We have imaged the fields of 35 pulsars with the VLA at 8.4 GHz to search for evidence that the pulsars have deposited energy into their surroundings in the form of an extended wind nebula. Candidates have both a wide range of transverse velocities and rotational energy losses. Of the 35 pulsars that we imaged, no pointlike or extended emission was detected toward 21 sources, to a typical rms surface brightness sensitivity of 40 μJy beam~(-1), for a beam size of 0.8″. Pointlike emission was detected toward 14 pulsars with peak flux densities between 0.2 and 3 mJy. After a careful examination of each source we conclude that these detections are simply the time-averaged pulsed flux density from the pulsars themselves, and thus no new pulsar wind nebulae were discovered: From the surface brightness limits and an estimate of the size of the region where the pulsar wind is first shocked, we deduce that the fraction of the pulsars' spindown energy going into producing a compact radio nebula must be less than 10~(-6). In contrast, this fraction is 2 orders of magnitude larger for the six pulsars which are known unambiguously to have a radio pulsar wind nebula around them. Several explanations are given for this behavior, and we conclude that only young pulsars with high rotational energy losses produce an observable radio nebula.
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