ISSN 00380946, Solar System Research, 2015, Vol. 49, No. 2, pp. 123–138. © Pleiades Publishing, Inc., 2015. Original Russian Text © E.N. Slyuta, S.A. Voropaev, 2015, published in Astronomicheskii Vestnik, 2015, Vol. 49, No. 2, pp. 131–147.
Gravitational Deformation of Small Bodies of the Solar System: History of the Problem and Its Analytical Solution E. N. Slyuta and S. A. Voropaev Vernadsky Institute of Geochemistry and Analytical Chemistry, Russian Academy of Sciences, ul. Kosygina 19, Moscow, 119991 Russia email:
[email protected] Received March 27, 2014
Abstract—We consider in retrospect the problem of gravitational deformation of small bodies of the Solar System and the transition observed between small and planetary bodies, which is closely related to the history of concepts of the shape of the solid Earth. It has been shown that these concepts in geology and comparative planetology developed in two main competing directions—thermal and gravitational. We consider an analyt ical solution for gravitational deformation of a nonequilibrium shape of small solid bodies of the Solar System and show that the linear theory of elasticity can be applied to estimate of the value and distribution of stresses in real small bodies of various composition that have the ultimate strength and yield strength under triaxial gravitational compression. From the performed analysis, it has been found that the value and distribution of stresses depend on the chemical and mineralogical composition of the small bodies and are determined by such main parameters as the mass of a body, its density, size, shape, yield strength, and the Poisson ratio. Keywords: small bodies, asteroids, cometary nucleus, Kuiper objects, gravitational deformation DOI: 10.1134/S0038094615010086
INTRODUCTION All solid bodies of the Solar System can be divided into two main classes in terms of their external mor phologic features, i.e., depending on their shape. The first one is a class of small bodies of irregular shape, i.e., they look like fragments or boulders. Among them are small satellites of the planets, asteroids, comets, and small objects of the Kuiper belt. Due to their insufficient mass, these bodies never developed into planetary bodies. The other class comprises planetary bodies that are characterized by an equilibrium spher ical figure. Among them are the terrestrial planets, large planetary satellites, asteroids Ceres and Vesta, and large objects of the Kuiper belt. A spherical shape of solid planetary bodies is formed due to hydrostatic equilibration of the body’s surface that is known in geotectonics as the isostatic compensation mecha nism or isostasy. The equilibrium shape of a planetary body is controlled by gravity which dominates the strength properties of the material, as if the body’s material were a gravitating incompressible fluid. Thus, the density differentiation onto a crust, mantle and core is only possible in a planetary body. Up to now, the problem of the observed transition between small and planetary bodies and the depen dence of this transition on the composition, mass, and sizes of the bodies, on physical, mechanical, and rheo logical properties of the material has remained a poorly studied field in cosmochemistry and planetol ogy. Many fundamental questions on the observed
transition between such different, externally and inter nally, objects as small and planetary bodies which are, starting from a certain time strength of their evolution, characterized by different forms of organization of the material and its different evolution are still lacking an answer. For example, is the hypothesis on creep in small bodies suggested long ago valid (Lichkov, 1965; Johnson and McGetchin, 1973; Simonenko, 1979) or, on the contrary, do they possess the ultimate strength and yield strength and how do these properties depend on composition and temperature? What is the maxi mum (critical) mass of small bodies of a specified composition, which, when exceeded, leads to gravita tional deformation of the body and what is the observed minimum mass of planetary bodies with analogous composition? What are the main parame ters determining the gravitational deformation of these bodies? These and other similar questions have not been answered yet, because, first of all, the data on chemical and mineralogical composition and mor phology of small bodies of the Solar System in required quality and quantity have appeared only dur ing the last 10–15 years. Such data appeared not only due to the use of the newest technologies in the remote sensing of small bodies carried out from the Earth, but, above all, due to the investigations of these objects (comets, asteroids, small planetary satellites) per formed directly from spacecraft. The essence of the creep hypotheses was the fol lowing. In small bodies, gravitational loading in the
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form of deviatoric stresses caused by the mass and nonequilibrium figure of the bodies and responsible for deformation is constant and, in fact, has existed starting from the moment of their formation (Slyuta and Voropaev, 1992, 1997). The change in the mechanical characteristics of rocks of small bodies subjected to longterm loadings, including those below the yield strength, is determined by their rheo logical properties. The longer the exposure of the rocks to the loading, the weaker the elastic properties of rocks, the lower the yield strength, and the more manifest their plastic properties. The phenomenon of the gradual increase in rock deformation at constant stress is known as creep (Rzhevskii and Novik, 1973). Outwardly, the creep phenomenon is similar to plastic yielding; however, while the latter occurs only beyond the yield strength, the creep appears under the long term loading and at stresses that do not exceed the yield strength. The initial elastic deformations gradu ally transform into plastic ones, leading to a gradual decrease in stress, i.e., relaxation, due to creep under constant deformation. Theoretically, the equipotential surface of a strengthmass gravitational field is a sphere. Conse quently, the irregular figure of a small body subjected to creep should gradually transform into a spherical one. The larger the mass of a small body compared to other bodies of the same composition, the stronger the deviatoric stresses responsible for deformation, and the closer its shape to a sphere. Thus, if there is no yield strength in small bodies and the main mechanism of gravitational deformation of small bodies is creep, the transition between small and planetary bodies should be gradual, i.e., gradual relaxation of the figure of small bodies in dependence on time and mass should be observed (Lichkov, 1965; Johnson and McGetchin, 1973; Simonenko, 1979). The invalidity of the creep hypothesis for small bodies of the Solar System was shown by Slyuta (2013a, 2014) on the basis of a thorough analysis of the shape, mass, and sizes of numerous small silicate bod ies containing ordinary and carbonaceous chondrites, metallic asteroids, small icy bodies composed mostly of water ice, and small Kuiper objects that contain not only silicate components and water ice, but also a sub stantial amount of exotic ices of other volatiles. The observed transition between small and planetary bod ies turned out to be abrupt, rather than gradual, which is a direct consequence of the absence of creep in small bodies. All of small bodies of the Solar System regard less their composition (from icy to metallic), being under triaxial gravitational compression and gravita tional deformation, have the ultimate strength and yield strength. It was also found that small bodies of different compositions are characterized by different shapes (Slyuta, 2014). Gravity is the only force that is able to overcome the barrier of fundamental strength of a small body and to transform its irregular figure to the equilibrium,
