Oscillating White Dwarf Stars

1. Background on White Dwarfs

2. Oscillating White Dwarfs

3. The Whole Earth Telescope

Two nice review articles will be placed on the webpage.

Basic Properties of White Dwarfs

Temperatures: 4 000 – 150 000 K

Nuclear Burning: None all from thermal cooling

Pressure Support: Electron degeneracy pressure

Mass: No larger than the Chandrasekhar limit ~ 1.4 Mּס

At this mass the gravitational support comes from the electron degeneracy pressure. For larger masses the star goes on to become a neutron star, even larger a supernova

Radius: 0.008 and 0.02 R 2.2 – 0.88 : ּס REarth

Surface Gravity: log g ~ 8.4 or 10000 larger than the Sun

Magnetic field: ~ 106 gauss

Note: White Dwarfs are the end fate of our Sun

Spectral Types – White Dwarfs

DA: Hydrogen lines in spectrum

DB: Neutral helium lines dominate

DO: Ionized helium strongest

DZ: Metal lines dominate

DQ: Carbon Features

DX, DXP: Unidentified features (polarization)

Example: DBAQ4 = Star showing He I, H, and C features (in order of decreasing strength) near Teff=12.600 K

+ temperature index = 50400/Teff

This is the spectrum of a DA white dwarf. It looks like an A-type star except the lines are even more pressure broadened due to the high gravity. The few and broad lines means that most studies of pulsating white dwarfs are done with photometry rather than radial velocities

From A. Kawaka

From J. Kaler

The H-R Diagram with Real Stars

White Dwarf as a Chronometers

White dwarfs are excellent „clocks“ for determining the age of galactic regions:

1. The represent the general population of stars: most stars will become white dwarfs.

2. They are homogenous and cover a narrow range of stellar masses: 0.15 < M/M 1.36 > ּס and with a mean mass of 0.593 ± 0.016 Mּס

3. They have the same simple structure, a Carbon/Oxgen core and thin layers of Hydrogen and Helium

4. They do not burn fuel, there luminosity comes entirely through thermal cooling. Thus the temperature of a white dwarf indicates its age.

Log (cool) ≈ Const. – log(L/Lּס)

Age luminosity relationship for WDs:

5

7

Internal Structure of White Dwarfsjayabarathan.wordpress.com

All white dwarfs have the same internal structure: a carbon/oxygen core

The only differences are in the thin layer of gas surrounding the core.

So where did all the hydrogen in the envelope go?

From A. Kawaka

The central star of planetary nebula are on their way to becoming white dwarfs. Clearly there is a lot of mass loss

The Space motion of Sirius A and B

Sirus B is a white dwarf

Mass = 0.98 Mּס

So why do WDs have a Carbon/Oxygen Core?

Answer: Helium Burning

4He + 4He 8Be

8Be + 4He → 12C +

Helium burning, often called the triple alpha process occurs above temperatures of 100.000.000 K. 8Be is unstable and decays back into He in 2.6 × 10–16 secs, but in the stellar interior a small equilibrium of 8Be exists. The 8Be ground state has almost exactly the energy of two alpha particles. In the second step, 8Be + 4He has almost exactly the energy of an excited state of 12C. This resonance greatly increases the chances of Helium fusing and was predicted by Fred Hoyle.

12C + 4He → 16O +

As a side effect some Carbon fuses with Helium to form Oxygen:

So you have nuclear burning that generates Carbon and Oxygen

Typical Internal Structure Model of a WD

Prototype Atmosphere Teff

ZZ Ceti Hydrogen ~12000

V777 Her Helium ~25000

GW Vir He/C/O ~120000

The Classes of White Dwarf Pulsators

There are 3 classes of white dwarf pulsators, divided according to temperature and thus composition of the atmosphere.

The Cepheids lie in the high luminosity end of the instability strip.

The instability strip crosses the main sequence where the Scuti variables are

Extending this further down to lower luminosities this crosses the DA white dwarfs→ from the H-R diagram alone one could have predicted the existence of pulsting white dwarfs

White Dwarfs in the H-R Diagram

ZZ Ceti Stars - Discovery

• In 1968 Arlo Landolt gathered light curves for the star HL Tau as part of other science. He noted a hot star that showed multi-periodic variations with a quasi-period of ~750 secs.

• Surveys of white dwarfs found other pulsating objects (possible due to the new technique of rapid photometry).

