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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. somewhat smaller intrinsic brightness than the whitish stars in
-7 - the outer spiral arms". It was not before 1943 that this question was answered. Baade then succeeded in resolving the central part of M 31. After that, the resolution of other nearby galaxies followed rapidly (Baade, 1944).
At the same time Baade discovered in M 31 some largr emission regions.
Emission regions had been interpreted by Stromgren (1938) as local regions of gas, ionized by the radiation of hot stars. Their presence could generally be correlated with the presence of young bright stars. Baade decided to make a new series of observations of M 31. From this survey he was able to draw the
following conclusions (Baade ^.nd Mayall, 1949):
1. Most of the emission regions (which turned out to be very numerous) are
found in the spiral arms.
2. Many of these objects are strongly reddened.
3. Dust is strongly concentrated in the spiral arms. These conclusions led
Baade to the general picture that spiral arms are mainly made up of gas,
dust and young stars, all concentrated into a thin disk. (Baade, 1963).
From the observations of the stars in the Galaxy, it was found, moreover, that
such young stars represented only a small fraction of the total number* of
stars. Moreover, the observations of the distribution of different stellar
populations in the Galaxy revealed that the distribution of the older stars ;
also shows a pronounced concentration towards the disk, although in a lesser i
degree than the young stars (Oort, 1965). This observation, together with the 'i
optical appearence of edge-on galaxies (fig. 1.1) indicated that in spiral f
galaxies a large fraction of the mass is strongly concentrated towards the
equatorial plane.
With the discovery of the differential rotation of the Galaxy (Oort,
1927a) and the related theoretical developments in the dynamics of stellar
systems (Lindblad, 1927a; Oort, 1927b) it became possible to estimate the mass*
- 8 - ^ Fig. l.l The spiral galaxy KGC 4565 seen edge OJI. of the Galaxy rather reliably. In a first attempt Oort (1927b) estimated this mass of the Galaxy as at least 8 x 10 M . This estimate was improved, mainly through a better determination of the solar distance to the centre, by Flasket and Pearce (1936). They obtained a solar distance of 10 kpc and a mass of
1.6 x 10" M . e
Although it had been evident for a long time that most of the spiral
galaxies showed rapid rotation (Slipher, 1915; Pease, 1916) the first realty
detailed observations of the internal motions in extragalactic systems were
made by Babcock (1939). He derived a rotation curve of M 31 to a distance of
80' from the nucleus, by measuring the Doppler shift of absorption lines in
the integrated spectra of the stars.
An even more extensive study was undertaken by Mayall and Aller (1942),
who measured emission lines from gaseous nebulae in M 31 and M 33. The
observations of M 33 were of special interest because in its outer parts
differential rotation was observed, the first extragalactic analogue to the
differential rotc!"ic~ of oar own Galaxy. From these new measurements the 11 ° total masses were found to be = 10 x 10 M and 1,8 x 10 M for M 31 and e e
M 33, respectively. For both galaxies the adopted distance was 220 kpc (Wyse
and Mayall, 1942 and Allar, 1942).
More detailed information on the masses and rotational velocities in
spiral galaxies has been obtained in recent years with the aid of radio-
astronomical techniques. The possibility of observing the 21-cm line of
neutral hydrogen was predicted by Van de Hulst in 1945. With this technique
it became possible for the first time to trace the galactic spiral structure
as well as to study the dynamical properties in the disk of the Galaxy away
from the solar neighbourhood. In !957 the first 2l-cm line observations of
M 31 were published (Van de Hulst etal., 1957). Since then information about
- 10 - neutral hydrogen, passes and rotation curves of extragalactic systems has accumulated. These data were summarized by Roberts (1969), who determined statistically some integral properties of spiral galaxies. He found that the ratio of hydrogen mass to total mass varies systematically with galaxy type, from 0.01 for the early type galaxies, Sa, to 0.10 for the late type spirals,
Sc.
Recently, Piddington (1973) has questioned Baade's (1963) generally accepted conclusion that the gas is concentrated in the arms. In order to illustrate his arguments Piddington quoted the results of., Rogstad and Shostak
(1971) who measured the H I-distribution in M 101. He justly concluded that the observed distribution, does not reveal a spiral structure. However, the resolution in these observations is too poor to show any structure on a scale of the order of the width of a spiral arm. High-resolution observations of
this galaxy, made with the Westerbork telescope in 1973, show on the contrary
that the neutral hydrogen is indeed strongly correlated with the spiral
structure as outlined by the H II regions and the dust. {Allen et al«,
1973). The same can be concluded for M 51 and M 81 and a number of other
galaxies, all observed with the Westerbork telescope (Rots, 1974; Shane and
Bajaja, 1975) (see fig. 1.2). This proves unambiguously that Baade's original
hypothesis is correct.
The developments sketched above have given us .•: fairly good idea of sow •
important properties of spiral galaxies. The key facts that.are important for
the present study can be summarized as follows:
1. A large fraction of the (stellar) mass in spiral galaxies is concentrated
into a thin disk. • •'••••
2. The neutral hydrogen gas represents only a small fraction of the total mass
of the galaxy. .
3. The spiral structure is found to exist mainJ.;' in. the dust, neutral hydrogen
- 11 - Fig. 1.2 Neutral hydrogen distribution in M SI measured with the Synthesis Radio Telescope at Westerbork. Half power beam width 24" in right- as sens-on and 32" in destination. 20 —2 Contour interval 3.75 x 10 atom am (Shane and BaQaga). .::. and H II regions.
4. Only a small fraction of the total mass of the galaxy is contained in the
spiral arms.
1.2 Theories of spiral structure.
The increasing knowledge of the physical properties of flat stellar systems did not immediately contribute to a better understanding of the spiral structure, the most conspicuous feature of these galaxies.
Theoreticians were led astray by Baade's conclusion that the spiral arms consist mainly of gas, dust and young stars. In the decade following Baade's discovery most emphasis was put on hydromagnetic theories of spiral structure.
Most of these theories suggested a mechanism by which the spiral arms were held together by magnetic fields. We shall not give a detailed account of these theories since "the precise mechanisms remain in doubt partly because the analysis of somewhat realistic models is so extremely difficult" (Woltjer,
1965). For details of these theories we refer the reader to reviews given by
Wentzel (1963) and Woltjer (1965).
The discovery of the differential rotation and the observation that the . mass concentrated in spiral arms represents only a email fraction of the total mass, apparently made it more difficult to understand how.spiral structure could be maintained during the life of a spiral galaxy. This problem was . clearly outlined by Oort (1962): "...... ,, if the'gas moved in circular paths,
the spiral arm would radically change its structure in a time of the order of
one revolution of the system. This, would give a.very implausible picture. For
the galactic disk has existed for at least 50 such revolutions: Ve can hardly
imagine that the present spiral structure originated in the last 1 to 2 .
percent of the life of the disk and that it is to be dissolved in the next .
-' • -•'.•
- 13 -::• few percents of this life. It may be remarked in this conneccion that the total mass of the gas in the galactic plane is only about 3 percent of its entire mass so that a priori it dees not seem likely that the gravitational attraction of the arms themselves could have helped to maintain them in a spiral pattern for such a long time as appears to be required for the permanence of this structure".
In the same paper Oort also remarks: "But we do not know with any certainty whether or not stars contribute in an important measure to the mass of the arms". *
Two years later, a new theory of spiral structure was put forward by
Lin and Shu (1964). They showed that the selfgravity of the spiral arms,
albeit very small, plays an important role in the dynamics of spiral structure.
Later developments showed that the main contribution to the mass of the arms
is by the stars and not by the gas. We shall return to this particular theory
in more detail in the next section.
Lin and Shu were not the first to suggest a gravitational theory of
spiral structure. Most interesting studies along this line were carried out by
the late B. Lindblad . His pioneering work in this field is the more
impressive when judged by the intricate difficulties which only appeared
clearly in the later developments. i
Lindblad persisted in proposing gravitational theories of spiral j
structure for more than three decades, although his mechanisms show a certain
evolution. In our opinion his work can best be summarized by reviewing briefly
the different mechanisms he explored in order to explain the spiral structure.
His first idea was based on a kind of surface instability of a rapidly
rotating homogeneous spheroid, similar to the one that occurs for MacLaurin
ellipsoids. We can read in his paper of 1927 : "When the effective
eccentricity is above the limit of stability for sectorial harmonic waves,
- 14 - matter will be ejected from the system at maximum speed of rotation" and !lIt may be remarked that the arms evidently wind out frora the "mother-system" in
the same direction as the rotation of the system". The most important prediction
in this theory, namely that spiral arms are leading with respect to the sense of rotation, was the main reason that Lindblad could not convince his contemporaries, -although he took great pains to show that the observed spiral structure was indeed leading (see e.g., Lindblad, 1948). In order to determine whether the spiral structure is leading or trailing with respect to the sense of rotation, one has to determine this sense and to identify the near and the far side of a spiral galaxy. Lindblad assumed in his analysis that the dust possessed an intermediate degree of flattening and that the spiral structure belonged to the most flattened component of galaxy. This led him to the
conclusion that the observed spiral structure was leading. Most of his
colleagues disagreed with his assumption on the degree of flattening of the
dust and were inclined to believe that the dust had the same degree of
flattening as the constituents that made up a spiral arm. Their view turned
out to be correct, and Hubble (1943) shoved, at least for a. few,,clear,;. „ --
examples, that this implies that the spiral structure is Trailing.
Another interesting mechanism studiad by Lindblad' (J948)was that of the
socalled density wave. Lindblad and collaborators studied the self-consistent
betiaviour of perturbations in the-density distribution of what was called a
"quasi-spheroidal" statt. (See also Lindblad and Lahgebartel, 1953). In this
analysis, after many complicated manipulations in order to account, for self-
gravity in a self-consistent manner, a .wave equation for the. surface density
was obtained. The same idea was taken up twenty years later by: Lin and Shu,
and applied to the disk of a spiral galaxy. Lindblad1 s analysis .resulted, in a.
bar-like structure for the density waves, mainly resulting fron the particular
configuration that was studied. He also noted that the presence of. a bar could
•"•'.. :-" - 15 - . • • ' •-'.••-. • ;: .. • •• ... :.,; ,•:•••..:,. ., ; . "•• work as a generation mechanism of spiral structure (compare e.g., Lin and
Feldman, 1970). However, no conclusive statement could again be made on the leading or trailing character of the spiral structure thus generated.
In order to explain the spiral structure in the disks of normal galaxies still a third mechanism was studied. Already in 1927 Lindblad was aware
that the natural oscillations of the disk would influence the development of
the spir?l structure. This finally led him to a totally different concept of
this phenomenon. The underlying idea is the existence of socalled dispersion
orbits (Lindblad, 1957). The idea of a dispersion orbit is best explained by
considering the simple case of a two-dimensional harmonic oscillator, with
coordinates x and y and corresponding frequencies u and to . Although the x y
motion in both x and y directions is periodic, the complete motion need not
itself be periodic. If the ratio between the frequencies u> and u is not a
rational number the orbit is not a closed curve in the x - y plane but an
open =:Lissajous figure" (Goldstein, 1969). Closed orbits are obtained only for; md) + loi = 0 x y with m and 1 integers.
If a collection of oscillators with different ratios u / u is x y
considered,an initial cloud of such particles will be generally smoothedout in-
the course of time by phase-mixing, except when the range of frequencies is i
such that mw + lu % 0. Then the particles will during a long time remain i x y '>• concentrated in the neighbourhood of a "dispersion orbit" defined by s mu + lw = 0 . x y As Lindblad stated it: "A dispersion oTbit has the property that in case
of a condensation of matter like a stellar asscociation or a cloud of gas,
which dissolves by differential motion, the individual particles will stay in
the neighbourhood of the disperion orbit".
In order to explain the observed spiral structure Lindblad observed that *
- 16 - a typical stellar orbit in the plane of the galaxy is characterized by two typical frequencies. These are Q(R), rotation frequency, associated with the rotation of a star around the centre of the galaxy, and K(R), the. epicycle frequency, associated with the radial oscillations carried out by a star. ifereR is the distance of the star to the centre of the galaxy. The normal situation is that the ratio of fi(R) and tc(R) is not a rational number. There- fore, stellar orbits generally are not closed curves in the plane of the galaxy. However, Lindblad noted that it is always possible to find a unifonriLy rotating coordinate system (with angular velocity say H) from which the orbit of a star is observed as a closed curve. The value of JJ(R) is determined by the condition
(fi - H) + im.K = 0, with 1 and m again integers. Different values of 1 and m correspond tc different forms for the closed orbits.
Lindblad observed that a particular interesting case is obtained for m = 2 and 1 = -1. First of all in this case the closed orbits in the uniformly rotating coordinate system have a bisymmetrical character.. Secondly, the frequency fl(R) as defined above, is more or less constant over the disk, except in the inner region of the galaxy. Therefore, any stationary distribu-* tion of the stars or gaseous clouds along.closed orbits in the uniformly rotating coordinate system does not dissolve by differential rotationbut, when observed in the inertial frame, varies periodically in titoe with the more or
less constant frequency 52. Perturbations in the distribution of stars that
vary in this manner are sometimes called kinetBatical "density waves". .
This led Lindblad to the conclusion that bisymmetrical (o* two-armed)
structures should have the largest chance of survival.
We shall show later in this thesis (section 4.2) that Lindblad *s
- 17 - kinematical density waves become again of great interest, be it in a different context, if the self-gravity of these density waves is taken into account
"Natural oscillations" of the disk also play an important role in our present understanding of a spiral structure.
If a third periodic motion is imposed on the stars, say with frequency u>,
these natural oscillations will give rise to resonant effects if a = m JKR) +
+ lfi(R). Because m is considered as a constant, such resonant effects are
important only at certain distances, depending on the values of m and 1. In
practice, it is found that the most important cases occur for m = +2 and
1 = 0, +1. For m = +2 and 1 = -1 this distance is called the inner Lindblad
resonance radius, for m = +2,1 = +1, the outer Lindblad resonance radius and
for m = +2, 1=0 the corotation radius. This established terminology once
more reflects the importance of Lindblad's studies on spiral structure. We :
shall not try to summarize the complete work of Lindblad, which would require .
an elaborate study in itself. But it is worth noting that all problems that in
later developments turned out to be important in the theory of spiral j
structure, had, in one way or another, already been touched upon or even
studied by Lindblad.
The main problem in theories of spiral structure has always been to
explain the persistence of spiral structure during a considerable fraction,
of the life of a galaxy. The theory proposed by Lin and Shu was the first stej
in the theoretical development to the solution of this particular problem.
None of the other existing theories of spiral structure, such as the
generation of spiral structure by tidal interaction with a companion galaxy *
(Toomre, 1972) or by an explosion (Ambartsumian, 1961; Arp, 1969 and Oort, .
1973) give a satisfactory explanation of the long-term persistence of spiral
- 18 - structure.
We shall further leave these different theories aside and concentrate our
efforts on the problem of the persistence of spiral structure.
1.3 Outline of the present study.
In June 1970 the Westerbork Synthesis Radio Telescope became operational.
With it a detailed study of the dynamics of nearby spiral galaxies became
possible. By observing the 21-cm line emission of spiral systems not only is
knowledge obtained about the distribution of neutral hydrogen, but also of the
state of motion. Altogether this yields an enormous amount of information 6n;
the dynamical properties of these galaxies. Moreover, for iieairby galaxies like
M 51 and M 81, the resolution of the WSRT is such that a study bE cf the spiral structure, so typical for flat galaxies, can{be made»c Wifth these future prospects for observations in mind, it appeared expedient to obtain a better insight in the most, promising theory of spiral structure, the density wave theory, proposed by Lin and Shu in 1964. The main assumption in this theory is that the mass concentrations in the spiral arms are relatively small in comparison to the average mass density, so that the spiral arms can be considered as small disturbances in an otherwise axisymmetric mass distribution.. The:fundamental idea, then, is;,that such disturbances in mass are generated by disturbances id the graviational potential which in their turn are caused by .the mais disturbances themselves. Lin and Shu showed; that the observed spiral patterns cau be explained in terms of such oelfconsistent mass disturbances when considered as wave-like structures, socalled density WSVCB. The inmediate Bucr.es of the density wave theory, is based on two of its consequences:'. •.-:. * •'.••• '•••- '• •• . ' ::"-: '•' - 19 - 1. It explains in a convincing manner that spirals need not wind up owing to differential rotation. (A discussion of the so-called winding-up problem was given by Oort (1962).) 2. It explains qualitatively why and how the systematic motion of clouds of of gas deviates from a circular motion, as has been observed in our own Galaxy (Lin, 19701. A serious objection to the theory in its original form, however, was brought forward by Toomre (1969). He showed that the Lin waves, when combined into wave packets forming spiral arms, cannot be expected to be long-lived features. The wave packets propagate inwards untill the inner Lindblad resonance radius •s (see section 1.2) is reached, where they disappear owing to resonant effects. * The Lin waves, therefore, have to be regenerated either continuously or intermittently. The old problemof how to account for the persistence of spiral structure ' has reappeared, although in quite a different form. Among other questions not satisfactorily settled by the theory, the following may be mentioned: } i i) Are certain frequencies that characterize the wave-like perturbations \ preferred over others, and if so, how can we compute these? ~ j ii) Why is the observed spiral structure always trailing, whereas the theory "S does not distinguish between leading and trailing features? J One of the reasons for which the questions listed above can not be solved by Lin's approach, is that it is assumed a priori that stellar disks are stable with respect to non-axisymmetric perturbations. In fact, Lin's theory is far from complete and although very important in concept, cannot be considered as the final answer to the problem of spiral structure. It has to I be realized that thus far observations have not yielded conclusive evidence,- either against or in favour of the density wave theory. At the present stage - 20 - of development we feel that any observational atteu.pt to prove definitively the validity of the density wave theory is bound to fail. For neither the theory, nor the observations yield direct information about such relevant parameters as wave frequency and amplitude. The basic thread connecting theory and observations is a descpiption of the gravitational potential underlying the density perturbations, in sufficient detail to provide information which enables a decisive comparison. Such a description does not yet exist. The philosophy we have therefore adopted is to concentrate our work on the main processes which, in our opinion, influence the ultimate form of the gravitational potential. For this purpose we make the fundamental hypothesis that the spiral structure as a general phenomenon is an inherent property of flat stellar systems, resulting from certain instabilities peculiar to such gravitational systems. Our first aim was to study the stability of a slightly-perturbed disk consisting only of stars (chapter 4), For such a stability investigation it is ;ecessary to know properties of unperturbed disks. Therefore, after describing in chapter 2 the mathematical tools, in chapter 3 the properties (only partly known) of unperturbed axisymmetric, steady stellar disks are discussed. Although the analysis mc.de in chapter A is far from complete we feel that the arguments presented there are plausible enough to allow, us.to^ conclude that "cold" flat stellar disks, are unstable wi th - respect-co-non-: : . •'.'. axisymmetric1- perturbations, in-a certain restricted range, of/ frequencies. A. . spontaneous growth of this iype of perturbations was indeed detected in . numerical experiments on the .evolution of .stellar disks (Hohi ,1970, 1972). . There it was also found, however^ that, after some time the, stellar disk heats . up, and as a consequence the. spiral structure disappears. .