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Structure and stability of self-gravitating discs Davies, John Bruce 1980

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STRUCTURE AND STABILITY OF SELF—GEAVITATING DISCS by JOHN BRUCE DAVIES B.Sc. r University of Bales, 1963 A THESIS SU3MITTED IN PARTIAL FULFILBENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n THE FACULTY OF GRADUATE STUDIES Department of Geophysics and Astronomy We accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA n.s • r C a l i f o r n i a I n s t i t u t e o f Technology, 1968 March, 1979 John Bruce Davies, 1979 In present ing t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of the requirements f o r an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree t ha t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r re ference and study. I f u r t h e r agree that permiss ion f o r ex tens i ve copying of t h i s t he s i s f o r s c h o l a r l y purposes may be granted by the Head of my Department or by h i s r e p r e s e n t a t i v e s . I t i s understood that copying or p u b l i c a t i o n of t h i s t he s i s f o r f i n a n c i a l ga in s h a l l not be a l lowed wi thout my w r i t t e n permi s s ion . Department The U n i v e r s i t y of B r i t i s h Columbia 2075 Wesbrook P lace Vancouver, Canada V6T 1W5 Date DE-6 BP 75-51 1 E ABSTRACT Three centuries ago, the concept of the Solar Sytem being formed from an unstable disc was i n i t i a l l y proposed. This research examines this cosmogony by the classical technique of i n i t i a l l y obtaining the density structure of steady state discs, and gravitational i n s t a b i l i t y of such systems Is then investigated i n order to examine correlations between observed phenomena i n the Solar System and predictions of the theory. A f l u i d mechanical approach to the steadystate axisymmetric structure i s formulated for isothermal and polytropic gas systems, with uniform or r a d i a l l y dependent rotation. The equations are reduced to a single quasi-linear e l l i p t i c a l p a r t i a l d i f f e r e n t i a l equation governing density, and known external boundary conditions are necessary to yield an unique density solution. When the external density i s non-zero, flattened discs are possible solutions of the basic system. Two asymptotic techniques i n spherical and c y l i n d r i c a l coordinates are created to obtain approximate solutions of the steadystate structure. Both techniques show that a self-consistent disc i s composed of a high-density central bulge encircled by a low-density f l a t outer disc. Gravitational i n s t a b i l i t y i n gaseous discs i s now formulated by the linear perturbation of the fundamental variables, density, pressure, gravita-tional potential and velocity. As the Solar System i s essentially a planar structure, axisymmetric radial i n s t a b i l i t y along the equatorial plane of rotation i s examined. Such ring type modes are shown to be unstable to shear and tend to self-coverage. A dispersion relation i s obtained analytically which indicates that the wavelength between rings i s approximately inversely proportional to the square root of the steadystate density at marginal s t a b i l i t y . However for the pure gas disc, the wavelengths are too long for any correspondence with the present spacing of the Planets. i i i As usual, the presence of dust is invoked close to the equatorial plane. Radial instability in this gas-dust disc has a dispersion relation for the resultant wave in which the gas and dust move together such that the density term is multiplied by the dust-gas mass loading ratio. Thus the wavelengths at neutral stability will be correspondingly shorter and a correlation of ring density maxima with Planetary positions in the Solar System is obtained for reasonable values of three dimensionless parameters. If any planets exist outside Pluto the theory shows their distance apart can be expected to be similar to that of the Outer Planets, 10 a.u. Solar Systems formed by this type of instability in self-gravitating dust-gas discs can be expected to have linearly increasing planetary distances close to the central Sun "(Titius-Bode Law) with a more constant spacing further out as illustrated by our Solar System. Table of Contents Chapter One : Thesis Summary Introduction Aims Chapter Three Summary Chapter Four Summary Chapter Five Summary Conclusion Chapter Two Review of Rotating Systems Chapter Three : Steady State Disc Structure Introduction Steady State Theory Part 1 : Spherical Coordinate Expansions Analysis Linear Problem Disc Solutions Results Part 2 : Cylindrical Coordinate Expansions Fundamental Equations Asymptotic Discs Boundary Layer Theory -N Asymptotic Expansions Disc Region Expansion Central Bulge Expansions Matching Of Solutions Composite Expansion Solution Technique Numerical Solutions Chapter Four : Ring Pertubations In Gas Discs Gravitational Instability Of Discs Governing Equations Case 0-): M=0, Axisymmetry Analytical Approach Numerical Solutions Results 1 Chapter Five : Ring Perturbations In Dust-Gas Dusty Discs Governing Equations Instabilities Results Dust-Gas Mass Loading ...Implications and. Speculations A3 47 49 53 57 59 64 66 68 70 70 71 75 78 85 86 Discs 87 87 87 88 95 96 Conclusion Figures Bibliography Appendices Appendix A : Isothermal Gas Sphere Appendix B : Mathematical Properties Appendix C : Non-Isothermal Perturbations Appendix D : Linear Problem Appendix E : Disc Equation Analysis Appendix F : L i s t of Symbols and Flowcharts Appendix G : Computer Program List of Tables Table I : Bode Law Values And Planetary Distances Table II : True And Calculated Planetary Distances v i i L i s t of Figures Figure 1: Courtesy Of Chandrasekar (1969). 10tV Figure 2: Synthetic Iso Disc, 103 Figure 3: Synthetic Iso Disc, 104 Figure 4:" Synthetic Iso Disc, 105 Figure 5: Synthetic Polydisc, 106 Figure 6: Synthetic Polydisc, 107 Figure 7: Synthetic Polydisc, 108 Figure 8: Synthetic Iso Disc, 109 Figure 9: Iso Composite Disc, 110 Figute 10: Iso Composite Disc, 111 Figure 11: Iso Composite Disc, 112 Figure 12: Spherical Isothermal Density Solutions 113 Figure 13: Density in Isothermal Flat Disc Region, Fig 9 114 Figure 14: Oscillations, Gas Composite, Fig 10, L=0 115 Figure 15: Oscillations, Dust-Gas, Fig 9, L=50 116 Figure 16: Oscillations, Dust-Gas, Fig 10, L=99 117 Figure 17: Oscillations, Dust-Gas, Fig 11, L=49 118 Figure 18: Equatorial Density, Iso Composite, Fig 10 119 Figure 19: Equatorial Density, Iso Synthetic 120 Figure 20: Oscillations, Dust-Gas, Synthetic 121 Figure 21: Graph of Calculated Versus Real Positions 122 v i i i Acknowledgements I t i s a p l e a s u r e t o acknowledge the guidance and encouragement of ay t h e s i s a d v i s e r Dr. H. Ovenden and h e l p f u l d i s c u s s i o n s with Drs. G. Fahlman, R. H i u r a , F. ¥an, L. flysack, and fi. P u d r i t z . Thanks a l s o t o the members o f the committee f o r t h e i r c r i t i c i s m s and s u g g e s t i o n s r e g a r d i n g the r e s e a r c h . T h i s t h e s i s owes much to the i n d u s t r y and morale of my wife t a whom i t i s d e d i c a t e d , nonetary support o f t h e Governments of Canada i s g r a t e f u l l y acknowledged. 1 Chapter One Introduction That our solar-system i s unique has been a much argued conjecture. Whether other solar systems e x i s t , and the p o s s i b i l i t y of i n t e l l i g e n t e x t r a t e r r e s t r i a l l i f e , i s not only of major s c i e n t i f i c importance. The consciousness of mankind w i l l a l t e r d r a s t i c a l l y when such questions are answered unequivocally. Without hard observational evidence of other solar systems, i t becomes even more necessary f o r the theorist to examine the possible formation of such systems. Three centuries ago Kant and Laplace developed the hypothesis of formation of the solar system by contraction of a disc. This disc shed rings which collapsed to y i e l d planets and a c e n t r a l sun. Other theories , numerous and diverse have been hypothesized and analyzed. Now, our solar system exists on the outer disc of a s p i r a l galaxy. These disc galaxies populate the observed universe together with spheroidal galaxies. Discs thus play a v i t a l r o l e i n man's visi o n of h i s universe. Throughout our cosmogony, knowledge of the structure and s t a b i l i t y of discs i s necessary to the comprehension of our environment. To be applicable to solar system formation the consequences of i n s t a b i l i t i e s in such discs must correlate at least with the observed properties of our one piece of evidence, t h i s solar system. Even then, assumptions are beinq made that a l l solar systems are formed i n the same manner. Such i s quite probably not the case as other mechanisms have been proposed, and r a r e l y 2 does Nature depend on only one method. Aims The aim of t h i s research i s to obtain the structure of steady state s e l f - g r a v i t a t i n g discs, and investigate the r a d i a l i n s t a b i l i t y of such discs composed of gas and dust with emphasis on th e i r r e l a t i o n s h i p to solar system formation and the properties of Our Solar System. The property of most relevance to t h i s theory i s the spacing of planets i n the Solar System, which occur close to the eguatorial plane of rotation. This pattern was i n i t i a l l y analysed by T i t i u s and Bode whose law expresses the distance of the planets from the Sun i n terms of the Earth's distance (a. u) as 0.4*0.3*2 , Allen (1972). Table I Planet Bode 1s Law (a.u) Planetary Distance I (a.u) Mercury - oo 0.4 0.39 Venus 0 0. 7 0.72 Earth 1 1. 0 1.0 Mars 2 1. 6 1.52 Asteroids 3 2. 8 2.9 Jupiter 4 5. 2 5.2 Saturn 5 10.0 9.5 Uranus 6 19.6 19.2 Neptune 7 38.8 30. 1 Pluto 8 77.2 39.5 3 It can be seen that t h i s simple phenomological law breaks down i n the outer regions where the spacing of planets becomes approximately constant. Our research w i l l aim to explain t h i s planetary spacing in terms of r i n g type density waves along the equatorial plane of a s e l f - g r a v i t a t i n g disc which are unstable and self-converge . The density maxima are located at a fundamental spacing dependent on steady-state density structure and dust-gas mass loading. A Bode type law i s found close to the central rotation axis where the density increases rapidly toward the centre. Further out i t i s shown that the density w i l l approach a constant on the equatorial plane and consequently a constant spacing of rings i s obtained. With such goals i n mind, th i s thesis w i l l describe and analyse a self-consistent approach to astrophysical discs. In t h i s chapter, the scenario of the theory w i l l be described. The t h e s i s i s structured along the c l a s s i c a l l i n e s of analysis of t h e o r e t i c a l continuum models. I n i t i a l l y a steady state structure and i t s s t a b i l i t y properties w i l l be examined. Then an acoustic l i n e a r perturbation analysis w i l l be investigated. F i n a l l y , we s h a l l show the necessity of dust i n order to y i e l d a s i m i l a r distance pattern to that observed i n our solar system. In chapter two, a review of previous research relevant to this thesis w i l l be given. In chapter three, the equilibrium structure of the disc i s investiqated. In chapter four, using a v i r i a l approach, v i r i a l s t a b i l i t y conditions governing the whole system are described. Chapter f i v e investigates the acoustic modes of r a d i a l and azimuthal perturbations of the disc and 4 the i r l o c a l s t a b i l i t y c r i t e r i a . Chapter six w i l l extend the theory to dust-gas mixtures and show that such systems at i n s t a b i l i t y w i l l y i e l d a distance pattern which corresponds with the observed solar system • A more detailed summary of the chapters of o r i g i n a l research w i l l now be given. Chapter Three The t h i r d chapter incorporates the theory governing the equilibrium structure of discs. The eguilibrium w i l l be discussed for a c y l i n d r i c a l l y symmetric disc rotating about a central axis. F l u i d dynamical equations of continuity and conservation w i l l be used. Both s t e l l a r and gaseous systems have been treated previously by f l u i d dynamics with success. The eguilibrium disc w i l l be assumed to be composed of a single isothermal or polytropic f l u i d . The equation of state i s added to the conservation equation and the eguation governing s e l f -g r a v i t a t i o n . As s e l f - g r a v i t a t i n g bodies i n the universe appear to be embedded i n very low but non-zero density media, a f i n i t e external pressure w i l l be present as boundary condition. This system of equations i s reduced to a single p a r t i a l d i f f e r e n t i a l eguation governing density. Perturbation techniques and asymptotic expansions w i l l be used for solving the basic equation. The previous approaches to solution of the basic eguation i n s p h e r i c a l coordinates w i l l be analysed . In a l l these methods a zero density external surface i s assumed and consequently only s l i g h t l y deformed spheres were obtained. Bemoving t h i s 5 r e s t r i c t i v e boundary condition allows the construction of disc shaped solutions. Knowledge of the external boundary shape and associated conditions can yield an unique solution. The fundamental geometry of discs i s such that t h e i r thickness i s much less than the equatorial radius. Without detailed knowledge of the outer boundary t h i s geometrical information can be used i n an asymptotic expansion of the disc solution i n c y l i n d r i c a l coordinates i n order to obtain an approximate solution. On removing dimensions i n the basic equation , three basic dimensionless parameters are discovered. These are measures of the r a t i o of g r a v i t a t i o n a l potential energy to thermal energy (alpha) , the r a t i o of r o t a t i o n a l k i n e t i c energy to thermal energy (beta), and the square of the r a t i o of the thickness to the maximum radius of the disc (epsilon). As i n a l l f l u i d dynamical problems incorporating dimensionless numbers this allows the scale of the systems to vary widely. Thus, f o r d i s c s , the dimensionless parameter (6) i s expected to be much less than unity. I t i s found that t h i s parameter multiplies the r a d i a l d i f f e r e n t i a l operator i n the basic d i f f e r e n t i a l eguation. As epsilon i s expected to be small for a disc, the solution w i l l be approximated as an asymptotic expansion in powers of £. I t can be understood that the previous approaches to discs, wherein no r a d i a l e f f e cts were allowed i . e a semi-infinite slab, are obviously inconsistent. In fa c t such approaches ignore the r o t a t i o n a l v e l o c i t y e f f e c t which i s the fundamental physical cause of discs. Nayfeh (1973) has commented:" i t i s the r u l e , rather than 6 the exception, that asymptotic expansions i n terms of a small parameter are not uniformly v a l i d and break down in boundary layers". As Van Dyke (1975) remarks, when a small parameter multiplies one of the higher derivatives, i n a straight forward perturbation scheme, that derivative i s lost i n the f i r s t approximation and the order of the equation i s reduced. Thus, one or more of the boundary conditions must be abandoned and the approximation breaks down near where they were to be imposed, i.e the boundary layers. A boundary layer i s mathematically found to form around the central axis of the disc. I n t u i t i v e l y , i t can be expected that close to the central axis, the r o t a t i o n a l c e n t r i f u q a l forces w i l l be small and the c e n t r a l bulge geometry w i l l be approximately s p h e r i c a l , with a s i m i l a r central condensation as i n the spherical case. This i s confirmed mathematically f o r the r a d i a l extent of the central bulge w i l l be of the order of the central thickness. In the outer region where r o t a t i o n a l and g r a v i t a t i o n a l forces are of the same order, a f l a t t e n i n g occurs yielding a disc-shaped system. The solutions i n the two main regions, normal disc and central bulge, are combined i n t o a composite solution with emphasis on obtaining the solution on the eguatorial plane. Numerical techniques are used to obtain the zeroth order solutions. Both approaches yield a central bulge , with density increasing r a p i d l y toward the centre encircled by a low density f l a t disc region. 7 Chapter Four In order to examine l o c a l s t a b i l i t y , a lin e a r non-d i s s i p a t i v e perturbation scheme i s set up in the usual fashion. The axisymmetric isothermal perturbations are governed by l i n e a r wave eguations that are reduced to a single eguation governing the basic variable that i s related to rate of change of angular momentum. As the solar system seems to have the planets d i s t r i b u t e d close to the equatorial plane, pertubations only i n r a d i a l and azimuthal directions w i l l be considered. The ca l c u l a t i o n s w i l l be performed for the density given on the equatorial plane of an isothermal disc. The wave eguation i s solved for the case of marqinal s t a b i l i t y i . e when the square of the frequency i s zero • The resultant mode wavenumber i s a function of radius, due to the non-uniform density, and the non-dimensional parameters tY,^ 5, S. The simplest case of axisymmetric perturbations i s considered in d e t a i l in order to elucidate the general properties of the system. Consideration w i l l be paid to the causes of i n s t a b i l i t y of these modes from examination of the dispersion r e l a t i o n s . Rayleigh found that any rotating body w i l l be unstable i f there i s a maximum i n i t s rotation curve. Spiegel (1972) has remarked on the ambiguity of any cause of i n s t a b i l i t y - a small change i n basic conditions w i l l often r a d i c a l l y a l t e r the i n s t a b i l i t y c r i t e r i a . Thus, the wave patterns of modes w i l l be discovered and analysed. In the axisymmetric case , a simple argument shows the inherent i n s t a b i l i t y of these perturbations. 8 We f i n d at marginal i n s t a b i l i t y i n the axisymetric case, that rings are formed which, due to shear motions, are probably unstable and self-converge . It i s then shown that ring-type i n s t a b i l i t i e s i n pure gas discs w i l l have wavelengths too long to y i e l d a solar system distance pattern . Other researchers have found t h i s r e s u l t from d i f f e r e n t approaches,and i n order to y i e l d planetary systems they usually invoke the presence of dust. Chapter Five Thus we investigate disc structure for a dust-gas mixture. In the no-slip case the basic mathematics and physics are similar to the previous case but with an added equation governing the inte r a c t i o n of dust and qas close to the equatorial plane. Allowinq weak waves in t h i s dusty disc qives i n s t a b i l i t i e s as before but with one major difference. We f i n d , as i n a l l dust-gas problems,that a hierarchy of waves i s produced, and that the wave of longest duration i s that i n which the dust and gas move together. We show that t h i s wave has proportionally a much shorter wavelength than i n the pure gas case. When the density r a t i o of dust to gas i s of order 50 , we obtain a pattern of r e l a t i v e distances between rings that correlates well with the observed planetary distances. Close to the c e n t r a l axis, a l i n e a r l y increasing i n t e r v a l between rings i s expected while out i n the disc region a more constant i n t e r v a l i s predicted for a Solar System produced via these i n s t a b i l i t i e s . The d i s t r i b u t i o n of angular momentum between Sun and planets observed i n the Solar System i s consistent with the predictions of the theory. Even though the number of planets 9 outside Pluto i s unpredictable, t h e i r mean separating distances can be expected to be s i m i l a r to that of the Outer Planets, 10 a. u. Conclusion Our research has developed and elucidated the following main r e s u l t s , concerning axisymmetric s e l f - g r a v i t a t i n g discs. Density solutions for uniform and d i f f e r e n t i a l l y r o t a t i n g discs, using f l u i d dynamics with no d i s s i p a t i o n , have been created by two d i f f e r e n t perturbation methods. The dynamical importance of the three dimensionless parameters alpha,beta,epsilon i s shown together with the necessity of f i n i t e external boundary conditions for an unique solution. Both asymptotic techniques y i e l d s e l f -consistent disc solutions composed of a high density central spherical bulge with outwardly decreasing density encircled by a low density f l a t disc region. V i r i a l analysis indicates the i n s t a b i l i t y of such discs at large c e n t r a l to external density r a t i o s , such that they tend to increase t h e i r moment of i n e r t i a . Linear acoustic analysis shows that the basic variable i s related to the rate of change of angular momentum. Axisymmetric rings , formed at marginal s t a b i l i t y , tend to shear and s e l f -converge . Dust-gas discs are developed s e l f - c o n s i s t e n t l y . At marginal s t a b i l i t y , weak waves have wavelengths proportionately shorter than the pure gas case. For a reasonable dust to gas mass loading r a t i o the r e l a t i v e distance between rings i s s i m i l a r to the observed pattern of our solar system planets. Trans-10 Plutonian planets, i f they e x i s t , are expected to have mean semi-major axes separated by distances s i m i l a r to the spacings of the Outer Planets. Flow charts of the basic research system and a n a l y t i c a l technique are given i n Appendix F together with an appropriate l i s t of symbols. The square of the angular velocity (in the unit irGp) along the Maclaurin and the Jacobian sequences. The abscissa, in both cases, is the eccentricity of the (1, 3)-section. Figure I : Courtesy of Chandrasekhar,(1969). 