The second panel is the transformation of a synthetic noise-free light curve consisting of the six highest amplitude, best-fit signals in the first transformation. The second panel is a transformation of a synthetic light curve constructed from the three best-fit signals from the first panel.
Chapter 1
Introducing GW Librae
Chapter 2
CVology: the Cataclysmic Variable Stars
Chapter 3
Seismology: The White Dwarf Pulsators
Introduction
The oldest white dwarfs contain information about the first waves of star formation in the early universe. The four classes of pulsating white dwarfs are the PNNVs (the planetary nebula core variables, with temperatures near 100 OOOK), the DOVs (the hot DO prowhite).
The Basics of Asteroseismology
The gas dampens the radiation in the compression phase of the pulsation cycle (which heats it up and changes the ionization levels and thus the opacity), and releases radiation in the expansion phase (which cools it down, changes the ionization and opacity, and drives the next cycle). Non-radial g-modes can be thought of as sloshing motions on the surface of the star.
Case Studies
The observed amplitudes for the m = ±1 components of the l = 1 mode will be zero if the star is observed pole-on, and at full amplitude if observed equator-on. A very striking feature of the power spectrum is the multiplicity of linear combination modes.
Chapter 4
CV -seismology: The Linking of Two Fields of Research
Accreting White Dwarf Pulsators
- The DBVs and the AM Canum Venaticorum Stars
- Speculations
These questions lead Nitta and Winget to make a preliminary investigation of the effects of a clustering history on DBVs (none of which are known to exist in a binary system). The thermal inversion resulting from the addition will increase the periods and period spacings of the modes. If the star has a negligible magnetic field, the disk accumulates at the equator (if . the spin axes of the star and the disk are parallel), spinning up the equatorial layers.
If accretion affects only the upper layers of the photosphere, perturbations to the normal modes may be small, but if the effects of accretion extend deep into the non-degenerate envelope, the normal modes may be fundamentally altered. Accretion can be seen as a high-velocity, low-density wind blowing across the surface of the much denser white dwarf photosphere, and it can generate r-mode oscillations in the photosphere.
Chapter 5
Observations, Reductions and Analysis Techniques
5 .1 High Speed CCD Photometry: The UCT CCD Photometer
Data Reduction
The bias for each frame is calculated by measuring the bias in the chip's overscan region (which is an "imaginary" region of the chip subject to the same instrumental effects as the chip itself, created by clocking a few extra cycles after the real charge distribution has been read out). An average ADU for each row of the overscan is calculated and subtracted from the image. After the frames have been cleaned, the brightness of the stars is determined in two ways: aperture photometry and profile fitting.
In the latter method, a profile is fitted to the point spread function (PSF) of the star. The magnitudes in the sum files are then extracted, giving light curves for the selected stars.
The Observations
The light curves with the best signal-to-noise ratio can be subtracted from the light curve of the target star, removing first-order extinction variations and any variations due, for example, to a thin cloud. When observation was possible, the poor blue responses of the chips resulted in very poor data quality. The resolution of the amplitude spectrum3 is inversely proportional to the length of the data set.
This results in a large differential extinction effect over the length of the series (differential photometry only accounts for first-order extinction effects4). 4In principle, all extinction effects can be removed if accurate UBV photometry is performed on all stars in the image and the light curves of the comparison stars are individually corrected for extinction.
5 .3 Temporal Spectroscopy: The Investigation of Multiperiodic Signals
Some EAGLE Algorithms
I then use a least squares routine to fit a sine with these parameters to the data set, which is then subtracted from the data point by point. The Fourier transform of the residuals is then calculated and the next highest frequency is found and subtracted. A third signal is then located in the residuals and all three are simultaneously fitted to the data, first with a linear least squares routine to find better values for their parameters, which are then refined by fitting with the non-linear algorithm.
A linear least squares routine finds the best fit values of the parameters of a set of functions. In the nonlin routine, the frequencies as well as the amplitudes and phases are allowed to vary as the routine searches for the best fit.
Chapter 6
A small variation in the eigenmode frequency would cause the split we see in the peak, and would result in a phase drift in the 0-C diagram. If the June data has a slightly different frequency than the May data, then the phases of the 4237.5 f.LHZ signal (which is calculated from the combined data set) in the May data would show a systematic drift. However, in the case of the behavior of the 236-s region, there may be a simpler explanation.
The presence of this second signal neatly explains the asymmetric appearance of the dominant mode. There is one final point of concern that needs to be addressed in this region: at first glance, apart from its asymmetry, the splitting of the 236-s mode looks very similar to the splitting in the window pattern (Figure 6.2).
- A first hypothesis
- Analysing transforms of synthetic data
The top panel is the transformation of the true data, the bottom, the synthetic data. It is clear that the transformation of a light curve consisting of the best-fitting signals does not resemble the real transformation. The second panel is a transformation of a synthetic light curve composed of the three best-fitting signals from the first panel.
