Emission from elliptical streams of dusty debris around white dwarfs

White dwarfs are routinely observed to have polluted atmospheres, and sometimes significant infrared excesses, that indicate ongoing accretion of circumstellar dust and rocky debris. Typically this debris is assumed to be in the form of a (circular) disc, and to originate from asteroids that passed close enough to the white dwarf to be pulled apart by tides. However, theoretical considerations suggest that the circularisation of the debris, which initially occupies highly eccentric orbits, is very slow. We therefore hypothesise that the observations may be readily explained by the debris remaining on highly eccentric orbits, and we explore the properties of such debris. For the generic case of an asteroid originating at several au from the white dwarf, we find that all of the tidal debris is always bound to the white dwarf and that the orbital energy distribution of the debris is narrow enough that it executes similar elliptical orbits with only a narrow spread. Assuming that the tidal field of the white dwarf is sufficient to minimise the effects of self-gravity and collisions within the debris, we estimate the time over which the debris spreads into a single elliptical ring, and we generate toy spectra and lightcurves from the initial disruption to late times when the debris distribution is essentially time steady. Finally we speculate on the connection between these simple considerations and the observed properties of these systems, and on additional physical processes that may change this simple picture.