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% ^JJV^/W^K Sar^fcsj^ sm il 1 » STELLINGEN 1. In de buitenste delen van ^piraalstelsels zijn de rotafiefrequentie, de epicycle- frequentie, en de oscillatiefrequentie Jie de beweging loodrecht op het sym- metrievlak karakteriseert, nagenoeg gelijk. Een eenmaal ontstane asym- metrische afwijking van de gasverdeling ten opzichte van het symmetnevlak (warping) kan zich daarom in de buitenste delen van een spiraalstelsel gedurende lange tijd handhaven. 2. De suggestie dat door resonante effecten bij de binnenste Lindbiad resonantie balkachtige structuren kunnen ontstaan berust vooralsnog op wishful thinkmg. J. W. K. Mark. 1974. in The formation and dynamics of galaxies. I. A. U. Symp. 5B. Hd. Shakeshaft. J. R. (Reidel. Dordrecht). 3. Het is moeilijk een fysische betekenis toe te kennen aan kinematische modellen gebaseerd op dispersieringen, als die modellen worden toegepast op waar- nemingen in de buurt van de binnenste Lindbiad resonantie. S. C. Simonson and G. L. Mader, 1973. Astron. Astrophysics. 27. 33". R. B. Tully. thesis, University of Maryland. 1972. 4. In publicaties van waarnemingen van de verdeling en kinematica van neutrale waterstof in extragalactische stelsels dient naast het afgeleide snelheidsveld ook een efficiënte presentatie te worden gegeven van de gemeten lijnprofielen. A. H. Rots. dissertatie, Rijksuniversiteit Groningen, 1974 5. De bewering van Fernie dat Huggins in 1865 door een foutieve interpretatie van zijn gegevens tot de conclusie kwam dat nevels gasvormig zijn, is onjuist. J. D. Fernie. 1970. Pub. A. S. P. 82, 1189. 6. De toenemende mogenlijkheden om de weersomstandigheden te beinvl^eden maakt spoedig internationaal overleg gewenst om vast te leggen binnen welke grenzen deze beinvloeding toela2tba-: is en om de rechtspositie van door veranderde klimatologische condities getroffen personen vast te stellen. Het internationale overleg over het gebruik van de oceaan kan hier als voorbeeld dienen. 7. Het rendement van de omzetting van stromingstype-energie (zon, wind, water) in electrische energie blijkt, gemiddeld over een grote landoppervlakte, vrijwel onafhankelijk van de omzettingsmethode te zijn. 8. Een rechtvaardige verdeling van arbeid en vrije tijd is een noodzakelijke voor- waarde voor een rechtvaardige verdeling van inkomens. 9. Het vrijwel ontbreken van litteraire biografieën in het Nederlandse taal- gebied staat in schrille tegenstelling met het tot vervelens toe centraal stellen van autobiografisch materiaal door hedendaagse Nederlandse auteurs. 10. De veronderstelling van Dr. F. Brouwer dat alles wat leeft en groeit altijd weer boeit is aan bedenkingen onderhevig. SPIRAL STRUCTURE AND THE DYNAMICS OF FLAT STELLAR PROEFSCHRIFT ter verkrijging van de graad van Doctor in de Wiskunde en Natuurwetenschapper' aan de Rijksuniversiteit ce Leiden, op gezag van de Rector Magnificus Dr. A.E. Cohen, hoogleraar in de faculteit der Letteren,volgens besluit van het College van Dekanen te verdedigen op woensdag 16 april 1975. te klokke 15.15 uur door ELISABETH DEKKER geboren te Haarlem PROMOTOR: PROF. DR. H.C. VAN DE HULST Shearing and suppsrless the Hero sate3 Biasphem'd his Gods3 the Diaet and damn'd h£a Fate. Then gnaw'd his pen, then dash'd it on the ground^ Sinking from thought tc> thoughtj a vast profound! Ptmg'd for his sense^ but found no bottom theret Yet wrote and flounder'd on3 in •teye despair. Pope (The PfflCIADJ CONTENTS CHAPTER 1 General characteristics of spiral galaxies 5 1.1 A brief review of two centuries of observation 5 1.2 Theories of spiral strucLurs 13 1.3 Outline of tne present study 19 CHAPTER 2 Mathematical tools 23 CHAPTER 3 Dynamical properties of flat s'. ellar systems 30 3.1 Introduction 30 3.2 Stellar orbits 30 3.3 Distribution function 50 CHAPTER 4 Stability of slightly perturbed disks 61 4.1 Introduction 61 4.2 Mathematical formulati.on 62 4.3 Instabilities 75 4.3.1 The rate of ch.inge of angular momentum 75 4.3.2 Growing waves 79 4.3.3 Damped waves 84 4.3.4 Physical significance of the growth rate y 86 4.3.5 Astronomical implications 89 CHAFTER 5 Stabilization of density waves by the gas 95 5.1 Introduction 95 5.2 Stabilization mechanism 99 5.3 Discussion '04 -3- CHAPTER 6 Quasi-linear theory 107 1.I Introduction 107 6.2 Derivation of diffusion equation 107 6.3 Diffusion coefficients 115 6.4 The persistence of spiral structure 125 Appendix 130 CHAPTER 7 Conclusions and summary 134 References 138 Acknowledgement j42 Samenvatting i43 Studieoverzicht 148 -4- One of the chief beauties of the spiral as an imaginative conception is that it is always growing, yet never covering the same ground, so that it is not merely an explanation of the past} but is also a prophecy of the future; and while it defines and. illuminates what has already happened, iv is also leading constant- ly to new discoveries. Sir T.A. Cook in "The Curves of Life", London. 