A theory of the spectral "zebra" pattern of the Crab pulsar's high-frequency interpulse (HFIP) radio emission is developed. The observed emission bands are interference maxima caused by multiple ray propagation through the pulsar magnetosphere. The high-contrast interference pattern is the combined effect of gravitational lensing and plasma de-lensing of light rays. The model enables space-resolved tomography of the pulsar magnetosphere, yielding a radial plasma density profile of $n_{e}\propto r^{-3}$, which agrees with theoretical insights. We predict the zebra pattern trend to change at a higher frequency when the ray separation becomes smaller than the pulsar size. This frequency is predicted to be in the range between 42 GHz and 650 GHz, which is within the reach of existing facilities like ALMA and SMA. These observations hold significant importance and would contribute to our understanding of the magnetosphere. Furthermore, they offer the potential to investigate gravity in the strong field regime near the star's surface.
Pulsars are highly magnetized neutron stars that emit pulsed electromagnetic radiation. The properties and theories of pulsars are comprehensively reviewed by A. Philippov & M. Kramer (2022). Among the over 3,700 known pulsars, the Crab pulsar stands out as one of the brightest and most extensively studied. It emits two pulses per rotation period that are observed across the entire electromagnetic spectrum, from radio waves to Xrays. The temporal coincidence of the pulses at different wavelengths suggests that the observed emissions originate from the same physical location. It is now widely accepted that the Crab pulsar emissions originate from within the current sheet, outside the light cylinder (X.-N. Bai & A. Spitkovsky 2010). The mechanism underlying this process involves magnetic reconnection and subsequent violent interactions of plasmoids within the current sheet (A. Philippov et al. 2019). This model implies that an individual pulse emission, including giant flares, is composed of a large number of (blended in time) "nanoshots" -bright sub-pulses of a few nanosecond duration each -thereby explaining their high temporal variability and broadband spectra.
In this paper, we are interested in the radio emission of the object in the GHz frequency range. The observed broadband spectrum is a general feature of the Email: medvedev@ku.edu main pulse, low-frequency interpulse, and several emission components in the frequency range between approximately 1 GHz and 30 GHz, as studied in Refs. (T. H. Hankins & J. A. Eilek 2007;T. H. Hankins et al. 2015T. H. Hankins et al. , 2016)). In contrast, the spectral pattern of the highfrequency interpulse (HFIP), observed between approximately ν ∼ 5 GHz and ν ∼ 30 GHz, is remarkably different and represents a sequence of emission bands resembling the “zebra” pattern. This peculiar spectral pattern was first reported in 2007 and subsequently studied in great detail (T. H. Hankins & J. A. Eilek 2007;T. H. Hankins et al. 2016).
The following are the known properties of HFIP emission that any successful model should account for:
• The presence of peculiar spectral features in the form of regular emission bands.
• The proportional separation of the bands and the “6% rule”. The frequency difference between the nearest bands is proportional to the band frequency, resulting in ∆ν ≈ 0.057ν.
• The persistence of the pattern. There has not been observed a single HFIP without spectral bands.
• The stability of the pattern. The band positions can remain stable for as long as a day, although they can also vary from one pulse to the next.
• The HFIP is nearly 100% linearly polarized, and the position angle is stable and does not change over many pulses.
• The HFIP has a variable and often larger dispersion measure (DM) compared to the main pulse.
• No Faraday rotation within the system has been reported.
Despite substantial theoretical efforts over the subsequent fifteen years, no satisfactory mechanism has been proposed to elucidate the HFIP puzzle. The proposed models either involve emission mechanisms generating multiple harmonics (e.g., cyclotron or maser emissions) or invoke propagation effects (e.g., interference at a source, within a current sheet, or wave modulation instability). The former class of models predicts an incorrect uniform band spacing. The latter class necessitates extremely high wave phase coherence and source stability, which are unrealistic to anticipate in a highly turbulent pulsar wind medium. A comprehensive review and details on recent models can be found in Refs. (J. A. Eilek & T. H. Hankins 2016;M. Labaj et al. 2024;E. Sobacchi et al. 2021).
Recently, M. V. Medvedev (2024) proposed an attractive model that explains the “zebra” band structure as the spectral interference pattern resulting from multiple light propagation paths within the pulsar magnetosphere. In this paper, we further elaborate upon the model to incorporate the influence of general relativity, which appears to be of significant importance. Here, we calculate the light ray paths from the first principles and demonstrate that the pulsar system is equivalent to the Young’s double-slit setup, which produces the high-contrast interference fringe pattern with visibility of order unity, in contrast to diffraction. Furthermore, the deduced plasma density distribution within the magnetosphere exhibits an inverse-cube-law, which is anticipated for a dipolar magnetic field.
The remainder of the paper is organized as follows. Section 2 presents the details of the model. In Section 3, the light ray paths through the plasma in a Schwarzschild geometry are computed. In Section 4, the interference of these rays is studied. Section 5 presents important discussion and predictions. Section 6 summarizes major results of the paper.
The primary assumption of the model is that the broadband radio emission source of HFIP is located behind the pulsar, as illustrated in Fig. 1. The precise origin of the
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