  We test the effect of assumptions about stellar motion on the behavior of
gravitational instabilities in protoplanetary disks around solar-type stars by
performing two simulations that are identical in all respects except the
treatment of the star. In one simulation, the star is assumed to remain fixed
at the center of the inertial reference frame. In the other, stellar motion is
handled properly by including an indirect potential in the hydrodynamic
equations to model the star's reference frame as one which is accelerated by
star/disk interactions. The disks in both simulations orbit a solar mass star,
initially extend from 2.3 to 40 AU with a r^-1/2 surface density profile, and
have a total mass of 0.14 M_sun. The gamma = 5/3 ideal gas is assumed to cool
everywhere with a constant cooling time of two outer rotation periods.
  The overall behavior of the disk evolution is similar, except for weakening
in various measures of GI activity by about at most tens of percent for the
indirect potential case. Overall conclusions about disk evolution in earlier
papers by our group, where the star was always assumed to be fixed in an
inertial frame, remain valid. There is no evidence for independent one-armed
instabilities, like SLING, in either simulation. On the other hand, the stellar
motion about the system center of mass (COM) in the simulation with the
indirect potential is substantial, up to 0.25 AU during the burst phase, as GIs
initiate, and averaging about 0.9 AU during the asymptotic phase, when the GIs
reach an overall balance of heating and cooling. These motions appear to be a
stellar response to nonlinear interactions between discrete global spiral modes
in both the burst and asymptotic phases of the evolution, and the star's
orbital motion about the COM reflects the orbit periods of disk material near
the corotation radii of the dominant spiral waves. This motion is, in
principle, large enough to be observable and could be confused with stellar
wobble due to the presence of one or more super-Jupiter mass protoplanets
orbiting at 10's AU. We discuss why the excursions in our simulation are so
much larger than those seen in simulations by Rice et al. 2003a.