1914, CHAPTER 1 GENERAL CHARACTERISTICS OF SPIRAL GALAXIES 1.1 A brief review of two centuries of observations. In the seventeenth and eighteenth century the steady improvement of the telescope led to the discovery of feebly shining cloudlike spots and patches which were called nebulae. The interest of astronomers being mainly concentrated on the motions of the planets and their satellites., it took until 1785 before the first catalogue of nebulous objects was published (Messier, 1784), actually as a by-product of a systematic search for comets. The first systematic search for nebulae as such was carried out by' William Herschel. His telescopes were considered the. best available at the time. Already during his first attempt to investigate the objects of Messier's list, Herschel found that many of them could be resolved into stars.: I'C seemed logical to conclude that all nebulae were large aggregations of stars. The unresolved objects vere thought to be too distant for individual stars to be distinguishable. : It is clear fror. Hersctiel's papers presented to the .Royal Society thai: with passing time he considered.the general validity of this hypothesis as more doubtful (see e.g., Hbskin, 5963). important.in this, respect was his discovery of a starlike object in the otherwise diffuse nebulosity, NGC 1514. The question of the starry nature of the nebulae was taken up again some seventy years later. The Irish astronomer Lord Rosse then made a systematic survey of Herschel's nebulosities with a much better, 6-foot, telescope. Many objects listed by Herschel as planetary of diffuse, were found to be concentrations of faint stars, "No real nebulosity seems to exist among so many of these objects chosen without bias, all appeared to be clusters of sta s and every additional one which shall be resolved will be an additional argument against the-.existence of any such". (Robinson, 1845), In one of the nebulae, M 51, the well-known whirl-pool nebula in Canes Venatici, curved filaments were discovered which gave to the whole the appearence of a spiral. Within a few years a fairly large number of spiral nebulae had been discovered. In the decades following these discoveries, however, a number of develop- ments seemed to support the hypothesis that nebulae were of a gaseous nature. In particular, early spectroscopic observations revealed that some nebulae, such as the Orion nebula, had an emission line spectrum and others, such as M 31, showed only a continuous spectrum, in contrast to the absorption line spectra as observed in bright stars (Huggins, 1865). Especially these nebulae that exhibited spiral structure were thought to be planetary systems in the making, according to an old theoretical concepi proposed by Laplace. In an address about the nebular theory to the Royal Institution in England in the beginning of this centrury by Ball (1902) we cas read: "Indeed it is not too much to say that next to a fixed star itself, the spiral nebula is the most characteristic object in the heavens. The signifi- ; cance of this statement in connection with the Nebular Theory can hardly be overestimated. There can be little doubt that at one stage of the history of - 6 - the solar system che gradually evolving nebula must have formed an object of that type wnich we term spiral". The planetary hypothesis for spiral nebulae could not be maintained for long. First of all, Wolf (see e.g., King, 1955), using better spectrographic methods, obtained a solar type spectrum of the Andromeda nebula. Secondly, it seemed possible to estimate statistically the sizes and distances of the nebulae through velocity determinations and proper motion measurements (Slipher, 1915; Curtis, 1915). But, once icore, the search for the true nature of spiral nebulae led the astronomers astray (Fernie, 1970). The work on proper motions by Van Maanen (1916) seemed to yield evidence for large internal motions in spiral nebulae that was incompatible with the hypothesis that spiral nebulae were remote aggregations of stars, similar to bur own Milky Way. It took until 1925 before the nature of spiral nebulae could be firmly established. In that year, Hubble discovered Cepheid variables in the nebulae M 31 and M 33, which enabled him to determine their distances rather reliably, thereby confirming the extragalactic nature of these objects. A year later Rubble (1926) published his well-known-classification scheme of nebulae. Even now with our. increased knowledge of the physical properties of galaxies, this, classification in a somewhat :reyised .fo'nn.can. still be • '• regarded as standard (Sahdage, 1961). ' ' ... ~ :•.:;.. ^-' - V. ' • Although the extragalactic nature of spiral, nebulae was recognized, very little was yet known, about; their physical properties..- , .. ' -..;•• One of the great questions concerning the'eontent of the nebulae was, as Lindblad stated :in 1927: "if the inner, amorphous nebulosity is really of a gaseous nature, .or is'only apparently irresolvable while consisting in reality of stars of.