47726e+11 10065.4 116.709 1.39301e-01 9.7643e-02

140 0.26423 2.59090e+11 9878.0 118.415 9.55201e-02 8.9348e-02

160 0.27191 2.66843e+11 9772.9 120.348 5.97621e-02 8.2328e-02

180 0.27760 2.68984e+11 9798.7 123.583 3.05345e-02 7.5194e-02

3.3.2 Some thermodynamic variables

In this subsection I will briefly show the stratification inside a star of some of the fun-

damental thermodynamic variables, such as temperature and sound speed, as results of

calculations. The models considered in these plots are taken from the ones reported in

the previous tables.

Mean molecular weight An approximated expression for the mean molecular weight

is:

µ−1 ≃ 2X +3

4Y +

1

2Z (3.28)

During evolution on the main sequence the hydrogen mass fraction X decreases in the

core as the result of nuclear reactions, the different behaviour in 1.0 and 2.0 M⊙ models

is explained in the caption of Fig. 3.4.

Sound speed Since in the next chapters we will be studying stellar acoustic modes

of oscillation (p modes), one of the thermodynamic quantities that is more strongly

associated with the seismic properties of the star is the adiabatic sound speed. It is

defined as c ≡ (dPdρ )

1/2ad ≡ (Γ1

Pρ )1/2, when the approximated ideal gas law is applied, it

becomes:

c =

(

Γ1 kbT

µmu

)1/2

(3.29)

30


Figure 3.4: The variation of the mean molecular weight µ as a function of the fractional mass,

as a star evolves, is driven by the change in the hydrogen abundance X due to nuclear reactions

in the core. In the 2.0 M⊙ model the presence of a convective core and therefore of mixing on

a time-scale much shorter than the evolutionary time-scale, creates a steep change in µ, which

does not appear in the 1.0 M⊙ model.

31


The change of the sound speed, as shown in Figure 3.5, is clearly explained by the

expression above: its behaviour is mainly driven by the temperature, nonetheless, during

evolution it is particularly sensitive to the variation of the mean molecular weight µ, as

a result of the nuclear reactions in the core (see Fig. 3.5).

First adiabatic exponent The behavior of the sound speed, and of its derivatives,

could also be affected by features that require to estimate directly the sound speed as:

c =

√

Γ1P

ρ(3.30)

rather than the approximated expression (3.29), which assumes the ideal gas law as

equation of state.

In the regions near the surface of the star hydrogen and helium may not be fully ionized,

as a result Γ1 shows rapid variations (see Fig. 3.6); these changes are responsible for a

characteristic signature in the frequencies of oscillation, as presented in Section 6.2.

32


Figure 3.5: The adiabatic sound speed as a function of the distance from the center of the star,

in the evolved 2.0 M⊙ star the sound speed near the base of the convective region has a sharp

variation due to the fact that the core is fully mixed in the bounded region where convetion

occurs.

33


Figure 3.6: The scaling of Γ1 near the surface of a 1.0 M⊙ and 2.0 M⊙ star. In both figures

the dip in Γ1 due to the second helium ionization is present, whereas only in the 2.0 M⊙ model

also the first helium ionization zone generates a local minimum in Γ1.

34

Chapter 4

Modelling linear adiabatic

oscillations

In this chapter I will describe the basic theory underlying linear, adiabatic, non-

radial stellar oscillations. I will briefly review the physical nature of the modes

of oscillation starting from simple solutions of the perturbed ideal fluid equations,

underline the hypothesis we are assuming and further approximations we might take

in order to deal with analytic solutions. Most of the results presented in this chapter

are treated in details in reference books as [42], [9] or in review papers such as [11]. I

will finally present some numerical results showing some properties of the theoretical

oscillation spectra computed.

4.1 Hydrodynamics equations

Since the proper theoretical environment to set and develop the theory of stellar oscilla-

tions is hydrodynamics, I will first recall the fundamental equations of fluids, then linear

perturbation equations and finally simple examples of wave motion.

4.1.1 Basic equations

The continuity equation This equation is an expression of mass conservation, which

in a differential form could be written as:

∂ρ

∂t+ div(ρv) = 0 (4.1)

where ρv is the mass flux density.

The equation of motion The equation of motion considered is written under the hy-

pothesis of an ideal fluid, we are therefore considering no processes of energy dissipation,

35

CHAPTER 4. MODELLING LINEAR ADIABATIC OSCILLATIONS

which may occur in consequence of thermal conduction and internal friction (viscosity).

ρ∂v

∂t+ v · ∇v = −∇p + f (4.2)

where f is the body force per unit volume.

The only volume force considered hereafter is the gravitational force per unit volume

ρg, we then neglect the effects of magnetic fields and apparent forces due to stellar rota-

tion1. The gravitational acceleration g could be expressed by means of the gravitational

potential Φ:

g = ∇Φ (4.3)

where Φ satisfies Poisson’s equation:

∇2Φ = −4πGρ (4.4)

The energy equation The energy equation provides a thermodynamic relation be-

tween p and ρ, which will complete the set of equations previously presented.

For the purpose of calculating stellar oscillations, to a high degree of precision, we

can assume the motion being adiabatic. Therefore, all the complications sketched in

Section 3.1.3 while treating energy transport in stellar interiors or atmosphere, can be

neglected. This assumption can be justified in stellar interiors by an estimate of the

heating term in energy equation, see [28], when it is negligible we may write:

dp

dt=

Γ1p

ρ

dρ

dt(4.5)

or simply write the equation of entropy conservation:

∂s

∂t+ v · ∇s = 0 (4.6)

where s is the entropy per unit mass.

4.1.2 Perturbation analysis

The equations of hydrodynamics form a set of non-linear, coupled differential equations

which is often too complicated to be solved, even numerically. Nonetheless, stellar

oscillations can be regarded, in most of the cases, as small perturbations on a static

equilibrium structure. This is certainly true for pulsations in solar-type stars, where the

luminosity variations are of the order of micro-magnitudes.

1The first order effects of moderate rotation, treated as a perturbation term in the equations, are

presented in Section 4.5

36

4.1. HYDRODYNAMICS EQUATIONS

We then consider every variable as developed around the static equilibrium state, e.g.

pressure:

p(r, t) = p0(r) + p′(r, t) (4.7)

and we linearize the equations in the perturbations, neglecting terms of order higher

than one in the perturbed variables. We obtain, for the continuity equation:

∂ρ′

∂t+ div(ρ0v) = 0 (4.8)

for the equations of motion:

ρ0∂v

∂t= −∇p′ + ρ0g

′ + ρ′g0 (4.9)

for Poisson’s equation:

∇2Φ = −4πGρ′ (4.10)

and for the energy equation in the adiabatic approximation:

p′ + δr · ∇p0 =Γ1,0 p0

ρ0(ρ′ + δr · ∇ρ0) (4.11)

where velocity v is the local time derivative of the displacement δr.

4.1.3 Pressure waves

In a compressible fluid, an oscillatory solution with small amplitude can be found as-

suming spatial homogeneity, adiabatic motion and neglecting the perturbation in the

gravitational potential. Combining equations (4.8)-(4.11) we obtain the so-called wave

equation

∂2ρ′

∂t2− c2

0∇2ρ′ = 0 (4.12)

where

c20 = Γ1

p0

ρ0

is the squared adiabatic sound speed. Substituting a monochromatic plane wave solution

in the wave equation we obtain the dispersion relation for acoustic waves:

ω2 = c20 |k|2 (4.13)

The displacement δr is in the direction of the wave vector k, hence acoustic waves are

longitudinal waves.

37


4.1.4 Surface gravity waves

The free surface of a fluid in equilibrium in a gravitational field is generally an equipo-

tential surface, a sphere for a self gravitating object as a star, a plane if the field is

uniform and parallel. If, under the action of some external perturbation, the surface is

moved from its equilibrium position at some point, motion will occur.

These perturbations will affect the interior of the fluid as well, but less and less at greater

and greater depths.

To derive the equation for surface waves we consider a fluid at constant density ρ0,

incompressible and with the boundary condition for an ideal fluid at a free surface,

that is requiring the lagrangian pressure perturbation being zero on the surface. The

dispersion relation for surface gravity waves, in the “deep water” approximation, i.e.

with the additional requirement that the perturbation does not diverge at great depths,

is:

ω2 = g0 kh (4.14)

and a solution for a wave traveling in the x direction is:

x − x0 = −Ak

ωekz0 cos (kx0 − ωt) (4.15)

z − z0 = −Ak

ωekz0 sin (kx0 − ωt) (4.16)

The fluid particles describe circles about the points (x0, z0) with a radius which dimin-

ishes exponentially with increasing depth. These expressions can be easily generalized

considering boundary to a finite depth, and obtain dispersion relations useful to explain

some fundamental behaviours of sea waves for instance.

4.1.5 Internal gravity waves

There is another kind of gravity waves which can propagate inside the fluid; unlike surface

gravity waves which are generated by a discontinuity in density on the surface of a fluid,

internal gravity waves are caused by inhomogeneities due to the gravitational field. The

argument presented follows the one in [28], but similar results are also achieved in [26].

The fluid could be treated again as incompressible, even if it is stratified in pressure

under gravity, this means that we neglect density changes due to pressure perturbations

but we consider those due to entropy stratification.

The perturbations in the gravitational potential are also neglected.

The privileged direction, in spherical coordinates, of the gravitational field suggests

to separate both the displacement δr and the wave vector k into radial and horizontal

components:

δr = ξrar + ξh (4.17)

38

4.2. EQUATIONS OF STELLAR LINEAR OSCILLATIONS

k = krar + kh (4.18)

Looking for solutions in the form of plane waves, e.g. v = const × ei(k·r−ωt) we obtain

from the perturbed equation of continuity:

v · k = 0 (4.19)

the fluid velocity is everywhere perpendicular to the wave vector k, that is internal

gravity waves are transverse waves.

Considering small frequencies it is possible to take further approximations and deduce

the following dispersion relation:

ω2 =N2

1 + k2r/k

2h

(4.20)

where N2 is the buoyancy frequency, defined as:

N2 = g0

(

1

Γ1,0

d ln p0

dr− 1

Γ1,0

d ln ρ0

dr

)

(4.21)

When N2 is positive the motion is oscillatory, with a maximum frequency N2 which

corresponds to modes with infinitely small horizontal wavelength; It is instructive to

notice that the condition N2 > 0 implies convective stability of the equilibrium structure

(see 3.1.3).

4.2 Equations of stellar linear oscillations

The equations describing stellar linear adiabatic oscillations are obtained re-writing equa-

tions (4.8)-(4.11), using explicitly the spherical symmetry of the equilibrium structure.

These equations will describe nonradial oscillations, i.e. the perturbations themselves

are not assumed to be spherically symmetric, thus the case of radial oscillations is con-

tained as a special case.

The problem of finding solutions to this system of equations could be set, with an

amazing mathematical similarity to the Schrodinger equation for a particle in a central

potential 2, as finding the spectral properties of a differential operator in a properly

defined Hilbert space. This latter approach is particularly useful when considering in

the oscillations equations small effects of non-adiabaticity, rotation or small variations

in the equilibrium structure, and only in those cases I will choose this approach (see

Sections 4.3, 4.5, 4.6).

2The spherical symmetry of both central potential and the equilibrium structure of a star obviously

plays a fundamental role

39


4.2.1 Equations of stellar linear adiabatic oscillations

It is useful to separate the displacement δr into radial and horizontal components:

δr = ξrar + ξh (4.22)

If we take the divergence of the equation of motion for the horizontal displacement,

combine it with the continuity equation (Eq. 4.8), taking into account that the horizontal

gradients of the equilibrium quantities are zero, we obtain:

− ∂2

∂t2

[

ρ′ +1

r2

∂

∂r(r2ρ0ξr)

]

= ∇2h p′ + ρ0∇2

h Φ′ (4.23)

The radial component of the equation of motion is:

ρ0∂2ξr

∂t2= −∂p′

∂r− ρ′g0 + ρ0

∂Φ′

∂r(4.24)

and Poisson’s equation:

1

r2

∂

∂r

(

r2 ∂Φ′

∂r

)

+ ∇2hΦ′ = −4πGρ′ (4.25)

The equation of energy, in the adiabatic approximation, becomes:

ρ′ =ρ

Γ1pp′ + ρξr

(

1

Γ1p

dp

dr− 1

ρ

dρ

dr

)

(4.26)

4.2.2 Separation of variables

It is useful to notice that in the set of equations (8.3.1)-(8.3.1) the derivatives in respect

to the angular variables θ and φ only appear in the combination ∇2h; moreover the

dependence on time appears only in the linear perturbations, we may seek a solution for

the dependent variables in the form:

f(r, θ, φ, t) =√

4πf ′(r)Y ml (θ, φ)e−i ωt (4.27)

where Y ml (θ, φ) is a spherical harmonic, i.e. Y m

l (θ, φ) an eigenfunction of the angular

Laplace operator ∇2h:

∇2h(Y m

l (θ, φ)) = − l(l + 1)

r2Y m

l (θ, φ) (4.28)

Using equation (4.28), the radial dependence will be found by solving the following

set of equations, obtained from (8.3.1)-(8.3.1) substituting the density perturbation ρ′

from the energy equation (8.3.1) and the expression (4.27) for each perturbed variable

(the subscript 0 for the equilibrium variables has been suppressed):

dξr

dr= −

(

2

r+

1

Γ1p

dp

dr

)

ξr +1

ρc2

(

S2l

ω2− 1

)

p′ − l(l + 1)

ω2r2Φ′ (4.29)

40

4.2. EQUATIONS OF STELLAR LINEAR OSCILLATIONS

Figure 4.1: Representation of spherical harmonics Y ml . l varies with each column beginning

with l = 1 on the right, ending with l = 5 and m varies on each line, from top to bottom starting

with m = 0 ending with m = l. The polar axis has been inclined 30 relative to the plane of the

page.

41


dp′

dr= ρ(ω2 − N2)ξr +

1

Γ1p

dp

drp′ + ρ

dΦ′

dr(4.30)

1

r2

d

dr

(

r2 dΦ′

dr

)

= −4πG

(

p′

c2+

ρξr

gN2)

+l(l + 1)

r2Φ′ (4.31)

where I introduced the characteristic acoustic frequency or Lamb frequency S2l :

S2l =

l(l + 1)c2

r2(4.32)

and N2 is the buoyancy or Brunt-Vaisala frequency:

N2 = g0

(

1

Γ1,0

d ln p0

dr− d ln ρ0

dr

)

(4.33)

This set of equations is a fourth order system of differential equations, in four dependent

variables p’, ρ′, ξr, Φ′, in order to solve it boundary conditions are needed.

