Radio pulsars scintillate because their emission travels through the ionized interstellar medium along multiple paths, which interfere with each other. It has long been realized that, independent of their nature, the regions responsible for the scintillation could be used as 'interstellar lenses' to localize pulsar emission regions. Most such lenses, however, resolve emission components only marginally, limiting results to statistical inferences and detections of small positional shifts. As lenses situated close to their source offer better resolution, it should be easier to resolve emission regions of pulsars located in high-density environments such as supernova remnants or binaries in which the pulsar's companion has an ionized outflow. Here we report observations of extreme plasma lensing in the 'black widow' pulsar, B1957+20, near the phase in its 9.2-hour orbit at which its emission is eclipsed by its companion's outflow. During the lensing events, the observed radio flux is enhanced by factors of up to 70-80 at specific frequencies. The strongest events clearly resolve the emission regions: they affect the narrow main pulse and parts of the wider interpulse differently. We show that the events arise naturally from density fluctuations in the outer regions of the outflow, and we infer a resolution of our lenses that is comparable to the pulsar's radius, about 10 kilometres. Furthermore, the distinct frequency structures imparted by the lensing are reminiscent of what is observed for the repeating fast radio burst FRB 121102, providing observational support for the idea that this source is observed through, and thus at times strongly magnified by, plasma lenses .
We report on an unusually bright observation of PSR J2051−0827 recorded during a regular monitoring campaign of black-widow pulsar systems with the Effelsberg 100-m telescope. Through fortunate coincidence, a particularly bright scintillation maximum is simultaneous with the eclipse by the companion, enabling precise measurements of variations in the flux density, dispersion measure (DM), and scattering strength throughout the eclipse. The flux density is highly variable throughout the eclipse, with a peak 1.7 times the average away from the eclipse, and yet does not significantly decrease on average. We recover the flux density variations from the measured DM variations using geometric optics, with a relative velocity as the only free parameter. We measure an effective velocity of 470 ± 10 km s−1, consistent with the relative orbital motion of the companion, suggesting that the outflow velocity of the lensing material is low, or is directly along the line-of-sight. The $2{{\ \rm per\ cent}}$ uncertainty on the effective velocity is a formal error; systematics related to our current model are likely to dominate, and we detail several extensions to the model to be considered in a full treatment of lensing. This is a demonstration of the causal link between DM and lensing; the flux density variations can be predicted directly through the derivatives of DM. Going forward, this approach can be applied to investigate the dynamics of other eclipsing systems, and to investigate the physical nature of scintillation and lensing in the ionized interstellar medium.
The study of astrophysical plasma lensing, such as in the case of extreme scattering events, has typically been conducted using the geometric limit of optics, neglecting wave effects. However, for the lensing of coherent sources such as pulsars and fast radio bursts (FRBs), wave effects can play an important role. Asymptotic methods, such as the so-called Eikonal limit, also known as the stationary phase approximation, have been used to include first-order wave effects; however, these methods are discontinuous at Stokes lines. Stokes lines are generic features of a variety of lens models, and are regions in parameter space where imaginary images begin to contribute to the overall intensity modulation of lensed sources. Using the mathematical framework of Picard-Lefschetz theory to compute diffraction integrals, we argue that these imaginary images contain as much information as their geometric counterparts, and may potentially be observable in data. Thus, weak-lensing events where these imaginary images are present can be as useful for inferring lens parameters as strong-lensing events in which multiple geometric images are present.
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