Stellar stability and asteroseismologydupret/Stab-et-astero-ang.pdfand the internal physical...

Stellar stability andasteroseismology

SPAT0044

Marc-Antoine Dupret

email: ma.dupret @ uliege.beweb site : www.astro.ulg.ac.be/~duprettel: 04 3669732Bât. B5c, 1er étage

Goals of the course

Precise mathematical problem

2) Understanding the underlying mathematical formalism

Asteroseismology

3) Understanding the link between stellar oscillations (frequencies, ...),and the internal physical structure of stars

Stellar stability

4) Understanding of the main energy processesat the origin of stellar oscillations

Pulsating stars in the HR diagram

5) Knowledge of the main families of pulsating stars, their theoretical characteristics and seismic probing potential

Stellar oscillations

1) Understanding the physics of stellar oscillations

Lecture notes and others

Powerpoint slides

Lecture notes « Stabilité Stellaire » of Richard Scuflaire

Lecture notes « Stellar Oscillations » of J.C. Dalsgaard

Methodology

Numerous analogies with other cases

- simpler (acoustic modes in an organ pipe, ...)

- well known (Schrödinger theory in quantum mecanics)

What are stellar oscillations ?

r(m) r’(m) = x r(m)

(m) ’(m) = (m)/x3

Poids(m) Weight’(m) = Weight(m)/x4

P(m) P’(m) = P(m) (’/)1 = P(m)/x3

1

First, hydrostatic equilibrium:

Homologous adiabatic contraction of the gas P’/P = (’/)1

If 1 < 4/3 Contraction (x < 1) P’(m) < W’(m) Force downwards

Force and displacement in the same direction

Collapse or explosion, dynamically unstable

If 1 > 4/3 Contraction (x < 1) P’(m) > Weight’(m) Force upwards

Expansion (x > 1) P’(m) < Weight’(m) Force downwards

Restoring force opposite to the displacement

Oscillations , dynamically stable

What are stellar oscillations ?Reminder : How a star reacts to a global contraction or expansion ?


In reality …

- The oscillation amplitude depends on the depth (not homologous)

- There are radial and non-radial oscillations

Amplitudes exagerated by a factor 106 !

- 2 possible restoring forces:

gradient of pressure Pressure modes

Archimedes force Gravity modes


In reality …

- The oscillation amplitude depends on the depth (not homologous)

- There are radial and non-radial oscillations

Characteristic times

Stellar oscillations = dynamical phenomenon

Characteristic time = dynamical time :

2

1

r

mG

dr

dP - −=

Dynamical time ~ free fall time

(pressure suppressed) GM

3

dyn

R t =22

dynt

R

R

GM=

Order of magnitude :

Other characteristic times:

(Forces imbalance)

Helmholtz – Kelvin time: Thermal imbalance

Nuclear time : Evolution due to nuclear reactions

Radial acceleration

Dynamical time : Forces imbalance

Always : tdyn << tHK << tnuc

At the time scale of oscillations :

1) The gas does not have time for significant heat exchange % heat capacity :

dq = T ds = du + P dv

|dq| << u ~ cv T In most of the star :

Oscillations are adiabatic

dq : Heat exchanged per unit mass during one oscillation cycle

Virial theorem :

Sun : 26 min 107 years 1010 years


Helmholtz – Kelvin time: Thermal imbalance

Nuclear time : Evolution due to nuclear reactions

At the time scale of oscillations :

1) The gas does not have time for significant heat exchange % heat capacity :

2) The chemical composition does not have time to change :

In most of the star : oscillations are adiabatic

tdyn << tnuc

Period of oscillation (fundamental mode) ~ dynamical time

Dynamical times of some types of star :

Star

Neutron star

White dwarf

Sun

Red supergiant

1015 g/cm3

106 g/cm3

1.41 g/cm3

10-9 g/cm3

0.05 ms

2 s

26 min

2 ans


Stars play music !

Solar type stars

A zoo of pulsating stars of different types

Thousands of observed modes !

Pressure modes(acoustic)

excited by turbulent motions at the top

of the convective envelope.

Periods ~ 3-8 min.

Helioseismology allows to probe:

- the sound speed, …

- the rotation speed

differential in the envelope.

Sun :

Pressure modes (acoustic).

Periods ~ 2-15 min.

Small amplitudes :

dL/L ~ 1-20 ppm, speeds ~ 1-20 cm/s

Modes difficult to detect,

Dozens of observed modes

Less information than in the

Solar interior.

Other stars (ex : ® Cen) :


Solar type stars

Averaging over the disk

g Doradus

Gravity modes

Long periods ~ 0.5 - 3 days

Space revolution (Kepler, …):

Dozens of identified modes

Asteroseismology allows us to probe

their deep regions:

- Extension of the convective core,

“age”, core rotation, internal

composition, …


d Scuti

Around the fundamental mode

Periods ~ 0.5 - 6 hours

1-80 modes (ground) hundreds (space)

Mode identification very difficult:

forest of modes without structure,

fast rotations.

Seismic probing is difficult

Mixed Modes allowing to probe

the core and superficial regions:

extension of the convective core, …


Rapidly oscillating Ap

High order pressure modes

Periods ~ 5.5 - 20 minutes

Strong magnetic field

Magneto-acoustic modes

Asteroseismology constrains

the internal magnetic field

Chemical peculiarities(diffusion process, …),Strong magnetic field


Slowly Pulsating B

Gravity modes

Long Periods ~ 0.8 - 3 days


Space revolution (Kepler, …):

Dozens of identified modes

Asteroseismology allows us to probe

their deep regions:

- Extension of the convective core,

“age”, core rotation, internal

composition, …

Cephei

Around the fundamental mode

Periods ~ 2 - 8 hours

1-10 detected and identified modes

Successful asteroseismology

Mixed modes allowing us to probe:

- Deep regions :

extension of convective core,

internal rotation, …

- External regions : Opacities, chemical

composition, rotation, …


Red giants

1) Radial mode(s) high amplitude :

Mira, irreg.

Periods ~ year(s)

2) Space revolution (Kepler, …):

Radial and non-radial modes (dozens) of

low amplitude excited by convection (~ sun)

Periods ~ 3-30 days

Asteroseismology : Detailed probing of the

superficial layers (p-modes) and the core

(thanks to mixed modes !)

Not enough for seismic probing


High amplitude radial mode(s)

Periods :

RR Lyrae : 8 – 12 hours

Cepheids : 1 – 50 days

1 (or 2) modes

Classical instability strip :

Cepheids (d Cephei), RR Lyrae

Not enough for asteroseismology

Period – luminosity relation

Distance determination


1) Near-fundamental modes

Periods : 1-10 minutes

2) Gravity modes (Betsy stars)

Periods : 0.5 - 3 hours

Stars having lost their envelope after the helium flash (binarity ?)

Asteroseismology :

Possibility of probing the interior of evolved

stars : internal chemical composition,

evolution channel of binary systems, …

Extreme horizontal branch(sub-dwarfs of type B, sdB)


White dwarfs

Gravity modes

Different types :

- DOV (GW Vir) : Teff ~ 105 K,

Periods ~ 5 - 80 min.

- DBV (V777 Her) : Teff ~ 3 104 K

Periods ~ 2 – 15 min.

- DAV (ZZ Ceti) : Teff ~ 104 K

Periods ~ 0.5 – 25 min.

Asteroseismology :

Allows us to probe the interior of compact stars:

- Chemical composition of each layer,

- Equation of state (crystallization, degeneracy)


Stellar hydrodynamics reminder

Lagrangian and Eulerian formalisms

2 types of spatial coordinates in fluid mechanics :

The 3 usual space coordinates (spherical coordinates r, µ, Á in stars)

1) Eulerian coordinates :

System of coordinates fixed in space, independent of motion

But not ideal to relate velocity and acceleration,

to follow changes of state and chemical composition

One notes /t the partial derivative % time in Eulerian coordinates.

2) Lagrangian coordinates :

System of coordinates (field of labels ) associated to each material

element of the fluid and following the motions.

