The relationship between X-ray luminosity and duty cycle for dwarf novae and their specific frequency in the inner Galaxy

We measure the duty cycles for an existing sample of well-observed, nearby dwarf novae (DNe) using data from American Association of Variable Star Observers, and present a quantitative empirical relation between the duty cycle of DNe outbursts and the X-ray luminosity of the system in quiescence. We have found that log DC = 0.63(±0.21) × (log L_X(erg s^-1) - 31.3) - 0.95(±0.1), where DC stands for duty cycle. We note that there is intrinsic scatter in this relation greater than what is expected from purely statistical errors. Using the DN X-ray luminosity functions from Pretorius & Knigge and Byckling et al., we compare this relation to the number of DNe in the Galactic Bulge Survey which were identified through optical outbursts during an 8-d-long monitoring campaign. We find a specific frequency of X-ray-bright (L_X ≳ 10³¹ erg s^{- 1}) cataclysmic variables (CVs) undergoing DNe outbursts in the direction of the Galactic bulge of 6.6± 4.7{×} 10^{-5} M_{{*sun;}}^{-1}. Such a specific frequency would give a solar neighbourhood space density of long-period CVs of ρ = 5.6 ± 3.9 × 10^-6 pc^-3. We advocate the use of specific frequency in future work, given that projects like Large Synoptic Survey Telescope will detect DNe well outside the distance range over which ρ ≈ const.