spherical figure of a planetary body. Plastic deforma tion of small bodies occurring under gravity is known as gravitational deformation (Slyuta and Voropaev, 1997). For the viscousplastic bodies having no ulti mate strength and no yield strength, this effect results in a set of complex shapes of a selfgravitating homo geneous viscous fluid, e.g., the Maclaurin spheroids, the Jacobi ellipsoids, etc. (Chandrasekhar, 1969). Gravitational deformation of the irregular figure of a small solid elastic body of the Solar System, demon strating no creep and having the ultimate strength and yield strength, can be correctly considered in terms of elasticity theory, which is the main purpose of the present study. The modern classical theory of elasticity was mainly developed and used for solving the two dimensional planar problems in different technical applications (Novozhilov, 1955; Muskhelishvili, 1966; Lur’e, 1970). The main problem is that, due to, prob ably, the complexity of natural objects and the lack of necessity, a threedimensional problem of elasticity theory in a complete analytical form, in terms of the gravitational potential for celestial bodies, has not yet been considered (Zharkov, 1983; Slyuta and Voropaev, 1997). This triggers the next question that requires an answer in the course of analytical study: whether the specified threedimensional problem can be correctly solved with the linear theory of elasticity. The gravita tional attraction forces are mass forces (they are active in the volume of a whole body) and influence the final shape of the body itself. Because of this, from the strength of view of mathematical physics, the spatial, or threedimensional, problem of elasticity theory under the conditions of gravitational compression is a feedback problem, where any change in the shape influences the gravitational potential and vice versa. HISTORY OF THE PROBLEM The history of the problem on selfgravitation and inelastic deformation of solid small and planetary bodies of the Solar System is closely connected with the history of the concepts of a shape of the solid Earth in terms of the Newton theory of gravity. The history of these concepts in geology and comparative plane tology developed in two main competing directions— thermal (thermalhydrostatic) and gravitational. The thermalhydrostatic direction was based on the solution of a classical problem for an equilibrium shape of selfgravitating fluid bodies under the influ ence of gravitational, centrifugal, and tidal forces. I. Newton (1687) was the first who ascertained that, under the influence of gravitational and centrifugal forces, the spherical fluid Earth should transform into a weakly oblate spheroid of revolution. C. Maclaurin (1742) (Maclaurin, 1962) developed this study for self gravitating rotating homogeneous bodies and discov ered the existence of ellipsoids of revolution with a high degree of ellipticity that were later called “the Maclaurin spheroids”. Following Macluarin, C.G.L. Jacobi (1834) SOLAR SYSTEM RESEARCH
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demonstrated the existence of equilibrium threeaxial ellipsoids for selfgravitating fluid bodies that were correspondingly called “the Jacobi ellipsoids”. Some what later, M.E. Roche (1850), while considering the action of tidal forces on an ellipsoidal fluid equilib rium body orbiting about a massive central body, found that there are no equilibrium solutions for a shape of the satellite of a specified mass inside a spherical vol ume of some limiting radius around the central body, and the satellite, consequently, falls to pieces. This critical radius is now known as “the Roche limit”. H. Poincaré (1885) discovered the nonellipsoidal pyri form equilibrium figures. Finally, in his wellknown book “Ellipsoidal Figures of Equilibrium”, S. Chan drasekhar (1969) thoroughly analyzed all of the above mentioned papers and all of the equilibrium states of ellipsoids of different types. In terms of the gravitation hypothesis, an equilib rium spherical shape of a body is the final step in the process of transformation of the irregular (i.e., non equilibrium) shape of a solid elastic body under the action of its own mass, i.e., selfgravitation. In this context, the abovementioned problem of the equilib rium shape of a selfgravitating fluid body is correct only at the stage of the planet or planetary body already formed, which is an independent and separate research direction. However, the most interesting strength in the gravitation hypothesis is the stage of gravitational deformation and rheology of a solid body having the fundamental strength (the yield strength). THERMAL HYPOTHESIS In 1743, A.C. Clairaut (1713–1765) published a book on the theory of the Earth’s shape based on the principles of hydrostatics (Clairaut, 1743). Clairaut was the first to introduce the planetary concept for the Earth due to its spherical shape. He explained the planetary sphericity of the Earth’s shape by the suppo sition that “the Earth’s shape should obey the laws of hydrostatics”. Since the Earth is solid now, Clairaut believed that the actual shape of our planet is inherited from its remote past, when the Earth was fluid. Clair aut compared the present spherical figure of the Earth to the surface of water “frozen after it had taken the shape corresponding to the equilibrium conditions”. To support this hypothesis, he also stressed that “the height of the highest mountains is negligible as com pared to the Earth’s diameter” (Clairaut, 1947). In 1755, I. Kant (1724–1804) published, first anonymously, the book “Universal Natural History and Theory of Heaven”, where he formulated the ori gin of the planetary system from a hot nebula as the main statement of the early history of the Earth and planets: after their formation, they gradually cooled down and solidified, keeping their cores hot (Kant, 1755). In his paper published earlier, in 1754, he wrote that, due to friction between the lithosphere and the hydrosphere, the tides caused by the influence of the Moon and the Sun should slow down the rotation of SOLAR SYSTEM RESEARCH
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the Earth about its axis. Kant considered his ideas in keeping with the development of the concept of gravity and qualified his monograph with the note “according to Newtonian principles”. In 1796, the book that proved a great success, “The System of the World” by P.S. Laplace (1749–1827) appeared (Laplace, 1825). In an enthralling and accessible form, the author expounded the analogous cosmogonic hypotheses based on the same principles as those of Kant; although he did not mention his equally great prede cessor. The KantLaplace hypothesis on the initially hot and fluid Earth and its subsequent thermal cooling, which explained the spherical shape of the Earth by hydrostatics of a melted fluid globe, dominated geol ogy for almost two centuries. The wellknown physi cist and naturalist W. Thomson (Lord Kelvin) (1824– 1907) supported this hypothesis and believed that the fluid Earth transformed to the rigid one through cool ing (Thomson, 1864). As a result of solidification, the Earth became constant in shape, where the effect of gravitational forces ceased and was replaced by fric tion. Thomson compared the Earth to “a steel globe of the same dimensions, without mutual gravitation of its parts”, i.e., if a large body is solid, the gravitational forces inside it, as in a small body, do not manifest themselves in any way. The studies of the pioneers of the longdominant tectonic theory of geology A. Heim (1849–1937) and E. Suess (1831–1914)—“Orogeny Mechanism” (Heim, 1878) and “Earth’s Face” (Suess, 1883)—also rest on the hypothesis of a formerly melted planet, the cooling and presumed subsequent compression of which served as a main source of energy for geologic processes. In the middle of the last century, this direc tion was also developed in the studies of the Soviet geologist M.A. Usov (1883–1939), who supposed, starting from the hypothesis of cooling and heating, that the selfdevelopment of the Earth is caused by alternation of the periods of compression and expan sion in the history of the planet (Usov, 1940). The hypothesis of the initially fluid bodies was also sup ported by the famous mathematician A.M. Lyapunov (1857–1918), who called it “the universally recog nized theory” and believed that the shapes of celestial bodies “should be shapes of the fluid mass, all particles of which are mutually attracting according to the Newtonian law and which is uniformly revolving about a fixed axis” (Lyapunov, 1903, 1930). After that, as a result of cooling, all of the bodies became solid and maintained “an invariant shape due to internal friction” (Lyapunov, 1930). GRAVITATION HYPOTHESIS The concept of gravity as a main driving geologic force, in opposition to the thermal hypothesis, was for the first time considered by the founder of geologic science J. Hutton (1726–1797) in his study “Theory of the Earth” (Hutton, 1795). He asserted that “gravita
