A Turn-over Scenario for Rotating Magnetic White Dwarfs: Models with Several Values of Mass, Angular Momentum, and Magnetic Field

We study a white dwarf model with differential rotation and magnetic field, for which the symmetry axis of the toroidal field, the magnetic axis of the poloidal field, and the principal axis I3 coincide permanently; the common axis defined this way is called “magnetic symmetry axis”. Furthermore, the magnetic symmetry axis inclines at a small angle χ relative to the spin axis of the model; this angle is called “obliquity angle” or “turn-over angle”. Such a model is almost axisymmetric and undergoes an early evolutionary phase of secular timescale, characterized by the fact that the moment of inertia along the spin axis, Izz ≃ I33, is greater than the moments of inertia along the (almost) equatorial axes, I11 = I22, since rotation and poloidal field (both responsible for the oblateness of the model) dominate over the toroidal field (responsible, in turn, for the prolateness of the model). During this early evolutionary phase, the model suffers from secular angular momentum loss due to weak magnetic dipole radiation activated by the poloidal field. Such an angular momentum loss leads gradually to a situation of dynamical asymmetry with I11 > I33. However, dynamically asymmetric configurations tend to turn over spontaneously, that is, to rotate about axis with moment of inertia greater than I33 with angular momentum remaining invariant. So, the fate of a dynamically asymmetric configuration is to become an oblique rotator and, eventually, a perpendicular rotator. During the so-called “turn-over phase”, the turn over angle, χ, increases spontaneously up to ∼ 90◦ on a “turn-over timescale”, tTOV, since the rotational kinetic energy of the model decreases from a higher level when χ ≃ 0◦ (aligned rotator) to a lower level when χ ≃ 90◦ (perpendicular rotator). At this lower level the model reaches the state of least energy consistent with its prescribed angular momentum and magnetic 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. In the present paper, we study numerically the so-called “turn-over scenario” (i.e., an evolutionary scenario, which takes into account the turn-over phase) for white dwarf models of several masses, angular momenta, and magnetic fields.