A Tight Relation between Spiral Arm Pitch Angle and Protoplanetary Disk Mass

We use two-dimensional Fourier transformation to measure the pitch angle ($\varphi$) of the dominant spiral Fourier mode of well-defined spiral arms in 13 protoplanetary disks, making use of near-infrared scattered-light images of AB Aur, SAO 206462, MWC 758, V1247 Ori, HD 142527, DZ Cha, LkH$\alpha$ 330, and HD 100453, and ALMA millimeter continuum images of Elias 2-27, IM Lup, AS 205, and HT Lup. We find that the measured pitch angle correlates strongly with disk mass ($M_{D}$), such that more massive protoplanetary disks have smaller pitch angles, following $|\varphi| = -(7.8\pm1.7)\log(M_{D}/M_{\odot})+(2.7\pm2.6)$. Interestingly, four disks with a known companion (HD 142527, HD 100453, AS 205, and HT Lup) share the same trend. Such a strong dependence of spiral arm pitch angle on disk mass suggests that the disk mass, independent of the formation mechanism, plays a fundamental role in determining the arm tightness of the observed spiral structure. The physical origin of the $\varphi-M_D$ relation is still not clear. The pitch angle of spiral arms in protoplanetary disks provides an independent constraint on the disk mass.