Studies in Hydrodynamics and Structure of Stars and Planets
From the first pages of
Jeremi Wasiutyński's doctoral dissertation
497 pages, 62 figures
Volume 4 in the series Astrophysica Norvegica
printed by Det norske Videnskaps-Akademi
Oslo 1946
Preface
Physical hydrodynamics is a young branch of science, and its application has been, as yet, chiefly confined to meteorological problems. In more recent times, however, the need for hydrodynamical research has become urgent also in astrophysics. In particular, the problem of the structure of stars and planets cannot be separated from the problem of their internal motions.
This fact does not always seem to have been realized by students. It is true that convection in stars has been given increased attention during the last two decades, but the majority of investigations on the subject have been concerned merely with a special type of convection due to gravitational instability, and confined to the influence of convection on the fields of temperature and pressure, and on the star as a whole, leaving the velocity field out of consideration.
These researches may be quite important. Thus, convection in central cores of the stars, demanded by our ideas on the source of stellar energy, decreases the compressibility of the stars, and affects their radii and luminosities. The question whether the stars may possess extensive outer convective layers, due to gravitational instability, is of primary importance for the problem of stellar evolution and of the influence of photospheric conditions on the equilibrium of the interior. But convection in stars and planets may also have many other causes, such as inertial or shearing instability, or the lack of thermal equilibrium – and all these are connected with rotation. The problem of rotation, in turn, is organically connected with that of internal circulation in general. Thus, the study of the velocity field itself cannot be excluded from investigations on the structure and evolution of stars and planets.
Besides this kind of questions, a number of hydrodynamical problems is encountered in astrophysics, such as stellar variability, solar activity and granulation, atmospherical phenomena on planets, etc. Even the surface features of planets, as e. g. the lunar craters, the Martian canals, and the terrestrial mountains, appear to be determined by hydrodynamical processes. In these circumstances it may really seem astonishing how little has been done in this field in comparison to what was achieved in applying atomic physics to astronomy.
The present work is devoted to problems of the kind specified above, and other allied problems. Although this is not a systematical treatise on astrohydrodynamics, it has in this study been considered useful, on suitable occasions, to review the work done on these and related subjects by other students, especially the results obtained by hydrodynamicists and theoretical meteorologists, which appear to be little known by astronomers. It is hoped that the present study will bring to light the importance and potentialities of hydrodynamical research in astronomy. As examples may serve the theories of lunar craters, of planetary orogenesis, of the atmospherical circulation on the major planets, and especially the new results and points of view in solar hydrodynamics, which have already proved useful in the physics of the stars in general, viz. in the theory of giant stars and the novae.
The order of presentation of the subject matter in this study does not correspond to the succession in which different sections were worked out. Chapter 7 and a large part of Chapter 8 were substantially elaborated as far back as 1938/39\[1]. A shorter version of this work was completed in 1942, but war conditions prevented its publication. In 1944/45 the manuscript was revised, and substantial additions were made to Chapters 3, 5, and 8.
In concluding, it is a pleasant duty to express my deep gratitude to Professor Svein Rosseland, for the hospitality he offered me at the Institute for Theoretical Astrophysics during four years devoted to the present studies, and for much valuable advice and many inspiring discussions. My acknowledgments are also due to Professor Bengt Strömgren for kindly having read a large part of the manuscript in its earlier version; to Dr. Einar Høiland, at the Institute for Theoretical Astrophysics, for many valuable discussions; to Professor Bertil Lindblad, for friendly interest in my work; and to Professor Halvor Solberg, who kindly advised me as to the literature on turbulence. I am also indebted to Professor K. Barth and Professor O. Holtedahl for discussions concerning the theory of lunar craters and terrestrial orogenesis, and to Professor V. M. Goldschmidt and Professor W. Werenskiold for advice concerning the literature on the stone polygons.
My best thanks are due to the Norwegian Academy of Sciences and Letters which bears the major part of the expenses connected with the publication of the present work. My acknowledgments are also due to other contributors: Mr. Egil Ekko, Professor Egil A. Hylleraas, and the Editorial Committee of “Fra fysikkens verden”. I wish particularly to thank Professor Hylleraas and Professor Bengt Strömgren for their kind interest in the publication of this study.
Oslo, September 1945. J. W.
