The present volume covers those parts of the theory of the stability of rotating gravitating liquids that seem to be of primary importance in determining the evolution of such systems. Apart from the intrinsic mathematical and dynamical interest of the subject, the problem is also of considerable interest from a cosmogonical standpoint, as its solution is the sole source of theoretical information on the question of how an isolated unstable rotating mass will develop. The important conclusion is reached, contrary to Jeans's views and to those still largely current amongst astronomers, that the dynamical evidence is entirely adverse to the so-called fission process of formation of binary systems. The work accordingly removes theoretical foundation from this process as playing any role in the evolution of binary systems. In this way the study indirectly assumes astrophysical value in that in disposing of the fission hypothesis it lifts what seems to the writer to have been one of the major obstacles to progress with the greater problem of stellar evolution.

My own interest in the subject of rotating gravitating liquids, as best I can remember, began twenty years ago with Professor H. F. Baker's lectures at Cambridge, when he used to conclude his course on celestial mechanics with a rapid survey of those parts of the present subject that could be dealt with by elementary methods. The astronomical importance of the problem became clear to me from studies of the origin of the solar system, and in particular it was from the problem of the origin of satellites that I was first led to suspect the validity of the fission process—this, incidentally, going to show how even conjectural studies can at times contribute suggestively to more precise matters. Investigation of the details of Jeans's researches soon disclosed numerous places where he had fallen into error, and I accordingly resolved to see if after rectifying these the subject could be brought into some more coherent form, not in conflict with the strong indications already available from other related fields.

The theory of the preceding chapter has shown how, in agreement with general ideas already arrived at by astronomers, comets consist of very large numbers of widely separated particles. With this picture of the internal structure of a comet, we come now to a simple consequence of the theory that throws considerable light on the whole question of the formation of tails. The difficulty has always been to understand why comets should more or less suddenly begin to emit material for tail formation and how they are able to go on repeating the cycle at each perihelion return more or less indefinitely.

It has been seen that as a result of gravitational forces within the accretion stream the cometary segments form initially at great distance from the sun and so give rise in the first place to long-period comets. Also, since by equation (19) of Chapter III the mass md depends on, R3/2 comets of greatest mass will tend to form at the greatest distance. Now as such a comet falls towards the sun its self-gravitation will remain unchanged, at any rate in order of magnitude, whereas the differential force on it due to the sun will increase like R-3. Since R decreases from several hundred astronomical units down to a few solar radii as the comet moves inwards, the effect of the sun's action increases by a very large factor. For example, we might have initially R = 500 a.u. = 105 solar radii, and if the comet eventually passes within about 2 solar radii of the sun's centre, the differential force would be increased as compared with the comet's internal self-gravitation by a factor of about 1014. This completely reverses the relative importance of the two effects; at great distance the internal gravitation predominates, though probably only by a moderate factor, whereas when close to the sun the internal gravitation is utterly negligible.

The same result holds good even though the comet is not a sun-grazer (i.e. q ∼ radius of the sun), and the point is of such importance to our argument that it may be worth while examining it more generally.

Introduction

The present chapter will be devoted to explaining how comets may come to be formed and also become dynamically associated with the sun. In brief, the proposed process is that of the accretion of interstellar dust through the gravitational action of the sun during passages through galactic dust clouds. It will be shown how this can result in the development of suitably compact aggregations of dust particles initially describing almost parabolic orbits round the sun. Estimates of the sizes and masses of these clusters of particles can be made mathematically from the theory of the mechanism, and also a rough calculation of the total number of such cometary aggregations likely to be brought into existence during the traverse of a single cloud. But before proceeding to the detailed discussion of the process, the analysis of which is somewhat intricate, it will be convenient to give first a short account of the evidence for interstellar dust, and the nature of its properties and distribution within the galaxy, since these things bear directly on the hypotheses of the theory.

Interstellar dust

The presence of obscuring matter in interstellar space seems first to have been recognized by Barnard, and that in fact it exists in the form of fine dust was established by Slipher from a study of the so-called reflexion nebulae often associated with dark regions. There are also numerous other observational factors that taken together have gradually confirmed the presence of such dust and further evidence is still accumulating. The frequent occurrence of dark lanes and patches, most of which are far too small and irregular in shape to be explicable as arising from the distribution of stars, provides direct evidence of material having strong obscuring power, a property highly characteristic of dust. Mass for mass, the more finely divided dust is, the greater area per unit mass can it screen, so long as the sizes of the subdivided particles do not fall much below the wave-length of the light concerned. For example, a centimetre cube of material could in the first place screen an area of 1 sq. cm. from light travelling perpendicular to one of its faces. But if it were divided into cubes 10-2 cm. in length and these all placed side by side, they would produce an area of 100 sq. cm.

