1. 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.

3. 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 M סּ 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 R Earth Surface Gravity: log g ~ 8.4 or 10000 larger than the Sun Magnetic field: ~ 10 6 gauss Note: White Dwarfs are the end fate of our Sun

4. 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 T eff =12.600 K + temperature index = 50400/T eff

5. 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

6. From J. Kaler The H-R Diagram with Real Stars

7. 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 סּ < 1.36 and with a mean mass of 0.593 ± 0.016 M סּ 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/L סּ ) Age luminosity relationship for WDs: 5 7

8. 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

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

10. The Space motion of Sirius A and B Sirus B is a white dwarf Mass = 0.98 M סּ

11. So why do WDs have a Carbon/Oxygen Core? Answer: Helium Burning 4 He + 4 He  8 Be 8 Be + 4 He → 12 C +  Helium burning, often called the triple alpha process occurs above temperatures of 100.000.000 K. 8 Be is unstable and decays back into He in 2.6 × 10 –16 secs, but in the stellar interior a small equilibrium of 8 Be exists. The 8 Be ground state has almost exactly the energy of two alpha particles. In the second step, 8 Be + 4 He has almost exactly the energy of an excited state of 12 C. This resonance greatly increases the chances of Helium fusing and was predicted by Fred Hoyle. 12 C + 4 He → 16 O +  As a side effect some Carbon fuses with Helium to form Oxygen: So you have nuclear burning that generates Carbon and Oxygen

12. Typical Internal Structure Model of a WD

13. PrototypeAtmosphereT eff ZZ CetiHydrogen~12000 V777 HerHelium~25000 GW VirHe/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.

14. 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

15. 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

16. 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 סּ ~ 2 x 10 33 gm R ~ 1 R earth ~ 6.4 x 10 8 cm Mean density ~ 2 x 10 6 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 d2Rd2R dt2dt2 = GM R2R2 – = G  R

17. 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.

18. 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.

19. 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.

20. 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 DB DO csep10.phys.utk.edu Note, these are for normal stars. White dwarfs have atmospheric pressures of 10 6 dynes/cm

21. Optical Light Curves of ZZ Ceti Stars

22. And their Power Spectra

23. 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.

24. 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

25. Optical Light Curves of V77 Her and GW Stars And sdB stars

26. Propagation Diagrams for WDs All stars have a mass of 0.6 M סּ 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.

27. Instability Strips for White Dwarfs

28. 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

29. 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

30. 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.

31. 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 T eff ~ 22.000 – 40.000 K Periods 100 – 250 secs Period of fundamental mode : P ~ 2860 (  סּ /  ) ½ s = 227 s. These are most likely radial modes and thus p-mode oscillations Existence predicted by pulsation theory (Charpinet et al. 1997)

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

34. Possible new class of pulsator: DQV

36. White Dwarfs in the gravity-temperature Diagram

37. Alias periods due to the Spectral Window: Undersampled periods appear as another period causing false peaks in the power spectrum

38.  f(u)  (x–u)du = f *  f(x):  (x): Window function: Convolution

39.  (x-u) a1a1 a2a2 g(x) a3a3 a2a2 a3a3 a1a1 Convolution is a smoothing function

40. 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

41. Time Domain = X = Simple sinc function * Complex: Several sinc functions Fourier Domain

42. 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

44. For more complicated signals the window function appears superimposed on every real peak in the Fourier spectrum 2 periods 2 real peaks The window function is superimposed on every real peak

45. Input Periods: 0.13 d, 0.34 d Output periods: 0.13 d, 0.34 d

46. 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.

47. 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

48. 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

49. The Whole Earth Telescope: Making the Window Function as Simple as Possible The idea of Ed Nather (left)

50. 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

51. A WET Power Spectrum

52. 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!

53. A WET Spectral Window

54. 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

55. 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 M סּ Helium Layer mass = 1.5 – 2.0 × 10 –2 M סּ 20% Carbon 80% Oxygen core out to 0.65 M * Mass = 0.65 M סּ Carbon ramp from core to pure carbon at 0.75 M * Effective temperature 11.450 – 12.100 K Rotation period = 58 hours

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

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

58. 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.

59. 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

60. Parabolic variations due to evolutionary period changes One example:

61. AgePeriodRVPhot ZAMS0.0484.4 min16 cm/s~10 –6 Now4.555 min23 cm/s~10 –6 Hottest Teff7.67 min31 cm/s~10 –6 H exhausted9.379 min40 cm/s~10 –6 Base RGB11.647.8 hours65 cm/s~10 –5 Tip RGB12.290 days760 m/s~1 mag HB12.210 hours~km/s~0.5 mag Central Star PN 12.3~30 min~km/s0.05 mag White Dwarf1315 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.