scholarly journals V. The gravitational stability of the Earth

1908 ◽  
Vol 207 (413-426) ◽  
pp. 171-241 ◽  
Keyword(s):  
Initial Stress ◽  
Gravitational Instability ◽  
Spherical Shape ◽  
Solid Sphere ◽  
Theory Of Elasticity ◽  
The Body ◽  
The Self ◽  
The Earth ◽  
Symmetrical Configuration ◽  
The Stability

If in a gravitating body there occurs a displacement which involves alteration of density, there must be a tendency for the material to move towards the places where the density is increased, and away from the places where the density is diminished. The effect of this tendency, if it were not held in check, would be to accentuate local alterations of density. In any body the tendency is partially held in check by the elasticity of the body, and, in particular, by the elastic resistance which the body offers to compression. If this resistance is sufficiently great, the body is stable, in spite of the tendency to instability which arises from gravitation. It is important to determine the conditions of stability for bodies of various forms and constitutions, with various distributions of density. The problem of the stability of spherically symmetrical configurations of a quantity of gravitating gas has been investigated by J. H. Jeans, and he has drawn from his investigations some interesting conclusions in regard to the course of evolution of stellar and planetary systems. In a subsequent memoir he proceeded to investigate a similar problem in regard to gravitating bodies of a more coherent character. A gravitating solid body, such as a planet may be conceived to he, might exist in a spherical shape with a spherically symmetrical distribution of density. In the absence of gravitation there could he no question of instability. The effect of any local condensation would be to set up vibrations, and the frequency of the vibration of any spherical harmonic type would depend upon the elasticity of the material. If the resistance of the material to compression is sufficiently high the stability persists in spite of gravitation. There are thus two competing agencies: gravitation, tending to instability, and the elasticity of the material, tending to stability. In a general way it is clear that, as the elasticity diminishes, the frequency of vibration of any type also diminishes; and, if the frequency can vanish for sufficiently small elasticity, the planetary body possessing such elasticity cannot continue to exist in the spherically symmetrical configuration. The problem is to determine the conditions as regards elasticity in which the instability occurs. A grave difficulty presents itself at the outset. In the equilibrium configuration the gravitating planet is in a state of stress; and, in a body of such dimensions as the Earth, this stress is so great that the total stress existing in the body when it vibrates cannot be calculated by the ordinary methods of the theory of elasticity. In that theory it is ordinarily assumed that the body under investigation is in a state so little removed from one of zero stress that the strain, measured from this state as a zero of reckoning, is proportional to the stress existing at any instant. In order that this assumption may he valid, it is necessary that the strain which is calculated by means of it should be so small that its square may be neglected. Now if we apply the equations of the ordinary theory to the problem of a solid sphere strained by its own gravitation, and if we take the sphere to he of the same size and mass as the Earth, and the material of which it is composed to possess moduluses of elasticity as great as those of ordinary steel, we find that the strains may be as great as and thus the strains are much too great for the assumption to he valid. The initial stress existing in the gravitating planet, the stress by which the self-attraction of the body is equilibrated, is much too great to perm it of the application of the ordinary theory. The same difficulty presents itself in every problem concerning the elasticity of a gravitating planet, for example, in the problem of tidal deformation or of the stress produced in the interior by the w eight of continents. In these problems the difficulty was turned by Lord Kelvin and Sir G. H. Darwin by taking the modulus of compression to be much greater than that of any known material, in other words, by taking the material to be incompressible. Their object was to determine the degree of rigidity which must be assigned to the Earth , and for that object it is permissible to turn the difficulty in this way. When the problem is that of gravitational instability this artifice cannot be adopted, because the whole question is that of the degree of compressibility which is admissible if the gravitating planet is to be stable in a spherically symmetrical configuration. The artifice adopted by Jeans (1903) consisted in annulling the initial stress by introducing an imagined external field of force to equilibrate the self-attraction of the planet.

scholarly journals The gravitational stability of the Earth

1907 ◽  
Vol 79 (529) ◽  
pp. 194-199 ◽  
Keyword(s):  
Initial Stress ◽  
Gravitational Instability ◽  
Direct Method ◽  
Elastic Solid ◽  
Frequency Equation ◽  
The Body ◽  
Additional Stress ◽  
Initial State ◽  
Planetary Body ◽  
The Earth