Example, spherical symmetric star (1D) : mass of the spheres: m

The partial derivative % time in a Lagrangian coordinates system

is called the material derivative , one notes it : D/Dt

It is the variation % time following the fluid motions.

Coordinates change :

fixéTrajectories : Velocity :

General relation : Acceleration :


Mass conservation :

Momentum conservation :

Poisson equation:

The Reynolds number associated with oscillations

is always very high the viscosity can be neglected


Conservation of energy : (negligible viscous dissipation)

Transport of energy by radiation :

Transport of energy by convection :

Much more complicate !Rigorously, one should follow all the turbulent motions 3D simulations …


Hydrodynamic simulations

Large Eddy Simulations (LES)

Main limitations :

Scales of 3D sim. >>> Dissipation scale

Age of stars >>> duration of 3D sim.

Real Reynolds nb. ~ 1010 >>> 104 in 3D sim.

Stellar convection

Analytical approaches: Mixing-length theory, …

Main difficulties in the description of convection :- Hydrodynamics involving a very large spectrum of space and time-scales :

~ 1cm (dissipation) up to the stellar size !

- Very very turbulent (Re ~ 1010 ) and non-linear

Very approximate

Introduction of free parameters …

Transport of energy by convection :

Much more complicate !Rigorously, one should follow all turbulent motions …

Analytical approach :

One separates : turbulent motions – average motion due to the oscillations

Turbulent motions lead to a new term in the energy equation,

the convective flux FC = cp Vc T (flux of enthalpy) :

Different approximate theories have been proposed relating the convective flux

to other quantities, for example the Mixing-Length Theory gives in the stationary

case without radiative loss :

But we don’t know how Fc reacts to oscillations…

Fc = cpT (P/)1/2

2(r - rad)

3/2


Stellar convection and oscillations

Main difficult questions :

What are the transfers of energy between turbulent motions and oscillations ?

How the convective motions and the convective flux vary with time due to the oscillations ?

One must thus build models of time-dependent convection, which is much more complicate than getting stationary models .

Equilibrium (stationary) spherical symmetric configuration

24 rdr

dm=

Conservationof mass

2r

mG

dr

dP −=

Hydrostatic equilibrium

3216

3

Trac

L

dr

dT

−=Radiation

P

T

r

Gm

dr

dT2ad-

=Convection

Transferof energy

)(4 2

gravnrdr

dL +=

Conservationof energy

One notes the equilibrium solutions:

P0(r) , T0(r) , 0(r) , L0(r), …

One cancels all derivatives % time and the velocity in hydrodynamic simulations.

Efficient convection

adiabatic stratification

Theory of small perturbations

We start from an equilibrium model:

Study of the oscillations around this equilibrium.

One assumes that the oscillation amplitudes are small % scale heights.

Consequently, one can linearize the problem.

Eulerian and Lagrangian perturbations

1) Eulerian perturbation of X : X’

P0(r) , T0(r) , ½0(r) , L0(r)

Difference between instantaneous and equilibrium value at a fixed point in space.

2) Lagrangian perturbation of X : dX

Difference between instantaneous and equilibrium value for a fixed material element.

1) Eulerian perturbation of X : X’

2) Lagrangian perturbation of X : dX

Relation between Lagrangian and Eulerian perturbations :

Displacement vector :

Field of average positions: Lagrangian coordinates

One assumes that the oscillation amplitudes are small

1st order Taylor development :


Useful relations :

Lagrange – Euler :

Gradients :

Time derivatives :

O(2)

Sums :


Products :

Functions :

Cas particulier :

Products of powers :


Useful relations :

Integration % time :

Conservation of mass :

Integrated form of the

continuity equation:

Adiabatic stellar oscillations

Lag. coord. cst., t0 = time when ½=½0

Method of small perturbations :

One takes the difference between the dynamical and equilibrium equation:

Conservation of momentum :

Equilibrium configuration :


Poisson equation

Adiabatic expansion of the gas :

At the time-scale of oscillations, the gas has no time for significant heat

exchange % its heat capacity :

In the main part of the star, oscillations are adiabatic.

Adiabatic relation between pressure and density variation :




Poisson equation:

Adiabatic relation:


Acoustic waves



Adiabatic relation :

Homogeneous medium, no gravity :

Divergence of the momentum equation :

Wave equation : Sound speed c :

Wave equation : Sound speed c :

Stationary waves

Eigenvalue problem : A x = s2 x s2 : Eigenvalue

x = P’ : Eigenfunction (vector)

: Linear operator

+ boundary conditions

In a rectangular box [0, L1] x [0, L2 ] x [0, L3]

Rigid boundary conditions at the edges:

Solution :

Countable spectrumof frequencies

Eigenmode

The knowledge of the spectrum gives L1 /c, L2 /c and L3 /c

Frequency

Acoustic standing waves

1 dimension dP’/dr(0) = P’(R) = 0

R ,

T c ~ (P/)1/2 / T1/2 ,

c constant(T constant)

Rr

Large separation

Boundary conditions

n + 1 : nbre of nodes

0Frequency

P’

Acoustic standing waves in a semi-open organ pipe

R ,

T c ~ (P/)1/2 / T1/2 ,

In stars :

Radial modes Dynamical time

Rr

Large separation 0Frequency


Rr

P’

C Non-constant

Temperature or section varying with r

What is the frequency spectrum ?

Particular case of a more general problem:

Sturm-Liouville problem

+ Homogeneous or periodic

boundary conditions:

p(x) > 0, (x) > 0 for a<x<b and continuous by part

In the organ pipe :



Examples :

Organ pipe :

Legendre equation:

Bessel equation:

1D Schrödinger equation :

…

+ Homogeneous or periodic

boundary conditions:

p(x) > 0, (x) > 0 for a<x<b and continuous by part

Theorems :

1) Space

is a Hilbert space,

for the scalar product :

and u verifies theboundary conditions

2) Linear operator

is auto-adjoint (hermitian) :

Eigenvalue problem :

+ Homogeneous or periodic boundary conditions:


3) Orthogonality : The eigenspaces are orthogonal.

Let u and v be eigenfunctions of two different eigenspaces:

4) The eigenvalues li are real and form a countable series :

l0 < l1 < l2 < … < ln < …

5) The eigenfunctions ui form a complete orthogonal base:

1


Theorems :


5) The eigenfunction ui form a complete orthogonal base:

6) The nodes of the eigenfunctions are interlaced:

For i < j (li < lj), there is a node of uj between 2 nodes of ui.

7) Variational principle : The function (Rayleigh ratio)

is stationary

on the eigenspaces and is equal to the eigenvalues there.

Theorems :



Properties of the acoustic modes spectrum ?

This is a Sturm-Liouville problem

Corresponding properties:

Orthogonality,for 2 eigenmodes P’1 (r) and P’2 (r) :

Countable set of frequencies : 0 < s0 < s1 < … < sn < …

…Completeness : The eigenmodes form a complete base.

…Variational principle : Rayleigh ratio

is stationary on the eigenspaces.


1

Radial adiabatic stellar oscillations

Conservation of momentum:

Conservation of mass:

Adiabatic relation:

Maintaining the spherical symmetry :

This is a Sturm-Liouville problem !

Maintaining the spherical symmetry :


Singular Sturm-Liouville problem (p(0) = p(R) = 0 )

Boundary conditions : Bounded solutions

bounded and regular at the center

0

At the center:

At the surface:



Corresponding properties:

- Orthogonality,for 2 eigenmodes x1 and x2 :

- Countable set of frequencies: s02 < s1

2 < … < sn2 < …

- Completeness: The eigenmodes form a complete base.

…- Variational principle:

is stationary on the eigenspaces and equal to si2 there.


Form of the eigenfunctions:

Fundamental mode

Polytrope n=3

1st overtone

Polytrope n=3

2nd overtone

Polytrope n=322th overtone

Sun

x = r/R

x = r/R

0.0 1.0

Not homologous !


