5 Discussion

The X-ray flux variations of Crab pulsar have a standard deviation of $\approx$0.6% to 0.8% of its mean value over time scales ranging from a period to almost a month; that this number is similar over such wide time scales may or may not be a coincidence. This is consistent with the Crab pulsar's behavior at optical and UV energies (Percival et al. 1993) and at IR energies (Lundgren et al. 1995). Therefore the Crab pulsar flux variations at higher energies are insignificant compared to those at radio wavelengths, where the modulation index $\mu$ is $\approx$1.0, most of which apparently comes from the giant pulses (Lundgren et al. 1995). Now, this causes problems for the conjecture that both the radio and the high energy emission are related through a common electric current. If the flux variations of Crab pulsar are due to temporal variation in the number of basic emitters, then the radio flux variations would also have been at the $\approx$0.7% level. If they are due to temporal variation in the coherence of the basic emitters, then the radio flux variations would have been at the $2 \times 0.7 \approx 1.4$% level, since the intensity of a coherent emission mechanism is proportional to the square of the number of basic emitters. If they are due to temporal variation in the angle of some elementary beams (Lundgren et al. 1995), one would have expected the radio flux variations to be smaller than those at high energy, since the common beams would be larger at the larger wavelengths. One could surely postulate variations in the angle of radio beams only, but that would have to be justified on the basis of some other independent physical mechanism. In summary the difference in the radio and high energy flux variations of Crab pulsar is difficult to explain, if the basic charged emitters at both wavelengths are somehow related.

Another method, of amplifying the very small flux variations of Crab pulsar at high energies to the very large variations at radio wavelengths, would be to somehow use a fraction of the $\approx$10⁷ amplification factor of particles in the gaps, due to cascading e⁺-e^- pair production (Ruderman & Sutherland 1975; Cheng et al. 1986a, 1986b). One could probably postulate that the radiation that is emitted by the later generation of charges in the pair cascade process suffers greater variation in its intensity, due to amplification of the variation in the number density of charges. This might also imply that one should see a monotonic increase in the modulation index as one observes at larger wavelengths. More detailed study of Cheng et al. (1986a, 1986b) and Romani & Yadigaroglu (1995) models is required for a reasonable solution. In any case, the results of this paper set very strong constraints on the explanation for the relative flux variations at the radio and X-ray energies.

The possible correlation of $\mu$ with the integrated profile in Fig. 4, if confirmed in future, might set very strong constraints on the basic emission mechanism of high energy emission from the Crab pulsar. In the framework of Cheng et al. (1986a, 1986b) and Romani & Yadigaroglu (1995) models, the two peaks of the integrated profile are cusps created by emission from different magnetic field lines, that add in phase along different directions due to relativistic aberration. A well defined relation between the mean flux and its variance at each point in the integrated profile (for example, $\mu$ might vary as $\langle I \rangle^\alpha$) will be an additional constraint, along with the exact shape $\langle I \rangle$ of the integrated profile, on the X-ray emission mechanism.

Further work on these data is in progress, that studies issues such as (a) verifying if the Crab pulsar shows at X-ray energies the three phenomenon that are often seen in several radio pulsars - "pulse nulling'' "systematic sub pulse drifting'' and "mode changing''; (b) looking for special behaviour in the X-ray integrated profile at the phase of the radio precursor, which is supposed to be different from the rest of the radio integrated profile; (c) comparison of the peak and the bridge X-ray emission of the Crab pulsar, which might further constrain the models of Cheng et al. (1986a, 1986b), Romani & Yadigaroglu (1995), and Cheng et al. (2000).

On the theoretical front, it is probably worth exploring the simultaneous modeling of $\langle I \rangle$ and $\sigma^2_I$ (or $\mu$) at X-rays for rotation powered pulsars, and more specifically for the Crab pulsar.

Acknowledgements

This research has made use of (a) High Energy Astrophysics Science Archive Research Center's (HEASARC) facilities such as their public data archive, and their FTOOLS software, and (b) NASA's Astrophysics Data System (ADS) Bibliographic Services. The author is thankful to them for their excellent services.