Contents
Abstract
Introduction
Chapter 1 Turbulence in Stars
§ I. Introduction
1.1. Turbulent Motions in Nature
1.2. The Problem of Stability of Flow
1.3. The Problems of Developed Turbulence
§ 2. The Theory of the Mixing Length
2.1. The Mean Values Determining the Mechanical and Thermal Effects of Turbulence
2.2. Taylor-Schmidt-Prandtl Theory of the Mixing Length
2.3. The Case of Rotating Fluid
2.4. Generalization of the Theory of Turbulent Friction to Three Dimensions
2.5. Generalization of the Theory of Turbulent Heat Transport to Three Dimensions
2.6. Interpretation of the Mixing Length
§ 3. Turbulent Viscosity by Rotation in Three Dimensions
3.1. General Formulae
3. 2. Cylindrical Rotation
3. 3. Rotational Evolution of Stars Embedded in Nebulae
§ 4. Heat Effects of Turbulence
4.1. The Flow of Thermal Energy
4. 2. Deviations from Adiabacy
4.3. The Work of Pressure on Turbulent Expansions
4. 4. The Equation of Heat in the Stellar Interior
§ 5. The Equation of Turbulent Energy
5. 1. The Scale of Turbulence and the Criterion of Turbulence
5.2. The Equation of Turbulent Energy
5.3. Criteria of Turbulence by Parallel Flow and by Cylindrical Rotation
5.4. Criterion of Turbulence and Distribution of Eddy Energy in the Stars
Chapter 2. Large-Scale Currents in Stars. Historical Survey and Generalities
§ 6. The Criterion of Mechanical Stability
6. 1. Schwarzschild’s and Rayleigh’s Criteria
6. 2. The Solberg-Høiland Criteria
6.3. Geometrical Interpretation of the Solberg-Høiland Criteria
§ 7. Historical Survey of the Ideas on Convection in Mechanically Unstable Regions of the Stellar Interior
7.1. Currents due to Gravitational Instability
7.2. Currents due to Rotational Instability in Homogeneous Incompressible Fluids
§ 8. Convection Currents in Mechanically Stable Layers
8.1. Convection Currents by Uniform Rotation and by Cylindrical Rotation
8. 2. Convection Currents by Arbitrary Law of Rotation
8.3. Analytical Treatment of the Problem of Convection in Mechanically Stable Layers
§ 9. The Flow by Vanishing Friction Force
9. 1. The Case of Pure Rotation
9. 2. Potential Streaming in Stars
Chapter 3. The Hydrodynamics of Solar Activity and Stellar Variability
§ 10. General Outlines of a Theory of Solar Activity
10.1. V. Bjerknes’ and Rosseland’s Theories of Sunspots
10. 2. Generation of Sunspots by Turbulence in a Gravitationally Stable Layer
10. 3. Explanation of the Zones of Sunspots and Prominences, and of their Movements in Latitude
10.4. The New Explanation of Sunspots Confronted with Further Observational Facts
§ 11. The Rotation of the Sun and Inertial Instability of the Sunspot-Layer at Low Latitudes
11.1. The Equatorial Acceleration of the Sun at Low Latitudes (in the Sunspot Zones)
11.2. The Rotation Law of the Deep Interior of the Sun
11.3. The Inertial Instability of the Sunspot-Layer
11.4. The Discontinuity of Opacity at the Inner Boundary of the Spot-Layer
§ 12. The Meridional Currents of Constant Direction and Turbulence in the Outer Unstable Layer of the Sun
12. 1. The Tuominen Currents
12.2. The Eddington Currents
12.3. Turbulence at High Latitudes (outside the Spot Zones) in the Unstable Layer of the Sun
12.4. The Equatorial Acceleration of the Sun at High Latitudes (outside the Spot Zones)
12.5. Comparison with Observation
12.6. Approximate Adiabacy of the Unstable Layer and Frequency of Gravoido-Inertial Oscillations of the Layer
12. 7. Dependence of Turbulence at High Latitudes on the Inertial Instability at Low Latitudes
12. 8. The Cause of Cyclical Activity
12:9. The Variation of the Equatorial Acceleration during the Activity Cycle.
§ 13. Solar Activity and Stellar Variability
13.1. The Origin of Faculae
13. 2. The Effect of Turbulence in the Outer Layer on the Sun as a Whole
13.3. The Cepheid- and Long-Period Variability
13.4. Secondary Waves. Irregular Variability
13.5. The Novae
Chapter 4. Convection Currents of the Bénard-Rayleigh and Allied Types, Especially in the Outer Layers of the Sun
§ 14. Historical Survey of the Work on Solar Granulation
14. 1. Visual Observation
14. 2. The Photospheric Net
14.3. Brightness and Area of the Granules
14. 4. The Life-Time and the Motions of the Granules
§ 15. Historical Survey of the Work on the Hydrogen Convection Layer
15. 1. The Origin of Granulation
15. 2. The Hydrogen Convection Layer on Other Stars
15. 3. The Granules are not “Turbulent Elements”
15. 4. Thermal Instability and the Nova-Phenomenon
§ 16. Historical Survey of the Work on the Bénard-Rayleigh Convection Currents
16.1. Bénard’s Experiments
16.2. Rayleigh’s Theoretical Treatment of the Problem
16. 3. Later Theoretical Investigations
16.4. Bénard’s Circulation of Turbulent Character
16.5. Later Experimental Investigations
16.6. Schmidt-Saunders and Chandra Régimes of Convection
§ 17. Interpretation of the Granular Forms of the Solar Photosphere
17. 1. The Relation of the Granules to the Vortices of Bénard
17. 2. Determination of the Thickness of the Convective Layer from the Diameter of a Granular Polygon
17.3. Computation of the Stream Velocity in a Granular Cell from the Contrast of Surface-Brightness
17.4. The Motions of the Granules. The Chains of Granules
17. 5. Tentative Explanation of the Photospheric Net
17.6. Explanation of the “Willow-Blades” and Other Similar Granular Forms
17.7. Occurrence in the Photosphere of the Chandra Régime of Convection
17. 8. The Cellular Structure of Faculae and Spots
17.9. Possible Cause of the Decay of the Granules
17. 10. The Value of Janssen’s Work on Granulation
§ 18. The Chromospheric Net and Chromospheric Granulations
18.1. The Chromospheric Calcium Net
18.2. The Hydrogen Net and the Contours of Filaments
18.3. The Minute Structure of the Flocculi
18.4. The Chromospheric Circulation
Chapter 5. Orogenic Currents in Planets
§ 19 The Origin of the Surface Features of the Moon
19. 1. Polygonal Forms on the Moon and Puiseux’s Net
19.2. Convection Currents in the Liquid Moon and the Formation of the Maria
19. 3. The Formation of Puiseux’s Net and of the Lunar Craters
19. 4. Explanation of Diverse Observational Facts
19. 5. The Internal Structure of the Moon