- • . : The observed spiral structure in.real galaxies, as well as tlie relatively small stellar velocity dispersion observed in our own Galaxy in the solar neighbourhood, both indicate that some stabilizing influence is at work on the perturbations; apparently the stellar disks actually stay relatively cool. Therefore, we investigate in chapter 5 the possible stabilization of the perturbations by the gaseous component of the disk. We further investigate the behaviour of the perturbations on large time scales taking into account the long term evolution of the stellar disk itself (chapter 6). Our conclusions are summarized in chapter 7. - 22 - CHAPTER 2 MATHEMATICAL TOOLS In this chapter we shall describe the basic equations that are used in the study of stellar systems. In chapters 3, 4 and 6 we shall deal with a model stellar system, consisting of stars only, all stars having equal masses m. The latter assumption is not strictly necessary but simplifies the description without vitiating the results. Further, it is assumed that external forces are absent. Therefore, every star moves under the influ- ence of the gravitational forces excerted on it by the other stars of the system. The most powerful method to describe such an assembly of stars still is (e.g., Jeans, 1915) to introduce a density function f(-r, "v, t) of posi- tion ~r, velocity ~v and time t in the six-dimensional phase space, so that f (r", v", t) drdv" is the number of stars that at time t has positions between 7 and r+dr", velocities between v and v+dv". By definition f (r\ 7, c) is non- negative. The concept of density function is especially useful-.if the time scales of interest are much smaller than t ., the relaxation time for rel' •.-...-. stellar encounters. This condition is fulfilled because for the galaxies studied here we are interested in timescales not.going beyond the life time of the system ( 1010 year) whereas t = 6.1O14-A.io'^ years (Lecar, 1970). Therefore, we may, as a good approximation, neglect in our model the effects'' of violent encounters and assume that the gravitational forces felt by. a .. star due to.the presence of all the other stars are derivable from a •••./ smoothed-out gravitational field V(r", t) of position r and time t. This smoothed-out gravitational field is readily found once the density. func- •' • • '.' ' ,' " '- 23 - '..••••••'•"/ • ••'•'•'•.•• ••:• '•• ••• •••-• tion f(r, v, t) is known, for by integrating f(r, v, t) over all velocities we find the mass density in normal space p{T,t)=mlii dvfCF, v, t), (2.1) from which the gravitational potential field V(r, t) is determined with Poisson's equation V2 V(r, t) = 4 it G pCr, t) . (2.2) Here G is the gravitational constant. It is clear that all information necessary to study the dynamical evolu- tion of the stellar system is contained in the phase space density function. The distribution function f(F, v, t) will change in time as the stars move through the six-dimensional phase space under the influence of a gravitation- al potential field V(r, t). Because it is assumed that no stars are created or destroyed the change in the number of stars in a volume element in phase ' space in a time interval is fully determined by the number of stars that has ; flown through the boundary of the volume element in that time interval. This : conservation of number density can be expressed as a continuity equation in ' the six-dimensional phase space i |f +-v.M +a.^-0. (2.3) ] 3F 3T \ A Here a = - .— , the acceleration arising from tha graviational potential 1 field. 2 The equations (2.1 - 3) can be considered as the basic equations of stel-4 lar dynamics, which fully determine, self-consistently, the dynamical evolu- I tion of the assembly of stars. When combined, eqs. (2.1 - 3) reduce to a non-"| linear integral-differential equation which is not soluble in general, but f only under very special assumptions that still need to be specified. Before doing this, we first review here briefly a few concepts of classical mechanics - 24 - that are especially useful in analysing the problems we intend to study. There are several forms of the equations of motion of a star in a gravi- tational field V(F, t). One possible form is to introduce the Lagrangian func- tion of space coordinates q., i = I, 2, 3, their total derivatives q. and the time t Uqv kif t) = TCqi5 q.) - V(q., t) . (2.4) In eq. (2.4) and all further equations a function F(q., c.) should be read as q q q q In eq (2 4 T q q is the ki F(q,, <32' 3' r 2' 3^* * " ^ ^ i> p netic energy and V(q., t) the potential energy, per unit mass. From a variational principle the equations of motions in terms of L can be obtained (see e.g. Goldstein, 1969) ii =0' A different form of the equations of motions, involving first derivatives only, is obtained by introducing the Hamiltonian function 3 qi Pi ~ L(qi> qi> where the generalized momenta p. are defined as p. = ft , i = 1, 2, 3. (2.7) Taking the differential of H and equating the coefficients of the differen- tials, using eq. (2.7), the Hamiltonian equations of motion are found p. =- J3 q. = (2.8) 1 91 i We shall see later that the choice of coordinates is an important tool to simplify the studies in stellar dynamics. If we wish to transform from one set of coordinates (q., p.) to a new set of coordinates ( q, = q, (q-', p., t), (2.9a) p. = p. (q., p., t). (2.9b) k k i i Clearly, only two of the four sets of variables p., q., p., q. (i = I, 2, 3) are independent. The generating function can therefore be written as a func- tion of independent variables in one of only four forms Vqi' V t} F2(qi' Pi'C ) F3(pi' qi't } F4(pi' pi't ) The choice of a particular generating function depends on the problem that is considered, and it is convenient here to choose F9. The transformation then is canonical if 3F2 pi = ^7 ' <2-'°a) (i = I, 2, 3) qi = if. (2.10b) and the new Hamiltonian H in terms of q. and p. is given by H ^ H + 3F2/3t. (2.11) The eqs. (2.10a) represent relations between p., q., p. and t. These can be solved for pi, thus yielding eqs. (2.9b). Once these relations are known, the - 26 - - I expression (2.9b) can be obtained with eq. (2.10a). Therefore, once a particu- lar generating function F. is given, for known q. and p., we can find p., q. and H immediately from eqs. (2.10 - 11). In terms of the new coordinates Hamilton's equations of motion read 3H i 3H p* ~ ~ —~ q- = —==• * (?.12) i 3q• I Sp. Canonical transformations as described above not only conserve the form of Hamiltcm's equations of motion. Other expressions are also invariant under these transformations, the most important ones for us being: i. the invariance of a volume in phase space, which is merely a special case of the more general integral invariants of Poincare', ii. the invariance of the socalled Poisson brackets, which are defined as 3 U, V I = > -r— • -T— - -r— " -r— , (-2.13) ' Jq,p . , 9q. 3p. 3p. 3q. ' ^ 1=1 ^i *1 ri Hi where u and v may be any functions of q., p. and t. With the above described concepts we have all tools necessary to proceed. We first note that if generalized coordinates q. and momenta p. are used to describe the orbits of stars in phase space, the equation of continuity eq.(2 -3) for the density function, now being a function fCq^, p^, t), can alternatively be written as • . . • where H is the Hamilton function defined in eq.. (2.6 ) and the brackets are Poisson brackets defined in eq. (2.13); We now return to the specific conditions under which we.sbalL.analyse the dy- namical behaviour of the stars. These are two-fold: - . - . • •'••. • •"•.•. -27 - ' -•'. •• •'.,'•• i) All stars are assumed to move in the same plane, ii) The gravitational potential is assumed to consist of two parts: a mean field, that is assumed to be axisymmetric and slowly varying in time, and perturbations that are assumed to be non-axisytnmetric and small compared to the mean field. Our first assumption is based on the fact that the spiral galaxies we intend to study appear to have most of their mass concentrated in a thin disk (see fig. 1.2). If so, the above assumption can be considered as a reasonable ap- proximation of real galaxies (Shu, 1970^. The second assumption is based on the observation that, although the optical pictures of spiral galaxies show a conspicuous non-axisymmetric structure, this structure contains only a small amount of the total mass in the galaxies. This is demonstrated by the obser- ved motions of the gas which generally deviate little from a pure rotational motion, suggesting thereby that the deviations from a mean gravitational po- tential field are small. With the assumption? made it is the obvious choice to describe the mo- ; tions of the stars in cilindric j. coordinates (R, 6, z) with the gravity centre of the stellar system at the origin and the plane z = 0 coincident with, the orbital plane of the stars. Because the motion of the stars is restricted { to one plane, it is sufficient to study the behaviour of the stars in the four' dimensional phase space. The density function thus is a function of R, 8, V , I Vfi and t. The gravitational potential, generated by the disk of stars, on the other hand, has to be considered as a function of R, 9, z and t and can still be described by Poisson's equation V2 V = ATTG 6(Z) O(R, 9, t). (2.15) Here 6(z),the delta function, expresses that all matter is concentrated in one olane. The surface density a a(R, 8, t) = m f dvR / dvQ f(R, B, vR, vQ, c). (2.16) Because we are only interested in the gravitational potential field in the disk we shall abbreviate it by V(R, 8, t); only if values outside the plane are considered the z-dependence will be explicitly mentioned. The equation of continuity in phase space is still given by eq. (2.14), but in the Poisson brackets the summation is now restricted to i = 1, 2. According to eq. (2.7) we find that the generalized momenta corresponding to the coor- dinates R and 8 are pD = vD and pn = Rv_. K. K. u 0 The Hamiltonian function, here being the total energy of a star per unit mass, in terms of these generalized coordinates and momenta, is given by 2 •) Pfl H(R, 6, p p , t) = HPp + -f ) + V(R, 6, t) . (2.17) R6 R R2 The gravitational potential according i:o our assumptions can symbolically be expressed as V(R, 9, t) = V°(R, at) + XV1(R, 6, t) In eq. (2.18) V°(R, at) represents the mean field and V (R, 6, t) the perturb^ ations. The quantity X is considered to be sirill. The quantity a is the time scale on which V°(R, at) varies considerably. It is considered long com- pared to the timescale on which V varies. Therefore, when we study the behav- iour of stellar disks, on time.scales small compared to n we may take a; ™ 0. Eqs. (2.14 - 18) will be the starting point of our investigations. In Chapter 3 we shall consider the.case that o = 0 and X =0, in chapter 4 the case a = 0, U 0, and chapter 6 will deal with the case that both a#>. and - 29 - • ••.•'••'•• CHAPTER 3 DYNAMICAL PROPERTIES OF FLAT STELLAR SYSTEMS 3.1 Introduction. We shall discuss some of the dynamical properties of a flat stellar syctern in a steady state (a = 0) with anaxisymmetric gravitational potential field V°(R) (X = 0)v. After discussing the orbits of the individual stars (section 3.2) we consider in section 3.3 the most important properties of such a stellar system in terms of the distribution function and its momants. 3.2 Stellar orbits. The most important characteristic of a stellar orbit in an axisymmetric gravitational potential field V°(R) is its periodicity. In systems with one degree of freedom we can generally distinguish between two kinds of periodic motion (see Goldstein, 1969). The first type f occurs whenever both the coordinate q and the generalized momentum p are i. periodic functions of time with the same frequency. Since p and q return to ; a their original starting values in one period, the orbit in phase space is f closed (fig. 3.1a). This type of motion is called a libration. The second * A. type of periodic motion occurs whenever the configuration of the system is thfr* j same for any two values of q, differing by a constant quantity, often taken t<£ be 2it (fig. 3.1b). The. motion then is called a rotation. Classic examples of a libration and a rotation are the oscillating pendulum and the rotating pendulum, respectively (with q the angle of deflection) . For more than one degree of freedom the motion is said tc be periodic if the projection of the particle's orbit on each (p., q.) plane is simply -30- Iteration rotation Fig. 3-1. The vuo types of periodic motion that can be distinguished in systems with one degree of freedom. periodic in the sense defined above. The complete motion of the particle need not itself be simply periodic. In fig. 3.2a, b. the projections of a typical stellar orbit on the pD - R plane and the pQ - 0 plane are shown, respectively. In this particul af case, the rotation in the p - 6 plane is a limiting case of the rotation- n • type of periodicity, to which an arbitrary period may be assigned reflecting the fact that 6 is a cyclic coordinate. The libration in the p_ - R plane (for fixed energy and angular momentum) K •• describes the oscillation that a star performs in radial direction. These., radial oscillations are fully determined by the so-called effective potential,. , which for a star with angular momentum p' is defined, as eff : P ) = V°(R) i g / P. (3.1) eff e + P and by the total energy of a star, H0,. which according to eqs."(2.17•• - 18) for the case of an axisyranetric, time independent gravitational potential, that is, -31- Fig. 3.2. Projection of a typical steVar orbit in an axisymmetvic, time- independent gravitational field on the pR - R plane (a) and on the p - e plane (b). for a. = 0 and A = 0,is given by H °= eff (3.2) As illustrated in fig. 3.3, a star with a given energy H° = E will oscillate back and forth between the radii R . and R defined as the zeros of the mm max equation E - V eff C3.3) For periodic motions, such as characterize a stellar orbit, it is often an advantage to describe the behaviour of the stars in terms of action and angle variables, as has been stressed by many authors (e.g., Kalnajs, 1971). In the first part of this section we shall describe the orbits of stars in terms of action and angle variables in a rather general way. In the second part we determine the orbit characteristics explicitly by introducing a -32- Veff 1 Rmin Rfflax 1 1 1 1 I 4 5 i E - \ Fig. 3.3. Sketch of V „„ (R3 pfiJ as a function of E for a particular choice of V°(E) and pQ. suitable approximation. Action and Angle variables...... ,.,:, , ;• The action variables J., corresponding to a periodic motion in the plane (p., q.) are defined as where the integration is to be carried out over a complete period of rotation or libration. In the limiting case, of the rotation type>of periodicity the integral must be evaluated from 0; to 2ir, the. natural'period of the • cyclic:;1.;. coordinate q.. . .••••• •'•••, ; ••'•-••.. '.'"•• '..' •' ': • ; ,-\ -.. •..•'•.• ••••• :', '••.• .'','• ..'• For a star moving ir. ah axisymmetric gravitational potential field.:. V°(R) we thus obtain:' .,-.•,..• -::': ••.".•. ':'''.••,• • •'• •.'•:.•••. :"'1 •'''"'...: -:\ ' •• •'.•'•: 33-i; ;,.,•-•. • . • . .• . • ';!-•. •' '•' • ' "'. "'• • >• V "•""'-" ' •' ."'."•• J. = 4r- de (3.! (3 6 J2= Tr. dR • >1 By solving eq. (3.2) for p we obtain ••$. 2 •as: = ± [ 2(H° - Veff (R, pe)) ] ~ (3. wh?Te the positive sign must be used if the star moves outward and the negative sign if the motion of the star is inward. i # Eqs. (3.5 - 7) describe a relation between J?, H and J. from which we can determine the energy H as a function of J. and J.. Thus, in a time- independent axisymmetric field the energy is a function of J. and J~ only: H° = , J2) . (3. In order to find the angle variables conjugated to the actions J, and > we introduce here the generating function F- described in chapter 2. R x 1 2 F2(R, e.-J,, J2) = / dR' (+) [2(H° -VBffeff (R , J,))] + + J. . (e - et). C3.9)j In eq. (3.9) H is considered as a function of J and J? by eq. (3.8) and R.-| and 9j are two arbitrary constants which may be chosen suitably. According to eqs. (2.10 - 11) we find 2 pR = BF2 / n = ± [ 2(H° - V (R, J ))] , (3.K -34- pe = 3F2 / 39 = Jj, (3.11) R [3H° / 3J - J /R'2]dR' w - 3F / 3J = e - e + / • , (3.12) 1 R, (±) [2(H° - Veff (RJ JJ))] * R aH° / 8J dR' , , 2 (3 13) w = 3F, / 3J = / - r ' ' 1 2 R, (±)[ 2(H° - veff (R; J,))] with J, / R'2 dR1 3H° / 3J, = 3H° / 3JO [2(H° - Veff(RJ J,))] (J,, J2), (3-14) dR 3H° / 3J, = U- 2 [ 2ir ±)[2(H° - veff (R; jj))] JJ = fl2 (J,, J2). •"""""(3.15) Both these symbols have a simple physical meaning: fi. is the angular velocity averaged over one complete radial oscillation.and fl2 " the frequency.of the . radial oscillation. • ' • ,.- . ; . ••• •: ••••'••• .•••;. . ••••.• .'•'•, . With the new variables thus defined the Hamilton, equations of motion read .••.:.'•• ' '••• : " •,' •;•"-• .: .' ' '•'... '• •" l i which have the solution.- •; J. = constant, (3.17f w. = n. (J , J?) t + constant, i = 1, 2. We see that the solutions of the equations of motion indeed take on a very simple form if action variables are used. This simplifies considerably the problem of analyzing the perturbations as we shall see in chapter 4. However, very little information is obtained about the orbit followed by a star in normal space, unless we find explicit expressions for the relations (3.6) and (3.12 - 15). This can only be done approximately. Two possibilities of doing so are discussed below. Approximations. The most popular approximation, the well-known epicycle approximation (Lindblad, 1959), amounts to a Taylor series expansion of the effective potential (eq. 3.1) in R around R up to second order. Here R , the so-called.. epicentre radius, is the radius at which the effective potential reaches a minimum, as defined by dVeff /dM R = R o If we introduce the angular velocity fi(R), not to be confused with Q and fi» introduced in eqs. (3.14) and (3.15), by dV° / dR 5 R fi2(R), (3. IS eq. (3.18) can alternatively be written as -36- which defines R as a function of J-. o 1 In the epicycle approximation one then obtains V ,_ (R, R ) = E + £ K2 (R ).(R - R eff ' o o 2 o c where E is the energy of a star moving in a circular orbit with radius R ° o 2 2 Eo = V° (Rc)+ I R n (Ro) , (3. and K is the epicycle frequency defined by 2 2 K (RQ) = 4fl (RQ) • { 1 + Kdlogfi / dlogR)R = } . (3.23) o For abbreviation we shall often write a for n(R ) and tc for K(R ). o o o o The epicycle approximation is applicable only to nearly circular orbits. We shall now consider a different approximation which can be applied to describe orbits that deviate more strongly from the circular shape. We shall call this approximation the Keplerian approximation because it amounts to a Taylor series expansion of the gravitational potential V°CR) (also up to second order) in R around R : o (R) - V (R b' " ld(l/R)JR = kR'V"R p. 2 .. ld(i/R) J_ = _ • ,.. .* Ro With the definition cf JJ(R) in eq. (3,19) ve have {dV/d(!/R)}R =R = -Ro V' • o • .••.,'. -:'7- 2 2 2 A 2 2 {d V/d(,/R)\ }_ R=R_ = R (K - n ) . (3.26) o We thus find n O O t o o~7\ V (R) - A - —y + ~^ , \i-t-l) R where A = E + | R 2 K 2, (3.28) o o o o B = R 3 K 2, (3.29) o o o C = i R 4 (K 2 - JJ 2). (3.30) o o o o With this expansion of V°(R) the effective potential eq. (3.1) becomes up to the second order of the approximation: B 2C + R Q V (R , R ) = A " jf + i ° ° — • (3.30 e o o R As an illustration we have in fig. 3.4 plotted" (1) V ff, (2) the Keplerian approximation according to eq. (3.31), (3) the epicycle approximation according to eq. (3.21) for a particular choice of V (R) and R V (R) has been chosen according to the Brandt model with n = 1 (Brandt, 1960). R has been chosen in such a way that R / R = 0.75, where R is the distance o o where the rotation curve R Q(R) reaches a maximum. As can be seen from this graph the Keplerian approximation and the epicycle approximation are , comparable for nearly circular orbits, but at higher values of E the two approximations diverge. In spite of the fact that in both approximations only three terms are retained in the Taylor series, the Keplerian approxi- -38- 1 1 1 1 (1,2) 2 :3 I 5 6 — R/Ro _--—— _—-—— (3l\ I Fig. 3.4. Curve (1): The dependence of V „„ on R. Curve (2): the dependence of V „„ on R'in the Keplerian approximation. Curve (3): the dependence of V »^. on R in the epicycle approximation. The particular choice of lf(R) and R are specified in the text. mat ion is valid to much higher energies than the epicycle approximation. The result for the epicycle approximation can be obtained from the results of the Keplerian approximation in the limit of nearly circular orbits. A more complete discussion of the applicability of the two approximations is given at the end of this section. We shall now calculate the action and angle variables in terms of energy, angular momentum, R and 6, using the Keplerian approximation. In order to obtain explicit expressions for the action and angle variables we note, first of all, that the turning points R . and R (eq. (3,3)) of a star with epicentre radius R and an energy E in the Keplerian approximation are given by R . i • ••: 13.32) mm max 1 - e Here r is defined by 2(E - e = (3.33) R2<2 o o With these expressions and choosing R = R 0, the expressions for mm J , w , w , f!. and n? follow by straightforward calculacions from the eqs.(3.6) (3.12 - 15). J2=Ro2 OU "e2)"* " 'I • (3.34^ 2i 3/2 (3.35) °,-«o 3/2 (3.36) n2 = KO { i - - } -.MM-2-(^"«)2l* + arcs in -i (3.37) arcsm (3.391 With these expressions the complete description of the orbit of a star i| can be given. For instance, we can find the geometrical form of the orbit, % relating R and e, by eliminating from w and w in eq. (3.17) the time: if -40- (3.39) From eqs. (3.37 - 39) we then obtain o = 1 + e cos 2 (3.40) R *r n •) • showing the well-known rosette-type character of the orbits (fig. 3.5). Of course, eq. (3.40) can be derived in a less complicated way. N f / 1 \ 1 \ 1 \ 1 \ ._„. 1 jj " \ 1 >\ s' \ 1 s \ \ / \ "^ \ V / \ \ / J* 1 V / • ' 1 / 1 ; \ / 1 \ / 1 V ( t \ t / \ 1 / ; / I I / / / / / / / / Fig. 3.5. Typical rosette-typs orbit in the plane c*" a galaxy according to eq. (3.40) far s ~ 0,5 and n« / fi» = 1.45. The detailed variation of R and 8 in time is obtained by solving R and fl from eqs. (3.37 - 38) as-functions of w and w2, and afterwards substituting the time dependence of the latter two as is given in eq. (3.17). From eq. (2.3©. we find . . . R I - £ cos (wo + Y(V,}) • . .. /• -41- where ••> i (3.42) Although no analytical solution of R in terms of w?, R and can be obtained we can apply an iteration procedure as follows. Inserting eq. (3.41) in eq. (3.42) we have w = w x( -.) E sin(wo + x( o)) t (3.43) which upon iteration gives XQ(W2) = e sin w2 , (3.44a) X.