How Bright are Planet-Induced Spiral Arms in Scattered Light?

Recently, high angular resolution imaging instruments such as SPHERE and GPI have discovered many spiral-arm-like features in near-infrared scattered light images of protoplanetary disks. Theory and simulations have suggested that these arms are most likely excited by planets forming in the disks; however, a quantitative relation between the arm-to-disk brightness contrast and planet mass is still missing. Using 3D hydrodynamics and radiative transfer simulations, we examine the morphology and contrast of planet-induced arms in disks. We find a power-law relation for the face-on arm contrast ($\delta_{\rm max}$) as a function of planet mass ($M_{\rm p}$) and disk aspect ratio ($h/r$): $\delta_{\rm max}\approx((M_{\rm p}/M_{\rm J})/(h/r)^{1.38})^{0.22}$. With current observational capability, at a 30 AU separation, the minimum planet mass for driving detectable arms in a disk around a 1 Myr 1$M_\odot$ star at 140 pc at low inclinations is around Saturn mass. For planets more massive than Neptune masses, they typically drive multiple arms. Therefore in observed disks with spirals, it is unlikely that each spiral arm originates from a different planet. We also find only massive perturbers with at least multi-Jupiter masses are capable of driving bright arms with $\delta_{\rm max}\gtrsim2$ as found in SAO 206462, MWC 758, and LkH$\alpha$~330, and these arms do not follow linear wave propagation theory. Additionally, we find the morphology and contrast of the primary and secondary arms are largely unaffected by a modest level of viscosity with $\alpha\lesssim0.01$. Finally, the contrast of the arms in the SAO 206462 disk suggests that the perturber SAO 206462 b at $\sim100$ AU is about $5-10M_{\rm J}$ in mass.