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| ISBN-13: | 9780486174662 |
|---|---|
| Publisher: | Dover Publications |
| Publication date: | 01/23/2013 |
| Series: | Dover Books on Physics |
| Sold by: | Barnes & Noble |
| Format: | eBook |
| Pages: | 208 |
| File size: | 6 MB |
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Theories of Figures of Celestial Bodies
By Wenceslas S. Jardetzky
Dover Publications, Inc.
Copyright © 2005 Dover Publications, Inc.All rights reserved.
ISBN: 978-0-486-17466-2
CHAPTER 1
Introduction
The theory of figures of celestial bodies has been developed by a number of most outstanding mathematicians, primarily by Maclaurin, Jacobi, Poincaré, and Liapounov. Their investigations, however, have been largely limited to the case of an isolated fluid mass in uniform rotation. This approach was justified for two reasons. First, the figures of celestial bodies actually differ but little from the figures given by the solutions of this problem (e.g., a sphere, ellipsoids of Maclaurin and Jacobi, figures of Poincaré and Liapounov), and second, the existence of such figures could be demonstrated by the most rigorous methods of mathematical analysis.
Some progress may still be possible in this area, but the development of an exact theory which would take into account conditions more closely approaching reality appears more interesting and important. Variations in figures and stratification of celestial bodies which result from internal movements should be considered as the next approximation. There are variations important primarily in the investigation of stellar structure. On the other hand, there are features observed in the Earth's crust and similar features undoubtedly existing in the crust of any other solidified planet which may be also explained by the departure from conditions of equilibrium. Thus, many problems arise in which the figures of celestial bodies have to be determined under conditions different from equilibrium.
The most exact methods used in the theory of figures of equilibrium and the most important results are discussed in the first six chapters. The second part of this book deals with some other problems concerning the figures of celestial bodies. Many of these problems still await their solution by equally rigorous methods as those developed in the theory of figures of equilibrium.
1.1. Fundamental Problem
The solution of the following problem of Hydrodynamics is being presented for the purpose of explaining the shape of the Sun and other celestial bodies. Let a fluid mass be rotating in space as a rigid body. Let us assume that it is isolated from other bodies and that its particles are subject to mutual attractions according to Newton's law of gravitation. What is the figure of this mass?
When the problem is given in this general form, not all conditions are taken into account. Nevertheless, we can make some important preliminary conclusions using the equations of motion of a fluid mass. We call z-axis the axis of rotation of the mass and take the origin of a rectangular system of axes Oxyz at the mass center. Let ω be the angular velocity. The velocity v of a point M is then given by the expression
(1.1.1)
v = ω × r
if the vector r(x, y, z) is equal to OM. The components of acceleration at M are: —ω2x, — ω2y, 0. Let fU be the gravitational potential, f the constant of gravitation and x and p the density and pressure of the fluid. We can write the equations of motion of a fluid mass either in the vector form
(1.1.2)
d2r/dt2 = grad fU - 1/x grad p
or in the scalar form
(1.1.3)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
If we assume that ω is constant, we obtain by (1.1.3)
(1.1.4)
fdU + ω2/2 d(x2 + y2) = 1/x dp
In two cases, namely, x = const. or x = φ(p) we have an integral of this equation in the form
(1.1.5)
fU + ω2s2/2 = ∫ dp/x + const.
where s = √x2 + y2 is the distance of a particle from the axis of rotation.
For Newton's law of gravitation we have
(1.1.6)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
where D is the distance of the particle at M from an element dV'.
If a fluid mass is isolated in space, the pressure at its boundary surface vanishes:
(1.1.7)
p = 0
Thus the boundary surface belongs to the set of surfaces of equal pressure. The general condition for this set may be immediately obtained by putting p = const. in (1.1.5)
(1.1.8)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
In the case x = φ(p) the surfaces p = const. coincide with the surfaces of equal density.
In general, the volume V of the fluid mass in the equation (1.1.8) is unknown and the shape of this mass must be chosen in such a way that this equation will be satisfied. Thus, the problem we have posed in its simplest form, leads to the solution of a functional equation (1.1.8), where the limits of integration in the first term are unknown.