Useful relations The mean square displacements, radial and horizontal, averaged on

time and on the stellar surface are:

δ2r =

1

2|ξ′(r)|2

δ2h =

1

2l(l + 1)|ξ2

h(r)|

Finally, if the oscillations are regarded locally as plane waves, ei(k x−ωt), where k =

krar + kh we may make the identificaton:

l(l + 1)

r2= k2

h (4.34)

where kh is the length of the horizontal component of the wave vector. Thus, for example,

the horizontal surface wavelength of the mode is given by

λh =2π

kh≃ 2πR√

l(l + 1)(4.35)

and l is then approximatively the number of wavelengths around the stellar circumfer-

ence. From Eq. (4.27) we may also identify m as the number of wavelength around the

equator. An example of the real part of some spherical harmonics is given in Fig. 4.1.

4.2.3 Boundary conditions

Four boundary conditions, discussed in detail in [42], have to be provided to the fourth

order set of equations (4.29)-(4.31).

42

4.3. FUNCTIONAL ANALYSIS

Conditions at the center By expanding the equations near the singular point r = 0,

it can be shown that ξ′r scales3 as rl−1, Φ′ and p′ instead as rl. From this expansion two

conditions are obtained:

ξr ≃ l ξh (4.36)

dΦ′

dr≃ l

rΦ′ (4.37)

Surface conditions Surface boundary conditions, as for the set of equations describ-

ing the equilibrium model of a star (see Section 3.1.5), are rather more complicated since

they involve directly how stellar atmospheres are modelled. I will only recall simple sur-

face conditions, see [8] for a more detailed treatment.

One condition is obtained requiring the continuity of Φ′ and of its derivative at the

boundary r = R, where the density is zero and Poisson’s perturbed equation can be

solved analytically:

dΦ′

dr+

l + 1

rΦ′ = 0 at r = R (4.38)

A second condition can be obtained, if a definite boundary is assigned to the star,

imposing zero lagrangian pressure perturbation, i.e. :

δp = p′ + ξrdp

dr= 0 at r = R (4.39)

4.3 Functional analysis of non-radial, adiabatic oscillations.

The oscillation equations can be presented as a linear eigenvalue problem in a properly

defined Hilbert space.

After separation of the time dependence, the perturbed equations of motions (4.9) can

be written as:

ω2δr = F (δr) (4.40)

where

F (δr) =1

ρ0∇p′ − g′ − ρ′

ρ0g0 (4.41)

It can be shown that F is a linear function of the displacement δr, and therefore rep-

resents a linear operator whose domain can be set in an Hilbert space with a suitable

inner product.

3In the case of radial perturbations, i.e. l = 0, ξr goes as r.

43


4.3.1 Effects on frequencies of a small change in the oscillations equa-

tions.

The operator F , in the case of adiabatic oscillations using zero pressure at the surface

as boundary condition, is hermitian (see [3] and [4]):

〈ξ, Fa(η)〉 = 〈Fa(ξ), η〉 , for ξ, η ∈ D(F ) (4.42)

From this property it follows that, if we consider a small perturbation on the oscillations

equations, and therefore a new operator Fa+δF , the perturbation on the frequencies can

be calculated, to the first order, using the eigenfunctions of the unperturbed operator

F , i.e.

δω2 =〈δr0, δF (δr0)〉

〈δr0, δr0〉(4.43)

where F (δr0) = ω2δr0.

The same property is used, for example, in stationary perturbations on the time-independent

Schrodinger’s equation, where generally to evaluate the n-th order perturbation on the

eigenvalue, the (n − 1)th order perturbation on the eigenfunction is needed (see [15]).

The perturbation term in the operator in Eq.(4.43) could arise from adding new terms

in the oscillations equations, or from considering small variations in the equilibrium

models. Taking into account moderate rotation in the star is an example of the former

case.

A similar analysis on Fl, the operator defining the space-separated oscillations equa-

tions, gives, when an eigenfunction ξnl = (ξr,nl, ξh,nl) is considered:

δωnl

ωnl=

1

2

δω2nl

ω2nl

=〈ξnl, δFl(ξnl)〉l2ω2

nl 〈ξnl, ξnl〉lwhere δFl(ξnl) = (φr(ξnl), φh(ξnl)). If we are concerned about the changes in the coef-

ficients of the oscillation equations generated by variations in the equilibrium structure

of the star, we can express them by a choice of two model variables, and the variations

in the frequencies written as:

δωnl

ωnl=

∫ R

0

[

Knlc2,ρ(r)

δrc2

c2(r) + Knl

ρ,c2(r)δrρ

ρ(r)]

]

dr (4.44)

(see [40] for an explicit expression for the kernels), this expression has been obtained with

no other assumptions but the adiabatic oscillations equations and boundary conditions

such as the lagrangian pressure perturbation being zero at r = R.

4.4 Properties of oscillation modes

The equations presented can be solved numerically, nonetheless analytic solutions are

useful to have physical insight into the characteristics of the modes.

44

4.4. PROPERTIES OF OSCILLATION MODES

The Cowling approximation A first approximation is to neglect the perturbation in

the gravitational potential Φ′, the fourth order system of differential equations reduces

to a second order system of coupled equations:

dξr

dr= −

(

2

r+

1

Γ1H−1

p

)

ξr +1

ρc2

(

S2l

ω2− 1

)

p′ (4.45)

dp′

dr= ρ(ω2 − N2)ξr +

(

1

Γ1H−1

p p′)

(4.46)

where

Hp = −(

d ln p

dr

)−1

(4.47)

is the pressure scale height, i.e. the distance over which the pressure changes by a factor

e.

This approximation can be qualitatively justified considering Equation (8.3.1) and esti-

matingd2

dr2≃ −n2

r2

thus Φ′ becomes

Φ′ ∼ − −4πGr2ρ′

n2 + l(l + 1)

and the approximation is accurate

- when l is large,

- when the radial order |n| is large.

A quantative justification of this basic approximation, which is also assumed in most

of the asymptotic treatments, (see Chapter 5), comes from the comparison between

frequencies calculated numerically both with the full set of equations and under Cowling’s

approximation (see [7]) .

4.4.1 Physical nature of the modes

Accurate approximations that lead to analytic solutions of the oscillations equations,

are presented in detail in Chapter 5, here I will recall a simple approximation (see [9]),

useful for a qualitative treatment and classification of the oscillation modes in stars.

A first crude approximation is to neglect, for oscillations of high radial order, the terms in

Equations (4.45) and (4.46) containing only derivatives of equilibrium quantities, which

are supposed to vary much more slowly than the perturbed variables.

The set of equations is then reduced to a single second order ordinary differential equa-

tion:d2ξr

dr2= −K(r)ξr (4.48)

45


where

K(r) =

(

1 − S2l

ω2

)(

1 − N2

ω2

)

(4.49)

The local solutions of such equation depend on the sign of K(r): when K(r) is positive

the solution is an oscillating function of r, that may approximatively be written as:

ξr(r) ∼ cos

(∫

K1/2 dr + φ

)

, K > 0

ξr has an exponential behaviour when K < 0

ξr(r) ∼ e±∫

|K|1/2 dr+φ, K < 0

The solution oscillates when

ω > |Sl| and ω > |N | (4.50)

or

ω < |Sl| and ω < |N | (4.51)

and behaves exponentially when:

ω < |Sl| and ω > |N | (4.52)

or

ω > |Sl| and ω < |N | (4.53)

A mode is then generally oscillating in regions where condition (4.50) or (4.51) is fullfilled

and will decay exponentially otherwise. The solution is said to be trapped between two

regions of exponential behaviour, the boundaries of the trapping region are generally at

points where

K(r) = 0

Low frequency modes, satisfying condition (4.51) are classified as g modes, whereas those

satisfying (4.50) are labelled p modes. Since both Lamb and buoyancy frequencies play

a fundamental role in determining the behaviour of the oscillations, in Fig. 4.2 I show

the behaviour of both Sl and N as a function of fractional radius.

p modes

The region where p modes are trapped is bounded by the surface of the star and by rt,

so that S2l (rt) = ω2, i.e., for a mode with given cyclic frequency ω:

c2(rt)l(l + 1)

r2t

= ω2 (4.54)

46

4.4. PROPERTIES OF OSCILLATION MODES

Figure 4.2: Lamb frequency S2

l and buoyancy frequency N2 as a function of the distance to the

center of a 1 M⊙, Xc = 0.30 star. S2

l is plotted for l = 1, 5 and 10.

The trapping point at the surface of the star is not contained in the simple analysis

presented in this section, but will be explained in Chapter 5.

Typically for p modes we have that ω ≪ N , so that K can be approximated by:

K(r) ≃ 1

c2(ω2 − S2

l ) (4.55)

This relation reveals the physical nature of these modes: their characteristics are de-

termined solely by the variation of the sound speed with r. These modes are acoustic

waves, and the restoring force is dominated by pressure, this is why they are classified

as pressure modes. From the dispersion relation for sound waves, Eq. (4.13), Eq. (4.35)

for the horizontal displacement and writing |k2| = k2r + k2

h we obtain:

k2r =

1

c2(ω2 − S2

l ) (4.56)

which is Eq. (4.55) once we have identified K with k2r .

The interior reflection of the p modes can be understood in terms of ray theory. As

wave propagate into the star they experience, generally, a higher sound speed and then

travel faster and bend from the radial direction, just as light does while propagating in a

47


medium with smaller refraction index. The dependence of the location of turning point

rt on the mode degree l and on the frequency is particularly interesting, since modes

with different degree and frequency sample different regions of the star.

From Eq. (4.55) it follows that K, and so the number of zeros in the eigenfunction,

increases with the frequency.

g modes

In the case of g modes the turning points are determined by the condition N = 0. The

presence and extension of convective regions in the star determine the trapping points of

these low frequency modes, as physically the condition of convective instability excludes

the presence of g modes, sought as oscillating perturbations around a stability condition.

As shown in Fig. 4.2 the trapping points of such modes are independent from l. For

high order g modes typically ω2 ≪ S2l , with an argument similar to those used above

for p modes and using the dispersion relation for internal gravity waves we obtain:

k2r =

l(l + 1)

r2

(

N2

ω2− 1

)

(4.57)

from this relation it is clear that the frequencies decrease with increasing order. As ex-

pected the frequency of a g mode cannot exceed the maximum of the buoyancy frequency

in the stellar interior.

4.4.2 Numerical results

The complete set of equations presented in the previous sections can be solved numer-

ically starting from an equilibrium model, eigenfrequencies and eigenfunctions can be

determined. As an example of these computations, in Figure 4.4 it is shown, for a model

of the sun, the behaviour of a p mode and of a g mode. As predicted from the simple

analysis of the trapping points presented in Section 4.4.1 g modes are generally trapped

in the regions where convection is absent, and have higher amplitudes near the center of

the star. p modes instead have higher amplitudes near the surface and penetrate more

deeply with increasing order and decreasing degree.

The presence of a convective core gives rise to a maximum in the buoyancy fre-

quency generated by a steep gradient in the mean molecular weight near the edge of

the convective region (see 3.1.4 and Figure 4.3). The consequences on the properties of

the oscillation modes, i.e. avoided-crossings and the appearance of modes which have a

mixed p and g character, are explained in [9].

48

4.5. ROTATION

Figure 4.3: Lamb frequency S2

l and buoyancy frequency N2 as a function of the distance to the

center of a 2 M⊙, Xc = 0.67 star. S2

l is plotted for l = 1, 5 and 10. The buoyancy frequency has a

maximum near the edge of the convective core due to the sharp gradient in chemical composition.

4.5 Rotation

So far we have described a star as a spherically symmetric structure, this is obviously

not true if a velocity field, such as the the one describing rotation, is present. The

effects of rotation on oscillation frequencies can be understood with the following simple

geometrical argument. The dependence of the perturbations on the longitude φ is in the

form cos (mφ − ωt) (see Eq. 4.27), we then consider a star rotating with angular velocity

Ω, and denote with ω0 the eigenfrequencies calculated in a reference frame integral with

the star; the oscillation behaves as cos(mφ−ω0t). The longitude φ′ in an inertial reference

frame is related to time and Ω by φ′ = φ−Ωt, therefore the perturbation observed from

the inertial frame is expressed by: cos (φ − ωmt) where

ωm = ω0 + mΩ (4.58)

Rotation, by breaking the spherical symmetry of the star, lifts the m degeneracy of the

frequencies.

49


Figure 4.4: Eigenfunction of a l = 20,n = 17 p mode (upper panel) and an eigenfunction of a

l = 2,n = −10 g mode (lower panel) for a solar model(from [9]).

4.5.1 Effects of slow rotation

The simple argument presented above assumes, unrealistically, that the star is rotating

as a solid-body and neglects a priori apparent forces in the rotating reference frame.

For a slowly rotating star4 the effects of rotation on the equilibrium structure can be

ignored, moreover the perturbed equation of motion is written to the first order in Ω, so

higher order contributions, such as centrifugal force for instance, can be neglected.

As described in Section 4.3.1, since it is possible to write the equations of oscillation as,:

ω2δr = F (δr) + δF (δr)

where δF (δr) represent a small correction, due to rotation, to the operator describing

oscillations, the effects on the oscillation frequencies can be found from equation (4.43).

The result can be written in the form:

ωnlm = ωnl0 + m

∫ R

0

∫ π

0Knlm(r, θ)Ω(r, θ)r drdθ (4.59)

where the kernels Knlm can be calculated from the eigenfunctions for the non rotating

model.

4The sun with its ∼ 2 km/s surface rotation rate is an example of a slow rotator.

50

4.6. EXCITATION AND DAMPING OF THE OSCILLATIONS

In the case of symmetric rotation, i.e. when Ω is independent from θ, the kernels are

independent from m and the frequency splitting may be written as:

ωnlm = ωnl0 + mβnl

∫ R

0Knl(r)Ω(r)dr

where βnl is a dimensionless parameter depending on the order and degree of the mode

(see [11]) and∫ R

0Knl(r)dr = 1

Finally if we consider uniform rotation we find:

ωnlm = ωnl0 + mβnlΩ (4.60)

which is the frequency splitting predicted by Eq. (4.58), multiplied by a factor which

physically is related to Coriolis force, and is approximatively one for high order or high

degree p modes.

4.5.2 Effects of moderate rotation on the oscillations equations

The treatment presented in the previous section is applicable to stars with low rotation

rates. On faster rotators the corrections in the frequencies have to be considered up to

higher orders in Ω, including effects in the equilibrium structure of the star.

Even under the assumption of uniform rotation, the treatment is rather complicated,

and, since not directly related to the problems addressed in this thesis, I will just refer

to [11] for an accurate treatment.

4.6 Excitation and damping of the oscillations

Stellar oscillations may be excited in two different ways: by being self-excited and by

being intrinsically damped but externally forced, typically by convection (as in our Sun).

The linear adiabatic theory presented so far predicts the characteristics of the resonances

of the star (eigenfrequencies and eigenfunctions), but gives no information whether the

mode considered is excited to an observable amplitude. Mode amplitudes and excita-

tion/damping rates can in principle be obtained by solving the full set of non-linear,

non-adiabatic equations; nonetheless linear growth rate of oscillation modes can be ob-

tained by a linear non adiabatic analysis presented in the next section.

A positive linear growth rate means that the mode is unstable and then, self excited, a

negative linear growth rate implies the stability of the mode, which could still be excited

by an external forcing.

