Exploring faint white dwarfs and the luminosity function with Subaru HSC and SDSS in Stripe 82

We present 5,080 white dwarf (WD) candidates selected from stars matching between the multi-band imaging datasets of the Subaru Hyper Suprime-Cam (HSC) Survey and the Sloan Digital Sky Survey (SDSS) in the Stripe82 region covering about 165~deg$^2$. We select WD candidates from the ``reduced proper motion'' diagram by combining the apparent magnitude in the range $i=19$ -- 24 and the proper motion measured from the datasets among a baseline of $\sim$ 14~years. We refine the WD candidates by fitting blackbody and template WD atmosphere models to HSC photometries for each candidate, enabling the estimation of distance and tangential velocity ($v_{\rm t}$). The deep HSC data allow us to identify low-temperature ($<4000$~K) and faint WD candidates down to absolute magnitude, $M_{\rm bol}\simeq 17$. We evaluate the selection function of our WD candidates using a mock catalogue of spatial and kinematic distributions of WDs in the (thin and thick) disc and halo regions based on a Galactic model. We construct samples of disc and halo WD candidates by selecting WDs with tangential velocity, $40<v_{\rm t}/[{\rm km}~{\rm s}^{-1}]<80$ and $200<v_{\rm t}/[{\rm km}~{\rm s}^{-1}]<500$, respectively. The total number densities of the disc and halo WDs are $(9.33 \pm 0.89) \times 10^{-3}$~pc$^{-3}$ and $(6.34 \pm 2.90) \times 10^{-4}$~pc$^{-3}$. Our luminosity functions (LF) extend down to fainter absolute magnitudes compared with previous work. The faint WDs could represent the oldest generation of building blocks over the past $\sim$10 billion years of the assembly history of our Milky Way.