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by Copyright Anjum Shagufta Mukadam 2004 The Dissertation Committee for Anjum Shagufta Mukadam certi es that this is the approved version of the following dissertation: Ensemble Characteristics of the ZZ Ceti stars Committee: D. E. Winget, Supervisor S. O. Kepler E. L. Robinson J. C. Wheeler J. Liebert H. C. Harris Ensemble Characteristics of the ZZ Ceti stars by Anjum Shagufta Mukadam, B.S., M.S., M.A. Dissertation Presented to the Faculty of the Graduate School of The University of Texas at Austin in Partial Ful llment of the Requirements for the Degree of Doctor of Philosophy The University of Texas at Austin August 2004 to you, Dad. Acknowledgments To teach someone is the most sel ess thing one can do. I am indebted to my teachers for bestowing upon me the honor of knowledge. I am grateful to my advisor, Don, for believing in me when I didn t and for helping me grow. Don taught me to be more optimistic, and is chie y responsible for developing my skills in scienti c writing. I have a Hubble fellowship today, and I owe a large part of it to Don. Don s enthusiasm for science is very contagious, and I will never forget all the intense discussions in the WET lab. These moments remind us of why we are there in the rst place. No words could express the gratitude I feel for Ed. I am honored to have assisted him in building Argos, the e cient frame transfer CCD photometer. I got my fair share of being yelled at whenever I made silly mistakes. :-) Had Argos not been the optical wonder that it is, my thesis would certainly not have been as good. I cannot thank Ed enough for his genius in conceiving Argos. Ed is solely responsible for my developing any computer skills at all, though I still have a long way to go. Ed and Don tried very hard that I should learn the di erence between models and reality, and have been partially successful. I think it will take me a few years more to truly graduate. Kepler is practically my co-advisor, who helped me immensely during the rst few years of my Ph.D. with a lot of code, when I did not know where to v turn. Kepler was always friendly enough that I did not hesitate to ask him a lot of silly questions, and his support helped me gain con dence. I also thank him for introducing me to Chef s orange beef at the Chinese restaurant, Tien Hong. Rob is the best teacher I have ever come across. He is exceptionally thorough, and possesses an impressive span of knowledge in diverse areas. I have fond memories of many arguments that I have had with Rob, and I was wrong on practically each of those occasions. I treasure a lot of admiration and respect for Rob, and I have certainly learnt a few lessons from him. Atsuko taught me how to observe, taught me how things operate in the WET lab, and is the one I have to thank for helping me develop a veteran attitude towards my work. When I was a baby graduate student, she helped me merge in a di erent culture in a strange country, and she was always a big sister watching out for me. I learnt a lot of the everyday programming from her, and I could not imagine how I would ever make it through when she graduated. Travis was the only other senior student at the time I joined the WET lab. I saw Travis as a cool, con dent computer-geek, who had assembled the cluster Darwin to crunch out white dwarf evolutionary models. Travis was the rst to teach me how to reduce data with IRAF, a useful skill. I enjoyed having lunch with Travis on the days when we did not have group lunch. During these times, we would discuss our day-to-day work and show each other plots about what we had been up to. This helped anticipate questions that other people may have, and I thank Travis for all those lunches together. Scot taught me how to observe well with the 3-channel photometer. It took me a few days after he had left the mountain to assimilate all that he had taught me, which was very di cult in his intimidating presence. Scot helped me along the way in my rst few years, but I got a chance to collaborate with him extensively when we started searching for variables in the Sloan Digital Sky vi Survey. I began to appreciate the big brother in all of Scot s feedback after a few months. Thanks for looking out for me. I would like to thank Antonio profusely for his data reduction algorithms, which I used extensively during my search for new ZZ Ceti stars and for their subsequent analysis. Antonio also helped during the commissioning run of Argos, and has been a warm and kind inspiration. I am very lucky to have observed extensively with Denis, a diligent observer and sound instrumentalist, who taught me quite a few things about observing. I had a lot of good discussions about my work with Denis during my last two years in Austin, as we were the only people working in the lab at the time. In stressful times, Denis always brought good cheer and helped me cope. He also tried, as did Ed, that I should learn to make better plots. In return, I tried to help them appreciate Indian folk music. :-) Ted arrived in our group during the last two years of my stay. He helped us in achieving a higher success rate in discovering ZZ Ceti stars than the photometric method we were using at the time. Ted has been a kind, and supportive uno cial committee member, who helped me during the commissioning run of Argos, and also with subsequent observations. Ted has tried hard to teach me the genteel way of communication, rather than the abrupt and blunt manner that I am used to. :-) I can t say that he has been successful. Mukremin taught me how to carry out spectroscopic observations at the 2.7 m telescope at McDonald Observatory, and also showed me how to reduce spectroscopic data. Mukremin is a dear brother to me, and he has always lent me a shoulder to cry on, specially during the last couple years of my dissertation. I have always found his calm con dent way of tackling problems, and his mature insights most admirable. I look up to him. I overlapped with Mikemon for a bit, when he was trying to graduate. I thank him for dropping me to the airport in his car lled with pizza boxes at vii that time. Mikemon returned to the WET lab six years later as a post-doc, only a few months before I was trying to graduate. I thank him once again for another ride to the airport in his new car devoid of pizza boxes. Times and people change. :-) Mikemon helped me get a physical insight about stellar pulsations, something I had failed to achieve even after six years in the WET lab. I am very grateful to him for helping me put together the fth chapter of my thesis. I feel sincere gratitude towards Dr. Frank Bash, without whose nancial contribution, Argos may have remained a design on paper, and this thesis would have been quite di erent from what it has turned out to be. I also thank him for sending a letter to the US Consulate in Mumbai, that certainly helped me in getting back to the United States to nish writing my thesis. I am grateful to my committee member, Craig, for all his advice and more importantly, nancial support for meetings and observing runs from the Cox funds. :-) I also thank Dr. Chris Sneden for bailing me out of tight spots by providing partial nancial assistance from the dept. funds. I also thank Pawan for his support, and encouragement during my last few years in Austin. It is unfortunate that we did not get a chance to work together on a project that we had planned. I owe Derek Wills a special thanks for loaning me $200 when I rst arrived in the dept. from India, and I had only $30 left to see me through the rst one month. I also owe a special thanks to Dr. David Lambert, who has always encouraged me and suggested that I should apply for a Hubble fellowship. I thank Phillip McQueen for helping me conceive the ba e design for Argos. I really appreciate all his time, knowing how tight his schedule is. I thank David Boyd and Gordon Wesley for the time and e ort put in designing the mount for Argos, and also for taking the time to teach me about various useful tools. I thank Jimmy and George for helping us x various mechanical problems with the Argos mount, for machining the ba es for Argos, and for viii building the mount for Cyclops, a camera used to track the location of the dome slit. I also thank Jimmy and David Boyd for giving me my rst lessons in using heavy machinery. I thank Gary Hansen and Doug Edmonston, who built a timing circuit for precise instrument timing. I also thank Fred Harvey for helping me with tasks concerning electronics. I have no words to thank David Doss enough, for his advice and his insights, that I have been utilizing ever since I started observing six years ago. Dave has helped us immensely in xing various problems with Argos, since the commissioning run of the instrument. His service and support remain unparalleled. I also thank Earl Green for a lot of discussions about instrument or telescope problems, that occur ever so often. I also thank Bob Franke, Kevin Meyer, Terry Wilemon, Mark Blackley, Darrin Cook, Marian Frueh, and Ed Dutchover for all their help and support. I also thank Patricia, Robert, and Miguel for making long runs bearable by providing good food, and for washing all the dirty dishes. I also thank my graduate coordinators, Elizabeth Korves and Stephanie Crouch. You were lifesavers for twits like me, who never knew what forms had to be lled or when the submission deadlines were. I bow down to the e ciency, patience, and perseverance shown by Cindy Thompson, without whom we would have missed a lot of deadlines we could not a ord to. I am also grateful for all the computing support given to me by Mark Cornell, Ron Wilhelm, Bill Spiesman, Martha Schaefer, and now Chris Wilkinson, Dario Landazuri, and Anita Cochran. I have to specially thank Peter Hoe ich for linux support at times when I was ready to tear my hair out. I thank Henry Cantu for making the mess, that are research grants, more legible to clueless selves like me and my advisor. I thank both Mae Collins and Bob Worley for helping with purchasing equipment, and Sherry for reimbursements. I thank Estela, then Melissa, and now Jamie Woods for helping with travel arrangements. I thank Yolanda, and ix now Estela, for making sure that I was paid. I thank Karen, Maegan, and Davis Winget for hosting a really nice graduation party. I am indebted to my friends Navin and Swati Palicherla, Don and Karen Winget, Srikant and Gauri, Randy and Gayathri, Anuroopa Shenoy, and Mikemon for a roof to stay under while working on my thesis. I would also like to thank all my batch mates, and all my friends and well-wishers in astronomy. You made the six years that I spent in Austin enjoyable, and I thank all the people with extra-soft shoulders for the times that weren t fun. I thank Mukremin, David Reaves, Atsuko, Scot, Kepler, Denis, Mikemon, Antonio, Mike Endl, Tommy Greathouse, Joe Tufts, Nairn, Marsha, Rica, Agnes Kim, Diane Paulson, Rob Hynes, Todd Watson, Bev and Derek Wills. I thank Priya for proofreading various manuscripts of mine, and Rohit Grover for incessantly advising me about proper formatting of my thesis and all my plots. :-) Last, but not the least, I thank my beloved MP for the immense support that you have given me ever since I arrived in Austin. I always poured all my troubles on you, and you soaked them dry till life was sunny for me again. And also thank you for re-doing my whole presentation in power point on the eve of my defense, at a time when I was struggling with low resolution PDF slides. Thanks for being there for six years. Anjum Shagufta Mukadam The University of Texas at Austin August 2004 x Ensemble Characteristics of the ZZ Ceti stars Publication No. Anjum Shagufta Mukadam, Ph.D. The University of Texas at Austin, 2004 Supervisor: D. E. Winget Global pulsations of stars can be used to probe their interiors, similar to the method of using earthquakes to explore the Earth s interior. This technique, called asteroseismology, is the only systematic way to study stellar interiors. White dwarf stars represent a relatively simple stellar end state for most main sequence stars like the Sun. This is because they are not expected to have any central nuclear fusion and their evolution is dominated by cooling. These stars are scienti cally interesting since they contain a fossil record of their previous evolution. Their high densities and temperatures make them good cosmic laboratories to study fundamental physics under extreme conditions. Besides, white dwarfs are not as centrally condensed as some other classes of variables, and hence the observed pulsations sample their interior better. Each pulsation mode is an independent constraint on the structure of the star. We can probe stellar structure and composition by nding a single star rich in pulsation modes, and/or by nding a large number of pulsators to use the method of ensemble asteroseismology. A fraction of white dwarf pulsators are observed to be extremely stable clocks; this property allows us to look for xi any orbiting planets. The drift rates of these stable clocks are expected to reveal the stellar cooling rate. Including this information in evolutionary white dwarf models allows us to determine the age of the star. Since most stars evolve into white dwarfs, we can use the distribution of white dwarf ages in di erent parts of the Galaxy to constrain the age of the Galaxy and its evolution. Variable white dwarfs can also be used as a means to measure Galactic distances. All these reasons motivate us to search for additional white dwarf pulsators. Four out of ve white dwarfs show hydrogen in their outermost layers and are classi ed as DAs. These are observed to pulsate in a temperature range of 11000 12000 K. I decided to search speci cally for DA white dwarf variables (DAVs), also known as ZZ Ceti stars. To substantially increase the sample of ZZ Ceti stars, I was forced to search at greater distances (or fainter magnitudes). This is because various research groups around the world have already examined the relatively nearby (or bright) candidates for variability. Hence, I helped Dr. R. E. Nather in building a high speed time-series CCD photometer for the prime focus of the 2.1 m telescope at McDonald Observatory. This CCD instrument allows us to obtain usable time-series data on 19th magnitude objects, as opposed to a limiting magnitude of 17 with our previous instrument. The combination of an e cient new instrument and a large amount of telescope time ( 100 nights/yr) gave me a unique opportunity to search extensively for new ZZ Ceti stars. Other members of my research group also contributed towards the 15 month long observations at McDonald Observatory, and helped me in data analyses. We pre-selected candidates by using the photometric and spectroscopic observations of the Sloan Digital Sky Survey. I present 35 new pulsating DA (hydrogen atmosphere) white dwarf stars discovered from the Sloan Digital Sky Survey (SDSS) and the Hamburg Quasar Survey (HQS). This increases the sample xii of 39 known ZZ Ceti stars to 74; the rst ZZ Ceti star was accidentally discovered in 1968. This is the rst time in the history of white dwarf variables that we have a homogeneous set of spectra acquired using the same instrument on the same telescope, and with consistent data reductions, for a statistically signi cant sample of ZZ Ceti stars. The homogeneity of the spectra reduces the scatter in the spectroscopic temperatures; we have essentially re-de ned the ZZ Ceti instability strip. We nd a narrow ZZ Ceti strip of width 1000 K, as opposed to the previous determination of 1500 K. We question the purity of the DAV instability strip as we nd several non-variables within. We present our best t for the red (cool) edge and our constraint for the blue (hot) edge of the instability strip, determined using a statistical approach. I also present the observed pulsation spectra of 67 ZZ Ceti stars with reliable spectroscopic temperatures. I verify the well-established relation of the increase in observed pulsation periods and amplitudes for the new ZZ Ceti stars, traversing from the blue to the red edge of the instability strip. The data on the new ZZ Ceti stars suggests that pulsation amplitude declines prior to the red edge. This means that ZZ Ceti pulsations do not shut down abruptly at the red edge of the instability strip. This is the rst possible detection of such an e ect. xiii Contents Acknowledgments Abstract Chapter 1 Introduction 1.1 White dwarfs: fossils of stars . . . . . . . . . . . . . . . . . . . . . . . . 1.2 White dwarf variables: the inside story . . . . . . . . . . . . . . . . . . 1.2.1 Short term seismology: structural constraints . . . . . . . . . 1.2.2 Long term seismology: drifting pulsation periods . . . . . . . 1.3 Motivation: why search for DAVs? . . . . . . . . . . . . . . . . . . . . . 1.3.1 Stellar structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.2 Stable clocks can be used to nd planets . . . . . . . . . . . . . 1.3.3 DAV seismology helps cosmochronometry . . . . . . . . . . . v xi 1 1 3 4 6 7 8 8 9 1.3.4 Seismological distances from a ickering candle . . . . . . . . 12 1.4 My dissertation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.4.1 Establishing the ZZ Ceti instability strip . . . . . . . . . . . . . 14 1.4.2 Ensemble seismology of the ZZ Ceti stars . . . . . . . . . . . . 15 Chapter 2 Ideal time-series photometer 16 2.1 Time-series photometry and ideal instrumentation . . . . . . . . . . 16 2.2 Argos: designed for time-series photometry . . . . . . . . . . . . . . . 19 xiv 2.2.1 The CCD camera . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.2.2 Wavelength response . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.2.3 Instrument timing . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.2.4 The prime focus mount . . . . . . . . . . . . . . . . . . . . . . . 26 2.2.5 Ba ing Argos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 2.3 Noise and measurement in photometry . . . . . . . . . . . . . . . . . . 30 2.4 Time keeping with stellar clocks . . . . . . . . . . . . . . . . . . . . . . 36 2.4.1 Uncertainty in period . . . . . . . . . . . . . . . . . . . . . . . . . 36 2.4.2 Uncertainty in phase . . . . . . . . . . . . . . . . . . . . . . . . . 37 2.4.3 Uncertainty in drift rate measurements . . . . . . . . . . . . . 38 2.5 Re ections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 Chapter 3 Search for new ZZ Ceti stars 42 3.1 Selection of DAV candidates from databases . . . . . . . . . . . . . . 42 3.1.1 Sloan digital sky survey . . . . . . . . . . . . . . . . . . . . . . . 42 3.1.2 Hamburg quasar survey . . . . . . . . . . . . . . . . . . . . . . . 43 3.2 Techniques to select the SDSS DAV candidates . . . . . . . . . . . . . 43 3.2.1 Photometric technique for the SDSS DAV candidates . . . . . 44 3.2.2 Equivalent width technique for the SDSS DAV candidates . . 45 3.2.3 Spectroscopic technique for the SDSS DAV candidates . . . . 46 3.3 Spectroscopic technique for HQS DAV candidates . . . . . . . . . . . 48 3.4 Data acquisition and analysis . . . . . . . . . . . . . . . . . . . . . . . . 50 3.4.1 Observing strategy . . . . . . . . . . . . . . . . . . . . . . . . . . 50 3.4.2 Data acquisition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 3.4.3 Data reduction and results . . . . . . . . . . . . . . . . . . . . . 51 3.5 Pulsation properties of the new ZZ Ceti variables . . . . . . . . . . . . 58 3.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 xv Chapter 4 Empirical DAV Instability Strip 63 4.1 Empirical instability strip . . . . . . . . . . . . . . . . . . . . . . . . . . 64 4.1.1 Biases in candidate selection . . . . . . . . . . . . . . . . . . . . 66 4.1.2 Non-uniform detection e ciency . . . . . . . . . . . . . . . . . 67 4.1.3 Uncertainties in temperature and log g determinations . . . 69 4.2 Probing the non-uniform DAV distribution using pulsation periods 72 4.3 Questioning the impurity of the instability strip . . . . . . . . . . . . 75 4.4 Narrow width of the ZZ Ceti strip . . . . . . . . . . . . . . . . . . . . . 77 4.5 Empirical blue and red edges . . . . . . . . . . . . . . . . . . . . . . . . 78 4.5.1 Technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 4.5.2 Estimating the uncertainties . . . . . . . . . . . . . . . . . . . . 82 4.5.3 Comparison with empirical edges . . . . . . . . . . . . . . . . . 83 4.5.4 Comparison with theoretical edges . . . . . . . . . . . . . . . . 83 4.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 Chapter 5 ZZ Ceti Ensemble characteristics 86 5.1 Pulsations in ZZ Ceti models . . . . . . . . . . . . . . . . . . . . . . . . 86 5.1.1 Growth of amplitude . . . . . . . . . . . . . . . . . . . . . . . . . 87 5.1.2 Fluid motions in the star . . . . . . . . . . . . . . . . . . . . . . . 88 5.2 Trends across the instability strip . . . . . . . . . . . . . . . . . . . . . 89 5.2.1 Observed pulsation periods . . . . . . . . . . . . . . . . . . . . . 89 5.2.2 Observed pulsation amplitudes . . . . . . . . . . . . . . . . . . 91 5.2.3 Mode density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 5.3 Drifting eigenmodes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 5.4 Why study ZZ Ceti stars as an ensemble? . . . . . . . . . . . . . . . . . 95 5.5 Teaching di erent DAVs to play the same tune . . . . . . . . . . . . . 96 5.5.1 Scaling the pulsation spectra . . . . . . . . . . . . . . . . . . . . 96 5.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 xvi Chapter 6 Summary and future work 110 6.1 Seismology of the ZZ Ceti stars . . . . . . . . . . . . . . . . . . . . . . . 110 6.1.1 Bene ts of an ensemble of DAVs . . . . . . . . . . . . . . . . . . 111 6.2 Ideal instrumentation for the DAV search . . . . . . . . . . . . . . . . 111 6.3 Fruitful search for the ZZ Ceti stars . . . . . . . . . . . . . . . . . . . . 113 6.4 Empirical ZZ Ceti instability strip . . . . . . . . . . . . . . . . . . . . . 114 6.5 Ensemble characteristics of the ZZ Ceti stars . . . . . . . . . . . . . . 