We present proper motion measurements for more than $0.55$ million main-sequence stars, by comparing astrometric positions of matched stars between the multi-band imaging datasets from the Hyper Suprime-Cam (HSC) Survey and the SDSS Stripe 82. In doing this we use $3$ million galaxies to recalibrate the astrometry and set up a common reference frame between the two catalogues. The exquisite depth and the nearly $12$ years of time baseline between HSC and SDSS enable high-precision measurements of statistical proper motions for stars down to $i\simeq 24$. A validation of our method is demonstrated by the agreement with the $Gaia$ proper motions, to the precision better than $0.1$ mas yr$^{-1}$. To retain the precision, we make a correction of the subtle effects due to the differential chromatic refraction in the SDSS images based on the comparison with the $Gaia$ proper motions against colour of stars, which is validated using the SDSS spectroscopic quasars. Combining with the photometric distance estimates for individual stars based on the precise HSC photometry, we show a significant detection of the net proper motions for stars in each bin of distance out to $100$ kpc. The two-component tangential velocities after subtracting the apparent motions due to our own motion display rich phase-space structures including a clear signature of the Sagittarius stream in the halo region of distance range $[10,\ 35]$ kpc. We also measure the tangential velocity dispersion in the distance range $5-20$ kpc and find that the data are consistent with a constant isotropic dispersion of $80\pm 10\ {\rm km/s}$. More distant stars appear to have random motions with respect to the Galactic centre on average.

中文翻译：

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使用斯巴鲁 HSC 和 SDSS Stripe 82 对高达 100 kpc 的恒星进行适当的运动测量

我们通过比较来自 Hyper Suprime-Cam (HSC) Survey 和 SDSS Stripe 82 的多波段成像数据集之间匹配恒星的天体测量位置，为超过 0.55 万美元的主序星提供了自行测量。在这样做时，我们使用价值 300 万美元的星系重新校准天体测量，并在两个目录之间建立一个共同的参考框架。HSC 和 SDSS 之间精致的深度和近 12 美元的时间基线能够对低至 24 美元的恒星统计自行进行高精度测量。我们方法的验证通过与 $Gaia$ 自身运动的一致性来证明，其精度优于 $0.1$ mas yr$^{-1}$。为了保持精度，我们基于与 $Gaia$ 自身运动与恒星颜色的比较对由于 SDSS 图像中的微分色折射造成的微妙影响进行了校正，这已使用 SDSS 光谱类星体进行了验证。结合基于精确 HSC 光度测量的单个恒星的光度距离估计，我们显示了对每个距离范围内恒星净自行的显着检测，距离高达 100 美元 kpc。由于我们自己的运动，在减去视运动后的二分量切向速度显示出丰富的相空间结构，包括距离范围 $[10,\35]$ kpc 的光晕区域中人马座流的清晰特征。我们还测量了 $5-20$ kpc 距离范围内的切向速度色散，发现数据与 $80\pm 10\ {\rm km/s}$ 的恒定各向同性色散一致。更远的恒星似乎平均相对于银河系中心有随机运动。