General case :

Initial conditions :

General solution :

with

Proof :

complex

Linear combination of eigenmodes


with

Proof To show :

complex

Because is linear


General case :

Initial conditions :

General solution :

Dynamical stability of a star

The star is dynamically

stable

Sum of the oscillating solutions(standing waves) The star is dynamically

unstable

Unstable term

imaginary

2 possibilities :

sj real

r(m) r’(m) = x r(m)

(m) ’(m) = (m)/x3

Poids(m) Weight’(m) = Weight(m)/x4

P(m) P’(m) = P(m) (’/)1 = P(m)/x3

1

First, hydrostatic equilibrium:

Homologous adiabatic contraction of the gas P’/P = (’/)1

If 1 < 4/3 Contraction (x < 1) P’(m) < W’(m) Force downwards

Force and displacement in the same direction

Collapse or explosion, dynamically unstable

If 1 > 4/3 Contraction (x < 1) P’(m) > Weight’(m) Force upwards

Expansion (x > 1) P’(m) < Weight’(m) Force downwards

Restoring force opposite to the displacement

Oscillations , dynamically stable

What are stellar oscillations ?Reminder : How a star reacts to a global contraction or expansion ?

A star is dynamically unstable

Variational principle :

For a star to be dynamically unstable, we must have in a « significant » part of it:

in a significant part of the star

But one can have locally 1 < 4/3 and dynamical stability.

Question : What determines the behavior of 1?


Rigorous definition :

slowly accretes matter from its red giant companion

1er scenario (most frequent): A white dwarf of C-O in a binary system

The C-O core reaches the Chandrasekhar’s limiting mass !

the C-O core grows in mass

2e scenario : During H and He shell burning,

Some case of dynamical instability …

1) C-O core reaching the Chandrasekhar’s limiting mass

The degenerated electrons reach relativistic speeds

e- capture by the nuclei : e- + p ! n + e

) e % ) P ¼ Pe (Ne) = Pe (/e) increases less ) 1 &


The Nova phenomenon

Nuclear reactions in a degenerated medium thermal runaway

the entire star burns everything in a few seconds

The star explodes ! Type Ia supernova L ~ 109-10 L¯

- General relativity effects

Moreover:

The core collapses and reaches extreme densities

1) C-O core reaching the Chandrasekhar’s limiting mass

The degenerated electrons reach relativistic speeds

- e- capture by the nuclei : e- + p ! n + e



Main scenario :

Core burning of Si, contraction,

Shell fusion of Si, contraction, …

) The Fe core contracts,

becomes degenerated and grows in mass.

When MFe ' Chandrasekhar’s mass

) collapse until the nuclei touch

each other ) rebound and chock wave

) Explosion, Type II Supernova

Onion-like structure

2) End of massive stars life


Physical factors accelerating the collapse:

a) e- capture by the nuclei and free protons: e- + p ! n + e


b) Iron Photo-disintegration

T % ) Equilibrium towards the right : endothermic

) P, T increase less ) 1 &

c) Coulomb effects

% ) the electrons decrease the Coulomb repulsion between nuclei

) P increases less ) 1 &

2) End of massive stars life


3) Hyper-massive stars ( )

The radiation pressure dominates

One shows that for ,

Very high temperature and low density

+ General relativity effects

The stars with are dynamically unstable

4) 2nd collapse of a proto-star

When Tc ~ 2000 K , dissociation : H2 H + H

Endothermic reaction 2nd collapse


Asymptotic behavior of the frequencies

Question : Shape of the frequency pattern towards high overtones ?

Link with the internal structure ?

dP’/dr(0) = P’(R) = 0

c constant(T constant)

Large separation :

Boundary conditions

n + 1 : nbre of nodes

0Frequency

Acoustic waves in a semi-open organ pipe

Equidistant frequencies

Same asymptotic behavior in the general case ?

C Non-constant Temperature or section varying with r

Solution - approximation JWKB :

Proof :

0

General form :

One assumes

small

Asymptotic behavior of the frequenciesAcoustic waves in a semi-open organ pipe

JWKB approximation:

Boundary conditions

For large n, the frequencies are equidistant.

The knowledge of 0 or immediately gives :

This is the acoustic radius.


C Non-constant


Non-radial adiabatic stellar oscillations

A stellar spectrum is often very rich, holding much more than radial modes only !

Some examples of non-radial oscillation modes …

Gradient :

One uses the usual spherical coordinates r, q, f

Hypothesis : The star oscillates around a spherically symmetric

equilibrium configuration

Small perturbations linear eigenmodes :

Useful reminder, differential operators in spherical coordinates:

Divergence :

Laplacian :




Spherical symmetryand hydrostatic equilibrium


Legendrian :

Horizontal motion Conservation of mass

Conservation of masse+

Horizontal motion

Poisson equation :


System of partial derivative equations to solve :

Method of variable separation, one searches solutions of the form:

Possible here because the derivatives % q and f appear within the same operator

The spherical harmonics are eigenfunctions of

Solutions of the form:


System of 4th order ordinary differential equations in r to solve :

Solutions of the form:

We switched from a 3D (r, q, f) problem to a 1D (r) problem !


Spherical harmonics

m : Number of meridional node lines

l : Total number of node lines

l = 5

m = 2

l = 5

m = 0

Legendre associated function


One uses the adiabatic relation and eliminates ‘ to have 3

equations and 3 unknowns (xr, P’, ‘)

N : Brunt-Väisälä frequency Ll : Lamb Frequency

Radial motion

Horizontal motion + continuity

Poisson equation


One defines :

Displacement vector:

Continuity equation :


Without proof …

At the center : P’(r), Ã’(r), … / rl

l = 0 : »r / r l = 1 : »r(0) = »h(0) l > 1 : »r, »h / rl-1

At the surface : Perfect reflexion of the waves dP(R) = 0

Analytical solution of the Laplace equation

bounded at infinity :

Boundary conditions : bounded solutions at the edges

If ½(R) = 0 :

If ½(R) > 0 :


Cowling approximation:

Eigenvalue problem of 2nd order :

Justified when the number of nodes of ‘(r) is large

When the number of nodes of the eigenmodes is very large :

Oscillation cavities, pressure and gravity modesIntegral solution of the Poisson equation :


NB : a less approximate reasonning shows that,

Under the Cowling approximation, the equation to solve is:

with :


Oscillation cavities, pressure and gravity modes

If s2 > N2 and s2

> Ll2

ors2 < N2 and s

2< Ll

2

k2 > 0 wave equation

Wave propagation cavity

Local wave number : k

s2 > N2 and s2

> Ll2

Pressure mode-wave Gravity mode-wave

s2 < N2 and s2

< Ll2



s2 > N2, Ll2

Pressure modes2 < N2, Ll

2

Gravity mode

k2 > 0 : Propagation cavity k2 < 0 : Evanescent zone

N2 < s2 < Ll2 or Ll

2 < s2 < N2s2 > N2 , Ll2 or s

2< N

2, Ll

2

Asymptotic solution (JWKB) : Asymptotic solution (JWKB) :



Approximative JWKB solution :

Schroedinger equation :

Quantum physics : Schroedinger theory

V(r) < E Propagation zone of the particle

V(r) > E Evanescent zone

V(r) 0

de Broglie wavelength

Crossing of the barrier by quantum tunneling


N / 2¼

The Sun :

p Cavity

L1 / 2¼

g Cavity

Radiative core Convective envelope

L5 / 2¼

L20 / 2¼



Pressure modes

s12 < s2

2 < … 1

p1 p2 …

s12 > s2

2 > … 0

g1 g2 …

Limiting case : s2 >> N2, Ll2

Sturm-Liouville pb. :

Gravity modes Limiting case : s2 << N2, Ll2

Sturm-Liouville pb. : s1-2 < s2

-2 < … 1

Countable infinity of frequencies tending

towards infinity for each l




towards zero for each l

Fundamental mode

For each l ¸ 2 , one find between the two countable infinities of g and p modes the fundamental mode, this is the only remaining mode in an incompressible sphere.

p1 p2 p3 …… g2 g1 f

Frequencies

0



Pressure modes

s12 < s2

2 < … 1

p1 p2 …

s12 > s2

2 > … 0

g1 g2 …

Limiting case : s2 >> N2, Ll2

Sturm-Liouville pb. :

Gravity modes Limiting case : s2 << N2, Ll2

Sturm-Liouville pb. : s1-2 < s2

-2 < … 1


towards infinity for each l


towards zero for each l

Pressure (acoustic) modes

s2 >> N2, Ll2

Like for standing acoustic waves in an organ pipe :

The pressure modes in stars are comparable toacoustic standing waves.

