Prograde spin-up during gravitational collapse

¹ Anton Pannekoek Institute, University of Amsterdam, Science Park 904, Amsterdam, The Netherlands
e-mail: r.g.visser@uva.nl
² Institute of Astronomy, University of Cambridge, Madingley Road, Cambridge CB3 0HA, UK
e-mail: mgb52@cam.ac.uk

Received: 7 April 2022
Accepted: 18 May 2022

Abstract

Asteroids, planets, stars in some open clusters, and molecular clouds appear to possess a preferential spin-orbit alignment, pointing to shared processes that tie their rotation at birth to larger parent structures. We present a new mechanism that describes how collections of particles, or “clouds”, gain a prograde rotational component when they collapse or contract while subject to an external, central force. The effect is geometric in origin, as relative shear on curved orbits moves their shared center-of-mass slightly inward and toward the external potential during a collapse, transferring orbital angular momentum into aligned (prograde) rotation. We perform illustrative analytical and N-body calculations to show that this process of prograde spin-up proceeds quadratically in time (δL_rot ∝ t²) until the collapse nears completion. The total rotational gain increases with the size of the cloud prior to its collapse, δL_rot /L_H ∝ (R_cl/R_H)⁵, and typically with distance to the source of the potential (L_H ∝ r₀). For clouds that form at the interface of shear and self-gravity (R_cl ~ R_H), prograde spin-up means that even setups with large initial retrograde rotation collapse to form prograde-spinning objects. Being a geometric effect, prograde spin-up persists around any central potential that triggers shear, even those where the shear is strongly retrograde. We highlight an application to the Solar System, where prograde spin-up can explain the frequency of binary objects in the Kuiper belt with prograde rotation.