Global Spiral Density Wave Modes in Protoplanetary Disks: Morphology of
  Spiral Arms
  We analyze two-armed global spiral density wave modes generated by
gravitational instability in razor-thin, non-viscous, self-gravitating
protoplanetary disks to understand the dependence of spiral arm morphology
(pitch angle $\alpha$ and amplitude) on various disk conditions. The
morphologies of the resulting spiral density wave modes closely resemble
observations. Their pitch angles and pattern speeds are insensitive to the
boundary conditions adopted. Gaussian disks exhibit more tightly wound spirals
(smaller pitch angle) than power law disks under the same conditions. We find
that at a fixed disk-to-star mass ratio ($M_d/M_*$), pitch angle increases with
average Toomre's stability parameter ($\overline Q$) or average disk aspect
ratio ($\overline h$). For a given $\overline Q$, density wave modes with
higher $M_d/M_*$ have larger pitch angles, while the behavior reverses for a
given $\overline h$. The interdependence between pitch angle and disk
properties can be roughly approximated by $\alpha\propto c_s^2/M_d$, where
$c_s$ is the sound speed. Our gravitational instability-excited spiral density
waves can be distinguished from planet-launched spirals: (1) massive cool disks
have spiral pitch angle falling with radius, while low-mass hot disks have
spiral pitch angle rising with radius; (2) the profile of spiral amplitude
presents several dips and bumps. We propose that gravitational
instability-excited density waves can serve as an alternative scenario to
explain the observed spiral arms in self-gravitating protoplanetary disks.