An increase of the pressure resulting from the gas compression acts as a restoring force producing these waves.

The spectrum of oscillation frequencies of the pressure modes mainly depends on the sound speed profile c(r).

Asteroseismology (stars) or helioseismology (Sun) based onpressure modes enables to probe

Sound speed :

Asteroseismology (stars) or helioseismology (Sun) of pressure modesenables to probe the sound speed profile :

T Boundaries of

convective zones

Mean molecular weight

chemical composition :

helium abundance, …

Partial ionization zone,

equation of state, helium abundance

Pressure modes (acoustic)

In partial ionization zones, endothermic bound-free transitions :

During gas compression,part of the released

gravitational energy is pumped by the reaction

La pression augmente moins

¡1


Sun

Hundred thousands of pressure modes !!

l = 0 l ~ 200 !



s2 > N2, Ll2

N

L5L20

P Cavity

L1

Propagation cavity: s > Ll (r) Evanescent zone : s < Ll (r)



Propagation cavity :

s > Ll (r)

Local representation (around ) of the mode as a plane standing wave:

Evanescent zone :

s < Ll (r)

Interpretation :


Spherical harmonics

m : Number of meridional node lines

l : Total number of node lines

l = 5

m = 2

l = 5

m = 0

Legendre associated function


Propagation cavity:

s > Ll (r)

Point where s2 = Ll2 (r) :

N

L1L5

L20

Evanescent zone where:

s < Ll (r)

Sun

This is the turning point of the waves (rt)

The waves propagate with

the local sound speed



Modes de pression (acoustiques)

N

L5L20L1

Different modes with different l, probe different layers of the star


p Cavity

s2 > N2, Ll2

DuvallRelation (1982)

JWKB approximation:

Boundary conditions :

At rt,


s2 > N2, Ll2

Duvall relation (1982)

The resolution of the Duvall equation for different l and n

gives a good approximation of the Solar p modes frequency spectrum.

Theory ObservationsSun

n =

n =

n =

n =


s2 > N2, Ll2

The Duvall relation shows very clearly the link between the

sound speed profile c(r) and the frequencies of p modes.

Knowing the observed frequency spectrum, one can probe the

sound speed profile through the inversion of this relation.


The resolution of the Duvall equation for different l and n

gives a good approximation of the Solar p modes frequency spectrum.


s2 > N2, Ll2

Very approximative asymptotic solution l << n :

Frequencyl = 0

l = 1

l = 2

Large separation ~ constant =

n

n

n

n+1

n+1

n+1



s2 > N2, Ll2

Small separation :

Very approximative asymptotic solution for l << n :

Large separationA better solution :

is large in the deep layers of a star,

the small separation informs us about these deep layers.


Solar-type oscillations ( Centauri A)

Power spectrum of the star Cen A (Bouchy & Carrier 2002) :dozens of detected modes

The seismic study of these stars is much easier because :

- Regular structure of the spectrumeasy mode identification

- Slow rotationsimple modeling

l=

0

l=

2

l=

1

l=

0

l=

2

l=

1

Echelle diagram of 16 Cyg A observed by Kepler:54 detected modes

l=

0

l=

2

l=

1

l=

3

Frequency modulo large separation

Solar-type oscillations (16 Cyg)

The seismic study of these stars is much easier because :

- Regular structure of the spectrum

easy mode identification

- Slow rotationsimple modeling

As the star evolves, in the core

center surface

in the core

center surface

c2 c2

The small separation is an

indicator of the

evolution stage of a star.

dc/dr > 0

s2 > N2, Ll2

Small separation

is large in the deep layers of a star,

the small separation informs us about these deep layers.


Seismic indicators :

Large separation :

Small separation :

Normalized small separation:

2nd order difference :

Independent of surface effects

Discontinuities of dc/dr :

Frontiers of convective

zones, …

s2 > N2, Ll2Pressure (acoustic) modes

The Sun

s2 > N2, Ll2

Thousands of modes

l = 0 l ~ 200 !

Seismic probing of the sound speed


s2 < N2, Ll2Gravity modes

If s2 > N2 and s2

> Ll2

ors2 < N2 et s

2< Ll

2

kr2 > 0 Propagation cavity,

Local wave number : kr

s2 > N2 et s2

> Ll2

Pressure mode-wave Gravity mode-wave

s2 < N2 et s2

< Ll2

Reminder : Cowling approximation, |n|

s12 > s2

2 > … 0g1 g2 …

s2 << N2, Ll2

Physical interpretation of gravity modes

1)

2)

3)

>

~

During phase 2, the element has gone up

and is heavier than the surrounding medium :

Archimedes force downwards.

But the pressure gradient ensures equilibrium

along the radial axis :

the horizontal

pressure gradient pushes back the element

towards its initial state.

~

>

Gravity modes

s2 << N2, Ll2

1)

2)

3)

>

~

The Archimedes force

is the restoring force

The pressure gradient acts as a

force transmitter: although the

restoring force is radial,

the motions are mainly horizontal.

~

>

Gravity modes

Physical interpretation of gravity modes

N

L5L20

The Sun

p cavities

L1

g Cavity

Radiative core Convective envelope

s2 < N2, Ll2

Gravity modes have low frequencies,

they propagate in the deep layers (radiative core) of stars

Gravity modes

The spectrum of gravity mode frequencies depends on the Brunt-Vaïsälä frequency N(r)

Mean molecular weight gradient

Very precise probing of the

internal chemical composition

Super-adiabatic gradient

Probing of the temperature gradient

Derivatives Precise and localized constraints on the internal structure

s2 << N2, Ll2Gravity modes

Spectrum of ~ equidistant periods

1

The main quantity measured by asteroseismology

of g-modes is :

s2 << N2, Ll2

JWKB approximation :

Boundary conditions : : Cavity where s2 < N2, Ll2

Gravity modes

Brunt-Väisälä frequency for different types of stars

1) Main sequence (M > 1.5 M0)

N tdyn

X

N tdyn

r/R

m/M

m/M

Large values of N2 in the mean molecular-weight gradient region above the receding convective core.

N/r sg

Contracting core

g-modes frequencies enable us to measure the size of the convective core and probe the mean molecular-weight gradient region just above it.

d Scuti, 2.2 M¯

s2 << N2, Ll2

The g-modes frequencies increase as the star evolves (contrary to p-modes).

Gravity modes

2) Post-main sequence

Huge values of N

Very dense core

(N tdyn)2

Very dense frequency spectrum

of non-radial modes

Huge

B supergiant



Discontinuities of chemical composition :(H He, He C-O)

Peaks of N

Seismic probing of the envelope’s chemical composition

N towards the center :

g-modes probe the envelope !

Degenerated core : ~ P()

dlnP/dln ~ ln P/ln |s

3) White dwarfs



The internal chemical composition of a white dwarf revealed by asteroseismology !

Giammichele et al. 2018, Nature


p7

p6

p5

p4

p3

p2

p1

fg1

g2

g3 g4 g5 g6 g7 g8 g9 g10

As the star evolves :

R (Expansion of the envelope)

sp / (s0R dr/c)-1 / R-3/2

N/r (Contraction of the core)

sg / s N/r dr

The spectra of p- and g-modes cross : Avoided crossings

d Scuti

Mixed modes

Frequencies of p-modes:

Frequencies of g modes :

l = 2

s2 > N2, Ll2 in the envelope Pressure mode behavior

s2 < N2, Ll2 in the deep layers Gravity mode behavior

Cephei, d Scuti, evolved stars, …

N

g mode

p mode

L1

Mixed modes

p7

p6

p5

p4

p3

p2

p1

fg1

g2

g3 g4 g5 g6 g7 g8 g9 g10

Because of avoided crossings and the presence of mixed modes :

Desorganised spectrum

One observed some modes, who are they ?

