  It has been suggested by Giddings that the origin of Hawking radiation in
black holes is a quantum atmosphere of near-horizon quantum region by
investigating both the total emission rate and the stress tensor of Hawking
radiation. Revisiting this issue in the exactly soluble model of a
dimensionally reduced Schwarzschild black hole, we shall confirm that the
dominant Hawking radiation in the Unruh vacuum indeed occurs at the quantum
atmosphere, not just at the horizon by exactly calculating the out-temperature
responsible for outgoing Hawking particle excitations. Consequently we show
that the out-temperature vanishes at the horizon and has a peak at a scale
whose radial extent is set by the horizon radius, and then decreases to the
Hawking temperature at infinity. We also discuss bounds of location of the peak
for the out-temperature in our model.
