The moon and the planets

, Volume 19, Issue 3, pp 341-398

Origin of the solar system

I: Gravitational contraction of the turbulent protosun and the shedding of a concentric system of gaseous laplacian rings
  • A. J. R. PrenticeAffiliated withDept. of Mathematics, Monash University

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A theory for the origin of the solar system, which is based on ideas of supersonic turbulent convection and indicates the possibility that the original Laplacian hypothesis may by valid, is presented.

We suggest that the first stage of the Sun's formation consisted of the condensation of CNO ices (i.e. H2O, NH3, CH4,...) and later H2, including He as impurity atoms, at interstellar densities to from a cloud of solid grains. These grains then migrate under gravity to their common centre of mass giving up almost two orders of magnitude of angular momentum through resistive interaction with residual gases which are tied, via the ions, to the interstellar magnetic field. Grains rich in CNO rapidly dominate the centre of the cloud at this stage, both giving up almost all of their angular momentum and forming a central chemical inhomogeneity which may account for the present low solar neutrino flux (Prentice, 1976). The rest of the grain cloud, when sufficiently compressed to sweep up the residual gases and go into free fall, is not threatened by rotational disruption until its mean size has shrunk to about the orbit of Neptune.

When the central opacity rises sufficiently to halt the free collapse at central density near 10−13 g cm−3, corresponding to a mean cloud radius of 104 R , we find that there is insufficient gravitational energy, for the vaporized cloud to acquire a complete hydrostatic equilibrium, even if a supersonic turbulent stress arising from the motions of convective elements becomes important, as Schatzman (1967) has proposed. Instead we suggest that the inner 3–4% of the cloud mass collapses freely all the way to stellar size to release sufficient energy to stabilize the rest of the infalling cloud. Our model of the early solar nebula thus consists of a small dense quasi-stellar core surrounded by a vast tenuous but opaque turbulent convective envelope.

Following an earlier paper (Prentice, 1973) we show how the supersonic turbulent stress\((\rho _t v_t ^2 ) = \beta \rho GM(r)/r\), where β is called the turbulence parameter, ρ is the gas density andM(r) the mass interior to radiusr causes the envelope to become very centrally condensed (i.e. drastically lowers its moment-of-inertia coefficientf) and leads to a very steep density inversion at its photosurface, as well as causing the interior to rotate like a solid body. As the nebula contracts conserving its angular momentum the ratio θ of centrifugal force to gravitational force at the equator steadily increases. In order to maintain pressure equilibrium at its photosurface, material is extruded outwards from the deep interior of the envelope to form a dense belt of non-turbulent gases at the equator which are free of turbulent viscosity. If the turbulence is sufficiently strong, we find that when θ→1 at equatorial radiusR e=R0, corresponding to the orbit of Neptune, the addition of any further mass to the equator causes the envelope to discontinuously withdraw to a new radiusR e>R0, leaving behind the circular belt of gas at the Kepler orbitR 0. The protosun continues to contract inwards, again rotationally stabilizing itself by extruding fresh material to the equator, and eventually abandoning a second gaseous ring at radiusR 1, and so on. If the collapse occurs homologously the sequence of orbital radiiR n of the system of gaseous Laplacian rings satisfy the geometric progression
$$R_n /R_{n + 1} = [1 + m/Mf]^2 = constant, n = 0, 1,2, \ldots ,$$
analogous to the Titius-Bode Law of planetary distances, wherem denotes the mass of the disposed ring andM the remaining mass of the envelope. Choosing a ratio of surface to central temperature for the envelope equal to about 10−3 and adjusting the turbulence parameter β∼~0.1 so thatR n/Rn+1 matches the observed mean ratio of 1.73, we typically findf=0.01 and that the rings of gas each have about the same mass, namely 1000M of the solar material. Detailed calculations which take into account non-homologous behaviour resulting from the changing mass fraction of dissociated H2 in the nebula during the collapse do not appreciably disturb this result. This model of the contracting protosun enables us to account for the observed physical structure and mass distribution of the planetary system, as well as the chemistry. In a later Paper II we shall examine in detail the condensation of the planets from the system of gaseous rings.