A 500 pc volume-limited sample of hot subluminous stars I. Space density, scale height, and population properties

We present the first volume-limited sample of spectroscopically confirmed hot subluminous stars out to 500 pc, defined using the accurate parallax measurements from the {\em Gaia} space mission data release 3 (DR3). The sample comprises a total of 397 members, with 305 ($\sim 77\%$) identified as hot subdwarf stars, including 83 newly discovered systems. Of these, we observe that 178 ($\sim58\%$) are hydrogen-rich sdBs, 65 are sdOBs ($\sim 21\%$), 32 are sdOs ($\sim 11\%$), and 30 are He-sdO/Bs ($\sim 10\%$). Among them, 48 ($\sim 16\%$) exhibit an infrared excess in their spectral energy distribution fits, suggesting a composite binary system. The hot subdwarf population is estimated to be 90\% complete, assuming that most missing systems are these composite binaries located within the main sequence (MS) in the \emph{Gaia} colour-magnitude diagram (CMD). The remaining sources in the sample include cataclysmic variables (CVs), blue horizontal branch stars (BHBs), hot white dwarfs (WDs), and MS stars. We derived the mid-plane density $ρ_{0}$ and scale height $\mathrm{h}_{z}$ for the non-composite hot subdwarf star population using a hyperbolic sechant profile (sech$^2$). The best-fit values are $ρ_{0}\,=\,5.17\pm 0.33 \times10^{-7}$ stars/pc$^{3}$ and $\mathrm{h}_{z} = 281 \pm 62$ pc. When accounting for the composite-colour hot subdwarfs and their estimated completeness, the mid-plane density increases to $ρ_{0}\,=\,6.15^ {+1.16}_{-0.53} \times10^{-7}$ stars/pc$^{3}$. This corrected space density is an order of magnitude lower than predicted by population synthesis studies, supporting previous observational estimates.