Where is the centre of a comet?

In discussing the orbits of comets it has been assumed more or less of necessity that a comet may be regarded as a simple gravitating particle in order that its motion relative to the sun may be describable by means of a point tracing out a curve. The assumption must contain some element of truth because of the degree of agreement with dynamical theory exhibited by the observed paths, but even so when we come to consider the internal structure of comets this assumption appears to be far from closely satisfied. If for the moment we pause to consider the motion round the sun of a large planet like Jupiter, it can be rigorously demonstrated as a matter of dynamics that there exists a certain point of the body—its centre of mass—whose motion round the sun is the same as if all the mass of the planet were concentrated at this point and all the external forces acted on it, exactly as if it were a particle of negligible size. The idea of centre of mass produces a remarkable simplification where the motion of the planet is concerned because Jupiter is effectively a rigid body despite its diameter of nearly 140,000 miles. The centre of mass remains fixed in it because it is rigid, and so the motion of this point adequately represents the general motion about the sun of the planet as a whole. (The rotatory motion of the planet about its centre is a secondary problem.) But where a comet is concerned there is no prior reason whatever for supposing that the line from the observer through the brightest point, or through the central point of the visible area, which varies with the means of observation, passes through the centre of mass of the comet as a whole. As will be explained later, the outline boundary of a comet is not always well defined, and the comet's position at any time is usually settled by the simple expedient of guiding on its brightest part, which does in fact sometimes appear as a definite point within the head.

In presenting the theory of the foregoing chapters I am aware that there is room for further investigation at many if not all points. But it is in the nature of science, and indeed the great merit of it, that when a substantially correct or valid hypothesis is obtained, instead of closing up the subject and completing our knowledge of it, the very opposite happens and we are confronted with numerous unanswered, though not necessarily unanswerable, questions that beforehand could not even be raised except perhaps in the vaguest terms. In attempting to assess at this stage the present theory of comets a number of considerations have to be borne in mind.

First to be remembered is the inherent difficulty of all cosmogonical problems, both from the mathematical point of view and from that of deriving valid hypotheses. The relevant processes must essentially involve questions of redistribution of the matter of the universe. A system beginning in a stable condition gradually evolves, or through external influence is caused to change, to some other state, and a passage from one stable form to some other form will result. Such a course of development must involve instability and dynamical processes of an irreversible nature, and nearly always will involve motion in the neighbourhood of neutral equilibrium conditions. As is well known, the mathematical difficulties associated with such questions are usually so great that apart from tracing the development up to the point where instability sets in, only considerations of a general nature can be advanced in attempting to decide the future course of the motion. The comet problem proves to be no exception to this. The motion of the cloud near the sun is, as we have seen, of a highly unstable character and subject to large possible changes in its details, but not in its general form, through any slight external disturbances such as must always be assumed to be present. Then again, within the accretion stream gravitational instability is an essential feature of the later development of comets. Thus at both its important stages we have in the process, what is indeed an essential requirement for the phenomena to be explained, a highly flexible mechanism subject to a large possible range of variation in the products that emerge from it, while all the time the mechanism itself preserving its same general character.

During recent years the subject of Comets has received little attention by astronomers, apart from the routine work of observation and computation of orbits. The theory of their origin has been almost completely neglected (of necessity, in the absence of hypotheses), and the obscurity attaching to the whole subject of Comets as a cosmogonical problem had come to be accepted as yet another of the numerous mysteries of astronomy. It has been one of the principal successes of the New Cosmology that, without having any idea of an attack on the cometary problem in view, nevertheless one of the fundamental processes discovered in connexion with stellar evolution has been found to lead quite naturally to a straightforward, and indeed a necessary, explanation of the presence of comets in the solar system, and also leads on to an understanding of many of their properties. This book represents an attempt to lay this theory before as wide a circle of astronomers as possible, in the hope that it will bring about renewed interest in the subject of Comets and thereby help to integrate astronomical theory into a united philosophical whole instead of remaining a closely guarded patchwork of disconnected, more or less taxonomic descriptions.