The problem of determining the conditions, as regards elasticity, in which a body of planetary dimensions can be gravitationally unstable, was attacked by J. H. Jeans. To turn the difficulty arising from the fact that such a body is necessarily in a state of “ initial stress,” by which gravitation is balanced throughout its interior, he introduced an artificial external field of force to balance gravitation in the undisturbed body, and he thus dispensed with initial stress. A more direct method of treating the problem has been advocated by Lord Rayleigh. In this method the stress at any point of the body, when disturbed from its equilibrium state, is taken to be com­pounded of two stress-systems ; an initial stress of the nature of hydrostatic pressure, balancing gravitation in the undisturbed body, and an additional stress connected with the strain (reckoned from the initial state) by the same formulæ as connect stress with strain in a homogeneous isotropic elastic solid body which is slightly strained from an initial state of zero stress. In the first part of the present paper the equations of vibratory motion of the planetary body, assumed to be initially spherical, are formed in accordance with this method, and they are solved by means of spherical harmonics. The frequency equation is obtained, and the conditions that it may be satisfied by a zero value of the frequency are found. When these conditions are fulfilled the body is in a state of gravitational instability.

‘A pinch of unseen, unguarded dust’: The World and Self in Thomas Hardy's Poems

2018 ◽  
Vol 8 (1) ◽  
pp. 49-66
Author(s):  
Monika Szuba
Keyword(s):  
The Body ◽  
The Self ◽  
Phenomenological Philosophy ◽  
The Earth ◽  
The World ◽  
The Universe ◽  
Recurrent Theme ◽  
Body Subject

The essay discusses selected poems from Thomas Hardy's vast body of poetry, focusing on representations of the self and the world. Employing Maurice Merleau-Ponty's concepts such as the body-subject, wild being, flesh, and reversibility, the essay offers an analysis of Hardy's poems in the light of phenomenological philosophy. It argues that far from demonstrating ‘cosmic indifference’, Hardy's poetry offers a sympathetic vision of interrelations governing the universe. The attunement with voices of the Earth foregrounded in the poems enables the self's entanglement in the flesh of the world, a chiasmatic intertwining of beings inserted between the leaves of the world. The relation of the self with the world is established through the act of perception, mainly visual and aural, when the body becomes intertwined with the world, thus resulting in a powerful welding. Such moments of vision are brief and elusive, which enhances a sense of transitoriness, and, yet, they are also timeless as the self becomes immersed in the experience. As time is a recurrent theme in Hardy's poetry, this essay discusses it in the context of dwelling, the provisionality of which is demonstrated in the prevalent sense of temporality, marked by seasons and birdsong, which underline the rhythms of the world.

Capillary instability of a streaming fluid-core liquid jet underself-gravitating forces

2007 ◽  
Vol 85 (8) ◽  
pp. 849-861
Author(s):  
A E Radwan ◽  
E E Elmahdy
Keyword(s):  
Capillary Force ◽  
Gravitational Instability ◽  
Axial Flow ◽  
The Self ◽  
Liquid Jet ◽  
Fluid Density ◽  
Fluid Core ◽  
The Stability ◽  
Stable Domains ◽  
Axisymmetric Disturbances

The stability criterion of a fluid cylinder (density ρ(1)) embedded in a different fluid (density ρ(2)) is derived and discussed. The model is capillary unstable in the domain 0 < x < 1 as m = 0 where x and m are the axial and transverse wave numbers, while it is stable in all other domains. The densities ratio ρ(2) / ρ(1) decreases the unstable domains but never suppress them. The streaming increases the unstable domains. Gravitationally, in m = 0 mode the model is unstable in the domain 0 < x < 1.0668 as ρ(2) < ρ(1), while as ρ(2) = ρ(1) it is marginally stable but when ρ(2) > ρ(1) the model is purely unstable for all short and long wavelengths. In m ≠ 0 modes, the self-gravitating model is neutrally stable as ρ(1) = ρ(2), ordinarily stable as ρ(2) < ρ(1), but is purely unstable as ρ(2) > ρ(1). The streaming destabilizing effect makes the self-gravitating instability worse and shrinks the stable domains. The stability analysis of the model under the combined effect of the capillary and self-gravitating forces is performed analytically and verified numerically. When ρ(2) < ρ(1) the capillary force and the axial flow have destabilizing influences but the ratio of the densities ρ(2) / ρ(1) has a stabilizing effect on the gravitating instability. If ρ(1) = ρ(2), the streaming is destabilizing but the capillary force is strongly stabilizing and could suppress the gravitational instability. When ρ(2) > ρ(1) the capillary force improves the gravitational instability and creates domains of much stability and moreover the instability of the self-gravitating force disappears in several cases of axisymmetric disturbances. PACS No.: 47.17+e