We obtain a particular case of the problem on considering figures of equilibrium of a fluid mass at rest. Since we assume now that ω = 0, these figures must be determined by the solutions of the equation
(1.1.9)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
We have to keep in mind that on deriving the integral (1.1.5) we have considered fluids as having the property that either χ = const., i.e., homogeneous and incompressible fluids, or x = φ(p), i.e., compressible fluids which are homogeneous by composition but can have a density varying with pressure. It is evident, however, that the equations (1.1.3) hold without these restrictions, and we can investigate also the figures of equilibrium of a fluid mass that is composed of a finite or even of an infinite number of incompressible fluids having different densities or figures of a compressible and inhomogeneous mass.
There are types of motion other than a uniform rotation of a fluid mass about a fixed axis which are also determined by equations (1.1.3). The problems concerning these more general movements will be discussed later.
1.2. Remarks Concerning Basic Assumptions
In order to solve a problem in the theory of figures of equilibrium or, in general, to determine the figures of a rotating fluid mass, certain basic assumptions are made concerning the physical properties of the fluid, the character of motion, the mass distribution, and others. Some of these assumptions will be discussed now.
For the motion of a fluid the condition of continuity can be written either in the form
(1.2.1)
dχ/dt + χ div v = 0
or
(1.2.1')
[partial derivative]χ/[partial derivative]t + v · grad χ + χ div v = 0
In case of a permanent motion, we have [partial derivative]χ/[partial derivative]t = 0, i.e., the density x does not depend on time explicitly. But, in some invariable figures of a fluid mass, the density of a particle will not vary along its path and, therefore, we have dx/dt = 0. By (1.2.1) and (1.2.1') we obtain div v = 0 and
(1.2.2)
v · grad χ = 0
Since the vector grad x is perpendicular to the surface of equal density x = const., the velocity v must be in the tangential plane passing through the given particle or point in the fluid.
If the motion of a fluid mass has the characteristics just mentioned, the equation of continuity has the form
(1.2.3)
div v = 0
for an incompressible fluid as well as for a compressible one. It is easy to see that, if the velocity v is given by the expression (1.1.1), the equation (1.2.3) will be satisfied either by ω = const. or by ω = F (x2 + y2, z ). It is, namely,
div v = - [partial derivative](ωy)/[partial derivative]x + [partial derivative](ωy)/[partial derivative]x + [partial derivative](ωx)/ [partial derivative]y = 0
in both cases. Thus, the assumption of a rotating figure of equilibrium is compatible with the condition (1.2.3).
The second assumption about the character of motion concerns the existence of an axis of rotation. It has been proved by Appell that a necessary condition of the existence of a figure of equilibrium of a fluid mass is that the rotation occurs with respect to an axis having a direction invariable in space and coinciding with one of the principal axes of inertia of this mass. Jardetzky has shown that this condition holds also for a more general kind of rotation, namely, for the so-called zonal rotation and for mixed systems, such as a body composed of solid and fluid parts.
It is not necessary to make special assumptions about the symmetry of figures of equilibrium. It is known (see, for example, Appell) that the axis of rotation can be but is not necessarily an axis of symmetry of a rotating fluid mass. It can be shown, for example, in case of a homogeneous mass that there can be a finite number of planes of symmetry passing through the axis. On the other hand, Lichtenstein [1] has proved that a figure of equilibrium must have a plane of symmetry perpendicular to the axis of rotation.
As to the assumptions concerning the physical properties of the fluid, few remarks have to be added. We have seen that we have to discuss the figures of a homogeneous or heterogeneous, compressible or incompressible, fluid mass. It is usually assumed that the fluid is isotropic. A theory of figures of an anisotropic fluid has not yet been developed.
In the most simple case, a fluid mass is considered incompressible and homogeneous in the sense that the density has equal values everywhere and that these values cannot be changed by a varying pressure. Then, the equation (1.1.8) takes the form
(1.2.4)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
The left side of this equation is an unknown function of coordinates determined by the shape of the boundary surface S of the volume V. If the value of the constant right-side member is taken for p = 0, the problem of figures of equilibrium is to find a set of surfaces S such that for the corresponding volume V the condition (1.2.4) will be satisfied, i.e., that (1.2.4) will be the equation of any of these surfaces.
The second problem in the theory of figures of equilibrium deals with the case where the density depends on the pressure only χ = φ([??]). In classical Hydrodynamics, these fluids are usually referred to as the inhomogeneous. To solve the problem in this case, we have to make use of equation (1.1.8) or (1.1.5), since the last equation yields the level surfaces. Under our assumption about the physical properties of the fluid, the surfaces p = const., χ = const., and the level surfaces (the surfaces of equal gravity potential in case of the Earth) will coincide if (1.1.5) is satisfied. In this equation, the density χ is the variable which will determine the stratification. The outer layer of the fluid corresponds to χ = φ(0).