19.6. The Temperature Gradient Necessary to Maintain a Circulation of the Velocity Required
§ 20. The Stone Polygons of the Terrestrial Subpolar Regions and Their Analogy to Lunar Craters
20.1. The Stone Polygons and Their Explanation
20. 2. Analogy with Lunar Craters
§ 21. The Evolution of the Surface Features of the Earth and Mars
21. 1. Mountain-building on the Earth
21. 2. The Convectional Theory of Mountain-building
21.3. The Evolution of Terrestrial Diastrophism
21. 4. The Origin of Tethys
21.5. The Origin of the Surface Features of Mars
§ 22. Sundry Mathematical Problems concerning Planetary Orogenesis
22. 1. The Cooling of a Planetary, Highly Conductive Core Surrounded by a Liquid Shell
22. 2. Convection Currents in a Liquid Planetary Shell: The Equations of the Problem
22.3. Expansion of the Coefficients of the Solution in a Power Series
22.4. Expansion of the Coefficients of the Solution in a Fourier Series
22. 5. The Boundary Problem. Solution in a Special Case
Chapter 6. Currents in Planetary Atmospheres and Stellar Envelopes
§ 23 Stable Atmospheres. Historical Introduction to the Theory of Unstable Atmospheres
23. 1. The General Circulation in the Atmosphere of the Earth
23. 2. The Atmospheric Circulation of Venus
23. 3. Historical Survey of the Work on Convection in Spherical Layers of Fluid
§ 24. Steady Flow in a Spherical Rotating Layer of Gas, Heated from Inside and Attracted towards the Centre
24. 1. Equations of the Problem
24. 2. Solution of the Fundamental Equations of the Problem
24. 3. Characteristic Equation for the Temperature Gradient
24. 4. Convergency of the Series of Legendre Polynomials
24.5. Numerical Solutions in the Case a << 1
24. 6. Stream-Function, Velocity of Rotation, Temperature, and Pressure
24. 7. Slowly Rotating Spherical Layer of Finite Thickness
24. 8. Effect of Rotation on Latitudinal Cellular Division
24. 9. Thin Spherical Layer of Gas and the Bénard-Rayleigh Problem
§ 25. The General Circulation in the Atmospheres of the Major Planets
25. 1. Observational Facts
25. 2. The Explanation Proposed
25. 3. The Hydrogen Envelopes and Fluidity of the Major Planets
Chapter 7. Gravitational Instability in Stars
§ 26. Instability of Radiative Equilibrium in Stars
26. 1. The Criterion of Stability
26. 2. Generalities on the Stability of Stars in Radiative Equilibrium
26. 3. The Case when there is Radiative Equilibrium in the Whole Region between the Layer under Consideration and the Boundary of the Star
26. 4. A Class of Stellar Models
26.5. Stability at the Centre of Arbitrary Models
26. 6. Density Minimum at the Centre of a Model Star
§ 27. The Equilibrium of a Star by Arbitrary Boundary Conditions
27. 1. Stability of Radiative Equilibrium by Arbitrary Boundary Conditions
27. 2. The Bearing of the Boundary Conditions on the Equilibrium of the Stellar Interior
27. 3. The Problem of the Outer Convective Layer
§ 28. Adiabatic Equilibrium in Stars
28. 1. Polytropes by Finite Radiation Pressure
28. 2. Adiabates of a Mixture of Ionizing Elements by Considerable Hydrogen- or Helium Content
Chapter 8. Stellar Structure and Evolution
§ 29. The Possible Stellar Models
29.1. The Classification of Stellar Models
29. 2. Physical Conditions Determining the Stellar Model
29. 3. The Helium Synthesis, Convection Currents, and the Evolution of the Structure of a Star
§ 30. The Stellar Model R R
30. 1. The Structure of the Convective Core
30. 2. The Source-Free Region in Radiative Equilibrium
30. 3. Determination of the Boundary of the Core
30. 4. Generalization of Tuominen’s Method
30.5. Application of Jeans’ Method
30. 6. The Boundary Criterion
30. 7. The Outer Layers of the Star
30. 8. Mass, Radius, and Luminosity of the R R-Model
30. 9. The Effect of Difference of Molecular Weight in the Core and in the Outside Region
30. l0. The Rate of Liberation of Energy in the Core
30. 11. A Numerical Example: The R R-Model with Jc = 0.25
§ 31. The Stellar Model C C
31.1. Formulation of the Problem
31. 2. The Case of an Outer Layer of Pure Hydrogen
31. 3. The Case of a Composite Hydrogen-Helium Layer
31.4. Approximate Treatment of the Case of a Mixture in Adiabatic Equilibrium by Considerable Hydrogen- or Helium Content
31.5. Approximate Treatment of the General Case of a Mixture in Adiabatic Equilibrium
31. 6. Exact Treatment of the General Case of a Mixture in Adiabatic Equilibrium
31. 7. The K1, K5- Relation
31.8. Instability in Hydrogen- and Helium Layers below the Levels of Ionization
31. 9. Instability in the Main Bulk of the Star
31. 10. The M-L-R Relation for Wholly Convective Stars
31. 11. The Structure of Convective Stars with High Content of the Heaviest Elements
§ 32. The Stellar Model R C
32.1. The Coefficient of Opacity of Highly Ionized Hydrogen and Helium
32. 2. The Structure of the Hydrogen Envelope in Radiative Equilibrium
32. 3. The Structure of the Helium Layer
32.4. Instability in the Main Mass of the Star. Scattering in the Hydrogen Envelope
32. 5. The Radius of the Model R C
32. 6. Giant Stars
§ 33. Stellar Evolution
33. 1. Evolution through Homologous Configurations
33. 2. Evolution through Non-Homologous Configurations
33. 3. The Evolution of the R R-Model with Finite Radiation Pressure
33. 4. The Evolution of the Hydrogen-Helium Convective Model
33. 5. Comparison with Observation. The Sub-Giants in Binary Systems
§ 34. The Collapse of a Wholly Convective Star
34. 1. General Features of the Collapsing Configuration
34. 2. Numerical Estimate of Luminosity and Radius
34.3. Numerical Estimate of the Time Scale
34.4. The Stellar Model with an Isothermal Core
Bibliography
Abstract
In Chapter 1, mechanical and thermal effects of turbulence in stars are studied on the authority of a physical interpretation of the notion of the mixing length. Formulae for the Reynolds stresses, as well as equations of heat and of turbulent energy are deduced, and a generalized criterion of turbulence is set forth. It is shown that the correctness of Richardson’s criterion of turbulence depends on the vortical character of turbulent motion. The spatial distribution of eddy energy in stars and the effects of turbulence on stellar rotation are discussed.