114 6.6 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 Bibliography Vita 118 129 xvii Chapter 1 Introduction 1.1 White dwarfs: fossils of stars White dwarf stars are the stellar remains of 98 99% of stars in the sky (Weidemann 1990), and contain an archival record of their main sequence lifetime. We can harness this archival record from white dwarfs relatively easily, instead of studying stars on the main sequence directly. White dwarf cores contain 99.99% of the star by mass. Masses of white dwarf stars exhibit a narrow distribution around 0.6M (e.g. Finley, Koester, & Basri 1997; Bergeron et al. 2001; Silvestri et al. 2001; Madej, Nale yty, & Alz thaus 2004). The high density in the core serves to ionize it completely, and the core is immersed in a sea of electrons. The degenerate electrons make the core isothermal. However, the ions in the core remain non-degenerate. Degeneracy pressure supports the electrons, while the massive ions are subject to the contracting force of gravity. Charge separation between the ions and the electrons then allows the forces of gravity and degeneracy pressure to balance each other. The core composition of a white dwarf is e ectively determined by nuclear reaction rates in the red giant stage. White dwarf cores do not undergo 1 any further nuclear fusion, and their composition remains unaltered. Therefore, white dwarfs allow us the opportunity to study nuclear reaction rates 12 C( , )16 O in red giant cores, without the inconvenience of central nuclear fusion (Metcalfe, Winget, & Charbonneau 2001; Metcalfe, Salaris, & Winget 2002; Metcalfe 2003). White dwarf spectra reveal that 80% of them have atmospheres domi nated by hydrogen (DAs; Fleming, Liebert, & Green 1986), and 20% have at- mospheres dominated by helium (DBs). This insulating outer layer controls the rate at which the residual thermal energy of the ions in the isothermal core is radiated into space. White dwarf evolution is dominated by cooling, leading to a simple relation between e ective temperature and age of the white dwarf, described approximately by Mestel theory (Mestel 1952; Van Horn 1971). Known white dwarfs at Tef f 4500 K are among the oldest stars in the solar neighborhood, and they serve as reliable chronometers. The exponential decrease in their cooling rate should cause a pile up of white dwarfs at lower temperatures, if we assume the star formation rate to be constant. The volume density of white dwarfs per unit absolute bolometric magnitude plotted as a function of their luminosity, i.e. the luminosity function, is expected to show more and more white dwarfs in lower temperature bins. However, the best current observational determinations of the white dwarf luminosity function for the disk indicate a turn-down in the space density of low luminosity stars (Liebert, Dahn & Monet 1988; Oswalt et al. 1996; Leggett, Ruiz & Bergeron 1998), interpreted to be a signature of the nite age of the disk; the luminosity where this turn-down occurs, in conjunction with theoretical cooling calculations, allowed Winget et al. (1987) to estimate the age of the Galactic disk. 2 1.2 White dwarf variables: the inside story Landolt (1968) accidentally discovered the rst variable white dwarf, HL Tau 76, while attempting to gather data on white dwarf standards. Subsequently, McGraw, Robinson, Lasker, and Hesser searched systematically for other variables; this led to the discovery of the rst signi cant sample of what came to be known as the class of DA variables (DAVs; Lasker & Hesser 1971; McGraw & Robinson 1975, 1976; McGraw 1976; Hesser, Lasker, & Neupert 1976; Robinson et al. 1978). Nather (1978) was the rst to distinguish these pulsating DA white dwarfs from cataclysmic variables. R 548 (ZZ Ceti ) was the rst white dwarf to be deciphered (Robinson, Nather, & McGraw 1976), and the DAVs were later called the ZZ Ceti stars (McGraw 1980), as the proto-type R 548. Pioneering e orts by McGraw (1979) and Greenstein (1982) established the location of the ZZ Ceti instability strip in temperature. Greenstein (1982) found ZZ Ceti stars in the range 10700 11700 K, on an extension of the Cepheid instability strip. However, the observed pulsation periods were two orders of magnitude longer than predicted for radial pulsations in white dwarf stars. Hence, Warner & Robinson (1972) and Chanmugam (1972) suggested that the observed pulsations were nonradial g-modes. However, Brickhill (1975) found the lowest order modes of non-radial adiabatic oscillations in his models, at the same temperatures as HL Tau 76 and G 44-32, to be smaller than the observed periods by a factor of two. Dolez & Vauclair (1981) initially suggested that DAs with a thin hydrogen layer (MH /M < 1.5 10 14 ) were pulsationally unstable in the range 11500 13500 K due to the mechanism in the helium partial ionization zone. They also suggested that DAs with a thicker hydrogen layer (MH /M > 1.5 10 14 ) pulsate between 10000 K and 11500 K, and in this case the mechanism in the hydrogen partial ionization zone was driving the pulsations. Dziembowski (1977), Keeley (1979), and Dziembowski & Koester (1981) 3 also found nonradial g-mode instabilities in their white dwarf models, driven by the He+ ionization zone. Winget et al. (1982a) carried out non-adiabatic calculations for a large grid of strati ed, evolutionary white dwarf models and found ZZ Ceti pulsations to be a result of hydrogen driving. Their models were pulsationally unstable for a wide range of H layer masses. Winget et al. (1982a) also claimed to nd a second instability strip at Te = 19000 K in their ZZ Ceti models with thin hydrogen layers (MH /M < 10 10 ) due to He driving. They suggested that He driving may also produce DB variables at Te 19000 K. The rst DB variable (DBV) was discovered soon after (Winget et al. 1982b), and there are now 13 DBVs known to date (see Bradley 1998, 2000; Nitta et al. 2004). There are comparatively 74 DAVs in the literature, including the new DA variables listed in this document. Apart from the DAV and DBV instability strips, there exist 15 other hot white dwarf variables. These are called the DO variables (DOVs, GW Vir stars) and the planetary nebula nuclei variables (PNNVs), found in an e ective temperature range of 80000 to 170000 K with log g 6. McGraw et al. (1979a, 1979b) discovered the rst hot white dwarf variable in this class, PG 1159-035. This hot instability strip is not well de ned in temperature; it is not clear whether these pulsators belong to a single strip or are indicating the presence of two distinct hot instability strips. 1.2.1 Short term seismology: structural constraints The purity of the instability strip led to the conclusion that pulsations are an evolutionary e ect in otherwise normal white dwarfs (Robinson 1979; Fontaine et al. 1985; Fontaine et al. 2001, 2003; Bergeron et al. 2004). Hence, we conclude that there is nothing unusual or unique about pulsating white dwarfs and 4 we can apply what we learn from them to white dwarfs at di erent temperatures. Each pulsation mode is an independent constraint on the structure of the star; the more modes we detect, the better our understanding of the stellar structure. Using an ensemble of modes gathered from di erent pulsators, we can draw general characteristics of white dwarf stars. This technique is bene cial to decipher pulsators that show only a few modes, such as the typical hDAVs. Each of the eigenmodes that can be excited in the star is described by a set of indices: k is the radial quantum number that gives the number of nodes between the surface and the center of the star, is the azimuthal quantum number that gives the number of nodes on the surface, and m is the number of nodes along the meridian; it is used to describe the frequency if the spherical symmetry is lost due to rotation or a magnetic eld. We use spherical harmonics (Y m) to describe these eigenmodes. This is a manifestation of the spherical gravitational potential, and is similar to the quantum numbers used to describe the state of an electron bound by the spherical electrostatic potential of the nucleus. The radial quantization results from boundary conditions that dictate an anti-node at the surface and a node at the center of the star in our models. The azimuthal quantization results from surface boundary conditions; nodes and anti-nodes along the surface have to add up to the circumference of the star. See the review by Winget (1998) for a detailed discussion. White dwarfs exhibit nonradial pulsations; this is easily understood when we consider that to pulsate radially, the star must expand and contract against gravity (log g 8), requiring a substantial amount of energy. Comparatively, it is much easier for the star to pulsate nonradially. All variable stars are expected to have larger pulsation amplitudes at the surface compared to the core due to the density pro le. White dwarfs are not as centrally condensed as some 5 other classes of variables, and hence the observed nonradial modes sample their interiors better (Montgomery & Winget 1999; Montgomery, Metcalfe, & Winget 2003). Their high densities and temperatures make these stars good cosmic laboratories to study fundamental physics under extreme conditions. 1.2.2 Long term seismology: drifting pulsation periods There are two competing internal evolutionary processes that govern the change in pulsation period with time (dP /dt = P ) for a single mode in the models of white dwarf variables. Winget, Hansen, & Van Horn (1983) found that cooling increased the periods in their models as a result of the increasing degeneracy, and residual gravitational contraction decreased the periods in their models. For white dwarfs with high e ective temperatures, contraction is still signi cant. The DOV star PG1159-035 revealed a drift rate of (13.0 2.6) 10 11 s/s for its main periodicity at 516 s (Costa, Kepler, & Winget 1999). This value of P agrees in sign with the theoretical models, but is an order of magnitude larger. Such a reliable drift rate has not yet been measured for the DBV white dwarfs. The unidirectional drift rate of the ZZ Ceti stars close to the blue edge has been constrained to be smaller than a few 10 15 s/s (O Donoghue & Warner 1987; Kepler et al. 2000a; Mukadam et al. 2003a). These drift rate measurements rely on the precision with which we can measure the arrival times of pulses from the stellar clock. Such a determination relies on the accuracy of instrument timing (see section 2.2.3), and the precision with which we can measure the pulsation period. Measuring the period with su cient accuracy to begin measuring the drift rate of a clock can greatly bene t from initial multi-site observations to distinguish between the true frequencies and aliases, and subsequent single-site observations to increase the baseline (refer to section 2.4 for more details). Constraining the evolution of these clocks 6 is not a simple task. We theoretically expect DAV evolution to be simple cooling at a nearly constant radius, and the measured drift rates are consistent with cooling rates in our models. However, di erent modes in the same star show drift rates di erent by a factor of 10 or more. Hence, we cannot be certain that the drift rate we have measured re ects cooling (see section 5.2.3). 1.3 Motivation: why search for DAVs? Since 80% of all white dwarf stars have atmospheres dominated by hydrogen (DAs; Fleming, Liebert, & Green 1986), to understand the DA variables (DAVs) is to understand the most common type of white dwarf. The ZZ Ceti pulsation periods range from 100 1200 s and are consistent with nonradial g-mode pulsations. The pulsation periods and amplitudes of the DAV stars show a distinct trend with temperature (see Clemens 1993). The hot DAVs (hDAVs) in the hotter half of the instability strip show relatively few pulsation modes, with low amplitudes ( 0.1 3%) and periods around 100 300 s. The cooler DAVs (cDAVs) show longer periods, around 600 1000 s, larger amplitudes (up to 30%) and greater amplitude variability (e.g. Pfei er et al. 1996; Kleinman et al. 1998). This well-established period-temperature and amplitude-temperature correlation (see section 5.2 for the theory) allows us to classify DAV stars meaningfully into hot DAVs (hDAVs) and cool DAVs (cDAVs). Understanding the structure and evolution of a statistically signi cant sample of DAVs has implications for other areas of astronomy, some of which are discussed below. 7 1.3.1 Stellar structure We can probe stellar structure and composition by nding a single star rich in pulsation modes, and/or by nding a large number of pulsators to use the method of ensemble asteroseismology. A unique model t to the observed periods can lead us to a determination of the stellar mass, core composition, age, rotation rate, magnetic eld strength, and distances using asteroseismology. Apart from our search, there are 38 DAVs in the literature (see Bergeron et al. 2004 and Warner & Woudt 2003); additional pulsators and additional modes will help us understand the DAVs as a class. Measuring the rotation period for DAVs and comparing it with other classes of white dwarf pulsators at di erent temperatures can give us clues about the evolution of angular momentum. Even though the mass distribution is centered around 0.6 M , we nd a small fraction of higher mass white dwarfs. BPM 37093 is a known M pulsator (Kanaan et al. 1992). Therefore, a search for a large number of DAVs is bound to yield extreme mass pulsators. Low mass (log g 7.6) DAVs could well be helium core white dwarfs; pulsating He core white dwarfs should allow us to probe their equation of state. We expect that these stars originate from interacting binaries, as the Galaxy is too young for single star evolution to result in white dwarfs of 0.5 M . High mass (log g 8.5) DAVs are potentially crystallized, paving the way for an empirical test of the theory of crystallization in stellar plasma (Winget et al. 1997; see section 1.3.3 for a detailed discussion). This should also have implications for models of neutron stars and pulsars, which are thought to have crystalline crusts. 1.3.2 Stable clocks can be used to nd planets The dominant modes (probably = 1, k = 2) in hDAVs like G 117-B15A, R 548 (ZZ Ceti), and L 19-2 have been found to exhibit extreme amplitude and fre8 quency stability (P 10 15 s/s), implying that these stars can serve as reliable clocks. To put this number in perspective, these clocks are expected to lose or gain one cycle in a few billion years. Should such stable clocks have an orbiting planet around them, their re ex motion around the center of mass of the system would become measurable, providing a means of detecting the planet (e.g. Kepler et al. 1991; Mukadam, Winget, & Kepler 2001; Winget et al. 2003). These hDAVs were once main sequence stars, suitable hosts for planet formation. Theoretical work indicates outer terrestrial planets and gas giants will survive the red giant phase (e.g. Vassiliadis & Wood 1993) with orbits stable on timescales longer than a fraction of a billion years (Duncan & Lissauer 1998). These timescales are comparable to the cooling time required by a newly formed white dwarf to reach the pulsational DAV strip. The success of a planet search around these stable clocks rests on nding a statistically signi cant number of hDAV stars. 1.3.3 DAV seismology helps cosmochronometry White dwarfs at Tef f 4500 K are among the oldest stars in the solar neighborhood; an average white dwarf may take up to several billion years to cool to 4500 K. We can use these chronometers to determine the ages of the Galactic disk and halo (e.g. Winget et al. 1987; Hansen et al. 2002). Renzini et al. (1996) and Zoccali et al. (2001) have also used this method to determine the ages of the globular clusters NGC 6752 and 47 Tucanae (NGC 104) respectively. While von Hippel & Gilmore (2000) and Kalirai et al. (2001) utilized this method to determine the ages of the open clusters NGC 2420 and NGC 2099 (M37) respectively. This method, known as white dwarf cosmochronometry, has a di erent source of uncertainties and model assumptions than main sequence stellar evolution. This is because white dwarfs spend most of their time on the cooling sequence. 9 A signi cant part of the theoretical uncertainty in the age estimation of white dwarfs comes from uncalibrated model cooling rates and uncertainties in the constitutive physics and basic parameters such as compositional strati cation, crystallization, and phase separation, used in calculating the cooling rates. The outer non-degenerate layers and the core composition play an important role in dictating the cooling rates. Although these cool white dwarfs (Tef f 4500 K) have not been observed to pulsate (Nitta et al. 2000a), we can still use asteroseismology to calibrate theoretical cooling curves, thus reducing the uncertainties in determining white dwarf ages. Massive pulsators help us study crystallization As a white dwarf star cools, theory suggests that the thermal energy of the ions becomes much smaller than the energy of the Coulomb interaction, and the interior of the white dwarf should begin to crystallize (Abrikosov 1960; Kirzhnitz 1960; Salpeter 1961). The model equation of state for matter, i.e. electrons and nuclei, at zero temperature and very large densities revealed that the nuclei form a lattice rather than a gas. For a 0.6 M model, the onset of crystalliza- tion begins at Te = 6000 K for a C core, and at Te = 7200 K for an O core (Wood 1992). Crystallization a ects the cooling rate of the white dwarf by releasing latent heat, and the outward moving crystallization front causes the periods to increase. In a 99% crystallized star, pulsation periods can increase over an evolutionary timescale by as much as 30% or more. Pulsation modes get excluded from the crystallized region, as the crystallization front represents a hard boundary. Van Horn (1968) rst showed that as the star crystallizes it releases latent heat, a ecting the cooling times by adding an additional energy source. In addition, theoretical calculations suggest that it does not freeze completely as an 10 alloy, some phase separation occurs between the C and O; this enhances the oxygen content in the crystallized region, while the overlying uid layer becomes carbon enhanced (Stevenson 1980; Segretain & Chabrier 1993; Montgomery et al. 1999; Isern et al. 2000). The phase separation is an additional energy source, as it releases gravitational binding energy from the star. Crystallizing temperatures for most white dwarfs are much cooler than the DAV instability strip from 11 000-12 500 K, implying that ordinarily e ects such as crystallization and phase separation cannot be studied with asteroseismology. However, because of the larger central pressures in massive pulsators (log g 8.6), they should be substantially crystallized even at 12 000 K (Winget et al. 1997; Montgomery & Winget 1999). These variables can provide the rst test of the theory of crystallization in stellar plasma. Metcalfe, Montgomery, & Kanaan (2004) present strong asteroseismological evidence that the massive DAV, BPM 37093, is 90% crystallized. Stable pulsators provide a calibration of white dwarf cooling curves Kepler et al. (2000a) conclude that the rate of cooling dominates the drift rate (P ) for the hDAV stars; the rate of contraction (R) of a model DAV with a radius of 9.6 108 cm is negligible 1 cm/yr. Cooling rates for a large sample of hDAVs with di erent masses and internal compositions will prove fruitful in calibrating the DA white dwarf cooling curves. Note that a stable clock with an orbiting companion will show both the parabolic cooling and the periodic variations due to the companion (see Kepler et al. 1991; Mukadam, Winget, & Kepler 2001); these e ects are discernible and will most likely have di erent timescales. We expect that a change in the gravitational constant (G) should alter the radius of the white dwarf, thus changing its rate of cooling. The observed drift rates of white dwarfs like G 117-B15A, R 548, and L 19-2 place a constraint on the 11 rate of change of G with time (Benvenuto, Garc a-Berro, & Isern 2004). Should the cooling rate alter as a consequence of a change in G, we should also be able to discern its e ects in the white dwarf luminosity function. The observed white dwarf luminosity function also places an independent upper limit on the rate of change of G (Isern, Garcia-Berro, & Salaris 2002). H & He layer masses from seismology help improve models Lack of knowledge of mass loss and details of thermal pulses lead to uncertainties in the H and He layer masses. The outer non-degenerate layers control the rate at which the residual thermal energy of the core is radiated into space. Poorly determined hydrogen and more importantly, helium mass fractions increase the uncertainty of white dwarf ages by 0.75 Gyr for an order of magnitude change in the He layer mass(Wood 1990, 1992). Although we can utilize pulsating white dwarfs to measure cooling rates directly, we can do so only in selected temperature ranges where pulsations occur because of the development of partial ionization zones. Hence, determining hydrogen and helium layer masses from asteroseismology will be helpful in improving evolutionary models and better determining white dwarf ages. 1.3.4 Seismological distances from a ickering candle Obtaining a unique model t to the pulsation modes allows us to determine an asteroseismological distance by matching the observed luminosity of the star with the model luminosity. Typically, the asteroseismological distance is more accurate than what we determine from measured parallax (e.g. Bradley & Winget 1994a; Bradley 2001). This also holds true for other kinds of pulsating stars, such as delta Scuti variables (e.g. Peterson & Hog 1998) and Cepheids (e.g. Kervella et al. 2004). Variable white dwarfs with a rich pulsation spec- 12 trum should therefore be of great interest for an independent calibration of the Galactic distance scale. Alternatively, Salaris et al. (2001) derived distances to globular clusters by tting a local template DA white dwarf sequence with accurate parallax measurements, and with Te between 10000 K and 20000 K, to the de-reddened cluster counterpart. The vertical shift applied to the local template sequence to t the cluster sequence provides a measure of the distance to the cluster. This white dwarf tting technique assumes that the average mass of the white dwarfs in the cluster is the same as the average mass of white dwarfs in the local DA template sequence. Uncertainties in this technique can be reduced via asteroseismology by measuring stellar masses and hydrogen layer thicknesses for DA pulsators and extending the results systematically to other DAs. 1.4 My dissertation Having established the importance of nding additional ZZ Ceti stars, we realized that most of the nearby bright DAVs had already been discovered. (For example, Kepler et al. (1995) searched for DAVs among the brighter candidates and found many non-variables.) Finding a signi cant number of new DAVs required observing newer candidates at fainter magnitudes. The 3-channel photometer in our possession (Kleinman, Nather, & Phillips 1996), based on photomultiplier tubes (PMTs), allowed us to obtain usable photometry on objects of magnitude 17 at the 2.1 m telescope at McDonald Observatory. This necessitated the building of an e cient time-series photometer that would allow us to observe fainter stars (B 19), enabling us to reach white dwarfs in a larger volume. The number density of DA white dwarfs brighter than g = 17.0 per square degree is 0.2, while the density of DA white dwarfs brighter than g = 19.0 per square degree is 2 (Fan 1999). 13 Assisting Dr. R. E. Nather in building Argos, an ideal time-series photometer, for the 2.1 m telescope at McDonald Observatory, constituted the rst part of my thesis. Carrying out the search to discover a statistically signi cant number of DAVs was the second step towards my project. Characterizing the shape and mass dependence (or lack of) of the instability strip was the third part of my thesis. Lastly, I hope to use the increased population of new DAVs towards ensemble seismology. Although this part of the thesis remains a work in progress, it certainly establishes the preliminary steps in the direction. 1.4.1 Establishing the ZZ Ceti instability strip Pulsation models indicate that the limits of the ZZ Ceti instability strip depend on the e ective temperature and mass of the star (Bradley & Winget 1994b). This was observationally con rmed by Giovannini et al. (1998) and Bergeron et al. (2004). Most model atmospheres of DAV stars treat convection with a mixing length prescription, assuming some parameterization, the choice of which can shift the edges of the instability strip in temperature by a few thousand Kelvin (Bergeron et al. 1995; Koester & Allard 2000). Determining the location of the red edge in theoretical models is di cult due to convective and nonlinear e ects (Winget et al. 1982a; Brickhill 1983; Bradley & Winget 1994b; Wu & Goldreich 1999, 2001). Kanaan et al. (2000a) and Kanaan, Kepler, & Winget (2002) nd the observed red edge for the ZZ Ceti instability strip at 11 000 K. They conclude it is not an observational selection e ect because their noise level was 50 times smaller than the expected amplitude at the red edge. Even =3 modes should be detectable at that S/N ratio. Finding more DAVs at di erent temperatures and masses will improve our observational determination of the edges of the ZZ Ceti strip, as well as determine its mass dependence. 14 1.4.2 Ensemble seismology of the ZZ Ceti stars The almost isothermal cores of these stars constitute 99.99% of the star by mass. White dwarf stars with masses in the range 0.55 1.1 M comprise chie y of carbon and oxygen (Iben 1990); the ratio of carbon to oxygen is determined during the main sequence lifetime by the astrophysically important, but experimentally uncertain, 12 C( , )16 O nuclear reaction rates (Metcalfe, Winget, & Char- bonneau 2001; Metcalfe, Salaris, & Winget 2002; Metcalfe 2003). Hence, we theoretically expect white dwarf cores to be similar; pulsation modes that probe the core better than others are expected to show a signature of this similarity. The ZZ Ceti stars are musical instruments, capable of ringing in thousands of modes. Ensemble seismology can use all the observed pulsation periods in all the similar ZZ Ceti stars to unravel their stellar structure. 