11 Chapter Two Review Of Rotating Systems In t h i s chapter we w i l l outline the previous research on s e l f - g r a v i t a t i n g , rotating systems. Much of t h i s work was inspired by the shape and figures of Earth, Planets, Stars and galaxies and, for special relevance to our work, disc systems, wherein the two main forces, gravity and rotation, act together. The fundamental Conservation Laws together with an appropriate Equation of State are additional constraints from which a continuum approach to the density structure of these bodies can be made. Most of the c l a s s i c a l research has concentrated mainly on bodies of constant density, even though they are p h y s i c a l l y impossible. The study tof the g r a v i t a t i o n a l equilibrium of homoqenous uniformly rotating masses began with Newton's inve s t i g a t i o n on the figure of the earth ( P r i n c i p i a , Book I I I , Propositions XVIII-XX). Newton showed that the e f f e c t of a small rotation on the figure must be in the d i r e c t i o n of making i t s l i g h t l y voblate ; and, further, that the equilibrium of the body w i l l demand a simple proportionality between the e f f e c t of r o t a t i o n , as measured by the e l l i p t i c i t y . eguatorial radius - polar radius the mean radius and i t s cause, as measured by mean g r a v i t a t i o n a l acceleration on the surface c e n t r i f u g a l acceleration at the equator 12 where G\ denotes the constant of gravitation and M i s the mass of the body. More precisely, Newton established the r e l a t i o n i n case the body i s homogeneous. An i n t e r e s t i n g h i s t o r i c a l observation may be made that t h i s appears to be one of the f i r s t uses of dimensionless numbers which play such an important part i n the continuum mechanics of such bodies. This subject i s usually s p l i t into two main branches; the steadystate density structures are constructed and then examined fo r secular and dynamical i n s t a b i l i t i e s . I n i t i a l l y the steady state density structure and shape of homogeneous, i s o t r o p i c rotating axisymmetric bodies was developed by MacLaurin (1742) who generalized Newton's r e s u l t to the case when the e l l i p t i c i t y caused by the rotation cannot be considered small, c f . Chandrasekhar ( 1 9 6 9 ) . MacLaurin's formula r<hf e 5 e> y where C i s e c c e n t r i c i t y defined by t 2"- J - Z A /R}~ ' A remarkable feature of Maclaurin's r e l a t i o n was noticed by Thomas Simpson ( 1 7 4 3 ) : for any angular v e l o c i t y less than a certain maximum value there are two and only two possible "oblata." This r e s u l t i s noteworthy i n that we cannot deduce from the f a c t of a small equatorial angular v e l o c i t y that the spheroid departs only s l i g h t l y from a sphere; for as J l - * 0 , we have two solutions: a solution which, indeed, leads to a spheroid of small e c c e n t r i c i t y and a second solution which leads to a highly flattened spheroid. Such a r e s u l t thus indicates that highly flattened discs 13 are possible solutions for the homogeneous case and when r e a l i s t i c equations of state are considered, the subsequent non-l i n e a r i t y can be expected to y i e l d these two families and possibly even others such as t o r o i d a l solutions. Recent numerical analysis by Harcus et a l (1977), has examined the stablest shapes of rotating axisymmetric s e l f - g r a v i t a t i n g homogeneous f l u i d s using a minimum energy c r i t e r i o n and found a variety of shapes possible though not necessarily stable. As physically r e a l bodies are non-linear, most of the c l a s s i c a l r esults are of a negative character, though s t i l l important, c f . Lebovitz (1967,1978) who stresses the following three important points. A necessary and s u f f i c i e n t condition for Jl_ to be independent of z i s that the surfaces of constant pressure coincide with surfaces of constant density,i.e., that p be a function of p only. Further in t h i s case the t o t a l potential i s also a function of p only. When i t e x i s t s , the equation of state and of energy consrvation may be thought of as determining the form of the p- p r e l a t i o n s h i p . Hence, by describing a p-yc r e l a t i o n s h i p , one avoids the complications of these further equations. Poincare ,s estimate.— The size of a rotating configuration i s l imited by the condition that the ce n t r i f u g a l accelaration not exceed the c e n t r i p e t a l acceleration of gravity. This leads , in the case of uniform rotation, to the following estimate by Poincare (1903) r e l a t i n g the angular velocity to the mean density: 14 L i c h t e n s t e i n 1 s Theorem— The shape of a rotating configuration i s constrained to have a plane of symmetry perpendicular to the axis of rotation . A c o r o l l o r y i s that a nonrotating configuration i s necessarily s p h e r i c a l . Jeans (1929) and Chandrasekhar (1933) employed expansion techniques to solve the more d i f f i c u l t problem of the shape and structure of isothermal and polytropic systems, the polytropic index being a measure of central condensation. These ideas w i l l be analysed i n depth l a t e r along with the further developement of expansion and perturbation theories. Another approach to the problem of rapidly rotating polytropes i s that of fioberts (1963) and Hurley 6 fioberts (1964, 1965). They have used a v a r i a t i o n a l p r i n c i p l e to f i n d the density s t r a t i f i c a t i o n under the assumption that the l e v e l surfaces are spheroids. In one case the e c c e n t r i c i t i e s of the spheroids were assumed a l l to be the same, the other case allowed them to vary and were determined by the v a r i a t i o n a l p r i n c i p l e along with the density s t r a t i f i c a t i o n . The more recent approach of numerical analysis and synthesis has been u t i l i s e d by James, Ostriker, Bodenheimer and collaborators i n a series of papers mainly concentrating on polytropes and small deviations from s p h e r i c i t y . A general result i s found for various angular momenta and polytropic equations of state such that above a c e r t a i n r a t i o of axes and k i n e t i c energy to potential energy ( 7J, ) that various i n s t a b i l i t i e s set i n . The generalized polytropic sequences bear close resemblance to the Maclaurin sequence im most respects. In p a r t i c u l a r , the generalised sequences do not terminate, they do 15 not r o t a t i o n a l l y eject matter, but they do reach points of b i f u r c a t i o n beyond which secular s t a b i l i t y probably ceases. Most research has concentrated on the shape of stars and thus on departure from sphericity due to rotation usually with only s l i g h t a x i a l changes. However the geometry of discs i s such that the thickness i s very much smaller than diameter. In recent years some e f f o r t has gone into the problem of determining a mass d i s t r i b u t i o n model which can account f o r the observed r o t a t i o n a l curves of galaxies. This has generally been c a r r i e d out by the method of spheroidal s h e l l s wherein a g a l a c t i c disk i s represented by a superposition of concentric spheroid s h e l l s of varying density and zero or f i n i t e e c c e n t r i c i t y . This technique has the advantage that the g r a v i t a t i o n a l f i e l d vanishes inside a spheroidal s h e l l , and i t has a r e l a t i v e l y simple form outside the s h e l l . Hunter (1963) has found i t convenient to represent the density and v e l o c i t y variations i n rotating disks by an expansion i n Legendre polynomials. This method works well for disks with r e l a t i v e l y f l a t density d i s t r i b u t i o n s , but for disks with a strong degree of concentration of mass toward the rotation axis, very high order terms dominate i n the Legendre polynomial expansions. In the following research, we w i l l employ a s i m i l a r yet more powerful technique in which t h i s convergence problem i s diminished. In order to obtain exact disc solutions, various s i m p l i f i c a t i o n s are made to the non-linear free-boundary problem. The cold pressureless disk has yielded many useful insights while the pressure disk has been solved a n a l y t i c a l l y i n 16 the simplest semi-infinite slab model. I n s t a b i l i t i e s i n both r a d i a l and thickness directions have teen examined including the influence of various s t r a t i f i c a t i o n s and rotations, c f . Hunter (1972). However, we have been able to obtain asymptotic solutions to the non-linear steadystate disc problem which show the conditions under which these approximations are useful. Note that various other shapes have been proposed as possible solutions for rotating s e l f - g r a v i t a t i n g bodies such as toroids with and without central bodies. Some of these observed figures and shapes may also be the result of the two main types of i n s t a b i l i t y which are possible in such systems secular and dynamic. Secular i n s t a b i l i t y i s the r e s u l t of mass motions that a l t e r the energy of a s e l f - g r a v i t a t i n g rotating body while keeping the mass and angular momentum constant. The energy i s dissipated as heat ( or g r a v i t a t i o n a l ) radiation and Hunter (1978) shows that the pa r t i c u l a r dissipation mechanism w i l l a ffect the onset of t h i s i n s t a b i l i t y . An example of thi s secular i n s t a b i l i t y i s the Jacobian sequence of homogeneous t r i a x i a l e l l i p s o i d s that can be obtained from the axisymmetric MacLaurin sequence i n a pa r t i c u l a r velocity and e l l i p t i c i t y ranqe but with lower enerqy. Chandrasekhar (1969) has pioneered the v i r i a l approach to the secular i n s t a b i l i t y analysis, and has used the dimensionless r a t i o of r o t a t i o n a l to gr a v i t a t i o n a l energy to determine a c r i t i c a l value above which secular i n s t a b i l i t y may be expected. Above a larger value of t h i s r a t i o , dependent on polytropic index, dynamical i n s t a b i l i t y sets i n and some natural 17 v i b r a t i o n a l mode of the system can grow in time. A form of i n s t a b i l i t y that i s peculiar to a s e l f -g ravitating system i s Jeans i n s t a b i l i t y (1929). The most elementary discusssion of i t comes from considering small disturbances to a uniform i n f i n i t e gravitating medium of constant volume density at r e s t . Sinusoidal disturbances with space and time dependence as exp.L ^ ( O t - f e . x ^ are possible provided the dispersion r e l a t i o n UOZ - kt. C.I - Ar<T <q f> , fe% /tel*" i s s a t i s f i e d where Q,is the velocity of sound appropriate to the medium. The e f f e c t of gravity i s to modify the sound waves that would otherwise occur and to make them unstable i f t h e i r wavelength exceeds the c r i t i c a l Jeans length, Vr B £U> (^J- The i n s t a b i l i t y arises because s e l f - g r a v i t y tends to cause a small increase in density to grow. This growth i s r e s i s t e d by pressure which tends to smooth out density perturbations but the d e s t a b i l i z i n g e f f e c t of s e l f - g r a v i t y tends to predominate at s u f f f i c i e n t l y large length scales. Ledoux's (1951) analysis of the isothermal gas sheet that i s s t r a t i f i e d i n plane p a r a l l e l layers symmetric with respect to the c e n t r a l plane shows that there i s no i n s t a b i l i t y to disturbances that are purely in the d i r e c t i o n of s t r a t i f i c a t i o n . This r e s u l t , which occurs i n the more general analysis helps j u s t i f y the thin-disk l i m i t i n showing that no i n s t a b i l i t y i s ignored when the d e t a i l s of the structure in the z direction are ignored. Rotation can add to the s t a b i l i t y . Chandrasekar (1961) has shown that the addition of a uniform rotation to Jeans' o r i g i n a l 18 analysis a f f e c t s s t a b i l i t y only for waves normal to the d i r e c t i o n of rotation. These waves are the ones of primary i n t e r e s t and possess the dispersion r e l a t i o n LO Goldreich has also investigated the simplest disks with Lynden-Bell (1964). Gravitational s t a b i l i t y of a s e m i - i n f i n i t e s t r a t i f i e d s e l f - g r a v i t a t i n g rotating sheet was investigated with inter e s t only i n waves i n the thickness d i r e c t i o n . Similar r e s u l t s to Ledoux(1951) were presented only for n o n - r e a l i s t i c discs. Other types of i n s t a b i l i t i e s have been considered by Spiegel and Zahn (1970) in reviewing i n s t a b i l i t i e s of d i f f e r e n t i a l r o t a t i o n . The amazing number of methods and conditions under which small environmental changes can induce d i f f e r e n t i n s t a b i l i t i e s i s well shown in t h i s a r t i c l e . The most widely discussed i n s t a b i l i t y i s that of a homogeneous i n v i s c i d f l u i d medium with the angular velocity a sp e c i f i e d function of , the distance from the axis of rotation. Bayleigh showed that i f the d i f f e r e n t i a l medium i s unstable. Remarkably enough, the condition for shear i n s t a b i l i t y of a plane p a r a l l e l i n v i s c i d flow was also given by Bayleigh who showed that i n s t a b i l i t y occurs when the velocity p r o f i l e has an i n f l e c t i o n point. The analogue of t h i s i n s t a b i l i t y c r i t e r i o n for the case of swirling flows with that when d o ^ o 19 at some CD , i n s t a b i l i t y occurs. When th i s condition i s met certain non-axisymetric disturbances are unstable. Now these considerations are modified when other e f f e c t s are considered as v i s c o s i t y normally plays a s t a b i l i z i n g role although exceptions can a r i s e . But, f o r most astrophysical s i t u a t i o n s , the v i s c o s i t y plays only a minor r o l e , Spiegel and Zahn ( 1 9 7 0 ) . The chief problem that must be confronted i s the e f f e c t of the known i n s t a b i l i t i e s which necessarily involves non-linear equations and i s of a higher order of d i f f i c u l t y than the s t a b i l i t y question i t s e l f . In the selection of correct models, a reasonable zeroth order approximation i s that the i n s t a b i l i t i e s tend to choke themselves off and keep the s i t u a t i o n at nearly c r i t i c a l or neutral i n s t a b i l i t y , just as in convectively unstable models where one often equates the temperature qradient to i t s value f o r neutral i n s t a b i l i t y . Much research has indicated the relevance of rotating gases to formation of solar systems. A number of conferences in recent years has concentrated on the cosmogony of solar systems. In the 1972 Nice symposium Cameron (1972) reviewed his and Pine's work on numerical models of the primitive solar nebula wherein a r b i t r a r y pressure and temperature d i s t r i b u t i o n s are used i n order to make the system tractable. However no exact mathematical treatment i s made of the equilibrium structure of th i s complex system or i t s i n s t a b i l i t i e s . General q u a l i t a t i v e r e s u l t s regarding formation of a solar system from a dust-gas rotating body are claimed. A more mathematical treatment of a dust-gas system and i t s relevance to solar system formation i s given by Spiegel (1972) 20 at the same symposium. His r e s u l t s follow that of standard workers in dust-gas systems who show that, in s u f f i c i e n t time, the gas and dust w i l l tend to move together due to drag forces. In more recent work, Goldreich and Hard (1974) have investigated the coagulation and formation of dust p a r t i c l e s toward the equatorial plane of th e i r sheet discs (i.e i n f i n i t e in radius ). Iheir main r e s u l t , l i k e that of other researchers, i s that the dust w i l l tend to form a layer or concentrated zone close to the equatorial plane. This work and i t s relevance to the author's results w i l l be examined in d e t a i l l a t e r . The o r i q i n a l hypothesis that the Solar system formed from a disc was put forward by Kant and Laplace who envisaqed a coolinq disc nebula that contracted, sped up due to angular momentum conservation and shed rings on the equatorial plane that were supposed to self-converge into planets. This scenario received much v a l i d c r i t i c i s m on account of the expected r e s u l t of t h i s theory of a fast rotatinq c e n t r a l Sun which i s not observed. Also rings formed by shedding are not capable of converging and are expected to be dissipated. This brought the disc approach into disrepute and a number of other theories were advanced. However the advantages of a disc o r i g i n are such as to s a t i s f y the observation of the planets a l l close to the plane of the e c l i p t i c , that they a l l revolve almost c i r c u l a r l y i n the same angular d i r e c t i o n around the Sun, and most rotate with vectors almost perpendicular to t h i s plane. These f a c t s must be s a t i s f i e d by any theory of Solar System formation together with the d i s t r i b u t i o n of angular momentum between Sun and planets, the distances of the planets, and t h e i r respective masses. 21 Chapter Three introduction The object of t h i s section i s to analyse previous research on s e l f - g r a v i t a t i n g rotating bodies , examine t h e i r domain of v a l i d i t y and create new solutions that w i l l describe discs. The p a r t i a l d i f f e r e n t i a l eguation governing steadystate density i s guasilinear e l l i p t i c a l thus necessitating known external boundary conditions for an unique solution. Two mathematical approaches, both involving asymptotic expansions i n a small parameter, w i l l be used to obtain approximate solutions. Steady, State Theory Even with the extensive work reviewed above no a n a l y t i c a l s e l f consistent theory of the steady state structure of discs has been created. Figures of rotating s e l f - g r a v i t a t i n g bodies have seemingly divided into two main categories; the s l i g h t l y deformed sphere with almost egual axes, and the disc with one axis (the thickness) small compared to the other two. These two apparently disparate systems are governed by the same fundamental mathematical equation. for the simplest case of incompressible systems, Simpson (1743) i n i t i a l l y noticed that i n McLaurin Spheroids for any angular ve l o c i t y less than a certa i n maximum there are two and only two 'possible' oblata, governed by the same eguation . The rel a t i o n s h i p i s shown in Figure 1 of Chandrasekhar (1969); the f u n c t i o n a l r e l a t i o n s h i p of the e c c e n t r i c i t y ( or r a t i o of minor to major axes ) to the angular velocity parameter being determined through use of the v i r i a l approach. 22 Because these incompressible systems are not r e a l i s t i c , attention has been directed to bodies with more r e a l i s t i c equations of state. In order to remove the temperature e x p l i c i t l y , the c l a s s i c a l equation of state i s used where pressure i s a function of density only. This i s also useful when allowing d i f f e r e n t i a l rotation with the anqular v e l o c i t y a function of c y l i n d r i c a l radius only. When pressure i s taken to be proportional to some power of density, such equations of state are termed polytropic. When the power i s unity the system i s isothermal and these are the systems that w i l l be discussed. The isothermal spherical body has the important property that i t s density only becomes zero at i n f i n i t y , while a f i n i t e size body has a non-zero external density. However, Nature appears to abhor a vacuum, and s e l f -g ravitating systems are taken to be embedded i n an environment of f i n i t e ncn-zero pressure and temperature. This implies that a f i n i t e s e l f - g r a v i t a t i n g body w i l l have a non-zero external density (pressure) boundary condition. E x p l i c i t l y , the problem to be solved i s to obtain models of the equilibrium density structure of s e l f - g r a v i t a t i n g , rotating discs obeying the above equations of state for f l u i d systems. In a l l but a p a r t i c u l a r case, the resultant p a r t i a l d i f f e r e n t i a l equation i s non-linear and no exact a n a l y t i c a l solutions are known. Two asymptotic expansion methods are used to obtain approximate solutions for the non-linear cases and the steps taken i n each method are outlined below. These w i l l be amplified and j u s t i f i e d i n the appropriate section. A : Spherical Expansion. 23 1. Derive the governing p a r t i a l d i f f e r e n t i a l eguation i n spherical coordinates for uniform rotation using the continuity equation, s e l f - g r a v i t a t i o n eguation and the appropriate equation of state. 2. Normalise variables with respect to c h a r a c t e r i s t i c values. 3. fiecognise the small parameter St = /2s^ as inhomogeneous term in the d i f f e r e n t i a l equation, where Jl_ i s the anqular velocity and i s the central density. 