In Chapter 2, the Solberg-Høiland general criteria of stability of equilibrium of a fluid rotating about a fixed axis, are rapidly deduced. Theoretical and experimental work on currents in rotating fluids due to the lack of thermal equilibrium or to instability, is reviewed in short. It is shown that by absence of meridional currents the angular velocity of rotation of a body in radiative or conductive equilibrium is determined by the mass distribution and boundary conditions. If angular velocity does not fulfil the corresponding condition, meridional currents must arise in the fluid. Internal motions in stars by vanishing friction force are studied.
Chapter 3 is devoted to the hydrodynamics of solar activity and stellar variability, and related problems. Bjerknes’ theory of solar activity is critically reviewed. A turbulence-theory of solar activity and a theory of internal structure of the sun are set forth. Sunspots are considered as traces at the level of the photosphere of deep-seated eddies; they are probably due not to the alleged pumping power of the eddies but to their magnetic fields. The eddies cannot arise in consequence of gravitational instability only, but must depend on instability of rotation – shearing or inertial – i. e. on spatial variation of angular velocity. It is pointed out that the convective core of the sun should rotate approximately as a rigid body, and that in the main bulk of the sun (or, generally, of a star), which is in radiative equilibrium, the angular velocity may be approximately a function of the distance from the sun’s (or the star’s) centre only. Rotational instability in the outer layer of the sun is not likely to arise, unless opacity abruptly changes at the inner boundary of the layer. Such an abrupt change or “discontinuity” of opacity may be due to chemical stratification of the solar matter and partially, perhaps, to ionization. The discontinuity of opacity may involve inertial instability at low latitudes. The angular momentum ωR2 at these latitudes must then be independent of the distance R from the axis of rotation. A law of equatorial acceleration follows, which is in fair accordance with the observations of the angular velocity of rotation in the whole spot zone, viz. up to 45° heliographic latitude, if the thickness of the layer is assumed equal 1/18 of the sun’s radius. It is concluded that the spots are due to violent turbulence arising by inertial instability. The angular velocity of rotation at the bottom of the layer is determined.
The rotational shear, from which inertial instability at low latitudes follows, is periodically smoothed out by turbulence and built anew by convection currents due to lacking thermal equilibrium (to the violation of a generalized von Zeipel condition). It is pointed out that instability is most rapidly built at mean latitudes, and consequently, that it is there turbulence must start at the beginning of an activity cycle. In the measure as the rotational shear is smoothed out at mean latitudes, and instability built in the adjoining belts, a wave of turbulent motion proceeds towards the equator, producing the known effect of a moving spot zone. Another, but weaker wave of turbulence due to shearing instability proceeds towards the pole, just as the high-latitude zone of prominences does. On the basis of these considerations a number of observational facts is explained; thus, the proper motions of leader spots, and the form of the curve of Wolf’s relative numbers with its secondary maxima and its anomalies. It is pointed out that the coexistence of two sunspot belts on each hemisphere at minimum activity must result in that currents due to inertial instability at low latitudes have opposite direction in consecutive cycles, and that, consequently, also the polarity of spots is opposite in consecutive cycles on the same hemisphere. By a similar reason, the polarity is opposite in the same cycle on different hemispheres. Besides these instability currents, other currents due to friction or lacking thermal equilibrium are flowing in the unstable layer. Their direction is constant and points from equator to pole at the surface of the sun at mean latitudes. Narrow belts of opposite direction of circulation may form in the nearest neighbourhood of the equator and the poles. It is conjectured that the circulation is identical with the circulation found by Tuominen from a statistical study of the motions of sunspots in latitude. The velocity of this circulation is probably less than the velocity of the instability currents, and in the second half of the activity cycle it is decreased by the previously mentioned currents (due to lacking thermal equilibrium) which rebuild the instability. These last currents have opposite direction, viz. from pole to equator at the sun’s surface. It is pointed out that differences between the curves of Wolf’s relative numbers for odd and even activity cycles can be explained by superposition of the currents of constant direction upon the instability currents.
