Expanding, homogeneous, ellipsoidal density perturbations. II. A simple theory for the acquisition of angular momentum by tidal torques

Abstract

AbstractA simple theory is built up for the acquisition of angular momentum by tidal torques in expanding, homogeneous, ellipsoidal density perturbations from the epoch when inhomogencities begin to grow to the present epoch. The expansion is described according to an analytical approximation already known in the literature, which well holds before the turnaround of a sphere with same mass and initial overdensity as the ellipsoid, and holds no longer after a singular configuration (i.e., with a null volume) is attained by the above mentioned sphere. Though calculations at latest stages turn out to be physically meaningless, it is inferred that the spin growth during the virialized phase is negligible with respect to the spin growth during the expansion and relaxation phase. The evolution of typical inhomogeneities with initial axis ratios ϵ2i = 0.7, ϵ3i = 0.5 and masses MB = 1012 MȮ is computed, leading to: (i) a spin parameter λ = 0.054 at the beginning of strong decoupling from the Hubble flow, in cloes agreement with both analytical and numerical attempts related to different theories; (ii) an angular momentum J = 1.62 – 3.17 · 1075 g cm2 s−1 at the turnaround of a sphere with same mass and initial overdensity as the ellipsoid, within the correct order of magnitude deduced for the Galaxy; (iii) an angular momentum J ≈ 3 JD and J ≈ 2JD at the singular configuration and at the turnaround of the above mentioned sphere, respectively, with JD angular momentum gained before the strong decoupling from the Hubble flow. The features (i) and (iii) are common to all masses. The evolution of inhomogeneities with rms overdensities, same initial volume and typical galactic mass MB = 1012 MȮ is computed for a wide range of initial shapes, leading to an non‐dimensional angular momentum Λ with the following behaviour: (i) it is null for both spherical and flat or oblong configuration; (ii) it increases passing from prolate to oblate configurations with same minor axis; (iii) it changes passing from flat to prolate configurations with same middle axis, increasing or increasing and then decreasing according if prolate configurations are spindle‐like enough or ball‐like enough, repectively; (iv) the larger non‐dimensional spin growth occurs for oblate configurations with e3i ≈ 0.6. The above features are common to all masses and peak heights. In conclusion it cannot be asserted, but it may safely be expected, that (contrary to what happens in many other theories) no arbitrary cut‐off on tidal torques has to be advocated in the current theory, to yield an amount of angular momentum close to what has been deduced from observation. A complete answer on this point might come only from a more refined description of the expansion.