  We argue that eclipses of radio emission from the millisecond pulsar A in the
double pulsar system J0737-3039 are due to synchrotron absorption by plasma in
the closed field line region of the magnetosphere of its normal pulsar
companion B. A's radio beam only illuminates B's magnetosphere for about 10
minutes surrounding the time of eclipse. During this time it heats particles at
r\gtrsim 10^9 cm to relativistic energies and enables extra plasma to be
trapped by magnetic mirroring. An enhancement of the plasma density by a factor
\sim 10^2 is required to match the duration and optical depth of the observed
eclipses. The extra plasma might be supplied by a source near B through B\gamma
pair creation by energetic photons produced in B's outer gap. Excitation of
pairs' gyrational motions by cyclotron absorption of A's radio beam can result
in their becoming trapped between conjugate mirror points in B's magnetosphere.
Because the trapping efficiency decreases with increasing optical depth, the
plasma density enhancement saturates even under steady state illumination. The
result is an eclipse with finite, frequency dependent, optical depth. After
illumination by A's radio beam ceases, the trapped particles cool and are lost.
The entire cycle repeats every orbital period. We speculate that the
asymmetries between eclipse ingress and egress result in part from the
magnetosphere's evolution toward a steady state when illuminated by A's radio
beam. We predict that A's linear polarization will vary with both eclipse phase
and B's rotational phase.