The criterion of turbulence deduced in Chapter 1 is applied to the high-latitude zone of the unstable layer, and a law of equatorial acceleration in high latitudes is deduced. This law is in fair agreement with the empirical law. The two laws of equatorial acceleration – at low and high latitudes – are found to fit each other at about 45° latitude, in accordance with observation. There, and at about 60° latitude, the equatorial acceleration is discontinuous, just as can be inferred from spectroscopic observations. It is shown that stationary turbulence in the unstable layer is impossible, because if it were established, rotational equilibrium would break down in a narrow belt at the common border of the two zones (of inertial and shearing instability), i. e. at about 45° latitude. It is suggested that this may be the true cause of the solar activity’s being cyclical. Turbulence at high latitudes is released by inertial instability at the border of the two zones. The turbulent exchange at high latitudes adjusts itself in the way as to make this border coincide with the parallel 45°, where instability is most rapidly built. The observed cyclical changes of angular velocity and the equatorial acceleration at the sun’s surface are detailedly interpreted. The structure of the unstable layer is shown to be practically adiabatic. The frequency of gravoido-inertial oscillations of the layer at high latitudes is determined from the coefficient of the empirical law of equatorial acceleration, and is found to coincide with the frequency of certain magnetic disturbances of apparently solar origin. The zones of faculae are tentatively explained by shearing instability of circulation in a relatively thin stratum of the unstable layer, where the temperature gradient is super-adiabatic, perhaps because of the ionization of helium. Currents in such a stratum have tendency to form a system of zonal cells.
In connection with this theory, an explanation of the supposed fluctuations of the solar radius and luminosity is proposed. A theory of stellar variability is sketched, founded on the model of a giant star (the “RC-model”) which has been suggested by the above study in hydrodynamics of solar activity, and investigated in Chapter 8. On the working hypothesis that the luminosity of a variable star increases with the intensity of turbulence in its outer layer (composed mainly of hydrogen and helium, and corresponding to the layer of sunspot-turbulence in the sun), the character of the light-variation and the variation of radial velocity of Cepheid, long-period, and irregular variables is explained. Such phenomena as secondary waves and flat minima of the light curves are included in the explanation. An attempt of explaining the phenomenon of a nova from the dynamics of the hypothetical outer hydrogen-helium layer of the star is made. It is suggested that the light outburst is due to radiation pressure’s sudden blowing up of the hydrogen-helium layer. It is shown that a suddenly released turbulence at the surface of chemical discontinuity (i. e. at the inner boundary of the hydrogen-helium layer) may cause an enormous increase of opacity and radiation pressure on the outside of this surface, sufficient (under certain not unplausible assumptions) for the blowing up of the outer layer, and adequate to the radial velocities observed.
Chapter 4 is chiefly devoted to granulation. The granules are considered as the products of subdivision of convection cells of the Bénard type in the layer of hydrogen ionization. The experimental and theoretical work done on the problem of convection currents of the Bénard-Rayleigh and allied types, as well as the work concerning granulation and the hydrogen convection layer in stars, are passed in review. The depth of the unstable layer is computed from the diameter of a cell, or a group of granules, and the stream velocity in a cell is determined from the contrast of surface brightness of the granules against their background. It is pointed out that many abnormal granular forms are unduly regarded as results of the spoliation of the photographic image by air turbulence. Their true cause may be either an abnormal local decrease of the temperature gradient in the hydrogen convection layer of the sun, or horizontal streaming in this layer. This is shown by laboratory experiments in which a multitude of abnormal granular forms were reproduced. Thus, the study of granulation gives us an important method of investigation of the motions in the outer layer of the sun. Especially the “photospheric net” appears to be a solar phenomenon due to local and temporary circulation in a thin stratum of the spot-generating layer (perhaps the same in which turbulence giving rise to the faculae takes place). Remarks are added on the cellular structure of spots and the possible cause of decay of the granules, and observational evidence for cellular circulations in the chromosphere is compiled.
In Chapter 5 it is attempted to show that various features of planetary crusts are due to hydrodynamical processes in the interior of the planet, before or after solidification. The origin of lunar “craters” and “maria” and of Puiseux’s net of rilles and dykes is studied first. It is shown that the existing theories of lunar craters are inadequate to explain the observational facts. On the other hand, these facts are naturally and in detail explained, if we assume that the surface features of the moon are conserved from the very time of the formation of a solid crust, and are secondary products of convectional circulation in the magma. We need only assume that on the moon, as on the earth, a relatively thin and light (granitic) superficial layer was formed (and partially solidified) on the surface of a still fluid basic magma which was the seat of circulation. Just before the solidification of the magma the circulation necessarily assumed the regular cellular régime investigated by Bénard, giving thus rise to regularly spaced centres of higher temperature and less viscosity. By solidification, contractional cracks were formed at the peripheries of the cells, partially conserved until now in Puiseux’s net. The last upheavals of the magma in the central parts of the cells were associated with exceptionally high stresses on the overlying granitic stratum, which was consequently drawn down into the magma near the peripheries of the cells. It is shown that the heat flux in the outer layer of the moon was sufficient to produce revolutions in the magma of the velocity required. The crater-bulwarks were formed from the granitic intrusions after the solidification of the magma, as the result of isostatic adjustment. The theory entails a number of consequences which are in accordance with observation. Thus, the height of the crater-bulwarks against the surrounding plateau can be predicted theoretically. The depth of the basaltic layer on the moon appears to be of the same order as on the earth. Statistical study of the diameters of the craters and maria may allow conclusions about structural discontinuities in the interior of the moon. In connection with the above theory of lunar craters, the geological phenomenon of the “stone-polygons” of the subpolar regions is discussed. The phenomenon is explained by instability of stratification of the muddy ground early in the subpolar summer. It is suggested that the instability may be due to the water content of the ground increasing with depth because of the melting of the tjæle-ice at the bottom of the layer of mud.
