The double pulsar PSR J0737-3039A/B displays short, 30 s eclipses that arise around conjunction when the radio waves emitted by pulsar A are absorbed as they propagate through the magnetosphere of its companion pulsar B. These eclipses offer a unique opportunity to probe directly the magnetospheric structure and the plasma properties of pulsar B. We have performed a comprehensive analysis of the eclipse phenomenology using multi-frequency radio observations obtained with the Green Bank Telescope. We have characterized the periodic flux modulations previously discovered at 820 MHz by McLaughlin et al., and investigated the radio frequency dependence of the duration and depth of the eclipses. Based on their weak radio frequency evolution, we conclude that the plasma in pulsar Bs magnetosphere requires a large multiplicity factor (~ 10^5). We also found that, as expected, flux modulations are present at all radio frequencies in which eclipses can be detected. Their complex behavior is consistent with the confinement of the absorbing plasma in the dipolar magnetic field of pulsar B as suggested by Lyutikov & Thompson and such a geometric connection explains that the observed periodicity is harmonically related to pulsar Bs spin frequency. We observe that the eclipses require a sharp transition region beyond which the plasma density drops off abruptly. Such a region defines a plasmasphere which would be well inside the magnetospheric boundary of an undisturbed pulsar. It is also two times smaller than the expected standoff radius calculated using the balance of the wind pressure from pulsar A and the nominally estimated magnetic pressure of pulsar B.
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