The angular momentum problem and magnetic braking during star formation: Exact solutions for an aligned and a perpendicular rotator

The problems of fragmentation, angular momentum, and magnetic flux during star formation are reviewed briefly. Then the resolution of the angular momentum problem through magnetic braking is studied rigorously. A disk-like interstellar cloud of uniform density ρ_cl is given an initial angular velocity Ω_o about its axis of symmetry, which is aligned with an initially uniform, frozen-in magnetic field. Torsional Alfvén waves transport angular momentum from the cloud to the external medium, which has a uniform density ρ_ext. The angular velocity of the cloud (Ω_cl) is determined analytically as a function of space and time for different ratios ρ_cl/ρ_ext (the only free parameter in the equations), representing different stages of contraction. Despite dissimilar transient response of the cloud (or fragment) structure to different initial conditions, the characteristic time for magnetic braking of the rotation of the cloud (or fragment) as a whole is remarkably insensitive to the initial conditions and independent of the stage of contraction. The latter conclusion is in agreement with an approximate result obtained recently (Mouschovias, 1978; 1979a). A cylindrical cloud (or fragment) of uniform density is also imparted an initial angular velocity about its axis of symmetry with respect to the external medium. The frozen-in magnetic field is now initially radial and perpendicular to the axis of symmetry. In this case magnetic braking becomes more efficient upon contraction. It is more efficient than the aligned rotator case typically by one order of magnitude. The angular momentum problem can be resolved in about 10⁶ yr during the early stages of cloud contraction. Planetary systems, such as the Sun-Jupiter pair, become dynamically possible. A stage exists in which a cloud (or fragment) is in retrograde rotation with respect to its surroundings. This provides the first and only observable prediction of magnetic braking in action. It also constitutes a natural explantation of retrograde rotation in stellar and planetary systems.