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UBC Theses and Dissertations

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UBC Theses and Dissertations

Structure and stability of self-gravitating discs Davies, John Bruce


Three centuries ago, the concept of the Solar Sytem being formed from an unstable disc was initially proposed. This research examines this cosmogony by the classical technique of initially obtaining the density structure of steady state discs, and gravitational instability of such systems Is then investigated in order to examine correlations between observed phenomena in the Solar System and predictions of the theory. A fluid mechanical approach to the steadystate axisymmetric structure is formulated for isothermal and polytropic gas systems, with uniform or radially dependent rotation. The equations are reduced to a single quasi-linear elliptical partial differential equation governing density, and known external boundary conditions are necessary to yield an unique density solution. When the external density is non-zero, flattened discs are possible solutions of the basic system. Two asymptotic techniques in spherical and cylindrical coordinates are created to obtain approximate solutions of the steadystate structure. Both techniques show that a self-consistent disc is composed of a high-density central bulge encircled by a low-density flat outer disc. Gravitational instability in gaseous discs is now formulated by the linear perturbation of the fundamental variables, density, pressure, gravitational potential and velocity. As the Solar System is essentially a planar structure, axisymmetric radial instability along the equatorial plane of rotation is examined. Such ring type modes are shown to be unstable to shear and tend to self-coverage. A dispersion relation is obtained analytically which indicates that the wavelength between rings is approximately inversely proportional to the square root of the steadystate density at marginal stability. However for the pure gas disc, the wavelengths are too long for any correspondence with the present spacing of the Planets. As usual, the presence of dust is invoked close to the equatorial plane. Radial instability in this gas-dust disc has a dispersion relation for the resultant wave in which the gas and dust move together such that the density term is multiplied by the dust-gas mass loading ratio. Thus the wavelengths at neutral stability will be correspondingly shorter and a correlation of ring density maxima with Planetary positions in the Solar System is obtained for reasonable values of three dimensionless parameters. If any planets exist outside Pluto the theory shows their distance apart can be expected to be similar to that of the Outer Planets, 10 a.u. Solar Systems formed by this type of instability in self-gravitating dust-gas discs can be expected to have linearly increasing planetary distances close to the central Sun "(Titius-Bode Law) with a more constant spacing further out as illustrated by our Solar System.

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