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. Based on a plausible geometric model, 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 � 10 9 cm to relativistic energies and enables extra plasma, beyond that needed to maintain the corotation electric field, to be trapped by magnetic mirroring. An enhancement of the plasma density by a factor ∼ 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 Bγ pair creation by energetic photons produced in B’s outer gap. Relativistic pairs cool by synchrotron radiation close to where they are born. Re-excitation of their 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
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