As to the equation of state χ = φ(p), it can determine a continuous function, or we can consider a system of a finite number of fluids differing in their physical properties superimposed on each other. For each layer, then, a particular equation of state will hold. A fluid for which no assumption such as χ = const. or χ = φ(p) is made can be also considered. Then a continuous stratification is postulated with a density expressed in terms of coordinates. For such fluids the conditions (1.1.4) and (1.1.5) do not always hold and, therefore, the discussion must begin with the equations (1.1.3).
In problems concerning the figures of equilibrium mentioned before, the stratification of an inhomogeneous fluid is not known in advance and is to be found. The figure of such a fluid is determined by the shape of the outer layer. Of course it is not always necessary to follow this direct way. One can make assumptions about the shape of the boundary surface and postulate certain stratification. Then, it must be verified that all other conditions are not violated by the assumptions made.
Volterra has proved that in case of equilibrium the layers of equal density cannot have the form of a surface of the second degree, and this proof was generalized for a zonal rotation (see, for example, Jardetzky,). However, when an approximate solution is considered, for example, for a slow rotation for which we know that the surfaces of equal density will differ but little from a spherical shape, an ellipsoidal stratification presents a certain degree of approximation.
It is usually assumed that in a figure of equilibrium the density is increasing towards its centrum. For example, the mass distribution in the Earth has been represented by the well-known laws of Levy, Lipschitz, Roche, and some others of a similar type.
We shall mention one more assumption, namely, that in the case of an inhomogeneous fluid the surfaces of equal density form a set of closed surfaces each enclosing the preceding one.
1.3. Density Distribution
In the investigations concerning the figures of equilibrium, it is usually assumed that the density is an increasing function of the distance from the surface of the body. However, for one case, at least, a proof of this fact can be given. Having demonstrated that a sphere is a unique figure of equilibrium of an isolated fluid mass at rest, Liapounov [10] had also shown that the density will increase from the surface toward the center because of the conditions of equilibrium. His proof is as follows.
Let A be the radius of a sphere which is the boundary surface of a fluid mass at rest and x the density. It is obvious that if an inhomogeneous fluid mass is in equilibrium, it has to be composed of concentric spherical layers. In general, the density can be a function of a parameter α. However, if the fluid is compressible and x = φ(p) it may be shown that x = ψ(α) cannot be an arbitrary function. By equation (1.1.9) we can put on a level surface
(1.3.1)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
It follows then by (1.2.4) and (1.1.5) that
(1.3.2)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
if ω = 0 and α0 corresponds to p = 0, i.e., to the free surface of the fluid. Thus, we obtain from this equation p = p (α) and, on inserting this function in the equation of state, it is x= ψ(α). In order to prove that the density is a function increasing toward the center, we shall transform the integral (1.3.1). Let UM be the potential at a point M (x, y, z) at a distance r from the center O of the fluid mass and the element χ'dV' taken at a point M' (x', y', z') at a distance r'. Taking the polar axis along OM, we assume that the angle θ corresponds to the colatitude and λ to the longitude. Let [theta'] be the angle between r and r', dσ' the element of the sphere having unit radius. Then d?' = sin θσ' = sin θ' dθ' dλ and D = √r2 + r'2 - 2rr' cos θ'. Now we can write
(1.3.3)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
since the volume element dV' at the distance r' is r'2dσ'dr'.
(Continues...)
Excerpted from Theories of Figures of Celestial Bodies by Wenceslas S. Jardetzky. Copyright © 2005 Dover Publications, Inc.. Excerpted by permission of Dover Publications, Inc..
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Table of Contents
Foreword by Otto StruvePart I. Figures of Equilibrium of a Rotating Fluid Mass
I. Introduction
II. Figures of Equilibrium; Inverse Method
III. Method of Poincaré
IV. Method of Liapounov
V. Lichtenstein’s Investigations
VI. Method of Wavre
Part II. Other Invariable or Varying Figures
VII. Zonal Rotation
VII. Varying Figures
IX. Systems Composed of Fluid and Rigid Parts
X. Fluid Mass and Centers of Attraction
XI. Figures of Compressible Masses
Bibliography
Index