15 Chapter 2 Ideal time-series photometer 2.1 Time-series photometry and ideal instrumentation To study phenomena variable at short timescales, we use a special technique called relative time-series photometry or high-speed di erential photometry. There is more to time-series photometry than meets the eye; it is not just relative photometry with a precise measurement of the observation epoch. A good time series photometer not only requires a precise measurement of the start time of an exposure, but also the duration of the exposure. Elements that cause a jitter in these measurements are undesirable. These include, but are not limited to, an undisciplined or drifting clock used for timing, a mechanical shutter, and an unregulated time-share data acquisition system. Besides accuracy in timing, a good time-series photometer must be able to provide su cient time resolution to sample the variable phenomena well. For example, to study the hot ZZ Ceti stars that exhibit pulsation periods in the range 100 300 s, we need a suitable time resolution of 5 10 s. This not only requires that the photometer should allow a short exposure time, but that it also allows an insigni cant dead time between consecutive exposures. Frame 16 transfer CCDs are ideal for time-series photometry as they can provide contiguous exposures with no dead time. Photometers with a chip readout time above 10 s are incapable of good time-series photometry, even if they allow a 1 s exposure time. Figure 2.1 shows the light curve of a pulsating white dwarf acquired using the low resolution spectrograph (LRS) in imaging mode on the 9.2 m Hobby-Eberly Telescope at McDonald Observatory (top panel). The bottom panel shows observations of the same star using Argos (see section 2.2) on the 2.1 m Otto Struve telescope. The data from the 9.2 m telescope exhibits uncertainties in period and phase larger by a factor of 1.5 than the 2.1 m data in the lower panel. q.e.d. Multi-site observations make it easier to distinguish between the true frequencies and aliases in our data1 . The Sun never rises on instruments like the Whole Earth Telescope (WET; Nather et al. 1990) used for the study of variable white dwarfs; WET comprises a collaboration of observatories around the globe. Increasing the precision of our frequency measurements of these pulsators requires data over a long timespan with accurate timing, acquired ideally from multiple sites with suitable time-series photometers. Each detected mode provides an independent constraint on the stellar structure, so maximizing the number of observed modes is essential for identifying a unique model- t to the data. High instrument detection e ciency, larger apertures, and smaller number of re ections will allow us to observe modes of small amplitude ( 0.1%), increasing the number of known modes for the star. The Fourier Transform (FT) of a sine curve of nite length shows aliases of amplitude comparable to, but smaller than, the amplitude of the true frequency. For noise-free data, this is mathematically expressed as the function sin(x)/x. 1 17 Figure 2.1: The top panel shows the light curve of a stable DAV G117-B15A (B=15.5) acquired with the Hobby-Eberly Telescope at McDonald Observatory (effective aperture 9.2 m), using the low resolution spectrograph in imaging mode. The bottom panel shows a light curve of the same star obtained using the 2.1 m Otto Struve telescope also at McDonald Observatory. In the second case, the data was acquired with an e cient instrument idealized for high speed timeseries photometry, a prime focus frame transfer CCD camera with the only optical surfaces between the star and the CCD being the primary mirror, a BG40 lter, and the glass window enclosing the vacuum around the CCD chip. 18 2.2 Argos: designed for time-series photometry R. E. Nather has designed a prime focus CCD photometer, optimized for high speed time-series measurements of oscillating white dwarf stars (Nather and Mukadam 2004), which he has named Argos. I have assisted Ed in making the instrument operational, and in the subsequent testing phase. After light re ects from the primary mirror, it focuses directly onto the small CCD chip without any intervening optics; lack of multiple re ections makes the instrument highly e cient. The combination of an e cient instrument and a large amount of telescope time ( 100 nights/yr) at the 2.1 m telescope has given me a unique opportunity to search for many pulsators. Note that this chapter is not meant to be a stand-alone description of Argos, but supposed to complement the publication by Nather & Mukadam (2004). 2.2.1 The CCD camera Argos is based on a commercial CCD camera made by Roper Scienti c, the Princeton Micromax 512 BFT NTE-CCD camera 2 . Its speci cations are shown in Table 2.1. We acquire an image scale of 3.05 pixels per arcsecond (F/3.9) for our 512 512 pixel CCD chip, and a eld of view of 2.8 arcmin on a side. Frame transfer, initiated by pulses from a GPS system, allows us to obtain contiguous exposures as short as 1 s. The CCD is back-illuminated for improved blue sensitivity and provides a quantum e ciency of 80% in the wavelength range 4500 6500 A. With thermoelectric cooling, we maintain the chip at a temperature of 45 C, and obtain a dark current of 1 2 ADU/s/pixel. The readout time for the entire chip with no binning is 0.28 s; the readout noise is less than 8 electrons RMS. 2 http://www.roperscienti c.com/micromax.html 19 Figure 2.2: Wavelength Response: The top panel shows the response of the Argos CCD chip convolved with atmospheric extinction. The middle panel shows the modi ed response, if we include a blue bandpass lter to reduce sky brightness and increase the measured pulsation amplitude of our hot pulsating white dwarfs. The lower panel shows the wavelength response of a PMT, a detector we used before Argos. The improvement in sensitivity is a factor of 9, as measured on the same telescope. 0.8 0.6 0.4 0.2 0 0.8 0.6 0.4 0.2 0 400 600 800 20 Table 2.1: Summary of Camera Speci cations Pixel size: 13 13 Pixel array size: 512 512, back illuminated On chip storage: 512 512, frame transfer operation Frame transfer time: 310 s Readout rate: 1 MHz, 16 bit A/D conversion Readout time: 0.28 s, full frame with no binning Thermoelectric + fan air exhaust Cooling: Chip temperature: -45 C Measured Read noise: 4 electrons RMS 2 electrons/ADU Gain Dark noise: 1 2 ADU/s/pixel Optical coating: broadband anti-re ection Quantum E ciency: 30% at 3500A, 80% 4500-6500A, 40% at 9000A Linearity: 1% below 40,000 ADU (saturation at 65,000 ADU) 2.2.2 Wavelength response We show the wavelength response of the CCD chip in the top panel of Figure 2.2, taking atmospheric extinction into account. Our chief interest lies in blue pulsating DA white dwarfs, whose pulsation amplitudes are a function of wavelength (Robinson et al. 1995; Nitta et al. 1998, 2000b). Including photons redward of the PMT cuto (ca. 650 nm), which are less modulated by the pulsation process, reduces the measured amplitude (Kanaan et al. 2000b). We nd that the pulsation amplitudes for our instrument, with and without a blue bandpass lter (Schott BG40 1 mm), are di erent by as much as 35 42% (see Figure 2.3). A blue bandpass lter is traditionally used to measure amplitudes comparable to blue-sensitive detectors like PMTs with bi-alkali photocathodes (see Kanaan et al. 2000b), that have been used to observe variable stars since the late sixties. Filterless observations of the ZZ Ceti star G 117-B15A yield an amplitude of 12.6 mma for the 215 s period, shown in the top panel of Figure 2.3. We measure noise in the FT to be 0.43 mma, average amplitude in the high frequency 21 region devoid of pulsations. This gives us a S/N ratio of 29.3 from an hour long lterless observing run on G 117-B15A (B=15.5) with Argos. Observing the same star for an hour with the BG40 lter gives us an amplitude of 21.8 mma and a noise level of 0.7 mma, measured in the same way as before. This yields a S/N ratio of 31.1, marginally better than lterless observations. However, this may not hold true for the same star in cloudy weather (also see Figure 2.4 and related discussions). We show the response of our instrument including a 1 mm BG40 Schott glass lter in the middle panel of Figure 2.2. Although the pulsation amplitude increases due to a blue bandpass lter, the middle panel indicates the loss of photons, which is close to a factor of two for hot white dwarfs, and a factor of six or so for cooler red stars in the eld. We nd an increase in measurement noise in the light curves of the target and comparison stars due to photon losses with the BG40 lter. To use a blue bandpass lter or not, is a decision that must be made individually for each pulsator, depending upon its magnitude and pulsation amplitude, as well as the number and relative magnitudes of the comparison stars to the target pulsator. A bright pulsator, surrounded only by a couple of relatively fainter comparison stars in the eld, will bene t if the blue lter is not used while observing. This is shown by the light curves of WD1345-0055 (g=16.7) in Figure 2.4. With the lter, we obtain an average noise of 1.0 mma, and 0.7 mma without. This translates into a S/N ratio of 10.4 with the BG40 lter, and 12.4 without the lter. Excluding the BG40 lter for stars like WD1345-0055 yields a better S/N ratio. We are unable to demonstrate the case for a faint pulsator in a similar manner; we did not nd a faint pulsator where the dominant mode is a singlet with a stable amplitude, ideal for the test. If there is any amplitude modula- 22 Figure 2.3: The top panel shows an hour of lterless observations of the optical clock, G 117-B15A, where the dominant mode has an amplitude of 12.6 mma. We acquired the data using Argos at the prime focus of the 2.1 m telescope at McDonald Observatory. The lower panel shows a Fourier Transform also based on one hour of data acquired with the same instrument using a BG40 lter. The dominant mode shows an amplitude of 21.8 mma with the lter, a di erence in amplitude of 42%. This di erence is also evident in the other modes. We compute a S/N ratio of 29 without the lter and 31 with the BG40 lter, marginally better. 23 tion, then the signal to noise ratio that we measure in our light curves changes correspondingly, and this change in S/N is not related to our detection system. We also do not have a good season of observations with and without a lter on any of the stars that show multiplet structure. It is our conjecture that using a lter will be helpful in observing faint stars with low pulsation amplitudes, surrounded by multiple bright comparison stars. But it remains to be tested. Observers should carefully think about their decision to use a blue bandpass lter or not, individually for each pulsator. The factors to consider are the loss of pulsation amplitude by 35 42% without the lter, the loss of photons for the white dwarf pulsator by a factor of two with the lter, and the loss of photons for each comparison star by a factor of six with the lter. A simple S/N calculation for each variable should help the observer in making an informed decision. The bottom panel of the Figure 2.2 shows a PMT (Photo Multiplier Tube, R 647 Hamamatsu) wavelength response for comparison, convolved with atmospheric extinction. We used a PMT photometer, called P3Mudgee, on the same telescope to gather data on pulsating white dwarfs before Argos (see Kleinman, Nather, & Phillips 1996 and Nather & Warner 1971). Argos is nine times more sensitive than P3Mudgee; we now obtain usable time-series photometry on objects of apparent magnitude 19, where the limit with P3Mudgee on the same telescope was B 17. The faintest known white dwarf pulsator to date, WD0947+0155, of apparent magnitude g=20 has been discovered with Argos. 2.2.3 Instrument timing A time-series photometer must know precisely when an exposure is started, and precisely how long it took. The Argos timing system is based on a GPS clock 3 3 http://www.trimble.com/thunderbolt.html 24 0.04 0 -0.04 0 1000 2000 3000 Figure 2.4: We show the reduced light curves of WD1345-0055 (g=16.7) with and without a BG40 lter. We nd an improved S/N ratio after excluding the BG40 lter, inspite of the reduction in signal by as much as 35 42%. 25 designed primarily for precision timekeeping. An antenna ensures that the clock remains synchronized with the GPS satellites. The claimed uncertainty for the GPS pulse is about 50 ns, considerably more precise than we need. We have assembled a simple timer card that plugs into the parallel port of the data acquisition computer. The timer card comprises a count-down register that accepts 1 Hz pulses from the GPS clock. When the user sets the exposure time in the acquisition program in the allowed range of 1 30 s, the register counts down from the integral exposure value and sends an output pulse to the CCD camera at the end of the exposure. The positive edge of the output pulse triggers a frame transfer in the CCD chip. The exposure intervals are thus contiguous, and determined directly from the clocking hardware. The jitter in the timing for this operation is hard to measure, but we certainly expect it is 100 s. Mechanical shutters introduce a timing jitter, which is not desirable in a good time-series photometer. They also cause non-uniform illumination of the eld and introduce a source of correlated noise in the light curve. Coherence of the pulsations over long timescales allows us to gain over random noise with time, but it is harder to eliminate non-Gaussian sources of noise. Besides, mechanical shutters have a lifetime of order a million cycles; at 5 s exposures we would have to replace the shutter every ten months of observing, if we assume observing 15 days every month with 10 hr observations every night. 2.2.4 The prime focus mount The Argos camera is shown on its prime focus mount in Figure 2.5, at the end of the 2.1 m telescope. A smaller CCD camera (Cyclops) with a wide angle eld of view (150 ) is attached to the Argos camera to capture regular images of the dome slit. The dome of the 2.1 m telescope does not track, and hence the images 26 Figure 2.5: We show the CCD photometer Argos on its prime focus mount at the 2.1 m telescope at McDonald Observatory. Attached to Argos is a smaller camera that captures images of the dome slit. These images allow the user to track the slit manually from the control room. prove useful to the observer in deciding when to move the dome. Design considerations for the prime focus mount necessitate that the optic axis pass through the center of the 6 mm 6 mm CCD chip, while the plane of the chip remains perpendicular to the optic axis. Since we cannot rely on machining precision alone to center the optic axis on the chip, we mount the camera on a metal plate that can move by half an inch in two orthogonal directions using x-y adjustment screws. These x-y axes align with the rows and columns of the CCD chip. The tip/tilt of the camera can be controlled by a pushpull arrangement of screws, to orient the CCD chip perpendicular to the optic 27 axis. The two-step alignment procedure involves centering the optic axis on the chip, and then adjusting the plane of the CCD chip till it is perpendicular to the optic axis. For the rst step, we point to a bright star of fourth or fth apparent magnitude, and defocus the telescope to get a donut-shaped stellar image. Non-uniform illumination of the donut breaks the circular symmetry and points in the direction of the optic axis. By moving the telescope, we place the donut-shaped image in di erent parts of the chip. Analyzing the images so obtained, we get an idea of the location of the optic axis on the chip. If it is not located at the center of the chip, we use the x-y alignment to move the camera relative to the optic axis. By successive iterations, we are able to align the optic axis to within the central fth of the chip ( 50 pixels). This level of precision is su cient for a small CCD chip such as ours. When properly aligned, the corners of the chip are 2 arcmin from the optical axis, where aberration from coma due to the parabolic primary mirror is calculated to expand a point image to about 1 arcsecond in diameter. We rarely experience sub-arcsecond seeing at the McDonald Observatory site, so we have not been able to verify this calculation. Argos mounts on the focus assembly of the 2.1 m telescope, and the entire photometer moves along the optic axis when we focus the telescope. In order to check if the plane of the CCD is perpendicular to the optic axis, we point to a crowded eld at the zenith in conditions of good seeing. We defocus the telescope, and then move in the direction of the focus in small steps. We continue past best focus, taking images at regular intervals. If the CCD is not perpendicular to the optic axis, then we expect to see stars in one part of the chip come to focus before stars in the diagonally opposite section. Performing this test on our instrument reveals that we need a negligible adjustment of the 28 push-pull alignment screws. We align the instrument as described above, every time the primary mirror is re-aluminized, and additionally as needed if the stellar images seem distorted. Our instrument can potentially be used by the observatory sta to ensure that the primary mirror is aligned well and in the same place as before. Argos is sensitive to misalignments of the optic axis by as small a value as 0.4 arcmin. 2.2.5 Ba ing Argos Scattered light initially posed a signi cant problem for Argos. The rst three nights of the commissioning run proved beyond doubt that the shiny aluminium surfaces of the mount had to be anodized. We chose hard black anodizing that does not corrode as easily as regular anodizing. Also, hard anodizing re ects less light than regular anodizing at a small or grazing angle of incidence due to its dull matt grey nish. We replaced the single crude ba e in the original design with a ve-stage ba e system consisting of two thin plates very close to the camera (henceforth, camera ba es), and three other ba es in the body of the mount (henceforth, mount ba es). The two camera ba es and the mount ba e closest to the camera have square shaped apertures with rounded corners and are derived by projecting the light beam backwards from the CCD chip. These are a few percent larger than the converging light beam from the primary. The other two mount ba es have circular apertures, which are 5 7% larger than the light beam. The edges of all the light ba es are at an angle of 45 with respect to the optic axis, so they may re ect light away from the CCD camera. The camera ba es play a crucial role in preventing scattered light from reaching the camera. The mount ba es serve as a mere support system, and 29 reduce scattered light from multiple re ections. Light can be incident on the CCD chip from a small annulus enclosing the primary without any intervention from the ba e system. Scattered light can also be incident on the chip from speci c angles after at least two re ections, but the dull black nish of the mount reduces the amount of stray photons at each re ection. CCD images acquired during twilight indicate that the ba e system is e ective. Constructing a special camera ba e that vignettes the CCD chip on all sides by a measured amount can also serve as a means of aligning the primary mirror of the telescope after aluminizing. This procedure can be done during the day time, and is much more precise than the method suggested in section 2.2.4. We should be able to detect changes 10 arcseconds in the orientation of the optic axis with ease, which would change the umbra of the vignetting pattern by 30 pixels. 2.3 Noise and measurement in photometry During data reduction, we subtract the sky background from all the stars in the eld, and use the constant stars to help divide out any modulations introduced by the Earth s atmosphere in the light curve of the target. We now discuss the precision of the extracted photometry and the dominant sources of noise as a function of stellar magnitude. We will not be discussing noise sources particular to an extraction algorithm or to an instrument. The measured counts for a star contain statistical photon noise, scintillation noise mainly due to turbulence high up in the atmosphere, seeing noise chie y due to turbulence close to the telescope, and modulations from varying atmospheric transparency. Sky counts also include statistical uctuations, modulations from changes in transparency, etc. Noise in the stellar and sky counts add in quadrature to give us the observed noise in the reduced light curve. 30 Using the scintillation power spectrum published in Dravins et al. (1998), we estimate a 0.4% scatter in the light curves of bright stars with exposure times 1 3 s. Scintillation noise arises due to atmospheric turbulence, which occurs at di erent spatial and temporal scales (Dravins et al. 1997). Although a large aperture allows us to average the high frequency components, we are still susceptible to the low frequency variations. Additional transparency changes occur due to changing mist, dust, etc. in the light path. Figure 2.6 shows a series of bright stars observed with Argos, along with the faint target (top panel, g=19.7). We can easily see the correlation in the light curves of the di erent bright comparison stars, spread over the chip at angular separations of 2 2.5 arcmin. We veri ed this correlation for a series of observing runs. We conclude that the cumulative e ect of low frequency scintillation and transparency changes is mostly correlated over the CCD chip. For faint stars, read noise and stochastic sky noise add in quadrature to swamp out scintillation e ects. We show the root mean square (RMS) scatter in the light curves of constant stars as a function of magnitude in Figure 2.7. We used 10 s exposures for all the data shown in the plot. RMS scatter is a good indicator of the di erent sources of noise discussed above. It depends chie y on the length of the exposure, weather conditions, extinction, and lters used. It can vary by as much as a factor of two for a given instrument set up, and for the same exposure time. Note that the magnitude on the x axis is not the true apparent magnitude of the star; it is an instrumental magnitude obtained by comparing the counts of the constant star with the target4 . The scatter in the plot comes from using a di erent set of comparison stars for di erent elds, and from a varied range of weather conditions and extinction values. Figure 2.7 shows a model of photon noise and sky noise. Photon noise is the dominant source of noise 4 When we use a faint target to determine the magnitude of a fairly bright unsaturated comparison star, our determination will be far from the true apparent magnitude of the star. 31 400000 300000 180000 150000 40000 30000 25000 20000 25000 20000 0 200 400 600 800 1000 Figure 2.6: We show the sky-subtracted light curves of a white dwarf (top panel, g=19.7) along with the comparison stars in the eld. The light curves of the bright comparison stars show that low frequency scintillation noise and transparency changes are mostly correlated over the chip, as these stars are at angular separations of 2.0 2.5 arcmin. 32 for bright stars, while read noise and sky noise dictate the noise level for faint stars5 . The magnitude at which this transition takes place helps characterize the sensitivity of an instrument. We determine this to be g=16.5 for Argos, with an uncertainty of order one magnitude arising from our imprecise calibration of instrumental magnitudes. Figure 2.7 also proves that the data reduction techniques we use are reliable even at the faint limits of what we can observe. Other sources of noise include deteriorating weather conditions or increase in cloud. We show our data on the new DA variable WD0949-0000 (Mukadam et al. 2004) obtained using Argos with 10 s exposures in cloudy conditions on 2 April, 2003 in the top two panels of Figure 2.8. The top panel of the gure shows the sum of a few comparison stars in the eld; the drop in the raw counts allows us to estimate that the clouds are about 10-20%. The second panel shows the reduced light curve of the faint (B 18.8) target star, after dividing by the summed comparison. Bad seeing conditions prove to be just as detrimental to faint stars, as do clouds. The lower two panels of Figure 2.8 show the light curves of the same star WD0949-0000, acquired on 27 March, 2003, when the seeing conditions deteriorated from 2 arcseconds to 3 arcseconds. These light curves were obtained using a constant aperture weighted with a Gaussian, and hence the change in seeing conditions causes a large change in the raw counts. This is not worrisome as the pattern divides out quite well. The lowest panel of the gure shows the reduced light curve of the target with a larger scatter than due to 20% cloud. Speci cations for the CCD camera indicate a read noise of 8 electrons RMS. This value makes the expected noise much higher than the observed noise for the faint stars. We measure a read noise of 4 electrons RMS by requiring that the quadrature sum of read noise and stochastic sky noise should match the observed noise for faint stars. 