51


4.6.1 The quasi adiabatic approximation

Taking into account the perturbed energy equation in the general non-adiabatic case,

after separation of the time dependence, it is possible to write the perturbed equation

of momentum as a linear operator, as in Section 4.3:

ω2δr = Fad(δr) + δF (δr) (4.61)

where Fad and δF are both linear operators on δr. The variation in frequencies, generally

represented by complex numbers in the non adiabatic case, can be expressed by means of

Eq. (4.43), where the integral can be evaluated on the eigenfunctions of the unperturbed

operator.

The explicit expression appearing in δF depends on the expression for the perturbed,

non adiabatic energy equation and I will then only refer to [28] for a complete expression.

4.6.2 Self excitation

From a qualitative point of view we can explain self-excitation processes looking at pe-

riodic thermodynamic changes happening in the regions of the star interested by oscilla-

tions: these variations can be considered as cycles, as the initial and final thermodynamic

state are the same, the oscillations will be excited when energy is extracted from that

cycle. In order to do so energy has to be stored during compression.

The known mechanisms driving self excited modes, described for example in [31], are

mainly:

• The γ−mechanism

• The κ−mechanism

in both cases a central role is played by the ionization regions of the star, there energy

can be extracted from the cycle during compression and stored in ionization energy,

or captured by a higher opacity. It is possible to estimate analytically (see [28]) the

conditions where these processes occur.

Another possible source of excitation is the ǫ−mechanism: the stability is controlled

by the density and temperature dependence of nuclear energy generation rates in the

cores, or burning shells, of a star; since the source of excitation energy is in the deeper

layers of the star it is reasonable to expect g-modes to be driven by this mechanism (no

observational evidence for such a class of pulsators has been found yet).

Stars with unstable modes tend to be found in well defined regions of the HR diagram,

e.g. the Cepheids instability strip.

52

4.6. EXCITATION AND DAMPING OF THE OSCILLATIONS

4.6.3 Stochastic excitation

In this case the energy required to excite oscillations is provided by turbulent motions

appearing in the convective regions of the star. Since turbulent convection produces

motions with a very broad spectrum, in terms of spatial and temporal frequency, it is

likely to excite oscillations over a broad range of frequencies, as observed in the Sun.

It is generally complicated to predict the amplitude of such oscillations, since interaction

between waves and convective motion has to be modeled. Numerical predictions on the

expected amplitudes of modes excited by convection in main sequence stars can be found

in [22]. Assuming that it is generally more likely to excite a mode if the excitation source

overlaps the region where the mode has higher amplitude, we expect primarily p modes

to be excited in stars with a convective envelope (as in the Sun). A stochastically excited

Figure 4.5: Power spectrum of a damped and externally forced oscillator for different damping

rates. The dotted lines represent the FWHL of the peak

mode will show in the power spectrum of time series a peak with a finite width inversely

proportional to the decay time of the mode (see Figure 4.5), as in a simple damped

harmonic oscillator, forced with a function whose power spectrum weakly depends on

the frequency.

Regardless the total amount of time of observations, this finite width represents an

intrinsic uncertainty while determining the frequencies of oscillation, and may hide the

fine structure of the spectrum generated by slow rotation.

53


54

Chapter 5

Asymptotic solutions of stellar

oscillations equations

Equations of linear adiabatic stellar oscillations can be solved numerically,

nonetheless analytic, but necessarily approximated, solutions provide a pow-

erful tool to reveal the fundamental physical parameters that influence the

characteristics of oscillation spectra.

I will first present the general asymptotic theory, and then mainly concentrate

on describing analytic expressions for low degree p-mode eigenfrequencies.

The results of the theory presented in this chapter have been widely applied

and investigated in this thesis, as will be presented in the next chapter.

5.1 Asymptotic theory of stellar oscillations

In this section I will present an asymptotic approximation of the equations of stellar

oscillations introduced in [16] and also reported in [28].

The starting point of the asymptotic treatment is the second order set of equations

(4.45)-(4.46). This set is obtained applying to the fourth order system (4.29)-(4.31) the

so-called Cowling approximation, that is, the perturbations in the gravitational potential

are neglected; this approximation, as described in Section 4.4, is accurate for high l and

n modes.

The next, somehow tricky, step is to rewrite the equations in the independent variable

X, related to the radial displacement by

X = c2 √ρdiv δr

where c and ρ represent the adiabatic sound speed and density of the equilibrium model.

After considerable manipulation and making the assumptions that the variations of g

55

CHAPTER 5. ASYMPTOTIC SOLUTIONS OF STELLAR OSCILLATIONS

EQUATIONS

and r are negligible in respect to those of the perturbations, it is possible to rewrite the

equations as one second order differential equation in X:

∂2X

∂r2+ K(r)X = 0 (5.1)

K(r) is expressed by:

K(r) =ω2

c2

[

1 − ω2c

ω2− S2

l

ω2

(

1 − N2

ω2

)]

(5.2)

and depends on the cutoff frequency ωc,

ω2c =

c2

4H

(

1 − dH

dr

)

(5.3)

where H is the local density scale height,

H−1 = −d ln ρ

dr

The sign of K(r), that can indeed change in different regions of the star, determines

whether the behaviour of the solution is exponential or oscillatory. The so-called trapping

points of a mode of oscillation, are determined as the location where the perturbation

from oscillatory decays exponentially (see Section 4.4.1).

Near the surface of the star, S2l is small due to its c/r scaling, K(r) is then:

K(r) ≃ ω2

c2

(

1 − ω2c

ω2

)

so for frequencies lower than ωc, K(r) is negative and the behavior of the perturbation

is exponentially decaying, that is the mode, with ω < ωc is “trapped” inside the star,

as it is shown in Figure 5.1; when ω > ωc the mode propagates to the atmosphere and

rapidly loses energy.

It is easier to derive some other qualitative properties of the solutions writing K as

K(r) =ω2

c2

(

1 −ω2

l,+

ω2

)(

1 −ω2

l,−

ω2

)

(5.4)

In the interior of a star:

ωl,+ ≃ Sl ωl,− ≃ N

we then find again the trapping conditions described in Section 4.4.1. Differently from

Section 4.4.1, we can find quantitative and analytic accurate expressions both for the

eigenfrequencies and eigenfunctions, to do so there are still a few assumptions we need

to take.

56

5.1. ASYMPTOTIC THEORY OF STELLAR OSCILLATIONS

Figure 5.1: The acoustic cutoff frequency ωc in the outer parts of a 1.0 M⊙, Xc = 0.30 star.

5.1.1 The JWKB approximation

JWKB (Jeffreys, Wentzel, Kramers and Brillouin) is a general method to find approx-

imated solutions of wave equations in the so-called “semi-classical” limit. It is, for

example, useful to obtain a semi-classical solution of the Schrodinger equation, reliable

in the short-wavelength limit (see [37] and [21] for the mathematical foundation of this

method). The assumption taken is that the solution varies rapidly compared to the

equilibrium variables, K(r) in the case of Equation (5.1), that is:

ξ′r(r) = a(r)e i Ψ(r) (5.5)

where a(r) varies slowly with r compared to Ψ(r), so that the local radial wave number

kr:

kr =dΨ

dr

is large. Substituting the expression above in the equations and neglecting terms of zero

order in kr, we find a(r) = |K(r)|−1/4, we can obtain local sinusoidal or exponential

solutions if K(r) is, respectively, positive or negative. To find a solution that holds

throughout the star we need to apply boundary conditions and require that exponential

and oscillatory solutions connect continuously and smoothly where K(r) = 0, which

represents a turning point of the wave.

57


EQUATIONS

In fact since rt, defined so that K(rt) = 0, is a singular point for the latter local

solutions, we need to find solutions of equation (5.1) near rt in another way, that is

developing K(r) to the first order in (r − rt) in Eq. (5.1). The result, achieved in-

troducing Airy functions and their asymptotic behaviour and considering two turning

points1 provide, after some manipulations (see [28]) both an analytic expression of the

eigenfunctions and∫ r2

r2

K(r)1/2dr ≃ π(n − 1/2) (5.6)

which represents a sort of dispersion relation, since it relates spatial and temporal peri-

odicities (K depends on ω, see Eq. (5.4)); r1, r2 are the trapping points of the mode.

5.2 Asymptotic theory for p modes

Since in this thesis I will mainly deal with p modes, it is useful to rewrite the asymptotic

expressions for the eigenfunctions and the dispersion relation introduced in the previous

section, assuming the further hypothesis ω2 ≫ N2 , so that we can neglect the terms

depending on the buoyancy frequency, which, as shown in Section 4.4.1, drives mainly g

modes.

5.2.1 The Duvall law

Equation (5.6) becomes:

ω

∫ r2

r1

(

1 − ω2c

ω2− S2

l

ω2

)1/2dr

c≃ π(n − 1/2) (5.7)

It is possible to show that we can derive a simpler relation, which becomes then of

practical use,

(n + α(ω))π

ω≃ F

(

ω

L

)

=

∫ R

rt

(

1 − L2c2

ω2r2

)1/2dr

c(5.8)

if we assume, near the outer turning point S2l ≪ ω2 and, near the inner turning point,

ω2c ≪ ω2, and we denote2 by L2 l(l + 1). The function of frequency α(ω) is determined

by the characteristics of the star near the p-mode upper turning point, that is near the

surface.

The aim of the approximations introduced is to find an explicit dispersion relation,

ν = ν(n, l), where n and l are the radial order and angular degree of the mode. This can

1The inner and the surface turning point in the case of p modes2A more careful analysis shows that L = l + 1/2

58

5.2. ASYMPTOTIC THEORY FOR p MODES

be accomplished by using different methods, as reported in the next two subsections;

the general expression found is of the form:

νn,l = ∆ν + F (n, l)∆ν2

ν+ O(

∆ν3

ν2)

It is useful, as will appear in the following paragraphs, to introduce the so-called large

and small frequency separations to characterize p-mode spectra:

∆νn,l ≡ νn,l − νn−1,l (5.9)

δνn,l ≡ νn,l − νn−1,l+2 (5.10)

5.2.2 Tassoul’s expression for δn,l.

The starting point of both Tassoul’s and Gough’s approximated expression for the eigen-

frequencies is the set of differential equations written for adiabatic oscillations in the

Cowling approximation, hence neglecting the perturbations in the gravitational poten-

tial Φ′.

The approach followed by Tassoul (see [39]) is to take the JWKB analysis of the oscil-

lation equations to higher order. This leads to:

νnl ≃ (n +l

2+

1

4+ α)∆ν −

(

AL2 − δ) ∆ν2

νnl(5.11)

where

A =1

4π2∆ν

[

c(R)

R−∫ R

0

dc

dr

dr

r

]

(5.12)

and

∆ν =

(

2

∫ R

0

dr

c

)−1

(5.13)

represents the inverse of twice the acoustic radius of the star, that is the time it takes

for a sound wave to propagate from the center to the surface of the star.

The oscillation spectrum can be characterized by the “large” frequency separation:

∆νn,l = ∆ν (5.14)

and the “small” frequency separation:

δνnl ≡ νnl − νn−1,l+2 ≃ −(4l + 6)∆ν

4π2νnl

∫ R

0

dc

dr

dr

r(5.15)

where the term in c(R) has been neglected, since the value of the sound speed at the

surface is small compared to the other quantities.

59


EQUATIONS

1000 2000 3000 4000 5000 ν (µHz)

0

100

200

300

400

Pow

er(ν

) (

cm2 s-2

µHz-1

)

Figure 5.2: The comb-like structure of the oscillation spectrum of the sun, the ordinate is

normalized to show velocity power per frequency bin. (from [17])

5.2.3 Gough’s expression for δn,l

The dispersion relation derived by Gough in [23] differs from Equation (5.15) only in the

following expression:

A =1

4π2∆ν

[

c(R)

R−∫ Rt

rt

dc

dr

dr

r

]

(5.16)

where the integral is extended in the l dependent trapping region of the mode, since rt

and Rt are respectively the inner and outer turning point of the mode.

Introducing again

D0 ≃ − 1

4π2x0

∫ Rt

rt

dc

dr

dr

r(5.17)

where x0 is an average ofνn,l

∆ν , and recalling its relation to δνnl

δνnl ≃ D0(4l + 6)

it is straightforward to see that, in the range where this expression is applicable, com-

paring “small separations” computed for different l values could give us information on

a localized region of the stellar interior, namely the region between the two inner turning

60


points of the modes considered.

If we assume that D0 depends on l only by the l-dependence of the inner turning point,

D0(l) ≃ − 1

4π2x0

∫ Rt

rt(l)

dc

dr

dr

r

a difference in the position of the inner trapping point could affect D0 by a significant

amount especially for a low-degree p-mode, whose turning point is located near the

center of the star where the sound speed has sharp3 variations (in particular for evolved

stars) and the integral defining D0 is more sensitive, due to its 1/r dependence.

5.2.4 Deviation from asymptotic expressions

One of the problems addressed in this thesis is to determine the information it is possible

to get from the deviations from the simple asymptotic predictions, of some properties of

oscillation spectra.

Sharp variations in the equilibrium structure

If a sharp variation occurs in the equilibrium structure of a star, such as in the ionization

regions and near convective borders, the basic hypothesis that led to the simple asymp-

totic dispersion relation may fail, since the wavelength of the mode is no longer much

shorter than the scales of variations of the internal stratifications of the star. Anyway,

if we assume that these variations are small, we are able to predict their consequences

as deviations from the asymptotic eigenfrequencies.

These deviations have generally the form of an oscillatory signal in the frequencies,

with a period inversely proportional to the acoustic depth of the location where the

sharp change occurs. I will discuss more quantitatively this result, in the particular case

of variations of the first adiabatic exponent in helium ionization regions, in Section 6.2;

here I will instead present a simple example, that could also be found in [12], to illustrate

the physical reason for the presence of such oscillatory signals.

The equation satisfied asymptotically by radial p-mode eigenfunctions can be written

approximatively as (see [12]):

d2Y (x)

dx2+[

ω2 − V 2(x)]

Y (x) = 0 (5.18)

where x varies on a limited interval, say [0, xt], and V0 is a potential that we may assume

constant.

We also require the boundary conditions:

Y (0) = Y (xt) = 0

3The sound speed variation should not happen on a scale smaller than the wavelength of the mode,

otherwise the expression for the eigenfrequencies used to define D0 would be invalid.

61


EQUATIONS

The dispersion relation for the eigenfrequencies, as for a vibrating string is easily ob-

tained:

ω20 − V 2

0 =

(

nπ

xt

)2

We want to consider the effect of a discontinuity in the potential on the dispersion

Figure 5.3: Example of a potential constant in the interval [0, xt] (dotted line) and with a step

at x = α xt (continuous line)

relation. To do so we introduce a step function potential (shown in Fig. 5.3) so that:

V (x) =

Va 0 ≤ x ≤ αxt

Vb αxt ≤ x ≤ xt

(5.19)

Requiring boundary conditions, and matching conditions for Y (r) and its first derivative

at x = αxt, we obtain a dispersion relation:

tan [∆b αxt] = −∆b

∆atan [∆a(xt − αxt)]

where ∆2i = (ω2 − V 2

i ), i = a, b; this expression developed to the first order in the small

quantities δV 2 = V 2a − V 2

b , ω2 − ω20 and V 2

i /ω2 becomes:

δω ∼ δV 2

4xtω20

sin (2∆aαxt)

62


which, as shown in Fig. 5.4 represents an oscillatory perturbation on the frequencies of

oscillation.

