Three dimensional Doppler tomography

Abstract: Doppler tomography is a method to compute the emissivity distribution within the co-rotating frames of binary stars from observations of their emission line profiles at multiple orbital phases. A key assumption of the method as it is usually applied is that all gas flow is parallel to the orbital plane of the binary. In this paper I examine the possibility of lifting this assumption to allow for motion parallel to the orbital "$z$" axis of the binary as well. I show that the problem is best considered in Fourier space, and that line profiles directly constrain the 3D Fourier transform of the 3D Doppler image in velocity space, but only over the 2D surface of a double-cone centred upon the origin, and aligned with the axis reciprocal to the $v_z$ velocity axis. Hence the full information needed for the recovery of the 3D emissivity distribution is simply not available. Despite this, an inversion method is presented and tested on a number of simulated images. While artefacts resulting from the missing information do appear, the tests suggest that there could be some value in applying 3D Doppler tomography to data from real systems, although considerable care is needed when doing so.