In continuation of the theory of planetary surface features, the mountain-building on the earth and associated geological processes, as well as the origin of the martian “canals” are dealt with. The hypothesis that mountain-building (orogenesis) is mainly due to convection currents in the rocky shell of the earth is adopted, and three different kinds of orogenic currents are considered: 1) cellular currents due to vertical instability of stratification and giving rise to intra-continental orogens; 2) subcrustal currents directed from the poles to the equator, due to a secularly increasing field of isosteric-isobaric solenoids and giving rise to the Tethydic (Mediterranean) orogen; 3) currents at the continental margins, due to the differences in temperature distribution below the continents and the oceans, and giving rise to marginal orogens, such as those of the Rockies and Andes. It is inferred from geological records that the type of orogenic processes on the earth continuously changed in the course of geological time: 1) the Archaean orogens seem to have shifted continuously over the whole surface of the earth, suggesting pronounced instability of vertical stratification in the shell; 2) in the Proterozoic and early Palaeozoic the displacements of the intra-continental orogens were but very slow, suggesting that the rocky shell was stabilized already at the beginning of this epoch; the end of the epoch is marked by the decline of intra-continental orogens, suggesting final solidification of the rocky shell; 3) the post-Variscian crustal processes seem to have been dominated by the fissuring of the primary continent along the solidifying meridional mobile belts associated with the Tethys (viz. the East-African and the Atlantic mobile belts), and by the following disintegration of this continent by subcrustal currents, as it has been reconstructed by A. Wegener. From the above facts it is inferred that the entire shell of the earth, with the exception of a thin crust, was liquid during the whole Archaean era. Its solidification began at the bottom and proceeded upwards; it was probably not accomplished before the Caledonian or even Variscian times. Geological facts supporting this view are surveyed, and the physical plausibility of such a course of thermal and hydrodynamical evolution of the earth is demonstrated. It is shown, that the flow of heat from the metallic core of the earth must have maintained convection in the shell during a time of the order of 109 years (i. e. possibly during the whole Archaean, as postulated above), by not unplausible assumptions concerning the conductivity of the core. Also, that the solidification of the shell must have lasted for a time of the same order of magnitude, in accordance again with the geological evidence discussed above. It is suggested that the Proterozoic and early Palaeozoic “paroxisms” of mountain-building were due to breakdowns of instability caused by the extrusion of gases by solidification at the bottom of the shell (a process analogous to that responsible for the formation of the lunar craters and the terrestrial stone-polygons). In the last phase of solidification of the shell convection seems to have proceeded only above the level of the density discontinuity now existing at ca. 475 km’s depth. It is suggested that the African net of swells and basins discovered by Krenkel might be one of the few still existing traces of this cellular circulation, forming counterpart of Puiseux’s net on the moon. Later “paroxisms” or “revolutions” were probably due to plastic flow of solidified rocks caused by a continually growing solenoid field (i. e. horizontal temperature gradient) at the continental margins and between the poles and the equator. This flow was periodic, because 1) it was started whenever the stress-differences due to growing temperature gradient overcome the strength of the rocks; 2) its initial velocity was greater than that corresponding to steady flow. The plastic currents preceding each revolution would bring masses from the bottom of the shell to its top; the temperature gradient in the solidified shell being super-adiabatic, this would each time result in a rise of subcrustal temperature and might be responsible for the melting of the basaltic layer and the volcanism following each revolution, according to geological evidence.
It is pointed out that a rotating planet cannot be in thermal equilibrium, but that zonal temperature differences must arise in it, the temperature being either maximum at the poles and minimum at the equator of a level surface, or vice versa. The rate of growth of such temperature differences is generally too low to be responsible for the formation of an equatorial orogen, such as the Tethys probably was. However, abnormally large vertical temperature gradients arising below the crust after each turn-over of the material of the shell would be associated with abnormally large zonal temperature perturbations; and these perturbations would be apparently sufficient to cause the plastic subcrustal flow of rocks from the poles to the equator which seems to have given rise to the Tethys.
The cellular flow in a liquid planetary shell is mathematically investigated. The “canals” of Mars are explained as a net of orogens in the phase between two paroxisms of mountain-building. The poor regularity of the pattern of terrestrial intra-continental orogens of Proterozoic times as compared with the pattern of martian canals is probably due to the disturbing influence of marginal orogens which do not exist on Mars. The belt of martian “maria” is interpreted as the counterpart of the terrestrial Tethys.
In Chapter 6, currents in planetary atmospheres in general, and in the outer layers of stars if gravitationally unstable, are investigated and discussed. The general circulation of the atmosphere of the earth is explained on the lines of the general theory of rotation and meridional circulation of a gravitationally stable gaseous envelope of a heavenly body. Stability conditions and currents in the atmosphere of Venus are discussed. The observational evidence seems to be contradictory, some reports making gravitational instability of the troposphere of Venus probable, others pointing towards conditions similar to those on the earth.