In the case of acoustic modes of a star calculations are far more complicated, since the

Figure 5.4: Perturbed vs. unperturbed frequencies, for clarity a continuous line is drawn even

if the frequency of a bounded system, as a star, are a discrete set.

“acoustic potential” is not constant and depends on the degree of the mode, therefore a

detailed analysis (Section 4.3.1) or numerical calculations (Section 6.2.2) are needed.

Nonetheless we can still, at least qualitatively, apply the previous argument if the “acous-

tic potential” is sought as a superposition of a smooth component and a sharp local vari-

ation which will play the role, respectively, of the constant term and of the discontinuity

in the potential (5.19).

Other corrections

JWKB is not the only approximation taken while deriving the asymptotic expressions

of the p-mode eigenfrequencies, in fact both perturbations in the gravitational potential

and buoyancy frequency have been neglected.

It is not an easy task to include these corrections keeping a simple dispersion relation,

which is generally the aim of the asymptotic treatment.

The accuracy of Cowling’s approximation can be investigated numerically (see [7]),

comparing eigenfrequencies calculated for the full (4.29)-(4.31) and from the simplified

63


EQUATIONS

second order set of equations (4.45)-(4.46). The contribution of gravitational perturba-

tions can also be estimated analytically by introducing a correction term, Fφ, in Duvall’s

law, that becomes:π(n + α)

ω= F

(

ω

L

)

+1

ω2Fφ

(

ω

L

)

where F is expressed in Equation (5.8).

More careful analysis show that also a term coming from the neglected buoyancy force

should be included, in analogy with the treatment of gravitational potential perturba-

tions (see [7] for references and for an analytic expression of Fφ).

These additional terms will obviously affect frequency separations too, in particular

in the case of the “small” separation the correction terms may become comparable with

the separation itself, making questionable the validity of both explicit expressions (5.17)

and (5.15), especially when considering low degree modes.

64

Chapter 6

Results on computed models

In this chapter I will first describe (Section 6.1) how the simple asymptotic

expression for the p-mode frequencies, within its limits, can help constraining

the mass and age of a star, and how deviations from that simple expression

are of potential interest. A sharp variation in the equilibrium structure of

the star, such as near the border of a convective zone or in the regions of

ionization, gives rise to periodic signals in the low degree p-mode frequencies.

In Section 6.2 I will show the information we can obtain by fitting an analytic

expression of these signals on the large frequency separation ∆νn,l(ν). In

Section 6.3 I will finally present a qualitative relation between the onset of

degeneracy in the core of small mass stars and the characteristics of their

p-mode spectra. All the calculations have been performed using low-degree

acoustic modes, a data set that will be provided, with sufficient accuracy, by

the forthcoming observations from space.

6.1 Asteroseismic HR diagrams.

The asymptotic theory, since provides us with simple analytic expressions, represents a

powerful tool to infer or constain physical parameters of stars.

The simple asymptotic relation concerning the p-mode oscillation frequencies, that I will

recall once more:

νasym = (n +l

2+

1

4+ α)∆ν −

(

AL2 − δ) ∆ν2

νnl(6.1)

suggests that both ∆ν and δνnl, calculated from the oscillation frequencies, may be

related to peculiar behaviours of the equilibrium variables in the stars (mainly the sound

speed and its derivative).

65

CHAPTER 6. RESULTS ON COMPUTED MODELS

The combined plot1 of averaged ∆ν and δνnl was first suggested in [6] as a possible

means to infer the mass and age of a star by asteroseismic data. While in literature

the first attempts to use C-D diagrams to infer properties of stars have been applied

to main-sequence low-mass stars (see [14] and [35]), it also seems possible to obtain

qualitative information on the cores of higher mass stars (see [41] and [27]).

The first problem that has to be faced in order to generate a C-D diagram is the

choice of representative values of the separations for a star, in order to make a meaningful

comparison with the values calculated for other stars. In fact both the separations

are expected to be frequency-dependent, as a result of the approximations used in the

asymptotic theory that led to expression (6.1). The comparison between the separations

computed from the frequencies (e.g. 〈∆νnl〉), and their expression which directly involves

variables computed in stellar models (e.g. ∆ν ≡ (2∫ R0

drc )−1 ), if successful, ensures the

validity of the asymptotic expression, if unsuccessful could give a qualitative relation to

the onset of the feature that caused its departure from asymptotics.

6.1.1 Large frequency separation.

Figure 6.1: Large separation ∆νnl for 2.0 M⊙ star shown as a function of the radial order n

and calculated for l = 0, 1, 2 and 3.

1Also known as asteroseismic HR diagram or C-D digram

66

6.1. ASTEROSEISMIC HR DIAGRAMS.

One of the easiest properties to determine from a solar-type star oscillation spectrum

is the regular spacing between modes of same degree and consecutive order,

∆νn,l ≡ νn+1,l − νn,l (6.2)

where νn,l is the frequency of the mode of order n and degree l. The so-called large

separation is mainly a measurement of the sound travel time between the surface of the

star and the center, since substituting expression (6.2) in Eq. (6.1) we obtain:

∆νn,l =

[

2

∫ R

0

dr

c

]−1

=1

2τ0(6.3)

where τ0 is the acoustic radius of the star. The basic scaling of ∆ν, with the mass and

age of a star, is through homology with t−1dyn, which also represents an estimate of the

inverse of the travel time of a sound wave in the star. Combining c2 = Γ1 P/ρ, τ0 ≃ R/c

and the estimate of P derived from the hydrostatic equilibrium equation (3.3) we obtain:

∆ν ≃ (GM)1/2

R3/2= (Gρ)1/2 (6.4)

As we consider stars with higher mass, or a star with a given mass evolving on the main

sequence, we would expect, from the homology scaling in Eq. (6.4) a decrease in ∆ν, as

it is shown in following figures of computed C-D diagrams.

Departures from the asymptotic theory When computing the large frequency

separation from a theoretical frequency spectrum, rather than a constant value, we find

a function of the frequency with a periodic component (see Fig. 6.1).

This dependence can be explained taking into account variations in the equilibrium

structure of the star that occur on a distance scale smaller than the wavelength of the

mode considered. These variations could be treated by means of the functional analysis

outlined in Section 4.3.1, assuming that the frequencies that follow the simple relation

(6.1) are generated by a smooth fictitious equilibrium model.

I will give a more detailed treatment of these periodic signals appearing in the large

separation in Section 6.2, where I will show, not only how they can be removed, but also

the information it is possible to extract from them.

It is possible to estimate numerically the accuracy between the large separation

obtained from the frequencies and the one computed from the equilibrium variables

of the models considered (by means of Eq.(6.3)). The results of this comparison are

presented in Tables 6.1 and 6.2: the percentage difference is typically less than 3%, this

demonstrates the relevance of the asymptotic theory at the first order.

Moreover the agreement is better for high mass stars, where the periodic signals have a

quasi sinusoidal behaviour that is substantially removed when taking an average of the

67


n M/M⊙ Xc ∆ν l = 2 1/(2τ0) %

5 1.0 5.76877e-01 154.2 155.0 -2.1

15 1.0 2.99063e-01 134.6 137.4 -2.0

30 1.0 8.05402e-02 118.4 120.9 -2.1

20 1.5 5.61206e-01 82.8 84.0 -1.6

80 1.5 2.60829e-01 63.0 64.1 -1.8

120 1.5 1.40914e-01 57.6 58.7 -1.8

20 2.0 5.60073e-01 73.1 74.0 -1.3

80 2.0 2.59374e-01 49.3 49.7 -1.0

120 2.0 1.39867e-01 41.7 42.1 -0.9

Table 6.1: Large separation computed from the frequency spectra and from Equation (6.3). ∆ν

represents ∆ν(ν) averaged on high order modes (n ≥ 20); the last column shows the percentage

difference between the two expressions; n identifies the model considered (see Chapter 3)

separation (as in Table 6.1). In order to choose a value of ∆ν which is representative of

the star I considered two options:

• taking an average (〈∆ν〉) of ∆ν(ν) on the range of frequencies considered, excluding

the lowest modes where the basic assumptions of the asymptotic theory fail,

• choosing the large separation evaluated at the same, high, radial order n.

In the following calculations and diagrams I chose the first option, as from Tables 6.1

and 6.2 〈∆ν〉 is generally in better agreement with the large separation predicted from

the asymptotic theory.

The dependence of the large separation on the degree l, as expected, is found to be

negligible, in all calculations I arbitrarily chose l = 2.

6.1.2 Small frequency separation

The small frequency separation is defined by:

δνn,l ≡ νn,l − νn−1,l+2 (6.5)

substituting the asymptotic expression for νn,l from equation (6.1), we obtain:

δνn,l ≃(4l + 6)∆ν

(2π)2νn,l

∫ R

0

(

−1

r

dc

dr

)

dr (6.6)

The small frequency separation is sensitive to the core structure of the star, because

of the 1/r dependence of the integrand in equation (6.6). The “small separation” is

68


n M/M⊙ Xc ∆ν l = 2 1/(2τ0) %

5 1.0 5.76877e-01 155.5 155.0 -1.6

15 1.0 2.99063e-01 135.0 137.4 -1.8

30 1.0 8.05402e-02 117.8 120.9 -2.6

20 1.5 5.61206e-01 82.7 84.1 -1.6

80 1.5 2.60829e-01 63.0 64.1 -1.8

120 1.5 1.40914e-01 56.9 58.7 -3.1

20 2.0 5.60073e-01 72.6 74.0 -2.0

80 2.0 2.59374e-01 48.9 49.7 -1.8

120 2.0 1.39867e-01 41.3 42.1 -1.7

Table 6.2: As in Table 6.1, here ∆ν represents ∆ν(n) evaluated at n = 25.

generally positive, since, as shown in Fig. 3.5, dcdr is negative over most of the star;

nonetheless in evolved stars the decrease of the mean molecular weight µ in the central

regions, caused by nuclear reactions, gives a significant negative contribution to the

integral: during evolution δνn,l decreases.

I will introduce the notation I will use in the following paragraphs:

dl,l+2 = D0 = 〈δνn,l〉n /(4l + 6)

where dl,l+2 takes into account the l dependence of D0, not predicted by Eq. (6.1), and

〈δνn,l〉n is an average of the small separation on the frequency.

Departures from the asymptotic theory I found it very useful to compare the

small separation calculated from the frequencies (Eq. (6.5)), with its asymptotic ex-

pression computed numerically from the equilibrium variables of the models (Eq. (6.6)).

This comparison is presented in Fig. 6.3 for a 1 M⊙ star, and in the subsequent Figures

6.4 and 6.5 for 1.5 M⊙ and 2.0 M⊙ stars.

It is clear from the figures that the agreement between the behaviour of δνn,l and

the asymptotic expression, while good in the case of the model closer to our sun, i.e.

M = 1 M⊙ and Xc=0.3, is questionable in other models, where the corrections to the

asymptotic expression outlined in Section 5.2.4 may become significant.

The asymptotic expression by Tassoul (see Section 5.2.2) also predicts that δν0,2 =35 δν1,3, i.e. the l dependence of the small separation is only in the l(l + 1) term in

Equation 6.6. The calculations show that this is not generally the case, in particular: in

models with M < 1.2M⊙ there is a systematic disagreement between the two separations;

in models with M > 1.2M⊙ the difference changes irregularly during the evolution of the

69


Figure 6.2: D0 as a function of frequency in a 1.5 M⊙ ZAMS model.

star on the main sequence. As presented later in Section 6.1.3 and 6.1.4 these departures

from the classical asymptotic predictions contain valuable information.

As in the case of the large separation, for each model considered I choose as the

representative value of d02 and d13 their average on high order modes (n ≥ 15).

6.1.3 Diagrams for main sequence models without convective cores

(M < 1.2 M⊙)

The stellar models considered in this section have a mass range between 0.9 and 1.1 M⊙,

starting from a chemical homogeneous model on the ZAMS, up to evolved models with

central hydrogen abundance Xc ≃ 0.04. Figure 6.6 shows the obtained (∆ν0,D0) di-

agram: the continuous lines represent the evolution tracks, the dotted lines connect

models with the same central hydrogen abundance.

The difference in the large separation between the evolution tracks for models of

different mass and the dependence of D0 on the evolutionary state of the star make this

diagram valuable to constrain the age and mass of a star by knowing only its acoustic

spectrum.

The diagrams presented here have been calculated from frequency spectra of models

whose initial hydrogen abundance, metallicity and mixing-length parameter are given.

It is interesting to study the sensitivity of the evolution tracks to these parameters since

70


Figure 6.3: δνn,l/l(l + 1) estimated from the computed frequencies (symbols) and from the

asymptotic expression (dotted line).

71


Figure 6.4: As in Fig. 6.3, the difference between δνn,l calculated for l = 0− 2 and l = 1− 3 in

the evolved model becomes significant.

Figure 6.5: As in Figure 6.3.

72


Figure 6.6: (∆ν0, D0) diagram, D0 is calculated using l = 0, 2 frequencies. The continuous

lines are evolution tracks, the dotted lines are lines of constant central hydrogen abundance

(isopleths).

they are generally unknown, or only partly constrained by other observational data such

as effective temperature and luminosity. The results of a partial investigation are shown

in Figure 6.7, where, for the limited set of models presented in Table 6.3, I compared

the evolution tracks for models of given mass obtained by changing X and α.

As pointed out in the previous section, D0 calculated from l = 0, 2 and from l = 1, 32

modes generally differ. Figure 6.8 shows that the difference d13 − d02 increases during

evolution.

Since Tassoul’s expression does not predict this behaviour, the latter may be gen-

erated by the inaccuracies of the asymptotic expression outlined in Section 5.2.4. A

possible explanation could also be based on Gough’s expression for the small separation,

Eq. (5.17), where d02 and d13 are predicted to differ since they are calculated using

modes that sample differently and at different depths the stellar interior. This qualita-

2Hereafter I will denote D0 calculated from l = 0, 2 and l = 1, 3 modes respectively by d02 and d13

73


M/M⊙ X0 Y Z α

0.9 0.7 0.28 0.02 2.0

0.9 0.7 0.30 0.02 1.8

0.9 0.68 0.30 0.02 2.0

1.0 0.7 0.28 0.02 2.0

1.0 0.7 0.30 0.02 1.8

1.0 0.68 0.30 0.02 2.0

Table 6.3: Initial characteristics of the models whose seismic properties are compared in

Figure 6.7. The evolution of each model is sampled at Xc = 0.6, 0.3 and 0.1.

tive explanation, which certainly needs further quantitative investigations, is based on

the l-dependence of the inner trapping point of p modes: d02 is expected to sound stellar

interiors at greater depths than d13.

