Astron. Astrophys. 333, 172-180 (1998)

2. An inverse Compton scattering (ICS) model

2.1. Assumption

Along the line of RS inner gap model (RS75), Qiao (1988a, 1992) presented a model for radio emission of pulsars. The basic assumptions of the model are as follows:

1. Neutron stars have dipole magnetic fields.

2. The radio emission observed of pulsars is produced in an ICS process: a low frequency wave (with angular frequency ) is scattered by high energy particles (with Lorentz factor , this is the energy of the secondaries, see RS75 and the ICS process of high energy particles off the thermal photon in or above the polar gap is taken into account, see Xia et al.1985, Bednarek et al. 1992, ZQ96, Luo 1996, Zhang et al. 1997a). The low frequency wave is produced in the inner gap sparking (see RS75; the gap continually breaks down on a time scale of a few microseconds, the angular frequency is , so is the angular frequency of the low frequency wave in the calculation below). The high energy particles are the secondary particles produced near the gap (ZQ96 for a self-consistent gap considering the ICS-induced process; also see Zhang et al. 1997a for a discussion of three modes of pulsar inner gap).

3. The low frequency waves can propagate near the neutron stars. A possible reason for this may be that large radiation pressure make particle density along the path of the emission substantially less dense than that predicted (e.g. Sincell and Coppi 1996), and the plasma frequency should be much lower if nonlinear effects are taken into account (Chian and Clemow 1975; Kotsarenko et al. 1996).

2.2. The luminosity of the radio emission

The efficiency of the ICS process is higher than that of the CR process, but as in the estimate bellow, incoherent radiation in the ICS process is inadequate in explaining pulsar radiation also. We can write the luminosity of the ICS process ¹:

Here is a constant in order of 1, is the magnetic field near the surface of the neutron stars, P is the rotational period of the neutron star, is the thickness of the gap, is the Lorentz factor of the particle. is the cross section of the inverse Compton process, and is the Thomson cross section.

If we take (see appendix), then

When the radiation take place at a higher position: for example, , where R is the radius of the neutron star. Using , where is the low frequency wave photon density at distance r (near the surface of the neutron star, that is, at distance R), we have

The luminosity observed of radio pulsars can be as large as to (Sutherland 1979, and see Eq. (C9)). This means that incoherent ICS radiation is inadequate in explaining pulsar radiation. A coherent mechanism should be involved.

Coherent emission mechanisms may be classified as: (1).maser mechanisms; (2).a reactive or hydrodynamic instability; or (3). to emission by bunches. Theories for these coherent emission processes are not as well developed as theories for incoherent emission processes (Melrose 1992).

The emission mechanism suggested in this paper are favorable to the coherent emission by bunches. In the ICS process, the outgoing photons are produced in a scattering process: the low frequency wave is scattered by particles moving along a bunch magnetic field lines. In the process, the outgoing photons produced by particles of a bunch are coherent; and those produced by particles in different sparking are not coherent. From this point of view, we can fit the linear and circular polarization observations of radio pulsars very well (Xu 1997; Qiao, Xu and Han 1997). If the number of the particles in each sparking is ,the coherent luminosity of the ICS process is:

and Eq. (C1) gives incoherent luminosity. If , the ratio of coherent luminosity to incoherent luminosity is: . Comparing Eq. (3) and (C9) as well as the ratio , we find that is enough to produce observed radio emission.

From (Goldreich & Julian 1969) and Eq. (C2), we get

Here , is the radius of inner gap, defines the angular width of polar cap from which all open field lines eminate, is the radius of the neutron stars, , and c is speed of light, e is the charge of an electron.

For typical parameters and , is about (, see appendix). Thus, , which means small proportion of the coherent particles is enough to produce the observed radio emission.

For coherent curvature radiation, there is a fundamental weakness in existing theoretical treatments which do not allow for any velocity dispersion of the particles (Melrose 1992). In the mechanism discussed here the weakness is much weaker. This is because the observed emission (including frequency and phase) is determined by the frequency of the incoming wave , the energy of the particles (Lorentz factor ) and the incoming angle , not only (see Eq. (6)).

2.3. The basic formulae for emission beams

For most pulsars, at points near or far from the surface of the neutron star. As the Lorentz factor , , and , in an ICS process of the outgoing high energy particles with the low frequency wave photons to give the outgoing photons (with the energy of ), we have (Xia et al. 1985; Qiao 1988a, b):

Here is the incoming (outgoing) angle between the direction of motion of a particle and the incoming (outgoing) photon, is the mass of electron, is the angular frequency of the low frequency wave produced in the inner gap sparking. In the Lab frame, most of the outgoing photons are emitted along the direction of motion of a particle within a small beam of width about .

For a dipole magnetic field line, we have

Here, r is the distance between a point Q and the center of the neutron star, (), is the radius of light cylinder, P is the pulse period of pulsar, R is the radius of the neutron star, is the polar angle at point with respect to the magnetic axis (see Fig. 1), is a constant for a dipole magnetic field line. For investigating the radiation, we only consider so-called open field lines.

Fig. 1. Geometry for the inverse Compton scattering process. Low frequency waves produced at point A are scattered by particles at point . Dipole field line is assumed.

