An Emission Model for GX 1+4

It has been shown (Kirk, Melrose and Peters, 1984, Melrose and Kirk 1986, Kirk and Melrose, 1986, Bussard et al. 1986) that the most important source of continuum photons (i.e. photons not near the cyclotron frequency) in X-ray pulsars of low to moderate luminosity is from two-photon cyclotron emission and the associated double compton scattering. In two-photon cyclotron emission an excited electron in an n=1 quantum state returns to the ground state by emitting two photons, the sum of whose frequency roughly equals the cyclotron frequency. The rate of two-photon cyclotron emission is highest when one photon has a frequency close to the cyclotron frequency and the other photon frequency is much lower. Kirk, Nagel and Storey (1986) calculated the X-ray emission intensity and resulting beam shapes for two-photon cyclotron emission for conditions existing in typical X-ray pulsars. The model applies to pulsars for which the frequencies observed are significantly less than the cyclotron frequency.

The observational and theoretical evidence that GX 1+4 has an unusually high magnetic field (implying that existing observations are at frequencies well below the cyclotron frequency) means that GX 1+4 is an ideal object for studying two-photon cyclotron emission. The Kirk, Nagel and Storey (1986) model calculations were compared with balloon observations of GX 1+4 (Greenhill et al. 1993) and it was shown that the model is consistent with the observed pulse shape up to energies of order 75 keV. The observed pulse at high frequencies was narrower than predicted by the two-photon cyclotron emission model alone, possibly due to resonant Compton scattering of X-ray photons by the accretion plasma above the emission region, or increased non-resonant Compton scattering (Greenhill et al. 1993).

The ASCA satellite observed GX 1+4 at energies between 3 and 10 keV. Compton scattering is less significant at frequencies far below the cyclotron frequency (Pavlov, Shibanov and Meszaros 1989) and so for the ASCA observations the observed pulse should correspond more exactly to the theoretically predicted pulse shape.

Low-frequency approximation to transition rates

In the highly magnetized emission region plasma the X-ray radiation is emitted in two natural modes and, in general, the natural modes are elliptically polarized. However, at frequencies well below the cyclotron frequency the modes can be approximated as having transverse linear polarization (Melrose and Kirk, 1986, hereafter MK86) with polarization vectors, indicating the polarization direction of the two natural modes of radiation, given by

The angle is the polar angle of emission of the photon with respect to the magnetic field and the angle is the azimuthal angle of the photon with respect to the magnetic field. The magnetic field B is in the z direction. The derivation of the transition rates for two-photon cyclotron emission in such a plasma, averaged over polarization and angle of emission is given in MK86. For the densities present in accretion streams (m), vaccuum polarization effects dominate plasma effects. An analysis of radiative transfer effects has been given in Kirk, Nagel and Storey (1986). The effect of radiative transfer through the emission region is not included in the calculation below.

The transverse polarization approximation is valid at very low frequencies compared to the cyclotron frequency and so applies to the ASCA observations of GX 1+4 if the pulsar has a cyclotron frequency . For two-photon cyclotron emission the probability of a soft photon of frequency being emitted in polarization mode , as a function of the frequencies and angles of emission of the two photons, and , the velocity of the emitting electron along B, is given by

where , the are matrix elements (MK86), is the wavevector of the emitted low frequency photon in the z direction and m is the mass of the electron. Units where h=c=1 are used throughout except where explicitly stated otherwise. Variables and refer to the second higher energy photon. The delta function approximation is discussed in Kirk, Nagel and Storey (1986) (see equation (3) of that paper).

In MK86 (36 a to d) expressions are given for the squares of the projected matrix elements, averaged over the azimuthal angles of the photons, and also averaged over the forward and backward directions, with respect to the magnetic field, of polar angles of the photons.

The results for the projected matrix elements are used to construct the transition probabilities for emission of a low frequency photon in either polarization mode as a function of its frequency and , the angle of emission with respect to the magnetic field. This is done by integrating over the frequency of the second photon using the delta function and averaging over the direction of emission with respect to the magnetic field of the second photon. In the low frequency limit, only the leading terms in need to be considered. Then the rate of emission of a soft photon of frequency has an angular dependence given by (after correcting some errors in the equations for the matrix elements in MK86)



where , superscript t indicates emission in the t polarization mode and superscript a indicates emission in the a polarization mode. For soft photons with emission in the t polarization mode dominates and the angular pattern for goes as . For terms in become important, and thus the equations above indicate that a change in polarization direction could be observed across a pulse. The sum of and leads to the total transition probability, independent of polarization.

Two-photon emission has been shown to be the dominant source of emission for moderate luminosity X-ray pulsars (Kirk and Melrose 1986). For such pulsars the deceleration region of the accretion plasma is thought to occur close to the surface of the star and the emission region is modelled as a thin slab of plasma on the polar cap of the neutron star (Kirk 1985).

The flux from the emission region F has an angular dependence given by

where the factor of takes into account the projected area of the slab. If the slab were radiating isotropically the pulse profile would have the form .

© Copyright Astronomical Society of Australia 1997