The sound speed near the center of the star, where the modes considered are trapped,

is decreasing with the distance to the center in ZAMS models and increasing for older

models (see Fig. 6.9). If we consider d13 and then compare it with d02 we have to extend

the domain of the integral (5.17) to an inner trapping point. This will initially generate a

positive contribution to d02, but later in evolution also a negative one which increases in

absolute value during the main sequence as the sound speed decreases with the distance

to the center. This argument could qualitatively explain the difference d13 − d02 during

the evolution.

6.1.4 Diagrams including models with convective cores (M ≥ 1.2 M⊙)

In models with convective cores the departures from the simple asymptotic expression

become substantial. If the large separation is still in good agreement with the analytic

expression, as underlined in Section 6.1.1, the small separation present changes in the

evolutionary tracks that are hardly explainable by means of the asymptotic theory.

The presence of a convective core, which is related to the increasing importance of the

CNO cycle in the core of stars with masses higher than the solar mass (see Section 3.2.4),

is responsible, as the star evolves, for a steep gradient in the mean molecular weight at

the edge of the core convective region. This sharp variation, which indeed influences also

other important variables such as the sound speed and the buoyancy frequency, affects

the modes of oscillation as described in Section 4.4.2. In the models with 1.2 and 1.5 M⊙

this feature is enhanced since the core convective region grows and shrinks during the

evolution (see Figure 6.10), generating a near-discontinuity in the chemical composition

and in the sound speed, as shown in Figure 6.11.

74


Figure 6.7: Evolutionary tracks computed from acoustic spectra generated by models of same

mass but slightly different input parameters (X0 and α as in Table 6.3). D0 is calculated using

l = 0, 2 modes.

As the small separation is an average of the first derivative of the sound speed,

weighted near the center of the star, we expect it to be sensitive to the presence of a

convective region.

Evolutionary tracks on a (∆ν0,D0) diagram have been computed for an extended set

of models, up to 2.0 M⊙ stars, in Figure 6.12 it is shown a diagram (∆ν0, d13).

I will present here, as a list, the main characteristics of these diagrams, emphasizing the

differences with the results obtained in the previous section:

• From Figure 6.12 it is clear that the separation in the evolutionary tracks according

to mass is reduced as models with higher mass are considered, this naturally would

decrease the effectiveness of such a diagram when constraining the mass of a star.

• As done in the previous section, it is useful to compute the asteroseismic HR

diagram both considering d02 and d13. Figure 6.13 shows how the irregularities

75


Figure 6.8: As in Figure 6.6, D0 is calculated both with l = 0, 2 (d02, continuous lines) and

l = 1, 3 (d13, dashed lines) frequencies.

in d02 are enhanced in respect to those in d13, in the 1.5 and 2.0 M⊙ models. A

general argument to explain qualitatively this feature would be, once more, that

d02 is calculated with modes that penetrate more deeply in the stellar interior and

then may sound regions of discontinuities (convective edge) ignored by d13. Figure

6.14 shows the rapid change in the behaviour of d02 in two 1.5 M⊙ models at two

close evolutionary states.

6.1.5 Solar data

In this section I show the large and small frequency separations calculated from low

degree solar frequencies determined by BiSON3 network (see [17]). Figures 6.15 and 6.16

confirm the behaviour predicted by the artificial spectra. I also reported the averaged

values of the separations on a C-D diagram (Figure 6.17). The mass and evolutionary

3Birmingham Solar Oscillation Network

76


Figure 6.9: The sound speed near the center of a 1 M⊙ star during the main sequence. The

increase of the mean molecular weight during hydrogen burning is responsible for the decrease

of the sound speed towards the center.

stage of the sun can be successfully estimated from d02. The value for d13 is indeed

higher than d02 as expected and described in the previous sections, nontheless it deviates

from the evolutionary track. More careful calculations, using models calibrated on solar

observations could explain the discrepancy.

77


Figure 6.10: The extension of the convective core during the main sequence evolution of 1.2,

1.5 and 2.0 M⊙ stars.

78


Figure 6.11: The behaviour of the sound speed near the center in 1.2, 1.5 and 2.0 M⊙ stars

at different evolutionary stages during the main sequence. The variation at the edge of the

convective core is nearly a discontinuity for the 1.2 and 1.5 M⊙ models.

79


Figure 6.12: As in Figure 6.6, D0 is calculated both with l = 0, 2 (d02, continuous lines) and

l = 1, 3 (d13, dashed lines) frequencies.

6.2 Seismic analysis of helium ionization zones.

Localized variations in the structure of the stars, as the one occurring at the base of

the convective envelope or in the region of the second ionization of helium, create a

characteristic signal in the frequencies of oscillation. The properties of such a signal are

related to the location and thermodynamic properties of the star at the layer where the

sharp variation occurs.

The signal generated at the base of the convective envelope has been identified in

solar frequency spectra and has provided valuable information on convective overshoot as

reported in [28] and [12]; the analysis of this signal has also been extended to simulated

spectra of solar-type stars ([13]). I will here concentrate on the signal generated by the

sharp variation of the first adiabatic exponent in the helium ionization zones.

By using the approach presented in Section 4.3 (see [30] and [29]) it is possible to

determine how the characteristics of the zone of ionization, considered as a localized per-

turbation on an otherwise “smooth” structure, are related to the parameters appearing

80

6.2. SEISMIC ANALYSIS OF HELIUM IONIZATION ZONES.

Figure 6.13: As in Figure 6.13, it is also shown the behaviour of d02 which generally presents

more or enhanced irregularities than d13 in the 1.2 and 1.5 M⊙ models.

Figure 6.14: d02 and d13 calculated as a function of frequency in two 1.5 M⊙ models, one slightly

more evolved than the other (compare the central hydrogen abundances reported in the figures).

81


Figure 6.15: Large frequency separation ∆νn,l determined from observed solar p-mode frequen-

cies. The upper figure shows the degree of each mode while the lower shows the error bars in the

separation (error bars in mode frequencies are negligible).

82


Figure 6.16: δνn,l/l(l + 1) calculated from BiSON solar data. The figure on the left shows the

values of d02 and d13 averaged on a range of modes with order between 18 and 24; the figure on

the right also includes error bars.

in the periodic signal.

In order to write an explicit expression of the changes in the frequencies generated by

a small variation of the equilibrium model, the Cowling approximation is applied to Eq.

(4.43) (see [29] for solar p-modes), therefore the gravitational potential is neglected in

δF . It follows that if we change the equilibrium configuration by small amounts δρ and

δc2, the eigenfrequency will change from the initial value ω0 to ω = ω0 + δω, according

to:

δω2 =δI

I1(6.7)

where

I1 ∼ 1

2τtξ

20

and

δI ∼∫ τt

0

[

(

δB1 +dδB0

dτ

)

ξ2r + δB2

dξ2r

dτ+ δB3

d2ξ2r

dτ2

]

dτ (6.8)

Here τt is the total acoustic depth of the star, ξ2r the normalized radial eigenfunction of

amplitude ξ20 , and δBi are functions of the equilibrium variables, linear in the perturba-

tions of the model (i.e. δp and δc2).

The explicit calculation, in the case of sharp variations in Γ1, is carried out by

Monteiro and Thompson [30], the main assumptions being:

• The asymptotic expression for the radial eigenfunction ξr,asym = ξ0 cos[

ω∫ τ0 (1 − ∆)1/2 + φ

]

is used as ξr in Eq. (6.8), valid in a region well inside the turning points

83


Figure 6.17: The location of the sun in a C-D diagram. The mass and evolutionary stage of the

sun is easily inferred, d03 is higher than d02 as predicted, a more careful analysis using “standard

solar models” to compute the evolutionary tracks could explain the difference between computed

and observed d13.

84


Figure 6.18: The thermodynamical coefficient Γ1 versus acoustic depth in a 1.0 M⊙ star, Xc =

0.30. The second helium ionization zone is approximatively located between 600 and 800 seconds.

• It is assumed that the variation of the sound speed, in the ionization zone, is mainly

determined by the changes in Γ1, this means that the terms in δρ and δp appearing

in the explicit expression of δB0, δB1, δB2 and δB3 are neglected.

• Since the analysis is restricted to localized variations, it is assumed that the model

differences are zero everywhere else.

• δΓ1 is modeled as follows:

δΓ1

Γ1= δd

(

1 + τ−τdβ

)

τd − (1 − α)β ≤ τ ≤ τd(

1 − τ−τdβ

)

τd ≤ τ ≤ τd + (1 + α)β

0 elsewhere

where 2β represents the width of the ionization zone, and α represents the asym-

metry of the bump.

The oscillatory signal finally becomes, taking into account the dominant contributions

(in terms of powers of ω and derivatives of the reference structure):

δωp ∼ a01 − 2∆

3

(1 − ∆)2

sin2[

βω(1 − ∆)1/2]

βωcos [2(ωτd + φ0)] (6.9)

85


where τd =∫ τd0 (1 − ∆)1/2 dτ , ∆ =

S2

lω2 and all quantities are evaluated at τ=τd.

The amplitude of the signal, a0, is given by

a0 = −3δd

2τt(6.10)

where τt is the total acoustic radius of the star and δd is proportional to the change in

Γ1 in the region where ionization takes place.

Far from the inner turning point ω2 ≫ S2l , thus we may assume ∆ ≃ 0 and the signal

becomes simply:

δωp(ω) ∼ a0sin2 [βω]

βωcos [2(ωτd + φ0)] (6.11)

6.2.1 Expected signal on first order differences.

One of the approximations used in the asymptotic theory is the slow variation of the

equilibrium variables relative to the eigenfunction describing the propagating wave4. We

may then assume that the frequency generated by the smooth model could be associated

with the one predicted by the asymptotic expression. Thus

ν ≃ νasym + δν

where

νasym = (n +l

2+

1

4+ α)∆ν −

(

AL2 − δ) ∆ν2

νnl

and 2πδν = δωp(ω2π ) (see Eq. (6.11)).

Since the amplitude of the oscillatory signal δν may be regarded as constant5 , we can

easily find the expected expression for the first order differences, ∆νn,l ≡ νn+1,l − νn,l,

expressing νn+1 ≃ νn + ∆ν in δν:

∆ν ≃ ∆νasym + 2 sin (2π∆ντd)δν (6.12)

The assumptions taken to obtain the former expression are questionable, as approxima-

tions other than the smooth behaviour of the equilibrium variable are used to obtain the

asymptotic expression and might be of the same order of magnitude as δν. Nonethe-

less in the subsequent analysis and fitting procedure, as in [27], the aim is not to verify

directly Eq.( 6.12), but to try to isolate the oscillatory signal in the first order differences.

The amplitude factor 2 sin (2π∆ντd) in Eq. (6.12) could enhance the signal if 1/3 <

τd∆ν < 5/3 or reduce its amplitude otherwise, this will naturally influence the detection

of the signal in different models.

4The JWKB approximation is used.5As it varies more slowly than cos [2(ωτd + φ0)].

86


6.2.2 Using numerical kernels.

The analytic expression for the periodic signal illustrated in Section 6.2 is a powerful

tool to get information on the second helium ionization zone, by performing a fit directly

on frequencies, as will be shown in Section 6.2.3. Nonetheless care must be taken while

applying Eq. (6.11) on an extended set of models with different masses and different

behaviours of δΓ1

Γ1.

A tool to check whether the assumptions taken in deriving Eq. (6.11) (especially the

way the “bump” in Γ1 is characterized) are still valid, is given by the general expression

Eq. (4.44), that we recall here, this time expressing the changes in the equilibrium

structure by Γ1 and u, where u = p/ρ:

δωnl

ωnl=

∫ R

0

[

KnlΓ1,ρ(r)

δΓ1

Γ1(r) + Knl

u,Γ1(r)

δru

u(r)]

]

dr (6.13)

where δω = ω − ωsmooth, δΓ1 = Γ1 − Γ1smooth and the “smooth” variables are referred

to a “smooth” models without bumps in Γ1.

As we are looking at the signal in the frequencies caused by sharp variations in Γ1,

and our aim is not to compare frequencies of two different models, we may neglect the

second term in the right member of Eq. (6.13). The periodic component that we expect

to clearly identify in δω can be compared with Eq. (6.11). From this comparison we

might be able to:

• Identify and distinguish, when present, signals coming from both helium ionization

zones (as shown in section 6.2.3)

• Determine by inversion δΓ1(r) and hopefully the parameters that might be sensitive

to the envelope helium abundance and EOS.

6.2.3 Results

In this section I will present how expression (6.11) can be successfully fitted on the large

separation determined from frequency spectra of moderate mass models (M ≤ 1.4M⊙)

whereas it needs to be generalized to include a second minimum of Γ1 when considering

stellar models with (M > 1.4M⊙).

The characteristics of the models considered in the following paragraphs are reported

in the tables at the end of Chapter 3. Figure 6.19 shows a typical behaviour of the large

frequency separation as function of frequency in a 1.2 M⊙ star. The are mainly two

parameters that drive the amplitude of the oscillatory signal, and make it detectable on

∆ν:

1. The amplitude a0 in Eq. (6.11), which depends on the ratio of the depth of the

ionization zone δd to the total acoustic radius of the star τt.

87


Figure 6.19: The large frequency separation in a 1.2 M⊙ star, Xc = 0.69 as a function of

frequency. The amplitude of the oscillatory signal decreases with ν, as predicted by Eq. (6.11).

2. The scaling factor 2 sin (2π∆ντd), that has to be taken into account since we are

looking at the oscillatory signal on the large frequency separation.

The combined behaviour of these factors determine the amplitude of the signal we are

aiming to detect.

The fitting procedure I apply to isolate and determine the characteristics of the

periodic signal is described in Appendix A.

Main sequence low mass stars (M ≤ 1.4M⊙).

The results of the fitting on main sequence stars with masses lower than 1.4 M⊙ are

shown in Table 6.4 and 6.5. The increase of the acoustic depth of the second helium

ionization zone with the age of the star, see Fig. 6.20, appears in the fitted values

too, which are in good agreement with the acoustic depth calculated directly from the

models.

Within this range of masses the smooth part seems to depend on l, the resulting

fit has been performed separately for each l value, the results of the separate6 fits are

compared in Table 6.4.

6An improvement on the fitting procedure could be to calculate the residuals for different l values

88


Figure 6.20: behaviour of Γ1 in the region of second helium ionization.

M⊙ Xc τmod(s) τfit(s) τstar(s) δ’ δ′/τ 2 sin (2π∆ντd)

l = 1 l = 2 l = 3

0.9 0.69 490 491 504 511 2688 0.009 3.35 1.06

0.9 0.45 560 580 585 589 2943 0.011 3.74 1.11

0.9 0.24 645 688 662 605 3319 0.017 5.12 1.13

Table 6.4: Results for 0.9 M⊙ star.

Plots of the fitted signal are reproduced in Fig. 6.21.

As explained in Section 6.2.2 it is possible to compare the expression of the signal

isolated from the frequencies with the one determined using numerical kernels. I applied

and then perform the non linear fit on all the residuals at once.