Impossible to know based on the frequencies alone …

Need of mode identification

techniques based on other

observables

(line profile variations, …).

l = 2

Mixed modes

Fitting~100 peaks

Fitting~2000 peaks

X10

X10

A delta Scuti star observed by the CoRoT satellite:

More than hundreds of detected modes without regularity.

These stars have a high seismic potential but their study is very difficult :

- No structure of the spectrum(mixed modes)

difficult mode identification.

- Very fast rotationcomplex modeling.

Mixed modes

How red giants differ from the Sun ?

l = 0

l = 1

l = 2

Regular frequency pattern ~ Asymptotic

Small separation Large separation

Sun

Modes de pression (acoustiques)

N

L5L20L1


p Cavity

Acoustic structure of a red giantVery high density contrast between the core and the envelope

Mixed modes

Evolution core contraction N

Very high restoring force (Archimedes)

The mode behaves

here as a gravity mode

The mode behaves here

as a pressure mode

The two cavities are coupled through the evanescent zone

Mixed modes exhibiting the p- and g-like behaviors

The size of the evanescent zone increases with l

Huge number of

nodes in the core

Eigenfunctions similar

to classical p-modes in

the envelope

From Christensen-Dalsgaard (2004)

Acoustic structure of red giants and mode trapping

Mode trapping in red giants

Mode trapped in the core

Mode trapped in the envelope

Inertia of a mode

M=2 M¯, Log(L/L¯)=1.8, pre-He burning

Mixed modes

M=2 M¯, Log(L/L¯)=1.8, pre-He burning

Modes trapped

In the core

Modes trapped

in the envelope


Mixed modes

Mixed modesMixed modes in a red giant

l=1 p-g avoided crossings

Observed

modes

From Bedding et al. (2011)


Mixed modes

Mixed modes in red giants

- Stade évolutif

- Masse du cœur d’hélium

- Composition chimique interne

-1

He burning

H-shell burning

Variational principle

Reminder : oscillation equations :

Radial motion

Horizontal motion

Mass conservation

Adiabatic relation

Poisson equation


, and are linear functions of

is a linear operator

One defines the scalar product :



Scalar product :

The operator is self-adjoint :

One defines the function :


The function is stationary

on the eigenspaces with S = si2



• The eigenvalues ¾i2 are real

• The eigenfunctions are orthogonal :

The variational principle will be very useful to establish the link

between the spectrum of oscillation frequencies { ¾i } and the internal

structure of stars : sound speed c(r), internal rotation (r), …



• The function is stationary

on the eigenspaces with S = si2

Question : Impact on the frequency spectrum { ¾i } ?

Consider a small modification of the internal equilibrium structure of a star :

c0(r) c0(r) + c(r)

0(r) 0(r) + (r)

0(r) 0(r) + (r) …

Let { ¾i0 , »i0 } be the frequencies and eigenfunctions of the initial model (c0(r), …)

and the operator such that

Modificaton of c(r), … The coefficients of the equations are modified

Modification of the frequencies :



Modification of the frequencies :

Demonstration :



Variational principle for te organ pipe

kernel

P0 constant, c02(r)

Inverse problem

Given measured frequencies : ¾i,obs

Given reference model with theoretical frequencies : ¾i,th

Question : Which change ¢c2(r) gives the agreement theory – observations

Answer : One has to solve the inverse problem :

Known Unknown function

Known

JWKB approximation:

Boundary conditions

For large n, the frequencies are equidistant.

The knowledge of 0 or immediately gives :

This is the acoustic radius.


C Non-constant


Inverse problems

General formulation :

For given functions b(x) and K(x,y), find f(y) such that

Examples :

Kernel

Deconvolution :

Earth seismology :

ds

r0

R(µ)

One mesures the arrival times of seismic waves

at different points of the globe .

One deduces the internal P and S waves speed

profiles by solving the inverse problem :

µ

For given functions b(x) and g(x), find f(y) such that

Inverse problem: non-radial stellar oscillations

p modes sound speed profile and internal chemical composition

Given measured frequencies : ¾n,l,obs

Given reference model with theoretical frequencies : ¾n,l,th

Inverse problem :

Given Sound speed and density profiles to determine

Equation of state : c2 = c2(½, P, Y, Z)

One can probe the internal composition profile and thus constrain the macroscopic

(rotation, …) and microscopic (diffusion, …) transport processes of chemicals.

small

From the equation of hydrostatic equilibrium, ¢P/P can be determined :

Helioseismology

Examples of kernels (l=0, n=21)¢c2/c2 and ¢½/½ obtained by seismic

inversion inversion from solar p-modes

Inverse problem: non-radial stellar oscillations

Signatures of sharp variations: the organ pipe

Effect of a local change of the sound speed:

JWKB approximation:

Seismic signature of the local change : oscillating component in

~

n

Dirac

Signatures of sharp variations: g-modes (SPB, g Dor)

Sharp variation of N at r

Periodicity of P(k) :


Amplitude of the discontinuity

Localisation of the transition

Indicator of Xc and thusthe evolution stage

m(r)

m(r0)

Constraints on extra-mixing due to rotation, g-modes (SPB, g Dor)

The turbulence induced by

differential rotation smoothens the

X profile above the convective core

Hence, it smoothens the Brunt-

Vaïsälä frequency profile

The turbulence induced

by differential rotation

decreases the amplitude

of the periodic signal in

P(k)

Constraints on extra-mixing due to rotation, g-modes (SPB, g Dor)

The turbulence induced

by differential rotation

smoothens the X profile

above the convective

core

Direct method in asteroseismology

For most stars, except the Sun, not enough detected and identified frequencies for using seismic inversion techniques.

Direct seismic methodObservational data :

- Observed frequencies : {ºobs,i}.- Non-seismic constraints, ex : L/L¯, Teff , log g (color indices,

spectroscopy), surface chemical composition (spectroscopy), R (interferometry), M (binarity), … with error bars !

Goal :To find a stellar model matching as well as possible the seismic and non-seismic measurements.

Method :

One reduces the solution space by assuming that the stellar structure models only depend on a finite number of free parameters {ai} such as : the stellar mass, age, initial mass fractions of hydrogen (X) and heavy elements (Z), mixing-length (®) and overshooting (®ov) parameters, …One determines the values of these parameters providing a model matching as well as possible the observations through the minimization of a merit function.

Set of free parameters

a1, a2, …, aM

Minimization of :

Theoretical frequencies Observed frequencies

Méthode :One reduces the solution space by assuming that the stellar structure

models only depend on a finite number of free parameters {ai} such as : the stellar mass, age, initial mass fractions of hydrogen (X) and heavy elements (Z),

mixing-length (®) and overshooting (®ov) parameters, …One determines the values of these parameters providing a model

matching as well as possible the observations through the minimization of a merit function.

Computation of the

corresponding structure model

Computation of the oscillation

frequencies by solving:

2 evaluation and choice of a new set of free parameters

Iteration

Direct method in asteroseismology

Rotation Reminder, oscillation equations without rotation :

- Spherical symmetry: the equilibrium quantities only depend on r .

- The only angular differential operator appearing in these equations is :

-

Partial derivative equations Variable separation (spherical harmonics)ordinary differential equations

Variable separation because :


Rotation

Analogy with quantum physics :

Bohr model of the hydrogen atom – Non-radial oscillations without rotation :

For a fixed l, the energy levels (resp. the oscillation frequencies) are identical for

the 2l+1 possible values of m (-l · m · l).

Without rotation, the equations and thus the solutions (frequencies, …) do

not depend on the azimutal order m. For a fixed l , they are the same for

the 2l+1 possible values of m (-l · m · l).

This degenerecy of the solutions comes from the spherical symmetry.


- Spherical symmetry: the equilibrium quantities only depend on r .

- The only angular differential operator appearing in these equations is :

-

Variable separation because :

Reminder, oscillation equations without rotation :

Rotation breaks the spherical symmetry

It lifts the m degeneracy

Rotation modifies the equations of the problem.