(w2) = e sin { w2 + E sin w2 } , (3.44b) X2(w2) = E sin {w + c(sin (w2+--))} , etc. (3.44c) In fig. 3.6 we have, for e = 0.5, plotted the actual dependence of x on w_, and w X (w?) and Xi( o)> from which it is seen that the second iteration already gives a fairly good approximation. In order to determine the dependence of R as a function of w« up to second order of e it is sufficient to use the zero-order solution of eq.(3.43) and we thus find - = 1 ~ e cos (w2 + E sin w2) _ &.k& R , 2 I o 1 - e ; In fig. 3.7, again for e = 0.5, we have plotted the actual dependence of R ini the Keplerian approximation on w« together with the epicycle approximation I 1 - e cos w and the approximation (3.45). | The dependence of 8 on w. and w. can be described by '% 4 9 = Wj - u(w2), (3- ® where "3 u(w( 0), = —°-1 I \ x(w -42- Fig. 3.6. The function x(uJ, shown here for a particular example together with two approximations, appears in eq. (S.fl) and measures the phase delay in the radial oscillations with respect to a harmonic motion in time. 2 - 1 - .7t/2 it — W2 , Fig. 3.7. The ratio of the distance of a star and its epicentre radius, R/Ro3 is shown as a function of the angle wg together with two approximations., for. a particular example. • • -43- 1 - W, Fig. 3.8. The function u{w~) shown here for a. particular example together with an approximation, appears in eq. (3.47) and measures the phase delay in azimuthal angle with respect to a rotation with a constant angular velocity. i x "J -43o- with 2 g(«2) = (1 - E ) { E cos(w9 + x) } / { 1 - ecos(w. (3.48) Upon expanding the arcsin in eq. (3.47) we obtain to second order of E fll u(w9) = — I 2e sin(w + x) + sin 2(w21x) (3>49) 1 - ecos(w2 + x) The second term in eq. (3.49) is in all relevant cases negligible because for 2 E << 1 it is of order E whereas if e approaches 1 it goes down instead of up. Together with the first iteration solution for x (e«l- 3.44a) we can as a good approximation write 2P o 3 u(w2) a _? . e sin (w2 + esin «2) , ( v?P) o ''''." fr • • '•'-"• where we have used the fact that in the Keplerian approximation eqs. (3.35 - 36) give £}, / n» = Q / K . In. fig. 3,8 the actual dependence of u(w_) in the 1 ZO O - £- Keplerian approximation and its approximate expression (3.50) are. plotted as a function of w7, again for E.= .0.5i -; .. . :• :' . The accuracy of the results given here, is sufficient for-all further ',-..' application because as we shall see below the Keplerian approximation only applies for e-values not exceeding: 0.5. For.e « I we find •. -|- = 1 - ecus w ' , !'V •••"• v---.- .",".. •-": ;(3;5i) O • • • • '••••..•..•.:..• ; ; : •2n '; ."•'>••:'•"'";". "' :.' \ •' • •" v "••'. •'" - •' ' ; ••-. • e = w - —° .;E sin w9 .", . ••; : •••• .. -..•••>. O-52) o' • : •":. •; : •; ' .:.,. .• : :• ..• ••• •• ...•••,'• : With • -:••. • -• -.7 •- .. • ••"• • '..: ' • : • "'-'"•••••"•••••• •' •••••••••' •...•••• w = n t + constant , w_ = K t + constant (3.53) ] o I o in agreement with the epicylce approximation. Validity of the approximations. The validity of the Keplerian and epicycle approximations, respectively, is examined by taking a particular example. Let us assume that the surface mass distribution in the disk is given by 2 2 a(R) = a(o)Va R . (3.54) For this Gaussian mass distribution the gravitational potential field is given by 2 2 iaR 2 2 V°(R) = Vfa) e~ Io(ia R ), (3.55) where I is the modified Bessel function of the first kind. From this o gravitational potential field we have determined the turning points R '/R and R /R as a function of e. This has been done for various values of max o R /R, where R is the distance at which the rotation curve reaches a maximum: o * jjjfHR n(R)) (3.56 R = R In fig. 3.9 we have plotted R . /R for two values of R /R as a function of min o o together with the dependence of R . /R as is described by the epicycle approximation, i.e. R . /R = 1 - e and by the Keplerian approximation (eq. 3.32). For e << 1 the Keplerian approximation and the epicycle . approximation give the same dependence and agree for all R > 0. It is clear^ however, that the Keplerian approximation agrees better with the actual model curves for higher values of c. This is because the strong variation of the effective potential for R < R comes mainly from thvj centrifugal force -45- 1.0 Fig. 3.9. Full curves: 8 . / R as a function of e for (a) R /.i? = Q.35 and (b) E / H =. 1.94. Dotted curves: R, . . /, E as a. function of e in the epicyale approximation^ (1 - eJ, and the Keplerian approximation, (1 + e) . (Gaussian mass model). Jj2 / 2R2 which in the epicycle approximation is expanded as a Taylor series around RQ, whereas in the Keplerian approximation it is taken fully "into account. We can therefore conclude that for/R.< R th« Keplerian approximation is always to be preferred over the epicjicle approximation, provided R ^-O. and that for E « 1 hcth approximation* are equivalent.1 In fig. 3.10 we have similarly plotted K / R for several values of R / R as a function of E max o , OK together with the dependence as is given by the epicycle' approximation, I1 R m,v / R - 1 + e, and by the Keplerian approximation («q. 3.32). A rather different picture is seen,here. For small values of tic epicentre radius the epicycle approximacion approaches better the act-oal oodej curve than the Keplerian one. When the epicentre radius increases the epicycle'approximation -46- Fig. 3.10. Full curves: R / R as a function of e for (a) R / R = 0,35, (b) RQ / R = 0.7, (a) Ro / R - 1.0 and (d) R /R=l,94 (Gaussian mass model). Dotted curves: R / R as a function of max o J e in the epicycle approximation, (l'+ z), and the Keplerian approximation, (1 - z) "*. becomes less and less good in the sense that the range of e to which it applies decreases whereas the Keplerian approximation becomes better. ^ For a full understanding of fig. 3.10 we consider the function e as a'j function of R where E is defined as (3.: -47- Clearly, z is the value of e corresponding by eq. (3.33) to E = 0. That is, for e •*• e the value of R will tend to infinity. Because in the Kepleriati o max r approximation R tends to infinity for £ -*• 1 and in the epicycle approxima- tion this only happens for e •+ oo , we can expect the Keplerian approximation to be preferred in the range of epicentre radii for which e = 1 whereas the o epicycle approximation is better in the range of epicentre radii for which E >•> 1. This conclusion is confirmed by fig. 3.1! in which we have, for the present mass model, plotted e as a function of R . 1 1 i 1 13.121 H aol- — \ 2.0 t- \ . - - 1 1.0 1- Fig. 3.11. E a3 a-function of E for1 a Gaussian mass\ rriodet;,'••'"•: '•" . ;.'". ' Fig. 3.12 c as a funation of RQ fov two Brandt modelsj_\='-l end-ii;-= 2.; •... In this particular model, e first decreases to reach a minimum .for R^ ./.R,= 1 (this corresponds Mith...the Ttftmost curve of fig., 3.10), theti.reacheB. a.. .• •• iHaxiniuai ana finally approaches unity i WhUe the -corresponding curve R^ / \ : approaches the curve (1 -E}-)» • : We thus see that in this particular model the Keplerian approximation is to be preferred over the epicycle approximation for R / R % 0.5, whereas the reverse is true for R / It £ 0.3. For intermediate values of the epicentre radius both approximations are comparable. Summarizing we conclude: i. The Keplerian approximation can be used for all epicentre radii R 5s 0 if e << 1. ii. For the larger epicentre radii the Keplerian approximation can be applied up to values of e = 0.5. iii. In order to obtain the best orbit description for the smaller epicentre radii at higher E values a more complicated description of the effective f potential seems advisable: it is suggested that the Keplerian approximation 'A is used for R < R and the epicycle approximation for R > R . | (3.131 _ 50 4.0 3.0 2.0 1.0 0.5 ID Fig. 3.13. The same as fig. 3.10 for a Brandt model with n = 1 for (a) RQ /R - 0.125, (b) R/R - 0.25, (a) R / R = 0,75 and (d) RQ / R = 1.5. ~i -49- Fig. 3.14. The same as fig. 3.10 for a Brandt model with, n = 2 for (a) RQ / F = 0.177, (b) RQ / R = 0.35, (a) RQ / R - 0,? and (d) RQ / R = 1,77. Although the above conclusions are based on the results of a particular model we find that a similar discussion can be given for other models. This may be inferred from figs.3.12,3.13 and3J4 where e (fig.3.12) and R / R o max o (figs.3.13,3.14) are plotted for the two Brandt models with n = 1 and n = 2 (Brandt, 1960). 3.3 Distribution functions. In the early days of stellar dynamics sorae attempts were made to find selfconsistent distribution functions that satisfied the combined set of equations (2.14 - 17) for flat axisymmetric stellar systems in steady state with differential rotation (Jeans, 1928 and Eddington, 1921). Lacking satisfactory results a semi-empirical approach was adopted in order to explain the observed motions of the stars in the solar neighbourhood. Already in 1907 Schwarzschild introduced a velocity distribution of maxwellian iype, suitably adapted to the conditions in stellar systems* Another well—known application of this cype of velocity distribution function was made by Oort.(1928) who succeeded in explaining many observed facts. Schwarzschild's distribution function is a special case of a whole' class of socalled ellipsoidal distribution functions. A complete and generalized description of• this : class of functions was given by Chandrasekhar ( 1942.)•From a theoretical point of view serious objections to the ellipsoidal hypothesis were raised;, in 1941. Caram (1941) showed that steady state ellipsoidal distribution functions do not satisfy Poisson'.s equation if simultaneously a realistic • model for the density distribution of the stellar system is1adopted.. This basic criticism was further explored by Kurth (1949) and by Fricke (1951). •; The letter author also showed that by abandoning the ellipsoidal form of the distribution function (arid replacing it by a different but selfconsistent form) it was still possible to explain the gross features of the motions of K, the stars in the solar neighbourhood. Moreover the then as yet unexplained f observed asymmetric distribution of the high velocity stars (Oort, 1928; could easily be accounted for. Since then several succesful attempts have been made to solve the combined set of equations (2.14 - 17). In this section we shall discuss some of these results and describe the particular distribution function we shall adopt in the remaining chapters of this thesis. Before doing so we shall first in more detail describe the basic equations (2.14 - 18) that apply to flat time-independent axisymnetric stellar systems with a gravitational potential V°(R), that is, for a = 0 and X = 0. Eq. (2.14) reduces to [F° , H°] = 0 • (3.58) where H is defined in eq. (3.2) and F is the distribution function for the steady axisymmetric state. According to eq. (2.16) we further have a°(R) = m fdv / dv F°(R, v v ), (3.59)t whereas finally a°(R) and V°(R) are related through Poisson's equation (2.15)^ IL is well-known that the solution of eq. (3.58) is given by any reasonably behaving function of the constants of motion, here E the energy and p the 6 angular momentum per unit mass: F° = F°(E, pe)'. (3.fl When E and pe are expressed in terms of R, vR and v we find that eq. (3.59) can be rewritten as -51- 2 2 0°(R) = m JdvR / dvQ F°(ivR + ivg + V°(R), Rv ) . (3.61) We shall assume that all stars in the system are bound. That is, ue assume that F =0 for E > 0. The integration in the velocity plane thus is restricted to the area enclosed by a circle with ra'dius (- 2V°(R) ) \ which simply means that only stars with vn + v < - 2V°(R) will be found at a R e distance R from the centre. Introducing these limits of integration in eq. (J.6I) and noting that F is a symmetric function of vD we find K (-2V° - vR ) 2 2 a°(R) = 2m j dv f dv F°(!v + Jv + V°(R), Rv )• (3.62) i K J e K e e 0 _(_2v°_VR2)i A further simplification of eq. (3.62) is obtained by introducing F » the symmetric part of F with respect to p_ , changing the integration over v_ 9 K. into an integration over x = ~(ivp + V°(R)) and that over v into an integ- ration over p = -£v. + x. After introducing z = -V (R) we obtain: o ( ' Eq. (3.63) can be considered as an integral equation for F once o (R) (and consequently also z) is known. Once F is known all higher order moments that are even in vD and v,. can be derived. From eq. (3.63) we can determine only K 8 • • ••..-. the symmetric part F of the distribution function. The surface density, combined with Poisson's equation therefore never suffices to detensine the distribution function F°. In order to determine the asymmetric part of F° with respect to p we have to prescribe the averaged rotational velocity cf the stars • • • '• -52- • ' •...'••• because for further application in this thesis we only need to know the I properties of F , we shall not consider this problem here. There are several % ways to solve equations of the type of oq. (3.63). "$• Often a special form of the function F (E, p ) is assumed at the very t beginning. As an illustration we consider the simplest case in which F is • assumed to depend on the energy only: f + F (E, pQ) = H(-E) for E (p ) < E < 0. 9 ° B (3.64) = 0 * for all other E. The function H(-E) is still unknown and has to be determined from eq. (3.63). Note that in this particular case F depends implicitly on p through the boundary E (p.), determined by eqs. (3.20) and (3.22). This cutoff in the o o 2 energy simply means that in any reasonably behaving system, for which K > 0 everywhere, a star with a given angular momentum p. has a lower limit on its energy, the circular orbit energy (Chandrasekhar, 1942). With our assumption eq. (3.64) on the form of F eq. (3.63) is now easily solved. We can determine the dependence of the left hand side of eq. t (3.63), thus o°(R), on z = - V°(R) with the aid of Poisson's equation. We shall write o (R) as a function P(z) of z and note that this function P(z) is. for a given surface density o°(R), indeed uniquely determined. The right hand side of eq. (3.63) depends on R only through z, and not through F . Therefore, once Poisson's equation has been used to determine ~ a (R) as a function of z, we can solve eq. (3.63) upon applying Abel's -~ inversion rule twice. We thus: find for the function H defined in eq. (3.64): (3.65) 1 For illustration we now apply this result to Toomre's model 1(Toomre, 1963) iu| -53.- which 2 2 V°(R) = VQ ( 1 + R / a ) ~* for z = 0, (3.66) 2 2 3/2 o°(R) =aQ ( 1 +R /a )' , from which we find VZ)= and thus T (f)2 • (3.68) Although the form assumed in eq. (3.64) for F is not very satisfactory, the example does show that solutions of eq. (3.63) exist and that moreover H is uniquely determined by the surface density. A more general method to solve eq. (3.63) is to expand F- in a double 2 seties with respect to E and pQ , a method first proposed by Fricke (1951). The right hand side of eq. (3.63) then reduces to a power series in z =~V°(R) 2 and R . If the left hand side is also expanded in a double power series in z 2 and R ("Frickes trick") one can determine the coefficients of the double 2 series in E and p by equating terms of equal powers. For flat stellar a systems Fricke's method was applied by Miyamoto (1971) with a surface density according to Toorare's model 1 (eq. (3.66)), Fricke (1951) applied this method to a three-dimensional mass distribution (with the assumption that F = F (E, p ), in which case the coefficients can be determined uniquely) .In y • • . • ...... flat stellar systems the expansion of the left hand side of equation.(3.63) 0 • • • • • • ' ' + in z and R is not unique. Therefore, when :-in a flat.system F also depends on p explicitly, an infinite number of solutions can be constructed for.a • given surface density.' . : • ' • ' • • • • '. • • ' • " " .. • The most urgent problem at present is.not how to construct a self- -54- consistent distribution function, but how to select from all physically acceptable solutions those that (albeit approximately) exhibit the characteristics of a real galaxy. Parenthetically, the same problem arises if in the 3-dimensional case F is assumed to depend also on the socalled third integral of motion. From - the observations the dependence of F on this socalled third integral seems very likely because for all type of populations the velocity dispersion 4 perpendicular to the plane of the Galaxy differs considerably from the .£ velocity dispersion in the radial direction in the plane (Oort, 1965). | The properties we shall accept as characteristic for flat stellar disks $ are best illustrated in the example of the disk population of the Galaxy. In fll the neighbourhood of the sun this population has a radial velocity % dispersion of 40 - 60 km/si , which is rather small compared to the averaged rotational velocity 200 - 250 km/s. , which in turn deviates only slightly from the circular velocity Rfi(R.). However, the characteristic motions of the stars in the solar neighbourhood are not sufficiently known to, provide a selection criterium for distribution functions. We, therefore, have to rely on theoretical arguments (or speculations). A possibility to find a selection criterion for the distribution functions is to study in detail the relaxation processess that have influenced the dynamical evolution of the galaxies leading to their present -^ form. This possibility was first considered by Lynden-Bell (1967). He * assumed that in the initial stages of the life of a galaxy a rapid relaxations cf the stell?r system has taken place owing to rapid large-scale fluctuations]? in the gravitational potential field. If it is assumed that these fluctuation^ are "violent" in the sense that they are comparable in order of magnitude witj some averaged gravitational field, one may argue that the memory of the stars ': -55- of their past history will soon be lost. As a consequence the relaxed state, obtained after the fluctuations are damped out, is expected to be one in which the degree of disorder is maximum (at least in the coarse grained sense). Lynden-Bell so arrives at what is called the Lynden-Bell statistics of stellar systems. The most probable distribution function turns out to be of the form: F°(E, p^) = Ae-aE+ ^6 , (3.69) where the factors A, a and 3 can Le determined from the total mass, the total energy and the total angular momentum. Distribution functions of this type and their macroscopic properties have been numerically investigated by Ng (1967), who introduced an energy Cut- off in order to include only bound stars. Although Ng succeeded in construct- ing exact self-consistent models for flat stellar disks none of these models satisfies the conditions typical for the solar neighbourhood. They are there- fore, to our opinion, unsuitable for the use of studying the properties of actual stellar disks. It has not been proven that no model at all of the type of eq. (3.69) can be constructed, which satisfies the solar neighbourhood conditions. Wa feel, however, that Ng's results contain a strong indication that the flat stellar systems actually observed are not fully relaxed in the sense assumed by Lynden-Bell. A study of partially relaxed disks was made by Shu (1969) who followed a similar line of reasoning, suitably modified for flat stellar disks. His most important modification is the assumption that- the only. relaxation, mechanism operative in a flat stellar disk is an axisymmetricvariant of the Jeans instability described by Toomre (1964). . Toomre showed that in order to avoid axisymmetric instabilities the velocity dispersion in radial direction < v_ > should exceed, a critical -56-- value given by = (0.2857)s 2TTGO/K, (3.70) ft where K is the epicycle frequency defined in eq. (3.23) and a is the surface % density of the stars. The critical velocity dispersion is small compared to g the rotational velocities in a stellar disk. As an illustration we have,for 'f 2 ^ Toomre's model 1 defined in eq. (3.66), plotted < v > / RK(R) as a t K crit %; function of R / R- (R is defined in eq. (3.56)) Sffefig. 3.15. r RK(R) 1 u 1 2 /- — R/R Fig. 3.15 The critical velocity dispersion in units of R K(R}J plotted as a function of R / ~R 3 for Toomre's model I, It is common practice to call a disk cold if the velocity dispersion Q ^ 1 where the often used parameter Q is defined by 2 Q = < v / < Vp >* . (3.71) R crit Stellar disks with Q values much larger than unity are said to be hot. For Q < 1 the disk is found to be unstable. The instabilities will usually cause ' -57- an increase in the velocity dispersion, or Q, thus leading to a relaxation of the disk. Shu (1969) argues that this relaxation process is mainly restricted to the disk of a stellar system. Once the velocity dispersions of the stars Uhece 2 i become larger than Clearly, since the detailed angular momentum is preserved no full relaxation of the disk can take place. Shu further assumed that the instabilities have been violent enough to bring the disk in its most probable state for a given detailed angular momentum distribution. With the above assumption Shu determines then the most probable state by minimizing the entropy of the stellar system. The distribution in this case has the form: aE F°(E, pe) = N(p6) e" , (3.72) where a is to be determined by the total energy of the system and M(p ) by the detailed angular distribution function. The next step then is to find self- consistent models for this type of distribution functions. In order to solve the integral equation (3.63) asymptotically for small velocity dispersions, Shu suggests a minor change in the form of the distribution function in eq. (3.71), F°(E, Pg) = P(pe) exp[- a(pe).(E --Eb)3 . " (3.73) In eq. (3.73) E is the energy of a star moving in a circular orbit whereas P(pft) and a(p ) are now assumed to be slowly varying functions of pg ... The only difference between the two eqs. (3.72) and (3.73) is the introduction of. the variation of a with p.Q. Shu tried to justify this change with the argument that the relaxation must be considered local rather than global and therefore one cannot expect a to lie'.a constant factor. Here the.term local is used to express the fact that for small velocity dispersions the orbits of the stars ' • '.'-. ' . '-SB- -• • • ' • •.'.'. : • v : \ " •- .•••.