STRUCTURE AND STABILITY OF SELF—GEAVITATING DISCS by JOHN BRUCE DAVIES n.s • r California B.Sc. r I n s t i t u t e o f Technology, 1968 U n i v e r s i t y o f B a l e s , 1963 A THESIS SU3MITTED IN PARTIAL FULFILBENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES Department of Geophysics and Astronomy We accept t h i s t h e s i s as conforming to the r e q u i r e d standard THE UNIVERSITY OF BRITISH COLUMBIA March, 1979 John B r u c e Davies, 1979 In p r e s e n t i n g 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 o f the r e q u i r e m e n t s f o r an advanced degree a t the U n i v e r s i t y of B r i t i s h C o l u m b i a , I agree t h a 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 r e f e r e n c e and s t u d y . I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s thesis f o r s c h o l a r l y purposes may be g r a n t e d by the Head of my Department o r by h i s r e p r e s e n t a t i v e s . I t i s understood t h a t c o p y i n g or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i t h o u t my written permission. Department The U n i v e r s i t y o f B r i t i s h Columbia 2075 Wesbrook P l a c e 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 d i s c was i n i t i a l l y proposed. T h i s research examines t h i s cosmogony by the c l a s s i c a l technique of i n i t i a l l y obtaining the density s t r u c t u r e of steady state d i s c s , and g r a v i t a t i o n a l i n s t a b i l i t y of such systems I s then investigated i n order to examine c o r r e l a t i o n s between observed phenomena i n the Solar System and p r e d i c t i o n s of the theory. A f l u i d mechanical approach to the steadystate axisymmetric structure i s formulated f o r isothermal and p o l y t r o p i c gas systems, with uniform r a d i a l l y dependent r o t a t i o n . or The equations are reduced to a s i n g l e quasi- l i n e a r 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 y i e l d an unique density s o l u t i o n . When the external density i s non-zero, f l a t t e n e d d i s c s are p o s s i b l e solutions of the b a s i c system. Two asymptotic techniques i n s p h e r i c a l and coordinates are created to obtain approximate solutions of the structure. Both techniques cylindrical steadystate show that a s e l f - c o n s i s t e n t d i s c i s composed of a high-density c e n t r a l bulge e n c i r c l e d by a low-density G r a v i t a t i o n a l i n s t a b i l i t y i n gaseous d i s c s i s now f l a t outer d i s c . formulated by the l i n e a r perturbation of the fundamental v a r i a b l e s , density, pressure, g r a v i t a t i o n a l p o t e n t i a l and v e l o c i t y . As the Solar System i s e s s e n t i a l l y a planar structure, axisymmetric r a d i a l i n s t a b i l i t y along the equatorial plane of r o t a t i o n i s examined. Such r i n g type modes are shown to be unstable to shear and tend to self-coverage. A dispersion r e l a t i o n i s obtained analytically which i n d i c a t e s that the wavelength between rings i s approximately i n v e r s e l y proportional to the square root of the steadystate density at marginal stability. However f o r the pure gas d i s c , the wavelengths are too long f o r any correspondence with the present spacing of the Planets. iii As usual, the presence of dust i s invoked close to the equatorial plane. Radial instability i n 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 i s multiplied by the dust-gas mass loading ratio. Thus the wavelengths at neutral stability w i l l be correspondingly shorter and a correlation of ring density maxima with Planetary positions i n the Solar System i s 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 i n 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 : C y l i n d r i c a l Coordinate Expansions Fundamental Equations Asymptotic Discs A3 Boundary Layer Theory 47 -N Asymptotic Expansions 49 Disc Region 53 Expansion Central Bulge Expansions 57 Matching Of Solutions 59 Composite Expansion 64 Solution Technique 66 Numerical Solutions 68 Chapter Four : Ring Pertubations In Gas Discs 70 G r a v i t a t i o n a l I n s t a b i l i t y Of Discs 70 Governing Equations 71 Case 0 - ) : M=0, 75 Axisymmetry Approach 78 Numerical Solutions 85 Analytical Results 1 Chapter Five : Ring Perturbations In Dust-Gas Discs 86 87 Dusty Discs 87 Governing Equations 87 Instabilities 88 Results 95 Dust-Gas Mass Loading 96 ...Implications and. Speculations 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 I I : True And Calculated Planetary Distances vii 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 i n Isothermal F l a t Disc Region, F i g 9 114 Figure 14: O s c i l l a t i o n s , Gas Composite, F i g 10, L=0 115 Figure 15: O s c i l l a t i o n s , Dust-Gas, F i g 9, L=50 116 Figure 16: O s c i l l a t i o n s , Dust-Gas, F i g 10, L=99 117 Figure 17: O s c i l l a t i o n s , Dust-Gas, F i g 11, L=49 118 Figure 18: Equatorial Density, Iso Composite, F i g 10 119 Figure 19: Equatorial Density, Iso Synthetic 120 Figure 20: O s c i l l a t i o n s , Dust-Gas, Synthetic 121 Figure 21: Graph of Calculated Versus Real Positions 122 viii Acknowledgements It is encouragement discussions a pleasure with their criticisms thesis owes much i s D r s . G. F a h l m a n , D r . H. the Ovenden R. H i u r a , guidance and regarding F. ¥ a n , L . the research t o t h e i n d u s t r y and m o r a l e o f my w i f e nonetary acknowledged. support o f t h e Governments and helpful flysack, a l s o t o t h e members o f t h e c o m m i t t e e and s u g g e s t i o n s dedicated, is gratefully acknowledge o f ay t h e s i s a d v i s e r and fi. P u d r i t z . T h a n k s it to for . This ta whom o f Canada 1 Chapter One Introduction That our s o l a r - s y s t e m i s unique conjecture. Whether other has solar been a systems exist, 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 , major s c i e n t i f i c importance. alter drastically The when much and the i s not o n l y of consciousness of such argued mankind questions are will answered unequivocally. Without hard o b s e r v a t i o n a l evidence it of other s o l a r systems, becomes even more necessary f o r the t h e o r i s t to examine p o s s i b l e f o r m a t i o n of such systems. Three c e n t u r i e s ago Laplace developed the hypothesis system by c o n t r a c t i o n of a d i s c . collapsed to yield of formation This disc shed galaxy. Kant of the solar rings which analyzed. our s o l a r system e x i s t s on the outer d i s c of a These disc galaxies populate the observed spiral universe together with s p h e r o i d a l g a l a x i e s . D i s c s thus play a v i t a l in man's vision of and p l a n e t s and a c e n t r a l sun. Other t h e o r i e s , numerous and d i v e r s e have been hypothesized and Now, the his universe. Throughout our cosmogony, knowledge of the s t r u c t u r e and s t a b i l i t y to the comprehension of our role of d i s c s is necessary environment. To be a p p l i c a b l e t o s o l a r system formation the consequences of i n s t a b i l i t i e s i n such d i s c s must c o r r e l a t e a t l e a s t observed p r o p e r t i e s of our one p i e c e system. Even then, assumptions of evidence, this the solar are beinq made t h a t a l l s o l a r systems are formed i n the same manner. Such not with is quite the case as other mechanisms have been proposed, probably and rarely 2 does Nature depend on only one method. Aims The steady aim their obtain relationship of to t h i s theory occur p a t t e r n was expresses the E a r t h ' s the Our solar system S o l a r System. The to initially the dust formation the analysed by Titius and (a. u) as 0.4*0.3*2 , A l l e n Planet Bode s Law 1 Bode emphasis the relevance System, whose law i n terms o f (1972). I (a.u) Planetary (a.u) I 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 - radial and Solar d i s t a n c e of the p l a n e t s from the Sun distance of e g u a t o r i a l plane of r o t a t i o n . T h i s Table Mercury with property of most i s the spacing of p l a n e t s i n close the to structure i n v e s t i g a t e the of such d i s c s composed of gas and properties which to s t a t e s e l f - g r a v i t a t i n g d i s c s , and instability on of t h i s r e s e a r c h i s Distance 3 It can be seen t h a t t h i s simple phenomological law down i n the outer regions where the spacing approximately constant. p l a n e t a r y spacing equatorial and and . spacing The becomes density this d e n s i t y waves along of a s e l f - g r a v i t a t i n g self-converge planets r e s e a r c h w i l l aim t o e x p l a i n i n terms of r i n g type plane fundamental Our of breaks d i s c which are maxima are the unstable located at a dependent on s t e a d y - s t a t e d e n s i t y s t r u c t u r e dust-gas mass l o a d i n g . A Bode type law i s found c l o s e t o the c e n t r a l r o t a t i o n a x i s where the d e n s i t y i n c r e a s e s r a p i d l y toward the c e n t r e . F u r t h e r approach a constant out constant spacing it is a that self-consistent thesis i s analysis of this an theoretical to acoustic yield along continuum i t s stability linear investigated. Finally, order will consequently thesis will describe a and approach to a s t r o p h y s i c a l d i s c s . structured s t a t e s t r u c t u r e and Then density of r i n g s i s obtained. t h i s c h a p t e r , the s c e n a r i o of the theory The the on the e q u a t o r i a l plane and With such goals i n mind, analyse shown w i l l be the described. classical lines models. I n i t i a l l y a properties w i l l perturbation In be analysis we s h a l l show the n e c e s s i t y of of steady examined. will be dust in a s i m i l a r d i s t a n c e p a t t e r n to t h a t observed i n our s o l a r system. In chapter this thesis two, will a review of p r e v i o u s r e s e a r c h be given. In chapter relevant to t h r e e , the e q u i l i b r i u m s t r u c t u r e of the d i s c i s i n v e s t i q a t e d . In chapter f o u r , using v i r i a l approach, v i r i a l s t a b i l i t y c o n d i t i o n s governing system f i v e i n v e s t i g a t e s the a c o u s t i c are described. modes of r a d i a l and Chapter azimuthal perturbations of the the a disc whole and 4 their local stability theory t o dust-gas instability will the observed A mixtures yield Chapter and show more that such the systems a d i s t a n c e p a t t e r n which corresponds at with detailed summary of the chapters chapter i n c o r p o r a t e s the theory of original be given. Three The third equilibrium structure of discs. discussed for a c y l i n d r i c a l l y central axis. conservation Fluid eguilibrium equations w i l l be used. Both s t e l l a r disc will fluid be assumed conservation governing eguilibrium will rotating of the be about continuity a and and gaseous systems have dynamics i s o t h e r m a l or p o l y t r o p i c f l u i d . The the The symmetric d i s c dynamical been t r e a t e d p r e v i o u s l y by to s i x w i l l extend s o l a r system • r e s e a r c h w i l l now Chapter criteria. with success. The to be composed of a s i n g l e equation of s t a t e is added equation and the eguation governing self- gravitation. As s e l f - g r a v i t a t i n g bodies i n the embedded in very low but non-zero e x t e r n a l pressure w i l l be present as system of equations i s reduced eguation governing asymptotic expansions density. will universe density boundary to a single used for to be media, a f i n i t e condition. partial Perturbation be appear This differential techniques solving the and basic equation. The p r e v i o u s approaches to s o l u t i o n of the basic eguation i n s p h e r i c a l c o o r d i n a t e s w i l l be analysed . In a l l these methods a zero d e n s i t y e x t e r n a l s u r f a c e i s assumed and consequently only slightly this deformed spheres were obtained. Bemoving 5 r e s t r i c t i v e boundary shaped solutions. c o n d i t i o n a l l o w s the c o n s t r u c t i o n Knowledge of the e x t e r n a l boundary a s s o c i a t e d c o n d i t i o n s can y i e l d an unique The fundamental thickness detailed is much knowledge geometry less of information can solution i n cylindrical approximate solution. of than the discs the are shape and is such that equatorial radius. outer boundary this their Without geometrical be used i n an asymptotic expansion of the d i s c coordinates On removing in order to obtain dimensions in the measures of the ratio an basic discovered. of g r a v i t a t i o n a l potential energy t o thermal energy (alpha) , the r a t i o of r o t a t i o n a l energy t o t h e r m a l energy (beta), and the square of the the disc solution. equation , t h r e e b a s i c d i m e n s i o n l e s s parameters are These of kinetic ratio of t h i c k n e s s to the maximum r a d i u s of the d i s c ( e p s i l o n ) . As i n all fluid dynamical problems i n c o r p o r a t i n g d i m e n s i o n l e s s numbers t h i s a l l o w s the s c a l e of the systems to vary widely. Thus, expected for to be discs, much the less dimensionless than parameter m u l t i p l i e s the r a d i a l basic (6) is u n i t y . I t i s found that this differential parameter operator in the d i f f e r e n t i a l e g u a t i o n . As e p s i l o n i s expected to be s m a l l f o r a d i s c , the s o l u t i o n w i l l be approximated as an asymptotic expansion i n powers of £. I t can be understood that the p r e v i o u s approaches t o d i s c s , wherein no r a d i a l e f f e c t s were allowed i . e a s e m i - i n f i n i t e are obviously slab, i n c o n s i s t e n t . In f a 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 is the fundamental physical cause of d i s c s . Nayfeh (1973) has commented:" i t i s the r u l e , r a t h e r than 6 the e x c e p t i o n , that asymptotic expansions parameter are i n terms of a not uniformly l a y e r s " . As Van Dyke (1975) multiplies of the higher d e r i v a t i v e s , i n a s t r a i g h t one p e r t u r b a t i o n scheme, approximation and that the small v a l i d and break down i n boundary remarks, when derivative is a small lost in parameter forward the first order of the equation i s reduced. Thus, one or more of the boundary c o n d i t i o n s must be abandoned and approximation breaks down near where they were to be the imposed, i . e the boundary l a y e r s . A boundary l a y e r i s mathematically central axis of the d i s c . I n t u i t i v e l y , i t can be expected c l o s e to the c e n t r a l a x i s , will be small approximately in found t o form around and the the rotational central bulge centrifuqal geometry the that forces will be s p h e r i c a l , with a s i m i l a r c e n t r a l condensation as the s p h e r i c a l case. T h i s i s confirmed mathematically f o r the r a d i a l extent of the c e n t r a l bulge w i l l be of the order central thickness. In the outer of r e g i o n where r o t a t i o n a l g r a v i t a t i o n a l f o r c e s are of the same order, a f l a t t e n i n g y i e l d i n g a disc-shaped The solutions c e n t r a l bulge, are emphasis Numerical on the and occurs system. in the combined obtaining the techniques are approaches yield two main r e g i o n s , normal d i s c into a composite solution used to solution on the e g u a t o r i a l obtain the and with plane. zeroth order solutions. Both i n c r e a s i n g r a p i d l y toward flat disc region. a central bulge , with d e n s i t y the c e n t r e e n c i r c l e d by a low density 7 Chapter Four In order to examine local stability, a linear d i s s i p a t i v e p e r t u r b a t i o n scheme i s s e t up i n the usual The non- fashion. axisymmetric i s o t h e r m a l p e r t u r b a t i o n s are governed by l i n e a r wave eguations that are reduced t o a s i n g l e eguation governing the b a s i c v a r i a b l e t h a t i s r e l a t e d t o r a t e of change of angular momentum. planets As the solar system seems to d i s t r i b u t e d c l o s e to the e q u a t o r i a l plane, radial and azimuthal calculations will e q u a t o r i a l plane The wave directions of an i s o t h e r m a l eguation be mode given in The on the disc. i s solved wavenumber only considered. density f o r the s t a b i l i t y i . e when the square o f the frequency resultant the pertubations will be performed f o r the have case of marqinal i s zero • The i s a f u n c t i o n of r a d i u s , due to the non-uniform d e n s i t y , and the non-dimensional parameters tY,^5, S. The simplest considered in case of detail in axisymmetric order to perturbations elucidate the is general p r o p e r t i e s of the system. C o n s i d e r a t i o n w i l l be paid t o the causes of i n s t a b i l i t y these modes Rayleigh is from examination of the dispersion relations. found t h a t any r o t a t i n g body w i l l be unstable a maximum i n i t s r o t a t i o n curve. Spiegel (1972) of if there has remarked on the ambiguity of any cause o f i n s t a b i l i t y - a s m a l l change i n basic conditions w i l l often radically alter the instability criteria. Thus, the wave patterns of modes w i l l be d i s c o v e r e d and analysed. In the axisymmetric case , a simple argument shows the inherent instability of these perturbations. 8 We f i n d that at marginal i n s t a b i l i t y in the axisymetric r i n g s a r e formed which, due to shear motions, unstable and s e l f - c o n v e r g e . I t i s instabilities in yield shown are probably that ring-type pure gas d i s c s w i l l have wavelengths too l o n g to y i e l d a s o l a r system d i s t a n c e have found then case, pattern . Other researchers 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 t o planetary systems they usually invoke the presence of dust. Chapter Five Thus we i n v e s t i g a t e d i s c s t r u c t u r e f o r a dust-gas In the similar no-slip case to previous the governing the equatorial basic case interaction mathematics but of with dust and an qas and p h y s i c s are added equation close plane. Allowinq weak waves i n t h i s dusty instabilities as i n the mixture. to disc as b e f o r e but with one major d i f f e r e n c e . We a l l dust-gas problems,that a hierarchy of the qives find, waves is produced, and t h a t the wave of l o n g e s t d u r a t i o n i s that i n which the dust and gas move together. We show t h a t t h i s wave has p r o p o r t i o n a l l y a much s h o r t e r wavelength than i n case. pure gas When the d e n s i t y r a t i o o f dust t o gas i s of order 50 , we obtain a pattern correlates of relative distances w e l l with the observed between expected while out in i n t e r v a l i s predicted f o r a instabilities. of Solar that disc region System between rings a more c o n s t a n t produced via these The d i s t r i b u t i o n of angular momentum between Sun and p l a n e t s observed predictions the rings p l a n e t a r y d i s t a n c e s . Close t o the c e n t r a l a x i s , a l i n e a r l y i n c r e a s i n g i n t e r v a l is the the i n the S o l a r System i s c o n s i s t e n t with theory. the Even though the number of p l a n e t s 9 outside can Pluto i s unpredictable, be their mean s e p a r a t i n g distances expected to be s i m i l a r to t h a t of the Outer P l a n e t s , 10 a. u. Conclusion Our research has developed main r e s u l t s , concerning Density created by two importance f o r uniform dynamics with different of the alpha,beta,epsilon asymptotic solutions composed shown a with outwardly d e c r e a s i n g flat disc region. Virial following no differentially rotating dissipation, together techniques of and dimensionless f i n i t e e x t e r n a l boundary c o n d i t i o n s Both the have p e r t u r b a t i o n methods. The three is elucidated axisymmetric s e l f - g r a v i t a t i n g d i s c s . solutions d i s c s , using f l u i d and with been dynamical parameters the n e c e s s i t y of f o r an unique s o l u t i o n . yield self-consistent disc high d e n s i t y c e n t r a l s p h e r i c a l bulge density e n c i r c l e d a n a l y s i s i n d i c a t e s the by instability a low density of such d i s c s at l a r g e c e n t r a l to e x t e r n a l d e n s i t y r a t i o s , such t h a t they tend increase t h e i r moment of inertia. L i n e a r a c o u s t i c a n a l y s i s shows t h a t the b a s i c related to t o the r a t e of change of angular variable is momentum. Axisymmetric r i n g s , formed at marginal s t a b i l i t y , tend t o shear and self- converge . Dust-gas d i s c s are developed s e l f - c o n s i s t e n t l y . At stability, weak than the pure gas loading the marginal waves have wavelengths p r o p o r t i o n a t e l y case. For a reasonable dust to shorter gas mass r a t i o the r e l a t i v e d i s t a n c e between r i n g s i s s i m i l a r to observed pattern of our solar system planets. Trans- 10 Plutonian planets, if they semi-major axes separated exist, are expected by d i s t a n c e s s i m i l a r t o t o have mean the spacings of the Outer P l a n e t s . Flow technique list of charts of the b a s i c r e s e a r c h system and are given i n Appendix symbols. F together with an analytical appropriate 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 Review Of In Rotating Systems this chapter self-gravitating, inspired by galaxies and, wherein the The on w i l l o u t l i n e the systems. shape and figures two main f o r c e s , are made. Most of the additional impossible. The study homoqenous uniformly the t of and rotation, act together. constraints structure on the f i g u r e of the figure s l i g h t l y o b l a t e ; and, v w i l l demand a simple must be further, masses earth that the the i t s cause, as measured concentrated mainly physically equilibrium began a can with of Newton's Book III, the e f f e c t of a s m a l l d i r e c t i o n of making i t equilibrium between