Comparison with Observations

The factor of in equation (3) means that at low frequencies the emission probability is reduced for emission nearly along the magnetic field. The results therefore indicate that at low frequencies the pulse has a central dip and thus a double-humped pulse is emitted, as has been observed for GX 1+4 (Greenhill et al. 1993). The opening angle of the central dip in the pulse, i.e. the angle at which the flux is a maximum, is given by the value of for which

Using (3), (4), and (5) and assuming a cyclotron energy of and an observing energy of 5 keV, the opening angle at the site of the emission of the double-humped pulse produced by two-photon cyclotron emission is . This is as would be observed on the surface of the pulsar. However, gravitational light bending significantly affects the beam shape. Leahy and Li (1995) have derived simple analytic formulae that adequately describe the effect of gravitational light bending due to general relativity for most neutron stars. Assuming a typical neutron star radius for GX 1+4 then the opening angle after gravitational light bending is related to the opening angle from the emission process alone by equation (4) from Leahy and Li (1995);

For an input angle of , this gives an opening angle including the effect of gravitational light bending of .

ASCA Observations

The folded light curves for different photon energies obtained from the ASCA satellite observations have been presented in Kotani (1996). The light curve for energies <3keV is likely to suffer from considerable contamination from galactic ridge emission and the light curve around 6.4keV is likely to be contaminated with unpulsed iron line emission.

  
Figure 1: Folded light curve for GX 1+4 summed over energies from 3-10 keV, excluding the data from around the iron line. Galactic ridge emission is not subtracted. For details of the ASCA observations see Kotani, 1996. The error bars in the above data are counts/unit time. The vertical bars are separated by the theoretically predicted opening angle of the double-humped pulse.

In Figure 1 is presented summed data for energies from 3keV to 10keV excluding the data from around the iron line. The data in Figure 1 are thus the most likely to give information on the emission process as they are the most likely to represent just pulsed emission. The pulse has a deep dip in emission at phase and there is also possibly a small dip at phase . It is not self-evident which dip represents the true edge of the pulse and which represents the central dip expected in the two-photon cyclotron emission model. The two-photon cyclotron emission model predicts a harder spectrum in the centre of the dip as the reduction in emission for small is greater for low frequency photons. In Table 1 we show the ratio [Counts/sec.(Energy = 7-10 keV)]/[Counts/sec.(Energy = 3-5 keV)] at phases 0.25 and 0.75 and the average value over all phases. There is slight evidence of spectral hardening at phase . However, the error bars are too large around the deep dip at phase to draw any conclusions about the spectral hardening here. Therefore, we do not consider that spectral hardening results conclusively determine which dip represents the central dip predicted by the two-photon model.

Table 1: Flux ratio as a function of phase

Phase	Counts/sec.(Energy = 7-10 keV)/Counts/sec.(Energy =3-5 keV)
0.25
0.75
average

Another feature of the pulse shape is that it is significantly asymmetric, with the leading half of the pulse brighter than the trailing half. The same asymmetry was observed in Greenhill et al. (1993). A possible mechanism causing the observed asymmetries in X-ray pulsar beams was outlined in Padden and Storey (1986). It was shown that the presence of a magnetic field in the neutron star whose dipole axis does not pass through the centre of the star can cause asymmetric pulse shapes when an accretion disk is present. The asymmetry makes it harder to estimate the opening angle of either the deep or shallow dip. The vertical bars in Figure 1 are separated by , the theoretical opening angle incorporating gravitational light bending. Taking the asymmetry of the pulse into account, the hypothesis that the dip at phase 0.25 is the two-photon dip is consistent with the observations. However, it is also possible that the deep dip at phase 0.75 is the two-photon dip and that there is another component in the beam. This is discussed further in Section 4.

Other Observations

Pulse profiles observed after the low intensity state in the 1980's tend to exhibit a double-humped profile and an asymmetry where the leading edge of the pulse is brighter than the trailing edge. The pulse peak separations on either side of the shallow dip in the pulse profiles from GX1+4 shown in Greenhill et al. (1993) at energies of 20-75 keV and Mony et al. (1991) at energies of 20-60 keV are consistent with the theoretically predicted maximum value of .

However, the light curves observed during the high state of the the 1970's tend to exhibit an asymmetry where the trailing edge of the pulse is brighter than the leading edge. The beams are much broader, there is often no clear double-humped structure and there is some evidence for additional components in the beam (eg White, Swank and Holt, 1983, Doty, Hoffman and Lewin, 1981). This implies that the emission geometry must have been significantly different during the high state.

© Copyright Astronomical Society of Australia 1997