• ••• with a particular value of p are confined to a small region around R , the epicentre radius, defined for a given p by eq. (3.18). We feel that all the arguments presented by Shu to justify his special choice of the form of the distribution function are questionable. First of all, it is assumed that the relaxation is violent enough to bring the stellar system into the most probable state for a fixed detailed angular distrxbution function. That stellar disks with Q < I are indeed violently unstable has become clear from numerical experiments on the behaviour of large numbers of stars. However, the same experiments made clear that the relaxation can neit-ier be considered as a local phenometcnnor that the detailed angular momentum distribution of the stars is conserved in this process. And it 7 seems unlikely to us that a system after having experienced rather large fluctuations in the gravitational potential field , can survive as a rather cold system of stars. Consequently we think that no complete destruction of ^ the dynamical memory of the stars can have taken place, as suggested by Shu. The stellar system can therefore not be expected to be now in the most probable state for a given detailed angular momentum distribution. a The conclusion does not rule out that dynamical relaxation has taken (and still does take) place. But, such a dynamical relaxation must be rather ' mild in the sense that the amplitudes of axisymmetric (and non-axisymmetric) _~_] perturbations in the gravitational potential are small when compared to some S suitably averaged potential. If so, no full relaxation, nor partial relaxatic in the sense assumed by Shu, has taken place and the concept of a steady distribution function becomes doubtful. It appears not unreasonable to assumes!) in thin case that the velocity dispersions in the disks of spiral galaxies ai of the order of the critical value, i.e., Q % 1. With the exception of the solar neighbourhood no observational evidence exists to justify the assuraptil -59- that the disk population of spiral galaxies is indeed cold in the sense as described above, although it is commonly assumed (see e.g., Miller, 1974) that this property (Q £ 1) is one of the striking characteristics of a disk of stars. For this reason none of the exact models constructed by Ng or the one by Miyamoto can be considered as acceptable, The only alternative left over at the present moment is the modified Schwarzschild distribution introduced by Shu (eq. (3.73). In spite of the fact that it is not even an, exact solution and of the theoretical objections mentioned above, this distribution function has the practical advantage tlat it relates the functions PCpQ) and a(pQ) in D y an analytical form to observable quantities by a (R ) a (R ) 2 a(RQ) = 1 / The dependence on p is found with eq. (3.20). Although the use of the modified SchwarzschiLd distribution is thus in itself questionable we feel that a correct description! of the properties; of the macroscopic quantities like surface density and velocity dispersion, is of- far greater importance.in actual applications than the precise form of the distribution function in phase space. We shalL tharefoT-1 use the modified, Schwarzschild distribution function with -Q'•'=•! as defining, the. characteristic properties of an axisynanetric stellar disk- model • ::•: . ; .: . :-\ "•;•'•"; •. . •.. •, •'. •'.•,'.'•' CHAPTER 4 STABILITY OF SLIGHTLY PERTURBED DISKS 4.1 Introduction In this chapter we shall discuss some properties of a disk of stars that is not in a steady axisymmetric state, but deviates only little from it. The motivation for such a study is twofold. First, there is the purely mathematic- al reason that this assumption makes it possible to linearize the equations which describe the stellar system and thus to reach a considerably simplifica- tion. The second reason is more fundamental. As was argued in chapter 3 we r surmise that stellar systems are not in statistical equilibrium. Already Eddington (1921) has stated: "it becomes an interesting question whether the stellar system is in approximate dynamical equilibrium or not", and "if it is not we are led to the further difficult question how tL° stars can have existed so long without arriving at it". Eddington's second remark was based * on the assumption that any irregular distribution of stars and stellar velo- cities should be smoothed out by a process of phase mixing (Lynden-Bell, 1962). This process should lead to a steady state (at least in the coarse grained sense) on a time scale of the order of the revolution period of an average star, that is in a few times 10 years. -C During the last decade it has become clear that a kinematical consider^ atioB as cited above caa be very misleading and chat the dynamical effects o£| the irregularities play a very important role. We shall show in the followiiggi that, at least for small perturbations, the self-gravity effects act againstS the smoothing process in the sense that these irregularities can be maintain* during ? much longer time than one revolution period. This does not* of "? ^ rourse, mean that the system should never approach towards a steady state at all. What actually will happen depends on whether the disk of stars is stable or not. The disk of stars is said to be stable if all perturbations tend to smooth out in the course of time, if not, the disk is called unstable. The importance of a stability analysis is demonstrated in section 4.2 , where the behaviour of a perturbed disk in the linear theory is formally de- scribed. Having done so, we examine in section 4.3 the stability of the disk by considering the constraint that is imposed by conservation of angular momentum. We shall make it plausible that stellar disks are unstable against non-axisymmetric perturbations. The physical mechanism will be discussed as well as the astronomical implications. 4.2 Mathematical formulation. •- We start here from eq, (2.18). For a = 0 , A =£ 0 the gravitational potential V(R,G,t) is given by: V(R,0,t) =•• V°(R) + XV^R.G.t) ; A « 1 , (4.1 ) and.the Hamiltoniatican be written as H = H° + AV^R.G.t) , • •• . •. l-h; ;-.- :•.. " ••" (4.2) where H° is defined in eq. (3.2.,*. This suggests a. distribution function of the form 1 3 f = F° + Af (vRlYBtRjeit> . <4- ? ?°, 11° and V° describe a steady axi .symmetric disk which we shall call the un- perturbed state, the properties of which are discussed in chapter 3. For the v time being we assume that F° , and consequently H° and V° are known. Sub- |; stituting now eqs. (4.2 ) and (4.3 ) in eq. (2.14 ), using the properties of § the unperturbed state and neglecting terms of the order of X , we obtain the «, linearised Vlasov equation which describes the evolution of the perturbation | in the distribution function f1 , in terms of F° , H° and V : (4.4 ) The potential perturbation V1 is assumed to be generated by the perturba- tions in the surface density given by 1 a^R.Q.t) = m f dv J dvQ f . (4.5 ) According to Poisson's equation we find that V1 and f1 are interrelated by 1 1 yZy = 4TT G 6(z)m fdvp / dvQ f . (4.6 ) The equations (4.4 ) and (4.6 ) fully determine the evolution of the perturba-~ 1 tions once the initial perturbation f (R,G,v ,vn,o) is known. R D Although no explicit analytical solution of these equations in terms of *"' the initial perturbation can be obtained we can gain insight in its nature by - considering the formal solution of this set of equations. This we shall obtain^ as follows. We first consider the solution of eq. (4.6 ) by which V1 is ex- 4. pressed in terms of f1 . Subsequently, we shall obtain a solution of eq. «* $ (4.4 ) to express f1 in terms of V1 and its initial distribution. The two ar solutions will then be combined into one/integral equation (4.23 ), the solu- tion of which is formally presented in eq. (4.30 ) . 63 The solution of eq. (4.6 ) is best described by using a Fourier series expansion of V1 and f1 in B : 2? m=-°° 2? m= where the coefficients are, as usual, given by VR'C) - / dee'^v'CR.e.t) , (4.9) o 2TT o We find, after substituting equations (4.7 ) and (4.8 ) in the integral re- presentation of eq. (4.6 ) that the Fourier coefficients if and 1? are m m related through Vm(R,t) = -Gwj RMR« Jdv^ Jdv^ni(R'fv^,t) Hm(R',R) , (4.11) where 27T H (R'.R) = j d0 . ., , ••.. "• '...... •/ ;•. . : .. (4.1.2 ) o (R1 + R2 -2R'R cos G) ' Eq. (4.11 ) describes the gravitational potential perturbation V1 that is generated by the perturbation in the distribution function f1 . '. .. Our next step is, reversely, to determine .the perturbation in the distribution function f1 that is caused by a gravitational potential V1 (eq. 4.4 ) . In order to solve the linearized Vlasov equation it is most con- venient to use the action and angle variables J, and w^ , introduced in chapter 3, because by using these coordinates the equation of motions obtain their simplest form. We can write eq. (4.4 ) alternatively as (4.i3 u° - . - where —— is the comoving derivative in the unperturbed state. Eq. (4.13 ) ~ simply tells us that in the linear theory the perturbation in the distribu- i tion function is obtained by integrating (adding) the changes in F° due to s the perturbing potential V1 along the unperturbed orbits. Assuming the 3 potential to be switched on at the time t = 0 and expressing the Poisson J bracket in terms of the action-angle variables we obtain ~S t - j f!(t) = f^OJ+fdt1 |y-|£- ', (k =1,2) . (4.14)1 o In eq. (4.14 ) and below the summation convention is to be applied when re- peated indices occur in products. We have used the fact that F° depends on the J.'s only. With the equations of motion (3.17) we find t 1 fi(t) = fi(0) + j}~J dt (§£•) (J.,w.(o) + Ojt'.t') . (4.15) o i This solution can further be simplified by expanding V1 and f1 in a v double Fourier series in w. and w . •• f- , _ i(«.iWi + «,2w2) ^ V»(J w t)) = -L- I V (J t) e , (4.16 JT 4TT2 65 __ i(£iw. + \vz) I ttili (J.,t) e , (4.17 ) where v r r -HM«l -r x,2w2; 1 £i£2 (J^t) = ' dwj I dw2 e V (Ji,wi,t) , (4.18 ) r r -i(£iwi + £2w2) 1 (Jj.t) = j dWj J dw2 e f (Ji,wi,t) . (4.19 ) o o The double sum in equations (4.16 ) and (4.18 ) runs for both indices &i and %i from -°° to -H» , Inserting eq. (4.16-17 ) in eq. (4.15 ) we find after some manipulation that the Fourier coefficients fn „ and Vn „ are interrelated by £i£2 £i&2 0 " (A.20 ) With eq. (4.20 ) f1 is fully determined once V! is known. We now proceed to combine the two solutions equations (A. 11 ) and (4.20 ). To this end we note that the relations between the Fourier coefficients £ and V , and f. and V , respectively, =ire given by m in. . *>m xrD o , i{£w2 + mu(w2,J.)} f f J e M 21) m = i } ta< i.t> •• •• ... ' : ^ -i{£w2 + mu(w2,J.)} ~ • . " • • •„•• 66 where u(w2,J.) is defined in eq. (3.47 ) . Substituting now consecutively eqs. (4.21 ), (4.20 ) and (4.22 ) into eq. (4.11 ) we find after some rearrangements of integrations the following integral equation t °° V (R,t) = D (R,t) + f dt-' [ dR' K (R,R',t-t?) V (R',t') . (4.23 ) m m J I IT m oo The quantity D (R.t) which we shall call driving term for reasons to be m * explained below, is determined by the initial conditions and the dynamical properties of the unperturbed state, according to 4" V*'0 = " f^ I / dJl / dJ2 Hl (R'Ji} .,o) e (4.24 with 2IT 1 H (R,J.) = f dw2 e H {R,R'(w2,J.)} , (4.25 ~ -1- / m x o Hm(R,R') being defined in eq. (4.12 ) . The kernel K^R.R'.t) in t.ie second term on the righthand side of eq. (4.23 ) is defined by = - g I | dpR / dpQ Hi (R,J.) e mu(w2,J.)} m with H^ defined in eq. C4.25 ) . The dependence of the kernel on R is 67 m explicit through the dependence of H* on R , whereas the dependence on R' is implicit through the dependence of the J. and W2 on R' . Eq. (4.23 ) is not yet what we really wish to obtain, namely an ex- pression which gives the gravitational potential perturbation in terms of the initial perturbation in f1 . We formally can find such an expression by taking first the Laplace transform of both sides of eq. (4.23 ) , by which it is reduced to 00 V (R,U)) = / dR' K (RsR',to) V (R',10) + D (F.to) . (4.27 ) m j m m m In eq. (4.27 ) V (R,(J)) , K (R,R* ,03) and D (R,co) are the Laplace trans- forms of, respectively, V (R,t) , K (R,R*,t) and D (R,t) . The formal solution of eq. (4.27 ) can now be written in terms of the Green function G (R,R* ,ui) as follows V (R,w) = D (R.Ui) + I dR' G (R,R' (U) D (R1 ,0)) , (4.28 ) m m ] m m o where G (R.R* ,oi) is the solution of the equation m 00 G (R.R'.ui) = K (R.R'.io) + I dR" K (R,R'» G (R",R\w) . (4.29 ) m m J o in o Taking the inverse Laplace transform of eq. (4,28 ) we finally have t « V (R,t) = D (R,t) + f dt1 I dR' G (R RI;t-tl) D (R',t') , (4.30 ) m m j J Tdfn f m o.. o where G (P,R',t) is obtained by taking the inverse Laplace transform of Gn(R,R',u) . 68 The form of eq. (4.30 ) gives rise to a number of crucial questions ". which will now be examined. Among these is the old question whether the self- gravity effects of the perturbations are indeed important. Further, the effects of instabilities, if present, should be noticeable. In order to answer these questions we first note.that the solution for the gravitational potential perturbations consist of two parts, to be under- stood in physical terms as follows, i) The kinematically evolving initial perturbation in the distribution function, that is the convection of fx(o) along the unperturbed orbits, causes a perturbation in the gravitational potential. This contribution to V1 is described by the driving term D (R,t) , the kinematically evolving part of the perturbation, ii) The driving term D (R,t) itself causes a perturbation in the distribution function, which in turn will cause a perturbation in the gravitational potential. This contribution to V1 is described by the second term on the right hand side of eq. (4.30.) and expresses the self-gravity effect of the perturbation. In contrast tc i) this term may- be called the dynamically evolving part of the perturbation. From this physical interpretation it is clear that if we had neglected the self-gravity effects of the perturbations the evolution of V1 would have been fully determined by the kinematicaliy evolving part. From the definition of D (R,t) in eq. (4.24 ) it is clear that this contribution ~ will, by phase mixing of the various components in the series in eq. (4.24 ), fade out in a time of the order of one typical rotation period. There are -*• exceptions for m = 2 , I = -1 and for m = -2 , 2. = 1 . The phases of these terms in the series on the right hand side of eq. (4.30) are, for near- ly circular orbits, proportional to Qo - KO/2 , which in actual galaxies is 69 approximately constant over the disk. Consequently, these terms can persist over many revolutions, as was noted by Lindblad (see section 1.6) already in 1957. The relative importance of the kinematically and dynamically evolving parts of the potential perturbations depend on whether the disk is stable or not. In order to show this, let us assume that the function G (R,R',U)) is analytic in the lower half plane of the complex co-plane, except in a finite number of isolated poles OJ, . We can then write: m io), (t-t1) G (R,R\t-f) = I gk (R,R') e k r i(l!(t-t') + I dm e G^R.R'.u) (4.31) m where the g (R,R!) are the contributions of the residus at the pole u = oj . The dynamically evolving part can now be separated into two terms: ito, t r m , -in) t' I e K I dR1 g (R,R') dt' e * V*''1^ ; (4.32a) k ' ' and f 10JC , . , -lWt . 1 ! I du! e j dR Gm(R,R',u) / dt e D^R'.t'). (4.32b) o Because the poles were assumed to lie in the lower half plane, of the com- plex u-plane, we have In (is, < 0 . Consequently, the first term ir» eq. (4.32) increases exponentially in time and will stand out from the 'kinefliatically evolving term as well an from the second term in equation (4.32j. Therefore, ft«r some initial transient period the solution eq. (4.30) becomes in good 70 approximation oo ioi t r m V (R,t) - I e dR' g (R,Rf) D (R1 ,(i ) , (4.33) m , j K m K. where it is assumed that the Laplace transform of D(R',w) is analytic in the lower half plane of the complex U)-plane, which for any reasonable initial distribution certainly will be the case. We thus see that if the function G (R,R' ,(i)) has poles in the lower plane of the complex w-plane, the dynamically evolving part contains a number of exponentially growing perturbations. In other words, the stability properties of the disk are fully described by the behaviour of the function G (R,R*,u) in the lower half plane of the complex cQ-plane. The properties of G (R,R',(JJ) depend only on the properties of the stellar disk in the un- perturbed state. The solution (4.33) is often said to represent a sum of un-~ stable modes, the relative importance of each of which is determined by the initial perturbation through D CR' ,u), ) . Clearly, if the disk is unstable, the kinematically evolving part of the potential perturbation can be totally disregarded. If, on the other hand, 0 (R,R', half plane of the complex to-plane, i.e. if the stellar disk is stable, the dynamically evolving part is determined by the second term in eq. (4.32) . In this case this part will only dominate the kinematically evolving term Dm(R,t) if the latter damps out rapidly in time. If so, the solution of eq. J (4.30) will be given by J fv +°° . oo ( not ™ ..«, V (R>t) 1 ! 1 m = J do) e j dR Gffl (R,R ,u>) Dm (R ,u>) , (4.34)", *J provided G (R.R1 ,ui) and D (R' ,o)) are regular functions at the real axis of the complex U)-plane, The perturbations are now fully selfsustained but will sooner or later damp out by phase mixing. In our discussion on the kinematically evolving term, D (R,t) , we have seen that in actual galaxies some components dominate over all others and can persist over many revolutions. In this case, the kinematically evolving part and the dynamically evolving part are equally important. The perturbation shows then a rather complicated behaviour in time, but also in this case, one can expect that the perturbations will damp out in the long run. It follows from the above discussion that a knowledge of the inherent stability of the stellar disk is of primary importance for the study of the behaviour of perturbations. The function G (R,R*,u)) contains in principle all information to investigate this stability in the linear theory* It is wellknown that the determination of the poles of G (R,R' ,w) is equivalent to the determination of the eigenvalues of the homogeneous equation of which G (R,R* ,ui) is the resolving kernel, that is, of eq. (4,27) mthout driving term: ••,• -•_•.-.. _ •. •, . • ••-._, .- ; ,90 7? V = / dR1 K (R,R» ,05) V (R', . (A.35) m J mm Eq. (4.35) represents an eigenvalue;problem with eigenvalues,, say,\-•..!!),_ and eigenfunctions V (R) and is sometimes referred to in the literature as the mode problem (compare e.g^ Shu,. 197Qa). ."-.. . " ••••'•...... ••••.'• . ;, \ Equations and solutions, of .this kind occur, in many branches of mathe-: natical physics, which make6. it. simple to use these common terms. Th« eigen- frequency u. .is (for a chosen value of m). the only frequency, or: one. of a few frequencies, in which the system described by;the integral equation can 72 perform a self-sustained motion that is periodic or exponential in time. The corresponding eigenfunctions in the general equation describing the system then give the full solution of all relevant physical quantities. Such a "permitted state of motion", corresponding to a particular eigenfrequency a), , usually is called a "mode". Often the modes have a wave character. A km mode in which a violin string can vibrate is in this terminology at the same time a standing wave. In order co describe a mode in a more physical way, we recall that V (OJ) is obtained by expanding V1(R,8,t) in a Fourier series in the azi- muthal angle 6 and subsequently taking the Laplace transform. A mode solution of eq. (4.35) with eigenfrequency ui, thus implies a contribution of the form i (u. t + m0) Re { V. (R) e } . (4.36a) km ' Writing now V. (R) = A, (R) exp { i $, . (R) 1 and a), = v, + iv, tccc tan l km ' km km ran this form can be expressed as Akm(R) e cos {vkmt + m8 + ^(R)} . (4.36b) The form (4.36b) shows that the mode solution represents a twodimensional ~ wave that propagates in azimuthal direction with an angular velocity V, /m,, ~ often called the pattern speed, and in radial direction with a phase velocity^ If » where k(R) is the wave number in radial direction defined by i . Tne solution exhibits a spiral structure, determined by the curves of con stant phase 73 m6 + 4, (R) = constant , whereas the number of spiral arms is determined by m , the "wave number" in angle 0 . The spiral waves here described for the gravitational potential are, of course, associated with corresponding waves in the surface density distribu- tion, and are therefore usually called density waves. Up till now, very little progress has been made to solve the integral equation (4.35) . Only one. numerical attempt has been made (Kalnajs, 1970) in which it was found that tht fastest growing mode in the case of a two- armed spiral (m = -2) has a frequency u> = 60 - 2i km/s.kpc „ This example, in which the observed rotation characteristics of M 31 were used, suggests strongly that the stellar disk is unstable. A less complete, but better known study of the properties of density waves was carried out by Lin and Shu (1964) and by Shu (197,0a,b) • It i& ;.'.-, difficult, however, to connect ^b°ir work with the mode problem described . above. In their analysis Lin and Shu aimed to determine, with the help; diE a WKBJ-like method,, a dispersion relation describing the dependence of the,; ..'• . -,-V:-' ~~t«-; Mii:.?^^-:^"itf^E^fefe..M^ wav,,—e numbe_._wr— i_n --radiaj;-1l direction,: k;vyon the: ifcaal); fte9ug"Bc!y~-ridV*»v-Spi^Csn>.iVi;j.";';;'-f' approach always leadsi to a continuous spectrum (Verhulst, 1973). and therefore can not yield a solutio:ion to the eigenvalue problem..We; ci.n;'only aiirmise.that their solutions, when combined for a whole range of (real). frequencies, de- scribe the second term:of, the dynamically evolving' part of'.the., distribution,.: function in eq. (4.32)..:.The- ; work'of Lib and Shu differs considerably fro^tb^ 1 approach used in the present study, because our main.::interest lias in the :: ; stabilitstability of the diskdisk-,,' that is, iri: the^igenyaluesthe eigenvalues;: y-"^isj.^-^ i.,ao;..and npnot Ih the.:the .. .; • ; firsfirst place in the eigetrf-anctionsk .- '•'} ."•.... •.;;• './-'.: ^ :•;' • ._..-• '•:.:. • / ... " . •;:' ; A direct determination of the eigenvalues u), is not possible by ana- i lytical methods. Therefore we shall, in the next section, present a different ^ method to study the stability of the disk. 4.3 Instabilities. 4.3.1 rftle_rate of_change of angular momentum. Some insight in the stability properties of the disk is obtained by con- sidering the constraints that are imposed by the conservation of angular momentum and energy. Because in the present context conservation of angular momentum implies conservation of energy (and vice versa) we need only con- sider the constraint imposed by one of these conservation laws (Kalnajs, 1970). We shall therefore consider only what consequences for the perturba- tions follow from the application of the conservation of angular momentum (Dekker, 1974). For this purpose it is useful first to consider the situa- tion in which the distribution function at time t = 0 is unperturbed, f (o) = 0 , but where the gravitational potential experiences, by an arbitrary external or internal cause, after time t = 0 the perturbation V^R.Ojt) . In this situation, and given a suitable form of V1 we shall calculate the rate of change of total angular momentum. Having completed this, we shall return to the problem of self-sustained gravitational potential perturbations in which case the total angular momentum is con- served. We assume that the gravitational potential perturbation has the wave-lii form: ra6> 75 where 0) and V (R) are assumed to be complex. TO We can write the rate of change of total angular momentu., if action-angle variables are used as ..tot 21T ** dJ, £—- - m JdJ, / dJ2 / dw, / Av2~£ U-.w^t) (4.38) where m is the mass of an individual star* Usiag now Hamilton's equation of motion dJl dt~ and subsequently substituting eq. (4.1) and (4.3) in eq. (4.38),; we find up to second order of A : .