Inferring (Sub)millimeter Dust Opacities and Temperature Structure in Edge-on Protostellar Disks From Resolved Multi-Wavelength Continuum Observations: The Case of the HH 212 Disk

(Sub)millimeter dust opacities are required for converting the observable dust continuum emission to the mass, but their values have long been uncertain, especially in disks around young stellar objects. We propose a method to constrain the opacity $\kappa_\nu$ in edge-on disks from a characteristic optical depth $\tau_{0,\nu}$, the density $\rho_0$ and radius $R_0$ at the disk outer edge through $\kappa_\nu=\tau_{0,\nu}/(\rho_0 R_0)$ where $\tau_{0,\nu}$ is inferred from the shape of the observed flux along the major axis, $\rho_0$ from gravitational stability considerations, and $R_0$ from direct imaging. We applied the 1D semi-analytical model to the embedded, Class 0, HH 212 disk, which has high-resolution data in ALMA Band 9, 7, 6, and 3 and VLA Ka band ($\lambda$=0.43, 0.85, 1.3, 2.9, and 9.1 mm). The modeling of the HH 212 disk is extended to 2D through RADMC-3D radiative transfer calculations. We find a dust opacity of $\kappa_\nu \approx $ $1.9\times 10^{-2}$, $1.3\times 10^{-2}$, and $4.9\times 10^{-3}$ cm$^2$ per gram of gas and dust for ALMA Bands 7, 6, and 3, respectively with uncertainties dependent on the adopted stellar mass. The inferred opacities lend support to the widely used prescription $\kappa_\lambda=2.3\times 10^{-2} (1.3 {\rm mm}/\lambda)$ cm$^2$ g$^{-1}$ advocated by Beckwith et al. (1990). We inferred a temperature of ~45K at the disk outer edge which increases radially inward. It is well above the sublimation temperatures of ices such as CO and N$_2$, which supports the notion that the disk chemistry cannot be completely inherited from the protostellar envelope.