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 logDC = 0.63(+/- 0.21) x (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 greater than or similar to 10(31) erg s(-1)) cataclysmic variables (CVs) undergoing DNe outbursts in the direction of the Galactic bulge of 6.6 +/- 4.7 x 10(-5) M-circle dot(-1). Such a specific frequency would give a solar neighbourhood space density of long-period CVs of. = 5.6 +/- 3.9 x 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 rho approximate to const.