Non-zero Inclination Research Articles

On the properties of curvilinear shock waves “smearing” in calculations by the particle-in-cell method

Abstract It is shown by the 2-dimensional numerical computations of the cylindrical shock wave propagation using the particle-in-cell method tha the width of a “smeared” shock wave zone reaches its minimum at some non-zero inclination angle of the velocity vector with respect to the x-axis in the “smeared” shock wave. Results of theoretical analysis of the characteristics of the characteristics of the smeared shock wave zone width dependence upon the flow orientation with respect to the rectangular computing mesh lines carried out by the use of the first differential approximation method agree well with calculation results.

Tidal friction and the early history of the Moon's orbit

The effect of strong tidal friction on the moon's orbit in its early history is examined. The results show that the moon could have formed in an equatorial orbit about the earth, whether by fission or accretion, provided that the rheology of the early earth was that of a highly viscous Newtonian liquid and the orbit suffered a perturbation out of the earth's equatorial plane. The present 5° inclination of the lunar orbit to the ecliptic can be explained if the moon's orbit was perturbed about 3° out of the earth's equatorial plane and the earth's viscosity was not less than 1018 P. It is also found that if the earth's viscosity were greater than 1016 P, then under certain conditions the radius of the moon's orbit might actually decrease temporarily and then increase and further, that an upper limit could be placed on the inclination of the moon's orbit to the earth's equator when the moon was slightly less than 3.83 RE from the earth, regardless of the moon's prior orbital history. These results depend critically upon a resonance that takes place at the earth-moon distance of 3.83 RE, where the moon's orbital period is twice the earth's rotational period. The O1 tide dominates the orbital evolution in the neighborhood of this distance, both the phase lag and the amplitude being large. The phase lags of the M2 and K1 tides remain near 90°, and their amplitudes near zero throughout this region. These conclusions differ from those of Gerstenkorn (1969) and Goldreich (1966), who found that the present nonzero inclination of the lunar orbit to the ecliptic implies a steep inclination of the orbit to the earth's equatorial plane in the early history of the earth-moon system, thereby ruling out an equatorial origin for the moon, such as one by fission or accretion. Their results are thus shown to be valid only for particular rheological models of the earth.