• McGraw (1979) for his Ph.D. Thesis, University of Texas, established that all discovered oscillating white dwarfs were isolated DA stars – ZZ Ceti stars

The Nature of the Observed Variations

The time scale for radial pulsations is roughly the sound travel time across the star, or as shown by Eddington, this is the same as the dynamical free fall time scale (about one hour for the sun)

dynamical ~ 1

(G ½

M ~ 1 M 2 ~ ּס x 1033 gm

R ~ 1Rearth ~ 6.4 x 108 cm

Mean density ~ 2 x 106 gm cm–3

dynamical ~ 3 secs!But the time scales of the observed oscillations are several hundred times longer → these cannot be radial pulsations and since the periods are longer than the radial mode (p-mode) they must be gravity modes

d2Rdt2 = GM

R2– = G R

The Nature of the Observed Variations

A good physical argument in favor of g-modes:

p-modes: most of the motion is in the vertical direction

g-modes: most of the motion is in the horizontal direction

White dwarfs have high surface gravities (log g ~ 8). It is very difficult for p-mode oscillations (e.g. radial modes) to overcome the strong gravity.

ZZ Ceti Stars – Driving Mechanism

• In 1981 Don Winget, for his Ph.D. thesis found the driving mechanism. Hydrogen in the outer envelope recombines from the ionized state at an effective temperature of ~12000 K. Hydrogen in going from ionized to neutral state increases its opacity. So, this is the classic mechanism.

V777 Her Stars (DB) – Discovery

• Winget also realized that the mechanism should also occur for He I ionization, or at an effective temperature of 25.000 – i.e. among DB stars.

• Winget predicted the pulsating DB stars, went to the telescope with help of his Texas colleagues and found the first pulsating DB star.

This is one of the few classes of stars that was predicted by theory before their discovery.

In retrospect this discovery should have been obvious!

From Cepheids we know that the mechanism occurs when He I is ionized, similar for Hydrogen. DA pulsators are hot and have hydrogen, so H I mechanism may be at work.

DB stars are hotter and have helium. They should also pulsate with the mechanism applied to He I/II just like in Cepheids

DA DBDO

csep10.phys.utk.edu

Note, these are for normal stars. White dwarfs have atmospheric pressures of 106 dynes/cm

Optical Light Curves of ZZ Ceti Stars

And their Power Spectra

GW Her (PG 1159) Stars – Discovery

• In 1979 McGraw and collaborators discovered pulsations in the WD PG 1159-035 that showed multi-periodic variations like in ZZ Ceti stars, but it was much hotter T~150000 K

• Post Asymptotic giant branch (AGB). A violent mixing event is induced by a helium flash in the post-AGB phase produces an envelope of helium, carbon, and nitrogen. Most likely caused by a mechanism caused by the ionization of K-shell electrons of carbon and oxygen.

Reminder: Post AGB stars have left the Asymptotic Giant Branch and are planetary nebula central stars on their way to becoming White Dwarfs

Post AGB

Optical Light Curves of V77 Her and GW Stars

And sdB stars

Propagation Diagrams for WDs

All stars have a mass of 0.6 Mּס

The solid line is the Brunt-Väisälä Frequency, N and the dashed line the Lamb frequency (sound speed) for l = 1.

Recall when the frequency is less than L, and N you have g-modes

When the frequency is greater than both L and N you have p-modes.

Instability Strips for White Dwarfs

A close up of the instability Strips for ZZ Ceti Stars (DA)

Filled circles are the pulsating stars, open circules are non-variable. Dotted horizontal curves are evolutionary tracks. This looks like a „pure“ instability strip: all Das will become ZZ Ceti pulsators as they cross the instability strip.

The diagonal lines are theoretical predictions of the „blue edge“ using two values of the mixing length parameter for convection

A close up of the instability Strips for V777 Her Stars (DB)

Filled circles: pulsating stars, open circles: constant stars. Horizontal lines are evolutionary tracks. Theoretical models depend on the amount of hydrogen. Left point: pure Helium, right point, pure Hydrogen. Diagonal lines are theoretical predictions for the blue edge of the instability strip fro different assumptions about convection. Bottom line: too few stars to say anything

Less is known about V777 Stars and only 17 are known compared to 136 ZZ Ceti stars. They are a small fraction of the WD population

A close up of the instability Strips for GW Vir Stars (DO)

The small filled circles are pulsating stars, open filled circles are constant, larger symbols are central stars of planetary nebulae (CSPN).

Note the difference to the ZZ Ceti stars as there is a mix of pulsating and non-pulsating stars in the same region of the H-R diagram.

Reason: spread in composition between stars, only the stars with the most carbon and oxygen can pulsate.

Subdwarf B Stars (sdB)

• sdB stars are believed to be core He-burning stars of 0.5 M on the extended horizontal branch that have lost their envelope

• Teff ~ 22.000 – 40.000 K

• Periods 100 – 250 secs

• Period of fundamental mode : P ~ 2860 (ּס/)½ s = 227 s. These are most likely radial modes and thus p-mode oscillations

• Existence predicted by pulsation theory (Charpinet et al. 1997)

Periods on the order of on hour, or 10 times longer than normal sdB stars. Thus these are most likely g-mode pulsators.

