Download Oscillating White Dwarf Stars Background on White Dwarfs

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 M‫סּ‬
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 סּ‬: 0.88 – 2.2 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)
+ temperature index = 50400/Teff
Example: DBAQ4 = Star showing He I, H, and C features
(in order of decreasing strength) near Teff=12.600 K
From A. Kawaka
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
The H-R Diagram with Real Stars
From J. Kaler
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 < סּ‬1.36 and with a mean mass of 0.593 ± 0.016 M‫סּ‬
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.
Age luminosity relationship for WDs:
Log (tcool) ≈ Const. –
5
7
log(L/L‫)סּ‬
Internal Structure of White Dwarfs
jayabarathan.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 M‫סּ‬
So why do WDs have a Carbon/Oxygen Core?
Answer: Helium Burning
4He
+ 4He  8Be
8Be
+ 4He → 12C + g
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.
As a side effect some Carbon fuses with Helium to form Oxygen:
12C
+ 4He → 16O + g
So you have nuclear burning that generates Carbon and Oxygen
Typical Internal Structure Model of a WD
The Classes of White Dwarf Pulsators
Prototype Atmosphere
ZZ Ceti
Hydrogen
Teff
~12000
V777 Her
Helium
~25000
GW Vir
He/C/O
~120000
There are 3 classes of white dwarf pulsators, divided according
to temperature and thus composition of the atmosphere.
White Dwarfs in the H-R Diagram
The Cepheids lie in the high
luminosity end of the instability
strip.
The instability strip crosses the
main sequence where the d
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
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)
d2R
dt2
GM
= –
R2
=GrR
tdynamical ~
1
(G r) ½
M ~ 1 M‫ ~ סּ‬2 x 1033 gm
R ~ 1Rearth ~ 6.4 x 108 cm
Mean density ~ 2 x 106 gm cm–3
tdynamical ~ 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
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 k mechanism.
V777 Her Stars (DB) – Discovery
• Winget also realized that the k 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!
Note, these are for normal stars. White dwarfs have atmospheric pressures of 106 dynes/cm
DA
DB
DO
csep10.phys.utk.edu
From Cepheids we know that the
k mechanism occurs when He I
is ionized, similar for Hydrogen.
DA pulsators are hot and have
hydrogen, so H I k mechanism
may be at work.
DB stars are hotter and have
helium. They should also pulsate
with the k mechanism applied to
He I/II just like in Cepheids
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 k mechanism caused by the ionization of K-shell electrons of
carbon and oxygen.
Post AGB
Reminder: Post AGB
stars have left the
Asymptotic Giant
Branch and are
planetary nebula
central stars on their
way to becoming
White Dwarfs
Optical Light Curves of V77 Her and GW Stars
And sdB stars
Propagation Diagrams for WDs
All stars have a mass of
0.6 M‫סּ‬
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)
The diagonal lines
are theoretical
predictions of the
„blue edge“ using
two values of the
mixing length
parameter for
convection
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.
A close up of the instability Strips for V777 Her Stars (DB)
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
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
A close up of the instability Strips for GW Vir Stars (DO)
Note the difference to the
ZZ Ceti stars as there is a
mix of pulsating and nonpulsating 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.
The small filled circles are pulsating stars, open filled circles are constant,
larger symbols are central stars of planetary nebulae (CSPN).
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 (r‫סּ‬/r)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
Window function: Convolution
 f(u)f(x–u)du = f * f
f(x):
f(x):
Window function: Convolution
f(x-u)
a2
a1
a3
g(x)
a3
a2
Convolution is a smoothing function
a1
Window function: Convolution
In Fourier space the convolution is just the product of
the two transforms:
Time Space
f*g
fxg
Fourier Space
F xG
F*G
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.
Time Domain
Fourier Domain
Simple sinc
function
Complex: Several
sinc functions
X
*
=
=
One-day aliases
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)
For more complicated signals the window function appears
superimposed on every real peak in the Fourier spectrum
2 real peaks
2 periods
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
1/0.34 +1 = 3.94 c/d → P= 0.25
Output periods: 0.25 d, 0.116 d, 0.17 d
1/0.13 +1 = 8.7 c/d → P= 0.116
One does not recover the true period, but an alias
A good window function is a pure sincfunction with low sidelobes (note: this is
amplitude spectrum, for power you would
square the values)
For infinite time
coverage your window
function is a deltafunction
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….
The Whole Earth Telescope: Making
the Window Function as Simple as
Possible
The 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
No
Do they agree?
yes
Real Model for WD
Change your model
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 M‫סּ‬
• Helium Layer mass = 1.5 – 2.0 × 10–2 M‫סּ‬
• 20% Carbon 80% Oxygen core out to 0.65 M*
• Mass = 0.65 M‫סּ‬
• 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
One example:
Parabolic variations
due to evolutionary
period changes
Overview of Pulsations in the Sun throughout its Life
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
p-modes
g-modes
15 min
~km/s
~0.01
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.