We consider the case that the low frequency wave is produced at the sparking point A near the boundary of the inner gap defined by the last open field lines, in this case, is taken in the calculation below. At any scattering point in a field line with , the incident angle (see Fig. 1) can be written as follows.

Here, is the vector of magnetic field, is the direction vector of incoming low frequency wave.

In a spherical coordinate system the dipole magnetic field will have components (Shitov,1985): , and , where m is the magnetic moment of neutron star.

In Cartesian coordinate system, the vector is:

Here and are the coordinate parameters of point A and Q respectively. The azimuthal angle ranges from 0 to for different field lines which are symmetric to the magnetic axis, and is the polar cap angle where the last open field line begins (see RS75), the azimuthal angle of the inner gap boundary can be from 0 to .

With Eq. (8), Eq. (9) we can get :

where , and .

When the emission regions are far from the surface (that is for , and ) and in the plane of a field line (that is ), we have

The angle between the radiation direction (in the direction of the magnetic field) and the magnetic axis, , has a simple relation with ,

The energy of high energy particles will be reduced when these particles come out along with the field lines owing to scattering with thermal photons and low frequency waves. It is assumed that

where reflects the energy lose of the particles and different pulsars have different and .

Using Eqs. (6), (10), (12) and (13), we have a numerical relation between the outgoing photon frequency and the beam radius . In the coordinate system with magnetic axis as z axis, we have . Finally, at any scattering point, we can get and the radiation direction defined by , .

2.4. Retardation and aberration effects

Both observations (Rankin 1983, 1990, 1993a, b) and calculations (this paper and Qiao et al. 1992) show that the core, "inner" cone and "outer" cone are emitted at different heights (see Fig. 5). This makes the apparent beams move their positions relative to each other. Two points are considered in this paper. First, it needs time for the low frequency wave photons to propagate to the points where they are scattered by relativistic particles. For the scattering process taking place at point (see Fig. 1), the emission point of the low frequency wave is not at point A, but at a point before that. As a result the incoming angle will be changed. In other words, for the scattering that takes place at this moment, the low frequency wave doesn't come in from the present gap but the gap in an earlier position. The angle difference is , where is the time for light to travel between the point where the low frequency wave is emitted and the point where it is scattered. Secondly, the core and two cones are emitted at different heights, hence a time delay between them would change the apparent positions. Thus, we can get a new and this makes that the beams are asymmetric to the magnetic axis: the central beam close to the trailing component of the cone, see Fig. 3a and b.

2.5. Basic results

The calculated results are shown in Fig. 2 to Fig. 6. Several main conclusions can be reached about the emission regions:

1. The theoretical emission beams we get have two or three parts, including "core" and "inner cone" and/or an extra "outer" cone at a given frequency. The angular radii of the beams are strongly related to the pulse period P and only P (see Fig. 2a and b and Fig. 3a and b). This result is in good agreement with the empirical figure of double-conal geometry for some pulsars (Rankin 1993a). The angular radius of central and conal beams are consistent with the conclusion from observations (Rankin 1983,1993a). Fig. 2a and b shows emission beams for a slow rotating pulsar with period , which have one core and double cones. Fig. 4 shows other emission beams for a fast rotating pulsar with , which only have one core and one cone and it is very difficult to produce an "outer" cone for this kind of short-period pulsars.
2. These three different emission components are emitted at different heights (Fig. 5). The core emission is emitted at a place very close to the surface of the neutron stars in a "pencil" beam and the "inner" cone is emitted at a lower height along the same group of peripheral field lines where the "outer" cone is produced. This is to say that radio emission with the same frequency can be emitted at different heights along a bunch of peripheral field lines, which is in agreement with the conclusion given by Rankin (1993a).
3. Considering the two effects of retardation and aberration, the theoretical beams for those pulsars with fast rotation will change their shapes. The apparent position of the core will move to later longitudes with respect to the position of the center of the cones.
4. In Fig. 6, at relatively low emission frequencies, core and inner cone can merge together (line B & line D) And at high frequencies, inner cone and outer cone, can merge together for slow rotating pulsars (line A). This can be seen clearly in Fig. 3a and b.
5. In some case (Fig. 3a and b and Fig. 4), the leading part of outer cone can be broken.

Fig. 2. Emission beams for a pulsar with , , , , , is the angle made by rotation axis with the magnetic axis; a. at (upper panel); b. at (lower panel). The beams are slightly asymmetric to magnetic axis (which is perpendicular to the paper plane at the center of cone). Retardation and aberration effects are weak for slow rotation pulsars.

Fig. 3. Emission beams with obvious retardation and aberration effects: a. at (upper panel); b. at (lower panel). , , , , . The beams are asymmetric to magnetic axis.

Fig. 4. Emission beams at for a fast rotation pulsar with , , , , .

Fig. 5. A relation between the frequency and the emission heights for a pulsar with the same parameters of Fig. 2a and b. When observing at a frequency, one should find that the core, inner cone and outer cone are emitted at different heights. The frequencies in the figures in this paper only have relative meaning.

Fig. 6. Figure for relation: a. with the same parameters of Fig. 2a and b (upper panel); b. with the same parameters of Fig. 4 (lower panel).

© European Southern Observatory (ESO) 1998

Online publication: April 15, 1998
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