# Presentation: 874352

## Using Pulsations to Test Stellar Evolution Models

***Joyce A. Guzik***
***Thermonuclear Applications Group, X-2***
**Los Alamos National Laboratory**
***April 15, 2002***

## Outline

**Pulsating star locations on the HR Diagram**
**Pulsation modes and driving**
- Period-mean density relation
- nonradial oscillations, p-modes and g-modes
- kappa/gamma effect driving
**Solar oscillations (helioseismology)**
** ****Attempts at asteroseismology (example:  delta Scuti star FG Vir)**

## From

- Christensen-
- Dalsgaard
- 2000

## Period-Mean Density Relation

- For homologous stars (same density stratification),
- oscillation frequencies scale approximately
- as sqrt(mean density)

- = dynamical frequency ~ 1 / free fall time over the distance
- of a stellar radius ~ fundamental model frequency

## Assume spherical symmetry, linear perturbations, sinusoidal time dependence
Allow both radial and horizontal motions
Solve coupled equations of continuity, Poisson, and energy
sometimes make adiabatic approx. and Cowling approx. (neglect gravitational potential perturbations) to simplify solutions
Solutions exist for discrete values of frequency (eigenvalues of equations)

- Allow both radial and horizontal motions
- Solve coupled equations of continuity, Poisson, and energy
- sometimes make adiabatic approx. and Cowling approx. (neglect gravitational potential perturbations) to simplify solutions
- Solutions exist for discrete values of frequency (eigenvalues of equations)

***Solutions of stellar pulsation equations (1)***

- See, e.g., Cox 1974; Hansen and Kawaler 1986;
- Christensen-Dalsgaard and Dziembowski 2000

## Solutions of stellar pulsation equations (2)

- Pressure perturbation, e.g., can be written in form with separated variables in spherical coordinates as:

- From equation of motion, solution for displacement vector has the form:

## Solutions of stellar pulsation equations (3)

- and

- are radial and horizontal displacements,
- and are functions of

- Lamb Frequency

- Brunt-Vaisala or
- buoyancy frequency

- These two frequencies determine the propagation regions
- for pressure (acoustic) and gravity waves within a star

## From J.P. Cox 1974

## p-and g-mode

- propagation
- regions for
- ZAMS solar model
- from Hansen and
- Kawaler 1994

## Spherical Harmonics

- From Toomre 1984

## Solar radial eigenfuction amplitudes

***Solar radial eigenfuction amplitudes***

## Pulsation driving by kappa/gamma effect (1)

- Radial and nonradial oscillations of most types of variable stars  are driven by the kappa/gamma (kappa as in opacity, gamma as in adiabatic indices) effect, due to the effects of ionization of H, He, C, O, or Fe in stellar envelopes.
- H, He  ionizes ~10,000 K
- He+ ionizes ~50,000 K
-  -enhancement due to Fe ionization ~ 200,000 K

- See, e.g., Bohm-Vitense 1992; Hansen and Kawaler 1986; Mihalas and Binney 1981

## Pulsation driving by kappa/gamma effect (2)

- Linearizing the diffusion equation, and assuming a power-law opacity:

- For a non-ionizing region, 3 = 5/3, s~3.5, n~1, and

- When region is compressed and heated, more radiation leaks out.

- For an ionization region, 3 --> 1, s < 0, and (for +C)

- Less radiation leaks out when a region is
- compressed and heated, and radiation is dammed up.

## Kappa Effect Driving 

- Fe

- T, 

- 

- He+

- H, He

- *

**Stellar Surface**

**Stellar Interior**

- Layer compressed

- Layer expands

- Radiation blocked,
- pressure increases

- T, 

- T, 

- Radiation escapes,
- pressure decreases

## Particular oscillation modes can be stable or unstable depending on whether:
region of star contributing significantly to mechanical response is in a driving region
driving in one region is not overcome by radiative damping in another region (integrated work/zone)
The kappa/gamma effect is weakened or disabled if convection can instead transport enough of the luminosity in the ionization region. 
	(Mechanisms such as “convective driving” and “convective blocking” have also been proposed to explain pulsations of white dwarf and  Dor stars)

- region of star contributing significantly to mechanical response is in a driving region
- driving in one region is not overcome by radiative damping in another region (integrated work/zone)
- The kappa/gamma effect is weakened or disabled if convection can instead transport enough of the luminosity in the ionization region.
- 	(Mechanisms such as “convective driving” and “convective blocking” have also been proposed to explain pulsations of white dwarf and  Dor stars)