5 33 Figure 2.7: We show the root mean square scatter in the light curves of the comparison stars or the photometric precision as a function of magnitude. We derive this relative magnitude by comparing the photon counts of the comparison star to the known white dwarf in the eld. The scatter in the plot is caused by di erent weather conditions and di erent comparison stars in each eld used for di erential photometry. Photon noise dominates the observed noise in bright stars, while the quadrature sum of read noise and sky noise essentially determines the noise in faint stars. 34 Figure 2.8: Non-photometric Weather conditions: The top two panels show data acquired in cloudy conditions, and the lower two panels show data obtained in bad seeing conditions. The top panel shows the summed light curve of a few comparison stars in the eld and the reduced counts indicate 10-20% cloud. The second panel shows the reduced light curve of the faint (B 18.8) new DA variable WD0949-0000 (Mukadam et al. 2003a, 2003b), after it has been divided by the summed comparison. The third panel shows the weighted comparison in bad seeing conditions (2-3 ), and the fourth panel shows the reduced light curve of WD0949-0000. Being able to extract usable data on faint stars in 20% cloud and 3 arcsecond seeing shows the limit of what is possible with Argos. 35 2.4 Time keeping with stellar clocks We cannot measure the absolute value of time, but we can compare two di erent clocks with each other. The O C diagram is a quantitative measure of the discrepancy of two clocks (e.g. Kepler et al. 1991), where O is the observed value of the phase for one clock and C is the calculated value based on a constant clock with the best t period. A linear trend in the O C diagram implies a correction to the best period, while a non-linear trend indicates that one clock is drifting with respect to the other. The (O C) technique can be used to improve the period estimates for any periodic phenomenon. Using the hot DAVs as stellar clocks for time keeping requires high signalto-noise data sets that resolve the pulsation spectrum well. The intervals between these seasonal data sets should be short enough to determine the number of clock ticks or cycles during the interval without an ambiguity. This phasing of data sets allows us to utilize the O C technique to improve our determination of the period, phase, and drift rate of the stellar clock. 2.4.1 Uncertainty in period When determining the period of a mono-periodic light curve without any gaps, we are essentially measuring the start time of the rst cycle and the end of the last cycle. Dividing this time interval by the total number of cycles yields the period. The uncertainty in period is then related only to the precision of measuring two points in time, as we know the number of cycles de nitively for continuous data. As we include additional cycles, the uncertainty in period reduces linearly with time. This reduction cannot continue inde nitely; it ceases when we reach the threshold set by the photometric precision of the light curves. Suppose we have multiple data sets of a mono-periodic clock, and the gaps between them allow bootstrapping (see Winget et al. 1985). This technique 36 helps us construct data to bridge the gaps using period and phase values of the observed data. In an ideal world, where mono-periodic clocks dwell in a noise-free environment, this technique will enable the uncertainty in period to reduce linearly with time. In practice, data with gaps imply aliases. If and only if we can discern between the true frequency and the aliases, can we overcome this source of non-Gaussian noise and acquire an improvement in the period, linearly with time. For the last case, we will consider multiple data sets that cannot be bootstrapped. We can obtain individual measurements of the period from each light curve and weight them according to their uncertainties. The uncertainty in period in this case reduces as the square root of the number of measurements, and also depends on the individual uncertainties. The top panels of Figures 2.9 and 3.0 show the uncertainties in period determined by a non-linear least squares t to light curves of di erent durations. Both stars shown in Figure 2.9, G 117-B15A and WD1354+0108, consist of a single dominant period, and can e ectively be treated as mono-periodic clocks. Their uncertainties scale inversely with time and also with pulsation amplitude. 2.4.2 Uncertainty in phase Measuring the period in a given light curve is a di erent process than using the light curve to determine the phase, and consequently the uncertainties in these quantities behave di erently. Phase is only meaningful when we can de ne the period of a phenomenon. Each cycle in the light curve provides an independent measure of the phase, and so we improve as the square root of the number of measurements. The uncertainty in phase reduces as square-root of time passes by. For multiple data sets that have been successfully bootstrapped, the un- 37 certainty in phase should reduce as the square-root of time only if we identify the true frequency correctly from among the aliases. For multiple data sets that cannot be bootstrapped, we cannot improve our phase value beyond that of a single set. The lower panels of Figures 2.9 and 3.0 show the corresponding plot for the uncertainty in phase. We can determine the uncertainty in phase for a mono-periodic star of the same brightness by scaling inversely with pulsation amplitude. 2.4.3 Uncertainty in drift rate measurements G 117-B15A and R 548 are the most stable optical clocks known; constraints on the drift rates of their dominant modes are smaller than a few times 10 15 s/s (Kepler et al. 2000a; Mukadam et al. 2003a). These constraints have been established with observations that span over three decades, using bootstrapped data sets. If there are no cycle count errors in the O C diagram, then we can expect that initial drift rate measurements will improve linearly with time. This is because the initial improvement in the drift rate comes from an improving measure of the period. After this initial era, the scatter in the O C diagram will inhibit further improvement in the period. Only after our measurement of the period is precise enough, can we start using the O C diagram to obtain meaningful constraints on the drift rate. Adding subsequent observations to the O C diagram should theoretically result in drift rate constraints that improve as the square of time goes by, until impeded by the scatter in the diagram. However, Kepler et al. (2000a) nd that drift rate uncertainties decrease linearly with time for G 117-B15A, possibly due to the 1.8 s scatter. We cannot verify this theoretical expectation for R 548 either because the O C diagram consists 38 Figure 2.9: The top panel shows the uncertainty in period reducing linearly with time for G 117-B15A and WD1354+0108. Both these stars can e ectively be thought of as mono-periodic clocks. Their uncertainties scale inversely with pulsation amplitude. The lower panel shows the uncertainty in phase improve with the square root of time. Changes in sky transparency and approaching twilight at the end of the observing run on WD1354+0108 do not allow the uncertainty in phase to improve any further. 39 4 3 2 1 0 5 10 15 20 Figure 2.10: The top panel shows the uncertainty in period and in phase for a mono-periodic clock WD1354+0108 over 17 days. The uncertainty in period reduces linearly with time and the uncertainty in phase reduces with the square root of time. 40 only of 10 points, and these observations were not acquired with telescopes of similar aperture (or more precisely, detection systems of comparable e ciency, i.e. comparable telescope apertures, instrument e ciencies, and weather conditions). 2.5 Re ections Argos saw rst light on 1 November, 2001, at the prime focus of the 2.1 m telescope at McDonald Observatory, six months after we purchased the CCD camera from Roper Scienti c. After struggling with initial problems concerning timing, scattered light, inexperienced observers, and so on, we slowly settled into an intensive program of observing 6 weeks every trimester. We discovered 35 new ZZ Ceti stars with Argos, mainly observing DAV candidates from the Sloan Digital Sky Survey. We discovered the faintest white dwarf pulsator known to date, a 20th magnitude DB variable, WD0947+0155 (Nitta et al. 2004; in preparation). Argos was also instrumental in the discovery of optical bursts from a low mass X-ray binary MS 1603.6+2600. The optical bursts con rmed that the compact object was a neutron star (Hynes, Robinson, & Je ery 2004). 41 Chapter 3 Search for new ZZ Ceti stars 3.1 Selection of DAV candidates from databases In order to nd a substantial number of ZZ Ceti stars, we required a large database for candidate selection; homogeneity of the database would ensure a high e ciency for our search. We decided to turn to the ongoing Sloan Digital Sky Survey (SDSS) and the Hamburg Quasar Survey (HQS). 3.1.1 Sloan digital sky survey Using a dedicated 2.5 m telescope with a CCD camera (see Gunn et al. 1998), the SDSS will ultimately result in ve band photometry of ten thousand square degrees in the north Galactic cap (York et al. 2000). It is a calibrated photometric and astrometric digital survey (see Hogg et al. 2001; Pier et al. 2003) with follow-up spectroscopy of selected objects, mainly targeting bright galaxies and quasars from the imaging survey. Potential white dwarfs are allocated bers for spectroscopy only when the number of higher priority targets is insu cient to ll the 640- ber spectroscopic plug plates (Kleinman et al. 2004). There have been three public data releases by the SDSS: the Early Date Release (EDR; 42 Stoughton et al. 2002), Data Release 1 (DR1; Abazajian et al. 2003), and Data Release 2 (DR2; Strauss et al. 2004). Kleinman et al. (2004) present 2561 spectroscopically identi ed white dwarfs from DR1; Harris et al. (2003) presented the initial survey of the SDSS white dwarfs. 3.1.2 Hamburg quasar survey Hagen et al. (1995) describe the HQS as a wide angle objective prism survey to nd bright quasars in the northern sky in an area of 14 000 square degrees using plates taken at the Calar Alto Schmidt telescope. Homeier & Koester (2001a) have produced a catalog of about 3000 DA white dwarfs in the temperature range of 9 000 30 000 K using an automated classi cation of the low resolution digitized photographic prism spectra, 1100 of which were found to have e ective temperatures close to the ZZ Ceti instability strip. Homeier et al. (1998) present Te and log g values from follow-up spectroscopy of 80 HQS DA white dwarfs. 3.2 Techniques to select the SDSS DAV candidates We outline below the di erent techniques that we used to select the SDSS DAV candidates along with the corresponding success rates. The success rate of discovering ZZ Ceti stars depends not only on the number of variables found, but also on our de nition of a non-variable, i.e., at what detection threshold do we stop pursuing a DAV candidate and call it a non-variable. The success rate for all search techniques is higher for brighter stars (14.5 B 17.5), for which we obtain a typical noise level of 1 3 mma in 1-1.5 hr runs with Argos on the 2.1 m telescope. However, most of our targets are fainter (18 B 19.5) and require a larger amount of telescope time to achieve the same noise level. 43 For such stars, our typical 2 hr observing runs lead to a detection threshold of 3 6 mma with Argos. 3.2.1 Photometric technique for the SDSS DAV candidates Greenstein (1982) acquired multichannel spectrophotometry for 14 DAVs and found that they lie in a narrow range in color space 0.41 G R 0.29. He concluded that the narrow band (G R) color is an excellent temperature indicator for DAs. The SDSS color system comprising the lters u, g, r , i, and z, calibrated by Smith et al. (2002), is a broadband color system like that of Johnson. Fontaine et al. (1982) show that DAV candidate selection based on the Johnson lter system yields a 30% success rate, for spectroscopically identi ed DA white dwarfs. Hence we expected to nd one pulsator for every 3 candidates observed at the telescope and started using the photometric technique in the initial stages of the project. Selection of candidates from the SDSS EDR (Stoughton et al. 2002) required us to calibrate the DAV strip in the SDSS colors as the EDR did not include any known DAVs. The original SDSS lter system u , g , r , i , & z is described in Fukugita et al. (1996). Stoughton et al. (2002) describe how the current adaptation u, g, r , i, & z di ers from the original lters. We utilized this technique in the early stages of the search and hence the following description alone is given in terms of the original SDSS lter system. Lenz (1998) derived synthetic colors in the SDSS lter system for white dwarfs with Multi-Channel Spectro-Photometric (MCSP) data from Greenstein & Liebert (1990). We used DA white dwarfs common to both papers and compared their MCSP G R colors to their synthetic SDSS g r colors and also U V colors to u g colors. Neglecting higher order terms, a best- t parabola to the resultant plots gave us the following transformations: 44 g r = a + b(G R) + c(G R)2 u g = d + e(U V ) + f (U V )2 (3.1) (3.2) where a = 0.0296 0.0057, b = 0.679 0.010, c = 0.000 0.024, d = 0.137 0.011, e = 0.776 0.029, and f = 0.013 0.021. Using this transformation, we chose spectroscopically identi ed DA white dwarfs in the color range 0.3 u g 0.6 and 0.26 g r 0.16 as our highest priority candidates. We achieved a success rate of 25% at the detection threshold of 1 3 mma for the candidates so chosen. We found the success rate to be 13% for the detection threshold of 3 6 mma. We found ve pulsators with this technique (Mukadam et al. 2003b) before moving to spectroscopic selection techniques with a higher yield. 3.2.2 Equivalent width technique for the SDSS DAV candidates Although the primary goal of the SDSS is extragalactic objects, spectroscopy of interesting stellar objects is also obtained. The SDSS program to target quasars and other blue objects resulted in many white dwarf spectra. DA white dwarf spectra show Balmer absorption lines, pressure broadened by the extremely high gravity. The SDSS spectra cover a wavelength range of 3800 9200 A and have a resolving power of 2000, su cient to resolve the absorption lines clearly. The equivalent widths of the H and H lines correlate well with the e ective temperature of the star, which essentially determines whether or not the star will pulsate. We have measured the equivalent widths of the H and H lines for all the observed variables and non-variables, which had been previously selected by the photometric method. We nd that the variables form a cluster in equivalent width space (except for the unusual pulsator WD2350-0054), as shown in Figure 3.1, suggesting a new technique to pre-select ZZ Ceti candidates. It is 45 not necessary to derive the absolute temperature of a DAV candidate for this relative method, but to compare its equivalent widths (for H & H ) to those from a homogeneous set of observed variables and non-variables. We nd a success rate of 56% at the detection threshold of 1 3 mma, and a success rate of 30% at the detection threshold of 3 6 mma for this technique. This is e ectively a low resolution spectroscopic technique and hence has a lower success rate compared to the following spectroscopic technique. Except for the opacity maximum, there are generally two temperature solutions for a given equivalent width of H and H (see Figure 4 in Bergeron et al. 1995), and this additionally explains the relatively low success rate of this technique. 3.2.3 Spectroscopic technique for the SDSS DAV candidates Our collaborators in the SDSS use Detlev Koester s atmosphere models that treat convection with ML2/ = 0.61 , best described in Finley, Koester, & Basri (1997) and references therein, to derive Te and log g ts for DA white dwarf spectra in the range 3870 A to 7000 A. Kleinman et al. (2004) give a detailed discussion of the method used to derive the temperatures and gravities for these white dwarfs2 Since we are establishing the ZZ Ceti strip empirically, we need not worry about any minor discrepancies between di erent theoretical models, as long as a consistent set of models are utilized for all candidates. We choose our high priority ZZ Ceti candidates between an e ective tem1 Various parameterizations of the mixing-length theory (MLT) are used to treat convection in models of ZZ Ceti stars. B hm & Cassinelli (1971) describe the ML2 version of convection, o assuming the ratio of the mixing length (l) to the pressure scale height (H), l/H = 1. Bergeron et al. (1995) analyzed optical and ultraviolet (UV) spectrophotometric data of ZZ Ceti stars and found that model atmospheres calculated using the ML2 version, assuming = 0.6, provide the best internal consistency between the optical and UV temperature estimates, the observed photometry, the trigonometric parallax measurements, and the gravitational redshift masses. 2 The DR1 white dwarf catalog in the public domain can be found at http://hello.apo.nmsu.edu/ sjnk/sdsswds/dr1cat/vac/wdDAS.table.DR1.html. The DR2 public website can be found at http://www.sdss.org/DR2. 46 Figure 3.1: Plot of equivalent widths of H and H lines for the observed SDSS DA white dwarfs: All but one of the SDSS pulsators are found in a small region in equivalent width space. In this region, we nd a success rate of 56% at a detection threshold of 1 3 mma. 47 perature of 12 500 K and 11 000 K. We achieve a success rate of 80% at a detection threshold of 1 3 mma for this technique and a success rate of 50% at a detection threshold of 3 6 mma. These rates are re ected in Figure 3.2. We can achieve a higher success rate of 90% by con ning our candidates to the temperature range 12 000 11 000 K, but being mainly interested in nding hDAV stars, we choose to include candidates in the temperature range 12 500 12 000 K in our observations. Our choice of candidates will also help in better establishing the blue edge of the ZZ Ceti strip. Note that we use the spectroscopic technique in conjunction with the equivalent width method during our search. It is therefore di cult to present meaningful statistics on these two techniques separately. However, we realize that equivalent width information is already contained in the line pro les; the equivalent width method is a low resolution spectroscopic technique. We concur with Fontaine et al. (2001, 2003) that the spectroscopic technique is the most fruitful way to search for these pulsators. 3.3 Spectroscopic technique for HQS DAV candidates The spectroscopic technique can be applied to photographic spectra as well, and has been used to obtain temperature estimates of DA candidates from the HQS (Homeier & Koester 2001b; Homeier 2001). Due to the signi cantly lower S/N ratio of the prism spectra, which also do not show resolved line pro les, errors in Te are typically 1 000 2 000 K. In some cases, the solutions can be on the wrong side of the Balmer line maximum (e. g. PG 1632+153). Also, Homeier (2003) estimate that 10 20% of this sample may not be white dwarfs. This collectively explains our low success rate at nding DAVs from this sample, and why we focused mainly on the SDSS white dwarfs. We achieve a success rate of 12.5% at the detection threshold of 1 3 mma and 9% for 3 6 mma in nding new 48 Figure 3.2: We show temperature and log g determinations of the observed SDSS DA white dwarfs using Koester s model atmospheres. By restricting our observations in the range 11 000 < Te < 12 500 K, we achieve a success rate of 80% in identifying new DAV stars with this method. 49 DAVs from the HQS. 3.4 Data acquisition and analysis 3.4.1 Observing strategy Searching for coherent signals in light curves helps in overcoming noise, as our signal-to-noise ratio improves with the time-base until we reach a limit set by photometric precision. In time series photometry, the signal-to-noise ratio can be calculated in Fourier space. Note that the signal-to-noise ratio depends not only on the magnitude of the star and the amplitude of pulsation modes, but also on the quality and duration of the data. We typically observe bright candidates (14.5 g 17.5) for 1 1.5 hr each and faint candidates (18 g 19.5) for about 2 hr each. We run online data extraction routines that allow us to plot the light curve and FT of the star in real time. If any of these show interesting features, we observe the target for longer. If we nd a pulsator, we observe it for a few hours, and at least twice to con rm its variability. We have been obtaining multiple four hour long data sets on the newly discovered hDAVs for our planet search project. A ZZ Ceti star may have closely spaced modes or multiplet structure, both of which cause beating e ects. A fraction of our low success rate with any technique can be attributed to our single-run investigations of most candidates; an apparent non-pulsator could well be a beating ZZ Ceti star or a low amplitude variable. For example, McGraw (1977) claim BPM 37093 to be non-variable, but Kanaan et al. (1992) show it to be a low amplitude variable with evident beating. Dolez, Vauclair, & Koester (1991) state that the non-variability limit of G 3020 is a few mmag3 , but Mukadam et al. (2002) found G 30-20 to be a beating 3 One milli-magnitude (mmag) equals 0.1086% change in intensity. 50 variable with an amplitude of 13.8 mma. Further observations of these stars are necessary to acquire in order to be certain of the purity of the instability strip. We will re-observe our apparent non-variables that lie in the empirically established ZZ Ceti strip with our collaborators in the coming year. 3.4.2 Data acquisition We have obtained high speed time series photometry with the prime focus CCD photometer Argos on the 2.1 m telescope at McDonald Observatory for 125 nights since February 2002, using the acquisition software written by R. E. Nather (see Nather & Mukadam 2004). During this time, we observed approximately 120 SDSS DA white dwarfs and 20 HQS stars. We used 5 15 s exposures for most of our targets. We have used a 1 mm thick Schott glass BG40 lter 4 for most of our observations (see 2.2.2). 