  Recent observations of protoplanetary disk have reported spiral structures
that are potential signatures of embedded planets, and modeling efforts have
shown that a single planet can excite multiple spiral arms, in contrast to
conventional disk-planet interaction theory. Using two and three-dimensional
hydrodynamics simulations to perform a systematic parameter survey, we confirm
the existence of multiple spiral arms in disks with a single planet, and
discover a scaling relation between the azimuthal separation of the primary and
secondary arm, $\phi_{\rm sep}$, and the planet-to-star mass ratio $q$:
$\phi_{\rm sep} = 102^{\circ} (q/0.001)^{0.2}$ for companions between Neptune
mass and 16 Jupiter masses around a 1 solar mass star, and $\phi_{\rm sep} =
180^{\circ}$ for brown dwarf mass companions. This relation is independent of
the disk's temperature, and can be used to infer a planet's mass to within an
accuracy of about 30% given only the morphology of a face-on disk. Combining
hydrodynamics and Monte-Carlo radiative transfer calculations, we verify that
our numerical measurements of $\phi_{\rm sep}$ are accurate representations of
what would be measured in near-infrared scattered light images, such as those
expected to be taken by Gemini/GPI, VLT/SPHERE, or Subaru/SCExAO in the future.
Finally, we are able to infer, using our scaling relation, that the planet
responsible for the spiral structure in SAO 206462 has a mass of about 6
Jupiter masses.