Possible new class of pulsator: DQV

White Dwarfs in the gravity-temperature Diagram

Alias periods due to the Spectral Window:

Undersampled periods appear as another period causing false peaks in the power spectrum

f(u)(x–u)du = f *

f(x):

(x):

Window function: Convolution

Window function: Convolution

(x-u)

a1

a2

g(x)a3

a2

a3

a1

Convolution is a smoothing function

In Fourier space the convolution is just the product of the two transforms:

Time Space Fourier Space f*g F x G

Window function: Convolution

The key to understanding the window function is to realize that convolution is symmetric: If you convolve two functions in the time space, you are multiplying in Fourier space. Likewise when you multiply in the time domain, you are convolving in the Fourier domain.

f x g F * G

Time Domain

=

X

=

Simple sinc function

*

Complex: Several sinc functions

Fourier Domain

Amplitude Spectrum of original data

Amplitude Spectrum of Data after removal of dominant frequency. Residual power due to window would not be seen in the noise (this is noise free data)

One-day aliases

For more complicated signals the window function appears superimposed on every real peak in the Fourier spectrum

2 periods2 real peaks

The window function is superimposed on every real peak

Input Periods: 0.13 d, 0.34 d

Output periods: 0.13 d, 0.34 d

In the presence of noise it is sometimes difficult to pick the correct peak. This is the amplitude spectrum of the 2 period data set with noise larger than the signal.

Input Periods: 0.13 d, 0.34 d

Output periods: 0.25 d, 0.116 d, 0.17 d

1/0.34 +1 = 3.94 c/d → P= 0.25

1/0.13 +1 = 8.7 c/d → P= 0.116

One does not recover the true period, but an alias

The window function introduces alias frequencies (periods) into the amplitude spectrum. In the presence of noise an alias peak my be higher than the real peak. The result is you recover an alias frequency, or have additional frequencies that are artifacts of the window function.

To minimize these effects you have to minimize the number of alias peaks (sidelobes). This can only done by „closing up the gaps“. However for stellar observations the sun gets in the way so you always have 1-day aliases.

Either you go into space, or….

A good window function is a pure sinc-function with low sidelobes (note: this is amplitude spectrum, for power you would square the values)

For infinite time coverage your window function is a delta-function

The Whole Earth Telescope: Making the Window Function as Simple as

PossibleThe idea of Ed Nather (left)

A WET Lightcurve

With luck you can get 24 hrs coverage

Sometimes the weather does not cooperate

And sometimes you do not get enough telescope time

A WET Power Spectrum

From the previous Fourier spectra I estimated frequencies off the graph and plotted their periods. Blue: data, Red: missing frequencies. These are all equally spaced in period → g-modes!

A WET Spectral Window

Modeling the Frequencies

Find a clever theorist. And what they will do:

Create a model for the white dwarf

Calculate the eigenmode frequencies

Compare to observations

Change your modelDo they agree?

Real Model for WD

yes

No

Modeling the Frequencies: GD 165

An example of what Asteroseismology of a White Dwarf can tell you (Bradley, 2001, ApJ, 552, 326)

• Hydrogen Layer mass = 1.5 – 2.0 × 10–4 Mּס

• Helium Layer mass = 1.5 – 2.0 × 10–2 Mּס

• 20% Carbon 80% Oxygen core out to 0.65 M*

• Mass = 0.65 Mּס

• Carbon ramp from core to pure carbon at 0.75 M*• Effective temperature 11.450 – 12.100 K

• Rotation period = 58 hours

More complicated: Fitting the light curve and not just the observed frequencies:

Modeling a light curve using nonlinear pulsation with convection: Good but not perfect

Using Pulsations to Search for Planets

The pulsations provide a clock

The difference in light travel time due to the barycentric motion causes shifts in the predicted maximum in the light curve, the so-called observed minus computed (O-C) diagram

Note: the same principle discovered the pulsar planets.

To search for planetary companions around pulsating stars you need:

1. Stable pulsations

2. A single mode as multiple modes may interfere with each other and this will look like changes in the predicted maximum

3. Worry about period changes due to other phenomenon:

1. Changes in rotation period

2. Contraction

3. Thermal Cooling

→ Evolutionary changes

Parabolic variations due to evolutionary period changes

One example:

Age Period RV Phot

ZAMS 0.048 4.4 min 16 cm/s ~10–6

Now 4.55 5 min 23 cm/s ~10–6

Hottest Teff 7.6 7 min 31 cm/s ~10–6

H exhausted 9.37 9 min 40 cm/s ~10–6

Base RGB 11.64 7.8 hours 65 cm/s ~10–5

Tip RGB 12.2 90 days 760 m/s ~1 mag

HB 12.2 10 hours ~km/s ~0.5 mag

Central Star PN

12.3 ~30 min ~km/s 0.05 mag

White Dwarf 13 15 min ~km/s ~0.01

Overview of Pulsations in the Sun throughout its Life

p-modes

g-modes

At some point in its life the Sun will undergo all the stellar oscillations we have seen in this class. Asteroseismology can thus tell us about the structure and fundmental parameters (mass, radius, etc) of a sun-like star during its life.