3.4.3 Data reduction and results We extract sky-subtracted light curves from the CCD frames using the IRAF script developed by A. Kanaan. O Donoghue et al. (2000) nd this technique of weighted circular aperture photometry to be one of the best extraction techniques. We use the average seeing during the observing to serve as the Full Width at Half Maximum (FWHM) for the weighting Gaussian function. Weighting minimizes the dependence of the S/N ratio of the reduced light curve on aperture size. To select the optimum aperture size, we employ two di erent methods. The rst method involves computing FTs for light curves extracted using di erent aperture sizes, but with the same weighting function. We compute the average noise of each FT, determined from the high frequency region devoid of pulsation modes. A plot of noise as a function of aperture radius then 4 The transmission of this lter can be found at http://www.besoptics.com 51 allows us to choose the optimal aperture. We check that the amplitude of the dominant mode remains the same for all the aperture sizes. In the rare instance that it does not, we compute the S/N ratio for each light curve to make our choice. Secondly, we use a technique suggested by M. Kilic, that involves subtracting all the light curves from a reference light curve extracted using the largest aperture. The set of residual light curves so obtained do not contain modulations due to pulsations, only noise added in quadrature from the original light curve and the reference light curve. We plot the Root Mean Square (RMS) scatter of these light curves as a function of aperture size, and the residual light curve with the lowest scatter serves as a good indicator of the optimal aperture size. This is a relative method, and the RMS scatter does not indicate an absolute measure of noise. If these two methods do not give the same answer, then the larger of the two apertures serves as the conservative choice. The light curves of the comparison stars are also extracted with the same aperture size as the target. This is done to ensure that we do not have systematic e ects from using di erent aperture sizes. We then correct for extinction variations and divide the light curve of the target star with the sum of one or more comparison stars; we prefer brighter stars for the division as their light curves have lower noise. After this preliminary reduction, we bring the data to the same fractional amplitude scale and convert the times of arrival of photons to Barycentric Coordinated Time (TCB; Standish 1998). We then compute a discrete FT for all the light curves. We present the new DAVs in Table 3.1, listing their coordinates, temperature and log g information, colors, equivalent widths, and magnitudes, along with identi cation numbers required to locate their spectra in the SDSS database. The SDSS spectral objects can be identi ed on the basis of a plate number, mod- 52 i ed Julian date (MJD) of observation, and a ber number. A single object may have multiple spectra, but the combination of a plate, MJD, and ber number will always lead to a unique observation. In Tables 3.2 and 3.3, we present similar information for our observed non-variables from the SDSS. The best re-reduced photometry and spectral parameters should be obtained from the SDSS directly. We plan to publish and maintain a table with all the pulsators, complete with the latest photometry and spectral ts on www.whitedwarf.org at a future date. We list our observed variables and non-variables with the non-variability limit from the HQS in Table 3.4, some of which may not necessarily be DA white dwarfs. We designate a ZZ Ceti candidate Not Observed to Vary as NOV and add the non-variability limit as a su x to this symbol. For example, NOV2 implies a DAV candidate not observed to vary at a detection threshold of 2 mma. If peaks in the FT of a DAV candidate seem to be less than or equal to twice the average amplitude, then these peaks are most probably consistent with noise. The highest white noise peak then de nes the detection threshold or the nonvariability limit. If the highest peak is also re ected in the FTs of the reference stars, then we do not use it to de ne the detection threshold. In that case, we would apply the same test to the second-highest peak, and so on. If a peak seems signi cantly higher than the average amplitude, then we re-observe the candidate to determine whether the peak is real or pure noise. Scargle (1982) gives a thorough discussion of the reliability of detecting a periodic signal in noisy data. 53 Table 3.1: New ZZ Ceti variables Object WD0102 0032a WD0111+0018 WD0214 0823 WD0318+0030a,d WD0332 0049d WD0815+4437 WD0825+4119 WD0842+3707 WD0847+4510 WD0906 0024d WD0923+0120 WD0939+5609 WD0942+5733d WD0949 0000a WD0958+0130 WD1015+0306 WD1015+5954 WD1056 0006d WD1122+0358d WD1125+0345 WD1157+0553 WD1345 0055 WD1354+0108 WD1417+0058d WD1443+0134b WD1502 0001d WD1524 0030c,d WD1617+4324d WD1700+3549d WD1711+6541 WD1724+5835a WD1732+5905a WD2350 0054 SDSS Name SDSS J010207.17 003259.4 SDSS J011100.63+001807.2 SDSS J021406.78 082318.4 SDSS J031847.09+003029.9 SDSS J033236.61 004918.3 SDSS J081531.75+443710.3 SDSS J082547.00+411900.0 SDSS J084220.73+370701.7 SDSS J084746.81+451006.3 SDSS J090624.26 002428.2 SDSS J092329.81+012020.0 SDSS J093944.89+560940.2 SDSS J094213.13+573342.5 SDSS J094917.04 000023.6 SDSS J095833.13+013049.3 SDSS J101548.01+030648.4 SDSS J101519.65+595430.5 SDSS J105612.32 000621.7 SDSS J112221.10+035822.4 SDSS J112542.84+034506.3 SDSS J115707.43+055303.6 SDSS J134550.93 005536.5 SDSS J135459.89+010819.3 SDSS J141708.81+005827.2 SDSS J144330.93+013405.8 SDSS J150207.02 000147.1 SDSS J152403.25 003022.9 SDSS J161737.63+432443.8 SDSS J170055.38+354951.1 SDSS J171113.01+654158.3 SDSS J172428.42+583539.0 SDSS J173235.19+590533.4 SDSS J235040.72 005430.9 Plate 396 694 668 413 415 547 760 864 763 470 473 556 452 266 500 503 559 276 836 836 841 300 301 304 537 310 815 820 350 366 366 386 MJD 51816 52209 52162 51821 51810 51959 52264 52320 52235 51929 51929 51991 51911 51630 51994 51999 52316 51909 52376 52376 52375 51666 51641 51609 52027 51990 52374 52433 51691 52017 52017 51788 Fiber 262 597 354 483 211 350 604 548 144 081 074 476 023 037 163 329 330 073 214 050 377 288 322 345 279 229 390 110 362 264 591 135 RA2000 01:02:07 01:11:01 02:14:07 03:18:47 03:32:37 08:15:32 08:25:47 08:42:21 08:47:47 09:06:24 09:23:29 09:39:45 09:42:13 09:49:17 09:58:33 10:15:48 10:15:20 10:56:12 11:22:21 11:25:43 11:57:07 13:45:51 13:55:00 14:17:09 14:43:31 15:02:07 15:24:03 16:17:38 17:00:55 17:11:13 17:24:28 17:32:35 23:50:41 Dec2000 00:32:59 +00:18:07 08:23:18 +00:30:30 00:49:18 +44:37:10 +41:19:00 +37:07:02 +45:10:06 00:24:28 +01:20:20 +56:09:40 +57:33:43 00:00:24 +01:30:49 +03:06:48 +59:54:31 00:06:22 +03:58:22 +03:45:06 +05:53:04 00:55:37 +01:08:19 +00:58:27 +01:34:06 00:01:47 00:30:23 +43:24:44 +35:49:51 +65:41:58 +58:35:39 +59:05:33 00:54:31 Te (K) 11050 100 11510 110 11570 090 11040 070 11040 070 11620 170 11820 170 11720 170 11680 110 11520 090 11150 70 11790 160 11260 070 11180 130 11680 060 11580 030 11630 110 11020 050 11070 080 11600 120 11050 050 11800 060 11700 050 11300 080 10830 150 11200 120 11190 100 11160 050 11310 040 11540 080 10860 100 10350 060 log g 8.24 0.08 8.26 0.06 7.92 0.05 8.07 0.05 8.25 0.06 7.93 0.09 8.49 0.06 7.73 0.09 8.00 0.07 8.00 0.06 8.74 0.06 8.22 0.07 8.27 0.05 8.22 0.11 7.99 0.03 8.14 0.02 8.02 0.06 7.86 0.03 8.06 0.06 7.99 0.07 8.15 0.04 8.04 0.03 8.00 0.02 8.04 0.05 8.15 0.20 8.00 0.08 8.03 0.07 8.04 0.04 8.64 0.03 7.89 0.05 7.99 0.08 8.31 0.06 EW (H ) (A) 54.66 1.14 52.19 1.56 54.80 1.04 56.09 0.92 53.16 1.03 52.79 1.83 55.10 1.33 56.82 1.48 53.99 1.20 52.94 0.95 46.82 1.20 52.46 1.53 50.91 0.82 51.42 1.65 51.96 0.60 54.61 0.34 54.48 1.11 49.36 0.85 51.71 1.12 53.86 1.10 50.22 0.87 56.51 0.54 54.52 0.47 55.83 1.07 EW (H ) (A) 34.31 0.82 35.52 1.13 32.88 0.77 30.27 0.62 29.33 0.73 32.19 1.33 32.91 0.98 33.95 1.06 34.73 0.86 35.47 0.67 30.39 0.91 34.46 1.12 30.10 0.60 29.88 1.27 38.22 0.42 35.50 0.24 34.58 0.81 32.41 0.60 34.21 0.78 37.60 0.78 32.75 0.63 33.51 0.38 32.32 0.33 31.67 0.76 u g 0.43 0.41 0.28 0.44 0.42 0.34 0.34 0.54 0.42 0.44 0.29 0.43 0.39 0.45 0.41 0.37 0.65 0.15 0.47 0.46 0.32 0.38 0.42 0.47 0.46 0.37 0.38 0.45 0.47 0.19 0.43 0.47 0.42 g r 0.04 0.19 0.14 0.18 0.11 0.06 0.11 0.18 0.22 0.18 0.16 0.17 0.13 0.13 0.23 0.10 0.31 0.20 0.01 0.12 0.04 0.17 0.17 0.22 0.12 0.14 0.23 0.19 0.16 0.11 0.19 0.10 0.11 g 18.21 18.76 17.92 17.81 18.18 19.30 18.50 18.75 18.32 17.73 18.34 18.70 17.43 18.80 16.70 15.66 17.95 17.52 18.13 18.07 17.59 16.70 16.36 18.03 18.72 18.68 16.03 18.33 17.26 16.89 17.54 18.74 18.10 54 52.36 1.26 30.70 0.88 51.81 1.22 50.10 0.85 54.32 0.63 58.33 0.77 51.91 1.39 45.81 1.23 33.06 0.88 33.70 0.62 28.54 0.46 30.85 0.54 32.36 0.96 27.16 0.90 a Multiple Spectra: WD0318+0030 (413 51929 494), WD0949 0000 (266 51602 31), WD1724+5835 (356 51779 271), WD1732+5905 (356 51779 584), WD1345 0055 (300 51943 282), WD1354+0108 (301 51942 324), WD1417+0058 (304 51957 338) b The SDSS spectrum of WD1443+0134 shows only half of the H its line; temperature and log g values are not reliable. c WD1524 0030 does not have a spectrum; Photometric ID information: Run=756, Rerun=8, Camcol=2, & Field ID=769 d Large pulsation amplitudes in cDAVs imply that the true uncertainty in their photometric magnitudes can be as high as 0.1 0.2. e The latest T e and log g ts should be obtained either from the SDSS website directly or from www.whitedwarf.org at a future date. Table 3.2: Not Observed to Vary (NOV) (mostly single 2 hr runs) at a detection threshold of 1 3 mma Object WD0040 0021 WD0152+0100 WD0210+1243 WD0222 0100 WD0257+0101 WD0318+0044 WD0733+2831 WD0740+2505 WD0746+3510 WD0747+2503 WD0751+4335 WD0814+4608 WD0827+4224 WD0946+5814 WD0949 0019 WD1136 0136 WD1138+6239 WD1235+5206 WD1243+6248 WD1302 0050 WD1444 0059 WD1529+0020a,b WD1659+6209 WD1659+6352 WD1718+5621 WD1735+5730 WD2326 0023 SDSS Object Name SDSS J004022.88 002130.1 SDSS J015259.20+010018.4 SDSS J021028.69+124319.0 SDSS J022207.04 010050.3 SDSS J025746.41+010106.1 SDSS J031802.34+004439.8 SDSS J073356.99+283123.8 SDSS J074033.49+250511.9 SDSS J074633.01+351022.8 SDSS J074724.61+250351.1 SDSS J075115.11+433513.9 SDSS J081451.28+460803.6 SDSS J082716.89+422418.7 SDSS J094624.31+581445.4 SDSS J094901.28 001909.5 SDSS J113604.01 013658.2 SDSS J113854.36+623903.4 SDSS J123541.62+520611.9 SDSS J124341.27+624836.3 SDSS J130247.98 005002.7 SDSS J144433.80 005958.9 SDSS J152933.26+002031.2 SDSS J165935.59+620934.0 SDSS J165926.58+635212.9 SDSS J171857.82+562150.2 SDSS J173513.30+573011.5 SDSS J232659.21 002348.0 Plate 392 402 428 406 410 413 754 857 542 857 434 441 761 453 266 327 776 885 782 294 308 314 351 349 367 366 383 MJD 51793 51793 51883 51817 51816 51821 52232 52314 51991 52314 51885 51868 52266 51915 51630 52294 52319 52379 52320 51986 51662 51641 51780 51699 51997 52017 51818 Fiber 063 523 138 252 578 466 226 388 476 625 445 280 476 124 026 535 511 231 360 293 256 354 372 520 416 053 111 RA2000 00:40:23 01:52:59 02:10:29 02:22:07 02:57:46 03:18:02 07:33:57 07:40:33 07:46:33 07:47:25 07:51:15 08:14:51 08:27:17 09:46:24 09:49:01 11:36:04 11:38:54 12:35:42 12:43:41 13:02:48 14:44:34 15:29:33 16:59:36 16:59:27 17:18:58 17:35:13 23:26:59 Dec2000 00:21:30 +01:00:18 +12:43:19 01:00:50 +01:01:06 +00:44:40 +28:31:24 +25:05:12 +35:10:23 +25:03:51 +43:35:14 +46:08:04 +42:24:19 +58:14:45 00:19:10 01:36:58 +62:39:03 +52:6:12 +62:48:36 00:50:03 00:59:59 +00:20:31 +62:09:34 +63:52:13 +56:21:50 +57:30:12 00:23:47 Te (K) 16160 060 12490 070 17160 090 12060 120 16580 210 18290 240 14610 290 18560 190 11050 110 19330 200 14450 230 16410 090 08940 020 10710 030 11710 070 14650 230 12140 100 11990 130 10640 030 15750 070 10490 060 12410 080 10410 030 12810 090 13410 160 10620 050 log g 7.88 0.01 7.84 0.02 7.90 0.02 8.12 0.05 8.29 0.04 7.87 0.04 7.80 0.06 8.28 0.04 7.93 0.08 8.11 0.03 7.60 0.06 7.92 0.02 8.26 0.04 8.15 0.03 7.96 0.04 7.89 0.05 8.06 0.04 8.05 0.06 8.30 0.02 8.05 0.02 7.91 0.06 8.11 0.03 8.17 0.03 8.12 0.03 7.86 0.03 8.33 0.04 EW (H ) (A) 53.51 0.25 59.59 0.47 55.81 0.58 56.56 1.03 57.61 0.86 48.82 1.13 56.63 1.44 53.51 0.93 51.12 0.57 53.53 1.01 48.61 1.23 50.83 0.90 50.14 0.82 27.02 0.89 46.75 0.57 57.00 0.89 53.76 1.24 57.76 0.63 57.44 0.98 46.18 0.55 56.14 0.45 48.96 1.04 56.04 0.45 45.65 0.93 55.53 0.73 61.44 0.48 45.98 0.87 EW (H ) (A) 32.64 0.17 37.60 0.33 30.04 0.36 38.63 0.71 35.93 0.57 28.10 0.74 34.77 1.01 32.11 0.64 33.29 0.40 33.15 0.71 31.91 0.82 32.16 0.62 31.85 0.56 17.62 0.66 26.12 0.43 34.54 0.63 39.95 0.87 35.37 0.45 34.56 0.71 26.70 0.39 33.90 0.32 37.55 0.71 37.83 0.33 26.38 0.65 37.75 0.50 35.10 0.33 29.95 0.63 u g 0.39 0.53 0.19 0.39 0.16 0.11 0.32 0.13 0.24 0.44 0.12 0.39 0.19 0.48 0.46 0.37 0.24 0.41 0.50 0.40 0.36 0.48 0.42 0.44 0.38 0.37 0.49 g r 0.13 0.16 0.37 0.16 0.27 0.29 0.25 0.35 0.30 0.12 0.33 0.20 0.29 0.04 0.16 0.19 0.28 0.22 0.22 0.13 0.18 0.12 0.19 0.14 0.21 0.25 0.09 g 14.83 16.43 16.86 18.04 17.66 18.35 18.83 17.83 16.69 18.39 18.38 17.79 17.44 17.39 16.51 17.76 18.38 16.87 17.85 16.55 16.22 18.21 16.25 17.88 17.47 16.51 17.52 NOV (mma) NOV1 NOV2 NOV3 NOV3 NOV3 NOV3 NOV3 NOV2 NOV2 NOV3 NOV3 NOV2 NOV3 NOV3 NOV3 NOV2 NOV3 NOV2 NOV3 NOV3 NOV2 NOV3 NOV2 NOV3 NOV3 NOV2 NOV2 55 a The star is a member of a DA4M binary system. b The non-variability limit of 3 mma comes from a half an hour long observing run and must be regarded with prudence. Table 3.3: Not Observed to Vary (NOV) (mostly single 2 hr runs) at a detection threshold of 3 6 mma Object WD0037+0031 WD0050 0023 WD0054 0025a WD0106 0014b WD0135 0057 WD0215 0015 WD0217+0058 WD0224+0038 WD0236 0038 WD0238+0049 WD0303 0808 WD0326+0018 WD0329 0007 WD0330+0024 WD0336 0006 WD0340+0106 WD0345 0036 WD0753+3543 WD0756+3803 WD0816+3307 WD0853+0005 WD0953 0051 WD1019+0000 WD1031+6122 WD1045 0018a WD1103+0037 WD1105+0016 WD1126+5144 WD1216+6158 WD1229 0017 WD1315 0131 WD1337+0104 WD1338 0023 WD1342 0159 WD1345+0328 WD1431 0012 WD1432+0146 WD1443 0006 WD1450+5543 WD1503 0052 WD1545+0321 WD1642+3824 WD1651+6334 WD1653+6254 WD1657+6244 WD1658+3638 WD1706+6316 WD1717+6031 WD1718+5909 WD1720+6350a WD1723+5546a WD1724+6205 WD1724+6323 WD1726+5331 WD1735+5356 WD2334 0014 WD2336 0051 WD2341+0032 WD2341 0109 WD2346 0037 SDSS Object Name SDSS J003719.13+003139.2 SDSS J005047.62 002316.9 SDSS J005457.61 002517.1 SDSS J010622.99 001456.3 SDSS J013545.62 005740.1 SDSS J021553.99 001550.5 SDSS J021744.29+005823.9 SDSS J022435.46+003857.5 SDSS J023613.64 003822.2 SDSS J023808.09+004908.8 SDSS J030325.22 080834.9 SDSS J032619.44+001817.5 SDSS J032959.57 000732.5 SDSS J033031.48+002454.9 SDSS J033648.33 000634.2 SDSS J034044.10+010621.9 SDSS J034504.20 003613.5 SDSS J075328.74+354304.9 SDSS J075607.77+380331.7 SDSS J081625.01+330740.4 SDSS J085325.55+000514.2 SDSS J095329.20 005100.7 SDSS J101911.51+000017.3 SDSS J103116.34+612232.6 SDSS J104517.79 001833.9 SDSS J110326.71+003725.9 SDSS J110515.32+001626.1 SDSS J112638.75+514430.9 SDSS J121613.37+615817.0 SDSS J122959.23 001714.5 SDSS J131557.18 013125.5 SDSS J133714.44+010443.8 SDSS J133831.75 002328.0 SDSS J134230.14 015932.8 SDSS J134552.01+032842.6 SDSS J143154.57 001231.2 SDSS J143249.11+014615.6 SDSS J144312.69 000657.9 SDSS J145040.50+554321.4 SDSS J150330.49 005211.3 SDSS J154545.35+032150.0 SDSS J164248.61+382411.2 SDSS J165128.84+633438.3 SDSS J165356.29+625451.4 SDSS J165747.03+624417.5 SDSS J165815.53+363816.0 SDSS J170654.01+631659.6 SDSS J171700.06+603141.8 SDSS J171853.27+590927.5 SDSS J172045.09+635031.7 SDSS J172346.69+554619.0 SDSS J172405.37+620501.4 SDSS J172452.91+632324.9 SDSS J172600.16+533104.1 SDSS J173536.49+535658.1 SDSS J233454.18 001436.2 SDSS J233647.01 005114.7 SDSS J234110.13+003259.9 SDSS J234141.64 010917.1 SDSS J234639.77 003716.0 Plate 392 394 394 396 400 703 405 406 407 408 458 414 414 414 415 416 416 757 543 862 468 267 272 772 275 277 277 879 779 289 340 298 298 912 529 306 536 308 792 310 594 818 349 349 349 820 349 354 366 350 367 352 352 359 360 384 384 385 385 386 MJD 51793 51876 51812 51816 51820 52209 51816 51817 51820 51821 51929 51869 51869 51869 51810 51811 51811 52238 52017 52325 51912 51608 51941 52375 51910 51908 51908 52365 52342 51990 51691 51662 51662 52427 52025 51637 52024 51662 52353 51990 52027 52395 51699 51699 51699 52433 51699 51792 52017 51691 51997 51694 51694 51821 51816 51821 51821 51877 51877 51788 Fiber 531 225 118 068 060 174 601 501 029 420 188 467 037 583 595 420 015 284 550 277 132 099 307 090 230 513 596 472 169 109 587 604 021 590 609 196 318 357 311 133 500 476 265 208 097 514 030 264 342 007 512 085 595 178 212 151 008 481 124 297 RA2000 00:37:19 00:50:48 00:54:58 01:06:23 01:35:46 02:15:54 02:17:44 02:24:35 02:36:14 02:38:08 03:03:25 03:26:19 03:30:00 03:30:31 03:36:48 03:40:44 03:45:04 07:53:29 07:56:08 08:16:25 08:53:26 09:53:29 10:19:12 10:31:16 10:45:18 11:03:27 11:05:15 11:26:39 12:16:13 12:29:59 13:15:57 13:37:14 13:38:32 13:42:30 13:45:52 14:31:55 14:32:49 14:43:13 14:50:41 15:03:30 15:45:45 16:42:49 16:51:29 16:53:56 16:57:47 16:58:16 17:06:54 17:17:00 17:18:53 17:20:45 17:23:47 17:24:05 17:24:53 17:26:00 17:35:36 23:34:54 23:36:47 23:41:10 23:41:42 23:46:40 Dec2000 +00:31:39 00:23:17 00:25:17 00:14:56 00:57:40 00:15:51 +00:58:24 +00:38:58 00:38:22 +00:49:09 08:08:34 +00:18:18 00:07:33 +00:24:55 00:06:34 +01:06:22 00:36:14 +35:43:05 +38:03:32 +33:07:40 +00:05:14 00:51:01 +00:00:17 +61:22:33 00:18:34 +00:37:26 +00:16:26 +51:44:31 +61:58:16 00:17:15 01:31:26 +01:04:44 00:23:27 01:59:33 +03:28:43 00:12:31 +01:46:16 00:06:58 +55:43:21 00:52:11 +03:21:50 +38:24:11 +63:34:38 +62:54:51 +62:44:18 +36:38:16 +63:17:00 +60:31:42 +59:09:28 +63:50:32 +55:46:18 +62:05:01 +63:23:25 +53:31:03 +53:56:58 00:14:36 00:51:15 +00:33:00 01:09:17 00:37:16 Te (K) 10960 050 11490 090 10100 060 14360 210 12570 500 15820 160 13600 240 09790 080 14280 440 13300 300 11400 110 12150 080 16690 330 14500 600 10400 040 12060 140 11430 150 16620 240 16040 250 15460 200 11750 110 10690 100 12760 150 11480 180 09540 040 10540 050 12850 060 11900 150 12200 180 13160 180 12550 210 11830 210 11650 090 11320 160 11620 140 12200 180 11290 070 11960 150 15010 120 11600 130 15050 370 18810 200 14190 150 13230 170 13610 250 11110 120 13140 100 13710 480 12430 350 11690 170 11730 120 13590 300 14530 390 11000 110 12940 280 13370 250 13250 250 13380 330 13090 170 12980 330 log g 8.41 0.03 8.98 0.03 8.02 0.07 7.50 0.05 7.80 0.10 7.85 0.04 7.94 0.04 8.11 0.12 7.65 0.09 7.88 0.06 8.49 0.06 8.09 0.03 7.82 0.07 7.75 0.09 8.26 0.04 8.06 0.05 7.76 0.09 8.28 0.04 7.84 0.05 7.75 0.05 8.11 0.06 8.64 0.11 8.35 0.06 7.68 0.11 8.09 0.06 8.22 0.05 8.26 0.02 8.03 0.07 8.19 0.07 7.89 0.04 8.12 0.08 8.39 0.11 8.08 0.05 8.42 0.09 7.80 0.08 7.92 0.07 8.23 0.06 7.87 0.07 7.48 0.03 8.21 0.07 7.88 0.07 8.40 0.04 7.68 0.03 7.94 0.05 7.77 0.05 8.36 0.09 8.51 0.03 8.17 0.14 7.94 0.11 8.08 0.09 8.07 0.06 7.81 0.06 7.79 0.09 8.23 0.08 7.85 0.07 7.86 0.05 7.86 0.05 7.90 0.08 7.92 0.04 7.97 0.08 EW (H ) (A) 51.11 0.84 55.53 1.39 50.75 1.37 48.98 1.20 56.67 1.30 51.88 1.42 57.35 0.86 37.19 1.65 56.92 1.68 61.18 1.43 51.61 1.43 60.16 0.76 49.62 1.71 54.75 1.51 44.31 0.95 57.95 1.10 54.81 1.29 53.20 1.30 54.61 1.37 51.92 0.92 56.19 1.14 43.13 1.53 59.71 1.13 52.79 1.46 49.30 1.19 46.22 0.96 59.82 0.29 53.61 1.27 56.49 1.13 58.84 0.78 55.85 1.22 54.18 1.56 54.89 0.68 48.05 1.57 51.58 1.23 57.84 1.22 53.62 0.86 56.96 1.35 49.33 0.83 55.35 1.12 54.82 1.45 52.73 1.15 54.33 1.09 62.35 1.30 59.75 1.40 50.04 1.85 58.84 0.81 56.41 1.88 59.89 1.28 50.79 1.34 49.54 1.40 59.10 1.24 52.60 1.59 49.55 1.37 60.11 1.34 56.55 1.28 56.61 1.23 57.18 1.99 58.98 1.07 56.63 1.34 EW (H ) (A) 30.54 0.61 31.15 0.99 30.84 0.92 35.95 0.83 36.91 0.92 34.60 0.95 38.99 0.58 26.94 1.16 29.72 1.19 38.66 1.00 30.33 1.03 34.24 0.53 29.20 1.18 35.38 1.01 27.49 0.67 40.45 0.76 30.30 0.91 35.20 0.89 34.33 0.95 35.19 0.64 33.77 0.83 25.88 1.10 37.16 0.83 34.85 1.03 30.40 0.81 29.22 0.70 37.05 0.21 34.85 0.91 37.49 0.80 40.72 0.54 35.85 0.89 33.14 1.15 34.48 0.50 34.00 1.14 36.92 0.89 35.27 0.85 33.21 0.63 34.07 0.96 32.91 0.58 36.85 0.77 37.91 0.98 31.70 0.82 33.08 0.76 39.68 0.91 37.82 0.96 27.03 1.38 32.88 0.58 38.09 1.41 35.29 0.91 30.68 0.89 38.52 0.96 38.11 0.84 32.63 1.06 27.98 1.01 34.77 0.95 41.01 0.91 40.05 0.88 36.38 1.38 38.34 0.75 36.16 0.95 u g 0.38 0.31 0.32 0.50 0.40 0.21 0.37 0.45 0.44 0.27 0.33 0.38 0.35 0.29 0.39 0.50 0.45 0.11 0.30 0.29 0.39 0.40 0.51 0.57 0.16 0.43 0.35 0.37 0.29 0.45 0.47 0.35 0.45 0.43 0.49 0.45 0.52 0.44 0.31 0.43 0.29 0.07 0.38 0.43 0.41 0.43 0.40 0.34 0.43 0.28 0.37 0.38 0.31 0.49 0.45 0.41 0.57 0.41 0.47 0.32 g r 0.07 0.10 0.16 0.22 0.20 0.28 0.22 +0.05 0.27 0.19 0.06 0.20 0.21 0.20 0.06 0.17 0.17 0.27 0.29 0.32 0.15 0.05 0.21 0.20 0.14 0.19 0.19 0.19 0.16 0.21 0.21 0.11 0.18 0.15 0.22 0.16 0.16 0.14 0.33 0.15 0.24 0.31 0.31 0.21 0.21 0.13 0.13 0.27 0.16 0.21 0.14 0.25 0.28 0.14 0.19 0.20 0.25 0.14 0.21 0.20 g 17.48 18.81 18.55 18.18 18.52 18.65 17.53 19.06 19.25 18.79 18.78 17.42 19.13 18.97 17.93 18.23 19.00 18.46 18.72 17.78 18.23 18.85 18.16 18.71 18.36 17.64 15.20 18.41 18.19 17.36 18.24 18.57 17.09 18.80 18.58 18.40 17.49 18.66 17.21 18.39 18.76 17.98 18.21 18.71 18.87 19.15 17.72 19.41 18.64 18.63 18.91 18.41 19.01 18.75 18.65 18.33 18.28 19.16 18.04 18.34 NOV (mma) NOV5 NOV6 NOV8 NOV9 NOV5 NOV8 NOV7 NOV4 NOV5 NOV5 NOV4 NOV5 NOV7 NOV8 NOV5 NOV5 NOV5 NOV6 NOV5 NOV4 NOV4 NOV4 NOV4 NOV4 NOV4 NOV6 NOV4 NOV4 NOV8 NOV5 NOV4 NOV4 NOV4 NOV4 NOV6 NOV7 NOV5 NOV5 NOV4 NOV4 NOV4 NOV4 NOV4 NOV8 NOV8 NOV4 NOV4 NOV10 NOV7 NOV8 NOV5 NOV6 NOV6 NOV7 NOV4 NOV6 NOV5 NOV6 NOV4 NOV6 56 a The star is a member of a DAM binary system. b Kleinman et al. (2004) give an interesting discussion of this most probable eclipsing star. Table 3.4: Observed stars from the Hamburg Quasar Survey Object HS0951+1312 HS0952+1816 HS0406+1700 HS0843+1956 HS0848+1213 HS0914+0424 HS0942+1416 HS0950+0745 HS1102+0032 HS1431+1521 HS1637+1940 HS1643+1423a PG 1643+143 HS1711+1716 HS2157+8152b HS2306+1303 HS2322+2040 HS0727+6915 HS0827+0334 HS0838+1643 HS0852+1916 HS0926+0828 HS0932+1731 HS0949+0823 HS1654+1927 a b c c Alternate Name NOV (mma) RA1950 hDAV cDAV LB 227 WD 0843+199 NOV1 NOV3 NOV2 NOV2 NOV2 PG 0950+077 NOV3 NOV2 PG 1431+153 NOV2 NOV2 NOV3 NOV2 NOV3 PG 2306+130 PG 2322+206 NOV1 NOV1 NOV4 NOV8 NOV4 LB 8888 NOV3 NOV6 NOV4 NOV4 NOV5 Dec1950 Te (K) BJ 16.7 16.2 15.4 16.4 16.6 16.7 16.9 15.6 14.7 15.7 16.7 09:51:03.1 +13:12:41 11000 09:52:25.4 +18:16:29 11000 04:06:37.0 +17:00:03 16000 08:43:42.5 +19:56:05 10000 08:48:22.0 +12:13:14 13000 09:14:18.2 +04:24:08 13000 09:42:04.4 +14:16:28 13000 09:50:20.4 +07:45:19 12000 11:02:41.4 +00:32:37 12000 14:31:44.5 +15:21:25 11000 16:37:58.9 +19:40:31 12000 16:43:21.5 +14:23:08 25450 260 15.5 17:11:41.1 +17:16:57 11000 21:57:18.5 +81:53:01 10700 40 23:06:00.3 +13:03:07 13000 23:22:05.4 +20:40:04 13000 16.7 B=16.0 15.2 15.5 07:27:34.6 +69:15:57 12100 250 B=16.9 08:27:38.5 +03:34:51 12000 08:38:18.7 +16:43:05 11000 08:52:40.1 +19:16:06 13000 09:26:56.8 +08:28:58 12000 09:32:05.5 +17:31:37 12000 09:49:17.5 +08:23:44 12000 16:54:17.0 +19:27:46 11000 16.7 17.1 15.7 16.5 16.8 16.6 16.2 log g = 7.79 0.05 from Finley, Koester & Basri (1997) log g = 8.71 0.08 from Homeier et al. (1998) log g = 8.29 0.14 from Homeier et al. (1998) 57 3.5 Pulsation properties of the new ZZ Ceti variables We nd that the hot DAV stars are mostly distinct from the cool DAV stars in terms of their pulsation characteristics, chie y the pulsation periods and the pulse shapes. Short pulsation periods typically in the range of 100 300 s are representative of hDAVs, while periods longer than 600 s are typically indicative of cDAVs. Some intermediate temperature DAVs show a rich pulsation spectrum with periods ranging from a few hundred seconds up to 500 s, exhibiting characteristics of both classes. For our purposes, we classify WD0111+0018, WD0214 0823, WD1015+0306, and WD1724+5835 as hDAVs since their dominant mode is representative of the hDAV class. We use the same basis to classify WD0906 0024 as a cDAV. WD2350 0054 is by far the coolest pulsator and is unusual because it is not expected to pulsate according to our empirical determination of the edges of the ZZ Ceti strip. The e ective temperature of WD2350 0054 derived from its SDSS spectrum places it 650 K below the cool edge of the instability strip. Its spectrum does not show any unusual features that we could attribute to a binary companion or contamination of any sort. Furthermore, it shows pulsation periods and pulse shapes characteristic of the hot DAV stars. The SDSS spectrum of WD1443+0134 has a missing section, and hence its temperature and log g values are not reliable. The hot ZZ Ceti stars show pulse shapes distinct from the cDAVs, e.g., see the light curve of WD0214 0823 compared to WD1524 0030. The brighter variables have well de ned pulse shapes, while the low amplitude faint variables do not. Among the hDAVs, only the light curves of the intermediate temperature DAVs like WD1015+5954 show pulse shapes distinct from the rest. The light curves of WD0923+0120 and WD1711+6541 show pulse shapes and amplitudes distinct from other cDAVs. Their low amplitude is a result of their high gravity (log g 8.6). The nonradial g-modes have a non-negligible radial component, 58 the amplitude of which scales inversely with stellar mass and plays a role in dictating the amplitude of the non-radial component. We plot the light curves and FTs of all the new variables in Figures 3.3 3.5. We present the period spectra of the new variables in Chapter 5. 3.6 Conclusions We conclude that the spectroscopic technique, determining temperatures and log g values by comparing the stellar spectra to a grid of atmosphere models, is the most fruitful way to search for ZZ Ceti pulsators, in accordance with Fontaine et al. (2001, 2003). We can achieve a 90% success rate by con ning our candidates between 12 000 K and 11 000 K with this technique at a detection threshold of 1 3 mma. But our interest in hDAVs and the blue edge of the instability strip leads us to choose candidates between 12 500 K and 11 000 K, reducing our success rate to 80%. With the discovery of 35 new DAVs, we almost double the current sample of 39 published ZZ Ceti stars discovered over three decades. 