4. When pressure i s proportional to the sguare of the density, a l i n e a r p a r t i a l d i f f e r e n t i a l eguation i s derived which yie l d s an exact solution of the form ( spherical s o l u t i o n oo •o^. ^2. m ). The i ^ ^ a r e multipole terms composed of Bessel ry\-o functions for r a d i a l dependence and Legendre polynomials for angular dependence. are arb i t r a r y c o e f f i c i e n t s to be determined. 5. For the general non-linear case, an approximate solution i s obtained by a regular asymptotic expansion of the solution i n powers of the small parameter £ c , the f i r s t two terms of which are the same form as the exact solution i n the l i n e a r case, ( oo ^, spherical solution + cY. • ^ ^ 7 ^ 2 . ^ )• 6. These are e l l i p t i c a l boundary value problems and thus a known external density and boundary shape w i l l uniquely determine the value of each of the c o e f f i c i e n t s . This l o g i c can be reversed such that i f we choose the values of the external density and the c o e f f i c i e n t s , then the shape of the external boundary w i l l be uniguely s p e c i f i e d . 7. As these asymptotic expansions are of the density only to OC^c) °f t n e central density, the external density must also 24 be of t h i s order O ^ S c ^ ) for a consistent solution and i s suitably chosen of t h i s order. 8. In order to obtain examples of disc models and comprehend thei r i n t e r n a l density structure and boundary shape, computations and density contour plots are generated by computer techniques. To obtain a disc shaped body, the programmer inputs p a r t i c u l a r c o e f f i c i e n t values and external density of 0(S cP t^that are obtained by t r i a l and error u n t i l the desired shape i s observed. From an edge-on view, the most important term for a disc i s the dipole component with c o e f f i c i e n t A^. A subset of disc-shaped systems can be generated by allowing only non-zero values for A^ . A^ and keeping higher c o e f f i c i e n t s zero, A ^ . Such r e s t r i c t i o n s do not allow the modelling of highly fl a t t e n e d disc systems which would require judicious choice of a much larger number of c o e f f i c i e n t s and i s thus not computationally feasible f o r t h e i r synthesis, and so a d i f f e r e n t asymptotic technique must be used. 9. Analysis of the density structure of these simple models shows that close to the c e n t r a l r o t a t i o n a l axis the density increases rapidly inward with correspondinq large density gradients, suggesting the a p p l i c a b i l i t y of a boundary layer approach as in the following asymptotic technique of solution. B. C y l i n d r i c a l Expansion. 1. The f i r s t step in t h i s solution technique i s to derive the governing p a r t i a l d i f f e r e n t i a l equation i n c y l i n d r i c a l coordinates. 2. Normalise the variables with respect to appropriate c h a r a c t e r i s t i c values d i f f e r e n t than those for the spherical 25 expansion method. 2. 3. fiecognxse that a small parameter £ } (disc thickness / radius)^ multiplies the terms governing the r a d i a l density gradient i n the p a r t i a l d i f f e r e n t i a l eguation. As £.-»• O the disc becomes highly flattened. 4. In regions where the r a d i a l density gradients are small, a solution corresponding to a f l a t slab i s the f i r s t approximation and i s c a l l e d the normal disc region. 5. In regions where the r a d i a l density gradients are large, the centre and outer r a d i a l edge, these terms cannot be ignored i n the d i f f e r e n t i a l eguation and boundary layers i n density w i l l e x i st. The solution i n t h i s central boundary layer, termed the central bulge, i s obtained and matched with the solution i n the normal disc region. 6. A composite solution composed of spherical c e n t r a l bulge and f l a t disc i s obtained for these inner two regions and i s of same form as the asymptotic solution i n the spherical method. 7. This approach does not assume an exact value of £, only that i t i s small, and thus s u f f i c i e n t conditions are not a v a i l a b l e for a t o t a l disc solution. The p a r t i c u l a r value of 6 can be related to invariants of the disc through use of the v i r i a l i n a s i m i l a r manner as that of Chandrasekhar (1969) for the homogeneous bodies. We s h a l l i n i t i a l l y examine the spherical coordinate expansion as c e r t a i n r e s u l t s and technigues from these solutions w i l l be used i n the subsequent c y l i n d r i c a l coordinate expansion v method. 26 Part 1, i Spherical Coordinate Expansion Let us begin by examining the fundamental equations governing these polytropic systems with r i q i d r o t a t i o n . 1. Hydrostatic Equation = " ^ JL • uZ> Cz0 where ^ i s pressure . 2. Equation of state r — P (_2-2.) where f\ i s polytropic index . 3. S e l f - q r a v i t a t i n g equation - - A - -7T ^  ^  c z . a ) where 5- i s g r a v i t a t i o n a l potential n^is constant SI i s constant angular velocity In spherical polar coordinates with a x i a l symmetry, cj^ denotes the r a d i a l coordinate and the cosine of the colatitude. The usual approach i s to normalise the density with respect to the c e n t r a l density and define the polytropic variables and Sr by and Note how the fundamental dimensionless parameter & c i s s i m i l a r i l y found to play a v i t a l role in the homogenous Maclaurin Spheroid. Combining (2.1) and (2.2) and integrating gives 55 4> + scf'{(» • ( I ' f | -f c o a s t e d 27 where ^  (jS)\s the second Legendre polynomial . From (2.5) and (2.3) , the fundamental eguation i s derived VZ <r = . S c - er n with boundary condition | = o ; cr= 1_ ( V c r = o When 8 c i s zero, the spherical Emden equation r e s u l t s the solutions of which have been well examined and tabulated . The Emden sphere i s defined by ^  = , where 19(^)^0, and represents the zero density surface of the non-rotating configuration. It i s assumed by a l l authors i n analysis of t h i s fundamental eguation (^2.7) that S>c,«'\ . Such a re s u l t i s expected f o r astrophysical objects. An asymptotic expansion yie l d s the density and i t s function 0~as an asymptotic series cr - &Cf) + J i L l ^ ) •+ ••• where S c i s s u f f i c i e n t l y small, O with ^ fixed. Previous research has r e s t r i c t i v e l y assumed that i n the reqion e x t e r i o r to the polytrope, the scaled g r a v i t a t i o n a l potential <j-> s a t i s f i e s Laplace's eguation f o r a zero density medium. The solution of which, with standard conditions at i n f i n i t y , i s Oo where X and V ^ 2 r n a r e constants . Due to symmetry about the equatorial plane only even harmonics i n the l a s t term are needed. 28 This equation (2.11) combined with (2.5) to y i e l d the mathematical density for the whole of the sphere exterior to the confiqurat ion, at zero density, T \y z ^ where X i s a constant. This equation (whenc7"=o) thus s p e c i f i e s the outer boundary of the polytrope. This equation (2.12) i s v a l i d for any polytropic equation of state as the i n t e r n a l — A / structure only affects the value of the c o e f f i c i e n t s and Cm Thus the zero density surface of a rotatinq polytrope i s given by CJ — O • The method thus used by Monaqhan and Eoxburqh (1965) to solve t h i s problem i s to use, i n the inner region, a f i r s t order expansion technique and i n the outer region to derive a f i r s t order approximation by i n i t i a l l y neglecting the mass of the outer l a y e r s , Monaqhan (1967) argues for the v a l i d i t y of t h i s approach which yields equations qoverninq the outer surface and the density^ of <S~- (^^j to f i r s t order i n Smith (1974) has developed a technigue of matched asymptotic expansions to solve t h i s problem to higher order approximations. Matching of the inner and outer solutions i s performed at an intermediate interface. Thus for the inner region, the expansion (2.9) i s substituted into the basic eguation (2.7) and c o e f f i c i e n t s of &c equated: 9 * ' 9* ; 7* 5 = / - n.®"''Z 29 To o b t a i n the s o l u t i o n of t h i s e q u a t i o n , i s d e v e l o p e d as a s e r i e s o f Legendre p o l y n o m i a l s rv-v- I E q u a t i n g c o e f f i c i e n t s o f m g i v e s y ^ X , ^ ^ Q . x _ x with the c e n t r a l boundary c o n d i t i o n s As $)($)is icnown, these e q u a t i o n s are r e a d i l y i n t e g r a b l e by n u m e r i c a l t e c h n i q u e s . I t i s i m p o r t a n t t o r e a l i s e i n t h i s e x p a n s i o n that a l l S^ y f u n c t i o n s w i t h m>0 a r e d e f i n e d by homogeneous e g u a t i o n s and t h u s the c o e f f i c i e n t s must be de termined from the boundary c o n d i t i o n s . A n a l y s i s However two main c r i t i c i s m s have been l e v e l l e d at t h i s e x p a n s i o n (2.9) and i t s a p p l i c a t i o n throughout the whole body . Monaghan (1967) has examined t h e s e and we summarise h i s arguments . The f i r s t c r i t i c i s m i s t h a t $ w i l l have a z e r o at the u n p e r t u r b e d Emden s u r f a c e . In t h i s o u t e r r e g i o n , where the second term i s l a r g e r than the f i r s t , and i t has been c o n j e c t u r e d t h a t t h i s i n d i c a t e s a breakdown i n t h e e x p a n s i o n . Monaghan (1967) a rgues t h a t the f i r s t o r d e r t h e o r y merely r e g u i r e s t h a t the sum of a l l terms h i g h e r than 0<X)be n e g l i g i b l e i n comparison with the sum of the f i r s t two. To determine t h a t the h i g h e r terms are unimportant i t i s n e c e s s a r y t o c a l c u l a t e them or show i t by comparison of the f i r s t o r d e r t h e o r y with 30 solutions obtained by other methods. This Monaghan (1967) and Monaghan and Roxburgh (1965) have done by comparing t h e i r r e s u l t s with those obtained numerically by James (1964). A more important objection i s due to the expansion of the term ( T i n a power series i n &c. I f (^o^.jEthen we may expand . vj/ .+ .... ( which yields the above equation (2.i3>) on equating powers of c^. However whenc^S^J , as occurs i n the outer region, then the power series expansion i s not correct as the expansion of 0 + /©+•••• T =~ I + n . S t 5 / 5 + -i s only v a l i d when . Monaghan (1967) argues that retention of t h i s expansion introduces a n e g l i g i b l e error i n ©A - l . . ensures that the term A©. J i s small and unimportant on the r i g h t hand side of . Monaghan neglects these terms, i n J i n the outer region and obtains the same massless approximation as Monaghan and Roxburgh in the outer region. Monaghan and Roxburgh (1965) have used the formulation (2.12) f o r o"~ as the f i r s t order approximation for the outer region i . e the surface layers. Monaghan (1967) by neglecting the small terms n . C J ' . ^ in the appropriate d i f f e r e n t i a l equations also obtains a si m i l a r functional dependence. By matching the inner solutions with t h i s outer solution at a chosen i n t e r f a c e (which can be taken as the Emden surface c f . Monaghan (1967) ) the constants ^#X^^ can be determined. These are derived for various values of H and l i s t e d i n Monaghan and Roxburgh (1967). An important general r e s u l t i s that by expanding ^.^^the matching shows that Ao =- CU — =. o »v\ ± o I Crr\ *w\ /r»o- I ' I 31 Thus a solution of the basic equation (2 .7 ) i s now for the inner region j < j» and for the massless region, for ^ - • T n e density &~ of the polytrope can thus be obtained. These r e s u l t s are not novel . Jeans (1929) had developed such an approach six t y years ago and from t h i s result had claimed that only systems with small e l l i p t i c i t i e s were possible for polytropes with index greater than n=0.8 . However the model on which t h i s argument i s based , and which was described above , r e l i e s on assumptions that- are not necessarily v a l i d for discs. The main assumption i s that a Roche envelope of zero or n e g l i g i b l e mass can be patched onto the central core. Only surface layers with a small deviation from a sphere can be expected to s a t i s f y such a constraint. It w i l l be shown i n subsequent analysis that a s e l f consistent disc w i l l not s a t i s f y such a stringent constraint, as the masses i n the f l a t disc and central bulge w i l l be of the same order. The other major assumption i s that the external boundary of the s e l f - g r a v i t a t i n g body i s at zero density. Only when the r a t i o of external to central density i s below a c r i t i c a l value i s t h i s assumption v a l i d . This i s confirmed by the simple analysis of examining the equation i n the region where <?~ =0(£t\ Substitution gives cn= £C.C such that, for n>2, the second term on the right hand side of 32 (2./5) i s n e g l i g i b l e , and the solution w i l l be that found above, the Roche solution. But t h i s Roche solution i s only valid i n the region where <5~- o ( S c ) a n d the density i s cr - O (&^\ I f the external density i s greater than t h i s , i t can be expected that other solutions are possible and that discs can exist. Linear Problem In order to elucidate solutions of the basic p a r t i a l d i f f e r e n t i a l equation with a non-zero external density we w i l l examine the l i n e a r problem, n=1, and i t s exact a n a l y t i c a l so l u t i o n . -2. V This has the exact solution where ^ ^ f ) Is the spherical solution that s a t i s f i e s the homogeneous equation / \ / , ^ ^ \ _ The solution i s . g) ^ A Suvx | / J> , where the other possible solution i s ignored due to i t s i n f i n i t e n e s s at the o r i g i n t £ =. O . The other term JJ2 obeys the inhomogeneous eguation •2 ~ ^ 7 • 5 - / - ^ V which i s decomposed, as usual, into Legendre function expansion. Then the solutions of which are ^ - ^ e , as before ignoring the other solution i n f i n i t e at the o r i g i n , and ^ . ( ^ ^ ^ vJ ^ l S ^ , which are spherical Bessel functions of the f i r s t kind, again ignoring the Bessel functions of the second kind which are i n f i n i t e at the o r i g i n . Thus the t o t a l solution i s and using the normalising i n i t i a l condition cX— I at j> = o gives A ~ A 0 - (- S c and thus As thas i s an e l l i p t i c a l boundary value problem, necessary and s u f f i c i e n t conditions to determine the c o e f f i c i e n t s uniquely are the s p e c i f i c a t i o n of the non-zero value of the density on the external boundary of a given shape. Such an approach would be of use i f we were to examine observationally a disc with t h i s particular eguation of state and measure i t s boundary shape and external density, solve for the values of the c o e f f i c i e n t s a n ^ u s e ^ e s e t o obtain the i n t e r n a l density structure. However, in t h i s work we are interested in s i m p l i f i e d models of discs and thus w i l l use the alternate approach of jud i c i o u s l y choosing values of AT and the external density which w i l l y i e l d an unique external boundary shape. This i s termed the synthesis of discs. We have used this l i n e a r problem to show the form of the solution as a sum of the spherical 'Emden' solution and a multipole expansion. This basic solution structure w i l l also be applicable to the non-linear case as we show next. 34 Disc Solutions The asymptotic expansion i s performed on the density C~ . <JA = <§> + £c- £ -t- °C&t) Consistent with the analysis above, we expand the density i t s e l f i n the small parameter hc f o r r i g i d rotation which y i e l d s as expected which has the Emden solution, and a n d ^ i s expanded as before where i j2 ~ with c e n t r a l i n i t i a l conditions The next order function obeys an eguation which i s also l i n e a r and homogeneous and arbitrary c o e f f i c i e n t s w i l l again multiply Legendre multipole terms whose value can be determined through external boundary conditions. Thus to order 0(.£^}this asymptotic solution of the non-l i n e a r case i s of the same form as the exact solution for the l i n e a r problem. cr" = ®(f) + Sc. S> (Aj,M) *IPOJ^S).?^) } *" As the expansion i s of the density upto and including terms of order 0(_£j) the external density must also be constrained to be of t h i s same order. 35 A s i m i l a r expansion and r e s u l t i s given f o r isothermal bodies , where the basic eguation i s 2_ and the dimensionless parameters are given by Expanding the density as before = c 9 L -h cf c- j j ; + - •• £ > ^ 0 gives s i m i l a r eguations governing the central spherical solution and the Legendre terms, c where t h i s has a s i m i l a r expansion . ^ K - I - i „ ® The isothermal asymptotic solution i s thus and to t h i s order i s of the same character as the exact solution i n the l i n e a r problem. Let us now recapitulate the steps to a solution for the density d i s t r i b u t i o n of a s e l f - g r a v i t a t i n g , rotating disc by the above asymptotic method. 1. Derive the governing p a r t i a l d i f f e r e n t i a l eguation i n spherical coordinates for uniform rotation from the hydrostatic equation, the s e l f - g r a v i t a t i o n eguation and the equation of 36 state. 2. Normalise variables with respect to the appropriate c h a r a c t e r i s t i c values. 3. Recognise the small parameter as inhomogeneous term. 4. Linear case gives the solution as the sum of the 1Emden' solution plus multipole terms with arb i t r a r y c o e f f i c i e n t s . 5. The non-linear case allows an asymptotic expansion i n powers of o° the f i r s t term of which i s the 'Emden1 solution and whose C 3 next term of 0[&J) incorporates multipole terms with a r b i t r a r y c o e f f i c i e n t s . 6. A v a l i d method to obtain an unique solution i s to judic i o u s l y choose the c o e f f i c i e n t s a n ( ^ external density and t h i s w i l l produce a system with a particular, unique shape. This i s the approach used herein to generate synthetic discs. 7. A subset of possible solutions can be generated i n which only the f i r s t few c o e f f i c i e n t s are non-zero and the remainder are made i d e n t i c a l l y zero for p r a c t i c a l generation of density structures and disc shapes by computer techniques. The j u s t i f i c a t i o n of these steps to a disc model follows : 1. For steps 1,2,3 t h i s i s standard practice, c.f. Jeans (1929), Chandrasekhar (1933), Monaghan and Roxburgh (1965). 2. The expansion of density to f i r s t order 0(6^), ignoring terms of 0 (jS*)t i s vali d f o r external densities constrained to be of 3. The asymptotic solution of 'Emden * sphere plus multipole terms to o c o i s o f i d e n t i c a l character to the exact solution of the l i n e a r problem. 4. Jeans (1.929) has shown :' when the pressure depends only on 37 the d e n s i t y , c o n f i g u r a t i o n s a f e q u i l i b r i u m may be s p e c i f i e d by t h e i r boundary a l o n e . ' 5. The f a c t t h a t c h o i c e of a p a r t i c u l a r boundary shape and e x t e r n a l d e n s i t y ( 0 ( £ t f p ) y i e l d s an unique a r r a y of the m u l t i p o l e c o e f f i c i e n t s a l l o w s r e v e r s a l of t h i s argument such t h a t c h o o s i n g the e x t e r n a l d e n s i t y and an a r r a y of c o e f f i c i e n t s y i e l d s an unique boundary shape. 6. R e s t r i c t i o n of t h i s a r r a y to n o n - z e r o v a l u e s f o r o n l y t h e f i r s t few c o e f f i c i e n t s y i e l d s a subset of p o s s i b l e s h a p e s . A h i g h degree of f l a t t e n i n g of the d i s c shape i s f o u n d t o n e c e s s i t a t e a l a r g e number of n o n - z e r o c o e f f i c i e n t s as i l l u s t r a t e d i n the f o l l o w i n g r e s u l t s . R e s u l t s For a d i s c shaped body i t can be expected t h a t the most i m p o r t a n t Legendre f u n c t i o n i n d e f i n i n g a d i s c w i l l be ^ (y"0 where ~ ^ ( 3 ^ i Z - - i ) • From an edge-on v i e w p o i n t one can imagine t h i s term as t h e d i p o l e component. T h i s can be seen from t h e d e n s i t y p l o t s i n f i g u r e s 2 — 6 where d i f f e r e n t v a l u e s of parameters are t a k e n and h i g h e r c o e f f i c i e n t s t h a n are z e r o . Thus these systems c o n s i s t e s s e n t i a l l y of the Emden sphere s o l u t i o n with a s m a l l adjustment from the term as the main c o n t r i b u t i o n c l o s e to the c e n t r e , t o g e t h e r w i t h t h e ^ term as t h e dominant term i n the f l a t o u t e r r e g i o n . I t must be emphasised t h a t t h e s e are s y n t h e t i c d i s c s whose c o n t o u r s a r e p l o t t e d i n u n i t s of the e x t e r n a l d e n s i t y . F o r the p o l y t r o p e s t h e e x t e r n a l d e n s i t y i s t a k e n as a f a c t o r S o f the c c e n t r a l d e n s i t y and f o r i s o t h e r m a l s as t w i c e t h i s v a l u e . Note 38 that the e c c e n t r i c i t y of the contour increases toward the external surface i n these cases, a r e s u l t consistent with the shape of rotating polytropes synthesized by Roberts(1963) and Hurley 6 Roberts (1964) by a numerical v a r i a t i o n a l technigue. Hunters (1963) expansion of disks i n terms of Legendre polynomials i s of s i m i l a r analytic character i n that as the degree of f l a t t e n i n g of the disc increases, higher order Legendre terms i n the series w i l l be more important. The fundamental difference between these two approaches however i s that our technigue yields a spherical Emden solution plus the Legendre expansion and thus avoids the fundamental d i f f i c u l t y of Hunter's expansion which needed a large number of terms to describe t h i s c e n t r a l bulge. These figures also show the concentration of density toward the centre for isothermal systems and polytropes of large index n. For the isothermal case i n Figure S the r a t i o of polar to eguatorial radius i s approximately 1/2.5 which i s more flattened than the Roche type solutions with zero density external boundary conditions. The shape of the density contours i s v i s u a l l y s imilar to that of spheroids, which have been used previously to construct and represent disks. A l l these solutions were found to have rapid l y increasing density as the centre i s approached with corresponding large density gradients in t h i s region. As the f l a t t e n i n g increases, the models derived by t h i s multipole expansion technique would need the s p e c i f i c a t i o n of many more multipole c o e f f i c i e n t s than the few used above, and so the following method must be used to describe simply such hiqhly flattened systems. 