The first panel is the transformation of a synthetic light curve containing the best-fit frequencies, amplitudes and phases for the three highest amplitude components of the structure I named a quintuplet. The second panel is a transformation of the same signals except that the relative phases have been changed to match the actual transformation.
Mode clamping can pull the modes closer together, but would have a problem explaining the coexistence of more than two modes in the same region of frequency space. The frequencies found in half of the first data set match the full data better than those found in half of the second data set. Because of the differences in results for half of the data sets, I would be cautious about trusting the values for the splits given in Table 6.5.
A single mode (with its m-split components) changing in frequency and amplitude could produce the jumbled forest we see in the 650-s region: sidebands produced by frequency and amplitude modulation could create a jumble of spurious signals. There are better techniques than Fourier analysis for analyzing modulated signals or non-linear effects (that is, non-sinusoidal signals), such as wavelet analysis, but ultimately there is no substitute for better data.
Linear Combinations and Harmonics
Of course, this can only be done if the signal-to-noise ratio of the harmonic and linear combination modes is high enough: these often have low amplitudes. In comparing the signals found in these regions with the eigenmodes, I initially only considered combinations containing the four highest-amplitude components of the 650-s mode(s) (labeled a1 to a4), the three resolved components of the 376-s quintuplets (b1 to b3), the center of the unresolved 376-s triplet (b4), and the highest amplitude component of the 236-s doublet. I assumed that the other signals would be too low in amplitude or too uncertain to have credible matches they contained in the low signal-to-noise regions of the combination modes.
Components of combined modes are likely to have different relative phases with respect to their parent modes (Kleinman 1998a, Yuille 1998b), so in principle they could provide us with a “different view” of the parent modes: parent mode components may have relative phases that make them unresolved, however, the relative phases of the components combined with another method may allow their resolution; the parent mode can be unresolved while its combination is resolved. The spacing between the components of the combinations of the two modes close to 376-s may indicate that the modes have the same l.
Devil's Advocate
Even a small modulation in data with a poor duty cycle can generate a terrible mess in the amplitude spectrum. In Figure 6.19 I show a transformation of a trio of signals separated by 1.2 ttHz with arbitrarily chosen amplitudes and phases, sampled with the spectral window of the real data. I modulated the frequencies and amplitudes of the three signals in the same way as described by equations (6.7) to (6.9), until I obtained transformations that resembled the real transformation in the 650-s region.
It does not require much modulation to produce synthetic light curves with transformations similar to the real one. First panel is the spectral window; second panel is the transformation of the real data in the 650s region; third panel is a triplet of arbitrary pure sinusoids; fourth panel is the same triplet modulated in frequency and amplitude.
Chapter 7
A Bigger Picture
7 .1 The 1997 Amplitude Spectra
Ideas and Speculations
If we ignore the hypothesis of the previous chapter, what interpretations can we give to the April structures. Any explanation will also have to take into account the simplicity of the 236-s signal in the 1998 data. As a DAY, GW Librae is too cool to do so, and its spectrum shows no sign of the warm accretion regions seen in IPs and polars.
Finding GW Librae's orbital period will help us place limits on the physical parameters of the. If only a few cycles of a period are present in a run, there is no way to tell whether the supposed period is a result of stochastic processes or if it is a true periodicity of the star.
Chapter 8
Future Work
The WET and the HST
As discussed in detail elsewhere in this thesis, the main method of identifying the (k, l, m) values of the modes is to accurately measure the period spacing between successive radial overtone modes and the fine-structure splitting caused by stellar rotation. Limb darkening has a strong wavelength dependence, and therefore (due to geometric cancellation) modes of different l have different amplitudes in the UY relative to those in the optical. This may be of great help in deciphering the mysterious 650-s and 376-s regions of the amplitude spectrum.
However, applying the limb dimming method to GW Librae can be difficult because, unlike other flickering white dwarfs, it has an accretion disk. The narrow emission spectrum of the disc should not be too difficult to separate from the broad spectrum of absorption lines of the white dwarf photosphere.
A Dearth of Modelling
Applying the limb dimming method to GW Librae could be very useful, but to be successful would require spectroscopy in the UY and optics of high enough resolution to separate the disc contribution from the white dwarf light to obtain an accurate amplitude pulsation. It might be useful to model the normal modes of DAVs, which are very different from currently known DAYs. What are the normal DAY modes with magnetic field strengths similar to those seen at the polar ones.
I sincerely hope that the discovery of the DAY nature of GW Librae will stimulate inquiry into these questions. GW Librae is a difficult star to study, but the potential rewards in the fields of Asteroseismology and Cataclysmic Variables make it worth the effort.
Kleinman, S.J., 1995, G29-38 and the Rise of Cool DAV Asteroseismology, PhD thesis, University of Texas at Austin.