59 Figure 3.3: Light curves and FTs of the new SDSS DAVs 60 Figure 3.4: Light curves and FTs of the new SDSS DAVs 61 Figure 3.5: Light curves and FTs of the new SDSS DAVs 62 Chapter 4 Empirical DAV Instability Strip This is the rst time in the history of white dwarf asteroseismology that we have a statistically signi cant homogeneous set of ZZ Ceti spectra, acquired entirely with the same detection system, namely the SDSS spectrograph on the 2.5 m telescope at Apache Point Observatory. All the spectra have been reduced and analyzed consistently using the same set of model atmospheres and tting algorithms, including the observed photometric colors (see Kleinman et al. 2004). This homogeneity should reduce the relative scatter of the variables in the T e log g plane, and possibly allow us to see emerging new features. The sample size of known DAVs is now almost twice as large since the last characterization of the instability strip by Bergeron et al. (2004). However, we will not include the previously known DAVs in our analysis with the exception of G 238-53, as these pulsators do not have SDSS spectra and will only serve to reduce the homogeneity of our sample. We list the Te and log g values of all the variables and non-variables we discovered within the SDSS data in Paper-I, along with their internal uncertainties. Note that we will not be considering WD2350 0054 in this paper as it may be a unique pulsator; it shows pulsation periods and pulse shapes characteristic 63 of the hot DAV stars, while the SDSS temperature determination places it below the cool edge of the instability strip. We focus on the general trends of the majority of the DAVs, and hence a discussion of WD2350 0054 is postponed to a future date. We will not be including WD1443 0054 either, as its temperature and log g determinations are unreliable due to a missing portion in its SDSS spectrum. We will be including G 238-53, the only previously known ZZ Ceti star with a published SDSS spectrum. 4.1 Empirical instability strip We show the empirical SDSS instability strip in Figure 4.1, as determined by 30 new ZZ Ceti stars and G 238-53. We plot histograms of the observed variables as a function of temperature and log g, and weighted histograms (see section 2.2) for the non-variables (Not Observed to Vary; NOVs). Figure 4.1 has two striking features: a narrow strip of width 950 K and non-variable DA white dwarfs within the instability strip. Pulsations are believed to be an evolutionary e ect in otherwise normal white dwarfs (Robinson 1979; Fontaine et al. 1985; Fontaine et al. 2003; Bergeron et al. 2004). Non-variables in the middle of the strip question this semiempirical premise, even if we use the uncertainties in temperature to justify the non-variables close to the edges. We also note that the DAV distribution appears to be non-uniform across the strip. The features of this plot are in uenced by various factors such as biases in candidate selection, non-uniform detection e ciency in the T e log g plane, and uncertainties as well as systematic e ects in spectroscopic temperature and log g determinations. We address these issues and their e ects on the DAV distribution in the next few sub-sections. 64 Figure 4.1: The distribution of new SDSS DAVs and NOVs (Mukadam et al. 2004) as a function of temperature and log g. We also include G 238-53 in this plot. The narrow width of the instability strip and the presence of non-variables within form the two prominent features of this gure. We also note the paucity of DAVs in the middle of the instability strip. 65 4.1.1 Biases in candidate selection We selected SDSS DAV candidates for high-speed photometry from those spectroscopically identi ed DA white dwarfs that lie in the temperature range 11000 12500 K. These temperature ts are derived by our SDSS collaborators using the spectral tting technique outlined in Kleinman et al. (2004). Paper-I gives a discussion of other candidate selection methods used in our search for ZZ Ceti stars prior to the spectral tting technique. Our various science goals lead to some biases in selecting DAV candidates for observation. The hot DAV (hDAV) stars exhibit extreme amplitude and frequency stability (e.g. Kepler et al. 2000a; Mukadam et al. 2003a). We plan to search for re ex motion caused by orbiting planets around such stable pulsators (e.g. Kepler et al. 1991; Mukadam, Winget, & Kepler 2001; Winget et al. 2003). These stable clocks drift at their cooling rate; measuring the drift rate in the absence of orbital companions allows us to calibrate our evolutionary models. These models are useful in determining ages of the Galactic disk and halo using white dwarfs as chronometers (e.g. Winget et al. 1987; Hansen et al. 2002). Therefore, we preferentially choose to observe hDAV candidates in the range 11700 12300 K to increase the sample of known stable pulsators with both the above objectives in mind. This bias is partially compensated for, as hDAVs are harder to nd (see section 2.2). We also preferentially observe DAV candidates of extreme masses. Low mass (log g 7.6) DAVs could well be helium core white dwarfs; pulsating He core white dwarfs should allow us to probe their equation of state. High mass (log g 8.5) DAVs are potentially crystallized (Winget et al. 1997; Montgomery & Winget 1999), providing a test of the theory of crystallization in stellar plasma. Metcalfe, Montgomery, & Kanaan (2004) present strong evidence that the massive DAV, BPM 37093, is 90% crystallized. 66 The distribution of chosen DAV candidates also depends on the distribution of available DAV candidates. We have an additional bias due to the SDSS criteria in choosing candidates for spectroscopy. But a histogram of the available DAV candidates is consistent with a random distribution and does not re ect any systematic e ects. The non-uniform nature of the DAV distribution does not appear to be a candidate selection e ect. However, we are in the domain of small number statistics since we observed only four DAV candidates in the range 11350 11500 K. Of these, two are massive and consequently expected to be low amplitude pulsators (see section 2.2), making detection di cult. Our data are suggestive of a bimodal DAV distribution in temperature. We hope to investigate this issue further by observing additional DAV candidates in the range 11350 11500 K with our collaborators. 4.1.2 Non-uniform detection e ciency The hDAVs show relatively few pulsation modes, with low amplitudes ( 0.1 3%) and periods around 100 300 s. The cooler DAVs (cDAVs) typically show longer periods, around 600 1000 s, larger amplitudes (up to 30%), and greater amplitude variability (Kleinman et al. 1998). Massive pulsators show low amplitudes as a result of their high gravity (log g 8.6). These distinct trends of the pulsation periods and amplitudes with temperature and log g imply that the detection e ciency must also be a function of Te and log g. The detection e ciency not only varies in the Te log g plane, but is also dependent upon weather conditions and the magnitude of the DAV candidate. Furthermore, a ZZ Ceti star may have closely spaced modes or multiplet structure, both of which cause beating e ects. Some of the non-variables in the instability strip could well be pulsators, that were in the destructive phase of their beating cycle 67 during the observing run. McGraw (1977) claimed BPM 37093 to be non-variable, but Kanaan et al. (1992) showed that it is a low amplitude variable with evident beating. Dolez, Vauclair, & Koester (1991) claimed the non-variability limit of G 30-20 to be a few mmag1 , but Mukadam et al. (2002) found G 30-20 to be a beating variable with an amplitude of 13.8 mma2 . In order to address these issues systematically, we simulate light curves of real pulsators for di erent conditions and compute the resulting Fourier Transform (FT) to see if the signal is detectable above noise. We utilize the real periods and amplitudes, with randomly chosen phases (to sample the beat period), to simulate two hour long light curves3 . We independently shu e the magnitudes and average seeing conditions of real data on the DAVs. This ensures a realistic distribution for both these parameters. We randomly select a magnitude and seeing value from these distributions to simulate white noise, the amplitude of which is determined using a noise table based on real data. We compute a FT of the light curve and check if the star can be identi ed as a pulsator or if the signal was swamped by noise. We repeat this procedure 100 times for each DAV for di erent phases, magnitudes, and seeing values. Note that our noise simulation is not completely realistic, as it does not include e ects due to variable seeing, gaps in the data due to clouds, and low frequency atmospheric noise. However, it does help us understand how the detection e ciency changes in the Te log g plane. We nd that we are able to rediscover almost all of the average and low mass cDAVs in the hundred simulated attempts. The high mass (log g 8.6) DAVs with a substantially lower amplitude are recovered with a 70% success rate. This implies that non-variables in Figure 4.1 in the region log g 8.6 have One milli-magnitude (mmag) equals 0.1086% change in intensity. One milli modulation amplitude (mma) corresponds to 0.1% change in intensity. 3 We generally observe the DAV candidates for two hours at a time when searching for new variables. 2 1 68 a 30% chance of being low amplitude variables. At the hot end of the instability strip, both low pulsation amplitude and beating can cause us to miss even the average or low mass hDAVs 35 out of 100 times. Table 1 lists the non-variables in the instability strip along with their temperature, log g, magnitude, and number of observing runs. The number after the NOV designation indicates the best non-variability limit in mma. Based on the simulations, we assign each non-variable a weight based on our estimate of the probability that the observed candidate is a genuine non-variable, and not a low-amplitude or beating pulsator. We use the non-variability limits to assign the weights 0.98, 0.95, 0.90, 0.85, 0.80, 0.70, and 0.60, for NOV1, NOV2, NOV3, NOV4, NOV5, NOV6, and NOV7 or higher, respectively. If the NOV is massive (log g 8.6), then we additionally multiply its weight by a factor of 0.7. If the NOV is close to the blue edge of the strip, then we multiply by a factor of 0.65 to account for low amplitude and/or beating pulsators. However if the NOV has been observed multiple times, then it is unlikely to have been missed as a result of beating. In such a case, we multiply its weight only by a factor of 0.8 instead of 0.65, to allow for a possible low amplitude variable. We utilize these weights in section 6 to compute best- t red and blue edges. 4.1.3 Uncertainties in temperature and log g determinations The true external uncertainties in the SDSS Te determinations are likely to be larger than listed in Paper-I. We expect that the external uncertainties are of the order of 300 K. However, the uncertainty that is relevant in determining the width and purity of the instability strip de ned by a homogeneous ensemble is the internal uncertainty. The low signal-to-noise of the blue end of the SDSS spectra reduces the reliability of the log g values. The H8 and H9 lines depend mostly on gravity 69 Table 4.1: Non-variables in the ZZ Ceti instability strip Object Limit Obs. runs SDSS Te (K) SDSS log g g Weight WD0037+0031 WD0050 0023 WD0222 0100 WD0303 0808 WD0345 0036 WD0747+2503 WD0853+0005 WD1031+6122 WD1136 0136 WD1337+0104 WD1338 0023 WD1342 0159 WD1345+0328 WD1432+0146 WD1443 0006 WD1503 0052 WD1658+3638 WD1726+5331 NOV5 NOV6 NOV3 NOV4 NOV5 NOV3 NOV4 NOV4 NOV2 NOV4 NOV4 NOV4 NOV6 NOV5 NOV5 NOV4 NOV4 NOV7 2 2 4 2 3 3 2 2 1 2 1 2 1 1 1 3 4 1 10960 050 11490 090 12060 120 11400 110 11430 150 11050 110 11750 110 11480 180 11710 070 11830 210 11650 090 11320 160 11620 140 11290 070 11960 150 11600 130 11110 120 11000 110 8.41 0.03 8.98 0.03 8.12 0.05 8.49 0.06 7.76 0.09 7.93 0.08 8.11 0.06 7.68 0.11 7.96 0.04 8.39 0.11 8.08 0.05 8.42 0.09 7.80 0.08 8.23 0.06 7.87 0.07 8.21 0.07 8.36 0.09 8.23 0.08 17.5 18.8 18.0 18.8 19.0 18.4 18.2 18.7 17.8 18.6 17.1 18.8 18.6 17.5 18.7 18.4 19.2 18.8 0.80 0.50 0.60 0.85 0.80 0.90 0.55 0.85 0.62 0.60 0.85 0.85 0.70 0.80 0.80 0.85 0.85 0.60 70 because neighboring atoms predominantly a ect higher energy levels (Hummer & Mihalas 1970), and their density depends on log g. The external uncertainties in log g for our ensemble may be as high as 0.1, twice that of the estimated uncertainty for the Bergeron et al. (2004) sample. We nd an average log g of 8.10 for our sample of 31 objects, while the 36 objects in Bergeron et al. 8.11. G 238-53 is common to both samples; Bergeron et (2004) average at al. (2004) derive Te =11890 K and log g=7.91, while the SDSS determination places G 238-53 at Te = 11820 50 and log g = 8.02 0.02. The temperature values agree within 1 uncertainties. Temperature is mainly determined by the continuum and the H , H , and H lines; the low S/N at the blue end of the SDSS spectra has a reduced e ect on temperature determinations. The well calibrated continuum, extending from 3800 9200 A provides an accurate temperature determination. The gradual change in mean mass as a function of temperature for the SDSS DA white dwarf ts is addressed in Kleinman et al. (2004), and Figure 7 of their paper shows a quantitative measure of this systematic e ect. The increase in log g across the width of the instability strip is only 0.02, and implies that our determinations of cDAV masses are negligibly higher. These systematic e ects are small in the range of the ZZ Ceti instability strip, and cannot produce either the narrow width or the impurity of the observed strip. We conduct a simple Monte Carlo simulation to estimate the internal T e uncertainties of our ensemble. Using the observed pulsation characteristics, we can separate the DAVs into two classes: hDAVs and cDAVs (see section 2.2). We show the observed distribution of the hDAVs and cDAVs in the lowest panel of Figure 4.2. These distributions are distinct, except for 3 objects. Based on the empirical picture, we conceive that the underlying DAV distribution may look similar to that shown in the topmost panel of Figure 4.2. We perform a Monte 71 Carlo simulation, drawing hDAVs and cDAVs randomly from the expected DAV distribution, and using Gaussian uncertainties with = 300 K. We show the resulting distribution in the second panel; the large uncertainties cause signi cant overlap between the cDAVs and hDAVs, swamping the central gap. We perform a similar simulation with =200 K (third panel), and it compares well with the observed distribution considering the small numbers of the empirical distribution. This suggests that the internal uncertainties in e ective temperature for our ensemble are 200 K per object, provided we believe that the hDAVs and cDAVs each span a range of at least 300 K. Note that the internal uncertainties for a few individual objects maybe as large as 250 300 K. 4.2 Probing the non-uniform DAV distribution using pulsation periods The mean or dominant period of a pulsator is an indicator of its e ective temperature (see section 2.2). This asteroseismological relation is not highly sensitive, but it provides a technique independent of spectroscopy to study the DAV temperature distribution. We show the distribution of the dominant periods of the SDSS DAVs in Figure 4.3. The top right panel in Figure 4.3 shows the number of DAVs per period interval and is suggestive of a bimodal distribution; this increases the likelihood that the dearth of DAVs near the center of the strip is real4 . 4 We made a similar plot using the dominant periods for the 36 previously known DAVs, but did not nd any evidence for a bimodal distribution. Determining the dominant period of the 36 ZZ Ceti stars in the literature proved to be di cult and quite inhomogeneous compared to our own data on the SDSS DAVs. 72 100 50 0 200 150 100 50 0 4 2 0 12000 11500 11000 Figure 4.2: We choose hDAVs and cDAVs from the distributions shown in the top panel, and use a Gaussian error function with = 300 K to compute the distributions shown in the second panel. We also similarly determine a DAV distribution with internal uncertainties of order 200 K, shown in the third panel. Comparing the empirical DAV distribution, shown in the bottom panel, to the synthetic computations, we conclude that the average internal uncertainty for our ensemble is 200 K. 73 1250 1000 750 500 250 0 4 0 12000 11500 11000 Figure 4.3: Period distribution of the SDSS DAVs as a function of temperature. The top left panel exhibits two distinct clumps consisting of the short period hDAVs and the long period cDAVs. The dominant period of a DAV is a seismological temperature indicator and the histogram shown in the top right panel is suggestive of a bimodal distribution. 74 4.3 Questioning the impurity of the instability strip Non-variables in the instability strip imply that all DA white dwarfs do not evolve in the same way. This notion has a severe implication: decoding the inner structure of a DAV will no longer imply that we can use the results towards understanding DA white dwarfs in general. Hence we question our ndings, and conduct simulations to verify our results. Although we estimate the internal T e uncertainties to be at most 200 K in section 2.3, we will conservatively assume = 300 K for all subsequent calculations. The SDSS spectra do not show any evidence of a binary companion for all the non-variables within the instability strip. Also, we used D. Koester s model atmospheres to ascertain that the SDSS algorithm had chosen a solution consistent with the photometric colors (u g, g r ) in every case. We now conduct a Monte Carlo simulation assuming a pure instability strip enclosed by non-variables, as shown in the top panel of Figure 4.4. Note that we have not included a log g dependence in our model, as we expect it to be a smaller e ect than what we are about to demonstrate. We choose nonvariables from outside the strip and add uncertainties chosen randomly from a Gaussian error distribution with =300 K to determine the NOV distribution shown in the middle panel. We nd that although non-variables leak into the strip, they are found mostly at the outer edges and their number tails o within the strip. The observed NOV distribution (bottom panel) does not show any signs of tailing o within the instability strip. Rather, it displays the same number of non-variables at the edges as in the center of the strip. This suggests that the instability strip is impure, and that all the NOVs within the instability strip did not leak in due to large Te uncertainties. We carried out these simulations several times to verify these results. We compute the likelihood that the instability strip is pure based on the 75 Figure 4.4: Assuming a pure instability strip as shown in the top panel, we use a Monte Carlo simulation assuming a Gaussian distribution for the internal uncertainties with =300 K to determine the expected distribution for nonvariables within the strip. The observed NOV distribution is at, and shows no signs of tailing o within the strip. The observed distribution shows the same number of non-variables at the edges as in the center of the instability strip. 76 following two criteria. There are two ways in which a non-variable can disappear from the instability strip: subsequent observations show it is a (low amplitude) variable or the internal uncertainties in Te prove to be large enough to allow the possibility that it may have leaked into the strip. Table 1 lists our estimates of the probabilities that the NOVs found within the strip are genuine non-variables. The chance that NOVs may have leaked into the strip due to large internal uncertainties = 300 K are: 0.35 for WD0037+0031, 0.18 for WD00500023, 0.13 for WD0303-0808, 0.04 for WD0345-0036, 0.25 for WD0747+2503, 0.42 for WD0853+0005, 0.15 for WD1031+6122, 0.38 for WD1136-0136, 0.31 for WD1338-0023, 0.11 for WD1342-0159, 0.28 for WD1345+0328, 0.13 for WD1432+0146, 0.25 for WD1503-0052, 0.20 for WD1658+3638, and 0.31 for WD1726+5331. The probability that each of the above non-variables disappear from the instability strip is then: 0.48, 0.59, 0.26, 0.23, 0.33, 0.68, 0.28, 0.62, 0.41, 0.24, 0.50, 0.30, 0.36, 0.32, and 0.59 respectively. Three or four of the above non-variables may have an inclination angle that reduces the observed amplitude below the detection threshold. Instead of calculating various permutations, we will evaluate the likelihood of the worst case scenario. Let four NOVs that have the least chance of disappearing from the instability strip be the ones that have an unsuitable inclination angle for observing pulsations. In that case, the chance that the instability strip is pure is 0.004%. The impurity of the instability strip suggests that parameters other than just the e ective temperature and log g play a crucial role in deciding the fate of a DA white dwarf, i.e., whether it will pulsate or not. 4.4 Narrow width of the ZZ Ceti strip Computing the width of the instability strip using the e ective temperatures of the hottest and coolest pulsators gives us a value, independent of our concep77 tion of the shape of the ZZ Ceti strip. Determining whether the blue and red edges continue to be linear for very high (log g 8.5) or very low (log g 7.7) masses is presently not possible with either our sample or the Bergeron et al. (2004) sample. The width of the instability strip calculated from the empirical edges at di erent values of log g involves additional uncertainties from our linear visualization of the edges. The empirical SDSS DAV instability strip spans from the hottest objects G 238-53 and WD0825+4119, both at Te = 11820 170 K, to the coolest object WD1732+5905 at 10860 100 K. This span of 960 200 K is considerably smaller than the 1500 K width in the literature (Bergeron et al. 1995; Koester & Allard 2000). The hottest pulsator in the Bergeron et al. (2004) sample is G 226-29 at 12460 K and the coolest pulsators are G 30-20 and BPM 24754 at 11070 K. The extent of the instability strip for the Bergeron et al. (2004) sample is then 1400 K. The drift rates of the stable ZZ Ceti pulsators give us a means of measuring their cooling rates (e.g. Kepler et al. 2000a, Mukadam et al. 2003a). Our present evolutionary cooling rates from such pulsators suggest that given a width of 950 K, a 0.6 M ZZ Ceti star may spend 108 yr traversing the instability strip. This agrees with theoretical calculations by Wood (1995) and Bradley, Winget, & Wood (1992). The narrow width constrains our understanding of the evolution of ZZ Ceti stars. 4.5 Empirical blue and red edges We draw blue and red edges around the DAV distribution that enclose all of the variables. This is shown in Figure 4.5 by the solid line for the blue edge and the line with dots and dashes for the red edge. These edges also include non-variables within the instability strip. 78 We now demonstrate an innovative statistical approach to nd the best t blue and red edges that maximize the number of variables and minimize the number of non-variables enclosed within the strip. To the best of our knowledge, no standard technique can be used to solve this interesting statistical problem. Our statistical approach has two advantages: we are accounting for the uncertainties in temperature and log g values and we are utilizing most of the variables and non-variables in our determination rather than just a handful close to the edge. This problem has essentially two independent sources of uncertainties: the uncertainties in temperature and log g that shift the location of a star in the Te log g plane and the uncertainty concerning the genuine nature of a non-variable. Pulsators masquerading as non-variables can signi cantly alter our determination of the blue and red edges. Hence, we assign di erent weights to DAVs and NOVs. Since the DAVs are con rmed variables, we assign them a unit weight. We use the non-variability limit to decide the weight of all the NOVs that lie outside the empirical ZZ Ceti strip, as in section 2.2, while we assign the weights listed in Table 1 for NOVs within the instability strip. 