In a corotating frame, 2 additional (fictious) forces

appear in the equation of momentum conservation :

the centrifugal force and the Coriolis force.


Rotation

1) The « Doppler effect » due to rotation

Rigid rotation models (Oscillations around the) corotating frame.

Approximation : One neglects the centrifugal and Coriolis forces

Same equations as before from the point of view of the corotating frame.

Inertial frame: Corotating frame :

Spherical coord. (r, µ, Á) Spherical coord. (r’, µ’, Á’) = (r, µ, Á - t)

Angular rotation speed

Same equations in the corotating frame

same solutions :¾0 : frequency

without rotationIn the inertial coordinates system :

From the observer’s point of view, the measured frequency is:


Rotation

1) The « Doppler effect » due to rotation

From the observer’s point of view, the measured frequency is :

Rotational splitting

triplet quintupletl = 1 l = 2

…Frequencies :

m = 1 0 -1 2 1 0 -1 -2

The rotational « splittings » give a

measure of the rotation


Rotation

1st order perturbative treatmentRotation :

Centrifugal force neglected spherical symmetry

Corotating frame where the frequency is ¾c

Coriolis acceleration taken into account:

Equation of motion (in the corotating frame) :

Without rotation With rotation (corotating frame)

Effect of rotation treated as a small perturbation of the problem to solve :

The variational principle can be used to determine the

effect of rotation (Coriolis) on the frequencies.


In the corotating frame (acceptable if rotation is rigid) :



Without rotation With rotation (corotating frame)

Effect of rotation treated as a small perturbation of the problem to solve :

The variational principle can be used to determine the

effect of rotation (Coriolis) on the frequencies.

From the point of view of the observer, the measured frequency is :

Ledoux constant

p modes :

g modes :

Differential rotation (r)

From the point of view of the observer :



In the corotating frame (acceptable if rotation is rigid) :

In the inertial frame, (r) :



triplet quintupletl = 1 l = 2

…Frequencies :

m = 1 0 -1 2 1 0 -1 -2

Inverse problem

Rotational multiplets : ¢¾i

One searches (r) such that



Sun, general case (r, µ)

Internal rotation profile in the Sun obtained by helioseismic inversion

• Differential rotation

in the envelope

• Rigid rotation in the

radiative core

Due to the transport of

angular momentum by

gravity waves

(Charbonnel & Talon 2005)

Inversion of rotation

Probing internal rotation :Kernels

HD 129929

Eridani

Cephei

2 observed multiplets but they probe

rotation in different layers because

their kernels are very different.

core ~ 3 surf

However, in other stars (ex: µ Oph),

rigid rotation cannot be rejected.

Action of the magnetic field ?

The determination of (r) provides contraints on angular

momentum transport processes.

Result here :

Effects of fast rotation

1. If rot2 << GM/R3

The star is significantly flattened by the centrifugal force

2. If rot << s

Significant impact of the Coriolis force on the

oscillations dynamics Gravito-Inertial modes

- The rotational multiplets are no

longer equidistant. 1st order

2nd order

- The dependence in µ, Á of the

eigenfunctions is no longer given

by spherical harmonics.

Effect of fast rotation on a multiplet

The traditional approximation (gravito-inertial modes)

One neglects the centrifugal force spherical symmetry

We take the corotating frame as reference where the (ang.) frequency is ¾c

We take the Coriolis acceleration into account:

Equation of motion (in the corotating frame) :

We consider gravito-inertial modes for which << sc

The perturbative approach is not justified for the modeling of

the effect of rotation on frequencies.

Non-perturbative approach required


Eq. of motion :

One considers gravito-inertial modes for which

radial displacements << horizontal displacements

- One neglects the radial component of the Coriolis force

- One neglects the radial displacement in the horizontal

component of the Coriolis force one neglects q

- One neglects ‘ (Cowling approximation)



One isolates xq and xf :

Substitutes them in the continuity equations:



Laplace tidal equation

Method of variables separation, one searches solutions of the form :

Solutions : Hough functions



Differential equations of the problem :

This is the nearly the same problem as before, l(l+1) has been replaced by l !

Same properties of the solutions for gravito-inertial modes :

Propagation cavity :

Local wave number :

Asymptotic periods :

Butis a function of and thus of the frequency as well as m



Why stars are pulsating ?

Why some modes are observed and others are not ?

What determines the amplitudes and phases of oscillations ?

What determines the growth or damping rates of oscillations ?

Can we get additional constraints from other observables

than the frequencies ?

Non-adiabatic asteroseismology enables to address these questions.

Energetical aspects of stellar oscillations

Questions :

Reminder, characteristic times

Helmholtz – Kelvin time-scale : Thermal desequilibrium

Nuclear time-scale : Evolution due to nuclear reactions

Dynamical time-scale : Forces desequilibrium

One has always : tdyn << tHK << tnuc

Global reasoning: At the time-scale of oscillations, the gas has no time

to exchange significant heat compared to the global heat capacity of the star.


|dQ| << u M ~ cv T M In the main part of the star :

oscillations are adiabatic

Q : Heat exchanged during one oscillation cycle

Virial theorem:

Sun : 26 min 107 ans 1010 ans


|Q| << u M ~ cv T M

Local reasoning: Same reasoning but for the superficial layers only.

We have seen that globally :

But in near-surface layers, the heat exchanged per unit mass is of the same order as

the internal energy:

The heat capacity of the superficial layers is of the same order (or smaller)

than the heat exchanged during one oscillation cycle.

Oscillations are thus non-adiabatic there.

Global reasoning: At the time-scale of oscillations, the gas has no time

to exchange significant heat compared to the global heat capacity of the star.


Q : Heat exchanged during one oscillation cycle

Virial theorem:

In the main part of the star :

oscillations are adiabatic

log tth: Thermal relaxation time

Quasi-adiabatic region :

¿th >> Period

In the main part of the star in term of mass,

oscillations are adiabatic.

The frequencies are accurately computed in

the frame of the adiabatic approximation.

Non-adiabatic

region :

¿th < Period

ds/cp 0

We will see that the main excitation

of the oscillation modes occurs in

the transition zone where the

thermal relaxation time and the

periods are of the same order.

I define the thermal relaxation time as :

In the superficial layers, the

oscillations are non –adiabatic :

To model oscillations there, one has

to solve the coupled

- Dynamical equations :

Conservation of momentum

and mass, Poisson equation.

- Thermal equations :

Conservation and transport

of energy.


Radial non-adiabatic oscillations

Equations of the problem

Oscillations around a model in thermal equilibrium :



:

Conservation of energy :


:

Unknowns :

Transfer of energy (radiative zone) :

Equations of state :

Nb. of equations : 4



The characteristic times associated to each intermediate reaction are very different.

Nuclear reactions :

g+→+ HeHH 3

2

2

1

1

1

HHeHeHe 1

1

4

2

3

2

3

2 2+→+

Examples : - Deuterium burning :

- He3 burning :

Very short reaction time tD ~ 1 sec << Oscillation period

Out of equilibrium, ¿He3 >> Périod of oscillation

During the oscillations, the reactions are out of equilibrium.

The variation of the abundances of nuclei depends on the oscillation period.

One cannot simply perturb the equilibrium equation :

The perturbation of ² depends on the frequency :


Solutions in the complex plane

i in the energy equation :

The solutions of the problem are complex :

¾ and dX(r) are complex

For classical pulsators, one expects that the observed modes are vibrationally unstable

Mean to confront theory and observations !

h is the mode’s growth rate:

− h > O The mode is vibrationally unstable.

− h < 0 The mode is vibrationally stable.

The vibrational instability is a form of thermal instability dynamical instability.

Associated time-scale = Helmholtz-Kelvin time-scale (GM2/(RL)) >> dynamical time.

h / sR ~ tdyn / tHK << 1


¾ and dX(r) are complex

f(r) is the phase of the considered quantity.

It varies as a function of depth and from a quantity to another

Phase shifts between the different quantities.

Examples :

- Phase-shift between the light curve (photometric variability) and

velocity curve (Spectroscopic Doppler effect).