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tion, and the “vis infita of matter, thus form the first two powers distinguishable in the operations of our system, and wisely adapted to the purpose for which they are employed.” He considered magnetism and electricity as additional and secondary forces. Hutton was also the first to introduce the notion of a huge duration of geological time. Gravitation, “vis infita of matter”, and “an immense period of (elapsed) time” are the main factors in the Hutton hypothesis that explain, as he supposed, all of the processes on the Earth, including its spherical shape (i.e., its planetary type). In his opinion, the Earth’s surface constantly and repeatedly changed during its long history and, due to the rotation of the Earth, always approached its equilibrium shape, i.e., the surface of an ellipsoid of revolution. Hutton distinguished between cosmogony and geology and stressed that the discovery of the method of the Earth’s origin connected with the prob lems of “the origin of things” is beyond the scope of geology. J. Playfair (1748–1819) was not satisfied with the completeness of the explanation given by Hutton and stressed in his “Illustrations of the Huttonian Theory of the Earth” that the rotation of the Earth itself should have brought the Earth’s shape to the equilib rium shape, which is the result of the balance between the gravitation and centrifugal forces (Playfair, 1822). The shape of the Earth should differ in dependence on the rotation velocity and structure, and its surface should be mainly perpendicular to the gravity force. Playfair also asserted that, to take a spherical shape, the Earth did not need to be fluid. The ideas of Playfair were supported by the famous astronomer J. Herschel (1792–1871), who stressed that, “in rotating around its axis, the Earth approaches its actual shape corre sponding to equilibrium, and it would have approached this shape even in the case, if it initially took another shape, say, by mistake” (Herschel, 1849). Ch. Lyell (1797–1875), one of the strongest fol lowers of Hutton’s ideas, also stressed that the spheric ity of the Earth’s shape certainly does not prove its ini tial fluid state (Lyell, 1830, 1838). It is connected with the centrifugal force action during the rotation of the planet. At the end of the 19th century, the geophysicist G. Darwin, a son of Charles Darwin, analyzed the tidal interaction of the Moon and the Earth and con cluded that “these attractions should induce alternat ing deformations of the Earth” (Darwin, 1898). Dar win wrote that our planet can be deformed by external forces. However, admitting the ability to be deformed by external forces, Darwin also supported the alterna tive strength of view expressed by his teacher in phys ics, W. Thomson, considering the Earth as “a solid body of high rigidity. The Earth is hard, and its rocks are hard for thousands of miles deep, hard as granite, and even stronger than granite” (Darwin, 1898). When developing the gravitation hypothesis that forms the basis for considering the Earth as a planet, A. Wegener (1880–1930), the founder of the plate tec
tonics theory, currently the dominating theory of the geology of the Earth, noted in his book “The Origin of Continents and Oceans”: “In the laboratory, a small steel model of a globe behaves in the same way as a solid body does. However, under the influence of its own attracting forces, a steel globe of the Earth’s size will flow, if not immediately, but at least when thou sands of years are available for this. Here, we observe the transition from the dominant molecular forces to the forces caused by masses.” (Wegener, 1966). It is evident that the model of “a steel globe of the Earth” of geologist A. Wegener now principally differs from the aboveconsidered model of “a steel globe of the Earth” of physicist W. Thomson. Somewhat earlier, a similar idea was also suggested by the geologist I.D. Lukashevich (1863–1928), who asserted that “a spherical shape of the Earth does not serve in itself as evidence of its fluid state in former times. If the Earth were solid and its shape were cubic, cylindrical, conical, or of any other angular type, its shape would be unstable for such a huge mass of the material as the Earth. Mutual attraction between par ticles would overcome their molecular cohesion… A spherical shape of large celestial bodies is not a play of chance, it is caused by molar, i.e., gravitational, forces rather than by molecular ones determining the shape of fluid droplets” (Lukashevich, 1908). He stressed that, “in such a huge accumulations of the material as our globe, the forces that are initiated by gravitation of the material, according to the Newto nian law, generate tremendous stress and induce the phenomena that are called the planetary processes. These forces are completely unnoticeable in small aggregates of the material, but they produce such effects that the simple analogies between small and large bodies fade into insignificance and become falla cious” (Lukashevich, 1911). According to Luka shevich, exactly the gravitational forces determine the strength of planetary bodies and are in large excess over the value of cohesion inside them, i.e., the molec ular forces, or, according to Hutton, the material strength, or, in currently used terms, the yield strength and ultimate strength of the material. Wegener illus trated this transition by the following clear example. “We cannot build a steel column of any height; we should restrict ourselves within some limits, beyond which the socle of the column will “flow”… If a solid globe is large, steel is no longer solid. Moreover, one may say that there are no longer solid bodies under such conditions: all of the bodies possess viscosity" (Wegener, 1966). Supposing that the height of terres trial mountains is controlled by gravity, Wegener cited A. Penck (1919) that “the height of mountains on our planet is not accidental: it is derivative of the interac tion of two groups of forces: molecular or cohesion forces and gravitational forces.” Further, it is worth mentioning the studies by the geophysicist V.A. Magnitskii (1915–2005), who devel oped the ideas of his teacher F.N. Krasovskii (1953) in SOLAR SYSTEM RESEARCH
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line with the gravitation hypothesis and showed that the slowing down of the Earth’s rotation should induce a deviation of the Earth’s shape from the grav itational equilibrium, which should be accompanied by the isostatic compensation on the surface and deep displacement of the material (Magnitskii, 1948, 1953). Finally, even in classical geology, one of the known theoreticians of the presentday geologic science, V.E. Khain (1914–2009), when classifying the tecto genesis factors in descending order of priority, namely, (a) compression, (b) differentiation—radioactive decay, (c) rotation of the Earth, and (d) isostasy, now explains their separate existence by three main kinds of energy—gravitational, thermal (radioactive decay), and mechanical (rotation of the Earth) (Khain, 1957). An ultimate result of the twohundredyear con frontation between the thermal and gravitation hypothesis was resumed in the papers by B.L. Lichkov (1888–1966), who also supposed that, “if belonging to the planetary category is a newly acquired form rather than an inherited one, and if it was acquired during the phase when the gravitational forces became dominant, and the cohesion and elasticity forces started to deter mine a general structure and shape of aggregates, the action of tectonics and dislocations is precisely the on going change in the structure and shape of the spher oid that has appeared” (Lichkov, 1960). The author stressed that, “to explain the shape of the Earth, spherical or ellipsoidal, close to spherical, no grounds from the theory of the origin of our planet are required. Regardless of any theory, the Earth, having its given size, should take this shape, and the geotectonic phe nomena cannot but start on the Earth” (Lichkov, 1965). So, only from the middle of the last century, the gravitation hypothesis on the membership of the Earth and other planets in the planetary category actually became a universally recognized theory. However, as soon as this happened, the next question, as one of the most important consequences of the gravitation hypothesis, has naturally emerged. Where is the lower limit of the planetary category membership or when does this membership start? This fundamental conse quence of the gravitational hypothesis was also consid ered for the first time by Lichkov (1960, 1965). Up to that time, it had been already well known that the small bodies represented by meteorites and asteroids have a close relationship and are characterized by an angular “cobblestone” shape. “The data volume sug gests that the smallest asteroids are huge meteorites, and large meteorites are the smallest asteroids. In interplanetary space, they both move approximately in the same orbits. These groups of celestial bodies dem onstrate the utmost unity, and there is no boundary between them” (Krinov, 1951). However, it was already evident that, while there is no boundary between meteorites and asteroids, the boundary SOLAR SYSTEM RESEARCH
