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 ($\sim 77\mathrm{\%}$) identified as hot subdwarf stars, including 83 newly discovered systems. Of these, we observe that 178 ($\sim 58\mathrm{\%}$) are hydrogen-rich sdBs, 65 are sdOBs ($\sim 21\mathrm{\%}$), 32 are sdOs ($\sim 11\mathrm{\%}$), and 30 are He-sdO/Bs ($\sim 10\mathrm{\%}$). Among them, 48 ($\sim 16\mathrm{\%}$) 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 ${\rho }_0$ 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 ${\rho }_0=5.17\pm 0.33\times {10}^{-7}$ 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 ${\rho }_0=6.15_{-0.53}^{+1.16}\times {10}^{-7}$ stars/pc${}^3$. This corrected space density is an order of magnitude lower than predicted by population synthesis studies, supporting previous observational estimates.