A Statistical Theory of Gravitating Body Formation in Extrasolar Systems A Statistical Theory of Gravitating Body Formation in Extrasolar Systems By Alexander Krot A Statistical Theory of Gravitating Body Formation in Extrasolar Systems By Alexander Krot This book first published 2021 Cambridge Scholars Publishing Lady Stephenson Library, Newcastle upon Tyne, NE6 2PA, UK British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library Copyright © 2021 by Alexander Krot All rights for this book reserved. No part of this book may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the copyright owner. ISBN (10): 1-5275-6222-0 ISBN (13): 978-1-5275-6222-6 Dedicated to my dear parents CONTENTS Introduction ............................................................................ 1 Part I: A Statistical Mechanics of the Formation of Gravitating Cosmogonical Bodies Chapter One ......................................................................... 23 On the Problems of the Origin of the Initial Gravitational Condensation of Spread Cosmic Matter 1.1 On Newton’s Universal Gravitation Law and the problem of finding the mass center of a spread cosmic matter under its initial gravitational condensation ................................. 25 1.2 The virial theorem .................................................... 53 1.3 On the gravitational instability of Jeans and the rotational instability of Rayleigh in a gravitating molecular cloud ... 60 1.4 Poincaré’s general theorem and Roche’s model apropos the equilibrium figure for rotating and gravitating continuous medium ........................................................... 75 1.5 On the fundamental difficulties of the theory of gravitational instability and the theory of gravitational condensation of an infinitely spread media ....................... 90 1.6 Fundamental principles and main problems of the statistical mechanics of a molecular cloud ...................... 109 1.7 On the evolutional equation in the statistical mechanics of molecular clouds ....................................... 135 Conclusion and comments ................................................... 142 viii Contents Chapter Two ....................................................................... 145 The Statistical Model of Initial Gravitational Interactions of Particles in a Molecular Cloud 2.1 The derivation of a function of particle distribution in space based on the statistical model of a molecular cloud ................................................................................ 147 2.2 The distribution of mass density as a result of the initial gravitational interaction of particles in a molecular cloud ............................................................... 157 2.3 The critical (threshold) value of mass density and gravitational condensation parameter .............................. 166 2.4 The strength and potential of the gravitational field of a sphere-like gaseous body formed by a collection of interacting particles ..................................................... 171 2.5 The potential energy of a gravitating sphere-like gaseous body ................................................................... 181 2.6 The probability interpretation of physical values describing the gravitational interaction of particles in a sphere-like gaseous body ................................................ 190 2.7 The statistical model of gravitation treated from the point of view of Einstein’s general relativity ............ 195 2.8 The pressure in a gravitating sphere-like gaseous body formed by a collection of interacting particles ................ 202 2.9 The internal energy of a gravitating sphere-like gaseous body ................................................................... 211 2.10 The Jeans mass and the number of particles needed for gravitational binding of a sphere-like gaseous body ....... 217 Conclusion and comments ................................................... 228 A Statistical Theory of Gravitating Body Formation in Extrasolar Systems ix Chapter Three .................................................................... 233 Formation of Cosmogonical Bodies based on a Statistical Model of a Rotating and Gravitating Spheroidal Body 3.1 Poincaré’s general theorem and Roche’s model in statistical interpretation for a slowly rotating and gravitating cosmogonical body ....................................... 235 3.2 The nonequilibrium particle distribution function for spatial coordinates in a sphere-like gaseous body during its initial rotation .................................................. 247 3.3 Derivation of the equilibrium distribution function of liquid particles in space and mass density functions based on the statistical model of a uniformly rotating and gravitating spheroidal body with a small angular velocity ............................................................................ 251 3.4 Derivation of the distribution function of the specific angular momentum value and angular momentum density for a uniformly rotating spheroidal body in a state of relative mechanical equilibrium ...................................... 269 3.5 The distribution function of particles in space for a rotating and gravitating spheroidal body from the point of view of the general relativity theory .................. 279 3.6 The strength and potential of the gravitational field of a uniformly rotating spheroidal body .......................... 288 3.7 The potential energy of a uniformly rotating and gravitating spheroidal body ............................................. 305 3.8 The mass density of a rotating flattened (disk-shaped) spheroidal body and the model of formation of a rotating disk ..................................................................... 311 Conclusion and comments ................................................... 329 x Contents Chapter Four ...................................................................... 337 Equations and State Parameters of a Forming Spheroidal Body in the Process of Initial Gravitational Condensation 4.1 The main anti-diffusion equation of initial gravitational condensation of a spheroidal body with a centrally symmetric distribution of masses from an infinitely spread matter ................................................................... 339 4.2 General differential equations for physical values describing the anti-diffusion process of an initial gravitational condensation of a centrally symmetric spheroidal body near mechanical equilibrium................. 352 4.3 Special cases of the basic equation of slow-flowing initial gravitational condensation and its solution near the state of mechanical equilibrium of a centrally symmetric spheroidal body ............................................................... 367 4.4 The gravity–thermodynamic relationship for a centrally symmetric gravitating spheroidal body ........................... 376 4.5 The mass density and internal energy flows for slow- flowing and initial gravitational condensation of a centrally symmetric spheroidal body .............................. 385 4.6 Dynamical states of a forming centrally symmetric spheroidal body near the points of mechanical equilibrium .................................................... 398 4.7 The derivation of the general anti-diffusion equation for a slowly evolving process of gravitational condensation of a rotating axially symmetric spheroidal body ............. 405 Conclusion and comments ................................................... 410 A Statistical Theory of Gravitating Body Formation in Extrasolar Systems xi Chapter Five ....................................................................... 417 On the Generalized Nonlinear Schrödinger-like Equation Describing the Evolution of a Forming Cosmogonical Body 5.1 The density of anti-diffusion mass flow and the anti-diffusion velocity into a gravitational compressible spheroidal body ............................................................... 419 5.2 The initial potential of an arising gravitational field, the initial gravitational strength induced by the anti- diffusion velocity, and the characterizing number K as a control parameter of dynamical states of a forming spheroidal body ............................................................... 427 5.3 The equilibrium dynamical states after the origin of a gravitational field inside a forming spheroidal body ...... 438 5.4 The dynamical states after the decay of a rotating spheroidal body and the formation of protoplanetary shells ................................................................................ 450 5.5 Interconnections of the proposed statistical theory of gravitating spheroidal bodies with Nelson’s statistical mechanics and Nottale’s scale relativistic theory ........... 464 5.6 The derivation of the generalized nonlinear Schrödinger- like equation in the statistical theory of gravitating spheroidal bodies ............................................................. 471 5.7 Some particular cases of the generalized nonlinear Schrödinger-like equation describing different dynamical states of a gravitating spheroidal body ............................ 476 5.8 Derivation of the reduced model in the state-space of a nonlinear dynamical system describing the behavior of the cubic generalized Schrodinger-like equation