We present a Chandra observation of the H II region RCW 89. The nebula lies 10' north from the central pulsar PSR B1509-58, and it has been suggested that the nebula is irradiated by the pulsar jet. We performed a spectral analysis of the seven brightest emitting regions aligned in a ``horseshoe'' shape and found that the temperature of the knots increases along the horseshoe in the clockwise direction, while, in contrast, the ionization parameter net decreases. This strongly supports a picture of energy transfer via the precessing pulsar jet. We examined the energy budget assuming that RCW 89 is powered by the pulsar jet and confirmed that the pulsar rotational energy loss is sufficient to drive the nebula. The rate of energy injection into RCW 89 by the jet was estimated from the synchrotron radiation flux. We obtained a heating timescale of 1400 yr, which is consistent with the pulsar characteristic age of 1700 yr. To explain the temperature gradient, we discuss the cooling process for plasma clouds in RCW 89. We argue that the plasma clouds can be cooled down by the adiabatic expansion within 70 yr and form the temperature gradient reflecting the sequential heating by the precessing pulsar jet. We also determined the velocities of the individual plasma clouds by spectral fitting. The plasma clouds in RCW 89 are moving away at 240-860 km s-1, which constrains the inclination angle of the pulsar spin axis i>50deg and the expanding velocity of the shell as vshell>1100 km s-1.
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