of the the body effect ellipticity. eguatorial and the which of these bodies (Principia, i n the proportionality r o t a t i o n , as measured by appropriate from gravitational rotating and systems, even though they are the was disc P r o p o s i t i o n s XVIII-XX). Newton showed t h a t rotation work on work, c l a s s i c a l r e s e a r c h has of constant d e n s i t y , on this of E a r t h , P l a n e t s , S t a r s gravity approach t o the d e n s i t y investigation of Conservation Laws together with an State bodies previous research Much f o r s p e c i a l relevance to our fundamental continuum we rotating the Equation of be Two r a d i u s - polar radius mean r a d i u s by centrifugal acceleration at the mean g r a v i t a t i o n a l a c c e l e r a t i o n on equator the surface of 12 where G\ denotes the constant of g r a v i t a t i o n and M i s the mass of the body. More p r e c i s e l y , Newton e s t a b l i s h e d the in case the observation body may is the usually continuum of the first numbers which play such an important part mechanics s p l i t i n t o two structures An i n t e r e s t i n g h i s t o r i c a l be made t h a t t h i s appears to be uses of dimensionless in homogeneous. are relation of such main branches; constructed and one bodies. the This subject i s steadystate density then examined f o r s e c u l a r and dynamical i n s t a b i l i t i e s . Initially the homogeneous, developed by steady s t a t e d e n s i t y s t r u c t u r e and isotropic MacLaurin rotating (1742) who axisymmetric shape bodies considered r< small, c f . hf e where C i s Chandrasekhar (1969). by Thomas Simpson certain maximum value This (1743): for there result is any 2 are two and spheroid departs have two solutions: spheroid ' noticed only two possible cannot deduce velocity that the only s l i g h t l y from a sphere; f o r as J l * 0 , - a solution of s m a l l e c c e n t r i c i t y and which, by v e l o c i t y l e s s than a noteworthy i n t h a t we from the f a c t of a s m a l l e q u a t o r i a l angular be y r e l a t i o n was angular to MacLaurin's formula t " - J - Z A /R}~ A remarkable f e a t u r e of Maclaurin's "oblata." cannot e> 5 e c c e n t r i c i t y defined was g e n e r a l i z e d Newton's r e s u l t the case when the e l l i p t i c i t y caused by the r o t a t i o n of indeed, leads a second s o l u t i o n which to we a leads to a h i g h l y f l a t t e n e d spheroid. Such a result thus i n d i c a t e s t h a t h i g h l y f l a t t e n e d d i s c s 13 are possible solutions f o r the homogeneous r e a l i s t i c equations of s t a t e are considered, linearity can possibly be even numerical expected others analysis stablest shapes homogeneous such by of to yield as toroidal Harcus et rotating f l u i d s using real bodies Lebovitz important A are necessary of and z not solutions. Recent self-gravitating necessarily non-linear, who stable. As most of the c l a s s i c a l though stresses sufficient i s that with s u r f a c e s function of p only. still the important, following Further three p- p relationship , to of c o n s t a n t be pressure that p be a i n t h i s case the t o t a l p o t e n t i a l i s may be thought of as determining the relationship. one surfaces f o r Jl_ When i t e x i s t s , the e q u a t i o n of s t a t e energy c o n s r v a t i o n the the condition of constant d e n s i t y , i . e . , of p only. also a function form of two f a m i l i e s and a l (1977), has examined the character, (1967,1978) coincide of these points. independent and when a minimum energy c r i t e r i o n and found a r e s u l t s are of a negative cf. and the subsequent non- axisymmetric v a r i e t y of shapes p o s s i b l e though physically case avoids Hence, by the complications describing a p-yc of these f u r t h e r equations. Poincare s estimate.— , The s i z e of a r o t a t i n g i s l i m i t e d by the c o n d i t i o n that the centrifugal configuration accelaration not exceed the c e n t r i p e t a l a c c e l e r a t i o n of g r a v i t y . T h i s l e a d s , in the Poincare density: case (1903) of uniform r o t a t i o n , t o the f o l l o w i n g estimate by relating the angular velocity to the mean 14 Lichtenstein s Theorem— 1 configuration is constrained perpendicular to the nonrotating Jeans techniques have a of a rotating plane of symmetry (1933) to solve the more d i f f i c u l t and employed expansion problem of the shape p o l y t r o p i c systems, the and polytropic being a measure of c e n t r a l condensation. These ideas be analysed i n depth l a t e r along o f expansion and Another polytropes i s that (1964, 1965). density perturbation approach under spheroids. them to vary more recent has (1963) the to of rapidly and Hurley assumption rotating 6 fioberts that the the level case the e c c e n t r i c i t i e s of be the same, the other the case were determined by the v a r i a t i o n a l density stratification. approach of numerical analysis and been u t i l i s e d by James, O s t r i k e r , Bodenheimer collaborators in a series polytropes problem In one and p r i n c i p l e along with the synthesis developement 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 are The further will theories. the s p h e r o i d s were assumed a l l allowed with the fioberts stratification surfaces to of and r e s u l t i s found small for a i s necessarily spherical. Chandrasekhar structure of isothermal index to shape a x i s of r o t a t i o n . A c o r o l l o r y i s t h a t configuration (1929) and The of papers deviations various mainly from angular and concentrating sphericity. momenta and A on general polytropic equations of s t a t e such t h a t above a c e r t a i n r a t i o of axes and kinetic energy to i n s t a b i l i t i e s set i n . The potential generalized energy ( 7J, ) that various p o l y t r o p i c sequences bear c l o s e resemblance to the M a c l a u r i n sequence im most r e s p e c t s . In particular, do the g e n e r a l i s e d sequences do not t e r m i n a t e , they 15 not r o t a t i o n a l l y e j e c t matter, but they do b i f u r c a t i o n beyond which s e c u l a r s t a b i l i t y Most research has thus on departure from reach probably points ceases. concentrated on the shape of s t a r s s p h e r i c i t y due to r o t a t i o n of usually and with only s l i g h t a x i a l changes. However the geometry of d i s c s i s such that the t h i c k n e s s i s very much s m a l l e r than diameter. years some effort has gone i n t o the problem of determining a mass d i s t r i b u t i o n model rotational curves In r e c e n t which can account for the observed of g a l a x i e s . T h i s has g e n e r a l l y been c a r r i e d out by the method of s p h e r o i d a l s h e l l s wherein a galactic disk is r e p r e s e n t e d by a s u p e r p o s i t i o n of c o n c e n t r i c s p h e r o i d s h e l l s of varying density technique has and the zero advantage or finite that vanishes i n s i d e a s p h e r o i d a l s h e l l , simple form o u t s i d e the Hunter (1963) density and expansion has in d i s k s with r e l a t i v e l y with a strong rotation axis, polynomial it variations Legendre and gravitational it has convenient to a In relatively in rotating disks by an polynomials. T h i s method works well f o r f l a t d e n s i t y d i s t r i b u t i o n s , but f o r d i s k s degree very of concentration of mass toward the high order terms dominate i n the Legendre expansions. In the f o l l o w i n g r e s e a r c h , we w i l l employ convergence i s diminished. order to simplifications are problem. field represent the a s i m i l a r yet more powerful technique i n which t h i s problem This shell. found velocity the eccentricity. The cold obtain made to exact the pressureless disc solutions, non-linear disk various free-boundary has y i e l d e d many u s e f u l i n s i g h t s while the pressure d i s k has been s o l v e d a n a l y t i c a l l y i n 16 the s i m p l e s t s e m i - i n f i n i t e s l a b model. Instabilities in both r a d i a l and t h i c k n e s s d i r e c t i o n s have teen examined i n c l u d i n g the influence of v a r i o u s s t r a t i f i c a t i o n s and r o t a t i o n s , c f . Hunter (1972). However, we have been able t o o b t a i n to the non-linear steadystate disc asymptotic problem c o n d i t i o n s under which these approximations that have various other shapes solutions for rotating with been solutions which are proposed show the useful. as possible as toroids and without c e n t r a l b o d i e s . Some of these observed figures and shapes s e l f - g r a v i t a t i n g bodies such Note may a l s o be the r e s u l t instability of the two main which are p o s s i b l e i n such systems types of s e c u l a r and dynamic. Secular i n s t a b i l i t y alter the energy of i s the a result of self-gravitating mass as heat (1978) shows that The energy the i s the particular dissipation mechanism Jacobian sequence i n a p a r t i c u l a r enerqy. Chandrasekhar ratio determine a c r i t i c a l be expected. Above of secular MacLaurin v e l o c i t y and e l l i p t i c i t y ranqe but with (1969) approach t o the s e c u l a r i n s t a b i l i t y dimensionless will sequence of homogeneous t r i a x i a l e l l i p s o i d s t h a t can be obtained from the axisymmetric lower is ( or g r a v i t a t i o n a l ) r a d i a t i o n and Hunter a f f e c t the onset of t h i s i n s t a b i l i t y . An example o f t h i s instability that r o t a t i n g body while keeping the mass and angular momentum c o n s t a n t . dissipated motions rotational has pioneered the v i r i a l a n a l y s i s , and has the to g r a v i t a t i o n a l energy t o value above which s e c u l a r a larger used instability may value of t h i s r a t i o , dependent on p o l y t r o p i c index, dynamical i n s t a b i l i t y s e t s i n and some n a t u r a l 17 v i b r a t i o n a l mode of the system can grow i n time. A form of gravitating elementary system space to a time Jeans of it uniform the dependence Z is satisfied medium. The - occur arises tends to suffficiently Ledoux's stratified central justify ignored comes considering from gravitating Sinusoidal most small medium of disturbances exp.L ^ ( O t - f e . x ^ C.I - Ar<T are with possible <q f> , fe% /tel*" v e l o c i t y of sound a p p r o p r i a t e and to to make Jeans them unstable effect out of density self-gravity that length, V r B £U> (^J- perturbations The small pressure but the tends t o predominate at l a r g e length s c a l e s . (1951) a n a l y s i s of the i s o t h e r m a l i n plane p a r a l l e l shows that there , which occurs i n the more thin-disk gas l a y e r s symmetric with is no t h a t are purely i n the d i r e c t i o n of the the i f their because s e l f - g r a v i t y tends to cause a smooth plane disturbances This r e s u l t self- The i n d e n s i t y t o grow. T h i s growth i s r e s i s t e d by destabilizing the a e f f e c t of g r a v i t y i s to modify the sound waves instability is to (1929). infinite wavelength exceeds the c r i t i c a l which peculiar instability as kt. where Q , i s the otherwise increase is dispersion relation UO would that volume d e n s i t y at r e s t . and provided is discusssion disturbances constant instability general sheet that respect instability to stratification. analysis helps l i m i t i n showing t h a t no i n s t a b i l i t y when the d e t a i l s of the to is s t r u c t u r e i n the z d i r e c t i o n are ignored. Rotation can add to the s t a b i l i t y . Chandrasekar shown t h a t the a d d i t i o n of a uniform (1961) has r o t a t i o n to Jeans' o r i g i n a l 18 analysis affects stability d i r e c t i o n o f r o t a t i o n . These i n t e r e s t and only for waves waves are the normal ones of to the primary possess the d i s p e r s i o n r e l a t i o n LO Goldreich Lynden-Bell stratified interest has also investigated the s i m p l e s t d i s k s with (1964). G r a v i t a t i o n a l s t a b i l i t y of a semi-infinite s e l f - g r a v i t a t i n g r o t a t i n g sheet was i n v e s t i g a t e d only in waves in the thickness direction. r e s u l t s to Ledoux(1951) were presented only for with Similar non-realistic discs. Other Spiegel types and Zahn differential conditions of instabilities (1970) rotation. in The have been reviewing amazing most widely homogeneous i n v i s c i d discussed of methods can and induce article. instability f l u i d medium with the s p e c i f i e d f u n c t i o n of of changes d i f f e r e n t i n s t a b i l i t i e s i s w e l l shown i n t h i s The by instabilities number under which small environmental considered is angular that of a velocity a , the d i s t a n c e from the a x i s of r o t a t i o n . B a y l e i g h showed t h a t i f the d i f f e r e n t i a l medium i s u n s t a b l e . Remarkably enough, the c o n d i t i o n f o r shear i n s t a b i l i t y plane parallel inviscid showed t h a t i n s t a b i l i t y inflection flow was a l s o given by B a y l e i g h occurs when the v e l o c i t y p r o f i l e p o i n t . The analogue of t h i s i n s t a b i l i t y the case of s w i r l i n g flows with d o o who has an criterion for t h a t when ^ of a 19 at some CD , instability c e r t a i n non-axisymetric occurs. When disturbances c o n s i d e r a t i o n s are modified are this condition i s unstable. Now normally can a r i s e . But, f o r most a s t r o p h y s i c a l s i t u a t i o n s , the plays a s t a b i l i z i n g r o l e although p l a y s only a minor r o l e , S p i e g e l and Zahn the these when other e f f e c t s are considered viscosity The met exceptions viscosity (1970). c h i e f problem t h a t must be c o n f r o n t e d i s the e f f e c t known instabilities equations and stability question i t s e l f . reasonable i s of a higher order neutral difficulty i s that the keep the instability, unstable models where one of than the In the s e l e c t i o n of c o r r e c t models, a z e r o t h order approximation or of which n e c e s s a r i l y i n v o l v e s n o n - l i n e a r tend to choke themselves o f f and critical as instabilities situation just as in at nearly convectively often equates the temperature q r a d i e n t to i t s value f o r n e u t r a l i n s t a b i l i t y . Much r e s e a r c h has i n d i c a t e d the r e l e v a n c e of r o t a t i n g to f o r m a t i o n of s o l a r systems. A number of conferences years has c o n c e n t r a t e d 1972 on models a r b i t r a r y pressure and order to make of (1972) reviewed the primitive temperature the system h i s and Pine's solar nebula distributions tractable. are However mathematical treatment i s made of the e q u i l i b r i u m this or i t s i n s t a b i l i t i e s . General complex system r e s u l t s r e g a r d i n g formation of a s o l a r system r o t a t i n g body are A i n recent on the cosmogony of s o l a r systems. In Nice symposium Cameron numerical gases the work wherein used no exact structure from in of qualitative a dust-gas claimed. more mathematical treatment of a dust-gas r e l e v a n c e to s o l a r system formation i s given by system and i t s Spiegel (1972) 20 at the same symposium. workers i n dust-gas the gas His results systems who show t h a t , i n sufficient and dust w i l l tend t o move t o g e t h e r due In more recent work, Goldreich i n v e s t i g a t e d the c o a g u l a t i o n and toward f o l l o w t h a t of standard formation Hard of forces. (1974) dust have particles e q u a t o r i a l plane of t h e i r sheet d i s c s (i.e i n f i n i t e i n r a d i u s ) . I h e i r main r e s u l t , l i k e t h a t of other researchers, is the and to drag time, that the dust w i l l tend to form a l a y e r or c o n c e n t r a t e d zone c l o s e to the e q u a t o r i a l plane. T h i s work and its the author's r e s u l t s w i l l be examined i n d e t a i l The put forward nebula that c o n s e r v a t i o n and supposed to much v a l i d theory later. by Kant and Laplace who contracted, shed up due t o angular momentum r i n g s on the e q u a t o r i a l plane that a fast expected on account of the expected r o t a t i n q c e n t r a l Sun to result which i s not be d i s s i p a t e d . T h i s brought i n t o d i s r e p u t e and a number of However the other of were advantages of a d i s c o r i g i n are such that they observed. and almost plane. the advanced. as t o s a t i s f y plane of to this most r o t a t e These with distribution of the facts vectors must s a t i s f i e d by any theory of S o l a r System f o r m a t i o n together the and a l l r e v o l v e 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, perpendicular this the d i s c approach theories the o b s e r v a t i o n of the planets a l l c l o s e to ecliptic, were self-converge i n t o planets. This scenario received criticism of sped envisaqed a c o o l i n q A l s o r i n g s formed by shedding are not capable of converging are to o r i q i n a l hypothesis t h a t the S o l a r system formed from a d i s c was disc relevance angular momentum between Sun be with and p l a n e t s , the d i s t a n c e s of the p l a n e t s , and t h e i r r e s p e c t i v e masses. 21 Chapter Three introduction The o b j e c t o f t h i s s e c t i o n i s t o analyse p r e v i o u s on self-gravitating rotating research bodies , examine t h e i r domain o f v a l i d i t y and c r e a t e new s o l u t i o n s t h a t w i l l d e s c r i b e d i s c s . partial differential guasilinear boundary eguation governing s t e a d y s t a t e d e n s i t y i s elliptical conditions thus necessitating f o r an approaches, both i n v o l v i n g parameter, The known unique s o l u t i o n . Two asymptotic expansions w i l l be used t o o b t a i n approximate external mathematical in a small solutions. Steady, S t a t e Theory Even with the e x t e n s i v e work reviewed above no a n a l y t i c a l s e l f c o n s i s t e n t theory of the steady s t a t e has been created. F i g u r e s of r o t a t i n g have seemingly d i v i d e d i n t o two main deformed axis sphere with fundamental for (1743) self-gravitating categories; the discs bodies slightly almost e g u a l axes, and the d i s c with one disparate systems are the s i m p l e s t case o f initially noticed incompressible that in two ' p o s s i b l e ' o b l a t a , governed r e l a t i o n s h i p i s shown i n F i g u r e functional relationship 1 by the systems, same through use of the v i r i a l Simpson McLaurin Spheroids f o r any maximum there a r e two and by the same e g u a t i o n . The of Chandrasekhar of the e c c e n t r i c i t y minor t o major axes ) t o the angular determined governed two mathematical e q u a t i o n . angular v e l o c i t y l e s s than a c e r t a i n the of (the t h i c k n e s s ) s m a l l compared to t h e other two. These apparently only structure velocity approach. (1969); ( or r a t i o o f parameter being 22 Because these a t t e n t i o n has incompressible been directed equations of state. explicitly, the c l a s s i c a l pressure i s a f u n c t i o n allowing differential function of c y l i n d r i c a l In systems to bodies order to equation of d e n s i t y rotation radius a r e not r e a l i s t i c , with more remove of the state realistic temperature i s used where only. T h i s i s a l s o u s e f u l when with the anqular velocity a only. When pressure i s taken to be p r o p o r t i o n a l t o some power density, power such e q u a t i o n s of s t a t e are termed p o l y t r o p i c . When the i s unity systems t h a t has infinity, system a The that finite i t s density size of f i n i t e systems and these are the isothermal body only spherical body becomes zero a t has a non-zero However, Nature appears t o abhor a gravitating vacuum, external and self- are taken t o be embedded i n an environment ncn-zero pressure and temperature. T h i s i m p l i e s t h a t a self-gravitating density i s isothermal w i l l be d i s c u s s e d . while density. body (pressure) boundary Explicitly, the the the important property finite of will have non-zero external condition. the problem t o be s o l v e d equilibrium density a i s t o o b t a i n models o f structure of s e l f - g r a v i t a t i n g , r o t a t i n g d i s c s obeying the above e q u a t i o n s of s t a t e f o r f l u i d systems. I n all but a p a r t i c u l a r case, t h e equation i s non-linear known. Two asymptotic approximate solutions and resultant partial differential no exact a n a l y t i c a l s o l u t i o n s are expansion methods a r e used f o r the n o n - l i n e a r to obtain cases and the s t e p s taken i n each method are o u t l i n e d below. These w i l l be a m p l i f i e d and justified i n the a p p r o p r i a t e A : Spherical Expansion. section. 23 1. Derive the spherical governing partial c o o r d i n a t e s f o r uniform r o t a t i o n e q u a t i o n , s e l f - g r a v i t a t i o n eguation of differential eguation in using the c o n t i n u i t y and the a p p r o p r i a t e e q u a t i o n state. 2. Normalise v a r i a b l e s with r e s p e c t t o c h a r a c t e r i s t i c v a l u e s . 3. the s m a l l parameter S = fiecognise term in t the differential v e l o c i t y and /2s^ equation, as Jl_ i s the anqular where i s the c e n t r a l d e n s i t y . 4. When pressure i s p r o p o r t i o n a l to the sguare linear partial exact •o^. ^2. o f the d e n s i t y , a d i f f e r e n t i a l eguation i s d e r i v e d which y i e l d s solution oo inhomogeneous m of ) . The the form ( spherical an solution i ^ ^ a r e m u l t i p o l e terms composed of B e s s e l ry\-o f u n c t i o n s f o r r a d i a l dependence and angular arbitrary dependence. are Legendre polynomials coefficients for to be determined. 