3 9 Part 2 i C y l i n d r i c a l C o o r d i n a t e E x p a n s i o n s The o b j e c t i n t h i s s e c t i o n i s t o d e s c r i b e the s t r u c t u r e and p r o p e r t i e s of s t e a d y - s t a t e d i s c s u s i n g an a s y m p t o t i c e x p a n s i o n i n c y l i n d r i c a l c o o r d i n a t e s . The approach used as b e f o r e w i l l be f l u i d d y n a m i c a l and s e l f - c o n s i s t e n t . The b a s i c e g u a t i o n g o v e r n i n g d e n s i t y d i s t r i b u t i o n i n the s i m p l e s t model w i l l be d e r i v e d u s i n g the c o n t i n u i t y e q u a t i o n s of m o t i o n , t h e e q u a t i o n o f s e l f - g r a v i t a t i o n , the e q u a t i o n of s t a t e t o g e t h e r with a x i s y m m e t r i c r o t a t i o n . On removing t h e d i m e n s i o n s i n the e g u a t i o n , t h r e e b a s i c d i m e n s i o n l e s s parameters a p p e a r , namely measures of the r a t i o o f g r a v i t a t i o n a l p o t e n t i a l energy t o thermal energy : a l p h a (o<) , the r a t i o of k i n e t i c energy of r o t a t i o n t o t h e r m a l energy : b e t a (p), and the square of the r a t i o of the t h i c k n e s s to the r a d i u s of the d i s c ; e p s i l o n ( £ ) . I t w i l l be shown t h a t the d e n s i t y , as a f u n c t i o n of r a d i u s and t h i c k n e s s , can be approximated by an a s y m p t o t i c e x p a n s i o n t e c h n i q u e . The approximate s o l u t i o n d e r i v e d by t h i s t e c h n i q u e w i l l approach the t r u e s o l u t i o n a s y m p t o t i c a l l y . For a d i s c where the t h i c k n e s s i s much s m a l l e r t h a n the r a d i u s , the s m a l l parameter used t o c r e a t e the a s y m p t o t i c s e r i e s i s taken to be e p s i l o n > €• . In the l i m i t i n g case when €-=0, the s o l u t i o n must be a f l a t s e m i - i n f i n i t e s l a b w i t h d e n s i t y v a r i a t i o n o n l y i n the t h i c k n e s s d i r e c t i o n . Such a f l a t s l a b has p r e v i o u s l y been used i n a s t r o p h y s i c a l approaches t o the d i s c problem even though i t i s p h y s i c a l l y u n r e a l i s t i c . The b a s i c g o v e r n i n g e g u a t i o n i s a q u a s i - l i n e a r p a r t i a l d i f f e r e n t i a l e g u a t i o n which q i v e s t h i s problem c e r t a i n s i m i l a r i t i e s w i t h the c l a s s i c a l T h i n A i r f o i l Problem, as s u g g e s t e d by P r o f . E. M i u r a . In t h i s a p p r o a c h , t h e 40 boundary i s expanded as a perturbation series and i n our case, a free boundary problem i s obtained , as suggested by Prof. F. Wan. For consistency, the angular velocity must also be expanded as an asymptotic series i n £• , the f i r s t term being uniform rotation at the central axis. This expansion technigue w i l l allow constraints to be placed on the external boundary. In the p a r t i a l d i f f e r e n t i a l equation i t i s found that the small parameter €• multiplies the terms containing density gradients along the c y l i n d r i c a l r a d i a l d i r e c t i o n . In the f l a t disc region these terms are small and can thus be neglected i n the f i r s t approximation. Close to the central axis these gradients increase r a p i d l y , as the previous technique and models showed, and so these terms become important and a boundary layer i n density i s expected,. Close to the central r o t a t i o n a l axis, qravity forces w i l l be large compared to c e n t r i f u g a l forces and a more s p h e r i c a l system i s i n t u i t i v e l y expected. Inside the normal f l a t disc region a boundary layer i s formed at the central axis of the d i s c , i n which the density increases r a p i d l y toward the centre, termed the " central bulge The basic f i n i t e disc can thus be expected to have a f l a t t i s h normal disc region with a spherical c e n t r a l bulge and an outer r a d i a l edge where the density decreases to the external value. We w i l l concentrate on the inner regions as these are of major astrophysical importance and w i l l be of use i n the i n s t a b i l i t y work that follows. Matching of the solutions i n the central bulge and f l a t disc regions w i l l be performed and a composite solution w i l l be derived which i s found to consist of the spherical central bulge and the f l a t disc solution. 4 1 Fundamental Equations The coordinate system i n which the equations are expressed w i l l be c y l i n d r i c a l polar with radius (r) ,' thickness [z. ) , and angle ( ) . As Spiegel (1972) has observed, most models of dis c systems tend to neglect d i s s i p a t i v e processes and thereby incur a lack of unigueness. This approach w i l l assume no v i s c o s i t y and t h i s w i l l allow us to choose the form of the rotation. Both polytropes and isothermal systems are considered. Isothermal bodies have the important property that only at i n f i n i t y does the density approach zero and t h i s constrains the external density to be f i n i t e f o r a f i n i t e body. The eguations of motion i n c y l i n d r i c a l polars C~,2-,£) governing the dependant variables: yO - density, P - pressure, - g r a v i t a t i o n a l potential, L*- ~ v e l o c i t y , i s given by - ^2.2.4-) where (6) i s time and the usual vector notation i s used. ' fe. V = u r .«% r - ^ Assuming axisymmetry about the central axis, and allowing no motions except in the (fi) d i r e c t i o n , we have 4 2 and, at steady-state, when the time derivative i s zero, the equations of motion reduce to - JC". r . - 1 af 3 . « ^ r 57 ' a The r e l a t i o n s h i p between d e n s i t y ^ ) and p o t e n t i a l i s expressed by Poisson's equation i n c y l i n d r i c a l polars with axisymmetry -.X T a#-7fr - -where(Gj) i s the Gravitational Constant. The c o n s t i t u t i v e equation qoverninq the dependant variables ( ) i s the equation of state. In the followinq derivation we s h a l l use the isothermal and polytropic equations w her e K. - i s qas constant, r '>t j J 'j j - i s temperature, bv\ - i s mean molecular weiqht of qas. These reduce to the fundamental equations i n the o r i q i n a l variables^ (r>-28~)> . 43 When rotation i s not present, J L=o, t h i s reverts to the spherical case. For the isothermal sphere, as the radius increases, the sol u t i o n , f i n i t e at the centre, i s known to approach and o s c i l l a t e around the singular solution i n which the density varies as the sguare of the radius, Figure 12. . A more complete analysis of the sphere i s given i n Appendix A. Notice that the v e l o c i t y term on the r i g h t side of the basic" equation i s also zero when SL oc r , which implies constant r o t a t i o n a l v e l o c i t y . As in a l l f l u i d mechanical problems involving normalisation , t h i s can be performed with respect to c h a r a c t e r i s t i c dimensions and values of the variables. These can be chosen for the convenience of the problem and i t s solution approach. A_sy_mp_totic Discs The basic density eguation for the polytropic and isothermal disc has been discussed i n the previous section above and shown that disc solutions exist depending on the external boundary shape and conditions expressed i n spherical coordinates . However, much ins i g h t can be gained into the structure of s e l f - g r a v i t a t i n g rotating discs by creating a small dimensionless parameter c h a r a c t e r i s t i c of discs i n which an asymptotic expansion can be performed for the density solution. The c h a r a c t e r i s t i c geometry of a disc i s that i t s equatorial radius i s much greater than i t s thickness Thus the obvious choice for a small parameter i s 3 D /R*_ or some multiple thereof . Thus, normalise the coordinates such that for the isothermal case r 44 and also normalise the angular velocity with respect to i t s value on the central axis and thus e {%+ i ] "> - £ s ' | s - ^ - «c — C^O where the dimensionless density i s C'/^= y° /^ >e . The dimensionless parameters are which i s a measure of the r a t i o of r o t a t i o n a l energy to thermal energy, which i s a measure of the r a t i o of g r a v i t a t i o n a l to thermal energy and x . £ - C / V ) ^ i s a geometrical dimensionless parameter, which w i l l be small for discs . S i m i l a r i l y for the polytrope, with dimensionless density cr -t^s 1- i d s j e^- rp<5sai i where and ^ = ^ r ( 5 ^ / £> -37 ) Thus asymptotic expansions using £ as a small parameter are an useful approach to obtain a solution of t h i s basic eguation for discs, as the smaller t h i s value of £. , the more d i s c - l i k e the body. 45 We have chosen t h i s method of normalisation f o r convenience i n the finding of the solution, but i t should be noted that the independent distance variables can be renormalised by dividing with OC , thereby obtaining a s i m i l a r equation as i n the previous section dealing with sp h e r i c a l coordinates. Vy ,• / ~ - ' gi - • ^. £ A / ^ i . Here the normalising density i s the external value where previously i t had been the c e n t r a l density and thus the two parameters are related by &i - £>c • f^/^ and where uniform rotation has also been assumed. Basic geometrical properties of t h i s isothermal system derived from Eiemannian manifold theory are detailed i n Appendix B. An important property of such equations as (02.31 ,a ?>S) with a small parameter £ can be shown by a coordinate transform S = S / e ^ and the operator on the l e f t hand side becomes and the parameter £. disappears completely from the problem. Such equations have been examined by Chanq (1961) i n a seminal paper on coordinate expansions who c a l l e d such parameters " a r t i f i c i a l " . Chang shows the important r e s u l t concerning asymptotic expansions i n such parameters that " i f £ i s an a r t i f i c i a l parameter then the expansion i s either not uniform, or the f i r s t term contains an exact s o l u t i o n . " Chang (1961) has also shown that "the ordinary technigues of parameter-type expansion in an a r t i f i c i a l parameter leads to 46 an indeterminacy" such that arbitrary constants are obtained i n the solu t i o n . Sometimes these can be found by r e l a t i n g them to certain i n t e g r a l properties of the system. Chang (1961) summarised the important, properties and problems of these constants i n the following quotation where p a r t i c u l a r reference i s made to his analysis of the Navier-Stokes e l l i p t i c a l problem of f l u i d flow around an ar b i t r a r y body. " 1. The constants are ar b i t r a r y . " " The p r i n c i p a l reason for thi s b e l i e f i s that there seems nothing in the expansion procedure which r e s t r i c t s the value of these constants." " 2. For any choice of constants and for any p a r t i a l sum there exists a related Navier-Stokes ( exact ) solution." " in other words, the related solution and the given approximation (be i t from the outer, inner or composite series) should agree uniformly, as C decreases to zero, to the order and the domain stipulated i n the approximation. I t (2) states e s s e n t i a l l y that our approximations make sense, that they are not grossly i n c o r r e c t . " Van Dyke (1975) has emphasised that "Perturbation problems i n which the small quantity i s a dimensionless combination involving the coordinates (space or time) rather than the parameters alone have certain s p e c i a l features ". S i m i l a r l y , Van Dyke (1975), "For e l l i p t i c equations, coordinate expansions usually provide only q u a l i t a t i v e r e s u l t s . One o r d i n a r i l y encounters a boundary value rather than an i n i t i a l - v a l u e problem. Then because of backward influence any l o c a l solution depends on remote boundary conditions and i t i s not possible to calculate successive terms of an expansion for small values of a coordinate, A l l than can be achieved i s to f i n d the form of the expansion , each term being indeterminate by one or more constants." Boundary Lay_er Theory The solution of d i f f e r e n t i a l eguations, i n which a small parameter multiplies one or more of the terms, has been extensively investigated within the l a s t few decades mainly by applied mathematicians working on f l u i d dynamical problems. The es s e n t i a l approach i s to create a perturbation technigue which w i l l y i e l d an asymptotic solution. An approximation of t h i s sort becomes increasingly accurate as the perturbation quantity tends to zero. In p r i n c i p l e , one can improve the r e s u l t by embedding i t as the f i r s t step in a systematic scheme of successive approximations; the resulting series i s by construction an asymptotic expansion. As general references to the subject, the following are of value: Van Dyke (1975), Cole (1968), Nayfeh (1973), O'malley (1974), Grasman (1971). The technigue which w i l l be u t i l i s e d i n the in v e s t i q a t i o n of the density equation i s termed the method of matched asymptotic expansions. This method has found i t s most extensive use i n cases where the dependant variable undergoes sharp changes i n some regions of the domain of the independant variables. This small i n t e r v a l across which the dependant variable changes rapid l y i s c a l l e d the 'boundary layer' i n f l u i d mechanics, the 'edge layer' i n s o l i d mechanics, and the 'skin layer* i n electrodynamics. Nayfeh (1973) has commented; i t i s the r u l e , rather than an 48 exception, that asymptotic expansions i n terms of a small parameter ( £ < < 1 ) are not uniformly v a l i d and break down i n regions c a l l e d "regions of non-uniformity" which are sometimes referred to as "boundary layers". To obtain uniformly v a l i d expansions, we must recognize and u t i l i z e the fact that the sharp changes are characterized by magnified scales which are d i f f e r e n t from the scale characterizing the behavior of the dependant variables outside the sharp-change regions. According to Van Dyke (1975), the c l a s s i c a l warning of such singular behavior i s fa m i l i a r from Prandtl's boundary-layer theory. A small parameter mult i p l i e s one of the highest derivatives i n the d i f f e r e n t i a l equations. Then i n a straight forward perturbation scheme that derivative i s l o s t i n the f i r s t approximation so that the order of eguations i s reduced. One or more of the boundary conditions must be abandoned, and .the approximation breaks down near where they were to be imposed. Grasman (1971) has discussed t h i s problem i n connection with e l l i p t i c p a r t i a l d i f f e r e n t i a l eguations of the general form va l i d i n a s t r i c t l y convex bounded domain . He shows that i n the neighbourhood of a point where the c h a r a c t e r i s t i c s of the operator {L^) , are tangent to the boundary, sharp changes i n can be expected leading to boundary layers. The c h a r a c t e r i s t i c s of the equation are given by the two f a m i l i e s of curves 0 - - ° ^ - vA»"-4-te-^dx-49 In each region, we can create an asymptotic expansion of the form The method of matched asymptotic expansions involves l o s s of boundary conditions. An outer expansion cannot be expected to s a t i s f y conditions that are imposed i n the inner regions, the boundary layer; conversely, the inner expansion w i l l not i n general s a t i s f y distant conditions. Hence, i n s u f f i c i e n t boundary conditions are generally available for either the outer or inner expansion. The missing conditions are supplied by matching the two expansions. The general matching p r i n c i p l e i s that the inner l i m i t of the outer expansion should, to appropriate orders, agree with the outer l i m i t of the inner expansion. Asy_m_gt.ot.ic Expansions The asymptotic techniques discussed above w i l l be applied to the basic c y l i n d r i c a l coordinate eguation (o?Sl) usinq epsilon, £ , as the small parameter. ooT£*e i - i p 50 The i n t e r e s t of thi s research i s s p e c i f i c a l l y concerned with discs wherein the thickness of the system i s expected to be much les s than the radius ie (£,<<1) . The basic eguation governing density i n dimensionless coordinates i s (2 .38 ) , for isothermal (a) and polytropes (b) , f -4- 1 I -+ 1 i 2, a,-S ^ s j d S ^ > a s ^ J P together with the necessary and s u f f i c i e n t boundary conditions, namely, •= o 1 ^ ) - i ? -on the external boundary^ C s ^  • Jeans (1925) has shown that," when the pressure depends only on the density, configurations of equilibrium may be s p e c i f i e d by t h e i r boundaries alone....... And that the density and normal density gradient are determined at every point of the boundary, such that the solution for density i s unigue." The asymptotic expansion for the density variable can thus be made CL. £ ^  = u>0 -h- g . xJt -f . . . . fe . cr 51 Obviously, as fc- when o , the solution must approach that of the se m i - i n f i n i t e f l a t slab which can have no r a d i a l s dependence and consequently v»6 =- >J0 6 3 ) a n d = only. As the zeroth order solution i s the se m i - i n f i n i t e f l a t slab, the outer boundary i s expanded also as an asymptotic series i n £ r as suggested by F. Wan. As the distance variable has been normalised with respect to the half-thickness of the semi-infinite f l a t slab then = 1. This i s a s i m i l a r approach to that used in the c l a s s i c a l e l l i p t i c a l problem of T h i n - A i r f o i l Theory. As the angular velocity i s dependent on the r a d i a l s variable, for consistency t h i s must also be expressed by an asymptotic seri e s i n €. , ^ ( s ; e ) = ^>0 + & • «o# (s) -h • • • where the f i r s t term i s taken to be the r o t a t i o n a l v e l o c i t y at the central axis and thus 1. This i s i n order that the f l a t slab zeroth order solution be r a d i a l l y independent. Inserting i n the basic isothermal eguation and eguating terms of the same order gives J (o? 3Q t1-The method of matched asymptotic expansions w i l l be applied to t h i s system. As i n such systems, boundary layers can be expected. Put the density eguation i n the form 52 Grasman (1971) , i n h i s d i s c u s s i o n of s i m i l a r q u a s i - l i n e a r e l l i p t i c p a r t i a l d i f f e r e n t i a l e q u a t i o n s has argued t h a t the c h a r a c t e r i s t i c s of the o p e r a t o r JLj. , te rmed the s u b c h a r a c t e r i s t i c s , determine the l o c a t i o n of boundary l a y e r s . For the above e g u a t i o n s the o p e r a t o r JLj, ± s i n d e p e n d e n t of the r a d i a l S c o o r d i n a t e and thus t h e s u b c h a r a c t e r i s t i c s are g i v e n by the s t r a i g h t l i n e s s=constant . These are tangent to the b o u n d a r i e s of the d i s c quadrant a t the c e n t r a l a x i s , s=0, and the out er e d q e , s=1. Us ing Grasman's arguments , t h i s i m p l i e s t h a t boundary l a y e r s can occur i n the neighbourhood of the c e n t r a l a x i s and the outer e d g e . Note t h a t Grasman's argument i s i n d e p e n d e n t of the v e l o c i t y term and any p o s s i b l e n o n - u n i f o r m i t y or o t h e r w i s e . However, when £\ =0, t h e r e i s no f u n c t i o n a l dependence on the r a d i a l s v a r i a b l e i n the f u n d a m e n t a l e g u a t i o n (P 3>S) as 6- =0 g i v e s e s s e n t i a l l y the s e m i - i n f i n i t e f l a t s l a b whose s o l u t i o n w i l l have no s dependence. T h i s s o l u t i o n w i l l not i n g e n e r a l s a t i s f y the boundary c o n d i t i o n s at the outer edge but does s a t i s f y t h e boundary c o n d i t i o n ^V^s - ° on the c e n t r a l a x i s . S u p e r f i c i a l l y i t c o u l d be argued t h a t a boundary l a y e r i s not needed a t the c e n t r a l a x i s f o r t h i s f l a t s l a b s o l u t i o n under u n i f o r m r o t a t i o n . However i t i s known t h a t t h i s i s a n e c e s s a r y but not s u f f i c i e n t c o n d i t i o n f o r a s o l u t i o n t o be v a l i d . R e c a l l C h a n g ' s (1961) remark c o n c e r n i n g these ' a r t i f i c i a l parameter • e x p a n s i o n s t h a t 11 the e x p a n s i o n i s e i t h e r not u n i f o r m , or the f i r s t term c o n t a i n s an exact s o l u t i o n . " Though t h e s e m i - i n f i n i t e s l a b i s an e x a c t s o l u t i o n to t h e o r i g i n a l e g u a t i o n , i t o b v i o u s l y cannot s a t i s f y r e a l i s t i c boundary c o n d i t i o n s f o r f i n i t e b o d i e s . 