Analytical Hot Spot Shapes and Magnetospheric Radius from 3D Simulations of Magnetospheric Accretion

We present an analytical formula for the position and shape of the spots on the surface of accreting magnetized stars in cases where a star has a dipole magnetic field tilted at a small misalignment angle Theta < 30 degrees about the rotational axis, and the magnetosphere is 2.5-5 times the radius of the star. We observed that the azimuthal position of the spots varies significantly when the position of the inner disc varies. In contrast, the polar position of the spots varies only slightly because of the compression of the magnetosphere. The azimuthal width of the spots strongly varies with Theta: spots have the shape of an arc at larger misalignment angles, and resemble a ring at very small misalignment angles. The polar width of the spots varies only slightly with changes in parameters. The motion of the spots in the azimuthal direction can provide phase-shifts in accreting millisecond pulsars, and the "drift" of the period in Classical T Tauri stars. The position and shape of the spots are determined by three parameters: misalignment angle Theta; normalized corotation radius, r_c/R_* and normalized magnetospheric radius, r_m/R_*, where R_* is the stellar radius. We also use our data to check the formula for the Alfv\'en radius, where the main dependencies are r_m \sim (\mu^2/\dot M)^{2/7}, where \mu is the magnetic moment of the star, and \dot M is the accretion rate. We found that the dependence is more gradual, r_m \sim (\mu^2/\dot M)^{1/5}, which can be explained by the compression of the magnetosphere by the disc matter and by the non-dipole shape of the magnetic field lines of the external magnetosphere.