5. For the g e n e r a l n o n - l i n e a r case, an approximate obtained by a r e g u l a r asymptotic expansion powers of the s m a l l parameter £ , the f i r s t c are the same form is of the s o l u t i o n i n two terms of which as the exact s o l u t i o n i n t h e l i n e a r case, ( oo spherical solution solution + cY. • ^, ^ ^ 7 ^ 2 . ^ )• 6. These a r e e l l i p t i c a l boundary value problems and thus a known e x t e r n a l d e n s i t y and boundary shape w i l l value of each of the r e v e r s e d such t h a t i f we coefficients choose d e n s i t y and the c o e f f i c i e n t s boundary w i l l be uniguely 7. to As these O C ^ c ) °f t n asymptotic uniquely determine the . values the T h i s l o g i c can be of the e x t e r n a l , then the shape of the e x t e r n a l specified. expansions are of the d e n s i t y only e c e n t r a l d e n s i t y , the e x t e r n a l d e n s i t y must a l s o 24 be of t h i s O^S ^) order for c a consistent solution and is s u i t a b l y chosen of t h i s order. 8. In order their to o b t a i n examples of d i s c models and internal density structure and comprehend boundary computations and d e n s i t y contour techniques. To obtain a d i s c shaped body, the programmer i n p u t s obtained by c trial and error From an edge-on view, the most important disc the disc-shaped term d i p o l e component with c o e f f i c i e n t A^. systems can be generated values f o r A^ . A^ t u n t i l the d e s i r e d shape i s observed. is by computer values and e x t e r n a l d e n s i t y of 0(S P ^that particular coefficient are p l o t s are generated shape, by a l l o w i n g for a A subset of only non-zero 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 h i g h l y f l a t t e n e d disc systems which would require l a r g e r number of c o e f f i c i e n t s and is feasible and for their technique must be synthesis, that increases close rapidly gradients, thus to the inward suggesting B. C y l i n d r i c a l governing computationally so a d i f f e r e n t the structure of asymptotic these simple models c e n t r a l r o t a t i o n a l a x i s the d e n s i t y with correspondinq applicability approach as i n the f o l l o w i n g asymptotic 1. The f i r s t not used. 9. A n a l y s i s of the d e n s i t y shows j u d i c i o u s c h o i c e of a much of large density a boundary l a y e r technique of s o l u t i o n . Expansion. step i n t h i s s o l u t i o n technique i s partial differential to derive the equation in cylindrical respect to appropriate coordinates. 2. Normalise characteristic the variables values with different than those f o r the s p h e r i c a l 25 expansion method. 3. fiecognxse multiplies the the partial highly 4. that a small parameter terms governing the differential corresponding to a f l a t i s c a l l e d the 5. In r e g i o n s where the centre and the £.-»• O s l a b i s the and in this central bulge, i s obtained and a approximation boundary l a y e r , matched with the asymptotic s o l u t i o n This approach does not i s small, similar first small, large, the ignored in termed solution i n will the the region. the and total r e l a t e d to are boundary l a y e r s i n d e n s i t y d i s c i s obtained f o r these i n n e r two a in d i s c becomes g r a d i e n t s are A composite s o l u t i o n composed of s p h e r i c a l form as gradient the gradients r a d i a l density eguation solution normal d i s c flat radius)^ normal d i s c r e g i o n . differential central for As density outer r a d i a l edge, these terms cannot be e x i s t . The it radial eguation. 2. (disc t h i c k n e s s / flattened. and 7. } In r e g i o n s where the r a d i a l d e n s i t y solution 6. £ assume an r e g i o n s and spherical disc solution. as of the that of The are i s of same £, only not Chandrasekhar of the that available p a r t i c u l a r value of d i s c through use and method. exact value of thus s u f f i c i e n t c o n d i t i o n s invariants manner i n the c e n t r a l bulge 6 can be virial in a (1969) for the homogeneous bodies. We shall initially examine expansion as c e r t a i n r e s u l t s and w i l l be method. used i n the the spherical technigues from these subsequent c y l i n d r i c a l c o o r d i n a t e coordinate solutions expansion v 26 Part 1, i S p h e r i c a l Coordinate Let us begin by Expansion examining the fundamental equations governing these p o l y t r o p i c systems with r i q i d rotation. 1. H y d r o s t a t i c Equation JL • uZ> " = ^ C0 z where ^ i s pressure . r 2. Equation of s t a t e — P (_2-2.) where f\ i s p o l y t r o p i c index . 3. S e l f - q r a v i t a t i n g equation where 5- i s g r a v i t a t i o n a l - A--7T ^^ c potential n^is constant SI i s constant angular In s p h e r i c a l denotes the polar radial velocity coordinates coordinate with and axial the symmetry, cj^ cosine of the colatitude. The u s u a l approach i s t o n o r m a l i s e the d e n s i t y with r e s p e c t to the c e n t r a l d e n s i t y and S r and d e f i n e the polytropic variables by and Note how the similarily found Maclaurin Spheroid. Combining 55 fundamental to play a dimensionless vital role in parameter the & is c homogenous (2.1) and (2.2) and i n t e g r a t i n g g i v e s 4> + cf'{(» • ( I ' f | s -f coasted 27 where ^ (jS)\s the second Legendre polynomial . From (2.5) and (2.3) , the fundamental eguation i s d e r i v e d V <r Z = .S - er c n with boundary c o n d i t i o n | =o ; cr= 1_ ( V c r = o When 8 i s zero, the s p h e r i c a l Emden equation results c the solutions The Emden of which have been w e l l examined and t a b u l a t e d . sphere represents the i s defined zero by ^ = density where 19(^)^0, , surface of the and non-rotating configuration. It is assumed fundamental expected by eguation a l l authors (^2.7) f o r astrophysical that in analysis S> ,«'\ . c objects. An Such asymptotic y i e l d s the d e n s i t y and i t s f u n c t i o n 0~as an asymptotic cr where &Cf) S is sufficiently c Previous research has to the potential satisfies this a result i s expansion series J i L l ^ ) •+ ••• small, reqion e x t e r i o r <j-> + of O with ^ fixed. restrictively polytrope, Laplace's the assumed t h a t i n the scaled eguation gravitational f o r a zero d e n s i t y medium. The s o l u t i o n of which, with standard c o n d i t i o n s at i n f i n i t y , i s Oo where X and ^ 2 V equatorial needed. a r e r n plane constants only even . Due to harmonics symmetry in the about the l a s t term are 28 T h i s equation (2.11) combined with (2.5) mathematical d e n s i t y f o r the whole of the sphere confiqurat ion, X \y ^ i s a c o n s t a n t . T h i s equation for the e x t e r i o r t o the z the outer boundary of the p o l y t r o p e . valid yield at zero d e n s i t y , T where to any polytropic (whenc7"=o) thus This equation equation of specifies (2.12) s t a t e as the i n t e r n a l — s t r u c t u r e only a f f e c t s the value of the is A/ coefficients and Cm Thus the CJ — given by The zero d e n s i t y s u r f a c e of a r o t a t i n q O polytrope i s • method thus used by Monaqhan and Eoxburqh (1965) solve t h i s problem i s t o use, i n the i n n e r r e g i o n , a f i r s t expansion technique order approximation outer layers, and by i n i t i a l l y Monaqhan developed this <S~- (^^j d e n s i t y ^ of a technigue problem i n n e r and interface. to outer Thus neglecting to qoverninq first mass of the outer s u r f a c e order i n of matched asymptotic solutions the 9 * ' 9* Smith expansions order approximations. is performed the at an ; and (1974) has to Matching solve o f the intermediate i n n e r r e g i o n , the expansion s u b s t i t u t e d i n t o the b a s i c eguation equated: the (1967) argues f o r the v a l i d i t y of t h i s higher for order i n the outer r e g i o n t o d e r i v e a f i r s t approach which y i e l d s equations the to (2.9) i s (2.7) and c o e f f i c i e n t s of & 7* 5 = / - c n.®"''Z 29 To obtain series of the solution Legendre 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 polynomials rv-v- I Equating coefficients of m gives y^X with the central , ^ ^ Q . x boundary As $)($)is icnown, numerical techniques. It is functions the these to are x conditions equations important w i t h m>0 _ are realise readily in this expansion d e f i n e d by homogeneous coefficients must be integrable determined that eguations from the by a l l S^y and thus boundary conditions. Analysis However expansion two main (2.9) and i t s Monaghan (1967) arguments. The f i r s t unperturbed the conjectured Monaghan that that comparison the higher or examined criticism is this the indicates with the it been is that that and sum o f l e v e l l e d at the we $will region, the first, than a breakdown the first the unimportant i t by c o m p a r i s o n of first is the in two. body. at his the where and i t has merely negligible To d e t e r m i n e to order been expansion. theory 0<X)be necessary first a zero the order than whole this summarise have outer sum o f a l l t e r m s h i g h e r terms are show these In t h i s larger argues (1967) in them has term have a p p l i c a t i o n throughout Emden s u r f a c e . second reguires criticisms that calculate theory with 30 s o l u t i o n s obtained by other methods. T h i s Monaghan results and Roxburgh have done (T i n a ( s e r i e s i n &. expansion then the of 0 + which y i e l d s the above i s only v a l i d when (1964). of equation . neglects a © . I + n.S 5/5 + negligible A-l - t ensures that error in t h a t the term on the r i g h t hand s i d e of . J i n the outer r e g i o n and approximation as Monaghan and Roxburgh these terms, o b t a i n s the same massless on i s not c o r r e c t as the T =~ introduces s m a l l and unimportant the (2.i3>) . Monaghan (1967) argues r e t e n t i o n of t h i s expansion Monaghan comparing t h e i r as occurs i n the outer power s e r i e s expansion /©+•••• and I f (^o^.jEthen we may expand c equating powers of c^. However w h e n c ^ S ^ J , region, (1967) o b j e c t i o n i s due t o the expansion power . vj/ .+ .... A©. J i s by with those obtained n u m e r i c a l l y by James A more important term (1965) Monaghan in i n the outer r e g i o n . Monaghan and Roxburgh (2.12) f o r o"~ as the (1965) first have also n.CJ'. ^ obtains a in the appropriate f o r the outer differential equations s i m i l a r f u n c t i o n a l dependence. By matching the chosen interface can be taken as the Emden s u r f a c e c f . Monaghan (1967) ) the c o n s t a n t s ^#X^^ can be determined. v a r i o u s values of H and l i s t e d An formulation (1967) by n e g l e c t i n g the i n n e r s o l u t i o n s with t h i s outer s o l u t i o n a t a (which the order approximation r e g i o n i . e the s u r f a c e l a y e r s . Monaghan small terms used important general matching shows t h a t These a r e d e r i v e d f o r i n Monaghan and Roxburgh (1967). r e s u l t i s t h a t by expanding Ao =- CU Crr\ *w\ =. — /r»o- o ^.^^the »v\ ± o I I ' I 31 Thus a s o l u t i o n of the b a s i c equation f o r the i n n e r r e g i o n and for f o r the ^ • now region, density T n e is j < j» massless - (2.7) &~ of the polytrope can thus be obtained. These results are such an approach s i x t y claimed that only for polytropes on not novel years . Jeans ago and (1929) from had this systems with s m a l l e l l i p t i c i t i e s with index greater that- result are not which was had were p o s s i b l e than n=0.8 . However the which t h i s argument i s based , and , r e l i e s on assumptions developed described necessarily model above valid for discs. The main assumption n e g l i g i b l e mass can surface layers be with is that a Roche envelope of zero patched a small expected to s a t i s f y such a onto the deviation constraint. central core. will be shown subsequent a n a l y s i s t h a t a s e l f c o n s i s t e n t d i s c w i l l not such a s t r i n g e n t c o n s t r a i n t , as the c e n t r a l bulge w i l l be of the same Only from a sphere can It masses i n the order. be in satisfy f l a t disc The or other and major assumption i s t h a t the e x t e r n a l boundary of the s e l f - g r a v i t a t i n g body is at zero density. Only when the r a t i o of e x t e r n a l c e n t r a l d e n s i t y i s below a c r i t i c a l value is this to assumption valid. This is equation i n the confirmed by the region such t h a t , f o r n>2, simple a n a l y s i s of examining where <?~ =0(£ \ S u b s t i t u t i o n g i v e s t the second term on the r i g h t hand the cn= £ .C side of C 32 (2./5) the i s n e g l i g i b l e , and the s o l u t i o n w i l l be t h a t found above, Roche s o l u t i o n . But this Roche solution <5~- o ( S ) a n d where the c density i s greater s o l u t i o n s are Linear is only density that in cr - O (&^\ is than t h i s , i t can p o s s i b l e and valid be d i s c s can the I f the expected that other exist. order to elucidate solutions of the basic d i f f e r e n t i a l equation with a non-zero e x t e r n a l d e n s i t y examine the linear T h i s has where problem, n=1, and its partial we will exact a n a l y t i c a l -2. solution. V the e x a c t s o l u t i o n ^ ^ f ) Is the spherical homogeneous equation solution is where external Problem In The region the / . other g) solution \ ^ A / Suvx | , solution origin £ =. O . The i n f i n i t e n e s s at the t ^^ / J> possible that satisfies \ the _ , is ignored other term due to JJ2 obeys its the inhomogeneous eguation 7 •2 ~ ^ •5 / ^ V which i s decomposed, as u s u a l , i n t o Legendre f u n c t i o n expansion. Then the solutions ignoring the of other which are solution ^ - infinite ^ e at the , as origin, before and ^ . ( ^ ^ ^ vJ^lS^, which are s p h e r i c a l B e s s e l f u n c t i o n s of the k i n d , again i g n o r i n g the Bessel f u n c t i o n s of the first 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 s o l u t i o n i s and using the normalising A ~A - gives (- S 0 and c i n i t i a l condition sufficient conditions boundary value to determine uniquely are the s p e c i f i c a t i o n of density on the external this the boundary approach would be of use i f we d i s c with coefficients ^ u s e problem, the necessary coefficients non-zero of of external density, a n at j> = o value of a given shape. Such the an were to examine o b s e r v a t i o n a l l y a p a r t i c u l a r eguation boundary shape and I thus As thas i s an e l l i p t i c a l and cX— ^ese t o state and solve f o r the measure its values of the o b t a i n the i n t e r n a l density s t r u c t u r e . However, i n t h i s work we are i n t e r e s t e d i n s i m p l i f i e d models of d i s c s and judiciously thus w i l l choosing values which w i l l y i e l d an termed the s y n t h e s i s of d i s c s . We have used s o l u t i o n as a sum multipole unique of use of the AT and approach the e x t e r n a l boundary density is t h i s l i n e a r problem t o show the form of the spherical and a expansion. T h i s b a s i c s o l u t i o n s t r u c t u r e w i l l a l s o be a p p l i c a b l e to the n o n - l i n e a r case as we 'Emden' shape. of This the external alternate solution show next. 34 Disc Solutions The asymptotic expansion i s performed on the d e n s i t y <§> + £ - £ <J = A Consistent °C&t) -t- c with the a n a l y s i s above, we i n the s m a l l parameter C~ . expand the d e n s i t y i t s e l f h for rigid rotation c which y i e l d s as expected which has the Emden s o l u t i o n , and and^ i s expanded as where i j2 ~ with c e n t r a l i n i t i a l The next order and before conditions f u n c t i o n obeys an eguation homogeneous and arbitrary which i s also c o e f f i c i e n t s w i l l again Legendre m u l t i p o l e terms whose value can be determined linear multiply through e x t e r n a l boundary c o n d i t i o n s . Thus to order 0(.£^}this asymptotic l i n e a r case i s of the same form as the exact linear the order solution for the problem. cr" = As s o l u t i o n of the non- ®(f) + S . S> (Aj,M) c *IPOJ^S).?^) } *" expansion i s of the d e n s i t y upto and i n c l u d i n g terms of 0(_£j) the e x t e r n a l d e n s i t y must a l s o be c o n s t r a i n e d to of t h i s same order. be 35 A s i m i l a r expansion and bodies , where the b a s i c result is given for isothermal eguation i s 2_ and the dimensionless parameters are given by Expanding the d e n s i t y as b e f o r e c9 = -h L cf - j j ; + c g i v e s s i m i l a r eguations governing £ > ^ 0 - •• the c e n t r a l s p h e r i c a l solution and the Legendre terms, c where t h i s has a s i m i l a r expansion ^ The K i s o t h e r m a l asymptotic - . I - i „ solution ® i s thus and to t h i s order i s o f the same c h a r a c t e r as the exact i n the l i n e a r Let us problem. now recapitulate the s t e p s t o a s o l u t i o n d e n s i t y 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 , r o t a t i n g above asymptotic method. 1. governing Derive spherical solution the partial differential c o o r d i n a t e s f o r uniform r o t a t i o n equation, the s e l f - g r a v i t a t i o n eguation d i s c by the eguation from t h e and the f o r the in hydrostatic equation of 36 state. 2. Normalise variables characteristic with respect to the appropriate values. 3. Recognise the s m a l l parameter as inhomogeneous term. 4. L i n e a r case g i v e s the s o l u t i o n as the sum of the 1 Emden' s o l u t i o n plus m u l t i p o l e terms with a r b i t r a r y coefficients. 5. expansion i n powers of The n o n - l i n e a r case a l l o w s an asymptotic o ° the f i r s t C term o f which i s the 'Emden 1 s o l u t i o n and whose 3 next term of 0[&J) i n c o r p o r a t e s m u l t i p o l e terms with arbitrary coefficients. 6. A v a l i d method t o o b t a i n an unique s o l u t i o n i s t o j u d i c i o u s l y choose the c o e f f i c i e n t s a n ( ^ e x t e r n a l density and t h i s produce a system with a p a r t i c u l a r , unique shape. This will i s the approach used herein t o generate s y n t h e t i c d i s c s . 7. A subset of p o s s i b l e s o l u t i o n s can be generated i n which only the few first made c o e f f i c i e n t s a r e non-zero and the remainder are identically zero for practical s t r u c t u r e s and d i s c shapes by computer The justification of these 2. 0 (jS*)t to 4. Jeans order (1929), (1965). 0(6^), i g n o r i n g terms i s v a l i d f o r e x t e r n a l d e n s i t i e s c o n s t r a i n e d t o be o f o c o of the l i n e a r techniques. (1933), Monaghan and Roxburgh 3. The asymptotic terms density p r a c t i c e , c . f . Jeans The expansion of d e n s i t y t o f i r s t of of steps t o a d i s c model f o l l o w s : 1. For s t e p s 1,2,3 t h i s i s standard Chandrasekhar generation s o l u t i o n of 'Emden i s * sphere plus multipole o f i d e n t i c a l c h a r a c t e r t o the exact s o l u t i o n problem. (1.929) has shown :' when the pressure depends only on 37 the density, their 5. boundary The f a c t external that that an high of few equilibrium of necessitate a illustrated i n the density may be s p e c i f i e d by to unique of shape array this and of the argument and an a r r a y non-zero yields flattening large boundary of such coefficients shape. array coefficients degree an allows reversal external this particular yields t unique boundary of a (0(£ fp) the Restriction first choice coefficients choosing yields af alone.' density multipole 6. configurations a subset of the number following values of disc of for only the p o s s i b l e shapes. shape non-zero is A found to coefficients as results. Results For a disc important Legendre where function ~^(3^i --i) • this This term can be different coefficients the seen than of from the term are the with the ^ the as the will most be viewpoint ^ (y"0 one can figures 2—6 component. density zero. plots are Thus these solution with a the d o m i n a n t term in taken main c o n t r i b u t i o n c l o s e term that a disc edge-on parameters Emden s p h e r e as an from t h e of expected in defining dipole values essentially together as c a n be From Z imagine where s h a p e d body i t and higher systems small to consist adjustment the i n the centre, flat outer discs whose region. It contours must are polytropes be e m p h a s i s e d plotted the that i n u n i t s of external density these are the is synthetic external density. taken a factor as For of S the the c central density and f o r isothermals as twice this value. Note 38 that the the contour e x t e r n a l s u r f a c e i n these cases, a result shape eccentricity of rotating of polytropes Hurley 6 Roberts (1964) by H u n t e r s (1963) a expansion numerical of of flattening Legendre terms fundamental that our in of the difference in terms character Hunter's expansion in the disc increases, series will be more between these two thus avoids which with the and technigue. of Legendre that as higher the order important. The approaches however i s technigue y i e l d s a s p h e r i c a l Emden Legendre expansion and the by Roberts(1963) variational disks toward consistent synthesized polynomials i s of s i m i l a r a n a l y t i c degree increases solution plus the the fundamental d i f f i c u l t y needed a large of number of terms to d e s c r i b e t h i s c e n t r a l bulge. These f i g u r e s a l s o show the c o n c e n t r a t i o n the centre n. For f o r isothermal the isothermal systems and of d e n s i t y polytropes of l a r g e toward index case i n F i g u r e S the r a t i o of p o l a r to e g u a t o r i a l r a d i u s i s approximately 1/2.5 which i s more f l a t t e n e d than zero the Roche boundary visually The shape s i m i l a r to t h a t of to c o n s t r u c t these with of spheroids, and r e p r e s e n t gradients is approached in t h i s region. the models d e r i v e d by t h i s m u l t i p o l e need the the density density which have external contours i s been used disks. s o l