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between asteroids and planets should be apparent, and it is exactly a planetary characteristic, which is charac teristic of the planets rather than asteroids that are “lumps or stones tens or more kilometers across” (Krinov, 1951). “CRYSTALLINE” (STRUCTURAL) AND PLANETARY FORMS OF THE MATTER ORGANIZATION From the strength of view of the thermalhydro static hypothesis, no special problems appeared with this boundary: according to this hypothesis, all the bodies of spherical shape were formerly melted, and asteroids and meteorites with angular shapes are only the fragments of solidified planetary bodies destroyed in collisions. However, from the strength of view of the gravitation hypothesis, this is a boundary at the transi tion between different forms of the matter organiza tion: “Unfortunately, no accurate data on the value of the asteroid’s mass, from which the angularity of its shape is lost or starts to disappear, are available; if this border were detected, it would be also the border between the gravitational and crystalline states of space. … During the transition of a body from the asteroid form to the planetary one, the state of space of this body changes; namely, it transits from the space of particulate forces to the space of gravitational forces. … There is no, and cannot be, similarity between the bodies obeying different states of matter because of a large difference in size” (Lichkov, 1965). While developing the gravitational concept of the transition between asteroid (small) and planetary bod ies, Lichkov correctly supposed that “there is certainly no tectonics in meteorites and asteroids, since the cohesion and elastic forces dominate in both of them. Tectonics appears in the body of a planet. … Tectonics and dislocation appear only in the body of gravity, because, only in such a body after its formation, the gravity forces begin to struggle with the cohesion forces, so completely dominant before” (Lichkov, 1960). Consequently, the bodies with different forms of matter organization should further evolve in cardi nally different ways: “after the formation of the plane tary Earth, not “a stable shape” has been achieved, but, vice versa, permanent systematic changes of the shape have started; there were no changes, while the asteroid existed, but they appear after the formation of the planet. … During this, the blurring (deformation) in the planetary body becomes a repetitive phenome non (depending on the rotation velocity), and the pro cess is regulated by maintaining the equilibrium of the body as a whole” (Lichkov, 1960, 1965). Lichkov also clearly distinguished between the internal (the body’s mass) and external (centrifugal and tidal) forces form ing and subsequently changing the Earth’s shape, because he well understood that, if the first ones do not exist, no inelastic effect of the second ones is possible: “If the Earth were a motionless body, this rebuilding
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would form a globe; however, since we observe a rotat ing body, this or that polar flattening should appear, which will result in the ellipsoid of a larger or smaller degree of flattening” (Lichkov, 1960). According to Lichkov, the character of the transi tion between small and planetary bodies should be gradual: “Angularity is large in meteorites, weaker in asteroids, and becomes negligible in planets, being greater in minor planets than in larger ones. The angu larity of shapes is larger in the Moon than in the Earth, so that the lunar mountains are absolutely and rela tively higher than the terrestrial ones. The smoothness of shape increases, if the aggregates grow in size” (Lichkov, 1960, 1965). The main statements of one of the fundamental consequences of the gravitation hypothesis in developing the geotectonic concept of the Earth—the transition between different forms of the matter organization, i.e., the transition between small and planetary bodies—formulated by Lichkov fell through the cracks and, as is usually the case, were forgotten. As the data on asteroid morphology accumulated, the problem of the transition between small and plan etary bodies has emerged again. However, as distinct from the already forgotten lower limit of membership in the planetary category, it has been revived now as the upper limit of the irregular shape of asteroids. To estimate the critical size of asteroids, the hydrostatic equilibrium equation (P = ρgh) was used as a basis for one of two or both models: (a) the static loading in dependence on the strength, density, and size of the body and (b) the deformation rate of the body in dependence on the material viscosity, temperature, and size of the body (Johnson and McGetchin, 1973; Simonenko, 1979; Farinella et al., 1982, 1983, 1985). In this process, the leading part was assigned to grav ity: “asteroids are so small that their gravity fails to form a globe, as in planets and their large satellites, by crumpling and “ramming” the material composing them” (Simonenko, 1979). Nevertheless, it was sup posed that, under the influence of radiogenic heat, the largest silicate asteroids and ice bodies may be melted with subsequent relaxation of the body’s shape and relief (Lewis, 1971; Johnson and McGetchin, 1973), i.e., this is the updated thermal hypothesis. The estimates obtained from the hydrostatic equi librium equation agreed with Lichkov’s suppositions on the gradual transition, including the gradual relax ation depending on time: “The relationship between the size of a body of known … composition and strength and its shape or the degree to which topogra phy can be supported on its surface is a problem of interest, because this bears on the support and lifetime of terrain features on planetary surfaces and the shape of smaller objects in the Solar System such as asteroids and satellites” (Johnson and McGetchin, 1973). It was supposed that the relaxation and gradual transi
tion are caused by creep appearing under the long term loading and stresses not exceeding the yield strength. The larger the asteroid’s mass, the higher the stresses responsible for deformation and the closer its shape to a globe. The partial internal relaxation was also an obvious consequence of the creep hypothesis and hydrostatic model. “Due to a small “weight” of stone, in the asteroids even 300–400 km across, … the yielding phenomenon is completely absent, while this process is extremely slow in the largest asteroids and occurs only deep in them. … Thus, only the bowels of several of the largest asteroids can be “rammed” by gravity” (Simonenko, 1979). However, the hydrostatic model is valid only for the case of rheology of a viscous fluid possessing no ultimate strength and no yield strength, as, for example, in the case of destruction of selfgravitating viscous bodies by tidal forces (Sridhar and Tremaine, 1992). Moreover, the hydrostatic pres sure itself results in no plastic deformation (Poirier, 1995). To the end of 80s and beginning of 90s of the last century, it was ascertained that asteroids and small sat ellites, contrary to the planets, are characterized by a nonequilibrium threeaxial shape, i.e., by the model of a nonequilibrium threeaxial ellipsoid (Soter and Har ris, 1977; Thomas, 1989). The