scholarly journals Gravitational instability and the figure of the Earth

1917 ◽  
Vol 93 (653) ◽  
pp. 413-417 ◽  
Keyword(s):  
Gravitational Instability ◽  
Elastic Solid ◽  
Additional Stress ◽  
Stable Configuration ◽  
Uniform Density ◽  
The Earth ◽  
Symmetrical Configuration ◽  
Additional Strain ◽  
Figure Of The Earth ◽  
Equal Density

1. In a paper published in 1902, “On the Vibrations and Stability of a Gravitating Planet,” I attempted to examine whether, owing to gravitational instability, an arrangement in concentric spherical shells might not be unstable for a planet of the size of our earth. It was not found possible to get exact results; progress only seemed possible by supposing that the problem would in its essentials be similar to a highly artificial problem in which gravitation was exactly annulled in the symmetrical configuration by the superposition of a countervailing field, so that in this state the matter was of uniform density and unstressed. The only possible justification for such an assumption was that I found myself unable to make any progress, either physically or mathematically, without it. Subject to this assumption, I found that, for a mass of the structure of our earth in its present state, the symmetrical configuration would undoubtedly be stable. It seemed likely, however, that in a past age in which the rigidity of the earth was considerably, although not enormously, less than it now is, the symmetrical configuration might have been unstable; the stable configuration would have been an unsymmetrical one, in which the surfaces of equal density were spherical but not concentric, so that the point of maximum density coincided neither with the centre of gravity nor the centre of figure of the earth’s surface. It was suggested that traces of this unsymmetrical configuration might still be found in the arrangement of oceans and plateaux on the earth’s surface. In 1906, Lord Rayleigh, in a paper “On the Dilatational Stability of the Earth,” drew attention to the wide departure from actuality implied in my assumptions, and suggested a set of assumptions which had more justification in nature. Lord Rayleigh’s suggestion was, in brief, that we might regard the symmetrical configuration as one in which the gravitational forces were balanced by hydrostatic pressures only. When disturbed, there is an additional stress superposed on to the hydrostatic pressure, and Lord Rayleigh suggested that this might be supposed connected with the additional strain— i. e ., the strain above and beyond that in the initial configuration by the ordinary elastic solid relations.

The body as first prior: Interoceptive predictive processing and the primacy of self-models

2018 ◽  
Author(s):  
Micah Allen ◽  
Manos Tsakiris
Keyword(s):  
The Body ◽  
The Self ◽  
Predictive Processing ◽  
Interoceptive Awareness ◽  
Self Awareness ◽  
Growing Body ◽  
Cortical Hierarchy ◽  
Privileged Status ◽  
The Stability

Embodied predictive processing accounts place the visceral milieu, its homeostatic functioning, and our interoceptive awareness thereof on the center stage of self-awareness. Starting from the privileged status that homeostatic priors have within the cortical hierarchy of an organism whose main imperative is to maintain homeostasis, we focus on the mechanisms that underlie interoceptive precision and its impact on embodiment and cognition. Beyond their privileged status for ensuring the stability of organism, this chapter considers the psychological importance that interoceptive priors and interoceptive precision have for self-awareness and the grounding of a coherent self-model. In a manner analogous to the role that interoception plays for homeostasis, interoception at the psychological level seems to contribute to the stability of self-awareness. This psychological role of interoception is illustrated by a growing body of research that considers the antagonism but also the integration between exteroceptive and interoceptive models of the self.