u t i o n s were found to have r a p i d l y i n c r e a s i n g d e n s i t y as the c e n t r e density solutions conditions. previously All type with corresponding large As the f l a t t e n i n g i n c r e a s e s , expansion technique would the s p e c i f i c a t i o n of many more m u l t i p o l e c o e f f i c i e n t s than few describe used above, and simply so the f o l l o w i n g method must be such h i q h l y f l a t t e n e d systems. used to 39 Part 2 i Cylindrical The object properties in in this of steady-state dynamical governing derived and density u s i n g the eguation, measures of thermal energy rotation to ratio the of It and is : can The is used €• . a flat of energy : to In beta the true as eguation model will much the limiting approaches suggested of as the square disc ; than the asymptotic case with density to the differential with by P r o f . series has disc previously to energy of of the (£). radius expansion technique the where small taken to only i n been eguation which classical the used In t h i s is qives Thin be must problem even though eguation E. M i u r a . namely solution variation governing the is the the a disc radius, when €-=0, slab in a f u n c t i o n of asymptotic of with epsilon For be energy of k i n e t i c the an The b a s i c similarities potential and by dimensions parameters appear, (p), be equation together asymptotically. Such a f l a t partial state density, smaller create the motion,the s o l u t i o n d e r i v e d by t h i s solution unrealistic. problem c e r t a i n basic The ratio be a p p r o x i m a t e d direction. quasi-linear . the radius the semi-infinite slab physically , expansion will gravitational (o<) and before the dimensionless structure used as of removing alpha to astrophysical Problem, On approximate thickness thickness in ratio the epsilon> . basic approach parameter equation be shown t h a t thickness, will The a p p r o a c h equations the an a s y m p t o t i c of thickness will using continuity thermal technique. be discs describe simplest rotation the to self-consistent the three is d i s t r i b u t i o n i n the self-gravitation, axisymmetric Expansions section c y l i n d r i c a l coordinates. fluid the Coordinate it a this Airfoil approach, the 40 boundary i s expanded as a p e r t u r b a t i o n s e r i e s and f r e e boundary problem i s obtained , as Wan. as For an c o n s i s t e n c y , the asymptotic rotation a t the allow c o n s t r a i n t s In the €• along i n £• , axis. the This placed on multiplies the the terms the approximation. Close gradients increase qravity r a p i d l y , as the flat The flattish disc region of the centre, basic is major central a flat neglected central in p r e v i o u s technique and models central a boundary layer rotational axis, expected. layer density and Inside i s formed at increases the the rapidly termed the " c e n t r a l bulge can disc region thus astrophysical density flat expected to disc s p h e r i c a l c e n t r a l bulge and decreases t o the inner importance work t h a t f o l l o w s . and be have with a s p h e r i c a l c e n t r a l bulge and will be of use in solutions in r e g i o n s w i l l be performed and the which i s found to c o n s i s t flat disc solution. a and external r e g i o n s as these are Matching of the composite s o l u t i o n w i l l be derived the density these boundary w i l l c o n c e n t r a t e on the bulge the axis d i s c , i n which the f i n i t e disc normal instability boundary. thus be the intuitively an outer r a d i a l edge where the value. We will w i l l be l a r g e compared t o c e n t r i f u g a l f o r c e s a more s p h e r i c a l system toward the can to i s expected,. Close t o the central axis technigue containing so these terms become important and forces normal F. uniform 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 small and i n density Prof. term being the e x t e r n a l these terms are showed, and first expansion disc region first by 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 t h a t s m a l l parameter gradients to be suggested case, a angular v e l o c i t y must a l s o be expanded series central i n our of the the a of 41 Fundamental The Equations coordinate system i n which the equations w i l l be c y l i n d r i c a l p o l a r with r a d i u s (r) angle ( ) . As S p i e g e l processes a lack of unigueness. T h i s approach w i l l will allow p o l y t r o p e s and bodies have us to systems important the d e n s i t y approach zero eguations governing of are property and and assume no thereby and incur viscosity motion considered. and Isothermal t h a t only at i n f i n i t y this d e n s i t y to be f i n i t e f o r a f i n i t e The t h i c k n e s s [z. ) , choose the form of the r o t a t i o n . Both isothermal the expressed (1972) has observed, most models of d i s c systems tend t o n e g l e c t d i s s i p a t i v e this ,' are constrains the does external body. in cylindrical p o l a r s C~,2-,£) the dependant v a r i a b l e s : yO - d e n s i t y , P- pressure, - gravitational potential, L*- ~ v e l o c i t y , ^2.2.4-) i s given by - where (6) i s time and ' . V fe = the usual vector n o t a t i o n i s used. u .«% r r - ^ Assuming axisymmetry about the c e n t r a l a x i s , no motions except i n the (fi) d i r e c t i o n , we have and allowing 42 and, at steady-state, when the time d e r i v a t i v e i s z e r o , the equations of motion reduce to - JC". r . - 1 r af 57 ' The r e l a t i o n s h i p between d e n s i t y ^ ) expressed by Poisson's equation « ^ . 3 in a and p o t e n t i a l i s cylindrical polars with axisymmetry .X T #-7f - - a r where(Gj) The ( i s the G r a v i t a t i o n a l Constant. c o n s t i t u t i v e equation qoverninq the dependant ) i s the equation of s t a t e . I n the f o l l o w i n q we s h a l l use the i s o t h e r m a l and p o l y t r o p i c variables derivation equations w her e K. - i s qas c o n s t a n t , j r 'j J '>t j - i s temperature, bv\ - i s mean molecular weiqht of qas. fundamental equations i n the o r i q i n a l These variables^ reduce (r -28~)> > t o the . 43 When i s not present, J L = o , t h i s r e v e r t s t o the rotation spherical case. increases, the For the solution, isothermal sphere, finite the at as the centre, radius i s known t o approach and o s c i l l a t e around the s i n g u l a r s o l u t i o n i n which the density complete analysis of the sphere i s given also when SL oc r zero A more i n Appendix A. N o t i c e that the v e l o c i t y term on the r i g h t s i d e of the is 12. . v a r i e s as the sguare of the r a d i u s , Figure basic" equation , which i m p l i e s c o n s t a n t r o t a t i o n a l velocity. As , in a l l fluid this can dimensions be mechanical problems i n v o l v i n g performed with respect to normalisation characteristic and values of the v a r i a b l e s . These can be chosen f o r the convenience of the problem and i t s s o l u t i o n approach. A_sy_mp_totic The basic isothermal and Discs shown density eguation d i s c has been discussed that the polytropic i n the previous s e c t i o n above expressed i n s p h e r i c a l . However, much i n s i g h t can be rotating gained into discs by d i m e n s i o n l e s s parameter c h a r a c t e r i s t i c of the characteristic equatorial radius the obvious choice multiple geometry i s much g r e a t e r for a small of a coordinates structure creating a discs which in asymptotic expansion can be performed f o r the d e n s i t y The and d i s c s o l u t i o n s e x i s t depending on the e x t e r n a l boundary shape and c o n d i t i o n s self-gravitating for disc small is3 an solution. i s that i t s than i t s t h i c k n e s s parameter of D Thus /R*_ o r some thereof . Thus, isothermal normalise case r the coordinates such that f o r the 44 and a l s o n o r m a l i s e the angular velocity with respect to its value on the c e n t r a l a x i s and e thus {%+ i where - ] "> the £ dimensionless '| - ^ s density C^O - «c — s is C' ^= / y° /^> e . The d i m e n s i o n l e s s 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 is a energy and measure of the r a t i o of g r a v i t a t i o n a l to thermal - C £ . x / V ) ^ i s a g e o m e t r i c a l dimensionless parameter, which will be small for discs . Similarily cr for the p o l y t r o p e , with d i m e n s i o n l e s s d e n s i t y - t^s - ids 1 j e^- r <5sai i p where and ^ = Thus asymptotic useful ^ r ( 5 ^ expansions / £>-37) using £ as a s m a l l parameter an approach to o b t a i n a s o l u t i o n of t h i s b a s i c eguation f o r d i s c s , as the s m a l l e r t h i s value of £. , the more body. are disc-like the 45 We in have chosen t h i s method of n o r m a l i s a t i o n f o r convenience the f i n d i n g of the s o l u t i o n , but i t should be noted independent with OC d i s t a n c e v a r i a b l e s can be renormalised , thereby obtaining a similar that the by dividing equation as i n the p r e v i o u s s e c t i o n d e a l i n g with s p h e r i c a l c o o r d i n a t e s . Vy Here ,• / ~ the - ' i - g normalising previously it had density been the where uniform geometrical rotation properties is of the £ A / ^ i . external value c e n t r a l d e n s i t y and &i - parameters are r e l a t e d by and ^. • has thus the important also been assumed. Basic t h i s i s o t h e r m a l system d e r i v e d from property of such equations as B. (02.31 ,a ?>S) with small parameter £ can be shown by a c o o r d i n a t e transform and the o p e r a t o r on the l e f t two f^/^ £>c • Eiemannian manifold theory are d e t a i l e d i n Appendix An where S = a S/e^ hand s i d e becomes and the parameter £. d i s a p p e a r s completely from the problem. Such equations on have been examined by Chanq coordinate "artificial". expansions Chang asymptotic expansions artificial parameter or the f i r s t Chang of shows who the (1961) i n a seminal called important i n such parameters then the expansion term c o n t a i n s an exact expansion such parameters result concerning that " if £ i s e i t h e r not is an uniform, solution." (1961) has a l s o shown t h a t "the parameter-type paper ordinary i n an a r t i f i c i a l technigues parameter l e a d s t o 46 an i n d e t e r m i n a c y " the such t h a t a r b i t r a r y constants are obtained s o l u t i o n . Sometimes these certain integral summarised the properties important, can be found of the by r e l a t i n g them t o system. properties and Chang problems made t o h i s a n a l y s i s of the Navier-Stokes of fluid flow around an a r b i t r a r y (1961) of c o n s t a n t s i n the f o l l o w i n g q u o t a t i o n where p a r t i c u l a r is in these reference elliptical problem body. " 1. The c o n s t a n t s are a r b i t r a r y . " " The p r i n c i p a l reason nothing these " for this i n the expansion belief procedure i s that there seems which r e s t r i c t s the value o f constants." 2. For any choice of constants and f o r any p a r t i a l exists a related Navier-Stokes " words, in other approximation sum there the given ( exact ) s o l u t i o n . " the r e l a t e d solution and (be i t from the outer, i n n e r or composite series) should agree u n i f o r m l y , as C decreases t o z e r o , t o the order and the domain stipulated in the approximation. e s s e n t i a l l y t h a t our approximations make sense, (2) It that states they are not g r o s s l y i n c o r r e c t . " Van in Dyke which the involving (1975) has emphasised t h a t " P e r t u r b a t i o n problems small the quantity coordinates is a (space dimensionless or time) combination rather than t h e parameters alone have c e r t a i n s p e c i a l f e a t u r e s ". Similarly, coordinate One Van Dyke expansions ordinarily initial-value "For e l l i p t i c equations, u s u a l l y provide only q u a l i t a t i v e encounters problem. (1975), a Then boundary value rather results. than an because of backward i n f l u e n c e any l o c a l s o l u t i o n depends on remote boundary c o n d i t i o n s and i tis not possible t o c a l c u l a t e s u c c e s s i v e terms of an expansion f o r s m a l l v a l u e s of a c o o r d i n a t e , A l l than can find the form of the expansion be achieved is to , each term being i n d e t e r m i n a t e by one or more c o n s t a n t s . " Boundary Lay_er Theory The s o l u t i o n of d i f f e r e n t i a l e g u a t i o n s , i n parameter multiplies one extensively investigated applied mathematicians or more of within the l a s t working on f l u i d the few terms, decades a small has been mainly by dynamical problems. The e s s e n t i a l approach i s t o create a p e r t u r b a t i o n will yield which technigue which an asymptotic s o l u t i o n . An approximation of t h i s becomes i n c r e a s i n g l y a c c u r a t e as the p e r t u r b a t i o n q u a n t i t y to it zero. In p r i n c i p l e , one can improve as the f i r s t approximations; step the in a series tends the r e s u l t by embedding systematic resulting scheme is of by successive c o n s t r u c t i o n an asymptotic expansion. As g e n e r a l r e f e r e n c e s t o the s u b j e c t , following are (1973), O'malley of value: (1974), Grasman The technigue which of the density Van Dyke is (1975), Cole (1968), utilised in termed the the in cases where the changes i n some regions variables. This small dependant of Nayfeh investiqation method asymptotic expansions. T h i s method has found i t s most use the (1971). w i l l be equation sort the interval variable domain across of which of matched extensive undergoes the the sharp independant dependant v a r i a b l e changes r a p i d l y i s c a l l e d the 'boundary layer' i n f l u i d mechanics, and the 'edge l a y e r ' i n s o l i d mechanics, the 'skin layer* i n electrodynamics. Nayfeh (1973) has commented; i t i s the r u l e , r a t h e r than an 48 exception, parameter regions that (£<<1) called asymptotic are not uniformly in valid terms and we must layers". recognize To break obtain and from the scale characterizing down uniformly scales s i n g u l a r behavior is small familiar parameter d e r i v a t i v e s i n the d i f f e r e n t i a l valid from one e q u a t i o n s . Then of such boundary-layer of in the highest a straight forward p e r t u r b a t i o n scheme that d e r i v a t i v e i s l o s t i n the approximation are regions. Prandtl's multiplies the which According to Van Dyke (1975), the c l a s s i c a l warning A in the behavior of the dependant v a r i a b l e s o u t s i d e the sharp-change theory. a small u t i l i z e the f a c t that sharp changes are c h a r a c t e r i z e d by magnified different of " r e g i o n s of n o n - u n i f o r m i t y " which are sometimes r e f e r r e d t o as "boundary expansions, expansions first so that the order of e g u a t i o n s i s reduced. One more of the boundary conditions must be abandoned, and or .the approximation breaks down near where they were t o be imposed. Grasman with e l l i p t i c (1971) operator {L^) discussed t h i s problem i n connection 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 s of the g e n e r a l form valid in a s t r i c t l y neighbourhood has of a convex bounded domain . He shows that i n the point where the characteristics of the , are tangent t o the boundary, sharp changes i n can be expected l e a d i n g t o boundary l a y e r s . The characteristics of the e q u a t i o n are g i v e n by the two f a m i l i e s of curves 0--°^ - vA»"-4-te-^dx- 49 In the each region, we can c r e a t e an asymptotic expansion form The method of matched asymptotic expansions involves loss of boundary c o n d i t i o n s . An outer expansion cannot be expected satisfy are imposed i n the i n n e r r e g i o n s , conditions that boundary l a y e r ; c o n v e r s e l y , general satisfy the inner expansion expansion. The outer agree with the Asy_m_gt.ot.ic The to missing expansions. The l i m i t of the the will not d i s t a n t c o n d i t i o n s . Hence, i n s u f f i c i e n t c o n d i t i o n s are g e n e r a l l y a v a i l a b l e f o r e i t h e r the two of conditions general matching p r i n c i p l e expansion outer should, l i m i t of the inner to the in boundary outer are s u p p l i e d by to or inner matching i s t h a t the appropriate the inner orders, expansion. Expansions asymptotic basic techniques d i s c u s s e d cylindrical e p s i l o n , £ , as the small coordinate above w i l l be eguation applied (o?Sl) usinq parameter. ooT£*e i-ip 50 The interest of this research with d i s c s wherein the t h i c k n e s s i s specifically concerned of the system i s expected t o be much l e s s than the r a d i u s i e (£,<<1) . The basic eguation f coordinates density I -+ i s (2.38) , for isothermal -4- S together governing 1 in (a) and p o l y t r o p e s 1 2, i dS^ ^ s j dimensionless (b) , a,- > a s ^ J with the necessary and s u f f i c i e n t P boundary c o n d i t i o n s , namely, •= o 1 ^ ) - on the e x t e r n a l boundary^ Jeans ( 1 9 2 5 ) i ? - C ^ • s has shown t h a t , " when t h e pressure on t h e d e n s i t y , c o n f i g u r a t i o n s o f e q u i l i b r i u m may by t h e i r boundaries a l o n e . . . . . . . density depends only be specified And t h a t t h e d e n s i t y and normal g r a d i e n t are determined at every point of the boundary, such t h a t t h e s o l u t i o n f o r d e n s i t y i s unigue." The asymptotic expansion f o r t h e d e n s i t y v a r i a b l e can be made CL. £ ^ fe . cr = u> 0 -h- g . xJ t -f . . . . thus 51 Obviously, as that of fc- when the s e m i - i n f i n i t e f l a t dependence and consequently only. As the zeroth series i n £ half-thickness v» =- >J 6 3 ) a 0 6 i s expanded n = d also as an flat asymptotic v a r i a b l e has been normalised with respect of the s e m i - i n f i n i t e f l a t s i m i l a r approach t o that problem of T h i n - A i r f o i l dependent s l a b which can have no r a d i a l s as suggested by F. Wan. r As the d i s t a n c e a , the s o l u t i o n must approach order s o l u t i o n i s the s e m i - i n f i n i t e s l a b , t h e o u t e r boundary is o on the s l a b then = 1. T h i s used i n the c l a s s i c a l Theory. As the angular t o the elliptical velocity is r a d i a l s v a r i a b l e , f o r c o n s i s t e n c y t h i s must a l s o be expressed by an asymptotic s e r i e s i n €. , ^ ( s e ) = ; where the f i r s t the ^> + & • «o (s) -h • • • 0 # term i s taken t o be the r o t a t i o n a l c e n t r a l a x i s and thus velocity at 1. T h i s i s i n order that the f l a t slab zeroth order s o l u t i o n be r a d i a l l y independent. I n s e r t i n g i n the basic isothermal eguation and eguating terms of the same order g i v e s (o? 3 Q J t1 The method of matched asymptotic expansions w i l l be a p p l i e d to t h i s system. As i n such expected. Put the d e n s i t y systems, boundary eguation i n the form layers can be 52 Grasman (1971) , elliptic partial in of subcharacteristics, the radial by above the straight boundaries of the disc outer edqe, that boundary l a y e r s axis independent or the the the Using velocity of JLj, ± that are central layers. axis, to the s=0, this Grasman's the given tangent and implies neighbourhood Note t h a t of are arguments, the the independent s the ,termed boundary These the in term argued JLj. Grasman's edge. quasi-linear subcharacteristics at occur outer has location operator quadrant can and t h e of the s=constant. s=1. similar operator and t h u s lines the central the eguations of equations determine coordinate S discussion differential characteristics For his of the argument is and any p o s s i b l e n o n - u n i f o r m i t y otherwise. However, the radial gives will s when variable essentially have satisfy satisfy the the needed a t the the it central sufficient This at first contains satisfy 11 for this is condition for that an e x a c t axis the flat expansion the - ° that flat is either whose not outer on the slab central is realistic the a is under Recall parameter • uniform, or the semi-infinite o r i g i n a l eguation, boundary c o n d i t i o n s f o r not necessary be v a l i d . an e x a c t s o l u t i o n . " Though t h e s o l u t i o n to layer does axis. solution this =0 general edge b u t 'artificial not 6- on solution in a boundary a s o l u t i o n to these (P 3>S) as slab will known t h a t concerning dependence eguation solution be a r g u e d However i t expansions cannot fundamental conditions remark (1961) no f u n c t i o n a l semi-infinite could but not term is b o u n d a r y c o n d i t i o n ^V^s rotation. is i n the boundary uniform slab there no s d e p e n d e n c e . Superficially Chang's £\ =0, it obviously finite bodies. 