89


Figure 6.21: The oscillatory and smooth component fitted on the large frequency separation

∆νn,l for l = 2 in 0.9 M⊙ star (Xc = 0.69, 0.45, 0.24) and 1.0 M⊙, Xc = 0.30.

this comparison to a subset of the models considered, namely the ZAMS chemically

homogeneous models for a range of masses between 1.0 and 1.4 M⊙.

In doing so, for each model considered, I proceed as follows:

• I consider, for each set of p modes, the kernels7 in Eq.(6.13): they are oscillatory

functions of r, the distance from the center of the star. See Fig. 6.22 as an example

of Kn,lΓ1,u(r) in 1.0 M⊙, the kernel is relative to a p mode with n = 15, l = 2.

• The other function appearing in the integrand is δΓΓ1

(r). In order to obtain δΓ1(r) =

Γ1,smooth(r) − Γ1(r) a fictitious, “smooth” model, with suppressed second helium

ionization zone is needed. An example of δΓΓ1

(r). is shown in Fig. 6.23.

7kindly provided by Dr. M.P. Di Mauro

90


M⊙ Xc τmod(s) τfit(s) τstar(s) δ’ δ′/τ 2 sin (2π∆ντd)

1.0 0.69 560 585 3023 0.018 5.9 1.07

1.0 0.42 630 650 3409 0.028 8.2 1.08

1.0 0.30 680 690 3639 0.031 8.5 1.08

1.0 0.21 720 728 3842 0.035 9.1 1.10

1.2 0.69 630 660 3980 0.07 17.6 0.94

1.2 0.53 690 694 4311 0.085 19.7 0.94

1.2 0.39 750 755 4679 0.10 21.4 0.94

1.2 0.19 930 912 5421 0.10 18.4 0.98

Table 6.5: Results for 1.0 and 1.2 M⊙ stars.

Figure 6.22: The kernel K15,2Γ1,u(r) in 1.0 M⊙, Xc = 0.70

• Once I have obtained δωnlωnl

and δν(ν) for each mode considered, I easily isolated its

periodic component shown in Figure 6.24.

• Since the fit is performed on the “large” frequency separation, it is useful to com-

pute the periodic signal expected on ∆ν itself, δ∆ν(ν) as δν(n+1, l)−δν(n, l) and

to compare it with the one derived by the fitting procedure.

The signal fitted directly on the large separation and the one determined using numerical

kernels agree if shifted in phase (φ′ ≃ 200µHz), as presented in Figure 6.25 for a ZAMS

1.2 M⊙ star. This agreement proves the validity of expression 6.11 as a powerful tool

to determine the characteristics of the first helium ionization zone in this mass range,

despite the simple way the bump in Γ1 is modeled. Nonetheless the phase difference φ′

91


Figure 6.23: The behaviour of Γ1 in the “real” model (continuous line) and in the fictitious

smooth model (dashed line). δΓ1

Γ1

= (Γ1,smooth − Γ1(r))/Γ1(r) computed using the functions

shown in the left figure.

Figure 6.24: The periodic signal isolated, by using the differential method, in the frequencies

of oscillation in 1.0 M⊙, Xc = 0.70.

arbitrarily introduced needs to be explained by further investigations.

Main sequence stars (M > 1.4M⊙).

The behaviour of Γ1 in near-surface layers in stars with masses higher than 1.4 M⊙ is

shown in Fig. 6.26.

In the mass range between 1.3 and 1.5 M⊙, both the smooth and periodic signal on

the large frequency separation change in shape and period, see Fig. 6.27. A first naive

attempt to apply the same fitting as in models of stars with lower masses gives no result,

but if we take a closer look at the behaviour of Γ1 = Γ1(r) we might get some hints. As

92


Figure 6.25: It is shown the comparison between the signal fitted on the large separation (dotted

line) and the one determined using numerical kernels (crosses). The period and amplitude of the

signal agree, but a shift in phase was introduced (φ′ ≃ 200µHz).

Figure 6.26: behaviour of Γ1 near the surface of a 1.5 and 2.0 M⊙ star.

we consider masses higher than 1.4 M⊙ another sharp variation in Γ1 appears near the

surface of the star, see Fig. 6.26, this other bump is related to the first helium ionization

zone.

It is clear that this new feature could no longer be modeled by a single “bump”, as

assumed when deriving the analytic expression Eq.(6.11).

To confirm this argument I compare the periodic signal in the large separation with

93


Figure 6.27: Large frequency separation for ZAMS stars with 1.30, 1.40, 1.47, 1.50 M⊙.

the one predicted using numerical kernels. The latter can be computed taking into ac-

count only the second or both first and second helium ionization zones. The result of

this comparison is encouraging: as shown in Fig. 6.29, if we model δΓ1

Γ1(r) considering

also as a deviation from a smooth model the dip due to the first helium ionization, δν(ν)

is consistent with the behaviour of the signal determined from the frequencies (Figure

6.28). The resulting signal, shown in Figure 6.29, displays an amplitude increasing with

frequency and hence deviates qualitatively from equation (6.11) where the amplitude

decreases with frequency. It seems likely, however, that the analysis leading to equa-

tion (6.11) can be generalized to take into account the effect of the dip at the first helium

ionization zone.

Taking a closer look at Kn,lΓ1,u(r) δrΓ1(r)/Γ1(r) for various n values we can infer in

which frequency range the contribution from the first helium ionization zone becomes

dominant. Looking at Figure 6.30 we expect, for a 2.0 M⊙ ZAMS model, the signal

from the first helium ionization zone to be dominant for modes with frequencies above

1.8 mHz.

94

6.3. ONSET OF DEGENERACY

Figure 6.28: Large frequency separation and Γ1 in a 2.0 M⊙ star, the latter shown as a function

of acoustic depth τ , compare this behaviour with Fig. 6.20.

Figure 6.29: The periodic perturbation in the frequencies of oscillation computed by using

numerical kernels in a 2.0 M⊙ ZAMS star.

6.3 Onset of degeneracy in the core of hydrogen shell burn-

ing phase low mass stars.

Stars with masses lower than 1.2 M⊙, at a certain point in evolution, show an increase

of the sound speed close to the center8, this could be interpreted as a consequence of

8Other than the increase due to temperature

95


Figure 6.30: The behaviour of Kn,lΓ1,uδΓ1(r) for modes corresponding approximatively to ν =

1100, 1500, 1900 µHz (2.0 M⊙ ZAMS star). The inner and outer maxima are due respectively

to second and first helium ionization zones.

M⊙ Xc τmod(s) τstar(s) 2 sin (2π∆ντd)

1.5 0.70 330 5405 0.37

2.0 0.70 370 5796 0.38

Table 6.6: The characteristics of the second helium ionization zone in M > 1.4 M⊙ stars.

the onset of degeneracy in the core. This feature of the sound speed may affect the

frequency separations of deeply internally trapped acoustic waves, and may be revealed

comparing small separations δνn,l calculated for different l values.

The set of models considered is described in Tables 6.7, 6.8 and 6.9, and their evo-

lution tracks are reported in Figure 6.31. In the following paragraphs I will refer to a

model of the set by its number n reported in the tables.

6.3.1 Sound speed in a partly degenerate stellar core.

A first simple approach, relevant to seismology, to see whether the gas in the stellar core

is partly degenerate is to compare the behaviour of the sound speed defined as

c2 ≡ Γ1p

ρ

with the sound speed calculated under the assumption of an ideal gas equation of state,

that is

c2 ≃ Γ1kbT

µmu

where µ is the mean molecular weight defined as

µ−1 =∑

i

XiZi + 1

Ai

96


Figure 6.31: HR diagram showing evolution tracks of a 1.0 (continuous line), 1.1 (dotted line)

and 0.9 M⊙ (dashed line) star. Each mark corresponds to a model listed in Tables 6.7, 6.8 and

6.9.

Xi being the mass fractions of the elements constituting the stellar gas.

In the plots we used an approximated9 expression for µ, µ ≃ 2X + 34Y + 1

2Z. As we

can see from Figure 6.32, at that point in evolution the core of the star is isothermal

and the mean molecular weight µ is constant in the region where hydrogen is exhausted

(there Y ≃ 1 − Z).

If we assume the ideal gas equation of state and a given temperature, both ions and

electrons contribute to the gas pressure by an amount proportional to their number per

unit volume,

Pgas = Pions + Pe =kbTρ

mu(µ−1

ions + µ−1e )

Taking into account the contributions of electrons and ions, treated as perfect gas par-

ticles, we underestimate the total pressure near the center of the star (see Figure 6.35).

At a temperature of 107K and density of 103g/cm3, anyway, we should start treat-

ing the thermodynamical properties of the gas with the general10 quantum statistics

expressions. There is no clean demarcation line on the T − ρ plane that distinguishes

degenerate from nondegenerate electrons, a precise idea of how the transition takes place

9This approximation assumes that the gas is fully ionized.10In the range of temperatures considered we can treat the gas non-relativistically.

97


Input Parameters

M/M⊙ X Z α

0.9 0.70 0.02 1.83

Output Parameters

n age (Gyr) R (cm) Teff (K) L/L⊙ Xc

15 7.41434 6.15316e+10 5464.9 0.626 3.20206e-01

35 12.34170 7.06132e+10 5577.3 0.894 4.84225e-02

45 13.50900 7.43355e+10 5584.7 0.996 3.65104e-03

50 14.05240 7.66087e+10 5585.0 1.058 5.80114e-04

65 15.32710 8.42339e+10 5555.1 1.252 1.00000e-10

70 15.69230 8.74437e+10 5529.8 1.325 1.00000e-10

75 15.89820 8.96061e+10 5508.9 1.370 1.00000e-10

80 16.11940 9.23261e+10 5478.1 1.422 1.00000e-10

Table 6.7:

Input Parameters

M/M⊙ X Z α

1.0 0.70 0.02 1.83

Output Parameters


15 5.03932 7.05897e+10 5783.6 1.033 2.99063e-01

35 7.57668 7.84775e+10 5810.0 1.300 4.06785e-02

45 8.29154 8.18171e+10 5806.4 1.410 3.98473e-03

50 8.64637 8.38492e+10 5802.9 1.477 9.76743e-04

65 9.56634 9.07899e+10 5774.8 1.698 2.36610e-06

70 9.81577 9.32949e+10 5758.7 1.774 1.00000e-10

75 10.03780 9.58659e+10 5739.1 1.847 1.00000e-10

80 10.23430 9.84895e+10 5715.7 1.918 1.00000e-10

Table 6.8:

98


Input Parameters

M/M⊙ X Z α

1.1 0.70 0.02 1.83

Output Parameters


15 3.26230 7.98182e+10 6037.6 1.569 2.94539e-01

35 4.62219 8.60241e+10 6021.6 1.803 4.02463e-02

45 5.09541 8.90186e+10 6013.6 1.920 5.41272e-03

50 5.33945 9.08264e+10 6009.0 1.993 1.57768e-03

65 6.01845 9.71219e+10 5984.1 2.241 1.85682e-05

70 6.21727 9.94507e+10 5971.4 2.330 2.75793e-06

75 6.40077 1.01874e+11 5956.2 2.420 1.00000e-10

80 6.56911 1.04383e+11 5938.4 2.511 1.00000e-10

Table 6.9:

requires explicit evaluation of Fermi-Dirac integrals. A first criterion to estimate when

this transition occurs (see [25]) is to compare the magnitude of the Fermi energy EF of

a completely degenerate electron gas and kbT .

In numbers EF ≃ kbT becomes

ρ

µe≃ 6.0 × 10−9T 3/2g cm−3

Applying this criterion to the electron gas near the center of the stars we are con-

sidering, we can say we are dealing with a partly degenerate electron gas. As shown

in Figure 6.34, during their evolution on and after the main sequence, low-mass stars

(in this case 0.9, 1.0 and 1.1 M⊙) move towards an isothermal degenerate core. The

assumption of a partly degenerate core may be strengthen by estimating the pressure

due to completely11 degenerate electrons

Pdeg = 1.004 × 1013(ρ

µe)5/3 dyne cm−2

As we can see from Figure 6.35, Pdeg has the same order of magnitude as the contributions

to pressure of ions and electrons treated as ideal gas particles.

11This approximation is questionable for a gas in a hot stellar core, where pressure still depends

strongly on temperature.

99


Figure 6.32: Temperature and mean molecular weight in the core of a 1 M⊙ star, the continuous,

dashed and dotted lines correspond respectively to n = 70, 45 and 35.

6.3.2 Expected consequences on “small separations” and possible as-

teroseismic inference on the presence of degeneracy in the stellar

core.

While Tassoul’s approximated expression for the eigenfrequencies predicts D0 being in-

dependent from l, at least by an amount of the same order of magnitude as D0 itself, we

see that, at a certain point in evolution, D0(0)12 and D0(1) differ one from the other.

On a plot D0(0) vs. D0(1) (Figure 6.36) at a certain age of the star considered (1 M⊙),

the points on the plot start to leave significantly the expected “x=y” line.

This behaviour is also shown for 0.9 and 1.1 M⊙ (see Figure 6.37), the age of the

“turning point” on the plot depends on the mass of the star but not on the radial order

n that we can, partly arbitrarily13, choose to calculate δνn,l and then D0(l).

The evolution parameter at which the turn occurs, seems to correspond to the appearance

of the increase of the sound speed near the center of the star, as we can see from Figure

6.38.

We may try to interpret this behaviour using Gough’s expression for D0, and its

dependence on the position of the inner trapping point rt = rt(l), that should also

result in different behaviours of D0(0) and D0(1) calculated directly from the computed

frequencies.

If we assume that D0(0) has a deeper sensitivity to the behaviour of the sound speed in

the core than D0(1), the sudden change in D0(0) vs. D0(1) could be related to the onset

of degeneracy, that causes the increase in the sound speed close to the center.

This feature in the sound speed it is not seen for a 1.2 M⊙ star, mainly because at

12(4l + 6)D0(l) ≃ δνnl13All the separations in these plots are calculated using n = 25

100


Figure 6.33: Sound speed in the core of a 1 M⊙ star, n = 75, the dotted line corresponds to

the ideal gas-approximated expression of the sound speed.

that mass stars start to develop a convective core which causes sharp variations in the

sound speed that invalidate the previous assumptions that led to the expression for the

eigenfrequencies (see Figure 6.39).

101


Figure 6.34: Temperature and density in the core of a 0.9, 1.0 and 1.1 M⊙ star, plotted for

different points in evolution. The background lines show the approximated demarcation line (as

defined in the text) between degenerate and non-degenerate gas, in the non-relativistic domain.

The continuous and dotted line correspond respectively to a gas with X=0.7 and X=0, the latter

is the relevant demarcation line the evolved stars considered, since hydrogen is exhausted in their

core (see Tables 6.7 and following).

102


Figure 6.35: Different contributions to the gas pressure in the core of a 1 M⊙ star, n=75, Pions

(dashed line), Pe (dotted line), Pdeg (dash-dot line), total gas pressure (continuous line).

