The rotation rates of accreting planetesimals

There are obtained upper limits for the relative velocity at infinity of accreting planetesimals for a nearly constant mass of the largest accreting planetesimal and also in the case of variable mass. We conclude, that while the larger planets cannot be brought to the stage of rotational instability by stochastic collisions, the asteroids could be brought. provided that the relative velocities in the asteroid belt were larger than about 2 km s^–1.     

On the angular momentum in star formation

We discuss the rotation of interstellar clouds which are in a stage immediately before star formation. Cloud collisions seem to be the principal cause of the observed rotation of interstellar clouds. The rotational motion of the clouds is strongly influenced by turbulence.Theories dealing with the resolution of the angular momentum problem in star formation are classified into five major groups. We develop the old idea that the angular momentum of an interstellar cloud passes during star formation into the angular momentum of double star systems and/or circumstellar clouds.It is suggested that a rotating gas cloud contracts into a ring-like structure which fragments into self-gravitating subcondensations. By collisions and gas accretion these subcondensations accrete into binary systems surrounded by circumstellar clouds. Using some rough approximations we find analytical expressions for the semi-major axis of the binary system and for the density of the circumstellar clouds as a function of the initial density and of the initial angular velocity of an interstellar cloud. The obtained values are well within the observational limits.     

Single close encounters in the planetary problem

We consider a large massM and two small massesm ₁ andm ₂ (m ₁ m ₂;m ₁,m ₂M). The orbit ofm ₁ is initially circular and the motion ofm ₂ hyperbolic with respect toM. The orbital elements of the small masses are strongly modified after a close, single encounter betweenm ₁ andm ₂.An approximative method, similar to the theory of stellar encounters, is used to determine the probabilities of collisions, hyperbolas, direct and retrograde ellipses, as well as the mean values of the semimajor axes and their root mean square deviation after the encounter.The results are close to those which are obtained if the massm ₂ is negligibly small, (Mm ₁m ₂;m ₂ 0), as should be also expected on general grounds.     

A new variant of the large-impact hypothesis for the origin of the Moon

The hypothesis is advanced that after collision of a Mars-sized impact with the Earth, collisions between debris particles themselves are able to place enough material into Earth orbit, to form the Moon. Collision probability estimates show that the collision frequency is high enough to place about one lunar mass into Earth orbit, if the average semimajor axis is of order of the Earth's Roche limit of 18 500 km.     

Trojan orbits with mass exchange of the primaries

The librational motion round the Lagrangian triangular points L₄, L₅ with mass exchange of the primaries is investigated according to Brown's theory. The results are the same as in the case of isotropic mass variation studied earlier (Horedt, 1974a): (i) The extrema of the elongations with respect to the small mass are unaffected by mass exchange. (ii) The equations for the extrema of the Trojan's distance from the Sun and for the libration period are formally the same as in the constant mass problem, but with the understanding that the masses are now time dependent quantities. A Trojan cannot leave the libration domain due to a mass variation of the primaries obeying the constraints from Equation (2.4), with a mass ratio of the primaries m/M≤0.0401.     

Two-and three-layer models of Uranus

We present simple two-layer models of Uranus with rocky core and polytropic envelope satisfying exactly the observed mass, radius and the gravitational moments. The models show that the value of the fourth order zonal harmonic isJ ₄ –38×10^–6, whileJ ₆ 10^–6. More elaborate threelayer models fail to satisfy the observational constraints of the ice/rock ratio and/or of the rotation period. We conclude that three-layer models with uniform chemical composition in each layer may be too restrictive. More realistic models should account for variable chemical composition within each layer.     

On seeliger's paradox

It is shown that there do not exist spatially-homogeneous and anisotropic Bianchi type-I cosmological models in Rosen's (1973) bi-metric theory of gravitation when the source of the gravitational field is a perfect fluid distribution. A vacuum cosmological model, in this theory, is presented and studied.     

Infinitesimal librations with variable primary masses

The motion around the libration points is studied with Charlier's first order theory assuming the isotropic mass variation of the primaries does not exceed the order of the small primary and the derivatives of the masses with respect to time are negligible second order quantities. The results are analogous to those obtained with constant primary masses.     

Cosmogony of the solar system

In Sections 1–6, we determine an approximate analytical model for the density and temperature distribution in the protoplanetary could. The rotation of the planets is discussed in Section 7 and we conclude that it cannot be determined from simple energy conservation laws.The velocity of the gas of the protoplanetary cloud is found to be smaller by about 5×10³ cm s^–1 in comparison to the Keplerian circular velocity. If the radius of the planetesimals is smaller than a certain limitr ₁, they move together with the gas. Their vertical and horizontal motion for this case is studied in Sections 8 and 9.As the planetesimals grow by accretion their radius becomes larger thanr ₁ and they move in Keplerian orbits. As long as their radius is betweenr ₁ and a certain limitr ₂ their gravitational interaction is negligible. In Section 10, we study the accretion for this case.In Section 11, we determine the change of the relative velocities due to close gravitational encounters. The principal equations governing the late stages of accretion are deduced in Section 12, In Section 13 there are obtained approximate analytical solutions.The effect of gas drag and of collisions is studied in Sections 14 and 15, respectively. Numerical results and conclusions concerning the last and principal stage of accretion are drawn in Section 16.