The angular momentum problem during star formation Magnetically linked, aligned rotators. I - an exact, time-dependent solution

We study the transfer of angular momentum J and energy E among (and away from the systems of) N fragments (or clouds or cores), modeled by uniform, rigidly rotating disks, embedded in a background ("external") medium of uniform density, and threaded along their common (z-)axis of symmetry by an initially uniform magnetic field B frozen in both media. An interior fragment is initially imparted an arbitrary angular velocity Ω ₀ with respect to the other fragments and the external medium. The torsional Alfven waves generated by this fragment's rotation bounce back and forth among fragments (with a crossing time τ₀ ≡ τ_A,ext,ext between consecutive fragments) setting the fragments into successive high and low spin states before they eventually carry the rotational kinetic energy of the system (of fragments plus interfragment medium) away into the medium beyond the outermost fragments. Explicit solutions are presented for the three-disk problem, which contains all the important ingredients of the general, N-disk problem. There is only one dimensionless free parameter σ in the problem, namely, the ratio of half the moment of inertia of a fragment I_fr and that of the medium between two consecutive fragments I_ext; it normally decreases as a fragment (of constant mass) contracts gravitationally. The exact MHD solution for σ > 1 can be represented well by a mechanical analog. Each disk executes a nearly simple harmonic rotation with a period τ_d = 2 π (2σ/3)^1/2 τ₀, modified only by the slow leakage of rms angular momentum out of the system with a characteristic time 3στ₀ at first and 12στ₀ at later times. The full MHD solution, consisting of three sets of wave packets (i.e., outgoing, incoming, and "transmitted"), is needed to represent accurately the cases with σ ≍ 1. In the limit σ < 1, however, the exact solution is represented accurately by the interaction of each fragment with one wave packet at a time. The net angular momentum of the system decreases to a negligible value in a time less than 2 τ₀, but considerable energy and rms angular momentum remain in the system for a much longer time. These quantities (normalized to their initial values) decrease in time in a step-function manner, with the mean behavior being represented by the power laws ≺E_sys(t)≻ = [2(t/τ₀) + 1]^-1/2 and J _{sys, rms}(t) = [2(t/τ₀) + 1]^{- 1/4} respectively. Solutions for specific values of σ are presented and applied to star formation in the accompanying paper.