Protoplanetary disks around magnetized young stars with large-scale magnetic fields I: Steady-state solutions

TL;DR

The paper develops a self-consistent 1+1D framework to study how a magnetized young star with a large-scale fossil field shapes the inner regions of protoplanetary disks. By coupling a 1D viscous hydrodynamic disk with a poloidal magnetic field and incorporating non-ideal MHD diffusion (Ohmic and ambipolar) through a conductivity-weighted vertical averaging, the authors track the evolution of the magnetic flux toward a stationary topology. They show that the inner disk is dominated by the stellar dipole with $B_z\propto r^{-3}$ while the outer disk up to several AU is influenced by diffusion and a fossil field, yielding a characteristic $eta_0$-dependent structure and an inclined field ($B_r^s/B_z$) that can enable MHD winds near a $\sim30^{\circ}$ inclination. The study further demonstrates how stellar parameters ($|B_*|$, $P_*$, $L_X$) and disk mass transport rate $\\dot M$ modulate the field topology, with implications for wind launching and disk evolution, and sets the stage for future long-term evolution simulations including MHD outflows over $\\sim 10^6$ years.

Abstract

Describing the large-scale field topology of protoplanetary disks faces significant difficulties and uncertainties. The transport of the large-scale field inside the disk plays an important role in understanding its evolution. We aim to improve our understanding of the dependencies that stellar magnetic fields pose on the large-scale field. We focus on the innermost disk region ($\lesssim$ 0.1 AU), which is crucial for understanding the long-term disk evolution. We present a novel approach combining the evolution of a 1+1D hydrodynamic disk with a large-scale magnetic field, consisting of a stellar dipole truncating the disk and a fossil field. The magnetic flux transport includes advection and diffusion due to laminar, non-ideal MHD effects, such as Ohmic and ambipolar diffusion. Due to the implicit nature of the numerical method, long-term simulations (in the order of several viscous timescales) are feasible. The large-scale magnetic field topology in stationary models shows a distinct dependence on specific parameters. The innermost disk region is strongly affected by the stellar rotation period and magnetic field strength. The outer disk regions are affected by the X-ray luminosity and the fossil field. Varying the mass flow through the disk affects the large-scale disk field throughout its radial extent. The topology of the large-scale disk field is affected by several stellar and disk parameters. This will affect the efficiency of MHD outflows, which depend on the magnetic field topology. Such outflows might originate from the very inner disk region, the dead zone, or the outer disk. In subsequent studies, we will use these models as a starting point for conducting long-term evolution simulations of the disk and large-scale field on scales of $\sim$ 106 years to investigate the combined evolution of the disk, the magnetic field topology, and the resulting MHD outflows.

Figures (13)

• Figure 1: Time evolution of the magnetic field strength over time (normalized in units of the diffusion timescale of the stationary model at the outer DZ boundary, $t_\mathrm{diff,stat}(r_\mathrm{DZ,out})$), where the color coding represents the magnetic field strength $|\vec{B}|$ at every point in space and time. The gray lines show magnetic flux transport over time, using the effective, vertically averaged magnetic field transport velocity $\bar{{\bar{u}}}_\mathrm{\psi}$. The dashed white lines denote the inner and outer boundary of the DZ, respectively, whereas the black full line denotes the diffusion timescale $t_\mathrm{diff,stat}$ of our stationary model. The inner DZ boundary is located very close to the inner disk rim.
• Figure 2: Timescales of our reference model (cf. Table \ref{['tab:ref_model']}) relevant to the long-term evolution of a PPD and a large-scale magnetic field. The solid blue, orange and green lines correspond to the viscous timescale $t_\mathrm{visc}$, the dynamical timescale $t_\mathrm{dyn}$ and the magnetic diffusion timescale $t_\mathrm{diff}$, respectively. The vertical, dashed black lines correspond to the corotation radius $R_\mathrm{cor}$ and the outer DZ boundary $r_\mathrm{DZ,out}$. Finally, the gray area marks the region between $1$ Myr and $4$ Myrs and depicts the expected lifetime of a PPD.
• Figure 3: The poloidal, stationary magnetic field profile of our reference model (cf. Table \ref{['tab:ref_model']}). Panel (a) shows both the vertical and radial magnetic field component at the surface, $B_\mathrm{z}$ and $B_\mathrm{r}^\mathrm{s}$, respectively. Panel (b) depicts the ratio of $B_\mathrm{r}^\mathrm{s} / B_\mathrm{z}$, which is a measure for the inclination of the field lines, whereas panel (c) shows the magnetic plasma beta $\beta_0$ at the disk midplane. The vertical dashed line correspond to the corotation radius $R_\mathrm{cor}$ and $r_\mathrm{DZ,out}$, respectively. The dotted, gray lines in panels (a) and (c) describe the corresponding values at $t=0$, whereas the horizontal dotted line in panel (b) denotes a surface angle of $30^\circ$ with respect to the midplane normal.
• Figure 4: Comparison of viscous timescale $t_\mathbf{visc}$ and the diffusion timescale $(t_\mathrm{diff,stat})$. The vertical dashed lines close to the inner rim and at $\sim 4$ AU represent $R_\mathrm{cor}$ and $r_\mathrm{DZ,out}$, respectively.
• Figure 5: For better understanding of the regimes of thermal and non-thermal ionization, the radial and vertical profiles of (a) ionization fraction $x_\mathrm{e}$, (b) the contribution of ohmic resistivity $\eta_\mathrm{O}$, (c) the contribution of ambipolar diffusion $\eta_\mathrm{A}$ and (d) the combined resistivity $\eta(r,z) = \eta_\mathrm{O}(r,z) + \eta_\mathrm{A}(r,z)$ are plotted for our reference model, which is explained in Sec. \ref{['sec:magnetic_field_evolution']} (cf. Table \ref{['tab:ref_model']}). The dashed gray lines in (a)-(d) correspond to specific heights above the disk midplane ($z = 0$, $z = H_\mathrm{p}$, $z = H_\mathrm{p} + \frac{1}{3}(z_\mathrm{B} - H_\mathrm{p})$, $z = H_\mathrm{p} + \frac{2}{3}(z_\mathrm{B} - H_\mathrm{p})$, and $z = z_\mathrm{B}$). In (e) the combined resistivity is plotted for those specific heights, as well as the resulting vertically averaged resistivity $\bar{{\bar{\eta}}}$ (thick blue line). For comparison the turbulent resistivity $\eta_\mathrm{t}$ corresponding to a Prandtl number of $1$ is also plotted, showing that $\eta_\mathrm{t} \ll \bar{{\bar{\eta}}}$ almost everywhere except for the very inner region.