Funnel Flows from Disks to Magnetized Stars

This work considers flows from an accretion disk corotating with the aligned dipole magnetic field of a rotating star. Ideal magnetohydrodynamics (MHD) is assumed with the pressure and density related as $p \propto \rho^\gamma$ and with $\rho {\bf v}^2 \ll {\bf B}^2/4\pi$, where ${\bf v}$ is the flow velocity. This limit corresponds to the Alfv\'en radius for the disk accretion larger than the corotation radius. Transonic flows, which go from subsonic motion near the disk to supersonic inflow near the star, are shown to be possible only for a narrow range of $R_d \sim r_c \equiv (GM/\Omega^2)^{1/3}$, where $R_d$ is the radius at which the dipole field line intersects the disk, $r_c$ the corotation radius, $M$ the mass of the star, and $\Omega$ its angular rotation rate. The transonic flows have very different behaviors for $\gamma > 7/5$ and $< 7/5$. In both cases, the plasma flow velocity ${\bf v}$ (which is parallel to ${\bf B}$) increases with decreasing distance $R$ from the star. However, for $\gamma >7/5$, the Mach number ${\cal M} \equiv |{\bf v}|/c_s$ (with $c_s$ the sound speed) initially increases with $R$ decreasing from $R_d$, but for $R$ decreasing from $\approx 0.22 R_d$ (for $\gamma=5/3$)the Mach number surprisingly {\it decreases}. In the other limit, $\gamma < 7/5$, ${\cal M}$ increases monotonically with decreasing $R$. Application of these results is made to funnel flows to rotating magnetized neutron stars and young stellar objects.