53 Thus two s e p a r a t e approaches expected the external at the little to it, outer the can be e x p e c t e d the central a sphere Disc Region The and axis of not as in disc shape of region, while the layer and t h e boundary in detail close to as the surrounded Physically centrifugal and t h e in this be region. boundary it is disc layer is of analyses. density will a This be e x a m i n e d bulge, the axis. subsequent layer the that primary i n t e r e s t the forces the , system by this will central the axis, normal central bulge be s m a l l c l o s e will tend to to that region. Expansion isothermal equilibrium later. to central termed central will boundary region, of are edge interest A termed around show case system will will now be d i s c u s s e d be s u b j e c t e d The d i m e n s i o n l e s s d e n s i t y substituting into the basic is to in detail acoustic expanded as this perturbations as d i f f e r e n t i a l equation gives ^o (J-39) 54 where we use the expansions The boundary c o n d i t i o n s are thus ^^CO I f t h i s cuter boundary could be integrated, expansion t r e a t e d as determined thereon. a known these s illustrated previously outer boundary has both w i n the uniform s p h e r i c a l (1925) above. Jeans density Thus | ^ } and normal — of the solution obtained. showed t h a t the density may a d d i t i o n a l boundary c o n d i t i o n or t o allow accuracy equations be qradient used as an confirmation The object of the i s to o b t a i n c o n s t r a i n t s on the terms i n the boundary expansion s u f f i c i e n t t o y i e l d the f u n c t i o n a l dependence on s boundary. Similar to the and y Thin-Airfoil variables Problem, of asymptotic expansion of the boundary n e c e s s i t a t e s t r a n s f e r of the conditions As to the zeroth Van Dyke carry (1975) remarks, with the p e r t u r b a t i o n out a s y s t e m a t i c boundary order boundary. " in boundary c o n d i t i o n i s imposed a t a s u r f a c e slightly the parameter such cases when the whose p o s i t i o n £ ,...... i n order procedure the boundary c o n d i t i o n expressed i n terms of q u a n t i t i e s e v a l u a t e d at the b a s i c varies to must be position 55 of the s u r f a c e corresponding t o £ = 0 . condition e f f e c t e d by using is The t r a n s f e r of a a knowledge of the way i n which the s o l u t i o n v a r i e s i n the v i c i n i t y the s o l u t i o n i s known t o be a n a l y t i c i n the c o o r d i n a t e s case the transfer is of t h e b a s i c s u r f a c e . accomplished S e r i e s about the values a t the b a s i c For our asymptotic Assuming density problem the basic i n which expanding i n a T a y l o r surface is y=1, with the density. the expansion terms are a n a l y t i c i n y at y=1, the boundary c o n d i t i o n gives i such t h a t . by Often surface." expansion f o r i s o t h e r m a l that boundary The normal d e n s i t y g r a d i e n t ^ - CM-J i ^ - - i s not known beforehand but i s represented by the asymptotic s e r i e s which by T a y l o r S e r i e s expansion ' 3» +< which y i e l d s The solution gives L v a l u e of the normal g r a d i e n t of the z e r o t h order f l a t be examined i n more d e t a i l . ^ *t> a J J=P 7 0 ^ i s obtained I * .4/ from the s l a b s o l u t i o n which w i l l now 56 The z e r c t h order eguation valid i n the d i s c is with boundary c o m d i t i o n s — Basic and E. restrictions on s o l u t i o n s an e x a c t e l l i p t i c a l i n t e g r a l Also discussed there been a n a l y s e d has of t h i s ^ i s the s i m p l e s t p r e v i o u s l y by S p i t z e r scheme will The partial with to integrate first order (1942) and Ledoux has (1951) and For the general eguation is and case of details on t h e c o m p o s i t e a c of this solution. non-homogeneous f o r the general ^ >0, linear case gradient <( solution which conditions As b o t h tf t ^ desired, c a s e o f ^>i = 0 numerically d i f f e r e n t i a l eguation and n o r m a l this a r e n o t known and system is not over a s t h e s y m b o l i c sum o f t h e non-homogeneous where on a x i s . be shown i n t h e s e c t i o n boundary system . i s the density i s necessary non-linear s o l u t i o n are given i n Appendix the s o l u t i o n where it o solution in fact determination is conditioned. Expressing the homogeneous the part and gives t h e f u n c t i o n a l dependence of the c o n s t a n t s o f integration 57 are shown. and thus Substituting three determination the flat first solution in unknown n o r m a l g r a d i e n t Central Bulge To the replacing axis, 3 by The d e r i v a t i v e the thickness the s an in of 5 will the is this disc the necessary yields <x. to ^ (.s) to within the "v^, . properties basic S to be t a k e n 'd boundary l a y e r is stretched i . e around S= S /^C^); eguations 0(0 if we t a k e - ^ = is 6 >2 . Thus Thus L £u-«) J to spherical parameter solve solve "*- V . 6 the of the the ° fii eguation i n the sense equation ( as and shows used in this original Expanding a s y m p t o t i c a l l y r7 of density boundary l a y e r original in order situation. region, the polytropes. "artificial" Obviously allows boundary p e r t u r b a t i o n J is t h r e e " unknowns gives order coordinate i n the and s i m i l a r l y f o r is with component the or This boundary c o n d i t i o n s Expansion elucidate central the equations of t h e disc into that £ by C h a n g ( 1 9 6 1 ) . inner equation. A region it Catch 22 before ~ <*iL ° <l L*.Sb) 58 as UD I =• -+• € • O, -4- with boundary c o n d i t i o n s and Similarily the normalised d e n s i t y and thus, everywhere density p = true ^e'to * shows t h a t solution the *\ •+• & • *\ -\ • • • w i t h i n the c e n t r a l bulge boundary, the t r u e ® • - He w i l l be concerned p r i m a r i l y with the z e r c t h order s o l u t i o n the - which will represent to accuracy ^(_r|fe\ Note t h a t t h i s expansion zeroth order eguation o r i g i n a l boundary l a y e r eguation is identical to the (2.So) with c o n s t a n t r o t a t i o n . The z e r o t h order s o l u t i o n i s d e f i n e d by the e q u a t i o n where This i s a boundary value problem which can be converted t o an i n i t i a l where value problem by chanqe o f v a r i a b l e c* i s c e n t r a l d e n s i t y . T h i s g i v e s c By doing t h i s , we now assume t h a t t h e c e n t r a l d e n s i t y ^ i s known and not the s u r f a c e d e n s i t y gxves . Putting 59 with from initial the gradient conditions centre at the out G Vrt^o; , at centre. w i l l y i e l d the d e n s i t y and s u r f a c e . Now the eguation by d i v i d i n g the whole eguation i s obtained coordinates, that approach around the outer discussed normalising with above yielded s=o{1). where formation yielding a negligible is of a closed effect a in The bulge, but also external only at the b a s i c s t r e t c h i n g technigue composite this on the i n n e r regions As this region region has and t h e i r s o l u t i o n s work near the detail. Solutions method s a t i s f y boundary and of matched asymptotic expansions i n v o l v e s l o s s artificial" conditions v i c e versa . The matching the two However and central of boundary c o n d i t i o n s . An outer expansion cannot be expected region, and region solution valid i n this boundary. a x i s , i t w i l l not be examined i n The spherical d e n s i t y boundary l a y e r not unimportant f o r a c o u s t i c p e r t u r b a t i o n Matching Of the C*^ , the same subsequent a n a l y s i s can be performed s i m i l a r l y f o r with thus governing the c e n t r a l bulge. c e n t r a l a x i s , the c e n t r a l edge and viz. Which i s the eguation This was normal d e n s i t y 0(^-0(1), parameter which i s expected i n a s t r o p h y s i c a l systems. By coordinates Integration , that are missing imposed in the to inner c o n d i t i o n s are s u p p l i e d by expansions. because parameter the € expansion is done in the 11 c e r t a i n problems a r i s e e s p e c i a l l y due t o the appearence of the o r i g i n a l eguation as the zeroth order 60 solution matching clear in the problem central bulge. The r e l e v a n c e o f t h i s t o the i s d e s c r i b e d by Chang (1961) : " i t is then t h a t the expansion at l a r g e r a d i u s (the outer expansion ) cannot be matched to an expansion near the body (the i n n e r expansion ) u n l e s s the exact s o l u t i o n near the body i s o b t a i n e d . Thus in p r i n c i p l e t h i s " matching " i n v o l v e s g e t t i n g the exact solution and very indeed." Chang's i s conseguently solution a of this impractical Catch procedure 22 dilemma, i n h i s c o o r d i n a t e expansion of the Navier-Stokes flow around bodies, was t o show that a f i n i t e arbitrary number of terms of the i n n e r and outer expansions can be matched t o the d e s i r e d order. expansions are the matching then combined t o produce i n t h i s d i s c problem a composite w i l l follow this These solution; same basic approach. We apply expansions an asymptotic matching i n each r e g i o n . Each procedure between the r e g i o n of space has i t s locally v a l i d asymptotic expansion. I f t h e r e e x i s t s an o v e r l a p r e g i o n i n which both are v a l i d , i t f o l l o w s , s i n c e both a r e i n p r i n c i p l e d e r i v e d from the same exact s o l u t i o n , v a l i d everywhere, two expansions must be i n d i s t i n g u i s h a b l e i n t h i s of validity. In particular each i n d i s t i n g u i s h a b l e t o the a p p r o p r i a t e cf. finite degree common partial of t h a t the region sum is approximation, Smith(1975), The basic eguation ( Z6Z ) i s f o r d e n s i t y in the normalised inner with central respect bulge r e g i o n to external value. C o n v e r t i n g t h i s eguation t o n o r m a l i s a t i o n with r e s p e c t t o c e n t r a l d e n s i t y gave (Q-GtS*) y 61 where S - S . <* • c ^ ^ = & c lo-"^-'" and using the g i v e s , as asymptotic before, \7* A • € ^ A = - c where the l o g a r i t h m has been ^oo expansion used. to region expansion However this ,which i n d i c a t e s the as the spherical expansion necessity solution order near the e x t e r n a l boundary, as singular solution jj^\^ )] L m T n expansion spherically is C o the symmetric s o l u t i o n and the s p h e r i c a l coordinate of when intermediate to this the i n t h i s inner region method using an being the dependence in <5 with ^ C angular O o . spherical term as a sum an asymptotic first i n the s m a l l parameter m u l t i p o l e terms i n Now n of down i s known to be of QO expansion e i s of the same form as derived i n uniform f\ breaks solution <^ - S - t j separable and terms is a thus function only of we can express the in spherical and the no| angular c o o r d i n a t e s as done i n the f i r s t s e c t i o n . \iy 5 Examine A^-D e now the s o l u t i o n i n the i n t e r m e d i a t e r e g i o n , where the s p h e r i c a l term i s of order O^. ~ I ia neighbourhood . Expand the d e n s i t y as &cT, ) * c o s e e ° St ( l + the i n t e r m e d i a t e c o o r d i n a t e s are d e f i n e d by 0 and the % ,_+ Sc -Vo ; of 62 which the on first s u b s t i t u t i o n i n t h e b a s i c e q u a t i o n (2..SZ) £p- term — yields for -jtvi _L_ ~- o which has t h e s o l u t i o n where Crj a r e undetermined f u n c t i o n s of c o s i n e o f colatitude s~ • Our matching procedure f o r two expansions each locally v a l i d i n r e g i o n s K and R^of space i s a c c o r d i n g t o t h e asymptotic ( matching p r i n c i p l e as d e f i n e d by Van Dyke (1964) •the m-term R,expansion of the n-term to the n-term R^expansion of the m-term R^expansion i s e q u a l R expansion•. ; We s h a l l c o n f i n e o u r s e l v e s t o t h e case of m=1,n=1 and match density related normalised by S c by the e x t e r n a l d e n s i t y . As t h e d e l t a s are c^e-^e/^the f i r s t term of the i n n e r central bulge s o l u t i o n i s which r e w r i t t e n i n i n t e r m e d i a t e v a r i a b l e s i s and expanded f o r m a l l y f o r small Now the f i r s t (pSl^) <jj^ g i v e s term i n t h e i n t e r m e d i a t e s o l u t i o n satisfies (<3.55) o r e q u i v a l e n t l y i n t h e o r i g i n a l s p h e r i c a l v a r i a b l e with s o l u t i o n where t h e c o e f f i c i e n t s a r e r e l a t e d by 63 Beverting density to density order then 1=0(1) which i m p l i e s of <5 Now i n o r d e r form i n t h e each = e to ^ (oJ.ifl ) the /** ~^/<y X the ^ (2.^2) which i s i e sol Putting ^ ''^"^'^p external when g=0(1) , inner — ^S . y' central < c 0 bulge expand for and t h i s "^ is colatitude 0 1 1 i n so expansion a f u n c t i o n of of 1 1 reguires to be this provided to the y variable relates outer disc only. to intermediate t f a e that the / variable region can be _ a n < * expanding for small 0^ g i v e s the first as using the solution This be f o u n d a s with the variable intermediate cosine t solution principle from as and to o c o - this matching expressed term respect yields We now match As these terms can f c / match indistinguishable solution of intermediate 6^ w h i c h The by with gives the small normalised is relationship in in this overlap therefore (e?.S9 ) region matched as a gives the f u n c t i o n of y intermediate only.„ w i t h and i n d i s t i n g u i s h a b l e f r o m the 64 outer disc solution which a order an is in {o24-i function £ of In f o r c e n t r a l b u l g e and obtained f o r the symmetric solution The where case = terms has and ^ & inner l i m i t of the disc be spherically expansions formed is solution. consistently, can solution and in the as i n s i m i l a r t o cover the whole been of the c e n t r a l shown expanded i n t o solution above, bulge that the a ^series solution in solution central 6^ the bulge first being i s the density i n the approaching and limit oscillating Thus i n is bulge, the a terms. around have t h e an solution solution „ ° excellent approximation throughout mosr when f^/n i s l a r g e . disc valid spherical the s i n g u l a r . x i s known t o be the c e n t r a l 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 as S - » ^ , we c2 solution exact consists of t h e d i s c outer l i m i t has c a n be expansion c o n t a i n s the Legendre of to zeroth have b e e n matched i n an asymptotic been a p p l i e d and which frPtjJa) disc region, the i t D, which matched A Now conditions (3^. Appendix where A disc t o g e t h e r with the f l a t of i s the Now, solution problem. a composite bulge ^ flat order i n whole system method isothermal central boundary Expansion the l i n e a r problems, with ) y o n l y . Thus t h e s o l u t i o n s overlap region to f i r s t Composite two of region, and approximation everywhere the singular f o r the s p h e r i c a l solution, 65 the composite In the c e n t r a l composite as solution solution ^ o But a s £> * w i l l te bulge region will , be J> _ ^ o next « = to the c e n t r a l •+ S,. *\ 1 point, + J the — j£ and composite will ( € will *t cs DO bulge next t o <jv=-rs , t h e + foQj) solution f o r the density solution composed t o zeroth order in )• composite a flat disc be t a k e n This solution as the s o l u t i o n throughout composite agrees composite approaches zero, of the f l a t For obtain by sum the regular of the spherical terms. T h i s slab a .solution a composite flat slab add these of s p h e r i c a l central bulge solution obtained consisted that ^> - be an a c c u r a t e bulge. £. , i n the c e n t r a l solution The and r\ solution three the with spherical f a r from solution system n e c e s s a r y t o know t h e e x a c t will edge r e g i o n we have o n l y assumed (196 1) considered, this that exact as approximate obtain composed o f However i n o r d e r a total value o f the geometric whereas Legendre axis. a n d o u t e r edge s o l u t i o n . to which i t would be n e c e s s a r y t o i n the outer components form o f approximation structure the c e n t r a l of the t o t a l central expansion and t h e d i s c i s o n l y a good and t h e d e n s i t y and the expected Emden s o l u t i o n solution disc £_<<1. As i n value can only solution i t is parameter the case to € , Chang be o b t a i n e d by 66 relating in the to next Solution — us recapitulate the first is will be done Recognise containing Use partial steps to taken in this a density method structure d i f f e r e n t i a l eguation variables values that for of a for non-uniform coordinates. with and the radial be i g n o r e d gradients the respect obtain the to three appropriate dimensionless ^ ; |S,"6 • parameters of and t h i s disc. the characteristic 4. approach. derived i n c y l i n d r i c a l Normalise can virial approximation governing rotation 3. system J flattened The 2. on t h e of the Technique obtaining highly invariants chapter —<r Let 1. the density parameter gradients. when Q <<1 e x c e p t especially asymptotic eguations small close to expansion £ multiplies This- i m p l i e s these i n r e g i o n s - with the in central powers terms large rotation of C to terms density axis. determine governing density d i s t r i b u t i o n i n normal f l a t of gradients, system disc region. 5. Because close large to central •central bulge. 6. Match expansion between i n the b u l g e and disc first and d e r i v e change asymptotic scale expansion in region for this 1 solutions 7. D e r i v e t h e to axis density so s m a l l parameter for 6^ i n each an .region by intermediate another region disc. composite order, obtained is expansion composed of v a l i d i n b u l g e and d i s c the 'Emden' sphere which, plus the flat solution. 8. These two solutions must be combined to satisfy not only constant the density external combination but a l s o c o n s t a n t normal d e n s i t y boundary. of These spherical restrictions 'Emden' gradient yield solution an plus on unigue flat slab solution. In the d i s c case, f o r a |2> integrating out y i e l d at t h i s d i s t a n c e particular to the external a normalised d e n s i t y of t h i s component of the normal d e n s i t y boundary i s then choice obtained of parameters boundary at y= 1 must o f u n i t y . The gradient a t the e x t e r n a l from the numerical s o l u t i o n i n the d i s c . For the s o l u t i o n on the c e n t r a l r o t a t i o n a x i s , composite and solution and where the i s the sum of the s p h e r i c a l Emden s o l u t i o n the above d i s c s o l u t i o n to some d i s t a n c e value y>1, such , both eguations are i n t e g r a t e d out that t h i s y i e l d s the value of y f o r the e x t e r n a l boundary on the c e n t r a l a x i s . However freedom o f c h o i c e of the s t a r t i n g c e n t r a l density i n the s p h e r i c a l Emden s o l u t i o n normal d e n s i t y gradient i s restricted boundary c o n d i t i o n that by the reguires . ... . *H a-. on this external boundary on the c e n t r a l a x i s y . c An unigue problem t o zeroth central bulge. composite solution associated Solutions central g r a d i e n t s a t the e x t e r n a l at most obtained f o r the order i n 6 i n the normal d i s c r e g i o n so 9,10,11, f o r v a r i o u s values o f and i s thus obtained are dimensionless densities. shown and the i n Figures parameters Comparison & , |S o f the d e n s i t y boundary f o r these s o l u t i o n s indicate an e r r o r of a few percent, although i t i s necessary t o generate and compare these computer solutions ad nauseam in 68 order to find Numerical composite equation we performed into solution requires the solution f o r the i s o t h e r m a l case. resort In pair. Solutions The so a correct to numerical on t h e e q u a t i o n s order t o do t h i s , two f i r s t order No e x a c t methods. governing the differential A similar is are known breakdown i s terms. initially equations ^ r solutions the Legendre system o f t h e Emden decomposed by p u t t i n g o o c3<T, u O O thus with initial conditions " /Vf s e There this pair a r e a number o f n u m e r i c a l through techniques, were such the range sometimes found technigue. A "stiff" can changes be found the technigues the for a system dependent of these standard numerical Hindmarsh ( 1 9 7 4 , 1 9 7 5 ) . The c o m p u t e r Appendix routines used technigues The for simpler predictor-corrector, to (*) became a "stiff" i s c o n s i d e r e d t o be one have l a r g e differences; i n which t h e r e a r e l a r g e variables. solving ). to resort of i t s Jacobian and in (^-/ f o r integrating when t h e d e n s i t y system developed listed to necessary differential expected i n one o f decade, (^=0) t o be u n s t a b l e which t h e e i g e n v a l u e s which techniques a s Eunge - K u t t a and b a s i c smallO{£). I t was t h u s in V ° *^ "° stiff Within program by last s y s t e m s have been solution developed the will Gear that be the (1970) and i s used F~ . The l o g - d e n s i t y i n t h e c e n t r a l is bulge. 69 obtained values by the s t i f f of tendencies larger (o^) numerical method, i s p l o t t e d i n f i g u r e (12) various f o r the Emden s o l u t i o n s . General can be observed i n the p a t t e r n of solution with a drop i n d e n s i t y from c e n t r e t o s u r f a c e with i n c r e a s i n g Indeed i t i s found t h a t f o r s u f f i c i e n t l y Q£,that for large the Emden s o l u t i o n o s c i l l a t e s with decreasing values of amplitude about the s i n g u l a r s o l u t i o n . The z e r o t h - o r d e r eguation v a l i d i n the d i s c - t y p e region i s with boundary c o n d i t i o n s "a It can be decomposed i n t o two f i r s t order eguations using -A then a Boundary c o n d i t i o n s are Numerical i n t e g r a t i o n s of t h i s system by the ' s t i f f gives the density d i s t r i b u t i o n i n large distance examples of these from and the parallel disc-type to region method at a the c e n t r a l a x i s . Some s o l u t i o n s are shown i n F i g u r e 13. 70 Chapter Four G r 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 r e s e a r c h we consistent hydrodynamic theory of i s o t h e r m a l and d i s c s with s o l i d - b o d y r o t a t i o n . such a disc has c e n t r a l bulge The and obtained steadystate and to (2.3!) , w i l l now density, be ( r , , Q) coordinates and functions set solution of in system detail. o b s e r v a t i o n s and Spieqel instability for These results cf., itself system sufficiently small will relation. axial will be be The the symmetry is compared to (1972) has emphasised the i n h e r e n t problem analysis such t h a t a s m a l l chanqe i n the stability just on the important as of case of any qoverninq instabilities arisinq . Thus marginal i n the well-known procedure of c o n v e c t i v e i n s t a b i l i t y theory where choke cylindrical p e r t u r b a t i o n s along with be previous r e s e a r c h work. t h i s approach w i l l concentrate neutral of radial c o n d i t i o n s can o f t e n r e s u l t i n new or same of l i n e a r d i f f e r e n t i a l equations e q u a t o r i a l plane of an i s o t h e r m a l d i s c examined the with the a s s o c i a t e d d i s p e r s i o n this of differential as the s t e a d y s t a t e s o l u t i o n . The perturbations. together structure p o t e n t i a l . These w i l l of under the assumption obtained p o l y t r o p i c gas analysed. w i l l be l i n e a r i s e d A self- be s u b j e c t e d to p e r t u r b a t i o n s pressure general a each major r e g i o n , the s t e a d y s t a t e system, governed by the b a s i c velocity, allowed been The the d i s c r e g i o n s , system f o r d e n s i t y in have shown d e t a i l s of the instability tends to o f f and keep the system c l o s e to n e u t r a l s t a b i l i t y S p i e g e l (1972) . 