The mathematical problem of Bénard-Rayleigh convection currents is generalized to spherical rotating layers of gas heated from within and attracted towards the centre. Approximate solutions for rotationally symmetric modes of flow of higher orders (i. e. corresponding to large eigen values of the temperature gradient) are given and computed numerically in the case of a rapidly rotating thin layer. The problem of convection currents in a spherical layer of gas of finite thickness is investigated in the case of vanishing rotation, and the fundamental equation of the problem is established. The theory is applied to the general circulation in the atmospheres of the major planets. The belts of Jupiter and Saturn are explained as formations of clouds (of ammonia or some other compound of similar behaviour) in zonal cells of atmospheric circulation. The situation of Jupiter’s belts is predicted by the theory with fair accuracy. Various phenomena observed on Jupiter find a natural explanation in periodic turbulent break-downs of the general circulation, and are detailedly discussed. It is pointed out that the explanation holds good only if the depth of Jupiter’s troposphere is of the order of 20 000 km, and the temperature at its bottom is of the order of several thousands degrees. Consequently the whole planet is probably fluid, and the circulation is maintained by heat flow from the planet’s interior. This is consistent with radiometric and spectroscopic determinations of Jupiter’s effective temperature. Similar conclusions are to be made concerning the internal state of Saturn and probably also the other major planets. It is conjectured that the tropospheres of these planets, i. e. the layers of belted circulation, are composed mostly of hydrogen and are the counterpart of the hydrogen layers of stars, whose existence has been inferred from the theory of equatorial acceleration of the sun, from the phenomenon of the novae and the existence of giant stars (cf. below), as well as other observational facts and theoretical considerations.
In Chapter 7 the conditions of stability of radiative equilibrium in stars are discussed. The limitations of the Solberg-Høiland stability criteria are pointed out, and gravitational stability is investigated in detail. A criterion for the stability at the centre of a model star is established. The dependence of stellar structure on conditions in a hypothetical outer layer composed mainly of hydrogen and helium is discussed. Formulae are deduced for adiabatic equilibrium by finite radiation pressure and complete ionization, as well as the corresponding approximate formulae for incomplete ionization.
Chapter 8 is devoted to stellar structure and evolution. Well-known theoretical considerations on the distribution of hydrogen in stars suggest that the concentration of hydrogen may be much higher in the outer layer of a star than in its interior. This is consistent with the conclusion arrived at in Chapter 3, that the outer layer of the sun, 1/18 of solar radius in thickness, has different constant of opacity than the material of the deeper interior of the sun. It is assumed that all stars have similar outer layers composed mainly of hydrogen (and possibly also helium). Physical conditions in the photosphere and small differences of chemical composition of the outer layer may be decisive for stellar structure. Four stellar models appear to be possible, all of which are convective and adiabatic in the central regions. They are labelled by pairs of letters R and C, – R referring to radiative equilibrium, C to convective and adiabatic. The first letter in a pair refers to the kind of equilibrium in the hydrogen-helium layer, the second to that just below this layer. The RR- and the CR-model have almost identical luminosities and radii for given masses; for small radiation pressure they have the same luminosities and radii as Cowling’s model. The chief difference between them is that the CR-model is “active” (the model of the sun) while the RR-model is “inactive”. Most stars of the main sequence can be probably represented by these models. The CC-model is convective throughout, except perhaps in the inner part of the region intermediate between the H-He layer and the core. This may be the model of a red dwarf. Finally, the RC-model can be used to represent giant, sub-giant, and super-giant stars.
These stellar models are investigated in detail, beginning with the RR-model (in general characteristics and in the structure of the deep interior identical with the CR-model). The model consists of a convective core and an outer source-free region in radiative equilibrium. Expansions of the structural variables in the core in power series of the distance from the centre are given, and various methods of approximate determination of the boundary of the core are developed. The effect of difference in the molecular weight of the core and the outer region is considered, and it is shown that, by equal central temperature and pressure, a diminution of the molecular weight in the outer region (due, e. g., to imperfect mixing of matter in this region) does not change the radius of the core, but increases the radius and mass of the star as a whole. The RR-model with a ratio J = 0.25 of radiation pressure to gas pressure at the centre is numerically computed.
In the model CC, photospheric conditions and the structure of the strata of ionization of hydrogen and helium control the luminosity and the radius of the star. It is shown that in spite of the apparent complications of the problem, formulae for the luminosity and the radius of the model can be established. A theory of adiabatic equilibrium of ionizing gas layers is developed. In the first place, the equations of equilibrium of a layer of ionizing hydrogen, resp. a mixture of ionizing hydrogen and helium, are integrated. In the second place, the case of a mixture of ionizing gases containing considerable amounts of hydrogen and helium, is approximately dealt with. Finally, the general equations of adiabatic equilibrium of an arbitrary mixture of ionizing gases are rigorously integrated. On the basis of this theory, the domain of convection in a CC-star is determined. Assuming adiabatic equilibrium about the level of 50 % ionization of hydrogen, the maximum temperature to which adiabatic equilibrium may extend in the H-He layer is found. This temperature proves sufficiently high for this kind of equilibrium reigning in the whole H-He layer. Under these conditions, radiative equilibrium below the H-He layer must be unstable, i. e. the star must be built on the CC-model. The domain of instability in the intermediate region of the star is also determined; it is found that, generally, it must stretch down to the convective core, i. e. the star must be convective throughout. Formulae for the luminosity and the radius of wholly convective stars of arbitrary chemical composition are established. They contain as parameters only the photospheric opacity, the mean molecular weight of the deep interior of the star, and the average numbers of free electrons per ion at the outer boundary of the intermediate region of the star and in the deep interior.