4.5.1 Technique We construct a grid in Te and log g space in the respective ranges 9000 14000 K and 6.0 10.5 with resolutions of 50 K and 0.05. For each point in the grid, we consider possible blue and red edges that vary in inclination angle relative to the temperature axis from 15 degrees to 165 degrees by half a degree with each successive iteration. For each point of the grid, and for each possible blue edge, we compute a net count as follows: DAVs on the cooler side of the edge count as +1 each and on the hotter side count as 1 each. NOVs on the hotter side of the edge 79 count as +w, and on the cooler side as w each, where w is the weight of the corresponding NOV. To determine the best blue edge, we consider all DAVs and NOVs that satisfy Te 11500 K. This ensures that the NOVs close to and beyond the red edge do not in uence the determination of the blue edge. If the DAV or NOV is within 3 of the edge, then we determine the net chance that it lies on the hot or cool side of the edge, assuming a Gaussian uncertainty distribution. We multiply this chance with the count for that object, before adding it to the total count. An e ect of this choice is that the best edge is determined by the global distribution of DAVs and NOVs, rather than the few close to the edge. Similarly, we determine the best red edge at each point of the grid by counting DAVs on the hotter side of the edge as +1 and NOVs on its cooler side as +w, and vice versa. We consider all DAVs and NOVs within the instability strip and cooler than 11820 K to compute the best red edge. If the DAV or NOV is within 3 of the red edge, then its contribution is a fraction of the above, depending on the probability that it lies on one side of the edge or the other. To test our statistical approach, we input the Te and log g determinations of the previously known DAVs from Bergeron et al. (2004) along with the SDSS NOVs. The resulting red and blue edges are fairly similar to those of Bergeron et al. (2004), and we attribute most of the di erence to using an independent set of NOVs5 . Figure 4.5 shows our best- t for the red edge and our constraint on the blue edge using our statistical approach. log g = 0.0043Te 43.48 log g = 0.0010Te 3.01 5 Best t blue edge Best t red edge (4.1) (4.2) We cannot use the same set of non-variables as Bergeron et al. (2004) as they did not publish the non-variable parameters or identi cations. 80 Figure 4.5: Statistical determination of the blue and red edges (solid lines) from the homogeneous set of 31 SDSS DAVs with an estimated 1 uncertainty (dotted lines; red edge coincides with one of the dotted lines). We also show a red edge inclusive of all DAVs (dash-dotted line). We show the empirical edges from Bergeron et al. (2004) as dashed lines, and the theoretical blue edge from Brassard & Fontaine (1997; ML2/ =0.6) for comparison. We show Mike Montgomery s computations (Montgomery 2004; private communication) of the theoretical edges assuming ML2/ =0.8 convection, based on the convective driving theory of Wu & Goldreich (1999). 81 4.5.2 Estimating the uncertainties The dominant e ect that dictates the uncertainties in the slope (log g dependence) and location (in temperature) of the edges arises as a result of the unreliable nature of the NOVs. Are they genuine NOVs or low amplitude pulsators? Our simulations in section 2.2 show that we miss 30% of high mass pulsators due to their low amplitude. We estimate this should introduce an uncertainty of order 0.2 in the total count for both the red and blue edges. The NOVs close to the blue edge, but within the instability strip, can introduce additional uncertainties in our determination. We add these independent sources of uncertainty in quadrature to obtain an estimated 1 uncertainty of 0.6 for the red edge and 0.4 for the blue edge. We show these as dotted lines in Figure 4.5. Our estimates of the 1 uncertainties clearly show that the red edge is well constrained, and the slope of the blue edge is not. For the blue edge, we determine: log g = 0.0016Te 10.64 log g = 0.0037Te 36.35 For the red edge, we determine: +1 away from the blue edge 1 away from the blue edge (4.3) (4.4) log g = 0.0012Te 4.73 log g = 0.0010Te 3.01 +1 away from the red edge 1 away from the red edge (4.5) (4.6) Note that we already account for the uncertainties in Te and log g in determining the red and blue edges. The unreliability of these uncertainties contributes towards an uncertainty in the slope of the edges; this turns out to be a negligible second order e ect. 82 4.5.3 Comparison with empirical edges We show the empirical blue and red edges from Bergeron et al. (2004) in Figure 4.5 for comparison. The slopes of the red edges from both samples agree within the uncertainties. But our constraint on the blue edge di ers signi cantly from that of Bergeron et al. (2004) and Kepler et al. (2000b), and suggests that the dependence on mass is less severe. The mean temperature of our sample is 11400 K, while the mean temperature for the Bergeron et al. (2004) sample is 11630 K. The observed extent of our instability strip de ned by 31 objects spans 10850 11800 K, while that of Bergeron et al. (2004) spans 11070 12460 K6 . We can consider these values to imply a relative shift of 200 K between our sample and that of Bergeron et al. (2004). We would also like to point out that our sample is magnitude limited and reaches out to g = 19.3. We are e ectively sampling a distinct population of stars as most of our sample lies between 200 250 pc, while the Bergeron et al. (2004) sample is within 100 pc. 4.5.4 Comparison with theoretical edges In Figure 4.5, we show the theoretical blue edge from Brassard & Fontaine (1997) due to the traditional radiative driving mechanism; they use a ML2/ =0.6 prescription for convection in their equilibrium models. We also show Mike Montgomery s computations (Montgomery 2004; private communication) of the blue and red edges from the convective driving theory of Wu & Goldreich (Brickhill 1991; Wu 1998; Wu & Goldreich 1999), assuming ML2/ =0.8 for convection. We see that the blue edges of the two theories are essentially the same, Excluding G 226-29, the Bergeron et al. (2004) sample spans a width of 1060 K from 11070 K to 12130 K. 6 83 and would nearly coincide if the mixing-length parameter were tuned. To obtain the red edge of Wu & Goldreich, Mike Montgomery made the following assumptions: (1) the relative ux variation at the base of the convection zone is no larger than 50%, (2) the period of a representative red edge mode is 1000 s, and (3) the detection limit for intensity variations is 1 mma. Within this theory, the convection zone attenuates the ux at its base by a factor of C , where C is the thermal response time of the convection zone, so we have adjusted C such that the surface amplitude 0.5/( C ) 10 3 , equal to the detection threshold. The observed distribution of variables and non-variables suggests that the mass dependence of the blue edge is less severe than predicted by the models. Both the slope and the location of the red edge we calculate are consistent with the observed variables and non-variables within the uncertainties. 4.6 Conclusion Using a statistically signi cant and truly homogeneous set of 31 ZZ Ceti spectra, we nd a narrow instability strip between 10850 K and 11800 K. We also nd non-variables within the strip and compute the likelihood that the instability strip is pure to be 0.004%. Obtaining higher signal-to-noise spectra of all the SDSS and non-SDSS DAVs as well as non-variables in the ZZ Ceti strip is crucial to a better determination of the width and edges of the instability strip, and in investigating the purity of the instability strip. This should help constrain our understanding of pulsations in ZZ Ceti stars. The DAV distribution shows a scarcity of DAVs in the range 11350 11500 K. After exploring various possible causes for such a bimodal, non-uniform distribution, we are still not entirely con dent that it is real. The data at hand are suggestive that the non-uniformity of the DAV distribution is real, and stayed hidden from us for decades due to the inhomogeneity of the spectra 84 of the previously known DAVs. However, we are in the domain of small number statistics and unless we investigate additional targets in the middle of the strip, we cannot be con dent that the bimodal distribution is not an artifact in our data. 85 Chapter 5 ZZ Ceti Ensemble characteristics 5.1 Pulsations in ZZ Ceti models Pulsations in ZZ Ceti models (and in stars) are self-driven oscillations. Stochastic noise frequencies coincident with the eigenfrequencies are ampli ed by the driving mechanism to observable amplitudes. The driving frequency is governed by the thermal timescale at the base of the hydrogen partial ionization zone. The blue edge of the ZZ Ceti instability strip occurs when the star is cool enough to have a hydrogen partial ionization zone, su ciently deep to excite global pulsations. The hydrogen partial ionization zone constitutes a region of recombination, where both free-free and bound-free opacities are signi cant. It is not only a region of higher opacity, but it has the ability to modulate the degree of ionization as it expands or contracts, more than any other layer in the star. A stochastic nonradial expansion, occurring locally in the partial ionization zone, will reduce the density, and hence the degree of ionization in that region. This will allow that portion of the star to cool easily. The restoring force of buoyancy will subsequently cause this region to contract, increasing the den- 86 sity. As the density is increasing, the degree of ionization continues to increase. This reduces the ow of energy through this region, causing it to heat up. 5.1.1 Growth of amplitude Why should the amplitude of such stochastic processes increase? A source of opacity provides a contracting region a method of stowing away energy, without a signi cant increase in temperature. This allows the region to contract some more. In the downstroke, as a region of the partial ionization zone is contracting, the degree of ionization increases. This enables it to continue contracting past density maximum, until it reaches pressure maximum. Similarly, when the region is expanding, the degree of recombination increases. The amplitude continues to grow with each passing cycle. This is similar to a child pumping energy into a swing; the child straightens his/her legs while swinging up, and folds his/her legs while swinging down, increasing the amplitude in both directions. The energy absorbed by the partial ionization zone between density maximum and pressure maximum is a measure of the energy pumped into the pulsation mode. This additional boost will cause the region to expand more during the upstroke. Recombination during expansion implies that pressure minimum will follow density minimum. The additional energy that ows through the region between the density and pressure minima serves as another boost, increasing the amplitude of pulsation. Is the presence of a partial ionization zone the only requirement for a star to pulsate? In order for a region to drive, it must absorb energy at maximum compression. If the expansion and contraction is adiabatic, then the region cannot pump energy into pulsation amplitudes. The partial ionization should be signi cantly above the adiabatic nonadiabatic transition zone, in the nonadiabatic regime, in order to drive. 87 Can any region in the nonadiabatic domain drive pulsations? A stochastic expansion or contraction will not modulate the degree of ionization as much for any region in the nonadiabatic domain, as for a region in the partial ionization zone. Hence, the same amount of energy ows essentially unmodulated and radially through the system. Higher pulsation amplitude is related to higher modulation of the degree of ionization, and hence to the size of the partial ionization zone. What limits the amplitude growth? If a region loses heat during compression, then it serves as an energy sink or a damping mechanism. In the downstroke, energy is radiated from the heated regions. Radiative damping then serves to limit the pulsation amplitude. 5.1.2 Fluid motions in the star Why are these pulsations non-radial? The star requires substantially more energy to expand against gravity for radial displacements compared to nonradial motions. The nonradial expansion necessitates that if some regions of the partial ionization zone are expanding, others must be contracting at the same time. The star has corresponding bright and dark regions on the surface. Frequencies with a phase mismatch cancel each other, and their amplitude do not grow beyond that of white noise. Are these uid motions restricted to the surface? Similar to ripples in water, these non-radial motions penetrate below the surface, although with reducing amplitude. Montgomery & Winget (1999) and Montgomery, Metcalfe, & Winget (2003) showed that pulsations probe up to the inner 99% of the star. Even though the amplitude in the core is extremely small, it is dense and massive enough that modes are sensitive to changes in the core conditions. We can learn more about the stellar interior from white dwarf variables than we can 88 from some other classes of variables, because white dwarfs are not as centrally condensed. 5.2 Trends across the instability strip The distinct behavior of the pulsation periods and amplitudes of the ZZ Ceti stars as a function of temperature has been discussed in several papers, e.g. McGraw et al. (1981), Winget & Fontaine (1982), and was systematically demonstrated for a signi cant sample of DAs by Clemens (1993) and more recently by Kanaan, Kepler, & Winget (2002). We now demonstrate and discuss these trends for the newly discovered DAVs. 5.2.1 Observed pulsation periods The driving frequency in ZZ Ceti models is governed by the thermal timescale at the base of the hydrogen partial ionization zone. As the star cools, the base of the partial ionization zone moves deeper in the star, and the thermal timescale increases. Hence, we expect cool ZZ Ceti stars to show longer periods compared to the hot ZZ Ceti variables. Figure 5.1 con rms this expectation; it shows a clear increase in the mean period, weighted by the observed amplitude, as the star cools across the instability strip. The bimodal distribution that we see in Figure 4.3 is not evident in this gure. This is perhaps because there are systematic di erences between the SDSS Te and log g determinations and those taken from Bergeron et al. (2004). Additionally, larger internal inhomogeneities in the Bergeron et al. (2004) sample could be swamping out the underlying bimodal pattern. 89 Figure 5.1: We show the weighted mean period of 67 ZZ Ceti stars vs their spectroscopic temperature; each period was weighted by the observed amplitude. The gure clearly indicates an increase in the pulsation period as DAVs cool across the ZZ Ceti strip. 90 5.2.2 Observed pulsation amplitudes As the base of the partial ionization zone moves deeper in the star, it increases in size. As a result, we expect an increase in pulsation amplitudes as the star cools across the DAV instability strip. Figure 5.2 shows the average amplitude of 67 DAVs as a function of their spectroscopic temperature. The outer envelope of these plots is a better indicator of the intrinsic pulsation amplitude, and shows an increase as we move from the blue edge to the red edge of the instability strip. The observed amplitude in a ZZ Ceti star can be lower than the intrinsic amplitude due to inclination angle e ects, geometric cancellation, and limb darkening. We are not able to resolve the disk of the star from Earth. Hence the observed amplitude of each pulsation mode is lower due to a disk-averaging e ect, depending on the angle of inclination. Therefore, we have a higher probability of detecting =1 modes than =2 modes, and so on. At ultraviolet (UV) wavelengths, the increased limb darkening decreases the contribution of zones near the limb. Hence, modes of higher compared to the low are canceled less e ectively in the UV modes (Robinson, Kepler, & Nather 1982). The intrinsic amplitude also depends on the mass of the star. We indicate the three massive pulsators, BPM 37093, WD0923+0120, and WD1711+6541 with hollow boxes to show that their high gravity is responsible for their low pulsation amplitudes. Nonradial g-modes have a non-negligible radial component, the amplitude of which scales with stellar mass and plays a role in dictating the amplitude of the nonradial component. Figure 5.3 shows the mean amplitude of the 30 SDSS DAVs as a function of their spectroscopic temperature. This plot exhibits a fairly homogeneous sample, with consistent data reductions, and we nd reduced scatter compared to Figure 5.2. Figure 5.2 shows a decline in amplitude near the red edge of 91 Figure 5.2: We show the square root of the total power of 67 ZZ Ceti stars vs their spectroscopic temperature. There is clearly an increase in the average power, as the ZZ Ceti stars cool towards the red edge of the instability strip. The scatter in this diagram comes chie y from di erent inclination angles and masses that alter the observed amplitude, and the outer envelope is indicative of the intrinsic average amplitude of the star. The points enclosed by a hollow square indicate the massive, and consequently low amplitude pulsators, WD1711+6541, WD0923+0120, and BPM 37093. 92 the instability strip, if we believe that the six low amplitude pulsators close to the red edge do not all have unfavorable inclination angles. If this is true, then this is the rst observational evidence that the red edge is not abrupt, and that pulsation amplitudes decline close to the red edge, before the star stops pulsating. 5.2.3 Mode density For a given , increasing the radial overtone increases the period. The period spacing for nonradial g-modes reduces asymptotically. The density of available eigenmodes as a function of period changes quite dramatically between the blue and red edges of the ZZ Ceti strip. At the red edge, multiple eigenmodes have a frequency close to the driving frequency and are excitable. While we expect to observe only a few modes close to the blue edge due to the relatively larger spacing. 5.3 Drifting eigenmodes Kepler et al. (2000a) conclude that the evolutionary P is dictated by the rate of cooling for the DAV stars, and contraction is not signi cant in the temperature range of the DAV instability strip. O Donoghue & Warner (1987), Kepler et al. (2000a), and Mukadam et al. (2003a) have constrained the unidirectional drift rate of the 192 s mode in L 19-2, 215 s mode in G 117-B15A, and 213 s mode in R 548 respectively to be smaller than a few 10 15 s/s. To put this number in perspective, these clocks are expected to lose or gain one cycle in a few billion years. These drift rates are consistent with evolutionary cooling in our models (e.g. Bradley, Winget, & Wood 1992; Bradley 1996). However, the 274 s mode in R 548 indicates a drift rate faster than the 93 Figure 5.3: We show the mean amplitude of the 30 SDSS ZZ Ceti stars vs their spectroscopic temperature. We see an increase in the mean amplitude as DAVs cool across the ZZ Ceti strip. 94 213 s mode by a factor of 100 (Mukadam et al. 2003a). The 270 s and the 304 s modes in G 117-B15A indicate a drift rate faster than the 215 s mode by factors of 10 and 20 respectively (Costa 2004). Di erent modes sample the star in a di erent way, and their drift rates could well be in uenced by changing conditions other than cooling. They could be undergoing an avoided crossing. It is also conceivable that the 270 s modes measure stellar cooling, while the 215 s mode constrains the stability of the mode trapping mechanism. The cDAV stars exhibit many pulsation modes, the amplitudes of which are observed to change signi cantly on timescales, orders of magnitude shorter than the evolutionary cooling (e.g. Pfei er et al. 1996; Kleinman et al. 1998). Although the eigenmodes of the cDAVs should also drift at the evolutionary rate, we have not yet been able to con rm this theoretical expectation. 5.4 Why study ZZ Ceti stars as an ensemble? Fundamental physics of degenerate matter governs the interior equation of state of white dwarf stars. White dwarf masses are distributed in a narrow range around 0.6 M . The ZZ Ceti variables have e ective temperatures con ned in the range 10850 11800 K. White dwarfs with masses in the range 0.55 1.1 M possess carbon-oxygen cores (Iben 1990), and their composition is dictated by the 12 C( , )16 O nuclear reaction rate (Metcalfe, Winget, & Charbonneau 2001; Metcalfe, Salaris, & Winget 2002; Metcalfe 2003). These fundamental similarities between various ZZ Ceti stars assure us that ensemble seismology may prove to be an e ective technique in probing the stellar interiors. Pulsation modes that probe the core better than others are likely to show a signature of this similarity. Ensemble seismology seems to be the only way to decipher pulsators that show relatively few modes, such as the typical hDAVs. 95 5.5 Teaching di erent DAVs to play the same tune Di erent ZZ Ceti stars have di erent masses, even though the dispersion in mass is small. Increasing the mass of a pulsating white dwarf causes the star to contract, and reducing the size of this resonating structure changes the eigenfrequencies. This is similar to changing the length of a vibrating string. Massive white dwarfs are also comparatively more dense than 0.6 M white dwarfs; this is similar to increasing the density of the vibrating string. As a pulsator cools, traversing across the instability strip, the periods of the eigenmodes increase at the rate of 4 10 15 s/s1 (Kepler 2004; private communication). Hydrogen and helium layer masses also play a role in tuning the eigenfrequencies of this musical instrument. Massive pulsators (log g 8.6) are expected to be substantially crystallized (see section 1.2.3). Pulsations are excluded from the crystallized region; the crystallization front serves as a hard boundary and this changes the pulsation periods substantially in our models of ZZ Ceti stars. In order to study these stars as an ensemble, i.e. in order to add the observed periods of di erent ZZ Ceti stars to a single set of modes, we will have to take their di erences into account. 