The frequency of gaseous debris discs around white dwarfs

1-3 per cent of white dwarfs are orbited by planetary dusty debris detectable as infrared emission in excess above the white dwarf flux. In a rare subset of these systems, a gaseous disc component is also detected via emission lines of the Ca II 8600A triplet, broadened by the Keplerian velocity of the disc. We present the first statistical study of the fraction of debris discs containing detectable amounts of gas in emission at white dwarfs within a magnitude and signal-to-noise limited sample. We select 7705 single white dwarfs spectroscopically observed by the Sloan Digital Sky Survey (SDSS) and $Gaia$ with magnitudes $g$ $\leq$ 19. We identify five gaseous disc hosts, all of which have been previously discovered. We calculate the occurrence rate of a white dwarf hosting a debris disc detectable via Ca II emission lines as 0.067$\pm$$_{0.025}^{0.042}$ per cent. This corresponds to an occurrence rate for a dusty debris disc to have an observable gaseous component in emission as 4$\pm$$_{2}^{4}$ per cent. Given that variability is a common feature of the emission profiles of gaseous debris discs, and the recent detection of a planetesimal orbiting within the disc of SDSSJ122859.93+104032.9, we propose that gaseous components are tracers for the presence of planetesimals embedded in the discs and outline a qualitative model. We also present spectroscopy of the Ca II triplet 8600A region for 20 white dwarfs hosting dusty debris discs in an attempt to identify gaseous emission. We do not detect any gaseous components in these 20 systems, consistent with the occurrence rate that we calculated.