Figure 6.36: D0(0) vs. D0(1) at different age for a 1 M⊙ star.

103


Figure 6.37: D0(0) vs. D0(1) at different age for a 1 M⊙ (dotted line), 1.1 M⊙ (continuous

line),0.9 M⊙(dotted line) star.

104


Figure 6.38: Sound speed in the core of 0.9, 1.0 and 1.1 M⊙ star, the lines correspond to

n=15,35,50, 65 and 80 (see Tables 6.7 and following). As we can see the described feature of

the sound speed appears (dotted lines) at different age , this parameters agree with the ones

corresponding to the turning points in the D0(0) vs. D0(1) plot.

105


Figure 6.39: D0(0)vs.D0(1) at different age for a 1.2 M⊙ star, we cannot recognize any smooth

behaviour in this plot.

106

Chapter 7

Conclusions and prospects

The forthcoming space missions (MOST, MONS, COROT and EDDINGTON) in the

next few years will provide accurate seismic data on solar-type stars. The limitations

imposed by observing from Earth will be finally overcome. The challenge will then

be how to infer the properties of the stellar structure from the oscillation frequencies

determined by observations. In order to achieve results we need to refine techniques

already used in helioseismology and develop new inversion procedures.

The work done in this thesis has been conceived under this perspective.

The C-D diagrams have been calculated for main sequence low- and moderate-mass

stars (0.9 M⊙ ≤ M ≤ 2.0 M⊙) and their relevance as a tool to constrain mass and age

of a star is described. It is also shown how the comparison between evolution tracks

calculated with l = 0, 2 and l = 1, 3 modes can qualitatively reveal details of the stellar

interior, such as the presence of a convective core or the onset of degeneracy in evolved

low-mass stars.

The behaviour of the large frequency separation, as a function of the mode frequency,

shows a periodic signal related to a local variation of the equilibrium structure of the

star in the helium ionization regions. A procedure to fit this signal and to infer the

properties of the second helium ionization zone is presented. In models of higher masses

the change in the characteristics of the periodic signal can be related to the influence of

the first helium ionization region.

A further and in-depth study of the techniques presented in this work promises to

provide a method based on seismic properties to constrain the envelope helium abun-

dance in low- and moderate-mass stars.

The failures of the asymptotic expression could be used to qualitatively suggest the

presence of a convective core. It is clear that more careful asymptotic relations need to be

derived in order to infer quantitatively the extent and properties of the inner convection

zone. The accuracy of the asymptotic theory is satisfying while considering the large

107

CHAPTER 7. CONCLUSIONS AND PROSPECTS

averaged separation, but it allows only qualitative considerations in the case of the small

frequency separation.

It is therefore clear that a further refinement of inversion methods applicable to a

wide range of stars has a primary importance in the development of asteroseismology,

as it will provide us with details of the inner structure of the stars from high quality

seismic data that will be gathered from space observations.

108

Chapter 8

Riassunto in italiano

Questo capitolo e un riassunto in italiano degli argomenti trattati nella tesi. Esso

presenta solo parzialmente la teoria utilizzata e i risultati ottenuti, pertanto per

approfondimenti e precisazioni si rimanda ai capitoli precedenti.

8.1 Introduzione

Lo studio della struttura ed evoluzione stellare ha un ruolo centrale in astrofisica. La

determinazione delle distanze, dell’eta e della composizione chimica delle stelle risulta

fondamentale per la comprensione della struttura e della storia della nostra galassia.

La possibilita di verificare le proprieta della materia stellare in condizioni difficilmente

riproducibili in laboratorio, inoltre, permette di sottoporre a verifica i diversi aspetti

della fisica utilizzata per calcolare i modelli.

Se la determinazione della luminosita e temperatura effettiva di una stella ne vin-

colano la struttura utilizzando informazioni sullo stato termodinamico degli strati es-

terni, l’astrosismologia, ovvero lo studio delle proprieta sismiche di una stella, promette

di fornire informazioni direttamente sulle regioni interne. L’astrosismologia si propone

dunque come promettente strumento utile a sottoporre a dettagliata verifica i modelli di

struttura ed evoluzione stellare. Le frequenze dei modi propri di oscillazione di una stella,

principalmente onde di pressione per stelle di tipo solare, grazie ad osservazioni da satel-

liti saranno determinati nei prossimi anni con un’accuratezza sufficiente da permettere

di vincolare massa, eta e caratteristiche degli interni stellari.

La tesi riguarda lo studio e l’estensione di tecniche che permetteranno di caratteriz-

zare le proprieta di stelle di tipo solare dai loro spettri di oscillazione.

109

CHAPTER 8. RIASSUNTO IN ITALIANO

8.2 Equazioni di struttura ed evoluzione stellare

Le equazioni utilizzate per modellare la struttura e l’evoluzione stellare coinvolgono

una sorprendente quantita di aspetti della fisica. Il sistema di equazioni differenziali e

descritto nella Sezione 3.1:

dP

dr=

1

4πρr2(8.1)

dP

dm= − Gm

4πr4(8.2)

dL

dm= ǫ − [

d

dt

u

ρ− P

ρ2

dρ

dt] (8.3)

dT

dm=

− 3κ4 a cT 3

L16π2r4 trasporto di energia radiativo

orΓ2−1Γ2

TP

dPdm trasporto di energia convettivo

(8.4)

dXi

dt= rXi , i = 1..I (8.5)

dove Γ2 e il secondo esponente adiabatico definito come:

Γ2

Γ2 − 1≡(

∂P

∂T

)

ad

Per trovare una soluzione al sistema di cinque equazioni accoppiate sopracitato in termini

delle variabili r ,P, L, T, X e necessario conoscere, oltre ad appropriate condizioni al

contorno, espressioni esplicite di ρ, T, u, Γ2, ǫ, κ in funzione del primo insieme di

variabili. Tali relazioni sono fornite dalla termodinamica, dalla fisica atomica e nucleare,

questo contributo fondamentale rappresenta il legame tra la fisica stellare e quella che

potremmo definire “micro-fisica”.

La cosidetta “macro-fisica” presenta semplificazioni nei calcoli: la trattazione della

convezione, per esempio, e generalmente approssimata utilizzando la teoria della mixing-

length, in alcuni modelli corretta tenendo conto dell’overshoot ai confini della regione di

convezione. La diffusione e la sedimentazione degli elementi pesanti dovuta alla gravita

non e solitiamente inclusa nei modelli, cosı come gli effetti dei campi magnetici vengono

ignorati.

Nel Capitolo 3, oltre ad una descrizione del sistema di equazioni e della fisica utilizzata

nei calcoli, sono presentate le caratteristiche dei modelli utilizzati. E inoltre descritto

l’andamento di alcune grandezze che influenzano direttamente le proprieta sismiche della

stella.

Il lavoro di tesi e stato svolto utilizzando spettri di oscillazione generati da mod-

elli stellari calcolati con il codice di struttura ed evoluzione stellare di J. Christensen-

Dalsgaard [5], le tracce evolutive di tali modelli sono riportate in Figura 8.1.

110

8.3. TEORIA DELLE OSCILLAZIONI STELLARI NON RADIALI

Figure 8.1: Un diagramma Hertzsprung-Russel che mostra le tracce evolutive dei modelli con-

siderati, ogni traccia e contrassegnata con la massa della stella in unita di massa solare.

8.3 Teoria delle oscillazioni stellari non radiali

8.3.1 Le equazioni di oscillazione

Le equazioni di oscillazione stellare lineari ed adiabatiche sono ottenute riscrivendo le

equazioni dei fluidi perturbate presentate nella sezione 4.1.2, usando esplicitamente la

simmetria sferica della struttura all’equilibrio. Le equazioni cosı ottenute descrivono

oscillazioni non radiali, ovvero non si assume che le perturbazioni siano sfericamente

simmetriche. Di conseguenza le oscillazioni radiali sono contenute come caso particolare.

Il sistema di equazioni che descrive le oscillazioni lineari adiabatiche, separando il

vettore spostamento δr in componenti radiali ed orizzontali δr = ξrar+ξh, e il seguente:

− ∂2

∂t2

[

ρ′ +1

r2

∂

∂r(r2ρ0ξr)

]

= ∇2h p′ + ρ0∇2

h Φ′

La componente radiale dell’equazione del moto e:

ρ0∂2ξr

∂t2= −∂p′

∂r− ρ′g0 + ρ0

∂Φ′

∂r

111


e l’equazione di Poisson perturbata:

1

r2

∂

∂r

(

r2 ∂Φ′

∂r

)

+ ∇2hΦ′ = −4πGρ′

L’equazione dell’energia, nell’approssimazione adiabatica, diventa:

ρ′ =ρ

Γ1pp′ + ρξr

(

1

Γ1p

dp

dr− 1

ρ

dρ

dr

)

Il problema di trovare soluzioni per tale sistema di equazioni puo essere posto come

determinazione delle proprieta spettrali di un operatore differenziale in uno spazio di

Hilbert propriamente definito. Questo approccio risulta particolarmente utile qualora si

vogliano includere nelle equazioni di oscillazione effetti di non adiabaticita, rotazione o

piccole variazioni nella struttura all’equilibrio (come presentato nelle sezioni 4.3, 4.5 e

4.6).

Separando le variabili angolari da quella radiale si trova che una generica pertur-

bazione scalare puo essere scritta nella forma:

f(r, θ, φ, t) =√

4πf ′(r)Y ml (θ, φ)e−i ωt

dove Y ml (θ, φ) e una funzione armonica sferica, i.e. Y m

l (θ, φ) e un’autofunzione dell’operatore

∇2h:

∇2h(Y m

l (θ, φ)) = − l(l + 1)

r2Y m

l (θ, φ)

La soluzione della parte radiale della perturbazione, ponendo le opportune condizioni

al contorno, risulta dipendere da un intero n. Un modo di oscillazione e caratterizzato

dunque da tre interi: l’ordine radiale n, il grado sferico l e l’ordine azimutale m. (l e

un intero non-negativo ed m un intero che puo assumere i valori 0,±1, . . . ,±l). Come

mostrato in Figura 8.2 vi sono |m| linee nodali perpendicolari all’equatore (θ = 90)

ed l − |m| linee nodali parallele all’equatore. Una pulsazione radiale naturalmente puo

essere vista come caso speciale di un modo non radiale dove l = m = 0.

I modi di oscillazione sono solitamente suddivisi in tre classi, a seconda della loro

natura fisica: modi p, g ed f. I modi p hanno frequenze relativamente alte e gli elementi di

massa oscillano prevalentemente radialmente. La forza che ristabilisce l’equilibrio nelle

oscillazioni e semplicemente la pressione, per questo motivo essi sono chiamati anche

modi acustici. I modi g hanno frequenze relativamente basse e gli elementi di massa

oscillano prevalentemente trasversalmente; per questi modi la forza di richiamo e data

dalla gravita attraverso la spinta idrostatica (vd. Sezione 4.4.1).

8.3.2 Approssimazione asintotica

Le equazioni che descrivono le oscillazioni stellari nell’approssimaziona adiabatica e lin-

eare possono essere risolte numericamente. Soluzioni analitiche, necessariamente ap-

112

8.3. TEORIA DELLE OSCILLAZIONI STELLARI NON RADIALI

Figure 8.2: Alcuni esempi della dipendenza angolare delle perturbazioni descritte dalle ar-

moniche sferiche. Partendo da sinistra si hanno modi con (l,m)=(5,0),(5,2) e (5,5)

prossimate, consentono tuttavia di rivelare esplicitamente quali parametri fisici influen-

zano le proprieta degli spettri di oscillazione.

La teoria asintotica sviluppata per i modi acustici (modi p) permette di arrivare ad

una semplice relazione:

νnl ≃ (n +l

2+

1

4+ α)∆ν −

(

AL2 − δ) ∆ν2

νnl(8.6)

dove

A =1

4π2∆ν

[

c(R)

R−∫ R

0

dc

dr

dr

r

]

(8.7)

e

∆ν =

(

2

∫ R

0

dr

c

)−1

(8.8)

rappresenta l’inverso del diametro acustico della stella, ovvero due volte il tempo impie-

gato da un’onda acustica per propagarsi dal centro alla superficie della stella.

E dunque naturale definire le seguenti:

grande separazione:

∆νn,l ≡ νn+1,l − νn,l = ∆ν (8.9)

e la piccola separazione:

δνnl ≡ νnl − νn−1,l+2 ≃ −(4l + 6)∆ν

4π2νnl

∫ R

0

dc

dr

dr

r(8.10)

dove il termine c(R) e stato trascurato dato che la velocia del suono sulla superficie

risulta trascurabile rispetto alle altre quantita.

Nella sezione 5.2.4 sono illustrati i limiti delle espressioni asintotiche e come essi

possano essere utilizzati per rivelare le caratteristiche di regioni in cui la struttura della

stella varia su scale di lunghezza minore della lunghezza d’onda dei modi considerati.

113


8.4 Risultati

8.4.1 Diagrammi C-D

La teoria asintotica, fornendo semplici espressioni analitiche, rappresenta uno strumento

per vincolare o inferire i parametri che descrivono una stella. La semplice relazione asin-

totica per i modi acustici (Equazione 8.10) suggerisce che sia ∆ν sia δν, determinate dagli

spettri di oscillazione, possano essere legate all’andamento delle variabili all’equilibrio

nelle stelle (principalmente la velocita del suono e la sua derivata).

Il diagramma costruito con i valori di ∆ν e δν mediati sulla frequenza1 dei modi e

stato suggerito in [6] come possibile metodo per dedurre la massa e l’eta di una stella

da dati sismici. Il primo problema da affrontare nel calcolo di un diagramma C-D e la

scelta di un valore significativo di entrambe le separazioni per ogni modello considerato.

In effetti entrambe le separazioni risultano essere dipendenti dalla frequenza come risul-

tato delle approssimazioni introdotte nella teoria asintotica che portano alla semplice

espressione 8.10.

Grande separazione ∆ν Una delle piu semplici proprieta di uno spettro di oscil-

lazione di tipo solare e la quasi-regolare spaziatura tra frequenze di modi di stesso

grado e di ordine successivo. La cosidetta grande separazione e principalmente una

misura del tempo impiegato dal suono per giungere dal centro alla superficie della stella.

L’andamento approssimato di ∆ν, considerando modelli di massa ed eta differenti, segue

l’inverso del tempo dinamico della stella, come presentato nella sezione 6.1.1:

∆ν ≃ (GM)1/2

R3/2= (Gρ)1/2

Calcolando la grande separazione da uno spettro di frequenze teorico, invece di un

valore costante si trova una funzione della frequenza con una componente periodica. Tale

dipendenza puo essere spiegata prendendo in considerazione le variazioni della struttura

della stella all’equilibrio che avvengono su una scala di lunghezze minore della lunghezza

d’onda del modo considerato.

E possibile confrontare numericamente l’accuratezza dell’espressione asintotica della

grande separazione con quella calcolata dalle frequenze utilizzando le varibili all’equilibrio

dei modelli. La differenza in percentuale tra le due espressioni, tipicamente inferiore al

2% (Tabella 8.1), dimostra la rilevanza della teoria asintotica al primo ordine.