It has seemed to me to be more than desirable to present also an account of the observational features of Comets, which do not appear to be by any means widely known, and this information I have culled from the vast literature of the subject. I have no direct observational experience, at any rate with a telescope, and I claim no originality for this material. I express my indebtedness to the numerous authors, most of whom are no longer with us, whose papers and writings I have found so absorbingly interesting; and I hope that too much of that element of interest has not disappeared as a result of my summarization and selection of their writings. The first two chapters of this book contain this account. There follow two chapters on the theory of the formation and structure of comets, and then by way of conclusion follows a short chapter showing the relation of the work to earlier attempts at theoretical explanations. An Appendix gives numerous references to the literature of Comets, but is not claimed to be exhaustive.

This book examines the foundational consistency of quantum mechanics incorporated within relativistic frameworks. Quantum physics remains a perplexing formalism that, although very successful in explaining physical phenomena, poses many philosophical and interpretational questions. Several of the subtleties of quantum physics become more manifest when quantum processes are described using relativistic dynamics. For instance, the successful connection of spin to quantum statistics is a consequence of the consistent incorporation of special relativity into the quantum formalism. There should be similar profound explanations awaiting discovery as gravitating phenomena are successfully incorporated into quantum formulations.

The common theme of this manuscript is the examination of the incorporation of relativistic behaviors upon the foundations of quantum physics. The approach is to keep all formulations as close to observed phenomena as possible, rather than to present a set of speculative models whose primary motivations are internal aesthetics. In the search for the most elegant models of physical phenomena, one must recognize that at its core, physics is an experimental science. The dimensional analysis of fundamental units, taught at the very beginning of introductory physics classes, demonstrates that phenomenology lies at the foundations of physics. Fundamental ideas such as correspondence, the principle of relativity, and complementarity provide direct contact with the physics used to guide this exploration. This manuscript is an elaboration and expansion on previously published work, but also contains some new material.

A synopsis of Big Bang cosmology

One of the fundamental principles guiding the expected behaviors of the cosmology is a generalization of ideas of Copernicus, known as the cosmological principle. This principle presumes that no non-rotating observer at rest to the CMB radiation is more special than any other. Since the observed universe has large-scale uniformity, the cosmological principle imposes an overall homogeneity and isotropy to the universe. In addition, the dynamics of most of the aggregate features in the universe can be described assuming that the energy-momentum content of the cosmology is consistent with being an ideal fluid.

The Friedmann-Lemaitre equations 8.6, which describe the dynamics of an ideal fluid cosmology, are spatially scale invariant (if the cosmological constant is negligible), but not temporally scale invariant. The form of those equations that govern a spatially flat (k = 0) expansion satisfies spatial scale invariance (at least to a very good approximation), due to the fact that the energy densities that drive the dynamics are intensive thermodynamic variables. However, there is apparently a beginning time t0 ≈ 13.7(±0.2) billion years ago, which represents the earliest backwards-looking extrapolation of the standard model expansion called the Big Bang. The physics during these earliest moments is an active field of research. Thus, the cosmological principle does not refer to the temporal evolution of the universe.

Fundamentals of quantum mechanics

Quantum mechanics remains one of the most successful, yet enigmatic, formulations of physics. Uncertainty and measurement constraints are incorporated at the core of this fundamental description of micro-physical dynamics. Quantum mechanics successfully describes the observed structures in chemistry and materials science. In particular, the impenetrability of matter due to Pauli exclusion is a consequence of incorporating special relativity into quantum mechanics. Microscopic causality, or the requirement that communications cannot propagate at greater than the speed of light, relates a particle's spin to its quantum statistics but does not exclude the space-like correlations associated with a coherent quantum state.

This section will examine some of the foundations of quantum physics. An emphasis will be placed upon how the microscopic fundamentals of quantum mechanics relate to the geometric fundamentals of gravitational mechanics.

Quantum formalism

There are several interpretations of quantum mechanics offering models of the underlying (often hidden) dynamics that generate the observed quantum phenomenology (for example, the Copenhagen interpretation, or the many-worlds interpretation). Since any interpretation consistent with quantum physics cannot be proved or disproved by experiment, only those aspects of quantum formalism directly connected to experimental observables will be developed, with minimal interpretation imposed. To establish the conventions utilized in what follows, a brief overview of the formalism that describes those calculable observables modeled by quantum physics will be given.