The radial disc structure around a magnetic neutron star: analytic and semi-analytic solutions

The radial structure of a thin accretion disc is calculated in the presence of a central dipole magnetic field aligned with the rotation axis. The problem is treated using a modified expression for the turbulent magnetic diffusion, which allows the angular momentum equation to be integrated analytically. The governing algebraic equations are solved iteratively between 1 and 10 4 stellar radii. An analytic approximation is provided that is valid near the disruption radius at about 100 stellar radii. At that point, which is approximately 60 per cent of the Alfvén radius and typically about 30 per cent of the corotation radius, the disc becomes viscously unstable. This instability results from the fact that both radiation pressure and opacity caused by electron scattering become important. This in turn is a consequence of the magnetic field which leads to an enhanced temperature in the inner parts. This is because the magnetic field gives rise to a strongly enhanced vertically integrated viscosity, so that the viscous torque can balance the magnetic torque. Accretion discs occur around strong magnetic stars in X-ray binary pulsars (Verbunt 1993), intermediate polar binaries (Warner 1995) and T Tauri stars (Basri & Bertout 1989). The stellar magnetic field interacts with the disc, significantly modifying its structure in its inner regions, and this in turn affects the spin evolution of the star. The early work on the problem is discussed in Campbell (1997). Campbell (1992) gave an analytic, reductio ad absurdum proof that, for realistic magnetic diffusivities, the disc cannot exist over a significant radial extent with jF mf j much larger than jF vf j, where F m and F v are the magnetic and viscous forces. Recently, Heptin-stall (1997) and Campbell & Heptinstall (1998) numerically integrated the magnetic disc equations throughout the disc. They found that the magnetic field causes the disc temperature to be elevated in its inner regions above values in an unperturbed disc. This causes the electron scattering opacity and radiation pressure to become important further from the star. When the radiation pressure becomes comparable to the gas pressure the density reaches a maximum and slightly closer to the star viscous instability occurs. Simultaneously the vertical scaleheight diverges as the disc ends. This work employed simple forms of magnetic diffusivity to represent the effects of turbulence and magnetic buoyancy. The present paper uses a form of magnetic diffusivity which enables more analytic progress to be …