Piccola separazione δν La piccola separazione in frequenza e sensibile alla struttura

delle regioni interne della stella, a causa del fattore 1/r nell’integrando dell’espressione

1Conosciuto come diagramma HR astrosismico o diagramma C-D

114

8.4. RISULTATI

n M/M⊙ Xc ∆ν l = 2 1/(2τ0) %

5 1.0 5.76877e-01 154.2 155.0 -2.1

15 1.0 2.99063e-01 134.6 137.4 -2.0

30 1.0 8.05402e-02 118.4 120.9 -2.1

20 1.5 5.61206e-01 82.8 84.0 -1.6

80 1.5 2.60829e-01 63.0 64.1 -1.8

120 1.5 1.40914e-01 57.6 58.7 -1.8

20 2.0 5.60073e-01 73.1 74.0 -1.3

80 2.0 2.59374e-01 49.3 49.7 -1.0

120 2.0 1.39867e-01 41.7 42.1 -0.9

Table 8.1: Grande separazione calcolata dagli spettri di frequenza e dall’espressione (8.10). ∆ν

rappresenta ∆ν(ν) mediata su modi di ordine radiale elevato (n ≥ 20); l’ultima colonna mostra

la differenza percentuale tra le due espressioni; n identifica il modello considerato.(vd. il Capitolo

3)

8.10. La piccola separazione e in generale una quantita positiva dato che dcdr e nega-

tiva nella maggior parte della stella; ciononostante, in stelle evolute, la diminuzione del

peso molecolare medio µ nelle regioni centrali, causato dalle reazioni nucleari di combus-

tione dell’idorgeno, contribuisce negativamente all’integrale: durante l’evoluzione δνn,l

decresce e puo dunque essere associato all’eta di una stella.

Il confronto tra il valore della piccola separazione determinato dalle frequenze e quello

calcolato dai modelli, e quantitativamente valido per modelli con caratteristiche simili

al nostro sole, ma diviene discutibile per modelli differenti. L’espressione asintotica di

Tassoul (8.10) predice inoltre che δν0,2 = 35 δν1,3, i.e. la dipendenza da l della piccola

separazione e racchiusa nel termine l(l + 1) nell’espressione (8.10). I calcoli mostrano

che cio in generale non e vero, in particolare: in modelli di massa M < 1.2M⊙ vi

e un disaccordo sistematico tra le due separazioni; in modelli di massa M > 1.2M⊙

le differenze cambiano irregolarmente durante l’evoluzione della stella sulla sequenza

principale. Come presentato nelle sezioni 6.1.3 e 6.1.4 queste deviazioni dalle previsioni

della teoria asintotica contengono preziose informazioni.

I diagrammi C-D sono stati calcolati per un insieme di modelli in un range di masse

da 0.9 a 2.0 M⊙. Per modelli senza nuclei convettivi (M < 1.2M⊙), come presentato in

Figura 8.4, la separazione tra le tracce evolutive per modelli di masse diverse diminuisce,

rendendo meno precisa la determinazione della massa e dell’eta di una stella. La dif-

ferenza tra tracce evolutive calcolate con modi l = 0, 2 ed l = 1, 3 puo essere spiegata

115


Figure 8.3: δνn,l/l(l + 1) calcolata dalle frequenze di oscillazione (simboli) e dall’espressione

asintotica (linea punteggiata) per un modello di 1 M⊙, Xc=0.3.

dalla maggior profondita di penetrazione dei modi di grado minore. In Figura 6.7 viene

mostrato come la variazione dei parametri utilizzati per generare i modelli all’equilibrio

(abbondanza di idorgeno iniziale, parametro di mixing-length) influisca sulla posizione

delle tracce evolutive in un diagramma C-D.

Per modelli di massa M ≥ 1.2 M⊙ la presenza di un nucleo convettivo e responsabile,

durante l’evoluzione di una stella, della comparsa di un rapido gradiente del peso moleco-

lare medio in prossimita del margine della regione convettiva; tale variazione, ancor piu

marcata in stelle di massa 1.2 e 1.5 M⊙2 influenza naturalmente altre variabili, tra le

quali la velocita del suono, la frequenza di Lamb e di Brunt-Vaisala che determinano

direttamete le propreta dei modi di oscillazione. Questo fenomeno potrebbe determinare

la deviazione dalla semplice espressione asintotica. Dal calcolo dei diagrammi C-D per

quest’ultimo insieme di modelli risulta chiaro che:

• la separazione delle tracce evolutive per modelli di massa differente si riduce pro-

gressivamente considerando stelle di massa maggiore, e naturalmente diminuisce

l’efficacia del diagramma C-D nel vincolare la massa di una stella,

• come per i modelli di massa inferiore, e interessante confrontare tracce evolutive

calcolate con separazioni di grado 0, 2 e 1, 3. Le irregolarita presenti nelle tracce

sono accentuate quando si considerano modi con l = 0, 2. Ancora una volta cio puo

essere spiegato considerando la maggior penetrazione di modi di basso grado l, ar-

gomento che puo essere reso quantitativo studiando l’andamento delle autofunzioni

2per le ragioni spiegate nel Capitolo 6

116

8.4. RISULTATI

Figure 8.4: Diagramma C-D in cui D0 e calcolato sia con modi di grado l = 0, 2 (d02, linee

continue) sia l = 1, 3 (d13, linee tratteggiate).

per i diversi modi considerati.

Nel Capitolo 6 vengono anche riportate le separazioni calcolate sulle frequenze dei modi

acustici solari di basso grado l, determinate da osservazioni.

8.4.2 Analisi sismica delle regioni di ionizzazione dell’elio

Variazioni localizzate nella struttura all’equilibrio delle stelle, come quella alla base

dell’envelope convettiva o nella regione di seconda ionizzazione dell’elio, generano un

segnale nelle frequenze di oscillazione. Le caratteristiche di tale segnale sono legate alla

posizione e alle proprieta termodinamiche della strato in cui avviene la rapida variazione

nella struttura all’equilibrio della stella. L’espressione esplicita di tale segnale, recente-

mente ricavata in [30] usando l’approccio presentato nella sezione 4.3.1, e la seguente:


βωcos [2(ωτd + φ0)] (8.11)

117


dove a0 e β descrivono le caratteristiche geometriche della variazione di Γ1 e τd ne

rappresenta la profonditaa acustica. Questa espressione e ricavata assumendo l’esistenza

di una variazione in Γ1 come presentata in Figura 8.5.

Figure 8.5: Il coefficiente termodinamico Γ1 rispetto alla profondita acustica in una stella di 1.0

M⊙, Xc = 0.30. La regione di seconda ionizzazione dell’elio e localizzata approssimativamente

tra 600 e 800 secondi.

Il segnale periodico previsto nelle frequenze e anche presente nella grande separazione

∆ν. E stata sviluppata una procedura di fitting che permette di isolare il segnale peri-

odico, di determinarne le caratteristiche e di dedurre quindi informazioni sulla regione

di seconda ionizzazione dell’elio per modelli stellari di masse M < 1.5 M⊙ (vd. Figura

??). Un’estensione ed approfondimento di questa tecnica promette di poter determinare

l’abbondanza di elio nelle envelope stellari utilizzando esculsivamente dati sismici.

Un’analisi piu generale degli effetti delle variazioni in Γ1 puo essere ottenuta notando

che piccole variazioni nella struttura all’equilibrio della stella generano cambiamenti nelle

frequenze nella forma (vd.. [40]):

δωnl

ωnl=

∫ R

0

[

KnlΓ1,ρ(r)

δΓ1

Γ1(r) + Knl

u,Γ1(r)

δru

u(r)]

]

dr (8.12)

con u = p/ρ, con δω = ω − ωsmooth, con δΓ1 = Γ1 − Γ1smooth, dove le variabili smooth

sono riferite ad un modello “liscio” che non presenta buche in Γ1.

118

8.4. RISULTATI

Figure 8.6: La sovrapposizione del segnale “liscio” e periodico ricostruiscono l’andamento della

grande separazione ∆ν(ν).

Oltre ad una componente “liscia”, sia nelle frequenze che in ∆ν, compare un segnale

periodico, come quello determinato in Figura 8.6 utilizzando la procedura di fitting

precedentemente presentata. Considerando invece modelli con massa M > 1.5 M⊙, nelle

regioni esterne della stella, e presente una seconda rapida variazione di Γ1 associata alla

regione di prima ionizzazione dell’elio (vd. Figura 8.7).

Il confronto tra il segnale periodico presente in ∆ν e quello determinato utilizzando

kernel numerici e tenendo conto del secondo minimo di Γ1 in δΓ1

Γ1(r) nell’espressione

8.12, permette di mettere in relazione il cambiamento dell’andamento del segnale con

l’esistenza della variazione associata alla regione di prima ionizzazione dell’elio.

Considerando in particolare le funzioni Kn,lΓ1,u(r) δrΓ1(r)/Γ1(r) per diversi valori

dell’ordine radiale n e possibile inferire in quale range di frequenze il contributo della

regione di prima ionizzazione dell’elio diviene dominante. Un successivo passo, seguendo

il medesimo approccio utilizzato in [30] per derivare l’espressione 8.11, sara quello di

ricavare un’espressione esplicita del segnale periodico che includa entrambe le variazioni

119


Model M/M⊙ Xc Age (Gyr) Teff (K) τstar (s) τfit (s)

1 0.9 0.70 0.0 5251 490 504

2 0.9 0.24 9.2 5514 645 663

3 1.0 0.70 0.0 5628 560 580

4 1.0 0.30 5.0 5784 680 690

5 1.2 0.70 0.0 6228 630 660

6 1.2 0.39 2.0 6269 750 755

7 2.0 0.70 0.0 9237

Table 8.2: Le proprieta dei modelli considerati, dove Xc e l’abbondanza di idrogeno nel centro

e Teff la temperatura effettiva. E anche riportata la profondita acustica della variazione di Γ1,

τstar ricavata dai modelli considerati e τfit determinata dalla procedura di fitting sulle frequenze.

di Γ1, in modo che sia possibile determinare le caratteristiche di tutte e due le regioni di

ionizzazione di stelle di massa moderata.

8.5 Conclusioni e prospettive

Le future missioni spaziali (MOST, MONS, COROT ed EDDINGTON) forniranno nei

prossimi anni dati sismici accurati riguardo stelle di tipo solare. Le limitazioni imposte

dalle osservazioni da terra saranno finalmente superate.

Affinche l’astrosismologia, al di la della determinazione delle caratteristiche dei modi di

oscillazione delle stelle, possa esprimere il suo potenziale e consentire di determinare le

proprieta degli interni stellari risulta di fondamentale importanza sviluppare ed estendere

tecniche che permettano di legare le caratteristiche delle frequenze proprie di oscillazione

ai parametri fisici che descrivono la struttura stellare.

Il lavoro svolto nella tesi e stato sviluppato in questa prospettiva.

I diagrammi C-D sono stati calcolati per un esteso set di modelli stellari comprendenti

stelle di sequenza principale di massa piccola e moderata (0.9 M⊙ ≤ M ≤ 2.0M⊙) e se

ne e mostrata l’importanza come strumento per vincolare la massa e l’eta di una stella

utilizzando esclusivamente dati sismici. Nella tesi e anche mostrato come il confronto tra

tracce evolutive calcolate con modi di grado l = 0, 2 ed l = 1, 3 riveli qualitativamente

dettagli degli interni stellari, come la presenza di un nucleo convettivo o lo stabilirsi di

un nucleo degenere in stelle evolute.

L’andamento della grande separazione, come funzione della frequenza dei modi di

oscillazione, mostra un segnale periodico legato alle caratteristiche della regioni di ion-

izzazione dell’elio in prossimita della superficie stellare. Un approfondimento delle tec-

niche elaborate ed estese in questa tesi promette di fornire un metodo per vincolare

l’abbondanza di elio nell’envelope di stelle di massa medio-piccola, a partire dalle loro

120

8.5. CONCLUSIONI E PROSPETTIVE

Figure 8.7: L’andamento di Γ1 vicino alla superficie di una stella di 2.0 M⊙, Xc=0.70.

proprieta sismiche.

Occorre sottolineare che per uno studio quantitativo delle proprieta dei nuclei con-

vettivi e necessario raffinare l’espressione asintotica di cui ora disponiamo.

121


122

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126

Appendix A

Fitting procedure.

The aim of the fitting procedure applied in Section 6.2 is to extract from the set

(ν,∆ν(ν)) the characteristics related to the region of helium ionization. The procedure

is written in IDL1.

Input data

For each model considered the frequency spectrum is calculated using the adiabatic

pulsation code; the large separation ∆ν(ν) is easily computed from the eigenfrequencies

stored in the ASCII obs output file. The acoustic depth of the helium ionization zone,

that will be compared to the one determined by the fit, is estimated from the equilibrium

variables of the star stored in the gong files, output of the stellar structure and evolution

code.

Procedure

The procedure consists of two steps:

1. Isolate in ∆ν(ν) the periodic signal δν(ν).

2. Fit expression


βωcos [2(ωτd + φ0)] (A.1)

on δν(ν).

The first step removes the “smooth” component of ∆ν(ν) by fitting a function of the

formm∑

i=0

bi1

νi(A.2)

1IDL Interactive Data Language, a registered trade mark of Research Systems Inc.

127

APPENDIX A. FITTING PROCEDURE.

Figure A.1: The smooth signal determined for a 1.2 M⊙ ZAMS star, using l = 2 acoustic modes

with radial order n ≥ 7.

using a standard polynomial least-squares fit, the signal is shown in Figure A.1.

On the residuals (∆ν(ν)−∆ν(ν)smooth) I then apply the non-linear fitting procedure

mpcurvefit.2 The result of the fit on the residuals is shown in Figure A.2.

The procedure is applied on a set of frequencies obtained excluding low order modes

where the asymptotic theory, used to derive Equation A.1, is invalid (see Section 6.2).

Two other parameters are changed during the procedure: n, the radial order correspond-

ing to the mode with the lowest frequency, and m, the highest exponent in expression

A.2. The set of parameters (a0, β, τd) with the lowest χ2 is chosen as the best-fit set.

2Freely downloadable from http://cow.physics.wisc.edu/

128

Figure A.2: The residuals determined as ∆ν(ν) − ∆ν(ν)smooth are fitted to expression A.1.

Figure A.3: The combination of the smooth and periodic signals reconstructs the behaviour of

∆ν(ν).

129

APPENDIX A. FITTING PROCEDURE.

130

Appendix B

Seismic analysis of the helium

ionization zones in low- and

moderate-mass stars.

To appear in the proceedings of Asteroseismology across the HR diagram, 1-5 July 2002

Porto, as an issue of Astrophysics and Space Science Series, Kluwer Academic Publishers.

131

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Andrea Miglio - uliege.beastrotheor3.astro.ulg.ac.be/miglio/tesi.pdf · Seismic analysis of solar-type stars Andrea Miglio Universit´a degli Studi di Milano Facolt´a di Scienze