71 Governing The Equations basic equations of motion governinq the perturbed dependent v a r i a b l e s , where S-P -density perturbation , - g r a v i t a t i o n a l potential perturbation , S P * -pressure perturbation , U.,—radial velocity r - " z" v e l o c i t y , -azimuthal velocity [r,^ S>J, coordinates r , are qiven, in the cylindrical by where £r V i s c o s i t y - has been ignored stability for small °/r; <9© ^ ^ as t h i s work perturbations. ^ ^ discusses neutral The p e r t u r b a t i o n s w i l l be taken, as i s o t h e r m a l as i t w i l l be assumed t h a t r a d i a t i o n w i l l be a sufficiently equation heat transport mechanism. of s t a t e becomes The equation and the eguation where rapid of s e l f - q r a v i t a t i o n V-U becomes of c o n t i n u i t y of mass i s - r (ru ) f +X£:u & -4'^, Thus, the 72 Removing above the equations governing set, guantities and the unperturbed linearising g r e a t e r than by ignoring system powers from t h e of small one, y i e l d s 3 art ^2: f £± (*s.o Using linear eguation differential [ $. £ ) i n [S'.S) a n d p u t t i n g i n t o a s e t o f e g u a t i o n s g i v e s t h e system: r H -J_3 =o (S.O>) y V7 "ft at Ik r$6> 73 In We t h i s form, t h i s s e t of e q u a t i o n s a r e h i g h l y do know however system i s t h a t zero. only that i n association under s o l i d - b o d y in space cylindrical central in equilibrium ( f"",^ ) . coordinates we w i l l distribution of r e l e v a n c e o f such a l l the From this the i s taken solve instabilities planets as a f u n c t i o n will constant only on on azimuth ccncerned functional dependence the t o Solar to the (2.) exp(imQ) as t h i s i s just the simple a linear system, set appropriate the in disc - the t h e r a d i a l (r) the equatorial ^will be of differential Also equilibrium separation being t o the d i f f e r e n t i a l Note plane o f dependence o f t h e plane. variable i n the System f o r m a t i o n . on t h i s p l a n e the with i n (£>) o f t h e v a r i a b l e s with solution as we a r e i n t e r e s t e d equatorial for of these region. l i e close the only be s p e c i f i e d a s a of Thus i n o r d e r t o make t h e g e n e r a l dependence orthogonal discs assumption as equilibrium density instabilities more t r a c t a b l e , we w i l l i g n o r e and the be s p e c i f i c a l l y and t h u s t h e s o l u t i o n interest. due t o self-consistent of the usual at equilibrium bulge and the d i s c type rotation with i n s t a b i l i t i e s s o l u t i o n i n t h e two main r e g i o n s Also, that concerned d i r e c t i o n , we know t h e d e n s i t y variables composite be t h e s y s t e m , o r , a t most, f u n c t i o n a l l y d e p e n d e n t fe) the density must (JU^^). r o t a t i o n . Thus, i n s t e a d where t h e m a t e r i a l matrix o c c u r s i n t h e above s e t with the o p e r a t o r perturbations f o r a s o l u t i o n of t h i s of the operator the operator be s p e c i f i c a l l y infinitesimal throughout a requirement the determinant Note a l s o We w i l l that intractable. most system system as there i s no density, the will be t a k e n a s of v a r i a b l e s f o r expressed system. by an 74 Also, system will it be assumed be p r o p o r t i o n a l t o (u>) i s c o m p l e x dependence will as this gives when value i s e s s e n t i a l l y perturbations, On s u b s t i t u t i o n ), part c a n become e x p o n e n t i a l l y marginal s t a b i l i t y , (o = 0 ) , the is qrowinq will then with t i m e . be o f s p e c i a l boundary of the Thus i f t h e f r e q u e n c y negative, between f o r the n o n - d i s s i p a t i v e of these assumptions, the following 4r t h e t i m e dependence (exp ( i ^ t ) ) . and t h e i m a g i n a r y of unstable that this The c a s e interest, stable and case. i n t o t h e system eguations CLo-ht^Sl -3 SI ^ tf? J ~ L r C O + L o SL y. C* lo) r r The for fundamental the c y l i n d r i c a l l y fir properties or a x i a l l y of t h i s ^0 system symmetric will special be d i s c u s s e d case (m=0). 75 C a s e ±2J_ 1 mfo T h i s i s the with simplest azimuthal cylindrically Putting dependence will exist (m=0 system %A f) ~f, ^ -r^.u This can be reduced eguation by introduction &fr & system differential defined where f r o m ^ . i z ) " " This the perturbations system will be ($1 lo) gives - P J l u r to a =o e single of the feO second-order variable (vfy as ° and and no symmetrical. ) i n t o the K yields where t h e has p o s s i b l e case wherein the ~ we r get " &u <*•>*> r differ.e<y\ \)^o boundary c o n d i t i o n been u s e d f ^ to derive } -4--^ . f~ ^ P-O (1 t O f t"5\%<^) (^.is) • } f ar 76 Before we proceed further with d i f f e r e n t i a l system, i t w i l l he changed the analysis of this i n t o the non-dimensional form. As i n b e f o r e we put and define The d i f f e r e n t i a l system then becomes where / ) Thus i s obtained from the s t e a d y s t a t e system the fundamental O variable in solution. this axisymmetric p e r t u r b a t i o n case i s of the form ( r a d i a l v e l o c i t y x d e n s i t y x radius) and the g r a d i e n t of which ^°/&c * i s °^ form (density p e r t u r b a t i o n x r a d i u s ) By e g u a t i o n (3". /*/-) the r a d i a l v e l o c i t y i s proportional the r a t e 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 variable is Louisville fundamental r e l a t e d to the r a t e of change of a n g u l a r momentum. This i s an e i g e n v a l u e problem. To f i n d possible to out more concerning i t s e i g e n f u n c t i o n s and v a l u e s (X) l e t us put i t i n Sturm - form. D i v i d e the whole system by S£ . Then the equation becomes 77 The boundary c o n d i t i o n s f o r t h i s t o d e f i n e Sturm - L o u i s v i l l e a self-adjoint problem i s one of the f o l l o w i n g at each boundary, i . e l i n e a r homogeneous boundary c o n d i t i o n s . Obviously these are satisfied by the boundary a t *=o. However the cuter boundary c o n d i t i o n a t ^ = ' i s more d i f f i c u l t t o comprehend, but homogeneous c o n d i t i o n s at gas i n t e r f a c e s a r e not usual. In p r a c t i c e i t i s boundary has little found as expected effect on the that this perturbations e q u a t o r i a l plane c l o s e to the c e n t r a l a x i s at n e u t r a l A simple a n a l y s i s that d i s c s do not allow In this order shows s e l f adjoint t o Rayleiqh's f - P 1 equation, quotient X that these on the stability. steadystate perturbations. t o e l l u c i d a t e the general Srurm-Liousville equation follows outer stability convert properties of this differential form cLc I n t e q r a t i n q by p a r t s : A " o €s -~ L € J [S*~ as from which we can deduce f a c t s about the s i q n s of I f £-^ r J A - O /C. /fr- . as w e l l as S and C , i s everywhere p o s i t i v e , and the boundary c o n d i t i o n s cause the v a n i s h i n g numerator of the Rayleigh of the l a s t term i n the q u o t i e n t , then obviously a l l r o o t s 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 t o a symmetric and p o s i t i v e d e f i n i t e matrix. In a l l other cases there 78 is at most only a f i n i t e number of negative e i g e n v a l u e s , the p o s i t i v e eigenvalues i n c r e a s e without while limit O b v i o u s l y , i n our case where the gradient and the term vary r a p i d l y from c e n t r a l bulge to f l a t d i s c and outer boundary c o n d i t i o n s (Goldreich & Lynden-Bell,1964) are such t h a t the system is probably not self-adjoint in physically s i t u a t i o n s . Thus we can expect n e g a t i v e values of as 'R. and / ^ a r e p o s i t i v e , imaginary (.<-=• t the time dependence i s C propagate will 2 , values of X ^ w i l l appear. an d As , t h i s i m p l i e s t h a t growing waves can i n t h i s d i s c system. N a t u r a l l y p h y s i c a l l y r e a l systems have expected = o -£ real dissipative to be processes modified. The where this effect of result can be the outer boundary c o n d i t i o n n i s thus minimal i n t h i s c o n t e x t , as n e u t r a l s t a b i l i t y i s c o n s i d e r e d and Analytical the e i g e n f u n c t i o n s and values are not needed. Approach In the d i s c - t y p e r e g i o n the l o g - d e n s i t y i s radius i n the l i m i t as 0 f o r the s t e a d y s t a t e s o l u t i o n on e g u a t o r i a l plane. Then eguation (J. 18) eguation as given by Korn And Korn As of the becomes; T h i s i s a s p e c i a l case of the g e n e r a l which has s o l u t i o n s independent Bessel differential (1968); , 79 where a,b,c functions are constants of the f i r s t (4119) a valid is of t h e second dispersion relation system C are the B e s s e l kind r e s p e c t i v e l y . s o l u t i o n due i t s s i n g u l a r i t y The disc function (>J#/v/) where and s e c o n d Thus t h e s o l u t i o n of where t h e B e s s e l and kind i s found not t o he at the o r i g i n . i s thus given for this part of the by ^ {*<~^*•-+ or a s p r o b a b l y more r e c o g n i z a b l e \ ^ E in original - 2 0 v a r i a b l e s , as where ^ - i s dimensionless ^ - i s normal C This waves given a Note t h a t , , velocity , z . dispersion relation constant density t h e wave number system at c r i t i c a l for axisymmetric with solid-body stability (<^=o ) by From t h i s be sound i s the f a m i l i a r through rotation. is or wavenumber - i s isothermal s wave number seen dispersion relation of Genken And (^.27), Safronov's t h e t r u t h can r e a d i l y (1975) assertion that v instability ( to <0) occurs first on t h e e g u a t o r r a l plane, as o f f 80 the plane of the d i s c the d e n s i t y Examination useful function »0. ) from { s\y eguation as and using and thus {f. using velocity - Cos - ( i l 24) f a r from ( Korn ' 33 - s . solution behave a l m o s t from ^D) yields the o r i g i n , some this s i n u s o i d a l l y i . e as And K o r n - X L+ (1968) 1 y 16) s the f a m i l i a r This intimate It ) will JJ^ -=- et radial \j, function Sufficiently and f o r ( S » * \ ) , (S ) now Bessel physical insights. Bessel ( S of t h i s i s l e s s than ^ (S.3-3 r e l a t i o n s h i p f o r Bessel and {£.25) functions. we g e t f relation (<*) c a n b e s t between be s e e n c a n be s e e n t h a t i n w a r d density perturbation i n the sketches of a perturbed (^>) and below . density peak, t h e 81 radial velocity i s positive the velocity axisymmetric rings is toward (m=0 while outward of the central constant d e n s i t y o u t s i d e of the density outer half ring of r i n g begins velocity. the rotational peak < ^ <=>tr to This has shown the rotate than the of the is special rotation A similarly steadystate solution, bulge. density of are can this rotational angular ring. For such a r i n g would able to the be offset this r i n g would self- readily understand the body. However Mestel(1969) retrograde motion under that the Toward the (£) equatorial solution and dispersion c e n t r a l bulge. In t h i s case we density c e n t r e the while toward the vary approximately i n v e r s e l y Thus the on expected t h a t interesting the on caused the certain laws. be found f o r the central original with c o n s e r v a t i o n of forces we inside inner edge h a l f of the the r e s u l t a n t can central the than thus i t would be faster Similarily, and agreement possibility i s opposite phase to is of the begins slower rotation in the rotate not, rings thus i s ne g a t i v e in concentric i s p o s i t i v e and o b j e c t s i t would be expected t h a t It system i s U.^ converge. From t h i s simple idea prograde the peak peak unstable u n l e s s d i s s i p a t i v e shear. between velocity. momentum. A shear e f f e c t physical As the d i s c r e g i o n . However, as ^ © to Uf, density axis. density ) , we c o u l d thus expect a system of to form, with egual d i s t a n c e eguilibrium the plane, and with the varies density surface relation know, from rapidly in the w i l l approach the the sguare of the density will radius. with comparison to mass and moment 82 of i n e r t i a this representation Let regime us find density at an been the two found to yield a fair extreme cases of t h i s s o l u t i o n approximate relation (s ->0 ) , the has of the d e n s i t y f u n c t i o n i n the c e n t r a l bulge. look and dispersion relation perturbation function i n each r e g i o n . Close to the density w i l l asymptotically . Thus assuming t h a t the the c e n t r a l a x i s i s constant at s — o _o density {^) s approach and central axis, the central i n the neighbourhood of gives 370*0-0 where or at critical stability, where {k = number c l o s e to the c e n t r a l a x i s . As will wavelength of d e n s i t y in the where outer ^ The perturbation w i l l thus be ^ <X ^< C bulge where disc-region. much larger case (X^ part of {^>>cy) on the e g u a t o r i a l plane. The behaviour i n t h i s r e g i o n [S.2o) becomes wave {fe ) d i s c than c l o s e to the c e n t r a l a x i s . In the approximate d e n s i t y then e g u a t i o n wavenumber {kj i n the most i n t e r e s t i n g s o l u t i o n occurs i n the central as"- (£>>^) i s the (p), t h i s be much l a r g e r than the wavenumber The the , and is outer 83 In order t o o b t a i n eguation, we shall some idea examine the of the solution of this g e n e r a l s o l u t i o n of e g u a t i o n (^"•"7) i n the form s where we hav case of c r i t i c a l s t a b i l i t y that (2 <£<<£>) which, from (\=0) and plane of the c e n t r a l bulge. Eguation(^'.?S?) i s a s p e c i a l case of where [<^> Relating assumed the n u m e r i c a l s o l u t i o n , i s a good approximation on the e g u a t o r i a l given by Watson have Lommel's eguation, as (1966), which i s are a r b i t r a r y f u n c t i o n s , and the s o l u t i o n i s given Lommel's eguation t o our g e n e r a l eguation when (m=1 ) , f o r i n t e g r a t i o n of the f i r s t (Sl3>%) g i v e s , bracket i n (5". i$) where (A) i s an i n t e g r a t i o n c o n s t a n t . If we take (^= constant) i n equating terms m u l t i p l y i n g Combining and [S'.tJ-') this solution, we get from t h e dependent v a r i a b l e i n (-^^9) , and (&</<z) y i e l d s the approximate relations thus from these two e q u a t i o n s , we can expect an approximate 84 solution of the form This result yields is a similar exact when t h e dispersion relation k - J^C This agrees regions with i n the procedure relation for applying it central a wavenumber K in this - is , CO found f o r constant media is a first to ^ the central bulge outer S (^) and I the s i m p l e media a s the K to -uniform ^ gives , solutions and c l o s e ncn the to the disc for density ^ > constant axis. The apply the usual dispersion approximation. region density Crudely where -7- region /<T. at critical stability J or Thus , in this approximately eguatorial maxima above below. inversely plane. will region Thus t h e increase approximate we can proportional wavelength linearly results expect at to or in this critical the wavenumber radius spacing outer along between central stability to are be the density bulge. The sketched 85 These approximate r e s u l t s w i l l by o b t a i n i n g an exact perturbation eguation numerical now be shown t o be solution of the governing on the e g u a t o r i a l plane.of the s o l u t i o n . S i m i l a r r e s u l t s f o r non-isothermal verified composite d i s c s are outlined i n the Appendix. Numerica1 S o l u t i o n Now, given the density distribution along the e g u a t o r i a l plane from the numerical approach f o r the s t e a d y s t a t e s o l u t i o n , t h i s allows us to compute the the p e r t u r b a t i o n eguation then we can o b t a i n two variables eguation numerical of (tf. 18). I f we d e f i n e first order d i f f e r e n t i a l eguations i n the eguivalent to the second-order (£". 18) . Thus, at c r i t i c a l s t a b i l i t y where the l o g - d e n s i t y ^u) solution differential (^>=0) we have i s g i v e n by the e g u i l i b r i u m numerical solution. Together at critical with the i n i t i a l c o n d i t i o n s stability r e a d i l y be seen j X ^ ^ o • S- o • (o=o), we can apply standard technigues t o t h i s system and o b t a i n a can X-^s-o numerical integration solution. It that these exact numerical s o l u t i o n s behave s i m i l a r l y t o the approximate a n a l y t i c a l s o l u t i o n o u t l i n e d above. As expected, until the the c r i t i c a l connection wavelength i n c r e a s e s from region between d i s c and c e n t r a l T h e r e a f t e r , the wavelength i s approximately exact a n a l y t i c a l result. the c o n s t a n t as centre bulge. in the 86 Results As the we c a n n o t solution pure gas for disc to pure case state yield maxima will . formed bulge i s Thus, if instability the gas disc obvious patterns system must at thus a hardly Planets disc seems t o dust. a point, against the It can for with l a r g e disc at radius be seen central spacings only a small bulge radial this region. eguations between of the phase that number effect plot for wave and polytropic The we interesting perturbation analysis of density of rings a non-uniform noticeable. are and as to the few and f a r be e n t e r t a i n e d . cause - density instability. the cf allows o n l y one peak i n t h e numerical similar perturbation comprehended. T h u s i n a p u r e gas be central pure allows wavenumbers 14. T h i s the be r e a d i l y similar exact density between velocity A the in figure relationship gas compute be obtained by number o f d e n s i t y between other The n e x t s e c t i o n will radial peaks i n a conditions in invoke the 87 Chapter Five Dusty D i s c s It has stability been shown above that the wavelength at marginal i n a pure gas d i s c i s i n c a p a b l e of y i e l d i n g a d i s t a n c e s t r u c t u r e such as found i n our s o l a r system. Such a r e s u l t i s not n o v e l . The c o n c l u s i o n of too wavelength has An obvious world been reached from other t h e o r e t i c a l e x t e n s i o n of our theory is to incorporate o b s e r v a t i o n s by i n f r a r e d and dust into optics of d i s c s i n d i c a t e the strong presence W.H. McCrea characteristic heavy and I.P. toward and conclude the physics. possible real Recent preplanetary o f dust. Williams (1965), have examined the times of g r a v i t a t i o n a l s e g r e g a t i o n of elements a directions. modelling the large light and t h a t d u s t - g r a i n s are important and necessary t o S o l a r System f o r m a t i o n . G o l d r e i c h and Ward (1973) have a l s o argued f o r a t h i n dust d i s c while L y t t e t o n (1972) a l s o sees, as a f i r s t stage, the formation of a t h i n dusty d i s c c l o s e to and the centre eguatorial plane. However, these analyses o f the motion of dust toward the e g u a t o r i a l by no various plane are means i n agreement i n a l l a s p e c t s . The one c o n c l u s i o n o f these numerable s t u d i e s i s t h a t a t h i n dust l a y e r collect will rapidly near the e g u a t o r i a l plane, governing Equations Marble mixtures. -(1970) has reviewed the b a s i c dynamics of dust-gas As b e f o r e , l e t the d e n s i t y , pressure and velocity of the gas be |D, Pandu . L e t there be hj dust g r a i n s per unit volume and, for simplicity l e t each dust g r a i n have the same mass 88 Let the the dust, usual considered assumption individual dust component becomes where is qravitational a "fluid" of drag have neglecting grains, the the of the equations the dust velocity random of on With velocities motion the . W for qas. each is 4> of the p o t e n t i a l q o v e r n e d by Neglecting conservation as of processes mass such a s grain formation and growth, gives (to The i s o t h e r m a l eguation o f s t a t e i s taJten, as usual, where the g r a i n s are assumed not t o c o n t r i b u t e t o pressure. Then Instabilities We a r e p r i m a r i l y i n t e r e s t e d i n r a d i a l i n s t a b i l i t i e s o f t h i s dusty gas d i s c . Marble waves one (1970) has reviewed t h i s theory of weak i n the dusty-gas. Two approaches have been c o n s i d e r e d : i n the detailed considered. In flow the about other, each the particle dusty-gas or sphere is f o r m u l a t i o n of the 89 acoustic problem, and e n e r g y of all linearised form the basis quantities from considered small. considering The This analysis than properties particle for is of temperatures of each four momentum exchange the gas The b a s i c and s moves The o t h e r is 2. the 3. t h e the sound s p e e d s The v e l o c i t y of the temperature of velocity but essential group. to two p h a s e s are and t e m p e r a t u r e are the of the but with of also the is to These heat and degrees of of the gas In t h i s wave only that f i x e d as if 'frozen . 