Long-term variations of X-ray pulse profiles for the Crab pulsar: data analysis and modeling

The pulse profiles of the Crab pulsar (as well as some other pulsars) vary with time. They can lead to a major source of intrinsic timing noise, which lacks a detailed physical model. The phase separation Δ between the first left peak (P1) and the second right peak (P2) is a key parameter that shows the variations of pulse profiles for the Crab pulsar. It was found that the evolution of A has a tendency with increasing rates of 0.82° ± 0.25°, 0.80° ± 0.54°, and 0.77° ± 0.28° per century for the 2–6, 6–15, and 15–60 keV bands, respectively. Furthermore, the flux ratios (P2/P1) of X-ray pulse profiles in the three bands were calculated, and the derived flux ratios were consistent with the radio and X-ray measurements of the Insight-HXMT. In addition to discovering the physical origin of the pulse changes, the high-SNR X-ray pulse profiles were simulated in the annular gap model, and two model parameters (e.g., the maximum emission heights of the two peaks) were observed to slightly affect the variations of peak separation. We fitted the long-term variations of emission heights of the two peaks and discovered that the emission heights showed increasing tendencies with time. Variations of these emission heights induced a characteristic period derivative, and a complete formula for both the magnetic dipole radiation and wind-particle-induced variations of the moment of inertia was used for the pulsar's spin-down to obtain the variation rate ά of the magnetic inclination angle, which was ‑1.60° per century. Intrinsic timing noise is observed to be mainly induced by the variations of pulse profiles, which might correlate with a characteristic spin period derivative arising from the fluctuations of the emission regions. This work will lay a foundation for understanding the origin of intrinsic timing noise and making high-precision timing models in the future.