- Phase-shifts between photometric variations in different passbands

(Photometric multi-color filters of Johnson, Strömgren, …)

Other means to confront theory and observations !


Solutions in the complex plane :

Work and energy

Energy of a mode :

Adiabatic oscillations :

Kinetic energy (/ u. vol.) :

Adiabatic oscillations are conservative

The energy of the mode is conserved in each layer.

Potential energy (/ u. vol.) :

Total energy (/ u. vol.) :

Global mode Energy (integrated over the full volume) :


Work performed during one oscillation cycle (/ u. mass) :Non-adiabatic

oscillationsphase-shifts

Global work performed during one oscillation cycle :

Mean power :


Work and energy

The work integral :

Equation of motion and integration :

Imaginary part of the result :


Work and energy

This result is easily understood :

With (the mean power = the increase of energy / u. time),

one recover the work integral.


Work and energy

The work integral :

The work integral, thermal analysis

The vibrational stability is associated to the non-adiabatic character of oscillations.


Work and energy

Thermodynamical cycle

Condition for vibrational instability: during the hot phase of pulsation :

the gas must receive heat from the surrounding : .

This is a well-known property of thermodynamical cycles :

Condition for a motor thermodynamical cycle :

Heat must be received by the system at the hot phase and

provided outside at the cold phase.

Classical example : the Carnot cycle.


Work and energy

Heat out at the Cold phase

Heat in at the Hot phasePositive work

Work = s P dV= Area

Carnot cycle

Isothermal :

PV = cst.

dQ = p dV

Adiabatic :

PVg

= cst.

-cv dT = p dV

The - mechanism

We focus now on the envelope where = 0 and examine the

conditions for at the hot phase.

For the - mechanism, the opacity variations produce the

« heat trapping » at the hot phase.

Term at the origin of the - mechanism


Condition for vibrational instability: during the hot phase of pulsation :

the gas must receive heat from the surrounding : .

Modes’ driving : The - mechanism

log T

κ

SurfaceCenter

Opacity peak due to a partial ionization zone :

He+ He++ (log T ~ 4.5, Cepheids, RR Lyrae, d Scuti)

Fe, level M (log T ~ 5.2, Cephei, SPB, sdB)

The opacity mainly depends on the temperature

T

κ

Hot phase

Surface

Hot phase : heat enters but cannot go out.



Center

Opacity peak due to a partial ionization zone :



κ

Surface

Cold phase : heat cannot enter but goes out easily.

T

Cold phase


Center


Opacity bump due to a partial ionization zone :



• Around the opacity bump:

Positive work excitation of the mode (input of mechanical energy)

Region having the largest weight (excitation or damping) :

Transition region where Thermal relaxation time ~ Pulsation period

• Out of the opacity bump :

Negative work damping of the mode (loss of mechanical energy)

Final result : Vibrational stability (damping of the mode)

or instability (excitation of the mode)

depending on the weight of the opacity bump region .


The heat capacity is very low compared

to the heat exchanges due to oscillations :

Superficial regions

Heat provided to the gas / u. time there :

does not vary significantly between the base of this region and the surface.

These superficial regions have a

small weight in the work integral,

provided heat negligible.

negligible work. Small near the surface

Thermal balance :




Deep regions

Large in the deep regions.

The deep layers have a small weight

in the work integral because the

variations of the heat flux due to

oscillations are small there. Small in the deep layers

The amplitude of the eigenfunctions is much smaller there than near the surface.

Ex : approx. JWKB :

In particular, and are very small there small heat exchanges,

quasi-adiabatic oscillations negligible work.




For efficient excitation (driving) of the oscillations, the

opacity bump must be located in the transition region.


• Around the opacity bump:

Positive work excitation of the mode (input of mechanical energy)

• Out of the opacity bump :

Negative work damping of the mode (loss of mechanical energy)



- mechanism: Blue border of the instability strip

Too hot model

κ

Centre

Tra

nsit

ion

Surface

κ

SurfaceCentre

Tra

nsit

ion

Model inside the instability strip Log Teff 4 3.5

He+ He++

Log Teff 4 3.5

In a Cephei,The Fe opacity bump is near

the surface it coincides

with the transition zone of

(low n) p-modes .

Frequency transition zone closer to the surface

g50

Log t

em

ps therm

ique

(Fe bump, layer M)

- mechanism : Pressure and gravity modes

Example : Cephei (pressure) – SPB (gravity)

Log Teff 4 3.5

g50

Log t

em

ps therm

ique

In a SPB, the Fe opacity bump

is deeper It coincides with

the transition zone of high-order

gravity modes.


(Fe bump, layer M)

Example : Cephei (pressure) – SPB (gravity)


Log Teff 4 3.5

Other similar example :

Subdwarfs of B type (sdB) :

Stars having lost their whole envelope

after the helium flash

below the main sequence.

g50

Log t

em

ps therm

ique

Fer opacity bump coincides also with thetransition zone

Pressure modes

Gravity modes



Modes’ excitation strongly depends on the height (and shape) of the opacity bump

The range of observed excited modes enables us to constrain the opacity.

How well do we know opacities ?

1. Uncertainties in opacity computations :

Huge number of transitions (bound-bound, bound-free), inaccuracies in cross-

sections computations (orbit-orbit, spin-orbit,spin-spin couplings, …).

Ex: Before 1990, impossible to explain the driving of Cephei pulsations because

the opacities due to Fe transitions were underestimated (neglected transitions).

Differences between opacity tables remain, inaccuracies are still present.

Non-adiabatic asteroseismology

Contraints on the micro-physics : matter-radiation interaction

- mechanism : non-adiabatic asteroseismology

2. Chemical composition : Abundance of absorbing ions

Ex: Cephei, SPB, sdB : Main contributors to the opacity at T~ 2£ 105 K :

Fe group elements (transitions from the M-layer): Fe, Ni, …

Constraints on chemical transport processes :

Radiative forces, gravitational settling, transport by gravity waves,

turbulence and meridional circulation due to differential rotation, …

Local accumulation of Fer, Ni, … Increases the opacity bump

The driving of the numerous observed modes in many sdB, Cep (in the LMC

& SMC, Z = 0.0025), … is not explained yet !

Modes’ excitation strongly depends on the height (and shape) of the opacity bump

The range of observed excited modes enables us to constrain the opacity.

How well do we know opacities ?

- mechanism : non-adiabatic asteroseismology

But … With a homogeneous chemical composition

= the observed surface one,

impossible to explain

the modes’ driving.Charpinet et al. 2001

– mechanism in the

Fe partial ionization zone

- mechanism, ex : Subdwarfs B (sdB)

damping

driving

Dynamical equilibrium

gravitation = radiative forces

Fe

Fe

Fe

Charpinet et al. 2001

Fe

Diffusive time¿

Evolution time

Consequence: local

accumulation of Fe

at T ~ 2 £ 105 K.

Opacity bump and thus

–mechanism reinforced

More excitation

- mechanism, ex : Subdwarfs B (sdB)

Very short wavelength oscillations

Large variations of the T gradient with heat loss at the hot phase and heat gain at the cold phase.

Strong radiative damping

Huge N2

huge

Non-radial modes are strongly damped in the core and cannot be observed

Except modes trapped in the envelope

In post-main sequence evolved stars:

Huge density constrast between the very dense

core and the very diluted envelope

Radiative damping mechanism

l = 1, f = 14.4 c/d ~ mode fondamental radial

xh

xr

2) Post-main sequence

Huge values of N

Very dense core

(N tdyn)2

Very dense frequency spectrum

of non-radial modes

Huge

B supergiant



dT> 0

dT<0

qq

qq

Very short wavelength oscillations

Large variations of the T gradient with heat loss at the hot phase and heat gain at the cold phase.

Strong radiative damping

huge

Non-radial modes are strongly damped in the core and cannot be observed

In post-main sequence evolved stars:

Huge density constrast between the very dense

core and the very diluted envelope

Except modes trapped in the envelope

Radiative damping mechanism

Huge N2

1. In g Dor stars, around the red border of the instability strip,

in solar-type stars, red giants, white dwarfs :

Transition zone inside the convective envelope.