direct proportionality between the relief height and the size observed for the bodies of irregular shape and, on the contrary, the inverse proportionality between these parameters for the planets also strengthed to their fundamental differ ence in structure and evolution (Croft, 1992). As a result, from the external morphologic characteristics, i.e., in dependence on the body’s shape, all solid bod ies of the Solar System were divided into two main classes (Slyuta and Voropaev, 1992, 1997). This is a class of small bodies of irregular shape, i.e., they look like fragments or lumps. They are small planetary sat ellites, asteroids, comets, and small objects from the Kuiper belt. Due to their insufficient mass, these bod ies never developed into planetary bodies. The other class comprises planetary bodies that are characterized by an equilibrium spherical shape. Among them are planets, large planetary satellites, asteroids Ceres and Vesta, and large objects from the Kuiper belt. The problem of the transition observed between small and planetary bodies and the dependence of this transition on the composition, mass, and size of the bodies, on the physical, mechanical, and rheological properties of the material has developed from a poorly investi gated field of planetology and cosmochemistry into an independent research direction that is now known as gravitational deformation of small bodies of the Solar System (Slyuta and Voropaev, 1997). SOLAR SYSTEM RESEARCH
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GRAVITATIONAL COMPRESSION The threeaxial shape of a small body is the largest wave heterogeneity of the gravitational field of a small body, i.e., the firstorder heterogeneity, which is the first to be compensated and to relax under gravita tional deformation. It is this heterogeneity, i.e., the nonequilibrium of the body’s shape, that determines the distribution of deviatoric stresses in a small body. The largest craters and relief of a small body are heter ogeneities of the second and weaker orders in depen dence on their relative sizes (regional and local). At the initial stage, triaxial gravitational compression is accompanied by densification the material of small bodies and occluding the pores and cracks, i.e., by “healing” the structure defects. The experimental data show that these processes are going on in rocks most intensively when the pressure grows from 0.1 to 50 MPa (Protod’yakonov et al., 1981). If a triaxial compression grows further in the small bodies of a sufficient mass and if there is no creep, the ultimate strength of small bodies will be further pro vided by the ultimate strength or yield strength of the “healed” rocks of coherent (monolithic) small bodies or coherent fragments of rocks of binary (multicom ponent) and of loose (rubble pile) small bodies regard less of their structure (Slyuta, 2014). In other words, in the present case, the yield strength will be determined only by the mineral composition and texture of the coherent rock of specified composition under speci fied temperature. The ultimate strength is determined by the value of the contraction (compressive strength) or stretching (tensilestrength) stresses, under which the rock is destroyed. The elastic strength or the yield strength is determined by the values of stresses under which the residual deformation is observed. For metallic bodies (e.g., iron meteorites), the yield strength is determined by the value of stresses under the residual deformation of 0.2% (Slyuta, 2013b). The yield strength of rocks usu ally either coincides or is somewhat less than the com pressive strength providing the ratio of these quantities is 0.8 : 1 (Slyuta and Voropaev, 1997). If the stress deviator exceed the yield strengt at a specified strength, a gravitational plastic deformation starts, which means that the destruction of rocks occurs beyond the elastic deformation limits—in the range of the plastic state characterized by the appear ance of substantial residual deformations in the rocks. A plastic deformation is accompanied by the develop ment of characteristic structures and static twinning of minerals depending on prevailing mechanism of deformation – between the grains (cataclase) and (or) inside (along the planes of the twinning and slip) (Garber et al., 1963; Nicolas, 1987; Slyuta and Voro paev, 1997). The preference for plastic deformations under large uniform pressure, which also include threedimensional gravitational compression, is explained by the fact that these conditions favor the easier manifestation of intragrain movement and dis SOLAR SYSTEM RESEARCH
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placement that do not lead to the destruction of conti nuity and initiation of fractures, i.e., to the breaking deformations (Rzhevskii and Novik, 1973; Poirier, 1995). The simplest rheological model reproducing the transformation of the shape of a small body to a spher ical one can be a plastic body (SaintVenant’s body) that is not deformed until the stress difference reaches a critical value—the yield strength, above which con tinuous deformation takes place. Since the loading duration is very long (in fact, this is a whole lifetime of the body from the moment of its origin), the body’s shape may be transformed even under small relative stresses, if they exceed the yield strength. A required value of the gravitational force is determined by the rest mass of the body. This leads to the necessity of introducing such a notion for small and planetary bod ies as the critical mass, at which the gravitational field strength of a body reaches the value required for over coming the yield strength of the body’s material by the planetary field of structural or tectonic stresses, under which an equilibrium shape of the planetary body is formed. Under the threedimensional gravitational com pression in the process of “healing” the structural defects and decreasing the porosity, the body’s density increases and becomes more uniform. If there is no internal thermal flux, the temperature deep in the small bodies of the Solar System will be constant and characterized by a uniform distribution over the body’s volume. Let us consider the structural stresses appearing in a small body with uniformly distributed density and temperature under the influence of the forces of its own gravitational field. The body’s shape, where the stresses will be analyzed, is a prolate ellip soid with semiaxes a > b = c, which is a somewhat sim plified version of the model of a threeaxial ellipsoid characteristic of small bodies (Fig. 1). The ellipsoid eccentricity is (b/а)2 = 1 – ε2, where ε > 0. Consequently, ε = 0 for a sphere. The body is assumed to be homogeneous and isotropic, its density is ρ0, and the temperature Т is uniformly distributed over the volume. For internal strengths of a homoge neous ellipsoid, the gravitational potential V is deter mined by the Poisson equation ΔV = – 4πρ 0 G 0 , where G0 is the gravitational constant. According to the potential theory (Tikhonov and Samarskii, 1977), the gravitational potential can be expressed as ∞ 2
V = πρ 0 G 0 b a
∫ 0
2
2
z D ( s ) ds, ρ – 1 – 2 2 s+b s+a
(b2 +
where D(s) = s)–1(a2 + s)–1/2. The ellipsoid sur face is determined by the equation 2
2
2
2
ρ 0 /b + z 0 /a = 1, where ρ2 = x2 + y2.
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The gravitational field strength F is determined by the gradient of the gravitational potential F = ρ0gradV(ε, r, θ).
Z a
n z0 θ
τ r b
O
Y ϕ ρ0
b
In the spherical coordinate system (Fig. 1), the dis placement vector u will be presented as the expansion u = erur + eθuθ, where ur, uθ, and uϕ = 0 are the radial, meridional, and azimuth components of the displacement, respec tively. Then, the equilibrium equation of an isotropic body in the gravitational field takes the form graddivu – 1/2rotrotu = –FG, 1 + ν ) ( 1 – 2ν) is the elastic constant, ν is where G = ( E ( 1 – ν ); the Poisson ratio, and Е is the Young modulus of the material of a small body. In this case, the deformation tensor is expressed as follows ⎫ ⎪ ⎪ ⎪ 1 e ϕϕ = ( u θ cot θ + u r ) ⎪ r ⎪ ⎬. ∂u r ⎪ e rr = ⎪ ∂r ⎪ ∂u θ ⎪ 1 ⎛ ∂u r ⎞ e θr = – u θ + ⎪ ⎠ ∂r ⎭ r ⎝ ∂θ 1 ∂u e θθ = ⎛ θ + u r⎞ ⎠ r ⎝ ∂θ
X
Fig. 1. A prolate ellipsoid with semiaxes a > b = c in spher ical coordinates. All of the designations of the deformation vector components are described in the text.