53 Thus two s e p a r a t e approaches show t h a t a boundary l a y e r i s e x p e c t e d around the c e n t r a l a x i s . T h i s r e g i o n , and the d i s c e x t e r n a l t o i t , a re o f pr imary i n t e r e s t whi le the boundary l a y e r at the o u t e r edge w i l l not be examined i n d e t a i l as i t i s o f l i t t l e i n t e r e s t t o t h e subsequent a n a l y s e s . A boundary l a y e r i n d e n s i t y c l o s e to the c e n t r a l a x i s , termed the c e n t r a l b u l g e , w i l l be surrounded by the normal r e g i o n , termed the d i s c r e g i o n . P h y s i c a l l y , t h i s c e n t r a l b u l g e can be e x p e c t e d as t h e c e n t r i f u g a l f o r c e s w i l l be s m a l l c l o s e t o the c e n t r a l a x i s and the shape o f t h e system w i l l t e n d to t h a t of a s p h e r e i n t h i s r e g i o n . D i s c Region Expansion The i s o t h e r m a l case w i l l now be d i s c u s s e d i n d e t a i l as t h i s e q u i l i b r i u m system w i l l be s u b j e c t e d t o a c o u s t i c p e r t u r b a t i o n s l a t e r . The d i m e n s i o n l e s s d e n s i t y i s expanded as and s u b s t i t u t i n g i n t o the b a s i c d i f f e r e n t i a l e q u a t i o n g i v e s ^o (J-39) 54 where we use the expansions The boundary conditions are thus I f t h i s cuter boundary ^ ^ C O w a s known these equations could be integrated, as i l l u s t r a t e d i n the uniform spherical expansion treated previously above. Jeans (1925) showed that the outer boundary has both density and normal density qradient determined thereon. Thus } | ^ — may be used as an ad d i t i o n a l boundary condition or to allow confirmation of the accuracy of the solution obtained. The object i s to obtain constraints on the terms in the boundary expansion s u f f i c i e n t to y i e l d the functional dependence on s and y variables of the boundary. Similar to the T h i n - A i r f o i l Problem, asymptotic expansion of the boundary necessitates transfer of the boundary conditions to the zeroth order boundary. As Van Dyke (1975) remarks, " in such cases when the boundary condition i s imposed at a surface whose position varies s l i g h t l y with the perturbation parameter £ ,...... i n order to carry out a systematic procedure the boundary condition must be expressed i n terms of quantities evaluated at the basic position 55 of the surface corresponding t o £ = 0 . The transfer of a boundary condition i s effected by using a knowledge of the way i n which the solution varies i n the v i c i n i t y of the basic surface. Often the solution i s known to be analytic i n the coordinates in which case the transfer i s accomplished by expanding i n a Taylor Series about the values at the basic surface." For our problem the basic surface i s y=1, with the asymptotic expansion for isothermal density. Assuming that the expansion terms are analytic i n y at y=1, the density boundary condition gives such that i - - CM-J i ^ -. The normal density gradient ^ i s not known beforehand but i s represented by the asymptotic ser i e s which by Taylor Series expansion gives which y i e l d s ' + <3» L ^ *t> a J 7J=P I * .4/ The value of the normal gradient 0 ^ i s obtained from the solution of the zeroth order f l a t slab solution which w i l l now be examined i n more d e t a i l . 56 The z e r c t h o r d e r e g u a t i o n v a l i d i n the d i s c i s with boundary c o m d i t i o n s — o B a s i c r e s t r i c t i o n s on s o l u t i o n s of t h i s n o n - l i n e a r system and an exac t e l l i p t i c a l i n t e g r a l s o l u t i o n are g i v e n i n Appendix E . A l s o d i s c u s s e d t h e r e i s the s i m p l e s t case of ^>i = 0 which has been a n a l y s e d p r e v i o u s l y by S p i t z e r (1942) and Ledoux (1951) and has the s o l u t i o n . where ^ i s the d e n s i t y on a x i s . For the g e n e r a l case of ^ c > 0 , i t i s n e c e s s a r y t o i n t e g r a t e n u m e r i c a l l y and d e t a i l s of t h i s scheme w i l l be shown i n the s e c t i o n on t h e c o m p o s i t e s o l u t i o n . The f i r s t o r d e r e g u a t i o n i s a non-homogeneous l i n e a r p a r t i a l d i f f e r e n t i a l e g u a t i o n f o r the g e n e r a l case with boundary c o n d i t i o n s and normal g r a d i e n t As b o t h tf<(t ^ are not known and i n f a c t d e t e r m i n a t i o n i s d e s i r e d , t h i s system i s not over c o n d i t i o n e d . E x p r e s s i n g the s o l u t i o n as the s y m b o l i c sum of the homogeneous p a r t and the non-homogeneous s o l u t i o n g i v e s where the f u n c t i o n a l dependence of the c o n s t a n t s o f i n t e g r a t i o n 57 are shown. S u b s t i t u t i n g i n t o the boundary c o n d i t i o n s g i v e s and t h u s t h r e e e q u a t i o n s w i t h t h r e e " unknowns a l l o w s the d e t e r m i n a t i o n of the f i r s t o r d e r boundary p e r t u r b a t i o n ^ (.s) to the f l a t d i s c s o l u t i o n i n t h i s d i s c r e g i o n , t o w i t h i n the unknown normal g r a d i e n t component "v^, . C e n t r a l Bulge E x p a n s i o n To e l u c i d a t e the p r o p e r t i e s of the boundary l a y e r around the c e n t r a l a x i s , the c o o r d i n a t e S i s s t r e t c h e d i . e S = S/^C^); r e p l a c i n g 3 by s i n the b a s i c d e n s i t y e g u a t i o n s >2 . Thus The d e r i v a t i v e i n 5 w i l l be t a k e n o f 0 ( 0 i f we take-^= 6 the t h i c k n e s s of the boundary l a y e r i s Thus or J L J £ u - « ) and s i m i l a r l y f o r p o l y t r o p e s . T h i s i s the o r i g i n a l s p h e r i c a l e g u a t i o n and shows t h a t £ i s an " a r t i f i c i a l " parameter i n t h e sense used by Chang(196 1 ) . O b v i o u s l y i n o r d e r t o s o l v e the e q u a t i o n i n t h i s i n n e r r e g i o n i t i s n e c e s s a r y to s o l v e the o r i g i n a l e q u a t i o n . A C a t c h 22 s i t u a t i o n . Expanding a s y m p t o t i c a l l y as b e f o r e y i e l d s r 7"*-<x. V . fii ~ <*i- <l 6 ' d ° ( L ° L*.Sb) 58 as UD =• I -+• € • O, -4-with boundary conditions and S i m i l a r i l y the normalised density - *\ & •+• • *\ -\ • • • and thus, everywhere within the c e n t r a l bulge boundary, the true density p = ^ e ' t o * ® • - He w i l l be concerned primarily with the zercth order solution which w i l l represent the true solution to accuracy ^(_r|fe\ Note that t h i s expansion shows that the zeroth order eguation i s i d e n t i c a l to the o r i g i n a l boundary layer eguation (2.So) with constant rotation. The zeroth order solution i s defined by the equation where This i s a boundary value problem which can be converted to an i n i t i a l value problem by chanqe of variable where c*c i s central density. This gives By doing t h i s , we now assume that the central density ^ i s known and not the surface density . Putting gxves 59 with i n i t i a l conditions G , Vrt^o; at centre. Integration from the centre out w i l l y i e l d the density and normal density gradient at the surface. Now the parameter 0(^-0(1), and thus which i s expected in astrophysical systems. By normalising the coordinates by dividing the whole eguation with C*^ , the same eguation i s obtained that was discussed above in spherical coordinates, v i z . Which i s the eguation governing the central bulge. This approach yielded a density boundary layer not only around the central axis, the central bulge, but also at the outer edge where s=o{1). The basic stretching technigue and subsequent analysis can be performed s i m i l a r l y for t h i s region with formation of a composite solution v a l i d i n t h i s region yielding a closed external boundary. As t h i s region has n e g l i g i b l e e f f e c t on the inner regions and t h e i r solutions and i s unimportant for acoustic perturbation work near the central axis, i t w i l l not be examined i n d e t a i l . Matching Of Solutions The method of matched asymptotic expansions involves l o s s of boundary conditions. An outer expansion cannot be expected to s a t i s f y boundary conditions that are imposed i n the inner region, and vice versa . The missing conditions are supplied by matching the two expansions. However , because the expansion i s done i n the 11 a r t i f i c i a l " parameter € certain problems arise e s p e c i a l l y due to the appearence of the o r i g i n a l eguation as the zeroth order 60 solution i n the central bulge. The relevance of thi s to the matching problem i s described by Chang (1961) : " i t i s then clear that the expansion at large radius (the outer expansion ) cannot be matched to an expansion near the body (the inner expansion ) unless the exact solution near the body i s obtained. Thus i n p r i n c i p l e t h i s " matching " involves getting the exact solution and i s conseguently a very impractical procedure indeed." Chang's solution of t h i s Catch 22 dilemma, i n h i s coordinate expansion of the Navier-Stokes flow around a r b i t r a r y bodies, was to show that a f i n i t e number of terms of the inner and outer expansions can be matched to the desired order. These expansions are then combined to produce a composite solution; the matching in t h i s disc problem w i l l follow t h i s same basic approach. We apply an asymptotic matching procedure between the expansions in each region. Each region of space has i t s l o c a l l y v a l i d asymptotic expansion. If there exists an overlap region i n which both are v a l i d , i t follows, since both are i n p r i n c i p l e derived from the same exact solution, v a l i d everywhere, that the two expansions must be indistinguishable i n t h i s common region of v a l i d i t y . In particular each f i n i t e p a r t i a l sum i s indistinguishable to the appropriate degree of approximation, c f . Smith(1975), The basic eguation i n the inner central bulge region ( Z6Z ) i s for density normalised with respect to external value. Converting t h i s eguation to normalisation with respect to cen t r a l density gave (Q-GtS*) y 61 where S - S . <*c • ^ ^ = & c l o - " ^ - ' " and u s i n g the a s y m p t o t i c e x p a n s i o n gives, as before, \7* A • € ^ A c = - ^oo where the logarithm expansion has been used. However t h i s expansion breaks down when to ,which indicates the necessity of an intermediate region as the spherical solution f\QO i s known to be of t h i s order near the external boundary, as n C o i s asymptotic to the singular solution jj^\^L)]m T n e expansion i n t h i s inner region i s of the same form as derived i n the f i r s t method using an uniform expansion in the small parameter <5C with ^ O o being the s p h e r i c a l l y symmetric solution and angular dependence i n multipole terms i n . Now the spherical solution i s a function only of the spherical coordinate <^  - S - t j and thus we can express the n o | term as a sum of separable terms i n spherical and angular coordinates as done in the f i r s t section. \5iey A^-D Examine now the solution i n the intermediate region, where the spherical term i s of order i a the neighbourhood of O^. ~ I . Expand the density as e ° - S t ( l 0 + &cT, ) and the intermediate coordinates are defined by % , _ + Sc-Vo ; * c o s e 62 which on substitution i n the basic equation (2..SZ) yields for the f i r s t term £p- - j -— t v i _ L _ ~- o which has the solution where Crj are undetermined functions of cosine of col a t i t u d e s~ • Our matching procedure for two expansions each l o c a l l y v a l i d in regions K( and R^of space i s according to the asymptotic matching p r i n c i p l e as defined by Van Dyke (1964) •the m-term R,expansion of the n-term R^expansion i s equal to the n-term R^expansion of the m-term R ;expansion•. We s h a l l confine ourselves to the case of m=1,n=1 and match density normalised by the external density. As the deltas are related by Sc- c^e -^e/^the f i r s t term of the inner c e n t r a l bulge solution i s which rewritten i n intermediate variables i s and expanded formally for small <jj^  gives (pSl^) Now the f i r s t term i n the intermediate sol u t i o n s a t i s f i e s (<3.55) or equivalently i n the o r i g i n a l spherical variable with solution where the c o e f f i c i e n t s are related by 63 B e v e r t i n g t o d e n s i t y n o r m a l i s e d w i t h r e s p e c t t o e x t e r n a l d e n s i t y g i v e s the o r d e r of each of these terms can be found as when g=0(1) , then 1=0(1) which i m p l i e s <5e = ^ f c / - o c o Now i n o r d e r to match t h i s s o l u t i o n with the i n n e r c e n t r a l b u l g e form i n the i n t e r m e d i a t e v a r i a b l e — ^Sc.<y'0 expand f o r s m a l l 6^ which y i e l d s The matching p r i n c i p l e r e g u i r e s t h i s to be i n d i s t i n g u i s h a b l e from (2.^2) and t h i s i s so p r o v i d e d t h a t We now match the i n t e r m e d i a t e e x p a n s i o n t o the o u t e r d i s c s o l u t i o n (oJ.ifl ) which i s a f u n c t i o n of y v a r i a b l e o n l y . As t h e c o s i n e of c o l a t i t u d e r e l a t e s t o the / v a r i a b l e by /**X~^/<y t ^ i e s o l 1 1 " ^ 0 1 1 i n t f a e i n t e r m e d i a t e r e g i o n can be e x p r e s s e d as _ P u t t i n g ^ ' ' ^ " ^ ' ^ p a n < * expanding f o r s m a l l 0^ g i v e s the f i r s t term as and u s i n g the r e l a t i o n s h i p i n (e?.S9 ) g i v e s the i n t e r m e d i a t e s o l u t i o n i n t h i s o v e r l a p r e g i o n as a f u n c t i o n of y o n l y . „ T h i s i s t h e r e f o r e matched with and i n d i s t i n g u i s h a b l e from the 64 outer d i s c s o l u t i o n of {o24-i ) with boundary c o n d i t i o n s frPtjJa) which i s a f u n c t i o n of y only. Thus the s o l u t i o n s to z e r o t h order i n £ f o r c e n t r a l bulge and f l a t d i s c have been matched i n an o v e r l a p r e g i o n t o f i r s t order i n (3^. Composite Expansion In the l i n e a r problem. Appendix D, an exact s o l u t i o n i s obtained f o r the whole system which c o n s i s t s of the s p h e r i c a l l y symmetric s o l u t i o n together with the f l a t d i s c s o l u t i o n . The method of matched asymptotic expansions i n the i s o t h e r m a l case has been a p p l i e d c o n s i s t e n t l y , and as i n s i m i l a r problems, a composite expansion can be formed to cover the whole c e n t r a l bulge and d i s c r e g i o n , where ^ i s the i n n e r l i m i t of the d i s c s o l u t i o n = ^ & the outer l i m i t of the c e n t r a l bulge s o l u t i o n Now, i t has been shown above, t h a t the c e n t r a l bulge s o l u t i o n can be expanded i n t o a ^ s e r i e s s o l u t i o n i n 6^ the f i r s t two terms being where A i s the d e n s i t y d i s t r i b u t i o n i n the s p h e r i c a l ^ case and A c o n t a i n s the Legendre terms. Now i n the l i m i t as S - » ^ , we have the s p h e r i c a l s o l u t i o n approaching and o s c i l l a t i n g around the s i n g u l a r s o l u t i o n c2 . x „ ° which i s known to be an e x c e l l e n t approximation throughout mosr of the c e n t r a l bulge, when f^/n i s l a r g e . Thus i n the d i s c r e g i o n , and everywhere the s i n g u l a r s o l u t i o n i s a v a l i d approximation f o r the s p h e r i c a l s o l u t i o n , 65 the composite s o l u t i o n w i l l te In the c e n t r a l bulge r e g i o n next to the c e n t r a l p o i n t , the composite s o l u t i o n w i l l be « = 1 •+ S,. *\ + J — as ^ o , J>o _ ^ j £ and But as £> * r\ , i n the c e n t r a l bulge next t o <jv=-rs , the composite s o l u t i o n ^>cs - * t D O + foQj) w i l l be an accurate s o l u t i o n f o r the d e n s i t y t o z e r o t h order i n ( € )• The composite s o l u t i o n composed of s p h e r i c a l c e n t r a l bulge and a f l a t d i s c s o l u t i o n w i l l be taken as the s o l u t i o n throughout the d i s c and c e n t r a l bulge. T h i s composite sum agrees with the expected form of s o l u t i o n obtained by the r e g u l a r s p h e r i c a l expansion which c o n s i s t e d o f the s p h e r i c a l Emden s o l u t i o n and the d i s c Legendre terms. T h i s composite s o l u t i o n i s only a good approximation as £. approaches z e r o , and the d e n s i t y s t r u c t u r e w i l l approximate that of the f l a t s l a b f a r from the c e n t r a l a x i s . For a .solution of the t o t a l system i t would be necessary t o ob t a i n a composite s o l u t i o n i n the outer edge r e g i o n composed of f l a t s l a b s o l u t i o n and outer edge s o l u t i o n . However i n order t o add these three components to o b t a i n a t o t a l s o l u t i o n i t i s necessary t o know the exact value of the geometric parameter € , whereas we have only assumed t h a t £_<<1. As i n the case Chang (196 1) c o n s i d e r e d , t h i s exact value can only be obtained by 66 r e l a t i n g t o the i n v a r i a n t s of the system and t h i s w i l l be done i n the next c h a p t e r on the v i r i a l a p p r o a c h . S o l u t i o n T e c h n i q u e —<r — J  L e t us r e c a p i t u l a t e the s t e p s t a k e n i n t h i s method o f o b t a i n i n g the f i r s t a p p r o x i m a t i o n t o a d e n s i t y s t r u c t u r e f o r a h i g h l y f l a t t e n e d d i s c . 1. The g o v e r n i n g p a r t i a l d i f f e r e n t i a l e g u a t i o n f o r n o n - u n i f o r m r o t a t i o n i s d e r i v e d i n c y l i n d r i c a l c o o r d i n a t e s . 2. N o r m a l i s e the v a r i a b l e s with r e s p e c t t o a p p r o p r i a t e c h a r a c t e r i s t i c v a l u e s and o b t a i n the t h r e e d i m e n s i o n l e s s parameters ^ ; | S , "6 • 3. R e c o g n i s e t h a t the s m a l l parameter £ m u l t i p l i e s terms c o n t a i n i n g r a d i a l d e n s i t y g r a d i e n t s . This - i m p l i e s these terms can be i g n o r e d when Q <<1 except i n regions- with l a r g e d e n s i t y g r a d i e n t s e s p e c i a l l y c l o s e t o t h e c e n t r a l r o t a t i o n a x i s . 4. Use a s y m p t o t i c e x p a n s i o n i n powers of C to determine system of e g u a t i o n s g o v e r n i n g d e n s i t y d i s t r i b u t i o n i n normal f l a t d i s c r e g i o n . 5. Because of l a r g e d e n s i t y g r a d i e n t s , change s c a l e i n r e g i o n c l o s e to c e n t r a l a x i s and d e r i v e a s y m p t o t i c e x p a n s i o n f o r t h i s • c e n t r a l b u l g e . 1 6. Match s o l u t i o n s so o b t a i n e d f o r each . r e g i o n by a n o t h e r e x p a n s i o n i n the s m a l l parameter 6^ i n an i n t e r m e d i a t e r e g i o n between bulge and d i s c . 7. D e r i v e the c o m p o s i t e expansion v a l i d i n bulge and d i s c w h i c h , to f i r s t o r d e r , i s composed of the 'Emden' sphere p l u s the f l a t d i s c s o l u t i o n . 8. These two s o l u t i o n s must be combined t o s a t i s f y not o n l y constant density but also constant normal density gradient on the external boundary. These r e s t r i c t i o n s y i e l d an unigue combination of spherical 'Emden' solution plus f l a t slab solution. In the disc case, for a p a r t i c u l a r choice of parameters |2> integrating out to the external boundary at y= 1 must y i e l d at t h i s distance a normalised density of unity. The value of t h i s component of the normal density gradient at the external boundary i s then obtained from the numerical solution i n the disc. For the solution on the ce n t r a l rotation a x i s , where the composite solution i s the sum of the spherical Emden solution and the above disc solution , both eguations are integrated out to some distance y>1, such that and t h i s y i e l d s the value of y for the external boundary on the cent r a l axis . However freedom of choice of the star t i n g c e n t r a l density in the spherical Emden solution i s r e s t r i c t e d by the normal density gradient boundary condition that reguires *H . ... . a-. on t h i s external boundary on the central axis y . c An unigue composite solution i s thus obtained for the problem to zeroth order i n 6 i n the normal disc region and the central bulge. Solutions so obtained are shown in Figures 9,10,11, f o r various values of dimensionless parameters & , |S and associated central densities. Comparison of the density gradients at the external boundary for these solutions indicate at most an error of a few percent, although i t i s necessary to generate and compare these computer solutions ad nauseam i n 6 8 order to f i n d a c o r r e c t p a i r . Numerical S o l u t i o n s The composite s o l u t i o n r e q u i r e s the s o l u t i o n of the Emden equation f o r the i s o t h e r m a l case. No exact s o l u t i o n s are known so we r e s o r t to numerical methods. A s i m i l a r breakdown i s performed on the equations governing the Legendre terms. In order to do t h i s , the system i s i n i t i a l l y decomposed i n t o two f i r s t order d i f f e r e n t i a l e q uations by p u t t i n g r ^ o o c3<T, u O O thus with i n i t i a l c o n d i t i o n s " /Vfe *^ s"° V ° There are a number of numerical techniques f o r i n t e g r a t i n g t h i s p a i r through the range (^=0) t o (^-/ ). The s i m p l e r t e c h n i q u e s , such as Eunge - Kutta and b a s i c p r e d i c t o r - c o r r e c t o r , were sometimes found t o be unstable when the d e n s i t y (*) became smallO{£). I t was thus found necessary t o r e s o r t t o a " s t i f f " t e c hnigue. A " s t i f f " d i f f e r e n t i a l system i s c o n s i d e r e d t o be one i n which the ei g e n v a l u e s of i t s Jacobian have l a r g e d i f f e r e n c e s ; which can be expected f o r a system i n which t h e r e are l a r g e changes i n one of the dependent v a r i a b l e s . Within the l a s t decade, t e c h n i g u e s of s o l v i n g these s t i f f systems have been developed and the r o u t i n e s used f o r s o l u t i o n w i l l be the standard numerical technigues developed by Gear (1970) and Hindmarsh (1974,1975). The computer program t h a t i s used i s l i s t e d i n Appendix F~ . The l o g - d e n s i t y i n the c e n t r a l bulge. 