Characterization of milkweed-seed gust response

2021 ◽  
Vol 16 (6) ◽  
pp. 066017
Author(s):  
Joshua N Galler ◽  
David E Rival
Keyword(s):  
Motion Tracking ◽  
Spherical Shape ◽  
Solid Sphere ◽  
The Body ◽  
Initial Flow ◽  
Flow Perturbation ◽  
Flow Acceleration ◽  
Porous Bodies ◽  
Aerodynamic Properties ◽  
Gust Response

Abstract Inspired by the reproductive success of plant species that employ bristled seeds for wind-borne dispersal, this study investigates the gust response of milkweed seeds, selected for their near-spherical shape. Gust-response experiments are performed to determine whether these porous bodies offer unique aerodynamic properties. Optical motion-tracking and particle image velocimetry (PIV) are used to characterize the dynamics of milkweed seed samples as they freely respond to a flow perturbation produced in an unsteady, gust wind tunnel. The observed seed acceleration ratio was found to agree with that of similar-sized soap bubbles as well as theoretical predictions, suggesting that aerodynamic performance does not degrade with porosity. Observations of high-velocity and high-vorticity fluid deflected around the body, obtained via time-resolved PIV measurements, suggest that there is minimal flow through the porous sphere. Therefore, despite the seed’s porosity, the formation of a region of fluid shear, accompanied by vorticity roll-up around the body and in its wake, is not suppressed, as would normally be expected for porous bodies. Thus, the seeds achieve instantaneous drag exceeding that of a solid sphere (e.g. bubble) over the first eight convective times of the perturbation. Therefore, while the steady-state drag produced by porous bodies is typically lower than that of a solid counterpart, an enhanced drag response is generated during the initial flow acceleration period.

‘Man Is Ill Because He Is Badly Constructed’: Artaud, Klossowski and Deleuze in Search for the Earth Inside

2011 ◽  
Vol 5 (1) ◽  
pp. 18-34 ◽  
Author(s):  
Rick Dolphijn
Keyword(s):  
Performance Art ◽  
Holy Spirit ◽  
The Body ◽  
Deleuze And Guattari ◽  
The Earth ◽  
The Holy Spirit ◽  
Life Itself ◽  
Body Without Organs ◽  
Ways Of Thinking ◽  
And Performance

Starting with Antonin Artaud's radio play To Have Done With The Judgement Of God, this article analyses the ways in which Artaud's idea of the body without organs links up with various of his writings on the body and bodily theatre and with Deleuze and Guattari's later development of his ideas. Using Klossowski (or Klossowski's Nietzsche) to explain how the dominance of dialogue equals the dominance of God, I go on to examine how the Son (the facialised body), the Father (Language) and the Holy Spirit (Subjectification), need to be warded off in order to revitalize the body, reuniting it with ‘the earth’ it has been separated from. Artaud's writings on Balinese dancing and the Tarahumaran people pave the way for the new body to appear. Reconstructing the body through bodily practices, through religion and above all through art, as Deleuze and Guattari suggest, we are introduced not only to new ways of thinking theatre and performance art, but to life itself.

THE CHARACTERISTICS OF ARTIFICIAL RICE WITH SEAWEED Eucheuma cottonii ADDITION AS A DIETARY FIBER SOURCE

2014 ◽  
Vol 6 (1) ◽  
Author(s):  
Natalia Prodiana Setiawati ◽  
Joko Santoso ◽  
Sri Purwaningsih
Keyword(s):  
Blood Glucose ◽  
Dietary Fiber ◽  
Dietary Fibre ◽  
Extrusion Process ◽  
The Body ◽  
Starch Digestibility ◽  
Food Commodities ◽  
The Stability ◽  
Extrusion Technology ◽  
Eucheuma Cottonii

The utilization of local food commodities such as corn and cassava with seaweed addition as a dietary fiber source for producing artificial rice through extrusion technology is an  alternative for food diversification. The research was carried out to find out the best composition (rice, corn, cassava, and seaweed) and temperature of extrusion process on making artificial rice and the influence of dietary fibre on sensory properties and physicochemical. The composition of rice, corn, and cassava in proportion  of 1:3:1 with 20% seaweed, Eucheuma cottonii, addition and temperature extruder of 90 °C were selected as the best product for artificial rice. The  sensory evaluation was 8.02±0.21 (people’s preference). In physicochemical properties, dietary fiber significantly affected on low bulk density and starch digestibility. This condition is very good for health especially in maintaining the stability of blood glucose in the body. Keywords: artificial rice, composition, extrusion, seaweed, dietary fibre, temperature

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