In a spherical coordinate system (Fig. 1), where x = rsinθcosθ, y = rsinθsinϕ, and z = rcosθ, the gravita tional potential can be written as 2
2
2 r r V ( ε, r, θ ) = A – 2 C + 2 cos ( θ ) ( C – B ), b b 2
a where A = B 0 2 A ( ε ); B = B0B(ε); C = B0C(ε); B0 = 2π(1 ⎯ b ε2)ρ0G0b2, 1+ε ln ⎛ ⎞ ⎝ 1 – ε⎠ 1 1 + ε⎞ , B ( ε ) = A ( ε ) = 1 ln ⎛ – 2 , 3 2ε ⎝ 1 – ε⎠ ε 2ε 1 + ε⎞ ln ⎛ ⎝ 1 – ε⎠ 1 C ( ε ) = – . 2 2 3 4ε 2ε ( 1 – ε ) In the linear approximation, when ε –σр,
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where τmax is determined with the intensity of deviatoric stresses τ (Novozhilov, 1958) 2
1 ( σ rr ( x, ε, r, θ ) – σ θθ ( x, ε, r, θ ) ) + ( σ rr ( x, ε, r, θ ) – σ ϕϕ ( x, ε, r, θ ) ) τ = 6 + ( σ θθ ( x, ε, r, θ ) – σ ϕϕ ( x, ε, r, θ ) ) 2 + 6σ rθ ( x, ε, r, θ ) 2 .
2
(3)
The stress tensor components in Eq. (3) are connected with the deformation tensor components by the Hooke law 1 σ rr ( x, ε, r, θ ) = e rr ( x, ε, r, θ ) 2μ
2 σ 2 ( x, ε, r, θ ) = τ sin ( ψ ( x, ε, r, θ ) ) 3 + 1 σ n ( x, ε, r, θ ), 3
+ ( x – 1 )e n ( x, ε, r, θ ),
2τ ⎛ ψ ( x, ε, r, θ ) + 4π σ 3 ( x, ε, r, θ ) = ⎞ sin ⎝ 3⎠ 3
1σ ( x, ε, r, θ ) = e ( x, ε, r, θ ) θθ θθ 2μ
1 + σ n ( x, ε, r, θ ), 3
+ ( x – 1 )e n ( x, ε, r, θ ),
where ψ is the stress pattern angle
1 σ ϕϕ ( x, ε, r, θ ) = e ϕϕ ( x, ε, r, θ ) 2μ + ( x – 1 )e n ( x, ε, r, θ ),
3 s 4 ( x, ε, r, θ )⎞ ψ ( x, ε, r, θ ) = 1 arcsin ⎛ . 3 ⎝ 2 ⎠ 3 τ
1 σ rθ ( x, ε, r, θ ) = e rθ ( x, ε, r, θ ), μ σ n ( x, ε, r, θ ) = σ rr ( x, ε, r, θ )
To analyze the gravitational deformation of a small body, the maximum stress deviator τmax is of the most interest; it is determined by the difference between the maximum σ1 and minimum σ3 main stresses
+ σ θθ ( x, ε, r, θ ) + σ ϕϕ ( x, ε, r, θ ), E 1 where the shear modulus μ = and σ n is the 2(1 + ν) 3 mean normal stress. The invariants of the stress tensor (Novozhilov, 1958) are determined by the expressions s 1 ( x, ε, r, θ ) = σ n ( x, ε, r, θ ), s 2 ( x, ε, r, θ ) = σ rr ( x, ε, r, θ )σ θθ ( x, ε, r, θ ) + σ θθ ( x, ε, r, θ )σ ϕϕ ( x, ε, r, θ ) 2
+ σ ϕϕ ( x, ε, r, θ )σ rr ( x, ε, r, θ ) – σ rθ ( x, ε, r, θ ) , s 3 ( x, ε, r, θ ) = σ rr ( x, ε, r, θ )σ θθ ( x, ε, r, θ ) 2
× σ ϕϕ ( x, ε, r, θ ) – σ ϕϕ ( x, ε, r, θ )σ rθ ( x, ε, r, θ ) , s 4 ( x, ε, r, θ ) = s 1 ( x, ε, r, θ )s 2 ( x, ε, r, θ ) 3 2 – s 1 ( x, ε, r, θ ) – 3s 3 ( x, ε, r, θ ). 9 According to the general theorem of tensor analysis (Zommerfel’d, 1954), the gravitational stress tensor, as a secondorder tensor, can be presented by three main stresses
2 2π σ 1 ( x, ε, r, θ ) = τ sin ⎛ ψ ( x, ε, r, θ ) + ⎞ ⎝ 3⎠ 3 1 + σ n ( x, ε, r, θ ), 3
σ1 – σ3 τ max = , 2 or, in the explicit form τ max = τ cos ( ψ ( x, ε, r, θ ) ) .
(4)
The obtained universal function (Eq. (4)) allows the gravitational deformation of a small body to be analyzed depending on the chemical and mineralogi cal composition and the shape parameters. The func tion is written in the form convenient for calculations τmax = σ0F(ε, ν), (5) 2
9 G 0M , G0 is the where the dimension factor σ0 = 2 8π a bc gravitational constant, М is the body’s mass 3 4 (M = πρ 0 R m , Rm is the mean radius of the equalvol 3 ume body), a, b, and с are the semimajor axes of the body’s shape, and F(ε, ν) is a dimensionless transcen dental function. The eccentricity of the shape εmean of the analyzed small body is calculated as εmean =
1 – bc 2 . a
The analysis of the rigorous expression for the devi atoric stress (Eq. (5)) reveals the dependence on the rock composition (the Poisson ratio), the shape of a SOLAR SYSTEM RESEARCH
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small body (the eccentricity), and the coordinates of a strength (the depth and latitude). Within the bound aries of a small body, the value of the deviatoric stress increases from the center to the surface (Fig. 2) and from the pole to the equator (Fig. 3). The distribution of deviatoric stresses in small isotropic bodies also depends on the elastic properties of the material (i.e., on the composition of small bodies) that are deter mined by the Poisson ratio (Figs. 4 and 5). 2
133
To estimate the shearing stress averaged over the angles 〈τ〉 within the limits of the maximum and min imum values, the expression 〈 τ〉 ≥ 0.633. 0.731 ≥ τ max can be used. Under the linear approximation for τ with eccentricities up to ε = 0.4, the following expres sion is valid within an error less than 10% 2
2
2 2 ε 32x – 41x + 12 2 r 2 – ε ( 1 + 3 cos ( 2θ ) ) + r τ ≅ ( 1 – ε ) 15 3 x 60x 3 ( 19x – 12 ) 60x 3 ( 19x – 12 ) , σ0 2 2 × [ ( 36x + 59x – 52 ) + 3 cos ( 2θ ) ( 36x – 93x + 44 ) ]
2 r where σ0 = 2π ρ 0 G0b2; r → . b
The distributions of deviatoric stresses described by the approximate solution (Eq. (6)) under small ellip soid eccentricities of a small body are close to the exact solution (Eq. (5)) and can be used in the engineering estimate of the approximate parameters of gravita tional deformation of small bodies (Fig. 6). For exam ple, for the material with the Poisson ratio ν = 0.31 and the ellipsoid eccentricity less than ε = 0.4, the differ
ence between the exact and approximate solutions does not exceed 10%. If the ellipsoid eccentricity of a small body is relatively large (ε > 0.4), which is charac teristic of most small bodies of the Solar System, the difference between the approximate and exact solu tions becomes substantial and reaches the value of the same or larger order of magnitude as that of the exact solution (Fig. 7). In this case, the linear approxima tion cannot be applied to estimating the gravitational deformation parameters, since this will lead to a large error and an incorrect result.