1 c a s e s when locally are more essential in different the particles is result medium is two p h a s e s in sound s p e e d s . speed correspond the approach velocities separately the of are the Not o n l y a r e phases greatest values (1970) summarises with d i f f e r e n t the dust disturbance previous Marble His propagation while the 1. by through between The motion disc. be p e r t u r b e d waves sound p r o p a g a t i o n C our gas constituent. represent alone, same a s perturbations. different eguilibrium. calculation. outlined allowed to of c o n t i n u i t y , uniform e q u i l i b r i u m problem these and gas the i n the as our eguations their the instabilities complicated obtain the equal, locally equal, two phases and are locally equal. In are the equal isothermal and is that perturbations propaqate qas slip and ^"tf = the C velocity, stage /(1+L). S and ; the case, sound as velocity An if moves then with these speed, the was initial disturbance finally , latter three C^-, at particles zero. It velocities which were f i x e d i n is found i m p u l s e moves w i t h t h e passes the the the that "frozen" t h r o u g h an intermediate complete equilibrium 90 L propagation v e l o c i t y , i o W e t « Comparison detailed shows of this approach work i n c o r p o r a t i n g f^/^ with flow • the around results individual of more particles t h i s dusty-gas theory to be c o m p l e t e l y adeguate f o r l e n g t h s l a r g e compared time dependence perturbation of with the a weak wherein the situation gas particle size. wave and the that w i l l be the system of e g u a t i o n s s i m p l i f i e s . assumption w i l l be made t h a t perturbations constant is a l l o w i n g an analytical equilibrium disc derived, this zones On the dust-gas mixture outside of comparing it discuss with dust-gas t r a n s i t i o n A similar of rv.=o w i l l before we our layer be carried w i l l assume an is that This will s h a l l see density radial that ratio gas for direction achieved. U n t i l the this complete i s capable of being shown a wider which the their model results but transition pure gas in and that e q u a t o r i a l plane, a t h i n pure layer of system e x i s t s more detail this that i s under examination. out for this angular v e l o c i t y axisymmetric dusty JL gas case disc. of the As unperturbed disc. For the gas we have - as later note here t h a t i t i s i s o t h e r m a l p e r t u r b a t i o n f o r the now we f l a t s l a b problem and i s formed, enveloped by shall gas long- must s u f f i c e though G o l d r e i c h dust d i s c before., We the r e s u l t to be formed. the wave- A l s o , an even more important s t r u c t u r e f o r dust and are dusty dust-gas along approximation a d i s c u s s e d and the (1973) have examined the three in Thus dust move t o g e t h e r . be Ward - C |^_E H- <2SL- u r - ^B/ p 60 91 Where the only d i f f e r e n c e from t h e p r e v i o u s case of pure gas i s the a d d i t i o n of the f o r c e s of i n t e r a c t i o n ^ dust. For the dust - = between gas and N^*^ The 2" component i s i g n o r e d as b e f o r e . of the s e l f - g r a v i t a t i o n K Similarily, perturbation eguation with the c o n t i n u i t y eguations f o r each component Assuming that the dust c o n t r i b u t e s n e g l i g i b l y t o the pressure term, the gas eguation of s t a t e can be used The above system has been s e t up f o r t h e most g e n e r a l case. The first assumption sufficiently From that that will be made is that after long time the gas and dust move t o g e t h e r . equations i t can be seen t h a t t h i s i m p l i e s i s zero i n t h i s l i m i t giving Using an argument elucidated by P r o f . L. S o b r i n o , i f we make no assumption the r a d i a l f o r c e (6.1a,b) we get suggested by and e l i m i n a t e t h i s ( (Mf*) +k - Prof. term W. McCrea between " \ 7 P + (f-tfdO • and about equations ft.la.) 92 for the r a d i a l component. P e r t u r b i n g t h i s eguation and removing the non-perturbed part yields J (ftp) which i s j u s t the momentum eguation ( 7 ^ ) - <*r f o r the W which no v e l o c i t y and are • density The gas-dust mix i n d i f f e r e n c e e x i s t s between the two components, thus equivalent to a single system with (f-t^<_0 • next fundamental assumption o b t a i n a simple a n a l y t i c a l s o l u t i o n t h a t we make i n order t o that w i l l give a fruitful i n s i g h t i s t h a t the r a t i o of the p e r t u r b a t i o n i n dust d e n s i t y t o the gas d e n s i t y p e r t u r b a t i o n remains constant : Z. = ^^/S^ For the equation of s t a t e t h i s g i v e s r o+o ,, J rx N and s u b s t i t u t i n g i n t o the momentum e q u a t i o n ^fo'V) y i e l d s 0-K-) or I o-ti-) f &r 1 e>r ' & Toqether with the v e l o c i t y equation fc-'O and the c o n t i n u i t y eguation f o r the gas i n t h e form t h i s g i v e s a s e t of 4 eguations qoverninq The qas d i f f e r e n c e b£tween t h i s s e t and t h a t used variables. f o r the pure can be seen i n the momentum equation where the dust and qas are c o n s i d e r e d as a s i n q l e unit as they s e l f - g r a v i t a t i o n equation effect case 4 dependent move toqether and i n the where the d e n s i t y i s i n c r e a s e d by the of the dust. I n t r o d u c i n g the same v a r i a b l e as i n the gas \> =- c ^ K t r g i v e s from ^fen)as before iJ> = - c O and d i f f e r e n t i a t i o n yields rpu f i fc-n) 93 lfo)gives which using The second cP~J> -—, d e r i v a t i v e i s thus 1\ c •= — <<<-or.s~.Sp <$r^ • dr • L LO S p — , f \ f4.20 and combining £ 6 0 a n d (fc2o)gives ,c Using ( V f t ^ i n the s e l f - g r a v i t a t i o n e g u a t i o n (/-e-u) • and i n t e g r a t i n g ^ - r £- - S3^) r (6.7.1.) using the boundary c o n d i t i o n ^ - o , r=-o Now using (& / 5 ) s u b s t i t u t e from ^ 2 / ) f o r the f i r s t . sl.&p (/ -ft.; ^ <3-r * <- I co • /^/V-L.") * r and f o r the second term use fa./qjto Again from (G-XV) «e have 0 and substituting term t o get - *~ crr^ ^ / 7 r &r J get / 1 — f - gives ^ co r these into /^./S") g i v e s , after multiplying — l<o<-[i-+Jk> through by This 'the fundamental e g u a t i o n i s s i m i l a r t c the form of p r e v i o u s e i g e n v a l u e problem f o r pure gas but with a v i t a l d i f f e r e n c e . The term m u l t i p l y i n g ^ i s g r e a t e r by the temperature, term factor and also e thermal, i s d i m i n i s h e d by a f a c t o r can expect t o f i n d a c o r r e s p o n d i n g i n c r e a s e 1. the in or and so we the wavenumber W ) 94 Converting to and c h a n g i n g 6 0 , we where , and obvious to format the used for the same v a r i a b l e S equilibrium as i n t h e gas gas structure problem when have as is as wavelength on t h e the before the „— „ log-density compared will , with of the be much s h o r t e r equatorial plane. the pure than gas. The c h a n g e gas the of relation. p u r e gas Thus case The new d i s p e r s i o n r e l a t i o n scale is is the when^-»1 95 Besults Some various ratio examples values isothermal and the usual composite rigid solution It terms the relationship ring are case Similar results as b e f o r e s e e n how wavelength the at density which The at normalised to ring shear Self-convergence of this expected. i t Thus axi-symmetric distance linearly solution function loading stability. with ratio The close i s compared for I I , where t h e c a l c u l a t e d r a d i a l c o o r d i n a t e s Z^= 1, planets in By n o r m a l i s i n g with the respect to mode l o c a t e d c l o s e 0-9^z* , e x a c t radius Solar to the locations i n Table I I . axi-symmetric susceptible mass pattern the i s formed from a r i n g disc comparable t h e Emden dust-gas the p o s i t i o n s of Neptune, are using Z-=4,in T a b l e as graphed i n F i g u r e d ' . outer steadystate F i g u r e s 1^,2-0 a r e shown i n marginal System, flat problem on t h e e g u a t o r i a l p l a n e i s maxima, i n d i m e n s i o n l e s s with loading perturbations associated t o our s o l a r systems b a s i c if* Ic? Of-^l-lf* compared isothermal The used and mass on t h e s i m p l e s t d i s c s with was a r e shown f o r u p t o and i n c l u d i n g t h e f o u r t h L e g e n d r e i s readily decreases He c o n c e n t r a t e that Iff. wave p a t t e r n parameters r o t a t i o n case. the disc synthesized Legendre the /i^/fe 7 /7. steadystate shown i n F i g u r e for this density of dimensionless i n Figures of of has motions ring shown generated of our i n c r e a s i n g spacing previously into single objects could gas d i s c , corresponds solar well system. of the rings to be by t h e p e r t u r b a t i o n s . i s seen t h a t o u r dust perturbations, patterns been then unstable with close to observed I t c a n be seen occurs be to that the 96 central rotation Saturn to axis Pluto, 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 . the wavelength corresponding ( /OA.U.) density disc region. model, the expected observed to is probable approximately origin We must now compare parameter and p r e d i c t e d values near the From constant the results constant of and i m p l i c a t i o n s this against information. D u s t ^ G a s Mass L o a d i n g Examination values mass of of found experimental constants I— o f 4> in the discs objects This r a t i o gives Solar observations preplanetary infrared wavelength dimensionless loading ratio pattern the been t h o u g h i n d i r e c t l y , by G o l d r e i c h semi*infinite proportions. settling the very dust this the will on and t h a t the layer The r e l a t i v e pure gas d i s c b a s e d on t h e S o l a r dust i s i n this the t h i c k n e s s a thin thickness of 5*10-3 f o r t h e r e l a t i v e If (1972) ,as that y e t , no number being for made of investigated, who c o n s i d e r e d a and gas plane . in of t h i s solar discuss T h e y show boundary pure layer layer dust 7*10-5, i t i s assumed layer mass to discs. of the t u r b u l e n t i s approximately turbulent dust-gas will that in a depend R e y n o l d ' s Number. At t h e c e n t r e o f zone System . a model a l l o w e d them t o in a turbulent critical are dust toward the e g u a t o r i a l concentrate is theoretically of slab that determined t h i s and Ward composed simplified dust time dust^gas exist. This of short crucially slab There t o be d u s t y has particular approximation observations which a p p e a r however a close have for ^ , s h o w s System. that though patterns is claimed gas-dust layer to using t h e i r numbers that of most and use i s made o f t h e i r of t o t a l to dust t o gas i n the the number Solar 97 System of , simple a r i t h m e t i c y i e l d s a dust-gas mass l o a d i n g ratio approximately 70 i n the t u r b u l e n t dust-gas l a y e r . G o i d r e i c h and Ward s t u d i e d i n s t a b i l i t y layer at the centre of p l a n e t e s i m a l s were formed the dust-gas in the layer pure dust and found that t h e r e i n . Such a r e s u l t can be expected as much s h o r t e r wavelengths w i l l be found i n the density than i n the dust-gas l a y e r . As t h e i r pure dust layer relative analysis discusses i n s t a b i l i t y i n t h i s extremely t h i n pure layer eguatorial ( pancake ) on the c o n c e n t r a t e s on the dust-gas l a y e r , high dust plane w h i l e t h i s work the two approaches are complementary . The r i n g modes i n the dust-'gas s e l f - c o n verge to shear ; such l a r g e s c a l e motions with the p l a n e t e s i m a l s so formed due can be expected t o i n t e r a c t i n the t h i n dust l a y e r . This i s o b v i o u s l y a n o n - l i n e a r and complex i n t e r a c t i o n but i t must be s o l v e d i n order to understand S o l a r Systems. I m p l i c a t i o n s And S p e c u l a t i o n s The most widely accepted c o l l a p s e and fragmentation of these have a small theory of s t a r f o r m a t i o n i s by spheroidal gas-dust amount of r o t a t i o n around a c e n t r a l c o n s e r v a t i o n of angular momentum i m p l i e s t h a t form, the near and bodies. disc shapes If axis, may as i n t u i t i v e l y i t can be expected t h a t p a r t i c l e s c l o s e to c e n t r a l a x i s can c o l l a p s e t o the c e n t r e while, further out the e g u a t o r i a l plane p a r t i c l e s have l a r g e r angular momenta will structure thus infall developed much for less. discs is The steadystate but one s o l u t i o n s and end r e s u l t s of c o l l a p s e . As unstable they of the p o s s i b l e are inherently , r i n g modes as d e s c r i b e d above can be formed. t h i s work has developed only the l i n e a r i n i t i a l density stage However of this 98 perturbation. under or what In above respectable other of planets factor was researched a c to the observed ~ fa*^ ft^*> Neptune Solar single belt. to the i s o t h e r m a l give i System The shown above value, and i t i s n e c e s s a r y observed that a distance dust-gas be i n a g r e e m e n t ^ s /O We w i l l gm/cc. c f . consider of the Solar theory mass with the t o examine central (1971), results and i s n a t u r a l here than off the order yield the compares The p a r a m e t e r and u s i n g t h e distance hydrogen density as external density of a r e o f t h e e x p e c t e d and (1973), with indicating observations approaches. the P l a n e t s and important Asteroid Belt i s a p r e c o n d i t i o n on any of determining the plane of the d i s c , plane hydrogen which 5 km/sec. , unity for rotational w i t h t h e model o t h e r i n our a p p r o a c h on t h e e q u a t o r i a l £ f o r molecular Larson theoretical That o are c o n s i s t e n t and compare System. an Such d e n s i t i e s Hieles s per second / c £ and ^ would t o the plane of the e c l i p t i c perturbations greater a gm/cc theoretical close this o-f o r d e r gives fO j ( of Neptune a t the d e d u c t i o n s o f o t h e r facts n /^i , 100°^ velocity /f\„ values, our a As of a s Sj,* approximately lie of ^> -\ I the ' of t h e o r d e r o f a k i l o m e t r e approximately and values and a s t e r o i d disc. temperature velocities with with into r e m a i n s t o be a n s w e r e d . Icf* , ^> = c^SO'^' ^ parameter axisymmetric and that whether and parameters. The an that parameters correlation distribution previously show n o n - l i n e a r regime modes c a n c o n t r a c t i s a guestion results dimensionless loading subsequent c o n d i t i o n s these r i n g multiple objects The C^ the radial as d e n s i t y and i n s t a b i l i t y w i l l is occur 99 initially The on t h i s e g u a t o r i a l plane, Genkin prograde the rotation & Safronov of the m a j o r i t y of Planets i s e x p l a i n e d by shear, c o n t r a c t i o n and s e l f - c o n v e r g e n c e of these r i n g modes although Mestel distributions motion. (1966) an Whether a p a r t i c u l a r open Belt. mode shown at velocity that The faced this location problem of other ring locations a either (1975). Solar System cosmogony mass the P l a n e t s , has only about distribution, i s the d i s t r i b u t i o n o f approximately 750 times one part i n two hundred o f the t o t a l a n g u l a r momentum o f t h e S o l a r System, A l l e n i s assumed that t h e s p e c i f i c angular through is between Mars and J u p i t e r and thus broke up, c . f . Ovenden any converge, appear t o have most important f a c t , a p a r t from d i s t a n c e by i s possible, of Planet X or that indeed i t d i d form but angular momentum. The Sun, with that and does not velocity i n retrograde distribution S o l u t i o n s of the l i n e a r formation subsequently special may be a p p l i c a b l e t o the o r i g i n o f the c o n d i t i o n s not a p p l i c a b l e to the prevented that mode i s l e f t i n t a c t guestion Asteroid ring has i n the o r i g i n a l d i s c can a l s o r e s u l t such t h a t a r i n g it (1975). momentum (1972). I f is conserved the f o r m a t i o n of Sun and P l a n e t s from these r i n g modes, what does the d i s t r i b u t i o n imply about t h e 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 r o t a t i o n 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 P l a n e t s from X t o a.u, Neptune's 30 orbit. g i v e s , as X<<30. This i s readily solved t o g i v e an approximate value f o r 100 X=0.08 a.u. I f we examine the r i n g mode p e r t u r b a t i o n s . Table I I , it can be seen that the proto-Sun corresponds with density peak half-way which to the extends first r a d i a l l y outward ring density peak the central to approximately corresponding to Mercury. Our t h e o r e t i c a l r e s u l t g i v e s t h i s as approximately 0.14 a.u, while the exact present l o c a t i o n o f Mercury would g i v e 0.2 a.u. As our agreement model can be but a rough i s remarkable and a l l o w s comprehension momentum d i s t r i b u t i o n i n the S o l a r System originated rotation approximation this of the angular i f the Sun's . matter from t h i s r e s t r i c t e d r a d i a l r e g i o n around the c e n t r a l axis. Our model and i t s r i n g mode p e r t u r b a t i o n s d i f f e r from the Kant-Laplace speeding-up and view shedding of a disc cooling, radically contracting, r i n g s , and t h e i r a s s o c i a t e d problems with r i n g d i s p e r s i o n and angular momentum are overcome by our system. Dust has been shown to be of v i t a l s i g n i f i c a n c e and t h a t the wavelength at critical stability c o n t r o l l e d by the dust-gas mass loading as f a r out as P l u t o i s factor. This implies t h a t the Outer P l a n e t s can be expected t o possess a d u s t - d e r i v e d component, a result which i s f a r from n o v e l . However the t o t a l mass o f each p l a n e t i s not given by this linear perturbation theory and i s d e r i v a b l e only by f o l l o w i n g the system through the nonlinear regime into the present c o n d i t i o n s of a many-body problem. To map from l i n e a r p e r t u r b a t i o n s o l u t i o n s subseguent nonlinearity, P r i n c i p l e of Least Action applied with through the mathematicians have used t h e success, Ovenden ['\97S) has d e r i v e d a P r i n c i p l e of Least I n t e r a c t i o n A c t i o n f o r t h e a n a l y s i s of the many-body S o l a r System problem and assuming Planet masses 10 1 has o b t a i n e d can be t h e i r L e a s t I n t e r a c t i o n l o c a t i o n s - Such an approach used to alternatively o b t a i n masses i f we assume the l o c a t i o n s given by the r i n g modes. The Neptune model and and i t s p a r t i c u l a r r e s u l t s . Table I I , imply P l u t o formed i n the r e g i o n where the c e n t r a l bulge merges i n t o the outer d i s c although predominant even the e f f e c t of the disc is at t h i s c l o s e to the a x i s as the d i s c c e n t r a l d e n s i t y i s much higher than the bulge-disc region, c . f . Figure greater that distances than bulge density in this outer 18. Whether more p l a n e t s e x i s t P l u t o i s not answerable by t h i s at linear s o l u t i o n but t h e i r t h e o r e t i c a l l o c a t i o n s can be expected t o have s i m i l a r d i s t a n c e spacings that solutions to the Outer Planets, 10 a.u. can be found t h a t g i v e r i n g modes i n the between the Sun and Note region Mercury. Alven and Arrhenius importance of magnetic (1975) have argued f o r c e f u l l y fields in the early for Solar the System. D i s p e r s i o n r e l a t i o n s f o r s e l f - g r a v i t a t i n g r o t a t i n g systems magnetic field have been d e r i v e d by Chandrasekhar boundless domain. For the dust-gas system we mass l o a d i n g f a c t o r completely relative have to the r o t a t i o n e f f e c t and also argued that (1961) i n a have seen t h a t the same can be expected i n magnetic be Bode theory form, a p p l i c a b l e t o the probably s o l v e such must be explainable Planetary non-isothermal problems. and effect. the d i s t a n c e d i s t r i b u t i o n i n n e r s a t e l l i t e s of the G i a n t P l a n e t s , which can law the dominates the d i s p e r s i o n r e l a t i o n comparison with the expected magnitude of the They with by distribution. technigues any Such are being of put the in a that i s systems were developed t o 102 Conclusion The steadystate d i s c s has been o b t a i n e d a xisymtnetric problem structure of structure. case. the two Future non-axisymmetry This will p e r t u r b a t i o n s of r i n g and simplest by problem has for different work will and its also self-gravitating rotating total solution e f f e c t s but relevance necessitate order the answering of one to galactic of s p i r a l type hut the above work on the i n d i c a t e d the method of s o l u t i o n . S i m i l a r l y the i n i t i a l stages which must i n c l u d e not only f o r more exact, comparison with question of non-isothermal subsequent evolution observation- As always, r e v e a l s numerous others. Table Planet the examination a l s o the n o n - l i n e a r regime and in in t a c k l e the more complex dust-qas s o l u t i o n d e r i v e d above i s but the techniques II 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 0.627 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 : ALPHAS. 