•'.' . , ., dJtot 2'ir 2ir " .-.:•..- ^: -~- = m /dJ, / dJ2 / di o A further reduetioii-of eq.' (4.40) xs-'Q^&ix^iS^:^^**^^ •« eq. (4.16-17) in e<|v (4.^^^d^aj^gr^^l; For a gravitational potential "perturbation!'''''I'll in '.lltthl 'e^ '• -^'torn1 - ;• of.;e(a>;A4-39) we C«B^' • •' write • ' '':\':^'M:?;K':A-\^:''''!:lr y'-,*i^:'/-:^::''^'^'v-?'':r\ ••.':'•'••' •••*•••• •••--I'jJ-t + a 5 . n '••£•••• -:• 76 J -i { &2W2 + mu (w2,J.) } Using eq. (A.20) we find that f. , (J.,t) can also be written as with 42., 9F°/3J^ , II -i («..«,k k, + w) t Substituting now eq. (4.44), (4.45) and (4.42) in eq. (4.41) we arrive at dJ. __ =, iZ5Z5i exp [ -2 Im(tu)t ] Im { G(u) + H(a),t) 1 (4.-46; where m 6F°/aj. + £ 9F°/9J £" ' and 3F° X . 3F° m 37^+ l 37; In eq. (4.46) we have separated the contributions due to dX. / dt into ~! two parts: the first part G(u)) describes the rate of change of angular momentum due to the force excerted on the stars, the second part arises 77 natural oscillations that are excited at the same time. In order to exhibit the time dependence of the latter part more clearly we write 1St exp [ -i (rafii + m2) t ] = / ds e 6 (s + mfti + m2) • (4.49) -co Upon inserting eq. (4.49) in eq. (4.48), changing the order of integration and using the property of the s') f(x) = f(0) 6(x) (4.50) we can write H(u),t) in the form r -i(d)-s)t e H(u,t) = - J ds (cu_s) g(s) , (4.51) where g(s) is defined by 2 g<"> = £l J «, f dJ2 { n gl ,1 gl HaJ Us * tf> - Uh) . * l 2 (4.52) Eq. (4.46), combined with equations (4.47) and (4.51-52) contains the answer to the initial problem we have, posed.' We now introduce the additional' assumption that the .perturbation in the potential is maintained by. the stars in the:disk itself. For such':seU-.' •••'••;;•:., sustained perturbations the:total:angular momentum isrcc.iserved and.the right hand side of eq. (4.46) must reduce to zero. Putting this constraint on' the.; perturbations enables us to derive an expression for the. inaginpry part of • .. Lhe frequency , Im(u) , in the;litnit of aiaulj varyirg amplitudes. In deriving this expression, the:behaviour of the ril&St hand Bide of e!q. (4.46) is:con- sidered in the time-asymptotic limit, where.the expression for growing waves differs from that for damped waves. Therefore we shall discuss these two cases separatively below (section 4.3.2 and 4.3.3 respectively). Although the analysis is different for growing and damped waves, the resulting expression for Im(w) turns out to be the same. The physical significance of this ex- pression is discussed in section 4.3.4 and some astronomical implications are mentioned in section 4.3.5 . 4.3.2 Qrowing_wayes. Growing waves are defined by Im(u)) < 0 , that is, we have a perturba- tion which increases exponentially in time. For such waves the function H(u,t) (eq. (4.51)) decays exponentially in time as exp { Im(u))} t so that after an initial transient period we have from eq. (4.46) : ..tot I ( 1 — - 2{lm(o))}t Im{G(w)} . (4.53) This expression was derived earlier by Kalnajs (1971). Conservation of total angular momentum now requires Im { G(ID) } = 0 , (4.54) where the frequency to is no longer arbitrarily but fully determined by the mode problem discussed in section 4.2 , that is, u) is one of the eigen- frequencies u tan ir Eq. (4.54) can also be derived from eq. (4.35) by multiplying both sides'^ by R a^ (R) md integrating over R ; a (R) is obtained by integrating eq. (4.10) over velocities, The result then is uu / RdR cj (R) Vm(R) = G(ID) (4.55) 79 1 and, since the left hand side of eq. (4.55) is real, we recover eq. (4.54) . In other words, if a) is an eigenvalue of eq. (4.35) and V (R) the k corresponding eigenfunction (with the a. 's determined, according to eq. (4.42), from V^(R)) > ecl» (4*54) is automatically obeyed. Although we thus have found here nothing but a confirmation of the con- servation of total angular momentum in the linearized equation (up to order •2) we still can use the property (4.53) to gain further insight in the problem, because it enables us to obtain an expression for the growth rate, ) , of the waves. km To this end we define the function dJ dJ 1 = £f J l / J22 0, + ^+in, j-toWl • (4.56) According to eq. (4.54) we thus have Im { G, (w. ) } = 0 . (4.57) km km - Writing now a), = d)° - ^Y^ > where &*£ and y, are real, we can ex- pand the function G, (OL ) around us,1 if V, /la? « 1 as! 1 km kin k-*i Km KI3 : 3G. km Since, according to eq.. (4.57) the imaginary:.p'art of 'either side af eq. (4.58) must be zero ve obtain ir. first order of y^_ ••'•'• •••:[ .; L . -. -;. ; 7 ••.-. ;."• ,;' •.;..; .:•.•.••;.•-.'•• ., • -/(4.59b) ^ :.••••• -•_ • ;• , • .. , • • I'.".,•••• '' • ' ••":. : • ' ' •.••. •.•" •• .'• •"•••. • •••'•• ••••,•:• ao-;: •..;-.-•• .•• • . •• • .••'••• ,• •••• • ••••'. The prime in eqs. (4.59a,b) denotes the derivative of G with respect to its complex argument w . For simplicity of notation we shall drop the indices of u>° and y , but keep in mind that the analysis giver in this subsection only applies for those values of OJ° and y which correspond to a particular eigenvalue u) Explicit expressions for the numerator and denominator of (4.59b) nay be found: by means of Plemeljs symbolic formulas (Montgomery and Tidman, 1964) 1 Lin 1 ' ^ 1 — -c-/ »\ ///-rt\ 1 _ n 7 r =—v = P T —rv + TTI 6(x - x ) . (4.60) Y - 0 (x - x1 + xy) (x - x') These formulae are called symbolic because they have a meaning only after multiplying them by a function of x' and integrating over x' . In eq. (4.60) the symbol P denotes Cauchy's principal value. As an illustration fig. 4.1 shows the real and imaginary part of the function (x - x1 - iy) for small y . Taking now the limit for Y * 0 and equating subsequently the imaginary parts on both sides of eq. (4.56) we obtain, applying eq. (4.60) : m Y 17 £=- Since G^m(o)) as defined is an analytic function, we may write the uenomina- tor of eq. (4.59) in the form which becomes upon applying eq. (4.60) -C' 81 T~l I I ! I I I I I I I I I I I Refx-x'-iy]"2 - % III I I [ i 1 I I I I I I i 1 I I I I I I I I I I I [ 1 I x-V x-Y Fig. 4.1. The real and the imaginary parts of the furcation (x - x1 - iy)~ for small y together with the function (x - x1) as a function of x' . Fig. 4.2. The real part of the function (x - x* - iy) together with the —2 ' '•'"'• '' ' function (x - x1) as a function of x" . >; (a) ktm ) r - ' • •• ••.. ... -;,v(A.62) dJ (a)0 The further reduction of eq.. (4.62) requires special care,,because a naive interchange of differentiation and integration'in this equaripn leads to. • nonsense, due to the occurrence of the principal .value of ths integral. This rroblem can be avoided by using a different definition of the Cauchy..; 82 principal value (Van Kampen and Felderhof, 1967) which is fully equivalent to the traditional one (see e.g. Whittaker and Uatson, 1969). The definition given by Van Kampen and Felderhof is obtained by adding the two eqs.(4.60) lim (4.63) (x - x') 5 y * 0 I (x - x' + iy) (x - x1 - iy) and leads upon differentiation to a new concept, the "principal value for an integral along the real axis whose integrand has a pole of the second order", -; symbolically written as * = i lim _-, + ^ . (4.64)1 ,72 2 y + 0 „—, T (x - x1) (x - x1 + iy) (x - x1 - iy) We have added the asterisk in the notation in order to warn the reader that -J ii- this definition does not coincide with the traditional one. f- • I With this definition we may now exchange the order of differentiation and ;-| integration in eq. (4.62) which then reduces to fo)} £ I | f2.(4.65) km 2 ; ' j ° S Ufa)2 -2 -i Fig. 4.2. shows the real part of the function (x - x' - iy) and the - _2 x "" function (x - x1) to demonstrate that by taking the principal value P as defined in eq. (4.64) instead of the traditional one, the divergent con- -' tribution near x = x' is removed. <- The final growth rate y is found from eq. (4.59b) as -*- 11 dJ J m + £ a 2 6 J. J , ]-d 2 1 ^~ 3T-) ! oJ fao + mHj + £fi2) % y = ^- (4.66)* / dJi / 83 ? Eq. (4.66) holds only for non-axisymmetric modes (m ^ 0) , because any ex- ponentially growing axisymmetric mode must have Re {OJ} = 0 (Kalnajs. 1971), which is if compatible with our assumption that -^j- « 1 . 4.3.3 Damp_ed waves• In order to discuss damped waves, we return again to eq. (4.46). Whereas the function H(u),t) between curly brackets on the right hand side of this equation was in the long run of no importance for growing waves, its con- tribution certainly remains important for damped waves ( Im(u)) > 0 ) and may even exceed that of G(co) . If we may assume that the function g(s) , appearing in the integrand of eq. (4.51), is a rather smooth function of s , and has an analytical continuation into some strip 0 < Ira(ai) < a in the upper half of the complex s-plane, and a > j Im(o)) | , wa can rewrite eq. (4.51) as , e H(u,t) = 2T7ig(to) + / g(s) ds . (4.67) The second term on the right hand side of eq. (4.67) decays as exp - { 3 - Im(o>) } t so that in the time-asymptotic limit we have H(ui,t + °°) = 2-n ig(u) . . •••-.. (4.68) Conservation of angular momentum for damped waves is therefore expressed by Im { G(oi) + 2-n ig(to) } ,.= 0 . . / ••'".'• ''"'-•: •...' / '••.--. ,:;•.• •• -.._..;• .C4.69) It is easy to see that, when following the sane expansion procedure of the. term between curly brackets as was done for growing'waves, that eq. (4.^6). is. recovered. Therefore, eq; (f.66) applieR to both growing aiid daaped wave*. The constraint on the damped wave obtained here (eq. (4,69)V differs from the constraint that follows from the integral equation describing the mode problem (see eq. (4.55)) by the term Im { 2TT ig(to) } . In a normal mode analysis this term is not considered, because such an approach takes into account only the "forced" oscillations imposed on the stars by the wave and not the "free" oscillations that are also excited. (The excitation of free oscillations always occurs at the natural frequencies of the physical system. A simple example is found in the case of an external force acting on an harmonic oscillator.) Therefore, the frequencies considered here need not ,: refer to the particular eigenvalues of the mode problem, u), . The damped ,; waves considered here do not refer to mode solutions. This is once more reflected in an apparent contradiction between the result obtained here and the prediction from the mode analysis, which states that the eigenvalues of eq. (4.35) do lie opposite to each other with respect to the real axis (Kalnajs, 1971). From our formula (4.66) it is found (and this will be dis- cussed in more detail in section 4.3.4) that for a given value of Re(w) the . value of Itn(iD) is either positive or negative. The prediction of the mode - analysis is a direct consequence of the invariance of the Vlasov equation for > time reversal. However, the time-reversal argument does not apply to the damped waves considered here, as can be explained as follows. The derivation of eq. (4.69) is based on the assumption that the function g(s) is ana- lytical in the upper half plane of the complex s-plane, which implies that the free oscillations give rise, through phase mixing, to an exponentially .$ decaying contribution, when integrated properties are considered. The phase %- mixing thus specifies an arrow in time for the macroscopic quantities of the "% 5 system. The perturbation in the distribution function itself does not decay R "i exponentially but is a rather weird function of time. Therefore, the motions H of the stars, which are themselves invariant to time reversal on a micro- J scopic level, lead to an irreversible behaviour on the macroscopic scaie, and the time reversal argument does not apply to the damped waves considered here. Although the above argument explains the apparent contradiction between the normal mode analysis and the formula (4.66), it does not clearly reveal the nature of the damped waves. We may possibly assume that'the damping here under discussion is equivalent to the Landau damping known from plasma physics (Van Kampen and Felderhof, 1967). It is certain that our description of the damped waves is yet far from complete and a more rigorous analysis apppars desirable. 4.3.4 In discussing the physical significance of the expression (4,66) for Y we shall refer from the onset only to growing waves. This eases the termino- logy and the discussion for damped waves would not be essentially different, The rate of change of total angular momentum for growing waves is given by eq. (4.53). Using the expansion made in eq. (A.58) we obtain dJtot • > _L_ = 1-S.l <**I o { ^.j j+ ig y e2Yt pp_{ ^.j } + 0(y2> . <4.?0) The expansion divides the contribution of the stars to the rate of chan&e of total angular moiren.tum into two parts; The first cenn on the right hand rtide of eq. (4.70) was fo.ind earlier by. Lynden-BeH and Kalnajs (1972). These authors studied the rate of change of angular momentum of the scars owing to an "external" gravitational potential perturbation iu the Hair of stationery amplitude, applying a.'method different from the one ve used in section 4.3.J, ^\ey found that this, term could be interpreted «s the result.of rescnant pheiiomena. Resonant effects ;:do-occur whene.ver periodic forces *ct en A. star. ...... » " .•"..: In a stellar disk the condition for resonant effects to be important is given by + 2 ( J J 1 coo + mfii O]tJ2) ^ - l' 2' *" ° ' where m is the number of spiral arms and Qi and fi2 are the "general- ized" rotation and epicycle frequency as used in eq. (3.17). This condition simply expresses that resonances do occur whenever the ratio of the doppler- shifted frequency coo + mfij and the epicycle frequency is a rational number. We shall call the stars for which the above condition holds, the resonant stars. Parenthetically, we note that for stars moving in nearly circular orbits the condition reduces to co + mfio + £ where the conventional rotation frequency f2o and the epicycle frequency KQ • have to be considered as functions of R , the epicentre radius. Lynden-Bellj and Kalnajs (1972) showed that some of these resonant stars lose and some of them gain angular momentum, in such a way that a net exchange of angular momentum between these stars and the perturbation takes place. For a thorough discussion on this interaction we refer the reader to the paper cited. In a linear theory, the rate of change of angular momentum of resonant stars can be expressed in terms of the gradients in the distribution function at exact resonance, which is expressed by the 6-function appearing in eq. (4.60). In the limit of slowly growing gravitational potential perturbations, not only- the resonant stars contribute to the rate of change of total angular momentum but so do the non-resonant stars, i.e. the stars for which the condition - to0 + nfli + l&z * 0 is not fulfilled. This contribution is expressed by the second term on the right hand side. of eq. (4.70), as is clear from eq. (4.65). By taicing the principal value of^; 87 the integral in eq. (4.65) in the sense as defined in eq. (4.64), the "divergent" contribution from the resonant stars is removed. Hence, for an arbitrary force, with period to0 and an exponentially increasing amplitude + -jr- Mar/ \ir) + o(Y -) . <4.7n res.stars non-res.stars If the periodic force is caused by a density wave maintained by the stars themselves, eq. (4.69) still holds, but conservation of total angular momentum implies: \dt res.stars v ' non-res.stars The non-resonant stars normally form the majority of the star population and regularly pass through the crest and valley of the wave in such a way that the wave is self-consistently sustained. Therefore, we shall regard, per definition, that it is the rate of change of angular momentum due to the non-resonant stars that essentially describes T'the rate of change of angular momentum of the wave .-..•' /dJ, \ /dJ. \ . •• . • (ar) ' HP )••- ••••.• .(4-73) non-res, stars / : •••:••••• Although this definition of.the total angular momentum of j:he wave nay. seem to be artificial, it is the only one,, in our opinion, which is physically, acceptable in the linear theory. The resonant stars do behave rather, singularly and.are therefore better treated as a separate subsystem. From eqs. (4.72-73) it eppears that:for slowly grcwing perturbations conservation of angular momentum implies that the angular momentun lost (or gained) by the wave is equal to the angular momentum gained (or lost) by the resonant stars res . stars Still with the assumption that y > 0 , upon integrating the second term on the right hand side of eq. (4.71) thus dJ /dt , from -°° up to t and assuming that 3 ( -°°) = 0 , we arrive at the following expression for the total angular momentum of the wave, up to zero order in y : 2Yt j = JE. e Re { G,' (0)°) } • (4.75) wave 8TT l km ' Note first that this can be positive or negative, depending on the actual value of to , and secondly that we obtain a finite value in the limit •y - 0 . Using eq. (4.72) the expression for y can be written as (dJ|/dt) -• - ~ res.stars _ (4_y6) J wave In other words, y is simply determined by the angular momentum gained or lost by the resonant sttrs at the expense of the wave, measured as a fraction of the total angular momentum of the wave. 4.3.5 As tronomical_im2lications. Although eq. (4.66) throws no new light on the existence of eigenvalues,' it provides plausible arguments why the real part of whatever existing eigen- value should lie within a limited range of frequencies. 89 We shall in the present section examine the question to which frequency intervals growing waves, if any, are confined in actual galaxies. The application of eq. (4.66) is restricted to non-axisymmetric modes. We shall further restrict the discussion to two-armed spirals (m = -2) . It is clear that eq. (4.66) (or (4.76)) permits two types of growing * wave s. i) Waves with negative angular momentum for which the net effect of the resonant stars is to absorb angular momentum. ii) Waves with a positive angular momentum for which the net effect of the resonant stars is to emit angular momentum. In order to determine the frequency intervals for which growing waves, if any, can occur we shall determine the sign of y as a function of ui° from eq. (4.66) . Me confine our discussion to cold stellar disks, that is to Q *» 1 (see section 3.3 for a definition of Q) . For such disks we may replace in eq. (4.66) Q\ by Q,^ and £^2 by K0 . Moreover, we may assume that which implies that for £ ^ 0 , the sign of the different terms in the sum of the numerator of eq. (4.66) is determined by the sign of -i. ( f>F/3J2 is negative for all reasonable stellar disk models). In actual-applications only the terms for £ = 0 •, _+ 1 , appearing in the sums in the numerator and denominator are important. Therefore; only these terms will be consider- ed here. . . ' • .' : •' .. • '-..-'''•• ~ •"• : ••'" •'• We define Rrrt as the epicentre radius-of those stars for which . ^° = 2J2 R (the. corotation resonance, m.a. -2 ;..S, = 0) , R™ . as the epi- centre radius of the stars for which <*)" = 2fl RI}J - K RJS (the inner 90 Lindblad resonance, m = -2 , I = 1) and R as the epicentre radius of the stars for which u° = 2S1 R UT + K0 P-OUT (the outer Lindblad resonance, m = -2 , I = -1) . Fig. (4.3.) shows RCf) , RQUT and RIN as a function of (JJ° for our own Galaxy, where we have assumed that the rotation characteristics are those as given by the 1965 Schmidt model. The participa- tion in the wave is limited to chose stars whose epicentre radius R falls roughly within the range (R , RonT) , the socalled principal range. Out- side this range the stars can not sustain the wave, since in order to main- tain it the doppler shifted frequency, here u)° - 2fi0 , should exceed the epicycle frequency Ko . This is because self-gravity effects always act to decrease the epicycle frequency Lynden-Beil and Kalnajs (1972) showed that for the cold disks con- sidered here, the resonant stars in the outer region of the galaxy (i.e. at the corotation av3 outer Lindblad resonance radius) absorb angular momentum, whereas the resonant stars in the inner regions of the galaxy (the inner Lindblad resonance radius) emit angular momentum. In order to determine the sign of the numerator, that is (dJ/dt) , in eq. (4.66), we now observe that for low enough frequencies (but of course sufficiently large to permit a wave), say OJ° = 25 km/s kpc , R,,- and R-,,™ are both very large. Since the surface mass density is very low in these outer parts, the contribution from these outer resonances to (dJ/dt) res. will be small compared to the contribution from the inner Lindblad resonance,* henre (dj/d1:) < 0 . However, if we increase the frequency 0)° R.,-. rt s. CO ifid RQJJT move rapidly inwards and the contribution from the outer resonance becomes more and more important. For sufficiently high frequencies we can ex- pect (dJ/dt)reg > 0 (see fig. 4.3.) . Therefore we can roughly divide the frequency range into 91 a) a range U)° < (1)1 ° for which (dJ/dt) < 0 b) a range to > uii for which (dJ/dt) > 0 . i—r~ ! I I -i-1 • I "I • I •] ••• M33- (kpc) (—) >0 - / BS 10 - < \ \ 8 \ - \ \ \ \ \ 6 J s. _ R -Rout - -A— in \ \ 4 + \ 0 — \I—Rco \ - 2 • ! I I i I, , I I , V 0 20 40 60 80 100 120 u.0'-m/s kpci 20 40 60 80 w° Fig. 4.3. The epicentre radii i?Jff .- RQUT and RCQ (see text for defini- tion) as a function of u>° for the Galaxy. The sign of the ex- change of angular momentum between resonant stars and the wave at the different resonances is indicated as well as the sign of the contributions of non-resonant stars to the total angular momentum of the waoe for different ranges of the. epicentre r-adius RQ of these stars. Fig. 4.4. Same as Fig. 4.Z. for MZZ . Another division of the frequency range can be made with regard to the bign of J , that is the denominator of eq. (4.66) - As also shown by 6 wave ' -..