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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 using data from AAVSO, and present a quantitative empirical relation between the duty cycle of dwarf novae outbursts and the X-ray luminosity of the system in quiescence. We have found that $\log DC=0.63(\pm0.21)\times(\log L_{X}({\rm erg\,s^{-1}})-31.3)-0.95(\pm0.1)$, 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 dwarf nova X-ray luminosity functions from \citet{Pretorius12} and \citet{Byckling10}, we compare this relation to the number of dwarf novae in the Galactic Bulge Survey which were identified through optical outbursts during an 8-day long monitoring campaign. We find a specific frequency of X-ray bright ($L_{X}>10^{31}\,{\rm erg\,s^{-1}}$) Cataclysmic Variables undergoing Dwarf Novae outbursts in the direction of the Galactic Bulge of $6.6\pm4.7\times10^{-5}\,M_{\odot}^{-1}$. Such a specific frequency would give a Solar neighborhood space density of long period CVs of $ρ=5.6\pm3.9\times10^{-6}\,$pc$^{-3}$. We advocate the use of specific frequency in future work, given that projects like LSST will detect DNe well outside the distance range over which $ρ\approx{\textrm const}$.