Accretion Disk Magnetic Braking
Kurt Liffman
TL;DR
This work investigates how a static axial magnetic field threading a protostellar disk can magnetically brake the disk and drive radial inflow. By treating the disk as inviscid with a scalar conductivity $\sigma(r)$, the authors derive analytic expressions for the infall speed $v_r$, mass accretion rate $\dot{M}_a$, disk energy dissipation, and the time evolution of the surface density $\Sigma(r,t)$ and the co-moving radial position $r(t)$, capturing the magnetic-torque coupling via a term proportional to $B^2$. The key finding is that $v_r$ and $\dot{M}_a$ both scale with $B^2$, yielding modest values for fiducial parameters, though stronger fields toward the protostar could enhance accretion locally; energy dissipation (disk luminosity) is small but likewise increases with $B^2$, and the radial evolution follows $r \approx r_0 / \sqrt{1 + (t-t_0)/\tau}$ with $\tau \approx 5\times10^5$ yr, suggesting magnetic braking alone is unlikely to drive whole-disk accretion but may influence inner-disk dynamics or gap formation. These analytic results provide a useful check on numerical simulations of protostar and disk formation and help clarify the potential role of magnetic braking in early disk evolution.
Abstract
A protostellar disk is threaded by a static magnetic field that is perpendicular to the disk-surface. The magnetic field acts to brake the protostellar disk and cause the disk material to move towards the protostar. General analytic equations are derived for the accretion speed, and mass accretion rate. Simplified analytic equations are also obtained for the disk energy dissipation, accretion timescale and the disk radial position plus disk surface density, as a function of time. In addition to providing physical insight, such equations might be useful as a check on computational models for protostar and protostellar disk formation.