69 obtained by the s t i f f numerical method, i s plotted for various values of (o^) i n figure (12) for the Emden solutions. General tendencies can be observed i n the pattern of solution with a larger drop i n density from centre to surface with increasing Indeed i t i s found that for s u f f i c i e n t l y large values of Q£,that the Emden solution o s c i l l a t e s with decreasing amplitude about the singular solution. The zeroth-order eguation v a l i d i n the disc-type region i s with boundary conditions "a It can be decomposed into two f i r s t order eguations using - A then a Boundary conditions are Numerical integrations of t h i s system by the ' s t i f f method gives the density d i s t r i b u t i o n i n the disc-type region at a large distance from and p a r a l l e l to the ce n t r a l axis. Some examples of these solutions are shown in Figure 13. 70 Chapter Four Gr a v i t a t i o n a l I n s t a b i l i t y Of fiiscs In the previous research we have shown d e t a i l s of a s e l f -consistent hydrodynamic theory of isothermal and polytropic gas discs with solid-body rotation. The steadystate structure of such a disc has been obtained and each major region , the central bulge and the disc regions, analysed. The steadystate system, governed by the basic d i f f e r e n t i a l system for density (2.3!) , w i l l now be subjected to perturbations in v e l o c i t y , density, pressure and potential. These w i l l be allowed to be general functions of the same c y l i n d r i c a l coordinates ( r , , Q) as the steadystate solution . The system w i l l be l i n e a r i s e d under the assumption of s u f f i c i e n t l y small perturbations. A set of l i n e a r d i f f e r e n t i a l equations w i l l be obtained together with the associated dispersion r e l a t i o n . The solution of t h i s system for r a d i a l perturbations along the equatorial plane of an isothermal disc with a x i a l symmetry i s examined in d e t a i l . These results w i l l be compared to observations and previous research work. Spieqel (1972) has emphasised the inherent problem of any i n s t a b i l i t y analysis such that a small chanqe i n the qoverninq conditions can often re s u l t in new i n s t a b i l i t i e s a r i s i n q . Thus th i s approach w i l l concentrate on the important case of marginal or neutral s t a b i l i t y just as in the well-known procedure of convective i n s t a b i l i t y theory where the i n s t a b i l i t y tends to choke i t s e l f off and keep the system close to neutral s t a b i l i t y c f . , Spiegel (1972) . 71 Governing Equations The basic equations of motion governinq the perturbed dependent variables, where S-P -density perturbation , - g r a v i t a t i o n a l potential perturbation , SP* -pressure perturbation , U. , — r a d i a l velocity r - " z" velocity , -azimuthal velocity , are qiven, i n the c y l i n d r i c a l coordinates [r,^rS>J, by where £r ^ °/r; <9© ^ ^ ^ Viscosity- has been ignored as t h i s work discusses neutral s t a b i l i t y for small perturbations. The perturbations w i l l be taken, as isothermal as i t w i l l be assumed that ra d i a t i o n w i l l be a s u f f i c i e n t l y rapid heat transport mechanism. Thus, the equation of state becomes The equation of s e l f - q r a v i t a t i o n becomes and the eguation of continuity of mass i s w h e r e V-U - r(ru f) +X£:u& -4'^, 72 Removing the equ a t i o n s governing the unperturbed system from the above s e t , and l i n e a r i s i n g by i g n o r i n g powers of s m a l l g u a n t i t i e s g r e a t e r than one, y i e l d s 3 art ^ 2 : f £± (*s.o Using eguation [ $. £ ) i n [S'.S) and p u t t i n g i n t o a s e t of l i n e a r d i f f e r e n t i a l eguations g i v e s the system: - J _ 3 V7 at Ik r$6> H r y "ft = o ( S . O > ) 73 In t h i s form, t h i s s e t of equations are h i g h l y i n t r a c t a b l e . We do know however t h a t a requirement f o r a s o l u t i o n of t h i s system i s t h a t the determinant of the operator matrix must be zero. Note a l s o t h a t the operator occurs i n the above s e t only i n a s s o c i a t i o n with the o p e r a t o r ( J U ^ ^ ) . We w i l l be s p e c i f i c a l l y concerned with i n s t a b i l i t i e s due t o i n f i n i t e s i m a l p e r t u r b a t i o n s i n e q u i l i b r i u m s e l f - c o n s i s t e n t d i s c s under s o l i d - b o d y r o t a t i o n . Thus, i n s t e a d of the u s u a l assumption where the m a t e r i a l d e n s i t y a t e q u i l i b r i u m i s taken as con s t a n t throughout the system, o r , at most, f u n c t i o n a l l y dependent only i n the fe) d i r e c t i o n , we know the d e n s i t y as a f u n c t i o n of these space v a r i a b l e s ( f"",^ ) . From the e q u i l i b r i u m s o l u t i o n i n c y l i n d r i c a l c o o r d i n a t e s t h i s d e n s i t y w i l l be s p e c i f i e d as a composite s o l u t i o n i n the two main r e g i o n s of the d i s c - the c e n t r a l bulge and the d i s c type r e g i o n . A l s o , we w i l l be s p e c i f i c a l l y ccncerned with the r a d i a l (r) d i s t r i b u t i o n of the i n s t a b i l i t i e s as we are i n t e r e s t e d i n the r e l e v a n c e o f such i n s t a b i l i t i e s t o S o l a r System f o r m a t i o n . Note th a t a l l the p l a n e t s l i e c l o s e t o the e q u a t o r i a l plane o f r o t a t i o n and thus the s o l u t i o n on t h i s plane ^ w i l l be of most i n t e r e s t . Thus i n order to make the ge n e r a l d i f f e r e n t i a l system more t r a c t a b l e , we w i l l i g n o r e the ( 2 . ) dependence of the system and s o l v e o n l y on the e q u a t o r i a l plane. Al s o as there i s no dependence on azimuth f o r the e q u i l i b r i u m d e n s i t y , the f u n c t i o n a l dependence i n (£>) of the v a r i a b l e s w i l l be taken as exp(imQ) as t h i s i s j u s t the simple s e p a r a t i o n of v a r i a b l e s f o r a l i n e a r system, with the v a r i a b l e being expressed by an orthogonal s e t a p p r o p r i a t e t o the d i f f e r e n t i a l system. 74 A l s o , i t w i l l be assumed t h a t the t ime dependence of the system w i l l be p r o p o r t i o n a l to (exp ( i ^ t ) ) . Thus i f the f r e q u e n c y (u>) i s complex and the i m a g i n a r y p a r t i s n e g a t i v e , then t h i s dependence can become e x p o n e n t i a l l y qrowinq with t i m e . The case of m a r g i n a l s t a b i l i t y , when (o = 0 ) , w i l l be of s p e c i a l i n t e r e s t , as t h i s v a l u e i s e s s e n t i a l l y the boundary between s t a b l e and u n s t a b l e p e r t u r b a t i o n s , f o r the n o n - d i s s i p a t i v e c a s e . On s u b s t i t u t i o n of t h e s e a s s u m p t i o n s , i n t o the system ) , g i v e s t h e f o l l o w i n g e g u a t i o n s 4 r r CLo-ht^Sl -3 SI tf? J ~ L C O + L SL r r fir y. ^0 ^ o C* lo) The f u n d a m e n t a l p r o p e r t i e s of t h i s system w i l l be d i s c u s s e d f o r the c y l i n d r i c a l l y or a x i a l l y symmetric s p e c i a l case (m=0). 75 Case ±2J_ 1 mfo T h i s i s the s i m p l e s t p o s s i b l e case wherein no p e r t u r b a t i o n s with azimuthal dependence w i l l e x i s t and the system w i l l be c y l i n d r i c a l l y symmetrical. P u t t i n g (m=0 ) i n t o the system ($1 lo) gives K%A&frf) ~f,&^ -r^.ur - P J l u e = o feO T h i s system can be reduced t o a s i n g l e second-order d i f f e r e n t i a l eguation by i n t r o d u c t i o n of the v a r i a b l e (vfy d e f i n e d as ° ~ r f ^ and where f r o m ^ . i z ) we get " " " r & u <*•>*> T h i s y i e l d s the differ.e<y\ where the boundary c o n d i t i o n \)^o } f~ P - O • f t"5\%<^)} (^.is) has been used t o d e r i v e -4--^ . ^ (1 tO f a r 76 Before we proceed further with the analysis of t h i s d i f f e r e n t i a l system, i t w i l l he changed into the non-dimensional form. As i n before we put and define The d i f f e r e n t i a l system then becomes where / ) i s obtained from the steadystate system solution. Thus the fundamental variable O in t h i s axisymmetric perturbation case i s of the form (radial velocity x density x radius) and the gradient of which ^°/&c * i s °^ form (density perturbation x radius) By eguation (3". /*/-) the r a d i a l v e l o c i t y i s proportional to the rate of change of r o t a t i o n a l v e l o c i t y . Thus t h i s fundamental variable i s related to the rate of change of angular momentum. This i s an eigenvalue problem. To find out more concerning i t s possible eigenfunctions and values (X) l e t us put i t i n Sturm -L o u i s v i l l e form. Divide the whole system by S £ . Then the equation becomes 77 The boundary conditions for t h i s to define a s e l f - a d j o i n t Sturm - L o u i s v i l l e problem i s one of the following at each boundary, i.e li n e a r homogeneous boundary conditions. Obviously these are s a t i s f i e d by the boundary at *=o. However the cuter boundary condition a t ^ = ' i s more d i f f i c u l t to comprehend, but homogeneous conditions at gas interfaces are not usual. In practice i t i s found as expected that t h i s outer boundary has l i t t l e effect on the perturbations on the equatorial plane close to the c e n t r a l axis at neutral s t a b i l i t y . A simple analysis that follows shows that these steadystate discs do not allow s e l f adjoint perturbations. In order to el l u c i d a t e the general s t a b i l i t y properties of th i s Srurm-Liousville equation, convert t h i s d i f f e r e n t i a l equation to Rayleiqh's quotient form P X cLc f - 1 Inteqratinq by parts: A " o L € s € - ~ J [S*~ a s J from which we can deduce facts about the siqns of A - O /C. /fr- . If £-^ r as well as S and C, i s everywhere p o s i t i v e , and the boundary conditions cause the vanishing of the l a s t term i n the numerator of the Rayleigh quotient, then obviously a l l roots are p o s i t i v e . The system i s then s e l f - a d j o i n t , and i s analaqous to a symmetric and positive d e f i n i t e matrix. In a l l other cases there 78 i s at most only a f i n i t e number of negative eigenvalues, while the p o s i t i v e eigenvalues increase without l i m i t Obviously, i n our case where the gradient and the term vary rapidly from central bulge to f l a t disc and outer boundary conditions (Goldreich & Lynden-Bell,1964) are such that the system i s probably not s e l f - a d j o i n t i n physically r e a l s i t u a t i o n s . Thus we can expect negative values of = o 2-£ , an d as 'R. and /^are p o s i t i v e , imaginary values of X^will appear. As (.<-=• t the time dependence i s C , t h i s implies that growing waves can propagate in t h i s disc system. Naturally physically r e a l systems w i l l have d i s s i p a t i v e processes where this r e s u l t can be expected to be modified. The e f f e c t of the outer boundary condition nis thus minimal i n t h i s context, as neutral s t a b i l i t y i s considered and the eigenfunctions and values are not needed. Ana l y t i c a l Approach In the disc-type region the log-density i s independent of radius in the l i m i t as 0 f o r the steadystate solution on the eguatorial plane. Then eguation (J. 18) becomes; This i s a s p e c i a l case of the general Bessel d i f f e r e n t i a l eguation as given by Korn And Korn (1968); A s , which has solutions 79 where a,b,c are con s t a n t s and where (>J#/v/) are the B e s s e l f u n c t i o n s of the f i r s t and second kind r e s p e c t i v e l y . Thus the s o l u t i o n of (4119) i s where the B e s s e l f u n c t i o n of the second kind i s found not to he a v a l i d s o l u t i o n due i t s s i n g u l a r i t y at the o r i g i n . The d i s p e r s i o n r e l a t i o n i s thus g i v e n f o r t h i s part of the d i s c system by C ^ {*<~^*•-+ \ E ^ - 2 0 or as probably more r e c o g n i z a b l e i n o r i g i n a l v a r i a b l e s , as where ^ - i s di m e n s i o n l e s s wave number , ^ - i s normal wavenumber , Cs - i s i s o t h e r m a l sound v e l o c i t y , z . T h i s i s the f a m i l i a r d i s p e r s i o n r e l a t i o n f o r axisymmetric waves through a constant d e n s i t y system with s o l i d - b o d y r o t a t i o n . Note t h a t the wave number at c r i t i c a l s t a b i l i t y (<^ =o ) i s given by or From t h i s d i s p e r s i o n r e l a t i o n (^.27), the t r u t h can r e a d i l y be seen of Genken And Safronov's (1975) a s s e r t i o n t h a t v i n s t a b i l i t y ( to <0) occurs f i r s t on the e g u a t o r r a l plane, as o f f 8 0 the plane of the d i s c the d e n s i t y i s l e s s than ^D) . Examination of t h i s B e s s e l f u n c t i o n s o l u t i o n y i e l d s some u s e f u l p h y s i c a l i n s i g h t s . S u f f i c i e n t l y f a r from the o r i g i n , t h i s B e s s e l f u n c t i o n { \j, ) w i l l behave almost s i n u s o i d a l l y i . e as ( S » 0 . ) and f o r ( S » * \ ) , from Korn And Korn (1968) 3 - X 1 3 - L+ y ( S ) s\y JJ^ - C o s ( s ' now from eguation {f. 16) -=- - s ^ and using the f a m i l i a r r e l a t i o n s h i p f o r B e s s e l f u n c t i o n s . a s (S.3-3 and thus using ( i l 24) and {£.25) we get et f T h i s i n t i m a t e r e l a t i o n between d e n s i t y p e r t u r b a t i o n (^>) and r a d i a l v e l o c i t y (<*) can best be seen i n the sketches below . I t can be seen t h a t inward of a perturbed d e n s i t y peak, the 81 r a d i a l v e l o c i t y i s p o s i t i v e while outward of the density peak the v e l o c i t y i s toward the central axis. As the system i s axisymmetric (m=0 ), we could thus expect a system of concentric rings to form, with egual distance between the rings in the constant density disc region. However, as ^ © i s opposite phase to Uf, outside of the density peak U.^ i s p o s i t i v e and thus the outer half of the ring begins to rotate faster than the eguilibrium r o t a t i o n a l v e l o c i t y . S i m i l a r i l y , inside of the density peak < ^ i s ne gative and the inner edge half of the <=>tr ring begins to rotate slower than the o r i g i n a l r o t a t i o n a l v e l o c i t y . This i s i n agreement with conservation of angular momentum. A shear e f f e c t i s thus caused on t h i s ring. For physical objects i t would be expected that such a ring would be unstable unless d i s s i p a t i v e forces are able to offset t h i s shear. It not, i t would be expected that the ring would s e l f -converge. From t h i s simple idea we can readily understand the prograde ro t a t i o n of the resultant body. However Mestel(1969) has shown the p o s s i b i l i t y of retrograde motion under c e r t a i n s p e c i a l rotation laws. A s i m i l a r l y i n t e r e s t i n g solution and dispersion r e l a t i o n can be found for the central bulge. In t h i s case we know, from the steadystate solution, that the density varies rapidly i n the central bulge. Toward the centre the density w i l l approach the central density (£) while toward the surface the density w i l l vary approximately inversely with the sguare of the radius. on the equatorial plane, and with comparison to mass and moment Thus 82 of i n e r t i a t h i s r e l a t i o n has been found to y i e l d a f a i r representation of the density function i n the c e n t r a l bulge. Let us look at the two extreme cases of t h i s solution regime and f i n d an approximate perturbation function and dispersion r e l a t i o n i n each region. Close to the central axis, (s ->0 ) , the density w i l l asymptotically approach the c e n t r a l density . Thus assuming that the density in the neighbourhood of the central axis i s constant at {^ ) gives s — o _o s 3 7 0 * 0 - 0 where or at c r i t i c a l s t a b i l i t y , where {k = , and (£>>^) i s the wave number close to the c e n t r a l axis. As (p), t h i s wavenumber {fe ) w i l l be much larger than the wavenumber {kj in the disc-region. The wavelength of density perturbation w i l l thus be much larger i n the outer disc than close to the central a x i s . In the case where ^ ^ <XC ^< ( X ^ The most i n t e r e s t i n g solution occurs i n the outer part of the c e n t r a l bulge where {^>>cy) on the eguatorial plane. The approximate density behaviour i n t h i s region i s then eguation [S.2o) becomes as"-83 In order to obtain some idea of the solution of t h i s eguation, we s h a l l examine the general solution of eguation (^ "•s"7) i n the form where we hav case of c r i t i c a l s t a b i l i t y (\=0) and have assumed that (2 <£<<£>) which, from the numerical solution, i s a good approximation on the eguatorial plane of the ce n t r a l bulge. Eguation(^'.?S?) i s a special case of Lommel's eguation, as given by Watson (1966), which i s where [<^> are arbitrary functions, and the solution i s given Relating Lommel's eguation to our general eguation (Sl3>%) gives, when (m=1 ), f o r integration of the f i r s t bracket i n (5". i$) where (A) i s an integration constant. If we take (^ = constant) i n t h i s s o l u t i o n , we get from equating terms multiplying the dependent variable i n (-^ 9^) , Combining [S'.tJ-') and (&</<z) yields the approximate r e l a t i o n s and thus from these two equations, we can expect an approximate 84 s o l u t i o n of the form T h i s r e s u l t i s exac t when t h e d e n s i t y (^ ) i s c o n s t a n t and y i e l d s a s i m i l a r d i s p e r s i o n r e l a t i o n I k - J^C , C O , ^ ^ > T h i s a g r e e s with t h e s o l u t i o n s found f o r the c o n s t a n t d e n s i t y r e g i o n s i n the d i s c and c l o s e t o the c e n t r a l a x i s . The u s u a l p r o c e d u r e f o r ncn - u n i f o r m media i s t o a p p l y t h e d i s p e r s i o n r e l a t i o n f o r the s i m p l e media as a f i r s t a p p r o x i m a t i o n . C r u d e l y a p p l y i n g i t t o the c e n t r a l bulge outer r e g i o n where - 7 -^ K S g i v e s a wavenumber i n t h i s r e g i o n at c r i t i c a l s t a b i l i t y K - /<T. J or Thus , i n t h i s r e g i o n we can expect the wavenumber to be a p p r o x i m a t e l y i n v e r s e l y p r o p o r t i o n a l t o r a d i u s a l o n g t h e e g u a t o r i a l p l a n e . Thus the wavelength or s p a c i n g between d e n s i t y maxima w i l l i n c r e a s e l i n e a r l y i n t h i s o u t e r c e n t r a l b u l g e . The above approximate r e s u l t s at c r i t i c a l s t a b i l i t y a re s k e t c h e d below. 85 These approximate re s u l t s w i l l now be shown to be v e r i f i e d by obtaining an exact numerical solution of the governing perturbation eguation on the eguatorial plane.of the composite so l u t i o n . Similar results for non-isothermal discs are outlined i n the Appendix. Numerica1 Solution Now, given the density d i s t r i b u t i o n along the eguatorial plane from the numerical approach for the steadystate solution, t h i s allows us to compute the numerical solution of the perturbation eguation (tf. 18). If we define then we can obtain two f i r s t order d i f f e r e n t i a l eguations i n the variables eguivalent to the second-order d i f f e r e n t i a l eguation (£". 18) . Thus, at c r i t i c a l s t a b i l i t y (^ >=0) we have where the log-density ^u) i s given by the eguilibrium numerical solu t i o n . Together with the i n i t i a l conditions X-^s-o j X ^ ^ o • S- o • at c r i t i c a l s t a b i l i t y (o=o), we can apply standard integration technigues to t h i s system and obtain a numerical solution. It can r e a d i l y be seen that these exact numerical solutions behave s i m i l a r l y to the approximate a n a l y t i c a l solution outlined above. As expected, the c r i t i c a l wavelength increases from the centre u n t i l the connection region between disc and c e n t r a l bulge. Thereafter, the wavelength i s approximately constant as in the exact a n a l y t i c a l r e s u l t . 86 R e s u l t s As we cannot compute exac t wavenumbers a t a p o i n t , we p l o t the s o l u t i o n f o r the d e n s i t y p e r t u r b a t i o n a g a i n s t r a d i u s f o r the pure gas d i s c i n f i g u r e 14. T h i s a l l o w s the i n t e r e s t i n g phase r e l a t i o n s h i p between the d e n s i t y p e r t u r b a t i o n wave and r a d i a l v e l o c i t y to be r e a d i l y comprehended. I t can be seen t h a t t h i s pure gas c a s e a l l o w s o n l y one peak i n the c e n t r a l b u l g e r e g i o n . A s i m i l a r n u m e r i c a l a n a l y s i s f o r p o l y t r o p i c e g u a t i o n s o f s t a t e y i e l d s i m i l a r p a t t e r n s with l a r g e s p a c i n g s between d e n s i t y maxima . Thus i n a pure gas d i s c o n l y a s m a l l number of r i n g s w i l l be formed at i n s t a b i l i t y . The e f f e c t of a n o n - u n i f o r m c e n t r a l b u l g e i s t h u s h a r d l y n o t i c e a b l e . T h u s , i f t h e P l a n e t s a re t o be o b t a i n e d by r a d i a l i n s t a b i l i t y cf a d i s c and as the number of d e n s i t y peaks i n a pure gas system seems t o few and f a r between o t h e r c o n d i t i o n s i n the d i s c must be e n t e r t a i n e d . The next s e c t i o n w i l l i n v o k e the o b v i o u s cause - d u s t . 