***Pulsation driving by kappa/gamma effect (3)***

## Example Work plot for  Doradus model g-mode

- Guzik et al. ApJL 542, L57 (2000)

## Helioseismology

**Solar Oscillations Discovered in 1960**
**	****(**Leighton, Noyes & Simon)
**Explained as global resonant acoustic modes of the Sun in 1970 ****(**Ulrich, Leibacher, & Stein)
**Extensive application to test physics of solar models and derive internal structure of Sun in 1980s (helioseismology)**
**Attempt to apply technique to other stars in 1990s (asteroseismology)**

## Doppler velocity observations of solar disk

- From Toomre 1984

## From Toomre 1984

## From

- Christensen-
- Dalsgaard 2000

## Slide 21

## Sound speed differences between model and seismic results using LLNL OPAL and LANL LEDCOP opacities

- Neuforge-Verheecke et al., ApJ 561, 450 (2001)

## From Christensen-Dalsgaard 

- From Christensen-Dalsgaard
- et al. 1991

## From Chaplin et al. 1999

- Solar Interior Rotation as inferred
- from inversions at 0°, 30° and 60° latitude

## What did we learn from helioseismology? 

**Convection zone helium abundance**
**0.248 ± 0.001 **(e.g. Basu 1998)
**Radius of convection zone base**
**0.7135 ± 0.0005 R******(e.g. Basu 1998)
**Sound speed versus depth**
**Agrees with modern standard models to within 1 %**
- (e.g. Basu et al. 1998)
**Evidence for diffusive settling of about 10% of initial helium from convection zone during Sun’s lifetime**
**Rotation profile versus latitude and depth**
**	****(e.g. Chaplin et al. 1999)**

## Evidence for Diffusive Helium Settling

**Diffusion calculations predict that about 10% of Helium by mass should settle out of solar convection zone during the Sun’s lifetime**
**Observational Evidence:**
- Convection zone He abundance (Y) determined from signature of He ionization = 0.248 ± 0.001 (Basu, 1997), whereas initial Y needed to calibrate solar model to present solar luminosity is 0.274.
- Much improved agreement between calculated and helioseismically inferred sound speed below solar convection zone (Gabriel 1997) when He diffusion included.
- Direct helioseismic inversion for hydrogen abundance profile (Kosovichev 1995).

## Solar composition profiles near convection zone base including diffusion

- From Christensen-Dalsgaard et al. 2000

## Sound speed differences for solar models with/without diffusion and extra mixing

- From Gabriel 1997

- No
- diffusion

## Inversions to determine solar H abundance gradient

- From Kosovichev 1995

## Helioseismology Results

- Latest opacities and EOS give excellent results in comparisons between ***calculated*** and ***observed**** *solar oscillation frequencies

- Agreement is to within 0.1%, but frequency observations are accurate to 0.01%!

- Guzik and
- Swenson 1997

## Results for  Scuti stars

- Example:  attempted detailed match to FG Vir
- 29 observed frequencies 106 to 395 μHz
-  Evolution models matching observational constraints 	(L, Teff, g, 140.5 μHz mode = radial fundamental) *nonunique*
*No* model found that matches all observed frequencies
- Frequency predictions are sensitive to
- composition and gradients
- extent of convective core and overshoot
- rotation rate and differential rotation
- More modes predicted than observed (problem worse for more evolved  Sct stars)

## FG Vir Evolution Models 

- Guzik et al. in Delta Scuti and related stars, ASP Vol. 210, 2000

## FG Vir Model Properties

- R (R)		2.26			2.31
- M/R3	               	 0.1575		0.1580
- Teff (K)		7368			7412
- log g			3.99			4.00
- log L/L	1.13			1.16
- Xcc			0.257		0.345
- Rcc/R	0.155		0.181
- Mcc/M		0.175		0.220
- Age (Gyr)		0.879		0.731

**1.82 M******** (Z=0.02)**

**1.95 M******** (Z=0.03)**

## 1.82 Mmodel frequencies vs. 1.95 M model frequencies

- Guzik et al. in Delta Scuti and related stars, ASP Vol. 210, 2000

## Observed and

- calculated
- FG Vir frequencies
- (observations
- from Breger et al.
- 1997)