τmax 0.10
0.08
0.06
0.04
0.02
0
0.2
0.4
0.6
0.8
r
Fig. 2. The distribution of deviatoric stresses τmax from the center to the surface (r) in the equatorial plane (θ = π/2) of the prolate ellipsoid of a small body calculated for the material with the Poisson ratio ν = 0.31. The models for the eccentricities of the shape a small body ε = 0.4, 0.6, and 0.8 are shown with solid, dotted, and dashed curves, respectively. SOLAR SYSTEM RESEARCH
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SLYUTA, VOROPAEV τmax 0.10
0.08
0.06
0.04
0.02 0
1.0
0.5
1.5 θ
Fig. 3. The distribution of deviatoric stresses τmax over the surface from the pole to the equator (θ) of the ellipsoid of a small body calculated for the material with the Poisson ratio ν = 0.31. The models for the eccentricities of the figure ε = 0.4, 0.6, and 0.8 are shown with solid, dotted, and dashed curves, respectively.
τmax 0.12
0.10
0.08
0.06
0.04
0.02 0
0.2
0.4
0.6
0.8
r
Fig. 4. The distribution of deviatoric stresses τmax from the center to the surface (r) in the equatorial plane (θ = π/2) of the prolate ellipsoid of a small body calculated for the eccentricity ε = 0.8 and different values of the Poisson ratio. The models for the Poisson ratio ν = 0.17, 0.28, 0.31, and 0.42 are shown with solid, dotted, dashed and dashdotted curves, respectively. SOLAR SYSTEM RESEARCH
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135
τmax 0.06
0.10 0.04
0.08
0.06 0.02 0.04
0.02 0
0.5
1.0
1.5 θ
0.2
0.4
0.6
r
0.8
Fig. 6. The distribution of deviatoric stresses τ max from the center to the surface (r) in the equatorial plane (θ = π/2) of the prolate ellipsoid of a small body with an eccentricity ε = 0.4; the Poisson ratio of the material is ν = 0.31. The models according to the exact (Eq. (5)) and approximate (Eq. (6)) solutions and for a sphere are shown with solid, dotted, and dashed curves, respectively.
Fig. 5. The distribution of deviatoric stresses τmax over the surface from the pole to the equator (θ) of a small body calculated for the eccentricity ε = 0.8 and different val ues of the Poisson ratio. The models for the Poisson ratio ν = 0.17, 0.28, 0.31, and 0.42 are shown with solid, dotted, dashed and dashdotted curves, respectively.
The behavior of the radial gradient of deviatoric stresses can be presented as a functionR(x, ε, θ) that is determined by the ratio of the maximum deviatoric s c stresses on the surface ( τ max ) and in the center ( τ max ) of the small body ellipsoid ⎛ τ smax⎞ R ( x, ε, θ ) = ⎜ ⎟ , ⎝ τ cmax⎠
0
(7)
and depends on the shape’s parameters (the eccentric ity) (Figs. 8 and 9). Along a major axis of the prolate ellipsoid of a small body, the ratio R decreases to 1 and, under some critical value of the eccentricity, the devi c atoric stress in the center τ max becomes larger than
τmax 0.3
0.2
0.1
s
that on the surface τ max (Fig. 8). For small bodies of different composition, the critical value of the eccen tricity, at which the inversion occurs, will be different (Fig. 8). Along a minor axis in the equatorial plane of the prolate ellipsoid of a small body, the radial gradient approaches 1 with increasing eccentricity, i.e., under s any eccentricity, the deviatoric stress τ max on the sur face at the equator of the prolate ellipsoid is always c larger than that in the center τ max (Fig. 9). The gravitational deformation rate of a small body depends on the value of the differential stress, temper SOLAR SYSTEM RESEARCH
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0
0.2
0.4
0.6
0.8
r
Fig. 7. The distribution of deviatoric stresses τ max from the center to the surface (r) in the equatorial plane (θ = π/2) of the prolate ellipsoid of a small body with an eccentricity ε = 0.8; the Poisson ratio of the material is ν = 0.31. The models according to the exact (Eq. (5)) and approximate (Eq. (6)) solutions and for a sphere are shown with solid, dotted, and dashed curves, respectively.
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R 4
R 6
3
5
2
4
1
3
0 0.6
0.7
0.8
0.9
ε
Fig. 8. The change of the ratio of deviatoric stresses R along a major axis of the prolate ellipsoid of a small body depending on the shape’s eccentricity ε. The models for the Poisson ratio of the material ν = 0.31 and 0.17 are shown with solid and dotted curves, respectively. The ratio of deviatoric stresses R = 1 is shown with a dashed curve.
ature, and dominant mechanism of deformation. The body will be deformed under any low temperature, if the deviatoric stress exceeds the yield strength (τmax > σp) under a specified temperature. Under the constant uniform pressure, the increase of the temperature or the decrease of the deformation rate diminishes the yield strength (Donath, 1990). The critical allowable shear stress in the glide plane of crystals usually decreases with increasing temperature, and the yield strength will decrease for rocks deformed by this mechanism. So, the increase of the temperature of a body will be equal to the increase of the stress differ ence, and, correspondingly, to the relative decrease of the value of the required critical mass of a body of a specified composition, and vice versa. Usually, under laboratory conditions, the deforma tion rates range from 10–3 to 10–8 s–1. In nature, the deformation rate obtained from the observational data on the tectonic deformation on the Earth is 10–14 s–1 (Handing, 1990). Under low temperatures, the rate of deformation of a body’s shape may be even lower. However, contrary to the temperature, the yield strength weakly depends on the deformation rate at low temperatures (Donath, 1990). For some types of rocks under low temperatures, deformation may be also be produced by cataclastic flow. Cataclasis is char acterized by mechanical granulation including the destruction and rotation of crystalline grains together with intergranular gliding. In cataclastic flow, the deformation rate has almost no influence at all on the yield strength (Donath and Fruth, 1971).
2
1 0.6
0.7
0.8
0.9
ε
Fig. 9. The change of the ratio of deviatoric stresses R along a short axis in the equatorial plane of the prolate ellipsoid of a small body depending on the shape’s eccen tricity ε. The models for the Poisson ratio of the material ν = 0.31 and 0.17 are shown with solid and dotted curves, respectively. The ratio of deviatoric stresses R = 1 is shown with a dashed curve.