-.. ••" . . - , Lytiden-Bell and Kalnajs (1972), stars with epicentre radius RQ < \Q give * negative contribution (AJwaye < Q) to the angular wmentum of .the wave, 92 • ' • • . • • •.. • • whereas the stars with epicentre radius RQ > R give a positive contribu- tion (AJ > 0) . For low-enough frequencies, say again U)° = 25 km/s kpc, wave the bulk of the non-resonant stars will have Ro < R and the angular momentum of the wave will be negative. However, if Ui° increases, RCQ moves inwards (see fig. A. 3.) and the contribution from stars with RQ < R decreases whereas the contribution from stars with RQ > R^_ increases. For sufficiently high frequencies J becomes positive. Hence we can now divide the frequency range into a) a range 'o0 < 0)2° for which J < 0 b) a range U)° > 0)2° for which J > 0 . We can now in principle distinguish the following cases: i) u)i° < (1)2° . Growing waves of type I can occur in the ii) uii ° > ui2 . Growing waves of type IT can occur in the range W2° < U)° < OJj° . % Clearly, in either case, growing waves can exist only in a limited range of •? frequencies. To convert these statements of principle into quantitative predictions re- quires very detailed calculations, which have not yet been performed. A first, rough impression may be obtained as follows. For our own Galaxy the value of the frequency u° , quoted to be the best fit to the observations (see e.g. Lin, 1971 and references cited there) - 0 is 25 km/s kpc (for a two-armed spiral the pattern speed £2 = -a- = P L = !2.5 km/s kpci . We have already shown that it is very likely that for this frequency J < 0 and (dJ/dt) < 0 . Such a wave must therefore #3v6 resrss. be damped. Growing waves can be expected to occur, however, for much higher pattern speeds (around Q = 30 km/s kpc) than the commonly accepted value 93 ^» •\*r mentioned above, This is in agreement with a prediction made by Kalnajs (1970) based on his numerical study of M31. As a rather different example we have drawn in fig. 4.4. RT^ , R and R as a function of (J)° for M33 . The rotation characteristics of this galaxy are those given by Shu et al. (1971). All frequencies are again given in units of km/s. kpc. From this figure it is clear that o)i° < 10 whereas one can expect 30 < W20 < 50 . Hence growing waves of type I (u)i° < (^2°) can be expected to occur somewhere in a range 10 < u)° < 40 . Clearly, in this galaxy growing waves may occur. Although we have not shown that growing modes do exist in flat cold stellar disks, the range of frequencies for which these waves can occur, to- gether with the numerical result obtained by Kalnajs, strongly suggest that unstable modes can indeed occur in cold flat stellar systems. 94: CHAPTER 5 STABILIZATION OF DENSITY WAVES BY THE GAS 5.1 Introduction. The conclusions about the stability of stellar systems reached in the j previous chapters only apply to cool systems as defined at the end of section 3.3. Results confirming the linear theory for cool stellar disks have been obtained by numerical experiments, in which the evolution of a disk-like distribution of a large number of stars was followed by step-by-step integration of the individual equations of motion (Miller and Prendergast, 1968; Hohl , 1972). In these experiments a spontaneous growth of spiral structure was obseived to occur in disks with a velocity dispersion equal to Toomre's (1964) critical value. However, it was also observed that during the ensuing evolution of the disks the stellar velocity dispersion increased considerably and that subsequently the spiral structure again disappeared. Much attention has been given to this "heating phenomenon" in the experiments (cf. Miller, 1970) and special experiments were carried out to elucidate it (Hohl , 1972). In a very recent study Miller (1974) concluded that the heating effect is not due to artificial numerical causes, but that it reflects a real physical phenomenon, directly related to the non-axi- _£ symmetric instability of cold disks. In other words, cold disks containing nothing but stars cannot remain cold. 4 The other feature numerically observed, namely the subsequent dis- appearance of the spiral structure, shows that stellar disks exhibit spiral ", structure only during a short time of their evolution. Clearly, hot stellar sr£ - 95 - & disks cannot support spiral structure. This suggests that there exists some stability criterium for non-axisymmetric perturbations, which depends on the kinetic energy stored in the random motions of the stars. No generally applicable criterium has thus far been obtained. Kalnajs' (1972a)analysis of uniformly rotating disks showed that if the random kinetic energy T b random exceeds a certain value, instability does no longer occur. This finding was confirmed in the 3-dimensional numerical experiments of Ostriker and Peebles (1973). They established the global criterium that non-axisymmetric in- stabilities will occur only if T / I $ I ^, 0.14, where $ is the total mean ' ' potential energy of the system and T is the kinetic energy stored in the mean (= non-random) motions of the stars. The latter is complementary to T , because by the virial theorem T + T , = J 1 $ I. random mean random ' ' In view of the preceeding discussion. i«; has now become imperative to examine what happens with the waves if their amplitude grows beyond the range in which the linear theory (chapter 4) may be applied. The numerical experiments mentioned above give us already an indication of what we might expect. Generally, in problems of this type, the further development of the system may take one of two main directionsJ i) The system evolves under the influence of non-linear effects from its originally unstable stats towards a different state which is stable, ii) The non-linear effects stabilize the amplitude of.the perturbation at.: - some finite value and the original state, is in thisvay stabilized. In the first case we must conclude that the originally assumedstationary ;. . state has in fact no chancelof survival in the physical world, whereas in the second case it has, Lf only with limited deviations, which are. inherently determined by the,system in the state in question itself. .;_: The results of thenumerical experiments cited aboya —.provided they indeed represent real features of a galaxy — clearly indicate that the first possibility is the correct one. In other words, disks containing nothing but stars must be hot and will not exhibit spiral structure, except for a short time. This conclusion contradicts observations because the disk populations of spiral galaxies are, as far as can be deduced from the measurements, relatively cold and the spiral structure appears to be rather long lived. We , tend therefore to conclude that it is the presence of another constituent ; besides the disk-population stars that has as essential influence on the •', actual physical'situation. t There remains a difference of opinion about the way in which a real % galaxy can differ from a purely stellar disk. v|j One possibility is to assume the presence of a hot halo of stars around 1 the system (Peebles and Ostriker, 1973). f: The addition of ii hot halo component to a stellar disk would considerably! increase the potential energy of the system, whereas the mean non-random • _ij kinetic energy would hardly change. Hence, according to the stability Jig criterium given above, the ratio T /|$|would decrease and instabilities :*ft mean ' ' -Vjg would be suppressed. Coupled to the experimental fact that the heating -S3 phenomenon is directly related to the presence of non-axisymmetric in- *|j stabilities (Miller, 1974), the addition of a halo might also explain why ttie|p disk population of a spiral galaxy remains (to soice extent) as cool as is ifff! observed. However, the halo idea does not at all explain the long-term 5J|| existence of spiral structure, because all perturbations will, by the stabilising influence of the halo, be damped out in time. We feel, therefore, that the addition of a halo component does not constitute an adequate way out of the difficulties. We shall assume instea^l^ that the essential difference between real galaxies and the stellar disitsi^^ - 97 - used in numerical experiments consists in the presence of a gaseous disk component in addition to the stars and.that a stellar disk without gas is truly unstable. The latter assumption does not necessarily exclude the possibility that a halo component also exists. It merely puts an upper limit to its mass. Up till now we have completely neglected the presence of gas, merely because it represents only a small fraction of the total mass of the stellar disk. The spiral wave is therefore mainly sustained by the stars. But gas, however small its mass, feels the presence of the perturbations in the gravitational potential as well as the stars do. And. whereas the behaviour of the stellar disk can still be regarded as linear, the gaseous rasp'jnse is essentially non-linear in character. This non-linear interaction between the stell?.r wave and the gaseous component works as a stabilizing agent, as we shall show in this chapter. We shall assume that the non-linear response of the gas can in? described by the stationary shock solutions proposed by Roberts (1969). We shall not discuss the details of these solutions,, and only mention litre one aspect, namely that in the shocks energy is dissipated. Kalnajs (J972b)showad that this dissipation acts at the expense of a density ~wsvc and, for the triiling type of wave, tends to damp it. The non-linear interaction bet-vt-an.stais and gas was reconsidered in detail by Roberts and Shu (1972), who confirmed Rilnajs conclusions. The non-linear interaction mentioned here would mdeed damp a stationary density wave in an extremely short time - of the order of 5.10 yrs. If the stellar disk population is essentially unstable, exponentially growing waves would r ccir in it. In this case the non-linear interaction owing to the gas might act as the required stabilizing agent in the disk. This idea then, as we shall show in the next section, leads to the type - 98 *--._" .-- t^ * behaviour of the system that we have mentioned earlier as possibility (ii): the character o£ the initial perturbation is not essentially altered, only its amplitude is limited. This theoretical possibility seems the more attractive to us since it explains en the one hand the persistence of spiral structure and on tae other hand creates some hope that also the observed coolness of the stellar disk population will be understood. 5.2 Stabilization mechanism. We shall now assume that the disks of spiral galaxies are truly unstable against rertain \ion-axisymmetric perturbations that initially grow in time like exp Ynt, with y_ > 0. It is well known that growing perturbations can often be stabilized by some (even weak) non-linear interaction. In the disks of spiral galaxies this is the interaction between the gas and the wave perturbation, which, as mentioned before, will tend to damp the wave. There- fore an initially growing wave will in fact grow at a somewhat different ratej-• say like exp (yn - Y )t> where y describes the damping of the wave due to o o this interaction. As long as the amplitude of the wave increases in time we <- can expect Y also to vary. The amplitude of the wave A(t) is then described c f 5 l A(t) -v exp / dt'(YQ - Yg(t )). < ' >t The interesting question now arises whether we can determine the variation o£ — Y with time. ? 8 i In order to attempt this we shall assume that the behaviour of the - "f ir' A" stellar disk can be described by the linear theory and that the wave is •? sustained by the stars only. We thus neglect the self-gravity effects of the _^- gas. The definition of the angular momentum of the wave (eq. (4.75) implies tE that X i 2 •• •-•• i J exchanged between the gas and the wave. Therefore, the angular momentum of the gaseous component in the disk, J , will vary in time. Conservation of gas total angular momentum for the star-gas system now implies stars T) «•« In section 4.3.3 we have shown that (dJ/dt) can be interpreted as partly arising from the socalled resonant stars, partly from the socalled non- resonant stars. Moreover, we have identified the rate of change of angular momentum of the non-resonant stars as the rate of change of angular mpnientum of the wave, J . Thus we can write wave _ / or, when eqs. (5.1 - 2) are applied: \dt/stars \dt/res. stars VJo "g v wave Sine* the wave is sustained by the stars only and the stars can.be described' by the linear theory, we may assume that y . is defined by eq..(A.?6)/With ;. this definition eq. (^.4bj reduces, to ygas _ •. ,'g .wave. •. ;•• , The dot indicates diffefentation with respect to time.. Combining eqs. (5. J - .2) and "(5.5) we thus fiiyl . t 2 T = (J / J (t )) exp I - 2 I dt'(Y ~ Y (t1)) . (5.6 'g gas wave o 1 J 0n go J t o This equation permits us, by differentiating it with respect to time, to deriv an equation for 7 (t): ^g " "2% " V \ + Tt (1°8 Jgas> ^g • (5'7 For a further reduction of eq. (5.7) we have to know the variation of J gas with time. In order to make a simple estimate we follow the line of thought expressed by Roberts and Shu (1972). They employed a theorem of Bernoulli (conservation of total enthalpy across a standing shock) in order to determi * J and thus found gas 2 2 J ^ /RdR maQa D(M ) . (5.8 In eq. (5.8) R is the distance to the centre of the Galaxy, o~ the average surface density of the gas, a the velocity of sound and m the number of spir 2 arms. The function D(M ) describes the jump in the Bernoulli constant and is given by D(Mn fM21) = !(M1 (\2 - Mj2 - 2 log M2), (5.9 2 where M is the ratio of the gas densities just after and just in front of t 2 shock. The function D depends through M on the amplitude of the wave A. If. A now increases in time, but slow enough for the gas flow to adjust its properties to the near-equilibrium state, we can write (J 2 ft 'gas^5I /^R(ma0a ) g , (5. if or -|RdR (mv2)° - 101 - In order to estimate dlogD / dlogA we have used the numerical calculations of Woodward (1974). In table I log D and log A are given for several values of the strength of the spiral field F, measured in units of the mean gravitational field, F : |k A(t) / dV° / dR|, where k is the wavenumber of the spiral wave in the radial di'ection. Table 1 Shock parameters for different values of Uie strength of the wave according to Woodward. F D( A* log A log D 0.02 2.75 0. 18 1 0 - 1.71 0.03 3.9 0. 46 1.5 0.40 - 0.77 0.05 7.7 1. 74 2.5 0.92 0.55 0.07 12.5 3. 7 3.5 1.25 1.30 A is measured in units of the amplitude that corresponds to a field strength of 2% of the mean field. From table 1 we find it permissible to adopt the average value dlogD / dlogA= 2.4. For the range of amplitudes considered here eq. (5.10b) thus reduces to dt Jgas VI0 'g' Jgas Introducing now the dimensionless variable u *= v / YQ. using eq. (5.11) we obtain from eq. (5.7) the following differential equation for u u = 0.4 YOU (1 - u) . (5.12) The solution of eq. (5.12) is vell-knovA: - 102 - u(t) = (5.13) exp[- 0.4YQ (t - where t is an arbritrary constant. Clearly, y increases in time until in the cime-asymptotic limit it equals yQ. In other words, the wave grows less and less until in the time-asymptotic limit it stabilizes at a finite amplitude. finally, we obtain from eq. (5.1) and eq. (5.13) the variation of the amplitude A(t) _, 1 2.5 (5.14) 1 + exp [- 0.4y0 (t - tQ)] where A is the time-asymptotic amplitude, as In fig. 5.1 u and A/A are shown as a function of Yr,(t - t ). 1U 1 1 ! ( 1 \____ _.- y ^ -' - 08 _ ^ ' z' 06 - V ''* 04 / S s s 02 ''' s s n lilt! -2 10 Fig. 5.1 The ratio u = y / y and the relative amplitude of the wave with respect to its time-asyrrptotio value as a function of time in unite -1 of y From the simple example worked out here we thus see that the non-linear - interaction between the wave and the gas is indeed capable of stabilizing the--, - 103 - *»*&- wave. More generally we can say that this kind of stabilization can always be expected to occur as long as the loss of angular momentum of the gas (or any other component) owing to its non-linear interaction with the wave varies somewhat faster with the amplitude of the wave (here power 2.4) than does the angular momentum of the wave itself (here power 2). 5.3 Discussion. Although the example worked out is only of illustrative significance it is nevertheless interesting to see what time scales and amplitudes are involved if we try to fit it to the real Galaxy. Then it would be reasonable to assume that the spiral structure as observed in the Galaxy is close to the time-asymptotic state, that is y - y and A = A . The value of a g u as characteristic time scale T for the damping of the wave (which is half of y defined here) was estimated by Kalnajs (1972b)and by Roberts and Shu 8 l 8 (1972) to be of the order of (3 - 6)x 10 years, thus yQ~ - (6 - 12) x IO years. This growth rate is of the same order of magnitude as that observed in numerical experiments (Bohl , 1972). From fig. 5.1 we then deduce that it takes between (4 - 8)x 10 years for the wave to increase its amplitude from 25% up to 90% of its asymptotic value. If we assume tnat the observed amplitude of the gravitational field of the galactic spiral structure is now of the order of 7% of the mean field, it amounted to at least '.% of the mean field some7 x 10 years ago. . . These estimated values of the time scales and the amplitudes are so promising that it seams worthwhile to discuss in nore detail the assumptions made in section 2. These are i) the stellar disk is unstable, ii) the wave is main- tained by the stars and its properties are dasc.ribed by the linear theory, iii) transient effects in the gas flow are "negligible. Can. we expect chese - 104 - ' . assumptions to hold over periods of several times ]0 years? We shall take up these assumptions in reverse ordi:r. Assumption iii) regarding the gas flow can be expected to hold as long as the amplitude of the wave varies little during the time it takes for the gas to pass fully through the spiral arms, that is as long as dlogA << 1 (5.15a) dt or yQ (1 - u) << 1. (5.15b) We recall that in these equations ft is the angular velocity of the gas and Q that of the spiral pattern. With the estimated value of - (6 - 12) x 10 years and u > J we find that this condition is fulfilled as long as lo - >> (1-2) km/s. kpc. (5.16) Clearly, our analysis cannot be applied in the neighbourhood of the corotarion resonance. Actually, we should have excluded this region from the very beginning because in this region the properties of the gas flow are not at all well determined. Away from the corotation resonance our assumption certainly holds good. Assumptions i) and ii) are far more difficult to judge. Small amplitudes^ can justify the use of the linear theory over large parts of the disk, but not in the neighbourhood of the various resonances that occur in spiral galaxies. There the time scale during which the linear theory applies is ~ severely limited by the amplitude of the wave. This is discussed on a quantitative basis in chapter 6. Fortunately, as will be shown there, the y,. results of the linear theory can, by a slight modification, be made applicableI* at longer time scales without affecting the line of reasoning obtaining Lne eqs. (5.1 - 5.5). We shall return to the assumptions i) and ii) in section 6.3. - 106 - CHAPTER 6 QUASI-LINEAR THEORY 6. 1 Introduction. * In this chapter we shall discuss the effects of a slowly varying "mean" state on the properties of the perturbations. We shall neglect the presence of the gaseous component but keep in mind that the gas is capable of limiting the growth of the perturbations. We wish to make it clear from the very beginning that a detailed account of all evolutionary effects cannot yet be given. We shall present a few considerations which were initiated by the problems mentioned in the beginning and end of chapter 5. We shall show in section 6.2 that under certain conditions the evolutionary effects of the perturbations on the mean state can be described by a diffusion equation. The most delicate problem then is to determine the diffusion coefficients; this problanis tackled in section 6.3. From the order of magnitude estimates made there we conclude that in the long run tightly wound waves have a chance to survive, and that only if the disk is cold and the density wave amplitudes are not excessive. i 6.2 Derivation of the diffusion equation. * In the presence of exponentially growing density waves the assumption /*_ of a time-independent, axisymmetric gravitational field becomes questionable. =1 From the discussion given in section 4.3.4. it is clear that .Che growth of T density waves is effected by a transport of angular momentum from the inner ~jL part of the stellar disk towards the outer parts. Lynden-Bell and Kalnajs (1972) showed that such an angular momentum transport can only be carried - 107 - by trailing waves, thus explaining the prevalence of this type of waves above leading waves. The same process of angular momentum transport will lead to a change in the mean state in tho course of time, and consequently also the properties of the density waves will change in time. In order to describe this kind of evolutionary effects we postulate that the gravitational potential field is given by eq. (2.18) with a j6 0 and X ^ 0. The potential will therefore be written in the form 1 V(R, e, t) = V°(R, at) + XV (R; 6, t) . (6,1) We recall that o is assumed to be large compared to the time scale on which V varies and that X « 1, if V and V° are taken to have the same order of magnitude. We shall describe in this section the equations that describe •: the behaviour of a stellar disk under these assumptions. Since V (R, at) depends on time we first have to reconsider the CGncept of action and angle variables, which are now defined in a slightly different way from the formulae given in chapter 3, Having done so, we are in a position to define more precisely what we shall call the nean state of the system. Further, we shall obtain a diffusion equation for the variation of this mean state with time. At the same tine, we obtain equations that describe.the evolution of the . ' .' perturbations. . . One of the advantages of the use of action, variables is that the actions are adiabatically^ conserved in a slowly varying gravitational field, that ist . if the field variations are slow compared to the period of a radial : oscillation, fl"1. Since the time variation of the perturbation V . is of order US' we have a/ SI- « 1. Therefore, the actions may. be assumed to, be ' . .: '••' •,-. ••'• ••'.-•'io8!-;v/--'-:''"';:-';-'-''-- V-.3 ,••'••..• .••.