4 : C/E^iO+6 112 113 Figure X2 coSI O Q "3 CDO o CO in 2 • CD C o ^ 1 r- 10~ 10" 10" 10" 10" 10 LOG RADIUS SPHERE 5 4 3 2 1 114 CD * F i g u r e 13 CD L O L U LTD CD • L U C M L O I 1 51 o 73 CD CD 0.0 DISC 0.33 0.66 THICKNESS 1 .0 (Z=l) LIT 118 119 Figure H lO" I 5 I |IM l l j I 4 I | IMIIj 10-* • 4 lO" L O G 1 I |I I 3 4 18 1 I | I llll) lO" R f i D I U S 3 4 lO" 1 I | I I III] 1 S P H E R E 4 10 1 I | Mill) 3 4 ID 1 120 F i g u r e 1.9 o ISOTHERMAL :.R2= - 1 5 : A L P H A = 1 0 X X 4 : BETA=2.0 : ^5 0-3 >- •7 CO LU CD CDS o o 1—1 d o 2 i 1 1 nni|—1 10" 4 10-« 11 I ini|—1 1 11 4 4 10" LOG 3 ID" R R D I U S 1 1 1n m | — 1 2 4 10- 1 S P H E R E i 1111111—1 4 10° 1 1111111 4 10 1 121 122 "Figure 21 123 Bibliography Allen,C.W. (1972) ; " A s t r o p h y s i c a l Q u a n t i t i e s ", London Alven,H. and G. Arrhenius, (1975) Press. ; " E v o l u t i o n of the S o l a r System ", N.A.S.A P r e s s . Cameron,W.A.G. (1971) ; Comm. A s t r . Space P h y s i c s Chandrasekhar,S. (1933) ; Mon. 3,2,59. Not. Roy. A s t r . Soc. , 93,391. Chandrasekhar,S. ( 1939); " S t e l l a r S t r u c t u r e ", Dover Press. Chandrasekhar,S. Instability 11 (1961); , Clarendon Chandrasekhar,S. " Hydrodynamic and Hydromagnetic Press. (1969); " E l l i p s o i d a l F i g u r e s Of E q u i l i b r i u m Yale U n i v e r s i t y P r e s s . Chandrasekhar, S. and N. L e b o v i t z , Chang,!. Ap. J n l . (1968) ; " P e r t u r b a t i o n •', Ginn B l a i s d e l l Gear,C.W. (1971) Methods i n Applied Mathematics Press. ; Differential " Numerical Initial Value Goldreich,P. Problems i n Equations ", P r e n t i c e - H a l l Press. Genken,I.L. 6 V. Safronov, (1975) ; Sov. Astron. Soc. 152,267. (1961) ; J n l . Math. Mechanics 10,6,811. Cole,J.D. Ordinary (1 968); S D. Lynden-Bell, (1964) ; Mon. 19,189. Not. Roy. Astr. 130,99. Goldreich,P. Grasman,J. Amsterdam & W. Ward, (1973) ; Ap. J n l . 130,293. (1971) ; " On the B i r t h of Boundary Layers Press. H i e l e s , C . e t a l (1971) ; Ann. Rev. Ast. Astrop. Hindmarsh,G. 9,293. (1975) ; UCID-30059-1, Lawrence Livermore Lab. 124 a Hunter,C. (1972) ; Ann. Rev. F l u i d Mechanics 4,219. Hunter,C. (1963) ; Mon. Not. Roy. A s t r . Soc. 126,299. Hurley,P.M. 6 P. Roberts, Jacobi,K. (1964) ; Ap. J n l . , 140,383. (1834) ; Ann. Physik und Chimie 33,229. James,I. (1964) ; Ap. J n l . 140,255 Jeans,J. (1929) ; " Astronomy and Cosmogony ", Cambridge P r e s s . Kazdan,J. & F. Warner, (1974) ; Ann. Math. 99,14. Korn,G. (1968) & T. Korn, ; " Mathematical Handbook f o r S c i e n t i s t s and Engineers " , McGraw-Hill Press. Lamb,H. (1945) ; " Hydrodynamics •', Dover Press. Larson,R.B. (1973) ; Ann. Rev. A s t . Astrop. 11,219. Lebovitz,N. (1967) ; Ann. Rev. A s t . Astrop. 5,465. Lebovitz,N. (1979) ; Ann. Rev. F l u i d Ledoux,P. Mech. 11, 229. (1951) ; Ann. d«App. 14,438. Lynden-Bell,D. S A. Wood, (1967) ; Mon. Not. Roy. A s t . Soc. 138,495. Lynden-Bell,D. (1967) ; Mon. Not. Roy. A s t . Soc. 136,101. Lynden-Bell,D. S P. O s t r i k e r , ( 1967) ; Mon. Not. Roy. Ast. Soc. 130,293. Lyttleton,R.A. Marble,F.E. (1972) ; Mon. Not. Roy. A s t . Soc. 151,463. (1970) ; Ann. Rev. F l u i d Marcus,I., W. Press & C. Teukolsky, Mechanics 2,397. (1977) ; Ap. J n l . 214,584. McCrea,W.H. (1957) ; Mon. Not. Roy. A s t . Soc. 117,562. McCrea,W.H. 6 I.P. W i l l i a m s , (1965) ; Proc. Roy. Soc. London, A 287,143. WcLaurin,C. Mestel,L. (1742) ; " A T r e a t i s e On F l u x i o n s " . (1966) ; Mon. Not. Roy. A s t r . Soc. 131,307. Monaghan,J. J. (1967) ; Z. Astrophys. 67,232. 125 Monaghan, J . J . & I.W. Roxburgh (1965) ; Mon. Not. Roy. A s t r . Soc. 131,13. Nayfeh,A.H. (1973) ; " P e r t u r b a t i o n 0'Malley,R.E Academic (1974) ;" I n t r o d u c t i o n Methods ", Wiley to S i n g u l a r Press. Perturbations", Press. O s t r i k e r , P . & P. Bodenheimer, (1973) ; Ap. J n l . 180,159. Ovenden,M. W. fioberts,P. (1975) ; V i s t a s i n Astronomy 18,473. (1963) ; Ap. J n l . , 138,809. Smith,B. (1975) ; Astrop. Spiegel,E.A. (1972) Space Science 35,223. ; " Symposium on O r i g i n of S o l a r System ", H. Reeves ed. S p i e g e l , E.A. S Zahn Spitzer,L. ( 1970) ; Comm. A s t r . Space P h y s i c s (1942) ; Ap. J n l . 95,329. Strom, A et a l . , (1975) ; Ann. Rev,. A s t . Astrop. Van Dyke,M.D. (1975); " P e r t u r b a t i o n ", Academic Watson, 13,187. Methods i n F l u i d Mechanics Press. (1966) University 2,5,178. Press. ; " Theory of Bessel Functions ", Cambridge 126 Appgn.dices Appendix A Isothermal Gas S p h e r e . 1 When t h e g a s e o u s s y s t e m ( gas £- =- / ). sphere The density w i t h no r o t a t i o n i s a sphere eguation (^^>) ( S ) which -^ i n the case gives of an i s o t h e r m a l becomes A.i This eguation and Wood (1968) have (1) r h e s i n g u l a r summarised its extensively and L y n d e n - b e l l solutions: solution (2) the i s o t h e r m a l the centre (3) h a s been a n a l y s e d ( gas sphere solutions with f i n i t e density at ) the general solutions whose densities tend to o as ( c^. —o < ) Chandrasekar analysed these illustrated and, f o r (<p solutions in around, gas s p h e r e his book and showed t h e i r how t h e i s o t h e r m a l and o s c i l l a t e d isothermal (1939) gas s p h e r e the s i n g u l a r on Stellar Structure homologous n a t u r e , solutions approached, solution for large solution i s readily s m a l l i s g i v e n by t h e power computed series. and (^). This mumerically 127 Among the fundamental p r o p e r t i e s of t h i s s o l u t i o n i s t h a t the d e n s i t y only approaches zero as t h e r a d i u s infinity . Thus a f i n i t e a surrounding comparison to medium a approaches non-zero density. This i s in gas spheres whose d e n s i t y can become zero a t a f i n i t e r a d i u s . Density shown i n F i g u r e \2- . ) r a d i u s i s o t h e r m a l gas sphere must have with pclytrcpic ( plots for various alpha are 128 Appendix B ^Hk§S§tiS§i Our example whose Properties basic of the equation qeneral properties mathematicians they solution to is eguation they f irst general the (I) to been c o n s i d e r e d ^ by a qroup of from the p o i n t of view (1973). In the the that be somewhere (K) positive of first of the be t a X e n (/) of (c) is operator eguations, at as this at ( ° £^ we < ^> at . are the From the expect a Thus t h e 2 (c=.-j_ is (<£) . can 0 Similarly which [V) on point. the which t h e r e (2), a solutions . (K) of on where symmetric symmetric value solutions will i n our c a s e rotationally case existence Kazdan and Warner show t h a t non-linear the particular condition for eigenvalue of a equation by Kazdan and Warner s o l v a b i l i t y of of is a necessary the the variation decades. obviously s a t i s f i e d that theory parameter few given a r o t a t i o n a l l y non-zero bifurcation last a non-existence observe obstructions is density investiqated More i m p o r t a n t l y , c there is This (# >>o). ), of (f) been c a s e has show t h a t manifold. ), the R i e m a n n i a n geometry O(o), for Density' Equation. non-linear e l l i p t i c have over The q e n e r a l of Of B a s i c range of 129 Appendix C Nonjlsothermal Discs The temperature p state ' enters the system _ through t h e eguation of jg?-j and >w i s mean molecular wieght. I n p h y s i c a l l y probable systems both / = f[fjs.) and ^ = ^T( »^) # r t*. = this ( ,£) may e x i s t . As they both r will be eguation f o r t h i s system used enter through as a new dependent v a r i a b l e . The i s d e r i v e d as b e f o r e : In the i s o t h e r m a l case a s p h e r o i d a l c e n t r a l bulge and flat outer d i s c were found. As the d e n s i t y decreased r a p i d l y outward {occ^) from the c e n t r e t h i s d i s c was o f much lower d e n s i t y than the c e n t r a l c o r e . Now the central s p h e r i c a l s o l u t i o n o s c i l l a t e d around and -7, approached r a p i d l y the singular solution In t h e more g e n e r a l case where from s p h e r i c a l case Thus constant In (2) , we get when 1^= r *i ^ ^ f • matching / orders i n the ,/L s- o « t h e i s o t h e r m a l case the s i n g u l a r s o l u t i o n dominated the c e n t r a l bulge. S i m i l a r i l y i t can be expected t h a t systems with 130 ^ D( f will have a d e n s i t y g r a d i e n t i n the outer c e n t r a l such t h a t - r f> Note that i n t h i s r e g i o n "f^J* ^ value or bulge of % tV and thus independent molecular weight been argued a I tXr that P*^ and of the i s v a l i d . Note t h a t i t has from chemical approaches =t> ^<xr i s independent d of the change of temperature i f this relation nebula. Thus n i n the p r e s o l a r , a not unusual value f o r 3 what may w e l l be a c o l l a p s i n g gas. This changing with r a d i u s may be expected t o a l t e r the acoustic perturbation solutions. However, an important r e s u l t i s obtained of the by consideration b a s i c d i s p e r s i o n r e l a t i o n found f o r r a d i a l along the e g u a t o r i a l plane. In many l i n e a r and propagation technigues the dipersion oscillations non-linear wave r e l a t i o n f o r the simple medium i s e x t r a p o l a t e d with success i n t o more complex domains . The governing d i s p e r s i o n r e l a t i o n i s LO^ = A-^&if> We have shown p r e v i o u s l y that at marginal ^ B\itp/ *- Thus dispersion in the <x f 1 stability. independent function • (Vf as -<-+>•) central i s expected but such k in outer hi Jl has been shown to be an 1 ; -t^si?~ , bulge where that \ « r a l l these cases. Thus the s o l u t i o n found f o r the i s o t h e r m a l 131 case c a n be e x p e c t e d C» = o) and z e r o e s of the This i s wave pattern stability, as also to be c l o s e to the perturbation more g e n e r a l c a s e (A> ) true p r o b l e m and and i n the for the density dusty maxima isothermal case. waves 0 can so be e x p e c t e d a similar at neutral 132 Appendix D Linear Problem When t h e linear which was linear case. For coordinates, the the first linear spherical this order basic case, and this solution V which i s has the exact perturbation equation becomes ~ solution expansion \7 @ yields ® - for - the non- ^ c> .3 A D.«-*c _ ——, such n=1 t h e . In s p h e r i c a l with p o l y t r o p i c index = ± decomposed as /- o.s ±- before into d o Legendre function expansion U -* that 4- and The coefficients and s u f f i c i e n t is the be composed of seen the linear for the the this spherical between problem. as from The c y l i n d r i c a l matching obtained boundary c o n d i t i o n s same p r o c e d u r e readily Aj^are disc The e g u a t i o n non-linear j$ expansion case. solution and t h e also region is satisfying on an e x t e r n a l exact term by t f This However i t t h a t the can disc is terms. comprehension and c e n t r a l c3 ^ ^ ^ - ' boundary. multipole allows necessary bulge f o r s'^J^^h^' ' of this " ^" 133 where the distances where 6^-o f o r with the ^"-^o - n o r m a l i s e d by zeroth-order exact Boundary are In the disc plane «^ =-o region, solution solution conditions on the eguatorial demand Thus o and as the As external before transforming, such t h a t we we i.e coordinates the w h i c h has the a % i central the , bulge ^ v /o<' eguation e by variable a ^ e o r i g i n a l eguation if n to exact d is expected such c o o r d i n a t e the However, solution a r solution i s , necessary then to covering As i n the stretching, s which r e v e r t s This obtain <Ti obtain a-? ( 1 9 6 1) boundary c o n d i t i o n i s as a b o v e , isothermal know a l l t h e solution is in the from the outer limit spherical analysis in spherical o f Chang for constants Z^** known : catch problem, coordinates exact D U t if matching it is t h e s e are known, 22. where is the in expansions. case, linear expressed 3 the spherical 134 as in the linear with the isothermal case. Now, t h e o t h e r t e r m i s g o v e r n e d by eguation, o . integral X? particular / and general solution satisfying This must solutions match with i n the stretched eguation 6 — ( t>.ti ) s o we l o o k f o r s ~ gives ° *- » ,€ solution where s=0 solution o f t h e form Substituting with the outer are integration , this implies Matching before, b=. o the =- As o on , and t h u s with t h e outer allows constants. solution determination by P r a n d t l ' s t e c h n i g u e , of A t h i s constant as and y i e l d s the s o l u t i o n This i s also note that dependent This and the t o t a l s o l u t i o n the form parameter total solution disc obtained solution of to the solution i s o"^ i s a f u n c t i o n composed i s o f same original i n the n o n - l i n e a r problem. as and v a l i d when t h e r o t a t i o n of r a d i a l velocity of spherical form eguation non-rotating the composite also. solution solution 135 Appendix E la this examined, its Appendix an e x a c t based with the once to C where f / ^ i s external f l y - gradient is integral form, with a simple is and stability techniques. c a n be integrated simple relation essentially an e x a c t indicates ^ on t h e / ^.2 C on y=0, on t h e the defined the form of that d integration ^ . / This On the and the yields the Examination of eguation solution. the plane. ~ normalised central density mast plane outward f o r a result. (Korn 6 K o r n , 1 9 6 8 ) s o l u t i o n regime. r is as integral y from t h e analysis central density a p h y s i c a l l y expected phase-plane constraints is gives with i n c r e a s i n g solution, zero normalised again denominator decrease is normalised density gradient Integrating which i s fy*/^ boundary the density + - g i v e n by the normal form together in (<P.<4I ) d i m e n s i o n l e s s parameters a l p h a , b e t a , the constant A obtained equation give As real disc eguation i (&jT ~ the discussed on P h a s e - P l a n e The b a s i c simple solution is implications analysis the P u t t i n g the also indicates eguation in the iT.t 136 indicates a critical As the gradient for no c r i t i c a l is physical zero points *l- a constraint point occurs on t h e solutions. is at some o n l y on t h e JU* central plane when the condition thus **** region i n > /?c alpha beta r parameter space for 137 Appendix F L i s t of Symbols O- P o l y t r o p i c n o r m a l i s i n g d i s t a n c e .A^Constants (2.4) of i n t e g r a t i o n . <X^ I s o t h e r m a l d i m e n s i o n l e s s number oC c Isothermal d i m e n s i o n l e s s number «p, P o l y t r o p i c dimensionless number £• r I s o t h e r m a l d i m e n s i o n l e s s number P o l y t r o p i c d i m e n s i o n l e s s number Sound v e l o c i t y f o r gas ^l^Sound v e l o c i t y i n dust-gas §. Dimensionless numbers £CJ G r a v i t a t i o n a l Energy mixture £ Sguare of r a t i o o f d i s c t h i c k n e s s t o r a d i u s € Newton's parameter €. E c c e n t r i c i t y ^ Normalised c e n t r a l bulge d e n s i t y and asymptotic Crj G r a v i t a t i o n a l Constants Constant of i n t e g r a t i o n ^£ Normal d e n s i t y g r a d i e n t and asymptotic terms X H a l f sum o f p r i n c i p l e moments o f i n e r t i a Bessel Functions Wavenumber ^ Normalised wavenumber IC Drag o f dust on gas <*Ll Isothermal parameter i n eguation of s t a t e P o l y t r o p i c parameter i n eguation of s t a t e terms 138 I— Dust-gas mass l o a d i n g Xj-jean's c r i t i c a l \ X factor wavelength Constants of i n t e g r a t i o n ( (Chap 5) X Eigenvalue M Newton's r a t i o of c e n t r i f u g a l a c c e l a r a t i o n to g r a v i t a t i o n a l potential M^j Mass o f dust grain IAA Mean molecular weight yJ Cosine of c o l a t i t u d e gS* Normalised i s o t h e r m a l ^ Mass o f c e n t r a l bulge M,-, Mass of outer f l a t A Polytropic density disc index A ° Density asymptotic normalised with respect to c e n t r a l density terms A/ Number of dust g r a i n s per u n i t Asymptotic volume terms i n expansion of \J Fundamental p e r t u r b a t i o n variable f o r gas \J' Fundamental p e r t u r b a t i o n variable f o r dust-gas J l Angular velocity Angular v e l o c i t y at c e n t r a l axis Asymptotic terms of normalised angular v e l o c i t y to Frequency of perturbation Pomega, c y l i n d r i c a l r a d i u s vector ^* P r e s s u r e Legendre polynomials Gravitational Potential <£> Normalised g r a v i t a t i o n a l potential and 139 First order expansion <y S p h e r i c a l r a d i a l term with angular dependence coordinate Radius of d i s c ^ Gas c o n s t a n t ^ Density ^ Central density ^ Constant ^ g d e n s i t y on outer d i s c a x i s Composite d e n s i t y solution TjT Mean outer d i s c d e n s i t y ^ c Mean c e n t r a l bulge density »^ E x t e r n a l d e n s i t y f: V S Cylindrical radial Normalised coordinate cylindrical radial coordinate <5~~ P o l y t r o p i c v a r i a b l e ff- P o l y t r o p i c d e n s i t y v a r i a b l e and asymptotic terms "7"" Temperature """("" K i n e t i c Energy C7 Time 6- Normalised density i n intermediate region R a t i o of k i n e t i c energy to g r a v i t a t i o n a l p o t e n t i a l energy u Thermal Energy U. Gas v e l o c i t y V/^ Volume of outer W Dust ^ Normalised disc velocity c y l i n d r i c a l thickness Outer boundary and asymptotic 2; C y l i n d r i c a l t h i c k n e s s terms coordinate 2 ^ Disc thickness at c e n t r a l axis coordinate 140 Disc thickness Normalised i n outer polytropic flat radius region 141 Self-consistent Discs Axisymmetric Fluid Dynamics Continuity Eguations Self-gravitation Laplace Eguation Eguation Of Isothermal, State Polytropic Differential Potation S t e a d y .S t a t e Density Virial Global S"t r u c t u r e Analysis Linear Instability Solar Perturbations Gas System Galaxies Formation ( Future ) & Dust-Gas 142 Steady S t a t e Spherical Uniform g/**' Coordinates Rotation Normalise Isothermal V V » Sc. Small Uniform 0 Density Parameter Asymptotic 6> Spherical Density - S Expansion <f .£ c Emden •' C e n t r a l «I Angular Bulge Disc Synthetic Finite Solutions External Central Bulge Density + Disc Legendre Terms : j j ^ 143 Steady State Cylindrical Density Coordinates Differential Rotation Normalise Three Parameters O Spherical Solution Axisymmetry Parameter Asymptotic Normal Outer "Flat £. Expansions Peg!on Axial Disc Boundary Central Composite Spherical 5 Restriction £.<^l Discs Small 2 <L yg < Solution Central Flat Disc Bulge Bulge Layer Perturbation Equations Linear Isothermal Non-dissipative Axisyrametric Eigenvalue Basic Bessel Dispersion Disc Feqion, Central Self-converae delations Gas Dust-Gas Marginal C o n s t a n t fe. k. : L Mass L o a d i n g R a t i o Stability Bulge Momentum Pattern Chan I r a s e h a r Marginal to Analysis "Function P i n as S h e a r Cf. Related of A n g u l a r Stability Pure Problem Variable K a t e o f Change Rings S"' , r — Stability Disc central Region, Bulge,
- Library Home /
- Search Collections /
- Open Collections /
- Browse Collections /
- UBC Theses and Dissertations /
- Structure and stability of self-gravitating discs
Open Collections
UBC Theses and Dissertations
Featured Collection
UBC Theses and Dissertations
Structure and stability of self-gravitating discs Davies, John Bruce 1980
pdf
Page Metadata
Item Metadata
Title | Structure and stability of self-gravitating discs |
Creator |
Davies, John Bruce |
Date Issued | 1980 |
Description | Three centuries ago, the concept of the Solar Sytem being formed from an unstable disc was initially proposed. This research examines this cosmogony by the classical technique of initially obtaining the density structure of steady state discs, and gravitational instability of such systems Is then investigated in order to examine correlations between observed phenomena in the Solar System and predictions of the theory. A fluid mechanical approach to the steadystate axisymmetric structure is formulated for isothermal and polytropic gas systems, with uniform or radially dependent rotation. The equations are reduced to a single quasi-linear elliptical partial differential equation governing density, and known external boundary conditions are necessary to yield an unique density solution. When the external density is non-zero, flattened discs are possible solutions of the basic system. Two asymptotic techniques in spherical and cylindrical coordinates are created to obtain approximate solutions of the steadystate structure. Both techniques show that a self-consistent disc is composed of a high-density central bulge encircled by a low-density flat outer disc. Gravitational instability in gaseous discs is now formulated by the linear perturbation of the fundamental variables, density, pressure, gravitational potential and velocity. As the Solar System is essentially a planar structure, axisymmetric radial instability along the equatorial plane of rotation is examined. Such ring type modes are shown to be unstable to shear and tend to self-coverage. A dispersion relation is obtained analytically which indicates that the wavelength between rings is approximately inversely proportional to the square root of the steadystate density at marginal stability. However for the pure gas disc, the wavelengths are too long for any correspondence with the present spacing of the Planets. 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. |
Subject |
Signal theory (Telecommunication) Imaging systems Signal processing |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2010-03-23 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0085496 |
URI | http://hdl.handle.net/2429/22348 |
Degree |
Doctor of Philosophy - PhD |
Program |
Astronomy |
Affiliation |
Science, Faculty of Physics and Astronomy, Department of |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
Aggregated Source Repository | DSpace |
Download
- Media
- 831-UBC_1980_A1 D38.pdf [ 6.15MB ]
- Metadata
- JSON: 831-1.0085496.json
- JSON-LD: 831-1.0085496-ld.json
- RDF/XML (Pretty): 831-1.0085496-rdf.xml
- RDF/JSON: 831-1.0085496-rdf.json
- Turtle: 831-1.0085496-turtle.txt
- N-Triples: 831-1.0085496-rdf-ntriples.txt
- Original Record: 831-1.0085496-source.json
- Full Text
- 831-1.0085496-fulltext.txt
- Citation
- 831-1.0085496.ris
Full Text
Cite
Citation Scheme:
Usage Statistics
Share
Embed
Customize your widget with the following options, then copy and paste the code below into the HTML
of your page to embed this item in your website.
<div id="ubcOpenCollectionsWidgetDisplay">
<script id="ubcOpenCollectionsWidget"
src="{[{embed.src}]}"
data-item="{[{embed.item}]}"
data-collection="{[{embed.collection}]}"
data-metadata="{[{embed.showMetadata}]}"
data-width="{[{embed.width}]}"
async >
</script>
</div>
Our image viewer uses the IIIF 2.0 standard.
To load this item in other compatible viewers, use this url:
http://iiif.library.ubc.ca/presentation/dsp.831.1-0085496/manifest