It is worth noting that the value of the yield strength is determined by rheology of a specified material and practically evades theoretical analysis. Because of this, for each of the considered materials, as a rule, the physical, mechanical, and rheological properties known from the experimental and observational stud ies are used. CONCLUSIONS The value and distribution of deviatoric stresses in the small bodies of the Solar System, possessing an ultimate strength and yield strength, depend on the chemical and mineralogical composition of these bod ies and are determined by such main parameters as the mass of the body, its density, size, shape, yield strength, and the Poisson ratio. A complete pattern of the distri bution of deviatoric stresses in a small body can be used both for estimating the main parameters of the natural or forced destruction of a solid small body and for estimating the physical and mechanical properties of the material and other unknown parameters of the small bodies. Within the boundaries of a small body, the devia toric stress increases from the body’s center to the sur face and from the pole to the equator. At some specific SOLAR SYSTEM RESEARCH
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critical values of the shape’s eccentricity of a small body, the inversion of the radial gradient of deviatoric stresses (the ratio of the maximum stresses on the sur s c face τ max and in the center τ max ) along the major axis of a small body is observed, when the maximum devi atoric stress in the center becomes higher than that on the surface. At small eccentricities of the shape of a small body, the value and distribution of deviatoric stresses obtained from the approximate solution are close to the exact solution and can be used in the engineering estimates of the approximate parameters of the gravi tational deformation of small bodies. For example, for the material with the Poisson ratio ν = 0.31 and a small ellipsoid eccentricity of the shape (ε = 0.4), the differ ence between the exact and approximate solutions does not exceed 10%. For a relatively large ellipsoid, eccentricities of the small body (ε > 0.4), which is characteristic for the dominant majority of small bod ies of the Solar System, the difference between the exact and approximate solutions becomes substantial and reaches the same or larger order of magnitude rel ative to the exact solution. In this case, the linear approximation is inapplicable to the estimation of the gravitational deformation parameters, since this will lead to a large error and an incorrect result. The centurylong competition with the thermal hypothesis has also been resolved, since it became one of the particular cases of a general analytic solution for the gravitational deformation of small bodies. In the small bodies, contrary to the planetary ones, there is no endogenous activity and internal thermal flux; con sequently, their temperature is determined by the inso lation level and depends on the location in the Solar System (Veeder et al., 1989). The change of the body’s temperature due to internal or external factors, e.g., in the case of radiogenic heat or dissipation or the tidal deformation energy, leads to the change in the yield strength of the material and is taken into account by the value of the threshold strength of deviatoric stresses, which, in turn, results in the change of all main parameters of the gravitational deformation. The cases of the thermal impact will be always notable for the anomalous values of the gravitational deformation parameters as compared to those characteristic of the temperatures in a specified region of the Solar System (Slyuta, 2013; Slyuta and Voropaev, 1997, 2014a, 2014b). For example, the transition observed between small and planetary bodies composed mostly of water ice is caused by the gravitational deformation of solid ice under low temperatures (Slyuta and Voropaev, 2014a). On the contrary, a substantial difference in the values of deviatoric stresses and yield strength of the silicate rocks of specific composition at corresponding tem peratures suggests that asteroid Vesta 4 was subjected to strong heating or, probably, to almost complete melting at the early stage of its life (Slyuta and Voro SOLAR SYSTEM RESEARCH
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paev, 2014b). Otherwise, Vesta would never have acquired a spherical equilibrium shape and, moreover, it would not have been differentiated. To remain a planetary body after the heat dumping and to maintain the isostasy mechanism, i.e., in essence, the gravita tional plastic deformation, working, as on other plan etary bodies, Vesta should have a mass exceeding its present value at the present temperature by more than an order of magnitude (Slyuta and Voropaev, 1997, 2014b). However, this is a subject of another paper. REFERENCES Chandrasekhar, S., Ellipsoidal Figures of Equilibrium, New Haven, CT.: Yale Univ. Press, 1969, p. 560. Clairaut, A., Theorie de la figure de la Terre. A Paris: J.B. Coignard, 1743. Croft, S.K., Proteus: geology, shape, and catastrophic destruction, Icarus, 1992, vol. 99, pp. 402–419. Darwin, G.H., The Tides and Kindred Phenomena in the Solar System, Boston, MA: Houghton, Mifflin and Co., 1898. de Laplace, P.S., Traite de mecanique celeste, Paris, 1825, vol. 2, 547 p. Donath, F.A. and Fruth, L.S., Dependence of strainrate effects on deformation mechanism and rock type, J. Geol., 1971, vol. 79, pp. 347–371. Donath, F.A., Deformations determination: experimental methods, in The Encyclopedia of Structural Geology and Plate Tectonics, Seyfert, C.S., Ed., N.Y.: Van Nos trand Reinhold Co., 1987, 876 p. Farinella, P., Paolicchi, P., Ferrini, F., Milani, A., Nobili, A.M., and Zappala, V., The shape of small Solar system bod ies: gravitational equilibrium vs. solidstate interac tions, in The Comparative Study of the Planets, Corad ini, A. and Fulchignoni, M., Eds., Dordrecht: D. Reidel Publ., 1982, pp. 71–77. Farinella, P., Milani, A., Nobili, A.M., Paolicchi, P., and Zappala, V., The shape of the small satellites of Sat urn—gravitational equilibrium vs solidstate strength, Moon Planets, 1983, vol. 28, pp. 251–258. Farinella, P., Milani, A., Nobili, A.M., Paolicchi, P., and Zappala, V., The shape and strength of small icy Satel lites, in The Ices in the Solar System, Klinger, J., et al., Eds., Dordrecht: D. Reidel Publ., 1985, pp. 699–710. Garber, R.I., Gindin, I.A., and Chirkina, L.A., Twinning and annealing of nonequilibrium ironnickel alloy (SikhoteAlin meteoritic iron), Meteoritika, 1963, no. 23, pp. 45–55. Heim, A., Untersuchungen über den Mechanismus der Gebirgsbildung. Basel: B. Schwabe, 1878. 592 p. Handin, J., Rocks rheology, in The Encyclopedia of Struc tural Geology and Plate Tectonics, Seyfert, C.S., Ed., N.Y.: Van Nostrand Reinhold Co., 1987, 876 p. Herschel, J.F.W., Outlines of Astronomy, Cambridge: Met calf and Co., 1849, 661 p. Hutton, J., Theory of the Earth with proofs and illustra tions. Edinbourgh: William Creech, 1975. V. I, 276 p. V. II, 547 p. Jacobi, C.G.J., Uber die figur des gleichgewichts, Poggen dorff Ann. Phys. Chem., 1834, vol. 33, pp. 229–238. Johnson, T.V. and McGetchin, T.R., Topography on satel lite surfaces and the shape of asteroids, Icarus, 1973, vol. 18, pp. 612–620.
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Translated by E. Petrova
SOLAR SYSTEM RESEARCH
Vol. 49
No. 2
2015