87 Chapter Five Dusty Discs I t has been shown above that the wavelength at marginal s t a b i l i t y i n a pure gas disc i s incapable of yi e l d i n g a distance structure such as found i n our solar system. Such a resu l t i s not novel. The conclusion of too large a wavelength has been reached from other t h e o r e t i c a l d i r e c t i o n s . An obvious extension of our theory toward modelling the r e a l world i s to incorporate dust into the physics. Recent observations by i n f r a r e d and optics of possible preplanetary discs indicate the strong presence of dust. W.H. McCrea and I.P. Williams (1965), have examined the c h a r a c t e r i s t i c times of g r a v i t a t i o n a l segregation of l i g h t and heavy elements and conclude that dust-grains are important and necessary to Solar System formation. Goldreich and Ward (1973) have also argued for a thin dust disc while Lytteton (1972) also sees, as a f i r s t stage, the formation of a thin dusty disc close to the centre and eguatorial plane. However, these various analyses of the motion of dust toward the eguatorial plane are by no means i n agreement i n a l l aspects. The one conclusion of these numerable studies i s that a thin dust layer w i l l r a p i d l y c o l l e c t near the eguatorial plane, governing Equations Marble -(1970) has reviewed the basic dynamics of dust-gas mixtures. As before, l e t the density, pressure and velocity of the gas be |D, Pandu . Let there be hj dust grains per unit volume and, for s i m p l i c i t y l e t each dust grain have the same mass 88 L e t the d u s t , c o n s i d e r e d as a " f l u i d " have v e l o c i t y W . With the u s u a l assumption o f n e g l e c t i n g the random v e l o c i t i e s o f i n d i v i d u a l dust g r a i n s , the e q u a t i o n s of motion f o r each component becomes where i s t h e drag of the dust on the q a s . 4> i s the q r a v i t a t i o n a l p o t e n t i a l qoverned by N e g l e c t i n g p r o c e s s e s such as g r a i n f o r m a t i o n and g r o w t h , c o n s e r v a t i o n of mass g i v e s (to The isothermal eguation of state i s taJten, as usual, where the grains are assumed not to contribute to pressure. Then I n s t a b i l i t i e s We are primarily interested in r a d i a l i n s t a b i l i t i e s of t h i s dusty gas dis c . Marble (1970) has reviewed t h i s theory of weak waves i n the dusty-gas. Two approaches have been considered: in one the detailed flow about each p a r t i c l e or sphere i s considered. In the other, the dusty-gas formulation of the 89 a c o u s t i c p r o b l e m , the l i n e a r i s e d e g u a t i o n s of c o n t i n u i t y , motion and energy form the b a s i s f o r t h e c a l c u l a t i o n . The d i s t u r b a n c e of a l l q u a n t i t i e s from t h e i r u n i f o r m e q u i l i b r i u m v a l u e s are c o n s i d e r e d s m a l l . T h i s i s the same as our p r e v i o u s approach i n c o n s i d e r i n g i n s t a b i l i t i e s i n the gas d i s c . The a n a l y s i s as o u t l i n e d by Marble ( 1 9 7 0 ) i s more c o m p l i c a t e d than our problem but summarises the e s s e n t i a l p r o p e r t i e s of t h e s e p e r t u r b a t i o n s . Not o n l y are v e l o c i t i e s of p a r t i c l e and gas a l l o w e d t o be p e r t u r b e d s e p a r a t e l y but a l s o the t e m p e r a t u r e s of each c o n s t i t u e n t . H i s e s s e n t i a l r e s u l t i s t o o b t a i n f o u r d i f f e r e n t waves w i t h d i f f e r e n t sound speeds . These r e p r e s e n t sound p r o p a g a t i o n through the medium with heat and momentum exchange between the phases i n d i f f e r e n t degrees o f e g u i l i b r i u m . The b a s i c p r o p a g a t i o n speed i s t h a t of the gas a l o n e , C s and i s the g r e a t e s t of the g r o u p . I n t h i s wave o n l y the gas moves w h i l e the dust p a r t i c l e s a re f i x e d as i f ' f r o z e n 1 . The o t h e r sound speeds c o r r e s p o n d to the cases when 1 . The v e l o c i t y of the two phases are l o c a l l y e q u a l , 2. the t e m p e r a t u r e of the two phases are l o c a l l y e q u a l , and 3. the v e l o c i t y and temperature of t h e two phases are l o c a l l y e q u a l . In t h e i s o t h e r m a l c a s e , , t h e s e l a t t e r t h r e e v e l o c i t i e s are e q u a l and i s t h a t sound s p e e d , C^-, a t which the p e r t u r b a t i o n s propaqate as i f the p a r t i c l e s were f i x e d i n the qas and the s l i p v e l o c i t y was z e r o . I t i s found t h a t "^tf = CS / ( 1 + L ) . An i n i t i a l i m p u l s e moves w i t h the " f r o z e n " v e l o c i t y , ; the d i s t u r b a n c e then passes through an i n t e r m e d i a t e s t a g e and f i n a l l y moves with the complete e q u i l i b r i u m 90 propagation v e l o c i t y , i o W e t « - L - f^/^ • Comparison of t h i s approach with the r e s u l t s of more detailed work incorporating flow around i n d i v i d u a l p a r t i c l e s shows t h i s dusty-gas theory to be completely adeguate for wave-lengths large compared with the p a r t i c l e s i z e . Thus the long-time dependence of a weak wave i n a dusty gas i s that perturbation wherein the gas and dust move together. This w i l l be the s i t u a t i o n that w i l l be discussed and we s h a l l see that the system of eguations s i m p l i f i e s . Also, an even more important assumption w i l l be made that the dust-gas density r a t i o for perturbations i s constant along the r a d i a l d i r e c t i o n t h i s allowing an a n a l y t i c a l r e s u l t to be achieved. U n t i l the complete equilibrium disc structure for dust and gas i s capable of being derived, t h i s approximation must s u f f i c e though Goldreich and Ward (1973) have examined the f l a t slab problem and shown that three zones are formed. On the equatorial plane, a thin pure dust disc i s formed, enveloped by a wider t r a n s i t i o n layer of dust-gas mixture outside of which the pure gas system e x i s t s as before., We s h a l l discuss t h e i r model i n more d e t a i l l a t e r comparing i t with our r e s u l t s but note here that i t i s t h i s dust-gas t r a n s i t i o n layer that i s under examination. A s i m i l a r isothermal perturbation for the axisymmetric case of rv.=o w i l l now be c a r r i e d out for t h i s dusty gas disc. As before we w i l l assume an angular ve l o c i t y JL of the unperturbed disc. For the gas we have - C | ^ _ E H- <2SL- u r - ^B/p 60 9 1 Where the only difference from the previous case of pure gas i s the addition of the forces of in t e r a c t i o n K between gas and dust. For the dust - ^ = N^ *^ The 2" component i s ignored as before. S i m i l a r i l y , perturbation of the s e l f - g r a v i t a t i o n eguation with the continuity eguations for each component Assuming that the dust contributes n e g l i g i b l y to the pressure term, the gas eguation of state can be used The above system has been set up for the most general case. The f i r s t assumption that w i l l be made i s that aft e r s u f f i c i e n t l y long time the gas and dust move together. From equations i t can be seen that t h i s implies that i s zero in t h i s l i m i t giving Using an argument suggested by Prof. W. McCrea and elucidated by Prof. L. Sobrino, i f we make no assumption about the r a d i a l force and eliminate t h i s term between equations (6.1a,b) we get ( (Mf*) + k - " \7P+ (f-tfdO • ft.la.) 92 for the r a d i a l component. Perturbing this eguation and removing the non-perturbed part yields W J ( f t p ) ( 7 ^ ) - <*r which i s just the momentum eguation for the gas-dust mix i n which no vel o c i t y difference e x i s t s between the two components, and are thus equivalent to a single system with • density (f-t^ <_0 • The next fundamental assumption that we make i n order to obtain a simple a n a l y t i c a l solution that w i l l give a f r u i t f u l insight i s that the r a t i o of the perturbation i n dust density to the gas density perturbation remains constant : Z. = ^^/S^ For the equation of state t h i s gives r o+o r x J ,  N and substituting into the momentum equation ^fo'V) yields 0-K-) or I o-ti-) f &r 1 e>r ' & Toqether with the velocity equation fc-'O and the continuity eguation f o r the gas in the form t h i s gives a set of 4 eguations qoverninq 4 dependent variables. The difference b£tween th i s set and that used f o r the pure qas can be seen i n the momentum equation where the dust and qas are considered as a sinqle unit as they move toqether and i n the s e l f - g r a v i t a t i o n equation where the density i s increased by the effe c t of the dust. Introducing the same variable as i n the gas case \> =- c ^ K t r gives from ^fen)as before iJ> = - c O r p u f i fc-n) and d i f f e r e n t i a t i o n y i e l d s 93 which using lfo)gives The second derivative i s thus cP~J> - 1\ c • -—, •= — <<<-or.s~.Sp — L LO S p , <$r^ • dr \ f f 4 . 2 0 and combining £6 , c0and (fc2o)gives Using ( V f t^in the s e l f - g r a v i t a t i o n eguation (/-e-u) • - ^ r £-r- S3^) (6.7.1.) and integrating using the boundary condition ^-o , r=-o gives Now using (& / 5)substitute from ^ 2 / ) for the f i r s t term to get . sl.&p * <- • - - ^ 7 / W ) (/ -ft.; ^  <3-r I c o /^ /V-L.") * r *~ crr^ r &r J and for the second term use fa./qjto get Again from (G-XV) «e have 1 / ^ 0 f — co r and substituting these into /^ ./S") gives, a f t e r multiplying — l<o<-[i-+Jk> through by 'the fundamental eguation This i s sim i l a r t c the form of previous eigenvalue problem for pure gas but with a v i t a l difference. The term multiplying ^ i s greater by the factor and also the thermal, or temperature, term i s diminished by a factor and so we can expect to f i n d a corresponding increase in the wavenumber 1 . e 94 C o n v e r t i n g t o the format used f o r the e q u i l i b r i u m gas s t r u c t u r e and c h a n g i n g to the same v a r i a b l e S as i n the gas problem when 6 0 , we have where , as b e f o r e „ — „ , and i s the l o g - d e n s i t y of the g a s . The change of s c a l e i s o b v i o u s as compared with the pure gas r e l a t i o n . Thus the wavelength w i l l be much s h o r t e r than the pure gas case w h e n ^ - » 1 on the e q u a t o r i a l p l a n e . The new d i s p e r s i o n r e l a t i o n i s 95 B e s u l t s Some examples of t h i s d e n s i t y wave p a t t e r n are shown f o r va r i o u s values of dimensionless parameters and mass l o a d i n g r a t i o i n F i g u r e s /i^/fe 7/7. He c o n c e n t r a t e on the s i m p l e s t problem of i s o t h e r m a l s t e a d y s t a t e d i s c s with i s o t h e r m a l p e r t u r b a t i o n s and the usual r i g i d r o t a t i o n case. The a s s o c i a t e d s t e a d y s t a t e composite s o l u t i o n t h a t was used on the e g u a t o r i a l plane i s shown i n F i g u r e Iff. S i m i l a r r e s u l t s are shown i n F i g u r e s 1^,2-0 f o r the d i s c s y n t h e s i z e d as before using the Emden s o l u t i o n with Legendre terms upto and i n c l u d i n g the f o u r t h Legendre f u n c t i o n I t i s r e a d i l y seen how the dust-gas mass l o a d i n g r a t i o decreases the wavelength at marginal s t a b i l i t y . The c l o s e r e l a t i o n s h i p to our s o l a r systems b a s i c p a t t e r n i s compared f o r the case if* Ic? Of-^l-lf* Z-=4,in Table I I , where the c a l c u l a t e d r i n g d e n s i t y maxima, i n dime n s i o n l e s s r a d i a l c o o r d i n a t e s Z^= 1, are compared with the p o s i t i o n s of the p l a n e t s i n the S o l a r System, as graphed i n F i g u r e d ' . By n o r m a l i s i n g with r e s p e c t t o Neptune, which i s formed from a r i n g mode l o c a t e d c l o s e to the f l a t o uter d i s c at normalised r a d i u s 0-9^z* , exact l o c a t i o n s are comparable i n Table I I . The axi-symmetric r i n g has been shown p r e v i o u s l y t o be s u s c e p t i b l e to shear motions generated by the p e r t u r b a t i o n s . S e l f - c o n v e r g e n c e o f t h i s r i n g i n t o s i n g l e o b j e c t s c o u l d then be expected. Thus i t i s seen that our dust gas d i s c , unstable t o axi-symmetric p e r t u r b a t i o n s , corresponds w e l l with observed d i s t a n c e p a t t e r n s of our s o l a r system. I t can be seen t h a t l i n e a r l y i n c r e a s i n g spacing of the r i n g s occurs c l o s e to the 96 c e n t r a l r o t a t i o n a x i s c o r r e s p o n d i n g t o Bode Law b e h a v i o u r . From S a t u r n to P l u t o , the wavelength i s a p p r o x i m a t e l y c o n s t a n t ( /OA.U.) c o r r e s p o n d i n g t o p r o b a b l e o r i g i n near the c o n s t a n t d e n s i t y d i s c r e g i o n . We must now compare the r e s u l t s of t h i s model , the e x p e c t e d parameter v a l u e s and i m p l i c a t i o n s a g a i n s t observed and p r e d i c t e d i n f o r m a t i o n . D ust^Gas Mass L o a d i n g E x a m i n a t i o n of the wavelength p a t t e r n s f o r p a r t i c u l a r v a l u e s of d i m e n s i o n l e s s c o n s t a n t s ^ , s h o w s t h a t a d u s t - g a s mass l o a d i n g r a t i o I— of 4> g i v e s a c l o s e a p p r o x i m a t i o n to t h a t p a t t e r n found i n the S o l a r System. There i s , a s y e t , no e x p e r i m e n t a l o b s e r v a t i o n s that have determined t h i s number f o r p r e p l a n e t a r y d i s c s though o b s e r v a t i o n s are b e i n g made o f i n f r a r e d o b j e c t s which appear t o be d u s t y d i s c s . T h i s r a t i o however has been t h e o r e t i c a l l y i n v e s t i g a t e d , though i n d i r e c t l y , by G o l d r e i c h and Ward (1972) who c o n s i d e r e d a s e m i * i n f i n i t e s l a b composed of dust and gas i n s o l a r p r o p o r t i o n s . T h i s s i m p l i f i e d s l a b model a l l o w e d them to d i s c u s s s e t t l i n g o f dust toward the e g u a t o r i a l plane . They show t h a t the dust w i l l c o n c e n t r a t e i n a t u r b u l e n t boundary l a y e r i n a very s h o r t t ime and t h a t the t h i c k n e s s of t h i s l a y e r w i l l depend c r u c i a l l y on the c r i t i c a l R e y n o l d ' s Number. At the c e n t r e o f t h i s d u s t ^ g a s l a y e r a t h i n zone of pure dust i s c l a i m e d to e x i s t . The r e l a t i v e t h i c k n e s s of the t u r b u l e n t g a s - d u s t l a y e r t o the pure gas d i s c i s a p p r o x i m a t e l y 7*10-5, u s i n g t h e i r numbers based on t h e S o l a r System . I f i t i s assumed t h a t most of the dust i s i n t h i s t u r b u l e n t l a y e r and use i s made o f t h e i r number 5*10-3 f o r the r e l a t i v e mass of t o t a l dust t o gas i n the S o l a r 97 System , simple arithmetic yields a dust-gas mass loading r a t i o of approximately 70 in the turbulent dust-gas la y e r . Goidreich and Ward studied i n s t a b i l i t y in the pure dust layer at the centre of the dust-gas layer and found that planetesimals were formed therein. Such a r e s u l t can be expected as much shorter wavelengths w i l l be found i n the r e l a t i v e high density pure dust layer than i n the dust-gas layer. As t h e i r analysis discusses i n s t a b i l i t y i n t h i s extremely thin pure dust layer ( pancake ) on the eguatorial plane while t h i s work concentrates on the dust-gas layer , the two approaches are complementary . The ring modes i n the dust-'gas self-con verge due to shear ; such large scale motions can be expected to i n t e r a c t with the planetesimals so formed in the th i n dust layer . This i s obviously a non-linear and complex i n t e r a c t i o n but i t must be solved i n order to understand Solar Systems. Implications And Speculations The most widely accepted theory of star formation i s by collapse and fragmentation of spheroidal gas-dust bodies. I f these have a small amount of rotation around a central axis, conservation of angular momentum implies that disc shapes may form, as i n t u i t i v e l y i t can be expected that p a r t i c l e s close to the central axis can collapse to the centre while, further out near the eguatorial plane p a r t i c l e s have larger angular momenta and w i l l thus i n f a l l much l e s s . The steadystate density structure developed for discs i s but one of the possible solutions and end results of collapse. As they are inherently unstable , ring modes as described above can be formed. However t h i s work has developed only the l i n e a r i n i t i a l stage of t h i s 98 p e r t u r b a t i o n . In the subsequent n o n - l i n e a r regime whether and under what c o n d i t i o n s these r i n g modes can c o n t r a c t i n t o s i n g l e or m u l t i p l e o b j e c t s i s a g u e s t i o n t h a t remains t o be answered. The above r e s u l t s show t h a t v a l u e s of the i s o t h e r m a l d i m e n s i o n l e s s parameters Icf* , ^> -\ I i g i v e a r e s p e c t a b l e c o r r e l a t i o n with the ' S o l a r System d i s t a n c e d i s t r i b u t i o n of p l a n e t s and a s t e r o i d b e l t . The d u s t - g a s mass l o a d i n g f a c t o r was shown above to be i n agreement with the p r e v i o u s l y r e s e a r c h e d v a l u e , and i t i s n e c e s s a r y t o examine the o t h e r p a r a m e t e r s . The parameter ^> = c^SO'^' ^ a n ( j ^ s o £ o r d e r u n i t y f o r an a x i s y m m e t r i c d i s c . As /^i , f o r m o l e c u l a r hydrogen and a temperature of 1 0 0 ° ^ t h i s would y i e l d r o t a t i o n a l v e l o c i t i e s of the o r d e r o f a k i l o m e t r e per second which compares with the o b s e r v e d v e l o c i t y of Neptune a t 5 km/sec . The parameter C ^ c ~ fa*^ ft^*> /f\„ ^ s o-f o r d e r / c £ , and u s i n g the d i s t a n c e t o Neptune as Sj,* g i v e s a c e n t r a l hydrogen d e n s i t y as a p p r o x i m a t e l y fO gm/cc and an e x t e r n a l d e n s i t y o f a p p r o x i m a t e l y / O g m / c c . Such d e n s i t i e s are o f the e x p e c t e d and observed v a l u e s , c f . H i e l e s (1971), L a r s o n (1973), i n d i c a t i n g t h a t our t h e o r e t i c a l r e s u l t s are c o n s i s t e n t wi th o b s e r v a t i o n s and the d e d u c t i o n s of o ther t h e o r e t i c a l a p p r o a c h e s . We w i l l c o n s i d e r and compare with t h e model o t h e r i m p o r t a n t f a c t s of t h e S o l a r System. That the P l a n e t s and A s t e r o i d B e l t l i e c l o s e t o the p l a n e of the e c l i p t i c i s a p r e c o n d i t i o n on any t h e o r y and i s n a t u r a l i n our approach of d e t e r m i n i n g the r a d i a l p e r t u r b a t i o n s on the e q u a t o r i a l p l a n e of the d i s c , as d e n s i t y i s g r e a t e r here t h a n o f f the p l a n e and i n s t a b i l i t y w i l l o c c u r 99 i n i t i a l l y on t h i s eguatorial plane, Genkin & Safronov (1975). The prograde rotation of the majority of Planets i s explained by the shear, contraction and self-convergence of these ring modes although Mestel (1966) has shown that s p e c i a l v e l o c i t y d i s t r i b u t i o n s i n the o r i g i n a l disc can also re s u l t in retrograde motion. Whether a par t i c u l a r v e l o c i t y d i s t r i b u t i o n i s possible, such that a rin g mode i s l e f t i n t a c t and does not converge, i s an open guestion that may be applicable to the origi n of the Asteroid Belt. Solutions of the l i n e a r problem appear to have a ring mode at t h i s location between Mars and Jupiter and thus conditions not applicable to the other ring locations either prevented formation of Planet X or that indeed i t did form but subsequently broke up, c.f. Ovenden (1975). The most important f a c t , apart from distance d i s t r i b u t i o n , faced by any Solar System cosmogony i s the d i s t r i b u t i o n of angular momentum. The Sun, with mass approximately 750 times that of the Planets, has only about one part i n two hundred of the t o t a l angular momentum of the Solar System, Allen (1972). I f i t i s assumed that the s p e c i f i c angular momentum i s conserved through the formation of Sun and Planets from these ring modes, what does the d i s t r i b u t i o n imply about the source of the Sun's matter. The s p e c i f i c angular momentum i s defined by C=R*fi*J"L and for a simple disc under uniform rotation gives C « B . Assuming that the matter now i n the Sun came from a r a d i a l region, centre to X a.u, and the Planets from X to 30 a.u, Neptune's o r b i t . This i s readily solved to give an approximate value for gives, as X<<30. 