  We analyze the response of an Unruh-DeWitt detector moving along an unbounded
spatial trajectory in a two-dimensional spatial plane with constant independent
magnitudes of both the four-acceleration and of a timelike proper time
derivative of the four-accelration. In a Fermi-Walker frame moving with the
detector, the direction of the acceleration rotates at a constant rate around a
great circle. This is the motion of a charge in a uniform electric field when
in the frame of the charge there is both an electric and a magnetic field. We
compare the response of this detector to a detector moving with constant
velocity in a thermal bath of the corresponding temperature for
non-relativistic velocities, and in two regimes: ultraviolet and infrared. In
infrared regime, the detector in the Minkowski space-time moving along the
spatially two-dimensional trajectory should move with a higher speed to keep up
with the same excitation rate of the inertial detector in a thermal bath. In
ultraviolet regime, the dominant modification in the response of this detector
compared to the black body spectrum of Unruh radiation is the same as the
dominant modification perceived by a detector moving with constant velocity in
a thermal bath.