5.5.1 Scaling the pulsation spectra We show all the pulsation periods of all the ZZ Ceti stars in the Tables 5.1 and 5.2. Figures 5.4 5.6 show the pulsation spectra of individual DAVs. We also show the grand sum of all the observed modes in Figure 5.7, and indicate a histogram of the observed periods. If we vary the bin size in the histogram, Note that this slow drift in the eigenmodes as a result of cooling is di erent from the increase in observed pulsation periods as the ZZ Ceti star traverses the instability strip (sections 1.1, 5.2). The increase in observed pulsation periods is due to the change in the driving frequency, which causes di erent eigenmodes to be excited in the star. 1 96 Sum 0 100 200 300 400 500 Figure 5.4: Pulsation spectra of the hDAVs we nd that various peaks rise and fall due to small number statistics. But we are convinced that the peaks marked in Figure 5.7 are real, as they are easily distinguished for various bin sizes. The histogram is suggestive that the peaks indicate modes with the same not be true for all cases. Assuming that all the periods observed close to 205 s in the ZZ Ceti stars correspond to the same mode, we determine a group mean period at 204.8 s. We and k values in di erent DAVs. But this may 97 Table 5.1: Pulsation periods and amplitudes of the known ZZ Ceti stars Object G 226 29 GD 165 L 19 2 WD1345 0055 WD1354+0108 WD0847+4510 G 238 53 LP 133 144 R 548 G 117 B15A G 185 32 WD0939+5609 GD 385 KUV 11370+4222 WD0958+0130 WD1125+0345 WD1015+0306 GD 66 WD1724+5835 WD0111+0018 WD0214 0823 WD2350 0054 GD 244 KUV 08368+4026 WD0842+3707 WD0815+4437 G 207 9 HS0507+0435 WD0949 0000 WD1015+5954 MCT 0145 2211 HL Tau 76 WD0923+0120 WD1711+6541 EC 14012 1446 Mass (M ) Period (s) and Amplitude (mma) 0.78 0.64 0.74 0.63 0.61 0.61 0.55 0.53 0.59 0.59 0.64 0.74 0.63 0.64 0.60 0.60 0.69 0.64 0.54 0.77 0.56 0.80 0.66 0.64 0.44 0.56 0.82 0.71 0.74 0.62 0.69 0.54 1.06 1.00 0.71 109.47, 120.36, 192.61, 195.20, 198.30, 201.00, 206.30, 209.18, 213.13, 215.22, 215.74, 181.90, 249.90, 256.13, 257.10, 264.40, 265.50, 270.00, 271.40, 279.50, 292.30, 297.50, 304.30, 307.00, 307.90, 309.30, 311.70, 317.97, 355.80, 365.20, 400.70, 462.20, 540.00, 595.20, 606.20, 610.00, 2.8; 109.28, 1.1; 109.09 2.5 4.8; 249.90, 0.6; 192.79, 0.8; 192.68, 2.3; 192.57, 1.9; 120.40, 1.8; 120.32, 1.4; 114.28, 0.7; 107.60, 0.3 5.8; 350.15, 1.0; 348.73, 0.4; 193.09, 1.0; 192.13, 0.7; 143.42, 0.6; 143.04, 0.3; 118.68, 1.1; 118.52, 1.8; 113.78, 2.1; 113.27, 0.6 5.5; 254.40, 2.4; 195.50, 3.9 6.0; 322.90, 1.9; 291.60, 2.2; 173.30, 1.1; 127.80, 1.50 7.3; 123.40, 2.50 8.10 10.0; 327.32, 4.0; 306.90, 5.0; 304.50, 4.0 6.7; 333.65, 1.0; 318.08, 0.9; 274.78, 2.9; 274.25, 4.1; 212.77, 4.1; 187.27, 1.0 19.8; 304.15, 8.8; 270.86, 7.1 1.9; 651.70, 0.7; 537.59, 0.6; 454.56, 0.4; 370.21, 1.6; 301.41, 1.1; 299.79, 0.9; 264.19, 0.5; 212.82, 0.5; 0.03; 148.45, 0.6; 141.24, 0.4; 72.92, 0.4; 72.54, 0.9; 70.93, 0.7 7.2; 48.50, 5.9 11.4; 256.33, 10.9 5.3; 463.00, 3.0 4.7; 203.70, 2.5 7.2; 265.80, 3.3; 208.60, 2.8 8.4; 255.70, 7.3; 194.70, 5.8 14.7; 810.40, 6.5; 301.40, 9.7; 196.70, 3.7 8.3; 337.90, 5.9; 189.20, 3.2 21.9; 255.30, 15.6; 146.50, 7.1 16.0; 348.10, 8.4; 347.10, 8.2; 263.50, 7.1; 149.20, 3.5 15.4; 391.10, 7.5; 273.00, 11.2; 143.50, 5.0 14.0; 294.60, 5.0; 256.30, 10.0; 203.30, 4.0 10.0; 778.80, 5.0; 519.90, 4.0; 400.10, 5.0; 362.50, 2.0; 257.30, 9.0 17.9; 212.30, 5.2; 154.10, 4.0 22.0; 787.50, 6.6; 511.50, 7.3; 311.30, 9.3; 258.30, 6.2 8.9; 738.55, 7.4; 557.41, 8.6; 291.97, 8.9; 259.07, 3.6 23.7; 743.40, 7.7; 557.60, 15.7; 556.60, 5.8; 555.30, 17.1; 446.10, 13.5; 445.30, 2.8; 444.60, 11.6; 355.40, 4.3; 354.90, 10.0 17.7; 711.60, 6.0; 634.20, 5.1; 516.60, 16.2; 364.00, 7.3; 363.20, 12.5; 213.30, 6.0; 181.10, 3.9 19.2; 1116.50, 12.8; 454.90, 17.9; 292.40, 8.5; 213.00, 13.0 25.0; 823.20, 15.0; 727.90, 18.0 29.0; 597.00, 14.0; 494.00, 26.0; 382.00, 16.0 7.4 5.3; 1248.20, 3.2; 690.20, 3.3; 234.00, 1.3 61.0; 937.00, 11.0; 724.00, 18.0; 530.00, 18.0; 399.00, 14.0 98 Table 5.2: Pulsation periods and amplitudes of the known ZZ Ceti stars (Cont d) Object G 29 38 BPM 31594 WD1502 0001 BPM 37093 WD0825+4119 PG 1541+650 HE 0532 5605 WD0942+5733 HE 1258+0123 WD0332 0049 WD0906 0024 BPM 30551 WD0318+0030 KUV 02464+3239 R 808 WD1122+0358 G 191 16 WD1617+4324 WD1700+3549 WD1417+0058 WD1157+0553 WD0102 0032 PG 2303 G 38 29 WD1056 0006 G 255 2 EC 23487 2424 GD 99 G 30 20 WD1732+5905 BPM 24754 GD 154 WD1443+0134 WD1524-0030 WD2350-0054 HS0951+1312 HS0952+1816 Mass (M ) Period (s) and Amplitude (mma) 0.69 0.68 0.61 1.11 0.91 0.67 0.91 0.77 0.63 0.76 0.61 0.75 0.65 0.66 0.63 0.64 0.64 0.63 0.63 0.63 0.70 0.76 0.66 0.55 0.52 0.71 0.67 0.66 0.58 0.60 0.63 0.70 614.00, 40.3; 915.44, 5.9; 894.04, 14.0; 859.64, 24.6; 809.45, 30.1; 770.75, 5.1; 678.40, 9.7; 648.70, 7.8; 551.90, 4.4; 499.30, 8.6; 401.23, 7.4; 400.43, 4.7; 399.66, 6.8; 354.94, 2.9; 283.87, 4.1 617.30, 24.2; 401.90, 9.4; 532.60, 5.0; 416.10, 5.0 629.50, 32.6; 687.50, 12.0; 581.90, 11.1; 418.20, 14.9; 313.60, 13.1 637.00, 1.6; 633.00, 1.3; 614.00, 1.1; 601.00, 0.7; 583.00, 0.9; 582.00, 1.1; 565.00, 1.2; 548.00, 1.1; 531.00, 1.2; 512.00, 0.7; 549.00, 0.8 653.40, 17.1; 611.00, 11.2 686.00, 45.0; 757.00, 14.0; 564.00, 15.0; 467.00, 0.7 688.80, 8.0; 586.40, 7.8 694.70, 37.7; 550.50, 12.2; 451.00, 18.4 744.60, 23.0; 1092.10, 14.0; 528.50, 10.0; 439.20, 10.0 767.50, 15.1 769.40, 26.1; 618.80, 9.1; 574.50, 23.7; 457.90, 9.5; 266.60, 7.6 823.00, 29.0; 744.70, 9.0; 682.70, 8.0; 607.00, 12.0 826.40, 21.1; 587.10, 10.1; 536.10, 10.6 831.60, 40.0 833.00, 50.0 859.10, 34.3; 996.10, 17.9 860.00, 70.0; 596.00, 10.0; 508.00, 10.0; 388.00, 10.0; 365.00, 5.0 889.60, 36.6; 626.30, 24.1 893.40, 54.7; 955.30, 20.4; 450.50, 19.3 894.50, 43.1; 812.50, 31.3; 522.00, 15.2 918.90, 15.9; 1056.20, 5.8; 826.20, 8.1; 748.50, 5.6 926.10, 37.2; 830.30, 29.2 936.30, 43.7; 1228.50, 7.7; 577.93, 11.6; 539.84, 5.1; 482.60, 6.2; 394.32, 8.8 938.00 28.00, 1024.00, 26.00;1020.00, 26.0; 910.00, 27.0 942.20, 62.3; 474.40, 22.9 985.22, 4.4; 898.47, 8.7; 855.43, 10.3; 819.67, 10.4; 775.19, 14.0; 773.40, 11.7; 681.20, 24.2; 607.90, 12.1; 568.51, 10.7; 325.42, 4.5 993.00, 37.7; 804.50, 19.3; 868.20, 12.8; 989.30, 11.0 1026.00, 8.1; 1071.70, 3.4; 1059.50, 5.2; 1004.80, 4.1; 1002.30, 2.1; 849.70, 2.2; 665.30, 1.5; 633.10, 2.0; 300.70, 1.4; 230.60, 1.6; 228.60, 4.5; 223.30, 2.7; 221.40, 1.7; 126.10, 2.1 1068.00, 13.8 1122.40, 8.0; 1248.40, 4.5 1176.50, 22.6; 1123.60, 6.5; 1098.90, 6.1; 1087.00, 13.2; 1052.60, 9.1; 980.40, 7.7 1186.50, 16.7; 1190.50, 6.3; 1183.50, 4.6; 1092.00, 3.0; 1088.60, 5.0; 1084.00, 5.6; 402.60, 2.7 968.90, 7.5; 1085.00, 5.2 873.20, 111.5; 434.00, 47.8 304.30, 15.4; 391.10, 7.5; 273.00, 11.2; 143.50, 5.0 208.00, 9.3; 281.60, 8.8; 258.60, 3.6 1159.70, 4.8; 1466.00, 4.5; 853.80, 3.9 99 Sum 300 600 900 1200 Figure 5.5: Pulsation spectra of the intermediate temperature DAVs 100 Sum 300 600 900 1200 Figure 5.6: Pulsation spectra of the cDAVs 101 Figure 5.7: Grand Sum of the pulsation spectra of all DAVs 102 Table 5.3: Periods observed close to 205 s as a function of log g Object SDSS Te (K) SDSS log g Pobs /P0 GD165 L19-2 WD1354+0108 WD0847+4510 WD1345-0055 G238-53 LP133-144 R548 G117-B15A GD244 WD1015+0306 WD1125+0345 WD0958+0130 GD66 WD0842+3707 G185-32 WD1724+5835 GD99 WD0949-0000 WD1015+5954 11980 200 12100 200 11750 050 11740 120 11840 060 11890 200 11800 200 11990 200 11630 200 11680 200 11620 030 11650 130 11730 060 11980 200 11860 210 12130 200 11650 080 11820 200 11190 130 11680 110 8.06 0.05 8.21 0.05 8.05 0.02 7.97 0.07 8.09 0.03 7.91 0.05 7.87 0.05 7.97 0.05 7.97 0.05 8.08 0.05 8.17 0.01 8.05 0.07 8.03 0.03 8.05 0.05 7.60 0.10 8.05 0.05 7.94 0.05 8.08 0.05 8.26 0.11 8.03 0.06 0.94 0.94 0.97 0.98 0.95 1.01 1.02 1.04 1.05 0.99 0.95 1.02 0.99 0.96 1.04 1.05 0.92 1.09 1.04 1.04 plot the ratio of the observed periods, divided by the group mean, as a function of log g in Figure 5.8. We indicate these values in Table 5.3. M. Montgomery (Montgomery; private communication) computed DA white dwarf models close to 0.6 M with a He layer mass of 2 10 2 M and a H layer mass of 2 10 4 M . Keeping the layer masses constant, he changed the mass of the star by 0.05 M in both directions and determined the corresponding frac- tional change in period. We show these points (squares) from the numerical models in Figure 5.8, and nd that the models agree with the observations for a signi cant fraction of the DAVs. Assuming that all the periods close to 260 s correspond to the same mode, we arrived at a group mean of 262.1 s. We show the observed periods 103 Figure 5.8: ZZ Ceti stars with periods around 205 s exhibit a group mean at 204.8 s. Dividing the pulsation period of each star by the group mean gives us a relative shift in individual periods, which we plot as a function of log g. The squares connected by the dotted line indicate values of the shift computed using DA white dwarf models around 0.6 M with a He layer mass of 2 10 2 M and a H layer mass of 2 10 4 M . 104 Table 5.4: Periods observed close to 262 s as a function of log g Object SDSS Te (K) SDSS log g Pobs /P0 GD165 WD1345-0055 R548 G117-B15A GD244 WD1015+0306 WD1125+0345 WD0958+0130 WD0939+5609 GD66 WD0111+0018 KUV11370+4222 GD385 WD0214-0823 WD1724+5835 WD2350-0054 G185-32 KUV08368+4026 WD0815+4437 G207-9 WD0906-0024 11980 200 11840 060 11990 200 11630 200 11680 200 11620 030 11650 130 11730 060 11770 160 11980 200 11430 100 11890 200 11710 200 11610 090 11650 080 10390 060 12130 200 11490 200 11630 170 11950 200 11570 090 8.06 0.05 8.09 0.03 7.97 0.05 7.97 0.05 8.08 0.05 8.17 0.01 8.05 0.07 8.03 0.03 8.27 0.07 8.05 0.05 8.27 0.06 8.06 0.05 8.04 0.05 7.99 0.05 7.94 0.05 8.30 0.06 8.05 0.05 8.05 0.05 7.93 0.09 8.35 0.05 8.00 0.06 0.95 0.97 1.05 1.03 0.98 0.98 1.01 1.01 0.95 1.04 0.97 0.98 0.98 1.01 1.07 1.04 1.01 0.98 0.99 0.99 1.02 divided by the group mean as a function of log g in Figure 5.9, and list the corresponding values in Table 5.4. Both Figures 5.8 and 5.9 show a reasonable agreement with models in the vicinity of log g = 8. But the periods observed for ZZ Ceti stars with relatively higher masses (log g 8.25) and lower masses (log g 7.8) do not agree with the models. There are many possible explanations, and we explore some of them below: Periods from heterogeneous modes: We assumed that all the periods close to 205 s and 260 s, respectively, correspond to the same modes; this need not be true. Preliminary investigations show that the periodicities close 105 Figure 5.9: ZZ Ceti stars with periods around 260 s yield a group mean of 262.1 s. Dividing the pulsation period of each star by the group mean gives us a relative shift in individual periods, which we plot as a function of log g. The points connected by the dotted line indicate values of the shift computed using DA white dwarf models around 0.6 M with a He layer mass of 2 10 2 M and a H layer mass of 2 10 4 M . 106 to 205 s and 260 s in the stars LP 133-44, G 238-53, WD0847+4510, and WD0815+4437 may well be = 2 modes. Erroneous log g determinations: The low signal-to-noise of the blue end of the SDSS spectra reduces the reliability of the log g values (see section 4.1.3) for the SDSS DAVs. However, massive DAVs with Te and log g determinations from Bergeron et al. (2004) also deviate signi cantly from the models. Hence, it is unlikely that erroneous log g determinations are causing the relative high and low mass ZZ Ceti stars to deviate from the models. E ects of Mode Trapping: Compositional strati cation occurs in white dwarf stars due to gravitational settling and prior nuclear shell burning; this alters the kinetic energy of oscillation for di erent modes in our models. A mechanical resonance is induced between the local g-mode oscillation wavelength and the thickness of one of the compositional layers (Winget, Van Horn & Hansen 1981). This mechanical resonance serves as a stabilizing mechanism in model calculations (Brassard et al. 1992; Montgomery 1998). Trapped modes are energetically favored, as the amplitudes of their eigenfunctions below the H/He interface are smaller than untrapped modes. Modes trapped in the envelope can have kinetic oscillation energies lower by a few orders of magnitude, as compared to the adjacent non-trapped modes (Winget et al. 1981; Brassard et al. 1992). Benvenuto et al. (2002) claim a marked weakening of mode trapping effects with a time-dependent element di usion in the DA white dwarf models with di erent thicknesses of the hydrogen envelope. Mode trapping can alter the period spacing for low k modes. The periods we observe close to 205 s are probably to 260 s are likely = 1, k = 2, and the periods close = 1, k = 3. It is therefore possible that the observed 107 periods in Figures 5.8 and 5.9 constitute trapped and untrapped modes. It could also be that the modes trapped in the star are untrapped in the models, and vice-versa. Di erent layer masses: Di erent H and He layer masses alter the trapping properties in a ZZ Ceti star, and change the observed periods. The models we show in Figures 5.8 and 5.9 have the same H and He layer mass. Perhaps some of the scatter in the plots arises from di erent H and He layer masses in the ZZ Ceti stars. Binary evolution: We observe a substantial deviation from the models for relatively high and low mass ZZ Ceti stars. Perhaps these stars originated in interacting binaries, and have substantially di erent H and/or He layer masses as a result of a di erent evolution. Missing physics in models: It is also conceivable that our models are incorrect in predicting pulsation periods for the relatively high and low mass ZZ Ceti stars. More than one of the above possibilities may be true; we hope to continue our explorations and analyses over the next few months. This chapter is a work in progress. 5.6 Conclusions We nd that the new DAVs conform to the well established trend of increasing pulsation periods and amplitudes, as we compare the cool ZZ Ceti stars to the hot DAVs. Our homogeneous sample with consistent data reductions suggests a decline in pulsation amplitudes close to the red edge, before pulsations shut 108 down entirely. This may be the rst observational evidence that the red edge is not an abrupt feature in the ZZ Ceti evolution. Clemens (1993) showed that all hDAVs are essentially identical, except for di erent stellar masses. He showed that their pulsation spectra could be aligned by translation, and a plot of the amount of shift vs log g was linear. In attempting to extend this idea to the new DAVs, we nd that although it works well around log g 8, it does not work for hDAVs with relatively higher masses (log g 8.25) and lower masses (log g 7.8). Perhaps high and low mass pulsators have di erent hydrogen and/or helium layer masses, since they went through a di erent evolutionary path. It is also possible that our assumption that all periods close to 205 s and 260 s belong to the same modes respectively, may not be true. We conclude that we can only use hDAVs with log g 8 for ensemble asteroseismology, until we can explain the observed di erences between the relatively massive and low mass variables. 109 Chapter 6 Summary and future work 6.1 Seismology of the ZZ Ceti stars Global pulsations in stars provide the only current systematic way to study their interiors. White dwarf stars are the stellar remains of 98 99% of stars in the sky, and contain an archival record of their main sequence lifetime. Pulsating white dwarf stars serve as e ective instruments to harness this archival record. Their high densities and temperatures make them good cosmic laboratories to study fundamental physics under extreme conditions. Known white dwarfs at T e 4500 K are among the oldest stars in the solar neighborhood, and they serve as reliable chronometers to date the Galactic disk and halo. The carbon-oxygen ratio in white dwarf cores contains the rate of the astrophysically important, but experimentally uncertain, 12 C( , )16 O nuclear reaction. Since 80% of all white dwarf stars are DAs, to understand the DAVs is to understand the most common type of white dwarf. Understanding the structure and evolution of a statistically signi cant sample of DAVs has implications for other areas of astronomy as well. We can use ensemble asteroseismology to probe the stellar structure; the ensemble of modes can be gathered from a single 110 DAV with a rich pulsation spectrum or from multiple DAVs with relatively fewer modes, if their structures are similar. 6.1.1 Bene ts of an ensemble of DAVs The hDAVs that show modes with long term stability (P 10 15 s/s) can be used as accurate clocks. These clocks serve as potential detectors of planetary systems, since any re ex motion will cause a periodic variation in their drift rate. In the absence of orbiting companions, the slow unidirectional drift rate of their period can help us constrain the white dwarf evolutionary sequences. These cooling curves are ultimately useful in cosmochronometry, constraining the age of the Galactic disk and halo using the age distribution of white dwarfs vs e ective temperature. Massive DAVs allow us to study crystallization in a stellar plasma, relevant for white dwarf cosmochronometry as well as neutron star crusts. The seismological distances for the DAVs are more accurate than distances from measured values of parallax. Pulsating DAVs are helpful in establishing a Galactic distance scale in more ways than one. 6.2 Ideal instrumentation for the DAV search Searching for a statistically signi cant number of DAVs seemed useful for all of the reasons listed above. However, since most of the nearby ZZ Ceti stars had been found, the only way to discover a large sample of new DAVs was to observe at fainter magnitudes. That required a new instrument; our previous photometer based on PMTs enabled us to observe targets brighter than B=17 on the 2.1 m telescope at McDonald Observatory. Thus, Argos was born and it saw rst light on 1 November 2001 at the prime focus of the 2.1 m telescope, six months after we purchased the CCD camera from Roper Scienti c. R. E. Nather single 111 handedly wrote the data acquisition program, operable in a Linux environment. Gordon Wesley and David Boyd designed the prime focus mount for the camera, and the parts were built in a commercial machine shop. Gary Hansen designed the timing card that serves as a crucial component for instrument timing. We obtained usable data with Argos on the very rst night, when we observed KUV 08368+4026 at the 2.1 m telescope at McDonald Observatory. We list the signi cant improvements to the instrument over the following two years: The commissioning run made us realize that the shiny aluminium surfaces of the mount were acting as a mirror for stray light. We bead-blasted the metallic parts of the mount, and had them hard anodized (Type II). The matt dark surface, resistant to corrosion, improved data quality substantially. We initially used GPS pulses from the adjacent 2.7 m telescope as input 1 Hz pulses for instrument timing. The S/N ratio was reduced due to the long length of the cable, so that the timing card sometimes counted noise pulses as genuine pulses. We then solved a large fraction of our timing problems by using 1 Hz pulses from our own GPS system at the 2.1 m telescope. The initial design for Argos consisted of a single ba e, 10% larger than the F/3.9 light beam. This allowed us to observe each car passing by the Observatory, and practically every glow worm in the dome. We substituted the dysfunctional ba e with an e ective 5-stage ba e system, designed with help and guidance from Phillip McQueen. The ba es were machined by George Barczak in the department. Dr. R. E. Nather made the next signi cant improvement to Argos; we stopped using the PC clock tuned by NTP for timing, and started using 112 information from the GPS system directly to tune the PC clock. The dome of the 2.1 m telescope does not track; an observer needs to guide the dome manually as needed. We purchased a new uncooled CCD camera to ride on the telescope, and take pictures of the dome. These images, transmitted to the control room, serve to inform the observer when it is time to move the dome. Dr. R. E. Nather wrote the data acquisition program for Cyclops, and Denis Sullivan helped us test the camera extensively, giving us valuable suggestions at every stage. Jimmy Welborn machined the mount for Cyclops. 6.3 Fruitful search for the ZZ Ceti stars We discovered 35 new ZZ Ceti stars with Argos, mainly observing DAV candidates from the Sloan Digital Sky Survey. After trying to select DAV candidates using u g colors, and then equivalent widths, we settled with the higher success rate of the spectroscopic technique. Our SDSS collaborators, namely Scot and Atsuko Nitta Kleinman, sent us Te and log g determinations for SDSS DA white dwarfs. We selected candidates for observation based on their Te values. Daniel Eisenstein used Detlev Koester s atmosphere models to derive these parameters, which were immensely useful to us. We could achieve a 90% success rate by con ning our candidates between 12 000 K and 11 000 K at a detection threshold of 1 3 mma. But our interest in hDAVs and the blue edge of the instability strip led us to choose candidates between 12 500 K and 11 000 K, reducing our success rate to 80%. With the discovery of 35 new DAVs, we almost doubled the sample of 39 published ZZ Ceti stars. 113 6.4 Empirical ZZ Ceti instability strip We used the 30 new variables with reliable Te and log g determinations discovered within the SDSS, along with G 238-53, to re-de ne the empirical ZZ Ceti instability strip. This was the rst time since the discovery of white dwarf variables in 1968 that we had a homogeneous set of spectra acquired using the same instrument on the same telescope, and with consistent data reductions, for a statistically signi cant sample of ZZ Ceti stars. The homogeneity of the spectra reduced the scatter in the spectroscopic temperatures and we found a narrow instability strip of width 950 K, from 10850 11800 K. We questioned the purity of the DAV instability strip due to the presence of several non-variables within. The slope of our best t for the red edge of the instability strip is well constrained, but the slope of the blue edge is not. The DAV distribution shows a scarcity of DAVs in the range 11350 11500 K. After exploring various possible causes for such a bimodal, non-uniform distribution, we are still not entirely con dent that it is real. The data at hand are suggestive that the non-uniformity of the DAV distribution is real, and stayed hidden from us for decades due to the inhomogeneity of the spectra of the previously known DAVs. However, we are in the domain of small number statistics and unless we investigate additional targets in the middle of the strip, we cannot be con dent that the bimodal distribution is not an artifact in our data. 6.5 Ensemble characteristics of the ZZ Ceti stars The new ZZ Ceti stars exhibit the well established trend of longer pulsation periods and larger amplitudes, as we traverse across the instability strip to the red edge. We may have found the rst observational evidence that the red edge 114 is not an abrupt feature in the ZZ Ceti evolution, and that pulsation amplitudes decline close to the red edge, before the variable stops pulsating. Clemens (1993) showed that all hDAVs are essentially identical, except for di erent stellar masses. We tried to extend his work to the new DAVs; we nd that ZZ Ceti stars with log g 8 do seem fairly identical to each other, and we should be able to apply the technique of ensemble asteroseismology to their pulsation spectra. But we nd that the relatively high (log g 8.25) and low (log g 7.8) mass ZZ Ceti variables cannot be scaled to the average mass hDAVs in a simple manner. Perhaps we are missing a relevant bit of physics in our models, or these stars have di erent H/He layer masses, composition, etc. re ective of their distinct evolution. 6.6 Future work Multiple projects emerge as a result of the work described here, some of which we list below: High quality homogeneous spectra with S/N 60 for all the 74 DAVs known to date, followed by consistent data reductions, should lead to a reliable determination of the shape of the ZZ Ceti instability strip. The location and edges of the instability strip could help us understand the role of convection. This may also shed more light on WD2350-0054, the hDAV whose SDSS Te determination places it 650 K below the cool edge of the instability strip. The impure instability strip de ned by our sample is most disconcerting, since we have to be prudent in applying the conclusions drawn from variable white dwarfs to non-variable white dwarfs. This result should be veri ed independently, with reliable log g determinations. Looking for 115 similarities between non-variables inside the ZZ Ceti strip may provide us with some explanations of why all DA white dwarfs do not evolve through the instability strip in the same way. Perhaps the blue edge depends not only on the temperature and mass of the star, but on additional parameters. Perhaps the non-variables have a strong magnetic eld or some other mechanism that shuts down pulsations. High signal to noise photometric observations of massive pulsators such as WD 1711+6541 (g = 16.9) and WD 0923+0120 (g = 18.3) may lead to additional modes. Each mode we observe brings us closer to obtaining a unique model- t to the star, which gives us an estimate of the crystallized mass fraction. Metcalfe, Montgomery, & Kanaan (2004) have already presented strong evidence that the massive DAV, BPM 37093, is 90% crystallized. Crystallization and phase separation represent the biggest sources of uncertainty in white dwarf cosmochronometry. This study is also relevant to neutron star crusts. The low mass pulsator WD0842+3707 could be a He core white dwarf, most probably conceived in an interacting binary system. Single star evolutionary models cannot produce such low mass white dwarfs in a Hubble time (Iben 1990). Using pulsations to probe the equation of state for a He core white dwarf could prove to be highly enlightening, as we do not really understand how bosons behave at high densities. WD0949-0000, WD1015+5954, GD 99, and possibly a few other DAVs show signs of three mode resonances, where two observed frequencies add to a third frequency. A resonance mechanism is di erent from linear combinations, because the third frequency is located at the site of a real mode, as in a parametric resonance. Such a mechanism might serve to stabilize 116 or de-stabilize the third frequency. Additional observations of WD09490000, WD1015+5954, and GD 99 may help us investigate this possibility further. Monitoring all the modes observed in the newly discovered hDAVs can shed more light on why we measure di erent drift rates for di erent periodicities in the same hDAV. It can also help us de-convolve the effect of cooling from other mechanisms that can alter the drift rate: trapping, avoided crossings, resonances, crystallization, and possibly magnetic elds to name a few. Such an understanding is crucial since we already use drift rate measures of hDAV pulsation periods to constrain our evolutionary models, and to search for orbiting planets. 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The macros used in formatting this dissertation were written by Dinesh Das, Department of Computer Sciences, The University of Texas at Austin, and extended by Bert Kay and James A. Bednar. 1A LT 130

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