The study of the RC-model is introduced by establishing a formula for the opacity of highly ionized hydrogen. If radiative equilibrium reigns in the stratum of ionization of a layer consisting of pure hydrogen, it extends over the whole layer. If helium is stratified below hydrogen, it is generally in convective equilibrium, but if the He layer is sufficiently thick, its lower strata are in stable radiative equilibrium. If the main mass of the star contains considerable amounts of elements heavier than helium, it is in convective equilibrium throughout, provided the extension of the H layer exceeds a certain minimum. Thus, a star whose outer layer is exceptionally poor in elements heavier than helium, is built on the RC-model, and its main mass is likely to be convective throughout. It is shown that the extension of the H envelope rapidly increases with its mass, tending to infinity when the mass approaches a certain value at most equal 13 % of the total mass of the star. Stars whose envelopes approach the critical mass, have giant dimensions. Their luminosities are found to be thousands of times larger than the luminosities of RR-stars of the same mass. Moreover, the luminosities of the RC-giants increase extremely rapidly with the total mass of the star. Giants of a mass 10 times larger than the mass of the sun, have a theoretical luminosity 5 × 108 greater than the sun. Actually, the range of variation of luminosity may be reduced by the intervention of scattering of radiation in the envelope. If the mass of the envelope exceeds the critical value, it must spread to infinity. It is suggested that planetary nebulae might arise in this way.
In a section devoted to stellar evolution, the RR-model with finite radiation pressure and a purely convective model CC composed chiefly of hydrogen and helium, are considered. Both these models pass through a series of non-homologous transformations in the course of their evolution. It is shown, however, that they have the same general trend of evolution as the homologous model investigated by Gamow. The deviations from homology of the RR-model by increasing stellar mass are found to be in agreement with the observed mass-luminosity relation. The anomalous sub-giant diameters of fainter components of tight binaries are explained as the consequence of the process of fissuring which gave rise to the binary. The fainter component is supposed to have taken the lion’s share of the H envelope of the primary star, and to be built on the RC-model.
In connection with the problem of evolution of the stars, the structural changes in a wholly convective star after the exhaustion of subatomic energy sources are studied. There is at first an increase in brightness proceeding at a very slow rate. The case of a star in which the material of the core does not mix with that of the outer region is investigated. Such a star, after the exhaustion of hydrogen in the core, would settle into a new state of equilibrium, characterized by a source-free isothermal core. It is shown, contrary to what was suggested by E. Öpik, that this model would have luminosity and radius of the same order as the totally convective model.
Introduction
Most of the matter in universe appears to be divided into isolated bodies which remain in a state of hydrostatic equilibrium under the action of their own gravitation and centrifugal force of axial rotation. However, the internal equilibrium of these heavenly bodies is only approximate, and whenever their surfaces are accessible for detailed observation, they show traces of actual or former internal motions, both around the axis of rotation and in the meridional planes. The investigation of these motions forms a methodically well-defined branch of astronomical research, which may be conveniently called astrohydrodynamics.
Bodies whose outer layers are fluid and whose surfaces can be detailedly observed, such as the sun and Jupiter, exhibit régimes of internal motion which are either approximately steady or periodic. There is reason to believe that this is the case with all heavenly bodies which have persisted in the fluid state for a sufficiently long time without being undercast perturbation from outside. In fact, whatever might have originally been their internal motions, these motions must be gradually annihilated by the forces of viscosity, unless they are maintained by thermal agencies; and these agencies tend to either a steady or periodic state together with the internal distribution of matter and temperature. A fundamental astrohydrodynamical problem might be formulated thus: Consider an isolated mass (of stellar order of magnitude), of given angular momentum and chemical composition; what are its possible steady or periodic states (secular changes due to exhaustion of subatomic energy sources being ignored) in respect to the spatial distribution of mass, chemical elements and temperature, and to internal motions? This general problem has not yet been solved, and we do not know whether the solution is unique or dependent upon the history of the body. Fortunately, astrohydrodynamical problems usually present themselves in a different form, e. g.: Given steady motions at the surface of a heavenly body (or even only the stream-lines at the surface, or the cellular division of stream-lines) and some structural parameters, such as the relative abundance of substances of which the body is composed; what internal motions and what structure of the body, e. g. what distribution of the composing substances, are consistent with them?
It is seen how intimately hydrodynamical and structural problems concerning heavenly bodies are interrelated. The solution of astrohydrodynamical problems of the type just defined gives usually, besides the explanation of the motion picture at the surface of the body, information about the stratification of material in its interior. Many problems dealt with in the present work can be used here as examples. Thus, the study of the solar equatorial acceleration leads to the conception of the outer hydrogen layer of the sun; the study of atmospheric phenomena on Jupiter – to a similar conception concerning the major planets. The study of the surface features of the moon discloses the structure of the lunar crust; that of the mountain ranges on the earth and martian “canals” – the state of the rocky shells of the earth (at different epochs) and Mars (actually), a. s. o. These results entail sometimes important consequences. They not only throw light on the past and future of the heavenly bodies directly concerned, but also on the nature of whole classes of cosmically important bodies. Thus, inferences concerning the hypothetical hydrogen envelopes of the sun and Jupiter mentioned above, lead to a revision of the theory of stellar structure and to an explanation of the phenomena of giant stars and the novae.
The above problems naturally present themselves in the course of our studies. We begin with preliminary considerations on turbulence in stars.
© Sissel Klokkhammer, Jeremi Wasiutyński’s literary estate.
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\[1] A preliminary note on certain results presented in Chapter 8 was published in 1940 in the Monthly Notices of the R. A. S., vol. 100.