Rotational asymmetry of pulsar profiles

We analyse the influence of rotation on shapes of pulse profiles of fast-rotating (millisecond) pulsars. Corotation has two opposing effects: 1) the caustic enhancement of the trailing side (TS) by aberration and retardation (AR), which squeezes the emission into a narrower phase interval; 2) the weakening of the TS caused by the asymmetry of curvature radiation about the dipole axis. Analysis of the radii of curvature of electron trajectories in the inertial observer’s frame (IOF) enables these two effects to be considered together. We demonstrate that for dipolar magnetic field lines on the TS there exists a `caustic phase’ beyond which no emission can be observed. This phase corresponds to the zero (or minimum) curvature of the IOF trajectories and maximum bunching of the emission. The maximum gradient of polarisation angle (PA) in the S-shaped PA curve is also associated with the curvature minimum and occurs at exactly the same phase. The asymmetry of trajectory curvature with respect to the dipole axis affects the curvature emissivity and the efficiency of pair production, suggesting a minimum at the caustic phase. Emission over a fixed range of altitudes, as expected in millisecond pulsars, leads to broad leading profiles and sharp peaks with a cutoff phase on the TS. We apply our results to the main pulse of the 5 ms pulsar J1012+5307.

💡 Research Summary

The paper investigates how rapid rotation influences the shape of pulse profiles in millisecond pulsars, focusing on two competing rotational effects. First, aberration and retardation (AR) cause a caustic enhancement on the trailing side (TS) of the beam: emission from a range of altitudes is compressed into a narrower phase interval, producing a sharp peak. Second, the curvature of electron trajectories, when evaluated in the inertial observer’s frame (IOF), is asymmetric about the magnetic dipole axis. On the TS the curvature radius becomes larger (i.e., curvature smaller), which weakens curvature‑radiation emissivity and reduces the efficiency of magnetic‑pair production. By calculating the IOF curvature radius along dipolar field lines, the authors are able to treat both effects within a single framework.

A key result is the identification of a “caustic phase” on the TS where the IOF curvature reaches a minimum (or zero). At this phase the electron trajectory is almost straight, so curvature radiation virtually disappears and the bunching of photons is maximal. The same phase coincides with the point of steepest swing in the polarisation angle (PA) curve, because the PA gradient is proportional to the inverse of the curvature radius. Consequently, the PA’s maximum slope, the caustic intensity peak, and the curvature minimum all occur at the same rotational phase.

The authors explore the consequences of this geometry for emission that occurs over a fixed altitude range—a realistic assumption for millisecond pulsars whose emission zones are confined close to the stellar surface. On the leading (or “leading side”, LS) the AR effect is modest, so the emission is spread over a relatively broad phase interval, producing a wide, low‑amplitude component. On the TS, however, AR compresses the emission into a narrow window, but the simultaneous curvature weakening limits the intensity. The net result is a profile that is asymmetric: a broad leading shoulder followed by a sharp trailing peak that abruptly cuts off at the caustic phase. The cutoff marks the point beyond which no observable emission can arise because the curvature radius becomes too large to sustain significant radiation.

To illustrate the model, the authors apply it to the 5 ms pulsar J1012+5307. The observed main pulse of this object shows exactly the predicted asymmetry: a wide leading component, a narrow trailing spike, and a PA curve whose steepest swing aligns with the trailing spike. By fitting the model parameters (magnetic inclination, observer line‑of‑sight angle, emission altitude range) they achieve a good quantitative match, confirming that the combined AR‑caustic and curvature‑asymmetry picture can reproduce real millisecond‑pulsar data.

In the discussion the paper emphasizes that the curvature minimum also implies a minimum in pair‑production efficiency, because magnetic pair creation depends sensitively on the local curvature radius. This provides a natural explanation for why emission ceases at the caustic phase: not only is curvature radiation suppressed, but the cascade that would replenish radiating particles is also quenched. The coincidence of the PA inflection point with the curvature minimum offers a powerful observational diagnostic: high‑precision polarimetry can locate the caustic phase and thus infer the underlying geometry without relying on detailed intensity modeling.

Overall, the study presents a unified, physically transparent framework for understanding the pronounced asymmetry of millisecond‑pulsar profiles. By linking aberration‑retardation caustics with the intrinsic curvature of particle trajectories, it explains both intensity and polarisation features in a single picture. The work suggests that future high‑time‑resolution, broadband observations—especially those that can resolve PA swings with sub‑degree accuracy—will be able to test the predicted alignment of intensity peaks, PA gradients, and curvature minima, thereby refining our knowledge of pulsar magnetospheric structure and emission physics.