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Self-gravitating system

A very important result can be obtained when an entire self-gravitating system is in equilibrium, i.e. if Eq. (17) is satisfied at all r, since it becomes possible to derive a simple relation between average internal pressure and the gravitational potential energy of the system. This is done by multiplying both sides of Eq. (17) by 47rr3 and integrating from r = 0 to r = R to obtain... [Pg.21]

The statement that the average pressure needed to support a self-gravitating system is one third of the stored gravitational energy is called the Virial Theorem. The physical meaning of pressure depends on the system itself, but it can be applied to clusters of galaxies as well as to individual stars. Let us consider two cases for gas in a star, where the equation of state relates the pressure of the gas (P) to the translational kinetic energy of the gas particles ( kin)-... [Pg.22]

Then for a self-gravitating system of volume V and gravitational energy Sgrav, the gravitational and kinetic energies are related by... [Pg.22]

If fxi and p2 refer to the same substance and this substance is in equilibrium throughout the system, so that m = fx2, the above equations would describe, say, a self-gravitating spherical mass of liquid water divided into two arbitrary volumes. The values of pi and ps in this case would be the external compressive stresses on the surfaces of each of these volumes, not the volumetric means of the principal stresses on the surfaces of the elementary volumes into which each might have been subdivided. If the p/s be the mean principal stresses on the elementary volumes of the whole of mass fluid mass, we would obtain... [Pg.260]

Disk systems composed of stars and presumably gas and which are self-gravitating are intrinsically unstable to the growth of a spectrum of longitu-... [Pg.502]

These results have not identified any limitation on the lower limit to sizes that can be assembled using capillary forces. In fact, since capillary forces should become stronger relative to other forces (primarily gravitational and electrostatic) at smaller size scales [ref. 64], we believe 3D MESA based on capillarity should apply to even smaller pieces (<0.5 pm). The inference that capillary forces may be appropriate for self-assembly in systems of very small components is compatible with other evidence (Table 4.1). [Pg.140]

Without this luxury Laplace set out to assess the natural stability of the system. When asked by Napoleon to clarify the role of God in this, he replied "I have no need of that hypothesis". The first important result, which he demonstrated mathematically, was that the irregularities in the eccentricities and inclinations of planetary orbits oscillate about fixed values, without amplification, and hence never deviate too far from the ideal orbits. He could therefore theorize that the solar system remains indefinitely stable. Like some self-correcting clockwork, driven by the universal force of gravitation, the solar system was concluded to be inherently stable and predictable. Laplace saw no reason why the whole universe should not be dynamically stable in the same sense. He claimed that ... [Pg.39]


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See also in sourсe #XX -- [ Pg.499 ]




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