100 X=0.08 a.u. If we examine the ri n g mode perturbations. Table I I , i t can be seen that the proto-Sun corresponds with the cen t r a l density peak which extends r a d i a l l y outward to approximately half-way to the f i r s t ring density peak corresponding to Mercury. Our t h e o r e t i c a l r e s u l t gives t h i s as approximately 0.14 a.u, while the exact present lo c a t i o n of Mercury would give 0.2 a.u. As our model can be but a rough approximation t h i s agreement i s remarkable and allows comprehension of the angular momentum d i s t r i b u t i o n i n the Solar System i f the Sun's . matter originated from t h i s r e s t r i c t e d r a d i a l region around the cen t r a l rotation axis. Our model and i t s ring mode perturbations d i f f e r r a d i c a l l y from the Kant-Laplace view of a disc cooling, contracting, speeding-up and shedding rings, and th e i r associated problems with ring dispersion and angular momentum are overcome by our system. Dust has been shown to be of v i t a l significance and that the wavelength at c r i t i c a l s t a b i l i t y as far out as Pluto i s controlled by the dust-gas mass loading factor. This implies that the Outer Planets can be expected to possess a dust-derived component, a resu l t which i s far from novel. However the t o t a l mass of each planet i s not given by t h i s l i n e a r perturbation theory and i s derivable only by following the system through the nonlinear regime i n t o the present conditions of a many-body problem. To map from l i n e a r perturbation solutions through the subseguent nonlinearity, applied mathematicians have used the Pr i n c i p l e of Least Action with success, Ovenden ['\97S) has derived a P r i n c i p l e of Least Interaction Action for the analysis of the many-body Solar System problem and assuming Planet masses 10 1 has obtained t h e i r Least Interaction locations- Such an approach can be used to a l t e r n a t i v e l y obtain masses i f we assume the locations given by the ring modes. The model and i t s particular r e s u l t s . Table I I , imply that Neptune and Pluto formed i n the region where the central bulge merges into the outer disc although the e f f e c t of the disc i s predominant even at t h i s close to the axis as the disc c e n t r a l density i s much higher than the bulge density i n t h i s outer bulge-disc region, c.f. Figure 18. Whether more planets exist at greater distances than Pluto i s not answerable by t h i s l i n e a r solution but t h e i r t h e o r e t i c a l locations can be expected to have simi l a r distance spacings to the Outer Planets, 10 a.u. Note that solutions can be found that give ring modes i n the region between the Sun and Mercury. Alven and Arrhenius (1975) have argued f o r c e f u l l y for the importance of magnetic f i e l d s i n the early Solar System. Dispersion relations for s e l f - g r a v i t a t i n g rotating systems with magnetic f i e l d have been derived by Chandrasekhar (1961) i n a boundless domain. For the dust-gas system we have seen that the mass loading factor completely dominates the dispersion r e l a t i o n r e l a t i v e to the rotation e f f e c t and the same can be expected i n comparison with the expected magnitude of the magnetic e f f e c t . They have also argued that the distance d i s t r i b u t i o n of the inner s a t e l l i t e s of the Giant Planets, which can be put in a Bode law form, must be explainable by any theory that i s applicable to the Planetary d i s t r i b u t i o n . Such systems were probably non-isothermal and technigues are being developed to solve such problems. 102 Conclusion The steadystate structure f o r s e l f - g r a v i t a t i n g rotating discs has been obtained by two d i f f e r e n t techniques i n the a xisymtnetric case. Future work w i l l t ackle the more complex problem of non-axisymmetry and i t s relevance to g a l a c t i c structure. This w i l l also necessitate examination of perturbations of ring and s p i r a l type hut the above work on the simplest problem has indicated the method of s o l u t i o n . S i m i l a r l y the dust-qas solution derived above i s but the i n i t i a l stages of the t o t a l solution which must include not only non-isothermal e f f e c t s but also the non-linear regime and subsequent evolution in order for more exact, comparison with observation- As always, the answering of one question reveals numerous others. Table II Planet Trjje Distance (a. u) Calculated Distance 2; JZA flercury 0. 39 0. .0C8 0 . 27 Venus 0. 72 0. ,015 0 . 50 Earth 1. 00 0. .031 0 . 97 Mars 1. 52 0. ,060 1. 9 Asteroids •> .9 0 . ,115 3. 6 Jupiter 5 . 2 0. .213 6 . 75 Saturn 9 . 5 0 . , 377 11 . 9 Uran us 19 . 2 0 . 6 2 7 19 . 9 Neptune 30 . 1 0. ,950 30 . 1 Pluto 39 .5 1. 32 4 1. 8 ISO : DELTA 10-3 : R2=-100 R4=10-2 i—1 o 105 F i g u r e _4 POLYTROPE : N=5.0 : DELTA=10-4 : fl2=-1000: R4=10 o <7\ ISO : DELTA 10-3 : A2=-70 A4^7X10-3 o COMPOSITE DISC : BETR=2 : RLPHR-2 : C/E=IO+4 COMPOSITE DISC : BETfl-1.3 : A L P H A S . 4 : C/E^iO+6 112 113 F i g u r e X2 c o S I Q O " 3 C D -O o C O C 2 o ^ 1 r-in • CD 10~5 10"4 10"3 10"2 10"1 10 L O G R A D I U S S P H E R E 114 C D * C D L O L U LTD C D • L U C M L O I 1 51 o 73 C D C D Figure 13 0.0 0 . 3 3 0 . 6 6 D I S C T H I C K N E S S (Z=l) 1 .0 LIT 118 119 F i g u r e 18 H I I | I M l l j I I | IMIIj 1 I | I I 1 I | I llll) 1 I | I I III] 1 I | Mill) l O " 5 4 10-* • 4 l O " 3 4 l O " 3 4 l O " 1 4 1 0 3 4 I D 1 L O G R f i D I U S S P H E R E 120 F i g u r e 1.9 o ^5 ISOTHERMAL :.R2= -15 : ALPHA=10XX4 : BETA=2.0 : 0 - 3 >-•7 CO LU CD CDS o o 1 — 1 d o 2 i 1 1 n n i | — 1 1 1 I i n i | — 1 1 11 1 1 1 n m | — 1 i 1111111—1 1 1111111 10" 4 10-« 4 10"3 4 ID" 2 4 10-1 4 10° L O G R R D I U S S P H E R E 4 10 1 121 122 "Figure 21 123 Bibliography Allen,C.W. 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(1963) ; Ap. J n l . , 138,809. Smith,B. (1975) ; Astrop. Space Science 35,223. Spiegel,E.A. (1972) ; " Symposium on Origin of Solar System ", H. Reeves ed. Spiegel, E.A. S Zahn ( 1970) ; Comm. Astr. Space Physics 2,5,178. Spitzer,L. (1942) ; Ap. J n l . 95,329. Strom, A et a l . , (1975) ; Ann. Rev,. Ast. Astrop. 13,187. Van Dyke,M.D. (1975); " Perturbation Methods in F l u i d Mechanics ", Academic Press. Watson, (1966) ; " Theory of Bessel Functions ", Cambridge University Press. 126 Appgn.dices Appendix A I s o t h e r m a l Gas S p h e r e . 1 When the gaseous system i s a sphere ( S - ^  ) which g i v e s ( £- =- / ) . The d e n s i t y e g u a t i o n i n the case of an i s o t h e r m a l gas sphere with no r o t a t i o n (^^>) becomes A.i T h i s e g u a t i o n has been a n a l y s e d e x t e n s i v e l y and L y n d e n - b e l l and Wood (1968) have summarised i t s s o l u t i o n s : (1) rhe s i n g u l a r s o l u t i o n (2) the i s o t h e r m a l gas sphere s o l u t i o n s with f i n i t e d e n s i t y a t the c e n t r e ( ) (3) the g e n e r a l s o l u t i o n s whose d e n s i t i e s tend to o as ( c^. —o < ) -C h a n d r a s e k a r (1939) i n h i s book on S t e l l a r S t r u c t u r e a n a l y s e d t h e s e s o l u t i o n s and showed t h e i r homologous n a t u r e , and i l l u s t r a t e d how the i s o t h e r m a l gas sphere s o l u t i o n s a p p r o a c h e d , and o s c i l l a t e d a r o u n d , the s i n g u l a r s o l u t i o n f o r l a r g e (^). T h i s i s o t h e r m a l gas sphere s o l u t i o n i s r e a d i l y computed m u m e r i c a l l y a n d , f o r (<p s m a l l i s g i v e n by t h e power s e r i e s . 127 Among the fundamental properties of th i s solution i s that the density only approaches zero as the radius ( ) approaches i n f i n i t y . Thus a f i n i t e radius isothermal gas sphere must have a surrounding medium with a non-zero density. This i s i n comparison to p c l y t r c p i c gas spheres whose density can become zero at a f i n i t e radius. Density plots for various alpha are shown i n Figure \2- . 128 Appendix B ^ H k § S § t i S § i P r o p e r t i e s Of B a s i c D e n s i t y ' E q u a t i o n . Our b a s i c e q u a t i o n f o r d e n s i t y v a r i a t i o n i s a p a r t i c u l a r example of the q e n e r a l n o n - l i n e a r e l l i p t i c e q u a t i o n whose p r o p e r t i e s have been i n v e s t i q a t e d by a qroup o f m a t h e m a t i c i a n s over the l a s t few decades . The q e n e r a l case has been c o n s i d e r e d from the p o i n t of view of Riemannian geometry by Kazdan and Warner (1973). In the case O ( o ) , they show t h a t a n e c e s s a r y c o n d i t i o n f o r the e x i s t e n c e of a s o l u t i o n t o (f) i s t h a t ( K ) be p o s i t i v e somewhere on the m a n i f o l d . T h i s i s o b v i o u s l y s a t i s f i e d i n our case where ( ^ > 0 ) , ( # c > > o ) . More i m p o r t a n t l y , Kazdan and Warner show t h a t at (c=.-j_ ) , t h e r e i s a n o n - e x i s t e n c e of r o t a t i o n a l l y symmetr ic s o l u t i o n s o f e g u a t i o n (I) g i v e n a r o t a t i o n a l l y symmetric (K) . S i m i l a r l y they o b s e r v e t h a t the f i r s t v a l u e of (c) at which t h e r e are o b s t r u c t i o n s to the s o l v a b i l i t y o f (/) i s ( 2 ) , which i s the f i r s t n o n - z e r o e i g e n v a l u e of the o p e r a t o r [V) on (<£) . From the g e n e r a l t h e o r y of n o n - l i n e a r e g u a t i o n s , we can expec t a b i f u r c a t i o n of the s o l u t i o n s at t h i s p o i n t . Thus the range o f t h e parameter ^ w i l l be taXen as ° £^ < 2 . 129 Appendix C Nonjlsothermal Discs The temperature ' enters the system through the eguation of state p _ jg?-j and >w i s mean molecular wieght. In physically probable systems both / = f[fjs.) and t*. = (r,£) may e x i s t . As they both enter through ^ = ^ T(r»^) # t h i s w i l l be used as a new dependent variable. The eguation f o r t h i s system i s derived as before: In the isothermal case a spheroidal central bulge and f l a t outer disc were found. As the density decreased rapidly outward {occ^) from the centre t h i s disc was of much lower density than the central core. Now the central spherical solution o s c i l l a t e d around and - 7 , approached rapidly the singular solution f • In the more general case where from matching orders i n the spher i c a l case (2) , we get when 1^= ^ ^ / ,/L s- o r * i Thus constant « In the isothermal case the singular solution dominated the central bulge. S i m i l a r i l y i t can be expected that systems with 130 ^ D( f w i l l have a density gradient i n the outer central bulge such that -f> r Note that in t h i s region "f^J* ^ a n d i s independent of the value of %tV and thus independent of the change of temperature or molecular weight i f t h i s r e l a t i o n i s v a l i d . Note that i t has been argued from chemical approaches that I tXr i n the presolar nebula. Thus =t> ^<xr and P* 3^ , a not unusual value for what may well be a collapsing gas. This changing with radius may be expected to a l t e r the acoustic perturbation solutions. However, an important r e s u l t i s obtained by consideration of the basic dispersion r e l a t i o n found for r a d i a l o s c i l l a t i o n s along the eguatorial plane. In many l i n e a r and non-linear wave propagation technigues the dipersion r e l a t i o n for the simple medium i s extrapolated with success into more complex domains . The governing dispersion r e l a t i o n i s LO^ = A-^&if> - t ^ s i ? ~ ; hi We have shown previously that at marginal s t a b i l i t y . ^ Jl B\itp/ has been shown to be an independent function (Vf as 1 *- • -<-+>•) Thus i n the outer c e n t r a l bulge where dispersion i s expected but such that k <x f1 , \ « r i n a l l these cases. Thus the solution found f o r the isothermal 131 case C» = o) can be e x p e c t e d t o be c l o s e t o the p e r t u r b a t i o n waves and z e r o e s of t h e more g e n e r a l case ( A > 0 ) T h i s i s a l s o t r u e f o r t h e dusty problem and so a s i m i l a r wave p a t t e r n and d e n s i t y maxima can be expected at n e u t r a l s t a b i l i t y , as i n the i s o t h e r m a l c a s e . 132 Appendix D L i n e a r Problem When the p o l y t r o p i c i n d e x n=1 the b a s i c e q u a t i o n becomes l i n e a r . In s p h e r i c a l c o o r d i n a t e s , t h i s has the e x a c t s o l u t i o n which was the f i r s t order p e r t u r b a t i o n e x p a n s i o n f o r the n o n -l i n e a r c a s e . For t h e l i n e a r c a s e , t h i s y i e l d s \7 @ - - ^ c> .3 with s p h e r i c a l s o l u t i o n ® ~ A D .«-*c ——, _ and V ± = / - ±- o.s which i s decomposed as b e f o r e i n t o Legendre f u n c t i o n e x p a n s i o n d o 4- U - * such t h a t and The c o e f f i c i e n t s Aj^are o b t a i n e d by s a t i s f y i n g n e c e s s a r y and s u f f i c i e n t boundary c o n d i t i o n s on an e x t e r n a l boundary . T h i s i s the same p r o c e d u r e as f o r the n o n - l i n e a r c a s e . However i t can r e a d i l y be seen from t h i s exac t s o l u t i o n t h a t the d i s c i s composed of the s p h e r i c a l term j$ and t h e m u l t i p o l e terms. The c y l i n d r i c a l e x p a n s i o n a l s o a l l o w s comprehension of matching between t h e d i s c r e g i o n and c e n t r a l b u l g e f o r t h i s l i n e a r p r o b l e m . The e g u a t i o n i s c3 ^ t ^^- ' f s ' ^ J ^ ^ h ^ ' ' " ^ " 133 where t h e d i s t a n c e s are n o r m a l i s e d by In the d i s c r e g i o n , where 6^-o f o r z e r o t h - o r d e r s o l u t i o n with the e x a c t s o l u t i o n Boundary c o n d i t i o n s on the e g u a t o r i a l plane «^  =-o demand ^ " - ^ o - Thus o and as the e x t e r n a l boundary c o n d i t i o n i s < T i , ^ v/o<'e As b e f o r e we o b t a i n the c e n t r a l bulge e g u a t i o n by t r a n s f o r m i n g , i . e s t r e t c h i n g , t h e v a r i a b l e such t h a t we o b t a i n a-? s a r a ^ e which r e v e r t s t o the o r i g i n a l e g u a t i o n i f expressed i n s p h e r i c a l c o o r d i n a t e s % i a n d i s e x p e c t e d from the a n a l y s i s of Chang (196 1) c o v e r i n g such c o o r d i n a t e e x p a n s i o n s . T h i s s o l u t i o n i s , as a b o v e , i n s p h e r i c a l c o o r d i n a t e s As i n the i s o t h e r m a l c a s e , f o r exact matching i t i s n e c e s s a r y t o know a l l the c o n s t a n t s Z ^ * * D U t i f t h e s e are known, then the e x a c t s o l u t i o n i s known : c a t c h 22. However, i n the l i n e a r p r o b l e m , where t h e s p h e r i c a l s o l u t i o n i s which has the o u t e r l i m i t 3 134 as i n the i s o t h e r m a l case. Now, the other term i s governed by the l i n e a r e guation, o . with p a r t i c u l a r i n t e g r a l X? / and g e n e r a l s o l u t i o n s a t i s f y i n g This must match with the outer s o l u t i o n ( t>.ti ) so we look f o r s o l u t i o n s o f the form S u b s t i t u t i n g i n the s t r e t c h e d eguation g i v e s 6 — s ~ ° *-,€» with s o l u t i o n where are i n t e g r a t i o n c o n s t a n t s . As =- o on s=0 , t h i s i m p l i e s b=. o , and thus Matching with the outer s o l u t i o n by P r a n d t l ' s technigue, as be f o r e , a l l o w s the determination of t h i s c o n s t a n t A and y i e l d s the s o l u t i o n T h i s i s a l s o the t o t a l s o l u t i o n to the o r i g i n a l eguation and note that the form of s o l u t i o n i s v a l i d when the r o t a t i o n dependent parameter o"^ i s a f u n c t i o n of r a d i a l v e l o c i t y a l s o . This t o t a l s o l u t i o n composed of s p h e r i c a l n o n - r o t a t i n g s o l u t i o n and d i s c s o l u t i o n i s of same form as the composite s o l u t i o n obtained i n the n o n - l i n e a r problem. 135 Appendix E l a t h i s Appendix the s i m p l e d i s c e q u a t i o n (<P.<4I ) i s examined, an exact s o l u t i o n i s o b t a i n e d i n i n t e g r a l f o r m , and i t s i m p l i c a t i o n s d i s c u s s e d t o g e t h e r w i t h a s i m p l e s t a b i l i t y a n a l y s i s based on P h a s e - P l a n e t e c h n i q u e s . The b a s i c e g u a t i o n with the d i m e n s i o n l e s s parameters a l p h a , b e t a , can be i n t e g r a t e d once to g i v e i (&jT ~ f l y - - + C ^ . 2 As the g r a d i e n t fy*/^ i s zero on y=0, t h e i n t e g r a t i o n c o n s t a n t C i s g i v e n by where f / ^ i s the n o r m a l i s e d d e n s i t y on t h e c e n t r a l p l a n e . On the e x t e r n a l boundary the n o r m a l i s e d d e n s i t y i s ~ / and the normal d e n s i t y g r a d i e n t i s d e f i n e d as . T h i s y i e l d s the s i m p l e r e l a t i o n I n t e g r a t i n g a g a i n g i v e s t h e i n t e g r a l e g u a t i o n which i s e s s e n t i a l l y an e x a c t form o f s o l u t i o n . E x a m i n a t i o n of the denominator i n d i c a t e s t h a t the n o r m a l i s e d d e n s i t y mast decrease with i n c r e a s i n g y from the c e n t r a l p l a n e outward f o r a r e a l s o l u t i o n , a p h y s i c a l l y expected r e s u l t . A p h a s e - p l a n e a n a l y s i s (Korn 6 Korn,1968) a l s o i n d i c a t e s c o n s t r a i n t s on the s o l u t i o n r e g i m e . P u t t i n g the e g u a t i o n i n the form ^ / d r ^ i T . t 136 i n d i c a t e s a c r i t i c a l p o i n t o c c u r s at some JU* when As the g r a d i e n t i s z e r o o n l y on the c e n t r a l plane the c o n d i t i o n f o r no c r i t i c a l p o i n t s i s thus *l- **** > / ? c a c o n s t r a i n t on t h e r e g i o n i n a l p h a r b e t a parameter space f o r p h y s i c a l s o l u t i o n s . 137 Appendix F L i s t of Symbols O- Polytropic normalising distance (2.4) .A^Constants of integration. <X^ Isothermal dimensionless number oCc Isothermal dimensionless number «p, Polytropic dimensionless number £• Isothermal dimensionless number r Polytropic dimensionless number Sound velocity for gas ^l^Sound velocity in dust-gas mixture §. Dimensionless numbers £CJ G r a v i t a t i o n a l Energy £ Sguare of r a t i o of disc thickness to radius € Newton's parameter €. E c c e n t r i c i t y ^ Normalised central bulge density and asymptotic terms Crj G r a v i t a t i o n a l Constant Constants of integration ^£ Normal density gradient and asymptotic terms X Half sum of p r i n c i p l e moments of i n e r t i a Bessel Functions Wavenumber ^ Normalised wavenumber IC Drag of dust on gas <*Ll Isothermal parameter i n eguation of state Polytropic parameter in eguation of state 138 I— Dust-gas mass loading factor X j -jean's c r i t i c a l wavelength \ ( X Constants of integration X Eigenvalue (Chap 5) M Newton's r a t i o of c e n t r i f u g a l accelaration to g r a v i t a t i o n a l p o t e n t i a l M^j Mass of dust grain IAA Mean molecular weight yJ Cosine of colatitude gS* Normalised isothermal density ^ Mass of central bulge M,-, Mass of outer f l a t disc A Polytropic index A ° Density normalised with respect to ce n t r a l density and asymptotic terms A/ Number of dust grains per unit volume Asymptotic terms i n expansion of \J Fundamental perturbation variable for gas \J' Fundamental perturbation variable for dust-gas J l Angular velocity Angular velocity at central axis Asymptotic terms of normalised angular ve l o c i t y to Frequency of perturbation Pomega, c y l i n d r i c a l radius vector ^* Pressure Legendre polynomials Gr a v i t a t i o n a l Potential <£> Normalised g r a v i t a t i o n a l p o t e n t i a l 139 F i r s t order expansion term with angular dependence <y Spherical r a d i a l coordinate Radius of disc ^ Gas constant ^ Density ^ Central density ^ Constant density on outer disc axis ^ g Composite density solution TjT Mean outer disc density ^ c Mean central bulge density »^  External density f: V C y l i n d r i c a l r a d i a l coordinate S Normalised c y l i n d r i c a l r a d i a l coordinate <5~~ Polytropic variable ff- Polytropic density variable and asymptotic terms "7"" Temperature """("" K inetic Energy C7 Time 6- Normalised density i n intermediate region Ratio of k i n e t i c energy to g r a v i t a t i o n a l potential energy u Thermal Energy U. Gas ve l o c i t y V/^  Volume of outer disc W Dust v e l o c i t y ^ Normalised c y l i n d r i c a l thickness coordinate Outer boundary and asymptotic terms 2; C y l i n d r i c a l thickness coordinate 2 ^ Disc thickness at central axis 140 D i s c t h i c k n e s s i n outer f l a t r e g i o n N o r m a l i s e d p o l y t r o p i c r a d i u s 141 S e l f - c o n s i s t e n t D i s c s Axisymmetric F l u i d Dynamics C o n t i n u i t y Eguations S e l f - g r a v i t a t i o n L a p l a c e Eguation Eguation Of State I s o t h e r m a l , P o l y t r o p i c D i f f e r e n t i a l P o t a t i o n Steady . De n s i t y S" State t r u c t u r e V i r i a l A n a l y s i s G l o b a l I n s t a b i l i t y L i n e a r P e r t u r b a t i o n s Gas & Dust-Gas S o l a r System Formation G a l a x i e s ( Future ) 142 Steady State Density Spherical Coordinates Uniform Rotation g/**' Normalise Isothermal Density V V » Sc. -Small Parameter S «I U n i f o r m A s y m p t o t i c E x p a n s i o n 6> - < f c . £ S p h e r i c a l Emden 0 •' C e n t r a l B u l g e A n g u l a r L e g e n d r e D i s c Terms : j j ^ S y n t h e t i c S o l u t i o n s F i n i t e E x t e r n a l D e n s i t y C e n t r a l B u l g e + D i s c 143 S t e a d y S t a t e D e n s i t y C y l i n d r i c a l C o o r d i n a t e s D i f f e r e n t i a l R o t a t i o n N o r m a l i s e T h r e e P a r a m e t e r s S p h e r i c a l S o l u t i o n O <L yg < 2 A x i s y m m e t r y R e s t r i c t i o n Discs £.<^l S m a l l P a r a m e t e r £. A s y m p t o t i c E x p a n s i o n s N o r m a l O u t e r Peg!on " F l a t D i s c A x i a l B o u n d a r y L a y e r C e n t r a l B u l g e C o m p o s i t e S o l u t i o n S p h e r i c a l C e n t r a l B u l g e 5 F l a t D i s c P e r t u r b a t i o n E q u a t i o n s L i n e a r I s o t h e r m a l N o n - d i s s i p a t i v e Axisyrametric Rings E i g e n v a l u e Problem B a s i c V a r i a b l e R e l a t e d to Kate of Change of Angular Momentum S t a b i l i t y A n a l y s i s B e s s e l "Function P a t t e r n Pin as Shear S e l f - c o n v e r a e D i s p e r s i o n d e l a t i o n s Pure Gas C f . Chan Irasehar M a r g i n a l S t a b i l i t y D i s c F e q i o n , Constant fe. C e n t r a l Bulge k. S"' Dust-Gas : Mass Loading L R a t i o M a r g i n a l S t a b i l i t y , r — Disc Region, c e n t r a l Bulge, 

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