Magnetic Forces in an Isopedic Disk

We consider the magnetic forces in electrically conducting thin disks threaded by magnetic fields originating in the external (interstellar) medium. We focus on disks that have dimensionless ratios λ of the mass to flux that are spatially constant, a condition that we term isopedic. For arbitrary distributions of the surface density Σ (which can be nonaxisymmetric and time dependent), we show that the magnetic tension exerts a force in the plane of the disk equal to -1/λ² times the self-gravitational force. In addition, if the disk maintains magnetostatic equilibrium in the vertical direction, the magnetic pressure, integrated over the z-height of the disk, may be approximated as (1 + η²)/(λ² + η²) times the gas pressure integrated over z, where η ≡ f_||/2πGΣ and f_|| is the component of the local gravitational field parallel to the plane of the disk. We apply these results to the problem of the stability of magnetized isothermal disks to gravitational fragmentation into subcondensations of a size comparable to the vertical scale height of the disk. Contrary to common belief, such dynamical fragmentation probably does not occur. In particular, the case of the magnetized singular isothermal disk undergoes not dynamical fragmentation into many subcondensations, but inside-out collapse into a single compact object, a self similar problem that is studied in a companion paper (Li & Shu 1997).