Magnetic braking and damping of differential rotation in massive stars

Fragmentation of highly differentially rotating massive stars that undergo collapse has been suggested as a possible channel for binary black hole formation. Such a scenario could explain the formation of the new population of massive black holes detected by the LIGO/VIRGO gravitational wave laser interferometers. We probe that scenario by performing general relativistic magnetohydrodynamic simulations of differentially rotating massive stars supported by thermal radiation pressure plus a gas pressure perturbation. The stars are initially threaded by a dynamically weak, poloidal magnetic field confined to the stellar interior. We find that magnetic braking and turbulent viscous damping via magnetic winding and the magnetorotational instability in the bulk of the star redistribute angular momentum, damp differential rotation and induce the formation of a massive and nearly uniformly rotating inner core surrounded by a Keplerian envelope. The $\text{core}+\text{disk}$ configuration evolves on a secular timescale and remains in quasistationary equilibrium until the termination of our simulations. Our results suggest that the high degree of differential rotation required for $m=2$ seed density perturbations to trigger gas fragmentation and binary black hole formation is likely to be suppressed during the normal lifetime of the star prior to evolving to the point of dynamical instability to collapse. Other cataclysmic events, such as stellar mergers leading to collapse, may therefore be necessary to reestablish sufficient differential rotation and density perturbations to drive nonaxisymmetric modes leading to binary black hole formation.

Figure 1

Meridional cut through a 3D volume rendering of the rest-mass density, normalized to its initial maximum value ${\rho }_{0,max}$ (log scale) for initial (left panels) and final configurations (right panels) for cases $\mathbf{N}\mathbf{29}$ (top row) and $\mathbf{N}\mathbf{295}$ (bottom row). For details see Table 1. White solid lines denote the magnetic field lines. Here $P_{\mathrm{c}}$ denotes the initial central rotation period, which is $P_{\mathrm{c}}=8193M$ for $\mathbf{N}\mathbf{29}$ and $P_c=10582M$ for $\mathbf{N}\mathbf{29}$, with $M=0.049(M/{10}^4{M}_{\odot })\mathrm{s}=1.47\times {10}^4(M/{10}^4{M}_{\odot })\mathrm{km}$.

Figure 2

Contours of the ${\lambda }_{\mathrm{MRI}}$ quality factor $q={\lambda }_{\mathrm{MRI}}/\mathrm{\Delta }x$ on the equatorial plane at $t=0$ for $\mathbf{N}\mathbf{29}$ (left panel) and $\mathbf{N}\mathbf{295}$ (right panel). Here $\mathrm{\Delta }x$ is the grid spacing. The fastest growing MRI mode is resolved initially by $\gtrsim 15$ gridpoints in the bulk of the star. The blue ring regions indicate the flip of the direction of the magnetic field. Smaller boxes represent successively higher levels of refinement (see Table 3).

Figure 3

Rest-mass density along with ${\lambda }_{\mathrm{MRI}}$ (white line) on the meridional plane at $t=0$ for $\mathbf{N}\mathbf{29}$ (left panel) and $\mathbf{N}\mathbf{295}$ (right panel). The fastest growing MRI mode is mostly embedded in the star.

Figure 4

Angular velocity profile on the equatorial plane at multiples of initial central rotation period for case $\mathbf{N}\mathbf{29}$ (left panel) and $\mathbf{N}\mathbf{295}$ (right panel). Curves are shown at equal time intervals in units of the initial period $P_{\mathrm{c}}=8193M$ for $\mathbf{N}\mathbf{29}$ and $P_{\mathrm{c}}=10582M$ for $\mathbf{N}\mathbf{295}$. Here $M=0.049(M/{10}^4{M}_{\odot })\mathrm{s}$. The red dashed curve displays the initial differential rotation profile. The final profile is shown by the dark black line. The thick, blue dashed curve at large $r/M$ shows the shape of a Keplerian profile. The spherical radius containing $1/2$ of the stellar rest-mass is denoted by the arrow.

Figure 5

Evolution of the various energy components for $\mathbf{N}\mathbf{29}$ (left panel) and $\mathbf{N}\mathbf{295}$ (right panel). All energies are normalized to the total binding energy at $t=0$ [see Eq. (19)]. The constituent energies are defined in Eqs. (20)–(23). The binding energy $E_{\mathrm{bind}}$ should be conserved. Some quantities are normalized by an additional numerical factor (as indicated) to ease visualization.