••" '^- adiabaticially conserved in Che slowly v .rying field V (P, at). In the present study we can keep J. as defined by eq. (3.5) but redefine ^he radial action J •- J2 = Tx tydR '(± ] In oq. (6.2) the prime with the integration sign denotes that the integration has to be carried out by keeping H and the time constant. In other words, we define in this case the radial action by the action the star would have if it would move with constant energy in a field V that is constant in time. From eq. (6.2) it is in principle possible to express H in the form H° = H° (Jj, J2, t) . (6.3) Thereby H is now also a slowly varying function or time. We now can proceed along the same lines as in chapter 3 in order to find the angles w. and «„. The result is that w , w~, ft. and &„ are described by the same expressions as given in eqs. (3.12 - 15) provided we add everwhere a prime to the integration sign. Since the transformation is now time-dependent the Hamiltonian in the new coordinate system becomes according to eq. (2.11) H = H(J., W., t) + -2 , (6.4) where 1 ( 3H-/3t - 0V-/3t) . - —n n 7 9 { ^6'5)^ (H° - V°(R\ t)) ~ J{/R'Z] * in which 3H° = 9J° U rk dR' (av°/3t) 1 ttL J, at - 109 - The equations of motion in terms of the new coordinates are A A 8H : 3H Wi " 8J. ' Ji = " 3w. > (6.7) x 1 whereas the evolution of the distribution fucntion is decribed by eq. (2.14): If + [f, H] - 0. (6.8) We now define a suitable mean state of the stellar system: . i 2i 2 V V°(R, z, t) = 4im G 6(z) / dvR ( dvo < f > . (6.10) For eq. (6.10) to be cons is tent with the assumption that V varies slowly in time, the mean state has to represent a slowly varying function of time. 1 -e definition of the mean staLe given in eq. (6,9) does include in principle noi. only a slowly varying part but also a part. that.varies rapidly in time. The rapidly varying part arises from axisymmetrir perturbations. Although the ::. presence of such1 perturbations can not a. priori be ruled out in real stellar disks, it is reasonable to assume that the level at which-any such perturbations exist is much smaller than that of the non-axisymmetric perturbations, because we assume thai: the velocity dispersion in the disk is sufficiently high to stabilize the.disk against axisyasnetric perturbations. Therefore, we shall assume here that axisynmetric perturbations are absent. s For this reason we.also can write - 110 - < H > = H°. (6.11) The evolution of the mean state is described by eq. (6.8) averaged over angles: 3y^ + < [f, H] > = 0. (6.12) The equation describing the evolution off = f - < f > can with the aid of eq. (6.8) be written in the following form: || + [f1, H] = [H, < f >] - 3 A further simplification is obtained by assuming that a/u>o « X , C6.U) where too is the characteristic time scale on which f varies and a on which _ the mean distribution function, can in equation (6.12) and (6.13) replace H by H whereas in eq. (6.13) we can also neglect the effects of 3 3f|> + < [f', V!]> = 0 , (6.15) •t. 3f'+ 1 _ 1 . J M where we have made use of H = 11°(J., t) + AV1 (J., w., t). Eqs. (6.15) and * (6.16) can be written in an alternative form if we expand V and f in Fourier JJf series in Wj and w-, In analogy with eqs. (4.16 - 17) we can write ^jjfe 1 1' 4ir2 111Z 1112 *' " » * . -• l ! 2 2 f'(J., w., t) = 1 ** f (j t) e (6.18 4TT l\L2 lli2 x * - ill - Substituting eqs. (6.17 - 18) in eq. (6.16) we find that f.. and V 1 2 l\ 12 are related to each other by (t) = i / dJ i (i Ir^K i (Ji(T)' T) • 1 1 k 3J 1 1 L l 2 tc I k = 1,2 \ k /J I 2 (6.19) In eq. (6.19) it is assumed that the gravitational perturbation, is switched on at time t , whereas the integration is made along the real orbit of the star, that is along the orbit described by the equations V-i. . *i- I. • <6>20> Inserting eqs. (6.17 - 18) in eq. (6.15) we find after averaging over angles: 8i£- - * T I i,: AM: The eqs. (6.19) and (6.21) Tepresent-:an---»1!;^ernkt'iyfe^n#^.^^e^^^^^l|^5sSi|-- and (6.15), -respectively. Once - the.;-gieia^£a^in>iis^t:(i^^^56^^^©9P^^^S^ fully determined by the properties of. the mean state f£> and describes "the.; deviations in the mean state that are created by V , :A1 though.£.-.. xs zero when, averaged over the angles ,w.. and w.j it. will^ inf luence the miaber .of-etarsV "' • that, per unit time, is forced1 :to flow t'hroughL, the boundary, of: a. vciluroe ;;-. •'= element in phase, space through the actipn'of ;V .Whenaveragedi'iofcer,^angles, this will lead t<>; a .change of the; mean state in! the course of tixoe. Becaiwe r^ 1 f' depends on the..mean state 1| : : : ; : : ; : ;: : ! : : •'•• • " .• ."..•'••• '. •••:••'•. -;: .' . •; ••!' '' ';^ ': -"' '-- ^ l:^^^ "''-'''''-^''' '-^y^' --'1'^- ' '-': "'' • ' '"' • " "•'• '• '•:.r;.:v:--;7 •!••••••> ••'•'•:'•'•:':'• '••v;V-^:i-X/-Kv:K!^;V:i'^-;'v;:^S;^iifisi-/!r-1'' gravitational potential perturbation V • We shall show that under certain conditions this non-linear equation reduces to a diffusion equation. Substituting »q. (6.19) in eq. (6.21) we obtain 3 -il (w (t) - w (T)) . V. . (x) e n n U . (6.22) 1L2 For shortness of notation the summation convention is adopted in eq. (6.22) and below. When the sum over repeated indices k, in and n in eq. (6.22) is carried out terms proportional to 1. and/or 1« are obtained which contribute to the double sum over 1. and 12- For the moment we postulate, that the contribution of (Aw2)2 1\12. lk lm ^l1!. (t) VllL2 -il (w (t) - w (T)) . e k, m = 1, 2 (6.23) in the integrand appearing in eq. (6.22) becomes small after a time T , shoit'*- o i compared to the time scale on which the conventional arguments used in the derivation of the Fokker-Elanck 5 equation, che classical example of a diffusion equation, and we shall discuss-^ this assumption in detail in the section 6>3. ,> r On this basis, we may assume that that we can write . »? t We thus can approximate the integral in eq. (6.23) by (6.24) by which the full eq. (6.22) becomes £ 3 8 \ Four diffusion coefficients are represented for k = 1; 2 and ra = 1, 2. The diffusion equation derived here is -valid for all kinds of internal or external perturbations. We are mainly interested in situations in which - the perturbations are self-consistently maintained. This means that V can not be chosen arbitrarily, but is related to f by Poisson's equation (see e.g. 4. f>). For the present analysis, however, the only important parameter is the time dependence of the perturbati ins, which will be assumed to vary like exp (iut), where,the frequency uis complex. Therefore, we shall leave tile full formulation for self-sustained perturbations aside and concentrate on the properties .that lean be derived from eq; (6.25). Before we discjss these properties in more detail we shall first l> mine some approximate expressions for the diffusion coefficients. 6.3 Diffusion coefficients. The fundamental assumption made in deriving the diffusion equation (6.25) is that ax << 1. We shall discuss this assumption in more detail in the present section. For simplicity we assume that a single density wave is present, for which 1 i A i V (R, 6,t) = Re { V(R) exp i(uit + m0) } , Re u < 0. (6.27; We write oi = w - iy , u and y real and y > 0, corresponding to growing o o o o o waves. We recall that we are interested in the behaviour of the expression (6.23) in time. For the wave defined in eq. (6.27) this expression becomes ,77172 £ \ lm IA1, 1,1 e 1 ; 1 2 l (6.28) where 2ir 2TT f f ~1U W + 1 W ) a = I dw / dw e ' ' 2 2 V(R)e'm . (6.29) *•!"1 "-O2 J ' J *• 1 l 0 0 If the phases of the terms appearing in the double sum in aq. (6.28) vary rapidly in time, that is, rapid when compared to a the use of the diffusion^ equation is justified. *~ In a linear analysis one assumes that the motion of the stars may be -1 approximated by their unperturbed orbits defined by ~^P" ~\~ Ji £ constant, «£=§§== a.^ (J.(Jj,, t)) .. (6.30 - 115 - In this approximation the phases of the terms in eq. (6.26) are given by oio + lfc f2k(t) }dt - YQ(t -T) + 2Y(Jt. (6.31) In this linear approximation the phases for the stars, which we have called non-resonant stars in section 4.3, vary rapidly, at a timescale of the order of(ojQ + l,n + 1 02 ) . For resonant stars, that is stars for which (ID + l.R. + l-Q^)*^ 0» tne phases vary very slowly. This difference in behaviour of these two groups of stars in the linear approximation implies that the diffusion rates will be different for these two groups of stars. Therefore, we shall treat these two cases separately. For non-resonant stars the phases of the terms in the double sum in eq. (6.28) can be approximated by eq. (6.31). Phase mixing of the various terms will cause it to vary on a time scale T , that is roughly the inverse of the phase difference of two consecutive, terms. This is of the order of a period of a radial oscillation. Since this period is assumed to be small compared to a we have otx « 1. The diffusion.^equation thus holds for the non-resonant stars. - \ • •-.'.-' The^coefficients D^_ for these stars, can be obtained l>y using (6.28) . and (6.30) and integrating eq. (6.26) in time. We find in the time-asymptotic limit: • . ..•':..• .• • • ' • ' • .;•.:'•• ' •.. • T ' '"••• '•-••ti'12-":- • : " ' lk 2 hh•" V'-i-:t;iIo1;*:i2a2i V' In the derivation of eq. (6.32) we have omitted some oscillating-terms because . •' • • .. -. -116 .>•: '•.-.. ... " -. '• •^ these will reduce to zero when averaged over a period of oscillation of the wave. In order to obtain an estimate of T for the resonant stars, we consider in more detail theorbitsof thesestars, as described by Lynden-Bell (1973). He studied the real orbits in a time-independent axisymmetric mean state with a wave perturbation of stationary amplitude superimposed. We shall first briefly review the main points of this analysis. In order to study a particular group of resonant stars it was noted by ; Lynden-Bell that an elegant description can be obtained by transforming to a new set of variables Js = Jl •J f = J2 4J. • (6"33> The generating function is given by + W } + J w (6 34) F2(JS, Jf, W), w2, t) = jg(w, - npt 5 2 f 2" ' with J2 , the pattern speed, defined by a»/(-m). P o The new angles are found to be w = w, - SJ + — w,, s 1 p m 2 ' wf = w2 , (5.351 whereas the new Hamiltonian is given by ~ je= H(Jf, Ja, wf, ws) - fip Jg . (6.36) Two important time scales can now be distinguished , one of order n 2 jf corresponding to the fast orbital motion of the stars, the other, of the order ~ of(iao + mJJj + lfl2) , corresponding to a slow drifting motion of a "resomni? *. orbit in rhe new coordinate system. Expressing the gravitational potential <^ - 117 - 1? also in terms of the new variables we have (for an m-armed spiral and Y =0: o v'(Js, Jf, ws> wf) = ^ I A1+n> m exp i{mws + nwf} . (6.37) In order to determine the orbits of the stars in the new coordinate system Lynden-Bell now argues as follows: all variables except wf slowly vary in time. Therefore, one can study the behaviour of all other variables by considering the forces averaged over the fast variation of w . The average potential then is found to be 1 K :'"mw Jf = 0 wf * il2 (6.39a) s r s In eq. (6.39) all variables are now average quantities. Note that J is _ S - • defined is such a manner that we have (|f°)J - fl ...... ;, (6.40, x s so r where J is the value of J at exact resonance • : •"'. • . so s • •. . •• •.'"...... Eqs. (6.38 - 40) form a closed set of equations, in the sense that for a given unperturbed Hamiltonian function -H and a given-gravitational potential perturbation V and proper starting values," it determines the full develop- ment in time, of the functions J,, w ', J and••'**„./ -. Lynden-Bell showed that the solutions for. J and w are of the. pendulum type. If J -J is small; librations occur with a frequency given by s so ..,,. .... •.. . • -118 - with I 2 s = - (a H°/3j2 ) . (6.42) | x ' s so j- The quantity s is fully determined by the unperturbed Hamiltonian, H° of the j I disk. I For larger J -values the motion becomes a rotation. The librating stars s are just those that are called the trapped stars. As a rough estimate one j finds that a star is trapped if j Ij -J l£ UUV1:*! s) * = AJ . (6.49) , 1 s so1 v ' ' smax ' Clearly, the larger the amplitude of the wave, the larger the number of J j trapped stars and the higher the frequency with which a trapped star lil rates j t i in the gravitational potential wave. Further more, the rate of change of angular momentum, here J , is zero when averaged over a libration period. s . i Therefore, trapped stars do not exchange angular momentum with the wave on . time scales larger than u> . Unly when the rate of change of angular momentum of these stars is considered on timescales short compare to o> , we find that some stars gain and other loose angular momentum, so that a net exchange of angular momentum between the stars and the wave can take place. , i This also explains why such an exchange does take place in the linear ] approximation (see section 4.3) . The infinitesimally small perturbations , -1 r ' considered there imply that u is assumed to be infinite. An extension of the linear theory to finite amplitude waves is only justified on time scales short compared to ui .In appendix we show that for typical values for the amplitude of the wave to is of the order of 5 x 10 years, a rather short *-.,. time indeed. : - 119 - . f -Jl With this brief account of Lynden-Bell's treatment of the behaviour of resonant stars in a constant amplitude spiral wave field superimposed on a time-independent axisymmetric gravitational potential field, we have prepared the ground for an estimate of T . Hence we can now tackle the problem under what conditions for these stars ax << 1, so that the diffusion equation is applicable. After having done so, we also wish to derive simple estimates for the diffusion coefficients. This will require an addition assumption to be introduced later. We start here with the estimate of T . For this purpose we consider the extension of Lynden-Bell's treatment to a slowly varying mean state. The timescale T now depends on the relaxation time of the mean state a and on th« amplitude of the wave. We shall show that the condition OT << 1 then^ leads to an upper limit of the strength of the perturbations. The main extension to be applied to the analysis made by Lynden-Bell, if a slowly varying mean field is present, is the introduction of the slow variation of H in time. As a consequence the value of J (at exact resonance) also changes slowly in time. In order to estimate this variation we note that J is defined by so J J t) =p C ^--ifs < .o-v -P '. • •. 1 ! ••.• where P. is the pattern speed of the wave. Clearly after some time At J is changed by the amount . . • so ... . (sn Idf) At AJ = -r-*7 : •..••-.- . ••.••• (6-45) so on/3Jssj) so ..-:.•..• '' We can now define in a physical way, the critical tiae TQ as the tine in which the gradual increase in JSQ approaches the boundary *JgmM of the, range of J. over which the stars are trapped (ec 6.13). Thus . - 120 T. On /at) x 4 3R /3t - aft = an . (6.47) \ s s p c Using eqs. (6.44, 38-39) we then find t I (ax ) - (to- /n ) . (6.48) I o T p * In order that at << 1, the fundamental assumption in deriving the diffusion | equations, it is required that uT/n << 1. Because u depends on the j amplitude of the wave this condition defines an upper limit to the strength j| of the gravitational potential wave, which we express by F (eq. A.7 ) . In | the appendix we have shown that I £ T ID g ~ = Bv^F . (6.49) ' I P \ For tighcly wound waves and a typical value of the pattern speed of the order | of 15 km/s. kpc we find that B "k. \ at the inner Lindblad resonance, and , j i B % 0.5 at the outer resonances. Therefore, as long as the waves are tighcly wound andF does not exceed 0.1, the necessary condition to use the diffusion. .equation at the resonances is indeed fulfilled. For open waves (k.R £ 1) the typical values of 8 are larger. Consequently, "- the constraint on the strength of the spiral field is more severe, especially ^,,1 W at the corotation resonance (F < 0.01). In actual situations typical values Z of F can be expected to be at least of the order of 1 percent. Hence, the 7- assumption in deriving the diffusion equation is not fulfilled for open - 121 - The present analysis is therefore restricted to tightly wound waves. We now wish to determine the diffusion coefficients for the resonant stars. A simple estimate is obtained if the condition ui i £ 1 holds, because then the time during which a star is trapped, T , is short compared to a libraticn period. A resonant star does not get time enough to carry out a full librauion. Therefore, if u T £ 1, it is justified to describe the orbit of a resonant star during the time T by its unperturbed orbit, characterized by eq. (6.30). The behaviour of the stars then is said to be quasi-linear. This permits the simplification that the phases in eq. (6.31) can be taken 0 during the time T , during which a star contributes to the integral in eq. (6.26). We thus find the diffusion coefficients: 2Y t (Air )2 ° where 1( and 1~ refer to the particular resonance studied and all oscillating terms at resonance have been left out. In order to see whether the time T is indeed short compared to the libration period of a "trapped" star, we estimate u T . The characteristic time scale T depends on the relaxation time, a 3 of the mean, state. In order to estimate a we note that a % .3 At the inner and the outer Lindblad resonance the.dominating contribution (for cold disks at least) in the numerator of eq. (6.5tj " S/8J2 U2? 3/3J2 We further assume that 3/3J2 sf> .% J?. ... we find in tlie limit of nearly circular orbitsj . r,;.|22-r"- '•• •-.: ? I 2 i 5 where Combining eq. (6.48) and eq. (6.54) we find P I (VQ) = 6/ {^T/np} * , (6.55) j- where f fi o << \V/ ^ > K f 2 p R O 0 O For the quasi—linear theory to be applicable we require to x >v I. This * TO - i. t condition puts a lower limt on the strength of the perturbations, since eq. _ | (6.55) depends en F only through (u> /tt ) in the denominator. We thus find j I ir * * • i At the inner Lindblad resonance we find for Q hi = 5, F it 0.0004 (6.58b) At the outer Lindblad resonance we find for SI /Q. - 0.5 Therefore np ^ " " (6.59a) or with eq. (6.49) for B = 0.5 F - 0.007 (6.59b) From eqs. (6.58-59) we conclude that at the inner and the outer Lindblad resonance the condition u> x £ ] holds for all cases of interest. The restraint at the corotation resonance is far more difficult to estimate because there only the deiivative with respect to J. appears in the diffusion equation. We can expect the relaxation time here to be much larger than at the Lindblad resonances. Consequently the condition something about the conditions at this resonance and we shall, therefore, not go into this matter further. ';..•' .". From the rough estimates presented in this section we thus conclude thct as long as the strength of the tightly wound waves, considered here is*, not excessively large, the variation in the mean state can be described.by a ... diffusion equation. Moreover, as long as the perturbations are not too small, a quasi linear behaviour of the disk of stars, expressed by the condition u T £ 1, can be expected to occur. An exception is found for the corotation TO' radius, but this exception does not seem to form a serious objection because 1) resonant effects at the corotation radius are generally an order of magnitude smaller in comparison to the effects at the inner and outer Lindblad resonances (Lynden-Bell and Kalnajs, 1972) and 2) transient effects in the resonance of the gas can be expected to suppress trapping phenomena. 6.4 The persistence of spiral structure. The diffusion equation (6.25) enables us in principle to determine the evolution of the mean state of a stellar system, once the mean state is specified at t = 0. The theory, thus far presented, is not sufficiently detailed to be applied for actual calculations. Nevertheless, the theory allows us to draw some conclusions on the long-term persistence of spiral structure. We shall first consider the growth rate of the wave and obtain an -1 s expression that is valid on time scales long compared to u) . Further we ^_ shall show that, whatever the ultimate form of the mean state, all growing waves become damped on time scales large compared to a ', the relaxation - time of the mean state. V A new expression for the growth rate is obtained by considering the 3- constraint imposed on the wave by conservation of total angular momentum as "^ we have discussed in section 4.3.1. After multiplying eq-. (6.25) with J. andJl; integrating over all phase space we find ^f" . Ttot "*c«* dJ t ^^ ' ^s&- j = ~~ i Qj / d*Jr* ^~* I in JJ ~*~~ I (fi 61i'» ^ ^- * I ~ ^- lc~ 1, /. mi£ o JL ^J^ ^ We can split up the right hand side of eq. (6.61) into two terms: --^ - 125 - 1). A contribution from trapped stars (or resonant stars): trap, stars trapped stars (6.62) 2). A contribution from non-trapped stars (or non-resonant stars): dJ tOt " Liiri:> non-trapped stars X\ |2 (6.63) In a similar way as was done in chapter 4 we can identify the angular momentum of the wave by J ) d t K wave J \Tt non-trapped stars -00 from which by requiring the conservation of total angular momentum it follows that ... . -(djfWt) . .:,.- ': . •. ..:•• •.•'•••••••' 2 vo - T wave In principle the contribution;ox. the trapped stars as well as the.angular V . momentum of the wave:will vary in time oving to the slow variation of the properties of the^ mean stated Therefore we also may expect the growth rate to vary slowly in time /accordingly. ••'•; 1 .. . . .-.,"• The essential difference between eq. (6.65) and. eq..(4.76).is that the ,, . . contribution of the tesfmant stars is replaced"by a cfmtributioa of trapped . . . stars, or in other words the delta function appearing in the numerator in eq. (4.66) is now replaced by the function R(J , J.) l (6.66) = 0 I T - J I > A| In a more detailed analysis one might expect, instead of eq. (6.66) a more complicated function to appear, but this function will again have a width in J of the order of T . s •. o In order to discuss the ultimate fate of the growing waves in the time' asymptotitotic limit, that is, on time scales large compacompar:e to a , we multiply eq. (6.25) by I^^/dJj dJ.D^/i^^ . (0.67) The left hand side of this equation is a positive function. Therefore, any time-asymptotic stationary state that might possibly result from the evolutionary effects caused by the perturbations has to be characterized by It is a very unlikely assumption that ;he first mentioned quantity becomes -' zero because this requires the distribution function to be independent af . \ s£ energy and angular momentum. Therefore we may conclude that any time-asympto-. tic solution of the diffusion equation will be characterized by D, =0. ^ , *• . km -'•'• The latter condition can only be fulfilled when the amplitudes of-thi. • ... V _. perturbations go to zero, that is, when the perturbations are damped in time," - Therefore, all initially growing waves must "evolve" towards damped waves, - that is, in the time-asymptotic limit y (t) must become negative. Clearly, * j the ultimate mean distribution function, - 127 - stable with respect to non-axisymmetric perturbations. As long as the amplitudes of the waves do not become excessive, the analysis above implies that the spiral structure in flat galaxies can persist on timescales at least — 1 — 1 of the order of a .A typical value of a is best estimated by the diffusion rate of non-resonant stars. We thus find: 1 2 S a" = The expression for Tt^1 reS' is given by eq. (6.32) for k = 2 and m = 2. We further note that, of all terms appearing in the double sum in eq. (6.32), the term which 1 =-2 and 1. = 1 is the most important because its denominator is smallest. For actual galaxies the denominator of this term is of the > _der of SI , the pattern speed. We find: • : -;• , ; ....-..; Y — \ 16 . 1 1 1 : ; '.'-•• ' • "•" " • ' '• •*•'.''.' ."' •"•' .'