Rotational Behavior of Magnetic White Dwarfs: Numerical Study of a Turnover Scenario

Abstract

We consider a white dwarf model with differential rotation and field, assuming that (1) the symmetry axis of the toroidal field, the axis of the poloidal field, and the principal axis I3 coincide permanently with each other (this common axis is called magnetic symmetry axis) and (2) the model declines slightly from axisymmetry, i.e., its symmetry axis is inclined at a small χ relative to its spin axis (this is called obliquity angle or angle). The latter assumption turns on the magnetic dipole radiation mechanism, which is fed by the rotational kinetic energy and causes emission of weak electromagnetic power since χ is assumed small; thus, the model suffers from secular angular momentum loss. This fact leads to a gradual decrease of the moment of inertia I33 along the principal axis I3 and, in turn, to a gradual increase of the moment of inertia I11 along the principal axis I1, since the toroidal field (tending to derive prolate configurations and thus to increase I11) becomes gradually more competitive against the combined action of both rotation and poloidal field (tending to derive oblate configurations and thus to increase I33). So, a dynamical asymmetry is established in the sense that, after a particular time, I11 becomes greater than I33. However, a dynamically asymmetric model tends to turn over spontaneously and thus to become oblique rotator with its angular momentum remaining invariant. As a consequence, the turnover increases spontaneously up to 90° on a timescale, tTOV, since the rotational kinetic energy of the model decreases from a higher level when χ 0° to a lower level when χ 90°; at this level the model becomes perpendicular rotator and reaches the state of least energy consistent with its prescribed angular momentum and field. The excess rotational kinetic energy due to differential rotation is totally dissipated due to the action of turbulent viscosity in the convective regions of the model. Thus, in the scenario the casting of the roles has mainly to do with rotation, poloidal field, toroidal field, and turbulent viscosity. In the present paper, we study in detail this scenario.