Convection plays a key role in the modes’ excitation and damping.

In all cases (except part of g Dor stars) :

Convective time-scales < ~ pulsation periods

2.

Important coupling between convection and oscillations

No good theory…

Comparaison with observations :

Frequency interval of excited modes,

Instability strips (red border) …

Contraints on convection

and its time-dependent

interaction with oscillations

Convection : effect on the modes’ excitation and damping

A simple case: Convective blocking

g Doradus

The transition regioncoincides with the base of the convective envelope.

The heat exchanges in this region are at the origin of the g-modes’ driving.

Radiative zone Convective envelope

At the hot pase of the oscillations, rT more heat enters in the

convective envelope.

Heat

Convection does not adapt instantaneously to the oscillations.

Heat flux is blocked

The heat provided to the base of the convective envelope is converted into mechanical energy of the g-modes.

Excitation

Type F, main sequence, g-modes

Instability regionsFor efficient driving, the transition region must coincide with the base of the convective envelope.

l = 1

The size of the convective envelope is not know accurately.

It can be constrained by comparing the theoretical and observed instability regions.

Mixing length parameter = 1, 1.5, 2

Teff Convective envelope’s size

Convective envelope’s size Instability region moves

to the red (Teff )

A simple case: Convective blocking

g Doradus Type F, main sequence, g-modes

Another “simple” case: White dwarfs

Teff = 19830 K

Teff = 26790 K

Teff = 22040 K

tconv (s)

Convective time-scales (~ 1 s)

Convection adapts quasi-instantaneously to the oscillations

Models of time-dependent convection

are simpler.

However, the red borders of

instability strips are still unexplained.

Convective time-scales

Periods of pulsation

Pulsation periods (g-modes, P ~ 500 s)<<

Stochastic excitation : Solar-like oscillations

Oscillations are damped (coherent interaction with convection)

h : damping rate, 1/h : mode’s lifetime

The Sun seen as a drum

M 2

2

h

PVs =

Turbulent convective motions at the top of the convective envelope

hit the drum (stochastic forcing).

P : power injected in the acoustic modes by turbulence

Power injected by turbulence

Damping rateMode « Mass » :

M

Mean quadratic speed of the mode

The problem of mode identificationIn all pulsating stars except solar-like stare :

mode identification (l, m, n ??) is difficult …

Phase-lags

Line-profile variations Amplitude ratios

Very sensitive to the non-adiabatic

modeling of time-dependent convection

Spectroscopy Photometry

Additional observables for mode identification

Monochromatic magnitude variationsdue to non-radial oscillations

• The distorsion of the stellar surface corresponds to the material

displacement at the photosphere

• Local variations of the

mono-chromatic flux:

• Limb darkening :

• Thermal equilibrium in the atmosphere

Hypotheses :

To integrate over the entire visible disk

• Non-radial modes described by spherical harmonics

OK for slow rotation

Integration of the intensity over the entire visible disk


Disc averaging factor

+

−

+

+

+ +−−

−=

+

+

)( cos ln

ln

ln

ln

) ( cos ln

ln

ln

ln )( cos )2)(1(

)(cos 10ln

2.5

e

e

eff

eff

eff

eff

tg

g

g

b

g

F

tT

T

T

b

T

Ft

bi Pm

T

m

sd

sd

s

d

ll

ll

ll

l

l

ll

ll

Inclination

Averaging over the visible disc

Small l only (l · 4)can be detected in photometry

Averaging over the visible disc

+

−

+

+

+ +−−

−=

+

+

)( cos ln

ln

ln

ln

) ( cos ln

ln

ln

ln )( cos )2)(1(

)(cos 10ln

2.5

e

e

eff

eff

eff

eff

tg

g

g

b

g

F

tT

T

T

b

T

Ft

bi Pm

T

m

sd

sd

s

d

ll

ll

ll

l

l

ll

ll


Non-adiabatic local variations of Teff

Distorsion of the stellar surface

From a grid of atmosphere modelsFilters Integration over the passband

Linear computations Amplitude ratios and phase shifts

l dependence Identification of l

Observations in

multi-color

photometry

Amplitude ratios

& pase shifts

Frequency range

Mode identification

Multi-color photometry and non-adiabatic asteroseismology

Better agreement

Stellar parameters

d Scuti

g Doradus

White dwarfs

Cephei

SPB

sdB

Convection

Chemical composition (Fe, Ni)

Atmosphere models - Limb darkening

Time dependence

Mixing-length ()

Non-adiabatic

computations

Equilibrium models,

Stellar evolution code

Thank you …

1 Monocerotis

Frozen convectionTime-dependent convection

Spectro-photometric amplitudes and phasesand mode identification

d Scuti

Simultaneous photometry and spectroscopy (Balona et al. 2001)

Time-dependent convection models

better agree with observations !

(Dupret et al. 2005)

Solar-type stars

A zoo of pulsating starswith very different physical characteristics

Pressure modes

Periods :3-8 min.

Large uncertainties

in the models of

coherent interaction

convection-oscillations

g Doradus


Gravity modes

Periods : 0.3-3 days

Role of turbulent viscosity ?

Role of rotation ?

Classicalinstability strip


Blue edge,

-mechanism (HeII): OK

Red edge : interaction

Convection-oscillations:

not fully understood

roAp


High frequencies (4-20 Min.)

“-mechanism”

“H”-ionization zone

Very sensitive to the

eigenfunctions shape.

Role of magnetic field ?

CepheiSlowly Pulsating B


-mechanism Fe M-shell

Possible constraints

on opacities,

Fe local abundances

Transport mechanisms

Subdwarf B

Red giants


1) High amplitude Radial modes

2) Moderate amplitude radial and

non-radial modes excited by

turbulent motions.

Superb discovery

of COROT !!

White dwarfs


DAV, DBV :Convective excitation

GW Virginis:−mechanism C-O K-shell

The blue edge location

depends on the Carbon and Oxygen abundances

The red edge depends onmass-loss-gravitational settling

Excitation mechanisms

• Instability strips

• Range of excited modes

(pressure or gravity modes ?)

Spectro-photometric

amplitude ratios

and phase-lags

Very useful for the

mode identification

Non-adiabatic oscillations

1. Excitation mechanisms

Self excitation

Stars as heat engines

Amplitude

until limit cycle reached

Cepheids, d Scuti, Cep, SPBs, g Dor, sdBs, …

• - mechanism

• Convective blocking

• mechanism

Stochastic excitation

Damped oscillation

+

Forcing term

(turbulent motions)

« Like a drum »

Solar-like oscillations

Excitation mechanisms : stars as heat engines

In a star (radial mode) :

Heat released atthe cold phase

Heat given atthe hot phase

Positive work

Mechanical energy of the

mode :

But … With homogeneous

chemical composition,

impossible to explain the

excitation of the modes.Charpinet et al. 2001

Main driving in the M-shell ionization zone of Fe

- mechanism : subdwarf B stars

damping

driving

Equilibrium between

radiative levitation and

gravitational settling

Fe

Fe

Fe

Charpinet et al. 2001

Fe

Diffusion time¿

Evolution time

Driving only possible if the

Fe is locally overabundant

at log T ~ 5.2

Thanks to radiative

levitation

Main driving in the M-shell ionization zone of Fe

- mechanism : subdwarf B stars

Huge number of

nodes in the core

Eigenfunctions similar

to classical p-modes in

the envelope

From Christensen-Dalsgaard (2004)

Acoustic structure of red giants and mode trapping

Seismic diagnostics from modes

trapped in the core

+ center

+ anticenterHe burning

CoRoT (Mosser et al. 2011)Kepler (Bedding et al. 2011)

Huge potential to probe the deepest layers :

- Determination of the evolutionary stage

- Measure of the He core mass (Montalban et al. 2012)

- Signatures of the -gradient in the core

H-shell burning

Period spacing :

Stellar stability and asteroseismologydupret/Stab-et-astero-ang.pdfand the internal physical...

Documents

Transcript of Stellar stability and asteroseismologydupret/Stab-et-astero-ang.pdfand the internal physical...