•'••• ". . .''ii' •, . • .' '•""•.. • ' "• " "fl •' • " ' :-' ' - • . Iti chapter 5 we havve estimated'.a'typical value -of Y %, 6.6 - U2 x \Q--. years. :;; Thus ' ' ... ''.; ;';•••; ••'.. ]'}:..':^'}:'"•• ''•'''. \. i: "•"?'';•••••''•'•• '• '•'•'•/'' '•''•'''•]:.''\\-'-v'^/- ''''••'. ':•. ^":'J:?:'.:. The estimates given.here are all very rough and indicate no, more than an • order of magnitude; but they allow .'jisf-'to .conclude.-that.! spiral^ structure can persist over a considerable^^.fraction of the life of a spiral galaxy, as long -. as the amplitudes do not.'grow eicessiveiy/large. • . . ... • '•;... ' •" •••••. .•;.• •'-;•:': • • •'•:.:.'• ": • '••"' •. • ;• •'••'..'•_ •'•. ..'. iv.',"'': '•" ••' '" •• ' • '• ' ' '-•.••• ';•;•, :'. " • '•.' " . i:^":'''[". ••'•'•':.:' :. •' '•'". '^ . .'-i •'. ' •'•'• • - -.••..• ••^i'^'1 :-"r-:-' ' -'•' ';':.•.. • ••:•• ••-•"v^^.;si- "••'•.• ••<••' • In chapter 5 we have shown that a density wave described by linear theory may be limited in annlitude as a consequence of the non-linear interaction with a gaseous component. The analysis just presented implies that exactly the same conclusion may be drawn if the density wave is described by quasi- linear theory (eq. (6.65)). The important consequence of this statement is that the conclusions, which on the basis of eq. (4.76) were warranted only for very short time scales, can now be taken to be valid for all time scales of interest. - 129 - * r-» APPENDIX In this appendix we shall estimate several quantities introduced in chapter 6. We start with the libration frequency, u , of a trapped star. For an m-armed density wave of the form defined in eq. (6.27) we have 2 2 , TT1 | uj = m 2 2 s =- {3 H°/3Js } j _ 3 (A.2) s so and the averaged amplitude of the wave at a particular resonance radius is v> l< l= I A. | / 4ir e (A.3) where A^ is defined in eq.(6.29) and 1 and m refer to a particular resonance. Following the method used by Lynden-Bell and Kalnajs (1972, appendix) we find in the limit for nearly circular orbits to which we shall restrict the present calculation: |A = 4IJ2 J (x) lm' l l l |VCR0>| fr.4) where J^ is the Bessel function of the first kind and |V(R)| the amplitude of the wave. All quantities have to be cous.i tic-red as a function of the epicentre radius. Furthermore x is defined by (A 5) X - k« For tightly wound waves k1 •.% k, with k the radial wave number. For open waves [ 2 j 7i I • •• • . k + m /R J *. We shall consider here only two-armed density waves, that is m = -2. .•..:.. ' ' ' .. ; . . • • ..•:.• .••• .' •• :• •-•i30'- .' ••.'.. - '••.."•.• With eqs. (A.3 - 4) we-can transform eq. (A.I) into: Y t s e ° . (A.6) Introducing F, the ratio of the spiral field and the mean field: Y t . F = k1 |V(R)|e / Ml . (A.7) we find at resonance 4sR (A-8> We now estimate s by noting that with the definitions of J and J, in eq. S I (6.53) we can write s= in where fi. and n_ are the generalized rotation and epicycle frequency, respectively. For nearby circular orbits we may write 3J, (A.10) __2 ^ _o J_ 1 - -2 (A. 11) 3J, % n '2 1 o R o and (A. 12) o o We thus obtain - 131 - s R i - >• - £ (A.13) o o o In table Al some typical values of the key quantities are given at the three resonances. We now determine some typical values of B, defined by (A. 14) From eq. (A-.8) it follws that rl'. 4sR (A. 15) The quantity (3 is different for tightly wound waves (k'R » 1) and open waves (k'R % 1). For the tightly wound waves we shall take k'R % 10. With a typical n 1 5 value of For the open waves we shall take k'R % I, consequently x % 0.1 and \'3 (x)] % 0.1 for 1 = ±1 and |JQ(x)| £ 1 for 1 = 0. Table A2 gives some typical values of 6 for open and tightly wound.'..Haves. The same table gives the corresponding values of us at the various resonances based on J2 % 15 km/s. kpc and F 3; 0.04. P Returning to eqs. (A.!), (A.3 - 4) ws find that u can also be written as. - 2 2 _ = m s •'..'.• (A.16) 4TV which for tightly wound waves, with~|jj(x)l'%-0.5, becomes with m.= -2, uT* = 2 s (A.17) Finally, the quantity n introduced in eq. (6.69) is givea by 2 ... • , . ..•••• ..- . "• (A.IS) - 132 - Typical values for the solar neighbourhood where fi /ft % 0.5 are K /fi ^ 1.4 and |J,(x)| ^0.5. We thus find 1 o n Table Al n /JJ s R o o °P o Inner Lindblad resonance (1 = 1, m = -2) 1.7 0.25 Corotation resonance (1 = 0, m = -2) 1.3 1.5 Outer Lindblad resonance (1 = -!, m = -2) • 0.5 Table A2 open waves tightly wound waves — I 8 u> (10 years) 6 ID (10 years) Inner Lindblad resonance L (1 = 1, m = -2) 1.6 1.1 3 Corotation resonance (1 = 0, m = -2) 2.5 1.33 0.55 Outer Lindblad resonance (1 = -1, m = -2) 0.78 0.55 - 133 - CHAPTER 7 CONCLUSIONS AND SUMMARY The main conclusions. In summarizing the main conclusions reached in this thesis we shall not redefine a number of concepts that have been fully explained in the literature of the past years. These are particularly: cold disk, Lindblad resonances, resonant stars, and "wave" as the collection of non-resonant stars. With these concepts considered as known, the main conclusions may be formulated as follows: (1) In a time-independent axisymmetric gravitational potential field V°(R) a good approximate description of the stellar orbits in the plane of the galaxy can be given by expanding V°(R) in a Taylor series up to second order in R~ around the epicentre radius R (section 3.2). This "Keplerian approximation" has a number of advantages over the well-Tcnowri epicycle approximation . (2) The evolution in time of a density pertufDatiori (or a p4ftiirbat£bn in the gravitational potential) is in/the linear perturbation-1 thepry ^determined by a dynamically and a kiriematitaliy"evolving ':plxltT1'iKly':if5%^'eirV£^liir'''dif1t"is'*-*1'"' unstable with respect to hon-axisymmetric perturbations the dynamically evolving part stands out clearly fromche kinematically evolving part. The' dynamically evolving part may be called a density wave. For stable disks Lindblad's conclusion (see e.g. Lindblad, 1957) that bi-symmetrical perturbations are least damped is confirmed (section 4.2).:. (3) The rate of growth of a density wave in the linear theory equals the rate of change of angular momentum of the resonant stars, divided by the total angular momentum of the wave. For cold stellar disks the real part of the frequency characterizing a growing wave can lie in a fairly wide range. When this result is combined with Kalnajs' (1970) numerical proof of the existence of at least one unstable mode, the frequency of which lies in this permitted range, the conclusion seems warranted that cold stellar disks are generally unstable to non-axisymmetric perturbations (section 4.3). (4) An unstable disk of stars will be stabilized by the presence of a gaseous component, still with the proviso that the properties of the stellar disk can be adequately described by the linear theory. This stabilization is brought about by the non-linear response of the gas on a "stellar" density wave (chapter 5). The same conclusion still holds if the proviso is changed into the assumption that the stellar disk can be described by quasi-linear theory (chapter 6). (5) The presence of growing density waves causes a long-term evolution of the mean state of a (cold) stellar disk. A quantitative examination of this process in chapter 6 shows that this evolution can be described by quasi- linear theory as long as the waves are tightly wound and their amplitude does not become excessively large. The variation in the mean state, which is slow in comparison to the time variation of the perturbations, results in a suppression of non-liaear effects at the resonances. Consequently, the definitions of growth rate and total angular momentum of the wave in the quasi-linear theory differ only little from the corresponding definitions in the linear theory. They are applicable over much longer time scales, however (chapter 6). (6) The mean state evolves in such a way that in the time-asymptotic limit it is stable with respect to non-axisymmetric perturbations. The ultimate fate of all perturbations is to be damped out. - 135 - •JS A coherent view of tightly wound spirals. The. conclusions form the basis of a theory of tightly wound spiral structures for flat galaxies which we shall now summarize. Spiral density waves are initially generated by a gravitational in- bcability that is inherent to cold stellar disks. They will, after some initial period, dominate over any other perturbations which may have been present at the same time. The driving mechanism of this instability lies in the stars that resonate with the wave. These socalled resonant stars absorb angular momentum at the expense of the non-resonant stars. These non-resonant stars are there- fore slowed down in respect of the rotation which these same stars would possess in the unperturbed disk. This is expressed by adding to the total angular momentum of these stars a negative term J . This term ° wave grows as more angular momentum is lost to the resonant stars. During this process angular momentum has to be transported by the wave from the inner regions of the disk to the outer parts. For tightly wound spiral waves such a transport can only be effected by a trailing density wave (Lynden-Bell and Kalnajs, 1972). , The transport of angular momentum (and energy) causes a gradual change in the mean state of the stellar disk in the course of time. As long as. the . anplitudeof a tightly wound spiral wave is. not excessively large, the evolution of this mean state is quasi-linear in the sense that: (i) the relaxation time of the mean state is much larger than a typical evolution period; (ii) the disk stays cool-during a relatively long time; (iii) non^linear effects at ... the resonances are kept insignificant.so that resonant stars continue to ... subtract angular momentum'from the. wave. • . ...; . • • ".• ..-•.' On time scales larger than y ~'» the inverse of the typical growth rate - 136 - of the wave, the limitation of the wave amplitude is effected by the gaseous component present in the disk of the galaxy. The response of the gas to the stellar wave is essentially non-linear. Shocks are formed that cause the gas to pile up at the bottom of the gravitational potential well of the wave, with as a result the subsequent formation of young stars and H II regions. These bright objects form the visible outline of the stellar densit/ wave structure which in itself is unobservable. The same nonlinear interaction between the gas and a trailing density wave leads to a stabilization of the disk. This is because the growth of the amplitude of the density wave slackens because in a trailing wave the angular 2.4 momentum gained by the wave from the gas (^ amplitude ) counteracts the 2 angularraoment-umlost by the wave to the resonant stars (^ amplitude ). The different dependence on amplitude of the wave makes it possible that in the end an equilibrium is reached in which as much angular momentum is gained by the wave from the gas as is lost by the wave to the resonant stars. The observed spiral density waves are therefore expected to have (real) frequencies that lie in the allowed range for growing waves. The final amplitudes are not expected to exceed about 10 percent of the local mean gravitational field. 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J. 95, 20. - 141 - ACKNOWLEDGEMENT At this place I want to express my indehtitude to the Netherlands Foundation for the Advancement of Pure Reserach (Z.W.O.) for their financial support during part of the time in which this research was done. I am grateful to ray promotor, Professor van de Hulst, to Professor J.H. Oort, to Dr. B. van Leer and Dr. W.W. Shane and to many others for their stimulating and continued interest in my work and their very useful advice. Through the kind and timely assistance of Miss Lenore de Leeuw and Mrs. Henny Tappermann, as well as of Mr. P. van den Hoed and Mr. J.F. Planken, this booklet could be prepared in time. I am sincerely indebted to them for their help. . •... . .:••<•'-.: .!,.'• .: SAMENVATTING Spiraalarmen werden voor het eerst waargenomen in 1845 in de Draaikolknevel, M 51 en later in vele andere extra-galactische stelsels, zo ook in ons eigen Melkwegstelsel. De aanwezigheid van differentiële rotatie in deze melkweg- stelsels vormde een groot probleem bij de interpretatie van de spiraalstruc- turen van deze stelsels: 6f er moet een mechanisme bestaan dat voorkomt dat de spiraalarmen door differentiële rotatie worden weggevaagd óf de spiraal- Q armen moeten vrij frequent (om de ca. 3.10 jaar) opnieuw worden opgewekt door een nog onbekend mechanisme. Hoewel in de vijftiger jaren veel aandacht aan het spiraalarm probleem werd besteed, zette pas in 1964 een belangrijke ontwikkeling in toen Lin en Shu de theorie van dichtheidsgolven presenteerden. Deze theorie berust op het idee dat dichtheidsverstoringen in melkwegstelsels in stand kunnenworden gehouden door de potentiaalverstoringen die door die dichtheidsverstcringen zelve teweegworden gebracht. Zulke (zelf-consistente) dichtheidsverstoringen vertonen een golf-karakter waarbij de dichtheidsgolf een spiraalvorm aanneemt die met een constante hoeksnelheid roteert in het :.ï melkwegstelsel. Essentieel in deze theorie is het besef dat deze golf voor- namelijk door de sterpopulatie in stand wordt gehouden en zich dus niet op- vallend manifesteert. Slechts doordat het eveneens aanwezige gas (en daarmee ook de uit het gas gevormde jonge sterren) zich onder invloed van de met de dichtheidsgolf geassocieerde potentiaalgolf ook opeenhoopt in een spiraal- vórmige structuur, wordt de golf waarneembaar, optisch zowel als bij radio- frequenties. Hiermee is het spiraalprobleem dus tot een sterdynamisch probleem teruggebracht. Hoewel met deze theorie een belangrijke vooruitgang werd ge- boekt bestonden er van de aanvang af van dit onderzoek enkele essentiële problemen. Het bleek dat ook met de door Lin en Shu voorgestelde golftheorie, - 143 - het bestaan van spiraalstructuren op lange termijn niet kan worden verklaard. Wanneer men een golfpakket bekijkt, opgebouwd uit Lin golven, dan verplaatst dit golfpakket zich naar binnen totdat het in het centrale deel van het melkwegstelsel wordt geabsorbeerd. Ook kan de Lin en Shu theorie geen voor- spelling doen omtrent de frequenties van de golf, een zeer belangrijke para- meter voor de interpretatie van de 21-cm lijn waarnemingen van extragalac- tische stelsels (bijv. met de SRT). Ons eigen onderzoek heeft zich daarom gericht op de volgende twee probleem- stellingen: a) Hoe kunnen we spiraalstructuren verklaren met levensduren langer dan 10 jaar. b) Waardoor wordt de frequentie van de waargenomen spiraalgolven bepaald. Allereerst hebben we in hoofdstuk 1 een kort overzicht gegeven van een aantal karakteristieke eigenschappen van melkwegstelsels. Daarna zijn in hoofdstuk 2 de nodige formules geciteerd en technieken uitgelegd, die "nodig zijn voor .de volgende hoofdstukken. Ten einde de stabiliteit van sterschijven te kunnen onderzoeken ïs liet bélaig- rijk om meer in detail enkele eigenschappen van sterrenstêlsëïö Inét ëèri'tijd-*- onafhankelijkeaxisymmetrischedichtheidsverdeling te beschrijven. In hoofdstuk 3 beschrijven we eerst de individuele sterbanen en vervolgens-geven we een korte discussie over enkele macroscopische eigenschappen zoals de'snelheids- • dispersies. In hoofdstuk 4 wordt vervolgens het gedrag van kleine versto- ringen op een tiidonafhaakelijke axisymmetrischedichtheidsverdeling bestu- deerd. Nadat we eerst formeel het probleem beschrijven als beginvoorwaarde - probleem, onderzoeken we vervolgens de stabiliteit van het sterrenstelsel door middel van een beperkende voorwaarde,; opgelegd door het . behoud van impulsmoment. Een uitdrukking voor dé groeisnelheid van dichtheidsgolven ..'••• • • . ' - 144.-. •• •"'•••• ' '••.••'• , ••" ••• •- wordt afgeleid, waaruit kan worden geconcludeerd dat, als schijfpopulaties instabiel zijn t.o.v. spiraalvormige verstoringen, de frequenties van deze instabiliteiten binnen bepaalde grenzen moeten liggen. Deze grenzen zijn zo- danig dat het aannemelijk lijkt dat deze instabiliteiten inderdaad optreden. Instabiliteit van sterschijven is ook geconstateerd in elders uitgevoerde numerieke experimenten met grote aantallen sterren. Hoewel de instabiliteit van de sterpopulatie op zichzelf een eenvoudige verklaring geeft voor de vorming van een spiraalstructuur, is hiermee het probleem nog niet opgelost. Een stabiliteits-analyse is slechts een eerste stap in de beschrijving van het gedrag van een fysisch systeem. Behalve een spontane groei van spiraal- structuur laten de boven vermelde numerieke experimenten zien dat bij een steeds toenemende golfamplitude de snelheidsdispersie van de sterren sterk toeneemt, hetgeen tenslotte leidt tot een stabilisering van de sterreschijf en het wederom verdwijnen van de spiraalstructuur. De waargenomen kleine snelheidsdispersies en langlevende spiraalstructuren in ons eigen melkweg- stelsel wijzen erop dat klaarblijkelijk andere dan tot nu toe beschouwde effecten de evolutie van de structuur van het melkwegstelsel beïnvloeden. In hoofdstuk 5 wordt het gas als mogelijke belangrijke invloedsfactor beschouwd. We tonen aan dat het gas door zijn interactie met de groeiende golf een stabiliserende invloed op deze golf uitoefent die tot een evenwicht leidt waarbij de spiraalgolf met een zekere eindige amplitude blijft bestaan. De 9 tijdschaal waarin dit evenwicht zich instelt ligt in orde van 5.10 jaar. i 8 '^ Op tijdschalen groter dan 5.10 jaar wordt de in hoofdstuk 4 aangenomen : J • p lineaire theorie en daarmee ook de in hoofdstuk 5 beschreven wisselwerking :5g niet het gas dubieus. Het is echter mogelijk onder bepaalde voorwaarden het :3 gedrag van de sterren en de golf op langere termijn te beschrijven met een ï; quasi-lineaire theorie. Deze wordt in hoofdstuk 6 ontwikkeld, geïnspireerd 00 - 145 - :5 soortgelijke ontwikkelingen in de plasmafysica. Hierbij uordt tevens een iets andere definitie van de groeisnelheid gegeven en worden de voorwaarden van geldigheid van deze theorie onderzocht. We tonen dan aan dat, zolang de amplitude van de golf niet al te groot, maar ook niet al te klein is, de quasi-lineaire theorie van toepassing is. Voor een typisch spiraalstelsel kan dit bijv. betekenen dat de potentiaal van de dichtheidsverstoring tussen 1 en 7 procent van de plaatselijke potentiaal van het ongestoorde stelsel moet liggen. De beschrijving van de golf door de quasi-lineaire theorie ver- andert de beschrijving van de stabiliserende invloed van het gas op de golf niet, zodat de conclusies uit hoofdstuk 5 kunnen worden gehandhaafd. De eindige amplitudes die worden bereikt zorgen omgekeerd dat de quasi-lineaire theorie toepasbaar is. Op lange termijn zullen de eigenschappen van het systeem gaan veranderen en daarmee ook de eigenschappen van de golf, die ten- slotte dan gedempt wordt. De tijdschaal hiervoor is echter groot, van de orde van 1-2 x 1010 jaar. De belangrijkste conclusies worden uitvoeriger samengevat in. hoofdstuk 7. Resumerend leidt or.s onderzoek tot een theorie van de spiraalstructuür van melkwegstelsels waarin deze structuur oorspronkelijk ontstaat door'een in- stabiïiteit van de sterpopulatie.'Naarmate dé'golf groeit wordt de infcèrScxie met het gas steeds belangrijker, totdat tenslotte een soort van evenwicht wordt bereikt. De stabilisatie van de golf door het gas voorkomt dat de amplitude van de golf te zeer toeneemt en voorziet in de mogelijkheid dat'de spiraalstructuür zich kan handhaven gedurende zeer lange perioden. De fre- quentie van de golf wordt volledig "bepaald door de eigenschappen van de ster- populatie, die het optreden van de instabiliteit beheersen. De uiteindelijke vorm en amplitude van de dichtheidsgolf hangt af van de dynamische eigen- schappen van de sterreschijf, zowel als van de structuur van de gasmassa's, - 146 - inclusief de massaverhouding'tussen gas en sterren. Zowel de waarnemingen van werkelijke spiraalstelsels, waaraan o.a. met de Synthese Radiotelescoop te Westerbork met man en macht gewerkt wordt, als ook nadere analytische en numerieke onderzoekingen, zullen hier uitsluitsel moeten geven. Hier ligt voor de toekomst een hoogst interessant terrein van onderzoek. - 147 - STUDIE OVERZICHT Op verzoek van de Faculteit geef ik hier enkele bijzonderheden omtrent mijn opleiding. Geboren te Haarlem in 1943, bezocht ik aldaar de lagere en middel- bare school en deed er eindexamen HBS-B in 1962. Daarna begon ik mijn studie in de wis- en natuurkunde aan de Rijksuniversiteit te Utrecht, waar ik het candidaatsexamen (a) met hoofdvakken wiskunde en natuurkunde en bijvak sterrenkunde aflegde in september 1966. Gedurende het schooljaar 1965/1966 was ik een jaar als lerares natuurkunde verbonden aan een lyceum te Rotter- dam. Tijdens mijn doctoraalstudie te Utrecht, met hoofdvak theoretische sterrenkunde en bijvak theoretische fysica, volgde ik de colleges van o.a. Prof. Dr. H.G. van Bueren en Prof. Dr. N.J. van Kampen. Mijn afstudeeronder- zoek omvatte een studie naar de multiplet factoren van optische overgangen in LS - koppeling in meer electronen systemen. Het doctoraalexamen werd (cum laude) in september 1969 afgelegd. Na een verblijf van een klein jaar aan de Universiteit van Maryland, waar ik werkte aan lijnverbredingsproblemen onder leiding van Prof. Dr. H. Griem, was ik acht maanden in dienst van de B.V. Noord-Hol lands che üitg. Mij, als bureau-redactrice. In januari 1971 kwam ik voor een gedeeltelijke dagtaak in dienst bij de Sterrewacht te Leiden, welk verband in 1972 werd vervangen door een tweejarig ZWO-subsidie voor het verrichten van theoretisch onderzoek over de dynamica van melkwegstelsels. Tenslotte kwam ik in 1974 in tijdelijke di ? 148 - ERRATA p. 42, line 3: Read w9, R and e can be obtained. p. 44, line 15: Read again for e = 0.5 and fi /ic0 = 1. R R p. 46, Fig. 33> vertical coordinate: Read min/ o instead of R p. 54, Eq.(3.67): Read P instead of Po. p. 66, line following eq. (4.19): Read (4.17) instead of (4.18). p. 70, line 2: Read section 1.2 instead of section 1.6. p. 100, line above eq. (5.5): Read this definition and eq (5.4b) we obtain from eq.(5.3) p. 109, eq. (6.5). The limits of integration are R and R. p. 119, Replace line 9 by = AJ (6.43) so' smax p. 125, line 8: Read response instead of resonance. p. 127, line 10 a-nd formula (6.67) Replace by: eq. (6.25) by non_»'es. + $2S1JL \Z Stars Stars The righthand side of eq. (6.67) is negative for growing waves. P. 127, line 10 from below: Instead of: or D^ = 0, k, m = I, 2, read: for all possible values of 1. and 1^ or e*p ^ £oCt')di' = o „ P. 127, line, 6 from below: Read expj (fo (t')dt' = 0 instead of Dkm « i-1 F i ( Hieibij nodig Ik u uit tot/iQt bijwonen war. mijn promotie op woensdag 16 april 1975 Cirn 15.15 uur in de Senaatskamer van de Rijksuniversiteit te Leiden, Academie- gebouw, Raponburfl 73, I.eiden. In verband met d« beperkte ruimte in de Senaatskamer, wordt van uw voornemen de promotie bij te wonen gaarne zo spoedig mogelijk bericht ontvangen. RECEPTIE NA AFlsOOP VAN DE PROMOTIE IN HET ACADEMIEGEBOUW RAPENBUROj73, LEIDEN